# Electro-Optic Control of Lithium Niobate Bulk Whispering Gallery Resonators: Analysis of the Distribution of Externally Applied Electric Fields

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

^{−1}[26]. However, air has a breakdown voltage of 3 kV mm

^{−1}only, which limits the maximum voltage that can be applied to the resonator [27]. The shortest path between the electrodes through the air can be enlarged and hence the maximum voltage until an electric breakdown occurs. Furthermore, insulators can be applied to further increase the breakdown voltage. However, smaller electrodes reduce the field at the rim for a given voltage and it is open which effect will win. (2) This gap can also stem from the fabrication process of the WGR. During the shaping and polishing of the resonator the electrode can be partially removed. By introducing the gap $\mathsf{\Delta}d$ we want to account for this, by either applying a gap just to the top electrode , in the following referred to as asymmetric case or to both electrodes, referred to as symmetric case in the following. Since the bulk WGRs are typically mounted on a post and therefore one side is protected, we consider the asymmetric case for a damage of the electrode during the fabrication process. The symmetric case is considered when applying an isolation layer on both electrodes.

## 3. Results

#### 3.1. Conventional Resonator Geometry

#### 3.2. Few-Mode Resonator

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Whispering gallery resonator | WGR |

## Appendix A. Bézier Curve

## References

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**Figure 1.**(

**a**) Artist drawing of a setup used in recent experiments of frequency conversion schemes involving bulk lithium niobate whispering gallery resonators involving the electro-optic effect. The laser light is coupled into the WGR via a coupling prism. The resonator is coated with a metal electrode. (

**b**,

**c**) The electric field distribution indicated by the green lines for a cylindrical resonator with an idealized homogeneous field and for a commonly used resonator geometry. The red area indicates the size and position of the optical mode, not to scale.

**Figure 2.**Geometries used for the simulation. We assume an axially symmetrical 2D model. The resonator has at the bottom and top a metal electrode. In (

**a**) the geometry with a constant ratio is illustrated. Here ${R}_{1}$ is the major radius, ${R}_{2}$ the minor radius and $\mathsf{\Delta}z$ the thickness. For this shape we also investigated a variation of the distance $\mathsf{\Delta}d$ between the rim of the resonator and the electrode. (

**b**) The few-mode geometry. The small bulge at the equator of the WGR is used to guide the light. Here ${r}_{\mathrm{s}}$ is the radius of the small circle and the distance $\mathsf{\Delta}r$ between the rim of the resonator and ${R}_{1}$.

**Figure 3.**Two examples of electric field distributions ${E}_{z}$ for different thicknesses. The distribution is plotted based on ${\u03f5}^{\mathrm{C}}$. The red area indicates where the intensity of the fundamental optical mode is larger than 10% of its maximum. The black bar has in (

**a**) a size of $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$ and in (

**b**) of $10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$. For better visibility the range of field values is limited such that the region of interest provides enough contrast.

**Figure 4.**(

**a**) The average electric field strength in the area where the intensity is larger than 10 of its maximal intensity of the fundamental optical mode for ${R}_{1}=1000$ μm and different thicknesses $\mathsf{\Delta}z$. The green line indicates the field strength ${E}_{z}$ according to ${E}_{z}={E}_{0}=U/\mathsf{\Delta}z$. (

**b**) Comparison between the electric field ${E}_{z}$ and the respective field in a plate capacitor ${E}_{0}$ for different $\mathsf{\Delta}z/{R}_{1}$. The error bars have point size.

**Figure 5.**(

**a**) The electric field strength in the area of the optical mode for different distances $\mathsf{\Delta}d$ of the electrode from the rim of the resonator between 0 and $150\mathsf{\mu}\mathrm{m}$. The major radius of the WGR is ${R}_{1}=1000\mathsf{\mu}\mathrm{m}$ and the thickness $\mathsf{\Delta}z=500\mathsf{\mu}\mathrm{m}$. (

**b**) Schematic of a possible configuration with an additional isolation layer with length $\mathsf{\Delta}l$ covering the top and the bottom electrode.

**Figure 6.**Two examples for different $\mathsf{\Delta}r$ in the case of a few-mode WGR with a semicircular bulge with radius ${r}_{\mathrm{s}}=5\mathsf{\mu}\mathrm{m}$, ${R}_{1}=1000\mathsf{\mu}\mathrm{m}$ and $\mathsf{\Delta}z=500\mathsf{\mu}\mathrm{m}$. The electric field strength is plotted using the unclamped dielectric constant. We simulate in (

**a**) $\mathsf{\Delta}r=1\mathsf{\mu}\mathrm{m}$ and in (

**b**) $\mathsf{\Delta}r=5\mathsf{\mu}\mathrm{m}$. The color code represents the electric field strength ${E}_{z}$. The red area indicates the area where the intensity of the fundamental optical mode is larger than 10% of its the maximum value. The black bar has a size of $5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{m}$. For better visibility the range of ${E}_{z}$ field values is limited such that the region of interest provides enough contrast.

**Figure 7.**(

**a**) The average electric field strength ${E}_{z}$ in the few-mode resonator with small radius ${r}_{\mathrm{s}}=5\mathsf{\mu}\mathrm{m}$ in the area where the normalized intensity of the fundamental optical mode is larger than $10\%$. The red data points represent the strength ${E}_{z}$ with ${\u03f5}^{\mathrm{C}}$ and black the one with ${\u03f5}^{\mathrm{UC}}$. The error bars represent $\mathsf{\Delta}{E}_{z}$. The inset show the highest possible optical mode. The yellow bar has a size of $5\mathsf{\mu}\mathrm{m}$. (

**b**) Comparison between the electric field ${E}_{z}$ and the respective field in a plate capacitor ${E}_{0}$. The gray bars as a measure for the field inhomogeneity are given by $\mathsf{\Delta}{E}_{z}/{E}_{0}$.

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**MDPI and ACS Style**

Minet, Y.; Zappe, H.; Breunig, I.; Buse, K.
Electro-Optic Control of Lithium Niobate Bulk Whispering Gallery Resonators: Analysis of the Distribution of Externally Applied Electric Fields. *Crystals* **2021**, *11*, 298.
https://doi.org/10.3390/cryst11030298

**AMA Style**

Minet Y, Zappe H, Breunig I, Buse K.
Electro-Optic Control of Lithium Niobate Bulk Whispering Gallery Resonators: Analysis of the Distribution of Externally Applied Electric Fields. *Crystals*. 2021; 11(3):298.
https://doi.org/10.3390/cryst11030298

**Chicago/Turabian Style**

Minet, Yannick, Hans Zappe, Ingo Breunig, and Karsten Buse.
2021. "Electro-Optic Control of Lithium Niobate Bulk Whispering Gallery Resonators: Analysis of the Distribution of Externally Applied Electric Fields" *Crystals* 11, no. 3: 298.
https://doi.org/10.3390/cryst11030298