Comparative Analysis of the Methods for Fiber Bragg Structures Spectrum Modeling

Round 1

Reviewer 1 Report

The work is dedicated to a comparative analysis of the Layer Sweep (LS) method and Transfer Matrix (TM) method for the FBG as a whole for the fiber Bragg grating (FBG) spectral response modeling.

The following issues are suggested for further revision:

1. The LP method in the last sentence of the penultimate paragraph of the introduction is not mentioned before. Please check whether it is a clerical error.

2. It is suggested to supplement the second paragraph of 3.1 on the selection of Bragg wavelength, refractive index and induced refractive index.

3. It is suggested to amplify the central wavelength of Figure 4 locally for comparison.

4. Why the deviations of the modeled FBG spectra across the whole relevant wavelength range can be considered negligibly small for most applications? Please add relevant evidence.

5. Why can we draw this conclusion that the RMSDLS converges to the value slightly lower than the RMSDTM from Figure 6

6. It is suggested to analyze and discuss the complete coincidence of the three curves in Figure 7.

Author Response

First of all, we would like to thank the Reviewer for the precious comments that allowed us to improve the quality of the paper.

The work is dedicated to a comparative analysis of the Layer Sweep (LS) method and Transfer Matrix (TM) method for the FBG as a whole for the fiber Bragg grating (FBG) spectral response modeling.

The following issues are suggested for further revision:

1. The LP method in the last sentence of the penultimate paragraph of the introduction is not mentioned before. Please check whether it is a clerical error.

Thank you, the typo was corrected.

2. It is suggested to supplement the second paragraph of 3.1 on the selection of Bragg wavelength, refractive index and induced refractive index.

Thank you for the suggestion. The following information was added to the second paragraph of 3.1: “The chosen Bragg wavelength corresponds to the C-band wavelength range, which is widely used in industry [20]. The refractive index of fiber core n₀ is chosen according to the specifications of the common optical fiber SMF-28 at 1550 nm [21], while the induced refractive index lies within the typical range for the FBG [12,22,23].”

3. It is suggested to amplify the central wavelength of Figure 4 locally for comparison.

The insert was added to Figure 4 showing the locally amplified spectra near the reflectance peak.

4. Why the deviations of the modeled FBG spectra across the whole relevant wavelength range can be considered negligibly small for most applications? Please add relevant evidence.

The following information was added to the manuscript (lines 265-268): “Nevertheless, the deviations of the modeled FBG spectra across the whole relevant wavelength range can be considered negligibly small for most applications, since the deviations do not exceed -40 dB, while the typical signal-to-noise ratio of the FBG interrogation device is between 30 to 40 dB [24].”

5. Why can we draw this conclusion that the RMSDLS converges to the value slightly lower than the RMSDTM from Figure 6

Thank you for the concern. This conclusion was drawn from the fact that the data points were approximated by the second order polynomial (blue line in Figure 6) with the following coefficients: 1.184, 1.311∙10^-3, 2.366∙10^-3, which at the Δz/Λ=0 is equal to 2.366∙10^-3, which is slightly lower than RMSD_TM=2.4∙10^-3 in that case.

6. It is suggested to analyze and discuss the complete coincidence of the three curves in Figure 7.

Although it may seem like a complete coincidence of the three curves in Figure 7, the curves, in fact, slightly differ, which is demonstrated by Figure 8, where their deviations are presented. In order to illustrate these deviations more clearly, we have added the insertion to Figure 7, similarly to Figure 4, amplifying the curves locally.

Reviewer 2 Report

After reviewing your article, the performance of this article is generally impressive. Some grammar issues must be revised, such as Line189 with “Which”. It should be “which”. In your figures, the unit of wavelength should be “µm”, not “m”. You must check all. How about the unit of z(?) in figures? Is it “(m)”?

In technology views, I have some concerns.

1.      In Fig. 1, you described the different characteristics of FPR, FBG, and FPR+FBG. It’s good. Please add more references or comments, especially for Fig. 1(c)and (d). how do you define b^k or give this b in the k^th layer? It’s not clear in the statement.

