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Oplical demultiplexer using a llolographic concave graring for POF-WDM systems LYUBOMYR V. BARTKIV, YAROSLAV V. BOBITSKI, HANS POISEL . . . . . . . . . . . . . . 59
Problems with cracking oJ AlxGa1_xN layers EwA DUMISZEWSKA, DARIUSZ LENKIEWICZ, WLODEK STRUPINSKI, AGATA JASIK, RAFAL S. JAKJELA, MAREK WESOŁOWSKI . . . . . . . . . . . . . . . . . . . III
Influence oj technological parameters on t he properties oJ sol-gel s i lica film s PAWEŁ KARASIŃSKI .... .. ... . ...... .. .. . .. .. .... . .... . ... . . . ....... .. II7
Semiconductor structures
Optical beam injection methods as a tool for analysis oJ semiconductor structures JAROSŁAW DOMARADZKJ, DANUTA KACZMAREK .. ....... . .. . .. . ....... . .. 129
Fast ion generation by a picosecond high-power laser
JAN BADZIAK1, PIOTR PARYS1, JERZY WOŁOWSKI1, HEINRICH HORA2,JOSEF KRÁSA3, LEOS LÁSKA3, KAREL ROHLENA3
1Institute of Plasma Physics and Laser Microfusion, ul. Hery 23, 00-908 Warszawa, Poland
2Department of Theoretical Physics, University of New South Wales, Sydney 2052, Australia
3Institute of Physics, A.S.C.R., Na Slovance 2, 182 21 Prague 8, Czech Republic
Recent progress in ultrashort-pulse high-power laser technology has resulted in the production ofextremely high light intensities approaching 1020 W/cm2. The great non-linear forces generatedby the laser pulse during its interaction with plasma can be used to accelerate electrons and ionsto energies from hundreds of keV to hundreds of MeV over distances of only microns. This createsthe prospect of construction of compact laser-based particle accelerators and their application inmaterial science, medicine, nuclear physics, and inertial confinement fusion. In this paper, theresults of our recent studies on fast ion generation in plasma produced by an intense 1-ps laserpulse, performed using the terawatt Nd:glass laser at Institute of Plasma Physics and LaserMicrofusion (IPPLM) in Warsaw, are briefly reviewed. The properties of fast proton beamsgenerated from thin foil targets of various structures as well as the heavy ion fluxes emitted frommassive high-Z targets are discussed. The possibility of producing picosecond ion beams ofultrahigh ion current densities (> 1010 A/cm2 close to the target) is considered. The most importantfeatures of fast ion generation in the plasmas produced by ultrashort (1 ps) and long (0.5 ns) laserpulses are also compared.
Keywords: fast ion, plasma, picosecond laser.
1. Introduction
The generation of fast particles using lasers was first observed in the sixties when thefirst experiments with a Q-switched laser irradiating a solid target were performed[1, 2]. Those experiments showed that plasma created on the target surface is a sourceof ions of energies (up to several keV) considerably higher than the mean energy ofions in the plasma. Along with the development of pulsed lasers with ever increasingpower, higher and higher ion energies have been achieved; in particular, anomalouslyfast ions in the MeV energy range have been recorded [3]. However, for many years,fast particles emitted from plasma were studied mainly from the point of view of theirimportance for laser-driven inertial confinement fusion, where they play a “negative”role as a source of significant energy loss and the cause of preheat of the fusion target.
6 J. BADZIAK et al.
More recently, however, some useful aspects of fast ion emission from laser-producedplasma have been emphasized. In particular, the application of laser-driven ionsources in accelerator technology [4, 5] and in ion implantation for modifying materialproperties [6] has been suggested and investigated using ns and sub-ns laser pulses[7–13].
A real breakthrough in the investigation of laser acceleration of charged particleswas the use of high power lasers generating ultrashort pulses (τL ≤ 1 ps). Theselasers can produce radiation of much higher intensity (currently up to ~1021 W/cm2
[14]), are smaller, and can work at pulse repetition rate orders of magnitude higherthan ns or sub-ns lasers with similar peak powers [15], which is of vital importancefor practical applications. Moreover, ultrashort-pulse lasers (USP-lasers) are capableof producing a short-lived hot plasma, which can emit intense beams of high-energyparticles in extremely short bursts of sub-ns or ps duration [16–18]. The above featurescreate prospects for applications of USP-laser-produced ion beams in medicine[19, 20], materials science [6, 13], inertial fusion [21], and nuclear physics on atabletop [22, 23].
In this paper, our recent studies of fast ion generation in the plasma produced byan intense 1-ps laser pulse, carried out using the terawatt Nd:glass laser at the Instituteof Plasma Physics and Laser Microfusion (IPPLM), Warsaw, are briefly reviewed. InSection 2, the essential differences in the properties of fast ion fluxes emitted from theplasmas produced by ultrashort (1 ps) and long (0.5 ns) laser pulses are presented anddiscussed. In Sections 3 and 4, some results of studies on laser generation of protons(Sec. 3) and heavy ions (Sec. 4) in ps-pulse-produced plasma are presented. Thepossibility of production of picosecond ion beams with ultrahigh ion current densities(>1010 A/cm2 close to the target) using picosecond lasers is considered in Sec. 5. Thelast section briefly summarizes the main results of the paper.
2. Comparison of the properties of fast ion fluxes produced by ultrashort and long laser pulses
In recent years both long (> 0.1 ns) and ultrashort (≤ 1 ps) laser pulses have beenused to produce intense ion fluxes with ion energies from hundreds of keV up to tensand hundreds of MeV (see Secs. 3 and 4). However, the characteristics of ion fluxesemitted from plasma depend on many other factors than just laser pulse duration andintensity (e.g., laser focusing conditions, laser prepulse parameters, target features).So, a detailed quantitative comparison of the ion characteristics from previouslong-pulse and ultrashort-pulse experiments, where different experimentalarrangements have been used, can be difficult or impossible. As a result, the potentialadvantages (or disadvantages) of the ion fluxes produced using USP-lasers over thosegenerated by long-pulse lasers have not been fully and unambiguously determined.
In experiments at IPPLM, the properties of fast ion fluxes emitted from the plasmacreated by interaction of a 1-ps laser pulse with a massive Au target were compared
Fast ion generation by a picosecond high-power laser 7
to the properties of the fluxes produced by a 0.5-ns laser pulse under the sameconditions. The comparison was carried out using 1-ps and 0.5-ns laser pulsesproduced by the same chirped-pulse-amplification (CPA) Nd:glass laser system[24, 25]. Excluding the pulse duration and intensity (up to 1017 W/cm2 for 1-ps andup to 2 × 1014 W/cm2 for 0.5-ns pulses), other parameters of the laser beam (energy,wavelength, focal spot diameter, etc.) were roughly identical for both pulses. Themeasurements were performed using the experimental set-up presented in Fig. 1 [26].Ion flux parameters were measured using ion collectors (ICs) and an electrostaticion-energy analyzer (IEA) [25, 27]. The IEA and the ring ion collector (IC1) recordedthe backward-emitted ion beam passing through the hole in the parabolic mirror alongthe target normal and the laser beam axis. To enable a rough estimate of the angulardistribution of ion emission, two additional ion collectors, viewing the target at anglesΘ of 26° and 34° with respect to the target normal, were used.
Figure 2 presents typical IC1 collector signals illustrating the time dependenceof backward ion emission (against the laser beam) from the Au target for the cases of1-ps and 0.5-ns laser pulses. The rough angular distributions of fast ion emission forthe ps and sub-ns pulses are presented in Fig. 3. These figures demonstrate the principaldifferences between the properties of fast ion fluxes emitted from plasmas producedby ultrashort and long laser pulses. In the long-pulse case, several groups of fast ions(subscript f in Fig. 2) are generated, the maximum of the fast ion current density doesnot coincide with the target normal (Θ = 0°), and the angular divergence of the fastion expansion is high.
In the ultrashort-pulse case, only a single fast ion group is generated and this groupis well separated in time from the thermal ion group. The fast ions expand with smallangular divergence and with a pronounced maximum along the target normal (atΘ ≥ 26° the ion current density was more than 100 times less than that recordedat Θ = 0°). The above features were observed independent of laser energy and laserfocus position with respect to the target surface (within ±0.8 mm).
Our IEA measurements showed that, for both the ultrashort-pulse and long-pulsecases, the fast ion groups contain highly charged (z ~ 20–30) Au ions as well ascontaminant ions (H, C and O). The maximum charge states and energies of the H, C,O and Au ions, measured using the IEA, were comparable for both cases.
Significant differences between the properties of the ion fluxes produced by ps andsub-ns laser pulses can also be observed, when the dependence on laser intensity ofthe maximum ion energy and the ion energy at the amplitude peak (≈ mean energy),
Fig. 2. IC1 collector signals for an Au targetirradiated by 1-ps (a) and 0.5-ns (b) laser pulses.The ion energies per nucleon at the amplitudepeaks are indicated (Ep).
a
b
0.60
0.40
0.20
0.0
0 1 2 3 4 5 6 7 8 9 10
0.10 0.08 0.06 0.04 0.02 0.0
0 2 4 6 8 10 12 14
Time of flight [µm]
Am
plitu
de [V
]
Ef /A = 20 keV/n p
Eth/A = 0.19 keV/n p
EL = 0.52 J
Ef1/A = 20 keV/n
Ef2/A = 10 keV/n
Eth/A = 0.04 keV/n Ef3/A = 2.6 keV/n
p
p
p p
EL = 0.46 J
τL = 0.5 ns
τL = 1 ps
Fig. 3. Rough angular distributions of the ioncurrent densities and velocities at the peak ofthe backward-emitted fast ion pulses driven by1-ps (a) and 0.5-ns (b) laser pulses.
a
b
4 3 2 1 0
1.0 0.8 0.6 0.4 0.2 0.0
0 10 20 30 40 50
0 10 20 30 40 50
Angle of measurements Θ [deg]
2.5 2.0 1.5 1.0 0.5 0.0
Pea
k cu
rren
t den
sity
of f
ast i
ons
at 1
m [m
A/c
m2 ]
Pea
k ve
loci
ty o
f fas
t ion
s [1
08 cm
/s]
EL = 0.52 J
EL = 0.49 J
τL = 1 ps
τL = 0.5 ns
2.5 2.0 1.5 1.0 0.5 0.0
Fast ion generation by a picosecond high-power laser 9
measured by IC1, are taken into account (Fig. 4). For sub-ns pulses, ion energies aredependent on approximately the square root of the intensity, which is consistent withthe Iλ2 hot electron temperature dependence obtained by GITOMER et al. [3], by meansof compiling data from various laboratories. However, for ps pulses, ion energiesincrease approximately linearly with laser intensity.
Contrary to the long-pulse case, where self-focusing of the laser beam in the plasmaand various plasma instabilities may contribute to the production of hot electrons andfast ion generation [3, 5], our ps laser pulse results can be qualitatively explained bya less complex physical picture. It assumes that the main role in the ion accelerationprocess is played by non-linear ponderomotive forces produced by a skin layerinteraction of the pulse with a thin (several λ) pre-plasma layer in front of the target[28, 29]. In such a case, the ps laser beam interacts most intensely with the plasma inthe skin layer near the critical electron density surface, nec = meω2/4πe2 where ω isthe laser frequency, and the geometry of the interaction is almost planar becauseLpre << dfoc, where Lpre is the pre-plasma thickness and dfoc is the focal spot diameter.The high plasma density gradient in the interaction region produces a non-linearponderomotive force acting nearly parallel to the target normal – if the laser beam isperpendicular to the target surface. For such geometry, the force density fNL can beexpressed approximately as the one-dimensional negative gradient of theelectromagnetic energy density of the laser field given by its (dielectric modified)electric and magnetic vectors E and H [28]: fNL = –(∂/∂x)(E2 + H2) /8π, where thex-direction is normal to the target. When the electron density ne(x) of the plasmaincreases monotonically in the direction of laser pulse propagation, the energy density(E2 + H2) /8π increases up to a maximum shortly before the density reaches nec andthen drops exponentially to zero in the overdense plasma (ne > nec) within the skindepth. The gradients of the energy density result in two opposite non-linear forces:one, in the underdense plasma, acting in the backward direction, and the other, in theoverdense plasma, acting in the forward direction. These forces break the plasma nearthe critical surface and drive the two plasma blocks (electron clouds with attached
Fig. 4. Maximum ion energy and ion energy at the amplitude peak, measured normal to the Au target, asa function of laser intensity for 1-ps (a) and 0.5-ns (b) laser pulses.
a b
10 J. BADZIAK et al.
ions) towards the vacuum and towards the plasma interior, respectively. Due to thealmost one-dimensional geometry of plasma acceleration, the fast ion beam emittedhas a small angular divergence, in agreement with observations (Fig. 3a).
The energy of ions accelerated by the ponderomotive force can be estimated from[28, 29] as
(1)
where s = S – 1 (S = 1/|n| is the dielectric swelling factor, n is the plasma refractiveindex), is the electron rest mass, z is the ion charge state and Irel ≈ 4.1 × 1018/λ2
[W/cm–2, µm2] is the relativistic intensity. At subrelativistic laser intensities (I << Irel),we have
(2)
in agreement with the nearly linear dependence Ei(I) shown in Fig. 4a.Summing up, our experiment has shown that the properties and plausible
mechanisms of fast ion generation are essentially different for plasmas produced byultrashort (≤ 1 ps) and long (> 0.1 ns) laser pulses. In the long-pulse case, severalgroups of fast ions are emitted into a wide solid angle and self-focusing of the laserbeam in plasma and various plasma instabilities seem to determine the ion fluxproperties. In the ultrashort-pulse case, only a single fast ion group with a small angulardivergence is generated and ions are likely to be accelerated by an axial ponderomotiveforce produced by the skin layer interaction of the pulse with a pre-plasma layer infront of the target. For a majority of applications of laser-driven ion fluxes, theultrashort-pulse laser-plasma interaction would be more favorable, due to bettertemporal and spatial characteristics of the ion flux and a higher fast ion current.
3. Fast proton generation in picosecond laser-produced plasma
The possibility of generating high-energy (0.1–1 MeV) proton fluxes from laserplasma was discovered in the eighties in experiments with ns and sub-ns lasers [3].In the past, however, high-energy (EL ~ 102–103 J) laser pulses were required, whichcould only be generated in large laser installations designed for research on laser-driven inertial fusion. Low proton generation efficiency, high angular divergence andinhomogeneity of the proton beam, and, most of all, the impossibility of such biglasers operating at sufficiently high pulse repetition rates did not create an encouragingprospect for practical application of laser-generated protons. The situation changedradically when lasers producing ps and sub-ps pulses began to be used to generateproton beams [16, 18, 22, 30–39]. Although the experiments were generally carriedout at lower pulse energies than with ns lasers (but with higher intensities), the
Eis2
------zmeoc
21
3IIrel
----------+
1 2⁄
1–≈
meo
Ei34
------szmeoc
2 IIrel
----------≈
Fast ion generation by a picosecond high-power laser 11
proton energies Ep obtained were considerably higher (Ep ~ 0.1–1 MeV, even withEL < 1 J [22, 32, 36]) and the beam was more homogeneous and had smaller angulardivergence.
As a rule, in proton generation experiments with ultrashort laser pulses, thin plastic(mylar, polystyrene) or metal foils are used. The target thickness dT should suit boththe energy and the intensity of the pulse so that, generally, dT ~ 1–100 µm. In the caseof plastic targets, the main source of protons is hydrogen from the chemicalconstituents of the target. In the case of metal targets, protons come from hydrogenadsorbed on the target surface (from contaminants always found on the target unlessit is specially processed). Protons can be generated both towards the laser (backwards;from the surface hit by the laser beam) and in direction of laser beam propagation(forwards; from the back of the target). More favorable energetic and spatial protonbeam characteristics are obtained for the case of a forward generated beam and thiswill be considered as the basic variation.
At IPPLM, a double-layer system for fast proton generation has been proposed[36]. The concept is presented in Fig. 5. An ultrashort laser pulse directly interactswith a high atomic number (Z) layer, creating a plasma at its surface. Ponderomotiveforces generated by the laser pulse and other mechanisms accelerate some of theelectrons to high velocities. These “hot” electrons have temperatures Th of~ 104–106 eV (at I ~ 1017–1020 W/cm2). Hot electrons are emitted both backwards(towards the laser) and forwards (towards the target back surface) in the form of a shortduration pulse τe ≈ τL. Electrons propagating forwards partly ionize the low-Zhydrogen containing target layer and, in particular, the hydrogen on the back surfaceof this layer. The cloud of hot electrons, flying out of the rear side of the target, createsa so-called Debye sheath at a distance λDh from the back surface of the target. Thesheath is negatively charged and functions as a virtual cathode. It produces a strong
Fig. 5. Double-layer system for laser generation of fast protons.
12 J. BADZIAK et al.
electric field, which additionally ionizes the hydrogen on the target back surface andaccelerates the protons formed at this surface. If the density gradient scale-length Liat this surface is smaller than λDh (the surface is smooth and undisturbed), the fieldstrength is
(3)
assuming a typical value for λDh of ~ 1 µm and the values for Th of ~ 104–106 eV givenearlier. Proton acceleration takes place during the time tac ~ 1 ps over a distanceLac ~ 10 µm [16, 40]. As a result, the protons acquire the energy
Ep ~ eεacLac ~ 0.1–10 [MeV] (4)
and are mainly generated perpendicular to the back target surface. As research at IPPLM has shown [36, 38], double-layer systems make it possible
to obtain higher proton energies and current densities than with single-layer systems.The investigations used a set-up similar to that in Fig. 1, with the parabolic focusingmirror replaced by an aspheric lens. The characteristics of forward emitted ion fluxesfrom various kinds of single-layer and double-layer targets at I ≤ 2×1017 W/cm2 weremeasured using ion collectors and an electrostatic ion-energy analyzer. Additionally,the amplitude of the hard (4–30 keV) X-rays emitted from the target was measuredusing semiconductor detectors.
In Figure 6, both the mean and maximum proton energies as well as the protoncurrent density at 1 m from the target are shown as a function of the amplitude of thehard X-ray signal [36]. Measurements were performed for targets of various thickness,
εac
Th
eλDh
--------------- 104–10
8[V/µm]∼≈ 10
8–10
10[V/cm]=
Fig. 6. Mean and maximum laser-generated proton energies and proton current density at 1 m from thetarget as a function of hard X-ray signal amplitude.
Fast ion generation by a picosecond high-power laser 13
atomic number Z, and structure. Layer composition and thickness are denoted asm1t1/m2t2, where m1 and t1 are the composition and thickness (in µm), respectively, ofthe first, or only, target layer and m2 and t2 those of the second layer of double-layertargets. Thus, Au0.05/PS2 designates a double-layer target with a 0.05-µm gold frontlayer and a 2-µm polystyrene back layer. As can be seen, the best proton beamparameters are obtained for the case of double-layer targets with a high-Z front layer(Au). In paper [36] it was proved that this is mainly caused by an increase in hotelectron generation efficiency with increasing target atomic number. Moreover, it hasbeen found that the proton beam parameters substantially depend on the totalthickness of the target [37, 38] (Fig. 7) and – in the case of double-layer targets – on
Fig. 7. Maximum and mean laser-generated proton energies and proton current density at 1 m from thepolystyrene target as a function of target thickness.
Fig. 8. Characteristics of proton beams emitted from Au/PS2 targets as a function of Au layer thickness.
a b
14 J. BADZIAK et al.
the thickness of the high-Z layer [38] (Fig. 8). With appropriate choice of targetparameters, the proton beam has low angular divergence (> 109 protons with energy> 0.1 MeV were recorded within a 3° angle [36]), which makes it possible to achieverelatively high proton current densities at a large distance from the target (Fig. 6). Onthe other hand, at a very small distance from the target (< 1 mm), proton current densitymay attain very high values, even at low laser pulse energies (EL < 1 J), and cansignificantly exceed the current densities achieved in conventional proton accelerators(see Sec. 5).
4. Heavy ion generation in picosecond laser-produced plasma
In the case of laser generation of heavy ions, i.e., ions of high atomic number (e.g.,Z > 50), we must deal with two problems: achieving high ion energies Ei and obtaininghigh charge states z. Depending on the potential application, the importance of eachof these problems could be different. For instance, for ion implantation ion energymatters most whereas the charge state is of minor importance. Conversely, when abeam of laser-generated heavy ions is used as an ion source for traditional accelerators,the issue of ion charge is essential.
The investigation of ion generation in laser-produced plasma, including heavy iongeneration, is inseparably linked to the investigation of laser-produced plasma ingeneral and laser-driven inertial fusion in particular, which is why it has been carriedout for over 30 years. Lasers generating ns and sub-ns pulses have usually been usedin these investigations, e.g., [3, 7–11, 41–46]. But, recently, ultrashort pulses have alsobeen used [25, 47–53]. For high-energy pulses, thick solid targets have been generallyused. As a result, only fluxes of backward-emitted (towards the laser) ions have beenmeasured. For ns or sub-ns laser pulses with energies of ~10–103 J and intensities of~1015–1016 W/cm2, maximum heavy ion (e.g., Ta, Au, Pb) energies are tens of MeVwhile maximum ion charge states in the far expansion zone (a large distance from thetarget) are ~50 [3, 8–11, 42–46]. For sub-ns laser pulses, the team of IPPLM and theInstitute of Physics, A.S.C.R. (Prague), among others, has obtained remarkable results.Experiments on this subject have been carried out in Prague, initially using the PERUNiodine laser (50 J/400 ps) [8–11, 41] and more recently with the kilojoule PALS iodinelaser (1.2 kJ/400 ps) [42–46].
As mentioned earlier, from the point of view of some future applications of laser-generated ion beams, lasers generating ultrashort pulses (≤ 1 ps) seem to be moredesirable than ns or sub-ns lasers. However, due to the short duration of the ultrashortpulse’s interaction with the target as well as rapid adiabatic cooling of the plasma,favoring the fast recombination of ions, achieving highly charged ion fluxes in the farexpansion zone with ultrashort pulses is harder than with long pulses.
The first experiment that demonstrated the possibility of producing a beam ofhigh-energy highly charged heavy ions using an ultrashort laser pulse of relativisticintensity was performed at the Rutheford Appleton Laboratory in England [49]. In this
Fast ion generation by a picosecond high-power laser 15
experiment, at I ≈ 5×1019 W/cm2, a flux of Pb ions with maximum charge statezmax = +46 and maximum energy ≈ 430 MeV was obtained. This is the highestion energy achieved so far using a laser.
The first experiment of this kind at non-relativistic intensities (≤ 5×1016 W/cm2),which produced Ta+38 and Au+33, was carried out at IPPLM [50]. This experiment wasperformed using an arrangement similar to that in Fig. 1 [50]. The source of theultrashort pulse (τL ≈ 1 ps) was a terawatt picosecond laser and the ion flux parameterswere measured using ion collectors and an electrostatic ion-energy analyser (IEA).The parameters of the ions produced from massive metal targets of medium (Al, Fe,Cu, Ag) and high (Ta, Au) atomic numbers were studied.
A diagram showing the maximum charge state of the ions produced as a functionof the Z number of the target, measured using the IEA (full circles), is presented inFig. 9 [50]. Fully stripped ions up to Z = 13 and highly charged ions of heavier elementsare visible. The measured charge states in the low-Z range of the diagram (Z ≤ 13) areconsiderably higher than those observed earlier [47] for high-contrast 170-fs laserpulses of 3×1017 W/cm2 intensity (thus of energy fluence on target close to that in ourexperiment). Besides longer pulse duration, which is usually advantageous forobtaining a higher degree of target ionization (both collisional and optical fieldionization rates are finite), the probable reason for this difference is the pre-plasmaformed by a prepulse on the target surface under the conditions of our experiment.This pre-plasma, whose thickness is estimated to be a few µm, can positively influencethe possibility of attaining high ion charge states in the far expansion zone, both forlow-Z and high-Z targets, for at least two reasons. First, it significantly increases theabsorption of light by the target [25, 54]. Second, it diminishes the cooling rate of theplasma [55], which, in turn, decreases the effectiveness of electron-ion recombination.
Figure 10 presents the maximum ion energy measured using the IEA, Emax, asa function of the Z number of the target [50]. Emax is not a smooth function of Z and alocal maximum occurs at low Z. The reason why low-Z ions from contaminants reach
Eimax
Fig. 9. Maximum charge state of ions measuredusing the IEA (•) as a function of target Z number.
16 J. BADZIAK et al.
higher velocities than heavier ions could be the higher z/A ratio of these ions (A is ionmass number).
We observed a tendency of the heavy ion current density to decrease with Z number,probably resulting from the increasing ion energy and the increasing ionization losseswith increasing Z-number. However, for both medium- and high-Z targets, quite highion current densities ≥ 1 mA/cm2 at 1 m from the target were recorded [50].
The most important reason for the high ion current density in the far expansionzone, in spite of the relatively low energy of the laser pulse, was the small angulardivergence of the ion beam. For all Z numbers, most of the high-energy ions, boththermal and fast, were confined within a cone angle less than 30º with the maximumof the emission normal to the target surface [50].
5. Production of ultrahigh current density ion beams
The important consequence of the fact that an ultrashort laser pulse interacting witha thin preplasma layer accelerates plasma with a density comparable to the criticaldensity nec (see Sec. 2), is the potential to produce very high fast ion current densitiesnear the target surface (in the laser focus) [56]. For a rough estimate of these currentdensities we can neglect the fact that, close to the critical surface, the ion density ni issomewhat higher in the overdense plasma region than in the underdense region andassume ni ≈ nic = nec /z. Taking j = zeniυi, where υi is the ion velocity, we obtain thefast ion current density close to the ion source
js ≈ enecυi = e(2/mp)1/2nec (Ei /A)1/2 (5)
where mp is the proton mass, Ei is the ion energy and A is the atomic mass number.For subrelativistic laser intensities (I << Irel), from (2) and (5) we obtain
Fig. 10. Maximum ion energy, measured using theIEA, as a function of the Z number of a target.Numbers near the points indicate the ion chargestates for which maximum energies were recorded.
Fast ion generation by a picosecond high-power laser 17
js ≈ 74(sz /A)1/2λ–1 I1/2 [A/cm2, µm, W/cm2]. (6)
For instance, at sz /A = 1, λ = 1 µm, and I = 1017 W/cm2, we get js ≈ 2.3×1010 A/cm2.To compare the absolute value of the fast ion current density at the ion source
obtained from the theoretical formula (5), with the value deduced from theexperiment we substitute Ei /A = 20 keV/nucl (Fig. 2a) in the formula, whichgives ≈ 3.1×1010 A/cm2 at λ = 1.05 µm. To roughly estimate on the basis ofour time-of-flight measurements, we use
(7)
where Qi is the total fast ion charge measured in the far expansion zone (Qi ≤ Qis, dueto possible recombination, where Qis is the fast ion charge exiting the ion source); τisis the duration of fast ion generation (roughly equal to the laser pulse duration τis ≈ τL[16], [18], [40]), and Ss is the area of the fast ion source. Since j (Θ ) was determinedwith a large error (Fig. 3a), we have calculated the fast ion charge passing throughthe IC1 collector aperture instead of Qi ( < Qi because the fast ions areemitted within an angle wider than the 3° seen by IC1) and we have taken Ss = Sf,where Sf is the area of the laser focal spot. Formula (7) then effectively becomes aformula for the lower limit of the current density
(8)
For ions emitted from the Au target, the lower limit of the current density in thefocus is ≈ 0.8×1010 A/cm2. So, the real value of for backward-emitted ionsshould be > 1010 A/cm2, which agrees with our theoretical predictions. A comparablevalue is expected for the current density of ions emitted forward (towards the targetinterior), as the lower velocity of these ions is compensated in part by their higherdensity.
Using the formula for we also calculated the fast ion current density at thelaser focus for the sub-joule 0.5-ns laser pulse and for high-energy (0.5 kJ)high-intensity (1016 W/cm2) 0.4-ns laser pulses from the PALS iodine laser system[46]. The 0.5-ns laser produces current densities 3 orders of magnitude lower andthe PALS laser at least one order of magnitude lower than the 1-ps pulse. The lowervalue of for a 0.4-ns pulse of a thousand times higher energy, compared to theenergy of the 1-ps pulse, can be understood if we realize that – apart from an over100 times longer ion generation time – in the sub-ns case the ions are pulled by hotelectrons from a plasma corona [3, 5] of relatively large thickness and low average iondensity << nic. A situation similar to that for the long-pulse case will occur forultrashort laser pulses with intense prepulses.
jsth
jsexp
jsth js
exp
j sexp
Qi τis⁄ Ss≈
QiIC1 Qi
IC1
js min,exp
QiIC1 τLSf.⁄≈
js min,exp
jsexp
jsexp
,
jsexp
jsexp
ni
18 J. BADZIAK et al.
The very high ion current density and ps duration of the fast ion pulse producedby the optimized skin layer interaction opens the prospect for fundamentally newexperiments in nuclear physics and inertial confinement fusion. In particular, a blockof DT plasma accelerated by a ps laser pulse towards the DT plasma interior seems tobe ideal for generation of a laser fusion ignition front. The plasma block acts like a(space charge neutral) DT ion beam for which the ignition condition in solid density(frozen) DT fuel of >1010 A/cm2 current density [57] for optimum ion energy (80 keV)can be easily achieved by ps laser pulses of I < Irel. For example, a ps laser pulse ofI ~ 0.3Irel can produce an ~ 80-keV DT ion flux with a current density ofjs ~ 5.5×1010 A/cm2 at λ = 1.05 µm or js ~ 5×1011 A/cm2 at λ = 0.35 µm. We see thatsuch extremely high ion current densities are attainable even at low energies (≤ 1 J)and subrelativistic laser pulse intensities, which can be easily generated at highrepetition rate. In particular, it opens a prospect for highly efficient DD or DT fusionin small-scale devices for, e.g., fast neutron production. Achievement of high (>>1)fusion gain through use of an optimized DT ion beam from the ps skin layer interactionin large-scale experiments can also be imagined, but it needs to be confirmed by furtherdetailed studies.
6. Summary
In this paper, the main achievements of our recent studies of fast ion generation inplasma produced by picosecond laser pulses of subrelativistic intensity have beenbriefly reviewed. In particular, we have:
– shown that essential differences exist between the properties and accelerationmechanisms of ion fluxes in plasmas produced by ultrashort (≤ 1 ps) and by long(> 0.1 ns) laser pulses,
– presented a novel, efficient method for USP-laser-driven fast proton generationutilizing double-layer targets,
– demonstrated the generation of intense fluxes of highly charged high-energyheavy ions, and
– demonstrated the production of ultrahigh ion current densities (> 1010 A/cm2)using low energy (< 1 J) ps laser pulses of subrelativistic intensity.
The results presented should be useful for development of highly efficient laser-driven ion sources for applications in accelerator technology, material science,medicine, inertial fusion science, and nuclear physics on a tabletop.
Acknowledgements – We would like to thank Dr. Frederick Boody of Ion Light Technologies GmbH forhis extensive editing of the manuscript. This work was supported in part by the International AtomicEnergy Agency in Vienna under Contract No. 11535/RO and by the State Committee for ScientificResearch (KBN), Poland, under Grants No. 2 PO3B 082 19 and No. 1 PO3B 043 26.
Fast ion generation by a picosecond high-power laser 19
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Received April 6, 2004in revised form September 17, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Near-IR diode laser-based sensor for remote sensing of methane leakage
HUANG WEI, GAO XIAOMING, LI XIAOYUN, LI WEIZHENG, HUANG TENG, PEI SHIXIN, SHAO JIE, YANG YONG, QU JUN, ZHANG WEIJUN
Environmental Spectroscopy Laboratory, Anhui Institute of Optics and Fine Mechanics, The Chinese Academy of Sciences, Hefei, Anhui, China 230031; e-mail: [email protected]
A tunable diode laser-based sensor for remote sensing of methane (CH4) leakage at a wavelengthof 1.65 µm was developed. The 1.65 µm distributed feedback (DFB) diode laser has a narrow linewidth and a wide tuning range. It was found that the sensor can detect a 10 cm3min–1 city-gas leakwith a sensor output equivalent to the range-integrated concentration over 100 ppm·m.
Tunable diode laser spectroscopy (TDLAS) [1–3] is a useful technique for the remotesensing of trace molecules in the atmosphere. In our laboratory, TDLAS systems formonitoring CO2 have been developed [4, 5].
Methane is a very important gas not only as one of the earth warming gases butalso as fuel. Natural gas contains mostly CH4 and is used widely in our daily life asprime energy-source material. From the viewpoint of safety, the detection of CH4leakage is important. Especially, a wide-area CH4 monitoring system is useful invarious places such as chemical plants, gas pipelines, and for preventing disasters inearthquake affected areas. TDLAS can determine the leak position and concentrationin a short time. Using the absorption band at 1.65 µm, a compact CH4 TDLAS systemcan be constructed with the desired performance for the detection of a gas leak withina radius of 200 m.
The absorption band around 3.39 µm is usually employed for the detection ofCH4 in the atmosphere (natural concentration level is about 1.6 ppm). The absorptionband around 1.65 µm is too small to detect atmospheric CH4. However, theconcentration level of CH4 in the atmosphere is dangerous only when it is more than5%. Therefore, the detection of 1000 ppm order is sufficient to avoid explosion. Insuch high-concentration monitoring, a very large absorption cross-section is not
24 H. WEI et al.
required. Though the absorption around 1.65 µm is 2% less than that at 3.39 µm,the sensitivity is sufficient for a gas sensor. The main attractive feature in the case of1.65 µm source is that a compact tunable near-IR light source such as distributedfeedback (DFB) diode laser is readily available. An additional advantage is that fastphotodetectors such as InGaAs PIN photodiode are also available in this region. Hence,this system can be used for the wide-area monitoring of CH4 leakage with a high spatialresolution (~15 m).
In the past tunable diode laser (TDL) sensors were based upon the use of multi-mode lead salt diode lasers in the mid- and far-IR spectral regions. These lasersaccessed the strong fundamental absorption bands of methane allowing sensitivedetection, but at a high cost in the complexity of the instrument. The lasers werecoupled to large monochromators to isolate individual lasing modes. Still morecomplex were to be the instruments used for liquid nitrogen cooling and in cooleddetectors. As the near-IR diode laser sources were futher developed and adopted bythe telecommunications industry it became possible to utilize the same sources andInGaAs detectors for near-IR overtone absorption spectroscopy. The near-IRdiode lasers were single mode devices and allowed the use of fiber optic couplingand transmission technology. The sensor we describe below is based upon thistechnology.
There are several commercially available, currently-o-the-shelf (COTS)technologies which we have taken advantage of in the development of this sensor.These technologies include telecommunications grade NIR DFB diode lasers, singlemode fiber optic components including splitters, collimators and patch cords. Thesensor architecture, calibration and resulting sensitivity are discussed below.
2. Diode laser-based absorption spectroscopy and wavelength modulation spectroscopy (WMS)
2.1. Absorption spectroscopy
The remote CH4 leakage sensor discussed in this publication is based on thequantitative absorption of a near-IR diode laser beam propagating through a gassample. The transmission is given by the Beer–Lambert relation:
(1)
where I(ν ) is the transmitted intensity at frequency ν after propagation through a pathlength, L (cm) is the initial intensity, S (cm/molecule) is the line strength of thetransition centered at ν0, g(ν – ν0) is the line shape function (cm) at pressure P andn is the number density of the absorbing species (cm–3). The peak absorptioncoefficient is given by:
(2)
TI ν( )I ν0( )-------------- Sg ν ν0–( ) g ν ν0–( ) n L–exp= =
α ν0( ) Sg ν ν0–( )n.=
Near-IR diode laser-based sensor... 25
The integrated absorption coefficient is:
(3)
Since by definition:
(4)
the line shape function g(ν – ν0), depends on the gas temperature and pressure and theabsorption line strength depends upon the gas temperature. The water concentration[CH4] in molecules/cm3 is given by:
(5)
2.2. Wavelength modulation spectroscopy (WMS)
The technology described above is called direct absorption measurements.Conventional direct absorption measurements have to resolve small changes in a largesignal. In comparison with direct spectroscopy, the benefits of modulationspectroscopy in TDLAS are two-fold. Firstly, it produces a difference signal which isdirectly proportional to the species concentration (zero baseline technique) and,secondly, it allows the signal to be detected at a frequency at which the laser noise issignificantly reduced. WMS has been used with tunable diode laser sources since theearly 1970s. The earliest TDL systems used a modulation frequency in the lowerkHz range and second harmonic detection. Today, a 50 kHz modulation with 100 kHzdetection is quite usual and, consequently, it is convenient to regard 100 kHz as thelimit of conventional wavelength modulated TDLAS. Modulation spectroscopy ischaracterized by ease with which diode lasers can be modulated.
Wavelength modulation spectroscopy involves modulation of a single mode laserbeam at a frequency much less than the width of the absorption line being measured.This modulated beam is then passed through a sample of the gas to be investigated andits intensity is measured using a photodetector. The photodetector output isdemodulated using a lock-in amplifier. For relatively weak absorption lines, Eq. (1)can be approximated as:
(6)
The time variation of the modulated laser frequency can be expressed as [4]
(7)
α Sn.=
g ν ν0–( )dν∞–
∞
∫ 1.=
n
I ν( )I ν0( )--------------- dνln∫–
SL------------------------------------------- .=
I I ν0( ) 1 Sg ν ν0–( ) n L– .=
ν t( ) ν0 δν ωmt( )cos+=
26 H. WEI et al.
where ωm is the modulation frequency and δν is the amplitude of the modulation.The time-dependent part of the intensity can then be expanded in a Fourier series [6]
(8)
where HN (ν0) is the N-th component of the Fourier series.After detection by the lock-in amplifier at the N-th harmonic of the modulation
frequency, the signal is proportional to [7]
(9)
where gN(ν ) is the N-th derivative of the line shape function. Due to the atmospherepressure used in our experiments, the line shape can, to a good approximation, beconsidered Lorentzian. The N-th derivative of a Lorentzian line can be written as [5]
(10)
where
(11)
(12)
Substituting the N-th derivative of a Lorentzian line into the expression for thesignal, given by Eq. (9), and evaluating this expression at line center, we find thatfor odd harmonics, the signal is equal to zero. These zero crossings at line center makeodd harmonic signals convenient for use as error signals to stabilize the laser frequency.While this null crossing can be predicted by considering the symmetry alone, theformulation above gives a quick and simple way of estimating the slope at these zerocrossings. The slopes are important in determining the precision with which thestabilization can be implemented. Other factors include noise levels in the feedbackloop and tuning characteristics of the laser. Most measurements in modulationspectroscopy are performed at 2ωm, and hence the term 2 f technology.
