Modeling of Subwavelength Gratings: Near-Field Behavior

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Chernyavsky et al report a method to model the subwavelength grating, with special attention paid to the near-field behaviour. The core of this work is the analytical expressions of the near field distribution by using 2-dimensional point dipole approximation and perturbation theory. Overall, I think this work is comprehensive and interesting and can be published on Photonics if the authors can address my following concerns.

Specifically,

1.     Numerical approaches such as FDTD have been widely adopted which can accurately capture the optical response of nanostructures. What is the very reason for deriving the analytical expressions? Any insights can be drawn that may benefit the structure design?

2.     The equations in the manuscript should be properly labelled. In its current form, it is really hard to understand the calculation process.

3.     Throughout the manuscript, the authors discussed the scattering under p-wave illumination. What would be the results for s-wave illumination?

4.     Page 3 Line 93: “The polarizability of a cylinder aligns well with the PDA and ultimately results in an effective lattice polarizability coefficient that differs from that of an individual cylinder.” This sentence sounds incorrect in grammar. Besides, this statement is not well documented.

5.     Page 4 Line 109: I do not understand how this equation was derived. For example, where comes the pi?

6.     Page 5 Line 138: The undocumented statements also include “It’s important to note that this model results in an overestimation of the field intensity near the point dipole itself.” I do not find any reason for this.

 

7.     Page 7 Line 183: “This minimum 182 results from strong reflection from the SWG.” This statement is not well supported by either literature or proper arguments.

Comments on the Quality of English Language

N.A.

Author Response

Chernyavsky et al report a method to model the subwavelength grating, with special attention paid to the near- feld behaviour. The core of this work is the analytical expressions of the near 
field distribution by using 2-dimensional point dipole approximation and perturbation theory. Overall, I think this work is comprehensive and interesting and can be published on Photonics if the authors can address my following concerns.
1. Numerical approaches such as FDTD have been widely adopted which can accurately capture the optical response of nanostructures. What is the very reason for deriving the
analytical expressions? Any insights can be drawn that may bene fit the structure design?
(We insert some text into the 3rd paragraph of the Introduction to explain why the new analytical approach is helpful for structure design.)
2. The equations in the manuscript should be properly labelled. In its current form, it is really hard to understand the calculation process.
( Now all the equation are numbered.)
3. Throughout the manuscript, the authors discussed the scattering under p-wave illumination. What would be the results for s-wave illumination?
( It is possible to get the result for s-wave, but we do not
want to overload the manuscript. We explain the motivation)
to study p-wave scattering in the 4th paragraph of the Introduction.)
4. Page 3 Line 93: "The polarizability of a cylinder aligns well with the PDA and ultimately results in an e ective lattice polarizability coeffcient that di ers from that of an individual cylinder." This sentence sounds incorrect in grammar. Besides, this statement is not well documented.
( The Referee is right; we have rewritten the sentence.
5. Page 4 Line 109: I do not understand how this equation was derived. For example, where comes the pi?
( In order to explain the appearance of π, we add several
words, Eq.(6), and Ref.[30].)
6. Page 5 Line 138: The undocumented statements also include "It's important to note that this model results in an overestimation of the  field intensity near the point dipole itself."
I do not  find any reason for this.
( The Referee is right. We exclude this statement.)
7. Page 7 Line 183: "This minimum 182 results from strong refection from the SWG." This statement is not well supported by either literature or proper arguments. (We delete this unsupported statement.)

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript, the authors report a point dipole approximation (PDA) method to characterize the subwavelength gratings. In general, the manuscript is well-written and presented. If my main concerns below are well addressed, I would like to recommend its publication.

 1.      In the introduction part, Discrete Dipole Approximation method should be introduced and compared with PDA method. The novelty of this work needs to be stressed.

 2.      The programming method should be indicated in the manuscript like Python or Matlab.

 3.      Parameter study should be done to give a clear comparison among PDA, perturbation theory, and numerical modeling methods. Especially, the authors also mention the importance of geometric parameters on the calculation accuracy of PDA.  

 4.      The authors mention the limitation of effective medium theory on the resonant behavior, however, transmission or reflection spectra are not presented at all.

 5.      Some references should be added for the readers to understand the derivation procedure of equations.

 6.      The authors need to give a summation on the scattering characteristics of subwavelength gratings by their investigation. Further development of PDA should be indicated in the conclusion part.

