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Physical Review A

Effects of injected atomic coherence on single-mode frequency locking in a cavity

Timothy Carty, Murray Sargent

Published Aug 1, 1990• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.1544

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World Scientific News•Jan 1, 2015•0 citations
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.
We investigate the semiclassical electric-dipole-allowed interaction of a unidirectional plane-wave electric field with a homogeneously broadened two-level medium prepared in a coherent superposition of energy eigenstates. Atomic level pumping and the effects of phase matching the new polarization contributions are discussed. The slowly varying polarization is derived and applied to the problem of the propagation of a weak field. The polarization of the medium acquires a direct (field-independent) contribution due to the injected coherence. We find that the slowly varying envelope propagation equation retains the standard single-mode absorption coefficient but develops additional driving terms. Phase matching in both time and space is crucial for the injected coherence to couple to a field. The phase-matched case of weak-field propagation proves to be a linear superposition of two processes: the usual Beer's law exponential decay (or growth) and a constant field supported by the medium. The injected coherence enables a weak electric field to persist in an absorbing medium.
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.
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.
Discussion begins with a brief account of atomic level-pumping and reasons why atomic coherence is typically not considered in cw work on optical interactions. This dissertation is divided into four parts: semiclassical treatments of one-photon electric- dipole atom-field single-mode interactions and multimode interactions, and corresponding treatments for the two-photon interaction. We present the effects of injected atomic coherence on the polarization of the medium, the slowly varying envelope wave equation, the single- and multiwave mixing coefficients, and weak field propagation in a homogeneously broadened two-level medium. Spatial and temporal phase matching of the injected coherence to a field mode is crucial throughout, since the field may not be able to remain in phase with the induced and injected polarizations. One-photon injected coherence contributes directly to the polarization at the atomic resonance frequency. The perfectly phase-matched case leads to a linear superposition of an exponentially decaying field (Beer's law) and a constant field driven by the injected coherence. The interaction of an injected coherence with a detuned field produces frequency-symmetric sidebands about the pump field polarization. The sideband spacing equals the atom-field detuning. To probe the injected coherence we inject a weak resonant field. The resulting three-wave mixing leads to multiwave mixing coefficients that are unaffected to first-order in the weak sidemodes, but the injected coherence adds inhomogeneous terms to the coupled-mode equations. For both single- and multimode interactions the injected coherence does not affect the exponential growth/decay of the sidemodes, but it supports a weak field that may propagate if properly phase matched. For two-photon media the injected coherence requires at least one field interaction in order to produce a polarization, which then appears in the single- and multiwave mixing coefficients. The exponential growth/decay rate is modified by the injected coherence. For a centrally-tuned pump the injected coherence contributes the standard multiwave mixing terms as well as additional effects. Four-wave mixing is discussed as a means of relaxing the spatial phase matching constraint on the injected coherence.
We propose a simple and convenient analytical model for the description of coherent passive mode locking in a two-section ring-cavity laser based on the generalized area theorem for the coherent pulse propagation in a resonant medium with inhomogeneous line broadening. This model is applied to theoretically analyze the possible regimes of the spatiotemporal dynamics of such a laser together with their stability properties. We examined the laser dynamics, when varying the energy and phase relaxation times in the media, the gain and absorption rate in the respective laser sections, as well as the ratio of the transition dipole moments of the absorber and gain media. As the result, the stability of the coherent mode locking as well as the large flexibility of the arising laser behavior were shown in wide parameter ranges.
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.