Symmetry Protected Invariant Scattering Properties for Arbitrary Polarizations
SSymmetry Protected Invariant Scattering Properties for Arbitrary Polarizations
Qingdong Yang, ∗ Weijin Chen, ∗ 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
Polarization independent Mie scattering of building blocks is foundational for constructions of optical systemswith robust functionalities. Conventional studies for such polarization independence are generally restrictedto special states of either linear or circular polarizations, widely neglecting elliptically-polarized states thatare generically present in realistic applications. Here we present a comprehensive recipe to achieve invariantscattering properties (including extinction, scattering and absorption) for arbitrary polarizations, requiring onlyrotation symmetry and absence of optical activities. It is discovered that sole rotation symmetries can effectivelydecouple the two scattering channels that originate from the incident circularly polarized waves of oppositehandedness, leading to invariance of all scattering properties for any polarizations on the same latitude circle ofthe Poincar´e sphere. Further incorporations of extra inversion or mirror symmetries would eliminate the opticalactivities and thus ensure scattering property invariance for arbitrary polarizations. In sharp contrast to previousinvestigations that rely heavily on complicated algebraic formulas, our arguments are fully intuitive and geometric,bringing to surface the essential physical principles rather than obscuring them. The all-polarization invariancewe reveal is induced by discrete spatial symmetries of the scattering configurations, underlying which thereare functioning laws of reciprocity and conservation of parity and helicity. This symmetry-protected intrinsicinvariance is robust against any symmetry-preserving perturbations, which may render extra flexibilities fordesigning optical devices with stable functionalities.
I. INTRODUCTION
Photonic devices that can function robustly for some prac-tical applications require polarization independent responseswhich are immune to perturbations that can easily perturb onepolarization state to another [1, 2]. For composite devices suchas those consisting of periodic, quasi-periodic or disorderedphotonic structures [3, 4], usually this can be reduced to anelementary Mie scattering problem [5]: its fundamental build-ing atom needs to exhibit invariant scattering properties fordifferent polarizations [6–9]. To get rid of the polarizationdependence actually constitutes a rather seminal problem inMie theory, for which discrete spatial symmetries [6, 10–16]and/or electromagnetic duality symmetry [12, 13, 17–20] canbe employed to secure the scattering invariance.A common limitation widely shared by previous studiesis that the polarization independence obtained covers onlysome specific polarization states (generally circular or lin-ear polarizations), occupying a rather small proportion of thewhole Poincar´e sphere that can represent all possible polariza-tions [1, 21]. To conduct comprehensive investigations into allpossible polarization states for realistic practical applications,restricting to some special polarizations is not sufficient consid-ering the following twofold reasons: (i) Scattering invariancefor some polarizations does not ensure the invariance for allpolarizations throughout the whole Poincar´e sphere. For exam-ple, even if the scattering properties are fully independent oflinear polarizations with arbitrary orientations, the scatteringvariance could still emerge for states that are elliptically polar-ized. (ii) In realistic photonic devices, those widely explored ∗ Authors contributed equally to this work. † [email protected] ‡ [email protected] circularly or linearly polarized states are not really absolutestable, which can be easily converted, by inevitable structuresdefects or external perturbations, into more generic ellipticallypolarized states.An extra problem for previous studies on symmetry pro-tected scattering invariance is that the arguments put forwardare heavily based on complicated algebraic formulas (see e.g. Refs. [11, 12, 14]), which has on one hand obscured