Abstract

In this work, we propose a method for designing optical devices described by coupled-mode equations. Following a commonly applied optimization strategy, we combine gradient-based optimization algorithms with an adjoint sensitivity analysis of the coupled-mode equations to obtain an optimization scheme that can handle a large number of design parameters. To demonstrate this adjoint-enabled optimization method, we design a silicon-on-insulator Raman wavelength converter. As structure, we consider a waveguide constructed from a series of interconnected and adiabatically-varying linear tapers, and treat the width at each interconnection point, the waveguide length, and the pump-Stokes frequency difference as independent design parameters. Optimizing with respect to these 1603 parameters results in an improvement of more than 10 dB in the conversion efficiency for a waveguide length of 6.28 cm and frequency difference 187 GHz below the Raman shift as compared to a converter designed by the conventional phase-matching design rule and operating at perfect Raman resonance. The increase in conversion efficiency is also accompanied by a more than 7 dB-improvement in the Stokes amplification. Hence, the adjoint-enabled optimization allows us to identify a more efficient method for achieving Raman conversion than conventional phase-matching. We also show that adjoint-enabled optimization significantly improves design robustness. In case of the Raman converter example, this leads to a sensitivity with respect to local variations in waveguide width that is several orders of magnitude smaller for the optimized design than for the phase-matched one.

© 2014 Optical Society of America

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References

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    [CrossRef]
  31. X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
    [CrossRef]
  32. E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
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    [CrossRef]
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2013 (3)

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol. 31, 2113–2121 (2013).
[CrossRef]

O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett. 25, 724–727 (2013).
[CrossRef]

2012 (3)

D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. 285, 3306–3311 (2012).
[CrossRef]

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. 10, 153–165 (2012).
[CrossRef]

J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express 20, 9227–9242 (2012).
[CrossRef] [PubMed]

2011 (4)

2010 (3)

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2010).
[CrossRef] [PubMed]

L. Jin, W. Jin, J. Ju, and Y. Wang, “Coupled local-mode theory for strongly modulated long period gratings,” J. Lightwave Technol. 28, 1745–1751 (2010).
[CrossRef]

2009 (2)

2007 (2)

2006 (3)

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[CrossRef]

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]

2005 (3)

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23, 2094–2102 (2005).
[CrossRef]

2004 (1)

2003 (4)

2002 (1)

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

1990 (1)

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

1987 (1)

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Agrawal, G. P.

Almeida, V. R.

Beccherelli, R.

D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. 285, 3306–3311 (2012).
[CrossRef]

Bennett, B.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Bergman, K.

Borel, P.

Cao, Y.

Y. Cao, S. Li, L. Petzold, and R. Serban, “Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,” SIAM J. Sci. Comput. 24, 1076–1089 (2003).
[CrossRef]

Chen, X.

Claps, R.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Dadap, J. I.

Debaes, C.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Y. Lefevre, N. Vermeulen, C. Debaes, and H. Thienpont, “Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism,” Opt. Express 19, 18810–18826 (2011).
[CrossRef] [PubMed]

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

Dianov, E.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Dimitropoulos, D.

Driscoll, J. B.

Dulkeith, E.

Elesin, Y.

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. 10, 153–165 (2012).
[CrossRef]

Fainman, Y.

D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2010).
[CrossRef] [PubMed]

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Fauchet, P. M.

Foster, M. A.

Frandsen, L.

Gaeta, A. L.

Golovchenko, E.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Green, W. M.

Grote, R. R.

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Han, Y.

Harpøth, A.

Hindmarsh, A. C.

R. Serban and A. C. Hindmarsh, “CVODES, the sensitivity-enabled ODE solver in SUNDIALS,” in “Proceedings of the 5th International Conference on Multibody Systems, Nonlinear Dynamics and Control, Long Beach, CA” (2005).

Hirayama, K.

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

Hsieh, I.-W.

Huang, W.-P.

Jalali, B.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23, 2094–2102 (2005).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003).
[CrossRef] [PubMed]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
[CrossRef] [PubMed]

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in “Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science” (2008), pp. 1–2.

Jensen, J.

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. 10, 153–165 (2012).
[CrossRef]

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

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

Jensen, J. S.

Jhaveri, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Jin, L.

Jin, W.

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Ju, J.

Koonath, P.

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in “Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science” (2008), pp. 1–2.

Kriezis, E.

D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. 285, 3306–3311 (2012).
[CrossRef]

Kriezis, E. E.

