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Physical Review A

Theory of a Zeeman ring laser: General formalism

David R. Hanson, Murray Sargent

Published Jan 1, 1974• 23 citations
Publisher

Abstract

A theory of a ring laser subject to a uniform, axial dc magnetic field is given in extension of the two-mirror standing-wave treatment by Sargent, Lamb, and Fork. The active medium consists of thermally moving atoms that have two electronic levels with arbitrary angular momenta. The electric field is treated classically for two circular polarizations of opposite sense in a cavity with any degree of cavity anisotropy. Losses due to backscattering are also included. In addition, the results of a generalized treatment are given which includes arbitrarily oriented magnetic field, a general state of electric field polarization, varying isotopic abundance, and hyperfine structure. The self-consistency requirement is used to obtain amplitude- and frequency-determining equations for multimode operation as functions of laser parameters. A general calculational technique, the "perturbation tree," is introduced in the calculation of the third-order component of the population matrix, greatly simplifying the algebra involved by allowing it to be abstracted in tabular form.

DOI
10.1103/physreva.9.466

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Theory of Magnetic Effects in Optical Maser Amplifiers and Oscillators

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On the theory of polarization effects in gas lasers

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Abstract Starting from a theory in which laser operation is described as the excitation of the natural modes of an open-resonator system, the case of a Fabry-Perot resonator with plane-parallel mirrors is treated. It is shown that, in the case of weakly anisotropic mirrors, the natural modes can be grouped in pairs, such that each pair corresponds to one scalar longitudinal mode. In that case, laser operation can and will be described in terms of these longitudinal modes, which then possess a vectorial field amplitude, in agreement with the usual formalism. The polarization preference in a single-mode gas laser operating on a transition between a singlet and a triplet state is investigated, taking into account differences in the relaxation constants for the various expectation values of atomic variables. A calculation of the effect on the relaxation constants due to interatomic collisions leads to the conclusion that theory and experiment are in good agreement with each other. Equations describing the time evolution of the polarization parameters of both modes in a two-mode operating gas laser are derived. It is found as a general rule that mode-mode interaction results in modes whose polarizations are mutually orthogonal. An exception to this rule is found in the case in which the single-mode preference is for circular polarization. The influence of two specific mirror anisotropies is discussed. The polarization behaviour of a single-mode gas laser in the presence of (partly) applied, transverse magnetic fields is extensively dealt with. It appears to be essential to distinguish between the case in which the magnetic field is applied only to a fraction of the active medium and the case in which the magnetic field is uniform in the entire medium. This distinction is shown to lead to a satisfactory theoretical description of a certain class of experiments. In the case of two other experiments, good agreement with the theory is obtained, by assuming the presence of mirror anisotropies of acceptable kind and size. Finally, the influence of small anisotropies on the degree of polarization of a single-mode gas laser is theoretically investigated, when the latter is polarized by means of a strong transverse magnetic field.

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Properties of Gaseous Optical Masers in Weak Axial Magnetic Fields

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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.

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Coherence Effects in Gaseous Lasers with Axial Magnetic Fields. I. Theoretical

W. Culshaw, J. Kannelaud

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.

Physical Review•Jan 7, 1966•51 citations
A theory of a ring laser subject to a uniform, axial dc magnetic field is given in extension of the two-mirror standing-wave treatment by Sargent, Lamb, and Fork. The active medium consists of thermally moving atoms that have two electronic levels with arbitrary angular momenta. The electric field is treated classically for two circular polarizations of opposite sense in a cavity with any degree of cavity anisotropy. Losses due to backscattering are also included. In addition, the results of a generalized treatment are given which includes arbitrarily oriented magnetic field, a general state of electric field polarization, varying isotopic abundance, and hyperfine structure. The self-consistency requirement is used to obtain amplitude- and frequency-determining equations for multimode operation as functions of laser parameters. A general calculational technique, the "perturbation tree," is introduced in the calculation of the third-order component of the population matrix, greatly simplifying the algebra involved by allowing it to be abstracted in tabular form.
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.
The general theory of a Zeeman laser given in part I is used to analyze laser operation in axial and transverse magnetic fields. The analysis is illustrated by graphs of intensities and beat frequencies versus cavity detuning and magnetic field strength. Among the topics discussed are frequency-locking effects due to cavity anisotropy, the dependence of competition between polarization modes on magnetic field strength for various atomic angular momenta, and the measurement of atomic decay rates and ggg values using a laser in a magnetic field.
Abstract Starting from a theory in which laser operation is described as the excitation of the natural modes of an open-resonator system, the case of a Fabry-Perot resonator with plane-parallel mirrors is treated. It is shown that, in the case of weakly anisotropic mirrors, the natural modes can be grouped in pairs, such that each pair corresponds to one scalar longitudinal mode. In that case, laser operation can and will be described in terms of these longitudinal modes, which then possess a vectorial field amplitude, in agreement with the usual formalism. The polarization preference in a single-mode gas laser operating on a transition between a singlet and a triplet state is investigated, taking into account differences in the relaxation constants for the various expectation values of atomic variables. A calculation of the effect on the relaxation constants due to interatomic collisions leads to the conclusion that theory and experiment are in good agreement with each other. Equations describing the time evolution of the polarization parameters of both modes in a two-mode operating gas laser are derived. It is found as a general rule that mode-mode interaction results in modes whose polarizations are mutually orthogonal. An exception to this rule is found in the case in which the single-mode preference is for circular polarization. The influence of two specific mirror anisotropies is discussed. The polarization behaviour of a single-mode gas laser in the presence of (partly) applied, transverse magnetic fields is extensively dealt with. It appears to be essential to distinguish between the case in which the magnetic field is applied only to a fraction of the active medium and the case in which the magnetic field is uniform in the entire medium. This distinction is shown to lead to a satisfactory theoretical description of a certain class of experiments. In the case of two other experiments, good agreement with the theory is obtained, by assuming the presence of mirror anisotropies of acceptable kind and size. Finally, the influence of small anisotropies on the degree of polarization of a single-mode gas laser is theoretically investigated, when the latter is polarized by means of a strong transverse magnetic field.
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.
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 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.