2.3. Sensitivity
For measurements made with the 2 f technique at frequencies below 100 kHz,low-frequency noise generally determines the detection limit. The minimum detectableabsorption is approximated by
g ν0( ) δν ωmt( )cos+ HN ν0( ) Nωmt( )cosN 0=
∞
∑=
SignalI0L
N!---------- δν N
nSgN ν( )=
gLN ∆ν
2π--------- 1–
N
yN 1+
--------------- C2k N,k 0=
N[ ] 2⁄
∑∆ν2
---------
2kν ν0–( )N 2k–
=
C2k N, 1–( )k N 1+( )!2k 1+
----------------------- N2k
,=
N2k N!
2k( )! N 2k–( )!---------------------------------------- .=
Near-IR diode laser-based sensor... 27
(13)
where B defines the frequency dependence of the laser excess noise and ranges between0.8 and 1.5, is the magnitude of the laser power fluctuations at 1 Hz in a 1 Hz detectionbandwidth: it is approximately proportional to the laser power and depends on theintrinsic noise of the diode laser and the measuring system. A value of Iex = I0 = 10–4
is typical.The minimum mixing ratio detectable in a given detection bandwidth is given by
(14)
where σ0 is the absorption cross-section at the centre of the monitoring line, L – thepath length, and Nt – the total molecular density of the gas. For the CH4 line selected,a detecting limit of a few ppm·m is estimated.
3. Sensor design
There are two basic spectroscopic parameters which govern the design of anyabsorption spectrometer when attempting to achieve a minimum number densitydetection sensitivity. The first is the absorption line strength S, at the anticipatedmeasurement temperature. The reported line strength value for the 1.65 µm absorptionfeature in the HITRAN database is 10–20 cm per molecule magnitude [8]. The secondimportant parameter is the collisional broadening parameter γ, for the bath gas ofinterest.
In the low pressure limit the absorption line shape is characterized by a Gaussiandistribution function with the temperature determining the Doppler width of theabsorption function. Doppler line shapes tend to be quite narrow with highly peakedabsorbance values. As the pressure of the sample gas rises collisional broadeningmechanisms begin to change the spectral width of the absorption line shape, addingLorentzian character. In the limit of low temperature and finite pressure the line shapewould be primarily Lorentzian. Lorentzian line shapes tend to be broad, with depressedpeak absorbance values. In a typical sensor operation the line shape falls somewherebetween Gaussian and Lorentzian, and is thus described by a Voigt line shape whichis a convolution of Gaussian and Lorentzian line shape functions.
3.1. Diode laser module
The diode laser module included a diode laser, a diode laser controller, a 1×2 fiberoptic splitter, a photodetector, and DC power supplies for the laser operation.
The laser is selected according to three criteria: absorption line sufficiently strongto obtain maximum sensitivity; mode characteristics of the laser emissions as cleanand narrow; absence of interference from other gases.
αmin
Iex
I0
--------- Bf
------ 1 2⁄
=
Mmin ppbv( )amin 10
9×σ0LNt
----------------------------=
28 H. WEI et al.
The diode laser operated at 1.65 µm and had a spectral line width of approximately2 MHz. The laser tuned at approximately 0.03 cm–1mA–1 and was typically scannedover 0.9 cm–1 to fully capture the methane lineshape. The laser injection currentand temperature were controlled by ilx LightWave current controller LDX-3100 andtemperature controller LDT-5100. The optically isolated fiber coupled output from thediode laser was split by a 70/30 single mode wavelength flattened fused fiber opticsplitter. The output from the 30% leg of the fiber splitter was coupled directly to anInGaAs reference photodiode located within the balanced detector. The 70% leg wasconnected to the fiber optic launch collimator to detect methane in air. After traversingthe absorption cell the signal beam was detected by a second, 3 mm in diameter,windowless InGaAs photodiode.
3.2. Computer module
The automated CH4 vapor sensor was controlled by a 1G MHz Pentium 3 industrialrack mounted computer. The system was outfitted with a 300 kHz data acquisition(DAQ) board AC6115 and a ADLink PCI-DIO-96 digital I/O (DIO) board. The DAQboard was used to control the laser diode controller and to digitize signals from thebalanced detector. The DIO board was used to implement software control of the digitalpotentiometer, LDX-3100 and LDT-5100. The sensor control software was developedusing National Instruments LabWindow/CVI software programming language. Thesoftware provided three separate graphical user interface (GUI) screens for sensoroperation including a main control and data presentation GUI, an oscilloscope GUI anda digital filter GUI.
3.3. Detector circuit
To decrease the effect of laser noise, we adopt difference balance detection scheme.Here we used a detection circuit which can reduce kinds of noises, background signal.The circuit has large dynamic range to meet the requirement of lock-in amplifer (LIA).A block diagram of the detection circuit is shown in Fig. 1. We established the circuitin our laboratory and compared it to New Focus 2011-FC high performance detector,and found that the performance of our home made detector is better than that of the2011-FC.
3.4. Schematic of sensor
The diagram of sensor is shown in Fig. 2. Tuning and modulation of the laser isaccomplished by changing injection current and temperature of the diode laser. Thefunction generator provides an output sine wave at a user specified amplitude andfrequency, which is used to modulate the laser beam. This sine wave is also used bythe LIA as an internal reference signal for synchronous detection of the signal. Themodulation frequency was chosen to be 3 kHz based on a noise analysis and the lasermodulation characteristics. The amplitude of the modulation was chosen to be at0.2 Vrms, which was found to provide the best SNR (signal-to-noise ratio). For
Near-IR diode laser-based sensor... 29
spectroscopic data a voltage ramp is also applied to sweep the laser across theabsorption lines being investigated. Part of the laser beam is sampled for wavelengthmeasurement with a Burleigh Wavemeter WA-1500.
4. Sensor performance as a gas-leak detector
In order to demonstrate the performance of the sensor as a practical gas-leak detector,we carried out an experiment concerning detection of simulated gas pipeline. Severalkinks of reflecting objects were selected as the target and were located within the rangeof 5 m with the angle of incidence 0°. We prepared a leak point using pure methanein front of the target and set the flow rate at 10 cm3min–1. This is the minimum flowrate to be detected in gas-leak detection. We found that the sensor can detecta 10 cm3min–1 gas leak with a sensor output equivalent to the range-integratedconcentration over 100 ppm·m.
We used an aluminum pipe, which is 30 cm-long and whose diameter is 25 mm,to simulate natural gas pipeline. For the main ingredient of natural gas is methane(90%), we chose methane as the main gas to be detected in our labortory. Theexperimental setup is shown in Fig. 4. The reflecting object we used in Fig. 4 includingbrick, wood plate, etc. We found that most reflecting objects have strong diffusereflection character and are not sensitive to incidental angle. The typical reflectingobjects we used are brick and wood board.
Fig. 3. Schematic of pipeline simulation.
Fig. 4. WMS signal of different reflecting objects: brick (a), wood board (b), bush leaf (c), cementplate (d), dry sod (e), humidity clay (f).
a b c
d e f
Near-IR diode laser-based sensor... 31
The 2 f signals of different reflecting objects are shown in Fig. 4. We can concludefrom the experiment and Fig. 4 that:
– If the reflectance of the reflector objects varies, the amplitudes and SNR of 2 fsignals are different, too. The higher the reflectance, the higher the SNR. For example,the reflecting signal of brick (Fig. 4a) and wood board (Fig. 4b) are greater than thatof hay (Fig. 4e) and clay (Fig. 4f).
– The amplitude of 2 f signal has strong relation to the reflectance of reflector andweak relation with the kinds of reflector.
– From the figure we can see that 2 f signals of different reflectors with the sameleak rate of methane differ from each other, so the reflectance of reflector is a keyfeature to natural gas pipe leakage detection.
5. Conclusions
We have developed a remote methane sensor using a 1.65 µm InGaAsP DFB laser.Wavelength modulation spectroscopy technology is used to get very high sensitivity.The sensor can detect a 10 cm3min–1 city-gas leak within a range of several meters.For certain applications, such as methane leaks in coal mines, the sensitivity is morethan adequate (i.e., orders of magnitude lower than the explosive limit can be detected).Applications for other gases can evidently be envisaged. If we change the diode laserto a diode laser at other wavelength, for example, at 1.39 µm, we can use the sensorto detect water vapor.
So, in fact, we developed a sensor that can be used to detect H2O (1.39 µm), CO2(1.57 µm), NH3 (1.53 µm) and NO2 (0.79 µm), etc.
Acknowledgements – This work was supported by National High Technology Project.
References
[1] MANTZ A.W., A review of the applications of tunable diode-laser spectroscopy at high-sensitivity,Microchemical Journal 50(3), 1994, pp. 351–64.
[2] MANTZ A.W., A review of spectroscopic applications of tunable semiconductor lasers, SpectrochimicaActa, Part A: Molecular Spectroscopy 51A(13), 1995, pp. 2211–36.
[3] WERLE P., A review of recent advances in semiconductor laser based gas monitors, SpectrochimicaActa, Part A: Molecular and Biomolecular Spectroscopy 54A(2), 1998, pp. 197–236.
[4] GAO X.M., HUANG W., LI Z.Y., FANG L., ZHANG W., Sensitive detection of CO2 molecule using nearinfrared diode laser absorption spectroscopy, Acta Optica Sinica 23(5), 2003, pp. 609–11.
[5] HUANG W., GAO X.M., ZHANG W.J., PEI S.X., QU J., SHAO J., LI X.Y., YANG Y., LEI L.Q., Study ofnear-infrared tunable diode laser spectrometer, Chinese Journal of Lasers 31 (Supplement), 2004,pp. 263–6.
[6] REID J., LABRIE D., Second-harmonic detection with tunable diode lasers-comparison of experimentand theory, Applied Physics B: Photophysics and Laser Chemistry B26(3), 1981, pp. 203–10.
32 H. WEI et al.
[7] DHARAMSI A.N., A theory of modulation spectroscopy with applications of higher harmonic detection,Journal of Physics D: Applied Physics 29(3), 1996, pp. 540–9.
[8] ROTHMAN L.S., BARBE A., CHRIS BENNER D., BROWN L.R., CAMY-PEYRET C., CARLEER M.R.,CHANCE K., CLERBAUX C., DANA V., DEVI V.M., FAYT A., FLAUD J.-M., GAMACHE R.R., GOLDMAN A.,JACQUEMART D., JUCKS K.W., LAFFERTY W.J., MANDIN J.-Y., MASSIE S.T., NEMTCHINOV V.,NEWNHAM D.A., PERRIN A., RINSLAND C.P., SCHROEDER J., SMITH K.M., SMITH M.A.H., TANG K.,TOTH R.A., VANDER AUWERA J., VARANASI P., YOSHINO K., The HITRAN molecular spectroscopicdatabase: edition of 2000 including updates through 2001, Journal of Quantitative Spectroscopy andRadiative Transfer 82(1-4), 2003, pp. 5–44.
Received September 18, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Application of Wigner transform for characterization of aberrated laser beams*
JAROSŁAW JAGUŚ, JAN K. JABCZYŃSKI, WALDEMAR ŻENDZIAN, JACEK KWIATKOWSKI
Institute of Optoelectronics, Military University of Technology, ul. Kaliskiego 2, 00-908 Warszawa, Poland; e-mail: [email protected]
The slit scan method was implemented for registration of intensity profiles along the caustics ofa laser beam. The inverse Radon transform of spatially normalized intensity profiles enables directcomputation of Wigner transform of real laser beam. The Rayleigh range, divergence angle, beamquality factor, global degree of coherence can be determined in such a simple way. A procedureenabling derivation of the shape of aberrated wavefornt and spherical aberration content waselaborated. This method was applied for investigation of the aberrated laser beams generated bycw and pulsed diode pumped lasers.
Quantitative characterization of spatial structure of a laser beam has been of vitalinterest to opticians and laser physicists since the advent of lasers. Because of inherentuncertainty of parameters of such a type of light source caused by its spatial andtemporal fluctuations as well as the state of coherence and polarization, it has been anattractive subject of intensive theoretical research as well as measurement andexperimental works. The well established simplest parameter describing the spatialproperties of laser radiation, i.e., the beam propagation factor M2 introduced bySIEGMAN [1], was accepted by ISO [2] as a measure of beam quality. The measurementsof M2 parameter can lead to some ambiguities, especially for untypical, asymmetricbeams, moreover it can be easily shown that quite different light beams can have thesame value of M2 parameter. Thus, additional parameters describing the state ofcoherence and wavefront aberrations should be defined.
To completely describe the properties of partially coherent light, the formalism ofmutual coherence function or cross-spectral density function can be applied [3]. The
*This work was presented at XIV Slovak-Czech-Polish Optical Conference in Nitra (Slovakia),13–17 September 2004.
34 J. JAGUŚ et al.
alternative approach (however, basically connected with the previous ones via Fouriertransform) is known as a Wigner distribution method (WDM), see [4–8]. The mainadvantage of WDM is the simultaneous access to intensity and phase distributions inthe far field (defined over u-angular frequency space) and in the near field (definedover x-spatial coordinate space). Moreover, there exist at least two, well-establishedexperimental procedures [6, 7] enabling direct access to Wigner distribution (WD)from experimental data. In the first one, Sagnac interferometer with inversion of fieldby means of Dove’s prism is applied [6], Wigner signal being the autocorrelation ofincidence electric field is collected on a wide area detector for the given x-spatial andu-angular positions of input mirror. In the second method, a typical set-up formeasurements of intensity distributions in the caustics can be applied. Basically, bothmethods require the measurements of intensity distributions in 2D space for 1Dgeometry of incident beam and 4D for 2D geometry. It was shown by EPPICH andRENG [7] that the Wigner distribution can be found as the inverse Radon transform ofintensity distribution in the caustics. Knowing Wigner distribution of a laser beam, thebeam quality parameter M2, the spatial coherence degree K2 and the coherence lengthcan be calculated [9, 10]. The WDM can also be applied to derive deterministicwavefront aberrations of a laser beam [8, 11, 12].
The goal of this work is to implement the Wigner transform for characterizationof aberrated beams generated by diode pumped lasers. The main properties of Wignertransform and method of wavefront retrieval are described in Sec. 2. In Section 3, theexperimental set-up for intensity measurements in caustics, based on the slit scanmethod is presented. The procedure of WDM was tested on several beams generatedby cw and Q-switched diode pumped lasers operating at 1064-nm and 1340-nmwavelengths.
2. Theory
2.1. Properties of the Wigner transform
For simplicity, our analysis will be limited to 1D geometry. The Wigner distribution(WD) function F(x, u) for partially coherent beam is defined as follows:
(1)
where Γ (x, s) is the mutual coherence function defined in the following way:
(2)
is the statistical averaging over time or ensemble, E(x) – the amplitude of electricfield at point x, s – the correlation spatial variable (x1 = x + s/2, x2 = x – s/2), u – theangular frequency, k = 2π/λ is the wavenumber and λ is the wavelength. A very useful
F u x,( ) λ 1– Γ x s,( ) iksu–( )dsexp∫=
Γ x s,( ) E x s 2⁄+( ) E* x s 2⁄–( )⟨ ⟩ E x1( ) E* x2( )⟨ ⟩= =
...⟨ ⟩
Application of Wigner transform... 35
property of WD is the simple transformation rule in the first order systems describedby ABCD matrix:
(3)
Thus, to completely describe propagation of partially coherent beam it is necessaryand sufficiently to know the WD in one, arbitrarily chosen, incidence plane. Theprojections of WD on x and u subspaces give the intensities in near and far fields,respectively:
(4)
The global coherence degree K2 can be defined as follows:
(5)
where P is the beam power given by The parameter
M2 can be defined in WDM as a product of beam radii in near and far fields [1]:
(6)
where is the n-th moment of intensity distribution.
2.1.1. Properties of Gauss–Schell model beam
For Gauss–Schell model (GSM) of partially coherent beam the mutual coherencefunction is given as follows:
(7)
where WGS is the radius of beam, ρGS is the coherence radius of beam. The parametersM2 and K2 for GSM beam are given as follows:
(8)
Fout x u,( ) Finp Dx Bu Cx– Au+,–( ).=
Inf x( ) F x u,( ) du,∞–
∞
∫=
Iff u( ) F x u,( ) dx.∞–
∞
∫=
K2
P2– Γ x s,( ) 2
dx ds∫∫=
P Γ x 0,( ) dx∫ I x( ) dx.∫= =
M2 π
λ------ 4σxσu,= σx x x⟨ ⟩–⟨ ⟩ 2
,= σu u u⟨ ⟩–⟨ ⟩ 2=
xn⟨ ⟩ P
1–x
nI x( ) dx∫=
ΓGS x s,( ) 2x
WGS
-------------
2
–12--- s
ρGS
-----------
2
–exp=
M GS2 WGS
ρGS
------------- ,= K GS2 ρGS
WGS
------------- .=
36 J. JAGUŚ et al.
The GSM beam is an “eigen”-function in transformation in the first order systems.The propagation law of GSM beam in free space is given by:
(9)
where W0, GS and ρ0, GS are the beam and coherence radii in waist plane, respectively,ZR, GS is the Rayleigh range given by:
(10)
where θGS is the divergence half angle of GSM beam TheWD function for GSM beam is given as:
(11)
For fully coherent GSM beam, i.e., we have the formulae describingthe properties of a Gaussian beam in terms of WDM. The GSM beam minimizes theproduct of beam quality M2 and coherence degree parameter K2:
(12)
2.1.2. Wavefront analysis in WDM
For real laser beams the following “laser beam optics principle” can be formulated [8]:
(13)
The laser source having for the given M2 the lowest value of the coherenceparameter K2 is the most “randomly” ordered and it has the smoothest profiles. In sucha case, the radiation is close to GSM beam and no deterministic phase deviations occur.On the other hand, when the M2K2 product is high, this means that some level ofdeterministic amplitude or phase modulation exists, which can be basically removed.Thus, from basic as well as practical points of view, it is important to determine thedeterministic wavefront aberration content for the given laser beam. The methods of
W GS2
z( ) W 0 GS,2
1z
ZR GS,-----------------
2
+
,=
ρ GS2
z( ) ρ 0 GS,2
1z
ZR GS,-----------------
2
+
=
ZR GS,W0 GS,
θGS
------------------πW 0 GS,
2
M GS2 λ
----------------------= =
θGS M GS2 λ πW 0 GS,⁄=( ).
FGS x u,( ) 2x
WGS
-------------
2
– 2u
θGS
------------
2
– .exp=
M GS2
1=
M GS2
K GS2
1.=
M2
K2
1.≥
Application of Wigner transform... 37
wavefront measurements can be divided, with respect to the principle, into two groups:interferometric and direct methods. The one of the well established representatives ofthe latter group, Hartmann–Shack method (see, e.g., [13]) gives the simultaneousdetermination of phase and amplitude distribution for a single shot beam. The WDMoffers an alternative way to wavefront analysis [8, 11, 12]. The basic principles of thisapproach, valid for 1D geometry of an incident beam are presented below. Thetransversal Poynting vector component St(x) for the given WD is defined as follows:
(14)
The ray aberration, i.e., the angle of ray with respect to the propagation axisU(x) ≅ sin(U(x)) can be found as a ratio of the Poynting vector St(x) to intensity in thenear field Inf(x) as follows:
(15)
Knowing the ray aberration vector U(x) we can calculate (see, e.g., [14]) smallangle approximation of the wavefront aberration φaber(x) as follows:
(16)
The feasibility of this approach was examined for a laser beam with a priori knownaberration by NEUBERT et al. [11, 12].
2.2. Eppich’s method of Wigner transform measurements
Firstly, we have to note that the analysis presented below is valid for 1D geometry(i.e., axially symmetric beam), however, the generalization to 2D geometry isstraightforward [8]. The main concept of Eppich’s method consists in application ofthe properties of Fourier and Radon transforms. He has shown [7] that the WD is theinverse Radon transform of specifically transformed 1D intensity distributionscollected for several zk locations in the caustics of a laser beam. The main idea of theprocedure is presented in Fig. 1. In the first step, the 1D intensity profiles as functionsof x-coordinate are measured for several zk locations along the propagation axis in thevicinity of caustics. The effective parameters of the beam, i.e., Rayleigh range, waistlocation and the M2 parameter are calculated according to ISO procedure [2]. For theset of intensity plots, the transformation to Gouy’s space is realized as follows:
(17)
St x( ) F x u,( )u du.∫=
U x( )St x( )Inf x( )----------------
F x u,( )u du∫F x u,( ) du.∫
----------------------------------= = .
U x( )∂φaber
∂x----------------- ,≅ φaber x( ) U t( ) dt.
∞–
x
∫≅
I x zk;( ) I x αk,( ),→ xx
wk
-------- ,= αk
zk z0–
ZR
------------------
,atan= wk
w0
αkcos---------------=
38 J. JAGUŚ et al.
where ZR is the Rayleigh range, z0 is the waist location, w0 is the beam radius in thewaist plane. It can be shown that the k-th normalized intensity plot corresponds tothe WD radial section in direction inclined at an angle αk to the x-axis (see Fig. 1).Thus, we have direct access to the values of WD data points defined in cylindricalsystem of coordinates. The return to the Cartesian system of coordinates (x, u) isperformed by means of the inverse Radon transform. Because of numericalimplementation of inverse Radon transform algorithm, the intensity data rows in anormalized Gouy’s space have to be equidistant with respect to Gouy’s angles αk.Thus, the appropriate values of zk locations should be chosen in the process of intensityregistrations, or the additional interpolation in Gouy’s space has to be done. Let usnote that we have no access to intensity in the far field (z/ZR → ∞ or α → 90°) withouttransformation through additional lens. It was found in experimental practice that toensure the sufficient accuracy in WD calculations, the number of zk sections shouldbe greater than 20, and the range of Gouy’s angles should be at least 135° correspondingto the range of ±3ZR in the distance along caustics.
3. Experiment
3.1. Laboratory set-up for WDM
The laboratory set-up is described in detail in paper [15]. Some brief information isgiven below. To ensure a satisfactory accuracy and reasonable size of laboratoryset-up, the laser beam under examination was focused to the 0.2–0.5 mm widthapplying thin, ideal lens of long focal length (typically, f = 300–500 mm). The beamwidths which we have to measure ranged from 0.2 up to 3 mm, the length of z-scanwas of a few dozens cm. We have decided to apply a slit scan method to measure the1D intensity plots, assuming axial symmetry of the beam examined. The slit witha variable width (10–20 µm) attached to large area detector was moved across the beamby means of a step motor with 2.5 µm resolution. The digitized signal with 12-bitresolution was sent via 841-Optel controller to PC computer. The knife edge definitionof a beam width with 10% clip level was applied to find the effective parameters of
Fig. 1. Idea of Wigner transform derivation from intensity in caustics measurements.
Application of Wigner transform... 39
convergent laser beam [2]. The typical values of Rayleigh ranges were of a fewdozens mm. The numerical procedure of WDM for experimental (as well astheoretical) series of 1D intensity data vectors was implemented in MATLAB v.5.3.
3.2. Measurements of laser beams
The main task of WDM set-up was to examine parameters of the beams generated bydiode pumped lasers. We have tested it at 1064-nm and 1340-nm wavelengths for cwand pulsed regimes of operation. Two examples of the near diffraction limited butaberrated laser beams are shown in Figs. 2, 3. The left contour map corresponds to 2Dintensity distributions in caustics as functions of horizontal x-axis and vertical z-axis.The middle picture presents the same beam after normalization and transformation toGouy’s space: x – horizontal axis, α – Gouy’s angle vertical axis, the right-hand picture
a b
c
Fig. 2. Intensity plot I(x:zk) in the caustics of laser beam, M2 = 1.16 (a). Normalized intensity plot I(x, αk)of laser beam in figure a (b). Wigner distribution F(x, u) of I(x:zk); inverse Radon transform of I(x, αk) (c).
Fig. 3. Intensity plots I(x:zk) in the caustics of laser beam, M2 = 1.52 (a). Normalized intensity plot I(x, αk)of laser beam in figure a (b). Wigner distribution F(x, u) of I(x:zk); inverse Radon transform of I(x, αk) (c).
a b
c
40 J. JAGUŚ et al.
presents the Wigner distribution of this beam (vertical axis corresponds to u-axis,horizontal to x-axis). The laser beam (M2 = 1.16, w0 = 0.105 mm, ZR = 28.2 mm)generated by Nd:YVO4 laser in V-type cavity at pump power Pp = 6.2 W is presentedin Fig. 2a–c. A different laser beam (M2 = 1.52, w0 = 0.082 mm, ZR = 13.9 mm),generated by the same laser at 18-W pump power is presented in Fig. 3a–c. Thecorresponding plots of intensity and wavefront aberration are given in Fig. 4a, b. Asa result of negative or negligible values of the calculated near field intensity, the valuesof ray and wavefornt aberrations can be undetermined at certain points in the wingsof a beam. A reliable method of a wavefront aberration derivation in WDM requireshigher accuracy in measurements and improvement in inverse Radon transformalgorithm.
4. Conclusions
The presented method and experimental set-up enable qualitative and quantitativecharacterization of aberrated laser beams. It was tested successfully on several beamsgenerated by diode pumped lasers. The shape of Wigner distribution can give someintuitive information on the deviation of the beam examined apart from Gauss–Schell
Fig. 4. Near field intensity (solid line), wavefront aberration (dashed line) vs. x, for the same beam as in:Fig. 2 (a) and in Fig. 3 (b).
a
b
Application of Wigner transform... 41
model. The wavefront analysis can lead to some ambiguities and requires furtherinvestigation. The WDM can offer an additional valuable tool for characterization oflaser beams.
Acknowledgements – This work was supported by the Polish Committee for Scientific Research (KBN)under the project 4T11B02724, PBZ-MIN-009/T11/2003. We would like to thank Dr. Z. Zawadzki forhis fruitful discussion on the results.
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WG1, ISO Geneva 1999.[3] MANDEL L., WOLF E., Optical Coherence and Quantum Optics, 2-nd Ed., Cambridge University
Press, England 1994.[4] WIGNER E., On the quantum correction for thermodynamic equilibrium, Physical Review 40, 1932,
pp. 749–59.[5] BAASTIANS M.J., Wigner distribution function and its application to first-order optics, Journal of the
Optical Society of America 69(12), 1979, pp. 1710–16.[6] MUKAMEI E., BANASZEK K., WAMSLEY I.A., DORRER C., Direct measurement of the spatial Wigner
function with area-integrated detection, Optics Letters 28(15), 2003, 1317–19.[7] EPPICH B., RENG N., Measurement of the Wigner distribution function based on the inverse Radon
transformation, Proceedings of the SPIE 2375, 1995, pp. 261–8.[8] EPPICH B., JOHANSSON S., LAABS H., WEBER H., Measuring laser beam parameters, phase and spatial
coherence using the Wigner function, Proceedings of the SPIE 3930, 2000, pp. 76–86.[9] EPPICH B., Definition, meaning and measurement of coherence parameters, Proceedings of the SPIE
4270, 2001, pp. 71–9.[10] EPPICH B., MANN G., WEBER H., Spatial coherence: comparison of interferometric and
non-interferometric measurements, Proceedings of the SPIE 4969, 2003, pp. 137–48.[11] NEUBERT B.J., HUBER G., SCHARFE W.-D., [In] Instruments and Standard Test Procedures for Laser
Beam and Optics Characterization, 2003, p. 44.[12] NEUBERT B.J., Measurements of the Wigner Distribution of Aberrated and Partially Coherent Laser
Beams, Cuvillier Verlag, Goettingen 2004.[13] SCHAFER B., MANN K., Determination of beam parameters and coherence properties of laser
radiation by use of an extended Hartmann-Shack wave-front sensor, Applied Optics 41(15), 2002,pp. 2809–17.
[14] BORN M., WOLF E., Principles of Optics, Pergamon Press, Oxford 1965.[15] JAGUŚ J., M.Sc. Thesis, Military University of Technology, Warszawa 2004 (in Polish).
Received September 29, 2004in revised form November 23, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Research of compound Nd:YAG phase-conjugate resonator with stimulated Brillouin scattering (SBS) cell
2Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China
How to improve the quality of output beam of phase conjugation cavity is a problem. In this paper,the output characteristic of compound phase-conjugate resonator with stimulated Brillouinscattering (SBS) cell was reported. The experimental results indicated it has high output energyand excellent stability. The output energy of the single pulse can reach 90 mJ, and the width of itis about 25 ns. The results of pulse widths, output energies and beam profiles were given.
Optics phase conjugation based on stimulated Brillouin scattering (SBS) is frequentlyapplied in pulsed solid-stated lasers to improve the beam quality through compensationof pump-induced phase distortions in the active material [1–4]. Such phase-conjugating SBS mirrors has been realized in gases, liquids, and solids. In general,the threshold of phase conjugation resonator (PCR) is high with the output energy ofit is lower. How to decrease the threshold and how to increase the output energy ofPCR are the tasks that people cared for. The way of decreasing the threshold of PCRis using optical feedback [5], using a multipass Herriott cell [6], or ring resonator [7, 8].Also, an internally tapered optical fiber can decrease the threshold to 15 µJ [9]. In thispaper, we have designed and experimented a novel compound SBS phase conjugationresonator with SBS cell on Nd:YAG laser which has not been reported before. Theexperimental results show that the cavity of this type can output stability selfQ-switched single pulse, while the quality and stability of the output beam is superiorto those provided by linearity cavity.
44 J. QU, W. ZHANG, X. GAO
2. Experiment setup and theory
The experimental arrangement of the cavity is shown in Fig. 1. The center wavelengthof coating of high reflecting mirror M1, M3, M4 is 1.064 µm, the bandwidth of whichbeing 100 nm. The reflectivity of mirror M2 is 20%, the F1 and F2 are thin lenses, the
focal length of them is 55 mm. The SBS cell of the length 100 mm is filled withacetone. The beam splitter BS has the splitting ratio 1:1 at the angle of π/4. The lengthof Nd:YAG rod is 70 mm, and its diameter is 6 mm. Xenon lamp pump were employed.The light path of the compound cavity is very complex, the assistant cavities of twoside couplings with each other.
3. Experimental results
At first, the beam was focused into 100 mm length SBS cell containing acetone bymeans of two 55 mm focal length lenses. The results of single pulse whose width isabout 25 ns appeared steady when the pump voltage is 800 V, and the length of phase
Fig. 2. Output energy of the compound cavity and its fluctuation.
Research of compound Nd:YAG phase-conjugate resonator... 45
conjugation resonator LPCR is nearly 700 mm at this time. The output energy is about20 mJ, and it increases with the pump voltage. The output energy and its fluctuationare shown in Fig. 2, and the single pulse output stability with the higher pump voltagein Fig. 3, respectively.
The characteristic of the linear cavity was studied at the absence of the BS in thesetup shown in Fig. 1. The experimental results indicated that the single pulse appearedas the pump voltage is 750 V, and the output energy increases with the pump voltage,but the stability of it was bad comparing to that for the compound cavity, and someburr was found in the single pulse at the high pump voltage, and also a few pulsesalternation by few microseconds appeared (Fig. 4). Higher and lower pump voltagecould cause the fluctuation of the output energy(Fig. 5). We have additionally capturedthe far field (3 m) spots of the cavity as the pump voltage was 900 V, and we can see
Fig. 3. Pulse of compound cavity as the pump voltage is 1000: expandedness of single pulse (a) and thepulse (b).
a b
Fig. 4. Pulse of line cavity as the pump voltage is 1000: the pulse (a); expandedness of single pulse (b).
a b
46 J. QU, W. ZHANG, X. GAO
that the divergence angle of the compound cavity is smaller than that of the linearcavity (Fig. 6).
It is very difficult to adjust the light path of compound cavity, since the light of themain cavity and the assistant cavity of two side should coaxal stringent. The outputenergy would decrease prodigious as the mirror M3 or M4 was taken away.
4. Summary
In the summary, compound SBS resonator is a novel PCR which has higher energyoutput along with steady passive Q-switched pulse. The width of it is 25 ns. It canoutput 90 mJ energy and the divergence angle of output beam is smaller than 1.5 mrad.
Acknowledgments – This research was supported by the Laser Technology Innovation Fundation of863 (No. 20030509) and the natural science foundation of Anhui provincial educational committee(No. 2005kj236).
Fig. 5. Output energy of the line cavity and its fluctuation.
Fig. 6. Profile of far-field spots: line cavity (a); compound cavity (b).
a b
Research of compound Nd:YAG phase-conjugate resonator... 47
Reference
[1] DAVYDOV M.A., KOSHEVNIKOVA I.N., Laser-pulse compression by stimulated brillouin scattering inliquids, Physics Letters A 127(6-7), 1988, pp. 345–6.
[2] GE CHUANWEN, ZHANG WEIJUN, CHEN CHANGSHUI, WANG PEI, SU HONG, WANG ZHEN, HANG YIN,Experimental investigation of broadband laser’s SBS based on frequency-band-dispersing method,Acta Opical Sinica 21(12), 2001, pp. 1454–7.
[3] QU J., ZHANG W., GAO X., LIU A., HUANG W., PEI S., SHAO J., YANG Y., Fluctuation of colliding-enhanced YAG phase-conjugate ring cavity in primary resonator stability, Optica Applicata 33(4),2003, pp. 649–53.
[4] QU J., ZHANG W., GAO X., LIU A., HUANG W., PEI S., The investigation of colliding-enhanced YAGphase-conjugate ring resonator, Acta Opica Sinica 24(4), 2004, pp. 495–8.
[5] WONG K.N., DAMZEN M.J., Investigations of optical feedback used to enhance stimulated scattering,IEEE Journal of Quantum Electronics 26(1), 1990, pp. 139–48.
[6] DUIGNAN M.T., FELDMAN B.J., WHITNEY W.T., Threshold reduction for stimulated Brillouin scatteringusing a multipass Herriott cell, Journal of the Optical Society of America B: Optical Physics 9(4),1992, pp. 548–59.
[7] SCOTT A.M., WHITNEY W.T., DUIGNAN M.T., Stimulated Brillouin scattering and loop thresholdreduction with a 2.1 um Cr, Tm, Ho:YAG laser, Journal of the Optical Society of America B: OpticalPhysics 11(10), 1994, pp. 2079–87.
[8] SCOTT A.M., WHITNEY W.T., Characteristics of a Brillouin ring resonator used for phase conjugationat 2.1 um, Journal of the Optical Society of America B: Optical Physics 12(9), 1995, pp. 1634–41.
[9] HEUER A., MENZEL R., Phase-conjugating stimulated Brillouin scattering mirror for low powersand reflectivities above 90% in an internally tapered optical fiber, Optics Letters 23(11), 1998,pp. 834–36.
Received May 12, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Highly sensitive diode laser absorption measurements of CO2 near 1.57 µm at room temperature
Anhui Institute of Optics and Fine Mechanics, The Chinese Academy of Sciences, 230031 Hefei, China;e-mail: [email protected]
The absolute absorption spectrum intensities of carbon dioxide sample have been recorded witha tunable diode laser spectrometer in the spectral range 6350–6364 cm–1, which is suitable for thein situ sensing of carbon dioxide in the lower stratosphere using a commercial telecommunication-type diode laser. It was found that the typical uncertainty of experimental line intensities is about1% compared with the values listed in the HITRAN database, which are calculated by directnumerical diagonalization (DND).
Carbon dioxide is one of the most important minor components in the atmosphere andthe second greenhouse gas after atmospheric water, and it contributes greatly to theglobal warming of the atmosphere [1]. As a result of human activity (in particular,combustion), its concentration has continuously increased during the last century [2].In order to retrieve the concentration profiles of this minor constituent in theatmosphere, accurate line parameters (i.e., positions, intensities and self-broadeningcoefficients) of carbon dioxide transitions are required [3, 4]. The improvement ofthese line parameters listed in the spectroscopic database is always the central interestin atmospheric spectroscopy [5]. Recently, we undertook an accurate line parametersmeasurement of CO2 around 1.31 µm and found a few new spectroscopy lines notreported in HITRAN database. The ability of this set of parameters in predictingunmeasured transitions has been discussed and has been found to have high sensitivityand resolution in absorption spectroscopy research [6].
The line parameters of the prominent CO2 absorption bands located near 1.57 µmare particularly useful for tropospheric studies. As a result, high-resolution laboratoryinvestigations of both their intensities and pressure broadening have been pursued in
50 J. SHAO et al.
this region [6–8]. Given the high accuracy necessary to be in compliance with thescientific objective, a precise set of CO2 molecular parameters is of particularimportance for the retrieval process. In order to confirm the accuracies of the pastresults and provide guidance for future improvements to the HITRAN database, in thispaper we revisit the line intensities of the CO2 transitions available in the lasertunability range from 6350 to 6364 cm–1. Our results are compared with those in[9, 10], which were obtained using a Fourier-transform spectrometer, and also withthose in the HITRAN database which are calculated by direct numericaldiagonalization (DND) [11].
In the WM technique, the signal detected from the lock-in can be expressed as [9]
(1)
where I0 is the incident power, ρ is the density of absorption species, S is the absorptionline intensity, L is the optical path length, υ is the frequency of laser emission andχ refers to the absorption profile.
