The adiabatic elimination principle [91, 92] is universal in the area of nonlinear dissipative dynamical systems, and its importance arises from the fact that it allows one to reduce the number of the equations which govern the dynamics of the system. We introduce it at this stage in the book in order to be able to describe the optical pumping mechanisms which can be exerted to attain population inversion. However, this principle will be applied repeatedly in the following chapters for other purposes. In the first subsection we describe the principle in general, while the following subsections concern the issue of optical pumping. We show first that it is not possible to obtain population inversion between the two levels of the lasing transition if the pump involves only these two levels. Next we illustrate the three-level pumping scheme and the four-level pumping scheme. We demonstrate that, by adiabatically eliminating the probabilities of the additional energy levels, one arrives at an equation identical to Eq. (4.12), which, for appropriate ranges of values of the parameters involved in the pumping, allows one to obtain population inversion. General formulation of the principle In a dissipative dynamical system the dynamical variables undergo relaxation processes associated with suitable rate constants such as, for example, the rate γ ⊥ for the normalized atomic polarization P and the rate γ ∥ for the normalized population difference D . The inverses of these rates provide the time scales which characterize the evolution of each variable. Therefore we can distinguish a set of fast variables , which evolve over short time scales and are therefore characterized by large relaxation rates, and a set of slow variables , which display small relaxation rates. A remarkable simplification in the analysis of dynamical systems is obtained by using the technique of adiabatic elimination of fast variables, which Haken considered as a basis for the discipline which he formulated and called Synergetics [91, 92].
The adiabatic elimination principle [91, 92] is universal in the area of nonlinear dissipative dynamical systems, and its importance arises from the fact that it allows one to reduce the number of the equations which govern the dynamics of the system. We introduce it at this stage in the book in order to be able to describe the optical pumping mechanisms which can be exerted to attain population inversion. However, this principle will be applied repeatedly in the following chapters for other purposes. In the first subsection we describe the principle in general, while the following subsections concern the issue of optical pumping. We show first that it is not possible to obtain population inversion between the two levels of the lasing transition if the pump involves only these two levels. Next we illustrate the three-level pumping scheme and the four-level pumping scheme. We demonstrate that, by adiabatically eliminating the probabilities of the additional energy levels, one arrives at an equation identical to Eq. (4.12), which, for appropriate ranges of values of the parameters involved in the pumping, allows one to obtain population inversion. General formulation of the principle In a dissipative dynamical system the dynamical variables undergo relaxation processes associated with suitable rate constants such as, for example, the rate γ ⊥ for the normalized atomic polarization P and the rate γ ∥ for the normalized population difference D . The inverses of these rates provide the time scales which characterize the evolution of each variable. Therefore we can distinguish a set of fast variables , which evolve over short time scales and are therefore characterized by large relaxation rates, and a set of slow variables , which display small relaxation rates. A remarkable simplification in the analysis of dynamical systems is obtained by using the technique of adiabatic elimination of fast variables, which Haken considered as a basis for the discipline which he formulated and called Synergetics [91, 92].
We reconsider the problem of the adiabatic elimination of selected dynamical variables in the description of nonlinear systems, with a special emphasis on the identification of suitable criteria for the global validity of this procedure. The problem is analyzed in detail using as a guideline the one-mode homogeneously broadened laser model, with an injected signal and an arbitrary population difference for added flexibility. We propose five conditions for the global validity of the adiabatic limit, after consideration not only of the relative size of the time scales involved, but also of the magnitude of all parameters, of the physical variables, and of their fluctuations. The adiabatic elimination is formulated in the context of the dressed-mode description, leading to a generalization of the procedure adopted in earlier studies of multimode absorptive bistability. In addition, we establish a rigorous link between the adiabatic-elimination procedure and the multiple-time-scale approach to dynamic systems. The scaling behavior of the variables, of the eigenvalues of the linearized equations, and of the dressed-mode amplitudes and eigenvectors with respect to the smallness parameter of the problem is studied in detail. The relevance of what we shall define as "normal" and "anomalous" scaling in the text to the applicability of the adiabatic-elimination process is clarified. We also develop an alternative adiabatic-elimination scheme for the special case where all the rate constants of the system have the same order of magnitude, but another smallness parameter can be identified. From our analysis, it should be clear that our main conclusions are model independent, and not at all restricted to the specific features of the dynamical system selected as a test case for our discussion.
In Chapter 1, we introduced the standard rate equations (SRE) for a laser containing two-level atoms between which stimulated emission is possible. But real lasers exhibit much more complicated energy level schemes. Throughout this chapter, we consider several models in which the light–matter interaction is described by population equations for all the levels involved in the laser operation. Although they are mathematically more complicated, we shall investigate these equations in the same way as in Chapter 1, i.e. by formulating dimensionless equations and by analyzing the stability properties of the steady states. The basic ingredients of the SRE model used up to here, namely pumping and relaxation processes, play key roles in the efficiency of lasers. But already during the pioneering days of the laser, it appeared important to investigate three- and four-level models in order to obtain more reliable information on quantities such as the power conversion efficiency or the response time. The common extensions of the two-level SRE typically consider an open two-level system, or three- or four-level systems, depending on the nature of the active medium. Restriction to as few as four levels is again a crude simplification of the complex population dynamics occurring in most lasers. But it is rather surprising to see how good these simple kinetic models are. As we demonstrate by studying specific examples, the solution of the three- or four-level rate equations differs only slightly from the solution of the SRE.
