Shear based gap control in 2D photonic quasicrystals of dielectric cylinders
Angel Andueza, Joaquin Sevilla, Jesus Perez-Conde, Kang Wang
aa r X i v : . [ phy s i c s . op ti c s ] F e b Shear based gap control in 2D photonic quasicrystals of dielectric cylinders
A. Andueza ∗ and Joaquin Sevilla Smart Cities Institute (SCI), Universidad P´ublica de Navarra, 31006 Pamplona, Spain
J. P´erez-Conde
Institute for Advanced Materials (INAMAT), Universidad P´ublica de Navarra, Campus de Arrosadia, 31006 Pamplona, Spain
K. Wang
Laboratoire de Physique des Solides, CNRS, Universit´e Paris-Saclay, 91405 Orsay, France (Dated: February 23, 2021)2D dielectric photonic quasicrystals can be designed to show isotropic band gaps. The systemhere studied is a quasiperiodic lattice made of silicon dielectric cylinders ( ε = 12) arranged asperiodic unit cell based on a decagonal approximant of a quasiperiodic Penrose lattice. We analyzethe bulk properties of the resulting lattice as well as the bright states excited in the gap whichcorrespond to localized resonances of the electromagnetic field in specific cylinder clusters of thelattice. Then we introduce a controlled shear deformation γ which breaks the decagonal symmetryand evaluate the width reduction of the gap together with the evolution of the resonances, for allshear values compatible with physical constraints (cylinder collision). The gap is reduced up toa 18 .
5% while different states change their frequency in different ways. Realistic analysis aboutthe actual transmission of the electromagnetic radiation, often missing in the literature, have beenperformed for a finite “slice” of the proposed quasicrystals structure. Two calculation proceduresbased on MIT Photonic Bands (MPB) and Finite Integration Technique (FIT) are used for the bulkand the finite structures finding an extremely good agreement among them.
I. INTRODUCTION
Photonic band gap (PBG) materials have been inves-tigated for more than three decades since the germinalpaper of Ohtaka [1, 2]. One of the main objectives of thepast PBG effort was the search of metamaterials withnew optical related properties like isotropic band gap [3],zero-refractive index [4], slow light manipulation [2], etc.These PBG allow to create waveguides [5], sensors [6],collimators [7] or to improve the existent solar cells [8],etc.Some realistic PBG materials were early proposed inthree [9] and two dimensions [10, 11] which were basedon periodic lattices with complete but anisotropic bandgaps (see for instance Ref. 12, specially chapter 5 for2D PC’s). Isotropic band gaps are preferred, though, forsome applications such as wavelength selective mirror orfilters [13] slow light technology [14], filters [15], collima-tors [7] or band filters [15].On the other hand, all-dielectric photonic crystalsavoid metals and therefore can be fabricated with onlydielectric components [16, 17]. Meta-optics based inMie-type resonances in all-dielectric nanostructures is anemerging field [17].The creation of photonic materials with complete bandgap is also important to control spontaneous emissionand Purcell effect [17]. All-dielectric photonic quasicrys-tals can be also thought as bandgap systems [18–20] oras zero-refractive index homogeneous materials [4]. ∗ [email protected] Photonic quasicrystals (PQC) have been proposed andbuilt with dielectric cylinders [21, 22]. Some of thesestructures presented an almost isotropic band gap whichis one of the required features to build most of the appli-cations. Since then many QC structures based on dielec-tric cylinders have been investigated [5, 18, 20, 23, 24].In these works high permittivity contrast (cylinder vssurrounding material) was needed to get a complete andsizable band gap as it was in periodic two dimensional PCcase [2]. Recently, low refractive index contrast 1 . ε = 12) arranged in a (regular) lattice of Penrosequasicrystal approximants, a generalization of previouslystudied in Ref. 19.In previous studies it has been shown the onsetof states in the gap due Mie-like localized resonances[18, 19, 24]. These resonances have been previously re-ported within a band structure context without descrip-tion of their the actual transmission properties. Here wecalculate their transmission output and their evolutionwith the shear of the underlying lattice together withthe band structure of the bulk lattice counterpart. Wealso analyze the group velocity of these resonances, whichis very small [18], an important feature for applicationswhere slow light technology is needed [2].Finally, the simulations have been realized assumingthat the cylinders are surrounded by air, although theresults should remain valid for cylinders in a dielectricmatrix if the permittivity contrast is kept. II. SYSTEM UNDER STUDY ANDCALCULATION PROCEDURES
We want to study, in one hand, the effect of symme-try braking by shear in the gap of photonic quasicrys-tals and, on the other, how this variation affects lighttransmission through the material. The gap calculationsare performed on an infinite periodical system, while thetransmission data must be obtained from a finite sam-ple in the incidence direction. In the following subsec-tions we describe how both systems are built (II A), howshear is applied (II B) and the numerical methods usedfor the calculations (II C), MIT Photonic Band (MPB)and CST Microwave Studio (CST) for photonic band gapand transmission spectra calculations, respectively.
