Electromagnetic scattering
Graduate Course Electrical Engineering (Communications) 1st Semester, 1390-1391 Sharif University of Technology Contents of lecture 2
� Lecture 2: Basic scattering parameters • Formulation of the problem • Scattering cross section • Absorption cross section • Scattering amplitude matrix • Unit vector systems • Expressions for the scattered field:
� Born approximation
� Relation with Fourier transform
Basic scattering parameters 2 Contents of lecture 2
� Complementary material: • Ishimaru – Chapter 10, pp. 278-288 • Tsang, Kong, Ding – Chapter 1, pp. 1-9
Basic scattering parameters 3 Introduction
� Often, one’s aim is to gain information on an ‘object’ far away by sending out a wave and analyzing the scattered wave
� At a sufficiently large distance such far field waves are seen as plane TEM waves
� So, we have to study the scattering of plane TEM waves by objects
� Scattering: polarization and conduction charges are set into motion by the incoming field
� Induced currents radiate an extra field (scattered field) Source
Basic scattering parameters 4 Introduction
� We have to calculate these induced currents and use them to compute the resulting far field, this is because the scattered field is also measured at a large distance for most applications
� But, first, we discuss a number of general parameters used in scattering theory
Source Measurement
Basic scattering parameters 5 Basic scattering parameters
� Consider an object with a permittivity different from the dielectric constant of the background. A plane wave with a known ˆ polarization propagates along k i and hits the objects.
� Far away from the object, the induced scattered field also ˆ behaves as a plane wave in each direction ks
� E� Eexp� �j k � r� Ei� E i,0 exp� �j ki � r� s s,0 s � p ()r E E s Scattered field, Incident field i V locally behaving ˆ ks as a plane wave ˆ in the far-zone ki H s
H i
Basic scattering parameters 6 Basic scattering parameters
� The incident electric field Ei ˆ Ei� E i,0 exp �jk ki � r E � � ˆ s ki r
� Far-zone scattered field ˆ Far field k� k rˆ exp�� jkr� s �EE� ˆ, ˆ k k s s,0 � s i � r
Amplitude vector of the scattered field up to a factor 1/r, also depends on the incoming electric field vector
� Scattering amplitude: � ˆ ˆ E Es,0 k s, k i ˆ ˆ s � � f�ks, k i � r� � EiE i,0
Basic scattering parameters 7 Basic scattering parameters
� Incident Poynting vector
2 E S� i,0 kˆ i2� i
� Poynting vector of the scattered wave
2 2 ˆ ˆ 2 E f �ks, k i � E S�s kˆ � i,0 k ˆ s2� s r2 2 � s
ˆ s�k�sin� s cos �s ,sin � s sin � s ,cos �s �
Basic scattering parameters 8 Scattering cross section
� Scattered power flowing through a small surface normal to the observed direction
2 ˆ ˆ ˆ dPs�S s � n dA � Si f � k s, k i � d�s
2 dPs ˆ ˆ �f�ks, k i� d � s Si
Differential scattering cross section d�s � sin� s d � s d � s
2 ˆ ˆ ˆ ˆ � d�k s,, k i � � f � ks k i �
Basic scattering parameters 9 Scattering cross section
� Differential scattering cross section has the dimension of area:
ˆ ˆ dPs� S i� d�k s , ki� d � s
� Power scattered into a small normal surface with a solid
angle d � s equals incident power passing through a surface normal to the incident wave with the area ˆ ˆ � d�k s, k i�d� s
� It is a measurable quantity if we know the intensity of the incident wave and the distance from the observation point to the scatterer
Basic scattering parameters 10 Scattering cross section
� Total scattering cross section:
2 P� S fkˆ, k ˆ d � s i� � s i� s 4�
P 2 s �fkˆ, k ˆ d � � � s i� s Si 4�
Scattering cross section d�s � sin� s d � s d � s P ��s � � kˆ, k ˆ d � s� d � s i� s Si 4�
Basic scattering parameters 11 Scattering cross section
� Scattering cross section: area of a surface normal to the incident wave which captures an incident power equal to the total power scattered
� Not equal to the geometrical cross section � g : area projected onto a plane perpendicular to the incident wave vector
� g kˆ i �s� � g
Basic scattering parameters 12 Absorption cross section
� Part of the power captured by the object may be dissipated due to polarization loss or conductivity
� Total power absorbed by the object (magnetic loss neglected):
�� �� 2 P� E ( r ) dV a � int 2 V Field inside the object
� Absorption cross section:
Pa � a � Si
Basic scattering parameters 13 Total cross section
� Total cross section
PPs a �t � � � �s � � a SSi i
� Albedo of the object
� � � �s � s �1 �t � s� � a
Basic scattering parameters 14 Scattering amplitude matrix
� So far, we only discussed the relations between the incident and scattered energy flow (power)
� But what about the fields themselves? What about their amplitudes and directions?
