Abstract

This paper demonstrates the efficiency of the differential method, a conventional grating theory, to investigate dielectric loaded surface plasmon polariton waveguides (DLSPPWs), known to be a potential solution for optical interconnects. The method is used to obtain the mode effective indices (both real and imaginary parts) and the mode profiles. The results obtained with the differential method are found to be in good agreement with those provided by the effective index method or finite elements. The versatility of the differential method is demonstrated by considering complex configurations such as trapezoidal waveguides or DLSPPWs lying on a finite width metal stripe.

©2008 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  13. K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” Wiley, New-York (2001).
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    [Crossref]
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    [Crossref] [PubMed]
  17. S.-D. Wu, T. K. Gaylord, E. N. Glytsis, and Y.-M. Wu, “Three-dimensional converging-diverging Gaussian beam diffraction by a volume grating,” J. Opt. Soc. Am. A 22, 1293–1303 (2005).
    [Crossref]
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    [Crossref]
  19. R. Zia, J. A. schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmons waveguides,” Phys. Rev. B 74, 165415 (2006).
    [Crossref]

2007 (5)

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[Crossref]

A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90, 211101 (2007).
[Crossref]

R. Kiyan, C. Reinhardt, S. Passinger, A. L. Stepanov, A. Hohenau, J. R. Krenn, and B. N. Chichkov, “Rapid prototyping of optical components for surface plasmon polaritons,” Opt. Express 15, 4205–4215 (2007).
[Crossref] [PubMed]

2006 (2)

R. Zia, J. A. schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmons waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

2005 (2)

2003 (1)

J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distribution of surface plasmon waveguide modes,” Phys. Rev. B 68, 115401 (2003).
[Crossref]

2001 (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite widths: Bound modes of asymmetric structures,” Phys. Rev. B 63, 125417 (2001).
[Crossref]

1999 (1)

E. Amenogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method andWavevector density method,” J. Light-wave Technol. 17, 929–941 (1999).
[Crossref]

1998 (1)

1996 (1)

1995 (1)

1994 (1)

P. Dawson, F. de Fornel, and J.-P. Goudonnet, “Imaging of surface plasmon propagating and edge interaction using a photon scanning tunnelling microscope,” Phys. Rev. Lett. 72, 2927 (1994).
[Crossref] [PubMed]

Amenogiannis, E.

E. Amenogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method andWavevector density method,” J. Light-wave Technol. 17, 929–941 (1999).
[Crossref]

Aussenegg, F. R.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Berini, P.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite widths: Bound modes of asymmetric structures,” Phys. Rev. B 63, 125417 (2001).
[Crossref]

Bouhelier, A.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Bozhevolnyi, S. I.

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[Crossref]

Brongersma, M. L.

R. Zia, J. A. schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmons waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

Chichkov, B. N.

Cho, M. H.

Colas des Francs, G.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Dawson, P.

P. Dawson, F. de Fornel, and J.-P. Goudonnet, “Imaging of surface plasmon propagating and edge interaction using a photon scanning tunnelling microscope,” Phys. Rev. Lett. 72, 2927 (1994).
[Crossref] [PubMed]

de Fornel, F.

P. Dawson, F. de Fornel, and J.-P. Goudonnet, “Imaging of surface plasmon propagating and edge interaction using a photon scanning tunnelling microscope,” Phys. Rev. Lett. 72, 2927 (1994).
[Crossref] [PubMed]

Dereux, A.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distribution of surface plasmon waveguide modes,” Phys. Rev. B 68, 115401 (2003).
[Crossref]

Ditlbacher, H.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Drezet, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Gaylord, T. K.

S.-D. Wu, T. K. Gaylord, E. N. Glytsis, and Y.-M. Wu, “Three-dimensional converging-diverging Gaussian beam diffraction by a volume grating,” J. Opt. Soc. Am. A 22, 1293–1303 (2005).
[Crossref]

E. Amenogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method andWavevector density method,” J. Light-wave Technol. 17, 929–941 (1999).
[Crossref]

Glytsis, E. N.

S.-D. Wu, T. K. Gaylord, E. N. Glytsis, and Y.-M. Wu, “Three-dimensional converging-diverging Gaussian beam diffraction by a volume grating,” J. Opt. Soc. Am. A 22, 1293–1303 (2005).
[Crossref]

E. Amenogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method andWavevector density method,” J. Light-wave Technol. 17, 929–941 (1999).
[Crossref]

Gonzàlez, M. U.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Goudonnet, J.-P.

P. Dawson, F. de Fornel, and J.-P. Goudonnet, “Imaging of surface plasmon propagating and edge interaction using a photon scanning tunnelling microscope,” Phys. Rev. Lett. 72, 2927 (1994).
[Crossref] [PubMed]

Grandidier, J.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Hohenau, A.

R. Kiyan, C. Reinhardt, S. Passinger, A. L. Stepanov, A. Hohenau, J. R. Krenn, and B. N. Chichkov, “Rapid prototyping of optical components for surface plasmon polaritons,” Opt. Express 15, 4205–4215 (2007).
[Crossref] [PubMed]

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Holmgaard, T.

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[Crossref]

Kawai, S.

S. Kawai, “Handbook of optical interconnects,” Marcel Dekker Inc., New-York2005.

Kawano, K.

K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” Wiley, New-York (2001).

Kim, P. S.

Kim, Y. S.

Kitoh, T.

K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” Wiley, New-York (2001).

Kiyan, R.

Kostuk, R. K.

Krasavin, A. V.

