The method of analytical regularization in the electromagnetic wave scattering by thin disks

THE

METHOD OF ANALYTICAL

REGULARIZATION IN

THE

ELECTROMAGNETIC

WAVE SCATTERING

BY

THIN

DISKS

M.V.

Balaban*, A.I. Nosich*, A. Altintast,

T.M. Bensontt

*Institute ofRadiophysicsand ElectronicsNASU, Ul.Proskury, 12, Kharkov61085,Ukraine tBilkent University, Electrical & Electronics Engineering Department, Bilkent, 06800-Ankara, Turkey

"George

Green Institute forElectromagneticsResearch, University of Nottingham, NottinghamNG72RD,UK Keywords: diffraction, disk, analyticalregularization.

(Ein1Hin)

Abstract

z

We consider the problem of diffraction of an arbitrary electromagnetic wave by a thin disk made from different materials and located in free space. Here we imply a

zero-thickness perfectly electrically conducting (PEC) disk, and also thinelectrically resistive (ER) and dielectric disks whose thickness is much smaller than the disk radius and the free

spacewavelength, and also much smaller than the skin-layer depthintheERdiskcase. The method used for themodeling is basedontheintegral equation(IE) technique andanalytical regularization. Starting with Maxwell's equations, boundary conditions and the radiation condition at infinitywe obtaina

set ofcoupled dual lEs (DIEs) for the unknowns and then reduce this set of equations to the coupled IEs of the Fredholm second kind. Toverify ourresultswecalculate the far field characteristics inthe case of the PEC disk with the incident field being the field of horizontal electrical dipole locatedonthe disk axis.

Figure1. Diskgeometry

The disk of the radius ais located in free space attheplane (z=0). We introduce the cylindrical coordinates (r,

P,z)

with the origin on the axis of symmetry of the disk and denote totalfieldas a sum of the fields scatteredby the disk and the incidentone:

E=

ESC

+

Ein,

H=Hsc +Hin

(1)

1

Introduction

Theproblem ofelectromagneticfield scattering byathin disk has been interested in since long ago. This is explained by

many different applications of this canonical shape. Besides of traditional applications in the printed disk antennas with

PEC orER disks, thin dielectric disk is used as a simplified

model of the tree leave [1]. Still besides, thin few-micron-radius disks are met as resonators of semiconductor lasers with ultralow thresholds [2].Manymethods ofcomputational electromagnetics have been used in their

analysis,

starting from empirical methods and finishing with the method of

moments (MoM) and finite difference time domain method (FDTD). However, the use of these methods leads to difficulties such as prohibitively large matrices and impossibility of estimation of the error of calculations. In contrast, by using the method of lEs with analytical regularization it ispossibletoescapethese difficulties.

2 Problem

statement

We consider the problem of diffraction of a given time-harmonic electromagneticfieldbyazero-thickness dielectric disk. Thegeometryof thisproblemis illustratedinFigure1.

The components of the scattered fieldmust satisfy the setof homogeneous Maxwell equations outside the disk, the 3-D

radiation condition, the condition of local integrability of

power (edge

condition),

and the generalized boundary conditions(see [3])onthe disksurface,at (z=0,r<a):

-[E; +Etg =ZoR* nx[H -Htg],

(<L) -S.nx E+ -

Et-g

.

Here, Z0 is the free-space impedance, and R and S arethe electric and magnetic resistivities. For a thin dielectric disk theyaregivenby

R=iZcot 8rirkT'

2 22/ )' S=-cot - 1 2Z 2 )'

Ir/'rI

>>1, T<<

4

.

(3)

8r/brI

>>1, 'T<<

4

.

(4)

ZO

Ht+g

+

Ht-g

: 2

Here, Z is the relative impedance of the disk material, k=co c is thewavenumber, Er is the relative permittivity,

P'r is therelative

conductivity,

20

is the

wavelength

infree

space,and r is the thickness of the disk.

For athinelectricallyresistive (ER) disk they are given by

R= 1

Zoo-r S=o0.

(5)

For azero-thickness PECdisk theyare given by Escr

R=0, S =°o (6)

Z

OHs+c-

r

On the rest part oftheplane (z=0) the components of the

iZoHJc

v

fieldarecontinuous. Here

Z Ox

inHC

= -o e

e0

;mZ()llK)d

(8)

where

e+,

(K) and h1+z(K) are the images of the normal to

the disk field components in the spectral domain. They are the unknown functions to be found. Thenthe tangentialtothe disk componentscan be written as follows:

°°m9 °°lYK

(KP)-/(K)r+

(K)) K 9

Z e'm9*

|e±'

(K)4Hm(KP)

z(

dlc

(10)

