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3.2.2 The transmissive Fréedericksz cell
ОглавлениеIn Figure 3.8 we continue the discussion of the transmissive cell begun with Figures 3.4(a) and 3.4(b). The incoming linearly polarized light enters at an angle α in Figure 3.8. The linear polarization occurs again according to Equation (3.52) at δ = 2π(Δn/λ)z = vπ, v = ±1, ±2, ..or for v = 1 at the smallest z-value
(3.56)
where λ0 is the pertinent wavelength.
Figure 3.8 Angles of polarizer and analyser for the Fréedericksz cell
At the output of the cell for z = d, where d is the thickness of the cell, the retardation is
This retardation is associated with a change of phase by δ = π after the wave has propagated the distance d through the cell. The cell operates as a λ/2-plate. Obviously, the retardation is the phase shift measured in parts of λ0.
For linear polarization with wavelength λ0 corresponding to the angular frequency ω0 of the electric field, the components in Equations (3.42) and (3.43) at z = d are
Equations (3.58) and (3.59) reveal the angle β of the linearly polarized light at z = d as
Due to Equations (3.46) and (3.47), we obtain on the other hand for the light at z = 0
Equations (3.60) and (3.61) indicate
as shown in Figure 3.8, where all important angles for a Fréedericksz cell are drawn.
The result for a wavelength λ0 at the output z = d of a cell without a voltage applied is linearly polarized light at an angle β = π − а, where α is the angle of the incoming linearly polarized light. If we place the analyser in the direction β = π − α the light can pass representing the normally white mode. The analyser perpendicular to β that is at an angle π/2 − α in Figure 3.8 blocks the light representing the normally black mode. We will investigate these two modes in greater detail.
We choose the angle γ for which the x′−y′ plane in Figure 3.8 is rotated from the x−y plane as γ = β = π − α. This provides, along with Equations (3.40) and (3.41),
or for the electrical field
(3.65)
and
For the wavelength λ = λ0 in Equation (3.57), we obtain at z = d
(3.67)
(3.68)
as expected, since we know already that at z = d light with wavelength λ0 is linearly polarized in the direction β = π − α. For this case, the Jones vectors provide the components of the electrical field as
(3.69)
(3.70)
Now we place the analyser perpendicular to the angle β = π − α that is in the direction with angle γ = π/2 − α in Figure 3.8. For this case Jzx′ is identical to − Jzy′ in Equation (3.64) and (3.66) and Jzy′ is identical with Jzx′ in Equations (3.63) and (3.64). Hence, we investigate Equations (3.63) through (3.66) for both cases. The intensity I′x = |Jdx′|2 for z = d is, with Equation (3.63),
(3.71)
or
For , Equation (3.64) yields
In order to learn how to choose a, we now consider the case of a large enough voltage across the LC cell to fully orient the LC molecules apart from two thin layers on top of the orientation layer, due to Δε > 0 in parallel to the electric field. The linearly polarized light coming in at angle a no longer experiences birefringence, as it is only exposed to the refractive index n┴. It reaches the plane z = d with the phase shift 2π(n┴d/λ). Its component Ep passing an analyser with the angle (π/2) − α in Figure 3.8 is
(3.74)
whereas the component Es passing an analyser with the angle π − α in Figure 3.8 is
(3.75)
The intensities belonging to Ep and Es are
(3.76)
and
(3.77)
The bar over the cos terms means the average over time needed for calculating the intensity. The maximum of Ip occurs for α = π/4, which (according to Figure 3.8) places the polarizer and analyser in parallel. Is assumes a maximum for α = 0, for which again the polarizer and analyser pertaining to Es are in parallel.
In Figures 3.9 and 3.10, the intensities Iy′ and Ix′ in Equations (3.73) and (3.72) are plotted versus x= (πdΔn/λ) and λ = (πdΔn/x). Iy′ in Figure 3.9 becomes zero at
(3.78)
Figure 3.9 The intensity Iy, of the Fréedericksz cell for two values of α in Equation (3.73)
Figure 3.10 The intensity Ix, of the Fréedericksz cell for two values of α in Equation (3.72)
or
from which
follows. The function cos2x is lowered around x = π/2 by the multiplication with sin22α. This is most welcome, as it enhances the black state, which is imperfect by the suppression of only λ0. This is demonstrated by two values for a in Figure 3.9. The intensity Ix′ in Equation (3.72) exhibits the same dependence on x as Iy′, and is plotted in Figure 3.10, demonstrating that at x = π/2
(3.81)
is the maximum intensity independent of α, which passes an analyser placed in the direction x′.
We are now ready to determine the contrast C(α) for the normally black and normally white cell as a function of α. C is defined as
where Lmax is the maximum luminance assumed to be proportional to the maximum intensity, whereas Lmin stands for the minimum luminance assumed to be proportional to the minimum intensity.
We first investigate the normally white mode. In the field-free state, the incoming linearly polarized light with angle α in Figure 3.11(a) again generates linear polarization for a wavelength λ0 at the angle β = π − α in Equation (3.62) with the intensity Ix′ given in Equation (3.72) representing the white state. If a large enough field is applied, the light reaches the analyser linearly polarized in the direction α independent of λ. Hence, the analyser in Figure 3.11(a) allows the component
Figure 3.11 The angles of the electric field and the polarizers in a normally white Fréedericksz cell with linearly polarized light at the output d = λ/2Δn. (a) Crossed polarizers; (b) parallel polarizers
to pass. This represents the black state. From Equations (3.72) and (3.83), we obtain the contrast in Equation (3.82) as
The optimum contrast is reached for a = π/4 for which C → ∞ because the denominator in Equation (3.84) is zero for all wavelengths. Further, for a = π/4 the numerator is maximum. The case a = π/4 is shown with dotted lines in Figure 3.11(a). The analyser is perpendicular to the linear polarized light at the output if a field is applied, and hence provides blocking of light independent of λ.
The normally black mode is shown in Figure 3.11(b). The analyser is perpendicular to the angle β = π − α, and allows the intensity Iy′ in Equation (3.73) to pass. This represents the black state. If a large enough field is applied, the light with the electrical field
independent of wavelength can pass the analyser according to Figure 3.11(b). This is the white state. The contrast is with Equations (3.85) and (3.73)
(3.86)
For the single wavelength λ0 in Equation (3.79), C is infinite as λ0 is blocked; this does not apply for other wavelengths in the light. Therefore, contrast in the normally black state is inferior to the contrast in the normally white state in Equation (3.84). An optimum C dependent on α does not exist. For α = π/4 the normally black cell has two parallel polarizers. This configuration will be used for reflective cells.
