An electromagnetic (EM) singularity refers to a point in space where a physical parameter of the EM field is undefined, which gives rise to nontrivial topologies of the field [1–6]. The EM singularities as topological defects have provided a unique perspective for connecting many fundamental concepts, such as angular momenta of light [5–7], bound states in the continuum [8–11], EM multipoles [12–14], and optical chirality [15,16], and thus have become a central topic that covers a broad range of the EM spectrum from microwaves, via terahertz, to optical frequencies, in the research field of wave engineering [6,12,17,18]. There are two typical categories of singularities: the scalar and the vector singularities, also known as phase and polarization singularities. The phase singularity occurs in a scalar field where the intensity is zero and the phase is undefined. In the paraxial regime, an EM wave with the phase singularity is aliased as a vortex mode that has a homogeneous polarization pattern and carries an orbital angular momentum (OAM) of lℏ (l is the order of phase singularity or topological charge; ℏ is the reduced Planck constant) [5,6,19]. The polarization singularity occurs in a vector field where a parameter characterizing polarization ellipse is undefined, such as orientation, handedness, or eccentricity [1,2,17]. An EM wave with the polarization singularity exhibits inhomogeneous polarization patterns. The manipulation of the polarization patterns with EM scatterers has been of importance in both classical and quantum optics and has fueled various applications, such as optical manipulation [20,21], optical information processing [22–24], engineering Bose–Einstein condensation [25], controlling single-quantum radiation [26], realizing optical quasi-particles [27–30], and designing EM devices beyond scalar optics limit [28,31–34].

In the study of phase and polarization singularities, the symmetries of EM scatterers always play a significant role. For example, rotational symmetries of the scatterers are closely related to the EM singularities [10,35–41]. This can be demonstrated by the fact that many devices with rotational symmetries have been proposed to generate scalar vortex modes with phase singularity and have facilitated a plethora of OAM-based applications ranging from classical to quantum regimes [42–45]. Accordingly, the relation between the symmetries and the phase singularity has been intensively studied [35–40]. Regarding the relation between the symmetries and the polarization singularities, symmetry analysis finds [46] that polarization singularities in one-dimensional (1D) and two-dimensional (2D) photonic crystals (PCs) can support bound states in the continuum (BICs). For this case, to realize the BICs, aside from the translation symmetries and possible rotational symmetries in the PCs, the PCs are further required to hold the C2zT and σz symmetries (in detail, the former is the rotation of 180 deg with respect to the axis perpendicular to the PCs together with the time reversal symmetry, while the latter is a reflection with respect to the mid-plane of the PCs). Also, a systematic study harnessing the group theoretical tool has been performed to study the polarization singularities in the momentum space of 2D photonic quasi-crystals [10]. However, both studies are carried out in the momentum space, and such relation in real space has not been clearly and systematically revealed.

In this work, we focus on a planar EM resonator with M rotational and M reflection symmetries (which form a DM group), and apply the group representation theory to investigate the relations between the symmetries and the polarization singularities. This work results in two key conclusions: the first is encoded in Eqs. (4) and (5) in Section 2 and the second, i.e., the “symmetry-matching condition” in Section 3. The former relates the topological features of the polarization distributions supported by the resonator with the resonator’s symmetries. This relation offers a neat way for designers interested in predicting accepted or forbidden topological charges of polarization singularities based on the symmetries of the underlying resonator. The latter provides a set of design rules that customizes the geometric parameters of a feeding network, so that the network can produce an incident field that excites a planar EM resonator to radiate polarization singularities with the desired topological charges. By this design rule, we design a microwave plasmonic resonator (MPR) with the D8 group symmetries. The numerical simulations demonstrate our theoretical considerations well. Our approach can be readily extended to other EM systems working in a wide EM spectrum, such as chiral nanoemitter arrays [36,37], microring resonators [38,39], photonic quasi-crystals [21], and circular antenna arrays [35]. It can provide not only a unique symmetry perspective on many effects in singular optics and topological photonics, but also a generic symmetry approach towards devising, e.g., optical skyrmions [27,47], plasmonic merons [29], and photonic spin textures [28,30].

