em scattering 6 new - sharif
TRANSCRIPT
Electromagnetic scattering
Graduate CourseElectrical Engineering (Communications)1st Semester, 1388-1389Sharif University of Technology
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
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
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
�
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
� � � �� � � �
� � � � �� �� ��
� � �� �� � �� �
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 ��
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� �� ��
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��
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
�
� � � �
�
� ��
�
�
��
� �
� �� � � �
��
�
� �
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
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� � � �
� � � � � �
��
� ��
� �� �
� �� � �
� ��
� �� �
�
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� �
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� �� � �� �� �
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� �� � � �� �� �
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� �
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
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.
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.
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
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
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
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
� �
�� ��
� � �
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)
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
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 �
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
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
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� � � � ���� � �
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� � �� �
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� �
�
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
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
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
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
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
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� �
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
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
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
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
�� � �
��
�
�
� �
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
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
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
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
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 �
���� �
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
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
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
�� �
�� � �
�
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
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
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 ��
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� � � � ���� � �
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
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
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
��
��
�� �
�� �
� � � �
�� �� �� � ��� �� �
�� � �
�
�� �� � �� �� �� �� �
���
�
���
�
���
�
���
�
�
�
�
� �
�
� �
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
�� �� �� ��� �� �
�
�
�
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
�� �
�� � �
�
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
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
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
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
�
�� �� �� �
�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������