Abstract

A comprehensive theory is developed for describing the nonlinear propagation of optical pulses through silicon waveguides with nanoscale dimensions. Our theory includes not only the vectorial nature of optical modes but also the coupling between the transverse electric and magnetic modes occurring for arbitrarily polarized optical fields. We have studied the dependence of relevant nonlinear parameters on waveguide dimensions and found a class of waveguide geometries for which self-phase modulation can have a dramatic impact on the polarization state of the optical field. Self-induced polarization changes are studied for both the continuous and pulsed optical fields propagating in silicon waveguides. We also discuss the possibility of using these effects for intensity discrimination and pulse compression.

© 2010 Optical Society of America

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2010

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[CrossRef]

2009

2008

J. I. Dadap, N. C. Panoiu, X. Chen, I. Hsieh, X. Liu, C. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008).
[CrossRef] [PubMed]

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[CrossRef]

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,” Nat. Photonics 2, 35–38 (2008).
[CrossRef]

2007

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
[CrossRef]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

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]

S. Ayotte, H. Rong, S. Xu, O. Cohen, and M. J. Paniccia, “Multichannel dispersion compensation using a silicon waveguide-based optical phase conjugator,” Opt. Lett. 32, 2393–2395 (2007).
[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]

2006

2005

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
[CrossRef]

2004

2003

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[CrossRef]

2002

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

1994

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

1987

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

Afshar V., S.

Agrawal, G. P.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[CrossRef]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

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]

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

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Arakawa, Y.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
[CrossRef]

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

Ayotte, S.

Bennett, B. R.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2005).

Boyd, R. W.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

Boyraz, O.

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990).

Chen, X.

Chou, C.

Chu, T.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
[CrossRef]

Cohen, O.

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990).

Cowan, A. R.

Dadap, J. I.

Dai, Y.

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

Dekker, R.

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[CrossRef]

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

Driessen, A.

Driscoll, J. B.

Dulkeith, E.

Ebendorff-Heidepriem, H.

Fauchet, P. M.

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

Först, M.

Foster, M. A.

Y. Dai, X. Chen, Y. Okawachi, A. C. Turner-Foster, M. A. Foster, M. Lipson, A. L. Gaeta, and C. Xu, “1 μs tunable delay using parametric mixing and optical phase conjugation in Si waveguides,” Opt. Express 17, 7004–7010 (2009).
[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,” Nat. Photonics 2, 35–38 (2008).
[CrossRef]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
[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]

Gaeta, A. L.

Y. Dai, X. Chen, Y. Okawachi, A. C. Turner-Foster, M. A. Foster, M. Lipson, A. L. Gaeta, and C. Xu, “1 μs tunable delay using parametric mixing and optical phase conjugation in Si waveguides,” Opt. Express 17, 7004–7010 (2009).
[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,” Nat. Photonics 2, 35–38 (2008).
[CrossRef]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
[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]

Garcia, H.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[CrossRef]

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,” Nat. Photonics 2, 35–38 (2008).
[CrossRef]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
[CrossRef]

Green, W. M.

Green, W. M. J.

Harpin, A.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

Hsieh, I.

Hsieh, I. -W.

Indukuri, T.

Ishida, S.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
[CrossRef]

Jalali, B.

Koonath, P.

Kuo, Y.

Lee, M.

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[CrossRef]

Liang, T. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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.

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

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

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

Lipson, M.

Y. Dai, X. Chen, Y. Okawachi, A. C. Turner-Foster, M. A. Foster, M. Lipson, A. L. Gaeta, and C. Xu, “1 μs tunable delay using parametric mixing and optical phase conjugation in Si waveguides,” Opt. Express 17, 7004–7010 (2009).
[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,” Nat. Photonics 2, 35–38 (2008).
[CrossRef]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
[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]

Liu, X.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

Lüsse, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[CrossRef]

McNab, S. J.

Monro, T. M.

Moormann, C.

Niehusmann, J.

Okawachi, Y.

Osgood, R. M.

Painter, O. J.

Paniccia, M.

Paniccia, M. J.

Panoiu, N. C.

Piredda, G.

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

Premaratne, M.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[CrossRef]

Quochi, F.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[CrossRef]

Raday, O.

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[CrossRef]

Raghunathan, V.

Rieger, G. W.

