Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane

158 J. Opt. Soc. Am. A/ Vol. 0, No. 8/ August 003 Y. Cai and Q. Lin ransformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane Yangjian Cai and Qiang Lin Institute of Optics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 31008, China Received October, 00; revised manuscript received March 4, 003; accepted March 7, 003 An analytical and concise formula is derived for the fractional Fourier transform (FR) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams. he corresponding tensor ABCD law performing the FR is obtained. he connections between the FR formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FR plane are studied in detail. he results show that the fractional order of the FR has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FR of twisted anisotropic GSM beams. 003 Optical Society of America OCIS codes: 030.1640, 070.590, 070.580. 1. INRODUCION he concept fractional Fourier transform (FR) proposed by Namias is the generalization of the conventional Fourier transform. 1 McBride and Kerr developed Namias s definition and made it more rigorous. Ozaktas, Mendlovic, and Lohmann introduced the FR into optics and designed optical systems to achieve the FR. 3 6 he connections between Fresnel diffraction and the FR 7 and between the Collins formula and the FR 8 have been discussed. he FR has been used widely in signal processing, 9 11 beam shaping, 1 beam analysis, 1,13 etc. Recently the discrete FR, 14 the scaled FR, 15 the fractional Hilbert transform, 16 the Fractional Hankel transform, 17 and multifractional correlation 18 have also been defined. Up to now, however, most efforts have been concentrated on deterministic light fields; i.e., the light beam was regarded as completely coherent. Only a few studies have been done on the fractional transform of the partially coherent beams based on the mutual intensity distribution 19 and the Wigner distribution function. 3 5 In fact, partially coherent beams are applied widely in practice, such as in improving the uniformity of intensity distribution in inertial confinement fusion 6 and in reduction of noise in photography. 7 herefore it is of practical significance to study the FR of partially coherent beams. Ever since the introduction of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams by Simon and Mukunda, 8 many studies have been done on the propagation and transformation properties of twisted anisotropic GSM beams by use of the Wigner distribution function. 9 33 But this is an indirect method. More directly, partially coherent beams can be described in terms of their cross-spectral density. Recently we introduced a new tensor method to treat the propagation and transformation of partially coherent twisted anisotropic GSM beams based on the cross-spectral density. 34 In the present study, the tensor method is used to apply the FR to partially coherent twisted anisotropic GSM beams and is based directly on the cross-spectral density. he relations between the FR formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed in detail as are the effects of the fractional order on transformation and spectrum properties in the fractional Fourier plane.. HEORY he two types of optical system for performing the FR are shown in Fig. 1. Assume a stationary quasimonochromatic source field E(r). he FR of E(r) achieved by the optical system as shown in Fig. 1 is given by 3,35 1 E p u Erexp ir u if sin f tan exp ir u d r, (1) f sin where r and u are the position vectors in the input plane and in the fractional Fourier plane, respectively. c/ is the optical wavelength, is the angular frequency, and c is the velocity of light in vacuum. he fractional order of the FR is given by p /. When p 4n 1, n being any integer, the FR goes back to being the conventional Fourier transform. In Eq. (1), a fac- 1084-759/003/08158-09$15.00 003 Optical Society of America

