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Electromagnetic scattering Graduate Course Electrical Engineering (Communications) 1 st Semester, 1388-1389 Sharif University of Technology

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Page 1: EM scattering 6 new - Sharif

Electromagnetic scattering

Graduate CourseElectrical Engineering (Communications)1st Semester, 1388-1389Sharif University of Technology

Page 2: EM scattering 6 new - Sharif

Scattering from spherical objects 2

Contents of lecture 6

� Contents of lecture 6:

• Scattering from spherical objects

• Scalar waves in spherical coordinates

� Spherical harmonics

� Spherical Bessel functions

• Vector wave equation

• Far field behavior of solutions

• Expansion of a plane wave

• Scattering by a perfectly conducting sphere

Page 3: EM scattering 6 new - Sharif

Scattering from spherical objects 3

Introduction

� The previously analyzed canonical problems were exactly solvable.

� But they were all related to objects which were infinitely extended in one (cylinder, wedge) or two (layered media) directions. (Although approximations can be made for finite structures.)

� A conducting or dielectric sphere is one of the (few) problems which are exactly solvable and involve a finite object

ik

Page 4: EM scattering 6 new - Sharif

Scattering from spherical objects 4

Introduction

� Since the system has spherical symmetry, we analyze the problem in terms of the ‘natural’ solutions of the wave equation in spherical coordinates

� We first consider the scalar wave equation in a homogeneous medium x

y

z

�

r

� �2 2 0k �� � �

22 2

2 2 2 2 2

1 1 1sin 0

sin sinr k

r r r r r

� � �� �

� � � � �� � � � �� � � �� � � �� � � �� � � � �� � � �

Wave equation in spherical coordinates

�

Page 5: EM scattering 6 new - Sharif

Scattering from spherical objects 5

Scalar waves in spherical coordinates

� This is a classic problem. Let us represent the solution as

� �( , , ) ( ) ,r f r Y� � � � ��

� �2 2 2( )( ) 0

d df rr k r f r

dr dr�� � � � �� �

� �

: constant to be specified�

� The functions should satisfy the equations

� � � � � �2

2 2

, ,1 1sin ,

sin sin

Y YY

� � � �� � � �

� � � � �� �� ��

� � �� �� � �� �

Page 6: EM scattering 6 new - Sharif

Scattering from spherical objects 6

Spherical harmonics

� The solutions of the first problem (eigenfunctions) are called the

spherical harmonics

� Since we are going to deal with fields which are periodic

functions of the angle �, we look for solutions of the type

� � � � � �, expY g jm� � � ��

2

2

1sin

sin sin

d dg mg g

d d� �

� � � �� � � � �� �� �

2 22

2 2(1 ) 2 0

1

d g dg mu u g g

du du u�� � � � �

�cosu ��

Page 7: EM scattering 6 new - Sharif

Scattering from spherical objects 7

Spherical harmonics

� This is the differential equation for Legendre functions

� Requiring analytic properties, solutions are only possible when

� � � �cosmg P� �� �

22 / 2( 1) ( 1)

( ) (1 )2 !

m mm m

m

d uP u u

du

�

�

� �� �

� �

� � ��

� �1� � �� � : nonnegative integer0,1, 2,�� �

� Corresponding solutions: associated Legendre functions of the

first kind (2nd kind is not analytic when

m� � �� �

0, or 1u� �� ��

Page 8: EM scattering 6 new - Sharif

Scattering from spherical objects 8

Spherical harmonics

� Note that m=0 corresponds to the so called Legendre

polynomials. Even values of m result in polynomials in general,

but odd values do not.

� Here are some examples:

00 (cos ) 1P � � 0

1 (cos ) cosP � ��

11

1(cos ) sin

2P � �� �

11 (cos ) sinP � �� �

12

1(cos ) sin cos

2P � � �� �

2 22

1(cos ) sin

8P � �� �

� �0 22

1(cos ) 3cos 1

2P � �� �

12 (cos ) 3sin cosP � � �� �

2 22 (cos ) 3sinP � ��

0�� 1�� 2��

Page 9: EM scattering 6 new - Sharif

Scattering from spherical objects 9

Spherical harmonics

� Note that for positive and negative values of m these functions

are not independent from each other

� �� �

!( ) ( 1) ( )

!m m mm

P u P um

� �� �

�� �

�

�

� Orthogonal properties of Legendre functions

1

1 0

( ) ( ) (cos ) (cos )sin

2 ( )!

