Abstract

A simple semi-analytical single mode model describing the mechanism of metal-insulator-metal gap plasmon mode excitation by a plane wave is proposed. The role of the insulator-metal-insulator (IMI) lattice mode is highlighted. Although many other works addressed this issue, the crucial role of this mode has never been demonstrated before. This mode appears as the missing link that ensures energy transfer between the incident plane wave and the metal-insulator-metal (MIM) gap plasmon mode. In this single mode model, the grating layer, the host layer of the IMI lattice mode, is viewed as an effective homogeneous medium, and the scattering matrix characterizing the interaction between the IMI lattice mode and the MIM gap plasmon mode is easily computed. The proposed simplified model allows us to grasp the physical origin of the modes of the system and to predict accurately the resonance frequencies of the 1D structure. These modes are classified in symmetric and antisymmetric modes. The incident field, by its symmetry properties, acts on the system as a selection rule, activating a class of modes with the same symmetry properties as itself.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2018 (3)

2017 (2)

K. Edee, “Polynomial modal method for analysis of the coupling between a gap plasmon waveguide and a square ring resonator,” J. Appl. Phys. 122, 153102 (2017).
[Crossref]

C. Lemaitre, E. Centeno, and A. Moreau, “Interferometric control of the absorption in optical patch antennas,” Sci. Rep. 26, 3004–3012 (2017).

2016 (1)

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (1)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

2013 (1)

C. Ciraci, J. B. Lassiter, A. Moreau, and D. R. Smith, “Quasi-analytic study of scattering from optical plasmonic patch antennas,” J. Appl. Phys. 114, 163108 (2013).
[Crossref]

2012 (3)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

J. Yang, C. Sauvan, A. Jouanin, S. Collin, J.-L. Pelouard, and P. Lalanne, “Ultra small metal-insulator-metal nano resonators: impact of slow-wave effects on the quality factor,” Opt. Express 20, 16880–16891 (2012).
[Crossref]

2011 (2)

2010 (1)

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[Crossref] [PubMed]

2007 (2)

1998 (1)

1992 (1)

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

1980 (1)

A. Hartstein, J.R. Kirtley, and J.C. Tsang, “Enhancement in the infrared absorption from molecular monolayers with thin metal overlayers,” Phys. Rev. Lett. 45, 201–204 (1980)
[Crossref]

Albreksten, O.

Bingham, C.

Bormann, D.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

Bour, D.

Bowen, P.

Bozhevolnyi, S.

T. Sondergaard and S. Bozhevolnyi, “Slow-plasmon resonant nanostructures: Scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[Crossref]

Bozhevolnyi, S. I.

Brendel, R.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

Capasso, F.

Centeno, E.

C. Lemaitre, E. Centeno, and A. Moreau, “Interferometric control of the absorption in optical patch antennas,” Sci. Rep. 26, 3004–3012 (2017).

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

Chilkoti, A.

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Ciraci, C.

C. Ciraci, J. B. Lassiter, A. Moreau, and D. R. Smith, “Quasi-analytic study of scattering from optical plasmonic patch antennas,” J. Appl. Phys. 114, 163108 (2013).
[Crossref]

Ciracì, C.

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Collin, S.

Corzine, S.

Crozier, K. B.

Cubukcu, E.

Diehl, L.

Djurišic, A. B.

Edee, K.

K. Edee, “Single mode approach with versatile surface wave phase correction for the extraordinary optical transmission comprehension of 1D period nano-slits arrays,” Opt. Soc. Am. Cont. 1, 613–624 (2018).

K. Edee, J.-P. Plumey, A. Moreau, and B. Guizal, “Matched coordinates in the framework of polynomial modal methods for complex metasurfaces modeling,” J. Opt. Soc. Am. A 35, 608–615 (2018).
[Crossref]

K. Edee, “Polynomial modal method for analysis of the coupling between a gap plasmon waveguide and a square ring resonator,” J. Appl. Phys. 122, 153102 (2017).
[Crossref]

K. Edee and J.-P. Plumey, “Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating,” J. Opt. Soc. Am. A 32, 402–410 (2015).
[Crossref]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[Crossref]

Elazar, J. M.

Esteban, R.

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[Crossref] [PubMed]

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

Gramotnev, D. K.

Granet, G.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

Greffet, J. J.

