Here are a few of the fractals we have come across in our research.

Sierpinski-triangle laser mode. One way to create fractals is to use a Multiple Reduction Copy Machine (MRCM), which creates multiple, reduced-size, copies of an image. When iterated, a fractal pattern results. We investigated the laser equivalent of such a MRCM, which we called Multiple reduction imaging lasers. The picture shows the (coloured-in) intensity cross-section through the eigenmode of a laser resonator designed to shape light into a diffraction-limited Sierpinski triangle.

• J. Courtial, Fractal multiple reduction imaging lasers, Opt. Commun. 174, 235-241 (2000)

Weierstrass-Mandelbrot laser mode. Around 1998, G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman discovered that all resonators in a specific, but common, class of laser resonators (namely unstable canonical resonators) have a mechanism built in to shape light circulating in the resonator into patterns with fractal statistics. We discovered that, in one particular plane, a mechanism is at work (which we called the monitor-outside-a-monitor effect) that gives the light not only statistical properties of fractals, but the shape of classic fractals (see below). The picture shows an intensity cross-section (intensity vs position) through a specific two-dimensional resonator (a "strip resonator") in this special plane. This curve is very closely related to the Weierstrass function, the first example of a curve that could not be differentiated, which makes it the granddaddy of all fractals.

• J. Courtial and M. J. Padgett, Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators, Phys. Rev. Lett. 85, 5320-5323 (2000)

Fractal video-feedback spiral. Investigating the monitor-outside-a-monitor effect we discovered in lasers (see above) in an analogous video-feedback system, we came across this fractal spiral reminiscent of the spirals in the Mandelbrot set.

• Visit the interactive page on our fractal video-feedback research at http://www.physics.gla.ac.uk/Optics/play/fractalVideoFeedback/research.html
• J. Courtial, J. Leach, M. J. Padgett, Image processing - Fractals in pixellated video feedback, Nature 414, 864 (2001)
• J. Leach, M. J. Padgett and J. Courtial, Fractals in pixellated video feedback, Contemp. Phys. 44, 137-143 (2003)

Koch snowflake laser mode. Our video-feedback experiment (above) provided the inspiration to go back to laser resonators and search for other classic fractals in the light. The picture shows a (coloured-in) intensity cross-section through the eigenmode of a (this time three-dimensional) resonator selected to shape light at the centre into a Koch snowflake (red).

• C. M. G. Watterson, M. J. Padgett, and J. Courtial, Classic-fractal eigenmodes of unstable canonical resonators, Opt. Commun. 223, 17-23 (2003)