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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Nonstationary noise propagation with sources

J. Ben-Benjamin, Leon W. Cohen

Published Jun 13, 2014• 0 citations
Publisher

Abstract

We discuss a number of topics relevant to noise propagation in dispersive media. We formulate the problem of pulse propagation with a source term in phase space and show that a four dimensional Wigner distribution is required. The four dimensional Wigner distribution is that of space and time and also wavenumber and frequency. The four dimensional Wigner spectrum is equivalent to the space-time autocorrelation function. We also apply the quantum path method to improve the phase space approximation previously obtained. In addition we discuss motion in a Snell’s law medium.

DOI
10.1117/12.2053113

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We discuss a number of topics relevant to noise propagation in dispersive media. We formulate the problem of pulse propagation with a source term in phase space and show that a four dimensional Wigner distribution is required. The four dimensional Wigner distribution is that of space and time and also wavenumber and frequency. The four dimensional Wigner spectrum is equivalent to the space-time autocorrelation function. We also apply the quantum path method to improve the phase space approximation previously obtained. In addition we discuss motion in a Snell’s law medium.
Abstract We show that phase space methods developed for quantum mechanics, such as the Wigner distribution, can be effectively used to study the evolution of nonstationary noise in dispersive media. We formulate the issue in terms of modes and show how modes evolve and how they are effected by sources.We show that each mode satisfies a Schrödinger type equation where the “Hamiltonian” may not be Hermitian. The Hamiltonian operator corresponds to dispersion relationwhere thewavenumber is replaced by the wavenumber operator. A complex dispersion relation corresponds to a non Hermitian operator and indicates that we have attenuation. A number of examples are given.
We apply quasi-distribution methods developed for quantum mechanics to the propagation of pulses in dispersive media with attenuation. We show that a Schrödinger type equation follows for propagation of the pulse for each mode. One then transforms the equation to obtain an equation of evolution in the phase space of position and wavenumber. In this paper we emphasize windowed wave functions and their corresponding phase space quasi-distributions. We obtain the time evolution equation, discuss possible approximations, and compare to the Wigner distribution approximation previously derived by Loughlin and Cohen by different methods.
We formulate a noise reverberation model that involves propagation effects. This is in contrast to the standard model where the amplitude and phase are taken to be random variables and propagation effects are not taken into account. The noise obtained is nonstationary and we show that an effective way to formulate the noise reverberation model is in phase space wherein the phase space function is taken to be the Wigner spectrum. The Wigner spectrum is the ensemble average of the Wigner distribution.
This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.