Liquid Crystal Displays. Ernst Lueder

Figure 3.9 The intensity Iy, of the Fréedericksz cell for two values of α in Equation (3.73)

Figure 3.10 The intensity Ix, of the Fréedericksz cell for two values of α in Equation (3.72)

      or

(3.79)

      from which

(3.80)

follows. The function cos²x is lowered around x = π/2 by the multiplication with sin²2α. This is most welcome, as it enhances the black state, which is imperfect by the suppression of only λ₀. This is demonstrated by two values for a in Figure 3.9. The intensity Ix′ in Equation (3.72) exhibits the same dependence on x as Iy′, and is plotted in Figure 3.10, demonstrating that at x = π/2

      (3.81)

      is the maximum intensity independent of α, which passes an analyser placed in the direction x′.

      We are now ready to determine the contrast C(α) for the normally black and normally white cell as a function of α. C is defined as

(3.82)

      where L_max is the maximum luminance assumed to be proportional to the maximum intensity, whereas L_min stands for the minimum luminance assumed to be proportional to the minimum intensity.

We first investigate the normally white mode. In the field-free state, the incoming linearly polarized light with angle α in Figure 3.11(a) again generates linear polarization for a wavelength λ₀ at the angle β = π − α in Equation (3.62) with the intensity Ix′ given in Equation (3.72) representing the white state. If a large enough field is applied, the light reaches the analyser linearly polarized in the direction α independent of λ. Hence, the analyser in Figure 3.11(a) allows the component

(3.83)

Figure 3.11 The angles of the electric field and the polarizers in a normally white Fréedericksz cell with linearly polarized light at the output d = λ/2Δn. (a) Crossed polarizers; (b) parallel polarizers

to pass. This represents the black state. From Equations (3.72) and (3.83), we obtain the contrast in Equation (3.82) as

(3.84)

The optimum contrast is reached for a = π/4 for which C → ∞ because the denominator in Equation (3.84) is zero for all wavelengths. Further, for a = π/4 the numerator is maximum. The case a = π/4 is shown with dotted lines in Figure 3.11(a). The analyser is perpendicular to the linear polarized light at the output if a field is applied, and hence provides blocking of light independent of λ.

The normally black mode is shown in Figure 3.11(b). The analyser is perpendicular to the angle β = π − α, and allows the intensity Iy′ in Equation (3.73) to pass. This represents the black state. If a large enough field is applied, the light with the electrical field

(3.85)

independent of wavelength can pass the analyser according to Figure 3.11(b). This is the white state. The contrast is with Equations (3.85) and (3.73)

      (3.86)

For the single wavelength λ₀ in Equation (3.79), C is infinite as λ₀ is blocked; this does not apply for other wavelengths in the light. Therefore, contrast in the normally black state is inferior to the contrast in the normally white state in Equation (3.84). An optimum C dependent on α does not exist. For α = π/4 the normally black cell has two parallel polarizers. This configuration will be used for reflective cells.

       3.2.3 The reflective Fréedericksz cell

A