Many books on polarization give tables of Mueller matrices. Here we give a table of Mueller matrices M , coherency matrices C , and coherency matrix factors F ...
A table of some coherency matrices, coherency matrix factors (CMFs), and their respective Mueller matrices Colin J. R. Sheppard1, Aymeric Le Gratiet1 and Alberto Diaspro1,2,3 1
Nanophysics Department, Istituto Italiano di Tecnologia, via Morego 30, 16163 Genova, Italy University of Genoa, 16145 Genoa, Italy 3 Nikon Imaging Center, Istituto Italiano di Tecnologia, via Morego 30, 16163 Genova, Italy 2
Many books on polarization give tables of Mueller matrices. Here we give a table of Mueller matrices M , coherency matrices C , and coherency matrix factors F for different polarization components. F is not given for some complicated cases. In many cases, though, F has a very simple form. The terms are defined in the Appendix. Free space
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
⎞ ⎛ ⎟ ⎜ ⎟ ,C = 2 ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎛ 1 ⎟ ⎜ 0 ⎟ ,F = 2 ⎜ ⎟ ⎜ 0 ⎟⎠ ⎜⎝ 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Diattenuating retarder Phase shift Δ . Diattenuation D = cosκ =
p12 − p22 , p12 + p22
p1 , p2 are the maximum and minimum amplitude coefficients. Oriented at an angle φ . Terminology as in [1] ⎛ 1 ⎜ ⎜ cosκ cos 2φ M = M 00 ⎜ sin 2φ ⎜ ⎜⎝ 0
cosκ cos 2φ
cosκ sin 2φ
cos 2 φ + cos Δ sinκ sin 2 φ
0
0
sin φ + cos Δ sinκ cos 2 φ
sin Δ sinκ sin 2φ
− sin Δ sinκ cos 2φ
2
⎞ ⎟ − sin Δ sinκ sin 2φ ⎟ , sin Δ sinκ cos 2φ ⎟⎟ ⎟⎠ cos Δ sinκ 0
⎛ 1+ cos Δ sinκ (cosκ − i sin Δ sinκ )cos 2φ (cosκ − i sin Δ sinκ )sin 2φ ⎜ 2 (cosκ + i sin Δ sinκ )cos 2φ (1− cos Δ sinκ )cos 2φ (1− cos Δ sinκ )sin 2φ cos 2φ C = M 00 ⎜ ⎜ (cosκ + i sin Δ sinκ )sin 2φ (1− cos Δ sinκ )sin 2φ cos 2φ (1− cos Δ sinκ )sin 2 2φ ⎜ ⎜⎝ 0 0 0 ⎛ 1+ cos Δ sinκ ⎜ (cosκ + i sin Δ sinκ )cos 2φ M 00 ⎜ F= 1+ cos Δ sinκ ⎜ (cosκ + i sin Δ sinκ )sin 2φ ⎜ 0 ⎝
0 ⎞ ⎟ 0 ⎟ , 0 ⎟⎟ 0 ⎟⎠
0 0 0 ⎞ 0 0 0 ⎟ ⎟. 0 0 0 ⎟ ⎟ 0 0 0 ⎠
1
Diattenuating retarder at 0 (horizontal) Phase shift Δ . Diattenuation D = cosκ =
p12 − p22 , p12 + p22
p1 , p2 are the maximum and minimum amplitude coefficients. Terminology as in [1] ⎛ 1 ⎜ cosκ M = M 00 ⎜ ⎜ 0 ⎜⎝ 0
cosκ 1 0 0
0 0 cos Δ sinκ − sin Δ sinκ
⎛ 1+ cos Δ sinκ ⎜ cosκ + i sin Δ sinκ ⎜ F= 0 1+ cos Δ sinκ ⎜ ⎜⎝ 0 M 00
⎞ ⎛ 1+ cos Δ sinκ ⎟ ⎜ cosκ + i sin Δ sinκ ⎟ ,C = M 00 ⎜ 0 ⎟ ⎜ ⎟⎠ ⎜⎝ 0
0 0 sin Δ sinκ cos Δ sinκ 0 0 0 0
0 0 0 0
0 0 0 0
cosκ − i sin Δ sinκ 1− cos Δ sinκ 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟, ⎟ ⎟⎠
⎞ ⎟ ⎟. ⎟ ⎟⎠
Elliptic diattenuator
D = cosκ =
p12 − p22 p12 + p22
The ratio between the strengths of components of the electric field = tan χ , ellipticity angle = χ . Oriented at an angle φ . Terminology as in [1]. ⎛ 2( p12 + p22 ) ⎜ 2 2 1 ⎜ 2( p1 − p2 )cos 2 χ cos 2φ M= ⎜ 2 4 ⎜ 2( p1 − p22 )cos 2 χ sin 2φ ⎜ ⎜⎝ 2( p12 − p22 )sin 2 χ
2( p12 − p22 )cos 2 χ cos 2φ p2 (3p1 + p2 ) + 2( p1 − p2 ) cos 2 χ cos 2φ 2
2
( p1 − p2 )2 cos 2 2 χ sin 4φ ( p1 − p2 )2 sin 4 χ cos 2φ
2
⎞ ⎟ ⎟ ⎟, p2 (3p1 + p2 ) + 2( p1 − p2 )2 cos 2 2 χ sin 2 2φ ( p1 − p2 )2 sin 4 χ sin 2φ ⎟ ⎟ ( p1 − p2 )2 sin 4 χ sin 2φ ( p1 + p2 )2 + ( p1 − p2 )2 cos 4 χ ⎟⎠ 2( p12 − p22 )cos 2 χ sin 2φ
2( p12 − p22 )sin 2 χ
( p1 − p2 ) cos 2 χ sin 4φ
( p1 − p2 )2 sin 4 χ cos 2φ
2
2
⎛ ( p1 + p2 )2 ( p12 − p22 )cos 2 χ cos 2φ ( p12 − p22 )cos 2 χ sin 2φ ( p12 − p22 )sin 2 χ ⎜ 2 2 2 2 2 2 2 2 2 ( p1 − p2 ) cos 2 χ cos 2φ ( p1 − p2 )cos 2 χ sin 2φ cos 2φ ( p1 − p22 )sin 2 χ cos 2 χ cos 2φ 1 ⎜ ( p1 − p2 )cos 2 χ cos 2φ C= ⎜ 2 2 2 2 2 2 ⎜ ( p1 − p2 )cos 2 χ sin 2φ ( p1 − p2 )cos 2 χ sin 2φ cos 2φ ( p1 − p2 )2 cos 2 2 χ sin 2 2φ ( p12 − p22 )sin 2 χ cos 2 χ sin 2φ ⎜ 2 2 2 2 2 2 ⎜⎝ ( p1 − p2 )sin 2 χ ( p1 − p2 )sin 2 χ cos 2 χ cos 2φ ( p1 − p2 )sin 2 χ cos 2 χ sin 2φ ( p1 − p2 )2 sin 2 2 χ ⎛ p1 + p2 ⎜ 1 ⎜ ( p1 − p2 )cos 2 χ cos 2φ F= 2 ⎜⎜ ( p1 − p2 )cos 2 χ sin 2φ ⎜⎝ ( p1 − p2 )sin 2 χ
⎞ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟⎠
0 0 0 ⎞ ⎟ 0 0 0 ⎟ . 0 0 0 ⎟ ⎟ 0 0 0 ⎠⎟
2
Elliptic diattenuator at 0
D = cosκ =
p12 − p22 p12 + p22
The ratio between the strengths of components of the electric field = tan χ , ellipticity angle = χ . Terminology as in [1].
