Model of holographic recording in thermoplastic materials

Istvan Banyasz

A method for the evaluation of images reconstructed from holograms recorded in thermoplastic materialsis reported. The method is based on the use of the experimental modulation transfer function andnonlinear holographic characteristics of the recording material. Calculations have been carried out forhigh-numerical-aperture holograms of a five-element Ronchi ruling. The quality of the reconstructedimage as a function of the recording parameters has been computed. The model predicts that it ispossible to optimize holographic recording in these materials. © 1998 Optical Society of America

OCIS codes: 090.0090, 090.2900, 050.1960.

1. Introduction

It was Urbach and Meier1 who published the firstresults on holographic recording in a bilayer of anorganic photoconductor and an isolating thermoplas-tic. A great number of papers have dealt with thevarious experimental characteristics @e.g., spectralsensitivity, modulation transfer function ~MTF!, andnonlinearity# of thermoplastic holographic recordingmaterials.2–8 Several theoretical papers on theirphysical properties can be found in the literature.9–12

Thermoplastic holograms have been applied in anumber of fields, ranging from holographic nonde-structive testing to holographic memories and, morerecently, to real-time optical processing13 and holo-graphic interconnections.14

The aim of the present research was to develop amodel that can be used for the evaluation of holo-graphic images reconstructed from thermoplasticholograms. The effects of the band-limited MTF ofthermoplastic materials were included in an earliermodel.15 The effects of the nonlinearity of the re-cording material are also included in the presentmodel, analogously to a method developed for theevaluation of images retrieved from nonlinear silverhalide holograms.16 The model is applied to theevaluation of images of microline test objects recon-structed from high-numerical-aperture ~N.A.! ther-moplastic holograms.

The author is with the Laboratorio de Holografıa, DepartamentoInteruniversitario de Optica, Universidad de Alicante, Apdo. 99,E-03080 Alicante, Spain.

Received 1 May 1997; revised manuscript received 29 September1997.

0003-6935y98y112081-06$15.00y0© 1998 Optical Society of America

2. Model

The recording and reconstruction geometry is shownin Fig. 1. The one-dimensional hologram H is lo-cated along the j axis. The object plane ~i.e., the uaxis! is inclined at an angle a0 to the hologram plane.The object is transilluminated perpendicularly by aplane wave. The separation of the hologram O andthe object G centers is R0, the angle of incidence of thereference wave is ar, and the hologram length is L.The real image of the object is reconstructed by theconjugate of the reference wave kc. The intensitydistribution of the reconstructed real image is calcu-lated along the x axis.

The diffraction-limited complex amplitude distri-bution of the reconstructed real image can be calcu-lated by the following double Fresnel–Kirchhoffintegral15,16:

t~x, z! 5 *j1

j2

*u1

u2

s~u!cos u cos r

r1 r2exp@ik~r1 2 r2!#dudj,

(1)

where s~u! is the complex amplitude of the objectwave in the object plane, u and r are the inclinationangles for the diffraction at the object and hologramplanes, respectively, and r1 and r2 are the separationsof the source and observation points at the first andthe second diffraction, respectively. It is assumedthat there is no wavelength change at reconstructionand that both the reference and the reconstructingwaves are plane waves and conjugates to each other.

The principal assumption of the model is that onecan take into account the effects of the recordingmaterial on the reconstructed image by multiplying

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the diffraction-limited reconstructed amplitude bythe following function:

s@n~j!, E0~j!, V~j!# 5 s1@n~j!#s2@E0~j!, V~j!#, (2)

where s1@n~j!# is the square root of the MTF of therecording material at the local spatial frequency nmeasured at point j in the hologram line and s2@E0~j!,V~j!# is the Lin function, i.e., the square root of thediffraction efficiency of a local elementary plane-wavehologram versus the local bias exposure E0~j! and thefringe visibility V~j!.

