Abstract

Tight confinement of light in subwavelength waveguides induces substantial dispersion of their nonlinear response. We demonstrate that this dispersion of nonlinearity can lead to the modulational instability in the regime of normal group velocity dispersion through the mechanism independent from higher order dispersions of linear waves. A simple phenomenological model describing this effect is the nonlinear Schrödinger equation with the intensity dependent group velocity dispersion.

© 2011 OSA

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    [CrossRef]
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    [CrossRef]
  3. 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]
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2010

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

2009

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. Photonics 3, 216–219 (2009).
[CrossRef]

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

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]

T. X. Tran and F. Biancalana, “An accurate envelope equation for light propagation in photonic nanowires: new nonlinear effects,” Opt. Express 17, 17934–17949 (2009).
[CrossRef] [PubMed]

N. C. Panoiu, X. P. Liu, and R. M. Osgood, “Self-steepening of ultrashort pulses in silicon photonic nanowires,” Opt. Lett. 34, 947–949 (2009).
[CrossRef] [PubMed]

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I.-W. 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]

M. Galili, J. Xu, H. C. Mulvad, L. K. Oxenløwe, A. T. Clausen, P. Jeppesen, B. Luther-Davies, S. Madden, A. Rode, D.-Y. Choi, M. Pelusi, F. Luan, and B. J. Eggleton, “Breakthrough switching speed with an all-optical chalcogenide glass chip: 640 Gbit/s demultiplexing,” Opt. Express 17, 2182–2187 (2009).
[CrossRef] [PubMed]

2007

2006

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

2004

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

2003

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[CrossRef]

2001

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the fermi-pasta-ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

1991

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

1987

Afshar V, S.

Agrawal, G. P.

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. Photonics 3, 216–219 (2009).
[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. Photonics 3, 216–219 (2009).
[CrossRef]

Biancalana, F.

T. X. Tran and F. Biancalana, “An accurate envelope equation for light propagation in photonic nanowires: new nonlinear effects,” Opt. Express 17, 17934–17949 (2009).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[CrossRef]

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. Photonics 3, 216–219 (2009).
[CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Chen, X.

Choi, D.-Y.

Clausen, A. T.

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Dadap, J. I.

Dai, X.

De La Rue, R. M.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

de Nobriga, C. E.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[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. Photonics 3, 216–219 (2009).
[CrossRef]

Ding, W.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[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. Photonics 3, 216–219 (2009).
[CrossRef]

Eggleton, B. J.

Emplit, Ph.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the fermi-pasta-ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

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. Photonics 3, 216–219 (2009).
[CrossRef]

Fan, D.

Fauchet, P. M.

Foster, M. A.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

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. Photonics 3, 216–219 (2009).
[CrossRef]

Gaeta, A. L.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Galili, M.

Gorbach, A. V.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

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

Gouveia-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Green, W. M.

Haelterman, M.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the fermi-pasta-ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

Hobbs, G. D.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Hsieh, I.-W.

Jeppesen, P.

Knight, J. C.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[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. Photonics 3, 216–219 (2009).
[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. Photonics 3, 216–219 (2009).
[CrossRef]

Lin, Q.

Lipson, M.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Liu, X.

Liu, X. P.

Luan, F.

Luther-Davies, B.

Madden, S.

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. Photonics 3, 216–219 (2009).
[CrossRef]

Moloney, J. V.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

Monro, T. M.

Mulvad, H. C.

Osgood, R. M.

Oxenløwe, L. K.

Panoiu, N. C.

Pelusi, M.

Potasek, M. J.

Rode, A.

Russell, P. St. J.

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Samarelli, A.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Schmidt, B. S.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Sharping, J. E.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Skryabin, D. V.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

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

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Sorel, M.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Staines, O. K.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Su, W.

Tang, Z.

Tran, T. X.

