wave polarimetrywave polarimetry · 2013. 8. 1. · complex polarisation plane. ℜ (ρ xˆ,yˆ)...

© E. Pottier, L. Ferro-Famil (01/2004)

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0

$x

$y( )rE z t,

$z

WAVE POLARIMETRYWAVE POLARIMETRYWAVE POLARIMETRY


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PROPAGATION EQUATIONPROPAGATION EQUATION

( )t,zEr

REAL ELECTRIC FIELD VECTOR

( ) ( )

( ) ( )( ) ( )( ) 0t,zB

t,zt,zDt,zJt,zHt

t,zBt,zE

T

=⋅∇=⋅∇

=∧∇

−=∧∇

r

r

rr

rr

ρ

∂∂

MAXWELL EQUATIONS

MAXWELL – FARADAY EQUATION

MAXWELL – AMPERE EQUATION

GAUSS THEOREM

( ) ( ) ( )

( ) ( )( ) ( )( ) ( )t,zEt,zD

t,zHt,zBt,zEt,zJ

tt,zDt,zJt,zJ

C

CT

rr

rr

rr

rr

εµσ

===

∂∂+=r

σ (Conductivity)µ (Permeability)ε (Permittivity)


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( ) ( ) ( )AAA ∇⋅∇−⋅∇∇=∧∇∧∇rrr

( ) ( ) ( ) ( )t

t,z1t

t,zEt

t,zEt,zE 2

22

∂ρ∂

ε∂∂µσ

∂∂µε −=−−∇

rrr

PROPAGATION EQUATION

( ) ( ) 0t

t,zEt,zE 2

22 =−∇

∂∂µε

rr

HELMOTZ PROPAGATION EQUATION

Source Free, Linear, Homogeneous, Isotropic,Dielectric and lossless Medium


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PROPAGATION EQUATIONPROPAGATION EQUATION

( ) ( )

ℜ= tjezEt,zE ωr

COMPLEX ELECTRIC FIELD VECTOR With:( )zE

HELMOTZ PROPAGATION EQUATION

( ) ( ) 0zEkzE 22 =+∇

SOLUTION: ( ) jkzeEzE −= With: E

=

=

z

y

x

joz

joy

jox

z

y

x

eEeEeE

EEE

δ

δ

δ

SINUSOIDAL PLANE WAVE

( ) 0z

E0t,zE z =

∂

∂⇒=⋅∇

r


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POLARISATION ELLIPSEPOLARISATION ELLIPSE

0

$x

$y( )rE z t,

$z


( )( )( )

=−−=−−=

=0E

kztcosEEkztcosEE

t,zE

z

yy0y

xx0x

δωδω

r


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z0

$y

$x

( )rE z t0 ,

0

$x

$y( )rE z t,

$z

THE REAL ELECTRIC FIELD VECTOR MOVES IN TIME ALONG AN ELLIPSE

( ) ( )δδ 2

2

y0

y

y0x0

yx2

x0

x sinEE

cosEEEE

2EE =

+−

xy δδδ −=With:


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A : WAVE AMPLITUDE

φ : ORIENTATION ANGLE τ : ELLIPTICITY ANGLE

0A

φα

τ

$x

$y

$,$zn

$y0

$x0( )rE z t, = 0

φ

α : ABSOLUTE PHASE

22πφπ ≤≤−

40 πτ ≤≤


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POLARISATION HANDENESSPOLARISATION HANDENESS

ROTATION SENSE: LOOKING INTO THE DIRECTION OF THE WAVE PROPAGATION

ANTI-CLOCKWISE ROTATION CLOCKWISE ROTATION

0

$x

$y

$z

+τ

−τ

LEFT HANDED POLARISATION RIGHT HANDED POLARISATION

ELLIPTICITY ANGLE : τ > 0 ELLIPTICITY ANGLE : τ < 0

44πτπ ≤≤−


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POLARISATION HANDENESSPOLARISATION HANDENESS

“…Circularly polarized waves have either a right-handed or left –handed sense, which is defined by convention. The TELSTAR satellite sent out circularly polarized microwaves. When it first

passed over the Atlantic, the British station at Goonhilly and the French station at Pleumeur Bodou both tried to receive its signals. The French succeeded because their definition of

polarization agreed with the American definition. The British station was set up to receive the wrong (orthogonal) polarisation

because their definition of sense…was contrary to ‘our’ definition…”

from J R Pierce “Almost Everything about Waves”, Cambridge MA, MIT Press, 1974, pp 130-131






Courtesy of Dr S.R. CLOUDE


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JONES VECTORJONES VECTOR


( )( )( )

=−−=−−=

=0E

kztcosEEkztcosEE

t,zE

z

yy0y

xx0x

δωδω

r

==

=y

x

joyy

joxx

eEEeEE

E δ

δ

( ) ( )

−ℜ= kztjeEt,zE ωr

With:

PHASOR = JONES VECTOR

GEOMETRICAL PARAMETERS

ABSOLUTE PHASE

xδα =

δφ cosEE

EE22tan 2

y02x0

y0x0

−=

2y0

2x0 EEA +=

δτ sinEE

EE22sin 2

y02x0

y0x0

+=

AMPLITUDE

ORIENTATION ANGLE ELLIPTICITY ANGLE

POLARISATION HANDENESS: Sign(τ)


