Guillermo Krötzsch

Geometry and dynamics in refracting systems

The geometric and dynamic postulates for rays in inhomogeneous optical media lead succinctly to the two Hamilton equations in regions where the inhomogeneity is smooth; at a surface of discontinuity between two smooth media, they lead to two conservation laws. One of these is the Ibn Sahl (-Snell-Descartes) law of finite refraction. The transformation due to finite refraction can be in general factorized into two simpler root transformations. These conclusions apply for mechanical as well as optical systems.

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. In the finite-N case, several options have been considered in the literature. We compare their fidelity in reproducing the classical harmonic motion of discrete coherent states.

Group-classified polynomials of phase space in higher-order aberration expansions

Polynomial functions of the classical phase variables p^2, pxq, and q^2, are used in higher-order perturbation expansions in various fields of physics, including geometric aberration optics of axially-symmetric lens and mirror systems. These polynomials participate in operations such as linear combination, multiplication, Poisson brackets, and a Baker-Campbell-Hausdorff compounding that corresponds to concatenation of optical elements. We are interested in handling the polynomials through structures and in bases where the above operations are as short as possible. The monomial basis is one obvious choice that performs efficiently under multiplication. For Poisson brackets and aberration-group products however, the symplectic basis that uses solid spherical, harmonics in three variables is shown to be a better choice. In the last operation, for seventh aberration order, we halve the computation complexity in the symplectic basis.

Canonical transformations to warped surfaces: Correction of aberrated optical images

The projection of optical images on warped screens is a canonical transformation of phase space between flat and warped evolution-parameter surfaces. In mechanics, the evolution parameter is time; in geometric optics it is the optical axis of coordinate space. We consider the specific problem of bringing to focus an axis-symmetric aberrating optical system by warping the output screen. The solution for the surface curvature coefficients is given in terms of the Lie aberration coefficients of the system; a linear optimization strategy applies.

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. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.

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.

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.

Finite optical Hamiltonian systems

In this essay we finitely quantize the Hamiltonian system of geometric optics to a finite system that is also Hamiltonian, but where signals are described by complex N-vectors, which are subject to unitary transformations that form the group U(N). This group can be decomposed into U(2)-paraxial and aberration transformations. Proper irreducible representation bases are thus provided by quantum angular momentum theory. For one-dimensional systems we have waveguide models. For two-dimensional systems we can have Cartesian or polar sensor arrays, where digital images are subject to unitary rotation, gyration or asymmetric Fourier transformations, as well as a unitary map between the two arrays.

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.

8th International Symposium on Quantum Theory and Symmetries (QTS8)

The Quantum Theory and Symmetries (QTS) international symposia are periodic biannual meetings of the mathematical physics community with special interest in the methods of group theory in their many incarnations, particularly in the symmetries that arise in quantum systems. The QTSs alternate with the International Colloquia on Group Theoretical Methods in Physics since 1999, when Professor Heinz-Dietrich Doebner organized the first one in Goslar, Germany. Subsequent symposia were held in Krak?ow, Poland (2001), Cincinnati, USA (2003), Varna, Bulgaria (2005), Valladolid, Spain (2007), Lexington, USA (2009), and Praha, Czech Republic (2011); the eighth QTS was awarded to Mexico (2013), and the next one (2015) will take place in Yerevan, Armenia. Further details, including committees and members, are available in the PDF