A. R. Cowan, G. W. Rieger, and J. F. Young, “Nonlinear transmission of 1.5 μm pulses through single-mode silicon-on-insulator waveguide structures,” Opt. Express 12, 1611–1621 (2004).
[CrossRef] [PubMed]

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5 μm pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84, 900–902 (2004).
[CrossRef]

Roberts, S. W.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

Rong, H.

Rukhlenko, I. D.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[CrossRef]

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,” Nat. Photonics 2, 35–38 (2008).
[CrossRef]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
[CrossRef]

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]

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P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
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H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

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,” Nat. Photonics 2, 35–38 (2008).
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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).
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P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
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G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5 μm pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84, 900–902 (2004).
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H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

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H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
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H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
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G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5 μm pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84, 900–902 (2004).
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A. R. Cowan, G. W. Rieger, and J. F. Young, “Nonlinear transmission of 1.5 μm pulses through single-mode silicon-on-insulator waveguide structures,” Opt. Express 12, 1611–1621 (2004).
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J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
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Adv. Opt. Photon.

Appl. Phys. Lett.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. 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]

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[CrossRef]

G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5 μm pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. 84, 900–902 (2004).
[CrossRef]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113 (2007).
[CrossRef]

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[CrossRef]

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J. Lightwave Technol.

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[CrossRef]

Jpn. J. Appl. Phys.

H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jpn. J. Appl. Phys. 44, 6541–6545 (2005).
[CrossRef]

Nat. Photonics

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[CrossRef]

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[CrossRef]

Nature

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]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2007).
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O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12, 4094–4102 (2004).
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O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
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[CrossRef] [PubMed]

Opt. Lett.

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2005).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the waveguide geometry employed.

Fig. 2
Fig. 2

γ 11 as a function of waveguide width and height at the 1550 nm wavelength.

Fig. 3
Fig. 3

EMAs, a ¯ 1 and a ¯ 2 , as a function of waveguide height for a fixed waveguide width of 500 nm at λ = 1550   nm .

Fig. 4
Fig. 4

η 11 , η 22 , and η 12 as a function of waveguide height for a fixed waveguide width of 500 nm at λ = 1550   nm .

Fig. 5
Fig. 5

LEFs and effective mode indices as a function of waveguide height for a fixed waveguide width of 500 nm at λ = 1550   nm .

Fig. 6
Fig. 6

Nonlinear parameters as a function of waveguide height for a fixed waveguide width of 500 nm at λ = 1550   nm .

Fig. 7
Fig. 7

Factors B 1 and B 2 as a function of waveguide height for a fixed waveguide width of 500 nm at λ = 1550   nm .

Fig. 8
Fig. 8

Stokes parameters as a function of the input power of a CW beam propagating through a 2-cm-long waveguide at λ = 1550   nm .

Fig. 9
Fig. 9

Stokes parameters versus time at the output of a 2-cm-long waveguide for a 15-ps input Gaussian pulse, polarized linearly at 45° and launched with 6.75 W peak power. The bottom plot shows optical power after the output pulse passes through a −45° linear analyzer. The center wavelength of the input pulse is 1550 nm.

Equations (55)

