# 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

## References

- Snyder, A.W.; Love, J. Optical Waveguide Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Birks, T.A.; Knight, J.C.; Russell, P.S.J. Endlessly single-mode photonic crystal fiber. Opt. Lett.
**1997**, 22, 961–963. [Google Scholar] [CrossRef] [PubMed] - Kumar, P.; Chen, J.; Voss, P.L.; Li, X.; Lee, K.F.; Sharping, J.E. Fiber-optic quantum information technologies. In Optical Fiber Telecommunications VA, 5th ed.; Elsevier: Amsterdam, The Netherlands, 2008; pp. 829–880. [Google Scholar]
- Wang, N.; Zeisberger, M.; Hübner, U.; Schmidt, M.A. Nanotrimer enhanced optical fiber tips implemented by electron beam lithography. Opt. Mater. Express
**2018**, 8, 2246–2255. [Google Scholar] [CrossRef] - Schmidt, M.A.; Argyros, A.; Sorin, F. Hybrid optical fibers—An innovative platform for in-fiber photonic devices. Adv. Opt. Mater.
**2015**, 4, 13–36. [Google Scholar] [CrossRef] - Liu, N.; Mukherjee, S.; Bao, K.; Li, Y.; Brown, L.V.; Nordlander, P.; Halas, N.J. Manipulating magnetic plasmon propagation in metallic nanocluster networks. ACS Nano
**2012**, 6, 5482–5488. [Google Scholar] [CrossRef] - Gerislioglu, B.; Ahmadivand, A.; Pala, N. Single-and multimode beam propagation through an optothermally controllable Fano clusters-mediated waveguide. J. Light. Technol.
**2017**, 35, 4961–4966. [Google Scholar] [CrossRef] - Meng, Y.; Chen, Y.; Lu, L.; Ding, Y.; Cusano, A.; Fan, J.A.; Hu, Q.; Wang, K.; Xie, Z.; Liu, Z.; et al. Optical meta-waveguides for integrated photonics and beyond. Light Sci. Appl.
**2021**, 10, 1–44. [Google Scholar] [CrossRef] - Saleh, B.E.; Teich, M.C.; Saleh, B.E. Fundamentals of Photonics; Wiley: New York, NY, USA, 1991; Volume 22. [Google Scholar]
- Winzer, P.J.; Leeb, W.R. Fiber coupling efficiency for random light and its applications to lidar. Opt. Lett.
**1998**, 23, 986–988. [Google Scholar] [CrossRef] - Edwards, C.A.; Presby, H.M.; Dragone, C. Ideal microlenses for laser to fiber coupling. J. Light. Technol.
**1993**, 11, 252–257. [Google Scholar] [CrossRef] - Cordero, E.; Latka, I.; Matthäus, C.; Schie, I.W.; Popp, J. In-vivo Raman spectroscopy: From basics to applications. J. Biomed. Opt.
**2018**, 23, 071210. [Google Scholar] [CrossRef] - Kennedy, K.M.; Kennedy, B.F.; McLaughlin, R.A.; Sampson, D.D. Needle optical coherence elastography for tissue boundary detection. Opt. Lett.
**2012**, 37, 2310–2312. [Google Scholar] [CrossRef] - Pahlevaninezhad, H.; Khorasaninejad, M.; Huang, Y.W.; Shi, Z.; Hariri, L.P.; Adams, D.C.; Ding, V.; Zhu, A.; Qiu, C.W.; Capasso, F.; et al. Nano-optic endoscope for high-resolution optical coherence tomography in vivo. Nat. Photonics
**2018**, 12, 540–547. [Google Scholar] [CrossRef] [PubMed] - Li, X.; Scully, R.A.; Shayan, K.; Luo, Y.; Strauf, S. Near-unity light collection efficiency from quantum emitters in boron nitride by coupling to metallo-dielectric antennas. ACS Nano
**2019**, 13, 6992–6997. [Google Scholar] [CrossRef] [PubMed] - Wang, N.; Zeisberger, M.; Hübner, U.; Schmidt, M.A. Boosting light collection efficiency of optical fibers using metallic nanostructures. ACS Photonics
**2019**, 6, 691–698. [Google Scholar] [CrossRef] - Yuan, S.; Riza, N.A. General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses. Appl. Opt.
**1999**, 38, 3214–3222. [Google Scholar] [CrossRef] - Gomez-Reino, C.; Perez, M.V.; Bao, C.; Flores-Arias, M.T. Design of GRIN optical components for coupling and interconnects. Laser Photonics Rev.
**2008**, 2, 203–215. [Google Scholar] [CrossRef] - Yermakov, O.; Schneidewind, H.; Hübner, U.; Wieduwilt, T.; Zeisberger, M.; Bogdanov, A.; Kivshar, Y.; Schmidt, M.A. Nanostructure-empowered efficient coupling of light into optical fibers at extraordinarily large angles. ACS Photonics
**2020**, 7, 2834–2841. [Google Scholar] [CrossRef] - Wang, N.; Zeisberger, M.; Hübner, U.; Schmidt, M.A. Nanograting-Enhanced Optical Fibers for Visible and Infrared Light Collection at Large Input Angles. Photonics
**2021**, 8, 295. [Google Scholar] [CrossRef] - Decker, M.; Staude, I.; Falkner, M.; Dominguez, J.; Neshev, D.N.; Brener, I.; Pertsch, T.; Kivshar, Y.S. High-efficiency dielectric Huygens’ surfaces. Adv. Opt. Mater.
**2015**, 3, 813–820. [Google Scholar] [CrossRef] [Green Version] - Palmer, C.; Loewen, E.G. Diffraction Grating Handbook; Newport Corporation: New York, NY, USA, 2005. [Google Scholar]
- Loewen, E.G.; Popov, E. Diffraction Gratings and Applications; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Bonod, N.; Neauport, J. Diffraction gratings: From principles to applications in high-intensity lasers. Adv. Opt. Photonics
**2016**, 8, 156–199. [Google Scholar] [CrossRef] [Green Version] - Saruwatari, M.; Nawata, K. Semiconductor laser to single-mode fiber coupler. Appl. Opt.
**1979**, 18, 1847–1856. [Google Scholar] [CrossRef] - Niu, J.; Xu, J. Coupling efficiency of laser beam to multimode fiber. Opt. Commun.
**2007**, 274, 315–319. [Google Scholar] [CrossRef] - Paniagua-Dominguez, R.; Yu, Y.F.; Khaidarov, E.; Choi, S.; Leong, V.; Bakker, R.M.; Liang, X.; Fu, Y.H.; Valuckas, V.; Krivitsky, L.A.; et al. A metalens with a near-unity numerical aperture. Nano Lett.
**2018**, 18, 2124–2132. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kanwal, S.; Wen, J.; Yu, B.; Kumar, D.; Chen, X.; Kang, Y.; Bai, C.; Zhang, D. High-efficiency, broadband, near diffraction-limited, dielectric metalens in ultraviolet spectrum. Nanomaterials
**2020**, 10, 490. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Plidschun, M.; Ren, H.; Kim, J.; Förster, R.; Maier, S.A.; Schmidt, M.A. Ultrahigh numerical aperture meta-fibre for flexible optical trapping. Light Sci. Appl.
**2021**, 10, 1–11. [Google Scholar] [CrossRef] - Wang, N.; Yan, W.; Qu, Y.; Ma, S.; Li, S.Z.; Qiu, M. Intelligent designs in nanophotonics: From optimization towards inverse creation. PhotoniX
**2021**, 2, 1–35. [Google Scholar] [CrossRef]

**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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**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