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Journal of the Optical Society of America

Quantum Mechanical Description of Maser Action at Optical Frequencies

Yoh‐Han Pao

Published Aug 1, 1962• 23 citations
Publisher

Abstract

This paper consists of a theoretical discussion of maser action at optical frequencies with emphasis on the differences between stimulated emission and absorption in coherent and incoherent fields. Use is made of a geometrical representation of the equations of motion of the density matrix to illustrate the conditions for continuous-wave operation, amplification, production of large pulses, and for the appearance of relaxation oscillations.

DOI
10.1364/josa.52.000871

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Applied Optics•Jan 1, 1962•2 citations
This paper consists of a theoretical discussion of maser action at optical frequencies with emphasis on the differences between stimulated emission and absorption in coherent and incoherent fields. Use is made of a geometrical representation of the equations of motion of the density matrix to illustrate the conditions for continuous-wave operation, amplification, production of large pulses, and for the appearance of relaxation oscillations.
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.
We discuss the quantum theory of a simple model of a maser oscillator, consisting of one radiation-field mode interacting with a large number of stationary three-level atoms. The field and the atoms also interact with separate heat reservoirs which represent dissipation mechanisms and an incoherent pumping mechanism. The model is sufficiently simple that some analytical progress can be made with the nonlinear quantal equations before further approximation is necessary. We start from the quantal equation of motion of the field-atom density operator. We make immediate use of a diagonal coherent-state expansion for the field part of the density operator and a somewhat similar expansion for the atom part. This yields an exact equation of the Fokker-Planck form for a ccc-number weight distribution, which retains all the significance of the original operator equation, and which has the semiclassical equation for the same model as a first, fluctuation-free approximation. We make use of our basic Fokker-Planck equation in a variety of ways. We discuss the reduction of the equation under conditions that the atomic decay constants are large (large atomic linewidth), arriving finally at an equation of motion for a field-only weight function which serves to demonstrate the basic coherence properties of a maser. We derive and discuss the equation of motion of the generalized Wigner density for the maser model. The generalized Wigner density is a smoothed version (a convolution) of our basic weight distribution, and from it we derive an equivalent classical model including noise sources. Finally, we discuss other useful weight distributions and the number representation for the field. The equations we derive in these discussions make contact with the rate equations of Shimoda, Takahasi, and Townes, as well as with the more recent work of Lax and Louisell, Lax, and of Scully and Lamb.
Maser action due to a mixing between unperturbed spin states, which is produced by a strong coherent pumping field at an angular frequency ω, is investigated by using rate equations of a spin density matrix referred to a rotating coordinate system. The relaxation of each state, which was introduced usually by phenomenological considerations, is obtained from an interaction Hamiltonian of the spin moments with local fields. The induced oscillating fields at 2ω±ω s and ω s , which are proportional to a small signal field at a frequency ω s , have the possibility of power emission (or absorption) by choosing suitably the pumping field. The components at 2ω±ω s are responsible for a frequency mixing between ω and ω±ω s occuring in the rotating coordinates. The emitted (or absorbed) line shape is not a simple Lorentzian in the case of the spin moment 1×ℏ1{\times}\hbar1×ℏ and considered to be a quantum affect in the three level scheme. The frequencies of the emitted (or absorbed) fields are expected to be shifted by varying the magnitude, or the frequency of the pumping field, or the static magnetic field.
The equations are generalized governing coherent emission from quantum mechanical amplifiers (microwave, infrared, and optical masers) using either electric or magnetic dipole transitions. Earlier analyses by Gordon, Zeiger and Townes, Feynman, Vernon and Hellworth, and Wang and Singer, are incorporated in this generalization. The principal new result of this analysis is that an amplitude modulation of the output is to be expected from all maser oscillators excepting those in which excited atoms are supplied at a notably greater rate than the depopulation rate due to coherent induced emission. This analysis also suggests that an energy population inversion of electric dipole moments can be performed using coherent light sources in the same manner as with magnetic dipole moments. It appears therefore that modulation of an optical or infrared maser (e.g., for communication purposes) may be easily accomplished by varying the population of excited states through control of the pumping energy. This modulation may be both by absorption and amplification. (Author)
An analysis of a three-level system using time-dependent perturbation theory shows that two distinct photon-emission processes take place within such a system when stimulated optical emission occurs: one-and two-photon transitions. The two-photon transition consists of the emission of a stimulated photon followed by the absorption of a coherent pump photon.In three-level systems which possess a narrow pump level, two-photon transitions are dominant at high pump powers while the maser line is split into two lines due to the modulation of the wave function at an angular frequency determined by the rate of pumping. This might lead to the possibility of frequency-modulating the maser line. These results are extended qualitatively to include optical masers which possess broad pump-bands.
An analysis of a three-level system using time-dependent perturbation theory shows that two distinct photon-emission processes take place within such a system when stimulated optical emission occurs: one-and two-photon transitions. The two-photon transition consists of the emission of a stimulated photon followed by the absorption of a coherent pump photon. In three-level systems which possess a narrow pump level, two-photon transitions are dominant at high pump powers while the maser line is split into two lines due to the modulation of the wave function at an angular frequency determined by the rate of pumping. This might lead to the possibility of frequency-modulating the maser line. These results are extended qualitatively to include optical masers which possess broad pump-bands.