Transcript MAS for Eigenvalue Problems of Plasmonic Structures

Tbilisi State University (LAE)
Method of Auxiliary Sources for Optical
Nano Structures
K.Tavzarashvili(1), G.Ghvedashvili(1), D. Kakulia(1), D.Karkashadze(1,2),
(1)Laboratory
of Applied Electrodynamics, Tbilisi State University, Georgia
(2) EMCoS, EM Consulting and Software, Ltd, Tbilisi, Georgia,
e-mail: [email protected]
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The Content and Purpose
•
Conventional interpretation of MAS applied to the solution of
electromagnetic scattering and propagation problems.
•
The general recommendations for the solution of these problems.
•
An application of the method to specific problems for the single body and a
set of bodies of various material filling.
•
The application areas of MAS, its advantages and benefits
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Mathematical Background of the Method of Auxiliary Sources
(MAS)
•
•
•
1.
2.
3.
4.
5.
6.
7.
8.
The name “MAS”, currently used, did not appear at once. The authors themselves adhered to the names:
“The Method of Generalized Fourier Series” [1-5]
“The Method of Expansion in Terms of Metaharmonic Functions” [6]
“The Method of Expansion by Fundamental Solutions [7,8]
V.D. Kupradze, M.A. Aleksidze: On one approximate method for solving boundary problems. The
BULLETIN of the Georgian Academy of Sciences. 30(1963)5, 529-536 (in Russian).
V.D. Kupradze, M.A. Aleksidze: The method of functional equations for approximate solution of some
boundary problems. Journal of Appl. Math. and Math. Physics. 4(1964)4, 683-715 (in Russian).
V.D. Kupradze, M.A. Aleksidze: The method of functional equations for approximate solution of some
boundary problems. Journal of Appl. Math. and Math. Physics. 4(1964)4, 683-715 (in Russian).
V.D. Kupradze: Potential methods in the theory of elasticity. Fizmatizdat, Moscow 1963, 1-472 (in
Russian, English translation available, reprinted in Jerusalem, 1965).
V.D. Kupradze: On the one method of approximate solution of the boundary problems of mathematical
physics. Journal of Appl. Math. and Math. Physics. 4(1964) 6, 1118 (in Russian).
I.N. Vekua: On the completeness of the system of metaharmonic functions. Reports of the Acad. of
Sciences of USSR. 90(1953)5, 715-717 (in Russian).
V.D. Kupradze, T.G. Gegelia, M.O. Bashaleishvili, T.V. Burchuladze: Three dimensional problems of
the theory of elastisity. Nauka, Moscow 1976, 1-664) (in Russian).
M.A. Aleksidze: Fundamental functions in approximate solutions of the boundary problems, Nauka,
Moscow 1991, 1-352 (in Russian).
“A common idea of these works is a basic theorem of completeness in L2(S) … of infinite set of
particular solutions, generated by the chosen fundamental or other singular solutions… ”[7] of
wave equation (D.K.)
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Electromagnetic Scattering and Propagation as the Boundary Problem
The main goal of problem is to find vectors of secondary electromagnetic field
in each bounded, simply connected domains
 m mM0 ,

   
Em , Bm , Dm , H m

M
m 0
confining with the set of smooth, closed
surfaces  S mn M
(interfaces between neighbouring m and n domains), while the primary field
m, n 0
   
E 0 , H 0 , D0 , B0 is given.


In corresponding domain secondary or primary electromagnetic field should satisfy:
a) Maxwell’s equation;






ˆ (E , B ) ;
b) some type of constitutive relation among field vectors – Dm  Fˆ (Em , Bm ), Hm  
m
m




 
c) boundary conditions Lˆmn ( Em , Hm , En , Hn )  0 on S mn
M
m, n 0
surfaces;
d) in unbounded free space Γ0 field vectors should satisfy the radiation condition.
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Method of Auxiliary Sources (MAS) (simple 2D case)
ΔE
scat
z
( x, y)  k E
2
scat
z
( x, y)  0, P( x, y)  0
r 
0 
n n 1 ,
r n0  S 0
2) Fundamental solutions of Helmholtz equation:
Ezscat ( xs , ys )  Ezinc ( xs , ys )  0
 
 
 
 n(r  rn0 )  k 2 n(r  rn0 )   (r  rn0 )
y


( Einc , H inc )
1) Everywhere dense points on the surface S0 -
 n(kR n0 ) 


