DAYS ON DIFFRACTION
INTERNATIONAL CONFERENCE
SaintPetersburg May June
ABSTRACTS
UNIVERSITAS
PETROPOLITANA MDCCXXIV
ORGANIZING COMMITTEE
Prof V M Babich Russia Prof V S Buldyrev Russia
Prof R Stone USA Dr N Zhu Germany Dr V P Smyshlyaev UK
Dr V E Grikurov The secretary Dr I V Andronov Pro ceedings Dr E V Novikova Visas
Prof M A Lyalinov Prof N Ya Kirpichnikova Prof A P Kiselev Prof A M Samsonov
ADDRESS
Prof V M Babich Prof V S Buldyrev
Mathematical Institute Institute in Physics St Petersburg University
Fontanka quay St Petersburg Ulianovskaja Ave Petro dvoretz Russia
Russia e mail buld math nw ru
e mail babich pdmi ras ru Fax
Dr V E Grikurov e mail grikurov math nw ru
Dr I V Andronov e mail iva list ru
Web site http math nw ru DD
DAYS on DIFFRACTION
FOREWORD
The Seminars Conferences Days on Di raction are annually held since in late Mayor
in June by the FacultyofPhysics of St Petersburg State University St Petersburg Branchofthe
Steklov s Mathematical Institute and Euler International Mathematical Institute of the Russian Aca
demy of Sciences
This b o oklet contains the abstracts of talks to b e presented at oral and p oster sessions in
days of the Conference Author index can b e found on the last page
The full texts of selected talks will b e published in the Pro ceedings of the Conference The texts in
A
L T X format are due by Septemb er to e mail iva list ru Format le and instructions
E
can b e found on the Seminar Web site at http math nw ru dd rules html The nal judgement
on accepting the pap er for the Pro ceedings will b e made by the Organizing Committee following the
recommendations of the referees
We are as always pleased to see in St Petersburg active researchers in the eld of Di raction
Theory from all over the world
Organizing Committee
The conference is sp onsored by
IEEE ED MTT AP
St Petersburg Chapter
Russian Foundation
Russian Academy of Sciences
for Basic Research
DAYS on DIFFRACTION
List of talks
Valeriy A Ab dulkadyrov Dmitry V Ab dulkadyrov
The di raction and disp ersion of waves in semiconductor di raction grating structure
Eugeny G Abramo chkin Vladimir G Volostnikov
Wigner distribution function and intensityintegral moments of Hermite Laguerre
Gaussian b eams
Thomas M Acho
Derivation of eigenvalues for the Sturm Liouville b oundary value problem with interior
singularities
K L Aghayan
Plane shear wave di raction in a comp osite elastic medium with partially b onded elastic
strip
Mikhail V Balaban Ayhan Altintas
Application of Ab el s integral transform to solving the electromagnetic wave di raction
byathindisk
M M Arakelyan K T Avetyan E A Nazaryan
Features of propagation and interaction of one dimensional top ological solitons in crystals
Agissilaos G Athanassoulis
Smo othed Wigner Transforms and homogenization of wave propagation
A G Bagdo ev A V Vardanyan S V Vardanyan A N Martirosyan
The analytic and numerical solution of second thrith fourth and sixth order Wienner
Hopf system for mixed b oundary elasticity dynamic problems
M A Bastos A Moura Santos
Perio dic di raction b oundary value problems
Pavel A Belov Yang Hao Mario G Silveirinha Constantin R Simovski
Subwavelength imaging at microwave terahertz and infrared frequencies
V V Belov S V Nikolaev
Integrable mo dels of the longitudinal motion of electrons in curved D nanotub es
Vladimir VladimirovichBelov Ekaterina Ivanovna Smirnova
Semiclassical solyton typ e solutions of the Hartree equation
D Bianchi B Tirozzi S Reutskiy
Prediction of the tra jectory of typho ons and the Maslov decomp osition
D Bianchi Brunello Tirozzi
Kohonen neural networks and genetic classi cation
Jianfeng Feng Mariya Shcherbina Brunello Tirozzi
Dynamical b ehavior of a large complex system
A Boag
Non uniform grid NG based compression of the metho d of moments matrices
Ya L Bogomolov E S Semenov A D Yunakovsky
Optimization of paraxial region for quasi optical electron accelerator
Victor V Borisov
The transientwaves pro duced byhyp erb olic motion of Gaussian s transverse sources
V A Borovikov A L Pop ov D A Chelub eyev
Sound eld excited by exural oscillations of elastic plate with round inclusion part
Serge P Boruhovich
Planar and three dimensional chirality measure mo del
V I Bronnikov
Di ractive mo del of scattering by a rough surface of radiation
DAYS on DIFFRACTION
I S Lobanov I Yu Pop ov
Two b o dy problem on p encil of lines
J Bruning S Yu Dobrokhotov R V Nekrasov A I Shafarevich
Propagation of the wavepackets in thin tub es with nonlinear integral p otential
Ya A Butko
Functional integrals for Schro edinger equation on a Riemannian manifold via Smolyanov
Weizsaecker approach
Volo dymyr O Byelobrov Alexander I Nosich
Dep endences of lasing thresholds for a layered structure with a quantum well on the
modesymmetry
A A Chernitskii
E ective Riemann space for wave propagation in nonlinear electro dynamics
V V Borzov E V Damaskinsky
Bessel Oscillator
M V Golovkina
Waveguide structure containing metamaterial slab with resistive lm
S Yu Dobrokhotov A I Shafarevich B Tirozzi
One representation of lo calized functions via the Maslov canonical op erator and its
application to asymptotic solutions of linear hyp erb olic systems
S Yu Dobrokhotov S O Sinitsyn
Lo calized asymptotic solutions of d wave equation with variable velo city implied by
time dep endentsource
A L Efros
The problem of a p erfect lens made of a slab with negative refraction
V G Daniele R D Graglia
Di raction of a p erfectly conducting half plane immersed in a gyrotropic medium
V A Es kin A V Kudrin T M Zab oronkova
Excitation of whistler mo des guided by a lossy anisotropic plasma cylinder
I V Donets V A Lerer A M Lerer S M Tsvetkovskaya
Investigation of multilayered and multislot resonant and p erio dical structures
Ali Ta jvidy Mohammad Nasermoghaddasi Ayaz Ghorbani
Multiple di raction loss calculation in micro cell environments
F F Gorbatsevich
Dep olarization of shear waves in anisotropic heterogeneous media
R D Graglia G Lombardi
A mo di ed Euler transformation for machine precision evaluation of p otential integrals
Valery E Grikurov
Windowed oscillating integral
Gaussian b eams metho d b eyond its limitations
Michael Katsav Ehud Heyman
Beam summation analysis of half plane di raction
Michael Katsav Ehud Heyman
A phase space b eam summation representation for D radiation from a line source
distribution
Ya A Ilyushin
The impact of the small scale uctuations of the ionospheric plasma on the ultra wide
band orbital ground p enetrating radar measurements
Hiromu Ishio
An attempt to improve semiclassical calculations of wave transp ort in a quantum wire
with Di raction
DAYS on DIFFRACTION
M Shatalov S V Joub ert W E Pretorius I Fedotov
Vibrations of inhomogeneous solid discs sub jected to an inertial rotation
P H Klopp ers T H Fay S V Joub ert
A probabilistic approachtobi variate function approximation
S V Joub ert M Shatalov C E Co etzee I Fedotov
Vibrations of a rotating solid elastic sphere lled with an inviscid uid
S V Joub ert T H Fay M Shatalov
Dynamics of a vehicle trailer system with a limited p ower supply
M V Karasev
Quantization of surfaces in magnetic eld
Karima Khusnutdinova
Nonlinear waves in coupled waveguides
N B Konyukhova M B Soloviev P M Lima M L Morgado
Singular nonlinear b oundary value problem for bubble typ e or droplet typ e solutions
in nonlinear physics mo dels
Yurii V Kiselev Vladimir N Troyan
The di raction tomography and iterative approach to restore the elastic parameters
and electrical conductivity
S A Kozlov P A Petroshenko
The nonparaxial self fo cusing of few cycle light pulses in dielectric media with disp ersion
P A Krutitskii
Boundary value problems for the Helmholtz equation in domains b ounded by closed
curvesandopenarcs
Valeri V Kucherenko
Semiclassic asymptotics for the vector Sturm Liouville problem with parameters
Andriy Kryvko Valeri V Kucherenko
Asymptotic solution of linear system with a turning p oint of high order
R S Makin
Averaging and chaotic mo des for nonautonomic nonlinear dynamic systems
D P Kouzov Yu A Solov eva
Di raction of nonstationary wavewithvarying along the front amplitude by a soft wedge
I S Nefedov
Sp ectra of mo des of waveguides loaded with wire media
M D Kovalev
The numb er of energy levels in MQW structure
Alexander G Kyurkchan Dmitrii B Demin
Solution scattering problems of electromagnetic waves from inhomogeneously layered
scatterers using pattern equation metho d
Alexander G Kyurkchan Nadezhda I Smirnova
Metho ds of the continued b oundary conditions and the pattern equations
A G Kyurkchan E A Skoro dumova
Mo deling the characteristics of scattering of electromagnetic waves by the b o dies of
complex geometry
A G Kyurkchan S A Manenkov E S Negorozhina
Mo delling of scattering characteristics of electromagnetic waves by group of b o dies using
the mo di ed metho d of discrete sources
Fedor N Litvinets Andrey Yu Trifonov Alexander V Shap ovalov
Berry phases for the Hartree typ e equation
V G Daniele G Lombardi
Factorization of Wiener Hopf matrices using Fredholm integral equations
DAYS on DIFFRACTION
V S Makin A Y Vorobyev
Mo del for the regular nanostructuring of transparent dielectrics and semiconductors by
femtosecond laser radiation
Raul Mahillo Isla Mar a Jesus Gonz alez Morales Carlos Dehesa Mart nez
Di raction of D complex b eams by a p erfect conductor half plane a sp ectral approach
R S Makin
Gap condition for nonlinear evolution equations
R S Makin
The numb er of zero es in invariant manifolds for evolution equations
V S Makin V V Trubaev
The comparison of elds excited by input grating coupler with those of SEW
G J McPhee P N Stavrinou D D C Bradley
On plasmonic and other mo des of double metal slab structures
V S Makin R S Makin Chunlei Guo A Y Vorobyev
UniversalityofFeigenbaum and dissipative microstructures for highly nonequilibrium
nonlinear systems
G Maximov
Biot s equations for multiphase media with di erent temp eratures on the basis of gen
eralized variational principle
StanislavI Maslovski
Re ection from a half space lled with wire medium the auxiliary source metho d
O V Motygin
On uniqueness in the two and three dimensional
Neumann Kelvin problem
Vassili A Fedotov Nikolay I Zheludev Pavel L Mladyonov Sergey L Prosvirnin
Electromagnetic prop erties of double p erio dic planar chiral array
K V Prozorov P A Krutitskii
The mixed value problem with the skew derivative
for the D Helmholtz equation outside cuts
G A Omel yanov
Interaction of sho ckwaves for genuinely nonlinear hyp erb olic systems
M Odit I Vendik O Vendik
D isotropic metamaterial based on dielectric resonant spheres
E G Papkelis H T Anastassiu P V Frangos
PO PTD near eld scattering and di raction metho d for path loss prediction in urban
mobile radio systems
Valeriy V Radchenko
Two layer spherical dielectric lens with re ector excited by a radial electric dip ole
D Bianchi F Raicich
Stability of p erturbation of ows and typho on generation
X Tr Tran N N Rosanov
Interaction of dissipative b er Bragg solitons
S V Fedorov N N Rosanov N A Veretenov
Complexes of weakly coupled D laser solitons
Al S Kiselev An S Kiselev N N Rosanov
E ect of Kerr nonlinearity on discrete dissipative optical solitons
Nataliya Sakhnenko Alexander Nerukh
Electromagnetic eld transformation in circular waveguiding and resonant structures
with time varying p ermittivity
DAYS on DIFFRACTION
S S Sautb ekov
Solving the problems of di raction by Wiener Hopf Fock metho d for nite ne structures
S Ya Sekerzh Zenkovich B I Volkov
On application of asymptotic solutions to calculation of tsunami piston mo del
Alexander A Chaplygin Victor I Sergeev
On the information transmission metho d using material ob jects far action reaction on
the external electromagnetic unconverted in uence
A Shanin E Shabalina
Numerical study of di raction phenomena in achitectural acoustics
V M Shelkovich
Singular solutions to systems of conservation laws connected with transp ortation and
concentration pro cesses
Yu A Shp olyanskiy A N Berkovsky M A Berlin S A Kozlov
Self fo cusing of few cycle light pulses in transparent disp ersive optical media with elec
tronic and electronic vibration cubic nonlinearities
Ilya A Shereshevskii
The scattering op erator in the stepwise waveguides
S M Sitnik
Explicit solutions to a singular di erential equation with Bessel op erator
Serguei A Nazarov
On continuous sp ectra of the elasticity problems
Alexander A Sidorov
Anisotropic approximation of radially layered elastic media
Irina I Simonenko
Transientwaves pro duced by a spherical source pulsating with di erent p erio ds
Y Smirnov E Sysova
Analysis of the TE wave propagation in nonlinear dielectric three layer planar waveg
uides with non Kerr nonlinearity
Y Smirnov D Valovik
Analysis of the TM wave propagation in nonlinear dielectric layer planar waveguides
with Kerr nonlinearity
O G Smolyanov
Cauchy problems in b ounded domains Feynman path integrals and difusions
Elena I Smotrova Alexander I Nosich Trevor M Benson Phillip Sewell
Lasing sp ectra and thresholds of a circular micro cavity laser emb edded in an annular
Bragg re ector
I V Sturova
The scattering of the lo calized surface waveby an elastic plate oating on shallowwater
of variable depth
T Tovstik
The waves propagation in the comp osed ro d
Sergey Yalunin
Casimir e ect and exactly integrable Hamiltonians
S A Tretyakov I S Nefedov
Transformation of electromagnetic elds using metamaterials
Andrei B Utkin
Pulsed radiation pro duced by a travelling exp onentially decaying bip olar current pulse
with high frequency lling
DAYS on DIFFRACTION
Roman O Rezaev Andrey Yu Trifonov Alexander V Shap ovalov
One dimensional nonlinear FPK equation with the p otential of a sp ecial form in semi
classical approximation
Andrey Yu Trifonov Roman O Rezaev Alexander V Shap ovalov
Symmetry op erators for the multidimensional FPK equation with quadratic nonlo cal
nonlinearity
S A Vakulenko
Asymptotics for nonlinear wave equations with external source
M V Ivakhnychenko E I Veliev T M Ahmedov
Fractional b oundary conditions in di raction problems on plane screens
Irina Vendik
Metamaterials based on structures with emb edded D resonant inclusions
A V Vozianova A G Nerukh
Resolvent op erator of maxwell equations for dimentional eld vector
A P Vinogradov A M Merzlikin A V Dorofeenko
Enhancement of magneto optic e ects with resonant structures
A M Merzlikin A V Dorofeenko A P Vinogradov
Distinct feature of magneto photonic crystals on formation of the Yeh band gap
A P Vinogradov A V Dorofeenko A M Merzlikin
A comparative analysis of mixing formula for SNG and DNG media
Alexander A Vinokurov
Analisys of applicability ranges of exact light scattering metho ds using spherical basis
M A Geyer T B Yanovskaya
Numerical approach for analysis of surface wave eld in the vicinity of caustics
N F Yashina T M Zaboronkova
Electromagnetic surface waves guided by the metal comp osite medium b oundary
Vasyl V Yatsyk
The problem of di raction of a plane electromagnetic waveonakerr typ e nonlinear
dielectric layeredstructure
S Yu Dobrokhotov P N Zhevandrov B Tirozzi
Asymptotic theory of the one dimensional linear scattering of the solitary waves on a
beach
Vera B Zon
Re ection refraction and transformation into photons of surface plasmons on metal
wedge
A A Zvyagintsev A I Ivanov D V Katkov
Whisp ering gallery eigenmo des of a surface with anisotropic surface imp edance
George V Filipp enko
Vibration of cylindrical shell partially submerged into the layer of liquid The liquid is
inside and outside the cylinder
F F Valiev
Pulse shap e of current moving along a straight line with sup erlightvelo city and time
derivative of Whittaker p otential in far region
Lev A Molotkov
Investigation of wave eld in e ective mo del of layered elastic medium with slide contact
on interfaces
V G Veselago
Ab out light pressure in negative refraction materials
DAYS on DIFFRACTION
The di raction and disp ersion of waves in
semiconductor di raction grating structure
Valeriy A Ab dulkadyrov Dmitry V Ab dulkadyrov
The Usikov s Institute of Radiophysics and Electronics of the NASU
Proskura str Kharkov Ukraine
e mail abdulkad ire kharkov ua
The wave phenomena in semiconductor structures have aroused considerable interest among physi
cists mathematicians and practicing sp ecialists The close relationship existing b etween the wave
phenomena and the fundamental prop erties of solids has spurred this interest for the knowledge of
the kind can contribute a lot to the improvement of the active and passive devices
In the rep ort the electromagnetic wave di raction by a di raction grating closely spaced
with a semiconductor layer backed by a dielectric substrate is considered relying up on Foigt s ge
ometry The structure of interest is shown in the gure The semiconductor eld is obtained
from the simultaneous solution
of Maxwell s equations and the
hydro dynamic equations of the
semiconductor plasma The pla
sma electrical prop erties are de
scrib ed by the tensor
ik
xx xy
yx yy ik
zz
After the elds have b een sub
ject to b oundary conditions at
all the structure interfaces one
gets the functional equation sys
tem reducible to the problem of
inhomogeneous conjugation of a
unit circle in the complex plane
which the Riemann Hilb ert
problem is The Fourier co e
cients of the obtained analytical function come from the in nite set of inhomogeneous linear equations
In view of the problem parameter smallness it is substituted by the reduced system which yields
the unknown co e cients of eld expansion to furnish us with the re ection and transmission factors
Having known the structure elds one gets the disp ersion equation from the b oundary value
problem The disp ersion equation solution yields the propagation constants of plasma p olaritons in
either isotropic or magnetoactive semiconductor layer closely spaced with the di raction grating The
behavior of the disp ersion curves has b een examined dep ending on the structure parameters
References
M C Steele and B Vural Waveinteraction in solid state plasmas Moscow Atomizdat
N I Muskhelishvili Singular integral equations Moscow Fizmatgiz
V P Shestopalov L N Litvinenko S A Masalov and V G Sologub Wave di raction on gratings
Kharkov Kharkov University
DAYS on DIFFRACTION
Wigner distribution function and intensityintegral
moments of Hermite Laguerre Gaussian b eams
Eugeny G Abramo chkin VladimirG Volostnikov
Coherent Optics Lab Samara branchofP N Leb edev Physical Institute of RAS Russia
e mail ega fian smr ru
Hermite Laguerre Gaussian HLG b eams are recently intro duced light b eam family that contains
Hermite Gaussian and Laguerre Gaussian b eams as particular representatives The Wigner function
and some other integral characteristics for HLG b eams are found
The Wigher distribution function plays an imp ortant role in quantum mechanics and optics It is
used for optical signals analysis and light elds characterization
In paraxial approximation the most famous examples of light elds with structural stability under
propagation in Fresnel zone are Hermite Gaussian HG and Laguerre Gaussian LG b eams The
Wigner functions for these elds were found in and resp ectively
Recently a unity of HG and LG b eam families was prop osed byintro ducing an additional
parameter The family of Hermite Laguerre Gaussian HLG b eams G x y j keep many im
n m
p ortant prop erties of HG and LG b eams In particular for any xed the family of HLG b eams is
an orthogonal basis of the space L R
In this work the Wigner function for HLG b eams is found with the help of generating function
technique Using this result it is shown that Fourier transform of an HLG mo de intensitymaybe
presented in terms of Laguerre p olynomials
ZZ
i x y
d d G j e
n m
R
x y x cos y sin x sin y cos
E exp L L
n m
n m
n m is the energy of an HLG mo de G x y j or the norm square of the mo de where E
n m
in the space L R
By di erentiation of eq the intensityintegral moments are found
ZZ
N M N M
hx y i G x y j x y dx dy
n m
R
For example for n andm the moments are
n m n m
h i E hxi hy i hxy i hx y i E hx y i E cos
The ab oveformulae are used for the calculation of some characteristics of HLG mo des Namely
b eam quality factor M see de nition in M G x y j n m
n m
phase space area see de nition in PSA G x y j n m
n m
References
Bastiaans M J Application of the Wigner distribution function in optics In The
Wigner Distribution Theory and Applications in Signal Processing eds W Mecklenbr auker
F Hlawatsch Elsevier Amsterdam pp
http www sps ele tue nl members M J Bastiaans pdfs wigner pdf
Simon R Mukunda N Optical phase space Wigner representation and invariant
quality parameters J Opt Soc Am A Vol pp
Tatarsky V I Wigner representation in quantum mechanics Soviet Physics Uspekhi
Vol pp
http www ufn ru ufn ufn Russian r b pdf in Russian
DAYS on DIFFRACTION
Gase R Representation of Laguerre Gaussian mo des by the Wigner distribution function
Quantum Electronics Vol pp
Abramo chkin E G Volostnikov V G Generalized Gaussian b eams J Optics A Pure
and Appl Optics Vol pp S S
http www iop org EJ article joa pdf
Sagha S Sheppard C J R The b eam propagation factor for higher order Gaussian
b eams Optics Commun Vol pp
Murphy J B Phase space conservation and selection rules for astigmatic mo de converters
Optics Commun Vol pp
Derivation of eigenvalues for the Sturm Liouville
b oundary value problem with interior singularities
Thomas M Acho
Department of Mathematics and Applied Mathematics
University of the Free State South Africa
Asymptotic solutions for the Sturm Liouville b oundary value problem with interior singulari
ties were obtained using asymptotic forms of the Whittaker functions for higher order mo des and
Titchmarsh Weyl m functions for low order mo des However these split interval techniques do not
readily provide the eigenrelations for low order mo des For the rst time with minimal constraints the
eigenvalues for the Sturm Liouville eigenproblem are obtained when the Titchmarsh Weyl m function
technique is employed
Plane shear wave di raction in a comp osite elastic
medium with partially b onded elastic strip
K L Aghayan
Institute of Mechanics ANAS RA
A comp osite elastic medium which consists of an elastic strip jxj jy j h jz j with the
width h and two identical half spaces y h y h is considered in a Cartesian co ordinate system
xy z The strip and the half spaces have di erent elastic prop erties It is assumed that along the half
panes x y h the strip and the half space are b onded full contact and along the half panes
x y h there is no contact b etween them and the half panes are free of stresses crack In
the elastic half spaces the plane shear wave
u x y exp ik x cos ik y sin
z
p
is incident from in nity at angle pi where k c is the wavenumber and c
is the velo city of shear waves in the half spaces It is assumed that the medium is in the state of
anti plane deformation and the task is to investigate the di racted wave eld in the strip and in the
half spaces
Representing the solution of the problem in the form of a sum of its even and o dd parts and using
the generalized Fourier transformations the problem is brought to solving two indep endent Wiener
Hopf functional equations
Analytical expressions are obtained for the amplitudes of the displacements in each part of the
comp osite medium Asymptotic formulas are presented expressing displacements in the far eld and
the distribution of stresses in the neighb orho o ds of the tips of contact zones tips of the cracks
Explicit expressions are obtained for the co e cients of transmission to the free part of the waveguide
DAYS on DIFFRACTION
Application of Ab el s integral transform to solving the
electromagnetic wave di raction by a thin disk
Mikhail V Balaban
Institute of Radiophysics and Electronics NASU Ul Proskury Kharkov Ukraine
Ayhan Altintas
Bilkent University Electrical Electronics Engineering Department
Bilkent Ankara Turkey
In this pap er we consider the di raction of arbitrary electromagnetic waveby thin PEC resistive
and dielectric disks and prop ose way to build a rigorous mathematical mo del of this group of prob
lems We start with Maxwell s equation generalized b oundary
conditions on the median cross section plane of the disk
E E Z R n H H
tg tg tg tg
Z H H S n E E
tg tg tg tg
and radiation and edge conditions Electrical and magnetic resis
tivities for a thin dielectric disk are given by
p p
R i Z cot k S i Z cot k j j
r r r r r r
We represent the normal to the disk scattering eld comp onents in terms of integral Hankel trans
form image functions
Z
X
im i
l
E e e e J d
m
sc z m z
m
Z
X
i im
l
Z H e h J d e
m
sc z m z
m
On substituting the tangential eld comp onents into the b oundary conditions we obtain a set or
two indep endent sets in the case of dielectric disk of coupled dual integral equations For the PEC
disk di raction problem they are
R
m
i j j i j j
J ka h e h J d i e e e
m m m m
m m
R
m
i j j i j j
e e e J ik a h e h J d
m m m m m
m
R
m
J h J d ik a e
m m
m m
R
m
ka e J i h J d
m m
m m
Weintegrate and decouple these equations and intro duce two or four constants of integration
By using Ab el s integral trans form technique for solving canonical dual integral equations with the
Bessel function kernels
R R R R
u J d f J d v J d f J d
R R
u J d v J d
we reduce our equations to a set of vector integral equations of the Fredholm second kind and
add equations for nding the con stants of integration The features of the Fredholm second kind
DAYS on DIFFRACTION
equations guarantee the existence of
solution and numerical con vergence
of the algorithm based on any reason
able discretization scheme
As an example weshow total far
eld radiation patterns for the PEC
disk excitation by a horizontal elec
tric dip ole Here the normalized disk
radius is ka normalized distance
from the dip ole to the disk is and
radiation patterns are plotted in the
planes of and
Features of propagation and interaction of
one dimensional top ological solitons in crystals
M M Arakelyan K T Avetyan E A Nazaryan
The Yerevan State University Armenia
e mail nazaryan ernest netsys am
The interaction of two analogous top ological solitons the Frenkel Kontorova dislo cations at joint
motion with similar velo cities has b een investigated in some sp eci ed velo cityinterval and the fea
tures of their motion have b een studied as well dep ending on the height of p erio dical p otential barrier
of lattice Based on a numerical exp eriment for soliton solutions of the nonlinear sinus Gordon wave
equation the following results were obtained the interaction b etween the dislo cations proved imp or
tantatlowvelo cities the displacement eld of some given dislo cation was strongly distorted under
the action of neighb oring dislo cation In our opinion in this case a transverse slide of the dislo cation
in question is observed The interaction force b etween these dislo cations decreases with increasing
velo city and in the limit when the dislo cation velo city tends to the velo city of sound the interaction
between the dislo cations tends to zero Having in view the interpretation of obtained results wehave
studied the distribution of stresses around the dislo cations in the velo cityinterval at issue Accord
ing to results of numerical exp eriment as the velo city increases the stress eld is contracted in the
direction of slide and grows in the normal direction by converging to One may assume thus that
the interaction force b etween the dislo cations is determined by the distribution width of stresses of
a ecting dislo cation It was then obtained that the b ehavior of dislo cation motion dep ended up on the
height of p erio dical p otential barrier of crystal lattice In case of low barriers the motion is transla
tional In the course of slide the dislo cation width is p erio dically changed At an increase of barrier
height the nature of motion changes In high barrier crystals the dislo cation as if executes oscillating
motion in the direction of slide The tail end ranges of dislo cations are distorted The width of dis
lo cation p erio dically changes remaining all the time less than that in low barrier crystals When the
allowance for the frictional force is made the motion of dislo cations slowed down the amplitude of
oscillations in the sliding direction is notably reduced
When the allowance for the frictional force is made the motion of dislo cations slowed down the
amplitude of oscillations in the sliding direction is notably reduced
The change of shap e of dislo cations in the course of slip has b een studied for b oth the high and
low barriers For that in the numerical exp eriment describ ed ab ove the distribution of displacement
derivatives was given for xed instants of time It was obtained that in the same time interval this
function p erio dically changed the amplitude of variation at higher barriers b eing more than for lower
ones Resp ectively the shap e of dislo cations for higher barriers is distorted stronger than for lower
ones that is esp ecially notable in the tail end regions
In all cases wegive detailed graphical results of the numerical exp eriment as well as made numerical estimates
DAYS on DIFFRACTION
Smo othed Wigner Transforms and homogenization of wave propagation
Agissilaos G Athanassoulis
The Program in Applied and Computational Mathematics Princeton University USA
e mail aathanas math princeton edu
The Wigner Transform WT has b een extensively used in the formulation of phase space mo dels
foravarietyofwave propagation problems including high frequency limits and the homogenization of
linear nonlinear and random waves The WT however is well known to present signi cant di culties
in the formulation of numerical schemes due to its so called interference terms
In this connection we prop ose the use of the smo othed Wigner Transforms SWT and derive
new exact equations for them covering a broad class of wave propagation problems The smo othing
i e the joint spatial sp ectral resolution at whichwecho ose to reformulate the original problem
now b ecomes part of the data of the phase space equations Equations for sp ectrograms and Husimi
Transforms are included as a sp ecial case
As an application a semiclassical numerical metho d is develop ed and analyzed making use of the
SWT It allows for the e cient and accurate recovery of dependendt slow scale observables e g the
envelop es in problems of semiclassical i e high frequency wave propagation
Working with concrete examples for the semiclassical linear Schro dinger equation it is seen that the
SWT approach is indeed signi cantly cheap er to work with than full numerical solutions of the original
equation in the semiclassical regime as well as than using the WT without smo othing Comparisons
with exact and indep endent numerical solutions are used to keep trackofnumerical errors
The analytic and numerical solution of second thrith fourth and sixth order
Wienner Hopf system for mixed b oundary elasticity dynamic problems
A G Bagdo ev A V Vardanyan S V Vardanyan A N Martirosyan
The antiplane plane and three dimensional unsteady elastic mixed problems for semi in nite cracks
in elastic plane and space and for stamps applied to elastic halfplane or halfspace are solved The
solution is obtained byintegral transformations metho ds Laplace on time and Fourier by co ordinates
The systems of Wienner Hopf equations are obtained from b oundary conditions are brought to Hilb ert
problem for vector functions of corresp onding order whichby metho d of Plemeli Hilb ert N P Vekua
is reduced to solution of the same order Fredholm integral equations system In mentioned problems
corresp onding matrix has singularities on in nity which are avoided and are obtained corresp onding
Wienner Hopf Hilb ert and furtehermore Fredgolm systems with continuous matrix The last ones
are solved numerically whichgives the technic of factorization of complex matrices Also the inverse
integral transformations are carried out and in all problems are obtained e ective forms of Smirnov
Sob olev for stresses on corresp onding parts of b oundaries The stress intensity co e cients near edges
of cracks and stamps are derived and calculated numerically
Perio dic di raction b oundary value problems
M A Bastos A Moura Santos
Dept of Mathematics I S T Technical University of Lisb on Lisb on Portugal
We consider a wave di raction problem by a p erio dic strip grating Two b oundary value problems
are studied in detail for an arbitrary geometry of the grating the oblique derivative and the classic
Neumann b oundary value problems After we formulate these problems as convolution typ e op erators
acting on Bessel p otential p erio dic spaces asso ciated op erators acting on spaces of matrix functions
de ned on comp osed contours are derived Within the last setting Fredholm prop erties and Fredholm
indices for the oblique derivative and the Neumann b oundary value problems are given Su cient
conditions for the invertibility of the Fourier symb ols of the asso ciated op erators are studied
DAYS on DIFFRACTION
Subwavelength imaging at microwave terahertz and infrared frequencies
Pavel A Belov Yang Hao
Department of Electronic Engineering Queen Mary University of London
London E NS UK
Mario G Silveirinha
Departamento de Eng Electrotecnica da Universidade de Coimbra Instituto de Telecomu
nicaco es Polo I I Coimbra Portugal
Constantin R Simovski
General Physics Department St Petersburg State University of Information Technologies
Mechanics and Optics Sablinskaya St Petersburg Russia
We study b oth theoretically and exp erimentally subwavelength imaging by a slab of le wire medium
two dimensional electromagnetic crystal formed by parallel conducting wires WM supp orts so
called transmission line mo des whichhave completely at isofrequency contours and exist at
very low frequencies as compared to the p erio ds of the lattice In order to achievesubwavelength
imaging wedonotinvolvenegative refraction and ampli cation of evanescent mo des We prop ose
to transform the most part of the spatial sp ectrum of the source radiation into the propagating
transmission line mo des of the crystal having the same group velo city directed along wires and
across the slab resp ectively and the same longitudinal comp onents of the wavevector The spatial
harmonics pro duced by a source propagating and evanescent refract into the crystal eigenmo des at
the frontinterface These eigenmo des propagate normally to the interface and deliver the distribution
of near eld electric eld from the frontinterface to the backinterface without disturbances This way
the incident eld with subwavelength details is transp orted from one interface to the other one The
problem of strong harmful re ection from the slab is solved bycho osing its thickness appropriately
so that it op erates as a Fabry Perot resonator In our case the Fabry Perot resonance holds for
all incidence angles and even for incidentevanescentwaves We call the describ ed regime as
canalization with subwavelength resolution The initial results on subwavelength imaging by wire
medium were published in In the presentwork we present recent results regarding bandwidth
of op eration and resolution of the imaging system We demonstrate transmission of images with
l resolution to the l distances Also we discuss opp ortunity to scale the structure so that it can
op erate at terahertz and infrared frequencies
References
P A Belov R Marques S I Maslovski I S Nefedov Silverinha M C R Simovski S A
Tretyakov Strong spatial disp ersion in wire media in the very large wavelength limit Phys
