Manipulating the Generation of Photonic Moiré Lattices Using Plasmonic Metasurfaces

Round 1

Reviewer 1 Report

The manuscript describes the design of a plasmonic metasurface of rotated nanoslits for generating focused moiré lattices, with interesting results.

The procedure is well described by providing a mathematical analysis of the interference by superposition of two sub-lattices adding to the formation of diverse moiré patterns and the results of simulations by the full finite-difference time domain (FDTD) method and theoretical results (not clear what theoretical methodology has been used) demonstrate feasibility of the proposed meta-surface design.

Although the mathematical analysis and the simulations are clearly described, the discussion is not well connected with the results. In particular, is not clear whether there is any experimental data/device where these simulations in the superposition of square and hexagonal  lattices have been experimentally reproduced. Furthermore the discussion is very short and look more an experimental setup description.

The discussion section must be completely reformulated and connected to the rest of the sections and the whole structure of the manuscript must be reorganised to better communication of the scientific and technological findings of this research.

The new version of the manuscript might be consider for publication.

Once the manuscript has been reworked, a language and grammatical revision of the text would be necessary. 

Author Response

Dear Editor:

We have finished the revisions of our manuscript and are submitting it to the Editorial Office. We first acknowledge our thanks to the editors for giving us the chance to revise our manuscript for further consideration. Particularly, we sincerely thank the reviewers and editors for their wide scope of knowledge, for their helpful and informative comments and for their contributions to our manuscript. In our revision, all the points of the reviewers have been covered, and we have tried our best to comply with all the suggestions of the reviewers. A point-by-point summary of our revisions and explanations according to comments of the reviewers is appended below.

We thank sincerely the Reviewers and Editors for the suggestions on both technical contents and presentations of the Manuscript. We thank them for their work and contributions to the improvement of our paper.

We hope our revisions have met the requirements of the Reviewers and the Editors.

Sincerely,

Zhanliang Mu and Yuqin Zhang, on behalf of all authors.

5/1/2024

-------------------------------------------------------------------------------------------------------

Summary of explanations and revisions according to Reviewers’ comments

----------------------------------------------

Point-by-point summary of the revisions according to the comments of Reviewer 1

We thank the reviewer for reading the manuscript so carefully, and for raising the nice questions, which give us the chance to improve the manuscript. The following is the response and revisions to the reviewer’s questions and suggestions for revision.

1) Comment 1: “Although the mathematical analysis and the simulations are clearly described, the discussion is not well connected with the results. In particular, is not clear whether there is any experimental data/device where these simulations in the superposition of square and hexagonal lattices have been experimentally reproduced. Furthermore, the discussion is very short and look more an experimental setup description.

The discussion section must be completely reformulated and connected to the rest of the sections and the whole structure of the manuscript must be reorganized to better communication of the scientific and technological findings of this research.”

Explanations: We thank Reviewer 1 for giving the suggestion of comment on the discussion section. We are sorry for not being able to give the detailed discussion connected to results that in the rest of sections. We realized this point after reading the reviewer’s this comment, so we express our particular thanks to the reviewer for giving us the chance to reformulated the discussion section. In the revised manuscript, we have added a new paragraph to provide an elaborate discussion. We present the fundamental technology for realizing and manipulating the generation of moiré lattices using the proposed metasurface, highlight the advantages of our methods over the previous works and outline the potential directions for future research. Next, we apologize again for not explicitly stating that our work primarily focuses on the theoretical and simulated results without experimental demonstration. We have made it clear in the revised manuscript.

