Fast algorithm for chirp transforms with zooming-in ability and its applications

Transcription

1 76 J. Ot. Soc. Am. A/ Vol. 17, No. 4/ Aril 000 Deng et al. Fast algorithm for chir transforms with zooming-in ability and its alications Xuegong Deng, Biin Bihari, Jianhua Gan, Feng Zhao, and Ray T. Chen Microelectronics Research Center, Deartment of Electrical and Comuter Engineering, The University of Texas, Austin, Texas Received July 16, 1999; revised manuscrit received December 7, 1999; acceted December, 1999 A general fast numerical algorithm for chir transforms is develoed by using two fast Fourier transforms and emloying an analytical kernel. This new algorithm unifies the calculations of arbitrary real-order fractional Fourier transforms and Fresnel diffraction. Its comutational comlexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the samling resolutions in both x and u sace and zoom in on any ortion of the data of interest. Comutational results are comared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10 1 for most cases. As an examle of its alication to scalar diffraction, this algorithm can be used to calculate near-field atterns directly behind the aerture, 0 z d /. It comensates another algorithm for Fresnel diffraction that is limited to z d / N [J. Ot. Soc. Am. A 15, 111 (1998)]. Exerimental results from waveguide-outut microcouler diffraction are in good agreement with the calculations. 000 Otical Society of America [S (00)01704-X] OCIS codes: , , , INTRODUCTION Resonses of many hysical systems can be described with chir transforms (ChT s). For examle, chirs on laser ulses, 1 scalar diffraction through a first-order otical system, 6 holograhic lenses, 7 and Fresnel transforms 8 (FnT s) are the most frequently reorted techniques used in scalar diffraction calculations. Recent develoments in otical interconnects surred by high-seed and huge-caacity otical communications include roblems similar to those just mentioned. 9 1 One examle of such a system is illustrated in Fig. 1, where the otical signal is transmitted from board to board and is detected by detectors located several tens of wavelengths away. The near-field diffraction attern at the detector ends directly affects the erformance of the interconnection. It is referable to obtain comlete information on the evolution of the otical signals along the interconnecting ath. In addition, it is easy to show that fractional-order Fourier transforms 13 (FrFT s) can also be treated as chir transforms. Therefore, an accurate, simle, and efficient numerical method will be beneficial in the use of these extensively emloyed formulas. Some studies have been done on the numerical evaluation of Fresnel diffraction; two examles are the use of fast Fourier transforms (FFT s) and convolution techniques 14 and a discrete-fourier-transform- (DFT-) like matrix method. 15,16 A number of algorithms are also secifically devised for FrFT s 17 1 whose roerties have been intensively investigated both mathematically 3 5, and hysically 3 5 in terms of their alications to otical beam roagation, 6 imaging, 4 6 diffraction, 7 and signal and image rocessing The relations among FnT s, DFT s, and FrFT s are also well established. 3,37,38 Our aim in this aer is to develo an efficient algorithm that unifies the evaluations of these formulas. In this aer we describe a fast numerical algorithm that is based on the chir-z transform 39,40 to calculate chir transforms. It emloys two FFT s with an analytical kernel, and its comutational comlexity is better than a fast convolution. In addition, one can freely choose the samling resolutions in both x sace (signal domain) and u sace (transformed, or resonse domain) and zoom in on any ortion of the data of interest. Zooming in may be very useful in studying fine structures of some chir systems, for examle, near-field diffraction. 41 The samling condition will also be addressed under the restriction of the Nyquist theorem. In Section, first we comare several hysical systems that can be classified as chir transforms and establish a general mathematical descrition. Then in Subsection 3.A we resent a discrete form of the transform. The samling condition and discussions on eeing any ortion of the data in u sace are mentioned at the end of the Subsection 3.A. Following the discretization of the integral form of the transform, we use similar techniques in the chir-z transform and develo the concrete fast algorithm in Subsections 3.B and 3.C. In Section 4 we give some numerical examles to demonstrate the effectiveness of this algorithm. Some closed-form transforms such as a Gaussian function and rect(x/a) are tested in Subsection 4.A. In Subsection 4.B the algorithm is alied to zooming in on the Fresnel diffraction of a rectangular window around the focal lane. Exerimental results are rovided for comarison. Near-field diffraction atterns from a 1-to-48-waveguide fan-out interconnection layer were measured, and they corresonded well to /000/ $ Otical Society of America