O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett. 25, 724–727 (2013).
[CrossRef]

Kristensen, M.

Lazarov, B.

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. 10, 153–165 (2012).
[CrossRef]

Lefevre, Y.

Li, S.

Y. Cao, S. Li, L. Petzold, and R. Serban, “Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,” SIAM J. Sci. Comput. 24, 1076–1089 (2003).
[CrossRef]

Lin, Q.

Lipson, M.

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Liu, X.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Ly Gagnon, D. S.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Mamyshev, P.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Miller, D. A. B.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Morita, H.

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

Mu, J.

Nishiwaki, S.

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 1999).
[CrossRef]

Nomura, T.

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

Ophir, N.

Osgood, J.

Osgood, R.

X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[CrossRef]

Osgood, R. M.

Painter, O. J.

Panepucci, R. R.

Paniccia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Panoiu, N.

X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[CrossRef]

Panoiu, N. C.

Petzold, L.

Y. Cao, S. Li, L. Petzold, and R. Serban, “Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,” SIAM J. Sci. Comput. 24, 1076–1089 (2003).
[CrossRef]

Pilipetskii, A.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Raghunathan, V.

Rong, H.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Salem, R.

Sato, K.

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

Serban, R.

Y. Cao, S. Li, L. Petzold, and R. Serban, “Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,” SIAM J. Sci. Comput. 24, 1076–1089 (2003).
[CrossRef]

R. Serban and A. C. Hindmarsh, “CVODES, the sensitivity-enabled ODE solver in SUNDIALS,” in “Proceedings of the 5th International Conference on Multibody Systems, Nonlinear Dynamics and Control, Long Beach, CA” (2005).

Shi, P.

Shoji, T.

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

Sigmund, O.

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. 10, 153–165 (2012).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (2011).
[CrossRef]

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

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Solli, D. R.

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in “Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science” (2008), pp. 1–2.

Soref, R.

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Sun, P. C.

D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2010).
[CrossRef] [PubMed]

Tan, D. T.

D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2010).
[CrossRef] [PubMed]

Thienpont, H.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol. 31, 2113–2121 (2013).
[CrossRef]

Y. Lefevre, N. Vermeulen, C. Debaes, and H. Thienpont, “Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism,” Opt. Express 19, 18810–18826 (2011).
[CrossRef] [PubMed]

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

Tsilipakos, O.

O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett. 25, 724–727 (2013).
[CrossRef]

Tsuchizawa, T.

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

Tsuji, Y.

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

Turner, A. C.

Van Erps, J.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Vermeulen, N.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol. 31, 2113–2121 (2013).
[CrossRef]

Y. Lefevre, N. Vermeulen, C. Debaes, and H. Thienpont, “Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism,” Opt. Express 19, 18810–18826 (2011).
[CrossRef] [PubMed]

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

Vlasov, Y. A.

Wahl, P.

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

Wang, F.

Wang, Y.

Watanabe, T.

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

Woo, J. C. S.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 1999).
[CrossRef]

Yamada, K.

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

Zhang, J.

Zografopoulos, D.

D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. 285, 3306–3311 (2012).
[CrossRef]

Zografopoulos, D. C.

O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett. 25, 724–727 (2013).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Electron. Lett. (1)

T. Shoji, T. Tsuchizawa, T. Watanabe, K. Yamada, and H. Morita, “Low loss mode size converter from 0.3 μm square Si wire waveguides to singlemode fibres,” Electron. Lett. 38, 1669–1670 (2002).
[CrossRef]

IEEE J. Quantum Electron. (3)

X. Chen, N. Panoiu, and R. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[CrossRef]

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

O. Tsilipakos, D. C. Zografopoulos, and E. E. Kriezis, “Quasi-soliton pulse-train propagation in dispersion-managed silicon rib waveguides,” IEEE Photon. Technol. Lett. 25, 724–727 (2013).
[CrossRef]

Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, and S. Nishiwaki, “Design of optical circuit devices based on topology optimization,” IEEE Photon. Technol. Lett. 18, 850–852 (2006).
[CrossRef]

J. Lightwave Technol. (3)

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

Laser Photon. Rev. (2)

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

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

Nat. Commun. (1)

D. T. Tan, P. C. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2010).
[CrossRef] [PubMed]

Nature (1)

