New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion.

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solution bases involving Legendre and Gegenbauer polynomials in nonorthogonal coordinates, close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch-Gordan coefficients.

Unitary rotations in two-, three-, and D-dimensional Cartesian data arrays.

Using a previous technique to rotate two-dimensional images on an N×N square pixellated screen unitarily, we can rotate three-dimensional pixellated cubes of side N, and also generally D-dimensional Cartesian data arrays, also unitarily. Although the number of operations inevitably grows as N(2D) (because each rotated pixel depends on all others), and Gibbs-like oscillations are inevitable, the result is a strictly unitary and real transformation (thus orthogonal) that is invertible (thus no loss of information) and could be used as a standard.

Analysis of digital images into energy-angular momentum modes.

The measurement of continuous wave fields by a digital (pixellated) screen of sensors can be used to assess the quality of a beam by finding its formant modes. A generic continuous field F(x, y) sampled at an N × N Cartesian grid of point sensors on a plane yields a matrix of values F(q(x), q(y)), where (q(x), q(y)) are integer coordinates. When the approximate rotational symmetry of the input field is important, one may use the sampled Laguerre-Gauss functions, with radial and angular modes (n, m), to analyze them into their corresponding coefficients F(n, m) of energy and angular momentum (E-AM).

Finite signals in planar waveguides.

We examine the evolution in phase space of an N-point signal, produced and sensed at finite arrays transverse to a planar waveguide within the framework of the finite quantization of geometric optics. We use the Kravchuk coherent states provided by the finite oscillator model to evince the nonlinear transformations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.

Mode analysis and signal restoration with Kravchuk functions.

When a continuous-signal field is sampled at a finite number N of equidistant sensor points, the N resulting data values can yield information on at most N oscillator mode components, whose coefficients should in turn restore the sampled signal. We compare the fidelity of the mode analysis and synthesis in the orthonormal basis of N-point Kravchuk functions with those in the basis of sampled Hermite-Gauss functions. The scale between the two bases is calibrated on the ground state of the field.

Unitary transformation between Cartesian- and polar-pixellated screens.

A unitary transformation between Cartesian and polar pixellations of finite two-dimensional images is obtained from the su(2) model for discrete and finite signals. This transformation analyzes the original image into its finite Cartesian "Laguerre-Kravchuk" modes (involving Wigner little-d functions) and synthesizes it back using a polar mode basis with the same set of mode coefficients. The polar basis is derived from the quantum angular momentum theory, and its modes are given by Clebsch-Gordan coefficients.

Rotation and gyration of finite two-dimensional modes.

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 SU(2) Fourier group that is represented by rigid rotations of the Poincaré sphere. In finite systems, where the emitters and the sensors are in NxN square pixellated arrays, one defines corresponding finite orthonormal and complete sets of two-dimensional Kravchuk modes.

Geometry and dynamics of squeezing in finite systems.

Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere.

Geometry and dynamics in the Fresnel transforms of discrete systems.

Free propagation in continuous optical and mechanical systems is generated by the momentum-squared operator and results in a shear of the phase space plane along the position coordinate. We examine three discrete versions of the Fresnel transform in periodic systems through their Wigner function on a toroidal phase space. But since it is topologically impossible to continuously and globally shear a torus, we examine a fourth version of the Fresnel transform on a spherical phase space, in a model based on the Lie algebra of angular momentum, where the corresponding Fresnel transform wrings the sphere.

Geometry and dynamics in the fractional discrete Fourier transform.

The N x N Fourier matrix is one distinguished element within the group U(N) of all N x N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N-->infinity contraction limit for the integral Fourier transform and its fractional powers.

Phase reconstruction from intensity measurements in linear systems.

The phase of a signal at a plane is reconstructed from the intensity profiles at two close parallel screens connected by a small abcd canonical transform; this applies to propagation along harmonic and repulsive fibers and in free media. We analyze the relationship between the local spatial frequency (the signal phase derivative) and the derivative of the squared modulus of the signal under a one-parameter canonical transform with respect to the parameter. We thus generalize to all linear systems the results that have been obtained separately for Fresnel and fractional Fourier transforms.

Hamiltonian orbit structure of the set of paraxial optical systems.

In the paraxial regime of three-dimensional optics, two evolution Hamiltonians are equivalent when one can be transformed to the other modulo scale by similarity through an optical system. To determine the equivalence sets of paraxial optical Hamiltonians one requires the orbit analysis of the algebra sp(4, R) of 4 x 4 real Hamiltonian matrices. Our strategy uses instead the isomorphic algebra so(3, 2) of 5 x 5 matrices with metric (+1, +1, +1, -1, -1) to find four orbit regions (strata), six isolated orbits at their boundaries, and six degenerate orbits at their common point.

Fractional Fourier transformers through reflection.

We show that an arbitrary paraxial optical system, compounded with its reflection in an appropriately warped mirror, is a pure fractional Fourier transformer between coincident input and output planes. The geometric action of reflection on optical systems is introduced axiomatically and is developed in the paraxial regime. The correction of aberrations by warp of the mirror is briefly addressed.