focusing capabilities into a planar antenna. At submillimeter‐wave frequencies, planar antennas suffer from high power loss due to the excitation of multiple surface wave modes on thick dielectric substrates. When a radiating source is placed on a dielectric surface, the rays propagate through the dielectric reaching the dielectric‐air interface. At that point, the rays that reach the substrate edge with an angle above the critical angle, defined as θc = a sin(1/nsubs) being nsubs the refraction index of the substrate, are reflected back into the substrate generating reflections along the transversal axis of the dielectric (see Figure 2.1a). These are trapped waves that do not radiate into free space; thus they represent an efficiency loss.
Figure 2.1 (a) Sketch of a planar antenna printed on a dielectric substrate. (b) Front‐to‐back ratio of an elementary dipole antenna as a function of the permittivity of the substrate [27].
Source: Modified from Rutledge et al.[27]; John Wiley & Sons.
One way to mitigate this loss is to reduce the thickness of the substrate until it is electrically thin (≈λsubs/100, being λsubs the wavelength in the substrate λsubs = λ0/nsubs) as in [28, 29] but at the cost of a poor front to back radiation. The antenna is integrated on a thin dielectric membrane which is typically less than 5 μm thick and realized on silicon, SiN, or SiO2 dielectrics. These antennas will couple weakly to the substrate and they will radiate the same amount of energy in the front and back hemispheres as if they were suspended in free space. In Ref. [30], a membrane of 1 μm of SiN was fabricated for an antenna working at 700 GHz.
Another approach is to use a thick substrate together with a lens of the same material to couple efficiently the radiation into free space as proposed by Kominami et al. [31]. The antenna is placed between two infinite mediums, one on the top and one on the bottom, and the top of the dielectric medium is curved in order to couple the radiation into a directive beam without having critical angle reflections. The front‐to‐back ratio of an elementary dipole planar antenna between the two mediums can be approximated as ηfront‐to‐back
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