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Local self-uniformity in photonic networks

Nature Communications

February 2017
8(1):14439

DOI:10.1038/ncomms14439

License
CC BY 4.0

Authors:

Weining Man

San Francisco State University

Shervin Sahba

University of Washington

Marian Florescu

University of Southampton

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.

| 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 3 b axis and made to perfectly overlap tree a. Similarly, two identical tetrahedral trees are labelled (b), and then the edges of tree b permuted. Reflection of tree B in the plane of edges 2 b and 4 b , followed by rotation around the new edge 3 b axis, brings the two trees into alignment. Two identical octahedral trees (c) can also be compared. We apply a permutation to tree B's 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.

…

| LSU and PBG forming ability. The LSU of amorphous gyroid networks, as measured by the mean of their F 22 distributions, is strongly correlated with PBG width (a). Similarly, the integrated DOS N(o) decreases smoothly with increasing F 22 (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 I 0 of h.

…

| 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 filtering 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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ARTICLE

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

Local self-uniformity in photonic networks

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

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 conﬁg-

urations. 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.

DOI: 10.1038/ncomms14439 OPEN

1Advanced Technology Institute and Department of Physics, University of Surrey, Guildford GU2 7XH, UK. 2Department 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: steven.sellers@surrey.ac.uk) or to M.F. (email: m.ﬂorescu@surrey.ac.uk).

NATURE COMMUNICATIONS | 8:14439 | DOI: 10.1038/ncomms14439 | www.nature.com/naturecommunications 1

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 circuits1–3.

However, although the complexity of PhC-based technologies

continues to grow, questions regarding the fundamental

mechanisms of PBG formation remain unresolved4–6.

The formation of PBGs is conventionally interpreted as a result

of coherent scattering by a PhC’s 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 PhC’s Brillouin zone (BZ). When this condition is met, a

pair of orthogonal standing wave modes, each possessing a

distinct electromagnetic ﬁeld proﬁle, is formed5,9,10. Energetic

interaction between the electric ﬁeld and the underlying dielectric

distribution then breaks the degeneracy of the standing wave

states. For the speciﬁc propagation direction under consideration,

the resulting forbidden spectral range deﬁnes 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 ﬁrst complete PBG thus focussed on face-centred

cubic crystals—the most isotropic of the three-dimensional (3D)

Bravais lattices—and 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 network—dubbed photonic amorphous

diamond (PAD)—exhibits a sizeable complete PBG4. This gap

exists despite PAD’s diffuse primary diffraction maximum which

spreads the structure’s 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 honey-

comb network of dielectric material15. Many glassy networks that

can be broadly styled as ‘hyperuniform disordered honeycombs’

have been found to possess sizeable TE PBGs6,16–20. 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 signiﬁcant LSU. We demonstrate the connection between

LSU and PBG forming ability by designing novel amorphous

gyroid (amorphous triamond) connected networks. Speciﬁcally,

amorphous gyroids can possess sizeable PBGs and an amorphous

gyroid’s 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 trans-

mission experiments. We explore also the possibility that

amorphous gyroid exists within the wing scales of butterﬂies. In

particular, we reveal that the microstructure in the scales of

Pseudolycaena marsyas possesses substantial amorphous gyroid

character and demonstrate that the butterﬂy’s reﬂectance

spectrum can be effectively reproduced by amorphous gyroid

microstructures.

Results

Local 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

state22–24. The disparate variety of bulk metallic glasses in

particular has challenged researchers to develop predictive

theories of an alloy’s glass-forming ability25. The existence of

sizeable PBGs in glassy networks exposes an analogous deﬁciency

in current understanding of a structure’s 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 PBG’s size

is usually measured as a dimensionless width given by Do/o

—

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 speciﬁes the central axis of a lobe of

material. As an illustration, Fig. 1a shows the fundamental unit

of a diamond network in its Wigner–Seitz (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 deﬁne 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°;wecallthisconﬁgurationatrihedral

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 CRN—an 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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tetrahedral units. Similarly, an amorphous gyroid is a CRN comp-

rising 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 aof a CRN is the set of edges and vertices that

can be reached from aby traversing no more than nedges. The

tree’s ‘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 deﬁne 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 ﬁrst, we consider two

trihedral 1-trees which we label T1

aand T1

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 (1

) and (1

). 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 T1

b. That is, we

physically pick up the tree and perform any combination of a

translation, a rotation and a reﬂection until the edges of both trees

overlap as desired29. Fourth, we score the extent to which T1

aand

bnow 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 T1

and T1

bfor all g! permutations of their root-edge alignment. We

deﬁne the spatial similarity statistic f

of these two trees as the

average result of the g! comparisons (see the Methods section).

This process can be applied without modiﬁcation 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

bis congruently transformed by a translation, a reﬂection 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 deﬁnes the spatial similarity

statistic f

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-trees—two identical

simple cubic network units (Fig. 2c), each with six edges about a

central vertex—and calculate their spatial similarity statistic. We

ﬁnd that perfect overlap can be achieved for speciﬁc 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 ﬁxed 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

ab c

de 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 deﬁne 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.

