Topologically Protected Complete Polarization Conversion · Yu Guo, Meng Xiao, and Shanhui Fan* Department of Electrical Engineering, Ginzton Laboratory, Stanford University, Stanford,

Topologically Protected Complete Polarization Conversion

Yu Guo, Meng Xiao, and Shanhui Fan*

Department of Electrical Engineering, Ginzton Laboratory, Stanford University, Stanford, California 94305, USA(Received 29 May 2017; published 18 October 2017)

We consider the process of conversion between linear polarizations as light is reflected from a photoniccrystal slab. We observe that, over a wide range of frequencies, complete polarization conversion can befound at isolated wave vectors. Moreover, such an effect is topological: the complex reflection coefficientshave a nonzero winding number in the wave vector space. We also show that bound states in continuum inthis system have their wave vectors lying on the critical coupling curve that defines the condition forcomplete polarization conversion. Our work points to the use of topological photonics concepts for thecontrol of polarization, and suggests the exploration of topological properties of scattering matrices as aroute towards creating robust optical devices.

DOI: 10.1103/PhysRevLett.119.167401

There is now significant interest in exploiting topologicalproperties in a wide variety of physical systems. In general,topologically nontrivial systems are characterized by topo-logical invariants that take integer values. Since an integercannot be continuously changed, physical quantities asso-ciated with the invariant can be robust against smallperturbations. Applying topological concepts to opticshas led to the development of topological photonics[1,2]. At present, most efforts in this field have beendevoted to the study of topological invariants such as theChern number in the photonic band structures of bulkphotonic crystal and metamaterial structures, where anonzero Chern number implies the existence of topologi-cally nontrivial edge states [3–13]. But there is alsoemerging interest in extending topological photonics con-cepts beyond band structure analysis. For example, it hasbeen recently noted that bound states in continuum [14,15],i.e., resonances with infinite quality factors, are topologicalin nature [16].In this Letter we seek to extend topological concepts to

the analysis of scattering matrices. The response of anylinear optical device is characterized by its scatteringmatrix, the elements of which are mode-to-mode trans-mission and reflection coefficients. Here as an illustrationwe consider the process of conversion between linearpolarizations for light reflected from a photonic crystalslab. Using both numerical studies and analytic theory, weshow that a photonic crystal slab can provide completepolarization conversion in a reflection process, which is atopological effect since the complex reflection coefficientshave a nonzero winding number in the wave vector space.Consequently, complete polarization conversion can beobserved over a wide range of frequencies. We also identifyan interesting connection between complete polarizationconversion and bound states in continuum [14–16]: thesebound states always lie on the critical coupling curve thatdefines the condition for complete polarization conversion.

Polarization is one of the most fundamental properties oflight. There have been significant recent efforts in usingvarious photonic structures to achieve polarization con-version [17–27]. Our result points to the use of topologicalphotonics concepts to achieve polarization control. Moregenerally, our work should motivate systematic studies oftopological properties of scattering matrices as a routetowards creating robust optical devices.Our structure consists of a photonic crystal slab patterned

with a square array of air holes introduced into thedielectric slab and a mirror underneath, as illustrated inFig. 1(a). The periodicity of the crystal is chosen such thatwithin the wavelength range of interest only zeroth orderdiffraction can occur in free space upon light incident fromdifferent angles. As a result, our structure is characterizedby the 2 × 2 reflection matrix,

R ¼�Rss Rsp

Rps Rpp

�: ð1Þ

Here Rσμ denotes the reflection coefficient of the μ-polarized incident wave reflected into a σ-polarized wave,where σ; μ ∈ fs; pg. The polarization is defined withrespect to propagation direction indicated by a unit vectork, that is, s ¼ z × k, p ¼ s × k, where z is the unit vectorperpendicular to the slab.We consider the lossless case first, where R is unitary.

As a result, complete polarization conversion, as describedby jRspj ¼ jRpsj ¼ 1, is equivalent to Rss ¼ 0. We focuson Rss since the zero of a complex function may betopological.We consider a structure with ϵ¼12, h ¼ 0.3a, r ¼ 0.3a,

where ϵ is the dielectric constant of the slab, a is theperiodicity, h is the thickness of the slab, and r is the radiusof the air hole. In Fig. 1(b), we plot the spectrum of Rsscomputed using the rigorous coupled wave analysis [28]for incident waves at two different parallel wave vectors

