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 three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.

Both geometric and wave optical models, as well as classical and quantum mechanics, realize linear transformations with matrices; for plane optics, these are 2×2 and of unit determinant. Students and some researchers could assume that the structure of this matrix group is fairly evident and hardly interesting. However, the properties and applications even of this lowest 2×2 case are already unexpectedly rich. While in mechanics they cover classical angular momentum, quantum spin, and represent ‘2+1’ relativity, in optical models they lead from the geometrical description of light propagation in the paraxial regime to wave optics via linear canonical transforms requiring a more penetrating view of their manifold structure and multiple covers. The purpose of this review article is to highlight the topological space of 2×2 matrices as it applies to classical versus quantum and wave models, to underline how the latter requires the double cover of the former, thus using 2×2 matrices as an alternative viewpoint of the quantization process, beside the traditional characterization by commutation and non-commutations of position and momentum.

The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of translations and orthogonal transformations; we replace the latter by the discrete dihedral group of N discrete rotations and reflections. The separation of variables in polar and elliptic coordinates is then used to define discrete Bessel and Mathieu functions, as approximants to the well-known continuous Bessel and Mathieu functions, as N -point Fourier transforms approximate the Fourier transform over the circle, with integrals replaced by finite sums. We find that these ‘discrete’ functions approximate the numerical values of their continuous counterparts very closely and preserve some key special function relations.

We present a straightforward discretization of the Bessel functions J_n(x) to discrete counterparts B^(N)_n(x_m), of N integer orders n on N integer points x_m ≡ m, that we call discrete Bessel functions. These are built from a Bessel integral generating function, restricting the Fourier transform over the circle to N points. We show that the discrete Bessel functions satisfy several linear and quadratic relations, particularly Graf's product-displacement formulas, that are exact analogues of well-known relations between the continuous functions. It is noteworthy that these discrete Bessel functions approximate very closely the values of the continuous functions in ranges n + |m| < N. For fixed N, this provides an N-point transform between functions of order and of position,f_n and f_m, which is efficient for the Fourier analysis of finite decaying signals.

Classical or quantum systems that stem from a basic symmetry are seen to be special in having several important properties. The harmonic oscillator and the Bohr system are such. Recent research into the Zernike system provides reasons to include it in this privileged class. Here we show that free motion on the 3-sphere can be projected down to produce classical orbits or complete and orthogonal bases for wavefronts in a circular pupil. This line of inquiry has been pursued in company with N.M. Atakishiyev, G.S. Pogosyan, C. Salto-Alegre, and A. Yakhno.

Systems that stem from projection of free motion on a manifold are the best candidates to exhibit remarkable symmetry properties. This is the case of free motion on the 3-sphere which, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system. This exhibits separability in a variety of coordinate systems, polynomial solutions, and interbasis expansion coefficients that are special Clebsch–Gordan coefficients and Hahn orthogonal polynomials.

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include separability in a variety of coordinate systems, polynomial solutions, and a particular subset of Clebsch-Gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the Askey scheme. Deriving these results from the initial formulation in spherical geometry provides the Zernike system with interest beyond its optical applications.

The Zernike system provides orthogonal polynomial solution bases on the unit disk that separate in coordinates that are generically elliptic. This is a superintegrable system whose optical realization is a scalar wavefield on the plane of a circular pupil. Here we describe the solution set in the trigonometric form of elliptic coordinates expressed in terms of special functions, and examine closely the two limits where the explicit form of the wavefunctions in elliptic coordinates reduce to wavefunctions in polar coordinates.

Meixner oscillators have a ground state and an ‘energy’ spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a difference operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,R). The reproducing kernel for the wavefunctions of these models is also found. PACS: 03.65.Bz, 03.65.Fd

The differential equation that defines the Zernike system, originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil, had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half-sphere. Here we find and examine the separation in the generic elliptic coordinate system on the half-sphere and its projected disk, where the solutions, separated in Jacobi coordinates, contain Heun polynomials.

Paraxial optical setups act through linear canonical transformations between the input and output optical phase spaces. In D-dimensional scalar wave models, this action is through unitary integral transforms that form a group with kernels that are oscillating Gaussian functions, and which can represented through 2D×2D symplectic matrices to simplify their composition and other computations. This group contains the Fourier subgroup of phase space rotations that include gyrations and fractional Fourier transforms. The general linear canonical transformations can be complexified in a range of its parameters to describe coherent states, or if symmetry permits, reduce to radial canonical transforms.

This contribution to the Proceedings bears the same title as the chapter by this author published in Progress in Optics, and recovers the basic construction starting from the compact algebras so(3) or so(4) for 1- and 2-dimensional finite pixellated-screen optics and their contraction to the Euclidean algebras, in which the geometric and wave models find their realization determined by two symmetry subalgebras, but with questions that may prompt further research. Here we follow and question the path from pixellated-screen optics to three-dimensional geometric optics by contraction between Lie algebras and groups.

We present a résumé of this year’s work on what we call the Zernike system. It stems from a differential equation proposed by Frits Zernike in 1934 to describe wavefronts at circular optical pupils through a basis of polynomial solutions on the unit disk and free boundary conditions. This system entails a classical model and a quantum model. The classical model leads to closed elliptic orbits while the quantum model yields bases of polynomial wavefunctions that separate in a manifold of coordinate systems, where only the polar one is orthogonal. Special functions that appear in the solutions and interbasis expansions include the Legendre, Gegenbauer, Jacobi, Hahn and Racah polynomials, as well as special Clebsch–Gordan and 6j coefficients. The underlying symmetry is a cubic Higgs superintegrable algebra.

The finite oscillator based on the Lie group of spin U(2) provides a model for finite one-dimensional (1D) arrays of N = 2 j +1 pixels, for j integer or half-integer which, as j → ∞, deforms to the continuous 1D model of geometric optics. Tra nslations, linear transformations and aberrations in the latter are ca nonical and have their N×N unitary counterparts in the former. Since in U(N) there are onlyN2 independent transformations, we identify the finite counte rparts of translations, linear transformations and aberra tions within the finite model, applicable to the correction of aber rated images or signals on N-pixel linear arrays.

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account, the quantum Zernike system, where that differential equation is seen as a Schrödinger equation with a potential, is special in that it has a potential and a boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates into two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schrödinger-type equations with Pöschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer, and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I) serves to define a classical system and a quantum system which have been found to be superintegrable. We have determined two new orthogonal polynomial solutions (indicated as II and III) that are separable and involve Legendre and Gegenbauer polynomials. Here we report on their three interbasis expansion coefficients: between the I–II and I–III bases, they are given by F23(⋯|1) polynomials that are also special su(2) Clebsch–Gordan coefficients and Hahn polynomials. Between the II–III bases, we find an expansion expressed by F34(⋯|1)’s and Racah polynomials that are related to the Wigner 6j coefficients.

We consider that the simultaneous development of the theory of linear canonical integral transforms among two quite distinct scientific communities, provides an interesting example of how a body of knowledge diffuses in applied—compared with theoretical—research fields.

We study the model of optics in which images are two-dimensional pixellated arrays of values on a screen, and in particular the unitary transformations that have direct correspondence with those in the paraxial geometric and wave optical models. This correspondence is established for the U(2) Fourier group that consists of rotations, gyrations, and two-dimensional Fourier transformations.

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The trajectories turn out to be closed ellipses. We show that this is due to the existence of higher-order invariants that close into a cubic Higgs algebra. The Zernike classical system thus belongs to the class of superintegrable systems. Its Hamilton-Jacobi action separates in three vertical projections of polar coordinates of a sphere, polar and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.