Y. Cai and Q. Lin Vol. 0, No. 8/August 003/J. Opt. Soc. Am. A 159 tor 1/(if sin ) in front of the integral ensures energy conservation after FR transform. Assume that the optical fields at the two arbitrary points r 1 and r in the input plane are E o (r 1 ) and E o (r ) and that the optical fields at the two arbitrary points u 1 W 0 r S 0 exp i r M i r, (7) c where M i isa4 4 matrix called a partially coherent complex-curvature tensor; it takes the form M M 11 M 1 M 1 M 11 * R ic I ic ic g g J ic g J R ic I ic, (8) g and u in the FR plane are E p (u 1 ) and E p (u ), respectively. he cross-spectral density in the input plane, W o (r ), and that in the FR plane, W p (ũ), are defined by W o r E o r 1 E o *r, W p ũ E p u 1 E p *u, () where the angular brackets denote an ensemble average, r (r 1 r ), ũ (u 1 u ). Substituting Eq. (1) into Eq. (), we get the relation between the cross-spectral density of a partially coherent beam in the FR plane and one in the input plane, as follows: W p ũ W 0 r cf sin exp ir 1 u 1 r u cf tan exp ir 1 u 1 r u dr 1 dr. (3) cf sin After rearrangement, Eq. (3) can be expressed in tensor form as follows, where I is a transverse spot-width matrix, g is a transverse coherence-width matrix, R is a wave-frontcurvature matrix, and is a scalar real-valued twist factor. I, g, and R are all matrices with transpose symmetry, given by I I11 I1 I1, I R R 11 R 1 g g11 R 1 R, g1 g1, g and J is a transpose antisymmetry matrix given by (9) J 0 1 0. (10) Substituting Eq. (7) into Eq. (4), after vector integration we obtain W p ũ W 0 r exp NR i cf sin c R d r, (4) where R (r ũ ) (r 1 r u 1 u ) is a 1 8 matrix and N isa8 8 matrix given by N N 11 N 1 N 1 N 11. (5) N 11 and N 1 are all 4 4 diagonal matrices given by N 11 1 f tan I 0 0 I, N 1 1 f sin I 0 0 I, (6) where I isa unit matrix. Equation (4) is applicable for the FR of any kind of partially coherent beams. Now we will use Eq. (4) to study the FR of partially coherent twisted anisotropic GSM beams. he cross-spectral density of the most general partially coherent twisted anisotropic GSM beam can be expressed in tensor form as follows 34 : Fig. 1. Optical system for performing the FR. (a) One-lens system, (b) two-lens system.

1530 J. Opt. Soc. Am. A/ Vol. 0, No. 8/ August 003 Y. Cai and Q. Lin W p ũ S 0 f sin detm i N 11 / exp i c ũ N 11 N 1 M i N 11 N 1 ũ. (11) o simplify Eq. (11), we introduce four matrices, Ā, B, C, and D, which take the following forms: Ā N 1 N 11 cos I 0 0 I, B N 1 f sin I 0 0 I, sin C N 11 N 1 N 11 N 1 I 0 f 0 I, D N 11 N 1 cos I 0 (1) 0 I. Ā, B, C, and D are the submatrices of the optical system that is performing the FR. We find that they satisfy Ā D, fc B /f. Substituting Eq. (1) into Eq. (11), we get W p ũ S 0 detā B M i exp / i c ũ M p ũ, (13) where M p C D M i Ā B M i. (14) M i and are the partially coherent complexcurvature tensors in the input and the FR planes, respectively. Equation (14) may be called the tensor ABCD law for partially coherent twisted anisotropic GSM beams passing through a FR optical system. Equations (13) and (14) provide powerful tools for treating the FRs of partially coherent twisted anisotropic GSM beams. M p 3. CONNECIONS BEWEEN HE FR FORMULA AND HE GENERALIZED DIFFRACION INEGRAL FORMULAS FOR PARIALLY COHEREN BEAMS Since Collins derived a diffraction integral formula for a complicated optical system based on the ray-transfermatrix method, 36 now known as the Collins formula, his formula has been used widely to treat the transformation and propagation of many kinds of laser beams through optical systems. he Collins formula was extended to axially nonsymmetrical optical systems by use of tensor algebra. 37 he generalized diffraction integral formula for coherent beams propagating through misaligned optical systems was also derived. 38 he connections between Fresnel diffraction and the FR 7 and the Collins formula and the FR 8 for coherent beams have previously been discussed. In this section we discuss the connections between the FR and the generalized diffraction integral formulas for partially coherent beams propagating through aligned optical systems and misaligned optical systems. From Eq. (1) we get N 11 B Ā D B, N 1 B D B Ā C. (15) Substituting Eq. (15) into Eq. (4), we have W p ũ S 0 detb / W 0 r exp i r B Ār r B ũ ũ D B ũ dr, (16) where Ā, B, C, and D are the submatrices in the FR system given by Eq. (1). Equation (16) is in the same form as the generalized Collins formula for partially coherent beams. 34 herefore Eq. (15) is the condition for an aligned optical system to perform FR. Any aligned optical system whose transfer matrices satisfy Eq. (15) can implement the FR for partially coherent beams. Now we discuss the relation between the FR formula and the generalized diffraction integral formula for partially coherent beams propagating through misaligned optical systems. Let us consider a misaligned optical system with multiple elements, as shown in Fig.. he misaligned optical system can be characterized by the 4 4 augmented matrix; S 1 and S represent the input and output reference plane, respectively. he generalized diffraction integral formula for a coherent beam propagating through misaligned optical systems is given by 38 E 1x, 1y i Fig.. b E 1 x 1, y 1 exp i b ax 1 y 1 x 1 1x y 1 1y d 1x 1y ex 1 fy 1 g 1x h 1y dx 1 dy 1. Misaligned optical system. (17)

