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The linear canonical transformations of geometric optics on two-dimensional screens form the group $Sp(4,R)$, whose maximal compact subgroup is the Fourier group $U(2)_F$; this includes isotropic and anisotropic Fourier transforms, screen…
Unitary rotations of polychromatic images on finite two-dimensional pixellated screens provide invertibility, group composition, and thus conservation of information. Rotations have been applied on monochromatic image data sets, where we…
Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su($3$) algebra of Cartesian three-dimensional oscillator multiplet states into so($3$) angular momentum submultiplets, we implement unitary rotations of…
The second part of the article is devoted to field transfers by diffraction that are represented by fractional Fourier transformations whose orders are complex numbers. The corresponding effects on the Wigner distributions associated with…
The one-dimensional (1D) fractional Fourier transform (FRFT) generalizes the Fourier transform, offering significant advantages in the time-frequency analysis of non-stationary signals. While various 2D extensions exist, such as the 2D…
In this paper we construct directionally sensitive functions that can be viewed as directional time-frequency representations. We call such a sequence a rotational uniform covering frame and by studying rotations of the frame, we derive the…
A group theoretical understanding of the two dimensional fractional supersymmetry is given in terms of the quantum Poincare group at roots of unity. The fractional supersymmetry algebra and the quantum group dual to it are presented and the…
A light ray in space is characterized by two vectors: (i) a transverse spatial-vector associated with the point where the ray intersects a given spherical cap; (ii) an angular-frequency vector which defines the ray direction of propagation.…
A novel effect of rotation in the optical image of structures in an opaque screen is presented. Theoretical calculations and experimental validation are presented. The features and the conditions of this effect are discussed.
Circular and hyperbolic fractional-order Fourier transformations are actually Weyl pseudo-differential operators. Their associated kernels and symbols are written explicitly. Products of fractional-order Fourier transformations are obtained…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
Fourier synthesis is one of the foundations of physical optics. Spatial Fourier optics is a basis for understanding optical imaging, microscopy, and holography. In conventional Fourier optics, the complex spatial field distribution in the…
Fourier optics, the principle of using Fourier Transformation to understand the functionalities of optical elements, lies at the heart of modern optics, and has been widely applied to optical information processing, imaging, holography etc.…
A point spread function of hexagonally segmented telescopes is derived by a new symmetrical formulation. By introducing three variables on a pupil plane, the Fourier transform of pupil functions is derived by a three-dimensional Fourier…
The first part of the paper is devoted to diffraction phenomena that can be expressed by fractional Fourier transforms whose orders are real numbers. According to a scalar theory, diffraction acts on the amplitude of the electric field as…
Digital cameras and displays utilise picture elements (pixels) that perform a single function: detecting or emitting light intensity. To exploit the full information content of electromagnetic waves, more advanced elements are required.…
Crystallographic groups describe the symmetries of crystals and other repetitive structures encountered in nature and the sciences. These groups include the wallpaper and space groups. We derive linear and nonlinear representations of…
Linear optical operations are fundamental and significant for both quantum mechanics and classical technologies. We demonstrate a non-cascaded approach to perform arbitrary unitary and non-unitary linear operations for N-dimensional…
A ray-rotation sheet consists of miniaturized optical components that function - ray optically - as a homogeneous medium that rotates the local direction of transmitted light rays around the sheet normal by an arbitrary angle [A. C.…
Modern perception increasingly relies on fisheye, panoramic, and other wide field-of-view (FoV) cameras, yet most pipelines still apply planar CNNs designed for pinhole imagery on 2D grids, where pixel-space neighborhoods misrepresent…