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CRC Press eBooks

SEMICLASSICAL LASER THEORY

Murray, Iii Sargent

Published Mar 5, 2018• 1 citation
Publisher

Abstract

In Chap. 5, the steady-state solution for the intensity in the ammonia beam maser was determined by energy conservation, that is, the energy given up by the molecules (gain) was equated to the losses in the maser cavity. Although attractive in its simplicity, this method provides no clue to the time depend­ ence of the intensity or to mode pulling (index of refraction) and cannot be used for treating multimode operation. These and other limitations can be avoided by describing maser action by a classical electromagnetic field gov­ erned by Maxwell’s equations and by demanding that the field be self-con­ sistent, that is, that the field E inducing polarization of the active medium be equal to the resulting field E ' as depicted in Fig. 8-1.

DOI
10.1201/9780429493515-15

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In Chap. 5, the steady-state solution for the intensity in the ammonia beam maser was determined by energy conservation, that is, the energy given up by the molecules (gain) was equated to the losses in the maser cavity. Although attractive in its simplicity, this method provides no clue to the time depend­ ence of the intensity or to mode pulling (index of refraction) and cannot be used for treating multimode operation. These and other limitations can be avoided by describing maser action by a classical electromagnetic field gov­ erned by Maxwell’s equations and by demanding that the field be self-con­ sistent, that is, that the field E inducing polarization of the active medium be equal to the resulting field E ' as depicted in Fig. 8-1.
To treat the laser in a realistic way, we need the density matrix concepts and formalism given in Chap. 7. This material makes it possible to incorporate dipole phase decays due, for example, to collisions and to treat multimode electric fields in a straightforward way. Nevertheless, it is gratifying to find that we can already determine the field intensity for a simplified model of the ammonia beam maser, the first device of the maser-laser genre. The maser was originally conceived in 1951 by Townes, who noted his thoughts on the back of an envelope during a period of contemplation on a bench in Franklin Park, Washington, DC.T The device was realized experimentally by Gordon, Zeiger, and Townes (1954). In this chapter, we treat the maser along the lines given by Lamb (1960). The analysis is not complicated by atomic (molecular) lifetimes, because they are long compared to the Rabi flopping time, which is the optimal transit time for molecules in the maser cavity. Furthermore, by considering single-mode, steady-state operation, we can assume by energy conservation (1) that the energy given up by the atoms due to stimulated emission is gained by the field, and (2) that this energy flow balances the cavity losses.
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.
In our semiclassical treatment of the laser in Chaps. 8-12, we “ Fourieranalyzed” the electric field. This frequency-domain method has much to commend it: explicit expressions for the field are often obtained, the calcu­ lations generally do not require much computer time, and considerable insight can be gained into the nature of the atom-field interaction which takes place in the laser. There are also shortcomings, however, to the frequency-domain picture. The number of modes which can be treated easily is small, perhaps as few as ten. In general, the mode amplitudes and phases cannot vary ap­ preciably in the atomic lifetimes (the total field, of course, can vary). Further­ more the strong-signal theories are rather complicated.
In this paper a completely classical model for laser action is discussed. An active medium consisting of classical anharmonic oscillators interacts with a classical electromagnetic field in a resonant cavity. Comparison with the case of a medium consisting of harmonic oscillators shows the significance of nonlinearities for producing self-sustained oscillations in the radiation field. The results for the classical model are found to be similar to those for a semiclassical model of the ammonia-beam maser. The conclusion is that laser action is not intrinsically a quantum-mechanical effect. The classical-laser theory as given in this paper can also be applied to the case of the electron-cyclotron maser.
A formal treatment of a traveling-wave optical maser is presented in which an assumed electromagnetic field in a rotating cavity, obeying Maxwell's equations, nonlinearly polarizes the moving gaseous atoms. The interaction is treated quantum mechanically in the frame of the moving atom. The resultant polarization, statistically summed over all velocity ensembles, is used as a source term in Maxwell's equations. The self-consistency gives a set of equations which determine the amplitudes and frequencies of oscillation of the modes of the independent oppositely directed traveling waves in terms of the parameters of the system. The results reduce to those obtained by Lamb for the case of a standing-wave optical maser with a stationary cavity. In addition, stability conditions on the oppositely directed waves are obtained for the cases where the active medium is a single isotope and a mixture of two isotopes.