Rate equation and population inversion In a material which has only two energy levels, χ ″ is always positive because ρ 11 > ρ 22 for E 1 < E 2 at thermal equilibrium. Prior to the invention of lasers, there was no known method to achieve ρ 22 > ρ 11 . However, we now know that a negative χ ″ in Eq. (5.41a) (i.e. ρ 22 > ρ 11 ) can be achieved by pumping processes that are available in materials that have multiple energy levels, as described in the following. Let there be many energy levels in the material under consideration, as shown in Fig. 6.1. Let there be a mechanism in which the populations at E 1 and E 2 , i.e. N 1 = Nρ 11 and N 2 = Nρ 22 , are increased by pumping from the ground state at pump rates R 1 and R 2 . In solid state lasers, the pumping action may be provided by an intense optical radiation causing stimulated transition between the ground state and other higher energy states, where the particles in the higher energy states relax preferentially into the E 2 state. In gas lasers, the molecules in the ground state may be excited into higher energy states within a plasma discharge; particles in those higher energy states then relax preferentially to the E 2 state. Alternatively, collisions with particles of other gases may be utilized to increase the number of particles in the E 2 state. Various schemes to pump different lasers are reviewed in ref. In order to obtain amplification, it is necessary to have R 2 ≫ R 1 .
Rate equation and population inversion In a material which has only two energy levels, χ ″ is always positive because ρ 11 > ρ 22 for E 1 < E 2 at thermal equilibrium. Prior to the invention of lasers, there was no known method to achieve ρ 22 > ρ 11 . However, we now know that a negative χ ″ in Eq. (5.41a) (i.e. ρ 22 > ρ 11 ) can be achieved by pumping processes that are available in materials that have multiple energy levels, as described in the following. Let there be many energy levels in the material under consideration, as shown in Fig. 6.1. Let there be a mechanism in which the populations at E 1 and E 2 , i.e. N 1 = Nρ 11 and N 2 = Nρ 22 , are increased by pumping from the ground state at pump rates R 1 and R 2 . In solid state lasers, the pumping action may be provided by an intense optical radiation causing stimulated transition between the ground state and other higher energy states, where the particles in the higher energy states relax preferentially into the E 2 state. In gas lasers, the molecules in the ground state may be excited into higher energy states within a plasma discharge; particles in those higher energy states then relax preferentially to the E 2 state. Alternatively, collisions with particles of other gases may be utilized to increase the number of particles in the E 2 state. Various schemes to pump different lasers are reviewed in ref. In order to obtain amplification, it is necessary to have R 2 ≫ R 1 .
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.
Introduction We were able to show in Chapter 3 that a medium in which we can obtain a population inversion (i.e. a situation in which the population density in the excited state is greater than that in the fundamental level) allows for optical gain of an electromagnetic wave having a frequency near to the resonant frequency of the system. By introducing feedback of the amplified signal into the medium, the system can be made to oscillate naturally, resulting in laser oscillations . To obtain this population inversion, we must introduce at least a third (and perhaps even a fourth) energy level into the system. (We saw how a two-level system under the influence of an intense pump beam will saturate with no resulting population inversion.) The aim then of this chapter is to introduce the concepts necessary to extend our two-level system into a working model capable of illustrating the phenomenon of laser oscillation. We will not spend too much time discussing atomic transition lasers as they do not figure readily in our treatment of quantum electronic properties of semiconductors. An exception will be made, however; we brush upon the particular topics of a diode pumped laser in Complement 4.E and a quantum cascade laser in Complement 13.H. Population inversion and optical amplification Population inversion We will show how population inversion can be achieved by carrier transfer from higher lying levels to the upper level of a two-level subsystem of interest.
Introduction When an electromagnetic wave propagates through amedium stimulated emissions increase the intensity of the wave, while absorptions diminish it. The overall intensity will increase if the number of stimulated emissions can be made larger than the number of absorptions. If we can create such a situation then we have built an amplifier that operates through the mechanism of stimulated emission. This laser amplifier , in common with electronic amplifiers, has useful gain only over a particular frequency bandwidth. Its operating frequency range will be determined by the lineshape of the transition, and we expect the frequency width of its useful operating range to be of the same order as the width of the lineshape. It is very important to consider how this frequencywidth is related to the various mechanisms by which transitions between different energy states of a particle are smeared out over a range of frequencies. This line broadening affects in a fundamental way not only the frequency bandwidth of the amplifier, but also its gain. A laser amplifier can be turned into an oscillator by supplying an appropriate amount of positive feedback. The level of oscillation will stabilize because the amplifier saturates. Laser amplifiers fall into two categories, which saturate in different ways. A homogeneously broadened amplifier consists of a number of amplifying particles that are essentially equivalent, whereas an inhomogeneously broadened amplifier contains amplifying particles with a distribution of amplification characteristics. Homogeneous line broadening All energy states, except the lowest energy state of a particle (the ground state) cover a range of possible energies.
A new version of obtaining short-wave lasing on transition between highly excited states and the ground state of active atoms in the buffer gas atmosphere is studied theoretically. The mechanism of obtaining population inversion on such a transition is associated with the establishment of the local Boltzmann distribution of populations in a group of highly excited levels due to frequent collisions. If the excitation of the upper-lying level is performed by two laser radiation sources with frequencies ω 1 and ω 2 , short-wave lasing can be obtained at a frequency close to the total frequency ω 1 + ω 2 . The conditions for the emergence of population inversion are analyzed and simple analytic formulas are derived. It is shown that collision-induced enhancement of short-wave radiation can occur for pumping intensities on the order of 100 W/cm 2 . For pumping intensities on the order of 1000 W/cm 2 , the amplification factor for short-wave radiation may attain values of 3 cm −1 (for an active atom concentration of N ∼ 10 15 cm −3 ), which is sufficient for the development of lasing per path through an active medium (superradiance condition) for a length of the active medium on the order of 10 cm.
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.
All optical oscillators and amplifiers can be analyzed as an electromagnetic structure, such as a cavity or a transmission line, which contains an amplifying medium. The operating characteristics of lasers are governed by both the electromagnetic properties of the structures and the properties of the amplifying medium. Stimulated emission and absorption of radiation from energy states are the physical basis of amplification in all laser materials. In order to study lasers, it is necessary to understand the energy levels of the amplifying medium and the stimulated transitions involving them. Quantum mechanical analysis of the energy states and the stimulated emission and absorption is presented in most books on lasers. However, quantum mechanical analyses cover only the analysis of individual atoms. We need to relate the effects of the quantum mechanical interactions to the macroscopic properties of the materials so that properties of the lasers may be analyzed. Therefore, the macroscopic susceptibility of materials related to stimulated emission and absorption is the focus of discussion in this chapter. It is not the purpose of this book to teach quantum mechanics. There are already many excellent books on this topic. The readers are assumed to have a fundamental knowledge of quantum mechanics. However, it is necessary first to review some of the major steps in order to understand precisely the notation used and the meaning of the results. This is presented in Section 5.1.