Resonant Transmission Through a Single Subwavelength Slit for Varied Metallic Permittivities and Its Modal Orthogonality

Round 1

Reviewer 1 Report

In this paper, resonant transmission through a single subwavelength slit is investigated by mode matching technique (MMT).

 

The transmission spectrum is investigated in the case that metal is perfect conductor, loss less material with negative permittivity, and realistic lossy material with complex permittivity. 

I think the results about transmission spectrum may be correct.

However, I think the results presented here related to the transmission spectrum do not include any novel fact in physics and useful information.

 

In the latter part of this paper, the orthogonality of lossy modes  in MIM waveguide is discussed and it is shown that the complete orthogonality is not satisfied in this type of waveguide.

However, I am not sure whether this is correct. First of all, I think the inner product of TM modes should be calculated as follows:

(1/2g)∫(1/n^2)Hy^n (Hy^m)^* dx

That is, I think that (1/n^2) is required.

Furthermore, in order to consider real power flow, real part of above inner product should be taken.

In addition, authors should more clearly show the significance about the fact that mode orthogonality is broken.

 

Author Response

We would like to thank the reviewers for their constructive comments and for taking time to review our paper. We have tried our best to revise the paper in accordance with the reviewers’ comments. Our responses are detailed below and specified by red in the revised manuscript.

 

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REVIEWERS' COMMENTS:

 

 

Response to Reviewer 1:

In this paper, resonant transmission through a single subwavelength slit is investigated by mode matching technique (MMT).

The transmission spectrum is investigated in the case that metal is perfect conductor, loss less material with negative permittivity, and realistic lossy material with complex permittivity. 

I think the results about transmission spectrum may be correct.

However, I think the results presented here related to the transmission spectrum do not include any novel fact in physics and useful information.

 

Reply and modifications:

 

We appreciate reviewer’s kind comments and believe that useful outcomes including physical meaning of our paper are as follows: First, we obtained the guide wavelengths for various slit geometries by the mode matching technique (listed in Tables 1 and 2) and systematically explained resonant transmission phenomena for each case. Then, periodic transmission peaks and attenuations were explained as the thickness of the plate increases. We have also looked at changes in guide wavelengths and resonant transmittances as the imaginary part of permittivities for the metal decrease, which cannot be obtained by other numerical techniques, such as the FDTD method or finite element method. Finally, the orthogonal relations were thoroughly studied for the lossless and lossy metallic plates. In our original manuscript, the novel aspects with useful outcomes have not been sufficiently described, so in the revised manuscript we addressed this issue in more detail as follows:

 

“The results using the MMT in this paper have several unique features compared to the previous studies in the following aspects. First, the resonant transmittance patterns are studied in detail as a function of the permittivity of the real metal when the metal is lossy. Then, periodic transmission peaks and attenuations are examined as the thickness of the plate increases. We also observe changes in guide wavelengths and resonant transmittances as the imaginary part of permittivities for the metal decreases, which cannot be obtained by other numerical techniques, such as the FDTD method or finite element method. Finally, the orthogonal relations are thoroughly studied for the lossless and lossy metallic plates.”

 

 

In the latter part of this paper, the orthogonality of lossy modes in MIM waveguide is discussed and it is shown that the complete orthogonality is not satisfied in this type of waveguide.

However, I am not sure whether this is correct. First of all, I think the inner product of TM modes should be calculated as follows:

(1/2g)∫(1/n^2)Hy^n (Hy^m)^*dx

That is, I think that (1/n^2) is required.

Furthermore, in order to consider real power flow, real part of above inner product should be taken.

 

Reply and modifications:

 

Reviewer-mentioned 1/n² is usually found in the study for the Green’s function [R1]. As we have checked, n is a modal index. When we inserted the modal index n into the Eq. (6), the orthogonal relations match for small n, but not for larger n, according to Eq. (6). On the other hand, the diagonal terms in the orthogonal relation by Eq. (6) of our manuscript are close to one. Therefore, we think our formulation for the orthogonality is correctly defined. When obtaining the orthogonal relation, we used [R2] as the reference.

