Local self-uniformity in photonic networks

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Received 13 Jul 2016 | Accepted 30 Dec 2016 | Published 17 Feb 2017

The interaction of a material with light is intimately related to its wavelength-scale structure. Simple connections between structure and optical response empower us with essential intuition to engineer complex optical functionalities. Here we develop local self-uniformity (LSU) as a measure of a random networks internal structural similarity, ranking networks on a continuous scale from crystalline, through glassy intermediate states, to chaotic congurations. We demonstrate that complete photonic bandgap structures possess substantial LSU and validate LSUs importance in gap formation through design of amorphous gyroid structures. Amorphous gyroid samples are fabricated via three-dimensional ceramic printing and the bandgaps experimentally veried. We explore also the wing-scale structuring in the buttery Pseudolycaena marsyas and show that it possesses substantial amorphous gyroid character, demonstrating the subtle order achieved by evolutionary optimization and the possibility of an amorphous gyroids self-assembly.

DOI: 10.1038/ncomms14439 OPEN

Local self-uniformity in photonic networks

Steven R. Sellers1, Weining Man2, Shervin Sahba2 & Marian Florescu1

1 Advanced Technology Institute and Department of Physics, University of Surrey, Guildford GU2 7XH, UK. 2 Department of Physics and Astronomy, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94132, USA. Correspondence and requests for materials should be addressed to S.R.S. (email: mailto:[email protected]

Web End [email protected] ) or to M.F. (email: mailto:[email protected]

Web End [email protected] ).

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Acomplete photonic bandgap (PBG) is frequency window within which a material, by virtue of its structure, supports no propagating electromagnetic modes. Typi

cally, structures which possess complete PBGs are periodic arrays of dielectric material; such arrays are called photonic crystals (PhCs). PhCs have the potential to play a key role in the development of next-generation photonic integrated circuits13. However, although the complexity of PhC-based technologies continues to grow, questions regarding the fundamental mechanisms of PBG formation remain unresolved46.

The formation of PBGs is conventionally interpreted as a result of coherent scattering by a PhCs lattice planes7,8. In this picture, a plane wave may be scattered onto its counter-propagating equivalent when the wavevector of the initial state lies on the edge of the PhCs Brillouin zone (BZ). When this condition is met, a pair of orthogonal standing wave modes, each possessing a distinct electromagnetic eld prole, is formed5,9,10. Energetic interaction between the electric eld and the underlying dielectric distribution then breaks the degeneracy of the standing wave states. For the specic propagation direction under consideration, the resulting forbidden spectral range denes a photonic stop gap.

To engineer a complete PBG, photonic stop gaps must open along all propagation directions. Further, these stops gaps must be spectrally aligned. Both these considerations can be addressed by designing PhCs to possess maximally spherical BZs11. The search for the rst complete PBG thus focussed on face-centred cubic crystalsthe most isotropic of the three-dimensional (3D) Bravais latticesand discovered a large complete gap in a diamond-like network of dielectric material12. In spite of the many PhC designs that have since been discovered, those based on the diamond network remain the champion, possessing the largest complete PBGs13,14.

There is, however, much evidence to suggest that PBG formation is governed by more than just coherent scattering processes. PhCs derived from the body-centred cubic single-network gyroid (SNG) structure (triamond) and low-symmetry diamond embeddings all possess near-champion PBGs in spite of their less spherical BZs13. Although based on a face-centred cubic lattice, the inverse opal network exhibits a complete PBG only one quarter the size of the champion diamond gap1. Most tellingly, a glassy 3D networkdubbed photonic amorphous diamond (PAD)exhibits a sizeable complete PBG4. This gap exists despite PADs diffuse primary diffraction maximum which spreads the structures coherent scattering power of a range of wavevectors5.

Similar evidence is found in two-dimensional structures that possess PBGs for light with a transverse electric (TE) polarization. Amongst these structures, the champion PhC design is a honeycomb network of dielectric material15. Many glassy networks that can be broadly styled as hyperuniform disordered honeycombs have been found to possess sizeable TE PBGs6,1620. As with PAD, these gaps exist despite the diminished coherent scattering power of each structure.

Here, we address the mechanisms governing PBG formation by re-formulating the ideal structural properties of a PBG-forming network. To achieve this, we introduce the concept of local self-uniformity (LSU). LSU measures the geometrical and topological similarities of the local environments in a connected network of uniform valency. We note that existing sizeable PBG networks possess signicant LSU. We demonstrate the connection between LSU and PBG forming ability by designing novel amorphous gyroid (amorphous triamond) connected networks. Specically, amorphous gyroids can possess sizeable PBGs and an amorphous gyroids LSU is strongly correlated with its PBG width. This correlation is explained by recognizing the contribution of spatially localized resonant scattering processes to PBG formation

in connected networks. Locally self-uniform ceramic 3D-printed amorphous gyroids are characterized through microwave transmission experiments. We explore also the possibility that amorphous gyroid exists within the wing scales of butteries. In particular, we reveal that the microstructure in the scales of Pseudolycaena marsyas possesses substantial amorphous gyroid character and demonstrate that the butterys reectance spectrum can be effectively reproduced by amorphous gyroid microstructures.

ResultsLocal self-uniformity. The exact structure of glasses has long been debated21. Recent research has demonstrated the complex interplay of ordered and disordered phases in the vitreous state2224. The disparate variety of bulk metallic glasses in particular has challenged researchers to develop predictive theories of an alloys glass-forming ability25. The existence of sizeable PBGs in glassy networks exposes an analogous deciency in current understanding of a structures PBG forming ability. Unlike silicate and metallic glasses, the structures of glassy PBG materials are designed. Nevertheless, the structural characteristics that render these glasses amenable to PBG formation remain mysterious. With the aim of clarifying these PBG forming characteristics, here we develop LSU a general measure of structural order in connected networks.

A typical PhC consists of a connected distribution of dielectric material surrounded by air. As an example, Fig. 1a,b show the champion PhC diamond; it is a connected network of dielectric cylinders (green) arranged as in a diamond crystal. A PBGs size is usually measured as a dimensionless width given by Do/o0

the frequency width divided by the central frequency. Figure 1c displays an amorphous version of the diamond network. The complete PBG in diamond has been shown to have a width of 30% for dielectric material of refractive index 3.6 (ref. 13). We focus our discussion of PBG properties hereafter in this high-refractive index regime.

To describe a general connected network, we decompose it into a set of vertices and edges. A vertex is a point at which two or more distinct lobes of material intersect. An edge is a vector between two vertices that species the central axis of a lobe of material. As an illustration, Fig. 1a shows the fundamental unit of a diamond network in its WignerSeitz (WS) cell. A vertex, which sits at the centre of the cell, has four edges connecting it to its nearest neighbour vertices. The four nearest neighbours sit at the corners of a tetrahedron and the four edges dene a tetrahedral unit. Similarly, Fig. 1d shows the fundamental unit of a SNG structure in its WS cell. The central vertex has three nearest neighbour vertices to which it is connected by three edges. These edges are of equal length and are separated by inter-edge angles of 120; we call this conguration a trihedral unit.

