Abstract

We theoretically and experimentally investigate the nonlinear evolution of two optical pulses in a silicon waveguide. We provide an analytic solution for the weak probe wave undergoing non-degenerate two-photon absorption (TPA) from the strong pump. At larger pump intensities, we employ a numerical solution to study the interplay between TPA and photo-generated free carriers. We develop a simple and powerful approach to extract and separate out the distinct loss contributions of TPA and free-carrier absorption from readily available experimental data. Our analysis accounts accurately for experimental results in silicon photonic crystal waveguides.

© 2015 Optical Society of America

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  1. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
    [Crossref] [PubMed]
  2. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
    [Crossref]
  3. Y. Zhang, C. Husko, J. Schröder, S. Lefrancois, I. H. Rey, T. F. Krauss, and B. J. Eggleton, “Phase-sensitive amplification in silicon photonic crystal waveguides,” Opt. Lett. 39, 363–366 (2014).
    [Crossref] [PubMed]
  4. F. Da Ros, D. Vukovic, A. Gajda, K. Dalgaard, L. Zimmermann, B. Tillack, M. Galili, K. Petermann, and C. Peucheret, “Phase regeneration of DPSK signals in a silicon waveguide with reverse-biased P-I-N junction,” Opt. Express 22, 5029–5036 (2014).
    [Crossref] [PubMed]
  5. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32, 2031–2033 (2007).
    [Crossref] [PubMed]
  6. H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
    [Crossref]
  7. C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. OFaolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express 18, 22915–22927 (2010).
    [Crossref] [PubMed]
  8. D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
    [Crossref]
  9. A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and kerr coefficients of silicon for 850–2200,” Appl. Phys. Lett. 90, 191104 (2007).
    [Crossref]
  10. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
    [Crossref] [PubMed]
  11. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J.-I. Takahashi, and S.-I. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13, 4629–4637 (2005).
    [Crossref] [PubMed]
  12. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, “Approximate expressions for estimation of four-wave mixing efficiency in slow-light photonic crystal waveguides,” J. Opt. Soc. Am. B 31, 366–375 (2014).
    [Crossref]
  13. D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
    [Crossref]
  14. X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. 11, 15–25 (2009).
  15. Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express 18, 5668–5673 (2010).
    [Crossref] [PubMed]
  16. P. Mehta, N. Healy, T. Day, J. Sparks, P. Sazio, J. Badding, and A. Peacock, “All-optical modulation using two-photon absorption in silicon core optical fibers,” Opt. Express 19, 19078–19083 (2011).
    [Crossref] [PubMed]
  17. L. Shen, N. Healy, C. J. Mitchell, J. SolerPenades, M. Nedeljkovic, G. Z. Mashanovich, and A. C. Peacock, “Two-photon absorption and all-optical modulation in germanium-on-silicon waveguides for the mid-infrared,” Opt. Lett. 40, 2213–2216 (2015).
    [Crossref]
  18. E.-K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Pulse compression and modelocking by using tpa in silicon waveguides,” Opt. Express 15, 6500–6506 (2007).
    [Crossref] [PubMed]
  19. Y. Yue, H. Huang, L. Zhang, J. Wang, J.-Y. Yang, O. F. Yilmaz, J. S. Levy, M. Lipson, and A. E. Willner, “UWB monocycle pulse generation using two-photon absorption in a silicon waveguide,” Opt. Lett. 37, 551–553 (2012).
    [Crossref] [PubMed]
  20. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
    [Crossref]
  21. S. Lefrancois, C. Husko, A. Blanco-Redondo, and B. J. Eggleton, “Nonlinear silicon photonics analyzed with the moment method,” J. Opt. Soc. Am. B 32, 218–226 (2015).
    [Crossref]
  22. Y. Zhang, C. Husko, J. Schröder, and B. J. Eggleton, “Pulse evolution and phase-sensitive amplification in silicon waveguides,” Opt. Lett. 39, 5329–5332 (2014).
    [Crossref]
  23. J. Li, L. O’Faolain, I. H. Rey, and T. F. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express 19, 4458–4463 (2011).
    [Crossref] [PubMed]

2015 (2)

2014 (4)

2012 (1)

2011 (3)

2010 (2)

2009 (1)

X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. 11, 15–25 (2009).

2008 (1)

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

2007 (4)

2006 (1)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

2005 (2)

D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J.-I. Takahashi, and S.-I. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13, 4629–4637 (2005).
[Crossref] [PubMed]

2002 (1)

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

1991 (1)

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[Crossref]

Agrawal, G. P.