The application of scalar diffraction theory to ther-moplastic holograms is justified by the fact that, be-cause of the thinness of the holograms and theirband-limited MTF’s, the ratio of the amplitude of thesurface modulation to the grating constant is suffi-ciently low. ~The criteria can be found, e.g., in Ref.17.! According to the measurements of Urbach andMeier,18 the maximum amplitude of deformation of athick ~d 5 1.1 mm! thermoplastic layer is approxi-mately 0.8 mm.

We have to evaluate integral ~1! in two steps.16

The complex amplitude of the object wave at thehologram is

S~j! 5 *u1

u2

s~u!cos u

r1exp ~ikr1!du. (3)

The complex amplitude of an off-axis reference waveat the hologram line is

P~j! 5 P~j!exp~2ikj sin ar!. (4)

The local bias exposure @E0~j!# and fringe visibility@V~j!# at point j can readily be obtained from Eqs. ~3!and ~4!.16 One can evaluate the actual s2@E0~j!,V~j!# function by substituting E0~j! and V~j! into Eq.~2!. The complex amplitude of the diffraction-limited first-order real image at the hologram line is

G~j! 5 M~j!P~j!S*~j!, (5)

where M is the complex amplitude of the reconstruc-tion wave and the asterisk stands for complex conju-gation.

Fig. 1. Diagram of the recording geometry.

2082 APPLIED OPTICS y Vol. 37, No. 11 y 10 April 1998

When the various diffraction orders do not overlapin the image position we obtain the complex ampli-tude of the reconstructed first-order real image bymultiplying the diffraction-limited complex ampli-tude @Eq. ~5!# by the function s@n~j!, E0~j!, V~j!# andperforming the second Fresnel–Kirchhoff integra-tion:

t~x! 5 *j1

j2

s1@n~j!#s2@E0~j!, V~j!#M~j!P~j!S*~j!cos r

r2

3 exp~ikr2!dj. (6)

The MTF ~hence its square root! of thermoplasticmaterials is often asymmetrical, but in most cases aGaussian function gives a good fit15:

s1~n! 5 expF2Sn 2 n0

c D2G , (7)

where s1 is the square root of the diffraction effi-ciency, n is the spatial frequency of the grating, n0 isthe center of the function, and c is its width at thevalue 1ye. The square root of the measured MTF ofa thermoplastic material ~after Lo et al.7! along witha fit made by Eq. ~7! can be seen in Fig. 2.

Lin functions of thermoplastic materials can be fit-ted by the same type of function as those for silverhalide recording materials16:

s2~E0, V! 5 f ~E0!@1 2 exp~2V!#expH2@V 2 V0~E0!#

2

w2~E0!J ,

(8)

where E0 is the bias exposure, V is the visibility of theinterference fringes, and f ~E0!, V0~E0!, and w~E0! are

Fig. 2. Square root of the measured MTF ~points! of a thermo-plastic material ~after Lo et al.7! and the fit ~curve! made with Eq.~7!.

parameter functions built up as products of sigmoidfunctions of the following form:

Par~E0! 5 ci013 1

exp Sci112 E0

ci12

D 1 1

1 ci1343 3 1

exp SE0 2 ci21

ci22

D 1 1

1 ci2343 3 1

exp Sci312 E0

ci32

D 1 1

1 ci334 , (9)

where Par stands for f, V0, and w, and cixx representthe three sets of constants ~i 5 f, V, W !. This s~E0,V ! function, fitted to the experimental characteristicsreported by Colburn and Tompkins,6 is shown in Fig.3. The parameters of the actual function are listedin Table 1.

Fig. 3. Lin curves of a thermoplastic material. Symbols, meas-ured by Colburn and Tompkins6; curves, fitted by Eq. ~8!. Thevisibility is V 5 1.0 ~filled circles!, 0.8 ~open squares!, 0.575 ~filledtriangles!, 0.309 ~open circles!. The parameters are shown inTable 1.

