Evolution and global charge conservation for polarization singularities emerging from nonhermitian degeneracies
EEvolution and global charge conservation for polarization singularities emerging fromnonhermitian degeneracies
Weijin Chen, Qingdong Yang, Yuntian Chen,
1, 2, ∗ and Wei Liu † School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China College for Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha, Hunan 410073, P. R. China
Core concepts in singular optics, especially the polarization singularity, have rapidly penetrated the surgingfields of topological and nonhermitian photonics. For open photonic structures with degeneracies in particular,the polarization singularity would inevitably encounter another sweeping concept of Berry phase. Severalinvestigations have discussed, in an inexplicit way, the connections between both concepts, hinting at that nonzerotopological charges for far-field polarizations on a loop is inextricably linked to its nontrivial Berry phase whendegeneracies are enclosed. In this work, we reexamine the seminal photonic crystal slab that supports thefundamental two-level nonhermitian degeneracies. Regardless of the invariance of nontrivial Berry phase fordifferent loops enclosing both exceptional points, we demonstrate that the associated polarization fields exhibittopologically inequivalent patterns that are characterized by variant topological charges, including even thetrivial scenario of zero charge. It is further revealed that for both bands, the seemingly complex evolutions ofpolarizations are bounded by the global charge conservation, with extra points of circular polarizations playingindispensable roles. This indicates that tough not directly associated with any local charges, the invariant Berryphase is directly linked to the globally conserved charge, the physical principles underlying which have all beenfurther clarified by a modified Berry-Dennis model. Our work can potentially trigger an avalanche of studiesto explore subtle interplays between Berry phase and all sorts of optical singularities, shedding new light onsubjects beyond photonics that are related to both Berry phase and singularities.
Pioneered by Pancharatnam, Berry, Nye and others [1–10],Berry phase and singularities have become embedded lan-guages all across different branches of photonics. OpticalBerry phase is largely manifested through either polarizationevolving Pancharatnam-Berry phase or the spin-redirectionBortolotti-Rytov-Vladimirskii-Berry phase [2, 4, 5, 11–15];while optical singularities are widely observed as singularitiesof intensity (caustics) [6], phase (vortices) [7] or polariza-tion [8–10]. As singularities for complex vectorial waves, po-larization singularities are skeletons of electromagnetic wavesand are vitally important for understanding various interferenceeffects underlying many applications [16, 17].There is a superficial similarity between the aforementionedtwo concepts: both the topological charge of polarization field(Hopf index of line field [18]) and Berry phase are defined ona closed circuit. In spite of this, it is quite unfortunate thatalmost no definite connections have been established betweenthem in optics. This is fully understandable: Berry phase isdefined on the Pancharatnam connection (parallel transport)that decides the phase contrast between neighbouring stateson the loop [3, 4]; while the polarization charge reflects accu-mulated orientation rotations of polarization ellipses, whichhas no direct relevance to overall phase of each state. Thisexplains why in pioneering works where both concepts werepresent [19–23], their interplay were rarely elaborated.Spurred by studies into bound states in the continuum, po-larization singularities have gained enormous renewed interestin open periodic photonic structures, manifested in differentmorphologies with both generic and higher-order half-integercharges [24–45]. Simultaneously, the significance of Berryphase has been further reinforced in surging fields of topologi-cal and nonhermitian photonics [1, 19, 22, 46–50]. In periodic structures involving band degeneracies, Berry phase and po-larization singularity would inevitably meet, which sparks theinfluential work on nonhermitian degeneracy [32] and severalother following studies [36, 39, 41] discussing both conceptssimultaneously. Though not claimed explicitly, those workshint that nontrivial Berry phase