The Ewald-Oseen Extinction Theorem in the Light of Huygens’ Principle

Introduction

When a light wave enters a material medium, it magnetizes and/or polarizes it. The excited medium re-irradiates light. In view of the linearity of the Maxwell equations for linear media, one could think that the total electromagnetic field in the medium is the sum of the original (incident, exciting) and the re-irradiated fields. Actually, the incident beam is not observed in the medium. The Ewald Oseen extinction theorem explains this absence in that the excited medium creates three waves: (i), one, which annihilates aka extincts the incoming wave, (ii), one that propagates trough the medium according to its optical properties, see Figure 1 (p. 8, Figure 4), and, (iii), the reflected wave [1-3].

Mansuripur comments, as if “the oscillating electrons conspire to produce a field that exactly cancels out the original beam everywhere in the medium” (p. 209). Now, it is hard to believe, that “electrons conspire” that way as this phenomenon has nothing to do with collective effects like Langmuir waves [4,5]. Huygens’ secondary wavelets consist of only one part which, however, depends on the local propagation conditions (double refraction needs additional considerations [6-9]). Of course, mathematically, due to the linearity of the Maxwell equations and the linear media usually considered in the literature, the electric and magnetic field quantities can be rather arbitrarily split and combined. Anyway, the theorem has been reconsidered several times and for various materials, e.g. to name a few, and it enters some textbooks, e.g. For a review of various interpretations of the extinction process, see [10,11].

Sein states, “. . . the extinction theorem is essentially an expression of Huygens’ principle for the incident field inside the medium.” According to Lian, if Huygens’ principle is mathematically formulated such that, in a vacuum or in homogeneous isotropic media, back- scattering is absent, “it must satisfy” the extinction theorem (p. 5 II). Using simple cases, we will show that the extinction theorem and Huygens’ principle are mathematically equivalent but not physically.

Thus, to clarify the physical content of the Ewald Oseen extinction theorem and, in particular, its relationship to Huygens’ principle, this article proceeds as follows. Section II sketches the most general representation of Huygens’ principle in terms of propagators due.
 
The incident pulse enters the medium without perturbation (red line), the atoms irradiate waves to Feynman. Section III presents a novel representation of the extinction theorem using that framework. For subsequent use, Section IV provides the Maxwell-Heaviside equations and sketches the macroscopic and microscopic approaches for describing the polarization as well as Hertz’s potential and the electric Hertz vector, where a novel hypothesis is proposed. Basing on that, Section V treats the dielectric half-space, commenting on Born & Wolfs and Zangwill’s treatments. Finally, Section VI summarizes and concludes this article.

It is a correction, clarification, and major extension of an earlier article by one of us [12-28].

Feynman’s Representation of Huygens’ Principle in Terms of Propagators
The Chapman-Kolmogorov Equation
According to Feynman, Huygens’ principle can be expressed through propagators P_ab as

where G is an appropriate propagator aka Green’s function [31,32]. It generalizes Huygens’ construction from sharp to spreading wave fronts, where the domain of sources of secondary wavelets is not necessarily a surface but may be a finite volume Vb. Sharp wave fronts correspond to a δ-function in G whence the volume integral is reduced to a surface integral. In the usual representations of Huygens’ construction, this surface is the location of the secondary sources.

Notice that the Chapman-Kolmogorov equation (2) is not fulfilled by the Green’s function of d’Alembert’s (Euler’s) wave equation in 3+1d. However, that does not mean that Huygens’ principle “is only approximately fulfilled in optics”. One has to transform a wave equation into two partial differential equations of first order in time and to analyze the corresponding matrix Green’s function.

Example
To illustrate that most general representation of Huygens’ principle, let us consider a very simple example, viz., a two-dimensional network of (ideally) lossless transmission lines, see Figure 2 [33].

A node connects 4 lines, say, in the directions West, North, East, and South. Each line has the impedance Z. A very short pulse of voltage 1 (arbitrary units) incident from the South is scattered as follows.

•    The pulse enters the node from a line with impedance Z. At the node, it meets three lines with impedance Z each. Because they are parallel, their common impedance at the node is 1/3 Z . For this, the reflection coefficient equals (a node works like a voltage divider).

The voltage impulses in that network are described by a set of difference equations of first order in the propagation (time) step. Its fundamental solution is a discrete matrix Green’s function which obeys a Chapman-Kolmogorov equation [37].

BTW, TLM networks are (idealized) physical realizations of correlated random walks [38]. This makes their algorithms in numerical
mathematics extremely stable.

The Ewald Oseen Extinction Theorem in Terms of Propaga Tors
Let a in formula (1) refer to a state (point) outside a medium (vacuum), c to one inside the medium, and b to one on its surface. Then, formula (1) reads

Mathematically, both formulae (8) and (10) are equivalent. Physically, however, they are quite different. For Huygens’ principle –formula (8) – states that the incoming ‘wave’ is absorbed at the boundary.

Remark 1 What should be the reason for the secondary sources to irradiate two different secondary ‘wavelets’ into the medium according
to formula (10)? (For double refraction, see.

Remark 2 Again, if the incident ‘field’ travels unperturbed through the medium, which ‘field’ is exciting the medium?