A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90, 211101 (2007).
[Crossref]

Krenn, J. R.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

R. Kiyan, C. Reinhardt, S. Passinger, A. L. Stepanov, A. Hohenau, J. R. Krenn, and B. N. Chichkov, “Rapid prototyping of optical components for surface plasmon polaritons,” Opt. Express 15, 4205–4215 (2007).
[Crossref] [PubMed]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Lacroute, Y.

J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distribution of surface plasmon waveguide modes,” Phys. Rev. B 68, 115401 (2003).
[Crossref]

Leitner, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Li, L.

Markey, L.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Massenot, S.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Nevière, M.

M. Nevière and E. Popov, “Light Propagation in Periodic Media, Differential Theory and Design,” Marcel Dekker Inc., New-York (2003).

Oh, C. H.

Park, S.

Passinger, S.

Popov, E.

M. Nevière and E. Popov, “Light Propagation in Periodic Media, Differential Theory and Design,” Marcel Dekker Inc., New-York (2003).

Quidant, R.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

Reinhardt, C.

Renger, J.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

schuller, J. A.

R. Zia, J. A. schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmons waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

Song, S. H.

Steinberger, B.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

A. Hohenau, J. R. Krenn, A. L. Stepanov, A. Drezet, H. Ditlbacher, B. Steinberger, A. Leitner, and F. R. Aussenegg, “Dielectric optical elements for surface plasmons,” Opt. Lett. 30, 893–895 (2005).
[Crossref] [PubMed]

Stepanov, A. L.

Weeber, J.-C.

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distribution of surface plasmon waveguide modes,” Phys. Rev. B 68, 115401 (2003).
[Crossref]

Wu, S.-D.

Wu, Y.-M.

Yeh, J.-H.

Zayats, A. V.

A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90, 211101 (2007).
[Crossref]

Zia, R.

R. Zia, J. A. schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmons waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90, 211101 (2007).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88, 094104 (2006).
[Crossref]

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[Crossref]

S. Massenot, J. Grandidier, A. Bouhelier, G. Colas des Francs, L. Markey, J.-C. Weeber, A. Dereux, J. Renger, M. U. Gonzàlez, and R. Quidant, “Polymer-metal waveguides characterization by Fourier plane leakage radiation microscopy,” Appl. Phys. Lett. 91, 243102 (2007).
[Crossref]

J. Light-wave Technol. (1)

E. Amenogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method andWavevector density method,” J. Light-wave Technol. 17, 929–941 (1999).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (4)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite widths: Bound modes of asymmetric structures,” Phys. Rev. B 63, 125417 (2001).
[Crossref]

R. Zia, J. A. schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmons waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

J.-C. Weeber, Y. Lacroute, and A. Dereux, “Optical near-field distribution of surface plasmon waveguide modes,” Phys. Rev. B 68, 115401 (2003).
[Crossref]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75, 245405 (2007).
[Crossref]

Phys. Rev. Lett. (1)

P. Dawson, F. de Fornel, and J.-P. Goudonnet, “Imaging of surface plasmon propagating and edge interaction using a photon scanning tunnelling microscope,” Phys. Rev. Lett. 72, 2927 (1994).
[Crossref] [PubMed]

Other (3)

S. Kawai, “Handbook of optical interconnects,” Marcel Dekker Inc., New-York2005.

K. Kawano and T. Kitoh, “Introduction to optical waveguide analysis,” Wiley, New-York (2001).

M. Nevière and E. Popov, “Light Propagation in Periodic Media, Differential Theory and Design,” Marcel Dekker Inc., New-York (2003).

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Figures (6)

Fig. 1.
Fig. 1. Illustration of the notations used by the differential method for a lamellar grating in conical diffraction.

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Fig. 2.
Fig. 2. Illustration of a dielectric loaded surface plasmon polariton waveguide, cross-section (a) and excitation scheme in ATR along the longitudinal axis of the waveguide (b).

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Fig. 3.
Fig. 3. Angular reflectivity curve (a) and its numerical derivative (b). Near-field intensity in the xy plane 5 nm above the top of the dielectric waveguide with a gaussian beam excitation (beam waist w 0=5 µm) (c) and along the longitudinal axis (x=0) (d).

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Fig. 4.
Fig. 4. Intensity mode profiles and confinement factors Γ for DLSPPWs having different structural parameters (thicknesses and widths).

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Fig. 5.
Fig. 5. Illustration of the studied trapezoidal DLSPPW.

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Fig. 6.
Fig. 6. (a): Illustration of a DLSPPW on a finite-width metal stripe; b: Evolution of the propagation length of the guided mode according the width of the metal stripe. The parameters are w=600 nm, t=600 nm and h=100 nm.

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Tables (2)

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Table 1. Comparison of the results obtained by the differential method (DM) to those provided by the Effective Index Method (EIM) and Finite Elements (FEM) for waveguides of different widths (t=600 nm, nw=1.535, nm=0.55+11.5ι and λ=1.55 µm)

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Table 2. Calculated effective index and propagation length for different trapezoidal DLSPP-Ws with the differential method

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Equations (3)

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E y ( i ) ( x , y , z ) = ∑ m = − N N A m ( i ) exp [ i ( k x , m ( i ) x + k y ( i ) y − k z , m ( i ) z ) ] + B m ( i ) exp [ i ( k x , m ( i ) x + k y ( i ) y + k z , m i z ) ]
d F dz = M ( z ) F
E inc = exp [ x 2 + y 2 w 0 2 ] = w 0 2 4 π ∫ ∫ exp [ − ( k x 2 + k y 2 ) w 0 2 4 ] exp [ i ( k x x + k y y ) ] dk x dk y

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