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Physical Review A

Multimode laser with an injected signal: Steady-state and linear stability analysis

Lorenzo M. Narducci, Jorge R. Tredicce, Luigi A. Lugiato, N. B. Abraham, Donna Kay Bandy

Published Sep 1, 1985• 32 citations
Publisher

Abstract

We discuss the steady-state and the linearized response of a multimode laser with an injected signal. The steady-state analysis is carried out for arbitrary values of the laser gain and resonator parameters, while the linearized stability calculation is limited to parameter ranges where the mean-field assumption is a reasonable approximation. For suitably large values of the polarization and population decay rates, we review the problem of the adiabatic elimination of the atomic variables and demonstrate explicitly the existence of possible pitfalls and how to avoid them.

DOI
10.1103/physreva.32.1588

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We discuss the steady-state and the linearized response of a multimode laser with an injected signal. The steady-state analysis is carried out for arbitrary values of the laser gain and resonator parameters, while the linearized stability calculation is limited to parameter ranges where the mean-field assumption is a reasonable approximation. For suitably large values of the polarization and population decay rates, we review the problem of the adiabatic elimination of the atomic variables and demonstrate explicitly the existence of possible pitfalls and how to avoid them.
This paper contains a description of the behavior of a multimode unidirectional ring laser with a homogeneously broadened active medium. Our formulation is based on the conventional Maxwell-Bloch (MB) equations, but is distinguished from other treatments by the inclusion of a finite mirror reflectivity and an arbitrary value of the gain parameter. We review the steady-state behavior of the system and analyze the longitudinal profile of the field and of the atomic variables. With an appropriate transformation of variables, we transform the boundary conditions of the ring cavity into standard periodicity type, even in the presence of a finite reflectivity, and derive an infinite hierarchy of coupled mode equations. We analyze exactly the linear stability of the system, and investigate the dependence of the instability domain on the reflectivity and gain parameters. A numerical study of the full MB equations for a parameter range of the type explored in the recent experiments by Hillman et al. [Phys. Rev. Lett. 52, 1605 (1984)] reveals similarities, but also considerable differences between the results of the theory and the main experimental signatures of their instability. However, the injection of numerical noise shows the presence of numerous coexisting basins of attraction which are likely to play a significant role in the dynamics of a noisy laser.
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 discuss the influence of injected atomic coherence on laser dynamics. The nonlinear equations of motion for the intensity and phase give the dependence on external parameters (inversion, amplitude of the atomic coherence, and cavity-field detuning). Depending on the inversion, there can be one or three steady-state solutions corresponding to a given value of the injected coherence. The situation is analogous to optical bistability, the atomic coherence playing the role of the injected field. For zero detuning the upper, laser-like branch starts at zero value of the injected coherence and a numerical stability analysis indicates that it is the only stable solution, the lower branch being everywhere unstable. For finite detuning the upper branch starts at a finite threshold value of the injected coherence. For coherences smaller than this threshold the system exhibits oscillatory behavior that corresponds to a stable limit cycle in the phase plane of the laser quadrature variables. The intensity fluctuations of the laser exhibit antibunching for sub-Poissonian pumping statistics.
We present an ab initio treatment of the steady-state of lasers with injected signals that describes a regime, valid for microlasers, in which the locking transition is dominated by cross saturation and spatial hole burning. The theory goes beyond standard approaches and treats multimode lasing with injected signals and finds the possibility of partially locked states and as well as repulsion of the free-running frequencies from the injected signal. The theory agrees well with exact integration of the full wave and matter equations for the system. It can also describe accurately complex modern lasers structures and is applied to the example of deformed disk lasers. We show that in the case of a one-dimensional cavity in the locked or regenerative amplifier regime the theory reduces to an improved version of the Adler equations in the appropriate limit.