Abstract

A simple method to obtain the coupling characteristics of a directional coupler consisting of two dielectric-loaded surface plasmon polariton waveguides is reported. The method is found to give accurate results in comparison with the widely used effective index method. Theoretical results are also found to match excellently with recently reported measurements on coupling lengths in such waveguides [Opt. Lett. 34, 310 (2009)].

© 2009 Optical Society of America

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References

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  1. T. W. Ebbessen, C. Genet, and S. I. Bozhevolnyi, “Surface plasmon circuitry,” Phys. Today 61, 44-50 (2008).
    [CrossRef]
  2. W. L. Barnes, “Surface plasmon-polaritons length scales: a route to sub-wavelength optics,” J. Opt. A Pure Appl. Opt. 8, S87-S93 (2006).
    [CrossRef]
  3. S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
    [CrossRef]
  4. Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Karasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34, 310-312 (2009).
    [CrossRef] [PubMed]
  5. A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguide,” Phys. Rev. B 78, 045425-8 (2008).
    [CrossRef]
  6. A. Hosseini, A. Niewouldt, and Y. Massoud, “Optimizing dielectric stripes over a metallic substrate for subwavelength light confinement,” IEEE Photon. Technol. Lett. 19, 522-524(2007).
    [CrossRef]
  7. B. Steinberger, A. Hohenau, H. Diltlbacher, 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]
  8. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phy. Rev. B 75, 245405 (2007).
    [CrossRef]
  9. B. Steinberger, A. Hohenau, H. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
    [CrossRef]
  10. A. Kumar and T. Srivastava, “Performance of effective index method in the modeling of a nanoscale rectangular apertures in a real metal,” Opt. Commun. 281, 4526-4529(2008).
    [CrossRef]
  11. A. Kumar and T. Srivastava, “Modeling of a nanoscale rectangular hole in a real metal,” Opt. Lett. 33, 333-335(2008).
    [CrossRef] [PubMed]
  12. A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguide: an accurate perturbation approach,” Opt. Lett. 8, 63-65 (1983).
    [CrossRef] [PubMed]
  13. A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
    [CrossRef]
  14. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

2009 (1)

2008 (4)

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguide,” Phys. Rev. B 78, 045425-8 (2008).
[CrossRef]

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

A. Kumar and T. Srivastava, “Performance of effective index method in the modeling of a nanoscale rectangular apertures in a real metal,” Opt. Commun. 281, 4526-4529(2008).
[CrossRef]

A. Kumar and T. Srivastava, “Modeling of a nanoscale rectangular hole in a real metal,” Opt. Lett. 33, 333-335(2008).
[CrossRef] [PubMed]

2007 (3)

A. Hosseini, A. Niewouldt, and Y. Massoud, “Optimizing dielectric stripes over a metallic substrate for subwavelength light confinement,” IEEE Photon. Technol. Lett. 19, 522-524(2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Diltlbacher, 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]

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

2006 (3)

B. Steinberger, A. Hohenau, H. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

W. L. Barnes, “Surface plasmon-polaritons length scales: a route to sub-wavelength optics,” J. Opt. A Pure Appl. Opt. 8, S87-S93 (2006).
[CrossRef]

S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
[CrossRef]

1986 (1)

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
[CrossRef]

1983 (1)

Aussenegg, F. R.

B. Steinberger, A. Hohenau, H. Diltlbacher, 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. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Barnes, W. L.

W. L. Barnes, “Surface plasmon-polaritons length scales: a route to sub-wavelength optics,” J. Opt. A Pure Appl. Opt. 8, S87-S93 (2006).
[CrossRef]

Bozhevolnyi, S. I.

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Karasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34, 310-312 (2009).
[CrossRef] [PubMed]

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

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

Chen, Z.

Dereux, A.

Diltlbacher, H.

B. Steinberger, A. Hohenau, H. Diltlbacher, 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. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Drezet, A.

B. Steinberger, A. Hohenau, H. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Ebbessen, T. W.

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

Genet, C.

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

Ghatak, A. K.

Goncharenko, A. M.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
[CrossRef]

Hohenau, A.

B. Steinberger, A. Hohenau, H. Diltlbacher, 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. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Holmgaard, T.

Hosseini, A.

A. Hosseini, A. Niewouldt, and Y. Massoud, “Optimizing dielectric stripes over a metallic substrate for subwavelength light confinement,” IEEE Photon. Technol. Lett. 19, 522-524(2007).
[CrossRef]

Karasavin, A. V.

Karpenko, V. A.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
[CrossRef]

Krasavin, A. V.

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguide,” Phys. Rev. B 78, 045425-8 (2008).
[CrossRef]

Krenn, J. R.

B. Steinberger, A. Hohenau, H. Diltlbacher, 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. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Kumar, A.

A. Kumar and T. Srivastava, “Performance of effective index method in the modeling of a nanoscale rectangular apertures in a real metal,” Opt. Commun. 281, 4526-4529(2008).
[CrossRef]

Kumar, A.

Kumar, A.

Leitner, A.

B. Steinberger, A. Hohenau, H. Diltlbacher, 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. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Maier, S. A.

