Nano-thick SiO2 channel for subwavelength plasmonic orbital angular momentum mode transmission

Zhishen Zhang; Xiaobo Heng; Shuai Gao; Li Zhang; Fei Lin; Weicheng Chen; Jiulin Gan

doi:10.3788/COL202523.013601

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Abstract

Orbital angular momentum (OAM) waveguides are critical for multi-channel photonic-on-chip applications. However, current large-mode-area waveguides pose a challenge for OAM device miniaturization. Here, a novel hybrid plasmonic waveguide is theoretically proposed to decrease the OAM mode area by two orders of magnitude. Benefiting from the Ag-As₂S₃-SiO₂-As₂S₃-Ag five-layer cylindrical structure, the guided OAM mode realizes a larger figure of merit of 88. Based on this waveguide, an OAM coupler with a record-breaking small footprint (0.68 µm × 5.7 µm) is designed. The proposed waveguide enables subwavelength OAM light transmission, which provides a key building block for high-density OAM photonics circuits.

Keywords

directional coupler hybrid plasmonic waveguide orbital angular momentum waveguide

1. Introduction

The nano-waveguide with non-diffraction-limited light transport has become increasingly critical for applications in many fields, such as chip-scale optical communication^[1], trace biochemical sensors^[2], and scalable quantum computation^[3]. However, a dielectric waveguide with a subwavelength diameter cannot achieve the mode confinement below the diffraction limit since the fraction of evanescent fields dramatically increases with the decreasing waveguide diameter^[4]. For constraining the light field into a smaller volume, both metallic waveguides^[5,6] and hybrid waveguides^[6,7] have been proposed. At the metal–insulator interfaces, the light is strongly coupled with the free electrons of the metal and excites the surface plasmon wave, which highly confines the light in nanoscale. Compared with all-metallic waveguides, the dielectric-metal hybrid waveguide^[6,7] tailors the penetration of surface plasmon modes in the dielectric layer and the metal layer, which reduces the ohmic loss and still confines light at smaller scales. Various kinds of nano-waveguides, such as metallic nanowires^[8] and grooves^[9], hybrid plasmonic waveguides^[10,11], and graphene-loaded structures ^[12], have been theoretically designed and experimentally demonstrated with the ability to confine light below the diffraction limit. However, the above waveguides are only applicable for dealing with the fundamental mode.

In comparison to the fundamental mode, orbital angular momentum (OAM) modes^[13] with a helical wavefront and doughnut spot have experienced significant growth in recent years due to their spatial orthogonality, which offers an additional degree of freedom for optical multiplexing^[14–17], multi-parameter sensing^[18], and spin–orbit interaction^[19]. However, OAM waveguides below the diffraction limit are still challenging, which seriously hamper the development of on-chip OAM modulations and communications^[20,21]. Currently, only a few strongly confining OAM waveguides have been proposed. In dielectric approaches, cylindrical microfibers^[22], rectangular waveguides^[23,24], and subwavelength-hole waveguides^[25] were designed for guiding OAM modes with negligible loss, and all achieved a mode area at the level of $λ^{2}$ . As the alternative metallic approaches, a plasmonic nanohole waveguide^[26] was proposed for propagating OAM modes with a loss of 2.73 dB/µm and mode area of $0.44 λ^{2}$ . The plasmonic helical nanowire^[27,28] was designed to confine OAM modes into the area of $0.03 λ^{2}$ . Further confining the OAM mode into nanoscale is much more challenging. For example, the OAM mode in a tapered metal waveguide^[29] can be focused into a nanotip but is rapidly scattered to the far-field. The main challenges arise from three aspects. First, because OAM modes are high-order modes, the size of the OAM waveguide is inherently larger than the size of the corresponding single-mode waveguide. Second, the annular OAM beam has higher stability in a circularly symmetric waveguide, which means the extremely sharp microstructural features in the most single-mode nano-waveguides cannot be used for reference. Third, a nanoscale OAM mode should be located in the dielectric region for achieving low loss, so a hybrid plasmonic waveguide needs to be designed with the optimal combination pattern of both dielectric layers and metal layers. Up to now, there is still no reported OAM waveguide synchronously achieving a nanoscale mode area and a micro-scale propagation range.

