Bubble optics: Leonardo's cross revisited-Part 3, nonparaxial analytical methods

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.