Table Of ContentSpringer Series in Optical Sciences 203
Vladimir Belyakov
Diffraction Optics
of Complex-
Structured Periodic
Media
Localized Optical Modes of Spiral Media
Second Edition
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Volume 203
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Vladimir Belyakov
Diffraction Optics
of Complex-Structured
Periodic Media
Localized Optical Modes of Spiral Media
Second Edition
123
Vladimir Belyakov
L. D.LandauInstitute for Theoretical
Physics
Russian Academy of Sciences
Moscow,Russia
ISSN 0342-4111 ISSN 1556-1534 (electronic)
SpringerSeries inOptical Sciences
ISBN978-3-319-43481-0 ISBN978-3-319-43482-7 (eBook)
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Preface to the Second Edition
Aquarterofacenturyhaspassed sincethepublicationoftheDiffractionOpticsof
Complex Structured Periodic Media, Springer-Verlag. During the past time, it has
beenachievedagreatprogresspracticallyinallfieldsofresearchrelatedtodifferent
typesofradiationinteractionwiththematterstudiedinthisbook.Withoutreferring
totheopticalwavelengthrangeoftheelectromagneticradiation,themainadvances
can be summarized as follows. In the X-ray range, the mentioned advances are
related to the emergence and improvement of synchrotron radiation (SR) sources
and X-ray free-electron lasers (XFELs), which make it possible to study the phe-
nomenaofpolarizationandmagneticscatteringintheX-raywavelengthrange(see,
for example, V. E. Dmitrienko, E. N. Ovchinnokova, S. P. Collins et al., Nature
Physics 10, 202–206 (2014) doi:10.1038/nphys2859). A special mentioning
deservestheprogressinMossbauroptics.Theimplementationofexperimentswith
an extremely high energy resolution has become accessible in practice by the
invention and the constructing the SR Mössbauer source (see G. V. Smirnov,
U. van Burk, A. I. Chumakov, A. Q. R. Baron, and R. Ruffer, Phys. Rev. B 55,
5811 (1997)). Major advances in neutron optics are obtained in research on
localizedneutronstates(inEarth’sgravity,seeV.V.Nesvizhevskii,Phys.Usp,47,
515 (2004), condensed media whispering gallery modes, see V. V. Nesvizhevskii,
A. Yu. Voronin et al. Nature Physics, v. 6, p. 114 (2010), and localized modes in
periodic condensed structures, see V. A. Belyakov, JETP, 124, 873 (2017)). In
order to take into account the achievements in the above-mentioned fields, the
corresponding chapters of the first edition of the book should be practically
transformedintoseparatemonographs.Therefore,inordertomaintainareasonable
size of the book, only the recent achievements in photonics research, i.e., in the
optical wavelength range of electromagnetic radiation, are presented in the second
edition.
A new field of optical research has emerged over the last three recent decades
which relates to the optics of periodic structures with the value of periodicity
comparable to the optical wavelength and which has been named photonics.
Photonics has revealed many important phenomena that are well applied in the
optical wavelength range. However, formally all phenomena studied by photonics
v
vi PrefacetotheSecondEdition
were known and present in X-ray diffraction. It is well known that the corre-
sponding X-ray researches began much earlier than the photonics emerged.
Nevertheless, a new term, photonics, proves to be quite adequate to the field of
optical research named by this term. The reason for this is related to the fact that
virtually all photonics-related effects are within the capacity of experimental
observation at the contemporary level of optical techniques while most of the
corresponding effects in the X-ray wavelength range are out of the capacity of
experimental observationatthepresentlevelofX-ray techniques.Now,duetothe
appearance of synchrotron and free-electron laser X-ray sources, the already
observedphotoniceffectsintheopticsseemtobeaccessible forobservationinthe
X-ray wavelength range in the near future.
Photonics researches show that many photonics-related optical processes are
moreefficientthanthecorrespondingprocessesinhomogeneousmedia,asaknown
example can be called the lasing. The photonics terminology proposes the term
“photoniccrystal”relatedtoaperiodic medium whoseperiodvalue isintherange
oftheopticalwavelengthstodistinguishthephotoniccrystalsfromtheconventional
crystals.Tothedate,anumberofreviewsandbookshavebeenpublishedonlinear
and nonlinear optics of photonic crystals (see, for example, J. D. Joannopoulus,
R. D. Meade, and J. N. Winn, Photonic Crystals: Moulding the Flow of Light,
Princeton University Press, second edition, 2011; K. Sakoda Optical Properties of
Photonic Crystal, Springer-Verlag, second edition, 2005; I. A. Sukhoivanov and
I.V.Guryev, Photonic Crystals, Springer, 2014). A common feature of these pub-
licationsis,practically,anabsenceofanalyticaltheoryofthephotoniccrystaloptics
which reflects the complexity ofphotonic crystals in ageneral case.