General Electromagnetic Formulas for Later Use
The Maxwell-Heaviside Equations
In what follows, we will consider an electromagnetic wave traveling through a vacuum and a linear, electrically neutral, homogeneous, and isotropic dielectric. To avoid confusions in the notations, let us begin with the Maxwell-Heaviside equations (SI units; cf. [39,40]).

Polarization
The relative dielectric constant ÃÃÃÃÂ???ÂÃÃÂ??ÂÃÂ?µr in eqs. (14) can also been represented by the polarization Pâ?? and the polarizability χ as

 Microscopic Theory
The polarization Pâ?? (18) is related to the bound charge and current densities as

The symmetry of the electromagnetic field is obvious; in particular, Π is independent of ÃÃÂ??. The vector and scalar potentials equal Aâ?? = (0, 0, −∂Π/∂t) and Φ = ∂Π/∂z, respectively. For this, the Hertz potential is also called a ‘super-potential’.

The boundary conditions used in eqs. (17) are not needed when looking for a solution in whole space. Such a solution can be obtained using the electric Hertz vector.

If that hypothesis holds true, it provides a reason for the fact that they have proven useful for solving many radiation problems (for a throughout analysis, see [44-46], for references of pedagogical purpose [47]). In contrast to the cumbersome traditional calculations (e.g., Section 2.4), the electric Hertz vector allows for a relatively direct treatment of the extinction theorem in non-magnetically, electrically neutral dielectrics (e.g. as will be discussed in what follows.

The Ewald-Oseen Extinction Theorem for a Dielectric Half- Space
Experimental Setup
Zangwill (Section 20.9) considers a dielectric as above which occupies the half-space z ≥ 0 (see his Figure 20.23 on p. 763). There is an incident plane wave of electric field strength Eâ?? (i) moving perpendicularly to the surface z = 0 of the dielectric,

The second line displays the extinction of the incident wave, while the last line describes the observed transmitted wave Eâ?? (t) (17) in the dielectric.

Our early recursion to the macroscopic theory in formulas (46) saves the calculations in (20.255) ff.

Obviously, that exposition is formally, mathematically correct. However, it contradicts Huygens’ principle, see Remark 4.

The second line displays the extinction of the incident wave, while the last line describes the observed transmitted wave Eâ?? (t) (17) in the
dielectric.

Our early recursion to the macroscopic theory in formulas (46) saves the calculations in (20.255) ff.

Obviously, that exposition is formally, mathematically correct. However, it contradicts Huygens’ principle, see Remark 4.

Summary and Conclusions

In our understanding, Huygens’ construction involves that the incident wave is completely extinguished by having excited the sources of the secondary wavelets; each of these sources irradiates one secondary wavelet according to its local propagation conditions (in case of double refraction, two secondary wavelets are irradiated).

On the contrary, the Ewald Oseen extinction theorem describes refraction this way: The incident wave moves through the refracting medium without any alteration;

There is a polarization and/or a magnetization of the medium creating three or four waves:

(a)    the reflected wave,
(b)    one wave which extinguishes the incoming wave,
(c)    one wave which corresponds to the observed one (in case of double refraction two waves).

Here, the question arises, how the medium is excited when the incident wave moves through it without alteration, i.e. without interacting with the medium? Hence, as the excitation of a medium has not any cause, the theorem is both physically and philosophically doubtful. Therefore, it resembles Vaihinger’s philosophy of ‘as if’ [50]. It can be useful to act “as if” it were physically (and philosophically) correct.

As an example, we have shown in Section III that an abstract propagator description is mathematically compatible with both points of view. This corroborates Feynman’s interpretation of Huygens’ principle in terms of the Chapman-Kolmogorov equation (2). However, both descriptions are mathematically but not physically equivalent.

Notice that all those treatments discard the interaction principle in that the incident wave acts upon the secondary sources, while there is no back-reaction from the secondary sources upon the incident wave (cf. end-note 9).

For the sake of completeness, let us remark this. The Ewald Oseen extinction theorem imagines that each secondary source in Huygens’ principle re-irradiates more than one secondary wavelet. This resembles the standard interpretation of the surface integral terms in Kirchhoff’s integral theorem in that it “involves two types of sources of varying strength.” [51,52]. For a criticism of that interpretation as well as considering Kirchhoff’s theorem as a representation of Huygens’ principle, see.

Acknowledgments
Parts of this work were performed at the U. Sultangazin Pedagogical Institute, Kostanai Akhmet Baitursynuly Regional University, Kazakhstan. The hospitality over there was overwhelming, comparable with the one described by J. M. Ziman in the preface to his well- known book on solid-state theory [53]. Moreover, one of us (PE) feels highly indebted to Masud Mansuripur for numerous enlightening explanations when writing the earlier version and Christian Vanneste for sharing his insights on scattering theory, which finally re- solved the local realization of Huygens’s principle within transmission line matrix modeling as well as his hospitality in Sophia Antipolis during the TLM conference over there. He also thanks Hassan Bolouri, Rudolf Germer, Jan Helm, Axel Kilian, and Bernd Steffen for a helpful discussion on the crucial issues of this article.

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