## Polarization of a Plane Wave

Electromagnetic fields are vector quantities, which have a direction in space. The polarization of a plane wave refers to this orientation of the electric field vector, which may be a fixed orientation (a linear polarization) or may change with time (a circular or elliptical polarization).

The electric field of a plane wave can be presented as a sum of two orthogonal components

where ux and Uy are the unit vectors in the x and y direction, respectively. In a general case this represents an elliptically polarized wave. The polarization ellipse shown in Figure 2.6(a) is characterized by the axial ratio Emax IEmi„, tilt angle t, and direction of rotation. The direction of rotation, as seen in

Figure 2.6 Polarization of a plane wave: (a) elliptic; (b) linear; (c) clockwise circular; and (d) counterclockwise circular.

the direction of propagation and observed in a plane perpendicular to the direction of propagation, is either clockwise (right-handed) or counterclockwise (left-handed).

Special cases of an elliptic polarization are the linear polarization, Figure 2.6(b), and circular polarization, Figure 2.6(c, d). If E1 ^ 0 and E2 = 0, we have a wave polarized linearly in the x direction. If both E1 and E2 are nonzero but real and the components are in the same phase, we have a linearly polarized wave, the polarization direction of which is in angle t = arctan (E2 /E1) (2.73)

In the case of circular polarization, the components have equal amplitudes and a 90° phase difference, that is, E1 = +jE2 = E0 (E0 real), and the electric field is

The former represents a clockwise, circularly polarized wave, and the latter a counterclockwise, circularly polarized wave. In the time domain the circularly polarized wave can be presented as (clockwise)

E(z, t) = E0 [ux cos (Mt — kz) + Uy cos (Mt — kz — t/2)] (2.76)

and (counterclockwise)

E(z, t) = E0 [ux cos (Mt — kz) — Uy cos (Mt — kz — t/2)] (2.77)

E(t) = E0 [ux cos Mt ± Uy cos (Mt — tt/2)] (2.78)

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