Simple interference experiments identify correlation functions which describe the response of maser amplifiers and the output of maser oscillators. General operator techniques are used to evaluate these functions in cavity maser systems for which the coupling of the maser electromagnetic field to external energy sources or sinks is mediated by systems whose short-term response depends primarily upon average or macroscopic system properties. The dielectric theory which results includes only those short-term non-linearities associated with the external pumping fields. In the long term both the external and the cavity fields can modify the susceptibility functions. The techniques utilized permit a detailed analysis of approximations and suggest generalizations for increased precision. As one important result of this type the theory indicates that macroscopic rate equations are valid if modulation frequencies are much less than the width of the narrowest coupling system spectral line.
Simple interference experiments identify correlation functions which describe the response of maser amplifiers and the output of maser oscillators. General operator techniques are used to evaluate these functions in cavity maser systems for which the coupling of the maser electromagnetic field to external energy sources or sinks is mediated by systems whose short-term response depends primarily upon average or macroscopic system properties. The dielectric theory which results includes only those short-term non-linearities associated with the external pumping fields. In the long term both the external and the cavity fields can modify the susceptibility functions. The techniques utilized permit a detailed analysis of approximations and suggest generalizations for increased precision. As one important result of this type the theory indicates that macroscopic rate equations are valid if modulation frequencies are much less than the width of the narrowest coupling system spectral line.
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.
Masers exhibit interesting transient behavior that cannot be completely understood on the basis of the rate equations. The use of the rate equations in most transient analyses is usually justified on a more or less intuitive basis and the implied assumptions are not always clear. In this paper, the macroscopic maser rate equations are derived systematically from the Boltzmann equation for the density matrix of the atomic systems and Maxwell's equations for the radiation fields. When the coherence linewidth (T2−1) of the atomic systems is much larger than the cavity linewidth and the natural linewidth (T1−1) of the atomic emission, and with a WKB approximation, in the lowest order of approximation one obtains the two widely used, coupled first-order nonlinear rate equations of Statz and deMars. On the other hand, if the cavity linewidth is much larger than the atomic linewidths (T1−1 and T2−1), one can use the so-called ``reaction-field principle'' of Anderson and obtain, again, two coupled first-order rate equations; however, only one of the equations is nonlinear. The ranges of validity of both approaches are discussed in some detail.
A simple maser model consisting of a single-mode field coupled to N identical two-level atoms is presented. The assumption of negligible statistical correlation between the atoms and the field, permits the Heisenberg equations of motion to be replaced by a self-consistent set of ordinary nonlinear differential equations. Relaxation terms and an energy source are introduced phenomenologically. The resulting equations exhibit a threshold, stable and unstable steady states, relaxation oscillations and, sometimes, spiking. The dynamic equations for the hydrogen maser and the rate equations of laser theory are derived as special cases. Also discussed are the maser amplifier, locking of a maser to an external signal, and the effects of cavity thermal noise.
This paper has two purposes: 1) to clarify the relationship between the quantum theory of radiation, where the electromagnetic field-expansion coefficients satisfy commutation relations, and the semiclassical theory, where the electromagnetic field is considered as a definite function of time rather than as an operator; and 2) to apply some of the results in a study of amplitude and frequency stability in a molecular beam maser. In 1), it is shown that the semiclassical theory, when extended te take into account both the effect of the field on the molecules and the effect of the molecules on the field, reproduces almost quantitatively the same laws of energy exchange and coherence properties as the quantized field theory, even in the limit of one or a few quanta in the field mode. In particular, the semiclassical theory is shown to lead to a prediction of spontaneous emission, with the same decay rate as given by quantum electrodynamics, described by the Einstein A coefficients. In 2), the semiclassical theory is applied to the molecular beam maser. Equilibrium amplitude and frequency of oscillation are obtained for an arbitrary velocity distribution of focused molecules, generalizing the results obtained previously by Gordon, Zeiger, and Townes for a singel-velocity beam, and by Lamb and Helmer for a Maxwellian beam. A somewhat surprising result is obtained; which is that the measurable properties of the maser, such as starting current, effective molecular Q, etc., depend mostly on the slowest 5 to 10 per cent of the molecules. Next we calculate the effect of amplitude and frequency of oscillation, of small systematic perturbations. We obtain a prediction that stability can be improved by adjusting the system so that the molecules emit all their energy h Ω to the field, then reabsorb part of it, before leaving the cavity. In general, the most stable operation is obtained when the molecules are in the process of absorbing energy from the radiation as they leave the cavity, most unstable when they are still emitting energy at that time. Finally, we consider the response of an oscillating maser to randomly time-varying perturbations. Graphs are given showing predicted response to a small superimposed signal of a frequency near the oscillation frequency. The existence of "noise enhancing" and "noise quieting" modes of operation found here is a general property of any oscillating system in which amplitude is limited by nonlinearity.
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)
A simple model of a maser or laser consisting of a single-mode field coupled to N identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.