2 Deng et al. Vol. 17, No. 4/Aril 000/J. Ot. Soc. Am. A 763 u, x R N, () where A() is the amlitude of the kernel and generally a comlex value, i 1, and (,, ) are the arameters of the transform, each of them being an N-dimensional vector. This transform mas the signal in x sace (signal domain) to u sace (chir sace, or resonse domain). All variables are dimensionless. To simlify the discussion, we will use the one-dimensional form of the above equations from now on. However, all the conclusions and results are alicable to the higherdimensional cases as well. In comarison, the Fresnel formula of scalar diffraction through a general Gaussian (first-order) otical system can be exressed with the Collins formula,,6,4 in which the transforming kernel is a secial case of chir transforms [Eq. (1)], namely, B M FnT u, x i B ex ik B Ax xu Du, (3) which is associated with a ray-transfer matrix,,4,43 M A B det M 1. (4) C D, In Eq. (3), ( )/k is the wavelength. This formula connects the araxial arameters and the Fresnel diffraction of the system. It can be rewritten as a secial case of chir transforms when the corresonding arameters are used:,, k A,, D, 4 B A i B. (5) Fig. 1. Schematic to view of the H-tree waveguide used in otical interconnections. The side view of one of its 48 microcoulers is illustrated at the right. Near-field diffraction atterns of the outcouling are critical to the couling of the otical signal to detectors in successive layers and hence to the erformance of the otical interconnections. the results simulated with this algorithm in Subsection 4.C. In Subsection 4.D the versatility of this algorithm is demonstrated in calculating arbitrary real-order FrFT s.. COMPARISON OF SEVERAL GENERAL CHIRP TRANSFORM SYSTEMS A chir transform of an arbitrary signal f(x) in an N-dimensional system is exressed by g ch u C f x and the integration kernel is B ch u, x A ex i l 1 N f x B ch u,x dx, (1) l x l l x l u l l u l, Similarly, the kernel of a real-value th-order FrFT, B FrFT, is another secial case of the chir transform and can be written as 44 B FrFT u, x ex i s / sin 1/ ex i u x cot ux csc, n, (6) in which ( /), s ( /)sign(sin ) and n is an integer. 45 Aarently, the corresonding ChT s arameters are,, 1 cot, csc, cot, 4 A ex i s / 1/. (7) sin Therefore a general algorithm for chir transform alies to comutations of all these cases of Fresnel diffraction, the Collins formula, and FrFT s. To reset a suitable form for numerical calculation of any ChT, in Subsection 3.A we will derive a discrete form of Eq. (1), and in Subsection 3.B we will concentrate on develoing the fast algorithm. 3. DEVELOPMENT OF A FAST NUMERICAL ALGORITHM FOR THE GENERAL CHIRP TRANSFORM A. Discrete Form of the Chir Transform For numerical calculation of the ChT of an arbitrary function f(x) excet those having analytical forms, first Eq. (1) will be digitized. For simlicity, we will use the simlest equidistant samling. The samle stes are denoted x for the signal sace and u for the chir sace. The samle numbers N x and N u for x sace and u sace must be limited; they are not necessarily equal. Therefore the discretization of Eq. (1) may be given as x n x n N x x, n 0, 1,,... N x 1, u k u k N u u, k 0, 1,,... N u 1,