FIG. 1. (a) Decagonal approximant in the original QC. Theunit cell is delimited by dashed lines is shown as well as the(b) dielectric cylinders (yellow circles) at the lattice nodes.
A. Quasicrystal approximants and Bravais lattices
The bulk quasicrystal photonic lattices are not suit-able to build a physical realization nor to perform trans-mission calculations on them. In this work we defineapproximants of the parent QC as the unit cell of anotherwise periodic lattice as was first proposed in Ref. 22. The approximant-based lattice reproduces most ofthe properties of the parent QC lattice, such as ten-fold rotational symmetries, long-range decagonal bond-orientational symmetries and photonic band gap. In ad-dition, it allows to use the standard computational meth-ods developed for periodic photonic crystals (see Fig. 1).When transmission computations are concerned, weneed a lattice which is finite in the propagation direc-tion to obtain a sizable output. As the signal intensitydecreases with the lattice width, we must therefore ar-rive to a compromise between contrast between gap andresonant states intensity, which needs a big enough crys-tal, and measurable transmission. On the other hand, weknow that at least three wavelength width in the propa-gation direction is necessary to get an adequate contrastbetween the in-gap states and the gap background in pre-vious work [6].The decagonal approximant produces an oblique 2 π/ w = a √ τ andheight h = a √ τ / √ τ , where a is the edge lengthof the tiles that build up the decagonal approximant em-ployed and τ is the golden number. The rod radius con-sidered is r = 0 .
25 a.
FIG. 2. Approximant and unit cells for decagonal approx-imant (brown) and computational sized h × w cell used intransmission CST calculations. The specific values of h, w, r are also given, where r = 0 .
25 a is the cylinder radius. Bluearrows are the normal to the QC surface and green arrowsindicate the range of possible different incoming light direc-tions.
The new rectangular unit size is correspondingly w × h as shown in Fig. 2 (b). The size of the lattice in the prop-agation direction, h , is wide enough to meet the previousrequirements of contrast between gap and in-gap statesand, on the other hand, it is thin enough to allow trans-mission measurements [6]. The band structure, though,is calculated as a whole 2D lattice with the MPB pack-age [26] with the rhombus unit cell containing 76 rods asshown in Fig. 2. The final approximant shapes and sizesused with MPB technique are depicted in Fig.2(c). B. Broken rotational symmetry by shear
We investigate the QC behavior as we shear the latticeand, therefore,when the local decagonal rotational sym-metry is broken, leaving unmodified the circular shape ofthe dielectric rods. Shear is one the three possible trans-formations that can be applied to a lattice. A genericstrain or deformation, D , in two dimensions can be imple-mented with a deformation applied to every ( x, y ) point,so that the new transformed point ( x ′ , y ′ ) can be writtenas [27], (cid:18) x ′ y ′ (cid:19) = D · (cid:18) xy (cid:19) = (cid:18) a bc d (cid:19) · (cid:18) xy (cid:19) = (cid:18) ax + bycx + dy (cid:19) (1)Any lattice deformation , D , can be decomposed ina dilatation, a rotation and a shear. We are interestedin deformations that can modify the band structure orthe lattice transmission spectrum. Also, although globaldilatation do produce a shift of the whole spectra and itcould be thought as a modification, this behavior can be FIG. 3. Unit cells of decagonal approximants in the originalQC and in the sheared case used with the MPB technique(top). Below we show the Bravais lattice where each pointrepresent an approximant for two values of shear γ = 0 . γ = 0 .
