Terahertz Generation through Coherent Excitation of Slow Surface Waves in an Array of Carbon Nanotubes

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the paper "Terahertz generation through coherent excitation of slow surface waves in an array of carbon nanotubes" by S.A. Afanas'ev et al., the authors propose a scheme for generating terahertz (THz) electromagnetic radiation. The underlying mechanism of the proposed scheme involves the interaction of a direct current (DC) with surface plasmon-polaritons (SPPs) excited in the double-walled carbon nanotubes (DWCNTs). The concept introduced in this work is the synchronization of SPPs in the DWCNT array, achieved through external laser radiation. In essence, the proposed generation scheme resembles a laser mechanism. The active medium here is the DWCNT array, wherein SPPs are excited by a DC. These SPPs are inherently incoherent. The external radiation from two laser sources, providing a frequency difference, establishes distributed feedback, synchronizing the initially incoherent plasmon waves in the DWCNTs.

 

This idea appears promising as existing literature on enhancing SPPs using a DC typically focuses on either graphene deposited on a substrate [T. A. Morgado and M. G. Silveirinha. Drift-Induced Unidirectional Graphene Plasmons. ACS Photonics 2018, 5, 11, 4253–4258], bilayer graphene [T. A. Morgado and M. G. Silveirinha, Phys. Rev. Lett. 119, 133901 (2017)] (both requiring intricate fabrication processes), or individual carbon nanotubes [A. M. Nemilentsau, G. Ya. Slepyan, and S. A. Maksimenko. Phys. Rev. Lett. 99, 14740] which have low emitted power.

 

In my opinion, the manuscript deserves publication in Photonics.

 

However, the following issues need to be addressed by the authors before the publication:

 

*  The authors employ a conductivity model describing flat graphene. How can this model be applied to DWCNTs?

*  How would the heterogeneity of a real DWCNT array affect the characteristics of the generated radiation?

*  The paper suggests the use of two laser beams to provide a frequency difference. What specific parameters of lasers can be used? What could be the range of this frequency difference and, consequently, the generated radiation?

*  The authors essentially employ a hydrodynamic approach, used in several works like [T. A. Morgado and M. G. Silveirinha, Phys. Rev. Lett. 119, 133901 (2017)]. In the cited work, an explicit expression for conductivity dependent on drift current is derived. In the presented paper, the conductivity expression lacks the drift velocity. How do the results of this study compare with findings from other works, especially with [T. A. Morgado and M. G. Silveirinha, Phys. Rev. Lett. 119, 133901 (2017)]?

Comments on the Quality of English Language

Minor proofreading is necessary

Author Response

In the paper "Terahertz generation through coherent excitation of slow surface waves in an array of carbon nanotubes" by S.A. Afanas'ev et al., the authors propose a scheme for generating terahertz (THz) electromagnetic radiation. The underlying mechanism of the proposed scheme involves the interaction of a direct current (DC) with surface plasmon-polaritons (SPPs) excited in the double-walled carbon nanotubes (DWCNTs). The concept introduced in this work is the synchronization of SPPs in the DWCNT array, achieved through external laser radiation. In essence, the proposed generation scheme resembles a laser mechanism. The active medium here is the DWCNT array, wherein SPPs are excited by a DC. These SPPs are inherently incoherent. The external radiation from two laser sources, providing a frequency difference, establishes distributed feedback, synchronizing the initially incoherent plasmon waves in the DWCNTs. 

This idea appears promising as existing literature on enhancing SPPs using a DC typically focuses on either graphene deposited on a substrate [T. A. Morgado and M. G. Silveirinha. Drift-Induced Unidirectional Graphene Plasmons. ACS Photonics 2018, 5, 11, 4253–4258], bilayer graphene [T. A. Morgado and M. G. Silveirinha, Phys. Rev. Lett. 119, 133901 (2017)] (both requiring intricate fabrication processes), or individual carbon nanotubes [A. M. Nemilentsau, G. Ya. Slepyan, and S. A. Maksimenko. Phys. Rev. Lett. 99, 14740] which have low emitted power. 

In my opinion, the manuscript deserves publication in Photonics.

However, the following issues need to be addressed by the authors before the publication:

1/ The authors employ a conductivity model describing flat graphene. How can this model be applied to DWCNTs?

