Method of Auxiliary Sources for Optical Nano Structures

Tbilisi State University (LAE)

David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 104/22/23

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]

Method of Auxiliary Sources for Optical Nano Structures



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



Mathematical Background of the Method of Auxiliary Sources (MAS)

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]

1. 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).

2. 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).

3. 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).

4. V.D. Kupradze: Potential methods in the theory of elasticity. Fizmatizdat, Moscow 1963, 1-472 (in Russian, English translation available, reprinted in Jerusalem, 1965).

5. 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).

6. 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).

7. 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).

8. 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.)



The main goal of problem is to find vectors of secondary electromagnetic field

in each bounded, simply connected domains confining with the set of smooth, closed

surfaces (interfaces between neighbouring m and n domains), while the primary field

is given.

In corresponding domain secondary or primary electromagnetic field should satisfy:

a) Maxwell’s equation;

b) some type of constitutive relation among field vectors –

c) boundary conditions on surfaces;

d) in unbounded free space Γ0 field vectors should satisfy the radiation condition.

,0

M

mm

M

nmnmS 0,

M

mmmmm HDBE 0,,,

0000 B, D, H, E

;),(ˆ),,(ˆmmmmmm BEHBEFD

Electromagnetic Scattering and Propagation as the Boundary Problem

0),,,(ˆ nnmmnm HEHEL

M

nmnmS 0,



Method of Auxiliary Sources (MAS) (simple 2D case)

Scatterer - S Auxiliary surface – S0

),( incinc HE

),( scatscat HE

Γ0Γ

x

y

02 ),(,0),(),( yxPyxEkyxE scat

zscatzΔ

0),(),( ssinczss

scatz yxEyxE

001

0 , Srr nnn

)()()( 0020nnnnn rrrrkrr

202000)1(

00 )()(),(

4)( nnnnnn yyxxRkRH

ikR

1) Everywhere dense points on the surface S0 -

2) Fundamental solutions of Helmholtz equation:

1

011

0 ),(),( nnsnnnnnsn rrWrr

3) Construction of the set of fundamental solutions of wave equation with radiation centers on 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).



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:

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:

N

nnsnsnss

scatz yyxxkHyxE a

1

2020)1(0 )()(),(

1

;M

mV

v v

m m s mr R = r r

0then,)(when

;),(

)()(),(2

1

2

2)(

1

2020)1(0

N

dSyxE

dSyyxxkHyxE

S

ssincz

S

N

nnsnsnss

incz a

N

nna 1

Corresponding discrepancy:(xs ,ys)

),( 00nn yx

S



Computational Procedures for Determination

MAS Unknown Coefficients

• Gram-Schmidt Orthogonalization approach;

• MAS-MoM-Galerkin approach;

• Method of Colocation;

• …..



Gram-Schmidt Orthogonalization Approach

;, k

S

kT

kmk

S

kT

m FdsGGCdsGG 0,

,

T Tm k m k

s m

m kds

F m k

1 1

2 2 1 21

3 3 1 31 2 32

1

1

( ) ( ),

( ) ( ) ( ) ,

( ) ( ) ( ) ( ) ,

. . . .

( ) ( ) ( ) ;

. . . .

m

m m k mkk

s G s

s G s s A

s G s s A s A

s G s s A

11

( ) ( ) , ( ) ( )M M

m m mmmG

m

m s m nn=1

R r L R = I R

11

1

, ( 0, );n

Tmn n nm nk k mk mn

k

A F C A F A A m n

1

( , , ) ( )M

s s s m mm

x y z s

H b

2

21

( ) ( ) min( )M

m mms

s s ds L

H b

From Fourier Theorem - The best expansion (in L2)

of given vector function H(xs,ys,zs) on surface S:

Scalar product definition:

Orthogonalization procedure

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).

