Complete Photonic Band Gaps in 2D Quasiperiodic Structures

Marian Florescu1,∗ Salvatore Torquato2,3, and Paul J. Steinhardt2,3

1 Department of Physics, Princeton University, Princeton, New Jersey, 08544, USA2Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA and

3Princeton Center for Theoretical Sciences, Princeton University, Princeton, New Jersey 08544, USA

(Dated: July 24, 2009)

Photonic materials exhibiting photonic band gaps (PBG), frequency ranges over which electro-magnetic wave propagation is prohibited for all directions and polarizations, are a new class artificialdielectric materials that enable controlling the generation and the flow of light. Usually, PBGs areassociated with photonic crystals, periodic systems, and the opening of the gap is thought to begoverned by the synergetic interplay between the micro-cavity resonances of individual dielectricscattering centers and the Bragg scattering resonances of the periodic dielectric array. Recentstudies have suggested that quasiperiodic photonic systems may also present photonic band gaps.However, most of the previous studies fail to account for both polarizations of the radiation or toaddress the nature of the band gap in terms of system size. Here we employ a novel optimizationmethod based on a space-tessellation scheme that we have recently formulated1 to show that is pos-sible to systematically design quasiperiodic photonic systems exhibiting TM, TE and full photonicband gaps in samples of arbitrary size. The quasiperiodic photonic systems exhibit large and robustband gaps that are highly isotropic, and may offer advantages for certain applications.

PACS numbers: 41.20.Jb, 42.70.Qs, 78.66.Vs, 85.60.Jb

I. INTRODUCTION

The field of photonic band gap materials2,3 has evolveddramatically in the last decade and their unusual prop-erties have found a wide range of applications includingefficient radiation sources4, telecommunications devices5,sensors6 and optical computer chips7. Usually, PBGopening in photonic crystals is governed by the syner-getic interplay between Bragg scattering resonances ofthe periodic dielectric array and the micro-cavity reso-nances of individual dielectric scattering centers8, whichare generated by the short- and long-range order enabledby the periodic arrangement of scattering centers.

FIG. 1: Protocol for mapping point patterns into tessellationdesigns for photonic materials (see text).

Similar scattering mechanisms are generated by

quasiperiodic short- and long-range order, and quasiperi-odic photonic structures have received a considerableamount of attention in the PBG materials field and it hasbeen shown that photonic structures based on them allowthe formation of photonic band gaps10–16. PBGs in pho-tonic quasicrystals are considerably more isotropic and ithas been argued that their opening requires a lower indexof refraction contrast than the periodic counterparts11.However, most of the QPC structures proposed are asso-ciated with a specific class of systems and/or with singlepolarization of the electromagnetic radiation, and a de-tailed understanding of the mechanisms dominating theband gap formation is still lacking. In general, the ar-chitectures required for the opening of the TM and TEband gaps are quite different (the TM PBG opening be-ing favored in structures based on a distribution of iso-lated dielectric inclusions, while the opening of TE PBGsbeing favored in structures based on connected networkarchitectures17) and in the case of quasicrystalline sys-tems (due to the presence of a rich variety of local en-vironments), the problem of finding an optimal compro-mise becomes more difficult. Consequently, until now ithas not been possible to systematically identify quasiperi-odic photonic structures displaying a sizable completephotonic band gap (for all directions of propagation andboth polarizations).

In this paper we consider photonic quasicrys-tals characterized long-range quasiperiodic transla-tional order and crystallographically forbidden rotationalsymmetries18,19. The examples illustrated here are basedon the vertices of a Penrose tiling20, a five-fold symmetricpattern composed of obtuse and acute rhombi and on thevertices of an octagonal tiling21 an eight-fold symmetric

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FIG. 2: Optimal photonic heterostructures and their Fourier transforms. Three designs of dielectric materials obtained usingthe protocol described in this paper and their diffraction patterns, S(k): a, design derived from a periodic 6-fold symmetricpoint pattern ; b, five-fold symmetric quasicrystalline design. c, eight-fold symmetric quasicrystalline design.