2.      In Section 2, you must add more references to illustrate Eqs.(1)-(3) because these equations are common.

3.      In Fig. 2, the precision of sampling technology will influence the n(RIU) performance. Please add some comments in this part.

4.      In Eq. (6), you all consider the error function. It’s great. Please add some comments or data in this part after simulation. Please expose the minimum error in consideration and the reason of this limitation or requirement.

5.      In Fig. 5, the y-axes in label are not clear. We didn’t know which one was the deviation or reflectance. Please illustrate this figure more in difference.

6.      In Figs.7 and 8, the reflectance behaviors are almost the same. The deviation is small. Thus, you need to explain the difference of reflectance causing the sensing capability or transmission error deeply, especially in error bit consideration.

      7. The references can be added more to enrich your article.

Author Response

We would like to thank the Reviewer for the thorough review of the paper and the valuable remarks.

After reviewing your article, the performance of this article is generally impressive. Some grammar issues must be revised, such as Line189 with “Which”. It should be “which”. In your figures, the unit of wavelength should be “µm”, not “m”. You must check all. How about the unit of z(?) in figures? Is it “(m)”?

Thank you for the concerns. The typo “Which” was corrected. The units of wavelength in figures are given in meters (m) multiplied by 10^-6, which is indicated in the bottom right corner of each figure. In the case of z in figure 10 and 11, the values are multiplied by 10^-3.

In technology views, I have some concerns.

1. In Fig. 1, you described the different characteristics of FPR, FBG, and FPR+FBG. It’s good. Please add more references or comments, especially for Fig. 1(c)and (d). how do you define b^kor give this b in the k^thlayer? It’s not clear in the statement.

Although it is not completely clear what is referred to as “b” by the Reviewer, the authors assume that the question is related to the refractive index definition of the particular layer for the LS method in the case of Figure 1(c) and (d). In Figure 1(c), a structure consisting of two sequential FBGs is shown. In this case, both FBGs are uniform, but they have different grating periods Λ. The refractive index of each layer of both FBGs is set according to the methodology presented in sub-section 2.1 (equations (4) – (8)). In Figure 1(d), two FBGs recorded one over the other are shown, which is also known as the moiré-recorded FBGs. Similarly to the previous case, the FBGs are uniform and have different grating periods. However, the resulting refractive index of the structure is actually a superposition of the periodic refractive index variations of both FBGs. Therefore, in equation (5), there will be two sinusoidal summands instead of one, each of them having its own period Λ, within the same range of z. As we mentioned in conclusions, modeling of the moiré-recorded FBGs will be the topic of further research.

2. In Section 2, you must add more references to illustrate Eqs.(1)-(3) because these equations are common.

Thank you for the suggestion. The references were added for the equations (1) and (2). Equation (3) follows from (2), where p is the relation of the FBG refractive index at the midpoint to its value at the edges of the FBG. The derivation is omitted in the text, since the authors believe it would be redundant. For your convenience, please, find the complete derivation of equation (3) below (see the attached file).

3. In Fig. 2, the precision of sampling technology will influence the n(RIU) performance. Please add some comments in this part.

As the Reviewer rightly noted, the precision of the refractive index sampling (i.e. the partition interval) does influence the resulting modeled spectrum. To highlight this, the following comment was added to the text (lines 105-106): “It must be noted that the length of the partition interval Δz = z_i+1 – z_i may significantly affect the performance of the modeled structure, which will be discussed in Section 3.” The influence of the partition interval on the FBG model performance is studied in sub-sections 3.2 – 3.4.

4. In Eq. (6), you all consider the error function. It’s great. Please add some comments or data in this part after simulation. Please expose the minimum error in consideration and the reason of this limitation or requirement.