produces nonzero polarizationcharge.Aiming to bridge Berry phase and polarization singularity,we reexamine the seminal photonic crystal slab (PCS) that sup-ports elementary two-level nonhermitian degeneracies. Despitethe invariance of nontrivial Berry phase, the corresponding po-larization fields on different isofrequency contours enclosingboth exceptional points (EPs) exhibit diverse patterns charac-terized by different polarization charges, including the trivialzero charge. It is further revealed such complexity of fieldevolutions is regulated by global charge conservation for bothbands, with extra points of circular polarizations ( C -points)playing pivotal roles. This reveals the explicit connection be-tween globally conserved charge and the invariant Berry phase,underlying which the physical mechanisms have been furtherclarified by a modified Berry-Dennis model [21]. Our studycan spur further investigations in other subjects beyond pho-tonics to explore conceptual interconnectedness, where boththe concepts of Berry phase and singularities are present.For better comparisons, we revisit the rhombic-lattice PCSin Ref. [32]: refractive index n , side length p , height h andtilting angle θ ; semi-major (minor) diameters are l ( l ); thewhole structure is placed in air background of n = 1 [Fig. 1(a);parameter values shown in the figure caption]. We have furtherdefined ϑ = (cid:77) l/l to characterize the mirror ( k y - k z plane)-symmetry breaking when air holes are partially filled. When ϑ = 0 , dispersion bands (in terms of real parts of complex a r X i v : . [ phy s i c s . op ti c s ] J un k x ★ ★ k y ω ˘ p h k x ( a ) ( d ) ( e ) (f) ( g )( b ) ★★ D l = D l / l l l q xy xzy ( c ) k y q =- =- =- =+ ★★ ★★ FIG. 1. (a) Unit cell of the rhombic-lattice PCS: index n = 1 . , p = 525 nm, h = 220 nm, l = 348 nm, l = 257 nm, θ = 114 . ◦ and ϑ = (cid:77) l/l . (b) Dispersion bands ( ϑ = 0 ) with two EPs ˘ ω =0 . , k x = 0 . , k y = ± . × − ) and two C -points onthe lower band ˘ ω = 0 . , k x = 0 . , k y = ± . × − .The polarization fields on a loop enclosing two C -points are shownin (d) with q = − . (c) Polarization fields for both lower (blue) andupper (red) bands, and three isofrequency contours are selected ( ˘ ω =0 . , . , . ), on which the polarization fields aresummarized in (e)-(g), with q = − / , + 1 / , − / , respectively. eigenfrequencies ˘ ω = ˘ ω + i ˘ ω for the Bloch eigenmodes cal-culated with COMSOL Multiphysics) are presented in Fig. 1(b).Throughout this work, both frequency and wave vector are nor-malized: ω → ωp/ πc ( c is light speed); k → k p/ π . Bothbranch cut (Fermi arc) and branch points (EPs) on the isofre-quency plane (position information shown in figure captions,as is the case throughout this work) are observed [marked alsorespectively in Fig. 1(c) by black curve and dots], confirm-ing the existence of nonhermitian degeneracies. On the lowerband, we have identified two C -points (marked by stars; thecorresponding eigenmodes are circularly polarized in the farfield) on the isofrequency plane (position information shownin figure captions, as is the case throughout this work). Po-larization fields (line fields in terms of the semi-major axisof the polarization ellipses) are projected on the Bloch vector k x - k y plane [Fig. 1(c)], with blue and red lines correspondingrespectively to the eigenmodes on the lower and upper bands(fields exhibiting mirror symmetry as required by the structuresymmetry). The representative eigenvalue-swapping feature isfurther confirmed in Fig. 1(b), where the polarization fields arecontinuous across the Fermi arc for opposite bands only [21].The coexistence of two C -points on the same band withequal charge q = − / (generic polarization singularities) isprotected by the mirror symmetry, decorated by typical star-like field patterns [51]. On a contour that encloses two C -points (without enclosing EPs), the polarization fields are shown inFig. 