15. The angles α and β are measured from the y -axisto the primitive vectors a and a respectively. also seen as a simple scaling of the lengths and frequenciesin the Maxwell equations and no symmetry change isimplied. We therefore restrict the possible deformationsto those of pure shear which break the rotational n-foldsymmetry of the QC. The shear transformations , S , canbe described by the matrix, S = (cid:18) γ x γ y (cid:19) , (2)where γ x , γ y are two real numbers which reflect theamount of the shear in the x, y directions respectively.The transformed points ( x ′ , y ′ ) from ( x, y ) are given fromequation 1, (cid:18) x ′ y ′ (cid:19) = (cid:18) γ x γ y (cid:19) · (cid:18) xy (cid:19) = (cid:18) x + yγ x y + xγ y (cid:19) (3)From now on, we study shear transformations with thesame deformation in both axes, γ x = γ y = γ , so that weonly need γ parameter to characterize a given shear. InFig. 3 the effects of the shear for the decagonal approxi-mant are shown, together with the corresponding originaland sheared Bravais lattices. FIG. 4. Brillouin zone of the approximant-based Bravaislattice (dashed blue line) together with the irreducible BZ(green) for three values of γ = 0 , .
05 and 0 .
15. The irre-ducible BZ is a fourth of the BZ for γ = 0 but when therotational symmetry is broke, γ = 0 is half of of BZ. The theprimitive vectors of the reciprocal lattice, b and b , are alsoshown. As the shear is applied to the whole quasicrystal theapproximants are transformed and the correspondingBravais lattices are also modified as was shown in Fig.3. We need to calculate the new sheared Brillouin zones(BZ), their irreducible part and the special symmetrypoints to compute the bands at these special values. InFig. 4 we show the original oblique Brillouin zone fromdecagonal approximant at γ = 0. We also illustrate the γ = 0 instance with two values of γ , 0.05 and 0.15, re-spectively. The loss of symmetry in these cases is clearlyseen as the irreducible BZ of the sheared lattice is half ofthe whole BZ, whereas the original unsheared irreducibleBZ’s for the decagonal lattice was a fourth of the wholeBZ. This is the expected behavior for the irreducible BZ:its size is always the 1 /g − th part of the first Brillouinzone (see Ref. 28 p. 281). Here g is the number of sym-metry operations of the original point group: C v , g = 4in the not sheared oblique periodic lattice and C , g = 2in the sheared oblique case. C. Numerical methods
We compute the periodic approximant-based latticesband by means of preconditioned conjugate-gradientminimization of the block Rayleigh quotient in a plane-wave basis, using MIT Photonic Bands package [26].The MPB software allows to inspect in detail the bandstructure, gap and resonant in-gap states due to lightlocalization. Those quantities describe a periodic andtherefore infinite lattice, however, in actual applications,we need to address the experimental accessibility of thephotonic lattice, in particular the transmission propertiesin a finite sample. We investigate this issue by meansof Finite Integration Technique [29] with the CST MI-CROWAVE STUDIO TM , a commercial code. The useof both methods allows, additionally, to cross check theresults.The numerical simulations for a finite version are per-formed on a two dimensional lattice of dielectric cylindersinfinite in the y direction and assuming infinite lengthcylinders in the x direction (infinite length rods). Theincoming radiation travels in the z direction as Trans-verse Electric and Magnetic (TEM) mode with the elec-tric field along the x-axis and the magnetic field alongthe y-axis. The incoming wave presents an incidence an-gle ϕ of 18 ◦ to the external surface of the structure asis indicated by the dashed green arrow in Fig. 2(c). Afrequency solver tool with a tetrahedral adaptive meshrefinement was used to calculate the power balance ofthe PQC in CST. The symmetry of the finite PQC andthe orientation of the electromagnetic field allow us tosimplify the problem restricting the calculation to a rect-angular unit cell. CST solves the Maxwell equations andanalyzes the weight of each diffraction order separately.A summation of the power density in all diffraction or-ders provides the total reflectance, R, and transmittance,T. As the refractive index employed for the calculationsis real, the absorptivity of the QC is not considered intothe power balance.We calculate the transmission spectra with the FITmethod for different lattice widths in the z direction,starting with the largest case ( h wide) as depicted inFig. 