Analysis of hybrid holographic gratings by thin-gratingdecomposition method
Hiroyuki Ichikawa, Jari Turunen, and Mohammad R. Taghizadeh
Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK
Received May 7, 1992; revised manuscript received December 1, 1992; accepted December 7, 1992
The thin-grating decomposition method is employed to evaluate the properties of hybrid holographic gratings,i.e., volume holograms recorded with a plane reference wave and an object wave produced by a periodiccomputer-generated hologram. The effects of the object wave period, defocusing, spatial filtering, referenceto object beam ratio, and off-Bragg incidence on the quality of the regenerated signal are investigated; hybridkinoform array generators are used as an illustration.
1. INTRODUCTION
A hybrid hologram' is a volume holographic optical ele-ment recorded with a plane reference wave and an objectwave front produced by a computer-generated hologram(CGH), as illustrated in Fig. 1. The hybrid hologramcombines the flexibility of the CGH (the ability to recon-struct a mathematically defined wave front) with the highdiffraction efficiency of a volume phase hologram. If theCGH is designed by means of the traditional coding meth-ods,2 the spatially filtered object wave front contains largeintensity modulations that the volume holographic mate-rial cannot record faithfully. As a result, the signalregenerated by the hybrid hologram has a lower signal-to-noise ratio than the signal regenerated by the CGH. Thesignal quality of the CGH, which is typically rather poor,can be recovered if one uses a high reference-to-objectbeam intensity ratio in the recording but only at theexpense of diffraction efficiency, which must be reducedto approximately 50%.i
The limitation on diffraction efficiency can be removedif one employs the hybrid kinoform technique.3 Here,first a phase-only CGH with an unrestricted phase profileis designed by means of one of the iterative coding methodreviewed in Refs. 4 and 5. Then a linear phase term isadded and the resultant phase profile is quantized to pro-duce a binary interferogram. When the interferogram isrealized in the form of a binary-amplitude transparency,the object wave front is a close approximation of the itera-tively designed kinoform phase profile. This object wave,which has virtually no intensity modulation, can be re-corded with high fidelity, and a high-quality signal appearsin the first diffraction order of the hybrid hologram, whichcan have a Bragg efficiency of almost 100%.
In this paper the fundamental performance limits of hy-brid holograms are evaluated by numerical simulations.We analyze the degradation of the signal quality caused byvolume diffraction effects and give lower bounds for thelateral dimensions of hybrid holograms. We also investi-gate the effects of the reference-to-object beam intensityratio, the width of the spatial filter and defocusing ofthe object wave in the recording setup, and the incidentangle dependence of the signal quality in regeneration.
Throughout the analysis we consider periodic hybrid holo-grams with a one-dimensional signal consisting of N adja-cent diffraction orders with equal diffraction efficiencies.The performance of these holograms can be characterizedby two figures of merit that are easy to visualize, i.e., thecombined diffraction efficiency of the N desired ordersand the uniformity of the signal beam array.
In the theoretical analysis of periodic hybrid hologramsnumerical implementations of the rigorous electromag-netic theory of diffraction gratings', would provide un-questionable answers. However, these methods cannotat present cope with practical hybrid hologram gratings,which have periods of at least several hundred wavelengthsand consequently generate hundreds or thousands ofpropagating diffraction orders. Therefore we must resortto approximate methods. Of the wide variety of suchtechniques, 7 we chose the thin-grating decompositionmethod (TGDM).5-l0 This method was implemented bymeans of a fast-Fourier-transform algorithm, as is cus-tomary in fiber and integrated optics,"- 3 where the tech-nique is known as the beam-propagation method.
In Section 2 we describe the refractive-index profileinside a hybrid holographic grating and present the nu-merical model for solving the diffraction problem. In Sec-tion 3 the range of validity and the accuracy of the modelare investigated, and the simulation results are given inSection 4.
2. NUMERICAL MODEL
We assume that the CGH in Fig. 1 is periodic and that itscomplex-amplitude transmission function is constant inthe y direction. The object wave front is then a super-position of plane waves corresponding to the diffractionorders m = M1,..., M2, with relative amplitudes Em,passed by the spatial filter SF We also assume that therecording material, which is confined between planes z =0 and z = HI, is surrounded by dielectric media with re-fractive index no equal to the real part of the complexrefractive index of the recording material. Then the com-ponents of the object wave are incident at angles 0
Reference wave intensity modulation is now given by
I(x, z) = IEr(x, z) + E0(x, Z)12.
