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© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
0
$x
$y( )rE z t,
$z
WAVE POLARIMETRYWAVE POLARIMETRYWAVE POLARIMETRY
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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)
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
( ) ( ) ( )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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARISATION ELLIPSEPOLARISATION ELLIPSE
0
$x
$y( )rE z t,
$z
REAL ELECTRIC FIELD VECTOR
( )( )( )
=−−=−−=
=0E
kztcosEEkztcosEE
t,zE
z
yy0y
xx0x
δωδω
r
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARISATION ELLIPSEPOLARISATION ELLIPSE
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:
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARISATION ELLIPSEPOLARISATION ELLIPSE
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 ≤≤
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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 ≤≤−
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
“…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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
JONES VECTORJONES VECTOR
REAL ELECTRIC FIELD 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)
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
JONES VECTORJONES VECTOR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
JONES VECTORJONES VECTOR
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
$x
40 πτ
θφ
+≤≤
=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
SU(2) MATRIX GROUPSU(2) MATRIX GROUP
Unitary Matrices Pauli Group
−=
=
−
=
=0j
j00110
1001
1001
3210 σσσσ
∗=− Ti
1i σσ 1)det( i =σProperties:
Multiplication Table:→⊗
0σ 1σ 2σ 3σ
0σ 0σ 1σ 2σ 3σ
1σ 1σ 0σ 3jσ 2jσ−2σ 2σ 3jσ− 0σ 1jσ
3σ 3σ 2jσ 1jσ− 0σ
Commutation Properties:
0ii σσσ =ijji σσσσ −=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
SU(2) MATRIX GROUPSU(2) MATRIX GROUP
[ ] p0p )sin(j)cos(
jeA σασα
ασ+==
Special Unitary Matrices Group
330
je)(sinj)cos(
)cos()sin()sin()cos( φσ
σφσφφφφφ −
=−=
−
220
je)(sinj)cos(
)cos()sin(j)sin(j)cos( τσ
στστττττ +
=+=
110
je)(sinj)cos(je0
0je ασσασαα
α +=+=
−
+
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
SU(2) MATRIX GROUPSU(2) MATRIX GROUP
[ ] [ ] ∗=− T1 AA [ ] 1Adet +=Properties:
1T
12T
23T
3j
ej
ej
ej
ej
ej
eασαστστσφσφσ +
=++
=++
=−
112233 je
*je
je
*je
je
*je
ασαστστσφσφσ −=
+−=
+−=
−
−=
+
++
+=
++
pqq
p
ppp
je
je
je
je
)(je
σασασ
σ
βσασσβα{ }
qp
321qp ,,,
σσ
σσσσσ
≠
∈with:
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
JONES VECTORJONES VECTOR
A=E xu
0$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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
JONES VECTORJONES VECTOR
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 ++−=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
JONES VECTORJONES VECTOR
0A
φα
τ
$y
$,$zn
$y0
$x0( )rE z t, = 0
φ$x
x123 u
je
je
jAeE
αστσφσ ++−=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
COMPLEX POLARISATION PLANECOMPLEX POLARISATION PLANE
0
( )( )y,xρℑ
( )( )y,xρℜ
( ) cstEE
x0
y0y,x ==ρ
( )( ) cstarg y,x == δρ
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
COMPLEX POLARISATION PLANECOMPLEX POLARISATION PLANE
0C ( )( )y,xρℜ
( )( )y,xρℑ
cst0
=>
φφ
( )φ2cscr =
( )( )0,2cot φ−
LOCI
FAMILY of ORTHOGONAL CIRCLES
cst=φ
cst0
=<
φφ
Left Circular
Right Circular
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
0
C
COMPLEX POLARISATION PLANECOMPLEX POLARISATION PLANE
( )( )y,xρℜ
( )( )y,xρℑ
cst0
=<
ττ
cst0
=>
ττ
( )τ2cotr =
( )( )τ2csc,0
Right Circular
Left Circularcst=τ LOCI
FAMILY of ORTHOGONAL CIRCLES
