On the Identification of Surface Waves in Numerical Studies

On the Identification of Surface Waves in Numerical Studies In a recent study published in this journal, in which surface waves were investigated numerically for a certain equichiral thin film, the criterion used to determine whether or not surface waves were excited is not adequately discriminating. Keywords Surface-plasmon-polariton wave · Dispersion relation · Cavity resonance In a recent study, the prism-coupled excitation of surface- configuration, whereas SPPWs are not when the thin film’s plasmon-polariton waves (SPPWs) was investigated numer- thickness is greater than a certain threshold [3, 4]. ically for the interface of a metal and an equichiral thin Accordingly, some of the absorptance peaks of magni- film (ETF) [1]. Since the constitutive properties of the ETF tude greater than 0.4 in Ref. [1] may well be attributable under investigation varied periodically in the direction of the to cavity resonances (and/or other mechanisms) but not to ETF’s thickness, more than one SPPW could be excited at a SPPWs as claimed in that paper. As a representative coun- given wavelength by changing the direction of the incident terexample, many absorptance peaks of magnitude greater light. The criterion used to identify an SPPW was based on than 0.4 were reported in Ref. [5] but these were not the magnitude of peaks in plots of absorptance versus angle attributable to SPPWs because they did not correspond to of incidence: if the absorptance peak was greater than 0.4, roots of the dispersion relation of the associated canonical then this peak was claimed to represent the excitation of an boundary-value problem. SPPW. No dispersion relation from the associated canonical To summarize, the only infallible way to identify an boundary-value problem [2] was solved in order to confirm SPPW is to solve the associated canonical boundary-value this claim. problem [5, 6]. Generally, in studies on SPPWs, many peaks may be In order to illustrate this matter further, the essentials observed in the plots of absorptance versus angle of of the associated canonical boundary-value problem are incidence. As is comprehensively reported in the literature presented here; for comprehensive details, readers are [2], some of these peaks may represent SPPWs, but this referred elsewhere [2]. The canonical boundary-value needs to be confirmed by checking that the wavenumbers problem is considered for electromagnetic-surface-wave agree with those that arise as roots of the dispersion relation propagation at the planar interface of two dissimilar of the associated canonical boundary-value problem. Some materials: one occupying the half-space z> 0 (labeled A) of the absorptance peaks may arise for other reasons such and the other occupying the half-space z< 0 (labeled B). as cavity resonances. These absorptance peaks should be The general formalism can accommodate many different carefully distinguished from the absorptance peaks that types of electromagnetic surface wave, not just SPPWs; for can be attributed to SPPWs. The wavenumbers for cavity examples, Fano waves, Zenneck waves, Dyakonov waves, resonances do not arise as roots of the dispersion relation for and Dyakonov–Tamm waves are also accommodated. In SPPWs. Furthermore, cavity resonances are highly sensitive the interests of generality, both partnering materials may be to the thickness of the thin film in the prism-coupled taken to be bianisotropic and periodically nonhomogeneous in the z direction [7]. Thus, the constitutive relations of these Tom G. Mackay two partnering materials may be expressed as T.Mackay@ed.ac.uk A A D(r) = ε (z) · E(r) + ξ (z) · H(r) ⎬ School of Mathematics and Maxwell Institute ,z>0(1) A A for Mathematical Sciences, University of Edinburgh, B(r) = ζ (z) · E(r) + μ (z) · H(r) ⎭ Edinburgh EH9 3FD, UK = = Plasmonics and is not restricted to metals and dielectric partnering materials (as applies in the case for SPPWs), but non-metals and B B D(r) = ε (z) · E(r) + ξ (z) · H(r) ⎬ many other types of material of recent interest [8]are also ,z< 0, (2) B B B(r) = ζ (z) · E(r) + μ (z) · H(r) ⎭ accommodated. Suppose that the electromagnetic surface = = ◦ ◦ wave propagates in the xy plane at an angle ψ ∈ [0 , 360 ] m m m m wherein ε (z), ξ (z), ζ (z),and μ (z),(m ∈ {A, B}), relative to the positive x axis. The electric field phasors may = = = are 3×3 constitutive dyadics. Consequently, the formalism then be expressed as A A A A C e (z) + C e (z) exp [iq (x cos ψ + y sin ψ )],z > 0 1 1 2 2 E(r) =   , (3) B B B B C e (z) + C e (z) exp [iq (x cos ψ + y sin ψ )],z < 0 3 3 4 4 A B wherein q is a complex-valued wavenumber, C and C appropriate credit to the original author(s) and the source, provide a 1,2 3,4 link to the Creative Commons license, and indicate if changes were are complex-valued amplitudes, and the two eigenvectors made. e (z) decay as z →∞ whereas the two eigenvectors 1,2 e (z) decay as z →−∞. By implementing the 3,4 usual boundary conditions at z = 0 (i.e., the tangential References components of the electric and magnetic field phasors must be continuous across the planar interface z = 0), the 1. Hosseininezhad SH, Babaei F (2018) Excitation of multiple sur- following matrix-vector equation arises: face plasmon-polaritons by a metal layer inserted in an equichiral ⎡ ⎤ ⎡ ⎤ sculptured thin film. Plasmonics https://doi.org/10.1007/s11468- 1 018-0701-y ⎢ ⎥ ⎢ ⎥ C 0 2. Polo Jr JA, Mackay TG, Lakhtakia A (2013) Electromagnetic ⎢ ⎥ ⎢ ⎥ Y (q) · =.(4) ⎣ ⎦ ⎣ ⎦ surface waves: a modern perspective. Elsevier, Waltham = 0 B 3. Motyka MA, Lakhtakia A (2008) Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. J Nanophoton 2:021910 The components of the 4×4 matrix Y (q) depend on the 4. Motyka MA, Lakhtakia A (2008) Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of constitutive dyadics of partnering materials A and B,aswell a metal and a sculptured nematic thin film. Part II: Arbitrary as on the propagation angle ψ. In order for a non-trivial incidence. J. Nanophoton. 3:033502 solution to Eq. 4 to exist, Y (q) must be singular, i.e., 5. Polo Jr JA, Mackay TG, Lakhtakia A (2011) Mapping multiple surface-plasmon-polariton-wave modes at the interface of a metal and a chiral sculptured thin film. J Opt Soc Am B 28(11):2656– det Y (q) = 0. (5) 6. Faryad M, Polo Jr JA, Lakhtakia A (2010) Multiple trains of same- Equation 5 constitutes the dispersion relation which can be color surface plasmon-polaritons guided by the planar interface solved in order to determine the wavenumber q. Numerical of a metal and a sculptured nematic thin film. Part IV: Canonical methods are generally required to extract q from Eq. 5. problem. J Nanophoton 4:043505 7. Mackay TG, Lakhtakia A (2010) Electromagnetic anisotropy and bianisotropy: a field guide. Word Scientific, Singapore Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// 8. Tang W, Wang L, Chen X, Liu C, Yu A, Lu W (2016) Dynamic creativecommons.org/licenses/by/4.0/), which permits unrestricted metamaterial based on the graphene split ring high-Q Fano- use, distribution, and reproduction in any medium, provided you give resonnator for sensing applications. Nanoscale 8:15196 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Plasmonics Springer Journals

