Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals

The association of structural chirality and liquid crystal anchoring in polymer stabilized cholesteric liquid crystals

Polymer stabilized cholesteric liquid crystals (PSCLCs) are electrically reconfigurable reflective elements. Prior studies have hypothesized and indirectly confirmed that the electro-optic response of these composites is associated with the electrically mediated distortion of the stabilizing polymer network. The proposed mechanism is based on the retention of structural chirality in the polymer stabilizing network, which upon deformation is spatially distorted, which accordingly affects the pitch of the surrounding low molar-mass liquid crystal host. Here, we utilize fluorescent confocal polarized microscopy to directly assess the electro-optic response of PSCLCs. By utilizing dual fluorescent probes, sequential imaging experiments confirm that the periodicity of the polymer stabilizing network matches that of the low molar-mass liquid crystal host. Further, we isolate distinct ion-polymer interactions that manifest in certain photopolymerization conditions.

Field-controlled dynamics of skyrmions and monopoles

Magnetic monopoles, despite their ongoing experimental search as elementary particles, have inspired the discovery of analogous excitations in condensed matter systems. In chiral condensed matter systems, emergent monopoles are responsible for the onset of transitions between topologically distinct states and phases, such as in the case of transitions from helical and conical phase to A-phase comprising periodic arrays of skyrmions. By combining numerical modeling and optical characterizations, we describe how different geometrical configurations of skyrmions terminating at monopoles can be realized in liquid crystals and liquid crystal ferromagnets. We demonstrate how these complex structures can be effectively manipulated by external magnetic and electric fields. Furthermore, we discuss how our findings may hint at similar dynamics in other physical systems and their potential applications.

Topological solitonic macromolecules

Being ubiquitous, solitons have particle-like properties, exhibiting behaviour often associated with atoms. Bound solitons emulate dynamics of molecules, though solitonic analogues of polymeric materials have not been considered yet. Here we experimentally create and model soliton polymers, which we call "polyskyrmionomers", built of atom-like individual solitons characterized by the topological invariant representing the skyrmion number. With the help of nonlinear optical imaging and numerical modelling based on minimizing the free energy, we reveal how topological point defects bind the solitonic quasi-atoms into polyskyrmionomers, featuring linear, branched, and other macromolecule-resembling architectures, as well as allowing for encoding data by spatial distributions of the skyrmion number. Application of oscillating electric fields activates diverse modes of locomotion and internal vibrations of these self-assembled soliton structures, which depend on symmetry of the solitonic macromolecules. Our findings suggest new designs of soliton meta matter, with a potential for the use in fundamental research and technology.

Scaling and spontaneous symmetry restoring of topological defect dynamics in liquid crystal

Topological defects-locations of local mismatch of order-are a universal concept playing important roles in diverse systems studied in physics and beyond, including the universe, various condensed matter systems, and recently, even life phenomena. Among these, liquid crystal has been a platform for studying topological defects via visualization, yet it has been a challenge to resolve three-dimensional structures of dynamically evolving singular topological defects. Here, we report a direct confocal observation of nematic liquid crystalline defect lines, called disclinations, relaxing from an electrically driven turbulent state. We focus in particular on reconnections, characteristic of such line defects. We find a scaling law for in-plane reconnection events, by which the distance between reconnecting disclinations decreases by the square root of time to the reconnection. Moreover, we show that apparently asymmetric dynamics of reconnecting disclinations is actually symmetric in a comoving frame, in marked contrast to the two-dimensional counterpart whose asymmetry is established. We argue, with experimental supports, that this is because of energetically favorable symmetric twist configurations that disclinations take spontaneously, thanks to the topology that allows for rotation of the winding axis. Our work illustrates a general mechanism of such spontaneous symmetry restoring that may apply beyond liquid crystal, which can take place if topologically distinct asymmetric defects in lower dimensions become homeomorphic in higher dimensions and if the symmetric intermediate is energetically favorable.

Tomographic measurement of dielectric tensors at optical frequency

The dielectric tensor is a physical descriptor of fundamental light-matter interactions, characterizing anisotropic materials with principal refractive indices and optic axes. Despite its importance in scientific and industrial applications ranging from material science to soft matter physics, the direct measurement of the three-dimensional dielectric tensor has been limited by the vectorial and inhomogeneous nature of light scattering from anisotropic materials. Here, we present a dielectric tensor tomographic approach to directly measure dielectric tensors of anisotropic structures including the spatial variations of principal refractive indices and directors. The anisotropic structure is illuminated with a polarized plane wave with various angles and polarization states. Then, the scattered fields are holographically measured and converted into vectorial diffracted field components. Finally, by inversely solving a vectorial wave equation, the three-dimensional dielectric tensor is reconstructed. Using this approach, we demonstrate quantitative tomographic measurements of various nematic liquid-crystal structures and their fast three-dimensional non-equilibrium dynamics.

