1. Introduction

Orbital angular momentum (OAM), together with the pseudospin characterized by spin angular momentum (SAM), are both significant properties of photons that are extensively investigated in many optical systems. In recent years, light beams structured to carry OAM have been applied to a wide range of domains, including but not limited to topological optics [1], vacuum slow light [2], quantum information processing and storage [3–9]. The interaction between photonic OAM and its intrinsic spin, called spin-orbit (SO) coupling, has resulted in various topological phenomena, photon transport and photonic Hall effect in both dielectric [10–12] and surface plasmonic systems [13–17]. One of the most promising on-chip platforms for realization of SO coupling effect is Fabry-Perot (F-P) microcavity system, which is usually constructed by metal mirrors or distributed Bragg reflectors (DBRs) [18,19]. The interaction between light and matter in such a system generates the quasiparticles, called exciton-polaritons with half-light, half-matter eigenstates. Moreover, the various types of mechanisms triggered by photonic polarization in cavity could serve as effective gauge fields acting on the photonic pseudospin and have resulted in significant physical achievements, i.e., the transverse-electric transverse-magnetic (TE-TM) splitting leads to the optical spin-Hall effect [20,21], dark-half solitons [22,23] and spin vortices [24], the optical activity (OA) generated by the material anisotropy [25] allows the appearance of Rashba-Dresselhaus SO coupling [26,27], helical polaritonic laser [28] and nontrivial topological bands and valleys [25–30]. We note that the photons and polaritons both propagate in two-dimensional (2D) space of the planar microcavities, i.e., F-P cavities, while in a photonic potential the photonic SO coupling may lead to many interesting optical phenomena. In laterally confined structures, for example the tunable open-access microcavity consisting of a concave and a planar cavity mirrors facing each other [31], several quantized zero-dimensional (0D) eigenstates for photons are generated and carry both SAM and Laguerre-Gaussian (LG) type OAM, due to the strong circularly-symmetric confinement induced by the concave mirror. In such a situation, the interaction between the SAM and the OAM, i.e., the SO coupling effect induced by the cavity TE-TM splitting, leads to new eigenmodes of photonic spin vortices [31,32] and optical skyrmions [33–36]. They have been respectively found in the first (OAM $|l |= 1$) and second (OAM $|l |= 0,2$) excited manifolds. Spin vortices can be realized by the coherent superposition of two states possessing opposite photonic pseudospin and anti-rotating OAM, whilst the skyrmions exhibiting the concentric alternating photonic spin in space can be constructed by linearly superimposing two orthogonal modes with the same total angular momentum (the sum of OAM and SAM) and the opposite photonics pseudospin [34]. Especially, optical skyrmions have been demonstrated in meta structures [37], free space [38,39] and surface evanescent electromagnetic fields [40]. Moreover, they are also proposed to realize many promising practical applications in optical information storage, precision metrology and sensing [41].

As the SAM has merely 2D basis, i.e., right- (${\sigma ^ + }$) and left- (${\sigma ^ - }$) circular polarizations, the main complexity of the confined system stems from the OAM, which in principle can reach a dimension of infinity. Although the eigenstates under the TE-TM SO coupling in the symmetrically confined potential has been well-studied for small OAM numbers, i.e., $|l |\le 2$, it is not yet clear about the situations with $|l |\ge 3$. In particular, do the resulting eigenenergies and eigenstates of vector fields exhibit universal rules with increasing $|l |$, which allow to predict the situations for all $|l |$s until approaching infinity? In addition, can the vector states of very high-orders be presented graphically by a diagram similar to the higher-order Poincare sphere (HOPS) [42] and be potentially applied to multi-dimensional information processing [43]? These questions are yet to be answered.

In this article, we report the eigenenergies and eigenstates up to the 6th-order manifolds derived by a systematic investigation based on the degenerate perturbation theory and obtain the universal rules of mode properties in circularly-confined systems with the TE-TM polarization splitting. We find that the SO coupling only leads to two-dimensional (2D) eigenvectors no matter how large the basis is, and forms only two types of eigenstates, i.e., degenerate skyrmions and non-degenerate spin vortices with $|l |$=1. The characters of these eigenstates are classified by the parity of the manifolds, while all the other possible superimposed states of vector LG modes are excluded despite the complexity of the system. The resulting eigenstates could be presented graphically by the SO hypersphere, which can be decomposed into a series of visualized three-dimensional (3D) HOPS thanks to the 2D nature of the eigenvectors. Our results constitute significant extension of the scope of photonic SO coupling and provide an efficient theoretical frame for potential applications in information processing with high-dimensional vector states.

