Polarization singularities in planar electromagnetic resonators with rotation and mirror symmetries

Jie Yang; Xuezhi Zheng; Jiafu Wang; Anxue Zhang; Tie Jun Cui; Guy A. E. Vandenbosch

doi:10.1364/PRJ.485625

Abstract

In this work, we apply the group representation theory to systematically study polarization singularities in the in-plane components of the electric fields supported by a planar electromagnetic (EM) resonator with generic rotation and reflection symmetries. We reveal the intrinsic connections between the symmetries and the topological features, i.e., the spatial configuration of the in-plane fields and the associated polarization singularities. The connections are substantiated by a simple relation that links the topological charges of the singularities and the symmetries of the resonator. To verify, a microwave planar resonator with the

D_{8}

group symmetries is designed and numerically simulated, which demonstrates the theoretical findings well. Our discussions can be applied to generic EM resonators working in a wide EM spectrum, such as circular antenna arrays, microring resonators, and photonic quasi-crystals, and provide a unique symmetry perspective on many effects in singular optics and topological photonics.

1. INTRODUCTION

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 $l ℏ$ ( $l$ 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 $C_{2}^{z} T$ and $σ_{z}$ 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 $D_{M}$ 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 $D_{8}$ 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].

Sign up for Photonics Research TOC. Get the latest issue of Photonics Research delivered right to you！Sign up now

2. EIGEN ELECTRIC FIELDS IN A RESONATOR WITH THE $D_{M}$ GROUP SYMMETRIES

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 $C_{M}$ group and the $D_{M}$ 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 $D_{8}$ 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.

A. Indexing the Irreps of the $C_{M}$ and $D_{M}$ Groups

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 $D_{M}$ group. For further discussions, we note that the rotations form the $C_{M}$ group, and the $C_{M}$ group is a subgroup of the $D_{M}$ group. For both the $C_{M}$ and the $D_{M}$ groups, $M$ is an integer larger than or equal to 2. An example of a resonator with the $D_{8}$ group symmetries is illustrated in Fig. 1.

Figure 1.Illustration of a resonator with eightfold rotation symmetries and eightfold reflection symmetries. These symmetries compose the $D_{8}$ group. In (a), the resonator (in orange) is assumed to be made of metal. The resonator in (a) is abstracted as eight points to demonstrate the symmetries [see the orange points in (b)]. In (b), three symmetries operations are given as examples: $C_{8}^{3}$ , is a rotation by $3 π / 8$ with respect to the $z$ axis (see the purple arrow); and $σ_{v}$ and $σ_{d}$ are reflections with respect to the planes made by the $z$ axis and the green line, and by the $z$ axis and the red line, respectively.

In the following, several important observations on the irreps of the $C_{M}$ group and the $D_{M}$ group are reiterated as follows [48]. For the $C_{M}$ group, the main points are: •When $M$ is odd, the irreps of the $C_{M}$ group have one 1D irrep, i.e., $A$ , and $(M - 1) / 2$ 2D irreps, i.e., $E_{h}$ , where $h$ is an integer spanning from 1 to $(M - 1) / 2$ .•When $M$ is even, the irreps of the $C_{M}$ group have two 1D irreps, i.e., $A$ and $B$ , and $(M / 2 - 1)$ 2D irreps, i.e., $E_{h}$ , where $h$ is an integer spanning from 1 to $M / 2 - 1$ .•Each dimension of the irreps of the $C_{M}$ 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 $E_{h}$ are indexed by $j = h$ and $j = M - h$ , respectively.

