Green Scalar Function Method for Analyzing Dielectric Media

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Authors presented a scalar Green's function approach to the problem of electromagnetic scattering by dielectric objects. The problem is of fundamental importance in numerical electromagnetics. Authors present the theory and implementation of the proposed method in a compact way.

My comments are as follows.

1. Applicability of the proposed method:

1) The proposed method is viable for the problems where the scattered wave has the same polarization as the incidence wave. Authors used dielectric cylinders illuminated by a vertically polarized planewave in their experiments where the above condition is met. If the incidence angle zeta is not 90 degrees, it appears that the method is not applicable.

Please add comments in Section 2 on the kind of problems that the proposed method is applicable.

2) Please add some comments on the extension of the proposed method to the vector scattering problem by applying the proposed method for each vector component. For example, is the proposed method applicable when the incident electric field is polarized in y direction?

2. Lines 151 and 152:

"A planar wave front with a wavelength of λ = 633 nm propagating in air impinges normally on the surface, ζ = π/2."

Please add a similar comment regarding Figures 1 and 2.

3. Segmentation and convergence:

1) Please add some comments on the segmentation of the problem domain. Do you use a rectangular grid or a triangular grid?

2) How many cells have used in Figures 2 and 3?

Author Response

 

Authors presented a scalar Green's function approach to the problem of electromagnetic scattering by dielectric objects. The problem is of fundamental importance in numerical electromagnetics. Authors present the theory and implementation of the proposed method in a compact way.

My comments are as follows.

1. Applicability of the proposed method:

1) The proposed method is viable for the problems where the scattered wave has the same polarization as the incidence wave. Authors used dielectric cylinders illuminated by a vertically polarized planewave in their experiments where the above condition is met. If the incidence angle zeta is not 90 degrees, it appears that the method is not applicable.

We have found that in the case of plane-wave illuminated dielectric cylinders there is a good agreement between the simulations performed for both polarizations when either the permittivity agreement between the simulations performed for both polarizations when either the dielectric permittivity of the cylinder is low or when the distance to the screen is high of the cylinder is low or when the distance to the screen is high.

 

Please add comments in Section 2 on the kind of problems that the proposed method is applicable.

We appreciate this comment and have considered adding a relevant sentence to it in the introduction section. Thus, scalar methods are particularly applicable in the analysis of scattering in homogeneous media (or those in which the refractive index varies slowly over the thickness) and in the characterisation of materials through techniques such as spectroscopy [Born, M., & Wolf, E. (1999). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press].

 

 

 

 

 

2) Please add some comments on the extension of the proposed method to the vector scattering problem by applying the proposed method for each vector component. For example, is the proposed method applicable when the incident electric field is polarized in y direction?

The method is scalar, so it is independent of polarization. It is applicable in cases where there are no there are no obvious differences between the two polarizations, as in the cases mentioned above.

 

2. Lines 151 and 152:

"A planar wave front with a wavelength of λ = 633 nm propagating in air impinges normally on the surface, ζ = π/2."

Please add a similar comment regarding Figures 1 and 2.

The sentences preceding Figure 1 explain the general scheme for any angle ζ. On the other hand, it has been added that the wavelength used in Figure 2 is 633 nm.

 

3. Segmentation and convergence:

1) Please add some comments on the segmentation of the problem domain. Do you use a rectangular grid or a triangular grid?

2) How many cells have used in Figures 2 and 3?

The segmentation is carried out assuming that the dielectric object is composed of small circular scatterers (in the 2D case). Assigning at each point a value of the dielectric constant and slightly varying the refractive index values to avoid convergence problems due to boundary conditions. In the case of Figure 2, 4925 cells have been used to form the cylinder with the largest radius. And for figure 3, the one with half the radius, 2521 cells. If the reviewer considers it necessary, we can comment on the number of cells used in the article. Nevertheless, we have added a sentence explaining the segmentation at the end of section 2.2.

