Abstract

Nanophotonic waveguides can be engineered in order to exhibit slow mode propagation thereby enhancing the nonlinear responses. In such waveguides, loss and nonlinear coefficients are strongly wavelength dependent, a property that must be considered when the signal to pump detuning is large. Exact formulas for the parametric gain and conversion efficiency, accounting for the dispersion of losses and nonlinearity, are derived here. They can be applied to any waveguide presenting such features; in particular they have been calculated for a III–V semiconductor photonic crystal waveguide, where narrow- and broad-band amplification are predicted. The asymmetry of losses causes major asymmetries in the gain and conversion efficiency, which are no longer simply related as in the case of waveguides in which loss do not depend on the wavelength.

© 2014 IEEE

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  1. T. F. Krauss, "Slow light in photonic crystal waveguides ," J. Phys. D, Appl. Phys. 40, 2666-2670 (2007).
  2. T. Baba, "Slow light in photonic crystals," Nature Photon. 2, 465-473 (2008).
  3. J. B. Khurgin, R. S. Tucke, Slow Light: Science and Applications (CRC Press, 2009).
  4. C. Monat, M. de Sterke, B. J. Eggleton, "Slow light enhanced nonlinear optics in periodic structures," J. Opt. 12, 288, 104003 (2010).
  5. B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, B. J. Eggleton, D. J. Moss, "Optical signal processing on a silicon chip at 640Gb/s using slow-light," Opt. Exp. 18, 7770-7781 (2010).
  6. L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, A. E. Willner, "Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation," Opt. Exp. 20, 1685-1690 (2012).
  7. M. Ebnali-Heidari, C. Monat, C. Grillet, M. K. Moravvej-Farshi, "A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration," Opt. Exp. 17, 18340-18353 (2009).
  8. V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. G. Someda, G. Vadalà, "Highly efficient four wave mixing in GaInP photonic crystal waveguides," Opt. Lett. 35, 1440-1442 (2010).
  9. M. Santagiustina, C. G. Someda, G. Vadalá, S. Combrié, A. De Rossi, "Theory of slow light enhanced four-wave mixing in photonic crystal waveguides ," Opt. Exp. 18, 21024-21029 (2010).
  10. C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. J. Eggleton, T. P. White, L. O’Faolain, J. Li, T. F. Krauss, "Four-wave mixing in slow light engineered silicon photonic crystal waveguides," Opt. Exp. 18 , 22915-22927 (2010).
  11. J. Li, L. O’Faolain, I. H. Rey, T. F. Krauss, "Four-wave mixing in photonic crystal waveguides: Slow light enhancement and limitations," Opt. Exp. 19, 4458-4463 (2011).
  12. B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O'Faolain, T. F. Krauss, B. J. Eggleton, "Ultracompact 160 Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide," Opt. Lett. 36, 1728-1730 (2011).
  13. P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, A. De Rossi, "Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides," Opt. Lett. 36, 2629-2631 (2011).
  14. I. Cestier, S. Combrié, S. Xavier, G. Lehoucq, A. D. Rossi, G. Eisenstein, "Chip-scale parametric amplifier with 11-db gain at 1550 nm based on a slow-light GaInP photonic crystal waveguide," Opt. Lett. 37, 3996-3998 (2012).
  15. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, P. I. Borel, "Photonic crystal waveguides with semi-slow light and tailored dispersion properties," Opt. Exp. 14 , 9444-9450 (2006).
  16. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Expr. 16, 6227-6232 (2008).
  17. P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, "Control of dispersion in photonic crystal waveguides using group symmetry theory," Opt. Exp. 20, 13108-13114 (2012 ).
  18. M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, T. F. Krauss, "Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law ," Phys. Rev. B 80, 195305-195310 (2009).
  19. S. Combrié, Q. Vy Tran, C. Husko, P. Colman, A. De Rossi, "High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption," Appl. Phys. Lett. 95, 221108 -221110 (2009).
  20. L. OFaolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovi, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, T. F. Krauss, "Loss engineered slow light waveguides," Opt. Exp. 18, 27627-27638 (2010).
  21. N. C. Panoiu, J. F. McMillan, C. W. Wong, "Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides," IEEE J. Sel. Top. Quantum Electron. 16, 257266 (2010).
  22. S. Roy, M.Santagiustina, P. Colman, S. Combri, A. De Ross, "Modeling the dispersion of the nonlinearity in slow mode photonic crystal waveguide," IEEE Photon. J. 4, 224-233 (2012).
  23. A. Melloni, F. Morichetti, M. Martinelli, "Four-wave mixing and wavelength conversion in coupled-resonator optical waveguide," J. Opt. Soc. Am. B 25, C87-C97 (2008).
  24. I. H. Rey, Y. Lefevre, S. A. Schulz, N. Vermeulen, T. F. Krauss, "Scaling of Raman amplification in realistic slowlight photonic crystal waveguides," Phys. Rev. B, Condens. Matter Mater. Phys. 84, 035306 (2011).
  25. S. Roy, A. Willinger, S. Combrié, A. D. Rossi, G. Eisenstein, M. Santagiustina, "Narrowband optical parametric gain in slow mode engineered GaInP photonic crystal waveguides," Opt. Lett. 37, 29192921 (2012).
  26. A. Willinger, S. Roy, M. Santagiustina, S. Combrié, A. D. Rossi, I. Cestier, G. Eisenstein, "Parametric gain in dispersion engineered photonic crystal waveguides," Opt. Exp. 21, 49955004 (2013).
  27. A. Willinger, S. Roy, M. Santagiustina, S. Combrié, A. D. Rossi, I. Cestier, G. Eisenstein, "Dual-pump parametric amplification in dispersion engineered photonic crystal waveguides," Opt. Exp. 21, 10440-10453 (2013).
  28. M. Karlsson, "Modulational instability in lossy optical fibers ," J. Opt. Soc. Amer. B 12, 2071-2077 (1995).
  29. M. E. Marhic, Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge Univ. Press, 2008).
  30. T. Tanemura, Y. Ozeki, K. Kikuchi, "Modulational instability and parametric amplification induced by loss dispersion in optical fibers," Phys. Rev. Lett. 93, 163902-163905 (2004).
  31. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  32. J. D. Joannopoulos (2008, Feb.). [Online]. Available: http://ab-initio.mit.edu/photons/.

