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 � Optical theorem 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.