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

Round 1

Reviewer 1 Report

Although this article may seem to be more of the same that is already in the literature, the results seem in fact to be new and interesting.

The numerical computations reveal phenomena that one might not guess from intuition and may inspire work into understanding them fundamentally.

 

 

Author Response

We you for his positive evaluation of our work.

 

Sincerely, Alexander Spiridonov

Reviewer 2 Report

This work investigated the on-threshold modes of microcavity laser with piercing holes through mathematical and numerical modeling. The topic is novel. Overall, the manuscript was well prepared. The presented results appear sound. The reviewer comments are given as follows.

The cited references should be fully discussed individually in terms of relevance to the current work. For example, “electromagnetic analysis … is very useful [1-8]. ” Page 1 Lines 31-33. It is necessary to discuss the limitations of the presented model and provide insight into future works for improvements. It is unclear to the reviewer how the modeling results were validated. Please clarify. In addition, the experimental measurements are strongly encouraged to be used for validation purpose. It is necessary to add a nomenclature section to define all symbols in equations and abbreviations for the convenience of readers. In the presented results, the piercing holes do not appear round as the author Please justify. The laser mode was investigated by varying the laser settings. However, the optimal setting has not been reported. The reviewer suggests discussing the optimization with a brief review with inverse analysis. Please review the following references for more information.https://doi.org/10.1007/s00170-019-03286-0; https://doi.org/10.1007/s00170-018-2508-6, etc.

Author Response

The cited references should be fully discussed individually in terms of relevance to the current work. For example, “electromagnetic analysis … is very useful [1-8]. ” Page 1 Lines 31-33.

Thank you for your interest in the topic of microcavities. The references [1-8] in our original text are either book chapters or massive review articles, plus one textbook. Therefore, it is difficult to impossible to discuss them in detail. In view of this, we have split these references to two groups, one dealing with circular-disk microcavity lasers [5-8], where the mode analysis was done with Maxwell equations and separation of variables, and the other – with non-circular cavities, where Geometrical Optics was applied. The corresponding comments have been added to Introduction.

 

It is necessary to discuss the limitations of the presented model and provide insight into future works for improvements.

Thank you for this comment. Surely, the presented in our manuscript LEP model for the analysis of on-threshold modes of microlasers has certain limitations. First of all, it is good only for the modes on the threshold. It cannot be applied to study the dynamics of laser as a source of light as it is formulated for the harmonic time dependence and does not account for any nonlinear phenomena. We have added this thought in Conclusion. We have also noted there that the presented model can be refined and improved along several lines. For instance, we had assumed that the gain index in the active region is a constant, while in reality it depends on the coordinates and on the frequency. This fact can be taken into account by introducing the gain index as a product of a known normalized function of coordinates or frequency (varying from 0 to 1) and an unknown number, which is now a part of two-component LEP eigenvalue. Still another improvement can be seen in the account of the effect bleaching of the cavity boundary that can be accounted for by introducing a narrow passive region along that boundary. We have placed these considerations in Conclusions.

 

It is unclear to the reviewer how the modeling results were validated. Please clarify. In addition, the experimental measurements are strongly encouraged to be used for validation purpose.

We are grateful to you for attracting attention to that matter. In fact, we had thoroughly validated the results, computed with our guaranteed-convergence numerical code. This was done using the 2-D microcavity shaped as a “kite,” for which the mode analysis had been performed using a different discretization scheme and published in [16]. Two results coincided with arbitrary number of digits, controlled by the order of discretization. This was, in fact, expected because the both algorithms had proven convergence. On the experimental validation, we regret to tell that we not able to do such experiments. Still, a reference to the paper [16] is useful, as there a comparison was presented of the far-field emission patterns computed with LEP for a kite-shaped laser and measured experimentally; this comparison showed good agreement. All these considerations around the validation of our results have been added to the manuscript.

 

It is necessary to add a nomenclature section to define all symbols in equations and abbreviations for the convenience of readers. 

We believe that all the notations used in the paper, and equally all abbreviations, are already carefully explained in the manuscript. It appears that this is a usual style in a mathematical work. If you notice some not-explained symbol or notation, we will be glad to correct our mistake.

 

In the presented results, the piercing holes do not appear round as the author Please justify.

Thanks indeed for that comment. In fact, we had been not very happy ourselves realizing that the piercing holes, in the near-field patterns, were too small to be clearly seen. Now, guided by your remark, we have added zooms of the hole’s vicinity to each near-field pattern. Thanks to this, the hole shape is clearly round and fine details of the pattern are visible.

 

The laser mode was investigated by varying the laser settings. However, the optimal setting has not been reported. The reviewer suggests discussing the optimization with a brief review with inverse analysis. Please review the following references for more information.https://doi.org/10.1007/s00170-019-03286-0;https://doi.org/10.1007/s00170-018-2508-6, etc.

We are grateful to you for that valuable suggestion. Right, every engineer keeps in mind rather a cavity optimization than purely mode analysis. To some extent, we have also performed optimization, although on the elementary level, when plotted the gain threshold value and the emission directivity as a function of the round hole coordinate or its radius. However, this can be done more systematically, if a dedicated optimization code is developed. Such a code can use our fast and accurate analysis code combined with global and local iterative algorithms, which minimize certain target function. In the case of a laser mode on threshold, the target function should be obviously the threshold value of gain index, or the inverse value of mode emission directivity, or a weighted sum of these two values. We have added these considerations to Conclusions, together with the suggested references.

 

We hope that now our manuscript is acceptable for publication in Axioms.

 

Sincerely, Alexander Spriridonov

Author Response File: Author Response.pdf