Asghari, M.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Badding, J.

Blanco-Redondo, A.

Boyraz, O.

X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. 11, 15–25 (2009).

E.-K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Pulse compression and modelocking by using tpa in silicon waveguides,” Opt. Express 15, 6500–6506 (2007).
[Crossref] [PubMed]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and kerr coefficients of silicon for 850–2200,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Cirloganu, C. M.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

Corcoran, B.

Da Ros, F.

Dalgaard, K.

Day, I.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Day, T.

Drake, J.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Ebnali-Heidari, M.

Eggleton, B.

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. OFaolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express 18, 22915–22927 (2010).
[Crossref] [PubMed]

D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

Eggleton, B. J.

Fishman, D. A.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

Foster, M. A.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Fu, L.

D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

Fukuda, H.

Gaeta, A. L.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Gajda, A.

Galili, M.

Geraghty, D. F.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

Grillet, C.

Hagan, D. J.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[Crossref]

Harpin, A.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Hasama, T.

Healy, N.

Huang, H.

Husko, C.

Hutchings, D. C.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[Crossref]

Ishikawa, H.

Itabashi, S.-I.

Kamalakis, T.

Kamei, T.

Kanakis, P.

Kawashima, H.

Kintaka, K.

Krauss, T.

Krauss, T. F.

Lefrancois, S.

Levy, J. S.

Li, J.

Liang, T.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Lin, Q.

Lipson, M.

Y. Yue, H. Huang, L. Zhang, J. Wang, J.-Y. Yang, O. F. Yilmaz, J. S. Levy, M. Lipson, and A. E. Willner, “UWB monocycle pulse generation using two-photon absorption in a silicon waveguide,” Opt. Lett. 37, 551–553 (2012).
[Crossref] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Littler, I.

D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

Mashanovich, G. Z.

Mehta, P.

Mitchell, C. J.

Monat, C.

Monroe, M.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

Mori, M.

Moss, D.

D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

Nedeljkovic, M.

O’Faolain, L.

OFaolain, L.

Ogasawara, T.

Okano, M.

Padilha, L. A.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

Painter, O. J.

Peacock, A.

Peacock, A. C.

Petermann, K.

Peucheret, C.

Qian, F.

Rey, I. H.

Roberts, S.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and kerr coefficients of silicon for 850–2200,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Sakakibara, Y.

Salem, R.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

Sang, X.

X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. 11, 15–25 (2009).

Sazio, P.

Schmidt, B. S.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Schröder, J.

Sharping, J. E.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Sheik-Bahae, M.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[Crossref]

Shen, L.

Shoji, T.

Shoji, Y.

SolerPenades, J.

Sparks, J.

Sphicopoulos, T.

Suda, S.

Takahashi, J.-I.

Takahashi, M.

Tien, E.-K.

X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. 11, 15–25 (2009).

E.-K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Pulse compression and modelocking by using tpa in silicon waveguides,” Opt. Express 15, 6500–6506 (2007).
[Crossref] [PubMed]

Tillack, B.

Tsang, H.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Tsuchizawa, T.

Turner, A. C.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Van Driel, H. M.

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and kerr coefficients of silicon for 850–2200,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Van Stryland, E. W.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[Crossref]

Vukovic, D.

Wang, J.

Watanabe, T.

Webster, S.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

White, T.

Willner, A. E.

Wong, C.

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Yamada, K.

Yang, J.-Y.

Yilmaz, O. F.

Yin, L.

Yue, Y.

Yuksek, N. S.

Zhang, L.

Zhang, Y.

Zimmermann, L.

Appl. Phys. Lett. (2)

H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and kerr coefficients of silicon for 850–2200,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Electron. Lett. (1)

D. Moss, L. Fu, I. Littler, and B. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Optoelectron. Adv. Mater. (1)

X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. 11, 15–25 (2009).