the funda-mental physical principles and on the other curbed the generalinterest by repelling mathematically the broader community inphotonics. This echoes what is widely believed true in math-ematics (and also in physics) [22, 23]: algebra is the offerfrom the devil to trade for our soul, stopping us from thinkinggeometrically and thus from grasping the underlying truth andreal meaning. Basically, comprehensive revelations, justifiedby intuitive geometric arguments, about symmetry dictatedinvariant scattering for all polarizations are pressingly desired,which is exactly what we aim to present here.In this study we show, by intuitive geometric reasoning, howto obtain symmetry-protected invariant scattering propertiesfor arbitrary polarizations, relying solely on rotation symmetryand optical activity elimination [identical responses for left-and right-handed circularly polarized (LCP and RCP) incid-net waves]. It is discovered that for a scattering configurationof more than two-fold rotation symmetry, the two scatteringchannels from the LCP and RCP waves are actually decoupled:there is effectively no contribution from their cross interfer-ences for scattering properties including extinction, absorptionor total scattering. Based on this discovery, we further re-veal subtle connections between scattering invariance and ahierarchy of discrete spatial symmetries: (i) Rotation sym-metries ( n -fold, n ≥ ) result in invariance of all scatteringproperties for polarizations on the same latitude circle of thePoincar´e sphere; (ii) Combined rotation-mirror (perpendicu-lar to the rotation axis) or rotation-inversion symmetries lead a r X i v : . [ phy s i c s . op ti c s ] J un (a) f q x z //ky E i o / R o t a ti on x y QQ QQQ Qqq (b) (d)(c) + =
LCP RCP EllipticalIncident Scattered Observed (q=0)
FIG. 1. (a) and (b): The geometric proof by contradiction that theelectrostatic force on the central charge has to be zero with threeidentical charges located on the vertexes of an equilateral triangle. (c)A scattering configuration with C symmetry placed in the sphericalcoordinate system parameterized by polar angle θ and azimuthal angle φ . (d) The geometric proof by contradiction for helicity preservationalong the forward direction: e.g. a LCP incident wave mixing with theforward scattered RCP wave would make the final observable stateof elliptical polarization with preferred ellipse orientation, which isforbidden by the rotation symmetry. to invariant extinctions for arbitrary polarizations, while thescattering and absorption are still variable; (iii) Combinedrotation-mirror (parallel to the rotation axis) symmetry ensuresinvariant extinction, scattering and absorption for all polariza-tions covering the whole Poincar´e sphere. Since underlyingthose apparent spatial symmetries there are hidden function-ing laws of reciprocity, parity and helicity conservation, thescattering invariance obtained is intrinsic and robust againstany non-symmetry-breaking perturbations, which can poten-tially enrich the toolbox of optical device designs and renderextra freedom for more flexible manipulations of light-matterinteractions. II. HELICITY PRESERVATION OF FORWARDSCATTERING FOR CONFIGURATIONS WITH n -FOLDROTATION ( C n , n ≥ ) SYMMETRY In this work, our discussions of Mie scattering with incidentplane waves are based on the circular basis L and R , whichcorrespond to LCP and RCP light, respectively. When circu-larly polarized (CP) waves are incident on a structure with C n symmetry (incident direction k is parallel to the rotationaxis l : k (cid:107) l ), the helicity is preserved along the forward direc-tion: the forward scattered waves are not only CP but alsoof the same handedness as that of the incident waves. Thishas already been rigorously proved by Ref. [12], through com-plicated algebraic formulas that to some extent obscure theunderlying physical principles. In this section, we provide apurely geometric formula-free proof that can directly confirmsuch helicity preservation, which is much simpler than that inRef. [12], without sacrificing any rigor. We emphasize herethat throughout this study, our arguments are restricted to - fold rotation symmetry, which can be quite