Answer:

The structure of multi-walled carbon nanotubes (MWCNTs) has been thoroughly studied using X-ray diffraction methods [1], revealing that MWCNTs are essentially cylindrical structures with diameters ranging from several to tens of nanometers and an interlayer distance of 0.34 nm. Strong covalent bonds connect the carbon atoms within each layer, while weaker van der Waals forces hold the layers together. These forces have minimal effect on the electron shells and band structure of the individual graphene layers that make up MWCNTs. It has been shown that the static conductivity along the layers is significantly higher than that between layers [2]. Furthermore, the curvature of MWCNTs walls (rolled graphene layers) has little effect on the band structure for nanotubes with 1 nm or larger diameters. Therefore, it is quite reasonable to use formulas for surface conductivity initially defined for flat graphene to model the conductivity of large-diameter carbon nanotubes (over 1 nm).

[1] Furuta H., et.al., Appl. Phys. Express (2010), 3, 105101, doi:10.1143/APEX.3.105101.

[2] Bourlon B., et.al., Phys. Rev. Lett. (2004), 93, 1–4, doi:10.1103/PhysRevLett.93.176806.

2/ How would the heterogeneity of a real DWCNT array affect the characteristics of the generated radiation?

Answer:

Our generator model assumes homogeneity of the CNTs for both optical excitation of SPPs (Eq. (4) in the manuscript) and SPP amplification by DC pumping (see Fig. 2(a) and its comments in the manuscript). Under these conditions, the SPP amplitude and the intensity of THz radiation emitted by the CNT array into the surrounding space will be maximized. The more homogeneous the CNT array, the closer our results will be to reality. Naturally, parameter spreading, including due to the non-identical nature of a real DWCNT array, will reduce the efficiency of excitation and amplification. However, our scheme allows for some parameter deviation from the "ideal" values while still maintaining SPP generation. For instance, there's some flexibility in the parameters of the laser beams used to excite SPPs in the CNT array (see Fig. 4 in the manuscript). The feature of SPP amplification by pumping current in such structures is that energy transfer from charge carriers to the surface electromagnetic wave occurs over a wide frequency range – see, for example, Fig. 2b in Ref. [Kadochkin, A.S.; Moiseev, S.G.; Dadoenkova, Y.S.; Svetukhin, V. V.; Zolotovskii, I.O. Surface Plasmon Polariton Amplification in a Single-Walled Carbon Nanotube. Opt Express (2017) 25, 27165, doi:10.1364/OE.25.027165] and also Fig. 5b and its comments in our manuscript. To account for array non-uniformity, it is necessary to introduce geometric parameters characterizing the degree of deviation (dispersion) of CNT sizes from the average (e.g., for length, radii), which could be the subject of future studies.

3/ The paper suggests the use of two laser beams to provide a frequency difference. What specific parameters of lasers can be used? What could be the range of this frequency difference and, consequently, the generated radiation?

Answer:

The main characteristics of the lasers, which have specific requirements, are as follows:

a) To control the frequency of excited SPPs and, consequently, the frequency of THz radiation, it is desirable for one of the laser's wavelengths to be tunable. However, there is also an additional possibility to control the radiation parameters of the structure by adjusting the incidence angles of the laser beams. The frequencies of the laser sources and their detuning are related by Eq. (4) (the phase-matching condition). The range of the frequency difference is ω < ω_max, where ω_max is given by Eqs. (5) and (6). For the example considered in the manuscript ω_max ~0.0057ω₁, i.e. ω_max= 2.19 THz for the wavelength λ₁= 0.775 µ

b) The laser radiation is assumed to be linearly polarized, with the electric vector having a longitudinal component (relative to the axes of the nanotubes) for effective excitation of currents on the nanotube walls.

4/ The authors essentially employ a hydrodynamic approach, used in several works like [T. A. Morgado and M. G. Silveirinha, Phys. Rev. Lett. 119, 133901 (2017)]. In the cited work, an explicit expression for conductivity dependent on drift current is derived. In the presented paper, the conductivity expression lacks the drift velocity. How do the results of this study compare with findings from other works, especially with [T. A. Morgado and M. G. Silveirinha, Phys. Rev. Lett. 119, 133901 (2017)]?

Answer:

We would like to note that we do not explicitly obtain the nanotube's conductivity considering the drift current, as done in the Refs. [1,2]. Instead, we consider the influence of the drift current on SPP dispersion by solving the system of Eqs. (7) and (8). This approach and those used in [2,3] allow us to describe SPP instability (see Fig. 5 in the manuscript) in the presence of drift current and negative Landau damping. Therefore, it is important to note that within the framework of our approach, the interaction between the drift current (with the corresponding drift velocity) and the SPP wave is correctly accounted for Eq. (2) of the manuscript relates to a system without drift current and should not be directly compared with conductivity expressions from [4-6].

