1. Introduction

Angular momentum can be divided into spin angular momentum (SAM) and orbital angular momentum (OAM) in paraxial beams, which are related to circular polarization and spatial distribution, respectively. In recent years, light beams carrying OAM (i.e. OAM modes), also referred to as vortex beams, have attracted more and more attention owing to their distinct characteristics such as the phase singularity at the beam center and the resultant doughnut shape intensity profiles [1, 2]. OAM modes with different topological charge values $l$, are theoretically unbounded and orthogonal to each other. Similar to other mode bases in free space or fiber, OAM modes are another mode basis with which to represent spatial modes. Different mode bases including OAM modes can be employed in mode-division multiplexing (MDM), which is a subset of space-division multiplexing (SDM). Very recently, OAM modes have shown great potential for MDM both in free space and fiber-based optical communications [3–8]. Vortex beams or OAM modes also provide unique opportunities for manipulation of micro-/nano-particles as they assert torque in addition to forces related to optical intensity gradient, giving rise to orbital and spin movements beyond trapping. Remarkably, for diverse applications of vortex beams in optical communications [3–8], optical tweezers [9], optical trapping [10] and quantum information technology [11], the successful generation of OAM modes is of great importance.

Driven by their distinctive properties and miscellaneous applications, there have been many attempts to generate and manipulate OAM beams. Such attempts include cylindrical lens mode converters, spiral phase plates, q-plates, and spatial light modulators (SLM) [12–15]. Recently, integrated approaches based on fibers or silicon devices are developed due to their outstanding features of small footprint, high speed, low-cost, and adaptation for various applications [16–25]. In 2011, a nanoscale V-shaped antenna meta-surface structure was proposed to generate single OAM beam by adjusting the wavefront parameters of scattered fields [16]. From then on, varieties of optical antenna, including metallic and dielectric metasurface arrays, have been used to generate OAM beams [17–24]. In [25], Cai et al. proposed and demonstrated an OAM beam emitter based on a silicon microring with azimuthally distributed second-order gratings. The operation principle of microring grating is based on the scattering fields, which are generated out-plane and propagated in free space. In addition, it also attracts much attention to fully explore and employ the unique feature of OAM beams in the region of guided optics [26, 27]. However, the generation of OAM beams in-plane on a silicon chip is still challenging and relevant works have not yet been reported much, as the conventional optical waveguide cannot support OAM modes.

In this paper, we present a scheme to generate OAM modes on silicon-on-insulator (SOI) chip by using a single-trench silicon waveguide. Trench structures have been previously employed to enable polarization rotator and mode converter [28–33]. A trench silicon waveguide can support two orthogonal LP-like modes whose optical axes are rotated by around 45° with respect to the horizontal and vertical directions. By combining two eigenmodes with different phase differences, one can selectively convert an input LP-like mode to OAM _{± 1} modes at the output terminal of the single-trench waveguide. The proposed trench structure for x- and y-polarizations are optimized, and high mode purity can be achieved over a wide wavelength range from 1.45 µm to 1.65 µm. As one typical example of applications, an on-chip OAM modes (de)multiplexer (OAM₀, OAM _{± 1}) for x-polarization is designed based on the tailored trench structure. Such an on-chip OAM modes (de)multiplexer has a footprint of <100 µm × 4 µm, and the average mode extinction ratio exceeds 20 dB over a wavelength range from 1.52 µm to 1.58 µm.

2. Concept and operation principle

The schematic three-dimensional (3D) structure and the cross-section of a typical single-trench waveguide for OAM generation are shown in Figs. 1(a) and 1(b). In order to generate OAM modes, the waveguide is designed with a trench, which can break the original rotation symmetry and split the mode degeneracy. As indicated in Figs. 1(a) and 1(c), the second-order mode (TE₀₁) in the typical rectangular waveguide can excite two orthogonal LP-like eigenmodes with different propagation constants (${\beta }_1$ and ${\beta }_2$) in the single-trench waveguide. The two LP-like eigenmodes can be further synthesized into OAM modes with topological charges of + 1 or −1 through different propagation distances, i.e. $L=\pi /2({\beta }_1-{\beta }_2)$ or $L=3\pi /2({\beta }_1-{\beta }_2)$. The intensity and phase evolutions of the combination of eigenmodes are illustrated in Fig. 1(c) and 1(d). When the phase difference between two LP-like eigenmodes is $\pi /2$ and $3\pi /2$, the second-order mode (TE₀₁) in the rectangular waveguide can be converted to OAM _{± 1} modes, respectively, at the output of the trench waveguide. Moreover, it could be also a mode rotator with a phase difference$\pi$ as displayed in [33]. In contrast, when the two LP-like eigenmodes are in phase (0 or$2\pi$), the output is still a second-order mode (TE₀₁). In this paper, we assume that the proposed OAM mode generator is based on a silicon chip with SiO₂ cladding and the refractive index for the SiO₂ cladding layer is set to be 1.445.

