Self-imaging, also known as the Talbot effect, is awell-known phenomenon in classical wave optics.In its original sense the effect consists in the repro-duction of the lateral field distribution at periodicspatial intervals along the axis of propagation of aperiodic wave field.1,2 The period along the propa-gation axis is called the Talbot distance. The Talboteffect has various interesting applications in manyfields, such as image processing and synthesis, photo-lithography, optical testing, optical computing, andoptical metrology.2 It has been studied extensivelyby means of Fresnel diffraction theory, angular spec-trum analysis, and other methods of modern waveoptics. Diffraction of a periodic object at distancesexpressed as fractions of the Talbot distance, resultsin a superimposition of shifted and complex weightedreplicas of the original object. This phenomenon isreferred to as the fractional Talbot effect.3 A usefulelement that incorporates this phenomenon is thearray illuminator, referred to as the Talbot arrayilluminator.4–7 Particular attention is given to thiselement in this paper.
Whereas the fractional Talbot effect is an interest-ing phenomenon that is useful for several applica-tions, periodicity is a required condition of the effect.Thus it is desirable to have a Talbot-based systemthat permits use of the fractional Talbot effect with-out the need for input periodic structures. The es-
H. Hamam �[email protected]� is with the Faculty of En-gineering, University of Moncton, Moncton, New Brunswick E1A3E9, Canada, and the School of Optometry, University of Montreal,Montreal, Quebec H3C 3J7, Canada.
sence of the technique is to have periodicitygenerated by the system itself. The operation thatcreates periodicity is referred to here as Talbot im-aging. It is also desirable to facilitate the oppositeoperation, namely, concentrating information into asingle period after profiting from the fractional Tal-bot effect. This operation is referred to as Talbotunification. Beam shaping is one interesting appli-cation that results from the direct use of Talbot uni-fication. All these techniques are developed in thepresent paper, and an illustration of Talbot imagingis given at the end.
2. Fractional Talbot Effect
For brevity of notation, we shall limit the analysis toone dimension and opt for the Fresnel approximation,given that we are interested here in relatively fardistances with respect to the object features. Hencethe diffracted field observed at a distance z is ex-pressed by the Fresnel �FR� transform, as follows8:
h� x, z� � FRz�h� x, 0��
�exp�i2�z���exp��i��4�
��zh� x� � fk� x, z�,
(1)
where h�x� � h�x, 0� is the initial field and � denotesconvolution. The Fresnel kernel is expressed as fol-lows:
fk� x, z� � exp�j�x2
�z� . (2)
For brevity of notation, the constant term of propa-gation exp�i2�z��� and the factor exp��i��4���zare ignored.
Let h�x� � h�x, 0� be the initial periodic field, andd its period. The diffracted field observed at a dis-tance z is expressed by the following sum:
h� x, z� � a�0
q�2�1
T�a, p, q�h�x �d2
�2ad
q � , (3)
where the Talbot coefficients are given by the follow-ing relation3:
T�a, p, q� �2q
b�0
q�2�1
exp��i��2b2
qp � b��
� exp�i�4ab
q � , (4)
with z � �p�q�ZT is the Talbot distance and ZT �2�d2��� �p and q are integers with no common factorand � is the illuminating wavelength�. Without lossof generality, we restrict the analysis to even valuesof q. Diffracted field h�x, z� can be expressed bymeans of a convolution product �Fig. 1�:
h� x, z� � kT� x, p, q� � h� x�, (5)
where the Talbot kernel is expressed by the followingsum:
kT� x, p, q� � m�0
m�q�2�1
T�m, p, q���x �d2
�2md
q � .
(6)
Thus diffracted field h�x, z� of a periodic object at afractional Talbot distance z � �p�q�ZT is the result ofsuperimposition of shifted and complex weighted rep-licas of the original object �Fig. 1 and Eq. �5��. TheTalbot coefficients are determined by the followingrelation �cst is a constant factor�5:
T�mp, p, q� � cst��1�m exp��i2�pq
m2� . (7)
Equations �3�–�7� are linked to the Huygens prin-ciple consisting of endless creation of nodes andwavelets. Equation �7� emphasizes the lateral–longitudinal link caused by diffraction. Parameterm of Eq. �7� stands for the lateral dimension, whereasp and q represent the longitudinal axis. The qua-dratic phase term of Eqs. �4� and �7�, which is equiv-alent to the transmittance of a lenslet if we replace mwith lateral variable x, is also present in the Huygensformalism because of the wavelets.
