Waves and Fields

Abstract

In a homogeneous lossless material at far distance from the source and in a limited angular range such that $\pmb{\mathcal{F}}(\vartheta,\varphi )$ in (3.33) is almost constant, the field is regarded as a function of distance r only, of the form

$$\displaystyle{\boldsymbol{E}_{\infty }(\boldsymbol{r}) = C\,\frac{\mathrm{e}^{-\mathrm{j}\kappa r}} {r} \,\pmb{\mathfrak{e}}_{0}{}_{\infty }\;,}$$

where the factor C includes the amplitude information, and the unit vector $\pmb{\mathfrak{e}}_{0}{}_{\infty }$ accounts for the polarization. In Earth observation, the distance r between the source (Sun, satellite, scattering object) and the region where the field is considered is generally quite large with respect to the dimension of the region itself. For instance, the distance from the Sun is large compared with the dimension of the elementary portion of the Earth’s surface being imaged by a space-based optical sensor. Analogously, the dimension of the antenna receiving the power scattered from the surface is much smaller than the distance between the observed surface and the platform on which the antenna is based. Therefore, within such limited angular ranges, the spherical wave surface Φ = κ r does not differ appreciably from a plane surface, neither the angular pattern changes. Again, this features apply to the solar radiation illuminating the scene instantaneously observed by a spectrometer, to the field locally created on the earth surface by a space-based radar, or to the wave reflected from an area of the surface and collected by the aperture of a satellite sensor. In such limited regions of space, in practice, the radial unit vector $\boldsymbol{\mathrm{r}}_{0}$ is regarded as constant.