We investigate the propagation of Γˆ(x1,x2,ν) the Fourier transform of the mutual coherence function Γ(x1,x2,τ) in nonlinear dielectrics. We assume that the classical phenomenological equations govern the electric field within the dielectric, and we restrict our discussion to the propagation of fields with narrow angular spread. The indeterminate character of the propagation equations is emphasized, and it is shown that a determinate equation results only if the nonlinear term is small. In the quasimonochromatic limit, we investigate for this latter case the dependence of the two-point coherence of the second harmonic on the full fourth-order coherence of the incident beam, a result implied by Ducing and Bloembergen (1964), and Akhmanov et al. (1966). We also show under what circumstances the Hanbury-Brown and Twiss experiment may be performed using a single nonlinear dielectric.
We investigate the propagation of Γˆ(x1,x2,ν) the Fourier transform of the mutual coherence function Γ(x1,x2,τ) in nonlinear dielectrics. We assume that the classical phenomenological equations govern the electric field within the dielectric, and we restrict our discussion to the propagation of fields with narrow angular spread. The indeterminate character of the propagation equations is emphasized, and it is shown that a determinate equation results only if the nonlinear term is small. In the quasimonochromatic limit, we investigate for this latter case the dependence of the two-point coherence of the second harmonic on the full fourth-order coherence of the incident beam, a result implied by Ducing and Bloembergen (1964), and Akhmanov et al. (1966). We also show under what circumstances the Hanbury-Brown and Twiss experiment may be performed using a single nonlinear dielectric.
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.
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.
In this paper we outline a method which allows one, in principle, to measure $〈{E}_{i}({\mathrm{x}}_{1}, t){E}_{j}({\mathrm{x}}_{2}, t){{E}_{k}}^{*}({\mathrm{x}}_{3}, t)\ifmmode\times\else\texttimes\fi{}{{E}_{l}}^{*}({\mathrm{x}}_{4}, t)〉$ (in general, x1\ensuremath{\ne}x2\ensuremath{\ne}x3\ensuremath{\ne}x4), for stationary quasimonochromatic light propagating principally in the forward direction. The procedure makes use of the properties of nonlinear dielectrics in a sequence of interference experiments. No correlation of photoelectron currents is used, and as a special case the contracted moment 〈∣Ei(x1,t)∣2∣Ej(x2,t)∣2〉 measured by Hanbury Brown and Twiss could be obtained in this manner. It is shown that it should be possible to measure the full fourth-order coherence function in the laboratory for laser light, using presently obtainable intensity levels.
An account is given of optical coherence theory as it existed in the 1950's when, essentially, only thermal sources and slow detectors were available and it sufficed to characterize the coherence and polarization properties of an electromagnetic field by second-order correlation functions. The necessity for a more sophisticated specification of the coherence properties of fields, involving the use of higher-order correlation functions, first hinted at by the Hanbury-Brown-Twiss experiments and emphasized by the advent of the laser, is then introduced. Both classical and quantum mechanical treatments are discussed, each leading, in almost all cases of practical interest, to similar results. It is seen that the development of fast detectors enabled the higher-order correlation functions to be measured using photon counting techniques and an account of experimental work verifying the complete theory using such techniques is given. The review concludes with a discussion of some developments in associated areas whose origin lies in the resurgence of interest in coherence theory during the last decade.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.