Bound States in a Continuum (BICs) are solutions of the wave equation that lie in the region of the continuous spectrum but are nevertheless uncoupled from the continuum and possess an infinite quality factor [1]. BICs are a research hotspot for topological materials in a condensed matter [2,3] and for topological photonics as an intriguing case of topological singularities [4,5]. In a Photonic Crystal (PhC) slab, the far field of BICs has to vanish in the momentum space, giving rise to a polarization singularity [6]. The far-field polarization winds around the singular point, giving rise to a quantized topological charge. Such singularities have been observed experimentally [7,8] and have a number of intriguing properties related to the conservation of topological charge, splitting of higher-order BICs, and generation and annihilation processes [9–11]. Upon symmetry reduction, a BIC can disappear giving rise to Circularly Polarized States (CPS) [9,12–14]. Such behaviors of polarization singularities can be usefully exploited in order to maximize optical chiralities [15–18]. Radiation–matter interaction can lead to exciton–polariton BICs with joint singularities for both the light and matter [19–21].

The fact that BICs are centers of polarization vortices in $k$ space seems to suggest that the polarization field of a vortex has to be linearly polarized. Indeed, the original case for the topological nature of BICs made reference to a symmetry argument: in the Hermitian system that is invariant under the $C_2^zT$ symmetry (where $C_2^z$ is a $\pi$ rotation around the $z$ axis and $T$ is the time reversal operator), the eigenstates should obey the same symmetry and be linearly polarized [6]. However, the argument does not apply to states lying above the light cone in the $k$–$\omega$ space, since such states couple to the radiation continuum and take the form of quasi-normal modes with a complex frequency [22]: in practice, even if the system is Hermitian and invariant under $C_2^zT$, the eigenstates in the continuum are not. In fact, it was shown later that the far-field polarization vector should be defined as the 2D projection of a full polarization state [9,23,24], which in general is elliptically polarized. Still, ellipticity in the very proximity of a BIC is not well understood, and its scaling laws have still to be clarified.

The goal of this work is twofold: first, we characterize the polarization states of the far-field emission in dielectric PhC slabs both near the topological singularities and in the whole $k$ space above the light cone for square lattices with unbroken and broken symmetries. Second, we relate the occurrence of singularities to far-field optical properties, specifically to the circular dichroism (CD) in reflectivity which is an easily accessible experimental quantity. We focus on the Stokes parameter $S_3$ as a measure of ellipticity and employ a finite-difference time domain (FDTD; Lumerical/Ansys) [25] as well as rigorous coupled-wave analysis (RCWA) [26] to get a full picture of far-field properties from two different computational tools.

We assume a PhC slab with a square lattice etched in a dielectric membrane with refractive index $n=3.48$ and thickness $d=0.3a$, where $a$ is the lattice period. The membrane is embedded in air on both sides. First we consider a symmetric basis in the unit cell, namely a circular air hole with the radius $r=0.3a$ (Fig. 1(a)), such that the 2D pattern has full $C_{4v}$ symmetry. The TE-like photonic band dispersion and $Q$ factors are shown in Fig. 1(b): we focus on the lowest photonic band above the light line (still below the diffraction cutoff), which displays a symmetry-protected BIC at $k=0$. The polarization field ${d}=(d_x, d_y)$ has nodal lines that cross at ${k}=0$ and a phase that winds around the singularity with topological charge $q=+1$ [6,8]: an analytic model is developed in Section S1 in Supplement 1. In Fig. 1(c) we show the $S_3$ parameter in the $k$ space above the light cone, while Fig. 1(d) shows a magnified view of the region close to the BIC. The ellipticity has a complex pattern with various features of interest: it is small in most of the $k$ space and it increases sharply toward the light cone. It vanishes along the symmetry directions of the square lattice and along all other directions it has a nodal point.

 

Fig. 1. (a) Sketch of a PhC slab with square lattice of circular air holes. (b) TE-like photonic band structure with the quality factor (Q) of each resonance. (c)–(d) Normalized Stokes parameter $S_3/S_0$ of the lowest resonance, obtained with the FDTD method.

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We are especially interested in the behavior of the polarization field close to the BIC, which we investigate in detail by means of the analytic treatment in Sec. S1 of the SM. The results for the normalized Stokes parameters $S_1/S_0, S_2/S_0, S_3/S_0$ are in very good agreement with those extracted from RCWA simulations; see Fig. S1 in Supplement 1. The ellipticity decreases quadratically for $k\rightarrow 0$:

The FDTD results shown in Fig. 1 have been obtained by first calculating the mode dispersion and then by exciting the discrete resonance at each wave vector ${k}$ with dipole emitters centered at the mode frequency to evaluate the near-field profile of the resonance, collected directly above the PhC slab. This profile is then projected to the far field in order to retrieve the complex amplitudes of the outgoing field. Within the RCWA, we derived the polarization field by means of temporal coupled-mode theory (TCMT) [27], as explained in Sec. S2 of Supplement 1. The two methods yield fully consistent results, as we show below.

