Seminar I

Minimal surfaces in biological systems

Faculty of Mathematics and PhysicsUniversity of Ljubljana

Author: Jure Zmrzlikarzmrzlikar.jure@gmail.com

Supervisor: doc. dr. Primoz Ziherl

December 2011

Abstract

Fabrication of materials with complete photonic band gap is a problem of intense research. In oursearch for such structures we encounter minimal surfaces. We introduce them as surfaces of zeromean curvature and present some examples.The most relevant are triply periodic minimal surfacessuch as: P-surface, D-surface and G-surface (gyroid). We describe photonic crystals and theiroptical properties. Finally we present a study on butterfly wing scales. Through TEM and SEMimages and computer simulations we obtain qualitative evidence that wing scales of C. Rubi andT. Imperialis possess a gyroid structure. We also find a partial photonic band gap in the scales ofbeetle L. Augustus. We discuss possible applications of these results to produce materials with acomplete photonic band gap.

Contents

1 Introduction 1

2 Minimal sufaces 22.1 What is a minimal surface? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Some examples of minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Triply periodic minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Photonic crystals 53.1 Photons vs. electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Photonic bandgap and photonic logic gates . . . . . . . . . . . . . . . . . . . . . . . 63.3 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Promising materials: hidden in scales of insects? 84.1 Methods of observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 TEM and SEM imaging of butterfly wing scales . . . . . . . . . . . . . . . . . . . . . 94.3 Photonic crystal structure in beetle scales . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Conclusion 11

1 Introduction

Our daily lives are permeated by semiconductor technology. Computers, mobile phones and allkinds of every-day accessories cling on them. The search for faster and more powerful devices is fullof diverse ideas. One of the most promising ones is to change the carrier of information: electronscould be replaced by photons. The key problem in the production of optical technology is themissing alternative to the transistor [1]. In this search the idea of photonic crystals has been born.Photonic crystals are structures with periodic modulation of the refractive index [2]. For somestructures, there may exist wavelengths of the order of the period of structure that can not passthrough the medium [3].

In the search for such structures we encounter minimal surfaces. These are surfaces with zeromean curvature [4]. Our wish is to find a material with forbidden frequencies in all directionsso the structures should have three-dimensional periodicity. Surfaces with such periodicity arecalled triply periodic minimal surfaces. They can be found in nature in many living beings such asbutterflies, beetles and other insects [5].

It was discovered that some butterfly and beetle species already possess triply periodic mini-mal surfaces [5, 6]. Because of potential of these structures to become the much-wanted photoniccrystals with certain properties, scientists started to study them. If they could imitate or somehowmake such structures, the unknown way to realization of optical technology could be revealed [7].Recently, there has been quite some attempts to find whether some species of insects have the neces-sary structure in their wing scales. It was found that the structure itself has the correct geometricalcharacteristics, but the dielectric contrast is not large enough [5]. If we could substitute the animaltissues by some artificial substance comercially available photonic crystals could announce a newera of technology [8].

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2 Minimal sufaces

2.1 What is a minimal surface?

In order to understand minimal surfaces, we shall start with some geometrical concepts. By uder-standing them, we will be able to define minimal surfaces more precisely.

Let T be the point on the surface S. We can imagine all the curves ci passing through the pointT . Every curve ci has its curvature ki, at T . From all ki-s, let κ1 be the maximal curvature andthe κ2 the minimal one. These curvatures (κ1 and κ2) are known as principal curvatures of S atpoint T . The mean curvature at point T is then the average of principal curvatures [9]:

H =1

2(κ1 + κ2) . (1)

Fig. 1 illustrates the mean curvature.

(a) (b)

Figure 1: Mean curvatures of paraboloid and hyperbolic paraboloid. In convex surfaces, meancurvature is positive. Paraboloid (panel a) is obviously not a minimal surface, since both of itsprincipal curvatures are positive. The saddle point is characterized by zero mean curvature (panelb). Black lines are curves corresponding to principal curvatures at point T . Plotted in Mathematica.