In this work, we propose a novel OAM nano-waveguide with a mode area of $0.0005 λ^{2}$ , which corresponds to decreasing the mode area by 2 orders of magnitude. Enlightened by the mode manipulation techniques in multi-cladding fibers, the light will be highly confined into a tiny dielectric region, which is surrounded by a high-index dielectric layer and a metal layer. The designed OAM waveguide has an ultrathin ${SiO}_{2}$ cylindrical layer acting as the OAM guiding channel, which is clamped at the middle of two ${As}_{2} S_{3}$ layers and two Ag layers. The OAM mode is more strongly confined in the ${SiO}_{2}$ layer with a relatively long propagation length (2 $λ$ to 11 $λ$ ) and a larger figure of merit ( $FoM = 88$ ). Based on the subwavelength OAM waveguide, a plasmonic OAM waveguide coupler is demonstrated with a record-breaking small footprint $0.68 μm \times 5.7 μm$ . Due to the efficient constraint mechanism of light fields, two tightly arranged waveguides had a low crosstalk of $- 28.14 dB$ . Our plasmonic OAM nano-waveguide holds great potential as a component for the on-chip OAM communications.

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2. Theoretic Analysis

The three-dimensional (3D) and cross-sectional schematic configurations of our proposed OAM nano-waveguide are shown in Figs. 1(a) and 1(b). The OAM nano-waveguide consists of a low-index dielectric ring-core, two high index dielectric claddings, and two metal claddings. The radius of the $i$ th layer (from inside to outside) is $r_{i}$ . It is worth noting that the evanescent field of the subwavelength dielectric waveguide is mostly located in the low-index cladding, instead of the high-index core. For example, the 216 nm diameter silicon wire^[30] at 1.5 µm wavelength only confines 10% energy inside the core and 90% energy in the air. Therefore, building the composite structure of the dielectric layer offers a new approach to control the mode distribution of a hybrid waveguide. For example, the light will be confined at the metal–dielectric interface due to the localization effect of the surface plasmon wave. By setting a low-index dielectric nano-structure near the metal–dielectric interface, most energy of the light will be compressed into this nano-structure. In our designed nano-waveguide, the guided modes are tightly restricted in the ultrathin low-index dielectric ring-core channel, which is sandwiched between two high-index dielectric layers and two metal claddings. Because most energy is located in the middle low-index dielectric layer, the mode propagation loss has a significant decrease, which enables a longer propagation range. Without loss of generality, we performed a proof-of-concept study in which ${SiO}_{2}$ , ${As}_{2} S_{3}$ , and Ag are chosen as the constructional materials of the dielectric ring-core, dielectric claddings, and metal claddings, respectively.

Figure 1.(a) Schematic diagram of the plasmonic OAM waveguide; (b) cross-section of the OAM waveguide; (c) energy percentage of the radial polarization component E_r of the surface plasmon mode SP₁₁. Inset is the intensity distribution of the SP₁₁ mode.

In order to investigate the deep regulation of OAM mode below the diffraction limit, the guiding properties have been studied by using the finite element method at the telecommunication wavelength of 1550 nm. The model equation is $\nabla \times (\nabla \times E) - k_{0}^{2} ε_{r} E = 0$ , where $E$ is the electric field, $k_{0}$ is the wave vector in vacuum, and $ε_{r}$ represents the permittivity distributions of simulation model. The perfectly matched boundary conditions are used. The inset of Fig. 1(c) shows the electrical field distributions of the surface plasmon mode ${SP}_{11}$ in the proposed OAM nano-waveguide with $r_{1} = 130 nm$ , $r_{2} - r_{1} = 10 nm$ , $r_{3} - r_{2} = 5 nm$ , $r_{4} - r_{3} = 10 nm$ , and $r_{5} - r_{4} = 50 nm$ . This result is consistent with the design expectation, that is, most of the mode energy is located into the ${SiO}_{2}$ channel. Less energy is distributed in two adjacent ${As}_{2} S_{3}$ layers, and no energy is diffused into the Ag layers. The ${SP}_{11}$ mode is radially polarized because the electric field component parallel to the metal–dielectric interface incurs great transmission loss. As shown in Fig. 1(c), when $r_{2} - r_{1} \leq 50 nm$ , the energy percentage of the radial polarization component $E_{r}$ of the ${SP}_{11}$ mode is always higher than 95% in the nano-waveguides with different $r_{1}$ . Higher-order SP modes also have similar radial polarization state in this cylindrical hybrid plasmonic waveguide.

