WAVE POLARIMETRYWAVE POLARIMETRY · 2013. 8. 1. · COMPLEX POLARISATION PLANE. ℜ (ρ xˆ,yˆ)...

SAPHIR

$y( )rE z t,

WAVE POLARIMETRYWAVE POLARIMETRYWAVE POLARIMETRY

SAPHIR

PROPAGATION EQUATIONPROPAGATION EQUATION

( )t,zEr

REAL ELECTRIC FIELD VECTOR

( ) ( )

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

t,zt,zDt,zJt,zHt

t,zBt,zE

=⋅∇=⋅∇

=∧∇

−=∧∇

∂∂

MAXWELL EQUATIONS

MAXWELL – FARADAY EQUATION

MAXWELL – AMPERE EQUATION

GAUSS THEOREM

( ) ( ) ( )

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

t,zHt,zBt,zEt,zJ

tt,zDt,zJt,zJ

εµσ

∂∂+=r

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

SAPHIR

( ) ( ) ( )AAA ∇⋅∇−⋅∇∇=∧∇∧∇rrr

( ) ( ) ( ) ( )t

t,zEt,zE 2

∂ρ∂

ε∂∂µσ

∂∂µε −=−−∇

PROPAGATION EQUATION

( ) ( ) 0t

t,zEt,zE 2

22 =−∇

∂∂µε

HELMOTZ PROPAGATION EQUATION

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

SAPHIR

PROPAGATION EQUATIONPROPAGATION EQUATION

( ) ( )

ℜ= tjezEt,zE ωr

COMPLEX ELECTRIC FIELD VECTOR With:( )zE

HELMOTZ PROPAGATION EQUATION

( ) ( ) 0zEkzE 22 =+∇

SOLUTION: ( ) jkzeEzE −= With: E

eEeEeE

SINUSOIDAL PLANE WAVE

( ) 0z

E0t,zE z =

∂⇒=⋅∇

SAPHIR

POLARISATION ELLIPSEPOLARISATION ELLIPSE

$y( )rE z t,

( )( )( )

=−−=−−=

kztcosEEkztcosEE

δωδω

SAPHIR

( )rE z t0 ,

$y( )rE z t,

THE REAL ELECTRIC FIELD VECTOR MOVES IN TIME ALONG AN ELLIPSE

( ) ( )δδ 2

x sinEE

cosEEEE

xy δδδ −=With:

SAPHIR

A : WAVE AMPLITUDE

φ : ORIENTATION ANGLE τ : ELLIPTICITY ANGLE

$x0( )rE z t, = 0

α : ABSOLUTE PHASE

22πφπ ≤≤−

40 πτ ≤≤

SAPHIR

POLARISATION HANDENESSPOLARISATION HANDENESS

ROTATION SENSE: LOOKING INTO THE DIRECTION OF THE WAVE PROPAGATION

ANTI-CLOCKWISE ROTATION CLOCKWISE ROTATION

LEFT HANDED POLARISATION RIGHT HANDED POLARISATION

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

44πτπ ≤≤−

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

Courtesy of Dr S.R. CLOUDE

SAPHIR

JONES VECTORJONES VECTOR

( )( )( )

=−−=−−=

kztcosEEkztcosEE

δωδω

eEEeEE

( ) ( )

−ℜ= kztjeEt,zE ωr

PHASOR = JONES VECTOR

GEOMETRICAL PARAMETERS

ABSOLUTE PHASE

xδα =

δφ cosEE

EE22tan 2

2x0 EEA +=

δτ sinEE

EE22sin 2

AMPLITUDE

ORIENTATION ANGLE ELLIPTICITY ANGLE

POLARISATION HANDENESS: Sign(τ)

SAPHIR

VERTICAL POLARISATION STATEHORIZONTAL POLARISATION STATE

rE z t( , )

ORTHOGONAL LINEAR POLARISATION STATELINEAR POLARISATION STATE

sincos

−=⊥

cossin

rE z t( , )

$ ′x$ ′y

rE z t( , )

$ ′x$ ′y

πθφ0=

SAPHIR

LEFT CIRCULAR POLARISATION STATE RIGHT CIRCULAR POLARISATION STATE

22πτ

πφπ

+≤≤−

22πτ

πφπ

+≤≤−0

rE z t( , )

