Before discussing nonlinear optics with quantized fields, it is useful to have a look at what happens with classical electromagnetic fields in nonlinear dielectric media. The theory of nonlinear optics was originally developed using classical fields by Armstrong, Bloembergen, Ducuing and Pershan in 1962, stimulated by an experiment by Franken, Hill, Peters and Weinreich in which a second harmonic of a laser field was produced by shining the laser into a crystal. This classical theory is sufficient for many applications. For the most part, quantized fields were introduced later, although a quantum theory for the parametric amplifier, a nonlinear device in which three modes are coupled, was developed by Louisell, Yariv and Siegman as early as 1961. In any case, a study of the classical theory will give us an idea of some of the effects to look for when we formulate the more complicated quantum theory. What we will present here is a very short introduction to the subject. Our intent is to use the classical theory to present some of the basic concepts and methods of nonlinear optics. Further information can be found in the list of additional reading at the end of the chapter. The discussion here is based primarily on the presentations in the books by N. Bloembergen and by R. W. Boyd. Linear polarizability We wish to survey some of the effects caused by the linear polarizability of a dielectric medium. When an electric field E ( r , t ) is applied to a dielectric medium, a polarization, that is, a dipole moment per unit volume, is created in the medium.
Before discussing nonlinear optics with quantized fields, it is useful to have a look at what happens with classical electromagnetic fields in nonlinear dielectric media. The theory of nonlinear optics was originally developed using classical fields by Armstrong, Bloembergen, Ducuing and Pershan in 1962, stimulated by an experiment by Franken, Hill, Peters and Weinreich in which a second harmonic of a laser field was produced by shining the laser into a crystal. This classical theory is sufficient for many applications. For the most part, quantized fields were introduced later, although a quantum theory for the parametric amplifier, a nonlinear device in which three modes are coupled, was developed by Louisell, Yariv and Siegman as early as 1961. In any case, a study of the classical theory will give us an idea of some of the effects to look for when we formulate the more complicated quantum theory. What we will present here is a very short introduction to the subject. Our intent is to use the classical theory to present some of the basic concepts and methods of nonlinear optics. Further information can be found in the list of additional reading at the end of the chapter. The discussion here is based primarily on the presentations in the books by N. Bloembergen and by R. W. Boyd. Linear polarizability We wish to survey some of the effects caused by the linear polarizability of a dielectric medium. When an electric field E ( r , t ) is applied to a dielectric medium, a polarization, that is, a dipole moment per unit volume, is created in the medium.
Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.
An introduction to the quantum theory of nonlinear optics This article is provides an introduction to the quantum theory of optics in nonlinear dielectric media. We begin with a short summary of the classical theory of nonlinear optics, that is nonlinear optics done with classical fields. We then discuss the canonical formalism for fields and its quantization. This is applied to quantizing the electromagnetic field in free space. The definition of a nonclassical state of the electromagnetic field is presented, and several examples are examined. This is followed by a brief introduction to entanglement in the context of field modes. The next task is the quantization of the electromagnetic field in an inhomogeneous, linear dielectric medium. Before going on to field quantization in nonlinear media, we discuss a number of commonly employed phenomenological models for quantum nonlinear optical processes. We then quantize the field in both nondispersive and dispersive nonlinear media. Flaws in the most commonly used methods of accomplishing this task are pointed out and discussed. Once the quantization has been completed, it is used to study a multimode theory of parametric down conversion and the propagation of quantum solitons.
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).
Introduction In 1961, just a few months after Maiman invented the ruby laser, Franken focused the pulses emitted from such a laser, of wavelength 694 nm, on a quartz plate, and examined the spectrum of the light transmitted using a simple prism (see Figure 7.1). He thus discovered that ultra-violet light of wavelength 347 nm was emerging from the quartz plate. Clearly, as it propagated through the quartz, the light of frequency ω had generated the second harmonic, of frequency 2ω. It thus transpires that in optics, as in any other part of physics, a system subjected to a strong enough sinusoidal excitation will leave the linear response regime. Nonlinearities cause harmonics of the excitation frequency to appear. But what intensity is needed before nonlinear effects will appear? One might think that a natural scale would be the electric field of the nucleus at the location of an atomic electron. In the case of the hydrogen atom in its ground state, this field is about, or 3 × 10 11 V.m −1 . (Here e is the charge of the electron, and a 0 the Bohr radius, of the order of 5 × 10 −11 m). In fact, experiment shows that, in the transparency zone of a dielectric material like quartz, a field of just 10 7 V.m −1 (corresponding to a light intensity of 2.5 kW.cm −2 ) is sufficient for nonlinear effects to appear perturbatively.
In Chapter 1, we treated the electromagnetic field as classical. Henceforth, we will want to treat it as a quantum field. In order to do so, we will first present some of the formalism of quantum field theory. This formalism is very useful in describing many-particle systems and processes in which the number of particles changes. Why this is important to a quantum description of nonlinear optics can be seen by considering a parametric amplifier of the type discussed in the last chapter. A pump field, consisting of many photons, amplifies idler and signal fields by means of a process in which a pump photon splits into two lower-energy photons, one at the idler frequency and one at the signal frequency. Therefore, what we would like to do in this chapter is to provide a discussion of some of the basics of quantum field theory that will be useful in the treatment of the quantization of the electromagnetic field. In particular, we will begin with a summary of quantum theory notation, and a discussion of many-particle Hilbert spaces. These provide the arena in which all of the action takes place. We will then move on to a treatment of the canonical quantization procedure for fields. This will allow us to develop a scattering theory for fields, which is ideally what we need. This relates the properties of a field entering a medium to those of the field leaving it, and this corresponds to what is done in an experiment.
There has recently been a rapid growth of activity in nonlinear optics. Effects such as frequency doubling, stimulated Raman scattering, phase conjugation and solitons are of great interest both for their fundamental properties and their many important applications in science and engineering. It is mainly these applications - especially in telecommunications and information processing - that have stimulated the recent surge of activity. This book is a self contained account of the most important principles of nonlinear optics. Assuming only a familiarity with basic mathematics, the fundamentals of nonlinear optics are fully developed from basic concepts. The essential quantum mechanical apparatus is introduced and explained. In later chapters the underlying ideas are illustrated by discussing particular experimental configurations and materials. This book will be an invaluable introduction to the field for beginning graduates in physics or engineering, and will provide an excellent overview and reference work for active researchers in the field.