From Turing instability to fractals
Huang, JG, Christian, JM, McDonald, GS, Jahanpanah, J and ChamorroPosada, P 2007, From Turing instability to fractals , in: Nonlinear Photonics Topical Meeting, 2nd – 6th September 2007, Quebec City, Canada.

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Abstract
Complexity focuses on commonality across subject areas and forms a natural platform for multidisciplinary activities. Typical generic signatures of complexity include: (i) spontaneous occurrence of simple patterns (e.g. stripes, squares, hexagons) emerging as dominant nonlinear modes [1], and (ii) the formation of a highly complex pattern in the form of a fractal (with comparable levels of detail spanning decades of scale). Recently, a firm connection was established between these two signatures, and a generic mechanism was proposed for predicting the fractal generating capacity of any nonlinear system [2]. The mechanism for fractal formation is of a very general nature: any system whose Turing threshold curves exhibit a large number of comparable spatialfrequency instability minima are potentially capable of generating fractal patterns. Spontaneous spatial fractals were first reported for a very simple nonlinear system: the diffusive Kerr slice with a single feedback mirror [3]. These Kerrslice fractals are distinct from both the transverse fractal eigenmodes of unstablecavity lasers [4], and also from the fractals found in optical solitonsupporting systems [5,6]. On the one hand, unstablecavity fractals may be regarded as a linear superposition of diffraction patterns with different scale lengths, each of which arises from successive roundtrip magnifications of an initial diffractive seed. On the other hand, fractals formed in the Kerr slice result entirely from intrinsic nonlinear dynamics (i.e. lightmatter coupling leading to harmonic generation and/or fourwave mixing cascades). These processes conspire to generate new spatial frequencies that, in turn, can produce optical structure on smaller and smaller scales, down to the order of the optical wavelength. Here we report the first predictions of spontaneous fractal patterns inside driven damped ring cavities containing a thin slice of nonlinear material. Both dispersive (i.e. diffusiverelaxing Kerr [3]) and absorptive (i.e. Maxwell Bloch saturable absorber [7]) are considered. New linear analyses have shown that the transverse instability spectra of these two cavity systems possess the requisite comparable minima that predict the capacity for the spontaneous generation of fractal patterns. Extensive numerical simulations, in both one and two transverse dimensions, have verified that both the dispersive and absorptive cavities do indeed give rise to nonlinear optical fractals in the transverse plane. Our results confirm that the mechanism for fractal formation has independence with respect to the details of the nonlinearity. An essential ingredient for the generation of fractals is the presence of a feedback mechanism [2]. Feedback drives the cascade process that is responsible for the creation of higher spatial wavenumbers, and which ultimately leads to the “structure across decades of scale” character of the fractal pattern. Cavity geometries are therefore ideal candidates as potential optical fractal generators. The simplest dispersive nonlinearity is provided by the relaxingdiffusing Kerr effect. The threshold curves possess the qualitative features necessary for the generation of spontaneous fractal patterns: successive and comparable spatial frequency minima. Rigorous simulations have shown that the Kerr cavity is indeed capable of generating fractal patterns. In a singleK configuration, where the filter attenuates all those spatial wavenumbers outside the first instability band, it is found that simple stripe patterns emerge. Once this stationary pattern has been reached, the spatial filter is removed to allow all waves to propagate. Energy is transferred to higher spatial frequencies, and the simple strip pattern acquires successive level of fine detail at a rate that depends upon the system parameters. By analysing the power spectrum P(K) it can be seen that a fractal pattern emerges relatively rapidly. Eventually, the system enters a dynamic equilibrium (within typically less than a hundred transits) where the average power spectrum remains unchanged even though the pattern continues to evolve in real space. When this statistically invariant state is attained, the pattern is referred to as a scaledependent fractal. An appreciable portion of the dynamic state is well described by a linear relationship ln P(K) = a + bK, where a and b are constants, and this type of behaviour is one of the characteristics of a fractal pattern [2]. We have recently found that a thinslice MaxwellBloch saturable absorber [7] can also generate fractal patterns. This system can be either purely absorptive or purely dispersive. Linear analysis, together with a generalized boundary condition (which allows for attenuation), yields the threshold condition for Turing instability. One finds that the threshold spectrum comprises a series of adjacent instability islands. Simulations have revealed that the MaxwellBloch system can also support fractals. The singleK patterns turn out to be hexagonal arrays, familiar from conventional pattern formation [1,3]. Once this state has been reached, the spatial filter is removed and one can observe a rapid transition toward a fractal pattern. The qualitative behaviour of fractals patterns in both dispersive and absorptive systems are found to be the same, confirming the assertion of independence with respect to nonlinearity. References: [1] J. B. Geddes et al., “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 5, 34713485 (1994). [2] J. G. Huang and G. S. McDonald, “Spontaneous optical fractal pattern formation,” Phys. Rev. Lett. 94, 174101 (2005). [3] G. D’Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537548 (1992). [4] J. G. Huang et al., “Fresnel diffraction and fractal patterns from polygonal apertures,” J. Opt. Soc. Am. A 23, 27682774 (2006). [5] M. Soljacic and M. Segev, “Selfsimilarity and fractals in solitonsupporting systems,” Phys. Rev. E 61, R1048R1051 (2000). [6] S. Sears et al., “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 19021905 (2000). [7] A. S. Patrascu et al., “Multiconical instability in the passive ring cavity: linear analysis,” Opt. Commun. 91, 433443 (1992).
Item Type:  Conference or Workshop Item (Lecture) 

Themes:  Energy Media, Digital Technology and the Creative Economy Subjects outside of the University Themes 
Schools:  Schools > School of Computing, Science and Engineering > Salford Innovation Research Centre (SIRC) 
Refereed:  Yes 
Depositing User:  JM Christian 
Date Deposited:  17 Oct 2011 15:04 
Last Modified:  29 Oct 2015 00:11 
URI:  http://usir.salford.ac.uk/id/eprint/18441 
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