( E scat , H scat )
i (1)
H 0 (kR n0 ), Rn0  ( x  xn0 ) 2  ( y  yn0 ) 2
4
3) Construction of the set of fundamental solutions of wave
equation with radiation centers on surface S0:


0
n (rs , rn )

n 1

 Wn  n1   n (rs , rn0 )n1

x
Γ
Scatterer - S
Γ0
Auxiliary surface – S0
Theorem: It can be shown [*], that for an arbitrary smooth surface
S (in the Lyapunov sense) one can always find the auxiliary
surface S0 such that the constructed set of functions is complete
and linearly independent on S in the functional space L2.
*M.A. Aleksidze: “Fundamental functions in approximate solutions of the boundary problems”.
Nauka, Moscow 1991, 1-352 (in Russian).
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r 
v M
m m 1
V ; Rm = rs  rmv
Tbilisi State University (LAE)
Method of Auxiliary Sources (MAS) (simple 2D case)
Any continuous function on S can be expanded in terms of the first N functions of the given set of
fundamental solutions:
N

Ezscat ( xs , y s )   an H 0(1) k ( xs  xn0 ) 2  ( ys  yn0 ) 2
n 1

Properties of this set of fundamental solutions guarantee existence of corresponding coefficients
providing the best in L2(S) mean-square approximation of any continuous function on S:
(xs ,ys)
( xn0 , yn0 )
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Corresponding discrepancy:

S
an nN1
N ( )

 E inc ( x , y )  a H (1) k ( x  x 0 ) 2  ( y  y 0 ) 2

s
s
n
0
s
n
s
n
 S z
n 1

2
inc
E
(
x
,
y
)
dS

z
s
s


S

when N ( )  , then  0
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
2
1
2

dS 

 ;



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Computational Procedures for Determination
MAS Unknown Coefficients
•
•
•
•
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Gram-Schmidt Orthogonalization approach;
MAS-MoM-Galerkin approach;
Method of Colocation;
…..
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Gram-Schmidt Orthogonalization Approach


M
Gm ( Rm )
m
  m ( rs )m1 , L  m ( Rm )  = I   ( Rn )
M
m 1
Orthogonalization procedure
 1 ( s )  G1 ( s ),
n=1
 2 ( s )  G2 ( s )   1 ( s )  A21 ,
From Fourier Theorem - The best expansion (in L2)
of given vector function H(xs,ys,zs) on surface S:
 3 ( s )  G3 ( s )   1 ( s )  A31   2 ( s )  A32 ,
. . . .
M
H ( xs , ys , zs )   m ( s)  bm
m 1
 m ( s )  Gm ( s )   k ( s )  Amk ;
m 1
 H ( s)  
s
m 1
k 1
2
M
m
( s)  bm ds  min( L2 )
Scalar product definition:

0, m  k
T
T



ds





 m k  
s  m k 

 Fm , m  k
. . . .
n 1


Amn  F  Cnm   AnkT  Fk  Amk  , ( Amn  0, m  n);
k 1


1
n
 G
T
m
S

 Gk ds  Cmk ,
 G
T
k

 Gk ds  Fk ;
S
bq  Fq1   q  H 
V.D. Kupradze: “On the one method of approximate solution of the boundary problems of mathematical physics”. Journal of
Appl. Math. and Math. Physics. 4(1964) 6, 1118 (in Russian).
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MAS-MoM-Galerkin Approach
Tn
n
ln
nn
Tn
n

n
3
4
2
rs
rn
n
O
1
MoM-like representation of current on the auxiliary surface
 ln

 2 A  n ,
 n
N
J (rs )   I n f n  rs  , f n (rs )   ln   ,
 2 A n
n 1
 n
0 ,

Triangulated auxiliary surface
Triangulated surface of the
scatterer
if
rs  Tn
if
rs  Tn
if
rs  Tn
Current basis function and testing function
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MAS+Collocation Approach (simple 2D case)
r 
0 N
n n 1
, r n0  S 0
N
a Z
n 1
n
nm
 rm Mm1 ,
rm  S
 Vm , (m  1,2,...,M )

