The aim of the present work is to obtain an integral representation of the field associated with the refraction of a plane wave by an arbitrary surface. To this end, in the first part we consider two optical media with refraction indexes n and n separated by an arbitrary interface, and we show that the optical path length, ϕ, associated with the evolution of the plane wave is a complete integral of the eikonal equation in the optical medium with refraction index n. Then by using the k function procedure introduced by Stavroudis, we define a new complete integral, S, of the eikonal equation. We remark that both complete integrals in general do not provide the same information; however, they give the geometrical wavefronts, light rays, and the caustic associated with the refraction of the plane wave. In the second part, using the Fresnel-Kirchhoff diffraction formula and the complete integral, S, we obtain an integral representation for the field associated only with the refraction phenomena, the geometric field approximation, in terms of secondary plane waves and the k-function introduced by Stavroudis in solving the problem from the geometrical optics point of view. We use the general results to compute: the wavefronts, light rays, caustic, and the intensity associated with the refraction of a plane wave by an axicon and plano-spherical lenses.

In this work, we assume that in free space we have an observer, a smooth mirror, and an object placed at arbitrary positions. The aim is to obtain, within the geometrical optics approximation, an exact set of equations that gives the image position of the object registered by the observer. The general results are applied to plane and spherical mirrors, as an application of the caustic touching theorem introduced by Berry; the regions where the observer can receive zero, one, two, three, and one circle of reflected light rays are determined. Furthermore, we show that under the restricted paraxial approximation, that is, when sin ⁡ ψ ≈ ψ and cos ⁡ ψ ≈ 1 , the exact set of equations provides the well-known mirror equation.

The aim of the present work is to introduce two monochromatic solutions
to the scalar wave equation in free space, characterized by a caustic with a singularity
of the hyperbolic umbilical type. The first solution, is a superposition of half-
Mathieu beams, and the second one, is a superposition of parabolic beams. Since
these solutions are determined by two particular complete integrals of the eikonal
equation in free space, we compute their geometrical wavefronts, the caustic regions,
and the corresponding Poynting vectors. Finally, we remark that, under certain
conditions, these solutions describe three-dimensional accelerating beams in free space,
propagating along semielliptical and parabolic paths, respectively