Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1283

Fresnel diffraction in fractionalTalbot planes: a new formulation

Jan Westerholm, Jari Turunen, and Juhani Huttunen

Department of Technical Physics, Helsinki University of Technology, 02150 Espoo, Finland

Received June 29, 1993; revised manuscript received October 15, 1993; accepted October 15, 1993

We consider field distributions in fractional Talbot planes behind a periodic two-dimensional complex-amplitude transparency that is illuminated by a unit-amplitude plane wave. In the paraxial approximationthe field in various fractional Talbot planes is expressed as a sum of contributions from a finite number ofpoints in the plane of the transparency, yielding compact algebraic formulas for the diffracted field. Giventhe desired intensity distribution in the fractional Talbot plane, we synthesize the transmission function fromnonlinear equations. An experimental illustration that uses a binary phase grating is given.

1. INTRODUCTION

In diffraction theory the determination of the field at agiven point P residing some distance z away from a planeA typically requires knowledge of the field at every pointwithin a bounded region R of A, assuming either thatthe field vanishes outside A or that R encloses a singleperiod of a periodic field. In the former case knowledge ofthe field at the boundary of R is in certain circumstancessufficient,' but contributions from an infinite number ofpoints are nevertheless necessary. One notable excep-tion to this is known: the Talbot effect.2 Within theparaxial approximation, a laterally periodic (period d x d)monochromatic field of wavelength A is also longitudinallyperiodic with period ZT = 2d2/A. In other words, thedetermination of the field at integer multiples of thedistance ZT from A, requires knowledge of the field atjust one point of -A. Analogy with paraxial imaging by alens gives rise to the widely used term self-imaging. Fora comprehensive review of this phenomenon, see Ref. 3.

In this paper we consider the case of a periodic field.We first show that the field at an arbitrary point P infractional Talbot planes Z = zT/4q (q an integer) canbe expressed as a sum of contributions from a finitenumber of points within the period d X d. The requirednumber of points is N = 4q2 . In Section 2 we deriveexplicitly the expression for the field in the ZT/ 4 plane.In Section 3 we rederive this result from a differentviewpoint, using translation operators in A. We alsogeneralize the expression for the field distribution in theZT/ 4 plane to that of the zT/4q plane. Similar resultsare given for ZT/ 3 , ZT/5, and ZT/6.

We then proceed to introduce the wave-front synthesisproblem: for a specified intensity distribution across aplane zT/4q, determine the field Uyq across .A. Withinthe thin-element approximation, knowledge of Uq speci-fies uniquely a diffractive optical element that converts aplane wave into USL. We thus have a synthesis problemthat belongs to the domain of diffractive optics. In Sec-tion 4 we discuss the constraints of this problem, usingour new formulation of the Fresnel diffraction theoryin fractional Talbot planes. Some exact and numerical

solutions of the synthesis problem are provided in variouszT/4q planes. In particular, in Section 5 we consider

the theory of weak focusing of light, comparing the per-formance of lens arrays that we constructed by usingconventional continuous profiles and the fractional Talboteffect. Finally, an experimental illustration is given inSection 6.

2. QUARTER-TALBOT PLANE

In this section we derive algebraic expressions for the fieldin the ZT, ZT/2, and ZT/4 planes in terms of the field inthe z = 0 plane, illustrating the main results of this pa-per. Consider a two-dimensional periodic thin-elementtransparency (period d X d) with a complex-amplitudetransmission function t(x,y) = A(x,y)exp[i0(x,y)], 0 -A(x, y) c 1, located in the z = 0 plane. We illuminatethe transparency by a unit-amplitude scalar plane waveUilc (x, y, z < 0) = 1 of wavelength A, propagating in thepositive z-axis direction. The field in the z = 0 plane,

U(x,y,z = 0) = tx,y)Ujic(x,y,z < 0), (1)

has the same periodicity as the transmission functiont(x,y) and may be expressed in the angular spectrumrepresentation as

U(x,y,z = 0) = Ei Tmn exp[2vi(mx + ny)/d], (2)m, n

where

Tmn = fd d dxdyU(x,y,z = 0)

x exp[-2iri(mx + ny)/d]

and m, n are integers. Each term in sum (2) representsa plane wave, which acquires a phase term 27z[A- 2

-

(m/d) 2 - (n/d) 2]12 on propagation in free space.4 Inthe paraxial approximation the diffracted scalar field

0740-3232/94/041283-08$06.00 ©1994 Optical Society of America

Westerholm et al.

