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Chaotic scattering problems with polygons and polyhedra: exit basins and uncertainty fractal dimension
Presentation / Conference
Christian, J. Chaotic scattering problems with polygons and polyhedra: exit basins and uncertainty fractal dimension. Poster presented at 4th IMA Conference on Nonlinearity and Coherent Structures, University of Loughborough (online)

The Gaspard-Rice (GR) problem provides a paradigm for studying ballistic scattering in a plane, and it exhibits the phenomenon of sensitive dependence on initial conditions. The classic incarnation comprises a point-particle projectile reflecting s... Read More about Chaotic scattering problems with polygons and polyhedra: exit basins and uncertainty fractal dimension.

Spontaneous patterns from Ablowitz-Ladik equations: cavity boundary conditions, instabilities, and mean-field theory
Presentation / Conference
cavity boundary conditions, instabilities, and mean-field theory. Presented at 4th IMA Conference on Nonlinearity and Coherent Structures, University of Loughborough (online)

In physics, the discrete nonlinear Schrödinger (dNLS) equation plays a key role in modelling wave propagation in periodic systems. Optical architectures typically involve light confined to a set of waveguide channels with nearest-neighbour coupling... Read More about Spontaneous patterns from Ablowitz-Ladik equations: cavity boundary conditions, instabilities, and mean-field theory.

Multi-Turing instabilities and spatial patterns in discrete systems : simplicity and complexity, cavities and counterpropagation
Presentation / Conference
Bostock, C., Christian, J., Leite, A., McDonald, G., & Huang, J. Multi-Turing instabilities and spatial patterns in discrete systems : simplicity and complexity, cavities and counterpropagation. Poster presented at 6th European Optical Society Annual Meeting (EOSAM 2016), Berlin Adlershof, Germany

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 spe... Read More about Multi-Turing instabilities and spatial patterns in discrete systems : simplicity and complexity, cavities and counterpropagation.

Vector waves with spatiotemporal dispersion and \chi(3) nonlinearity : transformations and relativity, solitons and stability
Presentation / Conference
Carter, L., Christian, J., & McDonald, G. Vector waves with spatiotemporal dispersion and \chi(3) nonlinearity : transformations and relativity, solitons and stability. Poster presented at 6th European Optical Society Annual Meeting (EOSAM 2016), Berlin Adlershof, Germany

A vector model, fully-second-order in both space and time, is proposed for coupled electromagnetic modes in nonlinear waveguides. Our formalism has strong overlaps with the special relativity. Exact two-component solitons are derived, asymptotic a... Read More about Vector waves with spatiotemporal dispersion and \chi(3) nonlinearity : transformations and relativity, solitons and stability.

Complex Ginzburg-Landau equations with space-time symmetry : attenuation and amplification, solitons and shockwaves
Presentation / Conference
Barrow, P., Christian, J., & McDonald, G. Complex Ginzburg-Landau equations with space-time symmetry : attenuation and amplification, solitons and shockwaves. Poster presented at 6th European Optical Society Annual Meeting (EOSAM 2016), Berlin Adlershof, Germany

We present an overview of our research into space-time-symmetrized complex Ginzburg-Landau equations, going beyond the traditional assumption of slowly-varying envelopes. Exact analytical solitary solutions are detailed, and their stability properti... Read More about Complex Ginzburg-Landau equations with space-time symmetry : attenuation and amplification, solitons and shockwaves.

Fresnel diffraction patterns from fractal apertures : boundaries and circulation, pentaflakes and islands
Presentation / Conference
Christian, J., McDonaldg, G., Kotsampaseris, A., & Huang, J. Fresnel diffraction patterns from fractal apertures : boundaries and circulation, pentaflakes and islands. Presented at 6th European Optical Society Annual Meeting (EOSAM 2016), Berlin Adlershof, Germany

We present a theoretical and numerical analysis of near-field diffraction patterns from hard-edged apertures whose shapes correspond to the iterations of (closed) classic fractal curves. The Fresnel (paraxial) area integral is transformed into a ci... Read More about Fresnel diffraction patterns from fractal apertures : boundaries and circulation, pentaflakes and islands.