The Talbot kernel, composed of uniformly spacedDirac functions, determines the number of replicasthat are created. For example, at distances z � 0,ZT�16, ZT�8, ZT�4, 3ZT�8, ZT�2, 3ZT�4, ZT we ob-serve, respectively, 1, 8, 4, 2, 4, 1, 2, and 1 replicas ofthe initial field. Thus the fractional Talbot effectbrings into play a continuous succession between con-structive and destructive interference. This meansthat the phase behaves in a particular fractional wayduring propagation such that constructive and de-structive interferences alternate. This behavior isuseful for Talbot imaging and unification, as we shallsee below.
Similarly, we can calculate h�x, 0� from h�x, z�:
h� x, 0� � kT� x, �p, q� � h� x, z�. (8)
Because the diffracted field is longitudinally periodic,with period ZT, we have
kT� x, �p, q� � kT� x, q � p, q�. (9)
Let us consider the following array of window func-tions, which represents the array of diffraction spotsthat the array illuminator should reconstruct:
Equation �14� formulates the periodic structure thatis required for the highest-compression-ratio Talbotarray illuminator �TAIL�, and Eq. �10� describes itsreplay field. The compression ratio is defined as theratio of spot spacing to spot size. A TAIL with thestructure of Eq. �14� offers a compression ratio of q�2.Other types of TAIL, such as that of a chessboardconfiguration, exist,6 which, however, offer lowercompression ratios. An analysis of these TAILs isnot made in this paper; the reader is referred to Ref.6 for such an analysis.
One can formulate the expression for the arrayilluminator in a different way by means of an inverseFresnel transform. Field I�x� � I�x, 0� is the inverseFresnel transform of I�x, z�:
I� x� � FR�z�rect� q2d
x� � n���
n� �
�� x � nd�� . (15)
Using Eq. �1�, we find that
I� x� � rect� q2d
x� � n���
n� �
fk� x � nd, �z� (16)
or
I� x� � n���
n� � �exp��j�� x � nd�2
�z � � rect� q2d
x�� .
(17)
If the signal in Eq. �10� is very narrow �small-signalextent�, the rect function in Eq. �17� can be approxi-mated by a Dirac function, and we obtain
I� x� � n���
n� �
exp��j�� x � nd�2
�z � . (18)
In the framework of this approximation, the TAIL�Eq. �18�� is identical to a microlens array in whicheach microlens has an infinite extent. The link be-tween the TAIL and the microlens array is not sur-prising because, as was mentioned above, the Talbotcoefficients include a quadratic phase term, as do themicrolenses.
Thus we can use the TAIL in a classical imagingsetup. We note that the rect function in Eq. �10� canbe approximated by a Dirac function if the TAIL pos-sesses a high compression ratio. The price for ob-taining a high compression ratio is the use of a bignumber of phase levels or a multilayer TAIL.5
In the general case expressed by Eq. �17�, a TAIL isthe result of the overlapping �sum� of infinitely ex-tended modified lenses separated laterally by periodd and expressed by the term in braces in Eq. �17�. Inits turn, a modified lens is the result of the denseoverlap of an infinity of infinitely extended lensescontinually shifted to one another �convolution withrect�. This dense overlap of lenses converges towardthe expression for a single lens when the rect functionin the sum of Eq. �17� is so narrow that it can beapproximated by a Dirac function. This conditionmeans that fractional order q is big. To be more
precise, given that the exponential functionexp��j��x2��z�� is periodic with respect to x2, theextent of the rect function in Eq. �17� must be largelysmaller �by a factor of 10, for example� than the firstperiod 2�z of this exponential function. Thus theapproximation in Eq. �18� is valid if q �� 2d�2�z.In other words, as �z is equal to the first radius ofthe Fresnel zone plate corresponding to quadraticphase term exp��j��x2��z��, fractional order q mustbe largely bigger than the ratio of the period of themicrolens array �Eq. �18�� to the first radius of thecorresponding Fresnel zone plate.