Oblique light incidence method to study topological defects in nematic layers with conical boundary conditions

A polarization microscopy method to investigate the orientational structures and boojums formed in the chiral and achiral nematic layers under conical (tilted) boundary conditions has been developed. Oblique light incidence on nematic layer is used, due to which the phase difference between the ordinary and extraordinary waves depends on the director's azimuthal angle. The phase difference gets maximal when the director azimuthal angle of achiral nematic [Formula: see text] and an azimuthal angle at the center of the chiral nematic layer [Formula: see text] independently of the total twist angle [Formula: see text]. It has been found that the [Formula: see text] boojums with the phase [Formula: see text] and [Formula: see text] are formed in achiral and chiral nematics, respectively, at the director tilt angle [Formula: see text] at the interface. In addition, the defectless structure of chiral nematic with the periodically variable azimuthal director angle on the substrates has been studied.

Thermally reconfigurable monoclinic nematic colloidal fluids

Fundamental relationships are believed to exist between the symmetries of building blocks and the condensed matter phases that they form¹. For example, constituent molecular and colloidal rods and disks impart their uniaxial symmetry onto nematic liquid crystals, such as those used in displays^1,2. Low-symmetry organizations could form in mixtures of rods and disks^3-5, but entropy tends to phase-separate them at the molecular and colloidal scales, whereas strong elasticity-mediated interactions drive the formation of chains and crystals in nematic colloids^6-11. To have a structure with few or no symmetry operations apart from trivial ones has so far been demonstrated to be a property of solids alone¹, but not of their fully fluid condensed matter counterparts, even though such symmetries have been considered theoretically^12-15 and observed in magnetic colloids¹⁶. Here we show that dispersing highly anisotropic charged colloidal disks in a nematic host composed of molecular rods provides a platform for observing many low-symmetry phases. Depending on the temperature, concentration and surface charge of the disks, we find nematic, smectic and columnar organizations with symmetries ranging from uniaxial^1,2 to orthorhombic^17-21 and monoclinic^12-15. With increasing temperature, we observe unusual transitions from less- to more-ordered states and re-entrant²² phases. Most importantly, we demonstrate the presence of reconfigurable monoclinic colloidal nematic order, as well as the possibility of thermal and magnetic control of low-symmetry self-assembly^2,23,24. Our experimental findings are supported by theoretical modelling of the colloidal interactions between disks in the nematic host and may provide a route towards realizing many low-symmetry condensed matter phases in systems with building blocks of dissimilar shapes and sizes, as well as their technological applications.

Review: knots and other new topological effects in liquid crystals and colloids

Humankind has been obsessed with knots in religion, culture and daily life for millennia, while physicists like Gauss, Kelvin and Maxwell already involved them in models centuries ago. Nowadays, colloidal particles can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be interlinked (or not) with colloidal particle knots, revealing the diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights into stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.

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.

Surface anchoring as a control parameter for stabilizing torons, skyrmions, twisted walls, fingers, and their hybrids in chiral nematics

Chiral condensed matter systems, such as liquid crystals and magnets, exhibit a host of spatially localized topological structures that emerge from the medium's tendency to twist and its competition with confinement and field coupling effects. We show that the strength of perpendicular surface boundary conditions can be used to control the structure and topology of solitonic and other localized field configurations. By combining numerical modeling and three-dimensional imaging of the director field, we reveal structural stability diagrams and intertransformation of twisted walls and fingers, torons, and skyrmions and their crystalline organizations upon changing boundary conditions. Our findings provide a recipe for controllably realizing skyrmions, torons, and hybrid solitonic structures possessing features of both of them, which will aid in fundamental explorations and technological uses of such topological solitons. Moreover, we discuss how other material parameters can be used to determine soliton stability and how similar principles can be systematically applied to other liquid crystal solitons and solitons in other material systems.

Topological solitons, cholesteric fingers and singular defect lines in Janus liquid crystal shells

Topological solitons are non-singular but topologically nontrivial structures in fields, which have fundamental significance across various areas of physics, similar to singular defects. Production and observation of singular and solitonic topological structures remain a complex undertaking in most branches of science - but in soft matter physics, they can be realized within the director field of a liquid crystal. Additionally, it has been shown that confining liquid crystals to spherical shells using microfluidics resulted in a versatile experimental platform for the dynamical study of topological transformations between director configurations. In this work, we demonstrate the triggered formation of topological solitons, cholesteric fingers, singular defect lines and related structures in liquid crystal shells. We show that to accommodate these objects, shells must possess a Janus nature, featuring both twisted and untwisted domains. We report the formation of linear and axisymmetric objects, which we identify as cholesteric fingers and skyrmions or elementary torons, respectively. We then take advantage of the sensitivity of shells to numerous external stimuli to induce dynamical transitions between various types of structures, allowing for a richer phenomenology than traditional liquid crystal cells with solid flat walls. Using gradually more refined experimental techniques, we induce the targeted transformation of cholesteric twist walls and fingers into skyrmions and elementary torons. We capture the different stages of these director transformations using numerical simulations. Finally, we uncover an experimental mechanism to nucleate arrays of axisymmetric structures on shells, thereby creating a system of potential interest for tackling crystallography studies on curved spaces.

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.