For synthesizing the OAM modes in the proposed nano-waveguide, the expression of $E_{r}$ of the $S P$ modes is analytically derived. The longitudinal electromagnetic fields ( $E_{z}$ and $H_{z}$ ) satisfy the following equations in a cylindrical coordinate system^[30]: $\frac{\partial^{2} Ψ}{\partial r^{2}} + \frac{1}{r} \frac{\partial Ψ}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} Ψ}{\partial φ^{2}} + {k^{2} {[n (r)]}^{2} - β^{2}} Ψ = 0,$ (1)where $Ψ$ , $k$ , $n (r)$ , and $β$ are the longitudinal electromagnetic fields ( $E_{z}$ or $H_{z}$ ) of the SP modes, wave-number, refractive index profile, and propagation constant, respectively. Based on the variable separation method, $E_{z}$ and $H_{z}$ are expressed by $E_{z} = A F_{l m} (r) f (l φ),$ (2) $H_{z} = C F_{l m} (r) g (l φ),$ (3)where $A$ and $C$ are the undetermined coefficients. $F_{l m} (r)$ is the radial dependence of the field profile. $l$ and $m$ are the azimuthal and radial mode orders. Due to the cylindrical symmetry, $f (l φ)$ can be the even and odd functions of the azimuth $φ$ , which represent the even and odd modes, respectively. Accordingly, $g (l φ)$ will be the odd and even functions of the azimuth $φ$ , $f (l φ) = {\begin{matrix} \cos (l φ) even modes \\ \sin (l φ) odd modes \end{matrix},$ (4) $g (l φ) = {\begin{matrix} - \sin (l φ) even modes \\ \cos (l φ) odd modes \end{matrix} .$ (5)

The transverse electromagnetic field ( $E_{r}$ ) can be obtained by $E_{z}$ and $H_{z}$ ^[16], $E_{r} = - \frac{j}{k^{2} {[n (r)]}^{2} - β^{2}} (β \frac{\partial E_{z}}{\partial r} + \frac{ω μ_{0}}{r} \frac{\partial H_{z}}{\partial φ}) .$ (6)

By substituting Eqs. (2) and (3) into Eq. (6), the $E_{r}$ expressions of the even and odd ${SP}_{l m}$ modes are $E_{r}^{e} = - \frac{j}{k^{2} {[n (r)]}^{2} - β^{2}} G_{l m} (r) \cdot \cos (l φ),$ (7) $E_{r}^{o} = - \frac{j}{k^{2} {[n (r)]}^{2} - β^{2}} G_{l m} (r) \cdot \sin (l φ),$ (8)where $G_{l m} (r) = β A \frac{\partial F_{l m} (r)}{\partial r} - \frac{ω μ_{0}}{r} C l F_{l m} (r) .$ (9)

By superimposing Eqs. (7) and (8), the helical phase $\exp (\pm j l φ)$ is obtained, $E_{r}^{e} \pm j \cdot E_{r}^{o} = - \frac{j}{k^{2} {[n (r)]}^{2} - β^{2}} G_{l m} (r) \cdot \exp (\pm j l φ) .$ (10)

Therefore, the OAM plasmonic mode synthetic formula is ${SP}_{\pm l, m}^{OAM} = {SP}_{l m}^{e} \pm j \cdot {SP}_{l m}^{o} .$ (11)