1(d) with the expected charge q = ( − / × − .Such a contour is not on an isofrequency plane and thus notquite feasible for direct experimental verifications. We thenproceed to isofrequency contours that are characterized by aninvariant π Berry phase [52–54]. Since both C -points locate onthe lower bands and on the isofrequency plane: for the upperband, there is no C -point enclosed by the contour; for the lowerband, the contour could enclose either zero or both C -pointssimultaneously. Polarization fields on three such contours [oneon the upper band (red dashed line) and two on the lower band(blue dashed lines)] are summarized in Figs. 1(e)-(g), with q = − / , +1 / , − / , respectively. The charge contrast of − between the two contours on the lower band are obviouslyinduced by C -points of total charge q = − .Though we have studied the same structure ( ϑ = 0 ) asthat in Ref. [32], our results presented in Fig. 1 are by nomeans mere reproductions, since the scenario of q = +1 / we demonstrate is missing in Ref. [32], where the key roles of C -points are also overlooked. We emphasize that though notexplicitly demonstrated, the case of q = +1 / was actually notforbidden by the arguments presented in Ref. [32]. Based onmode swapping and mirror symmetry properties, it was provedthere that the charge associated with the isofrequency contourhas to be a half-integer, accommodating both q = ± / .We then make a further step to investigate asymmetric struc-tures ( ϑ (cid:54) = 0 ). The polarization fields on the k x - k y plane fortwo scenarios ( ϑ = 0 . , . ) are summarized in Figs. 2(a)and (b), neither exhibiting mirror symmetry anymore. Withsymmetry broken, though one C -point on the lower band is rel-atively stable, the other can move to the Fermi arc [Fig. 2(b)] oracross to the upper band [Figs. 2(a)], with invariant q = − / (see Table I). When the two C -points locate on opposite bands[Fig. 2(a)], we choose two contours on the upper band (thecharge distribution on the lower band is similar): one enclosestwo EPs only and the other encloses also the C -point. Thepolarization fields on the contours are shown in Figs. 2(c) and(d), with q = 0 and − / , respectively. Despite this chargevariance, we emphasize that for any isofrequency contour,the Berry phase is an invariant π , regardless of whether thesymmetry is broken or not [52–54]. Basically, Fig. 2(c) tellsconvincingly that a nontrivial Berry phase does not necessarilyproduce a nonzero polarization charge.Except EPs, other points on the Fermi arc actually corre-spond to two sets of eigenmodes with equal ˘ ω while different ˘ ω . As a result, the C -point on the Fermi arc [Fig. 2(b)] isnot really shared by both bands (only EPs are shared), but stilllocate on the lower band, which can be confirmed by inspect-ing ˘ ω . With the absence of C -points, the charge distributionon the upper band would be identical to that in Fig. 1(b): anyisofrequency contour encloses two EPs only with q = − / .On the lower band, in contrast, an isofrequency contour canenclose either two EPs and inevitably a C -point on the Fermiarc, or two EPs and two C -points. Both scenarios are illus-trated in Figs. 2(e) and (f), with q = 0 and − / , respectively.Figure 2(e) reconfirms that Berry phase and partial polarization k x ( a ) ( b ) k y k x ★ ★ ★★ ( c )( e ) q =0 q =0 ( d )(f) ★★★★ q =- =- FIG. 2. (a) and (b) Polarization fields for two assymetric PCSs with ϑ = 0 . and . , respectively. The positions for the EPs are ˘ ω = 0 . , k x = 0 . , k y = ± . × − in (a) and ˘ ω = 0 . , k x = 0 . , k y = ± . × − in (b). Thepositons of the two C -points are: ˘ ω = (0 . , . , k x = (0 . , . , k y = (6 . × − , − . × − ) in(a) and ˘ ω = (0 . , . , k x = (0 . , . , k y = (6 . × − , − . × − ) in (b). In both (a) and (b), twoisofrequency contours are chosen, on which the polarizations fieldsare shown in (c)-(f), with q = 0 , − / , , − / and ˘ ω = 0 . + (5 . , . , . , × − , respectively. charge are not strictly interlocked.Charge distributions for all three structures are summarizedin Table I, with blank spaces corresponding to nonexistentscenarios. Table I clearly indicates that for both the upperand lower bands, the global charge (when the contour is largeenough to enclose both EPs and all C -points on the band) isinvariant ( q = − / ), irrespective of how the C -points aredistributed or whether the mirror symmetry is broken or not.In a word, there is a hidden order underlying the seeminglycomplex evolutions of polarization fields and their charges:the evolution is bounded by charge conservation. Consideringthe invariant π Berry phase for any isofrequency contours, itbecomes clear that the global polarization charge (rather thanpartial ones when the contours covers part of the singularities ofdegeneracies or C -points) is inextricably linked to this invariantBerry phase. Such a subtle connection is also manifest for notonly