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VERTICAL POLARISATION STATEHORIZONTAL POLARISATION STATE

=01

H

=10

V

02

=

=

τ

πφ0

rE z t( , )

$z

$y

$x

0

rE z t( , )

$z

$y

$x

00

==

τφ

ORTHOGONAL LINEAR POLARISATION STATELINEAR POLARISATION STATE

=θθ

sincos

L

−=⊥

θθ

cossin

L

0

rE z t( , )

$z

$y

$x

$ ′x$ ′y

θ

0

rE z t( , )

$z

$y

$x

$ ′x$ ′y

θ

02

=

+=

τ

πθφ0=

=τ

θφ


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LEFT CIRCULAR POLARISATION STATE RIGHT CIRCULAR POLARISATION STATE

4

22πτ

πφπ

+=

+≤≤−

=j1

21LC

−

=j

12

1RC

4

22πτ

πφπ

−=

+≤≤−0

rE z t( , )

$z

$y

$x0

rE z t( , )

$z

$y

$x

ORTHOGONAL ELLIPTICALPOLARISATION STATEELLIPTICAL POLARISATION STATE

′′

=⊥y

x

EE

E

04

2≤≤−

+=

τπ

πθφ

=

y

x

EE

E

0

rE z t( , )

$z

$y

$x

0

rE z t( , )

$z

$y

$x

40 πτ

θφ

+≤≤

=


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SU(2) MATRIX GROUPSU(2) MATRIX GROUP

Unitary Matrices Pauli Group

−=

=

−

=

=0j

j00110

1001

1001

3210 σσσσ

∗=− Ti

1i σσ 1)det( i =σProperties:

Multiplication Table:→⊗

0σ 1σ 2σ 3σ

0σ 0σ 1σ 2σ 3σ

1σ 1σ 0σ 3jσ 2jσ−2σ 2σ 3jσ− 0σ 1jσ

3σ 3σ 2jσ 1jσ− 0σ

Commutation Properties:

0ii σσσ =ijji σσσσ −=


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[ ] p0p )sin(j)cos(

jeA σασα

ασ+==

Special Unitary Matrices Group

330

je)(sinj)cos(

)cos()sin()sin()cos( φσ

σφσφφφφφ −

=−=

−

220

je)(sinj)cos(

)cos()sin(j)sin(j)cos( τσ

στστττττ +

=+=

110

je)(sinj)cos(je0

0je ασσασαα

α +=+=

−

+


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[ ] [ ] ∗=− T1 AA [ ] 1Adet +=Properties:

1T

12T

23T

3j

ej

ej

ej

ej

ej

eασαστστσφσφσ +

=++

=++

=−

112233 je

*je

je

*je

je

*je

ασαστστσφσφσ −=

+−=

+−=

−

−=

+

++

+=

++

pqq

p

ppp

je

je

je

je

)(je

σασασ

σ

βσασσβα{ }

qp

321qp ,,,

σσ

σσσσσ

≠

∈with:


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A=E xu

0$x

$y

$,$zn

( )rE z t, = 0

A

( ) ( )( ) ( )

−φφφφ

cossinsincos

A=E xu

0

A

φ$x

$y

$,$zn

$y0

$x0( )rE z t, = 0

φ

( ) ( )( ) ( )

−φφφφ

cossinsincos

A=E( ) ( )( ) ( )

ττττ

cossinjsinjcos

xu

0A

φ

τ

$x

$y

$,$zn

$y0

$x0( )rE z t, = 0

φ

( ) ( )( ) ( )

−φφφφ

cossinsincos

A=E( ) ( )( ) ( )

ττττ

cossinjsinjcos

−

α

α

j

j

e00e

xu

0A

φα

τ

$x

$y

$,$zn

$y0

$x0( )rE z t, = 0

φ


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Special Unitary Matrices Group and Jones Vector

−=

)sin(j)cos(

)cos()sin()sin()cos(jAeE )y,x( τ

τφφφφα

−

−=

01

je00je

)cos()sin(j)sin(j)cos(

)cos()sin()sin()cos(

AE )y,x( αα

ττττ

φφφφ

xj

ej

ej

AeE 123)y,x(

αστσφσ ++−=


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0A

φα

τ

$y

$,$zn

$y0

$x0( )rE z t, = 0

φ$x

x123 u

je

je

jAeE

αστσφσ ++−=


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POLARISATION RATIOPOLARISATION RATIO

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

joy

jox

y

x

ue00e

cossinjsinjcos

cossinsincos

A

eEeE

EE

Ey

x

−=

=

=

−

α

α

δ

δ

ττττ

φφφφ

COMPLEX POLARISATION RATIO

( )( ) ( ) ( )

( ) ( )τφτφρ δδ

tantanj1tanjtane

EE

EE

xyj

x0

y0

x

yy,x −

+=== −

INDEPENDENT OF AMPLITUDE AND ABSOLUTE PHASE


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POLARISATION RATIOPOLARISATION RATIO

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( )

xj

j*

2

xj

j

joy

jox

y

x

ue00e

11

1A

ue00e

cossinjsinjcos

cossinsincos

A

eEeE

EE

Ey

x

−

+=

−=

=

=

−

−

ξ

ξ

α

α

δ

δ

ρρ

ρ

ττττ

φφφφ

( ) ( )( ) ατφξ −= − tantantan 1With:

