Liquid Crystal Displays. Ernst Lueder

and , respectively.

      The two components can be represented by a column vector, which is called the Jones vector (Jones, 1941):

(3.23)

In a physical sense, J is not a vector as so far it consists only of two components and the vector product does not apply. It can also be termed a Jones matrix. According to Equation (3.10), the scalars associated with the Jones vector are

      (3.24)

      and

      (3.25)

In later investigations, the rectangular coordinates ξ and η will have to be rotated by an angle a into the new rectangular coordinates x and y, as shown in Figure 3.3. The vector E with the components E_ξ and Eη is transformed into the components Ex and Ey in the x-y-plane according to

(3.26)

(3.27)

if rotation by a is positive in the counter-clockwise direction. The roman numerals in Equations (3.26) and (3.27) indicate the sections marked in Figure 3.3, thus providing Equations (3.26) and (3.27). The matrix equation for (3.26) and (3.27) is

(3.28)

      R(α) is the rotation matrix, and R(−α) stands for a rotation in clockwise direction.

Figure 3.3 Rotation of the ξ−η coordinates by a into the x-y coordinates

3.2 Propagation of Polarized Light in Birefringent Untwisted Nematic Liquid Crystal Cells

       3.2.1 The propagation of light in a Fréedericksz cell

We investigate the liquid crystal cell according to Fréedericksz (Fréedericksz and Zolina, 1933; Yeh and Gu, 1999) with the top view in Figure 3.4(a) and the cross-section perpendicular to the top plane A in Figure 3.4(b). The vector E of the electric field in the plane A defined in the cartesian ξ−η- coordinates has the components using the notations in Equation (3.12)

Figure 3.4 (a) Top view of Fréedericksz cell with direction of LC molecules and vector E of electric field; (b) cross section of LC cell with parallel layers of molecules and wave vectors k, kx and ky

      (3.29)

      and

      (3.30)

with the Jones vector introduced in Equation (3.23)

      (3.31)

      J represents a plane harmonic wave entering the cell at z = 0 with wave vector k in Figure 3.4(b) perpendicular to A. The electric field E oscillates for all times t in a straight line; this is called a linearly polarized wave.

To obtain the components Jx and Jy of the Jones vector parallel and perpendicular to the director n of the parallel aligned LC molecules in the cell with thickness d in Figure 3.4(b), we rotate the ξ−η coordinates clockwise into the x−y plane according to Equation (3.28), which provides

(3.32)

      While propagating through the cell, the component Jx parallel to experiences the refractive index n_|| and the wave vector kx perpendicular to A or parallel to the unit vector z₀ in Figure 3.4(b), whereas Jy sees the refractive index n┴ and the wave vector ky parallel to z₀. The wave vectors are

(3.33)

      and

(3.34)

      where Equation (2.5) has been used.

      The Jones vector in the plane in the distance r = z × z₀ from the top plane in Figure