EM scattering 1 1390 - ee.sharif.eduee.sharif.edu/~emscattering_ms/Lecture 1.pdfIntroduction and...

Electromagnetic scattering

Graduate CourseElectrical Engineering (Communications)1st Semester, 1390-1391Sharif University of Technology

Introduction and fundamentals 2

General information

� Information about the instructor:

• Instructor: Behzad Rejaei

• Affiliation: Sharif University of Technology

• Room number: 620, EE Dept., Sharif University

• Email: [email protected]

• Research areas:

� Integrated passive microwave components

� Electromagnetic modeling

� Microwave magnetic devices

� Integrated artificial dielectrics

� Substrate integrated waveguides


General information

� Course structure: oral lectures + homework assignments

� Course material and references

• Lecture notes (download from eecourse/emscattering_ms)

• Advanced engineering electromagnetics; Constantine A. Balanis, 1989 Wiley

• Electromagnetic wave propagation, radiation, and scattering; Akira

Ishimaru, 1991 Prentice Hall

• Scattering of Electromagnetic Waves: Theories and Applications; Leung

Tsang, Jin Au Kong, Kung-Hau Ding, 2000 Wiley

� Pre-requisites:

• Electromagnetic theory, Microwave techniques (at the level of Pozar), Differential equations, Special functions


General information

� Homework assignments:

• Can be downloaded from download from “eecourse/emscattering_ms”

• Have to be returned before the homework class (these are not separate classes but are part of the usual classes on Sundays and Tuesdays)

• Bonus: points added to the final exam grade

� Times & dates: 1st semester, 1390-1391, every Sunday and Tuesday, 15:00-16:30


Contents of lecture 1

� Contents of lecture 1:

• Introduction & motivation

• Review of Maxell theory

• Green’s functions

• Far fields and Radiation


Introduction

� Electromagnetic scattering: active interdisciplinary area with applications in medical imaging, geo-science, remote sensing (weather, vegetation, etc.), radar

� Often the aim is to radiate the ‘object’ with a wave, and gain information about the object by analyzing the scattered wave


Introduction

� What does ‘scattering’ mean from a physical point of view?

• When the incoming (incident, probing) wave reaches the object, the electric charges inside the object are set into motion

• The oscillatory motion of those charges yield oscillatory currents

• These currents, in turn, radiate energy (like in an antenna). These radiated waves constitute the ‘scattered’ field.

� Two types of charge:

• Free charges which induce conduction currents

• Bound (polarization) charges which induce displacement currents

�

�

Positive ions

Electron cloud

Freely moving electrons

�,c c� ��

,p p

��


Introduction

� In this course we are interested in the phenomenon of scattering of EM waves by dielectric and conductive objects:

• Imagine a wave, generated by sources far away, hits an object

• These waves can be considered as plane waves

• How can we compute the scattered field ‘far away from the object’?

2DR

��


Introduction

� Broad overview of the course:

• Fundamentals

• Introduction to scattering parameters and some concepts

• Completely solvable cases (layered media, cylindrical objects, wedges, spherical objects*)

• General formulation of the scattering problem

• Scattering from ‘small’ objects (Rayleigh scattering)

• Short wavelength approximation techniques


Review of Maxwell theory: microscopic equations

� Microscopic Maxwell equations (in vacuum):

Electric current density (A/m2)

t

��

��

�

70 4 10� � ��

0�� 0 0 0t� ��

��

� ��

0

��

�

( )tr,�

( , )tr� Electric charge density (C/m3)

120 8.85 10�� Vacuum permittivity

Vacuum permeability


Review of Maxwell theory: macroscopic equations

� These equations are general and fundamental. But solving them inside true materials consisting of atomic charges and currents is almost impossible.

� Macroscopic approach: separate microscopic sources (bound to atoms and molecules) from macroscopic sources (free conduction electrons and their motion)

c p� ��

�

�

Positive ions

Electron cloud Freely moving

electrons

�,c c��

,p p��c p� ��



� Microscopic or polarization charges described by volume density of polarization

� Related to the formation of microscopic dipoles (separation of bound positive and negative charges) under influence of an electric field

( , )tr�

�

�

ip

1( , ) i

i

tV

�� r p�

r

V�

� Microscopic or polarization charges described by

p � ��



� 3rd Maxwell equation:

� � � �0 c p c�� c��

0� �� Electric flux density (C/m2)

� Associated polarization current p t

��

�

� 2nd Maxwell equation:

00 0

1 1p c ct t� �

� ��

� ��

� � � � ��



Magnetization

� Besides, there are microscopic currents due to rotation of electrons and their spins, unrelated to electric polarization.