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1284 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994

at a distance z > 0 in the Fresnel region behind thetransparency is then given by

U(x,y,z > 0) = exp(ikz) E Tmn exp[27ri(mx + ny)/d]m, n

X exp[-2iri(m 2 + n2)Z/ZT], (4)

where ZT = 2d2 /A is the self-imaging or Talbot distanceand k = 2/A. We obtain the corresponding formulafor a one-dimensional transparency, for example, periodicin the x direction and constant in the y direction, fromEq. (4) by letting n = 0.

We now consider Eq. (4) for various distances z > 0.At z = ZT the field from z = 0 is reproduced apartfrom a global phase factor exp(ikzT); hence the nameself-imaging distance. The plane z = z/2 is almostas trivial.5 Denoting for convenience Tmn exp[27ri(mx +ny)/d] Tmn(Xy), we observe that

Tmn(x + d/2,y) = Tmn(,y)exp(i7Tm),

Tmn(X,y + d/2) = Tmn(x,y)exp(i7rn). (5)

We then have

U(XYZT/2) = exp(ikzT/2) Tmn exp[27ri(mx + ny)/d]m, n

X exp(i7rm)exp(i rn)

= exp(ikzT/2)U(x + d/2,y + d/2,0), (6)

reproducing the field at z = 0 apart from a lateral shift ofhalf the period d in both the x and the y direction.

Consider now the z = ZT/4 plane. Notice that accord-ing to Eq. (4) the effect of a translation in the z directionon the diffracted field depends on the angular componentof the field. We have for z = Z4,

U(x,y,zT/4)exp(-ikZT/4) = I Tmn(XY)m, n

X exp[-27ri(m 2 + n2)/4].

(7)

Since the value of exp(-27rim 2 /4) depends only on mbeing even or odd (1 or -i), we split the summation over(m,n) into four parts:

U(x, y, zT/4)exp(- ikZT/4)

= Tmn(XY) -i F Tmn(XY)mevon modd-- ,on neven

-i I Tmn(Xy) - Tmn(XY)meven moddnodd nodd

(8)

The crucial observation now is that each of these partialsums over Tmn(x, y) may be constructed from the field inthe z = 0 plane, that is, from the total sum over Tmn(x, y),by a suitable choice of translations and judiciously chosenphase factors. As may be readily verified, the expression

U(x,y,0) + iU(x + d/2,y,0) + iU(x,y + d/2,0)+ i2U(X + d/2,y + d/2,0)

= 2iU(x,y,zT/4)exp(-ikZT/4) (9)

reproduces every sum in Eq. (8) with the correct factor.This interesting result shows that the field in the frac-tional Talbot plane ZT/4 at any given point {X,y,ZT/4}

depends only on the field values at four regularly spacedpoints in the z = 0 plane, not on, e.g., the integral overall the field values within a d x d period.

In view of Eq. (9) the field values in the ZT/4 planeare naturally combined into groups of four points. Theintensity IA(X, Y, ZT/4) may now be obtained from Eqs. (1)and (9):

IA(X,Y,ZT/4) = IU(XYZT/4)1 2

= 1/41A(x,y)exp[iq5(x,y)] + A(x + d/2,y)

x exp[io(x + d/2,y) + ir/2]+ A(x,y + d/2)exp[io(x,y + d/2) + i/2]+ A(x + d/2,y + d/2)x exp[io(x + d/2,y + d/2) + iV]12 . (10)

In the case of a phase-only transmission function, t(x, y) =exp[i0(x,y)], the intensity is given as

I(XYZT/4) = /4exp[ik (x,y)]

