Abstract

We characterize the nonlinear optical response of low loss Si0.6Ge0.4 / Si waveguides in the mid-infrared between 3.3 μm and 4 μm using femtosecond optical pulses. We estimate the three and four-photon absorption coefficients as well as the Kerr nonlinear refractive index from the experimental measurements. The effect of multiphoton absorption on the optical nonlinear Kerr response is evaluated and the nonlinear figure of merit estimated providing some guidelines for designing nonlinear optical devices in the mid-IR. Finally, we compare the impact of free-carrier absorption at mid-infrared wavelengths versus near-infrared wavelengths for these ultra-short pulses.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Nonlinear optical response of low loss silicon germanium waveguides in the mid-infrared

L. Carletti, P. Ma, Y. Yu, B. Luther-Davies, D. Hudson, C. Monat, R. Orobtchouk, S. Madden, D. J. Moss, M. Brun, S. Ortiz, P. Labeye, S. Nicoletti, and C. Grillet
Opt. Express 23(7) 8261-8271 (2015)

Self-phase-modulation in submicron silicon-on-insulator photonic wires

Eric Dulkeith, Yurii A. Vlasov, Xiaogang Chen, Nicolae C. Panoiu, and Richard M. Osgood
Opt. Express 14(12) 5524-5534 (2006)

Broadband mid-infrared supercontinuum generation in dispersion-engineered silicon-on-insulator waveguide

Hamed Saghaei and Vien Van
J. Opt. Soc. Am. B 36(2) A193-A202 (2019)

References

  • View by:
  • |
  • |
  • |

  1. R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
    [Crossref]
  2. B. Jalali, “Nonlinear optics in the mid-infrared,” Nat. Photonics 4(8), 506–508 (2010).
    [Crossref]
  3. F. K. Tittel, D. Richter, and A. Fried, “Mid-Infrared Laser Applications in Spectroscopy,” in Solid-State Mid-Infrared Laser Sources, (Springer Berlin Heidelberg, 2003).
  4. http://www.mirifisens-project.eu/ , “Mid infrared innovative lasers for improved sensor of hazardous substances
  5. http://optics.org/news/6/2/23 , “mirSense integrated QCLs target gas-sensing and biomed roles.”.
  6. G. C. M.Carras, G. Maisons, B. Simozrag, V. Trinite, M. Brun, S. Nicoletti, L.Orbe, “Advances toward monolithic broadly-tunable QCL sources,” in Proc. SPIE 8993 (INVITED) Quantum Sensing and Nanophotonic Devices XI, 2014.
  7. M. C. L. J. Orbe, G . Carpintero, G. Maisons, C. Gilles, F. Boulila, “MIR Photonic Integrated Circuits for Laser Spectroscopy,” in MIOMD - Mid-IR Optoelectronics : Materials and Devices (2014).
  8. B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
    [Crossref] [PubMed]
  9. N. Singh, D. D. Hudson, Y. Yu, C. Grillet, S. D. Jackson, A. Casas-Bedoya, A. Read, P. Atanackovic, S. G. Duvall, S. Palombo, D. J. Moss, B. Luther-Davies, and B. J. Eggleton, “Mid-IR supercontinuum generation from 2-6 um in a silicon nanowire,” Optica 2(9), 797–802 (2015).
    [Crossref]
  10. M. A. Ettabib, L. Xu, A. Bogris, A. Kapsalis, M. Belal, E. Lorent, P. Labeye, S. Nicoletti, K. Hammani, D. Syvridis, D. P. Shepherd, J. H. V. Price, D. J. Richardson, and P. Petropoulos, “Broadband telecom to mid-infrared supercontinuum generation in a dispersion-engineered silicon germanium waveguide,” Opt. Lett. 40(17), 4118–4121 (2015).
    [Crossref] [PubMed]
  11. B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
    [Crossref] [PubMed]
  12. A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
    [Crossref] [PubMed]
  13. S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
    [Crossref]
  14. X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
    [Crossref]
  15. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
    [Crossref]
  16. C. Grillet, L. Carletti, C. Monat, P. Grosse, B. Ben Bakir, S. Menezo, J. M. Fedeli, and D. J. Moss, “Amorphous silicon nanowires combining high nonlinearity, FOM and optical stability,” Opt. Express 20(20), 22609–22615 (2012).
    [Crossref] [PubMed]
  17. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
    [Crossref]
  18. X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
    [Crossref]
  19. T. Wang, N. Venkatram, J. Gosciniak, Y. Cui, G. Qian, W. Ji, and D. T. H. Tan, “Multi-photon absorption and third-order nonlinearity in silicon at mid-infrared wavelengths,” Opt. Express 21(26), 32192–32198 (2013).
    [Crossref] [PubMed]
  20. N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
    [Crossref]
  21. M. Brun, P. Labeye, G. Grand, J. M. Hartmann, F. Boulila, M. Carras, and S. Nicoletti, “Low loss SiGe graded index waveguides for mid-IR applications,” Opt. Express 22(1), 508–518 (2014).
    [Crossref] [PubMed]
  22. L. Carletti, P. Ma, Y. Yu, B. Luther-Davies, D. Hudson, C. Monat, R. Orobtchouk, S. Madden, D. J. Moss, M. Brun, S. Ortiz, P. Labeye, S. Nicoletti, and C. Grillet, “Nonlinear optical response of low loss silicon germanium waveguides in the mid-infrared,” Opt. Express 23(7), 8261–8271 (2015).
    [Crossref] [PubMed]
  23. K. Hammani, M. A. Ettabib, A. Bogris, A. Kapsalis, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, D. J. Richardson, and P. Petropoulos, “Optical properties of silicon germanium waveguides at telecommunication wavelengths,” Opt. Express 21(14), 16690–16701 (2013).
    [Crossref] [PubMed]
  24. K. Hammani, M. A. Ettabib, A. Bogris, A. Kapsalis, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “Towards nonlinear conversion from mid- to near-infrared wavelengths using Silicon Germanium waveguides,” Opt. Express 22(8), 9667–9674 (2014).
    [Crossref] [PubMed]
  25. G. P. Agrawal, Nonlinear Fiber Optics. (Academic Press, 2001).
  26. P. Ma, D. Y. Choi, Y. Yu, X. Gai, Z. Yang, S. Debbarma, S. Madden, and B. Luther-Davies, “Low-loss chalcogenide waveguides for chemical sensing in the mid-infrared,” Opt. Express 21(24), 29927–29937 (2013).
    [Crossref] [PubMed]
  27. I. W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14(25), 12380–12387 (2006).
    [Crossref] [PubMed]
  28. S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
    [Crossref]
  29. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15(25), 16604–16644 (2007).
    [Crossref] [PubMed]

