This chapter describes the formation, development, and major subjects of nonlinear optics and photonics. The semiclassical theory treats the light as a classical electromagnetic field governed by Maxwell equations, while the optical medium is a molecular system governed by quantum mechanics; the key parameters for this theory are various-order nonlinear susceptibilities. The quantum theory of radiation treats light as an ensemble of photons and the medium as a molecular system, both of which are described by quantum mechanics, and the key aspect is the concept of the so-called intermediate state represented by a virtual energy state level. The semiclassical theory gives a concise quantitative description of various nonlinear multi-wave mixing effects as well as those nonlinear processes related to the induced refractive-index changes. In contrast, the quantum theory of radiation can provide a conceptual interpretation for almost all known nonlinear optical effects and phenomena.
This chapter describes the formation, development, and major subjects of nonlinear optics and photonics. The semiclassical theory treats the light as a classical electromagnetic field governed by Maxwell equations, while the optical medium is a molecular system governed by quantum mechanics; the key parameters for this theory are various-order nonlinear susceptibilities. The quantum theory of radiation treats light as an ensemble of photons and the medium as a molecular system, both of which are described by quantum mechanics, and the key aspect is the concept of the so-called intermediate state represented by a virtual energy state level. The semiclassical theory gives a concise quantitative description of various nonlinear multi-wave mixing effects as well as those nonlinear processes related to the induced refractive-index changes. In contrast, the quantum theory of radiation can provide a conceptual interpretation for almost all known nonlinear optical effects and phenomena.
Abstract Laser radiation is an intense coherent light field that is characterized by its high monochromatic brightness and high photon degeneracy. Thus, it is capable of generating various stimulated scattering effects and a series of multiphoton excitation-induced opto-physical and opto-chemical effects. There are two parallel theoretical regimes. The first regime is the semiclassical theory that treats light as a classical electromagnetic field described by Maxwell’s equations. The second regime is the quantum electrodynamic theory, which treats light as an assembly of photons obeying Bose–Einstein statistics. Both regimes can be complementarily adopted to describe the interaction of light radiation with matter.
Introduction We have already encountered situations in which light of one frequency falling on an atomic system gives rise to light of different frequencies (cf. Section 15.6). The atom may here be considered to play the role of a (noisy) nonlinear transducer for the incident field. Even stronger nonlinear effects arise when one is dealing with a large number of atoms, or a nonlinear medium . Under these circumstances it is sometimes permissible to ignore the atomic structure and to treat the medium as a continuum, as in Maxwell's electromagnetic theory. The subject of the interaction of the incident field with the nonlinear medium is usually known as nonlinear optics. The subject had its beginnings in an experiment in which a strong beam of red light (wavelength 6943 Å) from a ruby laser was allowed to fall on a quartz crystal, and a faint beam of blue light at a wavelength of 3472 Å, the first harmonic of the red, was produced (Franken, Hill, Peters and Weinreich, 1961). The development of the subject of nonlinear optics into a mature field is due largely to the work of Bloembergen and his collaborators. In the following we shall consider only a few simple illustrative examples of phenomena in nonlinear optics. More topics and more detail can be found in the books by Bloembergen and others (Bloembergen, 1965; Yariv, 1967, Chap. 21; Shen, 1984; Schubert and Wilhelmi, 1986; Butcher and Cotter, 1990; Boyd, 1992).
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.
In this chapter we review a number of quantum features of optical fields associated with propagation of intense light through nonlinear, isotropic media. We begin with the classical description of field propagation, introducing the nonlinear polarization of the medium which enters as the source term into the approximate field equations obtained from the Maxwell equations in the slowly varying amplitude approximation. This establishes the classical background for further quantum considerations. Next, we quantize the field and construct the effective Hamiltonian from which we get equations of motion for the quantum fields. We discuss a number of quantum effects such as photon antibunching, squeezing, the formation of Schrodinger cats and kittens, changes in field polarization due to the quantum nature of the field, as well as the quantum description of the field phase. The characteristic feature of quantum evolution — the periodicity — is strongly affected by dissipation. We give exact analytical formulae describing the quantum evolution of the field, including dissipation. We have collected results illustrating various aspects of quantum evolution and we believe that this review, although far from complete, will be a useful source of information on the subject.
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.
The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.