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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 ﬁnd 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 deﬁne a CRN’s LSU distribution F

to depth nand

locality las the set of n-tree spatial similarities f

for all pairs of

trees whose root vertices are within ledges of one another. The

LSU distribution is a set of spatial similarities that describes the

extent to which the CRN’s geometry is uniform on the length-

scale l. For example, F

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 F

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, F

and F

, for amorphous diamond.

Amorphous networks have strong positional correlations at local

length-scales that fade with distance; as a result, trees become

Tree a

Tree b

1. Label 1a

Tree a

Tree b

1. Label

Tree a

Tree b

1. Label

2. Permute 1a

3b4b

2. Permute

3. Overlap

Perfect

Imperfect

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 bare then permuted. Tree bcan then be rotated around the edge 3

axis and made to perfectly overlap tree a. Similarly, two identical

tetrahedral trees are labelled (b), and then the edges of tree bpermuted. Reﬂection of tree B in the plane of edges 2

and 4

, followed by rotation around the

new edge 3

axis, brings the two trees into alignment. Two identical octahedral trees (c) can also be compared. We apply a permutation to tree B’s 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.

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more spatially dissimilar as their depths increase. We note that

strongly isotropic networks are the only CRNs whose F

comprise a single peak at unity for all tree depths and localities;

they are uniquely identiﬁed by their LSU distributions. This

property is a consequence of the deﬁnition 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 F

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 Bf¼0.98. The peak

signiﬁes 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-dimen-

sional 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 structures—the honeycomb and diamond networks in

two and three dimensions, respectively—are 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 signiﬁcant but imperfect LSU (see Fig. 3a).

Overall, evidence suggests that the extent of a network’s LSU

inﬂuences its PBG forming ability. Hence, we expect that a

hypothetical amorphous SNG, analogous to PAD4and 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 under-

take 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 modiﬁed 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 F

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 deﬁned 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 network’s

inability to support electromagnetic modes is strongly correlated

with its LSU (see also Supplementary Figs 2 and 3 and Supple-

mentary 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 geome-

trical 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 w¼p/2 representing a ﬂat trihedron (see Supplementary

Methods).

Figure 4c,d show typical frequency distributions of d,y,f

and wfor 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,fand wdistributions 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

conﬁguration. They possess gyroidal structural order on the

length-scale of a 2-tree unit—the fundamental building block of

SNG (Fig. 1d). Type-1 networks have 

F22 values around 0.72.

Type-2 networks have signiﬁcantly higher LSUs with 

F22s

0.18

Φ12

Φ22

Φ32 0.12

0.10

0.06

0.02

0.06

0.12

0.08

0.04

Relative frequency

0.10

0.06

0.02

Type-1

Type-2

0.7 0.8 0.9 1 0.7 0.8 0.9 1

Spatial similarity 

Figure 3 | Example LSU distributions. (a) The distribution of spatial

similarity statistics for tree comparisons of depth 2 and locality 2 (F

) for

an amorphous diamond network. (b) LSU distributions F

of amorphous

diamond for different tree depths n.(c)F

for a 216-vertex sample

comprising a single gyroid crystallite suspended in amorphous gyroid,

demonstrating LSU’s ability to differentiate between phases. (d)F

distributions for two characteristic types of amorphous gyroid network—

type-1 (1-tree local order) and type-2 (2-tree local order).

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around 0.89. Typical F

distributions for these networks are

showninFig.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 structure’s 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

andPBGformationinconnectednetworks.Inadditionto

the coherent scattering (Bragg) mechanism, there exists

aMiescatteringmechanism

5,32 of PBG formation. The Mie

mechanism is known to be the dominant formation process for

transverse magnetic (TM) polarization gaps in dielectric

cylinder arrays. Speciﬁcally, sizeable PBGs exist in periodic10,

quasicrystalline18 and random6cylinder 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 ﬁeld proﬁles 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 ﬁeld

nodes in the vicinity of the cylinder surfaces. Just above the gap,

ﬁeld nodes consistently bisect the dielectric cylinders. These two

node proﬁles derive from the interaction of a plane wave with an

isolated cylinder, and are associated with the ﬁrst and second Mie

resonances, respectively32. Just above the ﬁrst Mie resonance,

incident and scattered ﬁelds 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. Speciﬁcally, careful examina-

tion 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 ﬁeld nodes localized within the

dielectric network (conversely, ﬁeld anti-nodes focus within the

air cells). Just above the PBG, magnetic ﬁeld 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 signiﬁcance of spatially localized resonant

scattering processes.

Type-2

Type-1

f. (a.u.)

200

150

100

f. (a.u.)

200













150

100

d/d0

0.8 1.21

200

150

100

f. (a.u.)

–4

–8 –1.5

–0.5

–1

–1.5

–0.5

–1

–8 –4 0 4 8–8 –4 0 4 8

kya

–4

–8

f. (a.u.)

