Liquid Crystal Displays. Ernst Lueder

      and

      (3.91)

      The magnitude is constant, whereas the phase changes with the distance z from the input.

|Jzξ| from Equation (3.88) is plotted in Figure 3.14 versus α and z. The constant magnitude 1 for α = 0, (π/2) and π independent of z is visible as well as the maximum amplitude modulation for α = π/4. However, we want arc Jzξ to change with the voltage V across the cell. To this aim, we consider the Fréedericksz cell in Figure 3.15, where a voltage has been applied to tilt the molecules by an angle φ. The linearly polarized light E₀ stemming from the polarizer with angle α = 0 to the x-axis has to meet the boundary condition at the transition from the polarizer into the cell. The tangential components have to be equal on both sides, which means they are E₀ in Figure 3.15. This indicates that the light wave has the wave vector parallel to the z-axis. If no voltage V is applied, the LC molecules are parallel to the surface and the x-axis, yielding

Figure 3.14 |Jzξ| in Equation (3.88) plotted versus α and z

Figure 3.15 The linearly polarized light in parallel (α = 0) to the projection of the long axis of the LC molecules into the x-y plane

(3.92)

      With a large enough voltage V, all molecules are perpendicular to the surface or the x-axis, yielding

(3.93)

From Equations (3.92) and (3.93), we detect the voltage-induced change of the refraction index with n_|| for the lower voltage and n┴ for the higher voltage. Along with Equation (3.90), for a given wavelength λ₀ and for z = d, this provides

      (3.94)

      with

(3.95)

      How n changes with V cannot yet be determined. For that we need the propagation of light obliquely to the LC molecules, which will be discussed in Chapter 6. So far, arc Jzξ(V) is determined by measurements.

Figure 3.16 Measured phase-shift curves of a Fréedericksz cell

For λ₀ = 0.5 μ, n┴ = 14 and n_|| = 1.5 we notice that arc Jzξ may change from 2.8 × 2π × d to 3.2π × d. Obviously, the larger d the larger is the change of arc Jzξ, however, the necessary addressing voltage increases. For d = 3μ we obtain arc Jzξ= [8.4 × 2π, 9 × 2π], a change of 0.6 × 2π. Figure 3.16 shows in curve 1 with a = 0 a phase-only SLM with a maximum phase shift of 0.75π. Due to Equation (3.88), the other curves with a ≠ 0 cannot possess a constant magnitude, as is also visible in Figure 3.14.

      For a = π/2 the linearly polarized light in Figure 3.15 always encounters the refractive index n_┴ independent of the tilt, and hence is independent of V (curve 7 in Figure 3.16), which is of no use for a phase shifter.

       3.2.5 The DAP cell or the vertically aligned cell

The cell operating with the deformation of aligned phases, called the DAP cell (Glueck, 1995), is the inverse of the Fréedericksz cell. In the field-free state the LC molecules are perpendicularly (or in other words, homeotropically) aligned to the surface of both substrates, as depicted in Figure 3.17. This cell is also called a Vertically Aligned (VA) LCD. In this situation, incoming linearly polarized light with a wave vector in parallel to the z-axis in Figure 3.17 does not encounter birefringence, and arrives at the second substrate with an unchanged state of its polarization. If the analyser is parallel to the polarizer, the full light can pass representing the normally white state. If the analyser is crossed with the polarizer, the light is blocked at the output for all wavelengths and independent of d. This is the normally black state. The cell exhibits an extremely good black state, since the blocking is again independent of λ. Further, the molecules on the orientation layer are also, contrary to the Fréedericksz cell, vertically aligned. A low black value in the denominator of the contrast in Equation (3.82) is most beneficial for a high contrast. The main attraction of the DAP cell is this extremely high contrast, reaching values of more than 500: 1.

Figure 3.17 The DAP cell or Vertically Aligned (VA) cell in the field-free state

If an electrical field is applied, the LC molecules orient themselves perpendicularly to the field as Δε