 

Fig. 1 (a) Concept and principle of OAM beam generator based on a single-trench waveguide. (b) Cross-section of a single-trench waveguide. (c) The field distributions of two eigenmodes of a single-trench waveguide. (d) Intensity and phase evolutions of combination of eigenmodes for x-polarization.

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It is noted here that the dominant component of the electric field for TE mode is along x axis (or horizontal axis), while it is along y axis (or vertical axis) for TM mode. The polarization of the emitted OAM mode depends on the polarization of the input mode. Moreover, when the height and the width are equal (W = H), the second-order modes are hybridized and replaced by modes with spatially variant polarization. Therefore, to obtain a clear two-lobe pattern in the input waveguide of the OAM mode generator, the inherent symmetry of the square cross-section of the waveguide needs to be broken. Here, we choose W = 1.1 µm, H = 1 µm for x-polarization (TE mode) and W = 0.9 µm, H = 1 µm for y-polarization (TM mode).

3. Trench waveguide structure design and characterization

First, the dimensions of the waveguide in Fig. 1 should be such that the relevant high-order modes are supported. Next, the trench parameters would be required for the equal excitation of two orthogonal eigenmodes. Two parameters related to the trench such as the width w and the height h would be specially optimized. When the size of the trench is larger, the difference of the propagation constants of two eigenmodes also tend to be larger, leading to a shorter device length. However, crosstalk to undesired modes and undesired back reflections may become higher due to the mode field mismatch at the boundary between the waveguides with and without the trench. Meanwhile, the generated OAM modes may become nonuniform. Thus, w and h should be chosen to relatively small values in order to suppress the crosstalk to undesired modes. At the same time, the purities of the generated modes should be also considered in the design. To obtain more uniform spatial light distribution, the overlap integrals of two eigenmodes with input second-order mode should be both equal to 0.5. Figure 2(a) and 2(b) show w, h dependences of the normalized overlap integrals of two LP-like eigenmodes with the input second-order mode at a wavelength of 1.55 µm, where W = 1.1 µm, H = 1 µm for x-polarization. For y-polarization, the similar curves of w, h dependences are presented in Figs. 2(c) and 2(d).

 

Fig. 2 (a) Trench width w, and (b) height h dependences of normalized overlap integrals of two LP-like eigenmodes for x-polarization at 1.55 µm. (c) and (d) show width w, and height h dependences of normalized overlap integral of two LP-like eigenmodes for y-polarization at 1.55 µm, respectively.

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As shown in Fig. 2(a) and 2(b), when the trench width w and height h for x-polarization are chosen to be 0.34 µm and 0.12 µm, the overlap integrals for two eigenmodes with the input mode are close to 0.5 and 0.5, respectively. In such case, single-trench waveguide can support two approximately orthogonal LP-like modes whose optical axes are rotated by around ${45}^{\circ }$ with respect to the horizontal and vertical direction. When the structure parameters mentioned above are selected, two eigenmodes of single-trench waveguide in Fig. 1(c) will be excited equally and the generated OAM modes can possess high purity. Similarly, w = 0.11 µm and h = 0.31 µm are chosen for y-polarization in Figs. 2(c) and 2(d). Thus, two LP-like eigenmodes would be approximately orthogonal and excited equally.

By using the Finite Difference Eigenmode (FDE) method, the effective indices of two orthogonal LP-like modes of single-trench waveguide are depicted as a function of wavelength for x- and y-polarization, respectively. The effective indices are numerically calculated over a wide range of wavelength from 1.45 µm to 1.65 µm. In Fig. 3(a), the effective indices decrease gradually with the increase of wavelength for both polarizations. As shown in Fig. 3(b), the differences of effective indices increase with the wavelength from 1.45 µm to 1.65 µm for both polarizations. At the wavelength of 1.55 µm, the differences of the effective indices are 0.025 and 0.031 for x- and y-polarization, respectively. We first simulate the single-trench waveguide by input a second-order mode as shown in Fig. 1(a), whose lengths corresponding to the generation of OAM _{± 1} modes can be determined by the effective index differences,

 

Fig. 3 (a) The effective indices of orthogonal LP-like modes of single-trench waveguide as a function of wavelength for both x- and y-polarizations. (b) The effective index differences of orthogonal LP-like modes of single-trench waveguide as a function of wavelength for both polarizations.

