Wide class of new fractal light sources [Invited plenary paper]

Pattern emergence in Naturepsilas complex systems is mostly attributed to a classic Turing instability. There, a single length-scale becomes dominant and this defines a simple emergent structure (for example, a striped or hexagonal pattern). We have investigated whether a multi-Turing instability may result in another universal type of pattern: fractals. Fractals possess proportional levels of detail across decades of length-scale, and are thus inherently scale-less. Here, we make the first predictions of spontaneous spatial fractal patterns in nonlinear ring cavities. This will include the first reported spatial optical fractals arising from purely-absorptive nonlinearity. Analyses reveal characteristic (multi-Turing) spectral features for both dispersive and absorptive cavities. Simulations verify and quantify the fractal properties of the spontaneously-patterned light. Our findings greatly widen the scope for potential realization and exploitation of fractal light sources. In Nature-inspired device and system architectures, such sources are likely to play a pivotal role in developments

The role of boundary conditions in kaleidoscope laser modes

The complex character of transverse eigenmodes in one-dimensional (1D) unstable cavity lasers has been known for many years [1]. Early collaborations showed that the origin of such fractal (i.e., multiple spatial scale) structure lies in a subtle interplay between small-scale diffraction effects at the mirror edges and successive round-trip magnifications [2]. Kaleidoscope lasers are intuitive generalizations of the classic strip resonator to fully-2D geometries where the feedback mirror has a non-trivial transverse shape, such as a regular polygon [3]. The fundamental mechanism for fractal formation is preserved, but until recently these novel laser designs have remained largely unexplored. We will report on recent advances in our understanding of kaleidoscope lasers, made possible by new semi-analytical techniques. Aspects to be covered include mode patterns, eigenvalue spectra, convergence phenomena, and also the first calculations of fractal dimension for arbitrary cavity parameters

Kaleidoscope lasers - Complexity in simple optical systems

We present the first detailed account of modelling kaleidoscope laser modes where the equivalent Fresnel number Neq and magnification M may assume arbitrary values. The convergence toward circularity is also investigated through extensive numerical computations

Fresnel diffraction from polygonal apertures

We present, for the first time, two complementary analytical techniques for calculating the Fresnel diffraction patterns from a polygonal aperture illuminated by a plane wave. These frameworks are exact, in that they do not involve any further approximation beyond the (paraxial) Fresnel integral. Here, we consider regular polygonal apertures, but our results are readily extended to describe near-field diffraction from closed apertures of arbitrary shape. Our results are of fundamental importance and have specific applications where standard methods, such as Fast Fourier Transform (FFT) techniques, fail. For example, in unstable-resonator mode calculations, both (paraxial beam) ABCD matrix modelling and existing semi-analytical methods can only give accurate results in limited parameter regimes. Consequently, a complete and detailed study of optical fractal laser modes [1] has not previously been possible. A specific advantage of our formalisms is the ability to calculate and store the fine details of only a small portion of one, or many, complex diffraction patterns. Moreover, the explicit mathematical form of our results may also lend physical insight into a wide range of diffraction-related phenomena in physics. For example, insight into the physical origin of excess quantum noise in lasers, where the transverse symmetry of an aperturing element has been shown to play a central role in the observed phenomena [2]. While Fraunhoffer (far-field) diffraction patterns have been known for many years, there has been relatively little equivalent published work in the Fresnel (near-field)regime. The far-field approximation allows the expression of diffraction patterns and descriptions of derivative concepts (eg in holography, filtering, convolution and coherence) as simple Fourier integrals and transform theorems, respectively. Our new results permit the mathematical and physical expression of near-field diffraction patterns in terms of their elemental spatial structures (edge-waves). It is plausible that our results may also open future doors in the development of derivative concepts in Fresnel Optics. The Fresnel diffraction patterns consist of a plane-wave (uniform) component, plus a rapidly-varying interference contribution from boundary-diffracted waves from all edges of the aperture. This physical interpretation is present in the mathematical formulations of both the S-Function Method, which deals explicitly with edge-wave combinations, and also the Line-Integral Method, where the Fresnel integral over the aperture area is expressed as a circulation around its edge. Extensive computational investigations have verified that our two approaches are completely equivalent, and produce identical results. The amount of fine detail present in the pattern can be quantified by the Fresnel Number N = a2/λL + f(n), where a is the radius of a circle inscribing an n-sided polygon, L is the distance from the aperture to the observation plane (L^2 » a^2 is a paraxial approximation inherent to the near-field approximation), λ is the wavelength of the illuminating light, and f(n) = 0.30618n2 - 0.19533n - 0.68095 is a term allowing for the geometrical structure of the aperture. As n becomes larger, the level of detail increases. References [1] G.P. Karman, G.S. McDonald, G.H.C. New, and J.P. Woerdman, Nature 402 (1999) 138. [2] G.P. Karman, G.S. McDonald, J.P. Woerdman, and G.H.C. New, Appl Opt 38 (1999) 687

Instabilities & boundary conditions: fractal mode patterns in kaleidoscope lasers

