1. INTRODUCTIONSurface plasmon resonance (SPR) is a well-known tech-nique for characterization of linear optical parameterssuch as the thickness and the dielectric constants of films.It is based on the excitation of surface electromagneticwaves that propagate along the interface between twosemi-infinite media, one of which must have a negativedielectric constant. Because the dielectric constant of ametal is always negative below the plasma frequency, ametal–dielectric interface is typically used. Thesewaves, the existence of which was predicted by Sommer-feld as early as 1909,1 are often known as surface plas-mon waves (SPWs). The guided-mode nature of SPWsproduces strong enhancement of the intensity of light atthe interface, with an increase in intensity of as much as2 orders of magnitude with respect to the incoming beam.This narrow confinement makes SPW parameters, suchas amplitude, velocity, and damping, highly sensitivefunctions of the properties of the boundary and, conse-quently, makes SPR a powerful tool for measuring theintensity-dependent dielectric constants of thin and ultra-thin (nanometer-scale) layers of metal or dielectric mate-rials.
A major drawback of SPR is that thermal effects cannotbe excluded a priori. In addition, two more intrinsic limi-tations exist, namely, the compulsory use of TM propaga-tion and the need for a metal layer in contact with the di-electric film. However, generally speaking, thedrawbacks of SPR mentioned above are largely counter-balanced by several advantages, such as the ability ofSPR to characterize extremely thin dielectric layers witha high degree of insensitivity to their defects. Indeed,SPR provides good accuracy and resolution also forsamples that have poor optical quality (only a very smallarea is monitored, with a diameter usually of ,0.1 mm) 2
and large linear absorption.3 Moreover, the local field ef-fect associated with SPW propagation can be convenientlyexploited for several interesting applications, (i.e., devel-
opment of single-molecule microscopy,4,5 interferometricimaging,6 improvement of emission properties, andthreshold reduction of light-emitting diodes)7 that wereextensively reported in the literature and for the en-hancement of optical nonlinearities. In particular, in thefield of nonlinear spectroscopy the first experiments inthe use of SPR for the enhancement of optical mixing andcoherent anti-Stokes Raman spectroscopy date back tothe 1970s,8,9 whereas, more recently, surface-enhancedRaman scattering enhancement factors related to SPWexcitation in periodic silver structures have beencalculated.10 Moreover, SPR techniques have beenadopted for measuring the electro-optic coefficients ofnonlinear optical polymers,11 for the demonstration of all-optical switches based on photoinduced changes in theimaginary part of the complex refractive index of a photo-chromic dye,12 and for measuring the Kerr nonlinearity ofa polydiacetylene waveguide spun upon a metal-coateddiffraction grating.13
We investigated SPR as a method for the optical char-acterization of thin dielectric layers that exhibitintensity-dependent complex refractive indices. In thispaper we summarize the results of our study. The theo-retical approach adopted to model SPR measurementsand some characterizations of the sensitivity and accu-racy of the method under various experimental conditionsare described in Section 2. In Section 3 we describe theexperimental procedure and the results that we obtainedby using spun films of a conjugated polymer, the sol-uble polydiacety-lene 1,6-bis-(3,6-dihexadecyl-N-carba-zolyl)-2,4-hexadiyne (polyDCHD-HS),14 deposited uponsilver-coated glass plates.
2. THEORETICAL APPROACHA surface plasma wave guided at the boundary betweentwo semi-infinite media is characterized by a complexwave number:
2003 Optical Society of America
742 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Margheri et al.
Kp2 5 k0
2«r1«r3
«r1 1 «r3, Re~Kp! . 0, (1)
where «r1 and «r3 are the relative dielectric constants ofthe two media that form the interface and k0 is the wavenumber in vacuum. When the real part of the dielectricconstant of one of the two media (for example, medium 1)is negative, which means that medium 1 is generally ametal, guided propagation takes place, the magnetic fieldis parallel to the boundary, and, if uRe(«r1)u @ Im(«r1), theelectric field has a dominant component that is perpen-dicular to it. Because Re(«r1) and Re(«r3) have oppositesigns, Re(Kp) is always larger than the wave number ofthe freely propagating wave: This is why a SPW cannotbe excited directly. The most popular method of excitinga SPW at a metal–dielectric interface is known asKretschmann’s configuration15 and is analogous to theprism-coupling procedure widely used for opticalwaveguides.
With reference to Fig. 1, let us first consider a laserbeam that is totally reflected on the hypotenuse of a 45°–90°–45° glass prism that is in optical contact with a thinmetal layer. As a result of attenuated total reflection,the SPW at the metal–air interface is excited when thehorizontal component of the propagation constant of thetraveling field matches that of the SPW. One can detectthe SPW excitation by varying angle of incidence u andmonitoring the intensity of the reflected light: At the so-called plasma angle up , the conditions for SPR are estab-lished and the reflectivity exhibits a sharp dip (Fig. 1, in-set). Thickness d1 and complex dielectric constant «r1 ofthe metal film determine up and the shape of the reflec-tivity curve. In typical working conditions, up dependsessentially on Re(«r1), the reflectivity dip depends onmetal thickness d1 , and the FWHM of the curve dependsboth on the ratio Im(«r1)/Re(«r1) and on d1 . Usually, goodestimates of d1 and «r1 can be obtained from a best fit ofthe experimental reflectivity curve with its theoretical ex-pression.