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By using the 3-D finite-difference time-domain (FDTD) method, the proposed single-trench waveguide structure in Fig. 1(a) is simulated. The near fields of OAM_-1 modes for x-polarization and OAM₊₁ modes for y-polarization are both monitored and analyzed in Fig. 4. When the trench is located in one side of the silicon waveguide, OAM₊₁ mode can be generated at the end of the etched waveguide. On the contrary, OAM_-1 mode can be emitted with the trench in the other side of the silicon waveguide. That is to say, two ways (the position of trench, and the length of trench) can be used to generate OAM modes with opposite topological charges. Compared to low index materials [33–35], high index differences can lead to shorter trench lengths L, which are set to be 15.5 µm and 12.5 µm for x- and y-polarizations in the simulation, respectively. Intensity profiles and phase distributions of OAM₊₁ and OAM_-1 for x- and y-polarization are displayed in Figs. 4(a) and 4(c), respectively. One can see as expected the doughnut shape intensity profiles and helical phase fronts which successfully proves that it is accessible to employ single-trench waveguide to generate OAM modes. To qualify the quality of the generated OAM modes, the mode purity is calculated by overlap integral

 

Fig. 4 (a) Simulated near-field intensity profiles, phase distributions and (b) mode purity of OAM_-1 for x-polarization from 1.45 µm to 1.65 µm. (c) Simulated intensity profiles, phase distributions and (d) mode purity of OAM₊₁ for y-polarization from 1.45 µm to 1.65 µm. Insets are far-field intensity and phase profiles at 1.55µm.

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As presented in Figs. 4(b) and 4(d), single-trench waveguide can operate with high mode purity (close to 90%) for x- and y-polarization over a wide wavelength range from 1.45 µm to 1.65 µm. Meanwhile, it has been numerically confirmed that the undesired back reflection at the boundary between the waveguides with and without the trench is assessed to be less than −30 dB and could be ignored. The dominant insertion loss for OAM mode generation attributes to the facet reflection, which is assessed to be ~3 dB. On one hand, the near-field and far-field intensity profiles are both not very uniform due to high index contrast between silicon waveguide core and silica cladding. On the other hand, high index contrast contributes to decrease the size of the device. Therefore, we make a trade-off between mode conversion efficiency and device footprint in the proposed trench waveguide structure.

4. On-chip OAM modes (de)multiplexer

As one typical example of potential applications, we design an on-chip OAM modes (de)multiplexer (OAM₀, OAM _{± 1}) by using single-trench waveguides for x-polarization. Here we consider the on-chip OAM modes multiplexer and the demultiplexer similarly functions in an opposite way due to the reciprocity principle of light. At the first step, the second-order mode should be generated by adiabatic coupling. As mentioned above, single-trench waveguide can generate OAM modes over a wide wavelength range from 1.45 µm to 1.65 µm. Nevertheless, asymmetric directional coupler (TE₀₀-TE₁₀) has a limited bandwidth. Thus we design an on-chip OAM modes (de)multiplexer (OAM₀, OAM _{± 1}) based on the proposed trench waveguide structure for x-polarization over a wavelength range from 1.52 µm to 1.58 µm. The asymmetric adiabatic coupling (TE₀₀-TE₁₀) relies on the phase matching between the waveguides, i.e. the effective refractive index of TE₀₀ mode of the narrow waveguide should be equal to that of TE₁₀ mode of the wide waveguide at a center wavelength of 1.55 µm.

We calculate the dispersion relationship about the mode effective refractive index and width of the silicon waveguide at height of H = 1 µm. As shown in Fig. 5, width w = 0.335 µm for TE₀₀ mode of the narrow waveguide and width w = 0.7 µm for TE₀₁ mode of the wide waveguide are chosen since their effective indices have almost the same value$n_{eff}=2.696$.

 

Fig. 5 Structure dispersion relationship about the effective refractive index and width of silicon waveguide with height of H = 1 µm at a center wavelength of 1.55µm.

