is depicted in Figure 3.12(a) (Uchida, 1999; Lueder et al., 1998). The polarizer transmits linearly polarized light again at an angle of π/4 to the x-axis in Figure 3.12(b). The illumination is provided either by ambient light or by an external light source above the polarizer. After having travelled through the cell in Figure 3.12(a) with half the thickness d/2 of a transmissive cell, the light is reflected at the mirror usually made of Al and finally exits through the same polarizer. Thus, the reflective cell saves one polarizer. The reflective cell can be designed according to the general principles, which will be outlined now (Lueder et al., 1998). We first recall the operation of a Freedericksz cell with parallel polarizers as depicted in the left column of Figure 3.12(c). We know that linearly polarized incoming light in the direction α = π/4 to the x-axis is blocked at z = d for wavelength λ0. Due to Equation (3.80), the thickness is d = λ0/2Δn. We determine at which value z the light is circularly polarized.
With Equation (3.45) this happens for the first time for δ = π/2, reflecting in
As sin δ = sin(π/2) > 0 the light is right-handed circularly polarized seen against
Figure 3.12 The reflective Fréedericksz cell. (a) Cross-section; (b) top view; (c) explanation of the operation of a reflective cell in the field-free state
The normally white cell with crossed polarizers cannot be transformed into a reflective version as this version has only one polarizer able to realize only parallel polarizers. This fact, however, renders the reflective cell somewhat more economic as the added mirror is cheaper than the saved polarizer.
The surface of the mirror and the lower edge of the LC material in Figure 3.12(a) are supposed to be located at z = d/2, which is not exactly feasible because of the presence of the ITO and the orientation layers. As both layers are very thin, around 100 nm each, this does not show up in the performance of the cell.
3.2.4 The Fréedericksz cell as a phase-only modulator
So far we have treated the Fréedericksz cell as an amplitude modulator, as it is required for realizing grey shades in displays. The phase was of no importance. In coherent optical signal processing, the phase of the light wave is crucial; a voltage controlled phase shift without altering the amplitude is often required. A component with that performance belongs to the group of Spatial Light Modulators (SLMs). Another SLM is a pixellated optical multiplier in the form of an LCD placed behind a picture in Figure 3.13. Each area in front of the pixels of the multiplier is multiplied by the grey shade in those pixels. In other words, it is multiplied by a value 1 corresponding to a fully transparent pixel, a value 0 corresponding to a black pixel, and all values ε(0, 1) corresponding to the grey shades. As this multiplication occurs with the speed of light and with all pixels operating in parallel, extremely high processing speeds are feasible with the SLMs of an electro-optical processor (Lu and Saleh, 1990).
Figure 3.13 An LCD used as an SLM operating as a multiplier
The explanation of the SLM for phase-shifts starts with the most general Equations (3.40) and (3.41) of the Freedericksz cell containing the arbitrary angle a of the polarizer at the input and the pertinent angle γ of the analyser (Figures 3.4(a) and 3.8).
Equations (3.40) and (3.41) yield, for γ = α (that is, for parallel polarizers), the Jones vectors Jzξ and Jzη measured in the ξ−η coordinates in Figure 3.4(a)
(3.89)
The Jones vector Jz% of the light passing through the analyser parallel to the polarizer is, for a = 0, n and a = n/2 and for no voltage applied,