During the measurement of the absorption lines of gaseous molecules, there aretwo main broadening mechanisms, i.e., the Doppler broadening and the collidingbroadening. At low pressure, the mechanism of the Doppler broadening is dominant.The absorption line is then of the normalized Gaussian shape, which can be expressedas [9]
(2)
where γD is the half width of the Doppler broadening and υ0 is the central frequencyof absorption line. From Eq. (1), the second harmonic signal of the Gaussian lineshapefunction can be expressed as [9]
(3)
2. Experimental details
The experimental apparatus used in this work is schematically shown in Fig. 1. Themutilpass cell is of the White-type with the base path length of 8 m. The total opticpath length, which can be varied, ranges from 46 to 1159 m. In the experiment, a single-mode DFB diode laser was used, and the emission output of the pigtailed DFB InGaAs
laser diode, which is mounted in a butterfly package with a central emissionwavelength of 1.573 µm, sweeps over 6349–6365 cm–1, with the typical linewidth ofabout 2 MHz and a side mode suppression ratio greater than 30 dB. This linewidth isnegligible when compared to that of gas absorption, which is over 1 GHz. Thewavelength of laser, which is controlled by a laser-controller (TDS3724B,LightWave), varies with the laser current, whose magnitude and accuracy are about0.017 cm–1/mA and 0.01 mA, respectively. The output of the laser was directed to a1×3 fiber coupler. Some 10% of the laser power was directed to optical power meter(1830C, Newport) for monitoring the power, and another 10% was directed to opticalwavemeter (WA-1500 NIR, Buleigh) for monitoring the laser-frequency of the laser’semission. The remaining power from the diode laser was transmitted through a white-cell for absorption measurements. The transmitted laser intensity was monitored byan InGaAs detector. The transmission signals were then sent to a lock-in fordemodulation, and the output signals of lock-in were sent to a PC-based dataacquisition board (DAQ), which is capable of sampling at 20 kS/s with 16-bitresolution. Finally, the data were transferred to a personal computer and analyzedusing LabWindows/CVI programs. Each measured spectrum was recorded in a singlesweep of the laser without signal averaging.
3. Results and discussion
In this work, the CO2 spectra in the 1.57 µm region were recorded at high resolutionin the laboratory with tunable diode laser absorption spectrometer (TDLAS). Directabsorption spectroscopy and wavelength modulation absorption spectroscopy(WMAS) techniques have been adopted here for the measurements of spectraintensities. A low-frequency ramp at 1 Hz was used to scan the DFB diode over theselected absorption lines by the driving current.
Figure 2 features the eleven absorption lines of the 30012 ← 00001 band of carbondioxide, which are reachable in the tunability range. The signal was obtained from
Fig. 1. Sketch of experimental apparatus for the WM spectroscopy.
52 J. SHAO et al.
CO2 with 99.99% purity, under the condition that the theoretical absorption length was540.82 m and the CO2 pressure in the white cell was 0.5 torr. Figure 2 consists of thejuxtaposition of eleven experimental spectra. Each single spectrum was obtained byramping the driving current at an appropriate temperature to scan the laser emissionwavelength over the selected single CO2 transition. Table 1 gives the list of the linesinvestigated in this work. The experimental results of absorption line transition,positions, and the corresponding line intensities, are listed in the first, second and forthcolumns of Table 1, respectively. In this table, the values of the calculated lineintensities from the HITRAN database are also listed in the third column. Theexperimental results are compared with those in [9, 10], which were obtained with aFourier-transform spectrometer, and also with those in the HITRAN database, whichare calculated by DND [11]. From the values of ratio between the calculated and
Fig. 2. Experimental spectrum of the CO2 absorption lines of 31112 ← 01101 band around 1.57 µm.The pressure is 0.5 torr and the absorption path length is 540.82 m.
T a b l e 1. List of the lines investigated in this work; the molecular parameters are from papers [9, 10].
experimental line intensities, which are listed in the fourth column, we can see that thetypical uncertainty of line intensities is less than 1%.
The procedure of direct recording of the ratio of the transmitted signal with gas tothat without gas in the cell becomes increasingly problematic as the signal gets weaker.It is shown that the weak bands of CO2 (40012 ← 10001, 30011 ← 00001,31112 ← 01101, 41101 ← 00001, 32212 ← 02201) were not identified from Fig. 2.In order to improve signal-to-noise ratio we adopt wavelength modulation absorptionspectroscopy (WMAS) to detect the weak bands. An experimental result detected byWMAS is shown in Fig. 3. The signal was obtained from CO2 with 99.99% purity,under the condition that the theoretical absorption length was 540.82 m and the CO2pressure in the white cell was 0.5 torr. The temperature and center current of diodelaser were kept at 34°C and 70 mA, respectively. The modulation frequency andmodulation amplitude of lock-in were kept at 1.78 kHz and 30 mV, respectively. Theabsorption lines are the P(5) line of the CO2 (31112 ← 01101) band, the P(14) line ofthe CO2 (30011 ← 00001) band and the P(9) line of the CO2 (32212 ← 02201) band,respectively. The peak positions of these lines in the HITRAN2003 database are6352.311279 cm–1, 6352.116329 cm–1 and 6351.963537 cm–1, with correspondingintensities of 2.930×10–25 cm–1/(molecule⋅cm–2), 4.864×10–26 cm–1/(molecule⋅cm–2)and 1.748×10–26 cm–1/(molecule⋅cm–2), respectively.
From Eq. (1), we can see that the harmonic signals are directly proportional to I0and S for different absorption lines. Because I0 varies with the frequency of diode laser,in our experiment optical power meter was used to normalize it while detectingthe harmonic signals. Thus, the harmonic signals are only proportional to S, and theintensities of absorption line can be obtained by using a nonlinear least-squares fit ofthem to Eq. (3). Compared with HITRAN database, all the positions of spectral linesderived in the experiment can be ascertained accurately.
Fig. 3. Observed second harmonic signals of CO2 using the WM technique at an absorber pressure of0.5 torr and an absorption path length of 540.82 m.
54 J. SHAO et al.
The dashed curve in Fig. 3 represents the experimental signals detected byWMAS. The solid one is the fitted signal derived from a nonlinear least-squares fit ofthe data to Eq. (3), and then used to retrieve line intensity from the spectra.The calculated intensities of the line are 3.275×10–25 cm–1/(molecule⋅cm–2),4.61417×10–26 cm–1/(molecule⋅cm–2) and 1.6423×10–26 cm–1/(molecule⋅cm–2). ForP(9) line of the CO2 (32212 ← 02201) band, the absorption is 1.46586×10–7 and thesignal-to-noise ratio (SNR), which is the ratio of the peak amplitude of the fitted signalto this RMS deviation shown in Fig. 3, is about 19.66.
WMAS can yield a large improvement in SNR, as illustrated in Fig. 3, inwhich the 2nd harmonic signals in the vicinity of 6352 cm–1 by WMAS are presented.Figure 4 shows the experimental results of CO2 over 6350–6364 cm–1 spectral region.The signal was obtained from CO2 with 99.99% purity, under the condition that thetheoretical absorption length was 540.82 m and the CO2 pressure in the white cellwas 0.5 torr. The modulation frequency and modulation amplitude of lock-inwere kept at 1.24 kHz and 30 mV, respectively. In order to obtain higher sensitivity,
T a b l e 2. List of the lines investigated in this work; the molecular parameters are from the HITRAN.
Fig. 4. Observed second harmonic signals of CO2 using the WM technique at an absorber pressure of0.5 torr and an absorption path length of 540.82 m.
56 J. SHAO et al.
the 2nd harmonic signal was obtained by the average of 100 scans and the digitalfilter.
Table 2 shows the comparison of the spectral line positions and intensities fromour experiments with those from HITRAN database in the region 6350–6364 cm–1.From this table, we can see that the weakest line observed in our experiment isP2 transition of 30011 ← 00001 band, whose intensity, absorbance and SNR are1.116×10–26 cm–1/(molecule⋅cm–2), 9.96×10–8 and 4.53, respectively. The experimentalresults of absorption band, transition, line positions, and line intensities, are listed inthe first, second, third and fifth columns of Tab. 2, respectively. In this table, the valuesof the calculated line intensities from the HITRAN database are also listed in the fourthcolumn. In the sixth column of the table, the values of ratios between the calculatedand experimental line intensities are listed, from which it is easily obtained that thestandard uncertainty for the intensity varies from less than about 1% for the strongestlines to about 10% for the weakest lines.
4. Conclusions
This work presents the spectroscopic intensity measurements of the CO2 around1.57 µm based on DFB lasers, which are of primary importance for atmosphericapplications, and are helpful for further improvement of reliable retrievals of theconcentration profiles of this minor constituent in the atmosphere. Overtone absorptionlines of CO2 in the regions between 6350 and 6364 cm–1 have been examined by meansof tunable diode lasers in free-running mode. The diode laser emission wavelengthwas scanned around the gas resonances by simply sweeping its injection current,permitting a direct observation of the absorption line-shapes. Weak overtoneabsorption lines have been detected by using the wavelength modulation spectroscopywith the 2nd harmonic detection technique. The intensity of the weakest line detectedin this experiment is 1.116×10–26 cm–1/(molecule⋅cm–2) at the pressure of 0.5 torr, andthe corresponding absorption is 9.96×10–8 with SNR of 4.53.
Acknowledgments – This work has been supported by the National High Technology Research andDevelopment Programme of China under Grant No. 2002AA825100.
References
[1] POUCHET I., ZÉNINARI V., PARVITTE B., DURRY G., Diode laser spectroscopy of CO2 in the 1.6 µmregion for the in situ sensing of the middle atmosphere, Journal of Quantitative Spectroscopy andRadiative Transfer 83(3-4), 2004, pp. 619–28.
[2] DING Y., MACKO P., ROMANINI D., PEREVALOV V.I., TASHKUN S.A., TEFFO J.-L., HU S.-M., CAMPARGUE
A., High sensitivity cw-cavity ringdown and Fourier transform absorption spectroscopies of 13CO2,Journal of Molecular Spectroscopy 226(2), 2004, pp. 146–60.
[3] GOLDMAN A., STEPHEN T.M., ROTHMAN L.S., GIVER L.P., MANDIN J.-Y., GAMACHE R.R.,RINSLAND C.P., MURCRAY F.J., The 1 µm CO2 bands and the O2 (0–1)X3 –a1∆g and (0–1)X3 –b1 bands in the Earth atmosphere, Journal of Quantitative Spectroscopy and Radiative Transfer82(1-4), 2003, pp. 197–205.
[4] MALATHY D.V., CHRIS-BENNER D., RINSLAND C.P., SMITH M.A.H., Absolute rovibrational intensitiesof 12C16O2 absorption bands in the 3090–3850 cm–1 spectral region, Journal of QuantitativeSpectroscopy and Radiative Transfer 60(5), 1998, pp. 741–70.
[5] STEYERT D.W., WEVER M., SIROTA J.M., REUTER D.C., Absolute intensities for the Q-branch of the (581.776 cm–1) band in carbon dioxide, Journal of Quantitative Spectroscopy and
Radiative Transfer 54(5), 1995, pp. 815–18.[6] SHAO J., GAO X.-M., DENG L.-H., HUANG W., YANG Y., PEI S.-X., YUAN Y.-Q., ZHANG W.-J., Highly
Sensitive Tunable Diode Laser Absorption Spectroscopy of CO2 Around 1.31 µm, Chinese PhysicsLetters 21(10), 2004, pp. 1908–10.
[7] TASHKUN S.A., PEREVALOV V.I., TEFFO J.-L., LECOUTRE M., HUET T.-R., CAMPARGUE A., BAILLY D.,ESPLIN M.P., 13C16O2: Global treatment of vibrational-rotational spectra and first observation ofthe 2υ1 +5υ3 and υ1 + 2υ2 + 5υ3 absorption bands, Journal of Molecular Spectroscopy 200(2),2000, pp. 162–76.
[8] DEVI V.M., BENNER D.C., SMITH M.A.H., BROWN L.R., DULICK M., Absolute intensity measurementsof the 12O16O2 laser bands near 10 µm, Journal of Quantitative Spectroscopy and Radiative Transfer76(3-4), 2003, pp. 393–410.
[9] VALERO F.P.J., SUAREZ C.B., Measurement at different temperatures of absolute intensities, line half-widths, and broadening by Ar and N2 for the 3001II ← 0000 band of CO2, Journal of QuantitativeSpectroscopy and Radiative Transfer 19(6), 1978, pp. 579–90.
[10] HENNINGSEN J., SIMONSEN H., The (2201–0000) band of CO2 at 6348 cm–1: linestrengths, broadeningparameters, and pressure shifts, Journal of Molecular Spectroscopy 203(1), 2000, pp. 16–27.
[11] ROTHMAN L.S., BARBE A., CHRIS-BENNER D., BROWN L.R., CAMY-PEYRET C., CARLEER M.R.,CHANCE K., CLERBAUX C., DANA V., DEVI V.M., FAYT A., FLAUD J.-M., GAMACHE R.R., GOLDMAN A.,JACQUEMART D., JUCKS K.W., LAFFERTY W.J., MANDIN J.-Y., MASSIE S.T., NEMTCHINOV V.,NEWNHAM D.A., PERRIN A., RINSLAND C.P., SCHROEDER J., SMITH K.M., SMITH M.A.H., TANG K.,TOTH R.A., VANDER-AUWERA J., VARANASI P., YOSHINO K., The HITRAN molecular spectroscopicdatabase: edition of 2000 including updates through 2001, Journal of Quantitative Spectroscopyand Radiative Transfer 82(1-4), 2003, pp. 5–44.
Received September 26, 2004
4υ22
3υ23←
Optica Applicata, Vol. XXXV, No. 1, 2005
Optical demultiplexer using a holographic concave grating for POF–WDM systems
LYUBOMYR V. BARTKIV1, YAROSLAV V. BOBITSKI1, 2, HANS POISEL3
2Institute of Technology, University of Rzeszów, Rejtana 16a, 35-959 Rzeszów, Poland
3POF-AC, University of Applied Sciences, Wassertorstraße 10, 90489 Nuremberg, Germany; e-mail: [email protected]
Polymer optical fiber (POF) is one of the best transmission media for short-distance communications.To increase the transmission capacity of the fiber a wavelength-division-multiplexing (WDM)technique is commonly used. Several POF-WDM systems have been realized using interferencefilters and plane diffraction gratings as wavelength selective elements. In the present paper, forthe first time a concave diffraction grating is applied with POF. A holographic concave grating iscalculated and optimized for the use in an optical demultiplexer. The optical losses in the designeddemultiplexer are estimated theoretically.
Silica single-mode optical fiber is widely used in long-distance communicationsystems for high-speed data transmission (Gbit/s) because of its high bandwidth andlow attenuation coefficient. The use of this fiber for short-distance interconnections isnot preferred. This is because of the small core diameter of a single-mode fiber and,consequently, high requirements as regards adjustment. Therefore, in local systems itis favorable to use a multi-mode fiber, which has greater diameter. Nowadays, thepolymer optical fiber (POF) is the most promising solution as transmission mediumfor short-range communications. Its large core diameter (250–1000 µm) gives apossibility to use inexpensive polymer connectors and its flexibility enables bendingradii, which are by far more critical for glass fibers [1]. All this makes this fiberextremely suitable for in-home and LAN applications. The best developed at themoment is step-index POF (SI-POF). Because of large diameter and high numericalaperture this fiber has low bandwidth. One approach for increasing the transmissioncapacity of SI-POF is to use wavelength-division-multiplexing (WDM) techniques.
60 L.V. BARTKIV, Y.V. BOBITSKI, H. POISEL
To date several POF-WDM systems have been developed [2–5]. The devices forcombining and separating different wavelengths (multi/demultiplexers) are based oninterference filters and plane diffraction gratings. Unfortunately, all these requireadditional collimating and focusing optics that need alignment and lead to complicateddesigns. The most appropriate solution in this case can be a concave diffraction grating.A concave grating combines diffraction and focusing properties simultaneously in oneelement and, therefore, it does not need additional optics. Since such devices have notyet been developed especially for use with POF, in this paper we report on designinga concave grating demultiplexer for POF-WDM systems. At first the diffraction gratingwill be calculated and optimized for demultiplexing. Finally, a theoretical estimationof optical losses in the demultiplexer will be performed.
2. Design of holographic concave grating for demultiplexer
To build a concave grating demultiplexer, it is necessary to calculate a concavediffraction grating satisfying the specific requirements imposed on the demultiplexer.The main parameters of the demultiplexer are the wavelength band, number ofchannels, operating wavelengths and insertion losses. Since we intend to work withlow NA SI-POF, the operating wavelength band will be from 400 to 700 nm. This fiberhas three attenuation minima at 520, 570 and 650 nm, which are normally used fortransmission of three different signals. The main problem that arises is the separationof these wavelengths. For this purpose the diffraction grating with a correspondinglinear dispersion has to be used. Taking into account a fiber diameter d0 = 1 mm andminimal separation between adjacent channels ∆λ = 50 nm, the linear dispersiondx/dλ has to be
(1)
where f is the effective focal length and dθ /dλ is the angular dispersion. An opticallayout of the concave grating demultiplexer with classical grating is shown in Fig. 1.As can be seen from this figure, the parameter f is equal to Rcosβ and, therefore,Eq. (1) can be written as
(2)
where R is the radius of concave surface of the grating, β is the diffraction angle,m is the diffraction order and Λ is the groove spacing. Looking forward, the last valuehas to be taken bigger to get a good isolation between channels. Taking it into account,the last relationship will be written for the first diffraction order as:
(3)
dxdλ-------- f
dθdλ---------=
d0
∆λ----------≥ 20000=
dxdλ-------- f
dθdλ--------- R β m
Λ βcos-------------------cos
RmΛ
---------- 20000≥= = =
RΛ------ 40000.≥
Optical demultiplexer using a holographic concave grating... 61
Thus, if we know the radius of concave surface of the grating it is possible to findthe groove spacing at the grating center. These values have to be selected from practicalpoint of view making the grating applicable for demultiplexer use. Using our approachwe have selected optimal values for an incidence angle α, radius of concave surfaceR and groove frequency 1/Λ. They are 26 degrees, 35 mm and 1200 grooves/mm,respectively.
The size of the grating depends on the location and numerical aperture of inputfiber. For our demultiplexer the grating diameter should be approximately 20 mm.
In Figure 1, a general schematic of demultiplexer is shown. Usually, the classicalgratings are not used in a demultiplexer because of their large astigmatism. Therefore,a holographic concave grating is normally selected for this demultiplexer. To improvethe transmission performances of demultiplexer the grating aberrations have to beminimized and the grating efficiency has to be optimized.
2.1. Minimization of aberrations of holographic concave grating
The focusing properties of concave diffraction gratings are usually studied using anaberration function, which describes a difference between the optical paths of thecentral (AOB) and non-central (APB) optical rays and defines grating aberrations [6](see Fig. 2). By expanding this path-difference into a power series of w and l up to thefourth order the aberration function is simply expressed by
(4)
where Fij are the aberration coefficients. The last ones can be separated into two partsas follows:
(5)
δF Fijwil
j
i j∑F10w F20w
2F02l
2F30w
3F12wl
2F40w
4F22w
2l
2F04l
4+ + + + + + +
=
=
Fij Mijmλλ0
---------- Hij+=
Fig. 1. Optical layout of demultiplexer usinga classical concave grating.
62 L.V. BARTKIV, Y.V. BOBITSKI, H. POISEL
where Mij terms depend on the mounting scheme (r, r', α, β ), Hij terms depend on thegrating recording scheme (rC, rD, γ, δ ), λ is the wavelength of incident light and λ0 isthe grating recording wavelength. The explicit expressions for Mij and Hij, and alsomore comprehensive information on aberration function can be found in [7, 8].
The aberration function is useful for us because it gives the possibility to get thesize of an image formed by the concave grating. The width WS and height HS of theimage in spectral focus of the concave grating can be evaluated using expressionsobtained in [9]:
(6)
(7)
WS r' Φ 2⁄( )23 F30 F12+ ,=
HS r' 2Φ F02 Φ 2F12+
=
Fig. 2. Schematic diagram of the optical system with holographic concave grating.
T a b l e. Calculated parameters of holographic concave grating with minimized aberrations.
Parameter Value Unit
R 35 mm
r 34 mm
α 26 deg
rC 34.691 mm
rD 32.689 mm
γ –11.583 deg
δ 33.957 deg
λ0 632.8 nm
Optical demultiplexer using a holographic concave grating... 63
where Φ is the grating diameter. From Eqs. (6) and (7) it can be seen that the imagesize depends on aberrations of the concave grating. Thus, to reduce the image size and,consequently, insertion losses in the demultiplexer, it is necessary to minimize theaberrations. Minimization of aberrations for holographic concave grating is carriedout selecting the optimal location of recording sources C and D (see Fig. 2). For thispurpose, we have created a computer program in Delphi with variable parameters rC,rD, γ and δ. The obtained parameters of holographic concave grating are presentedin the Table. Spot diagrams from a point light source for operating wavelengths havealso been obtained using expressions derived in [10] and are shown in Fig. 3.
2.2. Optimization of grating efficiency
Since the concave grating demultiplexer consists of one single concave grating, itsperformance characteristics strongly depend on the grating efficiency. Usually, afterminimization of aberrations of the concave grating the optimization of gratingefficiency is carried out to further reduce the insertion losses in the demultiplexer. Themaximum efficiency can be obtained with gratings which have triangular grooveprofile, when the tilting angle of one of the groove facets is such that the directions oflight reflected from this facet and light diffracted from the grating for the sameincidence angle coincide [11]. Holographic gratings normally have a sinusoidal grooveshape and, consequently, lower diffraction efficiency. However, if it is necessary toget a higher efficiency the sinusoidal groove shape can be modified into other grooveshapes (quasi-triangular, etc.) using different techniques. Unfortunately, this makesthe grating cost grow and, therefore, is not suitable for our case.
The relative intensity of monochromatic light of wavelength λ diffracted intom-th order for sinusoidal reflection grating is expressed by:
(8)Im Jm2 π∆
λ--------- αcos βcos+( )
=
Fig. 3. Focal spot images from an on-axis point source for the three operating wavelengths in spectralfocus of the designed holographic concave grating.
64 L.V. BARTKIV, Y.V. BOBITSKI, H. POISEL
where Jm is the m-th order Bessel function, ∆ is the groove depth, α and β are the anglesof incidence and diffraction, respectively [12]. A portion of the light power diffractedinto m-th order can be calculated using the following expression:
(9)
where m– and m+ are the lowest and highest diffraction orders, respectively. As can beseen from Eqs. (8) and (9) the efficiency of a sinusoidal grating depends on the groovedepth modulation. Thus, for optimization of diffraction efficiency of a sinusoidalgrating it is necessary to optimize the groove depth. For this purpose, another programin Delphi has been composed. It was taken into account that recording beams havea Gaussian light distribution. As a result the groove depth has similar distribution overthe grating surface. The incoming light was also considered as Gaussian light beam.The analysis carried out for the efficiencies obtained for different values of ∆ hasshown that the maximum efficiency is obtained when the groove depth is 202 nm inthe grating center.
3. Theoretical evaluation of optical losses in demultiplexer
Optical losses are very critical in communication systems with POFs, which havea relatively high attenuation coefficient. To evaluate the performances of the concavegrating demultiplexer designed for use in such systems we applied a ray-tracingmethod for calculation of insertion losses for each wavelength channel. For thispurpose the following approach has been adopted. The incident light beam wasseparated into N rays with intensities IP falling on the grating surface. The full intensityof incoming light is the sum of intensities IP of all N rays. The intensity of each
ηm Im Ikk m–=
m+
∑1–
×=
Fig. 4. Spectral transmission of the demultiplexer based on holographic concave grating.
Optical demultiplexer using a holographic concave grating... 65
reflected ray is assumed to equal ηP IP, where ηP is the diffraction efficiency for eachincident ray with intensity IP. Thus, if M is the number of rays coupled into an outputfiber, the optical losses B in demultiplexer can be expressed by
(10)
Since POF has a large core diameter the fiber end face was considered as a numberof point light sources. Using the approach described above the optical losses for allchannels were obtained. Graphical results are presented in Fig. 4.
As can be seen from Fig. 4, the insertion losses for all the wavelength channels ofthe designed demultiplexer with holographic concave grating are approximately 2 dBand the channel isolation (crosstalk suppression) is more than 20 dB.
4. Conclusions
In this work, we present a design of the concave grating demultiplexer for use inPOF-WDM systems. For this purpose a holographic concave grating was calculatedand optimized to have minimal aberrations and maximum diffraction efficiency formultiplexed wavelengths. It has been shown that aberrations of holographic concavegrating are usually minimized by optimal allocation of recording sources anddiffraction efficiency for sinusoidal grating is optimized by the choice of optimalgroove depth. Values of optical losses obtained theoretically for wavelength channelsare quite low which allows this demultiplexer to be used in data transmission systemswith POFs. The channel isolation is good enough to avoid crosstalk and furthermoreto increase the number of channels that can be demultiplexed. The only problem, whichcan exist, is the demultiplexer cost that is normally defined by the cost of the grating.In any case, a considerable advantage of the concave grating demultiplexer is that noadditional optics is needed.
[2] DAUM W., KRAUSER J., ZAMZOW P.E., ZIEMANN O., POF – Polymer Optical Fibers For DataCommunications, Springer-Verlag, Berlin 2002.
[3] JUNGER S., TSCHEKALINSKIJ W., WEBER N., Proc. 11th Int. Conf. on Polymer Optical Fibers, POF’2002,Tokyo, Japan 2002, p. 69.
[4] TSCHEKALINSKIJ W., JUNGER S., WEBER N., Proc. 11th Int. Conf. on Polymer Optical Fibers, POF’2002,Tokyo, Japan 2002, p. 139.
[5] JUNGER S., TSCHEKALINSKIJ W., WEBER N., Proc. 12th Int. Conf. on Polymer Optical Fibers, POF’2003,Seattle, WA 2003, p. 237.
B 10
IP
N∑
ηP IP
M∑
----------------------- .log=
66 L.V. BARTKIV, Y.V. BOBITSKI, H. POISEL
[6] LOEWEN E.G., POPOV E., Diffraction Gratings and Applications, Marcel-Dekker, New York 1997.[7] NODA H., NAMIOKA T., SEYA M., Geometric theory of the grating, Journal of the Optical Society of
America 64(8), 1974, pp. 1031–6.[8] CHRISP M., [In] Applied Optics and Optical Engineering, Vol. X, Academic Press, New York 1987,
p. 391.[9] TSONEV L., POPOV E., Focal spot estimation for concave diffraction gratings, Optics Communications
90(1-3), 1992, pp. 11–15.[10] CHRISP M., Aberrations of holographic toroidal grating systems, Applied Optics 22(10), 1983,
pp. 1508–18[11] PALMER C., Diffraction Grating Handbook, 4-th Ed., Richardson Grating Laboratory, New York
2000.[12] BOIVIN L.P., Multiple imaging using various types of simple phase gratings, Applied Optics 11(8),
1972, pp. 1782–92.
Received July 8, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Assembly of a fast multi resolution spectrophotometer system for simultaneous measurement of absorption and luminescence spectra
In this work, the design and assembly of a dual spectrophotometer system capable of measuringthe absorption as well as time resolved luminescence spectra of liquid, gaseous, or solid samplesare reported. The system incorporates a 1024×376 elements coupled charge detector (CCD)capable of monitoring changes in optical spectra with time delays as short as few microsecondstime scale. Such a design reduced significantly the costs of purchasing two separate systemscontaining similar optical and electronic components. In addition, the design enabled furtherinvestigations on the photodegradation mechanism for a benzimidazole based pesticide.
Keywords: dual spectrophotometer, multiresolution spectrophotometer, low temperature fluorescence,benomyl, carbendazime.
1. Introduction
Absorption and luminescence spectrophotometers contain basically the sameelectronic and optical components. The only difference is the arrangement of theoptical components, i.e., the arrangement of the optical components determineswhether the system will be used for photo-absorption, photo-luminescence or Ramanmeasurements [1]. While in absorption measurements, the excitation source is locatedin line with the sample and detector. In luminescence, Raman, as well as othermeasurements, the excitation source is located at an angle normally 90° with respectto the sample and detector. Therefore, the requirement for obtaining absorption orluminescence spectra requires same optical components in addition to electroniccontrollers, amplifiers and data processors [2]. This work is considered as an extensionof the work [3], in which, the design and assembly of a compact and multifunctionspectrometer are presented. The same optical and electronic components are used forrecording the absorption, luminance, and time resolved spectra of solid, liquid, andgaseous samples. This design is expected to reduce significantly the cost ofspectrophotometric instruments. Moreover, the design enables tracing any changes in
68 Y.A. YOUSEF
the sample using absorption and luminescence spectra simultaneously, which is notpossible with the classical spectrophotometers. This can be achieved by recording theabsorption spectra which takes several microseconds followed by recording theluminescence spectra which can also be recorded in time scales as short as severalmicroseconds. The sequence can be repeated, and therefore in one-minute time scalethousands of absorption and luminescence spectra can be obtained without the needof removing the sample from its place.
The new system is designed in such a way as to enable easy replacement of thesample cell holder mentioned in reference [3] by a home-made, open cycle, liquidcryostat. To demonstrate the capabilities of the new system, the luminescence spectrafor a number of organic compounds at different temperatures and resolutions arepresented.
2. Experimental
A block diagram of the experimental arrangement is shown in Fig. 1. It consists offour major units: light source unit, spectrograph (3), optical 2D diode array detector(4), control electronics and data station (5, 6, 7).
Fig 1. Block diagram of the experimental set-up: 1 – electromechanical shutter, 2 – sample cell holder,3 – spectrograph, 4 – CCD unit, 5 – fast high voltage pulser, 6 – detector controller, 7 – delay generator,8 – computer, 9 – excitation monochromator, 10 – excitation monochromator drive unit, 11 – xenon lamp,12 – nitrogen laser, 13 – tungsten lamp, 14 – deuterium lamp, and 15, 16, 17 – 1 mm UV passive opticalfiber.
5
2
6
7
3
4
8 10
9 11
12
13
14
15
16
17
1
Power unit
Assembly of a fast multi resolution spectrophotometer system... 69
The light source consists of four main blocks: deuterium lamp (14) for thewavelength range (190–350 nm), tungsten lamp (13) for the wavelength range(350–900 nm), xenon lamp (11) for the wavelength range (190–900 nm), and nitrogenlaser (12) emitting at the wavelength of 337.1 nm.
The deuterium light beam is focused into the middle part of the sample cell (2)using a quartz lens. The power supply that ignites and supplies the deuterium lampwith constant current was designed and built in our laboratory.
A 100 W tungsten lamp operated by a DC stabilized power supply (Ealing, model021/026) is used as the visible radiation source. An optical fiber cable (15) is used totransfer the visible light beam to the lower part of the sample cell. Electromechanicalshutter (1), driven by UniBitz shutter controller Model SD-1000, is used to control theexposure of the sample to light.
A 250W air-cooled xenon lamp operated by a power supply consisting of lampigniter and DC source, (model: Muller SVX1530), is used as the excitation source forthe emission part of the system. The beam output is focused into the input slit of a0.3 m monochromator (9). The excitation monchromator drive unit (10) is interfacedto the computer enabling software selection of the excitation wavelength [4]. Theoutput of the monochromator is focused into a wide bore (1.0 mm UV passive) opticalfiber cable (17). The other end of the optical fiber cable is fitted into the sample cellholder at 90° to the deuterium and visible lamp. This configuration enables 90°excitation with respect to the detection surface.
A nitrogen laser, PRA Model LN 1000, with 337.1 nm wavelength and 1 mJ pulseenergy is used as an optional excitation light source where its output is transferred tothe sample via a wide bore 1 mm optical fiber (16). The nitrogen laser wavelength issuitable for exciting some of the samples used in our study, other lasers such as dyelasers can be used in place.
The sample holder block (2) was fabricated in the machine shop of the faculty [3].It is designed to accept standard (10×10×50 mm) sample cells. Two 5 mm holes weredrilled into the side perpendicular to the detection path as a modification on the cellholder used in reference [3]. The optical fiber cables, from the excitation light sources,are designed so as to fit into these holes, thus facilitative right angle sample excitationas well as facilitating other future applications for the system such as pulse probetechniques.
The spectrograph used in this setup is Chromex model 5001 [3]. It contains threegratings (75, 150, 300 grooves/mm) enabling the selection between three differentresolutions. All spectrograph operations such as the selection of grating, slit width aswell as wavelength calibration are fully controlled by Windows software.
The CCD detector, Princeton Instruments model LNCCD, consists of1024×376 elements. It has a broad spectral response (200–900 nm). It is different fromthe detector used in reference [3] in that the spectral window is wider since the numberof elements in the wavelength axis is double that in the ICCD unit. Moreover, thedetector is cooled using liquid nitrogen to maintain the temperature at –90°C, hence
70 Y.A. YOUSEF
lower dark current and better sensitivity. A detector controller (model ST-38) drivenby Windows 95 software is used to control and transfer the data to the computer.
A programmable delay generator model (DG-535) Stanford Research systems INCis used to synchronize the trigger of the diode array detector with the electromechanicalshutters (7). Upon the reception of an input pulse it outputs as many as 4 pulses witha programmable time delay between each of them. This is extremely important forsynchronization due to the large difference in response times between the electro-mechanical shutters and the electronic detectors.
The computer (8), DELL (P75t), includes a fast interface card driven by Windowssoftware. To start data acquisition, the computer triggers the detector controller viathe fast interface card installed on the computer motherboard. The detector controlleroutputs a trigger pulse that is fed into the programmable delay generator unit. Thelatter outputs two pulses with different time delays, the undelayed pulse is fed into theelectromechanical controller unit to activate the mechanical shutters while the delayedpulse is used to trigger the detector. The software fully controls all the functions ofthis generator, i.e., the gate pulse width, pulse delay, as well as the external or internalpulse trigger. The type of grating selected determines the spectral window, i.e., thewidth of the spectrum. The spectral window that can be achieved here is double thatmentioned in our earlier work [3]. For example, the 75 grooves/mm grating offereda spectral window of 150 nm while the spectral window for the same grating is 300 nm.The same idea applies to the other gratings. The wavelength selected by themonochromator is always the middle part of the spectrum. The whole spectrum iscollected during the selected exposure time of the detector. Exposure times as short asfew microseconds are possible, therefore a complete 300 nm spectrum can be obtainedin this time scale. The main benefits of this setup are:
1. The short time during which the sample is exposed to probe light, this time canbe as short as the detector exposure time. Since we use electromechanical shutters, weare limited to the shutter response time, which is of the order of milliseconds.
2. Two types of measurements can be obtained while the sample in its place andin a very short period, the only requirement is to control the electromechanical shuttersin front of the light sources.
3. The new system offered a spectral window that is double the width of that inreference [3].
A variable temperature sample holder with open cycle liquid nitrogen cryostat wasdesigned and fabricated in collaboration with the mechanical workshop to be used forlow temperature measurements. The system was designed for easy replacement ofthe sample holder block described above. A block diagram of the system is shown inFig. 2. It consists of a hollow aluminum cube 10×10×10 cm with 2-inch widow oneach of the 4 sides (1). The windows are designed to accept 2-inch quartz windowsfor optical excitation and detection. The top side of the cube is machined in such a wayas to accept Varian vacuum line adapters with 50 mm diameter (2). A steel vacuumcylinder, from Varian Vacuum Systems, is fitted to the top of the cube (5). A side tubeis attached to the topside of the steel cylinder to evacuate the space around the sample
Assembly of a fast multi resolution spectrophotometer system... 71
cell holder. A liquid nitrogen transfer line consists of two copper tubes fitted into eachother (4). The tube with smaller diameter (2 mm) is used for transferring liquid nitrogenfrom the liquid nitrogen dewar to the sample, while the tube with the wider diameter(1-inch) is used to shield the small diameter tube from the room temperature byevacuating the space between the tubes. One end of the transfer line is immersed intothe liquid nitrogen dewar. The other end is designed to fit into the middle of the steelcylinder and continues to the aluminum cube (3). A specially designed sample cellholder is fitted to the other end of the transfer line.
The thin tube in the transfer line continues to the outside of the system where anoiless vacuum pump is used to suck the liquid nitrogen from the dewar as is shown inFig. 3. The sample cell holder (1) is fabricated from a stainless steel metal block with
Fig. 2. Open cycle liquid nitrogen cryostat: 1 –aluminum block with quartz windows, 2 – Varian typeadapter, 3 – Rubber O-ring, 4 – metal block with drilledvolume to receive liquid nitrogen and to act as a heatexchanger, 5 – liquid nitrogen transfer line, 6, 7 –stainless steel cylinder with vacuum outlet, 8 – vacuumoutlet to evacuate the transfer line, 9 – capillary tubefor liquid nitrogen inlet.
5
7
6
4
2
8
9 3
1
Fig. 3. Liquid nitrogen transfer line and sample cell block: 1 – samplevolume drilled into the metal block with quartz window in the front,2 – metal block from stainless steel, 3 – metal block with drilled volume4 to receive liquid nitrogen to act as a heat exchanger, 4 – metal tubethrough which liquid nitrogen is removed, 5 – copper cylinderevacuated all over the transfer line, 6, 7 – liquid N2 inlet and outlet.
2
3 4
5
6 7
1
72 Y.A. YOUSEF
a 10 mm diameter half sphere drilled in its middle. A 5 mm hole is drilled in the bottomside of the block for filling the cell with the liquid sample. The front side of the blockis designed to accept a 1-inch quartz window for front surface optical excitation. Teflongaskets are used between the quartz window and the stainless steel block to preventthe leak of the liquid sample due to outside vacuum. A semiconductor siliconp-n junction is used to measure the temperature of the sample. The voltage acrossthe p-n junction is measured using a digital millivolt meter and calibrated totemperature. The sensitivity of the p-n junction was found to be as good as gold-copperthermocouple and easier to handle [5, 6]. A power resistor (10 Ω, 5 W) is attached tothe side of the sample cell holder for heating the sample by passing electrical currentthrough the resistor. An electronic circuit based on operational amplifier is used tocompare the actual temperature determined by the voltage across the diode and thevoltage set by a potentiometer which represents the required temperature. Theoperational amplifier outputs a voltage proportional to that difference to the powerresistor. A block diagram of the circuit is shown in Fig. 4.
3. Results and discussion
For testing the wavelength calibration and resolution, the spectral lines from a lowpressure standard mercury lamp (Ealing cat-no. 26-4812) at different resolutions wererecorded. Figure 5 shows the optimum resolution of the system where the intensemercury line at 546 nm and the closely spaced 576 nm and 579 nm lines are wellseparated. The resolution capabilities of the instrument can be changed easily byselecting a grating from a set of three gratings using the computer software. The easychange of instrument resolution is important for low temperature luminescencestudies where the fluorescence spectral lines get narrower with a decrease intemperature [7, 8].
Fig. 4. Temperature controller circuit diagram: 1 – semiconductor diode, 2 – operational amplifier,3 – power resistor, 4 – potentiometer.
