Flat silicon gradient index lens with deep reactive-ion-etched 3-layer anti-reflection structure for millimeter and submillimeter wavelengths

We present the design, fabrication, and characterization of a 100 mm diameter, flat, gradient-index (GRIN) lens fabricated with high-resistivity silicon, combined with a three-layer anti-reflection (AR) structure optimized for 160-355 GHz. Multi-depth, deep reactive-ion etching (DRIE) enables patterning of silicon wafers with sub-wavelength structures (posts or holes) to locally change the effective refractive index and thus create anti-reflection layers and a radial index gradient. The structures are non-resonant and, for sufficiently long wavelengths, achromatic. Hexagonal holes varying in size with distance from the optical axis create a parabolic index profile decreasing from 3.15 at the center of the lens to 1.87 at the edge. The AR structure consists of square holes and cross-shaped posts. We have fabricated a lens consisting of a stack of five 525 $\mu$m thick GRIN wafers and one AR wafer on each face. We have characterized the lens over the frequency range 220-330 GHz, obtaining behavior consistent with Gaussian optics down to -14 dB and transmittance between 75% and 100%.