is
With
and
we obtain the matrix equation by also inserting Jx and Jy from Equation (3.32):
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
and
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
and
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
where
and
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 = ±Ay, 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
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
A derivation for Equations (3.48) through