2 1 ( )!

m m m mP u P u du P P d

m

m

�

� � � �

�

� ��

�

�

��

� �

� �� � � �

��

�

� �

Page 10: EM scattering 6 new - Sharif

Scattering from spherical objects 10

Spherical harmonics

� Summarizing: � � � � � �, cos expm mY P jm� � � ��� � m� � �� �

� Visualization: here is the amplitude of these functions

� Negative m’s have not been shown as they have the same distribution as positive m’s

Page 11: EM scattering 6 new - Sharif

Scattering from spherical objects 11

Spherical harmonics

� Orthogonality of spherical harmonics:

2

4 0 0

( , ) ( , ) ( , ) ( , ) sin

4( 1)

2 1

m m m m

mmm

Y Y d Y Y d d� �

�

� � � � � � � � � � �

�� �

� �� �� �

� �

� �

� ��

� � �� � � �

���

� Completeness of spherical harmonics:

0

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

( ) (cos cos )

m m m

m

Y Y� � � �

� � � � � �

��

� ��

� �� �

� �� � �

� ��

� �� �

�

Page 12: EM scattering 6 new - Sharif

Scattering from spherical objects 12

Spherical Bessel functions

� Now, consider the equation for radial distance

� �2 2 2 1 0d df

r k r fdr dr� � � �� � � �� � � �� �

� �

� Solution: linear combinations of the spherical Bessel functions

1

2

( ) ( )2

j z J zz

��

���

1

2

( ) ( )2

y z Y zz

��

���

( ) and ( )j kr y kr� �

Page 13: EM scattering 6 new - Sharif

Scattering from spherical objects 13

Spherical Bessel functions

� Examples:

0

sin( )

zj z

z�

1 2

cos sin( )

z zj z

z z� � �

2 3 2

3 1 3( ) sin cosj z z z

z z z� �� � �� �� �

Page 14: EM scattering 6 new - Sharif

Scattering from spherical objects 14

Spherical Bessel functions

� Examples:

0

cos( )

zy z

z� �

1 2

cos sin( )

z zy z

z z� � �

2 3 2

3 1 3( ) cos siny z z z

z z z� �� � � �� �� �

Page 15: EM scattering 6 new - Sharif

Scattering from spherical objects 15

Spherical Bessel functions

� � � � � �( , , ) cos expmr f r P jm� � � � �� � �

� Overall solution

� �,mY � ��

� The function f may also be represented as a linear combination

of the spherical Hankel functions:

(1) (1)1/ 2( ) ( ) ( ) ( )

2h z j z jy z H z

z

��� � �� � � �

(2) (2)1/ 2( ) ( ) ( ) ( )

2h z j z jy z H z

z

��� � �� � � �

(1) (2)( ) and ( )h kr h kr� �

Page 16: EM scattering 6 new - Sharif

Scattering from spherical objects 16

Vector wave equation

� Aim: solutions of the vector wave equation

� Assume that �(r) is a solution of the ‘scalar’ wave equation; also

consider a known vector field a(r) whose curl is zero

� Then solutions of the vector wave equation are

� �1( ) ( ) ( )

k�� ��M r r a r

1( ) ( )

k� ��N r M r

� �2 0� ��� � � � � � �� �� �� �a a

� Provided that

� � 2 0k�� �� � �E E

0�� �a

Page 17: EM scattering 6 new - Sharif

Scattering from spherical objects 17

Vector wave equation

� It is customary to choose which satisfies both

equations (why?). It then follows that

� � �a r r

� �( ) ( ) ( )� ���� �� �M r r r r r

1( ) ( )

k� ��N r M r

Unlike cylindrical functions, the factor 1/k has not been included here in order to obtain dimensionless quantities.

� � � � � �, ( , , ) ,mm r f r Y� � � � � ��� �� � �r

We have used –m instead of m to preserve the convention used for the cylindrical case.

Page 18: EM scattering 6 new - Sharif

Scattering from spherical objects 18

Vector wave equation

� These two vector fields (M and N) are linearly independent from

each other as for the cylindrical vector functions.

� They satisfy the wave equation and have zero divergence.

� Similar to the cylindrical case, very solution of the vector wave

equation in a homogeneous medium which has zero divergence

can be written as a combination of these vectors for different

solutions of the scalar wave equation.