�
� p ()r E E s i V ˆ ks ˆ ki Hs
Hi
Basic scattering parameters 15 Scattering amplitude matrix
ˆ ˆ � The scattering amplitude f � k s , k i � was introduced without any reference to the polarization of the incident and reflected waves
ˆ Ei� E i,0 exp� �jk ki � r� Ei E kˆ r s exp�� jkr� i �EE� ˆ, ˆ k k s s,0 � s i � r ˆ Far field ks � k rˆ
� But, in reality, this function depends on these directions
Basic scattering parameters 16 Scattering amplitude matrix
� General formulation: choose a system of coordinates for the electric field of the incident wave and for the electric field of the scattered wave for each scattering direction ˆ � The unit vectors of the incident wave are perpendicular to k i ˆ and the unit vectors of the scattered wave are perpendicular to ks
aˆi aˆs Es Ei ˆ ki bˆ bˆ ˆ i s ks
Basic scattering parameters 17 Scattering amplitude matrix
aˆ b ˆ ˆ Ei��E i a i � E i bi�exp� � jkk i � r � exp�� jkr� �EEaˆ �Ea b ˆ b s� s s s s � r
a �ˆ ˆ ˆ ˆ � a �E � faa�k s,, k i� f ab � ks k i � �E � s � � � i �b � � � �b � �Es � fkˆ,, k ˆ f k ˆ k ˆ �Ei � �ba� s i � bb �s i � �
aˆi aˆ E s Scattering s amplitude E matrix i kˆ i ˆ ˆ bs kˆ bi s
Basic scattering parameters 18 Scattering amplitude matrix
� Note that the waves are not necessarily linearly polarized.
� Depending on their coefficients, they may have linear, circular or elliptic polarization in the plane perpendicular to the propagation direction
� Also note that the system of unit vectors for the scattered wave may depend on the scattering direction considered
� Finally, there are two main system of unit vectors which are usually used as we see next
Basic scattering parameters 19 Unit vector systems
� (1) System based on scattering plane: for every scattering ˆ ˆ direction, k i and k s lie on a plane with the normal vector
kˆ� k ˆ nˆ � s i choose aˆ� a ˆ � n ˆ si ˆ ˆ i s si ks� k i ˆ ˆ ˆ aˆ nsi bi� k i � aˆ i i ˆ ˆ ˆ ki aˆ bs� k s � aˆ s ˆ s bi ˆ bs kˆ � Vectors depend on scattering s direction considered
Basic scattering parameters 20 Unit vector systems
� (2) Vertical and horizontal polarization: often there is a preferred plane like the earth surface (in geographical systems). Let normal to this plane be the axis z
nd � Then: 2 unit vector chosen to lie in horizontal plane
bˆ ˆ ˆ i ˆ z� ki bi � ˆ zˆ� kˆ ki i bˆ ˆ s ks zˆ zˆ� kˆ bˆ � s s ˆ zˆ� ks
Basic scattering parameters 21 Unit vector systems
� The other unit vector follows:
ˆ ˆ ˆ ˆ aˆi� b i � k i aˆ s� b s � k i
� Terminology: horizontal bˆ bˆ i ˆ s and vertical polarization ki ˆ � Later we may use the ks aˆs notions TE and TM which zˆ aˆi should not be confused with waveguide modes
Basic scattering parameters 22 Unit vector systems
� In terms of cylindrical coordinate system:
1 z ˆ �k kcos� , k sin � , k ˆ i� i,,�i i � i i, z � k i k i, z k zˆ � kˆ i bˆ �i �� � sin� , cos � , 0 � � ˆ ˆ� k ˆ i ˆ i i i i zˆ � ki
ˆ ˆ y aˆi� b i � k i �i 1 x k � �ki,, zcos� i , k i z sin �i , � k i,� � i,� k
Basic scattering parameters 23 Unit vector systems
� In terms of spherical coordinate system:
ˆ i�k�sin� i cos �i ,sin � i sin � i , cos� i � ˆ ˆ bi�� �sin� i , cos �i , 0� � ˆ aˆi� �cos� i cos �i ,cos � i sin � i ,� sin � i� � θ i
ˆ s�k�sin� s cos �s ,sin � s sin � s , cos� s � ˆ ˆ bs�� �sin� s , cos � s , 0� � s ˆ aˆ s� �cos� s cos �s ,cos � s sin � s ,� sin � s� � θ s
Basic scattering parameters 24 Expressions for the scattered field
� So far we just discussed a number of parameters which are used in scattering problems
� But how can one express the scattered field in terms of what is happening inside the object?
