Exceptional Points through Variation of Distances between Four Coaxial Dielectric Disks
Round 1
Reviewer 1 Report
The authors proposed a structure consisting of four coupled dielectric disks to achieve multiple exceptional points. This is done by varying the coupling distances of the disks, which are relatively easy to implement experimentally. The authors conducted detailed theoretical study of the exceptional points in this system. In particular, they demonstrated complicated encirclings of different number of exceptional points. The paper is well written with clear explanations of the physics behind. The proposed structure can be a useful platform to study physics of higher-order exceptional points. I will be happy to recommend it for publication if the authors can address my following comments/suggestions:
- Numerical simulations of the modes of the disks can be tricky because these modes are “quasi-normal modes” with complex eigen frequencies. I suggest the authors add details about how the numerically simulations with COMSOL were conducted.
- The authors mention that “a change of a distance between disks in the dimer D effectively can be considered as a variation of aspect ratio of the dimer.” The physics behind this statement are not apparent to me. The aspect ratio is associated with single disk, while D is concerned with the coupling of two disks. Can the authors provide further explanations?
- Are the results presented in Fig. 3 and Fig. 4 (the eigenfrequencies) obtained by numerically simulations via COMSOL or analytical calculations using the effective Hamiltonian in Eq. (2)? If they are analytical results, how do they compare with full-wave numerical results?
- A large section of the paper is devoted to the discussion of encircling the exceptional points. The authors seem unaware of an important result presented in X.L. Zhang et al., Physical Review X 8, 021066 (2018) that the result of encircling exceptional points in fact depends on the “starting point” of the loop. This should be mentioned or briefly discussed in the paper. In addition, coupled multiple resonators have been employed to achieve higher-order and even arbitrary-order exceptional points which carry unique properties such as ultra-high sensitivity. The authors are suggested to add related discussions and literatures in the Introduction part. This will help the readers to understand the significance and impact of the results in the current paper.
Author Response
The responses to comments of the first Referee
We are grateful to the Referee for the positive evaluation of our paper and comments. We have taken account all of them.
The authors proposed a structure consisting of four coupled dielectric disks to achieve multiple exceptional points. This is done by varying the coupling distances of the disks, which are relatively easy to implement experimentally. The authors conducted
detailed theoretical study of the exceptional points in this system. In particular, they demonstrated complicated encirclings of different number of exceptional points. The paper is well written with clear explanations of the physics behind. The proposed structure can be a useful platform to study physics of higher-order exceptional points. I will be happy to recommend it for publication if the authors can address my following comments/suggestions:
Numerical simulations of the modes of the disks can be tricky because these modes are “quasi-normal modes” with complex eigen frequencies.
I suggest the authors add details about how the numerically simulations with COMSOL were conducted.
-------------------------------------------------------------
Our response:
In order for the numeric would sufficient we took 20 nodes per the wave length. The thickness of PML was chosen equaled to wavelength.
The authors mention that “a change of a distance between disks in the dimer D effectively can be considered as a variation of aspect ratio of the dimer.” The physics behind this statement are not apparent to me. The aspect ratio is associated with single disk, while D is concerned with the coupling of two disks. Can the authors provide further explanations?
------------------------------------------------
Our response:
The system under consideration consists of two dimers. Then a distance between disks in each dimer effectively can be considered as a height of the dimer, i.e., gives us the aspect ratio of the dimer. The distance between the dimers gives the second parameter
variation of which altogether with a thickness of the dimers provides the two-parametric space to reach EPs.
Are the results presented in Fig. 3 and Fig. 4 (the eigenfrequencies) obtained by numerically simulations via COMSOL or analytical calculations using the effective Hamiltonian in Eq. (2)? If they are analytical results, how do they compare with full-wave numerical results?
--------------------------------------------------------------------------------
Our response:
Discrete points of complex frequencies were calculated by COMSOL. But in order to present these points as trajectories, we used original MatLab codes.
We have no analytical results.
A large section of the paper is devoted to the discussion of encircling the exceptional points. The authors seem unaware of an important result presented in X.L. Zhang et al., Physical Review X 8, 021066 (2018) that the result of encircling exceptional points in fact depends on the “starting point” of the loop. This should be mentioned or briefly discussed in the paper. In addition, coupled multiple resonators have been employed to achieve higher-order and even arbitrary-order exceptional points which carry unique properties such as ultra-high sensitivity. The authors are suggested to add related discussions and literatures in the Introduction part. This will help the readers to
understand the significance and impact of the results in the current paper.
