The formation of noise gratings in various photo-sensitive media has been investigated over threedecades. These media include photographic silverhalide emulsions [1–4], all-organic photopolymers[5–10], holographic polymer-dispersed liquid crystals(HPDLCs) [11], photorefractive crystals [12–17], andactive dye solutions [18]. In these media, scattering ofincident coherent radiation may take place due torandomly distributed scattering centers such as sil-ver halide crystals of several-tens-of-nanometers sizein photographic emulsions, microscopic polymer do-mains (liquid crystal droplets) formed during thetransient phase-separation process in HPDLCs, andintrinsic impurities and inhomogeneities in photore-fractive crystals. As a result, randomly scatteredwaves interfere with the incident radiation, leadingto the formation of parasitic noise gratings in thesemedia. A common consequence of such coherent lightscattering, so-called holographic scattering, from thenoise gratings during recording or readout is the ap-pearance of scattering lines and�or rings on a screenplaced in the transmitted field. Because of academic
interests in such peculiar light scattering phenomenaand of practical needs for the suppression of holo-graphic scattering as harmful coherent noise, a lot ofstudies have been reported so far. Basic features ofholographic scattering under various incident condi-tions could be explained by the concept of the Ewaldsphere construction [10,13,17,19–22]. Effects of ma-terial’s swelling or shrinkage and light polarizationson holographic scattering have also been studied[9,10,23–25].
In this paper we report on an observation of holo-graphic scattering in nanoparticle-dispersed photo-polymer films that have already shown their highpotentiality as a holographic recording material withthe high refractive index change ��n�, high recordingsensitivity, and high mechanical stability [26–30]. Ithas been shown that the mutual diffusion of mono-mer molecules and nanoparticles plays a vital role inthe holographic grating formation [31,32]. Becausenanoparticles of the order of 10 nm or smaller areuniformly dispersed in monomer syrup, Rayleighscattering [33] in the visible spectral region is not adominant factor that restricts their holographicapplications [28,30]. However, it can initiate holo-graphic scattering due to the Bragg-matched self-amplification of scattering noise radiation via unshiftednoise gratings [34,35]. Such holographic scattering
may be prominent when a photopolymer film is thick(e.g., several hundreds of micrometers or more forholographic data storage applications [36]). Here, weinvestigate dependences of film thickness and nano-particle concentration on holographic scattering inSiO2 nanoparticle-dispersed photopolymer films anddiscuss its influence on the formation of a thick vol-ume hologram.
2. Experimental Results and Discussion
In our experiment, methacrylate monomer, 2-methyl-acrylic acid 2-4-[2-(2-methyl-acryloyloxy)-ethylsulfanylmethyl]-benzylsulfanyl-ethyl ester, wasused. Its refractive indices were 1.55 in the liquidphase and 1.59 in the solid phase, respectively, at awavelength of 589 nm. Inorganic SiO2 nanoparticles(their bulk refractive index of 1.46), acting as an in-organic guest material, were initially dispersed in asolution of methyl isobutyl ketone (MIBK). Such SiO2sol was uniformly dispersed in the monomer. Themonomer-SiO2 sol mixture was mixed with a titano-cene initiator (Irgacure 784, Ciba) in 1 wt.% withrespect to the monomer in order to provide photosen-sitivity in the green. Such multicomponent syrup wascast on a glass plate with polyester-film spacers hav-ing several thicknesses from 20 to 200 �m. It wasthen dried in an oven and was finally covered withanother glass plate. In this way several samples hav-ing different film thicknesses were prepared. ThisSiO2 nanoparticle-dispersed photopolymer systemwas used previously in our several experiments[28,31,32]. It was shown that the photopolymer sys-tem had low transmission losses in the green–redspectral region when the film thickness was below100 �m [28]. To record an unslanted transmissiongrating, we used two mutually coherent s-polarizedbeams with equal intensities from a Nd:YVO4 laseroperating at a wavelength of 532 nm. These beamshad their transverse widths of approximately 15 mmin the sample plane. A photoinsensitive 633-nmHe–Ne laser beam at s polarization was employed asa probe beam to quantitatively evaluate the opticalscattering loss without any absorption during andafter photopolymerization.