For the $D_{M}$ group, the main points are: •When $M$ is odd, the irreps of the $D_{M}$ group have two 1D irreps, i.e., $A_{1}$ and $A_{2}$ . They originate from the 1D irrep $A$ of the $C_{M}$ subgroup. Accordingly, the $A_{1}$ and the $A_{2}$ irreps inherit the index $j = 0$ from the $A$ irrep of the $C_{M}$ group.•When $M$ is even, the irreps of the $D_{M}$ group have four 1D irreps, i.e., $A_{1}$ , $A_{2}$ and $B_{1}$ , $B_{2}$ . The $A_{1}$ irrep and the $A_{2}$ irrep originate from the 1D irrep $A$ of the $C_{M}$ subgroup, while the $B_{1}$ irrep and the $B_{2}$ irrep are from the $B$ irrep of the $C_{M}$ subgroup. As a result, $A_{1}$ and $A_{2}$ are indexed by $j = 0$ , and $B_{1}$ and $B_{2}$ are indexed by $j = M / 2$ .•For both even and odd $M$ ’s, the 2D irreps $E_{h}$ of the $D_{M}$ group and their indices are the same as the $C_{M}$ group case.

For the sake of completeness, the irreps of the $C_{M}$ group and of the $D_{M}$ group are presented in Tables 1 –4 in Appendix A.

B. Singularities in Eigen Electric Fields

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 $D_{8}$ group symmetries as an example. For the out-of-plane component of the fields (see Fig. 2), i.e., $E_{z}$ , the $A_{1}$ and the $A_{2}$ irreps demonstrate a trivial scalar vortex mode, i.e., a vortex mode with topological charge of 0, and a hexadecapole; the $B_{1}$ and $B_{2}$ irreps correspond to two octopoles which can be obtained by a rotation by $π / 8$ of each other; and the $E_{1}$ , $E_{2}$ , $E_{3}$ irreps exhibit two scalar vortex modes with opposite topological charges of $- 1$ , $+ 1$ , $- 2$ , $+ 2$ , and $- 3$ , $+ 3$ , respectively.

Figure 2.Illustration of the magnitude and the phase distributions of the $z$ component of the eigen electric fields. The colors in the figure are coded from blue to red to denote the strength and the phase variations. Unless otherwise stated, the same color coding will be applied to the rest of this work.

For the in-plane components (see Fig. 3), i.e., $E_{∥}$ , various polarization distributions are exhibited. For the $A_{1}$ and $A_{2}$ 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 $B_{1}$ and $B_{2}$ 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 $B_{1}$ and $B_{2}$ irreps, eight lines along radial directions segregate the regions of right-handedness from the ones of the left-handedness. Lastly, for an $E_{h}$ ( $h = 1, 2, 3$ ) 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).

Figure 3.Illustration of the intensity and the polarization distributions of the in-plane components of the eigen electric fields. In the figure, ${| E_{∥} |}^{2} = E_{x}^{2} + E_{y}^{2}$ , and “Pol. Mor.” is an abbreviation of “polarization morphology.” To describe the polarization morphologies, three symbols, i.e., the black lines, the blue hollow ellipses, and the red solid ellipses, are introduced to represent the linear, the right-handed, and the left-handed polarization states, respectively. The topological charges carried by the polarization vortices are marked at the centers of the polarization distributions.

C. Relations between the Indices of the Irreps and the Topological Charges of Vortices

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 $D_{M}$ 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 $E_{z}$ ) is related to the index $j$ , $l_{z} = {\begin{cases} - j + M q, & j < M / 2 \\ M - j + M q, & j > M / 2 \end{cases},$ (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 $B_{1}$ and the $B_{2}$ 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 $E_{∥} = | E_{L} | e^{i l_{L} φ} \hat{L} + | E_{R} | e^{i l_{R} φ} e^{i β} \hat{R} .$ (2)

In the above equation, $\hat{L} = \frac{1}{\sqrt{2}} (\hat{x} + i \hat{y})$ and $\hat{R} = \frac{1}{\sqrt{2}} (\hat{x} - i \hat{y})$ are the left and the right circular bases, respectively, $l_{L}$ and $l_{R}$ 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] $I = (l_{R} - l_{L}) / 2 .$ (3)

