Optical Singularities in Photonic Microstructures with Rosette Symmetries: A Unified Theoretical Scheme

Jie Yang; Jiafu Wang; Xinmin Fu; Yueting Pan; Tie Jun Cui; Xuezhi Zheng

doi:10.1016/j.eng.2024.10.011

Engineering ›› 2025, Vol. 45 ›› Issue (2) :64 -75. DOI: 10.1016/j.eng.2024.10.011

Research Subwavelength Optics—Article

research-article

Optical Singularities in Photonic Microstructures with Rosette Symmetries: A Unified Theoretical Scheme

Jie Yang ^a^,^b^,^c
, Jiafu Wang ^a^,^b^,^d^,^*
, Xinmin Fu ^a^,^b
, Yueting Pan ^e
, Tie Jun Cui ^f^,^*
, Xuezhi Zheng ^c^,^g^,^*

Author information +

History +

PDF (3926KB)

Abstract

Optical singularities are topological defects of electromagnetic fields; they include phase singularity in scalar fields, polarization singularity in vector fields, and three-dimensional (3D) singularities such as optical skyrmions. The exploitation of photonic microstructures to generate and manipulate optical singularities has attracted wide research interest in recent years, with many photonic microstructures having been devised to this end. Accompanying these designs, scattered phenomenological theories have been proposed to expound the working mechanisms behind individual designs. In this work, instead of focusing on a specific type of microstructure, we concentrate on the most common geometric features of these microstructures—namely, symmetries—and revisit the process of generating optical singularities in microstructures from a symmetry viewpoint. By systematically employing the projection operator technique in group theory, we develop a widely applicable theoretical scheme to explore optical singularities in microstructures with rosette (i.e., rotational and reflection) symmetries. Our scheme agrees well with previously reported works and further reveals that the eigenmodes of a symmetric microstructure can support multiplexed phase singularities in different components, such as out-of-plane, radial, azimuthal, and left- and right-handed circular components. Based on these phase singularities, more complicated optical singularities may be synthesized, including C points, V points, L lines, Néel- and bubble-type optical skyrmions, and optical lattices, to name a few. We demonstrate that the topological invariants associated with optical singularities are protected by the symmetries of the microstructure. Lastly, based on symmetry arguments, we formulate a so-called symmetry matching condition to clarify the excitation of a specific type of optical singularity. Our work establishes a unified theoretical framework to explore optical singularities in photonic microstructures with symmetries, shedding light on the symmetry origin of multidimensional and multiplexed optical singularities and providing a symmetry perspective for exploring many singularity-related effects in optics and photonics.

Graphical abstract

Keywords

Optical singularity / Optical vortex / Photonic microstructures / Symmetries / Group representation theory

Cite this article

Download citation ▾

Jie Yang, Jiafu Wang, Xinmin Fu, Yueting Pan, Tie Jun Cui, Xuezhi Zheng. Optical Singularities in Photonic Microstructures with Rosette Symmetries: A Unified Theoretical Scheme. Engineering, 2025, 45(2): 64-75 DOI:10.1016/j.eng.2024.10.011

登录浏览全文

4963

注册一个新账户忘记密码

1. Introduction

Topology is one of the thematic melodies of physics today. As an essential concept spanning different areas of mathematics and physics, singularities are the key to understanding kaleidoscopic topological phenomena in physics [1], [2]. Singularities in a wave field refer to a point in space where some physical quantities describing the field are undefined. Since the revision of the connectivity of field configurations, singularities have been viewed as topological defects of wave fields. They have been extensively studied in many wave fields and identified as the topological manifestations of many fascinating phenomena, such as black holes in gravitational fields [3], topological elementary excitations in condensed matter [4], and equivalent magnetic monopoles in electronic wavefunctions [5]. Optical singularities are singularities in electromagnetic (EM) fields; they can be basically classified into three main categories: phase singularities in scalar fields, polarization singularities in two-dimensional (2D) vector fields, and topological defects in three-dimensional (3D) vector fields [6]. Phase singularities refer to a point in space where the phase of the EM fields is undefined. Polarization singularities refer to a point in space where a parameter describing the local polarization states of the EM fields is undefined. As 3D optical singularities, topological defects in 3D vector fields—such as optical skyrmions [7], [8], merons [9], [10], hopfions [11], [12], Möbius strips [13], [14], knots [15], and links [16]—can be viewed as the synthesis of the two aforesaid singularities. These multidimensional optical singularities introduce several new degrees of freedom that can be used to structure light and to manipulate light–matter interaction, leading to the creation of a new chapter of modern optics: singular optics. Many fascinating applications have been inspired by multidimensional optical singularities, including (deeply) subwavelength focusing or imaging [8], [17], [18], [19], micro/nanoscale plasmonic vortices for on-chip applications [20], [21], [22], [23], [24], [25], and photonic orbital angular momentum (OAM) multiplexing for high-capacity communications [26], [27], [28].

The generation and manipulation of EM singularities is a cutting-edge topic in optics and photonics. To this end, various photonic microstructures have been proposed, accompanied by specialized theoretical models to expound the working principles behind individual designs. For example, circularly arranged nanoemitters or antenna arrays have been proposed to engineer phase singularity [29], [30], [31], [32], [33], [34], which is explained from the perspective of antenna theory [34]; microring resonators have been proposed to engineer both phase and polarization singularities [35], [36], [37], [38], which is understood based on whispering gallery modes (WGMs) and the coupled mode theory [37], [38]; plasmonic vortex lenses have been proposed, with the experimental generation of phase singularities and optical quasi-particles being theoretically discussed within the framework of spin–orbit interaction (SOI) [9], [22], [21], [39], [40]; plasmonic polygon slits have been proposed to generate optical skyrmions in optical lattices [7], [17], [41], [42], which is theoretically interpreted as an interference phenomenon and explained by the Huygens principle [7], [25], [43]; point defects in photonic crystals have been proposed to engineer phase and polarization singularities [44], [45], with the theoretical discussions naturally falling in line with the band theory of photonic crystals [44], [45]; and spoof plasmonic resonators have been proposed to engineer phase and polarization singularities and optical quasiparticles [46], [47], [48], [49], [50], accompanied by equivalent medium theory and mode superposition principles to expound their working mechanisms [44], [45], [46], [48]. Since the proposed microstructures can be made of different materials, exhibit different geometries, and operate at different frequencies/wavelengths, the corresponding theories are only applicable to certain types of microstructures and thus are phenomenological, lacking wide applicability. Moreover, the proposed microstructures normally exhibit common geometric features, such as rotational and possible reflection symmetries. Most of the current theories still lack a systematic and quantitative study of these symmetry features, so they fail to reveal the connections between symmetries and multidimensional optical singularities. Very recently, efforts have been made to bridge this gap; however, such efforts emphasize the symmetry origins of a certain type of optical singularity [47], [49], [50], [51].

