Cataloged from PDF version of article.An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. A laboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested. (C) 1995 Optical Society of Americ

Bitran, Y.

Dorsch, R. G.

Ferreira, C.

Garcia, J.

Mendlovic, D.

Ozaktas, H. M.

Applied Optics

An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. A laboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested

Özaktaş, Haldun M.

Bilkent University Institutional Repository

Anamorphic fractional Fourier transform:optical implementation and applicationsDavid Mendlovic, Yigal Bitran, Rainer G. Dorsch, Carlos Ferreira, Javier Garcia,and Haldun M. OzaktazAn additional degree of freedom is introduced to fractional-Fourier-transform systems by use ofanamorphic optics. A different fractional Fourier order along the orthogonal principal directions isperformed. A laboratory experimental system shows preliminary results that demonstrate the proposedtheory. Applications such as anamorphic fractional correlation and multiplexing in fractional domainsare briefly suggested.Key words: Anamorphic systems, fractional Fourier transform, optical signal processing.r 1995 Optical Society of America1. IntroductionAnamorphic systems are well known and widely usedfor various special systems. The so-called astigmaticprocessor was exploited in optical data processing as adevice for one-dimensional 11-D2 Fourier transformingand imaging in two mutually perpendicular direc-tions.1,2 Later, the idea of obtaining a nonsymmetri-cal Fourier transform with crossed cylindrical lensesof different focal lengths working under plane-waveillumination was presented.3 Based on this idea, ananamorphic two-dimensional 12-D2 optical processorwas designed.4 In order to extend the performanceof the nonsymmetrical Fourier transformer, Andres etal.5 proposed the use of spherical-wave illumination.In this case, with a two-crossed-cylindrical-lensessetup it is possible to obtain an exact Fourier trans-form and a greater angular-magnification coefficient.Based on the nonsymmetrical Fourier transformerworking under spherical-beam illumination, amatched-filter anamorphic correlator with improvedangular discrimination was suggested6 and was alsoimplemented with multiple matched filters for remov-ing possible ambiguities in the recognition processD. Mendlovic, Y. Bitran, and R. G. Dorsch are with the Faculty ofEngineering, Tel Aviv University, Tel Aviv 69978, Israel. C. Fer-reira and J. Garcia are with the Departamento Interuniversitariode Optica, Universitat de Vale`ncia, Burjassot 46100, Spain. H. M.Ozaktaz is with the Department of Electrical Engineering, BilkentUniversity, Bilkent 06533, Turkey.Received 27 February 1995; revised manuscript received 17 July1995.0003-6935@95@327451-06$06.00@0.r 1995 Optical Society of America.1see, for example, Ref. 72. In Ref. 8, the anamorphicprocessor was applied for obtaining fine pseudocolor-ing encoding.We investigate the application of the anamorphicapproach to fractional-Fourier-transform 1FRT2 sys-tems. The FRT was recently applied to optical appli-cations by two different definitions: one based on theWigner-distribution transformation9 and the other onthe propagation in a graded-index medium.10 It wasshown that both definitions are equivalent.11 TheFRT operation offers a considerable number of newapplications. Some of them are discussed in Ref. 12.The combination of an FRT system and anamorphicoptics could result in a system that performs an FRToperation with two different fractional orders alongthe two main axes. This capability, which consider-ably extends the number of applications of FRTsystems, was cited in Ref. 9 but was not elaborated on.In Section 2 the basic details regarding the FRTdefinition according to the Wigner-distribution trans-formation are given. Section 3 shows the anamor-phic optical setup for performing a nonsymmetricalFRT; the relevant mathematical expressions for de-signing the optical setup are given. In Section 4 weshow computer simulations with laboratory experi-mental results that demonstrate the feasibility of thesuggested system. In Section 5 a list of applicationsis suggested.2. Fractional Fourier Transform and its Relation to theWigner DistributionBased on theWigner-distribution chart definition, theFRT with order P of an input function u1x2 was10 November 1995 @ Vol. 34, No. 32 @ APPLIED OPTICS 7451defined as follows.9 Performing the Pth FRT opera-tion corresponds to rotating the Wigner distributionof u1x2 by an anglef 5 Pp2112in the clockwise direction. On the basis of thisdefinition, Lohmann showed that the bulk-optics sys-tem of Fig. 1 performs FRT’s of order P provided thatthe following parameters are used:f 5f1Q, 122Z 5 Rf1, 132whereR 5 tan1f22 , 142Q 5 sin1f2, 152and f1 is a constant, which connects the spatialFig. 