Having characterized the far-field polarization, we now examine its relation to the optical properties. In Fig. 2 we show the reflectivity for light impinging on the PhC slab along the maximum chirality direction, as a function of polar angle, for various combinations of polarizations. The CD in reflectivity depends on the difference between left-circular polarization (LCP) and right-circular polarizations (RCP) and is defined as $CD_R =(R_{lcp}-R_{rcp})/(R_{lcp}+R_{rcp})$. All reflectivities display vanishing coupling to the BIC at $\theta =0^\circ$ [28]. For $\theta \neq 0$, the TE–TE reflectivity (Figs. 2(a) and S2 in Supplement 1) displays Fano resonances due to coupling to the first TE-like band [29]. Instead, the TE–TM or polarization-mixing reflectivity shows nearly Lorentzian peaks (Figs. 2(b) and S2 in Supplement 1). The difference between the two line shapes can be interpreted within TCMT discussed in Sec. S2, as the TE–TM reflectivity has only the scattering contribution from the resonant mode, while the TE–TE reflectivity has both direct and resonant scattering terms. The polarization-mixing reflectivity increases with the incidence angle and becomes substantial toward the light cone. The circularly polarized reflectivities in Figs. 2(c) and 2(d) are nearly identical for small angles, and then their difference increases with $\theta$ leading to interesting CD curves; see Fig. 2(e). These outcomes confirm that: (i) for nearly normal incidence, the outgoing field is characterized by an almost linearly TE-polarized plane wave; and (ii) at high incidence angles, the outgoing field has a polarization vector with both TE and TM components and by a CD in reflectivity, which is often called extrinsic chirality in the literature [30].

 

Fig. 2. Results for a PhC slab with a square lattice of circular air holes at an azimuthal angle $\phi =22.5^\circ$. (a)–(d) Reflectivity (R) spectra calculated by RCWA as a function of polar angle for different polarizations: (a) input TE, output TE; (b) input TE, output TM; (c) input LCP; and (d) input RCP. (e) Circular dichroism in reflectivity. (f) Normalized $S_3$ parameter calculated with both FDTD and RCWA.

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According to TCMT, a non-vanishing R(TE–TM) derives from nonzero TE and TM coupling constants to the resonant modes of the PhC slab. Thus, we focus our attention on the polarization states of the far-field emission from each eigenstate. In Fig. 2(f) we show the $S_3$ parameter extracted from FDTD and RCWA simulations, which are in very good agreement with each other. Starting from ${k}=0$, the ellipticity increases quadratically as described by Eq. (1), it changes sign, and it increases substantially toward the edge of the light cone. The overall behavior can be put in correspondence with the CD spectra in Fig. 2(e): indeed, it can be seen that at an incidence angle around $70^\circ$, the CD resonances are reversed. Thus the overall behavior of reflection CD and of $S_3$ is similar; however, the wave vector of sign reversal is different in the reflectivity CD and in the $S_3$ curves. We explain this difference by noticing that $S_3$ is a property of the eigenmode, while reflectivity is determined by coupling of the far field to the mode and by the direct scattering term (see Sec. S2). The interference between the two scattering channels implies that the CD in reflectivity does not fully correspond to the $S_3$ parameter. Instead, the CD in absorption or emission involves only the resonant mode and would correspond to the $S_3$ of the mode, as shown explicitly in Ref. [17].

Notice that the sign inversion of $S_3$ is driven by the phase difference $\delta =\mathrm {arg}(d_y/d_x)$ between the polarization components, which depends on the complex reflection and transmission coefficients of a uniform effective slab and must change sign at a finite angle, as shown in Ref. [23]. Indeed, since $S_3\propto 2|d_x||d_y|\mathrm {sin}\delta$, the vanishing of $S_3$ along a non-symmetry direction is due to the phase $\delta$ that equals $0$ or $\pi$, and can be called an accidental vanishing. The nodal line at finite wave vector reflects the square symmetry of the lattice.