pattern composed of squares and rhombi. 1

The Penrose and octagonal quasicrystalline point pat-terns considered in this work are constructed by the pro-jection and cut method23, which consists of projectingthe points of an n-dimensional hypercubic lattice intothe three dimensional space, where n = 4 for the octago-nal lattice and n = 5 for the Penrose lattice. In general,the lack of spatial periodicity makes impossible to definea unique unit cell able to generate these quasiperiodiclattices, and we overcome this problem by using rationalperiodic approximants of the quasicrystalline lattice. Inthe case of the Penrose lattice periodic approximants areconstructed by the same projection method by replacingby a Fibonacci ratio (τn = pn/qn = 1/1, 2/1, 3/2, 5/3, ...).

1 We note the quasiperiodic point patterns are a special subclassof hyperuniform point systems22. A pattern is hyperuniformif the number variance within a spherical sampling window ofradius R (in d dimensions) grows more slowly than the win-dow volume for large R, i.e., 〈N2

R〉 − 〈NR〉2 = ARp, where

p < d. A scale-invariant version of the surface-area coeffi-cient A can be employed to rank-order point patterns accordingto their density fluctuations and its values A5fold = 0.60052,A8fold = 0.59567 place the quasiperiodic systems among thepoint patterns with the highest degree of hyperuniformity. Forinstance, quasiperiodic patterns are less hyperuniform than theperiodic patterns (Atriangular = 0.508347, Asquare = 0.51640,

Ahoneycomb = 0.56702, AKagome = 0.5869922), but more hyper-uniform than disordered hyperuniform point patterns (see Ref.22

for the corresponding surface-area coeffcients of some hyperuni-form disordered structures).

The unit cell is rectangularly shaped and its area growsas n increases and the rational approximant approachesτ . In particular, for a rational approximant τn = pn/qn,the unit cell has dimensions Lx = 5(pn + qn(τ − 1), Ly =√

3 − τ(pnτ + qn) and contains N = 10pn(p + 2qn) ver-tices of the rhombic pattern. In the case of the octagonalquasilattices, we replace by its continuous fraction series(ρn = p/q=1/1, 3/2, 7/5, ...), and the resulting periodicapproximant has a square shaped unit cell of side L =pn+ρqn containing N = (pn+(ρ2+ρ)qn)(pn+(ρ2

−ρ)qn)vertices of the square-rhombic pattern. The periodic ap-proximants constructed this way results in tilings consist-ing of identical tiles as the original quasiperiodic tilingand transition from quasiperiodicity to periodicity is ac-commodated through the flipping of a certain number oftiles. These rational periodic approximants are knownto be the best approximants of the quasiperiodic latticeas they minimize the number of tile flipping defects inrapport to the perfect quasiperiodic tiling. We then em-ploy a supercell method and evaluate the band structureusing a conventional plane-wave expansion formalism24.

In the case of TM radiation (electric filed perpendic-ular on the pattern plan), we adopt the simplest opti-mization scheme and assume that the point patterns aredecorated with identical dielectric cylinders, and theirradius is varied until the structure exhibit a maximumband gap. Throughout this paper we assume the pho-tonic materials are composed of silicon (with dielectricconstant ǫ = 11.56) and air. For example, an optimizedquasiperiodic structure based on the n = 5/3 periodic ap-

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FIG. 3: Photonic band structure and density of states for5-fold and 8-fold symmetric photonic quasicrystals. (a) Pho-tonic band structure and the corresponding density of statesfor a structure based on a 2/1 periodic approximant of thePenrose quasilattice decorated through the centroidal tessel-lation protocol with dielectric circular inclusions of r/a=0.157and walls of thickness w/a=0.042. The structure presents afull photonic band gap (both polarizations) of 15.14% of itscentral frequency ωc/2πa = 0.43. (b) Photonic band struc-ture and the corresponding density of states for a structurebased on a 3/2 periodic approximant of the octagonal quasi-lattice decorated through the centroidal tessellation protocolwith dielectric circular inclusions of r/a=0.1613 and walls ofthickness w/a = 0.0136. The structure presents a full pho-tonic band gap (both polarizations) of 13% of the central fre-quency, ωc/2πa = 0.44.