[1] T. A. Morgado, M. G. Silveirinha, ACS Photonics 5, 4253–4258 (2018), doi:10.1021/ acsphotonics.8b00987

[2] T. A. Morgado, M. G. Silveirinha, Phys. Rev. Lett. 2017, 119, 133901 (2007), doi:10.1103/ PhysRevLett.119.133901

[3] D. Svintsov, Phys. Rev. B, 100, 195428 (2019), doi:10.1103/PhysRevB.100.195428

[4] D. Svintsov, PRB 97, 121405(R) (2018), doi:10.1103/PhysRevB.97.121405

[5] D. Svintsov, PRB 100, 195428 (2019), doi:10.1103/PhysRevB.100.195428

[6] T. Morgado, PRB 102, 075102 (2020), doi:10.1103/PhysRevB.102.075102

Reviewer 2 Report

Comments and Suggestions for Authors

The authors of this paper theoretically investigated the generation of this radiation in aligning CNT via surface plasmon polaritons amplification. 

Overall, the article shows a scientifically important result. All formulas are well reasoned via certain assumptions. Meanwhile. I would like to make some small comments on this paper:

1) In introduction different paragraphs are not logically linked. Therefore, I would suggest to rewrite it a bit. 

2) Generation of the THz radiation in CNT array is well known. Could authors provide the comperison between their parameters and parameters obtained by another methids (optical ractification, etc.). Some figure of merits are required to highlight the importance of the results. 

3) Also,  it is hard to read the article as thre is no introdction sentence to the formulas, why you need to use it, what you would like to show using these formulas?

Comments on the Quality of English Language

I am not able to fully judge the English quality. However, for my level of English it sounds good enough.  

Author Response

The authors of this paper theoretically investigated the generation of this radiation in aligning CNT via surface plasmon polaritons amplification. 

Overall, the article shows a scientifically important result. All formulas are well reasoned via certain assumptions. Meanwhile. I would like to make some small comments on this paper:

 1/ In introduction different paragraphs are not logically linked. Therefore, I would suggest to rewrite it a

Answer:

Following your suggestion, we have revised and restructured the "Introduction" section for better logical coherence. In the revised manuscript, the Introduction is organized as follows:

Paragraph 1 introduces the research object (CNT arrays) and their technical applications.

Paragraph 2 discusses the properties of CNTs that underpin these applications.

Paragraph 3 elaborates on the dual-beam THz generator scheme based on the CNT array.

Paragraph 4 explores the possibility of using the amplification effect of SPP in CNTs with DC pumping.

Paragraph 5 presents a new THz generation scheme that combines the two concepts above of excitation and amplification of SPP in CNTs.

Paragraph 6 outlines the structure of the manuscript.

The revised text is marked in red.

2/ Generation of the THz radiation in CNT array is well known. Could authors provide the comparison between their parameters and parameters obtained by another methods (optical ractification, etc.). Some figure of merits are required to highlight the importance of the results. 

Answer:

The approach proposed in our work significantly differs from the widely known methods of THz radiation generation [1-6], which are based on irradiating a target with short laser pulses. The basis of conventional laser pulse generation is "optical rectification" [7], where a wave at a near-zero difference frequency is formed due to quadratic nonlinearity. In contrast, the efficiency of the generation mechanism proposed in our manuscript is not directly related to energy conversion from external laser sources, which only serves to create distributed feedback. THz generation occurs due to the interaction of SPP with the drift current, acting as a pump, and this interaction can proceed continuously. Our amplification scheme inherently lacks the limitations associated with the low energy efficiency of nonlinear processes in conventional laser pulse generation. Additionally, since our proposed scheme utilizes the current pumping of SPP, the total power of the generated radiation can be high due to the high conductivity of CNTs.

Our manuscript includes an appropriate estimation of the emitted power.

