The two variational principles, the Maupertuis' and the Hamilton's principle, are discussed in conjunction with the Fermat's principle. These two principles are shown to describe two different aspects of waves, thus resulting in the different geometry of wave propagation, the treatment of which is thus called the stationary optics or the dynamical optics, respectively. Comparisons for the results obtained from these geometrical optics are given. Another new variational principle valid for the dynamical waves reflected/refracted at the inter-faces, which has not yet been discovered so far, is also derived.

We describe a geometrical optics approach for the analysis of dielectric tapered waveguides. The method is based on the ray-optical treatment for wave-normal rays defined newly to waves of light in open structures. Geometrical optics fields are represented in terms of two kinds of wave-normal rays: leaky rays and guided rays. Since the behavior of these rays is different in the two regions separated at critical incidence, the geometrical optics fields have certain classes of discontinuity in a transition region between leaky and guided regions. Guided wave solutions are given as a superposition of guided rays that zigzag along the guides, all of which are totally reflected upon the interfaces. By including some leaky rays adjacent to the guided rays, we obtain more accurate guided wave solutions. Calculated results are in excellent agreement with wave optics solutions.