⎛ 1 ⎜ 2 p1 ⎜ cosκ cos 2 χ M= 1+ cosκ ⎜ 0 ⎜ ⎜⎝ cosκ sin 2 χ ⎛ 1+ sinκ ⎜ p ⎜ cosκ cos 2 χ C= 1+ cosκ ⎜ 0 ⎜ ⎜⎝ cosκ sin 2 χ
0
sinκ + (1− sinκ )cos 2 2 χ
0
0 (1− sinκ )sin 2 χ cos 2 χ
sinκ 0
cosκ cos 2 χ (1− sinκ )cos 2 χ 0 (1− sinκ )sin 2 χ cos 2 χ
1+ sinκ 1− sinκ cos 2 χ 0 1− sinκ sin 2 χ
⎞ ⎟ (1− sinκ )sin 2 χ cos 2 χ ⎟ ⎟, 0 ⎟ sinκ + (1− sinκ )sin 2 2 χ ⎟⎠ cosκ sin 2 χ
⎞ ⎟ 0 (1− sinκ )sin 2 χ cos 2 χ ⎟ ⎟, 0 0 ⎟ ⎟⎠ 0 (1− sinκ )sin 2 2 χ 0
2
2 1
⎛ ⎜ p1 ⎜ F= .⎜ 1+ cosκ ⎜ ⎜⎝
cosκ cos 2 χ
cosκ sin 2 χ
⎛ 0 0 0 ⎞ p1 + p2 ⎟ ⎜ 1 ⎜ ( p1 − p2 )cos 2 χ 0 0 0 ⎟ ⎟= 2⎜ 0 0 0 0 ⎟ ⎜ ⎜ ( p − p ⎟ 0 0 0 ⎠ 1 2 )sin 2 χ ⎝
0 0 0 ⎞ ⎟ 0 0 0 ⎟ . 0 0 0 ⎟ ⎟ 0 0 0 ⎟⎠
Linear diattenuator Oriented at an angle φ .
D = cosκ =
p12 − p22 . p12 + p22
Terminology as in [1].
⎛ 1 cosκ cos 2φ cosκ sin 2φ 0 ⎜ 2 2 2 0 p1 ⎜ cosκ cos 2φ cos 2φ + sinκ sin 2φ (1− sinκ )sin 2φ cos 2φ M= 1+ cosκ ⎜ cosκ sin 2φ (1− sinκ )sin 2φ cos 2φ sin 2 2φ + sinκ cos 2 2φ 0 ⎜ ⎜⎝ 0 0 0 sinκ ⎛ 1+ sinκ cosκ cos 2φ cosκ sin 2φ ⎜ 2 2 (1− sinκ )cos 2φ (1− sinκ )sin 2φ cos 2φ p1 ⎜ cosκ cos 2φ C= 1+ cosκ ⎜ cosκ sin 2φ (1− sinκ )sin 2φ cos 2φ (1− sinκ )sin 2 2φ ⎜ ⎜⎝ 0 0 0 ⎛ ⎜ p1 ⎜ F= 1+ cosκ ⎜ ⎜ ⎜⎝
1+ sinκ 1− sinκ cos 2φ 1− sinκ sin 2φ 0
⎛ p1 + p2 0 0 0 ⎞ ⎟ ⎜ 1 ⎜ ( p1 − p2 )cos 2φ 0 0 0 ⎟ ⎟= 2⎜ ( p1 − p2 )sin 2φ 0 0 0 ⎟ ⎜ ⎜⎝ ⎟ 0 0 0 0 ⎠
⎞ ⎟ ⎟, ⎟ ⎟ ⎟⎠
0 ⎞ ⎟ 0 ⎟ , 0 ⎟⎟ 0 ⎟⎠
0 0 0 ⎞ ⎟ 0 0 0 ⎟ . 0 0 0 ⎟ ⎟ 0 0 0 ⎟⎠
3
Linear diattenuator at 0
χ = 0 , M 00 =
1 2
(p
2 1
)
+ p22 =
⎛ p2 + p2 1 2 ⎜ 1 ⎜ p12 − p22 M= ⎜ 2 0 ⎜ ⎜ 0 ⎝
p12 . Terminology as in [1]. 1+ cosκ
p12 − p22
0
p12 + p22
0
0
2 p1 p2
0
0
⎞ ⎟ 0 ⎟ p12 = ⎟ 0 ⎟ 1+ cosκ 2 p1 p2 ⎟⎠ 0
⎛ ( p + p )2 1 2 ⎜ 1 ⎜ p12 − p22 C= 2⎜ 0 ⎜ ⎜⎝ 0
0 0 ⎞ ⎟ p12 0 0 ⎟= ⎟ 0 0 ⎟ 1+ cosκ 0 0 ⎟⎠
( p1 − p2 )
⎛ p1 + p2 ⎜ 1 ⎜ p1 − p2 F= 2⎜ 0 ⎜ 0 ⎝
0 0 0 ⎞ ⎟ p1 0 0 0 ⎟= ⎟ 1+ cosκ 0 0 0 ⎟ 0 0 0 ⎠
p12 − p22 2
0 0
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
⎛ 1 ⎜ cosκ ⎜ ⎜ 0 ⎜⎝ 0
⎛ 1+ sinκ ⎜ cosκ ⎜ 0 ⎜ ⎜⎝ 0
1+ sinκ 1− sinκ 0 0
cosκ 1 0 0
cosκ 1− sinκ 0 0
0 0 sinκ 0 0 0 0 0
0 0 0 0
0 0 0 Δ sinκ
⎞ ⎟ ⎟, ⎟ ⎟⎠
⎞ ⎟ ⎟, ⎟ ⎟⎠
0 0 0 ⎞ ⎟ 0 0 0 ⎟. 0 0 0 ⎟ ⎟ 0 0 0 ⎠
Elliptic polarizer at 0 The ratio between the strengths of components of the electric field = tan χ , ellipticity angle = χ . Terminology as in [1].
⎛ 1 ⎜ p ⎜ cos 2 χ M= 2 ⎜ 0 ⎜ ⎜⎝ sin 2 χ
0 sin 2 χ cos 2 χ
⎛ 1 ⎜ cos 2 χ p F= 1 ⎜ 0 2⎜ ⎜ sin 2 χ ⎝
0 0 0 ⎞ 0 0 0 ⎟ ⎟. 0 0 0 ⎟ 0 0 0 ⎟⎠
cos 2 χ
⎞ ⎛ 1 ⎟ ⎜ 2 0 sin 2 χ cos 2 χ ⎟ cos 2 χ p ,C = 1 ⎜⎜ ⎟ 2 0 0 0 ⎟ ⎜ 2 ⎟⎠ ⎜⎝ sin 2 χ 0 sin 2 χ sin 2 χ
0
cos 2 χ 2
2 1
cos 2 χ
⎞ ⎟ 0 sin 2 χ cos 2 χ ⎟ ⎟, 0 0 ⎟ ⎟⎠ 0 sin 2 2 χ 0
cos 2 χ 2
0 sin 2 χ cos 2 χ
sin 2 χ
Linear total polarizer Oriented at an angle φ . Terminology as in [1].
⎛ 1 cos 2φ sin 2φ ⎜ 2 cos 2φ sin 2φ cos 2φ p ⎜ cos 2φ M= ⎜ 2 sin 2 2φ ⎜ sin 2φ sin 2φ cos 2φ ⎜⎝ 0 0 0
⎛ 0 ⎞ 1 cos 2φ sin 2φ ⎟ ⎜ 2 2 0 ⎟ cos 2φ cos 2φ sin 2φ cos 2φ p ,C = 1 ⎜ ⎟ ⎜ 2 0 ⎟ sin 2 2φ ⎜ sin 2φ sin 2φ cos 2φ ⎜⎝ 0 ⎟⎠ 0 0 0
2 1
⎛ 1 ⎜ cos 2φ p F= 1 ⎜ 2 ⎜ sin 2φ ⎜ 0 ⎝
0 ⎞ ⎟ 0 ⎟ , 0 ⎟⎟ 0 ⎟⎠
0 0 0 ⎞ 0 0 0 ⎟ ⎟. 0 0 0 ⎟ ⎟ 0 0 0 ⎠
Linear polarizer at 0
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 1 0 0
1 1 0 0
0 0 0 0
0 0 0 0
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 1 0 0
1 1 0 0
0 0 0 0
0 0 0 0
⎞ ⎛ ⎟ ⎜ ⎟ ,F = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
4
Linear polarizer at 45
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 1 0
⎛ ⎜ M=⎜ ⎜ ⎜⎝
⎛ 1 −1 0 0 ⎞ ⎟ ⎜ −1 1 0 0 ⎟ ,C = ⎜ 0 0 0 0 ⎟ ⎜ ⎜⎝ 0 0 0 0 ⎟⎠
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 −1 0
0 0 0 0
1 0 1 0
0 0 0 0
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 0 1 0
0 0 0 0
1 0 1 0
0 0 0 0
⎞ ⎛ ⎟ ⎜ ⎟ ,F = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 0 1 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Linear polarizer at 90
⎛ 1 −1 0 0 ⎞ ⎟ ⎜ −1 1 0 0 ⎟ ,F = ⎜ 0 0 0 0 ⎟ ⎜ ⎜⎝ 0 0 0 0 ⎟⎠
1 −1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
⎛ 0 −1 0 ⎞ ⎟ ⎜ 0 0 0 ⎟ ,F = ⎜ 0 1 0 ⎟ ⎜ ⎜⎝ 0 0 0 ⎟⎠
1 0 −1 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Linear polarizer at 135
⎛ 0 −1 0 ⎞ ⎟ ⎜ 0 0 0 ⎟ ,C = ⎜ 0 1 0 ⎟ ⎜ ⎜⎝ 0 0 0 ⎟⎠
1 0 −1 0
Circular diattenuator Diattenuation D = cosκ =
M 00 =
1 2
(p
2 1
)
+ p22 =
p12 − p22 , p12 + p22
p12 , 1+ cosκ
p1 , p2 are the maximum and minimum amplitude coefficients. Terminology as in [1].