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[Crossref] [PubMed]

Guizal, B.

K. Edee, J.-P. Plumey, A. Moreau, and B. Guizal, “Matched coordinates in the framework of polynomial modal methods for complex metasurfaces modeling,” J. Opt. Soc. Am. A 35, 608–615 (2018).
[Crossref]

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

Hartstein, A.

A. Hartstein, J.R. Kirtley, and J.C. Tsang, “Enhancement in the infrared absorption from molecular monolayers with thin metal overlayers,” Phys. Rev. Lett. 45, 201–204 (1980)
[Crossref]

Hill, R. T.

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Hofler, G.

Huang, Z.

Jia, X.

Jouanin, A.

Kirtley, J.R.

A. Hartstein, J.R. Kirtley, and J.C. Tsang, “Enhancement in the infrared absorption from molecular monolayers with thin metal overlayers,” Phys. Rev. Lett. 45, 201–204 (1980)
[Crossref]

Lalanne, P.

Lassiter, J. B.

C. Ciraci, J. B. Lassiter, A. Moreau, and D. R. Smith, “Quasi-analytic study of scattering from optical plasmonic patch antennas,” J. Appl. Phys. 114, 163108 (2013).
[Crossref]

Lemaitre, C.

C. Lemaitre, E. Centeno, and A. Moreau, “Interferometric control of the absorption in optical patch antennas,” Sci. Rep. 26, 3004–3012 (2017).

Liu, X.

Majewski, M. L.

Mock, J. J.

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Moreau, A.

K. Edee, J.-P. Plumey, A. Moreau, and B. Guizal, “Matched coordinates in the framework of polynomial modal methods for complex metasurfaces modeling,” J. Opt. Soc. Am. A 35, 608–615 (2018).
[Crossref]

C. Lemaitre, E. Centeno, and A. Moreau, “Interferometric control of the absorption in optical patch antennas,” Sci. Rep. 26, 3004–3012 (2017).

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

C. Ciraci, J. B. Lassiter, A. Moreau, and D. R. Smith, “Quasi-analytic study of scattering from optical plasmonic patch antennas,” J. Appl. Phys. 114, 163108 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Nielsen, M. G.

Omeis, F.

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

Pelouard, J.-L.

Plumey, J.-P.

Pors, A.

Rakiá, A. D.

Sauvan, C.

Smaali, R.

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

Smith, D. R.

X. Jia, P. Bowen, Z. Huang, X. Liu, C. Bingham, and D. R. Smith, “Clarification of surface modes of a periodic nanopatch metasurface,” Opt. Express 26, 3004–3012 (2018).
[Crossref] [PubMed]

C. Ciraci, J. B. Lassiter, A. Moreau, and D. R. Smith, “Quasi-analytic study of scattering from optical plasmonic patch antennas,” J. Appl. Phys. 114, 163108 (2013).
[Crossref]

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Sondergaard, T.

T. Sondergaard and S. Bozhevolnyi, “Slow-plasmon resonant nanostructures: Scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[Crossref]

Taliercio, T.

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

Teperik, T. V.

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[Crossref] [PubMed]

Tsang, J.C.

A. Hartstein, J.R. Kirtley, and J.C. Tsang, “Enhancement in the infrared absorption from molecular monolayers with thin metal overlayers,” Phys. Rev. Lett. 45, 201–204 (1980)
[Crossref]

Wang, Q.

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Wiley, B. J.

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Yang, J.

Yu, N.

Zhu, J.

Appl. Opt. (1)

J. Appl. Phys (1)

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

J. Appl. Phys. (2)

K. Edee, “Polynomial modal method for analysis of the coupling between a gap plasmon waveguide and a square ring resonator,” J. Appl. Phys. 122, 153102 (2017).
[Crossref]

C. Ciraci, J. B. Lassiter, A. Moreau, and D. R. Smith, “Quasi-analytic study of scattering from optical plasmonic patch antennas,” J. Appl. Phys. 114, 163108 (2013).
[Crossref]

J. Opt. Soc. Am. A (3)

Nat. Mater. (1)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref] [PubMed]

Nature (1)

A. Moreau, C. Ciracì, J. J. Mock, R. T. Hill, Q. Wang, B. J. Wiley, A. Chilkoti, and D. R. Smith, “Controlled-reflectance surfaces with film-coupled colloidal nanoantennas,” Nature 492, 86–89 (2012).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Soc. Am. Cont. (1)

K. Edee, “Single mode approach with versatile surface wave phase correction for the extraordinary optical transmission comprehension of 1D period nano-slits arrays,” Opt. Soc. Am. Cont. 1, 613–624 (2018).

