Liquid Crystal Displays. Ernst Lueder

As the denominator of the last equation is identical with lim_{ε → 0}k′/(2π/p) in Equation (4.35), we obtain

(4.36)

      ensuring

      (4.37)

The limit of M in Equation (4.31) is calculated in the following steps with φ = arctan as the elements mik in the unnormalized matrix M in Equation (4.28) and the magnitudes in Equations (4.29) and (4.30) tend to zero for ε → 0, we choose r₁ = r₂ 1/ε and evaluate the limits of the numerator and denominator in Equation (4.31) separately according to Hopital’s rule. By doing so, we obtain for the numerators

      (4.38)

      and

      With similar calculations as performed for D, we obtain the limit as

      (4.39)

      and in the same way, also

      (4.40)

For the magnitudes the evaluation of the limit value is performed at the square of the magnitudes in Equations (4.29) and (4.30), again leading with similar calculations as for D to

      (4.41)

      and

      (4.42)

      Hence, the normalized M is, for ε → 0,

(4.43)

      from which, as M is a unitary matrix, we obtain

(4.44)

Inserting Equations (4.29), (4.30), (4.43) and (4.44) into Equation (4.28) provides, for ε → 0,

(4.45)

with k′ in Equation (4.36). Performing the multiplications in Equation (4.45) results in

      (4.46)

      with

(4.47)

(4.48)

(4.49)

      and

(4.50)

The final result is, with Equation (4.15),

(4.51)

      given in the σ–τ coordinates in Figure 4.1, which are rotated by the twist angle −β from the x–y coordinates. An evaluation of the results in the x′ –y′ coordinates in Figure 4.1 requires a rotation by the angle ψ, resulting in

(4.52)

      This result will be discussed for three special cases, namely the Twisted Nematic cell (TN cell) with

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