Two- and three-step laser cascades have been recently detected experimentally in He-Ne mixtures and in pure neon. In the present paper a two-step cascade is treated in detail theoretically, starting with the density-matrix formulation. The corresponding equations can be simplified to rate equations under an assumption about the relative size of certain phase-memory times, which seems well justified in the present case. Because there are usually only minor differences in the results between a gaseous laser and a solid-state laser with an inhomogeneously broadened line, and because the mathematical treatment of the gas laser is more involved, the solid-state case will be treated here. The laser condition for both the lower and upper transitions is derived for the most general case, i.e., that all three levels are pumped. Finally, a special two-step cascade is treated in detail. The results are in qualitative agreement with the experimental findings, although a quantitative comparison is still impeded by the lack of an exact knowledge of some of the parameters, especially the radiative linewidth.
The semiclassical Lamb theory of gas lasers is extended to the case of lasers operating simultaneously on two coupled transitions. The theory is formulated in terms of the density-matrix equations for a three-level gaseous laser medium interacting with the field through electric-dipole transitions, so that there are two allowed transitions and one two-photon transition. It is assumed that there is a single resonator mode near each of the transition frequencies. Previous treatments of such three-level gas lasers either have used simple rate equations or have solved the density-matrix equations in an approximate fashion valid only for low intensities. Thus the numerical approach developed herein represents, to our knowledge, the first semiclassical theory of three-level gas lasers which is accurate for arbitrary detuning and intensity. Significant deviations from earlier, approximate treatments are found.
For pt.I see ibid., vol.20, p.251 (1987). Theoretical expressions for cascade light intensity as a function of time have been evaluated for the case of cascade Zeeman beats produced by two-step excitation with broad-line laser pulses, and the results are compared with experimentally observed cascade Zeeman beats in Fe I. The beats were detected in time-resolved fluorescence from intermediate levels populated by cascades from highly excited levels of Fe I, using three different stepwise-excitation-cascade sequences. In the case of coincident laser pulses, good agreement between theory and experiment is obtained over a range of magnetic fields when allowance is made for Larmor precessions during the excitation period, for the width and shape of the exciting laser pulses and for the contribution of impurity processes. The effects of precession during the excitation period are especially important for the excitation sequence a 5 D 4 to z 5 F degrees 5 to e 5 F 5 , for which short-pulse theories predict no quantum beats for the experimental geometry used (parallel linear polarisations in the two excitation steps). For non-coincident pulses a predicted oscillation in the amplitude of the cascade beats with magnetic field has also been observed.
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.
The density operator theory and method are used to extend the fully quantum-mechanical theory of the two-level single-mode laser advanced by Sargent, Scully and Lamb (1974) to the homogeneously broadened four-level atomic system so that the master equation of the three-mode laser operation for this system is obtained. The steady-state operation characteristics of the laser in different cases are discussed and compared. Especially, pictorial comparison and discussion of the quantum-statistical properties of the laser are carried out by means of numerical computation. Meanwhile some experimental results obtained by Otsuka (1978) are qualitatively explained.
The quantum theory of the three-energy-level two-mode laser in which all the levels are being pumped is investigated and the master equation of the laser operation is obtained. It contains two variables,
n1, the photon number of the first mode, and
n2, the photon number of the second mode. There appear four terms in the master equation which represent two-photon processes between the two modes, the absorption of a photon from one mode, and emission of a photon to the other. They are absent in the single-mode case and in the two-mode case with pumping only to the upper level. The master equation can be represented by a diagram of probability in two dimensions which can be extended to infinity, and each arrow of it represents a term on the right-hand side of the master equation. By summing each variable (
n1 or
n2) of the master equation, two equations can be obtained. Each equation contains only one variable, and can be expressed by a diagram of probability flow in one dimension. By considering the correspondence between macroscopic equilibrium and microscopic detailed balance the equations of motion in the steady state are obtained. By introducing a parameter
H, the equation of motion is simplified and a formal solution can be deduced which is dependent on
H. It appears that the parameter
H cannot be fully determined, but some properties of it can be deduced which prove that the introduction of
H is reasonable. The operation characteristics of the laser under different conditions are discussed on the basis of the formal solution: among them are the threshold condition, the condition for one-mode operation, and the change of the photon-statistical distribution. We especially discuss the laser output power curves under two-mode operation above threshold. The curve of one mode, which oscillates first, shows a bending-down phenomenon as the excitation increases when the other mode goes beyond its threshold as demonstrated experimentally by Otsuka. This phenomenon is explained qualitatively and is attributed to the two-photon processes between the two modes.
A theory of stepwise pulsed-laser excitation of cascade Zeeman quantum beats has been developed for a four-level system. The laser pulses are treated in the broad-line approximation and the full optical-pumping equations for the atomic-density matrix of each level have been obtained. These can be used to describe saturation and light shift effects. The weak-pumping theory solutions to the optical pumping equations have been used to obtain expressions for the cascade quantum-beat intensity versus time curves. Larmor precession, spontaneous-emission decay and the populating of the final cascade level during the excitation pulse are all allowed for. A Wiener-Levy phase diffusion model with slowly varying amplitudes is used for the laser pulses, and the final theoretical expressions for the quantum beats depend on the time-dependent intensities of the laser pulses as well as on their polarisations. Hyperfine effects have been ignored. The theory applies to both coincident pulses and to pulses with a time delay between them. An illustrative application to the case of two very short but coincident pulses (during which Larmor precession, spontaneous emission etc. can be ignored) is given. In an accompanying paper the theoretical expressions are further evaluated and successfully applied to the interpretation of cascade Zeeman quantum beat experiments in Fe I.