Multiple imaging with thin phase filters: A signal processingapproach

Pieter Matthijsse*Department of Elektrotechniek, M.I.L., Katholieke Universiteit Leuven, Kardinaal Mercierlaan 94, B-3030 Heverlee, Belgium

(Received 16 September 1977)

Starting from its behavior as an optical processing system some general characteristics of a multipleimaging system are derived. A unique expression for the coherent impulse response cannot be derivedbecause of the allowed arbitrary phase distribution in the output plane and the variation that ispermitted in the relative intensities of the multiple images. For a multiple imaging system based onamplitude division, several types of phase gratings are considered. These gratings, two of which wereinvestigated experimentally, result in high-efficiency systems with a wide range of multiplicationfactors. Finally, some new applications are proposed.

INTRODUCTION

In the design of an optical processing system the desiredtransfer function or impulse response is of primary impor-tance. Considering the various multiple imaging systemsproposed until now, however, we do not find one of these twofunctions as the starting point in the designing procedureanywhere. A start from the basic space-invariant processingsystem and a coherent illumination1' 2 make too-stringentdemands since both conditions, space invariance and coher-ence, are unnecessary. The imposed conditions get even morestringent by applying the sampling principle.2 -4 The signalincident upon the filter does not need to be band limited;neither does the output signal need to be periodic with respectto the phase distribution.

Another starting point can be the filter used in the multipleimaging setup, e.g., a fly's eye lens,5 a hologram,6 8 or a purephase filter.9' 10 Although this point is of great practical im-portance, it often prevents the derivation of general charac-teristics of a multiple imaging system. Therefore, our startingpoint will be the behavior as an optical processing system.

1. IMPULSE RESPONSE OF A MULTIPLEIMAGING SYSTEM

We restrict ourselves to the common case where the in-tensity distribution in the output plane forms a square arrayof (2M + 1) X (2M + 1) images. Each image is identified withthe label m,n. For simplicity we use the vector mi- (m/l)lx+ (n/l)ly, where lx and I, are unit length vectors in the x andy direction, and I is the unit length. Assuming that the inputsignal is confined to a square with side A, the period B of theimage array must be greater than the side Al V1 of one imageto avoid overlapping. Herein, V is the magnifying factor. Fora coherent input signal, described by the complex-valuedscalar function U1(71), 1

1 the output signal of image m,n is

Umn(- 2 ) = cmn expUlk'mn(T2 - ThB)] U1( 2 - B/V), (1)

where 7r1 - (xi,yi) and 72 (x2,y2) are the position vectors inthe input plane and the output plane, respectively. For eachimage an additional phase distribution mnn(72) is allowed.This phase distribution can be arbitrary except for some casesof parallel processing that are not considered. We define theratio I Cmn/cool2 as the relative intensity Imn of the imagem,n.

The transfer of the linear multiple imaging system is de-

scribed by the coherent impulse response h(F1 ,F2 ), definedby

U2(72) = pl a5 U1(7i)h(71,F2) dri. (2)

For an incoherent input signal, an analogous relation holdsfor the intensity distributions I,(7) and I2(7). As the impulseresponse in this latter case is proportional to Ih (W,22)I 2,12 thefunction hV(1,72) suffices to describe the transfer for coherentas well as for incoherent illumination. The relation stated inEq. (1) must be valid for all input signals with U(71) = 0 for Ixiior Iy11 > 1/2A. This yields for Ix11 and Iy11 < 1/2A

h(71,72) = E c,, expLj4mn(7iV)] i -( Ti -V2 1flB). (3)

Due to the additional phase function, the position vectors 71and 72 also occur in other than linear combinations. Fromthis, we can conclude that it is not necessary that the systembe space invariant.

For a simple space-variant system composed of a cascadeof a propagation length di, a thin filter [transmittance func-tion m(r)], and a propagation length d2 , the impulse responseis given by (Fresnel approximation)

h(71, 2 ) = K1 exp ( 2(d, + d 2 ) 172-r712) U(71),(4)

with d 1 dj 1 + d-1, 7' = (d/d1 )Fl + (d/d2 )72 , K1 is a con-stant, and where Ug(F) is defined as the signal at a distanced behind the filter that is illuminated with a perpendicularlyincident plane wave. Expression (4) clearly shows the closerelationship between the impulse response and the diffractedsignal US(F).