According to the widely known mean value theorem for definite integrals, if the function n(z) is continuous on the interval [z_i, z_i+1], there exists such value of z=z_MVT such that the integral of n(z) over the interval [z_i, z_i+1] is equal to n(z_MVT)∙(z_i+1 – z_i). In general case, the value of z_MVT may not coincide with the middle value z_M of the interval [z_i, z_i+1], which is used in equations (6) – (8). Therefore, we denote the difference between z_MVT and z_M as the error function o(Δz_i). The error o(Δz_i) is minimized using the method described in the sub-section 3.2, which implies that the partition interval is successively halved each time until the difference between the spectra obtained for two successive intervals becomes less than the pre-defined small value ε. In this work, we use the root-mean-square deviations (RMSD) of the obtained spectrum from the reference one (derived from the OptiGrating software), for the minimization of the error o(Δz_i). It must be noted that the values of o(Δz_i) are not considered directly, but minimized through the minimization of the spectral RMSD. The exact desired values of RMSD are defined by the requirements of particular application and also depend on the type of the modeled FBG structure. In our case, we consider the values of RMSD below 4∙10^-3 as acceptable, since at such RMSD values, the spectrum deviations do not exceed -40 dB, while the typical signal-to-noise ratio of the FBG interrogation device is between 30 to 40 dB (lines 267-268). Therefore, such difference would not be detected by interrogation device.

5. In Fig. 5, the y-axes in label are not clear. We didn’t know which one was the deviation or reflectance. Please illustrate this figure more in difference.

Thank you for the concern. The clarification on the y-axes can be found in the figure caption: “Figure 5. Deviations of the homogeneous FBG spectra from the reference (OptiGrating) spectrum (left axis): deviation of the Layer Sweep method (red line), deviation of the Transfer Matrix method (green line), reference OptiGrating spectrum (black line, right axis).” Similar notes were added to Figure 6 caption.

6. In Figs.7 and 8, the reflectance behaviors are almost the same. The deviation is small. Thus, you need to explain the difference of reflectance causing the sensing capability or transmission error deeply, especially in error bit consideration.

Indeed, the deviation of the reflectance is small for the results presented in Figs. 7 and 8. In this case, the deviations can be considered as negligible, since they do not exceed -40 dB and cannot be detected using a common FBG interrogation device with signal-to-noise ratio nearly 30-40 dB. However, in some cases presented in Figs. 9 and 10, the deviations of the spectrum obtained using the LS method can increase significantly, which is discussed further in the manuscript.

7. The references can be added more to enrich your article.

Thank you for the suggestion. As a result of the revisions, several references were added to the article.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors review two simulation methods for periodic optical structures, such as Bragg gratings, phase-shifted Bragg gratings, etc. The goal of the paper is to understand the simulation accuracy for the layer sweep (LS) (or layer-peeling) and transfer matrix (TM) method. The reviewer is confused about the novelty of the work. The TM and layer-peeling methods have been known for many years and are widely published and utilized. This appears to apply them to specific optical structures, but have these not been previously simulated as well?

The differences in the simulation methods appears to be extremely small for the cases studied. It is not clear when these differences are critical in the use or design of the structures. Could the authors give an example to show the importance of the simulation method?

In addition, simulation accuracy is one metric, but another is computational effort or time. The computational time required for the TM and LS methods are extremely different. How does this compare to the OptiGrating Spectrum method? Since the OptiGrating results are used as the reference, is there a large benefit to using the other simulation methods?

The reference for comparison for the simulations is plotted as "OptiGrating Spectrum". Assuming this is an independent code to predict the spectrum, what are the assumptions and mathematical modal use for that simulation. How is it known that this is an appropriate reference?

The authors state that the TM method cannot be used for the case in Figure 1(d) where two gratings are superimposed. Why is the case? The TM method does not require that each segment have the same length, therefore the structure could be represented by alternating longer and shorter segments.

Author Response

We would like to thank the Reviewer for the comprehensive review of the paper and the valuable remarks.