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ORTHOGONAL JONES VECTORORTHOGONAL JONES VECTOR
( ) ( )( ) ( )
( ) ( )( ) ( ) 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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ORTHOGONAL JONES VECTORORTHOGONAL JONES VECTOR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ORTHOGONAL JONES VECTORORTHOGONAL JONES VECTOR
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:
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION
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
+=
=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION
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
+=
=
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION
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)
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
STOKES VECTORSTOKES VECTOR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
STOKES VECTORSTOKES VECTOR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
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:
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
STOKES VECTORSTOKES VECTOR
JONES VECTOR
( ) ( )( ) ( )
( ) ( )( ) ( ) xj
j
ue00e
cossinjsinjcos
cossinsincos
AE
−=
−
α
α
ττττ
φφφφ
[U2(φ)] [U2(τ)] [U2(α)]
HOMOMORPHISM SU(2) - O(3)
(σp, σq) : Pauli Matrices
( )[ ] ( )[ ] ( )[ ]( )q2p*T
2q,p3 UUTr212O σθσθθ =
( ) ( )( ) ( )
( ) ( )
( ) ( )( ) ( )( ) ( )
xu
2cos2sin00
2sin2cos0000100001
2cos02sin00100
2sin02cos00001
000002cos2sin0
02sin2cos00001
2E gAg
= −
−−
αα
ααττ
ττ
φφ
φφ
STOKES VECTOR
[O4(2φ)] [O4(2τ)] [O4(2α)]
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
STOKES VECTORSTOKES VECTOR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
STOKES VECTORSTOKES VECTOR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION
SPECIAL UNITARY SU(2) GROUP
[ ] ( ) ( )( ) ( )
( ) ( )( ) ( )
−=
−
α
α
ττττ
φφφφ
j
j
2 e00e
cossinjsinjcos
cossinsincos
U
( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U
HOMOMORPHISM SU(2) - O(3)
(σp, σq) : Pauli Matrices
( )[ ] ( )[ ] ( )[ ]( )q2p*T
2q,p3 UUTr212O σθσθθ =
O(4) UNITARY GROUP
[O4(2φ)] [O4(2τ)] [O4(2α)]
[ ] ( ) ( )( ) ( )
( ) ( )
( ) ( )( ) ( )( ) ( )
= −
−−
αα
ααττ
ττ
φφ
φφ
2cos2sin00
2sin2cos0000100001
2cos02sin00100
2sin02cos00001
000002cos2sin0
02sin2cos00001
4O
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION
( )yu,xuEgE ⇒
REFERENCE BASIS ( )yx u,u
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
( )
( )
+=
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 $, $x y
A
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
0$y
$x
$z
RC
-45°
H
+45°
V
LCNORTHERN HEMISPHERE
LEFT ELLIPTICAL POLARISATIONS
SOUTHERN HEMISPHERERIGHT ELLIPTICAL
POLARISATIONS
NORTH POLELEFT CIRCULARPOLARISATION
EQUATORLINEAR
POLARISATIONS
SOUTH POLERIGHT CIRCULARPOLARISATION
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
2α
2 α
g2
g1
0
g3
Eg$z
$y
$x
2τ 2τ2φ
( ) ( )( ) ( )
( ) ( )
( ) ( )( ) ( )( ) ( )
xu
2cos2sin00
2sin2cos0000100001
2cos02sin00100
2sin02cos00001
000002cos2sin0
02sin2cos00001
2E gAg
= −
−−
αα
ααττ
ττ
φφ
φφ
=
=
ττφτφ
2sinA2cos2sinA2cos2cosA
A
gggg
g
2
2
2
2
3
2
1
0
E
2φ
[O4(2φ)] [O4(2τ)] [O4(2α)]
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
JONES VECTOR ORTHOGONAL JONES VECTOR
′′
=⊥y
x
EE
E
=
y
x
EE
EORTHOGONALITY CONDITIONS
( )
−=′+=′
ττφφτφ
π2, a
STOKES VECTOR ORTHOGONAL STOKES VECTOR
=
=
ττφτφ
2sinA2cos2sinA2cos2cosA
A
gggg
g
3
2
1
0
E
−−−
=
=⊥
ττφτφ
2sinA2cos2sinA2cos2cosA
A
gggg
g
3
2
1
0
E
ORTHOGONALITY = ANTIPODALITY
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE SPHEREPOINCARE SPHERE
LC
-45°
g3
PV
+45°g22τ
2φg1
H
P⊥
0
$z
$x
$y
=
=
ττφτφ
2sinA2cos2sinA2cos2cosA
A
gggg
g
3
2
1
0
E
−−−
=
=⊥
ττφτφ