The multi-scale - or fractal - nature of transverse modes in one-dimensional (1D) unstable cavity lasers has been known since the late 1990s [1]. Early collaborations with some members of our Group demonstrated that the origin of fractality (which demands both the presence and comparable strength of multiple spatial scales) lies in a subtle interplay between small-scale diffraction effects at the mirror edges and successive round-trip magnifications [2]. Kaleidoscope lasers are fully-2D generalizations of the more familiar 1D system, where the feedback mirror has a non-trivial transverse shape, such as a regular polygon [3]. The fundamental mechanism for fractal formation is the same as for 1D cavities, but until recently these highly geometric cavity designs have received relatively little attention. We will report on recent advances in our understanding of kaleidoscope lasers, facilitated by the development of new semi-analytical techniques. Key considerations include direct calculation of mode patterns, eigenvalue spectra, and convergence phenomena (e.g., as the feedback mirror tends towards the circular limit). We have also performed what appear to be the first computations of mode fractal dimension for arbitrary cavity parameters. Some surprising results have been uncovered. References: [1] G. P. Karman and J. P. Woerdman, Opt. Lett. 23, 1909 (1998). [2] G. H. C. New, M. A. Yates, J. P. Woerdman, and G. S. McDonald, Opt. Commun. 193, 261 (2001). [3] G. S. McDonald et al., J. Opt. Soc. Am. B 17, 524 (2000); Nature 402, 138 (1999)

Fractal laser sources: new analyses, results and contexts

A series of significant new extensions concerning fractal light generation are reported. Firstly, we summarise techniques and results from the first full analysis of the linear modes of ‘fractal lasers’ [1] – unstable-cavity geometries with arbitrary Fresnel number Neq and arbitrary round-trip magnification M. Secondly, simulations and analyses for new contexts of laser-driven ‘nonlinear fractal generators’ [2] – where analogous nonlinear processes spontaneously generate fractals – are presented. Finally, we outline why such fractal laser sources may play a pivotal role in future Nature-inspired devices and system architectures. Our discovery of fractal laser modes from unstable-cavity lasers [1] uncovered a general class of linear systems (with repeated magnification) that possess fractal eigenmodes. However, numerical or analytical analyses was limited to modes of either: very limited fractality, laser cavities with Neq ≈ O(1); or unlimited fractality, when Neq >> O(1). General properties of fractal modes from these two extremes are, perhaps unsurprisingly, different. Building on Fresnel diffraction theory developments [3], we report fractal mode characteristics in the important intermediate regime – corresponding to real-world systems with significant and exploitable fractality (see Figure 1). Figure 1. Lowest-loss eigenmode patterns for ‘kaleidoscope fractals lasers’ with Neq = 30 and M = 1.5. We further proposed fractal light generation through entirely-nonlinear mechanisms [2]. The context examined was a single configuration with a particular nonlinearity. Generalisation of this work to new contexts - with profoundly different nonlinearities and experimental configurations, such as ring cavities and cavity-less contexts – will be summarised. The huge spatial bandwidths associated with fractal sources have potential exploitation within novel technological contexts. We conclude with a brief account of such potential new technologies. References [1] Karman G P, McDonald G S, New G H C and Woerdman JP, Nature 402, 138 (1999). [2] Huang J G and McDonald G S, Phys. Rev. Lett. 94, 174101 (2005). [3] Huang J G, Christian J M and McDonald G S, J. Opt. Soc. Am. A 23, 2768 (2006)

Kaleidoscope lasers: Polygonal boundary conditions & geometrical instabilities

Kaleidoscope lasers are geometrically unstable cavities with a feedback mirror that has the shape of a regular polygon [1]. Early calculations of the transverse eigenmodes of these systems hinted toward a fractal (or multi-scale) characteristic, but computational-resource limitations of the day imposed fairly strong restrictions on the cavity parameter regimes that could be considered [2]. In this presentation, we will report on a new semi-analytical modelling approach for kaleidoscope lasers that combines a 2D generalization of Southwell’s classic virtual source method [3] with exact mathematical descriptions of constituent (Fresnel) edge-waves from polygonal apertures [4]. Key considerations include computations of mode patterns (see Fig. 1), eigenvalue spectra, and convergence phenomena (where the feedback mirror tends towards the circular limit). We have also performed what appear to be the first numerical calculations of (power spectrum) fractal dimension for arbitrary cavity parameters, uncovering some surprising results in the process. References [1] G. P. Karman, G. S. McDonald, G. H. C. New, and J. P. Woerdman, Nature 402, 138 (1999). [2] G. S. McDonald, G. P. Karman, G. H. C. New, and J. P. Woerdman, J. Opt. Soc. Am. B 17, 524 (2000). [3] W. H. Southwell, Opt. Lett. 6, 487 (1991); J. Opt. Soc. Am. A 3, 1885 (1986). [4] J. G. Huang, J. M. Christian, and G. S. McDonald, J. Opt. Soc. Am. A 23, 2768 (2006)

Kaleidoscope laser properties and new optical fractal contexts

We present the first detailed account of kaleidoscope laser modes where the equivalent Fresnel number Neq and magnification M may assume arbitrary values. Properties of these linear fractal eigenmodes are explored through extensive numerical computations. Considerations are extended to demonstration and analyses of new contexts for spontaneous nonlinear optical fractals

Multi-Turing instabilities and spatial patterns in discrete systems : simplicity and complexity, cavities and counterpropagation

The spontaneous pattern-forming properties of three discrete nonlinear optical systems are investigated, including the proposal of two new physical contexts for coupled-waveguide geometries. Linear analyses predict Turing threshold instability spectra with multiple minima, and simulations demonstrate emergent static patterns

Spontaneous spatial fractals: universal contexts and applications

We report on our latest research in the field of spontaneous spatial fractal patterns. New analyses, results and potential applications are reported for nonlinear ring cavities and kaleidoscope laser systems