Rotation and gyration of finite two-dimensional modes

Abstract

Hermite-Gauss and Laguerre-Gauss modes of a continuous optical field in two dimensions can be obtained from each other through paraxial optical setups that produce rotations in (four-dimensional) phase space. These transformations build the SU͑2͒ Fourier group that is represented by rigid rotations of the Poincaré sphere. In finite systems, where the emitters and the sensors are in N ϫ N square pixellated arrays, one defines corresponding finite orthonormal and complete sets of two-dimensional Kravchuk modes. Through the importation of symmetry from the continuous case, the transformations of the Fourier group are applied on the finite modes.

2D non-separable linear canonical transform (2D-NS-LCT) based cryptography

“…The 2D-NS-LCT can represent a wide variety of non-orthogonal, non-axially symmetric and anamorphic systems [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] . Among its special cases are the FT, FRT, and FST, gyrator transform, chirp transform, homogeous coordinate/affine transform (including e.g.…”

“…It however is widely accepted that the security of any encryption system depends on the keys used and therefore the larger the key-space the more the security will be 5 . In addition to these approaches, in this paper, for the first time, a novel 2D-NS-LCT [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] based optical DRPE system is proposed. This proposed system offers encrypting information in multiple (10) degrees of freedom (see Table 1).…”

Group Theory in Finite Hamiltonian Systems

“…Through symmetry importation, the images can be unitarily transformed under the four-parameter U(2)-Fourier group of geometric optics that includes isotropic and anisotropic Fourier-Kravchuk transforms, as well as rotations and gyrations [25]. The latter are the discrete counterparts of the transformation between Hermite-Gauss (HG) and Laguerre-Gauss (LG) wave-optical beams [26]. Finally, we also examine the relative advantages of using sampled values of HG and LG functions versus our discrete Kravchuk functions to analyze images into their mode [27] and angular momentum [28] components, and synthesize the original image faithfully.…”

Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals

“…These systems are collected under the general name of gyrators and are useful in 2D image processing, signal processing, mode transformation, etc. [15,[30][31][32][33]. As a result, the efficient and accurate digital computation of 2D-NS-LCTs is of importance in many areas of optics, optical signal processing, and general digital image processing.…”