Point-Defect-Localized Bound States in the Continuum in Photonic Crystals and Structured Fibers
Sachin Vaidya, Wladimir A. Benalcazar, Alexander Cerjan, Mikael C. Rechtsman
PPoint-Defect Localized Photonic Bound States in the Continuum
Sachin Vaidya, , ∗ Wladimir Benalcazar, Alexander Cerjan, , , and Mikael C. Rechtsman Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Sandia National Laboratories, Albuquerque, New Mexico 87123, USA Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque 87123, New Mexico, USA (Dated: February 3, 2021)We show that point defects in two-dimensional photonic crystals can support bound states inthe continuum (BICs). The mechanism of confinement is a symmetry mismatch between the defectmode and the Bloch modes of the photonic crystal. These BICs occur in the absence of bandgapsand therefore provide an alternative mechanism to confine light. Furthermore, we show that suchBICs can propagate in a fiber geometry and exhibit arbitrarily small group velocity which couldserve as a platform for enhancing non-linear effects and light-matter interactions in structured fibers.
Over the last three decades, photonic crystals (PhCs)have been shown to exhibit exceptional confinement andtransport properties due to their ability to host local-ized defect modes that can serve as high-Q resonatorsor waveguides [1]. These confined modes form the basisof many devices such as PhC fibers [2, 3], spectral fil-ters, and lasers [4]. To achieve such confinement, defectmodes are constructed to lie within photonic bandgapsso as to spectrally isolate them from the extended statesof the PhC. However, spectrally isolating such modes ne-cessitates the use of materials with a sufficiently high re-fractive index to open complete photonic bandgaps. Analternative mechanism for confinement could circumventthe need for bandgaps, enabling the use of many low-refractive index materials such as glasses and polymersas well as increasing design flexibility for the realizationof PhC-based devices.One possible way to achieve this is by using boundstates in the continuum (BICs). BICs are eigenmodes ofa system that, despite being degenerate with a contin-uum of extended states, stay confined - this confinementmay result from a variety of mechanisms [5]. For in-stance, modes of a PhC slab that lie above the light lineof vacuum and therefore could radiate, can remain per-fectly bound to the slab [6–12]. Previous designs withBICs have mostly shown confinement of a mode in onedimension lower than that of the environment. Recently,corner-localized BICs were predicted and observed intwo-dimensional chiral-symmetric systems with higher-order topology [13, 14]. However, chiral (sub-lattice)symmetry is, in general, strongly broken in all-dielectricPhCs. Indeed, confinement in the continuum has to thispoint not yet been achieved in point defects embeddedinside multi-dimensional PhCs.In this work, we predict the existence of bound statesin the continuum that are exponentially confined to pointdefects in a two-dimensional PhC environment. The de-fect cavity and bulk PhC are designed such that radiation * [email protected] FIG. 1. (a) The unit cell of the two-dimensional PhC con-sists of a circular disc with dielectric constant ε and radius r . The symmetry operators leaving the unit cell invariant arelabelled. These operators constitute the C v point group. (b)The Brillouin zone of the PhC showing its HSPs and the littlegroups under which the HSPs are invariant. The irreducibleBrillouin zone consists of all momenta that lie within the solidcolor. leakage is prohibited due to a symmetry mismatch be-tween the defect mode and the ambient continuum states.The BICs proposed here are protected by the simultane-ous presence of time-reversal symmetry (TRS) and thepoint group of the lattice and as such are robust as longas these symmetries are maintained. We also show howthese BICs can circumvent bandgap requirements and beused as propagating fiber modes with arbitrarily smallgroup velocity in a low-contrast slow-light PhC fiber.We draw a distinction between our BICs and the previ-ously reported defect modes degenerate with Dirac pointsin 2D PhCs [15–17] which have an algebraic spatial de-cay. In the latter case, the confinement of light to a defectsite is due to a vanishing density of states at the Diracpoint, which is where that confined mode’s frequency lies.Characteristically, such defect modes exhibit weak