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Physical Review A

Theory of a Wave Packet in a Laser Cavity

L. W. Davis

Published Jun 1, 1972• 3 citations
Publisher

Abstract

When describing the behavior of a multimode laser, it sometimes is convenient to express the cavity field in terms of an evolving wave packet that is reflected by the mirrors. We consider here the general relation between the normal-mode description of the field, where mode amplitudes and frequencies characterize the oscillations, and features of the corresponding wave packet. It is shown that in this problem it is natural to represent a packet as a linear superposition of monochromatic waves, integrating over wave number $k$. To illustrate some of the concepts, an example is treated that relates packet parameters to parameters of a theoretical model for the laser medium.

DOI
10.1103/physreva.5.2594

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When describing the behavior of a multimode laser, it sometimes is convenient to express the cavity field in terms of an evolving wave packet that is reflected by the mirrors. We consider here the general relation between the normal-mode description of the field, where mode amplitudes and frequencies characterize the oscillations, and features of the corresponding wave packet. It is shown that in this problem it is natural to represent a packet as a linear superposition of monochromatic waves, integrating over wave number kkk. To illustrate some of the concepts, an example is treated that relates packet parameters to parameters of a theoretical model for the laser medium.
In a previous paper we considered the linear problem of a one-dimensional laser having output coupling. In this paper we derive the nonlinear equation for the same model of the laser and solve it for steady state. The output coupling is exactly taken into account and the field is treated as having a continuous spectrum composed of the normal modes of the universe. Damping terms for the atomic polarization and for the population inversion are introduced phenomenologically. The fluctuating forces due to these damping terms and to the initial field are ignored. The resultant field inside the cavity consists of two oppositely travelling waves whose amplitudes are functions of the distance from the coupling surface. Outside the cavity we have a monochromatic, outgoing wave whose amplitude does not change with distance. The threshold inversion and the oscillation frequency are the same as those derived in the linear theory.
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.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.
We present a numerical and analytical study of a multimode Jaynes-Cummings model describing the spontaneous decay of a single atom in a high-Q cavity. The theory of cavity-modified spontaneous emission is discussed in terms of quantities that have clear physical interpretations: the photon wave packet radiated by the atom, its reflection by the cavity boundaries, and its reabsorption by the atom. Multimode corrections to the single-mode Jaynes-Cummings model are calculated. The multimode corrections are formulated in terms of quantities that may be calculated by solving the single-mode equations of motion.
Previous theories of laser radiation have described the electromagnetic field in terms of a discrete set of quasimodes (Fox and Li "modes") of the laser cavity. Each quasimode was assumed to have a finite quality factor Q. In steady state (when gain due to lasing atoms cancels the losses) and in the absence of noise sources, the semiclassical theory predicts a $\ensuremath{\delta}$-function line shape. In the present semiclassical analysis (fluctuations are ignored), this theory is generalized for a maser with a cavity having a semitransparent wall as one of the mirrors so that there are now many modes of the universe corresponding to each Fox-Li-type quasimode. It is demonstrated that the normal modes of the universe associated with a single Fox-Li "mode" may, under proper conditions, lock together and the $\ensuremath{\delta}$-function laser line shape may be regained. Specifically, we investigate the conditions under which the coupled multimode equations associated with laser oscillation can be reduced to an equation of the form found in the usual single-quasimode theory.
"We propose to study the random laser problem. As a lasing system one considers an active gain medium interacting with a light field where the damping of the latter is treated in the spirit of open systems theory, considering a multimode case with the spectral overlap of the modes in the losses term [11]. We approach the problem by using the density matrix formalism, deriving the master equation for the light field in order to find out the detailed description of the laser like emission. The specific character of the field damping and gain effects essentially modifies the master equation in comparison to that of the ordinary laser and thus contributes to the properties of the resulting laser emission. The derivation of the density operator for the field is the main ingredient for describing the quantum statistical properties of the lasing and such a progress represents a great interest for the study of random laser systems."