It is worth emphasizing that the synthetic formulas of the radially polarized OAM SP modes are different from the OAM modes in ring fiber. In the cylindrical hybrid plasmonic waveguide, the ${SP}_{11}$ mode can achieve the first-order OAM mode, and high-order SP modes ( ${SP}_{l m}$ ) can achieve OAM modes with the topological charge $\pm l$ . Figure 2 shows the intensity and phase distributions of ${SP}_{\pm 1, 1}^{OAM}$ mode, which are obtained by ${SP}_{11}^{e} \pm j \cdot {SP}_{11}^{o}$ . The uniform distributions of the doughnut-shaped intensity profile and the angular gradient phase indicate that the purity of the synthesized ${SP}_{+ 1, 1}^{OAM}$ mode is nearly 100%. The ${SP}_{- 1, 1}^{OAM}$ mode has similar results, as shown in Fig. 2. The modal effective index difference between the ${SP}_{11}^{e}$ and ${SP}_{11}^{o}$ modes is $Δ n_{eff} = 1 \times 10^{- 7}$ , where the related $2 π$ walk-off length is $L_{2 π} = λ / Δ n_{eff} = 15.5 m$ . The ${SP}_{\pm 1, 1}^{OAM}$ mode keeps stable during its propagation length of tens of micrometers.

Figure 2.Intensity and phase distributions of SP₁₁^e, SP₁₁^o, SP_+1,1^OAM, and SP_−1,1^OAM modes.

3. Results

To further understand the ${SP}_{+ 1, 1}^{OAM}$ mode in the proposed nano-waveguide, the effect of the waveguide size on the modal properties of ${SP}_{+ 1, 1}^{OAM}$ mode is investigated. In Fig. 3(a), the modal effective index $n_{eff}$ of ${SP}_{+ 1, 1}^{OAM}$ mode increases with the increase of the radii of inner Ag cladding ( $r_{1}$ ), which is consistent with the modal effective index of the fundamental mode in all-dielectric waveguides. When increasing the thickness of the ${As}_{2} S_{3}$ layer, the modal effective index decreases, which means the mode is less affected by the Ag cladding and becomes more concentrated in the dielectric layer.

Figure 3.Mode properties of SP_+1,1^OAM mode in the nano-waveguides with r₃ − r₂ = 5 nm, r₄ − r₃ = r₂ − r₁, and r₅ − r₄ = 50 nm. (a) Modal effective index; (b) normalized mode area; (c) propagation length; (d) FoM. The inset is the cross-section of the OAM waveguide, where the thickness of the As₂S₃ layer is denoted as r₂ − r₁.

The spatial constraint ability of ${SP}_{+ 1, 1}^{OAM}$ mode is evaluated by the normalized mode area $A_{norm}$ , which is defined as the ratio of the effective mode area $A_{eff}$ to the diffraction-limited area $A_{0}$ , $A_{norm} = A_{eff} / A_{0},$ (12) $A_{eff} = \iint P (r) d A / Max [P (r)],$ (13) $A_{0} = λ^{2} / 4,$ (14)where $P (r)$ is the energy flux density and $λ$ is the telecommunication wavelength of 1550 nm. As shown in Fig. 3(b), the ${SP}_{+ 1, 1}^{OAM}$ mode can be confined at a very small area of $0.002 A_{0} = 0.0005 λ^{2}$ , outperforming many other OAM waveguides. Due to the tiny mode area, the modal properties of the proposed OAM nano-waveguide will be sensitive to environmental variations. On one hand, the stability of the OAM waveguide and the associated photonic devices will be affected, but on the other hand, the waveguide has potential for achieving ultra-high sensitivity sensors. Actually, such strongly constrained ${SP}_{+ 1, 1}^{OAM}$ modes still suffer from optical loss because a small amount of energy locates at the Ag interface. The mode propagation loss is about 0.62–1.15 dB/µm, which is less than half that in existing OAM plasmonic waveguides^[23]. The propagation length of the proposed nano-waveguide is defined as $L = λ / [4 π \cdot Im (n_{eff})] .$ (15)

As shown in Fig. 3(c), the propagation length $L$ can increase from 3 µm to 17.5 µm ( $2 λ$ to $11 λ$ ) when the thickness of the ${As}_{2} S_{3}$ layer increases from 2 nm to 50 nm.