� Let us focus on an object which may be a dielectric and/or conductive in free space. This object will be characterized by a complex permittivity
� ()r j �0 p �()()()r�� r � � r p p � p V
Basic scattering parameters 25 Expressions for the scattered field
� When the incident field radiates the object, currents are induced. The sum of induced polarization and conduction currents is
�� � � Jp()()() r�j� � p r � 0 � E r Jc ( r )� � p ( r ) E ( r )
�� � �J()()() r � Jp r � J c r � j�� � p ()()r� 0 � E r
� What is the far field generated by this �
current? (This is the scattering field) � p ()r V
Basic scattering parameters 26 Expressions for the scattered field
� Far field expressions for the scattered field:
exp �� jkr � Ef () r �jk � rˆ � � r ˆ� F� r ˆ � � s 4� r � � jkexp �� jkr � Hf () r � �rˆ � F� r ˆ � s 4� r
F� rˆ ��� exp �jk r ˆ � r� � J (r � ) dV � V
� Note that in all directions the scattered wave is along the position vector ˆ ks � rˆ
Basic scattering parameters 27 Expressions for the scattered field
� (Far-zone) scattered electric field: for simplicity let us define jk� Q kˆ, k ˆ� � F k ˆ �s i � 4� �s � exp �� jkr � Ef () r � �kˆ � � k ˆ� Q k ˆ, k ˆ � s r s� s �s i � �
� Scattering from equivalent currents in a dielectric: k 2 Q kˆ, k ˆ � exp jk � r��� ()() r � E r � dV � �s i � � �s � p 4��0 V
This is the total field inside the ��p()()r�� � p r � ��0 object: incident plus the field generated by the object itself
Basic scattering parameters 28 Expressions for the scattered field
� Equivalently: exp�� jkr � Ef (), r� Q kˆ k ˆ s r � �s i �
Q kˆ,, k ˆ� Q k ˆ k ˆ ��k ˆ � Q kˆ, k ˆ � k ˆ � � s i� � s i� � s� s i� � s
� Compare with the definition of scattering amplitude:
E ˆ ˆs ˆ ˆ 1 ˆ ˆ f�ks,, k i � � r � f � ks k i � � Q� � ks, k i � Ei Ei,0
Basic scattering parameters 29 Born approximation
� In principle, we should know the total field inside the scattering object in order to be able to compute the scattered field
� But, if � p () r � � � 0 an approximation can be made
� In that case the field generated by the object is small and the total field is close to the incident field
E( r )� Ei ( r ) � E i,0 exp � � jki � r�
Basic scattering parameters 30 Born approximation
� Result: k 2 � � Q kˆ, k ˆ � exp �jk � k� r� ��� () r � dV � E �s i � �� ��s i � � p � i,0 4��0 �V �
exp �� jkr � Ef () r � �kˆ � � k ˆ� Q k ˆ, k ˆ � s r s� s �s i � �
eˆi ˆ ˆ es ki r ˆ ks
Basic scattering parameters 31 Born approximation
� Equivalently
exp�� jkr � Ef (), r� Q kˆ k ˆ s r � �s i �
k 2 � � Q kˆ, k ˆ � exp �jk � k� r� ��� () r � dV � � �s i � �� ��s i � � p � 4��0 �V � �E� E � kˆ k ˆ � �i,0� i ,0 s � s �
Basic scattering parameters 32 Born approximation
� So that
exp �� jkr � Ef () r�� E � E� kˆ k ˆ � S k ˆ, k ˆ s r �i,0� i ,0 s �s � � s i �
k 2 S kˆ, k ˆ � exp jk � k�� r� � �� () r � dV � �s i � � ��s i � � p 4� �0 V
ˆ � The polarization of the scattered field in the direction k s is given by the direction of the vector
ˆ ˆ Ei,0�� E i ,0 � k s � ks
Basic scattering parameters 33 Born approximation
� From the definition of the scattering amplitude it follows that
ˆ ˆ 1 ˆ ˆ ˆ ˆ f �ks, k i � � Ei,0 �� E i ,0 � k s � ksS � k s, k i � E i,0
� Note that the polarization of the scattered wave in this approximation does not depend on material properties
� Also the dependence of f on polarization is the same for all objects because S is independent of polarization.