"Please see the attachment."

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper presents the Green function method for analyzing light scattering based on scalar function. The scatterers considered in this paper are infinitely long cylinders with variant cross-sections. Correspondingly, the scalar Green function is formulated in 2D system. Numerical results validate the formulation. The work is interesting to me. Therefore, I recommend its acceptance for publishing, after the revisions have been made.

1.         Commercial software is now available, such as FDTD, COMSOL, HFSS and etc., based on the methods FDTD, DDA and so on. A comparison between the presented Green scalar function method and the above-mentioned methods is appreciated, both in the computation efficiency and the accuracy. The authors should give the CPU time for finishing the calculation of the examples in section 3.

2.         In all the calculations, the vector field (the electric field) has only one component and hence it is solved naturally by the scalar field. However, in case the incident field is arbitrarily polarized, can the Green scalar function method work properly? It is suggested that the authors discuss how large the error is induced in simplifying the vector problem to the scalar one, e.g. by comparing the results of the scalar Green function method and the tensor Green function method.

3.         In line 55, “…a plane z = cte…” I do not understand what it means.

4.         Please give a citation for the integral in equation (7).

5.         In lines 123-124, mistakes in grammar shall be corrected.

6.         In figure 1, what is the mobile screen? Is it a movable screen?

7.         In figure 1, it is shown that the incident light propagates in xz-plane with a tilted angle. Corresponding to this arrangement, the axis in figure 2a should be x, and that in figure 2b should be y, isn’t it? Please check figure 3 too.

8.         The axes in figure 4b are not correctly numbered, corresponding to figure 4a. Please check figure 5 too.

Comments on the Quality of English Language

Some typoes should be corrected, e.g. "...in Figure 1. Where a plane wave polarized perpendicular impinges on...", "analyse" and "analyze"...

Author Response

The paper presents the Green function method for analyzing light scattering based on scalar function. The scatterers considered in this paper are infinitely long cylinders with variant cross-sections. Correspondingly, the scalar Green function is formulated in 2D system. Numerical results validate the formulation. The work is interesting to me. Therefore, I recommend its acceptance for publishing, after the revisions have been made.

 

1. Commercial software is now available, such as FDTD, COMSOL, HFSS and etc., based on the methods FDTD, DDA and so on. A comparison between the presented Green scalar function method and the above-mentioned methods is appreciated, both in the computation efficiency and the accuracy. The authors should give the CPU time for finishing the calculation of the examples in section 3.

We really appreciate reviewer comment, and we will implement it, but currently until September our group will not return to the offices where we have some of these software together with the method proposed in this article. Thus, as soon as we return, we can check the CPU execution times for both the scalar method and the vector method and even in the case of commercial software such as COMSOL or FDTD and present it properly.

 

2. In all the calculations, the vector field (the electric field) has only one component and hence it is solved naturally by the scalar field. However, in case the incident field is arbitrarily polarized, can the Green scalar function method work properly? It is suggested that the authors discuss how large the error is induced in simplifying the vector problem to the scalar one, e.g. by comparing the results of the scalar Green function method and the tensor Green function method.

We have found that in the case of plane-wave illuminated dielectric cylinders there is a good agreement between the simulations performed for both polarizations when either the permittivity agreement between the simulations performed for both polarizations when either the dielectric permittivity of the cylinder is low or when the distance to the screen is high dielectric permittivity of the cylinder is low or when the distance to the screen is high, so in these cases the method would be applicable for an arbitrary method would be applicable for arbitrary polarization in these cases.

 

3. In line 55, “…a plane z = cte…” I do not understand what it means.

We have modified that line to z=const., referring to a constant plane whose normal vector has only component in z-axis.

 

4. Please give a citation for the integral in equation (7).

Following the link below, https://dlmf.nist.gov/10.9#ii,  it can be seen that this integral is proportional to the integral 10.9.10 and proportional to the one presented in reference [16].

