Liquid Crystal Displays. Ernst Lueder

were used.

      The further propagation of the light through the s − 1 remaining rotated slices is depicted in Figure 4.2 with the rotation matrices

(4.10)

      and the transmission matrices

(4.11)

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)

(4.14)

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

(4.15)

      For an easier calculation of the lim for ε → 0, we transform

(4.16)

      into a diagonal matrix

(4.17)

      where D contains the eigenvalues, and the columns of M are the eigenvectors of T(ε)R(ε) (Specht, 2000). The eigenvalues ξ_1,2 are obtained by |T (ε) R (ε) – ξI| = 0, where I the unity matrix, as

(4.18)

      providing

(4.19)

      Since |ξ₁| = |ξ₂| 1, the eigenvalues can be rewritten as

      (4.20)

      and

      (4.21)

      with

(4.22)

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 of this slice, we obtain

      (4.23)

      and

      (4.24)

      with

(4.25)

      This finally leads to

(4.26)

with k′ in Equation (4.25) and dε in Equation (4.5).

The eigenvectors V (V₁, V₂) with the components Vx_1,2 and Vy_1,2 are calculated from Equation (4.16) by solving

      as

(4.27)

with the two arbitrary constants r₁ and r₂. The transformation matrix M is, from Equation (4.27)

(4.28)

      with the magnitudes of the eigenvectors ||V₁|| and ||V₂|| given by

      and