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Applied Optics

Buildup of Laser Oscillations from Quantum Noise

M. Sargent, Marlan O. Scully, Willis E. Lamb

Published Nov 1, 1970• 37 citations
Publisher

Abstract

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.

DOI
10.1364/ao.9.002423

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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.
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.
We measure the statistical distribution of buildup times for the field of a single-mode CO2{\mathrm{CO}}_{2}CO2​ laser when cavity losses have rapidly changed from a large to a small value. The ratio between the average buildup time and its statistical spread is shown to be a sensitive test of the quantum fluctuations in the off state. This method operates as a ``statistical microscope,'' permitting the accurate determination of a small mean photon number in the infrared region by optical amplification, and a discrimination of the initial mean photon number from noise contributions along the amplification process. Following the buildup, the laser pulse decays on two time scales. The faster one, due to the radiation coupling, can be fitted by a Toda oscillator model; the slower one, due to the collisional dynamics of the excited population, provides a quantitative evaluation of two distinct population decay rates, one within the rotational manifold and the other one out of the vibrational band.
The evolution of the first and second moments of the photon density distribution for a two-mode laser system was studied by proposing appropriate transition rate expressions and using the Kubo equations developed for a Chapman-Kolmogorov system. A numerical analysis technique was used in solving the rate equations and computer plots were generated for various combinations of laser parameter values. Anomalously large fluctuations, similar to those occurring in single-mode lasers, were observed during photon buildup of the modes. When only one mode was oscillating, a photon buildup in the non-lasing mode was seen to cause similar fluctuations in the lasing mode. In all cases, these fluctuations were accompanied by a strong linear relationship between the modes. (Author)
The interaction of light and matter can be described by a set of coupled nonlinear evolution equations for the material and electromagnetic fields. In order to account for the effects of vacuum fluctuations, noise forces which cause spontaneous transitions have to be included. In these macroscopic systems various sources of noise besides fundamental quantum fluctuations exist which are of macroscopic origin. These cause fluctuations of the cavity dissipation as well as of the external pumping power. In the course of an adiabatic elimination procedure, these fluctuations appear as multiplicative noise in the equations of motion (e.g. for the field intensity, due to the nonlinearity of the interaction). While fluctuations of the cavity damping create noise linear in the field intensity, randomness in the pump strength causes fluctuations that depend nonlinearly on the light intensity. In this respect, the noise terms enter the equations of motion quite differently and have to be distinguished from spontaneous emission noise which is independent of the field strength, and is the basis of all traditional laser models.
A dynamic correspondence is set up between the density matrix of the electromagnetic field and an associated classical function. Means of one and two time quantum operators are shown to be obtainable from the corresponding averages of the associated classical random problem. When all atomic response rates are fast compared to photon rates, the equation of motion is obtained for the field density matrix. The associated classical problem is shown to obey a Fokker-Planck equation. Over a broad region that includes the threshold region, the Fokker-Planck equation is reduced to that for the rotating-wave van der Pol oscillator for which exact numerical solutions have been previously obtained by Hempstead and Lax.