For the topological charges $l_{L}$ and $l_{R}$ , we prove (see the proofs in Appendix B) that the topological charges of the left and right scalar vortices (associated with $E_{L}$ and $E_{R}$ ) are related to the index $j$ , $l_{L} = - (j + 1 + q_{L} M), l_{R} = - (j - 1 + q_{R} M),$ (4)where $q_{L}$ and $q_{R}$ are arbitrary integers. Equation (4) can be further illustrated by the numerical results (see Appendix D for the $D_{8}$ group case). As a result of Eqs. (3) and (4), the topological charge of the polarization singularity is $I = 1 + (q_{L} - q_{R}) \frac{M}{2} .$ (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 $j \neq 4$ , the choice of $M = 8$ and $q_{L} = q_{R} = 0$ leads to the topological charge of 1 (see Fig. 3); for the case $j = 4$ , the choice of $M = 8$ , $q_{L} = 0$ , and $q_{R} = 1$ leads to the topological charge of $- 3$ (see Fig. 3). Notably, in Fig. 3 both the $A_{1}$ and $A_{2}$ 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 $A_{1}$ irrep, the $E_{L}$ and $E_{R}$ components have no phase shift, i.e., $β = 0$ , which results in the final polarization distribution being radially polarized. For the $A_{2}$ irrep, the $E_{L}$ and $E_{R}$ components have $π$ phase shift, which results in the final polarization distribution being azimuthally polarized. Similar remarks apply to the $B_{1}$ irrep and the $B_{2}$ irrep as well (see Fig. 3). Notably, since $q_{L}$ and $q_{R}$ 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 $D_{7}$ group case is shown in Appendix E. There, the $B_{1}$ irrep and the $B_{2}$ irrep do not exist. As a result, when $q_{L}$ and $q_{R}$ are zero, the topological charge of the polarization vortices is 1 (see Fig. 13 in Appendix E). Besides, when $q_{R} - q_{L}$ 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 $M = 7$ , $q_{R} = - 1$ , and $q_{L} = 0$ , the topological charge $I$ is $- 5 / 2$ . This conclusion is illustrated for the $D_{7}$ group case (see Figs. 14 and 15 in Appendix E). As a note, the selection of the values of $q_{L}$ and $q_{R}$ 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 $D_{M}$ group reduces to the $C_{M}$ 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 $E_{z}$ 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 $E_{z}$ 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].

3. NUMERICAL VALIDATIONS

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.

A. Designs

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 $D_{8}$ 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., $r_{f 1}$ , 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 $Δ φ = n π / 4$ 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 $Δ d$ 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 $Δ φ$ and the length of the delay line $Δ d$ [see Fig. 4(f)].

B. Symmetry-Matching Condition

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 $A_{1}$ , the $B_{1}$ , the $E_{1}$ , the $E_{2}$ , and the $E_{3}$ irreps (Appendix F.1). For the two-port feeding case, the geometric angle $Δ φ$ and the dynamic phase $ψ$ can be tuned (see our previous work [51] for detailed derivations), $Δ φ = \frac{2 N_{1} + 1}{M - 2 j} π, ψ = 2 N_{2} π - j π \frac{2 N_{1} + 1}{M - 2 j},$ (6)so that an eigen electric field belonging to the dimension index by $j = h$ or $j = M - h$ of the $E_{h}$ irrep can be excited. It is worth noting that Eq. (6) is derived for a resonator with generic $D_{M}$ group symmetries, where $M$ can be any integer larger than or equal to $2$ ; and $N_{1}$ and $N_{2}$ are also arbitrary integers. In our design, since the geometric angle $Δ φ$ is assumed to be $n π / 4$ , $n$ is obtained as $\frac{M (2 N_{1} + 1)}{2 (M - 2 j)}$ .

Figure 4.Illustration of the devices with the one- and the two-port feeding networks. (a), (b), and (c) show the top, the bottom, and the side views of the device with the one-port feeding network. (d) and (e) show the side views of the device with the two-port feeding network. (f) shows the details of the two-port feeding network. In (a)–(c), the adopted parameters are: $r_{i} = 12 mm$ , $r_{o} = 24 mm$ , $r_{d} = 52 mm$ , $r_{f 1} = 26 mm$ , $r_{f 2} = 28 mm$ , $d_{r} = 4 mm$ . The width of the microstrip line is 1.1 mm to ensure that the input impedance of the microstrip line is 50 Ω. In (c) and (e), two additional vias marked by “via 3” and “via 4” are to connect the ground of the SMA connector with the fourth layer, i.e., the ground of the microstrip lines.