+ exp[i4(x + d/2,y) + i/2]+ exp[io(x,y + d/2) + i/2]+ exp[io(x + d/2,y + d/2) + i]1 2,

(11)

which can be reduced to a transcendental equation inqS(x,y) by use of the formula

N 2Y exp(iB31 ) = N + 2 Y cos(/3 -&)-u=1 15u<vsN

(12)

We finally have the intensity profile in the ZT/4 plane interms of the phase of a phase-only transmission function:

I(X,Y,ZT/4) = 1 + 1/2{sin[o(x,y) - (x + d/2,y)]

+ sin[Iot(x,y) - (x,y + d/2)]- cos[k(x,y) - (x + d/2,y + d/2)]+ cos[o(x,y + d/2) - (x + d/2,y)]+ sin[o (x + d/2,y) - (x + d/2,y + d/2)]+ sin[o(x,y + d/2) - (x + d/2,y + d/2)]}.

(13)

The one-dimensional analog of this result was derived inRef. 6.

From Eq. (13) we can easily calculate the intensity pro-file in the ZT/4 plane once the transmission phase functionis given. We will also use this expression from a syn-thesis point of view to reconstruct any desired intensityprofile in the ZT/4 plane by solving for the transmissionfunction. The question then arises of how arbitrary arethe profiles that one can obtain with, e.g., phase-onlytransmission functions, at least in principle. One canobtain a partial answer to this question by consideringthe sum of the intensities at the four points of interest:

I I (X, Y, ZT/4) + I (x + d/2, y, ZT/4)+ I(x,y + d/2,zT/4) + I(x + d/2,y + d/2,zT/4). (14)

One can easily obtain the expressions for the intensities

Westerholm et al.

Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1285

from I(X, Y, ZT!4 ) by translating the arguments of thephase function 0(x, y); we have, for instance,

I(x + d/2,y,zT/4) = 1/4lexp[iS(x + d/2,y)]

+ exp[iq0(x,y) + i/2]+ exp[ik(x + d/2,y + d/2) + ir/2]+ exp[i,0(x,y + d/2) + i]1 2. (15)

We evaluate I by using Eq. (12) and obtain the constant1/4(4 + 4 + 4 + 4) = 4 in addition to a collection of cosines.Consider now a particular phase difference, for example,,0(x,y) - 0(x,y + d/2)-oa. This difference appears fourtimes in I, once from each intensity:

cos(a - ir/2) + cos(a - 7r/2) + cos(-a - r/2)

+cos(a - 7r/2). (16)

This sum is identically zero, and it is straightforward toshow that all the other cosine terms may be similarlygrouped and evaluated to zero. The following relation-ship then holds true for the intensities, reflecting the factthat, according to our assumptions, no energy is reflectedor absorbed by the transparency in the z = 0 plane:

I(X, Y, ZT/4) + I(x + d/2, y, zT/4) + I(x, y + d/2, T/4)

+ I(x + d/2,y + d/2,zT/4) = 4. (17)

As is shown later in the paper, this property restricts toa certain degree the possible values of the intensities atvarious points in the ZT/

4 plane.

3. FRACTIONAL TALBOT PLANES zT/ 4 qThe main result of the calculations presented in Section 2is that in the paraxial approximation the scalar field ata particular point (x,y) in the ZT/4 plane in the Fresnelregion depends only on the field values at a finite numberof regularly spaced points in the z = 0 plane, according toEq. (9). We now generalize Eq. (9) to any ZT/4q plane,q an integer. This generalization is first presented for aone-dimensional field and subsequently is generalized totwo dimensions.

Consider the diffracted field in the Fresnel approxima-tion in one dimension:

U(x,z > 0) = exp(ikz) Y Tm exp(2irimx/d)m

(18)

Moving in the z direction by ZT/4q, the mth angularcomponent obtains a phase factor exp(-2rim 2 /4q) [inthe following, the global phase factor exp(ikz) will notaffect the arguments and will be included only whenneeded]. We wish to show that this phase factor canbe reproduced for every m by the combining of fieldvalues at regularly spaced points in the z = 0 plane,multiplied by appropriate m-independent phase factors.To this end we introduce the translation operator Pa in

the x direction:

Pj (x) = f(x + a).

