We present a general formalism for obtaining a coupled set of first-order rate equations for the optical field at a finite number of points in an optical resonator. We compare these local-field equations to an expansion in modes of the individual resonators and point out a shortcoming of the latter technique that has not been recognized before. We consider the special case of a set of two coupled resonators and derive the coupled rate equations for the fields in each of the two cavities with coupling coefficients. The resulting formulas are both simpler than those derived from coupled-mode theory and more accurate in the sense that they give a steady-state response which agrees with that calculated from the composite cavity modes. We show how the use of a single local-field rate equation can carry the same information as a full set of multimode rate equations in some cases, in particular, that of partial reflection back into a laser from a distant source.
We present a general formalism for obtaining a coupled set of first-order rate equations for the optical field at a finite number of points in an optical resonator. We compare these local-field equations to an expansion in modes of the individual resonators and point out a shortcoming of the latter technique that has not been recognized before. We consider the special case of a set of two coupled resonators and derive the coupled rate equations for the fields in each of the two cavities with coupling coefficients. The resulting formulas are both simpler than those derived from coupled-mode theory and more accurate in the sense that they give a steady-state response which agrees with that calculated from the composite cavity modes. We show how the use of a single local-field rate equation can carry the same information as a full set of multimode rate equations in some cases, in particular, that of partial reflection back into a laser from a distant source.
A dynamical analysis of nonlinear optical resonators is performed in the framework of a single-mode formalism in the plane-wave regime, taking the finite response time of the nonlinear medium into account. We demonstrate that analytically solvable generic equations can be derived for the time evolution of the output intensity, in the two limiting cases where the medium's response time is either very large or very small as compared to the cavity buildup time. The stability picture and the characteristic time scales, important for applications in the field of optical information processing, are investigated analytically. This study reveals that a surprising effect may occur where an increase in the material-response time leads to a decrease of the relaxation time of the nonlinear device. This effect is called the speeding-up phenomenon. We also present a numerical study of the full nonlinear dynamics.
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.
The quantum theory of a one-dimensional optical cavity is developed with emphasis on the presence of output coupling. First the resonant mode is defined and calculated classically. Then the field is decomposed into a sequence of modes by introducing an imaginary boundary at a large distance. The mode functions are proved to be orthogonal with respect to an integration with the dielectric constant as a weighting factor. The Hamiltonian of the field is shown to be equivalent to that of a collection of independent harmonic oscillators, the mass of which is a function of the frequency of oscillation. This equivalent of mass appears on normalizing the mode functions and proves to carry information on the structure of the cavity. Quantization of the field is carried out, and the commutation relation for the electric fields inside and outside the cavity is derived. The commutator is composed of an infinite set of derivatives of $\ensuremath{\delta}$ functions, which discloses the effect of the presence of output coupling and that of the small size of the cavity on the radiation field. Also, the commutator is shown to have some properties common with the classical resonant mode, i.e., the property of exponential decay and its rate.
A theory of the two-wave mixing in a photorefractive medium placed in a ring cavity is developped following the model first derived by Yariv and Kwong [1]: a steady refractive index involving the photorefractive effect is used as a non-linear source-term (PNL) in the Maxwell equations, and the cavity field is expanded on the transverse cavity modes. In this model we take into account the intensity change of both the pump and resonator beams inside the crystal due to the mutual power exchange: we have developed a multimode model which gives in the long-time limit the z-dependance of the fields (z is the optical axis) for a passage in the crystal, using the expansion in terms of the complete set of the cavity transverse modes. [2]
A transmitting window is used to weakly couple two resonant optical cavities. When active medium is placed in one cavity, the system describes a laser coupled to a resonant cavity, while active medium in both cavities describes two coupled lasers. Equations of motion for the fields in the two cavities are found. These equations are solved in the steady state when the cavities are sufficiently closely tuned that the lasers in the two cavities oscillate with the same frequency. In the locked situation, the second cavity tends to stabilize the frequency of operation against changes in length of the first cavity. When there is no active medium in the second cavity, and that cavity becomes very long, the laser acts as if it were looking into free space.
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.