Rev B vol
P A Belov Y Hao S Sudhakaran Subwavelength microwave imaging using an array of parallel
conducting wires as a lens Physical Review B vol
P A Belov C R Simovski P Ikonen Canalization of subwavelength images by electromagnetic
crystals Physical Review B vol
P A Belov Y Zhao S Sudhakaran A Alomainy Y Hao Exp erimental study of the sub
wavelength imaging by a wire medium slab Applied Physics Letters vol pp
M G Silveirinha P A Belov C R Simovski Subwavelength imaging at infrared frequencies
using an array of metallic nanoro ds Physical Review B vol pp
DAYS on DIFFRACTION
Integrable mo dels of the longitudinal motion
of electrons in curved D nanotub es
V V Belov S V Nikolaev
Moscow State Institute of Electronics and Mathematics Russia
In the framework of the adiabatic approach we nd several mo del examples of curved D
nanotub es and con gurations of a homogeneous magnetic eld for which the one dimensional matrix
equation can b e integrated exactly on a subregion of dimensional quantization For p erio dic tub es
with the Born Karman conditions we obtain explicit analytic formulas of the electron sp ectrum
and the time domain Green function of the corresp onding initially b oundary value problem In the
electron sp ectrum we distinguish the corresp onding contributions of the Aharonov Bohm and Berry
phases and the Maslov geometric p otential
References
V V Belov S Yu Dobrokhotov T Ya Tudorovskii Asymptotic solutions of nonrelativistic
equations of quantum mechanics in curved nanotub es Teor Mat Fiz
V V Belov S Yu Dobrokhotov T Ya Tudorovskii Generalized adiabatic principle for describ
ing the electron dynamics in curved nanotub es Usphekhi Fiz Nauk
Semiclassical solyton typ e solutions of the Hartree equation
Vladimir VladimirovichBelov Ekaterina Ivanovna Smirnova
The Moscow institute of electronics and mathematics
Moscow B Trehsvjatitelsky st Russia
e mail katerinasm bk ru
Concentrated asymptotic solutions in the form Haus packages are based on the ideas of complex
WKB Maslov metho d and covariant approach in the semiclassically approximation for the Hartree
equation with smo oth translation invariant p otential of self action These decisions are natural for
understanding as semiclassical solytons for convex p otentials in not resonant case not blurring wave
packages Haus structure
Prediction of the tra jectory of typho ons and the Maslov decomp osition
D Bianchi B Tirozzi
DepartmentofPhysics University La Sapienza Rome Italy
e mail danielabianchi virgilio it brunello tirozzi roma infn it
S Reutskiy
Institute of Magnethohydro dynamics National Academy of Sciences Kharkov Ukrain
e mail reutskiy bars net ua
We use the Maslov decomp osition for the solution of quasi linear hyp erb olic equations and the
numerical integration of the ShallowWater equations for isolated vortices to showhow to makea
forecast of the tra jectory of typho ons starting from the knowledge of the rst three p ositions of the
center We will show the results of the systematic application of this metho d to the a wide class
of typho ons from the tropical typho ons to the midlatitude ones
References
S Reutskiy B Tirozzi Numerical solutions of ShallowWater equation and typho on s propaga
tion to be publishedonRussian J of Mathematical Physics
DAYS on DIFFRACTION
Kohonen neural networks and genetic classi cation
D Bianchi Brunello Tirozzi
DepartmentofPhysics University La Sapienza Rome Italy
e mail danielabianchi virgilio it brunello tirozzi roma infn it
High throughput technologies such as genome sequencing and DNA microarrays are changing the
face of biology These recent advancements are pro ducing an enormous amount of data from which
biologically meaningful features need to b e extracted and interpreted However b ecause of the large
numb er of genes and complexity of biological networks it is di cult to interpret the resulting mass of
data so the clustering techniques b ecome essential in data mining pro cesses for identifying interesting
distributions and patterns in the underlying data Clustering algorithms have simpli ed the grouping
of genes with similar biological expression Co expressed genes found in the same cluster suggest
functional similarities There is a large literature on cluster analysis and its application to genetic
classi cation numerous approaches were prop osed on the basis of diRerent quality criteria and not all
the algorithms are well founded In addition the results of the algorithms dep end strongly on many
arbitrary choices for example on the initial conditions and the value of the threshold Therefore it
is imp ortanttohave a go o d control on the prop erties of clustering algorithms Wehaveinvestigated
the e cacy of Kohonen algorithm to cluster transcription pro les This algorithm was chosen since
its prop erties are well known and can b e controlled by the user In fact wehave clari ed the state
of the art of the convergence prop erty of the algorithm considered as a sto chastic pro cess and proved
rigorously that the rate of decay of the learning parameter which is most used in the application is a
su cient condition for almost sure convergence and wehavecon rmeditnumerically Many clustering
algorithms are unable to pro duce an unique nal result e g hierarchical clustering kmean graph
based clustering multidimensional SOM etc this problem has b een solved in our implementation
of mono dimensional Kohonen algorithm We apply our theorem and considerations to the case of
genetic classi cation which is a rapidly developing eld
References
D Bianchi R Calogero B Tirozzi Kohonen neural networks and genetic classi cation Vol
January Mathematical and computer mo delling
Dynamical b ehavior of a large complex system
Jianfeng Feng
Department of Computer Science and Mathematics Warwick University UK
e mail Jianfeng Feng warwick ac uk
Mariya Shcherbina
Institute for LowTemp erature Physics Kharkov Ukrain
e mail shcherbi flint ilt kharkov ua
Brunello Tirozzi
DepartmentofPhysics University La Sapienza Rome Italy
e mail brunello tirozzi roma infn it
The problem of synchronization of N neurons interacting through a random gaussian matrix is
discussed The synchronization prop ertyisformulated in a weaker sense via a generalized de nition of
the stability of the dynamics This prop erty is demonstrated by means of limit theorems of probability
and the self averaging prop erty of disordered systems
References
J Feng M Shcherina B Tirozzi Dynamical b ehavior of a large complex system Comm Pure and Applied Analysis in press
DAYS on DIFFRACTION
Non uniform grid NG based compression
of the metho d of moments matrices
A Boag
Scho ol of Electrical Engineering Tel Aviv University
Tel Aviv Israel
e mail boag eng tau ac il
The metho d of moments MoM discretization of the b oundary integral equations of computa
tional electromagnetics conventionally requires roughly ten unknowns p er linear wavelength Even
higher density of discretization is needed in the quasi static regime or for non smo oth geometries
On the other hand only two unknowns p er wavelength are su cient to describ e the number of
degrees of freedom of the scattered or radiated eld The contrast b etween the lo cal sampling re
quirements and the radiated eld prop erties that a ect the interaction b etween spatially separated
domains can b e exploited to reduce the complexity of solving the MoM equations An e cient
compression technique based on the construction of radiating and non radiating basis and testing
functions has b een prop osed in In this work we prop ose to construct lo cally and globally in
teracting basis and testing functions similar to those in via a numerically e cient algorithm
based on the non uniform grid NG sampling and interp olation of radiated elds As the rst
step the scatterer is decomp osed into sub domains of roughly equal size Then for each sub do
main an NG is constructed in a way that allows the computation of the eld pro duced bythe
currents con ned to that sub domain at anypoint of the scatterer byinterp olation This entails
that the lo cal and global radiating functions can b e constructed based on a rank revealing fac
torization of the matrix describing eld computation at the NG p oints The size of this matrix
is much smaller than the whole blo ck of the conventional MoM matrix describing the interaction
of a given sub domain with the rest of the scatterer Following the basis transformation the un
knowns corresp onding to the lo cal basis functions are expressed in terms of the global ones and
e ectively eliminated from the system of equations to b e solved using the Schur complement pro ce
dure
For arbitrary geometries the asymptotic computational complexity of the direct solver using the
NG based compression remains of O N where N is the numb er of unknowns however the constant
multiplier implicit in such complexity estimates is greatly reduced In fact thanks to the prop osed
matrix compression the complexitymultiplier is much smaller than unity In addition the prop osed
technique is geometrically adaptive and very high compression and truly fast direct solvers are p ossible
for geometries of reduced dimensionalities
This research is sp onsored in part byNATO s Public Diplomacy Division in the framework of
Science for Peace programme
References
F X Canning and K Rogovin A universal matrix solver for integral equation based problems
IEEE Antennas Propagat Mag vol no pp Feb
A Zhu R J Adams and F X Canning Schur factorization of the imp edance matrix in a
lo calizing basis J Electromagn Waves Appl vol no Jul pp
A Boag E Michielssen and A Brandt Non uniform Polar Grid Algorithm for Fast Field
Evaluation IEEE Antennas Wireless Propagat Lett vol no pp
DAYS on DIFFRACTION
Optimization of paraxial region for quasi optical electron accelerator
Ya L Bogomolov E S Semenov A D Yunakovsky
Institute of Applied Physics RAS
NizhnyNovgoro d Russia
For the version of an electron p ositron collider suggested in the inner surface of accelerating
structure is optimized to enlarge the accelerating gradient the RF eld b eing b elow the surface
disruption threshold Assuming the radial channels essentially narrower in comparison with the p erio d
of the structure it is p ossible to consider an axially symmetric metallic pip e with a corrugated surface
as an approximate mo del of the system Inside the accelerating channel an electromagnetic eld
represents the axially symmetric mo de of E typ e In the case considered the Maxwell equations
for the axially symmetric mo de are reduced to the Helmholtz equation with resp ect to the azimuth
comp onent of a magnetic eld To nd the solution of the problem the metho d of discrete sources is
applied
The following functionals are calculated accelerating gradient coupling resistance maxima of
the electric and the magnetic elds on the metallic surface normalized by the accelerating gradi
ent Dep endence of the functionals on the geometric parameters of the electro dynamical system is
investigated The results are mainly consistent with conclusions of
References
Petelin M I Quasi optical Collider Concept it Pro ceedings of the Advanced Accelerator Con
cepts Tenth Workshop p
Myakishev D NezhevenkoO Tarnetsky V Yakovlev V TW Accelerating Structures with
Minimal Electric Field Proc of Particle Accelerator Conference and Int ConferenceonHigh
Energy Accelerators Dal las p
The transientwaves pro duced byhyp erb olic
motion of Gaussian s transverse sources
Victor V Borisov
Fock Institute of Physics St Petersburg University
Ulyanovskaya Petro dvorets St Petersburg Russia
e mail victor borisov mail ru
The goal of the present rep ort is to construct two axisymmetric explicit solutions of the inhomo
geneous wave equation in the case of sub sup erluminal hyp erb olic motion of the Gaussian transverse
sources Possibilities of an adequate description of the generated waves due to the ab ove solutions
are discussed Application of the obtained wave structures to the description of the electromagnetic
waves is also investigated
Note that kindred solutions are given for the pulse sources starting at a xed instant of time and
moving along a straight line with an acceleration see and for an extended consideration
References
V V Borisov A B Utkin J Phys D Appl Phis
V V Borisov Days on Di raction
DAYS on DIFFRACTION
Sound eld excited by exural oscillations of
elastic plate with round inclusion part
V A Borovikov A L Pop ov D A Chelub eyev
Inst for Problems of Mechanics RAS Moscow Russia
e mail v borovikov sumail ru popov ipmnet ru
This problem was considered in the talk of V A Borovikov A L Pop ov and A L Timofeev Its
imp ortance is connected with the mathematical mo deling of non contact non destructive metho d of
testing of tub es This metho d consists of
excitation of exural oscillations of tub e
which in their turn excite the sound eld
outside of tub e Lo cal in clusions in tub e
lead to p erturbations of this eld which
can b e detected
In this talk we present the results of
numerical calculations of these p erturba
tions The example is shown on the g
ure where the sound pressure in Pa m
at distance cm ab ove the steel plate
with thickness mm is presented The
sound eld is excited by the incident the
plane exural wave at round xed inclu
sion with radius mm The frequency
of this wave corresp onds to wavelength cm Since the sound wave length for this frequency is larger
than cm non p erturb ed sound eld exp onentially decays while moving from the plate Therefore
all sound eld which is shown on the gure is due to the presence of inclusion its value is su ciently
large to b e detected in exp erimental measurements
The second item of our talk is the in uence of the insulation layer watertight coating on the
propagation of exural waves in thin elastic tub es and envelop es and on excited sound eld The
harmonic oscillations in such rubb er bitumen like coatings are describ ed usually by Helmholts equation
with complex wave number
We consider the harmonic oscillations of thin elastic plate covered by liquid layer with thickness H
and density and seek for the rst memb er of expansion in degrees of H of the equations of motion
l
of this plate In the absence of liquid layer this equation can b e written in the form a h
where a dep ends on thickness h and elastic prop erties of plate and is the density of plate so that the
co e cientwith is the plate surface density It turned out that the rst approximation for small
H of the equation of motion is a h H Inotherwords in the rst approximation
l
the only in uence of the thin liquid layer is the change of the plate surface density h by the total
surface density h H
l
This talk is supp orted by the grant RFFI No
References
V A Borovikov A L Pop ov A L Timofeev Di raction of exure waves in the in nite plate on a
round inclusion Proc III Russia conference on elasticity Rostov on Don Rostov on don
Novaya kniga p
Ya O Malakhov A L Pop ov On optimal p osition of sound absorbing layer in the problem of lo cal
source sound isolation by means of thin wall shell Prikl probl prochn i plastichn Moscow KMK p
DAYS on DIFFRACTION
Planar and three dimensional chirality measure mo del
Serge P Boruhovich
Institute of Radio Astronomy
Krasnoznamennaya st Kharkov Ukraine
e mail boruhovich inbox ru
During recentyears progress of b oth microwavetechnology and photonics has b een closely linked
to the topic of electromagnetic meta materials structures with unique characteristics amongst them
the branch of planar and d chiral materials The authors of Ref found that the e ectiveness of
optical activityby planar chiral surfaces is several orders of magnitude larger than that of bulk chiral
structures when op erating in the proximity of resonances Consequently b oth planar chiral materials
and monolayers consist of d chiral elements have at tracted much attention due to a numb er of new
p olarization phenomena asso ciated with them
The quantitativecharacteristic of the reason of these phenomena chirality measure is very
imp ortant It is obvious chirality measures can b e intro duced in a number of ways Here we suggest
new algorithms applicable to single chiral elements and their arrays returning a dimensionless valued
chirality measure It is scale indep endent and moreover limited to the interval
Here we provide algorithms for and dimensional chirality measure based on pure geometrical
approach They have b een employed to calculate chirality indices of some elements and structures
and b oth are simple in realization and converging stable and quickly
Comparison of these chirality indices with known data ab out p olarization transformation e ects
for light dif fraction on some structures has b een p erformed Close t has b een observed Further
researchisacurrent topic of presenter s work
References
Hecht L Barron L D Rayleigh and Raman optical activityfromchiral surfaces Chem
Phys Lett vol pp
Papakostas A Potts A Bagnall D M Prosvirnin S L Coles H J Zheludev N I
Optical manifestations of pla nar chirality Phys Rev Lett vol no p
Schwanecke A S Krasavin A Bagnall D M Potts A Zayats A V Zheludev N I
Broken time reversal of lightinteraction with planar chiral nanostructures Phys Rev Lett
vol no p
Krasavin A V Schwanecke A S Reichelt M Stroucken T Ko ch S W Wright E M
Zheludev N I Polariza tion conversion and fo cusing of light propagating through a small
chiral hole in a metallic screen Appl Phys Lett vol no p
Zhang W Potts A Papakostas A Bagnall D M Intensity mo dulation and p olarization
rotation of visible lightby dielectric planar chiral metamaterials Appl Phys Lett vol
no p
Potts A Bagnall D M Zheludev N I A new mo del of geometric chirality for two
dimensional continuous media and planar meta materials J Opt A vol pp
Boruhovich S P Prosvirnin S L Schwanecke A S Zheludev N I Multiplicative measure of
planar chirality for D meta materials Pro ceedings of the EuMA Vol March
DAYS on DIFFRACTION
Di ractive mo del of scattering by a rough surface of radiation
V I Bronnikov
Joint Company ISC Sun Energies Sosnovy Bor Leningrad region Russia
The scattering of waves on a real surface is of interest for various areas of mo dern physics Such
practical problems as de nition of intensity of the scattered eld computer mo delling of illuminated
ob jects visibility measurement of ob ject roughness parameters are p ermanent for physical optics
Used mathematical mo del of the scattering b eing synthesis of the theory of di raction and
mathematical statistics as well as application of the theory of indignations do not provide adequate
description of radiation scattered as a sp eckle structure In spite of the fact that the sp eckle structure
is a result of the scattering on random non uniformities of the ob ject roughness uctuations of the
refraction factor numerous works cover only investigations of its statistical prop erties not related
to parameters of phase non uniformities A di ractive mo del considering spatial distribution of the
radiation scattered by the rough surface is similar to the distribution of the radiation in case of
di raction of the wave on an ap erture with a shap e and size coincident with the illuminated area is
prop osed in the submitted presentation Intensity of the scattered eld is a sum of intensities of eld
comp onents where the intensity of the rst comp onent is de ned by a ro ot mean square roughness
at the base length equal to D n n numb er of the di ractive order of di raction on illuminated
area corresp onding to observation direction the second comp onentintensity at length D n the
third one at length D n and etc The reason is that the waves propagated in the observation area
have b een di racted on the structures of base lengths multiple D n and di ractive orders numb ers
divisible by n Approximations of big and minor compared to wavelength roughness are not used in
the prop osed mo del
There are numerous exp erimental validations for various wavelengths and ob jects typ es solid and
gaseous
References
L M Brehovskih Di raction waves on Rough Surf Ac Sc of USSR v No p
M A Isakovich Scatt of Waves USSR JETPh v No c
Beckmann P Spizzichino A Scattering of elec mag Oxford Pergamon Press p
F G Bass I M Fuks Scattering of waves on statistically rough surface M Science p
V I Bronnikov Pro ceeding of SPIE v
Two b o dy problem on p encil of lines
I S Lobanov I Yu Pop ov
St Petersburg State University of Information Technologies Mechanics and Optics
Kronverkskyy St Petersburg Russia
Scattering theory for a nonrelativistic quantum mechanical particle system on a quantum network
with n op en ends joined at a single vertex is considered The scattering is decomp osed into p otential
and geometrical contributions The Green functions of the involved Hamiltonians are calculated
The wave op erators and scattering matrices are obtained in the framework of stationary scattering
theory A detailed analysis is provided for inverse square particle interaction Some applications to
the quantum computing are discussed
The work is supp orted by RFBR grant
DAYS on DIFFRACTION
Propagation of the wavepackets in thin
tub es with nonlinear integral p otential
J Bruning
Department of Mathematics Humb oldt University Berlin Germany
e mail bruening mathematik hu berlin de
S Yu Dobrokhotov R V Nekrasov
Institute for Problems in Mechanics RAS Moscow Russia
e mail dobr ipmnet ru logno rambler ru
A I Shafarevich
Lomonosov Moscow State University Moscow Russia
e mail shafar mech math msu su
We consider the problem of propagation of the Gaussian wavepackets in quantum nonlinear wave
guides thin tub es describ ed by the D Hartree typ e equation
Z
b b
G r r j r j dr v x y i H H
int t
R
The kernel G r r is a smo oth function and the con nement p otential grows up when jy j
The last prop erty organizes the thin tub e wave guide with so called soft walls The well known
fact is that wavepackets spread during the motion along the wave guides We show that integral
nonlinearity can crucially change the situation and Gaussian wavepackets can move along the wave
guide preserving its form We apply this result to the problem of the ballistic transp ort of p erturbation
in a long protein molecules We compare the propagation of Gaussian packets with the propagation
of so called Davydov s solitons in long molecules
This work was partially supp orted by RFBR CNRS grant N and DFG RAS pro ject
DFG RUS
References
V M Babich V S Buldyrev I A Molotkov Space Time Ray Metho d Cambridge University
Press
V V Belov A Yu Trifonov A V Semiclassical Tra jectory Coherent Approximations of Hartree
Typ e Equations Theoretical and Mathematical Physics Springer
Functional integrals for Schro edinger equation on a
Riemannian manifold via Smolyanov Weizsaecker approach
Ya A Butko
Bauman Moscow State Technical University
Moscow Russia
Abstract We represent a new approach intro duced by Smolyanov and Weizsaecker to get solu
tions of evolutionary equations on Riemannian manifolds with the help of functional integrals We
obtain solution of Cauchy problem for Schro edinger equation on a Riemannian manifold in the form
of functional integrals with resp ect to Smolyanov Weizsaecker surface measures
DAYS on DIFFRACTION
Dep endences of lasing thresholds for a layered
structure with a quantum well on the mo de symmetry
Volo dymyr O Byelobrov Alexander I Nosich
Institute of Radiophysics and Electronics NASU ul Proskury Kharkiv Ukraine
e mail volodia byelobrov gmail com
Optical mo des and linear thresholds of lasing for a quantum well QW sandwiched b etween two
distributed Bragg re ectors DBRs are investigated as a sp eci c eigenvalue problem with active
imaginary part of the QW refractive index
QWs emb edded in epitaxially grown semiconductor micro
cavities are widely used in photonics We consider a layered
structure consisting of the cavity whose width is w sandwiched
c
between two DBRs In the centre of the cavity there is a QW
or activelayer of the width w Fig We mo del the active
a
layer by a complex valued refractive index with a complex value
i while adjacentlayers have the same real
c
values of this index which corresp onds to GaAS For
c
this structure weintro duce a lasing eigenvalue problem LEP
and reduce it to the following equation
i i w w
c a c
e R R
i b w w
c a c
R e R i
B T a
R R
a
i b w w
c a c
i
e R R
B T a
Figure Sketchoflayered structure
where R are re ectivities of the b ottom and top DBRs resp ectively We computed the LEP
B T
eigenvalues by iterations starting from their values for the QW o ccupying entire cavity and using a
two parameter Newton metho d Fig shows results of calculation of discrete pairs of the real valued
normalized frequency of lasing w and asso ciated threshold material gain forw w and
c c a
w w for a QW in the cavity lo cated in free space i e without DBRs In the latter case
c a
several of the rst o dd mo des are lost b ecause their thresholds are to o high The thresholds of the o dd
mo des are higher than those for even mo des until
the tenth mo de and vice versa until the twentieth
one etc We can supp ose that p erio dicity in ten
mo des relates to the fact that QW is one tenth of
cavity Unlike thresholds the lasing wavelengths
are very stable with resp ect to the QW width This
is b ecause they are determined mainly by the width
of the cavity
We discuss how placing the cavity equipp ed with
QW b etween two DBRs e ects the lasing wave
lengths and thresholds We also estimate the frac
tions of mo dal p ower leaking into the top and b ot
tom media through the DBRs as a function of the
layer pairs in re ectors
Figure Thresholds of Lasing
References
G M Yang et al J Appl Phys
X H Zhang et al Appl Phys Lett
E I Smotrova A I Nosich Optics and Quantum Electronics
E I Smotrova et al IEEE J Selected Topics in Quantum Electronics
DAYS on DIFFRACTION
E ective Riemann space for wave
propagation in nonlinear electro dynamics
A A Chernitskii
A Friedmann Lab oratory for Theoretical Physics
State Univ of Engineering and Economics
Marata str St Petersburg Russia
e mail AAChernitskii eltech ru
The characteristic equation for the nonlinear Born Infeld electro dynamics is considered This
equation has the form of the characteristic equation for the linear electro dynamics in some e ective
Riemann space The appropriate e ective metric includes the energy momentum tensor comp onents of
the electromagnetic eld The distortion of light b eams by the action of some distant electromagnetic
soliton solution is investigated in the framework of a p erturbation metho d This distortion corresp onds
to attraction with the solitons and lo oks like the gravitational distortion
The e ect under consideration is used for the p ossible solution of the problem for uni cation the
electromagnetism and the gravitation For more details see
References
A A Chernitskii Light Beams Distortion in Nonlinear Electro dynamics J High Energy Phys
No Pap er http arxiv org abs hep th
A A Chernitskii Dyons and Interactions in Nonlinear Born Infeld Electro dynamics J High
Energy Phys No Pap er http arxiv org abs hep th
A A Chernitskii Induced gravitation as nonlinear electro dynamics e ect Gravitation Cosmol
ogy Suppl http arxiv org abs gr qc
A A Chernitskii Born Infeld equations in Encyclop edia of Nonlinear Science ed Alwyn Scott
New York and London Routledge http arxiv org abs hep th
A A Chernitskii Gravitation and electromagnetism in theory of a uni ed four vector eld Grav
itation Cosmology No http arxiv org abs hep th
Bessel Oscillator
V V Borzov
Department of Mathematics St Petersburg UniversityofTelecommunications
Moika St Petersburg Russia
e mail vadim VB spb edu
E V Damaskinsky
Department of Mathematics University of Defense Technical Engineering
Zacharievskaya St Petersburg Russia
e mail evd pdmi ras ru
The approach to construction of oscillator like systems connected with given family of orthogonal
p olynomials on the real axis develop ed early by the authors is extended to the case of Bessel functions
of Bessel functions We We construct a generalized oscillator related with the family fZ x g
k
k
shaw that eigenvalue equation for the Hamiltonian of this oscillator is equivalent to the usual Bessel
equation We also discuss the p ossibility of construction of the family of coherent states for this
oscillator This investigation is partially supp orted by RFBR grant No
DAYS on DIFFRACTION
Waveguide structure containing metamaterial slab with resistive lm
M V Golovkina
Povolzhskaya State AcademyofTelecommunication and Informatics Samara Russia
e mail nauka yandex ru
The electromagnetic prop erties of arti cial media have attracted a great deal of theoretical and
exp erimental interest in recentyears Many authors haveinvestigated the waveguide structures con
taining a slab of negative index material whose b oth p ermittivity and p ermeability are negative
In this rep ort wehave considered theoretically the wave propagation in two layered closed waveguide
divided by a thin resistive lm Onlayer of thickness d is a negative index material and the other
one of thickness d is a usual material The thin resistive lm is consider as the b oundary condition
for the tangential comp onent of the magnetic eld at the interface b etween adjacent media divided by
this lm The disp ersion characteristics of TE and hybrid mo des are derived The prop erties of
n
eigenwaves in the waveguide are analyzed The dep endency of disp ersion parameters on the thickness
and the conductivity of the resistive lm is examined If the losses in the metamaterial slab increase
then the solution of the disp ersion relation disapp ears The considered waveguide structure can b e
applied as the lters with frequency resp onse
References
I S Nefedov S A Tretyakov Waveguide containing a backward wave slab Radio Science
H Cory A Shtrom Wave propagation along a rectangular metallic waveguide longitudinally
loaded with a metamaterial slab Microwave and optical technology letters
G I Veselov S B Raevskiy Laminated metal dielectric waveguide Moskow Radio i Swyaz
in Russian
One representation of lo calized functions via the Maslov canonical op erator
and its application to asymptotic solutions of linear hyp erb olic systems
S Yu Dobrokhotov
Institute for Problems in Mechanics RAS Moscow Russia
e mail dobr ipmnet ru
A I Shafarevich
Lomonosov Moscow State University Moscow Russia
e mail shafar mech math msu su
B Tirozzi
DepartmentofPhysics University La Sapienza Rome Italy
e mail brunello tirozzi roma infn it
We present a representation for the fast decaying functions having a form V x h h based
on an integral of the Maslov canonical op erator on some sp ecial Lagrangian manifolds We apply it
for a construction of asymptotic solutions to the Cauchy problem with lo calized initial data for a wide
class of linear hyp erb olic systems Our main examples are the wave equations with variable velo city
and the linearized Shallowwater equations
This work was partially supp orted by RFBR grant N and AgreementBetween Uni
versity La Sapienza Rome and Institute for Problems in Mechanics RAS Moscow
DAYS on DIFFRACTION
Lo calized asymptotic solutions of d wave equation with
variable velo city implied by time dep endentsource
S Yu Dobrokhotov
Institute for Problems in Mechanics RAS Moscow Russia
e mail dobr ipmnet ru
S O Sinitsyn
Moscow Institute of Electronic and Mathematics Moscow Russia
e mail sinitsyns mail ru
We construct e ective formulas for asymptotic solutions of D wave equation with variable velo city
and with time dep endent space lo calized right hand side lo calized the time dep endent source We
consider the wide range of sources and show that their di erent dep endence on time converts into
di erent space dep endence of the wave pro les of the solutions
This work was partially supp orted by RFBR grant N and AgreementBetween Uni
versity La Sapienza Rome and Institute for Problems in Mechanics RAS Moscow
References
S Dobrokhotov S Sinitsyn B Tirozzi Asymptotics of Lo calized Solutions of the One Dimensional
Wave Equation with Variable Velo city I The Cauchy Problem Russ Jour Math Phys v N
pp
The problem of a p erfect lens made of a slab with negative refraction
A L Efros
University of Utah Salt LakeCity UT USA
Recently there has b een growing interest in the creation of lenses with unusually sharp fo ci Those
lenses come in twotyp es
A The Veselago lens that is a slab of the material where due to some reason b oth electric p ermittivity
and magnetic p ermeability are negative at some frequency Such a medium is often called left handed
medium
B The quasistatic lens where only one of those two is negative Pendry claimed that in the absence
of absorption b oth typ es of lenses are p erfect It has b een shown that the sup ersolution prop osed by
Pendry do es not exist for any lens with a real fo cus After regularization by a small absorption the
solution strongly dep ends on a distance from the source to the slab In this presentation I show that
for small wavelength the image is always controlled by di raction Mathematically this means that
two limits wavelength and absorption do not commute In the rst limit one gets di raction
controlled image that is indep endent of absorption at small absorption while in the second limit one
gets absorption controlled image
I prop ose also a lens with a virtual fo cus VF and show that in some sense this fo cus is p erfect
The VF is lo cated either in front of the slab or inside the slab at a distance d from the source
It is p erfect in a sense that the radiation from the VF as it is observed b ehind the slab is exactly
the same as radiation from the source but the VF is shifted with resp ect to the source The slab
do es not intro duce any distortions b ecause of the ampli cation of the evanescentwaves exactly as
Pendry prop osed However in the case of the VF these arguments do not contradict to any general
theorems b ecause the elds do not haveany singularity in the VF The sup ersolution exists without
any absorption In fact there is no maximum of the eld in the VF itself but this fo cus maybeat
DAYS on DIFFRACTION
the p oint that is very close to the rare interface of the slab though still inside the slab Then the
eld at the interface will haveavery narrow subwavelength maximum The in uence of the small
absorption inside the slab is considered
References
J B Pendry Phys Rev Lett
Di raction of a p erfectly conducting
half plane immersed in a gyrotropic medium
V G Daniele R D Graglia
Dipartimento di Elettronica Politecnico Torino
Corso Duca degli Abruzzi Torino Italy
e mail vito daniele polito it roberto graglia polito it
The di raction of an electromagnetic waveby a b o dy immersed in a gyrotropic medium is a very
complex problem even when the b o dy has a simple shap e b ecause gyrotropic media supp ort two
mo des of propagation Nevertheless the di raction problem of a p erfectly conducting half plane has
b een solved in particular cases when one of the axes of the homogeneous gyrotropic material is a
symmetry axis of the structure for example it is parallel or orthogonal to the metallic half plane
Closed form solutions are available also for plane waves at oblique incidence The formulation
of these problems yields to Wiener Hopf WH equations By using the WH technique we are able to
obtain closed form solutions since the involved kernels are scalars or matrices that can b e explicitly
factorized b ecause of their particular form in fact they commute with p olynomial matrices In this
pap er we consider the general problem of the di raction of a skew plane waveby a p erfectly conducting
