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Physical Review

Theory of Double Resonance in Gaseous Lasers

W. Culshaw

Published Feb 4, 1966• 3 citations
Publisher

Abstract

The simultaneous action of a rf perturbation between the Zeeman sublevels of an atomic transition, which is also producing laser oscillation, is considered. The complete Hamiltonian of the system is made time-independent by appropriate unitary transformations, and the corresponding steady-state solutions of the density matrix are deduced using appropriate rate constants. Results for stationary atoms indicate a dependence of the threshold for oscillation on the rf perturbation, together with rf resonance effects which increase the laser intensity some two or three times for suitable transitions. The increase in intensity is greatest for rf perturbations between the upper Zeeman levels, which also have the smaller resonance width, because of the longer lifetimes involved. Such effects will be small unless there is a reasonable difference in the populations of the Zeeman levels in the absence of the rf perturbation. Appropriate laser transitions, cavity tuning, and polarization characteristics must thus be used to obtain maximum effect, and examples are given. The atomic motion is included in the theory in an approximate way, and similar conclusions are derived as regards the variations of laser intensity. However, no dependence of the threshold on the rf perturbation is then indicated, the former being determined by the relatively large Doppler width. Finally, some consideration is given to cases where more than one laser transition occurs between the Zeeman levels.

DOI
10.1103/physrev.142.204

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Double-Resonance Phenomena in the Gaseous Laser

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The theory of double-resonance phenomena, or the simultaneous action of optical transitions and rf perturbations between the Zeeman sublevels of an atom, is extended to the induced emission and absorption processes effected by the monochromatic radiation and the discrete resonances involved in the gas laser. Expressions for the probability amplitudes of the Zeeman sublevels under such conditions are derived from time-dependent perturbation theory, the lifetimes of the states being introduced in a phenomenological way. Zeeman splittings of both upper and lower states involved in the laser transition are considered, and an effective term diagram used to discuss the results. Small signal approximations are applied to consider some specific cases, which show that the light beats seen in spontaneous emission when the rf frequency ${\ensuremath{\omega}}_{0}$ equals the Larmor frequency, and in other instances, do not occur in induced emission unless such frequencies coincide with axial resonances of the laser cavity. Variations in the intensity and resonance effects are indicated at Zeeman splittings corresponding to ${\ensuremath{\omega}}_{0}$, with similar variations in the amplitude of any beat frequencies due to multiple axial resonances. Additional coherence effects occur where levels cross, which will give rise to changes in the intensity and polarization of the induced emission at the corresponding frequency.

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Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.

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Interaction of modes emitted due to adjacent transitions in gas lasers

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A theoretical investigation is made of the feasibility of simultaneous laser action due to adjacent transitions in atoms of a gaseous medium with level momenta j/sub b/..-->..j/sub c/ and j/sub b/..-->..j/sub a/. In the case of small relative excesses of the population inversions of the active levels over the threshold, an expression is derived for the region of simultaneous laser action with parallel and orthogonally polarized fields. The limiting cases of homogeneously and inhomogeneously broadened lines are studied. Calculations are made allowing for depolarizing atomic collisions. For a homogeneously broadened line and negligible degeneracy of the levels, a region of simultaneous laser action is found for an arbitrary relative excess of the population inversion over the threshold.

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Effects of Atomic Degeneracy and Cavity Anisotropy on the Behavior of a Gas Laser

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Expressions are developed in this paper which describe the behavior of a gas laser having generalized polarization characteristics. It is found that degeneracies of the atomic energy levels play an important part in determining the behavior of such a laser since significant terms occur in the nonlinear polarization which are attributable to an oscillatory mixing of these levels. As a result, it is found, for example, that for single-mode operation the field intensity is greatest for either plane or circular polarization depending upon whether a $\ensuremath{\Delta}j=\ifmmode\pm\else\textpm\fi{}1$ or $\ensuremath{\Delta}j=0$ atomic transition is involved in the laser action. For two-mode operation, on the other hand, the behavior depends in a complicated way both on the polarization states of the oscillations and on the degree of degeneracy of the energy levels. This behavior is discussed in a number of special cases.