Y. Cai and Q. Lin Vol. 0, No. 8/August 003/J. Opt. Soc. Am. A 1531 In Eq. (17) the phase factor exp(ikl 0 ) along the axis between the two reference planes has been omitted, and a, b, c, and d are the transfer-matrix elements of an aligned optical system. he parameters e, f, g, and h take the following form: e x x, (18) f y y, (19) g b d x b d x, (0) h b d y b d y, (1) where x, x, y, and y denote the two-dimensional misalignment parameters; x and y are the displacement elements in the x and y directions, respectively; and x and y are the tilting angles of the element in x and y directions respectively.,, and represent the misaligned matrix elements determined by 1 a, l b, c, d, () where l is the axial distance from the input plane and the output plane. For forward-going optical elements, is chosen for the plus sign; for backward-going ones, is chosen for the minus sign. Equation (17) can be rearranged into a more compact form by using the tensor method as follows, i 1/ E 1 Er 1 detb exp i r 1 B Ar 1 r 1 B 1 1 DB 1 exp i r 1 B e f 1 B g h dr 1, (3) where r 1 x 1 y 1 ; r x y ; e f e f ; g h g h; and A, B, C, and D take the following form: A a 0 0 a, B b 0 0 b, C c 0 0 c, D d 0 0 d. (4) By using the variable transformation 1e 1 1 e f, Eq. (3) can be transformed into the following form: E 1e i detb 1/ exp i 1e DB e f 1e B g h 1 e f B g h 1 4 e f DB e f Er 1 exp ik r 1 B Ar 1 r 1 B 1e 1e DB 1e dr 1. (5) Assume that the optical fields at the two arbitrary points r 1, r in the incident plane are E(r 1 ), E(r ) and that the optical fields at the two arbitrary points 1, in the output plane are E( 1 ), E( ), respectively. hen the cross-spectral density in the incident and output planes will be Wr Er 1 E*r, W E 1 E*, (6) where the angle brackets denote the ensemble average, r r 1 r, 1. Using Eq. (5), we can get the propagation formula for the cross-spectral density of partially coherent beams through misaligned optical systems as follows, W where ē f e f are defined as k 4 detb 1/ Wr exp ik r B Ār r B D B exp ik r B ē f B ḡ h dr, (7) e f ; ḡ h g h g h ; and Ā, B, C, and D Ā A 0 0 A, B B 0 0 B, C C 0 0 C, D D 0 0 D. Similarly, by using the variable transformation 1e 1 1 e f, e 1 e f, e 1 ē f, we can transform Eq. (7) into the following form: W e S 0 detb exp i 1/ e D B e f e B g h Wr exp ik r B Ār (8) r B e e D B e dr. (9) By comparing Eq. (16) and Eq. (9), we can find that the main difference is only that the term

153 J. Opt. Soc. Am. A/ Vol. 0, No. 8/ August 003 Y. Cai and Q. Lin exp i e D B e f e B g h existed in front of the integral in Eq. (9), which is irrelevant to the intensity distribution of the partially coherent beams in the FR plane. If we omit this term, then Eq. (9) can be expressed as W e S 0 detb 1/ Wr exp ik r B Ār r B e e D B e dr. (30) If we assume that Ā, B, C, and D in Eq. (30) satisfy Eq. (1), then the generalized diffraction integral formula for partially coherent beams through misaligned optical systems can be described in the form of the FR formula. But compared with the case of an aligned optical system, the cross-spectral density of a partially coherent beam passing through a misaligned optical system for performing the FR has a displacement, 1e 1 1 e f, e 1 e f, e 1 ē f. 4. BEAM PROPERIES IN HE FRACIONAL FOURIER RANSFORM PLANE In this section we study the transformation properties of partially coherent twisted anisotropic GSM beams in the FR plane by using the formulas derived in Section. Assume that the cross-spectral density and the corresponding partially coherent complex-curvature tensor in the input plane take the form of Eq. (7) and Eq. (8), respectively. Applying Eq. (8) and Eqs. (1) (14), we can easily study the transformation properties of twisted anisotropic GSM beams in the FR plane under different beam parameters and optical system parameters. Figure 3 shows contour graphs of the intensity distributions of the twisted anisotropic GSM beam in the FR plane with different fractional orders. he initial parameters used in the calculation are I 1 0. 0. 0.5 mm, 0.5 0. g mm 0. 0.33, R 0 0 0 0 mm, Fig. 3. Contour graphs of the intensity distribution of the twisted anisotropic GSM beam in the FR plane with different fractional orders. (a) p 0, (b) p 0.9, (c) p 1, (d) p 1..