2(b). Then, row after row, the cylinders are re-moved and the resulting transmission obtained for eachwidth value. A total of 15 configurations are obtainedin the process. The gap and resonances frequencies arethen obtained from the analysis of the resulting spectra.The bulk counterpart, as a periodic lattice of decagonalapproximants is shown in Fig. 2(c) for the original (un-sheared) and sheared lattices, is analyzed with the MPBpackage. We obtain the first 110 bands, which are enoughto capture the whole gap frequency range as well as theresonant states in the gap interval. III. RESULTS AND DISCUSSION
Results are presented in Fig. 5, 6 and 7 for differentvalues of shear γ =0, 0.05 and 0.1 respectively. The threefigures have the same structure, composed of differentimages that summarize all results.In the top left corner (labeled by a) there is a colormap that presents the transmission spectra (hotter colorshigher transmission) for different widths of the sample inthe horizontal axis. Frequency is presented normalizedto the lattice parameter a in the vertical axis (see figure1). Sample width, presented as h/a , is also normalized tothe same dimensional parameter. This representation ismade by adding several spectra as the one shown in thetop right figure (labeled by c) calculated with CST fordifferent values of sample width and interpolating to geta smoother color plot. The onset of a well-defined gap aswidth increases can be clearly seen in this representation.In addition, several bright states inside the gap are clearlyobservable.The band structure of the corresponding bulk lattice,calculated with MPB, is shown in the figures at top cen-ter (labeled by c), in the same frequency interval as thetransmission data, so that comparisons between the twocalculations can be readily made. The frequencies of gapedges and resonant states within the gap for the bulkstructure are very close to those observed in the trans-mission spectra for a finite sample (simulated with CST).We also show the electric field intensity distributionof the 5 bright states as identified in the bulk latticeband calculations (numbers 1 to 5 in the lower part ofthe figure). These data are calculated with MPB andpresented normalized to ± γ = 0 . h/a = 8.The gap range decreases with lattice deformation. Thefrequency range of low transmission is smaller as theshear increases. This happens because both gap bordersmove towards the center for higher values of shear. Thiscan be seen comparing representations of transmissionand band structures in figures 5, 6 and 7. The former(5(b), 6(b) and 7(b)), suggest that several bands thatare packed near the gap edges for the undistorted lat-tice begin to open and lose degeneration when symmetrystart to vanish due to deformation. In figure 8 we pro- FIG. 5. Original lattice, γ = 0. (a) Transmission spectra dependence on the normalized width, h/a (top left color map). (b)Band of the bulk lattice in the same frequency interval, around the gap (top center). The five resonances in the gap are labeledand their field distribution show below. (c) The transmission spectrum for the widest lattice is also shown h/a = 13 (top right).FIG. 6. Sheared lattice γ = 0 .
05. The same quantities as in Fig. 5. vide smooth curves of the evolution of the gap edges withshear, calculated for much more values of shear that thoseshown in previous figures.It is interesting to note that the bands inside the gapare significantly flatter than the bands present in the edgeof the band gap. This characteristic shows that the lightpropagates very slowly at the frequency of the excited states due to the slowing down of group velocity [24].This confirms that the low group velocity is essentiallyrelated to the resonant states on the local decagonal ringsof the lattice and other cylinder clusters, even when sheardisturbs the lattice [30].Bright states inside the gap can be clearly seen in fig-ures 5, 6 and 7. The states are present in the band struc-
FIG. 7. Sheared lattice γ = 0 .