half plane immersed in an arbitrary gyrotropic region The WH formulation of this problem is very
complex Di culties are overcome byintro ducing in the Fourier domain a circuit mo del given in terms
of vector transmission lines In this manner we establish a straightforward approach to obtain the
two times two WH matrices to b e factorized The form of these matrices p ermits one to infer the
cases when explicit factorization is p ossible Whenever explicit factorization is not p ossible or it is
not pursued we reduce the factorization problem to the solution of a Fredholm integral equation of
the second kind Numerical results for an arbitrary lossless gyrotropic medium will b e presented at
the Conference These results include also the evaluation of the di raction co e cients for particular
observation direction
This research is sp onsored byNATO s Public Diplomacy Division in the framework of Science
for Peace under the grant CBP MD SFPP
References
S R Seshadri and A K Ra jagopal Di raction by a p erfectly conducting semi in nite screen in
an anisotropic plasma IEEE Trans on Antennas and Propagat n pp
E V Jull Di raction by a conducting half plane in an anisotropic plasma Canad J Phys
vol pp
R A Hurd and S Przezdziecki Di raction by a half plane p erp endicular to the distinguished
axis of gyrotropic medium oblique incidence Canad J Phys vol pp
R A Hurd and S Przezdziecki Half plane di raction in a Gyrotropic medium IEEE Trans
on Antennas and Propagat AP n pp
V Daniele An intro duction to the Wiener Hopf Technique for the solution of elec
tromagnetic problems Rapp orto ELT Politecnico di Torino Sept
http www eln p olito it sta daniele
DAYS on DIFFRACTION
Excitation of whistler mo des guided bya
lossy anisotropic plasma cylinder
V A Es kin A V Kudrin
University of NizhnyNovgoro d
Gagarin Ave NizhnyNovgoro d Russia
e mail kud rf unn ru
T M Zab oronkova
Technical UniversityofNizhnyNovgoro d
Minin St NizhnyNovgoro d Russia
e mail zabr nirfi sci nnov ru
Whistler waves guided by cylindrical plasma structures aligned with an external dc magnetic eld
have attracted considerable interest in view of many applications ranging from space plasma re
search to radio frequency helicon plasma sources used in various plasma physics exp eriments
Recently several investigations have b een made involving the in uence of dissipative losses in a mag
netoplasma on the characteristics of guided mo des of a plasma cylinder lo cated in free space and
some interesting and unexp ected physical b ehaviors have b een found for such mo des In partic
ular conditions havebeenrevealed under which allowance for comparatively small losses results in
division of the guided waves into weakly and strongly damp ed mo des with signi cantly di erent eld
distributions across the plasma channel This should evidently lead to di erent excitation e ciencies
of the corresp onding mo des due to a given source It is the purp ose of the presentwork to study how
dissipative losses in a plasma channel can a ect the source excited eld
We consider the excitation of a lossy anisotropic cylindrical plasma column byaloopantenna with
uniform current distribution placed coaxially with the column near its surface Losses in the plasma
are accounted for byintro ducing electron collisions We calculate the total source excited eld and
determine the contribution to it from the discrete and continuous sp ectrum waves Conditions have
b een found under which the total eld is dominated by the discrete sp ectrum waves i e eigenmo des
of the plasma cylinder It is shown for this case that in the presence of two groups of eigenmo des with
essentially di erent damping rates the amplitudes of weakly damp ed mo des turn out to b e notably
greater than those of strongly damp ed ones Despite this fact the ma jor part of the p ower sp entbythe
lo op antenna go es to mo des with higher collisional losses b ecause their numb er considerably exceeds
the numb er of mo des with smaller damping As a result the eld near and far from the antenna is
determined bydi erent groups of mo des which stipulates the corresp onding variations in the total
eld distribution across the plasma column as a function of the distance from the source The results
obtained allow one to clarify factors resp onsible for the features of formation of the source excited
elds in lossy anisotropic plasma channels in the whistler frequency range
Acknow ledgments This work was supp orted by the Russian Foundation for Basic Research pro ject
Nos a and a and the Program for the State Supp ort of the Leading Scien
ti c Scho ols of the Russian Federation pro ject No NSh
References
I G Kondrat ev A V Kudrin and T M Zab oronkova Electrodynamics of Density Ducts in Mag
netized Plasmas Gordon and Breach Amsterdam
W E Amatucci Radio Sci Bul l No
F F Chen and R W Boswell IEEE Trans Plasma Sci
A V Kudrin and V A Es kin Phys Scripta
DAYS on DIFFRACTION
Investigation of multilayered and multislot
resonant and p erio dical structures
I V Donets V A Lerer A M Lerer S M Tsvetkovskaya
South Federal University Rostov on Don city Russia
Resonant and p erio dical multilayered and multislot structures are used as antennas elements
frequency selective devices and sp ecial waveguides
Cross section of investigated structure is shown at the gure Slots have nite length in z direc
tion Structure may b e p erio dical one with p erio d d along z direction
may include magnetic or electrical walls at the z z d also may
b e in nite in z direction Numb er of slots N and numb er of dielectric
layers P are arbitrary values
The system of pair sum equations in two dimensional Fourier trans
forms of magnetic current densities was obtained
Electromagnetic eld at the structure is sup erp osition of slots elds
Field of each slot is determined in the situation of absence of all other
slots These slots are replaced by p erfectly conducting screens Mag
netic current densities are unknown functions To nd them we satisfy
continuity b oundary conditions of tangential magnetic comp onents at
the surface of each slot H y H y As a result we
x z x z
obtain N equations with N unknown magnetic current densities at
Figure
the slots surface I
x z
Galerkin s metho d with basis in the form of Fourier transform of functions taking into account
eld singularities at the edges of rectangular area was used for solving obtained system of pair sum
equations SPSE
In SPSE the singular static part of ker
nel was extracted and transformed byan
alytical manner at the stage of numerical
calculation
Used basis functions provide fast inner
convergence of the metho d Tomakecal
culations with accuracy b etter than it
is necessary to takeinto account basis
functions for each current densitycompo
nentateach slot Usually the number of
sum terms after convergence improving not
exceeds
In the work was obtained following de
p endencies disp ersion curves of complex wa
veguides with insertions in the form of p eri
Figure
o dical slots of two kinds cross and Mal
tese cross Fig resonantcurves of multilayered and multislot resonators resonance curves of
complex shap ed slot resonators situated in b elow cuto waveguide
Comparing of obtained results with known theoretical and exp erimental data for some particular
cases of analyzed structures con rm accuracy of the investigation
References
Fei Ran Yang Kuang Ping Ma Yongxi Qian Tatsuo Itoh A novel TEM waveguide using uni
planar compact photonic bandgap UC PBG structure IEEE Trans on Microwave Theory and
Techniques p
DAYS on DIFFRACTION
Multiple di raction loss calculation in micro cell environments
Ali Ta jvidy Mohammad Nasermoghaddasi
Islamic Azad University Science and Research Branch Tehran Iran
Ayaz Ghorbani
Amirkabir UniversityofTechnology Department of Electrical Engineering Tehran Iran
This pap er will intro duce a mo del for prediction di raction loss from arbitrary numb ers of equal
height parallel buildings by using uniform theory of di raction UTD in urban environments Some
recentwork fo cused on plane wave approximation rather than spherical wave assumption even though
this may b e correct in macro cell but in micro cell since the transmitter and buildings are near
each other this approximation may not hold any more Therefore in this pap er we used spherical
wave approximation The presentwork takes into account only single di ractions over buildings
for calculations di raction loss Moreover b ecause of simplicity of the mo del it can b e applied to
area coverage and interference predictions in the planning of cellular systems For calculation the
di raction loss a typical urban cellular radio wave propagation is shown in Fig as can b e observed
in this Figure buildings could b e seen as an arrayofwedges with interior angles of radian joined
twobytwo forming the at ro ofed buildings and assumed to have the same height relativetothe
base station antenna height same thickness v and constantinter buildi ng spacing w The spherical
source which has an arbitrary height is considered to b e ab ove or level with the building s height and
lo cated at a near distance from the mentioned buildings so that a spherical wave impinges on them
with an angle of incidence Using a straightforward technique based on the nal UTD solution for
the di raction of plane waves by an array of dielectric wedges given in a new expression in terms
of UTD co e cients for the analysis of the di ractions pro duced byanarray of dielectric buildings
considering spherical wave incidence is prop osed The total eld at the observer p oint indicated in
Fig is calculated using the summation of pro duced elds by single di ractions If there will b e only
one building b etween transmitter and receiver the received eld at observed p ointisgiven
s
r v r exp jk r r r
D L exp jkv E E
x
r v r v r v
Figure Scheme of the prop osed environment
DAYS on DIFFRACTION ) 0 New Model
−5
−10
−15
−20
−25
−30 1 2 3 4 5 6 7
Excess Path Loss Over Free Space (dB Number of Buildings
Figure Excess path loss over free space for single di raction v m w m f MHz
and S m
r
Furthermore for n E n the following is denoted
s
n
jk r r
X n
m
s s
r e
y y
m
jks
y
E D L e E n
m y y
n r s s s s s s
n y y y y
y y
m
s
n
jk r r
n
p
X
r e
s s
p
z z
jks
z
Ep D L e
z z
r s s s s s s
n z z z z
z z
p
E is the eld reaching the rst corner of the ro ofs as shown Fig Hence for m E can b e
m m
presented as
s
m
jk r r
m
q
X
r e
s s
q
w w
jks
w
E D L e E
q w w m
m r s s s s s s
m w w w w
w w
q
s
m
jk r r
m
X
r
s r s e
v r v
jks
v
Er D L e
v v
r s s s s s s
m v v v v
v v
r
In these formulas r is distance b etween transmitter and di raction p oint also s is for calculation of
distance parameter and spreading factor The mo del was derived for buildings with same heights
m It was assumed that the height of transmitter antenna is equal m and distance b etween
transmitter and rst building is m The calculated results are shown in Fig The prop osed mo del
can nd application in the development of theoretical mo dels to predict more realistic path loss in
micro cells
References
J V Ro dr guez J M Molina Garc a Pardo and L Juan Ll acer An Improved Solution Expressed
in Terms of UTD Co e cients for the Multiple Buil di ng Di raction of Plane Waves IEEE An
tennas And Wireless Propagation Letters VOL
DAYS on DIFFRACTION
Dep olarization of shear waves in anisotropic heterogeneous media
F F Gorbatsevich
Geological Institute KSC RAS Fersman Str Apatity Russia
e mail gorich geoksc apatity ru
During propagating of shear waves through geological media an e ect observed connected with the
change of p olarization degree of these waves has b een observed Suchachange is accompanied bya
buildup in the elliptici ty of shear oscillations or by their dep olarization Following Aleksandrov
the p olarization phenomenon is accounted by the scattering of shear waves from small and large scale
inhomogeneities His conclusions were corrob orated by Crampin s exp eriments on inhomogeneous
cracked media A di erentinterpretation of the p olarization phenomenon was prop osed byObo
lentseva According to her theory dep olarization o ccurs in some solids due to their gyrotropic
or acoustical activity Whenashearwave propagates in such media its p olarization vector rotates
through an angle prop ortional to the distance travelled
To our opinion the dep olarization e ect is also appreciably manifested as shear waves propagate
in a medium consisting for example of di erently oriented elastic anisotropic layers Wehave anal
ysed some theoretical asp ects of the shear wave dep olarization SWD e ect mo deling results and
determinations of the e ect in crystalline ro ck samples
The mo deling results are shown in the Figure The acoustical p olarization metho d was used
to measure the amplitude of linear p olarized shear waves passing through two anisotropic plates The
mo del prepared consisted of two anisotropic plates of uniform thickness comp osed of the same elas
tically anisotropic ceramic mate
rial PZT Acoustop olarigrams
were obtained for di erent angles a
between the elastic symmetry ele
ments of the plates According to
Figure the VP diagrams show a gra
dual change over from a four p etal
pattern to a gure more and more
resembling a circle as the angle
increases The diagram for
almost coincides with that of an iso
tropic medium
The obtained results show
that this typ e of the SWD e ect is
rather frequently observed in aniso
tropic crystalline ro cks comp osed of
layers or grains whose elastic sym
metry is su ciently maintained for
example in two directions It is
necessary to note that the SWD ef
fect manifests itself on the backgro
und of the usual change in the de
gree of ellipticity of shear waves whi
le propagating in anisotropic media
Consequently the metho d used
Figure A Arrangement of the plates on the acoustical p olarisco
to detect the e ect must not b e sen
p e platform and B acoustical p olarization diagrams obtained for
di erent angles between the elastic symmetry elements of the
sitive to suchachange One such
plates Solid and dashed lines are for the parallel and crossed p ola
metho d is provided by the acousto
rization vectors resp ectively
p olariscopy
DAYS on DIFFRACTION
References
Aleksandrov S I Dep olarization of b o dy elastic waves scattered in a randomly heteroge
neous medium Fizika Zemli No pp in Russian
Crampin S Evaluation of anisotropyby shear wave splitting Geophysics Vol No
pp
Ob olentseva I R Seismic gyrotropy In Investigation of propagation of seismic waves in
anisotropic media Novosibirsk Nauka pp in Russian
Gorbatsevich F F Acoustic p olariscopyofrock samples Apatity KSC RAS p in
Russian
Lyamov V E Polarization e ects and anisotropyofinteraction b etween acoustic waves in
crystals Moscow Moscow State Univ Publ p in Russian
A mo di ed Euler transformation for machine
precision evaluation of p otential integrals
R D Graglia G Lombardi
Dipartimento di Elettronica Politecnico Torino
Corso Duca degli Abruzzi Torino Italy
e mail roberto graglia polito it guido lombardi polito it
The sup eriority of the cancellation metho d with resp ect to the subtraction metho d to numerically
compute singular and nearly singular p otential integrals with R singularities is discussed in
A new numerical technique to compute to machine precision singular and nearly singular p otential
integrals byworking directly in the parent reference frame is presented here Our integration scheme is
based on a new rational expression for the integrands obtained by a cancellation pro cedure and by use
of a mo di ed Euler transformation for the integration variables as well as on sp ecial quadrature rules
Gauss quadrature for rational functions together with classical Gauss Legendre quadrature The
technique can deal with static and dynamic p otentials on surface and volume elements In particular
in the static case of p olynomial source distributions our new cancellation pro cedure allows for the
exact integration of the p otential integrals The rules to establish the quadrature weights p oints
including their numb er to guarantee machine precision are rep orted in Several numerical results
for p otential integrals will b e presented at the conference
This research is sp onsored byNATO s Public Diplomacy Division in the framework of Science
for Peace under the grant CBP MD SFPP
References
M A Khayat and D R Wilton Numerical evaluation of singular and near singular p otential
integrals IEEE Trans Antennas Propag vol pp Oct
W Gautschi Algorithm GQRAT Gauss Quadrature for Rational Functions ACM Trans
Math Softw vol no pp
See also http doi acm org
R D Graglia and G Lombardi Machine precision evaluation of singular and nearly singular
p otential integrals by use of Gauss quadrature formulas for rational functions IEEE Trans
Antennas Propag submitted for publication
DAYS on DIFFRACTION
Windowed oscillating integral
Gaussian b eams metho d b eyond its limitations
Valery E Grikurov
Dept Math Comp Phys St Petersburg University Ulyanovskaya Petro dvorets
St Petersburg Russia
e mail grikurov math nw ru
High frequency asymptotic metho ds in wave propagation problems quasi classical in quantum
mechanics are known as physically based approaches to obtain in particular numerical results for
the range of parameters which are not accessible by nite deference or nite element techniques
However despite of the physical clearness of asymptotic metho ds their implementation is not always
straightforward for example in the simplest form known as the geometrical optics the approachis
said to b e non uniform and fails on manifolds carrying the singularities of the geometrical ob jects
connected with classical tra jectories rays
Starting the middle of th century a lot of work was done to understand the uniform generaliza
tions of the geometrical optics The Maslov s theory also known as the canonical operator method
provides the most common and uniform integral representation for a high frequency wave eld How
ever this representation is not e ective for numerics Another uniform integral representation known
as Gaussian beams method is highly promising b ecause of its simplicity and computational e ciency
but has certain internal limitations
In the talk we suggest the new uniform representation entitled windowed oscil lating integral WOI
The approach preserves all attractive features of Gaussian b eams metho d but can b e applied b eyond
the limitations of the latter WOI also is asymptotically equivalent to b oth Maslov s and Gaussian
b eams metho ds and thus serves as an asymptotic bridge b etween these two approaches The nu
merical e ciency of the prop osed approach will b e demonstrated
This research is sp onsored byNATO s Public Diplomacy Division in the framework of Science
for Peace under the grant CBP MD SFPP
Beam summation analysis of half plane di raction
Michael Katsav Ehud Heyman
Scho ol of Electrical Engineering Tel Aviv University
Tel Aviv Israel
e mail heyman eng tau ac il
Gaussian b eam summation GBS formulations are an imp ortanttoolinwave theory as they pro
vide a framework for ray based construction of sp ectrally uniform solutions in complex con gurations
In these formulations the eld is expanded into a sp ectrum of collimated b eam propagators that
emanate from the source domain in all directions and thereafter are tracked lo cally in the medium
and summed up at the observation p oints see for a recent review
In many applications such as in indo or or urban propagation the ambientenvironmentinvolves
edges and corners that cause the incident bream wave impinging near the edge to di ract in all
directions In order to b e used self consistently in GBS schemes this scattered eld needs to b e
expressed as a sum of GB s that emerge from the edge in all directions thereby describing the edge
by an equivalent GB to GB scattering matrix Various solutions for b eam scattering by edges have
b een derived in the past yet none of them has addressed the problem describ ed ab ove namely GBS
representation of the edge di racted eld
Following the discussion ab ovewe present a GBS representation for a half plane di raction of
an incident b eam wave The scattered eld is expanded into GB s that emerge from the edge with
DAYS on DIFFRACTION
appropriate excitation amplitudes Asymptotic expressions for the di racted b eams amplitudes are
derived and compared to exact numerical calculations Applications to GBS mo deling of indo or
propagation are discussed
This research is sp onsored byNATO s Public Diplomacy Division in the framework of Science
for Peace under the grant CBP MD SFPP
References
E Heyman and L B Felsen Gaussian b eam and pulsed b eam dynamics Complex source and
sp ectrum formulations within and b eyond paraxial asymptotics J Opt So c Am A
A phase space b eam summation representation for
D radiation from a line source distribution
Michael Katsav Ehud Heyman
Scho ol of Electrical Engineering Tel Aviv University
Tel Aviv Israel
e mail heyman eng tau ac il
Beam summation formulations are an imp ortanttoolinwave theory as they provide a framework
for raybased construction of sp ectrally uniform lo cal solutions in complex con gurations In these
formulations the eld is expanded into a phase space sp ectrum of collimated b eam propagators that
emanate from a set of p oints and directions in the source domain and thereafter are tracked lo cally
in the medium and their contributions at the observation p oint are summed up see a recent review
in
There are essentially two classes of GBS schemes for p oint sources and for distributed ap erture
sources In the former the eld is expanded as a angular sp ectrum of b eams that emerge from
the source in all directions whereas in the latter the eld is describ ed as a discrete phase space
sum of b eams that emerge from a set of p oints and directions in the source domain Here we
intro duce a hybrid scheme for radiation from a line source distribution involving a phase phase
space decomp osition of the source distribution along its axis and an angular sp ectrum of GB s in the
plane normal to the axis Our motivation for this new expansion scheme has b een the preparation
of the analytical framework for a GBS representation of D edge di raction as an extension of the
D expansion in The accuracy of this scheme and the choice of the expansion parameters are
illustrated numerically
This research is sp onsored byNATO s Public Diplomacy Division in the framework of Science
for Peace under the grant CBP MD SFPP
References
E Heyman and L B Felsen Gaussian b eam and pulsed b eam dynamics Complex source and
sp ectrum formulations within and b eyond paraxial asymptotics J Opt So c Am A
M M Pop ov A new metho d of computation of wave elds using Gaussian b eams Wave Motion
A Shlivinski E Heyman A Boag and C Letrou A phase space b eam summation formulation
for ultra wideband radiation IEEE Trans Antennas Propagat
M Katsav and E Heyman Gaussian b eam summation representation of a two dimensional
Gaussian b eam di raction by an half plane IEEE Trans Antennas Propagat submitted
DAYS on DIFFRACTION
The impact of the small scale uctuations of the ionospheric plasma on the
ultra wide band orbital ground p enetrating radar measurements
Ya A Ilyushin
Moscow State University Moscow Lengory Russia
e mail ilyushin phys msu ru
The propagation of the ultra wide band UWB signals of the ground p enetrating radars UWB
through the ionosphere with small scale irregularities is investigated Signal distortions due to joint
action of phase distortions in the regular layered disp ersive ionospheric plasma and the scattering on
the random small scale inhomogeneities is simulated numerically The impact of the di ractive e ects
on the function of the phase correction algorithms is investigated
It has b een stated that the plasma density uctuations intro duce signi cant distortions in the chirp
signal esp ecially at the frequencies close to the critical frequency of the ionosphere Satisfactory
quality of the signal compression can b e achieved at frequencies well exceeding the critical frequency
When the uctuations are strong the signal frequency must b e quite high compared to the critical
frequency of the ionosphere This conclusion is in agreement with the published exp erimental results
on planetary GPR sounding known at present
The op eration of the algorithms of correction of phase distortions in the regular ionosphere is
relatively little in uenced b e the plasma density uctuations at least when the uctuations are not
to o strong Thus the conclusion can b e made is that the regular and random comp onents if the
ionospheric distortions of the UWB chirp signals are indep endent The stability of the investigated
phase correction algorithms with resp ect to the scattering of the signals on the small scale uctuations
is shown
The investigation is in part supp orted by the RFFI grants and
References
Ilyushin Y A Kunitsyn V E Metho ds for correcting ionosphere distortions of orbital
ground p enetrating radar signals J Commun Technol Electron
An attempt to improve semiclassical calculations of
wave transp ort in a quantum wire with Di raction
Hiromu Ishio
N Scho ol of Engineering Nagoya University Nagoya Japan
Recently the systems of nanoscale wires made in semiconductor heterosurfaces are attracting
much attension b ecause of their p otential application to electronic devices They haveinteresting
features of quantum e ects such as quantization of conductance conductance uctuations etc All these
phenomena originate from the coherence of electronic quantum waves In our previous presentations in
the series of this seminar wehave shown an interesting phenomenon which had b een found numerically
and later explained semiclassically for coherent electronic transmission through a mesoscopic quantum
wire with a step structure The motion of free electrons in the wire is ballistic with sp ecular re ections
from the Dirichlet b oundaries The wire allows only a few mo des for electronic transmission This
time we rep ort an attempt to improve the semiclassical calculations of wave transp ort in sucha system
DAYS on DIFFRACTION
Vibrations of inhomogeneous solid discs sub jected to an inertial rotation
M Shatalov S V Joub ert W E Pretorius I Fedotov
Department of Mathematics and Statistics P B X Pretoria
Tshwane UniversityofTechnology South Africa
e mail mshatlov csir co za joubertsv tut ac za
The e ect of a vibrating pattern s precession in the direction of inertial rotation of a vibrating ring
was discovered byG Bryan in This e ect has several applications to navigational instruments
such as cylindrical hemispherical and planar circular disc rotational sensors A mo del of inhomoge
neous structures comp osed of concentric thin circular discs vibrating along the plane of the disks and
sub jected to inertial rotation is considered Dynamics of the disc gyroscop e are considered in terms
of linear elasticity It is assumed that the angular rate of inertial rotation of the discs is constant and
it has axial tangential orientation We also supp ose that the angular rate is much smaller than
the lowest eigenvalue of the discs and consequently anycentrifugal e ects that are prop ortional to
the square of the angular rate are neglected The mo del is formulated in terms of Novozhilov Arnold
Warburton s theory of thin shells The system of equations of motion of the discs are separated and
transformed to a pair of wave equations in p olar co ordinates A solution is obtained in terms of Bessel s
and Neumann s functions Various non axisymmetric mo des of the disc are considered and eigenvalues
as well as Bryan s constantisinvestigated for imp erfection of mass densities of the structure
A probabilistic approach to bi variate function approximation
P H Klopp ers T H Fay S V Joub ert
Department of Mathematics and Statistics P B X Pretoria
Tshwane UniversityofTechnology South Africa
e mail joubertsv tut ac za
For an arbitrary x the Gauss Bernstein approximation for a function f x known in the p oints
P
n
y p a nite weighted sum of y with weights x y f x j n is of the form k x
j j j j j j
j
p The weights are probabilities determined by the Gaussian distribution with mean x and standard
j
deviation x It is also p ossible to deriveascheme of cho osing x so that the Gauss Bernstein
approximation of a straight line passing through twopoints is exact that is the di erence b etween
f x and k x is guaranteed to b e exactly zero Ultimately it is then p ossible to uniformly approximate
any function of b ounded variation by piecewise straight line segments
In this pap er this idea will b e extended to approximate a bi variate function f x y
If a function z f x y is known in the p oints
f x y z f x y g i n j n
i j ij i j
it will b e shown that the Gauss Bernstein approximation is
X X
k x y z p
ij ij
i j
with p probabilities from a bi variate normal distribution with means x y variances
ij i j x y
and covariance
xy
Ascheme to select the variances and covariance and hence to calculate the probabilities will
b e explored
Appling the technique to approximate a surface plane it will b e shown that this technique is free of the Gibbs e ect
DAYS on DIFFRACTION
Vibrations of a rotating solid elastic sphere lled with an inviscid uid
S V Joub ert M Shatalov C E Co etzee I Fedotov
Dept of Mathematics and Statistics Tshwane Univ of Technology South Africa
e mail joubertsv tut ac za mshatlov csir co za
The dynamics of a solid elastic sphere lled with a compressible inviscid uid are considered in
the frame of a three dimensional theory Equations of motion are solved by means of a four p otential
metho d It is assumed that the system is sub jected to a small inertial rotation with regards to inertial
space This slow rotation means that the maximum of the absolute value of the inertial angular rate
is substantially smaller than the minimum eigenvalue of the system Consequently it is p ossible to
neglect e ects suchascentrifugal forces prestress of the sphere due to inertial rotation et cetera
that are prop ortional to the square of the angular rate It is shown that the gyroscopic e ect of the
vibrating pattern s precession dep ends on spheroidal mo des of vibrations The e ect of inertial rotation
of vibrating patterns in the direction of revolution that has a rate that is prop ortional to the inertial
angular rate is considered Various typ es of inertial rotation with constant and radial dep endence
of angular rates are examined These investigations are tentatively submitted as to ols to b e used in
order to understand some fundamental phenomena in seismology and b oth solar and stellar dynamics
Dynamics of a vehicle trailer system with a limitedpower supply
S V Joub ert T H Fay M Shatalov
Dept of Mathematics and Statistics Tshwane Univ of Technology South Africa
e mail joubertsv tut ac za mshatlov csir co za
A mo del of a vehicle trailer system having four degrees of freedom is considered The linear motion
of the vehicle is characterized bytwo degrees of freedom one degree describ es an angular lo cation of
the trailer and an additional degree of freedom is intro duced to describ e an engine which is considered
as a source with a limited p ower supply This means that the torque pro duced by the engine and
transformed to a driving force could opp ose a mean value for the frictional forces involved but cannot
opp ose p erio dic vibrations created by the p erturb ed motion of the vehicle trailer system This is why
a strong interaction b etween the engine and the vehicle trailer system exists and this interaction has
to b e taken into consideration in order to pro duce a realistic description of the dynamics of the system
It is shown that a simpli ed mo del of the vehicle trailer system can b e describ ed in terms of Mathieu s
equation and by means of which the critical regimes are found These regimes are then investigated
in a complete system with four degrees of freedom by an approximate p erturbation technique as well
as bymeansofanumerical solution of nonlinear equations of motion using Runge Kutta metho ds
Quantization of surfaces in magnetic eld
M V Karasev
Moscow Institute of Electronics and Mathematics Russia
We consider a p ointcharge on a surface in a su ciently strong magnetic eld The surface is
two dimensional smo oth oriented and arbitrarily curved as well the eld is smo oth and arbitrarily
inhomogeneous
In classical mechanics this is an imp ortant example of a natural system with two degrees of
freedom whichinvolves metric and gyroscopic forces only In quantum mechanics such a system
mo dels the electron in a thin lm for example on the surface of a nanotub e In this case the
surface curvature and the magnetic eld inhomogeneity manifest themselves already at atomic scales
The basic quantum Hamiltonian is the usual two dimensional Schro edinger op erator with metric and
DAYS on DIFFRACTION
magnetic eld If spin e ects are taken into account then the spin orbit and Pauli corrections are
added to the Hamiltonian
Our main goal is to demonstrate that in the framework of the nano momentum mechanics i e
on the class of states with nano scale kinetic momenta this nonintegrable system is asymptotically
reduced to a system with one degree of freedom which can b e e ectively analyzed in b oth the classical
and quantum cases
The original surface serves as the phase space for the reduced system The Poisson bracket on this
surface is determined by the inverse magnetic eld tensor In quantum case the Poisson bracket is
replaced bythecommutator and as a result the original surface b ecomes a quantum surface This
is an inhomogeneous generalization of the well known Heisenb erg typ e commutation relation b etween
co ordinates on the con guration plane whichwas prop osed byPeierls in th of the last century
in describing the Focklevel splitting of a charge in a homogeneous magnetic eld and a p erturbing
p otential In the general case studying arbitrary magnetic elds and curved surfaces we obtain a
rich family of examples of phase quantum manifolds of a complicated shap e in particular likethe
sphere the torus etc The symplectic area i e the magnetic ux around holes in the quantum
surface must b e multiple of the elementary London ux We stress that for the original problem such
quantized surfaces are just the usual con guration spaces
In the given mo del the motion of the charge can b e separated into oscillations of a light magnon
quantum vortex similar to the spin or hydro dynamic vortex and to dynamics of a heavy
quasiparticle representing the motion of the vortex center along the quantum surface The magnon
energy levels are degenerate Each of them is split into a cluster of energy levels of the quasiparticle
We note that the cause of this splitting is di erent from that in the well known Peierls scheme there
is no p erturbing p otential in the given problem at all
The Hamiltonian of the quasiparticle is a function on the surface which can b e expressed in
terms of the joint geometric invariants of the magnetic tensor and the metric This Hamiltonian
determines b oth the regular dynamics and the lo cal instabilities separatrices whichcharacterize
zones of irregularity quantum tunneling etc The b ehavior of the quasiparticle can b e analyzed
in particular by metho ds describ ed in whichallows us to solve in the nano momentum
approximation the original problem ab out the quantum charge on a surface in a magnetic eld
We conclude that the surface in a su ciently strong magnetic eld is strati ed by the quasiparticle
