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Introduction

Guang S. He

Published Feb 25, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780192895615.003.0001

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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.
Abstract The quantum electrodynamic theory, based on the quantization of the electromagnetic field, introduces the concept of photon and the related photon creation operator, photon annihilation operator, and photon number operator. It is used to describe stimulated Raman scattering, stimulated Kerr scattering, and various multiphoton excitation processes. Within this theoretical regime the light (as a photon field) and molecules are treated as a combined quantum mechanical system. The light emission or absorption is always accompanied by the quantum transition of the whole system from its initial eigenstate to the final eigenstate. The molecular emission or absorption probability rate is determined by the corresponding transition matrix elements of the interaction Hamiltonian operator.
Classical electromagnetics in free space leaves no room for accommodation of a light particle. To underline this, it is sufficient to remember that Einstein in order to reach his lightquantum (energy-quantum) hypothesis had to mix classical theoretical physics with a piece of experimental information that defies description in classical terms. In his 1905-paper Einstein concluded [60]:Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of mutually independent energy quanta of magnitude hν. The introduction of light quanta in the Wien regime (the high-frequency part of the Planck blackbody radiation law) is the first step toward the concept of electromagnetic radiation as a Bose gas of photons [180, 39, 68, 69]. Although a photon gas obeys Bose-Einstein statistics for all frequencies the statistical independence assumed by Einstein in 1905 is not true in general. Einstein used Boltzmann statistics when he conjectured the existence of light quanta. In the Wien regime Boltzmann and Bose-Einstein statistics lead to the same conclusion since the mean number of particles (photons) with a given energy is much less than one. The circumstance that the light quanta are not statistically independent in Bose-Einstein counting finds its explanation in quantum mechanics. Hence, the statistical correlation is induced by the requirement of totally symmetric wave functions.
From prehistoric times has come recognition that sunlight and firelight provide warmth, and that such illumination casts shadows. Expressed in more contemporary terms one would say that light travels in rays, and that this radiation has the potential to provide heat energy to absorbing material. From the time of Newton it has been known that light sources emit radiation comprising a distribution of colors. During the nineteenth century it was recognized that radiation had characteristics of transverse waves (with wavelength associated to color) but until the late twentieth century, when lasers became laboratory tools, it was hardly necessary to delve into the equations of electromagnetic theory to treat such experiments as were then possible; interest lay primarily with thermodynamic considerations of energy flow or with measurements of the dark or bright lines seen in the distribution of light that, after passing through a slit, was dispersed by a prism or grating into constituent colors. Although laser light sources are essential for the types of atomic excitation considered in this monograph, the legacy from thermal radiation still influences many interpretations of the interaction between radiation and matter, and it is therefore useful to summarize some of those concepts. The mathematics needed for describing laser radiation, or polarized light in general, has much in common with the mathematics of quantum theory discussed starting in Sec. 3.5 and specialized to two-level atoms in Sec. 5.6. In both cases one deals with two complex-valued functions – independent electric field amplitudes or probability amplitudes – whose absolute squares are measurable.
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’).