Since the mode localization and attenuation show the opposite trend, the FoM is used as an intuitive parameter for evaluating the guiding performances of waveguides. The FoM is defined as the ratio of propagation length to the diameter of the mode area, $FoM = \frac{L}{2} \cdot \sqrt{\frac{π}{A_{eff}}} .$ (16)

In Fig. 3(d), FoM has a non-monotonic behavior and reaches its maximum 81 with $r_{1} = 90 nm$ , $r_{3} - r_{2} = 5 nm$ , and $r_{2} - r_{1} = r_{4} - r_{3} = r_{5} - r_{4} = 50 nm$ .

Next, the effects of the thickness of the ${SiO}_{2}$ layer ( $r_{3} - r_{2}$ ) and outer ${As}_{2} S_{3}$ layer ( $r_{4} - r_{3}$ ) on the ${SP}_{+ 1, 1}^{OAM}$ mode’s performances are explored. Because most mode energy is concentrated in the ${SiO}_{2}$ layer, when decreasing the ${SiO}_{2}$ layer thickness, the normalized mode area and the propagation length of the ${SP}_{+ 1, 1}^{OAM}$ mode both decrease [Fig. 4(a)]. The FoMs of the nano-waveguides can exceed 70 [Fig. 4(b)] with different thicknesses of the ${SiO}_{2}$ layer. When increasing the outer ${As}_{2} S_{3}$ layer thickness, the mode energy located in the outer ${As}_{2} S_{3}$ layer increases, which increases the normalized mode area of ${SP}_{+ 1, 1}^{OAM}$ mode [Fig. 4(c)]. At the same time, the ${SP}_{+ 1, 1}^{OAM}$ mode will be less affected by the outer Ag cladding, which greatly decreases mode loss and achieves an FoM of 88 [Fig. 4(d)].

Figure 4.(a) Normalized mode area and (b) FoM of SP_+1,1^OAM modes in the nano-waveguides with r₄ − r₃ = r₂ − r₁ = 10 nm and r₅ − r₄ = 50 nm. Inset in (b) denotes the thickness of SiO₂ layer as r₃ − r₂. (c) Normalized mode area and (d) FoM of SP_+1,1^OAM modes in the nano-waveguides with r₂ − r₁ = 10 nm, r₃ − r₂ = 5 nm, and r₅ − r₄ = 50 nm. Inset in (d) denotes the thickness of the outer As₂S₃ layer as r₄ − r₃.

The FoMs of the proposed nano-waveguide are further compared with three kinds of OAM cylindrical plasmonic waveguides: the Ag nanowire (Model 1), the Ag nanotube (Model 2), and hybrid plasmonic waveguides with the dielectric ring-core channel (Model 3). As shown in Fig. 5, under conditions of the same radii of inner Ag layer, the ${SP}_{+ 1, 1}^{OAM}$ mode in our proposed waveguide has the largest FoM, which is about 4 times the FoM in Model 1 and 2 times the FoM in Model 3. In Model 2, the plasmonic OAM mode can only be propagated in the waveguide with the ${As}_{2} S_{3}$ core radii $r_{1} \geq 180 nm$ , which has disadvantage in waveguide miniaturization. These results illustrate that our proposed OAM nano-waveguide is highly promising for achieving the subwavelength OAM mode guiding and exhibits great potential in future on-chip OAM devices.

Figure 5.FoM of the SP_+1,1^OAM mode in four kinds of the nano-waveguides: the Ag nanowire (Model 1), the Ag nanotube (Model 2), the ring-core hybrid waveguide (Model 3), and this work.

