Abstract

Recently a surface plasmon polariton (SPP) source based on an electrically operated semiconductor laser has been demonstrated. Here we present a numerical investigation of the light-SPP coupling process involved in the device. The problem consists in the coupling via a diffraction grating between a dielectric waveguide mode – the laser mode – and a SPP mode. The issue of the coupling efficiency is discussed, and the dependence on various geometrical parameters of both the grating and the dielectric waveguide is studied in detail. A maximum coupling efficiency of ≈24% is obtained at telecom wavelengths, which could lead to a high-power integrated SPP source when combined to a laser medium.

© 2011 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  17. A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
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    [CrossRef]

2010 (2)

C. S. Kim, I. Vurgaftman, R. A. Flynn, M. Kim, J. R. Lindle, W. W. Bewley, K. Bussmann, J. R. Meyer, and J. P. Long, “An integrated surface-plasmon source,” Opt. Express 18(10), 10609–10615 (2010).
[CrossRef] [PubMed]

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

2009 (3)

S. Y. Park, J. T. Kim, J. S. Shin, and S. Y. Shin, “Hybrid vertical directional coupling between a long range surface plasmon polariton waveguide and a dielectric waveguide,” Opt. Commun. 282(23), 4513–4517 (2009).
[CrossRef]

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009).
[CrossRef]

2008 (3)

H. Ditlbacher, N. Galler, D. M. Koller, A. Hohenau, A. Leitner, F. R. Aussenegg, and J. R. Krenn, “Coupling dielectric waveguide modes to surface plasmon polaritons,” Opt. Express 16(14), 10455–10464 (2008).
[CrossRef] [PubMed]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

2007 (2)

2006 (1)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

2004 (1)

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

1997 (2)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

1996 (1)

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[CrossRef] [PubMed]

1983 (1)

Aussenegg, F. R.

Babuty, A.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

Bahriz, M.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Beaudoin, G.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Bewley, W. W.

Bielefeldt, H.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[CrossRef] [PubMed]

Boltasseva, A.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

Bousseksou, A.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Bozhevolnyi, S. I.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

Brucoli, G.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

Bussmann, K.

Chassagneux, Y.

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Colombelli, R.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

V. Moreau, M. Bahriz, R. Colombelli, R. Perahia, O. Painter, L. R. Wilson, and A. B. Krysa, “Demonstration of air-guided quantum cascade lasers without top claddings,” Opt. Express 15(22), 14861–14869 (2007).
[CrossRef] [PubMed]

Coudevylle, J. R.

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Cuisin, C.

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

Dagens, B.

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

De Wilde, Y.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

Decobert, J.

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Ditlbacher, H.

Doyen, I. M.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Emory, S. R.

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

Feld, M. S.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

Flynn, R. A.

Galler, N.

Garcia-Vidal, F. J.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface plasmon circuitry,” Phys. Today 61(5), 44–50 (2008).
[CrossRef]

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
[CrossRef]

Hecht, B.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[CrossRef] [PubMed]

Hohenau, A.

Hugonin, J. P.

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009).
[CrossRef]

Inouye, Y.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[CrossRef] [PubMed]

Itzkan, I.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

Kim, C. S.

Kim, J. T.

S. Y. Park, J. T. Kim, J. S. Shin, and S. Y. Shin, “Hybrid vertical directional coupling between a long range surface plasmon polariton waveguide and a dielectric waveguide,” Opt. Commun. 282(23), 4513–4517 (2009).
[CrossRef]

Kim, M.

Kneipp, H.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

Kneipp, K.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

Koller, D. M.

Krenn, J. R.

Krysa, A. B.

Lagay, N.

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
[CrossRef]

Lalanne, P.

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009).
[CrossRef]

Laruelle, F.

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

Leitner, A.

Lindle, J. R.

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007).
[CrossRef]

Liu, H. T.

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009).
[CrossRef]

Long, J. P.

Maradudin, A. A.

Martin-Moreno, L.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

Meyer, J. R.

Moreau, V.

Nie, S. M.

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

Novotny, L.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[CrossRef] [PubMed]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

Painter, O.

Park, S. Y.

S. Y. Park, J. T. Kim, J. S. Shin, and S. Y. Shin, “Hybrid vertical directional coupling between a long range surface plasmon polariton waveguide and a dielectric waveguide,” Opt. Commun. 282(23), 4513–4517 (2009).
[CrossRef]

Patriarche, G.

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Perahia, R.

Perelman, L. T.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[CrossRef]

Pohl, D. W.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[CrossRef] [PubMed]

Radko, I. P.

I. P. Radko, S. I. Bozhevolnyi, G. Brucoli, L. Martin-Moreno, F. J. Garcia-Vidal, and A. Boltasseva, “Efficiency of local surface plasmon polariton excitation on ridges,” Phys. Rev. B 78(11), 115115 (2008).
[CrossRef]

Sagnes, I.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Shin, J. S.

