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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 3.4(a) is

(3.35)

With

(3.36)

and

(3.37)

we obtain the matrix equation by also inserting Jx and Jy from Equation (3.32):

(3.38)

The evaluation of the result of Equation (3.38) in the x′ − y′ coordinates rotated from the x−y plane by the angle γ requires the components Jzx′ and Jzy′, given as

(3.39)

leading with Equations (3.36) and (3.37) to

(3.40)

and

(3.41)

From Equations (3.35) and (3.32), we derive the scalars according to Equation (3.10) for the components Ex and Ey in the distance z

(3.42)

and

(3.43)

Equations (3.42) and (3.43) represent the electric field in the x−y coordinates after having travelled the distance z through the Fréedericksz cell, whereas Equations (3.40) and (3.41) give the Jones vectors in the x′−y′ coordinates at distance z. These equations will be evaluated with respect to the amplitude modulation in the next two chapters, and the phase modulation in Section 3.2.4. In the remaining portion of this chapter, we investigate the polarization of the light while it travels through the Fréedericksz cell.

We start with Equations (3.42) and (3.43), and calculate the locus of the vector in the x−y plane as time evolves. To this aim, we eliminate τ = ωt + φ and, after some algebraic steps presented in Appendix 3, reach the result (Born and Wolf, 1980)

(3.44)

where

(3.45)

(3.46)

and

(3.47)

This is a conic. From Equations (3.42) and (3.43), it can be seen that the conic must lie within the rectangular region bordered by the lines x = ±Ax and y = ±A_y, as shown in Figure 3.5. Hence, the conic is an ellipse and light is called elliptically polarized. Equation (3.44) indicates that the principal axes of the ellipse are not parallel to the x- and y-axes; they are parallel to the ξ- and η- axes in Figure 3.5, in which the equation for the ellipse is

(3.48)

The principal axes a and b are given by

(3.49)

(3.50)

Figure 3.5 The ellipse as locus for the vector of the electric field

The angle Ψ for the rotation is determined by

(3.51)

A derivation for Equations (3.48) through (3.51) can be found in Born and Wolf (1980), and is repeated in Appendix 3.

The sense of revolution of the E-vector with increasing time is determined by Equations (3.42) and (3.43). If Ey is time-wise ahead of Ex, this happens for 0< δ < π, i.e., for sin δ > 0 with δ in Equation (3.45), then the rotation of E is right-handed elliptically polarized, as shown in Figure 3.6; if Ey lags Ex, that happens for π < δ < 2π or sin δ < 0, the rotation is left-handed elliptically polarized (see Figure 3.6). The polarization dependent upon δ is depicted in Figure 3.7. The sense of revolution may be verified by plotting Ex and Ey with increasing time until the change of one of these components alters its sign.

Two special cases are the linear and the circular polarization. For

(3.52)

Equation (3.44) degenerates into

(3.53)

Ex and Ey lie on a straight line in Figure 3.7. This is the case of linearly polarized light. For Ax=Ay and

(3.54)

Equation (3.44) provides the circle

(3.55)

which represents circularly polarized light as also depicted in Figure 3.7.

Figure 3.6 Right- and left-handed elliptically polarized light seen against the propagating wave with wave vector k. viewing against the arrow of k

Figure 3.7 Elliptical, circular and linear polarization for different phase differences δ = 2π(Δn/λ)z (Reproduced from Born and Wolf, 1980 with permission of Elsevier.)

Further characterizations of the polarization are given in Chapter 5. We are now ready to evaluate the results obtained so far for the Fréedericksz cell.