Z nm  H 0(1) k ( xm  xn0 ) 2  ( ym  yn0 ) 2
Log10(ε)
-0.30
-2.24

-4.18
Vm  E ( xm , y m )
inc
z
-6.12
(xm ,ym)
( xn0 , yn0 )
n=3
4
-8.06
5
6
7
kd
-10.00
0.50
1.80
3.10
4.40
5.70
7.00
Estimation of accuracy of solution
d
2 
E
S
inc
z
N
( xs , y s )   an H
n 1
(1)
0
E
k
inc
z
( xs  x )  ( y s  y )
0 2
n
2
0 2
n

2
dS
;
( xs , y s ) dS
S
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Method of Auxiliary Sources (problems)
Problem 1: unstability of linear system of algebraic equation
-0.21
Colocation points
(xm ,ym)
Log10(ε)
-0.30
b)
a)
n
-0.87
-2.24
5
6
7
Auxiliary Sources
( xn0 , yn0 )
Log10(ε)
-1.53
-4.18
-2.19
-6.12
-2.85
-8.06
n=3
d

nk0
-3.51
0.50
kd
1.80
3.10
4.40
5.70
7.00
 an  Z nm  Vm , (m  1,2,...,M )
n 1
 (a
n 1
n
1.80
kd
3.10
4.40
5.70
7.00
Accuracy of solutions versus relative distance kd
N
N
-10.00
0.50
4
5
6
7
  an )  ( Z nm  Z nm )  Vm  Vm
N
 m  am   an 
n 1
Z nm Vm

, (m  1,2,...,M )
Z11 Z11
Tikhonov-Arsenin → αm=
10-8 ÷10-10
αm - regularization parameter
Hadamard: for Znm  1012 and Vm  1012  an  an
A.N. Tikhonov, V.Ya. Arsenin: “The methods for solving non-correct problems”. Nauka, Moscow 1986, 1-288
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Method of Auxiliary Sources (problems)
Problem 2: scattered fields main singularities (SFMS)



E tot  E inc  E scat


E tot  H tot  0
From Uniqueness Theorem
The regular, in whole space, solution of Maxwell’s
equation satisfying the radiation condition at infinity
should be identically zero(*).
PEC
Scattered wave fields (both scalar and vector), which
are continuously extended inside the scatterer's domain
certainly has irregular points (singularities - SFMS).
d
q
h
q
R
Auxiliari charges
(unstable)
Auxiliari charges
(stabile)

 (r ,  ) 
1 
q
q

2
2
2

4 0
r  d  2rd cos
 r  ( R  d )  2r ( R  d ) cos
d




R2
Rq
; q  
Rh
Rh
V.D. Kupradze(*): “The main problems in the mathematical theory of diffraction”. GROL. Leningrad, Moscow 1935,1-112.
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Method of Auxiliary Sources (problems)
•
Problem 2: scattered fields main singularities (SFMS - conformal mapping procedure)
1.08
0.65
y
1.270.54 y
y
D1
L2
0.760.22
D1
0.22
0.25-0.11
-0.22
-0.25-0.43
-0.76-0.76
L1
S
-1.08
-2.16
L2
L2
D2
-0.65
L1
-1.29
-0.43
0.43
1.29
x
2.16
D2
S
-1.27-1.08
-1.38
-2.16 -0.83
-1.53
-0.29
-0.89
0.26
-0.26
0.81
0.37
x x
1.001.36
Image lines of primary source - L1 and L2;
“Auxiliary Sources” ( monopoles) surrounding the areas of SFMS concentration, imitating the
radiation from the images L1 and L2;
The lines of other possible distributions of auxiliary sources;
Some optimal distribution of collocation points on the main surface S.
D.Karkashadze: "On Status of Main Singularities in 3D Scattering Problems". Proceedings of VIth International
Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2001), Lviv,
Ukraine, September 18-20, 2001, pp. 81-84. http://www.ewh.ieee.org
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Method of Auxiliary Sources (problems)
•
Problem 3: Most general, linear form of constitutive relation (Bi-Isotropic medium)
Maxwell’s equations
 

I .   E  iB;
 
II.   B  0;
 

III.   H  iD;
 
IV.   D  0;
Constitutive Relations


 