ORTHOGONAL ELLIPTICALPOLARISATION STATEELLIPTICAL POLARISATION STATE

′′

2≤≤−

πθφ

rE z t( , )

40 πτ

+≤≤

SAPHIR

SU(2) MATRIX GROUPSU(2) MATRIX GROUP

Unitary Matrices Pauli Group

j00110

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 −=

SAPHIR

[ ] p0p )sin(j)cos(

jeA σασα

ασ+==

Special Unitary Matrices Group

je)(sinj)cos(

)cos()sin()sin()cos( φσ

σφσφφφφφ −

je)(sinj)cos(

)cos()sin(j)sin(j)cos( τσ

στστττττ +

je)(sinj)cos(je0

0je ασσασαα

α +=+=

SAPHIR

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

eασαστστσφσφσ +

112233 je

ασαστστσφσφσ −=

σασασ

βσασσβα{ }

321qp ,,,

σσσσσ

∈with:

SAPHIR

A=E xu

( )rE z t, = 0

( ) ( )( ) ( )

−φφφφ

cossinsincos

A=E xu

$x0( )rE z t, = 0

( ) ( )( ) ( )

−φφφφ

cossinsincos

A=E( ) ( )( ) ( )

ττττ

cossinjsinjcos

$x0( )rE z t, = 0

( ) ( )( ) ( )

−φφφφ

cossinsincos

A=E( ) ( )( ) ( )

ττττ

cossinjsinjcos

$x0( )rE z t, = 0

SAPHIR

Special Unitary Matrices Group and Jones Vector

)sin(j)cos(

)cos()sin()sin()cos(jAeE )y,x( τ

τφφφφα

je00je

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

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

AE )y,x( αα

ττττ

φφφφ

AeE 123)y,x(

αστσφσ ++−=

SAPHIR

$x0( )rE z t, = 0

x123 u

αστσφσ ++−=

SAPHIR

POLARISATION RATIOPOLARISATION RATIO

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

cossinjsinjcos

cossinsincos

ττττ

φφφφ

COMPLEX POLARISATION RATIO

( )( ) ( ) ( )

( ) ( )τφτφρ δδ

tantanj1tanjtane

yy,x −

+=== −

INDEPENDENT OF AMPLITUDE AND ABSOLUTE PHASE

SAPHIR

POLARISATION RATIOPOLARISATION RATIO

JONES VECTOR

( ) ( )( ) ( )

cossinjsinjcos

cossinsincos

ττττ

φφφφ

( ) ( )( ) ατφξ −= − tantantan 1With:

COMPLEX POLARISATION PLANE

SAPHIR

COMPLEX POLARISATION PLANECOMPLEX POLARISATION PLANE

0( )( )y,xρℜ

( )( )y,xρℑ ( )( ) ( )

( ) ( )τφτφρ

tantanj1tanjtan

y,x −+

( )( )( )

122sin

122tan

SAPHIR

( )( )y,xρℑ

( )( )y,xρℜ

( ) cstEE

y0y,x ==ρ

( )( ) cstarg y,x == δρ

SAPHIR

0C ( )( )y,xρℜ

( )( )y,xρℑ

( )φ2cscr =

( )( )0,2cot φ−

FAMILY of ORTHOGONAL CIRCLES

cst=φ

Left Circular

Right Circular

SAPHIR

( )( )y,xρℜ

( )( )y,xρℑ

( )τ2cotr =

( )( )τ2csc,0

Right Circular

Left Circularcst=τ LOCI

FAMILY of ORTHOGONAL CIRCLES

SAPHIR

ORTHOGONAL JONES VECTORORTHOGONAL JONES VECTOR

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

cossinjsinjcos

cossinsincos

ττττ

φφφφ

POLARISATION ALGEBRA

NORM OF A JONES VECTOR

SCALAR PRODUCT

ORTHOGONALITY

SAPHIR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

cossinjsinjcos

cossinsincos

ττττ

φφφφJONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) yj

cossinjsinjcos

cossinsincos

⊥ α

ττττ

φφφφ

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

cossinjsinjcos

sincoscossin

−−−

ττττ

φφφφ

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

22 ue00e

cossinjsinjcos

cossinsincos

−−−−

+++−+

⊥ α

ττττ

φφφφ

ORTHOGONAL JONES VECTOR

SAPHIR

ORTHOGONALITY CONDITIONS

−=′+=′

ττφφτφ

π2, a

CHANGE OF POLARISATION HANDENESS

$y0 $x0

φ’ $x

$x0τ’