con-finement due to the algebraic mode profile away from thedefect site. In contrast, the defect modes presented hereare bona fide symmetry protected BICs that are expo-nentially localized to the defect site.Consider a two-dimensional PhC consisting of a squarelattice of discs with dielectric constant ε and radius r a r X i v : . [ phy s i c s . op ti c s ] F e b embedded in vacuum. This PhC, as shown in Fig. 1(a) is invariant under 90 ◦ rotations ( C , C , C − ), andreflections along the x , y axes and two diagonals ( σ x , σ y , σ d , σ d ). These symmetry operations constitutethe C v point group. The irreducible Brillouin zone ofthis lattice contains three inequivalent high symmetrypoints (HSPs), namely, Γ = (0 , X = ( π/a,
0) and M = ( π/a, π/a ), as shown in Fig. 1 (b). The HSPs Γ and M are invariant under the full C v group, while X is invariant only under the little group, C v . Eigen-modes of the PhC at a HSP transform according to theirreducible symmetry representations (irrep) of the groupunder which the HSP is invariant. The X point has fourpossible one-dimensional irreps ( a , a , b , b ) with char-acter table as shown in Table I. Similarly, the Γ and M points have four one-dimensional irreps ( A , A , B , B )and one two-dimensional irrep ( E ) with character tableas shown in Table II [18]. C v I C σ x σ y a a − − b − − b − − C v point group. C v I C C σ x /σ y σ d /σ d A A − − B − − B − − E − C v point group. The mechanism for creating defect localized BICs is asfollows: The eigenmodes of a C v symmetric PhC thattransform according to the two-dimensional irrep ( E ) ofthe C v point group, commonly manifest as quadratictwo-fold degeneracies at Γ and M [20]. By changing thegeometric parameters of the lattice, the band dispersionof the bulk PhC can be designed such that the two-folddegeneracy at either Γ or M is spectrally isolated fromother bands. In other words, at the frequency of the two-fold degeneracy, there are no Bloch modes of the PhC forany other momenta. In a large system consisting of manyunit cells of such a PhC (a supercell), a single defect sitewith radius r d = r is introduced at the center. Thiscreates modes with a significant support on the defectsite which generally radiate by hybridizing with the bulkstates of the PhC, forming resonances. The frequency ofsuch modes can be tuned by changing the parameters ofthe defect site such as size or dielectric constant. Whenthe frequency of the defect mode exactly matches that ofthe spectrally-isolated two-fold degeneracy of the bulk,it becomes a perfectly confined BIC provided that thedefect mode transforms according to a one-dimensional irrep that is orthogonal to the two-dimensional irrep ofthe bulk.To demonstrate this, we simulate this system usingfinite-difference time domain method (FDTD) as imple-mented in MEEP [21]. The bulk band requirements areeasily met in a simple square lattice of discs with dielec-tric constant ε = 4 and radius r/a = 0 . a is thelattice constant in both x and y directions. The chosenvalues of ε and r/a allow the spectrally-isolated two-folddegeneracy to occur between TM bands 10 and 11 at the M point as shown in Fig. 2 (a). The photonic densityof states (DoS), also shown in the same figure, is givenby DoS( ω ) = P n R k ∈ BZ δ [ ω − ω n ( k )]d k , where ω n ( k ) isthe frequency eigenvalue at the momentum k and bandindex n . Since each band undergoes an extremum at thedegeneracy, the DoS exhibits a jump-discontinuity-typeVan Hove singularity between two finite and non-zerovalues. The non-vanishing set of states at the degen-eracy forms the continuum within which a BIC can becreated. In a large supercell, we now introduce a defectby changing the radius ( r d = r ) of a single disc in thecenter of the supercell. As we scan the values of r d , aBIC emerges for the specific value of the defect radiusthat corresponding to a mode with the exact frequencyof the bulk degeneracy. This is seen from the sharp di-vergence of the quality factor, Q = − Re( ω ) / ω ), ofthe defect mode as shown in Fig. 2 (b). Examining themode profile shown in the inset of Fig. 2 (c) reveals thatthe defect mode transforms according to the irrep A which is prevented from mixing with the basis modes ofthe orthogonal irrep, E of the bulk. Moreover, the modeshows very strong exponential localization to the defectsite which can be seen by plotting the intensity envelopeas shown in Fig. 2 (c). Another important feature ofthis BIC is its occurrence high up in the band structure,specifically, above ωa/ πc >