 

5. In lines 123-124, mistakes in grammar shall be corrected.

Now: “When a plane wave, polarized perpendicularly, impinges on the cylinder in the xz-plane with a tilt angle ζ the intensity pattern of the scattered light is measured on a screen.”

 

6. In figure 1, what is the mobile screen? Is it a movable screen?

Thank you for your comment. Indeed, now is corrected, showing movable screen.

 

7. In figure 1, it is shown that the incident light propagates in xz-plane with a tilted angle. Corresponding to this arrangement, the axis in figure 2a should be x, and that in figure 2b should be y, isn’t it? Please check figure 3 too.

No, following the coordinate system in Figure 1, we have that in the figure 2a we are moving the movable screen along the y-axis, making an axial scan, and in figure 2b we have in a point of the y-axis and we pick up in the x-axis, radial scanning.

 

8. The axes in figure 4b are not correctly numbered, corresponding to figure 4a. Please check figure 5 too.

We appreciate this comment and have considered changing the ticks labels of the figures and adding the names of the axes in the figures that did not have them.

 "Please see the attachment."

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Overall, the manuscript is well-written and study presented is interesting with sufficient references and bases. Thus, it can be accepted in its present form. 

I would however suggest that the authors include cases or examples to strongly support why the simplification to scalar form is advantageous over using vectors aside from complexity. It is also better what is/are compromised with this simplification. 

Author Response

Overall, the manuscript is well-written and study presented is interesting with sufficient references and bases. Thus, it can be accepted in its present form. 

I would however suggest that the authors include cases or examples to strongly support why the simplification to scalar form is advantageous over using vectors aside from complexity. It is also better what is/are compromised with this simplification. 

Thanks for the suggestion, we understand that the validity of the method has been checked using the case of a homogeneous cylinder only homogeneous cylinder only. In future research, further studies will be carried out with new geometries and compare the scalar method with the vector method in the possible cases.

"Please see the attachment." 

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

A formal method for solving electromagnetic scattering problems based on the scalar Green's function has been proposed. The electromagnetic scattering of media with different sizes, geometries, and refractive indices was discussed. Only theoretical calculations were performed, without any experiments.

The following issues exist:

1.     The Introduction section does not provide a sufficient background and description of previous work, and more papers from the past three years should be included. The Conclusion section is also missing.

2.     In lines 64 and 65, 'The procedure followed to obtain the scattered pattern is analogous to the one described in [16], except that we address the scalar case,' the reference [16] should be specified clearly. Similarly, in lines 99 and 100, 'In order to compare the numerical results, analytical solutions present in [17] are used,' the specific analytical results in reference [17] should be detailed.

3.     In line 100, the basis and reasons for using homogeneous cylinders should be clearly explained.

4.     In Figure 2, there is still a noticeable difference between the Green's function and the analytical solution curve proposed in Equation (17). Why does this phenomenon occur? An explanation is needed.

5.     The descriptions of Figures 2 and 3 should be clearer, such as specifying what the horizontal and vertical axes represent. Figures 4 and 5 are missing units on the axes.

6.     In line 157, 'Trying other geometries we opted for a prism with a rectangular base...' what exactly does this prism look like? A diagram should be provided for clarification.

7.     It is recommended to include a comparison with similar studies and a discussion on the advantages and disadvantages of other analytical methods.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The English expression meets the requirements and the sentences are clear and accurate.

Author Response

A formal method for solving electromagnetic scattering problems based on the scalar Green's

function has been proposed. The electromagnetic scattering of media with different sizes,

geometries, and refractive indices was discussed. Only theoretical calculations were performed,

without any experiments.

The following issues exist:

1. The Introduction section does not provide a sufficient background and description of previous work, and more papers from the past three years should be included. The Conclusion section is also missing.

