Closed-Form Relation Between the Scan Angle and Feed Position for Extended Hemispherical Lenses Based on Ray Tracing

This letter presents a closed-form relation between the scan angle and feed position for extended hemispherical lenses. This relation is derived using ray tracing, and it is valid for both large and small scan angles, in excellent agreement with full-wave simulations. It is demonstrated that the relation is linear in the small-angle limit, and the effective focal length determining the scan angle is presented. It is also demonstrated that the scan angle only depends on the geometrical configuration, and that it is independent of the lens material. To demonstrate the applicability of this scan angle relation to the design of focal plane arrays (FPAs), we demonstrate that it can be used to determine the FPA spacing that results in -3-dB overlap between switched beams. A comparison with full-wave simulations of lenses with varying materials and FPA elements demonstrates a root-mean-square (rms) accuracy of 0.27° for the scan angle estimation, and rms accuracy of 0.26 dB for the -3-dB overlap criterion between the central and adjacent beams. Finally, we present scaling rules, which show that the scan resolution is inversely proportional to the lens diameter, whereas the FPA spacing is independent of the total lens size.