On the Identification of Surface Waves in Numerical Studies

Plasmonics , Volume OnlineFirst – May 28, 2018
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Springer Journals
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Chemistry; Biotechnology; Nanotechnology; Biological and Medical Physics, Biophysics; Biochemistry, general
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10.1007/s11468-018-0770-y
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Abstract

In a recent study published in this journal, in which surface waves were investigated numerically for a certain equichiral thin film, the criterion used to determine whether or not surface waves were excited is not adequately discriminating. Keywords Surface-plasmon-polariton wave · Dispersion relation · Cavity resonance In a recent study, the prism-coupled excitation of surface- configuration, whereas SPPWs are not when the thin film’s plasmon-polariton waves (SPPWs) was investigated numer- thickness is greater than a certain threshold [3, 4]. ically for the interface of a metal and an equichiral thin Accordingly, some of the absorptance peaks of magni- film (ETF) [1]. Since the constitutive properties of the ETF tude greater than 0.4 in Ref. [1] may well be attributable under investigation varied periodically in the direction of the to cavity resonances (and/or other mechanisms) but not to ETF’s thickness, more than one SPPW could be excited at a SPPWs as claimed in that paper. As a representative coun- given wavelength by changing the direction of the incident terexample, many absorptance peaks of magnitude greater light. The criterion used to identify an SPPW was based on than 0.4 were reported in Ref. [5] but these were not the magnitude of peaks in plots of absorptance versus angle attributable to SPPWs because they did not correspond to of incidence: if the absorptance peak was greater than 0.4, roots of the dispersion relation of the associated canonical then this peak was claimed to represent the excitation of an boundary-value problem. SPPW. No dispersion relation from the associated canonical To summarize, the only infallible way to identify an boundary-value problem [2] was solved in order to confirm SPPW is to solve the associated canonical boundary-value this claim. problem [5, 6]. Generally, in studies on SPPWs, many peaks may be In order to illustrate this matter further, the essentials observed in the plots of absorptance versus angle of of the associated canonical boundary-value problem are incidence. As is comprehensively reported in the literature presented here; for comprehensive details, readers are [2], some of these peaks may represent SPPWs, but this referred elsewhere [2]. The canonical boundary-value needs to be confirmed by checking that the wavenumbers problem is considered for electromagnetic-surface-wave agree with those that arise as roots of the dispersion relation propagation at the planar interface of two dissimilar of the associated canonical boundary-value problem. Some materials: one occupying the half-space z> 0 (labeled A) of the absorptance peaks may arise for other reasons such and the other occupying the half-space z< 0 (labeled B). as cavity resonances. These absorptance peaks should be The general formalism can accommodate many different carefully distinguished from the absorptance peaks that types of electromagnetic surface wave, not just SPPWs; for can be attributed to SPPWs. The wavenumbers for cavity examples, Fano waves, Zenneck waves, Dyakonov waves, resonances do not arise as roots of the dispersion relation for and Dyakonov–Tamm waves are also accommodated. In SPPWs. Furthermore, cavity resonances are highly sensitive the interests of generality, both partnering materials may be to the thickness of the thin film in the prism-coupled taken to be bianisotropic and periodically nonhomogeneous in the z