Abstract

We propose a novel scheme for continuous-wave pumped optical parametric oscillation (OPO) inside silicon micro-resonators. The proposed scheme not only requires a relative low lasing threshold, but also exhibits extremely broad tunability extending from the telecom band to mid infrared.

© 2008 Optical Society of America

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  1. R. Soref, "The Past, Present, and Future of Silicon Photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006), and references therein.
    [CrossRef]
  2. 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]
  3. V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
    [CrossRef]
  4. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, "Four-wave mixing in silicon wire waveguides," Opt. Express 13, 4629-4637 (2005).
    [CrossRef] [PubMed]
  5. H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
    [CrossRef] [PubMed]
  6. M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12949-12958 (2007).
    [CrossRef] [PubMed]
  7. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, "Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires," Opt. Express 15, 1135-1146 (2007).
    [CrossRef] [PubMed]
  8. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, "Ultra-low power parametric frequency conversion in a silicon microring resonator," Opt. Express 16, 4881-4887 (2008).
    [CrossRef] [PubMed]
  9. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
    [CrossRef] [PubMed]
  10. M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys. Lett. 82, 2954-2956 (2003).
    [CrossRef]
  11. A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
    [CrossRef]
  12. 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. 90, 021111 (2007).
    [CrossRef]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2007).
  14. Although the waveguide shown in Fig. 1 is multimoded over a broad spectral range, higher-order modes have quite different mode profiles and dispersion properties compared with the fundamental quasi-TE mode. They are not likely to participate in the FWM process if the pump and signal waves propagate predominantly in the fundamental quasi-TE mode.
  15. The real and imaginary parts of χ(3) are related to Kerr nonlinearity and TPA, respectively [2]. An accurate description of SPM, XPM, TPA, and FWM requires complete information about χ(3)(-ωi;ωj, -ωk,ωl). However, current experimental knowledge is only available for χ(3)(-ωiωi,-ωi,ωi) [11, 12]. As cross-TPA involves the simultaneous absorption of two photons at ωi and ωj, we approximate χ(3)(-ωi;ωj, -ωj, ωi) ≈χ(3)(-ω- ;ω-, -ω-, ω-) where ω- = (ωi+ωj)/2. Similarly, FWM involves the annihilation of two pump photons tocreate a signal and idler photon, and we approximate χ(3)(-ωs;ωp,-ωi,ωp) ≈ χ(3)(-ωp;ωp,-ωp,ωp). Note also χ(3)(-ωi;ωp,-ωs,ωp)=χ(3)(-ωs;ωp,-ωi,ωp) = [χ(3)(-ωp;ωs,-ωp,ωi)]* because of the time-reversal symmetry.
  16. We fit each set of experimental data (1.2-2.2 µm) in Refs. [11, 12] with a fifth-order polynomial, and average them to obtain the silicon nonlinearity. TPA is zero and the Kerr nonlinearity is assumed to be constant for wavelength longer than 2.2 µm.
  17. R. A. Soref and B. R. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
    [CrossRef]
  18. For completeness, we have included all possible self- and cross-TPA and induced free carriers from all the three waves and their combinations in the numerical results of this paper.
  19. A. Y. H. Chen, G. K. L. Wong, S. G. Murdoch, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Widely tunable optical parametric generation in a photonic crystal fiber," Opt. Lett. 30, 762-764 (2005).
    [CrossRef] [PubMed]
  20. Y. Deng, Q. Lin, F. Lu, G. P. Agrawal, and W. H. Knox, "Broadly tunable femtosecond parametric oscillator using a photonic crystal fiber," Opt. Lett. 30, 1234-1236 (2005).
    [CrossRef] [PubMed]
  21. T. V. Andersen, K. M. Hilligsøe, C. K. Nielsen, J. Thøgersen, K. P. Hansen, S. R. Keiding, and J. J. Larsen, "Continuous-wave wavelength conversion in a photonic crystal fiber with two zero-dispersion wavelengths," Opt. Express 12, 4113-4122 (2004).
    [CrossRef] [PubMed]
  22. M. Borselli, T. J. Johnson, and O. J. Painter, "Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment", Opt. Express 13, 1515-1529 (2005).
    [CrossRef] [PubMed]
  23. T. J. Kippenberg, S.M. Spillane, and K. J. Vahala, "Kerr-Nonlinearity Optical Parametric Oscillation in an Ultrahigh-Q Toroid Microcavity," Phys. Rev. Lett. 93, 083904 (2004).
    [CrossRef] [PubMed]
  24. C. P. Michael, M. Borselli, T. J. Johnson, C. Chrystal, and O. Painter, "An optical fiber-taper probe for wafer-scale microphotonic device characterization," Opt. Express 15, 4745-4752 (2007).
    [CrossRef] [PubMed]
  25. In practice, a critical coupling over such a broad spectral region is difficult for a straight bus waveguide, but is possible by using a curved bus waveguide with a curvature similar to the resonator. SeeT . Carmon, S. Y. T. Wang, E. P . Ostby, and K. J. Vahala, "Wavelength-independent coupler from fiber to an on-chip cavity, demonstrated over an 850nm span," Opt. Express 15, 7677-7681 (2007).
    [CrossRef] [PubMed]
  26. I. T. Sorokina and K. L. Vodopyanov, eds., Solid-State Mid-Infrared Laser Sources, Top. Appl. Phys. 89 (2003).
    [CrossRef]
  27. B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
    [CrossRef]
  28. M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
    [CrossRef]
  29. H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, "A cascaded silicon Raman laser," Nat. Photonics 2, 170 (2008).
    [CrossRef]
  30. P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005).
    [CrossRef] [PubMed]

