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Figure 3.22 The electro-optical response to a square voltage pulse. (a) Of a TN cell with a prolonged relaxation; and (b) of a π cell with a fast relaxation

      (3.98)

where Δn(z) depends upon the angle Θ_e in Figure 3.21(b). We shall derive Δn(Θ_e) in Chapter 6, leading to Equation (6.29). For a π cell with crossed polarizers, the transmission T is obtained by a straightforward calculation with Jones vectors, yielding

      (3.99)

      A normally white cell has the first transmission maximum at (π/λ)R = (π/2), or

(3.100)

      corresponding to a λ/2-plate. We introduce an effective anisotropy Δn_eff for the entire optical path in the cell, which is given by

      (3.101)

providing, with Equation (3.100),

      (3.102)

The term Δn_eff renders this result similar to Equation (3.97).

      The wide viewing angle inherent to the π cell is discussed in Section 6.3.2.

       3.2.8 Switching dynamics of untwisted nematic LCDs

We assume that the LC molecules are anchored on the surface at z = 0 at an angle Θ₀ to the normal and at z = d under the angle Θ_d, as shown in Figure 3.23. In the field-free state the field of directors is defined by the equilibrium state with minimum free energy. After applying an electric field in the form of a step function the voltage V(t) across the cell has to exceed a threshold V_th before the molecules are able to rotate in order to assume the position imposed by the field. The threshold is caused by the intermolecular forces, which first have to be overcome by the forces of the field. The transition to the new voltage imposed field of directors is called the Fréedericksz transition. The dynamic of this transition (Degen, 1980; Priestley, Wojtowicz and Sheng, 1979) is governed by the interaction between the electric torques forcing the directors into positions parallel for Δε > 0 or perpendicular for Δε < 0 to the electric field and the mechanical torques trying to restore the field-free state. These torques are the only mechanical influences if the molecules do not undergo a translatory movement. A magnetic field is as a rule not applied in LC applications. The transient between the states of the director field is calculated by adding all free energies, and by taking the functional first derivative with respect to the angle Θ in Figure 3.23. The torques are related to splay, twist and bend with the elastic constants K₁₁, K₂₂, K₃₃ and the rotational kinematic viscosity η, as well as to the dielectric torque dependent on Δε. For the derivation of the results we refer to special publications (Labrunie and Robert, 1973; Saito and Yamamoto, 1978). In Saito and Yamamoto (1978), expressions for the rise time T_r and the decay time T_d for the reorientation of the LC molecules induced by a voltage step with amplitude V were derived. The results depend upon the tilt angles Θ₀ and Θ_d of the molecules. T_r and T_d translate directly in the rise time and the decay time of the luminance as it changes directly with the director field. The results for general angles Θ_d and Θ₀ are:

Figure 3.23 The anchoring of LC molecules at z = 0 and z = d

(3.103)

      with

      For

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