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Laser Technology

Derivation of the Lamb coefficients in four frequency ring laser

Yang Jian-qiang

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

In order to study the laser multi-modes coupling effects,theoretical deduction of four frequency ring laser at Ja=1Jb=2 transition was conducted with the third-order perturbation method from the Lamb semi-classical self-consistent theory.The Lamb coefficients in the non-Doppler limit general case including the population pulsation effects were derived.The analysis shows that the conclusion drawn can be generally applied,which is helpful to the quantitative study on the laser multi-modes coupling fine effects.

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In order to study the laser multi-modes coupling effects,theoretical deduction of four frequency ring laser at Ja=1Jb=2 transition was conducted with the third-order perturbation method from the Lamb semi-classical self-consistent theory.The Lamb coefficients in the non-Doppler limit general case including the population pulsation effects were derived.The analysis shows that the conclusion drawn can be generally applied,which is helpful to the quantitative study on the laser multi-modes coupling fine effects.
Aid or competition between two laser modes oscillating on different transitions of a three-level medium are studied following the method of Lamb. An arbitrary amount of Doppler broadening is allowed for. The "electric-quadrupole" term generated by back-to-back transitions of the two modes but having vanishing electric dipole moment is found to have little effect on the oscillation intensities in typically observed circumstances. This fact justifies, in part, the rate-equation approximation commonly used in multilevel analysis. For a bidirectional ring cascade configuration, however, this coherence term can be important because of constructive interference familiar from two-photon, Doppler-free spectroscopy. The ordinary (i.e., hole-burning) cross-saturation terms for cascade transitions produce mutual mode aid, rather than competition as in the competitive cases. A curious consequence of this difference is greater accuracy in the third-order theory of cascade transitions than in that of competitive transitions, as a result of partial cancellation of errors incurred through neglect of higher orders. The treatment is generalized to multilevel media (four or more levels) with numerical illustration and discussion of the method's relationship to laser oscillation involving diatomic molecules.
A Lamb semiclassical theory is developed for multimode, mode-locked laser operation. The equality of frequency spacings between adjacent modes causes population pulsations generated in the nonlinear atomic response to be integer multiples of the adjacent-mode spacing. This allows the atomic response to be written in terms of Fourier series, thereby reducing coupled differential equations of motion to algebraic equations ultimately expressible in terms of a continued fraction. A unidirectional ring configuration is assumed so that the more complicated spatial dependence of standing waves is avoided. Reductions are made to the rateequation limit and to the special two-mode cases of bidirectional ring operation and single-mode, standing-wave, gas-laser operation. Numerical analysis of the equations is motivated. Specific results are given in the following (companion) paper.
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 multimode strong-signal theory of the preceding paper is applied to a number of two- and three-mode cases. A two-mode stability analysis yields strong-signal counterparts to the third-order coupling parameter. Two-mode homogeneous-broadened operation leads to the discussion of population inversion gratings and associated Bragg scattering. Symmetry properties of the equations for the bidirectional ring are examined. Results for Doppler-broadened two-mode special cases and the three-mode locked unidirectional-ring operation are presented. The strong-signal results are compared to those of the third-order and rate-equation-approximation theories. For the bidirectional ring the historical third-order domain of relative excitation from 1.0 to 1.2 is divided into a valid third-order region and a less accurate region. The condition of bistable operation vs monostable operation is seen to be dependent on the relative excitation and the amount of homogeneous broadening. Two types of bistable operation are seen to exist, one where one mode is completely suppressed and another where both modes may have nonzero intensities.
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 dynamics of a multimode high-intensity gas laser including some velocity-changing-collision effects is investigated theoretically. Using a spatial Fourier expansion method for the density matrix of the active medium, one obtains the system of equations which contain only the Fourier components of the population inversion. To obtain the approximate solutions in the high-intensity case, the saturation terms due to hole burning and the population-pulsation terms are treated separately. The former are calculated exactly and the result is equivalent to the rate-equation approximation. The latter are treated by the perturbation method. The equations of motion for mode amplitudes have a similar form to those of Lamb's perturbation theory but can be applied to a high-intensity multimode laser. Experimental results of three-mode oscillation of a He-Ne laser are compared with the theory.