Minimal surface is a surface with zero mean curvature in all points [10]. Finding a minimalsurface for a specified boundary condition is a problem in the calculus of variations. Sometimes it isknown as Plateau’s problem [4]. Plateau was a 19th century Belgian physicist who was interested insoap films. He immersed wire frames of many different shapes in soap. By doing that, he obtained asoapy surface, where the wire was the boundary. The soap film is the solution to the problem [11].A plane is a trivial minimal surface, and the simplest nontrivial examples are the catenoid andhelicoid. They were found by Meusnier in 1776 [10].

Contrary to our intuition a sphere is not a minimal surface. It minimizes the surface-to-volumeratio, but it does not satisfy the mathematical definition of minimal surface [10]. In all points itsmean curvature is 1/R rather than 0.

A minimal surface parametrized in 3D as z = f(x, y) satisfies Lagrange’s equation [4]:

(1 + f2x)fyy − 2fxfyfxy + (1 + f2y )fxx = 0 (2)

2.2 Some examples of minimal surfaces

1. Plane (trivial case)

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2. Catenoidis a 3D minimal surface made by rotating a catenary around its symmetry axis. Except forthe plane it was the first discovered minimal surface [12]. Usually, we visualize it as a soapfilm spanned by two circular rings. See Fig. 2a.

3. Helicoidwas the third discovered minimal surface. Its name comes from similarity with helix. In fact,at every point in helicoid there exists a helix that passes through it and is contained in thehelicoid [13]. We usually visualize it as an Archimedes’ screw or stairs in a high castle tower.By cutting one turn of a helicoid and reconnecting the edges one can deform a helicoid suchthat it becomes a catenoid [12].

(a) (b)

Figure 2: Catenoid (panel a) and helicoid (panel b). The vertical axis of catenoid is the axis,around which the rotation of catenary is performed to make a catenoid. Reproduced from Refs.[12] and [13].

4. Enneper surfaceis a more complex example shown in Fig. 3a. Points on its surface satisfy the degree-9polynomial equation [14].

5. Costa’s minimal surfaceis also a more complex example. It was discovered in 1982 by Brazilian mathematician CelsoCosta. It is a thrice-punctured torus [15]. See Fig. 3b.

s

2.3 Triply periodic minimal surfaces

Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense ofthree dimensional periodicity. We refer to them as being triply periodic. We will present threefundamental examples:

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(a) (b)

Figure 3: Enneper surface and Costa’s minimal surface. Costa’s minimal surface is topologically athrice punctured torus. Reproduced from Refs. [14] and [15].

Primitive or P-surface

Primitive or P-surface is a triply periodic minimal surface of cubic symmetry. Hermann AmandusSchwarz, the 19th century analyst first described and parametrised the P-surface [16]. It consistsof two disjoint identical sub-volumes. Every cell connects to another via 6 chanells in the gometryof a cubic cell [16]. Its surface can be approximated with equation [5]:

cos(x) + cos(y) + cos(z) = t (3)

For some reason it is believed that physicists have named soft materials adopting this structure“plumbers nightmare phases” [16].

Diamond or D-surface

Diamond or D-surface is another representative of minimal surfaces with cubic symmetry. Everycell is connected to its four neighbors in the geometry of a tetrahedron [17]. It surface can beapproximated by [5]:

cos(z) sin(x+ y) + cos(x− y) sin(z) = t (4)

Gyroid or G-surface

The gyroid is an infinitely connected triply periodic minimal surface. It was discovered by AlanSchoen in 1970 [18]. Gyroid contains no straight lines, which was proven in 1986 by RobertOsserman [18]. Ten years later Karsten Große-Brauckmann and Meinhard Wohlgemuth also provedthat it has no mirror symmetries and that it is embedded so that it does not intersect with itself(contrary to Enneper surface). It is also the only known embedded triply periodic minimal surfacewith triple junctions. It divides the space in two disjunct regions, each of them having its ownhelicity: the topological symmetry of these sub-volumes is inversion [18].

