Page 594 - Introduction to Information Optics

10.2. A Brief Review of Types of Fiber-Optic Sensors 579
where A x and A y represent the amplitude for the x and y components,
respectively, while d x and 6 y represent the phase for the x and y components,
respectively. The transformation from one polarization state to another polar-
ization state is represented by a 2 x 2 transformation matrix with complex
components called the Jones matrix.
On the other hand, in the Mueller calculus, the polarization state is
represented by the Stokes vector, s = (s t,s 2, s 3, s 4) and transformation is
represented by a 4 x 4 transformation matrix with real components called the
Mueller matrix. The basic difference between the Jones calculus and the
Mueller calculus is that in the Jones approach, all the elements can be complex
numbers, while in the Mueller approach, all the elements are real numbers.
Table 10.1 illustrates the polarization state expression in terms of Jones
vectors and Mueller vectors. Table 10.2 illustrates the most commonly used
polarization state transformation matrices in terms of Jones matrices and
Mueller matrices.
With the Jones and Mueller calculi, the output polarization state can be
calculated from the input polarization state as well as the transformation
matrix. Assume that the light passes through a series of optical elements
consisting of polarizers and retarders. The output state can be obtained by
sequentially multiplying the vector for the input state by the matrix represen-
ting each optical element in the order that it encounters them. Mathematically,
this can be written as
7 0 - M / M / _ (10.6)

Table 10.1
2
is
Polarization for Light Vector E with E x = A xe'5\ £., = A ve >\ S = <5, — 6.., and A —
a x x y y ? x y ?
Status of polarization Jones vector Stokes (or Mueller) vector
li
[A xe *~\ f A 2
Linear polarization: Jones: a = 2
e
transmission axis L^y \J Mueller s- = A cos 2ff
2
at angle 0 to x axis /I sin 20
1 o
A 1 _ 2
A
Right { + ) and left (-) V ~
- / ^'^ ^
circular polarization: Jones, a = ,4 Mueller: S = 0
jl_ (? >(<5 JV ±rt/2) 0
Ax = Ay = /l..\/2, <5 = ±7t/2
2
_v''2 _+/! _
Mv
fv4 e l " ^ + X? "
Elliptical polarization Jones: a = x y
2
A,,e A - A 2
L J
* Mueller: s =
2A^A V cos d
_2A xA y sin<5