Appendix L SINE RELATION FOR HYPERBOLOIDAL/LENS CONCENTRATOR
With reference to Figure L.1 the lens images the input beams inclined at angles ±θ to the axis onto the edges FF′ of the virtual source. Therefore, by Fermat’s principle [QA′] + [A′F′] = [AF′], where the brackets denote optical path length, and similarly for the other beam. From the geometry of the hyperbola, [AF′] − [A′F′] = 2c. Also [QA′] = 2a sin θ, since QA is normal to the beam. The desired result follows: c = a sin θ.
Figure L.1 The thin lens at AA′ images the input beam inclined at angle θ to the axis onto F′