Fig. 1. Kretschmann’s configuration.
A thick dielectric layer deposited upon a metal film canbe viewed as a semi-infinite medium that replaces air. Inthis case, Kretschmann’s configuration still works, pro-vided that the conditions for total internal reflection be-tween prism and dielectric are fulfilled. In contrast, thedeposition of a thin dielectric overlayer (thicknessd2 5 1 –200 nm and dielectric constant «r2) upon themetal can be treated as a perturbation of the couplingconditions at the metal–air interface. If the thin over-layer is purely transparent, its first-order effect is that ofan increasing plasma angle2; if there is also some absorp-tion, this angular shift is accompanied by a reduction inthe energy transfer into the SPW, which gives rise to anincrease in the FWHM of the reflectivity dip. Because ofthe sharp nature of the SPR, even small variations insuch film parameters as thickness and refractive indexcan be detected, once the dielectric constant and thethickness of the metal are known.16
In the linear case the SPR characterization of amultilayer stack does not require an evaluation of thelight-intensity distribution inside the structure but needsonly suitable relationships for the reflectance.17 Let usrefer to Fig. 2, which represents a stack of four media: aprism, a metal film, a dielectric film, and a substrate.The first and the last media are semi-infinite. In thiscase the overall amplitude reflectivity rstack can be calcu-lated from Fresnel’s formulas and is expressed by the re-lationship
rstack 5r0,1 1 r ~1 ! exp~2ik0d1b1!
1 1 r0,1r ~1 ! exp~2ik0d1b1!, (2)
where
r ~1 ! 5r1,2 1 r2,3 exp~2ik0d2b2!
1 1 r1,2r2,3 exp~2ik0d2b2!, (3)
Fig. 2. Sketch of the four-layer stack with incident, reflected,and transmitted electromagnetic waves.
Margheri et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 743
rj,i 5«rib j 2 «rjb i
«rib j 1 «rjb i, (4)
b i 5 S 1 2«r0
«risin2 q D 1/2
, Re~b i! > 0, (5)
where i 5 0, 1, 2, 3 and «ri are the relative dielectric con-stants of the various media. The recursive application ofEq. (2) allows us to find rstack for any number of linear lay-ers.
Let us now consider nonlinear media, that is, when «r2depends on local intensity I of the electromagnetic field:
«r2~I ! 5 «r2,0 1 D«r2~I !, (6)
where «r2,0 is a low-intensity dielectric constant andD«r2(I) is the complex shift of the dielectric constant thatis due to nonlinearity. Because of the surface characterof the SPW, the fields in the nonlinear medium are inho-mogeneous, and this is reflected by a spatial variation of«r2 with coordinate z. As a consequence, the proceduresummarized by Eqs. (2)–(5) cannot be adopted, becausethe dielectric constants of the stack can be known only af-ter evaluation of the fields. Thus, knowledge of the in-tensity distribution inside the nonlinear layer is neces-sary and must be obtained from Maxwell’s equations.For monochromatic fields the equations to be solved are
¹ ∧ E 5 ivm0H, (7)
¹ ∧ H 5 2iv«0«r2~z !E, (8)
with the boundary conditions of continuity of the tangen-tial components of E and H. In Eqs. (7) and (8), «0 andm0 are the dielectric constant and the magnetic perme-ability of vacuum, respectively, and v is the angular fre-quency of the radiation. We assume (a) the TM solutionsof Eqs. (7) and (8) are in the form
E 5 @Ex~z !i 1 Ez~z !k#exp@i~kx 2 vt !#, (9)
H 5 Hy~z !exp@i~kx 2 vt !# j, (10)
where k is the propagation constant; (b) «r2 depends onlyon Ez(z):
«r2~z ! 5 «r2,0 1 D«r2@Ez~z !#; (11)
(c) the nonlinear medium is isotropic with respect to itslinear dielectric constants; and (d) the nonlinearity issmall:
D«r2@Ez~z !# ! «r2,0 . (12)
Then Eqs. (7) and (8) may be rearranged to give only onenonlinear equation for Hy(z). This equation can besolved analytically in the presence of the coupling prism,only when the Kerr medium is medium 3 and issemi-infinite.18,19 These analytical solutions have beenused, for example, to treat the optical bistability in a two-medium stratification or in the attenuated total reflectionconfiguration involving a nonlinear semi-infinitesubstrate.19,20 In contrast, in our experimental configu-ration the nonlinear medium is the perturbing dielectric,which has a thickness much smaller than the wavelength(1–200 nm). In this case analytical solutions can befound only for bounded waves, that is, when the couplingprism is absent.21 We must adopt a different approach
for calculation of the fields and of the induced modifica-tions of the dielectric constants. We solved Maxwell’sequations by using the matrix method described byWalpita.22 This method has been extensively adopted toyield the dispersion relations of linear, graded-index,uniaxial waveguides. For nonlinear waveguides the di-electric constants are field dependent and the method stillworks, provided that it is combined with a suitable itera-tive procedure.
