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Physical Review A

Optical instabilities in Fabry-Perot resonators

Andrea Li Bassi, Franco Prati, Luigi A. Lugiato

Published May 20, 2021• 7 citations
Publisher

Abstract

We start from the Maxwell-Bloch equations that govern the dynamics of Fabry-Perot lasers and, considering a doubled cavity, generalize our traveling wave formalism to the case in which the two-level medium does not fill the cavity but occupies only a portion of it. We linearize these equations and focus on the adiabatic elimination of both atomic fluctuations. Next, we restrict our attention to free running lasers under resonant conditions and analyze amplitude and phase instabilities. The examination of the unstable domain leads to the conclusion that the multimode Fabry-Perot instability arises near threshold only when the ratio of the longitudinal to the transverse atomic relaxation rate is substantially smaller than unity. This result agrees with our previous study in the limit of adiabatic elimination of the atomic polarization fluctuations only. We describe the self-pulsing behavior that arises from the multimode instability, and exhibits hysteretic behavior when the pump parameter is swept forward and backward. Finally, we investigate the single-mode instability and show that in the Fabry-Perot case there is no longer the correspondence between single-mode and multimode instabilities that is well known in the case of ring lasers. We confirm that in the case of resonant Fabry-Perot lasers the single-mode amplitude instability arises very far from threshold.

DOI
10.1103/physreva.103.053519

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We start from the Maxwell-Bloch equations that govern the dynamics of Fabry-Perot lasers and, considering a doubled cavity, generalize our traveling wave formalism to the case in which the two-level medium does not fill the cavity but occupies only a portion of it. We linearize these equations and focus on the adiabatic elimination of both atomic fluctuations. Next, we restrict our attention to free running lasers under resonant conditions and analyze amplitude and phase instabilities. The examination of the unstable domain leads to the conclusion that the multimode Fabry-Perot instability arises near threshold only when the ratio of the longitudinal to the transverse atomic relaxation rate is substantially smaller than unity. This result agrees with our previous study in the limit of adiabatic elimination of the atomic polarization fluctuations only. We describe the self-pulsing behavior that arises from the multimode instability, and exhibits hysteretic behavior when the pump parameter is swept forward and backward. Finally, we investigate the single-mode instability and show that in the Fabry-Perot case there is no longer the correspondence between single-mode and multimode instabilities that is well known in the case of ring lasers. We confirm that in the case of resonant Fabry-Perot lasers the single-mode amplitude instability arises very far from threshold.
In this chapter we outline the general scenario for optical temporal instabilities, starting from the standard Maxwell–Bloch equations. With the exception of Section 22.7, in Part II we will always consider the ring-cavity configuration because only in the case of the ultrathin-medium case considered in Section 22.7 is the treatment of instabilities in Fabry–Perot cavities not substantially more complicated than that in the ring-cavity configuration. Some authors have introduced special approaches in order to study temporal optical instabilities, exploiting the concepts of gain and of dispersion, which are natural in optical systems and therefore help our intuition. In particular, a hint of the mechanism that produces instabilities in optical systems is offered by the so-called “weak-sideband” approach or “gain-feedback approach” [245–249], whose validity can be justified rigorously with the help of the linear-stability analysis [250]. The central point of this method is the response of the system to a weak probe passing through a collection of two-level atoms driven by a strong field [251–253]. If the strong signal with carrier frequency ω 0 saturates the medium, we can find regions of the frequency axis, which are far removed from the carrier frequency of the signal, where the weak probe can experience gain instead of absorption even if the medium has no population inversion. In this book we follow the approach of [250], which is rigorously rooted in the linear- stability analysis and, in addition, establishes a general, useful and, in a sense, surprising connection between single-mode and multimode instabilities. We classify as single-mode a temporal instability that involves the resonant mode only, and classify as multimode a temporal instability that involves only off-resonance cavity modes. The discussion in this chapter is mainly focussed on this connection. In Section 20.1 the multimodal equations derived in Section 12.2 are reformulated by introducing a modal expansion also for the atomic variables. In this way one obtains a complex hierarchy of equations that couple the modal amplitudes for the field and for the atomic variables.
Abstract We formulate the dynamics of nonlinear optical Fabry–Perot (FB) cavities in terms of a model in which only one of the two counter-propagating electric field envelopes appear. Thus, the model is simpler than the standard ones but still exact. The field envelope which propagates in the opposite direction is expressed in a very simple way in terms of the envelope that appears in the model. Thus we generalize in the simplest way to the FB case the set of Maxwell–Bloch equations for the ring cavity. The boundary condition for the field envelope that appears in the model is a simple periodicity condition. This feature allows for expanding the variables of the model in terms of traveling waves instead of standing waves as it is customary, which implies noteworthy simplifications in the calculations. On the basis of the modal equations which arise from the model, we discuss the adiabatic elimination of the atomic variables and of the atomic polarization only.
We investigate the intensity stability of the superradiant laser. Our study focuses on the architecture where a continuous beam of atoms in an electronically excited state crosses the mode of a high-finesse Fabry-Perot cavity, which has been proposed as a new architecture of an active optical clock. We show that such superradiant laser can become unstable and develop chaotic behavior. We derive an analytical criterion for this instability and find that it may only occur when the lifetime of photons in the cavity is significantly shorter than the lifetime of atoms. This criterion allows for refining the necessary parameters to run a superradiant laser as a frequency reference in the optical domain. In particular, we point-out the consequences of the instability on intensity fluctuations and laser linewidth. On the other hand, we also point out that the superradiant laser, when in the unstable regime, can become an interesting playground for studying chaos. At the mean-field level, there is a direct mapping to the Bénard instability associated with fluid turbulence; however quantum fluctuations associated with photon out-coupling and atom re-filling substantially modify the expected behaviors. Finally, we point-out the existence of a regular self-pulsing regime at large atom numbers.
Instabilities are predicted for absorptive optical bistability in a nonideal Fabry-P\'erot cavity on the basis of Maxwell-Bloch theory. We show that the stability problem of the Maxwell-Bloch equations can be formulated in terms of a Riccati differential equation with boundary conditions. We have integrated this equation numerically. Our results crucially depend on the value of the transmission coefficient of the mirrors. For finite values of this quantity we find that the nearest side mode is responsible for two disconnected instability domains in the plane spanned by the output intensity and the medium response time. One of these domains can generate instabilities in the upper and the lower branch of the steady-state curve if the cooperation parameter is sufficiently large. Higher side modes can give rise to positive-slope instabilities as well. From our findings it can be understood why a recent experimental search for instabilities in absorptive optical bistability has led to a negative result. We finally demonstrate that the instability spectra of a Fabry-P\'erot cavity and a so-called equivalent ring cavity differ considerably.
The transmission characteristics of a Fabry-Perot cavity containing a resonant absorber are described analytically and graphically for internal fields of arbitrary intensity. The multimode instability considered by Bonifacio and Lugiato [Opt. Commun. 19 , 72 (1976)] for the unidirectional ring laser is treated for the standing-wave cavity. In particular, an instability may exist when the medium is located at one of the ends of the cavity, but not if the medium fills the cavity. The results are explained in terms of induced side-mode gain and spatial hole burning.