Waveguide Arrays: Interaction to Many Neighbors

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the submitted manuscript entitled “Waveguide arrays: Interaction to many neighbors”, the authors present a generalized theoretical framework for modeling long-range coupling in infinite optical waveguide arrays. By introducing a variable-order coupling model and employing generalized Bessel functions (GBF-N), the authors derive closed-form analytical solutions that extend beyond the conventional nearest-neighbor approximation. The manuscript is clearly structured and mathematically rigorous, with a strong emphasis on analytical derivation. I recommend it for publication in Dynamics the following concerns.

Q1: In practical photonic systems, the evanescent coupling strength between waveguides typically decays exponentially with distance. However, in Fig. 1, the authors appear to use linearly decreasing values for g_k (i.e., 0.8, 0.6, 0.4, 0.2), which deviates from the more realistic exponential behavior. Could the authors clarify whether this choice is intended for illustrative purposes or whether it reflects a specific physical model?

Q2: It is recommended to provide direct numerical simulations (e.g., FDTD or BPM) to benchmark the analytical results presented in Eq. (11) and Eq. (24). The amplitude curves are visualized under representative parameters to verify the accuracy and applicability of the GBF-N based approach.

Q3: While the paper focuses on infinite waveguide arrays, experimental implementations are necessarily finite. Could the authors discuss how boundary conditions (e.g., open or periodic) modify the solution and evaluate the resulting deviations at finite array scales? For instance, how does truncating the array to 21 or 41 waveguides affect the validity of the GBF-N-based expressions? Are features such as side lobes or beam divergence patterns sensitive to boundary reflections?

Q4: The paper explores a range of abstract coupling profiles, including logarithmic and polylogarithmic forms. Are these distributions physically achievable with current fabrication methods (e.g., femtosecond laser writing)? Are there practical constraints arising from material dispersion, mode overlap, or geometrical tolerances?

Q5: Some closely related references are suggested to be added and discussed, e.g.,

• org/10.1038/s41598-020-61149-1: Crosstalk reduction of integrated optical waveguides with nonuniform subwavelength silicon strips.
• doi: 10.29026/oes.2024.230036: High-intensity spatial-mode steerable frequency up-converter toward on-chip integration.
• doi: https://doi.org/10.37188/lam.2023.020:High-fidelity mode scaling via topological-optimized on-chip metalens
• doi:29026/oes.2023.220022: Spatio-temporal isolator in lithium niobate on insulator

Author Response

Attached the answer.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this work, the authors extend the analysis of waveguide arrays by incorporating higher-order coupling effects. They derive analytical expressions based on a generalized Bessel-like function and examine the resulting coupling dynamics. This study builds on previous work in coupled waveguide systems and should be of interest to researchers working in related areas. I recommend publication of this manuscript, provided that the following comments are adequately addressed in a revision.

 

The light evolution presented in this paper is calculated using Eqs. (11) and (24), which represent the core results of the study. I suggest that the authors perform beam propagation simulations on representative coupled waveguide structures by numerically solving the wave equation, in order to validate their theoretical predictions.

 

Building on this point, the authors should provide more detailed discussion regarding potential real-world implementations of the proposed higher-order couplings. While the manuscript briefly mentions the feasibility of fabrication via direct laser writing, it lacks specifics on the geometry, material platform, and design constraints required to achieve such interactions. The inclusion of a realistic structure, accompanied by supporting beam propagation simulations, would significantly strengthen the connection between theory and experimental feasibility.

 

Lastly, I recommend that the authors cite the following important references relevant to higher-order coupling and waveguide dynamics:

• Opt. Lett. 23(21), 1701–1703 (1998)

• Opt. Lett. 29(23), 2752–2754 (2004)

These references provide valuable context and would enhance the completeness of the literature review.

Author Response

Attached the answer.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 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.

Reviewer 2 Report

Comments and Suggestions for Authors

I recommend publication as is.