2008 (2)

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

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, "Ultra-low power parametric frequency conversion in a silicon microring resonator," Opt. Express 16, 4881-4887 (2008).
[CrossRef] [PubMed]

2007 (7)

A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (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. 90, 021111 (2007).
[CrossRef]

I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, "Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires," Opt. Express 15, 1135-1146 (2007).
[CrossRef] [PubMed]

C. P. Michael, M. Borselli, T. J. Johnson, C. Chrystal, and O. Painter, "An optical fiber-taper probe for wafer-scale microphotonic device characterization," Opt. Express 15, 4745-4752 (2007).
[CrossRef] [PubMed]

In practice, a critical coupling over such a broad spectral region is difficult for a straight bus waveguide, but is possible by using a curved bus waveguide with a curvature similar to the resonator. SeeT . Carmon, S. Y. T. Wang, E. P . Ostby, and K. J. Vahala, "Wavelength-independent coupler from fiber to an on-chip cavity, demonstrated over an 850nm span," Opt. Express 15, 7677-7681 (2007).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12949-12958 (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 (5)

R. Soref, "The Past, Present, and Future of Silicon Photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006), and references therein.
[CrossRef]

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
[CrossRef]

H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
[CrossRef] [PubMed]

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
[CrossRef] [PubMed]

2005 (6)

2004 (2)

2003 (1)

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

1987 (1)

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

Agrawal, G. P.

Andersen, T. V.

Barclay, P. E.

Bennett, B. R.

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

Borselli, M.

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. 90, 021111 (2007).
[CrossRef]

Brinkmeyer, E.

M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
[CrossRef]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Carmon, T

Chen, A. Y. H.

Chen, X.

Chrystal, C.

Claps, R.

Cohen, 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 (2008).
[CrossRef]

H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
[CrossRef] [PubMed]

Dadap, J. I.

Deng, Y.

Dimitropoulos, D.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
[CrossRef]

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]

Draheim, R.

M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
[CrossRef]

Fathpour, S.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

Fauchet, P. M.

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. 90, 021111 (2007).
[CrossRef]

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
[CrossRef] [PubMed]

Foster, M. A.

Fukuda, H.

Gaeta, A. L.

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]

Hansen, K. P.

Harvey, J. D.

Hilligsøe, K. M.

Hsieh, I-W.

Itabashi, S.

Jalali, B.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
[CrossRef]

Johnson, T. J.

Keiding, S. R.

Kippenberg, T. J.