Turner, A. C.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

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. Photonics 3, 216–219 (2009).
[CrossRef]

Van Simaeys, G.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the fermi-pasta-ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

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. Photonics 3, 216–219 (2009).
[CrossRef]

Wadsworth, W. J.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Wen, S.

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

Xiang, Y.

Xu, J.

Zhang, J.

Adv. Opt. Photon.

Appl. Phys. B

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

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. Photonics 3, 216–219 (2009).
[CrossRef]

Nature

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta , “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwave-length waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

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

Phys. Rev. E

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Phys. Rev. Lett.

G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the fermi-pasta-ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Guided mode of Al0.25Ga0.75As waveguide suspended in air: (a) profile of the dominant electric field component (ex ) at λp = 1.7μm for 500nm×300nm. Waveguide is indicated by solid lines; (b) calculated GVD, D = −2πcβ 2 2, white area indicates the range of normal GVD.

Fig. 2
Fig. 2

MI in Al0.25Ga0.75As waveguide: (a) δβ and Γ at λ p = 2πc/ωp = 1.7μm as functions of the signal/idler wavelengths. n 2 = 1.5 · 10−17m2/W, D = −0.07ps/nm/mm; (b) gain as function of pump power and signal/idler wavelengths, g > 0 within shaded areas with darker colours corresponding to larger values of g. Dashed lines show the threshold power Pth = δβ/(4 Γ); (c) gain as function of signal/idler wavelength for P = 800W.

Fig. 3
Fig. 3

Dynamics of the pump and signal waves for δβ > 0 (a) and δβ < 0 (b). Initial conditions: P(z = 0) = 800W, Ps (z = 0) = 0.8mW, Pi (z = 0) = 0, λp = 1.6μm, λs = 1.55μm (a), λp = 1.7μm, λs = 1.65μm (b). Solid/dashed line corresponds to pump/signal.

Fig. 4
Fig. 4

Fundamental mode of the silicon-polymer slot waveguide: a) profile of the dominant electric field component (ex ) at λp = 1.7μm for 500nm×220nm silicon waveguides, wall-to-wall separation 50nm, silica glass substrate and nonlinear polymer cladding. Geometry is indicated by solid lines; (b) calculated GVD.

Fig. 5
Fig. 5

The same as Figs. 2(a) and (b) but for the slot geometry, λp = 1.7μm (D = −0.0015), n 2, Polymer = 16.9 · 10−18 m 2/W, n 2, Silicon = 4 · 10−18 m2 /W, n 2, SiO 2 = 2.5 · 10−20 m 2/W.

Fig. 6
Fig. 6

Nonlinear parameters γ 0 (a) and γ 2 (b) for the waveguides as in Fig. 1 (solid lines) and Fig. 4 (dashed lines).

Equations (9)

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i z A n = k , l , m γ n k l m e i ( β k + β m β l β n ) z A k A l * m ,
γ n k l m = ε 0 ω n I n I k I l I m + χ n k l m ζ n k l m ( r ) d x d y ,
ζ n k l m = ρ [ ( e n * e m ) ( e k e l * ) + ( e n * e k ) ( e m e l * ) + ( e n * e l * ) ( e k e m ) ] + 3 ( 1 ρ ) i = x , y , z e n i * e k i e l i * e m i .
i z A p = γ p | A p | 2 A p ,
i z A s = 2 γ s p | A p | 2 A s γ 4 s A p 2 A i * e i δ β z ,
i z A i = 2 γ i p | A p | 2 A i γ 4 i A p 2 A s * e i δ β z ,
g = 1 2 Re ( 4 Γ + P δ β ) ( δ β 4 Γ P ) , Γ ± = ( γ s p + γ i p γ p ± γ 4 s γ 4 i ) / 2 .
4 Γ P < δ β < 4 Γ + P .
i z A = i γ 1 | A | 2 t A + 1 2 ( β 2 + γ 2 | A | 2 ) t 2 A γ 0 | A | 2 A .

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