~~c

~~~~kae+,

(K)

3 Basic equations

Weexpressthe scatteredfield components normal to the disk in terms of the azimuth-angle Fourier series and the radial

inverseintegral Hankeltransform:

E- = E elm"

e±lY()e±m

z(IC)Jml(icp)icdlc,

(7)

m=-oc 0

Hm

(KvP)

= mJIm(IKp)/(,p) IJl (IKo)

(1

1)

is the kernelmatrix function of the vector Hankel transform, p =rI

a,;

=z I a andy(K)=

C(ka)2

K2.

Onsubstituting (9) and (10) to (2) and continuity conditions outside the disk,we, ingeneral case, obtain the set of coupled DIEs for the four unknown functions in the spectral domain:

(O)~

_Ko

K p)(i/(C)(l(rIc)+

e';'(

*')0,°o (K)) +2Rikaum(K)d

o -ka

(v,

(K)+e ( IV0' (K))-2Ry(K)v (K))

(Kp)

ikaj

u

jKK=

( )

JHm

(Kp)

dlK = 0 K y-)(K)vZ

(K))

yO i0

V,i,j(K)(V,

(K)

+

eP

V4'0

'

(K))

+2Sikav,

(K)

fHm

(Kp)

ka

(u

(K)+

e (

)

°

IUrn'

(K))

+2S

I(K)u+

dl

Ic(

ikavm

(K)

(° JHm(K) dl I

oYY(K)0uZ+(K))

O

Here u+(K) X v-m(K) are linear combination of original

unknownimage function:

um~(K) =(emZ (K)+_ em,z(K))12(14

(14)

vm(K)=(hm1z(K) hm7z(K))12

The obtained set of DIEs can be written in the following abstractform: JHm(Kp)*SI(ka,K,p)*X(K)dK 0 = |Hm(Kp)*S2(ka, Kc, p)*Xo(Kc) dlK 0 (15)

where S12(ka,K,

p)

are singular diagonal matrix-functions

oftwo variables with the singular point at infinity (K=

o0),

X(K) is the unknown vector-function, XO(K) is the given vector-function. Note that the singularity of S2(ka,K,p)

leadstothelimitation ofX0(K) intermsof the allowedclass

p<l, (12) p>1, p<l, (13) p>1, c/co>> I

of functions. To solve (15), we use themethod ofanalytical

regularization [4]. It consists of the following two parts: First, we split the function

SI

(ka,K,p) into the singular and regular parts:

(16) S1(ka,K,p)=Z (ka,K,p) + P1(ka,K,p)

Then(15)canbe rewritten as follows:

0

|Hm

(Kvp)

* -1(ka,Kn,p)*

X(Kc)

dlc=

F(p,

X(...))

0

where theright-handpartis

F(p,

X(.))

f

Hm

(Kp).

(-PI

(ka,

K,

p)

*

X(K)

+

0

+S2 (ka,Kc,p)*X0

(Kc))

dlK

The solution of "canonical" IE is sought in the following

form:

X(2)

=

Km

(2,

p)*

F(p, X(...)) dp

(19)

0

Now we substitute expression (18) into (19) and reduce the

coupled DIEs (15) to the following Fredholm IE of the

second kind:

Xm(2)-JiW(K)Xm

(K)Kl)

(K,A)dK+ika 2m

AmIi)A-m

00 00

X(i)

+

fKm

(,

p)

fHm

(Kp).

P1

(ka,

K,

p)* X(K)

dK

dp

0 0

00 00

=

Km

(i,

p)

fHm

(Kp)

*

S2

(ka,

K,

p)

*

Xo(K)

dlK

dp

0 0

(20)

Extraction of the singular part of integral operators in (12),

(13) leads to two different-type DIEs in the case of PEC (R= 0 and S=ox) and dielectric or ER disks. Finding the (17) solutions in both cases with (m.0) is based on decupling

(17)

theDIEsby the integrationin p and introduction of four (in

general case)constantsof integration. This procedure leads to

theDIEs,whichcanbecast tothe following short form:

(p<1), (18) ₍₂₁₎ (p>1), {KU

U(-(K)-

f(K)}Jm (Kp)dK0= O |um(K)Jm(Kcp)dl =O 0

Such DIEs can be solved analytically by using the Abel

integral transform and inverse Hankel integral transform in

the case of 8=+1 orjust inverse Hankel integraltransform

in the case of 8 =0 . Additional equations for finding the constants ofintegration follow from the conditions of local integrabilityofpowernearthe diskedge.

As an example, we show the final integral equation of the

Fredholm second kind (m . 0) in case when the diffraction

by thePECdisk is considered.

-1/2(A+

1)-'

m+3 (2)+

+i2m!Bm 1 (2+1) Jm 1/2(2) J

)i

K l(K)(K+1)2elY(K)|;°

Ix'

(K)K()

(,A)dK 0

Ym(2)

JW(2)Ym(K)2Y

'()Kim

(K,2)dK+2mAJA2 2)7

-(2)(2++1)

J (2)+

0

00

+2mfkaIBW..(i)(K)d

+1kamAIi) 2mB_I

0~~~~~~~~~~~~~~~~~~

-JiW(K)Ym

(K)G(l

)'(K)dK

+ika2mAmIij

)-i

2m

BmI(

0

-

W(/C)ym

(1c)G(

1)

(1c)d1c+2m

AmI,3)

-ka2m

BmIm(4

Here Xm(A), Ym

(2)

arethe unknown functions, Am, Bm are

the unknown constants ofintegration,

Kj-l)