proximant of the Penrose lattice, displays a TM PBG of∆ω/ωC = 39%, and a similar optimized structure basedon the n = 7/5 periodic approximant of the octagonallattice displays a TM PBG of ∆ω/ωC = 42%, where ωC

is the gap central frequency. The optimal radius of thecylindrical inclusion is r/a = 0.177 for the Penrose latticeand r/a = 0.189 for the octagonal lattice, with a the sidelength of the corresponding tiling units of the underlyingpattern. To validate our supercell approximation we per-form a convergence analysis as a function of the periodicapproximation order and as shown in Fig. 4 the resultsare fully convergent. We also obtain that for the optimalstructures the band gap always occurs between bandsNP and NP + 1 where NP is the number of points of theunderlying periodic approximant point pattern. This isan analogous behaviour to the band folding that occursin periodic structures when analyzed with a super-cellapproach and underscores the importance of the singlescatterer properties in determining the photonic proper-

FIG. 4: a) Photonic band gaps for quasicrystals with five-fold symmetry: The fractional band gap size as a function ofthe number of particles in the periodic approximant, for TM(red circles), TE (orange squares) and complete (green dia-monds) band gaps. b) Photonic band gaps for quasicrystalswith eight-fold symmetry: The fractional band gap size as afunction of the number of particles in the periodic approxi-mant, for TM (red circles), TE (orange squares) and complete(green diamonds) band gaps.

ties of the structure.It was recognized before8 that the formation of the

PBGs in photonic crystals proceed as an interplay be-tween Mie and Bragg scattering effects. The dielectriccylinders support scattering resonances and, for frequen-cies above the lowest order resonance, the scattered radi-ation is out of phase with the incident wave and destruc-tive interference occurs thus prohibiting radiation propa-gation and favouring the opening of a complete TM gap.Since the point pattern displays quasiperiodic long-rangeorder, Bragg scattering also contributes to the PBG for-mation: whenever the wavevector of the incident radia-tion is directed along the effective Brillouin zone bound-ary, the reflected and refracted waves interfere to cancelthe incoming wave and prevent its propagation inside thestructure.

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FIG. 5: (a)-(d): Electric field distribution in five-fold sym-metric structures for TM polarized radiation. The structureconsists of dielectric cylinders of radius r/a = 0.177 and dis-plays a TM PBG of 36.5%. Lower (a) and upper (c) bandedge modes displaying a well defined degree of localization;Modes just below the lower band edge (b) and just above theupper band edge (d) displaying an extended character. (e)-(h)Magnetic field distribution in five-fold symmetric structuresfor TE polarized radiation. The structure consists of trihedralnetwork with wall thickness w/a = 0.102 and displays a TEPBG of 42.5%. Lower (e) and upper (g) band edge modesdisplaying a well defined degree of localization; Modes justbelow the lower band edge (f) and just above the upper bandedge (h) displaying an extended character.

FIG. 6: (a)-(d): Electric field distribution in eight-fold sym-metric structures for TM polarized radiation. The structureconsists of dielectric cylinders of radius r/a = 0.189 and dis-plays a TM PBG of 42%. Lower (a) and upper (c) band edgemodes displaying a well defined degree of localization; Modesjust below the lower band edge (a) and just above the up-per band edge (d) displaying an extended character.(e)-(h):Magnetic field distribution in eight-fold symmetric structuresfor TE polarized radiation. The structure consists of trihe-dral network with wall thickness w/a = 0.101 and displays aTE PBG of 40%. Lower (e) and upper (g) band edge modesdisplaying a well defined degree of localization; Modes justbelow the lower band edge (f) and just above the upper bandedge (h) displaying an extended character.

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Similarly, we find that the PBG formation for theTM polarized radiation proceeds through the estab-lishment of localized electromagnetic resonances at thedielectric cylinder location and that there is a strongcorrelation between the scattering properties of theindividual scatterers (dielectric cylinders) and the bandgap location (largest TM gap emerges when the spectrallocation of the first Mie resonance coincides with thelower edge of the PBG)16. As shown in Figs. 5 and 6we also find that the band gap size is determined by thedegree of localization of the photonic states, with thelower band-edge states well localized in the dielectriccomponent (lowering their frequency) and the upperband-edge states localized in the air fraction (increasingtheir frequency) of the quasiperiodic structure. Thehigh-index of refraction contrast of the structure makesthe photonic band gap more sensitive to the degree oflocalization exhibited by the lower band-edge states.

Due to the different boundary conditions at thedielectric-air interfaces, the formation of photonic bandcase for TE radiation (electric field oriented in the planeof the scatters) necessitates different dielectric architec-tures and its opening is favoured in connected networktopologies. The most used configuration employs in-verted structures, i.e., dielectric network formed by dec-orating the point pattern with non-overlapping air cylin-ders sitting in a dielectric background. However, in thecase of quasicrystalline structures we find this methodfails to provide optimal structures displaying sizable TEphotonic band gap. The main reason for this is that theresulting inverted structure presents a quite non-uniformdistribution of dielectric scattering domains and, as aresult, the resonances associated with these scatteringcenters present a broad distribution, which prohibits theexistence of well localized states in the dielectric fraction.To overcome this problem, we employ the general space-tessellation method introduced in Ref.1 able to identifynear optimal photonic structures that present sizable androbust TE photonic band gaps. As shown in Ref.1, themethod is general and is applicable to photonic structurebuilt around periodic, quasiperiodic or random point dis-tributions in one-, two- and three-dimensional spaces andprovides fully predictive band gap locations.

For the completeness sake, we briefly review this pro-tocol here. We begin by considering an arbitrary pointpattern denoted by blue disks in Fig. 1. For simplicity,we assume the point pattern is enclosed in a square do-main and employ periodic boundary conditions. First,a Delauney tessellation of the original point pattern isconstructed. This provides a triangular partitioning ofthe domain considered and we then associate with eachtriangular cell a uniquely defined point. Among the pos-sible choices, such as the triangle centroid, circumcenter,incenter, orthocenter, we opt here for the centroid as itprovides a natural minimum spatial separation betweenthe points of the resulting point distribution. Next, foreach point in the original point pattern, we use the cen-

troids of the triangles that have the point at one of itscorners to construct a cell surrounding the point. Analo-gous cells are constructed around all the points and theirwalls (assumed to have the same width) produce a tri-hedral connected network, as displayed in Fig. 1, whichconstitute the blueprint we are employing to optimizethe photonic band gaps for the TE radiation. The cellsof the connected network structure can be regarded asindividual scattering objects supporting electromagneticresonances, which become coupled to each other whenplaced into the connected network. We note that totalnumber of cells in the network is the same as the numberof points in the original point pattern and, even if thelocal environment around each point is different and re-sults in rectangular, pentagonal, hexagonal, heptagonaland octagonal cells, the trivalent connectivity is main-tained throughout the structure. The maximal TE bandgap is obtained by identifying the optimal value of thecell wall thickness.

This tessellation protocol also allows the identificationof an optimal structure exhibiting full photonic bandgaps, i.e., the TM and TE gaps are overlapping. Since theTM and TE gap formation requires apparently quite dif-ferent architectures, we use a compromise configurationproposed in Ref.1, in which the vertices of the trihedralnetwork constructed above are decorated with cylindri-cal inclusions. The number of such vertices is well deter-mined and is equal to two times the number of cells. Thefull band gap optimization proceeds then by identifyingthe optimal values of the two parameters characterizingthe resulting structure: cell wall thickness and cylinderradius. As shown in Fig. 2, the Fourier spectrum of thephotonic structures obtained through this centroidal tes-sellation protocol, decorated with circular dielectric in-clusions at placed at the pattern’s vertices and dielectricwalls along the network edges fully preserve the symme-try of the underlying point pattern.