We now claim that the operator

1 2q-1

S 2q-1 exp(i2i~s2 /4q)Psd2qr exp(i2 7r2/4q)

r=O

(19)

(20)

reproduces for every angular component m the phaseexp(-2viim 2 /4q). To prove our claim, we consider theeffect of S operating on the mth component of U(x, z = 0):STm exp(2rimx/d) = SmTm exp(27rimx/d), with

2q-1. exp[i27r(s + m) 2/4q]

Sm = exp(-27rim 2 /4q) 2q-1

Z exp(i2 7r 2/4q)r=O

(21)

As shown in Appendix A, the summation in the numeratoris independent of m and can be evaluated as

2q-1E exp(i2Vs 2/4q) = exp(i4)J2. (22)s=O

The two sums in Eq. (21) thus cancel, and our proof iscomplete. We can now write the one-dimensional gener-alization of Eq. (9) for the fractional Talbot plane ZT/4q:

U(X, ZT/4q) = exp(ikzT/4q) exp(-iv/4)2q-1

X I exp(i~gs2/2q)Psdl2qU(xz = O). (23)s=O

Equation (23) is easily generalized into two dimensions.From the angular spectrum representation in Eq. (4) wededuce that a shift in the z direction gives a phase factorto the m, nth angular component, which factorizes in m, xand n,y:

Tmn exp(21rimx/d)exp(-2 wim2z/zT)exp(27riny/d)

x exp(-2irin 2 z/zT) . (24)

As the diffracted field in the ZT/4q plane is reconstructedfrom the z = 0 plane through an operator S consistingof complex-number phase factors and translations, whichcommute for x and y, we may immediately write the fieldU(x, y, zT/4q), using the S operator twice: once in the xdirection and once in the y direction:

U(XyZT/4q) = exp(ikzT/4q) exp( i/2

2q-1x E exp(is2 r/2q)Psd/2qo

s=O

2q-1

x E exp(ir27r/2q)Po,rd12qU(x,y,z = 0)r=O

= exp(ikzT/4q) exp(iT/2)2q

2q-1X Y exp[i(s 2 + r 2)7r/2q]

r, s=O

X Psdl2q, rdl2q U(X, Y, Z = 0) (25)

where we have introduced the two-dimensional transla-

Westerholm et al.

X exp(-2,7rim 2ZIZT) -

1286 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994

tion operator Pa,13f(x,y) = f(x + a,y + 3). For ZT/4,Eq. (25) becomes

U(X,Y,ZT/4) = exp(ikz/4) (1 + iPd,2,0)

X (1 + iPodl 2 )U(XYZ = 0), (26)

generate totally arbitrary intensity patterns but onlythree almost arbitrary intensities such that, for example,

0 c I(x,y) + I(x + d/2,y) + I(x,y + d/2) 4,I(x + d/2,y + d/2) = 4 - [I(xy) + I(x + d/2,y)

+ I(x,y + d/2)].which for U(x,y,z = 0) = exp[io(x,y)] reproduces ourprevious result for ZT/4. One can also show that Eq. (17)generalizes to (see Appendix B)

2q-1 2q-1I I Psdl2qrd2qI(XyZT/4q) = (2q)2,s=O r=O

(27)

which restricts the possible intensity distributions thatcan be synthesized in the planes ZT/4 q.

In certain cases, results similar to those of Eq. (25) maybe found for other fractional Talbot planes. We have,e.g.,

U(x, ZT/3) =

U(XZT/5) =

U(XZT/6) =

exp(ikzT/3)2E exp(2vrir2/3)

r=O

2x E exp(2Vjs 2/3)PSd/3U(x,z = 0),

s=O

exp(ikzT/5)4

_ exp(-27rir 2 /5)r=O

4x exp(-2 ris2 /5)Psd/5U(x,z = 0),

s=O

exp(ikZT/6)2

r exp(-2rir2/3)r=O

2X Yj exp(-27ris 2/3)Pd/3+d2U(x, z = 0),

s=O

(30)

with obvious generalizations into two dimensions. Con-straints of the type of Eq. (27) may again be derived.