directly generalizedto cover all scenarios of n ≥ .As a first step, we turn to a seemingly unrelated problemsketched in Figs. 1(a) and (b): with three identical point chargeslocated on the vertexes of an equilateral triangle, what is theelectrostatic force on an extra point charge at the triangle cen-ter? Through algebraic calculations based on the Coulomblaw, we can get the answer that the force is zero. At the sametime, we can reach the same conclusion through pure intuitivegeometric considerations, without any detailed algebraic ma-nipulations: (i) Assume that there is a force on the centralcharge as indicated by a red vector in Fig. 1(a); (ii) Make a π/ rotation operation on the whole configuration, ending upwith what is shown in Fig. 1(b): both the force and the chargesare rotated accordingly; (iii) Charge distributions in Figs. 1(a)and (b) are identical due to the overall symmetry, requiring thatthe force in Fig. 1(b) (dashed red vector) is the same as that inFig. 1(a); (iv) The two forces (solid and dashed red vectors) inFig. 1(b) contradict each other, unless the force is zero. Thisconcludes our proof by contradiction.Now we turn back to our Mie scattering problem and thescattering configuration (exhibiting C symmetry) within aspherical coordinate system is schematically shown in Fig. 1(c).According to Mie theory [5], what is observed in the forwarddirection is mixed states of both incident and forward scatteredwaves. For incident CP waves, let us assume that the helicityis not preserved and thus there is CP scattered components ofopposite handedness along the forward direction. It is knownthat mixing CP waves of opposite handedness would produceelliptically polarized states with preferred orientation direc-tions of the polarization ellipses (in terms of its semi-majoror semi-minor axis): a detailed example is shown in Fig. 1(d)with an incident LCP wave mixing with the forward scatteredRCP wave. According to the same arguments presented abovefor Figs. 1(a) and (b), such a preferred orientation contradictthe overall rotation symmetry of the scattering configurationwith incident CP waves. When there is helicity preservation,the forward mixed state is still CP, for which there is no suchcontradiction as its ellipse orientation is not defined [24]). Thisconcludes our proof by contradiction for helicity preservationalong the forward direction.We note that the same symmetry arguments can be alsoemployed to verify the helicity flipping along the backwarddirection [12], which nevertheless is irrelevant to our followinginvestigations and thus would not be further discussed in detailhere. Moreover, there are actually subtle differences betweenthose shown in Figs. 1(a-b) and Figs. 1(c-d): the electrostaticforce is characterized by a vector that is variant upon the a π rotation, and thus C symmetry is sufficient to eliminate it;while the polarization ellipse orientation is characterized by aline that is invariant upon a π rotation, which means that morethan two-fold rotation symmetry is required to guarantee thehelicity preservation. III. INVARIANT SCATTERING PROPERTIES FORARBITRARY POLARIZATIONS INDUCED BY DISCRETESPATIAL SYMMETRIESA. General theoretical analysis with intuitive geometricarguments
In this study we aim to reveal scattering invariance for arbi-trary polarizations, to describe which we employ the widelyadopted Poincar´e sphere [characterized by Stokes vectors( S , S , S ), or location vector ( χ , ψ ) in terms of latitude andlongitude on a unit-sphere] as shown in Fig. 2(a) [1, 21]: lat-itude χ ∈ [ − π/ , π/ characterizes the eccentricities of thepolarization ellipses, with positive and negative χ (northernand southern hemisphere) corresponding to left and right hand-edness, respectively [see Figs. 2(b) and (c)]; LCP and RCPwaves locate respectively on the northern and southern poles,and all linear polarizations locate on the equator S = χ = 0 ;latitude ψ ∈ [0 , π ] characterizes the orientations of the polar-ization ellipses in terms of the semi-major axis [see Figs. 2(b)and (c)]. We emphasize here that the characterizing angles forthe polarization ellipses are half of those for Stokes vectors,since the polarizations are described by 