Revisions: 1) we added a new paragraph “For further discussion, we assume that the sum of the co-polarization components of the nanoslits in the focal plane of metasurface is approximately zero. This is achieved by introducing spin splitting and deflection through breaking the rotational symmetry of the metasurface. The cross-spin component, which carries geometric phase related to the rotation angle of the nanoslit, is primarily considered for generating wavefields. By modulating the geometric rotation of each individual nanoslit, the geometric phase carried by cross-polarization components can be manipulated and modified in a flexible manner. The modulation of the rotation angle of nanoslit facilitates the focus function in metasurface, accompanied by diverse phase distribution within the circle area and distinct phase differences between two sets round apertures. Manipulating the superposition of two square or hexagonal lattices with different twist angles and phase differences enables a wide range of diverse moiré patterns. Depending on the twist angle, the photonic moiré lattice exhibits different periodic or aperiodic structures. Our approach possesses several practical and technical advantages over other methods for producing the period photonic lattices, such as computer holography, polygon excitation slit, moiré nanolithography and Optical bilayer photonic crystal. Table 1 briefly compares the functionality and disadvantages of the proposed visualization techniques. In fact, the intensity profiles of moiré lattices can also be influenced by the relative amplitude of two sublattices. The primary focus of this work lies in demonstrating the functionality of phase modulation in metasurface through the equal-weights superposition. The capabilities of these metasurfaces can be enhanced by adjusting the radius and nanoslit distribution period of each around apertures, enabling amplitude modulation of two lattices and achieving unequal-weightings of superposition.” as the first paragraph in Discussion section in the revised manuscript.

2) According to the Reviwer1’s suggestions, we reorganized the manuscript. The first paragraph in Discussion section of the original manuscript was moved to become the last paragraph in Results section. To be clearer and in accordance with the reviewer concerns about the experimental data, we have added a brief description as follows “Although our work primarily focuses on the theoretical principles and design methods of metasurfaces, they have been validated through rigorous FDTD simulations and theoretical calculations. Here, the possible experimental measurements and demonstrations is suggested, which would not be too difficult to be realized” in the last paragraph of Result section in the revised manuscript.

Finally, we express our sincere thanks again to Reviewer 1 for the kind comments, suggestions, and contributions to our manuscript. We hope that we have understood the comments correctly and that our revisions have met Reviewer’s concerns and suggestions.

Supplementary revisions

 We find that there is a lack of the theoretical basis for the twist angle between the two square or hexagonal lattices in our original manuscript when we check the logic coherence of the article. For the two-dimensional moiré lattice rotated with respect to each other around a common lattice site, are periodic (commensurate) structures. Only when the rotation angle Δα satisfies cos (Δα) =a/c, sin (Δα) =b/c, where the positive integers (a, b, c) constitute a primitive Pythagorean triple a²+ b²= c², the photonic moiré lattices exhibit periodicity. Such angles are hitherto referred to as Pythagorean. For other rotation angles are non-Pythagorean, the lattice is aperiodic. Regarding the moiré lattices produced by the superposition of two hexagonal lattices, the rotation angles producing period patterns are given by the relation tan (Δα) = b √3 / (2a + b), where the integers a, b and c solve the Diophantine equation a² + b² + ab = c².

Revisions:

1) In order to strength the logic and the completeness of the manuscript, we have added the sentences “For the two-dimensional moiré lattices, composed of two square lattices rotating with each other, only when the rotation angle Δα satisfies cos (Δα) =a/c, sin (Δα) =b/c, where the positive integers (a, b, c) constitute a primitive Pythagorean triple a²+ b²= c², the photonic moiré lattices exhibit periodicity. Such angles are hitherto referred to as Pythagorean. For other rotation angles are non-Pythagorean, the lattice is aperiodic. The two examples are presented in Figures 2b1 and 2b2, corresponding to the superposition of two square lattices at Pythagorean angles Δα=arctan (3/4) ≈36° (Δθ =18°) and Δα=arctan (5/12) ≈22° (Δθ =11°), respectively. The resulting wavefield exhibit periodic moiré lattices and the period units are labeled with white squares. For non-Pythagorean angles Δθ =14° and 9° in Figures 2b3 and 2b4, the generated wavefields exhibit aperiodic structure (not disordered), and break the translational symmetry of the periodic lattices. However, it has long-range order and preserves the four-fold rotational symmetry of lattices.” at the end of second paragraph in Result section in the revised manuscript.