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

D. Zografopoulos, R. Beccherelli, and E. Kriezis, “Quasi-soliton propagation in dispersion-engineered silicon nanowires,” Opt. Commun. 285, 3306–3311 (2012).
[CrossRef]

Opt. Express (9)

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express 15, 12949–12958 (2007).
[CrossRef] [PubMed]

J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express 20, 9227–9242 (2012).
[CrossRef] [PubMed]

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17, 19134–19152 (2009).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[CrossRef] [PubMed]

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]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
[CrossRef] [PubMed]

Y. Lefevre, N. Vermeulen, C. Debaes, and H. Thienpont, “Optimized wavelength conversion in silicon waveguides based on “off-Raman-resonance” operation: extending the phase mismatch formalism,” Opt. Express 19, 18810–18826 (2011).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phot. Nano. Fund. Appl. (1)

Y. Elesin, B. Lazarov, J. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Phot. Nano. Fund. Appl. 10, 153–165 (2012).
[CrossRef]

Prog. Electromagn. Res. (1)

P. Wahl, D. S. Ly Gagnon, C. Debaes, J. Van Erps, N. Vermeulen, D. A. B. Miller, and H. Thienpont, “B-CALM: an open-source multi-GPU-based 3D-FDTD with multi-pole dispersion for plasmonics,” Prog. Electromagn. Res. 138, 467–478 (2013).
[CrossRef]

SIAM J. Sci. Comput. (1)

Y. Cao, S. Li, L. Petzold, and R. Serban, “Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution,” SIAM J. Sci. Comput. 24, 1076–1089 (2003).
[CrossRef]

Struct. Multidisc. Optim. (1)

J. S. Jensen, “Topology optimization of nonlinear optical devices,” Struct. Multidisc. Optim. 43, 731–743 (2011).
[CrossRef]

Other (7)

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in “Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science” (2008), pp. 1–2.

We employed the commercial software package MODE Solutions by Lumerical to calculate the dispersion characteristics and mode profiles of SOI waveguides.

ePIXfab, The silicon photonics website, http://www.epixfab.eu/ .

R. Serban and A. C. Hindmarsh, “CVODES, the sensitivity-enabled ODE solver in SUNDIALS,” in “Proceedings of the 5th International Conference on Multibody Systems, Nonlinear Dynamics and Control, Long Beach, CA” (2005).

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 1999).
[CrossRef]

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Figures (6)

Fig. 1
Fig. 1

A coupled-mode-based device is simulated by solving the coupled-mode equations for a given input vector A0, yielding the output vector Af. The input A0 as well as the coupled-mode equations depend on a set of design parameters θ1,...,θM, whereas the device’s performance figure G is calculated from the output Af. Designing the device consists of determining the parameter values θ̄ that optimize G.

Fig. 2
Fig. 2

Adjoint-enabled optimization optimizes the performance function G with respect to the design parameters θ by employing an iterative optimization algorithm consisting of five steps during each iteration. Note that, if G satisfies G / A f * = ( G / A f ) *, then λ = μ* so that only one adjoint vector has to be propagated in Step 4.

Fig. 3
Fig. 3

(a) The conversion efficiency Pa/Ps,0 and (b) Stokes amplification Ps/Ps,0 of that can be achieved with a 3 cm-long, non-varying, rectangular SOI waveguide [see inset (a)] with fixed height h = 220 nm depends strongly on its width w.

Fig. 4
Fig. 4

The variable-width waveguide proposed consists of a series of interconnected linear tapers with equal lengths of LTaper. The widths w1,...,wk, wk+1 ... at each interconnection point is taken as an independent design parameter.

Fig. 5
Fig. 5

Comparison between the evolutions in (a) waveguide width w, (b) conversion efficiency Pa/Ps,0, (c) Stokes amplification Ps/Ps,0, and (d) phase difference Δϕ for the initial Raman converter design derived from the phase-matching rule (dashed lines) and the design optimized by adjoint-enabled optimization (full lines). In (d), also the phases −ΔϕFWM,a and −ΔϕFWM,s of the anti-Stokes and Stokes FWM gains are shown, both for the initial frequency difference ΔΩ = ΩR (dash-dotted lines) and for the optimized one ΔΩ = ΩR − 187.0 GHz (dotted lines), to indicate at which Δϕ values the waves experience maximal gain.

