TUTORIAL ON SAR POLARIMETRY

Laurent Ferro-Famil

Description

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) ...

Informations

• Publié par: Viwyur
• Nombre de lectures: 18
• Langue: English

Extrait

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 ∇ r ∧E r (r r ,t)=− ∂B ∂ (r t ,t) ∇ r ∧H r (r r ,t)=J r T (r r ,t) + ∂D ∂ ( t r r ,t) 1)   ( r r r r ∇ ⋅D(r r ,t)= ρ(r r ,t)∇ ⋅B(r r ,t)=0 r r r r where E(r r ,t),H(r r ,t),D(r r ,t),B(r r ,t ) are the wave electric field, magnetic field, electric induction and magnetic induction respectively. r r The total current density, J T (r r ,t)=J a (r,t)+J c (r,t ) is composed of two terms. The first one, r J a (r r ,t ) , corresponds to a source term, whereas the conduction current density, r r J c (r r ,t)= σE(r r ,t ) , depends on the conductivity of the propagation medium, . The scalar field (r 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 r D(r r ,t)= εE(r r ,t)+P(r r ,t)B(r r ,t)= μH(r r ,t)+M(r r ,t) r r The vectorsP(r r ,t) andM(r r ,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(r r ,t)=P(r r ,t)=0 and J a (r r ,t)= 0 .

The equation of propagation is found by inserting r r r r r r r ∇ ∧(∇ ∧E(r r ,t))= ∇(∇ ⋅E(r r ,t))− ΔE(r r ,t is formulated as) and ΔE r (r r ,t ∂ με)− 2 ∂ E r t ( 2 r r ,t ) −μ ∂ σ E r ∂t r r ,t ) =− 1 ∂ ε r ∂ ρ∇ t ( r r ,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 t r 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(r r of the monochromatic time-space electric field, ) ,E(r r ,t) , defined as r r

E(r r ,t)= ℜE(r r )e jωt  

The propagation equation mentioned in (3) may then be rewritten as ΔE r ( r r ) +ω 2 ⎜ ⎝ εμ ⎛ 1−j ⎠ ⎞ ⎟ ωε σ E r ( r r ) =  ΔE r ( r r ) + k 2 E r ( r r )  with k ω =1−j σ    vωε

(4)

(5)

Here appears the concept of complex dielectric constant ε=ε ′ −j ε′′ =  ε−j then k ω =1−j− ′ = ε β ′ jα  (6) ωvε In a general way, a monochromatic plane wave , with constant complex amplitude, E r 0 =E à e jδ r , propagating in the direction of the wave vector, k  ,owinfoll ˆ g ht eah sel xocpm form r E r (r r )=E r 0 e −j k r ⋅r r withE(r r )⋅k ˆ =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 (x ˆ, y ˆ, zso that the direction of propagation ˆ) defined k ˆ zThe expression of the electric field ˆ .  = becomes E r (z)=E r 0 e −αz e −jβz withE 0z = )8 0  ( 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

E r (z,t) ⎢ ⎢ ⎢ = ⎣ ⎡ E E 0 0y x e e − − za za os( 0 c oc(s ω ωt t − − kz zkδ + + δ x y ) ) ⎥ ⎥ ⎥ ⎤   ⎦

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 ,

⎡ E 0x cos(ωt−kz E r (r,t)= ⎣ ⎢ ⎢ ⎢ E 0y cos(ωt 0 −kz

δ ⎤ + δ x ) ⎥  + y ) ⎦ ⎥ ⎥

1.1.3 Spatial evolution of a plane wave vector: helicoidal trajectory

(10)

At a fixed time, t=t 0 , the electric field is composed of two orthogonal sinusoidal waves with, in general, different amplitudes and phases at the origin.   $ y

0

E y

( z,t )

$ x

Figure 1Spatial evolution of a monochromatic plane wave components.

