Abstract

Using an asymptotic expansion of Maxwell equations and boundary conditions, we derive an amplitude equation for nonlinear TM modes in planar metal and dielectric waveguides. Our approach reveals that the physics of the significant enhancement of the nonlinear response in subwavelength waveguides with respect to their weakly guiding counterparts is hidden in surface effects. The corresponding enhancement factor is determined by the products of the surface discontinuities of the transverse field components and of the surface values of the longitudinal components of the electric field summed over all the interfaces. We present an insightful expression for the enhancement factor induced by the surface plasmon polaritons and discuss numerical and analytical results for the subwavelength dielectric and metal slot waveguides. Our theory also includes diffraction effects along the unbound direction in these waveguides.

© 2011 Optical Society of America

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  1. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
    [CrossRef]
  2. A. Di Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. 92, 083501 (2008).
    [CrossRef]
  3. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
    [CrossRef] [PubMed]
  4. E. Feigenbaum and M. Orenstein, “Plasmon-soliton,” Opt. Lett. 32, 674–676 (2007).
    [CrossRef] [PubMed]
  5. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express 16, 21209–21214(2008).
    [CrossRef] [PubMed]
  6. G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. 2, 125–135 (2008).
    [CrossRef]
  7. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Self-focusing and spatial plasmon-polariton solitons,” Opt. Express 17, 21732–21737 (2009).
    [CrossRef] [PubMed]
  8. A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
    [CrossRef]
  9. K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
    [CrossRef]
  10. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
    [CrossRef] [PubMed]
  11. G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
    [CrossRef]
  12. S. Afshar V., W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, “Small core optical waveguides are more nonlinear than expected: experimental confirmation,” Opt. Lett. 34, 3577–3579 (2009).
    [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  14. V. R. Almeida, Q. F. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004).
    [CrossRef] [PubMed]
  15. A. Marini and D. V. Skryabin, “Ginzburg–Landau equation bound to the metal–dielectric interface and transverse nonlinear optics with amplified plasmon polaritons,” Phys. Rev. A 81, 033850(2010).
    [CrossRef]
  16. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
    [CrossRef] [PubMed]
  17. S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009).
    [CrossRef] [PubMed]
  18. R. M. J. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. 1, 162–235 (2009).
    [CrossRef]
  19. B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
    [CrossRef]
  20. G. D. Valle and S. Longhi, “Geometric potential for plasmon polaritons on curved surfaces,” J. Phys. B 43, 051002 (2010).
    [CrossRef]

2010 (3)

A. Marini and D. V. Skryabin, “Ginzburg–Landau equation bound to the metal–dielectric interface and transverse nonlinear optics with amplified plasmon polaritons,” Phys. Rev. A 81, 033850(2010).
[CrossRef]

B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
[CrossRef]

G. D. Valle and S. Longhi, “Geometric potential for plasmon polaritons on curved surfaces,” J. Phys. B 43, 051002 (2010).
[CrossRef]

2009 (7)

2008 (3)

A. Di Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. 92, 083501 (2008).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express 16, 21209–21214(2008).
[CrossRef] [PubMed]

G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. 2, 125–135 (2008).
[CrossRef]

2007 (3)

E. Feigenbaum and M. Orenstein, “Plasmon-soliton,” Opt. Lett. 32, 674–676 (2007).
[CrossRef] [PubMed]

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

1987 (1)

Afshar V., S.

Agrawal, G. P.

Almeida, V. R.

Ashcom, J. B.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Barrios, C. A.

Benabid, F.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Boardman, A. D.

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Chen, X.

Cordeiro, C. M. B.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Couny, F.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Cruz, C. H. B.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Dadap, J. I.

Daniel, B. A.

Davoyan, A. R.

Di Falco, A.

A. Di Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. 92, 083501 (2008).
[CrossRef]

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Dulkeith, E.

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Ebendorff-Heidepriem, H.

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Feigenbaum, E.

Fragnito, H. L.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef] [PubMed]

Gattass, R. R.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Gorbach, A. V.

A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
[CrossRef]

Green, W. M.

He, S. L.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Hsieh, I.

Jacome, L.

Kivshar, Y. S.

Knight, J. C.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef] [PubMed]

Krauss, T. F.

A. Di Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. 92, 083501 (2008).
[CrossRef]

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef] [PubMed]

Lipson, M.

Liu, X.

Longhi, S.