� These are magnetization currents and induce no local charge. They lead to effective current

m � ��

Equivalent current

� Magnetization: volume density of ‘magnetic’ dipoles




0

1m c ct t�

� ��

� ��

� � � � �

ct

��

��

� �

0

1

�� Magnetic field (A/m)



� Macroscopic Maxwell equations (involving the free macroscopic sources only):

t

��

��

�

ct

��

��

� �

c��

0��



� Equations in frequency domain using phasor representation

j�� E B

j�� H D J

�� D

0�� B

� We have dropped the subscript ‘c’: all currents and charges are considered ‘free’ now. Note also that

0j�� J Charge conservation relation


Review of Maxwell theory: constitutive relations

� Constitutive relations: relations between ‘flux densities’ & ‘fields’

� In common materials the medium is linear and isotropic leading to simple linear relationships

� �0 0 0 1e e� �� P E D E P E E� � � �

� Dielectric constant

� � � �0 0 1m m� � � � �� M H B H M H H

� Permeability

� Dielectric const. and permeability may be frequency dependent and complex. Their imaginary parts are related to loss.



� Equivalent polarization and magnetization currents for linear, isotropic materials:

0/r �� Relative dielectric constant

� �0

1

1

m m m

m r

��

�

�

� ��

� ��

J M H H

J H

� �

0/r� � ��Relative permeability

� �

� �0 0

0 1

p p e

p r

j j jtj

� � � �

�

��

� � �

J P E E

J E

��

� �



� Most conductive media obey Ohm’s law:

��J E

Electric conductivity


j�

��

� �� Complex permittivity

jj j j

j

��

��

� ��

�

H D J E E E

E

� �


Review of Maxwell theory: boundary conditions

� Boundary conditions at the interface between two different media

1 1medium 1: ,� �

2 2medium 2: ,� �

n̂2 2,E D

1 1,E D

1 1,H B

2 2,H B

1 2ˆ ˆ� �n E = n E

1 2ˆ ˆ� �n B = n B

2 1ˆ ˆ s�� n D n D

2 1ˆ ˆ s� � � �n H n H J

points from medium 1 to 2ˆ : n

: surface (sheet) current and charge density, s s�J


Review of Maxwell theory: boundary conditions

� If medium 1 is a perfect conductor:

n̂2 2,E D

2 2,H B2ˆ 0� �n E

2ˆ 0� �n B

2ˆ s�� n D

2ˆ

s� �n H JPerfect conductor


Review of Maxwell theory: plane waves

� Maxwell equations in a linear, isotropic, homogeneous

medium without any conduction currents and charges:

j�� E H

j�� H E�

� Plane wave solutions:

� �0( ) exp j� � �E r E k r � �0( ) exp j� � �H r H k r

Wave vector Constant vectors


Review of Maxwell theory: plane waves

� It follows that 0 0�� k E H 0 0�� k H E�

� The electric and magnetic field are perpendicular to the

direction of propagation (wave vector), and to each other

k

E

H

� Furthermore: k � �� k �

1 ˆ��

� � � �k

H E k E

��

�Wave impedance ˆ

k�k

k


Review of Maxwell theory: energy and power

� Energy and power carried by the field:

� Complex Poynting vector:

� �*1ˆ ˆ

2S S

ds ds� � � � � �� rS n E H n� �

V

n̂rS

*1

2� �rS E H

� Total complex power ‘entering’ a volume through its surface:

S

� This power is partially stored in the volume and partially lost



� Complex power balance:

� �ˆ 2 M E l

S

ds j W W P�� rS n� V

n̂rS

S

2( ) ( ) ( )

4 4E

V V

W dV dV�� E r E r E r� �

2( ) ( ) ( )