E′′

′′

=⊥⊥ ρa

E10E,E

ρρ −==

⊥⊥ a

SAPHIR

JONES VECTOR

ORTHOGONAL JONES VECTOR

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

SAPHIR

ELLIPTICAL BASIS TRANSFORMATIONELLIPTICAL BASIS TRANSFORMATION

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

cossinjsinjcos

cossinsincos

ττττ

φφφφJONES VECTOR

ORTHOGONAL JONES VECTOR ( ) ( )( ) ( )

( ) ( )( ) ( ) yj

cossinjsinjcos

cossinsincos

⊥ α

ττττ

φφφφ

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

[ ]yxj

u,ue00e

cossinjsinjcos

cossinsincos

⊥ α

ττττ

φφφφ

ELLIPTICAL BASIS TRANSFORMATION

SAPHIR

ORTHOGONAL JONES VECTORS

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

[ ]yxj

u,ue00e

cossinjsinjcos

cossinsincos

⊥ α

ττττ

φφφφ

SU(2) : SPECIAL UNITARY TRANSFORMATION MATRIX

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

ττττ

φφφφ

cossinjsinjcos

cossinsincos

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

CONSERVATION OF THE WAVE ENERGY

ENSURES THE CORRECT PHASE DEFINITION

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

IUU 2D*T

SAPHIR

ORTHOGONAL JONES VECTORS

[ ] [ ]yxj

u,ue00e

⊥ ξ

SU(2) : SPECIAL UNITARY TRANSFORMATION MATRIX

( )[ ]ρ2U ( )[ ]ξ2U

CONSERVATION OF THE WAVE ENERGY

ENSURES THE CORRECT PHASE DEFINITION

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

IUU 2D*T

SAPHIR

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⊥⊥

UEE [ ]

[ ] [ ]

⊥⊥

SU(2) SPECIAL UNITARY ELLIPTICALBASIS TRANSFORMATION MATRIX

SAPHIR

LINEAR BASIS

( ) ( )( ) ( )

( )( )

φφφφα

sinjcos

cossinsincos

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

esincosesincos

2AeE πφ

ττττ

21U[ ]

CIRCULAR BASIS

Ernst LÜNEBURG(PIERS95 - Pasadena)

SAPHIR

POLARISATION SYNTHESISPOLARISATION SYNTHESIS

⊗ =E E eG Gj G

0E0DE0G

rE z t( , )= 0

$uG $uD

⊗ =E E eD Dj D

E u tG G$ ( )= 0

E u tD D$ ( )= 0

SAPHIR

STOKES VECTORSTOKES VECTOR

REAL REPRESENTATION OF THE POLARISATIONSTATE OF A MONOCHROMATIC WAVE

=⋅ *

EEEEEEEE

PAULI MATRICES GROUP

j00110

3210 σσσσ

−+−+

=+++=⋅1032

321033221100

ggjggjgggg

21gggg

21EE σσσσ

{ g0, g1, g2, g3 } STOKES PARAMETERS

SAPHIR

JONES VECTOR

eEEeEE

( )( )

ℑ−=ℜ=

EE2gEE2gEEg

STOKES VECTOR

WAVE POLARISATION STATE ESTIMATIONFROM INTENSITIES MEASUREMENTS

SAPHIR

STOKES VECTOR

( )( )

−=+=

ττφτφ

2sinAg2cos2sinAg2cos2cosAg

sinEE2gcosEE2g

EEgEEg

GEOMETRICAL PARAMETERS

δφORIENTATION ANGLE

ELLIPTICITY ANGLE

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

τττ

τττττσ

−=⇒

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

001)2(Oje0

0jeje 3

αααααα

αασ

SAPHIR

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

[ ] [ ][ ][ ]

123 je

je00je

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

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

)(U)(U)(UU

αστσφσ

ττττ

φφφφ

ατφ

++−=

[ ] [ ][ ][ ])2(O)2(O)2(OO 4444 ατφ=

= )2(O

4 χχwith:

