Читать книгу Introduction to the Physics and Techniques of Remote Sensing - Jakob J. van Zyl - Страница 23

An electromagnetic wave consists of a coupled electric and magnetic force field. In free space, these two fields are at right angles to each other and transverse to the direction of propagation. The direction and magnitude of only one of the fields (usually the electric field) is sufficient to completely specify the direction and magnitude of the other field using Maxwell’s equations.

The polarization of the electromagnetic wave is contained in the elements of the vector amplitude A of the electric field in Equation (2.10). For a transverse electromagnetic wave, this vector is orthogonal to the direction in which the wave is propagating, and therefore we can completely describe the amplitude of the electric field by writing A as a two‐dimensional complex vector:

(2.12)

Here we denote the two orthogonal basis vectors as for horizontal and for vertical. Horizontal polarization is usually defined as the state where the electric vector is perpendicular to the plane of incidence. Vertical polarization is orthogonal to both horizontal polarization and the direction of propagation, and corresponds to the case where the electric vector is in the plane of incidence. Any two orthogonal basis vectors could be used to describe the polarization, and in some cases the right‐ and left‐handed circular basis is used. The amplitudes, a_h and a_v, and the relative phases, δ_h and δ_v, are real numbers. The polarization of the wave can be thought of as that figure that the tip of the electric field would trace over time at a fixed point in space. Taking the real part of (2.12), we find that the polarization figure is the locus of all the points in the h‐v plane that have the coordinates E_h = a_h cos δ_h; E_v = a_v cos δ_v. It can easily be shown that the points on the locus satisfy the expression

(2.13)

This is the expression of an ellipse, shown in Figure 2.2. Therefore, in the general case, electromagnetic waves are elliptically polarized. In tracing the ellipse, the tip of the electric field can rotate either clockwise or counterclockwise; this direction is denoted by the handedness of the polarization. The definition of handedness accepted by the Institute for Electrical and Electronics Engineers, IEEE, is that a wave is said to have right‐handed polarization if the tip of the electric field vector rotates clockwise when the wave is viewed receding from the observer. If the tip of the electric field vector rotates counterclockwise when the wave is viewed in the same way, it has a left‐handed polarization. It is worth pointing out that in the optical literature a different definition of handedness is often encountered. In that case, a wave is said to have right‐handed (left‐handed) polarization when the wave is viewed approaching the observer, and tip of the electric field vector rotates in the clockwise (counterclockwise) direction.

Figure 2.2 Polarization ellipse.

In the special case where the ellipse collapses to a line, which happens when δ_h − δ_v = nπ with n any integer, the wave is said to be linearly polarized. Another special case is encountered when the two amplitudes are the same (a_h = a_v) and the relative phase difference δ_h − δ_v is either π/2 or −π/2. In this case, the wave is circularly polarized.

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:

(2.14)

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.

(2.15)

Note that for such a fully polarized wave, only three of the Stokes parameters are independent, since . Using the relations in (2.14) between the ellipse orientation and ellipticity angles and the wave amplitudes and relative phases, it can be shown that the Stokes parameters can also be written as

(2.16)

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 , with equality holding only for fully polarized waves. In the extreme case of an unpolarized wave, the Stokes parameters are S₀ > 0; S₁ = S₂ = S₃ = 0. It is always possible to describe a partially polarized wave by the sum of a fully polarized wave and an unpolarized wave. The magnitude of the polarized wave is given by and the magnitude of the unpolarized wave is . Finally, it should be pointed out that the Stokes parameters of an unpolarized wave can be written as the sum of two fully polarized waves

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.