C. Numerical Results

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 $S_{11}$ in Fig. 5(a) shows that there are five well-separated dips. The excited modes at the dips are marked by $m 1$ , $m 2$ , $m 3$ , $M 4$ , and $M 0$ , respectively. From the out-of-plane component, i.e., $E_{z}$ , at the dips [see Fig. 5(b)], it is observed that the $E_{z}$ component of the $m 1 - m 3$ modes demonstrates the dipole, quadrupole, and hexapole modes, respectively; and the $E_{z}$ component of the $M 4$ and the $M 0$ modes does the octopole mode and the zeroth-order mode (or trivial scalar vortex mode with topological charge of 0), respectively. Then, the $E_{z}$ component of the modes is projected onto each dimension of the irreps of the $D_{8}$ group. It is observed from Fig. 6 that the $m 1$ , the $m 2$ , and the $m 3$ modes only have nonzero projections along the dimensions indexed by $j = 1, 7$ of the $E_{1}$ irrep, the dimensions indexed by $j = 2, 6$ of the $E_{2}$ irrep, and the dimensions indexed by $j = 3, 5$ of the $E_{3}$ irrep, while the $M 4$ and the $M 0$ modes only have nonvanishing projections along the $B_{1}$ ( $j = 4$ ) and the $A_{1}$ ( $j = 0$ ) irreps. Hence, it is said that the $M 0$ , the $M 4$ , the $m 1$ , the $m 2$ , and the $m 3$ modes belong to the $A_{1}$ , $B_{1}$ , $E_{1}$ , $E_{2}$ , and $E_{3}$ 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 $A_{1}$ , the $B_{1}$ , the $E_{1}$ , the $E_{2}$ , and the $E_{3}$ 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 $M 4$ and $M 0$ modes and delay the discussions on the $m 1$ , the $m 2$ , and the $m 3$ modes to the two-port case. Since the $M 4$ and $M 0$ modes belong to the $A_{1}$ and $B_{1}$ 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., $q_{L} = q_{R} = 0$ are significant). This prediction is readily verified by evaluating the topological charges of the in-plane components of the $M 4$ and $M 0$ 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., $r_{f 1} = 4 mm$ , so that, in contrast to Fig. 5, only the eigen electric field belonging to the $A_{1}$ irrep is excited at the $M 0$ dip (see the reflection coefficient $S_{11}$ and the electric field in Appendix G).

Figure 5.Illustration of the simulation results for the one-port case in Figs. 4(a)–4(c). The reflection coefficient $S_{11}$ of the MPR is shown in (a). In (a), the dips in the reflection coefficient are marked by $m 1$ , $m 2$ , $m 3$ , $M 4$ , and $M 0$ (from low frequencies to high frequencies), respectively. In (b), the real parts of the $E_{z}$ component at the five dips are demonstrated. The fields are taken at the $z = 10 mm$ cut plane. In the inset of (b), a dashed black circle delineates the profile of the MPR; and a gray arrow points at the position of the via 0 in Fig. 4(c). In (b), the real part of the electric field is normalized to the range between $- 1$ and $+ 1$ .

Figure 6.Projections of the electric fields at the five dips [in Fig. 5(a)] along the dimensions of the irreps of the $D_{8}$ group. (a)–(e) correspond to the dips $m 1$ , $m 2$ , $m 3$ , $M 4$ , and $M 0$ , respectively. In each subplot, the projections along the dimensions of the irreps (i.e., the vertical axis) are plotted against the dimensions of the irreps (i.e., the horizontal axis). The insets illustrate the phase distributions of the $E_{z}$ components of the projected fields.

Figure 7.Illustration of the (a) and (c) intensity and the (b) and (d) polarization distributions of the in-plane components of the excited electric fields at the dips $M 4$ and $M 0$ . The topological charges are marked at the center of (b) and (d).