For the calculations we maintained assumptions (a)–(d) defined above. For polymeric films condition (c) isusually a first approximation. The optical anisotropy ofthe material could also admit of a TE excitation23,24: Inthis case, a much more complex procedure would be nec-essary. However, for the polyDCHD-HS films used in ourexperiments, condition (c) was actually satisfied, as con-firmed by previous tests.25 Assumption (b) is not strictlynecessary, but it permits a considerable reduction of thecomputational load, as discussed below.
With reference to the structure sketched in Fig. 2, weconsidered a p-polarized electromagnetic plane wave ofintensity I0 impinging upon the multilayer at a givenangle u, and we started the procedure by using the linearvalue of «r2 . According to Ref. 22, the solutions of Max-well’s equations for a stack of dielectrics can be repre-sented by a 2 3 2 matrix aij , which transforms the inputp-polarized magnetic field
by means of a linear modification of the two coefficientsA0 and B0 given by
S Aout
BoutD 5 Fa11 a12
a21 a22G S A0
B0D . (15)
In Eqs. (13) and (14), k 5 k0A«r0 sin(u), z2 is the posi-tive coordinate of the last interface, x is the longitudinaldirection of propagation of the wave, A0 is the amplitudeof the incident wave, and B0 is the amplitude of the re-flected wave. The coefficients pi , which in general arecomplex, are calculated as
pi 5 k0~2«ri 1 «r0 sin2 u!1/2, i 5 0, 1, 2, 3,
Re~ pi! > 0. (16)
The condition that no wave impinge from medium 3 re-quires that Bout 5 0. So the reflectivity of the stack atincidence angle u is given by
r 5 2a21
a225
B0
A0. (17)
When nonlinearity is absent, r 5 rstack , as defined by Eq.(11).
744 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Margheri et al.
Inasmuch as amplitude A0 of the impinging plane waveis a prescribed parameter, reflected amplitude B0 can beobtained from Eq. (17). The transformation of vector(A0 , B0) from the first interface through the subsequentinterfaces gives the field amplitude at the various posi-tions. For example, at the metal–dielectric interface theamplitudes of the progressive and the regressive waves,A2 and B2 , respectively, are obtained with the followingmatrix product:
S A2
B2D 5 F 1 1
2p2
«r2
p2
«r2
G 21
3 F exp~2p1d1! exp~ p1d1!
2p1 exp~2p1d1!
«r1
p1 exp~ p1d1!
«r1
G3 F 1 1
2p1
«r1
p1
«r1
G 21F 1 1
2p0
«r0
p0
«r0
G S A0
B0D . (18)
The expression for the magnetic-field amplitude in thenonlinear medium at coordinates x and z is readily foundfrom knowledge of A2 and B2 (Ref. 22):
After calculation of the electric field with Eq. (8), zeroth-order intensity I (0) is obtained by evaluation of the Poyn-ting vector in the nonlinear dielectric:
I ~0 !~z ! 5 1/2 Re~2EzHy* i 1 ExHy* k!, (20)
where * stands for complex conjugation. For surfaceplasmons, Eq. (20) may be simplified to considerablyshorten the computational load. In fact, in this case theinequality uEz(x, z)u @ uEx(x, z)u is valid,23 and the in-tensity can be approximated as
I ~0 !~z ! >1
2Re~2EzHy* ! 5
1
2Re
v«r2«0
kuEz~z !u2.
(21)
For simplicity, we also substituted I (0) for I (0)(z) over thethickness of the nonlinear layer. I (0) is the arithmetic av-erage of the maximum and minimum intensities at theboundaries of the nonlinear dielectric:
I ~0 ! 5I ~0 !~z1! 1 I ~0 !~z2!
2. (22)
Indeed, for small nonlinearities this procedure is ex-pected to have a negligible effect on the results, as wasconfirmed in Ref. 23. Substituting I (0) into Eq. (6) yieldsa first-order value «r2
(1) of «r2 . The procedure is then re-iterated by use of «r2
(1) to get the first-order distributionsof the fields, first-order intensity I (1), and the second-order value of «r2 , namely, «r2
(2). The whole process isrepeated until a steady-state solution is achieved, that is,until equilibrium values of «r2(I) and I are obtained.Amplitude reflectivity r is evaluated by use of these equi-librium values. Then the squared modulus of r, namely,
R, gives the quantity of experimental interest. In prac-tice, in our tests the convergence of the solutions was ob-tained after four or five iterations. This computation isrepeated for each angle of incidence.
