Cataloged from PDF version of article.Recently two optical interpretations of the fractional Fourier transform operator were introduced. We
address implementation issues of the fractional-Fourier-transform operation. We show that the original
bulk-optics configuration for performing the fractional-Fourier-transform operation 3J. Opt. Soc. Am. A
10, 2181 1199324 provides a scaled output using a fixed lens. For obtaining a non-scaled output, an
asymmetrical setup is suggested and tested. For comparison, computer simulations were performed.
A good agreement between computer simulations and experimental results was obtained

Bitran, Y.

Dorsch, R. G.

Lohmann, A. W.

Mendlovic, D.

Ozaktas, H. M.

Recently two optical interpretations of the fractional Fourier transform operator were introduced. We
address implementation issues of the fractional-Fourier-transform operation. We show that the original
bulk-optics configuration for performing the fractional-Fourier-transform operation 3J. Opt. Soc. Am. A
10, 2181 1199324 provides a scaled output using a fixed lens. For obtaining a non-scaled output, an
asymmetrical setup is suggested and tested. For comparison, computer simulations were performed.
A good agreement between computer simulations and experimental results was obtained

Özaktaş, Haldun M.

Bilkent University Institutional Repository

TECHNICAL NOTEFractional Fourier transform:simulations and experimental resultsYigal Bitran, David Mendlovic, Rainer G. Dorsch, Adolf W. Lohmann,and Haldun M. OzaktasRecently two optical interpretations of the fractional Fourier transform operator were introduced. Weaddress implementation issues of the fractional-Fourier-transform operation. We show that the originalbulk-optics configuration for performing the fractional-Fourier-transform operation 3J. Opt. Soc. Am. A10, 2181 1199324 provides a scaled output using a fixed lens. For obtaining a non-scaled output, anasymmetrical setup is suggested and tested. For comparison, computer simulations were performed.A good agreement between computer simulations and experimental results was obtained.Key words: Fourier optics, optical information processing, fractional Fourier transforms.Recently the fractional-Fourier-transform 1FRT2 opera-tor was described in terms of physical optics opera-tions.1–3 In this Note we address some experimentalaspects of the FRT operator.The first FRT definition1,2 is modeled by the varia-tion of the field during propagation along a quadraticgraded-index 1GRIN2medium by a length proportionalto the FRT order a. The eigenmodes of quadraticGRIN media are the Hermite–Gaussian 1HG2 func-tions, which form an orthogonal and complete basisset.4 Themth member of this set is expressed asCm1x2 5 Hm1˛2x@v2exp12x2@v22, 112whereHm is a Hermite polynomial of orderm and v isa constant that is connected with the GRIN-mediumparameters. Each function u1x2 can be expressed as alinear combination of Cm1x2, where the coefficient ofeach HG mode is denoted by Am. With the abovedecomposition the FRT of order a is defined asF a3u1x24 5 omAmCm1x2exp1ibmaL2, 122Y. Bitran and D. Mendlovic are with the Faculty of Engineering,Tel Aviv University, Tel Aviv 69978, Israel. R. G. Dorsch andA. W. Lohmann is with the Angewandte Optik, Erlangen Univer-sity, Erlangen 8520, Germany. H. M. Ozaktas is with the Depart-ment of Electrical Engineering, Bilkent University, Bilkent 06533,Ankara, Turkey.Received 16 March 1994; revised manuscript received 11 October1994.0003-6935@95@081329-04$06.00@0.r 1995 Optical Society of America.where L is the GRIN length that results in theconventional Fourier transform and bm is the propaga-tion constant for each HGmode.In Ref. 3 the FRT operation is defined alternativelyas what happens to the signal u1x2 when its Wigner-distribution function 1WDF2 is rotated by an anglef 5 ap@2. Because the WDF of a function can berotated with bulk optics, Lohmann suggested3 use ofthe bulk-optics system of Fig. 1 for implementing theFRT operator. In his paper,3 Lohmann character-ized this optical system using two parameters, Q andR:f 5 f1@Q, z 5 f1R, 132where f1 is an arbitrary fixed length, f is a variablefocal length of the lens, and z is the distance betweenthe lens and the input 1or the output2 plane. For afractional Fourier transform of order a, Q and Rshould be chosen asR 5 tan1f@22, Q 5 sin1f2, f 5 a1p@22. 