Chirality-Enhanced Periodic Self-Focusing of Light in Soft Birefringent Media.

We experimentally and numerically show that chirality can play a major role in the nonlinear optical response of soft birefringent materials, by studying the nonlinear propagation of laser beams in frustrated cholesteric liquid crystal samples. Such beams exhibit a periodic nonlinear response associated with a bouncing pattern for the optical fields, as well as a self-focusing effect enhanced by the chirality of the birefringent material. Our results open new possible designs of nonlinear optical devices with low power consumption and tunable interactions with localized topological solitons.

Physics-based multistep beam propagation in inhomogeneous birefringent media.

We present a unified theoretical framework for paraxial and wide-angle beam propagation methods in inhomogeneous birefringent media based on a minimal set of physical assumptions. The advantage of our schemes is that they are based on differential operators with a clear physical interpretation and easy numerical implementation based on sparse matrices. We demonstrate the validity of our schemes on three simple two-dimensional birefringent systems and introduce an example of application on complex three-dimensional systems by showing that topological solitons in frustrated cholesteric liquid-crystals can be used as light waveguides.

Playing the blues, the greens and the reds with cellulose-based structural colours.

Structural vivid colours can arise from the interference of light reflected from structures exhibiting periodicity on scales in the range of visible wavelengths. This effect is observed with light reflected from cell-walls of some plants and exoskeletons of certain insects. Sometimes the colour sequence observed for these structures consists of nearly circular concentric rings that vary in colour from Red, Orange, Yellow, Green, Cyan to Blue, from the periphery to the centre, similarly to the colour scheme sequence observed for the rainbow (ROYGB). The sequence of colours has been found for solid films obtained from droplets of aqueous cellulose nanocrystals (CNCs) suspensions and attributed to a "coffee ring" effect. In this work, coloured lyotropic solutions and solid films obtained from a cellulose derivative in the presence of trifluoroacetic acid (TFA), which acts as a "reactive solvent", are revisited. The systems were investigated with spectroscopy, using circularly and linearly polarised light, coupled with a polarised optical microscope (POM) and scanning electron microscopy (SEM). The lyotropic cholesteric liquid crystalline solutions were confined in capillaries to simplify 1D molecular diffusion along the capillary where an unexpected sequence of the structural colours was observed. The development and reappearance of the sequence of vivid colours seem consistent with the reaction-diffusion of the "reactive solvent" in the presence of the cellulosic chains. The strong TFA acts as an auto-catalyst for the chemical reaction between TFA and the hydroxyl groups, existing along the cellulosic chain, and diffuses to the top and bottom along the capillaries, carrying dissolved cellulosic chains. Uncovering the precise mechanism of colour sequence and evolution over time in cellulosic lyotropic solutions has important implications for future optical/sensors applications and for the understanding of the development of cellulose-based structures in nature.

Spotting plants' microfilament morphologies and nanostructures.

The tracheary system of plant leaves is composed of a cellulose skeleton with diverse hierarchical structures. It is built of polygonally bent helical microfilaments of cellulose-based nanostructures coated by different layers, which provide them high compression resistance, elasticity, and roughness. Their function includes the transport of water and nutrients from the roots to the leaves. Unveiling details about local interactions of tracheary elements with surrounding material, which varies between plants due to adaptation to different environments, is crucial for understanding ascending fluid transport and for tracheary mechanical strength relevant to potential applications. Here we show that plant tracheary microfilaments, collected from Agapanthus africanus and Ornithogalum thyrsoides leaves, have different surface morphologies, revealed by nematic liquid crystal droplets. This results in diverse interactions among microfilaments and with the environment; the differences translate to diverse mechanical properties of entangled microfilaments and their potential applications. The presented study also introduces routes for accurate characterization of plants' microfilaments.

Ray-based optical visualisation of complex birefringent structures including energy transport.

