An Elementary Theory for Beam Waveguides

An elementary theory for beam waveguides based upon geometrical optics as well as upon the HUYGENS undulatory theory is developed. A homogeneous FREDHOLM integral equation of the first kind is derived. It connects the field distribution as observed on a mathematical surface behind the n-th lens with the given initial distribution. If the spacing between the lenses is chosen such that the beam waveguide operates in a stable condition, then the eigendistributions are periodically reproduced along the beam waveguide. For these eigenmodes a homogeneous FREDHOLM integral equation of the second kind is formulated; the diffraction losses associated with the p<7-th eigenmode are determined by the corresponding eigenvalue. As pointed out by HEFFNER. the laser beam may be considered a special superposition of eigenmodes uniformly displaced in the direction of propagation. This superposition yields mode patterns which are periodically reproduced from lens to lens. It is suggested that the laser resonator be matched with its optically analogous beam waveguide for an optimum transmission. The general theory is applied to a beam waveguide whose lenses are a distance of d — 3 / apart. It is demonstrated that the theory presented is completely consistent with the classical laws in the zero-wavelength limit.