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Soviet Journal of Quantum Electronics

Coherence of the electromagnetic field in dielectric waveguides

I. A. Deryugin, Sherzod Shukhratovich Abdullaev, Ag T Mirzaev

Published Oct 31, 1977• 8 citations
Publisher

Abstract

An investigation is made of the coherence of the electromagnetic field in a plane dielectric waveguide excited by an external field with specific coherence properties. The quantum theory of an electromagnetic field in one-dimensionally inhomogeneous media is employed to derive an expression which relates the correlation functions of the field inside the waveguide to the external field. This expression is used to study the variation of the coherence of the radiation at various distances z from the area where the radiation is coupled into the waveguide. If the external field has a finite line width, the spatial coherence is a periodic function of z when z ≪ z cr and tends to a constant value when z ≫ z cr . When z ≪ z cr , the temporal coherence is the same as for the external source, whilst when z ≫ z cr , the temporal coherence as a function of τ has maxima which move away from the point τ = 0 with increasing z . It is found that the coherence tunes τ 1 and τ 2 determined following Wolf and Mandel behave differently when z ≫ z cr : τ 1 2 increases quadratically with increasing z , whereas τ 2 tends to a constant value. The form of the radiation modulation spectrum is deduced. Curves are plotted for the spatial and temporal coherence by way of a specific example.

DOI
10.1070/qe1977v007n10abeh012849

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Reviews of Modern Physics•Apr 1, 1965•1,174 citations
An investigation is made of the coherence of the electromagnetic field in a plane dielectric waveguide excited by an external field with specific coherence properties. The quantum theory of an electromagnetic field in one-dimensionally inhomogeneous media is employed to derive an expression which relates the correlation functions of the field inside the waveguide to the external field. This expression is used to study the variation of the coherence of the radiation at various distances z from the area where the radiation is coupled into the waveguide. If the external field has a finite line width, the spatial coherence is a periodic function of z when z ≪ z cr and tends to a constant value when z ≫ z cr . When z ≪ z cr , the temporal coherence is the same as for the external source, whilst when z ≫ z cr , the temporal coherence as a function of τ has maxima which move away from the point τ = 0 with increasing z . It is found that the coherence tunes τ 1 and τ 2 determined following Wolf and Mandel behave differently when z ≫ z cr : τ 1 2 increases quadratically with increasing z , whereas τ 2 tends to a constant value. The form of the radiation modulation spectrum is deduced. Curves are plotted for the spatial and temporal coherence by way of a specific example.
The electromagnetic field is quantized in a material whose dielectric function varies with frequency and one spatial dimension. The dielectric function is assumed to have a known form and to be real over the range of frequencies important for a particular application, for example the propagation of an optical signal. General properties of the mode functions are derived and employed in the quantization procedure. Expressions are obtained for the energy and momentum density and current operators, and these are shown to satisfy the appropriate conservation and continuity relations. The general formalism is illustrated by application to the examples of a homogeneous dielectric and two different dielectrics with a sharp interface.
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.
The relation between the retarded Green's function and second-order correlation tensor of electromagnetic potential for a pure electromagnetic radiation in thermal equilibrium is derived. This relation holds also for a long wavelength electromagnetic radiation in media, so long as the radiation is in thermal equilibrium. The electromagnetic second-order correlation tensors for blackbody radiation is obtained, using the relation mentioned above. Upon following Abrikosov, Gorkov and Dzyaloshinski the retarded Green's function for a electromagnetic radiation interacting with matter is expressed in terms of the complex dielectric constant which is necessarily introduced for carrying out a macroscopic formulation. With the assumption of spatial homogeneity of media the electromagnetic second-order correlation tensors for an optical field are obtained. These correlation tensors are also expressed in tems of the complex dielectric constant. This retarded Green's function is simply related to the temperature Green's function, so that the diagram analysis will possibly be applied in the theory of optical coherence for an interacting radiation with matter.
Using quantum theory, we study the propagation of an optical field in an inhomogeneous dielectric, and apply this scheme to traveling optical fields in a waveguide. We introduce a field-atom interaction Hamiltonian and derive the refractive index using quantum optics. We show that the transmission and reflection of optical fields at an interface between different materials can be described with normalized Fresnel coefficients and that this representation is related to the beam splitter operator. We then study the propagation properties of the optical fields for two types of slab waveguides: step-index and graded-index. The waveguides are divided into multiple layers to represent the spatial dependence of the optical field. We can evaluate the number of photons in an arbitrary volume in the waveguide using this procedure. Using the present method, the quantum properties of weak optical fields in a waveguide are revealed, while coherent states with higher amplitudes reduces to representation of classical waveguide optics.
A quantum-electrodynamic theory of spontaneous emission in presence of dielectrics and conductors is developed. The theory makes use of the master-equation techniques and the response-function formalism of part I of this series of papers. Various observable entities such as damping coefficients (lifetimes), Lamb shifts, and frequency shifts are related to the appropriate surface-dependent response functions. The results are valid for arbitrary geometries (involving linear dielectrics) and naturally contain, as a special case, the usual results of spontaneous emission in free space. As explicit examples we consider a two-level atom (also the multilevel atom) in presence of a plane dielectric interface and between two conducting plates. Formulas for the shifts and widths are given and their asymptotic behavior for large and short distances is discussed. The behavior when the atom is embedded inside the dielectric is different than when the atom is outside the dielectric. The origin of coherence in the present model is discussed and the coherence effects in this model are contrasted with those in Dicke's model. The results are also compared with those obtained by the image method. Exact expressions for the operator radiation-reaction fields are obtained in terms of the atomic polarization operators and response functions. Approximate results for such fields are also given. The far-zone behavior of the radiation fields is obtained in terms of response functions and the polarization operators. Some of the normally ordered correlation functions are also calculated. The connection of some of the theoretical results with a recent experimental work of Carniglia, Mandel, and Drexhage is discussed. The effect of anisotropy of the dielectric function on the lifetimes as well as on the far-field correlation functions is also considered. Finally the contribution to the shifts and widths due to the excitation of surface polariton modes is computed and the results are compared with those obtained by the quantization of surface polariton field. It is found that such surface modes contribute significantly to widths.
This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.