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 spatial-frequency 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 Kerr-slice fractals are distinct from both the transverse fractal
eigenmodes of unstable-cavity lasers [4], and also from the fractals found in optical soliton-supporting systems
[5,6]. On the one hand, unstable-cavity fractals may be regarded as a linear superposition of diffraction patterns
with different scale lengths, each of which arises from successive round-trip magnifications of an initial diffractive seed. On the other hand, fractals formed in the Kerr slice result entirely from intrinsic nonlinear dynamics (i.e. light-matter coupling leading to harmonic generation and/or four-wave 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. diffusive-relaxing 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 relaxing-diffusing 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 single-K 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 scale-dependent 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 thin-slice Maxwell-Bloch 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 Maxwell-Bloch system can also support fractals. The single-K 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, 3471-3485 (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, 537-548 (1992).
[4] J. G. Huang et al., “Fresnel diffraction and fractal patterns from polygonal apertures,” J. Opt. Soc. Am. A 23, 2768-2774 (2006).
[5] M. Soljacic and M. Segev, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048-R1051 (2000).
[6] S. Sears et al., “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902-1905 (2000).
[7] A. S. Patrascu et al., “Multi-conical instability in the passive ring cavity: linear analysis,” Opt. Commun. 91, 433-443 (1992)