We now turn to the properties of the square lattice with broken symmetry. We consider air holes with the shape of equilateral triangles (Fig. 3(a)) with the same area of the circular ones or $l=2(\pi /\sqrt {3})^{1/2}r=0.8081a$. The resulting lattice has $C_{1v}$ symmetry, i.e., the $yz$ plane is a mirror plane. The TE-like photonic band dispersion and $Q$ factors are shown in Fig. 3(b), indicating that the BIC has disappeared. In Fig. 3(c) we show the $S_3$ parameter in the whole $k$-space, while Fig. 3(d) shows a magnified view at small wave vectors: we can notice the appearance of CPSs with $S_3=\pm 1$ along the $\Gamma-X$ direction. An analytic treatment of the polarization field that describes the splitting of BIC into CPSs is developed in Sec. S3 of Supplement 1. The ellipticity to the lowest order is described by the following formula:

 

Fig. 3. (a) Sketch of a PhC slab with the square lattice of triangular air holes. (b) TE-like photonic band structure and $Q$ factors. (c) and (d) Normalized Stokes parameter $S_3/S_0$ of the lowest band obtained with the FDTD method (CPSs indicated with yellow dots).

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Finally, we discuss the optical properties of the broken-symmetry lattice along the [1,0] direction (azimuthal angle $\phi =0^\circ$). In Fig. 4 we show the reflectivities for various polarizations, the CD in reflectivity, and the $S_3$ parameter of the mode. The TE–TE reflectivity has sharp Fano line shapes for most angles, except those close to $\theta =0^\circ$. Here a surprising phenomenon occurs, namely the TM–TM reflectivity shows a strong excitation of the mode close to $\theta =0^\circ$ (see inset in Fig. 4(a)). This swapping of the polarization is fully explained by the analytic theory of Sec. S3, in particular by Eq. (S24) in Supplement 1, since at ${k}=0$ the polarization vector becomes ${d}=-ik_0\hat {x}$ and is fully $x$ polarized. The TE–TM mixing in reflectivity (Fig. 4(b)) vanishes at $\theta =0^\circ$, has two maxima around $\theta =\pm 6^{\circ }$, and increases again at large angles. The circularly polarized reflectivities are shown in Figs. 4(c) and 4(d). For an incident LCP (Fig. 4(c)), Fano resonances are clearly visible for $\theta <0$; instead, for small angles $\theta >0$, dispersion of the photonic band is barely appreciable and vanishes at $\theta \approx 6^\circ$. For incident RCP (Fig. 4(d)), an opposite situation occurs since the photonic band dispersion is appreciable for $\theta >0$ and weakens for $\theta <0$, vanishing at $\theta \approx -6^\circ$.

 

Fig. 4. Results for a PhC slab with the square lattice of triangular air holes at an azimuthal angle $\phi =0 ^\circ$. (a)–(d) Reflectivity (R) spectra calculated by RCWA as a function of the polar angle for different polarizations: (a) input TE, output TE; (b) input TE, output TM; (c) input LCP; and (d) input RCP. (e) Circular dichroism in reflectivity. (f) Normalized $S_3$ parameter calculated with both FDTD and RCWA.

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All these results indicate that CPSs are formed at an angle $\theta \simeq -6^\circ$ coupled to the LCP and $\theta \simeq +6^\circ$ coupled to the RCP light. This is confirmed by the CD in reflectivity (Fig. 4(e)), which reaches $+1$ ($-1$) for $\theta \simeq -6^\circ$ ($\theta \simeq +6^\circ$). The $S_3$ parameter (Fig. 4(f)) also reaches $\pm 1$ for $\theta \simeq \mp 6^\circ$. The FDTD and RCWA results are in very good agreement and confirm that a CPS with positive (negative) helicity is formed around $\theta =-6^\circ$ ($\theta =+6^\circ$). For larger values of $\theta$, the ellipticity changes sign and then increases again. Notice that the behavior of the CD is very similar to the dependence of $S_3$ on the incidence angle, except very close to the light cone. We can conclude that for the broken-symmetry lattice, the reflectivity CD is in close correspondence with the properties of the resonant mode.

In conclusion, we gave a full description of the far-field polarization in square lattices with unbroken and broken symmetries. The Stokes parameters close to the BIC (for the unbroken symmetry lattice) and related to the splitting of BIC into CPS (for the case of broken symmetry) are well described by an analytic treatment and are in full agreement with those extracted from FDTD and RCWA calculations via the temporal coupled-mode theory. The reflectivity CD can be related to the ellipticity of the mode but also depends on the direct scattering term, whose importance grows for increasing incidence angles toward the light cone. The results of the present work provide a deeper understanding of the far-field polarization close to topological singularities and can be used to design metasurfaces with desired chiroptical properties.