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The Twisted Nematic cell (TN cell) is the most widely commercially used LC cell. It was proposed by Schadt and Helfrich (1971), and is therefore also termed the Schadt–Helfrich cell. The theoretical investigation is based on Jones vectors. Solutions for the light exiting a TN cell were given by Yeh (1998), Yeh and Gu (1999), Grinberg and Jacobson (1976) and Rosenbluth et al. (1998). The derivation relies on rotating back the coordinate system to the original coordinates in twisted media and on the Chebychev identity of matrices (Bodewig, 1959). On the other hand, the derivation of the results presented here rotates the coordinates with the twist of the layers. The further calculation is based on Specht (2000).

The planar wave with wave vector at the input of the cell propagates along the z-axis in Figure 4.1. We investigate the propagation through the cell without considering the light reflected at the LC molecules or absorbed in the cell. The incoming light is linearly polarized in the ξ-direction at an angle α to the x-axis with the electrical field Eξ₀ giving the Jones vector

(4.1)

Figure 4.1 The general twisted nematic LCD with twist angle β

in the x–y coordinates. The LC molecules in the x–y plane are all anchored in the rubbing grooves parallel to the x-direction. The directors of all molecules in the cell are parallel to the x–y plane and form a helix with the z-direction as the axis, and with the linear twist angle

(4.2)

a pitch p given by

(4.3)

and a twist angle

(4.4)

at z = d. For calculations, the helix is cut into slices parallel to the x–y plane. In each slice, all molecules are assumed to be parallel. The slices are rotated from the previous slice by the angle ε. The angle ε corresponds to a thickness dε with

or

(4.5)

for each slice. The twist angle at z = d is from Equation (4.4) with α₀ in Equation (4.3)

(4.6)

whereas the number of slices in the cell is with Equation (4.5)

(4.7)

The Jones vector J₁ at the input is translated into the vector O₁ at the output of the first slice with thickness dε. Its component J_1x is parallel to the x-axis and the component J_1y is parallel to the y-axis, and hence (as already known from the Fréedericksz cell),without the need to rotate the input vector, we obtain

(4.8)

or

(4.9)

Figure 4.2 The propagation of light from the Jones vector J₁ at the input to the Jones vector O_s at the output through the transmission matrices Tv and the rotation matrices Rv

with

where and in Equations (3.33) and (3.34) were used.

The further propagation of the light through the s − 1 remaining rotated slices is depicted in Figure 4.2 with the rotation matrices

(4.10)

and the transmission matrices

(4.11)

The Jones vector Os at the output at z = d measured in the coordinates σ and τ with the angle β to the x-axis in Figure 4.1 is, with Equations (4.1) and (4.9)

(4.12)

or with Equations (4.10), (4.11), dε in Equation (4.5), s in Equation (4.7) and

(4.13)

(4.14)

We want to determine the lim for ε → 0 of this expression, providing an infinitesimally small thickness of the slices, and hence the exact solution. From Equation (4.14), we obtain

(4.15)

For an easier calculation of the lim for ε → 0, we transform

(4.16)

into a diagonal matrix

(4.17)

where D contains the eigenvalues, and the columns of M are the eigenvectors of T(ε)R(ε) (Specht, 2000). The eigenvalues ξ_1,2 are obtained by |T (ε) R (ε) – ξI| = 0, where I the unity matrix, as

(4.18)

providing

(4.19)

Since |ξ₁| = |ξ₂| 1, the eigenvalues can be rewritten as

(4.20)

and

(4.21)

with

(4.22)

On the other hand, ξ₁ and ξ₂ in Equation (4.19) describe the transmission of a wave through a slice with the thickness dε in Equation (4.5). Therefore, with the wave vector of this slice, we obtain

(4.23)

and

(4.24)

with

(4.25)

This finally leads to

(4.26)

with k′ in Equation (4.25) and dε in Equation (4.5).

The eigenvectors V (V₁, V₂) with the components Vx_1,2 and Vy_1,2 are calculated from Equation (4.16) by solving

as

(4.27)

with the two arbitrary constants r₁ and r₂. The transformation matrix M is, from Equation (4.27)

(4.28)

with the magnitudes of the eigenvectors ||V₁|| and ||V₂|| given by

and

or with ξ₁ and ξ_s in Equation (4.18),

(4.29)

and

(4.30)

with φ from Equation (4.22).

We continue with the normalized matrices M:

(4.31)

Based on Equation (4.17), we can represent T(ε)R(ε) with the known matrices D in Equation (4.26) and M in Equation (4.31) as

(4.32)

[T (ε) R (ε)]^2πd/pε in Equation (4.14) assumes the form

(4.33)

The evaluation of D^2πd/pε, with dε in Equation (4.5) and D in Equation (4.26), provides

(4.34)

K′ in Equation (4.25) leads to

As both the numerator and denominator tend to zero for ε → 0, the application of Hopital’s rule is needed, yielding

(4.35)

The repetition of Hopital’s rule provides

As the denominator of the last equation is identical with lim_{ε → 0}k′/(2π/p) in Equation (4.35), we obtain

(4.36)

ensuring

(4.37)

The limit of M in Equation (4.31) is calculated in the following steps with φ = arctan as the elements mik in the unnormalized matrix M in Equation (4.28) and the magnitudes in Equations (4.29) and (4.30) tend to zero for ε → 0, we choose r₁ = r₂ 1/ε and evaluate the limits of the numerator and denominator in Equation (4.31) separately according to Hopital’s rule. By doing so, we obtain for the numerators

(4.38)

and

With similar calculations as performed for D, we obtain the limit as

(4.39)

and in the same way, also

(4.40)

For the magnitudes the evaluation of the limit value is performed at the square of the magnitudes in Equations (4.29) and (4.30), again leading with similar calculations as for D to

(4.41)

and

(4.42)

Hence, the normalized M is, for ε → 0,

(4.43)

from which, as M is a unitary matrix, we obtain

(4.44)

Inserting Equations (4.29), (4.30), (4.43) and (4.44) into Equation (4.28) provides, for ε → 0,

(4.45)

with k′ in Equation (4.36). Performing the multiplications in Equation (4.45) results in

(4.46)

with

(4.47)

(4.48)

(4.49)

and

(4.50)

The final result is, with Equation (4.15),

(4.51)

given in the σ–τ coordinates in Figure 4.1, which are rotated by the twist angle −β from the x–y coordinates. An evaluation of the results in the x′ –y′ coordinates in Figure 4.1 requires a rotation by the angle ψ, resulting in

(4.52)

This result will be discussed for three special cases, namely the Twisted Nematic cell (TN cell) with α = 0 and β = π/2, the Supertwist TN cell (STN-cell) with α = 0 and β > π/2 and the Mixed mode TN cell (MTN-cell) with α ≠ 0 and β ≠ π/2.

The derivation of Equation (4.52) also contains a proof of the Chebychev identity. The full length of the derivation was presented in a detailed manner, as it leads to a core result for LCDs. Further, the considerations involved are useful for solving a variety of special display problems.