1
D  E  iB, H  iE   B
Isotropic magnetodielectrics:
Chiral medium:
Tellegen medium:
 =  = 0;
 =   0;
 =-  0;
Wave equation
any field vectors
2U  k 2U   (   )U  0, U  anyfild
vectors
Material parameters
Wave impedances
  r  r,
k r,
ˆ    ,  
,
r,


 
 r 
k r,  r,
r,
ˆ    ,  

 
 r
ˆ    ,
 
0 
 r,
0
,
0
2

0    2  (   )
4


  
2
1
 ,


r
r
Wave numbers
k
ˆ
k 
0

r
•
•
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0
k
 2 (   )2   
 r,

 , k  k     


4
2



 , k  k0


 r r
A.Lakhtakia: “Beltrami fields in chiral media”. World Sci. Publ. Co., Singapore 1994, 1-536
I.V.Lindell, A.H.Sihvola, A.A.Tretyakov, and A.J.Viitanen: “Electromagnetic waves in chiral and BiIsotropic media”. Artech House, Boston, London 1994, 1-332..
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Method of Auxiliary Sources (problems)
•
Problem 3: Fundamental Solution for Bi-Isotropic medium
Spinor basis
  r    v  i u 
  



i


v
u


 

l
Spinor of electromagnetic field
 F r   E  iη r H 
F  
 F   E  iη H 
  

Maxwell’s equation in the Majorana-Dirak form
Fundamental Solution for Bi-Isotropic medium

ˆ 0
 F  kF



0 

Gn 

Green function
Green function matrix
 Gnr
ˆ
Gn  
 0

Fn r ,    Gˆ n  n  1r ,  Gˆ n  n  kˆ  Gˆ n  n
k
Gn
r ,
r,
n
G
 ρ,ρ n   H 0(1)(k r ,    n ) - 2D case
 r , rn  
1
4
ik r , r rn
e
r rn
- 3D case
R.Penrose, W.Ringler: “Spinors and space-time”, vol. 1. Cambridge University Press, Cambridge (eng.), 1986
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General Concept of MAS
Problem formulation for most general form of
constitutive relations
Definition of fundamental or other
singular solutions of the wave equation
Geometry Analyzing for
Scattered Field Main Singularities (SFMS)
Finding the best placement and type
of the fundamental solutions (AS)
Definition of numerical method for evaluating
the amplitudes of the auxiliary sources
Method of collocation,
method of moments (MOM),
ortogonalization
Deriving the system of linear algebraic
equations
Post processing stage
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Computation of amplitudes of AS
Data processing and visualization
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MAS Applicatin to Some Particular Problems
•
MAS approach for electromagnetic scattering on the Bi-Isotropic bodies;
•
MAS simulation of wave propogation in Double-Negative medium;
•
MAS and MMP simulations of Finite Photonic Crystal (PhC) based devices;
•
MAS approach for electromagnetic scattering on Double-Periodic structures;
•
MAS Approach for Band Structure Calculation and eigenvalue search
problem;
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Electromagnetic Scattering Upon the Chiral Bodies
F.G. Bogdanov, D.D. Karkashadze, R.S. Zaridze. “The Method of Auxiliary Sources in Electromagnetic
Scattering Problems”. North-Holland, Mechanics and Mathematical Methods, A Series of Handbooks. First
Series: Computational Methods in Mechanics. Vol.4, Generalized Multipole Techniques for Electromagnetic and
Light Scattering, Chapter 7, pp. 143-172, 1999.
Fundamental solutin:
Boundary problem for spinor field:
 

ˆ
0
   F k F

Wˆ F(r )   s = f(r s ), M (r s )  S
r r
Required solution
N
N
 (N) 
 
 anr 0 


F (r )   aˆn Fn (r , rn ), r  D, aˆn  

 0 an  n 1
n 1



 

 r , 




1
ˆ
Fn    Gn  n  r ,  Gˆ n  n  kˆ  Gˆ n  n ;
 
k
r
 



ik r  r 



G

v  i u 
0
n 

 , G r , r  r  1 e





Gˆ  
,
n
      i 
 0 G 
4 r r
r
r ,


k
kˆ  
0

r
n  

0
k
v
 2 (   )2    
 r,
,
k

k




  
 , k  k0




4
2



u

 r r
Scattered electric field reconstracted from spinor fields
 


E sc (r )  Escr  Esc 
r N    
 N r  r  
an Fn (r , rn )
 an Fn (r , rn )   r   
 r    n1
n 1
Spherical shape chirolens
Constitutive relations
Eztot
Ez
Ezr


