# Wigner distribution function for Euclidean systems

@inproceedings{Nieto1997WignerDF, title={Wigner distribution function for Euclidean systems}, author={Luis Miguel Nieto and Natig M. Atakishiyev and Sergey M. Chumakov}, year={1997} }

Euclidean systems include poly- and monochromatic wide-angle optics, acoustics, and also infinite discrete data sets. We use a recently defined Wigner operator and (quasiprobability) distribution function to set up and study the phase-space evolution of these models, subject to differential and difference equations, respectively. Infinite data sets and two- dimensional monochromatic (Helmholtz) fields are thus shown by their Wigner function on a cylinder of (2 ) direction and location; the…

## 39 Citations

### Position and momentum bases for the monochromatic Maxwell fish-eye and the sphere

- Mathematics
- 2015

In geometric optics the Maxwell fish-eye is a medium where light rays follow circles, while in scalar wave optics this medium can only ‘trap’ fields of certain discrete frequencies. In the…

### Wigner functions for Helmholtz wave fields

- Physics, Mathematics
- 1999

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…

### Finite Hamiltonian systems on phase space

- Physics
- 2011

We construct and analyze a Hamiltonian system whose position coordinate takes values on the finite subset of contiguous integers Zj := {−j,−j+1, . . . , j}, with the purpose of applying the resulting…

### Linear transformations and aberrations in continuous and finite systems

- Mathematics, Physics
- 2008

In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum…

### Wigner functions for the pair angle and orbital angular momentum

- Physics
- 2016

The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle and angular momentum is solved. While a key element for the construction of Wigner…

### Finite optical Hamiltonian systems

- PhysicsInternational Commission for Optics
- 2011

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…

### Finite Hamiltonian systems: Linear transformations and aberrations

- Mathematics, Physics
- 2008

Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their…

### Finite signals in planar waveguides.

- PhysicsJournal of the Optical Society of America. A, Optics, image science, and vision
- 2011

The Kravchuk coherent states provided by the finite oscillator model are used to evince the nonlinear transformations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.

### Wigner distributions and quantum mechanics on Lie groups: The case of the regular representation

- Mathematics
- 2004

We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar…

## References

SHOWING 1-10 OF 20 REFERENCES

### Elements of euclidean optics

- Mathematics, Physics
- 1989

Euclidean optics are models of the manifold of rays and wavefronts in terms of coset spaces of the Euclidean group. One realization of this construction is the geometric model of Hamilton's optical…

### On the Quantum Correction for Thermodynamic Equilibrium

- Physics
- 1947

The behavior of any system at high enough temperatures approaches that of its classical counterpart. The probability of any configurational position is then proportional to exp—U/kT, with U the…

### Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations

- Mathematics
- 1981

We construct sesquilinear forms which are invariant under the similarity groups of the Klein–Gordon and Helmholtz equations. These give rise to positive definite inner products on subspaces of…

### Representation of Lie groups and special functions

- Mathematics
- 1991

At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the…

### Wigner distribution function for paraxial polychromatic optics Opt

- Commun
- 1997

### Wigner distribution function for finite signals

- PhysicsDefense, Security, and Sensing
- 1997

We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of…