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Parallel decom position of Mueller m atrices and polarim etric subtraction

José J. Gil

Universidad de Zaragoza, Spain. www.pepegil.es ppgil@unizar.es

Parallel decom position of Mueller m atrices and polarim etric subtraction

• 1. Concept of Mueller matrix
• 2. Parallel decompostions of a

Mueller matrix

• Spectral
• Trivial
• Arbitrary
• 3. Polarimetric subtraction

1 The concept of Mueller m atrix

Basic interaction: Jones description ε ε'

Intensity I = ε+ ⊗ ε DoP P = 1 χ azimuth ϕ ellipticity

T

ε´ = T ε

Jones matrix Jones vector

• Incident beam: P =1
• Single interaction P’ =1

y

x i y

A A e

δ

⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ε

ϕ χ

The “pure case”

Non-depolarizing system: for incident light whit

P= 1, emerging light has P’= 1

The system is equivalent to a serial combination of

two components:

A diattenuator (partial or total polarizer) A retarder

Only 7 independent physical quantities:

1 mean transmitance 3 from diattenuator 3 from retarder

Characterization of Jones matrices

Linear passive system T is a 2x2 complex matrix (7 physical parameters) T satisfies the transmittance condition (maximum gain ≤ 1)

( ) ( )

( )

( )

1 2 2

1 tr tr 4det 1 2

+ + +

⎧ ⎫ ⎡ ⎤ + + ≤ ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ⎩ ⎭ T T T T T T

11 12 21 22

,

ij

i ij ij

t t t t e t t

β

⎡ ⎤ ≡ = ⎢ ⎥ ⎣ ⎦ T

Basic interaction: Stokes-Mueller description s s'

cos2 cos2 s cos2 sin 2 sin 2 I IP IP IP χ χ ⎡ ⎤ ⎢ ⎥ ϕ ⎢ ⎥ ≡ ⎢ ⎥ ϕ ⎢ ⎥ ϕ ⎣ ⎦

I intensity P degree of polarization χ azimuth ϕ ellipticity

N

s´ = N s

Mueller-Jones matrix Stokes vector Incident beam: P =1 Single interaction P´ =1

Characterization of Mueller-Jones matrices

7 free parameters in T ⇒ 7 free parameters in N 1 Transmittance condition

• r

( )

( )

1 + −

= ⊗ N T L T T L

1 1 1 1 1 1 i i ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ ≡ ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ L

2 2 2 1 2 00 01 02 03

1, ( )

f f

g g n n n n ≤ ≡ + + +

2 2 2 1 2 00 10 20 30

1, ( )

r r

g g n n n n ≤ ≡ + + +

For pure systems

gf = gr

Macroscopic interaction: Synthesis of a Mueller matrix

Incident beam

s s ´

Irradiated area Emerging beam Incoherent superposition

Composed Mueller matrix

s s ´

Emerging beam

( ) ( ) ( ) ( )

1

( ) , 0, 1

i i i i i i i i i

p p p

∗ −

⎛ ⎞ ≡ ⎜ ⎟ ⎝ ⎠ ≡ ⊗ ≥ =

∑ ∑

M N N L T T L

Coherency matrix associated with a Mueller matrix

M H

3 ,

1 4

kl kl k l

m

=

= ∑ H E

Ekl set of 16 “Dirac matrices”

Coefficients mkl are 16 measurable quantities: the 16 elements of the Mueller matrix M associated with H

H represents univocally the Mueller matrix and vice-

versa

Coherency matrix H

* kl k l

= ⊗ E σ σ

σkl set of 4 “Pauli matrices”

Coherency matrix H(M)

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

02 12 20 21 22 33 00 01 03 13 30 31 23 32 10 11 02 12 22 33 20 21 00 01 03 13 23 32 30 31 10 11 20 21 22 33 02 12 00 01 30 31 23 32 03 13 10 11 22 33

1 4 m m m m m m m m i m m i m m i m m m m m m m m m m m m i m m i m m i m m m m m m m m m m m m i m m i m m i m m m m m m + + + + + + − + + − + + + − − − − + − + − − + − = + − − + + + + + + − − − + H

( ) ( ) ( )

20 21 02 12 00 01 23 32 30 31 03 13 10 11

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

M (H)