� Finally note that S is the Fourier transform of �� p for ˆ ˆ kd k s k i k��� ks k i � � �
Basic scattering parameters 34 Born approximation
� Example: sphere of radius R with constant�
� Using spherical coordinates, and choosing a linearly polarized incident wave propagating along z: k 2 �� � 1� SSkˆ, k ˆ � � � r �sink R� k R cos k R � �s i � � � 3 � �d � d �d �� kd � x kd� d �k2 k sin 2 ks
� Because of the symmetry of the ki � � structure we do not need to z consider other incident directions y
Basic scattering parameters 35 Born approximation
2 k �� p � S kˆ, k ˆ � exp jk � r� dV � Details of calculation: �s i � � �d � 4� �0 sphere
� Change coordinate system to one where z-is along kd k 2�� R �2 � S kˆ, k ˆ � p exp jk rcos� r2 sin � drd �d � �s i � � � � �d � 4� �0 0 0 0 k 2�� R � � p expjk r cos� r2 sin � drd � � � �d � 2�0 0 0 2 R 1 2 R k �� p k �� p sin �k r � � exp jk ru r2 drdu � d rdr � ��d � � 2�0 0� 1 �0 0 kd k2��kd R k 2 �� �p vsin vdv � p � sin k R� k Rcos k R � 3 � 3 � �d � d �d �� �0 kd 0 �0 kd
Basic scattering parameters 36 Born approximation R � � 0 � Forward scattering limit: k � 0 d ki ks ˆ ˆ 1 3 2 V s 2 S�ki, k� i � � Sf � R k �r �1 � � k ��r � 1 � 3 4� 4� R3 V � ˆ ˆ ˆ ˆ s 3 f�ki,, k i� S�� k i ki �
� � � � Backward scattering limit: ki
kd � 2 k ks �� � 1� SS�kˆ, k ˆ � � r � sin �2kR �� 2 kR cos � 2 kR �� �i i � b 8k � � ˆ ˆ ˆ ˆ f� ki,, k i � S��� ki k i � �
Basic scattering parameters 37 Born approximation
ks � As function of scattering angle �
ki
kR � 0.25 kR �1 kR � 5
S
Sf
� / 2 � / 2 � / 2
Basic scattering parameters 38 Born approximation
ks � Spheres small compared to wavelength � small change with angle �
� Large spheres show oscillatory ki behavior with decreasing envelop kR �15 � In all cases the maximum S is in forward direction � � 0 (different for S different radii) Sf V S�s k 2 � ��1 � f 4� r � / 2
Basic scattering parameters 39 Born approximation
� If the sphere is large compared to wavelength (kR>>1) then S is largest in the forward direction within the range given by
� � �� �k R � � � � � � d kR2 R kR �15
� S �� � R Sf ki
� / 2
Basic scattering parameters 40 Born approximation
� Note: in all these discussions, the scattering amplitude f follows from S by including
1 ˆ ˆ ˆ Ei,0�� E i ,0 � k s � ks � sin� � : angle between ks and E i Ei,0
� Forward, backward scattering: we look in a direction along ˆ ˆ incident wave ( k s � � k i ) so that the incident polarization is ˆ normal to k s and f and S are the same
� In other directions this is not the case, and we have to include the relative angle with respect to incident polarization to find the true scattering strength
Basic scattering parameters 41 Born approximation
� Let us consider some scattering parameters of a dielectric sphere in Born approximation
� Differential scattering cross section
2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ � d�k s, k i � �f� ks , k i � � sin � S �ks, k i �
� Total scattering cross section x
ks �� � kˆ, k ˆ d � s� d� s i � s k � 4� i � 2� � z ��kˆ, k ˆ sin �d � d � � � d� s i � 0 0 y
Basic scattering parameters 42 Born approximation
2� � 2 ��sin2 � S � sin � d� d� s � � � � 0 0
� Since incident wave is along z-direction, take its polarization
to be on the x-y plane making an angle � 0 with x-axis
� Then x k 2 � s 2 ˆ ˆ sin� � 1 ��ei � k s � � 2 2 0 � �1 � sin � cos �� � �0 � eˆi z y
Basic scattering parameters 43 Born approximation
� 2 �� � S �1 � cos2 � sin� d � s � � � � � 0
2 k �� p �� � � � �� � � � � S �� � � 3 �sin� 2kR sin �� � 2 kR sin �cos � 2kR sin �� �� � � �2 � � 2 � �2 �� �0 �2k sin � �2 �
� For a small sphere
2 3 k�� p R kR��1 � kd R �� 1 � S � � � ~ ��0
Basic scattering parameters 44 Born approximation
2 2 3 � �k�� p R � 2 �s � � � � �1 � cos� � sin �d � �� � �0 � 0
2 3 2 2 �k��p R � 8 2 4 3 � �� p � �� � � � k R� � V s ���0 � 3 9 � �0 �
Basic scattering parameters 45 Born approximation
� Why is the sky blue and the sunset red?
