2.2.2.4 Lens Reflection Efficiency
One of the main differences of lenses w.r.t. reflectors is the fact that in a reflector all the incident field is transformed into a transmitted field as the surface can be modeled as a perfect electric conductor (PEC). However, in the case of lenses, the radiation principle is based on the refraction law; thus some incident energy is transmitted to the air but some part is reflected inside of the lens toward the top focus (see Figure 2.7a) [34]. GO/PO field approximation only includes the effect in the far field of the first transmitted rays. Part of the energy is actually reflected in the lens interface. This energy will eventually be radiated to the far field via multiple reflections or it will be lost in the lens material. In the GO/PO field analysis, this reflected energy is simply considered a loss in efficiency. That is typically a good assumption since the multiple reflections do not usually contribute to the radiated field main beam, as they have a random phase, but to the far side‐lobes. It means that the broadside gain of the antenna can be related as the directivity multiply by the radiation efficiency:
Figure 2.7 (a) High transmission and high reflection region of a silicon made elliptical lens. (b) Elliptical lens with a matching layer and its equivalent transmission line representation.
(2.54)
where the radiation efficiency, ηr, is the ratio between the power radiated by the lens into the air and the power radiated by the primary source inside the lens
The amount of energy reflected inside the lens depends on how the lens feed illuminates the lens surface (see Figure 2.7b). We can identify two distinct zones as shown in Figure 2.7a. The top part of the lens is where the energy transmission is the highest, and therefore the most efficient part, whereas the lateral part leads to high reflected energy (even total at the critical angle) and therefore is the least efficient one [34]. That is why the lens feed should be designed to illuminate only the top part of the lens.
In order to improve the reflection efficiency of lens antennas, we can use a quarter wavelength impedance transformers. This impedance transformer is typically designed for broadside radiation (top part of the lens). It can be easily analyzed using a transmission line model for the TE and TM polarizations (corresponding to orthogonal and parallel polarizations, respectively), with the characteristic impedances of the lines representing the different mediums indicated with i defined as
The reflections coefficients of a silicon elliptical lens with and without anti‐reflection layer are shown in Figure 2.8 as a function of the lens feeder angle (as in Figure 2.4a). Total internal reflection (Γ = 1) occurs when the incident angle onto the lens surface is 90° corresponding to about 72° for the feed angle. Before that angle, there is a total transmission angle for the parallel polarization at the called Brewster's angle. The same figure shows the improved transmission coefficient of a lens made in silicon with an antireflective layer compared to the same lens without the antireflective layer. As it is shown, the transmission coefficient remains high until up 40° approximately with the use of an antireflective layer, and the parallel and perpendicular coefficients have now comparable values and therefore they will not introduce asymmetries in the azimuthal angle. It is important to mention that the introduction of an antireflective layer in lenses does not remove the problem of the critical angle. In both cases, the energy transmitted goes to zero at about 72°.
Figure 2.8 Parallel and perpendicular transmission coefficient for a lens made in silicon with and without matching layer for a silicon lens (εr ≈ 11.9).
Silicon is the dielectric material employed in integrated lenses and circuit substrates. It is an inherently strong and excellent material in terms of losses, due to its high resistivity (≥10 kΩ cm), which lowers the loss tangent to tan δ ≤ 10−4. It has also a large thermal conductivity, meaning that for cryogenic applications it will absorb little of the incoming heat load and conduct it efficiently to the edge where the heat sinks are mounted. Moreover, thanks to its uniformity and homogeneity, it is a material well suited for polarization measurements. The high permittivity of silicon (εr ≈ 11.9) allows designs with strong focusing and smaller thicknesses; however, it presents a major drawback in terms of reflection losses. At submillimeter‐wave frequencies, the plastic Parylene‐C is extensively used as an antireflective layer thanks to the close match to the ideal index and its deposition through pyrolysis [35]. Other plastics such as Cirlex have suitable permittivity, but they are glued to the silicon with lossy epoxy adhesive [36]. Other methods include the synthesis of artificial dielectric coatings through the patterning of subwavelength grooves through micro‐machining processes [37]. This artificial dielectric coating can synthesize the desired permittivity and behave as a continuous medium providing the closest match to the ideal antireflective coating.
2.3 Extended Semi‐hemispherical Lens Antennas
While elliptical lenses reach the highest possible directivity, extended hemispherical lenses are often employed to ease their fabrication. This alternative configuration can have a close performance to the elliptical when the primary feed is designed in such a way to excite only the upper part of the dielectric lens and a dense material is used. The dimensions of the lens are chosen to approximate the focusing properties of the elliptical lens when the feed is placed at the foci. The synthesis of the ellipse is achieved using a semi‐hemisphere of unit radius defined as:
(2.55)
and an extension defined by L (see Fig. 2.9). The distance b + c should be equal to L + 1 so both lenses are overlaid. Thus, the extension height L can be defined as:
(2.56)
It can also be synthesized the other way around, from an extended