The further propagation of the light through the s − 1 remaining rotated slices is depicted in Figure 4.2 with the rotation matrices
and the transmission matrices
The Jones vector Os at the output at z = d measured in the coordinates σ and τ with the angle β to the x-axis in Figure 4.1 is, with Equations (4.1) and (4.9)
(4.12)
or with Equations (4.10), (4.11), dε in Equation (4.5), s in Equation (4.7) and
(4.13)
We want to determine the lim for ε → 0 of this expression, providing an infinitesimally small thickness of the slices, and hence the exact solution. From Equation (4.14), we obtain
For an easier calculation of the lim for ε → 0, we transform
into a diagonal matrix
where D contains the eigenvalues, and the columns of M are the eigenvectors of T(ε)R(ε) (Specht, 2000). The eigenvalues ξ1,2 are obtained by |T (ε) R (ε) – ξI| = 0, where I the unity matrix, as
providing
Since |ξ1| = |ξ2| 1, the eigenvalues can be rewritten as
(4.20)
and
(4.21)
with
On the other hand, ξ1 and ξ2 in Equation (4.19) describe the transmission of a wave through a slice with the thickness dε in Equation (4.5). Therefore, with the wave vector
(4.23)
and
(4.24)
with
This finally leads to
with k′ in Equation (4.25) and dε in Equation (4.5).
The eigenvectors V (V1, V2) with the components Vx1,2 and Vy1,2 are calculated from Equation (4.16) by solving
as
with the two arbitrary constants r1 and r2. The transformation matrix M is, from Equation (4.27)
with the magnitudes of the eigenvectors ||V1|| and ||V2|| given by
and