# On-Chip Guiding of Higher-Order Orbital Angular Momentum Modes

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## Abstract

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## 1. Introduction

## 2. Decomposition of Higher-Order OAM Modes

_{l}represents a coefficient for amplitude equalization. One can see that l + 1 HG modes are required to represent l-th order LG mode. This implies that to support the OAM mode of topological charge l in the dielectric waveguides, l + 1 degenerate modes, whose profiles are similar to the corresponding HG modes, are required. This makes it difficult to guide higher-order OAM mode in the dielectric waveguides. To resolve this problem, we can rearrange (5) by grouping the odd- and the even-order terms separately [21]:

_{20}–HG

_{02}) is just a π/4 rotation of HG

_{11}. Thus, to generate l = ±2 OAM mode, we need two degenerate mode rather than three, and the required waveguide structure should have a π/4-rotation symmetry. The cases of l = 3 and 4 are represented in Figure 1b,c, respectively. In order to guide the l-th order OAM mode, the waveguide structure of π/2l-rotation symmetry should be designed so as to support two degenerate modes whose profiles are close to Equations (7a) and (7b). Therefore, the ideal two guided mode needed to support the l-th order OAM mode can be described as:

_{o}is the effective index of the guided mode and ${w}_{0}$ can be understood as an effective beam width.

## 3. Waveguide Design and Mode Analysis

#### 3.1. Waveguide Structure Simultaneously Supporting the l = ±1 and ±2 OAM Modes

_{01}, HG

_{10}, ${\mathrm{LG}}_{02}^{e}$, and ${\mathrm{LG}}_{02}^{o}$ (or HG

_{11}), and each pair of HG-similar guided modes forming the individual OAM mode should be degenerate for a long distance propagation with keeping its topological charge unchanged. As mentioned before, to support these conditions, the waveguide structure of π/4-rotation symmetry is required in principle. However, fabrication of such structures in an integrated waveguide form will not be so practical. Thus, to mimic the π/4-rotation symmetry, we designed a waveguide structure of a cross-shaped silicon core surrounded by a rectangular SiO

_{2}clad as depicted in Figure 2a. The refractive indices are, respectively, 3.4 and 1.45 for silicon and SiO

_{2}at 1.55 μm wavelength. To fulfill the degeneracy of HG

_{01}–HG

_{10}and ${\mathrm{LG}}_{02}^{e}$-${\mathrm{LG}}_{02}^{o}$ simultaneously, we optimized the structural parameters denoted as W

_{1}, L

_{1}, W

_{2}, and L

_{2}. Mode calculation was conducted with a finite-difference method (FDM)-based commercial software (Lumerical Mode Solutions, Lumerical Inc., Vancouver, BC, Canada). To facilitate the optimization, first we investigated the effect of each parameters on the effective indices of the modes using the linear regression, which is shown in Figure 2b. Based on this, we optimized the structure using the particle swarm optimization (PSO) method [24], resulting in the effective index differences of 4.5 × 10

^{−5}for HG

_{01}-HG

_{10}and 7.1 × 10

^{−5}for ${\mathrm{LG}}_{02}^{e}$-${\mathrm{LG}}_{02}^{o}$ with W

_{1}= 1.118 μm, L

_{1}= 0.921 μm, W

_{2}= 1.626 μm, and L

_{2}= 1.504 μm. Those four HG-similar guided modes in the designed waveguides are shown in Figure 3. Although there is slight effective index difference between two component modes for each OAM mode, it may not be a crucial problem in chip scale applications. In our designed waveguides, HG

_{01}and HG

_{10}modes are quite close to TE polarized modes whereas ${\mathrm{LG}}_{02}^{e}$ and ${\mathrm{LG}}_{02}^{o}$ modes are hybrid modes with TE polarization fractions for both modes close to 70%. Figure 4d, respectively, shows the field and phase distributions of the l = 1 (l = 2) OAM mode when the guided-modes of Figure 3a–d are simultaneously excited with a π/2 phase difference.

#### 3.2. Waveguide Supporting l = ±3 or ±4 OAM Modes

^{−5}for W

_{1}=1.315 μm, L

_{1}=1.315 μm, W

_{2}= 1.8 μm, and L

_{2}= 1.8 μm. The resulting component mode profiles are shown in Figure 5a,b. By combining those two component modes with π/2 phase difference, we obtained the field and phase profiles of l = ±3 OAM mode from the FDTD calculation, which are shown in Figure 5c,d, respectively. For this mode, an overlap-integral mode purity of 92.7% was achieved with l = 3 LG mode of a 0.49 μm beam waist, and its numerically calculated topological change was 2.587.