Three particular cases are generally discriminated:

• Linear polarization:

= y − x =0+m

 

E x ( z,t )

$ z

 

The electric field is then a sine wave inscribed within a plane oriented with an angle respect tox ˆ

 

r E(z 0 ,t)=

⎡ ⎤ E 0 2 x +E 2 0y ⎢ ⎢ ⎢ s ni oc s 0 ⎥ ⎥ φ ⎥ φ cos(ωt 0 −kz+ δ x )   ⎣ ⎦

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  and E 0x =E 0y  

x ( z,t ) $ z  

In this case, the wave has a constant modulus and is oriented with an angle to thex ˆ axis r 2 2 E(z,t 0 )=E 0x +E 0y and ( z)= ±(t à −kz+ x )

 

 

$ y

E y ( z,t )

Figure 3Spatial evolution of a circularly polarized plane wave.

The wave rotates circularly around thez ˆ axis.

• Elliptic polarization: Otherwise

The wave describes a helicoidal trajectory around thez ˆ axis.

4

r E ( z,t )

E x ( z,t )

(z respect) with

$ z

 

(12)

What is Polarization?

 

$ y

E y ( z,t )

Figure 4Spatial evolution of a elliptically polarized plane wave.

1.2

1.2.1

Polarization ellipse

Geometrical description

E r ( z,t )

E x ( z,t )

$ z

 

The former paragraph introduced the spatial evolution of a plane monochromatic wave and showed that it follows a helicoidal trajectory along the z  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=z 0 is generally preferred. $   y E r ( z, t )

$ y

0

Figure 5Temporal trajectory of a monochromatic plane wave at a fixed abscissa z=z 0 .

r ( z 0 ,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 the z  As time evolves, the wave propagates ˆ 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 of E(z 0 ,t)

5

What is Polarization?

) ⎛ ⎜ ( 0 , ) 2 ( 2 0  (, ) 0 so( , c) ⎜ ( 0 , ) ⎞ 2 ⎝ E x E z 0 x t ⎠ ⎞ ⎟ − E x z E 0 t x E E 0 y y z t δ y −δ x ⎝ ⎜ ⎛ + E y z E 0y t ⎟ ⎠ ⎟

=sin(δ y −δ x )

(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

E 0y

x ˆ

 

is called the ellipse amplitude and is determined from the ellipse axis as

2 A=E 0x +E 2 0y  

(14)

- ⎢ ⎣ ⎡ −∈φ 2 π 2 , ⎥ ⎦ ⎤ π orientation and is defined as the angle between the ellipse majoris the ellipse axis andx ˆ   

tan 2 2 E 0x E 0y 2 s φ= E 2 0x −E 0y coδ with

=

− y

  x

- ⎡ ∈ ⎢ ⎣ τ0, π 4 ⎦ ⎤ ⎥ is the ellipse aperture, also called ellipticity, defined as

sin 2τ =2 E 2 0x E 0y 2 sinδ   E 0x +E 0y

6

(15)

(16)

What is Polarization?

1.2.2 Sense of rotation r As time elapses, the wave vector E(z 0 ,t  in the () rotatesx ˆ, y ˆ) to  describe the polarization ellipse. The time-dependent orientation of E(z 0 ,t ) with respect tox ˆ ,  named (t) is shown in Figure 7.

 

zˆ

r Figure 7Time-dependent rotation of E(z 0 ,t) .

ˆ

r E(z 0 ,t)

ξ(t)

x ˆ

 

The time-dependent angle may be defined from the components of the wave vector in order to determine its sense of rotation. E y  Ez t y cot kz y   tanξ(t)= E x ( (z 0 0 , , t) ) = E 0 0 x ( (s socωt−kz 0 0 + δ x (1 ) 7   ) )

The sense of rotation may then be related to the sign of the variable ∂ ξ ∂ ( t t ) ∝ − sin δ ⇒sign ⎛ ⎜ ⎝ ∂ξ ( t t ) ⎟ = ⎞ ⎠ −sign () τ  2with siτ2 E 0x E 0y sinδ   ∂ n= E 2 +E 2 0x0y

(18)

By convention, the sense of rotation is determined while looking in the direction of propagation. A right hand rotation corresponds then to ∂ ∂(t t ) >0⇒(τ,δ)<  a left0 whereas hand rotation is characterized by ∂ ∂(t t ) <0⇒(τ,δ)>0 .