1. This implies that thelattice constant of the bulk PhC is larger than the wave-length of the BIC mode, a property which could proveuseful for fabrication, because features sizes would neednot be subwavelength.To conclusively show that this BIC is indeed symme-try protected, we change the defect site from a disc toa filled ellipse, which reduces the symmetry of the su-percell from C v to C v . The two states of the bulk de-generacy which previously transformed according to thetwo-dimensional irrep E of C v now transform accord-ing to the one-dimensional irreps b and b of C v inthe supercell. As before, we vary the defect size to tunethe frequency of the defect mode and find a maximum Q ∼ indicating that the mode is not a BIC but aleaky resonance. Indeed, the field pattern of the defectmode as shown in the inset of Fig 2 (d), transforms ac-cording to b , which coincides with one of the irreps ofthe bulk. Thus, the defect and bulk modes are able tomix to create a leaky resonance with a finite Q (see Sup-plementary Material for plot of Q vs. defect size for this FIG. 2. (a) The TM bands and photonic DoS of a square lattice of dielectric discs of ε = 4 and r/a = 0 .
275 calculated usingMPB [19]. The spectrally-isolated two-fold degeneracy occurs between TM bands 10 and 11 and is marked with an arrow. TheDoS has a jump discontinuity at the frequency of the degeneracy (dotted line) due to band extrema. (b) Quality factor (Q) ofthe defect mode as a function of defect radius ( r d ). The sharp divergence in Q indicates the existence of a BIC at r d /a = 0 . r d . (c) The E -field intensity envelope of the BIC showingexponential localization as a function of distance (along the y -axis) from the defect site. The inset shows the z -component of the E -field of the BIC, extracted from FDTD simulations. (d) The E -field intensity envelope of the resonance when the symmetryof the supercell is reduced from C v to C v . The inset shows the z -component of the E -field of the resonance, extracted fromFDTD simulations. case).The symmetry mismatch between the defect mode andbulk bands requires the existence of a spectrally-isolatedtwo-fold degeneracy in the bulk PhC so the question nat-urally arises: how easy is it to design this bulk band re-quirement? It is clear from our findings that even simplesquare lattices consisting of only circular discs are able tosatisfy the requirements for reasonably low dielectric con-trast and in fact, the feature in the TM bands of the PhCdiscussed in Fig. 2 (a) persists down to ε = 3 for a slightlysmaller value of r/a . Furthermore, such quadratic de-generacies can also occur at the Γ point in C v and C v symmetric lattices, forming two-dimensional irreps of therespective point groups. In the Supplementary Material, we outline a method for finding optimized structures withtunable parameters that exhibit such degeneracies.For traditional defect modes in 2D PhCs, it sufficesto have a bandgap for one polarization, either TE orTM, since they constitute orthogonal subspaces that donot mix. However, for applications such as PhC fibers,(i.e., where the 2D pattern described above is extrudedin the third direction, z , and k z = 0 generally), the dis-tinction between TE and TM is lost and one requiresan overlapping bandgap for both polarizations to confinedefect modes. In particular, slow-light PhC fibers relyon the existence of a complete bandgap at k z = 0 whichpersists for a small range of k z [22–24]. The arbitrar-ily small group velocity of the propagating modes comes FIG. 3. (a) The k k -band structure of the defect-free PhC fiber at k z = 0 .
18 (2 π/a k ). The spectrally-isolated two-fold degeneracyis marked with an arrow. (b) D -field intensity profile of a solid-core fiber BIC mode that occurs at k z = 0 .
18 (2 π/a k ). (c) D -field intensity profile of a hollow-core-like fiber BIC mode. from operating near the k z = 0 band edge. These slowly-propagating modes can then be used to strongly enhanceinteractions of light with either the dielectric materialitself or an infiltrated material [25, 26], depending onwhether the fiber hosts a solid or hollow core. Thus, thedesign of these fibers requires a high dielectric contrastto open complete a bandgap at k z = 0. To the best ofour knowledge, the smallest contrast for which a com-plete bandgap exists for 2D PhCs is for ε = 4 .
