1Npntfpmaacbf

wyjcapt

2764 J. Opt. Soc. Am. A/Vol. 23, No. 11 /November 2006 Aleksander Sokołowski

Consequences of sampling an image and Fourierplanes in a numerical light propagationmodel based on the Helmholtz–Kirchhoff

approximation

Aleksander Sokołowski

Rzeszów University of Technology, ul. W. Pola 2, 35-959 Rzeszów, Poland

Received January 3, 2006; revised April 23, 2006; accepted May 18, 2006; posted June 13, 2006 (Doc. ID 66999)

The numerical model of the light propagation phenomenon based on the Helmholtz–Kirchhoff approximationhas been analyzed. The model is based on the fast Fourier transform algorithm. It is proved that the fast Fou-rier transform algorithm limits the class of describing phenomena to the Talbot effects. © 2006 Optical Soci-ety of America

OCIS codes: 070.6760, 070.2590, 260.1960.

was(oFmatnttt

2Tc

. INTRODUCTIONumerical methods for studying optical phenomena com-rise a collection of robust cognitive tools. In the case ofumerical modeling of the light propagation phenomenon,he most frequent approximation in use is the Fresnel dif-raction model. The more general approximation of thishenomenon is the Helmholtz–Kirchhoff model supple-ented by Sommerfeld conditions.1 On the basis of the

bove model, Montgomery2,3 determined the conditionsppearing in the self-imaging phenomenon. The numeri-al model of light propagation tested in this paper isased on this model. Its mathematical formulation is asollows:

a2�x2,y2;z� =1

2��

−�

� �−�

�

a1�x1,y1;0�

�exp�ik��x2 − x1�2 + �y2 − y1�2 + z2�

��x2 − x1�2 + �y2 − y1�2 + z2

��ik −1

��x2 − x1�2 + �y2 − y1�2 + z2

�� z

��x2 − x1�2 + �y2 − y1�2 + z2dx1dy1, �1�

here x1, y1 denote the coordinates of the object plane; x2,2 denote the coordinates of the plane parallel to the ob-ect plane but lying at a distance z from it; a1 denotes theomplex amplitude of the light field in the object plane;nd a2 is the complex amplitude of the light field afterropagation over a distance z. Equation (1) can be easilyransformed into the following form3:

1084-7529/06/112764-4/$15.00 © 2

a2�x2,y2;z� =1

2��

−�

� �−�

� �−�

� �−�

�

a1�x1,y1;0�

�exp�− i2��x1�x + y1�y��dx1dy1

� expi2�z� 1

�2 − �x2 − �y

2�exp�i2��x2�x + y2�y��d�xd�y, �2�

here �x, �y denote the coordinates of the Fourier planend � is the wavelength. The plane-wave expansion of thepherical wave4 has been used to pass from Eq. (1) to Eq.2). Equation (2) is very useful from the numerical pointf view. The first step of the algorithm is to calculate theourier transform of a1. Next, the Fourier transform isultiplied by the distribution exp�i2�z�1/�2−�x

2−�y2�,

nd the calculation of the inverse Fourier transform ofhe above function is the last step of the algorithm. Fi-ally, the inverse Fourier transform is the complex ampli-ude, a2, of the light field after propagation over a dis-ance z. The properties of this algorithm are described inhis paper.

. THEORETICAL CONSIDERATIONShe numerical approach to the above propagation modelan be represented in the following form:

a2�q,r� =1

2�

1

�2M + 1��2N + 1� �m=−M

M

�n=−N

N

�k=−K

K

�l=−L

L

a1�k,l�

�exp�− i2��k�x1m��x + l�y1n��y��

� exp�i2�z

��1 − �2�m2��x

2 + n2��y2��

�exp�i2��q�x m�� + r�y n�� ��, �3�

2 x 2 y

006 Optical Society of America

wtar=nisccitsrdgtTaaprtr

z

wd

Tf=

wr�t

Ttsl

Taf

w

wmectt

itctc

wca

Trnd

Vo

mmmmmm

Aleksander Sokołowski Vol. 23, No. 11 /November 2006 /J. Opt. Soc. Am. A 2765

here a1�k , l� and a2�q ,r� are the sampled complex ampli-udes of the light field at the object plane and at the planet a distance z from it, respectively. To apply the fast Fou-ier transform (FFT) algorithm, the conditions �x1=�x2�x and �y1=�y2=�y have to be established.5 It is notecessary to assume that �x=�y, although this condition

