Abstract

We have investigated the nonlinear optical response of low loss Si0.6Ge0.4 / Si waveguides in the mid-infrared wavelength range from 3.25- 4.75μm using picosecond optical pulses. We observed and measured the three and four-photon absorption coefficients as well as the Kerr nonlinear refractive index. The dynamics of the spectral broadening suggests that, in addition to multiphoton absorption, the corresponding higher order nonlinear refractive phenomena also needs to be included when high optical pulse intensities are used at mid-infrared wavelengths in this material.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Mid-infrared nonlinear optical response of Si-Ge waveguides with ultra-short optical pulses

L. Carletti, M. Sinobad, P. Ma, Y. Yu, D. Allioux, R. Orobtchouk, M. Brun, S. Ortiz, P. Labeye, J. M. Hartmann, S. Nicoletti, S. Madden, B. Luther-Davies, D. J. Moss, C. Monat, and C. Grillet
Opt. Express 23(25) 32202-32214 (2015)

Mid-infrared octave spanning supercontinuum generation to 8.5  μm in silicon-germanium waveguides

Milan Sinobad, Christelle Monat, Barry Luther-davies, Pan Ma, Stephen Madden, David J. Moss, Arnan Mitchell, David Allioux, Regis Orobtchouk, Salim Boutami, Jean-Michel Hartmann, Jean-Marc Fedeli, and Christian Grillet
Optica 5(4) 360-366 (2018)

Graded SiGe waveguides with broadband low-loss propagation in the mid infrared

J. M. Ramirez, Q. Liu, V. Vakarin, J. Frigerio, A. Ballabio, X. Le Roux, D. Bouville, L. Vivien, G. Isella, and D. Marris-Morini
Opt. Express 26(2) 870-877 (2018)

References

  • View by:
  • |
  • |
  • |

  1. B. Jalali, “Nonlinear optics in the mid-infrared,” Nat. Photonics 4(8), 506–508 (2010).
    [Crossref]
  2. R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
    [Crossref]
  3. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
    [Crossref]
  4. C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
    [Crossref] [PubMed]
  5. 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]
  6. 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]
  7. 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]
  8. G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express 19(8), 7112–7119 (2011).
    [Crossref] [PubMed]
  9. P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
    [Crossref]
  10. 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]
  11. R. Shankar, I. Bulu, and M. Lončar, “Integrated high-quality factor silicon-on-sapphire ring resonators for the mid-infrared,” Appl. Phys. Lett. 102(5), 051108 (2013).
    [Crossref]
  12. 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, 1–11 (2013).
  13. M. Muneeb, X. Chen, P. Verheyen, G. Lepage, S. Pathak, E. Ryckeboer, A. Malik, B. Kuyken, M. Nedeljkovic, J. Van Campenhout, G. Z. Mashanovich, and G. Roelkens, “Demonstration of Silicon-on-insulator mid-infrared spectrometers operating at 3.8 μm,” Opt. Express 21(10), 11659–11669 (2013).
    [Crossref] [PubMed]
  14. C. Grillet, P. Ma, B. Luther-Davies, D. Hudson, C. Monat, S. Madden, D. J. Moss, M. Brun, P. Labeye, S. Ortiz, and S. Nicoletti, “Low loss SiGe waveguides in the MID-IR,” in 2013 Conference on Lasers and Electro-Optics - International Quantum Electronics Conference (Optical Society of America, 2013), paper CK_2_6.
    [Crossref]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007).
    [Crossref] [PubMed]
  22. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
    [Crossref] [PubMed]
  23. B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Lett. 35(7), 1073–1075 (2010).
    [Crossref] [PubMed]
  24. C. Husko, S. Combrié, Q. V. Tran, F. Raineri, C. W. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorption in III-V photonic crystal waveguides,” Opt. Express 17(25), 22442–22451 (2009).
    [Crossref] [PubMed]
  25. B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1(1), 67–72 (1984).
    [Crossref]
  26. 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]
  27. D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
    [Crossref]

2014 (3)

2013 (6)

2012 (1)

2011 (4)

G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express 19(8), 7112–7119 (2011).
[Crossref] [PubMed]

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]

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

2010 (5)

B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Lett. 35(7), 1073–1075 (2010).
[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]

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

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

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

2009 (4)

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]

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

C. Husko, S. Combrié, Q. V. Tran, F. Raineri, C. W. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorption in III-V photonic crystal waveguides,” Opt. Express 17(25), 22442–22451 (2009).
[Crossref] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

2007 (1)

1984 (1)

Agrawal, G. P.

Baets, R.

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, 1–11 (2013).

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]

Ben Bakir, B.

Bettiol, A.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Bogris, A.

Boulila, F.

Breese, M. B. H.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Brun, M.

Bulu, I.

R. Shankar, I. Bulu, and M. Lončar, “Integrated high-quality factor silicon-on-sapphire ring resonators for the mid-infrared,” Appl. Phys. Lett. 102(5), 051108 (2013).
[Crossref]

Carletti, L.

Carras, M.

