Abstract

We propose a coherent-form energy conservation relation (ECR) that is generally valid for the elastic transmission and reflection of a guided mode in a symmetric scattering system. In contrast with the classical incoherent-form ECR, |τ|2 + |ρ|2≤1 with τ and ρ denoting the elastic transmission and reflection coefficients of a guided mode, the coherent-form ECR is expressed as |τ + ρ|≤1, which imposes a constraint on a coherent superposition of the transmitted and reflected modes. The coherent-form ECR is rigorously demonstrated and is numerically tested by considering different types of modes in various scattering systems. Further discussions with the scattering matrix formalism indicate that two coherent-form ECRs, |τ + ρ|≤1 and |τρ|≤1, along with the classical ECR |τ|2 + |ρ|2≤1 constitute a complete description of the energy conservation for the elastic scattering of a guided mode in a symmetric scattering system. The coherent-form ECR provides a common tool in terms of energy transfer for understanding and analyzing the scattering dynamics in currently interested scattering systems.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
  3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
    [CrossRef]
  4. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
    [CrossRef] [PubMed]
  5. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
    [CrossRef] [PubMed]
  6. L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
    [CrossRef]
  7. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008).
    [CrossRef] [PubMed]
  8. J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
    [CrossRef] [PubMed]
  9. K. J. Huang, S. Y. Yang, and L. M. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt.46(9), 1429–1434 (2007).
    [CrossRef] [PubMed]
  10. X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011).
    [CrossRef]
  11. G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
    [CrossRef]
  12. Here the guided mode is defined as a propagative waveguide mode, which obeys an exponential propagation rule exp(ikz) with the propagation constant k being real or approximately real.
  13. For instance, the derivation of the resonance condition in [7] and [8] requires |τ + ρ|≈1 for the elastic scattering of SPPs, which is just the coherent-form ECR under the energy conservative condition.
  14. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999).
    [CrossRef]
  15. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A22(9), 1844–1849 (2005).
    [CrossRef] [PubMed]
  16. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  17. Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express19(4), 3251–3257 (2011).
    [CrossRef] [PubMed]
  18. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005).
    [CrossRef] [PubMed]
  19. L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997).
    [CrossRef]
  20. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express15(18), 11042–11060 (2007).
    [CrossRef] [PubMed]

2012 (1)

L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
[CrossRef]

2011 (4)

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
[CrossRef] [PubMed]

X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011).
[CrossRef]

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express19(4), 3251–3257 (2011).
[CrossRef] [PubMed]

2008 (2)

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

2007 (2)

2005 (3)

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005).
[CrossRef] [PubMed]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A22(9), 1844–1849 (2005).
[CrossRef] [PubMed]

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

1999 (1)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999).
[CrossRef]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

1997 (1)

Alameh, K.

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

Brongersma, M. L.

X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011).
[CrossRef]

Cai, L.

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Eisler, H. J.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Garcia-Vidal, F. J.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Guo, X.

L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
[CrossRef]

Han, Z. H.

Hecht, B.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Huang, K. J.

Huang, X. P.

X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011).
[CrossRef]

Hugonin, J. P.

Lalanne, P.

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
[CrossRef] [PubMed]

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008).
[CrossRef] [PubMed]

G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express15(18), 11042–11060 (2007).
[CrossRef] [PubMed]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005).
[CrossRef] [PubMed]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A22(9), 1844–1849 (2005).
[CrossRef] [PubMed]

Lecamp, G.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Li, G. Y.

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Li, L. F.

Liu, H. T.

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
[CrossRef] [PubMed]

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008).
[CrossRef] [PubMed]

Lou, J. Y.

L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
[CrossRef]

Martin, O. J. F.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Mühlschlegel, P.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Pendry, J. B.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999).
[CrossRef]

Pohl, D. W.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Porto, J. A.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999).
[CrossRef]

Rodier, J. C.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005).
[CrossRef] [PubMed]

Sauvan, C.

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Tong, L. M.

L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
[CrossRef]

K. J. Huang, S. Y. Yang, and L. M. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt.46(9), 1429–1434 (2007).
[CrossRef] [PubMed]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Xiao, F.

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Xu, A. S.

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Yang, J.

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
[CrossRef] [PubMed]

Yang, S. Y.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zi, F.

L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Nature (3)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008).
[CrossRef] [PubMed]

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008).
[CrossRef] [PubMed]

New J. Phys. (1)

G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011).
[CrossRef]

Opt. Commun. (1)

L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011).
[CrossRef]

Phys. Rev. Lett. (3)

J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999).
[CrossRef]

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005).
[CrossRef] [PubMed]

Science (1)

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Other (5)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Here the guided mode is defined as a propagative waveguide mode, which obeys an exponential propagation rule exp(ikz) with the propagation constant k being real or approximately real.

For instance, the derivation of the resonance condition in [7] and [8] requires |τ + ρ|≈1 for the elastic scattering of SPPs, which is just the coherent-form ECR under the energy conservative condition.

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Figures (3)

Fig. 1
Fig. 1

Numerical test of the coherent-form ECR |τ + ρ|≤1 for different types of modes in various symmetric scattering systems. (a)-(b) Scattering of a SPP at a periodic chain of infinite-depth square holes (a) or at an infinite-depth slit (b) in gold substrate with an air cladding (refractive index 1). The chain period is 940nm, and the hole side length and the slit width are 266nm. (c)-(d) For the fundamental SPP mode on a gold nano-wire in air scattered at an air gap with a width of 30nm (c) and 100nm (d). The nano-wire has a square cross section with a side length of 40nm. (e)-(g) For the fundamental SPP mode in a gold/air/gold waveguide (air-gap width 100nm), which is scattered at an infinite-depth slit (slit width 50nm) (e), at a finite-depth groove (groove width 50nm and depth 1μm) (f), and at an orthogonal corner (g). (h) Coupling of the fundamental mode of two tightly contacted silica nano-wires (diameter 400nm, refractive index 1.46 at wavelength λ = 633nm) in air. The considered fundamental mode is polarized in the plane determined by the two wire axes.

Fig. 2
Fig. 2

Theoretical demonstration of the coherent-form ECR for a symmetric scattering system. As shown in (a), the considered guided mode with a unitary coefficient is transmitted with a coefficient τ and reflected with a coefficient ρ. (b) Related scattering problem for the demonstration, in which two identical modes with a unitary coefficient are incident coherently from both ports of the system. (c) General description for the elastic scattering of a guided mode in a symmetric scattering system.

Fig. 3
Fig. 3

Numerical test of the coherent-form ECR for the transmission (with a coefficient t) and reflection (coefficient r) of a normally incident plane wave at a metallic hole array drilled in a gold membrane in air (inset). The plane wave is polarized in one periodic direction of the grating. |t + r| for different wavelengths is shown. (a) For a real lossy metal of gold. (b) For an artificial lossless metal. The array period is 940nm, the side length of square holes is 266nm, and the membrane thickness is 200nm.

Equations (2)

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|τ+ρ | 2 +|τ+ρ | 2 1 2 + 1 2 ,
S=[ τ ρ ρ τ ],

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