� Because the scattering cross section is proportional to the 4th order of frequency (k). Blue light is scattered more than red by small particles in the atmosphere (dilute gas)
� Light received from direction other than that of the incident beam reaches us by scattering. Thus is shifted to blue.
� Direct light tends to be red as it is the light which did not undergo scattering processes in the atmosphere.
Basic scattering parameters 46 Born approximation*
� Scattering amplitude matrix (in the scattering plane)
exp �� jkr � Ef () r�� E � E� kˆ k ˆ � S k ˆ, k ˆ s r �i,0� i ,0 s �s � � s i �
E�Ea aˆ � E b bˆexp � jkk ˆ � r i� i i i i� � i � aˆi nˆsi
� f f,, a f b exp�� jkr� ˆ ˆ ˆ ki aˆs s��EE s s �Eas s � b ˆ r bi ˆ ˆ bs ks �faa f ab � �1 0 � � � � S � � �fba f bb � �0 cos � �
Basic scattering parameters 47 Born approximation*
�EEa ��1 0 � � a � � Note that s S i �b �� � � � b � �EEs ��0 cos � � � i �
� Component normal to scattering plane (depends on observation direction) gets multiplied by S. The other component first projected on the unit axis of the incident wave.
a ˆ Ei nsi
� b kˆ a Ei i Es
Eb s ˆ ks
Basic scattering parameters 48 Born approximation
� So, if Born approximation applies, it seems that if we can measure the scattering amplitude, we can reconstruct the permittivity profile by an inverse Fourier transform: 4� � dk �� (),r� 0 S kˆ k ˆ exp �� jk � r � d p k 2 � �s i � d (2� ) 3
� But the range of observation in k-space is limited:
ˆ ˆ kkˆ kdk�� k s k � i � s � k �k2k sin � d d 2 ˆ kki 0�k d � 2k
Basic scattering parameters 49 Born approximation
� Let us be more specific by considering an example. We had
k 2 S kˆ, k ˆ � exp j k � r�� () r dV �s i � � �d � p 4� �0 V
� Let us consider a dielectric L ˆ slab with an internal dielectric kki y constant only depending on x, L
x ��p()()r � �� p x ˆ kks z w
Basic scattering parameters 50 Born approximation
� Then k 2 w/ 2 L / 2 L / 2 S kˆ, k ˆ � expjk� r �� ( x ) dxdydz �s i � � � � �d � p 4� �0 �w/ 2 � L / 2 � L / 2
2 w / 2 k sin�kd, y L / 2 � sin � kd, z L / 2 � � expjk x�� ( x ) dx � �d, x � p � �0 kd,, y k d z �w / 2
� Hence, we can extract the following function from measurement
w / 2 I k� exp jk x�� ( x ) dx �d, x �� �d, x � p � w / 2
ˆ ˆ ˆ kd, x� k �k s � ki � � x � �2k� kd, x � 2 k
Basic scattering parameters 51 Born approximation
� We apply the inverse Fourier transform to this function, neglecting the values outside the range of observation:
1 2 k e( x )� I k exp � jk x dk � �d,, x � � d x � d, x 2� �2 k w / 2 � �x � x��� () x � dx � � � � p �w / 2 1 2 k �x � x� � exp �� jk x � x� � dk � � � �d, x � � � d, x 2� �2 k sin 2k� x�� x� � � � � � � �x� x� �
Basic scattering parameters 52 Born approximation
� The reconstructed function e(x) is not the same as ��. It is a smoothened version averaged over the effective width of the sinc function w / 2 sin 2k� x�� x�� � e() x � � � �� ()x� dx � � p � w / 2 � �x� x� �
� Width of the sinc function
�� p � � �x � � 2k 4
sin� 2k� x� x�� � x� � �x� x� �
Basic scattering parameters 53 Relation with Fourier transform