1-spinors, which areeffectively the square roots of Stokes vectors [25, 26].An arbitrarily polarized incident wave [denoted by E i ; lo-cated at ( χ i , ψ i ) on the Poincar´e sphere] can be expressed incircular basis ( L , R ) as: E i = cos( π − χ i − i ψ i / L + sin( π − χ i i ψ i / R . (1)The scattered waves (denoted by E s ) along different directions[characterized by ( θ, φ ) as shown in Fig. 1(b)] are linearlyrelated to the incident waves through the following relation [5]: E s ( θ, φ ) = ˆ T ( θ, φ ) E i =cos( π − χ i )e − i ψ i / E Ls ( θ, φ ) + sin( π − χ i )e i ψ i / E Rs ( θ, φ ) , (2)where ˆ T is the scattering matrix; E Ls and E Rs are scatteredwaves with incident LCP and RCP waves, respectively.For scattering configurations with more than two-fold rota-tion symmetry [the scattering structure exhibits C n ( n ≥ )symmetry with the incident wave propagating along the ro-tation axis, as is the case throughout this work], the he-licity preservation along the forward direction requires that E Ls ( θ = 0) and E Rs ( θ = 0) are respectively LCP and RCPwaves, between which there is thus no interference. Accordingto the optical theorem [5], the extinction is only related tointerferences between incident and scattered waves along theforward direction [ E i and E s ( θ = 0) ]. Meanwhile, due to thehelicity preservation, there is no cross interference between L and E Rs ( θ = 0) , or between R and E Ls ( θ = 0) . As a result,the extinction cross section for arbitrarily polarized incidentwaves can be expressed as: C ext = cos ( π − χ i Lext + sin ( π − χ i Rext , (3)where C Lext and C Rext are extinction cross sections for incidentLCP and RCP waves, respectively. According to Eq. (3), the o ψ s s χ s ψ /2 χ /2 xy ψ /2 χ /2 xy χ >0 χ < (b) (a) (c) FIG. 2. (a) Representations of arbitrary polarizations by the Poincar´esphere parameterized by Stokes parameters S , , , latitude χ andlongitude ψ . (b) and (c) The corresponding polarization ellipses arecharacterized by half angles of those for the Stokes vectors: ψ/ indicates the ellipse orientation and χ/ describes ellipse eccentricity;polarizations of left (right) handedness locate on the northern (south-ern) hemisphere with χ > ( χ < ); linear polarizations locate onthe equator with χ = 0 . extinction has nothing to do with ψ i (the orientation of inci-dent polarization ellipse) and is invariant: (i) for the incidentpolarizations that locate on the same circle of latitude ( χ i isconstant); or (ii) for arbitrary incident polarizations when thereis no extinction activity C Lext = C
Rext [16].Then we turn to scattering cross sections ( C sca ) for anyincident polarizations, the calculation of which appears tobe more demanding than extinction since integrations of all-angular scattering intensities [ I s ( θ, φ ) = | E s ( θ, φ ) | ] haveto be implemented [5]. According to Eq. (2), the angularscattering intensity can be explicitly expressed as: I s ( θ, φ ) = cos (cid:0) π − χ i (cid:1) I Ls + sin (cid:0) π − χ i (cid:1) I Rs + sin( χ i ) cos ( ψ i ) (cid:0) E ∗ Ls · E Rs + E Ls · E ∗ Rs (cid:1) , (4)where ∗ means complex conjugation. Though along the for-ward direction E ∗ Ls and E Rs are orthogonal as required byhelicity preservation ( E ∗ Ls · E Rs = 0 when θ = 0 ), they aregenerally not orthogonal along other directions and thus the lastinference term ( I int s = E ∗ Ls · E Rs + E Ls · E ∗ Rs ) cannot be directlydismissed. This seemingly further adds to the complexities forgeneral discussions of scattering cross sections.Now we proceed to check in detail the scattered field dis-tributions in terms of both phase and amplitude. For incidentCP waves, the symmetry ( C for the following specific discus-sions) of the scattering configuration ensures that E L,Rs ( θ, φ ) and E L,Rs ( θ, φ + 2 mπ/ are interconnected rather than in-dependent for m = 1 , . It could be taken for granted that E L,Rs ( θ, φ ) = E L,Rs ( θ, φ + 2 mπ/ , as they are seeminglyequivalent and inter-convertible through a simple coordinatesystem rotation by mπ/ along the z -axis [see Fig. 1(b)].This is actually wrong, because such a coordinate rotation doesnot really leave the scattering configuration as it was, but wouldrather ultimately change it through introducing extra phase. Tobe specific, with the coordinate rotation, the incident CP waveswould transform as follows [refer to the Feynman Lectures(Volume III, Chapter 11) for more details] [27]: L → e − mπ/ L , R → e mπ/ R , (5)and the scattered fields would also transform accordingly: E Ls → e − mπ/ E Ls , E Rs → e mπ/ E Rs . (6)Equations (4) and (6) lead to the following transformation forthe interference term of the scattering intensity: I int s → cos(4 mπ/ I int s . (7)It is easy to conclude that the interference term would can-cel each other when integrated along all scattering directions,since π/ π/
