will compute the radiated field from the leaky‐wave feed inside the infinite silicon for different ρ. After this, we will calculate the taper edge angle θf for a certain taper field value in the 10–14 dB region. The field taper is chosen according to the tradeoff between the spillover and the taper efficiency, equivalent to the tradeoff in reflector antennas. This computation of the field at different ρ is necessary because, for the dimension of the lens aperture, the feed is placed on the near field of the antenna. This dependency can be seen in Figure 2.17b, when we move toward the far‐field region this dependency fades, and the curves in Figure 2.17b get flatten (θf becomes a constant with ρ). The optimal ρ and θf value will be given by the intersection of this curve with . Figure 2.18a shows the optimal ρ and θf values for different diameters ranging from 4 − 11λ0. And Figure 2.18b shows the optimum distance W defined as where the shallow lens surface should be placed.3 The shallow lens surface is optimized to maximize the antenna directivity. The lens curvature is defined by the radius R and the height H, or equivalently by the extension height L (see Figure 2.19a). For planar antennas, the optimum L and R are known, see Section 2.3. However, as stated previously, because the phase center of the leaky wave is below the waveguide aperture, this makes the optimum lens surface differ from the standard cases [33]. Then, we will vary the height and radius of the lens until the optimum is achieved. The variation of the directivity as a function of the extension height L is shown in Figure 2.18b. To perform this optimization, full‐wave simulators are not recommended as the computational time and complexity of the global structure is considerably. Instead, the PO techniques described previously provide a good compromise between quality of the results and computational time which makes it fundamental for integration into optimization of lenses. The optimum at different field tapers for each diameter are summarized in Figure 2.20a and b.Figure 2.19 (a) The shallow lens of diameter D is defined by a corresponding H and R. (b) Taper and aperture efficiency as a function of the lens extension height L.
4 Following these two steps, we can obtain the highest directivity for a desired aperture diameter D for the designed leaky‐wave feed. The directivity attainable with this architecture is shown in Figure 2.21a. Note that for the chosen field tapers of −10 dB to −14 dB the spillover remains below 0.25 dB. The Gaussicity, calculated with the expressions on Section II is shown in Figure 2.21b. The beamwaist and the phase center of the lens have been adjusted to maximized the Gaussicity for each diameter. As it is shown very high Gaussicity values can be achieved across all the diameters due to the high symmetry and shape of the leaky‐wave field.
2.5 Fly‐eye Antenna Array
The main advantage of having a lens antenna with a waveguide feed is its seamless integration in an array configuration for waveguide‐based receivers at submillimeter‐wave frequencies. As explained in previous sections, the leaky‐wave feed illuminates a small aperture of diameter D of the lens, making the resulting effective lens surface a very shallow lens. These shallow lenses can be packed together forming an array and it can be fabricated using silicon micromachining techniques. The feed and lens array can be entirely fabricated over a few silicon wafers and stacked to the rest of the instrument (see Figure 2.22).
The micromachining techniques used for the fabrication of heterodyne receivers and sources at submillimeter‐wave frequencies are based on CNC metal block machining, as well as wafer‐level processing using photolithographic techniques. Compared with metal block machining, wafer‐level processing allows a higher integration of the whole receiver, which reduces the volume, mass, and losses. The different waver level processing solutions in literature, from the use of SU‐8 [36] machining to lithography, electroplating, and molding (LIGA)‐based processes [37] that use thick resist and electroplating processes to form the waveguide walls, suffer from limitations in terms of multi‐etching capabilities and non‐uniformity problems. Silicon micromachining based on DRIE allows the fabrication of high aspect ratio features while maintaining straight sidewalls and smooth multi‐depth surfaces. This technology is very attractive for the development of heterodyne receivers and sources as the technology allows the fabrication of complex circuit features that require multiple waveguides of various depths, especially as the wavelength decreases, when machining tolerances become a concern.
Figure 2.20 (a) Radius R and (b) height H of the lens as a function of the diameter D for different field tapers.
Figure 2.21 (a) Directivity and (b) Gaussicity achieved for the shallow lens antenna Reflection coefficient centered at a central frequency f for the dimensions shown in the table.
As previously stated, the presented antenna composed by a leaky‐wave feed and the shallow lens can be fabricated in four silicon wafers using DRIE processes. The first wafer consists of the iris and the waveguide feature. The second wafer contains the air cavity. The third wafer contains dummy silicon wafer to achieve the correct lens thickness. The fourth wafer contains the lens surface. The four different wafers (i.e. iris air gap, bulk wafers, and lens) will be assembled together on a single wafer stack. Note that high resistivity silicon wafers, i.e. 10 kΩ cm, are required to reduce the dielectric absorption loss. And, the wafers need to be double‐sided polished in order to have good surface contact between all the wafers in the stack, avoiding air gaps. The alignment, which is a challenge when working at such high frequencies, has been solved by using a silicon compressive pin that is slightly larger than the pocket and can be compressed slightly when put into place [26].
Figure 2.22 Reflection coefficient centered at a central frequency f for the dimensions shown in the table.
2.5.1 Silicon DRIE Micromachining Process at Submillimeter‐wave Frequencies
In order to achieve the depth accuracy and smoothness of the features on silicon, Silicon‐on‐insulator (SOI) wafers are commonly employed, since thin layers of SiO2 act as etch stops of the silicon. For example, [26] presented an iris of the leaky‐wave antenna at 1.9 THz fabricated on a 2.5 μm silicon membrane. Compared with a thermal oxide membrane, the crystalline silicon provides more robustness to compressive stress. In that case, a SOI wafer with a 2 μm device layer was used because, together with the metallization, the membrane would reach the desired 2.5 μm. And a handling wafer of 400 μm defined the