+ –
+VCC
–VCC
1
3 4
2
Assembly of a fast multi resolution spectrophotometer system... 73
The speed and sensitivity of the system is demonstrated in Fig. 6 where the lowresolution, fluorescence spectrum of 1 µg/l pyrene/cyclohexane solution at room tem-perature was acquired using a single 15 nsec, 1 mJ nitrogen laser pulse. The spectrumcontains identical spectral characteristics of pyrene published in [9].
Figure 7a shows the low resolution fluorescence spectrum of benzo(a)pyrene inn-heptane at room temperature. The spectrum was acquired using 20 nitrogen laserpulses each of 15 nsec duration. The spectrum was compared with the data publishedin reference [7] and found to contain similar spectral characteristics. The lowtemperature high resolution fluorescnece spectrum of the same sample is shown in
Fig. 5. Mercury spectral lines acquired using the high resolution grating (300 grooves/mm).
5 4 0 5 6 0 5 8 0 0 E + 0
5 E + 3
1 E + 4
2 E + 4
2 E + 4
3 E + 4
Wavelength [nm]
Rel
ativ
e in
tens
ity
Fig. 6. Fluorescence spectrum of 1.0×10–6 g/l pyrene in cyclohexane at room temperature obtained witha single 15 nsec duration laser pulse at 337.1 nm using N2 laser excitation source.
6E+3
6E+3
5E+3 Rel
ativ
e in
tens
ity
400 440 480 520
Wavelength [nm]
74 Y.A. YOUSEF
Fig. 7b. The results indicate the spectral capabilities of the system including variabletemperature measurements, selection of excitation light sources and variation ofresolution according to the experiment requeirments.
Figure 8a shows the absorption spectrum of a benzimidazole based derivative(benomyl) in CH3CN solution while Fig. 8b shows the fluorescence spectrum for thesame compound in CH3CN solution excited at 290 nm. Knowing that some molecules,such as benomyl are very unstable in solution [10], the possibility of simultaneuousmeasurement of absorption and fluorecsence can be considered of high practicalimportance for monitoring the degradation process. The degradation process was
Fig. 7. Room temperature low resolution fluorescence spectrum (a) and low temperature (–150°C) highresolution fluorescence spectrum (b) of 2.0×10–5 g/l benzo(a)pyrene in n-heptane (λexc = 337 nm).
a b
Fig. 8. Absorption spectrum (a) and fluorescence spectrum (b) of 1×10–3 M benomyl in CH3CN,(λexc = 290 nm).
a b
Assembly of a fast multi resolution spectrophotometer system... 75
monitored using the fast monitoring absorption technique and published in a previouswork [3]. In this work, the fluorescence technique is used to monitor the degradationprocess and shown in Fig. 9. The spectrum provides a conclusive evidence about theinstability of benomyl in solution.
Acknowledgments – I should like to thank Prof. T. Akasheh for fruitful discussions and suggestions,Eng. K. Allawneh for designing the electronic temperature controller circuit. Mr. M. El-Jabali andA. Shuhadah for machining the metal components. This work was supported by the Jordanian HigherCouncil for Science and Technology, and by the Deanship of Scientific Reaserch and graduate Studiesat Yarmouk University.
References
[1] WEHRY E.L., Modern Fluorescence Spectroscopy, Vol. 1-4, Plenum Press, New York 1975–1981.[2] BERLMAN I.B., Fluorescence Spectra of Polyaromatic Molecules, Academic Press Inc, 1971.
Fig. 9. Fluorescence spectrum of benomyl CH3CN solution at different times of photolysis(λexc = 290 nm).
76 Y.A. YOUSEF
6[3]YOUSEF Y.A., FATAFTAH Z., AKASHEH T.S., RAWASHDEH A.M., Design and assembly of a fastspectrophotometer system for monitoring chemical reactions, Optica Applicata 31(3), 2001,pp. 563–70.
[4] YOUSEF Y.A., SHADERMA M.M., ABU-HASSAN L.H., ABU EL-HAIJA A.J., ROUSAN A.A., Design andconstruction of a microcomputer based interface/controller to drive and process spectrophotometerdata, Optica Applicata 26(1), 1996, pp. 61–5.
[5] ABU-ZEID M.E., KORDIA H.A., YOUSEF Y.A., An interactive implementation of IBM-PC forprocessing and simulation of photoacoustic spectrometer data, Journal of MicrocomputerApplications 15(2), 1992, pp. 89–102.
[6] GUPTA R., BOSE G., Design and fabrication of bulk silicon-based temperature sensor, Journal of theIndian Institute of Science 81(5), 2001, pp. 557– 61.
[7] ABU-ZEID M.E., YOUSEF Y.A., Molecules in Physics Chemistry and Biology, Vol. 2, KluwerAcademic Publishers, 1988, pp. 365–89.
[8] YOUSEF Y.A., ABUZEID M.E., KURDIA H.A., J. Abhath Al- Yarmouk, Vol. 6, (1997), pp. 81-91.[9] Spectral Atlas of Polycyclic Aromatic Compounds, [Ed.] W. Karcher, D. Reidel Publishing
Company, Boston 1985.[10] CHIBA M., SINGH R., High-performance liquid chromatographic method for simultaneous
determination of benomyl and carbendazim in aqueous media, Journal of Agricultural and FoodChemistry 34(1), 1986, pp. 108–12.
Received March 4, 2003in revised form August 16, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Deflective signal analysis in photothermal measurements in the frame of complex geometrical optics
ROMAN J. BUKOWSKI, DOROTA KORTE
Institute of Physics, Silesian University of Technology, ul. Bolesława Krzywoustego 2, 44-100 Gliwice, Poland
The influence of one dimensional plane thermal wave on probing Gaussian beam phase anddeflection by complex geometrical optics methods has been analyzed in the work. The probingbeam detection by quadrant photodiode has been investigated. The dependence of photodiodecurrent signal on the probing beam diameter, its waist, sample position, angular modulationfrequency and the height of the beam over the sample has been studied.
Nowadays, investigations of solid state thermal properties have great importance,especially for different nonhomogeneous layered systems. Among them the mostessential are those photothermal methods that are based on differences betweenthermal properties of different parts of that layered system. Temperature changes insuch a system are measured directly or indirectly and on this principle we can concludeabout its structure.
One of the indirect methods for measuring sample surface temperature changes isphotodeflective method. In this method, the periodically heated sample changes thetemperature of surrounding gas, and next it changes the gas refraction index. The lastchanges are detected by probing light beam with known light intensity distributionpassing through the heated gas layer. Changes in the gas refraction index causedeflection and phase change in the probing beam.
At present two theoretical methods for description of these phenomena are used[1, 2]. The first one is the ray method. It is based on the small shift of light beam(deflection) in optically nonhomogeneous media. There is also a generalization of thatmethod to wide probing beams [3, 4]. The second method is the wave one [2]. In thiswork, the wave equation was solved for the probing beam propagation but only itsphase change was taken under consideration.
78 R.J. BUKOWSKI, D. KORTE
A complete (with arbitrary accuracy) description of light beam propagation inoptically nonhomogeneous medium can be found in the frame of geometrical opticsmethod by the use of Debye’s expansion [5–7]. The proper analysis using complexgeometrical optics methods and taking into account only the phase change of probinglight beam, caused by thermal waves, was presented in [7, 8].
All of the works quoted assumed that deflection of probing beam is registered bythe use of quadrant photodiode, from which we can obtain two signals: normal andtangential one. The first one responds to illumination difference between upperand lower and the second one between left and right photodiode halves (definedrelatively to the “horizontal” surface of the sample under investigation). In realexperimental situation the thermal field in the gas over the sample because the pumpingbeam or sample heterogeneity is 3-dimensional. This type of thermal field waspresented in some works, e.g., [3]. In this situation, both normal and tangential partsare present in the deflective measurements. In this work, we present a new method forcalculating the influence of thermal field on the probing Gaussian beam parameter.For good presentation of this method we use a rather simple shape of thermal field. Ifthe sample stimulation is much wider than the width of the probing beam, thetheoretical description is one-dimensional and only the normal signal is important.
2. Gaussian beam in an optically homogeneous medium
From work [6] it follows, that the electric field distribution in Gaussian beam withradius a and wavelength λ (wave number k = 2π/λ ) which propagates in homogeneousmedium with refraction index n0 can be written as:
(1)
where
(2)
The beam enters the system on the plane z = 0 and propagates in the plus direction ofthe OZ axis, and its waist is placed on the plane z = L. E0 is the electric field intensityin the middle of the waist. The parameter zR = ka2n0 is called Rayleigh’s length, thequantity ψ0 – wave eikonal and A0 – its amplitude (of zero order). The beam raycoordinates are defined by equations:
u x y z, ,( ) A0 r( ) ikψ0 r( ) exp=
A0 r( ) E0 1i z L–( )
zR
---------------------+
1–
,=
ψ0 r( ) z L–( )n0 in0x
2y
2+
2zR
-------------------- 1i z L–( )
zR
---------------------+
1–
.+=
r τ( ) x τ( ) y τ( ) z τ( ), ,[ ]=
Deflective signal analysis in photothermal measurements... 79
(3)
where [ξ, η] are the ray start point coordinates from the plane z = 0 (XY),zRC = zR – iL and τ is the running coordinate (in general case the complex one) alongthe ray. It is easy to notice that relations (3) represents straight lines in 6-dimensionalcomplex space.
For a given observation point (e.g., point on the detectionplane) we need to find all rays coming to that point. For this purpose, a solution of theset of Eq. (3), relatively to “rays” variable [ξ, η, τ ], needs to be found (the so-calledgeometrical optics reversal problem). After linearization we get:
(4)
From this solution it follows, that for this simplification we have a particularly simplesituation – to all observation points there comes only one ray. Equations (4) defineexactly the start point of the ray (ξD, ηD) when its observation point (xD, yD, zD) isknown.
3. Gaussian beam in a thermally disturbed medium
Let us consider a standard experimental set-up scheme for the solid state photothermalinvestigation with photodeflective detection (Fig. 1). Modulated light beam is incidenton the sample and gives it periodically specified energy flux. As a result, the sample
x τ( ) ξ iξn0τzRC
----------- ,+=
y τ( ) η iηn0τzRC
----------- ,+=
z τ( ) n0τ 1ξ 2 η 2
+
zRC2
----------------------+=
rD xD yD zD, ,[ ]=
ξD xD 1izD
zRC
----------+
1–
,≅
ηD yD 1izD
zRC
----------+
1–
,≅
τD
zD
n0
-------- 1ξ 2 η2
+
2zRC2
----------------------– zD
n0
-------- .≅ ≅
80 R.J. BUKOWSKI, D. KORTE
and surrounding gas (e.g., air) are heated and in stationary state we deal with periodicaltemperature changes in time and space, so-called thermal waves. In typical photo-thermal measurements, the total increase of sample temperature is of the order offraction of Kelvin and changes in temperature amplitude are about an order smaller.The temperature field of the thermal wave in gas above the sample can be written as:
(5)
where ag – constant rise of gas temperature, bg – amplitude of the sample surfacetemperature, ϕg – phase shift between the sample surface temperature and stimulationbeam, – wave number of the thermal wave (equal to its attenuationcoefficient), κg – thermal diffusivity of gas, Ω – angular frequency of modulation. Thequantities ag, bg and ϕg depend on some thermal and geometrical parameters of thegas and the sample and are not considered here.
These thermal waves cause changes in the gas refraction index [9–11], which givesrise to modification of the probing beam parameters. In the first approximation we canassume that
T0 = const(r) (6)
where n0 – gas refraction index at temperature T0, – refraction index
ϑg x t,( ) ag bg kgx–( ) Ωt kgx– ϕg+( )cosexp+=
kg Ω 2κg⁄( )1 2⁄=
n T( ) n0dndT---------
T0
T T0–( )+≅ n0 n0sT T T0–( ),+=
sT1n0
------- dndT--------
T0
=
Detector
Stimulation
Sample
0
x
Laser + optics system L zl zp
z
T = T(r,t)
zD
Fig. 1. Experimental set-up scheme for solid state investigation by photothermal method withphotodeflectional detection. The gas heated region have the width ∆z = zp – zl and its left edge distancefrom the set-up beginning (light beam “input”) is equal zl. We assume that the heated region width alongthe OY axis is much more greater then the probing light beam diameter. The light beam radius in theirwaist is equal a and it is placed at distance L from the “input”. The screen (detector) is placed at distancezD from the light beam “input”.
Sample
Deflective signal analysis in photothermal measurements... 81
thermal sensitivity. In such a situation, the dielectric constant of the medium in thethermally changed region is expressed by the equation
(7)
Changes in the probing beam are expressed by ν.
4. Gaussian beam deflection in the thermal wave field
The Gaussian beam deflection is one of the effects of its propagation in the thermallydisturbed medium. This means that the ray trajectory change in the thermal wave fieldmust be found. Because of the fact that temperature distribution is merely a functionof variable x, only this coordinate of the ray trajectory is changed. The other raycoordinates are described by the second and the third equations in (4). The firstcorrection to the ray x coordinate is given by [5]:
(8)
which gives (after regarding (7) and boundary conditions) correction to x ray-coordinate(see Appendix 1):
(8a)
(8b)
(8c)
Above, we use H(τ ) – Haeviside’s step function. Finally, the perturbed ray coordinatesare then
5. Gaussian beam phase change in the thermal wave field
The Gaussian beam eikonal can be written as [5]:
(10)
where integration is carried along the corrected probing beam ray (9). Finally, correction to the eikonal is given in the form:
(11)
where ψ1d – the component (proportional to P(ξ0) that arises from Gaussian beamdeflection in the thermal wave field, ψ1f – the component resulting from the thermalfield influence on the Gaussian beam phase (two terms proportional to ag and bg,respectively).
6. Gaussian beam ray amplitude change in the thermal wave field
The Jacobian of the transition from Cartesian to ray coordinates in the region τ > τp isas follows (in paraxial approximation):
(12)
x ξ τ,( ) x0 ξ τ,( ) x1 ξ τ,( )+ ξ0 1 in0τzRC
-----------+
P ξ( ) τ τs–( )τpl,+= =
y ξ τ,( ) y0 ξ τ,( ),=
z ξ η τ, ,( ) z0 ξ η τ, ,( ).=
ψ τ( ) ψ 0 τ( ) ε r τ'( )( )dτ '0
τ
∫+
ψ 0 τ( ) ε 0 r τ '( )( )dτ'0
τ
∫12----- ν r τ '( )( )dτ '
0
τ
∫+ +≅
ψ0 ψ1+
=
=
ψ1 zn0
xD
zRC2
----------- τD τs–( ) P ξ0( ) 1 izD
zRC
-----------+
2–
τpl ag n02sT τpl
n02
sT bg τpl kg x ξ τs,( )–( )exp Ωt kg x ξ τs,( ) ϕg+–( )cos
+
+ ψ1d ψ1 f+
=
=
D τ( ) ∂x∂ξ--------- ∂y
∂η--------- ∂z
∂τ-------- .≅
Deflective signal analysis in photothermal measurements... 83
Using (9) we are given the probing beam amplitude changes expressed by therelation:
(13)
7. Normal signal from quadrant photodiode
The current normal signal from the photodiode under reverse bias is proportional tothe light intensity incident on it (see Appendix 2). In such a case, the signal analyzedarises from the illumination difference between the upper and lower photodiode halves:
(14)
where Kd – photodetector constant (its sensitivity). In expression (14), it has beenassumed that the sample curtains a part of the photodiode (in the region –∞ < xD < –h).
In our case, when a lock-in nanovoltmeter is used for the diode currentmeasurements, only IV (rD) is measured. We have three different contributions to thissignal, and due to this
(15)
After calculating proper integrals (see Appendix 2) we obtain
(16)
The total deflective part of the signal is the sum of two last expressions in (16):
(17)
A τ( ) A 0( ) D 0( )D τ( )
---------------
1 2⁄
E0
zR
zRC
----------- 1 in0τzRC
------------+
1–
112-----
τ τs–( ) ∂P∂ξ--------- τpl
1 in0τzRC
-----------+
-------------------------------------– .≅=
Sn Kd d yDh–
0
∫–0
∞
∫
d xD I rD( )∞–
∞
∫=
Sn SndA Sndf Snf+ + Snd Snf.+= =
Snf Af Ωt ϕg ϕf–+( ),cos=
SndA AdA Ωt ϕg ϕdA+ +( ),cos=
Sndf Adf Ωt ϕg ϕdfπ4-----–+ +
.cos=
Snd SndA Sndf+ Ad Ωt ϕg ϕd+ +( ),cos= =
84 R.J. BUKOWSKI, D. KORTE
(18)
(19)
Finally, the full signal is the sum of the deflective and phasial components and itcan be written in the form
(20)
where
(21)
(22)
8. Numerical calculations and graphs
Calculations were done for the interaction of He-Ne laser beam with thermal wavesin air. The values of the parameters taken into account were as follows: sample widthzp – zl = 5 mm, sample left edge position zl = 0.5 m, sample right edge positionzp = 0.505 m, beam waist position L = 0.5 m, detector position zD = 1.5 m, total powerof probing beam Pl = 1.0 W (“normalized” value), wavelength of probing beamλ = 636 nm, gas refraction index at temperature T0 was n0 = 1.0, refraction indexthermal sensitivity sT = 1.0 K–1 (“normalized” value), our sample was silicon withthermal diffusivity κs = 6.7×10–5 m2/s and thermal conductance λs = 110 W/(mK).The results of numerical calculations (formulas (21), (22) and (A2.17), (A2.18),(A2.23)) are presented in graphs with photothermal signal amplitude Ak [arb. u.] andadditional phase shift ϕk [rad] (i.e., relative to the temperature phase on the samplesurface) dependence on some experimental set-up parameters. Here, k = d, f, trelative to deflective or phasial component and total signal.
In the signal analyzed, as was mentioned in theoretical part of this work, twodifferent parts were marked out – phasial and deflective ones. From experimental point
Ad ϕg ϕd+( )cos Af ϕg ϕf–( )cos+------------------------------------------------------------------------------------- .=
Deflective signal analysis in photothermal measurements... 85
of view this differentiation is not essential and difficult to determine. But fromtheoretical point of view it is important, because as mentioned in the Introduction, twotypes of theoretical descriptions for deflective experimental analysis are used inliterature. In all our calculations we determine both parts as well as the total signal andcompare them. This enables us to determine the validity of the theory presentedand those used in literature.
In typical photothermal measurements, the amplitude Ak(h) and phase ϕk(h) ofphotothermal signal dependence on distance between probing beam axis andilluminated sample (Fig. 2) are investigated. With an increase of the height of probingbeam over the sample the signal dramatically decreases, which results from the factthat attenuation of the thermal wave is very strong in the medium. Analysing the graphsit can be concluded that the course of these curves is strongly dependent on both theprobing beam diameter a and angular modulation frequency Ω. Figure 2b presentsthe shape of deflective and phasial components of photothermal signal.
Figure 3 presents the amplitude and phase of photothermal signal dependence onprobing beam diameter for different angular modulation frequencies and heights ofthe probing beam over the sample. There is a value of the beam diameter for whichthe signal amplitude reaches a maximum. For small values of probing beam height
a
b
Fig. 2. Quadrant photodiode signal amplitude and additional phase shift changes relatively to probingbeam height over the sample for: Ω = 600 rad/s, a = 50 µm (curve 1); Ω = 2000 rad/s, a = 500 µm(curve 2); Ω = 600 rad/s, a = 500 µm (curve 3); others parameters are zD = 1.5 m, zp = 0.505 m, zl = 0.5 mand L = 0.5 m – a. Components of the total normal signal: deflectional (curve 1) and phasial (curve 2) forΩ = 2000 rad/s (a = 500 µm, zD = 1.5 m, zp = 0.505 m, zl = 0.5 m and L = 0.5 m); curve 3 presents thetotal signal – b.
86 R.J. BUKOWSKI, D. KORTE
over the sample a minimum is also seen. The beams of small radius are whollydisturbed by thermal wave, while the beams of large a pass through the thermal fieldin different phase, which causes their phase shift. Because of that current signals fromquadrant photodiode are also in different phase and they can extinguish each other. Onthe amplitude of photothermal signal dependance on probing beam diameter it can beseen as a minimum of the signal. For much larger values of a we enter the region inwhich the thermal wave is being declined, which means that the probing beam is notdisturbed and the value of the signal decreases.
As can be seen from Fig. 4, the signal from quadrant photodiode rapidly decreaseswith an increase of angular modulation frequency for some values of h and a (curves 1and 2). This arises from the fact that with an increase of frequency the thermal waveattenuation also increases. With an increase of frequency the temperature field gradientalso increases, which causes an increase of deflective component of the totalphotothermal signal. This effect can be seen in curves 3 and 4. It is worth mentioningthat the range of angular modulation frequency changes is determined by those valuesof Ω that are possible to attain using a mechanical modulator. With an increase ofangular modulation frequency the thermal waves attenuation increases, and that is whyvery high values of Ω are not used in photothermal measurements.
Fig. 3. Quadrant photodiode signal amplitude and additional phase shift changes relatively to probingbeam diameter: Ω = 60 rad/s, h = 200 µm (curve 1); Ω = 60 rad/s, h = 800 µm (curve 2); Ω = 600 rad/s,h = 200 µm (curve 3); others parameters are zD = 1.5 m, zp = 0.505 m, zl = 0.5 m and L = 0.5 m – a.Components of the total normal signal: deflectional (curve 1) and phasial (curve 2) for Ω = 600 rad/s,h = 200 µm (others parameters are zD = 1.5 m, zp = 0.505 m, zl = 0.5 m and L = 0.5 m); curve 3 presentsthe total signal – b.
a
b
Deflective signal analysis in photothermal measurements... 87
Figure 5 presents the dependence of photothermal signal on detector coordinatefor different angular modulation frequencies, different radii of the probing beam and
Fig. 5. Quadrant photodiode signal amplitude and additional phase shift changes relatively to detectorposition: a = 50 µm, Ω = 60 rad/s, h = 200 µm (curve 1); a = 50 µm, Ω = 600 rad/s, h = 200 µm(curve 2); a = 50 µm, Ω = 600 rad/s, h = 1200 µm (curve 3); a = 500 µm, Ω = 600 rad/s, h = 1200 µm(curve 1); pthers parameters are zp = 0.505 m, zl = 0.5 m and L = 0.5 m – a. Components of the totalnormal signal: deflectional (curve 1) and phasial (curve 2) for a = 50 µm, Ω = 60 rad/s, h = 200 µm(zp = 0.505 m, zl = 0.5 m and L = 0.5 m); curve 3 presents the total signal.
a
b
Fig. 4. Quadrant photodiode signal amplitude changes to modulation angular frequency: a = 50 µm,h = 200 µm (curve 1); a = 50 µm, h = 1200 µm (curve 2); a = 500 µm, h = 1200 µm (curve 3);a = 500 µm, h = 2400 µm (curve 4); others parameters are zD = 1.5 m, zp = 0.505 m, zl = 0.5 m andL = 0.5 m – a. Components of the total normal signal: deflectional (curve 1) and phasial (curve 2) fora = 50 µm, h = 200 µm (others parameters are zD = 1.5 m, zp = 0.505 m, zl = 0.5 m and L = 0.5 m);curve 3 presents the total signal – b.
a b
88 R.J. BUKOWSKI, D. KORTE
its height over the sample. There is a rapid drop of the signal when the beam waist isover the detector and a rise when the beam waist is a bit in front of or behind it.
As can be seen from Fig. 6, for small values of the probing beam radii there is dropof the signal when the beam waist is over the detector, and rise when the beam waistis a bit in front of or a bit behind it. No such property of the signal is observed for largevalues of the probing beam radii. In this case, the curve increases monotonically.
Figure 7 shows the signal amplitude and additional phase shift dependence on thesample position of probing beam. The value of the signal decreases when we approachthe detector. Phasial component gets smaller when the sample approaches the detector,but deflective component reaches a maximum when the beam waist is over the sample,and then decreases again.
On the graphs with phase change of photothermal signal from quadrant photodiodesome discontinuities can be seen. They are the result of only “partial” phasenormalization of the ambiguous component of the signal phase to the range (0, 2π).
9. Conclusions
The influence of different experimental set-up parameters on signal value determinedin photothermal investigation with mirage effect has been analysed in the work. Thesignal dependence on such parameters as probing beam radius, waist position, heightover the sample surface and detector position was considered. The theory worked out,based on complex geometrical optics methods, offers the possibility of taking intoaccount many other parameters (e.g., probing beam modulation frequency), which areimportant for interpretations of measurement results. The so-called phasial anddeflective components of normal signal created as a result of phase change anddeflection of the beam, which is probed the one-dimensional field of the thermal wavepropagated in the gas over the sample exited by harmonically modulated pumping
Fig. 6. Quadrant photodiode signal amplitude changes to beam waist position: a = 50 µm, Ω = 60 rad/s,h = 200 µm (curve 1); a = 50 µm, Ω = 600 rad/s, h = 200 µm (curve 2); a = 50 µm, Ω = 60 rad/s,h = 1200 µm (curve 3); a = 500 µm, Ω = 600 rad/s, h = 1200 µm (curve 4); others parameters arezD = 1.5 m, zp = 0.505 m, zl = 0.5 m – a. Components of the total normal signal: deflectional (curve 1)and phasial (curve 2) for a = 50 µm, Ω = 600 rad/s, h = 200 µm (zD = 1.5 m, zp = 0.505 m, zl = 0.5 m);curve 3 presents the total signal.
a b
Deflective signal analysis in photothermal measurements... 89
beam, were considered. Quadrant photodiode detection has been analysed. Results arepresented in analytical form and in the form of graphs and can be used for experimentalset-up optimization. This means that by proper choice of the probing beam radius,waist position, height over the sample surface and detector position, probing beammodulation frequency we are able to increase the total normal signal or, if necessary,one component of it (phasial or deflective one) can be eliminated.
In the theory presented, we take into account for first time simultaneously twoprocesses that are essential for photothermal signal creation – probing beam deflectionand probing beam phase change. The dividision of the normal signal into deflectiveand phasial parts has rather theoretical meaning. Elimination of these parts frommeasurement signal will be very difficult or impossible. We present this dividision inorder to make it possible for our readers to compare our theory with other ones, whichtake into account only one of the processes presented above. We use a new methodfor this type of calculation – complex geometrical optics method – and the resultsobtained are not fully equivalent to those of other calculations. From the graphspresented we can conclude that in many cases the results are qualitatively comparablewith those of other theories.
The analytical formulas obtained are rather complicated, but, in our opinion, thisis mainly due to the assumed method of detection, namely by a quadrant photodiode.
Fig. 7. Quadrant photodiode signal amplitude and additional phase shift changes relatively to sampleposition: a = 50 µm, Ω = 60 rad/s, h = 200 µm (curve 1); a = 500 µm, Ω = 600 rad/s, h = 1200 µm(curve 2); a = 50 µm, Ω = 60 rad/s, h = 1200 µm (curve 3); others parameters are zD = 1.5 m andL = 0.5 m – a. Components of the total normal signal: deflectional (curve 1) and phasial (curve 2) fora = 50 µm, Ω = 60 rad/s, h = 200 µm (zD = 1.5 m, L = 0.5 m); curve 3 presents the total signal.
a
b
90 R.J. BUKOWSKI, D. KORTE
It needs additional integration of the probe beam intensity (expression (14)). Of course,it is possible to use other methods of detection. Some of them were analyzed in [8],but for phasial component of the normal signal only.
10. Appendix 1
The thermal wave really exists only over the sample surface, so relation (5) is to berewritten as
(A1.1)
Because of the small value of the thermal wave amplitude bg in standardexperimental situation (of an order of mK) in relation (8) we neglect derivation overthe z coordinate. This means that we neglect additional phase shifts caused by inputand output of the probing beam into and out of the region with thermal wave. In sucha situation, we obtain from relation (8):
(A1.2)
where from relations (3) and (4) in paraxial approximation we have
(A1.3)
Because of a linear character of relations (A1.3) the integral in (A1.2) is elementaryone but it has very large and complicated shape. However, our calculations are notexact – we use paraxial approximation and perturbation calculus. So, if we have(zp – zl) << zD instead of exact value of the integral in (A1.2) we can apply its simplifiedvalue obtained by middle point method:
(A1.4)
ϑg x t,( ) ag bg kg x–( )exp Ωt kg x– ϕg+( )cos+ H z zl–( ) H zp z–( )– .=
x1 ξ τ,( ) 2 bgkgn02sT–=
τ τ'–( ) kgx τ '( )– Ωt kgx τ '( ) ϕg+–
π4-----–
sinexp
0
τ
∫×
H z τ '( ) zl– H zp z τ '( )–
–
× dτ '
x τ'( )zRC in0τ'+
zRC izD+------------------------------ ,= z τ '( ) n0τ '.=
f τ τ ',( )dτ 'τ1
τ2
∫ f τ τs,( )τpl,≅ τs
τ1 τ2+
2------------------- ,= τpl τ2 τ1.–=
Deflective signal analysis in photothermal measurements... 91
In our case, because of Heaviside’s step functions in integral (A1.2), quantitiesτs and τpl depend on zl and zp (see formulas (8c)). It is clear that in this case the accuracyof the simplified integration can be easily controlled.
The same simplified method of integration is used in other places when integrationover the τ ' is needed. It is worth noting here that in some cases integrals are notelementary.
11. Appendix 2
The amplitude and the phase of probing beam on the detector surface (quadrantphotodiode) can be written as:
(A2.1)
(A2.2)
where A0(zD), ψ0(zD) are expressed by Eq. (2) and ψ1(zD) is described by Eq. (11);a1(zD) from (A2.1) can be written as:
(A2.3)
where P(ξ ) is expressed by (16b) and zs = n0τ s. Finally, the electric field distributionof probing beam on the surface of detector (quadrant photodiode) can be written as
(A2.4)
We are allowed now to calculate the intensity distribution on that detector:
(A2.5)
Correction to the phase and amplitude is complex, so it can be written as:
I0g(rD) is the undisturbed Gaussian beam intensity distribution, but IV (rD) representschanges in the probing beam intensity caused by its interaction with the thermal wavefield. From relations (A2.8) we have
(A2.9)
This means that we can divide IV (rD) into two parts: deflective and phasial ones.Deflective component of Gaussian beam intensity changes has the shape:Igd(rD) = IgA(rD) + Igψd(rD) where IgA(rD) – deflective component resulting from theGaussian beam amplitude changes in the thermal wave field, Igψd(rD) – a part ofthe deflective component arising from the Gaussian beam phase change connectedwith ray path moving. The last term in (A2.9) is a phasial component of the Gaussianbeam intensity changes arising from the Gaussian beam phase change in the field ofthermal wave.
In our case (lock-in measurements) only IV (rD) is measured. Taking into account(A2.7) and (A2.8) we get:
(A2.10)where
(A2.11)
I rD( ) I0g rD( ) 1 2kψ 1I zD( ) 2a1R zD( )+–≅ Ig0 rD( ) IV rD( ),+=
Deflective signal analysis in photothermal measurements... 93
and Pl – the total power of undisturbed probing beam. Moreover, we need relations(A2.3) and (11), where we have:
(A2.12)
(A2.13)
In all calculations, we assume that zD > zp, i.e., quadrant photodiode is placedbehind the sample.
11.1. Phasial component of the normal signal
Taking into account relation (A2.9) we have
(A2.14)
where (comp. (11))
(A2.15)
We neglect here constant component proportional to ag – this component is notmeasured by lock-in technique. After calculating the proper integrals we are given:
(A2.16)
where
(A2.17)
(A2.18)
P ξ0( ) 2 n02
sT bg kg CxxD– Ωt CxxD ϕg+–
π4
------–
,sinexp=
Cx kg 1 izs
zRC
-----------+
1 izD
zRC
-----------+
1–
.=
Snf 2KdImh–
0
∫–0
∞
∫
d xD kψ 1 f zD( )IVx rD( ) .–=
ψ 1 f zD( ) 12-----n0
2sT bgτpl e
i Ω t ϕg CxxD–+( )e
i– Ω t ϕg CxxD–+( )+ e
CxxD–.=
Snf Af Ωt ϕg ϕf–+ cos=
Af12----- π
f----- Kd n0 sT bg k Im zp zl–( ) F1 l F2 l+( )2
F2R F1R–( )2+ ,=
ϕftanF2R F1R–
F1I F2I+--------------------------- ,–=
94 R.J. BUKOWSKI, D. KORTE
(A2.19)
(A2.20)
and erf(ξ ) is the error function.
11.2. Deflective component of the normal signal
Deflective part of the normal signal in our case has two components, the first onecomes from the diversion of the Gaussian beam in the thermal wave field (relations(13), (A2.1) and (A2.3)), and the second arises from the Gaussian beam phase changeconnected with ray path moving (relations (11), (A2.8) i (A2.9)).
For the first component calculation we have:
(A2.21)
Taking into account relation (A2.3) we obtain:
(A2.22)
(A2.23)
where
F1
1 i+( )Cx
2 f-------------------------
2
1 2erf1 i+( )Cx
2 f------------------------- erf
1 i+( )Cx
2 f------------------------- h f–+–
exp=
F1R iF1I,+=
F2
1 i–( )Cx
2 f-------------------------
2
1 2erf1 i–( )Cx
2 f------------------------- erf
1 i–( )Cx
2 f------------------------- h f–+–
exp=
F2R iF2I,+=
SndA 2KdReh–
0
∫–0
∞
∫
dxDa1 zD( )IVx rD( ) .=
SndA AdA Ωt ϕg ϕdA+ +( ),cos=
AdA 2Kd G1R G2R–( )2G1I G2I+( )2
+ ,=
ϕdAtanG1I G2I+
G1R G2R–-----------------------------–=
Deflective signal analysis in photothermal measurements... 95
Deflective signal analysis in photothermal measurements... 97
References
[1] MURPHY J.C., AAMODT L.C., Photothermal spectroscopy using optical beam probing: mirage effect,Journal of Applied Physics 51(9), 1980, pp. 4580–8.
[2] GLAZOV A., MURATIKOV K., Photodeflection signal formation in thermal wave spectroscopy andmicroscopy of solids within the framework of wave optics. “Mirage” effect geometry, OpticsCommunications 84(5-6), 1991, pp. 283–9.
[3] AAMODT L.C., MURPHY J.C., Photothermal measurements using a localised excitation source,Journal of Applied Physics 52(8), 1981, pp. 4903–14.
[4] BUKOWSKI R.J., BODZENTA J., MAZUR J., KLESZCZEWSKI Z., Parameter Estimation in PhotothermalMeasurements with Photodeflective Detection, [In] Nondestructive Characterization ofMaterials VII, Part 1, Proceedings of the Seventh International Symposium on NondestructiveCharacterization of Materials held in Prague, Czech Republic, June 1995, Transtec Publications1996, pp. 295–302.
[5] KRAWCOW JU.A., ORŁOW JU.I., Geometrical Optics of the Nonhomogeneous Media, WNT, Warszawa1993 (in Polish).
[6] BUKOWSKI R.J., Geometrical optics application in description of gaussian beam propagation in anoptically homogenous medium, [In] Proceedings of the 2nd National Conference “Physical Basisof the Nondestructive Investigations”, Gliwice Chapter of the Polish Physical Society and Instituteof Physics of the Silesian University of Technology, Gliwice’97 (in Polish).
[7] BUKOWSKI R.J., Mirage effect description in the frame of the complex rays optics Proceedings of theSPIE 3581, 1998, pp. 285–92.
[8] BUKOWSKI R.J., Complex geometrical optics application in different methods of detection analysisin photothermal measurements, Zesz. Nauk. Pol. Śl., Seria: Matematyka–Fizyka, z87, pp. 37–54(1999) (in Polish).
[9] YARIV A., YEH P., Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley,New York 1984.
[10] CARSLAW H.S.. JAEGER J.C., Conduction of Heat in Solids, Oxford University Press, Oxford 1959.[11] Bodzenta J., Bukowski R., Kleszczewski Z., Mazur J., Pustelny B., Thermal waves application in
Received July 3, 2004in revised form October 1, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Influence of oil dispersed in seawater on the bi-directional reflectance distribution function (BRDF)
ZBIGNIEW OTREMBA
Gdynia Maritime University, ul. Morska 81-87, 81-225 Gdynia, Poland
The bidirectional reflectance distribution function (BRDF) of the sea areas polluted by oil-in-wateremulsion has been studied at the ends and in the centre of the visible light spectrum. The MonteCarlo code was applied to model water leaving radiance for the entire upper hemisphere. Solarirradiance was represented by 1 billion virtual photons which reach the sea surface at an angle of40 deg. The results are displayed as the BRDF vs. two variables: zenith and azimuth angles. Thestrong impact of wavelength on the BRDF has been revealed while, the size distribution of oildroplets has an insignificant impact. The presence of oil emulsion modifies the shape of the BRDFsignificantly, and the latter depends on the oil type. Irradiance reflectances and radiancereflectances derived from BRDFs obtained are also presented.
Keywords: oil, seawater, phase function, inherent optical properties (IOP), radiance, bidirectional reflectance distribution function (BRDF), modelling.
1. Introduction
The sea, lake or river areas polluted by oil substances can modify the above abovewater radiance distribution. It has been found that oil film affects the angulardistribution of water leaving radiance [1]. Also oil present in the sea environment, inthe form of emulsion, can modify the light flux leaving sea surface. Introductory resultsof the modelling of visibility of oil emulsion in seawater have already been described[2]. However, only roughly derived phase function (PF) of spherical oil dropletssuspended in seawater has been applied. Results of PF modelling using a well-verifiedcode for Mie solution were reported lately [3]. Phase function and both scattering andabsorption coefficients of oil-in-water for various oil droplets size distribution togetherwith optical properties of natural components of the seawater have opened up thepossibility of constructing an optical model of sea environment polluted by oilemulsion and subsequently, the possibility of studying radiative transfer of oil pollutedsea area.
The bidirectional reflectance distribution function (BRDF) was first applied toremote investigation of land areas [4]. It is a strictly defined physical parameter, which
100 Z. OTREMBA
describes the ability of the surface of land, or aquatic areas to reflect light. Simply, ifincident angular (for the whole upper hemisphere) radiance Li is known, the reflectedradiance Lr can be determined using the following integral:
(1)
where: Lr(θr , ϕr) – reflected radiance, r (θi, ϕi, θr, ϕr) – BRDF, Li(θi, ϕi) – incidentradiance, θi – nadir angle for incident photons (0 < θ < π/2), ϕi – azimuth angle forincident photons, θr – nadir angle for reflected photons ( π/2 < θ < π), ϕr – azimuthangle for reflected photons.