The main conclusions of this section are Eqs. (4) and (5). To derive the conclusions, we start with reiterating several key observations on the irreducible representations (irreps) for the CM group and the DM group. Here, most importantly, we give an index, i.e., an integer j, to each dimension of the irreps of the groups. Then, we point out that each dimension of the irreps defines an eigen electric field distribution. By using a resonator with the D8 group symmetries, we illustrate the polarization morphologies and the topological features of the eigen electric field distributions. Last but not least, because each dimension of the irreps 1) is given an integral index, i.e., j, and 2) defines an eigen electric field distribution holding very specific topological features, i.e., topological charges, we rigorously establish a relation between the indices j’s and the topological charges in the eigen electric fields, i.e., Eqs. (4) and (5).

To streamline the text, the mathematical background and the mathematical derivations are put in the appendices. The mathematics behind Subsection 2.A is given in Appendix A, and the proofs for the relations are given in Appendix B.

Without loss of generality, we consider a planar metal resonator. The resonator has M-fold rotation and M-fold reflection symmetries. The rotations and the reflections form the DM group. For further discussions, we note that the rotations form the CM group, and the CM group is a subgroup of the DM group. For both the CM and the DM groups, M is an integer larger than or equal to 2. An example of a resonator with the D8 group symmetries is illustrated in Fig. 1.

In the following, several important observations on the irreps of the CM group and the DM group are reiterated as follows [48]. For the CM group, the main points are: •When M is odd, the irreps of the CM group have one 1D irrep, i.e., A, and (M−1)/2 2D irreps, i.e., Eh, where h is an integer spanning from 1 to (M−1)/2.•When M is even, the irreps of the CM group have two 1D irreps, i.e., A and B, and (M/2−1) 2D irreps, i.e., Eh, where h is an integer spanning from 1 to M/2−1.•Each dimension of the irreps of the CM group can be indexed by an integer j (see how the indices are stipulated in Appendix A). For the case that M is odd, A is indexed by j=0. For the case that M is even, A and B are indexed by j=0 and j=M/2, respectively. For both even and odd M’s, the two dimensions of Eh are indexed by j=h and j=M−h, respectively.

For the DM group, the main points are: •When M is odd, the irreps of the DM group have two 1D irreps, i.e., A1 and A2. They originate from the 1D irrep A of the CM subgroup. Accordingly, the A1 and the A2 irreps inherit the index j=0 from the A irrep of the CM group.•When M is even, the irreps of the DM group have four 1D irreps, i.e., A1, A2 and B1, B2. The A1 irrep and the A2 irrep originate from the 1D irrep A of the CM subgroup, while the B1 irrep and the B2 irrep are from the B irrep of the CM subgroup. As a result, A1 and A2 are indexed by j=0, and B1 and B2 are indexed by j=M/2.•For both even and odd M’s, the 2D irreps Eh of the DM group and their indices are the same as the CM group case.

For the sake of completeness, the irreps of the CM group and of the DM group are presented in Tables 1–4 in Appendix A.

Based on the group representation theory [49], each dimension of the irreps defines an electric field distribution. The electric field distributions corresponding (belonging) to different dimensions of the irreps are orthonormal. Since the field distribution is only dependent on symmetries, it is called an “eigen electric field” (see the end of Appendix A for details).

To illustrate the eigen electric fields, we use the resonator (see Fig. 1), with the D8 group symmetries as an example. For the out-of-plane component of the fields (see Fig. 2), i.e., Ez, the A1 and the A2 irreps demonstrate a trivial scalar vortex mode, i.e., a vortex mode with topological charge of 0, and a hexadecapole; the B1 and B2 irreps correspond to two octopoles which can be obtained by a rotation by π/8 of each other; and the E1, E2, E3 irreps exhibit two scalar vortex modes with opposite topological charges of −1, +1, −2, +2, and −3, +3, respectively.