1. Optical setup for performing a FRT.7452 APPLIED OPTICS @ Vol. 34, No. 32 @ 10 November 1995frequency with the spatial coordinate in the FRTplane.By analyzing the propagation of the input signalu1x2 through the optical system of Fig. 1, one couldwrite explicitly the FRT operation of order P, F P,which is the amplitude distribution in the planedefined for the x1 axis, asu1x12 5 F P5u1x265 e u1x2exp1ip x12 1 x2T 2exp12i2p xx1S 2dx, 162whereS 5 l f1 sin1f2, 172T 5 l f1 tan1f2, 182Note that although the above derivations are givenfor 1-D objects, the generalization to 2-D objects canbe done as shown in Ref. 9.3. Anamorphic Fractional-Fourier-Transform SystemThe bulk-optics optical system of Fig. 1 was extendedfor performing nonsymmetrical FRT operations byuse of anamorphic optics. We assumed that thesystem should perform an FRT of order Px along thex-axis direction and an FRT of order Py along the yaxis. For simplicity, it was also assumed thatPy , Px. 192The suggested system for performing the nonsym-metrical FRT is shown in Fig. 2. It contains asymmetrical part for performing the FRT of order Py,which is exactly the same as that in Fig. 1. Then ananamorphic part was added. In this part, along the xaxis, a FRT of order Px 2 Py is performed, while alongthe y axis, two imaging operations are done. NoteFig. 2. Anamorphic optical setup for performing FRT’s of different orders in the x and the y axes.that an imaging operation means a FRT of order 2;thus two imaging operations gives a FRT of order 4,which is the self-Fourier order.13Now we deal with the anamorphic part of thesystem. For performing the x-direction FRT, a cylin-drical lens with focal length fx is placed as shown inFig. 2. This lens is active only along the x axis. Inthe y direction two imaging operations are obtainedwith two identical lenses with a focal length fy. Onecan notice that imaging with lateral magnificationequal to 1 is obtained only if the distance between theoutput of the symmetrical system and the outputplane is 8fy. The system can be shortened to 4fy if alateral magnification of 21 is admitted.Analogous to Eq. 112, we havef8 5 1Px 2 Py2p@2. 1102Thus, owing to Eq. 132, one can write4fy 5 Rx f1 5 f1 tan1f82 2 . 1112Equations 122 and 132 lead tofx 5 f1@sin1f82, 1122Zx 5 Rx f1. 1132Note that f1 is the same constant that was used forcalculating the symmetric part of the system. Thisprovides the same spatial coordinates scale in bothaxes.Based on the knowledge of f1 and f8, and given thelens parameters fy and fx, the output of the anamor-phic system when the object u1x, y2 is placed at theinput plane isu1x2, y22 5 F xPx F yPy5u1x, y265 ee u1x, y2exp31ip x22 1 x2Tx 2 1 1ip y22 1 y2Ty 243 exp12i2p xx2Sx 2 i2pyy2Sy 2dxdy, 1142and Tx, Ty, Sx, and Sy are calculated according to Eqs.172 and 182withfx 5 Pxp@2, 1152fy 5 Pyp@2. 11624. Simulation and Experimental ResultsThe suggested anamorphic processor was simulatedby computer and experimentally demonstrated. Forthe computer simulations, MATLAB software for simu-lating the integral of Eq. 1142 was written. Note thatit involves a 2-D integral that contains a chirp factor,which requires a huge number of calculations. Thusthe estimation of this integral was done by use of amore efficient procedure that is based on the gradient-index medium definition of the FRT.10 Figure 31a2shows the input pattern that was used for the simula-tion. The order Px 5 0.667 and Py 5 0 was used.Figure 31b2 depicts the corresponding simulation re-sult.An optical setup similar to the one in Fig. 2 wasconstructed. In order to eliminate a cylindrical lens,we used a setup that performs Py 5 2. That is, alongFig. 3. 