s often established. The aim of this paper is to demon-trate that the digitization of the light propagation pro-ess with the above conditions causes restriction of thelass of the phenomena described by Eq. (3) to the self-maging effects. Montgomery stated the necessary condi-ion for the self-imaging phenomenon3: that the objectpatial frequencies lie on concentric rings of well-definedadii. The occurrence of the self-imaging phenomenon forifferent configurations of Montgomery rings was investi-ated by Indebetouw.6 For a specific self-image distance z,he configuration of Montgomery rings has a precise form.here is the ring with the highest radius. The frequencyt which the rings appear is high for large values of radiind decreases for smaller values of radii. If the regularoint grid is projected onto such a structure of rings, someings should be omitted because no grid point lies onhem. Some grid points can lie outside the Montgomeryings, and this situation can degrade self-imaging.

The self-imaging condition for the light field, such asero spatial frequency, has the following form2:

2�z��1/��2 − �x2 − �y

2 = 2�u, �4�

here u proceeds over the finite set of integers in accor-ance with the condition

0 � u � z/�. �5�

he above condition results from the limits of the spatialrequencies in the propagation process. If now ��x=��y��, then

z

��1 − �����2�m2 + n2� = u, �6�

here m and n denote the �m ,n� point lying on the Fou-ier plane. If the assumption that maximum value of�2�m2+n2� is greater than or equal to 0.01/�2 has been

aken into account, then

Table 1. Values of Function (12) for Sthe Propagation Distance Is E

aluesf m, n n=0 n=1 n=

=0 1 −1 �2 + i�3�2 −1 �2 +=1 −1 �2 + i�3�2 −�3�2 − i �2 −�3�2=2 −1 �2 + i�3�2 −�3�2 − i �2 −�3�2=3 1 −1 �2 + i�3�2 −1 �2 +=4 −1 �2 + i�3�2 −�3�2 − i �2 −�3�2=5 −1 �2 + i�3�2 −�3�2 − i �2 −�3�2

�1 − �����2�m2 + n2� � 1 −1

2�����2�m2 + n2�. �7�

he acceptance of the above assumption indicates thathe distance between neighboring pixels in the recordingystem is not smaller than 10�. Now Eq. (6) takes the fol-owing form:

z

��1 − �����2�m2 + n2�� = u. �8�

he first term on the left-hand side of Eq. (8), being thergument of the exponential function, gives the constantactor. Therefore Eq. (8) should have the following form:

z���2�m2 + n2� = v, �9�

here v denotes an integer. If z is equal to

z =2p

���2 , �10�

here p=1,2, . . ., then Eq. (9) is fulfilled for each value ofand n. This means that each image sampled with an

quidistant point grid has the self-imaging property. Be-ause of approximation (7), it would be more proper to sayhe quasi-self-imaging property.6 z is called the Talbot dis-ance if p=1.

If the propagation distance of a self-imaging light fields equal to the rational fraction of the Talbot distance,hen the fractional Talbot effect occurs.7–12 This effectonsists of the superposition of laterally shifted replicas ofhe object, modulated by certain complex factors of theonstant modulus. Then

z =s

t

2

���2 , �11�

here s / t is the above rational fraction and s and t areoprime integers. Then the exponential function with thergument of the right-hand side of Eq. (9) takes the form

exp�2�iz���2�m2 + n2�� = exp�2�is

t�m2 + n2�� . �12�

he summation over m and n causes the exponent on theight-hand side of Eq. (12) to fluctuate over a certainumber of values. The number of values in a single cycleepends on the value of s / t. There are a certain number of

bsequent Values of m and n, Whereto 1Õ3 of the Talbot Distance

n=3 n=4 n=5

1 −1 �2 + i�3�2 −1 �2 + i�3�2−1 �2 + i�3�2 −�3�2 − i �2 −�3�2 − i �2−1 �2 + i�3�2 −�3�2 − i �2 −�3�2 − i �2