For the in-plane components (see Fig. 3), i.e., E∥, various polarization distributions are exhibited. For the A1 and A2 irreps, the vector fields in the distributions are radially and azimuthally oriented, respectively. Both distributions have V-type polarization singularities (i.e., V-point). By definition, a V-point is an intensity null point in a linearly polarized field [17]. At the V-point, the polarization azimuth is undefined. Further, the B1 and B2 irreps demonstrate vectorial octopole patterns. Both polarization distributions show a mix of the linear, the left-circular and the right-circular polarization states. Especially, for the B1 and B2 irreps, eight lines along radial directions segregate the regions of right-handedness from the ones of the left-handedness. Lastly, for an Eh (h=1,2,3) irrep, the eigen electric fields are always left- (corresponding to the dimension indexed by j=h) and right-elliptically polarized (corresponding to the dimension indexed by j=M−h and here M=8). The fields always exhibit the so-called C-point singularities at which the orientation of the major axis of the polarization ellipse is undefined. The topological charges of the polarization distributions (marked in Fig. 3) can be evaluated by the synthesized Stokes fields of the in-plane components (see Appendix C).

In the above example, the eigen electric fields clearly demonstrate topological features, i.e., the EM singularities and the associated topological charges. Since the eigen electric fields are always associated with the dimensions of the irreps of the DM group, and each dimension of the irreps is indexed by an integer j, it is natural to establish a relation between the index j and the order of the topological charge. For the out-of-plane component, we have demonstrated in our previous work [50] that the topological charge of the scalar vortex (associated with Ez) is related to the index j, lz={−j+Mq,j<M/2M−j+Mq,j>M/2,(1)where q is an arbitrary integer. Equation (1) applies to both the odd M and the even M cases. Especially, when M is an even integer, the j=M/2 case exhibits the M-pole (see the B1 and the B2 cases in Fig. 2).

For the in-plane components carrying polarization singularities, considering that an EM wave with polarization singularity can be seen as the superposition of two scalar vortices in orthogonal circular bases [17], we can express the field E∥ as E∥=|EL|eilLφL^+|ER|eilRφeiβR^.(2)

In the above equation, L^=12(x^+iy^) and R^=12(x^−iy^) are the left and the right circular bases, respectively, lL and lR are the topological charges associated with the left and the right circular components, and β is the constant phase shift between the left and the right circular components. The order of the polarization singularity (or the topological charge) superposed by the left and the right circular vortices can be evaluated as [17] I=(lR−lL)/2.(3)

For the topological charges lL and lR, we prove (see the proofs in Appendix B) that the topological charges of the left and right scalar vortices (associated with EL and ER) are related to the index j, lL=−(j+1+qLM),lR=−(j−1+qRM),(4)where qL and qR are arbitrary integers. Equation (4) can be further illustrated by the numerical results (see Appendix D for the D8 group case). As a result of Eqs. (3) and (4), the topological charge of the polarization singularity is I=1+(qL−qR)M2.(5)

When M is even, the main conclusions in Eq. (5) can be readily confirmed by the example in Fig. 3. That is, for the cases where j≠4, the choice of M=8 and qL=qR=0 leads to the topological charge of 1 (see Fig. 3); for the case j=4, the choice of M=8, qL=0, and qR=1 leads to the topological charge of −3 (see Fig. 3). Notably, in Fig. 3 both the A1 and A2 irreps are indexed by j=0. But there is a difference between their polarization distributions. This difference originates from the constant phase shifts β in Eq. (2). For the A1 irrep, the EL and ER components have no phase shift, i.e., β=0, which results in the final polarization distribution being radially polarized. For the A2 irrep, the EL and ER components have π phase shift, which results in the final polarization distribution being azimuthally polarized. Similar remarks apply to the B1 irrep and the B2 irrep as well (see Fig. 3). Notably, since qL and qR are always integers, a fractional-order polarization singularity is strictly forbidden for an even M.