T. J. Kippenberg, S.M. Spillane, and K. J. Vahala, "Kerr-Nonlinearity Optical Parametric Oscillation in an Ultrahigh-Q Toroid Microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Knight, J. C.

Knox, W. H.

Krause, M.

M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
[CrossRef]

Kuo, Y.

Larsen, J. J.

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 (2008).
[CrossRef]

Leonhardt, R.

Lin, Q.

Lipson, M.

Liu, A.

Lu, F.

McNab, S. J.

Michael, C. P.

Murdoch, S. G.

Nielsen, C. K.

Osgood, R. M.

Ostby, E. P

Painter, O.

Painter, O. J.

Paniccia, 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 (2008).
[CrossRef]

H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
[CrossRef] [PubMed]

Panoiu, N. C.

Piredda, G.

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. 90, 021111 (2007).
[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 (2008).
[CrossRef]

Raghunathan, V.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
[CrossRef]

Renner, H.

M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
[CrossRef]

Rong, H.

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

H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
[CrossRef] [PubMed]

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Russell, P. St. J.

Salem, R.

Shoji, T.

Shori, R.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

Sih, V.

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

Soref, R.

R. Soref, "The Past, Present, and Future of Silicon Photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006), and references therein.
[CrossRef]

Soref, R. A.

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

Spillane, S.M.

T. J. Kippenberg, S.M. Spillane, and K. J. Vahala, "Kerr-Nonlinearity Optical Parametric Oscillation in an Ultrahigh-Q Toroid Microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Srinivasan, K.

Strafsudd, O.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

Takahashi, J.

Takahashi, M.

Thøgersen, J.

Tsuchizawa, T.

Turner, A. C.

Vahala, K. J.

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 nm," Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Vlasov, Y. A.

Wadsworth, W. J.

Wang, S. Y. T.

Watanabe, T.

Wong, G. K. L.

Xu, S.

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

Yamada, K.

Zhang, J.

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. 90, 021111 (2007).
[CrossRef]

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
[CrossRef] [PubMed]

Electron. Lett. (1)

M. Krause, R. Draheim, H. Renner, and E. Brinkmeyer, "Cascaded silicon Raman lasers as mid-infrared sources," Electron. Lett. 42, 1224-1225 (2006).
[CrossRef]

Appl. Phys. Lett. (3)

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

A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (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. 90, 021111 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

IEEE J. Sel. Top. Quantum Electron. (2)

R. Soref, "The Past, Present, and Future of Silicon Photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006), and references therein.
[CrossRef]

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour,D. Dimitropoulos, and O. Strafsudd, "Propests for silicon mid-IR Raman lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

J. Lightwave Technol. (1)

Nat. Photonics (1)

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

Opt. Express (12)

T. V. Andersen, K. M. Hilligsøe, C. K. Nielsen, J. Thøgersen, K. P. Hansen, S. R. Keiding, and J. J. Larsen, "Continuous-wave wavelength conversion in a photonic crystal fiber with two zero-dispersion wavelengths," Opt. Express 12, 4113-4122 (2004).
[CrossRef] [PubMed]

P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper," Opt. Express 13, 801-820 (2005).
[CrossRef] [PubMed]

M. Borselli, T. J. Johnson, and O. J. Painter, "Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment", Opt. Express 13, 1515-1529 (2005).
[CrossRef] [PubMed]

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

H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
[CrossRef] [PubMed]

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
[CrossRef] [PubMed]

I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, "Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires," Opt. Express 15, 1135-1146 (2007).
[CrossRef] [PubMed]

C. P. Michael, M. Borselli, T. J. Johnson, C. Chrystal, and O. Painter, "An optical fiber-taper probe for wafer-scale microphotonic device characterization," Opt. Express 15, 4745-4752 (2007).
[CrossRef] [PubMed]

In practice, a critical coupling over such a broad spectral region is difficult for a straight bus waveguide, but is possible by using a curved bus waveguide with a curvature similar to the resonator. SeeT . Carmon, S. Y. T. Wang, E. P . Ostby, and K. J. Vahala, "Wavelength-independent coupler from fiber to an on-chip cavity, demonstrated over an 850nm span," Opt. Express 15, 7677-7681 (2007).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12949-12958 (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]

