Geometric efficiency for a circular detector and a ring source of arbitrary orientation and position

Abstract

Two distinct axisymmetric radiation vector potentials are derived for a circular ring source, both of which are given in terms of elliptic integrals. In combination with Stokes's theorem these potentials reduce the surface integral for the geometric efficiency of a ring source and a general detector to a line integral, though only a circular detector is analyzed in detail here. One of the potentials is bounded as the axis of symmetry is approached and it can also be expressed as an integral of Bessel functions. It is used to derive the ring source analogue of Ruby's formula and its noncoaxial generalization. A trigonometric integral is given for the general noncoaxial case and closed form solutions are given for the coaxial case and the case where the ring source is in the detector plane. Numerical data is given for these cases. The second potential is singular along the entire ring axis and Stokes's theorem must be modified when using it whenever the ring axis intersects the detector or its boundary. This potential is used to derive a trigonometric integral for the geometric efficiency of the ring source and a circular detector in the general case where the source and the detector have arbitrary relative position and angular orientation. Numerical data is given for selected geometric parameters when the planes of the ring and disk are perpendicular and for the general case. The intersection of some results for point source radiation vector potentials with vector potentials arising in diffraction theory is given in an Appendix.