Equations are given describing the beat-frequency variation and mode competition in a gaseous optical maser operated in a magnetic field parallel to the maser axis. The equations include only lowest order nonlinear terms. Important terms in the amplitude- and frequency-determining equations are shown to arise from an induced atomic precession. These terms have a character similar to those describing the effects of selective depletion of the velocity distribution or "hole burning." It is shown that the induced atomic precession causes parametric conversion of an optical field of one circular polarization into one of the other polarization with a frequency shift equal to the rate of precession. This process tends to make the competition between modes of different polarizations important. An additional feature, not found in the scalar theory, is that, for sufficiently large magnetic fields, competition can be important between modes separated in frequency by several Doppler-broadened linewidths.
Equations are given describing the beat-frequency variation and mode competition in a gaseous optical maser operated in a magnetic field parallel to the maser axis. The equations include only lowest order nonlinear terms. Important terms in the amplitude- and frequency-determining equations are shown to arise from an induced atomic precession. These terms have a character similar to those describing the effects of selective depletion of the velocity distribution or "hole burning." It is shown that the induced atomic precession causes parametric conversion of an optical field of one circular polarization into one of the other polarization with a frequency shift equal to the rate of precession. This process tends to make the competition between modes of different polarizations important. An additional feature, not found in the scalar theory, is that, for sufficiently large magnetic fields, competition can be important between modes separated in frequency by several Doppler-broadened linewidths.
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.
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.
The Lamb theory of the optical maser is applied to circularly polarized atomic transitions, and used to consider the beat frequencies and the coherence properties of such orthogonal fields when axial magnetic fields are applied to the gaseous laser. The beat frequency approaches zero in near-zero magnetic fields and synchronization can then occur between the right- and left-handed circularly polarized oscillations. For a resonator with no undue polarization constraint, such a strong coupling gives rise to a linearly polarized output, and to a rotation of the plane of polarization with increasing magnetic field. A self-consistent expression is derived for this rotation under steady-state conditions, and a maximum rotation of $\ifmmode\pm\else\textpm\fi{}\frac{1}{4}\ensuremath{\pi}$ with magnetic field is indicated before the synchronization breaks down and circularly polarized beat phenomena appear. The rotation with magnetic field depends on the laser intensity, on the anisotropy in the cavity losses, and on the position of the cavity resonance within the Doppler linewidth. Also, the angle of rotation is indeterminate unless such anisotropy is present. Other regions of such coherence can occur at higher magnetic fields, where the beat frequency again approaches zero. These depend on the detailed shape of the various dispersion curves of the laser medium. The results derived from the theory used are in general agreement with experimental observations on the 1.153-\ensuremath{\mu} He-Ne laser transition.
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.
The lowest order nonlinear effects due to saturation in the simultaneous resonant absorption of several oscillating fields by a low-pressure gas are considered. The analysis is based on a perturbation solution of the Boltzmann equation for the density matrix of the gas in a quantum statistical description. Within certain ranges of the parameters involved, it is shown that: (1) When the gas is irradiated by two oscillating linearly polarized fields of different frequencies well within the Doppler-broadened width of the same absorption line, maximum absorption of both fields occurs when the two linear polarizations are perpendicular to each other. (2) When the gas is in a static magnetic field and irradiated by two right-circularly polarized fields at ${\ensuremath{\omega}}_{0}$ and ${\ensuremath{\omega}}_{0}+\ensuremath{\Delta}\ensuremath{\omega}$, and two left-circularly polarized fields at ${\ensuremath{\omega}}_{0}$ and ${\ensuremath{\omega}}_{0}\ensuremath{-}\ensuremath{\Delta}\ensuremath{\omega}$, all well within the Doppler width of the respective Zeeman components ($\ensuremath{\Delta}m=+1 \mathrm{and} \ensuremath{-}1$) of the same absorption line, there exists a definite phase relationship among the fields whereby absorption of all four fields is simultaneously at a maximum. Although these effects are quite small in normal spectroscopic work, manifestation of both effects is readily observable in gaseous He-Ne optical masers.
A theoretical model is developed for the behavior of an optical maser medium in a static magnetic field of arbitrary strength in the Z direction and an electromagnetic field which is composed of traveling waves with both states of polarization and with both +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 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 or 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.