Ideally, the refractive-index modulation inside the record-ing material after the exposure would be a linear functionof the local intensity, i.e.,
CGH LI SF L2 RM
fi L -fi f2 f2
Fig. 1. Setup for recording hybrid kinoforms. CGH, a binaryamplitude computer-generated hologram; Li, lens with focallength fi; L2, lens with focal length f; SF, spatial filter, RM,recording material.
space and T is the period of the object wave front in thex direction. The object wave interferes with a plane ref-erence wave, incident at an angle ,, which is at least afew times larger than the angular spread 10M1 - OMJI of theobject wave components. Finally we assume that
T sin Or = pAo/no, (1)
where p is an integer. This assumption ensures that theperiod of the interference pattern is the same as the periodof the object wave.
A. Grating FormationWhen dynamic effects during the recording process'4"5
are neglected, they components of the electric fields of theTE-polarized reference and object waves are invariantduring the exposure and can be written in the form
Er(x, z) = EA exp[ik(xsx + ZSr,z)]X (2)
M2
E0 (x, z) = > Em exp[ik(xsmx + zsm,z)], (3)m=Ml
respectively. Here k represents the complex wave numberof the recording material before exposure, i.e.,
k = 27rh/Ao = 2X(no + iK)/Ao, (4)
where K is the extinction coefficient. The amplitudes EAand Em are normalized such that
IEA12 + > IEm2 = 1, (5)m=Ml
and we define a quantity
M2 -
= IEAI 2 E IEm 2 (6)m=Ml
to describe the intensity ratio of the reference and the ob-ject waves.
In many recording materials, absorption during re-cording is not negligible, and the formalism of Ref. 16(pp. 615-616) should be used to describe the inhomoge-neous wave field inside the recording material.' 7 How-ever, we assume for simplicity that the recording materialis dielectric. Then we can write k = 2,7no/Ao, Sm x =sin Om, Sm,z = COS
0m, Sr, = sin r, and srz = cos Or. The
where at is a constant and T is the exposure time. In prac-tice, however, the index modulation may saturate withhigh exposure, and it is reasonable to write'8
where nm and Wo are constants.The geometry of the index modulation structure may
change during the processing steps after exposure. Theeffects of fringe bending'9 can be taken into account if oneshifts the calculated values of the local refractive indexin the x direction by an amount that depends on thez coordinate. Likewise, a change of the grating thicknesscan be modeled if one simply scales the profile in thez direction.
B. Summary of the Thin-Grating Decomposition MethodThe TGDM can be described as follows. First, the truerefractive index modulation of the volume grating, givenby Eq. (8) or (9), is replaced by an artificial modulationstructure consisting of J infinitely thin gratings locatedin planes zj = jd, j = 1,.. , J, separated by slabs of homo-geneous material of refractive index no and thicknessd = H/J. Each thin grating has a complex-amplitudetransmission function that approximates the effect that asingle slab (j - 1)d c z < jd of the volume gratingwould have on an incident plane wave. Therefore thefirst thin grating produces a multiplicity of plane wavesthat propagate into the plane of the second grating, whereeach wave is divided into a new set of plane waves. Thissuccessive diffraction process is carried out until the endsurface of the modulated region is reached. The resultsof the TGDM converge when the number of slabs is in-creased.
C. Diffraction by a Thin GratingMathematically the complex-amplitude transmission func-tion of a single thin grating may be defined as
tj(x) = exp[ikO r An(x, z)dz],f(j-l)d;or as
where ko = 27rno/Ao. We use the latter, more approximateexpression, because Eqs. (10) and (11) converge toward thesame result when the number of slabs J is increased toproduce an overall convergence. The optical field imme-diately behind the jth thin grating, E+(x,jd), is now re-lated to the field just in front of that grating, E-(x, jd), bythe expression
E+(x,jd) = tj(x)E-(xjd). (12)
(7)
An(x, z) = JfTI(x, z), (8)
An(x, z) = nm{1 - exp[-TI(x, z)/%0]}, (9)
(10)
tj(x) = exp{ikoAn[x,(j - 1)d]d}, (11)
Ichikawa et al.
