Liquid Crystal Displays. Ernst Lueder

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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

(3.58)

(3.59)

Equations (3.58) and (3.59) reveal the angle β of the linearly polarized light at z = d as

(3.60)

Due to Equations (3.46) and (3.47), we obtain on the other hand for the light at z = 0

(3.61)

Equations (3.60) and (3.61) indicate

(3.62)

      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),

(3.63)

(3.64)

      or for the electrical field

      (3.65)

      and

(3.66)

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

(3.72)

      For , Equation (3.64) yields

(3.73)

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