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We provide a general expression and different classification schemes for the general two-dimensional canonical integral transformations that describe the propagation of coherent light through lossless first-order optical systems. Main theorems for these transformations, such as shift, scaling, derivation, etc., together with the canonical integral transforms of selected functions, are derived.

Based on the definition of the instantaneous fre quency (signal phase derivative) as a local moment of the Wigner distribution, we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional Fourier transform (fractional Fourier transform power spectrum) with respect to the angle parameter. We show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra. It per mits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term, if only two close fractional power spectra are known. The proposed technique is noniterative and noninterferometric. The efficiency of the method is demonstrated on several examples including monocomponent, multicomponent, and noisy signals. It is shown that the proposed method works well for signal-to-noise ratios (SNRs) higher than about 3 dB. The appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees. Other applica tions of the angular derivative of the fractional power spectra for signal analysis are discussed briefly. The proposed technique can be applied for phase retrieval in optics, where only the fractional power spectra associated with intensity distributions can be easily measured.

The paper starts with some preparatory work on three types of transformations often used in information processing: a convolution, a matrix-vector multiplication, and a general linear integral transformation. In particular we represent these transformations in a diagonalized form. We then focus on cyclic transformations, and consider as examples the Hilbert transfromation, and the ordinary Fourier transformation and its direct relatives: the cosine, sine, and Hartley transformations. Ways to fractionalize transformations that can be diagonalized, are described in the main part of the paper. As an example of a convolution operator, we consider the fractional Hilbert transformation. The canonical fractional Fourier transformer, as it is usually applied in optics, is the main example of the fractionalization procedure of a matrix-vector multiplication with - in this case - the ray transformation matrix of such a transformer. With the fractional Fourier transformation properly defined, we can then immediately consider its relatives: the canonical fractional cosine, sine, Hartley, and Hankel transformations. More general ray transformation matrices will then be introduced, in particular ortho-symplectic matrices. It is shown that all first-order optical systems whose ray transformation matrices can be diagonalized, are similar (in the sense of matrix similarity) to the separable fractional Fourier transformer. This, once again, stresses the important role of the fractional Fourier transformation. A relatively short treatment of general integral transformations, in particular transformations with a complete set of eigenfunctions, concludes this main part. The last part of the paper deals with fractionalization without the necessity of diagonalization, by expressing the fractionalized operator in the form of a Fourier series. The procedure is applied again to the Fourier transformation, leading in a different way to the same results that were derived before.

Starting from Hermite-Gaussian beams, we generate a general class of rotationally symmetric beams. These beams are Laguerre-Gaussian beams, parameterized by two parameters h and g, representing the curvature and the width of the beam, respectively. The Wigner distribution of each member of this class is readily derived from the Wigner distribution of the Hermite-Gaussian beam from which it is generated. If these Laguerre-Gaussian beams propagate through an isotropic abcd-system, they remain in their class, while the propagation of the complex beam parameter h+ig (or h-ig) satisfies the well-known abcd-law.

Any first-order optical system (or ABCD-system) can be associated with a linear canonical integral transformation, described by Collins integral as long as the submatrix B is non-singular. To avoid the singular case, Moshinsky and Quesne have shown a decomposition of a symplectic ABCD-matrix with a singular B, as a cascade of two matrices that do not have such a singularity; the way to find these matrices, however, is not easy. In this paper we will show an alternative way to avoid possible difficulties that may arise from a singular submatrix B. Starting with the Iwasawa decomposition of a first-order optical system as a cascade of an ortho-symplectic system (a system that is both symplectic and orthogonal), a magnifier, and a lens, a further decomposition of the ortho-symplectic system is considered for the practically important case that the submatrices A, B, C, and D have dimensions 2 x 2. We propose a decomposition of the ortho-symplectic system in the form of a separable fractional Fourier transformer embedded in between two rotators. The resulting decomposition of the entire first-order optical system then shows a physically attractive way to overcome the singular case in the Collins integral. In particular, we will be able to present the linear canonical integral transformation (whether or not with a singular submatrix B) in the basic form of a separable fractional Fourier transformation; this Fourier transformer then acts on rotated input coordinates, and is followed by a further rotation of the output coordinates, by a magnifier, and by a multiplication with a quadratic phase function (a lens).