Table 1. Parameters of the s~E0, V! Function of the ThermoplasticMateriala

Index ~xx! cfxx cVxx cwxx

01 0.349 0.437 0.44411 4.8 2.7 14.512 1.0 2.0 6.013 0.0 0.0 0.521 250,000.0 18.0 12.022 120,000.0 30.0 50.023 0.0 0.80 0.8031 150.0 1700.0 1200.032 2300.0 4000.0 30,000.033 1.14 0.80 0.85

aSee Eqs. ~8! and ~9!.

3. Numerical Simulations

A. Recording Parameters

The model was applied to the evaluation of holo-graphic images of micro-objects reconstructed fromhigh-N.A. thermoplastic holograms. The object wasa five-element Ronchi ruling consisting of 1.0-mm-wide lines. The wavelength of the light was 632.8nm. The hologram width was L 5 84 mm, the objectwas located on the hologram normal at 32 mm fromthe hologram center, and the reference wave made anangle of ar 5 240° with the hologram normal.Plane auxiliary waves were assumed. The Lincurves shown in Fig. 3 were used in all the followingcalculations.

Because no MTF data were provided along with thenonlinear characteristics in Ref. 6, n0 was estimatedfrom the thickness of the thermoplastic layer on thebasis of previous publications2,6,18:

n0 @linesymm# 5382

d @mm#, (10)

where d is the thickness of the thermoplastic layer.This function, fitted to the experimental data of Lo etal.,7 is shown in Fig. 4.

In our calculations n0 was set to 1000 linesymm ~a0.38-mm-thick layer!. As for the parameter c, Ur-bach and Meier18 found that the bandwidth of theMTF of a photoconductor–thermoplastic material isof the order of 1y6 d, where d is the thickness of thethermoplastic layer in millimeters and transmissionhologram rams are recorded. The parameter c wasset to 750 linesymm in the present calculations.

B. Characterization of the Reconstructed Image

The quality of the reconstructed image of the Ronchiruling is described by the following quantities:

1. Contrast of the image, defined as

C 5 ItryIop, (11)

where Itr is the integral of the reconstructed intensityover the transparent object lines and Iop is that over

Fig. 4. Spatial frequency belonging to the maximum of the ma-terial MTF as a function of the thickness of the thermoplasticlayer. Points, experimental data published in Ref. 7; curve, a fitby Eq. ~9!.

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the opaque ones, including two opaque lines at theedges of the test pattern.

2. Total intensity or brightness IT of the recon-structed image, defined as the integral of the recon-structed intensity along the whole object.

3. Fluctuation of the reconstructed image:

D 5 F(i51

n

~Ii 2 Iav!2

n 2 1G1y2

, (12)

where Ii is the reconstructed intensity over the ithtransparent object line, n is the number of transpar-ent object lines, and Iav is the average of the Ii values.

4. Results

Reconstructed holographic images of the test objecthave been computed for a wide range of maximumbias exposures and minimum beam ratios. Con-trast, brightness, and fluctuation of the reconstructedimage of the test object as functions of the maximumbias exposure, at a minimum beam ratio of R 5 0.1,are shown in Fig. 5. The contrast has a sharp max-imum at E0 5 34 mJycm2, the brightness has a broadmaximum near 120 mJycm2, and the fluctuation has

Fig. 5. Contrast, brightness, and fluctuation of the reconstructedimage of a five-element Ronchi ruling versus the maximum biasexposure. R is the minimum beam ratio, 0.1.

Fig. 6. Contrast, brightness, and fluctuation of the reconstructedimage of a five-element Ronchi ruling versus the maximum biasexposure. R is the minimum beam ratio, 1.0.

2084 APPLIED OPTICS y Vol. 37, No. 11 y 10 April 1998

two minima, at E0 5 3 mJycm2 and at E0 5 34mJycm2. It is obvious that the quality of the recon-structed holographic image could be optimal with asetting of E0 5 34 mJycm2.

