Group of Optical Polarimetry. Achievements 2003 -Eng- of Optical Polarimetry... · Optical polarimetry. The Group of Optical Polarimetry Zaragoza University J. J. Gil1, J. M. Correas2,

Optical polarimetry. The Group of Optical Polarimetry

Zaragoza University

J. J. Gil1, J. M. Correas2, C. Ferreira2, I. San José3, P. A. Melero2, J. Delso4

1 ICE, Universidad de Zaragoza2 Departamento de Matemática Aplicada, Universidad de Zaragoza

3 Instituto Aragonés de Estadística4 Instituto Tecnológico de Aragón

VII Reunión Nacional de Óptica. Santander (Spain) 8-11 Septiembre 2003

Polarimetry Applications and problems to be solved

History andinterest of thisresearch

Problems to be solved




With a pieceof the

polarimeter

The solution forthe polarimetric subtraction was

found!

Goals

Obtainment of a proper andcomplete mathematical model forthe adequate representation of allthe possible polarimetric behaviorsof material media.

Construction of adequate modelsfor the exploitation of thepolarimetric measures



Polarization algebra:Generalized coherency matrix

1

0

1 n

i ii

cn

−

=

= Θ∑A

n: order of the matrix

n = 2 2D light. Fixed directionof propagation

n = 3 3D light. Fluctuatingdirection of propagation

n = 4 Polarimetric properties ofthe medium

ci measurable physical quantities

Q i basis of trace-orthogonal Hermitian matrices: generators of SU(n) group + identity matrix


Polarized light: 2D case

X

Y

Z

E

EY

EX

AY

AX

Elipse de polarización


Partial polarization:Polarization matrix

( ) ( )t t+= ⟨ ⊗ ⟩P ε ε

( )2

( ) 2

( ) ( ) ( )

( ) ( ) ( )

y

y

i tx x y

i tx y y

A t A t A t e

A t A t e A t

δ

δ

−⎛ ⎞⟨ ⟩ ⟨ ⟩⎜ ⎟⎜ ⎟⟨ ⟩ ⟨ ⟩⎝ ⎠

positive semidefinite Hermitian matrix


Coherency matrix and Stokes parameters

3

0

12 i i

is

=

= ∑P σ

0 1 2 3

1 0 1 0 0 1 0, , ,

0 1 0 1 1 0 0i

i−⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠σ σ σ σ

( )Tr , 0,1,2,3i is i= =Pσ


Degree of polarization

( )

12

241 DetGTr

⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠

PP

( )( )

12 2

2

21

TrG

Tr

⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠

PP

0 1G≤ ≤


3D case: Fluctuating polarization ellipse

X

Y

Z

E EY

EX

AY

AX

Polarization ellipse is inside a changing plane


Generalized polarization matrix( ) ( )t t+= ⟨ ⊗ ⟩R ε ε

Generalized Stokes parameters8

, 0

13 j i

i jq

== ∑R Ω

( )Tr , 0,1,...,9i iq i= =RΩ


Basis composed of the normalized Gell-Mann matrices plus the identity matrix

0 1 2

1 0 0 0 1 0 0 0 13 30 1 0 1 0 0 0 0 02 2

0 0 1 0 0 0 1 0 0

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Ω Ω Ω

3 4 5

0 0 1 0 0 0 0 03 1 30 0 0 1 0 0 0 12 220 0 0 0 0 2 0 1 0

ii

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Ω Ω Ω

6 7 8

0 0 0 0 0 1 0 03 3 30 0 0 0 0 0 1 02 2 2

0 0 0 0 0 0 0

ii

i i

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Ω Ω Ω


Degree of polarization purity

( )( )

12 2

(3) 2

31 12

TrG

Tr

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥= −⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

R

R


Indices of polarimetric purity

1 20 1P P≤ ≤ ≤

12 2 2

(3) 1 21 32

G P P⎡ ⎤= +⎣ ⎦

Región factible de los índices de pureza

P1

P2

(0,0)

