Alessandro Zannotti

Dynamics of the optical swallowtail catastrophe

Perturbing the external control parameters of nonlinear systems leads to dramatic changes of its bifurcations. A branch of singular theory, the catastrophe theory, analyses the generating function that depends on state and control parameters. It predicts the formation of bifurcations as geometrically stable structures and categorizes them hierarchically. We evaluate the catastrophe diffraction integral with respect to two-dimensional cross-sections through the control parameter space and thus transfer these bifurcations to optics, where they manifest as caustics in transverse light fields. For all optical catastrophes that depend on a single state parameter, we analytically derive a universal expression for the propagation of all corresponding caustic beams. We reveal that the dynamics of the resulting caustics can be expressed by higher-order optical catastrophes. We show analytically and experimentally that particular swallowtail beams dynamically transform to higher-order butterfly caustics, whereas other swallowtail beams decay to lower-order cusp catastrophes.

Pearcey solitons in curved nonlinear photonic caustic lattices

Controlling artificial Pearcey and swallowtail beams allows realizing caustic lattices in nonlinear photosensitive media at very low light intensities. We examine their functionality as 2D and 3D waveguiding structures, and show the potential of exploiting these lattices as a linear beam splitter, which we name a Pearcey-Y-splitter. For symmetrized Pearcey beams as auto-focusing beams, the formation of solitons in focusing nonlinearity is observed. Our original approach represents the first realization of caustic photonic lattices and can directly be applied in signal processing, microscopy and material lithography.

Caustic light-based fabrication of advanced photonic structures (Conference Presentation)

Nature holds situations in which sudden changes are caused by smooth alterations. The famous Airy and Pearcey beams represent diffraction patterns of corresponding fold and cusp bifurcations, respectively. When classified in a hierarchical order, they are subsequently followed by swallowtail and butterfly beams. These beams are generally characterized as cuspoid beams, accelerated on bent trajectories. They lead to various applications, among them the realization of curved waveguides. Their umbilic counterparts, however, characterized by even more complex diffraction patterns, have up to now only been characterized, but not yet been utilized as functional fabrication templates for applications in photonics. In this contribution, we present our results on embedding higher-order cuspoid and umbilic catastrophes in tailored light. These light structures show versatile curved strands of high intensity during propagation. The elliptic umbilic beam even morphs from a hexagonal transverse intensity pattern to a beam with a single central hot spot to become again the original hexagonal pattern. We thus exploit the dynamics of these caustics to optically induce corresponding photonic lattices in nonlinear media and demonstrate light propagation in elliptic umbilic lattices. Our approach enables fabricating continuously transforming lattices with varying band structure, paving the way for advanced topological photonics.

Embedding umbilic catastrophes in artificially designed caustic beams

Catastrophe science [1] is a branch of bifurcation theory in the study of dynamical systems; and it is also a special case of more general singularity physics. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in dynamical behaviour arising from small changes in external control parameters, analysing how the qualitative nature of solutions depends on these control parameters. In singular optics, these dramatic changes manifest as geometrically stable caustics, which, as natural phenomena, are associated with the arcs close to rainbows, or may occur as high-intensity networks on the floor of shallow waters. Similar to their formation behind refractive index lenses with imperfections, the formation of corresponding structures has been observed for numerous kinds of designed lenses with significant importance in advanced optical instrumentation, astrophysics and surface analytics.

Realizing curved nonlinear photonic caustic lattices by tailored optical catastrophes

Controlling higher-order cusp and swallowtail catastrophes allows realizing caustic lattices in nonlinear materials. We examine their light guiding features and functionality as linear beam splitter, and observe the formation of Pearcey solitons.

Control of light in complex aperiodic and random photonic lattices

Engineering photonic systems is ideal for investigating wave phenomena and to control light in random or artificially fabricated deterministic aperiodic photonic lattices. Disorder in photonic systems fosters weak and strong light localization, known as Anderson localization and coherent backscattering, respectively, evolving from multiple scattering on randomly distributed scattering centers. On the other hand, tailored light propagation by designing distinctive band gap properties of deterministic aperiodic structures, such as golden-angle Vogel spirals, and Fibonacci lattices, is highly appealing. We present paradigm-shifting techniques for the optical realization of these functional structures and investigate light control by these engineered material modifications. Recent developments in photonics shift investigations from periodic lattices to random structures and deterministic aperiodic order to investigate light control in these tailored scattering or band structure designed media. In order to study light localization phenomena as pure wave effects in random potentials, photonic systems are ideal due to the lack of secondary effects such as matter-matter interactions. Still, the principle is universal: Time-reversed pairs of scattering paths owing zero phase difference result in a constructive wavelet interference contribution that survives among statistical background which average out in an extensively large ensemble. Established classifications characterize localization in distinctive regimes: Strong localization, formally known as Anderson localization, originates from closed scattering paths and localization consequently occurs around the input position. Weak localization in contrast, describes the inversion of an incoming wave vector to its negative direction, as it is called coherent backscattering. Designing the optical properties of materials by tailoring their band structures is an active field of research. Nowadays, particular interest turned towards deterministic aperiodic structures showing distinctive band gap and Fourier spectrum properties. These aperiodic structures offer highly isotropic band diagrams due to the lack of rotational and translational symmetry. In particular, the golden-angle Vogel spiral has attracted much attention since its topology remarkably differs from many other lattices, implying effects like optical angular momentum-bearing discrete diffraction. In this contribution, we describe the preparation of ensembles of random photonic potential landscapes in order to experimentally investigate light localization. We established an elaborate platform to examine transverse scattering effects in an ensemble of optically induced disordered photonic structures. Owing to the peculiar power spectrum,we not only examine both weak and strong light localization separately but in particular their mutual transition from Anderson localization to coherent backscattering. Expanding photonic lattices to aperiodic structures, we present a new holographic fabrication technique for the pixel-wise optical induction of prominent aperiodic structures, based on Bessel beams as fundamental single-site entities. We accomplished this by means of optical induction which allows to realize a huge class of two-dimensional photo refractive index landscapes, including our demonstrations of deterministic aperiodic golden-angle Vogel spirals as well as Fibonacci lattices.

Caustic diffraction catastrophes: Optical swallowtail and butterfly beams

We transfer higher order caustic catastrophes to optics, thus extending Airy and Pearcey by paraxial swallowtail and butterfly beams. Their caustics provide outstanding light trajectories for optical induction of refractive index structures.

Discrete vortex propagation in three-dimensional twisted waveguide arrays

Twisted waveguide arrays are realized as refractive index structures using tailored three-dimensionally modulated light fields. Probed with discrete vortices of opposed topological charges, this system provides distinctive output states, additionally controllable by the writing power.