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IEEE Journal of Quantum Electronics

Theory of a distributed feedback laser

M. Sargent, William H. Swantner, James W. Thomas

Published Apr 1, 1980• 47 citations
Publisher

Abstract

This paper treats distributed feedback laser operation in which a classical electromagnetic field obeying Maxwell's equations interacts with an active medium according to the laws of quantum mechanics. The theory describes arbitrarily intense two-mode operation of fundamental and higher modes in both index and gain grating configurations. Spacial hole burning and finite atomic linewidth are included in the analysis and are shown to give marked changes in some laser intensity profiles.

DOI
10.1109/jqe.1980.1070490

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This paper treats distributed feedback laser operation in which a classical electromagnetic field obeying Maxwell's equations interacts with an active medium according to the laws of quantum mechanics. The theory describes arbitrarily intense two-mode operation of fundamental and higher modes in both index and gain grating configurations. Spacial hole burning and finite atomic linewidth are included in the analysis and are shown to give marked changes in some laser intensity profiles.
A theoretical model for the behavior of an optical maser is presented in which the electromagnetic field is treated classically, and the active medium is made up of thermally moving atoms which acquire nonlinear electric dipole moments under the action of the field according to the laws of quantum mechanics. The corresponding macroscopic electric polarization of the medium acts as a source for an electromagnetic field. The self-consistency requirement that a quasistationary field should be sustained by the induced polarization leads to equations which determine the amplitudes and frequencies of multimode oscillation as functions of the various parameters characterizing the maser. Among the results obtained are: threshold conditions, single-mode output as a function of cavity tuning, frequency pulling and pushing, mode competition phenomena including frequency locking, production of combination tones, and population pulsations. A more approximate discussion of maser action using rate equations is also given in which the concept of "hole burning" plays a role.
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.
An investigation is made of a higher-order distributed feedback which appears in a medium characterized by nonharmonic modulation of the parameters due to different radiation diffraction processes. The averaging method is used to obtain reduced equations for the amplitudes of opposite waves and expressions for the gain (absorption coefficient) as well as for the coupling between the waves, subject to all the gratings induced in a medium. Competition between the contributions of various diffraction processes to distributed feedback of arbitrary order is studied. An analysis is made of the coupling coefficient of waves in a dye solution laser with a second-order distributed feedback due to gratings formed by the populations of levels participating in a resonance transition and also thermal gratings induced in the active medium because of saturation of spatially inhomogeneous pumping. A calculation is made of the parameters of the medium for which the competition between the various diffraction processes makes the greatest contribution to a higher-order distributed feedback
The theory of the laser linewidth is formulated to account for arbitrarily large output couplings and spatial hole burning. We show explicitly that the linewidth can be interpreted in terms of either spontaneous-emission noise or the amplification of vacuum field modes leaking into the cavity, depending on the ordering of operators in the correlation function determining the laser spectrum. This allows us to derive the Petermann K factor associated with ``excess spontaneous-emission noise'' in a physically transparent and mathematically simple way, without the need to introduce adjoint modes of the resonator. It also allows us to straightforwardly include spatial-hole-burning effects, which are found to increase the K factor and the linewidth in high-gain systems appreciably.
Peculiarities of the implementation of a superradiant generation mode in a laser with a distributed feedback of the waves and an inhomogeneously broadened spectral line of the active medium are investigated. Based on a numerical solution of the semiclassical Maxwell-Bloch equations for the amplitudes of counterpropagating electromagnetic waves, polarization, and inversion of a two-level medium population, it is shown that a sequence of coherent high-power pulses is generated by the Bragg structure in a wide range of the laser parameters as a result of the selected mode and the cw pumping.