1. The same quantities as in Fig. 5. ture (shown as b in the three figures) and are also visi-ble as frequencies of increased transmission in the colormaps (labelled a in the figures) presenting a significantgood match. These states are due to local resonances ofa cylinder clusters of the structure as shown in the lowerpart of figures 5 to 6 and 7. The states are labeled 1through 5 in coincidence with their naming in the bandstructure.Resonance labelled 5 corresponds to a full symmet-ric resonance of the decagonal ring in the center of thestructure, with five maxima and five minima placed ineach cylinder of the ring. Resonances 1 and 2 are alsoheld in the decagonal ring but are composed of only fourmaxima and minima, not fitting so well with the cylin-ders and allowing two configurations (90 degree from eachother). These two geometrical configurations are totallyequivalent and therefore their energy levels are the same,appearing degenerate in the band structure of the undis-torted lattice (figure 5). However, this degeneration isbroken when shear is introduced, and symmetry is lost.Resonances 3 and 4 are due to incomplete decagonalrings (“C” shaped half ring), apart from the decagonalring. Two families of incomplete rings deform differentlywhen shear is introduced, separating their energy levelsthat were degenerated in the undistorted lattice. How-ever, these two geometrical configurations remain verysimilar, so their energy is quite close. There are otherpossible resonant configurations of substructures of thelattice (close to the edge of the gap), but their energy liesoutside the bandgap and cannot be considered as isolatedstates.As abovementioned, the states in the gap appear ashigh transmission frequencies in the spectrum (figures 5(a), 6(a) and 7(a)). When radiation in open spacereaches the edge of a finite sample of the structure, it ex-cites the resonance of the cylinder clusters for the appro-priate frequencies. This excitation is transmitted fromcluster to cluster until the other side of the sample isreached and the sample radiates again to the free space.Therefore, the radiation able to excite resonances cantraverse the sample end to end metaphorically jumpingin the clusters, in the same way that a person can crossa river jumping in stones placed in its course. This is thereason why the transmission at the frequency of the statesremains high as the sample width is increased, while theone of other frequencies decreases much more quickly.This generates a significant contrast between the trans-mission of the states and its background for widths graterthat h/a =4. A study of the optimum width to improvecontrast in this kind of systems was published elsewhere[6].In order to follow more closely the evolution of thestates in the gap with shear, we calculated the transmis-sion of samples of more deformation values than thosepresented previously as figures 5(c), 6(b) and 7(c). In allcases calculations are performed on finite samples wideenough to have the gap fully developed ( h/a = 13). Theresults are presented in figure 8.There are many interesting features that can be de-rived from this figure such as the decrease in the gapwidth (mainly reduced by the decreasing of the upperedge) or the increase in the number of modes that appearas deformation rises (mainly departing from the lowergap edge).Most of the states excited inside the band gap showa strong dependence on shear, increasing their frequency
FIG. 8. Frequency evolution of the resonances obtained fromtransmission calculations for h/a = 13 with γ the shear value.Solid red lines represent resonances with an average transmis-sion intensity higher than -20 dB, solid green lines to reso-nances with an intensity between -20 and -40 dB and solidblue lines to resonances lower than -40 dB. Resonances arelabeled from 1 to 5 as it was previously made in figures 5,6 and 7. The edges of the band gap are represented by thegrey area.Vertical dashed lines highlight the states shown infigures 6 and 7. position with it. Exceptions are states 3 and 4 (that remain unaltered for certain intervals) and state 1, theonly that decreases in frequency. The different slopesof these curves lead to crossings as the one observed forshear around 0.08, where states 2, 3 and 4 change theirrelative positions. States 1 and 2, degenerate in the un-deformed system, quickly diverge in opposite directionsfor low values of shear. The initial frequency decrease ofstate 1 ends for a shear value of 0.1, and from this pointon, merges with others that where rising in frequency. Itis important to note that some of these states, especiallystates 1, 2 and 3, are quite bright, presenting a significantcontrast with respect to the background, what could leadto interesting applications. We have explored elsewhere(ref. [31]) the possible construction of a strain sensorbased on the frequency difference between states 1 and2. IV. CONCLUSIONS