Figure 8provides a graphical description of the rotation sense convention.

7

What is Polarization?

 

 

y ˆ

zˆ

ˆ

x ˆ

x ˆ

 

zˆ

 

 

zˆ

y ˆ

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

x ˆ

x ˆ

 

zˆ

A wave polarization is completely defined by two parameters derived from the polarization ellipse

 -

- 

its orientatio ⎡ −∈ ππ n, φ ⎢ ⎣ 2 , 2 ⎥ ⎦ ⎤  

⎡ π its ellipticity −∈ ⎣ τ ⎢ 4 π 4 , ⎤ ⎥ ⎦ , 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 = y − x , E Ox , E Oy  

 •

0, =

 

8

What is Polarization?

The polarization is linear since

=0 and φ= −atan ⎜ ⎛ E 0y ⎞ ⎟ if   ⎝ E 0x ⎠

•   ±δ = 2 a nd E 0x =E 0y  

= the orientation angle is given by0 and φ=tan −1 ⎛ ⎜ E 0y ⎟ ⎠ ⎞ if ⎝ E 0x

=

 

The polarization is circular, since  ±τ = ontis  i  oftaro evigyb n 4 sn 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 axesx ˆ and y generally referred to as the horizontal, ˆ are hcatil n ˆ d aer vv ˆ 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 r with r r j t E=E(0)E(z,t)= ℜE(z)e ω  (19) r From the formulation of E(z,t  in (10),) givenEcan be written as

= ⎢ ⎢ ⎣ ⎡ E 0x e jδ x ⎥ ⎦ ⎤ ⎥ E Ee jδ y   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 e j ⎡ cos cosjsin in A   = α ⎣ ⎢ sin φ φcos+ τ τ − jcos φ φins s ⎤ ⎦ ⎥ τ τ

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=Ae jα ⎢ ⎣ ⎡ isn ocsφ −sni cso ⎡ ⎣ ⎢ φ ⎥ ⎦ ⎤ jni ocsτ ⎤ ⎦ ⎥   φ φsτ

Orthogonal polarization states and polarization basis

Orthogonal Jones vectors

(21)

(22)

Two Jones vectors,E 1 andE 2 scalar product is equal to 0,are orthogonal if their hermitian i.e.

E † 1 E 2 =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 ofE 1 andE 2 satisfy

φ 2 = φ 1 + a dn  2 2 = − 1  

(24)

One may remark that the orthogonality condition does not depend on the absolute phase term of each Jones vector, 1 and 2 , i.e. ifE 1  andE 2 are orthogonal, thenE 1  and E 2 e jψ are 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 (x ˆ, y ˆ) as E=E x x ˆ +E y y )25   ( ˆ

A Jones vector defined in the basis (x ˆ, y) , ˆ E (x ˆ, y ˆ) in the may defined from the unitary vector associated to the horizontal direction,x ˆ

x   E (x ˆ, y ˆ) =Ae jα ⎡ ⎣ ⎢ ins cosφ − s nsi oc ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ φ sni csoτ τ j ocs isn ⎤ ⎦ τ ⎥ τ ˆ φ φj

This expression may be further developed e j E (x ˆ, y ˆ) = A ⎡ ⎣ ⎢ isn ocs−φ φso sni c ⎣ φ ⎡ ⎢ ⎤ ⎥ φ ⎦ jsin cosτ τ j cos sin ⎢ ⎥ τ ⎦ ⎣ ⎡ ⎤ τ − 0 α The orthogonal Jones may be expressed from (24) as

11

0 ⎤ ⎥ x ˆ e jα ⎦  

(26)

(27)