1178 J. Opt. Soc. Am. A/Vol. 10, No. 6/June 1993
The first thin grating, located in the plane z = d, is illu-minated by a unit-amplitude plane reference wave
ER (x, z) = exp[iko no (x sin 0R + z cos OR)] . (13)
Here, OR is the angle of incidence, and we set E-(x,d) =
ER(X, d).
D. Propagation through a Homogeneous SlabTo complete the mathematical description of the algo-rithm, we need to relate the fields E+(x,jd) and E-[x,(j + 1)d]. The propagation of a scalar wave field betweenthe planes z = jd and z = (j + 1)d in a homogeneous ma-terial may be described exactly by means of the angularspectrum representation; for a field of period T, we have
E-[x,(j + 1)d] = > Am(jd)exp[i27r(umx + wind)],M-
(14)
where the coefficients
rT
Am( jd) = T' JE+(x, jd)exp(- i27rumx)dx (1
represent the amplitudes of the diffracted plane waves,
Um = mIT, (16)
_ [(no/Ako)2- m2] 112 when UM2 (n,/Ak )2Wm = jij[um2 - (no/Ao)2]"12 when Um2 > (no/A0)2
(17)
This formulation of the propagation model assumes thatone of the diffracted plane waves propagates along thez axis; i.e., the incident angle of the reconstruction wavemust satisfy the condition
T sin OR = qA,/no, (18)
where q is an integer.20
The propagation model presented above is consistentwith the definition of the transmission function, Eq. (10)or (11), only if the paraxial approximation
Wm no/Ao - Um2 Ao/2no (19)
is made in Eq. (17). Then the diffraction efficiencies
71m = Am(d)12 cos Om/cOs OR (20)
of the diffraction orders produced by the hybrid hologramreduce to
7 Im= IA.(d)12 , (21)
and energy is conserved in the formalism.For numerical analysis the summation in Eq. (14) must
be truncated: we retain L diffraction orders m = -L/2,.... ,L/2 - 1. If one also assumes that the integral inEq. (15) is evaluated by sampling the integrand at L pointsxi = TIL, I = 0,... , L - 1, Eqs. (14) and (15) may be writ-ten as
L/2-1E-[lT/L,(j + I)d] = 2 Am(jd)exp(i2wmd)
m--L12
X exp(i2ir1m/L), (22)
where
L-1Am( jd) = L1 E+(lT/L,jd)exp(-i2i1dm/L). (23)
1=0
The coefficients Am( jd) can be evaluated if one calculatesthe discrete Fourier transform (DFT) of the sequenceE+(lT/L,jd). Similarly, the electric field in the corre-sponding sample points xi = ITIL in the plane z = (j +1)d is the inverse discrete Fourier transform (IDFT) ofthe sequence Am jd)exp(i2rwmd).
If one defines the DFT and IDFT as
L-1fq = DFT(fp)= L- 2 f, exp(-i27rpq/L),
p-0
q = 0,1 ... ,L -1, (24)L-1
fp = IDFT(fq) = L-1/2 > fq exp(i2 7rpq/L),q=0
p = 1, L - 1, (25)
and replaces the parameter m by m' = m + L/2, Eqs. (22)5) and (23) take the forms
E-lT/L,(j + l)d]= (-l)L" 2 X IDFT[Am'(jd)exp(i27rwWm'd)], (26)
Am (jd) = L-12 X DFT[(-1)'E+(IT/L,jd)]. (27)
Standard fast-Fourier-transform (FFT) algorithms suchas the one in the Numerical Algorithm Group library 2'can be used to evaluate the DFT and the IDFT.
3. VALIDITY AND ACCURACY OF TGDMThe TGDM algorithm is known to give accurate results ifthe sampling in both the x and z directions is adequate(Subsection 3.B), the amplitudes of the reflected diffrac-tion orders are negligible, and the angular range in whichthe transmitted orders have significant amplitudesis much less than rad. 2
13 However, in our case, the
reference beam angle is too large for the latter conditionto be valid. The accuracy of our implementation of theTGDM under such circumstances is investigated in Sub-section 3.C.
