TUTORIAL ON SAR POLARIMETRY
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TUTORIAL ON SAR POLARIMETRY

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What is Polarization? 1. WHAT IS POLARIZATION? 1.1 Propagation of a monochromatic plane electromagnetic wave 1.1.1 Equation of propagation The time-space behavior of electromagnetic waves is ruled by the Maxwell equations set defined as r rr rr r r r rr ∂B(r,t) r r ∂D(r,t)∇ ∧ E(r,t) = − ∇ ∧ H (r,t) = J (r,t) +T (1)∂t ∂tr r r rr r r∇ ⋅ D(r,t) = ρ(r,t) ∇ ⋅ B(r,t) = 0r r r rr r r rwhere E(r,t), H (r,t), D(r,t), B(r,t)are the wave electric field, magnetic field, electric induction and magnetic induction respectively. r r rr r rThe total current density, J (r,t) = J (r,t) + J (r,t) is composed of two terms. The first one, T a cr rJ (r ,t), corresponds to a source term, whereas the conduction current density, ar rr rJ (r,t) = σ E(r,t) , depends on the conductivity of the propagation medium, σ . The scalar c rfield ρ(r,t) represents the volume density of free charges. The different fields and induction are related by the following relations r r r r r rr r r r r rD(r,t) = ε E(r,t) + P(r,t) B(r,t) = μ (H (r,t) + M (r,t) ) (2)r rr rThe vectors P(r,t) and M (r,t)are called polarization and magnetization, while ε and μ stand for the medium permittivity and permeability. In the following, we shall consider the propagation of an electromagnetic wave in a linear medium (free of saturation and hysteresis), free of sources. These hypothesis imposes that rr r r rr r rM (r,t) = P(r,t) = 0 and J (r,t) = 0 . aThe equation of propagation is found by inserting (1) ...

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What is Polarization?
1 . W H AT I S P O L A R I Z AT I O N ?
1.1 wave
1.1.1
Propagation of a monochromatic plane electromagnetic
Equation of propagation
The time-space behavior of electromagnetic waves is ruled by the Maxwell equations set
defined as rEr(rr,t)=B(rt,t)rHr(rr,t)=JrT(rr,t)+ ∂D(trr,t)1) ( r r r r ∇ ⋅D(rr,t)= ρ(rr,t)∇ ⋅B(rr,t)=0 r r r r whereE(rr,t),H(rr,t),D(rr,t),B(rr,t) are the wave electric field, magnetic field, electric induction and magnetic induction respectively. r r The total current density,JT(rr,t)=Ja(r,t)+Jc(r,t) is composed of two terms. The first one, r Ja(rr,t) , corresponds to a source term, whereas the conduction current density, r r Jc(rr,t)= σE(rr,t) , depends on the conductivity of the propagation medium, . The scalar field (rr,t) represents the volume density of free charges.
The different fields and induction are related by the following relations r r r r r r D(rr,t)= εE(rr,t)+P(rr,t)B(rr,t)= μH(rr,t)+M(rr,t) r r The vectorsP(rr,t) andM(rr,t) are called polarization and magnetization, while stand for the medium permittivity and permeability.
(2)
 and
In the following, we shall consider the propagation of an electromagnetic wave in a linear medium (free of saturation and hysteresis), free of sources. These hypothesis imposes that r r r r r M(rr,t)=P(rr,t)=0 andJa(rr,t)=0 .
The equation of propagation is found by inserting r r r r r r r ∇ ∧(∇ ∧E(rr,t))= ∇(∇ ⋅E(rr,t))− ΔE(rr,t is formulated as) and ΔEr(rr,tμε)2Ert(2rr,t)μσErtrr,t)=1εrρt(rr,t)  
1
(1)
and (2) into
(3)
What is Polarization?
1.1.2
Monochromatic plane wave solution
Among the infinite number of solutions to the equation of propagation mentioned in (3), we will study the special case of constant amplitude monochromatic plane waves which is adapted to the analysis of a wave polarization. r r
ρr tr The monochromatic assumption implies that the right hand term of (3) is null(,)=0  , t i.e. the propagation medium is free of mobile electric charges (e.g. is not a plasma whose charged particles may interact with the wave).
The propagation equation expression can be significantly simplified by considering the r r complex expression,E(rrof the monochromatic time-space electric field, ) ,E(rr,t) , defined as r r
E(rr,t)= ℜE(rr)ejωt 
The propagation equation mentioned in (3) may then be rewritten as ΔEr(rr) +ω2εμ1jωεσEr(rr) = ΔEr(rr) +k2Er(rr) withkω=1jσ  vωε
(4)
(5)
Here appears the concept of complex dielectric constant ε=εjε′′ = ε−jthenkω=1j=εβjα (6) ωvε In a general way, a monochromatic plane wave , with constant complex amplitude, Er0=Eàejδr, propagating in the direction of the wave vector,k ,owinfollˆg ht eah sel xocpm form r Er(rr)=Er0ej krrrwithE(rr)kˆ=0 (7)
One may verify that such a wave satisfies the propagation equation given in (5). Without any loss of generality, the electric field may be represented in an orthonormal basis (xˆ,yˆ,zso that the direction of propagationˆ) defined kˆzThe expression of the electric fieldˆ . = becomes Er(z)=Er0e−αzejβzwithE0z=)80 ( It may be observed from (8) that acts as the wave number in time domain, while corresponds to an attenuation factor. Back to time domain, this expression becomes in vectorial form
Er(z,t)=EE00yxeezazaos(0coc(sωωttkzzkδ++δxy)) 
2
(9)
What is Polarization?
The attenuation term is common to all the elements of the electric field vector and is then unrelated to the wave polarization. For this reason, the medium is assumed to be loss free, = in the following0 ,
E0xcos(ωtkz Er(r,t)=E0ycos(ωt0kz
δ ⎤ +δx) +y)
1.1.3 Spatial evolution of a plane wave vector: helicoidal trajectory
(10)
At a fixed time,t=t0, the electric field is composed of two orthogonal sinusoidal waves with, in general, different amplitudes and phases at the origin.  $ y
0
Ey
(z,t)
$ x
Figure 1Spatial evolution of a monochromatic plane wave components.
Three particular cases are generally discriminated:
 Linear polarization:
=yx=0+m
 