Page 19: EM scattering 6 new - Sharif

Scattering from spherical objects 19

Vector wave equation

� We next use spherical coordinates

1 1ˆ ˆˆsinr r r

� � ��

� � �� � �

� � � �� � �r θ

, ,, ,

, ,

1 ˆ ˆsin

ˆ ˆsin

m mm m

m mjm

� ��

� � �� �� �

� ��� � � �

� ��

� ��

� �� �

� �

M r θ

θ

θ

ˆr

�

� r

Page 20: EM scattering 6 new - Sharif

Scattering from spherical objects 20

Vector wave equation

� The 2nd solution

� �

2, ,

, ,

,,

1 1ˆ sin

sin sin

1ˆ ˆsin

m mm m

mm

m

k kr

jmr r

kr r kr r

� ��

� � � �

��

� �

� ��� ��� �� � � �� �� �� �� �� �

� � �� �� �� � �� �� � �� �� �

� �� �

��

N M r

θ

� Leads to

� � � �, ,, ,

1 1ˆ ˆˆsin

m mm m

jmr r

kr kr r kr r

� ��

� �� � � �� �� �

� � � �� �� � �� �� �

� �� �

� �N r θ

� � � � � � � � � �use �� � � �� � �� � �� � ��a b a b b a b a a b

Page 21: EM scattering 6 new - Sharif

Scattering from spherical objects 21

Vector wave equation

� Note that if the electric field is given by then the magnetic

field is necessarily given by

� Similarly if the electric field is given by then the magnetic

field is necessarily given by

� Note that also in this case the electric and magnetic fields are

normal to each other for each mode (why?)

� �, ,, , , , 0

sinm m

m m m m

jmr r

kr r r

� �� �

� � ��� � �� �� �

� � � �� �� �� � � �� �� �

� �� � � �M N

,m�M

,( / ) mj � �N

,m�N

,( / ) mj � �M

0� � �E H

Page 22: EM scattering 6 new - Sharif

Scattering from spherical objects 22

Far field behavior of solutions

� For later use, let us consider the vector functions when

� �(2) (2)1/ 2( ) ( )

2f r h kr H kr

kr

��� �� � �

� Consider the functions at a large distance from the center of the

sphere. By this we mean that

� Asymptotic relation for the spherical Hankel function

kr � �

� � � �1

(2) (2)1/ 2 ( ) exp

2

jh kr H kr jkr

kr kr

� �

�� ��

� � �

Page 23: EM scattering 6 new - Sharif

Scattering from spherical objects 23

Far field behavior of solutions

� The fields have the asymptotic behavior:

� �1,

exp ˆ ˆsin

m m

m

jkr jmY Yj

kr � �

� �� � � ��

�� ��� �M θ� � �� �

� �,

exp ˆ ˆsin

mm

m

jkr Y jmj Y

kr � �

��� � ��

�� ��� �N θ� �� ��

� For each mode the resulting Poynting vector is along the radial

direction: these are spherical waves (in far field)

Page 24: EM scattering 6 new - Sharif

Scattering from spherical objects 24

Far field behavior of solutions

� Note that for the particular case of m=0 we have linearly

polarized electric fields in both cases

� � 01

,0

exp ˆjkr Yj

kr �� � �

��

� �� �M

� � 0

,0

exp ˆjkr Yj

kr �� �

�� �

� �N θ

� But, in general, the fields have more complicated polarizations.

They are elliptically polarized on the plane described by the unit

vectors and . Hence, it is wrong to talk of horizontal and

vertical modes like for a cylinder.

ˆ θ

Page 25: EM scattering 6 new - Sharif

Scattering from spherical objects 25

Far field behavior of solutions

� On these planes the elliptic polarization is determined by the

ratio between

θ

ˆr

H

E

Wave front

�

� �cos

sin

mP ��

�� � �cosmdP

d

��

��

� It does not depend on the

angle �

Page 26: EM scattering 6 new - Sharif

Scattering from spherical objects 26

Expansion of a plane wave

� We have analyzed the ‘natural’ solutions to the vector wave

equation in spherical coordinates.

� These solutions behave as TEM waves at large distance

� But the actual problem we are interested is not the scattering of

these waves, but the scattering of a simple plane wave such as

� �0( ) expi i ij� � �E r E k r

i k�k ik

Page 27: EM scattering 6 new - Sharif

Scattering from spherical objects 27

Expansion of a plane wave

� As for cylindrical waves we have to find the appropriate plane

wave expansion for the incident wave vector

� � � �, , ,, , sin cos ,sin sin ,cosi i x i y i z i i i i ik k k k � � � � �� �k

x

y

z

i�i�

ik

� Again, 1st consider a scalar plane wave which

can be expanded as

0

exp( ) ( ) ( , ) ( , )m mi m i i

m

j w j kr Y Y� � � ��

�

� ��

� � � � ��

� � � �� �

k r

( 1) ( ) (2 1)mmw j� � � ��� �

� �sin cos ,sin sin ,cosr � � � � ��r

�

�

r

Page 28: EM scattering 6 new - Sharif

Scattering from spherical objects 28

Expansion of a plane wave

� Now, look at the following;