A. Object Wave and Recording ParametersAs an illustration of hybrid holograms, we consider hy-brid kinoform array generators.3 First the phase-onlycomplex-amplitude transmission function of an on-axiskinoform is optimized to generate an even number N ofequal-intensity diffraction orders m = - N/2 +1... , N/2.If N Ž 8, the diffraction efficiency into the N orders is7 > 0.95, the remaining light being distributed amongthe higher diffraction orders. The theoretical uniformityerror AR = (71max - 7min)/(7max + 7min) of the array is be-low 0.01. Adding a linear phase term 2Qx (where Qis an integer and the grating period T is normalized tounity) to the original phase profile of the kinoform shiftsthe signal array to orders m = Q - N/2 + 1,...,Q + N/2.Quantization of the continuous phase profile into twopermitted levels introduces uniformity error, but AR <0.06 if Q Ž 4N. When the binary-phase interferogram isrealized by a binary-amplitude transparency, the effi-
Ichikawa et al.
Vol. 10, No. 6/June 1993/J. Opt. Soc. Am. A 1179
Table 1. Performance of On-Axis KinoformDesigns and the Binary CGH's Used in This Work
N' 7YDb QC AROd
8 0.960 250 0.02528 0.960 128 0.0144
16 0.970 128 0.013932 0.971 128 0.0520
aNumber of beams.bEfficiency of the original kinoform solution.'Slope of linear phase term.dUniformity error of the object beam array.
0.95
* 0.94
. 90.93
0
0.04
0.92 AZ8 16 32 64
Number of slabsFig. 2. Convergence of the TGDM in terms of the number ofsample points and thin gratings. Solid curves, L = 8192; dashedcurves, L = 4096.
ciency into the N signal orders is reduced to ql/~r2. If thespatial filter is adjusted to pass the orders m = 1, .. ,2Q,and order m = Q is normally incident upon the recordingplate, the phase profile of the original kinoform is repro-duced rather faithfully in the entrance plane z = 0 of therecording material.'
Throughout the analysis, the thickness of the recordingmaterial is assumed to be H = 10 ,um, its bulk refractiveindex no = 1.5, and A = 488 nm. Except in Subsec-tion 3.C, the incident angle of the reference wave 0 , istaken to be 19.5°. Thus the angle between the referencewave and the order m = Q of the object wave front is 300in free space, and the period of the carrier grating is0.977 ,um. The period of the object wave is T = 1 mm,the coefficient Tr = 0.048 in Eq. (8), and y = 1 unlessother values are given. The ideal refractive-index modu-lation, Eq. (8), is assumed except in Subsection 4.D.
The design efficiences of the on-axis kinoform solutionsand the uniformity errors of the signal arrays are listed inTable 1.
B. Sampling and ConvergenceThe number L of sampling points in the x direction andthe number J of slabs in the z direction must both be largeenough to ensure that no further increase changes the re-sults given by the TGDM by a significant amount. Calcu-lation of the transmission functions tj (x) of the thingratings is a rather heavy computational task, especially ifa large number of diffraction orders are passed by the spa-tial filter. Similarly, the computational effort needed topropagate the optical field through the hybrid hologramcan be heavy; the effort depends on the number of samplingpoints as O(L log2 L) when the FFT algorithm is used.Therefore it is important to find the minimum number ofsampling points, in both the x and the z directions, that
can be used in the analysis without sacrificing numericalconvergence.
In Fig. 2 we give the results of the TGDM applied to theeight-beam array generator with Q = 128, for J = 8, 16,32, and 64 slabs, and for L = 212 = 4096 (dashed curves)and L = 213 = 8192 (solid curves) sampling points perperiod. The results for L = 2' = 16,384 sampling pointscoincide with those obtained with 8192 sampling pointswithin the resolution of the figure.
In view of the results of Fig. 2, J = 32 slabs and L =
8192 sampling points ensure the numerical convergence ofthe TGDM in the present circumstances; the value L =8192 corresponds to eight sampling points per carriergrating period.
C. Comparison with Rigorous Diffraction TheoryOwing to the large reference beam angle in a practicalhybrid kinoform recording setup, the validity of theparaxial approximation, Eq. (19), is in doubt. In Fig. 3 wecompare the result of the TGDM with those of rigorouscoupled-wave theory.22 The diffraction efficiencies oforders -1, 0, +1, and +2 of a sinusoidal phase grating areplotted, with the same parameters as in Figs. 2 and 3 ofRef. 22, i.e., SXT = 0.1815 and y = 1. In the TGDM analy-sis the slab thickness is T/10, and one grating period isrepresented by eight sample points. We analyze both anunslanted grating (reference and object beam angles are100 and -10°) and a slanted grating (200 and 400). Ac-cording to Fig. 3(a), the results of the TGDM agree satis-factorily with rigorous theory, at least up to the peak ofthe first-order efficiency curve. However, for the slantedgrating in Fig. 3(b) the paraxial TGDM fails.
