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IEEE Journal of Quantum Electronics

Phase and amplitude fluctuations of the laser oscillator

Herman Pauwels

Published Mar 1, 1966• 11 citations
Publisher

Abstract

A quantum theory for noise in the steady state of the laser oscillator above threshold is presented. The concept of operator noise source is introduced. A simple laser model operating in a single mode with resonant frequency tuned to the center of the material line is considered. The resonance linewidth of the inverted material is collision-broadened, and assumed to be much broader than the cold-cavity width. We restrict ourselves to one type of collision. The quantum description of this laser model leads to nonlinear operator equations. These are linearized, but this linearization is justified only for operation "sufficiently" above threshold. The first- and second-order Glauber functions and the expectation value of the field commutator are derived for the field inside the laser cavity. These results are compared with the predictions of a "semiclassical" theory in which the classical equations contain noise sources such that properties of the field below threshold are correctly predicted. Our results differ by certain terms from the results of this semiclassical theory. These terms are too small to have been detected experimentally.

DOI
10.1109/jqe.1966.1073779

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Physical Review A•Jul 1, 1978•153 citations
A quantum theory for noise in the steady state of the laser oscillator above threshold is presented. The concept of operator noise source is introduced. A simple laser model operating in a single mode with resonant frequency tuned to the center of the material line is considered. The resonance linewidth of the inverted material is collision-broadened, and assumed to be much broader than the cold-cavity width. We restrict ourselves to one type of collision. The quantum description of this laser model leads to nonlinear operator equations. These are linearized, but this linearization is justified only for operation "sufficiently" above threshold. The first- and second-order Glauber functions and the expectation value of the field commutator are derived for the field inside the laser cavity. These results are compared with the predictions of a "semiclassical" theory in which the classical equations contain noise sources such that properties of the field below threshold are correctly predicted. Our results differ by certain terms from the results of this semiclassical theory. These terms are too small to have been detected experimentally.
In a previous paper we considered the linear problem of a one-dimensional laser having output coupling. In this paper we derive the nonlinear equation for the same model of the laser and solve it for steady state. The output coupling is exactly taken into account and the field is treated as having a continuous spectrum composed of the normal modes of the universe. Damping terms for the atomic polarization and for the population inversion are introduced phenomenologically. The fluctuating forces due to these damping terms and to the initial field are ignored. The resultant field inside the cavity consists of two oppositely travelling waves whose amplitudes are functions of the distance from the coupling surface. Outside the cavity we have a monochromatic, outgoing wave whose amplitude does not change with distance. The threshold inversion and the oscillation frequency are the same as those derived in the linear theory.
The influence of injected atomic coherence on a laser's operation is discussed by analyzing the semiclassical equations of motion of this system. Both stationary-steady-state and time-dependent regimes of operation are investigated. We present the dependence of the stationary laser intensity and phase on the external parameters (population inversion, amplitude of atomic coherence, and detuning). For small cavity-field detuning and large atomic coherence, both frequency and phase locking occurs. The laser frequency is locked to the atomic frequency, and the phase is locked to a value determined primarily by the phase of the atomic coherence. Under certain conditions, in the inverted regime, the output intensity is an ssS-shaped function of the atomic coherence, leading to the possibility of a bistable behavior. However, only a portion of the bistable curve gives stable stationary operation, and in critical points (the location of which depends crucially on the external parameters) time-dependent instabilities branch away. When the detuning is larger than a critical value there is neither phase nor frequency locking. We find that the time-dependent behavior of the laser intensity in this case is oscillatory (quasiperiodic), and show that there is a stable limit cycle in the phase plane of the quadratures. There is a small parameter region where a stationary steady state and an oscillatory state may coexist. We also consider the nonlinear quantum theory of a laser with injected atomic coherence, and include the effect of pumping statistics. We derive the Fokker-Planck equation for the P representation, and express the noise in terms of moments. We find that in the steady state the intensity noise can be suppressed below the shot-noise limit but that the phase fluctuations are not affected by pump regularity. For nonzero detuning, we find that transient squeezing of the phase fluctuations is possible.
A semiclassical, nonlinear theory of a single-mode laser is developed so that the transition around threshold can be studied. The field is expressed in the frequency domain which allows us to emphasize the role of the Fabry-P\'erot cavity. A generalized Airy function is obtained for the laser line shape: it contains the source line shape and the empty cavity line shape. It describes the laser both above and below threshold, including the transition region where there is an abrupt increase of intensity and decrease of the laser linewidth. The general arguments are illustrated by detailed numerical calculations for the 3.39-\ensuremath{\mu}m line of the He-Ne laser. The intensity of the spontaneous emission source which plays a central role is computed from first principles.
The single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.
Treating the laser medium quantum mechanically and the laser electric field classically, a mathematical model was developed to study a laser field inside a single mode optical cavity by numerical and analytical techniques. The simulations for threshold population, population inversion and average population with electric field frequency for 500 kHz resonance frequency of the atom are presented. The gain coefficient and the pump parameter with the frequency of the cavity electric field for 1000 kHz resonance frequency of the atom were simulated. The threshold population is a minimum when the frequency of electric field is equal to the resonance frequency of the atom. The population inversion varies sinusoidal with time and the frequency of the electric field and is a maximum at the threshold frequency of the atom. The saturated laser intensity at steady state increases linearly with the pump parameter and for larger values of the pump parameter, a smaller time is needed to reach the saturation of the laser field.
We study the properties of the laser field in the case where the cavity contains two cells, one which amplifies and one which absorbs the radiation. We adopt the simplest possible model, in which there is only one running mode in the cavity and all atoms are fixed and homogeneously distributed in space. We first set up a semiclassical description. In the frame of the stationary semiclassical (SC) theory the most striking feature is the occurrence of a domain in which three solutions coexist. A linear-stability analysis shows that two of these solutions are stable. This gives rise to an hysteresis cycle. Most of the results in this section are analytic. In the frame of a purely quantum description, we derive a generalized Fokker-Planck equation which is exactly solved in the stationary case. This yields a stationary quasiprobability distribution function for the field. In the domain where the SC theory predicts three solutions, the field-distribution function has two maxima and one minimum whose positions are precisely given by the SC intensities. With this distribution function we study the intensity and the intensity fluctuations: the intensity shows a rather abrupt increase when crossing the narrow transition region; in this region the intensity fluctuations are extremely high. By means of an approximate method to solve the time-dependent Fokker-Planck equation we study the linewidth and show that it drastically decreases above threshold. Finally we consider the linearized and quasilinearized Fokker-Planck equations which can be solved analytically. We show that the domain in which neither of these equations fits the results of the generalized Fokker-Planck equation is extremely small. The properties of the metastable states are analyzed in terms of the generalized and asymptotic Fokker-Planck equations.