2) We added the sentence “For such lattices, the rotation angles producing period patterns are given by the relation tan (Δα) = b √3 / (2a + b), where the integers a, b and c solve the Diophantine equation a² + b² + ab = c²” as the second sentence of the fourth paragraph in the revised manuscript.

 3) We added the sentences “For Δθ=11°, which satisfies the Diophantine principle, the coherence of two simple hexagonal lattice fields with identical phases forms a periodic intensity pattern. Each unit exhibits a central focus spot and six alternating lobes in the outer circle. Interestingly, when comparing Figure 4b1 with Figure 4b2, with only slight azimuthal bias around approximately 30°, both moiré fields' structures coincide. As Δθ further increases to 23°, the hexagonal lattices emerge as prominent features with period unit pattern as depicted in Figure 4b3. When the value of twist angle fails to observe the Diophantine principle, in Figure 4b4 with Δθ =16°, it exhibits aperiodic structure (not disordered), and break the translational symmetry of the periodic lattices.” in the middle of fourth paragraph in Results section in the revised manuscript.

4) In order to be consistent with the added text, we have rearranged the order of figures in Figures 2-5.

Style Corrections

Our revision: We have checked carefully and modified the figure according to the MDPI style settings in the revised manuscript. Here we express our thanks for the Editors in checking the manuscript.

Additionally, some other minor modifications are also made to cope with the corrections, revisions and style corrections.

Again, we thank sincerely the Reviewers and Editors for the suggestions on both technical contents and presentations of the Manuscript. We thank them for their work and contributions to the improvement of our paper.

We hope our revisions have met the requirements of the Reviewers and the Editors.

Author Response File: Author Response.pdf

Reviewer 2 Report

I have the following suggestions:

1) The alphabetical order of figure 1 should be fixed. First show figure 1a, then b and c. 

2) Line 302, spelling error of 'metasurfae.'

3) The author said "the metasurface samples can be fabricated by fo- 337 cusing ion‐beam etching (FIB) using the metasurface obtained from FDTD simulations." This statement is not correct. From the simulations you can estimate the device performance, however, for fabrication, you need a device layout which should be made by a special program, then the layout has to be provided to an E-beam lithography, and then comes the etching. I suggest the authors make proper corrections. 

4) Throughout the paper, the author has significantly used the word, "we have", "we can", "we consider", "we now", etc. I suggest improving the sentence formation and making it more scientific. 

5) The author should focus on the potential applications of the control and manipulation of moire lattices. Where can it be used and explain it with practical examples?

6) The author should bring diversity to the cited references. The majority of the papers are cited from one geographical region. 

7) A device performance comparison table should be added at the end of the paper to highlight the vital features of the proposed device with the existing ones. 

none

Author Response

Dear Editor:

We have finished the revisions of our manuscript and are submitting it to the Editorial Office. We first acknowledge our thanks to the editors for giving us the chance to revise our manuscript for further consideration. Particularly, we sincerely thank the reviewers and editors for their wide scope of knowledge, for their helpful and informative comments and for their contributions to our manuscript. In our revision, all the points of the reviewers have been covered, and we have tried our best to comply with all the suggestions of the reviewers. A point-by-point summary of our revisions and explanations according to comments of the reviewers is appended below.

We thank sincerely the Reviewers and Editors for the suggestions on both technical contents and presentations of the Manuscript. We thank them for their work and contributions to the improvement of our paper.

We hope our revisions have met the requirements of the Reviewers and the Editors.

Sincerely,

Zhanliang Mu and Yuqin Zhang, on behalf of all authors.

5/1/2024

-------------------------------------------------------------------------------------------------------

Summary of explanations and revisions according to Reviewers’ comments

----------------------------------------------

Point-by-point summary of the revisions according to the comments of Reviewer 2

We thank Reviewer 2 for kindly recommending publication with the questions being addressed. We also thank the reviewer so much for carefully reading the manuscript. In particular, we thank the reviewer’s helpful comments and suggestions on the figure, the information about experimental sample fabrication and discuss about the which to our point of view, have greatly improved the quality of the manuscript.

