Optigami

In the design of optical systems that incorporate multiple reflections of converging and diverging beams, tedious graphic layout procedures are followed to assure that mirror dimensions are correct and that structural elements do not interfere with the ray bundles. A change in dimension, location, or angle of any one of the system elements requires a complete redrawing of the system, making this a time-consuming and costly process.

An example of a simple optigami simulation.

Thus began Jon Myer, a scientist at the Hughes Research Laboratories, in describing a technique for optical layout using origami. Nowadays, the physical layout of mirrors and lenses is done automatically by computer programs that analyze the optics by ray-tracing through the system of lenses and mirrors. In an earlier time, this analysis was carried out by hand. In complex optical systems, rays of light are bent, refracted, focused, and folded through contorted pathways, and ensuring that no bundle of light is inadvertently clipped off can be a nontrivial feat. In the days before personal computers were on every desktop, such ray-tracing was an elaborate paper-and-pencil exercise.

Myer recognized that the origami reverse fold, applied to a narrow triangle of paper, accurately reproduced several important features of optical ray-tracing. By folding a narrow triangle of paper and reverse-folding it at each reflection, one could simulate the contortions of the beam of light as it traversed the system with a concrete object for visualization. As a simulation tool, the reverse fold embodies several features that accurate simulate the tracing of rays of light:

• The reversal of color that accompanies a reverse-fold simulates image reversal that occurs at each reflection off of a mirror;
• The angle of the reverse fold automatically gives the proper angle for orienting a mirror at the point of reflection;
• By cutting and “feathering” the edges of the triangle, not only can focusing be simulated, but the rough curvature of the focusing mirror is also obtained.

A further advantage of the optigami technique is that one can alter the angles and locations of mirrors without altering the total optical path length—an advantage in many optical design problems.

Although simple and elegant, optigami was ultimately superseded by the rapid growth and availability of personal computers and computerized ray-tracing programs. Nowadays, for less than $1000, a computerized ray-tracer can simulate reflections refractions, and vignetting from flat and curved mirrors with considerably greater speed and accuracy than origami approximations. Nevertheless, as an aid to visualization, or for quick-and-dirty calculations — such as the proverbial back-of-the-napkin sketch — optigami still has its place in the optical designer’s arsenal of tools.

An example of the latter usage may be found in a type of laser described by US Patent 6,542,529, titled, appropriately enough, a folded-cavity laser. This family of lasers was developed by Swedish engineer Mats Hagberg and myself, working at laser company SDL, Inc. The folded-cavity laser addresses a fundamental problem in a type of laser called an index-guided broad area semiconductor laser — a laser microchip used in thermal printing and green laser pointers, among other places. The beam coming out of these lasers tends to spread across a broad range of angles. Mats showed that by taking a conventional laser and effectively folding the cavity in two places, it was possible to collapse the broad output beam down to a much narrower, much brighter beam, without losing efficiency.

Schematic diagram of the folded-cavity laser. The left and center figures show how a convention cavity is folded in two places; the right figure shows the configuration of the folded cavity.

For further information, see:

• Jon H. Myer, Optigami—A Tool for Optical System Design, Applied Optics, vol. 8, no. 2, p. 260, 1969.
• Mats Hagberg and Robert J. Lang, US Patent 6,542,529, Folded-Cavity, Broad Area Laser Source, April 1, 2003.

For another interesting take on the “optigami” concept, I recently learned of a project with that very name! Bart Millikan is an inventor who has developed folded reflectors for collimation and illumination, which he calls “optigami” on his website, optigami-design.com (no longer active, archived version here). Bart has also very kindly provided this downloadable pattern for a collimating reflector, if you want to give it a try.

And if you’d like to try designing your own foldable reflector and have the program Mathematica, I’ve written a design-your-own rotationally symmetric folded figure as a Mathematica Demonstration project; feel free to give it a whirl!