Equations (3) and (4) must be identical for Ix11 and Iy 111/2A. After some laborous calculations this identity directlyleads to (i) V = -d 2 /d 1 , (ii) Imn(7) = (k/2dV)Iyl 2 - (kB/d 2 )(m-r ) + Kmn, and (iii)

I k d 2m(r) = _ c'n expj-| r-m-B- ,

mn 2d d2(5)

where all constants are inserted in c'mn. Thus, the choice ofour setup determines both the magnifying factor and the ad-ditional phase distribution. Only a constant phase delay Kmn

may be added to each image. From expression (5) we see thatin our setup the only condition imposed on the filter is thatit must divide a perpendicularly incident plane wave into a

733 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 0030-3941/78/6806-0733$00.50 � 1978 Optical Society of America 733

number of wave fronts that converge to points at a distanced behind the filter and at distances dB/d 2 from each other.There are two basic methods of wave front division, i.e., spatialdivision and amplitude division. In the first, each point ofthe incident wave front contributes to one outgoing wave frontonly; in the second it contributes to all outgoing wave fronts.Spatial wave front division is accomplished by applying anarray of focusing elements, e.g., a lens,5 a pinhole,' 3 a pointhologram,14"15 or a tiny zone plate.16 In this paper we willconsider amplitude division only.

Prior to a further investigation, we define in Sec. II foursystem parameters to qualify a multiple imaging system.

11. SYSTEM PARAMETERS

In contrast to the literature published until now, we do notrequire the condition that the relative intensity of all imagesmust be unity. We define for a block of images the variationin relative intensity as

m(F) = exp ( k I 2) *PD ()

x E_ dmnexp (-kB OF - ))m,n ( id2 I

(7)

with dmn = c'mn expj[kdB 2 (m 2 + n2)/(2d2')], we can immedi-ately conclude that the filter can be realized as a cascade ofa thin lens with focal distance d and a periodic filter with pe-riod W = Xd2/B and Fourier coefficients dmn. One of thesetwo elements usually determines the aperture PD(F). Thelens constitutes the imaging element whereas the periodicfilter behaves as a diffraction grating. From here on we willno longer consider the lens as being part of the filter.

Because of the presence of the lens, the signal US(F) is givenby the product of a quadratic phase factor and a function thatis proportional to the Fourier transform of the transmittancefunction g(F) of the filter12

US(r) = K 2 exp (i k Ir 2)

R = (Irax - I'min)/(Nnax + Ir in)- (6)

For each application the value of R that is permitted must bespecified. If necessary, the actual variation may be reducedby cascading the multiple imaging system with a densitymask.17

The multiplication factor Q is given by the number of usefulimages. The usefulness of an image depends primarily on thevalue of R for the block of images to which this imagebelongs.

The efficiency q is defined as the ratio between the radiantpower in the useful part of the output signal and the radiantpower in the signal that is incident upon the filter aperture.

For a multiple imaging system of finite extent the impulseresponse will consist of a two-dimensional series of pulses atdistances B along the x2 or Y2 axes and distances B/I VI alongthe x1 or y, axes. Assuming for these pulses a width P in theT, plane, we can conclude from Eq. (2) that the imaging pro-cess will smooth out details in the input signal with a smallerdiameter. Therefore, the ratio BIPI 1I is a measure for theresolving power 0 of the system. If the object size A is notmatched to the system, i.e., if A < B/ VI, the actual resolvingpower will be proportionally smaller. Here it is assumed thatfor all values of T2 the phase variation in the integrand of Eq.(2) is small (e.g., <1/27r rad) when the position vector F, tra-verses a region with width P. If this assumption is not valid,the output signal will decrease due to that phase variation andthe imaging process will depend on the position in the outputplane. To get a good multiplication nevertheless, the phaseof the input signal may be varied with time (rotating groundglass) or an incoherent illumination may be used.

III. MULTIPLE IMAGING BASED ON AMPLITUDEDIVISION

A. General propertiesAssuming a square aperture, the transmittance function of

the filter is given by the product of Eq. (5) and the aperturefunction PD(F) with PD(F) = 0 for Ixl or IYI > 1/2D and PD(F)

= 1 elsewhere. By writing this product as

X _ dmrn sincD (m -r - (8)

with sinc = (sincx).(sincy) and sincx = (sin7rx)/7rx. FromEqs. (4) and (8) we find that the pulses constituting the im-pulse response have a periodic distance Xdl/W and a widthXdl/D,' 8 either along the xi or along the y, axis. Conse-quently, the resolving power is given by

( 9 D/W, (9)

i.e., equal to the number of periods along one axis of the filter.For each pulse the greatest phase variation is approximately7rA/D (for IxI or Iy11 - 1/2A) when the position vector F, in Eq.(2) traverses the distance P e Xd,/D. As usual AID << 1; thisphase variation will not disturb the imaging process. Thisimplies that either coherent or incoherent illumination maybe used. Note that this also applies for a holographic fil-ter.