4. SYNTHESIS PROBLEM INFRACTIONAL TALBOT PLANES

The formulas derived in Sections 2 and 3 will now beused to synthesize specified intensity distributions inthe ZT/

4q planes. As an example, consider a phase-

only transmission function, the ZT/4 plane, and Eqs. (11)

and (17). Given the desired intensity I(X,Y,ZT/ 4 ), weseek the values of q(x,y) in the z = 0 plane that willgive rise to these intensities. According to Eq. (13)the problem separates into four equations with the un-knowns ,b(x,y) for each quadruplet {I(x,y),I(x + d/2,y),I(x,y + d/2),I(x + d/2,y + d/2)}, where (x,y) E[O,d/2) x [O,d/2).

Two constraints in our synthesis problem are immedi-ate. The first constraint is related to the conservation ofenergy: according to Eq. (17) the four intensities understudy are interrelated. Therefore, it is not possible to

(31)

Therefore decreasing the energy at one point means in-creasing the energy at one or several other points. As itturns out, this constraint is necessary but not sufficient:there are intensity profiles that satisfy this constraint butnevertheless cannot be synthesized.

The second constraint follows from the observationthat only phase differences at z = 0, not the absolutelevel of the phase, will affect the diffraction result. Thisis reflected in Eq. (13), where only phase differences arepresent. Thus, without loss of generality, we may set,+(x + d/2,y + d/2) = 0. These two constraints areeasily generalized to any ZT/

4q plane.

In order to find the values {q0(x,y), +(x + d/2,y),qS(x,y + d/2)}, we employ both exact and numericalmethods.

A. Exact Results(28) In certain cases we can solve Eq. (13) exactly. Since we

can always set b(x + d/2,y + d/2) = 0, and as Eq. (17)makes I(x + d/2,y + d/2) known exactly from the otherthree intensities, we have for every triplet of phases{,b(x,y),qS(x + d/2,y),q0(x,y + d/2)} {01,q02,q03} threecoupled equations from the intensities:

(29) sin(02) + sin(¢ - 03) = I(x,y) + I(x + d/2,y) - 2,

sin(0 - 2) + sin(03) = I(x,y) + I(x,y + d/2) - 2,

cos(01) - cos(02 - 3) = I(x + d/2,y) + I(x,y + d/2) - 2.(32)

We have considered the following special cases:

Casel: 0I(x,y)'4,I(x,y+d/2) =I(x+d/2,y)=0. For arbitrary I(x,y) we have the solution

ck(x,y) = ,

,b(x,y + d/2) = (x + d/2,y) = arcsin[/2I(x,y) - 1].

(33)

Case 2: I(x,y) + I(x + d/2,y) = 4, I(x,y + d/2) =I(x + d/2,y + d/2) = 0. All intensity profiles withinthese restrictions may be synthesized. The solution is

f(x,y) = arccos[1/2I(x + d/2,y) - 1],,(x + d/2,y) = r/2,d(x,y + d/2) = k(x,y) - v/2. (34)

Case 3: I(x, y, ZT/4 q) - 4q2, all other I(x + sd/2q, y +rd/2q) = 0 in the general ZT/

4q plane. Case 3 is par-

ticularly interesting: now all the energy of the incidentunit-intensity wave within a period d d may be concen-trated into an area of fractional size 1/4q2 , within which

Westerholm et al.

Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1287

the intensity is uniform with value 4q2. This makes itpossible to focus the incoming wave very effectively withtheoretically no sidelobes within the period in the ZT/ 4 qplane. One obtains the desired transmission function bychoosing the translated phases 0(x + sd/2q,y + rd/2q)in Eq. (25) in such a way that they compensate for thephase that multiplies the translation, exp[i(s2 + r2 ) /2q].Thus

0 (x + sd/2q,y + rd/2q)= _(S2 + r2)>/2q. (35)

This important result was derived in a different mannerin Ref. 7.

