are two of the materials that are used for the fabrication of integrated lenses due to their low dielectric loss and still high front‐to‐back ratios. As shown in Figure 2.1b, when using a dielectric of silicon (εr ≈ 11.9) the power radiated to the silicon is around 97.5% of the power radiated to the air, while if the lens is quartz (εr ≈ 4), the power radiated is around 87.5%. The lens is put on the backside of the antenna on the substrate radiates most of their power into the dielectric side making the pattern unidirectional and also providing thermal and mechanical stability.
2.2.1 Elliptical Lens Synthesis
In the following example, we will derive the canonical geometry (elliptical lens) that achieves a planar wave front at the aperture, starting from a spherical wave at its focus. This demonstration is based on a geometrical optics (GO) approach, or also called ray tracing. We start by imposing that all the rays in Figure 2.2a have the same electrical length:
Figure 2.2 Geometrical parameters of an elliptical lens.
where k0 and
And we can also equate the projection in the z‐axis of these rays:
(2.3)
using (2.1) and (2.2) we conclude on the following equation:
(2.4)
On the other hand, we know that an elliptical lens geometry can be described in polar coordinates using the following expression:
where a and e are the semi‐major axis and eccentricity, respectively (see Figure 2.2b).
By recognizing in (2.5) that:
(2.6)
we can conclude that a spherical wave front produced by an antenna at the focus point of a lens of with an ellipsoidal shape will be transformed into a planar waveform. The antenna is placed at the second focus of an ellipse defined by the equation:
where b is the semi-minor axis of a ellipse. The eccentricity of the lens e, relates the geometric focus ellipse to the optical focus of the lens with the following relationship:
(2.9)
using Eqs. (2.7) and (2.8) we can derive the foci of the ellipse, defined as c as:
(2.10)
and the semi‐minor axis b as:
(2.11)
The radiation pattern obtained by elliptical lenses is the one that reaches the highest possible directivity when illuminated with a spherical phase front generated by the feeding antenna.
2.2.2 Radiation of Elliptical Lenses
For electrically moderate and large lenses, the use of high frequency analysis techniques is typically the preferred approach. These techniques are based on Physical Optical (PO) where the field scattered by the lens is evaluated from an equivalent surface currents approximated using the incident field. These currents can be evaluated directly either at the lens surface [32] or over a planar aperture, just above the lens as shown in Figure 2.3a. In both cases, the incident field coming from a known feed is propagated until these considered surfaces using a GO approach. Then, the equivalent current densities are integrated with the free space Green's function to obtain the field radiated by the lens antenna.
Figure 2.3 (a) Equivalent aperture on top of the elliptical lens antenna. (b) Scheme of the ray tracing of the elliptical lens antenna at a point Q in the surface.
In this section, we will derive an approximate analytical expression of these equivalent currents over a planar surface on top of the lens. This expression is valid only when the feed phase center is located in the lower focus of the ellipse and the lens surface is in the far field of the feeder. This expression can be used to perform a fast optimization of the lens and planar feeder. In order to explain this analysis method we will first evaluate the equivalent current distribution over a plane parallel to the lens aperture (i.e. that lies outside of the lens, see Figure 2.3a). The reference system used to evaluate the radiation patterns of the lens antenna is the one shown in Figure 2.3a. There are three approximations to calculate these equivalent surface currents:
First