3. Talbot Imaging
Let us now consider a nonperiodic object s�x� placed adistance z1 before the array illuminator, as depictedin Fig. 2. The subsystem in the rectangular dashedbox is ignored for the moment. We place a screen ata distance z2 such that
1z2
�1z1
�1z
, (19)
where z � �p�q�ZT. If we illuminate object s�x� by aplane wave, we obtain in the image plane the follow-ing diffracted field �see Appendix A�:
s�� x� � exp�i�x2
�� z2 � z�� n���
n� �
exp��j�
2qp
n2�� s��
z1
z2x � n
z2
zd� . (20)
Figure 2 presents an illustration of Eq. �20�. Theminus ��x� means that image s��x� is an invertedversion of s�x�. Factor z2�z1 is a scaling factor. Inother words, it represents the ratio of size of oneimage to object size G. Images may overlap in theimage plane at z2 behind the TAIL. To prevent over-lapping, the following relation must be satisfied:
z2
z1G � d� �
z2
zd �
z1 � z2
z1d (21)
Fig. 2. Talbot imaging setup: The object is reproduced in sev-eral replicas at distance z2. The phase plate and the lens in thedashed rectangular box transform the divergent beam into a planewave. �x, permitted lateral shift.
Equation �20� points out that diffracted field s��x� isnot strictly a periodic structure because of the twophase terms. The quadratic phase term outside thesum represents the transmittance of a divergent lensfor z2 � z �or z1 � 0�. The replicas in the image planepossess the same intensity profile but not the samephase profile because of the first term in the sum ofEq. �20�. To compensate for the effect of those phaseterms we need a convergent lens and a multilevelphase plate, as shown in the dashed box of Fig. 2.This issue is treated in Section 4 below. Note that,for overlapping of replicas, the phase term in the sumcannot be compensated for by a phase plate.
4. Talbot Unification
By “unification” is meant transforming a periodicstructure into an image of a single pattern �period� ofthis structure. In other words, we intend to developa system that operates in a way opposite that of thesetup of Fig. 2. The idea is to start from Eq. �20� andfollow the analysis of Section 3 backward until itfinally converges toward object s�x� of Fig. 2 by usingthe principle of reversibility of light. This meansthat we illuminate the image plane of Fig. 2 from theright. Figure 3 illustrates the principle.
Our object now is periodic structure t�x�. Thephase term in the sum of Eq. �20� can be compensatedfor by the phase element of Fig. 2 �dashed box�.Thus, according to the principle of reversibility oflight, we need the same phase element for Fig. 3:
ph� x� � n���
n� �
exp�j�
2qp
n2�rect��z1
z2x � n
z2
zd� .
(23)
The quadratic phase factor of Eq. �20� can be com-pensated for by a convex lens with focal length
fL � z2 � z. (24)
The second term in the sum of Eq. �20� is the periodicobject itself t�x�. In the image plane of Fig. 3 weobtain an image t��x� of one period of t�x�.
The unification processes may be analyzed by thecontributions of the spatial periods of TAIL. Themany contributions of the spatial periods to thecentral period, image t��x�, add constructively,whereas the many contributions to the other peri-ods vanish mutually as a result of destructive in-terference. As was mentioned above, constructiveand destructive interference is naturally involvedin the fractional Talbot effect because of the behav-ior of the phase. By using multilevel phase plateph�x� of Fig. 3, one may favor constructive interfer-ence in the central period and destructive interfer-ence in the other periods of the output plane atdistance z1. Thus the performance of the systemdepends on the precision of the phase levels of
ph�x�. The amount of light energy in the residualperiods increases with the phase error in the dif-fractive phase element �ph�x��.