 7.      Some recent and related references on metasurfaces and near-field imaging, may help enrich the introduction, e.g. doi: 10.1364/OE.422112; doi: 10.29026/oea.2022.220058; doi: 10.29026/oes.2022.210003

Author Response

1. In the introduction part, Discrete Dipole Approximation method should be introduced and compared with PDA method. The novelty of this work needs to be stressed.
( Now, in the 2nd and 3rd paragraphs of Section 2, we introduce and compare the PDA method with DDA.)
2. The programming method should be indicated in the manuscript like Python or Matlab.
( We employed three computer platforms for calculations of
our plots. They are FORTRAN (BEM) [31], COMSOL (FEM)
[32], and Wolfram Mathematica for analytical expressions.)
3. Parameter study should be done to give a clear comparison among PDA, perturbation theory, and numerical modeling methods. Especially, the authors also mention the importance
of geometric parameters on the calculation accuracy of PDA.
( We use dimensionless variables to show the levels in theoretica Fig. 2. All the physical and geometrical parameters in figures 3 − 5 are ε = 2.25, λ = 1.512 μm, a = 50 nm, L = 150 nm. In Fig. 6 the radius is a = 70 nm (narrow slit). We declare them in the captions. Calculation parameters are following:
360 points along the circle in BEM, 221 angular harmonics in the perturbation theory. For FEM, the COMSOL constructed
a grid of triangles with a side of 2 nm. We have included the
calculation parameters into the first paragraph of Section 4.)
4. The authors mention the limitation of e ective medium theory on the resonant behavior,
however, transmission or re ection spectra are not presented at all.
( We have calculated the reflection and transmission spectra
within the frames of effective medium theory earlier, see [23].
The near-field intensity distribution was the only unsolved
problem. In the present paper, we try to solve it by comparing
PDA, the Born approximation, BEM, and FEM.)
5. Some references should be added for the readers to understand the derivation procedure of equations.
( We have now numbered all the equations and added corresponding references to Section 2.)
6. The authors need to give a summation on the scattering characteristics of subwavelength gratings by their investigation. Further development of PDA should be indicated in the
conclusion part.
( We have added the paragraph about the further development into the end of conclusions.)
7. Some recent and related references on metasurfaces and near- eld imaging, may help enrich the introduction, e.g. doi: 10.1364/OE.422112; doi: 10.29026/oea.2022.220058; doi:
10.29026/oes.2022.210003
(We add all these references at the end of the first paragraph
in the Introduction. We also have edited the manuscript in order to improve English.)

Reviewer 3 Report

Comments and Suggestions for Authors

In this paper, the Authors present an analytical method to calculate the near field of a subwavelength cylindrical plasmonic grating. They also propose a perturbative approach based on Fourier decomposition to estimate the field inside the metallic cylinders. Although the mathematical treatment of the problem seems rigorous and the presented results correct, the main problem of this work is, in my opinion, the benefits that such an analytical method holds. The presented method is derived only for a single polarization under normal incidence and for a grating consisting of cylindrical elements. What about other element configurations, different E-field polarization or oblique incidence? Furthermore, the Authors do not explain why the proposed analytical method is better than the established computational methods like FDTD, BEM or FEM. All three methods can routinely reproduce the near field of an element for such a simple 2D configuration, and quite fast I’d rather add.

Having said all that, I believe that, scientifically speaking, the Article worth publication but I do not believe that it will be appreciated, at least not without addressing the issues mentioned above. Determining the range of applicability and the benefits of a new (computational) approach is equally important with the development of the method itself.

Comments on the Quality of English Language

There are a few problems with the language and an extensive proof reading is in order.

Author Response

1. In this paper, the Authors present an analytical method to calculate the near  field of a subwavelength cylindrical plasmonic grating. They also propose a perturbative approach based on Fourier decomposition to estimate the  field inside the metallic cylinders. Although the mathematical treatment of the problem seems rigorous and the presented results correct, the main problem of this work is, in my opinion, the bene ts that such an analytical method holds. The presented method is derived only for a single polarization under normalincidence and for a grating consisting of cylindrical elements.
(We insert some text into the 3rd paragraph of the Introduction
to explain why the new analytical approaches are quite
an urgent problem now.)
2. What about other element con gurations, different E- field polarization or oblique incidence? Furthermore, the Authors do not explain why the proposed analytical method is better than the established computational methods like FDTD, BEM or FEM. All three methods can routinely reproduce the near  field of an element for such a simple 2D con guration, and quite fast I'd rather add. Having said all that, I believe that, scienti cally
speaking, the Article worth publication but I do not believe that it will be appreciated, at least not without addressing the issues mentioned above. Determining the range of appli-
cability and the bene ts of a new (computational) approach is equally important with the development of the method itself.
(We add two sentences into the 4th paragraph of Introduction
about the other state of polarization.)

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The Authors did not convince me regarding the necessity or the advantages of their developed computational method. However, I do not have a strong scientific objection so I can propose rejection for such reasons. The work can be published despite my feeling that it won’t be appreciated.