lens and the extended hemispherical lens varies. Three examples are shown in Figure 2.9 where both lenses of silicon (εr ≈ 11.9), quartz (εr ≈ 4.5) and polytetrafluoroethylene (PTFE) (εr ≈ 2). The rays from a broadside plane wave incident onto the extended semi‐hemispherical lens, as opposed to the elliptical lens, do not come to a point focus, thus, which means that the lens introduces an aberration to the radiated fields. But as we will see in the following sections, even if it does not couple well to a planar equi‐phase front, the extended hemispherical lens will couple well to a Gaussian‐beam system.
2.3.1 Radiation of Extended Semi‐hemispherical Lenses
In Section 2.2, we computed the radiation pattern from an elliptical lens antenna by deriving the currents on a planar aperture above the lens and, from them, we obtained the radiated fields from the lens antenna. In this section, we will explain how to compute the radiated fields of the extended hemispherical lens antenna from the currents evaluated on the lens surface. A PO method provides an approximation of these surface currents over a lens of several wavelengths. Using again the Love's Equivalence Principle, the radiated fields from the antenna feed obtained previously are used to compute the equivalent magnetic and electric sheet currents outside of the lens surface:
(2.60)
(2.61)
where
Figure 2.10 Sketch of the extended semi‐hemispherical lens antenna parameters.
(2.62)
(2.63)
where the τ‖ and τ⊥ are Fresnel transmission coefficients for a dielectric lens of permittivity εr (2.27 and 2.28). This time the incident angle is evaluated using the normal vector corresponding to the hemispherical lens. The propagation vectors of the incident and transmitted fields are defined as follows:
(2.64)
(2.65)
(2.66)
Once the PO surface currents are evaluated via the transmitted fields, one can obtain the far‐field patterns. Those patterns can be obtained using the reference system shown in Figure 2.8, and integrating the PO surface currents over the lens hemispherical surface, as follows:
(2.67)
(2.68)
Figure 2.11 Directivity of an elliptical lens and an extended hemispherical lens as a function of the feed illumination. A feed illumination of a f(θ) = cosnθ is used in the example to illuminate a silicon lens of diameter 7.65 λ and an extended hemispherical lens of L = 0.375.
where R is the radius of the hemisphere of the lens,
Figure 2.11 shows the directivity of a silicon elliptical lens and a hemispherical lens (i.e. synthesized from the elliptic geometry) as a function of the subtended angle by the feed. As it is shown in the figure, the elliptical lens provides the highest directivity compared to the hemispherical lens, however, the difference is only noticeable when using a feed with low directivity (low n). As the directivity increases, the performance of the elliptical and extended semi‐hemispherical is equivalent.
2.4 Shallow Lenses Excited by Leaky Wave/Fabry–Perot Feeds
As mentioned in Section 2.3, the amount of energy reflected inside of the lens depends on how the lens feed illuminates the lens surface. The top part of the lens is the most efficient part of the lens since it is where the transmitted energy is the highest, and the least efficient area is the lateral part, leading to high reflected energy. Thus, in order to have a highly efficient lens antenna, we need a feed that illuminates only the top part of the lens, a.k.a. a shallow dielectric lens.
The use of shallow dielectric lenses, as the lens antenna is shown in Figure 2.12, presents great advantages in terms of fabrication and electrical performances at submillimeter‐wave frequencies. Lower cost and better surface accuracies