Let us have a look at the limitations of the system.Because the Talbot unification system is a direct ap-plication of the principle of reversibility of light on aTalbot imaging system, the limitations of the unifi-cation setup are linked to those of the imaging sys-tem. If object t�x� of Fig. 2 is laterally shifted by dx,we merely observe a shift, in the opposite direction, ofthe replicas at distance z2. Now, if it is desired tocompensate for the effect of the two phase terms ofEq. �20� we need a convergent lens and a multilevelphase plate, as shown by the dashed box of Fig. 2. Inthis case, shift dx is harmless so long as no replicamoves to the next phase level of the multilevel phaseplate. A lateral shift of the replicas of �x is permit-ted: �z2�z1�dx � �x �Fig. 2�. Similar reasoning isvalid for a shift dx of the phase plate: dx � �x.Thus, to improve the robustness of the Talbot unifi-cation or of the imaging system in terms of the phaseplate shift we should increase the separation betweenthe replicas.
Now if we laterally shift the convergent lens of Fig.2 by dx, we observe a prism effect at the outputbecause the quadratic phase term outside the sum ofEq. �20� will be not completely compensated for. Aresidual linear phase term, depending on the amountof shift dx, remains:
exp�i�x2
�� z2 � z��exp��i�� x � dx�2
�� z2 � z��Ç
shifted lens
�exp�i�
dx2
�� z2 � z��Ç
cons tan t
exp��i2�xdx
�� z2 � z��Ç
prism
.
(25)
Fig. 3. Talbot unification setup: t�x�, periodic structure. In theimage plane we obtain an image of one period of t�x�. t1�x� is thewave field just behind structure t�x�. t1�x, z2� is the diffractionfield of t1�x� at distance z2. ph�x� is the transmittance of the phaseelement.
Thus the effect of the lens shift is merely an angulardeviation of the output wave. If we slightly rotatethe lens about one of the lateral axes �X or Y� we alsoobserve a prism effect. Now, if we laterally shift theTAIL itself, then according to the analysis in Appen-dix A we should observe a combination of both previouseffects: a shift of the replicas and a prism effect.
5. Applications
As was mentioned in Section 1, the fractional Talboteffect is used in many applications.2 For example, itcan be used to perform logical and arithmetic opera-tions as well as image processing. However, peri-odic structures are necessary. Periodicity, whichmay be seen as a kind of data redundancy, may be astrong constraint and a serious source of energy lossin several applications. For example, when one isusing spatial light modulators it is preferable to useall pixels of these programmable diffractive elementsas effective data. Because these pixels are signifi-cantly larger than those of fixed diffractive elements,it is desirable not to replicate data in the spatial lightmodulator. Thus the Talbot imaging setup is usefulin such situations.
Moreover, it is important to concentrate the outputsignal into a single spatial period that represents thedata carrier. The output signal itself may be theinput signal of a second system that does not requireperiodicity. Thus it is desirable to concentrate theoutput energy of the first system before treating thesecond system. Using the system of Fig. 4, which iscomposed of three subsystems, S1, S2, and S3, is sug-gested. The fractional Talbot effect is used, al-though neither at the input nor at the output is aperiodic structure imposed. The input pattern �ob-ject� is replicated by Talbot imaging �S1� to yield aperiodic structure. One then uses this structure toperform optical operations that require periodicity,such as Talbot-based logic operations �S2�. The re-
sult is then entered into the Talbot unification sub-system �S3� to concentrate energy into a singlepattern.
Beam shaping is another interesting direct appli-cation of Talbot unification �Fig. 3�. Let us take anarbitrary input beam such as that of Fig. 5. Thebeam is not uniform in terms of intensity profile, andit has an arbitrary shape. We know from the liter-ature9,10 that small deviations from periodicity arealleviated by the fractional Talbot effect. Thus theuniformity of the beam will be improved by the Talbotunification setup �Fig. 5�. Indeed, the quasi-periodicobject �input beam� can be decomposed into a periodicobject and a nonperiodic pattern. The intensity dis-tribution in the fractional Talbot plane is formedmainly by the periodic field disturbed by the noisethat results from the nonperiodic diffracted field.We also know that the fractional Talbot effect can beused for object restoration. Some peripheral cells ofthe beam do not represent complete periods �Fig. 5�.The effect of these cells will be alleviated by the frac-tional Talbot phenomenon.