Figure 8.Illustration of the out-of-plane and the in-plane components, i.e., $E_{z}$ and $E_{∥}$ of the excited fields in the two-port feeding designs in Figs. 4(d)–4(f). The fields are taken on a cut plane at $z = 20 mm$ . (a) and (b), (c) and (d), (e) and (f) show the $E_{z}$ and the $E_{∥}$ belonging to the two dimensions indexed by $j = 1, 7$ of the $E_{1}$ irrep, to the two dimensions indexed by $j = 2, 6$ of the $E_{2}$ irrep, and to the two dimensions indexed by $j = 3, 5$ of the $E_{3}$ irrep, respectively. For the case of $j = 1$ and $j = 7$ , the lengths of the delay lines $Δ d$ are chosen as $- 27.5 mm$ and 27.5 mm, so that $- 90 °$ and 90° phase delays between two ports are achieved at the targeted frequency 1.72 GHz (in the final simulation, 1.73 GHz). For the cases of $j = 2$ and $j = 6$ , the lengths of the delay lines are chosen as $- 20.8 mm$ and 20.8 mm, so that $- 90 °$ and 90° phase delays between two ports are achieved at the targeted frequency 2.28 GHz (in the final simulation, 2.30 GHz). For the case of $j = 3$ and $j = 5$ , the lengths of the delay lines $Δ d$ are chosen as $- 18.7 mm$ and 18.7 mm, so that $- 90 °$ and 90° phase delays between two ports are achieved at the targeted frequency 2.54 GHz (in the final simulation, 2.56 GHz).

The one-port feeding network simultaneously excites the two eigen electric fields belonging to the two dimensions of the $E_{1}$ , the $E_{2}$ , and the $E_{3}$ irreps. To selectively excite the eigen electric field belonging to a specific dimension of the $E_{1}$ , the $E_{2}$ , or the $E_{3}$ irreps, the two-port feeding network is designed. According to Eq. (6), we tune the geometric angle $Δ φ$ , and the dynamic phase $ψ$ (which is thus the length of the delay line $Δ d$ ) to maximally excite the dimension indexed by $j = 1$ of the $E_{1}$ irrep. By the use of Eq. (6), we can obtain $Δ φ = 90 °$ and $ψ = - 90 °$ ( $N_{1} = 1$ , $N_{2} = 0$ , and $n = 2$ ). The length of the delay line, i.e., $Δ d$ , is calculated to be $- 27.5 mm$ so that a $- 90 °$ phase delay between two ports is achieved at 1.72 GHz, where the linear mode $m 1$ 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., $E_{z}$ and $E_{∥}$ , are shown in Fig. 8. For the out-of-plane component, i.e., $E_{z}$ , it can be observed that the scalar vortex modes with topological charges of $\mp 1$ , $\mp 2$ , and $\mp 3$ (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 $E_{L}$ , the $E_{R}$ , and the Stokes fields in Appendix H). The resonant frequencies in Fig. 8 are retrieved from the simulated reflection coefficient $S_{11}$ 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 $A_{2}$ and $B_{2}$ 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 $D_{8}$ 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 $E_{h}$ 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.

4. CONCLUSIONS

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

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.

APPENDIX A: SYMMETRIES

1.<inline-formula><math display="inline" id="m396" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>C</mi><mi>M</mi></msub></mrow></math></inline-formula> Group and <inline-formula><math display="inline" id="m397" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>D</mi><mi>M</mi></msub></mrow></math></inline-formula> Group

z = 0

2.Irrep of the <inline-formula><math display="inline" id="m416" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>C</mi><mi>M</mi></msub></mrow></math></inline-formula> Group and the <inline-formula><math display="inline" id="m417" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>D</mi><mi>M</mi></msub></mrow></math></inline-formula> Group

Γ

C_{M}

M

M

{(ε^{j})}^{m}

D_{M}

M

M

Diagonal Elements of the Irreps of the $C_{M}$ Group ( $M$ Is an Odd Integer)

	$j$	$r^{m}$
$A$	$j = 0$	${(ε^{j})}^{m}$
$E_{h} [h = 1, \dots, (M - 1) / 2]$	$j = h$	${(ε^{j})}^{m}$
$E_{h} [h = 1, \dots, (M - 1) / 2]$	$j = M - h$	${(ε^{j})}^{m}$

Table 2.