The WS cell is the basic building block from which an extended periodic network can be assembled. Stacking the WS cells of diamond and SNG produces extended diamond (Fig. 1b) and gyroid (Fig. 1e) networks, respectively. An extended periodic network of this type is a highly ordered case of a continuous random network (CRN). A general CRN is a collection of vertices, connected by edges, which are arbitrarily positioned in space26. CRNs do not typically possess translational periodicities as the diamond and gyroid crystals do. Instead, Fig. 1c illustrates a classic CRNan amorphous diamond network, studied extensively as a model of amorphous silicon27,28. Every vertex in amorphous diamond is tetravalent and has four nearest neighbours that describe the points of a deformed tetrahedron. The resulting network is a connected assembly of deformed

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a b c

d e f

Figure 1 | Strongly isotropic networks and their amorphous derivatives in three dimensions. The WS cell of a diamond PhC (a) contains a 1-tree of dielectric material (green); the edges of the tree dene a tetrahedral unit. Stacking of the WS cells generates the champion diamond PhC (b). Amorphous diamond (c) consists of deformed tetrahedral units connected in a CRN. Similarly, the WS cell of a single-network gyroid PhC contains a 2-tree (d), comprising dielectric material (blue), in which each vertex is a trihedron. When stacked, the WS cells generate a single-network gyroid PhC with a near-champion PBG (e). Amorphous gyroid (f) comprises deformed trihedral units connected in a CRN.

tetrahedral units. Similarly, an amorphous gyroid is a CRN comprising deformed trihedral units as depicted in Fig. 1f.

We now introduce the concept of a tree within a CRN. An n-tree on a vertex a of a CRN is the set of edges and vertices that can be reached from a by traversing no more than n edges. The trees root edges are understood to mean the edges belonging to vertex a. To illustrate this, the diamond WS cell (Fig. 1a) contains a 1-tree. The SNG WS cell (Fig. 1d) contains a single 2-tree. A tree thus describes a small network unit within a CRN.

We dene LSU as a property of a CRN that describes the extent to which its component n-trees have shapes similar to that of nearby n-trees within the network. For the moment, we consider comparing some example 1-trees. Such trees consist of a central vertex with a number of edges g. At rst, we consider two trihedral 1-trees which we label T1a and T1b.

To measure the extent to which these two trees have similar shape we follow the process shown in Fig. 2a. First, we label the root edges of both trees as (1a, 2a, 3a) and (1b, 2b, 3b). Second, we specify a permutation of the labels of tree b, say [213]. Third, we attempt to align the two trees such that the root edges overlap in the prescribed permutation. This alignment is achieved by performing a congruent transformation on tree T1b. That is, we physically pick up the tree and perform any combination of a translation, a rotation and a reection until the edges of both trees overlap as desired29. Fourth, we score the extent to which T1a and

T1b now overlap by taking the scalar product of the edge vectors in pairs as determined by the permutation. The result of this comparison is normalized such that perfectly overlapping trees receive a maximum score of 1.

Finally we repeat steps 1 through 4 and score the overlap of T1a and T1b for all g! permutations of their root-edge alignment. We dene the spatial similarity statistic fab of these two trees as the average result of the g! comparisons (see the Methods section).

This process can be applied without modication to determine the spatial similarity statistic of two identical tetrahedral 1-trees, as shown in Fig. 2b. As before, we label the root edges of the two

trees and specify some arbitrary permutation of the root edges of tree b; Fig. 2b shows the permutation [2413]. Once permuted, tree b is congruently transformed by a translation, a reection and a rotation to maximally overlap with tree a. The quality of the overlap is then measured. The average overlap quality for all 4! root-edge alignment permutations denes the spatial similarity statistic fab.

In Fig. 2a,b, we note that that the two trihedra and tetrahedra could be made to overlap exactly for their respective root edge permutations. In fact, pairs of either trihedra or tetrahedra will overlap exactly for all root edge permutations and have a spatial similarity statistic of unity. This property of trihedral and tetrahedral units is a corollary of work on strong isotropy29. There exist only three strongly isotropic networks in three dimensions or less. In three dimensions, these networks are diamond (Fig. 1b) and single gyroid (Fig. 1e). In two dimensions, the honeycomb network is the only strongly isotropic network.

We consider now a third set of example 1-treestwo identical simple cubic network units (Fig. 2c), each with six edges about a central vertexand calculate their spatial similarity statistic. We nd that perfect overlap can be achieved for specic root edge permutations, but in most cases there is no congruent transformation to align the two trees correctly; we call such a permutation inaccessible (this is depicted in Fig. 2c). As a result, the simple cubic crystal does not possess a strong isotropy property. In the remainder of this paper, we focus on trees comprising trivalent and tetravalent vertices only.

To build a robust measure of network structural order, we require a formalism that can compare the trees of arbitrarily disordered networks. We thus build upon the measurement of tree spatial similarity statistics in the following ways. We stipulate that the CRN from which the trees are drawn comprises only vertices with a xed number of edges. We then generalize the measurement of tree spatial similarities to n-trees of any size. In this case, the correct edge overlap pairings are not known beyond the root edges. To solve this, we have developed an algorithm

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1. Label

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Figure 2 | Comparison of 1-trees illustrated with trihedra, tetrahedra and octahedra. Two identical trihedral trees (a) are labelled by their edges. The edges of tree b are then permuted. Tree b can then be rotated around the edge 3b axis and made to perfectly overlap tree a. Similarly, two identical tetrahedral trees are labelled (b), and then the edges of tree b permuted. Reection of tree B in the plane of edges 2b and 4b, followed by rotation around the new edge 3b axis, brings the two trees into alignment. Two identical octahedral trees (c) can also be compared. We apply a permutation to tree Bs edges.

When overlapped, the two trees are now mismatched; their yellow and purple edges are not aligned and cannot be made so by any congruent transformation without creating a new mismatch.

which determines the natural edge overlaps; this is detailed in the Methods section. We also allow the similarities of arbitrarily disordered trees to be measured. Here we acknowledge that the root edges of two trees will not necessarily overlap perfectly. Instead, for each permutation we nd a transformation that yields an approximate overlap. We then quantify the quality of this overlap according to a metric of spatial similarity (see Methods section).

We now dene a CRNs LSU distribution Fnl to depth n and locality l as the set of n-tree spatial similarities fab for all pairs of trees whose root vertices are within l edges of one another. The LSU distribution is a set of spatial similarities that describes the

extent to which the CRNs geometry is uniform on the length-scale l. For example, F22 describes the spatial similarities for all pairs of trees of depth 2 separated by two edges or less.