We appreciate reviewer comment, and we have added 5 more current references to give a better context of the previous work done by the scientific community in the field of electromagnetic scattering and the numerical methods used. Following the instructions of the journal in relation to the conclusions section: “This section is not mandatory but can be added to the manuscript if the discussion is unusually long or complex.” Nevertheless, we have considered your comment and differentiated two sections, discussions and conclusions.

 

2. In lines 64 and 65, 'The procedure followed to obtain the scattered pattern is analogous to the one described in [16], except that we address the scalar case,' the reference [16] should be specified clearly. Similarly, in lines 99 and 100, 'In order to compare the numerical results, analytical solutions present in [17] are used,' the specific analytical results in reference [17] should be detailed.

We have detailed in which sections the equations consulted in the literature for this work are to be found.

 

3. In line 100, the basis and reasons for using homogeneous cylinders should be clearly

explained.

We have added an explanatory sentence saying that since both numerical and analytical results with cylindrical symmetries are available in the literature, we have chosen these geometries.

 

 

 

 

4. In Figure 2, there is still a noticeable difference between the Green's function and the

analytical solution curve proposed in Equation (17). Why does this phenomenon occur? An explanation is needed.

We really appreciate reviewers’ comment, and we have added this very pertinent point to the discussion section. We have to notice that the main difference between figures 2 and 3 are the dimensions of the dielectric cylinder, the one in figure 2 has twice the radius of figure 3. This means that the diffraction pattern is different, since according to the nature of the light it is logical to expect that the larger the size the more scattering modes are available and the zones derived from Fresnel propagation are subdivided into more regions, so that a more complex interference is obtained when it comes to picking up the scattered pattern. Therefore, there are some abrupt peaks in Figure 2 A in the axial representation derived from the resolution of the matrix system, as the dimensionality factor is considered in the equations. On the other hand, figure 3 A shows a much more smoothed profile that is much closer to the analytical result due to the size and its dependence on the system of equations.

 

 

5. The descriptions of Figures 2 and 3 should be clearer, such as specifying what the horizontal and vertical axes represent. Figures 4 and 5 are missing units on the axes.

We thank reviewer for this appreciation. In figures 2 A and 3 A the x-axis corresponds to the y-axis of the schematic in figure 1. Which is the axial distance from the cylinder to the screen where the scattered pattern is collected. On the other hand, the x-axis in Figures 2 B and 3 B refer to the x-axis of the coordinate system in Figure 1. That would be the radial distance or width of the screen where the diffracted pattern is collected. In all Figures 2AB and 3AB the y-axis corresponds to the normalized scattered intensity and in the title shows information about the dielectric medium and its position in the axes shown in Figure 1. On the other hand, Figures 4 and 5 have been modified by adding the axes and labels.

 

6. In line 157, 'Trying other geometries we opted for a prism with a rectangular base...' what exactly does this prism look like? A diagram should be provided for clarification.

We appreciate your comment and the need for clarification. Because the method is based on the 2D, planes have chosen to reduce the problem to the surface section of the dielectric medium. Therefore, a prism with a surface area of 20x30 micrometres of arbitrary height is studied by sticking to the plane of incidence of light.

 

7. It is recommended to include a comparison with similar studies and a discussion on the advantages and disadvantages of other analytical methods.

We appreciate this comment from the reviewer. Thus, in view of reviewer's comment number 2 and 4, we can add this comparison in a future article or revision. Comparing numerical methods and CPU execution times to show the advantages of this work, among other aspects. However, we do not return to our offices until the beginning of September for satisfying this requirement.

"Please see the attachment."

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

·       Line 30: I do not understand why to refer to a research work [15] to back up the idea that “any source can be described by the linear superposition of point sources”. This statement can be found in any textbook on mathematical physics or partial diff. equations. By the way, from such textbook it will be evident that there should be no minus sign in Eqs.(3) and (5).

·       Line 55: z=cte. Please correct.

