Holographic assembly of quasicrystalline photonic heterostructures

Quasicrystals have a higher degree of rotational and point-reflection symmetry than conventional crystals. As a result, quasicrystalline heterostructures fabricated from dielectric materials with micrometer-scale features exhibit interesting and useful optical properties including large photonic bandgaps in two-dimensional systems. We demonstrate the holographic assembly of two-dimensional and three-dimensional dielectric quasicrystalline heterostructures, including structures with specifically engineered defects. The highly uniform quasiperiodic arrays of optical traps used in this process also provide model aperiodic potential energy landscapes for fundamental studies of transport and phase transitions in soft condensed matter systems.

Quasicrystals have long-ranged orientational order even though they lack the translational periodicity of crystals. Not limited by conventional spatial point groups, they can adopt rotational symmetries that are forbidden to crystals. The resulting large number of effective reciprocal lattice vectors endows quasicrystals' effective Brillouin zones with an unusually high degree of rotational and point inversion symmetry [1]. These symmetries, in turn, facilitate the the development of photonic band gaps (PBG) [2] for light propagating through quasicrystalline dielectric heterostructures [3,4,5], even when the dielectric contrast among the constituent materials is low. Photonic band gaps have been realized in one- [6] and two-dimensional [7] lithographically defined quasiperiodic structures. Here we demonstrate rapid assembly of arbitrary materials into two-and threedimensional quasicrystalline heterostructures with features suitable for photonic device applications.
Our approach is based on the holographic opti- cal trapping technique [8,9,10] in which computergenerated holograms are projected through a highnumerical-aperture microscope objective lens to create large three-dimensional arrays of optical traps. In our implementation, light at 532 nm from a frequency-doubled diode-pumped solid state laser (Coherent Verdi) is imprinted with phase-only holograms using a liquid crystal spatial light modulator (SLM) (Hamamatsu X8267 PPM). The modified laser beam is relayed to the input pupil of a 100× NA 1.4 SPlan Apo oil immersion objective mounted in an inverted optical microscope (Nikon TE-2000U), which focuses it into traps. The same objective lens is used to form images of trapped objects, using the microscope's conventional imaging train [10]. We used this system to organize colloidal silica microspheres 1.53 µm in diameter (Duke Scientific Lot 5238) dispersed in an aqueous solution of 180 : 12 : 1 (wt/wt) acrylamide, N, N ′ -methylenebisacrylamide and diethoxyacetophenone (all Aldrich electrophoresis grade). This solution rapidly photopolymerizes into a transparent polyacrylamide hydrogel under ultraviolet illumination, and is stable otherwise. Fluid dispersions were imbibed into 30 µm thick slit pores formed by bonding the edges of #1 coverslips to the faces of glass micro- scope slides. The sealed samples were then mounted on the microscope's stage for processing and analysis.
Silica spheres are roughly twice as dense as water and sediment rapidly into a monolayer above the coverslip. A dilute layer of spheres is readily organized by holographic optical tweezers into arbitrary two-dimensional configurations, including the quasicrystalline examples in Fig. 1. Figures 1(a), (b) and (c) show planar pentagonal, heptagonal and octagonal quasicrystalline domains [11], respectively, each consisting of more than 100 particles. Highlighted spheres emphasize each domain's symmetry. These structures all have been shown to act as two-dimensional PBG materials in microfabricated arrays of posts and holes [3,12,13,14]. As a soft fabrication technique, holographic assembly requires substantially less processing than conventional methods such as electron-beam lithography, and can be applied to a wider range of materials. Unlike complementary optical fabrication techniques such as multiple-beam holographic photopolymerization [14,15,16,17], assembly with holographic optical traps lends itself to creating nonuniform architectures with specifically engineered features, such as the channel embedded in the octagonal domain in Fig. 1(d). Similar structures of comparable dimensions have been shown to act as narrow-band waveguides and frequency-selective filters for visible light [12,13,18,19].
Holographic trapping's ability to assemble free-form heterostructures extends also to three dimensions. The sequence of images of a rolling icosahedron in Fig. 2 shows how the colloidal spheres' appearance changes with distance from the focal plane. This sequence also recalls earlier reports [20,21] that holographic traps can successfully organize spheres into vertical stacks along the optical axis, while maintaining one sphere in each trap.
The icosahedron itself is the fundamental building block of a class of three-dimensional quasicrystals, such as the example in Fig. 3. Building upon our earlier work on holographic assembly [22], we assemble a threedimensional quasicrystalline domain by first creating a two-dimensional arrangement of spheres corresponding to the planar projection of the planned quasicrystalline domain, Fig. 3(a). Next, we translate the spheres along the optical axis to their final three-dimensional coordinates in the quasicrystalline domain, as shown in Fig. 3(b). One icosahedral unit is highlighted in Figs. 3(a) and (b) to clarify this process. Finally, the separation between the traps is decreased in Fig. 3(c) to create an optically dense structure. This particular domain consists of 173 spheres in roughly 7 layers, with typical inter-particle separations of 3 µm.
The completed quasicrystal was gelled and its optical diffraction pattern recorded at a wavelength of 632 nm by illuminating the sample with a collimated beam from a HeNe laser, collecting the diffracted light with the microscope's objective lens and projecting it onto a chargecoupled device (CCD) camera with a Bertrand lens. The well-defined diffraction spots clearly reflect the quasicrystal's five-fold rotational symmetry in the projected plane.
Holographic assembly of colloidal silica quasicrystals in water is easily generalized to other materials and solvents. Deterministic organization of disparate components under holographic control can be used to embed gain media in PBG cavities, to install materials with nonlinear optical properties within waveguides to form switches, and to create domains with distinct chemical functionalization. The comparatively small domains we have created can be combined into larger heterostructures through sequential assembly and spatially localized photopolymerization. In all cases, this soft fabrication process results in mechanically and environmentally stable materials that can be integrated readily into larger systems.
Beyond the immediate application of holographic trapping to fabricating quasicrystalline materials, the ability to create and continuously optimize such structures provides new opportunities for studying the dynamics [10] and statistical mechanics [23] of colloidal quasicrystals. The optically generated quasiperiodic potential energy landscapes developed for this study also should provide a flexible model system for experimental studies of transport [24] through aperiodically modulated environments.
We are grateful to Paul Steinhardt, Paul Chaikin and Weining Man for illuminating conversations. Support was provided by the National Science Foundation through Grant Number DMR-0451589.