In our revision, we add sentences or paragraph according to these comments, and we are happy for the improvement of the manuscript on completing the related modifications based on these suggestions. Again, we express our sincere thanks to the reviewer for the precious suggestions.

1) Comment 1: “The alphabetical order of figure 1 should be fixed. First show figure 1a, then b and c.”

Response: We thank Reviewer 2 so much for kindly giving the suggestions of comment on the Figures 1(a), 1(b), and 1(c). Indeed, we have realized that the arrangements of the Figures were not in proper order. We do feel very sorry for that.

Revisions: We have rearranged Figure 1 to start from Figure1(a).

2) Comment 2: “Line 302, spelling error of 'metasurfae.'”

Response: We sincerely thank Reviewer 2 for careful reading our manuscript. We are so sorry for our carelessness in writing the manuscript and for making such mistake. Thank you so much

Revisions: We have corrected “metasurfae” into “metasurface” and carefully checked spellings of other instances of “metasurface”

3) Comment 3: “The author said "the metasurface samples can be fabricated by focusing ion‐beam etching (FIB) using the metasurface obtained from FDTD simulations." This statement is not correct. From the simulations you can estimate the device performance, however, for fabrication, you need a device layout which should be made by a special program, then the layout has to be provided to an E-beam lithography, and then comes the etching. I suggest the authors make proper corrections.”

Response: We thank Reviewer 2 for the concern for the description about sample fabrication and give the kindly suggestions. We indeed neglect many essential production steps. In the revision, we supplemented necessary production process with concise descriptions.

Revisions: According to the Reviewer2’s suggestion, we added the sentences “Regarding the experimental performance, a flat quartz of thickness 0.5mm could be used as substrate, on which a 200nm-thick gold film would be deposited by magnetron sputtering. The layouts of metasurface samples are exported as picture formats and next the pictures are imported to the program. Then the sampling nanoslits may be etched in the gold film by a focused ion beam etcher (FIB)” in the last paragraph of Results section in the revised manuscript.

4) Comment 4: “Throughout the paper, the author has significantly used the word, "we have", "we can", "we consider", "we now", etc. I suggest improving the sentence formation and making it more scientific.”

Response and Revisions: We apologize for the poor language of our manuscript and thanks reviewer for kindly suggestion. We tried our best to improve the language quality and avoid to use the unprofessional words “we have", "we can", "we consider", "we now”, etc. We have worked on both language and readability and have also involved native English speakers for language corrections. We really hope that the flow and language level have been substantially improved.

5) Comment 5: “The author should focus on the potential applications of the control and manipulation of moire lattices. Where can it be used and explain it with practical examples?”

Response and revisions: We thank Reviewer 2 so much for the nice idea and kindly concern for potential applications of the control and manipulation of moiré lattices. In introduction section, we mentioned the relevant research on moiré lattices in two-dimensional materials and some properties of photonic moiré lattices. Indeed, we did not clearly point out the potential applications. In the revision, according to the reviewer’s suggestion, we have added the sentences “The utilization of photonic moiré patterns provides an opportunity to explore the topological properties of light and to investigate phenomena that are relevant to other branches of physics, such as manipulating ultracold atoms, creating gauge potentials, forming the optical solitons and particularly condensed matter, which may be challenging to directly explore. The potential applications can also be expanded to atomic physics, specifically Bose-Einstein condensates.” in the last paragraph in Conclusion section.

6) Comment 6: “The author should bring diversity to the cited references. The majority of the papers are cited from one geographical region.”

Response and revisions:  We sincerely appreciate the valuable comments. We have checked the literature carefully and added more references from different geographical regions. The references were newly added as Reference [6], [10], [11], [18], [23], [43] and [44] into different parts of the Introduction section in the revised manuscript.

6. Geisenhof, F. R.; Winterer, F.; Seiler, A. M.; Lenz, J.; Xu, T.; Zhang, F.; Weitz, R. T. Quantum anomalous Hall octet driven by orbital magnetism in bilayer graphene. Nature. 2021, 598, 53-58.