1a2 Input pattern used in experiments, 1b2 anamorphicFRT of orders Px 5 0.667 and Py 5 0 calculated digitally, and1c2 experimental optical result.10 November 1995 @ Vol. 34, No. 32 @ APPLIED OPTICS 7453the y direction, we perform an imaging with inversion.In this way, the optical setup contains only twocylindrical lenses. In our case, the lenses were withfocal lengths of 304.8 mm for the x direction and76.2 mm for the y direction. Again, the pattern ofFig. 31a2 was used as input. The obtained output ispresented in Fig. 31c2. A fair agreement between thesimulation and the laboratory experiment is obtained.Nevertheless, note that the aspect ratio of the com-puter simulation and the experimental results is notthe same.5. ApplicationsA. Anamorphic Fractional CorrelationBased on the symmetric FRT, various definitions forthe fractional correlation operation were suggested inRef. 14. It was shown that the fractional correlationis a useful tool for shift-variant pattern recognitionand for object localization. Below, we suggest theextension of the fractional correlation definition alsoto anamorphic fractional orders. If we take intoaccount the conclusion of Ref. 14 that the amount ofthe shift-variance property could be controlled by thefractional order, the motivation of this anamorphicdefinition might be a system that has different shift-invariant properties along the two main axes. Forinstance, when letters written in an x direction lineshould be recognized, the shift-invariant propertyalong the y direction is not necessary, and we can saveit for improving the noise performance of the system.Figure 4 shows a block diagram of the algorithm forperforming an anamorphic fractional correlation.It contains the filter generation procedure and thefractional correlation itself. Note that this proce-dure is based on the first definition that was given inRef. 14. Anamorphic implementations of the otherdefinitions can be done in a similar way.Fig. 4. Block diagram showing the fractional correlation proce-dure.7454 APPLIED OPTICS @ Vol. 34, No. 32 @ 10 November 1995B. Efficient MultiplexingIn many practical cases a single transmission line isused to transmit several signals packed together.Multiplexing techniques permit the signal compres-sion for the transmission. An analogous situationarises when several signals are to be stored in arecording medium. In these cases it is important touse in the most efficient way the space–bandwidthproduct of the recording medium or the transmissionline, which permits the transmission of the maximumpossible amount of information. Two typical waysfor multiplexing are in the space and in the frequencydomains. In the first case the signals are split intospatial 1or temporal2 pieces, which are interleaved forseveral signals in space 1time2 sequence. For fre-quency multiplexing the process is based on adding abias frequency that is different for every signal, whichensures that the spectra do not overlap. In this caseseveral signals can share the same space withoutcross talk. The procedure may be easily extended toboth space- and frequency-domain multiplexing.The useful area of the transmission line, defined bythe aperture of the system and the bandwidth, may becovered by signals of smaller extent.The multiplexing can be easily illustrated for 1-Dsignals in the Wigner-distribution space. This jointrepresentation involves simultaneously the space andthe frequency domains. The space and the fre-quency cutoff of the system or transmission linedefines a rectangular area 1DXTotal, DnTotal2, which lim-its the signals that are input into the system.Analogously, every signal may be circumscribed in arectangle of dimensions 1Dx, Dn2. The packing ofsignals in the Wigner space consists of shifting everysignal, creating an array inside the bandwidth of thesystem. An example is shown in Fig. 5 for the 1-Dcase.Nevertheless, this packing procedure, which usesthe space and the frequency bandwidths of the signal,is not always the most efficient one. In many cases,Fig. 5. Space and frequency multiplexing of the signals when themain axes of their domains are aligned with the 1x, n2 axes.the domain of the signal does not fully cover therectangle defined by 1Dx, Dn2. The previously de-scribed procedure for packing will result in a wastingof