In this work, we develop a unified theoretical scheme by systematically exploiting the intrinsic connections between the symmetries of a microstructure and its multidimensional optical singularities. In particular, we focus on isolated microstructures with rosette (i.e., rotation and, possibly, reflection) symmetries, which covers most of the planar scatterers engineered to generate multidimensional optical singularities. In contrast to our previous work, which followed a pure geometric approach [49] (i.e., only symmetry arguments were employed in determining the topological features of polarization singularities), in this work, we emphasize the interplay between symmetries and the scattering theory of EM waves. Firstly, within the frameworks of EM scattering theory and group representation theory, we demonstrate that the eigencurrents supported by a symmetric microstructure are categorized by the symmetries and that the eigencurrents belonging to the same category share the same symmetry features. This observation prompts us to investigate, for a specific category of eigencurrents, the simplest possible case—that is, a set of dipoles and the electric field radiated by the dipoles—that is, the eigenmode. Next, we theoretically study the topological features of the eigenmode and find that the eigenmodes of a symmetric microstructure can support multiplexed phase singularities [25] in their components—that is, the out-of-plane component, the radial and azimuthal components, and the left- and right-handed circular components. The topological invariants related to the found phase singularities are protected by the symmetries of the microstructures. Based on the phase singularities, more complicated optical singularities can be synthesized, such as C points, V points, L lines, and Néel- and bubble-type optical skyrmions. Furthermore, we discuss the possibility of forming optical lattices due to multiplexed phase singularities. Finally, we clarify the general condition—that is, the symmetry-matching condition—for the excitation of a specific type of optical singularity, which may serve as a general principle for future derivations of selection rules controlling the process of photonic SOI [43], [52], [53]. Our work sheds light on the symmetry origins of multidimensional and multiplexed optical singularities in various photonic microstructures, which can be useful in exploring new types of optical singularities (e.g., optical knots and links). Our work is also helpful in the exploration of other related topics, including optical chirality [54], photonic SOI [53], and geometric phases [6], and inspires the investigation of optical singularities in parametric or synthesized spaces, such as polarization singularities in momentum space [55] and exceptional points in parameter spaces [56].

2. Theoretical framework for light–microstructure interaction by symmetry principles

2.1. Electric field integral equation (EFIE) formalism for light–microstructure interaction

Fig. 1 [7], [17], [19], [22], [35], [36], [37], [39], [41], [47], [48], [50], [52], [53], [57], [58], [59], [60], [61] lists several representative microstructures that have been widely applied to generate multidimensional optical singularities. They can be made of metals, semiconductors, or dielectrics, and can operate at different frequency bands, including the microwave, terahertz, infrared, and optical frequency bands. The interaction of the microstructures and light can be described by a unified operator equation given by the EFIE formalism [62], [63], [64]. The operator equation reads as follows:

(1)

Z \{J (r^{'})\} = V_{inc} (r)

where

Z

is the impedance operator of a microstructure,

J

is the induced current flowing in/on the microstructure,

V_{i n c}

stands for the incident field, and

r^{'}

and

r

denote the source and observation points, respectively. In Cartesian and cylindrical coordinate systems,

r^{'} = (x^{'}, y^{'}, z^{'}) = (ρ^{'}, φ^{'}, z^{'})

and

r = (x, y, z) = (ρ, φ, z)

. Here,

x^{'}, y^{'}, z^{'}

and

ρ^{'}, φ^{'}, z^{'}

are the Cartesian and the polar components of

r^{'}

, respectively; (

x, y, z

) and (

ρ, φ, z

) are the Cartesian and the polar components of

r

, respectively; φ is the azimuthal angle of the vector

r

. Although the detailed expressions of

Z

and

V_{i n c}

may differ for structures made of different materials and operating at different frequency bands, the formulation in Eq. (1) is always applicable (details are provided in Appendix A Section S1). The

Z

operator describes the EM properties of the structure and defines an eigenvalue problem,

Z \{j (r^{'})\} = κ j (r^{'})

, where

κ

and

j

are the eigenvalue and eigenfunction (i.e., eigencurrent) of the

Z

operator, respectively. The electric field radiated by the eigencurrent, denoted as the eigenmode of the microstructure, can be evaluated by the following:

(2)

E (r) = E_{z} \hat{z} + E_{‖} = i ω μ_{0} \int_{V / S} \bar{G} (r, r^{'}) \cdot j (r^{'}) d r^{'}

where E is the electric field at a field point

r

\hat{z}

is the unit vector along the

z

direction, i is the imaginary unit, ω is the angular frequency, μ₀ is the vacuum permeability, and V/S refers to that the integration domain can be a three-dimensional (3D) volume

V

or a two-dimensional (2D) surface

S

\bar{G}

is the homogeneous space dyadic Green’s function.

E_{z}

and

E_{‖}

denote the out-of- and in-plane components of the eigenmodes, respectively. The in-plane components of the fields can be expressed in different bases, such as Cartesian, cylindrical and circular bases, where

E_{‖} = E_{x} \hat{x} + E_{y} \hat{y} = E_{ρ} \hat{ρ} + E_{φ} \hat{φ} = E_{+} \hat{L} + E_{-} \hat{R}

, respectively. Here,

(\hat{x}

\hat{y})

(\hat{ρ}, \hat{φ}),

and

(\hat{L}, \hat{R})

are the unit vectors of the Cartesian, the cylindrical and the circular bases, respectively. Correspondingly,

E_{x}, E_{y}, E_{ρ}, E_{φ}

E_{+}, {a n d E}_{-}

are the components along the unit vectors. We further adopt the convention

\hat{L} = | + 〉

and

\hat{R} = | - 〉

, where

| σ 〉 = (\hat{x} + i σ \hat{y}) / \sqrt{2}

and

σ = \pm 1

(

σ

marks the spin quantum number of light). The transformation relations of the electric field components among different bases are given below.

(3)

E_{σ} = \frac{1}{\sqrt{2}} (E_{x} - i σ E_{y}) = \frac{1}{\sqrt{2}} (E_{ρ} - i σ E_{φ}) e^{- i σ φ}

where E_σ denotes

E_{+}

E_{-}

according to the value of

σ

. Suppose the microstructure is placed in a non-magnetic homogeneous background that has a relative permittivity

ε_{r}

but a unit relative permeability, that is,

μ_{r} = 1

; for the on-substrate cases, it is only necessary to further include the reflected dyadic Green’s function [64].) For the sake of our subsequent discussions, we can rewrite the dyadic Green’s function as follows (details are provided in Appendix A Section S2):

(4)

\bar{G} (r, r^{'}) = \frac{i}{4 π} Rot (φ) \cdot [\sum_{n = - \infty}^{+ \infty} e^{i n (φ - φ^{'})} \cdot {\bar{I}}_{n} (ρ, z, ρ^{'}, z^{'})] \cdot {Rot}^{T} (φ^{'})

where

{\bar{I}}_{n}

is an auxiliary dyadic function and is independent of the azimuthal angles

φ

and

φ^{'}

n

is an integer and represents the cylindrical waves orders (Section S1).

R o t (φ)

and

R o t (φ^{'})

denote two rotation matrices that describe counter-clockwise rotations of

φ

and

φ^{'}

with respect to the z axis, respectively (Fig. 1).

2.2. Group representation theory for rosette symmetries

Rosette symmetries are another name for 2D point group symmetries; they include two categories of symmetries: 2D rotational and reflection symmetries. These symmetries can form two types of 2D point groups: the cyclic group

C_{M}

of rotational symmetries and the dihedral group

D_{M}

of rotational and reflection symmetries, where

M

denotes the number of rotations involved in the group. The generator of the

C_{M}

group is

r

, which is a rotation of an angle

θ_{0} = 2 π / M

(where M is the number of rotations involved in the group) with respect to the rotational symmetry axis (e.g., the

z

axis in Fig. 1(g)). All rotations of the

C_{M}

group can be written in a set of

\{1, r, . . ., r^{M - 1}\}

. The generators of the

D_{M}

group are

r

and

s

, where

s

is a reflection with respect to a reflection plane (the plane makes an angle of

θ_{0} / 2

with respect to the

xoz

plane in Fig. 1(e)). All reflections of the

D_{M}

group can be written in a set of

\{s, s ◦ r, . . ., {(s ◦ r)}^{M - 1}\}

, where s◦r reprensents a reflection whose reflection plane makes an angle of

θ_{0}

with respect to the z axis. The

D_{M}

group is the union of the above two sets.