Y. Cai and Q. Lin Vol. 0, No. 8/August 003/J. Opt. Soc. Am. A 1533 0.00001 mm, f 000 mm. From Fig. 3 we can we can find that the fractional order of the FR system has strong effects on the intensity distribution of twisted anisotropic GSM beams in the FR plane. he ratio between the long axis and the short axis of GSM beams varies with change of the fractional order. As shown in Fig. 3(b), the intensity distribution in the FR plane evolves to circles when the fractional order is properly chosen. What is more, we can also see that the orientation of the beam spot varies with changes of the fractional order. Figure 4 shows the dependence of the transverse spotwidth matrix element I11 of the output GSM beam with different initial transverse coherence-width matrix elements g11 on the fractional order p in the FR plane. he initial parameters used in the calculation are g1 5 mm, g 3 mm, 63.8 nm, Fig. 4. Dependence of the transverse-spot-width matrix element I11 of the twisted anisotropic GSM beam with different transverse-coherence-width matrix elements g11 on the fractional order p in the FR plane. Curve (a) g11 0.1 mm, curve (b) g11 0.4 mm, curve (c) g11 mm. f 000 mm, I 1 0. 0. 0.5 mm, R 0 0 0 0 mm, 0.00001 mm. From Fig. 4 we can find that the transverse spot width in the FR plane is closely related to the fractional order of the FR system. he evolution of the transverse spotwidth matrix element I11 along with the fractional order p is periodic, with period. When p n 1, the transverse spot-width matrix element I11 has a minimum value, and when p n, it has a maximum value. From Fig. 4 we can also find that the transverse spot width in the FR plane is closely related to the transverse coherence width. When p n, the GSM beam has the same transverse spot width for different transverse coherence widths, and when p n, it decreases with increase of the transverse coherence-width matrix element g11. Why is the beam spot width of a partially coherent beam with higher degree of coherence more focusable in the FR plane? he physics underlying this phenomenon is that the function of a lens is to change the phase of wave front of the beam. he phase of a lens is fixed or fully correlated. he phases of different parts of the partially coherent beam, however, are partially correlated, so the wave front of a partially coherent beam cannot be fully matched by the phase of a lens. his mismatch is less serious for a beam with higher coherence. herefore the beam with higher coherence is more focusable in the FR plane. o illustrate the relations between transverse coherence width that makes the distribution in the FR plane more focusable and the fractional order p, we calculate the dependence of the transverse coherence-width matrix element g11 of the twisted anisotropic GSM beam on the fractional order p in the FR plane, as shown in Fig. 5. he initial parameters used in the calculation are 0.5 0. g mm 0. 0.33, f 000 mm, Fig. 5. Dependence of the transverse-coherence-width matrix element g11 of the twisted anisotropic GSM beam on the fractional order p in the FR plane. I 1 0. 0. 0.5 mm, R 0 0 0 0 mm, 0.00001 mm, 63.8 nm. From Fig. 5 we can find that the evolution of the transverse coherence-width matrix element g11 along with the fractional order p is also periodic, with period. When p n 1, the transverse coherence-width element g11 has a minimum value; when p n, it has a maximum value. 5. SPECRUM PROPERIES IN HE FRACIONAL FOURIER RANSFORM PLANE Since Wolf first revealed that a partially coherent light beam undergoes spectral shift during its propagation in free space, 39,40 this phenomenon has become an attractive topic in optics, both theoretically 41,4 and experimentally. 43,44 Recently Palma et al. studied the spectral

