Table Of ContentPartially Ordered Systems
Editorial Board:
J. Charvolin . W. Helfrich . L. Lam
Advisory Board:
J.D. Litster . D.R. Nelson' M. Schadt
Partially Ordered Systems
Editorial Board: 1. Charvolin • W. Helfrich • L. Lam
Solitons in Liquid Crystals
Lui Lam and Jacques Prost, Editors
Bond-Orientational Order in Condensed Matter Systems
Katherine J. Strandburg, Editor
Diffraction Optics of Complex-Structured Periodic Media
V.A. Belyakov
V.A. Belyakov
Diffraction Optics of
Complex-Structured
Periodic Media
With 86 Illustrations
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
V.A. Belyakov
All-Union Surface and Vacuum Research Center
Andreevskaya nab. 2
117334 Moscow
Russian Federated Republic
Editorial Board:
Jean Charvolin Wolfgang Helfrich Lui Lam
Institut Max von Laue- Institut fUr Theorie der Department of Physics
Paul Langevin kondensierten Materie San Jose State University
Avenue des Martyrs Fachbereich Physik One Washington Square
38042 Grenoble Cedex Freie Universitiit Berlin San Jose, CA 95192
France Arnimallee 14 USA
1000 Berlin 33
Germany
Advisory Board:
John D. Litster David R. Nelson Martin Schadt
Francis Bitter National Department of Physics Department ZFE/RLC
Magnet Laboratory Harvard University F. Hoffman-La Roche
Massachusetts Institute of Cambridge, MA 02138 &Co.
Technology USA CH-4002 Basel
Cambridge, MA 02139 Swi tzerland
USA
Library of Congress Cataloging-in-Publication Data
Belyakov, V. A. (Vladimir Alekseevich)
Diffraction optics of complex-structured periodic media 1 V.A.
Belyakov.
p. cm. - (Partially ordered systems)
Revised and augmented translation of 1988 Russian edition.
Includes bibliographical references and index.
ISBN-13: 978-1-4612-8755-1 e-ISBN-13: 978-1-4612-4396-0
DOl: 10.1007/978-1-4612-4396-0
(Germany: alk. paper)
1. Diffraction. 2. Liquid crystals-Optical properties.
3. Matter-Effect of radiation on. 4. X-rays-Diffraction.
I. Title. II. Series.
QC415.B37 1992
535.4-dc20 91·22809
Printed on acid-free paper.
Original Russian edition: Difraktsionnaya optika periodycheskykh sred slozhnoy struktury. Moskva:
Nauka, 1988.
© 1992 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover I st edition 1992
All rights reserved. This work may not be translated or copied in whole or in part without the written
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Photocomposed from aLArE X file.
987654321
Preface to the English Edition
Little time has elapsed since 1988 when a Russian edition of this book was
published, but most of the problems touched on here were under intensive
investigation then. This indicates that the subject of the book was and
is quite urgent, and as a consequence, a number of new publications on
the subject have appeared since the first edition. It is why in the course
of preparing a new edition, I decided that many new results ought to be
reflected in it. .
Because I wanted to retain the general style of the original book and
its overall features, the problem of integrating new material was solved in
the following way. Three chapters, devoted to the fields of the most signifi
cant achievements, were added: on polarization phenomena in X-ray optics,
on X-ray magnetic scattering, and on M6ssbauer filtration of synchrotron
radiation. Only minor alterations were made to the other chapters; their
end-of-chapter reference sections were also updated.
I hope that the English edition of this book will give easier access to my
Western colleagues to the original Soviet results. Soviet authors contributed
quite significantly to the subject of this book and look forward to developing
joint investigations of the coherent interaction of different kind of radiations
with matter. I would also like to express my gratitude to the Springer-Verlag
editing staff for collaboration in this project.
Moscow V.A. Belyakov
Preface to the Russian Edition
Rapid progress in experimental techniques in the past 15-20 years, cou
pled with the ever-increasing variety of the objects of investigation, have
advanced the study of a series of phenomena which were beyond the scope
of researchers and which are very important for various applications.
Among such phenomena are the numerous unusual effects observed
in coherent interaction of various electromagnetic and corpuscular radia
tions with regular (spatially periodic) media having complicated structures.
