Theory of unstable laser modes: edge waves and fractality

Abstract

For large Fresnel numbers N, unstable laser modes are highly irregular and resemble fractals. To explore this, we derive an explicit formula for the lowest-loss mode of a one-dimensional laser (i.e. where the cavity is two dimensional) in terms of edge-diffracted waves, and demonstrate its accuracy for large N. Between the size a of the mirror (outer scale), and the inner scale a/ N, there is no distinguished scale, and the graph of mode intensity has a fractal dimension close to 2. Near the inner scale, the scaling is scale dependent, and the crossover is described by an explicit formula for a `local fractal dimension' D( K), describing the mode on scales near Δx= a/(2 πNK). As K increases through the inner scale K=1, D( K) decreases from 2 when K≪1 to 1 when K≫1 (reflecting the smoothness of the mode on fine scales).