2013

A. Willinger, S. Roy, M. Santagiustina, S. Combrié, A. D. Rossi, I. Cestier, G. Eisenstein, "Parametric gain in dispersion engineered photonic crystal waveguides," Opt. Exp. 21, 49955004 (2013).

A. Willinger, S. Roy, M. Santagiustina, S. Combrié, A. D. Rossi, I. Cestier, G. Eisenstein, "Dual-pump parametric amplification in dispersion engineered photonic crystal waveguides," Opt. Exp. 21, 10440-10453 (2013).

2012

S. Roy, A. Willinger, S. Combrié, A. D. Rossi, G. Eisenstein, M. Santagiustina, "Narrowband optical parametric gain in slow mode engineered GaInP photonic crystal waveguides," Opt. Lett. 37, 29192921 (2012).

S. Roy, M.Santagiustina, P. Colman, S. Combri, A. De Ross, "Modeling the dispersion of the nonlinearity in slow mode photonic crystal waveguide," IEEE Photon. J. 4, 224-233 (2012).

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

I. Cestier, S. Combrié, S. Xavier, G. Lehoucq, A. D. Rossi, G. Eisenstein, "Chip-scale parametric amplifier with 11-db gain at 1550 nm based on a slow-light GaInP photonic crystal waveguide," Opt. Lett. 37, 3996-3998 (2012).

2011

J. Li, L. O’Faolain, I. H. Rey, T. F. Krauss, "Four-wave mixing in photonic crystal waveguides: Slow light enhancement and limitations," Opt. Exp. 19, 4458-4463 (2011).

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O'Faolain, T. F. Krauss, B. J. Eggleton, "Ultracompact 160 Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide," Opt. Lett. 36, 1728-1730 (2011).

P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, A. De Rossi, "Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides," Opt. Lett. 36, 2629-2631 (2011).