Ex(z,t)
$ z
 
The electric field is then a sine wave inscribed within a plane oriented with an angle respect toxˆ
 
r E(z0,t)=
⎡ ⎤ E02x+E20ysniocs0φφcos(ωt0kz+ δx)  ⎣ ⎦
3
with
(11)
What is Polarization?
 
$ y
0
$ x
r E(z,t)
Figure 2Spatial evolution of a linearly (horizontal) polarized plane wave.
 
 
Circular polarization:
=
y
x=0+m/ 2 andE0x=E0y 
x(z,t) $ z  
In this case, the wave has a constant modulus and is oriented with an angle to thexˆ axis r2 2 E(z,t0)=E0x+E0yand (z)= ±(tàkz+x)
 
 
$ y
Ey(z,t)
Figure 3Spatial evolution of a circularly polarized plane wave.
The wave rotates circularly around thezˆ axis.
 Elliptic polarization: Otherwise
The wave describes a helicoidal trajectory around thezˆ axis.
4
r E(z,t)
Ex(z,t)
(z respect) with
$ z
 
(12)
What is Polarization?
 
$ y
Ey(z,t)
Figure 4Spatial evolution of a elliptically polarized plane wave.
1.2
1.2.1
Polarization ellipse
Geometrical description
Er(z,t)
Ex(z,t)
$ z
 
The former paragraph introduced the spatial evolution of a plane monochromatic wave and showed that it follows a helicoidal trajectory along thez From a practical point of view,ˆ axis. three-dimensional helicoidal curves are difficult to represent and to analyze. This is why a characterization of the wave in the time domain, at a fixed position,z=z0is generally preferred. $  y Er(z, t)
$ y
0
Figure 5Temporal trajectory of a monochromatic plane wave at a fixed abscissaz=z0.
r(z0,t) E
$ x
$ z
 
The temporal behavior is then studied within an equiphase plane, orthogonal to the direction of propagation and at a fixed location along thez As time evolves, the wave propagatesˆ axis. "through" equi-phase planes nd describe a characteristic elliptical locus as shown inFigure 5. The nature of the wave temporal trajectory may be determined from the following parametric relation between the components ofE(z0,t)
5
What is Polarization?
) (0, )2(20 (, )0so(,c)(0, )2 ExEz0xtExzE0txEE0yyz tδy−δx+EyzE0yt
=sin(δy−δx)
(13)
The expression in (13) is the equation of an ellipse, called the polarization ellipse, that describes the wave polarization.
The polarization ellipse shape may be characterized using 3 parameters as shown inFigure 6. ˆ  
Figure 6Polarization ellipse.
- 
zˆ
A
E 0x
τ
φ
E0y
xˆ
 
is called the ellipse amplitude and is determined from the ellipse axis as
2 A=E0x+E20y 
(14)
-φ2π2,πorientation and is defined as the angle between the ellipse majoris the ellipse axis andxˆ   
tan 2 2E0xE0y2s φ=E20xE0ycoδwith
=
y
 x
-τ0,π4is the ellipse aperture, also called ellipticity, defined as
sin 2τ =2E20xE0y2sinδ E0x+E0y
6
(15)
(16)
What is Polarization?
1.2.2 Sense of rotation r As time elapses, the wave vectorE(z0,t in the () rotatesxˆ,yˆ) to describe the polarization ellipse. The time-dependent orientation ofE(z0,t) with respect toxˆ , named (t) is shown in Figure 7.
 
