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American Journal of Physics

On the buildup of laser oscillation from noise

Peter W. Milonni

Published Feb 1, 1989• 2 citations
Publisher

Abstract

It is well known that laser oscillation is initiated by spontaneous radiation ‘‘noise.’’ Evidence for this is often based on the complete theory of laser oscillation, including the quantization of the electromagnetic field. In this article, the buildup of laser oscillation from quantum noise is demonstrated using the most elementary classical equation describing the amplification of laser intensity.

DOI
10.1119/1.16107

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It is well known that laser oscillation is initiated by spontaneous radiation ‘‘noise.’’ Evidence for this is often based on the complete theory of laser oscillation, including the quantization of the electromagnetic field. In this article, the buildup of laser oscillation from quantum noise is demonstrated using the most elementary classical equation describing the amplification of laser intensity.
Buildup of laser oscillation is expressed in terms of the probabilities p(n)(t) that there are n photons in the laser cavity. These probabilities obey a coupled set of difference-differential equations of motion. The method and results of numerical integration of these equations are given here for the initial condition that no photons exist (radiation field vacuum). Corresponding equations of motion for the moments of the photon distribution are derived, and their time dependences are given. A movie showing the time development of the p(n) from an initial vacuum is given on the corners of the November 1970 issue of Applied Optics and can be seen by flipping through the pages.
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]
In this chapter, laser rate equations were used to derive another set of simpler rate equations that couple cavity photon number and the population of excited active atoms as the dynamical variables describing the laser. These equations were used to explore the buildup of laser oscillations from noise in a laser cavity and the remarkable growth of cavity photon number as the laser crosses threshold of oscillation. These modified rate equations were used to derive a Master equation for the cavity photon number probabilities. The steady-state solution of this equation was used to discuss equally remarkable transformation of photon statistics of laser light near threshold. The effect of spontaneous emission on setting the ultimate limit on laser linewidth was also discussed.
The buildup of radiation from noise is considered in the case of the small-signal regime of a swept-gain amplifier with small Doppler width, of the sort indicated in recent schemes for an x-ray laser. The treatment uses the fully quantized electromagnetic field in order to take into account the spontaneous emission. The pulse is seen to grow nonexponentially with a rate that vanishes in the limit of large distances. The gain is generally much smaller than would be expected on the basis of the usual development. Many of the features normally associated with the small-signal regime of an amplifier, such as threshold conditions and spectral narrowing, are not seen. Instead we find spectral broadening, saturation, and the formation of steady states.
Many processes, including absorption and stimulated emission occurring in lasers, can be handled using a semi-classical model for the atom–radiation interaction, in which the matter is given a quantum description, but the radiation is represented as a classical electromagnetic field (see Chapter 2). There are other phenomena that cannot be adequately described without quantizing the radiation. For example, it has been known since the 1930s that spontaneous emission can only be treated correctly using a fully quantum framework for the interaction, in which both the matter and the radiation are quantized, as we shall see in Chapter 6. However, it was not until the 1970s that situations were found in which a free electromagnetic field , far from sources, exhibited properties and behaviour that could not be described by a classical field, but which could be perfectly well interpreted in terms of a quantized field. This chapter is devoted to the quantization of the free electromagnetic field, far from the charges and currents sourcing it. This free electromagnetic field will be called radiation , and in Chapter 6 we shall specify exactly what is meant by radiation when sources are present. The canonical quantization procedure used here starts from a description of the classical dynamics of the field in the framework of the Hamiltonian formalism, the basic features of which are discussed in Section 4.1.
A photon, unlike an electron, can stimulate an excited atom to emit a second photon into the same electromagnetic mode. This stimulated emission process is the foundation for light amplification and oscillation (i.e., self-generation). Initially, the term LASER referred to Light Amplification by Stimulated Emission of Radiation. Nowadays laser often means Light Oscillation by Stimulated Emission, which should literally be called "€œLOSER"€ instead of "€œLASER." To distinguish the above two devices, the former is called a laser amplifier, the latter a laser oscillator (Siegman 1986). Laser oscillation occurs when the photon generation rate exceeds the photon loss rate in a system. If gain saturation were absent, the photon number in a laser oscillator would diverge in time. In other words, the rate equation for the photon number would acquire an unstable solution above the oscillation threshold. In reality, gain saturation reduces the photon generation rate to the photon loss rate so that the number of photons in the oscillator remains at a finite value.