H qqq Fb 1



MAS-MoM-Galerkin Approach

, if2

( ), if

2

0 , if

nn s n

n

n s nn s n

n

s n

lr T

A

f r lr T

A

r T

1

( ) ,N

s n n sn

J r I f r

MoM-like representation of current on the auxiliary surface

Triangulated surface of the scatterer

Triangulated auxiliary surface

nT

nT

O

n

n

nl

nn

nn

nr

1

2

3

4

sr

Current basis function and testing function



MAS+Collocation Approach (simple 2D case)

001

0 , Srr n

N

nn Srr m

M

mm ,1

),...,2,1(,1

MmVZ mnm

N

nna

),(

)()( 2020)1(0

mminczm

nmnmnm

yxEV

yyxxkHZ

;

),(

)()(),(2

2

2

1

2020)1(0

S

ssincz

S

N

nnsnsnss

incz

dSyxE

dSyyxxkHyxE a

(xm ,ym)),( 00

nn yx

d

0.50

-0.30

1.80

-2.24

3.10

-4.18

4.40

-6.12

5.70

-8.06

7.00

-10.00 kd

n=3

4

5

6

7

Log10(ε)

Estimation of accuracy of solution



Method of Auxiliary Sources (problems)

Problem 1: unstability of linear system of algebraic equation

),...,2,1(,1

MmVZ mnm

N

nna

mmnmnmn

N

nn VVZZaa

)()1

(

nnmnm aaVZ 1212 10and10for:Hadamard

A.N. Tikhonov, V.Ya. Arsenin: “The methods for solving non-correct problems”. Nauka, Moscow 1986, 1-288

0.50

-0.21

1.80

-0.87

3.10

-1.53

4.40

-2.19

5.70

-2.85

7.00-3.51

n 5 6 7

Log10(ε)

kd0.50

-0.30

1.80

-2.24

3.10

-4.18

4.40

-6.12

5.70

-8.06

7.00-10.00 kd

n=3 4 5 6 7

Log10(ε)

),...,2,1(,11111

MmZ

V

Z

Z mnmN

nnmm aa

(xm ,ym)),( 00

nn yx

d

Colocation pointsAuxiliary Sources

0kn

a) b)

αm - regularization parameter Tikhonov-Arsenin → αm= 10-8 ÷10-

10

Accuracy of solutions versus relative distance kd




Problem 2: scattered fields main singularities (SFMS)

scatinctot EEE

PEC

From Uniqueness Theorem

The regular, in whole space, solution of Maxwell’s equation satisfying the radiation condition at infinity should be identically zero(*).

Scattered wave fields (both scalar and vector), which are continuously extended inside the scatterer's domain certainly has irregular points (singularities - SFMS).

V.D. Kupradze(*): “The main problems in the mathematical theory of diffraction”. GROL. Leningrad, Moscow 1935,1-112.

hR

Rqq

hR

Rd

;

2

cos2cos)(2)(4

1),(

2220 rddr

q

dRrdRr

qr

h

R

dqq

Auxiliari charges(stabile)

Auxiliari charges(unstable)

0 tottot HE




• Problem 2: scattered fields main singularities (SFMS - conformal mapping procedure)

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.

y

x

0.54

0.22

-0.11

-0.43

-0.76

-1.08 -2.16 -1.53 -0.89 -0.26 0.37 1.00

y

x

1.27

0.76

0.25

-0.25

-0.76

-1.27-1.38 -0.83 -0.29 0.26 0.81 1.36

D2

D1

L2

L2

L1

S

L1

L2

D1

S

y

x

1.08

0.65

0.22

-0.22

-0.65

-1.08 -2.16 -1.29 -0.43 0.43 1.29 2.16

D2

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



2, 2

0

( ),

4 2

0ˆ , 0

r

rr r

kk k k k k

k

Wave numbers

Wave impedances1

2, 2

0

00

0

( ), ,

4 2

,

ˆr

r

r

r

,

,,

, ,,

ˆ , ,

ˆ ,

r rr

r

r r rr

k

k

Material parameters

2 2 ( ) 0, any fild vectorsU k U U U Wave equation

any field vectors


• Problem 3: Most general, linear form of constitutive relation (Bi-Isotropic medium)

• 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 Bi-

Isotropic media”. Artech House, Boston, London 1994, 1-332..