_{1}=1.85 μm, L

_{1}= 1.55 μm, W

_{2}= 2.326 μm, and L

_{2}= 2.204 μm, resulting in an effective index difference of 2.97 × 10

^{−3}between the component modes (similar to ${\mathrm{LG}}_{04}^{e}$ and ${\mathrm{LG}}_{04}^{o}$ modes) whose profiles are shown in Figure 6a,b. The field and phase profiles of the l = ±4 OAM mode numerically realized from those two component modes with π/2 phase difference are shown in Figure 6c,d, respectively. For this mode, an overlap-integral mode purity of 90.6% was achieved with the l = 4 LG mode with a 0.57 μm beam waist, and its numerically calculated topological change was 3.596.

#### 3.3. Fabrication Process of the Proposed Waveguide

_{2}for the lower and the upper clad layers can also be deposited using the LPCVD. In this fabrication process, there are three patterning and etching processes. However, we may only need two photomasks by sharing the same photomask for the first patterning of the lower SiO

_{2}clad etching and the third patterning of the Si upper corner etching if the photoresists (PR) for these photolithography processes are properly chosen: the negative PR for the first process and the positive PR for the third process or vice versa. All the etching processes will be conducted using the dry etching such as the reactive ion etching (RIE) or the inductively coupled plasma (ICP)-RIE. It is desired to employ the same dry etching methods both for the second and the third etching processes (Si etching) for the reproducible etch rate control.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Decomposition of LG modes into HG modes: (

**a**) l = 2; (

**b**) l = 3; and (

**c**) l = 4. In each case, HG modes of the same azimuthal symmetry are grouped and dubbed ${\mathrm{LG}}_{0i}^{e}$ and ${\mathrm{LG}}_{0i}^{o}$ according to their symmetry.

**Figure 2.**(

**a**) Waveguide structure for simultaneously guiding l = ±1 OAM mode and l = ±2 OAM modes; and (

**b**) mode effective index dependency on waveguide parameters. Optimal design parameters are W

_{1}= 1.118 μm, L

_{1}= 0.921 μm, W

_{2}= 1.626 μm, and L

_{2}= 1.504 μm.

**Figure 3.**HG-similar mode field distributions in the designed waveguide. The mode effective indices (n

_{eff}) of (

**a**) HG

_{01}; (

**b**) HG

_{10}; (

**c**) ${\mathrm{LG}}_{02}^{o}$; and (

**d**) ${\mathrm{LG}}_{02}^{e}$ are 3.215525, 3.215482, 3.05968, and 3.059751, respectively.

**Figure 4.**Field (Ex, horizontal component) and phase distributions of the OAM modes with the designed component guided modes: (

**a**) electric field and (

**b**) phase distributions for the l = ±1 OAM mode; and (

**c**) the electric field and (

**d**) phase distributions for l = ±2.

**Figure 5.**HG-similar component mode field distributions in the designed waveguide for l = ±3 OAM mode: (

**a**) ${\mathrm{LG}}_{03}^{e}$ and; (

**b**) ${\mathrm{LG}}_{03}^{o}$ modes. Field (Ex, horizontal component) and phase distributions of the resulting OAM mode: (

**c**) electric field and (

**d**) phase distributions.

**Figure 6.**HG-similar component mode field distributions in the designed waveguide for l = ±4 OAM mode: (

**a**)${\text{}\mathrm{LG}}_{04}^{e}$ and;(

**b**) ${\mathrm{LG}}_{04}^{o}$ modes. Field (Ex, horizontal component) and phase distributions of the resulting OAM mode: (

**c**) electric field and (

**d**) phase distributions.

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**MDPI and ACS Style**

Lee, I.J.; Kim, S.
On-Chip Guiding of Higher-Order Orbital Angular Momentum Modes. *Photonics* **2019**, *6*, 72.
https://doi.org/10.3390/photonics6020072

**AMA Style**

Lee IJ, Kim S.
On-Chip Guiding of Higher-Order Orbital Angular Momentum Modes. *Photonics*. 2019; 6(2):72.
https://doi.org/10.3390/photonics6020072

**Chicago/Turabian Style**

Lee, In Joon, and Sangin Kim.
2019. "On-Chip Guiding of Higher-Order Orbital Angular Momentum Modes" *Photonics* 6, no. 2: 72.
https://doi.org/10.3390/photonics6020072