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, "Ultra-low power parametric frequency conversion in a silicon microring resonator," Opt. Express 16, 4881-4887 (2008).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

T. J. Kippenberg, S.M. Spillane, and K. J. Vahala, "Kerr-Nonlinearity Optical Parametric Oscillation in an Ultrahigh-Q Toroid Microcavity," Phys. Rev. Lett. 93, 083904 (2004).
[CrossRef] [PubMed]

Other (6)

I. T. Sorokina and K. L. Vodopyanov, eds., Solid-State Mid-Infrared Laser Sources, Top. Appl. Phys. 89 (2003).
[CrossRef]

For completeness, we have included all possible self- and cross-TPA and induced free carriers from all the three waves and their combinations in the numerical results of this paper.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2007).

Although the waveguide shown in Fig. 1 is multimoded over a broad spectral range, higher-order modes have quite different mode profiles and dispersion properties compared with the fundamental quasi-TE mode. They are not likely to participate in the FWM process if the pump and signal waves propagate predominantly in the fundamental quasi-TE mode.

The real and imaginary parts of χ(3) are related to Kerr nonlinearity and TPA, respectively [2]. An accurate description of SPM, XPM, TPA, and FWM requires complete information about χ(3)(-ωi;ωj, -ωk,ωl). However, current experimental knowledge is only available for χ(3)(-ωiωi,-ωi,ωi) [11, 12]. As cross-TPA involves the simultaneous absorption of two photons at ωi and ωj, we approximate χ(3)(-ωi;ωj, -ωj, ωi) ≈χ(3)(-ω- ;ω-, -ω-, ω-) where ω- = (ωi+ωj)/2. Similarly, FWM involves the annihilation of two pump photons tocreate a signal and idler photon, and we approximate χ(3)(-ωs;ωp,-ωi,ωp) ≈ χ(3)(-ωp;ωp,-ωp,ωp). Note also χ(3)(-ωi;ωp,-ωs,ωp)=χ(3)(-ωs;ωp,-ωi,ωp) = [χ(3)(-ωp;ωs,-ωp,ωi)]* because of the time-reversal symmetry.

We fit each set of experimental data (1.2-2.2 µm) in Refs. [11, 12] with a fifth-order polynomial, and average them to obtain the silicon nonlinearity. TPA is zero and the Kerr nonlinearity is assumed to be constant for wavelength longer than 2.2 µm.

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Figures (5)

Fig. 1.
Fig. 1.

Group velocity dispersion for a straight waveguide with a geometry shown in the inset, simulated by the finite element method (FEMLAB, COMSOL).

Fig. 2.
Fig. 2.

Parametric gain spectra at three pump wavelengths inside a waveguide shown in Fig. 1. Parameters are given in the text.

Fig. 3.
Fig. 3.

Phase-matched signal and idler wavelengths as a function of pump wavelength for the waveguide geometry in Fig. 1. Solid curves show the case with a pump power of 3 W. Dotted curves shows the case of purely linear phase matching (Δk=0) where the pump is absent. Their closeness to the solid lines in the normal-dispersion regime indicates the effect of high-order dispersion [19]. The black dashed line shows the location of the ZDWL.

Fig. 4.
Fig. 4.

Frequency mismatch of the resonant modes for a microring resonator with a diameter of 50.01 µm and a cross section of 1.7×0.4 µm at various pump wavelengths. The frequency mismatch is defined as (2ω 0p-ω 0s-ω 0i)/2π for angular-momentum conserved pump, signal, and idler (2mp =ms +mi ). The pump mode is located at 2308.4 nm (mp =195, blue), 2382.9 nm (mp =187, red), 2431.9 nm (mp =182, green), 2462.3 nm (mp =179, cyan), and 2493.5 nm (mp =176, pink), respectively. The resonant modes are simulated with the finite element method (FEMLAB, COMSOL).

Fig. 5.
Fig. 5.