� � ,0

exp( ) ( , , ) ( , )J mi m m i i

m

j w r Y� � � ��

� ��

�� � � �� ��

� � �� �

r k r M

� �� �

exp( ) exp( )

exp( ) exp( )i i

i i i

i k i k i i

j j j

j j

�� � � � � � � � �

�� � � � �� � � �

r k r k r k r

k k r k k r

� We have defined

� � , ,, ,

ˆ ˆsin

J Jm mJ J

m m

jm� ��

� ��

� �� � ��

M r θ � �� �

� � � �, ( , , ) ,J mm r j kr Y� � � � ���� � �

Page 29: EM scattering 6 new - Sharif

Scattering from spherical objects 29

Expansion of a plane wave

� Partial integration yields

� �, ,

4

( , , ) , exp( )4

lJ

m m i i i i

jr j d

�

� � � ��

� � � ��� �M C k r

1ˆ ˆsinii k i i

i i i� � �� �

�� � �� �

k θ

� It follows from the orthogonality of spherical harmonics that

,

4

( , , ) ( , ) exp( )4 i

lJ m

m i i i k i i

jr Y j d

�

� � � ��

�� �� � � ��� �M k k r

sini i i id d d� � �� �

Page 30: EM scattering 6 new - Sharif

Scattering from spherical objects 30

Expansion of a plane wave

� Note that

ˆ ˆ ˆ0 cos i i i i

i i

�� �� �

� �� �

θ

� �,

( , ) ( , )1 ˆ ˆ, sin

( , ) ( , )i i

m mi i i i

m i i i ii i i

m mk i i i k i i i

Y Y

Y Y

� � � �� �

� � �

� � � �

� �

� �

� �� �

� �

� �� � � � � �� �

C θ

k k

� ��

� �

� �,

1 ˆ, ( , )sin

1 ˆ ( , ) sinsin

mm i i i i i

i i

mi i i i

i i

Y

Y

� � � �� �

� � �� �

�

�

� � �� � ��� � �� � ��

C θ� �

�

Page 31: EM scattering 6 new - Sharif

Scattering from spherical objects 31

Expansion of a plane wave

� It then follows that

� �

, ,

1

,

4

1( , , ) ( , , )

, exp( )4

J Jm m

m i i i i

r rk

jj d

�

� � � �

� ��

�

� ��

� � � ��

� �

�

�

N M

B k r

� � � �, ,

ˆ, , ( , )

( , ) ( , )1ˆ ˆ sin

i

mm i i i m i i k i i

m mi i i i

i ii i i

k Y

Y Y

� � � � � �

� � � �� � �

�

� �

� � � �

� �� �

� �

� � �

� �

B k C

θ

Page 32: EM scattering 6 new - Sharif

Scattering from spherical objects 32

Expansion of a plane wave

� The vectors B and C are called

vector spherical harmonics

� They are functions defined on a

sphere in the k-space with the radius

k, i.e., they are functions of

� At each point on this sphere they are

normal to wave vector, hence are

tangential to sphere

� And they are normal to each other for

each mode

,i i� �

iθ

i

ik

i�

i�

K-space

Page 33: EM scattering 6 new - Sharif

Scattering from spherical objects 33

Expansion of a plane wave

� Now, we introduce a new vector spherical harmonic in the k-

space. Consider the radial vector function

� �,ˆ, ( , )m

m i i i i iY� � � ���� �A k

� From the scalar plane wave expansion it follows that

1

, ,

4

( )( , , ) ( , ) exp( )

4J

m m i i i i

jr j d

�

� � � ��

�

� � � ���

� �L A k r

, ,

1J Jm mk

�� �� �L

Page 34: EM scattering 6 new - Sharif

Scattering from spherical objects 34

Expansion of a plane wave

� The vector spherical harmonics constitute an orthogonal set on

the space of vector functions on the k-sphere

� � � � � �, ,

4

4 1, , ( 1)

2 1m

m i i m i i i mmd�

�� � � � � �� � � ��

�� � � �

�� � � ��

� �

�C C

� � � � � �, ,

4

4 1, , ( 1)

2 1m

m i i m i i i mmd�

�� � � � � �� � � ��

�� � � �

�� � � ��

� �

�B B

� � � �, ,

4

4, , ( 1)

