Liquid Crystal Displays. Ernst Lueder

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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

or with ξ₁ and ξ_s in Equation (4.18),

(4.29)

      and

(4.30)

with φ from Equation (4.22).

We continue with the normalized matrices M:

(4.31)

Based on Equation (4.17), we can represent T(ε)R(ε) with the known matrices D in Equation (4.26) and M in Equation (4.31) as

      (4.32)

      [T (ε) R (ε)]^2πd/pε in Equation (4.14) assumes the form

      (4.33)

      The evaluation of D^2πd/pε, with dε in Equation (4.5) and D in Equation (4.26), provides

      (4.34)

      K′ in Equation (4.25) leads to

As both the numerator and denominator tend to zero for ε → 0, the application

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