The authors review two simulation methods for periodic optical structures, such as Bragg gratings, phase-shifted Bragg gratings, etc. The goal of the paper is to understand the simulation accuracy for the layer sweep (LS) (or layer-peeling) and transfer matrix (TM) method. The reviewer is confused about the novelty of the work. The TM and layer-peeling methods have been known for many years and are widely published and utilized. This appears to apply them to specific optical structures, but have these not been previously simulated as well?

As the Reviewer rightly noted, the methods presented in the current work are based on the known approaches for the FBG modeling. Thus, we use our implementation of the layer-peeling method, referred to as the layer sweep (LS) method, which is based on the reflectance and transmittance determination for the plane waves propagating through layered structures, which results in the solution of a system of linear equations for the transmittance and reflectance of each layer using the sweep method. Although our implementations of the modeling methods use the known techniques, the findings related to the phase-shifted FBG modeling using the LS method were not reported in the literature, as far as the authors are concerned. In particular, it was established that the asymmetry of the refractive index profile partition near the phase shift relative to its center causes the asymmetry of the spectral response of the simulated grating, resulting in the increased simulation error. Therefore, in the case of phase-shifted FBG, in order to increase the simulation accuracy, it is not enough to decrease the partition interval of the LS method, which is generally accepted, but it is also required to fulfill certain conditions of the refractive index definition in the model, which are discussed in the manuscript.

The differences in the simulation methods appears to be extremely small for the cases studied. It is not clear when these differences are critical in the use or design of the structures. Could the authors give an example to show the importance of the simulation method?

Indeed, for the cases presented in Figures 4 – 5 and 7 – 8, the deviations between the spectra can be considered negligibly small for most applications, since they do not exceed -40 dB, while the typical signal-to-noise ratio of the FBG interrogation device is between 30 to 40 dB. The latter statement was added to the text (lines 266-268). Examples of critical deviations of the spectra are given in Figures 9 and 10, where it is shown that at certain partition intervals of the LS method, the deviations significantly increase. Thus, as it can be seen from Figure 10(a), the spectrum in such cases becomes distorted, and the displacement of transparency window of the phase-shifted FBG is comparable to its FWHM, which is not acceptable, since such structures are often used for high-resolution sensing (added in lines 328-330).

In addition, simulation accuracy is one metric, but another is computational effort or time. The computational time required for the TM and LS methods are extremely different. How does this compare to the OptiGrating Spectrum method? Since the OptiGrating results are used as the reference, is there a large benefit to using the other simulation methods?

According to the simulations performed by the authors, OptiGrating demonstrates computational time comparable to the one of the TM method. However, OptiGrating software has limitations in terms of refractive index definition, e.g. it does not allow to set the refractive index to be lower than the refractive index of the fiber core, which is sometimes required for the modeling of Fabry-Perot interferometers and in other applications, and it does not allow to model FBGs recorded one over the other, i.e. the superimposed gratings.  

The reference for comparison for the simulations is plotted as "OptiGrating Spectrum". Assuming this is an independent code to predict the spectrum, what are the assumptions and mathematical modal use for that simulation. How is it known that this is an appropriate reference?

The results obtained via OptiGrating software were taken as reference, since it is a commercial software widely used in industry. Although the original code of the OptiGrating is not available, the authors suppose that it is generally based on the Transfer Matrix method.

The authors state that the TM method cannot be used for the case in Figure 1(d) where two gratings are superimposed. Why is the case? The TM method does not require that each segment have the same length, therefore the structure could be represented by alternating longer and shorter segments.

As the Reviewer fairly noted, the superimposed grating can be modeled as a combination of short segments. However, in this case, each of the segment would have its own refractive index, represented using the corresponding transfer matrix. And in such case, this method would be identical to the layer peeling method, except for the slightly different formulation, since the number of matrices would be equal to the number of layers. Therefore, the authors refer the TM method to the approach, in which the transfer matrix of an FBG represents several periods of uniform variation of refractive index.