According to the analysis in Section 4, if the left-hand side beam of Fig. 5 is entered into the system ofFig. 3, we obtain at the output the square beam of Fig.5 �the square is magnified in Fig. 5�. To obtainnonsquare shapes such as rectangles and parallelo-grams we should use periodic two-dimensional ob-jects with nonorthogonal axes of periodization.7,11
Aberration reduction might be another applicationof the Talbot unification technique. Phase variationin a light beam might be seen as an error of period-icity. The fractional Talbot effect can be used toreduce this phase variation �aberration�. However,because phase information is much more of a deter-minant of the quality of the reconstructed image12
than amplitude information, the performance of theTalbot unification technique in terms of aberrationreduction decreases with the amount of aberration.
Fig. 4. System that uses Talbot imaging and unification.Whereas neither at the input nor at the output is a periodic struc-ture imposed, the system profits from the fractional Talbot effect.The input, a single pattern, is made periodic by the Talbot unifi-cation subsystem �S1�. Then any operation that uses the frac-tional Talbot effect, such as logic operations �S2�, can be performed.The periodic structure is passed through a Talbot unification sub-system �S3� to produce a single pattern at the output.
Fig. 5. Talbot unification system used for beam shaping: �a� anonuniform beam is entered; �b� this beam can be considered aquasi-periodic two-dimensional array of small cells �periods�.When this beam goes through the Talbot unification subsystem, allcells are added constructively to the central cell and we obtain thesquare cell at the output �magnified in this figure�.
I designed a two-dimensional TAIL with a squarestructure. Each period contains 128 pixels in eachdirection �x and y�, where each pixel is 8 �m large.
The TAIL provides a two-dimensional array of peri-odically distributed square spots, with a period of1024 �m � 1024 �m, at a fractional Talbot distancez � �1�256�ZT � 12.96 mm �ZT � 2d2�� � 3318 mm,d � 1024 �m, and � � 0.632 �m�. Each spot is 8 �mlarge, which yields a compression ratio of 128 in eachdirection. I simulated the setup of Fig. 2 and usedthe nonperiodic object of Fig. 6. To simulate thediffraction phenomenon I used the Fresnel transformas defined in the scalar theory of diffraction.12 Atfractional distance z� � �1�8�ZT� � 1659 mm �ZT� �2d�2�� � 13273 mm, d� � �z2�z�d � 2048 �m, andz2 � z1 � 25.92 mm�, we can observe the periodicEscher-like figure shown in Fig. 7. The Escher-likefigure results from the properties of the one-eighthTalbot plane.6 For both Figs. 6 and 7 the field am-plitude is considered. More illustrations are givenin Ref. 13.
Given that z � �1�256�ZT, 129 phase levels arerequired for the TAIL. We can use fewer phase lev-els, which will result, however, in a lower compres-sion ratio.6 Observe that better image quality wasobtained if more phase levels were used. To obtaina higher compression ratio while using fewer phaselevels, one can use multilayer Talbot array illumina-tors.5
Alternatively, one may quantize the phase profile
Fig. 6. Input object composed of four small segments spread in alarge dark zone.
Fig. 7. At the one-eighth Talbot plane, the four segments of Fig. 6 are replicated and fill all dark zones of the original to form anEscher-like figure. Only four periods are shown.
of the TAIL into equidistant phase levels. However,in theory, doing this will introduce a quantizationnoise, especially if few phase levels are used.14 Useof 16 levels ensured good image quality. To measureimage quality I used one of the uniformity criteria,namely, the complement of the rms error between theoriginal object and the reconstructed image �unifor-mity � 1 � rms�. To measure energy losses I optedfor the criterion of diffraction efficiency. It is definedas the ratio of light energy over the useful zone of theimage to the total energy in the image plane. Weknow from the literature that 16 phase levels offer adiffraction efficiency of 98.7%.14 Moving up to 32phase levels did not bring any significant improve-ment either in terms of image quality or in terms ofdiffraction efficiency. The uniformity is near 68%,84%, 94%, 96.5%, 98.5%, and 98% for phase levels 4,8, 16, 32, 64, and 128, respectively. For the resultsof Fig. 7 I quantized the phase profile of the analyt-ically calculated TAIL into 16 equidistant phase lev-els.
