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.
Read some 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.
Fractal laser experiments
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).
Further developments at the University of Salford have included proposal and verification of a mechanism for the spontaneous generation of optical fractals in nonlinear systems . We have also been studying various novel configurations and effects in our Video Feedback Laboratory .
Some research papers (fractal light from lasers)
1. MA van Eijkelenborg, GS McDonald, GP Karman, JP Woerdman and GHC New, Polygon Modes of Unstable-Cavity Lasers (Invited), European Quantum Electronics Conference, Glasgow, UK, Sept 13-18, 1998
2. GHC New, MA van Eijkelenborg, GS McDonald and JP Woerdman, Spatial Confinement Picture of Excess Quantum Noise, European Quantum Electronics Conference, Glasgow, UK, Sept 13-18, 1998
3. GS McDonald, GP Karman, JP Woerdman and GHC New, Excess-Noise Dependence on Intra-Cavity Aperture Shape, Quantum Electronics and Laser Science Conference, Baltimore, USA, May 23-28, 1999
4. GS McDonald, GHC New and JP Woerdman, Excess Noise in Low Fresnel Number Unstable Resonators (Full-length Article), OPT COMMUN 164, 285, 1999
5. GP Karman, GS McDonald, JP Woerdman and GHC New, Excess-Noise Dependence on Intra-Cavity Aperture Shape, APPL OPT 38, 6874, 1999
6. GP Karman and JP Woerdman, Fractal structure of eigenmodes of unstable-cavity lasers, OPT LETT 23, 1909, 1998
7. GS McDonald, GHC New, GP Karman and JP Woerdman, The Kaleidoscope Laser, Fourteenth National Quantum Electronics Conference, Manchester, UK, Sept 6-9, 1999
8. GS McDonald, GHC New and JP Woerdman, The Two-Dimensional Virtual Source Method, Fourteenth National Quantum Electronics Conference, Manchester, UK, Sept 6-9, 1999
9. JP Woerdman, GP Karman, GS McDonald and GHC New, Unstable-Cavity Lasers Produce Fractal Mode Patterns (Invited), Second Euroconference on Control of Complex Behaviour in Optical Systems and Applications, Munster, Germany, Oct 7-10, 1999
10. O Graydon (Science Editor), Fractals discovered in laser modes, OPTO & LASER EUROPE, October 1999. Lead News Story and Front Cover Article
11. GP Karman, GS McDonald, GHC New and JP Woerdman, Fractal Modes in Unstable Resonators, NATURE 402, 138, November 1999
12. GS McDonald, GHC New and JP Woerdman, Higher-order Modes of the Kaleidoscope Laser, Quantum Electronics and Laser Science Conference, San Francisco, USA, May 7-12, 2000
13. GS McDonald, GHC New, GP Karman and JP Woerdman, Higher-Order Fractal Laser Modes, International Quantum Electronics Conference, Nice, France, Sept 10-15, 2000
14. GS McDonald, GHC New and JP Woerdman, Lasers, Kaleidoscopes and Fractals, OPTICS AND PHOTONICS NEWS, November 2000
15. Lasers That Look Like Child's Play, NEWSLINE, Journal of the EPSRC, Issue 15, 2000. Front Cover Article and News Story
16. GS McDonald, GP Karman, GHC New and JP Woerdman, The Kaleidoscope Laser, J OPT SOC AM B 17, 524, 2000
17. GHC New, MA Yates, JP Woederman, GS McDonald, Diffractive origin of fractal resonator modes, OPT COMMUN 193, 261, 2001
18. MV Berry, C Storm, and W van Saarloos, Theory of unstable laser modes: edge waves and fractality, OPT COMMUN 197, 393-402, 2001
19. MV Berry, Fractal modes of unstable lasers with polygonal and circular mirrors, OPT COMMUN 200, 321-330, 2001
20. J Courtial, and MJ Padgett, Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators, PHYS REV LETT 85 (25), 5320, 2000
21. J Courtial, J Leach, and MJ Padgett, Image Processing: Fractals in pixellated video feedback, NATURE 414, 20-17 December, 864, 2001
22. A Forrester, M Lonnqvist, MJ Padgett, and J Courtial, Why are the eigenmodes of stable laser resonators structurally stable?, OPT LETT 27 (21), 1869, 2002
23. CMG Watterson MJ Padgett, and J Courtial, Classic-fractal eigenmodes of unstable canonical resonators, OPT COMMUN 223 (1-3), 17, 2003
24. J Leach, MJ Padgett, and J Courtial, Fractals in pixellated video feedback, CONTEMPORARY PHYS 44 (2), 137, 2003
25. JA Loaiza, ER Eliel, and JP Woerdman, Experimental observation of fractal modes in unstable optical resonators , arXiv:physics/0304046, v1 11 Apr 2003
26. Javier Augusto Loaiza, Experimental demonstration of the fractal nature of unstable-resonator modes (PhD thesis), Institute of Physics, Faculty of Mathematics and Natural Sciences, Leiden University, 23 Nov 2005
27. MA Yates, and GHC New, Fractal dimension of unstable resonator modes, OPT COMMUN 208 (4-6), 377-380, 2002
28. GHC New, and T Albaho, Spatial spectrum of intensity profiles diffracted by a slit, J MOD OPT 50 (2), 155-160, 2003
29. MA Yates, GHC New, and T Albaho, Calculating higher-order modes of one-dimensional unstable laser resonators, J MOD OPT 51 (5), 657-667, 2004
31. JG Huang, JM Christian, GS McDonald, Fresnel Diffraction from Polygonal Apertures , CLEO/Europ Quant Electr Conf (EQEC), Munich, Germany, June, 2005
32. JG Huang, JM Christian, GS McDonald, Fresnel diffraction and fractal patterns from polygonal apertures , J OPT SOC AM A 23, 2768, 2006
33. JG Huang, and GS McDonald, Spontaneous Optical Fractal Pattern Formation, PHYSICAL REVIEW LETTERS 95, Art No 174101, 2005
34. GS McDonald, Joule Physics Laboratory - Video Feedback Laboratory, Salford Science Spotlight prog 3 part 1, University of Salford, YouTube, 2008