⎛ 1 p12 ⎜ 0 ⎜ M= 1+ cosκ ⎜ 0 ⎜⎝ cosκ ⎛ ⎜ p1 ⎜ F= 1+ cosκ ⎜ ⎜ ⎝
0 sinκ 0 0
0 0 sinκ 0
⎞ ⎟ p12 ⎟ ,C = 1+ cosκ ⎟ ⎟⎠
cosκ 0 0 1
⎛ p1 + p2 0 0 0 ⎞ ⎟ ⎜ 1 ⎜ 0 0 0 ⎟ 0 = ⎜ 0 0 0 ⎟ 0 2 ⎟ ⎜ p −p 0 0 0 ⎠ 2 ⎝ 1
1+ sinκ 0 0 1− sinκ
⎛ 1+ sinκ ⎜ 0 ⎜ 0 ⎜ ⎜⎝ cosκ
0 0 0 0
0 cosκ 0 0 0 0 0 1− sinκ
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
⎞ ⎟ ⎟, ⎟ ⎟⎠
0 0 0 ⎞ ⎟ 0 0 0 ⎟ . 0 0 0 ⎟ 0 0 0 ⎟⎠
Circular polarizer, left
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 1
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 −1
0 0 0 0
0 0 0 0
1 0 0 1
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 0 0 1
⎞ ⎛ ⎟ ⎜ ⎟ ,F = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
0 0 0 0
0 0 0 0
1 0 0 1
1 0 0 1
0 0 0 0
⎛ 1 0 −1 ⎞ ⎜ 0 0 0 ⎟ ⎟ ,C = ⎜ 0 0 ⎟ ⎜ 0 ⎜⎝ −1 0 1 ⎟⎠
0 0 0 0
⎛ 1 0 −1 ⎞ ⎜ 0 0 0 ⎟ ⎟ ,F = ⎜ 0 0 ⎟ ⎜ 0 ⎜⎝ −1 0 1 ⎟⎠
0 0 0 0
Circular polarizer, right
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
5
Rotator Circular retarder with retardance = 2θ .
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 cos 2θ 0 − sin 2θ 0 0
⎛ cosθ ⎜ 0 F= 2⎜ 0 ⎜ ⎜⎝ i sin θ
0 0 0 0
0 sin 2θ cos 2θ 0 0 0 0 0
0 0 0 0
0 0 0 1
⎛ 2 cos 2 θ ⎞ ⎜ ⎟ 0 ⎟ ,C = ⎜ 0 ⎜ ⎟ ⎜ i sin 2θ ⎟⎠ ⎝
0 0 0 0
0 −i sin 2θ 0 0 0 0 0 2sin 2 θ
⎞ ⎟ ⎟, ⎟ ⎟ ⎠
⎞ ⎟ ⎟. ⎟ ⎟⎠
Elliptic retarder Phase shift between the two eigenstates = Δ , The ratio between the strengths of components of the electric field = tan χ , ellipticity angle = χ . Oriented at an angle φ . Terminology as in [1]. ⎛ ⎜ ⎜ ⎜ 1⎜ M= 4⎜ ⎜ ⎜ ⎜ ⎝
1
0 0 0 Δ 2 Δ 2 2 2 Δ 2 2 Δ 0 cos − sin (sin 2 χ − cos 2 χ cos 4φ ) sin cos 2 χ sin 4φ + sin Δ sin 2 χ sin sin 4 χ cos2φ − sin Δ cos2 χ sin 2φ 2 2 2 2 Δ Δ Δ Δ 0 sin 2 cos2 2 χ sin 4φ − sin Δ sin 2 χ cos2 − sin 2 (sin 2 2 χ + cos2 2 χ cos 4φ ) sin 2 sin 4 χ sin 2φ + sin Δ cos2 χ cos2φ 2 2 2 2 Δ Δ 2 Δ 2 Δ 0 sin sin 4 χ cos2φ + sin Δ cos2 χ sin 2φ sin sin 4 χ sin 2φ − sin Δ cos2 χ cos2φ cos2 − sin 2 cos 4 χ 2 2 2 2 2
⎛ Δ 2 cos2 −i sin Δ cos2 χ cos2φ −i sin Δ cos2 χ sin 2φ −i sin Δ sin 2 χ ⎜ 2 ⎜ Δ Δ Δ ⎜ i sin Δ cos2 χ cos2φ 2sin 2 cos2 2 χ cos2 2φ sin 2 cos2 2 χ sin 4φ sin 2 sin 4 χ cos2φ 1⎜ 2 2 2 C= ⎜ 4⎜ 2 Δ 2 2 Δ 2 2 2 Δ i sin Δ cos2 χ sin 2 φ sin cos 2 χ sin 4 φ 2sin cos 2 χ sin 2 φ sin sin 4 χ sin 2φ ⎜ 2 2 2 ⎜ Δ Δ Δ ⎜ i sin Δ sin 2 χ sin 2 sin 4 χ cos2φ sin 2 sin 4 χ sin 2φ 2sin 2 sin 2 2 χ ⎜⎝ 2 2 2 ⎛ Δ cos ⎜ 2 ⎜ Δ ⎜ i sin cos2 χ cos2φ 1 ⎜ 2 ⎜ F= Δ 2⎜ ⎜ i sin 2 cos2 χ sin 2φ ⎜ Δ ⎜ i sin sin 2 χ ⎜⎝ 2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟⎠
⎞ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟ ⎟. 0 0 0 ⎟⎟ ⎟ 0 0 0 ⎟⎟ ⎠
6
Elliptic retarder at 0 Phase shift between the two eigenstates = Δ , The ratio between the strengths of components of the electric field = tan χ , ellipticity angle = χ . Terminology as in [1].
⎛ ⎜ M = ⎜⎜ ⎜ ⎜⎝
1 0
0 cos 2 χ + cos Δ sin 2 2 χ
0 sin Δ sin 2 χ
0
− sin Δ sin 2 χ
cos Δ
2
0 (1− cos Δ)sin 2 χ cos 2 χ
⎛ Δ cos 2 ⎜ 2 ⎜ Δ Δ ⎜ i sin cos cos 2 χ C = 2⎜ 2 2 ⎜ 0 ⎜ ⎜ Δ Δ ⎜ i sin cos sin 2 χ 2 2 ⎝ ⎛ Δ cos ⎜ 2 ⎜ Δ ⎜ i sin cos 2 χ F= 2⎜ 2 ⎜ 0 ⎜ ⎜ Δ ⎜ i sin sin 2 χ 2 ⎝
− sin Δ cos 2 χ
Δ Δ cos cos 2 χ 2 2 Δ sin 2 cos 2 2 χ 2 0 Δ sin 2 sin 2 χ cos 2 χ 2
−i sin
⎞ 0 ⎟ (1− cos Δ)sin 2 χ cos 2 χ ⎟ ⎟, sin Δ cos 2 χ ⎟ sin 2 2 χ + cos Δ cos 2 2 χ ⎟⎠
Δ Δ cos sin 2 χ 2 2 Δ sin 2 sin 2 χ cos 2 χ 2 0 Δ sin 2 sin 2 2 χ 2
0 −i sin 0 0 0
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠
⎞ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟ . ⎟ 0 0 0 ⎟ ⎟ 0 0 0 ⎟ ⎠
Linear retarder Phase shift Δ , Oriented at an angle φ .Terminology as in [1].