We present a set of example LSU distributions in Fig. 3. Figure 3a shows F22 for an amorphous diamond network. The trees in amorphous diamond are non-identical and present a broad distribution of spatial similarity statistics. However, the spatial similarities remain relatively large and show that the local geometries in the CRN are similar. Fig. 3b compares three LSU distributions, F12, F22 and F32, for amorphous diamond.

Amorphous networks have strong positional correlations at local length-scales that fade with distance; as a result, trees become

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0.12

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Figure 3 | Example LSU distributions. (a) The distribution of spatial similarity statistics for tree comparisons of depth 2 and locality 2 (F22) for

an amorphous diamond network. (b) LSU distributions Fn2 of amorphous diamond for different tree depths n. (c) F22 for a 216-vertex sample comprising a single gyroid crystallite suspended in amorphous gyroid, demonstrating LSUs ability to differentiate between phases. (d) F22

distributions for two characteristic types of amorphous gyroid network type-1 (1-tree local order) and type-2 (2-tree local order).

more spatially dissimilar as their depths increase. We note that strongly isotropic networks are the only CRNs whose Fnl

comprise a single peak at unity for all tree depths and localities; they are uniquely identied by their LSU distributions. This property is a consequence of the denition of strong isotropy29. The LSU distributions are thus naturally interpreted as a continuous measure of the extent to which a network is strongly isotropic.

LSU distributions can also be used as a diagnostic of the phases present in a CRN. Figure 3c presents F22 for an amorphous gyroid sample containing a gyroid crystallite surrounded by an amorphous network. The distribution comprises a background spectrum from which a clear peak emerges Bf 0.98. The peak

signies the existence of network regions comprising a strained crystalline gyroid phase. The LSU distribution is thus diagnostic of separate phases within a system and is useful for probing the local order of complex systems undergoing phase transitions22.

Champion photonic bandgap architectures. 3D architectures which exhibit large complete PBGs, together with two-dimensional architectures with large TE PBGs, possess a number of shared structural characteristics. First, successful architectures are connected networks of dielectric material5,18,30. Second, the champion structuresthe honeycomb and diamond networks in two and three dimensions, respectivelyare both strongly isotropic. SNG, which is the only other strongly isotropic network, exhibits a near-champion PBG of width 28% for a refractive index contrast of 3.6:1 (see Supplementary Fig. 1). Third, amorphous derivatives of both honeycomb and diamond networks exhibit PBGs which are sizeable but smaller than the gaps of their parent crystalline networks. Crucially, both networks possess signicant but imperfect LSU (see Fig. 3a).

Overall, evidence suggests that the extent of a networks LSU inuences its PBG forming ability. Hence, we expect that a hypothetical amorphous SNG, analogous to PAD4 and disordered honeycomb6, should possess both a high degree of LSU and a sizeable complete PBG. To the best of our knowledge, amorphous gyroids have not been observed in any context. We thus undertake the design of amorphous gyroids as a means of testing the relationship between LSU and PBG forming ability. To achieve this, we apply the Wooten-Winer-Weaire (WWW) algorithm to anneal amorphous gyroids from random seed networks27,28. In order to yield faithfully gyroidal local geometries, networks are annealed using a modied potential energy function that is distinct to the regular Keating energy27 (see also Supplementary Methods).

We generated a set of amorphous gyroid models of varying total size as measured by their number of component vertices N. In particular, we generated an ensemble of 57, 216-vertex models across a spectrum of disorder. PBGs were probed by numerical solution of the Maxwell equations via a plane wave expansion method31. A refractive index contrast of 3.6:1 was used for all calculations. Bandgap widths were measured as a percentage of their central frequency.

We demonstrate that amorphous gyroid networks can possess sizeable PBGs. The average PBG measured for a set of high-quality 1,000-vertex networks was 16%. The largest single gap observed was for a well-annealed 216-vertex network and had a size of 21%. These gap widths compare favourably with the 18% gap in PAD at the same refractive index contrast4.

We investigated our ensemble of 216-vertex networks closely, calculating the LSU F22 distribution for each network. In Fig. 4a we plot the mean value

F22 of each distribution against PBG width for all 57 networks. We see that gap width is strongly correlated with network LSU. We expect PBGs in amorphous gyroid to open within the gap region of a SNG PhC of equivalent index contrast. We thus dened the SNG gap as a critical frequency region, and counted the number of photonic bands that each of our 216-vertex networks support within this window. The number of bands within the critical frequency region can be interpreted as an integrated density of states (DOS). We plot the integrated DOS against

F22 in Fig. 4b. We see that a networks inability to support electromagnetic modes is strongly correlated with its LSU (see also Supplementary Figs 2 and 3 and Supplementary Notes 1 and 2). Together these results demonstrate the power of LSU as a predictor of PBG forming ability.

We now discuss two characteristic types of amorphous gyroid in detail; we call these type-1 and type-2 networks. We characterize the networks through histograms of their geometrical properties. In particular, we measure edge lengths (d), inter-edge angles (y), dihedral angles (f) and skew angles (w). Note that the skew angle measures the coplanarity of a trihedral unit, with w p/2 representing a at trihedron (see Supplementary

Methods).

Figure 4c,d show typical frequency distributions of d, y, f and w for type-1 networks. They are characterized by strongly peaked edge-length and inter-edge angle distributions, but have non-uniform dihedral and skew angles. Type-1 networks can thus be considered high-quality amorphous networks of trihedral 1-tree units. Figure 4f,g show the same distributions for a typical type-2 network. Its d, y, f and w distributions are all peaked around the ideal values for gyroidal vertices. Compared with type-1 networks, the local geometries of type-2 structures are much closer to an ideal strongly isotropic conguration. They possess gyroidal structural order on the length-scale of a 2-tree unitthe fundamental building block of SNG (Fig. 1d). Type-1 networks have

F22 values around 0.72.

Type-2 networks have signicantly higher LSUs with

F22s

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a c f

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Figure 4 | LSU and PBG forming ability. The LSU of amorphous gyroid networks, as measured by the mean of their F22 distributions, is strongly correlated with PBG width (a). Similarly, the integrated DOS N(o) decreases smoothly with increasing F22 (b). Fit lines (cubic polynomials) are for visualization purposes only. Approximate LSU regions for type-1 and type-2 networks are indicated (a, orange). We present also the edge length frequency (f)

distributions (c,f), inter-edge (y), dihedral (f) and skew (w) angle frequency distributions (d,g) and structure factor slices (e,h) for typical type-1 and type-2 networks, respectively. Structure factor intensities I are plotted on logarithmic colour scales and data in both panels is normalized to the maximum intensity

I0 of h.

around 0.89. Typical F22 distributions for these networks are shown in Fig. 3d.

