Liquid Crystal Displays. Ernst Lueder

which as is shown in our discussion of Equation (4.76), yields a normally white state with an excellent black state. For α = π/4, Equations (4.80) and (4.81) yield

      (4.82)

      and

      (4.83)

      The intensity Isx′ of the light passing through the polarizer in the x′ direction in the normally white state with α = −ψ = π/4 is

(4.84)

      with

      Isx_′ is plotted in Figure 4.10 versus a for β = π/2; zeros are at a = 0, whereas maxima occur at The maximum at leads to . The fully on state offers an excellent black independent of λ. The normally black state with parallel polarizers possesses a black state for Osy′ = 0, that is, for γ(λ) = π. More on MTNs is presented in the following two chapters.

4.2.4 Reflective TN cells

      The basic structure of a reflective cell is depicted in Figure 3.12(a). After the light has passed the polarizer placed at an angle α to the x-axis, it is reflected at the rear mirror. Contrary to the untwisted case, we place the mirror at any distance z, and not only at z = d/2. Then the transmission from the input to the mirror is given by the Jones vectors Osσ and Osτ in

      Figure 4.10 The reduced intensity of a mixed mode TN display

      Equation (4.51) as

(4.85)

      with the transition matrix

(4.86)

The σ–τ coordinates are shown in Figure 4.1 with the twist angle β from the x-axis. The components 0_sσ and 0_sτ are reflected at a mirror placed at z = d in Figure 4.1. If the mirror is metallic it cannot sustain an electric field. For this reason, an incident elliptically polarized wave in Figure 4.11 generates the electric field shown with dashed lines in Figure 4.11. This field is reflected and propagates in the z-direction towards the input. The dashed field has a phase shift from the incident field. This phase shift is omitted in the future calculation as it is of no importance for the intensity to be calculated. Therefore, neglecting the phase shift caused by the reflection, we assume the vector with the components Osσ and Osτ at z ≠ 0 as the input for a wave travelling in the opposite direction towards the input. The pertinent transmission matrix T_r is, according to the principle of reciprocity in physics (Yeh and Gu, 1999),

Figure 4.11 Incident (—) and reflected (- - -) elliptically polarized light at the metallic mirror of a reflective cell

      (4.87)

      where T′ stands for the transpose of T. With T′ we obtain

(4.88)

      where Osx and Osy are the reflected Jones vectors at the input in the x–y coordinates. To obtain the component passing through the polarizer, we have to rotate the coordinates by α, resulting in

(4.89)

      Osξ and Osη are the components in the ξ−η coordinates in Figure 4.1.

Insertion of Equations (4.85) and (4.88) into Equation (4.89) yields

(4.90)

      With T from Equation (4.86), we obtain

(4.91)

and with Equation (4.90)

where the abbreviations in Equation (4.91) have been used. The intensity Iξ passing through the polarizer is