Gyroid surface can be approximated by [5]:

cos(x) sin(y) + cos(y) sin(z) + cos(z) sin(x) = t (5)

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(a) (b)

Figure 4: Unit cell of primitive or P-surface (panel a) and diamond or D-surface (panel b). Oneunit cell of P-surface connects with its 6 neighbors in the cubic lattice pattern, while D-surfaceconnects with its 4 neighbors in a diamond lattice pattern [19]. This property also gives surface itsname. Reproduced from Ref. [19].

Figure 5: Gyroid surface from two points of view. One can recognize the triple periodicity andtriple junctions of channels. Right panel shows that one can draw a set of straight lines parallel tothe body diagobal of unit cell such that they do not intersect with the gyroid surface. Reproducedfrom Ref. [20].

3 Photonic crystals

Photonic crystals - periodic structures of materials with dissimilar refractive indices [1] - havebeen studied for more than 100 years in one or another form, but it was only in 1987 that EliYablonovitch and Sajeev John have published two papers which mark the begining of the era ofphotonic crystals. After that, the number of research papers on photonic crystals began to increaseexponentially [21].

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3.1 Photons vs. electrons

We cannot imagine our daily life without semiconductor technology. In the last decades the high-speed performance and miniaturization of integrated electronic circuits has increased dramatically.Making things smaller and faster has certainly contributed to the development in many fields. Butlike always, there is a price to pay: miniaturization causes increased resistance and higher powerdissipation. Moreover, making things faster demands a greater sensitivity in signal synchronization.Nowadays, one option for a desired solution is to change the carrier of information. Light isconsidered the best alternative to electrons [1].

Light has some essential advantages. Passage of light through a dielectric material could bemuch faster than the speed of electron in a wire. It can also carry larger amount of informationper second. The bandwidth of dielectric materials is of order ∼ THz, whereas that of electronicsystems (like telephone) is only ∼ 100 kHz. Additionally, light particles (photons) interact withmatter far less than electrons. This is essential when it comes to energy losses.

Despite all this, optical circuits have not yet been widely used. The main problem is to designa multipurpose optical component, similar to electronic transistor. A new class of material, knownas photonic crystals is the most promising answer.

Manipulation of light traditionally relied mainly on the mechanism of total internal reflection.In this case, light propagates through a medium with a large dielectric contrast. When it hits theborder with a medium with low refractive index, it is reflected from the surface. This mechanismhas a great limitation. The size of a medium with constant refractive index must be much biggercompared to the wave length of light. That is certainly a big obstacle in miniaturization. Contraryto this, photonic crystals offer a completely different mechanism [1].

3.2 Photonic bandgap and photonic logic gates

In a semiconductor there is a natural periodic potential: atomic lattice. In photonic crystal theperiodic potential is ”made” of a lattice of alternating domains with different refractive indices.Contrary to semiconductor where atomic lattice is already given, photonic crystals have to befabricated. This represents a big challenge, especially if we want the periodicity to be in the rangeof visible light [1]. If the geometry of this lattice and the strength of potential (refractive contrast)satisfy certain conditions, very interesting and useful effects are observed [2]. In our context themost interesting one is the photonic band gap.

The concept of the band gap is very much the same as in semiconductors. The lattice geometryand the contrast of dielectric media determine how the frequency ω depends on the wavevector k[21]. We represent this relation in a band structure diagram (Fig. 6). Under certain conditions,there is a gap: for some interval of frequencies/energies there are no corresponding wave vectors.This means that crystal is opaque for certain wave lengths. In this case photonic crystal works likea filter for particles with certain frequencies/energies [2]. If photonic crystal is opaque for certainfrequencies in all direction for all kinds of polarizations we speak of a complete photonic band gap(complete PBG). If this is true only for some directions, the band gap is incomplete [1].

Once we have a photonic crystal with a gap we can introduce a defect to attempt to localize thelight. If line defect is used, light can be guided from one location to another. The main differencebetween traditional fibre-optic cables is when fibre-optic cable curves tightly around the angle.When this happens, the angle of incidence is too large for total internal reflection and light escapesfrom the cable. However, this is not true for waveguide (linear defect) in otherwise perfect photoniccrystal [1].