Diagonal Elements of the Irreps of the $C_{M}$ Group ( $M$ Is an Even Integer)

	$j$	$r^{m}$
$A$	$j = 0$	${(ε^{j})}^{m}$
$B$	$j = M / 2$	${(ε^{j})}^{m}$
$E_{h} [h = 1, \dots, M / 2 - 1]$	$j = h$	${(ε^{j})}^{m}$
$E_{h} [h = 1, \dots, M / 2 - 1]$	$j = M - h$	${(ε^{j})}^{m}$

Table 3.

Diagonal Elements of the Irreps of the $D_{M}$ Group ( $M$ Is an Odd Integer)

	$j$	$r^{m}$	$s r^{m}$
$A_{1}$	$j = 0$	${(ε^{j})}^{m}$	${(ε^{j})}^{m}$
$A_{2}$	$j = 0$	${(ε^{j})}^{m}$	$- {(ε^{j})}^{m}$
$E_{h} [h = 1, \dots, (M - 1) / 2]$	$j = h$	${(ε^{j})}^{m}$	0
$E_{h} [h = 1, \dots, (M - 1) / 2]$	$j = M - h$	${(ε^{j})}^{m}$	0

Table 4.

Diagonal Elements of the Irreps of the $D_{M}$ Group ( $M$ Is an Even Integer)

	$j$	$r^{m}$	$s r^{m}$
$A_{1}$	$j = 0$	${(ε^{j})}^{m}$	${(ε^{j})}^{m}$
$A_{2}$	$j = 0$	${(ε^{j})}^{m}$	$- {(ε^{j})}^{m}$
$B_{1}$	$j = M / 2$	${(ε^{j})}^{m}$	${(ε^{j})}^{m}$
$B_{2}$	$j = M / 2$	${(ε^{j})}^{m}$	$- {(ε^{j})}^{m}$
$E_{h} [h = 1, \dots, M / 2 - 1]$	$j = h$	${(ε^{j})}^{m}$	0
$E_{h} [h = 1, \dots, M / 2 - 1]$	$j = M - h$	${(ε^{j})}^{m}$	0

3.Concept of Projection Operators and Basis Functions

Γ

D_{M}

Equation (A2) defines the transformation rule when applying the transformation operator to a scalar and a vector function.

P_{i}^{Γ} f (r) = f_{i}^{Γ} (r), P_{i}^{Γ} f (r) = f_{i}^{Γ} (r) .

i

E_{h}

APPENDIX B: BASIS FUNCTIONS FOR THE CM GROUP and the DM GROUP

1.<inline-formula><math display="inline" id="m594" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>C</mi><mi>M</mi></msub></mrow></math></inline-formula> Group

C_{M}

j

f_{j}^{L} (r)

r = (ρ, θ)

P_{R} f_{j} (r) = e^{i j θ_{0}} [f_{j}^{L} (ρ, θ) \hat{L} + f_{j}^{R} (ρ, θ) \hat{R}] .

f_{j}^{L} (ρ, θ - θ_{0}) = e^{i (j + 1) θ_{0}} f_{j}^{L} (ρ, θ),

f_{j}^{L} (ρ, θ - θ_{0}) = e^{i (j + 1 + q_{L} M) θ_{0}} f_{j}^{L} (ρ, θ),

{(e^{i M θ_{0}})}^{q_{L}} = {(e^{i M θ_{0}})}^{q_{R}} = 1

u_{j}^{L}

f_{j}^{R} (r)

u_{j}^{R}

j

j

2.<inline-formula><math display="inline" id="m631" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>D</mi><mi>M</mi></msub></mrow></math></inline-formula> Group

D_{M}

A_{1}

f_{0}^{A_{1}} (r)

s^{- 1} \cdot r = (\begin{matrix} \cos θ_{0} & \sin θ_{0} \\ \sin θ_{0} & - \cos θ_{0} \end{matrix}) \cdot (\begin{matrix} ρ \cos θ \\ ρ \sin θ \end{matrix}) = (\begin{matrix} ρ \cos (θ_{0} - θ) \\ ρ \sin (θ_{0} - θ) \end{matrix}),