⎛ ⎜ M=⎜ ⎜ ⎜ ⎜⎝
1 0
0 cos 2φ + cos Δ sin 2 2φ
0 (1− cos Δ)sin 2φ cos 2φ
0 (1− cos Δ)sin 2φ cos 2φ
sin 2 2φ + cos Δ cos 2 2φ
0
2
sin Δ sin 2φ
⎛ Δ cos 2 ⎜ 2 ⎜ Δ Δ ⎜ i sin cos cos 2φ C = 2⎜ 2 2 ⎜ Δ ⎜ i sin cos Δ sin 2φ ⎜ 2 2 ⎜ 0 ⎝ ⎛ Δ cos ⎜ 2 ⎜ Δ ⎜ i sin cos 2φ F= 2⎜ 2 ⎜ ⎜ i sin Δ sin 2φ ⎜ 2 ⎜ 0 ⎝
− sin Δ cos 2φ
Δ Δ cos cos 2φ 2 2 Δ sin 2 cos 2 2φ 2 Δ sin 2 sin 2φ cos 2φ 2 0
−i sin
⎞ 0 ⎟ − sin Δ sin 2φ ⎟ , sin Δ cos 2φ ⎟ ⎟ ⎟⎠ cos Δ
Δ Δ cos sin 2φ 2 2 Δ sin 2 sin 2φ cos 2φ 2 Δ sin 2 sin 2 2φ 2 0
−i sin
⎞ 0 ⎟ ⎟ ⎟ 0 ⎟ , ⎟ 0 ⎟ ⎟ ⎟ 0 ⎠
⎞ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟ . ⎟ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎠
7
Linear retarder at 0 Phase shift Δ
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0
⎛ Δ cos 2 ⎜ 0 0 ⎞ 2 ⎜ 0 0 ⎟ Δ Δ ⎟ ,C = 2 ⎜⎜ i sin cos cos Δ sin Δ ⎟ 2 2 ⎜ − sin Δ cos Δ ⎟⎠ 0 ⎜ ⎜⎝ 0
0 1 0 0
⎛ Δ ⎜ cos 2 ⎜ Δ ⎜ F = 2 ⎜ i sin 2 ⎜ 0 ⎜ ⎜⎝ 0
Δ Δ cos 2 2 Δ sin 2 2 0 0
−i sin
⎞ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟, ⎟ 0 0 ⎟ 0 0 ⎟⎠
⎞ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟. ⎟ 0 0 0 ⎟ 0 0 0 ⎟⎠
Quarter-‐wave retarder at 0
⎛ ⎜ M=⎜ ⎜ ⎜⎝
0 0 1 0
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
⎛ 1 −i 0 0 ⎞ ⎟ ⎜ i 1 0 0 ⎟ ,F = ⎜ 0 0 0 0 ⎟ ⎜ ⎜⎝ 0 0 0 0 ⎟⎠
1 i 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
⎛ 0 −i 0 ⎞ ⎟ ⎜ 0 0 0 ⎟ ,F = ⎜ 0 1 0 ⎟ ⎜ ⎜⎝ 0 0 0 ⎟⎠
1 0 i 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
1 0 0 0
0 0 1 0 0 0 0 −1
1 0 0 0
0 0 0 1
⎛ 0 0 ⎞ ⎟ ⎜ 0 −1 ⎟ ,C = ⎜ 1 0 ⎟ ⎜ ⎜⎝ 0 0 ⎟⎠
1 0 i 0
1 0 0 0
0 1 0 0
⎛ 0 0 ⎞ ⎟ ⎜ 0 0 ⎟ ,C = ⎜ 0 −1 ⎟ ⎜ ⎜⎝ 1 0 ⎟⎠
1 i 0 0 −i 1 0 0 0 0 0 0 0 0 0 0
Quarter-‐wave retarder at 45
⎛ ⎜ M=⎜ ⎜ ⎜⎝
Quarter-‐wave retarder at 90
⎛ ⎜ M=⎜ ⎜ ⎜⎝
⎞ ⎛ ⎟ ⎜ ⎟ ,F = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 −i 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
⎛ 0 i 0 ⎞ ⎟ ⎜ 0 0 0 ⎟ ,F = ⎜ 0 1 0 ⎟ ⎜ ⎜⎝ 0 0 0 ⎟⎠
1 0 −i 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Quarter-‐wave retarder at 135
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0 0 0 0 −1
0 0 1 0
0 1 0 0
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 0 −i 0
8
Circular retarder Phase shift Δ .
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0 cos Δ sin Δ 0 − sin Δ cos Δ 0 0 0
⎛ Δ ⎜ cos 2 ⎜ 0 ⎜ F= 2⎜ 0 ⎜ ⎜ i sin Δ ⎜⎝ 2
⎛ Δ cos 2 ⎜ ⎞ 2 ⎜ ⎟ 0 ⎟ ,C = 2 ⎜⎜ 0 ⎟ ⎜ ⎟⎠ Δ ⎜ i sin cos Δ ⎜⎝ 2 2
0 0 0 1
⎞ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟⎠
Δ Δ cos 2 2 0 0 Δ sin 2 2
0 0 −i sin 0 0 0 0 0 0
⎞ 0 0 0 ⎟ ⎟ 0 0 0 ⎟ . 0 0 0 ⎟ ⎟ 0 0 0 ⎟⎟ ⎠
Quarter-‐wave retarder circular left
⎛ ⎜ M=⎜ ⎜ ⎜⎝
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
1 0 0 i
0 0 0 0
⎛ 1 0 −i ⎞ ⎜ 0 0 0 ⎟ ⎟ ,F = ⎜ 0 0 ⎟ ⎜ 0 ⎜⎝ i 0 1 ⎟⎠
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
⎛ 0 i ⎞ ⎟ ⎜ 0 0 ⎟ ,C = ⎜ 0 0 ⎟ ⎜ ⎜⎝ 0 1 ⎟⎠
1 0 0 −i
0 0 0 0
⎛ 0 i ⎞ ⎟ ⎜ 0 0 ⎟ ,F = ⎜ 0 0 ⎟ ⎜ ⎜⎝ 0 1 ⎟⎠
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
1 0 0 0 0 1 0 −1 0 0 0 0
0 0 0 1
Quarter-‐wave retarder circular right
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 −i
0 0 0 0
1 0 0 −i
Half-‐wave retarder (pseudorotator) Oriented at an angle φ .
