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Physical Review A

cw instability and steady-state pulses in a ring laser with intracavity parametric amplification

Mandaar Bhaskar Pande, Subhasish Dutta Gupta

Published Dec 1, 1992• 2 citations
Publisher

Abstract

A homogeneously broadened two-level ring laser with an intracavity parametric amplifier is studied. The presence of the parametric amplifier is shown to lead to two distinct nontrivial cw solutions compared to the one in its absence. The conditions for the stability of these cw solutions are obtained. It is shown that in the context of the resonant modes one cw solution is always phase stable, whereas the other is always phase unstable. For pump parameters beyond a critical value (the second threshold), both these cw solutions become amplitude unstable, and self-pulsing can set in. The shape and the velocity of these pulses are evaluated numerically. It is shown that two distinct types of steady-state pulses beyond the second laser threshold are possible due to the phase sensitivity introduced by the parametric amplifier. This is in sharp contrast to the case without the parametric amplifier, where only Risken-Nummedal pulses [see H. Risken and K. Nummedal, J. Appl. Phys. 39, 4662 (1968)] are possible. It is shown that the parametric amplifier can lead to the narrowing or broadening of the Risken-Nummedal pulses. Moreover, it is shown that the ring laser with or without the parametric amplifier allows for a pulsed solution distinct from the above-mentioned one, which, however, turns out to be unstable.