Reviewer 4 Report

The paper entitled “Comparative Analysis of the Methods for Fiber Bragg Structures Spectrum Modeling” presents a systematic comparison of various FBG spectrum models based on layer sweeping and transfer matrix methods. The work analyzed the FBGs response with different characteristics comprehensively and the mathematical demonstration is convincing. However, there is some minor comments for this paper:

1)    To investigate the deviation of the obtained FBG spectrum, it is suggested to variate effective index, the resolution of the grating period, and other modeling parameters with normal distribution. Then, analyze how the FBG spectrum will shift, broaden, and distort.

2)    It is better to use dB scale for the spectrum. It is hard to distinguish the difference among different modelling methods if using linear scale.

3)    For RMSD definition, it is hard to get an intuitive which range is good and which is not. Please explain this well. For example, in figure 9, is 0.04 of RMSD significantly better than 0.08? Can the difference be obviously reflected in obtained FBG spectrum figure?

4)    Please reorganize the figures and their captions (fig10&11). It is not well formatted as provided template.

Overall, the paper is well presented.

Author Response

We would like to thank the Reviewer for the comprehensive review of the paper and the valuable remarks.

The paper entitled “Comparative Analysis of the Methods for Fiber Bragg Structures Spectrum Modeling” presents a systematic comparison of various FBG spectrum models based on layer sweeping and transfer matrix methods. The work analyzed the FBGs response with different characteristics comprehensively and the mathematical demonstration is convincing. However, there is some minor comments for this paper:

1)    To investigate the deviation of the obtained FBG spectrum, it is suggested to variate effective index, the resolution of the grating period, and other modeling parameters with normal distribution. Then, analyze how the FBG spectrum will shift, broaden, and distort.

Thank you for the suggestion. Indeed, it would be interesting to study the influence of the FBG parameters deviations on its spectrum, however, the current article is dedicated to the analysis and comparison of the different modeling methods’ performance at the same fixed parameters of the FBGs. Nevertheless, the Reviewer has suggested a very good idea for further research, which will be the topic of our subsequent works.

2)    It is better to use dB scale for the spectrum. It is hard to distinguish the difference among different modelling methods if using linear scale.

Please, find an example of the spectrum in dB scale below (see the attached file). In this case, the data of Figure 7 are shown with the same color denotations.

As it can be seen, the difference is also hard to distinguish in such representation, except for the lowest values of reflectance, which are not relevant. Therefore, the authors decided to maintain the linear scale for the figures, but also added the inserts in Figure 4 and 10 amplifying the spectra near the reflectance peaks for better visibility. In addition, we would like to note that the deviations of the spectra are given separately in Figures 5 and 8.

3)    For RMSD definition, it is hard to get an intuitive which range is good and which is not. Please explain this well. For example, in figure 9, is 0.04 of RMSD significantly better than 0.08? Can the difference be obviously reflected in obtained FBG spectrum figure?

We consider the values of RMSD below 4∙10^-3 as acceptable, since at such RMSD values, the spectrum deviations do not exceed -40 dB, while the typical signal-to-noise ratio of the FBG interrogation device is between 30 to 40 dB (the statement was added in lines 267-268). Therefore, such difference would not be detected by interrogation device. Examples of critical deviations of the spectra are given in Figures 9 and 10, where it is shown that at certain partition intervals of the LS method, the deviations significantly increase. Thus, as it can be seen from Figure 10(a), the spectrum in such cases becomes distorted, and the displacement of transparency window of the phase-shifted FBG is comparable to its FWHM, which is not acceptable, since such structures are often used for high-resolution sensing (added in lines 328-330).

4)    Please reorganize the figures and their captions (fig10&11). It is not well formatted as provided template.

We thank the Reviewer for the suggestion. However, the authors believe that the current format provides optimal compactness and the readability of the mentioned figures, therefore, we would like to leave it to the discretion of the editors.

Overall, the paper is well presented.

Thank you for the appreciation of our manuscript.

Author Response File: Author Response.pdf

Reviewer 5 Report

This paper reports a comparative analysis of the methods for Fiber Bragg Structures spectrum modeling. The paper is interesting and can be potentially utilized for many sensing applications using FBG-based designs. I have the following comments/concerns about the submitted manuscript that could improve its overall quality and readability.