Figure 1 shows the transmittance variation at aprobe wavelength of 633 nm as a function of expo-sure fluence under uniform illumination of coherent(532-nm laser) and incoherent (365-nm LED) lightsources. The transmittance was defined as the ratioof the transmitted power to the incident power withFresnel correction to two air–glass substrate bound-aries. Two thick samples dispersed with SiO2 nano-particles of 34 vol.% were used. Note that thisconcentration gives the maximum change in �n forsamples thinner than 100 �m [28]. It can be seen thatwhen the incoherent light source is used for photopo-lymerization, the transmittance slightly decreases atthe initial stage of photopolymerization and then in-creases to the steady-state value of approximately98%. The initial decrease may be attributed to thetransient formation of supermolecular structures(i.e., spatially inhomogeneous polymeric microdo-
mains) during photopolymerization [6] because theprobe beam is scattered by such structures actingas transient scattering centers. Nevertheless, thesteady-state transmittance is very high, indicatingthat the optical loss due to incoherent Rayleigh scat-tering from uniformly dispersed SiO2 nanoparticlesis as small as �2%. On the other hand, the trans-mittance under coherent illumination monotonicallydecreases during the photopolymerization. The steady-state transmittance is approximately 84%, that is,approximately a one-order-of-magnitude increase inboth the optical loss and the turbidity [37] as com-pared with the case of incoherent illumination whena difference in the film thickness between the twosamples is taken into account. As observed in otherphotopolymer systems [6,7,10], such a significant in-crease in the optical loss can be attributed to theformation of parasitic noise gratings in the sample: atthe initial stage of photopolymerization the incidentcoherent light is scattered by inhomogeneous poly-meric microdomains as well as by SiO2 nanoparticles,inducing noise gratings formed by the coherent inter-ference of the incident and randomly scatteredwaves. Such noise gratings are fixed after the pho-topolymerization is completed. The partial recoveryof the transmittance toward the steady state may becaused by the sample’s shrinkage that results in cou-pling mismatch of noise gratings as suggested byFrantz et al. [10].
Typical temporal evolution of the transmitted lightpattern for a 200-�m thick sample dispersed withSiO2 nanoparticles of 34 vol.% is shown in Fig. 2. Atwo-beam recording setup with the intersection angleof 31° corresponding to a grating spacing of 1 �m wasused. The two recording beams had an intensity of50 mW�cm2 each. The screen was placed at a dis-tance of approximately 10 cm from the back glasssubstrate of the sample. It can be seen that as therecording process progresses, the bright spots of thetwo transmitted beams tend to “diffuse” to form onering pattern associated with two vertical lines pass-ing through the two bright spots. Similar scatteredring�line patterns were previously reported in pho-torefractive lithium niobate [13].
Fig. 1. Transmittance changes as a function of exposure fluenceunder incoherent (dotted curve) and coherent (solid curve) illumi-nation. Mechanical film thicknesses of the two samples under theformer and latter illumination were 109 and 96 �m, respectively.
The formation mechanism of such a scattering pat-tern shown in Fig. 2 may be explained by the conceptof the Ewald sphere construction [13,19,20]. Figure 3illustrates four Ewald spheres, where two solid cir-cles correspond to two primary image loci 1 and 2,respectively, for �Ks�� ks � ki� vectors of noise grat-ings written by each of the recording beams and itsscattered waves, and two dotted circles are the cor-responding two conjugate image loci 1 and 2, res-
pectively. Here, ks denotes a grating vector for ascattered wave derived from the ith recording beamhaving a wave vector ki�i � 1, 2�, and �Ks originatesat the intersection point O of k1 and k2. When the tworecording beams interfere with each other in the sam-ple, a refractive index grating having a grating vectorK�� k2 � k1� builds up. At the same time, noisegratings with different �Ks’s also form. The intersec-tion points A and O of the primary image loci 1 withthe conjugate image loci 2 form a circle in the three-dimensional space. This circle describes the locus ofBragg-matched scattered waves derived from the re-cording beam having k1. Likewise, the intersectionpoints A� and O describe the loci of Bragg-matchedscattered waves derived from the recording beamhaving k2. These two loci correspond to the samesingle diffraction cone and give a circle on the screenbehind the sample as seen in Fig. 2. In addition, theother two diffraction cones corresponding to the locuspassing through the intersection points B and O (seeFig. 3) are orthogonal to the cone of the forward-goingscattered radiation and give the two lines on thescreen as seen in Fig. 2.