Stringent demands must be made on the wavelength spreadAX of the light source. This spread will be of no influence onlyif the shift AXd2(m2 + n2 )1/2 /W of the smallest image detailis much smaller than the transverse dimension B/C of thatdetail. This results in the condition

AX/X << [(M2 + n 2 )1/2 0]'-. (10)

The amplitude of each pulse from the impulse response isproportional to dmn, so the relative intensity of an imageequals the ratio ldmn/docj2 . Since the Fourier coefficients dmnare samples of the Fourier transform G (f) of g(F), i.e., dmn =W- 2 G [N(1/W)], this means that the relative intensity dis-tribution is given by a sampling of the function I G(f)l2. Fora filter with D -- the efficiency q is given by

,1 = Z Idmn 2.usefulimage,

(11)

Assuming that the image distortion due to the finite resolvingpower is small, this expression can also be applied to a finiteaperture filter.

We derived some general properties for a very simple sys-tem. However, the same procedure can be followed for morecomprehensive systems. Since this does not add to the un-

734 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 Pieter Matthijsse 734

Q V7

IN LENS FILTER OUT

FIG. 1. Multiple imaging setup based on amplitude division.

derstanding, we suffice with some remarks about those sys-tems. For practical reasons a distance d3 between the imaginglens and the filter (Fig. 1) is often necessary. Neglecting thelens aperture, the impulse response is then given by

hWl,-2) = K3 -* Z* dmnD2

m,n

X sinD [-1 - d2/ r2 + r)W d2 -d3 62 di

with

=[ 121' Ir,12ex k2Ld 2 - d3 di

X d2d3 _ 1 .(di(d2 -d3) /

(12)

Disassembling the filter from the lens decreases the imagedistance but does not change the resolving power which is stillgiven by Eq. (9). If the filter is put in the focal plane of thelens, i.e., if d 3 = d, the function W(bT 2 ) no longer depends onT1. The signal in the filter plane is now, apart from a qua-dratic phase term, proportional to the Fourier transform ofthe transmittance function of the input mask. It is unfortu-nate that this proportionality is often suggested as beingnecessary for good multiple imaging. Note that because ofthe quadratic phase term the signal in the filter plane is notband limited. This implies that a description based on thesampling theorem cannot be used here.

The setup can be extended further by adding a second lensbetween the filter plane and the output plane, or by illumi-nating the input mask with a converging beam. In this wayseveral setups are possible."1' 7 All of these can be proven tobe equivalent, with regard to their multiple imaging charac-teristics, to the setup of Fig. 1. This equivalence can bedemonstrated very elegantly by using the description methodproposed by Butterweck.19

IV. FILTER DESIGN

Since the use of absorption filters results in a poor effi-ciency, we restrict ourselves to phase filters with g(r) =exp[-j4(x) -exp[-j((y)], where t(x) is a symmetric function.This implies that the filter can be realized by cascading twocrossed identical one-dimensional filters. If dm is the Fouriercoefficient belonging to the transmittance functionexpj-j#(x)] this gives dmn = dm-dn and dm = d-m. Thevariation in relative intensity is greatest for images on thediagonal in the output plane where Imm = I dmId 0 4.

The Fourier coefficients dm must meet the requirementthat for the useful images the distribution of Iddm1 is relativelyflat, i.e., Idol _ Id+, I d+21 -- -. Besides that, the squared

d,

2.87

-fW 0 lW

3.64

/ -X

-7W 0 WW

2.01

L' -X-1W -1W O W Iw

0.81

0.72

075

7r(2N.1) 0.40 -

©~ --- (2N+1, 0.48

iW -TN -T, 0 T. TN fW

FIG. 2. List of previously proposed thin phase filters. (a) Sinusoidal phasedelay (Ref. 10), (b) triangular phase delay (Ref. 10), (c) rectangular phasedelay (Ref. 10), (d) binary phase delay (Ref. 9). Only the results for N = 1,3, 5, 7, and 9 are reported in Ref. 9.

sum of their squares, i.e., the efficiency -q, must be as high aspossible. The phase of dm is also subject to restrictions,however, since only certain forms of the phase delay t(x) canbe realized. For this reason, this phase delay will be ourstarting point in the design of the filter.