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k∥ ¼ ðkx; kyÞ. The blue curve shows jRssj2 at k∥;1 ¼ð0.0522; 0.0240Þ2π=a, the red curve shows jRssj2 atk∥;2 ¼ ð0.0717; 0.0345Þ2π=a. We observe Rss ¼ 0 atω0;1 ¼ 0.388 × 2πc=a and ω0;2 ¼ 0.385 × 2πc=a, respec-tively, indicating complete polarization conversion. InFigs. 1(c) and 1(d), we show the amplitudes of reflectioncoefficients Rss as a function of the parallel wave vector k∥at ω0;1 and ω0;2, respectively. At both frequencies, weobserve Rss ¼ 0 at two isolated wave vectors related by thepoint group symmetry of the lattice. Numerically, whenvarying the incident frequency from ω ¼ 0.40 × 2πc=a toω ¼ 0.35 × 2πc=a, we observe the wave vectors at whichRss ¼ 0 shift from the Γ point towards the light cone. The

effect of complete polarization conversion is thereforerobust with respect to frequency variation.We now numerically demonstrate that the zeros in Rss

have nontrivial topological properties. Since Rss is complex,we consider a vector field of (ℜðRssÞ;ℑðRssÞ), and computethe winding number of the vector field along a closed path inthe ðkx; kyÞ space around the wave vectors where Rss ¼ 0.To visualize the winding number, in Figs. 1(e) and 1(f), weplot the direction flow of the (ℜðRssÞ;ℑðRssÞ) vector fieldat ω0;1 and ω0;2, respectively. We observe a saddle pointthat has a winding number of −1, at k∥;1, k∥;2 in Figs. 1(e)and 1(f), respectively. We also observe a source point thathas a winding number of þ1 in the upper triangular regionwith ky > kx. The source point is connected with the saddlepoint by the mirror operation with respect to the x ¼ y plane.Such a mirror operation flips the sign of the topologicalcharge. The nonzero winding numbers associated with thezeros of Rss indicate that the effect of complete polarizationconversion is topological in nature.Motivated by the numerical observation above, we now

present an analytic theory that provides insight into thephysical mechanism through which the nontrivial topo-logical feature is generated. The results shown in Fig. 1occur when the incident wave excites guided resonances[29]. From the temporal coupled mode theory [30,31], thereflection matrix R in the vicinity of a nondegenerateguided resonance can be expressed as [32]

R ¼ −σz�I −

dd†

iðω0 − ωÞ þ γ

�C; ð2Þ

where C ¼ diagðCss; CppÞ is the background reflectionmatrix, σz ¼ diagð1;−1Þ, I is the 2 × 2 identity matrix,d ¼ ðds; dpÞT , in which ds, dp are the coupling constant ofthe resonance to s- and p-polarized waves, respectively, γ isthe radiation loss rate of the resonance, and ω0 is theresonant frequency. At ω ¼ ω0, if jdsj ¼ jdpj ≠ 0, thenRss ¼ 0, which implies complete polarization conversion.Since in our scenario, the resonance can only couple to twolinear polarization radiation channels, the condition ofjdsj ¼ jdpj is equivalent to the condition of Q matchingto achieve perfect transmission in a resonant filter, or thecondition of critical coupling to achieve complete absorp-tion in a resonant absorber. Below, we refer to the conditionjdsj ¼ jdpj as the critical coupling condition.The guided resonances form photonic bands. On each

band, to achieve complete polarization conversion, we needto find the critical coupling curve on which jdsj ¼ jdpj.We demonstrate that the existence of the critical couplingcurve is guaranteed by the C4v symmetry of the structure.We plot the band diagram in Fig. 2(a). The color in theband diagram reflects the parity of the modes, red for oddmodes, green for even modes, where the correspondingmirror plane is y ¼ 0 for wave vectors along ΓX and x ¼ yfor wave vectors along ΓM. For the lowest band, the

FIG. 1. (a) Schematic of the structure. A dielectric photoniccrystal is sitting on top of a perfect mirror. We denote the dielectricconstant of the slab by ϵ, the periodicity by a, the thickness of theslab by h, the radius of the air hole by r. For the other subplots, theparameters are ϵ ¼ 12, h ¼ 0.3a, r ¼ 0.3a. (b) Blue curve:reflection spectrum at k∥;1 ¼ ð0.0522; 0.0240Þ2π=a. Red curve:reflection spectrum at k∥;2 ¼ ð0.0717; 0.0345Þ2π=a. The completeconversion where Rss ¼ 0 occurs at ω0;1 ¼ 0.388 × 2πc=a andω0;2 ¼ 0.385 × 2πc=a, respectively. (c), (d) jRssj at the frequency(c) ω0;1, (d) ω0;2. (e), (f) Direction flow of vector field(ℜðRssÞ;ℑðRssÞ) at the frequency (e) ω0;1, (f) ω0;2.

