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Nonlinear Forms of Laser Beams and White Light

Patterns of clouds in the sky, wind-sculpted striations in the desert sand, spots and stripes of animal skin are examples of patterns that emerge spontaneously in nature. The physical laws that govern spontaneous pattern formation in these dissimilar systems are strikingly similar. Our research examines such nonlinear phenomena when a system simultaneously undergoes optical and chemical changes. Because the optical and chemical properties of such systems can be precisely controlled, they provide convenient pathways to examine complex nonlinear processes. This research, while consistent with studies of nonlinear light propagation established by optical physicists and engineers, contribute new chemical perspectives to this field, which spans almost half a century.

Nonlinear waves in photochemical systems

Nonlinear waves propagate without dissipating over long distances in space or time. They exist in spectacularly varied forms and dimensions, and play vital roles in processes such as excitations along polymer chains, chemical waves in reaction-diffusion systems, thermal solitons driving biochemical cycles, vibrations along proteins, pulses along nerves and within the heart, sound, ocean waves, clouds and space plasma.

We study nonlinear optical waves such as self-trapped beams and self-trapped filaments under the precise parameters of an optics experiment. The tremendous surge of creativity and progress in this field in the past 15 years is motivated in part by the promise of intelligent photonics without preconfigured circuitry; here the generation and interactions (e.g. fusion, fission, repulsion) of self-trapped beams are harnessed to manipulate and process light signals.

Consider the example of a narrow beam of light. Regardless of whether it originates from the sun or a laser, the beam will naturally broaden (diffract) while travelling through most transparent materials including air, glass and water. In certain types of materials however, the same beam of light can trigger a transformation – such as a chemical reaction – that causes the refractive index along the path of the beam to increase. A nonlinear situation arises in which the natural tendency of the beam to diffract is always opposed by refraction, which forces the beam to focus. Our research has revealed that these conditions elicit a richly diverse suite of nonlinear phenomena including spontaneous pattern formation.

Our findings include self-trapped beams of light, which travel without broadening in certain photochemical systems over unexpectedly long distances. Self-trapping occurs when the beam becomes entrapped within the narrow channel of increased refractive index inscribed along its path of propagation. We have established a comprehensive understanding of the dynamics and properties of self-trapped beams and provided first experimental confirmation of several theoretical predictions of their behaviour. We have found self-trapped beams of both laser and separately, of white light and learnt that their properties are fundamentally different. Waves composing a laser beam possess the same wavelength and are strongly correlated to each other. Exactly the opposite is the case for white light, which is said to be incoherent because it is composed of an entire spectrum of wavelengths and has extremely poor correlation. Once self-trapped, a laser beam exhibits different and discrete optical modes – like the vibrational modes of a drum skin or the harmonics of a violin string. The optical modes of a self-trapped white light beam however fluctuate so rapidly that only a time-averaged or smoothed profile of the entire beam can be observed. (It is therefore particularly remarkable that such a rapidly fluctuating and incoherent wavepacket collectively self-traps and propagates as a single entity.) By applying our knowledge of the competition between diffraction and refraction in self-trapping behaviour, we have recently been able to demonstrate the spontaneous formation of patterns in large uniform beams of white light.

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The sequential excitation of high order modes during self-trapping of a laser beam (Villafranca et al)

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Self-trapped (a) bright (Zhang et al) and (b) dark incoherent light (Kasala et al)

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Diffraction rings due to self-phase modulation of a laser beam in a photopolymerizing medium (Villafranca et al)

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The spontaneous transformation of a diffraction ring into a self-trapped laser beam leaves a permanent imprint of a conical microstructure containing a cylindrical waveguide centre in the photopolymer medium (Villafranca et al)

Interactions of self-trapped beams

Scientists have always sought to understand the interactions between particles – whether they be atoms or planets. Like these bodies of matter, self-trapped beams of light can also interact with each other. Under specific conditions, a pair of self-trapped beams can attract, repel and even appear to orbit about each other. Because we can now generate self-trapped beams in photochemical systems, we can also study their interactions. Our objectives are to establish the basic principles that govern self-trapped beam interactions through experiments that ask questions such as (a) is there any exchange of energy between interacting beams? (b) are there any changes to the fundamental properties of a self-trapped beam such as its spectrum, coherence and modes? (c) what type of structures are inscribed in the photochemical medium as a result of these interactions? (d) are self-trapped beams restricted to pairwise interactions or do interactions extend to multiple (> 2) self-trapped beams? (e) how do these interactions compare with interactions of particles in other systems?

Spontaneous pattern formation: is coherence important?

Our discovery that a uniform white light beam transforms spontaneously into an ordered pattern – although explicable through the mathematics of nonlinear light propagation – is counterintuitive. Periodic patterns of light when two or more laser beams, which are mutually coherent, interfere. Even in his seminal experiment, Young used a pair of slits to introduce mutual coherence into two beams of sunlight, which in turn produced an interference pattern and revealed the wave-nature of light. An understanding of optical pattern formation processes could bridge the two extremes of periodic interference patterns and periodic patterns that form spontaneously across a single broad beam of incoherent white light. Studies are planned to systematically tune the (spatial) coherence of a beam from excellent to weak and to monitor the resulting effects on properties of the pattern such as symmetry, periodicity and degree of order.

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Modulation instability-induced self-trapped filaments in a photopolymer (Burgess, Shimmell et al)

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Interactions of self-trapped filaments of white light in a hexagonal lattice

Theory development and modelling

Our experimental research, while confirming several predictions of light propagation models, also revealed new forms of nonlinear phenomena (higher order optical modes, white light phenomena, spontaneous pattern formation). We plan to develop theoretical models to simulate these phenomena; initial approaches will seek numerical solutions to problems cast through the nonlinear Schrödinger equation for nonlinear light propagation.

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Simulations of the nonlinear propagation of a laser beam at different intensities in a photopolymer (Villafranca et al)