Group representation theory is a standard mathematical tool used to study how the symmetries of a physical system influence the physical properties of that system. At the core of group representation theory are irreducible representations (irreps), which are essentially matrices [65], [66], [67]. Diagonal elements of the irreps of the

C_{M}

and

D_{M}

groups are given in Table 1, Table 2, respectively. It can be observed from Table 1 that ① when

M

is even, the

C_{M}

group has two one-dimensional (1D) irreps (

A

and

B

) and

(M / 2 - 1)

2D irreps (

E_{h}

); and ② when

M

is odd, the

C_{M}

group has one 1D irrep (

A

) and

(M - 1) / 2

2D irreps (

E_{h}

). All irreps of the

C_{M}

group can be indexed by a parameter

j

(

j = 0, . . ., (M - 1)

), which is referred to as the irrep index. The corresponding relation between the irreps and the irrep index is given in Table 1. It can be observed from Table 2 that ① when

M

is even, the

D_{M}

group has four 1D irreps (

A_{1}

A_{2}

B_{1},

and

B_{2}

) and

(M / 2 - 1)

2D irreps (

E_{h}

); and ② when

M

is odd, the

D_{M}

group has two 1D irreps (A₁ and A₂) and

(M - 1) / 2

2D irreps (

E_{h}

). The irreps of the

D_{M}

group can also be indexed by the irrep index

j

, since the

D_{M}

group can be viewed as the direct product of the

C_{M}

group and the reflection group

D_{1}

[66]. The corresponding relation between the irreps of the

D_{M}

group and the irrep index is given in Table 2. To be complete, in this work, we only consider the vector irreducible representations of the groups and the projective representations will be addressed in our future work.

Based on the diagonal elements of the irreps of the

C_{M}

D_{M}

group, we can construct so-called projection operators, which are an important and useful concept provided by group representation theory. We mark an irrep with the Greek letter

Γ

. The projection operator

P_{i}^{Γ}

is defined for the

i t h

dimension (or the ith row) of the irrep [65], [66]:

(5)

P_{i}^{Γ} = \frac{d (Γ)}{N} \sum_{R} Γ_{i i}^{*} (R) \cdot P_{R}

where

N

is the dimension of the group (for the

C_{M}

group,

N = M

; for the

D_{M}

group,

N = 2 M

), and

d (Γ)

is the dimension of the irrep

Γ

. In Eq. (5), the summation is done with respect to

R

; that is, the summation is done over all the symmetries in the group. For each symmetry operation

R

, there is a matrix

Γ (R)

, and

Γ_{ii} (R)

is the diagonal element on the ith dimension (or row/column) of the matrix.

P_{R}

is the transformation operator of the symmetry operation

R

. The action of the projection operator in Eq. (5) on arbitrary scalar and vector functions (denoted by f(r) and f(r), respectively), and the resultant projected functions (denoted by

f_{i}^{Γ} (r)

and

f_{i}^{Γ} (r)

, respectively) are defined as follows [66]:

(6)

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

Since the projection operator is defined for the ith dimension (or row) of the irrep

Γ

, the projected functions

f_{Γ}^{i}

and

f_{Γ}^{i}

are said to belong to the ith dimension (or row) of the irrep

Γ

. It can be proven that the projected functions belonging to different rows of each irrep or belonging to different irreps are orthogonal and complete, and thus form an orthogonal and complete basis for expanding a generic function [65]. In this sense, the projected functions are also called “basis functions.” As a remark, as shown in Table 1, Table 2, for an

E_{h}

irrep, it can be seen that the diagonal elements of the irrep are zero for reflection symmetry operations. This indicates that the reflection symmetries play no role in the construction of the projection operator defined for the irrep. In addition, for a specific dimension of an

E_{h}

, the projection operator for the

D_{M}

group is the same as that for the

C_{M}

group.

2.3. Interplay of symmetries and light–microstructure interaction

This subsection focuses on the interplay between symmetries and light–microstructure interaction. It can be proven that each projection operator of the

C_{M}

D_{M}

group in Eq. (5) and the

Z

operator in Eq. (1) is commutative [68], [69], [70]:

(7)

[P_{i}^{Γ}, Z] = P_{i}^{Γ} Z - Z P_{i}^{Γ} = 0

Resulting from the commutative relation, the

Z

operator and the projection operators share a common set of eigenfunctions:

(8)

Z \{j_{i}^{Γ} (r^{'})\} = λ_{i}^{Γ} j_{i}^{Γ} (r^{'}), P_{i}^{Γ} j_{i}^{Γ} (r^{'}) = j_{i}^{Γ} (r^{'})

With Eq. (8), the eigencurrents and eigenvalues of the impedance operator

Z

are categorized by the dimensions of the irreps. The categorization is done by the projection operators, and the categories are defined and indexed by the irreps. Since the projection operators and the irreps are only related to the symmetries, it is readily concluded that the eigencurrents belonging to the same category—that is, a specific irrep—should demonstrate common symmetry features only defined by the irreps, which are purely determined by the symmetries of a microstructure and are thus independent of the geometric details of a structure (e.g., the length of the arms in Fig. 1(c)).

To summarize, by applying the integral relations in Eq. (2) to both sides of the second equation of Eq. (8) (e.g., Ref. [70]), we can demonstrate that the electric field radiated by the eigencurrents belonging to a specific irrep—that is, the eigenmode

E_{i}^{Γ} (r)

—belongs to the same irrep; that is,

(9)

P_{i}^{Γ} \{E_{i}^{Γ} (r)\} = E_{i}^{Γ} (r)

3. Optical singularities in the fields radiated by symmetry-classified eigencurrents

3.1. An electric dipole model for extracting common symmetry features

To extract the common symmetry features held by the eigencurrents belonging to an irrep, it is necessary to solve the eigenvalue problem in the second equation of Eq. (8). Furthermore, since the common symmetry features are independent of the geometry of a microstructure, we can consider the most rudimental case: a set of discrete points. The points are related by reflection and rotation symmetries as in a rosette group. At the points, electric dipoles are imposed, which can be seen as the most elemental eigencurrent distribution, in the sense that a generic current distribution can always be constructed from the dipole distribution. Thus, the set of dipoles is also called an “eigen dipole distribution.” For completeness, since the practical structures we consider in this work have a large cross-section-to-height ratio, the considered dipoles are restricted to having in-plane components. Nevertheless, the discussions and the related conclusions can be readily extended to cases where the out-of-plane component is significant, such as chromophore arrays [31], [32], [33].