One can also create imperfections to trap light at a point in the crystal. We can either removesome of the dielectric material or add it. This leads to a strongly localized state [1]. This is

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considered to be essential for the production of optical switches and other devices. It is a field ofintense research, but today realization of a ”photnic transistor” has not been seen yet. The ideasfor such device include complicated nonlinear optics and revealing it is nor in the range of ourknowledge nor the topic of this paper.

Figure 6: Dispersion relation of photons in a square lattice (as indicated in inset) of alumina rods(ε = 8.9). Theoretical model and measurements are depicted for wave vectors k from (0, 0) to(π/a, 0), or equivalently from Γ to X. There is a big band gap between first and second band forTM modes, marked with yellow stripe. However, there is no band gap for TE modes, which makesthis band gap incomplete. Reproduced from Ref. [1].

3.3 Fabrication

Production of one dimensional photonic crystals is not problematic since it is just a simple multilayerstructure. The bigger challenge is production of two- or three-dimensional ones. The problem isto work with sufficient precision and making the procedure commercially available at the sametime [1].

Potential method of fabrication for two-dimensionally periodic photonic crystals is a photonic-crystal fiber. It can be made by using production techniques developed for communications fibres[21]. Another promising method is the so-called photonic crystal slab. These structures consistof a slab of material (such as silicon) which can be patterned using techniques borrowed fromthe semiconductor industry. Such chips offer the potential to combine photonic processing withelectronic processing on a single chip [21].

For three-dimensionally photonic crystals various techniques are being used. Some of photo-lithography and etching techniques similar to those used for integrated circuits are already com-mercially available [21]. Several alternative approaches have been explored too. One idea is to growphotonic crystals as self-assembled structures from colloidal crystals [1].

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3.4 Applications

Photonic crystals are a topic of great interest in optics. One dimensional photonic crystals are usedin thin-film optics [2]. Their applications are low- and high-reflection coatings on lenses or mirrors,color changing paints and inks etc. . . There is a great interest in two- and three-dimensionalphotonic crystals. The two-dimensional ones are already spreading into commercial applications.They are available in the form of photonic-crystal fibers. Microscale structure is used to limitlight with radically different characteristics compared to conventional optical fiber characteristics.Nowadays, three-dimensional photonic crystals are still far from commercialization [21]. Whenthey do reach it, they will enable new device concepts like optical computers replacing the task ofsemiconductors [1].

4 Promising materials: hidden in scales of insects?

In this section we will look at some experiments and simulations, which will convince us that theconcepts and observed phenomena in insects’ wing scales could be used for production of photoniccrystals.

The color of all living beings is a result of spectrally selective reflection of incoming light. Thereflected color is determined by:

1. presence of absorbing pigments or/and

2. structural variance of order ∼ 100 nm which causes interference, difraction and scattering.

In the first case we refer to the color as pigmentary (chemical) and in the second as structural(physical). Although pigmentary colors are far more common, we can find many structural colorsin plants and animals. These colors play an important role in the life of some butterflies as theyare used in intersexual signaling, communication, warning signs, etc. [8].

It is known that organismal structural colors in the animal and plant kingdom are result ofspecial nanostructure of the materials in their surface. Light interacts with a specific biomaterial,which normally has a three-dimensional modulation of the refractive index with a period similarto wave length of visible light. This kind of structure is called a biological photonic crystal [5].The most simple example of periodic structure is a multilayer. Multilayers are responsible for themetallic color in some beetle species [8]. More complex (triply periodic) structures are found inthe wing scales of many butterflies like Callophrys Rubi, P. sesostris, T. Imperialis . . . (see Fig. 7)belonging to the families of papilionid and lycaenid [5].

The cells in wing scales grow and connect to each other in a certain way by chitin. Chitin isa long-chain polymer. Its chemical structure is similar to glucose. It is the main component inexoskeletons of shrimps, shells and insects [22]. It is reasonable that cells tend to have the highestrobustness to minimal amount of material ratio. If so, it is not surprising that the structure of theframework is minimal surface! Which kind . . . We will find it out in next pages.