1 −1 �2 + i�3�2 −1 �2 + i�3�2−1 �2 + i�3�2 −�3�2 − i �2 −�3�2 − i �2−1 �2 + i�3�2 −�3�2 − i �2 −�3�2 − i �2

ix Suqual

2

i�3�2− i �2− i �2i�3�2− i �2− i �2

vfcbatrvs

w

At

a

wtvvrEaTteaa

wvofif(

Bm−−tai

2766 J. Opt. Soc. Am. A/Vol. 23, No. 11 /November 2006 Aleksander Sokołowski

alues of function (12) in Table 1 for s / t=1/3. It is evidentrom the table that every third column and row are repli-ating, which suggests that the image at z=1/3 of the Tal-ot distance consists of a 3�3 superposition of verticallynd horizontally shifted original images. This means thathe summations over m and n in Eq. (13) can be sepa-ated into as many independent sums as the number ofalues in particular cycles. Equation (3) can then be pre-ented in the following form:

a2�q,r� =1

2�bMN �

k=−K

K

�l=−L

L

�m=−M

M

�n=−N

N

a1�k,l�

�exp − i2��m�x�k − q���x + n�y�l − r���y��

� expi2�

�zexp�i2�

s

t�m2 + n2�� , �13�

here

bMN =1

�2M + 1��2N + 1�. �13a�

fter separation into the independent sums over m and n,he above equation takes the form

2�q,r� =

expi2�

�z

2�bMN �

k=−K

K

�l=−L

L

a1�k,l�

� �c=0

Cx−1

�d=0

Dy−1

ecd �m=−Mc

Mc,Cx

�n=−Nd

Nd,Dy

�exp i2���q − k�m�x�� + �r − l�n�y����, �14�

y

I=sss

3Tiv

tiq[F

here the sums over c and d run over a single cycle forhe x and y axis, respectively, and ecd is an individualalue from such a cycle. Cx and Dy denote the number ofalues in a single cycle for the x and the y axis,espectively. For simplicity, we assume that inq. (14), �2M+1� and �2N+1� completely divide by Cxnd Dy, respectively. Then Mc=M−c and Nd=N−d.he superscripts in the summations over m and n signify

hat the summation steps are not equal to 1 but arequal to Cx and to Dy for m and n, respectively. Then Mcnd Nd can be calculated according to the followinglgorithms:

Mc = �integer part of �M/Cx�� � Cx + mod�M,Cx�,

�15a�

Nd = �integer part of �N/Dy�� � Dy + mod�N,Dy�,

�15b�

here the mod function denotes the remainder of the di-ision of the first argument by the second one. The sumsver m and n in Eq. (14) are in fact the summations ofnite geometric series.13 When the well-known formulaor the summation of the geometric series is applied to Eq.14), then the sums over m and n take the form

exp�i2��q − k��− Mc��x��� − exp�i2��q − k��Mc��x���exp�i2��q − k�Cx�x���

1 − exp�i2��q − k�Cx�x���

�exp�i2��r − l��− Nd��y��� − exp�i2��r − l��Nd��y���exp�i2��r − l�Dy�y���

1 − exp�i2��r − l�Dy�y���. �16�

oth above fractions become simpler when the first one isultiplied by exp�−i��q−k�Cx�x��� /exp�−i��qk�Cx�x��� and the second one by exp�−i��rl�Dy�y��� /exp�−i��r− l�Dy�y���. Then, both numera-

ors and denominators became real sinusoidal functions,nd the distribution described by Eq. (3) takes the follow-ng form:

a2�q,r� =1

2�expi

2�

�z �

k=−K

K

�l=−L

L

a1�k,l�

� �c=0

Cx−1

�d=0

Dy−1

ecd

sin���2Mc + Cx��q − k��x���

sin��Cx�q − k��x���

�sin���2Nd + Dy��r − l��y���

sin��D �r − l��y���, �17�

t follows from the above formula that at the distance zdTs / t, where dT denotes the Talbot distance, one can ob-erve replicas of the original object. The experimentaletup corresponding to the above model has been de-cribed in Refs. 14–16.