⎛ ⎜ M=⎜ ⎜ ⎜ ⎝
1 0 0 cos 4φ
0 sin 4φ
0
sin 4φ
− cos 4φ
0
0
0
⎛ 0 ⎜ cos 2φ F= 2⎜ ⎜ sin 2φ ⎜ 0 ⎝
⎛ 0 ⎞ ⎟ ⎜ 0 ⎟ ,C = ⎜ ⎜ 0 ⎟ ⎟ ⎜ −1 ⎠ ⎝
0 0 0 1+ cos 4φ
0 sin 4φ
0
sin 4φ
1− cos 4φ
0
0
0
0 ⎞ 0 ⎟ ⎟, 0 ⎟ ⎟ 0 ⎠
0 0 0 ⎞ 0 0 0 ⎟ ⎟. 0 0 0 ⎟ ⎟ 0 0 0 ⎠
Half-‐wave retarder 0
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0
⎛ 0 0 0 ⎞ ⎟ ⎜ 1 0 0 ⎟ ,C = 2 ⎜ 0 −1 0 ⎟ ⎜ ⎜⎝ 0 0 −1 ⎟⎠
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
⎞ ⎛ 0 ⎟ ⎜ 1 ⎟ ,F = 2 ⎜ ⎟ ⎜ 0 ⎟⎠ ⎜⎝ 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
9
Half-‐wave retarder 90 ( M,C are same as for 0 )
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0
⎛ 0 0 0 ⎞ ⎟ ⎜ 1 0 0 ⎟ ,C = 2 ⎜ 0 −1 0 ⎟ ⎜ ⎜⎝ 0 0 −1 ⎟⎠
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
⎞ ⎛ 0 ⎟ ⎜ −1 ⎟ ,F = 2 ⎜ ⎟ ⎜ 0 ⎟⎠ ⎜⎝ 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Normal reflection from a dielectric Refractive index = n
⎛ ⎛ n − 1⎞ ⎜ ⎜ M=⎜ ⎝ n + 1 ⎟⎠ ⎜ ⎜⎝ 2
⎛ n − 1 ⎛ ⎞⎜ ⎜ F= 2⎜ ⎝ n + 1 ⎟⎠ ⎜ ⎜⎝
1 0 0 0
⎛ 0 0 0 ⎞ 2⎜ ⎟ ⎛ n − 1⎞ 1 0 0 ⎟ ,C = 2 ⎜ ⎜ ⎝ n + 1 ⎟⎠ ⎜ 0 −1 0 ⎟ ⎜⎝ 0 0 −1 ⎟⎠ 0 −1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 1 0 0
0 0 0 0
⎞ ⎟ ⎟, ⎟ ⎟⎠
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Half-‐wave retarder 45
⎛ ⎜ M=⎜ ⎜ ⎜⎝
⎛ 1 0 0 0 ⎞ ⎟ ⎜ 0 −1 0 0 ⎟ ,C = 2 ⎜ 0 0 1 0 ⎟ ⎜ ⎜⎝ 0 0 0 −1 ⎟⎠
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 0
⎞ ⎛ 0 ⎟ ⎜ 0 ⎟ ,F = 2 ⎜ ⎟ ⎜ 1 ⎟⎠ ⎜⎝ 0
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Half-‐wave retarder 135
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 −1
⎞ ⎛ ⎟ ⎜ ⎟ ,C = 2 ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 0
⎞ ⎛ 0 ⎟ ⎜ 0 ⎟ ,F = 2 ⎜ ⎟ ⎜ −1 ⎟⎠ ⎜⎝ 0
⎛ ⎜ M=⎜ ⎜ ⎜⎝
⎛ 1 0 0 0 ⎞ ⎟ ⎜ 0 −1 0 0 ⎟ ,C = 2 ⎜ 0 0 −1 0 ⎟ ⎜ ⎜⎝ 0 0 0 1 ⎟⎠
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 1
⎞ ⎛ 0 ⎟ ⎜ 0 ⎟ ,F = 2 ⎜ ⎟ ⎜ 0 ⎟⎠ ⎜⎝ 1
⎛ ⎜ M=⎜ ⎜ ⎜⎝
1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 1
⎞ ⎛ ⎟ ⎜ ⎟ ,C = 2 ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 1
⎞ ⎛ 0 ⎟ ⎜ 0 ⎟ ,F = 2 ⎜ ⎟ ⎜ 0 ⎟⎠ ⎜⎝ −1
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Half-‐wave retarder left
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
Half-‐wave retarder right
0 0 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟. ⎟ ⎟⎠
10
Aligned linear diattenuation and retardance
⎛ ⎜ M=⎜ ⎜ ⎜⎝
⎛ A+C A B 0 0 ⎞ ⎟ ⎜ B + iD B A 0 0 ⎟ ,C = ⎜ 0 0 C D ⎟ 0 ⎜ ⎜⎝ 0 0 −D C ⎟⎠ 0
B − iD A−C 0 0
0 0 0 0
0 0 0 0
⎞ ⎟ ⎟, ⎟ ⎟⎠
⎛ A+ R A− R (B − iD) (C − R) (B − iD) ⎜ (C + R) 2R(R + C) 2R(R − C) ⎜ ⎜ (A + R)(B 2 + D 2 ) (A − R)(B 2 + D 2 ) F=⎜ ⎜ 2R(R + C) 2R(R − C) ⎜ 0 0 ⎜ ⎜⎝ 0 0
⎞ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟, ⎟ ⎟ 0 0 ⎟ 0 0 ⎟⎠
A ≥ C, R = B 2 + C 2 + D 2 . Medium of scattering particles obeying reciprocity and with a plane of symmetry [2]
⎛ A0 + B0 ⎜ C M=⎜ ⎜ 0 ⎜ 0 ⎝
⎛ A0 + A C − iD ⎞ ⎜ ⎟ C + iD B0 + B A+ B 0 0 ⎟ ,C = ⎜ ⎜ ⎟ 0 A− B D 0 0 ⎜ ⎟ 0 −D A0 − B0 ⎠ ⎜⎝ 0 0 C
0
0
⎞ ⎟ 0 0 ⎟ ⎟, B0 − B 0 ⎟ 0 A0 − A ⎟⎠ 0
0
A0 ≥ A ≥ 0, B0 ≥ B ≥ 0. Rotationally symetric medium of asymmetric scatterers in the direct forward scattering direction [2]
⎛ A0 + B0 ⎜ 0 M=⎜ ⎜ 0 ⎜ I−F ⎝
0 A −J 0
⎛ A0 + A 0 ⎞ ⎜ ⎟ 0 B0 J 0 ⎟ ,C = ⎜ ⎜ ⎟ A 0 0 −iF ⎜ 0 A0 − B0 ⎟⎠ ⎜⎝ I + iJ 0 0
F+I
0 iF B0 0
I − iJ ⎞ ⎟ 0 ⎟ ⎟, 0 ⎟ A0 − A ⎟⎠
⎛ A0 + R A0 − R (I − iJ ) (A − R) (I − iJ ) 0 ⎜ (A + R) 2R(R + A) 2R(R − A) ⎜ ⎜ i B+F ⎜ 0 0 ⎜ 2 F=⎜ B+F ⎜ 0 0 ⎜ 2 ⎜ ⎜ (A0 + R)(I 2 + J 2 ) (A0 − R)(I 2 + J 2 ) 0 ⎜ ⎜⎝ 2R(R + A) 2R(R − A)
⎞ ⎟ ⎟ ⎟ i B−F ⎟ − ⎟ 2 ⎟, B−F ⎟ ⎟ 2 ⎟ ⎟ 0 ⎟ ⎟⎠ 0
A0 ≥ A ≥ 0, B0 ≥ 0, R = A 2 + I 2 + J 2 .
11
Medium of asymmetric scatterers in the exact backscattering direction [2]
⎛ A0 + B C L I ⎜ C A + B J K M=⎜ ⎜ −L −J A− B D ⎜ I K −D A 0 −B ⎝
⎞ ⎛ A0 + A C − iD ⎟ ⎜ 2B ⎟ ,C = ⎜ C + iD ⎟ ⎜ 0 0 ⎟ ⎜ I + iJ K + iL ⎠ ⎝
I − iJ ⎞ ⎟ 0 K − iL ⎟ , ⎟ 0 0 0 A0 + A ⎟⎠ 0
A0 ≥ A ≥ 0, B ≥ 0. Medium of asymmetric scatterers in the non-‐exact backscattering direction, with rotational symmetry of the medium [2]
⎞ ⎟ ⎟ ,C = ⎟ ⎟ ⎠
⎛ A0 + A1 ⎜ 0 1⎜ ⎜ 2 0 ⎜ ⎜⎝ 2I
A0 + A1 + 2I
A0 + A1 − 2I
0
0
0
0
0
0
A0 + A1 + 2I
− A0 + A1 − 2I
0
⎛ A0 + B 0 0 I ⎜ 0 B 0 0 M=⎜ ⎜ 0 0 −B 0 ⎜ I 0 0 A 1−B ⎝ ⎛ ⎜ 1 ⎜⎜ F= 2⎜ ⎜ ⎜ ⎝
0
0
A0 − A1 + 4B
0
0
A0 − A1
0
0
⎞ ⎟ 0 ⎟ ⎟, 0 ⎟ A0 + A1 ⎟⎠ 2I
⎞ ⎟ ⎟ 0 ⎟ , A0 ≥ A1 ≥ 0, B ≥ 0. 2(A0 − A1 ) ⎟ ⎟ ⎟ 0 ⎠ 0
2(A0 − A1 + 4B)
Medium of asymmetric scatterers in the exact backscattering direction, with rotational symmetry of the medium [2]
⎛ A0 + B 0 0 I ⎜ 0 B 0 0 M=⎜ ⎜ 0 0 −B 0 ⎜ I 0 0 A 0 −B ⎝ ⎛ ⎜ 1 ⎜ F= ⎜ 2⎜ ⎜ ⎝
⎞ ⎛ ⎟ ⎜ ⎟ ,C = ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝
A0 + I
A0 − I
0
0 0
0 0
2 B 0
A0 + I
− A0 − I
0
A0 0 0 I
I ⎞ ⎟ 2B 0 0 ⎟ , 0 0 0 ⎟ 0 0 A0 ⎟⎠ 0
0
0 ⎞ ⎟ 0 ⎟ , A ≥ 0, B ≥ 0. ⎟ 0 0 ⎟ 0 ⎟⎠
G-‐antisymmetric Mueller matrix [3]
⎛ ⎜ ⎜ M=⎜ ⎜ ⎜⎝ ⎛ ⎜ ⎜ F=⎜ ⎜ ⎜ ⎜⎝
a
p1
p2
p1
a
p6
p2
p6
a
p3
p5
− p4
(a + r)3/2 a + r ( p1 + ip4 ) a + r ( p2 + ip5 ) a + r ( p3 + ip6 )
⎛ p3 ⎞ 2a ⎟ ⎜ − p5 ⎟ ⎜ p1 + ip4 ,C = ⎜ ⎟ p4 p + ip5 ⎟ ⎜ 2 ⎟ ⎜ a ⎠ ⎝ p3 + ip6
p1 − ip4
p2 − ip5
0
0
0
0
0
0
p3 − ip6 ⎞ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟⎠ 0
0 0 ⎞ ⎟ a − r ( p1 + ip4 ) 0 0 ⎟ ⎟. a − r ( p2 + ip5 ) 0 0 ⎟ ⎟ a − r ( p3 + ip6 ) 0 0 ⎟⎠ (a − r)3/2
r = a 2 + p12 + p22 + p32 + p42 + p52 + p62 ,a ≥ r.