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

01 10 00 11 00 11 01 10 22 33 22 33 23 32 23 32 01 10 00 11 00 11 01 10 22 33 22 33 23 32 23 32 03 30 02 20 02 20 03 30 13 31 13 31 12 21 12 21 02 20

i h h h h h h h h h h h h h h i h h i h h h h h h h h h h h h h h i h h i h h h h h h h h h h h h h h i h h i h h − − + − + + + + − + + − − − − + − + − − − + − − + − = − − + + + + + − − + + + − − + M

( ) ( ) ( ) ( ) ( )

02 20 03 30 03 30 12 21 13 31 13 31 12 21

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

Characterization of Mueller matrices

4 Eigenvalue Conditions 2 Transmittance Conditions

, 0,1,2,3

i

i λ ≤ =

Characterization theorem A real 4x4 matrix M is a Mueller matrix if, and only if, the four eigenvalues of H(M) are non-negative and M satisfies the transmittance conditions

( ) ( )

1 2 1 2 2 2 2 2 2 2 00 01 02 03 00 10 20 30

1, 1 ,

f r f r

g g g m m m m g m m m m ≤ ≤ ≡ + + + ≡ + + +

2 Parallel decom positions

Spectral Trivial Arbitrary

Parallel decomposition

N1 Ni Nn

S´ S

′ = s Ms

1 n i i=

=∑ s s

i i i

′ = s N s

1 1 n n i i i i i = =

′ ′ = =

∑ ∑

s s N s

1

s

i

s

n

s

1

′ s

i

′ s

n

′ s

Spectral decom position

Spectral decomposition

( )

1 2 3

, , , λ λ λ λ

+

= H UD U

( ) ( ) ( ) ( )

1 3 2

tr ,0,0,0 0,tr ,0,0 tr tr 0,0,tr ,0 0,0,0,tr tr tr λ λ λ λ

+ + + +

= + + + H UD H U UD H U H H UD H U UD H U H H

Spectral decomposition of H as a convex linear combination of four systems with equal mean transmittances

1 1 2 2 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 λ λ λ λ λ λ λ λ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = + + + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Spectral decomposition

( ) ( ) ( ) ( )

1 3 2

tr ,0,0,0 0,tr ,0,0 tr tr 0,0,tr ,0 0,0,0,tr tr tr λ λ λ λ

+ + + +

= + + + H UD H U UD H U H H UD H U UD H U H H

each term in the sum is affected by its corresponding eigenvector ui

( ) ( )

3 3

,

i i i i i i

p p

= =

= =

∑ ∑

H H N H N H

( )(

)

3

tr , , 1 tr

i i i i i i i

p p λ

+ =

≡ ⊗ ≡ =

∑

H H u u H

Trivial decom position

Trivial decomposition

( )

1 2 3

, , , λ λ λ λ

+

= H UD U

( ) ( ) ( )

1 1 2 2 3 3

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 λ λ λ λ λ λ λ ⎡ ⎤ ⎢ ⎥ − + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ − + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ − + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

Trivial decomposition

Trivial decomposition of H as a convex linear combination of four systems with equal mean transmittances

( ) ( ) ( ) ( ) ( )

1 2 3 1 2 3 3 1 2 1 2 3 1 2 3

, , , 2 3 4 tr tr tr tr 1 tr 1,0,0,0 , tr 1,1,0,0 , 2 1 1 tr 1,1,1,0 , tr 1,1,1,1 3 4 λ λ λ λ λ λ λ λ λ λ λ

+ + + + +

= = − − − = ⎡ ⎤ ⎡ ⎤ ≡ ≡ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ≡ ≡ ⎣ ⎦ ⎣ ⎦ H UD U A + B + B + B H H H H A H UD U B H UD U B H UD U B H UD U

Trivial decomposition

Trivial decomposition of the Mueller matrix M as a convex linear combination of four systems with equal mean transmittances

( ) ( ) ( ) ( )

1 1 2 1 1 2 3 2 2 3 3 3

tr 2 tr 3 tr 4 tr λ λ λ λ λ λ λ − = − + − + + M N A H M B H M B H M B H

Dim.

