# Investigations on Grating-Enhanced Waveguides for Wide-Angle Light Couplings

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## Abstract

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## 1. Introduction

## 2. Concept of Grating-Assisted Waveguide Couplings

**${n}_{air}$**are the RI of the inner core and ambient air, respectively. The produced diffraction coefficient is designated as ${a}_{q}$ (${\theta}_{i}$) for the qth order, i.e., ${a}_{-1}$ (${\theta}_{i}$), ${a}_{0}\left({\theta}_{i}\right)$, and ${a}_{+1}\left({\theta}_{i}\right)$. Here, the working principle can be intuitively understood as follows. The maximum coupling occurs in cases of normal incidences. With the help of grating structures, the qth-diffracted light at ${\theta}_{q}$ could be additionally coupled to the waveguide. Especially when ${\theta}_{q}$ turns to 0, the deflected light with a proportional power of ${a}_{q}$ turns to normal incidences, thereby leading to a solid enhancement of light-coupling efficiencies. As shown in Figure 1b, the binary gratings are occupied to actively tune diffracted angles and efficiencies. Resembling the experimental circumstances, other associated optical constants and geometry sizes in Figure 1 were applied.

## 3. Theoretical Model

**$d=0$**and ${x}_{0}=0$), Equation (2) can be rewritten to

## 4. Binary Coupling Grating

## 5. Gratings under Large Inputs

## 6. Coupling Efficiency of Grating-Based Waveguides

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Grating-enhanced waveguides for wide-angle light collections. (

**a**) Schematic showing a microstructure-modified waveguide excited by a focused Gaussian beam. (

**b**) A close look at a binary grating. (

**c**) The calculated coupling efficiencies based on a 2D analytical model.

**Figure 3.**Diffraction efficiencies of binary gratings under normal incidences. Two vertical groups refer to the H at 1.575 $\mathsf{\mu}$m and a half pitch, respectively. Horizontal pairs (

**a**,

**d**), (

**b**,

**e**), and (

**c**,

**f**) correspond to grating efficiencies of ${a}_{0}$, ${a}_{1}$, and ${a}_{2}$, separately. The red star-shaped markers in (

**b**,

**c**,

**e**,

**f**) are four gratings selected for the next studies.

**Figure 4.**Binary grating diffraction efficiencies under varied incident angles. (

**a**–

**d**) Four parameter combinations, as values of H, $\Lambda $, and FF are suggested in the corner.

**Figure 5.**Coupling efficiencies of the grating-enabled waveguides computed by FEM (separated points) and the analytical model (solid lines). From (

**a**–

**d**), each graph relates to the grating configuration in Figure 4. The light gray dots and vertical dashed lines indicate the bare waveguide-coupled values and diffraction order angles, respectively.

Structure (Gaussian Beam Excitation) | Max $\mathit{\eta}\left(\mathit{\theta}\right)$ | |
---|---|---|

Analytical Model | FEM | |

Seven-ring ($\Lambda $ = 1575 nm) | N.A. | 0.16 (70${}^{\circ}$) |

Grating ($\Lambda $ = 1875 nm) | 0.485 (55${}^{\circ}$) | 0.286 (50${}^{\circ}$) |

Grating ($\Lambda $ = 2325 nm) | 0.459 (40${}^{\circ}$) | 0.384 (40${}^{\circ}$) |

Grating ($\Lambda $ = 2925 nm) | 0.055 (32${}^{\circ}$), 0.15 (77${}^{\circ}$) | 0.028(35${}^{\circ}$), 0.017 (80${}^{\circ}$) |

Grating ($\Lambda $ = 3075 nm) | 0.053 (33${}^{\circ}$), 0.23 (77${}^{\circ}$) | 0.025 (30${}^{\circ}$), 0.074 (75${}^{\circ}$) |

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## Share and Cite

**MDPI and ACS Style**

Gu, Y.; Wang, N.; Shang, H.; Yu, F.; Hu, L.
Investigations on Grating-Enhanced Waveguides for Wide-Angle Light Couplings. *Nanomaterials* **2022**, *12*, 3991.
https://doi.org/10.3390/nano12223991

**AMA Style**

Gu Y, Wang N, Shang H, Yu F, Hu L.
Investigations on Grating-Enhanced Waveguides for Wide-Angle Light Couplings. *Nanomaterials*. 2022; 12(22):3991.
https://doi.org/10.3390/nano12223991

**Chicago/Turabian Style**

Gu, Yitong, Ning Wang, Haorui Shang, Fei Yu, and Lili Hu.
2022. "Investigations on Grating-Enhanced Waveguides for Wide-Angle Light Couplings" *Nanomaterials* 12, no. 22: 3991.
https://doi.org/10.3390/nano12223991