orbits actually each orbit is presented by a pair of nearby fermion sub orbits if the spin is taken into
account In the stability domain the orbits are closed and carry the discretized magnetic ux Now
by considering a collective motion of electrons along the surface we can state that a vortex free ow
of a quantum charged liquid of quasiparticles app ears on the surface The ow is generated by
geometric ob jects magnetic and metric tensors only This can b e regarded as one of p ossible origins
of the dimensional sup erconductivity e ect
The work is supp orted by the Russian Foundation for Basic Research under grant
References
L Magarill A Chaplik M Entin Sp ectrum and electron kinetics in curvilinear structures
Usp ekhi Fiz Nauk N
V Kozlov General Theory of Vortices Springer Verlag
M Karasev Magneto metric Hamiltonians on quantum surfaces in the con guration space Russ
J Math Phys v N
V Maslov Theory of Perturbations and Asymptotical Metho ds Moscow State Univ
V Babich V Buldyrev Asymptotical Metho ds in Problems of Short Wave Di raction Nauka
Moscow
H Cycon R Fro ese W Kirsch B Simon Schro edinger Op erators with Applications to Quantum
Mechanics and Global Geometry Springer Verlag
M Karasev V Maslov Nonlinear Poisson Brackets Geomtery and Quantization Nauka
Moscow
DAYS on DIFFRACTION
Nonlinear waves in coupled waveguides
Karima Khusnutdinova
Department of Mathematical Sciences Loughb orough University
Loughb orough LE TU UK
e mail K Khusnutdinova lboro ac uk
A system of coupled Klein Gordon equations cKG is prop osed as a mo del for one dimensional
nonlinear wave pro cesses in two comp onent media e g long longitudinal waves in elastic bi layers
where nonlinearity comes only from the b onding material We discuss general prop erties of the mo del
Lie group classi cation conservation laws invariant solutions and sp ecial solutions exhibiting an
energy exchange b etween the twophysical comp onents of the system To study the latter we
consider the dynamics of weakly nonlinear multi phase wavetrains within the framework of two pairs
of counterpropagating waves in a system of two coupled Sine Gordon equations and obtain a hierar
chy of asymptotically exact coupled evolution equations describing the amplitudes of the waves We
then discuss mo dulational instability of these weakly nonlinear solutions and its e ect on the energy
exchange Our metho dology is generic and can b e applied to other systems and other multi phase
wavetrains
The simple lattice mo del leading to the cKG equations has b een used to study some features
of propagation of nonlinear waves in bi layers with delamination The mo del can b e naturally
mo di ed to takeinto account nonlinear interactions in the layers and other degrees of freedom
for example by considering chains of interacting mechanical dip oles instead of chains of p oint
masses It can also b e easily two dimensionalised We exp ect such mo dels to b e useful for the
mo delling of nonlinear waves in a large class of bi layers We discuss some results in this direc
tion
The research is partially supp orted by the EPSRC grant EP D
References
I Sh Akhatov V A Baikov and K R Khusnutdinova Non linear dynamics of coupled chains of
particles J Appl Maths Mechs
S D Gri ths R H J Grimshaw K R Khusnutdinova The in uence of mo dulational instability
on energy exchange in coupled Sine Gordon equations Theor Math Physics
S D Gri ths R H J Grimshaw K R Khusnutdinova Mo dulational instabilityoftwo pairs of
counterpropagating waves and energy exchange in a two comp onent system Phys D
K R Khusnutdinova and V V Silb erschmidt Lattice mo delling of nonlinear waves in a bi layer
with delamination Pro c Estonian Acad Sci Phys Math
K R Khusnutdinova Nonlinear Waves in a Double RowParticle System Vestnik Moskov Univ
Ser I Mat Mekh
I V Semenova G V Dreiden A M Samsonov and K R Khusnutdinova Nonlinear bulk elas
tic waves in layered solid waveguides Pro ceedings of International Conference on Exp erimental
Mechanics ICEM Alexandroup olis Greece to app ear
DAYS on DIFFRACTION
Singular nonlinear b oundary value problem for bubble typ e or
droplet typ e solutions in nonlinear physics mo dels
N B Konyukhova M B Soloviev
Doro dnicyn Computing Centre RAS
ul Vavilova Moscow Russia
e mail nadja ccas ru
P M Lima
Centro de Matem atica e Aplica c oes Instituto Sup erior T ecnico
Av Rovisco Pais Lisb on Portugal
e mail plima math ist utl pt
M L Morgado
Departamento de Matem atica Universidade de Tr as os Montes e Alto Douro
Apartado Vila Real Portugal
For a second order nonlinear ordinary di erential equation a singular b oundary value problem
BVP is investigated which arises in hydromechanics and nonlinear eld theory when the static
centrally symmetric bubble typ e or droplet typ e solutions are sought The main theoretical results
of this talk are describ ed in detail in for the mo dels see e g Posed and partially studied
in for a capillary uid mo del with a simpli ed form of the chemical p otential of the medium
this singular BVP is the same for the scalar Higgs eld with two distorted vacua In the normalized
variables the considered singular nonlinear BVP has the form
N
r
r
j lim r j lim r r
r r
lim r lim r
r r
Here all magnitudes are real N and are the parameters
Eq has a regular singularityasr and an irregular one as r Wegive the restrictions
to the parameters for a correct mathematical statement of the limit b oundary conditions and
their accurate transfer into neighb orho o ds of singular p oints using some results for singular Cauchy
problems and stable initial manifolds Due to the approach of and certain results of the
necessary and su cient conditions for existence of bubble typ e or droplet typ e solutions are discussed
in the form of additional restrictions to the parameters and some estimates are obtained A priori
detailed analysis of singular nonlinear BVP including the representation of one parameter
sets of solutions in the neighb orho o ds of the singular p oints leads to e cient sho oting metho ds for
the numerical solution of this BVP Some results of numerical exp eriments are displayed and their
physical interpretations are discussed In particular wegive a comparison of di erent de nitions of the
bubble radius which in uence imp ortantcharacteristics of the bubble such as the minimal nucleation
radius surface tension and interface thickness
N B Konyukhova and M B Soloviev acknowledge supp ort from RFBR pro ject No
P Lima and L Morgado acknowledge supp ort from FCT pro ject POCTI MAT
References
N B Konyukhova P M Lima M L Morgado and M B Soloviev Bubbles and Droplets in Nonlin
ear Physics Mo dels Analysis and Numerical Simulation of Singular Nonlinear Boundary Value Problem prepared
DAYS on DIFFRACTION
F dell Isola H Gouin and P Sepp echer Radius and Surface Tension of Microscopic Bubbles by
Second Gradient Theory C R Acad Sci Paris Serie I Ib
V A Rubakov Classical Gauge Fields Editorial URSS Moscow in Russian
P M Lima N B Konyukhova A I Sukov and N V Chemetov Analytical Numerical Investigation
of Bubble Typ e Solutions of Nonlinear Singular Problems J Comput Appl Math Elsevier
Science
G H Derrick Comments on Nonlinear Wave Equations as Mo dels for Elementary Particles J
Math Phys
F Gazzola J Serrin and M Tang Existence of Ground States and Free Boundary Problems for
Quasilinear Elliptic Op erators Add Di Equ
The di raction tomography and iterative approach to restore
the elastic parameters and electrical conductivity
Yurii V Kiselev Vladimir N Troyan
Institute of Physics St Petersburg University
Ul yanovskaya St Petersburg Russia
The results of numerical simulation on restoration of electrical conductivity and elastic parameters
Lame parameters and mass density of the lo cal inhomogeneities corresp ondingly with the help of
electromagnetic and elastic sounding signals are considered
The direct problems for the Maxwell and Lame equations are solved by the nite di erence metho d
Restoration of the desired parameters is implemented by the di raction tomography metho d in the
time domain using the rst order Born approximation Restoration of the electrical conductivity
is studied in the low frequency case
We consider restoration of the inhomogeneity parameters with the help of the iterative pro cedure
Each step of the iterative pro cedure includes solution of the direct problem and correction of the
desired parameters by the di raction tomography metho d For this purp ose the algebraic metho ds
with di erent regularization schemes are used
The numerical simulation for D problems in elastic and electromagnetic cases is implemented
for the lo cal inhomogeneities with a simple and complex geometry and with a contrast ab out
relatively to the reference medium Main attention is given to study the convergence and accuracy of
the considered algorithms
This work was supp orted by RFBR grant
References
Ryzhikov G A Troyan V N Tomography and inverse sounding problems St Petersburg
St Petersburg University in Russian
Kiselev Yu V Troyan V N Numerical study of restoration of lo cal elastic inhomo
geneities by iterative approaches based on the di raction tomography metho d Proc of the In
ternational Seminar Day on Di raction St Petersburg Russia June July
FacultyofPhysics SPbU pp
DAYS on DIFFRACTION
The nonparaxial self fo cusing of few cycle light
pulses in dielectric media with disp ersion
S A Kozlov P A Petroshenko
St Petersburg State University of Information Technologies Mechanics and Optics
Saint Petersburg Kronverksky Ave Russia
e mail kozlov mail ifmo ru pavel petroshenko com
In the last decade laser systems that allow pulses comprising only a few cycles of a light eld to b e
obtained were created in many lab oratories The concept of an envelop e loses its physical meaning for
such extremely short pulses ESPs therefore for describing propagation of these pulses the equations
of evolution of envelop es that are conventional in nonlinear optics are no longer appropriate The
theory of nonlinear propagation of ESPs in various media is based as a rule on equations describing
the dynamics of no the pulse envelop e but its eld directly The rigorous theory of the self action of
ESPs in a bulk media should b e nonparaxial since in the calculation of the space time evolution of
pulses with small longitudinal size the p ossibility of o ccurrence of transverse inhomogeneities of the
same scale in the pulse structure should also b e taken into account
In the presentwork the solutions of the equation describing the nonparaxial dynamics of the space
time sp ectrum of an ESP in a homogeneous isotropic dielectric medium with an arbitrary sp ectral
dep endence of the linear refractive index and nonresonant electron nonlinearity deduced in is
illustrated bynumerical mo deling
The gure illustrates the temp oral sp ectrum and the corresp onding electric eld of two dimensional
ESP of TE p olarized radiation at di erent distances passed by the pulse in the fused silica The
disp ersion of the fused silica id describ ed by the dep endence n N a where N
s c m a
Figure The dynamics of the temp oral sp ectrum g and the electric eld E of initially fo cused few
cycle titanium sapphire laser pulse with input p eak intensityisI W cm in the fused silica
The gure shows that during self action of ESP the highly intensivelightkern with the transverse
size comparable with the central wave length is generated As the gure indicates the inhomogeneous
broadening of the sp ectrum into the red and the blue regions including the generation of tripled
comp onents takes place mainly at the exp ense of the energy of the time sp ectrum at high spatial
frequencies
References
Kozlov S A PetroshenkoP A Self Division of a Pulse Consisting of Several Light Field Oscilla
tions in a Nonlinear Medium with Disp ersion JETP Letters Vol No pp
DAYS on DIFFRACTION
Boundary value problems for the Helmholtz equation in
domains b ounded by closed curves and op en arcs
P A Krutitskii
KIAM Dept Miusskaya Sq Moscow Russia
e mail krutitsk math phys msu su
Boundary value problems for the Helmholtz equation are considered in planar domains with com
plicated b oundary The b oundary of a domain consists of b oth closed curves and op en arcs cracks or
cuts Formulation of b oundary value problems in these domains implies that the b oundary condition
is sp eci ed on the whole b oundary i e on b oth closed curves and op en arcs Boundary value prob
lems in such domains were not actively treated b efore though they have many applications Similar
domains mo del cracked solid b o dies or they mo del several obstacles and screens in a uid However
problems in domains b ounded by closed curves and problems in the exterior of op en arcs were treated
separately b ecause di erent metho ds were used in their analysis In the present talk we consider b oth
the Dirichlet problem and the Neumann problem for the Helmholtz equation in domains b ounded
by closed curves and op en arcs We study either the dissipative Helmholtz equation in interior and
exterior domains or propagative Helmholtz equation in exterior domains The rigorous mathematical
formulation of the b oundary value problems are given the uniqueness theorem is proved With the
help of p otentials the problems are reduced to the integral equation on the b oundary of a domain
By means of some transformations this equation is reduced to the Fredholm integral equation of the
second kind and index zero It is proved that this integral equation is uniquely solvable in the appro
priate Banach space In this way the classical solvability of the problems is proved and the integral
representation for a solution is obtained The e ective computational scheme for nding numerical
solution is suggested The Dirichlet and Neumann problems outside op en arcs in a plane have b een
studied in The Neumann problem for the dissipative Helmholtz equation in b oth interior and
exterior cracked domains has b een studied with Neumann b oundary condition in and with Dirich
let b oundary condition in The Neumann problem for the propagative Helmholtz equation in
exterior of several obstacles and screens has b een studied in in case of non resonance obstacles
and in in general case The Dirichlet problem for the propagative Helmholtz equation in an
exterior cracked domain has b een studied in The Helmholtz equation outside cuts in a plane
with the skew derivative b oundary condition generalizing the Neumann b oundary condition has b een
treated in
References
Krutitskii P A Comp Maths Math Phys v No p
Krutitskii P A Comp Maths Math Phys v No p
Krutitskii P A Zeit fuer Analysis und ihreAnwend v p
Krutitskii P A Int J Maths Math Sci v No p
Krutitskii P A J Math Kyoto Univ v No p
Krutitskii P A etall Computers and Math with Appl v p
Krutitskii P A ZAMM v No p
Krutitskii P A Mathematics and computers in simulation v p
Krutitskii P A Geophys Astrophys Fluid Dynamics v p
Krutitskii P A Rendiconti di Matematica Roma Serie v p
DAYS on DIFFRACTION
Krutitskii P A Mathematika London v p
Krutitskii P A Applicable Analysis v p
Krutitskii P A Nonlinear Analysis TMA v No p
Krutitskii P A ZAMM v No p
Krutitskii P A Hiroshima Mathematical Journal v No p
Semiclassic asymptotics for the vector
Sturm Liouville problem with parameters
Valeri V Kucherenko
IPN ESFM Mexico
e mail valeri esfm ipn mx
Consider asymptotics of linearly indep endent solutions of the matrix Sturm Liouville equation in
some vicinity W fx jx x j g
d
ih y A x u y Ey x W
dx
with resp ect to a small parameter h Here y C h h and u u u are some parameters
n
Let u V fu juj g The matrix Sturm Liouville problem o ccurs in manyphysical problems
Let A x u b e a self adjoint matrix function of class C W V Supp ose that the matrix
A x has the form A x U x k x k U x where U x k x k are the C W matrix
j ij j ij
function and the matrix U x is unitary Consider the case when eigenvalues x in the vicinity
j
W coincide only at the p oint x and it holds that
d x x
inf E x x x only at x x
j
dx W
x x
where j It is proved that in the domain W V for su ciently small the function
det A x u has the following representation
det A x u x u x x u jd x u j j u j
where inf x u x u u C V and x x Wecallx u the
W V
turning p oint function and the function u we call the e cient parameter It is easy to verify
q
x x u j u j h for any xed the application of WKB that in the domain
metho d provide asymptotic solutions of equation We will construct the asymptotic expansion for
the solutions of equation in the domain
j u j h jx x u j h for some xed
in mo di ed p representation and obtain representations of these asymptotics as the WKB asymp
totics near the b oundary of this domain i e for h jx x u j h By this approach
the connection formulas b etween WKB asymptotics of solutions of problem in domain x x u
and WKB asymptotics of these solutions in domain x x u are obtained
References
M V Fedoruk Asimptoticheskiie meto di dla lineinih obiknovennih di erensialnih uravnenii
Moscow Nauka p
DAYS on DIFFRACTION
Asymptotic solution of linear system with a turning p oint of high order
Andriy Kryvko Valeri V Kucherenko
IPN ESFM Mexico D F
e mail valeri esfm ipn mx
Linearizing a nonlinear hyp erb olic system of partial di erential equations for example the system
of equations of ideal compressible uids system of Maxwell equations with the current J E non
linearly dep endentonthecharge density and the electricity eldE one obtains a linear hyp erb olic
system with three or more coinciding characteristics at the p oints of multiplicity One dimensional
mo del of this case is the following system with a small parameter at the derivative
dy
ih A t y hB t y
dt
r
where y C r and h is a small parameter We consider the asymptotics of the
linearly indep endent solutions of system in a vicinity U ft jtj g Let B t C U bea
r r matrix function and A t b e a matrix function of the form
Ia t
b t d A t
c t
where I is r r identity matrix d R anda t b t c t are C U functions such that
a b c Ab ovementioned system can not b e reduced to the Webb er equation In the
present pap er we construct the asymptotics of this problem In the domain jtj h
the asymptotics of equation can b e constructed by the standard WKB metho d In the domain
jtj h for the solution of problem the asymptotics of the form
t
Z Z
i
A
p
w p p dp tp b d exp y t h
h
h
is obtained Here p is a patch function and is a contour that go es along the real axis and
bypasses the p oint p from b elowover the circule of radius h
In the domain jtj h an expansion of asymptotics y t h is obtained by the
saddle p oint metho d
Averaging and chaotic mo des for
nonautonomic nonlinear dynamic systems
R S Makin
Ulyanovsk State University Dimitrovgrad Branch
Dimitrovgrad Ulyanovsk region Russia
e mail rmak vinf ru
The Krylov Bogolyub ovaveraging theorem has b een considered the lo cal existence of p erio dical
orbits in some tasks ab out forced oscillating of nonlinear nonautonomic dynamic systems has b een
determined with its help Some general features particularly the existence of homo and hetero clinic
orbits have b een determined Such orbits can b e found with the help of Melnikov metho d when the
task has the saddle connection The discussion of resulting chaotic movementatavailabilityofsuch
orbits is given the examples are given
DAYS on DIFFRACTION
Di raction of nonstationary wave with varying
along the front amplitude by a soft wedge
D P Kouzov Yu A Solov eva
St Petersburg Russia
Exact analytic solution of new D problem of di raction byawedge with dirichlet b oundary
condition on its surface is found The incident eld is a plane wavewith typ e pro le The amplitude
of this wavevaries linearly along its front The solution is constructed by Smirnov Sob olev metho d
the metho d of functionally invariant solutions
Sp ectra of mo des of waveguides loaded with wire media
I S Nefedov
Radio Lab oratory SMARAD Helsinki UniversityofTechnology TKK
P O Box FI TKK Finland
e mail igor nefedov tkk fi
Wave propagation in rectangular waveguides loaded with single and double wire media is studied
Both TE and TM mo des are considered Propagation of TE mo des is quite trivial b ecause the spatial
disp ersion causes only reduction of the transversal wavenumber by the plasma wavenumb er TM
mo des p ossess more interesting disp ersion including propagation b elow cuto of an emptywaveguide
existence of backward waves etc
Following features of sp ectrum of mo des propagating in a waveguide lled with double wire media
were discovered
DominantTE mo de cannot propagate
The dominant mo de of WMW can propagate even b elow cuto of the TE mo de of the empty
waveguide
In pass band o ccupied byTE mo de in the emptywaveguide now propagates a backward
wave
Both low frequency forward and backward waves have p olarization similar to TM mo de of the
emptywaveguide
No waves propagate b etween high frequency cuto of this backward wave and TE TM
mo des b oth lowest mo des in WMW propagate in single mo de regime
There is di erence b etween geometries with connected and non connected wires If to connect
mutually p erp endicular wires the rst forward mo de disapp ears The features describ ed ab ove
take place indep endentl y whether the plasma frequency is b eloworabove the cuto frequency of the
emptywaveguide Decrease of the plasma frequency causes shift of the lowest pass band to lower
frequencies
The waveguides lled with wire media can nd applications as size reduced transmission lines
mo de lters and controllable devices if to load the wires bycontrolling elements
References
J Esteban C Camacho Penalosa J E Page T M Martin Guerrero and E Marquez Segura
IEEE Trans Microwave Theory Tech vol no pp April
I S Nefedov X Dardenne C Craeye and S A Tretyakov Micr Opt Tech Lett vol No pp
DAYS on DIFFRACTION
The numb er of energy levels in MQW structure
M D Kovalev
MGTU Moscow Russia
e mail kovalev math mtu net ru
A formula for the numb er of energy levels of a quantum particle in one dimensional piecewise
constant p otential eld of certain kind is obtained This eld we call a MQW multiple quantum well
structure It consists of an arbitrary numb er but more than of layers that is segments of constant
p otential We treat a sp ecial typ e of MQW structures consisting of wells and walls The p otential of
eachwell is equal to and the p otential of eachwall is equal to U The widths of wills and walls
are arbitrary b esides external walls having in nity width The deduction of the formula is based on
the analysis of the recently obtained multilayer equation allowing to calculate eigenvalues of
energy E of a quantum particle in MQW structure The multilayer equation lo oks like F E
i
where F E is a rather complex function built bya given MQW structure and dep ending on the
i
index i of an arbitrary chosen preferred b ounded layer In the previous work the author proved that
nontrivial zero es of the function F E are indep endent of index i of the preferred layer The formula
i
obtained for the numb er of energy levels is valid only for a general p osition case It means that in
some cases one can pick up widths of wells and walls in suchway that the formula ceases to b e valid
But this collections of widths are very rear
Besides the problem treated here the multilayer equation is extremely imp ortant in the theory of
optical waveguides It p ermits to calculate the eigenvalues of e ective index of refraction for the mul
tilayer waveguide The formula discussed here can b e rewritten to give the numb er of the eigenvalues
of e ective index of refraction Note that the multilayer equation may b e applied in the di erent situ
ations This situations and problems now are of signi cant practical interest for nanotechnologies
References
A A Maer and M D Kovalev Disp ersion equation for eigenvalues of the e ective refractive index
in a multilayered waveguide structure
A A Maer The equation for energy eigenvalues in MQW structures with an arbitrary number
of quantum wells Doklady Physics p
A A Maer The equation for energy eigenvalues in MQW structures with particles having di erent
e ective masses in di erentlayers Doklady Physics p
Solution scattering problems of electromagnetic waves from
inhomogeneously layered scatterers using pattern equation metho d
Alexander G Kyurkchan Dmitrii B Demin
MoscowTechnical University of Communication and Informatics
Aviamotornaya street A Moscow Russia
e mail kyurkchan yandex ru
This pap er presents the generalization of the pattern equation metho d PEM which had b een
applied b efore to solve electromagnetic scattering problems for conducting ob jects coated with dielec
tric materials utilized to solve scattering problems of electromagnetic waves from inhomogeneously
layered dielectric scatterers The core of layered ob ject represents the dielectric obstacle
By analogy with the previous works the basic p oint of the PEM is to reduce an initial b ound
ary problem to solving an in nite system of algebraic equations with resp ect to unknown expansion
co e cients of the scattering pattern and also elds inside the dielectric layer and core of scatterer
into series in terms of vector angular spherical harmonics that comp ose the orthogonal basis in the
DAYS on DIFFRACTION
Fig Scattering pattern for sphere
Fig Scattering pattern for spheroid
spherical co ordinates Under the certain restrictions on geometry of the problem which can b e strictly
established the received in nite linear system of the algebraic equations is solved by the metho d of a
reduction In the framework of the metho d the analytical solution of the scattering problem for sphere
as the Mie series with resp ect to the expansion co e cients of the scattering pattern has b een obtained
Under the develop ed numerical algorithm of PEM we carried out researches of accuracy of nu
merical calculations for scattering characteristics Moreover we carried out comparisons of our results
with the results obtained by PEM for imp edance and dielectric scatterers and by other metho ds also
The class of scatterer s geometry was limited to the b o dies of revolution that signi cantly simpli es
the numerical algorithm and reduces the surface integrals for matrix elements in the system of PEM
to single integrals
Some of the results obtained are shown here We consider the axial incidence of a plane wave
directed along the symmetry axis of the scatterer which coincides with the z axis The bistatic
RCS s of the dielectric sphere coated with a dielectric spherical layer are shown in Fig The
geometry of sphere coincides that was presented in The internal sphere has a radius and its
coating spherical layer is of thickness There are two cases presented in rst case the materials
of the internal sphere and the covering coincide curve relative p ermittivity and p ermeability
in second case materials of core and the covering are equal to
r r r r
and i resp ectively Both curves completely are agreed to what are given
r r
in We consider also a dielectric prolate spheroid coated by dielectric material One consists of
two concentric spheroids with the semiaxes of internal spheroid ka kc and the external
spheroid ka kc The medium in internal spheroid and its coating layer have the relative
p ermittivities and resp ectively The results plotted in Fig show the di erential
r r
scattering cross sections DSCS normalized by c for the coated spheroid curve E plane curve
E E
H plane DSCS is calculated from jF j where F is a pattern of electrical eld Our numerical
results agree very well with results that are obtained by the null eld metho d with discrete sources
The metho d prop osed here will b e extended to the solution of the electromagnetic wave scattering
problems bymultilayered scatterers
This work was supp orted by the Russian foundation of basic researches pro ject No
References
A G Kyurkchan D B Demin N I Orlova Solution scattering problems of electromagnetic waves
from ob jects with dielectric covering using pattern equation metho d Radiotec and Elect
N pp in Russian
J Y Li and L W Li Electromagnetic scattering by a mixture of conducting and dielectric ob jects
analysis using metho d of moments IEEE Trans on vehicular technology
A Doicu T Wriedt Null eld metho d with discrete sources to electromagnetic scattering from
layered scatterers Comput Phys Commun
DAYS on DIFFRACTION
Metho ds of the continued b oundary conditions and the pattern equations
Alexander G Kyurkchan Nadezhda I Smirnova
MoscowTechnical University of Communication and Informatics Russia
e mail kyurkchan yandex ru
For almost years the pattern equations metho d PEM has b een successfully applied to the
solution of the broad sp ectrum of wave di raction and propagation problems However essential lim
itation of the metho d is that in it s strict formulation it is not applicable to the solution of di raction
problems on b o dies with non analytical in particular piece smo oth b oundary caused by divergence
of Sommerfeld Weil integral representation in singular p oints of a wave eld The metho d of continued
b oundary conditions MCBC suggested recently allows to overcome this limitation The trick is that
according to MCBC the b oundary condition is satis ed not on b oundary S of scatterer but on some
surface S covering S and separated from it by some su ciently small distance It leads to the
approximate statement of a problem however as a result all di culties related to singular p oints of a
wave eld on scatterer b oundary in case it has breaks corners edges etc as well as di culties related
to singularity of the corresp onding integral equation kernels are removed Computational algorithm
thus b ecomes signi cantly simpler and practically universal It has b een shown that in framework
st
MCBC the b oundary problem can b e reduced to the solution of Fredholm integral equation of the
nd
and kind with smo oth kernel In particular in case of Dirichlet b oundary condition MCBC gives
the following equation
Z
ik R
u r e u r u r
ds R j r r j
n n n n R
S
S S S S
where u is a known function an incidentwave By parametrizing surfaces S and S and as
suming that in p oints corresp onding to the same parameter values the derivatives u r nj and
S
u r n j are equal whichisreasonable given the small distance b etween S and S the equa
S
nd
tion can b e treated as Fredholm integral equation of the kind relative to the unknown func
ik sin sin cos cos cos
tion u r nj u r n j Bymultiplying equation by e
S S
integrating on S and taking advantage of generalized Sommerfeld Weil representation for function
ik R
e R we obtain the following integro op erator equation of PEM relative to the scattering pattern g
i
Z Z Z Z
k g cos sin ik sin g g g
ik
g exp ik cos cos sin d d d d
sin
R
u r
k
exp ik cos ds is the generalized scattering pattern and Hereg
n
S
cos sin sin cos cos cos cos sin sin cos cos cos sin cos
sin sin sin sin sin cos cos sin cos Generally sp eaking the equation
is approximate since at its derivation it was assumed that g g However this equation
is now applicable to the di raction problems on b o dies with non analytical b oundary If b oundary
S is analytical the obtained equation b ecomes exact It is interesting to note that integro op erator
nd
equation of PEM cannot b e derived from standard currentFredholm integral equation of a kind
even for b o dies with analytical b oundary b ecause of the simple layer p otential normal derivative jump
This work was supp orted by the Russian foundation of basic researches pro ject No
References
A G Kyurkchan N I Smirnova Solution of di raction problems using the metho ds of continued
b oundary conditions and discrete sources J Commun Technol Electr
Do Dyk Tkhang A G Kyurkchan E cient metho d for solving the problems of wave di raction
by scatterers with broken b oundaries Acoustical Physics
DAYS on DIFFRACTION
Mo deling the characteristics of scattering of electromagnetic
waves by the b o dies of complex geometry
A G Kyurkchan E A Skoro dumova
MoscowTechnical University of Communication and Informatics Russia
The pattern equations metho d has app eared to b e the high e cient metho d on a series of di raction
problems in particular at solving of an acoustic problem of di raction for scatterers with a complex
con guration In this pap er this technique is extended to an electromagnetic case
Consider the problem of the waves scattering of primary mono chromatic electromagnetic eld E
H on a scatterer of a complex geometry with latter to b e presented as a combination of ob jects of
the more simple structure Let s consider two b o dies case
h i
are set on surfaces n n H Z Let the imp edance b oundary conditions n E
j j j j
S S
j j
are secondary di raction H S j E E E E H H H H is the total eld E
j
j j
elds which satis es to a homogeneous system of Maxwell s equations everywhere outside of S and
j
also to a condition of radiation on in nity
Reduction of an initial b oundary problem to solving an algebraic system of equations is based on
using the series expansion of the scattering patterns in vector spherical harmonics
Examination of the velo city of the convergence of the prop osed metho d for scatterers with an
analytical spheres and a nonanalytic cylinders b oundaries has shown that the algorithm of PEM
remains well converging when scatterers are coming close together up to their contact It allows
the extending the PEM to solving the problem of the wave di raction for b o dies with a complex
con guration by their representation as a combination of ob jects with a simple geometry at minimum
consumption of the computer resources
We carried out the examination of this tech
nique based on comparison of scattering patterns
of sup erellipsoids with scattering patterns of a group
of twohalves of these sup erellipsoids comp osing to
gether the larger sup erellipsoi d
Figure shows an example of such a comparison
for two p erfectly conducting sup erellipsoi ds with
parameters ka kc larger semiaxis
is along axis z and a single sup erellipsoid with pa
rameters ka kc The case of a longi
tudinal falling of a wavewas considered It is clear
from gure that the di erences of these character
istics are rather insigni cant
One of the metho ds of a validity estimation of
the solution of the problem of the di raction is verifying of ful llment of the optical theorem
The results obtained are showed that the accuracy of ful llment of the optical theorem for b o dies
with an analytical b oundary spheres spheroids is very high valid signi cant gures irresp ective
sizes of scatterers and the accuracy for the ob jects with a nonanalytical b oundary cylinders is
noticeably b elow valid signi cant gures but also is still acceptable
This work was supp orted by Russian Foundation for Basic Research Pro ject No
References
A G Kyurkchan E A Skoro dumova Solution of the Three Dimensional Problem of Wave Di rac
tion byanEnsemble of Ob jects Acoustical Physics Vol No pp
C Bohren and D Hu man Absorption and Scattering of Lightby Small Particles A Wiley
Interscience Publication New York Toronto
DAYS on DIFFRACTION
Mo delling of scattering characteristics of electromagnetic waves by
group of b o dies using the mo di ed metho d of discrete sources
A G Kyurkchan S A Manenkov E S Negorozhina
MoscowTechnical University of Communication and Informatics Russia
e mail kyurkchan yandex ru
The mo di ed metho d of discrete sources MMDS o ered in work subsequently has b een
applied to the solution of a wide class of problems of the theory of di raction and in all cases high
e ciency of a metho d has b een shown The main idea of the metho d consists in uniform wayof
construction of the carrier of discrete auxiliary sources by means of analytical deformation of b order
