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Journal of Modern Optics

Reverberation noise in phase-space

Leon W. Cohen

Published Jul 7, 2010• 2 citations
Publisher

Abstract

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.

DOI
10.1080/09500340.2010.495462

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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.
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 use the Wigner distribution to study systems subjected to random forces. We define the instantaneous spectrum as the ensemble average of the Wigner distribution, and we write the differential equation whose solution gives us the time-varying spectrum of the state variable. We consider the cases of both constant and time-varying coefficients. We apply the method to study the instantaneous spectrum of a harmonic oscillator driven by Gaussian noise, with both constant and time-varying coefficients. In the latter case our method clearly reveals the nonstationarity of the power spectrum and we confirm our result by numerical simulations.
We obtain an explicit equation for the time-frequency Wigner spectrum of momentum governed by the Langevin equation when the random driving term is quantum noise. The equation is solved exactly and includes both the transient and the stationary part. The time-dependent Wigner spectrum generalizes the result of Wang and Uhlenbeck wherein they showed that for the white noise driving force the power spectrum in the stationary state regime is Lorenzian. We show that our solution reduces to the classical solution when the parameters of the quantum noise are such that the white noise limit is approached and when the long time limit is taken.
We have recently derived a phase space approximation for wave propagation, and we have shown that this approximation is accurate especially compared to the stationary phase approximation. The method gives an approximate Wigner distribution, which raises the question, is the approximate Wigner distribution representable? If it is, then we can invert it to obtain the corresponding wave function. If it is not, can we obtain an approximate wave function from it, and how good is the approximation? Although the answer to the first question is no, we show that from the Wigner approximation one can recover the exact spatial spectrum magnitude and derivative of the phase. We also consider other methods for obtaining approximate wave functions from the Wigner approximation.