4 4H

V V

W dV dV� ��

� � �� H r H r H r

Averaged stored electric energy

Averaged stored magnetic energy

j� � ��

j� �� Note: here we have ‘not’ included conductivity into the dielectric constant



� Dissipated power:

Polarization loss

Magnetization loss

2

2

2

2

2

2

l

V

V

V

P dV

dV

dV

�

��

�

��

��

�

�

�

�

E

H

E

�

Conduction loss

� Using complex permittivity :

2 2

2 2l

V V

P dV dV��

� �� E H j� � ��

�


Review of Maxwell theory: vector wave equation

� Consider sources in a homogeneous medium

� Combining Maxwell equations leads to:

j�� E H

j�� H E J� ,�JsV

Volume containing sources

� � 2k j�� E E J

2 2k � �� Vector wave equation

,��


Review of Maxwell theory: vector potential

� Consider the same problem of sources in a uniform medium

� But now we treat the problem using the vector potential

j�� E H

j�� H E J�

� Introduce:

�� B H AVector potential

,�JsV

Volume containing sources

,��



� Electric field:

j� �� E AScalar potential

� From 2nd and 3rd Maxwell equations:

� �2 2 j� � � � � �� A A A J� �

2 j�

� �� A�



� Gauge freedom: vector potential is not unique, the transformation below yields the same electromagnetic field

j f� � ��

� This allows us to impose additional requirements on the vector potential

� In Lorentz gauge we demand:

f� ��A A

j� �� A �

� In Coulomb gauge:

0�� A



� Equations in Lorentz gauge:

2 2k �� A A J

2 2k�

� ��

� Equations in Coulomb gauge:

2 2 j� � � � � �� A A J� �

2 ��

�

2 2k � ��


Green’s function

� We restrict ourselves to the Lorentz gauge:

2 2k �� A A J

2 2k�

� ��

� The components of the vector potential and the scalar potential basically satisfy the same equation (Helmholtz)

� Consider the Green’s function satisfying:

� �2 2( , ) ( , )G k G �� r r r r r r

2 2k � ��


Green’s function

� This function gives the field at r generated by a point source at r’ (current or charge)

� The solution in infinite space is:

�-function point source

�r

r

Observation point

R �� r r

� �exp( , )

4

jkG

�

��

��

r rr r

r r

� Solution for potentials:

( ) ( , ) ( )sV

G dV� � � �� A r r r J r

1( ) ( , ) ( )

sV

G dV� �� r r r r�


Green’s function

� Expression for the electric field:

� Using charge conservation, partial integration, and including the surface charges, one gets:

( ) ( , ) ( )sV

j dV�� E r G r r J r

2

1( , ) ( , ) ( , )G G

k� � � �� G r r r r I r r

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

s sV V

j G dV G dV�� E r r r J r r r r�

Dyadic Green’s function (matrix)


Review of Maxwell theory: vector potential in 2D

� What if the system is uniform in one direction (e.g. z)

,�J sS

Cross section of the cylindrical region containing sources

x

y1

j� �

�

� � ��

� ��

E A

H A

1

x x

y y

yxz

E j Ax

E j Ay

AAH

y x

��

��

�

��

��

� � ��

��

0z

��

�

1

1

z z

zx

zy

E j A

AH

y

AH

x

�

�

�

� ��

��

� ��

TEz: no electric field along z

TMz: no magnetic field along z



� For the first set:

2 22

2 2 z z zA k A Jx y

��

� � � ��

2 22

2 2k

x y

��

� ��

2 2k � ��

2 22

2 2

2 22

2 2

x x x

y y y

A k A Jx y

A k A Jx y

�

�

� ��

� ��

� For the 2nd set (no potential)



� These sets are independent because the sources are decoupled. Continuity equation:

0yxJJ

jx y

��

� � ��

� The source of the TMz is Jz constant along z

� Thus the 2D problem may be decomposed in two separate problems: one for Jz and one for the other components of the current density


Green’s function in 2D

� The Green’s function for problems uniform in z-direction satisfies the equation

�-function point source

�ρ

ρ

Observation point

R �� ρ ρ

� �(2)2 0

1( , )