We propose an efficient method to simulate light propagation in lossless and non-scattering uniaxial birefringent media, based on a standard ray-tracing technique supplemented by a newly-derived transport equation for the electric field amplitude along a ray and a tailored interpolation algorithm for the reconstruction of the electromagnetic fields. We show that this algorithm is accurate in comparison to a full solution of Maxwell's equations when the permittivity tensor of the birefringent medium typically varies over a length much bigger than the wavelength. We demonstrate the usefulness of our code for soft matter by comparing experimental images of liquid crystal droplets with simulated bright-field optical micrographs, and conclude that our method is more general than the usual Jones method, which is only valid under polarised illumination conditions. We also point out other possible applications of our method, including liquid crystal based flat element design and diffraction pattern calculations for periodic liquid crystal samples.

Liquid-crystalline half-Skyrmion lattice spotted by Kossel diagrams.

Skyrmions are swirl-like topological entities that have been shown to emerge in various condensed matter systems. Their identification has been carried out in different ways including scattering techniques and real-space observations. Here we show that Kossel diagrams can identify the formation of a hexagonal lattice of half-Skyrmions in a thin film of a chiral liquid crystal, in which case Kossel lines appear as hexagonally arranged circular arcs. Our experimental observations on a hexagonal lattice of half-Skyrmions and other defect structures resembling that of a bulk cholesteric blue phase are perfectly accounted for by numerical calculations and a theoretical argument attributing strong reflections yielding Kossel lines to guided mode resonances in the thin liquid crystal film. Our study demonstrates that a liquid crystal is a model system allowing the investigation of topological entities by various optical means, and also that Kossel techniques are applicable to the investigation of thin systems with non-trivial photonic band structures including topologically protected optical surface states.

Reflection spectra and near-field images of a liquid crystalline half-Skyrmion lattice.

We investigate numerically the optical properties of a hexagonal half-Skyrmion lattice exhibited by a highly chiral liquid crystal confined between two parallel plates. Our study focuses on the near and far-field reflection for normally incident light with different polarizations. We show that, when the wavelength of the incident light is longer than a threshold value, the reflectivity is almost insensitive to the polarization of the incident light, although the intensity profiles of the reflected light, in particular in the near-field regime, depend significantly on the polarization. The former property is attributable to the quasi two-dimensional nature of the half-Skyrmion lattice, that is, almost uniform orientational order along the direction normal to the confining plates. Our results for the intensity of reflected light generated by evanescent as well as propagating contributions suggest that direct evidence of the formation and structure of half-Skyrmions could be provided by near-field optics with resolutions higher than that of conventional optical microscopy.

Liquid microlenses and waveguides from bulk nematic birefringent profiles.

We demonstrate polarization-selective microlensing and waveguiding of laser beams by birefringent profiles in bulk nematic fluids using numerical modelling. Specifically, we show that radial escaped nematic director profiles with negative birefringence focus and guide light with radial polarization, whereas the opposite - azimuthal - polarization passes through unaffected. A converging lens is realized in a nematic with negative birefringence, and a diverging lens in a positive birefringence material. Tuning of such single-liquid lenses by an external low-frequency electric field and by adjusting the profile and intensity of the beam itself is demonstrated, combining external control with intrinsic self-adaptive focusing. Escaped radial profiles of birefringence are shown to act as single-liquid waveguides with a single distinct eigenmode and low attenuation. Finally, this work is an approach towards creating liquid photonic elements for all-soft matter photonics.

Sensing and tuning microfiber chirality with nematic chirogyral effect.

Microfibers with their elongated shape and translation symmetry can act as important components in various soft materials, notably for their mechanics on the microscopic level. Here we demonstrate the mechanical response of a micro-object to imposed chirality, in this case, the tilt of disclination rings in an achiral nematic medium caused by the chiral surface anchoring on an immersed microfiber. This coupling between chirality and mechanical response, used to demonstrate sensing of chirality of electrospun cellulose microfibers, is revealed in the optical micrographs due to anisotropy in the elastic response of the host medium. We provide an analytical explanation of the chirogyral effect supported with numerical simulations and perform an experiment to test the effect of the cell confinement and fiber size. We controllably twist the microfibers and demonstrate the response of the nematic medium. More generally the demonstrated study provides means for experimental discrimination of surface properties and allows mechanical control over the shape of disclination rings.