3 764 J. Ot. Soc. Am. A/ Vol. 17, No. 4/ Aril 000 Deng et al. and N x 1 g k g ch k u n 0 f n B D k, n x, B D k, n A ex i n x n k x u k u, (8) where f n f(n x) and B D (k, n) B ch (k u, n x). It can be seen that the arameterized ChT of f(x) is equivalent to the Fourier transform of the modified function f m (x) f(x)ex( i x ), with a simultaneous variable scaling in u sace excet for the difference of a comlex hasor. This could be inferred from Eq. () and written as g ch u g v F f m x F f x ex i x, (9) where v u and F denotes a Fourier transform. 46 This is a well-known technique and was used in deriving several roerties of FrFT (for examle, see Refs. 13, 37, and 38). Therefore, according to the Nyquist samling theorem, if f m (x) is band limited (i.e., there exists a minimum value v m, g(v) 0, v v m 0), the samling ste in x sace must satisfy x 1 1. (10) v m u m This is one imlicit form of the Nyquist samling theorem in ChT saces. 34,47 Another exlicit form of the samling condition will be derived in Subsection 3.B. Under this restriction, one may correctly calculate the numerical transform of a given function. If one is interested in the data eeing in u sace and wants to scrutinize any region of g ch (u), it may be instructive to look at the shifting rule of Fourier transforms. For any nontrivial ChT, 0, and any interested data window center c in u sace, g ch u c g v c F f m x ex i cx. (11) The imlementation for the data eeing is straightforward and needs no further discussion. In Subsection 3.B we will resent the concrete fast rocedure for numerical evaluation of the ChT. B. Fast Numerical Algorithm for the Chir Transform The techniques used in the chir-z transform 39,40 are also useful for efficient calculation of the ChT of a given function f(x). Substituting the following exression into Eq. (8), we get n k 1 n k n k, (1) B D k, n A P n Q k B convol n k, where the hasorlike P n and Q k, as well as the modified kernel B convol (n k ) are P n ex i x x u n, Q k ex i u x u k, B convol n k ex i n k x u. Substituting Eq. (13) into Eq. (8), one can obtain N x 1 g k A x n 0 (13) f m n B convol n k Q k, (14) where f (m) n f n P n is the modified discrete signal. Therefore the discrete ChT of f n can be efficiently calculated through a fast convolution algorithm by use of FFT s such as the techniques in Refs. 17 and 39 which are straightforward and simle to imlement. We can write the rocedure symbolically as g k A F 1 F f m n F xb convol n Q k. (15) F denotes a FFT and F 1 an inverse FFT. However, this is not the most efficient method. The transforming kernel is actually a generalized Gaussian function whose Fourier transform has a closed form, 48 i.e., F ex ax a ex u, R a 0, a (16) which, when alied to Eq. (15), becomes Bˆ convol k F xb convol n i x ex u i u x u, R i 0. (17) For the discrete transform, suose that the samling number in x sace is L x N x and the Fourier transform of the kernel on the u grids is Bˆ convol k i x u ex i 1 x u k L x, k k L x, k 0, 1,,..., L x 1. (18) The discrete ChT of Eq. (15) can be rewritten as g k A F 1 F f m n Bˆ convol k Q k. (19) For imlementation of this algorithm, the samling of the Bˆ convol(k) inu sace must be so dense that the kernel is well aroximated. In ractice, this condition may be satisfied by 1 x u 1 L x 1. (0) It has been assumed that the FFT s of the functions involved do exist. However, these functions can be a distribution (for examle, the Dirac delta function) if we take the Fourier transforms in a general sense. Therefore one need not resort to mathematically rigorous conditions of the fast algorithm in ractice by abiding by the samling condition mentioned above. An alternative is to exlicitly imose the restrictions of Eq. (17) on the arameter.

4 Deng et al. Vol. 17, No. 4/Aril 000/J. Ot. Soc. Am. A 765 One can easily verify that all the Fourier transforms are valid if the chir transform in Eq. (1) exists. The comlexity of numerically evaluating Eq. (19) is about the same as that of two FFT s and no more than six sequential comlex multilications (including the ossible data-eeing oeration), namely, L x log(l x ) O(1) (L x is the number of FFT samles). Comared with the revious chir-z techniques, 39,40 such as those used in Ref. 17, whose time comlexity is exactly as that of Eq. (15), namely, 3L x log(l x ) O(1), the imrovement in comutation time is about one third. Besides, with an analytical form, the accuracy will be much better. Some examles will be resented in Section 4. By using this algorithm, one can freely set the samling stes in both x and u sace, which is very useful under some circumstances such as examining near-field diffraction atterns and interolation of sarse data in u sace. 15,39 One can also choose different numbers of samles in x and u sace by means of the imlementation techniques in Subsection 3.C. C. Imlementation of the Fast Algorithm To imlement the algorithm, the most convenient and efficient way is robably to use the standard FFT s. Different numbers of samles in x and u sace