2sinA2cos2sinA2cos2cosA
A
gggg
g
3
2
1
0
E
STOKES VECTOR
ORTHOGONAL STOKES VECTOR
RC
ORTHOGONALITY = ANTIPODALITY
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE PLANISPHEREPOINCARE PLANISPHERE
0
0 $y$x
$z
POINCARE SPHERE : 3D SPACE
GNOMONIC PROJECTION(Elliptical Aitoff-Hammer Projection)
(Meridional Lambert Projection)
=
−=
)2
cos(2)2sin(y
)2
cos(
)2(sin)(sin2x
P
22
P
ατ
ατα
)2cos()cos()cos( τφα =With:
POINCARE PLANISPHERE2D SPACE
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POINCARE PLANISPHEREPOINCARE PLANISPHERE
0V+45°-45°V H
LC
y
2φ π=2 23
φ π=
22
φ π=
23
φ π=2φ π= − 2 2
3φ π
= −
22
φ π= −
23
φ π= −
26
τ π= −
22
τ π= −
23
τ π= −
26
τ π=
22
τ π=
23
τ π=
x
RC
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
CARTESIAN BASIS
POLARIZATION
STATE
( )E x y$ , $
( )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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
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
∝
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
( )yu,xuEgE ⇒
REFERENCE BASIS ( )yx u,u
[ ]
−=
1111
21U[ ]
=1jj1
21U
( )
( )
+−=
−=
yxR
yxL
EEj2
1E
EjE2
1E ( )
( )
+−=
+=
°
°
yx135
yx45
EE2
1E
EE2
1E
CIRCULAR BASIS LINEAR 45°/135° BASIS
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
( )yu,xuEgE ⇒
REFERENCE BASIS ( )yx u,u
( )[ ]
−
∗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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
CANONICAL BASISCANONICAL BASIS
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)
© E. Pottier, L. Ferro-Famil (01/2004)
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARIZATION MATCHINGPOLARIZATION MATCHING
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 =
−−
===
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARIZATION MATCHINGPOLARIZATION MATCHING
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 −=
+−
===
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARIZATION MATCHINGPOLARIZATION MATCHING
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
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARIZATION MATCHINGPOLARIZATION MATCHING
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====
© E. Pottier, L. Ferro-Famil (01/2004)
SAPHIR
SAPHIR
POLARIZATION MATCHINGPOLARIZATION MATCHING
“…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
“…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
© E. Pottier, L. Ferro-Famil (01/2004)
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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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
{ }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 ++≥
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
[ ] ( ) [ ] [ ] [ ]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
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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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
© E. Pottier, L. Ferro-Famil (01/2004)
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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:
© E. Pottier, L. Ferro-Famil (01/2004)
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PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES
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
© E. Pottier, L. Ferro-Famil (01/2004)
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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
© E. Pottier, L. Ferro-Famil (01/2004)
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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
© E. Pottier, L. Ferro-Famil (01/2004)
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Questions ?Questions ?