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Journal of the Optical Society of America B

Properties of a Lorentz-broadened single-mode unidirectional ring laser

Paul Mandel

Published Jan 1, 1985• 24 citations
Publisher

Abstract

We study some properties of a single-mode unidirectional ring laser characterized by a Lorentzian inhomogeneous broadening. The emphasis is on the derivation of analytic results. First we determine the various classes of stationary solutions and demonstrate the existence of low-intensity and high-intensity optical bistability. Both types of optical bistability require a finite detuning and inhomogeneous broadening. Next we study the linear stability of the steady solutions and determine the conditions for the occurrence of a Casperson instability in various limits but on resonance.

DOI
10.1364/josab.2.000112

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Annual Meeting Optical Society of America•Jan 1, 1991•0 citations
We study some properties of a single-mode unidirectional ring laser characterized by a Lorentzian inhomogeneous broadening. The emphasis is on the derivation of analytic results. First we determine the various classes of stationary solutions and demonstrate the existence of low-intensity and high-intensity optical bistability. Both types of optical bistability require a finite detuning and inhomogeneous broadening. Next we study the linear stability of the steady solutions and determine the conditions for the occurrence of a Casperson instability in various limits but on resonance.
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.
The characteristics of the mode intensity and conditions for occurrence of optical bistability in a ring laser with a saturable absorber are considered. The rate equation approximation method is used to find analytic expressions for these characteristics. It is shown that changes in such physical parameters as the homogeneous and inhomogeneous broadening, coefficients of relaxation of amplifying and absorbing cells, etc. cause changes in the range of existence of optical bistability or a hysteresis cycle. The influence of inhomogeneous broadening and of spontaneous emission near the lasing threshold on the appearance of optical bistability is demonstrated for the single-mode regime.
The instabilities in homogeneously broadened lasers, which we described in the previous chapter, have a paradigmatic importance because they arise in models which allow one to perform the linear-stability analysis in a simple analytic form. The treatment of the corresponding instabilities in the case of in homogeneously broadened lasers is substantially more complex in general. However, it is very interesting because, as we shall see, the instabilities arise in parametric domains that are much more accessible from the experimental viewpoint. A related issue is that of the possibility of multimodal emission in a laser. It is common knowledge that in order to attain multimode emission it is necessary to have a sort of in homogeneity in the gain medium, because in a completely homogeneous configuration all modes interact with all atoms, and the arising competition among modes implies that only one of them survives. There are basically two main kinds of in homogeneity, the first is of spatial nature and the other is of spectral nature. The first kind is realized in Fabry–Perot lasers, because different cavity modes interact with atoms at different spatial positions in the cavity thanks to the standing-wave configuration of the field and to the consequent phenomenon of spatial hole- burning (see Section 14.2 and Fig. 14.1). This circumstance is capable of weakening the mode–mode competition and allowing multimode operation. In unidirectional ring lasers, on the other hand, there is an “inhomogeneous” configuration in the case of inhomogeneous broadening, because different groups of atoms have different transition frequencies and therefore interact with different modes thanks to the spectral hole-burning (see Section 15.2 and Fig. 15.3). Again, this weakens the mode–mode competition and allows multimode operation. However, these are qualitative considerations, and the only reliable and quantitative method to determine the possibility of multimodal operation is the linear-stability analysis of the single-mode stationary solutions.
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.
We consider the dynamics of 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 model for a single-longitudinal mode in the paraxial approximation, using the uniform field limit with cylindrical symmetry and a Gaussian profile for inhomogeneous broadening. Several domains of solutions are found for the resonant case in the good cavity limit. The laser threshold is found to be dependent on the mode spacing as well as on the gain and scaled Doppler broadening. Hysteresis between stable and unstable solutions is obtained in realistic operating conditions for both increasing (decreasing) gain and increasing (decreasing) mode spacing (i.e., for changes in the initial conditions).