will be used in the following rocedure. Suose that the original number of samles in x sace is N x and the required samling number in u sace is N u, which can be different from N x. Thus the length of a noncircular convolution is L convol N x N u 1, which is the minimum length that should be used for a FFT in Eq. (19). The sequence of calculations is summarized below. in which (L x N u ) should be an even number. This constraint is not necessary, though it could be met easily in ractice. Similar rocedures with minor modifications can be develoed for arbitrary N u. To simlify our discussion below, the equivalent requirement that N x and N u be of the same arity will be used. The erformance of the algorithm will be evaluated with some alication examles in Section APPLICATION EXAMPLES To demonstrate the erformance of this fast algorithm, we examine several chir systems. Different samlings and resolutions are used for these systems. In addition, to unify our discussions, we define a scaling factor, zoom u x, zoom 0, (4) which defines the finesse of the zooming in. Its recirocal, 1/ zoom, is the magnification of the data window. For alications of data eeing or interolation, 0 zoom 1, whereas zoom-out effects occur when zoom 1 under roer samling; this toic will be addressed below. 1. Find a ositive number L x L convol that makes (L x N x ) an even number. L x usually can take the lowest value that satisfies this requirement.. Then comute f n and zero-ad f (m) n : Z f m n n 0 0 m f n Lx N x / 0 n L x N x L x N x L x N x n L x N x n L x 1. (1) 3. Comute the analytical kernel Bˆ convol(k) by means of Eq. (18) for k k L x /, k 0, 1,,..., L x Perform the fast convolution oeration in Eq. (19): C l F Z f m n Bˆ convol k, l 0, 1,,..., L x 1. () 5. Discard the first and last (L x N u )/ data of C(l), and for 0 k N u 1 let g k A Q k C k L x N u, (3) Fig.. (a) Errors e max e(x) of the ChT s of a Gaussian function comared with the analytical transforms for different zoom factors zoom. (b) Tyical error curve e(x) ( zoom ).

5 766 J. Ot. Soc. Am. A/ Vol. 17, No. 4/ Aril 000 Deng et al. the transform is much lower than that for a Gaussian, e However, this does not decrease the validity and erformance of the algorithm. Comarisons of Fig. (b), Fig. 3(b), and Fig. 4 indicate that most of the errors are accumulated in the FFT s used in the calculations. Therefore one can otimize the FFT arameters to achieve higher accuracy. 50 B. Fresnel Diffraction of a Rectangular Window The diffraction of a first-order otical system is erhas one of the simlest chir transforms but is not a trivial chir one. In this section we calculate the well-known near-field diffraction of a rectangular window, rect(x/a, y/b), behind an ideal focusing lens at different distances. The system is illustrated in Fig. 5(a), whose sace bandwidth roduct is discussed in Ref. 49. The elements of its ray-transfer matrix M are A 1 l /f, B l 1 A l, C 1/f, and D 1 l 1 /f. In the simulations N x N u 104, 0.85 m, a 400 m, b 00 m, l m, f m, and l were varied from l 1 to l 1 f while zoom changed corresondingly. Fig. 3. (a) Errors max(e) of the ChT s of a rect(x/a) function comared with the analytical transforms for different zoom factors zoom. (b) Tyical error curve e(x) ( zoom ). A. Analytical Chir Transforms There are many functions whose ChT s are analytically available, such as a generalized Gaussian in Eq. (16). For simlicity, 0 and R x N x were used. The ChT becomes a Fourier transform, and its sace bandwidth roduct 47,49 is N x. We tested the fast algorithm with a Gaussian function for which a N x /(R x ), where x R x /,R x /, N x N u 51, and L x N x N u 104. A normalized error measurement e g k F ex ax max g k (5) between the numerical result and the analytical one is used to check the accuracy. For different values of zoom, the ChT s and the corresonding e max(e) are lotted in Fig. (a), and a tyical error curve for the ChT is lotted in Fig. (b). For 0.01 zoom.0, corresonding to a magnification of the data window of , e A square function, rect(x/a), was used to test the algorithm. Its Fourier transform also has the closed form, F rect(x/a) sin( au)/ u. For the same scaling factors, a R x /4, the ChT s as well as the accuracy measurement e are resented in Fig. 3(a). A tyical error curve for the ChT is lotted in Fig. 3(b). The accuracy of Fig. 4. Tyical error curve of the FFT for (a) a Gaussian function, (b) rect(x/a).