12
G-‐symmetric Mueller matrix [3]
⎛ 0 ⎜ ⎜ − p1 M=⎜ − p2 ⎜ ⎜⎝ − p3
p1
p2
0
p6
p6
0
p5
p4
⎛ p3 ⎞ ⎟ ⎜ p5 ⎟ ⎜ ,C = ⎜ p4 ⎟ ⎟ ⎜ ⎜⎝ 0 ⎟⎠
0 0
0 0
0 p6 + ip3
0
p6 − ip3
0
0
p5 + ip2
p4 − ip1
⎞ 0 p5 − ip2 ⎟ ⎟. p4 + ip1 ⎟ ⎟ ⎟⎠ 0
Diagonal Mueller matrix (canonical Mueller matrix,Type1)
M = diag ( d0 + d1 + d2 + d3 ,d0 + d1 − d2 − d3
d0 − d1 + d2 − d3 ,d0 − d1 − d2 + d3 ) ,
C = 2 diag ( d0 ,d1 ,d2 ,d3 ) , F = 2 diag
(
)
d0 , d1 , d2 , d3 .
Canonical Mueller matrix, Type 2 (Ossikovski)[4]
⎛ 2d0 ⎜ ⎜ d0 M=⎜ 0 ⎜ ⎜⎝ 0 ⎛ ⎜ ⎜ F=⎜ ⎜ ⎜ ⎜ ⎝
−d0
0
0
0
0
d2
0
0
⎛ d0 + d2 0 ⎞ ⎟ ⎜ 0 ⎟ 0 ⎜ ,C = ⎜ ⎟ 0 0 ⎟ ⎜ ⎟ ⎜ d2 ⎠ 0 ⎝
d0 + d2
0
0
0
d0 − d2
0
0
0
−i d0
0
0
0
0
d0 − d2
0
0
d0
0
id0
⎞ ⎟ 0 ⎟ , −id0 ⎟ ⎟ d0 ⎟⎠ 0
0 ⎞ ⎟ 0 ⎟ ⎟. 0 ⎟ ⎟ 0 ⎟⎠
d0
Canonical Mueller matrix, Type 2 (Bolshakov) [5, 6]
⎛ 2d0 ⎜ ⎜ d1 − d0 M=⎜ 0 ⎜ ⎜⎝ 0 ⎛ ⎜ ⎜ F=⎜ ⎜ ⎜ ⎜ ⎝
d0 − d1
0
2d1
0
0
d2
0
0
⎛ d0 + d1 + d2 0 ⎞ ⎟ ⎜ 0 ⎟ 0 ⎜ ,C = ⎟ ⎜ 0 0 ⎟ ⎜ ⎜⎝ d2 ⎟⎠ 0
d0 + d1 + d2
0
0
0
d0 + d1 − d2
0
0
0
i d0 − d1
0
0
d0 − d1
0 d0 + d1 − d2 0 0
⎞ ⎟ 0 0 ⎟ , d0 − d1 i(d0 − d1 ) ⎟ ⎟ −i(d0 − d1 ) d0 − d1 ⎟⎠ 0
0
0 ⎞ ⎟ 0 ⎟ ⎟ ,d0 ≥ d1 ,d0 + d1 − d2 ≥ 0,d0 + d1 + d2 > 0. 0 ⎟ ⎟ 0 ⎟⎠
13
Parameterized deterministic mueller matrix [7] ⎛ 1 ⎜ ⎜ ⎜ 2φ ⎜ l sin φ cosθ1 − 2mnsin 2 sin Δθ1 M = M 00 ⎜ ⎜ m sin φ cosθ − 2nl sin 2 φ sin Δθ 2 2 ⎜ 2 ⎜ ⎜ nsin φ cosθ − 2lm sin 2 φ sin Δθ 3 3 ⎜⎝ 2
φ l sin φ cosθ1 + 2mnsin 2 sin Δθ1 2 2 2φ cosφ + 2l sin 2 2φ −nsin φ sin θ 3 + 2lm sin cos Δθ 3 2 2φ m sin φ sin θ 2 + 2nl sin cos Δθ 2 2
φ φ m sin φ cosθ 2 + 2nl sin 2 sin Δθ 2 nsin φ cosθ 3 + 2lm sin 2 sin Δθ 3 2 2 2φ 2φ nsin φ cosθ 3 + 2lm sin cos Δθ 3 −m sin φ cosθ 2 + 2nl sin cos Δθ 2 2 2 2 2φ 2φ cosφ + 2m sin l sin φ cosθ1 + 2mnsin cos Δθ1 2 2 2φ 2 2φ −l sin φ sin θ1 + 2mnsin cos Δθ1 cosφ + 2n sin 2 2
l + m + n = 1 , Δθ i = (θ i+1 − θ i−1 ) in cyclic order, so that Δθ1 + Δθ 2 + Δθ 3 = 0 . 2
2
2
⎛ ⎜ ⎜ ⎜ ⎜ C = 2M 00 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
cos 2 1 2
φ 2
l sin φ eiθ1
1 2
m sin φ eiθ 2
1 2
nsin φ eiθ 3
1 2
l sin φ e−iθ1 l 2 sin 2
φ 2
φ lm sin 2 eiΔθ 3 2 φ nl sin 2 e−iΔθ 2 2
⎛ φ cos ⎜ 2 ⎜ φ iθ1 ⎜ ⎜ l sin 2 e F = 2M 00 ⎜ ⎜ m sin φ eiθ 2 ⎜ 2 ⎜ φ ⎜ nsin eiθ 3 ⎜⎝ 2
1 2
m sin φ e−iθ 2
φ lm sin 2 e−iΔθ 3 2 φ 2 m sin 2 2 φ mnsin 2 eiΔθ1 2
1 2
nsin φ e−iθ 3
φ nl sin 2 eiΔθ 2 2 2 φ −iΔθ1 mnsin e 2 φ n 2 sin 2 2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟⎠
⎞ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟ ⎟. 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟⎟ ⎠
14
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠
Uniform deterministic medium [8, 9]
L = LB − iLD , L ′ = LB′ − iLD′ and C = CB − iCD , are the generalized retardances of the medium, where the linear diattenuation in the x-y and 45 directions are LD and LD′ , respectively, the circular diattenuation is CD , the corresponding −1 retardances are LB, LB′,CB , all real and [m −1 ] , the amplitude absorption parameter is α / 2[m ]
(
T = (L, L ′,−C)T , T ⋅ T∗ = L + L ′ + C , T = L2 + L ′2 + C 2 2
2
2
)
1/2
2
2
τ = − argTz, tan τ =
= TB − iTD .
tan( 12 Tz) T TD 2 = = p . p is the homogeneity. κ = arg cos ( 12 Tz ) is , β = − arg tan ( 12 Tz ) , 2 ∗ tan (φ / 2) TB T⋅T
(
)
the geometric phase.
