Geometrical Optics of Weakly Anisotropic Media

A A Fuki, Yu A Kravtsov, O N Naida

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9781351445306
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This book explores recent developments in QIA and describes the application of the theory to different branches of wave physics, from plasma physics, quantum physics, and ionospheric radio wave propagation to acoustics, optics, and astrophysics.
This is an up-to-the-minute exposition of the latest developments in an important new area, written by authors of outstanding reputation. A rich source of both theoretical methods and practical applications, it covers a wide range of problems of general physical significance.
Until recently, there was no effective method for describing waves in weakly anisotropic inhomogeneous media. The method of quasi-isotropic approximation (QIA) of geometrical optics was developed to overcome this problem. The QIA approach bridges the gap between geometrical optics of isotropic media (Rytov method) and that of anisotropic media (Courant-Lax approach), thus providing a complete picture of the geometrical optics of inhomogeneous media.

1

Introduction

1.1 Vector field polarization transformation in weakly anisotropic media

The problem of the propagation of electromagnetic, elastic or other waves in anisotropic media makes up a substantial part of wave physics. The propagation of electromagnetic waves of different bands in a weakly magnetized plasma (laboratory, ionospheric, solar, interstellar), the influence of weak anisotropy of magnetic condensed matter, piezo- and ferromagnets on the transmission of electromagnetic waves, light waves in chiral media and in liquid crystals, polarization phenomena in deformed light guides, propagation of acoustic and seismic waves in weakly anisotropic and/or weakly deformed elastic media, splitting of beams of particles with spin in magnetic fields, that is the list of questions, which is by no means complete, connected with the theory of wave phenomena in weakly anisotropic media.

Melrose and McPhedran (1991), renowned for their contribution to electromagnetic wave theory, define a medium as a weakly anisotropic one if the effect of anisotropy can be treated as a perturbation breaking the degeneracy of the transverse states of polarization. The authors of this book use the the same definition.

Needless to say, the weakness of the anisotropy does not imply the polarization transformation of the electromagnetic field being small. Rather, the contrary holds true: it is precisely media with sufficiently weak anisotropy that are capable of controlling the field polarization structure. To mention a high-school example, let us consider crystal plates capable of converting the linearly polarized light into light with arbitrary elliptical polarization. In particular, a quarter wave plate, i. e., a plate of thickness d in which the difference of optical paths |n ₁ – n ₂|d between two linearly polarized waves equals λ₀/4,

$|n_1-n_2|d=\frac{{\displaystyle {\lambda }_0}}{{\displaystyle 4}}=\frac{{\displaystyle \pi }}{{\displaystyle 2k_0}}=\frac{{\displaystyle {\pi }_c}}{{\displaystyle 2_w}},$

can convert linearly polarized light into circularly polarized light.

Another example of fundamental change of polarization is furnished by the Faraday rotation occurring in natural active media or in media placed in a magnetic field, for example in a magnetized plasma. In either case intrinsic normal waves are polarized circularly. If n# and denote the refractive indices, respectively, for the right and left circular polarizations, the medium slab of thickness d rotates the electromagnetic strength vector through the angle

$\psi =-\frac{{\displaystyle 1}}{{\displaystyle 2}}{k}_0(n_R-n_L)d$

(Landau and Lifshitz, 1960; Melrose and McPhedran, 1991).

In both examples, even a weak anisotropy,

$\begin{array}{ccc}|n_1-n_2|\ll 1& \text{or}& |n_R-n_L|\ll 1,\end{array}$

may be responsible for substantial changes in the polarization of a vector field across a distance on the order of

$\begin{array}{ccc}d\approx \frac{{\lambda }_0}{|n_1-n_2|}& \text{or}& d\approx \frac{{\lambda }_0}{|n_R-n_L|}\mathrm{.}\end{array}$

In the examples considered above, the radical transformation of the polarization structure of the field only occurs at boundaries bounding the homogeneous medium. It is there that the degeneracy of transverse polarization states breaks.

Contrary to that, in a weakly anisotropic media whose parameters vary continuously, mutual transformation of normal waves happens at each point with the resultant polarization being determined by the magnitude of anisotropy and the rate at which the medium parameters vary.

The main feature of inhomogeneous weakly anisotropic media is their capability of substantially transforming the polarization structure of vector fields. Polarization transformation in weakly anisotropic regions of an inhomogeneous medium owes its existence to a strong interconversion of normal waves: it is in these regions that the polarization degeneracy smoothly arises or lifts. In essence, in a weakly anisotropic medium, the polarization structure of a wave field undergoes a drastic transformation: while in anisotropic medium the electromagnetic field is represented by a superposition of two normal waves, each characterized by a definite, intrinsic polarization, in an isotropic medium transverse magnetic waves are polarization-degenerate and the state of their polarization is not defined.

On entering an anisotropic medium, a transverse electromagnetic wave ceases to be polarizationally degenerate and experiences a substantial change of polarization. The conversion (reconstruction, “pere- stroyka”) of the polarization structure occurs in a weakly anisotropic layer that matches isotropic and anisotropic media, as shown in Figure 1.1). The final result of the polarization conversion (“polarization per- estroyka”) is the transformation of the polarization-degenerate transverse wave into a superposition of independent normal waves.

The polarization transformation of waves in an anisotropic medium is a physical problem of general character, which is important both theoretically and practically. This book is dedicated to its analysis.

1.2 Basic methods of wave field description in weakly anisotropic media

1.2.1 The Budden method

The final polarization state, or, as it often called, the “limiting” polarization of a vector wave field depends on the character of the inhomogeneities encountered in a weakly anisotropic medium. Describing the process of wave transformation in such a medium presents a rather complex mathematical problem which has not been completely solved up to now.

The theory of the linear interaction of normal modes in a planelayered plasma was elaborated by Budden (1952, 1961) (the Budden method). In a layered medium, Maxwell’s equations convert to a system of four coupled ordinary deferential equations. The Budden method, devised for smoothly inhomogeneous media, allows one to reduce substantially the order of that system: it becomes second-order and involves a pair of coupled first-order equations. It is these two equations that describe the linear interaction of normal waves in the framework of the Budden method. The method is not constrained by the requirement of weak anisotr...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication Page
• Table of Contents
• Preface
• 1 Introduction
• 2 Geometrical optics of inhomogeneous media
• 3 Modifications and generalizations of QIA
• 4 Electromagnetic waves in an inhomogeneous plasma in a weak magnetic field
• 5 Optical effects in weakly anisotropic media
• 6 Geometrical acoustics of weakly anisotropic media
• 7 Quasi-isotropic approximation in quantum mechanics
• 8 Conclusions
• References
• Index