A generalization of the continued fraction expansion on semiclassical laser theories

Arturo Bambini

The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.

M. Sargent

A. K. Popov

Stig Stenholm

This paper solves a special case of the equations for Lamb's semiclassical laser theory. The results are used to discuss the asymptotic behavior of the solution for large electromagnetic field and the power-series expansion in the field amplitude of the population inversion. A comparison with the continued-fraction method used earlier is made.

Kazuo Kuroda, Iwao Ogura

It is proved that the continued fractions as the solution of Lamb's semiclassical laser equation are convergent and satisfy the boundary condition required for the convergence of the Fourier expansion of the density matrix. Using the content of the proof the asymptotic behavior of a high intensity single-mode gas laser is investigated. The intensity increases proportionally to the relative excitation asymptotically and the rate of increase is independent of the detuning.

Krishnan Thyagarajan, Ajoy Kumar Ghatak