The relaxation rates κ , γ ⊥ and γ ∥ which appear in the single-mode model are irrelevant for the stationary state. Instead, their values become critical when we consider dynamical, i.e. time-dependent, phenomena. In particular, they control the possibility of eliminating adiabatically some of the variables in play, thus reducing the complexity of the problem. In this chapter we first illustrate, in Section 19.1, the useful laser classification introduced by Arecchi. Next, in Section 19.2, we discuss the dynamical behavior of the solutions of a model obtained from the single-mode model (derived in Chapter 12) by performing the adiabatic elimination of the atomic variables, which procedure is called the good-cavity limit in the literature. Section 19.2.1 is devoted to the case of the free-running laser ( class-A laser ) and discusses in particular, at the end, the cubic single-mode laser model which is the simplest model capable of describing a laser. On the other hand, in Section 19.2.2 we discuss the case of optical bistability in the good-cavity limit. If we eliminate adiabatically the atomic polarization only, we obtain, instead, the single- mode rate equations which are discussed in Section 19.3. In particular, Section 19.3.1 concerns the case of class-B lasers and focusses especially on the phenomenon of relaxation oscillations. Special attention is paid to demonstrating the equivalence of the model which describes class-B lasers to the rate-equation laser model derived in Chapter 1 and the semiconductor laser model obtained in Chapter 16. In Section 19.3.2 we describe the generation of a giant pulse in class-B lasers, a phenomenon which is called active Q- switching . The final section, Section 19.4, deals with the model obtained by adiabatically eliminating the field variable in the so-called bad-cavity limit . Classification of lasers Arecchi [243] identifies three especially important kinds of laser, which he calls class-A, class-B and class-C lasers, respectively. (1) Class-A lasers are characterized by the condition κ ≪ γ ⊥ , γ ∥ . In this case the atomic variables can be adiabatically eliminated. The He–Ne laser, argon laser and quantum-cascade laser belong to this class.
POPULATION RATE EQUATIONS From the discussion in the preceding chapter, it is clear that population inversion is the basic condition for an optical gain. For any system in its normal state in thermal equilibrium, a low-energy level is always more populated than a high-energy level, hence there is no population inversion. Population inversion in a system can only be accomplished through a process called pumping by actively exciting the atoms in a low-energy level to a high-energy level. If left alone, the atoms in a system relax to thermal equilibrium. Therefore, population inversion is a nonequilibrium state that cannot be sustained without active pumping. To keep a constant optical gain, continuous pumping is required to maintain population inversion. This condition is clearly consistent with the law of conservation of energy: amplification of an optical wave leads to an increase in optical energy, which is possible only if the required energy is supplied by a source. Pumping is the process that supplies energy to the gain medium for the amplification of an optical wave. There are many different pumping techniques, including optical excitation, electric current injection, electric discharge, chemical reaction, and excitation with particle beams. The use of a specific pumping technique depends on the properties of the gain medium being pumped. The lasers and optical amplifiers of particular interest in photonic systems are made of either dielectric solid-state media doped with active ions, such as Nd:YAG and Er:glass fiber, or direct-gap semiconductors, such as GaAs and InP. For a dielectric gain medium, the most commonly used pumping technique is optical pumping using either an incoherent light source, such as a flashlamp or a light-emitting diode, or a coherent light source from another laser. A semiconductor gain medium can also be optically pumped, but it is usually pumped by electric current injection. In this section, we consider the general conditions for pumping to achieve population inversion. Detailed pumping mechanisms and physical setups are not addressed here because they depend on the specific gain medium used in a particular application. The net rate of increase of the population density in a given energy level is described by a rate equation. As we shall see below, pumping for population inversion in any practical gain medium always requires the participation of more than two energy levels.
During six decades of quantum electronics, a vast majority of new types of quantum generators have been developed. Although the principle of population inversion has united different ranges of electromagnetic spectra (and respective quantum generators), the existence of the title laser without the population inversion, makes that the exception had confirmed the rule, i. e. that this title deserves to be discussed further. Developing of formalisms describing the operation of quantum generators, by now have produced several approaches, which must have a quantum mechanics base. For the practical reasons, negative coefficient of absorption is acquired using classic electromagnetics as well, however for the population purposes, quantum representation must be entered. A few levels of formalisms will be set in this paper, linked to quantum generators accenting the optical portion of the spectra. The lowest level descriptions are based on lumped circuits. This could be expanded to equivalents of other physical problems, using program packages developed for the electrical engineering application purposes (Spice, etc.). Schematics are defined at the macro as well as micro equivalent levels (atomic – electronic levels). The kinetic equations with simpler approach will be considered as well as simplified laser equations based on quantum/ semi-quantum approach. The use of Fourier analysis or other appropriate transformations leads to formulating the main five laser equations which serve as the base for various working regimes of quantum generators and amplifiers (free generation regime, Q switch, synchronization, operation with filters, two modes regime, regime with losses, etc.). The Lyapunov stability theorem has to be included here, etc. For some of the chosen types of quantum generator, analytical modeling will be analyzed as well as the results of program packages developed for the lasers dynamics, regimes and parameters. The systems pumped with electronic beams (relativistic) will be considered and the nuclear physics statements discussed which must be included at the beginning, in order to consider further necessary parts of the condensed – solid state theory and laser techniques, after slowing down towards thermal energies. Existing program packages provide fast modeling and visualization of laser energy distribution, temperature, modes, etc. in active material with or without the resonator. A modeling will be performed for the specified geometries and a temperature distribution in active material will be captured during operation of a chosen laser system. Different pump geometries will be compared. Contemporary lasers with the shortest existing pulse durations demand new formalisms. Areas of nonlinear optics and quantum electrody-namics, Glauber states and similar, are areas that have to be included. Two main formalisms thermodynamical and quantum mechanical with transition probabilities using perturbation methods and secondary quantization naturally had to be complemented if the Brillouin, Raman, Compton, soliton, fiber and other lasers are included more generally.
This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.