Maxwell’s equations

;0.IV;0.II

;.III;.I

DB

DiHBiE

BEiH ,BiED

1 Isotropic magnetodielectrics: = = 0; Chiral medium: = 0; Tellegen medium: =- 0;

Constitutive Relations



l

Maxwell’s equation in the Majorana-Dirak form

ˆ 0F kF ��

,,

1 ˆˆ ˆ ˆrn n n n n n nrF G G k G

k

��

Green function matrix

0ˆ0

rn

n

n

GG

G

Green function

)(kHρ,ρ G nr

nr

n ,)1(0

,

,

, 1,r

nrn n

n

ik r r

4 r r e G r r

- 2D case

- 3D case

SpinorSpinor of electromagnetic field

Fundamental Solution for Bi-Isotropic medium

rv u

v u

i

i

��

�

SpinorSpinor basis

R.Penrose, W.Ringler: “Spinors and space-time”, vol. 1. Cambridge University Press, Cambridge (eng.), 1986


• Problem 3: Fundamental Solution for Bi-Isotropic medium



Geometry Analyzing for Scattered Field Main Singularities (SFMS)

Geometry Analyzing for Scattered Field Main Singularities (SFMS)

Finding the best placement and type of the fundamental solutions (AS)

Finding the best placement and type of the fundamental solutions (AS)

Definition of numerical method for evaluating the amplitudes of the auxiliary sources

Definition of numerical method for evaluating the amplitudes of the auxiliary sources

Method of collocation, method of moments (MOM),

ortogonalization

Method of collocation, method of moments (MOM),

ortogonalization

Post processing stagePost processing stage Data processing and visualizationData processing and visualization

Definition of fundamental or other singular solutions of the wave equation

Definition of fundamental or other singular solutions of the wave equation

Problem formulation for most general form of constitutive relations

Problem formulation for most general form of constitutive relations

Deriving the system of linear algebraic equations

Deriving the system of linear algebraic equations Computation of amplitudes of ASComputation of amplitudes of AS

General Concept of MAS



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;



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.

S)r(M)r(f)r(FW sssrr

� ,= ˆ

0 ˆ FkF�� Boundary problem for spinor field: ;ˆˆˆ1ˆ

,

,nnnnrnn

rn GkG

kGF ��

uv

uv

n

rn

n i

i

�

��

,

0

0ˆ

G

GG

r

,,

1,

4 rr

rrikr

rerr G

2, 2

0

( ),

4 2

0ˆ , 0

r

rr r

kk k k k k

k

Fundamental solutin:

,ˆN

1n

)( Dr)r,r(Fa)r(F nnnN

�� ,

N

nn

rn

na

aa

10

0ˆ

Required solution

),(),()(11

nr

n

N

n

rnrnn

N

nnr

r

scrsc

sc rrFarrFaEErE��

Scattered electric field reconstracted from spinor fields

125.0thickness,6.0heght,400

,389.1,0.3,120

3.0

0

00

1

tdk

BEiH ,BiED

Constitutive relations

Spherical shape chirolens

totzEzErzE

x



Plane Wave Excitation on Ciral Sphere

rn110.8

77.9

44.9

12.0

-20.9

-53.9150 180 210 240 270 300

GHz

Im(n)

Re(n)

3.22

2.58

1.94

1.30

0.66

0.02150 180 210 240 270 300

n

GHz

Re(n)

Im(n)

f GHz

2Log

a

Scattering cross-section versus frequency.MAS approach: γ=0; γ=0.0001; T-matrix approach: γ=0.0001

Refractive index versus frequancy forRight-hand and Left-hand components

;)/();1.00.5(;2.0 1200 imma

• A.Lakhtakia, V.K.Varadan, V.V.Varadan: “Scattering and absorption characteristics of lossy dielectric, chiral, nonspherical objects”. Appl. Opt. 24(1985) 23, 4146-4154.