Phase delays proposed until now are listed in Fig. 2. Thedesign is based on the selection of a specific form and thecondition R = 0. For N free parameters this results in Q =(2N + 1) X (2N + 1). As deviations from the optimum phasedelay are inevitable, the sensitivity for these deviations mustbe determinated. This is only done in Ref. 10. In our ap-proach the condition R = 0 is not stated and we chose for themore laborious way of calculating the multiple imagingcharacteristics for any combination of the N free parameters.This method has the important advantages that (i) the mul-tiplication factor is not limited to (2N + 1) X (2N + 1), and(ii) the sensitivity to deviations from the optimum parametervalues can be determinated directly.

Some specific examples of the phase delays that we haveconsidered are given in Fig. 3. The corresponding best resultsare given in Table I. In a first approach the phase delay iscomposed of a sum of sinusoidal terms of different spatial

a,

0'/-jW 0 iW

t(x)

N= I

-jW 0 1WI

N=2 2a

-w 0 2W

ar d

-)W-T, -T,O T. T. fW -"W 0 IW

FIG. 3. Phase delays for a multiple imaging phase filter. For results seetext and Table I. Note that the phase delay in case c for N = 2 is notsymmetrical.

735 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978

363

3X3

3X3

Pieter Matthijsse 735

TABLE I. Numerical results for multiple imaging with two crossedphase filters with phase delays as sketched in Fig. 3. Application ofa density mask in case a gives n = 0.65. In case d it gives n = 0.42; Q= 27 X 27 and nt = 0.46; Q = 65 X 65 for a = 87r rad and a = 18r rad,respectively.

Case R Q 7/

a al = 2.59 rad 0.13 7 X 7 0.76a2 = 3.26 rad

b rl =0.02W 0 5X5 0.6072 = 0.37W

c N = 1;a = 2.04irrad 0.22 7 X 7 0.75N = 2; a = 7r rad 0.39 9 X 9 0.69N = 2; a = 1.987r rad 0.44 21 X 21 0.81N = 3; a = 27r rad 0.47 45 X 45 0.90

d a = 1.187rrad 0 3 X3 0.65a = 8irrad 0.50 21 X 21 0.44a = 187r rad 0.50 55 X 55 0.59

frequencies. Good multiple imaging occurs, e.g., if the phasedelay is formed by the sum of two harmonic terms (case a).The efficiency remains high when a density mask is appliedto make R =0. If the parameters al and a2 vary +10%, thevariation R keeps below 0.5. A design based on the conditionR = 0 would result in (al,a2 ) = (2.50,3.10) rad, where Q = 5 X5 and -q = 0.46.

The second approach involves a phase delay composed ofrectangular functions. The most interesting case, a two-levelphase delay, has been partly described in Ref. 9 [see Fig. 2(d)].Although the results are good, we believe that an improvementis possible by also considering delay differences other than 7rrad and even values of N. This can be illustrated with thedelays given in Figs. 2(c) and 3(b) for N = 1 and N = 2, re-spectively. Since a phase delay composed of rectangularfunctions of different heights is less interesting from a prac-tical point of view, we considered a phase delay with multipleslopes (case c). Good results appear for an odd number ofslopes with equidistant slope values ai = 27ri/r (i ='0,+ 1, ... ,±N), and equal segment widths r = WI(2N + 1). An at-tractive property of this phase delay is that the slope sequenceand the segment heights hi only have a small influence on themultiple imaging characteristics. If the maximum permittedvalue for R is 0.5 the multiplication factor is given by

Q c [2N(2N + 1) + 1 X [2N(2N + 1) + 1]. (13)

After choosing the number of slopes for a given value of Q thephase delay can be further optimized by considering varioussegment sequences or by allowing small variations to thevalues of ai, hi, or the segment width ri. The efficiency andthe multiplication factor will not change very much but thevalue of R, which in this case only applies for the farthestimages on the diagonal in the output plane, can decreaseconsiderably (see Sec. VI for N = 1).