These are polarization characteristics of light scattering, the unusually high
gyrotropy of chiralliquid crystals, which changes its sign depending on fre
quency, nuclear scattering of gamma and synchrotron radiation sensitive
to magnetic order in crystals, various anomalies in angular and frequency
distributions of the coherent radiation of fast charged particles, and similar
effects in nonlinear generation of high-order optical harmonics in periodic
media, etc.
It should be emphasized that since specific objects, types of radiation,
and conditions under which the above effects manifest themselves are very
diverse and, at first glance, have quite a different nature, they are often
considered unrelated phenomena. In many studies, such separate consid
erations of these phenomena are detrimental, and instead of reinventing
the wheel and ignoring the experience and results of other distant fields of
research, investigators have only to examine, in fact, closely related ones. It
is worthy to remind the reader here of the words written of L. Brillouin and
M. Parodi some 30 years ago in the foreword to their now famous Wave
Propagation in Periodic Structures. "No matter how diverse the problems
are, their mathematical treatment results in equations having a series of
common properties and similar solutions. This deep relation between quite
different problems was very well known already to Kelvin and Rayleigh.
But then it has been forgotten and only in recent works this very impor
tant statement has attracted attention anew." Unfortunately, the above
affirmation sounds as urgent today, in application to modern problems, as
it did then.
This book presents from one standpoint and, in fact, within the context
of one approach a theoretical description of various phenomena--diffraction
of M6ssbauer gamma radiation by magnetically ordered crystals, optics of
chiralliquid crystals, coherent radiation of fast charged particles in periodic
media with a complicated structure, some problems of nonlinear optics for
viii Preface to the Russian Edition
periodic media, and dynamical diffraction of thermal neutrons by perfect
magnetically ordered crystals. The main experimental results, if any, are
explored to the extent necessary for the illustration of general qualitative
effects and their comparison with theoretical calculations.
This book also includes introductory material on wave propagation in
one-dimensional periodic media and the approximations describing the in
teraction of radiation with periodic media. The main focus of the book is
the application of the kinematical and dynamical approximations of the
theory of scattering to the above-mentioned phenomena, which permits
one to obtain an analytical solution of the problems under consideration.
The numerical methods for solving the corresponding problems are beyond
the scope of this book, although the results obtained by such methods
(with corresponding references) are given in some instances to illustrate
the experiment or theory under consideration. This is explained not only
by the fact that for most problems, the above-mentioned methods yield
results with sufficient accuracy, but also by the fact that these methods
most clearly illustrate the common physical nature of the phenomena un
der consideration. We do not concentrate on the scattering of low-energy
electrons in crystals, which, in fact, is closely to the problems considered
in our book. Space limitations decided this, but more important, the above
approximation fails to describe with sufficient accuracy the interaction be
tween low-energy electrons and crystals. Also, extensive literature, reviews
and monographs already exist on this problem.
I believe that the consideration of all the above problems in one book will
be useful for both specialists in the field and beginners, since at present cor
responding data are scattered in numerous original publications or highly
specialized monographs. The bulk of this book presents a first attempt to
generalize the problems of diffraction optics for various media with periodic
ities of a nonscalar nature, but at the same time, some problems considered
here may also be found in recent monographs. Thus, some problems of the
dynamical theory of diffraction of Mossbauer gamma radiation are stated
by V.G. Baryshevskii in his monograph, Nuclear Optics of Polarized Me
dia (Minsk, Izd-vo BGU, 1976) and by M.A. Andreeva and R.N. Kuz'min
in Mossbauer Gamma Optics (Moscow, Izd-vo MFU, 1982). Problems con
cerning the optics of chiral cholesteric liquid crystals in an experimental and
theoretical context may be found in Optics of Cholesteric Liquid Crystals,
written by the author together with A.S. Sonin (Moscow, Nauka, 1982).
The radiation of uniformly moving charged particles in media with the
periodicity of scalar characteristics not associated with channeling is con
sidered by G.M. Garibyan and C. Yang in X-Ray Transition Radiation
(Erevan, Izd-vo AN Arm. SSR, 1983) and V.L. Ginzburg and V.N. Tsy
tovich in Transition Radiation and Transition Scattering (Moscow, Nauka,
1984).
Quite a large part of this book is based on the results obtained at the
Theoretical Laboratory of the All-Union Research Institute for Physical-
Preface to the Russian Edition ix
Technical and Radiotechnical Measurements and in the Laboratory of Solid
State Theory of the All-Union Surface and Vacuum Research Centre and
published at different times in coauthorship with Yu. M. Aivazian, R.Ch.