 

[R1] C. A. Balanis, Advanced engineering electromagnetics, Chap. 14. Green’s function, 1989.

[R2] G. V. Eleftheriades, A. S. Omar, L. P. B. Katehi, and G. M. Rebeiz, “Some important properties of waveguide junction generalized scattering matrices in the context of the mode matching technique,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 10, pp. 1896 – 1903, 1994.

 

 

In addition, authors should more clearly show the significance about the fact that mode orthogonality is broken.

 

Reply and modifications:

 

Modal orthogonality is one of the important factors in the MMT when computing the electromagnetic phenomena as well as the slit transmittances. The accuracy of the resonant transmittance through the slit may be degraded if the orthogonality is not satisfied. For the geometry in this paper, although the orthogonality is not completely satisfied, it is not enough to affect the accuracy of transmittance. Despite this incomplete orthogonality, the transmission characteristics mainly determined by the fundamental mode, so that the accurate results can still be maintained. In contrast, the field profiles by higher order modes change more, but the higher modes do not significantly affect the slit transmission. We added this issue in the revised manuscript as follows:

 

“Although the orthogonality for the real metal case is not completely satisfied, it is not enough to affect the accuracy of transmittance. Despite this incomplete orthogonality, the transmission characteristics are mainly determined by the fundamental mode, so that the accurate results can still be maintained. The field profiles by higher order modes change more, but the higher modes do not significantly affect the slit transmission.”

 

Author Response File: Author Response.pdf

Reviewer 2 Report

In this manuscript, the authors provide results of modeling optical transmission through a real metal (i.e. not a perfect conductor) parameterized in terms of the plate thickness, imaginary part of permittivity, and slit width.  Numerical results are verified in terms of mode orthogonality.

Overall, the science content is difficult to follow, primarily because of a too-brief introduction.  Some discussion/description of the 'mode matching technique' (MMT) must be provided, preferably including some reference(s).  As it stands, it's unclear how the results were generated by the authors.

Similarly the authors use technical jargon without clear definitions: specifically, I don't understand the terms 'PS' (point spectrum) and 'DCS' (discretized continuous spectrum): I don't understand Figure 4(d) and 4(e).

Possibly related to this confusion is Figure 5; I don't understand why the upper table, corresponding to a perfect conductor, does not present exact delta-function orthogonality.  I suspect this is simply a problem of poor communication rather than any error.

A possibly related concern is the content of the first paragraph on page 6 ('It is necessary...'). I appreciate that the authors directly address the discrepancy. However, while the authors give a brief explanation for the differences, I think a better and stronger approach would be to include some additional details about the different models and their assumptions: again, a discussion of MMT versus 'brute force' may be all that is required.

Lastly, I would like to mention that overall, the writing is of high quality; I could only find one typographical error, Eqn 4 (line 93, page 4) should be 'dimensionless'.

In conclusion, I think the authors can greatly strengthen their manuscript simply by providing a better explanation about MMT.

 

Author Response

We would like to thank the reviewers for their constructive comments and for taking time to review our paper. We have tried our best to revise the paper in accordance with the reviewers’ comments. Our responses are detailed below and specified by red in the revised manuscript.

 

===============================

REVIEWERS' COMMENTS:

 

 

Response to Reviewer 2:

In this manuscript, the authors provide results of modeling optical transmission through a real metal (i.e. not a perfect conductor) parameterized in terms of the plate thickness, imaginary part of permittivity, and slit width. Numerical results are verified in terms of mode orthogonality.

Overall, the science content is difficult to follow, primarily because of a too-brief introduction. Some discussion/description of the ‘mode matching technique’ (MMT) must be provided, preferably including some reference(s). As it stands, it’s unclear how the results were generated by the authors.