P_{s} f_{0} (r)

f_{0}^{A_{1}} (r) = [u_{0}^{L} (ρ, θ) e^{- i (q_{L} M + 1) θ} + u_{0}^{R} (ρ, - θ) e^{i (q_{R} M - 1) θ}] \hat{L} + [u_{0}^{R} (ρ, θ) e^{- i (q_{R} M - 1) θ} + u_{0}^{L} (ρ, - θ) e^{+ i (q_{L} M + 1) θ}] \hat{R},

q_{L}

A_{1}

C_{M}

B_{1}

f_{4}^{B_{1}} (r)

C_{M}

D_{M}

APPENDIX C: SYNTHESIZED STOKES FIELDS

S_{0} = I_{x} + I_{y} = {| E_{x} |}^{2} + {| E_{y} |}^{2},

I_{x}

σ_{12} = \frac{1}{2 π} \oint \nabla ϕ_{12} \cdot d s .

I_{C} or I_{V} = \frac{σ_{12}}{2} .

D_{8}

Figure 9.Illustration of the magnitude distribution and the phase distribution of the Stokes field $S_{12}$ (corresponding to Fig. 3 in the main text).

Figure 10.Illustration of the normalized Stokes parameters $S_{3} / S_{0}$ (corresponding to Fig. 3 in the main text).

APPENDIX D: D8 GROUP CASE

D_{8}

Figure 11.Illustration of the magnitude and phase distributions of the $E_{L}$ and the $E_{R}$ components of the in-plane eigen electric fields for all the irreps of the $D_{8}$ group.

APPENDIX E: D7 GROUP CASE

D_{7}

Figure 12.Illustration of the magnitude and phase distributions of the $E_{z}$ component of the eigen electric fields corresponding to the irreps of the $D_{7}$ group.

Figure 13.Illustration of the intensity and the polarization distributions of the in-plane components of the eigen electric fields corresponding to the irreps of the $D_{7}$ group.

Figure 14.Illustration of the magnitude and the phase distributions of two circular components of the in-plane eigen electric field corresponding to the second dimension (row) of the $E_{3}$ irrep (marked by $j = 4$ ) of the $D_{7}$ group. It can be observed that the $E_{R}$ and the $E_{L}$ exhibit two scalar vortex modes with topological charge of $- 3$ and 2, respectively.

Figure 15.Illustration of the magnitude and the phase distributions of the Stokes field $S_{12}$ , and the intensity and the polarization distributions of the in-plane eigen electric field corresponding to the second dimension (row) of the $E_{3}$ irrep (marked by $j = 4$ ) of the $D_{7}$ group. It can be observed that the Stokes field $S_{12}$ exhibits a scalar vortex with a topological charge of $- 5$ , and thus the in-plane eigen electric field carries a polarization singularity with a topological charge of $- 5 / 2$ .

APPENDIX F: SYMMETRY-MATCHING CONDITION

1.One-Port Feeding Network

D_{8}

Figure 16.Projections of the electric field radiated by the electric dipole oscillating at 1.72 GHz along the dimensions of the irreps of the $D_{8}$ group. The projections along the dimensions of the irreps (i.e., the vertical axis) are plotted against the dimensions of the irreps (i.e., the horizontal axis). The insets illustrate the phase distributions of the $E_{z}$ components of the projected fields. Notably, here for the sake of compactness, we only demonstrate the projected fields at the first resonant point (i.e., 1.72 GHz). For other resonant frequency points (e.g., 2.28, 2.54, 2.63, and 3.78 GHz), similar behaviors can be observed as well.

Figure 17.Illustration of a spatial configuration of a 16-port feeding system to excite the $A_{2}$ irrep.

Figure 18.Illustration of the spatial configuration of a 16-port feeding system to excite the eigen electric field belonging to the $B_{2}$ irrep.