The procedure that we have just described can beadopted in the common situation of Kerr behavior of a di-electric film. Then the intensity-dependent refractive in-dex n of medium 2 depends on the intensity according tothe relation
n~I ! 5 n0 1 ~n2R 1 in2I!I, (23)
where n2R and in2I represent nonlinear refraction andnonlinear absorption, respectively, and are related to thereal and the imaginary parts of third-order susceptibilityx (3) by simple formulas.13 In this hypothesis we per-formed theoretical simulations to assess the potentials ofthe method and to establish the best conditions for non-linear experiments. In particular, we evaluated thechanges in plasma angle Dup and in the reflectivity at itsminimum DR with thickness and nonlinearity of the di-electric layer. All the calculations were carried out at1064 nm for a 43-nm-thick Ag film («r1 5 248 1 2i) anda dielectric with a real refractive index of 1.557.
Figure 3(a) illustrates the results for purely dispersivenonlinearity. It shows the behavior of uDupu as a functionof the absolute value of the real refractive-index shift
Fig. 3. (a) Theoretical variation of plasma angle and (b) percent-age change of reflectivity at the position of maximum slope of theplasma resonance versus a real refractive-index shift at 1064 nmfor d1 5 43 nm and several values of d2 : 40, 80, 100, 120, 160,and 200 nm.
Margheri et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 745
uDnRu for thicknesses d2 of the dielectric layer. d2 rangesfrom 40 to 200 nm and increases in an upward direction.The behavior of uDupu versus uDnRu remains linear, atleast until uDnRu 5 0.01 (not shown in the figure).
Detection of the intensity-dependent shift of theplasma angle can be difficult and requires the acquisitionof complete reflectivity scans near up with subsequentdata fitting procedures. In many cases, and also in viewof the application of SPR techniques to the developmentof devices (e.g., all-optical switches or optical limiters), itcan be more convenient to fix the angle of incidence and todetect the reflectivity variations that are due to the angu-lar shift in the most sensitive position, which correspondsto the maximum slope of the curve (R 5 RF). Figure3(b) shows the percentage changes in reflectivity at thisposition, DR/RF , under the same conditions as for Fig.3(a): The dependence of DR/RF on uDnRu is linear and,moreover, also the slope of the straight lines exhibits alinear behavior that depends on d2 , at least up to 100 nm,with a tendency to saturation for larger values of d2 .
For a purely absorptive nonlinearity, no appreciable an-gular shift in the plasma resonance is expected. In thiscase the parameter of interest is the variation in reflec-tivity at plasma angle DR. Figure 4 shows the changesin DR versus the imaginary refractive-index shift DnI forthe same conditions as for Fig. 3(a). Whereas uDupu var-ies linearly with uDnRu, DR exhibits a saturating behav-ior. In particular, for a 160-nm-thick dielectric layer, nofurther changes in DR are expected for DnI . 0.004.Moreover, the initial slope of the curves of Fig. 4 variesrapidly with d2 , with a dependence that is close to qua-dratic. For example, the slope is 11.8 for d2 5 80 nmand 40 for d2 5 160 nm. In the general case of refrac-tive and absorptive nonlinearities, the absorptive effecthas only a small influence on Dup . Figure 5 shows thecorrection to Dup versus DnI with respect to the purely re-fractive case of Fig. 3(a) for a 120-nm-thick dielectriclayer and several values of DnR . In this case the correc-tion becomes appreciable only for large values of DnR andDnI : for instance, for DnI 5 uDnRu 5 0.003 a compari-son with Fig. 3 a shows that the variation in up is only17% of that produced by the dispersive nonlinearityalone, whereas for smaller values of DnR and DnI it canhardly be detected.
Fig. 4. Theoretical variation of the reflectivity dip at plasmaangle versus imaginary refractive-index shift at 1064 nm for d15 43 nm and several values of d2 : 40, 80, 120, 160, and 200 nm.
The results of Figs. 3–5 were obtained with d1 5 43nm; however, Dup and DR also depend on d1 . Figure6(a) illustrates the dependency of uDupu on d1 for a 100-nm-thick dielectric layer and uDnRu 5 2 3 1024: The an-gular shift rapidly decreases with increasing d1 . In con-trast, the behavior of DR with d1 , which is shown in Fig.6(b), is more complicated and is not monotonic. DRchanges sign with d1 , and the best working conditionscorrespond to d1 near 30 nm (DR , 0) and 60 nm (DR. 0).
Fig. 5. Theoretical variations of plasma angle versus imaginaryrefractive-index shift with respect to the purely absorptive caseof Fig. 3(a) for d2 5 120 nm and several values of DnR : 20.001,20.003, 20.0066, 20.009.
