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Physical Review

Stability of Traveling Waves in Lasers

John A. White

Published Mar 15, 1965• 21 citations
Publisher

Abstract

The equations of motion for a fully quantized system of atoms interacting with traveling waves is solved to terms of fourth order in the field strengths. From these, criteria are developed for the stability of traveling waves in solid and gaseous media. Expressions are given for the power outputs for both traveling and standing waves. The results for the latter agree with those obtained by Lamb in his semiclassical treatment of standing waves.

DOI
10.1103/physrev.137.a1651

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The equations of motion for a fully quantized system of atoms interacting with traveling waves is solved to terms of fourth order in the field strengths. From these, criteria are developed for the stability of traveling waves in solid and gaseous media. Expressions are given for the power outputs for both traveling and standing waves. The results for the latter agree with those obtained by Lamb in his semiclassical treatment of standing waves.
A formal treatment of a traveling-wave optical maser is presented in which an assumed electromagnetic field in a rotating cavity, obeying Maxwell's equations, nonlinearly polarizes the moving gaseous atoms. The interaction is treated quantum mechanically in the frame of the moving atom. The resultant polarization, statistically summed over all velocity ensembles, is used as a source term in Maxwell's equations. The self-consistency gives a set of equations which determine the amplitudes and frequencies of oscillation of the modes of the independent oppositely directed traveling waves in terms of the parameters of the system. The results reduce to those obtained by Lamb for the case of a standing-wave optical maser with a stationary cavity. In addition, stability conditions on the oppositely directed waves are obtained for the cases where the active medium is a single isotope and a mixture of two isotopes.
A previous calculation of forces on atoms in external (dc) fields and in homogeneous traveling-wave laser fields is modified for the case of a standing-wave laser field. The case of intense laser fields, where the Rabi frequency is much greater than the natural decay rate of the atom, is described. A general result for any Doppler shift of the laser frequency due to the atomic motion is obtained. It is specialized to the two cases in which this shift is either much smaller or larger than the natural decay rate. In the first case, we get the same result as that of a previous calculation. The second case has not been discussed previously.
The interplay between the polarization state of light and transverse effects in lasers is analyzed through an amplitude equation description of an atomic transition between spin sublevels. Linearly polarized traveling waves are found, whose stability is restricted by a phase instability associated with the direction of polarization. The instability persists for polarization stabilized lasers. Novel states of laser light such as standing waves with a spatially periodic linear polarization or coexisting traveling waves with different wave numbers and circular polarizations are also found.
Heisenberg's equations were used to formulate equations of motion for the operators of the electro-magnetic field and its interaction with the electrons of atoms, where second quantization is used. In order to take into account the effect of reservoirs, and to secure quantum mechanical consistency, damping terms and stochastic forces must be added. The averaged equations yield those of semiclassical laser theory. The solution of fully quantum mechanical equations lead to my prediction that below laser threshold the light source emits uncorrelated wave tracks, while above it laser light consists of a stabilized classical amplitude, on which small quantum fluctuations and phase diffusion are superimposed.
Theoretical studies of standing and traveling wave lasers are presented. The spectral behavior of the amplified spontaneous emission (superradiance) in a pencil-shaped column is investigated theoretically and experimentally. Spectral narrowing followed by rebroadening has been observed as predicted by the theory. A geometrical representation of an interacting two-level atomic system has been developed and applied to an optical laser. A semi-classical model for a standing wave laser has been examined using the rotating wave approximation, and compared to the exact solution. Depth and width of the Lamb dip are determined theoretically and experimentally. The effect of radial profile of the laser field has been studied using the self-consistent-field approach, and the radius of the beam was measured at various positions from the window.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.