⎛ sin ( 1 Tz ) ⎞ . κ − β + τ = arg ⎜ 1 2 ⎝ 2 Tz ⎟⎠ ⎧⎪ 2 2⎫ T ⋅ T∗ ⎪ 1 M 00 = e−α z ⎨ cos ( 12 Tz ) + 2 sin ( 2 Tz ) ⎬ , T ⎩⎪ ⎭⎪ 2 ⎧⎪ ⎫ sin ( 12 Tz ) 2 2 ⎪ M 11 = e−α z ⎨ cos ( 12 Tz ) − T ⋅ T∗ − 2 L ⎬ , 2 T ⎪⎩ ⎪⎭ 2 ⎧⎪ ⎫ sin ( 12 Tz ) 2 2 ⎪ M 22 = e−α z ⎨ cos ( 12 Tz ) − T ⋅ T∗ − 2 L ′ ⎬ , 2 T ⎪⎩ ⎪⎭ 2 ⎧⎪ ⎫ sin ( 12 Tz ) 2 2 ⎪ ∗ M 33 = e−α z ⎨ cos ( 12 Tz ) − T ⋅ T − 2 C ⎬. 2 T ⎩⎪ ⎭⎪
(
⎧ M 01 ⎫⎪ − α z ⎪ sinTz ⎬ = −e ⎨ M 10 ⎪ Tz ⎩⎪ ⎭ ±
±
±
+
(
)
2
z2 ⎫
[ LB′.CD − LD′.CB ] 2 ⎪⎬, ⎭⎪
2
z2 ⎫
[CB.LD − CD.LB ] 2 ⎪⎬, ⎭⎪
[CD cos(β − τ ) + CBsin(β − τ )] z
sin ( 12 Tz ) 1 2 Tz
M 12 ⎫⎪ −α z ⎧⎪ sinTz ⎬ = e ⎨± M 21 ⎪ Tz ⎩⎪ ⎭
)
[ LD′ cos(β − τ ) + LB′ sin(β − τ )] z
sin ( 12 Tz ) 1 2 Tz
M 03 ⎫⎪ −α z ⎧⎪ sinTz ⎬=e ⎨ M 30 ⎪ Tz ⎩⎪ ⎭
(
[ LD cos(β − τ ) + LBsin(β − τ )] z
sin ( 12 Tz ) 1 2 Tz
⎧ M 02 ⎫⎪ − α z ⎪ sinTz ⎬ = −e ⎨ M 20 ⎪ Tz ⎩⎪ ⎭
)
2
z2 ⎫
[ LB.LD′ − LD.LB′ ] 2 ⎪⎬, ⎭⎪
[CB cos(β − τ ) − CD sin(β − τ )] z
sin ( 12 Tz ) 1 2 Tz
2
z2 ⎫
[ LB.LB′ + LD.LD′ ] 2 ⎪⎬, ⎭⎪
15
M 13 ⎫⎪ −α z ⎧⎪ sinTz ⎬ = e ⎨± M 31 ⎪ Tz ⎪⎩ ⎭ −
sin ( 12 Tz ) 1 2 Tz
M 23 ⎫⎪ −α z ⎧⎪ sinTz ⎬ = e ⎨± M 32 ⎪ Tz ⎪⎩ ⎭ −
[ LB′ cos(β − τ ) − LD′ sin(β − τ )] z 2
z2 ⎫
[CB.LB + CD.LD ] 2 ⎪⎬, ⎭⎪
[ LD sin(β − τ ) − LB cos(β − τ )] z
sin ( 12 Tz ) 1 2 Tz
2
z2 ⎫
[ LB′.CB + LD′.CD ] 2 ⎪⎬. ⎭⎪
⎛ 2 iL∗ sin 12 T ∗z ⎜ cos ( 12 Tz ) T∗ ⎜ 2 ⎜ 2 L iL 1 ⎜ − sin ( 12 Tz ) 2 sin ( 2 Tz ) T T ⎜ C = 2e−α z ⎜ 2 L ′L∗ ⎜ iL ′ 1 1 − sin Tz ( ) 2 2 sin ( 2 Tz ) ⎜ T T ⎜ ⎜ iC 2 CL∗ ⎜ sin ( 12 Tz ) − 2 sin ( 12 Tz ) ⎜⎝ T T
(
⎛ φ cos 2 ⎜ 2 ⎜ ⎜ ⎜ φ iL e−i( β −τ ) ⎜ − sin 2 ∗ 1/2 T ⋅ T ⎜ = 2M 00 ⎜ φ iL ′ ⎜ e−i( β −τ ) ⎜ − sin 2 ∗ 1/2 T ⋅ T ⎜ ⎜ iC ⎜ sin φ e−i( β −τ ) ⎜ 2 T ⋅ T∗ 1/2 ⎝
( (
(
sin
iL ′ ∗ sin T∗
)
(
1 2
−
L′
2
T
2
sin ( 12 Tz )
)
L φ 2 ( T ⋅ T∗ )1/2
sin 2
φ LL ′ ∗ 2 ( T ⋅ T∗ )1/2
)
sin 2
φ L ′L∗ 2 ( T ⋅ T∗ )1/2
sin 2
L′ φ 2 ( T ⋅ T∗ )1/2
⎛ cos ( 12 Tz ) ⎜ ⎜ − iL sin 1 Tz (2 ) ⎜ T − α z/2 ⎜ F = 2e ⎜ − iL ′ sin ( 1 Tz ) 2 ⎜ T ⎜ ⎜ iC sin ( 12 Tz ) ⎜⎝ T
0 0 0 0
− sin 2
φ CL∗ 2 ( T ⋅ T∗ )1/2
2
− sin 2
φ CL ′ ∗ 2 ( T ⋅ T∗ )1/2
⎛ φ cos eiκ ⎜ ⎞ 2 0 0 ⎜ ⎟ φ iL ⎜ ei(κ −β +τ ) ⎜ − sin 2 0 0 ⎟⎟ ∗ 1/2 T⋅T ⎜ ⎟ = 2M ⎜ 00 φ iL ′ 0 0 ⎟ ⎜ − sin ei(κ −β +τ ) ⎟ 2 T ⋅ T∗ 1/2 ⎜ ⎟ ⎜ 0 0 ⎟⎟ φ iC ⎜ ei(κ −β +τ ) ⎠ ⎜ sin 2 ∗ 1/2 T ⋅ T ⎜⎝
(
−
2
(
)
(
)
)
iC ∗ sin T∗
(
1 2
T ∗z
)
−
2 LC ∗ 1 2 sin ( 2 Tz ) T
−
2 L ′C ∗ 1 2 sin ( 2 Tz ) T
2 CL ′ ∗ 1 2 sin ( 2 Tz ) T
φ iL∗ φ iL ′ ∗ i( β −τ ) e sin ei( β −τ ) 2 ( T ⋅ T∗ )1/2 2 ( T ⋅ T∗ )1/2 2
)
2 LL ′ ∗ 1 2 sin ( 2 Tz ) T
sin 2
)
T ∗z
C
2
T
2
sin ( 12 Tz )
2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠
⎞ φ iC ∗ i( β −τ ) e ⎟ 2 ( T ⋅ T∗ )1/2 ⎟ ⎟ ∗ ⎟ LC 2φ − sin 1/2 ⎟ ∗ 2 (T ⋅ T ) ⎟ ⎟. ∗ L ′C ⎟ 2φ − sin ⎟ 2 ( T ⋅ T∗ )1/2 ⎟ ⎟ 2 C φ ⎟ sin 2 ⎟ 2 ( T ⋅ T∗ )1/2 ⎠
− sin
⎞ 0 0 0 ⎟ ⎟ ⎟ 0 0 0 ⎟ ⎟ ⎟. 0 0 0 ⎟ ⎟ ⎟ ⎟ 0 0 0 ⎟ ⎟⎠
Appendix We review the basic definitions and terminology [2, 10]. The electric field vector is e (complex). The polarization matrix (sometimes called the coherency or coherence matrix) is
P = e e† ,
(1)
16
where denotes an average (time or space) and the dagger means complex conjugate transpose. The Stokes matrix S is the polarization matrix represented in the Pauli basis. Written as a column vector it becomes the Stokes vector s . The Mueller matrix describes the transformation of the Stokes vector from s to s′ :
s′ = Ms.
(2)
In a similar way, the Mueller matrix in the Cartesian (standard) representation, N , describes the transformation of the polarization vector p (the polarization matrix written in column vector form) to p′ [11-‐13]:
p′ = Np.
