The two circular polarizations also span the two-dimensional space of polarization. The right- and left-handed orthogonal unit vectors defined in terms of linear components are
The electric field in the polarization plane can be expressed in terms of these new unit vectors:
When projecting a vector onto one of these unit vectors, it is necessary to use the complex conjugate in the scalar (dot) product:
When we project ˆR onto itself, we obtain
Similarly,
The right- and left-handed circular (RHC and LHC) components are orthonormal. A circular polarization ratio can be defined from the equation
Let us look at a predominately left-handed circularly polarized wave when time and space combine to a phase of zero for EL. We draw the polarization as two circles (Figure 1). The circles rotate at the rate ωt in opposite directions (Figure 2), with the center of the right-handed circular polarization circle moving on the end of the vector of the left-handed circular polarization circle. We calculate the phase of the circular polarization ratio from the complex ratio of the right- and left-handed circular components. Maximum and minimum electric fields occur when the circles alternately add and subtract as shown in Figure 1. Scale 1 shows the relationship between circular cross-polarization and axial ratio:
The tilt angle of the polarization ellipse τ is one-half δc, the phase of . Imagine time moving forward in Figure 2. When the LHC vector has rotated δc/2 CW, the RHC vector has rotated δc/2 CCW and the two align for a maximum.
FIGURE 1 Polarization ellipse LHC and RHC components. (After J. S. Hollis, T. J. Lyons, and L. Clayton, Microwave Antenna Measurements, Scientific Atlanta, 1969, pp. 3–6. Adapted by permission.)
FIGURE 2 Circular polarization components. (After J. S. Hollis, T. J. Lyons, and L. Clayton, Microwave Antenna Measurements, Scientific Atlanta, 1969, pp. 3–5. Adapted by permission.)
SCALE 1 Circular cross-polarization/axial ratio.
Polarization of Antennas ---- Huygens Source Polarization
When we project the currents induced on a paraboloidal reflector to an aperture plane, Huygens source radiation induces aligned currents that radiate zero cross-polarization in the principal planes. We separate feed antenna radiation into orthogonal Huygens sources for this case. To calculate the far-field pattern of a paraboloid reflector, we can skip the step involving currents and integrate over the Huygens source fields in the aperture plane directly. We transform the measured fields of the feed into orthogonal Huygens sources by
where Ec is the φ = 0 direction of polarization in the feed pattern and Ex is the φ = 90◦ polarization. This division corresponds to Ludwig’s third definition of crosspolarization. The following matrix converts the Huygens source polarizations to the normal far-field components of spherical coordinates:
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