Bokun, A.G. Grigorian, V.E. Dmitrienko, A.A. Grigor'ev, LV. Maslov, S.M.
Osadchii, V.P. Orlov, E.V. Smirnov, and N.V. Shipov. The author is very
grateful to all of these individuals.
The author also wishes to thank V.E. Dmitrienko and M.V. Fedorov,
who have read the manuscript, for their valuable remarks.
Contents
Preface to the English Edition v
Preface to the Russian Edition vii
Symbol Index xvii
Introduction xix
1 Waves in Media with One-Dimensional Periodicity
(Exact Solution) 1
1.1 Layered Medium 1
1.1.1 Harmonic Modulation of Media 1
1.1.2 Properties of Eigenwaves 4
1.2 Optics of Cholesterics (Exact Solution) 5
1.2.1 Dielectric Properties of Cholesterics 6
1.2.2 Eigenwaves 6
1.2.3 Properties of Eigensolutions 8
1.3 Solution of Boundary Problem 10
1.3.1 Planar Cholesteric Texture 11
1.3.2 Reflection from Thick Layers 13
1.3.3 The Case Where Medium and Cholesteric Have
Equal Dielectric Constants 15
1.3.4 Dependence of Reflection on Polarization 17
1.3.5 Effect of Dielectric Boundaries 17
1.3.6 Method of Characteristic Matrices 20
References 23
2 Approximate Description of Interaction of
Radiation with Regular Media 25
2.1 Kinematical Approximation 26
2.1.1 Scattering Cross Section 26
2.1.2 Polarization Characteristics 27
2.1.3 Scattering of Light in Cholesterics 29
2.1.4 Kinematical Approximation Limitations 32
2.2 Dynamical Theory 33
2.2.1 Set of Dynamical Equations 33
2.2.2 Dispersion Surfaces 35
2.2.3 Solution of the Boundary Problem 37
References 39
xii Contents
3 Diffraction of Mossbauer Radiation in
Magnetically Ordered Crystals 40
3.1 Diffraction of Mossbauer Radiation 40
3.2 Amplitude of Coherent Mossbauer Scattering 44
3.2.1 Amplitude of Elastic Scattering 45
3.2.2 Isotope and Spin Incoherence 48
3.2.3 Coherent Amplitude in Limiting Cases of
Completely Split and Unsplit Lines 49
3.3 Kinematical Theory of Mossbauer Diffraction 50
3.3.1 Structure Amplitude 50
3.3.2 Phase Determination 52
3.3.3 Diffraction at Magnetically Ordered Crystals 52
3.3.4 Resolved Zeeman Splitting of a Mossbauer Line 55
3.3.5 Crystals with Mossbauer Nuclei at Sites
with Inhomogeneous Electric Field (EFG) 58
3.4 Dynamical Theory of Mossbauer Diffraction 59
3.4.1 Unsplit Mossbauer Line 60
3.4.2 Suppression of Inelastic Channels of Nuclear
Reactions 62
3.4.3 Bragg Reflection from Crystals 64
3.4.4 Solution of the Dynamical Equations for
Hyperfine Splitting of Mossbauer Line 66
3.4.5 Polarization-Independent Amplitude of
Scattering at Zero Angle 70
3.4.6 Completely Resolved Zeeman Splitting of
Mossbauer Line 71
References 74
4 Optics of Chiral Liquid Crystals 80
4.1 Optics of Cholesteric Liquid Crystals (CLC) 80
4.1.1 The Fundamental Equations 81
4.1.2 Eigensolutions 83
4.1.3 The Solution of the Boundary Problem 85
4.1.4 High-Order Reflection for Oblique Incidence 90
4.1.5 A Planar Layer 95
4.1.6 Optical Rotation 98
4.2 Absorbing Cholesteric Liquid Crystals 105
4.2.1 The Dielectric Tensor of Absorbing CLCs 105
4.2.2 Suppression of Absorption 105
4.2.3 Optical Rotation 107
4.2.4 Borrmann Effect for Oblique Incidence 107
4.3 Chiral Smectic Liquid Crystals 108
4.3.1 Dielectric Permittivity Tensor of a C* 109
4.3.2 Second-Order Diffraction Reflection 110
4.3.3 First-Order Reflection 110