7. Discussion
Like the microlens array, the Talbot array illumina-tor, used in a Talbot imaging setup, provides replicasof the nonperiodic input object. The main differencebetween the two array illuminators hangs on the na-ture of the contribution of the spatial periods. Im-aging through the TAIL is the result of interactionamong all illuminated spatial periods of this diffrac-tive element, whereas, for the microlens array, thereplicas of the original nonperiodic object are pro-vided separately by each illuminated microlens.Therefore, replicas provided by the microlens arrayare less uniform because each replica observed in theback focal plane of this array is locally produced byone microlens. As a consequence, compared to thosein the microlens array, local defects on the TAIL haveless effect on image quality. This behavior may beused to reduce the effect of quantization noise.
Given that the behavior of a TAIL is based on thephenomenon of diffraction, it benefits from the ad-vantages of the diffractive optical elements.15 Toperform Talbot imaging requires only one diffractiveelement, namely, the TAIL. In this case the outputbeam is spherical and diverging. However, if theapplication imposes a beam with a zero vergency, twoadditional elements are required, as shown by thedashed box in Fig. 2. Alternatively, one may replacethe subset of the refractive lens and the phase ele-ment of Fig. 2 with a single diffractive phase elementthat is a modified version of the Fresnel zone plate.One would then obtain a fully diffractive system com-posed of two spatially separated diffractive elements,which form what is referred to as a multilayer dif-fractive element.15 Talbot unification and beamshaping can also be performed by such an expandeddiffractive element.
Appendix A
Fresnel transform equation �1� can be expressed in anintegral form8:
h� x, z� � FRz�h� x��
� exp�i�x2
�z� ���
�
h� x1�exp�i�x1
2
�z�� exp��i2�
x1 x�z �dx1. (A1)
When the Fourier transform �FT� is used, Eq. �A1� isequivalent to
h� x, z� � exp�i�x2
�z�FT�h�u�exp�i�u2
�z�� u��x���z�
.
(A2)
After calculation of the Fourier transform, spatialfrequency u� is replaced by x���z�.
If we illuminate object s�x� by a plane wave, weobtain at distance z1 the following diffracted field�Fig. 2 without the dashed box�:
s�x, z� � exp�i�x2
�z1�FT�s�u�exp�i�
u2
�z1��
u��x���z1�
.
(A3)
Just behind the TAIL, we obtain field s1�x� if weconsider Eq. �18�:
s1� x� � s� x, z� n���
n� �
exp��j�� x � nd�2
�z � . (A4)
Using Eqs. �A3�, �A4�, and �19�, we obtain
s1� x� � exp��i�x2
�z2�
n���
n� �
exp��j�n2d2
�z �� exp�j�
2xnd�z �FT�s�u�
� exp�i�u2
�z1��
u��x���z1�
. (A5)
In the image plane at a distance z2 behind the TAIL,the diffracted field is �from Eq. �A2��
s1� x, z2� � exp�i�x2
�z2�FT{ n���
n� �
exp��j�n2d2
�z �� exp�j�
2xnd�z �FT�s�u�
� exp�i�u2
�z1�� u��x���z1�}
u��x���z2�
. (A6)
Two successive Fourier transforms are involved inEq. �A6�. The first term in the sum has no effectbecause it is a constant phase term. Given that the
second term in the sum is a linear phase term, weobtain
s�� x� � s1� x, z2�
� exp�i�x2
�� z2 � z�� n���
n� �
exp��j�n2d2
�z �� s��
z1
z2x � n
z2
zd� . (A7)
Because z � �p�q�ZT and ZT � �d2���, we finallyobtain
s�� x� � exp�i�x2
�� z2 � z�� n���
n� �
� exp��j�
2qp
n2�s��z1
z2x � n
z2
zd� . (A8)
The author thanks Duc Phi for help in preparingthis paper.
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