Abstract

We present a numerical tool that searches an optimal cross section geometry of silicon-on-insulator waveguides given a target dispersion profile. The approach is a gradient-based multidimensional method whose efficiency resides on the simultaneous calculation of the propagation constant derivatives with respect to all geometrical parameters of the structure by using the waveguide mode distribution. The algorithm is compatible with regular mode solvers. As an illustrative example, using a silicon slot hybrid waveguide with 4 independent degrees of freedom, our approach finds ultra-flattened (either normal or anomalous) dispersion over 350 nm bandwidth in less than 10 iterations.

© 2014 Optical Society of America

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  1. N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
    [CrossRef]
  2. 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]
  3. A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904–1908 (2010).
    [CrossRef]
  4. X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
    [CrossRef]
  5. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
    [CrossRef]
  6. R. Halir, Y. Okawachi, J. S. Levy, M. A. Foster, M. Lipson, and A. L. Gaeta, “Ultrabroadband supercontinuum generation in a CMOS-compatible platform,” Opt. Lett. 37, 1685–1687 (2012).
    [CrossRef]
  7. A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret, and M. V. Andrés, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001).
    [CrossRef]
  8. X. Liu, W. M. J. Green, X. Chen, I.-W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R. M. Osgood, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008).
    [CrossRef]
  9. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
    [CrossRef]
  10. S. Mas, J. Caraquitena, J. V. Galán, P. Sanchis, and J. Martí, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010).
    [CrossRef]
  11. L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010).
    [CrossRef]
  12. L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012).
    [CrossRef]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge, 2007).
  14. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
    [CrossRef]
  15. J. Riishede and O. Sigmund, “Inverse design of dispersion compensating optical fiber using topology optimization,” J. Opt. Soc. Am. B 25, 88–97 (2008).
    [CrossRef]
  16. E. Silvestre, T. Pinheiro-Ortega, P. Andrés, J. J. Miret, and A. Coves, “Differential toolbox to shape dispersion behavior in photonic crystal fibers,” Opt. Lett. 31, 1190–1192 (2006).
    [CrossRef]
  17. J. J. Miret, E. Silvestre, and P. Andrés, “Octave-spanning ultraflat supercontinuum with soft-glass photonic crystal fibers,” Opt. Express 17, 9197–9203 (2009).
    [CrossRef]
  18. S. V. Afshar 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]
  19. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef]
  20. D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
    [CrossRef]
  21. E. Silvestre, T. Pinheiro-Ortega, P. Andrés, J. J. Miret, and A. Ortigosa-Blanch, “Analytical evaluation of chromatic dispersion in photonic crystal fibers,” Opt. Lett. 30, 453–455 (2005).
    [CrossRef]
  22. B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23, 4477–4485 (1984).
    [CrossRef]
  23. L. Zhang, C. Bao, V. Singh, J. Mu, C. Yang, A. M. Agarwal, L. C. Kimerling, and J. Michel, “Generation of two-cycle pulses and octave-spanning frequency combs in a dispersion-flattened micro-resonator,” Opt. Lett. 38, 5122–5125 (2013).
    [CrossRef]
  24. S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
    [CrossRef]
  25. R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340–343 (1939).
    [CrossRef]
  26. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  27. E. Silvestre, M. V. Andrés, and P. Andrés, “Biorthonormal-basis method for the vector description of optical-fiber modes,” J. Lightwave Technol. 16, 923–928 (1998).
    [CrossRef]
  28. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

2013 (1)

2012 (4)

L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012).
[CrossRef]

R. Halir, Y. Okawachi, J. S. Levy, M. A. Foster, M. Lipson, and A. L. Gaeta, “Ultrabroadband supercontinuum generation in a CMOS-compatible platform,” Opt. Lett. 37, 1685–1687 (2012).
[CrossRef]

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

2011 (1)

S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

2010 (4)

2009 (2)

2008 (2)

2007 (1)

2006 (3)

2005 (1)

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

2001 (2)

1998 (1)

1984 (1)

1982 (1)

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

1939 (1)

R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340–343 (1939).
[CrossRef]

Afshar, S. V.

Agarwal, A. M.

Agrawal, G. P.

Andrés, M. V.

Andrés, P.

Aspnes, D. E.

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

Baba, T.

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

Baets, R.

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[CrossRef]

Bao, C.

Beausoleil, R. G.

Bogaerts, W.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[CrossRef]

Caraquitena, J.

Chen, X.

Coves, A.

Dadap, J. I.

Dumon, P.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[CrossRef]

Ferrando, A.

Feynman, R. P.

R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340–343 (1939).
[CrossRef]

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge, 2007).

Foster, M. A.

Gaeta, A. L.

Galán, J. V.

Green, W. M. J.

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

X. Liu, W. M. J. Green, X. Chen, I.-W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R. M. Osgood, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008).
[CrossRef]

Halir, R.

Hayakawa, R.

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

Hosoi, R.

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

Hsieh, I.-W.

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Ishikura, N.

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Jensen, S.

S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef]

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef]

Kimerling, L. C.

Kuyken, B.

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

Levy, J. S.

Lin, Q.

Lipson, M.

Liu, X.

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

X. Liu, W. M. J. Green, X. Chen, I.-W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R. M. Osgood, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008).
[CrossRef]

Manolatou, C.

Martí, J.

Mas, S.

Michel, J.

Miret, J. J.

Monro, T. M.

Mu, J.

Okawachi, Y.

Ortigosa-Blanch, A.