3) = 0 . It immediately becomesclear that C symmetry does not gurantee the interference can-cellation and thus there is no polarization independence, as cos(0) + cos(2 π ) (cid:54) = 0 . Consequently, the scattering cross sec-tion for arbitrarily polarized incident waves can be simplifiedas (for more than two-fold rotation symmetry): C sca = cos ( π − χ i Lsca + sin ( π − χ i Rsca , (8)where C Lsca and C Rsca are scattering cross sections for incidentLCP and RCP waves, respectively. Similar to Eq. (3), Eq. (8)confirms that the scattering is invariant: (i) for the incidentpolarizations that locate on the same circle of latitude ( χ i isconstant); or (ii) for arbitrary incident polarizations when thereis no scattering activity C Lsca = C
Rsca [16]. It now becomesclear that the all-angle integration actually simplifies ratherthan complexifies the expressions of C sca , but for scatteringconfigurations with rotation symmetry only.Up to now, we have discussed only extinction and scatter-ing, and properties of absorption can be simply deduced asthe absorption cross section can be obtained through the fol-lowing realtion C abs = C ext − C sca , as secured by the opticaltheorem [5]. According to Eqs. (3) and (8), this leads to: C abs = cos ( π − χ i Labs + sin ( π − χ i Rabs , (9)where C Labs and C Rabs are absorption cross sections for incidentLCP and RCP waves, respectively. The principles we haverevealed here for C symmetry can be directly extended toany rotation symmetry C n ( n > ). We emphasize that thephase term introduced by coordinate system rotation [shownin Eq. (5)-(6)] is only observable through the interferenceterm in Eq. (4). When it is CP incident waves, there is nosuch interference and then such a phase is not observable,confirming that the arguments we presented proving the helicitypreservation are still valid and not affected by the presence ofsuch a phase. B. Invariant scattering properties induced by sole rotationsymmetries
Equations (3), (8), and (9) are the core results of our study,requiring the only precondition that the scattering configura-tion is of more than two-fold rotation symmetry. They indicate (cid:25) (cid:26) (cid:27)(cid:20)(cid:19) (cid:16)(cid:26) (cid:19)(cid:19)(cid:17)(cid:24)(cid:20) (cid:20)(cid:19) (cid:16)(cid:20)(cid:24) (cid:25) (cid:26) (cid:27)(cid:20)(cid:19) (cid:16)(cid:26) (cid:19)(cid:19)(cid:17)(cid:24)(cid:20) (cid:20)(cid:19) (cid:16)(cid:20)(cid:24) l(m m ) l(m m ) (cid:25) (cid:26) (cid:27)(cid:20)(cid:19) (cid:16)(cid:26) (cid:19)(cid:19)(cid:17)(cid:24)(cid:20) (cid:20)(cid:19) (cid:16)(cid:20)(cid:24) l(m m ) ° Au x yzunit:nm c r o ss s ec ti on ( a . u . ) (60°,30°) (60°,160°) extinction absorption (60°,75°) (c)(a) (d)(b) c r o ss s ec ti on ( a . u . ) c r o ss s ec ti on ( a . u . ) (0,75°) extinction absorption (0,30°) (0,160°) extinction absorption (60°,60°) (-80°,0) FIG. 3. (a) A scattering configuration consisting of two touchinggold particles (geometric parameters specified in the figure only) thatexhibits C symmetry only. The extinction and absorption spectra aresummarized in (b)-(d) for: linear polarizations of different orientationsin (b); elliptical polarizations of different orientations while on thesame lattitue circle of the Poincar´e sphere in (c); two randomly chosenpolarizations not on the same latitude circle in (d). The labeling anglescorrespond to the location vector ( χ i , ψ i ). that for extinction, absorption and total scattering, there is noeffective contribution from the interferences between the twoscattering channels with incident LCP and RCP