For N > 4 the phase delay with multiple slopes as describedabove, can be approximated by a parabolic phase delay (cased) V(x) = 4a(x/W) 2 for lxl 1 1/2 W. The relation between theparameters a and N is given by a ir(N2 + N). The multi-plication factor can now reach extremely high values with amoderate value of the efficiency (Fig. 4). Note that applica-tion of a density mask increases the multiplication factor asmore images become useful.

FIG. 4. Multiple imaging with a low-quality array of 8 X 8 plastic lenslets.Only a part of the multiple image is reproduced.

It will be clear that besides the phase delays described inthis section many more delays are possible. An importantcriterion in choosing a specific delay, however, is the possi-bility of realization. This will be the subject of Sec. V.

V. REALIZATION

A simple method to realize a filter with a transmittancefunction expj~(x) is to provide a transparent plate with acorresponding thickness profile s(x). If n, and n2 are therefractive indices to both sides of the surface profile therelation between ~ (x) and 8 (x) is given by

i W = k(n i - n2)S(X). (14)

This relation holds if the filter meets the conditions for athin-filter description that we derived in Ref. 20 (see theAppendix).

The phase delay from Fig. 3(a) can be realized by cascadingtwo pairs of crossed sinusoidal phase gratings which arecommercially available. Another possibility is to illuminatea suitable emulsion, possibly two sided, repeatedly with dif-ferent interference patterns of two interfering plane waves.After that, the phase delay can be realized by chemical action.The same holds for the phase delay from Fig. 3(b) where theilluminating pattern is made by using a suitable mask (as inRefs. 9 and 10). The phase delay composed of multiple slopesis especially suited to be made mechanically by ruling Vgrooves. Here we can make use of the enormous experiencegained by the manufacturers of classical diffraction gratings.Gratings with two or more different V grooves are very wellpossible.2 ' It must be noted, however, that most of thesegratings must be considered as thick filters, showing, e.g., fromthe polarization dependence. Some modification will benecessary, resulting primarily in much greater angles for theruling diamond. The parabolic phase delay can also be madeby ruling or milling action. If a high resolving power is notnecessary, a high quality lenticular screen or lenslet array canalso be useful. Note that these latter arrays can also be usedfor spatial wavefront division. The resolving power is thengiven by , 1.3a - 2.la3 2 .

736 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 Pieter Matthijsse 736

R. .1,;* O.#J

o 0.20

-5 -4 -3 -2 -1 0 I 2 3 4 -

q 0FIG. 5. Multiple imaging with two crossed phase gratings with uniformV grooves. The efficiency in this region varies from q - 0.72 (p 1.03)to c i 0.76 (p 0.99).

VI. EXPERIMENTS

After some introductory measurements with pinhole arraysto check Sec. III, we concentrated on the phase delay of Fig.3(c) for N = 1. Comparison with Fig. 2(b) shows that an in-crease of the multiplication factor from Q = 3 X 3 to Q = 7 X7 is possible by simply adding a region with constant phasedelay. To optimize this phase delay, the parameters p andq are introduced. They are defined as the maximum numberof periods within the phase delay, i.e., p = In, - n21 d/X, andthe relative width of the region with constant phase delay, q= (W - 2r)/r. So the groove depth and the groove width ared and 2T, respectively. Results are given in Fig. 5. The op-timization results in a decrease of the variation from R = 0.22[Table I, (p,q) = (1.02,1.00)] to R = 0.11 for (p,q) =(1.005,1.10), whereas the efficiency hardly varies. Applicationof a density mask results in R = 0 and 77 = 0.64.

We acquired from the firm of Jobin-Yvon, France, a one-dimensional grating with (i) a groove angle X of 1100, (ii) nj= 1.57, n2 = 1, and (iii) 80 grooves/mm. Figure 6 shows thatthe grooves are highly finished. Owing to some misunder-standing in the ordering procedure, this grating did not cor-respond to (p,q) = (1.005,1.10) but to (p,q) - (2.4,1.3) (Ref.22) for X = 632.8 nm and n2 = 1. By choosing different im-mersion liquids or wavelengths, however, different values ofp could be obtained (Fig. 7). For p n 2.4 (no immersion), thegrating behavior appeared to be polarization dependent. Thisindicated that for this value of p we were on the verge of theapplicability of a scalar theory. Although an exact interpre-