the transmission capacity of the system. An ex-ample is depicted in Fig. 6. In this situation arectangle oblique to the 1x, n2 axes would include thesignal domain more accurately. This can be accom-plished by rotation of the Wigner distribution of eachsignal before multiplexing. Then, adding a bias fre-quency and shifting every signal, one would obtain anarrangement analogous to that of Fig. 5. As men-tioned in Section 1, a FRT of order P produces arotation of the Wigner distribution by an angle f 5P1p@22. The procedure for demultiplexing would in-volve a FRT of order 2P for every signal. Thegeneralization for 2-D signals is straightforward.For the case of 2-D signals, if a symmetric opticalsystem is used for performing the FRT, the order ofthe transformation is the same for both axes. Amuch greater flexibility is provided if an anamorphicsystem is used. In this case, it is possible to render adifferent-order FRT for each variable. The conse-quent four-dimensional rotation in the Wigner spacemay adjust the domain of the signal to the axis of thetransmission line in amore accurate way. It is worthnoting that the rotation can be distinct for everysignal. Extension to nonequally spaced distribu-tions of signals in the Wigner space is also possible.C. Anamorphic Chirp FilteringA common use of the Fourier transform is its use inspatial filtering. A simple case is that with an imagecorrupted with an additive single-frequency noise.The filtering of the Fourier transform with two smallstops removes this noise without significantly alter-ing the image. More complicated is the case in whichthe noise does not consist of a single frequency. Aparticular case that arises in some practical situa-tions is the chirp noise. It consists of a signal with aspatially variable linear frequency, represented, forinstance, by u1x2 5 sin1kx22. It may appear as aFig. 6. Inefficient multiplexing of signals with the domains notaligned with the 1x, n2 axes.result of the interference of two spherical beams withdifferent curvatures. This signal is not well locatedin either space or frequency domains, being, in prac-tice, impossible to remove in any of these spaces byspatial filtering without the image being severelyaltered. A FRT can solve this problem. The reasonis that a chirp signal has a Wigner distribution that isconcentrated in a line in the 1x, n2 space. Rotating itso that the line is perpendicular to the x axis, we canagain obtain a spatial filtering by using a small stop.Recovery of the filtered image would involve aninverse FRT of the same order.In the 2-D case, chirp noise with distinct character-istics in both spatial axes may appear. If the param-eters of the chirps are not the same, a differentrotation of the Wigner distribution may be needed tofilter out each chirp noise. In a symmetric system,this would involve filtering in two different planescorresponding to different fractional orders. The useof anamorphic optics introduces the possibility ofperforming a different FRT for two orthogonal axes.The system can simultaneously filter two chirp noiseswith different characteristics.6. ConclusionsIn this paper the anamorphic fractional Fourier trans-form was optically implemented by means of a setupthat contains cylindrical lenses. Such an anamor-phic transform can increase the flexibility of theoptical system. The degree of fractional Fouriertransform can be set independently for two perpen-dicular directions. This provides the possibility ofacting in different fractional domains in these axes.Experimental resultsmatchwell with computer simu-lations. Finally, some applications of anamorphicfractional systems such as anamorphic fractionalcorrelation and multiplexing in the fractional domainare suggested.This work was partially supported by the Comisio´nInterministerial de Ciencia y Tecnologı´a, Spain, undergrant TAP93-0667-C03-03. Rainer Dorsch is onleave from Angewandte Optik, Physikalisches Insti-tut, Erlangen University, Germany. Fruitful discus-sions with Adolf W. Lohmann are gratefully acknowl-edged.References1. J. W. 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Anamorphic Fractional Fourier Transforming--Optical Implementation and Applications

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