Liquid Crystal Displays. Ernst Lueder

rel="nofollow" href="#ulink_2188d751-27e0-538b-83da-9b1f713270b7">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 α = 0 and β = π/2, the Supertwist TN cell (STN-cell) with α = 0 and β > π/2 and the Mixed mode TN cell (MTN-cell) with α ≠ 0 and β ≠ π/2.

The derivation of Equation (4.52) also contains a proof of the Chebychev identity. The full length of the derivation was presented in a detailed manner, as it leads to a core result for LCDs. Further, the considerations involved are useful for solving a variety of special display problems.

4.2 The Various Types of TN Cells

       4.2.1 The regular TN cell

      This cell is the most widely used active matrix LCD(Schadt and Helfrich, 1971; Yeh, 1988), and represents a special case of the general result in Equation (4.51). The linearly polarized light is incident at an angle

      (4.53)

      in Figure 4.1 that is parallel to the director of the LC molecules on top of the orientation layer. The twist angle is

(4.54)

      leading to a pitch in Equation (4.49) of p = 4d. Hence, at z = d the helix has reached a quarter of a turn. From these values follow a in Equation (4.48) as

(4.55)

and γ in Equation (4.47) as

(4.56)

In Equation (4.51), the σ-axis is rotated by β = π/2 from the x-axis, whereas the τ-axis is parallel to the x-axis. The Jones vectors in the σ-τ axes assume, after insertion of Equations (4.54), (4.55) and (4.56) into Equation (4.51), the form

(4.57)

      and

      (4.58)

For parallel polarizers the analyser lies in the τ-axis. The intensity Iτ passing this analyser is (Gooch and Tarry, 1974)

(4.59)

      The reduced transmission is plotted in Figure 4.3 versus a = 2dΔn/λ. Zeros occur for and hence for . In the first minimum we obtain, with from Equation (4.55), the optical retardation

      (4.60)

      and the thickness d for a given λ = λ₀ as

(4.61)

      The d value is larger by a factor