-8 -6 -4 -2 0 2 4 6 8-- m

-10 -5 0 5 to

FIG. 7. Multiple imaging with a one-dimensional phase grating with uniformV grooves. (a) Water immersion, X = 632.8 nm, p = 1.05. (b) Waterimmersion, X = 434 nm, p = 1.53. The upper record is made on anemulsion with a steep characteristic. (c) No immersion, X = 632.8 nm,p = 2.5. Note the distortion due to the registration on a plane photographicsheet.

tation was difficult, it also appeared that the condition (A -1) for a thin-filter description was not met. With water as animmersion liquid (p 1), the grating no longer showed ameasurable polarization dependence. Since Eq. (Al) wasnearly met, we did not expect considerable thickness effects.in accordance with this we found that the relative intensitydistribution agreed very well with the expected distribution,both with respect to the efficiency and the variation in relativeintensity.2 3 The resolving power increased linearly with thefilter aperture until D c 16 mm (diffraction-limited lens withan aperture of 5 cm,f = 20cm, d3 = 5cm, and d2 = 2m). Herewe had to take care that neither the vignetting effect nor thelens limited the resolving power. The resolved lines wereabout 0.008 mm, which corresponded to a resolving power ashigh as 1300.

VII. DISCUSSION

From the preceding sections it appears that a multipleimaging system based on amplitude division can be very

FIG. 6. Detail of a proof gratingwith uniform V grooves and amultiple image obtained with thegrating described in Sec. VI.

737 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978

ItpQ.7x7

I I I I � I I I I . I- t t I . - -

to.

:: . . . , , I

oust

Pieter Matthijsse 737

simple. One imaging element in combination with a dif-fraction grating suffices. We considered some new phasegratings that make possible low as well as high values of themultiplication factor, thus increasing the capability of amultiple imaging system. Besides for the multiplication ofimages, application can also be considered for beam splittingpurposes, e.g., in data storage and fiber communicationtechniques. In an optical data storage system one incidentbeam must illuminate an array of positions.24 Usually, thesepositions are illuminated sequentially. Application of a highefficiency diffraction grating might favor a parallel illumi-nation.2 5 Besides this, multiple imaging can be used to pro-duce holographic lenslet arrays for this goal.15

In fiber communication networks, the proposed methodsof power division in a star coupler are mainly based on theprinciple of spatial wave front division.26 The inherentdrawbacks are the loss due to the packing density and thedependence on the mode structure. Quite recently the ap-plication of the amplitude division method by means of abeamsplitter 2 7' 28 or a point hologram2 9 was proposed. Thiscan be extended to the application of any phase grating thatis useful in a multiple imaging system.30 Note that the gratingmust be polarization independent for this purpose.

ACKNOWLEDGMENTS

The author wishes to thank Professor G. E. Francois (Ka-tholieke Universiteit Leuven), A. Diekema (Dr. Neher Lab-oratory), and G. Strohschein (University of Windsor, Canada)for critically reading the manuscript and suggesting valuableimprovements.

APPENDIX

Consider a phase filter bounded by the planes z = 0 and z= d and consisting of two layers with thicknesses s (x) and d-s (x), where 0 < s (x) < d. In Ref. 20 we derived two suffi-cient conditions for a thin-filter description of this filter. Inthis derivation we assumed that the thickness d is largecompared to the wavelength of the incident light and that ascalar description is allowed. As the maximum phase delayfor the grating with uniform V grooves [Fig. 3(c) for N = 1] isonly about 2ir rad, the applicability of the conditions isquestionable. If the conditions are derived using geometricaloptics only, the condition d >> X is not stated explicitly but theapplication of geometrical optics is questionable here too.

We nevertheless apply the conditions hoping that at leastan indication whether or not thickness effects can be expectedmight result. Note that no alternatives to solve this problemhave been presented until now.

For the grating with uniform V grooves the conditions resultin

cotg 1/20 < 1/4 n2 - XF (Al)(pn,- n2 l / In,- n21

a lower limit for the groove angle, and

XF < 1/4 Inl n21 nin 2 ) 1/2 (A2)nl+n2 p

which gives an upper limit to the cutoff frequency F of the

incident signal. This frequency depends on the position ofthe filter in the multiple imaging setup. In the object spacethe cutoff frequency is primarily determined by the smallestdetail in the object mask. Suppose that for a one-dimensionalsignal this frequency is F,. If the filter is placed in the spacebehind the lens the cutoff frequency of the signal incidentupon the filter is23

F2 = Fill MI + A/2Xd + 2(XAI VI )-1/2 (A3)

*Present address: Dr. Neherlaboratorium, Netherlands Postal andTelecommunications Services, St. Paulusstraat 4, Leidschendam,The Netherlands.