First-optical systems (or ABCD-systems) with a singular submatrix B are considered. Starting with the Iwasawa-type decomposition of a first-order optical system as a cascade of a lens, a magnifier, and an ortho-symplectic system (a system that is both symplectic and orthogonal), a further decomposition of the ortho-symplectic system in the form of a separable fractional Fourier transformer embedded in between two spatial-coordinate rotators is proposed. The resulting decomposition of the entire firstorder optical system leads to a representation of the linear canonical integral transformation, which is valid also in the case of a singular submatrix B. Some examples of ABCDsystems with a singular submatrix B are given.

Based on the recently introduced orthonormal Hermite-Gaussian-type modes, a general class of sets of nonorthonormal Gaussian-type modes is introduced, along with their associated bi-orthonormal partner sets. The conditions between these two bi-orthonormal sets of modes have been derived, expressed in terms of their generating functions, and the relations with W¨unsche’s Hermite two-dimensional functions and the two-variable Hermite polynomials have been established. A closed-form expression for Gaussiantype modes is derived from their derivative and recurrence relations, which result from the generating function. It is shown that the evolution of non-orthonormal Gaussiantype modes under linear canonical transformations can be described by the same mechanism as used for the evolution of orthonormal Hermite-Gaussian-type modes, when, simultaneously, the associated bio-orthonormal modes are taken into account.

We present a short overview on the application of fractional cyclic and linear canonical transformations to optical signal processing and dedicate some of the discussions to the particular features found in the fractional Fourier transform domain.

We analyze the evolution of the vortex part of the orbital angular momentum during its propagation through separable first-order optical systems. We obtain that the evolution of the vortex part depends only upon the parameters ax, ay, bx, and by of the ray transformation matrix, and that isotropic systems with the same ratio b/a produce the same change of the vortex part of the orbital angular momentum. Finally, it is shown that when light propagates through an optical fiber with a quadratic refractive index profile, the vortex part of the orbital angular momentum cannot change its sign more than four times per period.

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After the introduction of the Wigner distribution [1] for the description of coherent and partially coherent optical fields in a phase space, it became an important tool for optical signal/image analysis and beam characterization. The Wigner distribution completely describes the complex amplitude of a coherent optical field (up to a constant phase factor), or the two-point correlation function (or cross-spectral density) of a partially coherent field. Since the Wigner distribution of a two-dimensional optical field is a function of four variables, it is difficult to analyze. Therefore, the optical field is often represented not by the Wigner distribution itself, but by its global moments. Beam characterization based on the second-order moments of the Wigner distribution, for instance, thus became the basis of an International Organization for Standardization standard. [2] In previous papers, the special but important case of rotational symmetry has been studied extensively; we mention twisted Gaussian-Schell model light and the characterization of rotationally symmetric light in terms of second-order moments. In this paper we study the particular case of rotationally symmetric partially coherent light, and the constraints that this kind of symmetry imposes on the second- and higher-order moments. We will see that only even-order moments remain, and that the number of parameters that we need to describe these even-order moments reduces drastically compared to the general case: whereas in general we have (N+1)(N+2)(N+3)/6 different moments of order N, this number reduces to (1+N/2)^2 in the case of rotational symmetry. A way to measure the moments as intensity moments in the output planes of (generally anamorphic but separable) first-order optical systems is presented. [3] [1] E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev., vol. 40, pp. 749-759, 1932. [2] ISO Technical Committee/Subcommittee 172/SC9, "Lasers and laser-related equipment - test methods for laser beam parameters - beam widths, divergence angle and beam propagation factor," ISO Doc. 11146, International Organization for Standardization, Geneva, 1999. [3] M. J. Bastiaans and T. Alieva, "Wigner distribution moments in fractional Fourier transform systems," J. Opt. Soc. Am. A, vol. 19, pp. 1763-1773, 2002.

We show how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of fractional Fourier transform systems (generally anamorphic ones), and we derive the minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments. The results follow directly from the general relationship that expresses the intensity moments in the output plane of an anamorphic fractional Fourier transform system in terms of the moments in the input plane and the two angles. They can also be derived by formulating rotation-type input-output-relationships between certain combinations of moments. The latter method yields, as a by-product, a number of moment combinations that are invariant under anamorphic fractional Fourier transformation.

