Asymmetrical Three-Dimensional Conformal Imaging Lens

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

    The authors discussed the design of absolute instruments via transformation optics, especially the three-dimensional conformal mappings. The authors are experts in this field, and this work shows an elegant way to construct novel optical lens. I think this work is of general interest and suitable for publication in this journal. The following are some suggestions that might make this work easier to be understood.

 

1. As the authors say, there are no Cauchy-Riemann equations in 3D, so they use Eq. (2) to  establish the coordinate correspondence between the two spaces. I think some more explanations are needed for the expressions of Eq. (2). For example, the three formulas all have positive signs on the right sides, while Eq. (1), i.e., the Cauchy-Riemann conditions, has a minus sign in the second formula. The authors are probably making an analogy to the two-dimensional Cauchy-Riemann condition, but what is the reason for the choice of signs? In addition, conformal mappings means keeping the orientation and angles of curves. Can Eq. (2) meet this criterion? Or the conformality here means something else, e.g., leading to no extra anisotropy?

 

2. Based on Eq. (2), x, y, z, u, v, w seem to dimensionless coordinates. When calculating the optical path, does the missing length dimension have any impact on the final result?

 

3. In lines 66-67, the authors wrote, “For the inverse conformal transformation w=1/z”. I am not quite sure what w and z refer to. Are they the complex numbers in 2D, or are one of the 3D Cartesian coordinates in the physical space and the virtual space, respectively?

 

4. In line 84, why the optical path should be unchanged after the mapping?

 

5. In lines 130-131, the authors wrote “Here we implement the ray tracing calculations based on the Hamiltonian optics.” Can the authors provide some references about the ideas of Hamiltonian ray dynamics? I am a little confused here because comparing the optical path seems to be a Lagrangian way, and how the Hamiltonian method is introduced here.

 

6. In lines 186-189, the authors mentioned making the analogy of harmonic oscillations. What is the detailed meaning of this analogy? 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, the author design two types of three-dimensional non-spherically symmetric absolute instruments based on conformcniques, which provided valuable insights for 3D imaging methods.Here are my comments:

1. How does the performance of 2D and 3D Luneburg and Lissajous lenses compare in practical applications? Are there specific performance metrics that can quantify how good or bad they are?

2.  May I ask how this research should be processed for testing? Is it feasible for actual testing?

3. What are the potential practical applications of these 3D co-forming image lenses in the future?

4. In the introduction part, These two articles could be added to complete the discussion:10.1002/adpr.202300156 and 10.1109/TGRS.2023.3286103.

Comments on the Quality of English Language

The English writing of the article is OK, and the reading experience is good.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The comments and suggestions are in the following Word

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This authors design three-dimensional (3D), non-spherically symmetric absolute instruments (AIs) using conformal inverse transformations. Traditionally, research on AIs has focused on two-dimensional (2D) and rotationally symmetric configurations, with less emphasis on their three-dimensional (3D) counterparts. By leveraging conformal mappings, specifically inverse transformations, the authors introduce 3D AIs derived from the Luneburg and Lissajous lenses. The study demonstrates the form invariance of the scalar Helmholtz equation under 3D inversion transformations and uses geometric ray tracing to analyze the imaging characteristics of these new lens designs. The work expands the family of AIs, moving from 2D to 3D and from symmetric to asymmetric configurations. Authors mention that the proposed AIs have potential applications in optical, acoustic, and elastic wave imaging. I authors should address the following comments:

1.     Including a wave-optics analysis of the designed AIs is essential to comprehensively demonstrate their behavior and potential applications. This analysis will provide deeper insights into the optical performance and practical utility of the AIs. This could also reveal details about diffraction effects and potential limitations not captured by ray tracing.

2.     Discuss the possible implementation methods of these AIs. What would be the advantages of 3D AIs considering the required sophisticated fabrication techniques?

3.     Mentioning the application of AIs in integrated photonics would be interesting for readers. For instance:

- 10.1364/JOSAB.392644

- 10.1088/2040-8986/ab4fa3

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have responded to my comments.