4DG H kj

� �� ρ ρ ρ ρ

� 2D Green’s function in homogeneous space

2 22

2 22 2( , ) ( , ) ( )D DG k G

x y�

� �� ρ ρ ρ ρ ρ ρ

x

y

� �,x yρ


Green’s function in 2D

2( ) ( , ) ( )s

D

S

G dx dy� � � � �� A ρ ρ ρ J ρ

2

1( ) ( , ) ( )

s

D

S

G dx dy� �� ρ ρ ρ ρ�

� The potentials in 2D


Far fields

� Consider again sources in an infinite medium

� Quite often we are interested in fields generated by these sources far away from themselves

J

sV�r

r

Observation point� Remember that:

j� �� E A

( ) ( , ) ( )sV

G dV� � � �� A r r r J r

1( ) ( , ) ( )

sV

G dV� �� r r r r�


Far fields

� Lets us inspect the first term with vector potential

J

sV�r

r

Observation point

( ) ( , ) ( )sV

G dV� � � �� A r r r J r

� �exp( , )

4

jkG

�

��

��

r rr r

r r

22 2 ˆ

2 1 2r

r r rr r

� ��

r rr r r r

2 2ˆ ˆ1 11

2 2

rr r r

r r r

� ��

�r r r r

r r


Far fields

� Keeping terms up to the first order in

J

sV�r

r

Observation point

� � � �

� � � � 2

ˆexp exp( , )

4 ˆ1

exp 1ˆexp

4

jkr jkG

rr

jkrjk O

r r

�

�

��

� �

� � ��

r rr r

rr

r r

/r r�

ˆ1rr

��

rr r r ˆ

r�r

r

� � � �expˆ( ) exp ( )

4sV

jkrjk dV

r

��

� � �� A r r r J r

Depends only on direction of r


Far fields

� The vector potential term of the electric field drops as 1/r

� We can repeat the same procedure for the scalar potential, but instead we use a different approach which is faster

� Since we have the far field vector potential, we have the far magnetic field

� � � �

� � � �

� � � �

exp1ˆ( ) ( ) exp ( )

4

expˆexp ( )

4

expˆexp ( )

4

s

s

s

V

V

V

jkrjk dV

r

jkrjk dV

r

jkrjk dV

r

� �

�

�

� ��

� ��

� � ��

��

�

�

�

H r A r r r J r

r r J r

r r J r


Far fields

� It can be shown that

� � � �

� �

2

exp exp 1ˆ

4 4

1ˆexp

jkr jk jkrO

r r r

jk Or

� ��

� ��

r

r r

� Then:

� � � � 2

exp 1ˆ ˆ( ) exp ( )

4sV

jk jkrjk dV O

r r��

� ��H r r r r J r

Far magnetic field


Far fields

� What about the electric field?

� Then a similar procedure shows that

� � � �

1

expˆ ˆ exp ( )

4sV

j

jkrjk dV

r

�

��

� � �

� ��

� ��

E H

r r r J r

�

� � � � 2

exp 1ˆ ˆ ˆexp ( )

4sV

jkrjk jk dV O

r r�

�

� ��

�E r r r r J r

Far electric field


Far fields

� Let us summarize the results, but first look at

� Then the far fields are

� � � �ˆ ˆexp ( )sV

jk dV� � �� F r r r J r

� � � �expˆ ˆ ˆ( )

4f jkr

jkr

��

� � �� E r r r F r

� This vector only depends on direction of observation

� � � �expˆ ˆ( )

4f jk jkr

r��

� � �H r r F r

J

sV�r

r̂

Observation point

Current form factor


Far fields

� In terms of the component of F normal to the position vector:

� � � �expˆ( )

4f jkr

jkr

��

�� E r F r

� � � �expˆ ˆ( )

4f jk jkr

r� �

�� H r r F r

J

sV�r

r̂

�FfH

fE

� The far fields are perpendicular to the direction vector and to each other (TEM)r̂

� �1ˆ ˆ( )f f

�� H r r E r

F

� � � � � �ˆ ˆ ˆ ˆ ˆ� � � �� F r F r r F r r


Far fields

� The far field behaves like a TEM wave propagating in the direction of observation with the wave vector

� � � �expˆ( )