'Sun Lu, "Generating multiple images for integrated circuits byFourier transform holograms," Proc. IEEE 56, 116-117 (1968).

2A. Kalestyniski, "Enhancement of multiple images by samplingspatial filtering using additional auxiliary sampling," J. Opt. Soc.Am. 65, 1443-1447 (1975).

3A. Kalestyfiski and B. Smolifiska, "Micropattern array multiplicationby sampling spatial filtering," Wave Electron. 3, 69-78 (1977).

4A. Kalestyfiski and B. Smolifiska, "Spatial frequency sampling byphase modulation as a method of generating multiple images,"Appl. Opt. 16, 2261-2263 (1977).

5E. E. Rudge, W. E. Harding, and W. E. Mutter, "Fly's eye lenstechnique for generating semiconductor device fabrication masks,"IBM J. Res. Devel. 2, 146-150 (1963).

6G. Groh, "Holographische Methoden in der IC-Technologie," Reporton the Fourth Microelectronics Congress, November, 1970,Mdnchen, Germany (unpublished).

7N. Aebischer and A. B. Agbani, "Grilles multiplicatrices hologra-phiques: simplifications successives des principes de fabrication,"Nouv. Rev. Opt. 6, 37-47 (1975).

8A. Kalestyfiski, "Multiplying lensless Fourier holograms recordedusing a multibeam reference light field," Appl. Opt. 15, 853-855(1976).

9H. Dammann and K. Gbrtler, "High-efficiency in-line multipleimaging by means of phase holograms," Opt. Commun. 3, 312-315(1971).

'0L. P. Boivin, "Multiple imaging using various types of simple phasegratings," Appl. Opt. 11, 1782-1792 (1972).

"We assume a time factor exp(-iwt).12

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill,New York, 1968).

13p. A. Newman and V. E. Rible, "Pinhole array camera for integratedcircuits," Appl. Opt. 5, 1225-1228 (1966).

14 G. N. Buinov et al., "Image multiplication using uniaxial holo-grams," Opt. Spectrosc. 37, 196-197 (1974).

15E. Spitz, J. P. Huignard, and 0. Royer, U.S. Patent No.3,941,450.

16 R. G. Olsson, "Generation of micro-circuit pattern arrays by dif-fraction imaging," Report on the Topical Meeting of the OpticalSociety of America on the use of Optics in Microelectronics, Jan.1971, Las Vegas (unpublished).

17B. J. Thompson, "Multiple imaging by diffraction techniques,"Appl. Opt. 15, 312 (1976).

18 Herein, the width of a sinc function is assumed to be unity.19H. J. Butterweck, "A general theory of linear, coherent optical

data-processing systems," J. Opt. Soc. Am. 67, 60-70 (1977).2 0

p. Matthijsse, "Sufficient conditions for a thin-filter descriptionof thick phase filters," J: Opt. Soc. Am. 65, 1337-1341 (1975).

2 1 Private correspondence with several firms.22An exact value of (p,q) cannot be given here as the groove-depth

varied slightly with the position on the grating surface (d2.58-2,76 /m).

23p. Matthijsse, "Optische Patroonvermenigvuldiging," PhD dis-sertation, (Katholieke Universiteit Leuven, Belgium, 1976) (un-published).

2 4D. Chen and J. D. Zook, "An overview of optical data storage tech-nology," Proc. IEEE 63, 1207-1230 (1975).

2 5See, e.g., B. Hill et al., "Polycube optical memory: a 6.5 X 107 bitread-write and random access optical store," Appl. Opt. 14,2607-2613 (1975), where an array of 8 X 8 parallel beams is obtainedby a cascade of six beamsplitters.

26See, e.g., F. Auracher et al., "Verzweigungseinrichtungen fur

738 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 Pieter Matthijsse 738

Lichtwellenleiter," Siemens Forschungs und Entwicklungs Berichte5,47-53 (1976).

2 7Y. Suzuki and H. Kashiwagi, "Concentrated-type directional couplerfor optical fibers," Appl. Opt. 15, 2032-2033 (1976).