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f ̃ ( ω ) = f ( t ) e i ω t d t ,
× E ̃ = i ω μ 0 H ̃ ,
× H ̃ = i ω ε 0 n 2 ( x , y ) E ̃ i ω P ̃ NL ,
E ̃ ( k ) ( r , ω ) = e ( k ) ( x , y , ω ) exp [ i β ( k ) ( ω ) z ] ,
H ̃ ( k ) ( r , ω ) = h ( k ) ( x , y , ω ) exp [ i β ( k ) ( ω ) z ] ,
( E ̃ ) + 2 E ̃ + ω 2 μ 0 ε 0 n 2 ( x , y ) E ̃ + ω 2 μ 0 P ̃ NL = 0.
E ̃ ( r , ω ) a 1 ( z , ω ) e ( 1 ) ( x , y , ω 0 ) e i β ( 1 ) ( ω ) z + a 2 ( z , ω ) e ( 2 ) ( x , y , ω 0 ) e i β ( 2 ) ( ω ) z ,
d a k d z = i ω 2 N k e ( k ) ( x , y ) P ̃ NL ( x , y , z ) e i β ( k ) ( ω ) z d x d y ,
N k = Re [ ( e ( k ) × h ( k ) ) z ̂ d x d y ] .
A ̃ k ( z , ω ω 0 ) = N k 2 a k ( z , ω ) exp { i [ β ( k ) ( ω ) β 0 ( k ) ] z } ,
A k z = [ n = 1 i n + 1 β n ( k ) n ! n t n ] A k + exp [ i ( β 0 ( k ) z ω 0 t ) ] i ω 0 2 2 N k e ( k ) ( x , y ) P NL ( x , y , z , t ) d x d y ,
E ( r , t ) 2 N 1 A 1 ( z , t ) e ( 1 ) ( x , y ) exp [ i ( β 0 ( 1 ) z ω 0 t ) ] + 2 N 2 A 2 ( z , t ) e ( 2 ) ( x , y ) exp [ i ( β 0 ( 2 ) z ω 0 t ) ] .
P NL ( r , t ) = P ( 3 ) ( r , t ) + P ( f c ) ( r , t ) .
P μ ( 3 ) ( r , t ) = 3 ε 0 4 α , β , γ χ μ α β γ ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 ) E α ( r , t ) E β ( r , t ) E γ ( r , t ) .
χ μ α β γ ( 3 ) = χ c [ ρ 3 ( δ μ α δ β γ + δ μ β δ α γ + δ μ γ δ α β ) + ( 1 ρ ) q R q μ R q α R q β R q γ ] ,
( R 11 R 12 R 13 R 21 R 22 R 23 R 31 R 32 R 33 ) = 1 2 ( 1 0 1 1 0 1 0 2 0 ) .
χ c = 4 3 ε 0 c n 0 2 n 2 ( 1 + i r ) ,     r = β TPA 2 k 0 n 2 ,
Δ α f c = σ a N ,
Δ n f c = σ n e N ( σ n h N ) 4 / 5 ,
P ( f c ) ( r , t ) = 2 ε 0 n 0 [ Δ n f c + ( i / 2 k 0 ) Δ α f c ] E ( r , t ) .
A k z = [ n = 1 i n + 1 β n ( k ) n ! n t n ] A k + T 3 o k + T f c k ,
T 3 o k = l m n 3 i ω 0 ε 0 4 ( N k N l N m N n ) 1 / 2 A l A m A n   exp ( i Δ β k l m n z ) μ α β γ χ μ α β γ ( 3 ) e μ ( k ) e α ( l ) e β ( m ) e γ ( n ) d x d y ,
T f c k = l i ω 0 ε 0 n 0 ( N k N l ) 1 / 2 A l   exp [ i ( β 0 ( l ) β 0 ( k ) ) z ] [ Δ n f c + ( i / 2 k 0 ) Δ α f c ] e ( k ) e ( l ) d x d y ,
A 1 z = [ n = 1 i n + 1 β n ( 1 ) n ! n t n ] A 1 + i γ 11 ( 1 + i r ) | A 1 | 2 A 1 + i γ 12 ( 1 + i r ) | A 2 | 2 A 1 + i γ 12 ( 1 + i r ) A 2 2 A 1 e 2 i k 0 Δ n z + i k 0 Γ 1 n 0 n ¯ 1 ( Δ n 1 f c + i 2 k 0 Δ α 1 f c ) A 1 α 1 2 A 1 ,
A 2 z = [ n = 1 i n + 1 β n ( 2 ) n ! n t n ] A 2 + i γ 22 ( 1 + i r ) | A 2 | 2 A 2 + i γ 12 ( 1 + i r ) | A 1 | 2 A 2 + i γ 12 ( 1 + i r ) A 1 2 A 2 e 2 i k 0 Δ n z + i k 0 Γ 2 n 0 n ¯ 2 ( Δ n 2 f c + i 2 k 0 Δ α 2 f c ) A 2 α 2 2 A 2 ,
γ 11 = n 0 2 Γ 1 2 n ¯ 1 2 a ¯ 1 η 11 k 0 n 2 ,     γ 22 = n 0 2 Γ 2 2 n ¯ 2 2 a ¯ 2 η 22 k 0 n 2 ,