6 Deng et al. Vol. 17, No. 4/Aril 000/J. Ot. Soc. Am. A 767 For nonsearable cases, one can directly use the onedimensional ChT s and the techniques of multidimensional FFT s. 51 Two calculated diffraction atterns are deicted in Figs. 5(b) and 5(c). With the zoom-in factor changing with each distance, it is convenient to examine the fine structures near the focal regions as in Fig. 5(c) by using this fast algorithm. C. Near-Field Outcouling Patterns from a Waveguide In another imortant alication, we examine the nearfield atterns of the oututs from the microcoulers in a 1-to-48 H-tree fan-out waveguide otoelectronics interconnection layer for massive clock signal distribution in a Cray T-90 suercomuter. 9 The otical layer of the system is illustrated in Fig. 1, where a side view of the structure of one of the 48 outcouling oints is schematically resented at the right. The olyimide waveguide is fabricated on a silicon substrate with - m-thick silicon dioxide as bottom cladding. The waveguide is 65 m wide 5 and 10 m thick. The refractive index of the olyimide is at m for TE-olarized waves. The outcouling window is a 45 slanted rectangular mirror. The mode structure of the waveguide was calculated; its first ten eigenmodes in the 10 m direction are lotted in Fig. 6. Matching the excitation boundary conditions for the waveguide indicate that multile modes do exist in the waveguide. This finding validates the choice of a quasi-lane wave for the outcouling wave at the outcouling window in our calculations. For m, N x N u 104, and R x N x x 10 3 m, simulated results from the fast algorithm are deicted in Figs. 7(a), 7(b), 7(c), and 7(d) that corresond to the distances z 10 8, 100, 1000, and 5000 m, resectively, surface normally above the waveguide. The zoom factor, zoom, were also changed along with the roagation distance. As a result of the divergence of the diffraction along the 10- m side of the window, all zoom factors are greater than unity. The diffracted atterns were measured at z 100, 1000, and 5000 m; they are resented in Figs. 7(e), 7(f), and 7(g), resectively. A detailed comarison of the exerimental data with the calculations confirms our assumtions on the mode excitation of waveguide. The Fig. 5. Fresnel diffraction atterns of different zoom factors at different distances behind the aerture. (a) Layout of the otical system. Diffraction atterns for (b) zoom 0.35, l 4500 m and (c) zoom 0.50, l 5000 m. A little trick can be emloyed for the x y-searable cases like this one. In Cartesian coordinates, we can just use outer roduct of two one-dimensional ChT s in the x and y directions, i.e., u M diff u x, u y C M rect x/a, y/b C M rect x/a t C M rect y/b, (6) in which the one-dimensional transform is regarded as a row vector and the suerscrit t denotes its transose. Fig. 6. First ten normalized eigenmodes in the 10 m direction of the outcouling oints of the 1-to-48 fan-out H-tree olyimide waveguide. A slab waveguide model is used in the calculation because the width or the thickness of the waveguide is much larger than the wavelength.

7 768 J. Ot. Soc. Am. A/ Vol. 17, No. 4/ Aril 000 Deng et al. Fig. 7. Near-field outcouling atterns from the waveguide as deicted in Fig. 1. Simulated results at (a) z 10 8 m, zoom 1.000; (b) z 100 m, zoom 1.10; (c) z 1000 m, zoom 1.600; (d) z 5000 m, zoom Exerimental images corresonding to (e) z 100 m, (f) z 1000 m, and (g) z 5000 m.