We have studied the effect of shear in the photonicband structure and transmission properties of 2D dielec-tric photonic quasicrystals based on cylinders with highdielectric permittivity ( ε = 12). The band structure ofthe bulk lattice (and field distribution of the resonances)was calculated with MPB, while the transmission of finitesamples of different widths was calculated with CST soft-ware. The system presents an isotropic bandgap and fivebright states inside the gap. These features can be ob-served (with a remarkable match) both, in the finite sam-ple transmission and in the bulk photonic band structure.The photonic states in the bandgap are due to Mie likeresonances of local clusters of cylinders and are associ-ated with very flat bands, which generates very low groupvelocity of the transmitted radiation. The quasicrystalundergoes a remarkable electromagnetic response vari-ation induced by shear, gap bandwidth decreases, andthe distortion of the position of the cylinders, breaks en-ergy degeneration of excited states into the bandgap andshifts the energy values of all of them. Besides, it exits acomplex evolution of the frequency position of the stateswith shear. This set of characteristics is interesting inthe design of practical devices such as sensors. [1] K. Ohtaka, Phys. Rev. B , 5057 (1979).[2] M. Notomi, Rep. Prog. Phys. , 096501 (2010).[3] W. Man, M. Florescu, K. Matsuyama, P. Yadak, G. Na-hal, S. Hashemizad, E. Williamson, P. Steinhardt,S. Torquato, and P. Chaikin, Opt. Express , 19972(2013).[4] S. V. Boriskina, Nat. Photonics , 422 (2015).[5] T. Priya Rose, G. Zito, E. Di Gennaro, G. Abbate, andA. Andreone, Opt. Express , 26056 (2012).[6] ´A. Andueza, K. Wang, J. P´erez-Conde, and J. Sevilla, J.Appl. Phys. , 083101 (2016).[7] A. F. Matthews, Opt. Commun. , 1789 (2009). [8] I. Cornago, S. Dominguez, M. Ezquer, M. J. Rodr´ıguez,A. R. Lagunas, J. P´erez-Conde, R. Rodriguez, andJ. Bravo, Nanotechnology , 095301 (2015).[9] E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. ,1950 (1989).[10] P. R. Villeneuve and M. Pich´e, Phys. Rev. B , 4913(1992).[11] R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D.Joannopoulos, Appl. Phys. Lett. , 495 (1992).[12] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D.Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Prenceton, N.J, 2008). [13] K. Edagawa, Sci. Technol. Adv. Mater. , 034805(2014).[14] T. Baba, Nat. Photonics , 465 (2008).[15] N. B. Ali and M. Kanzari, Phys. Status Solidi , 161(2011).[16] K. Edagawa, Nature Nanotechnol. , 23 (2016).[17] I. Staude, T. Pertsch, and Y. S. Kivshar, ACS Photonics , 802 (2019).[18] K. Wang, Phys. Rev. B , 085107 (2007).[19] K. Wang, Phys. Rev. B , 045119 (2010).[20] Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photon-ics , 177 (2013).[21] M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J.Baumberg, and M. C. Netti, Nature , 740 (2000).[22] K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz,Phys. Rev. B , 2095 (2003).[23] J. Romero-Vivas, D. Chigrin, A. Lavrinenko, and C. So-tomayor Torres, Opt. Express , 826 (2005). [24] K. Wang, Phys. Rev. B , 235122 (2006).[25] T. Weiland, in Computational Electromagnetics , editedby P. Monk, C. Carstensen, S. Funken, W. Hackbusch,and R. H. W. Hoppe (Springer Berlin Heidelberg, Berlin,Heidelberg, 2003) pp. 183–198.[26] S. Johnson and J. Joannopoulos, Opt. Express , 173(2001).[27] R. W. Ogden, Non-Linear Elastic Deformations (DoverPublications, Mineola, N.Y, 1997).[28] A. W. Joshi,
Elements of Group Theory for Physicists (New Age International, Mineola, N.Y, 1997).[29] M. Clemens and T. Weiland, PIER (265).[30] J. G. P. Berman and J. F. M. Izrailev, Physica D , 445(1983).[31] ´A. Andueza, J. P´erez-Conde, and J. Sevilla, Opt. Express29