2015 (5)

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

L. Carletti, P. Ma, Y. Yu, B. Luther-Davies, D. Hudson, C. Monat, R. Orobtchouk, S. Madden, D. J. Moss, M. Brun, S. Ortiz, P. Labeye, S. Nicoletti, and C. Grillet, “Nonlinear optical response of low loss silicon germanium waveguides in the mid-infrared,” Opt. Express 23(7), 8261–8271 (2015).
[Crossref] [PubMed]

M. A. Ettabib, L. Xu, A. Bogris, A. Kapsalis, M. Belal, E. Lorent, P. Labeye, S. Nicoletti, K. Hammani, D. Syvridis, D. P. Shepherd, J. H. V. Price, D. J. Richardson, and P. Petropoulos, “Broadband telecom to mid-infrared supercontinuum generation in a dispersion-engineered silicon germanium waveguide,” Opt. Lett. 40(17), 4118–4121 (2015).
[Crossref] [PubMed]

N. Singh, D. D. Hudson, Y. Yu, C. Grillet, S. D. Jackson, A. Casas-Bedoya, A. Read, P. Atanackovic, S. G. Duvall, S. Palombo, D. J. Moss, B. Luther-Davies, and B. J. Eggleton, “Mid-IR supercontinuum generation from 2-6 um in a silicon nanowire,” Optica 2(9), 797–802 (2015).
[Crossref]

2014 (2)

2013 (4)

2012 (1)

2011 (2)

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

2010 (5)

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
[Crossref]

B. Jalali, “Nonlinear optics in the mid-infrared,” Nat. Photonics 4(8), 506–508 (2010).
[Crossref]

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[Crossref]

2009 (2)

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

2007 (1)

2006 (1)

Agrawal, G. P.

Alic, N.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Atanackovic, P.

Baets, R.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

Belal, M.

Ben Bakir, B.

Boggio, J. M. C.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Bogris, A.

Boulila, F.

Brun, M.

Cardenas, J.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Carletti, L.

Carras, M.

Casas-Bedoya, A.

Chen, X.

Choi, D. Y.

Coen, S.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Colman, P.

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

Combrie, S.

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

Corcoran, B.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

Cui, Y.

Dadap, J. I.

De Rossi, A.

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

Debbarma, S.

Divliansky, I. B.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Duvall, S. G.

Eggleton, B. J.

N. Singh, D. D. Hudson, Y. Yu, C. Grillet, S. D. Jackson, A. Casas-Bedoya, A. Read, P. Atanackovic, S. G. Duvall, S. Palombo, D. J. Moss, B. Luther-Davies, and B. J. Eggleton, “Mid-IR supercontinuum generation from 2-6 um in a silicon nanowire,” Optica 2(9), 797–802 (2015).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

Ettabib, M. A.

Fain, R.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Faolain, L. O.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

Fedeli, J. M.

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[Crossref]

Gaeta, A. L.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Gai, X.

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

P. Ma, D. Y. Choi, Y. Yu, X. Gai, Z. Yang, S. Debbarma, S. Madden, and B. Luther-Davies, “Low-loss chalcogenide waveguides for chemical sensing in the mid-infrared,” Opt. Express 21(24), 29927–29937 (2013).
[Crossref] [PubMed]

Gosciniak, J.

Grand, G.

Green, W. M. J.

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

Griffith, A. G.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Grillet, C.

Grosse, P.

Hammani, K.

Hänsch, T. W.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Hartmann, J. M.

Holzner, S.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Hon, N. K.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

Hsieh, I. W.

Hudson, D.

Hudson, D. D.

Husko, C.

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

Ideguchi, T.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Jackson, S. D.

Jalali, B.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

B. Jalali, “Nonlinear optics in the mid-infrared,” Nat. Photonics 4(8), 506–508 (2010).
[Crossref]

Ji, W.

Kapsalis, A.

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[Crossref]

Krauss, T. F.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

Kuyken, B.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

Labeye, P.

Lau, R. K. W.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Lee, Y. H. D.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Leo, F.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[Crossref]

Lin, Q.

Lipson, M.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Liu, X.

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

Lorent, E.

Luther-Davies, B.

Ma, P.

Madden, S.

Madden, S. J.

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

McNab, S. J.

Menezo, S.

Mohanty, A.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Monat, C.

Mookherjea, S.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Moro, S.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Moss, D. J.

Nicoletti, S.

Okawachi, Y.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Orobtchouk, R.

Ortiz, S.

Osgood, R. M.

Painter, O. J.

Palombo, S.

Panoiu, N. C.

Park, J. S.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Petropoulos, P.

Phare, C. T.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Picqué, N.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Poitras, C. B.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Price, J. H. V.

Qian, G.

Radic, S.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Read, A.

Richardson, D. J.

Roelkens, G.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

Shepherd, D. P.

Singh, N.

Soref, R.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
[Crossref]

Syvridis, D.

Tan, D. T. H.

Tran, Q. V.

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

Van Campenhout, J.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

Venkatram, N.

Verheyen, P.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

Vlasov, Y. A.

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

I. W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14(25), 12380–12387 (2006).
[Crossref] [PubMed]

Wang, T.

White, T. P.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

Xu, L.

Yan, M.

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

Yang, Z.

Yu, M.

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Yu, Y.

Zlatanovic, S.

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

Appl. Phys. Lett. (1)

S. Combrie, Q. V. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95(22), 221108 (2009).
[Crossref]

J. Appl. Phys. (1)

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011).
[Crossref]

Laser Photonics Rev. (1)

X. Gai, Y. Yu, B. Kuyken, P. Ma, S. J. Madden, J. Van Campenhout, P. Verheyen, and R. Baets, “Nonlinear absorption and refraction in crystalline silicon in the mid-infrared,” Laser Photonics Rev. 7(6), 1–11 (2013).
[Crossref]

Nat. Commun. (2)

B. Kuyken, T. Ideguchi, S. Holzner, M. Yan, T. W. Hänsch, J. Van Campenhout, P. Verheyen, S. Coen, F. Leo, R. Baets, G. Roelkens, and N. Picqué, “An octave-spanning mid-infrared frequency comb generated in a silicon nanophotonic wire waveguide,” Nat. Commun. 6, 6310 (2015).
[Crossref] [PubMed]

A. G. Griffith, R. K. W. Lau, J. Cardenas, Y. Okawachi, A. Mohanty, R. Fain, Y. H. D. Lee, M. Yu, C. T. Phare, C. B. Poitras, A. L. Gaeta, and M. Lipson, “Silicon-chip mid-infrared frequency comb generation,” Nat. Commun. 6, 6299 (2015).
[Crossref] [PubMed]

Nat. Photonics (6)

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010).
[Crossref]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O. Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic- crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
[Crossref]

B. Jalali, “Nonlinear optics in the mid-infrared,” Nat. Photonics 4(8), 506–508 (2010).
[Crossref]

Opt. Express (10)

I. W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14(25), 12380–12387 (2006).
[Crossref] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15(25), 16604–16644 (2007).
[Crossref] [PubMed]

B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. J. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011).
[Crossref] [PubMed]

C. Grillet, L. Carletti, C. Monat, P. Grosse, B. Ben Bakir, S. Menezo, J. M. Fedeli, and D. J. Moss, “Amorphous silicon nanowires combining high nonlinearity, FOM and optical stability,” Opt. Express 20(20), 22609–22615 (2012).
[Crossref] [PubMed]

K. Hammani, M. A. Ettabib, A. Bogris, A. Kapsalis, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, D. J. Richardson, and P. Petropoulos, “Optical properties of silicon germanium waveguides at telecommunication wavelengths,” Opt. Express 21(14), 16690–16701 (2013).
[Crossref] [PubMed]

P. Ma, D. Y. Choi, Y. Yu, X. Gai, Z. Yang, S. Debbarma, S. Madden, and B. Luther-Davies, “Low-loss chalcogenide waveguides for chemical sensing in the mid-infrared,” Opt. Express 21(24), 29927–29937 (2013).
[Crossref] [PubMed]

T. Wang, N. Venkatram, J. Gosciniak, Y. Cui, G. Qian, W. Ji, and D. T. H. Tan, “Multi-photon absorption and third-order nonlinearity in silicon at mid-infrared wavelengths,” Opt. Express 21(26), 32192–32198 (2013).
[Crossref] [PubMed]

M. Brun, P. Labeye, G. Grand, J. M. Hartmann, F. Boulila, M. Carras, and S. Nicoletti, “Low loss SiGe graded index waveguides for mid-IR applications,” Opt. Express 22(1), 508–518 (2014).
[Crossref] [PubMed]

K. Hammani, M. A. Ettabib, A. Bogris, A. Kapsalis, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “Towards nonlinear conversion from mid- to near-infrared wavelengths using Silicon Germanium waveguides,” Opt. Express 22(8), 9667–9674 (2014).
[Crossref] [PubMed]

L. Carletti, P. Ma, Y. Yu, B. Luther-Davies, D. Hudson, C. Monat, R. Orobtchouk, S. Madden, D. J. Moss, M. Brun, S. Ortiz, P. Labeye, S. Nicoletti, and C. Grillet, “Nonlinear optical response of low loss silicon germanium waveguides in the mid-infrared,” Opt. Express 23(7), 8261–8271 (2015).
[Crossref] [PubMed]

Opt. Lett. (1)

Optica (1)

Other (6)

F. K. Tittel, D. Richter, and A. Fried, “Mid-Infrared Laser Applications in Spectroscopy,” in Solid-State Mid-Infrared Laser Sources, (Springer Berlin Heidelberg, 2003).

http://www.mirifisens-project.eu/ , “Mid infrared innovative lasers for improved sensor of hazardous substances

http://optics.org/news/6/2/23 , “mirSense integrated QCLs target gas-sensing and biomed roles.”.

G. C. M.Carras, G. Maisons, B. Simozrag, V. Trinite, M. Brun, S. Nicoletti, L.Orbe, “Advances toward monolithic broadly-tunable QCL sources,” in Proc. SPIE 8993 (INVITED) Quantum Sensing and Nanophotonic Devices XI, 2014.

M. C. L. J. Orbe, G . Carpintero, G. Maisons, C. Gilles, F. Boulila, “MIR Photonic Integrated Circuits for Laser Spectroscopy,” in MIOMD - Mid-IR Optoelectronics : Materials and Devices (2014).