2 1m

m i i m i i i mmd�

�� � � � � �� � � ��� � � �

�� � � �� �A A

Page 35: EM scattering 6 new - Sharif

Scattering from spherical objects 35

Expansion of a plane wave

� They are also mutually orthogonal

� � � �, ,

4

, , 0m i i m i i id�

� � � �� ��� � �� � �B C

� � � �, ,

4

, , 0m i i m i i id�

� � � �� ��� � �� � �B A

� � � �, ,

4

, , 0m i i m i i id�

� � � �� ��� � �� � �A C

Page 36: EM scattering 6 new - Sharif

Scattering from spherical objects 36

Expansion of a plane wave

� Consider a vector field defined in the k-space on a sphere with

the radius k, i.e., a vector function of the angles

� Then the vector field can be expanded in the spherical vector

functions which are also defined on this sphere

� In other words: spherical vectors are complete in this space

,i i� �

Page 37: EM scattering 6 new - Sharif

Scattering from spherical objects 37

Expansion of a plane wave

� �0( ) expi i ij� � �E r E k r

� Consider the incident field

� Now, let us keep constant while

considering the direction of k as a variable

(of course this field does not represent a

true plane wave any more)

� The incident wave is now a vector function

defined on the k-sphere since

0iE

� �sin sin cos cos cosi i i ikr � � � � � �� � � �� �� �k r

iθ

i

ik

i�

i�

K-sphere

Page 38: EM scattering 6 new - Sharif

Scattering from spherical objects 38

Expansion of a plane wave

� Expansion in terms of vector spherical harmonics:

� �� � � � � � � � � � � �

0

, , ,,

exp

, , ,

i i

m m i i m m i i m m i im

j

a b c� � � � � �

� � �

� �� �� �� � � � � � ��

E k r

r A r B r C

� � � � � �0,

4

2 1( 1) , exp

4m

m i m i i i ia j d�

� �� �

�� �� � � � � �� �� � �� �

�r E A k r

� � � � � � � �0,

4

2 1( 1) , exp

4 1m

m i m i i i ib j d�

� �� �

� ��� � � � � �� ��� �

�� �

�

� �r E B k r

� � � � � � � �0,

4

2 1( 1) , exp

4 1m

m i m i i i ic j d�

� �� �

� ��� � � � � �� ��� �

�� �

�

� �r E C k r

Page 39: EM scattering 6 new - Sharif

Scattering from spherical objects 39

Expansion of a plane wave

� Result:

� � � �1 0,( 1) ( ) 2 1 ( , , )m J

m i ma j r � ���� �� � � � �� �

�� ��r E L

� � � �1 0

,

2 1( 1) ( ) ( , , )

1m J

m i mb j r � ���

� ��� � � �� ��� �

�� �

�

� �r E N

� � � �0

,

2 1( 1) ( ) ( , , )

1m J

m i mc j r � ��

� ��� � � �� ��� �

�� �

�

� �r E M

Page 40: EM scattering 6 new - Sharif

Scattering from spherical objects 40

Expansion of a plane wave

� It follows that

� �( ) 1, ( 1) ( ) 2 1a mmv j �� � � ��� �

� �( ) 1,

2 1( 1) ( )

1b mmv j � �� � �

��

�

�

� �

� � � �

� �� �

0 0 ( ), , ,

,

( ), , ,

( ), , ,

exp ( , , ) ,

( , , ) ,

( , , ) ,

a Ji i i m m m i i

m

b Jm m m i i

c Jm m m i i

j v r

v r

v r

� � � �

� � � �

� � � �

�

�

�

�� � � � �

�

�� �

� � � ��

� � �

� � �

E k r E L A

M B

N C

� �( ),

2 1( 1) ( )