B. Numerical MethodsIn cases in which the intensities in at least three quad-rants of the period in the ZT/4 plane are nonzero, we solveEqs. (32) numerically by employing multidimensionalNewton-Raphson techniques.8 These methods yield, inaddition to the results presented in Subsection 4.A, otherinteresting intensity patterns in the ZT/

4 plane. Oneimportant pattern from an application point of view seemsimpossible: we were unable to find a phase profile o (x, y)that would give I(x,y) = I(x + d/2,y) = I(x,y + d/2) =4/3,I(x + d/2,y + d/2) = 0.

5. WEAK FOCUSING OF LIGHT

In many areas of modern optics it is of interest togenerate regular two-dimensional arrays of light spotsfrom an incident plane wave.9 The need for this isencountered, e.g., in biasing arrays of optical logic devices,in matching the fill factor of focal plane arrays, and inincreasing the efficiency of dynamic diffractive elementswith partially opaque pixels.7"10-' 3 Large arrays of spotsmay be generated with a variety of techniques, includingmicrolens arrays and the fractional Talbot effect. In theformer case, assuming a perfect lens-array fill factor,the complex-amplitude transmission function may bewritten as

t(x,y) = exp[-AF (x - d/2]

X exp- AF (Y -d/2

We also consider the spot generated by a single lens with acomplex-amplitude transmittance of the form of Eq. (36):

I(x,y,zT/4q) = (2q)2 sinc2[2q(x - d/2)/d]X sinc2[2q(y - d/2)/d], (38)

where sinc(x) = sin(irx)/7rx. The results are shown inFig. 1 for q = 1,... ,4; since Eqs. (35), (37), and (38) areseparable, we consider the one-dimensional case.

Inspection of Fig. 1 reveals clearly that the diffractiveTalbot-type array illuminators are superior to arrays ofrefractive lens arrays even if 100% lens-array fill factor

(a)2.0

'N 1.5

1.0

0.5

n n

4.0

i.-N

3.0

2.0

1.0

0.0

(b)

U.U _

0.0

N

(36)

with 0 x < d and 0 y < d. In the latter case,assuming that z = zT/ 4 q, the optimum phase profile maybe constructed with use of Eq. (35).

We proceed to compare the performance of Talbot-typearray illuminators and lens arrays of the form of Eq. (36),with F = ZT/4q. The focal-plane intensity profile of thelens array may be evaluated by insertion of Eq. (36) intoEq. (25):

I(x, y, ZT/4q) =( 2

\\q) Y (-1)s-r+15r<s52q

X sin2 [ir(s - r)x/d] 15u<v52q

X sin2 [7(v - u)x/d].

(- 1)Vu

N

1.0 c)

0.8

0.6

0.4

0.2u IU.U

0

.1 n, .u

0.8

0.6

0.4

0.2

0.00~

0.2 0.4 0.6 0.8 1.0x/d

0.2 0.4 0.6 0.8x/d

1.0

.0.0 0.2 0.4 0.6 0.8 1x/d

(d)

.0 0.2 0.4 0.6 0.8 1.0x/d

Fig. 1. Comparison of focal-plane intensity patterns of diffrac-tive Talbot-type array illuminators (dashed lines), arrays ofrefractive lenses (solid curves), and single refractive lenses (dot-

u+1 ted curves) in fractional Talbot planes ZT/4

q with q = 1,...,4.In (c) and (d) the profiles have been truncated [the peaks areI(0,ZT/1 2 ) = 6 and I(0,ZT/1 6 ) = 8, respectively] so that they

(37) show the structure of the sidelobes more clearly.

: I. I ,~~/

fL~f\

- - 7 - - - - III . II I I

I..III

I I

.

.

