Waveguide Arrays Interaction to Second Neighbors: Semi-Infinite Case

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents an analytical framework for describing the propagation of light in waveguide arrays. Although this topic traces its origins back to the pioneering work of Fock, it has recently regained significant attention due to its capacity to precisely control light propagation. Such control enables high gain and resolution, making these systems particularly relevant for modern telecommunication applications. Moreover, waveguide arrays have emerged as powerful platforms for simulating complex physical systems and implementing quantum information protocols, as they allow for the realization of arbitrary unitary transformations. More importantly, these systems can be readily implemented in the laboratory, providing a realistic and experimentally accessible testbed.

The two senior authors have made notable contributions to this area of research, and their expertise is reflected in the depth and rigor of the present work. The paper considers both infinite and semi-infinite configurations of waveguide arrays, and using a Schröddinger equation approach, are able to write in a compact way the solutions for both cases resorting to a smart trick involving ladder operators of the Sussking-Glogower type.

The paper  is written in a clear and engaging manner. The results are expected to be of interest to the photonics and quantum optics communities. That said, I have a few comments and suggestions which, if properly addressed, would further improve the clarity, completeness, and impact of the manuscript:

1.- The authors formulate the optical dynamics using the Schrödinger equation, which is the same as assuming the validity of the paraxial approximation in the optical domain. However, this assumption is not always strictly satisfied in realistic waveguide systems. Indeed, nonparaxial light propagation is widespread in natural photonic structures, with phenomena such as spin–orbit interactions, nonparaxial Airy beams, and other accelerating beam solutions. Recent studies have highlighted the relevance of nonparaxiality in processes such as third-harmonic generation and asymmetric topological pumping.

While I understand that the study of nonparaxial effects in waveguide arrays is still in its early stages, it would be valuable for the authors to comment on whether nonparaxiality could play a constructive role or open new directions in this context.

2.- The couplings analyzed in the paper are linear. Could the authors comment on possible extensions of their formalism to account for nonlinear effects? Nonlinear coupling could significantly enrich the dynamics and broaden the applicability of the proposed model, particularly in the context of photonic lattices.

3.- A key conceptual ingredient in the authors theoretical formulation is the role of the Newton–Susskind–Glogower operator. However, it is important to distinguish between the versions of this operator: when the summation extends from  − \infty  to  + \infty, it corresponds to the Newton operator, which is unitary; when the summation runs from 0  to  + \infty, it defines the Susskind–Glogower operator, which is nonunitary. This distinction has physical implications, as in the semi-infinite case considered by the authors, the corresponding propagator is not strictly unitary. The authors should clarify how this affects the validity or interpretation of their results.

4.- Both the infinite and semi-infinite cases discussed in the paper represent useful mathematical idealizations. However, in practical implementations, the number of waveguides is always finite. It would strengthen the work if the authors could provide an estimate of how many waveguides are required for the system to approximate the ideal limits closely, or to what extent finite-size effects modify the main predictions.

In summary, this is a sound paper that addresses an important topic. Once the above points are adequately discussed and clarified, I would be pleased to recommend the paper for publication.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents a rigorous analytical framework for the problem of light propagation in waveguide arrays with second-neighbor interactions. The authors successfully extend the known solutions for infinite arrays (based on generalized Bessel functions) to the more challenging semi-infinite case. The article is well-structured, starting from the familiar cases of infinite and semi-infinite arrays with first-neighbor interactions, and progressively transitioning to the core contribution – the solution for the semi-infinite case with second-neighbor interactions. This is complemented by numerical simulations and applications to coherent state initial conditions to validate and demonstrate the theoretical results. This work possesses clear theoretical value and potential application significance in the fields of waveguide photonics and quantum optics simulation. Although the research topic is quite attractive, several key issues need to be addressed before publication can be considered.

1. The term -g₂ |0⟩⟨0| is introduced in the article, but the physical and mathematical meaning of this operator is not explained. Its role and significance should be explicitly interpreted, preferably in the Methods section or an appendix.

1. The article mentions both semi-infinite and infinite cases at the beginning but lacks a comparative analysis of these two scenarios. The differences, relationships, and their impact on the results should be discussed.
2. The article contains minor spelling errors; for example, the Section 2 title "Interaction to first neighbor's" should be "Interaction to first neighbors," and in Section 3, "studding" should be "studying." A comprehensive language check is recommended to correct minor spelling and grammatical errors, enhancing the overall professionalism of the writing.
3. Has the proposed simulation model been compared with commercial simulation software? If so, what is the error (e.g., in modal effective refractive index, field distribution)? Please clarify these details.
4. What material is used for the waveguide herein? Is the design in this manuscript compatible with different material systems—such as metal surface plasmon waveguides (with high absorption and strong confinement) or high-refractive-index dielectric waveguides?
5. What is the unit of propagation distance in the figure? What is the duration of the simulation?
6. Some related works are suggested to be added and discussed, e.g., DOI: 10.29026/oes.2024.240003, DOI: 10.29026/oes.2024.230036, DOI: 10.29026/oes.2023.220022, and doi.org/10.1038/s41598-020-61149-1,=-19572

 

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have appropriately dealt with all the concerns raised in my report. In my view, the paper has considerably improved. I have no hesitation in recommending publication.

Reviewer 2 Report

Comments and Suggestions for Authors

I'm glad that the revised version has solved the reviewers' concerns and the manuscript quality has been greatly improved. I suggest it can be accepted now.