COMPLEX POLARISATION PLANE


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COMPLEX POLARISATION PLANECOMPLEX POLARISATION PLANE

0( )( )y,xρℜ

( )( )y,xρℑ ( )( ) ( )

( ) ( )τφτφρ

tantanj1tanjtan

y,x −+

=

( )( )( )

( )( )( )

2y,x

y,x2

y,x

y,x

122sin

122tan

ρ

ρτ

ρ

ρφ

+

ℑ=

−

ℜ=


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0

( )( )y,xρℑ

( )( )y,xρℜ

( ) cstEE

x0

y0y,x ==ρ

( )( ) cstarg y,x == δρ


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0C ( )( )y,xρℜ

( )( )y,xρℑ

cst0

=>

φφ

( )φ2cscr =

( )( )0,2cot φ−

LOCI

FAMILY of ORTHOGONAL CIRCLES

cst=φ

cst0

=<

φφ

Left Circular

Right Circular


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0

C


( )( )y,xρℜ

( )( )y,xρℑ

cst0

=<

ττ

cst0

=>

ττ

( )τ2cotr =

( )( )τ2csc,0

Right Circular

Left Circularcst=τ LOCI

FAMILY of ORTHOGONAL CIRCLES


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ORTHOGONAL JONES VECTORORTHOGONAL JONES VECTOR

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

joy

jox

y

x

ue00e

cossinjsinjcos

cossinsincos

A

eEeE

EE

Ey

x

−=

=

=

−

α

α

δ

δ

ττττ

φφφφ

POLARISATION ALGEBRA

0A,A

BAB,A

EEE*T

2y0

2x0

=

=

+=

⊥

NORM OF A JONES VECTOR

SCALAR PRODUCT

ORTHOGONALITY


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( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

ue00e

cossinjsinjcos

cossinsincos

AE

−=

−

α

α

ττττ

φφφφJONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) yj

j

ue00e

cossinjsinjcos

cossinsincos

AE

−=

−

⊥ α

α

ττττ

φφφφ

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

ue00e

cossinjsinjcos

sincoscossin

A

−

−

−−−

=−

α

α

ττττ

φφφφ

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

22

22 ue00e

cossinjsinjcos

cossinsincos

AE

−−−−

+++−+

=−

⊥ α

α

ππ

ππ

ττττ

φφφφ

ORTHOGONAL JONES VECTOR


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ORTHOGONALITY CONDITIONS

( )

−=′+=′

ττφφτφ

π2, a

CHANGE OF POLARISATION HANDENESS

0A

φ

τ

$x

$y

$,$zn

$y0 $x0

0

φ’ $x

$y

$,$zn

$y0

$x0τ’

x

yE

y

x

EE

EE

E′′

=

′′

=⊥⊥ ρa

x

yE

y

x

EE

EE

E =

= ρa

*E

E10E,E

ρρ −==

⊥⊥ a


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JONES VECTOR

xj

j

E

*E

2E

ue00e

11

1

AE

−

+=

−

ξ

ξ

ρρ

ρ

ORTHOGONAL JONES VECTOR

yj

j

E

*E

2E

xj

j

E

*E

2E

ue00e

11

1

A

ue00e

11

1

AE

−

+=

−

+=

−

−

⊥

⊥

⊥

⊥

ξ

ξ

ζ

ζ

ρρ

ρ

ρρ

ρ

( ) ( )( ) ( ) ( )( ) ατφζατφξ −=−= −−− tantantantantantan 111With:


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ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

ue00e

cossinjsinjcos

cossinsincos

AE

−=

−

α

α

ττττ

φφφφJONES VECTOR

ORTHOGONAL JONES VECTOR ( ) ( )( ) ( )

( ) ( )( ) ( ) yj

j

ue00e

cossinjsinjcos

cossinsincos

AE

−=

−

⊥ α

α

ττττ

φφφφ

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

[ ]yxj

j

u,ue00e

cossinjsinjcos

cossinsincos

AE,E

−=

−

⊥ α

α

ττττ

φφφφ

ELLIPTICAL BASIS TRANSFORMATION


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ORTHOGONAL JONES VECTORS

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

[ ]yxj

j

u,ue00e

cossinjsinjcos

cossinsincos

AE,E

−=

−

⊥ α

α

ττττ

φφφφ

SU(2) : SPECIAL UNITARY TRANSFORMATION MATRIX

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

−=

−

α

α

ττττ

φφφφ

j

j

e00e

cossinjsinjcos

cossinsincos

U

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

CONSERVATION OF THE WAVE ENERGY

ENSURES THE CORRECT PHASE DEFINITION

[ ][ ] [ ][ ]( ) 1Udet

IUU 2D*T

+=

=


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ORTHOGONAL JONES VECTORS

[ ] [ ]yxj

j

E

*E

2E

u,ue00e

11

1

AE,E

−

+=

−

⊥ ξ

ξ

ρρ

ρ

SU(2) : SPECIAL UNITARY TRANSFORMATION MATRIX

[ ]

−

+=

−

ξ

ξ

ρρ

ρ j

j

E

*E

2E

e00e

11

1

1U

( )[ ]ρ2U ( )[ ]ξ2U

CONSERVATION OF THE WAVE ENERGY

ENSURES THE CORRECT PHASE DEFINITION

[ ][ ] [ ][ ]( ) 1Udet

IUU 2D*T

+=

=


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yyxx uEuEE +=

REFERENCE BASIS ( )yx u,u

⊥⊥+= AAAA uEuEE

ELLIPTICAL BASIS ( )⊥AA u,u

( ) [ ]( )yxA,AAA u,uUu,u⊥⊥

=⊥⊥

+= BBBB uEuEE

ELLIPTICAL BASIS ( )⊥BB u,u

( ) [ ]( )yxB,BBB u,uUu,u⊥⊥

=

[ ]

