scattering by a perfectly conducting infinite cylinderee.sharif.edu/~emscattering_ms/lecture...

Scattering from cylindrical objects 49

Scattering by a perfectly conducting infinite cylinder

� Remember that this is the full solution everywhere. We are

actually interested in the scattering in the far field limit.

� We again use the asymptotic relationship

� � � �(2)0

exp exp2( 1) exp

4

m m

m

j jk j jkjH k jC

k k

� ��

� �

� ��

� � �

� ��

� �

� We find that in the far field limit the fields are parallel to:

,ˆ( 1) TE

s ik � � � �E,,

, ˆ ˆ( 1) ii zTMs i

kkk

k k�

� ��

� �E z

Same as θ̂



� More specifically:

� � � �

� ��

, 0 , ,, 0

,

,

(2),

exp( )ˆ ˆˆ

exp

i i i zi zTMs i i

i

m i

im m i

k jC jk jk zk

k k k

J k Rjm

H k R

� �

�

�

�

�

�

� ��

��

� � ��

� �

��

E z E v

� ��

� �

00

,

(2),

exp( )ˆ ˆ

exp

zTEs i i

m i

im m i

jC jk jk z

k

J k Rjm

H k R

�

�

�

�

�

�

� ��

��

� ��

��

E E h



� The interesting point here is the behavior of far field as

function of the angle � , which is governed by the functions

� ��

� �,

(2),

expm i

im m i

J k Rjm

H k R

�

�

� ��

��

��

� Remember that

(2) (2)( 1) ( 1)m mm m m mH H J J� ��

� �� ,

(2),

expm i

im m i

J k Rjm

H k R

�

�

� ��

��

��

TE

TM


TE Scattering by a thin conducting cylinder

� To get some insight let us consider the limit of a thin cylinder

, 1ik R kR� � �

� For small arguments (sufficient to consider positive or zero m’s)

� � � �1

0 1( )2 ( 1)!

m

m m m m

m zJ z J z J z

z m

�

� ��

�� 0 2

zJ z� ��

� �(2)0

2jH z

z�� (2) (2) (2)

0 1 1

2 !( )

m

m m m m

m j mH z H z H z

z z��



� To the lowest order we have for the TE case

� ��

� �

20,0

,

, ,

ˆˆ( , , )

4

exp( ) 1 2cos

i i iTEs

i

i i z i

k RCz

k

jk jk z

�

�

�

��

�

� � �

��

� � � ��

E hE

� ��

� �

� � � �

,

(2),

2

,

exp

1 2cos4

m i

im m i

i

i

J k Rjm

H k R

k Rj

�

�

�

� �

��

�

��

��

� � ��

�

�



� Consider the scattered electric field. It’s phase does not

depend on angle. It’s amplitude is given below

, , ,ˆ( , , ) ( ) exp( )TE TE

s s i i zz E jk jk z� �� E

� ��

20,0

,

,

ˆ( ) 1 2cos

4

i i iTEs i

i

k RCE

k

�

��

��

�

��

E h

i� ��ik

x

y

� � i�

,TEsE �



� Note that scattering is largest when

� �cos 1i i� � � � ��

i� � ��

x

y

� � i�

,TEsE �

� This is the backward scattering case

� It is zero when 2i

��


Numerical experiments

� Let us return to the exact result for TE case and again

focus on the azimuthal component of the electric field

� We would like to compare this with the asymptotic result

� This a difficult function to compute because of ‘bad’ behavior of

Bessel functions of large order

� � � ��

� ��

0, ,

, (2),(2)

,

ˆ( ) exp

( ) exp

TEs i i i z

mm i

m i im m i

E j jk z

j J k RH k jm

H k R

�

��

�

� � ��

��

� � � �

��

r E h



� So let us consider the exact numerical result for the azimuthal

component of the electric field

� We plot this function when

� �, , , ,0 ,0, ,i i i x i z i i xk k k k�� k

� � � �0 0, , , ,

ˆ ˆ( ) exp expi i y i i y i x i zE j jk x jk z� � � � � �E r y k r E y

0 0,

ˆ ˆ ˆˆ i i i i i yE� � � �h y E h

x

y

,i xk

iE

,TEsE �

�



� For a thin cylinder we have the amplitude profile

, 0.25ik R� �

,i xk

iE / 3�



� It is also instructive to look at constant phase fronts, at which

, 0.25ik R� �

( ) 0phase Q �



� So in the far field, the phase fronts are almost circular in the x-

y plane: there is no angle-dependence of the phase of the

scattered field, as found from the thin-film approximation

� Also the amplitude is largest in case of backscattered waves,

as found from the same approximation

� The amplitude becomes nearly zero when

� Again this is in line with what we found before

� So this approximation is quite accurate

3i

��


TM Scattering by a thin conducting cylinder

� For the TM case from a thin cylinder we have

, 1ik R kR� � �

� For small arguments (consider positive or zero m’s)

� � 1

! 2

m

m

zJ z

m� ��

� � �(2)0

2ln

jH z z

�� (2)

0

2 ( 1)!m

m m

j mH z

z��

��

� Lowest order in : ,ik R�

� ��

,

(2), ,

exp2ln

m i

im m i i

J k R jjm

H k R k R

�

� �

��

�

��

��



� Resulting far scattered field for a thin conducting cylinder

� � � �

,,, , ,

0 0,

,,

ˆ ˆ( , , ) exp( )

ˆ2 ln

ii zTM TMs s v i i z

TMs v i i

ii

kkz E jk jk z

k k

CE

k Rk

��

��

� � �

��

� ��

� �

� �

E z

E v

� In the lowest order approximation there is no angle

dependence for the TM scattered wave!