1c mmv j

�� � �

��

�

�

� �

Page 41: EM scattering 6 new - Sharif

Scattering from spherical objects 41

Expansion of a plane wave

� Since this holds for any constant vector we must have

� � � �

� � � �

( ), , ,

,

( ) ( ), , , , , ,

exp ( , , ) ,

( , , ) , ( , , ) ,

a Ji m m m i i

m

b J c Jm m m i i m m m i i

j v r

v r v r

� � � �

� � � � � � � �

�

� �

�� � � ��

�� �

� � � ��

� � � � � �

I k r L A

M B N C

0iE

� � � �

� �� �

0 ( ) 0, , ,

,

( ) 0, , ,

( ) 0, , ,

exp ( , , ) ,

( , , ) ,

( , , ) ,

a Ji i m m m i i i

m

b Jm m m i i i

c Jm m m i i i

j v r

v r

v r

� � � �

� � � �

� � � �

�

�

�

�� � � ��

� �

�� � �

� � � ��

� � �

� � �

E k r L A E

M B E

N C E

Page 42: EM scattering 6 new - Sharif

Scattering from spherical objects 42

Expansion of a plane wave

� Now, for a true plane wave the electric field has no component

along the wave vector so that

� � � �

� �

0 ( ) 0, , ,

,

( ) 0, , ,

exp ( , , ) ,

( , , ) ,

b Ji i m m m i i i

m

c Jm m m i i i

j v r

v r

� � � �

� � � �

�

�

�� � � ��

�� � �

� � � ��

� � �

E k r M B E

N C E

� �,

( , ) ( , )1ˆ ˆ, sin

m mi i i i

m i i i ii i i

Y Y� � � �� �

� � ��

� �� �

� �� �

�B θ

� �,

( , ) ( , )1 ˆ ˆ, sin

m mi i i i

m i i i ii i i

Y Y� � � �� �

� � ��

� �� �

� �� �

�C θ

Page 43: EM scattering 6 new - Sharif

Scattering from spherical objects 43

Expansion of a plane wave

� Example: wave propagating along +z-

axis:

� This case is general enough as the

sphere is symmetric

� Specify the angle �i before taking the

limit, otherwise the direction of unit

vectors cannot be determined

� Any angle will do, again because the

sphere is symmetric, we can later take

0i� �

iθ

i

ik

i�

0i� �

ˆ ˆ ˆ0 , i i i� � � � �θ x y

x

y

Page 44: EM scattering 6 new - Sharif

Scattering from spherical objects 44

Expansion of a plane wave

� We then use the relationships (m>0)

� � � � � �,1 1

( )1 ˆ ˆ exp1 i i i

u

dP uj j

du�

�

� � ��

�� � �B θ

� � � �, 1 1

( ) ˆ ˆ expi i iu

dP uj j

du��

�

� � ���B θ

� It can be shown that all B’s become zero except

� � / 22 ( )( ) ( 1) 1

mmm m

m

d P uP u u

du� � � �

�

� �� �

� �� � � � / 22! ! ( )

( ) ( 1) ( ) 1! !

mmm m m

m

m m d P uP u P u u

m m du� � �

� � � �� �

�� �

� �

� �

Left polarized

Right polarized

Page 45: EM scattering 6 new - Sharif

Scattering from spherical objects 45

Expansion of a plane wave

� Similarly, all C’s become zero except

� � � �, 1 1

( ) ˆ ˆ expi i iu

dP uj j j

du��

�

� � ���C θ

� � � � � �,1 1

( ) ˆ ˆ exp1 i i i

u

dP ujj j

du�

�

� � � ��

�� � �C θ

Left polarized

Right polarized

� All vector functions are zero for 0��

� Also, note that � �1

1( )

2u

dP u

du �

���� �

Page 46: EM scattering 6 new - Sharif

Scattering from spherical objects 46

Expansion of a plane wave

� Collecting the results:

� � � �, 1 ˆ ˆ1 2

jj� � � � �� � �C x y

� �,1 ˆ ˆ2

jj� � ��C x y

Left polarized

Right polarized

� �0i� �

� � � �, 1

1ˆ ˆ1

2j� � � � �� � �B x y

Left polarized

Right polarized

� �,1

1ˆ ˆ

2j� ��B x y

Page 47: EM scattering 6 new - Sharif

Scattering from spherical objects 47

Expansion of a plane wave

� Returning to the plane wave

� �

� � � �

� � � �� �

0

0 1,1 ,1

1

0 1, 1 , 1

1

exp

1ˆ ˆ ( ) 2 1 ( , , ) ( , , )

2

2 11ˆ ˆ ( ) ( , , ) ( , , )

2 1

i

J Ji

J Ji

jkz

j j r r

j j r r

� � � �

� � � �

��

�

��

� ��

� �

� �� � � � � �� �

�� �� � � �� ��

�

�

�� �

�

�� �

�

�

�

� �

E

x y E M N

x y E M N

Page 48: EM scattering 6 new - Sharif

Scattering from spherical objects 48

Expansion of a plane wave

� More specifically

� �� � � � � �

� �� � � � � � � �

1,1

ˆ ˆexp cos1 sin

ˆ ˆexp cos cos1

J j kr jj P

j krj j

� �� �

� � � � �

��� �� � �� �� �� �

� �� � �� ��

�� �

�� �

� �

� �

M θ

θ

� � � � � �

� � � � � � � �

(2) 1, 1

(2)