1534 J. Opt. Soc. Am. A/ Vol. 0, No. 8/ August 003 Y. Cai and Q. Lin changes in Gaussian-cavity lasers and the spectral shift of a GSM beam beyond a thin lens. 45,46 he phenomenon of spectral switch of a partially coherent beam has been studied in detail both theoretically and experimentally. 47 49 However, up to now, most studies have been restricted to the spectrum properties of GSM beams, because there has been a lack of convenient methods to study the spectrum properties of twisted anisotropic GSM beams. Recently the spectrum properties of twisted anisotropic GSM beams propagating in free space 50 in dispersive and absorbing media have been studied in detail by using the newly introduced tensor method. 51 In this section we study the spectrum properties of partially coherent twisted anisotropic GSM beams in the FR plane. he spectrum of a twisted anisotropic GSM can be obtained by setting u 1 u in Eq. (13), i.e., Su 1, Wu 1 u,. (31) Let us assume that the initial spectrum S 0 () isofthe Lorentz type, i.e., S 0 S 0 0, (3) where S 0 is a constant, 0 is the central frequency of the initial spectrum, and is the half-width at half-maximum of the initial spectrum. In the following, the initial parameters used in the calculations are 0 3. 10 15 s, 0.6 10 15 s, S 0 1. Assume that the cross-spectral density and the corresponding partially coherent complex-curvature tensor in the input plane take the form of Eq. (7) and Eq. (8), respectively. hen applying Eqs. (1) (14) and Eqs. (31) and (3), we can obtain the normalized on-axis spectrum of the twisted anisotropic GSM beam in the FR plane with different fractional orders p. he results are shown in Fig. 6. he initial parameters used in the calculation are 0.00001 mm, I 1 0. 0. 0.5 mm, g 10 3.3 mm, R 0 0 0 0 mm, f 000 mm. From Fig. 6 we can find that the shape of the normalized on-axis spectrum of twisted anisotropic GSM beams in the FR plane is similar to that of the source spectrum, but its peak position is blueshifted, and the spectral shift varies with the fractional order p. he phenomenon of spectral changes of the partially coherent beam in propagation is the so-called Wolf effect. he Wolf effect is caused by the correlations between the source fluctuations within the source region and is closely related to the spatial-coherence properties of source. 39 o learn more about the effects of the fractional order on the spectral shift in the FR plane, we will study the relative central-frequency shift of the beam during propagation. Assume that m is the central frequency of the spectrum in the FR plane; then the relative central frequency shift can be expressed by / 0 m 0 / 0. (33) Figure 7 shows the dependence of the on-axis relative central-frequency shift of GSM beams with different transverse coherence-width matrix elements g11 on the fractional order p in the FR plane. he parameters used in the calculations take the same value as those used in Fig. 6. From Fig. 7 we can find that the dependence of relative central-frequency shift along with the fractional order p is periodic, with period. When p n 1, the relative central-frequency shift has a maximum value, and when p n, it has a minimum value of zero. We can also find from Fig. 7 that the relative central-frequency shift is closely related to the transverse-coherence width. With increase of transverse coherence-width matrix element g11, the relativefrequency shift decreases. his implies that for higher degrees of coherence of the twisted anisotropic GSM beams, the relative-frequency shift is smaller. he derived result is in accordance with the results reported in Fig. 6. Normalized on-axis spectrum of the twisted anisotropic GSM beams in the FR plane with different fractional orders p. Curve (a) Source spectrum, curve (b) p 0.5, curve (c) p 0.8, curve (d) p 1. Fig. 7. Dependence of the on-axis relative central-frequency shift of twisted anisotropic GSM beams with different transverse-coherence-width matrix elements g11 on the fractional order p in the FR plane. Curve (a) g11 0.1 mm, curve (b) g11 0.6 mm, curve (c) g11 1mm.

