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
from which
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
where Lmax is the maximum luminance assumed to be proportional to the maximum intensity, whereas Lmin 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
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
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
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