=

−

⊥

⊥ y

x1A,A

A

A

EE

UEE [ ]

=

−

⊥

⊥ y

x1B,B

B

B

EE

UEE

[ ] [ ]

=

⊥

⊥⊥

⊥

−

A

AA,A

1B,B

B

B

EE

UUEE

SU(2) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX


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LINEAR BASIS

( ) ( )( ) ( )

( )( )

−=

ττ

φφφφα

sinjcos

cossinsincos

AeE j

( ) ( )( )( ) ( )( ) ( )

−+

= −

−

2j

jj

esincosesincos

2AeE πφ

φα

ττττ

[ ]

=1jj1

21U[ ]

−

=jj

112

1U

CIRCULAR BASIS

Ernst LÜNEBURG(PIERS95 - Pasadena)


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POLARISATION SYNTHESISPOLARISATION SYNTHESIS

⊗ =E E eG Gj G

0δ

0

δD

0E0DE0G

δG0

0101

rE z t( , )= 0

$x

$y

$x

$y

$x

$y

$x

$y

$x

$y

$uG $uD

⊗ =E E eD Dj D

0δ

E u tG G$ ( )= 0

E u tD D$ ( )= 0

⊕

$q$p


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STOKES VECTORSTOKES VECTOR

REAL REPRESENTATION OF THE POLARISATIONSTATE OF A MONOCHROMATIC WAVE

=⋅ *

yy*xy

*yx

*xx*T

EEEEEEEE

EE

PAULI MATRICES GROUP

−=

=

−

=

=0j

j00110

1001

1001

3210 σσσσ

{ }

−+−+

=+++=⋅1032

321033221100

*T

ggjggjgggg

21gggg

21EE σσσσ

{ g0, g1, g2, g3 } STOKES PARAMETERS


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JONES VECTOR

==

=y

x

joyy

joxx

eEEeEE

E δ

δ

( )( )

ℑ−=ℜ=

−=

+=

=

*yx3

*yx2

2y

2x1

2y

2x0

E

EE2gEE2gEEg

EEg

g

STOKES VECTOR

WAVE POLARISATION STATE ESTIMATIONFROM INTENSITIES MEASUREMENTS


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STOKES VECTOR

( )( )

====

=

==

−=+=

=

ττφτφ

δδ

2sinAg2cos2sinAg2cos2cosAg

Ag

sinEE2gcosEE2g

EEgEEg

g

23

22

21

20

y0x03

y0x02

2y0

2x01

2y0

2x00

E

GEOMETRICAL PARAMETERS

0

32

y02x0

y0x0

1

22

y02x0

y0x0

ggsin

EEEE

22sin

ggcos

EEEE

22tan

=+

=

=−

=

δτ

δφORIENTATION ANGLE

ELLIPTICITY ANGLE


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O(4) UNITARY ROTATION ROUPO(4) UNITARY ROTATION ROUP

HOMOMORPHISM SU(2) - O(3)

(σp, σq) : Pauli Matrices

( )[ ] ( )[ ] ( )[ ]( )q2p*T

2q,p3 UUTr212O σθσθθ =

[ ]

−=⇒

−=

−

1000)2cos()2sin(0)2sin()2cos(

)2(O)cos()sin()sin()cos(j

e 33 φφ

φφφ

φφφφφσ

[ ]

−=⇒

=+

)2cos(0)2sin(010

)2sin(0)2cos()2(O

)cos()sin(j)sin(j)cos(j

e 32

ττ

τττ

τττττσ

[ ]

−=⇒

−

+=

+

)2cos()2sin(0)2sin()2cos(0

001)2(Oje0

0jeje 3

1

αααααα

αασ


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O(4) UNITARY ROTATION ROUPO(4) UNITARY ROTATION ROUP

[ ] [ ][ ][ ]

123 je

je

je

je00je

)cos()sin(j)sin(j)cos(

)cos()sin()sin()cos(

)(U)(U)(UU

αστσφσ

αα

ττττ

φφφφ

ατφ

++−=

−

+

−=

=

[ ] [ ][ ][ ])2(O)2(O)2(OO 4444 ατφ=

[ ]

= )2(O

000

0001

)2(O3

4 χχwith:


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JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

j

ue00e

cossinjsinjcos

cossinsincos

AE

−=

−

α

α

ττττ

φφφφ

[U2(φ)] [U2(τ)] [U2(α)]



( )[ ] ( )[ ] ( )[ ]( )q2p*T


( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

xu

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

2E gAg

= −

−−

αα

ααττ

ττ

φφ

φφ

STOKES VECTOR

[O4(2φ)] [O4(2τ)] [O4(2α)]


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SAPHIR


VERTICAL POLARISATION STATEHORIZONTAL POLARISATION STATE

0

rE z t( , )

$z

$y

$x

02

== τπφ

=

0011

g H

−

=

001

1

gV0

rE z t( , )

$z

$y

$x

00 == τφ

ORTHOGONAL LINEAR POLARISATION STATELINEAR POLARISATION STATE

0

rE z t( , )