Figures 4(a) and 4(b) show film-thickness depen-dences of the steady-state scattering loss at a wave-length of 633 nm for several concentrations of SiO2
nanoparticles when curing was made under incoher-ent [Fig. 4(a)] and coherent two-beam [Fig. 4(b)] ex-
Fig. 2. (Color online) Temporal evolution of cross section of the transmitted light when the two-beam exposure started at 0 s. Two brightspots observed at 1 s correspond to the two transmitted beams.
Fig. 3. (Color online) Schematic explanation of observed holo-graphic scattering in terms of the concept of the Ewald sphereconstruction.
posures, respectively. The scattering loss was definedas unity minus the transmittance for the incoherentexposure and as unity minus the ratio of the sum ofthe transmitted and the first-order Bragg diffractedpowers to the incident power for the coherent expo-sure, respectively, both with Fresnel correction. InFig. 4(a) the results of the least-squares fits of theRayleigh scattering formula [33,37] to the data withthe nanoparticle diameter as a fitting parameter areplotted as dashed, dot-dashed, and solid curves forthe SiO2 nanoparticle concentrations of 11, 20, and 34vol.%, respectively. It can be seen that the scatteringprocess is more or less dominated by Rayleigh scat-tering regardless of the film thickness and the SiO2nanoparticle concentration. The estimated values forthe nanoparticle diameter were 26, 22, and 18 nm forthe SiO2 nanoparticle concentrations of 11, 20, and 34vol.%, respectively, when the bulk refractive index of1.46 was used for SiO2 nanoparticles. These esti-mated values are somewhat larger than the averagesize of 13 nm estimated from a transmission electron-microscope image of SiO2 nanoparticles deposited oncarbon-coated grids after evaporating an MIBK sus-pension of the SiO2 sol. Such a difference may becaused partly by partial aggregation of SiO2 nanopar-ticles during photopolymerization. Another possiblereason is attributable to the SiO2 nanoparticle’s re-fractive index used in the curve fitting. It is reportedthat the refractive index of a nanoparticle tends to
decrease with a decrease of its size [38]. Indeed, theestimated values for the nanoparticle diameter were16, 13, and 11 nm for the SiO2 nanoparticle concen-trations of 11, 20, and 34 vol.%, respectively, whenthe refractive index of 1.32 (a 10% reduction from thebulk value) was used for SiO2 nanoparticles. Thisresult suggests the necessity of measuring the effec-tive refractive index of SiO2 nanoparticles when oneacurately estimates the average size of SiO2 nanopar-ticles from the scattering loss measurement.
It can be seen from Fig. 4(b) that while the scat-tering losses are more or less 1% for the samplesthinner than 100 �m, they increase significantly forthe samples thicker than 100 �m. Such a drastic in-crease in the scattering loss can be attributed tothe onset of holographic scattering in the thick��100 �m� samples. It can also be seen from Fig. 4(b)that the film-thickness dependence tends strongerwith an increase of the SiO2 nanoparticle concentra-tion. We also examined the effect of light reflectionfrom the substrate–air boundary on holographic scat-tering by measuring the scattering loss with the an-tireflection treatment to the glass substrates. It wasfound that a factor-of-2 reduction in the scatteringloss was possible for a sample of 200-�m thicknesswith the SiO2 nanoparticle concentration of 34 vol.%.But such a treatment would not substantively reduceholographic scattering since holographic scattering isessentially a volume effect. Because the turbidity forRayleigh scattering has a cubic dependence on thediameter of spherical scatters [33,37], the observedresult suggests the use of SiO2 nanoparticles smallerthan 10 nm to suppress holographic scattering with-out decreasing �n in a thick ��100 �m� film. Indeed,we found that the use of 3-nm ZrO2 nanoparticlesgave a very low scattering loss [30] so that no observ-able holographic scattering pattern was observedeven for a thick �� 200 �m� film.
