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Physical Review A

Statistical Analysis of Randomly Modulated Laser Light

Lee E. Estes, John D. Kuppenheimer, Lorenzo M. Narducci

Published Mar 1, 1970• 9 citations
Publisher

Abstract

A modified Michelson interferometer in which one of the mirrors is driven by a piezoelectric crystal connected to a Gaussian random-noise generator has been used to phase modulate a light beam from a Spectra Physics 119 model laser. The recombination of the modulated and the unmodulated beam reflected by a stationary prism generates a randomly varying interference pattern whose intensity distribution has been analyzed by means of a standard photon-counting technique. The experimental data agree with the theoretical semiclassical predictions in the range of linearity of the piezoelectric crystal. Deviations from the theory have been detected and correlated with the hysteresis of the crystal for large root-mean-square voltages applied to it.

DOI
10.1103/physreva.1.710

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A modified Michelson interferometer in which one of the mirrors is driven by a piezoelectric crystal connected to a Gaussian random-noise generator has been used to phase modulate a light beam from a Spectra Physics 119 model laser. The recombination of the modulated and the unmodulated beam reflected by a stationary prism generates a randomly varying interference pattern whose intensity distribution has been analyzed by means of a standard photon-counting technique. The experimental data agree with the theoretical semiclassical predictions in the range of linearity of the piezoelectric crystal. Deviations from the theory have been detected and correlated with the hysteresis of the crystal for large root-mean-square voltages applied to it.
Results were recently reported on statistical properties of phase-modulated laser light. A detailed analysis is presented here of random-phase modulation. In particular it is shown that there are two kinds of such modulation: modulation by phase fluctuations, in which the instantaneous phase is a stationary stochastic process, and modulation by phase diffusion in which the phase is a diffusion process with an always increasing variance. In the case of a Gaussian modulation we show that the optical fields in the two preceding cases are very different, and we calculate their coherence functions. This difference can be detected by an interference experiment, but more easily by photon coincidence or intensity-correlation experiments on a beat between a reference beam and the modulated beams. We present a theoretical analysis of such experiments in the two cases of modulation.
In this analysis the quantum statistics of the received signal of a frequency modulated laser communication system are derived. In particular the detection statistics for heterodyne and electric field detection of the frequency modulated laser beam are determined. These detection statistics are used to define a measure of the communication system performance in terms of a signal to noise ratio. The development of the quantum electromagnetic field in terms of the photon annihilation eigenstates is the basis for determining these statistics. The derivation used accounts for fluctuations in the carrier, fluctuations due to background radiation introduced after modulation, fluctuations due to the local laser oscillator, and zero point field fluctuations. The results of this analysis indicate the probability distribution for the optical heterodyne detection is not a familiar one of classical communication theory, but it is of the form of probability distribution that satisfies experimentally measured photocount statistics of unmodulated laser radiation. The probability distribution for the electric field intensity detection of the frequency modulated laser beam is the familiar Gaussian distribution. (Author)
The statistics of the optical phase of the light emitted by a semiconductor laser diode when subject to periodic modulation of the applied bias current are theoretically analyzed. Numerical simulations of the stochastic rate equations describing the previous system are performed to describe the temporal dependence of the phase statistics. These simulations are performed by considering two cases corresponding to random and deterministic initial conditions. In contrast to the Gaussian character of the phase that has been assumed in previous works, we show that the phase is not distributed as a Gaussian during the initial stages of evolution. We characterize the time it takes the phase to become Gaussian by calculating the dynamical evolution of the kurtosis coefficient of the phase. We show that, under the typical gain-switching with square-wave modulation used for quantum random number generation, quantity is in the ns time scale; that corresponds to the time it takes the system to lose the memory of the distribution of the initial conditions. We compare the standard deviation of the phase obtained with random and deterministic initial conditions to show that their differences become more important as the modulation speed is increased.
A quantum statistical analysis of phase-locked coupled light beams is presented with the coupling mechanism between the individual modes provided by the time-modulated dielectric constant of the crystal inserted in the cavity. The intense electric field that drives the modulator is treated classically. A Hamiltonian including the modulator interaction is formulated and the Heisenberg equations of motion of the mode operators are solved utilizing a generating function of the mode operators. The evolution in time of the phase-space distribution functions for single and multiple modes is derived under various initial conditions. Initially coherent oscillating modes are shown to conserve their coherent character. Below threshold, the modes are assumed to be chaotic Gaussian. It is shown that when all modes have the same average number of photons, they remain uncorrelated Gaussian. When the average photon numbers are nonidentical, the field remains a pure noise field, but the various modes are correlated. The analysis of the resonant-case mode of operation, properly interpreted, leads directly to the case of nonresonant modulation.
An interference pattern is obtained with a Michelson interferometer; the intensity distribution in the pattern is determined by counting the rate of photons by means of a photomultiplier. It is shown that the pattern obtained for very low intensities of light does not differ outside the margin of experimental error from the pattern obtained for normal intensities. At low intensities about 106 photons enter the interferometer per second thus for these low intensities in average at any time much less than one photon is contained inside the interferometer.
An interference pattern is obtained with a Michelson interferometer; the intensity distribution in the pattern is determined by counting the rate of photons by means of a photomultiplier. It is shown that the pattern obtained for very low intensities of light does not differ outside the margin of experimental error from the pattern obtained for normal intensities. At low intensities about 10 photons enter the interferometer per second thus for these low intensities in average at any time much less than one photon is contained inside the interferometer.