$z

$y

$x

$ ′x$ ′y ( )( )

=

02sin2cos

1

g L θθ

0

rE z t( , )

$z

$y

$x

$ ′x$ ′y

02

=+= τπθφ

( )( )

−−

=

02sin2cos

1

g L θθ

0== τθφ


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SAPHIR


LEFT CIRCULAR POLARISATION STATE RIGHT CIRCULAR POLARISATION STATE

0

rE z t( , )

$z

$y

$x

422πτπφπ +=+≤≤−

=

1001

g LC

0

rE z t( , )

$z

$y

$x

0

rE z t( , )

$z

$y

$x

ORTHOGONAL ELLIPTICALPOLARISATION STATE

042

≤≤−+= τππθφ

422πτπφπ −=+≤≤−

−

=

1001

g RC

−−−

=⊥

3

2

1E

ggg1

g

ELLIPTICAL POLARISATION STATE

0

rE z t( , )

$z

$y

$x

40 πτθφ +≤≤=

=

3

2

1E

ggg1

g


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SAPHIR


SPECIAL UNITARY SU(2) GROUP

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

−=

−

α

α

ττττ

φφφφ

j

j

2 e00e

cossinjsinjcos

cossinsincos

U

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U



( )[ ] ( )[ ] ( )[ ]( )q2p*T


O(4) UNITARY GROUP

[O4(2φ)] [O4(2τ)] [O4(2α)]

[ ] ( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

= −

−−

αα

ααττ

ττ

φφ

φφ

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

4O


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SAPHIR


( )yu,xuEgE ⇒


ELLIPTICAL BASIS ( )⊥AA u,u

( ) [ ]( )yxA,AAA u,uUu,u⊥⊥

=( )⊥

⇒A,A uuEgE

ELLIPTICAL BASIS ( )⊥BB u,u

( ) [ ]( )yxB,BBB u,uUu,u⊥⊥

=( )⊥

⇒B,B uuEgE

( ) ( )[ ] ( )[ ]( )⊥⊥⊥⊥

−=A,AAABBB,B uuEu,u4

1u,u4uuE gOOg

O(4) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX


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SAPHIR

POINCARE SPHEREPOINCARE SPHERE

STOKES VECTOR

( )( )

( )( )

=

−+

=

ℑ−ℜ

−

+

=

=

ττφτφ

δδ

2sinA2cos2sinA2cos2cosA

A

sinEE2cosEE2

EEEE

EE2EE2EE

EE

gggg

g

2

2

2

2

y0x0

y0x0

2y0

2x0

2y0

2x0

*yx

*yx

2y

2x

2y

2x

3

2

1

0

E

{g0} TOTAL WAVE INTENSITY

{g1, g2, g3 } POLARISED WAVE INTENSITIES

23

22

21

20 gggg ++= WAVE FULLY POLARISED

{g1, g2, g3 } Spherical Coordinates of a

point P on a sphere with radius g0


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SAPHIR


P

B

0

α T

θ T

A xQ

yQ

( )( )ℑ ρ $, $x y

( )( )ℜ ρ $, $x y

$y

$x

$z COMPLEX POLARISATION PLANE

Q

STEREOGRAPHIC PROJECTION


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SAPHIR


( )

( )

+=

ℑ

+=

ℜ

=

)2cos()2cos(1)2sin(m

)2cos()2cos(1)2sin()2cos(e

:Q

y,x

y,x

φττρ

φτφτρ

=

ττφτφ

2sin2cos2sin2cos2cos

:P

STEREOGRAPHICPROJECTION

$z

$x

$y

0B

P

Q

( )( )ℑ ρ $, $x y

( )( )ℜ ρ $, $x y

A


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SAPHIR


0$y

$x

$z

RC

-45°

H

+45°

V

LCNORTHERN HEMISPHERE

LEFT ELLIPTICAL POLARISATIONS

SOUTHERN HEMISPHERERIGHT ELLIPTICAL

POLARISATIONS

NORTH POLELEFT CIRCULARPOLARISATION

EQUATORLINEAR

POLARISATIONS

SOUTH POLERIGHT CIRCULARPOLARISATION


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SAPHIR


2α

2 α

g2

g1

0

g3

Eg$z

$y

$x

2τ 2τ2φ

( ) ( )( ) ( )

( ) ( )

( ) ( )( ) ( )( ) ( )

xu

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

2E gAg

= −

−−

αα

ααττ

ττ

φφ

φφ

=

=

ττφτφ


A

gggg

g

2

2

2

2

3

2

1

0

E

2φ

[O4(2φ)] [O4(2τ)] [O4(2α)]


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SAPHIR


PE

0

H Px= $

δ

$z

$x $y

2γ

2φ

2τ

( )( )

= δγ

γδjesin

cosjAeE x

( )( ) ( )( ) ( )

=

δγδγ

γ

sin2sinAcos2sinA

2cosAA

Eg

2

2

2

2STOKES VECTOR

DESCHAMPS PARAMETERS ( γ, δ )

JONES VECTOR

( ) ( ) ( )( ) ( ) ( )

==

δγτδγφ

sin2sin2sincos2tg2tg ( ) ( ) ( )

( ) ( )( )