200

150

100

66666

π5π4π3π2ππ

d/d0

0.8 1.21

666

Angle Angle

0π5π4π3π2ππ

kxa

0.7 0.75 0.8 0.85 0.9

N () (a.u.)

kxa

Φ22

0.7 0.75 0.8 0.85 0.9

Δ

0

acf

Figure 4 | LSU and PBG forming ability. The LSU of amorphous gyroid networks, as measured by the mean of their F

distributions, is strongly correlated

with PBG width (a). Similarly, the integrated DOS N(o) decreases smoothly with increasing F

(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 Iare plotted on logarithmic colour scales and data in both panels is normalized to the maximum intensity

of h.

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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 ﬁelds are in antiphase and

interfere destructively, localizing a ﬁeld 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). Speciﬁcally, 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 inﬂuence 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 mechan-

isms contribute to PBG formation. The largest PBGs are obtained

by optimization of a structure’s dielectric ﬁll 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 reﬂect the

presence of local order that, depending on its LSU, may favour

gap formation. This observation clariﬁes the relationship between

LSU and work on PBG formation in hyperuniform structures6.

Architectures derived from disordered hyperuniform point

patterns possess signiﬁcant 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

k¼0 in the structure factor of networks with signiﬁcant LSU

(Fig. 4h). The association of LSU with PBG formation parallels

the proof that amorphous materials with well-deﬁned atomic

connectivity can possess an electronic bandgap35.

Microwave experiments with amorphous gyroid.Toverifyour

theoretical calculations, we fabricated millimetre-scale amor-

phous 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 (Al

whose permittivity was experimentally determined to be

Er¼(9.5±0.3) at frequencies in the microwave Kband

(18–26.5 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 D¼2.03 mm, was

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

Norm. trans. (dB)

Raw trans. (dB)

–10

27 17 f (GHz)

–25

–35

–45

–55

–20

–30

Norm. trans. (dB)

–10

–20

–30

–40

–50

15 20 25 30

Freq. (GHz)

SNG [111]

Cal. PBG

Norm. trans. (dB)

–10

–20

–30

–40

–5015 20 25 30

Freq. (GHz)

A-Gyroid

Cal. PBG

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 ﬁltering 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

a¼(3.13±0.05) 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 deﬁned. 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 ﬁgure.

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 signiﬁcant 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 efﬁciency 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 cylinder’s 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 ﬁnite-difference time-domain (FDTD) electro-

magnetic simulations. Speciﬁcally, we place a number of dipole

sources inside our amorphous gyroid cylinder, and record the

power ﬂux 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

colour38–41. Study of these architectures can inspire the develo-

pment of industrial scale fabrication techniques for next-

generation PhC-based technologies42–44.

In this regard, self-assembly is a particularly attractive

fabrication method. The three strongly isotropic networks have

all been observed to self-assemble45–48. 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 butterﬂies47,51.

We then present a disordered network structure in the scales of

the Cambridge Blue butterﬂy and model its reﬂectance spectrum

with an amorphous gyroid structure52.

Many butterﬂy 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). Micro-

scopy 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 inter-

face 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 ﬁll fractions. Accounting for these obser-

vations, 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 ﬁt the experimental results, we generate

a number of distinct crystallites across a range of chitin

ﬁll fractions and sum their diffraction patterns incoherently,

optimizing their weights in the summation to minimize the

Reitveld weighted proﬁle Rfactor, R

, of the ﬁt54. We calculate

two types of ﬁts, described in detail in the Methods section. In the

ﬁrst 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. Speciﬁcally, 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.R

values for

our level-set and PDG ﬁts 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) reﬂections. 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 ﬁt is most evident for high (hkl) values; here,

the overall proﬁle 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 butterﬂies within which to search for an

amorphous gyroid.

We now turn our attention to the Cambridge Blue butterﬂy

(Ps. marsyas, Theclinae, Fig. 6b). Its wing scales contain a unique

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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 ﬁll fraction of 25% strongly

resemble the butterﬂy 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 band—this 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 reﬂectance measurements. We employ a number of

amorphous gyroid CRNs with 

F22s around 0.88. We calculate

their far ﬁeld reﬂectance spectra by FDTD solution of the

Maxwell equations.

We estimated the scale thickness from FIB sections to be

1300±200 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 coefﬁcient 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 117±6 nm; this

corresponds to an effective SNG primitive cell parameter of

a¼166±8 nm. The reﬂectance 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 reﬂectance suggests a¼169 nm. The theoretical

reﬂectance presented (Fig. 6g) is an average over six amorphous

gyroid models, all scaled by an avalue of 166 nm.

Given the assumptions made in modelling the pigmentation,

the general agreement between theoretical and experimental

reﬂectance 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 butterﬂy scale across a 300 to 700 nm range; future studies

should measure this directly. The small divergence at red

wavelengths is attributable to reﬂections from unmodelled

melanized ground scales.

Two other neotropical hairstreaks, Arcas imperialis (Theclinae)

and Evenus coronata (Theclinae)—close relatives of the

Cambridge Blue—have previously been surveyed through micro-

scopy56. 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 reﬂectance data and the existence of topological

matching defects between gyroid grains in green hairstreak scales

suggests that the production of amorphous gyroid is devel-

opmentally possible.

