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Oxford University Press eBooks

Laser gain and some of its consequences

Warren Nagourney

Published May 8, 2014• 0 citations
Publisher

Abstract

The laser gain equation is derived in two ways. First, the optical Bloch equations are derived, based upon a semiclassical treatment of a two-level atomic system and a radiation field. As a visualization aid, there follows a discussion of the formally identical classical Bloch equations, which describe the behavior of a magnetic moment in a combined static and rotating magnetic field. It is shown that laser gain is proportional to the imaginary part of the linear susceptibility, which is obtained from the semiclassical treatment. The second viewpoint uses a slightly modified form of Einstein’s A and B coefficients adapted to describing interactions between atoms and monochromatic fields. Various damping and broadening mechanisms are introduced in a reasonable but non-rigorous fashion, and the crucial distinction between homogeneous and inhomogeneous broadening is introduced.

DOI
10.1093/acprof:oso/9780199665488.003.0005

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•Jan 1, 1987•1 citation
The laser gain equation is derived in two ways. First, the optical Bloch equations are derived, based upon a semiclassical treatment of a two-level atomic system and a radiation field. As a visualization aid, there follows a discussion of the formally identical classical Bloch equations, which describe the behavior of a magnetic moment in a combined static and rotating magnetic field. It is shown that laser gain is proportional to the imaginary part of the linear susceptibility, which is obtained from the semiclassical treatment. The second viewpoint uses a slightly modified form of Einstein’s A and B coefficients adapted to describing interactions between atoms and monochromatic fields. Various damping and broadening mechanisms are introduced in a reasonable but non-rigorous fashion, and the crucial distinction between homogeneous and inhomogeneous broadening is introduced.
This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.
We briefly review the classical model of a Lorentz atom interacting with an optical field. We then ask how the equations describing the atom’s behavior should be modified to include atomic energy level quantization. Simply discussing what modifications make sense provides a strong plausibility argument for the optical Bloch equations. Contrasted with the standard derivation, this argument is less rigorous but has a certain pedagogical appeal: It simply assumes two atomic states with energy difference ℏω, rather than invoking the Schrödinger equation. Also, because our goal is more-or-less to guess the correct equations with minimal formalism, this discussion focuses on the physical meaning of the Bloch equations.
The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.
Interaction of two coherent fields of arbitrary intensity with a general homogeneously broadened three-level molecular system for both an amplifier as well as a laser oscillator under general conditions of pumping is discussed. The amplifier solution is obtained through the time-averaged density-matrix formalism with two coherent fields treated classically. A detailed consideration of both real and imaginary parts of the electric susceptibility at pump and emission frequencies, corresponding to gain and dispersion in the active medium has been taken into account. The results obtained are used to study a number of situations for weak and strong pumping in the laser amplifier. Subsequently these are applied to a mode analysis of a unidirectional single-mode laser oscillator to determine the output characteristics of the laser and to establish conditions under which emission becomes dynamically unstable resulting in sustained oscillatory emission. The results obtained are found to be in good agreement with those of a full numerical analysis of the atomic and field coupled equations describing the system.