ˆ ˆexp cossin

ˆ ˆexp cos cos

J j dj kr j P

d

j kr j j

� �� �

� � � � �

�� �� � �� �

� �� � �� �

� � �

� � �

M θ

θ

� � � �1 coscos

sin

P �� �

�� � �

� � � � �1 coscos

dP

d

�� �

�� � �

�

Page 49: EM scattering 6 new - Sharif

Scattering from spherical objects 49

Expansion of a plane wave

� � � � � �

� � � � � �

, 1

1ˆexp

ˆ ˆ cos cos

J j j krkr

d rj krj

krdr

�

� � � �

�

��� ��

��� � �� � � �� �� ���

� �

�

� �

� �N r

θ

� �� �

� � � �

� � � � � �

,1

exp 1ˆ

1

ˆ ˆ cos cos

J jj kr

kr

d rj krj

krdr

�

� � � �

� ��� � ��� �

�� � �� � � �� �� ���

� �

�

� �

� �

� �N r

θ

Page 50: EM scattering 6 new - Sharif

Scattering from spherical objects 50

Scattering by a perfectly conducting sphere

� We now consider the scattering of a plane wave by a perfectly

conducting sphere

� When the incident wave hits the

cylinder, surface currents (and

charges) are induced

� These currents create the ‘scattered’

field. At any point, the total electric

field is

� �0( ) expi i jkz� �E r E

( ) ( ) ( )i s� �E r E r E r ik

Page 51: EM scattering 6 new - Sharif

Scattering from spherical objects 51

Scattering by a perfectly conducting sphere

� We saw how the incident plane wave can be represented in

terms of spherical vector solutions

� The scattered field (outside the sphere) can also be expanded

in terms of those solutions

� But: for the scattered field we should use vectors with the right

condition at the infinity

� We should use the spherical Hankel function of the 2nd kind for

these waves which satisfy the radiation condition (behave as

outgoing waves at infinity)

r ��

Page 52: EM scattering 6 new - Sharif

Scattering from spherical objects 52

Scattering by a perfectly conducting sphere

� Expansion of the scattered field:

� � , ,0

( , , ) ( , , )h hs m m m m

m

b r c r� � � ��

� ��

� �� �� �� ��

� � � �� �

E r M N

� � � � � �,, , ,

1 1 1ˆ ˆˆsin

hmh h h

m m m

jmr r

kr kr r kr r

�� �

� �� � � �� �� � � �� � �� �

�� � �

� �N r θ

,, ,

ˆ ˆsin

hmh h

m m

jm ��

� ��

� ���

� �M θ

� � � �(2), ( , , ) ,h mm r h kr Y� � � � ���� � �

Page 53: EM scattering 6 new - Sharif

Scattering from spherical objects 53

Scattering by a perfectly conducting sphere

� More specifically

� � � � � �(2),

ˆ ˆexp cossin

h mm

jmh kr jm P� �

� ���� �� � �� ��� �

� � �M θ

� �� � � � � � � �

,

(2) (2)

exp

1 1 ˆ ˆˆ cossin

hm

m

jm

jmh kr rh kr P

kr kr r

�

�� �

�

� �

�� �� �� �� ��� �� �� �� �� �� �

�

� � �

� �

N

r θ

� From matching at the surface of the sphere, it directly follows

that only the m = -1 and m = +1 terms can contribute to the

series for the scattered field

Page 54: EM scattering 6 new - Sharif

Scattering from spherical objects 54

Scattering by a perfectly conducting sphere

� It then follows that

� �� �

0 1, , 1 0 , 1 0

1

, 1 , 1 0 , 1 , 1 01

2 1ˆ ( ) ( , , ) ( , , )

1

ˆ ( , , ) ( , , )

J Ji

h h

E j R R

b R c R

� � � �

� � � �

��

� � ��

�

� � � ��

�� �� � � �� ��

� �� � �� �

�

�

�� �

�

� � � ��

�

� �r M N

r M N

� �0 1, ,1 0 ,1 0

1

,1 ,1 0 ,1 ,1 01

ˆ ( ) 2 1 ( , , ) ( , , )

ˆ ( , , ) ( , , )

J Ji

h h

E j R R

b R c R

� � � �

� � � �

��

��

�

�

� �� � � � �� �

� �� � �� �

�

�

�� �

�

� � � ��

�r M N

r M N

� �0 0,

1ˆ ˆ

2i iE j� � � �x y E � �0 0,

1ˆ ˆ

2i iE j� � � �x y E

Page 55: EM scattering 6 new - Sharif

Scattering from spherical objects 55

Scattering by a perfectly conducting sphere

� It can be shown

that M and N

functions do not

mix up

� Besides, different values of � do not

mix up

� � � �� �

� � � �

� �� �� �

� �� �

� �� �

� �

� �

00 1,1 , (2)

0

0 00 1,1 ,

(2)0 0

00 1, 1 , (2)

