RESUMEN
A new, to the best of our knowledge, method for designing a thick-lens achromatic doublet based on the concept of a caustic surface to correct both third- and fifth-order spherical aberration is presented. We consider two different wavelengths brought into coincidence at the back focal length instead of the effective focal length as it is usually done, to calculate the radii of curvature assuming predefined values for axial thicknesses and their indices of refraction for both lenses. Alternatively, we apply Taylor's series around the optical axis, and to vanish the approximate caustic surface, we obtain the values for the conic constants, which reduce at third- and fifth-order spherical aberration. Two designs for cemented doublets are provided assuming that the lenses are cemented. Finally, we propose a method to qualitatively test doublet lenses by using null screens, considering to place the detection plane at arbitrary positions.
RESUMEN
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.
RESUMEN
We have implemented an exact ray trace through a plano-freeform surface for an incident plane wavefront. We obtain two caustic surfaces and provide the critical points related to the ray tracing process. Additionally, we study the propagation of the refracted wavefronts through the plane-curved surface. Finally, by using the Ronchi-Hartmann type null screen and placing the detection plane within the caustic region, we have evaluated the shape of a plano-freeform optical surface under test, obtaining an RMS difference in sagitta value of 6.3 µm.
RESUMEN
We study the formation of caustic and wavefront surfaces produced by a tilted plane wavefront propagating through spherical positive lenses. The shape of the caustic surface is a function of the indices of refraction, the geometrical parameters of the lens involved in the process of refraction, and the obliquity angle with respect to the optical axis, as we expect. We provide exact and approximate analytic equations for tangential and sagittal focal surfaces and also for Petzval field curvature considering arbitrary lenses.
RESUMEN
We study the formation of caustic surfaces produced by conic lenses, considering a plane wavefront propagating parallel to the optical axis. The shape of the caustic can be modified by changing the parameters of the lens in such a way that if we are able to vanish the caustic, the optical system produces the sharpest diffraction-limited images. Alternatively, caustic surfaces with a large area can be applied to the design of non-imaging optical systems, with potential applications such as diffusers of light for illumination or solar concentrators. Here, we provide analytic equations for the conic constants, principal surfaces, and caustic surfaces, and also approximations at the third and fifth orders formed by conic lenses, in order to reduce the spherical aberration at these orders.
RESUMEN
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.
RESUMEN
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 ronchigrams exhibit closed-loop fringes when the grating is placed at the caustic region.
RESUMEN
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.
RESUMEN
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.
RESUMEN
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. It is shown that the expression for the caustic is the same as that--reported in the literature--obtained by using an exact ray tracing. The general results are applied to a parabolic mirror. 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 reflected wavefront at the caustic region locally has singularities of the cusp ridge and swallowtail types.
RESUMEN
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.