DOI
10.1103/physreva.46.7260

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A homogeneously broadened two-level ring laser with an intracavity parametric amplifier is studied. The presence of the parametric amplifier is shown to lead to two distinct nontrivial cw solutions compared to the one in its absence. The conditions for the stability of these cw solutions are obtained. It is shown that in the context of the resonant modes one cw solution is always phase stable, whereas the other is always phase unstable. For pump parameters beyond a critical value (the second threshold), both these cw solutions become amplitude unstable, and self-pulsing can set in. The shape and the velocity of these pulses are evaluated numerically. It is shown that two distinct types of steady-state pulses beyond the second laser threshold are possible due to the phase sensitivity introduced by the parametric amplifier. This is in sharp contrast to the case without the parametric amplifier, where only Risken-Nummedal pulses [see H. Risken and K. Nummedal, J. Appl. Phys. 39, 4662 (1968)] are possible. It is shown that the parametric amplifier can lead to the narrowing or broadening of the Risken-Nummedal pulses. Moreover, it is shown that the ring laser with or without the parametric amplifier allows for a pulsed solution distinct from the above-mentioned one, which, however, turns out to be unstable.
Analytical traveling-wave self-pulsing solutions of a homogeneously broadened ring laser are presented in the limit that the dipole relaxation rate is much greater than the atomic inversion relaxation rate. The phase velocity v(\ensuremath{\Lambda}) as a function of the normalized pump \ensuremath{\Lambda} is given explicitly. This function has a simple relation with the upper boundary ${\ensuremath{\alpha}}_{\mathrm{max}(\mathrm{\ensuremath{\Lambda}}}$) of the Risken-Nummedal-Graham-Haken (RNGH) instability domain in the (\ensuremath{\alpha},\ensuremath{\Lambda}) plane, where \ensuremath{\alpha} is the wave vector of the perturbations. The self-pulsing may appear in different ways, depending on how the stationary solution becomes unstable. If the instability of the stationary solution is caused by an unstable mode touching upon the upper boundary of the RNGH-instability domain in the (\ensuremath{\alpha},\ensuremath{\Lambda}) plane, the self-pulsing is supercritical; if the unstable mode lies on the lower boundary, a subcritical self-pulsing arises, and the system is bistable. This rule provides new signatures of the self-pulsing phenomena and will be helpful to experimental identification of the self-pulsing arising from the RNGH instability. The linear stability analysis reveals two kinds of instabilities for these self-pulsing solutions. One kind of instability is rooted in the RNGH instability of the stationary solution and may occur even when the amplitude of the self-pulsing solution approaches zero. The other kind of instability occurs when the oscillating amplitude of the self-pulsing solution becomes large. Above this large-amplitude instability threshold no traveling-wave self-pulsing can be stable any longer. The results for the corresponding Lorenz model are also presented.
We provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as a coupling between them via scattering from a spatial grating formed in the population inversion. If the grating is a sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistability between the two modes when the cavity is tuned to resonance is recovered, a result that persists, in part, even when the losses between the two modes are different.With detuning, the unidirectional steady-state solutions can be destabilized, creating regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lasing threshold. Asymptotic limits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzian line shape and for arbitrary inhomogeneous linewidths. Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to CO2{\mathrm{CO}}_{2}CO2​ lasers show that there are broad regions in the parameter space of gain and detuning for which the behavior is dynamically chaotic in the form of nearly ``square-wave'' alternation between the counterpropagating modes of the laser with irregular switching times. In other regions of that parameter space we find more irregular pulsations and pulse alternations. Regions of periodic pulsations also have been found.
We consider an inhomogeneously broadened, unidirectional ring laser formulated in the traditional Maxwell–Bloch configuration of a two-level atom with transverse modes. We numerically analyze the steady state for a single longitudinal mode in the paraxial approximation, using the uniform-field limit with cylindrical symmetry and Gaussian profile for inhomogeneous broadening. Several domains of steady-state solutions are found for the resonant case in the good-cavity limit. Both single- and multiple-transverse mode solutions are observed; the expected TEM00 mode is seen in the limit of single-mode operation. The laser threshold is found to depend on the mode spacing (A1) as well as on the gain (2C) and the scaled Doppler broadening (σD). This means that as A1 is increased, the laser requires a larger gain to operate, up to a critical gain; beyond this critical gain, the laser threshold is independent of A1. Hysteresis between zero- and nonzero-intensity solutions at the laser threshold is obtained under realistic operating conditions for both increasing and decreasing 2C and A1 (i.e., for changes in the initial conditions). These distinct, but overlapping, domains of parameter space where both single- and multiple-mode steady states exist are qualitatively the same for increasing values of σD ≥ 1 and may indicate the presence of competing attractors.
In 1966 Haken and Risken, Schmid and Weidlich(1) showed that under proper conditions the stationary state of a homogenously broadened, single mode ring laser can become unstable. The instability requires a "bad cavity" situation, i.e. the cavity damping constant must be larger than the atomic decay rates. Later, Haken(2) proved that such an instability coincides with the well known Lorenz instability, which leads to chaotic behaviour. On the other hand, Risken and Nummedal and independently Graham and Haken(3) analyzed the homogeneously broadened ring laser taking into account all the longitudinal cavity modes. They showed that under appropriate conditions in the good cavity case some cavity modes, different from the one resonant with the atomic system, can become unstable. In this case, one has the formation of a pulse that travels in the cavity. In 1978, Bonifacio and Lugiato(4) showed that a similar self-pulsing behaviour in good cavity conditions occurs in optical bistability in a ring cavity filled with a homogeneously broadened atomic sample.
We present an examination of the stability and time-dependent behavior of a semiclassical model of the ring laser with a saturable absorber, in the vicinity of multiple steady-state solutions. We consider particularly the case in which the cavity is tuned away from the resonance frequency of the gain medium. The model considers the absorbing and amplifying media to consist of two-level atoms. Only homogeneous spectral broadening and unidirectional propagation of the light are considered. We are able to treat the steady state with an approximate mean field solution and also with its exact longitudinal dependence. The latter is found numerically from an implicit function solution to the Maxwell-Bloch equations in the steady state. There may be as many as four steady-state solutions for a given set of parameter values. When there are four solutions, we find that three solutions are of one intensity class (high or low) and that one is of a second intensity class (low or high). When the absorbing and emitting atoms are in resonance and the cavity is tuned to their line-center frequency, two of the members of the three-solution class have the same intensity. This degeneracy is broken by detuning.
This paper presents new results for the steady states of a detuned ring laser with a saturable absorber. We employ a semiclassical model which assumes homogeneously broadened two-level atoms. We proceed by solving the Maxwell-Bloch equations for the longitudinal dependence of the steady states of this system, and then simplify our solution by use of the uniform-field approximation. We present uniform-field results for squared electric field versus operating frequency, and for each of these versus cavity tuning and laser excitation. Various cavity linewidths and both resonant and nonresonant amplifier and absorber line-center frequencies are considered. The most notable finding is that cavity detuning breaks the degeneracies found in the steady-state solutions of the fully tuned case. This leads to the prediction that an actual system will bifurcate from the zero-intensity solution to a steady-state solution as laser excitation increases from zero, rather than to the small-amplitude pulsations found for the model with exactly resonant tuning of the cavity and the media line centers. Other phenomena suggested by the steady-state results include tuning-dependent hysteresis and bistability, and instability in both intensity and frequency due to the appearance of one or more new steady-state solutions as tuning is varied. These effects of detuning are being tested by a linearized stability analysis whose results will be reported separately.