A group of students at EPFL have started a company called Rayform to make objects with customized caustics (both reflection and refraction), using materials such as PMMA, aluminum, and glass to form their “caustic generators”. This turns out to be a centuries-old Japanese art form known as “magic mirrors”; the originals were copper mirrors hand-scraped to produce customized caustics under the influence of copper’s elasticity and a varying material thickness, legendarily so that underground Japanese Christians could escape religious persecution; the principle by which these “diaphanous mirrors” or “makkyo” worked was not understood until the 1960s, although they had been manufactured in Japan and also China for thousands of years. This prompted a couple of thoughts from me tonight. First, what about sunlight automicroscopy? Second, how about cheaper fabrication technology?
See also files Gauzy shit and Caustic simulation.
Sunlight reflecting off a convex surface projects a magnified image of whatever colors or patterns are on the surface; this is easily seen with, for example, a Red Bull can. Sunlight is about 100 kilolux, while sunlight shadows can vary but are typically around 3–10 kilolux. So even if the reflected light is spread over a 20× larger surface than the reflective convex object, it still has brightness comparable to the ambient light. And the projected image is substantially larger than the patch on the surface that it is projected from — in the aluminum-can case, all the spread is in a single dimension, so it can be on the order of 20× larger, while the correspondingly bright reflection from a sphere would have a linear magnification factor of only 4–5 before becoming undesirably dim.
However, the caustic-shaping technique can potentially rescue the method — in the geometrical-optics approximation with point-source illumination, it can focus the light from an arbitrary area onto a point or line of zero area, thus achieving infinitely bright illumination. Rayform’s demo videos seem to show focusing of more than one order of magnitude.
Point-source light reflecting from a surface whose normal varies over some angle θ will in turn vary over the angle 2θ. The sun subtends about half a degree, so the reflection from a curved surface patch will subtend about half a degree more than the curved surface, which blurs the projected image somewhat. However, this is a limit on the angular resolution of the microscopy method, not its spatial resolution. And narrowing the sunbeam by passing it through a pinhole that subtends less than half a degree from the point of view of the generator.
The nonzero angular size of the sun also provides the limit on the brightness increase available by focusing: the projected focus spot will have, at minimum, the same angular size as the sun, as viewed from the point on the generator that is generating it.
Another limit on this technique is the diffraction limit: as the concave facet producing the focused spot becomes smaller, the produced beam has a larger divergence. I think that roughly to achieve half a degree divergence — the best you can do with a half-degree-wide sun light source — you need a facet of diameter roughly λ/(½sin(½°)). This works out to about 126 μm for 550-nm light. This is a spatial resolution limit.
The facet can be a concave paraboloid section, in which case it produces a point caustic, but if it is less concave in one direction, it will spread out its light to produce a line caustic subtending some arbitrary angle at some arbitrary rotation.
Setting the spatial and angular resolution limits equal, maybe we would like 126 μm to subtend about ¼° of the curve of the convex surface, which gives us a radius of about 29 mm. Spheres or cylinders with a diameter smaller than 58 mm will have an unnecessarily coarse angular automicroscopy resolution limit, larger than ¼°, imposed by diffraction of light from their facets; those with a larger diameter will have an unnecessarily coarse spatial automicroscopy resolution limit, larger than 126 μm, imposed by the apparent size of the sun.
If we want these 126 μm facets to project pixels at about 72dpi — 350 μm, a lower limit for comfortably readable text, although older computer terminals and printers used a slightly coarser 8 vertical pixels per 6lpi line, giving 530 μm — then we want 350 μm to subtend ½° as seen from the surface of the mirror. This gives a projection distance of 40 mm, which seems rather small to me, so maybe 100 mm would be better, which gives blurry 870-μm projections. Since each facet can project an arbitrarily oriented line, rather than just a point, you only need about, say, 5 of them per letter. This means our 29-mm-radius shiny sphere with its 10500-mm² surface area, holding about 850 000 facets of 0.0124 mm² each, can project about 170 000 letters, about 40 or 50 pages’ worth of text.
This may not be reasonable — the pixels may be too crowded together. Consider that if the facets are all just directly pointed away from the center, the spots they project will be ½° apart as seen from the surface — which is to say, they will all kind of blur together, unless you stack some of them on top of each other, which is of course what Rayform does.
How bright are they by default? Consider a mm² in the center of the sunbeam, which is reflecting its light onto a screen in shadow positioned a negligible angle to the side of it. This mm² subtends 34 milliradians (1.98°) and so its projection will subtend 78 mrad (4.0°), which means that at 100 mm it covers an area 6.9 mm × 6.9 mm, which is 48 mm². So it will be 48× dimmer than the direct sunlight: 2100 lux, visible on a 10 kilolux shadow background but far from overwhelming. But areas illuminated by several times this 126-micron-wide minimum will be considerably brighter than the shadow.
How much spatial precision do we need to make the surface reflect like this? Suppose we’re willing to tolerate ¼λ deviations. Well, at 555 nm, that works out to 139 nm (≈ 5.5 micro-inches). This is a relative radius error of 4.8 parts per million. Regular ABMA bearing balls of grade 100 have a surface finish smoothness of 5.0 micro-inches (127 nm), but to get roundness of 5 micro-inches, you have to go down to grade 5. That isn’t even the lowest grade, but grade-5 bearing balls don’t come as large as 58 mm; they only go up to 2 inches, which is 50.8 mm.
Turning a spherical surface with a radius of 29 mm into a 127-micron-wide paraboloidal facet with its focus at 100 mm requires changing the curvature radius from +29 mm to -200 mm. At +29 mm, the middle of the facet would be 69 nanometers proud of planar; at -200 mm, it would be 10 nm below. This seems like I must have some kind of calculation error, since it seems inconceivable for such a small difference to produce a precise focus.
First off, what about using sugar glass instead of PMMA or soda-lime glass for the refractive pieces? Instead of polishing it with rouge, you could polish it with water.
Second, how about using electropolishing to remove tiny, precisely controlled amounts of metal, leaving a smoothly varying surface, while leaving a mirror finish? Electroplating at around 1000 A/m² deposits chromium at something like 100 nm/minute, so it seems like thickness control down to the level of less than a monolayer should be feasible. This is also potentially useful for making large mirrors out of invar or similar, then aluminizing or silvering them.