Obtaining information about the phase throughdiffraction experiments has always been a problemwhen acquiring one image (naturally two dimen-sional) through an optical system. The third dimen-sion (containing the phase shift) information isusually obtained by interferometric techniques [1],where light diffracted by or scattered from an objectinterferes with a reference wave. The unknownphase shift related to the object is encoded in a sys-tem of interference fringes. The problem is simplifiedif a priori information about the shape of the phaseobject is given. One possible way to overcome theexperimental difficulties encountered with interfero-metric techniques (where the information is obtainedusing only one acquisition) is to exploit the intensitychanges in several diffraction patterns resultingfrom different experimental parameters. One ofthe parameters can be the position of the sampleinside an optical system in a tight focus geometricconfiguration. In nonlinear optics, such an ideawas introduced by the Z-scan technique to study,
principally, the third-order nonlinearities [2]. In thismethod we measure the transmittance of a focusedGaussian beam into a nonlinear medium througha circular aperture placed in the diffracted far field.The variation of the output signal as a function of z,the sample position with respect to the focal plane,provides information about the instantaneous phaseshift induced into the material in the nonlinearregime (for more details, see [2]). In [3] and morerecently in [4] it has been shown that Z-scan tracescan also reveal the permanent change of the phaseshift inside thin films due to photoinduced effects(ablation or photodarkening). Here, we present anew contribution to measurements of the permanentphase shift inside the specimen. Particularly, weanalyze the influence of a rectangular phase line em-bedded in thin films on the far field normalizedtransmittance using a Z-scan closed aperture config-uration. We show that in certain conditions (detailedbelow), we can obtain a peak–valley shape of theZ-scan profile as in the nonlinear measurement.The difference between the maximum (peak) andthe minimum (valley) of the transmittance (usuallycalled ΔTpv) is related linearly to the permanentphase shift, therefore providing a simple formula
for optical parameter characterization, as n, thelinear refractive index, or t, the thickness. Severaloptical techniques allow one to determine thesequantities; see, for example, [5] and references there-in. But, the interferometric and the ellipsometricsystems [1,6] are the most popular due to their non-contact character and high sensitivity along withtheir imaging derivatives [7–11]. Although all ofthese methods are very precise, they require rela-tively complex and expensive experimental appara-tus, contrary to the technique reported here, whichoffers simplicity in both optical setup and relatedmeasurement formulas.
2. Theoretical Model
It is assumed that scalar diffraction theory is suffi-cient to describe image formation using a 4f system[12] (Fig. 1). For self-consistency, we briefly recall thetheoretical model we used (see [4,13,14] for moredetails). Since we are concerned with the imageintensity, the temporal terms will be omitted. Theamplitude electric field distribution at the objectplane (front focal plane of lens L1) is Gaussian,Eðx; yÞ ¼ E0 exp½−ðx2 þ y2Þ=ω2
e �, where x, y are thespatial coordinates, E0 denotes the on-axis ampli-tude, and ωe is the beam waist at the entry of the set-up. Let Sðu; vÞ be the spatial spectrum of Eðx; yÞ:
Sðu; vÞ ¼ ~F ½Eðx; yÞ�
¼Zþ∞
−∞
Zþ∞
−∞
Eðx; yÞ exp½−j2πðuxþ vyÞ�dxdy; ð1Þ
where ~F denotes the Fourier transform operation,and u and v are the normalized spatial frequencies.Instead of propagating the field in the spatial domainand in order to reduce the computing time, we choseto propagate the spectrum of the object over a dis-tance z0, by taking into account the transfer functionof the wave propagation phenomenon (see Chap. 3 in[12]), Hðu; vÞ ¼ expðj2πz0
λ is the wavelength. The field amplitude at z0 after
the free propagation is obtained by computingthe inverse Fourier transform, Eðx; y; z0Þ ¼~F−1½Sðu; vÞHðu; vÞ�. To calculate the output beamafter passing through a lens of focal length f , we ap-ply the phase transformation related to its thicknessvariation in the paraxial approximation: tLðx; yÞ ¼exp½−jπðx2 þ y2Þ=λf �.
