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

Abstract

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.

Geometric treatment of the radiation captured by the Large Millimeter Telescope (LMT/GTM) and ronchigrams at the caustic region for a static source

“…We find that in general, the caustic is a two-dimensional surface with a stable singularity of the hyperbolic umbilic type [5]. The present work can be considered a natural extension of the work developed in [6,7]. It is important to remark that the results we present here can be used for any parabolical mirror.…”

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

“…The fact that one branch of the caustic in general is a segment of a line is an intrinsic property associated with a spherical mirror. If the spherical mirror is deformed, for example, to a parabolical mirror, then when the point light source is off the optical axis the two branches of the caustic are two-dimensional surfaces that locally have stable singularities of well-known types [15].…”

“…The general results were applied to a spherical mirror when the point light source is located on the optical axis. In this work we use our general equations obtained in [12,14,15] to study the structure of the reflected wavefronts, caustics, and ronchigrams for a spherical mirror when the point light source is located at an arbitrary position of space. More explicitly, we obtain analytical expressions for the reflected wavefronts and the caustic when the point light source is located on and off the optical axis.…”

Towards the Ronchi test for gravitational lenses: the gravitoronchigram

“…In practice, it is common to avoid any closed loop fringe, which is possible only when the grating is located outside the caustic region associated with the refracted light rays [4]. In a series of works [7][8][9][10], we have remarked that the existence of the closed loops can be explained by using the caustic-touching theorem introduced by Berry [11], which gives the conditions to obtain multiple images associated with a single extended source. Recently, we have realized that not only the caustic plays a major role in the description of the ronchigram but also the interference region associated with the refracted light rays [12].…”