or
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
and for
with
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
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
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
providing
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
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