(3)
We take the Pauli matrices in their usual order in optics, but also orthonormalize them ( {σ µ , σν } = Tr{σ †µ σν } = δ µν ) [11, 13]:
σµ =
1 ⎛ 1 0 ⎞ 1 ⎛ 1 0 ⎞ , , 2 ⎜⎝ 0 1 ⎟⎠ 2 ⎜⎝ 0 −1 ⎟⎠
(4)
1 ⎛ 0 1 ⎞ 1 ⎛ 0 −i ⎞ , . 2 ⎜⎝ 1 0 ⎟⎠ 2 ⎜⎝ i 0 ⎟⎠
ˆ µ is the unitary matrix Λ (so ΛΛ = I 4 , the 4 × 4 unit matrix) Then the 4 × 4 matrix formed from the four Pauli vectors σ †
given by [11]
Λ=
( σˆ
0
σˆ 1
σˆ 2
σˆ 3
)
⎛ 1 ⎜ ⎜ = 2⎜ ⎜⎝
1 1 0 0 0 0 1 −1
0 0 ⎞ 1 −i ⎟ ⎟. 1 i ⎟ 0 0 ⎟⎠
(5)
The Mueller matrix M represents transformation of the Stokes vector s to s′ : s′ = Ms, where the 16 elements of the Mueller matrix are real. An Hermitian matrix C , called the coherency matrix, can be generated from the Mueller matrix by [14, 15]
C = ∑ M µν Γ µν
(6)
µ ,ν
where the 16 × (4 × 4) (i.e. sixteen 4 × 4 ) unitary coherency basis matrices Γ µν are defined by
Γ µν = Λ† (σ µ ⊗ σν∗ )Λ.
(7)
Here ⊗ means Kronecker (outer) product [14]. We have the relationships Γ †µν = Γ µν ( Γ µν is Hermitian), and
Γ νµ = Γ ∗µν = Γ Tµν , where † signifies conjugate transpose, T signifies transpose and ∗ signifies complex conjugate. Then
p = Λs.
(8)
If the Pauli matrices are trace-‐normalized, {σ µ , σν } = Tr{σ †µ σν } = δ µν , then Λ is unitary. Then
N = ΛMΛ† ,M = Λ†NΛ.
(9)
For a deterministic system, (i.e. one that can be described by a Jones matrix), the electric field vector is transformed from e to e′ by the Jones matrix J , with column vector j . Then
e′ = Je.
(10)
The Jones vector written in the Pauli representation c is related to j by
j = Λc.
(11)
N = J ⊗ J∗ ,
(12)
M = Λ† (J ⊗ J∗ )Λ.
(13)
Then
so that
We define
17
K = [Λ ⊗ (Λ−1 )T ]
(14)
where ⊗ means Kronecker (outer) product. If M and N are written as column vectors m,n , then [16]
n = Km.
(15)
If Λ is unitary, ΛΛ = I 4 , the 4 × 4 identity matrix, and †
K = (Λ ⊗ Λ∗ )
(16)
is also unitary, so m = K n . N can be converted into an Hermitian matrix H by a partial rearrangement of rows operation [12, 13]. H is the covariance matrix, also called the standard Hermitian matrix. In terms of column vectors this can be written †
h = Rn. The matrix R is both orthogonal and unitary, so that R
−1
(17)
= R , and so n = Rh . R can be written
R = I2 ⊗ A 2 ⊗ I2 ,
(18)
−1
where (A 2 ) = I 4 , so that also A 2 = A 2 . It is found that [17] 2
3
(
)
A 2 = ∑ σ µ ⊗ σ †µ . µ =0
(19)
i.e. equal to the trace of the matrix of the basis matrices σ µ ⊗ σν† . From Eqs. 15 and 17, we have
h = RKm = Ψm,
(20)
Ψ = RK,
(21)
H = jj† = j ⊗ j∗ .
(22)
H = Λ(cc† )Λ† = Λ(c ⊗ c∗ )Λ† .
(23)
where
is unitary. For a deterministic system,
This can also be written
The coherency matrix C , also called the Pauli Hermitian matrix, is given by
C = Λ†HΛ, H = ΛCΛ† ,
(24)
C = cc† = c ⊗ c∗ .
(25)
[13-‐15, 18]. For a deterministic system
We see that Eqs. 23 and 25 are consistent with Eq. 24. Also, the similarity of Eqs. 9 and 24, allows us to write in column vector form, using Eq. 15,
h = Kg,
(26)
where H written in column vector form is h , and C written in column vector form is g . From Eqs. 20 and 26, we then have
g = K†h = Γm,
(27)
Γ = Ψ †RΨ = K†RK,
(28)
where
and ΨΓΨ † = ΨΓ † Ψ † , so Γ = Γ −1 is an involutory matrix. Finally, we have the relationship
g = Ψ †n = K†Rn.
(29)
Hence the matrices, written in column vector form, can be calculated from each other using vector multiplications by R and K . The polarization matrix is transformed according to
P′ = JPJ † .
(30)
18
An arbitrary, non-‐deterministic polarization matrix can be decomposed by summing over the four eigenvalues (called the Kraus decomposition [19], like a coherent mode decomposition) 3
P′ = ∑ Jα PJα† ,
(31)
α =0
Jα = λa Jα , where Jα is a normalized Jones matrix. This allows a combination of Jones matrices to be used to represent a non-‐deterministic system. A similar form for the Stokes matrix, i.e. the Stokes vector written in matrix form, rather than the Jones matrix, seems not to exist [20]. For a deterministic system, the coherency matrix factorizes, C = cc † (Eq. 25). The coherency matrix is non-‐negative definite, and any non-‐negative definite Hermitian matrix can be factorized into the product of a matrix and its conjugate transpose . So we can factorize any coherency matrix, whether deterministic or not, into the product of a matrix F and its conjugate transpose [21]. We have 3
3
3
(
C = UDU † = ∑ λα Cα = ∑ UDα U † = ∑ U Dα α =0
α =0
Cα = uα uα† ,Fα = U Dα , F = U diag
(
α =0
)( U
Dα
) = ∑F F †
3
α =0
α
† α
= FF† ,
(32)
)
λ0 , λ1 , λ2 , λ 3 ,
where λα are the eigenvalues of C , D is the diagonal matrix of eigenvalues, Dα is the matrix of a single eigenvalue,
uα are the normalized eigenvectors of C , and U the matrix of eigenvectors. The square root of a diagonal matrix is equal to the matrix of the square roots of the elements. We call F the coherency matrix factor (CMF). It consists of the four component c-‐vectors, weighted by the square roots of the coherency matrix eigenvalues, and arranged in column vector form. Each column of F has an arbitrary phase, which cancels upon multiplication to give C . The first row of F can be taken as real and positive. In the same way that we can factorize the coherency matrix, we can factorize the covariance matrix H , to give the covariance matrix factor (the matrix of weighted Jones-‐vectors), E [21]: 3
3
3
(
H = VDV † = ∑ λα Hα = ∑ VDα V † = ∑ V Dα α =0
α =0
Hα = v α v α† ,Eα = V Dα , E = V diag
(
α =0
)( V
Dα
) = ∑E E †
3
α =0
α
† α
= EE† ,
(33)
)
λ0 , λ1 , λ2 , λ 3 ,
where λα are the eigenvalues of H , equal to the eigenvalues of C , v α are the normalized eigenvectors of H , and
V the matrix of eigenvectors. We call E the covariance matrix factor. It consists of the four component Jones vectors, weighted by the square roots of the covariance matrix eigenvalues, and arranged in column vector form.
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12. R. Simon, "The connection between Mueller and Jones matrices of polarization optics," Optics Comm. 42, 293-‐297 (1982). 13. A. Aiello and J. P. Woerdman, "Linear algebra for Mueller calculus," ArXiv:math-‐ph 0412061 (2006). 14. S. R. Cloude, "Group theory and polarization algebra," Optik 75, 26-‐36 (1986). 15. S. R. Cloude, "A review of target decomposition theorems in radar polarimetry," IEEE Trans. Geoscience and Remote Sensing 34, 498-‐518 (1996). 16. D. G. M. Anderson and R. Barakat, "Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix," J. Opt. Soc. Am. A 11, 2305-‐2319 (1994). 17. C. Rakotonirina, "Expression of a tensor commutation matrix in terms of the generalized Gell-‐Mann matrices," International Journal of Mathematics and Mathematical Sciences 2007, 20672 (2007). 18. S. R. Cloude, "Depolarization by aerosols: Entropy of the Amsterdam light scattering database," J. Quantitative Spectroscopy and Radiative Transfer 110, 1665-‐1676 (2009). 19. K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer, 1983). 20. C. J. R. Sheppard, M. Castello, and A. Diaspro, "Expressions for parallel decomposition of the Mueller matrix: erratum," J. Opt. Soc. Am. A 34, 813 (2017). 21. C. J. R. Sheppard, A. Le Gratiet, and A. Diaspro, "Factorization of the coherency matrix of polarization optics," Optics Letters, submitted.
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