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Journal of Applied Physics

On Maser Rate Equations and Transient Oscillations

Carson L. Tang

Published Oct 1, 1963• 60 citations
Publisher

Abstract

Masers exhibit interesting transient behavior that cannot be completely understood on the basis of the rate equations. The use of the rate equations in most transient analyses is usually justified on a more or less intuitive basis and the implied assumptions are not always clear. In this paper, the macroscopic maser rate equations are derived systematically from the Boltzmann equation for the density matrix of the atomic systems and Maxwell's equations for the radiation fields. When the coherence linewidth (T2−1) of the atomic systems is much larger than the cavity linewidth and the natural linewidth (T1−1) of the atomic emission, and with a WKB approximation, in the lowest order of approximation one obtains the two widely used, coupled first-order nonlinear rate equations of Statz and deMars. On the other hand, if the cavity linewidth is much larger than the atomic linewidths (T1−1 and T2−1), one can use the so-called ``reaction-field principle'' of Anderson and obtain, again, two coupled first-order rate equations; however, only one of the equations is nonlinear. The ranges of validity of both approaches are discussed in some detail.

DOI
10.1063/1.1729098

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Physical Review•Nov 15, 1965•63 citations
Masers exhibit interesting transient behavior that cannot be completely understood on the basis of the rate equations. The use of the rate equations in most transient analyses is usually justified on a more or less intuitive basis and the implied assumptions are not always clear. In this paper, the macroscopic maser rate equations are derived systematically from the Boltzmann equation for the density matrix of the atomic systems and Maxwell's equations for the radiation fields. When the coherence linewidth (T2−1) of the atomic systems is much larger than the cavity linewidth and the natural linewidth (T1−1) of the atomic emission, and with a WKB approximation, in the lowest order of approximation one obtains the two widely used, coupled first-order nonlinear rate equations of Statz and deMars. On the other hand, if the cavity linewidth is much larger than the atomic linewidths (T1−1 and T2−1), one can use the so-called ``reaction-field principle'' of Anderson and obtain, again, two coupled first-order rate equations; however, only one of the equations is nonlinear. The ranges of validity of both approaches are discussed in some detail.
The transient response of a three-level optical maser is calculated for a density of homogeneously broadened atoms coupled to one homogeneously broadened cavity mode. The atoms are treated in terms of the density matrix and the cavity mode is treated classically. Solutions are found analytically for the steady state and for the linearized form of two simplifications of the general equations. These special solutions have been previously obtained by Bloom and Statz and de Mars. Numerical solutions are found for a certain set of parameters. The response linewidth is found assuming one cavity mode coupled to a density of inhomogeneously broadened atoms.
We discuss the quantum theory of a simple model of a maser oscillator, consisting of one radiation-field mode interacting with a large number of stationary three-level atoms. The field and the atoms also interact with separate heat reservoirs which represent dissipation mechanisms and an incoherent pumping mechanism. The model is sufficiently simple that some analytical progress can be made with the nonlinear quantal equations before further approximation is necessary. We start from the quantal equation of motion of the field-atom density operator. We make immediate use of a diagonal coherent-state expansion for the field part of the density operator and a somewhat similar expansion for the atom part. This yields an exact equation of the Fokker-Planck form for a ccc-number weight distribution, which retains all the significance of the original operator equation, and which has the semiclassical equation for the same model as a first, fluctuation-free approximation. We make use of our basic Fokker-Planck equation in a variety of ways. We discuss the reduction of the equation under conditions that the atomic decay constants are large (large atomic linewidth), arriving finally at an equation of motion for a field-only weight function which serves to demonstrate the basic coherence properties of a maser. We derive and discuss the equation of motion of the generalized Wigner density for the maser model. The generalized Wigner density is a smoothed version (a convolution) of our basic weight distribution, and from it we derive an equivalent classical model including noise sources. Finally, we discuss other useful weight distributions and the number representation for the field. The equations we derive in these discussions make contact with the rate equations of Shimoda, Takahasi, and Townes, as well as with the more recent work of Lax and Louisell, Lax, and of Scully and Lamb.
Simple interference experiments identify correlation functions which describe the response of maser amplifiers and the output of maser oscillators. General operator techniques are used to evaluate these functions in cavity maser systems for which the coupling of the maser electromagnetic field to external energy sources or sinks is mediated by systems whose short-term response depends primarily upon average or macroscopic system properties. The dielectric theory which results includes only those short-term non-linearities associated with the external pumping fields. In the long term both the external and the cavity fields can modify the susceptibility functions. The techniques utilized permit a detailed analysis of approximations and suggest generalizations for increased precision. As one important result of this type the theory indicates that macroscopic rate equations are valid if modulation frequencies are much less than the width of the narrowest coupling system spectral line.
A simple model of a maser or laser consisting of a single-mode field coupled to NNN identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.
A simple maser model consisting of a single-mode field coupled to N identical two-level atoms is presented. The assumption of negligible statistical correlation between the atoms and the field, permits the Heisenberg equations of motion to be replaced by a self-consistent set of ordinary nonlinear differential equations. Relaxation terms and an energy source are introduced phenomenologically. The resulting equations exhibit a threshold, stable and unstable steady states, relaxation oscillations and, sometimes, spiking. The dynamic equations for the hydrogen maser and the rate equations of laser theory are derived as special cases. Also discussed are the maser amplifier, locking of a maser to an external signal, and the effects of cavity thermal noise.
A theoretical model is developed for the behavior of an optical maser medium in a static magnetic field of arbitrary strength in the ZZZ direction and an electromagnetic field which is composed of traveling waves with both states of polarization and with both +z+z+z and $\ensuremath{-}z$ directions. An integral formulation of a truncated phenomenological density matrix is introduced to treat the degeneracy of the atomic energy levels within the natural width. This integral is an exact solution of the phenomenological differential equation for the density matrix which was introduced by Wilcox and Lamb. The inherent degeneracy in the electromagnetic field is included by treating the field in a classical manner. The integral formulation is iterated to yield the first- and third-order contributions to the electric polarization vector P in a spatial element drd\mathrm{r}dr, and the electric field and the electric polarization are made self-consistent in this spatial element by using coefficients in Maxwell's equations which are slowly varying functions of position and time. Atomic motion, hyperfine, isotope, and other line-broadening effects are included in the polarization vector P. The general linear-amplifier, linear-oscillator, and closed-path-oscillator or photon-rate-gyroscope problems are formulated and in the simplest case require the analysis of eight nonlinear equations. These reduce to four equations for the linear oscillator with an axial magnetic field, and these examples are discussed in detail and compared with experiment. An anomalous situation arises for the ja=jb{j}_{a}={j}_{b}ja​=jb​ or Fa=Fb{F}_{a}={F}_{b}Fa​=Fb​ linear amplifiers or linear-oscillator masers, and the development indicates that these systems are stable near line center only for either right or left circular polarization in zero or weak magnetic fields. A magnetic field "dip" near line center is discussed. Special examples which require four or fewer nonlinear equations are discussed.