Westerholm et al.

n ^

1288 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994

(C)

1"W

Fig. 2. (a) One period of the desired intensity pattern, (b) one period of the binary-phase Talbot element (with a phase delayof r rad between the black and the white regions) that generates the (a) pattern in the quarter-Talbot plane, (c) experimentalreconstruction of the intensity profile.

is achieved, which is difficult. The difference is particu-larly obvious for array focusing at the lowest numericalapertures, i.e., for the smallest values of q. In the planez/4 the fractional Talbot approach delivers spot arrays

with a compression ratio of 2 and no sidelobes, whereasthe lens array can generate only a sinusoidal pattern.When the focusing characteristics of the refractive lensarray are compared with those of a single refractive lens(that has a square aperture), the locations of the minimaand maxima coincide, which may be seen analyticallyfrom Eqs. (37) and (38). The number of sidelobes insidethe period increases linearly with q, and the intensityprofile of a single lens remains below that generated bythe lens array, reflecting the fact that the single lens alsohas sidelobes outside x/d E (0, 1).

6. EXPERIMENTAL DEMONSTRATION

As an illustration of wave-front synthesis in fractionalTalbot planes, we consider a binary-phase transparency,which gives rise to a binary intensity profile in the planeZT/4. The desired intensity profile shown in Fig. 2(a) isan example of an Escheresque pattern.' 4 We wish toproduce this pattern, starting from a plane incident wave,without any loss of energy. In view of Eqs. (32), thistask may be accomplished by use of a binary-phase profileof the form of Fig. 2(b), provided that the phase delay is,r rad.

The required binary-phase diffractive element wasfabricated by the following procedure. First, a binary-amplitude mask was generated by a Micronic LRS-18

Westerholm et al.

Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1289

direct-write laser-beam pattern generator. We chose agrating period d = 500 ,um, which implies that ZT/4

198 mm for A = 633 nm. The mask was subsequentlycontact copied on a chrome-covered fused silica substrate,which was wet etched in hydrofluoric acid to create thesurface-relief profile.

The binary-phase element, with a 35-mm patternedaperture, was placed in the path of a collimated He-Nelaser beam, and the pattern in the ZT/4 plane was mag-nified with a microscope objective to produce the photo-graph shown in Fig. 2(c). The observed pattern differsslightly from the theoretical result shown in Fig. 2(a):the boundaries of the pattern exhibit fringing effects,and there is an array of undesired vertical lines. Thesedefects may be attributed to walk-off effects (only ordersIml, Inl < 70 contributed to the pattern) and to departuresfrom paraxial propagation rather than to errors in thelithography.

7. CONCLUSIONS

The main result of this paper is that, within the paraxialapproximation, the diffraction of a laterally periodic fieldfrom the plane z = 0 to planes z = zT/ 4 q may be describedby simple algebraic expressions, which connect the fieldat a point P in the observation plane with the field valuesat a finite number (N = 4q2

) of points within the periodd X d of the field in the plane z = 0. This situation is incontrast to the usual formulations of diffraction theory,which require contributions from an infinite number ofpoints in the plane z = 0. The new Fresnel diffractionformula, valid in fractional Talbot planes, was shownto be useful in determining the constraints and findingsolutions of a particular synthesis problem: one finds afield distribution in the plane z = 0, which gives rise to aspecified intensity distribution in the plane z = zT/4 q.

APPENDIX A

We prove Eq. (22). Denoting

2q-1B(m,q) = _ exp[iv(s + m)2 /2q],

s=O(Al)

we first show that B(m, q) is independent of m and maytherefore be evaluated at m = 0. Writing B(m + 1, q)and using a new summation index s' = s + 1, we have

2qB(m + 1, q) = > exp[i-7(s' + m) 2/2q]

s1=l

2q-1= . exp[iir(s' + m)2 /2q] - exp(i Vm

2/2q)s8=0

+ exp[i-g(2q + m)2/2q]

- B(m,q) + exp(iirm2/2q)

X [-1 + exp(iv2m)exp(i 7r2q)]

= B(m,q), (A2)

may be converted into a sum of double the length by

2q-1 2q-12 cos( rs2 /2q) = Y cos( rs2/2q)

s=O s=o

4q-1+ E cos[X(s - 2q)2/2q]

s=2q

4q-1= E cos(irs2 /2q).

s=O(A3)

After similar manipulations to the imaginary part, weobtain

4q-1 1 4q-1

B(q) = 2 Y cos(2rs 2 /4q) + i - > sin(2Vs2 /4q)2s=O 2 =

= 4 [1 + cos(4q7r/2) + sin(4qgr/2)]

+ i [1 + cos(4qsr/2) - sin(4qv/2)]. (A4)

In the last equation we used well-known formulas (seeRef. 15). We finally have

B(q) = 1/ (1 + i) q, (A5)

which proves that B(m, q) = exp(iv/4) ff.