of a scatterer Thus the a priori information on prop erties of analytical continuation of di raction
eld inside the scatterer is materially used
Represents the big interest a question on how it is necessary to apply MMDS to the solution of
problems of di raction of waves on a group close lo cated b o dies of revolution The problem at issue
is that as a result of di raction interaction of b o dies the picture of arrangement of singularities of
analytical continuation of wave eld inside of each scatterer can di er from that whichtakes place in
case of single b o dy rather essentially At a close arrangement of the scatterers singular p oints start to
be multiply that is the singularities inside of one b o dy generate the singularities inside of another
We shall consider for example a problem of di raction of waves on two b o dies attened spheroid and
smo othed double cone Axis cross section of the b o dies is shown in Fig Here at a close arrangement
of b o dies the singular p oint in the edge of double cone is represented in a spheroid generating an
additional singular p oint whichwould not b e if the problem ab out di raction of a plane wave on this
single spheroid was solved Standard technique MMDS do es not consider this circumstance
In the presentwork some mo di cation of MMDS is realized allowing to make this metho d e ective
at the solution of problems of wave di raction on a group close lo cated b o dies The essence of this
idea consists that the carrier of discrete sources for each of the b o dy is constructed using usual scheme
of MMDS but in addition to the basic sources the sources surrounding singular p oints which app ear
b ecause of interaction of the scatterers are added As a result as have shown researches accuracy
of calculations increases approximately for two orders at the same sizes of corresp onding algebraic
systems This fact is illustrated in Fig in which the distribution of the residual on the contour of the
cross section of the attened spheroid is shown see the geometry in Fig In some cases co ordinates
of the images of singular p oints are found analytically however in the general situation it can b e
necessary to use numerical metho ds to search of these co ordinates For a nding of co ordinates of
singular p oints in this work the e ectivenumerical algorithm based on a metho d of continuation on a
parameter is o ered E ciency of the metho d is illustrated by a lot of numerical results
Figure Figure
DAYS on DIFFRACTION
This work was supp orted by the Russian foundation of basic researches pro ject No
References
A G Kyurkchan S A Minaev A L Solovejchik A mo di cation of the metho d of discrete sources
based on prior information ab out the singularities of the di racted eld Radiotekh Electron
A P Anyutin A G Kyurkchan S A Manenkov S A Minaev Ab out D solution of di raction prob
lems by MMDS J Quantitative Spectroscopy Radiative Transfer
V F Ap elcin A G Kyurkchan Analytical prop erties of wave elds MSU
A G Kyurkchan B Yu Sternin V E Shatalov Singularities of wave elds UFN
Berry phases for the Hartree typ e equation
Fedor N Litvinets Andrey Yu Trifonov
Department of Mathematical Physics Tomsk Polytechnic University Tomsk Russia
e mail litvinets mph phtd tpu edu ru trifonov phtd tpu edu ru
Alexander V Shap ovalov
Department of Theoretical Physics Tomsk State University Tomsk Russia
e mail shpv phys tsu ru
The Hartree typ e equation
f i H t g x t
t
Z
H t H z s d y y t V z w s y t
n
R
is considered Here pseudo di erential op erators H z s and V z w s b eing functions of non com
mutative op erators
n
z i r x w i r y x y R
x y
dep end on slow time s t T whereT is a p erio d of adiabatic evolution The op erators H z
and V z w are determined by their smo oth symbols H z and V z w resp ectively A function
is complex conjugate to is a real parameter is a small parameter
We construct semiclassical solutions of the Cauchy problem for Eq in adiabatic approximation
taking eigenfunctions of the nonlinear op erator H as an initial condition The solutions are
obtained in the class of semiclassically concentrated functions using the Maslov s complex germ
metho d The leading term of the asymptotics in the parameter T is shown to remain an eigenfunction
of the nonlinear op erator H at every time and only gains a phase factor Following Berry we
decomp ose the phase into the dynamic and the adiabatic Berry phases In the linear case
the expressions obtained corresp ond to the well known results
Acknowledgements The work was supp orted in part by President of the Russian Federation Grant
No SS DFG Litvinets F was supp orted in part by the nonpro t DynastyFoundation
References
Belov V V Trifonov A Yu and Shap ovalov A V The tra jectory coherent approximation
and the system of moments for the Hartree typ e equation Int J Math Math Sci
Berry M V Quantum phase factors accompanying algebraic changes Proc Roy Soc Lon
don A
DAYS on DIFFRACTION
Factorization of Wiener Hopf matrices using Fredholm integral equations
V G Daniele G Lombardi
Dipartimento Elettronica Politecnico Torino Corso Duca degli Abruzzi Torino Italy
e mail vito daniele polito it guido lombardi polito it
The main problem in solving vector Wiener Hopf equations is the factorization of the n n matrix
kernel G This problem has b een considerably studied in the past Nowadays closed form factoriza
tions for particular classes of problems are available in literature but no metho d to factorize a general
n n matrix kernel is still known Consequently it is necessary to develop approximate techniques
of factorization To this end all the p owerful metho ds develop ed in functional analysis i e iterative
metho ds moment metho ds regularization metho ds and so on can b e e ectively used for solving W H
equations or for obtaining approximate factorized matrices Since it is p ossible to factorize rational
matrices the factorization problem could b e faced byintro ducing rational approximants of the entries
of the matrix kernel for instance Pade approximants These authors exp erienced all these technique
Some of them are very cumb ersome and p erhaps not easy to b e applied in engineering applications
In the framework of approximate techniques these authors suggest a new e cienttechnique for
the factorization of Wiener Hopf matrices This metho d is based on the reduction of the factorization
problem to the solution of a Fredholm integral equation of the second kind The aim of this pap er is
to discuss the theory of this metho d Several numerical applications concerning matrices of order
two three and four will b e presented in the oral presentation and in the full pap er
Acknowledgment This work is supp orted byNATO North Atlantic Treaty Organisation in the
framework of the Science for Peace Programme under the grantCBP MD SFPP
References
V G Daniele An intro duction to the Wiener Hopf Technique for the solution of electromagnetic
problems Rapp orto ELT Politecnico di Torino Sept
http www eln polito it staff daniele
V G Daniele and G Lombardi Wiener Hopf Solution for Imp enetrable Wedges at Skew Inci
dence IEEE Transactions on Antennas and Propagation Sept pp
Mo del for the regular nanostructuring of transparent dielectrics and
semiconductors byfemtosecond laser radiation
V S Makin
Reseach Inst for Complex Testing of Opto Electronic Devices
Sosnovy Bor City Leningrad district Russia
A Y Vorobyev
Institute of Optics UniversityofRochester New York USA
Exp erimental results ab out the interaction of linear p olarized femtosecond pulses of laser radiation
with wide band dielectrics quartz glass calcium uoride zinc selenide and silicon were analyzed It
is shown that at laser p ower density of the order W cm fs the nonequilibrium plasma
having concentration n cm is pro duced and the nonthermal phase transition o ccurs In
this situation at the b oundary nonequilibriu m plasma dielectric the conditions for surface plasmon
p olaritons excitation by incident radiation arises Mutual surface plasmon p olaritons interference and
one with the incident laser radiation cause the standing interference ngers of nonequilibrium electron
density formation Then the dielectric volume breakdown o ccurred the near symmetric channel surface
plasmon p olaritons interference is the cause of the volume nanostructures formation
As a next step the di erentways of the time dep endentevolution of mo dulated electron density
into the remnant p erio dic nanostructures of material are considered
The mechanism for nanostructures formation on the transparent semiconductor surface is the same
DAYS on DIFFRACTION
Di raction of D complex b eams by a p erfect
conductor half plane a sp ectral approach
Raul Mahillo Isla Mar a Jesus Gonz alez Morales Carlos Dehesa Mart nez
Departamento de Teor a de la Senal y Comunicaciones e Ingenier a Telem atica
Universidad de Valladolid Spain
The classical approach when tackling radiation and di raction problems dealing with Complex
b eams is to use an analytical continuation of the cartesian co ordinates in the isotropic line source
solution Since this can b e applied to the exact solution of a particular problem with the isotropic
cylindrical wave the obtained solutions are exact as well The analytical continuation of asymptotic
solutions can b e also p erformed but the region where the approximation is go o d can b e di erent from
that considered when the continuation is not carried out as such in the radiation problem of a D
Complex b eam happ ens
The di raction problem of a D Complex b eam by a half plane made of p erfect conductor is
discussed in by using the analytical continuation of the asymptotic solution of the cylindrical
incidentwave The authors prop osal is di erentfrom since other approach to the problem is used
The way the problem is treated is similar to the isotropic line source in the solution is constructed
by sup erp osition of plane waves solutions of the half plane problem In order to set out the problem
the plane wave sp ectrum of Complex b eams will b e carried out
Totackle the problem in this way provides a closed expression for the exact solution from which
the asymptotic forms arise Likewise the asymptotic solutions are go o d enough in certain regions
that will dep end on parameters de ned during the early steps taken in order to solve the problem
The pap er will present a detailed derivation of the asymptotic expressions for the problem and a rst
characterization of the regions where asymptotic solutions are valid
References
P C Clemmow The Plane Wave Sp ectrum Representation of Electromagnetic Waves Oxford
University Press IEEE Press Oxford New Jersey st reissued edition
L B Felsen Complex source p oint solutions of the eld equations and their relations to the
propagation and scattering of gaussian b eams Symp osia Mathematica
E Gago Ribas M J Gonz alez Morales and C Dehesa Mart nez Analytical Parametrization of
a D Real Propagation Space in Terms of Complex Electromagnetic Beams IEICE Transactions
on Electronics E C
M J Gonz alez Morales C Dehesa Mart nez and E Gago Ribas Ab out Complex Extensions
and Their Application in Electromagnetics In Complex Computing Networks Brain like and
Waveoriented Electro dynamic Algorithms I C G oknar and L Sevgi editors Springer Verlag
Berlin st edition
G A Suedan and E V Jull Two Dimensional Beam Di raction by a Half Plane and Wide Slit
IEEE Transactions on Antennas and Propagation
Gap condition for nonlinear evolution equations
R S Makin
Ulyanovsk State University Dimitrovgrad Branch Russia
e mail rmak vinf ru
Conditions of non p erio dical solutions existence for a class of nonlinear evolution equations have
b een considered The conceptions of invariant stable and unstable manifolds inertial manifolds are
mainly used during studying their prop erties The conditions where inertial manifolds of nite fractal
dimension exist in the task are determined gap condition is the main of them The examples of this
condition for classic tasks of di raction and also for transp ort equation are given
DAYS on DIFFRACTION
The number of zeroes in invariant manifolds for evolution equations
R S Makin
Ulyanovsk State University Dimitrovgrad Branch Russia
e mail rmak vinf ru
The prop erties of stable and unstable invariant manifolds for nonlinear evolution equations have
b een considered As a mo del task one dimensional evolution equation of parab olic typ e has b een
considered where the main memb er is elliptic op erator the secondary memb er is nonlinear op erator
with certain conditions Detailed prop erties thus numb er of zero es numb er of laps stable and
unstable manifolds and also its dimensions and asymptotical prop erties hyp erb olic solutions of
original nonlinear task are determined The examples of prop osed approach application for more
complicated di raction tasks are given
The comparison of elds excited by input
grating coupler with those of SEW
V S Makin
JSC NanoLab Dekabristov str Saint Petersburg Russia
V V Trubaev
Moscow State UniversityofCommunication Ways
Obraztsov str Moscow Russia
Surface electromagnetic waves SEW are the eigenmo des on interface of two media one of which
has negative real part of dielectric constant for example metal and the other has p ositive one for
example vacuum or air The electromagnetic elds asso ciated with SEW are lo calized in the vicinity
of the interface and damp out on b oth sides of it As SEW propagates along the surface it s intensity
decreases exp onentially b ecause of dissipative losses in metal In the middle and far infrared the
SEW propagation length is ab out some centimeters and it increases with wavelength In this case the
pro cesses of SEW launching byincident radiation free SEW propagation along the surface and SEW
recovery backinto bulk radiation may b e separated A great numb er of studies in the middle and far
infrared SEW propagation along metal surface have b een made It was supp osed that exactly SEW
propagates along a free part of surface b ehind the input coupler However wehave discovered that if
input grating coupler is used to excite SEW the generated wave is di ered essentially from SEW
This wave is conventionally named pressed wave by us
Wehave calculated the elds generated by input grating coupler for the radiation CO laser The
metals used are copp er gold and aluminum This selection of metals and wavelength p ermits us to
compare calculated results with exp erimental ones In this work we compare the elds of pressed wave
with that of SEW We examine the evolution of pressed wave eld on the metal surface with distance
from input grating coupler and compare that with analogous dep endence for SEW It is shown that
unlike SEW the intensity of pressed wave decreases not exp onentially while it propagates along the
surface The pressed wave b ehavior may b e the reason to re examine the results of early exp eriments
of SEW s propagation lengths Also wehave calculated the pressed wave eld as a function of distance
from the surface for numb er p ositions along the wave propagation Wehave got that unlike SEW the
pressed wave eld pro le is mo di ed as it propagates along the surface It follows from our calculations
that as the distance from the input grating is increased more and more the discrepancy b etween the
elds of pressed wave and SEW decreases
References
V V Trubaev V S Makin Pressed waves Surface bulk electromagnetic eigenmo des of metal
surface International Journal of Infrared and Millimeter Waves v No pp
DAYS on DIFFRACTION
On plasmonic and other mo des of double metal slab structures
G J McPhee P N Stavrinou D D C Bradley
The Blackett Lab oratory Imp erial College London UK
e mail graeme mcphee imperial ac uk
The electromagnetic mo des of planar optical systems comprising two metal slabs b ounded by
dielectrics have b een studied on several o ccasions in the past Original studies by Economou
considered implicit disp ersion relations and schematic energy versus momentum plots for ideal metals
in a range of con gurations The disp ersion of the mo des of a double slab system were later studied
by Stegeman and Burke and more recently byRevelli and co workerset al in the context of
Organic Light Emitting Dio de structures where the metal plates act as the electro des In the present
work we re examine and extend these analyses to consider a much broader range of system parameters
and nd some previously unrep orted features of the general system
Figure a The geometry of the system and the schematic form of the electromagnetic mo des b The
imaginary part of the propagation constant of the electromagnetic mo des of the fully symmetric double
Aluminium slab as a function of slab separation divided bywavelength in medium
In common with Stegeman four surface plasmon mo des exist that corresp ond to the four p ossible
symmetries gure b black traces There is however a sp ectrum of weakly guided mo des grey
traces similar to parallel plate waveguide mo des which can p enetrate to great depths into the outer
dielectric layers and which cut o at a particular maximum value of slab separation As in the
case of the plasmonic mo des the prop erties of these mo des are very sensitive to system parameters
Furthermore whereas Stegeman found that one of the surface plasmon mo des SAS exhibits a cut o
below a certain slab separation as in gure b wehave found that for many cases where the metal
thickness is varied the lowest order weakly guided mo de WG evolves into the SAS mo de which
has the same symmetry gure a and neither is cut o The evolution of the mo des can b e b est
appreciated by mapping their lo ci in the complex plane which for in some cases extends into other
sheets of the Riemann surface We show that for a range of p ermittivityp ermittivities and thicknesses
the rather complicated b ehaviour can b e rationalised and understo o d in terms of the basic symmetries
Some interesting features emerge For example it is p ossible by design to increase greatly the long
range plasmon propagation lengths by making very small changes change in the p ermittivity
of the central dielectric slab
Our ndings suggest that structures such as these provide many desirable features which could b e utilised in the design of nano scale optical devices
DAYS on DIFFRACTION
References
E N Economou Surface Plasmons in Thin Films Phys Rev
G I Stegeman J J Burke Long range Surface Plasmons in Electro de Structures Appl Phys
Lett
J F Revelli L W Tutt B E Kruschwitz Waveguide analysis of organic light emitting dio des
fabricated on surfaces with wavelength scale p erio dic gratings J Appl Opt
UniversalityofFeigenbaum and dissipative microstructures
for highly nonequilibrium nonlinear systems
V S Makin
Research Institute for Complex Testing of Opto electronic Devices
Sosnovy Bor City Leningrad region Russia
R S Makin
Research Institute of Atomic Reactors
Dimitrovgrad CityUlyanovsky region Russia
Chunlei Guo A Y Vorobyev
Institute of Optics Ro chester University New York USA
The discovered phenomenon of step like resonant microrelief p erio d reducing of titanium surface
under the action of series of femtosecond laser pulses is discussed The physical mo del of phenomenon
is prop osed based on interference pro cesses with surface plasmon p olaritons participation
On the base of physical picture of the phenomenon the mathematical mo del is given involving
logistical mapping It is shown that functional dep endence of total electromagnetic eld intensity
for considered system has quadratic dep endence of the height of formed resonant microrelief This
functional dep endence changes in the pro cess of sequential irradiation of surface by series of femtosec
ond laser pulses allows to explain exp erimentally observed changes of microrelief p erio d Following
the Feigenbaum the bifurcation diagram of cascade of the p erio d doubling for quadratic mapping is
created So the simple example of highly dissipative nonlinear system with the transition from simple
p erio dic to complex nonp erio dic state by the reverse microrelief p erio d doubling mo de is demonstrated
One more exp erimentally realized example of reverse cascade p erio d doubling is analyzed for
microrelief of quartz glass surface under the action of scanned linear p olarized radiation of continuous
waveCO laser
Biot s equations for multiphase media with di erent
temp eratures on the basis of generalized variational principle
G Maximov
Moscow Engineering Physics Institute State university Russia
e mail maximov dpt mephi ru
The system of generalized Biot s equations describing waves propagation in multiphase or multi
comp onents media in presence of heat exchange b etween phases is derived on the basis of generalized
variational principle Maximov G A DD p It is shown that at presence of N phases
there are N eigen mo des in this system At high frequencies N mo des have the wavetyp e of motion
and N mo des have the di usive heat typ e of propagation Atlow frequencies there is the single
acoustical wave mo de and the rest N mo des p ossess the di usion heat typ e of b ehavior
For the two comp onent medium without temp erature exchange the develop ed approach is reduced
to the well know Biot s mo del Account of temp erature elds gives generalized Biot s mo del for two
comp onents medium
DAYS on DIFFRACTION
Re ection from a half space lled with wire
medium the auxiliary source metho d
Stanislav I Maslovski
Radiophysics Dept St Petersburg State Politechnical University
St Petersburg Russia
e mail stanislav maslovski gmail com
Solving wave re ection problems for slabs of materials that exhibit spatial disp ersion is not triv
ial Amethodtoattack such problems will b e presented The metho d op erates in wave vector
space and requires k of the material to b e known some hints how to correctly de ne k
for spatially disp ersive media will b e also given It will b e shown that byintro ducing auxiliary
sources into unb ounded media it is p ossible to simulate nite slabs or material half spaces Earlier
Henneb erger in prop osed a somewhat similar source based metho d however his approachwas in
complete and even erroneous Another approach op erating in the wave vector space was develop ed
in
We will apply our metho d to the problem of plane wave re ection at the interface of free space
and the wire medium with wires oriented orthogonally to the interface The problem will b e the
same as in where it was rst solved microscopically In contrast to that our solution will b e
completely macroscopic and will make use of the spatially disp ersive p ermittivity of wire medium that
was derived in It will b e shown that the results of our metho d are in agreement with Another
calculation of the re ection and transmission co e cients for slabs of wire media was p erformed in
The authors of used Henneb erger s metho d directly and obtained results di erent from ours and
of
References
K Henneb erger Additional b oundary conditions An historical mistake Physical Review Letters
D F Nelson and B Chen Comment on Additional b oundary conditions An historical mistake
Physical Review Letters
B Chen and D F Nelson Wave vector space metho d for wave propagation in b ounded media
Physical Review B
B Chen and D F Nelson Wave propagation of exciton p olaritons bywave vector space metho d
Physical Review B
M G Silveirinha Additional b oundary condition for the wire medium IEEE Transactions on
Antennas and Propagation
P A Belov R Marques S I Maslovski I S Nefedov M Silverinha C R Simovski and S A
Tretyakov Strong spatial disp ersion in wire media in the very large wavelength limit Physical
Review B
I Nefedov A Viitanen and S Tretyakov On re ection from interfaces with some spatially
disp ersive metamaterials Journal of Magnetism and Magnetic Materials e e
DAYS on DIFFRACTION
On uniqueness in the two and three dimensional
Neumann Kelvin problem
O V Motygin
Institute of Problems in Mechanical Engineering
V O Bol shoy pr St Petersburg Russia
e mail o v motygin gmail com
We consider the uniqueness question for the classical Neumann Kelvin problem of the linear theory
of ship waves The problem describ es motion of b o dies in an ideal incompressible heavy uid W
with a free surface F The ships move through the still water with a constantspeed U along the
horizontal x axis the co ordinate system is attached to the b o dies and y axis is directed upwards We
consider b oth D and D problems and the case when the contours of ships S are totally submerged
The mathematical problem consists in nding the velo city p otential u which is a harmonic function
r u intheunb ounded domain W satis es the Neumann condition on S and the b oundary
u on the unp erturb ed free surface F fy g where g U and g denotes u condition
y x
acceleration due to gravity The problem is completed with the condition sup jruj and the
W
R
jru x z j d z D asx condition at in nity upstream jruj D
A feature of the problem under consideration is that solutions to the homogeneous problem can
have in nite energy due to wave propagation to in nitydownstream There are quite few uniqueness
R
theorems not assuming niteness of energy jruj d V F Ursell proved uniqueness for one
W
circular cylinder D problem see e g x B R Vainb erg and V G Maz ya see e g
ch proved unique solvability of the problem for b o dies of arbitrary shap e and for all values of U
except some nite set of values which cannot b e de ned within the metho d thus the uniqueness
cannot b e guaranteed for anygiven interval of U
In this work weprove a new uniqueness theorem valid for b o dies of arbitrary shap e we only
demand S C without any assumptions on niteness of energy For this we use a
mo di cation of the technique suggested in Being based on Green s identity maximum principles
and study of solution s and Green function s b ehaviour at in nity the approach leads us to the
R
conclusion that if the inequalitysup G Q P d s holds then the homogeneous
P
Q F
n P S
problem has only the trivial solution G is the Green function of the problem
The latter inequalityallows us to nd very simple b ounds for the set of parameters for whichthe
uniqueness is guaranteed For example for the three dimensional problem uniqueness parameters are
given by the inequality
Z
y
e jy j
d s y
x y z
j yj
S
and it can b e easily deduced also referring to ch for the solvability arguments that the problem
has a unique solution for any system of b o dies when they are su ciently deeply submerged the value
of the depth can b e found from the latter formula Results of numerical computation of the b ounds
for various geometries will b e presented
The work is supp orted byINTAS YSF grant no
References
Kuznetsov N G Maz ya V G Vainb erg B R Linear water waves A Mathematical Approach
Cambridge University Press
Maz ya V G Vainb erg B R On ship waves Wave Motion Vol P
Motygin O V McIver P A uniqueness criterion for linear problems of wave b o dy interaction
IMA Jl Appl Math Vol P
DAYS on DIFFRACTION
Electromagnetic prop erties of double p erio dic planar chiral array
Vassili A Fedotov Nikolay I Zheludev
EPSRC NanoPhotonics Portfolio Centre Opto electronics ResearchCentre
University of Southampton SO BJ United Kingdom
Pavel L Mladyonov Sergey L Prosvirnin
Institute of Radio Astronomy of NAS of Ukraine Kharkov Ukraine
e mail mladyon rian kharkov ua
We rep ort the rst numerical and exp erimental results on electromagnetic prop erties of a new
planar chiral metamaterial The metamaterial is formed by a periodic array of uninterrupted crossed
wavy shap ed metal strips placed on a dielectric substrate see inset to Fig The metal dielectric
p erio dic structure is a mo di cation of a grating of wavy strips or a sh scale structure
Crucial di erences of the considered metamaterial are its fold symmetry and a planar chirality due
to an arrangementofnonchiral shap ed strips in the array The inset to Figure shows one of two
enantiomorphic array fashions
Doubly p erio dic planar structures of complex shap ed strips are attractive for microwave applica
tions b ecause of their resonance prop erties in the frequency band of a single wave regime In recent
years a great numb er of pap ers was published on arrays of separated metal strip elements Neverthe
less prop erties of arrays of uninterrupted complex shap ed crossed strips are not considered su ciently
Numerical analysis of electromagnetic wave di raction by the structure was carried out bythe
integral equation metho d describ ed in The metal pattern was treated as a p erfect conductor
while the substrate was assumed to b e a lossy dielectric In the exp eriments we used the structure
that was etched from a m copp er lm on a b erglass PCB material The overall size of the
sample was approximately mm mm We found a very go o d agreement of the numerical and
exp erimental results
Because to a fold symmetry of the array the re ection and transmission prop erties of the struc
ture don t dep end on a p olarization of normal incident plane wave In contrast to arrays of separated
metal particles the structure is non transparentinlow frequencies The rst frequency of a high reso
Figure Normal incidence re ection a and transmission b of linearly p olarized waveby the pla
nar metamaterial Solid lines and separate symb ols denote the exp erimental and numerical results
resp ectively The inset shows a fragment of the array of crossed wavy shap ed copp er strips patterned
on a dielectric substrate The dashed line b ox indicates a mm mm elementary translational
cell of the structure Wavy shap ed strips consist of semicircles The width of strips is mm The
thickness of substrate is mm Substrate relative p ermittivity is i
DAYS on DIFFRACTION
nance transparency is considerably lower than corresp onding frequency of an array of straight crossed
wires arranged with the same spacing A planar chirality of the metamaterial leads to di erence in
di raction of left and right circular p olarized waves at an oblique incidence This prop ertymaybe
useful for FSS applications of the metamaterial in setups op erated with circular p olarized wave b eams
At last control electron devices may b e incorp orated into structure at the cross p oints of strips and
b oundaries of translational cells in order to transform the array of uninterrupted strips into an array
of separated gammadion shap ed particles and vice versa
References
S L Prosvirnin S A Tretyakov P L Mladyonov Electromagnetic wave di raction by planar
p erio dic gratings of wavy metal strips J Electrom Waves and Appl pp
V A Fedotov P L Mladyonov S L Prosvirnin N I Zheludev Planar electromagnetic metama
terial with a sh scale structure Phys Rev E
The mixed value problem with the skew derivative
for the D Helmholtz equation outside cuts
K V Prozorov P A Krutitskii
Russia
We study the mixed value problem outside cuts in a plane for the Helmholtz equation with atten
uation The similar problem for the Helmholtz equation describing waves has b een studied in
under di erent conditions
By a simple op en curvewe mean a smo oth nonclosed arc without self intersections In the plane
x x x R we consider simple op en curves of class C so that they
N
S
N
We assume that each curve is parameterized do not have common p oints We put
n n
n
by the arc length s
fx x x s x s x s g s a b n N
n n n
so that a b a b Therefore p oints x and values of the parameter s are in
N N
S
N
one to one corresp ondence Below the set of intervals a b on the s axis will b e denoted by
n n
n
also
The tangent and normal vectors to at the p oint x s are denoted by x s x s and
x
n x s x s resp ectively
x
We consider as a set of cuts The side of which is on the left when the parameter s increases
will b e denoted by and the opp osite side will b e denoted by
Wesay that the function u x b elongs to the smo othness class K if
u x C R n C R n and u is continuous at the ends of
S
N
R n nX where X is a p oint set consisting of the end p oints of X x a ru C
n
n
x b
n
in the neighb ourho o d of any p oint x d X for some constants c the inequality
jruj cjx x d j holds where x x d and d a or d b n N
n n
By C R n we denote functions which are continuously extended on cuts from the left and
right but their values on from the left and right can b e di erent so that the functions mayhavea
jump on Let us formulate the mixed value problem for the Helmholtz equation in R n
Problem U Find a function u x which b elongs to the class K ob eys the Helmholts equation
u x k u x x R n arg k f g
x x
satis es the b oundary condition
s F u x j
x s
u u
F s ij j const j j
n
x x
x s
DAYS on DIFFRACTION
Zommerfeld radiation conditions if Im k and the following conditions if Im k
ju x j o jxj jruj o jxj jxj
All conditions of the problem must b e satis ed in a classical sense
On the basis of the energy equalities we can prove that the problem U has at most one solution The
problem U is studied by p otential theory and the b oundary integral equation metho d The problem
is reduced to the Cauchy singular integral equation and then to the Fredholm integral equation of the
second kind which is uniquely solvable The integral representation for a solution of the problem U
is obtained in the form of p otentials
Theorem If arcs areofclassC F s C F s C then the
solution of the problem U exists it is unique and is given in the form of potentials The density in
potentials is a unique solution of the Fredholm integral equation of the second kind and index zero
References
P A Krutitskii The skew derivative problem for the Helmholtz equation outside cuts in a
plane Rendiconti di Matematica Roma Serie P
P A Krutitskii The oblique derivative problem for the Helmholtz equation and scattering tidal
waves Pro ceedings of the Royal So ciety of London ser A P
N I Muskhelishvili Singular integral equations No ordho Groningen
P A Krutitskii Neumann problem for the Helmholtz outside cuts in a plane Comp Maths
Math Phys P
Interaction of sho ckwaves for genuinely nonlinear hyp erb olic systems
G A Omel yanov
Universidad de Sonora Mexico
e mail omel hades mat uson mx
We consider strictlyhyp erb olic systems of conservations laws
u f u
x R t
t x
where u u u f f f Let R b e the eigenvalues and the eigenvectors of the matrix
i i
R uniformly in u Let f u i The assumption of genuine nonlinearity means that
i
iu
the initial data for b e a sup erp osition of two jumps
uj u A H x x A H x x
t
and H x is the Heaviside function We consider the problem x were u and A are constants x
i
for arbitrary values of the amplitudes A assuming that the initial jumps generate two
i
sho ckwaves which remain stable until the time instant of the sho ckwaves paths intersection
To describ e uniformly in time the pro cess of the sho ckwaves interaction we pass to a smo oth
regularization of the problem and use the weak asymptotics metho d Since centered rarefaction
waves can app ear as a result of the interaction the asymptotic anzatz contains many free parameters
Resp ectively there app ears a very complicated dynamical system to describ e the time evolution of
these parameters However it is p ossible to avoid a detailed analysis of the system supplementing to
the anzatz small in the weak sense corrections Therefore we do not need to sp ecify the nonlinearity
We consider the uniform in time description of the centered rarefaction app earance phenomena as
amostinteresting result of this research
The researchwas supp orted by CONACYT under grant Mexico
DAYS on DIFFRACTION
D isotropic metamaterial based on dielectric resonant spheres
M Odit I Vendik O Vendik