Phys. Rev. B (1)

T. Sondergaard and S. Bozhevolnyi, “Slow-plasmon resonant nanostructures: Scattering and field enhancements,” Phys. Rev. B 75, 073402 (2007).
[Crossref]

Phys. Rev. Lett. (2)

A. Hartstein, J.R. Kirtley, and J.C. Tsang, “Enhancement in the infrared absorption from molecular monolayers with thin metal overlayers,” Phys. Rev. Lett. 45, 201–204 (1980)
[Crossref]

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[Crossref] [PubMed]

Prog. Electromagn. Res. (1)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2012).
[Crossref]

Sci. Rep. (2)

C. Lemaitre, E. Centeno, and A. Moreau, “Interferometric control of the absorption in optical patch antennas,” Sci. Rep. 26, 3004–3012 (2017).

R. Smaali, F. Omeis, A. Moreau, T. Taliercio, and E. Centeno, “A universal design to realize a tunable perfect absorber from infrared to microwaves, ” Sci. Rep. 6, 32589 (2016).
[Crossref] [PubMed]

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Figures (5)

Fig. 1
Fig. 1 Figure (1(a)) presents the schematic of the metasurface based on a periodic array of silver-nanoribbons deposited on a dielectric spacer layer, itself deposited on a dispersive thick gold substrate. Figure (1(b)) is the illustration of the coupling between the IMI plasmon mode and MIM gap plasmon mode. The gap under each nanoribbon is viewed as a Fabry-Perot cavity which both ports are excited by the fundamental mode of the silver grating.
Fig. 2
Fig. 2 Figure 2(a) shows a comparison between the reflectance of the structure and the normalized backward power spectrum | r 0 ( 1 ) | 2associated with the eigenmode γ 0 ( 1 ). The modulus of the magnetic field at the resonance wavelength λ = 1.01μm is also presented in this figure. Figure (2(b)) presents the modulus of the eigenfunction H 0 y ( 1 ) ( x ) associated with γ 0 ( 1 ) for different wavelengths. Numerical parameters: θ0 = 0°, d = 250nm, a = 75nm.
Fig. 3
Fig. 3 Figures (3(a)) and (3(b)) show a comparison between |Δ(λ)|, |S11(λ) − S12(λ)|, |S11(λ) + S12(λ)| and the reflectance of the structure obtained from PMM (solid line) for θ = 0° and θ = 40°. Figures (3(c)) and (3(d)) present the real part of magnetic field Hy(x, z) for θ = 40° at resonance wavelengths λ = 1.01μm (Fig. (3(c))) and λ = 0.672μm (Fig. (3(d))). For λ = 1.01μm a symmetrical gap plasmon cavity mode is excited by the incident plane wave while an anti-symmetrical gap plasmon cavity mode is excited at λ = 0.672μm. Numerical parameters: θ0 = 40°, d = 250nm, a = 75nm.
Fig. 4
Fig. 4 Figure (4(a)) shows the spectrum of |Δ(λ, k″0)| for k″0 ∈ [0, 1]rad /μm while in Fig. (4(b)) this spectrum is computed for three relevant values of k″0: k″0 = {0.1, 0.2, 0.5}rad /μm. In Fig. (4(b)), for the resonance wavelength λ = 1.01μm, the modulus of Δ decreases to a minimum value very close to zero and the optimal value of k″0 for the current example is k″0 = 0.2 radm. Numerical parameters: θ0 = 40°, d = 250nm, a = 75nm.
Fig. 5
Fig. 5 Figures (5(a)) and (5(b)) show a comparison between the reflection of the structure, |S11(λ) − S12(λ)|, |S11(λ) + S12(λ)| and |Δ(λ)| for θ0 = 0°, θ0 = 40°. The period d is set to 300nm and a = 150nm. Since the geometrical parameters are larger than the previous case, higher resonance frequencies of the Perot-Fabry MIM cavity can be obtained. Three resonance wavelengths which positions are denoted 1, 2, 3 are clearly distinguished in these figures. Figures 5(c) and 5(d) present the real part of the magnetic field Hy(x, z) corresponding to points 2 and 3 of Fig. 5(b). Point 2 corresponds to an anti-symmetrical cavity mode resonance while at point 3 a symmetrical cavity gap-plasmon mode is excited.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