1.       The authors need to provide the main contribution and novelty of their proposed technique over the already reported in the literature.

2.       The authors should support their argument that the Layer Sweep method offers more flexibility than other reported methods in the literature.

3.       The authors should cite the references from where equations 2, 3, and 4 are obtained.

4.       The authors should also discuss the possible techniques for the enhancement of the LS method’s computational performance.

5.       In Figure 9, the authors should explain in detail why there is so much variation in the RMSD values for various partition intervals.  

6.       In line 90 of the manuscript, the M should be subscript in Z_M. Also, in line 214 the refractive index should have 0 as a subscript. (n0 = 1.4682)

7.       More references to recent developments in FBG-based structural spectrum modeling techniques should be included.

8.       Finally, the authors may consider adding a recommendation to propose possible improvements in their modeling as future work.

9.       There are a few typos in the manuscript that need to be fixed.

Author Response

We would like to thank the Reviewer for the comprehensive review of the paper and the valuable remarks.

This paper reports a comparative analysis of the methods for Fiber Bragg Structures spectrum modeling. The paper is interesting and can be potentially utilized for many sensing applications using FBG-based designs. I have the following comments/concerns about the submitted manuscript that could improve its overall quality and readability.

1. The authors need to provide the main contribution and novelty of their proposed technique over the already reported in the literature.

The methods presented in the current work are based on the known approaches for the FBG modeling. Thus, we use our implementation of the layer-peeling method, referred to as the layer sweep (LS) method, which is based on the reflectance and transmittance determination for the plane waves propagating through layered structures, which results in the solution of a system of linear equations for the transmittance and reflectance of each layer using the sweep method. Although our implementations of the modeling methods use the known techniques, the findings related to the phase-shifted FBG modeling using the LS method were not reported in the literature, as far as the authors are concerned. In particular, it was established that the asymmetry of the refractive index profile partition near the phase shift relative to its center causes the asymmetry of the spectral response of the simulated grating, resulting in the increased simulation error. Therefore, in the case of phase-shifted FBG, in order to increase the simulation accuracy, it is not enough to decrease the partition interval of the LS method, which is generally accepted, but it is also required to fulfill certain conditions of the refractive index definition in the model, which are discussed in the manuscript.

2. The authors should support their argument that the Layer Sweep method offers more flexibility than other reported methods in the literature.

As we stated in Conclusions (lines 387-391), “the Layer Sweep method offers more flexibility in terms of the modeled fiber optic structure configuration, since it can simulate any refractive index profile of the fiber medium, such as two or more FBGs recorded one over the other (also known as the moiré recording) [4], which is not possible to model using the transfer matrix (TM) approach”. The OptiGrating software also has limitations in terms of refractive index definition, e.g. it does not allow to set the refractive index to be lower than the refractive index of the fiber core, which is sometimes required for the modeling of Fabry-Perot interferometers and in other applications, and it also does not allow to model the moiré FBGs.

3. The authors should cite the references from where equations 2, 3, and 4 are obtained.

The references were added to the equation (2). Equation (3) follows from (2), where p is the relation of the FBG refractive index at the midpoint to its value at the edges of the FBG. The derivation is omitted in the text, since the authors believe it would be redundant. For your convenience, please, find the complete derivation of equation (3) below (see the attached file).

Equation (4) follows from the first mean value theorem for definite integrals, which is widely known and represents the mean value of n(z) over the interval [z_i, z_i+1].

4. The authors should also discuss the possible techniques for the enhancement of the LS method’s computational performance.

The enhancement of the LS method’s computational performance can be achieved, for instance, by applying the variable partition interval. The statement was added to the Conclusions (lines 397-398). The authors would not like to discuss it in detail before its testing and verification. Nevertheless, the idea is that the partition interval would decrease in the areas of the structure where homogeneous FBG is used, while it would decrease in the areas of non-homogeneous elements, such as phase shifts.

