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Thus we see that the aperture field distribution is analogous to the angular distribution of the source ("feed") pattern as denoted in Eq. (3.14). Because of the paraboloid's geometry, the term exp(i k p)/ p does not influence the phase function over the aperture compared to the feed pattern. The amplitude distribution over the aperture is however somewhat weaker toward the edge compared to the feed pattern, because the spherical wave from the feed travels a longer path to the reflector edge than to its center. This effect is known as the "free-space taper" of the illumination function. We shall return to this in more detail in the next chapter.

We continue now with the detailed treatment of the aperture integration method. Let us choose as the aperture plane the (x, y)-plane through the focal point O (Fig. 3.1 and 3.2). The aperture A is thus the projection of the reflector rim onto this plane. The electric field in this plane is given by Eq. (3.24). We now have y =0 and hence cos y' — sin q cos (^ — f). Also the distance from the field point P to a point in the aperture follows from Eq. (3.20) as r — R — b sin q cos(f — C + -jr , where b is the radial coordinate in the aperture, thus b = d a / 2.

We recall that the method assumes that the aperture and the distance to the field point are both large with respect to the wavelength and that the calculation is restricted to relatively small angles 0 about the beam axis. In practice reliable results are obtained up to angles incorporating several sidelobes. For sufficiently large values of the distance R the second term in the exponent can be ignored. In that case we are dealing with Fraunhofer diffraction and we refer to the resulting field as the farfield pattern of the antenna. In cases where this term cannot be neglected, we are in the region of Fresnel diffraction and the point P is said to be in the nearfield region of the antenna. In the following sections we examine these two cases in more detail. We maintain the second order term here and arrive at the simplified form of Eq. (3.23)

J0 J0 F(a, Xexp[ik{-d2a sin0 cos(x-f) + ^R-}] a dadx. (3.26)

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