The OAM waveguide coupler within a small footprint is highly desired for chip-scale OAM applications. Based on the proposed subwavelength waveguide, the compact OAM coupler is composed of two parallel identical hybrid plasmonic waveguides, where the Ag claddings are fused together with a gap between two annular ${As}_{2} S_{3}$ layers. Without losing generality, the coupling characteristics of ${SP}_{+ 1, 1}^{OAM}$ mode are investigated in the plasmonic coupler with $r_{1} = 100 nm$ , $r_{2} - r_{1} = 10 nm$ , $r_{3} - r_{2} = 5 nm$ , $r_{4} - r_{3} = 10 nm$ , and $r_{5} - r_{4} = 50 nm$ . The mode match method^[31,32] is used for calculating the lossy high-order mode propagation through the coupling region. The key is that the two parallel waveguides are treated as a new uniform waveguide [Fig. 6(a)] and the coupling process is considered as the interference of the guide modes, which is decided by the overlapping coefficient of this mode and the input electrical field at the starting point of the coupling region, i.e., the injecting mode in the single input waveguide [Fig. 1(a)]. The coupling efficiencies of modes in the output waveguide are calculated at the end of the coupling region, which is similarly decided by the overlapping coefficients of the interference results and each mode in the single output waveguide.

Figure 6.(a) Schematic diagram of the plasmonic OAM waveguide coupler. (b) Propagating electric field of the SP_+1,1^OAM mode coupler, where the white arrows represent the direction of optical energy flow. The upper dashed block diagram is the electric field and phase distribution of input SP_+1,1^OAM mode. The lower dashed block diagram is the electric field and phase distribution of coupling mode with the maximum coupling efficiency. (c) Coupling efficiencies of different modes in the SP_+1,1^OAM mode coupler. (d) Maximum coupling efficiency and corresponding coupling length of the SP_+1,1^OAM mode coupler with different center-to-center distances.

Figure 6(b) shows the propagating electric field of the OAM plasmonic coupler with the center-to-center distance $d = 270 nm$ . When injecting the annular ${SP}_{+ 1, 1}^{OAM}$ mode, the cross-sectional electric fields are two parallel trajectories with reciprocating optical energy flow. The optimal coupling length is only 5.7 µm ( $L = 3.68 λ$ ). As shown in the lower dashed block diagram of Fig. 6(b), the coupled mode in the adjacent waveguide has a counterclockwise spiral phase, which means that the input ${SP}_{+ 1, 1}^{OAM}$ mode has been coupled into the ${SP}_{- 1, 1}^{OAM}$ mode. The reason is that both ${SP}_{+ 1, 1}^{OAM}$ and ${SP}_{- 1, 1}^{OAM}$ modes are the superposition of the ${SP}_{11}^{e}$ and ${SP}_{11}^{o}$ modes, but with a different phase difference. Due to the form birefringence of the coupling region, the ${SP}_{11}^{e}$ and ${SP}_{11}^{o}$ modes have different propagation constants, which induces an additional phase difference of $π$ for turning the ${SP}_{+ 1, 1}^{OAM}$ mode into the ${SP}_{- 1, 1}^{OAM}$ mode. As shown in Fig. 6(c), the peaks of the coupling efficiency curves decrease exponentially with the increase of coupling length $z$ because of the transmission loss characteristics of the hybrid plasmonic waveguides. When the coupling length is 5.7 µm, the adjacent waveguide has a maximum coupling efficiency 57.37%. It is worth noting that, although the fundamental mode ( ${SP}_{01}$ mode) is excited due to the strong coupling effect, the mode purity of ${SP}_{- 1, 1}^{OAM}$ mode in the adjacent waveguide is still up to 95.22% by controlling the center-to-center distance. The presented design of the OAM plasmonic coupler has achieved an extremely small footprint of $0.68 μm \times 5.7 μm$ , which shows an improvement of over 2 orders of magnitude compared to traditional OAM devices, such as the OAM microfiber coupler (10 µm × 25,000 µm)^[31] and the OAM waveguide generator ( $4 μm \times 100 μm$ )^[33]. It is also noteworthy that the proposed coupler can be further expanded to the mode-selective coupler, which is widely used for efficiently generating the OAM mode from the fundamental mode, and converting the light to other OAM modes in waveguides.

The crosstalk between two adjacent OAM plasmonic waveguides is investigated in Fig. 6(d). When increasing the center-to-center distance $d$ , the coupling effect is weakened, and the critical coupling length is increased. Considering that the plasmonic waveguides are highly lossy, the maximum coupling efficiency has decreased dramatically. For tightly arranged waveguides with $d = 350 nm$ , the maximum coupling efficiency is only 0.15%, and the crosstalk is $- 28.14 dB$ . This result confirms that the proposed OAM nano-waveguide can be further applied to achieve high-density photonic circuits.