The integral (1) has a closed reference with the definition of the BRDF firstlyproposed by NICODEMUS [5]:
(2)
The numerator in Eq. (2) expresses the infinitesimal reflected radiance observedfrom the direction created by angles θr and ϕr caused by the infinitesimal incidentirradiance (denominator) from direction determined by angles θi and ϕi. As regardsactual measurement of BRDF the denominator expresses irradiance from only onedirection (θi, ϕi) while the rest of hemisphere must be dark, the numerator being theradiance measured from defined direction (θr , ϕr). Measurements of the BRDF ofterrestrial areas are performed using special radiometers [6]. Unfortunately, in theocean areas measurement of the BRDF is impossible as yet. However, there are popularoperational reflectances (possible to measure in environment but dependent on solarlight condition) in ocean optics, namely: irradiance reflectance RE and radiancereflectance RL called also remote sensing reflectance Rrs . Those operationalreflectances can be derived using the BRDF:
(3)
Lr θr ϕr,( ) r θi ϕi θr ϕr, , ,( ) Li θi ϕi,( ) θisin θidθidϕicos0
π2---
∫0
2π
∫=
r θi ϕi θr ϕr, , ,( )d Lr θr ϕr,( )
Li θi ϕi,( ) θisin θi dθidϕicos------------------------------------------------------------------------ .=
RE
Er 0+( )
Ei 0+( )
--------------------=
r θi ϕi θr ϕr, , ,( ) Li θi ϕi,( ) θisin θidθidϕicos0
In the above equations Er(0+) denotes the water-leaving irradiance just above thewater surface, Ei(0
+) – the solar irradiance just above the water, Lr(θr = 0, 0+) – waterleaving radiance measured perpendicularly to the surface [7]. Notation 0+ indicatesthat the defined parameter is measured just above the surface (analogously 0– relatesto just below the surface). Indexes i (incident) or r (reflected) [8, 9] are sometimesreplaced respectively by d (downwelling) and u (upwelling) [7, 10], especially whenthe defined parameters are related to nadir angle θ equal 0 or π only.
The BRDF of the sea area can be modelled using the Monte Carlo method, in whichoptical model of aquatic environment is applied to simulate the radiative transfer byanalyzing the virtual migration and definitive destination of photons which are directedto sea surface on the trot. The Monte Carlo code has been just successfully used in theBRDF modelling when sea surface is covered by a thin oil film [1]. In this paper theattention is mainly focused on the scale of the influence of oil-in-water emulsionsphase function on the shape of BRDF of sea area polluted by oil immersed in the bulkof water.
2. Model of the environment under study
Optical features of seawater are described by absorption coefficient a, scatteringcoefficient b and by phase function of light scattering p(θ ). Values of absorption andscattering coefficients have been chosen just as in the paper on the BRDF of sea areascovered by oil film [2], namely for typical turbid coastal case II sea water appropriatefor the Gulf of Gdańsk (Poland) in the southern Baltic Sea (see the Table), and thephase function pn(θ ) characteristic of turbid water after PETZOLD [11] was applied.Optical parameters a and b of water from oil polluted area, as well as water from
RL
Lr θr = 0 0, +( )
Ei 0+( )
--------------------------------------=
r θi ϕi θr = 0, ,( ) Li θi ϕi,( ) θicos θisin dθi dϕi0
T a b l e. Optical parameters of reference seawater.
Optical parameter 420 nm 550 nm 700 nm
Light absorption coefficient a [m–1] 0.7 0.27 0.8
Light scattering coefficient b [m–1] 0.7 0.5 0.55
102 Z. OTREMBA
reference sea area were represented by the same values. However, it was assumed thatfor polluted sea area contribution of those parameters of oil-in-water emulsion and ofnatural sea components is the same. The phase function po(θ ) of oil component wasderived and already presented [3]. In this paper, a series of various phase functions foroil emulsion have been applied. They relate to:
– two types of oil: Petrobaltic extracted from southern Baltic in the PolishEconomic Zone, Romashkino from reservoir in the Tatarstan (country in a formerSoviet Union);
– three wavelengths (420 nm, 550 nm, 700 nm); – two oil droplets size distributions (one adequate for 1 week old and one – for
2 weeks). Totally – 12 cases were analyzed.
Optical features of two kinds of oil mentioned above are described in the earlierpaper [12] (the Petrobaltic oil represents oil types of relatively low values of the realand imaginary parts of the complex refractive index, whereas the Romashkino oil hasopposite parameters). Tests were carried out for oil concentration of 0.5 ppm (whereasadmissible oil concentration in vessel waste water is 15 ppm).
3. Monte Carlo model settings
The Monte Carlo sampling method has been used for sea environment simulation. AllMonte Carlo simulations were carried out using 1 billion incident photons, which playthe role of a virtual model of incident solar irradiance. Photons were directed on theplane sea surface at an angle of θi = 40 deg. The Monte Carlo input data, i.e., theprobability of photon absorption pa and photon scattering ps on defined length ofphoton run, are determined from values of an absorption coefficient a (Eq. (5)), anda scattering coefficient b (Eq. (6)), respectively, whereas the cumulative function ofprobability of photon scattering under defined angle d(θ ) is derived from the phasefunction p(θ ) (Eq. (7)).
(5)
(6)
(7)
Analogously, other environment properties like the Fresnel equations (whichdescribe air-water boundary optical properties) or bottom light reflection have been
pa 1 a–( ),exp–=
ps 1 b–( ),exp–=
d θ( )
p θ( )dθ0
θ
∫
p θ( )dθ0
π
∫
--------------------------- .=
Influence of oil dispersed in seawater... 103
converted to adequate probability densities. One thousand eight hundred and thirtythree virtual receivers of photons – counting photons entering solid angles of variousvalues – cover the upper hemisphere. One thousand two hundred and ninety sixreceivers capture photons in sectors of size of 0.004363323130 sr; 252 – were2.5 times less; 144 – 5 times less; 144 – the narrowest and closest to the zenith sectorsof a size 10 times less. Thus, it was possible to obtain the values of the BRDF for all1836 directions covering the whole upper hemisphere as the output data. The accuracyof the Monte Carlo simulation depends on the number of incident photons, on an angleθr and on a value of solid angle photon receivers. The settings listed above give arelatively small error of the output data, which is not greater than 1% for angles lessthan 60°. Further statistical spread of output data increases and above 80° the error canreach even 50%. The negative aspects of simulations based on such settings are thatthey are time-consuming. It appears that in order to obtain the BRDF values for eachof 1836 hemisphere sectors with satisfactory accuracy, 8–12 hours of the PC-computerwork is necessary. The time-period of one simulation grows with an increase ofscattering coefficients or a decrease of absorption coefficient (because average lifetimeof photons in the bulk of water grows) and inversely gets less when scatteringcoefficient decreases or absorption coefficient increases.
4. Results
Simulations of radiative transfer were carried out for 15 cases: 12 cases for sea areascontaining oil-in-water emulsion and 3 – for natural seawater. The output data ofsimulations were given in the form of matrices (size 55×36) representing the BRDFsfor 1836 upper hemisphere directions. The BRDFs obtained were smoothed tostatistical noise reduction [13], and presented in cylindrical coordinates in which radialcoordinate r expresses zenith angle θr, angle coordinate ϕ – azimuth angle ϕr,
Fig. 1. Matrix of the BRDF obtained from Monte Carlo simulation (a) and the same after smoothing (b).The same results are displayed in cylindrical coordinates in Fig. 6b.
a b
104 Z. OTREMBA
coordinate z – value of BRDF. Figure 1 shows an example of rough and smoothedresults obtained from one of the Monte Carlo simulations. This visualizations of aBRDF matrix relates to the reference sea area at wavelength λ = 550 nm. One can
Fig. 2. BRDFs of sea areas polluted by oil-in-water emulsion made from Petrobaltic type crude oil forthree wavelengths: 420 nm (a), 550 nm (b), 700 nm (c). Emulsion was aged for one week. White pointsindicate position of specular reflection (θr = θi = 40°).
a b c
Fig. 3. BRDFs of sea areas polluted by oil-in-water emulsion made from Petrobaltic type crude oil forthree wavelengths: 420 nm (a), 550 nm (b), 700 nm (c). Emulsion was aged for two weeks.
a b c
Fig. 4. BRDFs of sea areas polluted by oil-in-water emulsion made from Romashkino type crude oil forthree wavelengths: 420 nm (a), 550 nm (b), 700 nm (c). Emulsion was aged for one week.
a b c
Influence of oil dispersed in seawater... 105
compare both visualizations of the BRDF: in cartesian coordinates (Fig. 1) and incylindrical ones (Fig. 6b).
Results for oil-polluted sea area are divided into three cases for the followingwavelengths: 420 nm, 550 nm and 700 nm. Figure 2 shows the BRDFs for sea areapolluted by Petrobaltic crude oil as one-week-old in-water emulsion. Figure 3 – thesame as in Fig. 2 but for a two-week-old emulsion. Analogical cases are presented inFigs. 4 and 5 but for Romashkino crude oil. Figure 6 shows the BRDFs for a referencesea area.
5. Discussion
Both solar light wavelength and the type of oil have an impact on the BRDF shape.If a wavelength is considered, one can notice that the values of the BRDF arerelatively small at the ends of light spectrum in comparison with the BRDF obtainedin the center of the spectrum. Generally, the presence of oil emulsion in the watercauses a decrease of the BRDF as well as the modification of the shape of this function.It is characteristic that if a certain sea area is polluted by an oil emulsion the peak
Fig. 5. BRDFs of sea areas polluted by oil-in-water emulsion made from Romashkino type crude oil forthree wavelengths: 420 nm (a), 550 nm (b), 700 nm (c). Emulsion was aged for two weeks.
a b c
Fig. 6. BRDFs of reference sea areas (without oil pollution) for three wavelengths: 420 nm (a),550 nm (b), 700 nm (c).
a b c
106 Z. OTREMBA
representing light backscattering appears. Such a peak does not exist in the BRDFs ofreference seawater. The altitude of a backscattering peak is bigger in the BRDFs relatedto Petrobaltic transparent oil in comparison with the BRDF related to Romashkino oil.At the short wave end of the spectrum the BRDFs for Romashkino have very smallpeaks. This phenomenon occurs because the absorption coefficient of Romashkino oilemulsion has a relatively high value in relation to the scattering coefficient.
The above mentioned backscattering peaks for the reference sea area are invisiblefor any wavelength, however a small increase of the BRDF is noticeable around solarlight direction (especially for λ = 550 nm). The phenomenon of the existence ofbackscattering peak for sea area polluted by oil-in-water emulsion shows thepossibility of identifying oil pollution in the sea using the shape of the BRDF beinganalyzed. Comparing the BRDFs for one-week old (Figs. 2 and 4) and two-week-old(Fig. 3 and 5) emulsions one can notice that the oil droplet size distribution only slightlyaffects the BRDF.
The next benefit of modelling the BRDF is in the potential for the derivation ofpractically measurable reflectances like the irradiance reflectance or radiancereflectance. Those reflectances were numerically calculated using Eqs. (3) and (4).The radiance reflectance is higher in the middle than at the ends of the spectrum andalso is higher for the reference water than for the polluted one. As mentioned in Sec. 2,the optical parameters a and b of polluted and reference water have the same values.Therefore, one should interpret the differences in RL as a consequence of dissimilarityof phase functions. Namely, the differences in values of RL observed in Fig. 7 arecaused by the differences between phase functions of oil droplets and phase functionof natural seawater components. Comparing values of RL for two kinds of oil one cannotice that when oil has low transparency (Romashkino-type) RL takes lower value incomparison with more transparent oil (Petrobaltic). However, this differencedisappears at long-wave end of light spectrum. On the other hand, if the irradiance
Fig. 7. Radiance reflectance of the sea areas polluted by oil-in-water emulsion (gray bars) and the samefor the reference sea area (white bars). Bright gray bars relate to Petrobaltic-type oil, dark gray ones – toRomashkino-type oil.
Influence of oil dispersed in seawater... 107
reflectance RE is considered (Fig. 8), values of RE are higher for oil of low transparency– inversely to those for RL (but only at short-wave end of light spectrum, because inthe middle and long-wave end values of RE are the same). More distinct differencesin RE are observed when photon receiver is positioned just below the sea surface,because the dazzle phenomenon induced by mirror-like reflection of the sun is thenavoided. To estimate this effect also for such a case the underwater irradiancereflectance R(0–) was also determined (Fig. 9).
Values of R(0–) are almost 2 times higher for reference water than for the pollutedwater. It is worth noticing that during the investigation in real sea environment suchmeasurement would not be possible by any distant method, because a sensor of R(0–)has to be immersed in the water. Additionally, in such a situation problems withself-shading of irradiance meter must be solved [14]. High accuracy of virtually
Fig. 8. Irradiance reflectance of the sea areas polluted by oil-in-water emulsion (gray bars) and the samefor the reference sea area (white bars). Bright gray bars relate to Petrobaltic-type oil, dark gray ones – toRomashkino-type oil.
Fig. 9. Just below the sea surface irradiance reflectance of the sea areas polluted by oil-in-water emulsion(gray bars) and the same for the reference sea area (white bars). Bright gray bars relate to Petrobaltic-typeoil, dark gray ones – to Romashkino-type oil.
108 Z. OTREMBA
measured RL (Fig. 7) and RE (Figs. 8 and 9) is achieved owing to the Monte Carlomodel settings described in Sec. 3; it was confirmed that the dissipation of the resultsdid not exceed 0.5%.
The only changes of the BRDF caused by oil emulsion were analyzed ininvestigation described in this paper. Analogous studies can be carried out also forother permanent or incidental components of sea areas, such as some plankton cells,air bubbles, sand-dust and unexpected contaminants. The efficiency of modellingthe BRDF and consequently the modelling of the BRDF derivatives, i.e., radiancereflectance and irradiance reflectance will depend on good knowledge of the inherentoptical properties of all substances suspended in the bulk of water. These inherentoptical properties include: absorption coefficient a, scattering coefficient b and phasefunction p(θ ). At the same time it should be mentioned that the attenuation coefficientof water or a Secchi disc depth are insufficient for the BRDF modelling.
Methods presented in this paper can also be very useful for interpretation ofthe images of water areas, and will help with the reverse problem solutions(determination of water components using the shapes of the BRDFs). Investigationdescribed in this paper could be conducted in estuary, lake and river waters, but undercondition that the knowledge on inherent optical properties (IOPs) of such waters ismore developed.
References
[1] OTREMBA Z., PISKOZUB J., Modelling the bidirectional reflectance distribution function (BRDF)of seawater polluted by an oil film, Optics Express 12(8), 2004, pp. 1671–6; http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1671
[2] OTREMBA Z., PISKOZUB J., Modeling the remotely sensed optical contrast caused by oil suspended inthe sea water column, Optics Express 11(1), 2003, pp. 2–6; http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1671
[4] WANG Z., ZENG X., BARLAGE, M., DICKINSON R., GAO, F., SCHAAF C., Using MODIS BRDF and albedodata to evaluate global model land surface albedo, Journal of Hydrometeorology 5(1), 2004, pp. 3–14.
[5] NICODEMUS F., Reflectance nomenclature and directional reflectance and emissivity, Applied Optics9(6), 1970, p. 1474–5.
[6] SANDMEIER, S., MIDDLETON E., DEERING D., QIN W., The potential of hyperspectral bidirectionalreflectance distribution function data for grass canopy characterization, Journal of GeophysicalResearch 104(D8), 1999, pp. 9547–60.
[7] OLSZEWSKI J., DARECKI M., Derivation of remote sensing reflectance of Baltic waters from above-surface measurements, Oceanologia 41(1), 1999, pp. 99–111.
[8] SNYDER W., Definition and invariance properties of structured surface BRDF, IEEE Transactions onGeoscience and Remote Sensing 40(5), 2002, pp. 1032–7.
[9] STRÖMBECK N., Water quality and optical properties of Swedish lakes and coastal waters in relationto remote sensing, Acta Universitatis Uppsaliensis, Series: Comprehensive summaries of Uppsaladissertation from the Faculty of Science and Technology 2001, p. 27.
Influence of oil dispersed in seawater... 109
[10] HALTRIN V., Diffuse reflection coefficient of a stratified sea, Applied Optics 38(6), 1999, pp. 932–6.[11] PETZOLD T., Volume scattering functions for selected ocean waters, Rep. to Contract No. N62269-
-71-C-0676, SIO (Scripps Institute of Oceanography, La Jolla, USA) Series, Ref. 72–78, 1972; alsoin: Light in the Sea, [Ed.] J.E.Tyler, Dowden, Hithinsons and Ross Inc., Strounburg, Pennsylvania1977, p. 384.
[12] OTREMBA Z., The impact on the reflectance in VIS of a type of crude oil film floating onthe water surface, Optics Express 7(3), 2000, pp. 129–34; http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-129.
[13] O’HAVER T.C., BEGLEY T., Signal-to-noise ratio in higher order derivative spectrometry, AnalyticalChemistry 53(12), 1981, pp. 1876–8.
[14] PISKOZUB J., Effect of ship shadow on in-water irradiance measurements, Oceanologia 46(1), 2004,pp. 103–12.
Received May 11, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Problems with cracking of AlxGa1–xN layers
EWA DUMISZEWSKA1, 2, DARIUSZ LENKIEWICZ1, 2, WLODEK STRUPINSKI1, AGATA JASIK1, 2, RAFAL S. JAKIELA1, 3, MAREK WESOLOWSKI1
1Institute of Electronic Materials Technology, ul. Wólczynska 133, 01-919 Warszawa, Poland
2Warsaw University of Technology, Plac Politechniki 1, 00-661 Warszawa, Poland
3Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warszawa, Poland
AlxGa1–xN is a wide band-gap material, which can be used for manufacture of UV detectors.Unfortunately, there are problems with the cracking of those layers occurring above some criticalthickness, which is a bit smaller from the one used for detectors (about 1 µm). Our investigationconcentrated on the causes of crack formation. To avoid it we used so-called special AlN nucleationlayer, which was to stop the relaxation. We obtained a strained layer free of cracking, but witha very big number of dislocations. We compared dislocation densities of strained and relaxedAl0.4Ga0.6N layers. The first one was characterized by a higher dislocation density than the secondone. We also investigated the problem with cracking occurring in Al0.4Ga0.6N epitaxial layersduring the doping, and how to control this process. The relaxation of the layers started for verylow impurity densities and went on when we increased the amount of the dopant.
Keywords: Al0.4Ga0.6N, GaN, Si doping.
1. Introduction
AlxGa1–xN is a very interesting wide band-gap material because it can be used forfabrication of many optoelectronic devices such as blue-green light emitting diodes,laser diodes, and visible-blind photodetectors [1]. AlxGa1–xN grown on GaN layer isthe most often used component in all of these devices in order to ensure a very goodperformance. Unfortunately, a big lattice mismatch (2.16%) between GaN and AlNlayers results in generation of a big number of dislocations. This causes the crackingof AlxGa1–xN layers with either big thickness or high Al content, which material isused for preparation of UV detectors. This cracking seems to deteriorate performanceof the device [2].
According to Qu et al. [3], increasing AlxGa1–xN thickness makes more and moredislocations generate for relaxation. The propagation of cracks will not occurdislocations are generated easily enough. If some processes which impede thegeneration of misfit dislocations exist, cracks will generate to relax the strain at some
112 E. DUMISZEWSKA et al.
thickness. After that, the cracks will propagate with increasing the thickness and whenthe latter reaches the critical value, the will propagate to the surface of the film. Theinsertion of LT-interlayer between high temperature grown layers is very promisingfor the improvement of the AlxGa1–xN/GaN heterostructure. The LT-interlayer isthought to have similar effect as that of the LT buffer layer, acting as a kind ofnucleation layer. This method succeeded with thick AlxGa1–xN layer (about 3 µm) freeof cracks [3].
The problem with cracking occurs also during the growth of intentionally dopedn-type AlxGa1–xN layers. Si doping changes the growth mode of AlxGa1–xN layer.CANTU et al. [4] showed that it is either the composition or the strain that changesduring the layer growth. These effects cause relaxation of the strain and finallycracking.
In this work we investigated the effect of the behavior of cracks and then itssuppression by applying so-called special AlN nucleation layer as proposed byQU et al. [3]. In the second part we studied the behavior of AlxGa1–xN layers duringthe n-type doping.
2. Experiment
Both experiments were carried out in metal organic vapor phase epitaxy low-pressurereactor. The heterostructures were grown on c-plane (0001) sapphire substrates.Trimethylgallium (TMGa), trimethylaluminium (TMAl), ammonia (NH3) and silane(SiH4) were used as precursors of Ga, Al, N and Si, respectively. High purity H2 wasused as a carrier gas.
First, Al0.4Ga0.6N layer of about 1 µm in thickness was grown directly on GaNhigh temperature layer (1.5 µm thick), on GaN nucleation layer (20 nm thick) ata temperature of 1100°C and pressure of and 75 mBar. The flows of TMGa, TMAland NH3 were kept at 4.4×10–5, 1.2×10–4 and 6.7×10–2 mol/min, respectively. Then,for comparison, special AlN nucleation layer (about 20 nm thick) was deposited ata temperature of 550°C between GaN and Al0.4Ga0.6N high temperature layers. Bothlayers were etched in melted eutectic of KOH, NaOH in order to check the dislocationdensity.
The next step was n-type doping of Al0.4Ga0.6N layers (about 1.5 µm thick) on AlNnucleation layer grown for 6 min (20 nm thick) at a temperature of 550°C and pressureof 70 mBar. The amount of dopant was varied from 0.1 to 6 nmol/min in order to getthe carrier concentration as high as possible.
3. Results and discussion
Figure 1 shows the surface morphologies of about 1 µm thick Al0.4Ga0.6N layersgrown without (Fig. 1a) and with special AlN nucleation layer (Fig. 1b) obtained bymeans of SEM.
Problems with cracking of AlxGa1–xN layers 113
The first layer is characterized by hexagonal cracks typical of the layer grownjust on GaN buffer layer. The Al0.4Ga0.6N layer grown on special LT AlN layer hasthe mirror-like surface without any defects such as cracks and hillocks. We obtainedthe same results as QU et al. [3]. LT AlN interlayer prevented the layer from cracking.It reduced the strain caused by growing Al0.4Ga0.6N directly on GaN layer and actedas a filter against threading dislocations, which have screw components and make thegrown layer crack. On the other hand, additional pure edge dislocations were generatedinstead [5].
Figure 2 shows the etched surfaces of both Al0.4Ga0.6N layers without (Fig. 2a)and with (Fig. 2b) LT AlN layer. Each pit, which is present on the surfaces occurredon one dislocation [6].
The pit densities are 4×107 and 5×109 counted for Al0.4Ga0.6N layer grown withoutand on special nucleation layer, respectively. Similar results are given by other authorsfor layers grown with special AlN or Al0.4Ga0.6N low temperature nucleation layers.The relaxation of strain of the first layer made it crack and lowered the dislocationdensity (EPD) about two orders of magnitude. The special AlN nucleation layer
Fig. 1. AlxGa1–xN layer surface morphology made without (a) and with (b) special nucleationlayerobserved with scanning microscope.
a b
Fig. 2. Etched surface of AlxGa1–xN layer grown without (a) and with (b) LT AlN layer observed withSEM.
a b
114 E. DUMISZEWSKA et al.
reduced the strain of Al0.4Ga0.6N layer and prevented it from cracking, but many newedge dislocations appeared instead. We can say that special LT AlN improves themorphology of grown Al0.4Ga0.6N layer but it does not reduce its EPD.
Figure 3 shows the surface morphologies of n-type Al0.4Ga0.6N layers (1.5 µm)deposited on sapphire and AlN nucleation layer (20 nm thick) grown for 6 minobserved by a scanning microscope. The layers were doped with 6, 4, 3 and0.1 nmol/min of silicon (Fig. 3a–d, respectively).
The surfaces of three of these samples were characterized by typical hexagonalcracks made by the intentional doping. The density of cracks depended on the amountof the dopant. Decreasing the amount of silicon made the grown Al0.4Ga0.6N layer lessand less cracked. The process of appearing of the cracks was connected with therelaxation of Si doped layer that was grown on sapphire. On the surface of the probedoped with the least amount of silicon only small holes occurred. They seemed to bethe intersection of dislocations with the sample surface as shown by CANTU et al. [4]for probes of a smaller thickness and bigger amount of dopant. In the case of our layersthe relaxation went on with the thickness, and finally caused cracking of layers.
4. Summary
We investigated the behavior of Al0.4Ga0.6N layers during the growth either on GaNbuffer layer or the n-type doping. Depositing the layer just on GaN layer made thegrown layer crack. Inserting special LT AlN layer between both buffer layers stopped
Fig. 3. The n-type AlxGa1–xN layer morphology doped with 6 (a), 4 (b), 3 (c) and 0.1 nmol/min (d) ofsilane observed with SEM.
a b
c d
Problems with cracking of AlxGa1–xN layers 115
the relaxation and prevented grown Al0.4Ga0.6N layer from cracking. We measured thedislocation density for both Al0.4Ga0.6N grown just on GaN layer and on special AlNLT layer. The first probe was characterized by EPD being two orders of magnitudelower than that grown on special AlN LT layer. Doping of Al0.4Ga0.6N layer alsocaused cracking of the layer. It started from very low amount of dopant causing holesand went on with increasing the dopant causing hexagonal cracks. The main reason ofcracking seems to be the process of relaxation of the layer with its thickness. For thesmaller amount of the dopant there appeared only intersection of dislocations with thesample surface (small holes), which finally led to cracking with increasing the amountof silicon.
References
[1] NAKAMURA S., FASOL G., The Blue Laser Diode, Springer, Berlin 1997, p. 277.[2] JUNG HAN, CRAWFORD M.H., SHUL R.J., HEARNE S.J., CHASON E., FIGIEL J.J., BANAS M., Monitoring
and controlling of strain during MOCVD of AlGaN for UV optoelectronics, MRS Internet Journal ofNitride Semiconductor Research 4S1, 1999, G7.7.
Si doping effect on strain reduction in compressively strained Al0.49Ga0.51N thin films, AppliedPhysics Letters 83(4), 2003, pp. 674–676.
[5] AMANO H., AKASAKI I., Critical issues in AlGa1–xN growth, Optical Materials 19(1), 2002, pp. 219–22. [6] WEYHER J.L., TICHELAAR F.D., ZANDBERGEN H.W., MACHT L., HAGEMAN P-R., Selective photoetching
and transmission electron microscopy studies of defects in heteroepitaxial GaN, Journal of AppliedPhysics 90(12), 2001, pp. 6105–9.
Received July 6, 2004in revised form October 14, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Influence of technological parameters on the properties of sol-gel silica films
PAWEŁ KARASIŃSKI
Silesian University of Technology, Institute of Physics, ul. Bolesława Krzywoustego 2, 44-100 Gliwice, Poland; e-mail: [email protected]
The paper presents the influence of selected technological parameters on the properties of silicafilms produced in sol-gel processes using the dip-coating method. The influence of water contentand catalyst content was investigated as well as the influence of aging time of sols. It wasdemonstrated that the conditions of film deposition and the content of catalyst have a greatinfluence on the thickness of the films produced. The greatest influence on the refractive index isexerted by molar ratio TEOS:H2O. Together with the rise of aging time of the sol its propertiesare changing, and the changes depend on pH.
Keywords: sol-gel, silica film, dip coating.
1. Introduction
Due to their properties, silica films are particularly attractive for planar optics. Theadvantages of silica films are attributed to their high thermal stability and chemicalresistance. Silica films, directly on silicon substrates, can be produced in the processesof thermal oxidation of silicon. In other cases, silica films can be produced with theapplication of vacuum techniques, such as chemical vapor deposition, sputtering orflame hydrolysis. The sol-gel method can be an alternative for those techniques. Itdoes not require expensive technological installations and its biggest advantage overthe conventional deposition methods is that the microstructure of the film beingdeposited can be modified accordingly [1]. In the sol-gel method glass or ceramics areproduced from liquid phase at ambient temperature [1, 2]. Dielectric films producedwith the use of sol-gel method are characterized by high homogeneity, and theirproperties (refractive index, porosity, thickness) are formed by an appropriate selectionof technological parameters. Using the sol-gel method we can produce silica films ofporous structure, which are characterized by lower refractive indices than dense silica.Such films can be applied in production of sensitive films for chemical amplitudeevanescent wave sensors [3–5]. In such cases silica plays the role of a matrix in whichindicators are bound. The properties of such sensors are dependant on their porosity
118 P. KARASIŃSKI
and on the indicator applied. Porosity has the influence on the response time andregeneration time of the film. Sensor properties of such layers have been discussed inmany publications [5–10], but the influence of technological parameters on theirproperties has not been thoroughly investigated in the literature.
In the present paper, we present the results of investigations involving the influenceof selected technological parameters on the properties of the silica films produced. Thefilms were produced with the application of dip-coating method. The investigationscovered the influence of water amount, catalyst and aging time of sol on the thicknessof the obtained films and their refractive index.
2. Sol-gel processing
2.1. Chemical basis
The sol-gel technique can be defined as a chemical production method of glass andceramics from liquid phase. The following stages can be distinguished in sol-gelprocesses [1, 2]: formation of colloidal system, hydrolysis and condensation,deposition of sol film on the substrate as well as drying and heating of the films beingproduced. When tetraethoxysilane (TEOS) is applied as silica precursor, the reactionsof hydrolysis and condensation are carried out in the following way [2]:
The reactions of condensation start before the reactions of hydrolysis are finished.In theory, to ensure a full hydrolysis reaction of TEOS it is enough to ensure the ratioH2O:TEOS at the level of R = 2. However, even in excess water (R >> 2), the reactionof TEOS hydrolysis does not go completion [2]. The proportions of the applied outputcomponents, kind and amount of the applied catalyst as well as parameterscharacterizing particular stages of the technological process have the influence on theproperties of the films obtained.
2.2. Film formation by dip coating methods
In sol-gel technique the films are produced with the application of three methods: spincoating method, dip coating method and meniscus coating method [2]. In the studiespresented the dip coating method was applied, in which substrate withdrawal speedfrom the sol is the basic parameter having the influence on the thickness of the film.When the sol shows the properties of Newtonian liquid and its viscosity and substratewithdrawal speed are high enough to reduce the curvature radius of meniscus, then thedependence of the thickness d of the sol film being deposited on the substratewithdrawal speed v can be defined by means of the following expression [1, 2]:
Influence of technological parameters... 119
(4)
where c1 ≈ 0.8, η is the dynamic viscosity of sol, ρ is the concentration of sol andg is the acceleration due to gravity. When the movement of substrate does not resultin the reduction of meniscus curvature, the dependence of thickness d of the film beingobtained on the speed v of the substrate is expressed by the expression of Landau andLevich [1, 2]:
(5)
where σL, V is the liquid-vapor surface tension. When deriving the above relations the evaporation of solvents was not considered.
The sol films deposited on the substrate are then subjected to drying and heating. Inthe above processes the films get concentrated whereby their thickness is considerablysmaller. This effect can be allowed for by complementing the above expression witha contraction factor [11]. Taking into consideration expressions (4) and (5), thedependence of final film thickness on substrate withdrawal speed from the sol can bewritten in the following form:
(6)
where ξ = 1 (cm/min)–α is a scaling factor (the speed is expressed in cm/min). Fora given technological process the factor of proportionality a, and exponent α can bederived empirically. Frequently the relation d (v ) is presented in the literature in thelogarithmic scale and hence the exponent α is referred to as a slope. With Eqs. (4)and (5) taken into consideration, we can expect that its value is within the range from0.50 to 0.66.
2.3. Film fabrication
Tetraethoxysilane (TEOS) was the precursor of the silica films produced, andhydrochloric acid (HCl) was used as a catalyst. In all processes described here TEOSwas dissolved in ethyl alcohol (EtOH) in constant molar ratio TEOS:EtOH = 1:4. Afteradding the appropriate volume of water and catalyst the hydrolysis of TEOS wascarried out. For all solutions the reactions were carried out for 3 h in a closed glassvessel at a temperature of 50°C with ultrasonic stirring being used. After cooling thesol to the room temperature (18°C) the films were deposited on the glass substrates.The tests were carried out with respect to solutions of different content of water anddifferent amount of the catalyst. In the solutions under investigation the molar ratioH2O:TEOS was R = 2, 4, 8 and 16, respectively. And the molar ratio TEOS:HCl variedfrom 0.01 to 0.38. The films were deposited on glass substrates by means of controlled
d c1ηvρg---------
1 2⁄
=
d 0.94ηv( )2 3⁄
σ L V,1 6⁄ ρg( )
1 2⁄-----------------------------------=
d aξ vα
=
120 P. KARASIŃSKI
withdrawal from the sol. For that purpose, a mechanical system was constructed whichallowed the withdrawal of the substrates at a controllable rate. In the investigationsthe withdrawal rate from 2 to 10 cm/min was applied. The accuracy of withdrawal ratemeasurement was ∆v = 0.02 cm/min for v = 2.00 cm/min and ∆v = 0.12 cm/min forv = 10.00 cm/min but the uniformity of the withdrawal rate was considerably better.Microscope slides (Menzel–Glaser) of the dimensions 76×26×1 mm were applied assubstrates. Substrate glass was subjected to cleaning procedure which involved thefollowing: mechanical washing in water with detergent, rinsing in deionized water,soaking in the solution of ammonia water, rinsing in deionized water, rinsing in acetoneand drying. The sol in which the substrates were immersed was in a beaker, the wholebeing shielded by a glass cylinder. The application of such a procedure preventedaccidental movement of air and, in consequence, ensure homogeneity of the filmsobtained. The fabricated films were then dried and in each case they were annealedfor 2 h at a temperature of 150°C.
3. Measuring method
The thickness and refractive indices of silica films were measured in the ellipsometricway. The ellipsometric method consists in changing the polarization state, whichhappens to the light beam reflected from the investigated sample [12]. The basicellipsometric equation has the following form:
(7)
with Rp and Rs indicating the reflection coefficients respectively for the light polarizedin parallel (subscript p) and perpendicularly (subscript s) to the plane of incidence.The angles Ψ and ∆ are referred to as ellipsometric angles and they generally dependon the film parameters, substrate and the ambient medium. From the measurements ofellipsometric angles, film parameters are determined. The experiments were carriedout for the wavelength of λ = 632.8 nm with the application of a monochromaticellipsometer Sentech SE400 (Germany). During the measurements the relativehumidity in the room was about 40%.
4. Experimental results and discussion
4.1. Effect of water:precursor ratio
The influence of water :precursor ratio R on the properties of the sol obtained wasinvestigated, and then consequently on refractive index and thickness of the silica filmsproduced. Technological processes were carried out in which different water:precursorratio R was used to prepare the sols. The sols were prepared according to the procedurepresented in Sec. 2.3., in which the molar ratio of water to TEOS was R = 2, 4, 8 and12, respectively. The molar ratio TEOS:HCl was 1 :0.02 (pH > 2). On the same day
ρRp
Rs
--------- Ψtan i∆( ),exp= =
Influence of technological parameters... 121
the films were fabricated which, on the next day, were subjected to annealing for 2 hat a temperature of 150°C. The parameters of the films were measured in theellipsometric way. The dependences of the final thickness of films on the speed ofsubstrate withdrawal from the sol are presented in Fig.1. In the figures the experimentalpoints have been marked. The curves have been determined from the approximationof experimental data using relation (6).
The films were fabricated in room conditions when the sol was placed in a beakershielded by a glass cylinder, which reduced the movement of air around the beaker.In such a system, there is a high concentration of solvent vapours (mainly of EtOH)in the air inside the cylinder. It has the influence on their evaporation rate from thedeposited sol film, and hence on the thickness of the fabricated films. Thecharacteristics reflecting such conditions are additionally marked with the symbol A.It can be seen from the dependences presented in Fig. 1 that the content of water inthe sol has the influence on their run. For R > 2, for a given substrate withdrawal speedfrom the sol, the thickness of films decreases with the rise of R. The solution with thelowest water content (R = 2) exhibited different properties. Then, the runcorresponding to the highest water content (R = 16) is characterized by high dispersionof experimental points. The sols, for which R < 16, yielded homogeneous films of goodquality. The films fabricated from the sol for which R = 16 had heterogeneousthickness. And hence the run corresponding to them is characterized by the highestdispersion of points. This is caused by the fact that with such a high amount of waterthe surface tension of sol is rising considerably and due to that the wettability ofsubstrate deteriorates.
The technological processes presented here were repeated in the same way, withthe conditions of film deposition being slightly changed. The change consisted inraising the cylinder shielding the beaker with sol by a few millimeters to ensure themovement of air around the beaker. In this way, the concentration of solvents in the
Fig. 1. Dependence of film thickness on the substrate withdrawal speed for different water content R(TEOS:EtOH:HCl = 1:4:0.02).
2 3 4 5 6 7 8 9 1 0
v [cm/min]
400 300 200
d [n
m]
A, R = 2
A, R = 4
A, R = 8
A, R = 16
122 P. KARASIŃSKI
vicinity of the beaker with sol decreased, which, in consequence, had influence ontheir evaporation rate from the films being deposited. The results obtained in suchconditions were marked with symbol B. For the sake of comparison, the selectedcharacteristics, corresponding to different conditions of film deposition are presentedtogether in Fig. 2, and the parameters of matching curves are given in the Table. Wecan clearly see the influence involving the change of film deposition conditions on thecharacteristics obtained. The biggest differences can be observed for R = 2. For R = 4the influence involving the change of film deposition conditions is smaller, and forR = 8, as can be seen from Fig. 2, the characteristics practically overlap. For R = 16the parameters of matching curves are almost the same. The growing differencebetween the characteristics with R being decreased is probably caused not only by thechange of concentration of solvents present in the atmosphere surrounding the sol butalso by water being transferred to the layer of sol by the passing air. Additionally, theflow of air around the film, increasing the evaporation of the catalyst from the sol film,can increase its pH, accelerating in this way the condensation process. Also, in thecase of the repeated process, for R = 16 the films obtained were of the worst quality.It can be observed from the comparison of matching curves that only for R = 4 and forthe process A, the slope α is within the expected range of values, and the filmscorresponding to it are characterized in terms of visual assessment by the best quality.For the remaining cases the slope α has the values below expectation. This is indicatedby the fact that the sols in the deposited films do not show the properties of Newtonianliquids. It can be seen from the parameters presented in the Table that the flow of airin the vicinity of the beaker with sol has the influence on the properties of sols in thedeposited films. The flowing air changes the evaporation rate of solvents (mainlyEtOH), but also by transferring water to the film, it affects the processes of hydrolysisand condensation taking place in the film. All results presented below correspond withthe conditions of film deposition when the flow of air around the sol was reduced. Theinfluence of water content R in the sol on refractive indices of films is illustrated inFig. 3. Two runs are presented which correspond with different times of film dryingmarked in the figure. We can observe from the runs that the lowest refractive indexcorresponds to R = 2. For R > 4 the refractive index rises inconsiderably with the riseof water content in the sol. We can also observe that lower values of refractive indexcorrespond also with longer times of film drying. During the drying process the
T a b l e. Parameters of matching curves.