APPENDIX G: OPTIMIZATION OF THE ONE-PORT NETWORK FOR THE MAXIMAL EXCITATION OF THE M0 MODE

r_{f 1} = 4 mm

Figure 19.Illustration of the simulated reflection coefficient $S_{11}$ and the excited electric field. (a) shows the reflection coefficient. The magnitude and the phase of the out-of-plane component, i.e., $E_{z}$ , are plotted in (b) and (c), while the intensity and the polarization distributions of the in-plane components, i.e., $E_{| |}$ , are plotted in (d) and (e).

APPENDIX H: FURTHER DISCUSSIONS ON Figs.?7 and 8

E_{L}

Figure 20.Illustration of the simulated magnitude and phase distributions of the $E_{L}$ and the $E_{R}$ components of the in-plane electric fields. The figure includes eight columns. The columns marked by $A_{1} (j = 0)$ and $B_{1} (j = 4)$ correspond to Fig. 7 in the main text. The rest of the columns correspond to Fig. 8 in the main text. The figure includes two subplots. In each subplot, there are two panels. The upper panel and the lower panel correspond to the $E_{L}$ and the $E_{R}$ components of the in-plane eigen electric fields, respectively. The first row and the second row of the upper (lower) panel are the magnitude and the phase distributions of the $E_{L}$ ( $E_{R}$ ) component of the in-plane electric fields. All the phase distributions are zoomed-in plots in the squares encircled by white lines in the $B_{1} (j = 4)$ column.

Figure 21.Illustration of the magnitude [ $abs (S_{12})$ ] and the phase [ $angle (S_{12})$ ] distributions of the Stokes field $S_{12}$ . Similar to Fig. 20, the figure consists of eight columns. The columns marked by $A_{1} (j = 0)$ and $B_{1} (j = 4)$ correspond to Fig. 7 in the main text. The rest of the columns correspond to Fig. 8 in the main text. Further, in the column marked by $E_{1} (j = 7)$ , a white dashed circle and two solid circles mark three loop paths. And, in the same column, a black square denotes a square region where the intensity and the polarization distributions in Fig. 23 are plotted. Additionally, in the columns marked by $E_{1} (j = 1)$ , $E_{2} (j = 2)$ , and $E_{2} (j = 6)$ , the arrows point at the C-points whose topological charge is 1/2. Lastly, all the phase distributions are plotted in the square encircled by the white line in the column marked by $B_{1} (j = 4)$ .

Figure 22.Illustration of the normalized Stokes parameters $S_{3} / S_{0}$ corresponding to Fig. 8 in the main text. Two white arrows point at the locations where the $S_{3} / S_{0}$ equals 1.

Figure 23.Illustration of the $E_{| |}$ in the black square in Fig. 21. (a) Intensity; (b) polarization distribution. In (b), the black lines denote the streamlines of the local polarization states, from which it can be observed that there are two C-points with a topological charge of 1/2, as pointed at by the green arrows, and one (quasi-)L line.

APPENDIX I: REFLECTION COEFFICIENTS FOR DIFFERENT TWO-PORT FEEDING NETWORK DESIGNS

The reflection coefficients corresponding to Fig. 8 in the main text are shown in Fig. 24 .

Figure 24.Reflection coefficients $S_{11}$ of different two-port feeding networks. The reflection coefficients $S_{11}$ in (a)–(f) correspond to (a)–(f) in Fig. 8 of the main text. The design parameters of the two-port feeding networks can be found in the caption of Fig. 8 of the main text.

APPENDIX J: ELECTRIC FIELDS ON DIFFERENT z CUTS ABOVE THE RESONATOR

The in-plane and the out-of-plane components of the electric field radiated by the resonator in Fig. 4 are illustrated in Fig. 25 .

Figure 25.Illustration of the out-of-plane and the in-plane components of the electric field radiated by the resonator in Fig. 4 of the main text. Here, the electric fields are plotted at 1.73 GHz on two cut planes. (a) $z = 0.5 λ$ and (b) $z = λ$ (where $λ$ is the vacuum wavelength corresponding to 1.73 GHz), and the electric fields belong to the $E_{1} (j = 7)$ irrep. In (a) and (b), the intensities of the $E_{z}$ are normalized by the maximal intensities of the $E_{| |}$ , respectively. “Pol. Mor.” is the abbreviation of “polarization morphology.”