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.
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.
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.
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 the previous chapters we considered the Maxwell–Bloch equations with the boundary condition appropriate for a unidirectional ring cavity and calculated the stationary solutions for laser and optical bistability, respectively. Next, we showed that the exact stationary solutions simplify appreciably in the low-transmission limit. In this chapter we want, first of all, to apply the low-transmission approximation on the dynamical equations directly, beyond the stationary case. We start from the Maxwell–Bloch equations (4.35)–(4.37) and the boundary condition (8.36). The latter is complex and includes the retardation Δ t and some parameters such as the transmissivity and reflectivity coefficients T and R , the input field y and the cavity detuning δ 0 . We introduce a transformation of the coordinates z and t and a transformation of the variables F and P [121–123]. As a result, the boundary condition reduces to a simple periodic boundary condition, and the electric-field equation incorporates the parameters which previously appeared in the boundary condition. The virtue of the periodic boundary condition is that it allows one to expand the electric field in terms of modal amplitudes and to recast the field equation in the form of a set of time-evolution equations for the modal amplitudes. All of this is shown in Section 12.1. In Section 12.2 we apply the low-transmission approximation and the Maxwell–Bloch equations assume a form that is a generalization of a set of equations commonly used in the literature to describe the ring laser [124]. Together with the Bloch equations for the atomic variables, the modal equations provide a model that lends itself ideally to a numerical resolution of the equations. In Section 12.3 we show that, if we assume that only the resonant cavity mode is active, this model reduces to a simple single-mode model consisting in three coupled ordinary differential equations, which will be analyzed in Part II of this book. Next, in the same section we derive the single-mode model in a more direct way, without passing through the multimodal equations.
A formal treatment of a traveling-wave optical maser is presented in which an assumed electromagnetic field in a rotating cavity, obeying Maxwell's equations, nonlinearly polarizes the moving gaseous atoms. The interaction is treated quantum mechanically in the frame of the moving atom. The resultant polarization, statistically summed over all velocity ensembles, is used as a source term in Maxwell's equations. The self-consistency gives a set of equations which determine the amplitudes and frequencies of oscillation of the modes of the independent oppositely directed traveling waves in terms of the parameters of the system. The results reduce to those obtained by Lamb for the case of a standing-wave optical maser with a stationary cavity. In addition, stability conditions on the oppositely directed waves are obtained for the cases where the active medium is a single isotope and a mixture of two isotopes.
In this work a theoretical model treating the optical bistability in a two state dissipative atomic medium inside a Fabry-Perot cavity is presented.The model is based on a semiclassical point of view to explain the field-matter interaction.The atoms of the medium are described by the "quantum polarization equation of motion" while the field is described by the "classical Maxwell's field equations".Applying the boundary conditions to the fields inside the Fabry-Perot cavity, the occurrence of optical bistability is deduced between the incident field intensity and both the intracavity and transmitted field intensities.Also the effect of building standing and nonstanding waves inside the cavity on bistability is discussed.
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.