C.)0.8
U.!0
5 0.6
0* 0.4
00.2
A
Thickness (HI)(a)
0 1 2 3 4 5
Thickness (H/T)(b)
Fig. 3. Diffraction efficiencies of sinusoidal gratings for (a) anunslanted grating and (b) a slanted grating. Solid curves, TGDMwith paraxial approximation; dashed curves, rigorous theory.
* i I u.uz
-- - -- - - -- - -- - - -- - -
Ichikawa et al.
-4
I
I
1180 J. Opt. Soc. Am. A/Vol. 10, No. 6/June 1993
.)0.8-
06-4
2 00 l 2 3 4 s 6
Thickness (H5r)(a)
0 ~ ~~~~~~~~+ 0.
~0.6.4-
0
0 1 2 3 4 6
Thickness (H/T)
(a)
Fig. 4. Diffraction efficiencies of sinusoidal gratings for (a) anunslanted grating and (b) a slanted grating. Solid curves, TGDMwith evanescent waves; dashed curves, rigorous theory.
Figures 4(a) and 4(b) give corresponding results, but,following Ref. 13, we employ the nonparaxial Eq. (17) forwm in the beam-propagation model, but we still use Eq. (21)for the diffraction efficiencies. In the unslanted casethe agreement is now satisfactory for the whole thicknessrange shown in Fig. 4(a), and in the slanted case theTGDM results are reasonably accurate, at least if HIT <4. In our hybrid kinoform simulations the slant angleand the refractive-index modulation are approximatelyone third and one half of the values used in Fig. 4(b), re-spectively. Therefore the accuracy of the hybrid kino-form simulation results is better than in Fig. 4(b).
4. SIMULATION RESULTS
A. Grating PeriodReduction of the dimensions of optical systems that in-clude hybrid holographic elements requires that the periodof the object wave be reduced. We expect volume effectsto increase the array uniformity error if the phase of theobject wave in the plane z = 0 changes substantially overa distance H. Such volume effects are investigated inFig. 5. Here the diffraction efficiency and the uniformityerror are calculated for 8-, 16- and 32-beam array genera-tors with Q = 128.
If the period is large, the results converge toward thevalues given in Table 1 (the Q value of 128 is not largeenough to give a low uniformity error for the 32-beamdesign). When the period is reduced below 0.1-0.2 mm,however, the uniformity error begins to increase rapidly.The efficiency remains essentially unaffected until the
uniformity error has grown to a level unacceptable inmost applications.
If we consider a uniformity error of 0.1 as satisfactory,the angular spread of the signal orders is obtained fromthe corresponding grating periods. They are 2.0°, 3.5,and 2.7° for the 8-, 16- and 32-beam arrays, respectively.
B. DefocusingAs shown in Fig. 1, the recording setup of hybrid holo-grams uses a two-lens imaging system, and thus the spa-tially filtered image of the CGH should be focused on thesurface of the recording material. However, this is notan easy task in practice, partly because the field in therecording plane has an almost uniform intensity profileand partly because the intensity level is often too weak topermit an accurate visual judgment of the focus. Theeffect of defocusing on the performance of the hybridhologram can be investigated by means of the TGDM ifone simply shifts the entrance boundary of the gratingfrom z = 0 to z = Az.
The diffraction efficiency is not significantly affectedby defocusing. The uniformity error of an eight-beamarray with Q = 250 and Q = 100 component waves is plot-ted in Fig. 6 as a function of Az. The uniformity error isnot affected critically even if Az is much larger than thegrating thickness of 10 jkm. This may be explained quali-tatively as follows. The focal depth of the recording setupof Fig. 1 is given by (f 2/a)2 Ao/2, where a is the radius ofthe aperture. When 100 component plane waves aretaken, the angular spread of the object wave front is 1.9°,which implies a fringe displacement of only 3% of the grat-ing period even with 1-mm defocusing.
Fig. 6. Effect of defocusing on the uniformity error.
I I
Ichikawa et al.
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Vol. 10, No. 6/June 1993/J. Opt. Soc. Am. A 1181
C. 0.8
C.).6
00.4
m 0.2a
u 10 20
I __ I
II --------------------% - - - - - -
1.II
I ,, II
I
.