We now apply this protocol to construct a tessellationwhose edges define the structures that produce a nearlyoptimal TE band gap. We find that in the case of thePenrose quasilattice, the optimal structure has dielectricmaterial width w/a = 0.103 along each edge; the TEband gap is ∆ω/megaC = 42.3%, the largest ever re-ported for a quasicrystalline structure. For the octagonalquasilattice, the optimal structure has dielectric materialwidth w/a = 0.106 along each edge, which produces a TEband gap of ∆ω/ωC = 39.2%. The TE band gap forma-tion proceeds analogous to the TM case as an interplaybetween the radiation scattering on individual cells andthe Bragg scattering of the quasiperiodic arrangement ofscattering planes. As shown in Fig. 5 we also find thatthe band gap size is determined by the degree of localiza-tion of the photonic states, now with the magnetic fieldof the lower band-edge states well localized in the sircomponent and the corresponding magnetic field of theupper band-edge states localized in the dielectric fractionof the quasiperiodic structure. Similar to the case of TMpolarized radiation, we also obtain that for the optimal

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structures the band gap always occurs between bands NP

and NP + 1 where NP is now the number of cells in thestructure (which coincides with the number points of theunderlying periodic approximant point pattern).

Finally, for the complete band gap, we find that the op-timal structure based on the Penrose point pattern con-sists of placing along each edge of trihedral intersection ofthe network a circular cylinder of radius r/a = 0.157 andsetting the cell wall thickness to w/a = 0.042, whereasin the case of the octagonal point patterns the optimalparameters are r/a = 0.167 and w/a = 0.014. The scat-tering properties of the individual scattering centers andcells are again essential in the band gap opening and thecomplete band gap occurs always between bands 3NP

and 3NP + 1, where NP is the number of points in theunderlying periodic approximant point pattern (there are3NP total scattering units in the system, 2NP dielectricdisks and NP dielectric cells). The resulting optimal five-fold symmetric structure displays complete (TM and TE)photonic band gaps of 16.5% — the first complete bandgap ever reported for a photonic quasicrystal with fivefold symmetry and comparable to the largest band gap(20%) found for photonic crystals with the same dielec-tric contrast25, and the eight-fold symmetric structurepresents a full photonic band gap of 13.5%. As shown inFig. 4, the band gap size obtained by considering a con-tour along the Brillouin zone is fully validated by densityof states calculations, and convergence tests for depen-dence on system size, illustrated in Fig. 3, show that theband gap size show little variation with the size of the

periodic approximant considered.

To sum, we have show that it is possible to systemat-ically expand the spectrum of dielectric materials withsizeable complete band gaps to include quasiperiodicphotonic structures. Although photonic crystals havelarger complete band gaps, quasicrystalline PBG mate-rials offer advantages for many applications. In the caseof quasicrystalline structures the PBGs are significantlymore isotropic, which is advantageous for use as highly-efficient isotropic thermal radiation sources26. The prop-erties of defects and channels useful for controlling theflow of light are different for crystal and quasicrystalstructures. Radiation with frequencies above or belowthe band edges are propagating modes that are trans-mitted through photonic crystals but are likely localizedmodes in the case of quasiperiodic patterns, which givethe former advantages in some applications, such as lightsources9. On the other hand, due to the lack of trans-lational symmetry, photonic band gap structures con-structed around quasiperiodic point patterns can providea large number of inequivalent local environments and assuch can support a rich variety of localized modes. Theselocalized modes can be employed to design laser systemswith highly unusual field patterns with possible applica-tions to biological sensing27.

Acknowledgments

This work was supported by National Science Founda-tion under Grant No. DMR-0606415.

∗ Electronic Address:Marian.Florescu@princeton.edu1 M. Florescu, S. Torquato and P. J. Steinhardt, unpub-

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