direction [7]. Thus, the constitutive relations of these Tom G. Mackay two partnering materials may be expressed as T.Mackay@ed.ac.uk A A D(r) = ε (z) · E(r) + ξ (z) · H(r) ⎬ School of Mathematics and Maxwell Institute ,z>0(1) A A for Mathematical Sciences, University of Edinburgh, B(r) = ζ (z) · E(r) + μ (z) · H(r) ⎭ Edinburgh EH9 3FD, UK = = Plasmonics and is not restricted to metals and dielectric partnering materials (as applies in the case for SPPWs), but non-metals and B B D(r) = ε (z) · E(r) + ξ (z) · H(r) ⎬ many other types of material of recent interest [8]are also ,z< 0, (2) B B B(r) = ζ (z) · E(r) + μ (z) · H(r) ⎭ accommodated. Suppose that the electromagnetic surface = = ◦ ◦ wave propagates in the xy plane at an angle ψ ∈ [0 , 360 ] m m m m wherein ε (z), ξ (z), ζ (z),and μ (z),(m ∈ {A, B}), relative to the positive x axis. The electric field phasors may = = = are 3×3 constitutive dyadics. Consequently, the formalism then be expressed as A A A A C e (z) + C e (z) exp [iq (x cos ψ + y sin ψ )],z > 0 1 1 2 2 E(r) =   , (3) B B B B C e (z) + C e (z) exp [iq (x cos ψ + y sin ψ )],z < 0 3 3 4 4 A B wherein q is a complex-valued wavenumber, C and C appropriate credit to the original author(s) and the source, provide a 1,2 3,4 link to the Creative Commons license, and indicate if changes were are complex-valued amplitudes, and the two eigenvectors made. e (z) decay as z →∞ whereas the two eigenvectors 1,2 e (z) decay as z →−∞. By implementing the 3,4 usual boundary conditions at z = 0 (i.e., the tangential References components of the electric and magnetic field phasors must be continuous across the planar interface z = 0), the 1. Hosseininezhad SH, Babaei F (2018) Excitation of multiple sur- following matrix-vector equation arises: face plasmon-polaritons by a metal layer inserted in an equichiral ⎡ ⎤ ⎡ ⎤ sculptured thin film. Plasmonics https://doi.org/10.1007/s11468- 1 018-0701-y ⎢ ⎥ ⎢ ⎥ C 0 2. Polo Jr JA, Mackay TG, Lakhtakia A (2013) Electromagnetic ⎢ ⎥ ⎢ ⎥ Y (q) · =.(4) ⎣ ⎦ ⎣ ⎦ surface waves: a modern perspective. Elsevier, Waltham = 0 B 3. Motyka MA, Lakhtakia A (2008) Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of a metal and a sculptured nematic thin film. J Nanophoton 2:021910 The components of the 4×4 matrix Y (q) depend on the 4. Motyka MA, Lakhtakia A (2008) Multiple trains of same-color surface plasmon-polaritons guided by the planar interface of constitutive dyadics of partnering materials A and B,aswell a metal and a sculptured nematic thin film. Part II: Arbitrary as on the propagation angle ψ. In order for a non-trivial incidence. J. Nanophoton. 3:033502 solution to Eq. 4 to exist, Y (q) must be singular, i.e., 5. Polo Jr JA, Mackay TG, Lakhtakia A (2011) Mapping multiple surface-plasmon-polariton-wave modes at the interface of a metal and a chiral sculptured thin film. J Opt Soc Am B 28(11):2656– det Y (q) = 0. (5) 6. Faryad M, Polo Jr JA, Lakhtakia A (2010) Multiple trains of same- Equation 5 constitutes the dispersion relation which can be color surface plasmon-polaritons guided by the planar interface solved in order to determine the wavenumber q. Numerical of a metal and a sculptured nematic thin film. Part IV: Canonical methods are generally required to extract q from Eq. 5. problem. J Nanophoton 4:043505 7. Mackay TG, Lakhtakia A (2010) Electromagnetic anisotropy and bianisotropy: a field guide. Word Scientific, Singapore Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// 8. Tang W, Wang L, Chen X, Liu C, Yu A, Lu W (2016) Dynamic creativecommons.org/licenses/by/4.0/), which permits unrestricted metamaterial based on the graphene split ring high-Q Fano- use, distribution, and reproduction in any medium, provided you give resonnator for sensing applications. Nanoscale 8:15196

Journal

PlasmonicsSpringer Journals

Published: May 28, 2018

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