Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Higher-Order Perturbation Theory in Gaseous Lasers

W. Culshaw

Published Dec 10, 1967• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.164.329

Related papers

Higher Order Calculation of the Lamb Dip in the Output of an Optical Maser

Kiyoji Uehara, Kōichi Shimoda

A higher order calculation of the theory of an optical maser is carried out for the purpose of obtaining more accurate theoretical expressions of the Lamb dip. The amplitude of maser oscillation is calculated in the approximations up to the fifth order and the calculated power vs. frequency characteristics by the third and the fifth order approximations are compared. The Lamb dip becomes considerably shallower if the fifth order term is included. The fifth order calculation should therefore be employed for a more accurate determination of the linewidth parameters of the active atoms from observed shapes of the Lamb dip. It is shown, however, that the correction due to the fifth order calculation is small when the relative excitation is very close to the threshold. The fifth order calculation in the case when the “Doppler limit” does not apply is also discussed as well as in the case of the “Doppler limit”.

Japanese Journal of Applied Physics•Nov 1, 1965•31 citations

Theory of Double Resonance in Gaseous Lasers

W. Culshaw

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.

Physical Review•Feb 4, 1966•3 citations

Double resonance in gaseous lasers

W. Culshaw

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.

IEEE Journal of Quantum Electronics•Dec 1, 1968•6 citations

Fifth-, Seventh- and Ninth-Order Calculations of the Lamb Dip in Lasers and Saturated Absorption

Kōichi Shimoda, Kiyoji Uehara

The semi-classical theory of a single-mode gas laser has been extended to include higher-order terms by a simpler method than previous ones. The present theory takes not only the lifetimes of the two states but also the effect of phase-perturbing collisions into consideration by using three phenomenological decay constants in the two-level system. Explicit expressions in the general case of nonlinear susceptibility of the laser medium or the saturable absorber are presented for the fifth and seventh orders, while the ninth-order expression is shown for a special case of a very long longitudinal relaxation time.

Japanese Journal of Applied Physics•Apr 1, 1971•13 citations

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

Double-Resonance Phenomena in the Gaseous Laser

W. Culshaw

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.

Physical Review•Jul 20, 1964•11 citations
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 higher order calculation of the theory of an optical maser is carried out for the purpose of obtaining more accurate theoretical expressions of the Lamb dip. The amplitude of maser oscillation is calculated in the approximations up to the fifth order and the calculated power vs. frequency characteristics by the third and the fifth order approximations are compared. The Lamb dip becomes considerably shallower if the fifth order term is included. The fifth order calculation should therefore be employed for a more accurate determination of the linewidth parameters of the active atoms from observed shapes of the Lamb dip. It is shown, however, that the correction due to the fifth order calculation is small when the relative excitation is very close to the threshold. The fifth order calculation in the case when the “Doppler limit” does not apply is also discussed as well as in the case of the “Doppler limit”.
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 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.
The semi-classical theory of a single-mode gas laser has been extended to include higher-order terms by a simpler method than previous ones. The present theory takes not only the lifetimes of the two states but also the effect of phase-perturbing collisions into consideration by using three phenomenological decay constants in the two-level system. Explicit expressions in the general case of nonlinear susceptibility of the laser medium or the saturable absorber are presented for the fifth and seventh orders, while the ninth-order expression is shown for a special case of a very long longitudinal relaxation time.
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.
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.