It is critical to understand the nature of the propagation medium in terms of the available resonances which couple to the electromagnetic field. The detailed specification of the complex index of refraction as a function of frequency is the subject of spectroscopy. Since propagation media cover all phases of matter, the spectroscopy of gases, solids, and liquids is introduced in this chapter. Topics are not exhaustively covered, but rather with sufficient depth to support the practical applications that will come later. The chapter begins with a formal review of topics covered in stationary-state quantum mechanics necessary for the development of spectroscopy. The next section develops models to calculate spectral line positions, which are necessary to characterize a spectral line. The full development of spectral line parameters is not only the goal of this chapter but the next two as well. The remaining sections address the essential elements of the spectroscopy of gases, solids and liquids. Both classical and quantum models are used. Time-independent quantum mechanics allows the description of the quantized energy level structure of matter. This is the central topic of this chapter. Time-dependent quantum mechanics is necessary for the development of a theory covering transitions between stationary-state energy levels and is the topic of Chapter 5. The classical concept of light is that of an oscillating continuous wave field, as presented in the first two chapters. This is consistent with the classical electrodynamics based on Maxwell’s equations. However, the work by Planck on blackbody radiation and Einstein on the photoelectric effect showed that optical fields also have a quantized or particle-like nature. Planck, in 1900, suggested that the energy of light, E, be quantized according to . . . E = h f [J] (3.1) . . . where f is frequency (sec−1) and h is Planck’s constant (h = 6.6260755(40) × 10−34 J-sec). This allowed a theoretical description of blackbody radiation for the first time. However, it was Einstein, who, a few years later, made Eq. 3.1 more credible by applying it to explain the photoelectric effect.
It is critical to understand the nature of the propagation medium in terms of the available resonances which couple to the electromagnetic field. The detailed specification of the complex index of refraction as a function of frequency is the subject of spectroscopy. Since propagation media cover all phases of matter, the spectroscopy of gases, solids, and liquids is introduced in this chapter. Topics are not exhaustively covered, but rather with sufficient depth to support the practical applications that will come later. The chapter begins with a formal review of topics covered in stationary-state quantum mechanics necessary for the development of spectroscopy. The next section develops models to calculate spectral line positions, which are necessary to characterize a spectral line. The full development of spectral line parameters is not only the goal of this chapter but the next two as well. The remaining sections address the essential elements of the spectroscopy of gases, solids and liquids. Both classical and quantum models are used. Time-independent quantum mechanics allows the description of the quantized energy level structure of matter. This is the central topic of this chapter. Time-dependent quantum mechanics is necessary for the development of a theory covering transitions between stationary-state energy levels and is the topic of Chapter 5. The classical concept of light is that of an oscillating continuous wave field, as presented in the first two chapters. This is consistent with the classical electrodynamics based on Maxwell’s equations. However, the work by Planck on blackbody radiation and Einstein on the photoelectric effect showed that optical fields also have a quantized or particle-like nature. Planck, in 1900, suggested that the energy of light, E, be quantized according to . . . E = h f [J] (3.1) . . . where f is frequency (sec−1) and h is Planck’s constant (h = 6.6260755(40) × 10−34 J-sec). This allowed a theoretical description of blackbody radiation for the first time. However, it was Einstein, who, a few years later, made Eq. 3.1 more credible by applying it to explain the photoelectric effect.
Thus far, we have developed the properties of the electromagnetic field at optical frequencies, based on Maxwell’s equations. These equations further give a classical macroscopic perspective on the coupling of the propagation media to the field, as presented in Chapter 2. The macroscopic properties of a medium are based on averaged microscopic properties. The microscopic energy structure of matter was presented in Chapter 3, covering gases, solids, and liquids by employing mostly quantum models. We now proceed to the next level of development, the dynamic description of the interaction between the optical field and the propagation medium as a function of the field frequency and propagation media variables (e.g., energy structure, temperature, and pressure). In this chapter, the classical electromagnetic field is coupled to discrete frequency oscillators via Newton’s equation of motion. This approach leads to the popular classical oscillator model, often presented in introductory books on lasers. The classical oscillator model is an incomplete theory and can be only a semiempirical model. In the next chapter, a more detailed and comprehensive approach, which also includes statistical and quantum mechanics, is used leading to robust semiclassical and quantum oscillator models. This chapter and the next are the basis for the applied models presented in Part II of this book. Classical electrodynamics is based on Maxwell’s equations, as given in Chapter 2, and the Lorentz force relation, as given below: . . . F = q[E + (v × B)]. (4.1) . . . These equations cover the classical description of the interaction of light and matter. The first term in Eq. 4.1 represents coupling of the electric field to the medium. As discussed in Chapter 2 (Section 2.2), the leading mechanism for this is the electric dipole moment. To see that this is the coupling mechanism in the first term, consider the potential function driving this force, F = −∇V(r) = −∇(−qr · E). The above expression contains the dipole moment, as defined in Chapter 2. The second term in Eq. 4.1 represents coupling of the magnetic field to the medium.
