# Discovery of Fractal Lasers

Within an Anglo-Dutch research collaboration (involving the University of Salford, Imperial College London, and the University of Leiden), we discovered that certain types of laser designs output fractal light patterns. This work concerns fractal formation in linear systems, and is quite distinct from our later studies predicting the emergence of spontaneous patterns in nonlinear systems.

Popular accounts of the fractal laser research:

- Fractal modes in unstable resonators
- Lasers that look like child’s play
- Fractals discovered in laser modes
- Lasers, kaleidoscopes and fractals

## Science background of fractal light discovery

The original aims of this work were focused on the excess noise properties of microlasers [1-5]. A scaled-up experimental prototype of a microlaser is shown in figure 1. Such small lasers benefit from optimising their light amplification by employing so-called ‘unstable cavities’. Within this type of cavity geometry, the circulating light expands to allow high overlap between the amplifying medium and the light itself – see part b) of figure 1. Since the light is repeatedly magnified inside the laser, there will be strong aperturing effects. This is because, as the light beam gets wider, some part of the laser cavity will act as an aperture on the circulating light. The role of the shape of this aperturing part is thus expected to be important for the microlaser output characteristics – see part c) of figure 1.

**Figure 1** – a) Experimental configuration of the fractal laser, b) magnification of the circulating light, c) shapes of the aperturing element.

For each shape of cavity aperturing, there was a comparison of theory and experimental results for the laser output. We were rather surprised when a detailed study of the light intensity patterns was undertaken. The columns of figure 2 show transverse laser light profiles when the aperturing element has the following shapes: triangular, rhomboid, pentagonal, hexagonal and octagonal. Looking down each column, one sees progressive development of additional small-scale details as the aspect ratio of each cavity is increased. Some of these output laser modes looked strikingly like ‘laser snowflakes’!

## Fractal laser modes

**Figure 2** – Cross-sections of laser beam profiles. Colour-coding is used to distinguish different light intensities.

Of course, real snowflakes are fractals (patterns with proportional levels of more detail when one looks closer and closer). Some further analyses of the laser patterns confirmed that these cavities did indeed result in fractal laser modes [6-19]. In fact, the underlying principle of fractal linear eigenmodes, in systems with magnification greater than one, may have much wider applications. For example, workers at the University of Glasgow were quick to spot that a similar principle could be employed to generate fractal patterns in video feedback systems, and that exact self-similar fractals were possible [20-24].

## Further developments (fractal light research)

In more recent developments, the experimental group in Leiden extended their studies to examine cavity designs that permit a very wide range of fractality – i.e. a greater extent of smaller-scale details [25,26]. At Imperial College London, further investigations have included gaining a much better understanding of the role of the cavity and the mode characteristics in determining the fractal dimension of the resulting laser light [27-30]. A key limitation in our earlier theoretical investigations was that both semi-analytical and full-numerical modelling had been limited to relatively small aspect-ratio cavities (and hence examination of limited ranges of the fractality of the light). The central problem there was in the mathematical description of diffraction from many two-dimensional apertures of widely-varying size.

## Magnification of fractal modes

**Figure 3** – New compact formulations of Fresnel diffraction now allow us to calculate fractal laser modes to an arbitrary level of accuracy. This figure shows: cross-sections of laser beam profiles (top row), and magnified central portions of these mode profiles (bottom row).

At the University of Salford, we subsequently derived two new compact formulations of Fresnel diffraction arising from closed apertures of arbitrary shape. This overcame the earlier modelling limitations, and has allowed us to calculate fractal laser mode patterns with an arbitrary level of detail [31,32]. The detail permitted is only limited by the Fresnel conditions themselves. Figure 3 shows results from some sample calculations, involving the following aperture shapes: triangle, pentagon, hexagon, and decagon. In each case, one can magnify the central details of the laser mode to see even more of the structure present (shown in the frames below).