To further confirm the above conclusions for the odd M case, the D7 group case is shown in Appendix E. There, the B1 irrep and the B2 irrep do not exist. As a result, when qL and qR are zero, the topological charge of the polarization vortices is 1 (see Fig. 13 in Appendix E). Besides, when qR−qL is nonzero, the topological charge I can be fractional. That is, the odd-fold rotationally symmetric resonators can indeed support the fractional-order polarization singularities. For example, when M=7, qR=−1, and qL=0, the topological charge I is −5/2. This conclusion is illustrated for the D7 group case (see Figs. 14 and 15 in Appendix E). As a note, the selection of the values of qL and qR is closely related to the geometries of resonators. Detailed theory on this aspect will be deliberated in our future work. For the cases where the mirror symmetries are broken, the DM group reduces to the CM group. But, even in this case, Eq. (5) is still valid (see proofs in Appendix B).

Lastly, it is noted that polarization singularities can be only properly defined when the out-of-plane component, i.e., the Ez component, is negligible or significantly smaller than the in-plane components. In the current section (e.g., in Fig. 2) and also in the section that follows (e.g., in Fig. 8), the discussions on the Ez component are only included to demonstrate that the symmetries of a scatterer define the topological properties of all three components of the eigen electric fields as a whole. This observation can be useful for constructing vectorial EM field configurations in the near-field regime of the scatterer, e.g., the EM skyrmion in Ref. [47].

In this section, we present two categories of designs to numerically demonstrate the symmetry-based arguments in the previous section. Both categories are designed based on the symmetry-matching conditions in Section 3.B.

The first category is a device [see Figs. 4(a)–4(c)] excited by a one-port feeding network. The first layer of the device is an MPR with the D8 group symmetries. The MPR is etched on a dielectric plate made of Rogers 4530B (with the relative permittivity 3.48, the loss tangent 0.0037, and the thickness 0.508 mm), which serves as the second layer. The same dielectric plate is used for the third and the fifth layers. The fourth layer is made of metal film and serves as the ground of the device. Two microstrip lines are etched on the lower surface of the fifth layer and the upper surface of the third layer, respectively, and are connected by a “via” through from the third to fifth layer, as shown in Fig. 4(c). Due to the metal ground (the fourth layer), the incident field is dominated by the field radiated by the microstrip line on the upper surface of the third layer, as shown in the top right-hand panel of Fig. 4(c). The third, the fourth, and the fifth layers establish the feeding system. Especially, we have a parameter for tuning the one-port feeding system, i.e., rf1, the length of the microstrip line on the lower surface of the fifth layer.

The second category is a device excited by a two-port feeding network [see Figs. 4(d)–4(f)]. The feeding systems in the third, the fourth, and the fifth layers are redesigned. In detail, two microstrip lines related by a rotation of geometric angle Δφ=nπ/4 are etched on the upper surface of the third layer and serve as two ports, as shown in Fig. 4(d). The two ports are linked by two vias (marked by “via 1” and “via 2”) to a 1-to-2 power divider etched on the lower surface of the fifth layer, as shown in Figs. 4(e) and 4(f). The input power is fed equally to two ports by a power divider. The dynamic phase difference ψ between two ports is realized by a delay line of the divider with the length of Δd in one branch, as shown in Fig. 4(f). Therefore, we have two tunable parameters for tailoring the two-port feeding system, i.e., the geometric angle Δφ and the length of the delay line Δd [see Fig. 4(f)].