Sensing surface morphology of biofibers by decorating spider silk and cellulosic filaments with nematic microdroplets.

Probing the surface morphology of microthin fibers such as naturally occurring biofibers is essential for understanding their structural properties, biological function, and mechanical performance. The state-of-the-art methods for studying the surfaces of biofibers are atomic force microscopy imaging and scanning electron microscopy, which well characterize surface geometry of the fibers but provide little information on the local interaction potential of the fibers with the surrounding material. In contrast, complex nematic fluids respond very well to external fields and change their optical properties upon such stimuli. Here we demonstrate that liquid crystal droplets deposited on microthin biofibers--including spider silk and cellulosic fibers--reveal characteristics of the fibers' surface, performing as simple but sensitive surface sensors. By combining experiments and numerical modeling, different types of fibers are identified through the fiber-to-nematic droplet interactions, including perpendicular and axial or helicoidal planar molecular alignment. Spider silks align nematic molecules parallel to fibers or perpendicular to them, whereas cellulose aligns the molecules unidirectionally or helicoidally along the fibers, indicating notably different surface interactions. The nematic droplets as sensors thus directly reveal chirality of cellulosic fibers. Different fiber entanglements can be identified by depositing droplets exactly at the fiber crossings. More generally, the presented method can be used as a simple but powerful approach for probing the surface properties of small-size bioobjects, opening a route to their precise characterization.

Particles with changeable topology in nematic colloids.

We show that nematic colloids can serve as a highly variable and controllable platform for studying inclusions with changeable topology and their effects on the surrounding ordering fields. We explore morphing of toroidal and knotted colloidal particles into effective spheres, distinctively changing their Euler characteristic and affecting the surrounding nematic field, including topological defect structures. With toroidal particles, the inner nematic defect eventually transitions from a wide loop to a point defect (a small loop). Trefoil particles become linked with two knotted defect loops, mutually forming a three component link, that upon tightening transform into a two-component particle-defect loop link. For more detailed topological analysis, Pontryagin-Thom surfaces are calculated and visualised, indicating an interesting cascade of defect rewirings caused by the shape morphing of the knotted particles.

Knot theory realizations in nematic colloids.

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

Colloidal nanoparticles trapped by liquid-crystal defect lines: a lattice Monte Carlo simulation.

Lattice-based Monte Carlo simulations are performed to study a confined liquid crystal system with a topological disclination line entangling a colloidal nanoparticle. In our microscopic study the disclination line is stretched by moving the colloid, as in laser tweezing experiments, which results in a restoring force attempting to minimize the disclination length. From constant-force simulations we extract the corresponding disclination line tension, estimated as ∼50 pN, and observe its decrease with increasing temperature.

Generation of vector beams with liquid crystal disclination lines.

We report that light beams, guided along liquid crystal defect lines, can be transformed into vector beams with various polarization profiles. Using finite-difference time-domain numerical solving of Maxwell equations, we confirm that the defect in the orientational order of the liquid crystal induces a defect in the light field with twice the winding number of the liquid crystal defect, coupling the topological invariants of both fields. For example, it is possible to transform uniformly polarized light into light with a radial polarization profile. Our approach also correctly yields a zero-intensity region near the defect core, which is always present in areas of discontinuous light polarization or phase. Using circularly polarized incident light, we show that defects with noninteger winding numbers can be obtained, where topological constants are preserved by phase vortices, demonstrating coupling between the light's spin, orbital angular momentum, and polarization profile. Further, we find that an ultrafast femtosecond laser pulse traveling along a defect line splits into multiple intensity regions, again depending on the defect's winding number, allowing applications in beam steering and filtering. Finally, our approach describing the generation of complex optical fields via coupling with topological defect lines in optically birefringent nematic fluids can be easily extended to high-intensity beams that affect nematic ordering.

Quasicrystalline tilings with nematic colloidal platelets.