 

Reply and modifications:

As the reviewer mentioned, we added how to obtain the slit transmittances by the MMT, unique features of this paper, and related references in the revised manuscript as follows:

 

“In this research, the procedure obtaining the transmittances by the MMT is as follows. First, the modes constituting each region for the free space and the metal-insulator-metal waveguide need to be obtained in advance. To calculate the modal eigenvalues in the MIM waveguide, the dispersion equation should be applied [9]. Then, the tangential incident, reflected, and transmitted electromagnetic fields are set to be continuous at each junction. The unknowns in the electromagnetic field equations can be solved though the system of equations, and the reflection and transmission coefficients can be determined. The overall reflected and transmitted electromagnetic fields of the single slit geometry can be obtained by using multiregion problem. The entire computing procedure is listed and explained in [10] in detail.

The results using the MMT in this paper have several unique features compared to the previous studies in the following aspects. First, the resonant transmittance patterns are studied in detail as a function of the permittivity of the real metal when the metal is lossy. Then, periodic transmission peaks and attenuations are examined as the thickness of the plate increases. We also observe changes in guide wavelengths and resonant transmittances as the imaginary part of permittivities for the metal decreases, which cannot be obtained by other numerical techniques, such as the FDTD method or finite element method. Finally, the orthogonal relations are thoroughly studied for the lossless and lossy metallic plates.

These resonant transmission or extraordinary transmission can be applied to practical devices like band selective spatial filter [17,18] and chemical sensor [19,20]. In addition, the controllable dual transmissions have attracted significant attention to the research societies because these phenomena can be practically applicable to the research areas such as switching, sensing, polarimetry microscopy, hyperspectral imaging, and security encryption [21–26].”

 

 

 

Similarly, the authors use technical jargon without clear definitions: Specifically, I don't understand the terms ‘PS’ (point spectrum) and ‘DCS’ (discretized continuous spectrum): I don’t understand Figure 4(d) and 4(e).

 

Reply and modifications:

 

The point spectrum (PS) and the discretized continuous spectrum (DCS) are explained in the revised manuscript as follows.

 

“The PS can describe the sinusoidal electromagnetic field patterns mainly in the ‘insulator’, while the DCS depicts the sinusoidal field patterns mainly in the ‘metal’ region [9,10].”

 

“Figure 4 (d) shows the field profiles for the first mode of the PS and may correspond to the first field profile of Figure 4 (b). The field shape in the white insulator region is almost maintained, however, the field shape in the sky-blue metal region is different. The field at the boundary between insulator and metal penetrates into the metal region because the metal is no longer PEC. Figure 4 (e) describes field profiles of the first, second, and third modes of the DCS, respectively. The field profiles represent sinusoidal harmonics in the metal region, and various field patterns in the metal region can be described by superposition of the modes in DCS.”

 

 

Possibly related to this confusion is Figure 5; I don’t understand why the upper table, corresponding to a perfect conductor, does not present exact delta-function orthogonality. I suspect this is simply a problem of poor communication rather than any error.

 

Reply and modifications:

As the reviewer pointed out, the orthogonal relation in Figure 5(a) is not the computational results by the PEC case. The results in Figure 5 (a) are obtained when ε_r is -14.88. If we consider a parallel plate waveguide surrounded by the PEC, orthogonal relations become 1 for diagonal elements and 0 for off-diagonal elements, respectively.

 

 

A possibly related concern is the content of the first paragraph on page 6 (‘It is necessary...’). I appreciate that the authors directly address the discrepancy. However, while the authors give a brief explanation for the differences, I think a better and stronger approach would be to include some additional details about the different models and their assumptions: again, a discussion of MMT versus ‘brute force’ may be all that is required.

 

Reply and modifications:

As the reviewer suggested, we modified the paragraph starting of ‘It is necessary ~’. We emphasized our solving procedure that is different from other numerical methods. We also added useful outcomes that cannot be obtained by other methods in the revised manuscript as follows:

 

“The results using the MMT in this paper have several unique features compared to the previous studies in the following aspects. First, the resonant transmittance patterns are studied in detail as a function of the permittivity of the real metal when the metal is lossy. Then, periodic transmission peaks and attenuations are examined as the thickness of the plate increases. We also observe changes in guide wavelengths and resonant transmittances as the imaginary part of permittivities for the metal decreases, which cannot be obtained by other numerical techniques, such as the FDTD method or finite element method. Finally, the orthogonal relations are thoroughly studied for the lossless and lossy metallic plates.”

