# A Theoretical Description of Node-Aligned Resonant Waveguide Gratings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Definition of Geometry and Symmetry Parameters

#### 2.2. Geometry with ${n}_{g1}={n}_{g2}$

#### 2.3. Geometry with ${n}_{g1}\ne {n}_{g2}$ with Variation in Symmetry Parameters

#### 2.4. Sensitivity to Asymmetric Refractive Index Changes

## 3. Conclusions

## 4. Methods

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A waveguide grating with an eigenmode of large propagation length ${L}_{prop}$ and, thus, a small angular divergence $\Delta \theta $. ${\theta}_{0}$ is the mean angle under which light is emitted; (

**b**) a shorter propagation length ${L}_{prop}$ and larger $\Delta \theta $ under refractive index tuning ($n\to n+\Delta n$ ); (

**c**) the definition of the geometry parameters of the proposed waveguide grating. TE-polarized light is considered. Further details are provided in the text.

**Figure 2.**The distributions of the normalized electric field $Re\left({E}_{y}\right)$ and normalized intensity $I\propto {\left|{E}_{y}\right|}^{2}$ as well as filling factors ($FF$) of two exemplary waveguide grating geometries with ${t}_{d1}+{t}_{d2}=0.632\lambda $, ${t}_{g}=0.079\lambda $, ${n}_{s}=1.0$, ${n}_{d1}={n}_{d2}=1.5$ (${\chi}_{n}=1.0$) and ${n}_{g1}={n}_{g2}=1.275$: (

**a**) TE

_{0}mode at ${\chi}_{gp}=0$; (

**b**) TE

_{1}mode at ${\chi}_{gp}=0.0$; (

**c**) TE

_{1}mode at ${\chi}_{gp}=1.0$; (

**d**) TE

_{1}mode at ${\chi}_{gp}=1.0$.

**Figure 3.**Variation in the asymmetry parameter ${\chi}_{gp}$ of a waveguide grating with ${n}_{g1}=1.0$, ${n}_{g2}=1.5$ and $D=0.5$ under a constant value of ${t}_{d1}+{t}_{d2}={t}_{d}=0.632\lambda $. All other parameters are identical to the ones discussed in Figure 2: (

**a**) the normalized propagation length ${L}_{prop}/\lambda $ and (

**b**) the divergence angle of the $T{E}_{0}$ mode (black) and $T{E}_{1}$ mode (red); (

**c**–

**f**) normalized electric fields $Re\left({E}_{y}\right)$ and intensities $I\propto {\left|{E}_{y}\right|}^{2}$ of the $T{E}_{0}$ mode and $T{E}_{1}$ mode for ${\chi}_{gp}=0.0$ and ${\chi}_{gp}=1.0$.

**Figure 4.**(

**a**) The normalized propagation length ${L}_{prop}/\lambda $ and (

**b**) the divergence angle of the $T{E}_{0}$ mode (black) and $T{E}_{1}$ mode (red) with variation in the normalized grating thickness ${t}_{g}/\lambda $ for ${\chi}_{gp}=0.0$ (dashed lines) and ${\chi}_{gp}=1.0$ (solid lines) and a fixed normalized waveguide grating thickness ${t}_{d1}+{t}_{d2}=0.632\lambda $; (

**c**,

**d**) corresponding plots with variation in the normalized waveguide grating thickness ${t}_{WG}/\lambda $ and a fixed normalized grating thickness ${t}_{g}=0.079\lambda $.

**Figure 5.**(

**a**) The dispersion relations and normalized propagation lengths of the TE

_{0}mode for ${\chi}_{gp}=0.0$ and the TE

_{1}mode for ${\chi}_{gp}=1.0$ under fixed ratios of all geometry parameters as used in Figure 3 (only the wavelength is varied with respect to the geometry) and ${\chi}_{n}=1$; (

**b**) dispersion relations and normalized propagation lengths for an asymmetric geometry with ${\chi}_{gp}=0.91$, ${\chi}_{n}=1.0\overline{3}$ for both the $T{E}_{0}$ mode and $T{E}_{1}$ mode with $\frac{{t}_{g}}{{t}_{d1}+{t}_{d2}}=0.045$.

**Figure 6.**(

**a**) The normalized propagation length, (

**b**) $FoM$, and (

**c**) ${S}_{\Delta n}$ of a waveguide grating with ${t}_{g}=1.5\times {10}^{-3}\lambda $ and otherwise identical parameters to the ones in Figure 3 and Figure 4 with asymmetric refractive index changes (${n}_{d1}$ is varied, and ${n}_{d2}$ is fixed at a value of 1.5).

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**MDPI and ACS Style**

Meudt, M.; Henkel, A.; Buchmüller, M.; Görrn, P. A Theoretical Description of Node-Aligned Resonant Waveguide Gratings. *Optics* **2022**, *3*, 60-69.
https://doi.org/10.3390/opt3010008

**AMA Style**

Meudt M, Henkel A, Buchmüller M, Görrn P. A Theoretical Description of Node-Aligned Resonant Waveguide Gratings. *Optics*. 2022; 3(1):60-69.
https://doi.org/10.3390/opt3010008

**Chicago/Turabian Style**

Meudt, Maik, Andreas Henkel, Maximilian Buchmüller, and Patrick Görrn. 2022. "A Theoretical Description of Node-Aligned Resonant Waveguide Gratings" *Optics* 3, no. 1: 60-69.
https://doi.org/10.3390/opt3010008