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Journal of the Optical Society of America A

Stability theory of class-C (far-infrared) lasers with an injected signal

J. Jahanpanah

Published Apr 1, 2005• 15 citations
Publisher

Abstract

The behavior of a single-mode class-C laser in the presence of an injected signal is investigated theoretically in detail. We have determined the ranges of input signal strength and the frequency detuning for which the oscillations of the cavity electric field and the atomic induced dipole moment are simultaneously stable and have only one oscillation component with the frequency equal to that of the input signal (injection-locked state). One can reproduce all the previous results corresponding to class-A and class-B lasers from our results by using their relevant conditions. Finally, the energy conservation law is demonstrated for both the below-threshold and the injection-locked states of class-C laser amplifiers.

DOI
10.1364/josaa.22.000680

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The behavior of a single-mode class-C laser in the presence of an injected signal is investigated theoretically in detail. We have determined the ranges of input signal strength and the frequency detuning for which the oscillations of the cavity electric field and the atomic induced dipole moment are simultaneously stable and have only one oscillation component with the frequency equal to that of the input signal (injection-locked state). One can reproduce all the previous results corresponding to class-A and class-B lasers from our results by using their relevant conditions. Finally, the energy conservation law is demonstrated for both the below-threshold and the injection-locked states of class-C laser amplifiers.
The theory of the laser amplifier is developed for conditions in which the strength of the input signal is increased from small values, where the amplification is linear, to larger values, where the amplification becomes nonlinear. The below-threshold laser amplifier oscillates at a single frequency equal to that of the input signal, and its properties are found by solution of the nonlinear equation of motion for the single excitation amplitude. For the above-threshold laser amplifier, the effects of the nonlinear behavior are to shift the laser frequency from its free-running value and to transfer intensity from the laser line to the signal frequency and to a range of satellite lines, whose frequency detunings are integer multiples of the signal detuning. The intensities of the various emission lines of the laser are calculated by power-series expansions of the field amplitudes up to terms of fourth order in the input signal strength. The onset of injection locking is determined by the conditions for which the intensity at the shifted free-running laser frequency falls to zero. The injection-locked state is characterized by a single excitation frequency equal to that of the input signal, and its properties are found by solution of the same nonlinear equation of motion as for the below-threshold amplifier. The ranges of input signal strength and detuning are determined for which the injection-locked state is stable. The energy conservation properties of the laser amplifier are considered for each of its operating states. \textcopyright{} 1996 The American Physical Society.
The coupling of an electric field to an atomic beam prepared in a phase-locked coherent superposition of energy eigenstates is examined for a ring-laser cavity. The active medium is assumed to be a collection of noninteracting homogeneously broadened two-level atoms whose transition is electric-dipole allowed. The atom-field interaction is treated semiclassically and the atomic coherence injected into the cavity is assumed to be phase matched to the unidirectional plane-wave cavity mode. The polarization is derived for the case in which the electric-field-mode envelope has at most linear phase variations in time and is applied to the problem of frequency locking in a cavity. The presence of the atomic coherence causes the active medium to possess an inherent polarization, i.e., a field-independent contribution. This enables an optical cavity to support laser oscillation over the entire range of relative excitations, in particular, even for strongly absorbing media. A simple linear stability analysis is performed on the electric-field envelope by describing the medium with a slowly varying polarization valid when the time scales of the atomic response are much smaller than those of the variations in the envelope. The stability of frequency-locked oscillation depends on the detuning of the cavity from line center and the strength of the injected coherence relative to that of the injected population difference.
The class-C lasers are well recognized to have very narrow atomic transition bandwidths. Therefore, a minor mismatch between the cavity resonance and atomic transition frequencies (CAF) is normally led to the large modifications in the gain and noise profiles. The motion of class-C lasers is described by the three variables of cavity electric field, atomic population inversion, and dipole moment of atoms that are coupled to each other by the optical Maxwell-Bloch equations. The fluctuating features of these three variables are completely elaborated by solving their equations of motion in the presence of the cavity Langevin force. The results peculiarly indicate that the noise profile of the cavity electric field exactly mimics the amplification profile of an input signal by the laser in both below and above threshold states. The both gain and noise profiles simultaneously tend to infinity at the normalized pumping rates and CAF mismatches determined by the laser stability theory. The noise flux profiles are finally confirmed by illustrating the flux conservation.
The theory of small-signal laser amplification is developed for homogeneously broadened systems in which the three main decay rates, for the collective atomic dipole moment, the population inversion, and the field in the laser cavity, have arbitrary relative magnitudes. The calculations extend previous work on class-AAA and -BBB lasers, where the dipole decay rate greatly exceeds the other two rates, to class-CCC lasers, where the dipole decay rate is comparable to the others. The free-running laser is assumed to excite a single longitudinal mode of the cavity, whose frequency generally differs from that of the coupled atomic transition. The linear gains of the laser are derived for input signals whose frequencies are close to that of the lasing mode or to one of its adjacent nonlasing longitudinal modes. Divergences in the gain that occur for these two arrangements are associated with the previously studied single-mode and multimode instabilities of the free-running laser, respectively.
We consider a single-mode unidirectional ring laser containing a homogeneously broadened two-level medium whose atoms are injected in a coherent superposition of states. Provided that appropriate phase-matching conditions are met, the injected coherence contributes directly to the polarization of the medium and hence to the laser gain. We assume that the field is classical and perfectly phase matched with the injected coherence. We find that laser oscillation can occur below the usual laser threshold, and even in the absence of an inversion, with a frequency locked to the atomic line center. Mode locking is unstable for sufficiently strong relative excitation.
We consider a single-mode unidirectional ring laser containing a homogeneously broadened two-level medium whose atoms are injected in a coherent superposition of states. Provided appropriate phase matching conditions are met, the injected coherence contributes directly to the polarization of the medium and hence to the laser gain. We suppose the field is classical and perfectly phase matched with the injected coherence. We find that laser oscillation can occur below the usual laser threshold and even in the absence of an inversion. In these cases the steady state laser frequency is locked to the atomic line center. As the excitation increases, the ability of the injected atomic coherence to lock the laser frequency decreases. Slightly above the usual laser threshold, the system unlocks and produces an output weakly modulated in time.