2 D 3 D 4 D

Coh. matrix Purity quantities Limits Global purity

Indices of polarimetric purity

1

tr P λ λ − = Φ

1 1 1 2 2

tr 2 tr P P λ λ λ λ λ − = + − = R R

3

1 2

i i i

s

=

= ∑ Φ σ

8

1 3

i i i

q

=

= ∑ R Ω

( )

1 2

tr P P λ λ − ≡ = Φ

2 2 (3) 1 2

1 3 2 P P P = +

1 P ≤ ≤

1 2

1 P P ≤ ≤ ≤

3 ,

1 4

ij ij i j

m

=

= ∑ H E

1 1 1 2 2 1 2 3 2

tr 2 tr 3 tr P P P λ λ λ λ λ λ λ λ λ − = + − = + + − = R R R

1 2 3

1 P P P ≤ ≤ ≤ ≤

( )

2 2 2 2 1 2 3 4

1 2 1 2 3 3 3 P P P P ⎛ ⎞ = + + ⎜ ⎟ ⎝ ⎠

Indices of purity and trivial decomposition

1 1 2 1 2 3 1 2 3

2 3 , , tr tr tr P P P λ λ λ λ λ λ λ λ λ − + − + + − ≡ ≡ ≡ H H H

Indices of polarimetric purity

( )

2 2 2 2 1 2 3 4

1 2 1 2 3 3 3 P P P P ⎛ ⎞ = + + ⎜ ⎟ ⎝ ⎠

1 2 3

1 P P P ≤ ≤ ≤ ≤

( )

1 2 3 4

1 P P P P = = = =

Pure

( )

1 2 3 4

P P P P = = = =

Equiprobable mixture

Degree of polarimetric purity

Physical interpretation of the trivial decomposition

in terms of the indices of polarimetric purity

( ) ( ) ( ) ( )

1 2 3 1 2 1 1 3 2 2 3 3

, , , 1 P P P P P P λ λ λ λ

+

= = − − − H UD U A + B + B + B

A pure component (rank 1) B1 non-pure component (rank 2) B2 non-pure component (rank 3) B3 non-pure component (rank 4), perfect depolarizer

Arbitrary decom position

Arbitrary decomposition: existence

There exist decompositions of M into pure components, other than the spectral decomposition?

Arbitrary decomposition

Given N1, N2… N4 pure elements, we can place them as a parallel combination

N1 N2 N3 N4

4 1 i i=

=∑ s s

i i i

′ = s N s

4 4 1 1 i i i i i = =

′ ′ = =

∑ ∑

s s N s

1

s

2

s

3

s

4

s

1

′ s

2

′ s

3

′ s

4

′ s

Arbitrary decomposition

Given N1, N2… N4 arbitrary pure elements, we can construct Obviously, Ni are arbitrary, and not necessarily coincide with the “spectral” components

4 1 i i i

p

=

≡ ∑ M N

4 1

1

i i i

pure Mueller matrices p

=

=

∑

N

Arbitrary decomposition

How many “arbitrary decompositions” do exist?

Arbitrary decomposition

3 00 i i i

l m

=

= ∑ H A

00

same tran tr tr smittance

i

m = = A H

( )

pure rank c 1

• m

s ponent

i =

A

3 00

incoherent c 1

• nvex sum

i i

l m

=

=

∑

( )

3 00 00

independent

, 1,

i i i i i i

l m m

+ =

⎡ ⎤ = ⊗ = ⎣ ⎦

∑

H v v v v

[ ]

( )

3 00 00 00

,

i i i i

l m m

=

= =

∑

M N N

3 Polarim etric subtraction

Statement of the problem

Given the coherency matrices of:

• the sample as a whole (H)
• a known pure component (A)

Find α > 0 such that

H-αA is the coherency matrix of the rest

H

A

Data Test Is A compatible to be inside H?

Subtraction test Subtraction is possible if, and only if, the only eigenvector of A with non-null eigenvalue lies in the subspace generated by the eigenvectors of H with non-null eigenvalues

Subtraction procedure

If rank (H) = 4, the test is always positive

Simultaneous diagonalization of H, A If rank (H) < 4, and the test is positive Zeros framing of H , A Simultaneous diagonalization of H, A

• J. J. Gil, J. M. Correas, P. A. Melero, C. Ferreira, Monogr. Semin. Mat.

García Galdeano 31, 161 (2004)

• J. J. Gil, Eur. Phys. J. Appl. Phys. 40, 1-47 (2007)

H A

Data Calculate the concentration α of A in H Test ? Yes

H A

Data

αA

H A

Data Subtraction proceduure

αA

H- αA

Thank you!

ppgil@unizar.es www.pepegil.es