α a [nm]
A B A B
R = 2 0.404±0.004 0.118±0.004 146.04±0.98 290.58±1.52
R = 4 0.514±0.004 0.422±0.004 119.51±0.64 142.84±0.79
R = 8 0.485±0.002 0.441±0.005 118.07±0.37 126.47±0.83
R = 16 0.428±0.010 0.436±0.011 112.14±2.01 117.78±1.67
Influence of technological parameters... 123
structure of the film material is stiffening, and during the heating process the structureof the film material condenses. The stiffer structure is less susceptible to condensation,and hence the films, which were subjected to drying were characterized by higherthickness and lower refractive index, as presented in Fig. 2.
The results presented above show that when the hydrolysis is carried out inconditions with water excess and pH > 2, then with the rise of R, for a given substratewithdrawal speed, the films obtained get thinner and thinner. The excess of water inweakly acidified silica sols accelerates hydrolysis and slows down the condensation,effecting the formation of more furcated polymer networks [13]. The rise of R resultstherefore in higher dissolution of sol and lower condensation level, and thenconsequently reduced viscosity of sol and in effect smaller thickness of the fabricatedfilms. MCDONAGH et al. [14] have shown that for sols obtained in the conditions ofacid hydrolysis (pH < 2) the thickness of films increases with the rise of R.
4.2. Effect of catalyst:precursor ratio
The influence of the amount of catalyst on the dependence of film thickness of thefabricated film on the substrate withdrawal speed from the sol is presented in Fig. 4.The results were obtained for the films fabricated from sols of the molar ratioTEOS:EtOH:H2O = 1:4:4. The molar ratios HCl:TEOS is marked in the figure. Thecharacteristics presented correspond with the films, which were fabricated on the nextday, after the sols had been prepared, i.e., when the sols had been subjected to agingprocess for 24 h. The dependence of film thickness on the amount of catalyst, for thesubstrate withdrawal speed from the sol v = 6 cm/min is presented in Fig. 5. Thedependences correspond with the films obtained on the day the sol was prepared(dependence A) and with the films prepared on the next day (dependence B). Higher
400
300 200
2 3 4 5 6 7 8 9 10
v [cm/min]
B, R = 8
A, R = 8
A, R = 2
B, R = 2
d [n
m]
1.450
1.440
1.430
1.420
0 4 8 12 16 R
n
Drying time
1 h
20 h
Fig. 2. Influence of film deposition conditions (A – with flow of air around the beaker, B – without flowof air around the beaker).
Fig. 3. Refractive index vs. water content R (TEOS:EtOH:HCl = 1:4:0.02).
124 P. KARASIŃSKI
content of the catalyst in the sol corresponds with higher thickness of films. For thesol subjected to aging for 24 h the dependence of film thickness on log(HCl:TEOS) islinear. The influence of catalyst content on refractive indices of the films, for twoselected aging times of sols, is presented in Fig. 6. We can observe a weak dependenceof refractive index on the amount of catalyst in the sol. The values of refractive indexslightly decrease with the rise of molar ratio HCl:TEOS. These changes are within themeasurement error. BRINKER et al. [1] have shown that the pH value of the sol,influencing the processes of hydrolysis and condensation, affects the slope α. Theproblem will be investigated in the next part of the paper.
400
300 200
3 4 5 6 7 8 9 10
d [n
m]
v [cm/min]
0.01
0.02
0.04
0.38
HCl:TEOS
A
B
340
320 300 280
260 0.01 0.10 1.0
HCl:TEOS
d [n
m]
Fig. 4. Film thickness vs. substrate withdrawal speed for different molar ratio HCl:TEOS(TEOS:EtOH:H2O = 1:4:4).
Fig. 5. Film thickness vs. molar ratio HCl:TEOS. Withdrawal speed v = 6 cm/min, A – the films fabricatedon the day the sol was prepared, B – the films fabricated on the next day.
Fig. 6. Refractive index vs. molar ratio HCl:TEOS (TEOS:EtOH:H2O = 1:4:4).
1.450
1.440
1.430
1.420
0.01 0.10 1.0
HCl:TEOS
n
Aging time
24 h
73 h
Influence of technological parameters... 125
4.3. Influence of aging time of the sol
The results presented in Sec. 4.2 show that together with aging time of the sol itsproperties undergo changes. The set of characteristics, which were obtained fordifferent aging times of sols and for different amounts of catalyst are presented inFig. 7a and b. From the approximation of experimental points with the relation (6) theslope α was determined whose dependence on aging times of sols of different catalystconcentrations is presented in Fig. 8. We can see three different types of dependence,which are due to the changes taking place in the microstructure of particular sols. Thesmallest changes are observed for the sol for which the molar ratio TEOS:HCl = 0.04.
Fig. 7. Dependences of film thickness on the substrate withdrawal speed: a – TEOS:EOH:H2O:HCl == 1:4:4:0.01, b – TEOS:EOH:H2O:HCl = 1:4:4:0.04.
400 300 200
3 4 5 6 7 8 9 10
v [cm/min]
d [n
m]
Aging time
1 h
26 h
49 h
118 h
400 300 200
3 4 5 6 7 8 9 10v [cm/min]
Aging time
3 h
24 h
50 h
118 h
d [n
m]
a b
0.65 0.60
0.55
0.50 0.45
0 48 96 144 Aging time [h]
α
TEOS: HCl 1:0.38
1:0.04
1:0.02
1:0.01
400 350 300 250
0 48 96 144 Aging time [h]
d [n
m]
TEOS: HCl 1:0.38
1:0.04
1:0.02
Fig. 8. Slop a vs. aging time for different amounts of catalyst (TEOS:EtOH:H2O = 1:4:4).
Fig. 9. Influence of aging time on refractive index for different molar ratio TEOS:HCl, v = 6 cm/min.
126 P. KARASIŃSKI
In this case, for a fresh sol α = 0.516±0.001 and for the sol subjected to aging for119 hours α = 0.534±0.003. For this sol pH~2 was measured and hence its properties.For silica sols catalyzed with hydrochloric acid, when pH~2, weakly branched silicasols are deposited and the condensation rate is minimized [2]. And hence the propertiesof the sol, for which the molar ratio TEOS:HCl = 1:0.04, were inconsiderablychanging in time. In effect we have small changes of slope α. In the sol with highconcentration of catalyst, for which the molar ratio TEOS:HCl = 1:0.38 (pH < 2, acidcatalysis) the hydrolysis process ran very fast, and the condensation process yieldedweakly branched structures, resulting in gradual rise in viscosity and, in consequence,gradual decrease of slope α. In the remaining sols, for which pH > 2, the process ofhydrolysis ran slowly and that of condensation was running fast. So, the increase ofaging time of the sol resulted in the increase and aggregation of particles. The resultsobtained show a pronounced change of the sol properties. The slope values show thatthey have different properties from those of Newtonian liquids. However, after 24 hof aging, their properties changed considerably; slope surpassed the value of 0.5, whichmight mean that they have the properties of Newtonian liquid. Further elongation ofaging time of the sol results in a gradual increase of its viscosity and the reduction ofslope α.
The influence of aging time of the sol on the thickness of films for a set substratewithdrawal speed from sols v = 6 cm/min is presented in Fig. 9. With the rise of agingtime of the sols the thickness of films is rising with a substrate withdrawal speed fromthe sols. However, this increase is not uniform. The biggest rise can be observed inthe first phase of sol aging. For sols with the highest content of catalyst, the strongestdependence is observed, which is then getting weaker with the rise of aging time ofthe sol. For smaller concentrations of the catalyst the dependence of film thickness onaging time of the sol is becoming a weak dependence when these times are longer than24 hours. It may be indicative of the fact that the properties of sols get somehowstabilized. The investigations carried out have not proved that there is any influenceof aging time of the sols on the refractive index of films.
4.4. Uniformity of films
All films produced from sols, whose molar ratio water :precursor was R < 16 werecharacterized by high homogeneity of thickness. Figure 10 presents the dependenceof the thickness d on the location x for the films produced from the sol for which themolar ratio TEOS:EtOH:H2O:HCl = 1:4:2:0.2. During the production of films, thebeaker with sol was shielded by a glass cylinder, owing to which any incidental flowof air around the coated layer was eliminated. The relations presented were obtainedfrom the measurement of film thickness carried out along the sample. The distancebetween the measurement points was 1 mm. In each case the width of the interval ofthickness d, within which the measurement points are located is lower than 3 nm. Forthe films presented here, the values of standard deviation of the thickness are
Influence of technological parameters... 127
respectively: ∆d = 0.90 nm for film A, ∆d = 0.86 nm for film B and ∆d = 0.92 nm forfilm C. As can be seen from the above, by the application of dip-coating method insol-gel technology, we can produce silica films which are characterized by thicknesshomogeneity of the order of 1 nm. However, it must be emphasized that the acquisitionof such homogeneity of thickness was only possible when we ensured constantwithdrawal rate and when any incidental flow of air around the coated film waseliminated.
5. Conclusions
The paper presents the results of investigations involving the influence of selectedtechnological parameters on the properties of silica films fabricated in sol-gel process.We presented the influence of water amount, the influence of the amount of the appliedcatalyst, the influence of substrate withdrawal speed from the sol and the influence ofthe conditions of film deposition and aging time of sols on the final parameters of thefilms fabricated. The results of the investigations show that the application of too muchwater has a negative influence on the uniformity of the film thickness, but at the sametime the processes with the application of high content of water are least sensitive tothe changes of conditions involving film deposition. The reduction of water contentin the sols as well as longer time of film drying result in the reduction of refractiveindex of the films. The rise of catalyst amount has influence on the rise of the filmthickness with a given substrate withdrawal speed from the sol, but it hasinconsiderable influence on the lowering of refractive index. The sols investigatedbecome stable after at least 24 hours. By ensuring constant withdrawal rate and by theelimination of incidental air flow around the sol during the coating process, we can
Fig. 10. Distributions of thickness for fabricated silica films.
A
B
C
300
280
260
240
0 10 20 30 x [mm]
d [n
m]
128 P. KARASIŃSKI
produce silica films of the thickness homogeneity at the level of 1 nm. The results ofinvestigations can be applied in the sensitive film technology for planar chemicalevanescent waveguide sensors.
Acknowledgements – The work has been co-sponsored by the Polish State Committee for ScientificResearch (KBN) under the grant KBN 7 T08D 013 21.
References
[1] BRINKER C.J., FRYE G.C., HURD A.J., ASHLEY C.S., Fundamentals of sol-gel dip coating, Thin SolidFilms 201(1), 1991, pp. 97–108.
[2] BRINKER C.J., SCHERER G.W., Sol-Gel Science, Academic Press, San Diego 1990.[3] BOISDE G., HARMER A., Chemical and Biochemical Sensing With Optical Fibers and Waveguides,
Artech House, Boston 1996.[4] KARASIŃSKI P., ROGOZIŃSKI R., OPILSKI A., Analysis of waveguide planar chemical sensor with
amplitude modulation, Proceedings of the SPIE 4516, 2001, pp. 218–22.[5] KLEIN R., VOGES E., Integrated-optic ammonia sensor, Sensors and Actuators B: Chemical
B11(1-3), 1993, pp. 221–5. [6] KARASIŃSKI P., Sol-gel derived sensitive films for ammonia evanescent wave sensors, Optica
Applicata 33(2-3), 2003, pp. 477–87.[7] BUTLER T.M., MACCRAITH B.D., MCDONAGH C., Leaching in sol-gel-derived silica films for optical
pH sensing, Journal of Non-Crystalline Solids 224(3), 1998, pp. 249–58.[8] MCDONAGH C., BOWE P., MONGEY K., MACCRAITH B.D., Characterisation of porosity and
sensor response times of sol-gel-derived thin films for oxygen sensor applications, Journal ofNon-Crystalline Solids 306(2), 2002, pp. 138–48.
[9] LECHNA M., HOLOWICZ I., ULATOWSKA A., PODBIELSKA H., Optical properties of sol-gel coatings forfiberoptic sensors, Surface and Coatings Technology 151-152, 2002, pp. 299–302.
[10] KLEIN L.C., Sol-Gel Optics, Processing and Applications, Kluwer Academic Publishers, 1994,pp. 279–302
[11] STRAWBRIDGE I., JAMES P.F., The factors affecting the thickness of sol-gel derived silica coatingsprepared by dipping, Journal of Non-Crystalline Solids 86(3), 1986, pp. 381–93.
[13] FARDAD M.A., YEATMAN E.M., DAWNAY E.J.C., GREEN M., HOROWITZ F., Effects of H2O on structureof acid-catalysed SiO2 sol-gel films, Journal of Non-Crystalline Solids 183(3), 1995, pp. 260–7.
[14] MCDONAGH C., SHERIDAN F., BUTLER T., MACCRAITH B.D., Characterisation of sol-gel-derived silicafilms, Journal of Non-Crystalline Solids 194(1-2), 1996, pp. 72–7.
Received April 29, 2004in revised form December 7, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Optical beam injection methods as a tool for analysis of semiconductor structures
JAROSŁAW DOMARADZKI, DANUTA KACZMAREK
Faculty of Microsystem Electronics and Photonics, Wrocław University of Technology, ul. Janiszewskiego 11/17, 50-372 Wrocław, Poland
Optical beam injection methods, such as an optical beam induced current (OBIC) one, have severaladvantages. Such methods enable a comprehensive analysis of photocurrent generated at themicroregion of a semiconductor material or a device by focused light beam. In the paper, examplesof applications of the OBIC method for : i) examination of the silicon p-i-n diodes used in a scanningelectron microscope (SEM) as a detector and ii) localization of electrically active regions at theinterface of the new transparent oxide semiconductor (TOS)–semiconductor structure have beenoutlined.
Manufacturing of today’s microelectronic devices needs the use of novelnon-destructive techniques for optical and electrical characterization of the deviceparameters with good spatial resolution. Suitable methods for such requirement arebased on focused beam which causes induction of excess carriers in a localized region.The analysis of the current collected in the external circuit can provide importantqualitative and quantitative information of the local transport properties of the deviceunder test (DUT) [1].
Up to now, methods in which the focused light is used in order to generate carriersfor semiconductor investigations have been applied in many variations. There alsoexist many names such as: light/laser beam induced current (LBIC) [2], optical beaminduced current (OBIC) [3], infrared beam induced current (IRBIC) [4], mono-chromatic beam induced current (MBIC) [5] and others, but all of them describe thesame method. From experimental point of view, of major importance is the way howthe induced current is collected. As has been presented in Fig. 1, there is a commondifference depending on whether the signal measured was taken at a short (contacts Aand C) or an open (contacts A and B) junction. In both cases focused and modulatedlight beam is scanned across the semiconductor sample and generates electron-holepairs in localized area. If this localized region has any electrical field (i.e., junction or
130 J. DOMARADZKI, D. KACZMAREK
electrically active region caused by defects), the charges will be separated and theinduced current will flow between two remote contacts.
A short circuit current configuration has been especially applied in investigationsof p-n junction based devices, such as: solar cells [6], photodetectors [7], and siliconwafers [8]. Based on the photocurrent measured it is possible to calculate quantumefficiency of the device as well as other parameters such as: diffusion length, surfacerecombination velocity or lifetime of minority carriers [9].
Although the open circuit configuration was first applied in the middle of the1980’s [10], up to now there has not been developed any consistent theory which wouldallow electrical parameters to be determined on the basis of photocurrent measured inthis case. Few exceptions, for example, include measurement of the junction depth, aswas reported by the group of MUSCA et al. [11]. A great advantage of the open circuitconfiguration was that it allowed simultaneous investigation of multiple structuresfabricated at the same wafer using a single connection.
In the present work, tests of the p-i-n diodes were carried out in order to enable thechoice of detectors with similar electrical parameters. These detectors are most com-monly used in multidetector system for backscattered electrons (BSE) in a scanningelectron microscope (SEM).
Additionally, the examination of electrically active areas at the interface of thetransparent semiconducting oxide–semiconductor by applying the OBIC method hasbeen outlined. The semiconducting thin film oxide was based on the titanium dioxidelattice doped with transition metals and manufactured by a modified magnetronsputtering method [12, 13]. As a substrate well conducting (n-type) silicon wafers havebeen used.
2. Experimental set up
In our experimental set up (Fig. 2) an OPTEL monochromator, remotely controlled byPC equipped with a 450 W xenon lamp has been used as a light source. Owing to thisthe experiment can be done at different wavelength of incident light beam rangingfrom 2×10–7 m to 24×10–7 m with the minimum wavelength step of 1.5×10–10 m.
B
C
A
Light beam IphB IphA
p
n
Fig. 1. Schematic diagram of beam induced current phenomena in the case of planar p-n junction; A, B,C – ohmic contacts, Iph – photocurrent.
p
n
A B
C
Optical beam injection methods... 131
Besides the power and the wavelength of the incident light beam also the spot size hasto be taken into account [14]. At first, the light is chopped and illuminates a 3×10–4 mpinhole. Then, the beam is focused with the help of optical lenses onto the samplesurface of about 35×10–6 m (or less) in diameter.
The optical system has been equipped with a digital CCD camera connected withthe host computer via USB (universal serial bus) interface. Thanks to this, both thescanning area selected in the experiment and the actual position of the scanning beamcan be observed directly on the computer screen. A digital image of the light reflectedfrom the surface of the device under test (DUT) can be achieved and stored. DUT isplaced on a remotely controlled x-y-z stage. To obtain an image, the current inducedmust be measured step by step at each point of selected area. A high signal to noiseratio is assured by a lock-in technique application. The lock-in type amplifier (EG&GPARC 5301A type) measures the magnitude of generated photocurrent and the phase-shift between this current and reference signal. The digitized values are sent via theGPIB (IEEE-488.2) interface to PC and stored in a file as a function of incident lightposition. The minimum step size in the applied system amounts to 2.5×10–6 m.The data is plotted as 2D gray scale or colour images with custom palette. Both themagnitude and phase-shift images can be obtained. The control system wasimplemented in the TestPoint environment working under Windows. The set ofequipment used allows one to make a full-automated measurement and dataacquisition.
Fig. 2. Schematic block diagram of the measurement system used in our OBIC method.
132 J. DOMARADZKI, D. KACZMAREK
3. OBIC application
Focused optical beam injection methods enable comprehensive analysis of photocurrentgenerated in the microregion of semiconductor and on the basis of collected data 2Dgrey scale or colour images can be created [14, 15]. This may be useful in detectionof defects with very poor electrical activity [3, 16].
3.1. Analysis of SEM detectors
In Figures 3 and 4, exemplary distribution maps of the magnitude of induced currentmeasured on silicon p-i-n type photodiode in the case of different connectionconfigurations have been presented. In Fig. 3, the induced current was obtained in thecase of the open circuit current collected between A and B contacts. The magnitudeof collected current strongly depended on the position of incident light beam vs. remotecontacts. Near the centre of the structure being analysed photocurrent decreased to
B A IphA – IphB
A B IphB – IphA
III 2phB
2phAph +=
A B
Fig. 3. Distribution of the magnitude of induced current obtained at silicon p-i-n detector in the case ofopen circuit connection configuration between: A and B (a), B and A (b) contacts and the magnitude Iph
calculated as (c).Iph I phA2
I phB2
+=
138
79
0 nA
-93
-187
a
178
89
0 nA
-93
-186
b
c 256
192
127 nA
65
1
200 µm
Pho
tocu
rren
tP
hoto
curr
ent
Pho
tocu
rren
t
x [µm]
x [µm]
x [µm]
Optical beam injection methods... 133
zero. The polarity of the current measured depends on whether a signal was collectedat A or B contact. In Fig. 3c the magnitude of collected current calculated as
is presented.In Figure 4, a short junction configuration is presented and the current is collected
between A (or B) and C contacts. The photocurrent has quite a uniform distributionover the area under investigation. The visible darker spots are due to the surfaceimpurities of the structure being tested.
Figure 5 presents a distribution of phase shift between measured photocurrent andthe reference signal in the case of different connection configuration in the OBICmethod. Similarly to the image presented in Fig. 4, Fig. 5b shows a uniformdistribution of measured quantity over investigated area. Pinholes correspond to thesame surface impurities which can also be observed in Fig. 4. The phase shift in Fig. 5ahas changed near the centre of the structure, which indicates change of the directionof collected current (see Fig. 3).
Experiments described above in the case of open and short circuit current wereperformed without any additional external electric field. The current could be observedthanks to the photogenerated excess minority carriers separated directly at the p-njunction of the detectors being tested. Besides the nature of collected current it is worth
Iph I phA2
I phB2
+=
A(B)
C
Iph – IphA(B)
Fig. 4. Distribution of the magnitude of the induced current obtained at silicon p-i-n detector in the caseof short junction connection configuration.
480
355
230 µA
105
19
200 µm
Pho
tocu
rren
t
x [µm]
-90
200 µm 200 µm
Fig. 5. Distribution maps of phase shift between measured current and reference signal in the case of theconnection configuration as: a – in Fig. 3c, and b – in Fig. 4.
-114
-131
-147 deg
-180
-165
180
90
0 deg
-180
-90
134 J. DOMARADZKI, D. KACZMAREK
noting that measured signal in Fig. 4 was about two orders of magnitude higher thanin the case of signal in Fig. 3.
3.2. Analysis of the transparent semiconducting oxide–semiconductor interface
A new kind of structure with nanocrystalline titanium dioxide-based semiconductingthin films was manufactured using a low-pressure hot target reactive magnetronsputtering process [12]. As dopants a V and Pd sheets have been co-sputtered fromTi target in order to obtain Ti-V-Pd oxide compositions. The amount of dopants hasbeen estimated at V = 10.6 at.%, Pd = 6.67 at.%. The thickness of the fabricated films,measured by optical interference method with Hg (551 nm) filtered lamp was 395 nm.
In order to make electrical measurements, the Ag/Ti10W90 electrode with adiameter of 3 mm was evaporated through the mask in the layer under examination.At the backside of silicon wafer an ohmic contact has been made using In-Ga eutecticalloy. A schematic drawing of the structure under test is presented in Fig. 6.
In Figure 7a, characteristics obtained from the measurement of iph(x) photocurrentgenerated during a single scan of light beam through the range of analysed structure:electrode–thin oxide film–silicon substrate (Me/(Ti-V-Pd) oxide/Si) are presented.Below the characteristics, a general view of the structure is shown.
The measurements were carried out at room temperature at the wavelengthλ = 660 nm, for which the thin film is permeable at different values of supply voltage(Ubias). The increase in the amplitude of photocurrent, registered in the junction range:thin oxide film–silicon substrate, confirms the presence of the built-in potential (spacecharge), ensuring the separation of generated current carriers. Based on thecharacteristics shown in Fig. 7 it can be noticed that the location of electrically activerange corresponds to dimensions of analysed structure. Similarly to the case ofconventional semiconductor junctions, the increase in bias voltage results in extensionof space charge range. This is confirmed by the fact that at the boundary: thinTi-V-Pd oxide film–silicon substrate, the junction has been formed. Additionally, the
Fig. 6. Schematic drawing of the structure under test.
Optical beam injection methods... 135
increase in the value of measured photocurrent with increase in bias voltage ispresented in Fig. 7b.
Distribution of space charge can also be observed on 2D maps of measuredphotocurrent, as is shown in Fig. 8. The maps were obtained by point-by-pointmeasurement of photocurrent generated in the selected area scanned with the lightbeam upon selected polarisation bias.
Fig. 7. Characteristics of photocurrent iph(x): a – obtained in effect of single scan of light beam acrossthe area of Me/(Ti-V-Pd)/Si structure and b – vs. polarisation bias (Ubias). Light beam parameters:diameter ca. 35 µm, frequency of modulation f = 182 Hz. The characteristics were taken at roomtemperature with the wavelength of λ = 660 nm in the case of short junction connection configuration.
0.60
0.45
0.30 [nA]
0.15
0
1.2
0.9
0.6 [nA]
0.3
0
Fig. 8. Map of photocurrent distribution at the boundary: semiconducting oxide–Si of Me/(Ti-V-Pd)oxide/Si structures determined at the bias voltage: a – Ubias = 0 V, b – Ubias = 1 V.
0.60
0.45
0.30 [nA]
0.15
0
1.2
0.9
0.6 [nA]
0.3
0
a b
136 J. DOMARADZKI, D. KACZMAREK
Photocurrent presented in Figs. 7 and 8, was collected in the short circuit currentconfiguration. This type of connections allowed us to collect the photogeneratedcurrent in a simple way with a high sensitivity (a high signal to noise ratio).
The results described above seem to be very attractive for novel applications ofthis kind of structures in microelectronics. For example, as we have already shown in[13], they could be applied in a new kind of photodevices.
4. Conclusions
Optical beam induced current method (OBIC) has proven to be very useful in thediagnostics of typical semiconductor detectors as well as analysis of the new kind ofstructures with the electrically active region at the interface of semiconducting metaloxide thin film deposited on silicon substrate.
Characteristics scanned “in-line” allow us to indicate the electrically active regionswhich were equal to physical dimensions of the structure analysed. Similar to theconventional junction based structures, the distribution of measured photocurrent atthe fabricated heterojunction of semiconducting oxide–semiconductor was found tobe dependent on external polarisation bias.
The photoelectric effect revealed on the boundary of thin semiconducting oxidelayer with silicon substrate by OBIC method seems to be very interesting. From thesubject literature [17] it follows that thin semiconductor films produced on the basisof oxides are highly desired. In the the authors’ opinion, the connection of the filmswith conventional semiconductor materials, like silicon, which are widely used inelectronics, should lead in the nearest future to the fabrication of new types ofmicroelectronic devices.
Acknowledgment – This work was supported by the State Committee for Scientific Research (KBN) inthe years 2004–2005.
References
[1] CHAN D., PHANG J., CHIN J., KOLACHINA S., Single contact beam induced current phenomena– a review, Diffusion and Defect Data Pt. B: Solid State Phenomena 78-79 (2001), pp. 11–8.
[2] ACCIARRI M., BINETTI S., RACZ A., PIZZINI S., AGOSTINELLI G., Fast LBIC in-line characterization forprocess quality control in the photovoltaic industry, Solar Energy Materials and Solar Cells 72(1-4),2002, pp. 417–24.
[3] CASTALDINI A., CAVALLINI A., POLENTA L., Optical beam induced current investigations of particledetectors, Physica Status Solidi B 222(1), 2000, pp. 245–50.
[4] ASTAFIEV O.V., KALINUSHKIN V.P., YURYEV V.A., Scanning mid-IR-laser microscopy: An efficienttool for materials studies in silicon-based photonics and photovoltaics, Journal of Crystal Growth210(1-3), 2000, pp. 361–5.
[5] SHIMOKAWA R., TAJIMA M., WARASHINA M., KASHIWAGI Y., KAWANAMI H., Correspondence amongPL measurement, MBIC measurement and defect delineation in polycrystalline cast-Si solar cells,Solar Energy Materials and Solar Cells 48(1-4), 1997, pp. 85–91.
Optical beam injection methods... 137
[6] GALLOWAY S.A., EDWARDS P.R., DUROSE K., Characterisation of thin film CdS/CdTe solar cells usingelectron and optical beam induced current, Solar Energy Materials and Solar Cells 57(1), 1999,pp. 61–74.
[7] CASTALDINI A., CAVALLINI A., POLENTA L., NAVA F., CANALI C., Electric field distribution inirradiated silicon detectors, Nuclear Instruments and Methods in Physics Research, Section AAccelerators, Spectrometers, Detectors and Associated Equipment 476(3), 2002, pp. 550–5.
[8] MARTINUZZI S., HENQUINET N.G., PERICHAUD I., MATHIEU G., TORREGROSSA F., Efficiency of cavitygettering in single and in multicrystalline silicon wafers, Materials Science and Engineering B: SolidState Materials for Advanced Technology B71(1-3), 2000, pp. 229–32.
[9] GEIGER P., HAHN G., FATH P., BUCHER E., Comparing improved state-of-the-art to former EFGSi-ribbons with respect to solar cell processing and hydrogen passivation, Solar Energy Materialsand Solar Cells 72(1-4), 2002, pp. 155–63.
[10] BAJAJ J., BUBULAC L.O., NEWMAN P.R., TENNANT W.E., RACCAH P.M., Spatial mapping of electricallyactive defects in HgCdTe using laser beam-induced current, Journal of Vacuum Science andTechnology A: Vacuum, Surfaces, and Films 5(5), 1987, pp. 3186–9.
[11] MUSCA C.A., REDFERN D.A., SMITH E.P., DELL J.M., FARAONE L., BAJAJ J., Junction depthmeasurement in HgCdTe using laser beam induced current (LBIC), Journal of Electronic Materials28(6), 1999, pp. 603–10.
[12] DOMARADZKI J., PROCIOW E., KACZMAREK D., Ti Zr dielectric layers deposited by hot target reactivemagnetron sputtering, [In] ASDAM’02. Conference Proceedings. Fourth International Conferenceon Advanced Semiconductor Devices and Microsystems, [Eds.] Breza J., Donoval D., IEEE,Piscataway, NJ, USA 2002, pp. 47–50.
[13] DOMARADZKI J., PROCIOW E., KACZMAREK D., BERLICKI T., KUDRAWIEC R., MISIEWICZ J.,MIELCAREK W., Structural, optical and electrical characterization of Co-Pd doped TiO2semiconducting thin films sputtered on silicon, Optica Applicata 33(4), 2003, pp. 661–668.
[14] DOMARADZKI J., Light-beam-induced current (LBIC) technique for semiconductors and ICs testing,Proceedings of the SPIE 5064, 2003, pp. 269–74.
[15] DOMARADZKI J., KACZMAREK D., WĘGRZECKI M., WĘGRZECKA I., BUDZYŃSKI T., KRZEMIŃSKI S.,GRABIEC P., Detectors of optical and nuclear radiation examined by the light-beam-induced current(LBIC) method, Proceedings of the SPIE 5064, 2003, pp. 275–80.
[16] BERLICKI T., OSADNIK S., PROCIÓW E., Thermal sensors with controlled sensitivity, [In] Proceedingsof XXIV International Conference IMAPS’00, Rytro 2000, p. 365.
[17] VAN DE KROL R., TULLER H.L., Electroceramics – the role of interfaces, Solid State Ionics, Diffusionand Reactions 150(1-2), 2002, pp. 167–79.
Received September 23, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Efficient calculations of dispersive properties of photonic crystals using the transmission line matrix method
G. ROMO, T. SMY
Carleton University, Department of Electronics, Ottawa K1S 5B6, ON, Canada; e-mail: [email protected]
In this paper, we present an analysis of the accuracy and efficiency of different approaches for thesimulation of photonic crystals using the transmission line matrix method. The approaches that wepresent can be divided into two categories: complex- and real-valued algorithms using a uniformmesh, and complex- and real-valued algorithms using a multigrid mesh. The advantages anddisadvantages of each approach are discussed and a brief comparison between these methods ismade from the points of view of computational expense and accuracy. It is found that a combinationof a real-valued method in a multigrid mesh results in the most efficient algorithm. However, whilethe complex-valued formulation is valid for the analysis of any photonic crystal, the applicabilityof the real-valued formulation is limited by structural constraints requiring cell symmetries. It isalso found that a multigrid approach can considerably reduce the computational cost required forsimulating photonic crystals and our results indicate that a good compromise between accuracyand computational cost can be found. Various photonic crystals are simulated by applying theseapproaches, and the results are validated using alternative methods.
The unique property of photonic crystals (PCs) to inhibit the propagation of radiationof certain frequencies, has driven the interest of many researchers worldwide. Thus,a great deal of effort has been set upon the characterization of these structures, asknowledge of the frequencies that will and will not propagate, is indispensable for theirpractical applications. There exist several methods to obtain this information, and theygenerally involve the computation of the photonic band structure (or dispersionrelation) of the periodic structure.
One of the most widely used methods to compute the photonic band structure ofPCs, is the plane wave expansion (PWE) method. This method takes advantage of thefact that both, the solution (electromagnetic fields) and dielectric structure are periodicby expanding them with a Fourier series [1]. This allows Maxwell’s equations to be
140 G. ROMO, T. SMY
recast as an eigenvalue problem which is then solved by using standard techniques.Among the advantages of the PWE method are its relative simplicity, accuracy andefficient implementation using fast algorithms [2]. The main disadvantages of thismethod are that it scales as O(N3) where N is proportional to the size of the problem,and that it cannot deal with lossy dielectric materials.
Another alternative for the computation of the photonic band structure of PCs isto use time domain methods such as finite differences time domain (FDTD) and thetransmission line matrix (TLM) method. Unlike the PWE method, time domainmethods allow for the straightforward incorporation of material losses into thesimulations. In addition, unit cells of arbitrary shapes can be simulated withoutadditional computational effort. In a time domain method, the time required fora calculation scales linearly O(N) with the number of points used in the discretization[3, 4]. Despite of this, the main drawback of these methods is their long executiontime, since very little optimization can be done to reduce the number of operationsinherent to the method. Among the time domain methods, the FDTD method is by farthe most widely used and has extensively been applied for the analysis and computationof the photonic band structure of PCs [3–9]. The TLM method, on the other hand, hasbeen given very little attention in this respect, despite its increasing popularity for thesimulation of general electromagnetic problems [10, 11].
In this paper, we present a number of different approaches for the computation ofthe photonic band structure of PCs using the TLM method. First, the complex- andreal-valued formulations of the TLM method are applied for simulating PCs usinga uniform mesh. These formulations are briefly described and their basic advantagesand limitations discussed. Next, the complex- and real-valued formulations arerevisited using a non-uniform mesh, and the new trade-offs that come into play arealso discussed. Thus, the rest of this paper is organized as follows: in Sec. 2, thecomplex- and real-valued formulations are briefly presented and the limitations ofthe real-valued formulation are described in detail. In Sec. 3, the procedure for usingthe non-uniform or multi-grid mesh used in this paper is presented. In Sec. 4, the resultsof the simulations of various PCs using these different approaches are presented anddiscussed. Finally, Section 5 presents some conclusions.
2. Problem formulation
In this section we describe the basic mathematical concepts governing the simulationof PCs within the formulation of a time method such as TLM. Rather than attemptinga rigorous mathematical derivation, the equations are reproduced here only for the sakeof completeness. For more details see, for example [12].
From basic solid state physics theory we know that periodic problems havesolutions of the form
(1)ψ r( ) uk i k r⋅–( )exp=
Efficient calculations of dispersive properties... 141
where the function uk is periodic in space and k is a propagation vector. Equation (1)is known as Bloch’s theorem, although it is usually referred to as Floquet’s theoremin one-dimensional problems. If T describes the spatial period of the structure, that is,if it represents a lattice constant vector, then
(2)
Note that Eqs. (1) and (2) imply that
(3)
thus, the points r and r + T have the same physical properties and the functions differfrom each other only by a phase factor. The application of Eq. (3) to the modelling ofPCs within the formulation of the TLM method leads to equations of the form [13]
(4)
relating the incident and reflected pulses from the boundaries of the simulation domain.In Equation (4) V i and V r represent the incident and reflected voltage pulses of thenodes on the edge of the simulation domain, respectively. Note that in this equation,the time dependency is not shown, but implicit.
2.1. Formulation of a complex-valued TLM algorithm
From Equation (3) we can see that enforcement of the periodicity condition of thefields across the boundaries of the unit cell requires a complex-valued network. In theTLM method, this issue can be dealt with by breaking the periodicity condition intoreal and imaginary voltage pulses. These voltage pulses are then treated separatelythroughout the simulation domain and coupled only at the boundaries [14].
When this approach is taken, the periodic boundary conditions in Eq. (4) can bewritten in terms of the real Vreal and imaginary components Vima of the voltagepulses as:
(5)
uk r T+( ) uk r( ).=
ψ r T+( ) i k T⋅–( )ψ r( ),exp=
Vi r T+( ) V
r r( ) i k T⋅–( ),exp=
Vi r( ) V
r r T+( ) i k T⋅( ),exp=
V reali r T+( ) V real
r r( ) k T⋅( )cos V imar r( ) k T⋅( ),sin–=
V imai r T+( ) V real
r r( ) k T⋅( )sin V imar r( ) k T⋅( )cos ,+=
V reali r( ) V real
r r T+( ) k T⋅( )cos V imar r T+( ) k T⋅( ),sin+=
V imai r( ) V– real
r r T+( ) k T⋅( )sin V imar r T+( ) k T⋅( )cos .+=
142 G. ROMO, T. SMY
In these equations, the superscripts i and r stand for incident and reflected voltagepulses of nodes adjacent to the cell boundaries, respectively. Also note that Eqs. (5)are neither subject to approximations nor simplifications, thus its validity holds forunit cells of arbitrary shape and spatial periodicity.
2.2. Formulation of a real-valued TLM algorithm
In a real-valued TLM algorithm, the computational effort is reduced by half byrestricting the computations to either the real or imaginary part of the complex-valuedalgorithm. The derivation of a real-valued TLM algorithm was presented in detail in[15] and [16] and, therefore, only the relevant equations will be reproduced here.Within this formulation, the periodicity condition (Eq. (4)) is replaced by:
(6)
(7)
Note that in these equations, the underscripts corresponding to the voltage pulses havebeen dropped, since all the pulses are known to be real. In addition, v1 is a phase factorpreviously introduced in [16].