To excite an eigen electric field, the so-called symmetry-matching condition needs to be satisfied (see our previous work [51]). That is, the incident field must have a nonvanishing projection along the dimension of the irrep that defines the eigen electric field. This argument is due to the orthonormality of the eigen electric fields discussed in Section 2 [48]. In this spirit, for the one-port feeding case, it can be demonstrated that the design given in Figs. 4(a)–4(c) can excite the eigen electric fields belonging to the A1, the B1, the E1, the E2, and the E3 irreps (Appendix F.1). For the two-port feeding case, the geometric angle Δφ and the dynamic phase ψ can be tuned (see our previous work [51] for detailed derivations), Δφ=2N1+1M−2jπ,ψ=2N2π−jπ2N1+1M−2j,(6)so that an eigen electric field belonging to the dimension index by j=h or j=M−h of the Eh irrep can be excited. It is worth noting that Eq. (6) is derived for a resonator with generic DM group symmetries, where M can be any integer larger than or equal to 2; and N1 and N2 are also arbitrary integers. In our design, since the geometric angle Δφ is assumed to be nπ/4, n is obtained as M(2N1+1)2(M−2j).

We numerically simulate the designs with the one- and the two-port feeding networks in CST Microwave Studio (CST). For the one-port case, the reflection coefficient S11 in Fig. 5(a) shows that there are five well-separated dips. The excited modes at the dips are marked by m1, m2, m3, M4, and M0, respectively. From the out-of-plane component, i.e., Ez, at the dips [see Fig. 5(b)], it is observed that the Ez component of the m1–m3 modes demonstrates the dipole, quadrupole, and hexapole modes, respectively; and the Ez component of the M4 and the M0 modes does the octopole mode and the zeroth-order mode (or trivial scalar vortex mode with topological charge of 0), respectively. Then, the Ez component of the modes is projected onto each dimension of the irreps of the D8 group. It is observed from Fig. 6 that the m1, the m2, and the m3 modes only have nonzero projections along the dimensions indexed by j=1,7 of the E1 irrep, the dimensions indexed by j=2,6 of the E2 irrep, and the dimensions indexed by j=3,5 of the E3 irrep, while the M4 and the M0 modes only have nonvanishing projections along the B1 (j=4) and the A1 (j=0) irreps. Hence, it is said that the M0, the M4, the m1, the m2, and the m3 modes belong to the A1, B1, E1, E2, and E3 irreps. This observation immediately confirms the prediction from the symmetry-matching condition (see Appendix F.1). That is, the designed one-port network excites the eigen electric fields belonging to the dimensions of the A1, the B1, the E1, the E2, and the E3 irreps. Further, by comparing Fig. 6 to Fig. 2, it is seen that the projected modes share the same topological features as the corresponding eigen electric fields. For example, for the case j=1, both the projected modes and the eigen electric fields hold a topological charge of −1. For the in-plane components, i.e., E∥ (see Fig. 7), we focus on the polarization distributions of the M4 and M0 modes and delay the discussions on the m1, the m2, and the m3 modes to the two-port case. Since the M4 and M0 modes belong to the A1 and B1 irreps, they should demonstrate the topological charge of +1 and −3 as predicted by Eq. (5) and shown in Fig. 3 (we tailor the geometry of the resonator so that the lowest-order q’s, i.e., qL=qR=0 are significant). This prediction is readily verified by evaluating the topological charges of the in-plane components of the M4 and M0 modes (see Fig. 7), respectively. Lastly, by using the symmetry-matching condition as an optimization criterion, we find the optimal length of the microstrip line on the lower surface of the fifth layer, i.e., rf1=4mm, so that, in contrast to Fig. 5, only the eigen electric field belonging to the A1 irrep is excited at the M0 dip (see the reflection coefficient S11 and the electric field in Appendix G).