The first propagation is performed on a distancez0 ¼ f 1, where the beam enter lens L1, and after thattL is applied with f ¼ f 1. Then we propagate thebeam up to the sample located at the algebraic valuez, using z0 ¼ f 1 þ z inH, the optical transfer function.Note that z ¼ 0 corresponds to the back focal point ofthe lens L1. The transmission of the rectangularphase line of size Lx and Ly along x and y axes,respectively (Lx ≫ Ly), is defined as Tðx; yÞ ¼exp½−jφðx; yÞ�, where φðx; yÞ ¼ ½2πðn̂ − 1Þt × rectðx=LxÞ×rectðy=LyÞ�=λ is the phase shift. Here n̂ ¼ n − jκ isthe complex refractive index, with n denoting thelinear refractive index, and κ is the extinction coeffi-cient. The rectangular function rectðxÞ is defined asequal to one if jxj ≤ 1=2 and zero elsewhere. Next, weperform propagation of the electric field at the exitface of the sample over a distance z0 ¼ f 2 − z and aphase transformation due to lens L2, and the finaldiffraction is calculated with z0 ¼ f 2 at the outputof the 4f system. The intensity is calculated bysquaring the amplitude field. The open aperturenormalized Z-scan transmittance is obtained byintegrating spatially over the entire image plane.The closed aperture is obtained by integrating overa circular aperture [2] with radius ra ¼ωo½0:5 lnð1=ð1 − SÞÞ�1=2, where S is the closed aper-turelinear transmission, and ωo ¼ ωef 2=f 1 is the outputbeam waist of the Gaussian beam in the absenceof any diffracting phase elements.
3. Experiment and Results
Usually, our experiment is used for nonlinear opticalmeasurements. That is why the excitation isprovided here by a Nd:YAG laser delivering 17ps sin-gle pulses at 1064nm with a 1Hz repetition rate.Nevertheless, a more energy stable CW laser canbe beneficially used. The focal length of the twolenses composing the 4f system is f 1 ¼ f 2 ¼ 20 cm.The photoreceptor is a 1000 × 1018 pixel cooled(−30 °C) CCD camera with fixed linear gain. Thecamera pixels have 4095 gray levels, and each pixelis 12 × 12 μm2. The laser intensity is kept in low re-gime in order to avoid any nonlinear optical phenom-ena and destruction of the material. A beam splitter(BS1) at the entry of the setup allows one to monitorthe fluctuations (through lens L3) occurring in the in-cident laser beam independently from the absorptionthat might occur inside the material. Open andclosed aperture Z-scan normalized transmittancescan be numerically processed from the acquiredimages, by integrating over all the CCD pixels inthe first case and over a circular numerical filterin the second one. We choose a linear transmission
Fig. 1. Schematic of the 4f coherent system imager. The sample ismoved around the focal region. The labels refer to the amplitudefield Eðx; yÞ, lenses (L1–L3), mirrors (M1, M2), and beam splitters(BS1, BS2).
of the filter S ¼ 0:4 since it is a good compromisebetween having a large signal and a relatively highsensitivity [2]. The sample is a thin film of poly-methyl methacrylate (PMMA) deposited on a glasssubstrate by the spin coating method. The materialis completely transparent in the infrared region(κ ¼ 0), and its refractive index is n ¼ 1:48 [15].The rectangular shape phase line in the film wasmade by carefully scratching the film with a sharpedge over a distance Lx long enough (2 cm) in compar-ison to the focused beam waist in order to neglect itsdiffracting effects along the x direction. The profile ofthe scratch (Fig. 2) was measured by a probing me-chanical stylus giving a deepness t ¼ 120� 10nmand a width Ly ¼ 55� 5 μm. We aligned very care-fully (i) the motor translation stage (Z-scan axis) tobe parallel with the direction of the light propaga-tion; (ii) using the image given by the CCD at differ-ent z positions of the sample, the groove line wascentered in the middle of the beam in order to havea symmetrical diffracted intensity distribution on itsborder. In Fig. 3(a) one can see the experimental ac-quisition of the image at the output of the 4f systemfor a sample location at z ¼ −45mm and forωe ¼ 2:52mm. The latter value gives a beam waistin the focal plane (z ¼ 0) ωf ¼ 27 μm. Negative z isrelated to the sample position close to the first con-verging lens (L1). In Fig. 3(b) we can see the corre-sponding simulated image obtained with the sameexperimental parameters. In order to define the nu-merical parameters of the phase object, we used thewidth and the deepness given by the profilemeter.The qualitative good agreement between these twoimages was the first step to validate our numericalcalculations.The Z-scan measurements were performed within
a range of z ¼ �45mm, displacing the sample with a1mm step and acquiring one image for every posi-tion. In Fig. 4 we can see a comparison betweenthe experimental closed aperture normalized trans-mittance and its numerical simulation obtained for
S ¼ 0:4 and the previous mentioned experimentalparameters (ωf ¼ 27 μm, Lx ¼ 1:2mm, Ly ¼ 60 μm,and φð0; 0Þ ¼ φ0 ¼ 0:34). The remarkably good agree-ment between experiment and theory validatestotally our numerical simulation. The prefocal trans-mittance maximum (peak) followed by postfocal
Fig. 2. Scan made by mechanical profilemeter showing thegeometrical parameters of the rectangular groove (width anddeepness).
