Kurt Bernardo Wolf

Fractional Fourier–Kravchuk transform

We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N→∞) reproduces Fourier paraxial optics.

Canonical transforms. I. Complex linear transforms

Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x and p to a pair η and ζ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η, ζ can be realized in the Schrodinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square‐integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well‐known Bargmann transform is the special case when η and iζ are the harmonic oscillator raising and lowering operators. The Moshinsky‐Quesne transform is regained in the limit when...

Structure of the set of paraxial optical systems

The set of paraxial optical systems is the manifold of the group of symplectic matrices. The structure of this group is nontrivial: It is not simply connected and is not of an exponential type. Our analysis clarifies the origin of the metaplectic phase and the inherent limitations for optical map fractionalization. We describe, for the first time to our knowledge, an image girator and a cross girator whose geometric and wave implementations are of interest.

Canonical Transformations and the Radial Oscillator and Coulomb Problems

In a previous paper a discussion was given of linear canonical transformations and their unitary representation. We wish to extend this analysis to nonlinear canonical transformations, particularly those that are relevant to physically interesting many‐body problems. As a first step in this direction we discuss the nonlinear canonical transformations associated with the radial oscillator and Coulomb problems in which the corresponding Hamiltonian has a centrifugal force of arbitrary strength. By embedding the radial oscillator problem in a higher dimensional configuration space, we obtain its dynamical group of canonical transformations as well as its unitary representation, from the Sp(2) group of linear transformations and its representation in the higher‐dimensional space. The results of the Coulomb problem can be derived from those of the oscillator with the help of the well‐known canonical transformation that maps the first problem on the second in two‐dimensional configuration space. Finally, we mak...

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.

Foundations of a Lie algebraic theory of geometrical optics

We present the foundations of a new Lie algebraic method of characterizing optical systems and computing their aberrations. This method represents the action of each separate element of a compound optical system —including all departures from paraxial optics— by a certain operator. The operators can then be concatenated in the same order as the optical elements and, following well-defined rules, we obtain a resultant operator that characterizes the entire system. These include standard aligned optical systems with spherical or aspherical lenses, models of fibers with polynomial z-dependent index profile, and also sharp interfaces between such elements. They are given explicitly to third aberration order.

Wigner functions for Helmholtz wave fields

We investigate a general form of the Wigner function for wave fields that satisfy the Helmholtz equation in two-dimensional free space. The momentum moment of this Wigner function is shown to correspond to the flux of the wave field. For a forward-propagating wave field, the negative regions of the Wigner function are seen to be associated with small regions of backward flux in the field. We also study different projections of the Wigner function, each corresponding to a distribution in a reduced phase space that fully characterizes the wave field. One of these projections is the standard Wigner function of the field at a screen. Another projection introduced by us has the added property of being conserved along rays and is better suited to the description of nonparaxial wave fields.

Canonical transformations and accidental degeneracy. I. The anisotropic oscillator

The problem of accidental degeneracy in quantum mechanical systems has fascinated physicists for many decades. The usual approach to it is through the determination of the generators of the Lie algebra responsible for the degeneracy. In these papers we want to focus from the beginning on the symmetry Lie group of canonical transformations in the classical picture. We shall then derive its representation in quantum mechanics. In the present paper we limit our discussion to the anisotropic oscillator in two dimensions, though we indicate possible extensions of the reasoning to other problems in which we have accidental degeneracy.

Lie algebras for systems with mixed spectra. I. The scattering Pöschl–Teller potential

Starting from an N‐body quantum space, we consider the Lie‐algebraic framework where the Poschl–Teller Hamiltonian, − 1/2 ∂2χ +c sech2 χ+s csch2 χ, is the single sp(2,R) Casimir operator. The spectrum of this system is mixed: it contains a finite number of negative‐energy bound states and a positive‐energy continuum of free states; it is identified with the Clebsch–Gordan series of the D+×D− representation coupling. The wave functions are the sp(2,R) Clebsch–Gordan coefficients of that coupling in the parabolic basis. Using only Lie‐algebraic techniques, we find the asymptotic behavior of these wave functions; for the special pure‐trough potential (s=0) we derive thus the transmission and reflection amplitudes of the scattering matrix.

Fractional Fourier transforms in two dimensions

We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle S1) in the one-dimensional case to U(2) (the four-parameter group of 2 x 2 unitary matrices) in the two-dimensional case [rather than simply U(1) x U(1)]. Our treatment clarifies the situation in the N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2 = S1 x S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 coordinates. FrFTs corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite-Gaussian beams and the Laguerre-Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, R) first-order system is essentially a FrFT.