Chen, X.

Choi, D. Y.

Clark, A.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

Cohen, P. I.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

Collins, M.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

Combrié, S.

Corcoran, B.

Crnjanski, J.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Cui, Y.

De Rossi, A.

Debbarma, S.

Ebnali-Heidari, M.

Eggleton, B. J.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Lett. 35(7), 1073–1075 (2010).
[Crossref] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

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.

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.

Feldman, L. C.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

Freude, W.

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

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, 1–11 (2013).

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]

Giusca, C.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

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]

Gregory, J. M.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

Grillet, C.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[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]

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]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

Grosse, P.

Hammani, K.

Hartmann, J. M.

Headley, W.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

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]

Hu, Y.

Husko, C.

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.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Lett. 35(7), 1073–1075 (2010).
[Crossref] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

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.

Labeye, P.

Lepage, G.

Leuthold, J.

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

Li, J.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[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]

Loncar, M.

R. Shankar, I. Bulu, and M. Lončar, “Integrated high-quality factor silicon-on-sapphire ring resonators for the mid-infrared,” Appl. Phys. Lett. 102(5), 051108 (2013).
[Crossref]

Luther-Davies, B.

Ma, P.

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]

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, 1–11 (2013).

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, 1–11 (2013).

Malik, A.

Mashanovich, G. Z.

Menezo, S.

Miloševic, M. M.

Monat, C.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[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]

B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Lett. 35(7), 1073–1075 (2010).
[Crossref] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

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]

Moss, D. J.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[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]

B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Lett. 35(7), 1073–1075 (2010).
[Crossref] [PubMed]

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]

Muneeb, M.

Nedeljkovic, M.

Nicoletti, S.

O’Faolain, L.

Osgood, R. M.

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]

Owens, N.

Pathak, S.

Pelusi, M.

Petropoulos, P.

Pudo, D.

Qian, G.

Raineri, F.

Reed, G. T.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Richardson, D. J.

Roelkens, G.

Ryckeboer, E.

Schroeder, J.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

Seo, D.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

Shankar, R.

R. Shankar, I. Bulu, and M. Lončar, “Integrated high-quality factor silicon-on-sapphire ring resonators for the mid-infrared,” Appl. Phys. Lett. 102(5), 051108 (2013).
[Crossref]

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]

Stankovic, S.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Syvridis, D.

Tan, D. T. H.

Teo, E. J.

G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express 19(8), 7112–7119 (2011).
[Crossref] [PubMed]

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Thomson, D.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Tolk, N. H.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

Tran, Q. V.

Van Campenhout, 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, 1–11 (2013).

M. Muneeb, X. Chen, P. Verheyen, G. Lepage, S. Pathak, E. Ryckeboer, A. Malik, B. Kuyken, M. Nedeljkovic, J. Van Campenhout, G. Z. Mashanovich, and G. Roelkens, “Demonstration of Silicon-on-insulator mid-infrared spectrometers operating at 3.8 μm,” Opt. Express 21(10), 11659–11669 (2013).
[Crossref] [PubMed]

Venkatram, N.

Verheyen, P.

M. Muneeb, X. Chen, P. Verheyen, G. Lepage, S. Pathak, E. Ryckeboer, A. Malik, B. Kuyken, M. Nedeljkovic, J. Van Campenhout, G. Z. Mashanovich, and G. Roelkens, “Demonstration of Silicon-on-insulator mid-infrared spectrometers operating at 3.8 μm,” Opt. Express 21(10), 11659–11669 (2013).
[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, 1–11 (2013).

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]

Wang, T.

Wherrett, B. S.

White, T. P.

Wong, C. W.

Xiong, B.

Xiong, C.

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

Yang, P. Y.

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

Yang, Z.

Yin, L.

Yu, Y.

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, 1–11 (2013).

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]

Appl. Phys. Lett. (1)

R. Shankar, I. Bulu, and M. Lončar, “Integrated high-quality factor silicon-on-sapphire ring resonators for the mid-infrared,” Appl. Phys. Lett. 102(5), 051108 (2013).
[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]

J. Mater. Sci. Mater. Electron. (1)

P. Y. Yang, S. Stankovic, J. Crnjanski, E. J. Teo, D. Thomson, A. Bettiol, M. B. H. Breese, W. Headley, C. Giusca, G. T. Reed, and G. Z. Mashanovich, “Silicon photonic waveguides for mid- and long-wave infrared region,” J. Mater. Sci. Mater. Electron. 20(S1), 159–163 (2009).
[Crossref]

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

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, 1–11 (2013).