Figure 4e,h show planar slices through the structure factors of type-1 and type-2 networks, respectively. The structure factor is a quantity that characterizes a structures coherent scattering power as a function of wavevector. We see that the peak scattering power of both network types is distributed in a circular ring on account of the average isotropy of the structures; wavevectors that lie on this ring are strongly scattered. We averaged the full structure factor across all propagation directions and measured the peak scattering power of type-2 networks to be approximately 1.6 times greater than type-1. In spite of this small increase, type-2 networks take the PBG width from practically zero to a maximum of 21%. We attribute these radically different PBG widths to the formation of locally self-uniform trees within type-2 networks.

We now demonstrate the natural connection between LSU and PBG formation in connected networks. In addition to the coherent scattering (Bragg) mechanism, there exists a Mie scattering mechanism5,32 of PBG formation. The Mie mechanism is known to be the dominant formation process for transverse magnetic (TM) polarization gaps in dielectric cylinder arrays. Specically, sizeable PBGs exist in periodic10, quasicrystalline18 and random6 cylinder arrangements and are observed to be of the same origin in each case32.

The gap originates from resonant scattering by the Mie modes of a single cylinder. For all TM-gap cylinder arrays, it is clear

from the electric eld proles at the edges of the fundamental PBG that Mie scattering mediates light propagation6,10,18. Just below the gap, modes are characterized by localization of eld nodes in the vicinity of the cylinder surfaces. Just above the gap, eld nodes consistently bisect the dielectric cylinders. These two node proles derive from the interaction of a plane wave with an isolated cylinder, and are associated with the rst and second Mie resonances, respectively32. Just above the rst Mie resonance, incident and scattered elds are in antiphase at the cylinder surface; this creates a localized standing wavefront which inhibits propagation and leads to PBG formation32.

Existing evidence suggests that PBG formation in two-dimensional and 3D dielectric networks is governed by a similar mechanism of resonant scattering. Specically, careful examination of the gap-edge eigenmodes of all honeycomb-derived networks presents a consistent picture of the nature of these scattering resonances. In both crystalline and hyperuniform disordered honeycombs6,18, modes just below the fundamental PBG are characterized magnetic eld nodes localized within the dielectric network (conversely, eld anti-nodes focus within the air cells). Just above the PBG, magnetic eld nodes pass between air cells, cutting the inter-vertex dielectric walls almost normally. These gap-edge node characteristics are consistent across honeycomb-derived trivalent networks and, by analogy to the TM case, evidence the signicance of spatially localized resonant scattering processes.

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We therefore view CRNs as a connected ensemble of distinct scattering units (n-trees) which, in isolation, exhibit a number of resonant electromagnetic modes. For frequencies just above a resonance, scattered and incident elds are in antiphase and interfere destructively, localizing a eld node in the vicinity of the scatterer and suppressing propagation through it.

A potential champion PBG structure comprises geometrically identical scattering units. All scattering centres thus possess degenerate electromagnetic resonances, and the spectral ranges for which each scatterer inhibits transmission are maximally aligned. Fixed valence networks comprising non-identical scattering units exhibit smaller PBGs than their crystalline precursors. Structural deformation of the scattering centres breaks the degeneracy of their scattering resonances; the resulting PBG is thus narrowed by imperfect overlap of the spectral ranges for which each scattering centre suppresses transmission.

The set of networks comprising geometrically identical scattering units exhibits a clear hierarchy of PBG size33 (see Supplementary Table 1). Specically, networks built from vertices with a low coordination number possess the largest PBGs. Accordingly, the diamond and SNG architectures are champion and near-champion, respectively; these networks are strongly isotropic due to the simplicity of their vertices and comprise scattering units which are perfectly superimposable under permutation. This combinatorial symmetry has a strong inuence on the PBG width. We argue that symmetry under permutation minimizes the number of distinct scattering resonances that a scattering centre supports. As a result, the frequency gaps between scattering resonances are maximized, together with the width of the spectral region above resonance for which transmission is suppressed.

Fundamentally, both Bragg and resonant scattering mechanisms contribute to PBG formation. The largest PBGs are obtained by optimization of a structures dielectric ll fraction to overlap the spectral range associated with the two mechanisms. We note, however, that strong diffraction rings in the structure factor of

amorphous materials do not directly lead to PBGs, but reect the presence of local order that, depending on its LSU, may favour gap formation. This observation claries the relationship between LSU and work on PBG formation in hyperuniform structures6. Architectures derived from disordered hyperuniform point patterns possess signicant local structural correlations and local geometrical order; these characteristics have proven essential in establishing sizeable PBGs6. However, hyperuniform point patterns must be tessellated in an ad-hoc way to produce viable PBG-forming networks6,34. This tessellation protocol naturally creates nearly-optimal network topologies, but these networks are successful only because they possess locally self-uniform structural order. In contrast to hyperuniformity, LSU measures both geometrical and topological order simultaneously and is thus an effective measure of PBG forming ability. Hyperuniformity and LSU remain compatible; we note the emergence of a hyperuniform-like exclusion domain around k 0 in the structure factor of networks with signicant LSU

(Fig. 4h). The association of LSU with PBG formation parallels the proof that amorphous materials with well-dened atomic connectivity can possess an electronic bandgap35.

Microwave experiments with amorphous gyroid. To verify our theoretical calculations, we fabricated millimetre-scale amorphous gyroid samples and experimentally characterized their PBGs. Samples were produced at the Fraunhofer Institute for Ceramic Technologies and Systems using a 3D ceramic printing technique. The samples were made from alumina (Al2O3),

whose permittivity was experimentally determined to be Er

(9.50.3) at frequencies in the microwave K band (1826.5 GHz). Two types of sample were made: cuboidal samples of SNG (Fig. 5c) and amorphous gyroid (Fig. 5a,d), and a cylindrical sample of amorphous gyroid (Fig. 5b). The internal network, comprising cylinders with diameter D 2.03 mm, was

well formed (Fig. 5c,d). The gyroid primitive cell parameter was

Norm. trans. (dB)

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Figure 5 | Microwave experiments with single network and amorphous gyroid structures. Alumina prototypes of amorphous gyroid, both a single piece cuboid (a) and a compound cylinder (b). View down the (111) axis of our single-network gyroid sample (c), and along an arbitrary axis of amorphous gyroid (d), showing the network quality. Comparison of transmission between single-network gyroid (SNG [111]) and amorphous gyroid (A-Gyroid) cuboidal samples (e,f, respectively), with gap edges predicted by plane wave expansion (vertical dashed lines); measured data was smoothed by Fourier ltering for clarity. Measured (g) and simulated (h) polar false-colour maps of transmission for the amorphous gyroid cylinder, with gap edges calculated by band structure overlaid as black and white rings, respectively. Scale bars in (c,d) 5 mm.

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designed to be 3.08 mm and experimentally measured to be a (3.130.05) mm.