 

 

Lastly, I would like to mention that overall, the writing is of high quality; I could only find one typographical error, Eqn 4 (line 93, page 4) should be ‘dimensionless’.

 

Reply and modifications:

As the reviewer noted, we corrected the equation as follows:

τ [dimensionless] = ~~~~(in the revised manuscript).           (4)

 

 

In conclusion, I think the authors can greatly strengthen their manuscript simply by providing a better explanation about MMT.

 

Reply and modifications:

As the reviewer mentioned, we have supplemented the explanations of the PS and the DCS, and how to obtain the slit transmittances in the revised manuscript.

Author Response File: Author Response.pdf

Reviewer 3 Report

ID: applsci-674297
Authors: Jong-Eon Park 1, Hosung Choo 2 and Young-Ki Cho 3

Title: Resonant Transmission Through A Single
Subwavelength Slit for Varied Metallic Permittivities
And Its Modal Orthogonality

The authors compute the transmission properties through a
sub-wavelength metallic sub-wavelength aperture including the
properties of a complex
permittivity of the metal. They claim to use a standard multi-scale
computing scheme. The first author references his/her own publication
"Analysis of deep-subwavelength Au and Ag slit transmittances at
terahertz frequencies", doi: 10.1364/JOSAB.33.001355 (2016)
The manuscript has indeed been cited 12 times, mainly self-citations,
one of them is

"Modal Analysis of Point and Discretized Continuous Spectra for
Metal-Insulator-Metal Waveguides in the Terahertz Region"
J Hur, H Choo, JE Park - Journal of Electrical Engineering &
Technology, 2018, these papers are not cited.

which seems to be the very same setup as is presented in this work here.

The authors should put "brute-force" as a non-scientific term in
quotation marks. The scientific language is not really standard in
this community, neither is it a textbook language.

The computation of a single slit as such is no computationally of
numerically complicated problem, neither is it novel, the complexity
of the solution however usually  depends on the used numerical
platform for 3D computation. While the authors describe in words what
the claimed to happen physics is, it would be great to have a 3D
computing result of it, which is more intuitive to most readers in the
field.

The manuscript is definitely short on references - and as such it is
overaged. This research field is definitely extremely active and there
is a whole industry of experimentalists and theoretical physicists,
optical engineers and quantum electronics engineers working on similar
problems. Thus the manuscript is not adequately embedded in the
research field, also the claimed merits have not been benchmarked
against existing work.

It would also be good to have some results for novel setups, which
have not been investigated in broad detail by the experimentalists
before.

As presented the manuscript will not attract a larger readership.

As such I believe the manuscript has severe shortcomings, and it
should be definitely revised before one could think of publication.

Author Response

We would like to thank the reviewers for their constructive comments and for taking time to review our paper. We have tried our best to revise the paper in accordance with the reviewers’ comments. Our responses are detailed below and specified by red in the revised manuscript.

 

===============================

REVIEWERS' COMMENTS:

 

 

Response to Reviewer 3:

The authors compute the transmission properties through a sub-wavelength metallic sub-wavelength aperture including the properties of a complex permittivity of the metal. They claim to use a standard multi-scale computing scheme. The first author references his/her own publication “Analysis of deep-subwavelength Au and Ag slit transmittances at terahertz frequencies,” doi: 10.1364/JOSAB.33.001355 (2016) The manuscript has indeed been cited 12 times, mainly self-citations, one of them is “Modal Analysis of Point and Discretized Continuous Spectra for Metal-Insulator-Metal Waveguides in the Terahertz Region,” J Hur, H Choo, JE Park - Journal of Electrical Engineering & Technology, 2018, these papers are not cited, which seems to be the very same setup as is presented in this work here.

 

Reply and modifications:

As the reviewer mentioned, the paper entitled “Analysis of deep-subwavelength Au and Ag slit transmittances at terahertz frequencies” has already been cited as [10], however, the paper titled “Modal analysis of point and discretized continuous spectra for metal-insulator-metal waveguides in the Terahertz region” is not included in the references. We appreciate the reviewer’s comment and have added this paper as the reference [11].