hermitian degeneracies [21, 23, 41], but also scenarioswith the degeneracies removed by further perturbations [21,39].As the final step, we employ the local Berry-Dennis modelproposed in Ref. [21] to clarify the underlying mechanisms.The corresponding Hamiltonian of this model in linear basis is: H ( k x , k y ) = ( k x + iγ ) σ z + k y σ x + κσ y , (1)where k x,y are real; σ x,y,z are Pauli matrices; κ and γ are theplanar chirality and radiation loss terms, respectively [55]. ThisHamiltonian matrix is indeed a rather ordinary × nonhermi-tion matrix, except that Berry and Dennis view its eigenvectorsas Jones vectors [56] for generally elliptically polarized light inlinear basis, thus establishing an effective connection between ϑ = 0 ϑ = 0 . ϑ = 0 . − / − / − / − / Two EPs +1 / − / − / Two EPs + 1C − / − / − / − / Global − / − / − / − / − / − / TABLE I. Charges for C -points and different isofrequency contours( L : lower band; U : Upper band). Blank spaces correspond to nonexis-tent scenarios. the Hamiltonian matrix and the electromagnetic polarizationfields (see Supplemental Material (SM) [55] for justificationsof this connection and the incorporation of κ ). With this con-nection and the complex eigenvector denoted as x = ( x ; x ) :when κ = 0 , EPs are chiral points with degenerate eigenvec-tors satisfying x ± ix = 0 , overlapping with C -points; when κ (cid:54) = 0 , EPs are nonchiral and thus separated from C -points[21, 57–59]. Since for all the scenarios discussed above (seeFigs. 1 and 2) the EPs do not overlap with C -points, the intro-duction of chirality term κ is inevitable, which is missing inRef. [32].For convenience of analysis, to directly locate C -points inparticular, the Hamiltonian can be converted into a circular-basis form as [21]: H c ( k x , k y ) = ( k x + iγ ) σ x + k y σ y + κσ z = (cid:32) κ k x − i k y + i γk x + i k y + i γ − κ (cid:33) , (2)since such conversion would transform σ x,y,z in linear basisto σ y,z,x in circular basis [55]. After this conversion, thechiral points now correspond to points of linear polarizations,while circular-basis eigenvectors of x c x c = 0 correspond to C -points. Identical to the linear basis case, the EPs correspond tocircular (noncircular) polarizations with the chirality term κ =0 ( κ (cid:54) = 0 ). The superiority of this circular-basis Hamiltonianresides in that the positions of C -points can then be directlyidentified by setting the off-diagonal terms of the matrix equalto zero: k x − i k y + i γ = 0 and k x + i k y + i γ = 0 . Their roots k x = 0 , k y = γ and k x = 0 , k y = − γ are the positions of C -points on the lower and upper bands, respectively [21]. Thismodel can explain the charge distributions shown in Fig. 2(a)(also summarized in Table I with ϑ = 0 . ) with the two C -points located on opposite bands, except that in this modelthe topological charge of the C -point and the global chargefor either band is +1 / rather than − / (see SM [55]). Toaccount for these discrepancies, we modify the Hamiltonianas: H c ( k x , k y ) = ( k x + iγ ) σ x − k y σ y + κσ z , (3)by adding a minus sign before the σ y term in Eq. (2). This issimilar to substituting the Hamiltonian of the K-valley for thatof the K (cid:48) -valley in graphene, which would induce a π jump k x ( a ) ( c ) ( d )( e ) (f)( b ) k y k x ★ q =- q =0 q =- =+1/2 ★★ ★★ ★★ λ = . λ = . λ =- . λ =- . cc c c FIG. 3. (a) and (b) Polarization fields extrated from the model re-spectively in Eq. (3) and Eq. (4), with γ = 1 and κ = 0 . . Twoiso-eigenvalue contours are selected in (a) and (b), on which the po-larizations fields are shown in (c)-(f), with q = − / , , , − / ,respectively. of Berry phase from π to − π [60, 61] (see SM [55]). ThoughBerry phases of π and − π are effectively the same (phaseis only definable modulo π ), the corresponding polarizationfields and charge distributions are contrastingly different (seeSM [55] for the connections between Berry phase and po-larization charges), with opposite sings for both the C-pointcharge and the global charge of both bands ( q = +1 / ver-sus q = − / ). The polarization fields extracted from thismodified model ( γ = 1 and κ = 0 . ) are shown in Fig. 3(a),which are topologically equivalent to those in Fig. 2(a): foreach C -point q = − / ; for iso-eigenvalue ( λ c ) contours thatenclose both EPs, q = 0 and q = − / with and without theextra C -point surrounded, respectively [see Figs. 