8 Deng et al. Vol. 17, No. 4/Aril 000/J. Ot. Soc. Am. A 769 Fig. 8. FrFT s of a rectangular aerture, zoom 1.0. (a) 0.1; (b) 0.4; (c) 0.7; (d) 0.9. surious smaller dimensions in the exerimental icture at z 5000 m, i.e., Fig. 7(g), result from the fact that only the central lobe of the diffracted beam is visible, owing to background noise. A careful measurement of the lobe size as well as the sidelobe ositions reveals the correct corresondence between the calculations and the exeriments. For examle, the vertical sot size is 68 m as determined from exeriments, and the calculated vertical sot size is 80 m (the central gray region). Other characteristics such as nonzero regions from the central lobe to the first-order sots in the horizontal central region are also correctly located. It is worth ointing out that the calculations for the above two alications are carried out for distances ranging from essentially zero behind the aerture to the focal lane or the far field; these calculations comensate another algorithm for Fresnel diffraction. 15 In Ref. 15 the alicable distance is limited to z d / N, where d N x x is the aerture size and N is the samling number. However, the samling requirement of our algorithm, namely, Eq. (0), is equivalent to 1 B x ul x N x x zoom L x /N x 1, (7) in which we have lugged in Eq. (5). For a free-sace system, B z. The alicable distance range comensates the distance range in Ref. 15, z (N x x) / zoom (L x /N x ) R x /. The associated errors for scalar diffraction and its validity are analyzed in Refs. 8 and 53, and references therein. D. Fractional Fourier Transform of Arbitrary Orders The transverse distribution of the light roagating in one kind of graded-index fiber can be described with continuous order FrFT s. 30,54 As in Subsection 4.B, the incident wave is chosen as a lane wave on the rectangular aerture. For the fractional orders of 0.1, 0.4, 0.7, 0.9, the x and u sace are normalized so that the windows for calculation are R x N x x R u N u u N x. The size of the transarent aerture is 0.R x 0.5R x. The intensity of the FrFT s of the field is lotted in Fig. 8. Note that for 1.0 the FrFT is just an ordinary Fourier transform. One may refer to Refs. 15 and 30 for comarisons. In this case the domain of interest for most cases is in the same range excet for those with n 1. Therefore we have set the zoom-in factor zoom 1.0 regardless of the ossible need to examine the details of the transformation. On the free choice of the resolutions, there is an uncertainty rincile in addition to the exlicit restrictions such as Eq. (0). The least roduct of sace bandwidth for a FrFT system satisfies 3,38 x u 1/ 1 4. (8)

9 770 J. Ot. Soc. Am. A/ Vol. 17, No. 4/ Aril 000 Deng et al. 5. CONCLUSIONS Emloying some techniques formerly used in chir-z transforms, we have derived a general fast algorithm for the numerical evaluation of chir transforms. This algorithm emloys two fast Fourier transforms (FFT s) and an analytical Gaussian kernel. The new algorithm is also suitable for the calculation of arbitrary real-order fractional Fourier transforms and scalar-field diffraction in first-order otical systems. Its comutation comlexity is L x log L x O(1) and is better than a fast convolution using Fourier transforms. In addition, one can freely choose the samling resolutions in both x and u sace under the restriction of the Nyquist samling theorem and zoom in on any ortion of the data of interest. Comutational results are comared with analytical ones. The errors are limited by the accuracy of the FFT s. Extensions of the algorithm to higher dimensions is straightforward. Several alication examles are resented to demonstrate the versatility of the algorithm. Exerimental results from an H-tree waveguide for otical interconnections confirmed the simulations. The new algorithm is caable of calculating near-field atterns directly behind the aerture, 0 z d /. It comensates another algorithm for Fresnel diffraction that is limited to z d / N. We believe that it is a handy tool for many general scalar diffraction roblems. ACKNOWLEDGMENTS Xuegong Deng thanks Yonging Li (University of Science and Technology of China, Hefei, China), Yongkang Guo, Jingqin Su, and Yixiao Zhang (University of Sichuan, Sichuan, China) for their rivate communications regarding one of our former studies on FrFT s. Thanks also go to Yuri Rzhanov (Heriot-Watt University, Edinburgh, UK) for his careful review of the aforementioned literature and for discussions on several existing imlementations of FrFT s. This work is artly suorted by the Defense Advanced Research Projects Agency, Ballistic Missile Defense Organization, the U.S. Air Force Office of Scientific Research, Cray Research, 3M Foundation, the Texas Advanced Technology Program, and the Otoelectronics Industry Develoment Association Joint U.S. Jaan Otoelectronics Projects. Corresonding author Ray T. Chen can be reached at the address on the title age or by hone, ; fax, ; or , raychen@uts.cc.utexas.edu. REFERENCES AND NOTES 1. A. E. Siegman, Lasers (Mill Valley, Calif., 1986).. S. A. Collins, Lens-system diffraction integral written in forms of matrix otics, J. Ot. Soc. Am. 60, (1970). 3. S. Abe and J. T. Sheridan, Otical oerations on wave functions as the Abelian subgrou of the secial affine Fourier transformation, Ot. Lett. 19, (1994). 4. S. Abe and J. T. Sheridan, Almost-Fourier and almost- Fresnel transformations, Ot. Commun. 113, (1995). 5. S. Abe, J. T. Sheridan, Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an oerator aroach, J. Phys. A 7, (1994). 6. X. Deng, Y. Li, D. Fan, and Y. Qiu, Proagation of araxial flattened Gaussian beams in a general otical system, Ot. Commun. 140, 6 30 (1997). 7. A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, Chired gratings in integrated otics, IEEE J. Quantum Electron. QE-13, (1977). 8. G. W. Forbes, Validity of the Fresnel aroximation in the diffraction of collimated beams, J. Ot. Soc. Am. A 13, (1996). 9. B. Bihari, J. Gan, L. Wu, Y. Liu, S. Tang, and R. T. Chen, Otical clock distribution in suercomuters using olyimide-based waveguides, in Otoelectronic Interconnects VI, J. P. Bristow and S. Tang, eds., Proc. SPIE 363, (1999). 10. J. W. Goodman, F. I. Leonberger, S. Y. Kung, and R. A. Athale, Otical interconnections for VLSI systems, Proc. IEEE 7, (1984). 11. P. Cinato and K. C. Young, Jr., Otical interconnections within multichi modules, Ot. Eng. 3, (1993). 1. J. Jahns, Planar integrated free-sace otics, in Micro- Otics: Elements, Systems and Alications, H. P. Herzig, ed. (Taylor & Francis, London, UK, 1997), ; W. Singer and K. H. Brenner, Stacked micro-otical systems, V. Namias, The fractional order Fourier transform and its alication to quantum mechanics, J. Inst. Math. Al. 5, (1980). 14. C. Ko and P. Meyrueis, Near-field Fresnel diffraction: imrovement of a numerical roagator, Ot. Commun. 158, 7 10 (1998). 15. S. B. Tucker, J. Ojeda-Castañeda, and W. T. Cathey, Matrix descrition of near-field diffraction and the fractional Fourier transform, J. Ot. Soc. Am. A 16, (1999). 16. V. Arizón and J. Ojeda-Castañeda, Fresnel diffraction of substructured gratings: matrix descrition, Ot. Lett. 0, (1995). 17. X. Deng, Y. Li, D. Fan, and Y. Qiu, A fast algorithm for fractional Fourier transforms, Ot. Commun. 138, (1997). 18. F. J. Marinho and L. M. Bernardo, Numerical calculation of fractional Fourier transforms with a single fast-fouriertransform algorithm, J. Ot. Soc. Am. A 15, (1998). 19. H. Haman and J. L. de Bougrenet de la Tocnaye, Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction, J. Ot. Soc. Am. A 1, (1995). 0. B. W. Dickinson and K. Steigletz, Eigenvectors and functions of the discrete Fourier transform, IEEE Trans. Acoust. Seech Signal Process. ASSP-30, 5 31 (198). 1. S. C. Pei and M. H. Yeh, Imroved discrete fractional Fourier transform, Ot. Lett., (1997).. G. S. Agarwal and R. Simon, A simle relation of fractional Fourier transform and relation to harmonic oscillator Green s function, Ot. Commun. 110, 3 6 (1994). 3. H. M. Ozaktas and D. Mendlovic, Fourier transforms of fractional order and their otical imlementation, Ot. Commun. 101, (1993). 4. A. W. Lohmann, A fake zoom lens for fractional Fourier exeriments, Ot. Commun. 115, (1995). 5. S. Liu, J. Xu, Y. Zhang, L. Chen, and C. Li, General otical imlementations of fractional Fourier transforms, Ot. Lett. 0, (1995). 6. L. M. Bernardo and O. D. D. Soares, Fractional Fourier transform and imaging, J. Ot. Soc. Am. A 11, 6 66 (1994). 7. P. Pellat-Finet, Fresnel diffraction and the fractional Fourier transform, Ot. Lett. 19, (1994). 8. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, and H. M. Ozaktas, Chir filtering in the fractional Fourier domain, Al. Ot. 33, (1994). 9. D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms and their otical imlementations: I, J. Ot. Soc. Am. A 10, (1993). 30. H. M. Ozaktas and D. Mendlovic, Fractional Fourier trans-

