A lumographic ("light-drawing") lens makes a picture by concentrating and diverging light rays to make bright and dark patches in the image. It produces the same hypnotic play of light that you see under moving water on a sunny day, and the same way -- an undulating surface bends light rays -- but with quite different results:
The trick is to find a lens shape that rearranges light rays just right to make a desired picture. More than one lens shape will work, so I find the smoothest one, which will be the easiest to fabricate. The result is a little like a funhouse mirror:
Speaking of which, one can also make lumographic mirrors:
As well as internally focusing lenses:
And multiple-refraction lenses:
With very high-end machining, detail and contrast can be photographic-quality:
This project got started at the beach while musing on the patterns of light in the water. How could that be harnessed to make a picture? I ultimately got an answer by setting up lens design as a problem in Optimal Mass Transport. That's the study of rearrangements; I wanted to rearrange light rays. OMT has an excellent backstory: Lore has it that it was originally inspired by the Napoleons' propensity for generating extensive piles of rubble, whose removal/recycling was an urgent problem. The 18-19th century geometer Monge framed it thus: Given two piles of dirt, how to change one into the other by moving the fewest shovelfuls? Two world wars later, Kantorovich, faced with Stalin's even more formidable rubble-making habit, developed the modern formulation: What is the simplest map between two continuous distributions of stuff? Solving this problem is an active area of research today. To get a solution for picture-forming lenses, I recast Kantorovich's problem so that the map (of light rays) between source and target is constrained be a (nonlinear) function of the lens surface. With a little mathematical nudging, this reformulation yielded an algorithmic one-liner that tells me, in a matter of seconds, how to shape a lens to make any target image.
Physical fabrication requires ~50 nanometer-accurate machining, but such equipment was military-grade when I first went looking and is still basically out of reach for artists. In recent years — and with some useful tips from the fellows at the local body shop — I've been able to coax reasonable results from tools and machinery made for the automotive industry.
See another invention, sheet-metal holography: knots nature humans surfaces motion 720 etc .
© 2008-2010 Matt Brand. All rights reserved. Trademark & patents pending.