Fig. 6. (a) uDupu versus d1 for a 100-nm-thick dielectric layerand uDnRu 5 2 3 1024. (b) DR versus d1 for d2 5 100 nm andDnI 5 2 3 1024 and for d2 5 40 nm and DnI 5 1023.
746 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Margheri et al.
3. EXPERIMENTAL TESTSA. Sample Preparation and Experimental SetupOur procedure for measuring the SPR of thin-film nonlin-earities was tested on spin-coated films of the solublepolydiacetylene polyDCHD-HS that were deposited uponglass substrates previously coated with silver. In fact,silver is the most used metal in SPR experiments becausethe small value of Im(«r1) («r1 5 248 1 i1.8 in bulk at1064 nm) sharpens the widths of the SPR curves and in-creases the sensitivity of detection of the plasma angle.When the results of Section 2 (Fig. 6) are taken into ac-count, the choice of metal thickness requires a trade-offbetween optimization of the sensitivity and angular reso-lution of the experimental setup. In practice, good oper-ating conditions with silver films at 1064 nm correspondto a value of d1 in the 30–60 nm range.26 Therefore, sil-ver films, which were 30–60 nm thick, were depositedupon microscope glass slides by electron-gun-assistedevaporation equipment. Subsequently, we spunpolyDCHD-HS layers onto the slides under clean-roomconditions by starting from dilute toluene solutions. Wecould tune the thicknesses of the polymer films in the 10–200-nm range by varying the concentration (4–20 g/L)and the spinning velocity (3000–6000 rpm). Figure 7 il-lustrates the effect of the polymer layer on the reflectivitycurve of a silver film: It shows a 49-nm-thick silver filmwith «r1 5 234.5 1 2.3i (circles) and the same film plus a20.3-nm-thick polyDCHD-HS layer with n 5 1.561 0.148i (triangles), as observed with a cw laser diode at849 nm. The prism used for Kretschmann’s coupling ofFig. 7 is made from BK7 Schott glass, with refractive in-dex n 5 1.5099 at 849 nm.
The nonlinear experiments were carried out at 1064nm and with p-polarized light. The laser that we em-ployed to test the samples was a 10-Hz oscillator–amplifier, mode-locked Nd:YAG (EKSPLA ModelPL2143A). The pulse energy fluctuation was ,2% over100 pulses. The pulse duration was measured by stan-dard autocorrelation techniques every day before the non-linear tests were started. In typical working conditionsand with a fresh dye solution, the pulse duration was 256 1 ps FWHM (assuming a sech2 pulse shape). Theshort-pulse duration and low repetition rate were chosento minimize thermal contributions to the observed nonlin-earities. To the same end, some tests were also per-
Fig. 7. SPR curves for a 49-nm thick silver film with «r15 234.5 1 2.3i (circles) and of the same film plus a 20.3-nm di-electric layer with n 5 1.56 1 0.148i (triangles), as observedwith a cw laser at 849 nm.
formed with a reduced repetition rate (5 Hz). However,apart from considerable deterioration in the quality of thereflectivity curves as a result of the increased laser insta-bilities with this mode of operation, no significantchanges in the nonlinear behavior of the samples were de-tected. The energy of the laser beam impinging upon thesamples under test varied from 0.8 to 35 mJ, correspond-ing to energy densities of as much as 4 3 107 J/m3. Weobtained the energy variations by acting directly on theoscillator–amplifier lamp delay of the laser head: In fact,in our operating conditions we verified that this methoddid not introduce any detectable modification to the beam(i.e., in direction, intensity distribution, or pulse dura-tion). We measured laser energy by using a pyroelectricdetector.
Our experimental setup is illustrated in Fig. 8. Wecoupled the samples that were being tested to the hypot-enuse of a 45° total internal reflection BK7 prism bymeans of index-matching oil to achieve Kretschmann’sconfiguration. The laser beam was filtered and then col-limated with lens L, which had an 80-cm focal length. Asmall percentage of the incident beam was sampled andsent to a reference silicon photodiode (PhR); we filteredthe remaining part through filter A2 to select the centrallobe of the Airy pattern and then sent it on to the prism’shypotenuse. The light reflected at the prism’s base wasthen collected with a second silicon photodiode (PhS). A16-bit analog-to-digital (A/D) converter simultaneouslyacquired the signals of the photodiodes, which were thenstored and processed on a PC. The prism–sample struc-ture was mounted upon a motorized rotation stage with0.01° resolution and was controlled by the same PC. Foreach angular position along the scan range we measuredreflectivity R and averaged it over several laser pulses(typically 35).
B. Experimental ProcedureThe experimental procedure required several steps:
1. Deposition of silver film.2. SPR measurement of the thickness and the complex
dielectric constant of the silver film at the lowest detect-able level of intensity.
3. High-intensity SPR measurements of the silver filmto check any nonlinear behavior within the irradiation re-gime used in the subsequent nonlinear tests.