Microwave Micro electronic Lab oratory Dept of Micro electronics Radio Engineering St
Petersburg Electro technical University Prof Pop ov St St Petersburg Russia
e mail MAOdit mail eltech ru
Arti cial metamaterials with negative p ermittivity and p ermeability double negative material
DNG are very promising for designing novel electromagnetic comp onents with interesting and useful
prop erties Most of mo dels intro duced are anisotropic however isotropic DNG material should b e
more practical for many applications Earlier in the mo del of isotropic double negative structure
was intro duced A structure considered consists of two sublattices of dielectric spherical particles of
high p ermittivity and di erent radii emb edded in a dielectric matrix of smaller p ermittivity It has
b een shown that such a comp osite medium reveals prop erties of an isotropic double negative media
DNG in a limited frequency range when resonance oscillations of H mo de in one kind of particles
and E mo de in another kind of particles are excited simultaneously
Consideration of scattering of the plane electromagnetic wave on a sphere allows intro ducing a
di raction mo del to de ne e ective DNG medium parameters Mo dules of elds inside and outside the
spheres were analytically calculated The calculation shows a sizeable attenuation of the electromag
netic eld outside the spheres therefore in a rst approachwe can neglect mutual interactions b etween
particles in case of appropriate distance b etween the particles and to consider each particle as a single
indep endent sphere Our next step was taking into consideration this interaction In order to do that
analytical di raction mo del was improved Firstly for calculating resonant frequencies of dielectric
spheres the real p ermittivity of the host material was used In the next iteration we consider that
spheres are placed in a host medium with e ective frequency dep endent p ermittivity The e ective
p ermittivity results obtained are close enough to the previously calculated This implies the fact that
the interaction b etween particles is weak enough
In previous pap ers the structure containing three single cells with typ e of spheres with
p ermittivity and radii r mm r mm was simulated by the full wave analysis
More accurate analysis taking into account symmetry of the structure has b een implemented It has
b een shown that there is a stop band near the resonant frequency in case of negative p ermittivityor
p ermeability only But for mixture medium with b oth set of spheres there is a narrow pass band near
the resonant frequency The frequency range of the electromagnetic wave transmission corresp onds
to the double negativecharacteristics of the structure Evidently the resonance frequency is close to
that calculated by the di raction mo del Pass band width is ab out MHz This value is close to the
analytically calculated
Also exp eriment has b een p erformed with four samples with di erent radii mm mm
mm and mm These dimensions were selected to obtain the resonant frequency near
GHz Ferro electric spherical particles have b een placed by turns into metallic waveguide connected
with the network analyser Firstly one waveguide p ort was shorted with a moveable electric wall
Moving the electric wall we obtain the electric eld distribution in the waveguide with its maximum
in the place where the sphere is situated That allows to get maximum re ection level from the sphere
at the resonant frequency In the case of bigger sphere inside the waveguide wehave a magnetic dip ole
resonance That means that we should use a magnetic wall to get maximum re ection With regard
to imp ossibility to create a real magnetic wall we also use real electric wall placed at distance
from the plane where the virtual magnetic wall should b e In the next exp eriment the waveguide
with a matched termination have b een used and the transmission co e cientwas measured Resonance
nature of S and S parameters was observed near theoretically predicted resonance frequency
Exp erimental veri cation proves validity of di raction mo del The results obtained allow to con
sider p ossible realization of arti cial double negative material
DAYS on DIFFRACTION
References
I B Vendik O G Vendik M A Odit Isotropic arti cial media with simultaneously negative
p ermittivity and p ermeability Sp ecial Issue of Microwave and Optical Technology Letters pp
vol
J A Stratton Electromagnetic Theory McGraw Hill Bo ok Co Inc
PO PTD near eld scattering and di raction metho d for
path loss prediction in urban mobile radio systems
E G Papkelis H T Anastassiu P V Frangos
Division of Information Transmission Systems and Materials Technology Departmentof
Electrical and Computer Engineering National Technical UniversityofAthens
Iro on Polytechniou Str Zografou Athens Greece
e mail pfrangos central ntua gr
The near Fresnel zone eld calculation of radio wave scattering and di raction is very imp ortant
for the accurate path loss prediction in cellular mobile radio systems since in a typical urban street
grid environment the scattered and di racted near Fresnel zone eld from building walls and wedges
o ccupies a large p ercentage of the surrounding area In recent publications wehavedevelop ed a two
dimensional simulation mo del for the radio coverage prediction in urban area sites This mo del is based
on the analytical metho ds of Physical Optics PO and Physical Theory of Di raction PTD and takes
into account the electromagnetic elds at the receiving antenna due to rst and high order propagation
mechanisms In contrast to other published deterministic metho ds where the radio coverage analysis is
limited to the far eld scattering this mo del includes near Fresnel eld calculations since in a typical
urban environment the near or Fresnel eld scattering area covers most of the street canyons region
due to the large dimensions of the building walls relative to the wavelength of the radiation Even
though this mo del was proved to have satisfactory prediction accuracy for MHz and GHz
as compared to other high frequency metho ds and measurement data it is quite time consuming
mainly due to the near eld computations which are based on numerical integration over the surface
PO currents and subsequentnumerical di erentiation of the vector p otential In this pap er a time
e cient metho d for the calculation of the scattered eld from plane scatterers rectangular plates for
the D case and plane strips of nite length and in nite height for the D case in the near and
Fresnel zone is presented The metho d whichwecall NeartoFar Field Transformation Metho d
is based on the analytical metho d of Physical Optics PO and uses the technique of segmentation
of the scattering surface into an appropriate numb er of small rectangles cells By such a division
of the large scatterer an observation p oint which is originally lo cated in the near or Fresnel zone
of the scatterer is now relo cated in the far zone of the smaller cells As a result the PO far
eld approximations can b e invoked in order to calculate the electromagnetic eld at the receiver
due to the induced PO currents at each cell The total scattered eld at the receiver s p osition is
calculated by complex vector addition of all signal comp onents received The incorp oration of the
prop osed metho d in the simulation co de by replacing the numerical integration and di erentiation
op erations mentioned ab ove decreases drastically the calculation time and at the same time provides
even more accurate results since it avoids p ossible numerical errors due to the di erentiation of the
vector p otential Finally extensive research studies are currently undertaken in order to implement
the Near to Far Field Transformation Metho d for D di raction from the building wedges In
particular the di racted eld near the building wedges is calculated by applying a combination of
the prop osed metho d describ ed ab ove with the accurate Incremental Length Di raction Co e cients
theory PTD which has b een intro duced by Mitzner for the D di raction problem in the far eld area
DAYS on DIFFRACTION
Two layer spherical dielectric lens with
re ector excited by a radial electric dip ole
Valeriy V Radchenko
Institute of Radiophysics and Electronics NASU
Ul Proskury Kharkov Ukraine
In this pap er we study the prop erties of the double layered spherical dielectric lens with re ector
It is considered that the layered sphere excited by a radial electric dip ole lo cated near the surface of
the p erfectly conducting zero thickness re ector shap ed as a spherical disk These antennas are used
for multi b eam scanning at millimeter and microwave frequencies in mobile communication systems
remote sensing and radio astronomy In the geometrical optics approximation these typ es of lenses
transform the feed s spherical wavefrontinto a plane wave The attractive features of these antennas
are the op eration over a broad band of frequencies and the spherical symmetry of the lens that allows
multi b eam scanning by placing an array of feeds around the lens There are many pap ers considering
this typ e of lenses Normally they are analyzed by the FDTD metho d which has b oth advantages and
disadvantages such as accuracy loss in the presence of resonances Toovercome this problem we use
well develop ed metho d of analytical regularization MAR
Toformulate the electromagnetic wave radiation problem
for a dip ole and lens we use all conditions necessary for the
uniqueness theorem From b oundary conditions on the spher
ical surface we obtain the dual series equations in terms of the
asso ci ated Legendre functions and further use MAR which
is e cient for numerical analysis It is based on the analyti
cal inversion of the static problem for a p erfectly conducting
spherical disk in free space The nal result has a form of
in nite set of linear alge braic equations
h i
X X
m
X X Q n n Q
m n n n
nm n nm
n n
This set is of the Fredholm second kind and therefore has guaranteed convergence and controllable
accuracy even in the presence of resonances
The main aim of our researchis
the lens prop erties study to optimize
them in future The plots of the ra
diation resistance radiation pattern
and directivity in the wide band of
parameters variation are presented in
this pap er for di erentlenslayers pa
rameters and di erent angular sizes
of re ector Besides we compare the
prop erties of the layered dielectric lens
in the presence of re ector and with
out it As one can supp ose the pres
ence of re ector decreases the level of
the back side lob es in the radiation
pattern
a c s s
DAYS on DIFFRACTION
Stability of p erturbation of ows and typho on generation
D Bianchi
DepartmentofPhysics University La Sapienza Rome Italy
e mail danielabianchi virgilio it
F Raicich
Institute of Marine Sciences CNR Trieste Italy
e mail fabio raicich ts ismar cnr it
We analyze the velo city eld of the atmospheric ows involvedintheoriginofatypho on According
to a recent theoretical result the lo cal p erturbation of a given ow conserves its form if the Cauchy
Riemann conditions of the velo city eld of the main ow are veri ed Thus we analyze the velo city
elds in the region where lo cal p erturbations are generated starting from the velo city distribution of
the wind measured in such situations and check these conditions
References
S Yu Dobrokhotov B Tirozzi A Shafarevich Cauchy Riemann conditions and lo calized asymp
totic solutions of the linearized shallowwater equation Applied Math Mech V N
Interaction of dissipative b er Bragg solitons
X Tr Tran N N Rosanov
St Petersburg State University of Information Technologies Mechanics and Optics
St Petersburg Russia
e mail nrosanov yahoo com
The interaction of dissipative optical solitons in b ers with saturable gain and absorption and linear
Bragg grating is investigated The initial phase di erence b etween solitons plays a crucial role in their
interaction Dep ending on this value the solitons can attract or rep el each other at the b eginning
stage of their interaction As illustration Figure shows the interaction of two dissipative Bragg solitons
that are in phase initially then these initially motionless solitons transform into two solitons moving
with fairly large sp eed in the opp osite
directions
Beyond the approximation of slow
ly varying amplitudes wehave found
that single solitons are lo cated near
maxima of refractive index of Bragg
grating This feature results in the
p ossibilityoflargenumb er of coupled
states of motionless dissipative Bragg
solitons We demonstrate stable b o
und states of two motionless dissipa
tive Bragg solitons whose nal phase
di erence is nearly equal to or
whereas similar pairs with phase
di erence near and are unstable
Figure Interaction of two dissipative Bragg initially in phase solitons
DAYS on DIFFRACTION
Complexes of weakly coupled D laser solitons
S V Fedorov N N Rosanov N A Veretenov
Scienti c Industry Corp oration S I Vavilov State Optical Institute
Birzevaya line Saint Petersburg Russia
e mail nrosanov yahoo com
We consider complexes of D dissipative solitons in a continuous medium with frequency disp ersion
and nonlinear gain and absorption describ ed by the generalized complex Ginzburg Landau equation
for the electric eld envelop e E
E a E E g E
i d i d f jE j E f jE j
k
z x y
jE j b jE j
Here we use the dimensionless Cartesian longitudinal co ordinate z transverse co ordinates x and y
t z v t is time v is the group velo city g and a are small signal gain and absorption co e cients
g g
resp ectively b is the ratio of
intensities of saturation for
gain and absorption non
negative co e cients d rep
resent the medium dichro
ism d and sp ectral lter
ing d The governing
k
equation has the most sym
metric form in the sp ecial
case d d In Fig we
k
presentsometyp es of sta
ble soliton complexes and
Fig Stable complexes of D laser solitons in phase and antiphase pairs left
Fig demonstrates a vari
cub e in phase motionless tetrahedron centre and moving in line
ant of these complexes col
pyramid with one antiphase soliton in the vertex right the eld phase
is co ded in grey scale on the left
lision
Fig Collision of twomoving pyramid like complexes and formation of a rotating soliton complex
DAYS on DIFFRACTION
E ect of Kerr nonlinearity on discrete dissipative optical solitons
Al S Kiselev An S Kiselev N N Rosanov
St Petersburg state university of information technologies mechanics and optics
Kronverskiy pr St Petersburg Russia
e mail andreys kiselev yahoo com nrosanov yahoo com
We consider propagation of intensive radiation in an arrayofweakly coupled single mo de b ers
with fast nonlinearity of gain absorption and refractive index Kerr nonlinearity The goal is to
analyze the e ect of Kerr nonlinearityoncharacteristics of spatial discrete dissipative solitons in the
array
The analysis of one dimensional spatial discrete dissipative solitons with weak coupling on the
nearest neighb oring b ers bases on the following equations of the coupled mo de theory
i h
E g a
n
i jE j i E f jE j f jE j C E E
n n n n n n
z
jE j b jE j
n n
Here z is the longitudinal co ordinate E is mo de amplitude of the n th b er C is coupling co e cient
n
is Kerr co e cient g and a are small signal gain and absorption co e cients resp ectively b is the ratio
E E
h h of intensities of saturation for gain and absorption In the continual limit E E
n n
x
x
where h is the spacing b etween the b ers these equations come to the generalized complex Ginzburg
Landau equation where dissipative solitons are well known in the literature At the same time more
interesting are highly lo calized structures not describ ed by the continual mo del
Numerical simulation of radiation propagation along the array of b ers results in the following
conclusions on the e ect of Kerr nonlinearity see Fig Self fo cusing nonlinearity gives the
ultimate narrow radiation lo calization and extremely stable discrete solitons On the contrary self
defo cusing nonlinearity results in soliton pro le widening and decreases its stability
Figure Evolution of intensity distribution in an arrayofweakly coupled single mo de b ers with
nonlinear gain and absorption Kerr nonlinearitytyp e is self fo cusing left zero centre and self
defo cusing right
Simulations show also the p ossibility of formation not only transversely motionless but also moving
discrete dissipative solitons with pro le de ned by the system parameters It is p ossible to control
the soliton transverse motion intro ducing gain or absorption inhomogeneity These regimes present additional to ols for optical information pro cessing
DAYS on DIFFRACTION
Electromagnetic eld transformation in circular waveguiding
and resonant structures with time varying p ermittivity
Nataliya Sakhnenko Alexander Nerukh
Kharkov National University of Radio Electronics Lenin Ave Kharkov Ukraine
e mail n sakhnenko yahoo com
This pap er presents the analytic results of comparison of mo des transformation in time varying
structures on examples of circular waveguide and circular whisp ering gallery mo de resonator The sit
uation is that a natural mo de propagates inside the structure at the moment when the refractive index
changes abruptly in time inside it A closed form solution is obtained for the electric eld comp onent
by applying the Laplace transform directly to the wave equation and enforcing the corresp onding
initial and b oundary conditions
The basic interest is to gain insightinto details of mo de evolution during the transient p erio d and in
steady state regime Wereveal the common features that underlie these phenomena in b oth structures
Time changing of the dielectric p ermittivityinunb ounded space leads to change of a wave fre
quency and conservation of a wavenumb er In the presence of b oundaries this pro cess is more
complex and it dep ends qualitatively on con guration of the initial eld For the case of initial eld
without external source e g mo de of the resonator changing of refractive index leads to the changing
of frequency and wavenumb er but near eld pattern conservation If the initial eld has external
source e g p oint source then the temp oral change of dielectric p ermittivity inside the resonator
leads to complicated transient pro cess caused by excitation of all mo des of the resonator in its new
state and transformation of the near eld pattern Possibility of the source near eld pattern control
in D circular resonator by adjusting in time of media parameters is discussed
The analytical solutions of circular waveguide mo des transformation due to abrupt time change
in dielectric p ermittivityisconsideredaswell It is revealed that initial mo de evolves into time
transmitted and time re ected mo des of the structure in its new state These mo des havethe
same eld pattern as initial one Exception is the fast time decayofwaveguide higher mo des when
dielectric p ermittivity in the core decreases
Acknowledgements The authors wish to thank the supp ort of this pro ject under a NATO Reinte
gration Grant CBP EAP RIG
References
F R Morgenthaler IRE Trans Microwave Theory and Techn MTT
N K Sakhnenko T M Benson P Sewell A Nerukh Optical Quantum Electron
Solving the problems of di raction by
Wiener Hopf Fock metho d for nite ne structures
S S Sautb ekov
Euroasian National University Munaitpassov str Astana Kazakhstan
e mail sautbek mail ru
This pap er deals with further development of Wiener Hopf Fock metho d for solving the wave s
di raction on ne strip slotted structures problems The classical problem on di raction of plane
wave on a strip is o ered as a key problem The b oundary value problem is consecutively solved by
reducing to the system of singular b oundary integral equations then to the system of the second kind
Fredholm equations A few way constructing the analytic solutions of this system are prop osed Using
the integral op erator of di raction from intro duced edges sources the solutions are presented in the
form of di raction series and also with use of sp ecial functions From solutions for the strip slotted
structures the characteristic equations and resonance conditions are obtained
DAYS on DIFFRACTION
On application of asymptotic solutions to
calculation of tsunami piston mo del
S Ya Sekerzh Zenkovich
Institute for Problems in Mechanics RAS Moscow Russia
e mail seker ipmnet ru
B I Volkov
DepartmentofPhysics Lomonosov Moscow State University Moscow Russia
e mail volkovbmeister gmai com
The conditions are derived at which asymptotic solutions of the Cauchy problem for the wave
equation are applicable to calculation of the tsunami piston mo del in an uniform depth basin It
is done by comparing the results of numerical mo deling with those obtained using the asymptotic
solutions for the initial b ottom disturbances in the form of the gaussian exp onents The question of
the wave propagation is discussed
This work was partially supp orted by RFBR grant N
References
Dobrokhotov S Sekerzh Zenkovich S Tirozzi B Tudorovski T Description of tsunami propa
gation based on the Maslov canonical op erator Doklady Mathematics pp
On the information transmission metho d using material ob jects far action
reaction on the external electromagnetic unconverted in uence
Alexander A Chaplygin Victor I Sergeev
Russia
The aim of this pap er is to describ e a p ossible metho d of one bit information transmission using a
far action reaction i e instantaneous action in distance of material ob jects on external electromag
netic unconverted in uence
The initial information is created by means of simultaneous formation of an electromagnetic soli
ton and an electromag netic instanton resp ectively Then electromagnetic soliton and instanton are
united into electromagnetic soliton instantonic breather SIB After b eing formed the SIB is for
warded to a certain rst resonator and it excites the rst resonator Then the SIB lo cated in the rst
resonator is exp osed to the in uence of the electromagnetic eld with pre set parameters to cause a
soli ton instantonic tremor The rst resonator excitation by the electromagnetic breather results in
induction of an extreme fast decaying EMP in the second resonator Both resonators should have
a pass band not smaller than the SIB sp ectrum in opp osite case the breather will disintegrate into
ordinary waves group
The rst and the second resonator can have di erentoverall dimensions In case the second
resonator admits equal or larger amplitude values of p ower characteristics in comparison with the rst
resonator then in the second resonator the EMP not larger than the rst resonator EMP is inducted
In case the second resonator admits amplitude characteristics lesser than the rst resonator signal
amplitude characteristics then in the second resonator the maximum p ossible EMP but not larger
than the rst resonator EMP is inducted The rst and the second resonators are oriented in space
accordingly one another
In case the said conditions are violated information transmission by use of the describ ed metho d is imp ossible
DAYS on DIFFRACTION
It has b een p ointed that during the signal trip through dielectric media in signal a reactive
comp onent app ears whichischaracterized by the presence of an imaginary comp onent in analytical
description of a non linear signal Thus in case the soli ton is formed with simultaneous propagation
of this signal through dielectric media the splitting of the signal to soliton and to instanton takes
place Both parts are shown as an intertwined structure the soliton instantonic breather Therefore
in this case the supplementary uni cation of the soliton and the instanton to breather is not needed
The said metho d of information transmission is a result of an actual transference of the rst
part of Einstein Po dolsky Rosen paradox statements to macro level where the resonance systems are
represented not by elementary particles but by macro resonators in singlet connection In lightof
the ab ove stated it is obvious that the singlet connection b etween macro resonators is realized due
to their coherence and corresp ondence including identical orientation in space Besides the breather
tremor corresp onds to the e ect of parametric absorption of electromagnetic waves energy by material
ob jects
Consequently the formation of solitons non linear signals that excite the rst resonance system
is crucial for realization of the information transmission metho d using a far action
The linear signal represented by an electromagnetic sine wave oscillation or impulse signal is formed
The linear signal is fed to a non linear b o dy In general the frequency of the linear signal carrier wave
do es not in uence a p ossibility of a nonlinear signal formation with the given non linearitytyp e In
fact for every frequency range of a linear signal carrier wave there is only one optimal choice of a
non linear b o dy formation
A non linear b o dy may b e represented by a three p ole unit two inputs and one output One of
the p ole unit s inputs receives a linear signal that should b e transformed into non linear signal and
the other input receives a pumping signal which can manage the non linear parameters of a non linear
body The output yields a non linear signal which is a result of parametric addition of a linear signal
and a pumping signal in a non linear b o dy
The non linear parameters management is p erformed at the exp ense of the in uence of a strong
pumping signal and or linear signal since the non linear parameters are a temp oral functional of the
pumping signal and or linear signal variation law
In whole the algorithm of the pumping signal and or linear signal parameters de nition is as
follows the typ e of the non linearity of the resulting nonlinear signal is know a priori is given
The equation is resolved with one or more unknown quantities each of which is a time function that
describ es required signal or signals The choice of a resp ective non linearitytyp e of a nonlinear signal
in uences the choice of pumping signal and linear signal parameters that forces us to a priori de ne
the nonlinear prop erties nonlinear parameters of a nonlinear b o dy
The utilization of an arti cial nonlinear b o dy lo cated in the zone of nonlinear signal propagation
in the soliton formation device can b e considered a realization of the describ ed metho d of a nonlinear
signal formation a soliton Such nonlinear signal is passed through a nonlinear b o dy as a parametric
converter The nonlinear b o dy receives a pumping signal The resulting nonlinear signal will b e
a result of parametric addition of a linear signal and a pumping signal under conditions of a non
linear b o dy The required pumping signal parameters are de ned as a result of a parametric addition
equation solution according to ab ove mentioned algorithm of pumping and or linear signal parameters
de nition
According to the describ ed metho d of the information transmission which is based on the hyp oth
esis of a far action of material ob jects reaction on unconverted external electromagnetic in uence
an excitation of the rst resonator is carried out under xed small p ower exp enditures The second
resonator s can b e lo cated at maximum distance That is utilization of the describ ed metho d of the
information transmission allows to increase the distance of information transmission under xed small
power exp enditures and also ensures the information transmission in complicated electromagnetic
conditions since according to the rst part of the Einstein Po dolsky Rosen paradox statements the
resonance systems in singlet connection will react inde p endently from the distance and electromag
netic conditions of the circumambient media
DAYS on DIFFRACTION
Numerical study of di raction phenomena in achitectural acoustics
A Shanin E Shabalina
Moscow Russia
The aim of the research is to estimate the in uence of typical di raction phenomena on acoustical
prop erties of ro oms halls etc As the main prop ertyofaroomwetake its reverb eration time Thus
we develop a metho d of numerical estimation of the reverb eration time accurate enough to contain
the information on the di raction e ects
Toachieve this ob jective weuseaveraging in frequency domain to de ne the reverb eration time
This averaging enables us to suppress the in uence of global interferention pattern As a result
weintro duce a family of dep endencies of reverb eration time vs frequency having the bandwidth of
averaging lter as a parameter A prop er choice of the bandwidth is discussed in the talk
The program for numerical mo deling in D case is describ ed The results are presented and
discussed
Singular solutions to systems of conservation laws connected
with transp ortation and concentration pro cesses
V M Shelkovich
St Petersburg State Architecture and Civil Engineering University Russia
e mail shelkv VS spb edu
n
There are nonclassical situations when systems of conservation laws admit and sho ck
typ e solutions They are singular solutions such that their comp onents maycontain the Dirac delta
functions and their derivatives up to the order n n For the theory of sho cks see
n
and the references therein An absolutely new theory of sho cks was established in
Using our previous results we study the Cauchy problems admitting sho cks for multidi
mensional zero pressure gas dynamics systems in conservativeform
r U U r U U
t t
and non conservative form
r U U U r U
t t
Using our results we study the Cauchy problems admitting sho cks for the system
u f u v f u v w f u v f u w
t x t x t x
where f u f u f u f u are smo oth functions
These typ es of solutions are related with transportation and concentration processes We derive
and sho ck balance relations related with mass momentum and surface transp ortation to singular
fronts
References
S Alb everio V M Shelkovich On the delta sho ck front problem in Analytical approaches to
multidimensional balancelaws ch Ed O S Rozanova Nova Science Publ Inc
V G Danilov V M Shelkovich Delta sho ckwavetyp e solution of hyp erb olic systems of conser
vation laws Quart Appl Math
E Yu Panov V M Shelkovich De nition of sho ckwaves typ e solutions and the Rankine
Hugoniot conditions Proc Days on Difraction St Petersburg
E Yu Panov V M Shelkovich Sho ckwaves as a new typ e of solutions to systems of conserva
tion laws J Di erential Equations
V M Shelkovich The Riemann problem admitting sho cks and vacuum states the vanishing
viscosity approach J Di erential Equations
DAYS on DIFFRACTION
Self fo cusing of few cycle light pulses in transparent disp ersive optical
media with electronic and electronic vibration cubic nonlinearities
Yu A Shp olyanskiy A N Berkovsky M A Berlin S A Kozlov
St Petersburg State University of Information Technologies Mechanics and Optics Russia
Optics of femtosecond fs pulses containing only few eld cycles is a hot topic of a mo dern laser
physics Here we analyze theoretically propagation of few cycle wavepackets in bulk optical media
with normal disp ersion and cubic nonlinearity of electronic and electronic vibration Raman nature
For the purp oses of this work let us lab el pulses with ten or less eld cycles as extremely short pulses
ESP
Theoretical mo dels of ESP propagation having b een actively discussed in recentyears di er from
conventional mo dels based on the slowly varying envelop e approximation whichwas initially suggested
for quasimono chromatic pulses with many eld oscillations Nonlinear equations of ESP propaga
tion are written more and more often directly for the pulse electric eld or its sp ectrum b ecause
the envelop e approach b ecomes unnecessary for few cycle regimes b oth analytically and numerically
Following the technique suggested in wehavederived unidirectional equation for ESP sp ectrum G
Z Z
e e
i T G z G z G z
G i n i
G d d
z c c n n n n
ic
G
n
where z is the propagation co ordinate is the radiation frequency c is the lightvelo cityinvacuum
is the transverse Laplacian and x y are transverse co ordinates Medium resp onse
x y
is de ned by the arbitrary dep endence of the linear refraction index on frequency n cubic sus
e e
ceptibilitie s of electronic and electronic vibration nature and aswell as parameters and
T which quantitatively characterize disp ersion of electronic vibration nonlinearity This is the most
general equation of unidirectional paraxial evolution of pulses with continuum sp ectra in transparent
media with electronic and electronic vibration nonlinearities
Equation has b een solved numerically using Fourier Split Step Technique for axisymmetric
Gaussian likewave packets with few cycle duration and continuum sp ectra centered at the fundamental
wavelength of a Ti S laser and propagating in bulk fused silica We demonstate regimes leading to the
formation of ultrashort light structures like dumbb ells or bubbles dep ending on the initial intensity
An example of dumbb ell li ke distribution is given in Figure A bubble which is not presented here
Figure Self fo cusing of fs ESP with intensity W cm in fused silica medium input z
left and z mm right
DAYS on DIFFRACTION
includes a center area with emptiness where the eld is practically nulli ed We considered the
pro cess of the pulse tail steep ening and subsequent turnover of the envelop e sho ckwave and illustrated
dramatic eld changes and asso ciated sup ercontinuum generation This regime is b eyond the scop e of
the envelop e approachby de nition
It is shown that the account of electronic vibration nonlinearity of fused silica do es not change the
sel o cusing scenario qualitatively the di erences app ear to b e quantitative and b etter seen in the
sp ectral domain We found that the central frequency of generated sup ercontinuum is moved to lower
frequencies due to Raman e ect to compare to simulations were electronic vibration nonlinearityis
ignored Formation of evident Stokes or anti Stokes features has not b een observed in computations
We b elieve that this is due to consideration of ultrabroad sp ectra whichfaroverlap Stokes and
anti Stokes shifts and comparatively low combination activity of fused silica
References
Th Brab ec F Krausz Rev Mo d Phys
S A Kozlov S V Sazonov JETP
A V Husakou J Herrmann Phys Rev Lett
M Kolesik J V Moloney Phys Rev E
S A Izyurov S A Kozlov JEPT Letters
G P Agrawal Nonlinear b er optics Academic San Diego
The scattering op erator in the stepwise waveguides
Ilya A Shereshevskii
Institute for Physics of Microstructures RAS Nizhnii Novgoro d Russia
e mail ilya ipm sci nnov ru
The problem of mo des transformation in non homogeneous waveguides has the very long history
and extensive list of physics and mathematics publications was devoted to this problem during last
century Nevertheless the well known and widely used term scattering matrix seems to b e not yet
mathematically well de ned in the waveguide problems and corresp ondingly the mathematical prop
erties of the scattering op erator in waveguide in any sense are not well describ ed Such description
is however necessary to understanding the pro cesses in the quantum electro dynamics and acoustics
waveguides and in particular to developing go o d numerical algorithms for waveguides simulation
I consider in this work the mathematical mo del of the stepwise waveguide and demonstrate that
the investigation of this mo del is in essential the problem of the theory of the selfadjoint extensions
of the symmetric op erators in the Hilb ert spaces In such approach the scattering op erator app ears
naturally as the parameter of the selfadjoint extensions of some symmetric op erator and hence the
