This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω 0 equal to the experimentally determined absorption frequency. Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave. The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.
This first complement is devoted to a completely classical approach of lightmatter interaction which was proposed by Lorentz at the end of the nineteenth century, before the advent of quantum mechanics, but after the discovery of the electron. Lorentz' phenomenological model is based on the experimental fact that atoms have well-defined and sharp absorption lines: he assumed that atoms behaved like harmonic oscillators, in which the electrons are bound to the atomic nucleus by a restoring force which varies linearly with its displacement (from its equilibrium point close to a nucleus), and makes them oscillate at a given frequency ω 0 equal to the experimentally determined absorption frequency. Within the frame of this model, we first calculate the electromagnetic field radiated by an oscillating electron. We show that in the absence of an externally applied force the free oscillations of the electron are damped, because electromagnetic energy is radiated at the expense of mechanical energy. We then study the characteristics of the radiation that is emitted when the oscillations are forced by the application of an external oscillatory electromagnetic field of angular frequency ω. We characterize the different regimes of this scattering of the incident electromagnetic wave and finally determine the polarization induced in the atomic medium by the incident electromagnetic wave. The Lorentz model can be considered as a lowest order approximation to a description of the light–matter interaction, a better approximation being the semi-classical treatment presented in Chapter 2, and the rigorous treatment being the completely quantum mechanical model presented in Chapter 6.
We investigate in this chapter the general problem of the interaction of an atom (or a molecule) and a classical electromagnetic field. Its importance derives firstly from the fact that a large part of our knowledge of atoms is obtained from a study of the radiation they absorb or emit (we shall consider equally visible radiation and radio-frequency and X-rays) and, secondly, that the interaction with matter modifies the propagation of the electromagnetic field itself, notably through absorption, refraction or scattering. The interactions of atoms with light therefore encompass a vast range of physical effects that we could not hope to cover adequately in a single chapter. In this chapter we shall therefore present the fundamental features of the interaction of an atom, which will be treated quantum mechanically, with a classical electromagnetic field, that is an electromagnetic field described by real electric and magnetic vectors obeying Maxwell's equations. A rigorous description of the atom–light interaction would have to take into account the quantum nature of light. This we shall leave to Chapter 6. It turns out, nevertheless, that many important results can be obtained from the semi-classical viewpoint adopted here (although it is perhaps more appropriate to refer to this treatment as ‘semi-quantized’).
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.
Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.
Chapter 12 introduces light–matter interaction at the atomic scale and at interaction energies less than, or comparable to, the chemical bond. The classical electrodynamics treatment for fields is retained, but a very simple quantum atomic structure with electric dipole transitions among atomic and molecular internal states is introduced. The correspondence between classical and quantal electron orbital angular momentum states in atoms is developed in detail for the hydrogen atom, and the difference between quantised orbital states and real, space-directed orbitals is explained. A wave mechanical quantum representation brings out the analogies between classical light waves, quantum matter waves, classical dipole radiation, and atomic radiative emission. The intrinsic electron spin, that has no classical analogue, is introduced and discussed. The chapter ends with a discussion of the rigid rotor, harmonic oscillator model to interpret rotational and vibrational diatomic molecular spectra. Finally rovibronic states complete the picture with the introduction of diatomic molecular electronic transitions.
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 presents the electron oscillator model of light-matter interaction. It models atomic response to the fields in an electromagnetic wave after a classical driven charged (dipole) oscillator. It is used to calculate atomic polarization and dielectric susceptibility and derive an equation of motion for atomic polarization for a collection of atoms. By replacing the oscillator number density in this equation with the level population density difference and interpreting oscillator frequency with transition frequency and decay rate with spontaneous emission rate a quantum mechanically correct equation for atomic polarization is obtained. Poynting theorem on energy exchange between light and atoms in an EM wave then yields quantum mechanically correct rate equations for population densities. These equations together with the equation for atomic polarization provide a complete description of atomic response to light. Modifications to the atomic response due to finite atomic lifetimes, collisions, atomic motion, local variation of resonance frequency and other factors are also discussed in terms of lineshape function and cross-section.