4. Spin coating of the polymer film.
Fig. 8. Experimental setup.
Margheri et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 747
5. Low-intensity measurement of reflectivity R of themultilayer structure.
6. Increase in the intensity in small steps and mea-surements of R.
7. Low-intensity checks before each increment of theintensity to exclude any light-induced permanent damageto the dielectric film.
Step 3 is a key point. In fact, because of the existenceof a nonnegligible absorption coefficient @Im(«r1) . 1#, in-tensity dependence of the dielectric constant of a silverfilm cannot be excluded a priori. Therefore this point de-serves careful investigation. According to the SPR ex-periments reported in Ref. 27, the absorption of laserlight induces a temperature increase at the silver-air in-terface; the temperature of the silver film stabilizes afterfew picoseconds, whereas the subsequent heat releaseinto the neighboring dielectrics takes place on a nanosec-ond time scale. Thus, because of the low absorption at1064 nm and the small heat diffusion constant (;3 ordersof magnitude lower than the metal constant) we can rea-sonably assume negligible heating of the polymer filmduring the laser pulse (25 ps) and complete cooling be-tween two adjacent pulses (0.1 or 0.2 s), thus excludingany thermal influence on the pulse’s nonlinear behavior.As we point out below, this description of the polymer’sbehavior was confirmed by the reversibility of the SPRcurves at different energy levels. Moreover, evaluation ofthe maximum energy per pulse that can be absorbedwithin the silver layer in our experimental conditionsgave a temperature increase of the metal that was always,4 K. Nevertheless, any dependence of «r1 on tempera-ture even if it is small, must be taken into account whenone is simulating the nonlinear experiments performedon metal–polymer structures. There are reports in theliterature of temperature variations of «r1 . In particu-lar, according to Ref. 28, d Re(«r1)/dT 5 8.5 3 1024 K21
and d Im(«r1)dT 5 1.5 3 1023 K21. In contrast, Ref. 29describes a method that is based on Drude’s model of met-als and on electron–phonon interactions, which permitsan evaluation of d Re(«r1)/dT and d Im(«r1)/dT for bulk sil-ver: The value of d Re(«r1)/dT obtained is in agreementwith that reported in Ref. 28, whereas d Im(«r1)/dT is ap-proximately six times larger. Because such differences ind Im(«r1)/dT can affect the simulation of our data, we per-formed SPR measurements with a 34-nm-thick silver filmand incident energy densities in the range 2.5 3 106 –43 107 J/m3, to establish d Im(«r1)/dT for our silver films.These energy densities were of the same order as orlarger than those used in the nonlinear tests with poly-mer layers. We found that the variations in «r1 with en-ergy did not exhibit a well-defined trend but were statis-tically distributed about the average value «r15 (249.90 6 0.01) 1 i(1.478 6 0.002), as illustrated inFigs. 9(a) and 9(b) for the real and the imaginary parts of«r1 , respectively. This result was in agreement with thedata reported in Ref. 28. Indeed, by taking into account(i) the volume heat capacity of silver (2.4 3 106
J m23 K21), (ii) the estimated maximum temperature in-crease of the metal film in our experimental conditions,and (iii) the rates of change of «r1 with temperature re-ported in Ref. 28, we should obtain the maximum ex-
pected variations of «r1 in the experiment of Fig. 9: 0.003for the real part and 0.004 for the imaginary part. Thesevalues are well within the experimental errors for Re(«r1)and are just at the limit of detection for Im(«r1), and forthis reason, we neglected temperature-induced variationsof «r1 in the simulation of subsequent experimental data.
C. Experimental ResultsThe results of our measurements, performed on a 43-nm-thick polyDCHD-HS layer deposited upon silver, are sum-marized in Fig. 10. In particular, Figure 10(a) shows re-flectivity versus incident angle for several averageintensities ranging from 18 to 75 MW/cm2, correspondingto incident energy of 0.058–0.22 mJ. The data were reg-istered in a narrow range about the plasma angle that, inthis case, was uP 5 44.55°. By ‘‘average intensity’’ wemean the pulse intensity at a metal–polymer interfaceaveraged both on the Airy spot of the incoming beam andon the polymer film according to Eq. (22). In this case,because of the limited thickness of the dielectric layer, thedifference between the average intensity and its peakvalue at the metal–polymer interface was small (,13%).The most apparent feature in Fig. 10(a) is the rapid in-crease and subsequent saturation of the reflectivity withan overall excursion of ;5%. The two curves that corre-spond to I 5 18 MW/cm2 were registered at the beginning
Fig. 9. (a) Re(«r1) and (b) Im(«r1) of a 34-nm-thick silver film.Solid lines represent average values; circles and squares corre-spond to measurements performed with energy densities in the2.5 3 106 –1 3 107 and 2 3 107 –4 3 107 J/m3 ranges, respec-tively. The abscissa shows the order number of the measure-ments.
748 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Margheri et al.