SAPHIR

JONES VECTOR

( ) ( )( ) ( )

( ) ( )( ) ( ) xj

cossinjsinjcos

cossinsincos

ττττ

φφφφ

[U2(φ)] [U2(τ)] [U2(α)]

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

( ) ( )( ) ( )

( ) ( )

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

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

2E gAg

−−

ααττ

STOKES VECTOR

[O4(2φ)] [O4(2τ)] [O4(2α)]

SAPHIR

VERTICAL POLARISATION STATEHORIZONTAL POLARISATION STATE

rE z t( , )

== τπφ

rE z t( , )

00 == τφ

ORTHOGONAL LINEAR POLARISATION STATELINEAR POLARISATION STATE

rE z t( , )

$ ′x$ ′y ( )( )

02sin2cos

g L θθ

rE z t( , )

$ ′x$ ′y

=+= τπθφ

( )( )

−−

02sin2cos

g L θθ

0== τθφ

SAPHIR

LEFT CIRCULAR POLARISATION STATE RIGHT CIRCULAR POLARISATION STATE

rE z t( , )

422πτπφπ +=+≤≤−

rE z t( , )

ORTHOGONAL ELLIPTICALPOLARISATION STATE

≤≤−+= τππθφ

422πτπφπ −=+≤≤−

−−−

ELLIPTICAL POLARISATION STATE

rE z t( , )

40 πτθφ +≤≤=

SAPHIR

SPECIAL UNITARY SU(2) GROUP

[ ] ( ) ( )( ) ( )

( ) ( )( ) ( )

ττττ

φφφφ

2 e00e

cossinjsinjcos

cossinsincos

( )[ ]φ2U ( )[ ]τ2U ( )[ ]α2U

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

O(4) UNITARY GROUP

[O4(2φ)] [O4(2τ)] [O4(2α)]

[ ] ( ) ( )( ) ( )

( ) ( )

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

−−

ααττ

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

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

SAPHIR

POINCARE SPHEREPOINCARE SPHERE

STOKES VECTOR

( )( )

ℑ−ℜ

ττφτφ

2sinA2cos2sinA2cos2cosA

sinEE2cosEE2

EE2EE2EE

{g0} TOTAL WAVE INTENSITY

{g1, g2, g3 } POLARISED WAVE INTENSITIES

20 gggg ++= WAVE FULLY POLARISED

{g1, g2, g3 } Spherical Coordinates of a

point P on a sphere with radius g0

SAPHIR

( )( )ℑ ρ $, $x y

( )( )ℜ ρ $, $x y

$z COMPLEX POLARISATION PLANE

STEREOGRAPHIC PROJECTION

SAPHIR

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

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

φττρ

φτφτρ

ττφτφ

2sin2cos2sin2cos2cos

STEREOGRAPHICPROJECTION

( )( )ℑ ρ $, $x y

( )( )ℜ ρ $, $x y

SAPHIR

LCNORTHERN HEMISPHERE

LEFT ELLIPTICAL POLARISATIONS

SOUTHERN HEMISPHERERIGHT ELLIPTICAL

POLARISATIONS

NORTH POLELEFT CIRCULARPOLARISATION

EQUATORLINEAR

POLARISATIONS

SOUTH POLERIGHT CIRCULARPOLARISATION

SAPHIR

2τ 2τ2φ

( ) ( )( ) ( )

( ) ( )

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

2cos2sin00

2sin2cos0000100001

2cos02sin00100

2sin02cos00001

000002cos2sin0

02sin2cos00001

2E gAg

−−

ααττ

ττφτφ

[O4(2φ)] [O4(2τ)] [O4(2α)]

SAPHIR

H Px= $

( )( )

= δγ

γδjesin

cosjAeE x

( )( ) ( )( ) ( )

δγδγ

sin2sinAcos2sinA

2cosAA

2STOKES VECTOR

DESCHAMPS PARAMETERS ( γ, δ )

JONES VECTOR

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

δγτδγφ

sin2sin2sincos2tg2tg ( ) ( ) ( )

( ) ( )( )