S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
[CrossRef]

Markey, L.

Massoud, Y.

A. Hosseini, A. Niewouldt, and Y. Massoud, “Optimizing dielectric stripes over a metallic substrate for subwavelength light confinement,” IEEE Photon. Technol. Lett. 19, 522-524(2007).
[CrossRef]

Mogilevich, V. N.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
[CrossRef]

Niewouldt, A.

A. Hosseini, A. Niewouldt, and Y. Massoud, “Optimizing dielectric stripes over a metallic substrate for subwavelength light confinement,” IEEE Photon. Technol. Lett. 19, 522-524(2007).
[CrossRef]

Palik, E. D.

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

Sotskii, A. B.

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
[CrossRef]

Srivastava, T.

Srivastava, T.

A. Kumar and T. Srivastava, “Performance of effective index method in the modeling of a nanoscale rectangular apertures in a real metal,” Opt. Commun. 281, 4526-4529(2008).
[CrossRef]

Steinberger, B.

B. Steinberger, A. Hohenau, H. Diltlbacher, 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. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Stepnov, A. L.

B. Steinberger, A. Hohenau, H. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

Thyagarajan, K.

Zayats, A. V.

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Karasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34, 310-312 (2009).
[CrossRef] [PubMed]

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguide,” Phys. Rev. B 78, 045425-8 (2008).
[CrossRef]

Appl. Phys. Lett. (2)

B. Steinberger, A. Hohenau, H. Diltlbacher, A. L. Stepnov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguide,” Appl. Phys. Lett. 88, 094104 (2006).
[CrossRef]

B. Steinberger, A. Hohenau, H. Diltlbacher, 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]

IEEE J. Sel. Top. Quantum Electron. (1)

S. A. Maier, “Plasmonics: the promise of highly integrated optical devices,” IEEE J. Sel. Top. Quantum Electron. 12, 1671-1677 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Hosseini, A. Niewouldt, and Y. Massoud, “Optimizing dielectric stripes over a metallic substrate for subwavelength light confinement,” IEEE Photon. Technol. Lett. 19, 522-524(2007).
[CrossRef]

J. Appl. Spectrosc. (1)

A. M. Goncharenko, V. A. Karpenko, V. N. Mogilevich, and A. B. Sotskii, “Methods of approximate variable separation in the theory of weakly inhomogeneous optical waveguides (review),” J. Appl. Spectrosc. 45, 663-671 (1986).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

W. L. Barnes, “Surface plasmon-polaritons length scales: a route to sub-wavelength optics,” J. Opt. A Pure Appl. Opt. 8, S87-S93 (2006).
[CrossRef]

Opt. Commun. (1)

A. Kumar and T. Srivastava, “Performance of effective index method in the modeling of a nanoscale rectangular apertures in a real metal,” Opt. Commun. 281, 4526-4529(2008).
[CrossRef]

Opt. Lett. (3)

Phy. Rev. B (1)

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

Phys. Rev. B (1)

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguide,” Phys. Rev. B 78, 045425-8 (2008).
[CrossRef]

Phys. Today (1)

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

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Schematic of the cross section of (a) coupled DLSPPW, (b) structure characterized by ε ( x ) , and (c)  ε ( y ) .

Fig. 2
Fig. 2

Transverse field profile of the vertical component E y of the electric field (the structural parameters are d = 100 nm , t = 500 nm ).

Fig. 3
Fig. 3

Variation of (a)  the real part of mode effective index Re ( n eff ) of symmetric and antisymmetric modes and (b) coupling length L c with respect to ridge width w for S = 50 nm and t = 600 nm , obtained with EIM1, EIM2, SVM, and FEM.

Fig. 4
Fig. 4

Variation of coupling length L c with respect to the wavelength obtained with the SVM and experimental measurements.

Equations (9)

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

L c = π Re ( β s β a ) ,
ε ( x , y ) = ε ( x ) + ε ( y ) ε 3 ,
   ε ( x ) = { ε 4 | x | < S / 2 , | x | > ( S / 2 + w ) ε 3 S / 2 < | x | < ( S / 2 + w ) } , ε ( y ) = { ε 1 y < 0 ε 2 0 < y < d ε 3 d < y < ( d + t ) ε 4 y > ( d + t ) } .
d 2 ψ d x 2 + d 2 ψ d y 2 + [ k 0 2 ε ( x , y ) β 2 ] ψ = 0 ,
1 X d 2 X d x 2 + [ k 0 2 ε ( x ) β 2 ] = 0 ,
1 Y d 2 Y d y 2 + [ k 0 2 ε ( y ) β 2 ] = 0 ,
β 2 = β 2 + β 2 k 0 2 ε 3 ,
tanh ( γ S / 2 ) = ( κ γ ( tan ( κ w ) γ / κ 1 + ( γ / κ ) tan ( κ w ) ) ) ± 1 ,
i tan κ 3 t = m 2 m 3 ( m 1 tanh γ 2 d ) ( 1 m 1 tanh γ 2 d ) m 3 ( 1 m 1 tanh γ 2 d ) m 2 ( m 1 tanh γ 2 d ) ,

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