 

Hur, J.; Choo, H.; Park, J.E. Modal analysis of point and discretized continuous spectra for metal-insulator-metal waveguides in the terahertz region. J. Electr. Eng. Technol. 2018, 13, 1644-1654.

The authors should put “brute-force” as a non-scientific term in quotation marks. The scientific language is not really standard in this community, neither is it a textbook language.

 

Reply and modifications:

As the reviewer suggested, the term brute-force seems not to be a scientific language, although it is sometimes used to describe those numerical simulation methods. As the reviewer suggested, we put the quotation marks for the term “brute-force” in the revised manuscript.


The computation of a single slit as such is no computationally or numerically complicated problem, neither is it novel, the complexity of the solution however usually depends on the used numerical platform for 3D computation. While the authors describe in words what the claimed to happen physics is, it would be great to have a 3D computing result of it, which is more intuitive to most readers in the field.

 

Reply and modifications:

The reviewer's opinion is reasonable and we fully agree with the need for 3D analysis. Our opinion, however, is that after understanding of 2D slit transmission in detail, it is necessary to move on to the next step of the 3D problem in order to more correctly understand the physical phenomenon. In this paper, periodic resonant transmittances are obtained by deriving guide wavelengths for each 2D slit geometry, and the attenuated transmittance patterns are also fully explained for the lossy metallic slit cases. We believe these are very useful study and should be considered first in the 2D geometry. The transmittances for the 3D slit geometry are considered to be our future research topic.

The manuscript is definitely short on references - and as such it is overaged. This research field is definitely extremely active and there is a whole industry of experimentalists and theoretical physicists, optical engineers and quantum electronics engineers working on similar problems. Thus the manuscript is not adequately embedded in the research field, also the claimed merits have not been benchmarked against existing work.
It would also be good to have some results for novel setups, which have not been investigated in broad detail by the experimentalists before.
As presented the manuscript will not attract a larger readership.
As such I believe the manuscript has severe shortcomings, and it should be definitely revised before one could think of publication.

 

Reply and modifications:

As the reviewer pointed out, we included the more relevant references in the revised manuscript and also modified the introduction to emphasize novel aspects compared to the previous research, as follows:

 

“The results using the MMT in this paper have several unique features compared to the previous studies in the following aspects. First, the resonant transmittance patterns are studied in detail as a function of the permittivity of the real metal when the metal is lossy. Then, periodic transmission peaks and attenuations are examined as the thickness of the plate increases. We also observe changes in guide wavelengths and resonant transmittances as the imaginary part of permittivities for the metal decreases, which cannot be obtained by other numerical techniques, such as the FDTD method or finite element method. Finally, the orthogonal relations are thoroughly studied for the lossless and lossy metallic plates.”

 

“These resonant transmission or extraordinary transmission can be applied to practical devices like band selective spatial filter [17,18] and chemical sensor [19,20]. In addition, the controllable dual transmissions have attracted significant attention to the research societies because these phenomena can be practically applicable to the research areas such as switching, sensing, polarimetry microscopy, hyperspectral imaging, and security encryption [21–26].”

 