(K,A),

Gmcj)

(K)

arethekernelfunctions, w(K)istheweight function,

x5,

(A)

(22) i(Kc+I)2ei

Ymc)4'

(,

(2A)dK

(2) =iK

)'y(K)(K

+1)2

elr(K)I0 Ix

(K)G(1)

(K)dK

0

') =-|i(K+1)2

eiY(K)0I4

0y

(K)Gm1)(K)dK 0

and

yo

(2)

are the given functions determined by the

4

Numerical results

The Fredholm second kind nature of the derived equations guarantees the existence of solutions of (22) and the

convergence of numericalalgorithm basedonanyreasonable

discretization scheme. For example, it can be a simple

projection scheme with the basis functions taken as the

so-calledstepfunctions:

(>I) Ee[s(n-1),sn)

To verify the obtained equations we calculate the main

characteristics inthe case of the PEC disk with the incident

fieldbeing the field ofa horizontal electrical dipole located onthe disk axis andorientedalongthe line

(p

=0).Theyare

the far-field radiation patterns and the total radiated power.

Besides, we calculate truncation and discretization errors in

thecomputation of thetotal radiatedpower.

1,45- =z /a=1.600 0=~~~~~~~~~~~~~~~~~~~~~~~~~_1,40- 1l^11step=0.0250 1,35 trunc=ka +k5.0 . 1,30

1(3

1,25-1,20 (U1,15-o1,10 1,05-1,00 0,95 0,0 2 4 6 1

1'2 1'4 1'6 1'8

2'0

ka

Figure2a.Normalized radiationpowervs.thenormalized disk radius. ci) 0 cr_ 0 1,50 1,45 1,40 1,35 1,30 1,25 1,20 1,15 1,10 1,05 1,00 0,95 0,90 2 4 6 8 10 12 14 16 18 20 22 24 ka

Figure2b. Normalized radiationpower vs.the normalized

disk radius.

Figures 2a-2b show the normalized radiation power vs. the

frequency normalized by the disk radius.

Here;

is the normalized distance from the dipoletothe disk center, trunc is the truncation value of the domain ofintegration in (22) that is adapted to the frequency as ka+5. Figures 3a-3b show the relativecomputationerrorsgiven by:

Errcurr = Pcurr-Pprev / Pprev (24)

where Pprev is the normalized total power at the previous computation point,

Pcurr

is the normalized totalpoweratthe current point. Figure 4 shows the dependence of P vs.

;,

which is the normalized distance between disk and dipole. Figure 5 shows the far field radiation patterns for some

characteristic values ofparameters.

a kz LU (0a cIQ-0) 1-0,1 -0,01 a=dla=0.00 ;=zo/a=1.00 trunc=7.900 ka=2.9 , 312...5...I. ..5.... .0.,25 0,03 125 0,0625 0,1125 0,25 0...0,5 1 ...1 step

Figure3a.Discretizationerrorvs.thegridstep.

0,01563 0,00781 0,00391 S 0,00195 *

0,00098

!

LU _0,00049

0,00024

U 3(10 0 i* U 0,00012 0,0001|6 | 2 3 4 5 6 7 8 9 10 11 12 trunc

Figure 3b. The computationerror vs.the value of truncation of theintegration domain.

0 0= 0 0 CD _ CD -T_CD m 2r 1,8 -1,6 -1,4 -1,2 - 1,0-0,8 -0,6 -0,4 -0,2 -8=dla=0.000 step=0.0250 ka=2.9 trunc=7.9

1 g~~~~e-~

/

2 3 4 5 6 7 8 =zola 0,0-i

Figure4.Normalized radiationpower vs. the distance from the disktothedipole.

1,0 -0,8 -0,6- 300 0,4- 0,2-0,0-270 0,2- 0,4-0,6- 240 0,8 -1,0- 210 180 8=dla = 0.00 30 ;=z0/a= 1.00 step=0.025 ka=1.50 60 90 120 -- Total, v=00 - Total,9=900 150 - - Incident,C=0° Incident, 9=90° Figure 5b. Farfield radiation pattern ka=1.5.

10° -0,8 0,6 0,4 -0,2 0,0 270 0,2 0,4 0,6 0,8 1,0- 210 180 5=dla= 0.00 30 ;=zo/a 1.00 step=0.025 ka=1.00 60 90 120 -- Total C=00 - Total, C=900 150 - Incident, C= 0 Incident, (9=900

Figure5a. Farfield radiationpattern for ka=1.

1,0 -0,8 -0,6- 300 0,4- 0,2-0,0-270 0,2- 0,4-0,6- 240 0 0 ,8 ---1,O 210 150 180

Figure5c. Farfield radiationpattern ka==

S=dla=0.00 ;=zo/a=1.00 step=0.025 ka = 2.9 60 !90 120 -- Total,v=0° - Total, 9=900 -- Incident,9=00 Incident,9=900 2.9.

5

Acknowledgements

This work was supported by the National Academy of Sciences of Ukraine viaprojectPORIG #36/07H, the Turkish Council for Research in Science and Technology and the Royal Society,UKviajoint projects.

imperfect thin screens", J. Telecommunications and InformationTechnology,Warsaw: NITPress,2001,no3,

pp. 72-79.

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