� Remember that A() r�� G r � r� J r � dV � 0� 0 � � � � V
� Assume that these currents are
inside the object. Consider two x infinite planes on the two sides of the object z y
�0, � 0
Basic scattering parameters 54 Relation with Fourier transform
� Let us apply a Fourier transform on a plane
� � A( r )exp jk x� jk y dxdy � � � �x y � �� �� � G� z� z�| k , k exp jk x � � jk y�J r � dV � 0� 0 �x y � � x y � � � V
� � G� z| k , k� G r exp jk x � jk y dxdy 0 �x y � � � 0 � � �x y � �� �� exp � jk z Not necessarily real! �z � 2 2 2 But should satisfy �, kz � k� k x � k y 2 jk z Re�kz� � 0,Im� k z � � 0
Basic scattering parameters 55 Relation with Fourier transform
� For the plane on the right: z� z �
� exp�� jk z� A�(z | k , k ) � 0 z exp jkR � r� J r � dV � x y � � � � � 2 jkz V
R �k(,,)kx k y k z
� On the left: z� z �
� exp� jk z� A�(z | k , k ) �0 z exp jkL � r� J r � dV � x y � � � � � 2 jkz V
L �(,,)kx k y �k z
Basic scattering parameters 56 Relation with Fourier transform
� Inverse Fourier transform
� � dk dk ARL, ( r )� A� (zRL, | k , k )exp � jk x� jk y x y � � x y� x y � 2 �� �� �2� �
� Magnetic field
1 � � dk dk HRL, ( r )� �� A� (zRL, | k, k )exp � jk x � jk y x y � � x y� x y � 2 �0 �� �� �2� �
� Fourier transform of the magnetic field
RL, � RL, jk � RL, �H(z | kx , k y ) � � � A ( z |kx , k y ) �0
Basic scattering parameters 57 Relation with Fourier transform
� Similarly, Fourier transform of the electric field
1 � RL, RLRLR.,,�� LRL, � �E(z | kx , k y ) � � k�� k � A (z | kx , k y ) � j��0 � 0 � RL, exp�jkz z � RLRL.,� RL, � �k � � k��exp�j k �r� � J� r � � dV �� 2kz��0 � V �
Basic scattering parameters 58 Relation with Fourier transform
2 2 2 RL, � Consider the case where kx k y k��: � realk vectors
F� kˆ RL, � �� exp� j kRL, � r��J� r � � dV � V
RL, k� exp�� jkz z � �E� (zRL, | k , k ) � kˆRLRL.,�� k ˆ � F k ˆRL, � x y �� � � 2kz
RL. � Let ˆ ˆ and use k� ks
exp �� jkr � Ef () r � �jk� kˆ � � k ˆ� F k ˆ � s 4� r s� s� s � �
Basic scattering parameters 59 Relation with Fourier transform
� Hence, with ˆ ˆ RL, rˆ � ks � k
RL, exp�� jkr � jkzexp�� jk z z � f,, R L () � �Er E� (zRL, | k , k ) s r 2� s,, x s y
� The far electric field along any direction in space is proportional to the 2D Fourier transform of the electric field on an arbitrary plane (between the object and the observer) with the Fourier components:
kx� k s,, x, k y � k s y
Basic scattering parameters 60 Relation with Fourier transform
� From the definition of the scattering amplitude it follows that k f(kˆ , k ˆ )� z E� ( z | k ,k ) s i 2� s,, x s y
� Also note that
k� exp � jk z RL, � RL, � z � ˆ RL, E (z | ks,, x , k s y ) � � F� � k � 2k z exp �� jkr � Ef,, R L () r � jk� F kˆ RL, s 4� r � � �
Basic scattering parameters 61 Optical theorem
� Now, consider the scattering problem when the incident wave is along z, normal to the fictitious planes
� Total field: x Not necessarily far-zone, but the EEE�i � s field generated by k� k zˆ HHH� � induced currents i i s at any point z
�0, � 0
Basic scattering parameters 62 Optical theorem x