Any further search for a natural amorphous gyroid should

not be limited to butterﬂies. A thorough survey of avian

feather barbs has revealed an abundance of colour-producing

channel-type architectures. Several species—Diglossa cyanae58

(Thraupidae), Passerina cyanea58 (Cardinalidae) and Alcedo

10–3

10–2

10–1

100

h2 + k2 + l2

PDG

Level-set

C. gryneus SAXS

Reflectance

400 500 600

200 300 400 500 600 700 800

 (nm)

kra

S(kr)

Smax

246810121416

1.7

1.5

0.1

0.7

0.6

0.5

0.4

0.3

0.2

0.1

P. marsyas exp.

A-Gyroid model

n Model (inset)

k Model (inset)

Figure 6 | LSU in butterﬂy scales. Comparison with SAXS data of level-set and partially disordered gyroid models of wing scales in the butterﬂy 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

reﬂectance 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 Ve

´rtesy et al.52. TEM, transmission electron microscopy.

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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.

Discussion

We have introduced LSU as means of measuring the extent

to which local environments in a CRN of ﬁxed 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 network’s LSU decreases from its

maximum it becomes glassy and eventually chaotic. LSU can

thus classify all CRNs of ﬁxed 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 electro-

magnetic resonances of a network’s 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 network’s 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 butterﬂy 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 compar-

ison method is a powerful framework for controlling the LSU of

the scattering centres in a CRN. Sculpting a network’s 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

Deﬁnition of the LSU distributions.Consider a CRN Cwith a set of deﬁned

vertex positions and inter-vertex connectivities, in which each vertex in Chas

exactly gedges. We deﬁne an n-tree Ta

non vertex aof Cas the set of vertices within

nedges of a. The root edges of Ta

nare all the edges of Ta

1. Computationally, all

information regarding Ta

ncan be obtained by performing a breadth-ﬁrst graph

search to depth nfrom vertex a.

We consider now two n-trees of equal depth Ta

nand Tb

n. We label the root edges

of the trees {1

} and {1

... g

}, respectively. The spatial similarity

statistic f

of these two trees is deﬁned as

fab¼1

g!X

i¼1

n;Tb

n;si



;ð1Þ

where fis a similarity measure which grades the overlap of Ta

nand Tb

nwhen they

are maximally aligned in a root edge permutation s

. 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 ftakes and the method that is used to maximally align the two trees for

each permutation. We describe our choice of fand the alignment procedure we

follow in the next section.

The LSU distributions of the CRN are particular sets of spatial similarity

statistics. We deﬁne a new tree Ta

lwith depth lon vertex a. The set of vertices in Ta

is the local neighbourhood of ato depth l. The LSU distribution F

can then be

written as

Fnl¼fab

:b2Ta

l;a2C:ð2Þ

is thus the set of spatial similarity statistics for all trees of depth nwhose root

vertices are within ledges of one another. F

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 F

Calculation of the overlap between two trees.Here we discuss in detail the exact

form of our similarity measure fand the process by which two trees are maximally

overlapped. We note that the choice of fis an important user-controllable degree of

freedom. In general, fmust be maximal when two trees can be overlapped perfectly;

this deﬁnes uniquely the maximally self-uniform conﬁguration. However, the exact

manner in which overlap is calculated can be deﬁned 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 over-

lapped in pairs with their most natural partners.

Consider now an n-tree Ta

nin the CRN C. We label its Ta

vertices with index j,

subject to the constraint that indices {1, 2 ... g} represent the tree’s root edges. The

branches of the tree are deﬁned by its edge vectors. For Ta

nwe denote these as ra

which deﬁnes the vector to vertex jfrom its parent vertex as deﬁned by a breadth-

ﬁrst graph search starting on a.

Consider now two n-trees Ta

nand Tb

n. 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 s

. First, Tb

nis translated such that vertices aand boverlap. Second, Tb

is rotated until rb

si1ðÞand ra

1are parallel; this is always possible. Third, Tb

nis rotated

about the ra

1axis until rb

si2ðÞand ra

2are maximally parallel, deﬁned to be the

conﬁguration 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 sufﬁcient to maximally align the two trees.

For tetravalent networks it is necessary to introduce a fourth alignment step. In

this case, Tb

nis reﬂected in the plane deﬁned by ra

1and ra

2so as to bring ra

3into

maximal alignment with rb

si3

ðÞand ra

4with rb

si4

ðÞ

. This step is performed only if the

alignment between Ta

nand Tb

nis improved, as measured by an increase in the value

of ra

3rb

si3ðÞ

þra

4rb

si4ðÞ

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

aligned.

Once maximally aligned, we deﬁne our similarity measure ffor grading the

overlap of two trees as

n;Tb

n;si



¼1

1X

j2Ta

jrb

kðjÞ

4ra

þrb

kðjÞ





2:ð3Þ

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 Ta

1 edge pair comparisons are summed and

averaged such that fyields 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-ﬁrst sense, greatly

simplifying the process of determining natural pairings. At a particular point in the

algorithm’s execution, it has reached some vertex pair jand k(j) by comparison of

the natural edge pair ra

jwith rb

kjðÞ

. 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 rb

kjðÞ

, to reﬂect that

the choice of edge vector in tree bis a function of the edge vector ra

jto which it is

being compared.