A precise description of optical and structural properties of these biological photonic crystalsis crucial for understanding of their evolution and function. Additional information could also beused to design color producing biomimetic devices. After all, nature has developed these structuresthrough millions of years; can we hope to do it better on a time scale of years?

4.1 Methods of observation

When investigating properties of photonic crystal there are three most commonly used methods:TEM, SEM and SAXS. We will briefly describe each of them.

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(a)

(b)

(c)

Figure 7: Wing scales (panel a and b) and a Callophyris Rubi. Butterflies are marked by the color oftheir scale-covered wings. The scales come off easily without harming the butterfly [24]. In panel aone can observe wing scales oriented in a specific direction while panel b shows the micro-structureof a single wing scale. C. Rubi - Green Hairstreak (panel c) is well-known for its green wing scales.It lives mainly in Europe [23]. Reproduced from Refs. [24] and [25].

Transmission electron microscopy (TEM) is a technique where a beam of electrons is transmittedthrough an ultra thin sample. An image is formed from the interaction of the electrons which aretransmitted. Image is then magnified and focused onto an imaging device. Most commonly TEMis used to investigate the inner structure and contours of objects (tissues, cells, viruses) [26].

Scanning Electron Microscopy (SEM) is a type of electron microscope. It images the sampleby scanning it with a high-energy beam of electrons in a raster scan pattern. It gives informationabout the sample’s surface topography and composition. It is normally applied to visualize thesurface of tissues, macromolecular aggregates and materials [27].

Small angle X-ray scattering (SAXS) is a technique where the elastic scattering of X-rays(wavelength between 0.1 and 0.2 nm) is observed on a sample which has inhomogeneities in the nmrange. Scattering is recorded at very low angles (typically 0.1 - 10o). It is used for the determinationof the structure of samples. It can give information about the average particle sizes, shape, andsurface-to-volume ratio. The materials can be either solid or liquid. They can also contain solid,liquid or gaseous subsections of another material. The method is accurate, non-destructive andusually requires little of sample preparation [28]. SAXS is capable of investigating structures ofsize with between 1 and 50 nm, while photonic crystals are bigger (150–350 nm). Only recentlySAXS has been improved to observe structures in the size of our interest. But for larger latticeparameters procedure requires even smaller scattering angles and advanced X-ray optics [8].

4.2 TEM and SEM imaging of butterfly wing scales

Recently, Michielsen and Stavenga [5] used TEM to investigate images of various structurallycolored wing scales. To better understand the results they simulated TEM pictures with gyroidcomputer model and reported good agreement. We will discuss this comparison in detail.

TEM images are usually random cuts through the sample (thickness ≈ 60-100 nm). Thesecuts can lead to complicated patterns in the results. The patterns depend on the orientationof the section, thickness and viewing angle. Besides that, they can look very similar for manyspace groups. It is obvious that a complete determination of the space group from only one cut

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is impossible: combining the results is a must. When comparing TEM micrographs to computersimulations, the structure can be identified with considerable certainty.

The assumption of Michielsen and Stavenga is that they are modeling cubic air-cuticle structuresgiven with Eqs. (3), (4) and (5). They introduced function that generates the three-dimensionalmodel of butterfly wing scales. The model consists of voxels, and value 1 denotes filling with chitinwhile value 0 denotes filling with air. Varying the parameter t changes the amount of cuticle (cuticleto air ratio). The function itself can be written as:

g(x, y, z) =

{1 if f(x, y, z) < t

0 if f(x, y, z) ≥ t. (6)

Cuts in specific directions ares simulated just like when preparing samples for TEM studies. Chitinis considered as completely opaque for the electron beam. By doing so, computing two-dimensionalprojections is considerably simplified. We sum all the values of voxels alog the line parallel toviewing direction. If the sum is greater than 0, we assign 1 to the corresponding pixel (otherwisein the value of the pixel is 0). The results are presented in Figs. 8 and 9.