2. Results and discussion

As widely reported, the TE-TM splitting in F-P microcavities serves as an “effective gauge field” acting on the photonic spin [25], which can be described by the Hamiltonian written in the circular polarization basis, i.e., ${({1\; 0} )^T}$ and ${({0\; 1} )^T}$,

 

Fig. 1. Sketch of the microcavity system with TE-TM splitting. (a) Diagram of the TE-TM splitting characterized by $\beta $ and wave vector k. (b) Sketch of the tunable open-access microcavity with the top concave mirror providing the laterally confinement V.

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The eigenvalues and eigenstates of Hamiltonian in Eq. (1) can be derived via the degenerate perturbation theory if the TE-TM SO coupling is much weaker than the confined potential V, as in the tunable open-access microcavities [31]. Analytical solutions can be derived by treating the terms of TE-TM splitting, i.e., $\beta {k^2}{e^{ {\pm} 2i\varphi }}$ in the off-diagonal elements as perturbations. A detailed description of the degenerate perturbation theory with proper examples can be found in Supplement 1. We start with the eigenstates of the unperturbed system by setting $\beta = 0$, the solution of which, when considering circular symmetry, are obviously the LG modes denoted as $L{G_{p,l}}{\sigma ^ \pm }$, where ${\sigma ^ + }$(${\sigma ^ - }$) is associated with right (left) circularly-polarized photons, p is the radial quantum number and l is the azimuthal quantum number related to the photonic OAM. Herein we define the total quantum number $n = 2p + |l |$ according to the phase degeneracy of LG beams [44]. The LG-type eigenstates in the $n$th manifold of the cavity system are all $2({n + 1} )$-fold degenerate and the eigenenergies are denoted as $E_n^{(0 )}$, corresponding to a $2({n + 1} )$-dimensional orthogonal basis as shown in Table 1. In the situation of nonzero TE-TM splitting term, i.e., $\beta \ne 0$, the degenerate perturbation theory that is applied to the Hamiltonian in Eq. (1) yields new eigenstates as a result of the SO coupling, which partially lift the degeneracy. The calculated new eigenenergies and eigenstates of the lowest 5 manifolds are presented in the following.

Table 1. $2({n + 1} )$-dimensional orthogonal basis in the lowest five manifolds

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For $n = 0$,

For $n = 1$,

For $n = 2$,

For $n = 3$,

For $n = 4$,

(i) The central states, which refer to the states located at the energy spectral center. These states are not affected by the TE-TM SO coupling, still being the bare LG modes and keeping their energies unchanged at $E_n^{(0 )}$. According to Rule 2, these states exhibit $p = 0$ and thereby the highest possible OAM $|l |= n$. They have the mathematical form of $\mathit{\Psi} _{na}^0 = L{G_{0,n}}{\sigma ^ + }$ and $\mathit{\Psi} _{nb}^0 = L{G_{0, - n}}{\sigma ^ - }$, which display the same sign for OAM and SAM. These states are basically phase vortices, but can be coherently superposed due to certain types of symmetry breaking and form spin antivortices with a topological charge n, showing a $2\pi n$ winding of linear polarization angle counter-rotating with the real-space azimuthal angle. For example, the polarization textures of $\mathit{\Psi} _{1a}^0$ and $\mathit{\Psi} _{1b}^0$ in Fig. 2 (c-2) and Fig. 2 (c-3) exhibit such features, and the coherent superposition of the two states is shown in Supplement 1. We don’t make further discussion on these states as they are not the results of SO coupling, which is, however, the main interest of this work. Meanwhile, one should note that the situation of $n = 0$ is an exception as no OAM can be yet involved.