OPO lasing threshold for the micro-ring resonator used in Fig. 4. The pump wavelength is fixed at 2308.4 nm (dashed line) and critical coupling is assumed at all three waves. The red spot shows the frequency-matched signal and idler modes at 1562.2 and 4419.6 nm, respectively (see Fig. 4). Note that the two kinks at 2103 and 2559 nm are numerical artifacts because of the kink in TPA coefficient at 2.2 µm (see Ref. [16]). In practice, the lasing threshold would change smoothly with signal wavelength.

Equations (34)

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A p z = { i ( β p + β p f ) α lp 2 } A p + i ( γ p P p + 2 γ ps P s + 2 γ pi P i ) A p + 2 i γ pspi A s A i A p * ,
A s z = { i ( β s + β s f ) α ls 2 } A s + i ( γ s P s + 2 γ sp P p + 2 γ si P i ) A s + i γ spip A p 2 A i * ,
A i z = { i ( β i + β i f ) α li 2 } A i + i ( γ i P i + 2 γ ip P p + 2 γ is P s ) A i + i γ ipsp A p 2 A s * ,
γ ijkl = 3 ω i η ijkl χ ( 3 ) ( ω i ; ω j , ω k , ω l ) 4 ε 0 c 2 a ¯ ijkl n i n j n k n l ,
η ijkl F i * F j F k * F l dxdy { Π v = i , j , k , l F v 4 dxdy } 1 4 ,
κ = Δ β + Δ β f + 2 P p Re [ γ sp + γ ip * γ p ] ,
d A p d t = i Δ ω p A p A p 2 τ tp + i ω p n ~ p f n 0 p A p + i ( γ p c U p + 2 γ ps c U s + 2 γ pi c U i ) A p + 2 i γ pspi c A s A i A p * + i A in τ ep ,
d A s d t = i Δ ω s A s A s 2 τ ts + i ω s n ~ s f n 0 s A s + i ( γ s c U s + 2 γ sp c U p + 2 γ si c U i ) A s + i γ spip c A p 2 A i * ,
d A i d t = i Δ ω i A i A i 2 τ ti + i ω i n ~ i f n 0 i A i + i ( γ i c U i + 2 γ ip c U p + 2 γ is c U s ) A i + i γ ipsp c A p 2 A s * ,
γ ijkl c = 3 ω i η ijkl c χ ( 3 ) ( ω i ; ω j , ω k , ω l ) 4 ε 0 n 0 i n 0 j n 0 k n 0 l V ijkl , with η ijkl c si d r ( ε ri ε rj ε rk ε rl ) 1 2 E ~ i * E ~ j E ~ k * E ~ l { Π v = i , j , k , l si d r ε rv 2 E ~ v 4 } 1 4 ,
Ω K 2 U p Re ( γ sp c + γ ip c * γ p c ) ,
Ω f = ω s n fs n 0 s + ω i n fi n 0 i 2 ω p n fp n 0 p ω p 2 n fp ( 1 ω s n 0 s + 1 ω i n 0 i 2 ω p n 0 p ) ,
g 2 = γ spip c γ ipsp c * U p 2 ( Ω 0 + Ω K + Ω f + i Δ ) 2 4 γ spip c γ ipsp c * U p 2 ( Ω 0 + Ω K ) 2 4 ,
4 U p γ spip c γ ipsp c * 2 U p γ spip c γ ipsp c * Ω K < Ω 0 < 2 U p γ spip c γ ipsp c * Ω K 0 .
4 γ spip c γ ipsp c * U p 2 = ( 1 τ ts + c α fs n 0 s + 2 β Tsp c U p ) ( 1 τ ti + c α fi n 0 i + 2 β Tip c U p ) ,
