Provided by the author(s) and University College Dublin Library in accordance with publisher policies. Please cite the published version when available. Title Fast linear canonical transforms Author(s) Healy, John J.; Sheridan, John T. Publication date 2010-01-01 Publication information Journal of the Optical Society of America A, 27 (1): 21-30 Publisher Optical Society of America Link to online version http://dx.doi.org/10.1364/JOSAA.27.000021 Item record/more http://hdl.handle.net/10197/3296 information This paper was published in Journal of the Optical Society of America. A, Optics and image science and is made available as an electronic reprint with the permission of OSA. The paper Publisher's can be found at the following URL on the OSA website: statement http://www.opticsinfobase.org/josaa/abstract.cfm?uri=josaa-27- 1-21. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law. Publisher's version (DOI) http://dx.doi.org/10.1364/JOSAA.27.000021 Downloaded 2018-11-14T04:52:14Z The UCD community has made this article openly available. Please share how this access benefits you. Your story matters! (@ucd_oa) Some rights reserved. For more information, please see the item record link above. J. J. Healy and J. T. Sheridan Vol. 27, No. 1/January 2010/J. Opt. Soc. Am. A 21 Fast linear canonical transforms John J. Healy1,2,3,4 and John T. Sheridan1,2,3,* 1UCD Communications and Optoelectronic Research Centre, College of Engineering, Mathematical and Physical Sciences, University College Dublin, Belfield, Dublin 4, Ireland 2SFI Strategic Research Cluster in Solar Energy Conversion 3School of Electrical, Electronic and Mechanical Engineering, College of Engineering, Mathematical and Physical Sciences, University College Dublin, Belfield, Dublin 4, Ireland 4Complex Adaptive Systems Laboratory, University College Dublin, Belfield, Dublin 4, Ireland *Corresponding author: [email protected] Received August 25, 2009; accepted November 2, 2009; posted November 4, 2009 (Doc. ID 116174); published December 3, 2009 The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretiza- tion, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande–Tukey fast Fourier transform. Results calculated with an implementation of this algorithm are presented and compared with the corresponding analytic functions. © 2010 Optical Society of America OCIS codes: 070.4560, 080.2730, 100.2000, 200.2610, 200.3050, 200.4560, 200.4740. 1. INTRODUCTION tion), and magnification (scaling). We restrict our discus- The Fresnel transform provides a well-known mathemati- sion here to lossless systems and therefore to LCTs with cal model of paraxial free space propagation [1]. Discrete real parameters. However, LCTs with complex param- approximations to it are used for reconstruction of digital eters are also of interest, e.g., the Laplace transform and holograms [2]. The linear canonical transform (LCT) can the Gauss–Weierstrass transform [7]. be used to model systems composed of lenses and free The continuous LCT has been used in a number of ap- space and other quadratic phase systems, e.g., graded- plications, e.g., in optical systems to measure tilt and index media [3,4]. Named for its area-preserving, linear translation [10–12], to provide additional keys in double- coordinate transforming effect on the phase space descrip- random phase encoding optical encryption schemes tion of a wave field (e.g., its Wigner–Ville distribution [13,14], in analysis of speckle size [15], and in noninter- function (WDF) [5,6]), the LCT is a parameterized linear ferometric phase extraction schemes [16,17]. However, integral transform. The parameters of the transform are there are many situations where it is desirable to numeri- related to the ABCD or Collins matrix characterization of cally approximate the transform. A comparison may be the system used in ray-tracing calculations [1]. The LCT’s made with the utility of the ubiquitous fast Fourier trans- origins are in quantum mechanics; a brief overview may form (FFT) algorithms for numerically approximating the be found in [7]. Although it was considered for such use FT. Fast, accurate, and simple numerical tools for ap- much earlier, e.g., [8], the LCT’s application in scalar dif- proximating the LCT are particularly useful in situations fraction theory has gained further recognition since the where discrete data from a digital camera must be pro- fractional Fourier transform (FRT) was first introduced in cessed, e.g., in digital holography. Such a discretization of optics. The LCT has also been discussed as a generaliza- the transform appears to have been first undertaken by tion of the FRT [3,9]. A review of the history of the LCT in Pei and Ding [18], and while