Parallel decom position of Mueller m atrices and polarim etric - PowerPoint PPT Presentation

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 T ε ε ' Jones vector � I ntensity I = ε + ⊗ ε ϕ ⎛ ⎞ A � DoP P = 1 x = ⎜ ⎟ ε ⎜ ⎟ δ i χ � χ azimuth A e y ⎝ ⎠ y � ϕ ellipticity ε ´ = T ε • Incident beam: P =1 • Single interaction P’ =1 Jones matrix

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 t β ≡ = i T 11 12 ⎢ ⎥ , t t e ij ij ij ⎣ ⎦ t t 21 22 � T satisfies the transmittance condition (maximum gain ≤ 1) ⎧ ⎫ ( ) 1 2 ( ) ( ) ( ) ⎡ ⎤ 1 tr 2 + + + + + ≤ T T T T T T ⎨ ⎬ tr 4det 1 ⎢ ⎥ ⎣ ⎦ ⎩ ⎭ 2

Basic interaction: Stokes-Mueller description N s s' Stokes vector ⎡ ⎤ I � I intensity ⎢ ⎥ ϕ χ IP cos2 cos2 � P degree of polarization ⎢ ⎥ ≡ ⎢ s ⎥ ϕ χ � χ azimuth IP cos2 sin 2 ⎢ ⎥ ϕ ⎣ ⎦ IP sin 2 � ϕ ellipticity s´ = N s Incident beam: P =1 Single interaction P´ =1 Mueller-Jones matrix

Characterization of Mueller-Jones matrices � 7 free parameters in T ⇒ 7 free parameters in N ( ) ( ) = ⊗ + − ⎡ ⎤ N T L T T L 1 1 0 0 1 ⎢ ⎥ − 1 0 0 1 ⎢ ⎥ ≡ ⎢ L ⎥ 0 1 1 0 ⎢ ⎥ − ⎣ ⎦ 0 i i 0 � 1 Transmittance condition ≤ ≡ + + + 2 2 2 1 2 g 1, g n ( n n n ) f f 00 01 02 03 For pure systems � or g f = g r ≤ ≡ + + + 2 2 2 1 2 g 1, g n ( n n n ) r r 00 10 20 30

Macroscopic interaction: Synthesis of a Mueller matrix Irradiated Incident beam area s Incoherent superposition Emerging beam ´ s

Composed Mueller matrix Emerging beam s ´ s ⎛ ⎞ ∑ ( ) ≡ ⎜ M N i p ⎟ i ⎝ ⎠ i ∑ ( ) ( ) ( ) ∗ − ≡ ⊗ ≥ = N i L T i T i L 1 ( ) , p 0, p 1 i i i

Coherency matrix associated with a Mueller matrix M H

Coherency matrix H H represents univocally the Mueller matrix and vice- versa = ∑ 3 1 H E m kl kl 4 = k l , 0 σ kl set of 4 “Pauli matrices” = ⊗ E σ σ * kl k l E kl set of 16 “Dirac matrices” Coefficients m kl are 16 measurable quantities: the 16 elements of the Mueller matrix M associated with H

Coherency matrix H ( M ) + + + ⎛ + ⎞ m m m m m m m m 02 12 20 21 22 33 00 01 ⎜ ⎟ ( ) ( ) ( ) + + + + − + + − ⎜ m m i m m i m m i m m ⎟ 10 11 03 13 30 31 23 32 ⎜ ⎟ ⎜ ⎟ + − − − m m m m m m m m ⎜ ⎟ 02 12 22 33 20 21 00 01 ( ) ( ) ( ) ⎜ ⎟ − + + − − + − − i m m m m i m m i m m 03 13 10 11 23 32 30 31 ⎜ ⎟ 1 = H ⎜ ⎟ 4 + − − + ⎜ ⎟ m m m m m m m m 20 21 22 33 02 12 00 01 ⎜ ⎟ ( ) ( ) ( ) + + + + − − + − i m m i m m m m i m m ⎜ ⎟ 30 31 23 32 10 11 03 13 ⎜ ⎟ ⎜ ⎟ + − − − m m m m m m m m ⎜ 22 33 20 21 02 12 ⎟ 00 01 ⎜ ⎟ ( ) ( ) ( ) − − + − − − − + ⎝ i m m i m m i m m m m ⎠ 23 32 30 31 03 13 10 11

M ( H ) ( ) ⎛ − − ⎞ + − + i h h h h h h h h 01 10 ⎜ 00 11 00 11 01 10 ⎟ ( ) + + + − + + − − ⎜ ⎟ h h h h h h i h h 22 33 22 33 23 32 23 32 ⎜ ⎟ ⎜ ⎟ ( ) − − + − + i h h ⎜ ⎟ h h h h h h 01 10 00 11 00 11 01 10 ⎜ ⎟ ( ) − − − + − − + − h h h h h h i h h ⎜ ⎟ 22 33 22 33 23 32 23 32 = M ⎜ ⎟ ( ) ⎜ ⎟ − − + + + i h h h h h h h h ⎜ ⎟ 03 30 02 20 02 20 03 30 ( ) + + − − + + + − ⎜ ⎟ h h h h h h i h h 13 31 13 31 12 21 12 21 ⎜ ⎟ ⎜ ⎟ ( ) ( ) ( ) − − − + ⎜ ⎟ i h h i h h i h h h h 02 20 02 20 03 30 03 30 ⎜ ⎟ ⎜ ⎟ ( ) ( ) ( ) − − + − − − − − h h i h h i h h i h h ⎝ ⎠ 12 21 13 31 13 31 12 21