4. Discussion

The proposed nano-waveguide can be fabricated as follows. The silver nanowires can be synthesized via a two-step polyol process^[34]. The ${As}_{2} S_{3}$ and ${SiO}_{2}$ layers can be prepared via the Stober method^[34]. By changing the concentration of the As and Si ion solution, the dielectric layer thickness can be controlled on a nanoscale. The outer Ag layer can be prepared by an ion sputtering process. Additionally, the key to the proposed nano-waveguide is the metal–dielectric–metal structure, which means that the composite materials can be easily replaced according to the realizable preparation conditions and practical applications.

Considering the possible fabrication imperfections, such as the offset and elliptical transformation of the ${SiO}_{2}$ channel in OAM waveguide, the performances of ${SP}_{\pm 1, 1}^{OAM}$ mode in 10 µm waveguides [parameters as shown in the inset of Fig. 1(c)] have been studied. The modes still have doughnut-shaped intensity profiles with high FoMs (68 to 80). However, an additional phase $Δ Φ$ between the ${SP}_{11}^{e}$ and ${SP}_{11}^{o}$ modes will be induced due to the form birefringence. By taking $Δ Φ \leq 0.05 π$ as the criterion, the tolerances of the ${SiO}_{2}$ channel are the offset $Δ x \leq 9 nm$ and the elliptical axis difference $- 2 nm \leq Δ r \leq 2 nm$ .

The proposed waveguide can support higher-order OAM mode by simply increasing its size. For example, the OAM nano-waveguide ( $r_{1} = 150 nm$ , $r_{2} - r_{1} = 10 nm$ , $r_{3} - r_{2} = 5 nm$ , $r_{4} - r_{3} = 10 nm$ , and $r_{5} - r_{4} = 50 nm$ ) supports OAM modes with $l = \pm 2$ , as shown in Fig. 7. This ${SP}_{+ 2, 1}^{OAM}$ mode has a mode area of $0.012 λ^{2}$ and a propagation length of $2.43 λ$ . The parameter sweep analysis method will further improve waveguide performances. It is also worth noting that the proposed waveguide also keeps the small mode area of ${SP}_{\pm 1, 1}^{OAM}$ modes and $FoM \geq 60$ in a broader range of wavelengths (from 900 to 2300 nm). Due to the tiny mode area, the proposed OAM nano-waveguide has a larger nonlinearity, which has potential for constructing the nonlinear device and generating the OAM mode with a new frequency.

Figure 7.Intensity and phase distributions of SP₂₁^e, SP₂₁^o, SP_+2,1^OAM, and SP_−2,1^OAM modes in the waveguide with r₁ = 150 nm.

In conclusion, we have proposed a novel cylindrical hybrid plasmonic waveguide for propagating the OAM mode beyond the diffraction limits. The OAM nano-waveguide consists of a ${SiO}_{2}$ ring-core, two ${As}_{2} S_{3}$ claddings, and two Ag claddings. The plasmonic OAM modes are tightly restricted in the ${SiO}_{2}$ ring-core channel and simultaneously realize the radial polarization state, a small mode area ( $0.0005 λ^{2}$ ), and a long propagation range ( $2 λ$ to $11 λ$ ). Compared with traditional OAM plasmonic waveguides, the ${SP}_{\pm 1, 1}^{OAM}$ mode in the proposed waveguide has a larger FoM of 88. Based on this waveguide, a directional coupler with an extremely small footprint ( $0.68 μm \times 5.7 μm$ ) is designed, which has a maximum coupling efficiency of 57.37%. Due to the strong mode confinement, the crosstalk between two tightly arranged waveguides is down to $- 28.14 dB$ . The waveguide also can realize a transmission of the higher-order OAM mode in a broadband wavelength range. These results illustrate that our proposed OAM nano-waveguide is highly promising for achieving the high-performance of OAM mode guiding and exhibits great potential in future on-chip OAM devices.