41 [27].We now extend the idea of point-defect localized BICsto propagating slow-light fiber modes which circumventsthe requirement for a complete bandgap.The fiber design that we propose is identical to anextruded version of the 2D PhC discussed before, nowconsisting of cylinders extended along the direction ofpropagation in the fiber. However, since the distinctionbetween TE and TM polarizations is lost, the spectrally-isolated two-fold degeneracy of the bulk needs to occurin the full band structure in order to create a BIC. Thisis easily achieved in our structure for a range of k z valuesaround 0. For instance, Fig. 3 (a) shows the band struc-ture of the fiber with ε = 4, r/a = 0 . k z = 0 . π/a k ), where a k is the lattice constant in the x, y plane.As before, we introduce a defect site and tune the radius r d and find a BIC at r d /a = 0 .
230 for this particularvalue of k z . Alternatively, one could think of the BICmode as occurring at the intersection of the k z -dispersioncurves of the bulk degeneracy and the defect mode. Thefield profile of the BIC is plotted in Fig. 3 (b), forming asolid-core mode and displaying strong confinement to thedefect site. Since the spectrally-isolated two-fold degen-eracy persists down to k z = 0, the group velocity of thisBIC along the length of the fiber ( v g z = d ω/ d k z ) can bemade arbitrarily small with an appropriate choice of r d .It is also possible to create a hollow-core-like fiber modewhere the BIC has reasonable support in the air region.To achieve this, we omit the central defect site and in- stead tune the radius of the nearest eight sites uniformlyso as to maintain C v and find a BIC as shown in Fig. 3(c).In conclusion, we have presented the discovery ofBICs that are exponentially localized to defects beyondbandgaps in both 2D PhCs and structured fibers. ThePhC slow-light fiber implementation relaxes the need forbandgaps at k z = 0 and thus allows for a wider range ofmaterials to be used for their implementation. The re-sults presented here have consequences for the general de-sign of PhC-based devices since the requirement for find-ing bandgaps could potentially be replaced with findingisolated degeneracies at HSPs, which occur more com-monly, at lower dielectric contrast and higher up in theband structure. For example, the principles discussedhere could be applied to create high-Q nanocavities ingapless PhC slabs where vertical leakage is unavoidablebut in-plane leakage could be suppressed through thesymmetry mismatch mechanism. Such isolated degen-eracies are also known to occur in 3D PhCs which couldlead to true gapless confinement of light in all directionssuch as those that are precursors to structures havingWeyl points [28–30]. Furthermore, due to the structuralsimilarity between PhC fiber modes and ‘hinge modes’in the context of higher-order topological insulators [31–33], it is possible that the BIC mechanism [13, 14] canbe used to realize topological-protected hinge modes inhigher-order photonic topological insulators.M. C. R. acknowledges support from the Office of NavalResearch (ONR) Multidisciplinary University ResearchInitiative (MURI) grant N00014-20-1-2325 on RobustPhotonic Matertials with High-Order Topological Pro-tection as well as the Packard Foundation under fellow-ship number 2017-66821. W.A.B. is grateful for the sup-port of the Eberly Postdoctoral Fellowship at the Penn-sylvania State University. This work was performed, inpart, at the Center for Integrated Nanotechnologies, anOffice of Science User Facility operated for the U.S. De-partment of Energy (DOE) Office of Science. 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Rechtsman Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Sandia National Laboratories, Albuquerque, New Mexico 87123, USA Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque 87123, New Mexico, USA
Q-FACTOR FOR C AND C SUPERCELLS