Although it is not the intention of this paper to compare the performance ofthe FDTD and TLM methods, it should be noticed at this point that the complex-valuedformulation of both methods has a very close resemblance. As a matter of fact, theFDTD counterpart of Eq. (5) can be obtained simply by replacing voltage pulses byelectric and magnetic fields in those equations. However, the real-valued TLMalgorithm presented in this section proves advantageous, since, to the best of theauthors knowledge, it has no straightforward unconditionally stable counterpart inthe FDTD method. This is the subject of ongoing research.
In principle, the recursive application of Eqs. (6) and (7) at every time step issufficient to single out the frequency modes that satisfy the Bloch condition out ofa general excitation. However, the use of the real-valued formulation must be donewith care due to inherent assumptions.
2.3. Limitations of the real-valued algorithm
As it was previously indicated, by using the real-valued TLM algorithm presented inthe previous section to calculate the photonic band structure of PCs, the computationaleffort is reduced by half. However, this advantage is achieved at the expense ofintroducing some limitations in the definition of the unit cell under simulation.
Consider the one dimensional form of the phase factor presented in [16] such thatr =(x1, y1, z1) and r + T =(x1 + tx, y1, z1). If the spatial period of the structure is
Efficient calculations of dispersive properties... 143
normalized to one (tx = a = 1) and the phase factor vx, 1 is graphed as a function of thewave vector value kx, the plot shown in Fig. 1 is obtained. Note that the normalizationcondition of the unit cell implies that the propagation vector numerically equals thephase difference between symmetry planes and can only take values between 0 and π.
Now consider what happens when is substituted in Eqs. (6) and (7) forthree different phase values. For a phase difference of zero, Fig. 1 indicates that v1 = 1.Thus Eqs. (6) and (7) reduce to V i(r) = V r(r + T) and V i(r + T) = V r(r) so that thevoltage pulses are just wrapped around the cell without any alterations. Similarly, whenthe phase difference is equal to π, Fig. 1 indicates that v1 = –1 and Eqs. (6) and (7)reduce to V i(r) = –V r(r + T) and V i(r + T) = –V r(r) such that the voltage pulses arenow wrapped around but with opposite signs. These results are somewhat intuitive andpose no difficulty to our simulation method. Consider however what happens whenthe phase difference varies between these two extreme values. Take for examplea phase difference of π/2; for this case, v1 = 0 and Eqs. (6) and (7) reduce toV i(r) = V r(r) and V i(r + T) = –V r(r + T). The fact that the reflected and incidentvoltage pulses at one end of the simulation domain become equal, implies a mirrorsymmetry plane (or magnetic wall) at that end. Note that this condition holds trueregardless of how the unit cell has been defined. In other words, for the results of thesimulation to be valid, the unit cell cannot be defined arbitrarily, but it has to be definedsuch that it is bounded by planes of mirror symmetry. This was made explicitly in [15]where the real-valued formulation was implemented with a magnetic wall on the sideof the unit cell.
To illustrate these ideas, consider for example the two-dimensional chessboard PCshown in Fig. 2. Note that for this structure the unit cell can be defined in different,but equivalent ways. For example, the dashed lines in the figure show two possibleways of defining the unit cell.
v1 vx 1,≡
Fig. 1. Phase factor as a function of the normalized propagation vector (phase difference betweensymmetry planes).
144 G. ROMO, T. SMY
Based on the previous analysis, the flexibility of defining the unit cell in differentways is only true for the complex-valued formulation. In the real-valued formulation,different unit cell definitions are, in general, not equivalent. This is illustrated in Fig. 3.The fact that the equations imposing the periodicity condition intrinsically contain theassumption that a perfectly symmetric scattering problem (with respect to the edgeof the simulation domain) is taking place on “adjacent” cells, is in agreement only withthe unit cell definition shown on the right hand side of Fig. 3. Note that this constraintmakes the real-valued TLM algorithm not applicable to some PCs because some unitcells do not posses mirror symmetry in all directions (i.e., triangular lattice). However,there is a wide number of PCs and microwave waveguides structures which satisfy thesymmetry conditions to be well posed within its formulation.
Fig. 2. Simulated two-dimensional chessboard structure. The dashed lines illustrate two possible ways ofdefining the unit cell.
ya
a
x
Fig. 3. Unit cell definitions in the complex and real-valued formulations; a – alternative definitions areequivalent in a complex-valued formulation; b – the definition of the unit cell is restricted to be boundedby mirror symmetry planes in the real-valued formulation.
Wrong definition Correct definition
Mirror symmetryplanes
a
b
Efficient calculations of dispersive properties... 145
3. Multi-grid approach
In the previous section, the real-valued TLM method was presented as an alternativefor reducing the computational effort when computing the photonic band structure ofPCs. However, the analysis of the equations enforcing the periodicity conditionshowed that such an approach can only be applied to unit cells satisfying certainsymmetry conditions. Thus, a more general way to proceed toward an efficientcomputation of the photonic band structure of PCs, is to reduce the number of blocksused for the discretization of the simulation domain. There are two basic ways ofaccomplishing this: i) to increase the discretization step while maintaining a uniformmesh, and ii) to combine regions with different discretization steps in a multi-gridfashion.
The choice of i) or ii) depends on several factors. While increasing thediscretization step results in fewer blocks and therefore faster simulations, it decreasesthe cut-off frequency of the mesh and increases node dispersion. In addition, theresolution of the mesh is decreased and fine structural details of the cell, cannot beproperly represented. In a multigrid mesh, the cut-off frequency is still limited by thebiggest block present in the simulation domain, however, it provides the flexibility toproperly represent localized structural details where required. This approach isparticularly important for the simulation of three-dimensional unit cells of arbitraryshape and large complex structures, where the use of a uniform mesh is computationallyprohibited.
The multigrid mesh for the simulation of PCs was implemented using an in-housecomputer program (Atar). The basic idea behind the model building is to create a complexmesh geometry using rectangular blocks of varying sizes. The model is created suchthat a block and its neighbors form a relatively simple topology. Each block in themesh can have either two or four blocks on any side in what is called a quad tree mesh.The mesh is automatically generated based on a set of input parameters such as theminimum and maximum block size, number of levels and location of refinements, etc.A constraint that is imposed during the building process for a periodic structure, is thatthere is a one-to-one correspondence between blocks on opposite sides of thesimulation domain. This facilitates the connection of these blocks when enforcingthe periodicity condition.
In terms of the TLM method, the small and big internal faces of the blocks areconnected using a similar procedure to that described in [17]. The connectionprocedure is based on ideal matching transformers which preserve the unconditionalstability of the TLM method. The basic difference between that formulation and ours,is that in the aforementioned reference, the incident and reflected pulses at the interfaceare related by circuit-type equations. In our implementation, we relate the pulses bymeans of a scattering matrix instead. This allows to reduce the number of operationsinvolved in the connection procedure.
146 G. ROMO, T. SMY
4. Simulations and evaluation
In this section, we validate the previously described approaches by simulating variousPCs. For obtaining the photonic band structures, we used the general proceduredescribed in [16] and more details about the actual implementation can be found inthis reference. In all cases, our results were compared with those predicted by the PWEmethod to verify their validity. Two- and three-dimensional structures were analyzed.For the case of a uniform mesh, the unit cell was discretized into 32×32(×32) uniformlyspaced mesh points. In all cases, the lattice constant a was normalized such that a = 1.In addition, only TM modes were considered in the two-dimensional structures. Thatis, the only non-zero field components were the electric field along the infinitedimension and the two magnetic field components transverse to it.
4.1. Complex- and real-valued simulations using a uniform mesh
The complex- and real-valued formulations of the TLM method using a uniform meshwere previously presented and validated in [14] and [16], so only the relevant featureswill be emphasized here.
The first simulated PC was the chessboard structure shown in Fig. 2. This structurehas the same symmetry properties as a conventional square lattice. The chessboardstructure consisted of an air-dielectric composite and the value of the dielectric materialwas set to εr = 11.7 which corresponds to square pillars of Si connected by the cornersand embedded in air. This PC has previously been simulated using the complex-valuedTLM method and will be used here, to illustrate the limitations of the real-valuedformulation.
Fig. 4. Dispersion relation of the PC shown in Fig. 3. Circles (filled-squares): simulation results usingthe real-valued formulation and the unit cell definition shown on the right (left) hand side of Fig. 3. Solidline: PWE method. The inset shows the first Brillouin zone and the symmetry points used for thecalculation.
Efficient calculations of dispersive properties... 147
Figure 4 shows a comparison of the dispersion relations of the chessboard structureshown in Fig. 2 obtained by applying the real-valued formulation. The unit cell wasdefined in two different ways as shown in Fig. 3. The circles correspond to the pointspredicted by the real-valued formulation using the correct cell definition as shown onthe right hand side of Fig. 3, whereas the filled-squares correspond to the pointspredicted by the same formulation using the incorrect cell definition presented on theleft hand side of Fig. 3. In addition, the solid lines show the predictions of the PWEmethod. As the figure shows, the TLM real-valued formulation and the PWE methodsare in good agreement provided that the unit cell is defined such that it is bound bymirror symmetry planes. In the figure, the frequency has been normalized with respectto 2πc/a and the inset to it, shows the first Brillouin zone of the chessboard lattice andthe symmetry points used for the calculation of the dispersion relation.
It should be emphasized at this point that when the appropriate cell definition isused, the results predicted by the real- and complex-valued formulations are practicallyindistinguishable from each other. Obviously, this also requires that the two methodsbe used under the same circumstances (time and discretization steps, number ofiterations, TLM-node, etc.). Thus, there is a factor of two in the computational effortrequired by the real and complex-valued formulations when computing the photonicband structure of PCs.
4.2. Complex- and real-valued simulations using a multigrid mesh
In previous sections, it was mentioned that another way to proceed toward acomputationally efficient simulation of PCs, is to reduce the number of blocks usedfor discretizing the simulation domain using a multigrid mesh. In principle, both thecomplex and real-valued formulations of the TLM method can be applied in a multigridmesh. However, given that the complex-valued formulation represents the mostgeneral scenario, such is the case considered in this section. It should be bear in mind,however, that the results can be extended to the real-valued formulation in the sameway as it was done for the case of a uniform mesh.
To illustrate the advantages and disadvantages of a multigrid approach forsimulating PCs, we used as a first example, a square lattice of circular rods embeddedin air. The dielectric constant of the cylinders was set to εr = 9.0 which correspondsto alumina rods embedded in air. The radius of the rods was set to 0.38a, with arepresenting the lattice constant. The corresponding unit cell and first Brillouin zoneare shown in Fig. 5. In addition, the multi-grid mesh utilized for discretizing the unitcell is shown in Fig. 6. Note that an averaging procedure was applied at the interfacebetween the air and dielectric rods to smooth the transition between the two media.The criterion applied for the discretization of the simulation domain was to limit therefinement to no more than two levels. That is, the multigrid mesh consisted ofrectangular blocks formed by the combination of three basic lengths: ∆L, 2∆L, and4∆L (where ∆L is the step size used for the discretization of the cell using a uniformmesh).
148 G. ROMO, T. SMY
Figure 7 shows a comparison of the dispersion relations of the square lattice ofcircular pillars obtained by applying three different methods. The dots correspond tothe frequency points predicted by the complex-valued formulation using a uniformmesh. In addition, the filled-squares represent the frequency points predicted by thecomplex-valued formulation using the multigrid mesh of Fig. 6. In the figure,the frequency has been normalized as before. As a way of comparison, the solid linesshow the results predicted by the PWE.
As Fig. 7 shows, the overall agreement between the three different approaches isfairly good. However, the introduction of a coarser mesh has slightly degraded theaccuracy of the TLM method as expected. The accuracy of the results were mainlyaffected by two sources of error: node dispersion and meshing error. Node dispersionrelates to the fact that short wavelengths (compared with the block size) are notproperly represented by the mesh. As a result, the wave velocity becomes dispersiveand dependent on the direction of the propagation vector. Meshing error refers to thedegree of accuracy with which structural details of the unit cell and dense fields arerepresented in the mesh. Naturally, the multigrid mesh can be devised so that the
Fig. 5. Unit cell definition (a) and first Brillouin zone (b) of the two-dimensional PC.
Symmetry planes
a bSymmetry planes
a = 1
x
y
2πa
ky
M
Γ X kx
2πa
Fig. 6. Multi-grid mesh used for the discretization of the unit cell.
Efficient calculations of dispersive properties... 149
structural details of the cell are properly represented in the mesh. However, there isno guarantee that the resulting mesh will properly represent the field modes in allregions. As a consequence, field meshing error is still present in the multigrid mesh.
Note that, while it is generally true that in a multigrid mesh the maximum erroroccurs at higher frequencies, our present example shows that this is not necessarilythe case. For the present photonic band structure, the worst discrepancy betweenfrequency modes predicted by the uniform and multigrid meshes occurred near theends of the sixth band, with an error of approximately 2.7%. Also note thatthe predictions of the multigrid mesh are in good agreement for frequencies well abovethe theoretical cut-off frequency of the mesh (~0.26 units of normalized frequency).For this example, the discretized structure consisted of 220 blocks. Thus, whencompared with a 32×32 uniform mesh, the computational efficiency is increased bya factor of 4.65.
The efficiency of time-domain methods is directly related to the number of blocksused to build the model. Thus, a multigrid mesh is a natural approach to reduce thenumber of blocks and hence the computational effort. However, there is a large numberof ways in which a multigrid mesh can be defined to discretize a given PC. Thus, whileit is not possible to characterize all the different multigrid meshes individually, theycan be characterized collectively by the minimum and maximum block size present inthe mesh. The minimum block size determines the spatial resolution and the maximumblock size, the cut-off frequency of the mesh.
The real advantage of a multigrid mesh is better appreciated in larger problems.To illustrate this point, the next simulated example is a three-dimensional one. For thiscase, we chose a structure consisting of dielectric spheres embedded in air. Thedielectric constant of the spheres was set to 12.0 and their radius to 0.3125a. To alsoillustrate the trade-offs that come into play when multi-griding is used, two different
Fig. 7. Dispersion relation of the PC shown in Fig. 3. Dots: simulation results using the R-V formulationin a uniform mesh. Filled-squares: simulation results using a C-V formulation in a multi-grid mesh. Solidline: PWE method.
150 G. ROMO, T. SMY
meshes are looked at. Figure 8 shows the multigrid meshes that were used for thediscretization of the unit cell. In Figs. 8a and 8b, one and two levels of refinementwere allowed respectively during the generation of the mesh. Thus, while the spatialresolution is the same for both meshes (equal to 1/32), the cut-off frequency of themesh in Fig. 8b is half that of the mesh in Fig. 8a. In what follows, we will denotethese meshes as Multigrid-(1/16) and Multigrid-(1/8), respectively, where the numberin parentheses denotes the maximum block size (remember that the lattice constanthas been normalized to a = 1). Similarly to the previous example, an averagingprocedure was applied to smooth the transition between the dielectric spheres andthe air. In addition, Figs. 8a and b show that three-dimensional structures favor theformation of cubic, rather than rectangular blocks during the building process.
Figure 9a shows the photonic band structure of the cubic lattice of dielectricspheres. The corresponding first Brillouin zone and symmetry points used as a referencefor the computation of the band structure are shown in Fig. 9b. The solid lines wereobtained by using a computer program based on the PWE method in a uniform mesh [2].The open circles represent the frequency points predicted by the complex-valued TLMformulation using the Multigrid-(1/16) for discretizing the unit cell. Similarly, thefilled-squares represent the frequency points predicted by the complex -valued TLMformulation when the Multigrid-(1/8) is used. For the Multigrid-(1/16), the simulationdomain consisted of 5776 blocks. This indicates that the computational efficiency isincreased by a factor of 5.7 when compared with a 32×32×32 uniform mesh.
In terms of accuracy, Fig. 9a shows that the agreement between the uniform andmultigrid meshes is fairly good up to the frequency shown in the figure. This is truedespite the fact that, strictly speaking, the cut-off frequency of the Multigrid-(1/16) isonly ~0.46 units of normalized frequency. In calculating this frequency, a minimumof ten nodes per wavelength has been used as a reference. In addition, it has been
Fig. 8. Multigrid meshes used for the discretization of the unit cell. The maximum block has been limitedto: a – 1/16 and b – 1/8.
a b
Efficient calculations of dispersive properties... 151
calculated with respect to the dielectric sphere, where the shorter wavelengths (fora fixed frequency) are expected.
When the Multigrid-(1/8) was used to discretize the unit cell, the resulting structureconsisted of 3928 blocks. In this case, the computational efficiency is increased bya factor of 8.3 when compared with a 32×32×32 uniform mesh. As expected, theaddition of bigger blocks in the simulation domain has further degraded the accuracyof the results. However, we can see from Fig. 9a that although the cut-off frequencyof this mesh is only ~0.23 units of normalized frequency, the results are in goodagreement up to at least two times this value. We can also see that the worst discrepancybetween the uniform and multigrid methods occurred near the ends of the upper mostbands. That is, when one (or more) of the components of the propagation vector isequal to zero. This is again, an expected result, since it is known that block dispersionis worst under this scenario [18]. It should be pointed out that these results wereobtained by using a stub-loaded node and are expected to improve, due to superiornode dispersion properties, by using hybrid or super condense nodes [19].
Figure 10 shows the computational time associated with the various methodspresented in this paper as a function of the spatial resolution of the mesh (minimumblock size). In the figure, the computational time has been normalized with respect tothe complex-valued TLM method applied in a uniform mesh, which is the slowest, butmost accurate scenario. This method is indicated by the dashed line in the figure.Similarly, the dot-dashed line represents the computational time of the real-valuedTLM method also applied in a uniform mesh. As it was explained before, this methodreduces the computational effort by half as compared with the complex-valuedformulation while retaining the same accuracy. The two remaining lines in the figureshow the results of the multigrid approach. As the figure indicates, two cases wereconsidered. The solid and dotted lines correspond to multigrid meshes where the
Fig. 9. Dispersion relation of the simple cubic lattice; filled-squares: Multigrid-(1/16) mesh; open circles:Multigrid-(1/8); solid lines: PWE method using a uniform mesh (a) and first Brillouin zone of the simplecubic lattice (b).
a
b
152 G. ROMO, T. SMY
resolution is varied while the maximum block size is limited to 1/16 and 1/8,respectively. In the figure, the corresponding locations of the multigrid meshes shownin Fig. 8 are also indicated.
Figure 10 shows that the computational efficiency of a multigrid approach isenhanced toward higher resolutions (smaller block size). This is particularly importantfor the simulation of PCs because it is in this region where the cut-off frequency ofa uniform mesh is unnecessarily high (>1 unit of normalized frequency). This translatesinto the possibility of finding a good compromise between computational time andaccuracy. It should be noted that the computational time associated with the multigridmethod could further be reduced by half, if the geometry of the unit cell allowed forthe real-valued formulation to be used.
The results presented in this section indicate that a multigrid approach canconsiderably reduce the simulation time that is required for computing the photonicband structure of PCs while maintaining a good accuracy.
5. Conclusions
In this paper, different approaches for the simulation of PCs using the TLM methodwere presented. These approaches were divided into complex- and real-valuedalgorithms using a uniform mesh, and complex- and real-valued algorithms usinga multigrid mesh. The advantages and disadvantages of each approach were discussedand a comparison between these methods was made from the points of view ofcomputational expense and accuracy.
Fig. 10. Computational time as a function of minimum block size for the various methods. Dashed(dot-dashed) line: complex (real)-valued formulation in a uniform mesh; solid and dashed lines: complex-valued formulation in a multigrid mesh, the maximum block size is limited to 1/16 and 1/8, respectively.
Efficient calculations of dispersive properties... 153
It was found that a combination of a real-valued method in a multigrid mesh resultsin the most efficient algorithm. However, unlike the complex-valued formulation, itsapplicability is limited to unit cells satisfying certain geometrical details. It was alsofound that while a multigrid approach can considerably reduce the computational effortrequired for simulating PCs, it also reduces the accuracy of the results. In this respect,our results indicate that a good compromise between execution time and accuracy canbe found when using the multigrid approach. In addition, it was observed thatthe multigrid meshes presented in this paper had a good performance, well above thetypical theoretical cut-off frequency of the mesh.
Acknowledgements – This work was supported by the Natural Sciences and Engineering Research Councilof Canada (NSERC), the Microelectronics Network within the Canadian Federal Networks of Centers ofExcellence Program (MICRONET), The Canadian Institute for Photonic Innovations (CIPI) and EMPWR.
References
[1] SAKODA K., Optical Properties of Photonic Crystals, Springer, Germany 2001.[2] JOHNSON S.G., JOANNOOULUS J.D., Block-iterative frequency-domain methods for Maxwell’s
equations in a planewave basis, Optics Express 8(3), 2001, pp. 173–190. [3] CHAN C.T., YU Q.L., HO K.M., Order-N spectral method for electromagnetic waves, Physical
Review B: Condensed Matter 51(23), 1995, pp. 16635–42.[4] WARD A.J., PENDRY J.B., A program for calculating photonic band structures, Green’s functions
and transmission/reflection coefficients using a non-orthogonal FDTD method, Computer PhysicsCommunications 128(3), 2000, pp. 590–621.
[5] YAMADA S., WATANABE Y., KATAYAMA Y., COLE J.B., Simulation of optical pulse propagation ina two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domainalgorithm, Journal of Applied Physics 93(4), 2003, pp. 1859–64.
[6] THÉVENOT M., REINEIX A., JECKO B., FDTD approach for modelling PBG structures, Journal ofOptics A: Pure and Applied Optics 1(4), 1999, pp. 495–500.
[7] MEKIS A., FAN S., JOANNOPOULOS J.D., Absorbing Boundary Conditions for FDTD Simulations ofPhotonic Crystal Waveguides, IEEE Microwave and Guided Wave Letters 9(12), 1999, pp. 502–4.
[8] CHUTINAN A., NODA S., Waveguides and waveguide bends in two-dimensional photonic crystal slabs,Physical Review B: Condensed Matter 62(7), 2000, pp. 4488–92.
[9] QIU M., AZIZ K., KARLSSON A., SWILLO M., JASKORZYNSKA B., Numerical studies of mode gaps andcoupling efficiency for line-defect waveguides in two-dimensional photonic crystals, PhysicalReview B: Condensed Matter and Materials Physics 64(15), 2001, pp. 155113/1–5.
[10] ABU EL-HAIJA A.J., Analysis of the optical properties of thin films using the transmission line method,Optica Applicata 27(2), 1997, pp. 121–41.
[11] JACQUIN O., HDAGIJIMANA F., CACHARD A., BENECH P., Application of the TLM technique to integratedoptic component modelling, International Journal of Numerical Modelling: Electronic Networks,Devices and Fields 14(2), 2001, pp. 95–105.
[12] COLLIN R.E., Field Theory of Guided Waves, 2nd Ed., NJ: IEEE Press, Piscataway 1991.[13] CELUCH-MARCYSIAK M., GWAREK W.K., Spatially looped algorithms for time-domain analysis of
periodic structures, IEEE Transactions on Microwave Theory and Techniques 43(4), 1995, pp. 860–5. [14] ROMO G., SMY T., Dispersion relation calculation of photonic crystals using the transmission line
matrix method, International Journal of Numerical Modelling: Electronic Networks, Devices andFields 17(5), 2004, pp. 451–9.
154 G. ROMO, T. SMY
[15] WALTER M., PERTZ O., BEYER A., A contribution to the modeling of longitudinally periodicwaveguides by the help of the TLM method, IEEE Transactions on Microwave Theory andTechniques 48(9), 2000, pp. 1574–6.
[16] ROMO G., SMY T., Modeling of photonic crystals using a real-valued transmission line matrix method,Journal of Applied Physics 94(4), 2003, pp. 2177–82.
[17] WLODARCZYK J., New multigrid interface for the TLM method, Electronics Letters 32(12), 1996,pp. 1111–12.
[19] TRENTIK V., CHRISTOPOULOS C., BENSON T.M., Development of a general symmetrical condensednode for the TLM method, IEEE Transactions on Microwave Theory and Techniques 44(12), 1996,pp. 2129–35.
Received November 4, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Research on absorption spectroscopy of CH4 around 1.315 µm
1Laboratory of Atmospheric Optics, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Science, Hefei 230031, China
2Environmental Spectroscopy Laboratory, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Science, Hefei 230031, China
Measurements have been made about the absorption features of CH4 around 1.315 µm by usingthe second harmonic detection technique of tunable diode laser with multipass cell (white). Studiesconcerning the second harmonic detection technique of CH4 around 1.65 µm have been widelyreported currently, but reports about the absorption features in NIR spectra of CH4 around 1.315 µmcan scarcely be found. We have performed a detailed study about the absorption features of CH4around 1.315 µm and we present the results here. We give the line positions, the line intensitiesand self-broadening coefficients near 1.315 µm at a pressure of 0.077 torr.
Keywords: modulation spectroscopy, diode lasers, the second harmonic detection.
1. Introductions
Wavelength-modulation (WM) methodologies are frequently used for detection ofspecies in absorption spectrometry, primarily with diode lasers due to their excellenttunability [1–3]. This experimental technique is not only widely used to monitor weaksignals for the trace gas detection but can also be used to study the parameters ofabsorption lines [2, 4, 5].
Tunable diode laser absorption spectroscopy was used with second harmonicdetection, which has the merits of high sensitivity, real-time detection, etc. It hasbecome an advanced technique applied to optical sensors of polluting gases and hasrecently aroused wide interest. Diode lasers are characterized by low-cost, smallvolume, tunability, fiber-optic compatibility and can detect many characteristicqualities of different places and different gases synchronously.
Methane, the main component of natural gas, is one of the most important energyresources used in everyday life. It is very dangerous when its concentration in theatmosphere exceeds 5% [6]. The detection of concentration of CH4, H2O, CO2, CO, etc.,
156 J. SHAO et al.
is always related to combustion and propulsion processes. Their concentrations aredirectly connected with the capability parameters of combustion and propulsionprocesses. So, the diagnostic method of detecting the concentration of CH4, and H2Oin the combustion and propulsion processes has a good applicational future. A thoroughinvestigation of the absorption spectra of CH4 is of vital importance. There arenumerous reports on the detailed absorption lines of H2O around 1.315 µm, but thoseconcerning CH4 are scarce. Our research results present detailed absorption spectrallines of CH4 around 1.315 µm obtained by using tunable diode laser absorptionspectroscopy (TDLAS) with a long-path absorption cell (white). We obtain detailedspectral features of pure CH4 in the range of 7602–7617 cm–1 under the low pressurecondition and offer the experimental testimony to detecting CH4 and H2O synchronouslyusing the same laser as gas sensors.
2. Theory
TDLAS usually scans over an isolated absorption line of the species underinvestigation using a single narrow laser line. To achieve the highest selectivity,the sample is made at low pressure, where the absorption lines are not substantiallybroadened by pressure [2, 7]. The TDLAS technique has developed into a verysensitive and general technique for monitoring most atmospheric trace species.
The intensity of monochromatic laser radiation of frequency ω transmitted througha sample cell containing an absorbing species is given by Beer’s law:
(1)
where I0 and I are referred to as the incident and transmitted radiation intensities,respectively, σ – the absorption cross-section of sample gases, L – the optical pathlength within the cell, ρ – density of absorbing species.
In order to improve the sensitivity, wavelength modulation spectroscopy withtunable diode laser sources has been used since the early 1970s [7]. When the laser ismodulated around its center frequency υ0 at a frequency ωm, the instantaneousfrequency is υ = υ0 + δυcos(ωmt), where δυ is the modulation amplitude. The intensityI(υ) of the radiation transmitted through the absorption cell can then be expressed asa cosine Fourier series:
(2)
The individual harmonic components An (for n > 0) can be measured with a lock-inamplifier and are given by:
I υ( ) I0 σ Lρ–( )exp=
I υ t,( ) An υ( ) nωmt( ).cosn 0=
∞
∑=
Research on absorption spectroscopy of CH4 around 1.315 µm 157
(3)
where θ has been substituted by ωmt. In the limit of low absorbance (ρLσ << 1), whichis true for trace gas detection at ppb level, this equation becomes:
(4)
Thus, each harmonic component is directly proportional to the species density ρ. Inthis case, a Taylor series expansion of σ (υ)
(5)
Here, the n-th harmonic component is proportional to the n-th derivative of χ (υ) andthe technique is known as derivative or wavelength modulation spectroscopy. At thesame time, one can see that the harmonic components are directly proportional tothe density ρ of the sample, so the technique can be used to detect the density of gases.
During the measurement of the absorption lines of gaseous molecules, there arethree main broadening mechanisms, which are natural broadening, Doppler broadeningand collision broadening. Because of the relatively narrow width of naturalbroadening of gaseous molecules compared to the linewidth of diode lasers, thenatural broadening will not be considered in the TDLAS technique. At low pressure,Doppler broadening is of great importance. The absorption has the normalizedGaussian lineshape:
(6)
where γD is the half-width of Doppler broadening, and
(7)
(υ0 – the central frequency of absorption line, T – the temperature in Kelvin andM – the molecular weight).
At high pressure, the collision broadening is of importance and the absorptionlineshape is given by the normalized Lorentzian function.
Here, γL is the half-width of collision broadening, γL, 0 – the half-width of collisionbroadening under the standard atmospheric pressure (T0, p0).
The second harmonic signal of Gaussian lineshape function is:
(10)
The second harmonic signal of Lorentzian lineshape function is:
(11)
When the Doppler linewidth is comparable to Lorentizian linewidth, the absorptionlineshape is the convolution of a Lorentizian and a Gaussian lineshape function.
3. Experimental apparatus
The experimental apparatus used in this work is shown schematically in Fig. 1. Forthis experiment the NEL 13563TG DFB laser has been adopted as the source.The current and temperature were controlled by a laser-controller (TDS3724B), whichperformed the scan of the emission wavelength by the driving current with a lowfrequency sawtooth signal or by scanning the driving temperature. After splitting, asmall portion of light from diode laser was sent to a wavemeter (WA-1500 NIR,Buleigh) through optic coupler for wavelength monitoring, the majority of lighttransmitted the mutilpass cell through coupling system and then was detected by adetector (New Foucs). The signal from detector was sent to a lock-in and demodulatedto extract the second harmonic signals. A DAQ was used to acquire the harmonic signalfrom the lock-in. A GPIB card was used with the computer to control the laser
χL υ( ) 1π-----
γL
υ υ0–( )2 γ L2
+--------------------------------------- ,=
γL γL 0,pp0
--------T0
T--------
1 2⁄
.=
SL2 υ( ) η
I0SρLγL
2π------------------------
γ L2
3 υ υ0–( )2–
υ υ0–( )2 γ L2
+3
-----------------------------------------------
δ2υ–=
SG2 υ( ) η
I0SρL
2γ L5
------------------- 2 2υ υ0–
γD
------------------
2
ln– γ D2
2 2 υ υ0–( )2ln– δ2υ.expln–=
Research on absorption spectroscopy of CH4 around 1.315 µm 159
controller and read the wavenumber from the wavemeter. The mutilpass cell iswhite-type geometry with an 8 m base path length and can achieve total path lengthof 46–1159 m.
4. Results and discussion
As an example, a record of the low-pressure (0.077 torr) Doppler broadened CH4 at7605.54656 cm–1 is presented in Fig. 2 together with the residual of the multispectrumnonlinear least-squares fitting to the Gauss lineshape model. Figure 2, illustrates theabsorption spectra with the peak at 7605.54656 cm–1 of CH4 within a current scan
Fig. 1. Schematic diagram of experimental set up.
Fig. 2. Absorption line observed by WM spectroscopy and the 2nd harmonic detection technique for thegas pressure of 0.077 torr and the lock-in time constant of 3 ms.
[cm–1]
160 J. SHAO et al.
cycle. The dashed line represents the signal detected by the second harmonic. The solidone is the fitted signal by using wavelength modulation theory and non-linear leastsquare fit. During the wavelength scan period we keep the temperature and centercurrent of diode laser at 27°C and 70 mA, respectively, with the optical path of602.68 m. The modulation frequency and modulation amplitude of lock-in are1.78 kHz and 40 mV, respectively. The calculated SNR is about 4, and thecorresponding absorption is 3.04×10–6.
In our experiment, many CH4 absorption lines have been detected by TDLAS.There are tens of absorption lines found in the 7602–7617 cm–1 range. Theirwavenumbers and line intensities have been obtained from the measured wavenumbersby comparing to the reference signal of CO2 absorption lines. The experimental dataobtained are given in Fig. 3. The figure shows a plot of the 7602–7617 cm–1 spectralregion as recorded with a sample pressure of 0.183 torr 99.99% pure CH4 and the
Fig. 3. Signal measured spectral absorbance of pure CH4 by WM and the 2nd harmonic detectiontechnique. The cell pressure was 0.183 torr, optical length was 602.68 m.
Fig. 4. Position of spectral line and the intensity of CH4 obtained by means of WM spectroscopy andthe 2nd harmonic detection technique.
Research on absorption spectroscopy of CH4 around 1.315 µm 161
absorption path of 602.68 m. The spectra positions and relative intensity are illustratedin Fig. 4.
The experimental data for the line intensity and self-broadening of CH4 near1.315 µm are compiled in the Table.
T a b l e. Measured spectra positions, line intensities and self-broadening coefficients.
We obtained detailed spectral features of pure CH4 in 7602–7617 cm–1 region at roomtemperature by using NIR TDLAS technique, which gives the experimental basis fordetecting CH4 and H2O synchronously by just using a laser. We give line intensity andself-broadening coefficient of unknown lines from the experimental data. At the sametime, we give nearly tens of strong absorption lines of CH4, which points out thedirection of investigating the gas sensor of CH4 within the region.
References
[1] GAO XIAOMING, HUANG WEI, LI ZIYAO, FANG LI, ZHANG WEIJUN, Sensitive detection of CO2 moleculeusing near infrared diode laser absorption spectroscopy, Acta Optica Sinica 23(5), 2003, pp. 609–11.
[2] KLUCZYNSKI P., GUSTAFSSON J., LINDBERG A.M., AXNER O., Wavelength modulation absorptionspectrometry – an extensive scrutiny of the generation of signals, Spectrochimica Acta Part B: AtomicSpectroscopy 56(8), 2001, pp. 1277–354.
[3] AXNER O., KLUCZYNSKI P., LINDBERG A.M., A general non-complex analytical expression for the nthFourier component of a wavelength-modulated Lorentzian lineshape function, Journal of QuantitativeSpectroscopy and Radiative Transfer 68(3), 2001, pp. 299–317.
[4] LI QIAN, XIAO LIANTUAN, LI CHANGYONG, JIA SUOTANG, ZHOU GUOSHENG, Theoretical andexperimental investigation of low-frequency wavelength modulation spectroscopy, Acta Optica Sinica21(3), 2001, pp. 317–19.
[5] YELLESWARAPU C., SHARMA A., A simple technique for indirect measurement of absorption line widthsin Voigt profile regime, Journal of Quantitative Spectroscopy and Radiative Transfer 72(5), 2002,pp. 733–40.
[6] IKUTA K., YOSHIKANE N., VASA N., OKI Y., MAEDA M., UCHIUMI M., TSUMURA Y., NAKAGAWA J.,KAWADA N., Differential absorption lidar at 1.67 µm for remote sensing of methane leakage, JapaneseJournal of Applied Physics, Part 1 Regular Papers, Short Notes and Review Papers 38(1A), 1999,pp. 110–14.
[7] WERLE P., A review of recent advances in semiconductor laser based gas monitors, SpectrochimicaActa, Part A Molecular and Biomolecular Spectroscopy 54A(2), 1998, pp. 197–236.
Received May 31, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Teaching optics
Set-up for spontaneous and induced birefringence measurements
AGNIESZKA CIŻMAN, RYSZARD POPRAWSKI
Institute of Physics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland; e-mail: [email protected]
The paper presents a simple polarization-interference set-up for demonstration of temperaturedependence of birefringence and both electrooptical phenomena which can be used for the studentslaboratory and scientific investigation.
Keywords: birefringence, ferroelectric material.
1. Introduction
A polarized light beam passing through a linearly birefringent crystal is divided intotwo beams linearly polarized in perpendicular planes and then propagates withdifferent velocities.
When a light beam is passing through a crystal, the phase shift between the fastand the slow light waves is related to the linear birefringence by:
(1)
where λ0 – wavelength of the light in vacuum, L – length of the light path through thesample, D – linear birefringence [1–3].
When the light exits the crystal, both interfering beams give in effect either a linear,circular or elliptical beam depending on the phase shift.
For the crossed arrangement of a polarizer and an analyzer the intensity of lightleaving the system is given by
(2)
where I0 – the intensity of light leaving the system when sin(2ϕ) = ±1 andsin(∆γ /2) = ±1, ϕ – the azimuth of the first eigenvector related to the analyzer [4].
∆γ 2πλ0
---------- ∆n L=
I I0 sin2
2ϕ( ) sin2 ∆γ
2---------
=
164 A. CIŻMAN, R. POPRAWSKI
Using formula (2) we can calculate the phase shift between the fast beam and theslow one. The maximum change of the light intensity exiting the system is achievedwhen ϕ = ±45°, then formula (2) is given by
(3)
Birefringence ∆n of the medium can be changed, e.g., by temperature, externalelectric or magnetic fields (the electro- or magneto-optical effects). Especially,spontaneous birefringence changes are interesting in ferroelectric crystals in the phasetransition temperature region.
Methods and measurement results of the spontaneous birefringence within theferroelectric phase transition as well as the linear and quadratic electrooptic effects(the Pockels and the Kerr effects) are presented and discussed.
2. Spontaneous birefringence in ferroelectric materials
In centrosymmetric ferroelectric crystals in paraelectric phase the spontaneousbirefringence has the nature of the spontaneous Kerr effect:
(4)
where: r – electrooptical coefficient, Ps – spontaneous polarization [1, 2].It is known that the temperature dependence of the spontaneous polarization Ps in
ferroelectrics exhibiting the 2-nd order phase transition can be described as [5, 6]:
(5)
where: α, β – free energy expansion coefficients, Tc – the Curie–Weiss temperature.A subject of this work, i.e., (NH2CH2COOH)3·H2SO4 (TGS) crystal exhibiting the
second-order ferroelectric phase transition at about 322 K, is one of the most importantand best known ferroelectrics. Above this temperature, it belongs to centrosymmetricalpoint group 2/m of rohmbohedral system, but below the transition point it isferroelectric and belongs to the point group 2 of monoclinic system, therefore the linearbirefringence in ferroelectric phase has the Kerr phenomenon character.
