SPT-3G secondary mirror geometry

TL;DR

This paper details the geometry of the secondary mirror of the SPT-3G, a telescope system for cosmic microwave background observations, including a geometric theorem related to prolate spheroids and the Dragone condition.

Contribution

It provides a detailed geometric analysis of the secondary mirror, including a proven theorem linking spheroid intersections and cone constructions relevant to off-axis mirror design.

Findings

Prolate spheroid intersection with planes forms ellipses.

Lines from spheroid points to foci form right circular cones.

Cones from two foci intersect outside the spheroid.

Abstract

SPT-3G is a detector system for the 10m diameter South Pole Telescope, comprising 16,000 millimeter-wave bolometers. It is used for a deep Cosmic Microwave Background survey of the Southern sky. This paper describes the geometry of the secondary mirror, which is a section of a prolate spheroid, in several useful coordinate systems. There is application to off-axis mirrors in general. A geometric theorem is proven, relating to the Dragone condition: the intersection of a prolate spheroid and any plane is an ellipse; the lines connecting points on that ellipse to either focus compose a right circular cone; the central axes of the two cones from the two foci intersect outside the interior of the spheroid.