I. H. Rey, Y. Lefevre, S. A. Schulz, N. Vermeulen, T. F. Krauss, "Scaling of Raman amplification in realistic slowlight photonic crystal waveguides," Phys. Rev. B, Condens. Matter Mater. Phys. 84, 035306 (2011).

2010

L. OFaolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovi, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, T. F. Krauss, "Loss engineered slow light waveguides," Opt. Exp. 18, 27627-27638 (2010).

N. C. Panoiu, J. F. McMillan, C. W. Wong, "Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides," IEEE J. Sel. Top. Quantum Electron. 16, 257266 (2010).

V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. G. Someda, G. Vadalà, "Highly efficient four wave mixing in GaInP photonic crystal waveguides," Opt. Lett. 35, 1440-1442 (2010).

M. Santagiustina, C. G. Someda, G. Vadalá, S. Combrié, A. De Rossi, "Theory of slow light enhanced four-wave mixing in photonic crystal waveguides ," Opt. Exp. 18, 21024-21029 (2010).

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. J. Eggleton, T. P. White, L. O’Faolain, J. Li, T. F. Krauss, "Four-wave mixing in slow light engineered silicon photonic crystal waveguides," Opt. Exp. 18 , 22915-22927 (2010).

C. Monat, M. de Sterke, B. J. Eggleton, "Slow light enhanced nonlinear optics in periodic structures," J. Opt. 12, 288, 104003 (2010).

B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, B. J. Eggleton, D. J. Moss, "Optical signal processing on a silicon chip at 640Gb/s using slow-light," Opt. Exp. 18, 7770-7781 (2010).

2009

M. Ebnali-Heidari, C. Monat, C. Grillet, M. K. Moravvej-Farshi, "A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration," Opt. Exp. 17, 18340-18353 (2009).

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, T. F. Krauss, "Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law ," Phys. Rev. B 80, 195305-195310 (2009).

S. Combrié, Q. Vy Tran, C. Husko, P. Colman, A. De Rossi, "High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption," Appl. Phys. Lett. 95, 221108 -221110 (2009).

2008

T. Baba, "Slow light in photonic crystals," Nature Photon. 2, 465-473 (2008).

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Expr. 16, 6227-6232 (2008).

A. Melloni, F. Morichetti, M. Martinelli, "Four-wave mixing and wavelength conversion in coupled-resonator optical waveguide," J. Opt. Soc. Am. B 25, C87-C97 (2008).

2007

T. F. Krauss, "Slow light in photonic crystal waveguides ," J. Phys. D, Appl. Phys. 40, 2666-2670 (2007).

2006

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, P. I. Borel, "Photonic crystal waveguides with semi-slow light and tailored dispersion properties," Opt. Exp. 14 , 9444-9450 (2006).

2004

T. Tanemura, Y. Ozeki, K. Kikuchi, "Modulational instability and parametric amplification induced by loss dispersion in optical fibers," Phys. Rev. Lett. 93, 163902-163905 (2004).

1995

M. Karlsson, "Modulational instability in lossy optical fibers ," J. Opt. Soc. Amer. B 12, 2071-2077 (1995).

Opt. Exp.

L. OFaolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovi, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, T. F. Krauss, "Loss engineered slow light waveguides," Opt. Exp. 18, 27627-27638 (2010).

Phys. Rev. Lett.

T. Tanemura, Y. Ozeki, K. Kikuchi, "Modulational instability and parametric amplification induced by loss dispersion in optical fibers," Phys. Rev. Lett. 93, 163902-163905 (2004).

Appl. Phys. Lett.

S. Combrié, Q. Vy Tran, C. Husko, P. Colman, A. De Rossi, "High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption," Appl. Phys. Lett. 95, 221108 -221110 (2009).

IEEE J. Sel. Top. Quantum Electron.

N. C. Panoiu, J. F. McMillan, C. W. Wong, "Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides," IEEE J. Sel. Top. Quantum Electron. 16, 257266 (2010).

IEEE Photon. J.

S. Roy, M.Santagiustina, P. Colman, S. Combri, A. De Ross, "Modeling the dispersion of the nonlinearity in slow mode photonic crystal waveguide," IEEE Photon. J. 4, 224-233 (2012).