Although the primarily phenomenological theory of absorption and refraction of light by matter, based on classical models as presented in Chapter 4, is very useful, it is incomplete and often inadequate. A more complete and accurate picture of electrodynamics is given by the theory of quantum optics, and that is the topic of this chapter. The models developed in this chapter are more detailed and therefore more complicated than the phenomenological models of Chapter 4. The most robust models, which are applied in Part II, are presented in this chapter. The quantum models accurately represent experimental data and allow extrapolation and interpolation of such data. Many practical computer based models concerning optical propagation are based on this theory. The theory of elastic scatter as presented in Chapter 4 is consistent with quantum optics and is not presented again. (However, inelastic scatter must address the quantum nature of the scattering medium.) Quantum optics is not completely covered in this chapter. Entire textbooks are devoted to this diverse and comprehensive topic covering optics (see Refs. 5.1–5.3). The emphasis of this book is on absorption and reflection spectroscopy. Now details of internal structure of the medium impacting light–matter interaction are examined. The classical oscillator model is upgraded by semiclassical radiation theory and a quantum oscillator model is developed. Semiclassical radiation theory is based on a quantized medium coupled to a classical field. It is often applied to laser theory, where near-line-center stimulated emission dominates. The quantum oscillator model again utilizes the quantized medium and classical field, but with more attention to detailed balance between absorption and emission. It satisfies causality and the fundamental symmetry relationships established in Chapter 2. These quantum optics models are more complete formalisms and provide solutions to the shortcomings of classical electrodynamics. Of particular interest to propagation in gaseous media is the line shape in the far wing. To achieve long path lengths, propagation near line center of a resonance must be avoided. Line shape models in quantum optics accurately represent much of the frequency and temperature dependence observed in experimental data.
This chapter provides a basis for an insight into the nature of quantum effects. The Newtonian approach cannot explain diffraction phenomena. The wave theory of the nature of light proposed by Huygens was able to account for these phenomena as well as for geometric optics. Planck was not only able to overcome the difficulties encountered by his predecessors, but included their findings as special cases of his analysis. Planck considered that the radiating surface was composed of electric dipole oscillators, and that a relationship should exist between the energy of the radiation and the frequency of the oscillators. According to classical theory a continuous spectrum is expected when matter is excited by an arc, or spark, as in spectroscopy. It is found experimentally that only “lines”, or discrete radiations, occur when the light is diffracted. Bohr’s revolutinary ideas provided a significant portion of the bases for the modern understanding of spectra and, consequently, of atomic structure.
The behavior of light and its interaction with matter is certainly one of the most interesting phenomena of physics. It has puzzled thinkers and scientists in ancient Greece (Archimedes) and through the Middle Ages, at least since the times of Robert Grosseteste (1168–1235), right up to the theories of Galileo Galilei and Isaac Newton. The classical electromagnetic theory of light (see Tab. 13.1) was beautifully established in the second half of the nineteenth century with the Maxwell equations. Quantum optics is nothing other than the quantum theory of light. It finds its roots in Planck's discovery (1900) that the assumption of energy quantization for a harmonic oscillator allowed us to solve the black-body problem (see Subsec. 1.5.1). The successive early steps that led to the quantization of the electromagnetic field include Einstein's explanation of the photoelectric effect with the introduction of the quantum of light (see Subsec. 1.2.1) and the interpretation of atomic spectra (see Subsec. 1.5.4). Quantum optics is usually not covered by conventional textbooks on quantum mechanics. We include its discussion here for two main reasons: first, because it is a fundamental and advanced application of the basic framework of quantum mechanics, and, second, because in the last 30 years quantum optics has played (and is still playing) a major role in the advancement and understanding of quantum mechanics.
In this introductory chapter we focus on the interaction of optical fields with matter because it forms the basis of signal amplification in all optical amplifiers. According to our present understanding, optical fields are made of photons with properties precisely described by the laws of quantum field theory [1]. One consequence of this wave–particle duality is that optical fields can be described, in certain cases, as electromagnetic waves using Maxwell's equations and, in other cases, as a stream of massless particles (photons) such that each photon contains an energy h ν, where h is the Planck constant and ν, is the frequency of the optical field. In the case of monochromatic light, it is easy to relate the number of photons contained in an electromagnetic field to its associated energy density. However, this becomes difficult for optical fields that have broad spectral features, unless full statistical features of the signal are known [2]. Fortunately, in most cases that we deal with, such a detailed knowledge of photon statistics is not necessary or even required [3, 4]. Both the linear and the nonlinear optical studies carried out during the last century have shown us convincingly that a theoretical understanding of experimental observations can be gained just by using wave features of the optical fields if they are intense enough to contain more than a few photons [5]. It is this semiclassical approach that we adopt in this book. In cases where such a description is not adequate, one could supplement the wave picture with a quantum description.
The first objective of this chapter is to briefly review some of the basic ideas about the structure of matter, in particular the concepts of microscopic physics, in order to recall the knowledge gained in previous physics (and chemistry) courses and make it more precise. Our review will be very concise, and most statements will be made without any proof or detailed discussion. A second objective is to give a brief description of some of the crucial stages in the early development of quantum physics. We shall not follow the strict historical order of this development or present the arguments used at the beginning of the last century by the founding fathers of quantum mechanics; rather, we shall stress the concepts which we shall find useful later on. Our last objective is to give an elementary introduction to some of the basic ideas, like those of a quantum particle or energy level, that will reappear throughout this text. We shall base our review on the Bohr theory, which provides a simple, though far from convincing, explanation of how energy levels are quantized and how the spectrum of the hydrogen atom arises. This chapter should be reread later on, once the basic ideas of quantum mechanics have been made explicit and illustrated by examples. From the practical point of view, it is possible to skip the general considerations of Sections 1.1 and 1.2 at the first reading and begin with Section 1.3, returning to those two sections later on as needed.