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eigenmode at the Γ point has a one-dimensional irreduciblerepresentation B1 [33], where the eigenvalues of operationsC4 (rotation by π=2 around the z axis), σv (mirror withrespect to y ¼ 0), and σd (mirror with respect to x ¼ y) are−1, 1, −1, respectively, as seen in Fig. 2(b). From thecompatibility relations [33], wave vectors along ΓX belongto the A representation of the C1h point group, where theeigenmode is even with respect to the mirror plane y ¼ 0[Fig. 2(c)], and wave vectors along ΓM belong to the Brepresentation of the C1h point group, where the eigenmodeis odd with respect to the mirror plane x ¼ y [Fig. 2(d)].Clearly, odd modes can only couple to s-polarized waves

and even modes can only couple to p-polarized waves. Forthe lowest band shown in Fig. 2, along ΓX, the couplingrate to s-polarized wave ds is zero, whereas along ΓM, thecoupling rate to p-polarized wave dp is zero. Thus for apoint Δ on ΓX, the coupling constants of the lowest bandsatisfy jdpðΔÞj ≥ jdsðΔÞj ¼ 0, whereas for a point Σ onΓM, jdsðΣÞj ≥ jdpðΣÞj ¼ 0. On a line segment connectingthese two points Δ and Σ inside the reduced Brillouin zone,there must be at least one point where jdsj ¼ jdpj. At thispoint we have complete polarization conversion unlessjdsj ¼ jdpj ¼ 0.To verify our theory, we compute the coupling constants

of the lowest band [34], as shown in Fig. 3. ds and dp areindeed zero along ΓX and ΓM, respectively, as argued

above and shown in Figs. 3(a) and 3(b). And we indeedobserve the critical coupling curve in k space shown as thedashed curve in Fig. 3(c). Also, with the coupled modetheory [Eq. (2)] and the numerical results of the couplingconstants, we have confirmed that at each frequency withinthe lowest band, the zeros of Rss on the lower (upper)critical coupling curve in Fig. 3(c) indeed have a windingnumber of −1 (þ1), consistent with the direct numericalcalculations presented above [35].We note an interesting connection between complete

polarization conversion, and bound states in the continuumas discussed in Refs. [14,16]. The Γ point lies on the criticalcoupling curve as shown by the dashed curve in Fig. 3(c).For this particular band, at the Γ point the resonance issingly degenerate and cannot couple to external radiationmodes, and hence has a diverging quality factor shown inFig. 3(c). Therefore, this resonance at the Γ point is a boundstate in continuum. In general, the bound state in con-tinuum always lies on the critical coupling curve, since at abound state both ds and dp vanish, we have jdsj ¼ jdpj. Ofcourse, exactly at a bound state there is no polarizationconversion. But in the immediate vicinity of a bound statethere is always a wave vector where complete polarizationconversion occurs. The frequency bandwidth over whichsignificant polarization conversion occurs vanishes as oneapproaches a bound state.In Fig. 3(c), the bound state at the Γ point is protected by

symmetry. On the other hand, there is a recent discoveryof a new type of bound states in continuum that is notprotected by symmetry [14,16]. The argument presentedabove applies to both types of bound states. Thus, weexpect that the non-symmetry-protected bound state incontinuum should appear on the critical coupling curve aswell. As an illustration, we consider a photonic crystal slabstructure with the parameters ϵ ¼ 4, h ¼ 0.9a, r ¼ 0.4a.Figure 4(a) shows the band diagram for this structure.Examining the quality factor of the lowest order guided

FIG. 3. The parameters are ϵ ¼ 12, h ¼ 0.3a, r ¼ 0.3a, whichare the same as those in Fig. 1 and Fig. 2. (a) Amplitude ofcoupling constant to s-polarized waves of the lowest band,denoted by jdsj. jdsj is zero along ΓX. (b) Amplitude of thecoupling constant to p-polarized waves of the lowest band,denoted by jdpj. jdpj is zero along ΓM. (c) The backgroundshows quality factor of the lowest order resonances in alogarithmic scale. The dashed curve denotes the critical couplingcurve on which jdsj ¼ jdpj.

FIG. 2. (a) Band diagram of the structure in Fig. 1(a) withϵ ¼ 12, h ¼ 0.3a, r ¼ 0.3a. The dashed lines denote the lightcone. (b), (c), (d) Electric field (ℜðEzÞ) distribution of the lowestorder guided resonance at the interface between photonic crystalslab and mirror at (b) Γ point, (c) ðkx; kyÞ ¼ ð0.1; 0.0Þ2π=a onΓX, (d) ðkx; kyÞ ¼ ð0.1; 0.1Þ2π=a on ΓM, where red colordenotes positive values and blue color denotes negative values.