The eigen dipole distributions corresponding to the irreps of a group can be constructed by applying the projection operator in Eq. (5) to a harmonic electric dipole positioned at

r_{0}^{'}

(Fig. 2(a)), where

r_{0}^{'} = {(ρ_{0}, 0, 0)}^{T}

and ρ₀ is the distance from radial distance between the dipole and the origin. Without loss of generality, we assume that the electric dipole makes an angle of

α

with respect to the

x

axis (Fig. 2(a)). The dipole moment of the dipole (p) can be expressed as follows:

(10)

p (r_{0}^{'}) = p δ (r^{'} - r_{0}^{'}) {(\begin{matrix} \cos α & \sin α \end{matrix})}^{T}

where

p

is the magnitude of the dipole moment

(A ∙ m)

and

δ

is the Dirac function

(m^{- 3})

. The time factor

e^{- i ω_{d} t}

(where

ω_{d}

is the angular frequency and the corresponding wavelength is

λ

) is assumed but suppressed in the following discussions. Applying the projection operator of the ith dimension of the irrep

Γ

defined in Eq. (5) to the dipole

{p (r}_{0}^{'})

, we can obtain the eigen dipole distribution corresponding to the

i t h

dimension of the irrep

Γ

, that is,

P_{i}^{Γ} \{p (r_{0}')\} = p_{i}^{Γ} (r^{'})

. We take the

D_{M}

group with even

M

as an example. The eigen dipole distribution belonging to an irrep of the group is as follows,

(11)

p_{i}^{Γ} (r^{'}) = \frac{d (Γ)}{N} \sum_{m = 0}^{M - 1} [Γ_{i i}^{*} (r^{m}) (\begin{matrix} \cos (θ_{m} + α) \\ \sin (θ_{m} + α) \end{matrix}) + Γ_{i i}^{*} (s ◦ r^{1 - m}) (\begin{matrix} \cos (θ_{m} - α) \\ \sin (θ_{m} - α) \end{matrix})] p δ (r^{'} - r_{m}^{'})

where

θ_{m} = m θ_{0} = m \frac{2 π}{M}

. Using Eq. (11), we can conveniently obtain the eigen dipole distribution belonging to the irrep by plugging the diagonal elements of a specific irrep (Table 2) into Eq. (11) (expressions are provided in Appendix A Section S3). Eq. (11) is not limited to the

D_{M}

group with even

M

, but can also be applied to other 2D point groups—that is, the

D_{M}

group with odd

M

and the

C_{M}

groups with odd and even

M

To illustrate Eq. (11), we take the

D_{6}

group as an example and plot out the eigen dipole distributions belonging to the irreps of the group in Figs. 2(b)–(f). From Fig. 2, it can be observed that, for the

A_{1}

irrep, all dipoles are radially oriented and in phase; for the

A_{2}

irrep, all dipoles are azimuthally oriented and in phase; for the

B_{1}

irrep, all dipoles are radially oriented and in phase, and the two neighboring dipoles are out of phase; for the

B_{2}

irrep, all dipoles are azimuthally oriented, and the two neighboring dipoles are out of phase; and, for the

E_{h}

irrep, all dipoles are circularly arranged, with the same oriented angle of

α

with respect to the radial direction.

3.2. Optical singularities in the radiation of the eigen dipole distributions

Next, based on Eqs. (2), (4), we evaluate the electric field radiated by the eigen dipole distributions in Eq. (11):

(12)

E_{i}^{Γ} (r) = {(\begin{matrix} E_{i, ρ}^{Γ} & E_{i, φ}^{Γ} & E_{i, z}^{Γ} \end{matrix})}^{T} = \frac{i ω^{2} μ_{0} p}{4 π} \sum_{n = - \infty}^{+ \infty} {\bar{I}}_{n} (ρ, z, ρ_{0}) \cdot F_{i}^{Γ} (n) \cdot e^{i n φ}

In Eq. (12), the cylindrical components of the radiated electric field are

E_{i, ρ}^{Γ}

E_{i, φ}^{Γ}

, and

E_{i, z}^{Γ}

. According to Eq. (4) and Section S2,

{\bar{I}}_{n}

can be interpreted as a 3 × 3 matrix in the current coordinate bases.

F_{i}^{Γ}

is a vectorial function (the detailed form of

F_{i}^{Γ}

is provided in Appendix A Section S4) and can be interpreted as a 3 × 1 vector in the current coordinate bases. In particular, we note that

F_{i}^{Γ}

is a function of the summation index

n

. We demonstrate in Section S4 that

F_{i}^{Γ}

is nonzero if and only if

n = - j + M q

q \in Z,

where

Z

marks the set of integers. Hence, Eq. (12) can be recast as follows:

(13)

E_{i}^{Γ} (r) = \frac{i ω^{2} μ_{0} p}{4 π} \sum_{q = - \infty}^{+ \infty} {\bar{I}}_{Mq - j} (ρ, z, ρ_{0}) \cdot F_{i}^{Γ} (Mq - j) \cdot e^{i (Mq - j) φ}

The summation in Eq. (13) indicates that the cylindrical components of the radiated electric field—that is,

E_{i, ρ}^{Γ}

E_{i, φ}^{Γ}

and

E_{i, z}^{Γ}

—are superposed by a set of partial waves indexed by

q

. The qth partial wave has an

e^{i (q M - j) φ}

dependency, suggesting that the qth partial wave holds a helical wave front and thus carries a phase singularity with the order

Mq - j

. As a result, there are phase singularities in the cylindrical components

E_{i, ρ}^{Γ}

E_{i, φ}^{Γ}

, and

E_{i, z}^{Γ}

and the orders

l_{ρ}

l_{φ}

, and

l_{z}

are as follows:

(14)

l_{ρ} = M q_{ρ} - j, l_{φ} = M q_{φ} - j, l_{z} = M q_{z} - j

where

q_{ρ}, q_{φ}, and q_{z}

are arbitrary integers.

The product of the auxiliary dyadic function

{\bar{I}}_{Mq - j}

and the vectorial function

F_{i}^{Γ}

determines the weight of the qth partial wave. For a fixed observation point

r = (ρ, φ, z)

, we can see from the product that the weight is dependent on

ρ_{0}

. As will be shown later, tuning

ρ_{0}

makes it possible to control the desired order of the phase singularity.

In addition to expressing the radiated field in terms of cylindrical components (i.e.,

E_{i, ρ}^{Γ}

E_{i, φ}^{Γ}

, and

E_{i, z}^{Γ}

), we express the electric field in terms of circular bases—that is,

E_{i, +}^{Γ}

E_{i, -}^{Γ},

and

E_{i, z}^{Γ}

. From Eq. (3), it can be seen that

E_{i, +}^{Γ}

and

E_{i, -}^{Γ}

are the linear combinations of

E_{i, ρ}^{Γ}

and

E_{i, φ}^{Γ}

; they can be seen as the superpositions of helical partial waves, as suggested in Eq. (13), and thus carry phase singularities. But the order of the phase singularity carried by

E_{i, +}^{Γ}

(

E_{i, -}^{Γ}

) is always one order less (more) than the radial and azimuthal components. That is, the order of the phase singularity carried by a circular component

E_{i, σ}^{Γ}

is as follows:

(15)

l_{σ} = M q_{σ} - (j + σ)

where

q_{σ} \in Z

l_{σ}

denotes the topological charge of scalar vortex in the circular component

E_{i, σ}^{Γ}

Above, we demonstrate that the components of the radiated electric field carry phase singularities. However, not all of them can be termed as scalar vortex modes due to the requirement of homogeneous polarization distribution. Only

E_{i, z}^{Γ}

E_{i, +}^{Γ}

, and

E_{i, -}^{Γ}

are scalar vortex modes, as their bases are not dependent on spatial coordinates. At the same time,

E_{i, ρ}^{Γ}

and

E_{i, φ}^{Γ}

are radial and azimuthal polarization vortices, since their bases—that is, the unit vector along the radial and azimuthal directions in a cylindrical coordinate system—vary from one point to another. The polarization singularities carried by the two polarization vortices are V points, and their topological charges are always 1 [58]. Therefore, we obtain three scalar vortex modes in

E_{i, z}^{Γ}

E_{i, +}^{Γ}

, and

E_{i, -}^{Γ}

and two polarization vortices in

E_{i, ρ}^{Γ}

and

E_{i, φ}^{Γ}

. It is worth noting that the scalar vortex modes in

E_{i, +}^{Γ}

and

E_{i, -}^{Γ}

are crucial to studying geometric phase or SOI in photonic and optic systems [6].