Nanoscale hyperspectral imaging of tilted cholesteric liquid crystal structures

Ongoing research on chiral liquid crystals takes advantage of the peculiar behavior of twisted structures subject to curvature. We demonstrate the fine tunability of the characteristics of the bandgap of a cholesteric structure in which the orientation of the helix axis spatially changes. To date, the spectral resolution of the order of 6 nm, herein reached by hyperspectral imaging, has not been solved in tilted helices. A correlation between spectral shifts and spatial twists is thus made possible.

Super-resolution stimulated emission depletion microscopy of director structures in liquid crystals

Extending the optical imaging of director structures in liquid crystals (LCs) beyond the diffraction limit is poised to provide insights into previously elusive LC physics at the nanoscale. Here, we develop and characterize super-resolution stimulated emission depletion microscopy with molecular orientation sensitivity and apply it to reveal spatially localized director structures in chiral nematic and smectic LCs. As examples, we demonstrate director imaging and reconstruction of nanoscale LC configurations, including solitonic Bloch walls and two-dimensional skyrmions, both observed in sub-micrometer-thick strongly confined LC films, and focal conic domains in smectic LCs. The ≲100  nm resolution of our orientation-resolved STED imaging is three times better than that of fluorescence confocal polarizing microscopy and polarized nonlinear imaging techniques, but can be potentially improved even further.

Liquid Crystals under Confinement in Submicrometer Capsules

Liquid crystal (LC) ordering and phase transition behavior under confined conditions have attracted extensive attention and enabled many applications. However, the ordering and phase transition behavior of LCs in submicrometer capsules have seldom been studied, primarily due to the lack of proper capsulizing and visualization approaches to such small LC microcapsules. Herein, we achieve submicrometer LC capsules with the sizes down to 100 nm by using emulsion-based interfacial sol-gel reaction. The behavior of LCs under the submicrometer confinement conditions is investigated while the sizes and chemical composition of the microcapsule shell surface are tuned in a controllable way. The phase transition temperatures of LCs in the submicrometer capsules shift from those of bulk LCs due to the surface-induced ordering of LCs under the strong confinement conditions, which causes formation of topological defects and alters the order parameter. Using nonlinear optical imaging technology, we explore the structures of director field of LCs that arise as a result of the competition between the surface boundary conditions and LC elasticity. The results show that the nanoscale encapsulation can significantly influence the structural configurations of the director and phase transitions of LCs under various confinement conditions.

Repulsion-attraction switching of nematic colloids formed by liquid crystal dispersions of polygonal prisms

Self-assembly of colloidal particles due to elastic interactions in nematic liquid crystals promises tunable composite materials and can be guided by exploiting surface functionalization, geometric shape and topology, though these means of controlling self-assembly remain limited. Here, we realize low-symmetry achiral and chiral elastic colloids in the nematic liquid crystals using colloidal polygonal concave and convex prisms. We show that the controlled pinning of disclinations at the prism edges alters the symmetry of director distortions around the prisms and their orientation with respect to the far-field director. The controlled localization of the disclinations at the prism's edges significantly influences the anisotropy of the diffusion properties of prisms dispersed in liquid crystals and allows one to modify their self-assembly. We show that elastic interactions between polygonal prisms can be switched between repulsive and attractive just by controlled re-pinning the disclinations at different edges using laser tweezers. Our findings demonstrate that elastic interactions between colloidal particles dispersed in nematic liquid crystals are sensitive to the topologically equivalent but geometrically rich controlled configurations of the particle-induced defects.

Squirming motion of baby skyrmions in nematic fluids

Skyrmions are topologically protected continuous field configurations that cannot be smoothly transformed to a uniform state. They behave like particles and give origins to the field of skyrmionics that promises racetrack memory and other technological applications. Unraveling the non-equilibrium behavior of such topological solitons is a challenge. We realize skyrmions in a chiral liquid crystal and, using numerical modeling and polarized video microscopy, demonstrate electrically driven squirming motion. We reveal the intricate details of non-equilibrium topology-preserving textural changes driving this behavior. Direction of the skyrmion’s motion is robustly controlled in a plane orthogonal to the applied field and can be reversed by varying frequency. Our findings may spur a paradigm of soliton dynamics in soft matter, with a rich interplay between topology, chirality, and orientational viscoelasticity.

A skyrmion is a topological object originally introduced to model elementary particles and a baby skyrmion is its two-dimensional counterpart which can be realized as a defect in liquid crystals. Here the authors show that an electric field can drive uniform motion of baby skyrmions in liquid crystals.

Nonsingular defects and self-assembly of colloidal particles in cholesteric liquid crystals

Cholesteric liquid crystals can potentially provide a means for tunable self-organization of colloidal particles. However, the structures of particle-induced defects and the ensuing elasticity-mediated colloidal interactions in these media remain much less explored and understood as compared to their nematic liquid crystal counterparts. Here we demonstrate how colloidal microspheres of varying diameter relative to the helicoidal pitch can induce dipolelike director field configurations in cholesteric liquid crystals, where these particles are accompanied by point defects and a diverse variety of nonsingular line defects forming closed loops. Using laser tweezers and nonlinear optical microscopy, we characterize the ensuing medium-mediated elastic interactions and three-dimensional colloidal assemblies. Experimental findings show a good agreement with numerical modeling based on minimization of the Landau-de Gennes free energy and promise both practical applications in the realization of colloidal composite materials and a means of controlling nonsingular topological defects that attract a great deal of fundamental interest.

