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Phoamtonic Designs Yield Sizeable 3D Photonic Band Gaps

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Phoamtonic Designs Yield Sizeable 3D Photonic Band Gaps Phoamtonic designs yield sizeable 3D photonic band gaps Michael A. Klatta,1, Paul J. Steinhardta,1, and Salvatore Torquatoa,b,c,d,1 aDepartment of Physics, Princeton University, Princeton, NJ 08544; bDepartment of Chemistry, Princeton University, Princeton, NJ 08544; cPrinceton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544; and dProgram in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544 Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 7, 2019 (received for review July 23, 2019) We show that it is possible to construct foam-based heterostruc- In this paper, we consider phoamtonics, the potential applica- tures with complete photonic band gaps. Three-dimensional tion of these foam-based architectures to photonic materials. For foams are promising candidates for the self-organization of large this purpose, the essential question is whether the resulting mate- photonic networks with combinations of physical characteristics rials have complete photonic band gaps (PGBs) that prohibit that may be useful for applications. The largest band gap found is the propagation of light in all directions and for all polarizations based on 3D Weaire–Phelan foam, a structure that was originally for a substantial range of frequencies. If so, they would expand introduced as a solution to the Kelvin problem of finding the 3D the range of 3D heterostructures available for photonic appli- tessellation composed of equal-volume cells that has the least sur- cations beyond photonic crystals (22–24), quasicrystals (25–27), face area. The photonic band gap has a maximal size of 16.9% and amorphous networks (28–30). (at a volume fraction of 21.6% for a dielectric contrast " = 13) Here we demonstrate that a 3D photonic network based on and a high degree of isotropy, properties that are advantageous the Weaire–Phelan foam results in a substantial complete PBG in designing photonic waveguides and circuits. We also present (16.9%) with a high degree of isotropy, an important property results for 2 other foam-based heterostructures based on Kelvin for photonic circuits. We find substantial PBGs for the Kelvin and C15 foams that have somewhat smaller but still significant and C15 foam, but the Weaire–Phelan network has the largest band gaps. phoamtonic band gap. This provides impetus for further studies of foam-based photonic materials. Plateau’s laws for dry foam j Weaire–Phelan foam j TCP structures and There are 2 motivations for phoamtonics. First, Plateau’s laws Frank–Kasper phases j self-organization j complete photonic band gap guarantee exclusively tetrahedral vertices. The latter are gen- erally thought to be advantageous for photonic networks, as explained below. This naturally relates photonics to the Kelvin oams are both of fundamental interest (1) and important problem, that is, the search for the 3D tessellation composed Ffor a host of applications ranging from chemical filters to of equal-volume cells that has the least surface area—a prob- heat exchangers (2). They locally minimize the surface area of lem of fundamental interest in both mathematics and physics. their cells subject to volume constraints. In the limit of vanishing Its connections to famous packing and covering problems (31) liquid fraction, the so-called dry foam is completely character- relate foams, for example, to Frank–Kasper phases (6, 14, 32–34) ized by Plateau’s laws, where each cell vertex is tetrahedral; and their dual clathrate structures (35–37). The self-assembly∗ of that is, 4 edges meet with bond angles equal to arccos(−1=3) ≈ 109◦. These laws were first observed empirically (3) and later Significance proven rigorously for area-minimizing tessellations with volume constraints (4, 5). Kelvin posed the fundamental question: What tessellation with Foams appear ubiquitously in many fields of science and cells of equal volume has the least surface area? He conjec- technology. However, applications to photonics have not tured that the solution is a relaxation of the Voronoi tessellation been possible because until now, foam-based 3D heterostruc- of the body-centered cubic (BCC) lattice—now known as the tures with substantial complete photonic band gaps have Kelvin foam (1). More than 100 y later, Weaire and Phelan not been identified. Here we provide explicit examples based disproved the conjecture by finding a foam with a slightly smaller on the edges of dry-crystalline foams, including the famous surface area (6). Their numerical finding was later rigorously Weaire–Phelan foam. More generally, foam networks based confirmed (7). The Weaire–Phelan foam can be obtained by on Frank–Kasper phases offer new, promising approaches to relaxing a weighted Voronoi tessellation of the A15 crystal. the rapid self-organization of large photonic networks. More- Importantly, the foam has been not only studied numerically over, foams are