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
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
With a large enough voltage V, all molecules are perpendicular to the surface or the x-axis, yielding
From Equations (3.92) and (3.93), we detect the voltage-induced change of the refraction index
(3.94)
with
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
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