Osgood, R. M.

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

X. Liu, W. M. J. Green, X. Chen, I.-W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R. M. Osgood, “Conformal dielectric overlayers for engineering dispersion and effective nonlinearity of silicon nanophotonic wires,” Opt. Lett. 33, 2889–2891 (2008).
[CrossRef]

Pinheiro-Ortega, T.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge, 2007).

Riishede, J.

Roelkens, G.

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

Salem, R.

Sanchis, P.

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]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[CrossRef]

Selvaraja, S. K.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[CrossRef]

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]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[CrossRef]

Shinkawa, M.

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

Sigmund, O.

Silvestre, E.

Singh, V.

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Tamanuki, T.

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

Tatian, B.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge, 2007).

Thourhout, D. V.

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[CrossRef]

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]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[CrossRef]

Turner-Foster, A. C.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge, 2007).

Vlasov, Y. A.

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Willner, A. E.

Yan, Y.

Yang, C.

Yin, L.

Yue, Y.

Zhang, L.

Am. J. Phys. (1)

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012).
[CrossRef]

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

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Sub-nanometer linewidth uniformity in silicon nano-photonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).
[CrossRef]

J. Lightwave Technol. (1)

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

Laser Photon. Rev. (1)

S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

Nat. Photonics (1)

X. Liu, B. Kuyken, G. Roelkens, R. Baets, R. M. Osgood, and W. M. J. Green, “Bridging the mid-infrared-to-telecom gap with silicon nanophotonic spectral translation,” Nat. Photonics 6, 667–671 (2012).
[CrossRef]

Nature (1)

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]

Opt. Express (9)

A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret, and M. V. Andrés, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane-wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006).
[CrossRef]

S. V. Afshar 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]

J. J. Miret, E. Silvestre, and P. Andrés, “Octave-spanning ultraflat supercontinuum with soft-glass photonic crystal fibers,” Opt. Express 17, 9197–9203 (2009).
[CrossRef]

A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904–1908 (2010).
[CrossRef]

L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010).
[CrossRef]

S. Mas, J. Caraquitena, J. V. Galán, P. Sanchis, and J. Martí, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010).
[CrossRef]

L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. (1)

R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340–343 (1939).
[CrossRef]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Other (2)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, 3rd ed. (Cambridge, 2007).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Outline of the slot waveguide under study [11]. Comparison of the derivatives of the effective refractive index (b) and the group-velocity dispersion parameter (c) with respect to the geometrical parameters h, w, hs, and ds, shown in (a), calculated numerically (dashed curves) and by Eq. (5) (crosses).

Fig. 2.
Fig. 2.

Five examples of optimization starting from the same geometry (dotted black curve, see details in the text) and with five different flattened dispersion profiles as a goal (solid blue line); namely, zero (a), low and anomalous (b), low and normal (c), high and anomalous (d), and high and normal (e). Dispersion profiles between the first iteration (dotted–dashed green curve) and the last one (dashed red line) are omitted.

Fig. 3.
Fig. 3.

Plot of the evolution of the geometrical parameters during the optimization procedure of the same five examples shown in Fig. 2, normalized to its starting values (solid colored curves, left-hand axis). Dashed black curves represent the evolution of the square root of the merit function, the standard deviation of β2 with respect to the target in the design frequency range (right-hand axis).

Fig. 4.
Fig. 4.

Last four target (solid blue lines) and optimized (dashed red curves) dispersion profiles shown in Fig. 2 together with those obtained under the restriction hs45nm (dotted green curves).

Fig. 5.
Fig. 5.

Plot of the dispersion profiles corresponding to the optimized geometry for the system described in Fig. 2(b) (solid black curve, see details in the text); and the same geometry with each one of the design parameters (h, w, hs, and ds) increased (red lines) or decreased (green lines) by 2 nm.

Tables (1)

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Table 1. Geometrical Parameters and Merit Function after Convergence for the Five Dispersion Targets (Fig. 2)

Equations (17)

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χ2(p)=1Nωk=1Nω(β2(p;ωk)β2target(ωk))2,
β2lin(p;ω)=β2(p(m);ω)+pβ2(p(m);ω)·(pp(m)),
χloc2(p)=1Nωk=1Nω(β2lin(p;ωk)β2target(ωk))2.
pβ=ϵ0ω2S(et·(pϵtt)et+(pϵzz)ez2)dSS(et×ht)·z^dS,
pβ=ϵ0ω2S((pϵnn1)dn2(pϵTT)eT2+(pϵzz)ez2)dSS(et×ht)·z^dS,
Lht={ttT+ηϵttηTω2c2ηϵttt[ϵzz1tTη]}ht=β2ht,
(β2)p=h˜t|(pL)hth˜t|ht,
h˜t|(pL)ht=SdS(ω2c2etT(pϵtt)ηThtetT(pϵtt)t[ϵzz1tTηht]etTϵttt[(pϵzz1)tTηht]).
×H=iϵ0μ0ωcϵE,
×E=iμ0ϵ0ωcH,
·D=0.
tTηht=iϵ0μ0ωcϵzzez,
tez=iμ0ϵ0ωcηht+iβet,
tT(ϵttet)=iβϵzzez,
SdSetT(pϵtt)t[ϵzz1tTηht]=SdSetT(pϵtt)[ω2c2ηht+ϵ0ωβet].
SdSetTϵttt[(pϵzz1)tTηht]=ϵ0ωβSdS(pϵzz)ez2.
ε^=11/εP+ε(IP),

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