Fig. 6
Fig. 6

The sensitivity of the Raman converter performance Pa(zf)−1∂Pa(zf)/∂wk with respect to local variations in waveguide width wk is several orders magnitude smaller for the design optimized by adjoint-enabled optimization (full line) than for the initial design derived from the phase-matching rule (dashed line). The inset shows a close-up of the sensitivity Pa(zf)−1∂Pa(zf)/∂wk between −1·10−5 nm−1 and 1·10−5 nm−1.

Equations (27)

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E ( r , t ) = Re [ j = 1 N A j ( z ) e j ( x , y ) e i ω j t ] .
A = [ A 1 A 2 A N ] .
A z = F ( z , A , A * , θ ) ,
A ( z 0 ) = A 0 ( θ ) .
μ z = ( F A ) μ ( F * A ) λ , λ z = ( F A * ) μ ( F * A * ) λ ,
μ ( z f ) = ( G A ) , λ ( z f ) = ( G A * ) ,
μ = [ μ 1 μ 2 μ N ] , λ = [ λ 1 λ 2 λ N ] .
d G d θ k = G θ k + μ ( z 0 ) A 0 θ k + λ ( z 0 ) A 0 * θ k + 2 z 0 z f μ F θ k + λ F * θ k d z .
μ z = ( F A ) μ ( F * A ) μ * ,
μ ( z f ) = ( G A ) ,
d G d θ k = G θ k + 2 Re [ μ ( z 0 ) A 0 θ k ] + 2 z 0 z f Re [ μ F θ k ] d z ,
A p z = ( i β 0 , p α p 2 ) A p + j = p , s , a ( i γ K , p ( 2 δ p j ) Γ j p p j K + i γ R , p Γ j p p j R ) S j S p | A j | 2 A p + ( i ω p c n f , p α f , p 2 ) Γ p f S p A p + i γ R , p j = s , a Γ j p p j R H R ( ω p ω j ) S j S p | A j | 2 A p + i ( 2 γ K , p Γ s p a p K + γ R , p Γ s p a p R [ H R ( ω s ω p ) + H R ( ω a ω p ) ] ) S p S s S a A s A a A p * ,
A s z = ( i β 0 , s α s 2 ) A s + j = p , s , a ( i γ K , s ( 2 δ s j ) Γ j s s j K + i γ R , s Γ j s s j R ) S j S s | A j | 2 A s + ( i ω s c n f , s α f , s 2 ) Γ s f S s A s + i γ R , s Γ p s s p R H R ( ω s ω p ) S p S s | A p | 2 A s + i ( γ K , s Γ s p a p K + γ R , s Γ s p a p R H R ( ω p ω a ) ) S p S s S a A p 2 A a * ,
A a z = ( i β 0 , a α a 2 ) A a + j = p , s , a ( i γ K , a ( 2 δ a j ) Γ j a a j K + i γ R , a Γ j a a j R ) S j S a | A j | 2 A a + ( i ω a c n f , a α f , a 2 ) Γ a f S a A a + i γ R , a Γ a p p a R H R ( ω a ω p ) S p S a | A p | 2 A a + i ( γ K , a Γ s p a p K + γ R , a Γ s p a p R H R ( ω p ω s ) ) S p S s S a A p 2 A s * .
γ K , j = ω j c n 2 + i β T 2 ,
γ R , j = ω j ω ref g R , ref Γ R Ω R ,
H R ( Δ Ω ) = Ω R 2 Ω R 2 Δ Ω 2 2 i Γ R Δ Ω .
α f , j = ( ω r ω j ) 2 14.5 × 10 18 N f ,
n f , j = ( ω r ω j ) 2 ( 8.8 × 10 4 N f 8.5 N f 0.8 ) × 10 18 .
N f = i , j = p , s , a τ 0 ( 2 δ i j ) β T S i S j Γ i j j i K | A i | 2 | A j | 2 ( ω i + ω j ) A wg .
z k + 1 = z k + L Taper ,
w ( z k z z k + 1 ) = w k + w k + 1 w k L Taper ( z z k ) .
F Δ Ω = F ω a F ω s ,
F ( z k z z k + 1 ) w k + 1 = F w w w k + 1 = F w z z k L Taper ,
F ( z k z z k + 1 ) w k = F w w w k = F w L Taper ( z z k ) L Taper .
G z f = 2 Re [ G A f A f z f ] = 2 Re [ μ A z ] z = z f .
G FWM , j = i ( γ K , j Γ s p a p K + γ R , j Γ s p a p R H R ( ω j ω p ) ) S p S s S a

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