-z |
__ I I-------
I T__
1,
II % %I
11 11
11 11
11 11
III II
11 11 11
11 ,,
11 11
- 11 I
11 11I II 11
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Ichikawa et al.
Figure shows the results for 8-, 16- and 32-beam hybridkinoform array generators with Q = 128. It is seen thatat least 4N diffraction orders need to be passed throughto produce a high-quality N-beam array.
I
0.8
0.6
0.4
0.2
D. Beam RatioBartelt and Case pointed out that one can improve thereconstruction fidelity of the hybrid hologram if one in-creases the beam ratio y1 This effect has also been ob-served when binary-phase gratings have been copied in
21volume holographic material, in particular when only thesignal diffraction orders are passed through the spatialfilter in an attempt to cut off the unwanted higher dif-fraction orders. We investigated the effect of the beamratio on the performance of hybrid kinoforms with bothideal and saturated refractive index modulation. Theparameters of the saturated modulation [see Eq. 9)]were chosen as nn = 0054, coo = 5900 j/M2 , and 56 =13,000 j/M2 to give the same maximum refractive-indexmodulation as in the ideal model (Fig. 9). The results foran eight-beam array with Q = 250 are given in Figs. 0and 11.
In general, a high beam ratio can improve the unifor-mity error at the expense of diffraction efficiency.Figure 11 shows some examples of the power spectrum.When a large number of orders are passed through thespatial filter, the uniformity error is good regardless ofthe beam ratio, but a high value of y reduces the efficiencyconsiderably. If only the signal orders are passed through,the uniformity error at y = 1 is high, and the higher ordersreappear with efficiencies that are actually higher thanwould be obtained without filtering. However, this situ-ation changes completely at high values of y: the un-wanted diffraction orders surrounding the signal ordersdisappear almost completely. Therefore it is particularlyuseful to record with a high beam ratio in applications in
25which nonsignal diffraction orders should vanish.The effect of the beam ratio is investigated more sys-
tematically in Figs. 10(a) (ideal refractive-index modula-tion) and 10(b) (saturated refractive-index modulation).It is interesting that while the ideally modulated kinoformwith 8 component waves needs a beam ratio of 50 to reachthe uniformity error of a hybrid kinoform with 100 com-ponent waves, the saturated modulation requires a beamratio of only 20. Moreover, in the case of saturated modu-lation,. the diffraction efficiency does not depend signifi-cantly on the number of component waves.
..500 100 5
.50 loo 200
Number of component wavesFig. 7 Effect of spatial filtering. Solid curves, hybrid kino-form; dashed curves, Dammann gratings.
-. . ...............0.8- 0.8
)0.6- 0.6
0.4- 0.4
A 0. 2 - 0.2
...................................
0 16 32 64 128Number of component waves
Fig. 8. Effect of spatial filtering and array size. Solid curves,8-beam array, dashed curves, 16-beam array, dotted curves, 32-beam array.
C. Spatial FilteringEvery phase-only complex-amplitude transmission func-tion generates an infinite number of Fourier coefficients,each of which has a nonvanishing amplitude that mani-fests itself as the appearance of higher, noise diffractionorders outside the signal. If one cuts off higher orders bymeans of spatial filtering, amplitude modulation is intro-duced in the object wave field, but the effect is not seriousif Q >> M and orders m = 2Q are passed through.Despite the high efficiency (>0.96) of continuous on-axisphase holograrns,' the modulation of the object wave fieldin recording of the hybrid kinoform becomes highly sig-nificant when only the signal orders m = Q - N12 +
Q + N12 are passed through. If the phase profile ofthe CGH is restricted by some constraints (e.g., binary-value),13 the efficiency is reduced, and the cutting of thehigher orders has an even more pronounced effect in theamplitude modulation of the object wave.
Figure 7 shows the diffraction efficiency and the uni-formity error of two hybrid holograms that generate aneight-beam array: the hybrid kinoform with Q = 250and a binary-phase grating with q - 0.80. Not surpris-ingly, the performance of the hybrid binary-phase gratingis satisfactory only if the number of the diffraction orderspassed by the spatial filter is large (of the order of 500-1000), which means that the lenses used in the recordingsetup must have high numerical apertures. The perfor-mance of the hybrid kinoform also deteriorates with spa-tial filtering, but the effects are clearly visible only whenfewer than 20-50 orders are passed through.