Measurements were made with microwave radiation using an established experimental set-up9,19. Each sample was placed between two facing microwave horn antennae connected to an HP-8510C vector network analyser (VNA). Figure 5e,f present the transmission as a function of frequency along the (111) crystal axis of SNG and an arbitrary amorphous gyroid axis, respectively. Both samples show wide transmission gaps, with up to 35 dB of attenuation. Propagation through the periodic sample (Fig. 5e,f) is predominantly ballistic9; photons are weakly scattered both above and below the PBG, and the gap edges are well dened. As a result, the measured transmission gap for the SNG sample agrees very well with the simulated PBG, highlighted by vertical dashed lines in the gure.

The measured transmission gap for the amorphous gyroid sample (Fig. 5f) is also centred at the frequency range calculated by simulation (shown as dashed lines), although it appears wider; we attribute this to the limited dynamic range of our experiment (see Methods section) and strong diffusive scattering9. Scattering processes are signicant for the amorphous network. They scatter radiation both away from the detector and into different polarization states. As a result, strongly scattered radiation has a low coupling efciency into the receiver horn antenna; this prevents the transmission from recovering its peak value for frequencies above the PBG, thus widening the perceived transmission gap.

We employed the amorphous gyroid cylindrical sample (Fig. 5b) to investigate the isotropy of the PBG. The sample was rotated around its cylindrical axis and a transmission spectrum was recorded every 2. The resulting set of transmission spectra is presented in Fig. 5g as a polar, false-color map in which the radial coordinate represents frequency and the angular coordinate records the cylinders rotation angle36,37. Here, the rotational isotropy of the PBG is clear; the blue-and-green ring represents an isotropic transmission gap. The expected PBG edges, as predicted by band structure calculation, are overlaid as black solid curves. We corroborate the PBG isotropy by performing nite-difference time-domain (FDTD) electromagnetic simulations. Specically, we place a number of dipole sources inside our amorphous gyroid cylinder, and record the power ux some distance from the cylinder using a circular array of detectors (see Supplementary Methods). This result is presented as a second polar false-colour map in Fig. 5h; the expected PBG edges, according to the band structure, are overlaid as white solid curves. Although the transmission contrast of the experimental data is noise-limited (see Supplementary Note 3), the frequencies of the experimentally measured PBGs accord well with the results of both band structure and FDTD simulations.

Self-uniformity by evolution. Many plants and animals have evolved wavelength-scale microstructures as means of producing colour3841. Study of these architectures can inspire the development of industrial scale fabrication techniques for next-generation PhC-based technologies4244.

In this regard, self-assembly is a particularly attractive fabrication method. The three strongly isotropic networks have all been observed to self-assemble4548. Interestingly, amorphous honeycomb and diamond have also been observed in the natural world49,50; it is thus clear that a self-assembly pathway capable of producing complex short-range order exists. Here, we explore evidence that an amorphous gyroid could be similarly self-assembled. First, we demonstrate that topological defects exist in the gyroidal microstructures of green hairstreak butteries47,51. We then present a disordered network structure in the scales of

the Cambridge Blue buttery and model its reectance spectrum with an amorphous gyroid structure52.

Many buttery species are known to derive colouration from gyroidal networks of chitin within their scales, most famously Parides sesostris (the emerald-patched cattleheart) and numerous species of the genus Callophrys (the green hairstreaks). Microscopy has demonstrated that the scales contain numerous crystallites of a well-ordered network of chitin in air51,53. Small angle X-ray scattering (SAXS) has assigned the symmetry group of SNG to the structures47.

However, tomographic reconstruction of the chitin/air interface in Callophrys rubi shows that the network is not everywhere a perfect topological gyroid51. Evidence suggests that the chitin network passes continuously between adjacent gyroid grains, and that this grain-matching is facilitated by topological network defects51. Further, different crystallites within the same scale exhibit different chitin ll fractions. Accounting for these observations, we now show that SNG crystallites incorporating small amounts of topological and positional disorder are better models of the scale structure in the green hairstreaks than a perfect gyroid crystal.

To model the SAXS results, we approximate the scale as an ensemble of separate crystallites which contribute to the diffraction spectrum independently. The positional correlations between adjacent crystal domains are destroyed by the matching defects at the grain boundaries; the SAXS spectrum can thus be considered an incoherent superposition of the diffraction patterns of distinct crystallites. To t the experimental results, we generate a number of distinct crystallites across a range of chitin ll fractions and sum their diffraction patterns incoherently, optimizing their weights in the summation to minimize the Reitveld weighted prole R factor, Rwp, of the t54. We calculate

two types of ts, described in detail in the Methods section. In the rst case, we model the scales as perfect gyroids, employing the level-set approximation of the gyroid minimal surface (dotted orange curve in Fig. 6a). In the second case, we model the tomographic data51 using partially disordered gyroid (PDG) networks (dashed blue line in Fig. 6a). PDG is generated by introducing a small number of topological defects into a perfectly ordered gyroid net. Specically, one defect is introduced for every 100 network vertices together with a small amount of vertex positional disorder.

Note that, at publication, no SAXS data for C. rubi was available. Instead we compare our models to SAXS results for Callophrys gryneus (solid green line in Fig. 6a); we expect the microstructure in C. gryneus to comprise gyroid grains connected by topological matching defects, just as in C. rubi. Rwp values for our level-set and PDG ts are 980 and 645, respectively, suggesting that PDG is a superior model of the green hair-streak scale structure. The pure level-set model produces overly prominent peaks, particularly between the (110) and (211), and (400) and (420) reections. The inclusion of topological and positional disorder dampens the network correlations and reduces this contrast across the whole spectrum. The quality of the disordered PDG t is most evident for high (hkl) values; here, the overall prole of the pattern, in particular the double peak of the (321) and (400) planes, is well-captured.

Taken together, the tomographic data51 and the results of our scale modelling (Fig. 6a) suggest that topological imperfections form, to a limited degree, at the grain boundaries in the gyroidal microstructures of the green hairstreaks. The existence of these topological defects renders the hairstreaks (sub-family Theclinae) a promising family of butteries within which to search for an amorphous gyroid.