J. Opt.

C. Monat, M. de Sterke, B. J. Eggleton, "Slow light enhanced nonlinear optics in periodic structures," J. Opt. 12, 288, 104003 (2010).

J. Opt. Soc. Am. B

A. Melloni, F. Morichetti, M. Martinelli, "Four-wave mixing and wavelength conversion in coupled-resonator optical waveguide," J. Opt. Soc. Am. B 25, C87-C97 (2008).

J. Opt. Soc. Amer. B

M. Karlsson, "Modulational instability in lossy optical fibers ," J. Opt. Soc. Amer. B 12, 2071-2077 (1995).

J. Phys. D, Appl. Phys.

T. F. Krauss, "Slow light in photonic crystal waveguides ," J. Phys. D, Appl. Phys. 40, 2666-2670 (2007).

Nature Photon.

T. Baba, "Slow light in photonic crystals," Nature Photon. 2, 465-473 (2008).

Opt. Lett.

S. Roy, A. Willinger, S. Combrié, A. D. Rossi, G. Eisenstein, M. Santagiustina, "Narrowband optical parametric gain in slow mode engineered GaInP photonic crystal waveguides," Opt. Lett. 37, 29192921 (2012).

Opt. Exp.

A. Willinger, S. Roy, M. Santagiustina, S. Combrié, A. D. Rossi, I. Cestier, G. Eisenstein, "Parametric gain in dispersion engineered photonic crystal waveguides," Opt. Exp. 21, 49955004 (2013).

A. Willinger, S. Roy, M. Santagiustina, S. Combrié, A. D. Rossi, I. Cestier, G. Eisenstein, "Dual-pump parametric amplification in dispersion engineered photonic crystal waveguides," Opt. Exp. 21, 10440-10453 (2013).

P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, "Control of dispersion in photonic crystal waveguides using group symmetry theory," Opt. Exp. 20, 13108-13114 (2012 ).

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, P. I. Borel, "Photonic crystal waveguides with semi-slow light and tailored dispersion properties," Opt. Exp. 14 , 9444-9450 (2006).

B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O'Faolain, T. F. Krauss, B. J. Eggleton, D. J. Moss, "Optical signal processing on a silicon chip at 640Gb/s using slow-light," Opt. Exp. 18, 7770-7781 (2010).

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

M. Ebnali-Heidari, C. Monat, C. Grillet, M. K. Moravvej-Farshi, "A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration," Opt. Exp. 17, 18340-18353 (2009).

M. Santagiustina, C. G. Someda, G. Vadalá, S. Combrié, A. De Rossi, "Theory of slow light enhanced four-wave mixing in photonic crystal waveguides ," Opt. Exp. 18, 21024-21029 (2010).

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. J. Eggleton, T. P. White, L. O’Faolain, J. Li, T. F. Krauss, "Four-wave mixing in slow light engineered silicon photonic crystal waveguides," Opt. Exp. 18 , 22915-22927 (2010).

J. Li, L. O’Faolain, I. H. Rey, T. F. Krauss, "Four-wave mixing in photonic crystal waveguides: Slow light enhancement and limitations," Opt. Exp. 19, 4458-4463 (2011).

Opt. Expr.

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Expr. 16, 6227-6232 (2008).

Opt. Lett.

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O'Faolain, T. F. Krauss, B. J. Eggleton, "Ultracompact 160 Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide," Opt. Lett. 36, 1728-1730 (2011).

Opt. Lett.

Phys. Rev. B

M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, T. F. Krauss, "Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law ," Phys. Rev. B 80, 195305-195310 (2009).

Phys. Rev. B, Condens. Matter Mater. Phys.

I. H. Rey, Y. Lefevre, S. A. Schulz, N. Vermeulen, T. F. Krauss, "Scaling of Raman amplification in realistic slowlight photonic crystal waveguides," Phys. Rev. B, Condens. Matter Mater. Phys. 84, 035306 (2011).

Other

J. B. Khurgin, R. S. Tucke, Slow Light: Science and Applications (CRC Press, 2009).

M. E. Marhic, Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge Univ. Press, 2008).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

J. D. Joannopoulos (2008, Feb.). [Online]. Available: http://ab-initio.mit.edu/photons/.

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