0

0 00 1, 1 ,

(2)0 0

( ) 2 1

( ) 2 1

2 1( )

1

2 1( )

1

i

i

i

i

j kRb E j

h kR

kR j kRc E j

kR h kR

j kRb E j

h kR

kR j kRc E j

kR h kR

��

��

�� �

�� �

� � � �

�� �� �� � ��� �� �

�� � �

�

�� �� � �� �� �� �� �

���

�

���

�

���

�

���

�

�

�

�

� �

�

� �

Page 56: EM scattering 6 new - Sharif

Scattering from spherical objects 56

Scattering by a perfectly conducting sphere

� Collecting the results, the far field behavior is given by

� � � � � �� �

� � � �

� � � �

� � � �

� �

1

0,

0,

0,

0,

exp 2 1

1

ˆ ˆcos cos exp( )

ˆ ˆ cos cos exp( )

ˆ ˆcos cos exp( )

ˆ cos

s

i

i

i

i

jkr

jkr

E j j

E j j

E j j

E

� � � � �

� � � � �

� � � � �

� �

�

�

�

�

�

�

� ��

�

� �� � �� �� �� � �� �

� �� � �� �

�

��

� � �

� � �

� � �

� �

�

� �E r

θ

θ

θ

θ � �ˆ cos exp( )j j� � �� � �� ��

� �� �

0(2)

0

j kR

h kR� � ��

�

� �

� �0 0

(2)0 0

kR j kR

kR h kR

�� �� �� ��� �� �

�

�

�

Page 57: EM scattering 6 new - Sharif

Scattering from spherical objects 57

Scattering by a perfectly conducting sphere

� Example:

� � � � � �� �

� � � ��� � � � �

01

exp 2 1

1

ˆ ˆ cos cos cos sin

ˆ ˆ cos cos cos sin

s

jkrE

jkr

� � � � � �

� � � � � �

�

�

� ��

�

� �� � �� �

� �� �� �

��

� � �

� � �

�

� �E r

θ

θ

0 0 00 , , 0

1ˆ

2i i iE E E E� �� � � �E x

� � � �1 coscos

sin

P �� �

�� � �

� � � � �1 coscos

dP

d

�� �

�� � �

�

Page 58: EM scattering 6 new - Sharif

Scattering from spherical objects 58

Scattering by a perfectly conducting sphere

� Components of the scattered far field

� � � � � �� � � � � �, 0

1

exp 2 1cos cos cos

1s

jkrE E

jkr� � � � � ��

�

� �� �� � ��� ��� � � � �

�

�

� �r

� � � � � �� � � � � �, 0

1

exp 2 1sin cos cos

1s

jkrE E

jkr� � � � � ��

�

� �� �� � � ��� ��� � � � �

�

�

� �r

Page 59: EM scattering 6 new - Sharif

Scattering from spherical objects 59

Scattering by a small conducting sphere

� To get some insight let us consider

the limit of a small sphere

0 1kR �

� �� �

� �� �

� �� � � �

2 1

0 0 02(2)

0 0 1.3.5 2 1 2 1

j kR j kR kRj j

h kR y kR

�

� � � �� �� �� �

�

� ��

� � � � �

� �

� �� �� � � �

2 10 0 0

2(2)

0 0

1

1.3.5 2 1 2 1

kR j kR kRj

kR h kR

��� � �� �� � � �� � �� �� � � �� �

��

�

�

�

� � � �

ik

0R

Page 60: EM scattering 6 new - Sharif

Scattering from spherical objects 60

Scattering by a small conducting sphere

� Keeping the lowest order terms:

� �3

1 03

jkR� � � �3

1 0

2

3

jkR� � �

� � � �11

1

coscos 1

sin

P �� �

�� � � � � � �1

11

coscos cos

dP

d

�� � �

�� � �

� � � � � �

� � � �

3

0 0

exp

2ˆ ˆ cos 1 2cos sin 2 cos

s

jkrE kR

kr

� � � �

�

� �� � �� �

�E r

θ

Page 61: EM scattering 6 new - Sharif

Scattering from spherical objects 61

Scattering by a small conducting sphere

� Consider the amplitude of the scattered electric field

ik

iE

sk

�

�

� � � � � �4 6

2 2 2 22 200 2

cos 1 2cos sin 2 cos4s

k RE

r� � � �� �� � �� ��E r

� Differential cross section

� � � �4 6

2 22 20 cos 1 2cos sin 2 cos4d

k R� � � � �� �� � �� ��

� Total cross section

4 60

4

10

3d

k Rd

�

�� �� �� �

Page 62: EM scattering 6 new - Sharif

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