Fundamentals of Terahertz Devices and Applications. Группа авторов

of a certain taper, for example −14 dB, with a phase of , where is the distance from the phase center of the waveguide feed to the fabricated lens surface and is the distance from the fabricated lens surface to the lens aperture plane. The resulting radiation pattern is obtained by calculating the Fourier transform of this resulting field distribution. From these fields, the directivity and Gaussicity loss can be computed to have a sense of how our antenna would perform. In this example, the small lens of 2.6 mm aperture would result in a directivity loss of 0.2 dB and a Gaussicity loss close to 2%. Figure 2.25b shows the radiation patterns of the fabricated lens at 1.9 THz for a 2.6 mm diameter lens using the method explained compared with the measured radiation pattern.

Figure 2.26 Photographs of different lens antenna prototypes fed by leaky‐wave feeds (a) at 550 GHz with lens laser micro‐machined [33].

      Source: Alonso‐DelPino et al. [33]; IEEE.

      (b) At 550 GHz with lens fabricated using DRIE silicon micromachining [25].

      Source: Llombart et al. [25]; IEEE.

      (c) At 1.9 THz integrated with tripler all in silicon micromachining package [26].

      Source: Alonso‐delPino et al. [26]; IEEE.

      (d) At 550 GHz integrated with a piezo‐electric motor in order to perform beam‐scanning [48].

      Source: Alonso‐delPino et al. [48]; IEEE.

      2.5.2 Examples of Fabricated Antennas

      In this section, we will show different example implementations of leaky‐wave antennas feeding silicon lenses. Figure 2.18 shows the photograph of the different prototype examples we will comment on.

Figure 2.26a and b show two prototypes of leaky‐wave microlens antennas fabricated at 550 GHz. The first lens was fabricated using laser micromachining, while the second lens was fabricated using DRIE silicon micromachining. Even though both antennas obtained a good agreement with the expected results, the DRIE silicon micromachined lens portrayed some aberrations that were visible in in the radiation pattern measurements showing an increase in the secondary lobes and a small tilt.

Figure 2.26c shows a microlens antennas at 1.9 THz integrated with a 1.9 THz Schottky‐based tripler all in the same wafer. The antenna was synthesized on 12 resistivity silicon wafers and the tripler waveguide circuit was synthesized on three wafers. All the wafers, including the lenses, were fabricated using DRIE silicon micromachining process, aligned using the silicon compressive pin technique, and finally glued together. An alignment better than 2 μm was achieved across the wafer stack [26]. Even though in this case, the lenses suffered from aberrations caused by the lack of the surface control in the fabrication process, well‐focused beam patters very similar to the design ones were achieved.

      On another note, new efforts are now being investigated to enable lens beam‐scanning capabilities in the system front end. Figure 2.26d shows a highly integrated beam scanning lens‐antenna using a piezo‐electric motor demonstrated operating at 550 GHz presented in [48]. A hemispherical lens was glued on top of a silicon wafer containing alignment marks processed using DRIE silicon micromachining. The piezoelectric motor displaced the lens around ±1 mm from the center position of the lens, providing a beam scan of ±25°. Not only this method can be employed for improving the alignment of the lens with the feed, but it also has the potential to enable beam‐scanning capabilities on the system front end for future terahertz imaging systems.

      The results achieved so far show a great potential to use these dielectric lens antennas in the development of future focal plane arrays at terahertz frequencies. By using the leaky‐wave waveguide feed, we only need a small part of the surface of the lens, which reduces the reflection losses and phase errors that these type of lenses suffer. But most of all, it allows the use of photolithographic process when fabricating the lens. The fabrication of the lenses using photolithographic process reduces the cost, with the same performance achieved with other fabrication methods, such as laser micromachining.

Exercises

      E2.1 Derivation of the Transmission Coefficients and Lens Critical Angle

      In Section 2.2, we defined the Fresnel reflection coefficients τ^⊥(Q), τ^∥(Q) as:

      (2.69)

      (2.70)

      where . Similarly, the transmission coefficients can be obtained from the reflection coefficients τ^⊥ = 1 + Γ^⊥ and . Thus, they are represented as:

Note that the angle of refraction θ_t increases as the incident angle θ_i inside of the lens increases (θ_t > θ_i). Total internal reflection (Γ = 1) occurs when the angle of refraction is 90° and an incident angle known as the critical angle . After this angle there is no transmission, therefore, this angle defines the domain of the equivalent aperture. As this angle lies at the center distance between the two focuses, the radius of the equivalent aperture is therefore the minor axis b. The incident angle that creates zero reflection in the parallel reflection coefficient is called Brewster's angle. An example of the reflection and transmission coefficients for a lens made in silicon are shown in Figure 2.27a and b, illustrating the critical and Brewster angles.

Figure 2.27 (a) Reflection and (b) transmission coefficient for a silicon lens as a function of the angle. The reflection coefficient becomes 1 for angles larger or equal than the critical angle. The transmission coefficient becomes one for the Brewster