Self-assembly of colloidal particles in deformation landscapes of electrically driven layer undulations in cholesteric liquid crystals

We study elastic interactions between colloidal particles and deformation landscapes of undulations in a cholesteric liquid crystal under an electric field applied normal to cholesteric layers. The onset of undulation instability is influenced by the presence of colloidal inclusions and, in turn, layers' undulations mediate the spatial patterning of particle locations. We find that the bending of cholesteric layers around a colloidal particle surface prompts the local nucleation of an undulations lattice at electric fields below the well-defined threshold known for liquid crystals without inclusions, and that the onset of the resulting lattice is locally influenced, both dimensionally and orientationally, by the initial arrangements of colloids defined using laser tweezers. Spherical particles tend to spatially localize in the regions of strong distortions of the cholesteric layers, while colloidal nanowires exhibit an additional preference for multistable alignment offset along various vectors of the undulations lattice. Magnetic rotation of superparamagnetic colloidal particles couples with the locally distorted helical axis and undulating cholesteric layers in a manner that allows for a controlled three-dimensional translation of these particles. These interaction modes lend insight into the physics of liquid crystal structure-colloid elastic interactions, as well as point the way towards guided self-assembly of reconfigurable colloidal composites with potential applications in diffraction optics and photonics.

Edge pinning and transformation of defect lines induced by faceted colloidal rings in nematic liquid crystals

Nematic colloids exhibit a large diversity of topological defects and structures induced by colloidal particles in the orientationally ordered liquid crystal host fluids. These defects and field configurations define elastic interactions and medium-mediated self-assembly, as well as serve as model systems in exploiting the richness of interactions between topologies and geometries of colloidal surfaces, nematic fields, and topological singularities induced by particles in the nematic bulk and at nematic-colloidal interfaces. Here we demonstrate formation of quarter-strength surface-pinned disclinations, as well as a large variety of director field configurations with splitting and reconnections of singular defect lines, prompted by colloidal particles with sharp edges and size large enough to define strong boundary conditions. Using examples of faceted ring-shaped particles of genus g=1, we explore transformation of defect lines as they migrate between locations in the bulk of the nematic host to edge-pinned locations at the surfaces of particles and vice versa, showing that this behavior is compliant with topological constraints defined by mathematical theorems. We discuss how transformation of bulk and surface defect lines induced by faceted colloids can enrich the diversity of elasticity-mediated colloidal interactions and how these findings may impinge on prospects of their controlled reconfigurable self-assembly in nematic hosts.

Hexadecapolar colloids

Outermost occupied electron shells of chemical elements can have symmetries resembling that of monopoles, dipoles, quadrupoles and octupoles corresponding to filled s-, p-, d- and f-orbitals. Theoretically, elements with hexadecapolar outer shells could also exist, but none of the known elements have filled g-orbitals. On the other hand, the research paradigm of ‘colloidal atoms' displays complexity of particle behaviour exceeding that of atomic counterparts, which is driven by DNA functionalization, geometric shape and topology and weak external stimuli. Here we describe elastic hexadecapoles formed by polymer microspheres dispersed in a liquid crystal, a nematic fluid of orientationally ordered molecular rods. Because of conically degenerate boundary conditions, the solid microspheres locally perturb the alignment of the nematic host, inducing hexadecapolar distortions that drive anisotropic colloidal interactions. We uncover physical underpinnings of formation of colloidal elastic hexadecapoles and describe the ensuing bonding inaccessible to elastic dipoles, quadrupoles and other nematic colloids studied previously.

Colloidal systems can form bulk phases such as liquid and crystals, but they also exhibit interesting behaviours that have no atomic analogues. Here, by dispersing solid polymer microspheres in a nematic liquid crystal, Senyuk et al. demonstrate spontaneous formation of hexadecapolar nematic colloids.

Colloidal spirals in nematic liquid crystals

One of the central experimental efforts in nematic colloids research aims to explore how the interplay between the geometry of particles along with the accompanying nematic director deformations and defects around them can provide a means of guiding particle self-assembly and controlling the structure of particle-induced defects. In this work, we design, fabricate, and disperse low-symmetry colloidal particles with shapes of spirals, double spirals, and triple spirals in a nematic fluid. These spiral-shaped particles, which are controlled by varying their surface functionalization to provide tangential or perpendicular boundary conditions of the nematic molecular alignment, are found inducing director distortions and defect configurations with non-chiral or chiral symmetry. Colloidal particles also exhibit both stable and metastable multiple orientational states in the nematic host, with a large number of director configurations featuring both singular and solitonic nonsingular topological defects accompanying them, which can result in unusual forms of colloidal self-assembly. Our findings directly demonstrate how the symmetry of particle-generated director configurations can be further lowered, or not, as compared to the low point group symmetry of solid micro-inclusions, depending on the nature of induced defects while satisfying topological constraints. We show that achiral colloidal particles can cause chiral symmetry breaking of elastic distortions, which is driven by complex three-dimensional winding of induced topological line defects and solitons.