scattering op erator is from the very b eginning the op erator in appropriate Hilb ert space namely
the de ciency space of the initial symmetric op erator This allows us to investigate the prop erties of
the scattering matrix in frame of the op erator theory
Besides the construction of the scattering op erator we obtain in this way the description of some
of its imp ortant prop erties Some of them seem to b e rather unexp ected as for example the fact
that this op erator may b e in general unb ounded in the Hilb ert space of the sequences of mo des
amplitudes with usual scalar pro duct Also it is interesting that under some conditions the scattering
op erator may b e approximated in some well de ned sense by nite dimensional op erators whichis
scattering op erators for appropriate nite dimensional waveguide This approximation conserves
most of imp ortantpropertiesofsuch op erators such as the ow conservation low and can b e used to
numerical calculation of the scattering matrix for various typ es of waveguides
I consider also some algebraic prop erties of the family of scattering op erators which allowusto
construct new op erators from existing ones These prop erties are well known in the nite dimensional
case and are very useful for investigation of multi step waveguides
DAYS on DIFFRACTION
Explicit solutions to a singular di erential equation with Bessel op erator
S M Sitnik
Russia
Consider a problem of nding explicit solutions to a singular di erential equation with the Bessel
op erator B and a p otential function q x
B u r q r u r u r
via solutions of simpler homogeneous equation B v r v r in the next integral form
Z
P r t v r dr u r v r
r
Here P r t isa kernel v r is the Bessel functions of some kind and B is Bessel op erator of the
form B u u u p otential function is integrable at in nity and satis es condition
r
jq r s j p r s r with some nonnegative p r
Theorem Under ab ove conditions the representation exists with the next kernel function
estimate
Z
y t r t r t
P jp y j dy jP r t j
r tr y
Z
y t r t r t r
expf P jp y j dy g
tr y
t r
here P z is the Legendre function
If q r C then P r t C t r and istwice di erentiable For the kernel also
explicit integral equations are found Many imp ortant p otentials such as strongly singular Bargman
and Jukawa are covered by this theorem Equivalent problems are nding of transmutation inter
twining op erator suchasS B q B S or the generalized translation op erator The case with
t r
integrals in over interval is also studied In sp ecial case we arrive at the explicit
formula for the mixed problem to the equation
p y u q x u u u u
yy xx
References
Levitan B M Razlozheniya p o funktsiyam Besselyavryadi i intagrali Fourier UMN V
P
Carroll R Transmutations scattering theory and sp ecial functions North Holland
Sitnik S M Factorization and estimates of the norms of Buschman Erde lyi op erators in weighted
Leb esgue spaces Soviet Math Dokl Vol No P
Sitnik S M Op erator preobrazovaniya i predstavleniya Josta dlya uravnenia s singulyarnym p o
tentsialom PreprintIAPUDVO RAN Vladivostok P
Katrakhov V V Sitnik S M Otsenki reshenii o dnomernogo uravnenia Schro dingera s singul
yarnym p otentsialom Doklady RAN V N P
Katrakhov V V Sitnik S M Komp ozitsionniy meto d p ostro eniya B elliptichi skikh B
parab olicheskikh i B hyp erb olicheskikh op eratorov preobrazovaniya Doklady RAN V
N P
DAYS on DIFFRACTION
On continuous sp ectra of the elasticity problems
Serguei A Nazarov
Institute of Mechanical Engeneering Problem RAS St Peterburg Russia
It will b e demonstrated that for a p eak shap ed solid with the surface free of traction the sp ectrum
of the op erator b ecomes continuous and a certain information on the structure of the sp ectrum can b e
obtained This observation leads to several conclusions on asymptotics of solutions to sp ectral elasticity
problems for solids with singularly p erturb ed b oundaries In particular concentration of eigenvalues
in the ultra low frequency range and lo calization of the corresp onding eigenmo des were discovered
for elastic b o dies with blunted p eaks and for heavy and hard elastic p eak shap ed inclusions As a
result of these consideration the following result is obtained Under a few symmetry assumptions on a
comp osite elastic cylinder any preassigned numb er of linearly indep endent trapp ed mo des solutions
to the homogeneous elasticity problem decaying at in nity can b e found on the continuous sp ectrum
byvarying the physical prop erties of inclusions This result provides the e ect of decrement damping
of certain packets of elastic waves
Anisotropic approximation of radially layered elastic media
Alexander A Sidorov
Institute of Physics St Petersburg State University Russia
e mail asidorov earth phys spbu ru
Approximate solution of elasticity equation for the cylindrical pip e consisting of multiple isotropic
homogeneous layers was obtained
Following L Molotkovtechnique we replaced multilayered pip e by e ective anisotropic pip e with
radial anisotropy Thus we got one equation for anisotropic pip e instead of several equations for
eachlayer interconnected by b oundary conditions Then formal solutions for e ective equations were
found as series All these considerations are valid only for low frequencies and under some geometrical
conditions
As the next step we conidered solutions obtained at zero frequency limit only rst terms of series
were taken and got formula for tub e wavevelo cityinmultilayered cylindrical pip e with water inside
Velo cities computed by the formula obtained are in a go o d agreementwithnumerical computations
Transientwaves pro duced by a spherical
source pulsating with di erent p erio ds
Irina I Simonenko
Institute of Physics St Petersburg University
Ulyanovskaya Petro dvorets St Petersburg Russia
e mail simon IS spb edu
The solutions of the initial value problem to inhomogeneous wave equation are constructed in terms
of spherical harmonics for sources distributed on a spherical surface p erio dically expanding with the
velo city of light The sphere has the xed radius at the initial moment of time The time of sphere
expansion is xed The numb er of expantions is nite Perio ds b etween two consequent expansions
are not equal to the time of sphere expansion as it was considered in The wavefunctions are
constructed for the di erent p erio ds with the help of the Smirnov metho d of incomplete separation of
variables and the Riemann Formular Separating the p olar angle variable we represent the solution
DAYS on DIFFRACTION
as the Legendre p olynomial series whose co e cients are function of the radial and time variables The
analytical expressions for the ab ove co e cients are obtained by means of the Riemann formula
We apply the constructed solution in the cases when the source is distributed on the circle b elonging
to the pulsate sphere and moving p oint source on the circle Weinvestigate the space time structure of
formed waves in dep endence of pulsation p erio ds Applicatiom of the scalar solution to the description
of electromagnetic waves is also discussed
References
Simonenko I I Proceedings Day on Di raction St Petersburg p
Simonenko I I Proceedings Day on Di raction St Petersburg p
SmirnovVI Dokl Akad Nauk SSSR
Analysis of the TE wave propagation in nonlinear dielectric
three layer planar waveguides with non Kerr nonlinearity
Y Smirnov E Sysova
Department of Mathematics and Mathematical Mo deling Penza State University
ul Krasnaya Penza Russia
Propagation of electromagnetic waves in linear media along three layer planar dielectric waveguides
is a relevant topic of classical electro dynamics Cylindrical waveguide structures consisting of
nonlinear media is also an ob ject of intense studies However until recently any rigorous pro ofs of
the existence of mo des in nonlinear dielectric waveguides have not b een obtained as well as the correct
mo de classi cation Planar structures consisting of several plane parallel layers lled with nonlinear
Kerr law media is another well studied family of waveguides see Whereas for example in the
sp ecial case of a three layer planar waveguide with Kerr nonlinearity in the layers the eld can b e
describ ed in terms of elliptic functions
In this work we consider propagation of TE p olarized electromagnetic waves in a three layer
planar waveguide lled with a non Kerr typ e nonlinear nonabsorbing nonmagnetic and isotropic
dielectric We lo ok for time harmonic propagating solutions and reduce the problem to a transmission
and b oundary value problem for quasilinear di erential equations The di erential equations are solved
analytically by elliptic functions Numerical results are also presented
References
Borgnis F and Papas C Electromagnetic Waveguides in HandbuchderPhysik Editor
Flugge S Springer Verlag Berlin
Snyder A and Love J Optical Waveguide Theory Chapman and Hall London
G Agrawal Nonlinear Fib er Optics Second Ed New York Academic Press
Eleonski V M et al Cylindrical Nonlinear Waveguides Sov Phys JETP
Chen Y Nonlinear Fib ers with Arbitrary Nonlinearity J Opt So c Am B
Schurmann H W Serov V and Shestopalov Yu TE p olarized waves guided by a lossless
nonlinear three layer structure Phys Rev E
Smirnov Yu Schurmann H W and Shestopalov Yu Integral Equation Approach for the
Propagation of TE Waves in a Nonlinear Dielectric Cylindrical Waveguide J Nonlin Math
Phys no
Schurmann H W Smirnov Yu and Shestopalov Yu Propagation of TE waves in cylindrical
nonlinear dielectric waveguides Phys Rev E
DAYS on DIFFRACTION
Analysis of the TM wave propagation in nonlinear
dielectric layer planar waveguides with Kerr nonlinearity
Y Smirnov D Valovik
Department of Mathematics and Mathematical Mo deling Penza State University
ul Krasnaya Penza Russia
Propagation of electromagnetic waves in linear media along three layer planar dielectric waveguides
is a relevant topic of classical electro dynamics Waveguide structures consisting of nonlinear
media is an ob ject of intense studies However until recently any rigorous pro ofs of the existence
of mo des in nonlinear dielectric waveguides have not b een obtained as well as the correct mo de
classi cation Planar structures consisting of several plane parallel layers lled with nonlinear Kerr
law media is well studied family of waveguides see Whereas for example in the sp ecial TE case
of a three layer planar waveguide with Kerr nonlinearityinthelayers the eld can b e describ ed in
terms of elliptic functions
In this work we consider propagation of TM p olarized electromagnetic waves in a two layer and
three layer planar waveguides lled with a Kerr typ e nonlinear nonabsorbing nonmagnetic and
isotropic dielectric We lo ok for time harmonic propagating solutions and reduce the problem for
Maxwell s equations to a transmission and b oundary value problem for quasilinear di erential equa
tions Integration yields an equation so that one eld comp onent can formally b e expressed in terms
of the other leading to the p ossibility of nding one eld comp onentbyintegration Evaluation of the
b oundary conditions for the layer structure leads to the disp ersion relation The di erential equations
are solved analytically by elliptic and hyp erelliptic functions Numerical results are also presented
References
Borgnis F and Papas C Electromagnetic Waveguides in HandbuchderPhysik Editor
Flugge S Springer Verlag Berlin
Snyder A and Love J Optical Waveguide Theory Chapman and Hall London
G Agrawal Nonlinear Fib er Optics Second Ed New York Academic Press
Eleonski V M et al Cylindrical Nonlinear Waveguides Sov Phys JETP
Chen Y Nonlinear Fib ers with Arbitrary Nonlinearity J Opt So c Am B
Schurmann H W Serov V and Shestopalov Yu TE p olarized waves guided by a lossless
nonlinear three layer structure Phys Rev E
Leung K M P p olarized nonlinear surface p olaritons in materials with intensity dep endent di
electric functions Physical Review B
Joseph R I Christo doulides D N Exact eld decomp osition for TM waves in nonlinear media
Optics Letters
Cauchy problems in b ounded domains
Feynman path integrals and difusions
O G Smolyanov
Lomonosov Moscow State University Moscow Russia
One investigates some relations b etween Cauchy problems in b ounded domains Cauchy problems
in the whole space Feynman path integrals over tra jectories in Riemannian manifolds and di usions
on such manifolds
DAYS on DIFFRACTION
Lasing sp ectra and thresholds of a circular micro cavity
laser emb edded in an annular Bragg re ector
Elena I Smotrova Alexander I Nosich
Institute of Radio Physics and Electronics NASU Kharkov Ukraine
e mail elena smotrova yahoo com
Trevor M Benson Phillip Sewell
George Green Inst for Electromagnetics Research Univ of Nottingham NG RD UK
Key elements of compact integrated circuits are micro cavity lasers due to their small size and very
low threshold op eration Ring and disk lasers use whisp ering gallery WG mo des that exp erience
almost total internal re ection at the cavity rim These mo des op erate under extremely low thresh
old pumping energy Further reduction of micro cavity size leads to the increasing of mo de threshold
as far as low order lasing mo des are non WGmodes Here we analyze a circular micro disk laser
emb edded into an annular Bragg re ector ABR Figure
This structure has p otential to supp ort low threshold mo des
with low azimuth index as far as ABRs have the advantage of
strong in plane radial optical con nement In this pap er
we study D mo del of micro cavity and fo cus on the H po
z
larized mo des Our study is based on the Lasing Eigenvalue
Problem LEP which enables one to nd not only the
lasing frequencies but the thresholds as well
Several factors bring fundamental in uence on thresholds
of lasing mo des The rst is the distance from the activecav
ity to ABR the second is the numb er of pairs of layers of
lower and higher refractive indices in ABR and the third is
the thicknesses of pairs in re ector We discuss a p ossible op
Figure Micro disk inside ABR in D
timal con guration of the microlaser to achievethelowering
of the thresholds
References
A Jebali et al J Appl Phys
E I Smotrova et al IEEE J Selected Topics in Quantum Electronics
The scattering of the lo calized surface wavebyanelastic
plate oating on shallowwater of variable depth
I V Sturova
Lavrentyev Institute of Hydro dynamics SB RAS Novosibirsk Russia
The wave forcing of a oating thin plate can b e used to mo del a wide range of physical systems
for example very large oating structures sea ice o es and breakwaters In mathematical mo delling
such structures are often treated as thin elastic plates At present there have b een considerable
numb ers of studies on the hydro elastic b ehavior of the oating elastic plate Main interests of the
previous researches have b een con ned in the hydro elastic resp onse of a thin plate on at sea b ottom
In reality the seab ed is not uniform in depth To our knowledge the consideration of a variable
water depth was made only for di raction problem by solving the linear hydro elastic problem for
DAYS on DIFFRACTION
a single frequency Unsteady b ehavior of b eam plate oating on shallowwater of variable depth is
considered in this pap er This problem has b een chosen b ecause the sea b ottom e ects b ecome more
signi cantinshallowwater than that in deep water Prop osed metho d maybeusedforany unsteady
D problem of linear shallow water theory but here the motion of the elastic b eam plate is considered
for a travelling lo calized surface wave
Figure
An elastic b eam of width L oats freely on the surface of an inviscid incompressible uid layer
The surface of the uid that is not covered with the plate is free The uid region S is divided into three
parts S jxj L S x L S x L Without the plate the uid depth is equal to H x in
S and the uid depths in the left and right hand domains of constantdepth S and S are equal to H
and H resp ectively With the plate the uid depth in S is equal to h x H x d where d is the
draft of the plate It is assumed that the maximal depth of the uid is small in comparison with the
lengths of surface waves and exural gravity ones and the shallowwater approximation is used The
velo city p otentials describing the uid motion in the regions S are denoted by x t j
j j
A de ection of an elastic plate w x t is describ ed by the equation
D w x m w t g w t x S
where D is the exural rigidity of the plate m is the mass p er unit length of the plate is the uid
density and g is the gravity acceleration The draft of the plate is equal d m
According to linear shallow water theory the following relation is valid
w t h x x x x S
In the free water regions the velo city p otentials x t and x t satisfy the equations
t gH x x S t gH x x S
The displacements of the free surface x t and x t are determined in the regions S and S
from the relations
g t x S j
j j j
If jxj L the matching conditions continuity of pressure and mass should b e satis ed At the edges
of the b eam the free edge conditions are satis ed which imply that the b ending moment and shear
force are equal to zero
It is assumed that at the initial time the plate and uid in the regions S and S are at rest In
p
region S the lo calized displacement of the free surface x gH t travels to the right At t
the displacementreaches the left edge of the plate and the plate b egins to undergo a complex b ending
motion in resp onse to the incoming wave Consequently the initial conditions have the form
w t t x t g x t
The transient elastic deformations of a b eam is computed by means of time domain di erential
equation metho d with elastic de ections expressed by a sup erp osition of mo dal functions with time
dep endent unknown amplitudes The plate de ections and wave motions of the uid for various
b ottom top ographies have b een calculated
The researchwas supp orted by the Foundation Leading Scienti c Scho ols Grant No NSh
and the Integration Program No SB RAS
DAYS on DIFFRACTION
The waves propagation in the comp osed ro d
T Tovstik
Institute for Problems in Mechanical Engineering RAS St Petersburg Russia
e mail tptovstik mail ru
In this pap er weinvestigate the waves propagation in the comp osed materials Let us consider the
mo del problem deal with the one dimensional longitudinal waves in the comp osed ro d We consider
the comp osed ro d as two connected ro ds and intro duce the elastic and viscous linear interaction
between p oints of ro ds At the initial moment the left end of ro ds is under action the force impulse
or the displacement impulse of this end is given The right side of ro ds is free we consider also the semi in nite ro ds
f(t) f(t) x
-L 0L
Figure The mo del of the b o dy
Longitudinal displacements are describ ed by the system of equations
u u u u
S R u u B E S
x t t t
u u u u
E S S R u u B
x t t t
where u x t are the longitudinal displacements of the p oints of ro ds i E S are the
i i i i
parameters of ro ds Young mo dules transverse areas and mass densities R B are the co e cients
of the elastic and of the viscous interaction of ro ds This problem is a mo del of the longitudinal wave
propagation in a ro d of sandwichtyp e consisting of the alternating hard and soft lowers
In contrary to the single ro d here the wavevelo city dep ends on the wave length and therefore the
disp ersion equation is investigated By using the Laplace transform in time the stress strain state near
the frontofwaveisinvestigated For comparison the solutions are constructed by using the Fourier
transformation and also by the nite elements metho d The in uence of the problem parameters in
partial the viscous forces on the motion is discussed
Casimir e ect and exactly integrable Hamiltonians
Sergey Yalunin
Theoretical Physics Department Kant State University
A Nevsky st Kaliningrad Russia
e mail yalunin bk ru
Application of the dimensional regularization to a certain class of multi dimensional manifolds
with b oundaries is considered Simple formula for regularized vacuum energy is obtained Examples
for some integrable Hamiltonians are considered including Hamiltonians constructed via Darb oux
transformation The Casimir force results from the alteration by the b oundaries of the vacuum
energy Applications to a certain class of integrable mo del of eld theory is discussed This approach
can b e useful in the numeric calculations
DAYS on DIFFRACTION
Transformation of electromagnetic elds using metamaterials
S A Tretyakov I S Nefedov
Radio Lab oratory SMARAD Helsinki UniversityofTechnology TKK
P O Box FI TKK Finland
e mail sergei tretyakov tkk fi
In this review pap er we discuss various approaches to electromagnetic control and transformation
of electromagnetic elds with engineered metamaterials including a novel concept of eld transforming
metamaterials These arti cial media change the elds in the volume o ccupied by the medium in a
prescrib ed fashion Weshow what electromagnetic prop erties of transforming medium are required if
the transformed elds are arbitrary linear functions of original elds
It has b een recently realized that metamaterials arti cial electromagnetic materials with engi
neered prop erties can b e designed to control electromagnetic elds in rather general ways More
precisely it has b een shown that transformation of spatial co ordinates accompanied by the corre
sp onding transformation of electromagnetic elds can b e mimicked byintro ducing electromagnetic
materials with sp eci c electromagnetic prop erties into the domain where the co ordinates have b een
transformed As an example of such co ordinate transforming device an invisibility cloak has
b een suggested Not only spatial co ordinates can b e transformed and the result simulated bysome
metamaterial but also the time co ordinate can b e transformed with a similar interpretation It has
b een known for a long time that the material relations of an isotropic magneto dielectric transform
into certain bi anisotropic relations if the medium is moving with a constantvelo city e g It
was suggested that also this e ect of time transformation can p ossibly b e mimicked by a metamaterial
In this presentation we discuss opp ortunities for eld control from a more general p osition in
tro ducing the concept of eld transforming metamaterials In this concept we start directly from a
desired transformation of electromagnetic elds and nd the material prop erties that are necessary to
realize this transformation Wewillshow that an arti cial medium with the bianisotropic constitutive
relations of the form
j j
D E rG H B H rF H
G F
transforms electric and magnetic elds E r and H r into
E r t F r t E r t H r t G r t E r t
Here and are the relative p ermittivity and p ermeability of the host medium resp ectively
Materials with the constitutive relations b elong to the class of moving omega media Of
course this material is physically at rest but the elds see it as a moving material This is also a ma
terial which is required to physically realize the eld transformation corresp onding to transformations
of time Possible realizations of such media will b e discussed in the presentation
References
J B Pendry D Shurig D R Smith Science vol pp
U Leonhardt Science
nd
L D Landau E M Lifshits Electrodynamics of continuous media ed Pergamon Oxford
England
S A Tretyakov Microwave and Guided Wave Letters vol no pp
A N Serdyukov I V Semchenko S A Tretyakov A Sihvola Electromagnetics of bi anisotropic
materials Theory and applications Amsterdam Gordon and Breach Science Publishers
U Leonhardt T Philbin New J of Physics vol p
DAYS on DIFFRACTION
S A Tretyakov A H Sihvola A A So chava C R Simovski J of Electromagnetic Waves and
Applications vol no pp
Pulsed radiation pro duced by a travelling exp onentially
decaying bip olar current pulse with high frequency lling
Andrei B Utkin
INOV Inesc Inova c ao Rua Alves Redol Lisb on Portugal
e mail andrei utkin inov pt
Investigation of electromagnetic radiation accompanying propagation of pulses with high frequency
lling moving along a line segment is stimulated by the problem of launching directional scalar and
electromagnetic waves as well as by results of exp erimental investigation of sup erradiation waveforms
see e g Egorov V S et al Sov J Quantum Electronics p In this work the sp eci c
envelop e function an exp onentially decaying bip olar pulse a sum of two exp onentially decaying pulses
of opp osite p olarity is considered as b eing one of the simplest but feasible approximation of many
real source currents
Generation of the electromagnetic waves is describ ed by solving the inhomogeneous Maxwell equa
tions directly in the space time domain rst the electric and magnetic eld vectors are expressed in
terms of one scalar function and then the resulting hyp erb olic typ e PDE is solved using incomplete
separation of variables and the Riemann formula
Being an extension of a well known description of the travelling waveantenna this mo del can
explain p eculiarities of some non stationary electromagnetic waves pro duced by traditional arti cial
and natural line radiators in particular frequency transform and b eatings The choice of the sp eci c
envelop e function allows deriving observable and easy to analyse expressions related to the space
time structure of the emanated waves whichmay b e used for characterizing natural phenomena and
optimizing parameters of man made radiators
One dimensional nonlinear FPK equation with the p otential
of a sp ecial form in semiclassical approximation
Roman O Rezaev Andrey Yu Trifonov
Department of Mathematical Physics Tomsk Polytechnic University
Tomsk Russia
e mail rezaev tpu ru trifonov phtd tpu edu ru
Alexander V Shap ovalov
Department of Theoretical Physics Tomsk State University Tomsk Russia
e mail shpv phys tsu ru
Solution of the Cauchy problem is constructed for the Fokker Planck equation
P x t U x t P x t
t x x
x x
U x t hx t ix
Here prime denotes derivative with resp ect to x in semiclassical approximation accurate to O
DAYS on DIFFRACTION
is a constant is a small parameter The moment hx t i is de ned by expression
Z
dxxP x t hx t i
The solution of eq is constructed in the class of semiclassically concentrated functions in
the framework of the Maslov complex germ metho d The nonlinear evolution op erator is found in
explicit form that admits to construct a parametric family of symmetry op erators
The results obtained are discussed in the context of solutions found in
Acknowledgements The work was supp orted in part by President of the Russian Federation Grant
No SS DFG Rezaev O was supp orted in part by the nonpro t DynastyFoundation
References
Belov V V TrifonovA Yu and Shap ovalov A V The Tra jectory Coherent Approximation
and the System of Moments for the Hartree Typ e Equation Int J Math and Math Sci USA
Drozdov A M and M Morillo Solution of nonlinear Fokker Planck equation Phys Rev E
Symmetry op erators for the multidimensional FPK
equation with quadratic nonlo cal nonlinearity
Andrey Yu Trifonov Roman O Rezaev
Department of Mathematical Physics Tomsk Polytechnic University Tomsk Russia
e mail rezaev tpu ru trifonov phtd tpu edu ru
Alexander V Shap ovalov
Department of Theoretical Physics Tomsk State University Tomsk Russia
e mail shpv phys tsu ru
A metho d of construction of semiclassical asymptotics for for the Fokker Planck Kolmogorov
equation FPKE of sp ecial form with a quadratic nonlo cal nonlinearity based on the Maslov s complex
germ theory is applied to the multi dimensional FPKE with the quadratic p otential
Z
o n
W x y t u y t d y u x t V x t
t
x
n
R
where
V x t K x W x y t K x K y
n n
Here t R x x x R y y y R are indep endentvariables x x
n n n
means a transp ose to a vector or a matrix d x dx dx the dep endentvariable u x t isa
n
real smo oth function decreasing as k xk K K K are arbitrary constant matrices of order
n n and are real parameters t x is a gradient op erator with resp ect to x
t
x
n
P
x is a Laplace op erator
x x
i
i
The Cauchy problem for the Fokker Plank Kolmogorov equation is reduced to a similar prob
lem for the corresp ondent linear equation The relation b etween symmetry op erators of the linear
and nonlinear Fokker Plank Kolmogorov equations is considered Illustrative examples of the one
dimensional symmetry op erators are presented
Acknowledgements The work was supp orted in part by President of the Russian Federation Grant
No SS DFG Rezaev O was supp orted in part by the nonpro t DynastyFoundation
DAYS on DIFFRACTION
Asymptotics for nonlinear wave equations with external source
S A Vakulenko
Institute of Problems of Mechanical Engineering RAS SaintPetersburg Russia
n
This pap er considers the Cauchy problem in R for some nonlinear wave equations with external
sources for example for sin Gordon equation
u u a sin u f x t
tt
Such systems are imp ortant for many applications for example for physics chemistry phase transi
tions In these applications one often observes wave solutions
It is well known that for integrable systems n solutions for small can b e studied see for
example and references in it
Moreover in general case n such asymptotics can b e obtained under essential restrictions to
initial data when these data are close to a wave solution for example a soliton or a kink
We study formal asymptotics for large with a strong external source Under some conditions
on the initial data they should b e close to a wave solution of the usual linear wave equation such
asymp otics can b e obtained by standard linear metho ds However justi cation of these formal
solutions is a di cult problem
References
Babich V M Buldyrev V S Short Wavelength Di raction Theory Asymptotic Approach
Springer Berlin
Molotkov I A and Vakulenko S A Lo calized nonlinear waves in Russian St Petersburg
State University Petersburg
R F Bikbaev and S B Kuksin A p erio dic b oundary value problem for the Sine Gordon equa
tion St Petersburg Math J
Fractional b oundary conditions in di raction problems on plane screens
M V Ivakhnychenko E I Veliev
IRE NAS of Ukraine Kharkov Ukraine
T M Ahmedov
IMM NAS of Azerbaijan Baku Azerbaijan
Last fteen years N Engheta develop ed the metho d of fractional op erators in solving wide class of
problems in electromagnetic theory N Engheta Fractional Paradigm in Electromagnetic Theory
achapter in IEEE Press chapter pp Fractional derivative and integral and also
fractional curl op erator are some of the examples of fractional op erators In this pap er fractional
derivatives de ned by Riemann Liouvill e integral are considered N Engheta intro duced the concept
of Intermediate state in electro dynamics fractional paradigm It means the following if the
function and its rst derivative describ es two canonical cases of some electromagnetic eld then
f y corresp onds to intermediate situation fractional derivative of this function D
y
of the eld b etween two canonical original cases Engheta investigated suchintermediate situations
as new waves intermediate b etween plane and cylindrical waves new sources of eld intermediate
between sheet and line sources etc In this pap er developing the fractional paradigm new fractional
b oundary conditions BC are intro duced which act as intermediate b etween known BC of Dirichlet
and Neumann In some sense fractional BC can b e treated as imp edance BC Utilizing fractional BC
wesolve di raction problem on in nite thin strip with imp edance of sp ecial kind It is shown that
for the solution can b e expressed in analytical form for anyvalues of frequency For the strip
with sp ecial imp edance suchphysical characteristics as radiation pattern cross sections are studied
DAYS on DIFFRACTION
Metamaterials based on structures with
emb edded D resonant inclusions
Irina Vendik
St Petersburg Electrotechnical University
Prof Pop ov Str St Petersburg Russia
e mail IBVendik eltech ru
A classical structure exhibiting prop erties of medium with negative p ermittivity and p ermeability
within a limited frequency range was suggested in and then exp erimentally examined in where
the negative refraction phenomenon was observed The structure suggested in is a combination
of two lattices a lattice of the split ring resonators SRR and a lattice of in nitely long parallel
wires This structure is anisotropic and reveals the negative p ermittivity and p ermeability ifthe
propagation direction of the electromagnetic wave is orthogonal to the axes of wires Though the
theory and exp erimental investigation of such structures are still under interest of many scientists
isotropic DNG structures are very attractive for practical use In this case structures with emb edded
D resonant inclusions are very promising
In this presentation di erentways to create the D isotropic and DDNG medium based on a
regular lattice of resonant inclusions are reviewed
Single negative or double negative metamaterial formed by a rectangular lattice of isotropic cubic
unit cells of particles Each unit cell is formed by putting p erfectly conducting resonant particles
on the faces of a cubic unit cell Di erent particles are used simple and complicated SRRs
W particles or combination of SRR and wire medium
DNG medium formed by a regular lattice of spherical resonant inclusions providing excitation
of electric and magnetic dip oles at the same frequency In this case the magneto dielectric
particles or dielectric particles in double spherical structures are used
DNG medium formed byanarray of dielectric ro ds with di erent radii which is D analogue of
arti cial media based on spherical particles
Clustered dielectric particle metamaterials which are constituted by the p erio dic rep etition of
a molecule li ke cluster of dielectric atom like particles
References
D R Smith W J Padilla D C Vier S C Nemat Nasser and S Schultz A Comp osite Media
with Simultaneously NegativePermeability and Permittivity Appl Phys Lett No pp
May
J B Pendry Negative refraction makes a p erfect lens Phys Rev Letters pp
C R Simovski and S He Frequency range and explicit expressions for negative p ermittivity
and p ermeability for an isotropic medium formed by a lattice of p erfectly conducted particles
Physics Letters A pp
Th Koschny L Zhang C M Soukoulis Isotropic three dimensional left handed metamaterials
Phys Rev B R
L Jelinek J D Baena R Marques and J Zehentner Howtomake a fully isotropic magnetic
metamaterial th International Student Seminar on Microwave Applications of Novel Physical
Phenomena St Petersburg Russia Octob er Seminar Pro c pp
C Holloway E Kuester A double negative comp osite medium comp osed of magneto dielectric
spherical particles emb edded in a matrix IEEE Trans on AP Octob er
O G Vendik M S Gashinova Arti cial Double Negative DNG Media Comp osed byTwo
Di erent Dielectric Sphere Lattices Emb edded in a Dielectric Matrix Pro c EuMC Paris
France Octob er p
DAYS on DIFFRACTION
I Vendik O Vendik M Gashinova Arti cial dielectric medium p ossessing simultaneously neg
ative p ermittivity and magnetic p ermeability Tech Phys Lett pp May
Irina Vendik Orest Vendik Odit Mikhail Marina Gashinova and Igor Kolmakov Isotropic arti
cial media with simultaneously negative p ermittivity and p ermeability Microwave and Optical
Techn Lett accepted for publication in
I A Kolmakov L Jylha S A Tretyakov S Maslovski Lattice of Dielectric particles with
Double Negative Resp onse th GA URSI New Delhi India pap er BCD p df Octob er
A Rennings C Caloz M Coulomb e Unusual wave phenomena in a guiding radiating clustered
dielectric particle metamaterial CDP MTM Pro c EuMC Manchester pp
Resolvent op erator of maxwell equations for dimentional eld vector
A V Vozianova A G Nerukh
Kharkov National University of RadioElectronics
