Nonparaxial accelerating waves as a superposition of nondiffracting Bessel-lattice optical fields.

In the first part of this work, we introduce a monochromatic solution to the scalar wave equation in free space, defined by a superposition of monochromatic nondiffracting half Bessel-lattice optical fields, which is determined by two scalar functions; one is defined on frequency space, and the other is a complete integral to the eikonal equation in free space. We obtain expressions for the geometrical wavefronts, the caustic region, and the Poynting vector. We highlight that this solution is stable under small perturbations because it is characterized by a caustic of the hyperbolic umbilical type.

Exact mirror equation via Berry's caustic touching theorem: plane and spherical mirrors.

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.

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.

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.

Paraxial and tightly focused behaviour of the double ring perfect optical vortex.

In this paper we compare the intensity distributions in the paraxial and tightly focused regimes corresponding to a double ring perfect optical vortex (DR-POV). Using the scalar diffraction theory and the Richards-Wolf formalism, the fields in the back focal plane of a low and high (tight focusing) NA lens are calculated. In the paraxial case we experimentally observed a DR-POV whose rings enclose a dark zone thanks to the destructive interference introduced by a π phase shift.

The vector Durnin-Whitney beam.

We show that $(\textbf{E},\textbf{H})=({\textbf{E}_0},{\textbf{H}_0}){e^{i[{k_0}S(\textbf{r})-\omega t]}}$(E,H)=(E,H)e is an exact solution to the Maxwell equations in free space if and only if $\{{\textbf{E}_0},{\textbf{H}_0},\nabla S\}${E,H,∇S} form a mutually perpendicular, right-handed set and $S(\textbf{r})$S(r) is a solution to both the eikonal and Laplace equations. By using a family of solutions to both the eikonal and Laplace equations and the superposition principle, we define new solutions to the Maxwell equations. We show that the vector Durnin beams are particular examples of this type of construction.

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.

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)=ce(ϕ,q)+ise(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)=ce(ϕ,q) and A(ϕ)=se(ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable.

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 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.

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 H, H, and H 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.

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).

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.

Wavefronts, caustics, and ronchigrams of a spherical wave reflected by a spherical mirror.

The aim of the present work is twofold: first we obtain analytical expressions for both the wavefronts and the caustic associated with the light rays reflected by a spherical mirror after being emitted by a point light source located at an arbitrary position in free space, and second, we describe, in detail, the structure of the ronchigrams when the grating or Ronchi ruling is placed at different relative positions to the caustic region and the point light source is located on and off the optical axis. We find that, in general, the caustic has two branches: one is a segment of a line, and the other is a two-dimensional surface. The wavefronts, at the caustic region, have self intersections and singularities.

The point-characteristic function, wavefronts, and caustic of a spherical wave refracted by an arbitrary smooth surface.

The aim of this paper is to obtain expressions for the k-function, the wavefront train, and the caustic associated with the light rays refracted by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in a three-dimensional homogeneous optical medium. The general results are applied to a parabolic refracting surface. For this case, we find that when the point light source is off the optical axis, the caustic locally has singularities of the hyperbolic umbilic type, while the refracted wavefront, at the caustic region, locally has singularities of the cusp ridge and swallowtail types.

Describing the structure of ronchigrams when the grating is placed at the caustic region: The parabolical mirror.

In this work we use the geometrical point of view of the Ronchi test and the caustic-touching theorem to describe the structure of the ronchigrams for a parabolical mirror when the point light source is on and off the optical axis and the grating is placed at the caustic associated with the reflected light rays. We find that for a given position of the point light source the structure of the ronchigram is determined by the form of the caustic and the relative position between the grating and the caustic. We remark that the closed loop fringes commonly observed in the ronchigrams appear when the grating and the caustic are tangent to each other.

Wavefronts and caustic of a spherical wave reflected by an arbitrary smooth surface.

The aim of the present work is to obtain expressions for both the wavefront train and the caustic associated with the light rays reflected by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in free space. To this end, we obtain an expression for the k-function associated with the general integral of Stavroudis to the eikonal equation that describes the evolution of the reflected light rays. The caustic is computed by using the definitions of the critical and caustic sets of the map that describes the evolution of an arbitrary wavefront associated with the general integral.

Exact computation of image disruption under reflection on a smooth surface and Ronchigrams.

We use geometrical optics and the caustic-touching theorem to study, in an exact way, the change in the topology of the image of an object obtained by reflections on an arbitrary smooth surface. Since the procedure that we use to compute the image is exactly the same as that used to simulate the ideal patterns, referred to as Ronchigrams, in the Ronchi test used to test mirrors, we remark that the closed loop fringes commonly observed in the Ronchigrams when the grating, referred to as a Ronchi ruling, is located at the caustic place are due to a disruption of fringes, or, more correctly, as disruption of shadows corresponding to the ruling bands. To illustrate our results, we assume that the reflecting surface is a spherical mirror and we consider two kinds of objects: circles and line segments.

Computation of the disk of least confusion for conic mirrors.

We use geometrical optics to compute, in an exact way and by using the third-order approximation, the disk of least confusion (DLC) or the best image produced by a conic reflector when the point source is located at any position on the optical axis. In the approximate case we obtain analytical formulas to compute the DLC. Furthermore, we apply our equations to particular examples to compare the exact and approximate results.