Refracting and reflecting interfaces transforming a given wavefront into another one.

The aim of this work is threefold. First, following Luneburg and using our own notation, we review the Cartesian ovals. Second, we obtain analytical expressions for the reflecting and refracting surfaces that transform a prescribed smooth two-dimensional wavefront into a spherical one. These results are applied to show that the reflecting surface that connects a plane wavefront to a spherical one is a parabolical surface, and we design a lens, with two freeform surfaces, that transforms a spherical wavefront into another spherical one. These examples show that our equations provide the well-known solution for these problems, which is given by the Cartesian ovals method. Third, we present a procedure to obtain exact expressions for the refracting and reflecting surfaces that connect two given arbitrary wavefronts; that is, by assuming that the optical path length between two points on the prescribed wavefronts is given by the designer the refracting and reflecting surfaces we are looking for are determined by a set of two algebraic equations, which in the general case have to be solved in a numerical way. These general results are applied to compute the analytical expressions for the reflecting and refracting surfaces that transform a parabolical initial wavefront into a plane one.

Classical characterization of quantum waves: comparison between the caustic and the zeros of the Madelung-Bohm potential.

From a geometric perspective, the caustic is the most classical description of a wave function since its evolution is governed by the Hamilton-Jacobi equation. On the other hand, according to the Madelung-de Broglie-Bohm equations, the most classical description of a solution to the Schrödinger equation is given by the zeros of the Madelung-Bohm potential. In this work, we compare these descriptions, and, by analyzing how the rays are organized over the caustic, we find that the wave functions with fold caustic are the most classical beams because the zeros of the Madelung-Bohm potential coincide with the caustic. For another type of beam, the Madelung-Bohm potential is in general distinct to zero over the caustic. We have verified these results for the one-dimensional Airy and Pearcey beams, which, according to the catastrophe theory, have stable caustics. Similarly, we introduce the optical Madelung-Bohm potential, and we show that if the optical beam has a caustic of the fold type, then its zeros coincide with the caustic. We have verified this fact for the Bessel beams of nonzero order. Finally, we remark that for certain cases, the zeros of the Madelung-Bohm potential are linked with the superoscillation phenomenon.

Paraxial optical fields whose intensity pattern skeletons are stable caustics.

We construct exact solutions to the paraxial wave equation in free space characterized by stable caustics. First, we show that any solution of the paraxial wave equation can be written as the superposition of plane waves determined by both the Hamilton-Jacobi and Laplace equations in free space. Then using the five elementary stable catastrophes, we construct solutions of the Hamilton-Jacobi and Laplace equations, and the corresponding exact solutions of the paraxial wave equation. Therefore, the evolution of the intensity patterns is governed by the paraxial wave equation and that of the corresponding caustic by the Hamilton-Jacobi equation.

Wavefronts and caustics associated with Mathieu beams.

In this work we compute the wavefronts and the caustics associated with the solutions to the scalar wave equation introduced by Durnin in elliptical cylindrical coordinates generated by the function A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q), with ν being an integral or nonintegral number. We show that the wavefronts and the caustic are invariant under translations along the direction of evolution of the beam. We remark that the wavefronts of the separable Mathieu beams generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q) is stable because locally its caustic has singularities of the fold and cusp types. To show this property, we present the wavefronts and the caustics for the Mathieu beams with characteristic value aν=0 and q=0,0.2,0.3,0.5. For q=0, we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the z axis. For q≠0, the wavefronts are deformations of conical ones, and the caustic surface, for some values of q, has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value aν=0 and 0≤q<1 has associated a caustic with singularities of the swallowtail type, which is structurally stable. Therefore, we conclude that this type of Mathieu beam is more stable than plane waves, Bessel beams, parabolic beams, and those generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q). To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order m=5, and the Mathieu beam of order m=5 and q=50 with complex transversal amplitude given by Ce5(ξ,50)ce5(η,50)+iSe5(ξ,50)se5(η,50), where (ξ, η) are the elliptical coordinates on the plane.

Wavefronts, caustic, and intensity of a plane wave refracted by an arbitrary surface: the axicon and the plano spherical lenses.

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 n1 and n2 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 n2. 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.

Curvatures of the refracted wavefronts and Ronchigrams for a plano arbitrary lens.

In this work we obtain the equations for curvatures of refracted wavefronts for a plano arbitrary lens. The functions H0, H1, and H2 that determine the caustic also determine the curvature of these wavefronts. The analysis performed in these calculations allows us to study the behavior of the Ronchigrams for the case of plane incident wavefronts. We apply this procedure for a plano-spherical lens, and we discover that it is possible to describe the behavior of the Ronchigrams based on the τ function, which labels the refracted wavefronts of the optical system.

Internal structure of an optical null Ronchi grating test for a plano-parabolic lens.

In this work we use geometrical optics and the caustic touching theorem, introduced by Berry, to describe the internal structure of the null Ronchi grating for a plano-parabolic lens illuminated by a point light source placed on the optical axis. The aim of this work is to explain the role of the caustic region in the process of morphology change between image and object in computing the null Ronchi grating. To this end, we obtain the analytic expression of the null Ronchi grating, and after that we deeply study the change in morphology between a single straight fringe image at the Ronchigram and the multiple curve rulings that can generate it (one open and one closed). We analyze exactly how multiple rulings generate the same straight image fringe, or how an entire ruling collapses into a single point image. For this analysis, we take different observation planes at different positions with respect to the caustic region. Finally, we characterize this topological change as one of two possible kinds depending on the relative position between the observation plane and the caustic region.

Wavefronts, caustic, ronchigram, and null ronchigrating of a plane wave refracted by an axicon lens.

The aim of this work is threefold: first we obtain analytical expressions for the wavefront train and the caustic associated with the refraction of a plane wavefront by an axicon lens, second we describe the structure of the ronchigram when the ronchiruling is placed at the flat surface of the axicon and the screen is placed at different relative positions to the caustic region, and third we describe in detail the structure of the null ronchigrating for this system; that is, we obtain the grating such that when it is placed at the flat surface of the axicon its associated pattern, at a given plane perpendicular to the optical axis, is a set of parallel fringes. We find that the caustic has only one branch, which is a segment of a line along the optical axis; the ronchigram exhibits self-intersecting fringes when the screen is placed at the caustic region, and the null ronchigrating exhibits closed loop rulings if we want to obtain its associated pattern at the caustic region.