(open circles) and at the end (filled diamonds) of the ex-periment. Their good agreement demonstrated that nodamage had occurred during the measurements and, atthe same time, gives an idea of the reproducibility of theexperimental data. We fitted the data illustrated in Fig.10(a) assuming that «r1 5 247.38 1 i1.664 and d1 5 33nm for the silver layer, i.e., the values that were mea-sured before polymer deposition (step 2). The thickness
Fig. 10. Experimental results obtained with a 43-nm-thickpolyDCHD-HS film spun onto a 33-nm-thick silver layer: (a) re-flectivity versus incident angle; (b) real and (c) imaginary refrac-tive indices versus average intensity. Insets, (b) refractive and(c) absorptive nonlinearity obtained from each reflectivity curvein the second run of the fitting procedure.
of the polymer film was measured independently with acw diode-pumped Nd:YAG laser, which was alignedwithin 0.01° of the path of the pulsed beam. In a first ap-proximation, the small variations (,7%) of the intensityalong each angular scan was disregarded, so each reflec-tivity curve of Fig. 10 returned a value of n. The valuesof refractive index versus average intensity at plasmaangle are plotted in Fig. 10(b) for the real part and in Fig.10(c) for the imaginary part of the refractive index.Intensity-dependent behavior of the refractive index wasclearly observed. However, it was not simply linear, asexpected in the case of true Kerr behavior. In particular,the imaginary part exhibited saturation and subsequentbleaching, after an initial increase with intensity. To ac-count for these features we best fitted the experimentaldata of Figs. 10(b) and 10(c) by assuming a third-ordernonlinearity with saturation:
nR 5 n0R 1n2RI
1 1 I/IsR, nI 5 n0I 1
n2II
1 1 ~I/IsI!2 .
(24)
The unknown parameters of Eqs. (24), namely, the linearindex of refraction n0 5 (n0R 1 in0I), the real and imagi-nary components of the nonlinear index of refraction n2(n2R and n2I , respectively), and the saturating intensi-ties IsR and IsI , were determined with a best fit. Thebest-fit curves are represented by continuous curves inFigs. 10(b) and 10(c) and correspond to n0 5 (1.57911 i1025), n2R 5 21.3 3 10214 m2/W, and n2I 5 1.73 10214 m2/W. The saturating intensity was 30MW/cm2 for the real part of the nonlinearity; for theimaginary part, the bleaching effect appeared for intensi-ties larger than IsI 5 60 MW/cm2. The higher-ordersaturating behavior of nI was assumed to account for theobserved bleaching effect of the imaginary part and wasjustified by statistical x2 tests. In contrast, an attemptto fit the real part with the same second-order saturatingbehavior, in this case, did not improve the quality of thefitting, whereas for thicker samples, such as the 157-nm-thick polymer film described here and prepared as de-scribed below, it gave a x2 value that was smaller by 1 or-der of magnitude.
Fig. 11. Experimental data (triangles and squares) and theoret-ical best fits (continuous curves) for the scans registered with 43-nm-thick polyDCHD-HS films at 18 and 30 MW/cm2, respec-tively.
Margheri et al. Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. B 749
As a refinement of the previous procedure, we per-formed a second simulation of the reflectivity curves,starting from the values found with the first approxima-tion as described above. Here (i) we took into account theintensity and refractive-index variations along eachcurve; (ii) we assumed for n the functional dependence re-ported in Eqs. (24) and the low-intensity value obtainedfrom the best fit of Figs. 10(b) and 10(c); (iii) we left n2 ,IsR , and IsI as free parameters. With this procedure,each curve returned a value of n2 . The results are illus-trated in the insets of Figs. 10(b) and 10(c). The fluctua-
Fig. 12. Experimental results obtained with a 157-nm-thickpolyDCHD-HS film spun upon a 41-nm-thick silver layer: (a)reflectivity versus incident angle; (b) real and (c) imaginary re-fractive indices versus average intensity. Insets, (b) refractiveand (c) absorptive nonlinearity obtained from each reflectivitycurve in the second run of the fitting procedure.
tions of n2 correspond to an error of 15%. Figure 11shows two examples of the accuracy of the fitting proce-dure that we achieved. It shows experimental data (tri-angles and squares) and theoretical curves (continuouscurves) for the scans registered at 18 and 30 MW/cm2, re-spectively. Although the measurement procedure wasquite delicate, the agreement is fairly good.