waves, respec-tively. This is to say, the two scattering channels are effectivelydecoupled in terms of the scattering properties we discus inthis work. As a result, all scattering properties for arbitrarilypolarized incident waves can be obtained by direct summa-tions of the contributions from LCP and RCP incidences. Itis clear from those equations that all scattering properties areinvariant for polarizations on the same circle of latitude ( χ i is constant), a special case of which is the linear polarization( χ i = 0 ) independence discussed in Ref. [11]. Despite thatour conclusion in this work is more general (not limited tolinear polarizations), the proof we have presented here is su-perior: the former proof in Ref. [11] relies on the coupleddipole approximation (thus non-intrinsic) and extremely heavyalgebraic manipulations (thus less accessible for general inter-est) that significantly obscure the physical principles; whilehere we provide an intrinsic symmetry-based proof assisted byintuitive geometric reasoning, which brings to the surface thekey mechanisms and thus more comprehensible for the broadcommunity in photonics.To verify what has been claimed above, we show in Fig. 3(a)a composite scattering configuration exhibiting sole C symme-try: the two touching scatterers are made of gold, permittivityof which is taken from Ref. [28]; the geometric parameters arespecified in the figure and numerical results are obtained usingCOMSOL Multiphysics, as is the case throughout this work.Despite the invariance of all scattering properties (only extinc- Au x yz (cid:25) (cid:26) (cid:27)(cid:20)(cid:19) (cid:16)(cid:26) (cid:19)(cid:19)(cid:17)(cid:24)(cid:20) (cid:20)(cid:19) (cid:16)(cid:20)(cid:24) (cid:24) (cid:26) (cid:28)(cid:20)(cid:19) (cid:16)(cid:26) (cid:19)(cid:19)(cid:17)(cid:27) (cid:20)(cid:19) (cid:16)(cid:20)(cid:24) unit:nm unit:nm l(m m ) c r o ss s ec ti on ( a . u . ) extinction absorption l(m m ) c r o ss s ec ti on ( a . u . ) extinction absorption (60°,60°) (-20°,30°) (a) (d)(b) (80°,60°) (-70°,30°) Au ° (c) x yz FIG. 4. Two C -symmetric scattering configurations consisting ofgold particles that exhibit extra perpendicular-mirror symmetry in(a) and inversion symmetry in (c) with two identical particles. Theextinction and absorption spectra are summarized respectively in(b) and (d), for two randomly chosen polarizations not on the samelatitude circle of the Poincar´e sphere. The labeling angles correspondto the location vector ( χ i , ψ i ). tion and absorption spectra are shown with respect to wavelenth λ ) for linear polarizations [see Fig. 3(b)], we also demonstratein Fig. 3(c) such invariance for elliptic polarizations on thesame latitude circle of the Poincar´e sphere ( χ i = 60 ◦ ). Forpolarizations not on the same latitude circle, such invariance isimmediately lost, as is shown in Fig. 3(d). C. Invariant scattering properties induced by combinedrotation-mirror (perpendicular to the rotation axis) orrotation-inversion symmetries
According to Eqs. (3), (8), and (9), to obtain invariant ex-tinction, scattering or absorption for arbitrary polarizations(not limited to the same latitude circle on the Poincar´e sphere),we only have to extinguish the corresponding scattering ac-tivities to ensure equal responses for incident LCP and RCPwaves [16]: C Lext , sca , abs = C Rext , sca , abs , respectively.It is recently proved that the law of reciprocity and parityconservation can intrinsically eliminate the extinction activity(but the scattering and absorption activities are still present)when there is mirror (perpendicular to the incident direction) orinversion symmetry [16]. Two such scattering configurationsare shown in Figs. 4(a) and (c), which besides rotation symme-try also exhibit perpendicular mirror and inversion symmetry,respectively. The scattering spectra (in terms of extinction andabsorption) are shown respectively in Figs. 4(b) and (d) fortwo randomly chosen polarizations (not on the same latitudecircle). As is clearly shown there is extinction invariance butno such invariance for absorption or scattering [the two absorp-tion spectra in Fig. 4(d) are quite close, but definitely different,which is most visible close to the wavelength