The one-port feeding network simultaneously excites the two eigen electric fields belonging to the two dimensions of the E1, the E2, and the E3 irreps. To selectively excite the eigen electric field belonging to a specific dimension of the E1, the E2, or the E3 irreps, the two-port feeding network is designed. According to Eq. (6), we tune the geometric angle Δφ, and the dynamic phase ψ (which is thus the length of the delay line Δd) to maximally excite the dimension indexed by j=1 of the E1 irrep. By the use of Eq. (6), we can obtain Δφ=90° and ψ=−90° (N1=1, N2=0, and n=2). The length of the delay line, i.e., Δd, is calculated to be −27.5mm so that a −90° phase delay between two ports is achieved at 1.72 GHz, where the linear mode m1 appears. This procedure can be applied to other indices j’s as well (see the caption of Fig. 8 for the selected parameters of the two-port feeding networks corresponding to different indices j). The simulated out-of-plane and in-plane components, i.e., Ez and E∥, are shown in Fig. 8. For the out-of-plane component, i.e., Ez, it can be observed that the scalar vortex modes with topological charges of ∓1, ∓2, and ∓3 (corresponding to the dimensions of the irreps indexed by j=1,7, j=2,6, and j=3,5) are excited by the designed two-port feeding networks. For the in-plane components, i.e., E∥, the polarization distributions indexed by j=1,2,3 and j=7,6,5 exhibit right-handed and left-handed elliptical polarizations and hold C-point singularities with topological charges +1 (see the EL, the ER, and the Stokes fields in Appendix H). The resonant frequencies in Fig. 8 are retrieved from the simulated reflection coefficient S11 of each two-port feeding network (see Appendix I). Comparing the frequencies in Fig. 8 to the ones in Fig. 5, it can be observed that there are small frequency shifts. The differences are due to the feeding networks: that is, in Figs. 5 and 8, one-port and two-port feeding networks are used. Finally, it is noted that for the A2 and B2 irreps, the corresponding eigen electric fields cannot be excited by the designed one- or two-port feeding system. More complex feeding systems are needed (see discussions in Appendix F.2). As a caveat, it is noted that Fig. 8 focuses on the near-field regime of the resonator, where both the out-of-plane and the in-plane components of electric fields are significant. However, this is only to validate the theoretical arguments in Section 2, i.e., Eqs. (1) and (5), and to demonstrate that the symmetries of the resonator indeed define the topological singularities in all three field components. To be in line with the definition of in-plane polarization singularities (see the end of Section 2), we plot the electric field corresponding to the j=7 case on different cuts above the resonator in Appendix J. There, the out-of-plane component is weak enough so that in-plane polarization singularities can be properly defined.

Last but not least, it is worth mentioning that, since the one- and the two-port feeding networks do not hold the D8 group symmetries of the resonator, adding them as excitations indeed breaks the symmetries of the resonator. This results in a splitting of the C-type polarization singularities in Fig. 8 (see details in Appendix H). Especially, for the two-port feeding network case, it has to be emphasized that the design rules in Eq. (6) aim to maximally but not exclusively excite the eigen electric fields belonging to a specific dimension of an Eh irrep. That is, the feeding network can still excite eigen electric fields belonging to other dimensions of the irreps but only in a parasitic manner. To exclusively excite a specific dimension, a feeding network that fully matches the symmetries of the resonator is needed. This feeding network requires 16 ports and is envisaged in Appendix F.2.

In conclusion, we systematically investigate the relation between the symmetries of an EM resonator and the topological features of the in-plane polarization distributions supported by the resonator. The relation is consolidated: 1) the morphologies of polarization distributions are categorized by the irreps determined by the symmetries of the resonator, and 2) the topological charges of the polarization distributions are directly linked with the irreps through Eqs. (5) and (6). Although our discussions focus on the relation between the in-plane electric fields supported by the resonator, and the discrete rotation and reflection symmetries of a planar microwave resonator, the same methodology can be applied to the discussions on the vector fields defined on other spaces, e.g., the momentum space, and the other types of symmetries, e.g., the translational symmetries in a lattice. This line of reasoning can be very helpful in searching for the symmetry origins of many photonic effects.

Acknowledgment. T. J. Cui acknowledges the support by the National Natural Science Foundation of China. X. Zheng and G. A. E. Vandenbosch would like to thank the KU Leuven internal funds: the C1 project C14/19/083, the IDN project IDN/20/014, and the small infrastructure grant KA/20/019; and the Research Foundation of Flanders (FWO) project G090017N. X. Zheng is also grateful for the IEEE Antennas and Propagation Postdoctoral Fellowship.