� The amplitude in all directions is the same



� Let us compare the scattering strength in the two cases

� ��

20,0

,, ,

ˆ( ) 1 2cos

4

i i iTEs i

i i

k RCkE

k k

�

��

��

�

��

E h

� �00

,

, ,

ˆ2 ln

TMs v i i

i i

CE

k k R� �

��

� �E v

� For equal horizontal and vertical components of the incident

field we have

� � � �2,

, ,,,

( ) 3ln

2

TEs

i iTMis v

E kk R k R

kE

��

�

��



� This ratio is quite small for thin cylinders

� But this result is to be expected: a

vertically polarized incident electric field

has a component along the ‘wire’ and

easily induces electric currents along the

wire. These currents, in turn, generate the

scattered field.

� A horizontally polarized incident wave has

no longitudinal components and cannot

excite such currents

TEiE

TMiE


TE Scattering by a thick conducting cylinder

� Now, let us investigate the other limit, that of a thick conducting

cylinder which satisfies

2 2, , 1i i zk R R k k� � � �

� Remember: scattered electric field

� � � ��

� �,0 ,0

(2), ,

exp( )ˆ ˆ expm ii zTE

s i i imi m i

J k RjC jk jk zjm

k H k R

��

� �

��

�

�

��

�� E E h

� �� ,

,

,(2),

( ) ( , , )i z

m m iTE Hs m k

m m i

u J k Rz

H k R

�

�

� ��

��

��

��E r M

� Far field limit



� ��

� �

� � � �

, ,,(2)

,

,

exp exp2 4

( ) exp

m i ii i

m m i

mm i i

m

J k R k R jjm j jk R

H k R

j J k R jm

� ��

�

�

� ��

� �

�

��

�

��

� � ��

��

�

�

�

� � � �cos exp cos ( ) ( ) expmm

m

j jz j J z jm� � ��

��

��

� Next, use the relation

� Let us approximate the denominator

� �(2) (2) 1 21 ( ) exp

4m

m m

jz H z jH z j jz

z

��

� � ��

� � �



� � � �

� �

0

, , ,

ˆ ˆ( , , ) cos

exp ( ) cos

TEs i i i

i i i i z

Rz

jk R jk R jk z� �

� � � ��

� � �

� � �

� ��

E E h�

� The TE far scattered field becomes

� Remember that this result is also based on the far-field behavior

of the vector solutions (used here), which in turn, was based on

the asymptotic behavior of the Hankel function when

1k��


Numerical results for a thick cylinder

� So let us again look at some numerical results for the azimuthal

component of the electric field

� We again plot this function when

� �, , , ,0 ,0, ,i i i x i z i i xk k k k�� k

� � � �0 0, , , ,ˆ ˆ( ) exp expi i y i i y i x i zE j jk x jk z� � � � � �E r y k r E y

0 0,

ˆ ˆ ˆˆ i i i i i yE� � � �h y E h

x

y

,i xk

iE

,TEsE �

�



� The amplitude profile is shown below

, 10i xk R �

,i xk

iE



� Phase fronts:

, 10i xk R �,i xk

iE



� Note that in front of the cylinder the scattered wave looks

normal, it has a circular phase front

� But behind the cylinder, at a not too far distance, the phase

fronts are flat!

� Besides, its amplitude is much larger than the waves scattered

back (to the left).

� But this is just a misconception. To understand this point, one

should realize that the total field is

( ) ( )TEi s�E r E r



� Let plot the � component of

the total field

� The amplitude of the

scattered wave behind the

cylinder is large because it

has to partially cancel the

incident wave, to form a

shadow region



� Nevertheless, the numerical results are very different from the

asymptotic result we found earlier if we only demand

� To understand why, consider the exact electric field

� � � ��

� ��

0, ,

, (2),(2)

,

ˆ exp

( ) exp

TEs i i i z

mm i

m i im m i

E j jk z

j J k RH k jm

H k R

�

��

�

� � ��

��

� � � �

��

E h

� Asymptotic behavior of Hankel function depends on whether

argument is larger, smaller, or comparable to order m

1k��



� For a thin cylinder only small m’s contribute (|m|=0,1)

� Then, far field asymptotic is valid since

, 1ik R� �

� ��

,

(2),

m i

m i

J k R

H k R

�

�

�

�

m

, 1ik R� �

, 10ik R� �

, 0.25ik R� �

, 1ik � � �

� �2 (2), , ,

,

2( ) exp / 4m

i m i ii

k m H k j jk jk� � �

�

� � � ��

� � ��

� The correct asymptotic result is



� But, for a thick cylinder, all orders with have a

contribution so that large orders are also important

� The used asymptotic is then only accurate when

,im k R��

� �2

2

, ,,

2i i

i

Rk k R� �

�

��

��

� This is similar to the ‘definition’ of the far field region (why?)

� So our earlier result (and the earlier discussion of the far-field

behavior of vector wave solutions) is, strictly speaking, only

applicable in this limit



� Similar discussion applies to TM scattering: whereas in the far

field limit the scattered wave behaves as TEM waves with a

conical wave front (circular in x-y plane), the behavior outside

this limit can be totally different

� One can have flat wave fronts, a shadow region, etc

� In particular, note that when the wavelength is very short

compared to the dimensions of the object, the far field result is

only valid so far away that it may be useless for practical

applications


Scattering cross section (scattering width)

� We saw in the beginning how a scattering cross section is

defined for a finite scatterer in terms of the scattered power

� An infinite cylinder, however, is not a finite object

� The field radiated by sources inside the cylinder (scattered

field) does not drop as , but as

� We have to reformulate our definition of scattering cross

section for infinite cylindrical objects

1/ r 1/ �

��

scattering by a perfectly conducting infinite cylinderee.sharif.edu/~emscattering_ms/lecture...

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