142By analyzing the optical configuration of Fig. 1, onemay writeF a3u1x24 5 C1 e u1x02exp1ip x02 1 x2lf1 tan f23 exp12i2p xx0lf1 sin f2dx0, 152where l is the wavelength and C1 is a constant.Equation 152 defines the FRT for one-dimensional10 March 1995 @ Vol. 34, No. 8 @ APPLIED OPTICS 1329functions. Generalization for two-dimensional func-tions is straightforward.3 This FRT integral defini-tion is fully equivalent to themodal definition given inEq. 122, as shown in Ref. 5.Unlike the conventional Fourier-transform opera-tion, which is scale invariant 1scaling the input objectresults in a reciprocal scaling of the output2, thegeneralized FRT is scale variant.1,2 Thus if one usesthe wrong scaling factor at the input plane, it will beimpossible to get the correct FRT output simply byscaling the output plane. In other words it is manda-tory to use the correct scaling factor at the inputplane.The influence of changing the scale factor on theHG representation can be found by substitution of thefirst HG mode C01x2 in Eq. 152. The result is thefollowing relation between f1 and theHGparameterv:v2 5 lf1@p. 162As is mentioned below, computer simulations weredone based on the HG representation. Thus theabove relation is important to adjust the scale factorof the computer simulations and the bulk-optics ex-periments.Equation 152 implies a scale factor of 1l f121@2 for boththe input and the output. However, f1 5 f sin31ap2@24 isa function of the FRT order. Thus to keep the scaleconstant, one needs a zoom lens, which is inconve-nient.To avoid the scale changes when performing theexperiments, with the same lens for all fractionalorders, we suggest an asymmetrical configuration.Figure 2 shows the optical setup including the twofree-space propagation lengths Af1 and Bf1 and thefixed-lens focal length f 5 f1@Q. After some deriva-tion, the output field due to the input u01x02 isu1x2 5 e u01x02exp3 iplf1 1x02 1 2 QBB 1 A 2 QAB1 x21 2 QAB 1 A 2 QAB22xx0B 1 A 2 QAB24dx0.172We want this output equation to match the FRTdefinition. By inspecting the coefficients of x2 andx02, one notes that the output of the asymmetricalsetup is multiplied by a quadratic phase term. ThusFig. 1. Setup for performing a two-dimensional fractional Fouriertransform according to the WDF definition.1330 APPLIED OPTICS @ Vol. 34, No. 8 @ 10 March 1995only the absolute value can be matched to the FRTdefinition,0u1x20 5 0 F a3u01x0240 , 182by choice ofA 5sin f 2 11 2 cos f2@Qcos f, B 5 11 2 cos f2@Q.192Fig. 3. Input function 1a2 cross section and 1b2 two-dimensional,separable function obtained with a pair of different one-dimen-sional functions in outer-product form.Fig. 2. Setup for performing the fixed-scale asymmetrical FRT.Fig. 4. FRT experimental result obtained with the optical symmetrical setup of Fig. 1 with the parameters of Table 1. The FRT order is1a2 a 5 0.25, 1b2 a 5 0.5, 1c2 a 5 0.75, and 1d2 a 5 1.A convenient selection for Q is Q 5 1, which impliesf 5 f1.To acquaint ourselves with the FRT operator, weperformed computer simulations. Based on the abovetwo definitions, two approaches were suggested: oneis based on the HG definition 3Eq. 1224, and the secondis based on the WDF definition 3Eq. 1524. Eq. 152suggests the performance of a Fourier transformationon the input function multiplied by a quadratic phaseterm 1chirp term2. Because of the high frequenciesnecessary to represent truly the chirp term, theresolution necessary to represent the quadratic phaseterm is much higher than that of the input, requiringa high number of sampling points.The other FRT computer-implementation approachis based on the HG definition. With the modalnotation,2 Eq. 122 can be written asua 5 F a3u04 5 CbaC21u0, 1102where ba is a diagonal matrix having exp12ipam@22 asits mth diagonal element, and the C matrix is con-structed from the HG functions Cm as its rows. EachHG function is normalized by 1hm21@2 to ensure that Cis orthonormal. We found the HG approach to be thefaster simulation procedure, and thus it was usedhere for performing the computer experiments.A MATLAB subroutine was written based on Eq. 1102.First, the HG modes C were calculated and stored asa random-access-memory variable. Now each FRTof a vector consists of the product of the input vectorby CbaC21. C21 5 CT, and ba is diagonal. Thusthe number of operations is 2N2 for the two matrix–vector multiplications, whereas ba1C21u02 is a vector–vector multiplication 1only N operations2. C is com-Table 1. Parametersa Used for Performing the SymmetricalSetup ExperimentsFRTOrder a f° R Q f1 1mm2 Z 1mm20.25 22.5 0.199 0.383 95.75 19.050.50 45 0.414 0.707 