 


D  E  iB, H  iE   1B
0.3 

  
,  3.0,  1.389,
0
120  0
k 0 400, heght  d  0.6,
thickness t  0.125
x
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Plane Wave Excitation on Ciral Sphere
Log

 a2
a  0.2 mm;    0 (5.0  i0.1);   (0   2 /  )1;     
110.8
nr
Im(n)
77.9
44.9
12.0
-20.9
-53.9
150
3.22
fGHz
GHz
Re(n)
180
210
GHz
240
270
300
n
2.58
Re(n)
Scattering cross-section versus frequency.
MAS approach:
γ=0;
γ=0.0001;
T-matrix approach:
γ=0.0001
•
•
1.94
1.30
0.66
A.Lakhtakia, V.K.Varadan, V.V.Varadan: “Scattering and absorption
Im(n)
characteristics of lossy dielectric, chiral, nonspherical objects”. Appl. Opt. 0.02
GHz
300
150
180
210
240
270
24(1985) 23, 4146-4154.
F.G.Bogdanov and D.D.Karkashadze: “Conventional MAS in the problems of Refractive index versus frequancy for
Right-hand and Left-hand components
electromagnetic scattering by the bodies of complex materials”. Proc. of the 3-rd
Workshop on Electromagnetic and Light Scattering, Bremen 1998,133-140.
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Kirchhoff-Kotler formula + MAS for Field Reconstruction in DoubleNegative Medium (wave front reversal approach)
k  k0 1  1
y
r =1; r=1
r =−1; r=−1
y
r =1; r=1
r =−1; r=−1
x
x
Gaussian beam
λ
λ
Gaussian beam illuminated transparent object
The total electric field Ez component reconstruction
from space x<0 to spece x>0. Total electric field Ez
component was reconstructed from EM field
tangential components on the YOZ plane.
Electric field distribution from a Gaussian beam
source left x<0 (actual) and MAS predicted
electric field distribution in a virtual DNM half
space right x>0 (reconstructed from EM field
tangential components on YOZ plane).
David Karkashadze, Juan Pablo Fernandez, and Fridon Shubitidze: “Scatterer localization using a
left-handed medium”. OPTICS EXPRESS 9906, (C) 2009 OSA, 8 June 2009 / Vol. 17, No. 12
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MAS and Multiple Multipole Method (MMP) Simulations of
Finite Photonic Crystal (PhC) Based Devices
•
E. Moreno, D. Erni, Ch. Hafner: “Modeling of discontinuities in photonic crystal waveguides with
the multiple multipole method”. Phys. Rev. E 66, 036618, 2002
•
D. Karkashadze, R. Zaridze, A. Bijamov, Ch. Hafner, J. Smajic, D. Erni: “Reflection
compensation scheme for the efficient and accurate computation of waveguide discontinuities in
photonic crystals”. Applied Computational Electromagnetics Society Journal, Vol. 19, No. 1a,
March 2004, pp. 10-21
•
D. Karkashadze, R. Zaridze, A. Bijamov, Ch. Hafner, J. Smajic, D. Erni: “MAS and MMP
Simulations of Photonic Crystal Devices”. Extended Papers of Progress In Electromagnetic
Research Symposium (PIERS-2004). March 28-31, 2004, Pisa, Italy. pp. 29-32
7/6/2015
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“Filtering T-junction” Design (MMP, MAS)
IWGAInput
DFDI
DFDI
IWGARight
IWGALeft
TM-polarization. Lattice constant a=1μm , base
rods permittivity ε=11.4, base rods radii r=0.18a
f1=1.038 1014Hz
Comparison (without optimization)
MMP simulation
Case
MAS simulation
R (%)
Tl (%)
Tr (%)
 (%)
R (%)
Tl (%)
Tr (%)
 (%)
Left
35.37
63.38
0.41
99.16
36.38
63.71
0.42
100.51
Right
36.51
0.11
63.24
99.86
36.02
0.11
63.76
99.89
Results of Filtering T_Junction optimization
f2=1.230 1014Hz
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f1=1.038 1014 Hz
f1=1.230 1014 Hz
Rup=0.73%
Rup=0.76%
Tright=0.16%
Tright=97.77%
Tleft=98.59%
Tleft=0.88%
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Coupling a Slab with a PhC Waveguide: fa/c=0.38 (MMP, MAS)
Optimizing Inclusion ro=0.15R
Crossing waveguide operating with
different frequency in each channels
Before optimization: SWR_WGMAS= 1.72;
SWR_WGMMP=1.66
After optimization : SWR_WG = 1.08
hD_WG = 3.379; hPhC_WG = 1.644;
7/6/2015
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3D Double-Periodic Green Functions
2
2