. NUMERICAL RESULTShe small bright square has been selected for numerical

nvestigations. Such a choice allows more readable obser-ations of the effects described in Section 2.

Figure 1 shows the results of computer simulations ofhe light-field propagation model. Figure 1(a) shows thenitial object. It can be seen that the model describes theuasi-self-imaging phenomenon at the Talbot distanceFig. 1(b)] and at a fraction of it [Fig. 1(c) for z=1/3dT andig. 1(d) for z=99/100d ]. Only rational numbers are

T

ubanobq

4AoosoaT

rf

R

1

1

1

1

1

1

1

Fta

Aleksander Sokołowski Vol. 23, No. 11 /November 2006 /J. Opt. Soc. Am. A 2767

sed as the propagation distances (because only they cane inserted into computer memory), which means that thebove model can describe only the quasi-self-imaging phe-omenon at the Talbot distance multiple or at a fractionf it as the fractional self-imaging phenomenon. This isecause each rational number can be presented as theuotient of two coprime integers.

. CONCLUSIONSnalysis of the numerical light propagation model basedn the Helmholtz–Kirchhoff approximation with the helpf the FFT algorithm shows that the model does not de-cribe the propagation phenomenon in general. The gridf equidistant sampling points is a frequently acceptedssumption, because it allows the FFT algorithm to apply.

ig. 1. (a) Initial object, (b) image of the object at the Talbot dis-ance, (c) image of the object at 1/3 of the Talbot distance, (d) im-ge of the object at 99/100 of the Talbot distance.

he acceptance of the equidistance point grid at the Fou-

ier plane causes the reduction of the applicability areaor the above model to only self-imaging phenomenon.

A. Sokolowski’s e-mail address is alex5@prz.edu.pl.

EFERENCES1. M. Born and W. Wolf, Principles of Optics (Cambridge U.

Press, 1999), Sec. VIII.2. W. D. Montgomery, “Self-imaging objects of infinite

aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).3. W. D. Montgomery, “Algebraic formulation of diffraction

applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124(1968).

4. P. C. Clemmow, The Plane Wave Spectrum Representationof Electromagnetic Fields (Pergamon, 1996), Section 2.

5. J. W. Cooley and J. W. Tookey, “An algorithm for themachine calculation of complex Fourier series,” Math.Comput. 19, 297–302 (1965).

6. G. Indebetouw, “Quasi-self-imaging aperiodic sequences,”J. Opt. Soc. Am. A 9, 549–558 (1992).

7. J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structuredetermination of phase objects,” Opt. Acta 18, 677–682(1971).

8. V. Arrizón and J. Ojeda-Castañeda, “Multilevel phasegratings for array illuminators,” Appl. Opt. 33, 5925–5931(1994).

9. C. Zhou and L. Liu, “Simple equations for the calculationsof a multilevel phase grating for Talbot array illumination,”Opt. Commun. 115, 40–44 (1995).

0. M. Testorf and J. Ojeda-Castañeda, “Fractional Talboteffect: analysis in phase space,” J. Opt. Soc. Am. A 13,119–125 (1996).

1. V. Arrizón and G. Rojo-Velázquez, “Fractional Talbot fieldof finite gratings: compact analytical formulation,” J. Opt.Soc. Am. A 18, 1252–1256 (2001).

2. K. Banaszek, K. Wódkiewicz, and W. P. Schleich,“Fractional Talbot effect in phase space: a compactsummation formula,” Opt. Express 2, 169–172 (1998).

3. A. Papoulis, Systems and Transforms with Application inOptics (McGraw-Hill, 1968), Section 4.

4. A. Kalestynski and B. Smolinska, “Multiple imageformation by sampling Fraunhofer diffraction pattern,”Opt. Acta 24, 1115–1124 (1977).

5. A. Kalestynski, “Improvement of the impulse response of adeigmaidolon optical system,” Opt. Acta 27, 863–867(1980).

6. K. Patorski, “The self-imaging phenomenon and itsapplications,” in Progress in Optics, Vol. XXVI, E. Wolf, ed.

(North-Holland, 1989), pp. 1–108.