� Poynting’s theorem applied to the k� k zˆ surface consisting of two infinite i planes normal to the z-axis, and z surrounding the object nˆ nˆ
� � 1 * S L S R �Re��E � H � � nˆdS � � Pl 2 �� �SSLR� � =0 why? � � �0, � 0 � � � � 1 * 1 * � �Re ��Ei � H i �� nˆdS � �Re ��Es � H s �� nˆdS � 2 ��2 � � �SSLR� � �SSLR� � � � � � 1 * 1 * �Re ��Ei � H s � � nˆdS � �Re ��Es � H i � � nˆdS � 2 ��2 � � �SSLR� � �SSLR� �
Basic scattering parameters 63 Optical theorem
� Result: Scattered power Dissipated power � � 1 * Re ��Es� H s � � nˆdS � � Pl � 2 �� �SSLR� � � � � � 1 * 1 * � Re ��Ei � H s � � nˆdS � �Re ��Es � H i � � nˆdS � 2 ��2 � � �SSLR� � �SSLR� �
Basic scattering parameters 64 Optical theorem
� Using the definition of a Fourier transform:
E� H* � nˆdS � zˆ � exp jkz R EH� * dS ��� s i � � � � �� s i,0 � SSLR� SR �zˆ � exp jkzL E � H * dS �� � ��s i,0 � SL RR� * LL� * �exp�jkz �zˆ � Es � z ;0,0 � � H i,0 �exp�jkz �zˆ � Es � z ;0,0 � � H i,0 �H* � zˆ � �expjkzRR E� z ;0,0� exp jkzLLE� z ;0,0 � �i,0 �� � � s � � � �s � �� 1 * � RR�LL � � �EEi,0 � exp�jkz �s � z ;0,0 �� exp � jkz�Es � z ;0,0 � � � �
1 * � k� LL� � �Ei,0 �� � F� � zˆ � � exp�jkz �Es � z ;0,0 �� � � 2kz �
kz�0 since k x � k y � 0
Basic scattering parameters 65 Optical theorem
� Using the definition of a Fourier transform:
E� H* � nˆdS � zˆ �exp � jkz R EH� * dS ��� i s � � � � �� i,0 s � SSLR� SR �zˆ � exp �jkzL E � H * dS �� � ��i,0 s � SL R � * R L � * L �exp� �jkz �zˆ � Ei,0 � H s � z ;0,0�� exp � �jkz � zˆ � Ei,0 � H s � z ;0,0 � �EH ��exp � jkzRR� * z ;0,0 �zˆ �exp � jkzLLH� * z ;0,0 � zˆ� i,0 � � �s � � � �s � � � 1 � RR� * LL� * � �EEi,0 �exp� � jkz �s � z ;0,0 � �exp� � jkz �Es � z ;0,0 � � � � 1 � k� � Plus sign because * LL� * it is the returning �Ei,0 � � F� � zˆ � � exp�� jkz �Es � z ;0,0 � � � wave � � 2kz �
Basic scattering parameters 66 Optical theorem
� Collecting the results:
� � � � 1 * 1 * PPs� l � �Re ��Ei � H s � � nˆdS � �Re ��Es � H i �� nˆdS � 2 ��2 � � �SSLR� � �SSLR� �
1 � * � � LL � � �ReE � � F zˆ � exp jkzE� z ;0,0 � i,0 � � � � � �s � �� 2� � � 2 �
� � * LL* �� �E � � F zˆ �exp � jkzE� z ;0,0 i,0 � � � � � �s � ��� � 2 �� 1 �Re �E* � F� zˆ � � 2 �i,0 � �
Basic scattering parameters 67 Optical theorem
� Conclusion: the total scattered plus absorbed power is related to the forward scattering amplitude because if we look at the scattered field along the incident field exp�� jkr� f, R () �Erjk� � zˆ � F s 4� r �
� In terms of scattering cross section:
P 1 Total � � l �* ˆ � t� s � � Re �Ei,0 � F� � z � � cross section SSi2 i
Scattering Absorption cross section cross section � a Basic scattering parameters 68 Optical theorem
� Rewriting this equation:
2 E � S �i,0 �� � Re �E* � F� zˆ � � i 2� t 2 �i,0 � � Ei,0
� Same result could have been easily obtained using the conservation of energy inside the object!
Basic scattering parameters 69 ��������������������������������������������������������������������������� ��������������������������������������������������������������������������������� �����������������������������������������������������