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
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
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.
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.
3.2.2 The transmissive Fréedericksz cell
In Figure 3.8 we continue the discussion of the transmissive cell begun with Figures 3.4(a) and 3.4(b). The incoming linearly polarized light enters at an angle α in Figure 3.8. The linear polarization occurs again according to Equation (3.52) at δ = 2π(Δn/λ)z = vπ, v = ±1, ±2, ..or for v = 1 at the smallest z-value
(3.56)
where λ0 is the pertinent wavelength.
Figure 3.8 Angles of polarizer and analyser for the Fréedericksz cell
At the output of the cell for z = d, where d is the thickness of the cell, the retardation is
This retardation is associated with a change of phase by δ = π after the wave has propagated the distance d through the cell. The cell operates as a λ/2-plate. Obviously, the retardation is the phase shift measured in parts of λ0.
For linear polarization with wavelength λ0 corresponding to the angular frequency ω0 of the electric field, the components in Equations (3.42) and (3.43) at z = d are
Equations (3.58) and (3.59) reveal the angle β of the linearly polarized light at z = d as
Due to Equations (3.46) and (3.47), we obtain on the other hand for the light at z = 0