In this second edition of the book, a special type of photonic crystal is studied,
namelyaspiralmedium,allowingthedevelopmentofananalytical theory,asnew
main objects of studies are chosen localized optical modes in spiral media, the
so-called edge mode (EM) and defect mode (DM). The dispersion equations
determining frequencies of localized modes for EM and DM are presented and
analyzed. It is demonstrated that many optical phenomena occurring at the fre-
quencies of localized modes reveal unusual properties which can be used for effi-
cientapplicationsofthecorrespondingphenomena,forexample,efficientfrequency
conversion, low-threshold lasing. The influence of phenomenological parameters
on the localized mode characteristics of the structures under consideration is, for
certainty, analyzed for chiral liquid crystals (CLCs). The cases of non-absorbing,
absorbing (amplifying) CLC layers are considered in detail for EM and DM. The
cases of active defect layers for defect modes are also considered. Options for
suppressingabsorptionatlocalizedmodefrequencyforCLCwithlocalanisotropic
absorption (Borrmann effect) are also investigated. The theory presented in the
book allows to describe the known experimental results on the efficient
low-threshold distributed feedback (DFB) lasing at the localized mode frequencies
in chiral liquid crystals in the framework of a single approach, to find localized
mode frequencies in their connection with the parameters of the object under
consideration and, some time, to predict an unknown effect. Main experimental
results related to the developed theory are also discussed. The presented analytic
PrefacetotheSecondEdition vii
resultsrelated,forexample,tothelow-thresholdDFBlasingandtotheabnormally
strong absorption at the localized mode frequencies, have as a direct application
ofthecorrespondingresultstorealphysicalobjectssuchaschiralliquidcrystals,so
can also be applied to the conventional photonic crystals (not spiral) as a first
approximation. No doubt that the analytical results presented in the book can be
used as a useful guide for the understanding of unusual optical properties of pho-
tonic crystals in general.
Whatisconcernedwiththestructure ofthebookitcontains,withtheexception
of new chapters devoted to the properties of the localized mode, also introductory
chapters devoted to the “prephotonic era” in the optics of spiral media. The
inclusioninthesecondeditionofthebooktheknown“prephotonicera”materialis
justified as by the intention to present an autonomous from many scattered over
various journals original papers on the subject book as well by the fact that in
English there is no easy accessible book on the optics of spiral media (a great
review on the subject by Belyakov, V. A.; Dmitrienko, V. E. Optics of Chiral
Liquid Crystals; In Soviet Scientific Reviews/Section A: Physics Reviews. (Ed.)
Khalatnikov, I. M. Harwood Academic Publisher: London, UK, 1989 is not
available in an electronic version).
The author expresses his sincere gratitude to S. V. Semenov, the author’s
co-authorinnumerousarticles onthesubjectofthebook,andisgrateful toSofiya
Tarshina, Ivan Belyakov, and Dariya Chufarova for the valuable technical assis-
tance in the preparation of the manuscript.
Moscow, Russia Vladimir Belyakov
December 2017
Preface to the First Edition
Little time has elapsed since 1988 when a Russian edition of this book was pub-
lished,butmostoftheproblemstouchedonherewereunderintensiveinvestigation
then. This indicates that the subject of the book was and is quite urgent, and as a
consequence,anumberofnewpublicationsonthesubjecthaveappearedsincethe
firstedition.Itiswhyinthecourseofpreparinganewedition,Idecidedthatmany
new results ought to be reflected in it.
Because I wanted to retain the general style of the original book and its overall
features,theproblemofintegrating newmaterial wassolvedinthefollowingway.
Three chapters, devoted to the fields of the most significant achievements, were
added: on polarization phenomena in X-ray optics, on X-ray magnetic scattering,
and on Mössbauer filtration of synchrotron radiation. Only minor alterations were
made to the other chapters; their end-of-chapter reference sections were also
updated.
IhopethattheEnglisheditionofthisbookwillgiveeasieraccesstomyWestern
colleagues to the original Soviet results. Soviet authors contributed quite signifi-
cantly to the subject of this book and look forward to developing joint investiga-
tionsofthecoherentinteractionofdifferentkindofradiationswithmatter.Iwould
also like to express my gratitude to the Springer-Verlag editing staff for collabo-
ration in this project.
Moscow, Russia Vladimir Belyakov
ix
Preface to the Russian Edition
Rapidprogressinexperimentaltechniquesinthepast15–20years,coupledwiththe
ever-increasingvarietyoftheobjectsofinvestigation,haveadvancedthestudyofa
series of phenomena which were beyond the scope of researchers and which are
very important for various applications.
Amongsuchphenomenaarethenumerousunusualeffectsobservedincoherent
interaction of various electromagnetic and corpuscular radiations 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 frequency, 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
conditionsunderwhichtheaboveeffectsmanifestthemselvesareverydiverseand,
at first glance, have quite a different nature, they are often considered unrelated
phenomena.Inmany studies, suchseparateconsiderations ofthese phenomenaare
detrimental, and instead of reinventing the wheel and ignoring the experience and
resultsofotherdistantfieldsofresearch,investigatorshaveonlytoexamine,infact,
closely related ones. It is worthy to remind the reader here of the words written of
L.BrillouinandM.Parodisome30yearsagointheforewordtotheirnowfamous
Wave Propagation in Periodic Structures. “No matter how diverse the problems
are, their mathematical treatment results in equations having a series of common
propertiesandsimilarsolutions.Thisdeeprelationbetweenquitedifferentproblems
wasverywellknownalreadytoKelvinandRayleigh.Butthenithasbeenforgotten
and only in recent works this very important statement has attracted attention
anew.”Unfortunately,the aboveaffirmation sounds asurgent today, inapplication
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 Mössbauer
gamma radiation by magnetically ordered crystals, optics of chiralliquid crystals,
xi
