J. A. Arnaud

Bell System Technical Journal, Volume 49, Issue 9, pages 2311–2348, November 1970


The modes of propagation in optical systems which do not possess meridional planes of symmetry (nonorthogonal systems) are investigated in the case where the effect of apertures and losses can be neglected. The fundamental mode of propagation is obtained with the help of a complex ray pencil concept. An integral transformation of the field, based on a quasi-geometrical optics approximation and a first-order expansion of the point characteristic of the optical system, is given; it shows that the complex (three-dimensional) wavefront of the fundamental mode is transformed according to a generalized “ABCD law.” A simple expression is also obtained for the phase-shift experienced by the beam. The higher order modes of propagation are obtained from a power series expansion of the fundamental mode. These higher order modes are expressed, in oblique coordinates, as the product of the fundamental solution and finite series of Hermite polynomials with real arguments. In the special case of systems with rotational symmetry, these series reduce to the well-known generalized Laguerre polynomials. The theory is applicable to media such as helical gas lenses and optical waveguides suffering from slowly varying deformations in three dimensions. Nonorthogonal resonant systems are also investigated. An expression for the resonant frequencies, applicable to any three-dimensional resonator, is derived. Numerical results are given for the resonant frequencies and the resonant field of a twisted path cavity which exhibits interesting properties: the usual polarization degeneracy is lifted and the intensity pattern of all of the modes possesses a rotational symmetry.