How do I design a terahertz antenna using a silicon lens with an integrated planar feed?

Designing a terahertz antenna using a silicon lens with an integrated planar feed combines a small planar antenna element (fabricated on a silicon or quartz substrate using photolithography) with a macroscopic dielectric lens (typically high-resistivity silicon) to create a directive beam at THz frequencies. This approach is the standard for THz systems because: at THz frequencies (0.1-10 THz): the wavelength is 0.03-3 mm, making conventional horn antennas very small and difficult to machine, while planar antennas on a substrate suffer from substrate mode trapping (most of the radiated power goes into the dielectric substrate rather than into free space; for silicon with Er = 11.7: approximately 96% of the power is radiated into the substrate). The silicon lens solves this by: collecting the power radiated into the substrate (placing a hemispherical or elliptical silicon lens on the substrate side of the planar antenna captures the substrate-mode radiation and focuses it into a directive beam), providing high gain (a 10 mm diameter silicon lens at 300 GHz provides approximately 25 dBi directivity), and simplifying the antenna design (the planar feed can be a simple dipole, bowtie, log-spiral, or slot antenna, fabricated using standard photolithography; the lens provides the directivity, not the feed). The design procedure: select the lens material (high-resistivity silicon (rho greater than 10 k-ohm-cm) is the standard: Er = 11.7, low loss (tan_delta less than 0.001 at THz), mechanically robust, and available in high quality), choose the lens shape (elliptical: focuses the on-axis radiation to a collimated beam with minimal aberration; the eccentricity of the ellipse is determined by the refractive index: e = 1/n = 1/sqrt(Er) = 1/3.42 for silicon; hemispherical with extension length: simpler to manufacture, the extension length L = R/n for optimal focusing), and design the planar feed (a double-slot antenna or log-periodic antenna is commonly used; the feed is positioned at the focus of the lens on the flat surface of the silicon hemisphere).