zˆ
r Figure 7Time-dependent rotation ofE(z0,t).
ˆ
r E(z0,t)
ξ(t)
xˆ
 
The time-dependent angle may be defined from the components of the wave vector in order to determine its sense of rotation. Ey Ez tycot kzy  tanξ(t)=Ex((z00,,t))=E00x((ssocωtkz00+ δx(1)7 ))
The sense of rotation may then be related to the sign of the variable ξ(tt)∝ −sinδ ⇒signξ(tt)=sign()τ 2with siτ2E0xE0ysinδ  n=E2+E2 0x0y
(18)
By convention, the sense of rotation is determined while looking in the direction of propagation. A right hand rotation corresponds then to(tt)>0(τ,δ)< a left0 whereas hand rotation is characterized by(tt)<0(τ,δ)>0 .
Figure 8provides a graphical description of the rotation sense convention.
7
What is Polarization?
 
 
yˆ
zˆ
ˆ
xˆ
xˆ
 
zˆ
 
 
zˆ
yˆ
ˆ
(a) (b) Figure 8(a) Left hand elliptical polarizations. (b) Right hand elliptical polarizations.
1.2.3
Quick estimation of a wave polarization state
xˆ
xˆ
 
zˆ
A wave polarization is completely defined by two parameters derived from the polarization ellipse
 -
- 
its orientatioππ n,φ2,2 
⎡ π its ellipticityτ4π4,, withsign(
) indicating the sense of rotation
The ellipse amplitude A can be used to estimate the wave power density.
The following procedure provides a quick (calculation free) way to roughly estimate a wave polarization. Three cases may be discriminated from the knowledge of=yx,EOx,EOy 
 
0, =
 
8
What is Polarization?
The polarization is linear since
=0 andφ= −atanE0yif  E0x
  ±δ =2a ndE0x=E0y 
= the orientation angle is given by0 andφ=tan1E0yif E0x
=
 
The polarization is circular, since ±τ =ontis  i  oftaro evigyb n4sn e ees dht nasign(
If< the 0 ,polarization is right circular, whereas for
If
 Otherwise
< the polarization is right elliptic, whereas for0 ,
1.2.4
Canonical polarization states
) .
>0 the polarization is left circular.
>0 the polarization is left elliptic.
In practice the axesxˆ andy generally referred to as the horizontal,ˆ arehcatil nˆd aer vvˆ directions.
 
 (a) Figure 9(a) Horizontal polarization (b) Vertical polarization.
 
 
 
 
 (a) Figure 10(a) Linear + 45° polarization. (b) Linear - 45° polarization.   
 
9
(b) 
(b) 
 
 
What is Polarization?
 
 
 (a) Figure 11(a) Right circular polarization. (b) Left circular polarization.
 
 
(b) 
 (a) (b) Figure 12(a) Right elliptical –45 °polarization. (b) Left elliptical +45 °polarization.
1.3
1.3.1
Jones vector
Definition
 
 
The representation of a plane monochromatic electric field under the form of a Jones vector aims to describe the wave polarization using the minimum amount of information. r A Jones vector,E, is defined from the time-space vectorE(z,t) as rwithr rj t E=E(0)E(z,t)= ℜE(z)eω (19) r From the formulation ofE(z,t in (10),) givenEcan be written as
=E0xejδxEEejδy 0y
(20)
The definitions of a polarization state from the polarization ellipse descriptors or from a Jones vector are equivalent.
A Jones vector can be formulated as a two-dimensional complex vector function of the polarization ellipse characteristics as follows :
10
What is Polarization?
Where
E ejcos cosjsin in A  =αsinφφcos+ττjcosφφinssττ
is an absolute phase term.
The Jones vector may be written under a more effective matrix form
1.3.2
1.3.2.1 
E=Aejαisnocsφ −snicsoφjniocsτ φ φsτ
Orthogonal polarization states and polarization basis
Orthogonal Jones vectors
(21)
(22)
Two Jones vectors,E1andE2scalar product is equal to 0,are orthogonal if their hermitian i.e.
E†1E2=0
(23)
with † the transpose conjugate operator. From the definition of a Jones vector given in (22), it is straightforward to remark that the orthogonality condition implies that ellipse parameters ofE1andE2satisfy
φ2= φ1+a dn 22= −1 
(24)
One may remark that the orthogonality condition does not depend on the absolute phase term of each Jones vector,1and2, i.e. ifE1 andE2are orthogonal, thenE1 andE2ejψare orthogonal too, for any value of .
1.3.2.2 Polarization basis
According to the definition of a Jones vector from the time-space electric field given in (19), any Jones vector expressed in the orthonormal basis (xˆ,yˆ) as E=Exxˆ+Eyy)25 (ˆ
A Jones vector defined in the basis (xˆ,y) ,ˆE(xˆ,yˆ)in the may defined from the unitary vector associated to the horizontal direction,xˆ
x E(xˆ,yˆ)=Aejαinscosφ −snsiocφsnicsoττjocsisnττˆ φ φj
This expression may be further developed ej E(xˆ,yˆ)=Aisnocsφφsosnicφφjsincosττjcossinττ0α The orthogonal Jones may be expressed from (24) as
11
0xˆ ejα 
(26)
(27)