10. López, M. R.; Peñaranda, F.; Christensen, J.; San-Jose, P. Flat bands in magic-angle vibrating plates. Phys. Rev. Lett. 2020, 125, 214301.

11. Park, M. J.; Kim, Y.; Cho, G. Y.; Lee, S. Higher-order topological insulator in twisted bilayer graphene. Phys. Rev. Lett. 2019, 123, 216803.

18. Gómez-Urrea, H. A.; Ospina-Medina, M. C.; Correa-Abad, J. D.; Mora-Ramos, M. E.; Caro-Lopera, F. J. Tunable band structure in 2D Bravais–Moiré photonic crystal lattices. Opt. Commun. 2020, 459, 125081.

23. Ivanov, S. K.; Konotop, V. V.; Kartashov, Y. V.; Torner, L. Vortex solitons in moiré optical lattices. Opt. Lett. 2023, 48, 3797-3800.

43. Lubin, S. M.; Zhou, W.; Hryn, A. J.; Huntington, M. D.; Odom, T. W. High-rotational symmetry lattices fabricated by moiré nanolithography. Nano lett. 2012, 12, 4948-4952.

44. Tang, H.; Lou, B.; Du, F.; Zhang, M.; Ni, X.; Xu, W.; Mazur, E. Experimental probe of twist angle–dependent band structure of on-chip optical bilayer photonic crystal. Sci. Adv. 2023, 9, eadh8498.

7) Comment 7: “A device performance comparison table should be added at the end of the paper to highlight the vital features of the proposed device with the existing ones.”

Response: We thank Reviewer 2 so much for kindly raising this concern of the vital feature of the proposed method and the nice suggestion about providing the performance comparison table. We are also deeply grateful for you to give us the chance to address the issue, which we believe would contribute to the improvement of the manuscript.

Indeed, several methods have been proposed to generate and control the moiré lattice in the recent years. In principle, computer holography and spatial light modulator offer the opportunity to directly manipulate the generation of moiré lattices with diverse phase difference and periodic distributions. However, such an approach significantly increases the complexity and volume of experimental systems as well as require complex algorithms.  Specially, Tsesses et al. proposed polygon excitation slit to control over the topology of period photonic lattices as honeycomb, Kagome and hexagonal lattices in the sub-wavelength scale, yet this method cannot conduct on the superposition of these periodic lattices and generate the moiré lattice. Although recent studies demonstrated that the moiré nanolithography and optical bilayer photonic crystal can be used to control over the superposition of two square or hexagonal lattices and realize the generation of periodic or aperiodic moiré lattices, it remains challenging to adjust the phase differences between two superposed sub-lattices. Our proposed approach possesses several practical and technical advantages over these methods. The proposed structures can manipulate the superposition of two periodic sub-lattices at specific angles to generate the moiré lattice at the focal plane. The introduction of a geometric phase through the rotation of nanoslits' orientation enables the formation of an overall spiral phase, and facilitates a controlled phase shift between sub-lattices. The twist angle between two superposed sub-lattices is determined by the azimuth position of two sets of round apertures relative to the original point. The aforementioned properties facilitate the superposition of two periodic lattices with distinct twist angles and diverse phase differences, thereby allowing for flexible manipulation of different moiré lattices.

Revisions:  As suggested by Reviwer2, we made the revisions as follows:

• Added a Table in the Discussion section

Table 1. The comparison between the generation methods of photonic period lattices.

• In the revision based on this comment, we added the sentences “Our approach possesses several practical and technical advantages over other methods for producing the period photonic lattices, such as computer holography, polygon excitation slit, moiré nanolithography and Optical bilayer photonic crystal. Table 1 briefly compares the functionalities and disadvantages of the proposed generation techniques.” in the middle part of the first paragraph in Discussion section in the revised manuscript.

Finally, we express our sincere thanks again to Reviewer 2 for the kind comments, suggestions and contributions to our manuscript. We hope that we have understood the comments correctly and that our revisions have met Reviewer’s concerns and suggestions.