In the case of the 43-nm-thick polymer layer, variationsin the refractive index of the dielectric material were ofthe order of 1023. We obtained analogous results witheven thinner polymer layers (9–14 nm), as reported inRefs. 30 and 31. Although these variations were quitelarge, their effect on DR was relatively small and that onDup could only be inferred from numerical simulations.However, our theoretical calculations predicted a largersensitivity of the method with thicker dielectric films,with the best working conditions corresponding to a poly-mer thickness of 150–200 nm. For this reason we madefurther measurements with thicker samples to improvethe experimental accuracy. Figure 12 summarizes theresults obtained with a 157-nm-thick polymer film. Thedielectric constant and the thickness of the silver layer, inthis case, were «r1 5 248.4 1 i1.17 and d1 5 41 nm, re-spectively. Figure 12(a) shows the reflectivity measure-ments versus incident angle. Also in this case, an in-crease in the reflectivity dip and subsequent saturationwith intensity were observed but with an overall varia-tion smaller than 1%, which corresponded to a Dn of theorder of 1024. Nevertheless, as predicted by theory, theincreased thickness of the polymer film improved the sen-sitivity of the experiment and permitted the detection ofthis value of Dn, which was approximately ten timessmaller than those measured with thinner polymer
Fig. 13. (a) DnR and (b) DnI versus average intensity.Squares, 43-nm-thick polymer layer; circles and triangles, posi-tions on the 157-nm-thick polymer layer.
750 J. Opt. Soc. Am. B/Vol. 20, No. 4 /April 2003 Margheri et al.
samples. The simulation of the experimental data [Figs.12(b) and 12(c)] gave n0 5 (1.5771 1 i1025), n2R5 21.3 3 10215 m2/W, n2I 5 1.1 3 10215 m2/W, andIsR 5 IsI 5 40 MW/cm2. The error in n2 , which can beinferred from the insets of Figs. 12(b) and 12(c), was 30%and 20% for the real and imaginary parts, respectively.
To provide more evidence of the considerable differ-ences between the 43-nm-thick and the 157-nm-thickpolymer films, we show in Fig. 13 the behavior of DnR andDnI versus intensity in the two cases. The squares referto the thin polymer layer, and the circles and triangles re-fer to two sets of measurements of the thick polymerlayer. Although the functional behavior of DnR and DnIwas the same with both samples, not only were the peakvalues 1 order of magnitude smaller for the thick film butthe bleaching of the nonlinear absorption took place at alower intensity. An analysis of this behavior was dis-cussed elsewhere.32 It was related to the presence of thesilver film, whose nanostructured surface provides anelectromagnetic mechanism for the enhancement of non-linearity through local field effects. This interpretationwas confirmed by surface-enhanced Raman spectroscopyexperiments, which suggested the existence of an interac-tion between the p-conjugated system of the polymerchain and the silver surface.
4. CONCLUSIONSA measurement procedure based on surface plasma reso-nance has been described that permits characterization ofthe nonlinear behavior of thin and ultrathin (nanometersthick) dielectric layers. Even if the need for TM polariza-tion and the contact of the dielectric film with a metallayer prevent the SPR technique from being used as ageneral screening tool for new highly nonlinear materials,the examples reported in this paper nevertheless confirmthat SPR nonlinear tests can be highly attractive—if notirreplaceable—in several situations: in particular, forstudying the nonlinearity of thin films in the presence ofmetals. Indeed, the behavior of metal–dielectric inter-faces in high-irradiation regimes is attracting increasinginterest, in that it has been proved that metal nanostruc-tured surfaces provide electromagnetic mechanisms forenhancing nonlinearity not only near resonance but alsooff resonance, typically in the near infrared.33
Our numerical simulations and experimental testswith polyDCHD-HS have proved that SPR can be advan-tageously used for measuring the sign and the real andimaginary parts of the off-resonant n2 of dielectric layersof 10–200-nm thickness. This use of SPR permits a con-siderable reduction to be made in the material quantitieswithin a few milligrams that are necessary for completecharacterization of novel compounds. In comparison, theminimum film thickness required by well-establishednonlinear techniques, such as Z-scan and forward degen-erate four-wave mixing, are ;100 times larger. A draw-back of the SPR technique is that it relies on relativelylarge Dn (.1025) for achieving good sensitivity (manypolymers saturate for Dn near 1025). The superpositionof thermal and electronic effects, which is common toother one-beam nonlinear characterization methods, can
be avoided by reduction of laser pulse duration and rep-etition rates, as confirmed by our tests.
Among the possible applications of the SPR method, wewish to stress the monitoring of the nonlinearity of filmsof quasi-unidimensional long-chain materials, such aspolydiacetylenes, deposited upon nanostructured metalsurfaces and with a thickness that is comparable to oreven smaller than the chain length. SPR can give impor-tant information on such microscopic complex systems byrelating the nonlinearity, the local field effects that aredue to the metal layer, and the supramolecular organiza-tion of organic molecules.14,32,33
ACKNOWLEDGMENTSThis research was partially supported by the Italian Pro-getto Finalizzato MSTAII of the National Research Coun-cil of Italy. The authors are indebted to Francesco Giam-manco (University of Pisa, Italy) and GiovannaDellepiane (University of Genoa, Italy) for stimulatingdiscussions and to Dina Cavallo and Carlo Dell’ Erba(University of Genoa, Italy) for the preparation of thepolyDCHD-HS.
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