Fundamentals of Terahertz Devices and Applications

The equivalent current distributions on the aperture, using the incident field is defined as in ( 2.21and 2.22) and the phase relation derived in (2.1), can then be written as follows:

(2.50)

(2.51)

where r ′ ( θ ′) is given by (2.5)and tan θ ′ = ρ ′/ z ′ where . The vectorial components of these equivalent current distributions can be also expressed in Cartesian coordinates using and .

With these equations, we have determined the surface currents on the lens aperture, and thus, the far field patterns in the reference system used in Figure 2.6can be obtained using the following expressions:

(2.52)

Figure 2.6Reference system for the evaluation of the far fields radiated by the elliptical lens antenna.

(2.53)

where and ; and η 0is the intrinsic impedance of free space.

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, η r, 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 and with as shown in Figure 2.7b. Using this model, we can derive the condition for no reflection when Z in= ζ dwhich implies that the permittivity of the antireflective layer material is and its thickness .

The reflections coefficients of a silicon elliptical lens with and without anti‐reflection layer are shown in Figure 2.8as 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°.