Complex nematic fluids have the remarkable capability for self-assembling regular colloidal structures of various symmetries and dimensionality according to their micromolecular orientational order. Colloidal chains, clusters, and crystals were demonstrated recently, exhibiting soft-matter functionalities of robust binding, spontaneous chiral symmetry breaking, entanglement, shape-driven and topological driven assembly, and even memory imprinting. However, no quasicrystalline structures were found. Here, we show with numerical modeling that quasicrystalline colloidal lattices can be achieved in the form of original Penrose P1 tiling by using pentagonal colloidal platelets in layers of nematic liquid crystals. The tilings are energetically stabilized with binding energies up to 2500 kBT for micrometer-sized platelets and further allow for hierarchical substitution tiling, i.e., hierarchical pentagulation. Quasicrystalline structures are constructed bottom-up by assembling the boat, rhombus, and star maximum density clusters, thus avoiding other (nonquasicrystalline) stable or metastable configurations of platelets. Central to our design of the quasicrystalline tilings is the symmetry breaking imposed by the platelet shape and the surface anchoring conditions at the colloidal platelets, which are misaligning and asymmetric over two perpendicular mirror planes. Finally, the design of the quasicrystalline tilings as platelets in nematic liquid crystals is inherently capable of a continuous variety of length scales of the tiling, ranging over three orders of magnitude in the typical length (from ~ 10 nm to ~ 10 µm), which could allow for the design of quasicrystalline photonics at multiple frequency ranges.

Mutually tangled colloidal knots and induced defect loops in nematic fields.

Colloidal dispersions in liquid crystals can serve as a soft-matter toolkit for the self-assembly of composite materials with pre-engineered properties and structures that are highly dependent on particle-induced topological defects. Here, we demonstrate that bulk and surface defects in nematic fluids can be patterned by tuning the topology of colloidal particles dispersed in them. In particular, by taking advantage of two-photon photopolymerization techniques to make knot-shaped microparticles, we show that the interplay of the topologies of the knotted particles, the nematic field and the induced defects leads to knotted, linked and other topologically non-trivial field configurations. These structures match theoretical predictions made on the basis of the minimization of the elastic free energy and satisfy topological constraints. Our approach may find uses in self-assembled topological superstructures of knotted particles linked by nematic fields, in topological scaffolds supporting the decoration of defect networks with nanoparticles, and in modelling other physical systems exhibiting topologically analogous phenomena.

Topological zoo of free-standing knots in confined chiral nematic fluids.

Knotted fields are an emerging research topic relevant to different areas of physics where topology plays a crucial role. Recent realization of knotted nematic disclinations stabilized by colloidal particles raised a challenge of free-standing knots. Here we demonstrate the creation of free-standing knotted and linked disclination loops in the cholesteric ordering fields, which are confined to spherical droplets with homeotropic surface anchoring. Our approach, using free energy minimization and topological theory, leads to the stabilization of knots via the interplay of the geometric frustration and intrinsic chirality. Selected configurations of the lowest complexity are characterized by knot or link types, disclination lengths and self-linking numbers. When cholesteric pitch becomes short on the confinement scale, the knotted structures change to practically unperturbed cholesteric structures with disclinations expelled close to the surface. The drops with knots could be controlled by optical beams and may be used for photonic elements.

Janus Nematic Colloids with Designable Valence.

Generalized Janus nematic colloids based on various morphologies of particle surface patches imposing homeotropic and planar surface anchoring are demonstrated. By using mesoscopic numerical modeling, multiple types of Janus particles are explored, demonstrating a variety of novel complex colloidal structures. We also show binding of Janus particles to a fixed Janus post in the nematic cell, which acts as a seed and a micro-anchor for the colloidal structure. Janus colloidal structures reveal diverse topological defect configurations, which are effectively combinations of surface boojum and bulk defects. Topological analysis is applied to defects, importantly showing that topological charge is not a well determined topological invariant in such patchy nematic Janus colloids. Finally, this work demonstrates colloidal structures with designable valence, which could allow for targeted and valence-conditioned self-assembly at micro- and nano-scale.