(0,1) (1,1)

1 21

1 2 32

2

PTr

PTr

λ λ

λ λ λ

−=

+ −=

R

R


Purity space

P1

P2

(0,0)

(0,1) (1,1)


Polarimetric properties of

material media


Basic interaction

εε’

T

′ = Tε ε

+′ =P TPT


Mueller-Jones matrix

( )12kl k ln Tr += T Tσ σ

( )* 1−= ⊗N L T T L

′ =s Ns

, 0,1, 2,3k l =


Mueller matrix

( ){ }*( ) ( ) 1 ( )n n

i i ii i

i ip p−= ⊗ =∑ ∑M L T T L N

Incident light

s

s´

Irradiatedarea

Emerging light

Incoherentsuperposition s´= Ms


Coherency matrix of the medium

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

02 12 20 21 22 3300 01

10 11 03 13 30 31 23 32

02 12 22 33 20 2100 01

03 13 10 11 23 32 30 31

20 21 22 33 02 1200 01

30 31 23 32 10 11 03 13

22 33

14

m m m m m mm mm m i m m i m m i m m

m m m m m mm mi m m m m i m m i m m

m m m m m mm mi m m i m m m m i m m

m m

+ + +++ + + + − + + −

+ − −−− + + − − + − −

=+ − −+

+ + + + − − + −

+

H

( ) ( ) ( )20 21 02 12 00 01

23 32 30 31 03 13 10 11

m m m m m mi m m i m m i m m m m

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− − −⎜ ⎟⎜ ⎟− − + − − − − +⎝ ⎠


Coefficients of theexpansion

Expansion of H

Basis composed of themodifyed Diracmatrices

Coherency matrix H as an ensemble average

Coherency matrix and Mueller matrix: General characterization

*12kj k l e

h t t=

kl k l= ⊗E σ σ

3

, 0

14 kl kl

k l

m=

= ∑H E

( )kl klm Tr= E H


Degree of polarimetric purity of themedium

( )1

2 2

2

41 1( )3

TrG

Tr

⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠

HH

0 1G≤ ≤


Generalized degree of purity andpurity criteria

( )1

2 2

2

1 11 ( )n

nTrG

n Tr

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥= −⎨ ⎬− ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

AA

( ) ( )2 2 2( )Tr Tr nTr≤ ≤A A A


Indices of purity

( ) ( )

0 11

0 1 2 32

2 33

PTr

PTr

PTr

λ λ

λ λ λ λ

λ λ

−=

+ − +=

−=

H

H

H

( )1

2 2 2 21 2 3

1 2 23

G P P P= + +


Purity space

( )1 10, ,3 3

( )0,0,0

( )1,1,0

( )0,1,0

P2

P3

P1

Región factible de los índices de pureza


Modeling of the polarimetric properties through coherency matrices

Coherency matrix (nD)

Basis of nxn trace-orthogonal Hermitian matrices composed of the generators ofthe SU(n) group plus the identity matrix

The coefficients of the expansion of thecoherency matrix in this basis, are themeasurable quantities with physicalsignificance


Kernel matrix associated with a Mueller matrix

=M RZL

R, L Retarders

Z Kernel matrix:

Polarizance-diattenuance

Forward polarizance

Reverse polarizance

Depolarizance

Indices of purity


Decompositions into pure components

0

01

n

i ii

n

ii

p

p

=

=

=

⎛ ⎞=⎜ ⎟⎝ ⎠

∑

∑

A X

Xi pure components

Light: incoherentsuperposition of totallypolarized beams

Medium: Parallelcombination of purecomponents


H A

Data

Subtraction procedure

αAH- αA

Polarimetric subtraction


Overview of the group

Collaborators

Citations

Future


Group of Optical Polarimetry. Achievements 2003 -Eng- of Optical Polarimetry... · Optical polarimetry. The Group of Optical Polarimetry Zaragoza University J. J. Gil1, J. M. Correas2,

Documents