To investigate the significance of holographic scat-tering in the volume grating formation in a SiO2
nanoparticle-dispersed photopolymer film, we mea-sured the diffraction efficiency as a function of Bragg-angle detuning of a single volume grating recorded insamples of 46-�m and 197-�m thicknesses with theSiO2 nanoparticle concentration of 34 vol.%. It can beseen from Fig. 5(a) that the data are well described byKogelnik’s formula [39] for an unslanted transmis-sion grating. The estimated value for �n was 7.7� 10�3. This fitting result implies that the volumegrating was recorded uniformly along the thicknessdirection. It is consistent with our observation thatholographic scattering was not noticeable for thesamples thinner than 100 �m, as seen in Fig. 4. Onthe other hand, it can be seen from Fig. 5(b) that theBragg-angle detuning data for the thicker sampleexhibit overmodulation [27] with the apodized side-lobe variation that is typical for a nonuniform volumegrating (i.e., a volume grating having the linear tiltchange [40] or the varied grating modulation alongthe thickness direction [41,42]). Moreover, the slightasymmetry in the Bragg-detuning curve is seen. This
Fig. 4. (Color online) Steady-state scattering loss versus filmthickness for SiO2 nanoparticle-dispersed photopolymer samplesfor several concentrations (�, 11 vol.%; ●, 20 vol.%; X, 34 vol.%) ofSiO2 nanoparticles. (a) Curing was made under incoherent anduniform (365-nm LED) illumination. Each curve correspond to theleast-squares fit of the Rayleigh scattering formula to each data. (b)Curing was made under coherent two-beam illumination. The totalrecording intensity was 100 mW�cm2 for both incoherent and co-herent exposure cases.
can be attributed to the nonlinear bending of thevolume grating [40]. These linear and nonlinearphase variations of the initially unslanted volumegrating may be caused by polymerization shrinkagealong the lateral and transversal directions in a thickphotopolymer film [43]. We consider that the variedgrating modulation along the thickness direction isalso induced by the contrast degradation of theintensity-interference fringe pattern due to the for-mation of noise gratings. To examine this possibility,we performed the least-squares fit of Uchida’s for-mula [42] for an exponentially decaying volume grat-ing along the thickness direction to the data, which isshown as the solid curve in Fig. 5(b). This formulaassumes �n�z� given by �n�0�exp��z�leff�, where z isthe distance measured from the front surface alongthe thickness direction and leff is the effective modu-lation length. Good agreement with the data wasobtained with �n�0� � 6.0 � 10�3 and leff � 182 �m.The estimated value for �n�0� is close to that of the46-�m sample. The estimated value for leff is smallerthan the mechanical thickness of 197 �m and is con-sistent with our observation that the 46-�m samplehas a uniform volume grating. Although the asym-metry and the lack of well-defined null points that areseen in Fig. 5(b) may be partly attributed to the mul-tidimensional polymerization shrinkage, the lattertrend would also be attributable to the formation ofnoise gratings leading to holographic scattering. Wenote that the effect of absorption of the recordingbeams on the varied grating modulation is negligible
since the linear absorption coefficient of the initiatorat 532 nm is 4.5 cm�1, which corresponds to the skindepth of 2.2 mm. We also note that the degree of theobserved asymmetry in the Bragg-angle detuningcurve varied from one thick sample to another. Wespeculate that such low measurement reproducibilityis caused by uncontrolled mechanical stress takenplace in prepared samples as a result of the volumeshrinkage.
3. Conclusion
We have described holographic scattering in SiO2nanoparticle-dispersed photopolymer films. Depen-dences of film thickness and nanoparticle concentra-tion on the scattering loss have been evaluated. Wehave shown that the geometric feature of the holo-graphic scattering pattern in the two-beam recordingsetup can be satisfactorily explained by the Ewaldsphere construction. It is found that despite the lowincoherent light scattering nature of SiO2 nanoparticle-dispersed photopolymer films, films thicker than100 �m exhibit noticeable holographic scattering inthe two-beam recording setup. Although the gratingbending in a thick film can be avoided essentially bya substantive reduction of polymerization shrinkage,holographic scattering and the associated gratingnonuniformity would remain significant as long ascoherent light is used. Because Rayleigh scatteringfrom nanoparticles as well as from transient super-molecular structures is the main cause for such un-wanted phenomena in the single-grating recordingsetup, the use of nanometer-size nanoparticles is es-sential. On the other hand, it is reported that noisegratings are unlikely to form in the angularly multi-plex recording setup for holographic data storage [9].This result must also be examined for nanoparticle-dispersed photopolymer films. Our investigation inthis direction is currently under way.
This work was supported by the 21st CenturyCenter-of-Excellence (COE) program, the Universityof Electro-Communications, granted by Ministry ofEducation, Culture, Sports, Science and Technology,Japan.
References1. K. Biedermann, “The scattered flux spectrum of photographic
materials for holography,” Optik 31, 367–389 (1970).2. R. R. A. Syms and L. Solymar, “Noise gratings in photographic
emulsions,” Opt. Commun. 43, 107–110 (1982).3. L. Solymar and G. D. G. Riddy, “Noise gratings for single- and
double-beam exposures in silver halide emulsions,” J. Opt. Soc.Am. A 7, 2107–2108 (1990).