Characterization of Mueller matrices � 4 Eigenvalue Conditions ≤ λ = 0 , i 0,1,2,3 i � 2 Transmittance Conditions ≤ ≤ g 1, g 1 f r ( ) ( ) 1 2 1 2 ≡ + + + ≡ + + + 2 2 2 2 2 2 g m m m m , g m m m m f 00 01 02 03 r 00 10 20 30 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

2 Parallel decom positions Spectral Trivial Arbitrary

Parallel decomposition S ′ = s Ms S´ n = ∑ s s s i ′ s 1 N 1 i = 1 1 ′ = s N s s ′ i i i s i N i i s n n ′ ∑ ∑ s ′ ′ n = = s s N s n N n i i i = = i 1 i 1

Spectral decom position

Spectral decomposition Spectral decomposition of H as a convex linear combination of four systems with equal mean transmittances ( ) = λ λ λ λ + H UD U , , , 0 1 2 3 λ λ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎢ λ ⎥ ⎢ ⎥ ⎢ λ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = + + + 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ λ λ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ λ λ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 λ λ ( ) ( ) + + = + H UD H U UD H U 0 1 tr ,0,0,0 0,tr ,0,0 H H tr tr λ λ ( ) ( ) + + + + UD H U UD H U 2 3 0,0,tr ,0 0,0,0,tr H H tr tr

Spectral decomposition λ λ ( ) ( ) + + = + H UD H U UD H U 0 1 tr ,0,0,0 0,tr ,0,0 H H tr tr λ λ ( ) ( ) + + + + UD H U UD H U 2 3 0,0,tr ,0 0,0,0,tr H H tr tr 3 3 ∑ ∑ ( ) ( ) = = H H N H N H p , p i i i i = = i 0 i 0 each term in the sum is affected by its corresponding eigenvector u i λ ) ( ) 3 ∑ ( ≡ ⊗ + ≡ = H H u u i tr , p , p 1 i i i i i H tr = i 0

Trivial decom position

Trivial decomposition ( ) + = λ λ λ λ H UD U , , , ⎡ ⎤ 0 1 2 3 1 0 0 0 ⎢ ⎥ ( ) 0 0 0 0 λ − λ + ⎢ ⎥ 0 0 0 0 0 1 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 ⎡ ⎤ 1 0 0 0 ⎢ ⎥ ( ) 0 1 0 0 λ − λ + ⎢ ⎥ 0 0 0 0 1 2 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 ⎡ ⎤ 1 0 0 0 ⎢ ⎥ ( ) 0 1 0 0 λ − λ + ⎢ ⎥ 0 0 1 0 2 3 ⎢ ⎥ ⎣ ⎦ 0 0 0 0 ⎡ ⎤ 1 0 0 0 ⎢ ⎥ 0 1 0 0 λ ⎢ ⎥ 0 0 1 0 3 ⎢ ⎥ ⎣ ⎦ 0 0 0 1

Trivial decomposition Trivial decomposition of H as a convex linear combination of four systems with equal mean transmittances ( ) + = λ λ λ λ = H UD U , , , 0 1 2 3 λ − λ λ − λ λ − λ λ = A + B + B + B 0 1 1 2 2 3 3 2 3 4 1 2 3 H H H H tr tr tr tr 1 ( ) ( ) ⎡ ⎤ ⎡ ⎤ ≡ + ≡ + A H UD U B H UD U tr 1,0,0,0 , tr 1,1,0,0 , ⎣ ⎦ ⎣ ⎦ 1 2 1 1 ( ) ( ) ⎡ ⎤ ⎡ ⎤ ≡ + ≡ + B H UD U B H UD U tr 1,1,1,0 , tr 1,1,1,1 ⎣ ⎦ ⎣ ⎦ 2 3 3 4

Trivial decomposition Trivial decomposition of the Mueller matrix M as a convex linear combination of four systems with equal mean transmittances λ − λ ( ) = M N A 0 1 H tr λ − λ ( ) + M B 1 2 2 tr 1 1 H λ − λ ( ) + M B 2 3 3 H 2 2 tr λ ( ) + M B 3 4 tr 3 3 H

Indices of polarimetric purity 2 D 3 D 4 D Dim. = ∑ = ∑ 3 = ∑ 3 1 8 1 1 Φ σ H E R Ω Coh. s m q ij ij i i i i matrix 4 2 3 = = = i j , 0 i 0 i 0 λ − λ = 0 1 P λ − λ 1 R tr = 0 1 P λ + λ − λ λ − λ 1 R 2 tr = Purity = 0 1 0 1 2 P P λ + λ − λ quantities Φ 2 R tr 2 tr = 0 1 2 P λ + λ + λ − λ 2 R 3 tr = 0 1 2 3 P 2 R tr ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ 0 P P 1 0 P P P 1 Limits 0 P 1 1 2 1 2 3 λ − λ ⎛ ⎞ 1 1 2 1 Global ≡ = = + + = + 0 1 P P 2 2 2 2 2 2 ⎜ ⎟ P 2 P P P P 3 P P ( ) ( ) purity Φ 2 (3) 1 2 ⎝ 1 2 3 ⎠ 4 tr 2 3 3 3