Nature (1)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[Crossref] [PubMed]

Nature Photon. (2)

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. 2, 35–38 (2008).
[Crossref]

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. 5, 561–565 (2011).
[Crossref]

Opt. Express (8)

E.-K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Pulse compression and modelocking by using tpa in silicon waveguides,” Opt. Express 15, 6500–6506 (2007).
[Crossref] [PubMed]

Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express 18, 5668–5673 (2010).
[Crossref] [PubMed]

P. Mehta, N. Healy, T. Day, J. Sparks, P. Sazio, J. Badding, and A. Peacock, “All-optical modulation using two-photon absorption in silicon core optical fibers,” Opt. Express 19, 19078–19083 (2011).
[Crossref] [PubMed]

F. Da Ros, D. Vukovic, A. Gajda, K. Dalgaard, L. Zimmermann, B. Tillack, M. Galili, K. Petermann, and C. Peucheret, “Phase regeneration of DPSK signals in a silicon waveguide with reverse-biased P-I-N junction,” Opt. Express 22, 5029–5036 (2014).
[Crossref] [PubMed]

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

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[Crossref] [PubMed]

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[Crossref] [PubMed]

Opt. Lett. (5)

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Figures (5)

Fig. 1
Fig. 1 (a) Normalized output powers and (b) phases obtained from Eq. (3). (c) Logarithmic scale and (d) linear scale of output powers of pump and probe with (solid) and without (dashed) free carriers vs input pump power.
Fig. 2
Fig. 2 (a) Schematic of experimental setup. MLL: mode-locked laser, SPS: spectral pulse shaper, EDFA: erbium-doped fiber amplifier, PC: polarization controller, OSA: optical spectrum analyzer. (b) Measured linear transmission spectrum (fiber to fiber) and group index (ng) of the PhC waveguide, where arrows indicate the locations of the probe and pump.
Fig. 3
Fig. 3 (a) Input spectrum (light) and output spectra of the probe without pump(solid) and with pump (dashed) at zero and 10 ps delays. (b) Normalized probe output obtained from experiment (square) and numerical solution of Eq. (3) (solid with FCA and dashed without FCA in the probe) as a function of probe delay at a pump power of 4.4 W.
Fig. 4
Fig. 4 (a) Normalized pump and probe output at −1 ps delay and 10 ps delay obtained from Eq. (3) (lines) and experimental results (markers). (b) XTPA and TPA obtained from experiment (markers) using Eq. (4), numerical solution of Eq. (3) (solid) and analytic solution of Eq. (2) without free carriers (dashed) as a function of pump powers at −1 ps delay.
Fig. 5
Fig. 5 XTPA difference between analytic solution and numerical calculation in the presence of free carriers as a function of pump power and pulse duration. The dashed line indicates δ = 10%. The cross is our experimental regime, where δ = 20%.

Equations (16)

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A 1 z = ( i γ γ TPA 2 ) | A 1 | 2 A 1 α 2 A 1 + ( i k 0 n FC σ 2 ) N c A 1 ,
A 2 z = 2 ( i γ γ TPA 2 ) | A 1 | 2 A 2 α 2 A 2 + ( i k 0 n FC σ 2 ) N c A 2 ,
P 2 ( z , t ) = P 2 , in ( t ) e α z ( 1 + P 1 , in ( t ) z eff γ i ) 2 ,
ϕ 2 ( z , t ) = 2 γ γ TPA ln [ 1 + γ TPA P 1 , in ( t ) z eff ] ,
P 1 ( z , t ) = P 1 , in ( t ) e α z 1 + P 1 , in ( t ) z eff γ TPA ,
ϕ 1 ( z , t ) = γ γ TPA ln [ 1 + γ TPA P 1 , in ( t ) z eff ] .
P 1 ( z , t ) = P 1 , in e α z e γ TPA P 1 d z e σ N c d z ,
ϕ 1 ( z , t ) = γ P 1 d z + k 0 n FC N c d z ,
P 2 ( z , t ) = P 2 , in e α z e 2 γ TPA P 1 d z e σ N c d z ,
ϕ 2 ( z , t ) = 2 γ P 1 d z + k 0 n FC N c d z .
TPA = e γ TPA P 1 d z = P 2 P 1 , in P 1 P 2 , in ,
XTPA = e 2 γ TPA P 1 d z = ( P 2 P 1 , in P 1 P 2 , in ) 2 ,
FCA = e σ N c d z = P 1 2 P 2 , in P 1 , in 2 P 2 1 e α z ,
TPA = e γ TPA P 1 d z = ( P 2 P 1 , in P 1 P 2 , in e α 2 z e α 1 z ) 1 2 η 1 ,
XTPA = e 2 γ X TPA P 1 d z = ( P 2 P 1 , in P 1 P 2 , in e α 2 z e α 1 z ) 2 2 η 1 ,
FCA = e σ N c d z = P 1 2 η P 2 , in 2 η 1 P 1 , in 2 η P 2 2 η 1 e 2 ( η 1 ) α 1 z e ( 2 η 1 ) α 2 z ,

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