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Physical Review

Theory of a Traveling-Wave Optical Maser

Frederick Aronowitz

Published Aug 2, 1965• 158 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.139.a635

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Physical Review•Dec 10, 1967•87 citations
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 theoretical model for the behavior of an optical maser is presented in which the electromagnetic field is treated classically, and the active medium is made up of thermally moving atoms which acquire nonlinear electric dipole moments under the action of the field according to the laws of quantum mechanics. The corresponding macroscopic electric polarization of the medium acts as a source for an electromagnetic field. The self-consistency requirement that a quasistationary field should be sustained by the induced polarization leads to equations which determine the amplitudes and frequencies of multimode oscillation as functions of the various parameters characterizing the maser. Among the results obtained are: threshold conditions, single-mode output as a function of cavity tuning, frequency pulling and pushing, mode competition phenomena including frequency locking, production of combination tones, and population pulsations. A more approximate discussion of maser action using rate equations is also given in which the concept of "hole burning" plays a role.
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.
Abstract We formulate the dynamics of nonlinear optical Fabry–Perot (FB) cavities in terms of a model in which only one of the two counter-propagating electric field envelopes appear. Thus, the model is simpler than the standard ones but still exact. The field envelope which propagates in the opposite direction is expressed in a very simple way in terms of the envelope that appears in the model. Thus we generalize in the simplest way to the FB case the set of Maxwell–Bloch equations for the ring cavity. The boundary condition for the field envelope that appears in the model is a simple periodicity condition. This feature allows for expanding the variables of the model in terms of traveling waves instead of standing waves as it is customary, which implies noteworthy simplifications in the calculations. On the basis of the modal equations which arise from the model, we discuss the adiabatic elimination of the atomic variables and of the atomic polarization only.
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.
The theory of a gaseous optical maser in a magnetic field, as derived by Sargent, Lamb, and Fork, was compared with experiment for the case of a single-spatial-mode, internal-mirror maser in an axial magnetic field. Under these circumstances the twofold polarization degeneracy of the cavity mode is broken down, and the maser tends to oscillate in the two oppositely circularly polarized components of the mode. The intensities and frequency difference of these two components were measured as functions of cavity length and magnetic field for $J=1\ensuremath{\rightarrow}J=0$ and $J=1\ensuremath{\rightarrow}J=2$ transitions in He-Ne optical masers. It was found that all of the qualitative features of the results were correctly predicted by the theory, and in many cases excellent quantitative agreement was found between theory and experiment. Two areas of disagreement were found. (1) The coupling between the two polarizations is somewhat stronger than that predicted by theory. This is very clear for a $J=1\ensuremath{\rightarrow}J=0$ transition, and may be present in the $J=1\ensuremath{\rightarrow}J=2$ transition. No completely satisfactory explanation of this effect has been given. (2) The calculated frequency difference for the $J=1\ensuremath{\rightarrow}J=2$ transition appears to be smaller than that found in the experiment, although theory and experiment are in good agreement for the $J=1\ensuremath{\rightarrow}J=0$ transition. The effects, of small anisotropies in the cavity, on both the intensities and frequency difference were studied, and it was found that the theoretical results are in excellent agreement with experiment.