λ = 0 . µ m], (cid:19)(cid:17)(cid:27) (cid:20) (cid:20)(cid:17)(cid:21)(cid:20)(cid:19) (cid:16)(cid:25) (cid:19) (cid:20)(cid:19) (cid:16)(cid:20)(cid:24) Au x yz l(m m ) c r o ss s ec ti on ( a . u . ) extinction absorptionunit:nm (b)(a) (60°,60°) (-20°,30°) FIG. 5. (a) A scattering configuration consisting of two touchinggold particles that exhibit both rotation and parallel-mirror ( C )symmetry. The extinction and absorption spectra are shown in (b) fortwo randomly chosen polarizations not on the same latitude circle.The labeling angles correspond to the location vector ( χ i , ψ i ). since the scattering and absorption activities are not eliminatedby the extra mirror or inversion symmetry [16]. D. Invariant scattering properties induced by combinedrotation-mirror (parallel to the rotation axis) symmetries
Equations (3), (8), and (9) tell that to achieve invariancefor all scattering properties, all the scattering activities shouldbe eliminated simultaneously. This is possible for a config-uration with extra mirror (parallel to the incident direction)symmetry, where the law of parity conservation ensures iden-tical responses for incident LCP and RCP waves [6, 16]. Ascattering configuration exhibiting both rotation and parallelmirror symmetry (with broken perpendicular mirror symmetry)is shown in Fig. 5(a), with the scattering and absorption spectrafor two arbitrary polarizations shown in Figs. 5(b), confirmingthe invariance of all scattering properties.
IV. CONCLUSIONS AND DISCUSSIONS
In conclusion, we have revealed through intuitive geomet-ric reasoning, how to achieve invariant scattering properties(including extinction, scattering and absorption) for arbitrarypolarizations based on discrete spatial symmetries. It is dis-covered that sole rotational symmetries (more than 2-fold)can secure the scattering invariance for polarizations on thesame latitude circle on the Poincar´e sphere. To achieve invari-ance for all polarizations covering the whole Poincar´e sphere,besides rotation symmetry we can: (i) introduce extra perpen-dicular mirror symmetry or inversion symmetry to produceall-polarization invariant extinctions (but not scattering or ab-sorption); (ii) or introduce extra parallel mirror symmetry toobtain all-polarization invariant extinction, scattering and ab-sorption. Underlying those apparent spatial symmetries, thereare functioning laws of reciprocity, helicity and parity con-servation, which guarantee that the invariance obtained areintrinsic and robust against any symmetry-preserving structuredefects or perturbations.Here in this study, we have confined our discussions to fully-polarized incident waves on the Poincar´e sphere, but neglectthose unpolarized and partially polarized states that are withinthe Poincar´e sphere. Since our core results shown in Eqs (3),(8), and (9) have nothing to do with the cross interference termor relative phase difference between two different scatteringchannels, and thus all equations are valid for states within thePoincar´e sphere. It is worth noting that for unpolarized inci-dent light, the interference term is automatically cancelled andthus the validity of Eqs (3), (8), and (9) does not reside onthe rotation symmetry of the scattering configuration anymore.It is revealed that to obtain invariance for arbitrary polariza-tions, the absence of optical activities is crucial, which can beeither intrinsic (protected by symmetry and thus is broadbandas shown in this work) or accidental (activities are only elimi-nated at some specific wavelengths). The principles we haverevealed in this work and also the novel approaches we have employed in our intuitive geometric reasoning can shed newlight on not only optical device designs, but also on fundamen-tal explorations in photonics where the light-matter interactionsare dictated by symmetry.
ACKNOWLEDGEMENT
We acknowledge the financial support from National Nat-ural Science Foundation of China (Grant No. 11874026,11404403 and 11874426), and the Outstanding Young Re-searcher Scheme of National University of Defense Technol-ogy.Q. Yang and W. Chen contributed equally to this work. [1] A. Yariv and P. Yeh,
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