Suppose my selective electro-etching process works to produce caustic surfaces. Can I make myself a business card?
I'm not sure how big a business card is. If it’s 50 mm × 30 mm and has a 50 × 30 mm reflection, presumably the focal distance needs to be short enough that the sun’s ½° size gives adequate resolution to the letters. Minimally we need 5×8 and 5 lines of 20 letters, amounting to 4000 pixels in the caustic. The reflection could potentially be larger than the card itself, but if that’s going to be significant, it needs to be curved. A spherical bead would be cool, but let’s start by thinking about flat metal plates.
That’s about 2.7 pixels per square millimeter (of the image), or about 0.6 mm per pixel. Half a degree is about 9 milliradians, so the projection distance can’t be more than about 70 mm to get that resolution. This is feasible but uncomfortably tight.
That is: the sunlight, with ½° = 9 mrad divergence, comes in and illuminates the overall-flat shiny mirror-polished metallic 50 mm × 30 mm surface, producing a 50 mm × 30 mm reflection on a surface in shadow in the same direction as the sun, 70 mm away. Because of the divergence, there’s a fuzzy border around the reflection with a radius of 0.3 mm. Tiny variations from flatness across the mirror result in major variations in brightness across the reflection, in particular bright spots from caustics, which also have a fuzziness radius of 0.3 mm. These bright spots spell out my name, email address, and so on. They can reasonably be spaced some 0.6 mm apart, giving a resolution of some 83×50 “pixels”, which is 16 5×8 letters on each of 6 lines, a total of 96 character positions. This can be improved somewhat with subpixel positioning and proportional fonts but not a whole lot.
The bright spots reach their maximum brightness when the corresponding 0.6-mm-diameter spot on the mirror is concave parabolic with a spherical radius of curvature of 140 mm. This means that the center is depressed from the edge by 140 mm - √((140 mm)² - (0.3 mm)²) = 320 nm, about half a wavelength of light.
This is small enough that the geometrical optics approximation may not apply and we may have to consider diffraction. (We can perhaps improve the situation there a bit with a blue coating.) The usual Airy limit is sin θ = 1.220λ/D for a circular aperture; in this case our λ is about 555 nm and our D is about 600 μm, so this works out to sin θ = 0.0011, which is to say that our first diffraction null is about 1.1 milliradians in radius, 2.2 milliradians in diameter. This is substantially smaller than the 9 milliradians of the sun’s disk, so the diffraction effect is significant, but not dominant; we’re in good shape.