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Journal of the Optical Society of America B

Photon-number variance in laser diodes with quiet pumps

Jacques Arnaud

Published Sep 1, 1997• 4 citations
Publisher

Abstract

A previously reported expression [Opt. Quantum Electron.27, 63 (1995)] for the variance of photon number m in a laser diode driven by nonfluctuating currents is generalized to account for spontaneous carrier recombination. The semiclassical theory used here yields Q≡〈Δm2〉/m-1=b+(n/m)/(1+b)-1/2, where n is the average carrier number and b is the spontaneous-to-stimulated rate ratio. This expression is in exact agreement with a recently reported quantum-theory result [J. Opt. Soc. Am. B13, 761 (1996)]. At high power, b≈0 and m≫n imply that 〈Δm2〉=〈m〉/2, i.e. sub-Poissonian internal-field statistics. The Q factor that relates to carrier number is given by the same expression, except that the first b term is missing. Practical means of measuring the photon-number variance and the role of spectral-hole burning are briefly discussed.

DOI
10.1364/josab.14.002193

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A previously reported expression [Opt. Quantum Electron.27, 63 (1995)] for the variance of photon number m in a laser diode driven by nonfluctuating currents is generalized to account for spontaneous carrier recombination. The semiclassical theory used here yields Q≡〈Δm2〉/m-1=b+(n/m)/(1+b)-1/2, where n is the average carrier number and b is the spontaneous-to-stimulated rate ratio. This expression is in exact agreement with a recently reported quantum-theory result [J. Opt. Soc. Am. B13, 761 (1996)]. At high power, b≈0 and m≫n imply that 〈Δm2〉=〈m〉/2, i.e. sub-Poissonian internal-field statistics. The Q factor that relates to carrier number is given by the same expression, except that the first b term is missing. Practical means of measuring the photon-number variance and the role of spectral-hole burning are briefly discussed.
When light originating from a laser diode driven by non-fluctuating electrical currents is incident on a photo-detector, the photo-current does not fluctuate much. Precisely, this means that the variance of the number of photo-electrons counted over a large time interval is much smaller that the average number of photo-electrons. At non-zero Fourier frequency ΩΩΩ the photo-current power spectrum is of the form Ω2/(1+Ω2)Ω^2/(1+Ω^2)Ω2/(1+Ω2) and thus vanishes as Ω→0Ω\to 0Ω→0, a conclusion equivalent to the one given above. The purpose of this paper is to show that results such as the one just cited may be derived from a (semi-classical) theory in which neither the optical field nor the electron wave-function are quantized. We first observe that almost any medium may be described by a circuit and distinguish (possibly non-linear) conservative elements such as pure capacitances, and conductances that represent the atom-field coupling. The theory rests on the non-relativistic approximation. Nyquist noise sources (in which the Planck term ℏω/2\hbarω/2ℏω/2 is being restored) are associated with positive or negative conductances, and the law of average-energy conservation is enforced. We consider mainly second-order correlations in stationary linearized regimes.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]
Spectral-hole burning (SHB) profoundly affects the modulation and noise properties of laser oscillators and amplifiers at high optical power. The present semiclassical theory of SHB shows that the optical gain should be considered a function of the carrier number and photon rate (rather than photon number) plus a fluctuation at the shot-noise level (for full population inversion). Constant-voltage-driven laser diodes generate amplitude-squeezed light, a result not predicted by previous theories that treat gain compression in a formal way. Amplitude and phase noise of oscillators and amplifiers are considered.
The statistics of the beam of excited atoms that pumps a micromaser influence the photon number distribution of the cavity field. Of the cases studied so far, regular injection statistics minimize the number fluctuations and Poisson statistics maximize them. Here we extend these studies in two directions. Previous master equations describing sub-Poissonian injection statistics were derived by using standard reservoir theory. Recently, corrections to these equations have been found that involve the commutator of the gain and loss operators. Describing the system by a discrete map, we can avoid the approximations inherent in reservoir theory. The results are compared to those of the master equation. Differences are found to be small for large photon number but increasingly important for decreasing photon number and pump beam intensity, indicating the breakdown of reservoir theory in this regime. We give an approximate expression for the photon number distribution for weak damping and discuss its connection with semiclassical theory. Finally, we generalize our treatment for super-Poissonian pump fluctuations.
The direct measurement of the Mandel parameter of weak laser pulses, with 10 ns pulse duration time and the mean number of photon per pulse nˉ\bar nnˉ being approximately 0.1, is investigated by recording every photocount event. With the Hanbury Brown and Twiss detection scheme, and not more than one photon per pulse being detected during the sample time by single-photon counters, we have found that the single mode diode laser with driving current lower than the threshold yields a sub-Poissonian statistics. In addition, when the diode laser driving current is much higher than the threshold, it is validated that the Mandel parameterQ C of the Poissonian coherent state is nearly −nˉ/2 - \bar n/2−nˉ/2 . The experimental results are in good agreement with theoretical prediction considering the measurement error.
Single-mode cavity laser light statistics is considered within the framework of rate equations. According to this approach, fluctuations are caused by jumps in active and detecting atoms. The algebra is simple, allowing analytical expressions for the intracavity Fano factor and the photocurrent spectral density to be obtained. Poissonian, quiet, and optical pumps are considered. The results are verified by comparison with Monte Carlo simulations. An essentially exhaustive investigation of sub-Poissonian light generation by classical laser schemes, two-mode lasers, and semiconductor lasers is proposed.