# Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Regularization of the Generalized Complex-Frequency Eigenvalue Problem

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Corollary**

**1.**

## 3. Nyström Method

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

## 4. Numerical Results for LEP for Pierced Equilateral Triangle Laser

_{1}, n

_{2}, n

_{3}, e) for the even with respect to the ${x}_{1}$-axis mode. The indices n

_{1}, n

_{2}, and n

_{3}correspond to the numbers of maxima of the function |u/max(u)| along the upper left-hand side, the lower left-hand side, and the right-hand side of the equilateral triangle, respectively. This mode is one of the two modes, which have minimum thresholds and the normalized frequency of lasing, $ka$, laying in between 23.5 and 26.5.

## 5. Conclusions

_{1}axis in the cavity and the radius of the hole and measured the changes in the lasing frequencies, directionalities, and thresholds. Our numerical investigation has shown that a hole of a suitable radius and located at a certain place can lead to a notable growth of the directivity of lasing mode with the conservation of its low threshold. Hence, a small piercing hole’s radius and position in the 2-D equilateral triangular dielectric microcavity laser can be used as an engineering tool to control efficiently the directivity of emission.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Normalized near and far field patterns of the mode (9,9,10,o) of the fully active triangular microcavity laser (

**a**). Panel (

**b**) shows the near and far fields of the same mode with the hole position and the hole radius providing the maximum D. A zoom of the vicinity of the hole is also shown.

**Figure 3.**Dependence of the normalized frequency (

**a**), threshold gain index (

**b**), and directivity of emission (

**c**) of the mode (9,9,10,o) on the normalized position of the hole on the x-axis, x/a. Relative radius of the hole is r/a = 0.021.

**Figure 4.**Near and far fields of the mode (9,9,10,o) of the cavity with the hole, r/a = 0.021 and ox = −0.5 (

**a**), −0.25 (

**b**), 0 (

**c**), and 0.25 (

**d**).

**Figure 5.**Dependence of the normalized frequency (

**a**), threshold gain index (

**b**), and directivity of emission (

**c**) of the mode (9,9,10,o) on the relative radius of the hole located at (0, 0.17).

**Figure 6.**Normalized near and far fields of the mode (9,9,10,o) of the cavity with the hole, r/a = 0.01 (

**a**), 0.02 (

**b**), 0.03 (

**c**), ox = 0.17.

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Spiridonov, A.O.; Karchevskii, E.M.; Nosich, A.I.
Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes. *Axioms* **2019**, *8*, 101.
https://doi.org/10.3390/axioms8030101

**AMA Style**

Spiridonov AO, Karchevskii EM, Nosich AI.
Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes. *Axioms*. 2019; 8(3):101.
https://doi.org/10.3390/axioms8030101

**Chicago/Turabian Style**

Spiridonov, Alexander O., Evgenii M. Karchevskii, and Alexander I. Nosich.
2019. "Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes" *Axioms* 8, no. 3: 101.
https://doi.org/10.3390/axioms8030101