4. A. Fimia, R. Fuentes, and A. Beléndez, “Noise gratings inbleached silver halide diffuse-object holograms,” Opt. Lett. 19,1243–1245 (1994).
5. J. M. Moran and I. P. Kaminow, “Properties of holographicgratings photoinduced in polymethyl methacrylate,” Appl.Opt. 12, 1964–1970 (1973).
6. E. S. Gyul’nazarov, T. N. Smirnova, D. V. Surovtsev, and E. A.Tkhonov, “Light scattering in holograms written on photo-polymerizing compositions,” J. Appl. Spectrosc. 51, 728–733(1989).
7. A. Beléndez, A. Fimia, L. Caretero, and F. Mateos, “Self-
Fig. 5. (Color online) Steady-state diffraction efficiencies as afunction of Bragg-angle detuning for samples of (a) 46-�m and(b) 197-�m thicknesses with the SiO2 nanoparticle concentration of34 vol.%. The solid curve shown in (a) and (b) is the least-squaresfit of Kogelnik’s (Uchida’s) formula to the data. The total recordingintensity was 100 mW�cm2.
induced phase gratings due to the inhomogeneous structure ofacrylamide photopolymer systems used as holographic record-ing materials,” Appl. Phys. Lett. 67, 3856–3858 (1995).
8. L. Carretero, S. Blaya, R. Mallavia, R. F. Madrigal, and A.Fimia, “A theoretical model for noise gratings recorded inacrylamide photopolymer materials used in real-time hologra-phy,” J. Mod. Opt. 45, 2345–2354 (1998).
9. J. A. Frantz, R. K. Kostuk, and D. A. Waldman, “Coherentscattering properties of a cationic ring-opening volume holo-graphic recording material,” Proc. SPIE 4296, 267–273 (2001).
10. J. A. Frantz, R. K. Kostuk, and D. A. Waldman, “Model ofnoise-grating selectivity in volume holographic recording ma-terials by use of Monte Carlo simulations,” J. Opt. Soc. Am. A21, 378–387 (2004).
11. M. A. Ellabban, M. Fally, H. Uršic, and I. Drevenšec-Olenik,“Holographic scattering in photopolymer-dispersed liquid crys-tals,” Appl. Phys. Lett. 87, 151101 (2005).
12. W. Phillips, J. J. Amodei, and D. L. Staebler, “Optical andholographic storage properties of transition metal doped lith-ium niobate,” RCA Rev. 33, 94–109 (1972).
13. R. Magnusson and T. K. Gaylord, “Laser scattering inducedholograms in lithium niobate,” Appl. Opt. 13, 1545–1548(1974).
14. V. Voronov, I. Dorosh, Yu. Kuz’minov, and N. Tkachenko,“Photoinduced light scattering in cerium-doped barium stron-tium niobate crystals,” Sov. J. Quantum Electron. 10, 1346–1349 (1980).
15. R. A. Rupp and F. W. Dress, “Light-induced scattering in pho-torefractive crystals,” Appl. Phys. B 39, 223–229 (1986).
16. M. Imlau, Th. Woike, R. Schieder, and R. A. Rupp, “Holographicscattering in centrosymmetric Na2[Fe(CN)5NO] � 2H2O,” Phys.Rev. Lett. 82, 2860–2863 (1999).
17. M. Fally, M. A. Ellaban, R. A. Rupp, M. Fink, and J. Wolf-berger, “Characterization of parasitic grating in LiNbO3,”Phys. Rev. B 61, 15778–15784 (2000).
18. T. N. Smirnova and E. A. Tikhonov, “Conical scattering of laserbeams in active solutions,” Sov. J. Quantum Electron. 9, 93–97(1979).
19. M. R. B. Forshaw, “Explanation of the “Venetial blind” effect inholography using the Ewald sphere concept,” Opt. Commun. 8,201–206 (1973).
20. M. R. B. Forshaw, “Explanation of the two-ring diffractionphenomenon observed by Moran and Kaminow,” Appl. Opt. 13,2 (1974).
21. S. I. Ragnarsson, “Scattering phenomena in volume hologramswith strong coupling,” Appl. Opt. 17, 116–127 (1978).
22. G. D. G. Riddy and L. Solymar, “Theoretical model of recon-structured scatter in volume holograms,” Electron. Lett. 22,872–873 (1986).