their formulation remains optics is presented in [4], which also discusses the trans- the accepted one in praxis [19], it presents certain practi- form’s properties. cal inconveniences for users [20], particularly in relation There are a number of attractive facets to an LCT- to sampling issues [21]. A new definition of the discrete based discussion of optics: (i) it provides links to geomet- LCT (DLCT) has recently been proposed [22,23], along ric optics via the ray-tracing matrix that characterizes with a sampling methodology that appears to address the the optical systems in both; (ii) it has the potential to issues raised in [20,21]. We will review this methodology draw mathematical tools from the vast literature on Fou- in a subsequent section. We have chosen to use the older rier analysis; and (iii) it allows simple relationships to be definition of the DLCT [18] in this paper, but as we will drawn between phase space optics and time–frequency show in Section 2, there is little practical difference be- representations. Aside from the Fresnel transform and tween it and that of [22,23] other than the reconstruction the FRT, the LCT’s special cases also include the Fourier filter used. transform (FT), the effect of a thin lens (chirp multiplica- The focus of this paper is efficient, fast algorithms for 1084-7529/10/010021-10/$15.00 © 2010 Optical Society of America 22 J. Opt. Soc. Am. A/Vol. 27, No. 1/January 2010 J. J. Healy and J. T. Sheridan evaluating the DLCT. These are to the DLCT what the 2. LCT: DEFINITION, PROPERTIES, FFT is to the discrete FT (DFT). Numerical approxima- SAMPLING, AND DISCRETIZATION tion of the transformation is desirable for the develop- ment of simulation tools. Considering the ubiquitous use Given a system of lenses and free space, we may charac- of the FFT [the generic term for numerical algorithms for terize the system in the paraxial approximation using an calculating the DFT with O͑N log N͒ complexity], analo- ABCD matrix, or ray-transfer matrix [1]. In ray tracing, gous fast algorithms for the LCT may prove to be valuable this matrix relates the position and angle of rays at the outside of their obvious application in simulating optical input and output of a system. The matrix is symplectic. systems. The discrete calculation can be thought of as a The ABCD matrix for free space (and hence the Fresnel ͑ A B ͒ ͑ 1 ␭z ͒ ␭ matrix multiplication, fM =LMf, where f isa1ϫN vector transform) is M= CD = 01 , where is the wavelength of consisting of the samples of the input wave field, fM is the the propagating quasi-monochromatic light and z is the vector of samples of the output field, and LM is the N propagation distance. The ABCD matrix for a thin lens of ϫ ͑ 1 0 ͒ N discrete transform matrix for ABCD matrix M. The focal length f is −1/␭f 1 . If a system consists of several se- direct evaluation of this multiplication is of O͑N2͒ com- quential sections for which the ABCD matrices are plexity. Fast algorithms exploit redundancies in the ma- known, the total system matrix is given by the product of trix to iteratively break the multiplication into multiple the ABCD matrices of the subsections. The inverse prob- smaller calculations, reducing the overall complexity. The lem, i.e., decomposition of a system’s ABCD matrix, is first direct fast LCT (FLCT), a radix-4 mixed time- and relevent for algorithm analysis and design [24,25] and frequency-division algorithm, was proposed in [19]. It it- system design [26]. While the two matrices above are of eratively decomposed the matrix multiplication into four particular physical significance for optical system design, smaller ones. A second, alternative approach to numerical designers of numerical algorithms may prefer to employ approximation of the LCT is based on decomposing the different “building blocks.” Thus while chirp multiplica- transform or, equivalently, the discrete transform matrix, tion remains an important numerical operation, the FT, into a series of special cases for which fast algorithms are scaling, and even the FRT are more important for algo- known, e.g., the DFT and the discrete FRT. This avenue of rithm design than free space propagation, because ma- research has thus far culminated in [24]. It is not yet ture numerical algorithms for calculating them make de- clear which of these two types of algorithm, if either, is compositions using those transforms faster than the most useful. However, decompositions that use the FT do alternatives. benefit from the availability of highly optimized FFT We will now define the 1D LCT for a given optical sys- implementations. tem. Generalization to the 2D case is usually straightfor- In overview, this paper is organized as follows.