| H y ( k ) = | H y ( k ) + + | H y ( k )
H y ( k ) + ( x , z ) = q A q ( k ) e i k 0 γ q ( k ) ( z z k ) n H n q ( k ) P n ( x ) = q A q ( k ) e i k 0 γ q ( k ) ( z z k ) H q ( k ) ( x )
H y ( k ) ( x , z ) = q B q ( k ) e i k 0 γ q ( k ) ( z z k 1 ) n H n q ( k ) P n ( x ) = q B q ( k ) e i k 0 γ q ( k ) ( z z k 1 ) H q ( k ) ( x ) ,
( k ) ( ω ) | H q ( k ) ( ω ) = ( γ q ( k ) ( ω ) ) 2 | H q ( k ) ( ω )
( k ) ( x , ω ) = ( c ω ) 2 ε ( k ) ( x , ω ) x 1 ε ( k ) ( x , ω ) x + ε ( k ) ( x , ω ) .
r 0 ( 1 ) ( ω ) = A 0 ( 1 ) ( ω ) B 0 ( 1 ) ( ω ) e i k 0 ( ω ) γ 0 ( 1 ) ( ω ) h 1 .
α 0 ( 1 ) = ε ( 1 ) γ 0 ( 1 ) 2 ,
[ S 11 S 12 S 21 S 22 ] [ C 1 C 2 ] = [ D 1 D 2 ]
n ( ω ) = α 0 ( 2 ) ( ω ) / ε ( 2 ) ( ω ) α 0 ( 1 ) ( ω ) / ε ( 1 ) ( ω ) ,
{ S 11 ( ω ) = [ 1 n 2 ( ω ) ] [ 1 ϕ 2 ( ω ) ] [ 1 + n ( ω ) ] 2 [ 1 n ( ω ) ] 2 ϕ 2 ( ω ) S 12 ( ω ) = 4 n ( ω ) ϕ ( ω ) [ 1 + n ( ω ) ] 2 [ 1 n ( ω ) ] 2 ϕ 2 ( ω ) .
Δ ( ω ) = S 12 ( ω ) S 21 ( ω ) S 11 ( ω ) S 22 ( ω ) = [ S 11 ( ω ) S 12 ( ω ) ] [ S 11 ( ω ) + S 12 ( ω ) ] = 0
{ S 11 ( ω ) S 12 ( ω ) = 0 [ ϕ ( ω ) r 1 ( ω ) ] [ ϕ ( ω ) + 1 / r 1 ( ω ) ] = 0 or S 11 ( ω ) + S 12 ( ω ) = 0 [ ϕ ( ω ) + r 1 ( ω ) ] [ ϕ ( ω ) 1 / r 1 ( ω ) ] = 0
ϕ b ( ω ) = e i k 0 ( ω ) f ( θ 0 , d ) α 0 ( 1 ) ( ω ) b ,
f ( θ 0 , d ) = f ( θ 0 , d 2 ) f ( θ 0 , d 1 ) d 2 d 1 ( d d 1 ) + f ( θ 0 , d 1 ) .
{ ϕ + ( ω ) = ± 1 / r 1 ( ω ) ϕ + ( ω ) r 1 ( ω ) = ± 1 ϕ ( ω ) = ± r 1 ( ω ) [ ϕ ( ω ) ] 1 r 1 ( ω ) = ± 1 .
{ S 11 ( ω ) S 12 ( ω ) = 0 ϕ s + ( ω ) r 1 ( ω ) = 1 S 11 ( ω ) + S 12 ( ω ) = 0 ϕ a + ( ω ) r 1 ( ω ) = 1 .
{ S 11 ( ω ) S 12 ( ω ) = 0 corresponds to the dispersion relation of symmetrical mode S 11 ( ω ) + S 12 ( ω ) = 0 corresponds to the dispersion relation of anti-symmetrical mode ,
ϕ a = e i ( k 0 i k 0 ) α 0 ( 2 ) a ,
ϕ b = e i ( k 0 i k 0 ) α 0 ( 1 ) f ( θ 0 , d ) b .