Fig. 3. Images at the output of the 4f system for z ¼ −45mm:(a) experimental acquisition, (b) numerical simulation. The coordi-nates (x, y) are in pixels.
Fig. 4. Experimental (dots) closed aperture Z-scan normalizedtransmittance TðzÞ and the corresponding theoretical simulation(solid line).
transmittance minimum (valley) can be physicallyexplained by noting that the phase object has an ap-proximate sampled shape of a thin diverging lens[see Fig. 5(a)]. For a strip line deposited on the film[Fig. 5(b)] the simulation of the normalized Z-scantransmittance shows a minimum followed by a max-imum, therefore indicating a positive phase shift.The peak followed by a valley is similar to the oneprovided for Z-scan traces in [2], where a negativenonlinear coefficient (n2) characterizes the nonlinearmaterial. When the sample is far away from the fo-cus, the beam waist is large compared to the width ofthe phase line, only a small part of the beam is dif-fracted, and the transmittance remains relativelyconstant with respect to the position. When the sam-ple is brought closer to the focus, the phase objectstarts to act as a focusing or defocusing lens sincethe beam waist becomes smaller; more energy inter-acts with the area of the phase steps diffracting moreor less light outside or inside the aperture dependingon z position. When the sample is in the focal regionand for large Ly compared to ωf , there are no signif-icant changes in the transmittance because the fo-cused light does not encounter the phase step. InFig. 6 the dashed line (Ly ¼ 250 μm ≫ 2ωf ¼ 54 μm,φ0 ¼ 0:34) represents such a situation, revealing aflattened central part included between points Aand A0. In this figure we can see the simulated trans-mittance for two different widths of the phase line.One of them (the solid line) is the simulationreported in Fig. 4 and plotted here for comparisonpurposes. Using simple geometrical optics approxi-mation and calculating zAðzA0 Þ, the position of pointAðA0Þ, where the beam waist of the converging(diverging) beam is equal to the half-width of therectangular line, we obtain zA ¼ Lyf =2ωe (see Fig. 6,where the variations of the transmittance becomesignificant). For small on-axis phase shift(jφ0j < 1), the peak and valley occur at the same dis-tance with respect to the focus. Empirically we havefound that the maximum (minimum) is situated atapproximately twice the zAðzA0 Þ value. Introducing
the beam waist at the focus ωf ¼ λf =πωe, the maxi-mum and the minimum are separated by Δzpv ¼2πLyωf =λ. Numerically, we have found a more precisevalue (less than 2% error) valid for large Δzpv(≥10mm), Δzpv ≈ 2:4πLyωf =λ. Theoretically, everylateral size of the phase objects can be seen by thelight as a diverging or a converging lens dependingon the beam waist size with respect to z position.