23. L. Solymar and J. C. W. Newell, “Silver halide noise gratingsrecorded in dichromated gelatin,” Opt. Commun. 73, 273–276(1989).
24. R. K. Kostuk and G. T. Sincerbox, “Polarization sensitivity ofnoise gratings recorded in silver halide volume holograms,”Appl. Opt. 27, 2993–2998 (1988).
25. A. Beléndez, L. Carretero, and I. Pascual, “Polarization influ-ences on the efficiency of noise gratings recorded in silverhalide holograms,” Appl. Opt. 32, 7155–7163 (1993).
26. N. Suzuki and Y. Tomita, “Holographic recording in TiO2
nanoparticle-dispersed methacrylate photopolymer films,”Appl. Phys. Lett. 81, 4121–4123 (2002).
27. Y. Tomita and H. Nishibiraki, “Improvement of holographicrecording sensitivities in the green in SiO2 nanoparticle-dispersed methacrylate photopolymers doped with pyrro-methene dyes,” Appl. Phys. Lett. 83, 410–412 (2003).
28. N. Suzuki and Y. Tomita, “Silica-nanoparticle-dispersedmethacrylate photopolymers with net diffraction efficiencynear 100%,” Appl. Opt. 43, 2125–2129 (2004).
29. Y. Tomita, K. Furushima, K. Ochi, K. Ishizu, A. Tanaka, M.Ozawa, M. Hidaka, and K. Chikama, “Organic nanoparticle(hyperbranched polymer)-dispersed photopolymers for volumeholographic storage,” Appl. Phys. Lett. 88, 071103 (2006).
30. N. Suzuki, Y. Tomita, K. Ohmori, M. Hidaka, and K. Chikama,“Highly transparent ZrO2 nanoparticle-dispersed acrylatephotopolymers for volume holographic recording,” Opt. Ex-press 14, 12712–12719 (2006).
31. Y. Tomita, N. Suzuki, and K. Chikama, “Holographic manipu-lation of nanoparticle-distribution morphology in nanoparticle-dispersed photopolymers,” Opt. Lett. 30, 839–841 (2005).
32. Y. Tomita, K. Chikama, Y. Nohara, N. Suzuki, K. Furushima,and Y. Endoh, “Two-dimensional imaging of atomic distribu-tion morphology created by holographically induced masstransfer of monomer molecules and nanoparticles in a silica-nanoparticle-dispersed photopolymer film,” Opt. Lett. 31,1402–1404 (2006).
33. H. C. van de Hulst, Light Scattering by Small Particles (Dover,1957).
34. N. Kukhtarev, V. Markov, S. Odoulov, M. Soskin, and V.Vinetskii, “Holographic storage in electrooptic crystals. II.Beam coupling-light amplification,” Ferroelectrics 22, 961–964(1979).
35. A. P. Yakimovich, “Dynamic self-amplification of scatteringnoise in voume-hologram recording,” Opt. Spectrosc. 49, 191–193 (1980).
36. V. A. Barachevskii, “Photopolymerizable recording media forthree-dimensional holographic optical memory,” High. EnergyChem. 40, 131–141 (2006).
37. W. Heller, “Elements of the theory of light scattering. I. Scat-tering in gases, liquids, solutions, and dispersions of smallparticles,” Rec. Chem. Prog. 20, 209–233 (1959).
38. T. Kyprianidou-Leodidou, W. Caseri, and U. W. Suter, “Sizevariations of PbS particles in high-refractive-index nanocom-posites,” J. Phys. Chem. 98, 8992–8997 (1994).
39. H. Kogelnik, “Coupled wave theory for thick hologram grat-ings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
40. L. B. Au, J. C. W. Newell, and L. Solymar, “Non-uniformitiesin thick dichromated gelatin transmission gratings,” J. Mod.Opt. 34, 1211–1225 (1987).
41. D. Kermisch, “Nonuniform sinusoidally modulated dielectricgratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
42. N. Uchida, “Calculation of diffraction efficiency in hologramgratings attenuated along the direction perpendicular to thegrating vector,” J. Opt. Soc. Am. 63, 280–287 (1973).
43. J. E. Boyd, T. J. Trentler, R. K. Wahi, Y. I. Vega-Cantu, andV. L. Colvin, “Effect of film thickness on the performance ofphotopolymers as holographic recording materials,” Appl. Opt.39, 2353–2358 (2000).