[1] J. A. Fülöp, et. al., Opt. Lett. 37, 557-559 (2012), doi:10.1364/ol.37.000557

[2] S. Huang, et. al., Carbon, 132, 335-342 (2018), doi:10.1016/j.carbon.2018.02.067

[3] W.-M. Wang, et. al., Opt. Express 16, 16999-17006 (2008), doi:10.1364/OE.16.016999

[4] M. Hassani and F. Jahangiri, Opt. Express 29, 38359-38375 (2021), doi:10.1364/oe.442168

[5] J. Parashar, et. al., Physica E, 44, 2069-2071 (2012), doi:10.1016/j.physe.2012.06.013

[6] L.V. Titova, et. al., Nano Lett., 15, 5, 3267–3272 (2015), doi:10.1021/acs.nanolett.5b00494

[7] M. Bass, et. al., Phys. Rev. Lett. 9, 446 (1962), doi:10.1103/PhysRevLett.9.446

3/ Also, it is hard to read the article as there is no introduction sentence to the formulas, why you need to use it, what you would like to show using these formulas?

Answer:

This manuscript continues a series of our previous works [1-3], where the analytical methods and equations used were detailed. In this manuscript, only the main formulas are presented with necessary brief explanations. This approach allows us to sufficiently describe the methods used while focusing attention on the new research results.

[1] Kadochkin A.S., et. al., Surface Plasmon Polariton Amplification in a Single-Walled Carbon Nanotube. Opt. Express 2017, 25, 27165, doi:10.1364/oe.25.027165

[2] Dadoenkova Y.S., et. al., Surface plasmon polariton amplification in semiconductor–graphene–dielectric structure // Ann. Phys. (Berlin), V. 529, No. 5, pp. 1700037 (1-7) (2017) http://dx.doi.org/10.1002/andp.201700037

[3] Moiseev S.G, et. al., Generation of Slow Surface Plasmon Polaritons in a Complex Waveguide Structure with Electric Current Pump. Ann. Phys. 2018, 530, 1800197, doi:10.1002/andp.201800197.

Reviewer 3 Report

Comments and Suggestions for Authors

Afanas’ev et al. studied terahertz generation through coherent excitation of slow surface waves using an array of double-walled carbon nanotubes. They first go a good overview of the previous work on the field, then perform both theoretical calculation and numerical simulations to show that longitudinal surface plasmon polaritons (SPPs) can be coherently excited in double-walled carbon nanotubes with specific dimensions by using two near-infrared laser beams with slightly different frequencies. The amplification of these surface plasmon polaritons can be facilitated and controlled using a DC current flowing through the CNTs. They perform their analysis as a function of the angle of incidence of the laser beams.

They are also showing that the amplification of the SPPs requires that the phase velocity of the is closely matching the drift velocity of the charge carriers in the CNTs, a condition needed for efficient energy exchange between the current and the surface electromagnetic waves.

The paper is well written and is an extension of the work performed previously by some of the authors in this paper.

One main question the authors should address is how easily the proposed scheme could be realized in practice. The double-walled carbon nanotubes in the array will never be exactly identical and controlled the separation between them may be an impossible task. So, the authors should address how the non-uniformity in the CNT array would affect the claims made in the paper.

 

Author Response

Afanas’ev et al. studied terahertz generation through coherent excitation of slow surface waves using an array of double-walled carbon nanotubes. They first go a good overview of the previous work on the field, then perform both theoretical calculation and numerical simulations to show that longitudinal surface plasmon polaritons (SPPs) can be coherently excited in double-walled carbon nanotubes with specific dimensions by using two near-infrared laser beams with slightly different frequencies. The amplification of these surface plasmon polaritons can be facilitated and controlled using a DC current flowing through the CNTs. They perform their analysis as a function of the angle of incidence of the laser beams.

They are also showing that the amplification of the SPPs requires that the phase velocity of the is closely matching the drift velocity of the charge carriers in the CNTs, a condition needed for efficient energy exchange between the current and the surface electromagnetic waves.

The paper is well written and is an extension of the work performed previously by some of the authors in this paper.

1/ One main question the authors should address is how easily the proposed scheme could be realized in practice. The double-walled carbon nanotubes in the array will never be exactly identical and controlled the separation between them may be an impossible task. So, the authors should address how the non-uniformity in the CNT array would affect the claims made in the paper.

Answer:

Thank you for your insightful comments and the main question regarding the practical realization of our proposed scheme. We understand your concerns about the uniformity and alignment of double-walled carbon nanotubes (DWCNTs) in an array, and we appreciate the opportunity to address this important aspect.

Indeed, the higher the homogeneity of the CNT array and the consistency in the parameters of the CNTs, the closer our results will be to real-world applications. To account for the non-uniformity of the array, it is necessary to introduce geometric parameters that characterize the degree of deviation (dispersion) in the sizes of the CNTs from the average (for example, in terms of length and radius). This could be the subject of the following publications on this topic.

