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Physical Review

Theory of Magnetic Effects in Optical Maser Amplifiers and Oscillators

C. V. Heer, Ronald D. Graft

Published Nov 15, 1965• 63 citations
Publisher

Abstract

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 $d\mathrm{r}$, 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 ${j}_{a}={j}_{b}$ or ${F}_{a}={F}_{b}$ 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.

DOI
10.1103/physrev.140.a1088

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•Aug 12, 2015•1 citation
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\ensuremath{-}z\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.
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.
Using semiclassical radiation theory, a formalism similar to that used by Lamb in his Theory of an Optical Maser is developed for studying the amplification of vector traveling waves in a laser-type medium. The effect of the medium on the waves is given in terms of space (or time) dependent field amplitudes and phases and a nonlinear index of refraction. With particular emphasis on typical gaseous media, the effects of Doppler broadening are treated in detail for arbitrary ratios of natural to Doppler linewidths. Polarization and propagation vectors in various directions are considered, and the nonlinear effects are found to make an isotropic medium effectively anisotropic. Lowest order nonlinear effects (due to a polarization cubic in the field amplitudes) are studied extensively, and the frequency dependence of several of these processes is presented in graphical form. In particular, the introduction of fields at new frequencies and polarization effects are considered. The characteristics of these nonlinear processes peculiar to Doppler broadened lines are discussed, and the processes are interpreted in terms of saturation and coherent modulation of the population inversion density. Strong nonlinear effects are considered in a more approximate way and are found to consist of saturation of the various linear and nonlinear processes previously considered. These strong nonlinear effects should occur at low enough intensities to be easily observed in practice on a CW basis. With the present formalism, the analytical results of Gordon, White and Rigden regarding gain saturation in laser amplifiers are obtained, and the extension is made to include frequencies away from line center and the effects of multiple spectral components. Again, the introduction of fields at new frequencies is considered in detail. These results are also discussed in terms of saturation and coherent modulation of the populations and hole burning.
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.
Using classical fields and the density-matrix formalism, we analyze the nonlinear characteristics of a gas laser amplifier subject to an axial magnetic field, operating with two optical-frequency signals of arbitrary polarization. Perturbational solutions, valid for relatively weak intensities, are obtained which describe the saturation interaction of the signals and the generation of combination tones. An arbitrary amount of Doppler broadening is considered throughout. The simple $J=1\ensuremath{\rightarrow}J=0$ model is found convenient to describe the physical processes taking place. In general, however, the interactions are found to depend on the JJJ values of the levels. We find that only saturation and no combination-tone generation occurs for two opposite circularly polarized signals, while both effects are, in general, present for two linearly or elliptically polarized fields. The limiting case of zero magnetic field is examined separately. It is also found that in many cases the medium is made effectively anisotropic by the nonlinear interactions. Applications to measurements of atomic parameters such as JJJ values, ggg values, and lifetimes are discussed.
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.
Theoretical and experimental studies of a gas laser amplifier are presented, assuming the amplifier is operating with a saturating optical frequency signal. The analysis is primarily concerned with the effects of the gas pressure and the presence of an axial magnetic field on the characteristics of the amplifying medium. Semiclassical radiation theory is used, along with a density matrix description of the atomic medium which relates the motion of single atoms to the macroscopic observables. A two-level description of the atom, using phenomenological source rates and decay rates, forms the basis of our analysis of the gas laser medium. Pressure effects are taken into account to a large extent through suitable choices of decay rate parameters. Two methods for calculating the induced polarization of the atomic medium are used. The first method utilizes a perturbation expansion which is valid for signal intensities which barely reach saturation strength, and it is quite general in applicability. The second method is valid for arbitrarily strong signals, but it yields tractable solutions only for zero magnetic field or for axial magnetic fields large enough such that the Zeeman splitting is much larger than the power broadened homogeneous linewidth of the laser transition. The effects of pressure broadening of the homogeneous spectral linewidth are included in both the weak-signal and strong-signal theories; however the effects of Zeeman sublevel-mixing collisions are taken into account only in the weak-signal theory. The behavior of a He-Ne gas laser amplifier in the presence of an axial magnetic field has been studied experimentally by measuring gain and Faraday rotation of linearly polarized resonant laser signals for various values of input signal intensity, and by measuring nonlinearity - induced anisotropy for elliptically polarized resonant laser signals of various input intensities. Two high-gain transitions in the 3.39-μ region were used for study: a J = 1 to J = 2 (3s2 → 3p4) transition and a J = 1 to J = 1 (3s2 → 3p2) transition. The input signals were tuned to the centers of their respective resonant gain lines. The experimental results agree quite well with corresponding theoretical expressions which have been developed to include the nonlinear effects of saturation strength signals. The experimental results clearly show saturation of Faraday rotation, and for the J = 1 t o J = 1 transition a Faraday rotation reversal and a traveling wave gain dip are seen for small values of axial magnetic field. The nonlinearity induced anisotropy shows a marked dependence on the gas pressure in the amplifier tube for the J = 1 to J = 2 transition; this dependence agrees with the predictions of the general perturbational or weak signal theory when allowances are made for the effects of Zeeman sublevel-mixing collisions. The results provide a method for measuring the upper (neon 3s2) level quadrupole moment decay rate, the dipole moment decay rates for the 3s2 → 3p4 and 3s2 → 3p2 transitions, and the effects of various types of collision processes on these decay rates.