• F.G.Bogdanov and D.D.Karkashadze: “Conventional MAS in the problems of electromagnetic scattering by the bodies of complex materials”. Proc. of the 3-rd

Workshop on Electromagnetic and Light Scattering, Bremen 1998,133-140.

GHz



Kirchhoff-Kotler formula + MAS for Field Reconstruction in Double-Negative Medium (wave front reversal approach)

r =1; r=1 r =−1; r=−1

x

y

λ

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).

r =1; r=1 r =−1; r=−1

x

y

λ

Gaussian beam

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. 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

110 kk



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



““Filtering T-junction” DesignFiltering T-junction” Design (MMP, MAS)(MMP, MAS)

Comparison (without optimization)f1=1.038 1014Hz

f2=1.230 1014Hz

IWGARight

IWGAInput

IWGALeft

DFDIDFDI

TM-polarization. Lattice constant a=1μm , base rods permittivity ε=11.4, base rods radii r=0.18a

Tleft=0.88%Tleft=98.59%

Tright=97.77%Tright=0.16%

Rup=0.76%Rup=0.73%

f1=1.230 1014 Hzf1=1.038 1014 Hz

Results of Filtering T_Junction optimization

99.8963.760.1136.0299.8663.240.1136.51Right

100.510.4263.7136.3899.160.4163.3835.37Left

(%)Tr (%)Tl (%)R (%) (%)Tr (%)Tl (%)R (%)

MAS simulationMMP simulationCase



Coupling a Slab with a PhC Waveguide: Coupling a Slab with a PhC Waveguide: fa/c=0.38 (MMP, MAS)

Optimizing Inclusion ro=0.15R

Before optimization: SWR_WGMAS= 1.72;

SWR_WGMMP=1.66

After optimization : SWR_WG = 1.08

hD_WG = 3.379; hPhC_WG = 1.644;

Crossing waveguide operating withdifferent frequency in each channels



3D Double-Periodic Green Functions

22 2

22 2

exp, , exp( cos cos )

x y

x x y yn m

x y

ik x nd y nd zx y z iknd ikmd

ik x nd y nd z

Poisson Transformation

2 2

2 2

2 2

2 1, , exp( )

2 2cos , cos , , 0, 1, 2,... , 0, 1, 2,...

, if and ,

, if and

xq yp zpqp qx y z

xq x yp y p ypx y

p xq p xq yp

zqp zqp

xq p p xq yp

x y z ik x ik ik zd d ik pq

k k q k k p h k k p qd d

h k h k k kk k

i k h h k k k

2 2 , if p xq ypi h k k k

ikzpq

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



0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Ref

lect

ion

coef

fici

ent Г

p-pol.

s-pol.

20

0

Relative period d/λ

z

x

y

d

d

2a

ES

EP

k

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.

Oblique Incident of Plane Wave on Bi-Periodic Array of Dielectric Spheres



Dielectric layer with hexagonally bi-periodic dents: a) problem geometry and incidents plane wave orientation; b) transmission coefficient versus relative period.

a)

k

E

H

X direction

0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0N=2352N=1200

Relative period dx / λ

Tra

nsm

issi

on C

oeffi

cien

t Т b)

The Test for Convergence of MAS Results



0

0.2

0.4

0.6

0.8

1

x*dx

fa/c

TM TE

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 1014

a

fa/c

2D bi-periodic structure with arbitrary shaped dielectric scatterers and unit cell with distribution of auxiliary sources (left);Periodic Green’s functions for different angles of incidence a) =00, b) =450 (right).

Band structure for a perfect PhC made of dielectric rods (left); Band structure for a perfect PhC made of silver wires (right).

kx

ky

2/a

Γ

Μ

Χ

MAS and MMP Approach for Band Structure Calculation

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

Non periodic expansions

Unit cell

Sin

Sout

D1

D0

Periodic expansions

dy

x

y

dx

Ez

k



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.



Thank you for attention

[email protected]

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Method of Auxiliary Sources for Optical Nano Structures