4f jkr

jkr

��

�� E r F r

J

sV�r

ˆ ˆ�k r

fH

fE

ˆk�k r

� The polarization (electric field) given by

� Note: for each direction of observation the polarization and propagation direction of TEM wave are different


Radiation

� Consider now the far field Poynting vector

� �2

*1 1ˆ ˆ

2 2 2f f

� ��

f

f fr

ES E H E r E r

J

sV�r

r̂

�F

fH

fE

r̂

� � � � 22

2 2

ˆˆ ˆ

32

k

r

�

��r

F rS r r

� � � �ˆ ˆexp ( )sV

jk dV� � �� F r r r J r

� � � � � �ˆ ˆ ˆ ˆ ˆ� � � �� F r F r r F r r


Radiation

� Intensity of radiated power may be different in different directions, best representation by using spherical coordinates

J

sV

r̂

�FfH

fE

x

� � 22

2 2ˆ

32

k

r

� � �

��rF ,

S r

� �,� ��F F

� �ˆ sin cos , sin sin , cos� � � � ��r

y

z

�

�


Radiation

� Radiated power through a small surface far away

J

sV

rS

n̂

x

2ˆ ˆ sinr r rdP dA r d d� � �� S n S n

y

z

�

�

r

� �2

2

2ˆ ˆ sin

32r

kdP d d

��

� �� F , r ndA

Differential solid angle

sd�


Radiation

� Total radiated power: consider a spherical surface

rSn̂

x

y

z

�

�

r� �

222

20 0

sin32r

kP d d

� ��

� �� F ,

� �22

2

20 032r s

kP d

� ��

� �� F ,


Far fields (2D)

� Let us repeat the above derivation for a 2D configuration where the observation point is far away from the source

J

sS�ρ

ρ

Observation point

� The 2D Green’s function

� �(2)2 0

1( , )

4DG H kj

� �� ρ ρ ρ ρ

� One may proceed as before, but to get intuitive results we restrict ourselves to the case where the distance is so large that

��ρ ρ

1k �� ρ ρ


Far fields (2D)

� Under this condition:

2

1 2( , ) exp

4 4D

jG jk

j k

��

� �� ρ ρ ρ ρ

ρ ρ

� Next we use

ˆ1� � �

��

� ��

� � � �3/ 22

1 2ˆ( , ) exp exp

4 4D

jG jk jk O

j k

��

� ��

� �ρ ρ


Far fields (2D)

� Now first consider the TMz case:

� �2ˆexp

4 4fz z z

jE j A jk F

k

��

� ��

� �

2( ) ( , ) ( )

2ˆexp exp ( )

4 4

s

s

z D z

S

z

S

A G J dx dy

jjk jk J dx dy

j k

�

� ��

� �

� � � ��

� � � � � ��

�

�

ρ ρ ρ ρ

ρ

� The corresponding far-zone electric field

� �ˆzF


Far fields (2D)

� Far magnetic field best represented in cylindrical coordinates:

� �

� �

3 / 21

1 1 2ˆexp

4 4

f z

f zz

AH O

A k jH jk F

�

�

��

��

� � ��

��

�

� � ��

� The far field behaves like a TEM wave propagating in the direction of observation with the wave vector

� This is a cylindrical wave whose amplitude drops as

ˆk�k

�


Far fields (2D)

� Cylindrical wave:

zJ

H�

zE

ˆk�k

� Treatment of the TEz is a bit more complicated but the final result for the far field is the same:

� The far field again behaves as a locally TEM, cylindrical wave with the magnetic field directed along z and the far electric field directed along φ̂

TMz


Radiation (2D)

� Consider now the far field Poynting vector for the TMz case

� �2

2*1ˆ ˆ

2 2 16

fzf

z

E kF

��

� � �frS E H

� � � �

� �

� �

ˆ ˆexp ( )

exp cos sin ( )

s

s

z z

S

z

S

z

F jk J ρ dx dy

jk x y J ρ dx dy

F

� �

�

� � � ��

� � � � ��

�

�

�J

ρ̂

�

� Note that


Radiation (2D)

� Total radiated power (per unit length)

� �2

2

016 z

kP F d

��

�� r

��

EM scattering 1 1390 - ee.sharif.eduee.sharif.edu/~emscattering_ms/Lecture 1.pdfIntroduction and...

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