=

=

φτδ

τφγ

2sin2tgtg

2cos2cos2cos


SAPHIR

SAPHIR


JONES VECTOR ORTHOGONAL JONES VECTOR

′′

=⊥y

x

EE

E

=

y

x

EE

EORTHOGONALITY CONDITIONS

( )

−=′+=′

ττφφτφ

π2, a

STOKES VECTOR ORTHOGONAL STOKES VECTOR

=

=

ττφτφ


A

gggg

g

3

2

1

0

E

−−−

=

=⊥

ττφτφ


A

gggg

g

3

2

1

0

E

ORTHOGONALITY = ANTIPODALITY


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SAPHIR


LC

-45°

g3

PV

+45°g22τ

2φg1

H

P⊥

0

$z

$x

$y

=

=

ττφτφ


A

gggg

g

3

2

1

0

E

−−−

=

=⊥

ττφτφ


A

gggg

g

3

2

1

0

E

STOKES VECTOR

ORTHOGONAL STOKES VECTOR

RC

ORTHOGONALITY = ANTIPODALITY


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SAPHIR

POINCARE PLANISPHEREPOINCARE PLANISPHERE

0

0 $y$x

$z

POINCARE SPHERE : 3D SPACE

GNOMONIC PROJECTION(Elliptical Aitoff-Hammer Projection)

(Meridional Lambert Projection)

=

−=

)2

cos(2)2sin(y

)2

cos(

)2(sin)(sin2x

P

22

P

ατ

ατα

)2cos()cos()cos( τφα =With:

POINCARE PLANISPHERE2D SPACE


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SAPHIR

POINCARE PLANISPHEREPOINCARE PLANISPHERE

0V+45°-45°V H

LC

y

2φ π=2 23

φ π=

22

φ π=

23

φ π=2φ π= − 2 2

3φ π

= −

22

φ π= −

23

φ π= −

26

τ π= −

22

τ π= −

23

τ π= −

26

τ π=

22

τ π=

23

τ π=

x

RC


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SAPHIR

CANONICAL BASISCANONICAL BASIS

CARTESIAN BASIS

POLARIZATION

STATE

( )E x y$ , $

( )φ$ , $x y

( )τ$ , $x y

( )γ$ , $x y

( )δ$ , $x y

( )ρ$ , $x y

Horizontal

1

0

0

0

0

0

0

Vertical

0

1

π / 2

0

π / 2

0

∝

Linear +45° 1

2

1

1

π / 4

0

π / 4

0

1

Linear +135°

12

11

−

3π / 4

0

π / 4

π

-1

Left Circular 1

2

1

+

j

?

π / 4

π / 4

π / 2

j

Right Circular 1

2

1

−

j

?

-π / 4

-π / 4

-π / 2

-j


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SAPHIR


CIRCULAR BASIS

POLARISATION

STATE

( )RL u,uE

( )RL u,uφ

( )RL u,uτ

( )RL u,uγ

( )RL u,uδ

( )RL u,uρ

Horizontal 1

2

1

−

j

?

-π / 4

π / 4

-π / 2

-j

Vertical 1

2 1

−

j

?

π / 4

π / 4

π / 2

j

Linear +45° 1

2

1

1

−

−

j

j

π / 4

0

π / 4

0

1

Linear +135°

12

11

− −+

jj

3π / 4

0

π / 4

π

-1

Left Circular

1

0

0

0

0

0

0

Right Circular

0

1

π / 2

0

π / 2

0

∝


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SAPHIR


LINEAR 45°/135° BASIS

POLARIZATION

STATE

( )°+°− 13545 u,uE

( )°+°− 13545 u,u

φ

( )°+°− 13545 u,u

τ

( )°+°− 13545 u,u

γ

( )°+°− 13545 u,u

δ

( )°+°− 13545 u,u

ρ

Horizontal 12

1

1−

-π / 4

0

π / 4

π

-1

Vertical 1

2

1

1

π / 4

0

π / 4

0

1

Linear +45°

1

0

0

0

0

0

0

Linear +135°

0

1

π / 2

0

π / 2

0

∝

Left Circular 1

2

1

1

+

− +

j

j

?

π / 4

π / 4

π / 2

j

Right Circular 1

2

1

1

−

− −

j

j

?

-π / 4

π / 4

-π / 2

-j


SAPHIR

SAPHIR


( )yu,xuEgE ⇒


[ ]

−=

1111

21U[ ]

=1jj1

21U

( )

( )

+−=

−=

yxR

yxL

EEj2

1E

EjE2

1E ( )

( )

+−=

+=

°

°

yx135

yx45

EE2

1E

EE2

1E

CIRCULAR BASIS LINEAR 45°/135° BASIS


SAPHIR

SAPHIR


( )yu,xuEgE ⇒


( )[ ]

−

∗ℜ+

∗ℑ+

+

=

2R

2L

RL

RL

2R

2L

u,u

EE

EEe2

EEm2

EE

EgRL ( )[ ]

