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
While propagating through the cell, the component Jx parallel to
(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
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