Aside from the V points in

E_{i, ρ}^{Γ}

and

E_{i, φ}^{Γ}

, much richer singular features can be revealed in the in-plane components of the electric field radiated by the eigen dipole distribution. It is well known that a polarization vortex can be viewed as the superposition of two scalar vortex modes in orthogonal circular bases, and the order of polarization singularity (denoted as

I

) can be evaluated as

I = (l_{-} - l_{+}) / 2

, where l₋ and l₊ are the topological charges of scalar vortices in the right- and left-circular components, respectively [6], [71], [72]. This indicates that the in-plane components of the radiated field should exhibit polarization vortices and carry polarization singularities, since the in-plane components

E_{i, ‖}^{Γ}

are the superposition of the

E_{i, +}^{Γ}

and

E_{i, -}^{Γ}

components. Moreover, the orders of polarization singularity (i.e., the topological charges of polarization vortices), can be evaluated as follows:

(16)

I = 1 + \frac{M}{2} (q_{-} - q_{+})

where q₋ and q₊ are arbitrary integers.

Eq. (16) is a very powerful conclusion, in the sense that it predicts all possible topological charges of polarization vortices that can occur in the in-plane components of the electric field radiated by an arbitrarily shaped microstructure with rosette symmetries. This conclusion is consistent with the results derived by the cyclic Bloch theorem [49]. As a remark, when

M

is even, the topological charge

I

is always an integer; when

M

is odd and

(q_{-} - q_{+})

is nonzero, we notice that the topological charge

I

can be fractional—that is, odd-fold rotationally symmetric microstructures can support fractional-order polarization singularities. For example, when

M = 5

q_{-} = - 1

, and

q_{+} = 0

, the topological charge

I

- 3 / 2

(see an example in Fig. S1 in Appendix A Section S5).

3.3. Scalar vortices, polarization vortices, and 3D optical singularities in electrically small microstructures

To illustrate the above theoretical results, we numerically evaluate the electric field radiated by an eigen dipole distribution with the

D_{6}

group symmetries (i.e.,

M = 6

). The parameters of the model are set as follows: The magnitude of the dipole moment is

p = 1 A ∙ m

; the oscillating wavelength of the dipole is chosen to be

λ = 600 n m

;

ρ_{0} = λ / 8

; the observation plane is chosen at

z = λ

; and the observation region is chosen to be

7 λ \times 7 λ

. Such a setting would allow us to investigate the most general symmetry and the associated topological features in the electric field radiated by an electrically small microstructure. Also, since the structure is subwavelength, it can be readily seen from the weight (i.e., the product of the tensorial function

{\bar{I}}_{Mq - j}

and the vectorial function

F_{i}^{Γ}

in Eq. (13)) that the radiation is dominated by the lowest order singularities. That is, only the

q = 0

partial wave in Eq. (13) is prominent. The resultant phase and magnitude distributions of

E_{i, z}^{Γ}

E_{i, ρ}^{Γ}

E_{i, φ}^{Γ}

E_{i, +}^{Γ}

, and

E_{i, -}^{Γ}

are illustrated in Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, from which it can be observed that the orders of the phase singularities carried by the five components agree well with Eqs. (14), (15) (note that, in Eqs. (14), (15),

q

and

q_{σ}

are set to 0). Therefore, the numerical results well demonstrate our theoretical model. It is worth noting that our results regarding

E_{i, ρ}^{Γ}

and the

E_{i, φ}^{Γ}

can be applied to explain the on-chip generation of cylindrical vector vortices in ring resonators [58], [73] and to construct optical quasiparticles such as skyrmions and merons based on the Stokes vectors of local polarization states [10].

For the polarization singularities in the in-plane components

E_{i, ‖}^{Γ}

, we verify Eq. (16) using Fig. 6, Fig. 7, Fig. 8. We take the

j = 4

case as an example. It can be read from Fig. 6, Fig. 7 that

l_{-}

and

l_{+}

are 3 and 1, respectively. Because

I = (l_{-} - l_{+}) / 2

, the topological charge is 1. This is echoed by the

j = 3

in Fig. 8.

For a complete argument, we can use the scalar vortices in the

z

component (e.g., Fig. 3) and the polarization vortices in the in-plane components (e.g., Fig. 8) of the radiated field to construct 3D optical singularities, such as optical skyrmions, in microstructures. The 3D optical singularities can be viewed as the superposition of lower-order singularities (i.e., phase and polarization singularities). For example, the optical skyrmion can be viewed as the superposition of a scalar vortex with a topological charge of 0 in the

z

component of a field and a polarization vortex with a topological charge of 1 in the in-plane component of the field [7], [50]. We notice that the required scalar and polarization vortices coexist in the

A_{1}

irrep (see the

j =

0 case in Fig. 3, Fig. 8). Therefore, an optical skyrmion may be constructed under the

A_{1}

irrep (Fig. 9). From Figs. 9(a) and (b), we see that, at the center of the vectorial configuration, the vector of the electric field is up; then, with an increase in the radius, the vector gradually flips over and becomes down at the boundary denoted by the white dashed circle. This is further confirmed by Fig. 9(c). The shown vectorial configuration in Fig. 9 is the typical vectorial configuration of a Néel-type skyrmion. The topological invariant of the synthesized optical skyrmion (i.e., the skyrmion number) can be readily evaluated to be 1 [7], [8], [50]. Since the constructed skyrmion belongs to the

A_{1}

irrep of the

D_{6}

group, it is protected by the symmetries of the systems. This observation also applies to the

C_{M}

group (where the

A

irrep plays the same role as the

A_{1}

irrep) and can be further generalized to the 1D unitary Lie group

U (1)

[41]. It is worth noting that the average skyrmion number of the synthesized optical skyrmion is zero in one optical cycle, which is consistent with previous works [7], [17], [50]. The spin optical skyrmion with nonzero skyrmion numbers can be synthesized by the other types of EM field vectors, such as a photonic spin or Stokes vector [8], [10], [41].

3.4. Multiplexed optical singularities in electrically large microstructures

In Section 3.3, we consider the case where

ρ_{0}

is on the subwavelength scale, that is,

ρ_{0} = λ / 8

. This corresponds to electrically small microstructures that could practically be, for example, spoof plasmonic resonators [47], [52], [74] or metal/dielectric spheres [14], [75]. In this case, we only observe the lowest-order optical singularities (Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8). In this section, we consider another extremity,

ρ_{0} > λ

, which corresponds to an electrically large microstructure, such as plasmonic vortex lenses [21], [22], [53] or plasmonic polygon gratings [7], [41], [42]. These microstructures are usually used to generate multiplexed optical singularities [22] and optical lattices [42]. In the following discussions, we maintain all the parameters in Section 3.3 but use

ρ_{0} = 8 λ

and illustrate the effects of changing

ρ_{0}

from

λ / 8

8 λ

by the radiated field belonging to the

A_{1}

irrep (Fig. 10). As can be predicted from Eq. (13), when

ρ_{0}

becomes multiple wavelengths, the weights of the higher-order partial waves, that is,

q = \pm 1, . . .