--------------------------------------------------------------------------------
Our response:
We thank the Referee for this interesting paper about dynamic encircling which we missed. We briefly discussed this paper as well as the paper by R. Uzdin, et al ([46], [47] in the corrected list of the revised paper. Although the consideration of the dynamic encircling concerns in these papers the PT-symmetry systems, we think these considerations can applicable to dielectric systems.
Reviewer 2 Report
The authors propose a manuscript, whose tittle is " Exceptional points through variation of distances between four coaxial dielectric disks ". Concretely, their interest is in the variation of a refractive index and aspect ratio of isolated disk they achieve exceptional points (EPs) at which the resonant frequencies and resonant modes are coalesced. They indicate that they consider the way to avoid the problem by a substitution of two disk's dimers. "In each dimer variation of distance between disks is equivalent to variation of the aspect ratio of the dimer. Moreover variation of the distance between dimers provides the second parameter that gives rise to a vast number of EPs. " they recover the initial resonant eigenmode by an encircling multiple EP s, 2, 3 and 4 times in the 2 – dimensional parametric space of distances.
1 /. Remark: A remark concerning the way the authors cite their papers. They provide some references for their previous works on the subject such as [31], [34] and [45]. Reading their previous reference helps to understand this paper. That is why it is wise for the authors to have cited these papers and not to be considered as too much self-citation.
2 /. Remark: Now we can say a few words about the part 1 as an introduction of this proposed paper. The authors correctly gave the references of the teams that studied this subject. It is all relevant. So, no special question.
3 /. Question: Page 2, line 48. This is in fact my first question. Authors could say few words about Comsol multiphysics which use a method on finite elements. What is the size of the mesh adopted in your modeling? Why did you choose this mesh and this number of nodes? Is this number of nodes the result of an optimization?
4 /. Question. Page 2, line 58. Could authors say few words about the possible uncertainty on the given values? What could be th emain parameters with an influence on it?
About the rest of the manuscript, I must say that I find it quite well written and clearly explained, so I have no remarks, nor any additional questions to ask.
The opinion on this paper is very positive. The model presented is of interest. But there's still work to ensure a little more clarity in this paper on paragraph “2. Exceptional points in single disk”. It would be nice if the authors could improve their paper based on the comments, suggestions and questions mentioned in this review. To conclude, I tend to suggest that this paper should be accepted subject to minor corrections.
Author Response
The author responses to comments of the second Referee
1 /. Remark: A remark concerning the way the authors cite their papers. They provide some references for their previous works on the subject such as [31], [34] and [45]. Reading their previous reference helps to understand this paper. That is why it is wise for the authors to have cited these papers and not to be considered as too much self-citation.
Our response:
We are grateful the Referee for this comment.
2 /. Remark: Now we can say a few words about the part 1 as an introduction of this proposed paper. The authors correctly gave the references of the teams that studied this subject. It is all relevant. So, no special question.
3 /. Question: Page 2, line 48. This is in fact my first question. Authors could say few words about Comsol multiphysics which use a method on finite elements. What is the size of the mesh adopted in your modeling? Why did you choose this mesh and this number of nodes? Is this number of nodes the result of an optimization?
Our response:
We have taken 20 nodes per wavelength that is sufficient for numeric. The thickness of PML is chosen around the wavelength.
4 /. Question. Page 2, line 58. Could authors say few words about the possible uncertainty on the given values? What could be the main parameters with an influence on it?
---------------------------------------------------------------
Less than one percentage in parameters.
About the rest of the manuscript, I must say that I find it quite well written and clearly explained, so I have no remarks, nor any
additional questions to ask.
The opinion on this paper is very positive. The model presented is of interest. But there's still work to ensure a little more clarity
in this paper on paragraph “2. Exceptional points in single disk”. It would be nice if the authors could improve their paper based on
the comments, suggestions and questions mentioned in this review. To conclude, I tend to suggest that this paper should be accepted subject to minor corrections.
-----------------------------------------------------------------------------------------
Our response:
We are very grateful to the Referee for this encouraging evaluation of our paper.
