Applicability of the Effective Index Method for the Simulation of X-Cut LiNbO₃ Waveguides

Round 1

Reviewer 1 Report

In the review of the manuscript entitled Applicability of the effective index method for the simulation of X-cut LiNbO3 waveguides. The authors have provided a good description and the methodology is also fine. I would like to see this article publish but after some questions as follow;

1.     What are photonic integrated circuits (PIC) used for?

2.     What are PICs generally fabricated on?

3.     What is the promising material platform for the complex high-speed PICs?

4.     How should design and simulation of PICs on anisotropic materials be performed?

5.     Why are simulation methods based on approximations used during the PIC design?

6.     What is the effective index method (EIM) and what is it widely applied for?

7.     What is the focus of the study mentioned in the abstract?

8.     How are the results obtained by EIM compared in the study?

9.     What method is used to estimate radiation losses in waveguides with rough sidewalls?

10.  What are the results of the study regarding the applicability of EIM for simulation of anisotropic LNOI-based waveguides?

Comments for author File: Comments.docx

Author Response

Answers for reviewer 1

Dear reviewer, thank you for the review of the paper and for possibility to expand it. We will give the answers below for each question.

All changes in the manuscript are highlighted in yellow.

Question 1. What are photonic integrated circuits (PIC) used for?

Answer. The scope of the PIC covers optical metrology, sensing, optical signal processing, telecommunications, etc. For example, photonic integrated circuits can be used as microspectrometers for optical coherence tomography in the frequency domain or for optical signal processing to interrogate Bragg grating signals. The description is given  in the abstract (in lines 8-9); in addition, we added a description to the introduction (lines 27-28)

Question 2. What are PICs generally fabricated on?

Answer. There are several classical materials used for making PICs such as various glasses for low loss and low contrast waveguides for telecommunications, LiNbO3 for high-speed applications, Si₃N₄ as low loss and small bend radius waveguides for applications in quantum cryptography, LIDAR fabrication, Si as the high contrast material platform. In addition, InP is used for active devices. The corresponding text is in the abstract (in lines 9-10). We have also added a description to the article on lines 30-32.

Question 3. What is the promising material platform for the complex high-speed PICs?

Answer. There several points of view according to the best material platform for the complex high-speed PICs. Firstly, the thin film LiNbO₃ can be promising material platform according to strong electrooptic effect, small bending radii, potentially good possibilities for high performance microwave photonic devices fabrication. Secondly, LiTaO₃ which as well characterized by high electrooptic coefficients. Thirdly, different hybrid materials platforms as Si/ITO. In the introduction we write about LiNbO₃ on lines 32-36.

Question 4. How should design and simulation of PICs on anisotropic materials be performed?

Answer. Finite Difference Time Domain (FDTD), Finite Element Time Domain (FETD), Spectral Element Method (SEM), Rigorous Coupled Wave Analysis are the set of numerical methods can be used for anisotropic structures analysis. These methods provide excellent accuracy and possibilities for anisotropic structure analysis but could be characterized high memory and time consumption. The problem of design and simulation anisotropic devices should be divided by smaller and simple parts, which could be analyzed by simpler methods if it is possible. In the introduction we describe the process on lines 53-64.

Question 5. Why are simulation methods based on approximations used during the PIC design?

Answer. The designer should perform many iterations of simulation different PIC devices for scanning parameters of waveguides crossections, photonic devices topology. Thus, approximation methods are good choice for that task. Once the initial parameters are found, high precision methods can be used to check them. We describe it in the introduction on lines 60-64.

Question 6. What is the effective index method (EIM) and what is it widely applied for?

Answer. The EIM is a semianalytical method based on assumption of electromagnetic field transverse component separation. This means that two-dimensional wave equation could be separate to two simple one-dimensional wave equations, which could be solved separately. Widespread application of EIM is explained by their simplicity. In addition, EIM can reduce the wave propagation problem in the 3D photonic integrated device to 2D. We describe it in the material and methods on lines 111-117. Also the description of widespread application of EIM is added in the materials and methods on lines 118-119.

Question 7. What is the focus of the study mentioned in the abstract?

Answer. The modes properties of anisotropic waveguides such as the ridge waveguides on thin film LiNbO₃ normally should be calculated by proper numerical methods to take into account the anisotropy. The study presented in the paper focus on the study of the EIM applicability for modes calculation in the singlemode ridge waveguides on thin film LiNbO₃ when waveguides directed along one of the crystallographic axis. In addition, the study focus on the radiation losses calculation in the waveguides by Payne-Lance model if the EIM provide good estimation of effective mode index. The main task of the study is presented in the abstract on lines 17-19.

Question 8. How are the results obtained by EIM compared in the study?