Finally, the spatial similarity statistic f

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 ﬁnished 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 ﬁll fractions were determined using a water displacement method, yielding an

effective gyroid primitive cell parameter of a¼3.13±0.05 mm, and alumina

volume-ﬁll-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 Teﬂon 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 ﬂat sample surface,

and the frequency varied from 15 to 35 GHz. Note that frequencies beyond the

standard microwave K band (18–26.5 GHz) have very low efﬁciency 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

17–27 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 reﬂection and scattering of

radiation into the environment. The normalized transmission is then deﬁned as the

ratio between detected intensities with and without the sample in place. The

addition of microwave-absorbing material lowers the overall coupling-efﬁciency

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 a

SAXS experiment can be written as a Fourier transform of the electron density

function r(r):

IqðÞ¼

VZV

rðrÞ

r½eiqrdr



;ð4Þ

for qthe scattering vector in reciprocal space, r

the radius of the electron and Vthe

total sample volume65. We Fourier transform the difference between r(r) and its

volume average 

rto remove the forward scattering peak and access any diffraction

signatures that may be present at small scattering vectors. The r(r) functions for

our 3D network structures were generated by voxelization and were distributed on

[0, 1], representing the air and chitin distributions of the butterﬂy scale networks,

respectively. Fourier transforms were calculated using a fast Fourier transform

algorithm. Powder-like structure factors were calculated by azimuthal averaging of

the total scattered intensity. Structure factors were normalized to the intensity of

the (110) peak for comparison with experimental data.

To model the observed scattering from C.gryneus, we investigated both partially

disordered gyroid (PDG) networks and level-set surfaces as approximations to the

chitin network. In all cases, cubic crystallites of dimension 10a(1,000 vertex points)

were used. When f¼0, the function

fx;y;zðÞ¼sinðxÞcosðyÞþsinðyÞcosðzÞþsinðzÞcosðxÞð5Þ

deﬁnes the level-set approximation to the gyroid surface; it divides space into two

inter-penetrating labyrinths each with a 50% volume ﬁll. For fa0, it deﬁnes a

surface with the symmetry of SNG, and alters the relative ﬁlling fraction of the two

labyrinths. A set of level-set basis states was generated by sampling these different

ﬁll fractions via a critical value t, assigning unity to r(r) for f4tand zero otherwise.

PDG networks were generated by introducing geometrical and topological

disorder into SNGs. All PDG networks were topologically disordered by the

introduction of 10 Stone–Wales defects. This was performed using our

implementation of the WWW algorithm. Geometrical disorder was added by

perturbation of the vertex points by a random vector whose magnitude was

normally distributed with mean zero and s.d. sup to 40% of the gyroid edge

length. To increase the physicality of this disorder, perturbations were propagated

through the network with their magnitude decaying with depth; vertex positions

thus retained a local correlation. The ﬁnal set of disordered SNG basis states

sampled various degrees of vertex positional disorder, as deﬁned by their

perturbation standard deviations and rates of perturbation decay, each across a set

of ﬁll fractions.

Final diffraction spectra consist of a weighted sum over the diffraction patterns

of the appropriate basis states. A ﬁt to the experimental data was performed by

optimization of these weights so as to minimize the Rietveld weighted proﬁle factor

of the ﬁt.

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 butterﬂies, supporting this supposition. The radius

rof such a helix satisﬁes the approximate relation 2pr¼7.732l, for lthe SNG edge

length.

Pseudolycaena marsyas reﬂectance 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 a¼166±8 nm, corresponding to an edge

length of l¼117±6 nm. Approximating the scale’s reﬂectance as a result of

coherent scattering from (111)-plane-type reﬂections suggests a¼169 nm.

Six models were made, each of thickness 1,350 nm and scaled with an avalue of

166 nm. To model the apparent variation in cross-member thickness within the

scale’s network, amorphous gyroid point patterns were decorated with cylinders

having normally distributed radii r, with a mean fractional radius r/l¼0.44

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

The Maxwell equations were solved using the Lumerical’s 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 reﬂection. To model this, scales were

illuminated from above at normal incidence with a total ﬁeld scattered ﬁeld source.

The scattered ﬁelds were recorded by a monitor box, and were projected into the

far ﬁeld 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. Reﬂectance data was calculated across the wavelength range 360–670 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 butterﬂy 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 butterﬂy Pa. sesostris53 and many other structurally-coloured

butterﬂies67.