We now turn our attention to the Cambridge Blue buttery (Ps. marsyas, Theclinae, Fig. 6b). Its wing scales contain a unique

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h2 + k2 + l2

100

b c

101

S(kr)

Smax

102

d e

103

2 4 6 8 10 12 14 16

1.7 0.1

kra

Reflectance

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 200 300 400 500 600 700 800 (nm)

Figure 6 | LSU in buttery scales. Comparison with SAXS data of level-set and partially disordered gyroid models of wing scales in the buttery C. gryneus (a). Specimen of Ps. marsyas (b) and FIB section of one of its blue scales (c), imaged with 52 tilt in the FIB. Detail (d), taken from (c), compared with a projection of an amorphous gyroid model (e). Transmission electron micrograph of a Ps. marsyas scale cross-section (f). Comparison of experimental reectance data from a large wing area to an amorphous gyroid scale model (g). Scale bars, 1 cm (b); 1 mm (c); 500 nm (d,e); and 1.5 mm (f). The TEM section (f) is reproduced with permission from Vrtesy et al.52. TEM, transmission electron microscopy.

aperiodic photonic structuring from which they derive a blue brilliance. Focussed ion beam (FIB) sections reveal an intricately connected trivalent network, with fully 3D interconnectivity and an apparent LSU (Fig. 6c). Computational projections of amorphous gyroid with a volume ll fraction of 25% strongly resemble the buttery network (Fig. 6d,e). Transmission electron microscopy sections (Fig. 6f) reveal a mixture of interwoven layering and fully disordered network regions52. In cross-section, the lower half of the scale (bottom half of Fig. 6f) appears as a dark bandthis increased contrast can be associated with pigmentation55.

We now model scales of Ps. marsyas as amorphous gyroid CRNs and investigate the compatibility of these models with experimental reectance measurements. We employ a number of amorphous gyroid CRNs with

F22s around 0.88. We calculate their far eld reectance spectra by FDTD solution of the Maxwell equations.

We estimated the scale thickness from FIB sections to be 1300200 nm. The lower half of the scale was taken to be absorbing in accordance with the pigment distribution observed by transmission electron microscopy (Fig. 6f). We suggest a plausible model for the complex refractive index of the Ps. marsyas pigment, derived from the extinction coefcient of the pigment in Papilio nireus (see the Methods section). Amorphous gyroid edge lengths for our Ps. marsyas sample were estimated from electron micrographs to be 1176 nm; this corresponds to an effective SNG primitive cell parameter of a 1668 nm. The reectance of a large wing area of

Ps. marsyas was measured by a previous study52; scaling of the amorphous gyroid (110)-type SAXS peak to the wavelength of maximum reectance suggests a 169 nm. The theoretical

reectance presented (Fig. 6g) is an average over six amorphous gyroid models, all scaled by an a value of 166 nm.

Given the assumptions made in modelling the pigmentation, the general agreement between theoretical and experimental reectance spectra suggests that an amorphous gyroid with a UV-absorbing pigment is a plausible model of the structure in Ps. marsyas scales. The divergence at UV wavelengths is attributable to uncertainty in the exact complex refractive index of the buttery scale across a 300 to 700 nm range; future studies should measure this directly. The small divergence at red wavelengths is attributable to reections from unmodelled melanized ground scales.

Two other neotropical hairstreaks, Arcas imperialis (Theclinae) and Evenus coronata (Theclinae)close relatives of the Cambridge Bluehave previously been surveyed through microscopy56. Their structures appear to possess strong multilayer and gyroidal characteristics, respectively. It is therefore possible that amorphous gyroid has evolved in the Cambridge Blue by some small change in scale cell development conditions, leading to an evolutionary divergence. Indeed, such a divergence has been postulated in the case of the gyroid-containing Pa sesostris, whose-scale structuring has diverged from the perforated multilayers of many closely related Parides species57.

Our work indicates that the scale structuring in the Cambridge Blue is related to an amorphous gyroid. Electromagnetic modelling shows that amorphous gyroid models are consistent with observed reectance data and the existence of topological matching defects between gyroid grains in green hairstreak scales suggests that the production of amorphous gyroid is developmentally possible.

Any further search for a natural amorphous gyroid should not be limited to butteries. A thorough survey of avian feather barbs has revealed an abundance of colour-producing channel-type architectures. Several speciesDiglossa cyanae58 (Thraupidae), Passerina cyanea58 (Cardinalidae) and Alcedo

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atthis59 (Alcedinidae)are themselves a brilliant blue and possess microstructures with striking similarity to amorphous gyroid. Experimentally, it may be possible to produce amorphous gyroid or a similar architecture via block co-polymer self-assembly60. It is possible to generate mixed lamellar and gyroidal states that may resemble the Ps. marsyas structure61,62. An alternative pathway may be to quench the phase transition between the gyroid and metastable perforated multilayer phases62.

DiscussionWe have introduced LSU as means of measuring the extent to which local environments in a CRN of xed valency are spatially similar. When applied to strictly trivalent and tetravalent networks the states of maximum LSU are the strongly isotropic single gyroid and diamond networks, respectively. As a networks LSU decreases from its maximum it becomes glassy and eventually chaotic. LSU can thus classify all CRNs of xed valency by the extent of their internal order.

We designed LSU as a means of characterizing the optical properties of CRNs. In particular, we have shown that networks with maximal LSU possess champion PBGs in both 2D and 3D. Further, other known networks endowed with a complete PBG in 3D or a TE polarization PBG in 2D are characterized by high levels of LSU. PBGs result from scattering by the electromagnetic resonances of a networks local scattering units. When these units are spatially similar, their resonances are maximally degenerate and complete PBG formation is favoured. LSU is thus a predictive measure of a networks PBG forming ability. While here we focussed on trivalent and tetravalent networks, LSU can be generalized to include networks of arbitrary or mixed valency.

We have introduced designs of novel amorphous gyroid CRNs. Here we used ceramic 3D printing to fabricate amorphous gyroid samples in high-refractive index alumina and demonstrated their sizeable isotropic PBGs via microwave transmission experiments. The relevance of amorphous gyroid, and architectures which can be derived from it, is broad. In particular, its development, or that of a closely related CRN, appears to have occurred in the scales of the buttery Ps. marsyas. That it is possible to self-assemble such a structure is a prerequisite for its existence in natural systems. It may be possible to produce amorphous gyroid networks via block co-polymer experiments or equivalent self-assembly methods. This poses an interesting experimental challenge, the solution of which will facilitate the fabrication of advanced optical metamaterials for industrial applications43,60.

Fundamentally, we have demonstrated that the tree comparison method is a powerful framework for controlling the LSU of the scattering centres in a CRN. Sculpting a networks LSU distributions translates directly to advanced control over its optical properties. The optical properties controlled need not be limited to PBG forming ability; they could include structural colour, the scattering mean free path63 and random lasing64. Moreover, similar design principles may be employed to control other wave phenomena in electronic, phononic, elastic and acoustic materials.

Methods

Denition of the LSU distributions. Consider a CRN C with a set of dened vertex positions and inter-vertex connectivities, in which each vertex in C has exactly g edges. We dene an n-tree Tan on vertex a of C as the set of vertices within n edges of a. The root edges of Tan are all the edges of Ta1. Computationally, all information regarding Tan can be obtained by performing a breadth-rst graph search to depth n from vertex a.

We consider now two n-trees of equal depth Tan and Tbn. We label the root edges of the trees {1a, 2ayga} and {1b, 2b ... gb}, respectively. The spatial similarity

statistic fab of these two trees is dened as

fab

1 g !