In the main text, we argue that changing the symmetry of the supercell from C v to C v allows the defect modeto form a leaky resonance due to a change in the representation of the bulk degeneracy from E to b ⊕ b . Thissymmetry breaking is achieved by changing the defect site from a circular disc to an elliptical disc with semi-majorand semi-minor axes lengths of 0 . a × t d and 0 . a × t d respectively where a is the lattice constant and t d is a tuningparameter. Fig. S1 shows a comparison of quality factors of the defect mode for the two cases and clearly shows thelack of divergence of Q in the C v symmetric supercell, indicating that the defect mode is indeed a resonance. FIG. S1. (a) Quality factor of the defect mode (irrep A ) in a supercell with C v symmetry. The divergence in Q showsthe appearance of a BIC. (b) Quality factor of a defect mode (irrep b ) in a supercell with only C v symmetry. The lackof divergence indicates that the defect mode is a resonance. The insets show the defect mode profiles for parameter valuescorresponding to the maximum Q. FINDING SPECTRALLY ISOLATED DEGENERACIES IN PHCS USING MPB
The BICs presented in this work can only occur when the bulk band structure exhibits a spectrally-isolated quadratictwo-fold degeneracy. Such degeneracies can be found either at Γ or M in C v symmetric PhCs or at Γ in C v and C v symmetric PhCs. The software package MPB [1] outputs the bandgap along a given trajectory in k -space andprovides optimization routines to find bandgaps given some free parameters. Here we describe a method to findisolated degeneracies based on the use of this function. The idea is based on the fact that in a PhC where spectrally-isolated degeneracies occur at HSPs, simply detuning away from HSPs by a small amount ∆ k results in the openingof a small stop band proportional to ∆ k . The structural parameters of the PhC can then be optimized to find thesestop bands.To demonstrate this, we consider the PhC shown in Fig. S2 (a) which consists of three circular discs of radii r , r and r . The dielectric constant of the high-index material (gray) is ε = 2 . ε = 1. Using MPB, we run an optimization function on the radii to find the aforementioned stop bands bycomputing the band structure along the path ( Γ + ∆ k ) → X → ( M + ∆ k ) → ( Γ + ∆ k ) for some small ∆ k , ∆ k . a r X i v : . [ phy s i c s . op ti c s ] F e b stop band along the detuned path and hence the required degeneracy is found between TM bands 7 and 8 at Γ for r /a = 0 . r /a = 0 . r /a = 0 . FIG. S2. (a) The PhC design with three parameters: r , r , and r made out of a dielectric material with ε = 2 . r /a = 0 . r /a = 0 . r /a = 0 . Γ and is marked with an arrow. SUPERCELL BAND STRUCTURES FOR IDENTIFYING DEFECT MODES
Besides using FDTD, a second way to identify the presence and symmetries of the defect modes is by examining theband structure of a reasonably sized supercell of a PhC with periodic boundaries that contains a defect. To illustratethis, we consider the TM modes of a 2D PhC made of circular discs with r = 0 .
15 and ε = 6. This PhC exhibitsthe spectrally isolated degeneracy between its second and third bands. We introduce a defect in this supercell bydetuning the radius of the central disc and plot the band structure of this supercell going through the high symmetrypoints of its small Brillouin zone. Fig. S3 (a) shows that the bulk degeneracy can still be clearly seen at M in thefolded band structure. The defect modes are easily identified by characteristic flat bands in the dispersion of thissupercell. Moreover, as the defect radius is varied, the frequencies of the bulk states which have support on all sitesof the supercell are barely affected but the frequencies of the defect localized modes are strongly affected. We canthen see the effect of tuning the defect size in middle panel of Fig. S3 (a) where the symmetry mismatch between thebulk and defect modes allows for a fine-tuned three-fold degeneracy to occur. The absence of any avoided crossingsindicates the formation of a BIC due to a lack of mixing between the bulk and defect modes. IG. S3. (a) The band structure of a supercell consisting of 7x7 sites with periodic boundary conditions. The defect introducedin the center has radii 0.69a, 0.695a and 0.7a in the three sub-plots (left to right). The middle panel shows that the defect modecan be fine-tuned to be degenerate with the spectrally isolated two-fold degeneracy of the bulk. (b) z-component of E -field ofthe defect modes labelled 1 and 4 (irreps B and A respectively) and the two modes of the bulk degeneracy labelled 2 and 3(irrep E).[1] S. G. Johnson and J. D. Joannopoulos, Opt. Express8