Nonorthogonal Optical Waveguides and Resonators

An optical system, or a resonator, is called "nonorthogonal" when it is not possible to define two mutually orthogonal meridional planes of symmetry (Ref. 1, p. 240). The helical gas lens 2,3 is an example of a nonorthogonal lenslike medium. A conventional ring type cavity 2311 2312 T H E BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1070 generally ceases to be orthogonal when its path is twisted, i.e., becomes nonplanar.* Let us briefly review the major approaches in the theory of optical resonators. The field in a resonator can be expressed exactly in terms of known functions only for a few simple boundary surfaces. No exact solution is available for nonorthogonal systems. However, we are interested only in the high frequency operation of large resonators. In that limit, the waves have a tendency to follow closed curves in the resonator, either clinging to the concave parts of the boundary (whispering gallery modes 5 ) or connecting opposite points of the boundary (bouncing ball modes). One defines the axial mode number as the number of wavelengths existing along such closed curves. The nodes of the field in the transverse planes define the transverse mode numbers. More insight concerning the mode structure and the resonant frequencies can be gained by using a geometrical optics approximation, or a paraxial form of the Huygens diffraction principle. The geometrical optics approach was developed by Keller and Rubinow. 6 It consists of setting up in the resonator a manifold of rays tangent to a caustic.