Xg !i1f Tan; Tbn; si

;

where f is a similarity measure which grades the overlap of Tan and Tbn when they are maximally aligned in a root edge permutation si. We sum the similarities for all g! overlap permutations of the trees root edges and then take the average.

The value of the spatial similarity statistic will depend both on the form that the measure f takes and the method that is used to maximally align the two trees for each permutation. We describe our choice of f and the alignment procedure we follow in the next section.

The LSU distributions of the CRN are particular sets of spatial similarity statistics. We dene a new tree Tal with depth l on vertex a. The set of vertices in Tal is the local neighbourhood of a to depth l. The LSU distribution Fnl can then be written as

Fnl fab

f g : b 2 Tal; a 2 C: 2

Fnl is thus the set of spatial similarity statistics for all trees of depth n whose root vertices are within l edges of one another. Fnl can be plotted as histograms as in Fig. 3. In Fig. 3 the histogram frequencies are normalized through division by the total number of spatial similarity statistics in Fnl.

Calculation of the overlap between two trees. Here we discuss in detail the exact form of our similarity measure f and the process by which two trees are maximally overlapped. We note that the choice of f is an important user-controllable degree of freedom. In general, f must be maximal when two trees can be overlapped perfectly; this denes uniquely the maximally self-uniform conguration. However, the exact manner in which overlap is calculated can be dened to best suit the application and to possess meaningful properties as the network departs from maximal self-uniformity. Here, we adopt an intuitive framework in which tree edges are overlapped in pairs with their most natural partners.

Consider now an n-tree Tan in the CRN C. We label its Tan

vertices with index j, subject to the constraint that indices {1, 2 ... g} represent the trees root edges. The branches of the tree are dened by its edge vectors. For Tan we denote these as raj, which denes the vector to vertex j from its parent vertex as dened by a breadth-rst graph search starting on a.

Consider now two n-trees Tan and Tbn. Before they can be compared, they must be maximally overlapped. This process consists of determining a congruent transformation which maximizes the overlap of their root edges in some chosen permutation si. First, Tbn is translated such that vertices a and b overlap. Second, Tbn is rotated until rbs 1

and ra1 are parallel; this is always possible. Third, Tbn is rotated

about the ra1 axis until rbs 2

and ra2 are maximally parallel, dened to be the

conguration which minimizes their scalar product. For disordered trees, it is not usually possible to make these two vectors perfectly parallel. For trivalent networks, we deem these three steps sufcient to maximally align the two trees.

For tetravalent networks it is necessary to introduce a fourth alignment step. In this case, Tbn is reected in the plane dened by ra1 and ra2 so as to bring ra3 into maximal alignment with rbs 3

and ra4 with rbs 4

. This step is performed only if the

alignment between Tan and Tbn is improved, as measured by an increase in the value of ra3 rbs 3

ra4 rbs 4

. At this point, the two trees are considered to be maximally

aligned.

Once maximally aligned, we dene our similarity measure f for grading the overlap of two trees as

f Tan; Tbn; si

1 Ta

: 3

Overlap is calculated between edge pairs by taking their scalar product and normalizing it with the square of their mean norm; overlap of a single pairing is thus distributed on [ 1, 1]. The Tan

Xj2Traj rbkj 14 raj

1 2

rbkj

1 edge pair comparisons are summed and averaged such that f yields a maximum value of unity for perfectly overlapping trees.

Edge pairs are overlapped in the combination that is most natural. The whole overlap calculation is performed recursively in a depth-rst sense, greatly simplifying the process of determining natural pairings. At a particular point in the algorithms execution, it has reached some vertex pair j and k(j) by comparison of the natural edge pair raj with rbk j . Overlap of all (g 1)! possible sets of edge pair

comparisons is calculated. The set that maximizes the sum of edge pair scalar products is chosen as the set of natural pairings; this result is accepted, and the algorithm proceeds to calculate overlap for the edges around each of the (g 1)

child vertices. This decision process is captured in the notation rbk j , to reect that

the choice of edge vector in tree b is a function of the edge vector raj to which it is being compared.

Finally, the spatial similarity statistic fab is determined by repeating this process for all root edge permutations according to equation (1).

Microwave transmission measurement. Gyroid and amorphous gyroid models were printed and nished in the workshops of the Fraunhofer Institute for Ceramic Technologies and Systems, Dresden. The gyroid primitive cell parameter was

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designed to be to be 3.08 mm. As part of the printing process, the samples were sintered at high temperature, shrinking in side-length by B20% and introducing some uncertainty into their true sizes. Actual scaling values were thus measured and ll fractions were determined using a water displacement method, yielding an effective gyroid primitive cell parameter of a 3.130.05 mm, and alumina

volume-ll-fractions between 27 and 29%. Amorphous gyroid samples were fabricated preferentially. We also produced one SNG model. This model was designed for measurement of transmission along the [111] axis; this axis was chosen because of its symmetry (see Supplementary Note 4).

Characterization of the samples was performed with microwave radiation using an HP-8510C VNA. A single polarization mode was coupled through a pair of rectangular horn antennae and a pair of custom-made Teon microwave lenses. For the measurements shown in Fig. 5e,f, the beam was aligned along the short edge of the cuboidal samples (Fig. 5a,c) and perpendicular to a at sample surface, and the frequency varied from 15 to 35 GHz. Note that frequencies beyond the standard microwave K band (1826.5 GHz) have very low efciency coupling through the horns and waveguides. For the measurements shown in Fig. 5g, the cylindrical sample was aligned with the incident beam perpendicular to the cylinder axis, around which we rotated the sample and recorded the transmission every 2. This set of measurements was performed across a frequency range of 1727 GHz.

The total dynamic range of the HP-8510C VNA is from 0 to 65 dB

(measured dark noise). A large area of microwave-absorbing material, into which a window was cut to hold the sample, was used to prevent reection and scattering of radiation into the environment. The normalized transmission is then dened as the ratio between detected intensities with and without the sample in place. The addition of microwave-absorbing material lowers the overall coupling-efciency through the pair of horn-antennae. As a result, the actual accessible dynamic range through this measurement is only B35 dB, which limits the measured gap-depth.

This limited dynamic range is apparent in the amorphous gyroid transmission of Fig. 5f; the transmission bottoms out and becomes noisy for frequencies between 20 and 26 GHz. For the same reason, transmission results for the amorphous gyroid cylinder (Fig. 5g) appear noisy in comparison to the theoretical results (Fig. 5h). An increased dynamic range could be accessed by amplifying the source power.

SAXS pattern modelling. The total scattered intensity measured in aSAXS experiment can be written as a Fourier transform of the electron density function r(r):

I q

r2e

Pseudolycaena marsyas scale imaging. A dried male specimen of Ps. marsyas was acquired online from The Bugmaniac Insect Shop. The specimen was spread and mounted for sample photography. Single scales were removed, mounted and gold sputtered to produce a 4 nm coating. Scales were then imaged via electron microscopy, and sectioned using a FIB.