φτδ

τφγ

2sin2tgtg

2cos2cos2cos

SAPHIR

JONES VECTOR ORTHOGONAL JONES VECTOR

′′

EORTHOGONALITY CONDITIONS

−=′+=′

ττφφτφ

π2, a

STOKES VECTOR ORTHOGONAL STOKES VECTOR

ττφτφ

−−−

ττφτφ

ORTHOGONALITY = ANTIPODALITY

SAPHIR

+45°g22τ

ττφτφ

−−−

ττφτφ

STOKES VECTOR

ORTHOGONAL STOKES VECTOR

ORTHOGONALITY = ANTIPODALITY

SAPHIR

POINCARE PLANISPHEREPOINCARE PLANISPHERE

0 $y$x

POINCARE SPHERE : 3D SPACE

GNOMONIC PROJECTION(Elliptical Aitoff-Hammer Projection)

(Meridional Lambert Projection)

cos(2)2sin(y

)2(sin)(sin2x

ατα

)2cos()cos()cos( τφα =With:

POINCARE PLANISPHERE2D SPACE

SAPHIR

POINCARE PLANISPHEREPOINCARE PLANISPHERE

0V+45°-45°V H

2φ π=2 23

φ π=

φ π=2φ π= − 2 2

3φ π

φ π= −

τ π= −

τ π=

SAPHIR

CANONICAL BASISCANONICAL BASIS

CARTESIAN BASIS

POLARIZATION

( )E x y$ , $

( )φ$ , $x y

( )τ$ , $x y

( )γ$ , $x y

( )δ$ , $x y

( )ρ$ , $x y

Horizontal

Vertical

π / 2

Linear +45° 1

π / 4

Linear +135°

3π / 4

π / 4

Left Circular 1

π / 4

π / 2

Right Circular 1

-π / 4

-π / 2

SAPHIR

CIRCULAR BASIS

POLARISATION

( )RL u,uE

( )RL u,uφ

( )RL u,uτ

( )RL u,uγ

( )RL u,uδ

( )RL u,uρ

Horizontal 1

-π / 4

π / 4

-π / 2

Vertical 1

π / 4

π / 2

Linear +45° 1

π / 4

Linear +135°

− −+

3π / 4

π / 4

Left Circular

Right Circular

π / 2

SAPHIR

LINEAR 45°/135° BASIS

POLARIZATION

( )°+°− 13545 u,uE

( )°+°− 13545 u,u

Horizontal 12

-π / 4

π / 4

Vertical 1

π / 4

Linear +45°

Linear +135°

π / 2

Left Circular 1

π / 4

π / 2

Right Circular 1

− −

-π / 4

π / 4

-π / 2

SAPHIR

( )yu,xuEgE ⇒

21U[ ]

1E ( )

CIRCULAR BASIS LINEAR 45°/135° BASIS

SAPHIR

( )yu,xuEgE ⇒

( )[ ]

∗ℜ+

∗ℑ+

EgRL ( )[ ]

∗ℑ−

∗ℜ−

Eg13545

CIRCULAR BASIS LINEAR 45°/135° BASIS

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

DESCHAMPS VECTOR COMPONENTS

H = Px

2γ(x,y)

(x,y) +45 = P+45

LC = PL

2γ(L,R)

2γ(45,135) δ (45,135)

δ (L,R)

SAPHIR

POLARIZATION MATCHINGPOLARIZATION MATCHING

Voltage Equation *i

TiOC EhE,hV ==

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

Ex: Left Circulary Polarized Antennas

hEi’xR

Co-Ordinate Reversal

−=⇒

[ ] 0j1

j121EhE,hV '*

−−

SAPHIR

Ex: Left Circulary Polarized Antennas

hEi’’xR

Time Reversal

−−

[ ] 1j1

j121EhE,hV '*

iOC −=

SAPHIR

hEi’’

JONES Vector does not describe the direction of wave propagation

2 Separate Features to remember

−=Co-ordinates Reversal

Time Reversal ( )*'i

"i EE =

( )*jkzjkzjkz eee −+− =aWave propagation

SAPHIR

hEi’’

( )( )