The width of the spatial filter that one needs to obtainhigh image quality obviously depends on the array size.
I1�I0's5
.56
AC�
0.04
0.03
0.02
0.01
0 0o
A%I 1.50.5
Grating periods (x/T)
Fig. 9 Refractive-index profiles recorded with two plane waves.Solid curves, ideal modulation; dashed curves, highly saturatedmodulation.
1182 J. Opt. Soc. Am. A/Vol. 10, No. 6/June 1993
array with Q = 250 and Q = 100 component waves. Theefficiency curves follow closely those of a sinusoidal grat-ing with similar recording beam angles (0° and 19.5°).This behavior can be understood as follows: since theangular spread of the signal beams (1.90) is much smallerthan the reference beam angle (19.5°), the hologram actsessentially as a phase-modulated sinusoidal grating. Moreinterestingly, the uniformity error remains essentiallyconstant unless the array efficiency is close to zero, inwhich case the concept of array uniformity is meaningless.
5. DISCUSSION AND CONCLUSIONS
0.8 0.8.0 '-
0.6 0.6
0.4 0.4
0.2 : 0.2
0 1 2 5 10 20 50
Beam ratio(b)
Fig. 10. Effect of beam ratio for hybrid kinoforms (a) with idealmodulation and (b) with saturated modulation. Solid curves, 100component waves are passed by the spatial filter; dashed curves,8 component waves are passed.
Fig. 11. Power spectrum of eight-beam array produced by a hy-brid kinoform with ideal refractive-index modulation. The verti-cal scale of the beam ratio (=100) is magnified by 10.
E. Angular SensitivityA drawback of the hybrid hologram is the strict Bragg con-dition for the incident angle of the reconstruction wave.This is a problem if, for example, a hybrid kinoform arraygenerator is used as a space-invariant optical interconnectbetween two logic planes and input beams incident fromdifferent directions generate the arrays at different dif-fraction efficiencies.
The angular sensitivity of the zeroth-order efficiency,the total efficiency of the signal beams, and the array uni-formity error are analyzed in Fig. 12 for the eight-beam
In conclusion, the fundamental performance limits andseveral important factors in recording of periodic hybridholograms have been investigated by means of an FFT-based implementation of the thin-grating decompositionmethod that is versatile enough to cope with almost arbi-trary refractive-index modulation profiles. The mainresults of the analysis may be summarized as follows.The image quality of the CGH that generates the objectwave field can be reproduced, with a high diffraction effi-ciency and a satisfactory uniformity error (less than 0.01),if the angular spread of the desired object is less than 2.0°.An attempt to cut off noise outside the desired image issuccessful if a high reference-to-object beam ratio is used,but only at the expense of a considerable reduction of thediffraction efficiency into the desired image. Correct fo-cusing of the object wave front into the plane of the record-ing material is not an exceedingly critical factor. Theangular sensitivity of the diffraction efficiency of a hybridhologram is essentially the same as the first-order effi-ciency of a sinusoidal grating, but the signal-to-noise ratioof the image is not affected by off-Bragg incidence.
Finally, the efficiency of a hybrid kinoform is under nocircumstances higher than the theoretical efficiency ofthe continuous-phase CGH for the same signal. Thisappears to be the case in general, at least if we requirereconstruction fidelity, whatever optical recording tech-nique is used.
Although these results were demonstrated for periodichybrid holograms only, the conclusions are believed tohold for any computer-generated holograms copied inter-ferometrically in a volume phase material.
0.8.)
0.6
0-8 0.4
O0.2
0R
0.6k
0.4
lo0 15 20 25 30
Incident angle (degree)
Fig. 12. Angular sensitivity of diffraction efficiency and unifor-mity error of the hybrid kinoform eight-beam array generator.Solid curve, diffraction efficiency of the beam array; dashedcurve, zeroth-order diffraction efficiency; dotted curve, unifor-mity error.
0 0.80.)E 0.60
*8 0.4
0.2
0.
Beam ratio(a)
0 L
Ichikawa et al.
IU.
Vol. 10, No. 6/June 1993/J. Opt. Soc. Am. A 1183
ACKNOWLEDGMENT
The authors appreciate useful discussions with AnttiVasara, Helsinki University of Technology.
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