P th = τ ep 4 U p ( 1 τ tp + c α fp n 0 p + β Tpp c U p ) 2 ,
P th = τ ep 8 τ tp 2 ρ + + [ ρ 2 + γ spip c γ ipsp c * ( τ ts τ ti ) ] 1 2 γ spip c γ ipsp c * β Tsp c β Tip c ,
2 E ~ ( r , ω ) + ω 2 c 2 ε r ( r , ω ) E ~ ( r , ω ) = μ 0 ω 2 P ~ ( r , ω ) ,
E ~ ( r , ω ) = j A j ~ ( ω ) E j ~ ( r , ω 0 j ) .
d 2 A i d t 2 + ω 0 i 2 A i = d 2 d t 2 d r E ~ i * ( r , ω 0 i ) · P ( r , t ) ε 0 d r ε r ( r , ω 0 i ) E ~ i ( r , ω 0 i ) 2 .
d A i dt = i ( ω i ω 0 i ) A i A i τ ti + i ω i 2 ε 0 d r E ˜ i * ( r , ω 0 i ) · P ( r , t ) d r ε r ( r , ω 0 i ) E ˜ i ( r , ω 0 i ) 2 ,
P f ( r , t ) = ε 0 j χ f [ ω j , N ( r , t ) ] A j ( t ) e i ω j t E ˜ j ( r , ω 0 j ) ,
P ( 3 ) ( r , t ) = 3 4 ε 0 jkl χ ijkl ( 3 ) A j ( t ) A k * ( t ) A l ( t ) e i ( ω j ω k + ω l ) t E ˜ j ( r , ω 0 j ) E ˜ k * ( r , ω 0k ) E ˜ l ( r , ω 0l ) ,
d A i d t = i ( ω i ω 0 i ) A i A i τ ti + i ω i 2 n 0 i 2 χ f ( ω i , N i - ) A i + i jkl γ ijkl c A j A k * A l ,
V v { d r ε r ( r , ω 0 v ) E ˜ v ( r , ω 0 v ) 2 } 2 si d r ε r 2 ( r , ω 0 v ) E ˜ v ( r , ω 0 v ) 4 ,
N ¯ i ( t ) si d r N ( r , t ) ε r ( r , ω 0 i ) E ˜ i ( r , ω 0 i ) 2 d r ε r ( r , ω 0 i ) E ˜ i ( r , ω 0 i ) 2 .
d N ¯ i dt = G ¯ i N ¯ i τ 0 , with G ¯ i ( t ) si d r G ( r , t ) ε r ( r , ω 0 i ) E ~ i ( r , ω 0 i ) 2 d r ε r ( r , ω 0 i ) E ~ i ( r , ω 0 i ) 2 ,
P u ( 3 ) ( r , t ) = 3 ε 0 4 e i ω u t A u ( t ) E ~ u { χ uuuu ( 3 ) A u ( t ) 2 E ~ u 2 + 2 v , v u χ uvvu ( 3 ) A v ( t ) 2 E ~ v 2 } ,
u { 1 4 E u * ( r , t ) P u ( 3 ) ( r , t ) t + c . c }
= 3 ε 0 8 { u ω u Im ( χ uuuu ( 3 ) ) A u 4 E ~ u 4 + u , v , v u ( ω u + ω v ) Im ( χ uvvu ( 3 ) ) A u 2 A v 2 E ~ u 2 E ~ v 2 } ,
G ( r , t ) = 3 ε 0 8 h ¯ { 1 2 u Im ( χ uuuu ( 3 ) ) A u 4 E ~ u 4 + u , v , v u Im ( χ uvvu ( 3 ) ) A u 2 A v 2 E ~ u 2 E ~ v 2 } .
G ¯ i ( t ) = u c 2 β Tuu η iuu f U u 2 2 h ¯ ω u n 0 u 2 ( V iuu f ) 2 + u , v , u v c 2 β Tuv η iuv f U u U v h ¯ ω u n 0 u n 0 v ( V iuv f ) 2 ,
V v f { [ d r ε r ( r , ω 0 v ) E ~ v ( r , ω 0 v ) 2 ] 3 si d r ε r 3 ( r , ω 0 v ) E ~ v ( r , ω 0 v ) 6 } 1 2 ,
η iuv f si d r ε ri ε ru ε rv E ~ i 2 E ~ u 2 E ~ v 2 { Π j = i , u , v si d r ε rj 3 E ~ j 6 } 1 3 .

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