Guo, J.; Leong, H. Mode splitting of surface plasmon resonance in super period metal nanohole array gratings. Appl. Phys. Lett. 2012, 101, 241115. Azad, A.K.; O’Hara J.F.; Singh R.; Chen, H.-T.; Taylor, A.J. A review of terahertz plasmonics in subwavelength holes on conducting films. IEEE J. Sel. Top. Quantum Electron. 2012, 19, 8400416. Lee, D.; Kim, D.-S. Light scattering of rectangular slot antennas: parallel magnetic vector vs perpendicular electric vector. Sci. Rep. 2016, 6, 18935. Hur, J.; Choo, H.; Park, J.E. Modal analysis of point and discretized continuous spectra for metal-insulator-metal waveguides in the terahertz region. J. Electr. Eng. Technol. 2018, 13, 1644-1654. Rim, J.-W.; Koh, I.-S. SAR image generation of ocean surface using time-divided velocity bunching model. J. Electromagn. Eng. Sci. 2019, 19, 82-88. Seo, S. M. An IE-FFT algorithm to analyze PEC objects for MFIE formulation. J. Electromagn. Eng. Sci. 2019, 19, 6-12. Tareki, A.M.; Lindquist, R.G.; Kim, W.; Heimbeck, M.S.; Guo, J. Terahertz transparent electrode using tripod metal aperture array. IEEE Trans. Terahertz Sci. Technol. 2016, 7, 80-85. Zarei, S. A design to tune the frequency in a terahertz filter based on dual layered metallic slit arrays. Photon. Nanostruct. Fundam. Appl. 2019, 34, 5-10. Dhawan, A.; Gerhold M.D.; Muth, J.F. Plasmonic structures based on subwavelength apertures for chemical and biological sensing applications. IEEE Sens. J. 2008, 8, 942-950. Larson, S.; Carlson, D.; Ai B.; Zhao Y. The extraordinary optical transmission and sensing properties of Ag/Ti composite nanohole arrays. Phys. Chem. Phys. 2019, 21, 3771-3780. Alfalou, A.; Brosseau, C. Dual encryption scheme of images using polarized light. Opt. Lett. 2010, 35, 2185-2187. Wang, Y.; Tong, Y.; Zhang, X. Transmission properties of terahertz waves through asymmetric rectangular aperture arrays on carbon nanotube films. AIP adv. 2016, 6, 045304. Nakata, Y.; Urade, Y.; Okimura, K.; Nakanishi, T.; Miyamaru, F.; Takeda, M.W.; Kitano, M. Anisotropic Babinet-invertible metasurfaces to realize transmission reflection switching for orthogonal polarizations of light. Phys. Rev. Appl. 2016, 6, 044022. Pelzman, C.; Cho, S.-Y. Multispectral and polarimetric photodetection using a plasmonic metasurface. J. Appl. Phys. 2018, 123, 043107. Lee, G.; Maeng, I.; Kang, C.; Oh, M.-K.; Kee, C.-S. Strong polarization-dependent terahertz modulation of aligned Ag nanowires on Si substrate. Opt. Express 2018, 26, 13677-13685. Pattanayak A.; Rana, G.; Jain, R.; Bhattacharya A.; Duttagupta, S.P.; Gandhi, P.S.; Achanta, V.G.; Prabhu, S.S. Resonant THz transmission through asymmetric aperture array with polarization controlled resonant peaks and Q-factors. J. Appl. Phys. 2019, 126, 223103.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I think the authors have revised their manuscript to try to make the important things clear. However, I feel the novelty of this paper is still weak. On the other hand, I feel it is OK to be published if the editor think the revision of this paper is enough.

Followings are my comments.

I think the obtained results in this paper can be easily  predicted from the well-known results reported in the past research.

I think the mode orthogonality should be defined for Poynting vector. In that mean, Sz = Ex Hy^* ∝ (1/n^2) Hy Hy^* should be evaluated. In addition, the integral should be taken over entire waveguide cross-section instead of only gap region. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

ID: applsci-674297 Authors: Jong-Eon Park , Hosung Choo and Young-Ki Cho Title: Resonant Transmission Through A Single Subwavelength Slit for Varied Metallic Permittivities And Its Modal Orthogonality I carefully read the authors response as well as the revised version of the manuscript, and I found that the manuscript has not been sufficiently revised. It does not help to answer a reviewers question extensively without revising the paper equivalently. Further I have to admit that the revised passages do not contribute to the understanding of the manuscript, since the scientific language these authors use in this manuscript here differs profoundly from what is found in fundamental literature. Additionally, the presentation of the theory in general lacks clarity. The claimed to be found results - which according to the authors cannot be found by other computing techniques such as FDTD - are neither explained nor are they benchmarked against other techniques. The novelty of the results is not recognizable. This manuscript is not clear enough to be published.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf