# Van Der Waals Materials for Subdiffractional Light Guidance

^{*}

## Abstract

**:**

_{2}) and tungsten disulfide (WS

_{2}) claddings can operate in a transparency region slightly above (20%) the diffraction limit and even overcome it by 10% around 700 nm, providing an even better confinement than air cladding, but with excitonic losses. Further analysis reveals that van der Waals materials with an in-plane refractive index of about five or an out-of-plane index around two provide subdiffractional and lossless guidance. Therefore, our results establish the route for ultra-dense photonic integration based on layered materials.

## 1. Introduction

_{2}) and tungsten disulfide (WS

_{2}) [6,19]. Additionally, the diversity of vdW materials, with more than 5000 compounds [22], suggests that these may not be the record values. Hence, vdW materials are a perfect platform for light confinement inside waveguides. In this work, we comprehensively discuss the prospects of optical anisotropy for reducing the waveguide mode’s size and, thus, improving the integration density of the optical elements on a chip.

## 2. Results

_{2}and WS

_{2}as cladding materials (Figure 1c) since they are currently record holders regarding the highest optical anisotropy [6,19]. For the mode size calculations, we started with computations of the planar waveguide (Figure 1c) mode index ${n}_{\mathrm{eff}}$, which is presented as follows [26]:

_{2}and WS

_{2}as claddings and silicon (Si) as a core, with the results presented in Figure 2. It is worth noting that instead of thick layers of MoS

_{2}and WS

_{2}, one can leverage their monolayer form because of their similar anisotropic responses, but a monolayer thickness is not enough for a sufficient reduction in mode size. As expected, anisotropic materials’ claddings yield more than 40% higher light confinement than standard silicon dioxide (SiO

_{2}) for TM polarization (Figure 2). Notably, in the transparency region of MoS

_{2}and WS

_{2}, the normalized mode size is almost at the diffraction limit, but it is not reached. Of immediate interest is the region around 700 nm, where a subdiffraction regime is achieved owing to the enormous birefringence of $\Delta n~3.0$. This result is even better than that for a highly confined TE mode for an SiO

_{2}/Si/SiO

_{2}structure (Figure 2a). However, at these wavelengths, silicon has high optical losses, and so do MoS

_{2}and WS

_{2}because their maximum anisotropies are located at exciton resonances. The replacement of silicon with gallium phosphide (GaP) greatly diminishes optical losses since GaP is transparent down to 600 nm [28], but the mode size increases, as seen from Figure 2b. Moreover, the losses in MoS

_{2}and WS

_{2}still remain and prohibit the long propagation of the waveguide mode.

_{2}and WS

_{2}[6,19]. Figure 3a shows the decrease in the mode size with the increase in the in-plane component of the refractive index. From the graph, we conclude that a small increase in the in-plane refractive index by 1.0 is enough, and the required core refractive index range includes Si and GaP properties (Figure 3a). The situation is even more pronounced for the variation of ${n}_{z}$ (Figure 3b), where even a 0.5 change in ${n}_{z}$ corresponds to subdiffraction guidance. Surprisingly, the minimum of the normalized mode size behaves linearly with respect to the out-of-plane refractive index, as seen from the inset of Figure 3b. It is also worth mentioning that, again, the refractive index of GaP and Si belongs to the optimum interval of the core refractive index. As a consequence, the vdW material, with a slightly stronger in-plane or a slightly weaker out-of-plane optical response, is in high demand for an efficient degree of light confinement, while Si and GaP ideally complement the vdW materials’ optical anisotropy for lossless subdiffraction guidance.

## 3. Discussion

_{2}and WS

_{2}, leading to giant optical anisotropy. Indeed, our results show that MoS

_{2}and WS

_{2}enable a degree of light confinement up to the diffraction limit without optical losses. Moreover, we determined optimum values of the in-plane and out-of-plane refractive indices of layered materials equal to 5.0 and 2.0, respectively, for subdiffraction guidance. Although the currently used layered materials do not satisfy these criteria in their transparency regions, they have very close characteristics: 4.0 for in-plane and 2.5 for out-of-plane refractive indices. Nevertheless, their rich diversity [22], heterostructures [29], and modifications [30,31] provide the possibility to tailor optical responses and reach the desired parameters, which were both found in our work.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Optical constants of Si, GaP, SiO

_{2}, MoS

_{2}, WS

_{2}, and hBN, used in the work. (

**a**) Refractive index and (

**b**) extinction coefficient. The inset shows extinction coefficient in logarithmic scale.

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**Figure 1.**Variation of waveguide claddings. Planar waveguide with (

**a**) isotropic, (

**b**) anisotropic metamaterials, and (

**c**) vdW cladding.

**Figure 2.**Subdiffraction guiding in waveguides with MoS

_{2}and WS

_{2}claddings. (

**a**) Silicon core and (

**b**) gallium phosphide core. The results for SiO

_{2}/Si/SiO

_{2}and SiO

_{2}/GaP/SiO

_{2}planar waveguides for both polarizations (TM and TE) are included for comparison. Optical constants, used for calculation, are presented in Appendix section (Figure A1).

**Figure 3.**Parameter dependence of waveguide mode size. Optimal mode size for various (

**a**) in-plane refractive indices at a fixed out-of-plane component (${n}_{z}=2.5$) and (

**b**) out-of-plane refractive index at a fixed in-plane component (${n}_{xy}=4.0$). The insets show the minimum of normalized mode size dependence on in-plane and out-of-plane refractive indices.

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**MDPI and ACS Style**

Ermolaev, G.; Grudinin, D.; Voronin, K.; Vyshnevyy, A.; Arsenin, A.; Volkov, V.
Van Der Waals Materials for Subdiffractional Light Guidance. *Photonics* **2022**, *9*, 744.
https://doi.org/10.3390/photonics9100744

**AMA Style**

Ermolaev G, Grudinin D, Voronin K, Vyshnevyy A, Arsenin A, Volkov V.
Van Der Waals Materials for Subdiffractional Light Guidance. *Photonics*. 2022; 9(10):744.
https://doi.org/10.3390/photonics9100744

**Chicago/Turabian Style**

Ermolaev, Georgy, Dmitriy Grudinin, Kirill Voronin, Andrey Vyshnevyy, Aleksey Arsenin, and Valentyn Volkov.
2022. "Van Der Waals Materials for Subdiffractional Light Guidance" *Photonics* 9, no. 10: 744.
https://doi.org/10.3390/photonics9100744