On the diffraction of monsters : Weierstrass and Young, analysis and edge-waves
Presentation / Conference
Christian, J., & McDonald, G. On the diffraction of monsters : Weierstrass and Young, analysis and edge-waves. Presented at 6th European Optical Society Annual Meeting (EOSAM 2016), Berlin Adlershof, Germany

We report on recent research developments investigating the diffraction of fractal (i.e., multi-scale) light waves from simple hard-edged apertures. A bandwidth-limited Weierstrass function is used as a physical model for illumination, and a formal... Read More about On the diffraction of monsters : Weierstrass and Young, analysis and edge-waves.

Unstable resonators with Gosper-island boundary conditions : virtual-source computation of fractal eigenmodes
Presentation / Conference
Christian, J., & Huang, J. Unstable resonators with Gosper-island boundary conditions : virtual-source computation of fractal eigenmodes. Poster presented at 29th European Quantum Electronics Conference (CLEO Europe / EQEC), Munich, Germany

The Gosper island is a well-known fractal belonging to a family of self-similar “root 7” curves constructed from a simple iterative algorithm [1]. One begins with a regular hexagon (the initiator, corresponding to iteration n = 0) with sides of refer... Read More about Unstable resonators with Gosper-island boundary conditions : virtual-source computation of fractal eigenmodes.

Electromagnetic diffraction by fractal dusts, triangles and carpets : a Kirchhoff approach to circulation
Presentation / Conference
a Kirchhoff approach to circulation. Poster presented at 30th European Quantum Electronics Conference (CLEO Europe / EQEC), Munich, Germany

The diffraction of plane waves by perfectly-conducting thin screens is of fundamental physical and mathematical interest in electromagnetics [1]. Classic laser-optics experiments involve both open (single- and double-slit arrangements) and closed (ci... Read More about Electromagnetic diffraction by fractal dusts, triangles and carpets : a Kirchhoff approach to circulation.

Scattering of electromagnetic waves by cantor screens : Rayleigh-Sommerfeld integrals on complex domains
Presentation / Conference
Christian, J., & Middleton-Spencer, H. Scattering of electromagnetic waves by cantor screens : Rayleigh-Sommerfeld integrals on complex domains. Poster presented at 30th European Quantum Electronics Conference (CLEO Europe / EQEC), Munich, Germany

The scattering of light from fractal screens has been a topic of sustained interest in optics for many decades. A common thread weaving together much of the theoretical literature is the scalar approximation, wherein the polarization state of the ele... Read More about Scattering of electromagnetic waves by cantor screens : Rayleigh-Sommerfeld integrals on complex domains.

Discrete nonlinear Schrödinger equations for periodic optical systems : pattern formation in \chi(3) coupled waveguide arrays
Presentation / Conference
pattern formation in \chi(3) coupled waveguide arrays. Presented at 14th International Conference on the Mathematical and Numerical Aspects of Wave Propagation (WAVES 2019), Vienna University of Technology, Austria

Discrete nonlinear Schrödinger equations have been used for many years to model the propagation of light in optical architectures whose refractive index profile is modulated periodically in the transverse direction. Typically, one considers a modal... Read More about Discrete nonlinear Schrödinger equations for periodic optical systems : pattern formation in \chi(3) coupled waveguide arrays.

Electromagnetic scattering problems on perfectly-conducting complex domains: from Rayleigh-Sommerfeld integrals toward fractal screens
Presentation / Conference
Christian, J., & Middleton-Spencer, H. Electromagnetic scattering problems on perfectly-conducting complex domains: from Rayleigh-Sommerfeld integrals toward fractal screens. Presented at 14th International Conference on the Mathematical and Numerical Aspects of Wave Propagation (WAVES 2019), Vienna University of Technology, Austria

The diffraction of light by an aperture in an otherwise perfectly conducting plane screen of infinite extent is a phenomenon of fundamental interest in electromagnetics. Here, we consider classes of problems where the aperture domain is complex (p... Read More about Electromagnetic scattering problems on perfectly-conducting complex domains: from Rayleigh-Sommerfeld integrals toward fractal screens.