The polarization ellipse (see Fig. 2.2) can also be characterized by two angles known as the ellipse orientation angle (ψ in Fig. 2.2, 0 ≤ ψ ≤ π) and the ellipticity angle, shown as χ (−π/4 ≤ χ ≤ π/4) in Figure 2.2. These angles can be calculated as follows:
Note that linear polarizations are characterized by an ellipticity angle χ = 0.
So far it was implied that the amplitudes and phases shown in equations (2.12) and (2.13) are constant in time. This may not always be the case. If these quantities vary with time, the tip of the electric field vector will not trace out a smooth ellipse. Instead, the figure will in general be a noisy version of an ellipse that after some time may resemble an “average” ellipse. In this case, the wave is said to be partially polarized, and it can be considered that part of the energy has a deterministic polarization state. The radiation from some sources, such as the sun, does not have any clearly defined polarization. The electric field assumes different directions at random as the wave is received. In this case, the wave is called randomly polarized or unpolarized. In the case of some man‐made sources, such as lasers and radio/radar transmitters, the wave usually has a well‐defined polarized state.
Another way to describe the polarization of a wave, particularly appropriate for the case of partially polarized waves, is through the use of the Stokes parameters of the wave. For a monochromatic wave, these four parameters are defined as
Figure 2.3 Polarization represented as a point on the Poincaré sphere.
Note that for such a fully polarized wave, only three of the Stokes parameters are independent, since
The relations in (2.16) lead to a simple geometric interpretation of polarization states. The Stokes parameters S1, S2, and S3 can be regarded as the Cartesian coordinates of a point on a sphere, known as the Poincaré sphere, of radius S0 (see Fig. 2.3). There is therefore a unique mapping between the position of a point in the surface of the sphere and a polarization state. Linear polarizations map to points on the equator of the Poincaré sphere, while the circular polarizations map to the poles (Fig. 2.4).
In the case of partially polarized waves, all four Stokes parameters are required to fully describe the polarization of the wave. In general, the Stokes parameters are related by
Figure 2.4 Linear (upper) and circular (lower) polarization.
(2.17)
These two fully polarized waves have orthogonal polarizations. This important result shows that when an antenna with a particular polarization is used to receive unpolarized radiation, the amount of power received by the antenna will be only that half of the power in the unpolarized wave that aligns with the antenna polarization. The other half of the power will not be absorbed, because its polarization is orthogonal to that of the antenna.
The polarization states of the incident and reradiated waves play an important role in remote sensing. They provide an additional information source (in addition to the intensity and frequency) to study the properties of the radiating or scattering object. For example, at an incidence angle of 37° from vertical, an optical wave polarized perpendicular to the plane of incidence will reflect about 7.8% of its energy from a smooth water surface, while an optical wave polarized in the plane of incidence will not reflect any energy from the same surface. All the energy will penetrate into the water. This is the Brewster effect.
2.1.6 Coherency
In the case of a monochromatic wave of certain frequency ν0, the instantaneous field at any point P is well defined. If the wave consists of a large number of monochromatic waves with frequencies over a bandwidth ranging from ν0 to ν0 + Δν, then the random addition of all the component waves will lead to irregular fluctuations of the resultant field.
The coherency time Δt is defined as the period over which there is strong correlation of the field amplitude. More specifically, it is the time after which two waves at ν and ν + Δν are out of phase by one cycle; that is, it is given by:
(2.18)
The coherence length is defined as
(2.19)
Two waves or two sources are said to be coherent with each other if there is a systematic relationship between their instantaneous amplitudes. The amplitude of the resultant field varies between the sum and the difference of the two amplitudes. If the two waves are incoherent, then the power of the resultant wave is equal to the sum of the power of the two constituent waves. Mathematically, let E1(t)
