26
SCATTERING BY FINITE TARGETS: THE MUELLER MATRIX and eq. (71) can be written ˆik S2 S3 Ei (0) · e −f11 =i ˆi⊥ S4 S1 Ei (0) · e f21
−f12 f22
a c
b d
75
ˆ LF Ei (0) · y ˆLF Ei (0) · z
ˆik and e ˆi⊥ are related to y ˆ LF , z ˆLF by The incident polarization states e ˆik ˆ LF e cos φs sin φs y = ˆi⊥ ˆLF e sin φs − cos φs z
ˆ LF y ˆLF z
=
cos φs sin φs
sin φs − cos φs
ˆik e ˆi⊥ e
.
(77)
(78)
.
(79)
The angle φs specifies the scattering plane, with ˆLF = e ˆ2 · z ˆLF , cos φs = φˆs · z ˆ LF = −ˆ ˆ LF . sin φs = −φˆs · y e2 · y Substituting (79) into (77) we obtain ˆik S2 S3 Ei (0) · e −f11 =i ˆi⊥ S4 S1 Ei (0) · e f21
(80) (81)
ˆik Ei (0) · e ˆi⊥ Ei (0) · e (82) Eq. (82) must be true for all Ei (0); hence we obtain an expression for the complex scattering amplitude matrix in terms of the fml : S2 S3 −f11 −f12 a b cos φs sin φs =i . (83) S4 S1 f21 f22 c d sin φs − cos φs −f12 f22
a c
b d
cos φs sin φs
sin φs − cos φs
This provides the 4 equations used in subroutine GETMUELLER to compute the scattering amplitude matrix elements:
26.2
S1
= −i [f21 (b cos φs − a sin φs ) + f22 (d cos φs − c sin φs )] ,
(84)
S2
= −i [f11 (a cos φs + b sin φs ) + f12 (c cos φs + d sin φs )] ,
(85)
S3
= i [f11 (b cos φs − a sin φs ) + f12 (d cos φs − c sin φs )] ,
(86)
S4
= i [f21 (a cos φs + b sin φs ) + f22 (c cos φs + d sin φs )] .
(87)
Stokes Parameters
It is both convenient and customary to characterize both incident and scattered radiation by 4 “Stokes parameters” – the elements of the “Stokes vector”. There are different conventions in the literature; we adhere to the definitions of the Stokes vector (I,Q,U ,V ) adopted in the excellent treatise by Bohren & Huffman (1983), to which the reader is referred for further detail. Here are some examples of Stokes vectors (I, Q, U, V ) = (1, Q/I, U/I, V /I)I: • (1, 0, 0, 0)I : unpolarized light (with intensity I); • (1, 1, 0, 0)I : 100% linearly polarized with E parallel to the scattering plane; • (1, −1, 0, 0)I : 100% linearly polarized with E perpendicular to the scattering plane; • (1, 0, 1, 0)I : 100% linearly polarized with E at +45◦ relative to the scattering plane; • (1, 0, −1, 0)I : 100% linearly polarized with E at -45◦ relative to the scattering plane; • (1, 0, 0, 1)I : 100% right circular polarization (i.e., negative helicity); • (1, 0, 0, −1)I : 100% left circular polarization (i.e., positive helicity).