Supplementary revisions

 We find that there is a lack of the theoretical basis for the twist angle between the two square or hexagonal lattices in our original manuscript when we check the logic coherence of the article. For the two-dimensional moiré lattice rotated with respect to each other around a common lattice site, are periodic (commensurate) structures. Only when the rotation angle Δα satisfies cos (Δα) =a/c, sin (Δα) =b/c, where the positive integers (a, b, c) constitute a primitive Pythagorean triple a²+ b²= c², the photonic moiré lattices exhibit periodicity. Such angles are hitherto referred to as Pythagorean. For other rotation angles are non-Pythagorean, the lattice is aperiodic. Regarding the moiré lattices produced by the superposition of two hexagonal lattices, the rotation angles producing period patterns are given by the relation tan (Δα) = b √3 / (2a + b), where the integers a, b and c solve the Diophantine equation a² + b² + ab = c².

Revisions:

1) In order to strength the logic and the completeness of the manuscript, we have added the sentences “For the two-dimensional moiré lattices, composed of two square lattices rotating with each other, only when the rotation angle Δα satisfies cos (Δα) =a/c, sin (Δα) =b/c, where the positive integers (a, b, c) constitute a primitive Pythagorean triple a²+ b²= c², the photonic moiré lattices exhibit periodicity. Such angles are hitherto referred to as Pythagorean. For other rotation angles are non-Pythagorean, the lattice is aperiodic. The two examples are presented in Figures 2b1 and 2b2, corresponding to the superposition of two square lattices at Pythagorean angles Δα=arctan (3/4) ≈36° (Δθ =18°) and Δα=arctan (5/12) ≈22° (Δθ =11°), respectively. The resulting wavefield exhibit periodic moiré lattices and the period units are labeled with white squares. For non-Pythagorean angles Δθ =14° and 9° in Figures 2b3 and 2b4, the generated wavefields exhibit aperiodic structure (not disordered), and break the translational symmetry of the periodic lattices. However, it has long-range order and preserves the four-fold rotational symmetry of lattices.” at the end of second paragraph in Result section in the revised manuscript.

2) We added the sentence “For such lattices, the rotation angles producing period patterns are given by the relation tan (Δα) = b √3 / (2a + b), where the integers a, b and c solve the Diophantine equation a² + b² + ab = c²” as the second sentence of the fourth paragraph in the revised manuscript.

 3) We added the sentences “For Δθ=11°, which satisfies the Diophantine principle, the coherence of two simple hexagonal lattice fields with identical phases forms a periodic intensity pattern. Each unit exhibits a central focus spot and six alternating lobes in the outer circle. Interestingly, when comparing Figure 4b1 with Figure 4b2, with only slight azimuthal bias around approximately 30°, both moiré fields' structures coincide. As Δθ further increases to 23°, the hexagonal lattices emerge as prominent features with period unit pattern as depicted in Figure 4b3. When the value of twist angle fails to observe the Diophantine principle, in Figure 4b4 with Δθ =16°, it exhibits aperiodic structure (not disordered), and break the translational symmetry of the periodic lattices.” in the middle of fourth paragraph in Results section in the revised manuscript.

4) In order to be consistent with the added text, we have rearranged the order of figures in Figures 2-5.

Style Corrections

Our revision: We have checked carefully and modified the figure according to the MDPI style settings in the revised manuscript. Here we express our thanks for the Editors in checking the manuscript.

Additionally, some other minor modifications are also made to cope with the corrections, revisions and style corrections.

Again, we thank sincerely the Reviewers and Editors for the suggestions on both technical contents and presentations of the Manuscript. We thank them for their work and contributions to the improvement of our paper.

We hope our revisions have met the requirements of the Reviewers and the Editors.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors did a large revision of the manuscript. The resulting version has important improvements and is acceptable to be published, as the scientific topic is interesting and the contents are sound.

The text must be carefully revised for the English expression and orthographic typos.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

I am willing to accept the paper in its current form. 

none

Author Response

Please see the attachment

Author Response File: Author Response.pdf