Wave front equation, caustics, and wave aberration function of simple lenses and mirrors

Proof that spot diagram borders are always caustic curves and/or marginal rays II: annular (circular and elliptical) and single hexagonal exit pupils

In a previous paper [Appl. Opt. 61, C20 (2022)] it was proven that for a circular exit pupil and any optical path differences, the border of any spot diagram is integrated by the caustic surface and/or marginal rays. In this paper, the previous results are extended to annular (circular an elliptical) as well as hexagonal (single and segmented) exit pupils. Several examples of wavefronts affected by linear combinations of orthonormal Zernike aberrations are shown.

Borders of spot diagrams are always caustic surfaces and/or marginal rays

In this paper, it is proven that for any optical system with a circular exit pupil and wavefronts affected by any aberrations, the borders of all leaving rays are caustic surfaces and/or marginal rays. Several examples are shown for wavefronts affected by linear combinations of Zernike aberrations.

Caustics of the vortex beams generated by vortex lenses and vortex axicons

In this work, the propagation of vortex beams is treated using a catastrophe theory approach. Analytic expressions are deduced to describe caustic surfaces produced by vortex lenses and vortex axicons. The obtained analytics allow us to explain the formation of the shadow region along the optical axis for vortex beams using geometric optics (previously, the zero axial intensity was explained just by diffraction effects). Thus, the presence of a vortex eikonal leads to a fundamental change in the type of axial caustic. Another important distinction of the caustics produced by vortex beams from those produced by nonvortex radial beams has been shown to consist in wavelength-dependence. The results of numerical simulation show that the propagation operator defined using a geometrical optics approximation agrees well with the numerical simulation results obtained using a nonparaxial diffraction operator based on the conical wave expansion.

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.

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.

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. Furthermore, we find that the caustic locally has singularities of the purse or hyperbolic umbilic type, and the ronchigram obtained when the grating is located at certain specific positions at the caustic locally is of the serpentine type. The main motivation of this work is that nowadays a quantitative analysis of the Ronchi test is applied only when the grating is outside the caustic, and we claim that by working at the caustic, the sensitivity of the Ronchi test will be improved. Therefore, a clear understanding of the properties of the ronchigrams when the grating is placed at the caustic will be needed to extend the Ronchi test to that region.