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.

Metal nanoparticle dispersion, alignment, and assembly in nematic liquid crystals for applications in switchable plasmonic color filters and E-polarizers

Viewing angle characteristics of displays and performance of electro-optic devices are often compromised by the quality of dichroic thin-film polarizers, while dichroic optical filters usually lack tunability and cannot work beyond the visible part of optical spectrum. We demonstrate that molecular-colloidal organic-inorganic composites formed by liquid crystals and relatively dilute dispersions of orientationally ordered anisotropic gold nanoparticles, such as rods and platelets, can be used in engineering of switchable plasmonic polarizers and color filters. The use of metal nanoparticles instead of dichroic dyes allows for obtaining desired polarizing or scattering and absorption properties not only within the visible but also in the infrared parts of an optical spectrum. We explore spontaneous surface-anchoring-mediated alignment of surface-functionalized anisotropic gold nanoparticles and its control by low-voltage electric fields, elastic colloidal interactions and self-assembly, as well as the uses of these effects in defining tunable properties of the ensuing organic-inorganic nanostructured composites. Electrically tunable interaction of the composites may allow for engineering of practical electro-optic devices, such as a new breed of color filters and plasmonic polarizers.

Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals

Some of the most exotic condensed matter phases, such as twist grain boundary and blue phases in liquid crystals and Abrikosov phases in superconductors, contain arrays of topological defects in their ground state. Comprised of a triangular lattice of double-twist tubes of magnetization, the so-called ‘A-phase’ in chiral magnets is an example of a thermodynamically stable phase with topologically nontrivial solitonic field configurations referred to as two-dimensional skyrmions, or baby-skyrmions. Here we report that three-dimensional skyrmions in the form of double-twist tori called ‘hopfions’, or ‘torons’ when accompanied by additional self-compensating defects, self-assemble into periodic arrays and linear chains that exhibit electrostriction. In confined chiral nematic liquid crystals, this self-assembly is similar to that of liquid crystal colloids and originates from long-range elastic interactions between particle-like skyrmionic torus knots of molecular alignment field, which can be tuned from isotropic repulsive to weakly or highly anisotropic attractive by low-voltage electric fields.

Topological defects can be spontaneously generated to thermodynamically stabilize a variety of peculiar condensed matter phases for technological applications. Here, Ackerman et al. show electrically controllable self-assembly of knotted defects into periodic arrays in chiral liquid crystals.

Topological switching and orbiting dynamics of colloidal spheres dressed with chiral nematic solitons

Metastable configurations formed by defects, inclusions, elastic deformations and topological solitons in liquid crystals are a promising choice for building photonic crystals and metamaterials with a potential for new optical applications. Local optical modification of the director or introduction of colloidal inclusions into a moderately chiral nematic liquid crystal confined to a homeotropic cell creates localized multistable chiral solitons. Here we induce solitons that “dress” the dispersed spherical particles treated for tangential degenerate boundary conditions, and perform controlled switching of their state using focused optical beams. Two optically switchable distinct metastable states, toron and hopfion, bound to colloidal spheres into structures with different topological charges are investigated. Their structures are examined using Q-tensor based numerical simulations and compared to the profiles reconstructed from the experiments. A topological explanation of observed multistability is constructed.

Magnetically responsive gourd-shaped colloidal particles in cholesteric liquid crystals

Particle shape and medium chirality are two key features recently used to control anisotropic colloidal self-assembly and dynamics in liquid crystals. Here, we study magnetically responsive gourd-shaped colloidal particles dispersed in cholesteric liquid crystals with periodicity comparable or smaller than the particle's dimensions. Using magnetic manipulation and optical tweezers, which allow one to position colloids near the confining walls, we measured the elastic repulsive interactions of these particles with confining surfaces and found that separation-dependent particle-wall interaction force is a non-monotonic function of separation and shows oscillatory behavior. We show that gourd-shaped particles in cholesterics reside not on a single sedimentation level, but on multiple long-lived metastable levels separated by a distance comparable to cholesteric periodicity. Finally, we demonstrate three-dimensional laser tweezers assisted assembly of gourd-shaped particles taking advantage of both orientational order and twist periodicity of cholesterics, potentially allowing new forms of orientationally and positionally ordered colloidal organization in these media.

Towards template-assisted assembly of nematic colloids

Colloidal crystals belong to a new class of materials with unusual properties in which the big challenge is to grow large-scale structures of a given symmetry in a well-controlled and inexpensive way. Recently, template-assisted crystallization was successfully exploited experimentally in the case of colloidal particles dispersed in isotropic fluids. In liquid crystal (LC) colloids, particles are subjected to long-range anisotropic elastic forces originating from the anisotropic deformation of the underlying order parameter. These effective interactions are easily tunable by external electric or magnetic fields, light, temperature, or confinement and, thus, provide additional handles for better control of colloidal assembly. Here we use the coupling between microsculptured bounding surfaces and LC elasticity in order to guide the self-assembly of large-scale colloidal structures. We present explicit numerical calculations of the free energy landscape of colloidal particles in the presence of convex protrusions modeled as squared pyramids comparable to the size of the particles. We show the existence of strong trapping potentials that are able to efficiently localize the colloidal particles and withstand thermal fluctuations. Three-dimensional optical imaging experiments support the theoretical predictions.