3(c) and (d)];the global charge is constant ( q = − / ) for both bands.Since the model presented above is linear, there is only onesolution when either of the off-diagonal terms is setting to zero.This means that there is one and only one C -point on each band.As a result, this linear model would fail to account for whatis observed in Fig. 1(b), where there are two C -points on thesame band. Actually the linear model in Eq. (3) has broken the k x - k z mirror symmetry of the polarization fields (symmetriesof the Hamiltonian and constructed fields are different, due toinvolvements of construction basis [55]), as confirmed by thefield patterns in Fig. 3(a). To reflect the mirror symmetry ofthe structure and thus also the polarization fields, the linearmodel can be further modified as: H c ( k x , k y ) = ( k x + iγ ) σ x − k y σ y − k y σ z , (4)where the constant chirality term κ in Eq. (3) is replaced bythe variable − k y , which guarantees that the constructed polar-ization fields are symmetric with respect to the k y - k z plane(see SM [55] for detailed arguments concerning field symme-try). The symmetric fields [in contrast to asymmetric ones inFig. 3(a)] based on this model ( γ = 1 and κ = 0 . ) are shownin Fig. 3(b), which is topologically equivalent to Fig. 1(b): for each C -point q = − / ; for iso-eigenvalue contours on the up-per band q = − / ; iso-eigenvalue contours on the lower bandthat enclose both EPs have q = +1 / and q = − / , withand without the two C -points surrounded, respectively [seeFigs. 3(e) and (f)]; the global charge is an invariant q = − / for both bands. This reconfirms the claim in Ref. [32]: com-bined mirror symmetry and mode swapping produces half-integer charges. Here for simplicity we have confined to linearmodels only, aiming to explain topologically what has beenobserved in Figs. 1 and 2. To obtain more than one C -pointson the same band, besides introducing the variable chiralityterm as shown in Eq. (4), we can also incorporate nonlinearterms into the Berry-Dennis model, which nevertheless wouldchange the charge distribution both locally and globally (seeSM [55]).In conclusion, we revisit PCSs supporting nonhermitian de-generacies and establish a subtle connection between invariantBerry phase and the conserved global charge. It is revealedthat for any isofrequency contour enclosing both EPs, despitethe nontrivial π Berry phase invariance, the topological chargeis contrastingly variable, which could even be the trivial zerocharge. Such seemingly complex evolutions of charge distribu-tions are mediated by extra C -points, ensuring global chargeconservation for both bands that is synonymous with the Berryphase invariance.Our discussions are confined to fundamental two-level sys-tems, which can be extended to more sophisticated systemswith more complex EPs distributions [62–74]. We emphasizethat in this work, the Berry phase and polarization charge ac-tually characterize different entities of eigenvectors of Blochmodes and their projected far fields: Bloch modes are definedon the momentum-torus and can be folded into the irreducibleBrillouin zone; while far fields are defined on the momentum-sphere, due to the involvement of out-of-plane wave vectorsalong which there is no periodicity. It is recently shown theBerry phase for electromagnetic fields themselves on a contourcan be well defined [14, 75]. We expect that blending all thoseconcepts (non-hermitian degeneracies, Berry phase of their ma-trix eigenvectors, Berry phase and polarization singularities ofthe corresponding electromagnetic waves) would render muchmore fertile platforms to incubate new fundamental investiga-tions and practical applications, including the rare scenario ofBerry phase (for electromagnetic fields) with slaving param-eters (eigenvectors from which the electromagnetic fields areconstructed) themselves also having Berry phase. Acknowledgments : We acknowledge the financial supportfrom National Natural Science Foundation of China (GrantNo. 11874026 and 11874426), and several other ResearcherSchemes of National University of Defense Technology. W. L.is indebted to Sir Michael Berry and Prof. Tristan Needhamfor invaluable correspondences. ∗ [email protected] † [email protected][1] S. Pancharatnam, “The propagation of light in absorbing biaxialcrystals,” Proc. Indian. Acad. Sci. , 86–109 (1955).[2] S. 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