−=−=

ℑ−ℜ

EE2EE2EE

ThOC EhE,hVgg

SAPHIR

Courtesy of Dr S.R. CLOUDE

SAPHIR

PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVES

DETERMINISTIC SCATTERING RANDOM SCATTERING

COMPLETELY POLARISED WAVE PARTIALLY POLARISED WAVE

Polarisation Ellipse varies in timeAmplitude, Phase: Random processes

STATISTICAL DESCRIPTION

SAPHIR

{ }EJONES VECTORS

WAVE COVARIANCE MATRIX

EEEEEJ

gggjggjggg

PARTIALLY POLARISED WAVES

20 gggg ++≥

SAPHIR

WAVE COVARIANCE MATRIX

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

EIGENVALUES DECOMPOSITION

[ ] [ ] [ ] *T222

12 uuuuU

UJ λλλ

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

{ }{ }2

gggg21

++−=

λ2 REAL EIGENVALUES

SAPHIR

PARTIALLY POLARISED WAVES DESCRIPTORS

Degree of Polarisation

Anisotropy

[ ]( )[ ]( )

−=+−

=JTrace

Jdet41g

gggDoP 2

λλλλ

Polarised Wave PowerTotal Wave Power

Wave Entropy

( )∑=

−=2i

1ii2i plogpH

λλλ+

=With:

Degree of randomness, statistical disorder

SAPHIR

COMPLETELY POLARISED WAVES

[ ]( )

⇒=⇒=0H

1DoP00

0JdetEEEEEE2

λyMaximum Correlation Between andxE E

COMPLETELY UNPOLARISED WAVESAbsence of any Polarised Structure in the Wave

[ ]( ) [ ]( )

⇒=⇒=⇒

1H0DoP

4JTraceJdet

2x λλ

PARTIALLY POLARISED WAVES

[ ] [ ]( )

≤≤≤≤

≥≠≥

1H01DoP0

00Jdet

yCorrelation between andxE E

SAPHIR

WAVE DECOMPOSITION = WAVE DICHOTOMYBorn & Wolf Decomposition – Shandrasekar Decomposition

[ ]2D*T

11 Iuuuu =+

[ ]( ) [ ]2D2

*T1121

IuuuuuuJ

λλλλλ

+−=+=

COMPLETELYUNPOLARISED

COMPLETELYPOLARISED

PARTIALLYPOLARISED

SAPHIR

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

1121 JJIuuJ +=+−= λλλPARTIALLY POLARISED WAVE

COMPLETELY POLARISED WAVE

−+++

−+++=

gggggjg

gjggggg21J

COMPLETELY UNPOLARISED WAVE

++−=

CUgggg0

0gggg21J

SAPHIR

STOKES VECTOR

gggg 2

COMPLETELYUNPOLARISZED

COMPLETELYPOLARISED

PARTIALLYPOLARISED

SAPHIR

PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVESCOMPLETELY UNPOLARISED WAVE

gggg 23

−−−

2 ORTHOGONAL COMPLETELY POLARISED WAVES

SAPHIR

PARTIALLY POLARISED WAVESPARTIALLY POLARISED WAVESWAVE DECOMPOSITION = WAVE DICHOTOMY

−−−

′−

PARTIALLYPOLARISED

COMPLETELYUNPOLARISED

2 ORTHOGONAL COMPLETELYPOLARISED WAVES

44444 344444 21

COMPLETELYPOLARISED

20 gggg ++=′With:

SAPHIR

SphereP ∈

SphereP ⊂

CenterP =

COMPLETELY POLARISED WAVESThe total wave energy is polarised

g g g g0 1

2= + +H = 0 DoP = 1

PARTIALLY POLARISED WAVESA part of the total wave energy is polarised

g g g g0 1

2≥ + +0 < H < 1 0 < DoP < 1

COMPLETELY UNPOLARISED WAVESThe total wave energy is unpolarised

g g g1

2 0+ + =

H = 1 DoP = 0

SAPHIR

WAVE DESCRIPTORSWAVE DESCRIPTORSMONOCHROMATIC PLANE WAVES

COMPLEX DOMAIN REAL DOMAIN

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

WAVE DESCRIPTORSWAVE DESCRIPTORSPARTIALLY POLARISED PLANE WAVES

COMPLEX DOMAIN REAL DOMAIN

gSTOKES VECTOR

PLANE WAVES FULLY DESCRIBEDBY 4 INDEPENDANT PARAMETERS

{ }3210

g,g,g,g

E,EE,EE,E

[ ] *TEEJ =COVARIANCE MATRIX

WAVE POLARIMETRIC DIMENSION = 4

SAPHIR

Questions ?Questions ?