(a)

(b)

Figure 8: TEM micrograph of a cross section through the scale of P. sesostris (panel a). Thenumbers in upper part indicate different domains in the cuticle crystal. The vertical lines indicatethe boundaries. Scale bar is 2.5 µm. In the bottom part, computer-simulated projections for a G-structure with t = −0.3

[Eq (5).

]are shown. The projections are generated along various directions

and agree with experimental data very well. In panel b SEM micrograph of a scale of P. sesostris isshown. Scale bar is 1 µm. Circular holes ordered in a particular pattern can be seen in lower part.When comparing this picture to the Fig. 9a we see good agreement. Reproduced from Ref. [5].

4.3 Photonic crystal structure in beetle scales

Galusha and coworkers [6] investigated the structure of iridescent scales of the weevil Lamprocyphusaugustus (Fig. 10a). In nature, many of iridescent colours are a result of light propagation throughperiodically ordered low and high refractive index structures. Still, weevil L. augustus has unusuallyangle-independent coloration. Considering the relatively low refractive index of cuticular chitin-based material, this points to highly evolved 3D photonic structure.

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(a)

(b)

Figure 9: Eight unit cells of gyroid structure constructed from Eq. (5). For t = 0.0, t = −0.3and t = −1.0 the volume fractions of cuticle are 0.5, 0.4 and 0.17, respectively. Also shown areprojections of the gyroid structures along the direction

[0 0 1

]. Circular holes have a diameter

of approximately a/4, a/3 and a/2. The similarity with experimental image (panel b) is striking.TEM image showing a section through a green scale of C. Rubi on top of panel b. Labels indicatedifferent domains in the cuticle crystal. Scale bar is 1 µm. Bottom part of panel b: computer-simulated projections for a G-structure. The agreement is very good. Reproduced from Ref. [5].

Galusha and coworkers used a similar way of analysis as Michielesen and Stavenga describedin the previous subsection. By combining SEM imaging with theoretical modeling they discovereda natural three-dimensional photonic crystal. With SEM they investigated random cuts throughthe scales and so reconstructed the inner structure of the observed scales. They concluded thatthe best match is achieved by modeling it with the D-surface (Fig. 11). By knowing the refractiveindex of chitin and the lattice structure, they calculated the band structure and discovered somepartial band gaps (Fig. 10b).

5 Conclusion

We showed that the cuticular structure in the wing scales of Papilionidae and Lycaenidae speciescan be described by a gyroid structure. We also presented evidence that beetle L. Augustus hasa D-structure in its scales. In these species that we have investigated, the cuticle volume fractionvaries between 0.17 and 0.40.

Photonic band gap and caracteristics of photonic crystals depend on many features such asdielectric contrast between the composites, the symmetry and topology of the structure, and theratio between the volume occupied by each dielectric. From computer simulations we know thatfor dielectric contrasts (n/n′ ≈ 3.5), the gyroid structure has a complete PBG for filling factorsbetween 0.04 and 0.55. The largest gap is observed for a filling factor of ≈ 0.20. Structures inbutterfly wing scales have a refractive index contrast n/n′ ≈ 1.55 [5] and in beetle wing scalesthe contrast is n/n′ ≈ 2.5 [6]. We conclude that these structures are biological photonic crystalswithout a complete PBG. However, partial band gaps do exist for light propagation in some specificdirections or polarizations, as shown on the example of L. Augustus. It is expected that light

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(a)

(b)

Figure 10: L. Augustus: iridescent green color is clearly seen. Theoretically calculated photonicband diagram for L. Augustus (panel b). The diagram was calculated by assuming that the beetlescales have a D-surface structure. One can see band gaps in the directions Γ-W (210), W-K (110)and K-Γ (100). The band gaps are marked with yellow rectangles. Knowing the size of the unitcell and rescaling the results of calculation yields the wavelengths between 541 and 647 nm. Thiscumulative gap spans the entire green wavelength region [6]. Still these band gaps are only partial.Reproduced from Refs. [29] and [6]

Figure 11: SEM images of green scales of L. Augustus. Scale top surface with different domains isshown in panel a. Panels b, c and d are SEM images of the marked individual single-crystallinedomains from panel a. Black framed insets are the calculated dielectric functions. Reproducedfrom Ref. [6].

reflected from such structures depends on the angle of incidence. However, this effect is masked bythe fact that one scale consists of several domains with different orientations, producing almost anuniform color [5].