5. In Figure 9, the authors should explain in detail why there is so much variation in the RMSD values for various partition intervals. 

Thank you for the concern. In fact, the subsequent paragraphs of the sub-section 3.3 are dedicated to the explanation of the variation of the RMSD values noted by the Reviewer. In order to highlight this, we added the following to the last paragraph of sub-section 3.3 (lines 340-341): “…which results in the significant increase of the RMSD_LS values at certain partition intervals, as it was reported in Figure 9”.

6. In line 90 of the manuscript, the M should be subscript in Z_M.Also, in line 214 the refractive index should have 0 as a subscript. (n0 = 1.4682)

Thank you, we double checked the subscripts. In line 90, the capital letter M is the subscript of the lowercase letter z, which at this typeface may appear to be of the same size. Similar is observed for the n₀ in line 214.

7. More references to recent developments in FBG-based structural spectrum modeling techniques should be included.

Several references were added to the manuscript as a result of revisions.

8. Finally, the authors may consider adding a recommendation to propose possible improvements in their modeling as future work.

As we have responded to the concern No. 4, the enhancement of the LS method’s computational performance can be achieved, for instance, by applying the variable partition interval, which  was added to the Conclusions (lines 397-398).

9. There are a few typos in the manuscript that need to be fixed.

Thank you for the careful review. Several typos were corrected.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The revised article is good.

Author Response

Thank you for the appreciation of our article.

Reviewer 3 Report

The response to the reviewers concerns were reasonable in the additional response document. However, several of these responses were not incorporated into the manuscript and would be helpful to the general audience as well. For example, the authors use the OptiGrating spectrum as the baseline simulation. They should explain why they chose this in the manuscript. Also, if the authors state that it is based on the transfer matrix method, how is this different that the authors TM simulations?

Second, and perhaps more importantly, the authors stated in the response that "Although our implementations of the modeling methods use the known techniques, the findings related to the phase-shifted FBG modeling using the LS method were not reported in the literature, as far as the authors are concerned." So if this is the novelty of the paper, it should be clear from the introduction. As it reads now, there is a lot of the beginning material that just a repeat of previous work and the reader doesn't immediately see the value of the paper. So the introduction needs to clearly state that value and the conclusions, etc. should also focus on the new information gained.

Author Response

Thank you for the additional remarks. Please, find our replies below.

The response to the reviewers concerns were reasonable in the additional response document. However, several of these responses were not incorporated into the manuscript and would be helpful to the general audience as well. For example, the authors use the OptiGrating spectrum as the baseline simulation. They should explain why they chose this in the manuscript. Also, if the authors state that it is based on the transfer matrix method, how is this different that the authors TM simulations?

Thank you for the suggestion. The following was added to the text (lines 219-220): “The spectra simulated via the OptiGrating were taken as reference, since it is a commercial software widely used in industry.”

As we stated in our previous reply, the original code of the OptiGrating is not available, and the statement that the software is based on the transfer matrix method is our assumption, which we cannot confirm due to the above reasons. Therefore, we would like not to mention this in the manuscript.

Second, and perhaps more importantly, the authors stated in the response that "Although our implementations of the modeling methods use the known techniques, the findings related to the phase-shifted FBG modeling using the LS method were not reported in the literature, as far as the authors are concerned." So if this is the novelty of the paper, it should be clear from the introduction. As it reads now, there is a lot of the beginning material that just a repeat of previous work and the reader doesn't immediately see the value of the paper. So the introduction needs to clearly state that value and the conclusions, etc. should also focus on the new information gained.

In order to highlight the findings of the article, the following was added to Introduction (lines 82-88): “It was established that the asymmetry of the refractive index profile partition near the phase shift relative to its center causes the asymmetry of the spectral response of the simulated grating, resulting in the increased simulation error. Therefore, in the case of phase-shifted FBG, in order to increase the simulation accuracy, it is not enough to decrease the partition interval of the LS method, but it is also required to fulfill certain conditions of the refractive index definition in the model, which are discussed in the manuscript.”