28K. Takahashi and S. Nonaka, "Optical directional coupler," Report

on the Topical meeting on Optical Fiber Transmission II, February1977, Williamsburg (unpublished).

29G. Goldmann and H. H. Witte, "Holograms as optical branchingelements," Opt. Quant. Electron. 9, 75-78 (1977).

30P. Matthijsse, Dutch Patent Application No. 7714270.

Geometry of the half-symmetric imageOrestes N. Stavroudis, Ronald C. Fronczek*, and Rong-Seng Chang

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721(Received 12 January 1978)

Herzberger defined the half-symmetric image as the image formed when the manifold of diapointsdegenerates into a curve on the meridian plane. The caustic associated with the half-symmetric imageis here analyzed in terms of the W and the C, functions discussed in an earlier paper. It is shownthat two cases are obtained. In one, an aberration pattern is produced which resembles coma. In theother, an object point is imaged as a curve on the meridian plane and thus resembles astigmatism.However, true coma and true astigmatism both belong to the more general deformation error inwhich the diapont manifold covers a region of the meridian plane.

Herzberger defined a diapoint as the point of intersectionof a skew ray with the meridian plane. Strictly speaking,diapoints are defined only for rotationally symmetric opticalsystems. The meridian plane is the plane containing the axisof symmetry and the object point. A meridian ray is a rayfrom the given object point which is confined to the meridianplane. A skew ray is a ray from the given object point whichis not a meridian ray. A skew ray will intersect the meridianplane in image space in exactly one point. That point is calleda diapoint. A diapoint for a meridian ray can be defined bya limiting process. Consider any sequence of skew rays whichconverge to a meridian ray. Then the sequence of diapointsof the skew rays will converge to a point which we will call thediapoint of the meridian ray.

The totality of all rays from a fixed object point forms apattern of diapoints on the meridian plane in image space.Herzberger'sl- 3 study of the distribution of diapoints led himto a hierarchical image error theory in which each category ofaberration is contained in a more general class and containsall more specific classes. In the most general category, inwhich the set of diapoints cover an area on the meridian plane,the image is said to possess deformation error. Should thisarea degenerate into a curve on the meridian plane, the imageis said to be half-symmetric. Further, should the curve de-generate into a straight line, the image is called symmetric.Finally, should the straight line degenerate to a point theimage is said to be sharp.

In an earlier paper 4 we presented a description of wavefronts and caustics as vector functions of two parameters.These arise out of a general solution of the eikonal equation 5

and are in the following form. Here the general results arespecialized for a rotationally symmetric optical system inwhich the z axis is the axis of symmetry and the y,z plane isthe meridian plane. The wave front is given by

W(v,w;s) = Sq/n 2-K (1)

and the two sheets of the caustic surface by

C_(vw) = S(R + Y)/2n 2 - K. (2)

Both wave front and caustic possess bilateral symmetry withthe plane of symmetry being the meridian plane. Here S =(u,v,w), where u, v, and w are the three reduced direction co-sines of a ray in a medium in which the refractive index is n,and therefore S2 = n2. The parameter s represents distancealong a ray measured from some arbitrary starting point. Byholding u, v, and w fixed and allowing s to vary, W in Eq. (1)describes a straight line which is the ray path. On the otherhand, by holding s fixed and allowing u, v, and w to vary, Wdescribes the wave front located a distance s from the startingpoint. Thus W, interpreted one way, describes an orthotomicsystem of rays; in another way, the train of wave fronts asso-ciated with that system.

The remaining quantities in Eqs. (1) and (2) are as fol-lows:

q=ns-k+vkv+wkw, (3)

K = (0,kv,k.), (4)

R = (U2 + w 2)kv, + (u2 + v 2)kww - 2vwkhv (5)

T = kvvkw, - k2w (6)

,g2 = R2 - 4n2u 2 T. (7)

Here k is an arbitrary function of v and w which arises in thesolution of the eikonal equation. Subscripts denote partialderivatives with respect to these variables.

The function k is our primary concern. Its nature deter-mines entirely the structure and the form of the geometricimage. We shall show here the relationship between the hi-erarchy of image errors and the form of k.

The vector K defined in Eq. (4) obviously lies on the meri-dian plane and defines a single point for each skew ray be-longing to the system of rays. Clearly K depicts the manifoldof diapoints which will, in general, cover an area of the meri-dian plane.

739 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 0030-3941/78/6806-0739$00.50 � 1978 Optical Society of America 739