, become non-negligible; thus, the higher order singularities come into the picture. This is well-confirmed by Fig. 10. First, we focus on the multiplexed singularities in the components of the electric field, that is,

E_{z}, E_{φ}, E_{ρ}, E_{+}

, and

E_{-}

(Figs. 10(a)–(e)). There, in addition to the lowest order (the

q = 0

case),

q = \pm 1

appears, highlighted by the white solid circle in Fig. 10(a) and the black circles in Figs. 10(b)–(e). As predicted by Eq. (14) and confirmed by the weights in Fig. 10(h), the orders of the higher-order singularities in the

E_{z}

E_{φ}

, and

E_{ρ}

are

\pm 6

. At the same time, according to Eq. (15), the orders of the highlighted singularities in

E_{+}

and

E_{-}

are

+ 5

and

- 5

, respectively, determining the order of the higher-order polarization singularities to be

- 5

(Eq. (16) and Figs. 10(f) and (g)). Also, in Fig. 10(g), we observe a transition (the red dashed circle in the figure) at the interface between the polarization vortices with topological charges

+ 1

and

- 5

. This transition leads to the optical domain wall in Fig. 10(i), which can be viewed as a bubble-type optical skyrmion or skyrmion-like vectorial configuration rather than the Néel-type one in Fig. 9 [7]. The transition from the Néel-type optical skyrmion to a bubble-type one can be explained by comparing the magnitude distributions of

E_{z}

and

|E_{‖}|

in the central parts of Figs. 10(a) and (f), respectively. These numerical results indicate that, with an increase in the radial dimensions

ρ_{0}

, microstructures such as plasmonic vortex lenses [22], [53] can indeed support multiplexed optical singularities and even multiplexed 3D optical singularities, such as skyrmionium [76].

As a further remark, comparing Fig. 10(a) and the

j = 0

panel in Fig. 3 reveals that

E_{z}

tends to form an optical lattice with an increase in

ρ_{0}

. By this token, we further increase

ρ_{0}

from

8 λ

40 λ

. As a result, a hexagonal optical lattice has been formed in the

E_{z}

component in Figs. 11(a) and (b). This result can be useful in the Floquet engineering of quantum states of electrons or other particles, such as Bose-Einstein condensations [77]. The formation of the optical lattice can be explained from two perspectives. From one perspective, it can be explained as the superposition of multiplexed phase singularities in

E_{z}

. Fig. 10(h) indicates that the (quasi-)optical lattice with fewer unit cells in Fig. 10(a) is the superposition of three scalar vortices with topological charges of 0 and ±6. In addition, Fig. 11(c) indicates that the (quasi-)optical lattice in Fig. 11(a) is the superposition of nine scalar vortices with topological charges of 0, ±6, ±12, ±18, and ±24. From another perspective, as reported in Refs. [77], [78], [79], optical lattices can be viewed as the result of interfering spherical waves radiated by six electric dipoles. The optical lattice revealed here can be applied to construct more complex vectorial optical lattices, such as optical skyrmion lattices [7]. It is worth noting that the topological charges in Fig. 11(c) are equally spaced with the rotational-symmetry dimension

M

(

M = 6

here), which can be a useful property in applications of optical vortex nanosieves [25], [43]. Limited by the length of this work, the above discussions are restricted to the

A_{1}

irrep. Further results regarding other irreps will be reported later.

4. Excitation of optical singularities

For a complete argument, we focus on how to build an appropriate incident field in order to selectively excite a desired optical singularity supported by a symmetric microstructure, as discussed in Section 3. To proceed, we apply the projection operator in Eq. (5) to the main equation that controls the interaction of light with the microstructure in Eq. (1) and consider the commutative relation in Eq. (7):

(17)

Z \{j_{i}^{Γ} (r)\} = P_{i}^{Γ} \{E_{inc} (r)\}

Eq. (17) suggests that, to excite the eigencurrent

j_{i}^{Γ}

belonging to the ith dimension of the irrep

Γ

, a given incident field must have a non-vanishing projection along the same dimension of the same irrep:

(18)

P_{i}^{Γ} \{E_{inc} (r)\} \neq 0

In this sense, we say that the symmetry of the incident field matches with the irrep. In a special case, if the incident field is invariant under the action of a projection operator, that is,

(19)

P_{i}^{Γ} \{E_{inc} (r)\} = E_{inc} (r)

then the eigencurrent and the secondary field radiated by the eigencurrent (i.e., the eigenmode) belonging to the same dimension of the same irrep are exclusively induced by the incident field. The condition in Eqs. (18), (19) can be quickly applied to the process of SOI in the generation of plasmonic vortices in plasmonic vortex lenses. In this way, a symmetry-compatible angular momentum conservation relation can be found, which serves as a selection rule or the incident Laguerre–Gaussian modes to generate on-chip plasmonic vortex sources [43], [53]. A similar selection rule can also be derived in the transfer process of the OAM of light to plasmonic excitations in metamaterials [52].

To explicitly show the feasibility of our proposed theoretical scheme, we design a plasmonic polygonal microstructure similar to the one given in Fig. 1(b). Such a structure can generate plasmonic vortices and topological quasiparticles under the customized excitation of the Laguerre–Gaussian modes [7], [79]. The full-wave simulation results are given in Fig. S1 in Appendix A Section S5, which well demonstrates our theoretical scheme (details are provided in Section S5).

5. Conclusions

In summary, we presented a unified scheme for systematically studying multidimensional optical singularities in photonic microstructures with rosette symmetries. We discussed the interplay between the light-microstructure interaction and the symmetries of the structure, and classified the eigencurrents and eigenmodes of the structure according to their symmetry features. Then, we developed an electric dipole model to study the topological features of the eigenmodes. Multidimensional optical singularities were shown in the components of the eigenmodes, including the phase singularities carried by scalar vortex modes in

E_{z}

E_{φ}

E_{ρ}

E_{R}

, and

E_{L}

; the polarization singularities in the polarization vortices in

E_{‖}

; and the optical skyrmion or skyrmion-like vectorial configuration in

E

. An extended discussion was provided on a key parameter (the radius

ρ_{0}

) in the dipole model in order to understand the effects of the dimension of photonic microstructures on optical singularities (e.g., multiplexed phase and polarization singularities), the switch from a Néel-type optical skyrmion to a bubble-type one, and the formation of optical lattices. Finally, we reached a symmetry matching condition for the excitation of a specific optical singularity.

Not limited to optical/EM systems, the proposed scheme can also be applied to some other wave field systems (e.g., acoustic or quantum systems) to discuss singularities in the related wave fields in the spirit of the commutative relation between the projection operators of symmetry groups and the governing operators of these systems. As an example, we discussed the phase singularities in a quantum system by replacing the impedance operator in the current work with the Hamiltonian of the system in order to discuss the singularities emitted by the system [31], [32], [33]. Our work establishes an intrinsic connection between the symmetries of a microstructure and the topological features of its radiated fields, introduces a symmetry perspective on the process of generating multidimensional optical singularities via photonic microstructures, and is useful in the further exploration of interesting optical singularities such as optical knots and links, to name just a few. The proposed theoretical scheme can also be extended to investigate the generation of multidimensional singularities via defects in photonic crystals [44], [45] or in the process of spin-orbital coupling in nonlinear light-microstructure interactions [39].

Acknowledgments

This work was supported by the National Natural Science Foundation of China (62301596 and 62288101) and Shaanxi Provincial Science and Technology Innovation Team (23-CX-TD-48). Xuezhi Zheng would like to thank the KU Leuven internal funds: the C1 Project (C14/19/083), the Interdisciplinary Network Project (IDN/20/014), and the Small Infrastructure Grant (KA/20/019); the Research Foundation of Flanders (FWO) Project (G090017N, G088822N, and V408823N), and the Danish National Research Foundation (DNRF165).

Compliance with ethics guidelines

Jie Yang, Jiafu Wang, Xinmin Fu, Yueting Pan, Tie Jun Cui, and Xuezhi Zheng declare that they have no conflict of interest or financial conflicts to disclose.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.10.011.

References

Publishing order | Descend order by publishing year | Descend order by cited within

[1]	Liu W, Liu W, Shi L, Kivshar Y.Topological polarization singularities in metaphotonics.Nanophotonics 2021; 10(5):1469-1486.