Effect of plasmon-enhancement on photophysics in upconverting nanoparticles

Surface plasmon polaritons (SPP) waves have been shown to significantly affect the near-field photophysical phenomenon. In particular, strong Coulombic interactions can enhance nearby non-linear optics and energy transfer process, while SPP waves also affect other photophysical processes like quenching observed in fluorescent and excitonic systems. Here, using different plasmonic substrates, we show the effect of plasmon-enhancement on quenching, phonon-assisted non-radiative decay, weak Purcell effect or electromagnetic field enhancement, and energy transfer rates of upconverting doped-lanthanide nanoparticles. While the resonant plasmons enhance the local electromagnetic field and the rate of energy transfer leading to enhanced upconversion photoluminescence of infrared radiation to visible light, it can also increase the quenching and non-radiative decay rates of photoexcited electron-hole pairs leading to losses and lower efficiency. These results can guide the design of optimized substrate geometry for using surface plasmons to modulate the photophysics in other applications too.

Determining optimum operating conditions of the polarization-maintaining fiber with two far-lying zero dispersion wavelengths for CARS microscopy

Single femtosecond laser-based coherent anti-Stokes Raman scattering (CARS) microscopy, using a photonic crystal fiber (PCF) pumped in the near-IR to generate a supercontinuum for the Stokes source, is rapidly being adopted as a cost-effective approach. A PCF with two closely-lying zero dispersion wavelengths is a popular choice for the Stokes source, but it is often limited to imaging lipids. A polarization-maintaining PCF with two far-lying zero dispersion wavelengths offers important advantages for polarization CARS microscopy, and for CARS imaging in the fingerprint region. This PCF fiber, though commercially available, has limited use for CARS microscopy in the C-H bond region. The main problem is that the supercontinuum from this fiber is typically noisier than that from a standard PCF with two closely-lying zero dispersion wavelengths. To overcome this, we determined the optimum operating conditions for generating a low-noise supercontinuum out of a PCF with two far-lying zero dispersion wavelengths, in terms of the input parameters of the excitation pulse. We measured the relative intensity noise (RIN) of the Stokes and the corresponding CARS signal as a function of the input laser parameters in this fiber. We showed that the results of CARS imaging using this alternate fiber are comparable to those achieved using the standard fiber, for input laser pulse conditions of low average power, narrow pulse width with slightly positive chirp, and polarization direction parallel to the slow axis of the selected fiber.

Cell optical density and molecular composition revealed by simultaneous multimodal label-free imaging

We show how Raman imaging can be combined with independent but simultaneous phase measurements of unlabeled cells, with the resulting data providing information on how the light is retarded and/or scattered by molecules in the cell. We then show, for the first time to our knowledge, how the chemistry of the cell highlighted in the Raman information is related to the cell quantitative phase information revealed in digital holographic microscopy by quantifying how the two sets of spatial information are correlated. The results show that such a multimodal implementation is highly useful for the convenience of having video rate imaging of the cell during the entire Raman measurement time, allowing us to observe how the cell changes during Raman acquisition. More importantly, it also shows that the two sets of label-free data, which result from different scattering mechanisms, are complementary and can be used to interpret the composition and dynamics of the cell, where each mode supplies label-free information not available from the other mode.

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.

Plasmon-enhanced energy transfer for improved upconversion of infrared radiation in doped-lanthanide nanocrystals

Upconversion of infrared radiation into visible light has been investigated for applications in photovoltaics and biological imaging. However, low conversion efficiency due to small absorption cross-section for infrared light (Yb(3+)), and slow rate of energy transfer (to Er(3+) states) has prevented application of upconversion photoluminescence (UPL) for diffuse sunlight or imaging tissue samples. Here, we utilize resonant surface plasmon polaritons (SPP) waves to enhance UPL in doped-lanthanide nanocrystals. Our analysis indicates that SPP waves not only enhance the electromagnetic field, and hence weak Purcell effect, but also increase the rate of resonant energy transfer from Yb(3+) to Er(3+) ions by 6 fold. While we do observe strong metal mediated quenching (14-fold) of green fluorescence on flat metal surfaces, the nanostructured metal is resonant in the infrared and hence enhances the nanocrystal UPL. This strong Coulombic effect on energy transfer can have important implications for other fluorescent and excitonic systems too.

Rotational and translational diffusion of anisotropic gold nanoparticles in liquid crystals controlled by varying surface anchoring

We study translational and rotational diffusion of anisotropic gold nanoparticles (NPs) dispersed in the bulk of a nematic liquid crystal fluid host. Experimental data reveal strong anisotropy of translational diffusion with respect to the uniform far-field director, which is dependent on shape and surface functionalization of colloids as well as on their ground-state alignment. For example, elongated NPs aligned parallel to the far-field director translationally diffuse more rapidly along the director whereas diffusion of NPs oriented normal to the director is faster in the direction perpendicular to it while they are also undergoing elasticity-constrained rotational diffusion. To understand physical origins of these rich diffusion properties of anisotropic nanocolloids in uniaxially anisotropic nematic fluid media, we compare them to diffusion of prolate and oblate ellipsoidal particles in isotropic fluids as well as to diffusion of shape-isotropic particles in nematic fluids. We also show that surface functionalization of NPs with photosensitive azobenzene groups allows for in situ control of their diffusivity through trans-cis isomerization that changes surface anchoring.

