Meta-Heuristics Optimization of Mirrors for Gravitational Wave Detectors: Cryogenic Case

Round 1

Reviewer 1 Report

The Authors describe a multi objective optimization to have a multilayer coating that allows to have at the same time as low thermal noise and transmittivity as possible. They extend the study to a cryogenic detector. This is very important for next generation gravitational wave detector.

I think the paper can be published in MDPI after a few minor corrections and some English editing, especially in the introduction.  

- Ref 1 to 3 --> Maybe a published paper is better
- line 18: first direct evidence of black hole stars --> first direct evidence of black hole mergers
- line 19: the joint detection of three detectors with great precision in localization --> the joint detection achived with three detectors (which allowed great sky localization precision)
- Ref 14 has a typo: research --> research
- lines 68-70: could the Authors explain why the eta_H is the reference for the cryogenic case and in the room T case the situation is reversed?  
- Fig 3 why the numerical precision is reached only in the middle or at the beginning?
-234 Paoreto --> Pareto
- Fig 6: why upper bound? Shouldn't be the lower bound given that the lower the transmittivity and the thermal noise the better?

Author Response

Answer to referees

The authors thank the anonymous referees for valuable suggestions, all of which have been implemented. All changes done to the manuscript are indicated in bold in the attached revised version (see attached PDF with the highlighted manuscript).

Specifically referee 1

Ref: Ref 1 to 3 --> Maybe a published paper is better

Ans.: OK done, we add published paper to the cited website (see references 1-7 in the revised version)

 

Ref: line 18: first direct evidence of black hole stars --> first direct evidence of black hole mergers

Ans.: Ok done (line 19 revised version)

 

Ref: line 19: the joint detection of three detectors with great precision in localization --> the joint detection achieved with three detectors (which allowed great sky localization precision)

Ans.: OK done (line 19 revised version)

 

Ref: Ref. 14 has a typo: research --> research

Ans.: Ok done (see Ref 17 revised version)

 

Ref: lines 68-70: could the Authors explain why the eta_H is the reference for the cryogenic case and in the room T case the situation is reversed?

Ans: It is just a matter of definition. The authors prefer to take the lower of the two mechanical loss coefficients as a reference. In the cryogenic case, we assume that the material with a low refractive index has the greatest mechanical losses. This is the case not studied in the previous paper [35].

 

Ref: Fig 3 why the numerical precision is reached only in the middle or at the beginning?

Ans: Thank you for the opportunity to mention the issue of Pareto Front sampling uniformity.

Any multi-objective heuristic method might tend to concentrate on certain regions of the constraint space (focusing issue). This issue provides another motivation for the study done in this paper. In lines 217-222 of the revised version we have added a discussion of this point, giving also two additional references ([52] and [53]).

 

Ref: 234 Paoreto --> Pareto

Ans: Ok done (see below line 242)

-

Ref: Fig 6: why upper bound? Shouldn't be the lower bound given that the lower the transmittivity and the thermal noise the better?

Ans: Ok we agree with the anonymous referee it can be confusing, we replaced everywhere in the text “ upper bound” with “always lies above” (revised version caption Fig. 6 and line 270).

 

 

 

 

 

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

In this work, the authors explored the behavior of several optimization methods for reducing coating Brownian noise in the mirrors mainly designed for gravitational wave detectors.
The paper aims to answer the question of whether there is a unique criterion of optimal design in cryogenic detectors and elaborate on its possible physical interpretation.
The practical problem is to encounter the ideal design of the device to reproduce the behavior of a perfect mirror whose reflection coefficient equals unity.
This is implemented by coating the substrate with multiple layers of dielectric materials, mainly aimed at the cryogenic temperature environment.
Theoretically, each layer can be effectively modeled by a matrix.
The design aims to minimize the power transmittance, subjected to a constraint on the maximum thermal noise, of the resultant coating interface as a function of the normalized layer thicknesses, which is a standard problem in the optical design of layered dielectric structures.
In the context of multi-objective optimization, one essentially solves for the Pareto front regarding the underlying parameter space.
The authors considered, in particular, the case of a dielectric mirror made of two alternating materials.
By analyzing the results from various algorithms, which include NSGA2, NSGA3, GDE3, MOEA/D, Borg MOEA, and MOPSO, the universal properties of the Pareto front were studied.
The authors discussed the convergence of the common Pareto front among different approaches.
The relation between the common Pareto front and the Pareto front of period design is elaborated, and in particular, the origin of the bumpy structure in the common Pareto front is explored.

Although most algorithms and numerical studies have been performed otherwise in the literature, the present study is focused on the convergence and properties of the common Pareto front.
As a further study based on some of the authors' previous work on coating operating at room temperature, the present study deals with the cryogenic case.
Closely associated with its immediate application to the interferometry used in the ongoing effort of gravitational wave detection, the subject is indeed a worthy one.
I will give my recommendation for its publication, provided the authors appropriately address the following concerns.

(1) The present study is essentially an extension to Ref.~[30] from room temperature eta_H > eta_L to the cases eta_H \le eta_L.
As a matter of fact, the overall presentation of the two papers seems similar.
Therefore, rather than briefly mention it in the present manuscript, the authors should provide a more detailed comparison between the two studies regarding the reported results that are in common and emphasize the new findings.