2 
exp
ik
x

nd

y

nd

z




x
y






  x, y, z     exp(iknd x cos  x  ikmd y cos  y )
2
2
n  m 
ik  x  nd x    y  nd y   z 2
  x, y , z   
2
dxd y
k xq  k cos  x 
k zqp


Poisson Transformation
1
exp(ik xq x  ik yp  ik zpq z )


ik
pq
p  q  ik
z zpq
2
2
q, k yp  k cos  y 
p, h p 
dx
dy
 h 2  k 2 , if h  k and k  k
xq
p
xq
yp
 p

,
2
2
 i k xq  hp ,if hp  k xq and k  k yp
2
k 2  k yp
, p  0, 1, 2,... , q  0, 1, 2,...
k zqp  i hp2  k xq2 , if k  k yp
D. Kakulia, K. Tavzarashvili, G. Ghvedashvili, D. Karkashadze, and Ch. Hafner: “The Method of Auxiliary
Sources Approach to Modeling of Electromagnetic Field Scattering on Two-Dimensional Periodic Structures”.
Journal of Computational and Theoretical Nanoscience, Vol. 8, 1–10, 2011
7/6/2015
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Oblique Incident of Plane Wave on Bi-Periodic Array of Dielectric Spheres
1
Reflection coefficient Г
z
ES
EP
k
y
2a
d
x
d
0.8
  20
 0
0.6
s-pol.
0.4
p-pol.
0.2
0
0.6
0.7
0.8
0.9
1
Relative period d/λ
a) Problem geometry (radius of spheres - a, period – d, incident angle θ=20º, φ=0º, permittivity
of dielectric spheres ε=3.0, a/d=0.4);
b) Transmission coefficient versus relative period. The solid curve is for p-polarization and the
dotted curve is for s-polarization. Comparison of MAS approach with results presented in
• M. Inoue: “Enhancement of local field by two-dimensional array of dielectric spheres placed on the
substrate”. PHYSICAL REWIEV B, vol. 36 #5, 15 August 1987, p 2852-2862.
7/6/2015
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The Test for Convergence of MAS Results
1.0
Transmission Coefficient
Т
E
0.8
H
N=2352
N=1200
0.6
k
0.4
X direction
0.2
a)
0.0
0.6
b)
0.7
0.8
0.9
Relative period dx / λ
1.0
Dielectric layer with hexagonally bi-periodic dents: a) problem geometry and
incidents plane wave orientation; b) transmission coefficient versus relative period.
7/6/2015
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MAS and MMP Approach for Band Structure Calculation
y
Non periodic expansions
dx
D0
Sout
dy
D1
1 ,  1

x
k?
Sin
Periodic expansions
Unit cell
Ez
2D bi-periodic structure with arbitrary shaped dielectric scatterers and unit cell with distribution of auxiliary sources (left);
Periodic Green’s functions for different angles14of incidence a) =00, b) =450 (right).
x 10
1
5
4.5
0.8
4
3.5
0.6
fa/c
fa/c
3
0.4
2.5
ky
2
Μ
1.5
0.2
1
TM
TE
0


x*dx
Γ
Χ kx
0.5


0

2/a

a


Band structure for a perfect PhC made of dielectric rods (left); Band structure for a perfect PhC made of silver wires (right).
K. Tavzarashvili, Ch. Hafner, Cui Xudong, Ruediger Vahldieck, D. Kakulia, G. Ghvedashvili and D.
Karkashadze, “Method of Auxiliary Sources nd Model-Based Parameter Estimation for the Computation of Periodic
Structures”, Journal of Computational and Theoretical Nanoscience Vol.4, 1–8, 2007
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Tbilisi State University (LAE)
Conclusion
The procedure of solution of scattering problems by the
method of auxiliary sources (MAS) needs preliminary
considerations of various problems.
Correct solution of these problems can radically influence
efficiency of MAS.
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Tbilisi State University (LAE)
Thank you for attention
[email protected]
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