10 Deng et al. Vol. 17, No. 4/Aril 000/J. Ot. Soc. Am. A 771 forms and their otical imlementations. II, J. Ot. Soc. Am. A 10, (1993). 31. S. Granieri, O. Trabocchi, and E. E. Sicre, Fractional Fourier transform alied to satial filtering in the Fresnel domain, Ot. Commun. 119, (1995). 3. H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, Convolution, filtering, and multilexing in fractional Fourier domains and their relation to chir and wavelet transforms, J. Ot. Soc. Am. A 11, (1994). 33. D. Mendlovic, Y. Bitran, R. G. Dorsh, and A. W. Lohmann, Otical fractional correlation: exerimental results, J. Ot. Soc. Am. A 1, (1995). 34. D. Mendlovic, Z. Zalevsky, R. G. Dorsh, Y. Bitran, A. W. Lohmann, and H. M. Ozaktas, New signal reresentation based on the fractional Fourier transform: definitions, J. Ot. Soc. Am. A 1, (1995). 35. H. M. Ozaktas, and D. Mendlovic, Fractional Fourier otics, J. Ot. Soc. Am. A 1, (1995). 36. P. Pellat-Finet and G. Bonnet, Fractional-order Fourier transform and Fourier otics, Ot. Commun. 111, (1994). 37. A. I. Zayed, On the relationshi between Fourier transform and fractional Fourier transform, IEEE Signal Process. Lett. 3, (1996). 38. A. W. Lohmann and B. H. Soffer, Relationshis between the Radon Wigner and fractional Fourier transforms, J. Ot. Soc. Am. A 11, (1994). 39. S. Roose, B. Brichau and E. W. Stijns, An efficient interolation algorithm for Fourier and diffractive otics, Ot. Commun. 97, (1993). 40. L. R. Rabiner and B. Gold, Theory and Alications of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975). 41. M. Syek, Light roagation in the Fresnel region: new numerical aroach, Ot. Commun. 116, (1995). 4. P. A. Béleuger, Beam roagation and the ABCD ray matrix, Ot. Lett. 16, (1991). 43. A. Yariv, Otical Electronics (CBC College Publishing, New York, 1985), Cha., Y. B. Karasik, Exression of the kernel of a fractional Fourier transform in elementary functions, Ot. Lett. 19, (1994). 45. For n the kernel becomes a Dirac delta function, B (n) FrFT (u, x) u ( 1) n x, and the transform is straightforward and needs no further calculation. 46. To clarify the later results and be self-consistent, we will adot the following definition of Fourier transform in the discrete form: Given f(x), its Fourier transform is g u F f x f x ex ixu dx, which could be numerically aroximated by g u F f x g k N x 1 l 0 f n ex i n N x / k N u / x u x. We have assumed that the Fourier transform will ma f (x) from x (N x x)/, (N x x)/ to g(u) in the domain u (N u u)/, (N u u)/. If g k is given by a standard DFT or FFT, however, the maed domain will be u 1/( x), 1/( x) owing to the samling condition x u 1/N x. 47. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevshy, and C. Ferria, Sace bandwidth roduct of otical signals and systems, J. Ot. Soc. Am. A 13, (1996). 48. I. S. Granshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1985). 49. H. M. Ozaktas and H. Urey, Sace bandwidth roduct of conventional Fourier transforming systems, Ot. Commun. 105, 1 6 (1994). 50. L. Austander and F. A. Grünbaum, The Fourier transform and the inverse Fourier transform, Inverse Probl. 5, (1989). 51. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recies in C: the Art of Scientific Comuting (Cambridge U. Press, Cambridge, UK, 199). 5. The width of the mask used to fabricate the 10- m-thick waveguide is 50 m. Owing to the highly isotroic etching, the final width of the olyimide waveguide can be varied in the range of m deending on the recise control of exeriment environments. In our simulation, we used a tyical value of 65 m. 53. F. Deasse, M. A. Paesler, D. Courjon, and J. M. Vigoureux, Huygens Fresnel rincile in the near field, Ot. Lett. 0, (1995). 54. A. Yariv, Otical Electronics (CBC College Publishing, New York, 1985).