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. This general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

The fractional Fourier transform, which is a generalization of the ordinary Fourier transform (FT), was introduced 70 years ago, but only in the last decade has it been actively applied in signal processing, optics and quantum mechanics. The fractional FT gives a more complete representation of the signal in phase space and enlarges the number of applications of the ordinary FT [1]. In addition to the FT, the cosine and sine transforms (CT, ST), which are based on half-range expansions of a function over cosine and sine basis functions, respectively, are also important tools in signal processing. Despite of some lack of elegance in their properties compared to the FT, the CT and ST have their own areas of applications. The idea of fractionalization of the CT and ST was proposed in [2], where the real and imaginary parts of the fractional FT kernel were chosen as the kernels for a fractional CT and a fractional ST, respectively. Nevertheless, the authors note that their fractional transforms are not index additive and, in our point of view, cannot be considered as a fractional version of the CT and ST. In this paper we introduce fractional cosine and sine transforms that are additive on the index and preserve the similar relationships with the fractional FT as the ordinary CT and ST have with the FT. We derive the main properties of the fractional CT and ST and show, as examples, the fractional CT of some selected signals. Note that, although there are different ways for the fractionalization of cyclic transforms [3] like the FT, the CT, and the ST, in this paper we consider the fractional CT and ST in relation to the fractional FT, which is more useful for signal analysis because the fractional FT corresponds to a rotation of the Wigner distribution and the ambiguity function in phase space. 1. L.B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Signal Process., vol. 42, pp.3084-3091, 1994. 2. A.W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorch, "Some important fractional transformations for signal processing," Opt. Commun., vol. 125, pp.18-20, 1996. 3. T. Alieva and M.L. Calvo, "Fractionalization of cyclic transforms," J. Opt. Soc. Am., vol. 17, pp. 2330-2338, 2000.

It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of fractional Fourier transform systems (generally anamorphic ones). The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived.

It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of fractional Fourier transform systems (generally anamorphic ones). The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived.

The main properties of n-th order power filtering in the fractional Fourier domain and its relationship to the differentiation operation are considered. The application of power filtering for solving the phase retrieval problem is proposed.

Different types of joint time-frequency representations are used in signal processing. The advantages and disadvantages of most representations are well known. Thus, for example, the short-time Fourier transform (STFT) of multicomponent signals, although not stressing the auto-terms very well, is free from cross-terms, if the components do not overlap. On the other hand, the Wigner distribution (WD) of such signals is highly concentrated, but suffers from cross-terms, which may hide some of the auto-terms. In particular the S-method weighted WD (SWWD) of a multicomponent signal leads to a representation with significantly reduced cross-terms, while the auto-terms are close to or exactly the same as the ones in the WD. The SWWD is based on weighting the STFT, and two different forms of it have been proposed, based on either an averaging in the time direction or an averaging in the frequency direction. In this paper we introduce the SWWD in the fractional Fourier transform (FT) domain, where we have to calculate the STFT of the fractional FT of the signal, followed by an averaging with an appropriate weighting window. As the STFT of a fractional FT corresponds to the STFT of the signal itself, but with the window function being the fractional FT of the original one and with an additional rotation of the coordinate system, the STFT in a fractional domain can be calculated with roughly the same computational costs as for the normal STFT. In order to find the most appropriate fractional domain in which to apply the weighted averaging, the properties of fractional FT moments are used. In particular, we suppose that an optimal SWWD calculation direction corresponds to minimal signal width, i.e., minimal fractional second-order moment. Determination of this moment can be done analytically, based on three known moments for three different directions. Numerical simulations show a qualitative advantage in the time-frequency representation, when the calculation is done in this optimal fractional domain.

In this paper we review fractional linear integral transforms, which have been actively used during the last decade in optical information processing. The general algorithm for the fractionalization of the linear cyclic integral transforms is discussed and the main properties of fractional transforms are considered. It is shown that there is an infinite number of continuous fractional transforms related with a given cyclic integral transform. The optical fractional Fourier transform used for different applications such as adaptive filter design, phase retrieval, encryption, watermarking, etc., is discussed in detail. Other fractional cyclic transforms that can be implemented in optics, such as fractional Hankel, Sine, Cosine, Hartley, and Hilbert transforms, are investigated.