excellent candidates for designer materials (8) but also realized experimentally (9). Whether it is the with versatile multifunctional characteristics. globally optimal solution to Kelvin’s problem is still an open Author contributions: M.A.K., P.J.S., and S.T. designed research; M.A.K., P.J.S., and S.T. question. The Weaire–Phelan foam has 2 types of cells with dif- performed research; M.A.K. analyzed data; and M.A.K., P.J.S., and S.T. wrote the paper.y ferent pressures. So the Kelvin foam has still the least surface The authors declare no competing interest.y area among known foams with equal pressure in each cell. Kelvin cells have also been found in experiments (10–12). The A15 This article is a PNAS Direct Submission.y crystal (β-tungsten) belongs to a larger class of so-called tetrahe- Published under the PNAS license.y drally close-packed (TCP) structures (or Frank–Kasper phases) Data deposition: All data generated or analyzed for this study, including configura- tions, parameter files, raw output, and postprocessed data, are available at the Zenodo (13). This class includes, for example, the C15 crystal (cubic repository, https://zenodo.org/, as Databases 1–3 (DOI: 10.5281/zenodo.3401635). y Friauf–Laves phase). Relaxing its Voronoi diagram, subject to 1 To whom correspondence may be addressed. Email: [email protected], steinh@ the constraint that each cell has the same volume, defines the princeton.edu, or [email protected] C15 foam (14). Its surface area is larger than that of the Kelvin This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. foam. Here we study these 3 foams—Weaire–Phelan, Kelvin, 1073/pnas.1912730116/-/DCSupplemental.y and C15—as classic examples of crystalline foams. Their electri- First published November 6, 2019. cal, mechanical, and transport properties have been intensively *In this paper, we apply the term “self-assembly” to equilibrium systems, whereas “self- studied elsewhere (15–21). organization” also refers to athermal or nonequilibrium systems. 23480–23486 j PNAS j November 19, 2019 j vol. 116 j no. 47 www.pnas.org/cgi/doi/10.1073/pnas.1912730116 Downloaded by guest on September 30, 2021 Frank–Kasper phases in soft-matter systems has recently sparked tonic PBGs of up to 17%, comparable to or exceeding those of great interest (38). Second, the utility of foams for applica- typical examples of self-organizing photonic crystals like geomet- tions is proven by their widespread use in technology. Par- rically optimized synthetic opals (42). To experimentally realize ticularly promising for photonics are their self-organization, such thickened edges, a dry foam needs to be first solidified multifunctionality, and a high degree of isotropy among crystal and then coated by a dielectric material as recently suggested structures. by Ricouvier et al. (39). In fact, a similar procedure has already Recently, Ricouvier et al. (39) studied 2D dry and wet foams been used to convert a 3D polymer network into a photonic as a self-organizing PBG material. These authors demonstrated silicon-coated network with TiO2 cores designed for infrared that 2D amorphous dry foams exhibit PBGs only for transverse wavelengths (43). electric polarization. They found that the PBG is incomplete The somewhat reduced PBG compared to diamond photonic for the transverse magnetic polarization. The authors suggested crystals may be compensated by the additional advantages of continuing the quest for a complete PGB by considering foams phoamtonics compared to other photonic materials. For exam- in 3D. This is a nontrivial endeavor: dimensionality dramati- ple, the self-organization of foams may allow for rapid fabrica- cally changes the picture because in 3D, there are 2 polariza- tion of large samples. Also, due to the extensive previous studies tions that couple, in contrast to 2D. Nevertheless, stimulated of the physical properties of foams, such as their stiffness and by their suggestion, we pursued the 3D case and found that conductivity (15–18), photonic foam-based heterostructures may 3D foam-based heterostructures with substantial complete PBGs be excellent candidates for certain multipurpose applications. are possible. Furthermore, as we will demonstrate, Weaire–Phelan and C15 A reason why one might consider 3D foams to be promising networks have complete PBGs that are nearly isotropic, which is as photonic materials is that Plateau’s laws guarantee that the useful for applications ranging from highly isotropic and efficient edges form exclusively tetrahedral vertices. The latter are well thermal radiation sources (44) to more adjustable bending angles known to be favorable for photonic networks (24, 30, 40). For in waveguides (27, 45, 46) and photonic circuits. example, the network with the largest PBG of about 30% is the diamond crystal (40), where the gap size is measured by the ratio Geometry and Topology of Foam Edges of the frequency width ∆! of the gap and the midgap frequency Here we study dry foams that are defined as local minima of the !m .
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