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Schematic representation of one of the ridge waveguides. The color orange represent the Si0.6Ge0.4 core that is surrounded by crystalline Si (in yellow).
Fig. 2
Fig. 2 Normalized transmitted spectrum as a function of the coupled peak intensity for femtosecond pulses centered at wavelengths of 3.3 μm (a), (b) 3.75 μm (c), (d) and 4 μm (e), (f). Spectra (a), (b), (c) and (d) are measured using waveguide 1 whereas spectrum (e) and (f) is measured using waveguide 2.
Fig. 3
Fig. 3 (a) Spectral broadening as a function the coupled peak intensity. (b) Normalized transmitted power as a function of the coupled peak intensity measured using pulses centered at 3300 nm (downwards green triangles) and 3750 nm (upwards red triangles) from waveguide 1 and at 4000 nm from waveguide 2 (black dots). The two curves at 3750nm and 4000nm have been shifted vertically for clarity.
Fig. 4
Fig. 4 (a) Numerical calculation (Calc) of the transmission as a function of wavelength compared to experiment (Exp). (b) and (c) Numerical calculation of the transmitted spectra at increasing coupled peak intensity for a pulse centered at 4 µm from waveguide 2. In the calculations n2 = 5.25 × 10−18 m2/W, α3PA = 8.8 × 10−28 m3/W2, α4PA = 4.76 × 10−41 m5/W3 (d) The spectral broadening derived from the simulation spectra and from the measurements (Fig. 3(a)) is compared.
Fig. 5
Fig. 5 (a) Three- and (b) four-photon absorption coefficients as a function of wavelength estimated from the transmission measurements using optical pulses with a FWHM duration of 320 fs (this work) and of 7.5 ps [22].
Fig. 6
Fig. 6 Nonlinear refractive index as a function of wavelength estimated from the SPM measurements using optical pulses with a FWHM duration of 320 fs (this work) and of 7.5 ps [22].
Fig. 7
Fig. 7 (a) Nonlinear FOM as a function of wavelength calculated for a coupled peak intensity of 10 GW/cm2 using the results in Figs. 5 and 6. (b) Nonlinear FOM as a function of nonlinear phase shift ΔΦ = γP0Leff derived for λ = 3.75µm (circles) and λ = 4µm (triangles) using the results in Figs. 5 and 6. For ΔΦ = π the coupled peak intensity is 10.7GW/cm2 at λ = 3.75µm and 3.8GW/cm2 at λ = 4µm. The wavelength for which the maximum FOM is obtained is 4µm for ΔΦ<5π and 3.75µm for ΔΦ>5π. (c) Nonlinear losses due to multiphoton absorption as a function of ΔΦ derived for λ = 3.75µm (circles) and λ = 4µm (triangles) using the results in Figs. 5 and 6.
Fig. 8
Fig. 8 (a) Ratio r = FCA/3PA as a function of the pulse fluence at a wavelength of 3300 nm and 3750 nm. The dashed green line shows the ordinate r = 1. On the top of the plot the corresponding peak intensity is shown. (b) Normalized transmission as a function of the coupled peak intensity for a 320 fs pulse centered at 3300 nm from waveguide 1. The red continuous line is the results when considering only three-photon absorption and neglecting free-carriers effects, for the dotted black line three-photon absorption and FCA are considered and for the continuous green line also free-carrier dispersion (FCD) is considered. On the top of the plot the pulse fluence is shown.
Fig. 9
Fig. 9 (a) GVD of the dispersion engineered SiGe waveguide under consideration (in inset) (b) The evolution of the spectrum as a function of the coupled peak intensity into the waveguide as obtained from numerical modelling.

Tables (1)

Tables Icon

Table 1 Geometrical properties of the waveguides and mode properties extracted from simulations at the wavelengths used in the experiment.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

A z + i 2 β 2 2 A t 2 i 6 β 3 3 A t 3 + i 24 β 4 4 A t 4 + α 2 A= i k 0 n 2 A 3eff | A | 2 A α 3PA 2 A 5eff 2 | A | 4 A α 4PA 2 A 7eff 3 | A | 6 A σ 2 ( 1iμ ) N c A
N c t = α 3PA 3ω ( | A | 2 A 5eff ) 3 + α 4PA 4ω ( | A | 2 A 7eff ) 4 N c τ c
r= FCA 3PA = nσ E p 3 3 n 0 ω 0 A 5eff

Metrics