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Journal of Nonlinear Optical Physics & Materials

An Improved Laser Model based on the Nonlinear Coupling of Lightfield with Boson Elementary Excitation

Minggao He, Fa Ou

Published Dec 1, 1998• 1 citation
Publisher

Abstract

The dynamics of the laser model based on the nonlinear coupling of light-field with boson elementary excitation is improved. A rate equation of pumping system directly coupled with the boson matter is added to the original equations. The system as a whole, therefore, becomes a perfect self-organized one, but the character of mirrorless unchanges, which is a character essentially different from conventional lasers, and the laser threshold behavior of second-order-like transition also remains. There is some resemblance between the new revised laser model and Maxwell–Bloch two-level-atom laser model.

DOI
10.1142/s021886359800034x

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The dynamics of the laser model based on the nonlinear coupling of light-field with boson elementary excitation is improved. A rate equation of pumping system directly coupled with the boson matter is added to the original equations. The system as a whole, therefore, becomes a perfect self-organized one, but the character of mirrorless unchanges, which is a character essentially different from conventional lasers, and the laser threshold behavior of second-order-like transition also remains. There is some resemblance between the new revised laser model and Maxwell–Bloch two-level-atom laser model.
We study the lasing action due to nonlinear interactions of photons with boson elementary excitation, such as phonons and excitons, and point out the distinct features from the conventional laser model described by the two-level model. External feedback by means of cavity mirrors is unnecessary for the lasing action as well as for the increasing absorption optical bistability. It is shown that the Landau theory of the second-order phase transition is applicable to the threshold behaviour of the present mirrorless laser 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.
Bloch equations for the atomic population and the polarization/coherence and the equation of motion for the photon number in a laser are solved in steady state as a function of the pump rate. Two level atom and two modes of three levels atom are investigated. Close to threshold the usual linear dependence of the intensity on the pump rate in found for all cases. However, far above threshold strongly nonlinear dependence is encountered. In the cases for which the pump connects the lower lasing state to one of the excited states the character of the non-linearity differs crucially from the cases when the pump in not related directly to the lower lasing state. Non-monotonic dependence of laser intensity upon the pump rate is predicted. Detailed discussion of the nonlinear behavior is presented, including saturation and depletion effects.
Summary form only given.The behavior of a laser intensity as a function of a pump rate is studied. We consider a three-level lasing scheme with two different types of pumping. In the first case, the lower lasing level is depleted to the ground state, and then the pumping is used to excite the atom from the ground state to the upper lasing level. In the second case the pumping is used to excite the atom directly from the lower lasing level to one of the excited states, which is then depleted to the upper lasing level. We refer these two types of pumping as type 1 and type 2, respectively. It is shown that although these two schemes are equivalent mathematically, i.e., they are formally described by the same set of equations, physically their dynamical properties are completely different. It is noted that the pumping of the type 1 does not affect either of lasing levels directly, whereas the pumping of the type 2 does connect the lower lasing level with one of the excited states. It is this difference of the two ways of pumping which is responsible for the crucial difference in the dynamic behavior of the lasing schemes of types 1 and 2.
Abstract In this chapter we study with the tools developed in Chapter 3 the basic models that are the foundations of light–matter interaction. We start with Rabi dynamics, then consider the optical Bloch equations that add phenomenologically the lifetime of the populations. As decay and pumping are often important, we cover the Lindblad form, a correct, simple and powerful way to describe various dissipation mechanisms. Then we go to a full quantum picture, quantizing also the optical field. We first investigate the simpler coupling of bosons and then culminate with the Jaynes–Cummings model and its solution to the quantum interaction of a two-level system with a cavity mode. Finally, we investigate a broader family of models where the material excitation operators differ from the ideal limits of a Bose and a Fermi field.
A detectable nonlinear response between light and matter depends on one of two key ingredients: Either there is a resonance between the light wave and some natural oscillation mode of the material, or the light is sufficiently intense. In this chapter, the authors let the matter variables themselves play active roles and derive coupled nonlinear differential equations for the electric and matter fields. They outline the use of Schrodinger's equation to derive the Bloch equations, which describe the dynamics of the oscillators of matter and help to derive the Maxwell equation for the envelope of an almost monochromatic electromagnetic wave. The authors analyse the case of two distinct co- or counter-propagating waves interacting with a three-level medium where both waves excite separate quantum transitions sharing a common level. They also help to derive an effective two-level atom model that describes two-photon absorption and stimulated Raman scattering.