A spatially discretised Reduced Maxwell Bloch model is investigated, which models nonlinear optical propagation through a thin slab of resonant atomic material placed in a recirculating cavity. The model may be used to determine the stationary states of the cavity field, which may then be subjected to stability analysis. The analysis identifies the structure of the canonical integrable discrete model that approximates the electromagnetic model.
A spatially discretised Reduced Maxwell Bloch model is investigated, which models nonlinear optical propagation through a thin slab of resonant atomic material placed in a recirculating cavity. The model may be used to determine the stationary states of the cavity field, which may then be subjected to stability analysis. The analysis identifies the structure of the canonical integrable discrete model that approximates the electromagnetic model.
We study the Maxwell–Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves 6 + N(N + 1)/2 unknowns, the six-dimensional electromagnetic field plus the N(N + 1)/2 unknowns describing the state of matter, where N is the number of atomic energy levels of the considered material. We consider at once a high frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrödinger-rate model, involving 3 + N unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in ℂ3. It satisfies a Schrödinger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a N-dimensional vector describing the occupation numbers of each atomic level. It satisfies Einstein's rate equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two-dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an ad hoc Ansatz.
The dynamics of soliton pulses for use in nonlinear optical devices is mathematically modelled by Maxwell-Bloch systems of equations for the interaction of light with a uniform distribution of quantum-mechanical atoms. We study the Reduced Maxwell-Bloch (RMB) equations occurring when an ensemble of rotationally symmetric 3-level atoms is assumed. The model applies for on and off-resonance conditions and is completely integrable using Inverse Scattering theory, since it arises as the compatibility condition of a 3 x 3 AKNS-system. Furthermore this integrability remains valid for all timescales of the optical field because only the “one-way wave approximation” is required during the derivation. Solutions are constructed in two ways: 1. Darboux-Bäcklund transforms are applied, generating single soliton pulses of ultrashort (< 1ps) duration, and families of elliptically polarised 2-solitons not possible in lower dimensional problems. 2. A general Inverse Scattering scheme is developed and tested. The Direct Scattering Problem is dealt with first to obtain a complete set of scattering data. Subsequently the Inverse Problem is solved both formally and then in explicit closed form for the special case that the reflection coefficients vanish for real values of the spectral parameter. In this case the main result is a determined system of n linear algebraic equations which yield the n-soliton of our RMB-system. It is confirmed that the 1-solitons found by means of Darboux transform are precisely the same as those given by the full mechanism of Inverse Scattering. Finally we calculate the invariants of the motion for the RMB-equations, and derive an evolution equation giving the variation with propagation distance of the invariant functionals when the original RMB-system is modified by an arbitrary perturbing term. As an application dissipative effects on 1-solitons are considered.
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.
Nonlinear optics is a fairly young science, having taken off with the advent of the laser in 1960. Nonlinear optics (NLO) deals with the interaction of electromagnetic waves and matter in the infrared, visible, and ultraviolet domains. The frontiers of NLO are somewhat blurred, but microwaves and γ-rays are clearly outside its domain. This book is entirely devoted to a study of NLO in a resonant cavity and when the quantum nature of the electromagnetic field is not of prime importance. More precisely, we study those aspects of cavity NLO in which fluctuations in the number of photons and atoms are not relevant. This particular area of optics is dominated by the Maxwell–Bloch equations, which constitute its paradigm in the sense of T. S. Kuhn. The status of the Maxwell–Bloch equations is quite peculiar. From a fundamental viewpoint, they describe the laws of evolution of the first moments of a density operator, which verifies the von Neumann equation. However, to account for the finite lifetime of the atoms and of the field in the necessarily lossy cavity, some legerdemains have to be introduced to obtain the Maxwell–Bloch equations. Stated more explicitly, the von Neumann equation for a large but finite system does not explain irreversibility, whereas the Maxwell–Bloch equations fully include the irreversible decay of the atoms and of the cavity field. This problem is not specifically related to optics but reflects the general failure of statistical mechanics to explain convincingly the irreversible evolution of macroscopic systems.
The linear and nonlinear properties of propagation of an electromagnetic field through a medium may be derived by the effective elimination of the medium from the coupled Maxwell-Bloch equations. In this paper we suggest, on the other hand, the elimination of the field from the Maxwell-Bloch equations as a means to derive a nonlinear master equation for, e.g., an atomic sample in a laser field. This presents a straightforward and physically intuitive derivation of equations for ``nonlinear atom optics,'' obtained recently by other approaches.