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Figure 6 shows the schematic configuration for the generation and multiplexing of three modes including two OAM ones. The fundamental modes for TE polarization are launched at the inputs, and three modes with different topological charges can be observed at the outputs. From input1, the fundamental mode TE₀₀ is firstly coupled to the second-order mode TE₁₀ via an asymmetric directional coupler. Then a taper with length D is utilized between the narrow waveguide and the wide waveguide with trench to guarantee an adiabatic evolution of the modes. When the total length of the single-trench waveguide satisfies a certain phase difference in Fig. 1(d), the mode TE₁₀ is converted to the mode TE₀₁ through the first trench waveguide, and the mode TE₀₁ is further converted to the OAM_-1 mode by the second trench waveguide, i.e. input fundamental mode TE₀₀ to output OAM_-1 mode. From input2, the fundamental mode TE₀₀ transmits throughout the whole structure and also appears as the fundamental mode TE₀₀ (i.e. OAM₀ mode) at the output, i.e. input fundamental mode TE₀₀ to output OAM₀ mode. From input3, the fundamental mode TE₀₀ is coupled to the second-order mode TE₁₀ via an asymmetric directional coupler, and the mode TE₁₀ is further converted to the OAM₊₁ mode by the second trench waveguide shown in Fig. 6, i.e. input fundamental mode TE₀₀ to output OAM₊₁ mode. Consequently, the presented structure design shown in Fig. 6 can facilitate on-chip OAM modes multiplexer (OAM₀, OAM _{± 1}) with the inputs compatible with Gaussian modes (single-mode waveguides or single-mode fibers). Additionally, when launching multiplexed OAM modes (OAM₀, OAM _{± 1}) from the right side of the structure, the OAM modes demultiplexing is also achievable.

 

Fig. 6 Schematic structure of an on-chip OAM modes (de)multiplexer (OAM₀, OAM _{± 1}).

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In the simulations, the determined values of structure parameters are listed in Table 1. For the adiabatic coupler, w1 and w2 are set as 0.335 µm and 0.7 µm, which satisfy the dispersion relationship in Fig. 5. To obtain a high coupling efficiency, the length d of the coupler is set as 12.8 µm. The width w3 of the trench waveguide is 1.1 µm, and the length of the taper between narrow waveguide and trench waveguide is selected as 5 µm.

Table 1. Values of structure parameters of on-chip OAM modes (de)multiplexer.

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Using Eq. (3), the mode purity of the generated OAM modes is calculated by overlap integral. Despite ~3 dB insertion loss due to the facet reflection, the average extinction ratio up to 20 dB can still be achieved from 1.52 µm to 1.58 µm, as shown in Fig. 7. The obtained simulation results indicate favorable operation performance of efficient generation and multiplexing of OAM modes, which may find potential applications in OAM multiplexing communications for enhanced transmission capacity.

 

Fig. 7 Simulated mode purity and extinction ratio of (a) OAM₀, (b) OAM₊₁ and (c) OAM_-1 by calculating the overlap integrals of the generated OAM beams with corresponding Laguerre-Gaussian beams from 1.52 µm to 1.58 µm.

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By specially designing the trench waveguide, two orthogonal LP-like eigenmodes with different propagation constants could be synthesized into four different modes (TE₀₁, TE₁₀, OAM _{± 1}). Such a compact integrated structure provides a feasible approach to realize robust mode conversion. Compared to previous works [22, 26], the proposed device has a smaller footprint of 100 µm$\times$4 µm. In addition, the coupling processes by phase matching are greatly reduced. Limited by the bandwidth of asymmetric directional coupler and vertical coupling grating, the presented OAM modes generator and (de)multiplexer operate efficiently from 1.52 µm to 1.58 µm. With further improvement, OAM modes generator and (de)multiplexer could work more efficiently with a wider wavelength range by employing novel taper couplers [37–39] and edge coupling setup. Moreover, a higher mode conversion efficiency might be also realized by somehow sacrificing the device size with low refractive index materials such as silica, Si₃N₄ and polymer.

5. Conclusion

In summary, we propose an ultracompact integrated structure to generate optical OAM modes by specially designing a single-trench waveguide. OAM mode generators for x- and y-polarizations are both studied over a wide wavelength range from 1.45 µm to 1.65 µm. Moreover, an on-chip OAM modes (de)multiplexer (OAM₀, OAM _{± 1}) for x-polarization is established, which has a footprint of <100 µm × 4 µm. An average mode extinction ratio exceeding 20 dB can be obtained over a wavelength range from 1.52 µm to 1.58 µm. The presented on-chip OAM mode generator may provide chip-scale solutions to wide OAM-involved applications in optical communications, optical tweezers, optical trapping and quantum information processing.