176.75 73.170.75 67.5 0.668 0.924 231.0 154.311 90 1 1 250 250aThese parameters take into account different scale factors forthe various FRT orders.puted only once. Consequently each FRT calculationconsist mainly of two matrix–vector multiplications,which are performed relatively fast. For example,the number of operations for a 256-element inputvector 1N 5 2562 is roughly 2N2 5 132,000 operations.The input vector length is alwaysN; thus the requirednumber of operations is independent of the fractionalorder.The experimental demonstration of the FRT wasperformed with bulk-optics setups, both for the sym-metrical one 1Fig. 12 and the asymmetrical one 1Fig. 22.Although the GRIN approach could be useful forlaboratory experiments, the superiority of the bulk-optics system is apparent because of its much higherSW 1space bandwidth product2 performance and flex-ibility. The GRIN approach did serve for the com-puter simulations.Figure 3 shows the input function that was used forthe simulations and for the experiments. For sym-metrical-set-up laboratory experiments a single lenswith f 5 250 mm was used. The control of thefractional order was done by a change in the param-eter z; thus, as mentioned above, the scale of the inputand the output depends on the FRT order. The inputmask size was 3.75 mm. The distance z to obtaindifferent fractional orders a is given by a combinationof Eqs. 132 and 142:z 5 f tan f@2 sin f, 1112where f 5 a1p@22.Figures 41a2–41d2 show the experimental results ofthe intensity distribution for fractional orders a equalto 0.25, 0.5, 0.75, and 1, respectively. The opticalsetup of Fig. 1 was used with the parameters of Table1. A fixed lens was used. Thus the various orderexperiments were done with a different scale factor ofTable 2. Parametersa Used for Performing the AsymmetricalSetup ExperimentsFRT Order a f° Af11mm2 Bf11mm20.25 22.5 83.0 19.00.50 45 146.5 73.250.75 67.5 200.25 154.251 90 250 250af is 250 mm.10 March 1995 @ Vol. 34, No. 8 @ APPLIED OPTICS 1331Fig. 5. FRT experimental result obtained with the asymmetrical optical setup of Fig. 2 with the parameters of Table 2. The FRT order is1a2 a 5 0.25, 1b2 a 5 0.5, 1c2 a 5 0.75 and 1d2 a 5 1.the input and the output. This was taken intoaccount when the computer simulations were per-formed, and an excellent agreement of the experimen-tal and computer-simulation results was obtained.For brevity we omitted the computer-simulation plots.An additional set of experiments was performed inorder to explore the asymmetrical configuration.According to Fig. 2, the parameters of the setup werecalculated to provide the FRT orders 0.25, 0.5, 0.75,and 1. These parameters are presented in Table 2; fand l are the same as in the symmetrical experiment.Figure 5 shows the obtained results. The resultswere compared with the computer-simulation results,and again an excellent agreement was obtained.To conclude, we have introduced the various possi-bilities for implementing the fractional-Fourier-transform operation. For computer simulations theGRIN interpretation was used. A bulk-optics opticalimplementation was suggested in a configurationvery similar to the conventional 2-fFourier-transform-ing system according to the WDF interpretation.Two optical setups were introduced, a symmetrical1332 APPLIED OPTICS @ Vol. 34, No. 8 @ 10 March 1995and an asymmetrical one. It was shown that, withthe symmetrical configuration with a fixed lens, theinput and the output objects should be scaled. Theasymmetrical setup avoids this scale factor but intro-duces a quadratic phase distribution to the outputplane. Both setups were implemented optically, andexperimental results were demonstrated.References1. D. Mendlovic and H. M. Ozaktas, ‘‘Fractional Fourier transfor-mations and their optical implementation. Part I,’’ J. Opt. Soc.Am.A 10, 1875–1881 119932.2. H. M. Ozaktas and D. Mendlovic, ‘‘Fractional Fourier transfor-mations and their optical implementation. Part II,’’ J. Opt.Soc. Am.A 10, 2522–2531 119932.3. A. W. Lohmann, ‘‘Image rotation, Wigner rotation, and thefractional Fourier transform,’’ J. Opt. Soc. Am. A 10, 2181–2186119932.4. A. Yariv,Optical Electronics, 3rd ed. 1Holt, NewYork, 19852.5. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, ‘‘Graded-index fibers, Wigner distribution functions, and the fractionalFourier transform,’’ Appl. Opt. 33, 6188–6193 119942.

Fractional Fourier Transform- Simulations and experimental results

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