In our paper, we chose what we believe to be the most straightforward and relevant approach for modeling, assuming identical parameters for the nanotubes and their highly ordered arrangement. To our knowledge, there are already technological prerequisites for creating a structure similar to the one proposed in our paper. For instance, the alignment of ultra-high-density CNT arrays is achieved using the dielectrophoresis method [1]. In Ref [2], a method for obtaining well-aligned arrays of semiconductor CNTs (within 9 degrees) with adjustable density from 100 to 200 CNTs per micrometer on a 10-centimeter silicon wafer has been developed. Positioning CNTs using dielectrophoresis is simple and effective, and the technology is compatible with silicon, allowing for the fabrication of microplates for electronic devices based on CNTs [3]. Furthermore, promising results are emerging in creating CNT arrays with precisely defined transport properties [4, 5].

[1] Shekhar S., et. al., ACS Nano 2011, 5, 1739–1746, doi:10.1021/nn102305z.

[2] Liu L., et al. Science 2020, 368, 850–856, doi:10.1126/science.aba5980.

[3] Kimbrough J., et. al., Micromachines 2021, 12(1), 12, doi:10.3390/ MI12010012.

[4] Zhang R., et. al.,  Chem. Soc. Rev. 2017, 46, 3661–3715, doi:10.1039/C7CS00104E.

[5] He M., et. al.,  Chem. Rev. 2020, 120, 12592–12684, doi:10.1021/ACS.CHEMREV.0C00395.

 

Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript by S.A. Afanas’ev et. al.  presents a novel and intriguing approach to terahertz (THz) radiation generation using arrays of DWCNTs and a direct electric current. The study investigates the interaction between slow surface plasmon polariton (SPP) modes within DWCNTs and an applied electric current, shedding light on the potential for coherent THz radiation sources. I recommend to accept this manuscript with minor changes. Below, I provide a detailed assessment of the paper based on several aspects:

1. The paper introduces a novel approach to THz radiation generation, leveraging DWCNTs and their unique properties. The combination of slow SPP modes, coherent excitation, and the concept of DWCNTs as coherently emitting dipole antennas adds originality to the field. While the use of DWCNTs for THz generation has been explored previously, the coherent excitation aspect and the detailed analysis presented here contribute to the novelty.

2. The potential for compact, coherent THz sources is of great significance, given the wide range of applications in telecommunications, spectroscopy, and imaging. The paper addresses this significance by proposing a practical approach to achieve this goal. However, further discussion of specific applications and potential practical challenges would enhance the significance of the content.

3. The scientific soundness of the paper is well-maintained, with thorough numerical simulations and a detailed analysis of the interaction between SPP modes and electric currents in DWCNTs. However, additional experimental validation or discussions of potential experimental challenges could strengthen the scientific basis of the study.

4. The manuscript furnishes a mathematical description for the THz source. However, it would be advantageous to draw a more explicit comparison between the concepts underpinning THz generation in this manuscript and conventional laser pulse generation. Such a comparison would facilitate comprehension for a broader readership and underscore the manuscript's unique contributions.

Comments on the Quality of English Language

The quality of English language in this manuscript is generally excellent. The text is well-structured, and the language is clear and precise, which is crucial for effective communication in scientific writing.

Author Response

This manuscript by S.A. Afanas’ev et. al.  presents a novel and intriguing approach to terahertz (THz) radiation generation using arrays of DWCNTs and a direct electric current. The study investigates the interaction between slow surface plasmon polariton (SPP) modes within DWCNTs and an applied electric current, shedding light on the potential for coherent THz radiation sources. I recommend to accept this manuscript with minor changes. Below, I provide a detailed assessment of the paper based on several aspects:

1/ The paper introduces a novel approach to THz radiation generation, leveraging DWCNTs and their unique properties. The combination of slow SPP modes, coherent excitation, and the concept of DWCNTs as coherently emitting dipole antennas adds originality to the field. While the use of DWCNTs for THz generation has been explored previously, the coherent excitation aspect and the detailed analysis presented here contribute to the novelty.

Answer:

We appreciate your positive assessment of our novel approach to THz radiation generation using DWCNT arrays and direct electric current.