To make progress in modelling the reﬂectance of Ps. marsyas scales, we propose

a plausible complex refractive index for the pigmented lower layer. We derive the

extinction coefﬁcient k(o) of the ultraviolet-absorbing pigment in Papilio nireus

using the Beer–Lambert law55. The real component of the refractive index was

taken to be equal to that of the upper layer of the scale. Lumerical’s in-built

refractive index ﬁtting algorithm was used to ﬁt this combination of nand 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 reﬂectance 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 reﬂectance data shows that an amorphous gyroid network is a

plausible model for the structure in Ps. marsyas scales. The divergence between

theoretical and experimental reﬂectance 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 ﬁndings 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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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

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14439

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

regarding the microscopy of butterﬂy scales, and Zoﬁa Ve

´rtesy and Laszlo Biro

´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 Surrey’s 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 Foundation’s 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.

Additional information

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

naturecommunications

Competing ﬁnancial interests: The authors declare that British patent application no.

1601838.4 has been ﬁled 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 ﬁnancial interests.

Reprints and permission information is available online at http://npg.nature.com/

reprintsandpermissions/

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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NATURE COMMUNICATIONS | 8:14439 | DOI: 10.1038/ncomms14439 | www.nature.com/naturecommunications 13

Supplementary Information

Data

February 2017

Steven R. Sellers · Weining Man · Shervin Sahba · Marian Florescu

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Preprint

Sep 2016

Understanding the structural origins of the properties of amorphous materials remains one of the most important challenges in structural science. In this study we demonstrate that local 'structural simplicity', embodied by the degree to which atomic environments within a material are similar to each other, is powerful concept for rationalising the structure of canonical amorphous material amorphous silicon (a-Si). We show, by restraining a reverse Monte Carlo refinement against pair distribution function (PDF) data to be simpler, that the simplest model consistent with the PDF is a continuous random network (CRN). A further effect of producing a simple model of a-Si is the generation of a (pseudo)gap in the electronic density of states, suggesting that structural homogeneity drives electronic homogeneity. That this method produces models of a-Si that approach the state-of-the-art without the need for chemically specific restraints (beyond the assumption of homogeneity) suggests that simplicity-based refinement approaches may allow experiment-driven structural modelling techniques to be developed for the wide variety of amorphous semiconductors with strong local order.

Article

Full-text available

Jul 2024

Transport processes underpin a wide variety of phenomena, ranging from chemistry, to physics and ecology. Despite their pervasiveness, however, several distinctive features of these processes are still elusive, making it difficult to recognize and classify the associated transport regimes. Using light scattering as a probe to explore different propagation regimes, we report on the experimental observation of non-self-similar light transport through turbid membranes. Our results show that a breakdown of self-similarity can arise for light waves even in the presence of isotropic and homogeneous disorder, and can be tuned by varying the turbidity of the system. By introducing the concept of self-similarity for light propagation, we provide a unified framework for the classification of light transport regimes—overcoming the dichotomy between normal and anomalous diffusion—and show that non-self-similar propagation is a common and experimentally accessible phenomenon. This insight can help to understand and model other scenarios where light transport is dominated by rare propagation events, such as in nonlinear and active media, but also in other fields of research beyond optics. Published by the American Physical Society 2024

Thin film structural color is widespread in slime molds (Myxomycetes, Amoebozoa)

Article

Full-text available

Jan 2024
OPT EXPRESS

Brilliant colors in nature arise from the interference of light with periodic nanostructures resulting in structural color. While such biological photonic structures have long attracted interest in insects and plants, they are little known in other groups of organisms. Unexpected in the kingdom of Amoebozoa, which assembles unicellular organisms, structural colors were observed in myxomycetes, an evolutionary group of amoebae forming macroscopic, fungal-like structures. Previous work related the sparkling appearance of Diachea leucopodia to thin film interference. Using optical and ultrastructural characterization, we here investigated the occurrence of structural color across 22 species representing two major evolutionary clades of myxomycetes including 14 genera. All investigated species showed thin film interference at the peridium, producing colors with hues distributed throughout the visible range that were altered by pigmentary absorption. A white reflective layer of densely packed calcium-rich shells is observed in a compound peridium in Metatrichia vesparium, whose formation and function are still unknown. These results raise interesting questions on the biological relevance of thin film structural colors in myxomycetes, suggesting they may be a by-product of their reproductive cycle.

Stealthy and hyperuniform isotropic photonic band gap structure in 3D

Article

Sep 2024

In photonic crystals, the propagation of light is governed by their photonic band structure, an ensemble of propagating states grouped into bands, separated by photonic band gaps. Due to discrete symmetries in spatially strictly periodic dielectric structures their photonic band structure is intrinsically anisotropic. However, for many applications, such as manufacturing artificial structural color materials or developing photonic computing devices, but also for the fundamental understanding of light-matter interactions, it is of major interest to seek materials with long range nonperiodic dielectric structures which allow the formation of isotropic photonic band gaps. Here, we report the first ever 3D isotropic photonic band gap for an optimized disordered stealthy hyperuniform structure for microwaves. The transmission spectra are directly compared to a diamond pattern and an amorphous structure with similar node density. The band structure is measured experimentally for all three microwave structures, manufactured by 3D laser printing for metamaterials with refractive index up to n=2.1. Results agree well with finite-difference-time-domain numerical investigations and a priori calculations of the band gap for the hyperuniform structure: the diamond structure shows gaps but being anisotropic as expected, the stealthy hyperuniform pattern shows an isotropic gap of very similar magnitude, while the amorphous structure does not show a gap at all. Since they are more easily manufactured, prototyping centimeter scaled microwave structures may help optimizing structures in the technologically very interesting region of infrared.

Not only a matter of disorder in I-WP minimal surface-based photonic networks: Diffusive structural color in Sternotomis amabilis longhorn beetles

Article

Aug 2024

Light in correlated disordered media

Article

Nov 2023

The optics of correlated disordered media is a conceptually rich research topic emerging at the interface between the physics of waves in complex media and nanophotonics. Inspired by photonic structures in nature and enabled by advances in nanofabrication processes, recent investigations have unveiled how the design of structural correlations down to the subwavelength scale could be exploited to control the scattering, transport, and localization of light in matter. From optical transparency to superdiffusive light transport to photonic gaps, the optics of correlated disordered media challenges our physical intuition and offers new perspectives for applications. This review examines the theoretical foundations, state-of-the-art experimental techniques, and major achievements in the study of light interaction with correlated disorder, covering a wide range of systems: from short-range correlated photonic liquids to Lévy glasses containing fractal heterogeneities to hyperuniform disordered photonic materials. The mechanisms underlying light scattering and transport phenomena are elucidated on the basis of rigorous theoretical arguments. Ongoing research on mesoscopic phenomena such as transport phase transitions and speckle statistics and the current development of disorder engineering for applications such as light-energy management and visual appearance design are overviewed. Finally, special efforts are made to identify the main theoretical and experimental challenges to address in the near future.

Nanofabrication and coloration study of artificial Morpho butterfly wings with aligned lamellae layers

Article

Full-text available

Nov 2015

The bright and iridescent blue color from Morpho butterfly wings has attracted worldwide attentions to explore its mysterious nature for long time. Although the physics of structural color by the nanophotonic structures built on the wing scales has been well established, replications of the wing structure by standard top-down lithography still remains a challenge. This paper reports a technical breakthrough to mimic the blue color of Morpho butterfly wings, by developing a novel nanofabrication process, based on electron beam lithography combined with alternate PMMA/LOR development/dissolution, for photonic structures with aligned lamellae multilayers in colorless polymers. The relationship between the coloration and geometric dimensions as well as shapes is systematically analyzed by solving Maxwell's Equations with a finite domain time difference simulator. Careful characterization of the mimicked blue by spectral measurements under both normal and oblique angles are carried out. Structural color in blue reflected by the fabricated wing scales, is demonstrated and further extended to green as an application exercise of the new technique. The effects of the regularity in the replicas on coloration are analyzed. In principle, this approach establishes a starting point for mimicking structural colors beyond the blue in Morpho butterfly wings.

A predictive structural model for bulk metallic glasses

Article

Full-text available

Sep 2015

Great progress has been made in understanding the atomic structure of metallic glasses, but there is still no clear connection between atomic structure and glass-forming ability. Here we give new insights into perhaps the most important question in the field of amorphous metals: how can glass-forming ability be predicted from atomic structure? We give a new approach to modelling metallic glass atomic structures by solving three long-standing problems: we discover a new family of structural defects that discourage glass formation; we impose efficient local packing around all atoms simultaneously; and we enforce structural self-consistency. Fewer than a dozen binary structures satisfy these constraints, but extra degrees of freedom in structures with three or more different atom sizes significantly expand the number of relatively stable, 'bulk' metallic glasses. The present work gives a new approach towards achieving the long-sought goal of a predictive capability for bulk metallic glasses.

Unfolding the band structure of non-crystalline photonic band gap materials

Article

Full-text available

Aug 2015

Non-crystalline photonic band gap (PBG) materials have received increasing attention, and sizeable PBGs have been reported in quasi-crystalline structures and, more recently, in disordered structures. Band structure calculations for periodic structures produce accurate dispersion relations, which determine group velocities, dispersion, density of states and iso-frequency surfaces, and are used to predict a wide-range of optical phenomena including light propagation, excited-state decay rates, temporal broadening or compression of ultrashort pulses and complex refraction phenomena. However, band calculations for non-periodic structures employ large super-cells of hundreds to thousands building blocks, and provide little useful information other than the PBG central frequency and width. Using stereolithography, we construct cm-scale disordered PBG materials and perform microwave transmission measurements, as well as finite-difference time-domain (FDTD) simulations. The photonic dispersion relations are reconstructed from the measured and simulated phase data. Our results demonstrate the existence of sizeable PBGs in these disordered structures and provide detailed information of the effective band diagrams, dispersion relation, iso-frequency contours, and their angular dependence. Slow light phenomena are also observed in these structures near gap frequencies. This study introduces a powerful tool to investigate photonic properties of non-crystalline structures and provides important effective dispersion information, otherwise difficult to obtain.

High-Q Optical Cavities in Hyperuniform Disordered Materials

Article

Full-text available

Apr 2015

We introduce the first designs for high-Q photonic cavities in slab architectures in hyperuniform disordered solids displaying isotropic band gaps. Despite their disordered character, hyperuniform disordered structures have the ability to tightly confine the TE-polarised radiation in slab configurations that are readily fabricable. The architectures are based on carefully designed local modifications of otherwise unperturbed hyperuniform dielectric structures. We identify a wide range of confined cavity modes, which can be classified according to their approximate symmetry (monopole, dipole, quadrupole, etc.) of the confined electromagnetic wave pattern. We demonstrate that quality factors (Q)

Q>10^{9}

can be achieved for purely 2D structures, and that for three--dimensional finite-height photonic slabs, quality factors

Q>20,000

can be maintained.

Polymorphism in glassy silicon: Inherited from liquid-liquid phase transition in supercooled liquid

Article

Full-text available

Feb 2015

Combining molecular dynamics (MD) simulation and Voronoi polyhedral analyses, we discussed the microstructure evolution in liquid and glassy silicon during cooling by focusing on the fraction of various clusters. Liquid-liquid phase transition (LLPT) is detected in supercooled liquid silicon However, freezing the high-density liquid (HDL) to the glassy state is not achieved as the quenching rate goes up to 10(14) K/s. The polyamorphism in glassy silicon is found to be mainly associated with low-density liquid (LDL).

Bright-White Beetle Scales Optimise Multiple Scattering of Light

Article

Full-text available

Dec 2014

Whiteness arises from diffuse and broadband reflection of light typically achieved through optical scattering in randomly structured media. In contrast to structural colour due to coherent scattering, white appearance generally requires a relatively thick system comprising randomly positioned high refractive-index scattering centres. Here, we show that the exceptionally bright white appearance of Cyphochilus and Lepidiota stigma beetles arises from a remarkably optimised anisotropy of intra-scale chitin networks, which act as a dense scattering media. Using time-resolved measurements, we show that light propagating in the scales of the beetles undergoes pronounced multiple scattering that is associated with the lowest transport mean free path reported to date for low-refractive-index systems. Our light transport investigation unveil high level of optimisation that achieves high-brightness white in a thin low-mass-per-unit-area anisotropic disordered nanostructure.

Pigmentary and photonic coloration mechanisms reveal taxonomic relationships of the Cattlehearts (Lepidoptera: Papilionidae: Parides)

Article

Full-text available

Jul 2014
BMC EVOL BIOL

Background The colorful wing patterns of butterflies, a prime example of biodiversity, can change dramatically within closely related species. Wing pattern diversity is specifically present among papilionid butterflies. Whether a correlation between color and the evolution of these butterflies exists so far remained unsolved.ResultsWe here investigate the Cattlehearts, Parides, a small Neotropical genus of papilionid butterflies with 36 members, the wings of which are marked by distinctly colored patches. By applying various physical techniques, we investigate the coloration toolkit of the wing scales. The wing scales contain two different, wavelength-selective absorbing pigments, causing pigmentary colorations. Scale ridges with multilayered lamellae, lumen multilayers or gyroid photonic crystals in the scale lumen create structural colors that are variously combined with these pigmentary colors.Conclusions The pigmentary and structural traits strongly correlate with the taxonomical distribution of Parides species. The experimental findings add crucial insight into the evolution of butterfly wing scales and show the importance of morphological parameter mapping for butterfly phylogenetics.

Photonic Crystals: Molding the Flow of Light - Second Edition

Book

Oct 2011

Making photonic structures via cell culture: Morpho butterfly scales

Article

Mar 2015

The photonic structures that cause the electric blue colour of some Morpho butterflies are desirable within industry. They have been reproduced on very small areas to produce prototype biomimetic structures, yet there is currently no technique to accurately manufacture these structures at a commercial level. Here, tissue and cell culture techniques are explored as a possible method of producing large quantities of the blue scales of Morpho peliedes. This represents the first attempt to culture the wing scales of a Morpho butterfly, and many problems were encountered. These could be remedied in future studies, but it is concluded that cell culture is probably not a viable technique to provide the Morpho structure for industry in the near future. However, tissue culture experiments revealed important results of scale development.

Large-scale integration of wavelength-addressable all-optical memories on a photonic crystal chip

Article

May 2014

Photonic integration has long been pursued, but remains immature compared with electronics. Nanophotonics is expected to change this situation. However, despite the recent success of nanophotonic devices, there has been no demonstration of large-scale integration. Here, we describe the large-scale and dense integration of optical memories in a photonic crystal chip. To achieve this, we introduce a wavelength-addressable serial integration scheme using a simple cavity-optimization rule. We fully exploit the wavelength-division-multiplexing capability, which is the most important advantage of photonics over electronics, and achieve an extremely large wavelength-channel density. This is the first demonstration of the large-scale photonic integration of nanophotonic devices coupled to waveguides in a single chip, and also the first dense wavelength-division-multiplexing nanophotonic devices other than filters. This work paves the way for optical random-access memories and for a large-scale wavelength-division-multiplexing photonic network-on-chip.

Local self-uniformity in photonic networks

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