One of the advantages of the Z-scan method [2]comes from the simple expression relating ΔTpv tothe induced Gaussian nonlinear phase shift, allowingnonlinear optical characterization of the materialwithout resorting to complicated computer fits.Following the same way, we have to introduce a simi-lar expression for a permanent rectangular phaseshift line. Numerically, we have found that for smallon-axis dephasing, jφ0j ≤ 1, the following relation isfulfilled within less than 10% error:
ΔTpv ¼ αjφ0j; ð2Þ
where α is a proportionality factor. The linearity ofthis relationship becomes more and more accuratefor lower on-axis phase shift, and for jφ0j ≤ 0:3 theerror is negligible (less than 1%) [see Fig. 7(a)].Generally the proportionality factor is dependenton the aperture size (S), the beam waist (ωf ), andthe lateral geometrical dimensions (Lx and Ly) ofthe rectangular phase object. Since this problem isquite complicated and possesses many parameters,we fixed S ¼ 0:4 and ωf ¼ 27 μm. Let us focus onthe importance of the size of the rectangular linewith respect to the beam waist. In Fig. 7(a) we cansee ΔTpv versus jφ0j calculated with the size of thephase line close to that measured experimentally(Ly ¼ 60 μm and Lx ≫ Ly ≈ 2ωf ). The slope of thecurve is α ¼ 0:9. This slope defines the sensitivityof the method. In our case, with a pulsed laser havingrelatively higher energy fluctuation than a CW one,the resolution of the measurement is not optimized.
Fig. 5. The similarity between the cross section of (a) rectangulargroove embedded in thin film and the diverging lens (dashed line),and (b) rectangular strip layer on thin film and the converging lens(dashed line)
Fig. 6. Comparison of the Z-scan normalized transmittances. Thesolid line is the simulation reported in Fig. 4 for φ0 ¼ 0:34(Lx ¼ 1:2mm, Ly ¼ 60 μm). The dashed line is the simulation givenfor the same parameters with a larger width of the rectangularphase object, Ly ¼ 250 μm.
Nevertheless, we are able to resolve normalizedtransmittance changes as low as 3 × 10−3. Consider-ing a signal to noise ratio approximately equal to 3,we are able to measure more than approximately 1%transmittance change. According to Eq. (2), the mea-sured phase value will be equal to jφ0j ¼ 10−2=0:9 ¼ 11:1mrad. Thus the minimum optical pathlength that we can measure (corresponding to a wavefront distortion) is Δn × t ¼ 11:1 × 10−3 × λ=2π ≈ λ=550.The dashed line in Fig. 7(b) shows the variations of
α versus Lx for very large Ly (1:2mm). The highestvalue (α ¼ 0:9) is obtained for Lx ≈ 2ωf. Therefore,for a given one lateral dimension of the rectangularphase object, the maximum of the sensitivity appearswhen the second dimension is comparable to the focalbeam diameter. Moreover, by fixing one dimension atthis maximum (Ly ≈ 2ωf ) and by varying the otherone, α is constant when Lx ≫ Ly, as is shown bythe solid line in Fig. 7(b). The experimental acquisi-tions shown previously (see Fig. 4) correspond to thiscase. Note that when all the geometrical dimensionsare comparable (Lx ≈ Ly ≈ 2ωf ), α is not constant any
more and reaches a maximum as we can see at thebeginning of the solid line in Fig. 7(b). One shouldavoid this region where the diffraction occurs in bothdirections contributing to very large variation in theZ-scan signal with the lateral dimension of the phaseshift. The enhancement of the sensitivity will come atthe expense of a reduction in accuracy. Nevertheless,the order of magnitude of the measured phase shift ispreserved. It has to be added that the same experi-mental setup could be used to estimate the lateralsize (Lx and Ly) of the phase object by recordingan image of its diffracting borders when the objectis located at the entry of the 4f system.
This study is being extended for phase objects withnonrectangular shape, particularly for those havingGaussian profile induced during laser inscription ofoptical waveguides in order to optimize the fabrica-tion process. Our preliminary results show that theZ-scan method can be a valuable inexpensive tool toestimate the permanent change in the refractiveindex and to characterize as well the photodarkeningeffect.
4. Conclusion
We have extended the usefulness of the Z-scantechnique outside the field of the nonlinear optics.We have shown that the method can be used to inves-tigate small permanent change of the phase shiftinside thin films. The closed aperture Z-scan profilesreveal a peak–valley configuration of the normalizedtransmittance for a rectangular phase object allow-ing us to determine the sign of the dephasing. Wehave introduced a simple relation in order to deter-mine the permanent phase shift directly from theZ-scan data. The validity of this relation has beendiscussed taking into account the lateral geometricalsize of the object with respect to the focal beamwaist.The method shows high sensitivity that allows thick-ness as low as λ=550 to be measured. Moreover, sincethe Z-scan is a single-beam technique based ondiffraction phenomenon, its optical setup offerssimplicity in comparison with interferometric techni-ques generally used for this purpose.
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