The reconstruction of a signal - and in particular its phase - from the distributions associated with the instantaneous power of the signal or, more general, its fractional power spectra, is an important problem in signal processing. In spite of several successful iterative algorithms for phase reconstruction from the squared modulus of the signal and its power spectrum, the development of noniterative procedures remains an attractive research topic. The fractional power spectra, i.e., the squared moduli of the fractional Fourier transform, are now a popular tool in signal processing. As it is known, the fractional power spectra are equal to the projections of the Wigner distribution of the signal. Thus, by using a tomographic approach and the inverse Radon transform, the Wigner distribution - and therefore the signal itself, up to a constant phase factor - can be reconstructed by knowing all its projections. The method is based on the rotation in the time-frequency plane of the Wigner distribution under the fractional Fourier transform. It requires the measurements of the fractional power spectra in the entire angular region, which sometimes is impossible or very cost consuming. In the paper we first present the relationship between the fractional power spectra and the ambiguity function of a signal, and we establish the connection between the instantaneous frequency in a fractional domain and the angular derivative of the fractional power spectra. We then show that the instantaneous frequency is determined by the convolution of the angular derivative of the fractional power spectra and the signum function. From this relationship we derive a new method to reconstruct the phase of the signal from only two fractional power spectra. The new method significantly reduces the need for projections measurements and calculations; it is also direct and does not use iterative procedures. We discuss the discrete version of the proposed phase retrieval method and demonstrate its efficiency on some examples.

A Fourier transformation maps a one-dimensional time signal into a one-dimensional frequency function, the signal spectrum. Although the Fourier transform provides the signal's spectral content, it fails to indicate the time location of the spectral components, which is important, for example, when we consider non-stationary or time-varying signals. In order to describe such signals, time-frequency representations (TFRs) are used [1], which map one-dimensional time signal into a two-dimensional functions of time and frequency. In this paper we consider the fractional Fourier transform [2], which belongs to the class of linear TFRs, and establish its connection to the Wigner distribution, which is one of the most widely used quadratic TFRs in electrical engineering. In particular we will use the Radon-Wigner transform, which relates projections of TFRs to the squared modulus of the fractional Fourier transform. The application of the different TFRs often depends on how informative their moments are and how easily these moments can be measured or calculated. The established connection between the Wigner distribution and the Radon-Wigner transform permits to find an optimal way for the calculation of the known Wigner distribution moments and to introduce fractional Fourier transform moments which can be useful for signal analysis. We conclude that all frequently used moments of the Wigner distribution can be obtained from the Radon-Wigner transform. [1] F. Boudreaux-Bartels, "Mixed time-frequency signal transformations," in The Transforms and Applications Handbook, A.D. Poularikas, Ed., CRC Press, Alabama (1996) 887-962. [2] L.B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Signal Process. 42 (1994) 3084-3091.

The parameters of the transfer matrix describing a first-order optical system that is a cascade of k identical subsystems defined by the transfer matrix M, are determined from considering the subsystem's eigenfunctions. A condition for the cascade to be cyclic is derived. Particular examples of cyclic first-order optical systems are presented.

The structure and main properties of coherent and partially coherent optical fields that are self-reproducible under propagation through a first-order optical system are investigated. A phase space description of self-imaging in first-order optical systems is presented. The Wigner distribution function of the complex field amplitudes is invariant under linear transformations.

The Radon-Wigner transform, associated with the intensity distribution in the fractional Fourier transform system, is used for the analysis of complex structures of coherent as well as partially coherent optical fields. The application of the Radon-Wigner transform to the analysis of fractal fields is presented.

Representation of two-dimensional optical signals on the orbital angular Poincare sphere is useful for beam analysis, synthesis and comparison. This mapping is based on the measurement of the second-order moments, which are widely used for beam characterization. It is well known that two second-order moments invariants allow dividing two-dimensional signals into two classes: isotropic and anisotropic. Using the modified Iwasawa decomposition of the ray transformationmatrix and bringing the second-order moments matrix to its diagonalized form, we are able to associate the anisotropic signal with a certain point on the sphere. The latitude of this point describes the vorticity of the signal, while its longitude corresponds to the orientation of the beam's principal axes. Apart from that, the beam's scaling and its curvature can be defined. Before beam comparison, it is thus appropriate to perform first its normalization and mapping on the Poincare sphere. There are many very different beams associated with the same point and therefore this procedure makes sense for fine analysis of beams whose intensity distributions have similar forms. Moreover, every point on the sphere is associated with an orthonormal set of Hermite-Laguerre-Gaussian modes, which can be used for the corresponding beam decomposition that is important for its synthesis and analysis. The developed algorithm for the beam mapping is demonstrated on several examples.