Generating the Hopf fibration experimentally in nematic liquid crystals

The Hopf fibration is an example of a texture: a topologically stable, smooth, global configuration of a field. Here we demonstrate the controlled sculpting of the Hopf fibration in nematic liquid crystals through the control of point defects. We demonstrate how these are related to torons by use of a topological visualization technique derived from the Pontryagin-Thom construction.

Imaging of director fields in liquid crystals using stimulated Raman scattering microscopy

We demonstrate an approach for background-free three-dimensional imaging of director fields in liquid crystals using stimulated Raman scattering microscopy. This imaging technique is implemented using a single femtosecond pulsed laser and a photonic crystal fiber, providing Stokes and pump frequencies needed to access Raman shifts of different chemical bonds of molecules and allowing for chemically selective and broadband imaging of both pristine liquid crystals and composite materials. Using examples of model three-dimensional structures of director fields, we show that the described technique is a powerful tool for mapping of long-range molecular orientation patterns in soft matter via polarized chemical-selective imaging.

Active shape-morphing elastomeric colloids in short-pitch cholesteric liquid crystals

Active elastomeric liquid crystal particles with initial cylindrical shapes are obtained by means of soft lithography and polymerization in a strong magnetic field. Gold nanocrystals infiltrated into these particles mediate energy transfer from laser light to heat, so that the inherent coupling between the temperature-dependent order and shape allows for dynamic morphing of these particles and well-controlled stable shapes. Continuous changes of particle shapes are followed by their spontaneous realignment and transformations of director structures in the surrounding cholesteric host, as well as locomotion in the case of a nonreciprocal shape morphing. These findings bridge the fields of liquid crystal solids and active colloids, may enable shape-controlled self-assembly of adaptive composites and light-driven micromachines, and can be understood by employing simple symmetry considerations along with electrostatic analogies.

Topological colloids

Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids, a type of soft matter consisting of tiny particles dispersed in chemically distinct host media. Being abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses to fabrication of third-generation quantum-dot solar cells. Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth, their surfaces tend to have minimum-area spherical or topologically equivalent shapes such as prisms and irregular grains (all continuously deformable--homeomorphic--to spheres). Although toroidal DNA condensates and vesicles with different numbers of handles can exist and soft matter defects can be shaped as rings and knots, the role of particle topology in colloidal systems remains unexplored. Here we fabricate and study colloidal particles with different numbers of handles and genus g ranging from 1 to 5. When introduced into a nematic liquid crystal--a fluid made of rod-like molecules that spontaneously align along the so-called 'director'--these particles induce three-dimensional director fields and topological defects dictated by colloidal topology. Whereas electric fields, photothermal melting and laser tweezing cause transformations between configurations of particle-induced structures, three-dimensional nonlinear optical imaging reveals that topological charge is conserved and that the total charge of particle-induced defects always obeys predictions of the Gauss-Bonnet and Poincaré-Hopf index theorems. This allows us to establish and experimentally test the procedure for assignment and summation of topological charges in three-dimensional director fields. Our findings lay the groundwork for new applications of colloids and liquid crystals that range from topological memory devices, through new types of self-assembly, to the experimental study of low-dimensional topology.

Nonlinear photoluminescence imaging of isotropic and liquid crystalline dispersions of graphene oxide

We report a visible-range nonlinear photoluminescence (PL) from graphene oxide (GO) flakes excited by near-infrared femtosecond laser light. PL intensity has nonlinear dependence on the laser power, implying a multiphoton excitation process, and also strongly depends on a linear polarization orientation of excitation light, being at maximum when it is parallel to flakes. We show that PL can be used for a fully three-dimensional label-free imaging of isotropic, nematic, and lamellar liquid crystalline dispersions of GO flakes in water. This nonlinear PL is of interest for applications in direct label-free imaging of composite materials and study of orientational ordering in mesomorphic phases formed by these flakes, as well as in biomedical and sensing applications utilizing GO.

Plasmonic complex fluids of nematiclike and helicoidal self-assemblies of gold nanorods with a negative order parameter

We describe a soft matter system of self-organized oblate micelles and plasmonic gold nanorods that exhibit a negative orientational order parameter. Because of anisotropic surface anchoring interactions, colloidal gold nanorods tend to align perpendicular to the director describing the average orientation of normals to the discoidal micelles. Helicoidal structures of highly concentrated nanorods with a negative order parameter are realized by adding a chiral additive and are further controlled by means of confinement and mechanical stress. Polarization-sensitive absorption, scattering, and two-photon luminescence are used to characterize orientations and spatial distributions of nanorods. Self-alignment and effective-medium optical properties of these hybrid inorganic-organic complex fluids match predictions of a simple model based on anisotropic surface anchoring interactions of nanorods with the structured host medium.

Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams

Condensed matter systems with topological defects in the ground states range from the Abrikosov phases in superconductors, to various blue phases and twist grain boundary phases in liquid crystals, and to phases of skyrmion lattices in chiral ferromagnets and Bose-Einstein condensates. In nematic and chiral nematic liquid crystals, which are true fluids with long-range orientational ordering of constituent molecules, point and line defects spontaneously occur as a result of symmetry-breaking phase transitions or due to flow, but they are unstable, hard to control, and typically annihilate with time. Here we describe the optical generation of two-dimensional crystalline, quasicrystalline, and arbitrary ensembles of particlelike structures manifesting both skyrmionlike and Hopf fibration features--dubbed "torons"--composed of looped double twist cylinders and point defects embedded in a uniform director field. In these two-dimensional lattices, we then introduce various dislocations, defects in positional ordering of the torons. We show that the periodic defect lattices with and without dislocation are light- and voltage-tunable reconfigurable two-dimensional diffraction gratings and can be used to generate various controlled phase singularities in the diffracted laser beams. The results of computer simulations of optical images, diffraction patterns, and phase distributions with optical vortices are in a good agreement with the corresponding experimental findings.

Smectic shells

Confining a smectic liquid crystal between two spherical surfaces results in the formation of a complex defect structure characterized by a set of curvature walls that divide the sphere into crescent domains, causing the undulation of the smectic layers. In this paper, we examine in detail these smectic textures and discuss the various possible origins of the observed patterns.

Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities

Topological defect lines are ubiquitous and important in a wide variety of fascinating phenomena and theories in many fields ranging from materials science to early-universe cosmology, and to engineering of laser beams. However, they are typically hard to control in a reliable manner. Here we describe facile erasable “optical drawing” of self-assembled defect clusters in liquid crystals. These quadrupolar defect clusters, stabilized by the medium's chirality and the tendency to form twisted configurations, are shaped into arbitrary two-dimensional patterns, including reconfigurable phase gratings capable of generating and controlling optical phase singularities in laser beams. Our findings bridge the studies of defects in condensed matter physics and optics and may enable applications in data storage, singular optics, displays, electro-optic devices, diffraction gratings, as well as in both optically- and electrically-addressed pixel-free spatial light modulators.

Polarization holographic microscopy for extracting spatio-temporally resolved Jones matrix

We present a high-speed holographic microscopic technique for quantitative measurement of polarization light-field, referred to as polarization holographic microscopy (PHM). Employing the principle of common-path interferometry, PHM quantitatively measures the spatially resolved Jones matrix components of anisotropic samples with only two consecutive measurements of spatially modulated holograms. We demonstrate the features of PHM with imaging the dynamics of liquid crystal droplets at a video-rate.

Optically generated reconfigurable photonic structures of elastic quasiparticles in frustrated cholesteric liquid crystals

We describe laser-induced two-dimensional periodic photonic structures formed by localized particle-like excitations in an untwisted confined cholesteric liquid crystal. The individual particle-like excitations (dubbed "Torons") contain three-dimensional twist of the liquid crystal director matched to the uniform background director field by topological point defects. Using both single-beam-steering and holographic pattern generation approaches, the periodic crystal lattices are tailored by tuning their periodicity, reorienting their crystallographic axes, and introducing defects. Moreover, these lattices can be dynamically reconfigurable: generated, modified, erased and then recreated, depending on the needs of a particular photonic application. This robust control is performed by tightly focused laser beams of power 10-100 mW and by low-frequency electric fields at voltages ~10 V applied to the transparent electrodes.

Measurement of viscosity of lyotropic liquid crystals by means of rotating laser-trapped microparticles

We describe a simple microrheology method to measure the viscosity coefficients of lyotropic liquid crystals. This approach is based on the use of a rotating laser-trapped optically anisotropic microsphere. In aligned liquid crystals that have negligible effect on trapping beam's polarization, the optical torque is transferred from circularly polarized laser trapping beam to the optically anisotropic microparticle and creates the shear flow in the liquid crystalline fluid. The balance of optical and viscous torques yields the local effective viscosity coefficients of the studied lyotropic systems in cholesteric and lamellar phases. This simple yet powerful method is capable of probing viscosity of complex anisotropic fluids for small amounts of sample and even in the presence of defects that obstruct the use of conventional rheology techniques.

Healing of defects at the interface of nematic liquid crystals and structured Langmuir-Blodgett monolayers

We use Langmuir-Blodgett molecular monolayers and nematic liquid crystals as model two- and three-dimensional orientationally ordered systems to study the stability and healing of topological defects at their contact interfaces. Integer-strength defects at the monolayer induce disclinations of similar strength in the nematic that, however, do not propagate deep into the bulk, but rather form single- or double-split arch-shaped loops pinned to the interface. This behavior is qualitatively independent of the far-field director orientation and involves either half-integer singular or twist-escaped unity-strength nonsingular nematic disclinations. These two defect configurations can be selected by varying sample preparation given their comparable free energy, consistently with direct probing by use of laser tweezers.