Answer. Finite difference frequency domain (FDFD) method was chosen as the reference method for effective indices and mode polarization fraction calculation since it solves full-vector wave equation without any approximation with the exception of finite difference approximation of partial derivatives. The simulated waveguides were directed along Y-crystallographic axis, so the dielectric permittivity tensor contained only diagonal elements. The assumption full-vector wave equation divides into two semi-vectorial wave equations was accepted if the modes in the waveguide are polarized along the transverse axes. The polarization rotation terms in semi-vectorial equations were omitted because of strong polarization of modes. Thus, the semi-vectorial equations were solved by the EIM. Further, the relative errors of effective indices obtained by FDFD and EIM were calculated. A description of the reference method for modeling and comparing the results of calculating the effective refractive indices by the FDFD and EIM methods is added in lines 132-137.

Question 9. What method is used to estimate radiation losses in waveguides with rough sidewalls?

Answer. Payne-Lacey model based on EIM approximation was used for the estimation of radiation losses. We mention it in the text in lines 19-20.

Question 10. What are the results of the study regarding the applicability of EIM for simulation of anisotropic LNOI-based waveguides?

Answer. In the paper, we showed that effective indices of optical modes in waveguide can be calculated by the EIM with sufficient accuracy for fast parameters scanning, for building of simple models for circuit simulation based on transfer matrix method if the calculated waveguide is directed along the crystallographic axes. It was concluded that the EIM provides good result for the anisotropic ridge waveguides based on thin film LiNbO₃. The results showed that accuracy of EIM strongly depend on sidewall angle, etching depths and widths of the waveguide. Consequence of the EIM applicability for anisotropic LNOI waveguides is possibility to radiation losses estimation by Payne-Lacey model. The results regarding the applicability of the EIM are depicted on figures 4, 6 and 7 and discussed in the Results, Discussion and Conclusion.

Author Response File: Author Response.docx

Reviewer 2 Report

The manuscript entitled “Applicability of the effective index method for the simulation of X-cut LiNbO3 waveguides”, analyzed the applicability of EIM for simulations of anisotropic waveguides and radiation losses in waveguides with rough sidewalls. For the simulation of waveguide, the calculation time can be reduced by using applicable approximations. However, the manuscript about the applicability of the EIM for the X-cut waveguide, needs to give further explanation and verification, before it can be accepted.

 (1)  The manuscript claimed that “…which the difference between the results of calculation by finite difference method and by EIM does not exceeds 1 %.” However, the calculated effective index of 90 degree slope angle and 0.1 um waveguide width (in Fig 6) by FDFD and EIM seem seems to differ by more than ~0.04, this difference is relatively large. For lithium niobate, it is difficult to get a waveguide with 90 degree slope angle currently, the smaller sidewall angles (<60 degree) may make more sense.

 (2)  In the manuscript (Line 149) “…The waveguide width for each etching depth was chosen slightly less than width, at which the first-order modes appear.” Does it mean that the EIM can only be applied to the simulation of fundamental mode of anisotropic waveguide?

 (3)  In the manuscript (Line 206) “…the optical radiation losses could be evaluated with Payne-Lacey model…”, Does this optical radiation loss include the loss caused by the bending of waveguide?

(4)  For Figure 4 the author should give the detailed simulation parameters, such as waveguide width, slope angle, etch depth, etc.

Author Response

Answers for reviewer 2

Dear reviewer, thank you for the review of the paper and for possibility to expand it. We will give the answers below for each question.

All changes in the manuscript are highlighted in yellow.

 

Question 1. The manuscript claimed that “…which the difference between the results of calculation by finite difference method and by EIM does not exceeds 1 %.” However, the calculated effective index of 90 degree slope angle and 0.1 um waveguide width (in Fig 6) by FDFD and EIM seem seems to differ by more than ~0.04, this difference is relatively large. For lithium niobate, it is difficult to get a waveguide with 90 degree slope angle currently, the smaller sidewall angles (<60 degree) may make more sense.

Answer. Thank you for your remark this allowed us to improve the article in terms of consistency between different parts of the article.

In the previous paper [29] the LiNbO3 etching process was studied with the goal to transfer the technology from bulk LiNbO3 to thin film LiNbO3, according to obtained results the sidewall angle was 75⁰. In the present paper firstly FDFD and EIM simulation was carried out for the waveguide cross-section with geometrical parameters based on results from [29] (Fig.2 in present paper) as the waveguide of interest to us, and secondly for set of the waveguide parameters given in Table 1 «The parameters of cross-section of the studied waveguides based on X-cut LNOI». Figure 4a shows the effective indices of TE- and TM-modes calculated by FDFD and EIM for wavelengths range from 1.5 to 1.6 and in Fig. 4b is the corresponding relative error not exceeding 1% for the waveguide section shown in Fig. 2. Figure 6 shows the results of FDFD (Fig. 6a) simulations for waveguides with parameters from Table 1 and EIM simulation results (Fig. 6b) for the waveguide with the different etching depths, but with sidewall angle 90⁰. Figure 6 is presented to show how the etching depth and sidewall angles influences the error in the effective indices of TE- and TM-mode calculation by the EIM. If we look at the TM-mode (dashed lines on Fig.6a) at Figure 6a and Figure 6b we can see that for etching depth the difference in the effective indices is about 0.003 (the value from calculations) and it increases with increasing of the etching depth. We study the ridge waveguide on thin film LiNbO3 fabrication process and the preliminary results show that it is possible to get the waveguides with the slope of sidewall angle around 80⁰. Further, we intend to publish that result. We added the description in lines 232-235 and 366-370.