G. D. Valle and S. Longhi, “Geometric potential for plasmon polaritons on curved surfaces,” J. Phys. B 43, 051002 (2010).
[CrossRef]

Lou, J. Y.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

MacDonald, K. F.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
[CrossRef]

Maier, S. A.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Marini, A.

A. Marini and D. V. Skryabin, “Ginzburg–Landau equation bound to the metal–dielectric interface and transverse nonlinear optics with amplified plasmon polaritons,” Phys. Rev. A 81, 033850(2010).
[CrossRef]

Maxwell, I.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Mihalache, D.

Monro, T. M.

O’Faolain, L.

A. Di Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. 92, 083501 (2008).
[CrossRef]

Orenstein, M.

Osgood, R. M. J.

Panoiu, N. C.

Poulton, C.

Samson, Z. L.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
[CrossRef]

Seaton, C. T.

Shadrivov, I. V.

Shen, M. Y.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Skryabin, D. V.

A. Marini and D. V. Skryabin, “Ginzburg–Landau equation bound to the metal–dielectric interface and transverse nonlinear optics with amplified plasmon polaritons,” Phys. Rev. A 81, 033850(2010).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
[CrossRef]

Stegeman, G. I.

Stockman, M. I.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
[CrossRef]

Tong, L. M.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Twardowski, T.

Vallaitis, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Valle, G. D.

G. D. Valle and S. Longhi, “Geometric potential for plasmon polaritons on curved surfaces,” J. Phys. B 43, 051002 (2010).
[CrossRef]

Vlasov, Y. A.

Vorreau, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

Wiederhecker, G. S.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Wright, E. M.

Wurtz, G. A.

G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. 2, 125–135 (2008).
[CrossRef]

Xu, Q. F.

Zanoni, R.

Zayats, A. V.

G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. 2, 125–135 (2008).
[CrossRef]

Zhang, W. Q.

Zheludev, N. I.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

A. Di Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. 92, 083501 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

G. D. Valle and S. Longhi, “Geometric potential for plasmon polaritons on curved surfaces,” J. Phys. B 43, 051002 (2010).
[CrossRef]

Laser Photon. Rev. (1)

G. A. Wurtz and A. V. Zayats, “Nonlinear surface plasmon polaritonic crystals,” Laser Photon. Rev. 2, 125–135 (2008).
[CrossRef]

Nat. Photon. (3)

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photon. 3, 55–58 (2009).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216–219 (2009).
[CrossRef]

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007).
[CrossRef]

Nature (1)

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. A (2)

A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
[CrossRef]

A. Marini and D. V. Skryabin, “Ginzburg–Landau equation bound to the metal–dielectric interface and transverse nonlinear optics with amplified plasmon polaritons,” Phys. Rev. A 81, 033850(2010).
[CrossRef]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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Figures (2)

Fig. 1
Fig. 1

(a)–(c) Surface-induced nonlinearity enhancement factor g for three different waveguide geometries. The corresponding waveguides and electric field profiles are shown in (d)–(f). (a), (d) Single dielectric waveguide of the width w. (b), (e) Dielectric slot waveguide, i.e., two dielectric waveguides separated by the slot width d. (c), (f) Metal slot waveguide, i.e., two metal interfaces separated by the slot width d. All other notations and features are explained in the text.

Fig. 2
Fig. 2

Comparison between γ / L y (dashed curves) and the surface-enhanced ϒ / L y (solid curves) nonlinear parameters for the (a) silicon and (b) metal slot waveguides with the polymer-filled slot. n 2 for silicon and polymer are assumed to be the same 4 · 10 18 m 2 / W , while the dielectric constants are ϵ w = 12 and ϵ d = 3.2 , respectively. w = 0.7 , λ = 1.5 μm , and L y = 100 nm . In (b), the metal is assumed linear and ϵ m = 25 , ϵ d = 4 , λ = 0.8 μm .

