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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Wavelength multistability in ring and fabry-Pérot lasers: The effect of spatial hole burning

Antonio Pérez Serrano, J. Javaloyes, S. Balle

Published May 1, 2011• 0 citations
Publisher

Abstract

In this contribution, we investigate theoretically the multistability by means of a travelling wave model (TWM) for a homogeneously broadened medium with the appropriate boundary conditions, that allow us to model RLs and FP lasers. We take into account that the presence of counter-propagating fields creates a spatial modulation or grating of the population inversion (D2). We compute the monochromatic steady state solutions of the system via a shooting method and we numerically perform their linear stability analysis by diagonalising the jacobian obtained from the evolution operator of the discretized system.

DOI
10.1109/cleoe.2011.5942552

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In this contribution, we investigate theoretically the multistability by means of a travelling wave model (TWM) for a homogeneously broadened medium with the appropriate boundary conditions, that allow us to model RLs and FP lasers. We take into account that the presence of counter-propagating fields creates a spatial modulation or grating of the population inversion (D2). We compute the monochromatic steady state solutions of the system via a shooting method and we numerically perform their linear stability analysis by diagonalising the jacobian obtained from the evolution operator of the discretized system.
We are studying what can be considered as an intermediate situation between the unidirectional and the Fabry-Perot cavity configurations: the bidirectional ring laser. In it some spatial hole-burning appears because of the interference between the two counter propagating fields, which creates a shallow spatial modulation of the population inversion. Multimode emission in such systems has been investigated recently with a full Maxwell-Bloch model in which the uniform field approximation is not applied. With such a model only a few analytical results can be derived and the study of is thus mainly numeric. We address the analytical study of a mean-field bidirectional laser model that includes the possibility of backscattering.
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 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.
Homogeneously broadened bidirectional ring lasers show alternation of the lasing direction induced dynamically by a long-lived population grating and slight detuning of the laser cavity from the atomic resonance1. We show that the mode switching is accompanied by abrupt jumps by pi in the total relative phase of the system2. Our analysis reveals that the carrier frequencies of the two modes are not equal even though the two unidirectional steady state solutions have the same frequency2. When one mode dominates and the other is suppressed, the dominant mode has the steady state frequency. However, a mode growing in strength has a frequency closer to the atomic resonance while a mode falling in strength has a frequency further from the atomic resonance. The dynamically induced mode-pulling and mode-pushing can be as much as 25 % of the steady state frequency detuning. The shifting frequencies of the modes indicate that the standing wave interference pattern rotates in the laser ring cavity, alternately moving clockwise and counterclockwise with nearly equal and constant speeds. The longitudinal spatial hole-burning pattern consequently also rotates in the ring tending to wash out the population grating and keeping it from developing an amplitude of much more than one percent of the spatially averaged population inversion.
A stochastic, semiclassical model is developed for a multimode, homogeneously broadened laser with rapid-dipole dephasing (e.g., semiconductor or dye lasers). The theory is valid for systems with a free spectral range that is large compared to the decay rates of the population inversion and the field. Population dynamics including temporal beating effects and relaxation oscillations, spatial-hole burning, coherent-wave mixing, and quantum noise are all included self consistently. A principal restriction of the model, when applied to coupled-cavity systems, is that the modes satisfying the boundary conditions of the gain medium must form a subset of the modes defined by the cavity and that only modes in this subset are above threshold. For typical systems, the model is valid for pump rates up to several times threshold. The effects on the emission linewidth of coherent-wave mixing and the number of modes included in the simulation at various pump rates are discussed.
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.