# Alternative description of the linear canonical integral transformation

## Tatiana Alieva and Martin J. Bastiaans

Any first-order optical system (or ABCD-system) can be associated with a linear canonical integral transformation, described by Collins integral [1] as long as the submatrix B is non-singular. To avoid the singular case, Moshinsky and Quesne [2] 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 [3] 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).

1. S.A. Collins Jr., "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60 (1970) 1168-1177.
2. M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12 (1971) 1772-1780.
3. K.B. Wolf, Geometric Optics on Phase Space (Springer, Berlin, 2004) Section 9.5.

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