·       Line 72: “We can then treat this singularity separately, rewriting the equation (10)”. The following Eq.(11) lacks the term including the integration over this small volume, while this term appears in  the discretized Eq.(12).

·       Eq.(17): The E⁽³⁾ quantity has not been defined, so it is not clear how the Eq.(17) is derived.

·       Line 119: “As described by [16].” – incomplete statement? Or this is the final phrase of the precedent statement? Please correct.

·       Lines 171-185: This is a conclusion rather than discussion of the results.

·       Appendix, line 213: “we can replace the vector r by the scalar r and give a spherically symmetric homogeneous solution”. Of course, we cannot! To obtain Eq.(A5) one should explicitly take into account the spheric symmetry directly in Eq.(A4), i.e. leave only the radial part of Laplacian there. I would recommend to remove this appendix at all since its only purpose is to prove Eq(5), and such a proof can be found in any textbook.

Comments for author File: Comments.pdf

Author Response

Line 30: I do not understand why to refer to a research work [15] to back up the idea that “any source can be described by the linear superposition of point sources”. This statement can be found in any textbook on mathematical physics or partial diff. equations. By the way, from such textbook it will be evident that there should be no minus sign in Eqs.(3) and (5).

We appreciate this comment and understand that it is too much of a stretch to cite a reference just to mention the overlapping principle. We have therefore chosen to delete it.

On the other hand, by the very definition of the integral of Green's functions, a sign convention can be established. We have therefore opted to keep this negative sign so that the equations have the form we want. If the reviewer finds it clearer to opt for a formulation without negative sign, we can modify equations (3) and (5) so that readers do not worry.

 

Line 55: z=cte. Please correct.

We have corrected this error, and have put z=const., referring to a constant plane.

 

Line 72: “We can then treat this singularity separately, rewriting the equation (10)”. The following Eq.(11) lacks the term including the integration over this small volume, while this term appears in  the discretized Eq.(12).

We understand your comment but there is no missing term. When we integrate, we are excluding that part of the volume differential, which subsequently appears as the expression in equation (13), where we refer to the points on the mesh at which the volume differential tends to zero. And by the definition of the integral of equation (11) we can separate the integral of the volume into two terms, that of the volume excluding the differential and that of the differential, the latter of which would refer to equation (13).

 

Eq.(17): The E(3) quantity has not been defined, so it is not clear how the Eq.(17) is derived.

This equation is derived from equation (2.23) of reference [17]. We have mentioned before presenting this equation that the function E(3) refers to the scattered field.

 

Line 119: “As described by [16].” – incomplete statement? Or this is the final phrase of the precedent statement? Please correct.

We have added this sentence to the previous one instead of splitting it up.

 

Lines 171-185: This is a conclusion rather than discussion of the results.

We have modified this section and differentiated between the discussions and conclusions section.

 

Appendix, line 213: “we can replace the vector r by the scalar r and give a spherically symmetric homogeneous solution”. Of course, we cannot! To obtain Eq.(A5) one should explicitly take into account the spheric symmetry directly in Eq.(A4), i.e. leave only the radial part of Laplacian there. I would recommend to remove this appendix at all since its only purpose is to prove Eq(5), and such a proof can be found in any textbook.

We appreciate reviewer 5 comment and have opted to delete the appendix due to its limited relevance.

"Please see the attachment."

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors revised the manuscript and now it looks better. 
I think it very important to show more information about the efficiency of the proposed method in numerical calculation, as I suggested in point 1 of the comments. The authors promised doing so when they return the office. Therefore, I would like to see this in near future. 

 

Author Response

See attached document

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have answered all the questions. However, the last question about comparison is still unsettled, because the authors do not return to their office until the beginning of September for satisfying this requirement.

Comments on the Quality of English Language

Minor revision is required.

Author Response

See attached document

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

I appreciate much the detailed responce (which has not been sufficiently discussed in the discussion section of the paper).

But now I can recommend its acceptence for publishing.