James A. Lock

In about 1508, Leonardo da Vinci first noted and sketched a cross-shaped caustic produced by a plane wave diagonally incident on the circularly symmetric meniscus surrounding a bubble on the surface of a water-filled container. This caustic was analyzed in two companion papers using a numerical ray theory spot diagram approach, and paraxial analytical methods. In this study we employ nonparaxial analytical methods, which are necessary when the outer bubble meniscus rises high above the ambient water surface. We also use a simple exactly soluble approximation to the outer meniscus shape, and obtain a distorted astroid caustic. We show that one of the refraction astroid cusp points is blocked by an aperture effect for steeply diagonally incident light, and the missing cusp is approximately replaced by a new cusp due to light transmitted through the bubble into the water, and reflecting from the underside of the outer meniscus. (C) 2021 Optical Society of America

Around 1508, Leonardo da Vinci first noted and sketched a cross-shaped caustic produced by a plane wave diagonally incident on the meniscus surrounding a bubble on the surface of a water-filled container. This caustic was analyzed in a companion paper using a numerical ray theory spot diagram approach and is analyzed here using analytic ray theory approaches appropriate to the paraxial regime. We demonstrate that the caustic assumes the standard astroid form when the maximum meniscus slope is small, and the shape starts to distort as the maximum meniscus slope increases. We show that competition between generalized coma and generalized astigmatism, with generalized coma dominating, is responsible for the cusp points of the astroid in the plane of the incident light being reversed with respect to their images in the exit plane of the optical system. We also show that Leonardo’s cross is the result of the twofold symmetry of the astigmatic focusing of the diagonally incident plane wave by the circularly symmetric meniscus.

Small air bubbles on the rear inside surface of a water-filled cylinder, near its edges, appear horizontally elongated, joined in pairs, and take on color. Similarly, if an extended object is sufficiently close to the water-filled cylinder, three images of the object are seen when looking through the cylinder. The center image joins onto the left or right image as the observer moves his or her head back and forth in front of the cylinder. The first observation is explained in terms of glare points of light, and the real and virtual parts of the external caustic of the light transmitted through the water-filled cylinder. The second observation is explained as an example of Berry's caustic touching theorem which describes the topological method of fragmentation of an object's image into multiple images. For the situation studied here, an imaginary cylindrical aberration caustic of the water-filled cylinder decomposes object space into a three-ray region sandwiched between two one-ray regions. As an extended object crosses the caustic boundary from one of the one-ray regions into the three-ray region, an image-pair creation event occurs, which is followed by an image-pair disconnection event producing the three images. Similarly, when the extended object crosses the caustic boundary from the three-ray region into one of the one-ray regions, an image-pair merging event occurs, which is followed by an image-pair annihilation event producing the one remaining image. (C) 2020 Optical Society of America

A water drop hanging from a house siding board after a rain shower is near-normally illuminated by sunlight either shortly after sunrise or before sunset. A focusing caustic consisting of a bright V-shape or U-shape with a small bright elliptical shape immediately above it is frequently seen on the next lower siding board. In addition, there are two broad regions of illumination immediately above the caustic, fanning out to the upper left and upper right. This complicated pattern, composed of a bright V-shape or U-shape, and the bottom half of the small bright elliptical-shape immediately above it, is caused by the hyperbolic umbilic diffraction caustic near the condition of maximum focus. This can be observed because, by a stroke of good fortune, the distance between the lower edge of a siding board and the flat portion of the next siding board beneath it is nearly equal to the paraxial focal distance of the caustic. Blocking off the light incident on the top, bottom, left side, and right side of the drop was used to determine the portion of the drop responsible for different parts of the caustic. The results were found to match the predictions for the hyperbolic umbilic caustic. (C) 2020 Optical Society of America