Lenin Ave Kharkov Ukraine
e mail annavenus mail ru
Current developmentoftechnologies requires investigation of non stationary electromagnetic elds
in complex organic mediums e g chiral mediums moving mediums arti cial mediums metamateri
als etc Mo deling of corresp onding phenomena requires solution of complex initial b ound ary electro
magnetic problems with explicit accounting for time dep endence of elds that is usually nonharmonic
Moreover in such tasks it is necessary to account for the essentially vector structure of electromagnetic
eld Furthermore it is often imp ossible to separate material equations into indep endent electric and
magnetic comp onents In such cases it is necessary to consider essentially dimensional structure of
electromagnetic eld
Along with di erential approach to description of electromagnetic eld there exists a new p owerful
and rapidly growing to ol for temp oral mo deling of electromagnetic pro cesses in complex structures
integral equations For their solution there are used analytical as well as numerical metho ds
Regardless of solution metho d formulating of integral equations in time domain require knowledge
of explicit expression of space time Green function Such function is used in free space problems as well
as in problems for mediums three dimensional vector problems with complex mediums anisotropic
mediums problems of multilayer dielectric structures problems of structures with forbidden regions
Temp oral Green function for Maxwell equations in unrestricted space allows solving many electro dy
namic problems such as problems for non stationary sources problems of scattering in temp oral Born
approximation problems of electromagnetic oscillations in photonic crystals etc In work there was
considered full Green function of free space for Maxwell equations in time domain The Green function
was obtained using metho d of propagation op erators propagators that are common to quantum me
chanics but quite seldom used in classical electro dynamics There exist a limited number of works that
use such approach e g temp oral integral op erators of propagation were obtained in work for investi
gation of pulses propagation in non stationary uniform and isotropic dielectric and magnetic medium
In this work there is given general approachtoinvestigation of initial b ound ary problem for
Maxwell equations in uniform medium and in plasma in time domain by their reduction to inte
gral Volterra equation of the second kind for dimensinal eld vector This is attained using derived
Green function in dimensional formulation in time domain The integral equation is equivalentto
Maxwell equations and contains initial and b oundary conditions It gives uni ed de nition of eld in
the whole space including non stationary region of irregularityandambient space There was obtained
resolving op erator for this equation whichwas used for investigation of plane wave transformation and
emission of lump ed source in the medium with sharp time variation of parameters Arbitrary non
stationary of the medium can b e approximated by sequence of such sharp variations of its parameters
and for eachvariation there is calculated exact analytical solution obtained using resolvent metho d
DAYS on DIFFRACTION
Enhancement of magneto optic e ects with resonant structures
A P Vinogradov A M Merzlikin A V Dorofeenko
Russia
Magneto optical MO Faraday e ect which consists in rotation of p olarization plane of transmitted
wave in presence of magnetic eld can b e substantially ampli ed by means of di erent resonant
structures The examples are Fabry Perot resonance in homogeneous layer defect mo de of photonic
crystal and Tamm surface state on a b oundary of two photonic crystals with intersecting band gap
In all these cases a resonance is manifested as a p eak in the transmission co e cient In the presence
of magnetization the p eak splits into two ones corresp onding to b oth circular p olarizations Fig
thick lines The incidentwave b eing linearly p olarized the transmitted one has the same p olarization
at frequency of crossing of the thick curves in Fig The Faraday rotation angle is determined as
shown in Fig
We carry out an analysis of ampli cation of the Faraday rotation on the base of partial waves
Fig The large phase di erence which determines is a result of di erent sign of phases of
partial waves This mechanism has nothing in common with multiple passing the MO layers
Fig Transmission co e cient thick lines and Fig Two circularly p olarized waves presented
corresp onding phases thin lines for two circular as a sum of partial waves
p olarizations at the presence of magnetization
Distinct feature of magneto photonic
crystals on formation of the Yeh band gap
A M Merzlikin A V Dorofeenko A P Vinogradov
Russia
Systems of anisotropic layers play a signi cant role in a numb er of applications narrow band Solc
lter multistage electro optic mo dulators and p olarizer We consider the e ect of natural and
arti cial anisotropy and birefringence on magneto optical MO prop erties of photonic crystals PC
It is well known that the natural anisotropy usually masks the magneto optical e ects in homoge
neous media Indeed the di erence in the values of wavenumb ers of ordinary and extraordinary waves
as a rule exceeds the di erence app earing due to the external magnetic eld As a consequence con
trary to isotropic medium in naturally anisotropic media magneto optics e ects is quadratic in small
magneto optics parameter
In PC b eyond the natural anisotropy there is anisotropy due to anisotropy of primitive cell The
later e ect is more pronounced in D and D PC Thus in the general case to observe linear in the
magnetization MO e ects one needs working with directions where the Blo chwavenumb er of ordinary
DAYS on DIFFRACTION
and extraordinary waves are identical Along such directions the PC is e ciently isotropic
ABlochwavevector is determined up to the vector of recipro cal lattice Thus there is a series of
wavenumb ers characterizing the same Blo chwave Eachwavenumb er corresp onds to one of the plane
wave the Blo chwave is built of These plane waves di er in wavenumb ers as well as in amplitude
p olarization and phase of their phasors Considering MO e ects wehave to distinguish the case of
coincidence of wavenumb ers of ordinary and extraordinary Blo chwaves dep ending on the rank of the
corresp onding plane wave in formation of the Blo chwave If we deal with plane waves harmonics
each of whichgives the maximal contribution having maximal amplitudes to the corresp onding Blo ch
wave it means that the Blo chwaves b elong to the bands with the identical order numb ers intraband
In opp osite case the Blo chwaves b elong to the bands with di erent order numb ers interband case
It shown that the Faraday e ect can b e observed in intraband case only If the ordinary and
extraordinary Blo chwaves b elong to di erent bands then in the very direction of the optical axis the
Yeh s band gap forms A signi cant di erence in amplitudes as well as b eamsteering e ect
make it di cult to observe MO e ects On the other hand the formation of the Yeh s band gap in D
MPC under in uence of external magnetic eld allows us to create new typ e of magneto controlled
devices
References
A M Merzlikin A P Vinogradov M Inoue A B Khanikaev A B Granovsky JMMM
P Yeh J Opt So c Am
A M Merzlikin A P Vinogradov A V Dorofeenko M Inoue M Levy and A B Granovsky
to b e published in PhisikaB
A comparative analysis of mixing formula for SNG and DNG media
A P Vinogradov A V Dorofeenko A M Merzlikin
Russia
Recently a great deal of attention has b een paid to metamaterials which are as a rule comp osite
materials consisting of ingredients with negative p ermittivity and p ermeability double negative media
DNG or wit negative p ermittivity or p ermeability single negative media SNG In this connection the
problem app ears how to treat mixing formula The key moment is existence of the Bergman Milton
p oles of the sp ectral function on real negativeaxis
It is shown that in static case when even DNG media can b e treated as a SNG media we should
take care when extracting of the ro ot of function of two complex variables Firstly we need makea
prop er cat of the complex plane to obtain a unique analytical branch Secondly we need factorization
to reduce the function of two complex variables to two functions of single complex variable
Dealing with calculation of wavenumb er of a plane wavetraveling through a DNG media we
cannot choice a unique analytical branch of the ro ot to treat all p ossible events The reason is that
in electro dynamics we deal with two quantities namely with wavenumb er and imp edance As is
shown by our analysis calculation of the physically true values of quantities we need employ di erent
branches of the ro ot Thus wehavetotake addition assumption that for example a plane wave
transferring energy from left to right has a p ositivevalue of real part of imp edance After doing that
wecanchoice a prop er branch of the ro ot and determine the sign of the wavenumb er
DAYS on DIFFRACTION
Analisys of applicability ranges of exact light
scattering metho ds using spherical basis
Alexander A Vinokurov
St Petersburg University of Aero cosmic Instrumentation
Bol Morskaya St Petersburg Russia
e mail vinokurov dataart com
The light scattering metho ds expanding elds in terms of wave functions are widely applied due
to their high e ciency We analyse some of these metho ds namely the extended b oundary condition
EBCM separation of variables SVM and p ointmatching PMM ones considering their
theoretical and practical applicabili ty Though the metho ds lo ok alike b ecause of their use of the
same expansions of the elds it is found that these approaches di er in imp ortant asp ects
These approaches seem to b e very similar as they search for the problem solution in the form of the
same eld expansions with the expansion co e cients b eing derived from solution of systems of linear
algebraic equations The main di erence of the metho ds is their use of di erent problem formulation
which leads as a result to di erent systems In the EBCM the elds and Green function expansions
are substituted into the b oundary condition presented in a surface integral form in the SVM the eld
expansions are substituted into the b oundary condition written in a usual di erential form in the
PMM any of the ab ove forms can b e used but the system is derived from minimization of a residual of
the b oundary condition considered in selected p oints at the scatterer surface see for more details
From the theoretical p oint of view applicability of the approaches is related with convergence of
the eld expansions and solvability of the systems used to derive the expansion co e cients The
convergence of the expansions of the scattered and internal elds dep ends on the eld singularities
It is shown that in the far eld zone EBCM can b e formulated in terms of eld patterns
In this case its applicabili ty is restricted only by the solvability condition whichisweaker than the
convergence one For the SVM a similar analysis has not b een p erformed though at least the EBCM
system can b e obtained within the SVM The system arising in the PMM is always solvable as it
is p ositively determined see e g
For numerical analisys wehavedevelop ed a homogeneous set of co des based on the metho ds under
consideration Their accuracy in the far eld zone was studied by using the relative di erence of the
extinction and scattering cross sections jC C j C C calculated for nonabsorbing
ext sca ext sca
particles
For spheroids where all the three metho ds are mathematically correct the EBCM is generally
preferable to other metho ds The situation is a bit di erent for axisymmetric Chebyshev particles
Our calculations well con rm that EBCM provides p o or accuracy when its resolvability condition isn t
met However SVM and PMM still coverge
Thus despite the large similarity of the EBCM and SVM see the discussions in their
theoretical applicabili ty conditions di er in principle as the solvability of the SVM is not determined
by the EBCM solvability condition Our preliminary results of similar comparison of the metho ds
when the spheroidal functions were used for the eld expansions led to the same conclusions
Acknowledgements The work was supp orted by a grant of the RFFI V I acknowledges a supp ort
by the grants RNP and NSh
References
M I Mishchenko J Hovenier and L D Travis eds Light Scattering by Nonspherical Particles
Academic Press San Francisco
V G Farafonov V B Il in Single light scattering computational metho ds In A
Kokhanovsky Ed Light Scattering Reviews Springer Praxis Berlin pp
V G Farafonov A A Vinokurov and V B Il in Comparison of light scattering metho ds using
the spherical basis Opt Sp ectrosc N
DAYS on DIFFRACTION
Numerical approach for analysis of surface
wave eld in the vicinity of caustics
M A Geyer T B Yanovskaya
Sankt Petersburg State University Russia
Due to lateral heterogeneity of the Earth surface wave paths deviate from great circle arcs that
results in app earance of caustics on the hemisphere opp osite to the epicenter The caustics consist of
two branches forming the cusp in the p ointofcontact Between the branches three waves interfere
coming to and from one of the branches and coming to another branch Asymptotic representation
of the wave eld in the vicinity of such a caustic at high frequencies is expressed in terms of the
Pearcey integral which is a function of twovariables related to lo cation of the p oint where the eld
is considered resp ectively to the cusp and to the caustic Analysis of harmonic surface wave eld
in a simplest plane mo del on the basis of the Pearcey integral Yanovskaya has shown that
wave amplitude oscillates within the cusp in two direction along the caustic axis and in the direction
p erp endicular to the caustic The wave phase varies along the caustic axis unevenly that results
in sharp variations of the phase velo city However these results are true only for high frequencies
where the asymptotic approach can b e applied In the real Earth p erio ds of surface waves maybe
rather big up to s and corresp ondingly the wavelengths are big as well In such a case the
asymptotic approach for analysis of surface wave eld cannot b e applied Besides this metho d can b e
realized for plane surface and for an individual caustic while the surface of the real Earth is spherical
and we frequently observe sup erp osition of such caustics Another di culty is related to necessityto
calculate travel times of the sup erimp osed waves in the same p ointwithvery high precision that is
necessary to calculate parameters of the Pearcey integral At last application of this metho d needs
to extrap olate the Pearcey integral parameters outside the caustic So this approachisvalid only for
drawing qualitative conclusions ab out b ehavior of the wave elds near the caustic
In the present study we prop ose a numerical metho d for calculation of surface wave eld in the
region containing caustics The wave eld is considered as a sup erp osition of the waves arriving from
the circle dividing two hemispheres centered in the epicenter and in the antip o de So for calculation
of the wave eld on the hemisphere centered in the antipode we can use the representation theorem
The factor in the expression for surface wave eld describing variation of the eld due to con guration
of the rays on the surface so called geometrical factor satis es the wave equation with phase velo city
of the waves U c r We consider only the geometrical factor for which the representation
theorem b ecomes
Z
U u
G
U r u r S U S dS
G
n n
S
where u r r is the Green function The geometrical factor U on the contour S a circle b ounding
G
the hemisphere centered in the epicenter is calculated on the basis of the ray theory b ecause the rays
on this hemisphere are not intersected The Green function on the opp osite hemisphere is assumed the
same as on a laterally homogeneous spherical surface This limitation is not to o substantial b ecause
con guration of the rays on this hemisphere is mostly determined by the eld on the circle The integral
in the representation theorem is calculated numerically The result turns to b e dep endentonachoice of
the limits of integration A metho d is prop osed for estimating the integral indep endent of these limits
The results for some numerical mo dels as well as for the mo del of the real Earth are presented It
is shown that in case of two sup erimp osed caustics strong increase or decrease of amplitude may o ccur
in the intersection of the caustic axes Strong phase anomalies forestalling of the phase are observed
outside the cusps that results from arrival of the waves from the parts of the contour neighbortothe
caustic axis The main conclusion from numerical mo deling of surface waves on the real Earth is that
in a xed p ointwemay observe di erent amplitude anomalies for di erent p erio ds This shows that
sp ectrum of the wavemay b e strongly distorted
The researchwas supp orted by RFBR grant
DAYS on DIFFRACTION
Electromagnetic surface waves guided by
the metal comp osite medium b oundary
N F Yashina T M Zab oronkova
Technical UniversityofNizhnyNovgoro d Minin St NizhnyNovgoro d Russia
Over the past decade the impressing success in the development the pro ducing and the testing
of a comp osite materials has b een achieved In microwave range these materials demonstrate some
qualities whichareunusual for natural one As it is known the comp osite material consists of a metal
resonant elements placed in dielectric which sizes are smaller then wave lengths It is p ossible to create
isotropic as well as anisotropic magnetic media These materials are used for nonre ecting coverage
frequency multipliers transformers and so on Electro dynamics of new comp osite materials for the
Microwave technical equipmentaswell as propagation re ection and refraction of electromagnetic
waves have b een discussed in the literature
It is the purp ose of the presentwork to consider the p ossibility of existence of electromagnetic
surface waves on the metal comp osite medium b oundary Our investigations have shown that
it dep ends on the sign of comp onents of dielectric p ermittivity and magnetic p ermeability tensors
Anisotropic of any tensor leads to app earance of surface waves if one of diagonal comp onents is
negative The disp ersion prop erties of the surface waves are investigated for some cases of interest
bianisotropic chiral medium and so on We also considered the particular cases when parametric
instability of surface waves guided by the planar metal gyrotropic anisotropic medium b oundary can
arise due to the presence of intense external electric eld into the medium
Acknow ledgments This work was supp orted by the Russian Foundation for Basic Research pro ject
No ts
References
A A Zharov I G Kondratiev and A I Smirnov Radiophysics and Quantum Electronics
p
S A Tret yakov Journal of Communication Technology and Electronics p
N F Yashina and T M Zab oronkova Proc International Seminar Day on Di raction
St Petersburg State University St Petersburg p
The problem of di raction of a plane electromagneticwave
on a kerr typ e nonlinear dielectric layered structure
Vasyl V Yatsyk
Usikov Institute for Radiophysics and Electronics NASU
Ac Proskura Str Kharkov Ukraine
e mail yatsyk vk kharkov ua
The problem of di raction of a plane electromagnetic wave on transverse non homogeneous isotropic
not magnetic linearly p olarized weakly nonlinear dielectric layer and layered dielectric structures
Kerr like nonlinearity in resonant area of frequencies are considered The analysis of a problem
of di raction on weakly nonlinear layer layered dielectric structures without taking into account
multiple the frequencies generated in considered op en weakly nonlinear structure is lead
Numerical analysis of the di raction by the weakly nonlinear layer layered dielectric structures
is p erformed in the resonant frequency range when the parameter of cubic susceptibili ty takes p os
itive and negativevalues It is shown that as the excitation eld intensity increases the di raction
DAYS on DIFFRACTION
characteristics acquire essentially di erent prop erties for p ositive and negativevalues of susceptibili ty
For a layered structure consisting of layers with p ositive and negative susceptibility of the enveloping
medium the e ects inherenttoenvironments with p ositive and negativevalue of susceptibilityare
observed in certain areas of variation of the excitation eld intensity
The prop osed metho ds and results of computations can b e further applied to the analysis of various
physical phenomena including self in uence and interaction of waves determination of eigen elds
natural resonance frequencies of nonlinear ob jects and disp ersion amplitude phase characteristics
of the di raction elds description of evolution pro cesses in the vicinities of critical p oints and to the
design and mo deling of novel scattering transmitting and memory devices
References
Yatsyk V V Ab out a problem of di raction on transverse non homogeneous dielectric layer of
Kerr like nonlinearity Elektromagnitnie volny i elektronnie sistemy pp
Yatsyk V V Di raction byalayer and a layered structure with p ositive and negative suscepti
bilities of Kerr nonlinear media Uspehi sovremennoi radioelectroniki no pp
Asymptotic theory of the one dimensional linear
scattering of the solitary waves on a b each
S Yu Dobrokhotov
Institute for Problems in Mechanics RAS Moscow Russia
e mail dobr ipmnet ru
P N Zhevandrov
Escuela de Ciencias Fisico Matematicas Un Michoacana Morelia Mexico
e mail pzhevand zeus umich mx
B Tirozzi
DepartmentofPhysics University La Sapienza Rome Italy
e mail brunello tirozzi roma infn it
We discuss the formulas which describ es the amplitude of the solitary waves approaching a b eachin
the one dimensional case and in linear approximation These formulas are derived using the asymptotic
expansion in some small parameter of physical relevance of the solutions of the one dimensional wave
equation with non constantvelo city The case of a b each with a linear b ottom of constant slop e is
explicitly treated The scattering of the solitary waves on the b each is discussed also in connection
with the problem of the run up in the tsunami events The metamorphosis of the scattered solitary
wave is shown The event of the Algerian tsunami of will b e used as an example
This work was partially supp orted by RFBR grant N and AgreementBetween Uni
versity La Sapienza Rome and Institute for Problems in Mechanics RAS Moscow
References
T Vukasinac P Zhevandrov Geometric asymptotics for a degenerate hyp erb olic equation Rus
sian Journal of Mathematical Physics Vol N pp
G F Carrier and H P Greenspan Water Waves of nite amplitude on a sloping b each J Fluid
Mechanics V Pag
S Yu Dobrokhotov S O Synitsyn B Tirozzi Asymptotics of lo calized solutions of the one
dimensional wave equation with variable velo city I The Cauchy Problem Russian Journal of
Mathematical Physics Vol N p
DAYS on DIFFRACTION
Re ection refraction and transformation into
photons of surface plasmons on metal wedge
Vera B Zon
Voronezh State University University sq Voronezh Russia
Maluzhinets theory develop ed in the th for plasmons on wedge surface with a small imp edance
is used for the calculation of surface plasmons re ection refraction and transformation into photons
The metho dically simpli ed summary of Maluzhinets theory whichisvalid in the rst order on is
presented The approach is satisfactory for the well conductive metals up to optic frequencies Photons
radiation takes place primary in the direction of their motion along the wedge surface Angular
distribution of the radiated photons has Lorenzian form with width For the rectangular wedge
re ection refraction co e cients and di usion part of photons radiation are also
Di raction eld in wave zone
100 10
Figure Angle distribution of radiation for rectangular wedge with imp edance i and
i from left to right solid curve is directed radiation dashed curve is di usion radiation Di raction near eld
2
Figure Near eld for copp er at distances equal to from the wedge edge from left to
right Solid curve corresp onds to zeta i A dashed curve corresp onds to
i A
DAYS on DIFFRACTION
Whisp ering gallery eigenmo des of a surface
with anisotropic surface imp edance
A A Zvyagintsev A I Ivanov D V Katkov
Kharkiv National University Ukraine
e mail alexei i ivanov univer kharkov ua
An algorithm of nding high frequency whisp ering gallery WG eigenmo des of a curvilinear surface
with anisotropic surface imp edance is prop osed It is based on the metho d of reference problem
mo di ed to deal with three dimensional problems
Whisp ering gallery WG mo des are the waves usually propagating along a concave surface and
according to geometrical optics GO rep eatedly re ecting from it It has b een shown that GO
fails to describ e suchwaves if the numb er of re ections exceeds prede ned level Instead the metho d
of reference problem can b e used for this purp ose if the wavenumber k tends to in nity and can
b e considered as a large parameter We consider the problem of nding asymptotic solutions of the
Maxwell equations in complex form that are concentrated in a vicinity of a smo oth surface and b ehave
c
as WG mo des The b oundary condition on the surface is given by H W n E where H
s
c
E are tangential to the surface comp onents of the magnetic and electric eld strengths and W
s
is the surface imp edance tensor It is required that the p ermittivity and p ermeability of the outer
medium to b e slowvariable on a wavelength real functions and the e ective radius of curvature to b e
p ositive The problem is solved in co ordinate system n s t where n is a distance from surface to
the observation p oint s is a natural parameter of a geo desics passing through fo ot of a p erp endicular
dropp ed from the observation p oint to the surface and t is a natural parameter of a curveonthe
surface p erp endicular to the geo desics The Maxwell equations have b een reduced to system of the
second order for the electric eld strength It has b een shown that the system can b e reduced to
three Helmholtz equations by p erturbation metho d and the Helmholtz equations can b e solved by
the metho d of reference problem Each of the comp onents of the electric eld strength is obtained in
the form of asymptotic series
M M
X X
m m
E exp s t k v s t k
j m m
m m
where M is an integer numb er dep ending up on required accuracy v z is the Airy Fock function
nk and are known co e cients Some numerical results of applying the describ ed
m m
technique are presented and the prop erties of the solutions are discussed
References
V M Babich and V S Buldyrev Asymptotic metho ds in problems of short waves di raction
Nauka Moscow
A A Zvyagintsev A I Ivanov D V Katkov Whisp ering gallery eigenmo des of a curvilinear
imp edance surface analysis by the metho d of reference problem in Pro c th Internat Conf on Mathematical Metho ds in Electromagnetic Theory MMET Kharkiv pp
DAYS on DIFFRACTION
Vibration of cylindrical shell partially submerged into the layer
of liquid The liquid is inside and outside the cylinder
George V Filipp enko
Institute of Mechanical Engineering
Vasilievsky Ostrov Bolshoy Prosp ect St Petersburg Russia
e mail george gf spb edu
The problem of oscillations of elastic constructions partially submerged into liquid is one of the
actual problems of mo dern techniques It is imp ortant to estimate the amplitude of vibration and
acoustical elds in such systems as ships oil platforms di erent supp orts tub es protruding ab ovethe
surface of water and so on
The calculations of such b o dies vibrations are very complicated need great computational re
sources and havelow accuracy So it is useful to explore the pro cesses in these systems taking as an
example more simple mechanical systems The cylindrical shell is one of these often used mo dels
In this work the problem of cylindrical shell oscillations when the axis of the cylinder is orthogonal
to the liquid s surface is considered The liquid is inside and outside the cylinder The problem of oscil
lations of the cylindrical shell partially submerged into the layer of liquid is considered in the rigorous
mathematical statement The exact analytical solution of the problem is constructed The disp ersion
equation basing on the exact solution and the representation for the eigen functions are obtained
Pulse shap e of current moving along a straight line with sup erlight
velo city and time derivative of Whittaker p otential in far region
F F Valiev
V A Fock Institute for Physics St Petersburg State Univ St Petersburg Russia
The electromagnetic elds of sources moving with velo cities v equal or higher than velo cityof
lightinvacuum c have b een discussed in connection with an interest to the problem of lo calized
electromagnetic waves and Vavilov Cherenkov radiation particularly One way to do this is
to use p otential of Whittaker
In this work the calculation of derivative of Whittaker p otential U with resp ect to was p erformed
in the framework of linear current mo del Here ct is the time variable
In the general case the U is equal to the sum of several integrals dep ending on the interrelation
of the following parameters p osition of observation p oint z pulse duration velo city of the running
pulse and length of the line segment L
It was shown that correlation
p
Z
L
z z z f U
p
dz
z z
holds for the Cherenkovangle arccos c v where f z is a form of current pulse propagating
along a straight line
Results of computation haveshown similaritybetween di erent forms of running pulse Gaussian
distribution square pulse or distribution deduced from simulation by means of a computational exp eri
ment where the co de GAINT was used and derivatives of Whittaker p otential U in far region
Go o d agreementbetween U and f z is observed with in nitesimally close to
References
A Sezginer J Appl Phys vol No p
Borisov V V Utkin A B J Phys D Appl Phys
F F Valiev Tech Phys Lett v
F F Valiev Tech Phys No
DAYS on DIFFRACTION
Investigation of wave eld in e ective mo del of layered
elastic medium with slide contact on interfaces
Lev A Molotkov
St Petersburg Branch of V A Steklov Mathematical Institute Russia
For the medium consisting of two alternating elastic layers with slide contact the e ective mo del
was constructed and investigated In the wave eld of this mo del there are leading triangular
and back concavefronts In this mo del the wave eld exited by an impulse p oint source is represented
in a half plane as Fourier and Mellin integrals We transform the expressions of the wave eldby
the metho ds used in In the Mellin integral we replace contour of integration by a stationary
contours In the obtained expressions we rearrange the integrals and calculate the inner integral
The external integral is equal to two residues The corresp onding p oles are ro ots of two equations
of sixth order These ro ots are situated at the right half plane and can b e complex or real If these
ro ots go out at the imaginary axis then the corresp onding p oints of observation are on the fronts or
inside of triangular and back concave fronts In these cases the displacements are equal to zero The
obtained representation for the wave eld corresp onds to the expressions derived by the metho d of
Smirnov Sob olev For calculation of the complex and real ro ots determining the wave eld we suggest
an e ective metho d
The work was supp orted by RFBI grant
References
L A Molotkov Zap Nauchn Semin POMI
L A Molotkov Investigation of wave propagation in p orous and fractured media on the basis of
Biot e ective mo del and of layered media S Petersburg
L A Molotkov Days on di raction
Ab out light pressure in negative refraction materials
V G Veselago
Moscow Institute of Physics and Technology Russia
e mail v veselago relcom ru
As it was shown in for material with negative refraction NRM many formulas of electro dy
namics must b e used very carefully This in particular p ertains to well known formula
p hk h n c h v
ph
linking value of momentum of photon p with its wavevector k and or index of refraction n and phase
velo city v
ph
If formally follow this equation one might conclude that momentum of electromagnetic eld in
the NRM is directed opp osite to ow of energy of eld or group velo city v This means that in NRM
gr
instead of lightpressuremust exist the light attraction However it is easy to show following metho d
develop ed in that requirement of conservation of the centre of masses of system consisting of eld
spreading in NRM transmitter and receiver can b e not satis ed if acknowledge the equitable formula
For this case the conservation of the centre of masses p osition follows only from correlation
p h v c
gr
The details of contradiction b etween these formulas are discussed in rep ort
References
V G Veselago V G Appl Phys B
Einstein A Ann Phys
DAYS on DIFFRACTION
Author index
Merzlikin A M Ab dulkadyrov D V Sinitsyn S O Frangos P V
MladyonovP L Ab dulkadyrov V A
Sitnik S M
Geyer M A
Molotkov L A Abramo chkin E G
Skoro dumova E A
Ghorbani A
Morgado M L Acho T M
Smirnov Y
Golovkina M V
Motygin O V Aghayan K L
Smirnova E I
Gonz alez Morales M J
Ahmedov T M
Smirnova N I
Nasermoghaddasi M
Altintas A
SmolyanovO G
Nazarov S A
Gorbatsevich F F
Anastassiu H T
Smotrova E I
Nazaryan E A
Graglia R D
Arakelyan M M
Solov evaYu A
Nefedov I S
Grikurov V E
Athanassoulis A G
Soloviev M B
Negorozhina E S
Gua Ch
Avetyan K T
Stavrinou P N
Nekrasov R V
Sturova I V
Hao Y
Nerukh A Bagdo ev A G
Sysova E
Heyman E
Nerukh A G Balaban M V
Nikolaev S V Bastos M A
Ta jvidy A
Ilyushin Ya A
Nosich A I BelovP A
Ishio H
Tirozzi B
Belov V V
Ivakhnychenko M V
Tovstik T
Odit M
Benson T M
Ivanov A I
Tran X Tr
Omel yanov G A
Berkovsky A N
Tretyakov S A
Berlin M A
Joub ert S V
Papkelis E G
Trifonov A Yu
Bianchi D
PetroshenkoP A
Troyan V N
Boag A
Karasev M V
Pop ov A L
Trubaev V V
BogomolovYa L
Katkov D V
Pop ov I Yu
Tsvetkovskaya S M
Borisov V V
Katsav M
Pretorius W E
Borovikov V A
Khusnutdinova K
Prosvirnin S L
Utkin A B
BoruhovichS P
Kiselev Al S
Prozorov K V
Borzov V V
Kiselev An S
VakulenkoS A
Bruning J
Kiselev Yu V Radchenko V V
Valiev F F
Bradley D D C
Klopp ers P H RaicichF
Valovik D
Bronnikov V I
Konyukhova N B Reutskiy S
Vardanyan A V
ButkoYa A
Rezaev R O
Kouzov D P
Vardanyan S V
Byelobrov V O
Rosanov N N
Kovalev M D
Veliev E I
Kozlov S A
Vendik I
Chaplygin A A
Sakhnenko N
Krutitskii P A
Vendik O
Chelub eyev D A
Santos A M
Kryvko A
Chernitskii A A VeretenovN A
Sautb ekov S S
Kucherenko V V
Co etzee C E
Veselago V G
Sekerzh ZenkovichS Ya
Kudrin A V
VinogradovA P
Kyurkchan A G
Damaskinsky E V
Vinokurov A A
Semenov E S
Daniele V G
Sergeev V I
Volkov B I
Dehesa Mart nez C
Sewell P
Volostnikov V G
Lerer A M
Demin D B
Shabalina E
Vorobyev A Y
Lerer V A
Dobrokhotov S Yu
ShafarevichA I
Vozianova A V Lima P M
Shanin A
Litvinets F N
Donets I V
Yalunin S
Shap ovalov A V
LobanovI S
Dorofeenko A V
Yanovskaya T B
Lombardi G
Shatalov M Yashina N F
Efros A L
Mahillo Isla R
Shcherbina M
Yatsyk V V
Es kin V A
Makin R S
Shelkovich V M
Yunakovsky A D
Makin V S
Shereshevskii I A Fay T H
Zab oronkova T M
Manenkov S A
Shp olyanskiy Yu A Fedorov S V
Zheludev N I
Martirosyan A N
Sidorov A A Fedotov I
ZhevandrovP N
Maslovski S I
Silveirinha M G Fedotov V A
Zon V B
Maximov G
Simonenko I I Feng J
McPhee G J Simovski C R Filipp enko G V Zvyagintsev A A