∗ℑ−

−

∗ℜ−

+

=

°°

°°

°°

°°

°°

13545

2135

245

13545

2135

245

u,u

EEm2

EE

EEe2

EE

Eg13545

CIRCULAR BASIS LINEAR 45°/135° BASIS


SAPHIR

SAPHIR


STOKES VECTOR COMPONENTS

g1 g2 g3

Cartesian Basis

( ) ( )y,xy,xττφφ == cos( )cos( )2 2φ τ sin( )cos( )2 2φ τ sin( )2τ

( ) ( )y,xy,xδδγγ == cos( )2γ sin( )cos( )2γ δ sin( )sin( )2γ δ

Circular Basis

( ) ( )RLRL u,uu,uττφφ == −sin( )2τ sin( )cos( )2 2φ τ cos( )cos( )2 2φ τ

( ) ( )RLRL u,uu,uδδγγ == − sin( ) sin( )2γ δ sin( )cos( )2γ δ cos( )2γ

Linear (+45°,+135°) Basis

( ) ( )°+°+°+°+==

1354513545 u,uu,uττφφ −sin( )cos( )2 2φ τ cos( )cos( )2 2φ τ sin( )2τ

( ) ( )°+°+°+°+==

1354513545 u,uu,uδδγγ

− sin( )cos( )2γ δ cos( )2γ sin( )sin( )2γ δ


SAPHIR

SAPHIR


DESCHAMPS VECTOR COMPONENTS

PE

0

H = Px

δ

2γ(x,y)

(x,y) +45 = P+45

LC = PL

2γ(L,R)

2γ(45,135) δ (45,135)

δ (L,R)


SAPHIR

SAPHIR

POLARIZATION MATCHINGPOLARIZATION MATCHING

Voltage Equation *i

TiOC EhE,hV ==

yT

Ei

h

yR

xR

xT

iETh

2OCOC gg

21VP ==Power Equation

with: h: Complex Effective Height of AntennaEi: Incident Jones Vector on the receive antennagh, gE : Associated Stokes vectors

h and Ei must be expressed in the same reference coordinates system


SAPHIR

SAPHIR


Ex: Left Circulary Polarized Antennas

yT

Ei

hEi’xR

yR

yRxR

xT

Co-Ordinate Reversal

−=

−=⇒

=j1

21E

1001

Ej1

21E i

'ii

[ ] 0j1

j121EhE,hV '*

iT'

iOC =

−−

===


SAPHIR

SAPHIR


Ex: Left Circulary Polarized Antennas

yT

Ei

hEi’’xR

yR

yRxR

xT

Time Reversal

( )

−−

==⇒

−=

j1

21EE

j1

21E

*'i

"i

'i

[ ] 1j1

j121EhE,hV '*

iT'

iOC −=

+−

===


SAPHIR

SAPHIR


yT

Ei

hEi’’

yR

xR

yR

xR

xT

JONES Vector does not describe the direction of wave propagation

2 Separate Features to remember

i'i E

1001

E

−=Co-ordinates Reversal

Time Reversal ( )*'i

"i EE =

( )*jkzjkzjkz eee −+− =aWave propagation


SAPHIR

SAPHIR


yT

Ei

hEi’’

yR

xR

yR

xR

xT

( )( )

−=−=

==

=⇒

ℑ−ℜ

−

+

=

=

3"3

2"2

1"1

0"0

"E

*yx

*yx

2y

2x

2y

2x

3

2

1

0

E

gggg

gggg

g

EE2EE2EE

EE

gggg

gii

2"*i

T2"i

2OCE

ThOC EhE,hVgg

21P "

i====


SAPHIR

SAPHIR












Courtesy of Dr S.R. CLOUDE


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SAPHIR

PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES

0

$x$y

$z 0

$x$y

$z

DETERMINISTIC SCATTERING RANDOM SCATTERING

COMPLETELY POLARISED WAVE PARTIALLY POLARISED WAVE

Polarisation Ellipse varies in timeAmplitude, Phase: Random processes

STATISTICAL DESCRIPTION


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SAPHIR


{ }EJONES VECTORS

WAVE COVARIANCE MATRIX

[ ]

== 2

y*xy

*yx

2x*T

EEE

EEEEEJ

[ ]

−+

−+=

1032

3210

gggjggjggg

21J

PARTIALLY POLARISED WAVES

23

22

21

20 gggg ++≥


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SAPHIR


WAVE COVARIANCE MATRIX

[ ]

== 2

y*xy

*yx

2x*T

EEE

EEEEEJ

DIAGONAL ELEMENTS : INTENSITIES ON EACH OF THE 2 ORTHOGONALCOMPONENTS OF THE WAVE

OFF-DIAGONAL ELEMENTS : CROSS-CORRELATIONS BETWEEN THE 2ORTHOGONAL COMPONENTS OF THE WAVE

[ ]( ) 22y

2x AEEJTrace =+= TOTAL WAVE INTENSITY

THE WAVE COVARIANCE MATRIX IS A 2x2 HERMITIAN POSITIVE SEMI-DEFINITE MATRIX


SAPHIR

SAPHIR


EIGENVALUES DECOMPOSITION

[ ] [ ] [ ] *T222

*T111

12

2

12 uuuuU

00

UJ λλλ

λ+=

= −

[ ] [ ]212 u,uU =2 ORTHOGONAL EIGENVECTORS

{ }{ }2

32

22

102

23

22

2101

gggg21

gggg21

++−=

+++=

λ

λ2 REAL EIGENVALUES


SAPHIR

SAPHIR


PARTIALLY POLARISED WAVES DESCRIPTORS

Degree of Polarisation

Anisotropy

[ ]( )[ ]( )

−=+−

=++

=JTrace

Jdet41g

gggDoP 2

21

21

0

23

22

21

λλλλ

Polarised Wave PowerTotal Wave Power

Wave Entropy

( )∑=

=

−=2i

1ii2i plogpH

21

iip

λλλ+

=With:

