Fractal optics and beyond

Fractals, shapes comprised of self-similar parts, are not merely prescribed linear structures. A wide class of fractals can also arise from the rich dynamics inherent to nonlinear optics. I n 1967, Benoit Mandelbrot published a paper that gave birth to the study of fractals, entitled " How long is the coast of Britain? Statistical self similarity and fractional dimension " 1. According to Mandelbrot, " a fractal is a shape made of parts similar to the whole in some way " 2. One particularly spectacular example of a fractal in nature is the Romanescu broccoli (Fig. 1). Although there are a number of different fractal classification systems, one stands out rather distinctly: exact (regular) fractals versus statistical (random) fractals. An exact fractal is " an object which appears self-similar under varying degrees of magnification, in effect, possessing symmetry across scale, with each small part replicating the structure of the whole ". Taken literally, when the same object replicates itself on successively smaller scales, even though the number of scales in the physical world is never infinite, we call this object an 'exact fractal'. When, on the other hand, the object replicates itself in only its statistical properties, it is defined as a 'statistical fractal'. Perhaps the best known example of an exact fractal is the Cantor set fractal, a shape that can be explained by describing its generation. Starting with a single line segment, the middle third is removed to leave behind two segments, each with a length of one-third of the original. From each of these segments, the middle third is again removed, and so on, ad infinitum. At every stage of the process, the result is self-similar to the previous stage, which is identical upon rescaling. Of course, this 'triplet set' is not the only possible Cantor set. Any arbitrary cascaded removal of portions of the line segment may form the repetitive structure of an exact fractal. Other famous examples of exact fractals are the Sierpinski triangle and the Koch snowflake. Statistical fractals have been observed in many physical systems, ranging from material structures such as polymers, aggregation and interfaces, through to biology, medicine, electric circuits, computer interconnects, galactic clusters and stock market price fluctuations 3. Exact fractals, on the other hand, seem merely to be mathematical constructs, and it is not at all apparent that they exist in nature. In 1993 Mandelbrot won the prestigious Wolf Prize …