SNG net scaling parameters were estimated from micrographs under the assumption that network regions with a locally honeycomb-like appearance correspond to the nine-segment helices visible along the SNG (111) axis. The appearance of such regions is similar to views along the (111) axis of gyroid crystallites in green hairstreak butteries, supporting this supposition. The radius r of such a helix satises the approximate relation 2pr 7.732l, for l the SNG edge

length.

Pseudolycaena marsyas reectance modelling. Models of Ps. marsyas scales were generated using type-2 amorphous gyroid networks with

F22s around 0.88.

The effective single-network gyroid primitive unit cell dimension was estimated from micrographs of Ps. marsyas to be a 1668 nm, corresponding to an edge

length of l 1176 nm. Approximating the scales reectance as a result of

coherent scattering from (111)-plane-type reections suggests a 169 nm.

Six models were made, each of thickness 1,350 nm and scaled with an a value of 166 nm. To model the apparent variation in cross-member thickness within the scales network, amorphous gyroid point patterns were decorated with cylinders having normally distributed radii r, with a mean fractional radius r/l 0.44

and a 15% s.d. The resulting network has a mean volume ll fraction of B25%.

The Maxwell equations were solved using the Lumericals implementation of the FDTD method (Lumerical Solutions, Inc). The previously published experimental data was gathered through normal illumination of a large wing area and use of an integrating sphere in reection. To model this, scales were illuminated from above at normal incidence with a total eld scattered eld source. The scattered elds were recorded by a monitor box, and were projected into the far eld in the upper half-space, back along the axis of incidence. The total power scattered into the upper half-space was determined and normalized to the injected power. Reectance data was calculated across the wavelength range 360670 nm using a uniform 11.5 nm mesh.

Cylinders in the upper half of each scale were modelled as a dispersive material with the refractive index of chitin as measured by polarizing interference microscopy applied to the buttery Graphium sarpedon66. In accordance with an existing study55, the dark banding of the lower half of the Ps. marsyas scale was interpreted as pigmentation. The absorbance of this pigmentation was not measured experimentally. Rather, we suggest that the pigment is ultraviolet absorbing, as has been shown to be the case for Papilio nireus55, the gyroid-containing buttery Pa. sesostris53 and many other structurally-coloured butteries67.

To make progress in modelling the reectance of Ps. marsyas scales, we propose a plausible complex refractive index for the pigmented lower layer. We derive the extinction coefcient k(o) of the ultraviolet-absorbing pigment in Papilio nireus using the BeerLambert law55. The real component of the refractive index was taken to be equal to that of the upper layer of the scale. Lumericals in-built refractive index tting algorithm was used to t this combination of n and k; extra weight was given to the imaginary component to accurately model the absorption. The resulting complex refractive index used for the lower half of the scales is inset into Fig. 6g.

The reectance spectrum of Fig. 6g should be interpreted in light of the assumptions we have made in modelling the complex refractive index of the pigment. Given these assumptions, the agreement between the theoretical and experimental reectance data shows that an amorphous gyroid network is a plausible model for the structure in Ps. marsyas scales. The divergence between theoretical and experimental reectance at ultraviolet wavelengths is a result of uncertainty in the true refractive index of the Ps. marsyas pigment in this wavelength range. Further modelling of Ps. marsyas scales should be informed by a direct measurement of their complex refractive index55.

Data availability. The data underlying the ndings of this study are available without restriction. Details of the data and how to request access are available from the University of Surrey publications repository: http://dx.doi.org/10.15126/surreydata.00813094

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e iq rdr

; 4

for q the scattering vector in reciprocal space, re the radius of the electron and V the total sample volume65. We Fourier transform the difference between r(r) and its volume average

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Acknowledgements

We thank Uwe Scheithauer of Fraunhofer IKTS for technical advice and support related to the 3D ceramic printing. We thank Vlad Stolojan and David Cox for assistance

12 NATURE COMMUNICATIONS | 8:14439 | DOI: 10.1038/ncomms14439 | http://www.nature.com/naturecommunications

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14439 ARTICLE

regarding the microscopy of buttery scales, and Zoa Vrtesy and Laszlo Bir for providing data on Pseudolycaena marsyas. We also thank Miroslav Hejna for advice regarding implementation of the WWW protocol. This work was partially supported by the University of Surreys IAA award to M.F., the EPSRC (United Kingdom) DTG Grant No. EP/J500562/1, EPSRC (United Kingdom) Strategic Equipment Grant No. EP/ L02263X/1 (EP/M008576/1) and EPSRC (United Kingdom) Grant EP/M027791/1.

The work was also partially supported by the National Science Foundations award DMR-1308084 (USA) and the American Chemistry Society award ACS PRF 52644-UR6 to W.M.

Author contributions

S.R.S. initiated the project, designed the local self-uniformity measure and the algorithm for its measurement, performed simulations and wrote the paper. W.M. designed and performed the microwave experiments, analysed the data and wrote the paper. S.S. performed the microwave experiments and analysed data. M.F. initiated the programme, oversaw and directed the project, designed the local self-uniformity measure and wrote the paper.

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Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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Competing nancial interests: The authors declare that British patent application no. 1601838.4 has been led in the name of the University of Surrey. The contributing authors Steven Sellers and Marian Florescu are recorded as the co-inventors. Weining Man and Shervin Sahba declare no competing nancial interests.

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How to cite this article: Sellers, S. R. et al. Local self-uniformity in photonic networks. Nat. Commun. 8, 14439 doi: 10.1038/ncomms14439 (2017).

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Abstract

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The interaction of a material with light is intimately related to its wavelength-scale structure. Simple connections between structure and optical response empower us with essential intuition to engineer complex optical functionalities. Here we develop local self-uniformity (LSU) as a measure of a random network's internal structural similarity, ranking networks on a continuous scale from crystalline, through glassy intermediate states, to chaotic configurations. We demonstrate that complete photonic bandgap structures possess substantial LSU and validate LSU's importance in gap formation through design of amorphous gyroid structures. Amorphous gyroid samples are fabricated via three-dimensional ceramic printing and the bandgaps experimentally verified. We explore also the wing-scale structuring in the butterfly Pseudolycaena marsyas and show that it possesses substantial amorphous gyroid character, demonstrating the subtle order achieved by evolutionary optimization and the possibility of an amorphous gyroid's self-assembly.

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Title

Local self-uniformity in photonic networks

Author

Sellers, Steven R; Man, Weining; Sahba, Shervin; Florescu, Marian

Pages

14439

Publication year

2017

Publication date

Feb 2017

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms14439

ProQuest document ID

1869026718

Local self-uniformity in photonic networks

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