Applications of photonic crystals are mostly in the visible (400-700 nm) or near-infrared (500-

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1300 nm) wavelength range [5]. An affordable three-dimensional artificial photonic structures witha large complete PBG in these limits would be of great technological improvement. Nowadays,manufacturing such photonic crystals is a goal for both research and industry. One way to constructthree-dimensional photonic crystals in the optical range could be by using the gyroid/diamondcuticular structure. The insects scales could serve as a template for such fabrication. By coatingthe structure with a material having higher refractive index than chitin, one could produce aphotonic crystal with a large complete PBG.

Recently, a controlled replication was performed to produce aluminium oxide replicas of wingscales from a Morpho peleides butterfly [5]. Although the photonic structure in M. peleides is muchsimpler (and two-dimensional) than the ones we described, a similar technique could be used toproduce a replica of the scales of C. Rubi, P. sesostris or L. Augustus.

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[3] L. M. Moreno, F. J. Garcia-Vidal and A. M. Somoza, Self-Assembled Triply Periodic Minimal Surfacesas Molds for Photonic Band Gap Materials, Phys. Rev. Lett. 67, 2295 (1991).

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[5] K. Michielsen and D. G. Stavenga, Gyroid cuticular structures in butterfly wing scales: Biologicalphotonic crystals, J. R. Soc. Interface 5, 85 (2008).

[6] J.W. Galusha, L. R. Richey, J. S. Gardner, J. N. Cha and M. H. Bartl, Discovery of a diamond-basedphotonic crystal structure in beetle scales, Phys. Rev. Lett. E 77, 050904(R) (2008).

[7] H. J. Moon, Triply Periodic Bicontinuous Structures as Templates for Photonic Crystals: A Pinch-offProblem, Adv. Mater. 19, 1510 (2007).

[8] V. Saranathan, C. O. Osuji, S. G. J. Mochrie, H. Noh, S. Narayanan, A. Sandy, E. R. Dufresne and R.O. Prum, Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals inbutterfly wing scales, Proc. Natl. Acad. Sci. USA 107, 11676 (2010).

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[12] http://en.wikipedia.org/wiki/Catenoid (13. 12. 2011)

[13] http://en.wikipedia.org/wiki/Helicoid (13. 12. 2011)

[14] http://en.wikipedia.org/wiki/Enneper surface (13. 12. 2011)

[15] http://en.wikipedia.org/wiki/Costa%27s minimal surface (13. 12. 2011)

[16] http://epinet.anu.edu.au/mathematics/p surface (13. 12. 2011)

[17] http://epinet.anu.edu.au/mathematics/d surface (13. 12. 2011)

[18] http://en.wikipedia.org/wiki/Gyroid (13. 12. 2011)

[19] http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html (13. 12. 2011)

[20] http://www.bathsheba.com/math/gyroid/ (13. 12. 2011)

[21] http://en.wikipedia.org/wiki/Photonic crystal (13. 12. 2011)

[22] http://en.wikipedia.org/wiki/Chitin (19. 12. 2011)

[23] http://en.wikipedia.org/wiki/Green Hairstreak (26. 12. 2011)

[24] http://en.wikipedia.org/wiki/Butterfly (26. 12. 2011)

[25] http://vasutag.freeblog.hu/archives/2009/08/27/Callophrys rubi/ (20. 12. 2011)

[26] http://en.wikipedia.org/wiki/Transmission electron microscopy (20. 12. 2011)

[27] http://en.wikipedia.org/wiki/Scanning electron microscope (19. 12. 2011)

[28] http://en.wikipedia.org/wiki/Small-angle X-ray scattering (26. 12. 2011)

[29] http://www.computescotland.com/champion-photonic-structured-beetle-or-weevil-1249.php(26. 12. 2011)

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