S. Y. Park, J. T. Kim, J. S. Shin, and S. Y. Shin, “Hybrid vertical directional coupling between a long range surface plasmon polariton waveguide and a dielectric waveguide,” Opt. Commun. 282(23), 4513–4517 (2009).
[CrossRef]

Shin, S. Y.

S. Y. Park, J. T. Kim, J. S. Shin, and S. Y. Shin, “Hybrid vertical directional coupling between a long range surface plasmon polariton waveguide and a dielectric waveguide,” Opt. Commun. 282(23), 4513–4517 (2009).
[CrossRef]

Sirtori, C.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

A. Bousseksou, Y. Chassagneux, J. R. Coudevylle, R. Colombelli, C. Sirtori, G. Patriarche, G. Beaudoin, and I. Sagnes, “Surface-plasmon distributed-feedback quantum cascade lasers operating pulsed, room temperature,” Appl. Phys. Lett. 95(9), 091105 (2009).
[CrossRef]

Stegeman, G. I.

Tetienne, J.-P.

A. Babuty, A. Bousseksou, J.-P. Tetienne, I. M. Doyen, C. Sirtori, G. Beaudoin, I. Sagnes, Y. De Wilde, and R. Colombelli, “Semiconductor surface plasmon sources,” Phys. Rev. Lett. 104(22), 226806 (2010).
[CrossRef] [PubMed]

Thedrez, B.

J. Decobert, N. Lagay, C. Cuisin, B. Dagens, B. Thedrez, and F. Laruelle, “MOVPE growth of AlGaInAs–InP highly tensile-strained MQWs for 1.3μm low-threshold lasers,” J. Cryst. Growth 272(1-4), 543–548 (2004).
[CrossRef]

Vurgaftman, I.

Wallis, R. F.

Wang, B.

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009).
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[CrossRef]

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The commercial software package COMSOL Multiphysics was employed.

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Figures (6)

Fig. 1
Fig. 1

Scheme of the SPP source. The device comprises an active section – a distributed feedback laser – and a passivecoupler designed to inject SPPs into a metal-air interface mode.

Fig. 2
Fig. 2

Geometry of the configuration under study. The structure is assumed to be infinite in the y-direction. The frequency-dependent gold index nm is taken from [18]. The red curves represent the typical mode profiles of the two modes to be coupled.

Fig. 3
Fig. 3

(a) Calculated SPP coupling efficiency as a function of the wavelength. Different curves show data for various numbers N of periods composing the grating. Inset: Maximum SPP coupling efficiency versus N. (b) Calculated SPP coupling efficiency as a function of the grating period Λ for various wavelengths.

Fig. 4
Fig. 4

Calculated SPP coupling efficiency as a function of the grating duty cycle dc (a), and the metal thickness tm (b) for various grating periods Λ. The data in panel (b) are shown for both dc = 50% and dc = 60%.

Fig. 5
Fig. 5

Calculated SPP coupling efficiency as a function of the waveguide core thickness tc for various grating periods Λ and for both dc = 50% and dc = 60%.

Fig. 6
Fig. 6

Electric field distribution calculated for the problem described in Fig. 2 with the optimized parameters tc = 600 nm, N = 10, Λ = 555 nm, dc = 60%, tm = 100 nm. The wavelength of the impinging guided wave is λ0 = 1.3 μm. The data are plotted in linear color scale with an arbitrary unit. Inset: overall view of the simulated area.

Equations (9)

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k d = k s p p + p 2 π Λ ,
Λ = λ 0 n d n s p p ,
η = P s p p P d ,
H y ( x , z ) = H max d e i k d x { cos ( φ ) e α a z ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​     ​                                       ​                   f o r       z > 0 cos ( k z z + φ )                                                       f o r     t c < z < 0 cos ( k z d + φ ) e α s ( z + t c )           f o r       z < t c ,
{ k z d = tan 1 ( ε c α a ε a k z ) + tan 1 ( ε c α s ε s k z ) ε a k 0 2 = k d 2 α a 2 ε c k 0 2 = k d 2 + k z 2 ε s k 0 2 = k d 2 α s 2 ,
P d = d z 1 2 Re { ( E × H * ) e x } = | H max d | 2 k d 4 ω ε 0 [ cos 2 ( φ ) ε a α a + d ε c + sin ( 2 φ ) sin ( 2 k z d + 2 φ ) 2 ε c k z + cos 2 ( k z d + φ ) ε s α s ] .
H y ( x , z ) = H max s p p e i k s p p x { e α a ( z t m ) ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​     ​                                       ​                   f o r       z > t m e α m ( z t m )           ​ ​                                                 f o r       z < t m ,
{ k s p p = k 0 ε a ε m ε a + ε m ε a k 0 2 = k s p p 2 α a 2 ε m k 0 2 = k s p p 2 α m 2 .
P s p p = d z 1 2 Re { ( E × H * ) e x } = | H max s p p | 2 Re { k s p p 4 ω ε 0 [ 1 ε a Re { α a } + 1 ε m Re { α m } ] } .

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