WAVE POLARIMETRYWAVE POLARIMETRY · 2013. 8. 1. · COMPLEX POLARISATION PLANE. ℜ (ρ xˆ,yˆ)...

Documents

Παρακαταθήκη, τ.89

W06 naprezenia glowne[tryb zgodno [ci]wbia.pollub.pl/files/86/content/files/1439_naprezenia_glowne.pdf · Tensor naprężeń Naprężenia w stanie przestrzennym: τ σ τ σ τ τ

Elementul de bar` 2D supus numai la [ncovoiere (grinda) · 2002. 10. 10. · 1 Curs MEF A. Pascu yˆ, vˆ xˆ ˆ 2 , vˆ 2 F y M2,θ2 ˆ 1 , vˆ 1 F y L 12 M1,θ1 Elementul de bar`

2%3#4%’56/%&7#)8’9:/,%./ !#$%&’()$’*+,-)-.-+/’0)$%&1(,%& · 0 τ s (e − τ s /2 τ 0 − 1) − 2 τ 0 τ s λ s 2 λ 0 2 + λ s 2 1/2 e − τ s /2 τ 0 − 1 τ 0 temporal

OPTINĖSELEKTRONIKOS ĮTAISAI - el.vgtu.lt · sr sr nr vid 2π 1; 2π 1; 1 1 1; τ η τ τ τ τ τ τ η = = + + f = ∆F = d s Dvigubos heterosandūros aktyviojo sluoksnio storis

Μετ ε φ α | εδ α ατ -Τ εστ Κ α Οδ ε ~σ τ M912Netbook Α ... · • Π σ δ}σετετ Netbookσα ετ ε α, ε α - ε τε τ τ|σ ε ατ τ σα }ασα

Gas turbine based power generation - Stanford Universitycantwell/AA283_Course_Material/...Ideal Gas Turbine Power Generator. f= τ λ−τ λ τ f−τ λ Example, let Gas Turbines

Δ ε λ τ ί ο Σ Ε Β γ ι α τ η ν κ ρ ί σ η τ ο υ COV I D - 1 9 › Uploads › Documents › 52773 › 13-05... · 2020-05-13 · Έ κ τ α κ τ α μ έ τ

5τ’-’ 6.’-’ ˆ0+˚δε)φ-’, · ,’5τ’-’ 6.’-’ ˆ0+˚δε)φ-’, ˜)% τ%/ 5+’%/ ’δ’( / 8’τ.-8 / τ%/ ))%+’( / -0) / #’α τ-+ ))%+’σ5 τ%

Kapitel 6 — Folgen und Stetigkeit...Kapitel 6 — Folgen und Stetigkeit Deﬁnition 6.10 (Grenzwert einer Funktion) Es sei D eine Teilmenge und xˆ 2 D.Weiterseif : D \{xˆ}! eine

Αναιρέσεις, τ. 5

Τ τ ιτ οτ ατ ετ ιτιά όταν ιτ οτ ατ ετ · 2013. 1. 16. · Τ τ ιτ οτ ατ ετ ιτ οτ ατ ετ ιτιά όταν άτι έτος

Física 1 - Biólogos y Geólogos - users.df.uba.arusers.df.uba.ar/mclaudia/f1ByG2017/Parte1_Mecanica.pdf · ... D = 0 xˆ – 5 yˆ 2) Halle las componentes ... las componentes cartesianas

Η Ι σ τ ο ρ ί α τ ω ν Χ ρ ω μ ά τ ω ν

Τ’ άσπρα πουλήσαν τον Χριστό Τ’ άσπρα πουλούν κι εσένα · Τ’ άσπρα πουλήσαν τον Χριστό Τ’ άσπρα πουλούν

Αναιρέσεις, τ. 9

Τ' ΑΓΓΕΛΟΚΑΣΤΡΙΤΙΚΑ

Block15 τ.01

Παρακαταθήκη, τ. 80

Τ˚ Τα π τ˚˝ Παιχνιδ˚κρ τ˚ρα - Czech Games EditionΜέρες τώρα, μαζεύω και αποξηραίνω τα συστατικά με ανυπομονησία