APPENDIX B

To prove Eq. (27) we wish to show that the sum ofthe intensities at corresponding points in the fractionalTalbot plane ZT/4q for a phase-only transmission functiont(x,y) = exp[iq$(x,y)] is a constant depending only on q.To this end we evaluate the sum in the one-dimensionalcase, from which we can easily generalize to twodimensions:

2q-1

I = _ Psd2qI(XZT/4q)s=02q-1

Psd/2q2

2n-1X I exp(i 7rr2/2q)exp[i0(x

r=O

Using a slightly rewritten Eq. (12),

N-1 2_ exp(iI36)

U=o:z

05u, vsN-1

2

+ rd/2q)] (B1)

cos(3" - i3v)

= N + j cos(31 - 3),0OuiwsN-1

(B2)

we write I as

1 2q-1 2q-1I = Y Psd/2q Y cos[wu2/2q + q5(x + ud/2q)

2q = u, v=0

-irv2/2q - b(x + vd/2q)]12q-1 2q-1 2 2

= 2q + 1 Y Y cos{lr(u2 - v2)/2qq s=o uv=0

+(k[x + (s + u)d/2q] - [x + (s + v)d/2q]}. (B3)

since m, q are integers. We then consider the real andimaginary parts of B(m = 0,q) B(q). The real part Consider a particular pair of phases f[x + (s + u)d/2q3 -

Westerholm et al.

1290 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994

O[x + (s + v)d/2q] y; that is, we fix the values ofs + ua and s + v / 3modulo 2q. As s increases from0 to 2q - 1, u and v have to be chosen correspondinglysmaller in order for a and /3 to be kept fixed. At somestage u (and v) will become negative, in which case weincrease u (v) by 2q since this does not change the valueof the cosine terms:

cos[(u + 2q)27r/2q + y] = cos[u2%/2q + y]. (B4)

Every term in the summation over s in Eq. (B3) givesus the chosen pair twice, the second time with a relativeminus sign. Consider first the positive differences anddenote v - u- Au, an integer. We then have

i 2- v2 = 2sAvu + a2 - /32, (B5)

and the summation over the cosines in Eq. (B3) becomesa sum of the form

2q-1 2q-1Z cos(2sA,,?T/2q + c) = cos(c) _ cos(sA,,ir/q)

s=O s=O

2q-1- sin(c) _ sin(sA,,,v/q),

s=O

(B6)

where c = constant. These sums may be evaluated' 5 :

2q-1Y cos(sAvuu7/q) =

s=Osin(A, 7r)cos[(2q - I)A uu rl2q]l

sin(A r/2q),

2q-1I sin(sAuu 7T/q) = sin(Auir)sin[(2q - 1)Avuir/2q]/s=0

sin(Av7/2q) . (B7)

The contribution from the sum of the cosines for thepositive values of y vanishes. The same holds true forcosine terms in Eq. (B3) containing -y, and thereforeI = 2q.

In two dimensions we retain the arguments given in theproof above. The number of different summation indicesin Eq. (B3) increases to four, and the constant contribu-tion from the expression for the sum of the intensities intwo dimensions gives

2q-1 2q-1

I = E E PsdI2q,rdI2qI(XY,ZT/4 q)s=O r=O2q-1 2q-1 2q-1

r-O s-O u=v=O (2q)2= (2q)2 , (B8)

which completes the proof of Eq. (27).

ACKNOWLEDGMENTS

We thank M. Westerholm for the design of the Es-cheresque pattern, A. Salin of Terapixel, Inc., for fabri-cating the amplitude mask, and J.-P. Laine for skillfullyperforming the wet etching. The research of J. Huttunenwas funded by the Academy of Finland and the Jennyand Antti Wihuri Foundation.

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