The same characteristics of the reconstructed im-

Fig. 7. Contrast, brightness, and fluctuation of the reconstructedimage of a five-element Ronchi ruling versus the maximum biasexposure. R is the minimum beam ratio, 10.

Fig. 8. Contrast, brightness, and fluctuation of the reconstructedimage of a five-element Ronchi ruling versus the minimum beamratio. E0 is the maximum bias exposure, 14 mJycm2.

Fig. 9. Contrast, brightness, and fluctuation of the reconstructedimage of a five-element Ronchi ruling versus minimum beam ratio.E0 is the maximum bias exposure, 38 mJycm2.

age at a beam ratio of unity are presented in Fig. 6.Here both the contrast and the total intensity havelow values at the minimum of the fluctuation, but itis possible to obtain images of acceptable quality if E0is set somewhat above that minimum, e.g., to 14mJycm2.

If R is set to 10 ~Fig. 7!, the fluctuation has a sharpminimum at E0 5 2.0 mJycm2 and a flat one between

Fig. 10. Contrast, brightness, and fluctuation of the recon-structed image of a five-element Ronchi ruling versus minimumbeam ratio. E0 is the maximum bias exposure, 82 mJycm2.

70 and 150 mJycm2. However, at the first minimumthe total intensity is low, so the optimum exposure isat the second minimum. For the above results to bechecked, the characteristics of the reconstructed im-age were computed by the fixing of the bias exposureat the optimum values obtained previously and thechanging of the beam ratio. It can be seen from Fig.8 that indeed there is a minimum of the fluctuation atR 5 1.2 if E0 is set to 14 mJycm2. There is another~somewhat lower! minimum of the fluctuation at R 50.32, but the corresponding contrast and total inten-sity are halves of those that belong to R 5 1.2 ~Fig. 8!.

If we set E0 to 38 mJycm2, we can see from Fig. 9that the fluctuation has a minimum at R 5 0.1 andthere is a maximum in the contrast at the same valueof beam ratio. Although the maximum brightness isnear R 5 0.2, it is still rather high at the aboveminimum. The second minimum of the fluctuationis at R 5 0.7. It has a lower brightness and a sub-stantially lower contrast than the former one, so theoptimum R is 0.1.

Finally, characteristics computed for E0 5 82 mJycm2 are shown in Fig. 10. Here there are two min-ima in the fluctuation, one at R 5 0.03 and another atR 5 10. At R 5 0.03, D is approximately 10 timeshigher than at R 5 10, so the optimum value is theR 5 10, in spite of the lower values of It and C.

Fig. 11. Calculated reconstructed images of the test object. E0, maximum bias exposure; R, minimum beam ratio.

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A set of reconstructed images of the test object isshown in Fig. 11. The parameters were chosen toshow both high- and low-quality images. From thepreceding discussion, we find that good-quality im-ages can be obtained at R 5 1.0 and E0 5 14 mJycm2.A nontrivial result of the model is that a high-qualityimage can be reconstructed even if the beam ratio hasbeen reversed over a part of the hologram ~R 5 0.1and E0 5 38 mJycm2!. If R 5 10, one can reconstructimages of acceptable quality by setting the maximumbias exposure to either 38 or 82 mJycm2. Note thatthe effects of the recording material on the recon-structed image are, obviously, less pronounced formacroscopic objects ~see, e.g., the experimental re-sults in Ref. 19!.

5. Conclusion

A diffraction model of nonlinear holographic record-ing in photoconductor thermoplastic recordingmaterials has been developed. The numerical sim-ulations based on the model suggest that it is pos-sible to optimize holographic recording in thesematerials. Consequently, reducing the resolutionlimit in such applications as reconfigurable holo-graphic interconnections seems to be feasible.

The author was with the Hungarian Academy ofSciences when part of this research was performed.Support from the Hungarian Fund for Scientific Re-search under grant F-014352 is gratefully acknowl-edged. The author is also indebted to the SpanishMinistry of Education and Scientific Research forgranting him a scholarship.

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