Liquid Crystal Displays. Ernst Lueder

      or

(4.92)

The intensity perpendicular to the polarizer is

      or

      (4.93)

      An maximum for Iξ dependent on α is reached by d Iξ/d α = 0. This leads to

(4.94)

      and for

(4.95)

      with

(4.96)

      with γ from Equations (4.47) and (4.49) and a from Equations (4.48) and (4.49).

For 2α in Equation (4.95), the term in parentheses in Equation (4.92) vanishes, hence this provides the minimum intensity

(4.97)

whereas 2α in Equation (4.94) yields the maximum intensity

      (4.98)

      the largest intensity possible.

The minimum intensity in Equation (4.97) becomes zero for

(4.99)

For this value Equation (4.95) provides the pertinent angle α of the polarizer as

(4.100)

      which has a solution for a ≥ 1.

      From the results for the field-free reflective cell reached so far, we derive the conditions for a normally black cell. We recall that in a reflective cell in Figure 3.12(c), the polarizer and the analyser are identically oriented at an angle α to the x-axis.

The minimum intensity in Equation (4.97) with the angle of the polarizer in Equation (4.100) provides the black state. It can be zero for tan γ in Equation (4.99), that is for

(4.101)

      providing

(4.102)

Equation (4.101) must be solved numerically. A table of solutions can be found in Yeh and Gu (1999).

      Two examples are:

Once a in Equation (4.101) has been found, Δnd/λ and α can be calculated from Equations (4.96) and (4.102). The black state is achieved only for a given λ, which determines d in Equation (4.96); λ = 550 nm and Δn = 0.05 yield d = 3.19 μm in the first case and d = 18.26 μm in the second case in the table above.

      The solutions for normally black reflective cells with α ≠ 0 discussed so far belong to the mixed mode TN cells introduced in Section 4.2.3. For α = 0 Equation (4.81) requires a = l, and from Equation (4.99) tan γ = ∞ or leading to a twist angle . The regular TN cell with β = π/2 does not permit a solution for a reflective cell with Iξ = 0 at a given wavelength λ. Therefore, mixed mode TN cells are preferred for reflective TN displays.

      As normally white reflective TN cells are seldom used due to their poorer performance, they are not dealt with here. A description can be found in Yeh and Gu (1999).

4.3 Electronically Controlled Birefringence for the Generation of Colour

      We consider the mixed mode TN cell for the angle of the polarizer α = π/4 with the intensity

Figure 4.12 The reduced intensity of a mixed mode TN display versus λ = πdΔn/a