γ 12 = 2 n 0 2 Γ 1 Γ 2 η 12 k 0 n 2 n ¯ 1 n ¯ 2 ( a ¯ 1 a ¯ 2 ) 1 / 2 ,     γ 12 = n 0 2 Γ 1 Γ 2 η 12 k 0 n 2 n ¯ 1 n ¯ 2 ( a ¯ 1 a ¯ 2 ) 1 / 2 ,
a ¯ k = ( | e ( k ) | 2 d x d y ) 2 ( ( | e ( k ) | 2 ) 2 d x d y ) .
Γ k = | e ( k ) | 2 d x d y | e T ( k ) | 2 + ( β ( k ) ) 1 Im ( e T ( k ) T e z k ) d x d y .
η l m = μ α β γ χ c 1 χ μ α β γ ( 3 ) e μ ( l ) e α ( l ) e β ( m ) e γ ( m ) d x d y [ ( | e ( l ) | 2 ) 2 d x d y ( | e ( m ) | 2 ) 2 d x d y ] 1 / 2 ,
η 12 = μ α β γ χ c 1 χ μ α β γ ( 3 ) e μ ( 1 ) e α ( 2 ) e β ( 1 ) e γ ( 2 ) d x d y [ ( | e ( l ) | 2 ) 2 d x d y ( | e ( m ) | 2 ) 2 d x d y ] 1 / 2 .
Δ n k f c = Δ n f c ( x , y ) | e ( k ) | 2 d x d y ( | e ( k ) | 2 d x d y ) ,
Δ α k f c = Δ α f c ( x , y ) | e ( k ) | 2 d x d y ( | e ( k ) | 2 d x d y ) .
Δ n ¯ k f c = n 0 n ¯ k Γ k Δ n k f c n 0 n ¯ k Γ k Π k Δ n f c = B k Δ n k f c ,
Π k = Si | e ( k ) | 2 d x d y ( | e ( k ) | 2 d x d y ) .
δ = δ L + δ NL ,
δ L = ( β 0 ( 1 ) β 0 ( 2 ) ) L ,
δ NL = ϕ 1 ( L ) ϕ 2 ( L ) .
S 0 = | A 1 | 2 + | A 2 | 2 ,     S 1 = | A 1 | 2 | A 2 | 2 ,
S 2 = Re ( 2 A 1 A 2 e i δ L ) ,     S 3 = Im ( 2 A 1 A 2 e i δ L ) .
d N d t = G N τ f c ,
G = r A c ω 0 ( γ 11 | A 1 | 4 + γ 22 | A 2 | 4 + 2 γ 12 | A 1 A 2 | 2 ) ,
d P 1 d z = 2 γ 11 r P 1 2 2 γ 12 r P 1 P 2 B 1 Δ α f c P 1 α 1 P 1 ,
d P 2 d z = 2 γ 22 r P 2 2 2 γ 12 r P 1 P 2 B 2 Δ α f c P 2 α 2 P 2 ,
d δ NL d z = ( γ 11 γ 12 ) P 1 + ( γ 12 γ 22 ) P 2 + k 0 ( B 1 B 2 ) Δ n f c .
E ( k ) = E T ( k ) + E z ( k ) z ̂ ,     H ( k ) = H T ( k ) + H z ( k ) z ̂ ,
k = 1 M d a k d z [ 2 i β ( k ) E T ( k ) T E z ( k ) ( T E T ( k ) ) z ̂ ] = ω 2 μ 0 P ̃ NL ,
l d a l d z [ 2 i β ( m ) E T ( m ) E T ( l ) E T ( m ) T E l z T ( E T ( m ) E T ( l ) ) + E T ( l ) T E z ( m ) ] d x d y = ω 2 μ 0 E ( m ) P ̃ NL d x d y ,
( T E T ( l ) ) E z ( m ) = T ( E z ( m ) E T ( l ) ) E T ( l ) T E z ( m ) .
R T ( E z ( m ) E T ( l ) ) d x d y = R E z ( m ) E T ( l ) d .
l d a l d z [ 2 i β ( m ) E T ( m ) E T ( l ) E T ( m ) T E z ( l ) + E T ( l ) T E z ( m ) ] d x d y = ω 2 μ 0 E ( m ) P ̃ NL d x d y .
2 i β ( m ) E T ( m ) E T ( l ) E T ( m ) T E z ( l ) + E T ( l ) T E z ( m ) = i ω μ 0 [ E ( l ) × H ( m ) + E ( m ) × H ( l ) ] z ̂ .
l d a l d z [ e ( l ) × h ( m ) + e ( m ) × h ( l ) ] z ̂ d x d y = i ω e ( m ) P ̃ NL   exp [ i β ( m ) z ] d x d y .
[ e ( l ) × h ( m ) + e ( m ) × h ( l ) ] z ̂ d x d y = 2 N m δ l m ,
d a m d z = i ω 2 N m e ( m ) P ̃ NL   exp [ i β ( m ) z ] d x d y .

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