[2]	Mond D, Montaldi J.Singularity theory and its applications. Springer, Berlin (1991)

[3]	Ori A.Structure of the singularity inside a realistic rotating black hole.Phys Rev Lett 1992; 68(14):2117-2120.

[4]	Kosterlitz JM, Thouless DJ.Ordering, metastability and phase transitions in two-dimensional systems.J Phys C Solid State Phys 1973; 6(7):1181-1203.

[5]	Dirac PAM.Quantised singularities in the electromagnetic field.Proc R Soc Lond A Contain Pap Math Phys Character 1931; 133(821):60-72.

[6]	Bliokh KY, Alonso MA, Dennis MR.Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects.Rep Prog Phys 2019; 82(12):122401.

[7]	Tsesses S, Ostrovsky E, Cohen K, Gjonaj B, Lindner NH, Bartal G.Optical skyrmion lattice in evanescent electromagnetic fields.Science 2018; 361(6406):993-996.

[8]	Du L, Yang A, Zayats AV, Yuan X.Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum.Nat Phys 2019; 15(7):650-654.

[9]	Dai Y, Zhou Z, Ghosh A, Mong RS, Kubo A, Huang CB, et al.Plasmonic topological quasiparticle on the nanometre and femtosecond scales.Nature 2020; 588(7839):616-619.

[10]	Shen Y.Topological bimeronic beams.Opt Lett 2021; 46(15):3737-3740.

[11]	Sugic D, Droop R, Otte E, Ehrmanntraut D, Nori F, Ruostekoski J, et al.Particle-like topologies in light.Nat Commun 2021; 12:6785.

[12]	Shen Y, Yu B, Wu H, Li C, Zhu Z, Zayats AV.Topological transformation and free-space transport of photonic hopfions.Adv Photonics 2023; 5(1):015001.

[13]	Freund I.Optical Möbius strips in three-dimensional ellipse fields: I. lines of circular polarization.Opt Commun 2010; 283(1):1-15.

[14]	Bauer T, Banzer P, Karimi E, Orlov S, Rubano A, Marrucci L, et al.Observation of optical polarization Möbius strips.Science 2015; 347(6225):964-966.

[15]	Dennis MR, King RP, Jack B, O K’holleran, Padgett MJ.Isolated optical vortex knots.Nat Phys 2010; 6(2):118-121.

[16]	Zhong J, Liu S, Guo X, Li P, Wei B, Han L, et al.Observation of optical vortex knots and links associated with topological charge.Opt Express 2021; 29(23):38849-38857.

[17]	Davis TJ, Janoschka D, Dreher P, Frank B, Meyer zu Heringdorf FJ, Giessen H.Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution.Science 2020; 368(6489):eaba6415.

[18]	Yuan GH, Zheludev NI.Detecting nanometric displacements with optical ruler metrology.Science 2019; 364(6442):771-775.

[19]	Heeres RW, Zwiller V.Subwavelength focusing of light with orbital angular momentum.Nano Lett 2014; 14(8):4598-4601.

[20]	Kim H, Park J, Cho SW, Lee SY, Kang M, Lee B.Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens.Nano Lett 2010; 10(2):529-536.

[21]	Spektor G, Kilbane D, Mahro AK, Frank B, Ristok S, Gal L, et al.Revealing the subfemtosecond dynamics of orbital angular momentum in nanoplasmonic vortices.Science 2017; 355(6330):1187-1191.

[22]	Yang Y, Wu L, Liu Y, Xie D, Jin Z, Li J, et al.Deuterogenic plasmonic vortices.Nano Lett 2020; 20(9):6774-6779.

[23]	Ni J, Huang C, Zhou LM, Gu M, Song Q, Kivshar Y, et al.Multidimensional phase singularities in nanophotonics.Science 2021; 374(6566):eabj0039.

[24]	Tsai WY, Huang JS, Huang CB.Selective trapping or rotation of isotropic dielectric microparticles by optical near field in a plasmonic archimedes spiral.Nano Lett 2014; 14(2):547-552.

[25]	Jin Z, Janoschka D, Deng J, Ge L, Dreher P, Frank B, et al.Phyllotaxis-inspired nanosieves with multiplexed orbital angular momentum.eLight 2021; 1:5.

[26]	Krenn M, Handsteiner J, Fink M, Fickler R, Ursin R, Malik M, et al.Twisted light transmission over 143 km.Proc Natl Acad Sci USA 2016; 113(48):13648-13653.

[27]	Cheng W, Zhang W, Jing H, Gao S, Zhang H.Orbital angular momentum for wireless communications.IEEE Wirel Commun 2019; 26(1):100-107.

[28]	Ren Y, Li L, Xie G, Yan Y, Cao Y, Huang H, et al.Line-of-sight millimeter-wave communications using orbital angular momentum multiplexing combined with conventional spatial multiplexing.IEEE Trans Wirel Commun 2017; 16(5):3151-3161.

[29]	Carlon Zambon N, St-Jean P, Mili Mćević, Lema Aître, Harouri A, Le Gratiet L, et al.Optically controlling the emission chirality of microlasers.Nat Photonics 2019; 13(4):283-288.

[30]	Sala VG, Solnyshkov DD, Carusotto I, Jacqmin T, Lema Aître, Ter Hças, et al.Spin–orbit coupling for photons and polaritons in microstructures.Phys Rev X 2015; 5(1):011034.

[31]	Williams MD, Coles MM, Bradshaw DS, Andrews DL.Direct generation of optical vortices.Phys Rev A 2014; 89(3):033837.

[32]	Williams MD, Coles MM, Saadi K, Bradshaw DS, Andrews DL.Optical vortex generation from molecular chromophore arrays.Phys Rev Lett 2013; 111(15):153603.

[33]	Coles MM, Williams MD, Saadi K, Bradshaw DS, Andrews DL.Chiral nanoemitter array: a launchpad for optical vortices.Laser Photonics Rev 2013; 7(6):1088-1092.

[34]	Mohammadi SM, Daldorff LK, Bergman JE, Karlsson RL, Thid Bé, Forozesh K, et al.Orbital angular momentum in radio–a system study.IEEE Trans Antenn Propag 2010; 58(2):565-572.

[35]	Cai X, Wang J, Strain MJ, Johnson-Morris B, Zhu J, Sorel M, et al.Integrated compact optical vortex beam emitters.Science 2012; 338(6105):363-366.

[36]	Miao P, Zhang Z, Sun J, Walasik W, Longhi S, Litchinitser NM, et al.Orbital angular momentum microlaser.Science 2016; 353(6298):464-467.

[37]	Zhang Z, Qiao X, Midya B, Liu K, Sun J, Wu T, et al.Tunable topological charge vortex microlaser.Science 2020; 368(6492):760-763.

[38]	Zhang Z, Zhao H, Pires DG, Qiao X, Gao Z, Jornet JM, et al.Ultrafast control of fractional orbital angular momentum of microlaser emissions.Light Sci Appl 2020; 9:179.

[39]	Spektor G, Kilbane D, Mahro AK, Hartelt M, Prinz E, Aeschlimann M, et al.Mixing the light spin with plasmon orbit by nonlinear light–matter interaction in gold.Phys Rev X 2019; 9(2):021031.

[40]	Dai Y, Ghosh A, Yang S, Zhou Z, Huang C, Petek H.Poincaré engineering of surface plasmon polaritons.Nat Rev Phys 2022; 4(9):562-564.

[41]	Dai Y, Zhou Z, Ghosh A, Kapoor K, D Mąbrowski, Kubo A, et al.Ultrafast microscopy of a twisted plasmonic spin skyrmion.Appl Phys Rev 2022; 9(1):011420.