Equations (33)

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E x : Δ j [ ϵ E x + N x ] = 0 , E y : Δ j [ E y ] = 0 , H y : Δ j [ x E z e i β z z ( E x e i β z ) ] = 0 , E z : Δ j [ E z ] = 0 , H z : Δ j [ x E y y E x ] = 0 ,
Δ j [ f ] = lim δ 0 ( f ( x j δ ) f ( x j + δ ) ) .
N ( E x , E y , E z ) = 1 2 χ 3 ( | E | 2 E + 1 2 ( E · E ) E * ) ,
E x = A x ( ψ , x ) + B x ( ψ , x ) + O ( s 5 / 2 ) , E z = A z ( ψ , x ) + B z ( ψ , x ) + O ( s 5 / 2 ) , E y = C ( ψ , x ) + O ( s 2 ) , ψ = ψ ( z , y ) ,
L ^ A = 0 , E x : Δ j [ ϵ a A x ] = 0 , E z : Δ j [ A z ] = 0 ,
A ( A x A z ) , L ^ ( β 2 ϵ a i β x i β x x x 2 ϵ a ) .
e ( x ) = ( e x , e z ) T
P z = 1 4 k + ( A x H y * + A x * H y ) d x = I | ψ | 2 ϵ 0 c 2 β k Q ,
Q = + ϵ a | e x | 2 d x ,
x x 2 C ( β 2 ϵ a ) C = 0 , E y : Δ j [ C ] = 0 , H z : Δ j [ x C ] = I 1 / 2 y ψ Δ j [ e x ] .
L ^ B = I 1 / 2 J , E z : Δ [ B z ] = 0 , E x : Δ j [ ϵ a B x ] = Δ j [ ϵ b A x ] I 1 / 2 ψ | ψ | 2 Δ j [ n x ] ,
( e * · L ^ B ) d x = η P I 1 / 2 ( i β z ψ ) ,
( e * · J ) d x = i ( 2 + η ) β P z ψ + P ( 1 + η ) y y 2 ψ + ψ ϵ b | e | 2 d x + ψ | ψ | 2 ( e x * n x + e z * n z ) d x .
η = 1 β P j ( i e z ( x j ) ) * Δ j [ e x ] ,
P = | e | 2 d x .
i ψ ( z / k ) + 1 2 β k 2 ψ ( y / k ) 2 + α ψ + ϒ | ψ | 2 ψ = 0 ,
ϒ = g γ ,
γ = 2 k 2 3 β 2 P 2 ϵ a n 2 [ | e | 4 + 1 2 | e 2 | 2 ] d x .
g = 1 ( 1 + η ) 2 ,
α = k g 2 P β ϵ b | e | 2 d x .
L x = ( | e x | 2 d ( x / k ) ) 2 | e x | 4 d ( x / k )
g spp = ( ϵ d 2 + ϵ m 2 ) 2 ( ϵ d + ϵ m ) 4
ϒ spp = 2 k 2 ϵ d n 2 3 ϵ m 2 ( ϵ d + ϵ m ) 2 [ 1 + 1 2 ( ϵ m + ϵ d ϵ d ϵ m ) 2 ] .
e x { i β κ cos ( κ x ) | x | < w / 2 , i β q sin ( κ w / 2 ) e q ( | x | w / 2 ) | x | > w / 2 ,
e z = { sin ( κ x ) | x | < w / 2 , x | x | sin ( κ w / 2 ) e q ( | x | w / 2 ) | x | > w / 2 ,
η = 2 ( ϵ w ϵ d 1 ) × [ β 2 κ 2 1 + w κ ( β 2 κ 2 + 1 ) sin 1 ( κ w ) + ϵ w ϵ d ( β 2 q 2 + 1 ) ] 1 .
e x = i β [ 1 q d e q d x θ ( x ) 1 q m e q m x θ ( x ) ] ,
e z = e q d x θ ( x ) + e q m x θ ( x ) ,
η = 2 ϵ d / | ϵ m | 1 + ϵ d 2 / ϵ m 2 ,
γ = 2 k 2 ϵ d n 2 3 ( 1 ϵ d / | ϵ m | ) 2 [ 1 + 1 2 ( 1 ϵ d / | ϵ m | 1 + ϵ d / | ϵ m | ) 2 ] [ 1 + ϵ d 2 / ϵ m 2 ] 2 .
e x = { i β q d cosh ( q d x ) | x | < d / 2 , i β q m sinh ( q d d / 2 ) e q m ( | x | d / 2 ) | x | > d / 2 ,
e z = { sinh ( q d x ) | x | < d / 2 , x | x | sinh ( q d d / 2 ) e q m ( | x | d / 2 ) | x | > d / 2 ,
η = 2 ( 1 ϵ d ϵ m ) × [ β 2 q d 2 + 1 + d q d ( β 2 q d 2 1 ) sinh 1 ( q d d ) ϵ d ϵ m ( β 2 q m 2 + 1 ) ] 1 .

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