Physical Review•Jul 8, 1966•47 citations
The simultaneous action of a rf perturbation between the Zeeman sublevels of an atomic transition, which is also producing laser oscillation, is considered. The complete Hamiltonian of the system is made time-independent by appropriate unitary transformations, and the corresponding steady-state solutions of the density matrix are deduced using appropriate rate constants. Results for stationary atoms indicate a dependence of the threshold for oscillation on the rf perturbation, together with rf resonance effects which increase the laser intensity some two or three times for suitable transitions. The increase in intensity is greatest for rf perturbations between the upper Zeeman levels, which also have the smaller resonance width, because of the longer lifetimes involved. Such effects will be small unless there is a reasonable difference in the populations of the Zeeman levels in the absence of the rf perturbation. Appropriate laser transitions, cavity tuning, and polarization characteristics must thus be used to obtain maximum effect, and examples are given. The atomic motion is included in the theory in an approximate way, and similar conclusions are derived as regards the variations of laser intensity. However, no dependence of the threshold on the rf perturbation is then indicated, the former being determined by the relatively large Doppler width. Finally, some consideration is given to cases where more than one laser transition occurs between the Zeeman levels.
The theory of double-resonance phenomena, or the simultaneous action of optical transitions and rf perturbations between the Zeeman sublevels of an atom, is extended to the induced emission and absorption processes effected by the monochromatic radiation and the discrete resonances involved in the gas laser. Expressions for the probability amplitudes of the Zeeman sublevels under such conditions are derived from time-dependent perturbation theory, the lifetimes of the states being introduced in a phenomenological way. Zeeman splittings of both upper and lower states involved in the laser transition are considered, and an effective term diagram used to discuss the results. Small signal approximations are applied to consider some specific cases, which show that the light beats seen in spontaneous emission when the rf frequency ${\ensuremath{\omega}}_{0}$ equals the Larmor frequency, and in other instances, do not occur in induced emission unless such frequencies coincide with axial resonances of the laser cavity. Variations in the intensity and resonance effects are indicated at Zeeman splittings corresponding to ${\ensuremath{\omega}}_{0}$, with similar variations in the amplitude of any beat frequencies due to multiple axial resonances. Additional coherence effects occur where levels cross, which will give rise to changes in the intensity and polarization of the induced emission at the corresponding frequency.
The simultaneous action of a RF perturbation between the Zeeman sublevels of an atomic transition, which is also sustaining laser oscillations, is given further consideration. General equations are derived that describe the phenomena in a rotating effective magnetic field basis. These are valid for any level of the RF perturbation, being then solved by iteration to third order in the laser electric field and integrated exactly over the atomic velocity distribution. Results are given for single-π-mode laser operation on aJ = 1 \rightarrow 0transition and for single-mode operation on each of the well-resolved σ components of the Zeeman splitting. The extension to more complex transitions, such as aJ = 1 \rightarrow 2, is then given. A dominant resonance peak in the laser intensity occurs when the RF equals the Zeeman splitting of either the upper or lower states of the transition. Population differences amongst the upper sublevels are more effective than those between the lower sublevels and give the smallest resonance widths. For operation on the σ modes, the main effect is a reduction in the saturation coefficient β, together with smaller changes in the coupling coefficients θ between the oscillations. Such changes will affect the general behavior near the singular points of the nonlinear equations, and will change the shape of the curves of mode intensities as functions of cavity detuning and magnetic field.
Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.
The Lamb theory of the optical maser is extended to fifth and higher orders in the perturbation by the time-dependent iteration method, and by a rate-constant approach obtained by removing the time dependence in the Hamiltonian with a unitary transformation. The third- and fifth-order Fourier projections of the atomic polarization are integrated exactly over the atomic velocity distribution, assumed Maxwellian, and the results are then valid for any ratio of natural linewidth, or cavity detuning, to the Doppler width. Dominant fifth-order terms occur whose variation with cavity detuning depends only on the natural line-width of the transition. These produce an increase in laser intensity versus cavity detuning, including a reduction of the dip phenomena, and are effective at low levels of laser excitation. In addition, third-order and numerous fifth-order terms occur which involve sharper resonances near line center because of Doppler interference effects. Such terms involve the individual decay constants of the states in a complicated way. Owing to cancellation effects, the totality of such terms is in general small compared with the usual saturation term and the dominant fifth-order contributions. This considerably simplifies the deduction of higher-order perturbation terms. Collision processes would also reduce their effect as a result of their higher-order atomic response functions, but the theory is reasonably consistent with present experimental characteristics even without this possibility. The rate-constant approach facilitates the deduction of the various higher-order perturbation results and would be of even greater utility in discussing the Zeeman laser, particularly for axial magnetic fields and complicated transitions.
A theoretical investigation is made of the feasibility of simultaneous laser action due to adjacent transitions in atoms of a gaseous medium with level momenta j/sub b/..-->..j/sub c/ and j/sub b/..-->..j/sub a/. In the case of small relative excesses of the population inversions of the active levels over the threshold, an expression is derived for the region of simultaneous laser action with parallel and orthogonally polarized fields. The limiting cases of homogeneously and inhomogeneously broadened lines are studied. Calculations are made allowing for depolarizing atomic collisions. For a homogeneously broadened line and negligible degeneracy of the levels, a region of simultaneous laser action is found for an arbitrary relative excess of the population inversion over the threshold.
Expressions are developed in this paper which describe the behavior of a gas laser having generalized polarization characteristics. It is found that degeneracies of the atomic energy levels play an important part in determining the behavior of such a laser since significant terms occur in the nonlinear polarization which are attributable to an oscillatory mixing of these levels. As a result, it is found, for example, that for single-mode operation the field intensity is greatest for either plane or circular polarization depending upon whether a $\ensuremath{\Delta}j=\ifmmode\pm\else\textpm\fi{}1$ or $\ensuremath{\Delta}j=0$ atomic transition is involved in the laser action. For two-mode operation, on the other hand, the behavior depends in a complicated way both on the polarization states of the oscillations and on the degree of degeneracy of the energy levels. This behavior is discussed in a number of special cases.