[42]	Tsesses S, Cohen K, Ostrovsky E, Gjonaj B, Bartal G.Spin–orbit interaction of light in plasmonic lattices.Nano Lett 2019; 19(6):4010-4016.

[43]	Yang Y, Thirunavukkarasu G, Babiker M, Yuan J.Orbital–angular–momentum mode selection by rotationally symmetric superposition of chiral states with application to electron vortex beams.Phys Rev Lett 2017; 119(9):094802.

[44]	Chen MLN, Jiang LJ, Wei EI.Generation of orbital angular momentum by a point defect in photonic crystals.Phys Rev Appl 2018; 10(1):014034.

[45]	Zhao C, Gan X, Liu S, Pang Y, Zhao J.Generation of vector beams in planar photonic crystal cavities with multiple missing-hole defects.Opt Express 2014; 22(8):9360-9367.

[46]	Su H, Shen X, Su G, Li L, Ding J, Liu F, et al.Efficient generation of microwave plasmonic vortices via a single deep-subwavelength meta-particle.Laser Photonics Rev 2018; 12(9):1800010.

[47]	Yang J, Zheng X, Wang J, Zhang A, Cui TJ, Vandenbosch GAE.Customizing the topological charges of vortex modes by exploiting symmetry principles.Laser Photonics Rev 2022; 16(4):2100373.

[48]	Zhang Y, Zhang Q, Chan CH, Li E, Jin J, Wang H.Emission of orbital angular momentum based on spoof localized surface plasmons.Opt Lett 2019; 44(23):5735-5738.

[49]	Yang J, Zheng X, Wang J, Zhang A, Cui TJ, Vandenbosch G.Polarization singularities in planar electromagnetic resonators with rotation and mirror symmetries.Photon Res 2023; 11(6):936.

[50]	Yang J, Zheng X, Wang J, Pan Y, Zhang A, Cui TJ, et al.Symmetry-protected spoof localized surface plasmonic skyrmion.Laser Photonics Rev 2022; 16(6):2200007.

[51]	Shi P, Du L, Li M, Yuan X.Symmetry-protected photonic chiral spin textures by spin–orbit coupling.Laser Photonics Rev 2021; 15(9):2000554.

[52]	Arikawa T, Hiraoka T, Morimoto S, Blanchard F, Tani S, Tanaka T, et al.Transfer of orbital angular momentum of light to plasmonic excitations in metamaterials.Sci Adv 2020; 6:eaay1977.

[53]	Yang J, Feng P, Han F, Zheng X, Wang J, Jin Z, et al.Symmetry-compatible angular momentum conservation relation in plasmonic vortex lenses with rotational symmetries 2022.arXiv:2209.14735.

[54]	Chen W, Yang Q, Chen Y, Liu W.Extremize optical chiralities through polarization singularities.Phys Rev Lett 2021; 126(25):253901.

[55]	Che Z, Zhang Y, Liu W, Zhao M, Wang J, Zhang W, et al.Polarization singularities of photonic quasicrystals in momentum space.Phys Rev Lett 2021; 127(4):043901.

[56]	Peng B, ŞÖzdemir K, Liertzer M, Chen W, Kramer J, Y Hılmaz, et al.Chiral modes and directional lasing at exceptional points.Proc Natl Acad Sci USA 2016; 113(25):6845-6850.

[57]	Sebbag Y, Levy U.Arbitrarily directed emission of integrated cylindrical vector vortex beams by geometric phase engineering.Opt Lett 2020; 45(24):6779-6782.

[58]	Shao Z, Zhu J, Zhang Y, Chen Y, Yu S.On-chip switchable radially and azimuthally polarized vortex beam generation.Opt Lett 2018; 43(6):1263-1266.

[59]	Arikawa T, Morimoto S, Tanaka K.Focusing light with orbital angular momentum by circular array antenna.Opt Express 2017; 25(12):13728-13735.

[60]	Cogn KGée, Doeleman HM, Lalanne P, Koenderink AF.Generation of pure OAM beams with a single state of polarization by antenna-decorated microdisk resonators.ACS Photonics 2020; 7(11):3049-3060.

[61]	Yue F, Wen D, Xin J, Gerardot BD, Li J, Chen X.Vector vortex beam generation with a single plasmonic metasurface.ACS Photonics 2016; 3(9):1558-1563.

[62]	Chew WC, Tong MS, Hu B.Integral equation methods for electromagnetic and elastic waves. Springer International Publishing, Cham (2009)

[63]	Zheng X, Kupresak M, Verellen N, Moshchalkov VV, Vandenbosch GAE.A review on the application of integral equation-based computational methods to scattering problems in plasmonics.Adv Theory Simul 2019; 2(9):1900087.

[64]	Chew WC.Waves and fields in inhomogenous media. John Wiley & Sons, Hoboken (1999)

[65]	Tung WK.Group theory in physics. World Scientific, Singapore (1985)

[66]	Dresselhaus MS, Dresselhaus G, Jorio A.Group theory: application to the physics of condensed matter. Springer Science & Business Media, Berlin (2007)

[67]	Serre JP.Linear representations of finite groups. Springer, Berlin (1977)

[68]	Collins JT, Zheng X, Braz NV, Slenders E, Zu S, Vandenbosch GA, et al.Enantiomorphing chiral plasmonic nanostructures: a counterintuitive sign reversal of the nonlinear circular dichroism.Adv Opt Mater 2018; 6(14):1800153.

[69]	Kuppe C, Zheng X, Williams C, Murphy AW, Collins JT, Gordeev SN, et al.Measuring optical activity in the far-field from a racemic nanomaterial: diffraction spectroscopy from plasmonic nanogratings.Nanoscale Horiz 2019; 4(5):1056-1062.

[70]	Zheng X, Verellen N, Vercruysse D, Volskiy V, Van Dorpe P, Vandenbosch GA, et al.On the use of group theory in understanding the optical response of a nanoantenna.IEEE Trans Antennas Propag 2015; 63(4):1589-1602.

[71]	Senthilkumaran P, Pal SK.Phase singularities to polarization singularities. Int J Opt (2020), Article 2812803

[72]	Wang Q, Tu CH, Li YN, Wang HT.Polarization singularities: progress, fundamental physics, and prospects.APL Photonics 2021; 6(4):040901.

[73]	Schulz SA, Machula T, Karimi E, Boyd RW.Integrated multi vector vortex beam generator.Opt Express 2013; 21(13):16130-16141.

[74]	Zhang X, Cui TJ.Single-particle dichroism using orbital angular momentum in a microwave plasmonic resonator.ACS Photonics 2020; 7(12):3291-3297.

[75]	Garcia-Etxarri A.Optical polarization mobius strips on all-dielectric optical scatterers.ACS Photonics 2017; 4(5):1159-1164.

[76]	Shen Y, Zhang Q, Shi P, Du L, Yuan X, Zayats AV.Optical skyrmions and other topological quasiparticles of light.Nat Photonics 2024; 18(1):15-25.

[77]	Kouznetsov D, Deng Q, Van Dorpe P, Verellen N.Revival and expansion of the theory of coherent lattices.Phys Rev Lett 2020; 125(18):184101.

[78]	Zhang R, Zhang Y, Ma L, Zeng X, Li X, Zhan Z, et al.Nanoscale optical lattices of arbitrary orders manipulated by plasmonic metasurfaces combining geometrical and dynamic phases.Nanoscale 2019; 11(29):14024-14031.

[79]	Wang Y, Xu Y, Feng X, Zhao P, Liu F, Cui K, et al.Optical lattice induced by angular momentum and polygonal plasmonic mode.Opt Lett 2016; 41(7):1478-1481.