Y. Cai and Q. Lin Vol. 0, No. 8/August 003/J. Opt. Soc. Am. A 1535 Fig. 8. Relative central-frequency shift of the twisted anisotrpic GSM in the FR plane versus the transverse coordinate y p with different fractional orders. Curve (a) p 0.7, curve (b) p 0.8, curve (c) p 0.85. Refs. 44 and 46, where the effects of spatial coherence on the Wolf effect are analyzed in detail. Next we discuss the off-axis spectral shift of twisted anisotropic GSM beams in the FR plane and the effects of the fractional order on the off-axis spectral shift. he relative-frequency shift versus the transverse coordinate y p with different fractional orders in the FR plane is shown in Fig. 8. he initial parameters used in the calculation are 0.00001 mm, 1 1 0. 0. 0.5 mm, g 10 3.3 mm, R 0 0 0 0 mm, f 000 mm, x p 0. From Fig. 8 we can find that the relative centralfrequency shift for on-axis and off-axis points are different. With increase of transverse coordinate y p, the relative central-frequency shift decreases, and the spectral shift changes from blueshift to redshift. We can also find that the decreasing speed of the relative-frequency shift along with the transverse coordinate varies with the changes of fractional order p. For a certain value of the transverse coordinate, the spectral shift is zero. he derived result is in accordance with that found for the case of partially coherent GSM beam beyond a thin lens. 46 From what has been discussed above, we can easily conclude that we can conveniently control the intensity distribution, the elliptical beam s spot area and its orientation, and the spectrum properties of partially coherent twisted anisotropic GSM beams in the FR plane by properly choosing the fractional order of the FR system and the initial beam parameters. he derived results will have applications in material thermal processing by laser beams, where special intensity distribution is required. 6. CONCLUSIONS In conclusion, the fractional Fourier transform for partially coherent twisted anisotropic GSM beams was analyzed by using the tensor method. An analytical formula was derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through FR systems. he relationships between the FR formula and the generalized diffraction integral formulas for partially coherent beams were discussed. he derived formulas provide powerful tools for analyzing and calculating the FR of partially coherent twisted anisotropic GSM beams. he derived formulas were used to study in detail the intensity properties and the spectrum properties of twisted anisotropic GSM beams in the FR plane. he results show that the intensity and the spectrum properties of partially coherent beams are closely related to the parameters of the FR system and the initial beam parameters. We can conveniently manipulate the intensity distribution and the spectrum properties by properly choosing these parameters. ACKNOWLEDGMENS his work is supported by National Natural Science Foundation of China (60078003). Q. Lin s e-mail address is qlin@mail.hz.zj.cn. REFERENCES 1. V. Namias, he fractional Fourier transform and its application in quantum mechanics, J. Inst. Math. Appl. 5, 41 65 (1973).. A. C. McBride and F. H. Kerr, On Namia s fractional Fourier transforms, IMA J. Appl. Math. 39, 159 175 (1987). 3. A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, J. Opt. Soc. Am. A 10, 181 186 (1993). 4. D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their optical implementation: I, J. Opt. Soc. Am. A 10, 1875 1881 (1993). 5. H. M. Ozaktas and D. Mendlovic, Fractional Fourier transforms and their optical implementation: II, J. Opt. Soc. Am. A 10, 5 531 (1993). 6. A. W. Lohmann and D. Mendlovic, Fourier transform: photonic implementation, Appl. Opt. 33, 7661 7664 (1994). 7. P. Pellat-Finet, Fresnel diffraction and the fractional-order Fourier transforms, Opt. Lett. 19, 1388 1390 (1994). 8. Z. Liu, X. Wu, and D. Fan, Collins formula in frequencydomain and fractional Fourier transforms, Opt. Commun. 155, 7 11 (1998). 9. D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, New signal representation based on the fractional Fourier transform: definitions, J. Opt. Soc. Am. A 1, 44 431 (1995). 10. S. C. Pei, M. H. Yeh, and. L. Luo, Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform, IEEE rans. Signal Process. 47, 883 888 (1999). 11. S. Shinde and V. M. Gadre, An uncertainty principle for real signals in the fractional Fourier transform domain, IEEE rans. Signal Process. 49, 545 548 (001). 1. Y. Zhang, B. Dong, B. Gu, and G. Yang, Beam shaping in the fractional Fourier transform domain, J. Opt. Soc. Am. A 15, 1114 110 (1998). 13. X. Xue, H. Q. Wei, and A. G. Kirk, Beam analysis by fractional Fourier transform, Opt. Lett. 6, 1746 1748 (001). 14. C. Candan, M. A. Kutay, and H. M. Ozaktas, he discrete

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