(ii) The edge states, which refer to the states at the two edges of the energy spectrum. The edge states experience the largest energy shift or splitting induced by the TE-TM SO coupling (thereby with the largest value of i in $\mathit{\Psi} _{na,nb}^{ {\pm} i}$). The interesting phenomenon is that the nature of these states depends on the parity of the manifold. According to Rule 2, the edge states exhibit the smallest $|l |$, which results in a big difference between the situations of n being even and odd. For the manifold of an odd $n$, the smallest possible OAM $|l |= 1$ instead of 0, otherwise p would not be kept as an integer considering $n = 2p + |l |$. The TE-TM SO coupling then selects the form

For the manifold of an even n, the smallest possible $|l |= 0$. According to Rule 1, the new eigenstate is a superposition of two LG bare states. However, unlike the odd manifolds holding two LG states with opposite OAM $l ={\pm} 1$, there is always one LG state of $|l |= 0$ for a given even n, namely, $L{G_{n/2,0}}$ without considering the spin degree of freedom. Therefore, another LG state with a larger $|l |$ has to be involved. The fact $n = 2p + |l |$ tells that the smallest increase of $|l |$ has to be 2 instead of 1, otherwise p cannot be kept as an integer for an even n. In such a situation, the SO coupling selects the general form of the edge states to be

(iii) The intermediate states, which refer to the states between the central and edge ones in the energy spectrum. These states containing the intermediate values of p and $|l |$ exist only when $n \ge 3$ and have the general forms

One should note that Rules 1 and 2 are the necessary but not the sufficient conditions for deriving the physical properties of the eigenstates described above. The above points (i)-(iii) have to satisfy but are not deduced from Rules 1 and 2, and therefore constitute an independent rule that we summarize as Rule 3: the central states are double-degenerate bare LG modes; the edge states are single spin vortices for odd manifolds and double-degenerate solid skyrmions for even manifolds; all the intermediate states between the center and odds ones existing only when $n \ge 3$ are double-degenerate hollow skyrmions showing increasing radial quantum number p with increasing energy splitting.

 

Fig. 2. Eigenenergies and polarization textures of polaritons in different manifolds due to TE-TM splitting. (a) Structures of eigenenergies, in which (a-1) ∼ (a-5) correspond to the situation of $2p + |l |= 0\sim 4$. $E_n^{(0 )}$ ($n = 1\sim 4$) are the energies of the unperturbed LG modes and $\alpha $ is the parameter determining the mode size. $\mathit{\Psi} $ located at the left-side of their corresponding energy levels represent the new eigenstates in the main text. (b) Polarization textures of the new eigenstates, in which (b-1) ∼ (b-2), (c-1) ∼ (c-4), (d-1) ∼ (d-6), (e-1) ∼ (e-8) and (f-1) ∼ (f-10) respectively correspond to the states possessing $2p + |l |= 0,1,2,3$ and $4$. The following parameters are used: $\alpha = 1/0.65\mu {m^{ - 1}}$, $\beta = 0.065meV \cdot \mu {m^2}$ and $\varphi = 0.01\pi $.

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Although the rules are mathematically derived, it is important to understand the underlying physical origins. As the 2 × 2 Hamiltonian describes the coupling between two polarizations, it is natural that each resulting eigenstate contains two LG components with opposite polarizations. The SO coupling mechanism, which is based on the TE-TM splitting, always results in the largest energy splitting for the pure TE and TM modes, results in smaller energy splitting for the modes containing both TE and TM components which are not completely balanced, and results in no energy splitting for the modes consisting of completely balanced TE and TM components. As the radially and azimuthally polarized linear spin vortices are pure TE and TM modes, they exhibit the strongest lift of degeneracy and therefore form the edge states for odd n. Although the situation for even n is a bit different for the sake of the existence of $l = 0$ singlet states, each of its edge state (solid skyrmions) keeps the largest possible TE or TM components compared to the intermediate states, which are hollow skyrmions whose TE and TM components are further balanced in each mode with decreasing p and increasing $|l |$. Finally, the central states, whose linear superposition forms the hyperbolic spin antivortex [31] with polarization map shown in Supplement 1, possess completely balanced TE and TM components and therefore do not exhibit any lift of degeneracy.