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resonances in the wave vector space [Fig. 4(b)], we find twodistinct non-symmetry-protected bound states with infinitequality factors indicated by the blue circles, one on ΓX, theother on ΓM, in addition to the symmetry-protected boundstate at the Γ point. Figures 4(c) and (d) show theamplitudes of ds and dp of the lowest band, respectively.In addition to being zero due to mirror symmetry, both ds

and dp are zero at the wave vector where the non-symmetry-protected bound states in continuum occur, asindicated by blue circles. In Fig. 4(b), we also show thecritical coupling curve on which jdsj ¼ jdpj, and indeednon-symmetry-protected bound states in continuum lie onthis curve. Our results indicate an intriguing connectionbetween the two topological photonics effects of completepolarization conversion, and bound states in continuum.Our structure also exhibits unusual effects on circularly

polarized waves. We denote the left- and right-handedcircularly polarized waves by jLi and jRi, which can berepresented by ð1;∓ iÞT in the linear polarized s, p basis.Ordinary mirrors, where the reflection matrix R ¼½−1; 0; 0; 1�, reverse the handedness of the incident circu-larly polarized wave, e.g., RjLi ¼ −jRi and RjRi ¼ −jLi.

For our structure, due to reciprocity and inversion sym-metry, we find that RpsðkÞ ¼ −RspðkÞ, which fixes therelative phase of RpsðkÞ and RspðkÞ. Without loss ofgenerality, we can write the reflection matrix R at completepolarization conversion as ½0; 1;−1; 0�. Upon reflection,one finds RjLi ¼ −ijLi and RjRi ¼ ijRi. Therefore, incontrast to regular mirrors which flip handedness uponreflection, our structure can completely reflect the incidentcircularly polarized wave while preserving its handedness.The robustness of the complete polarization conversion

effect comes from the fact that the eigenmodes on ΓX andΓM possess opposite parities with respect to their corre-sponding mirror plane. Though in this work we focus onthe lowest band, it is clear that any bands where the Γpoint belongs to the B1 or B2 representation [33] of the C4vgroup should be able to achieve complete polarizationconversion as well. For a given structure, complete polari-zation conversion can occur over a broad range of frequen-cies. At each of these frequencies, there exist angles atwhich complete polarization conversion occurs. At a fixedincident frequency, the operating angular bandwidth isinversely proportional to the quality factor of the guidedresonance, which is tunable by varying the dielectricconstant or geometrical parameters [36–38].As a final remark, we briefly comment on the effects of

loss, with a more detailed discussion provided in theSupplemental Material [39]. We gradually increase the lossof the dielectric slab and examine how it affects the zeros ofRss. For illustration, we choose h ¼ 0.3a, r ¼ 0.3a, fix thereal part of ϵ to be 12, and vary the imaginary part of ϵ. Aswe increase the imaginary part of ϵ from zero, we observethat the two opposite charges of Rss move towards eachother, and annihilate on the diagonal line kx ¼ ky at ℑðϵÞ ≈0.04 [39]. The efficiency of polarization conversion,i.e., jRpsj2, generally decreases as the loss increases.Nevertheless, in the wave vector regions where the radiationloss rate of the guided resonance dominates the material lossrate, one can observe near perfect polarization conversioneven in the presence of realistic loss. On the other hand, theeffect of Rss ¼ 0 is topological, and hence there is always awave vector at which Rss remains exactly zero as a modestamount of loss is added in the system. Our results thus pointto the manifestation of an interesting effect that is topologi-cally protected against loss.To summarize, we identify a nontrivial topological effect

in the reflection matrix of a simple photonic crystal slab,which leads to a capability for controlling the polarizationof light. In this system, the nontrivial topology arises sincethe underlying map is from a two-dimensional momentumspace to a complex field. We expect a richer set oftopological phenomena as we consider maps from higherdimensional space to other aspects of the scattering matrixthat defines optical devices. Our work should point to afruitful avenue where we use the concepts of topology todesign optical devices.

FIG. 4. The parameters here are ϵ ¼ 4, h ¼ 0.9a, r ¼ 0.4a.(a) Band diagram of the structure. The dashed lines enclose theregion where only zeroth order diffraction can occur. The twoblue circles indicate guided resonances with infinite qualityfactor. (b) The background shows quality factor of the lowestorder resonances in a logarithmic scale. The dashed curve denotesthe critical coupling curve on which jdsj ¼ jdpj. The blue circlesindicate the wave vectors of the non-symmetry-protected boundstates in continuum where jdsj ¼ jdpj ¼ 0. (c) Amplitudes ofcoupling constants to s-polarized waves of the lowest band,denoted by jdsj. The blue circle indicates the wave vector on ΓMfor which jdsj ¼ 0. (d) Amplitude of coupling constant to p-polarized waves of the lowest band, denoted by jdpj. The bluecircle indicates the wave vector on ΓX for which jdpj ¼ 0.

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Y. G. is grateful to Yang Zhou, Yu (Jerry) Shi, Qian Lin,Dr. Alexander Cerjan, and Dr. Chia Wei Hsu for helpfuldiscussions. The authors acknowledge the support of theNational Science Foundation (Grant No. CBET-1641069),and the Air Force Office of Scientific Research (FA9550-16-1-0010, FA9550-17-1-0002).

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