Question 2. In the manuscript (Line 149) “…The waveguide width for each etching depth was chosen slightly less than width, at which the first-order modes appear.” Does it mean that the EIM can only be applied to the simulation of fundamental mode of anisotropic waveguide?

Answer. Thank you for the remark we added the mention that the EIM works for any modes. Below we explain why the parameters from Table 1 were chosen.

The reason to fabricate the waveguide with width slightly less than width, at which the first-order modes appear associated with the value of radiation losses. The optical mode confinement factor increases with increasing width (height). The larger the mode confinement factor, the weaker the roughness scatters the mode. That is why the parameters of cross-section from Table 1 were used. The EIM can be used to calculate any modes in the waveguide. However, the accuracy of the method strongly depends on proximity of the mode to cut-off. The description was added in the text in lines 362-366.

 

Question 3.  In the manuscript (Line 206) “…the optical radiation losses could be evaluated with Payne-Lacey model…”, Does this optical radiation loss include the loss caused by the bending of waveguide?

Answer. Payne-Lacey model is used only to evaluate the radiation losses due to the sidewall roughness.

Question 4.  For Figure 4 the author should give the detailed simulation parameters, such as waveguide width, slope angle, etch depth, etc.

Answer. The results on Figure 4 were obtained for the waveguide cross section depicted on Figure 2. Clarification about the cross-section was added to the article (Lines 232 - 235). Parameters of the calculation window and the grid are given in the part «Materials and methods» (Lines 139 - 141).

Author Response File: Author Response.docx

Reviewer 3 Report

The authors analyzed the applicability of EIM for the simulations of anisotropic waveguides. The results obtained by EIM are compared with the calculation results of rigorous finite difference frequency domain (FDFD) method for evaluation of the EIM applicability limits.

The work done in this manuscript looks interesting and worthy for publication if the authors are able to address the following comments:

1. The authors stated that Design and simulation of PICs on anisotropic materials should be performed by rigorous numerical methods based on Maxwell’s equations. 

In fact such anisotropic structures are designed using inverse problems that invovle maxwell's equations in modeling of the problem. The authors need to explore their statement. 

2. finite element or finite difference are low order methods in terms of accuracy. This point was the base on which the authors did their work. However, on the other hand, Spectral element method can be utilized in such problem with amazing accuracy. The authors need to attract the attention of readers that this method can be an alternative and refer to: Persistence of photonic nanojet formation under the deformation of circular boundary,   and  to,   On-and off-optical-resonance dynamics of dielectric microcylinders under plane wave illumination,   both were published in the Journal of the Optical Society of America B: Optical Physics

3. The authors need to present a comparison in terms of the computational cost (memory and time)

minor

Author Response

Answers for Reviewer 3

Dear reviewer, thank you for the review of the paper and for possibility to expand it. We will give the answers below for each question.

All changes in the manuscript are highlighted in yellow.

Question 1. The authors stated that Design and simulation of PICs on anisotropic materials should be performed by rigorous numerical methods based on Maxwell’s equations. 

In fact such anisotropic structures are designed using inverse problems that invovle maxwell's equations in modeling of the problem. The authors need to explore their statement. 

Answer. Thank you for the remark. The authors understand exact methods as methods that directly solve Maxwell's equations without any approximations such as FDTD, FDFD, FETD which are represented in the commercial software or more accurate methods such as spectral element method (SEM). The references were added as well [24, 25].

Question 2. Finite element or finite difference are low order methods in terms of accuracy. This point was the base on which the authors did their work. However, on the other hand, Spectral element method can be utilized in such problem with amazing accuracy. The authors need to attract the attention of readers that this method can be an alternative and refer to: Persistence of photonic nanojet formation under the deformation of circular boundary,   and  to,   On-and off-optical-resonance dynamics of dielectric microcylinders under plane wave illumination,   both were published in the Journal of the Optical Society of America B: Optical Physics

Answer. Thank you for the new information. The mention of the method are added in the introduction (lines 56-58) and the references were added as well [24, 25].

Question 3. The authors need to present a comparison in terms of the computational cost (memory and time)

Answer. That is a good remark. Comparison of FDFD and EIM for one iteration was added in the Results (lines 288-290).

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The authors have addressed all my concerns properly, and I have no further comments now.