Rules 1, 2 and 3 are the essential conclusions that we obtain for the confined vector eigenstates under the TE-TM splitting. The results show well the truth that simple rules can be applied to complicated systems, and also show that some kinds of vector states that are easy to be constructed by superposing circularly-polarized high-order LG modes, for example, the spin vortices with topological charges higher than 1, do not appear as eigenstates of the confined system with TE-TM SO coupling. All the skyrmion states, being double-degenerate, usually cannot be observed in spectroscopy individually. More often, a superposition of the two degenerate states with a given phase difference is no longer an optical skyrmion but still an eigenstate of the system, which could be observed due to the slight broken of perfect circular symmetry. Nevertheless, it is shown that single skyrmion could be picked up by employing the non-Hermitian properties of the photonic system. The details of such non-Hermiticity can be found in our recent work [34].

In addition, we address that the rules are drawn from the calculations up to the lowest 7 manifolds, which show sufficiently clear trend that higher manifolds would follow. Therefore, the set of rules we obtained can be viewed as a physical rule or a reasonable mathematical conjecture, but not a theorem. It becomes a pure mathematical problem to strictly prove that the rules can be applied to all values of n until infinity, which is, though out of the scope of this work, likely to be interesting for mathematicians who search for analytical solutions of eigenvalues in novel physical systems.

It is well-known that the vector-vortex states involving both SAM and OAM can be graphically presented by higher-order Poincare spheres (HOPSs) [42]. More recently, a concept of spin-orbit hypersphere was proposed for the graphical presentation of the high-dimensional states resulting from spin-orbit coupling [43]. As we know, the photonic pseudospin can be graphically described by Poincare sphere, on which each point represents a polarized state $\mathit{\Psi} $ constructed by coherently superposing right (${\sigma ^ + }$) and left (${\sigma ^ - }$) circularly-polarized photons as following,

Each element in this basis, i.e., $L{G_{0,4}}{\sigma ^ + }$, can be regarded as one pole of the fourth-order spin-orbit hypersphere and therefore it has ten poles. All the eigenstates of this system can find their own position on this hypersphere and are written as $\mathit{\Psi} = \vec{\mathit{\Psi} } \cdot {\vec{\sigma }_{LG}}$ with $\vec{\mathit{\Psi} } = ({{\mathit{\Psi} _1},{\mathit{\Psi} _2},{\mathit{\Psi} _3},{\mathit{\Psi} _4},{\mathit{\Psi} _5},{\mathit{\Psi} _6},{\mathit{\Psi} _7},{\mathit{\Psi} _8},{\mathit{\Psi} _9},{\mathit{\Psi} _{10}}} )$. Obviously, these high-dimensional hyperspheres cannot be graphically visualized. Nevertheless, since each eigenstate created by the TE-TM splitting consists of only two LG components, the hypersphere can be decomposed into a series of two-poled spheres in three-dimensional space, as exampled in Fig. 3, which are straightforward for geometrical visualization. It should be noted that the $n = 4\; $ hypersphere can be decomposed into a total of $C_{10}^2 = 45$ Centional spheres, and Fig. 3 shows only two of them as an example.

 

Fig. 3. Examples of higher-order Poincare spheres presenting the relation of the $n = 4$ manifold. (a) and (b) correspond to two different choices of LG basis components. The left panels display the spheres and the right panels represent the polarization textures of some of the points at the equators of the spheres.${P_1}$ and ${P_3}$ in both (a) and (b) are the skyrmion-like eigenstates formed by TE-TM splitting.

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3. Conclusions

In summary, we derived the eigenenergies and eigenstates of photons confined in a circularly-symmetric potential under the TE-TM SO coupling effect, and revealed the universal rules of the mechanism of eigenmode generation. Our results unambiguously demonstrate that the high-order eigenstates of confined vector light under SO coupling effect, which is conventionally considered as a very complicated electromagnetically problem, can be analytically solved by the perturbation theory in quantum mechanics, and show well-organized rules up to the order of infinity. It is thereby interesting and useful to further apply such principle to other polarization-associated systems for which analytical solution by using electromagnetics is difficult to obtain and the numerical solutions cannot completely correspond to the symmetry of the ideal system. The skyrmion-like and vector vortex states are promising to be experimentally realized in concave-planar and micropillar cavities, leading to skyrmion-like lasers and single photon emission. Finally, the strict mathematical proof of the eigenstates up to the order of infinity becomes a pure mathematical problem, which may bring remarkable insights to the area of mathematics dealing with eigenvalue issues in real physical systems.