(2) The convergence of the common Pareto front was investigated in this work by varying the evolution time and the number of layers N_H.
The latter is potentially interesting, as it generalizes the context by changing the underlying parameter space.
Unfortunately, the cases studied by the authors (mainly shown in Fig.5, 6, and 7) have been primarily carried out by decreasing the value of N_H.
Partly, this is because it was aimed to explore the effect and number of "bumps."
I would suggest further verifying how the convergence persists by increasing the parameter space (i.e., N_H, which is assumed to be a constant for a given search), for instance, by showing how the front evolves as a function of N_H to a numerically accessible extent.

(3) As the title of the manuscript pointed out, the main application of the present paper is the coated mirrors for GW detection.
The entire algorithm is aimed at a desirable noise level and refraction coefficient.
From a practical viewpoint, therefore, it is relevant to review quantitatively, in the first section, the noise level (particularly the essential floor noises and their specific orders of magnitude) with respect to that resulting from the coating technique in question.

Other minor issues:

(1) The term "black hole star" is utilized in the Introduction.
From the GR perspective, this is not a standard practice.
The word "star" is mainly used to refer to a spacetime metric without singularity or horizon.
Simply using "evidence of black holes" will be fine.

(2) Among others, the definition of $\mathcal{T}_{12}$ should be provided below Eq. (4) in terms of the quantities introduced beforehand.
Similarly, in Eq. (5), the definitions of n^{(0)} (though mentioned in the caption of Fig. 1) and n_l are also missing.

(3) The legends in the figures are not entirely consistent. For instance, Fig. 5 and 6 have adopted different ways to label the values of N_H.
This should be improved.

Author Response

Answer to referee 2

 

The authors thank the anonymous referees for valuable suggestions, all of which have been implemented. All changes done to the manuscript are indicated in bold in the attached revised version (see attached  PDF with the highlighted manuscript).

Ref: The present study is essentially an extension to Ref.~[30] from room temperature eta_H > eta_L to the cases eta_H \le eta_L.

As a matter of fact, the overall presentation of the two papers seems similar.

Therefore, rather than briefly mention it in the present manuscript, the authors should provide a more detailed comparison between the two studies regarding the reported results that are in common and emphasize the new findings.

Ans.: We thank the referee for pointing out this problem. To clarify the comparison between this paper and Ref. [35 ] (in the revised version) we add a few sentences (see conclusion below line 319).

In the first sentence we give the main motivation for this study, in the second sentence we say that the results obtained in the case of interferometers working at room temperature are also found in the cryogenic case.

Although not reported, merely so as not to duplicate results, the various heuristics used in the present paper applied to the room-temperature case provide the same results shown in Ref [35].

 

Ref: The convergence of the common Pareto front was investigated in this work by varying the evolution time and the number of layers N_H.

The latter is potentially interesting, as it generalizes the context by changing the underlying parameter space. Unfortunately, the cases studied by the authors (mainly shown in Fig.5, 6, and 7) have been primarily carried out by decreasing the value of N_H.

Partly, this is because it was aimed to explore the effect and number of "bumps."

I would suggest further verifying how the convergence persists by increasing the parameter space (i.e., N_H, which is assumed to be a constant for a given search), for instance, by showing how the front evolves as a function of N_H to a numerically accessible extent.

Ans.: We agree with the referee and we add some new formula eq. (13) and a discussion (close to line 243) on the convergence of Pareto fronts for high values of N_H, i.e. close to a realistic number for the applications. We change also the caption of Fig. 5.

 

Ref As the title of the manuscript pointed out, the main application of the present paper is the coated mirrors for GW detection.

The entire algorithm is aimed at a desirable noise level and refraction coefficient.

From a practical viewpoint, therefore, it is relevant to review quantitatively, in the first section, the noise level (particularly the essential floor noises and their specific orders of magnitude) with respect to that resulting from the coating technique in question.

Ans: Thanks to the referee for the suggestion. We add a brief review of the coating thermal noise issue in the wide band interferometric antenna (see below line 65 ) and the reference [29].

 

 

Referee 2 minor changes:

 

Ref: The term "black hole star" is utilized in the Introduction.

From the GR perspective, this is not a standard practice.

The word "star" is mainly used to refer to a spacetime metric without singularity or horizon.

Simply using "evidence of black holes" will be fine.

Ans: OK the terms star has been removed.

 

Ref: Among others, the definition of $\mathcal{T}_{12}$ should be provided below Eq. (4) in terms of the quantities introduced beforehand.

Similarly, in Eq. (5), the definitions of n^{(0)} (though mentioned in the caption of Fig. 1) and n_l are also missing.

 

Ans: Thanks to the referee we fix in the revised version all the mentioned problems.

 

Ref: The legends in the figures are not entirely consistent. For instance, Fig. 5 and 6 have adopted different ways to label the values of N_H.

This should be improved.

Ans: Ok done, we revise the fig.s. We use the same labeling for N_H in fig 5 and 6.

 

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have addressed most of my concerns in the previous report, and therefore it is ready to be accepted for publication in its present form.