Nat. Commun. (1)

C. Monat, C. Grillet, M. Collins, A. Clark, J. Schroeder, C. Xiong, J. Li, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Integrated optical auto-correlator based on third-harmonic generation in a silicon photonic crystal waveguide,” Nat. Commun. 5, 3246 (2014).
[Crossref] [PubMed]

Nat. Photonics (5)

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]

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]

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

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

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

Opt. Express (11)

G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express 19(8), 7112–7119 (2011).
[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]

M. Muneeb, X. Chen, P. Verheyen, G. Lepage, S. Pathak, E. Ryckeboer, A. Malik, B. Kuyken, M. Nedeljkovic, J. Van Campenhout, G. Z. Mashanovich, and G. Roelkens, “Demonstration of Silicon-on-insulator mid-infrared spectrometers operating at 3.8 μm,” Opt. Express 21(10), 11659–11669 (2013).
[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]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17(4), 2944–2953 (2009).
[Crossref] [PubMed]

C. Husko, S. Combrié, Q. V. Tran, F. Raineri, C. W. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorption in III-V photonic crystal waveguides,” Opt. Express 17(25), 22442–22451 (2009).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (1)

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83(19), 195203 (2011).
[Crossref]

Other (2)

C. Grillet, P. Ma, B. Luther-Davies, D. Hudson, C. Monat, S. Madden, D. J. Moss, M. Brun, P. Labeye, S. Ortiz, and S. Nicoletti, “Low loss SiGe waveguides in the MID-IR,” in 2013 Conference on Lasers and Electro-Optics - International Quantum Electronics Conference (Optical Society of America, 2013), paper CK_2_6.
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 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 (a) Scanning Electron Microscopy (SEM) image of a SiGe waveguide before encapsulation. A cross section of a SiGe waveguide after encapsulation is shown in the inset. (b) Schematic view of the experimental setup.
Fig. 2
Fig. 2 Propagation loss and coupling loss as a function of wavelength for a waveguide with a cross section of 1.4 x 2 µm2. In the inset the propagation loss of a waveguide with a 1.4 x 1.9 µm2 cross section is shown along with simulation results.
Fig. 3
Fig. 3 (a-d) Normalized transmitted spectrum as a function of the coupled peak intensity for picosecond pulses centered at wavelengths between 3.25μm and 4.75μm. (e) and (f) Spectral broadening and normalized mean wavelength shift (i.e. (λc0)/λ0 where λc is the mean wavelength of the spectrum and λ0 is the low power mean wavelength) as a function the coupled peak intensity.
Fig. 4
Fig. 4 (a) Output power from the Si/SiGe waveguide as a function of the coupled peak intensity for all the wavelengths used in the experiments. (b) Normalized transmission as a function of the coupled peak intensity.
Fig. 5
Fig. 5 (a-d) Nonlinear transmission deduced from the measurements as a function of the coupled peak intensity (circles) at different wavelengths. The nonlinear transmission predicted by solving the NLSE (continuous line) is shown along with the experimental results. (e) Comparison of three- and four-photon absorption coefficients of our SiGe waveguides (estimated in this work), along with that of Si [12], and Ge [27].
Fig. 6
Fig. 6 Experimental (continuous line) and numerical (dashed line) transmission spectrum due to SPM as a function of the coupled peak intensity for a pulse centered at a wavelength of 3250nm, (a), and 4750nm, (b). (c) Nonlinear refractive index as a function of wavelength.
Fig. 7
Fig. 7 Experimental (continuous line) and numerical (dashed line) transmission spectrum due to SPM as a function of the coupled peak intensity. For both scenarios n2 = 0.75x10−18 m2/W, then on the left kc = 1.78x10−26 m3 while on the right n4 = 5x10−33m4/W2 and kc = 5.95x10−27m3
Fig. 8
Fig. 8 (a) and (b) Mean wavelength shift and spectral broadening as a function of the coupled peak pulse intensity for the three simulation scenarios. For all the curves n2 = 0.75x10−18 m2/W. For the black curve kc = 5.95x10−27 m3 as in Fig. 6(a), for the red kc = 1.78x10−26 m3 as in Fig. 7(a), while for the green n4 = 5x10−33m4/W2 and kc = 5.95x10−27m3 as in Fig. 7(b).
Fig. 9
Fig. 9 (a) Experimental (continuous line) and numerical (dashed line) transmission spectrum as a function of the coupled peak intensity calculated considering the real parts of χ(5) and χ(7). The values used are n2 = 2x10−18 m2/W, n4 = 1x10−34 m4/W2, n6 = 5x10−46 m6/W3 and kc = 1.27x10−26 m3. (b) and (c) Mean wavelength shift and spectral broadening as a function of the coupled peak pulse intensity. For both curves n2 = 2x10−18 m2/W and kc = 1.27x10−26 m3. For the green curve n4 = 1x10−34 m4/W2 and n6 = 5x10−46 m6/W3.

Equations (5)

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

A z + i 2 β 2 2 A t 2 + 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
P(t)= ε 0 [ χ (1) E(t)+ χ (3) E 3 (t)+ χ (5) E 5 (t)+ χ (7) E 7 (t)+ ]
Δn= n 2 I+ n 4 I 2 + n 6 I 3
A z + i 2 β 2 2 A t 2 + α 2 A=i k 0 ( n 2 | A | 2 A 3eff + n 4 | A | 4 A 5eff 2 + n 6 | A | 6 A 7eff 3 )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

Metrics