The single crystals of TGS were grown from water solution by slow evaporationat constant temperature. Crystals of good optical-quality were obtained after a fewweeks. TGS samples for the birefringence investigation were cleaved perpendicularto the b-axis and cut perpendicular to a- and c-axis out of optically clear crystals andpolished to a thickness of about 3 mm.
The TGS crystals are recommended for spontaneous birefringence measurementscarried out in students laboratory. The reasons are numerous. Measurements in a wide
I I0 sin2 ∆γ
2---------
.=
δΓ rPs2∼
Psαβ
------ T Tc–( )–±=
Teaching optics 165
temperature range, including the phase transition temperature, can be carried outwithout the cryogenic liquid. The TGS crystals are easy to grow and exhibit a goodoptical quality with right orientation and are of a relatively large size.
3. Experimental and results
3.1. Temperature changes of spontaneous birefringence
We used a polarizing-interfering technique, presented in Fig. 1, for the birefringencemeasurements. The method has long been one of the basic techniques used in studyingof the spontaneous and induced birefringence in crystals.
A system of the spontaneous birefringence change measurement is shownschematically in Fig. 2. The optical thermostat is one of the basic elements of themeasuring system. The thermostat consists of two coaxial cylinders, a heater, a thermo-couple and a PT-100 temperature sensor. A polarizer, an analyzer, an He-Ne laser(the light source) and a photodetector are necessary elements of the measuring system.A 660-type UNIPAN temperature controller was used to ensure a continuoustemperature variation.
Fig. 2. Diagram of the birefringence measuring method.
Computer
Powersupply
He-Nelaser
Polarizer
mV
Thermostat
Analyzer
µA
Photodetector
Temp.sensorHeater
PT-100Heater
Thermocouple
166 A. CIŻMAN, R. POPRAWSKI
The birefringence measurements were carried out at temperature ranging from 295to 335 K. Each heating survey was carried out at a constant temperature rate of0.2 K/min. The sample was placed between two crossed polarizers.
The intensity changes promoted by the temperature variation are read by thephotodetector and recorded by a computer system.
In order to calculate an anomaly in the crystal spontaneous birefringence changesit is necessary to measure the system transmission and temperature in temperaturerange including the phase transition point, Fig. 3. The measuring system was notprotected aganist the changing daylight, which had an influence on the shape of thespectrum (see Fig. 3). These changes had not any significant impact on birefringenceresults.
The temperature dependence of the spontaneous birefringence changes of TGScrystals calculated from the results presented in Fig. 3a and 3b is shown in Fig. 4.
Fig. 3. Temperature dependence of the system transmission (see Fig. 2), i.e., the photodetector current.Samples are cut perpendicular to a (a) and b (b) axes.
a b
Fig. 4. Temperature dependence of the phase shift between the fast and the slow light waves in TGScrystals.
a bTp
Tp
∆γ [r
ad]
∆γ [r
ad]
T [K] T [K]
Teaching optics 167
The birefringence along the b-axis (Fig. 4a) and the a-axis (Fig. 4b) agrees well withthe previous spontaneous birefringence studies [6].
Using temperature dependence of the system transmission (Fig. 3a and 3b) onecan calculate the phase shift between the fast and the slow light waves and thespontaneous birefringence changes.
The temperature dependence of phase shift between the fast and the slow lightwaves for TGS crystals cut along the a- and the b-axis is presented in Fig. 4.
For appropriate temperatures in Fig. 3 we collected the minimum and maximumvalues of the intensity of light and based on this we calculated the temperaturedependence of the phase shift between fast and slow light waves (Fig. 4). From Fig. 4we can clearly see that the phase transition temperature point Tp is at 322 K.
Using formula (1) we can calculate changes of spontaneous birefringence in TGScrystals for a- and b-axis directions. The spontaneous birefringence changes ofTGS crystals as a function of temperature are presented in Fig. 5a and 5b.
3.2. Induced birefringence – electric field changes
The birefringence change induced by electric field is called an electrooptical effect.When the birefringence change δ(∆n) is a linear function of an electric field then thephenomenon is a linear electrooptical effect (the Pockels effect). The effect can bedescribed in a very simplified way as
δ(∆n) = cE. (6)
When the birefringence change is proportional to the square of the electric fieldintensity the phenomenon is called the Kerr effect
δ(∆n) = CE2. (7)
Fig. 5. Temperature dependence of the linear birefringence changes in TGS crystals.
a b
δ(∆n
) 10
–3T [K]T [K]
δ(∆n
) 10
–3
168 A. CIŻMAN, R. POPRAWSKI
Using Eqs. (1), (2) and (6) or (7) we can obtain relation between the intensity ofthe light changes after crossing the system presented in Fig. 6
(8)
and in the case of Kerr effect
(9)
The investigation of the Pockels and the Kerr effects consists in determiningchanges in the intensity of polarized light passing through the system as a function ofan electric field intensity.
A schematic diagram of the measuring set-up for linear and quadratic electro-optical effect studies is shown in Fig. 6. The system consists of a light source (a laserdiode), a power supply, a polarizer, either the Pockels or the Kerr cell, an analyzer,
I I0sin2 2π
λ0
--------- LcE
=
I I0sin2 2π
λ0
--------- LCE2
.=
Fig. 6. Schematic diagram illustrating a system for measuring Pockels (Kerr) effect.
Powersupply
Laserdiode
Pockels (Kerr)cell Photodetector
PolarizerϕP = 90°
Amplifier
AnalyzerϕA = 0°
µA
Fig. 7. Transmission of the system as a function of the Pockels cell voltage.
Teaching optics 169
a photodetector, an amplifier and an ammeter. During experiment the dependence ofthe intensity of light I on the voltage V applied to Pockels cell (Fig. 7) and Kerr cell
(Fig. 9) was measured. Using the appropriate formula we∆γ 2I Imin–
Imax Imin–------------------------------asin=
Fig. 8. Phase shift dependence on thevoltage for Pockels cell.
Fig. 9. Transmission of the system asa function of the Kerr cell voltage.
Fig. 10. Phase shift dependence on thevoltage for Kerr cell
170 A. CIŻMAN, R. POPRAWSKI
can obtain a phase shift as a function of the voltage for the Pockels cell (Fig. 8) andKerr cell (Fig. 10).
Electrooptical measurements were carried out in the voltage ranging from –1000to +1000 V. Results of linear and quadratic optical measurements are shown in Figs. 7and 8 for the Pockels and Fig. 9 and 10 the Kerr effects, respectively. As could beobserved, the data obtained clearly confirm the dependence of birefringence on theintensity of electric fields to be linear for Pockels effect (Fig. 8) and quadratic for Kerreffect (Fig.10). It is worth noting that all of the results presented were taken fromstudents reports.
The set-up presented is suitable for students laboratory, but it can be used forresearch purposes as well.
Reference
[1] SONIN A.S. VASILEVSKAYA A.S., Electrooptical Crystals, Atomizdat Publ., Moscow 1971 (in Russian).[2] LOMOVA L.G., SONIN A.S., REGULSKAYA T.A., Spontaneous electrooptic effect in the TGS crystals,
Kristallografiya 13(1), 1968, pp. 90–4 (in Russian).[3] RATAJCZYK F., Optyka ośrodków anizotropowych, Państwowe Wydawnictwo Naukowe PWN,
Warszawa 1994 (in Polish).[4] STRUKOV B.A., LEVANYUK A.P., Ferroelectric Phenomena in Crystals: Physical Foundations,
Springer, Berlin 1998.[5] ANDRIYEVSKY B.V., MYSHCHYSHYN O.YA., ROMANYUK M.O., Anisotropy of critical indices of
ferroelectric phase transition in TGS crystals by the optical interference investigation, CondensedMatter Physics 2(4), 1999, pp. 693–702.
[6] LINES M.E., GLASS A.M., Principles and Application of Ferroelectrics and Related Materials,Clarendon Press, Oxford 1977.
Received October 1, 2004
Optica Applicata, Vol. XXXV, No. 1, 2005
Accuracy improvement of bulk optical polarization interferometric sensors
PAWEŁ WIERZBA, BOGDAN B. KOSMOWSKI
Gdańsk University of Technology, Faculty of Electronics, Telecommunication and Informatics, Department of Optoelectronics, Narutowicza 11, PL-80952 Gdańsk, Poland
Interferometric sensors using bulk optical components exhibit very high measurement resolution.In order to attain high accuracy, these sensors are often implemented as polarizationinterferometers, in which stable and well-defined states of polarization are maintained. Unwantedphenomena degrading accuracy of this class of sensors are discussed in the paper. Signal processingtechnique which improves accuracy of polarization interferometric sensors is presented. Itsimplementation using analogue circuits is discussed and a method of improving its performanceis devised.
Interferometric measurement methods have been an indispensable measuring tool usedin an extensive range of applications (e.g., [1–4]). This is due to their advantages, suchas very high resolution and accuracy, ability to measure a broad range of quantitiesand relative ease of use. Setups performing interferometric measurements can bedivided, according to the type of components used in their sensing part, into twogroups: i) sensors with bulk optical components, ii) optical fibre sensors. Both groupsof sensors share the same operating principles and basic analytical description.However, due to the difference in the properties of corresponding components used insetups belonging to each group (e.g., bulk beamsplitters vs. 2×2 fibre couplers) and indescription of light propagation in optical fibres and in free space, more detaileddescriptions of these two groups differ considerably. Consequently, it is difficult toprovide a concise unified description of both groups of sensors. Since the descriptionof bulk optical sensors is more straightforward and is also the basis for description ofoptical fibre sensors, the following discussion covers interferometric bulk opticalsensors. For comprehensive discussion of all aspects of implementation and operationof interferometric fibre optic sensors, refer to, e.g., [4–6].
Often quoted advantages of interferometric sensors which use bulk opticalcomponents include: contactless operation, moderate requirements on the surfaces ofinvestigated objects (no special pre-treatment is required) and very high measurement
172 P. WIERZBA, B.B. KOSMOWSKI
resolution, which for displacement measurement is often better than 0.1 nm [7].Accuracy of interferometric sensors is usually much lower than their resolution – about1/100 of the operating wavelength λ. This is caused mainly by uncontrolled changesin the state of polarization of the source and drifts of polarization properties of opticalcomponents used in the interferometer.
Implementation of interferometric sensors using polarization interferometers, inwhich interfering beams have stable state of polarization, not only results in improvedaccuracy, but makes it possible to devise new configurations of increased sensitivity(cf. [8] and Fig. 7b) or improved tolerance to misalignment. In order to maintain stableand well-defined states of polarization, polarization interferometric sensors usepolarizing beamsplitters or birefringent prisms, optical components which do notmodify the states of polarization of interfering beams and the source which has stablepower, wavelength and state of polarization.
Polarization interferometric sensors can perform absolute [9] and relative [10]distance measurement, birefringence measurement [11], surface roughness measurement[12] (i.e., optical profilometry) and wavelength measurement [13]. Moreover, they canbe incorporated into other precise measuring instruments, such as atomic forcemicroscope (AFM) [14]. Finally, polarization interferometric sensors are often usedfor indirect measurement of several physical quantities (e.g., pressure, temperature,refractive index) whose changes can be converted into changes of optical path length.
An example of polarization interferometric sensor is presented in Fig. 1. Light froma single frequency laser, linearly polarized at 45° to the plane of the figure, is split bythe polarizing beamsplitter PBS into the measurement beam, polarized in the plane ofthe figure, and the reference beam, polarized perpendicularly to that plane.Subsequently, the beams are reflected by retroreflectors and recombined in PBS. Phasedifference between the beams is proportional to the optical path length differencebetween measurement and reference arms. On entering the detection setup each beam
Fig. 1. Displacement sensor using polarization interferometer and quadrature detection.
∆l Measured displacement Non-polarizing
beamsplitter BS
Laser
Analyser
λ/4
Detection electronics
U2 = RI0sin(k∆l)
Detection setup
D2
U1 = RI0cos(k∆l)
A1
D1
Q1
A2
Polarizing beamsplitter PBS
Reference retroreflector
Measurement retroreflector
Detection electronics
Teaching optics 173
is divided by the non-polarizing beamsplitter BS into two channels. In the first channelthe beams are brought to interference on detector D1 by polarizer A1 whose axis isaligned at 45° to the plane of the figure. In the second channel, additional phasedifference of 90° is introduced between the reference and measurement beams usingquarter-wave plate Q1 placed in front of polarizer A2 bringing the beams to interferenceon detector D2.
Intensity of light reaching detectors D1 and D2 can be expressed asI1 = I0 /2[1 + cos(k∆l)] and I2 = I0 /2[1 + sin(k∆l)], respectively, where I0 – maximumintensity on a detector, k – wavenumber (k = 2π/λ), ∆l – measured displacement. Bymeans of simple operations on analog signals from the detectors, two electrical signals,U1 and U2, are obtained which are functions of ∆l:
(1)
where R is sensitivity of the detection setup. Using U1 and U2 and knowingwavenumber k, value of ∆l can be calculated. As signals U1 and U2 are shifted by 90°with respect to each other, the direction of ∆l changes can always be determined.
In many particular applications custom polarization interferometric sensors mustbe developed which perform these measurements with high resolution and accuracy.In order to accomplish this task, it is vital to have a good understanding of factorsaffecting accuracy and stability of these sensors, as discussed in the following section.The possibility of appling correction methods that greatly reduce nonlinearity ofpolarization interferometric sensors, also discussed there is very beneficial, too. Thesemethods can be used with most polarization interferometric sensors, athough they wereoriginally developed for displacement measurement metrology. Because a largeportion of these sensors is implemented as homodyne interferometers, the followingdiscussion is restricted to that class of interferometers.
2. Nonlinearity sources in polarization interferometric sensors
The accuracy of homodyne polarization interferometric sensors is decreased as a resultof several parasitic phenomena, of which the most important are: i) polarizationcross-talk (Fig. 2a), ii) parasitic birefringence of components (Fig. 2b and c), iii) finiteextinction ratio of components (Fig. 2d) and iv) reflections at glass–air interfaces(Fig. 2e).
Polarization cross-talk (Fig. 2a) occurs when part of linearly polarized light beam(Ex or Ey) is coupled to the orthogonal beam (Exo or Eyo, respectively) becoming aparasitic component (Epx or Epy, respectively) of this beam. Present in waveplates,polarizing beamsplitters and polarizing prisms (e.g., Wollaston or Nomarski prisms),this phenomenon is caused by misalignment between the direction of the optical axisin these components and polarization plane of beams Ex or Ey. When Ex or Eyare reference and measurement beams of a polarization interferometer, polarization
U1 RI0 k∆l( ),cos=
U2 RI0 k∆l( )sin=
174 P. WIERZBA, B.B. KOSMOWSKI
cross-talk introduces a phase shift between them, which is an important source of errorsin polarization interferometric sensors.
Parasitic birefringence exists in optical elements, such as non-polarizingbeamsplitters, waveplates, retroreflectors or Dove prisms, affecting the state ofpolarization of light propagating through them. When circular birefringence (i.e.,optical activity) is present in an element, the plane of polarization of light incident onit is rotated through an angle α, as shown in Fig. 2b. When linear birefringence ispresent in an element and its optical axis is parallel to one of orthogonal componentsof the light beam, a phase shift is introduced between them, as shown in Fig. 2c.Otherwise, polarization cross-talk occurs, as described above.
Polarizing beamsplitters and polarizing prisms, such as Wollaston or Nomarskiprisms, should divide a light beam incident on them into two beams linearly polarizedin orthogonal planes. Therefore, the beam incident on the polarizing beamsplitter PBSshown in Fig. 2d should propagate across it. In fact, a small amount of power is coupledout to the other direction (downwards in Fig. 2d). Extinction ratio e can be defined asthe ratio of power transmitted in the right direction Pout to power Pparasitic coupled intothe other direction, i.e., e = Pout /Pparasitic. Ideal beamsplitters and polarizing prismshave infinite extinction ratio, due to Pparasitic = 0. In real components this parameterranges from 104 to 106.
Another source of inaccuracy of polarization interferometric sensors is Fresnelreflection at the surfaces of optical components (e.g., glass–air, glass–vacuum orplastic–air interfaces). When a beam having amplitude Eo crosses such an interface(either entering or leaving a component), part of its power is reflected back. Reflectedbeam, whose amplitude ER depends on refractive index difference between air and thematerial of the optical component, can combine with another beam EC propagatingin the same direction, as shown schematically in Fig. 2e. Assuming that the beams are
Pout
Pparasitic
PBS
Pin
∆ϕ = 0
∆ϕ ≠ 0
Optical axis
Ex
Exo
α
Epx
Eyo
Optical axis Epy
Ey Ex
Exo
Fig. 2. Phenomena affecting accuracy of homodyne polarization interferometers polarization cross-talk(a), parasitic birefringence (b and c), finite extinction ratio (d), reflections at glass-air interface (e).
a b
c d e
ER Es
Eo
EC
Teaching optics 175
coherent and have the same state of polarization, amplitude ES of their superpositioncan be expressed as:
(2)
where ϕ – phase difference between beams ER and EC. This phenomenon can be seenas an unwanted phase modification imparted on beam EC by reflected beam ER. Whenphase difference ϕ is a function of the measured quantity, this is an important sourceof errors in optical setups in which beams Eo and Ec overlap, e.g., in Michelsoninterferometers.
Apart from measurement errors caused by the phenomena present in opticalcomponents and described above, additional errors are introduced by the electronicpart of the detection setup. Most often these errors take the form of voltage offsetspresent at the outputs of the setup or gain mismatch between the two outputs.
All these phenomena manifest themselves as periodical nonlinearity of polarizationinterferometric sensors. According to description introduced in [15], distorted outputsignals U1d and U2d can be expressed in terms of undistorted output signals U1 and U2,given by Eq. (1), as:
(3)
where r – gain ratio of the two channels (r = G1/G2), p – offset in the first channel,q – offset in the second channel, α – quadrature error.
Lissajous figure of U1d and U2d is an ellipse, randomly distorted by the presenceof noise in U1d and U2d, as shown in Fig. 3. In an ideal system, gains of both channelsare equal (i.e., their ratio r = 1), no offsets are present (p = q = 0) and phase differencebetween the channels is 90° (i.e., α = 0), therefore Eq. (3) reduces to Eq. (1) andLissajous figure of U1d and U2d becomes a circle.
Performing the least-square fitting of Eq. (3) to measurement data acquired for therange of k∆l greater than 2π, values of r, p, q and α can be found. Subsequently,undistorted output signals U1 and U2 can be calculated from Eq. (3), rewritten as:
(4)
Correction procedure described above is a very time-consuming process, in whichthe most computation-intensive task is calculation of r, p, q and α by the least-squarefitting. Since Eq. (3) must remain fitted to data being processed, all parameters haveto be recalculated each time any one of them changes. In order to avoid frequent
ES EC ER jϕ( )exp+=
U1d U1 p,+=
U2d1r
----- U2 αcos U1 αsin–( ) q+=
U1 U1d p,–=
U21
αcos--------------- U1d p–( ) αsin r U2d q–( )+ .=
176 P. WIERZBA, B.B. KOSMOWSKI
recalculations, the sensor should have low drift of polarization properties of itselements, small dependence of these properties on the instantaneous value of measuredquantiy as well as low noise and drifts in detection electronics.
Correction of measurement data can be performed either off-line, when data areacquired and stored first and the correction process is performed afterwards, or on-line,i.e., during data acquisition. Off-line correction is performed in digital domain, usinga PC or dedicated DSP hardware, often after passing the data through a digital noisefilter. On-line correction, which is much more difficult to perform due to timeconstraints, can be carried out either in digital domain, using a high-throughput DSPsystem [16] or in a mixed-signal mode, by performing the fitting in digital domain,converting values of calculated parameters into voltages using D/A converters, andfeeding these voltages to an analog circuit performing the correction in analog domain[9]. The latter method can reduce the amount of calculations, and when frequentrecalculations of ellipse parameters are not required, it may obviate the need to usea high-throughput DSP system.
An example of circuit for correcting output signals U1d and U2d from a polarizationinterferometer is presented in Fig. 4. Its transfer function can be written as:
(5)
where K – scale factor of the multipliers, UGx and UGy – voltages controlling gain ofthe first and second channel, respectively, UOx and UOy – voltages controlling offsetof the first and second channel and Uφ – voltage controlling phase difference betweenthe two channels. Comparing Eq. (5) with Eq. (3), values of Uφ, UGx , UGy , UOx andUOy can be expressed in terms of ellipse parameters r, p, q and α.
U1
U1dUGx
K----------------------- UOx,+=
U2
UGy
K------------
U1dUφK
-------------------- kU1d U2d+– UOy+=
Fig. 3. Lissajous figure of ideal (—) and real (+)phase quadrature signals from a polarizationinterferometer.
1 U1d [V]
U2d [V]
1
Teaching optics 177
Nonlinearity reduction up to 18 times was demonstrated using the circuit fromFig. 4 by authors of [9]. Further reduction seems to be possible by decreasingnonlinearity of the correction circuit by addressing its most important source –multipliers. Since the range of Uφ, UGx and UGy is limited (i.e., ∆U/U ≤ 1) for allpractical circuits, it is possible to use an elegant solution due to PEASE [17] in whichmost of the gain is obtained with a linear amplifier, and a multiplier is used only toadjust it over required range. In the following section we present our solutionemploying this technique.
3. Improved analog correction circuit
First, let us consider the first channel of correction circuit presented in Fig. 4. We cansafely assume that required gain range in this circuit is 0.8 to 1.2 (i.e., ±20% changearound nominal value 1.0), that U1d varies from –5 to +5 V and that maximum voltageat any multiplier input can range from –10 to +10 V, while its output voltage can befrom –12 to +12 V.
We can replace the first channel of this circuit with the circuit presented in Fig. 5,where γ is the division ratio of the resistive divider. The transfer function of the lattercircuit can be expressed as:
(6)
Since the amplitude of input signal U1d is 5 V and maximum voltage allowed onthe multiplier is 10 V, G2 can be set to 2V/V. As in most cases |UGx /K| ≤ 1,division ratio γ needed to obtain ±20% gain change is 10. It should be noted that anynonlinearity observed at the output of the multiplier is also divided by 10. Therefore,
U1 1 G2
UGx
K------------ 1
γ-----+ U1d UOx.+=
Fig. 4. Example implementation of nonlinearity correcting circuit.
U2d
UOx
Uφ
U1d
UGx
UOy UGy
U2
U1
k = –0.5
First channel
Second channel
178 P. WIERZBA, B.B. KOSMOWSKI
a nonlinearity reduction by one order of magnitude, compared to the first channel ofthe circuit from Fig. 4, can be attained.
Second, let us start modification of the second channel of the circuit from Fig. 4by rewriting its transfer function, which can be expressed using Eq. (4) as:
(7)
The transfer function of the circuit from Fig. 6 is:
(8)
Comparing Eq. (7) with Eq. (8), we can write:
(9)
Let us assume that U1d and U2d vary from –5 to +5 V, required gain ratio r canchange from 0.8 to 1.2, |α | ≤ 15°, and that maximum voltage at any multiplier inputcan be between –10 and +10 V while its output voltage can range from –12 to +12 V.
Gains G1 and G2 can be set to 2V/V since the amplitude of input signals U1d andU2d is 5 V and maximum voltage allowed on the multiplier is 10 V. For the given range
U2 α U1dtanr
αcos--------------- U2d p αtan
rαcos
--------------- q+–+=
αtan U1dr
αcos--------------- U2d UOy.+ +=
U2 G1
UφK
--------- 1γ1
------- U1d 1 G2
UGy
K------------ 1
γ2
-------+ U2d UOy.+ +=
αtan G1
UφK
--------- 1γ1
------- ,=
rαcos
---------------- 1 G2
UGy
K------------ 1
γ2
------- ,+=
UOy p αtanr
αcos--------------- q+ .–=
Fig. 5. First channel of correcting circuit aftermodification.
+
UOx
U1d
UGx
U1
G2 γ
Teaching optics 179
of α we have |tanα | ≤ 0.268 and 0.800 ≤ (r/cosα) ≤ 1.242. Using these values inEq. (9), together with the fact that |UGy/K | ≤ 1 and |Uφ /K | ≤ 1, we arrive at:
(10)
from which division ratios γ1 and γ2 can be calculated:
γ1 = 7.463, γ2 = 8.764. (11)
It should be noted again that any nonlinearity present at the outputs of the multipliersis divided by γ1 and γ2. Therefore, in the worst case, nonlinearity reduction in thesecond channel is over seven times (γ1 = 7.46).
Finally, let us compare the complexity of the circuit from Fig. 4 with that of ourmodified circuit from Figs. 5 and 6. Assuming that summation points are implementedusing an operational amplifier, which is usually the case, the latter circuit containsthe same number of multipliers and only one operational amplifier more than theformer circuit. Therefore, the cost of the latter circuit is only marginally higher thanthat of the original one.
In conclusion, the circuit presented in Figs. 5 and 6 offers nonlinearity improvementwhich is over seven times compared to the original circuit from Fig. 4. Moreover,this considerable improvement is obtained with only small additional complication ofthe circuit.
The method discussed can be used in sensors employing other, more sophisticateddetection setups, such as the four-detector setup presented in Fig. 7 or the setup shownin Fig. 8 [18].
Operation of the balanced-quadrature detection setup from Fig. 7 is similar to thatof detection setup presented in Fig. 1. On entering the setup, planes of polarization of
G11γ1
------- 0.268,≤ G21γ2
------- 0.242,≤
Fig. 6. Second channel of correcting circuitafter modification.
Uφ
U2d
UGy
γ2 G2
+
UOy
U1d
U2
γ1
G1
180 P. WIERZBA, B.B. KOSMOWSKI
reference and measurement beams are rotated by 45° using the halfwave plate H1.Then, each beam is divided by the non-polarizing beamsplitter BS into two channels.In the first channel, the beams are brought to interference on detectors D1 and D2 bythe polarizing beamsplitter PBS1. In the second channel, additional phase differenceof 90° is introduced between the reference and measurement beams using a quarter-wave plate Q1 placed in front of the polarizing beamsplitter PBS2. This beamsplitterbrings the beams to interference on detectors D3 and D4. Signals from detectorsD1–D4 after I/U converters can be expressed as
(12)
Therefore, output signals of instrumentation amplifiers, giving directly the desiredinformation can be expressed as:
(13)
The detection setup presented in Fig. 8a [18] uses a Nomarski prism, an analyserand a set of three photodiodes D1, D2 and D3 shown in Fig. 8b. Reference andmeasurement beams entering the setup are expandend by the beam expander BEand are incident on the Nomarski prism NP, whose optical axis is parallel to the
Ud112
------ RI0 1 k∆l( )cos+ ,= Ud212
------ RI0 1 k∆l( )cos– ,=
Ud312
------ RI0 1 k∆l( )sin+ ,= Ud412
------ RI0 1 k∆l( )sin– ,=
U1 Ud1 Ud2– RI0 k∆l( ),cos= =
U2 Ud3 Ud4– RI0 k∆l( )sin .= =
Fig. 7. Balanced quadrature detection setup.
λ/2
D4
D3
Beamsplitter BS
U1 = RI0cos(k∆l)
I/U converter Instrumentation
amplifier IA1
Polarizing beamsplitter
PBS2
λ/4
H1
D2
I/U converter
I/U converter
I/U converter
U2 = RI0sin(k∆l)
Polarizing beamsplitter
PBS1
Input from interferometer
D1
Q1
Teaching optics 181
polarization plane of one of the beams. This prism changes propagation directions ofreference and measurement beams, so that a small angle (0.5–5°) α is introducedbetween the beams. The analyser A brings the two beams into interference, creatinga fringe pattern on the photodiodes, as shown in Fig. 8c. Positions of dark and brightfringes depend on phase difference between the two beams.
Assuming that maximum intensity of the fringe pattern Imax(x, y) is constant acrossthe surfaces of photodiodes (D1, D2, D3), it can be shown that their currents i1, i2 andi3 can be written as:
(14)
From Eqs. (14) it can be seen that in this case it is easier to obtain output signalsproportional to sin(k∆l + π/4) and cos(k∆l + π/4), rather than to sin(k∆l) and cos(k∆l).These signals are obtained by:
iref i1 i2 i3+ + const,= =
i1
iref
4--------- 1
2 2 Vπ
------------------ k∆lπ4-----+
cos– ,=
i2
iref
2--------- 1
2 2 Vπ
------------------ k∆lπ4-----+
sin+ .=
Fig. 8. Detection setup using Nomarski prism: view of the setup (a), dimensions of detectors (b), fringepattern illuminating detectors (c); h – fringe spacing.
a
b c
Detectors
Fringe pattern on detectors
Analyser A
Beams from interferometer
Nomarski prism NP
Beam expeder BE
α
h
D3 D1 D2
2 h h
4 h 4
h
h
182 P. WIERZBA, B.B. KOSMOWSKI
(15)
where V – visibility of fringes. An important advantage of this setup is its simplicity and compact layout.
Moreover, the setup does not use non-polarizing beamsplitters which can be animportant source of polarization cross-talk and birefringence in detection setups.
4. Polarization interferometric sensors
The technique for nonlinearity reduction described above is used in severalapplications, such as surface roughness measurement, wavelength measurement,distance-measurement metrology as well as in AFM. It can be extended to otherpolarization interferometric sensors, such as hydrostatic pressure sensors (Fig. 9a) orrelative displacement measurement sensors (Fig. 9b) [8]. Moreover, it can also beapplied to waveguide polarimetric sensors for complex refractive index measurement,biosensing and immunosensing (Fig. 9c) [19].
Operation of hydrostatic pressure sensor presented in Fig. 9a is similar to that ofthe Michelson interferometer. Light from the source is divided by the Wollaston prisminto the measurement beam, polarized in the plane of the figure and the reference beampolarized perpendicularly to the plane of the figure. Reflected from the pressure-sensing membrane, the beams return to the Wollaston prism, where they are combined.Phase difference between the orthogonal components of the resulting beam isproportional to the optical path length difference of reference and measurement arms.An important advantage of this sensor is that it is sensitive only to pressure-inducedmembrane deformation. Translation of the membrane, e.g., induced by change oftemperature, does not influence the measurement.
Relative displacement sensor, presented in Fig. 9b, which can also be used tomeasure hydrostatic pressure, is a four-pass sensor, i.e., light travels four times in itsarms. Input beam is divided by the Wollaston prism into the measurement beam,polarized in the plane of the figure and the reference beam polarized perpendicularlyto the plane of the figure. Reflected from the measured object, the beams pass througha quarter-wave plate, are reflected from a mirror placed behind it and pass againthrough the plate. The axes of the plate are aligned in such a way that the beams changetheir polarizations into orthogonal ones, i.e., the measurement beam becomes polarizedperpendicularly to the plane of the figure and the reference beam becomespolarized in that plane. The beams fall on the measured object, then are reflected fromit, and return to the Wollaston prism, where they are combined. An importantadvantage of this sensor is that it is sensitive only to relative displacement ∆x, rather
4i1 iref– iref2 2 V
π------------------ k∆l
π4-----+
,cos=
iref 4i2– iref2 2 V
π------------------ k∆l
π4-----+
sin=
Teaching optics 183
than to translation of measured object. Moreover, this sensor is not sensitive to smalltilts of the surface of measured objects.
Complex refractive index sensor, presented in Fig. 9c, can also be used as abiosensor, in which an antigen-antibody interaction affects refractive index. Polarizedlight from the source is coupled into the sensing waveguide, where two polarizationmodes xHE11 and yHE11 are excited with equal amplitudes. Propagation constants ofthese modes are functions of refractive index of the sample (or antigen layer in
LED
λ1 Sample
Waveguide
Substrate
Microscope objective Analyser Collimating
lens
Protective coating
λ/4
Detection electronics
D1 U1 = RI0[1 + sin(k∆l)]
Detection electronics
D2
U2 = RI0[1 + cos(k∆l)]
Beamsplitter
Fig. 9. Polarization interferometric sensors in which nonlinearity correction can be used: hydrostaticpressure sensor (a), relative displacement sensor (b), polarimetric waveguide sensor (c).
a
b
c
Hydrostatic pressure
Lens Wollaston prism
BS In
Out
Membrane
∆x
∆x
Wollaston prism
Mirror λ/4
Lens
184 P. WIERZBA, B.B. KOSMOWSKI
biosensors). Therefore, the value of refractive index can be obtained based onmeasured phase difference between polarization modes leaving the waveguide. Thismeasurement is accomplished by the detection setup, in which the polarization modesare brought to interference by analysers.
5. Implementation issues
While polarization interferometric sensors combined with nonlinearity reductiontechnique described above are an attractive highly-accurate measuring tool, theyexhibit certain disadvantages hindering their implementation. The most importantproblems are: excessive size of some optical setups, distortion of interferingwavefronts, difficult setting up due to reflections from the surfaces of lenses, difficultyto couple the sensors to a single-mode polarization-maintaining fiber as well as changeof the wavefront shape as a function of displacement.
Excessive size is often encountered in sensors using a Wollaston or Nomarski prismin which spacing between sensing and reference beam is above 1 mm. The primarycause of this problem is small (< 5–10°) splitting angle of the prism. A sensor similarto that from Fig. 9b, in which the transverse spacing of the beams is 2 mm, has overalllength of optical setup over 60 mm [20]. Using Wollaston/Nomarski prisms withhigher splitting angle (up to 30°) often requires more sophisticated focusing optics,such as multi-element assemblies containing aspheric lenses. Multiple lens surfacesgive rise to a number of reflections on glass-air interfaces, making these setups difficultto align. Moreover, in such setups, wavefronts of measurement and reference beamare often distorted in a different way, which decreases interference contrast and makesthese interferometers more sensitive to vibration.
Operation of polarization interferometric sensors may also be affected by changeof the wavefront shape, caused by the change in the shape of measured surface. Asillustrated in Fig. 9a, the measured hydrostatic pressure deforms the membrane in sucha way that its surface becomes convex, modifying focal length of the system. As aresult, the reflected measurement beam is no longer collimated (i.e., it does not havea flat wavefront), which may affect the operation of the detection setup.
6. Conclusions
The accuracy of polarization interferometric sensors, which is often degraded byunwanted optical and electrical phenomena, can be improved by using a correctiontechnique developed for metrology, based on adjustment of gain and removal of offsetsin detection setup. Employing this technique it is possible to provide a real-timecorrection of measured data using analog or digital signal processing. Analog correctioncircuits can achieve accuracy much higher than demonstrated to date, by complementingmultiplier-based variable gain stages with linear gain stages, as discussed in Sec. 3.Although some polarization interferometers are difficult to implement because of theirsize or complicated alignment, this class of sensors shows considerable applicationpotential. Sensors employing optical components made from new crystalline and liquidcrystal birefringent materials can potentially be made more compact and economical
Teaching optics 185
at the same time offering high measurement accuracy. However, futher research isneeded on implementation of polarization interferometric sensors.
References
[1] HARIHARAN P., Interferometry with lasers [In] Progress in Optics, [Ed.] E. Wolf, Vol. 24, Elsevier1987, p. 104.
[2] PLUTA M., Advanced Light Microscopy, Warszawa: PWN; Amsterdam, New York: Elsevier 1988.[3] BOBROFF N., Recent advances in displacement measuring interferometry, Measurement Science and
Technology 4(9), 1993, pp. 907–26.[4] JAROSZEWICZ L.R., The role of polarization and coherence in fibre-optic interferometry, MUT
Warszawa 1995 (in Polish).[5] WOLIŃSKI T.R., Polarimetric optical fibers and sensors [In] Progress in Optics, [Ed.] E. Wolf,
Vol. 15, Elsevier 2000, p. 1.[6] WOLIŃSKI T.R., Polarization phenomena in optical systems [In] Encyclopedia of Optical Engineering
[Eds.] R. Barry Johnson, R.G. Diggers, Marcel Dekker Inc., New York 2003. [7] LAWALL J., KESSLER E., Michelson interferometry with 10 pm accuracy, Review of Scientific
Instruments 71(7), 2000, pp. 2669–76. [8] DYSON J., Optics in a hostile environment, Applied Optics 7, 1968, p. 569.[9] EOM T., KIM J., JEONG K., The dynamic compensation of nonlinearity in a homodyne laser
interferometer, Measurement Science and Technology 12(10), 2001, pp. 1734–8.[10] LIU C., CLEGG W., LIU B., Ultra low head-disk spacing measurement using dual-beam polarization
interferometry, Optics and Laser Technology 32(4), 2000, pp. 287–91. [11] INDEBETOUW G., KLYSUBUN P., Measurement of induced birefringence in film samples using a
balanced polarization ring interferometer, Optics Communications 151(4-6), 1998, pp. 203–6.[12] GLEYZES P., LORIETTE V., SANT-JALMES H., BOCCARA A.C., Roughness measurements in the
picometric range using a polarization interferometer and a multichannel lock-in detection technique,International Journal of Machine Tools and Manufacture 38(5-6), 1998, pp. 715–7.
[13] JUNCAR P., PINARD J., Instrument to measure wave numbers of CW and pulsed laser lines:the sigmameter, Review of Scientific Instruments 53(7), 1982, pp. 939–48.
[14] SCHONENBERGER C., ALVARADO S.F., A differential interferometer for force microscopy, Review ofScientific Instruments 60(10), 1989, pp. 3131–4.
[15] HEYDEMANN P.L.M., Determination and correction of quadrature fringe measurement errors ininterferometers, Applied Optics 20(19), 1981, pp. 3382–4.
[16] DEI G., POHLENZ F., DANZEBRINK H.-U., HASCHE K., WILKENING G., Improving the performance ofinterferometers in metrological scanning probe microscopes, Measurement Science and Technology15(2), 2004, pp. 444–50.
[17] PEASE B., What’s all this multiplication stuff, anyhow?, Electronic Design, August 1991, p. 121.[18] WIERZBA P., Modelling of polarimetric sensors for applications in road traffic monitoring systems,
Ph.D. Thesis, Gdansk University of Technology 2000, p. 78 (in Polish).[19] GNYBA M., WIERZBA P., PLUCINSKI J., Application of sol-gel-developed integrated optic devices to
biochemical fiber optic sensors based on polarimetric interferometry, Proceedings of SPIE 5505,2004, pp. 78–83.
[20] WIERZBA P., Optical Hydrophones, VTT Technical Research Centre of Finland Internal ResearchReport, Oulu, Finland 2001.
Received August 18, 2004in revised form January 12, 2005
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