2/ The potential for compact, coherent THz sources is of great significance, given the wide range of applications in telecommunications, spectroscopy, and imaging. The paper addresses this significance by proposing a practical approach to achieve this goal. However, further discussion of specific applications and potential practical challenges would enhance the significance of the content.

Answer:

The proposed THz generator in our manuscript can potentially be used in fields demanding a compact source of coherent or at least narrowband THz radiation. These fields include biological and medical research, technical diagnostics, and 6G-communication. Plasmonic generators, where the amplification of surface plasmon polaritons (SPPs) is achieved through current pumping, allow for significantly greater SPP intensity and, consequently, higher intensity of free THz waves. The corresponding text is added into the conclusion section.

3/ The scientific soundness of the paper is well-maintained, with thorough numerical simulations and a detailed analysis of the interaction between SPP modes and electric currents in DWCNTs. However, additional experimental validation or discussions of potential experimental challenges could strengthen the scientific basis of the study.

Answer:

We do agree that the experimental realization of the proposed generator scheme is not trivial, but we are working in this direction. Several conditions must be met to implement a generator based on a CNT array with laser excitation and current pumping of plasmonic oscillations. 1) To achieve high energy conversion efficiency into free THz waves, it is necessary to fabricate a homogeneous array of CNTs with closely similar structural parameters, ideally identical chirality and wall diameter. Maximizing the density of the array is also highly desirable. 2) Electrical contacts must be connected to the CNT array. 3) The radiation of two lasers must be focused on the target – the CNT array – and the laser beams' frequencies, directions, and polarizations must be coordinated and stabilized. 4) Overheating of the sample could be a separate issue, requiring either efficient heat dissipation or transitioning the generator to a pulsed operating mode. Insufficient uniformity of the CNT array could partially be compensated by, as we believe, increasing the depth of modulation of the CNT array by the laser beams, which is one of the main ideas of this work. This will allow for generation, although it may reduce the energy efficiency of the proposed system. Overall, the primary goal of this work is to develop a THz generator scheme, with many specific technical issues still requiring further development and investigation.

4/ The manuscript furnishes a mathematical description for the THz source. However, it would be advantageous to draw a more explicit comparison between the concepts underpinning THz generation in this manuscript and conventional laser pulse generation. Such a comparison would facilitate comprehension for a broader readership and underscore the manuscript's unique contributions.

Answer:

The approach proposed in our work fundamentally differs from the widely known approach to THz radiation generation [1-6], based on irradiating a target with short laser pulses. In conventional laser pulse generation, "optical rectification" [7] occurs, where a difference frequency close to zero is formed due to quadratic nonlinearity. This phenomenon arises when irradiating solid [1], liquid crystal [4], or gaseous [3] media with femtosecond laser pulses. In the nonlinear medium, a kind of "frequency down-conversion" occurs, generating broadband THz pulses. In particular, several studies have considered CNT arrays as targets for laser pulses [2,4-6]. Typically, optical rectification in CNTs is also associated with quadratic nonlinearity. Using multiphoton processes for generating THz radiation entails inherently low energy efficiency. In contrast, the efficiency of the generation mechanism proposed in our work is not directly related to energy conversion from external laser sources, which only serves to create distributed feedback. THz generation occurs due to the interaction of SPP with the drift current, acting as a pump, and this interaction can proceed continuously. Our amplification scheme inherently lacks the limitations associated with the low energy efficiency of nonlinear processes. Moreover, since our proposed scheme utilizes the current pumping of SPP, the total power of the generated radiation can be high due to the high conductivity of CNTs.

The revised text and new Refs are marked in red.

[1] J. A. Fülöp, et. al., Opt. Lett. 37, 557-559 (2012), doi:10.1364/ol.37.000557

[2] S. Huang, et. al., Carbon, 132, 335-342 (2018), doi:10.1016/j.carbon.2018.02.067

[3] W.-M. Wang, et. al., Opt. Express 16, 16999-17006 (2008), doi:10.1364/OE.16.016999

[4] M. Hassani and F. Jahangiri, Opt. Express 29, 38359-38375 (2021), doi:10.1364/oe.442168

[5] J. Parashar, et. al., Physica E, 44, 2069-2071 (2012), doi:10.1016/j.physe.2012.06.013

[6] L.V. Titova, et. al., Nano Lett., 15, 5, 3267–3272 (2015), doi:10.1021/acs.nanolett.5b00494

[7] M. Bass, et. al., Phys. Rev. Lett. 9, 446 (1962), doi:10.1103/PhysRevLett.9.446