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Japanese Journal of Applied Physics

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

Kiyoji Uehara, Kōichi Shimoda

Published Nov 1, 1965• 31 citations
Publisher

Abstract

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”.

DOI
10.1143/jjap.4.921

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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 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.
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.
A simple model of a maser or laser consisting of a single-mode field coupled to NNN identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.
This paper extends the calculations of the semiclassical Lamb theory of a Doppler-broadened gas laser (optical maser) to arbitrary intensities for the case of single-mode operation. The coupled equations of the classical electromagnetic field and an ensemble of two-level atoms are set up, and a solution is obtained in the form of a continued fraction. This is used to compute intensity-detuning curves and atomic population inversion densities as functions of both the velocity and the position in the laser. When the laser is tuned to resonance, the velocity dependence of the inversion density shows a previously unknown fine structure consisting of a "bump" in the bottom of the hole. The calculations are compared to results obtained from a perturbation expansion in powers of the field and exact results known for atoms with zero velocity. The response of the laser output to a slow modulation and the buildup of oscillations are also discussed.
A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\omheld, and Oka.
A fully quantum theory for the general case of any higher order optical harmonic generation associated with resonance absorption of pump photons is presented. The treatment utilizes density-matrix formalism based on the Scully-Lamb model, both atomic system and radiation fields being quantized. Detuning, homogeneous and inhomogeneous broadening and relaxation parameters are included. Coherence characteristics and build up of the harmonic fields are discussed for different initial distributions. The Doppler limit restriction has been lifted and the possibility of a relatively efficient conversion of harmonic photons of higher orders is discussed.