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Cambridge University Press eBooks

Optical Amplification

Jia-Ming Liu

Published Aug 5, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9781316687109.009

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•Jan 1, 1992•1 citation
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.
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 .
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.
The most commonly used lasers operate with population inversion. This requires an incoherent pumping power (from lamps, electrical discharges, etc.) that rapidly scales with the frequency of the generated laser radiation. In practice, this is the main obstacle for the operation in the continuous-wave regime of short-wavelength lasers in the UV, X-rays, etc. But lasers can also operate without population inversion using atomic coherences excited in the laser medium by coherent pumping with another laser. We present in this paper a short review of the field of lasing without inversion (LWI). In particular, we summarize the basic physics of LWI in the framework of both semiclassical theory and quantumjump formalism to describe light-matter interaction. We also review the experimental state-of-the-art in LWI. Most proof-of-principle demonstrations of LWI have used atomic vapours, so far. But recently gain without inversion (a pre-requisite for lasing) has been observed also in a technologically relevant medium: semiconductor nanostructures that can be tailored to have “artificial atom” electron states.
Introduction Usage has reserved the expression optical bistability mostly for coherently driven passive systems. An atomic system is called passive when there is no population inversion. This is the case, for example, of a system at thermal equilibrium. On the contrary, the laser that we have described in Chapter 1 is a driven active system, because a population inversion is created. The laser equations we have studied so far describe an incoherently driven laser. However, nothing prevents driving a lasing cavity with a coherent field emitted by another laser. This is realized in a whole class of lasers that are used mainly as frequency converters. In optical bistability (OB), the driving field is usually a coherent field. Thus we have to account for two differences between the laser and the optically bistable system: (1) In OB, there is no inversion of population: 〈| A 2 |〉>〈| B 2 |〉, in the notation of Chapter 1. Hence, in the absence of interaction with a field, the population difference D relaxes toward a negative value. (2) The pumping is coherent, meaning that an external laser field is added to the cavity field. Using the single-mode equations (1.48)–(1.50), we have to change the sign of D and D a (which amounts to keeping D defined in (1.34) as it is but changing P into – P and A into – A ) and to add a source term in the equation for the complex field amplitude.
This thesis is concerned with the amplification of coherent radiation from an active medium under conditions which require no population inversion between the energy levels. Gain without population inversion may be obtained by creating a sufficiently strong correlation between pairs of atomic states, usually with the help of external sources of resonant electromagnetic radiation. Our main objective is to investigate two possible selections for the active medium: a three level atom in a lambda configuration and a Raman-driven four level model. For both we identify the conditions that lead to inversionless laser action, we calculate the small signal gain and investigate the physical mechanisms which are responsible for the emergence of gain. While the ultimate cause of the gain is quantum interference, we propose attractive alternative pictures based on the dressed states of the coupled atom-driving field systems. We address the important practical issue of the effect of atomic motion and related Doppler broadening on the inversionless gain profile and we also analyze the transient response of each laser model with the help of the appropriate Maxwell-Bloch equations. Finally we develop a fully quantum mechanical description of the lambda model and find that the radiated field can display unique quantum features which manifest themselves with sub-Poissonian photon fluctuations.