Degree of randomness, statistical disorder


SAPHIR

SAPHIR


COMPLETELY POLARISED WAVES

[ ]( )

==

⇒

=≠

⇒=⇒=0H

1DoP00

0JdetEEEEEE2

1*xy

*yx

2y

2x λ

λyMaximum Correlation Between andxE E

COMPLETELY UNPOLARISED WAVESAbsence of any Polarised Structure in the Wave

[ ]( ) [ ]( )

==

⇒=⇒=⇒

==

=

1H0DoP

4JTraceJdet

0EEEE

EE21

2

*xy

*yx

2y

2x λλ

PARTIALLY POLARISED WAVES

[ ] [ ]( )

≤≤≤≤

⇒

≥≠≥

⇒

=

1H01DoP0

00Jdet

EEE

EEEJ

212

y*xy

*yx

2x

λλ

yCorrelation between andxE E


SAPHIR

SAPHIR


WAVE DECOMPOSITION = WAVE DICHOTOMYBorn & Wolf Decomposition – Shandrasekar Decomposition

[ ]2D*T

22*T

11 Iuuuu =+

[ ]( ) [ ]2D2

*T1121

*T222

*T111

IuuuuuuJ

λλλλλ

+−=+=

COMPLETELYUNPOLARISED

WAVE

COMPLETELYPOLARISED

WAVE

PARTIALLYPOLARISED

WAVE


SAPHIR

SAPHIR


[ ] ( ) [ ] [ ] [ ]CUCP2D2*T

1121 JJIuuJ +=+−= λλλPARTIALLY POLARISED WAVE

COMPLETELY POLARISED WAVE

[ ]

−+++

−+++=

12

32

22

132

3212

32

22

1CP

gggggjg

gjggggg21J

COMPLETELY UNPOLARISED WAVE

[ ]

++−

++−=

23

22

210

23

22

210

CUgggg0

0gggg21J


SAPHIR

SAPHIR


STOKES VECTOR

++−

+

++

=

000

gggg

ggg

ggg

gggg 2

32

22

10

3

2

1

23

22

21

3

2

1

0

COMPLETELYUNPOLARISZED

WAVE

COMPLETELYPOLARISED

WAVE

PARTIALLYPOLARISED

WAVE


SAPHIR

SAPHIR

PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVESCOMPLETELY UNPOLARISED WAVE

++−

000

gggg 23

22

210

−−−

++−

+

++−

3

2

1

23

22

210

3

2

1

23

22

210

qqq

gggg

21

qqq

gggg

21

2 ORTHOGONAL COMPLETELY POLARISED WAVES


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SAPHIR

PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVESWAVE DECOMPOSITION = WAVE DICHOTOMY

−−−

′−

+

′−

+

′

=

3

2

1

00

3

2

1

00

3

2

1

0

3

2

1

0

qqq

gg

21

qqq

gg

21

gggg

gggg

PARTIALLYPOLARISED

WAVE

COMPLETELYUNPOLARISED

WAVE

2 ORTHOGONAL COMPLETELYPOLARISED WAVES

44444 344444 21

COMPLETELYPOLARISED

WAVE

23

22

21

20 gggg ++=′With:


SAPHIR

SAPHIR


g1

g2

g3

P

g1

g2

g3

P

g1

g2

g3

P

SphereP ∈

SphereP ⊂

CenterP =

COMPLETELY POLARISED WAVESThe total wave energy is polarised

g g g g0 1

2

2

2

3

2= + +H = 0 DoP = 1

PARTIALLY POLARISED WAVESA part of the total wave energy is polarised

g g g g0 1

2

2

2

3

2≥ + +0 < H < 1 0 < DoP < 1

COMPLETELY UNPOLARISED WAVESThe total wave energy is unpolarised

g g g1

2

2

2

3

2 0+ + =

H = 1 DoP = 0


SAPHIR

SAPHIR

WAVE DESCRIPTORSWAVE DESCRIPTORSMONOCHROMATIC PLANE WAVES

COMPLEX DOMAIN REAL DOMAIN

=

3

2

1

0

E

gggg

g

=

y

x

EE

EJONES VECTOR STOKES VECTOR

( ) ( ){ }321

xyx0x0

g,g,g,,A,,A

,E,Eor

⋅⋅

−=⋅δγτφ

δδδ

PLANE WAVES FULLY DESCRIBEDBY 3 INDEPENDANT PARAMETERS

WAVE POLARIMETRIC DIMENSION = 3


SAPHIR

SAPHIR

WAVE DESCRIPTORSWAVE DESCRIPTORSPARTIALLY POLARISED PLANE WAVES

COMPLEX DOMAIN REAL DOMAIN

=

3

2

1

0

E

gggg

gSTOKES VECTOR

PLANE WAVES FULLY DESCRIBEDBY 4 INDEPENDANT PARAMETERS

{ }3210

2y

*xy

*yx

2x

g,g,g,g

E,EE,EE,E

⋅

⋅

[ ] *TEEJ =COVARIANCE MATRIX

WAVE POLARIMETRIC DIMENSION = 4


SAPHIR

SAPHIR

Questions ?Questions ?

wave polarimetrywave polarimetry · 2013. 8. 1. · complex polarisation plane. ℜ (ρ xˆ,yˆ)...

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