Explicit models are derived for good cavity one- and two-photon lasers with an injected signal in a Fabry-Pérot cavity. The steady solutions and their stability properties are obtained analytically and compared with the corresponding ring cavity model ones. Only quantitative differences between both types of cavities are found. In particular we show that (i) the Fabry-Pérot cavity reduces significantly the domain of self-pulsing with respect to the ring cavity, and for the two-photon laser case (ii) larger output can be extracted from a Fabry-Pérot cavity than from a ring cavity under certain conditions, something impossible in free-running lasers. We conclude that ring cavity models are sensible models for describing qualitatively Fabry-Pérot experimental configurations.
Explicit models are derived for good cavity one- and two-photon lasers with an injected signal in a Fabry-Pérot cavity. The steady solutions and their stability properties are obtained analytically and compared with the corresponding ring cavity model ones. Only quantitative differences between both types of cavities are found. In particular we show that (i) the Fabry-Pérot cavity reduces significantly the domain of self-pulsing with respect to the ring cavity, and for the two-photon laser case (ii) larger output can be extracted from a Fabry-Pérot cavity than from a ring cavity under certain conditions, something impossible in free-running lasers. We conclude that ring cavity models are sensible models for describing qualitatively Fabry-Pérot experimental configurations.
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 Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.
By means of the semiclassical method, we discuss the bistability in a two-photon laser with injected signal for the homogeneously broadened single mode ring cavity. It is demonstrated that in the resonant case one can find bistable states in a two-photon laser with injected signal. In the limit of "good cavity" and "bad cavity", we give a simple description of bistability leading to a predictions for the transient behavior of the output light and the hysteresis cycle by adiabatically eliminating the fast variables. It is found that the approach to the stationary state is monotonic in the low-branch and is oscillatory in the high-branch, and a critical slowing down occurs where the output field changes discontinuousiy.
In this paper we analyze the steady-state and stability properties of bistable optical systems, lasers with an injected signal, and ordinary free-running lasers. The starting point of our study is a unified model of a ring cavity containing a finite-size cylindrical sample of homogeneously broadened two-level atoms, and capable of supporting a single field mode with a Gaussian transverse profile. After solving the equations of motion in the steady state, we carry out a linear stability analysis of the stationary solutions, identify the domains of unstable operation of each of the three systems, and compare in detail the results of this work with those of earlier plane-wave uniform-field calculations. In the case of the laser with an injected signal, we find a significant enhancement of the instability domain relative to the plane-wave limit. A similar conclusion holds for mixed absorptive and dispersive optical bistability, although the enhancement of the domain of instability is less pronounced. Unstable behavior is predicted to occur for experimentally accessible values of the bistability parameter, and to be favored by the absence of bistability and by the selection of atomic and cavity detunings having opposite signs. This configuration ensures that the instability will be responsible for the emergence of undamped output pulsations of the type that will make the bistable system behave as an optical clock.
With the exception of Chapter 1, up to this point we have considered cavityless optical systems. With this chapter, instead, we start focussing on the case that the atomic medium is located in an optical cavity. We describe in detail the case of a unidirectional ring resonator, the boundary condition which characterizes it and the relations between its output fields and the intracavity field (Sections 8.1–8.3). We discuss the transmission of an empty cavity (Section 8.4) and of a cavity that contains a linear medium (Section 8.5). Optical cavities An optical cavity or optical resonator is formed by at least two mirrors, appropriately arranged in such a way that the radiation beam follows a closed path. The beam splitters allow the radiation to escape from the cavity (output) and allow one to inject radiation into the cavity (input). In order to confine the radiation transversally with respect to the direction of propagation, the mirrors are usually spherical (see Fig. 1.2). The case of cavities with spherical mirrors will be described in Part III of this volume. On the other hand, an analytical treatment of the problems is possible only within the plane-wave approximation which we will use in the remainder of Part I and in Part II. In accord with this, we assume here that the mirrors are planar. The two most common types of optical resonator are the Fabry–Perot cavity and the ring cavity . The planar Fabry–Perot cavity consists in two beam splitters in a parallel configuration as shown in Fig. 8.1. Within the cavity there are two counterpropagating fields, E F propagating in the forward direction and E B propagating in the backward direction with respect to the input field E I . Because we have derived the Maxwell–Bloch equations for the unidirectional configuration, in this and the following chapters we focus on the case of a ring cavity, which allows unidirectionality. The Fabry–Perot case will be considered in Chapter 14.
Abstract : A semiclassical analysis of injected lasers with stable and unstable resonators and arbitrary outcoupling fractions was performed. Homogeneously broadened lasers with Fabry-Perot and with positive branch, confocal unstable resonators were modeled. Both time-dependent and steady-state analyses of free-running and injected lasers were performed. Because of deficiencies in the standing wave semiclassical and the simple saturable gain models, a new semiclassical theory of homogeneously broadened lasers were developed. The theory is predicted upon the following assumptions: the atoms are two-level, a single longitudinal mode exists in the cavity, the transverse mode structure can be adequately modeled by the geometric optics mode, and the laser electric fields can be described by a pair of counterpropagating planar, cylindrical, or spherical traveling waves with different amplitudes. A general semiclassical theory was first developed, and then specialized to the Fabry-Perot and positive branch, confocal unstable resonator cases. Both the Maxwell-Bloch and rate equation approximation forms of the theory were derived. The general theory is sufficiently broad to allow the modeling of the geometric optics mode of any standing wave resonator configuration. The theory was shown to reduce to the simple saturable gain and standing wave semiclassical models in the appropriate limits. The analysis of the injected lasers were performed with the new semiclassical theory. Both time-dependent and steady-state analyses were performed.