Abstract

We present a systematic experimental study of the linear and nonlinear optical properties of silicon-germanium (SiGe) waveguides, conducted on samples of varying cross-sectional dimensions and Ge concentrations. The evolution of the various optical properties for waveguide widths in the range 0.3 to 2 µm and Ge concentrations varying between 10 and 30% is considered. Finally, we comment on the comparative performance of the waveguides, when they are considered for nonlinear applications at telecommunications wavelengths.

© 2013 OSA

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2012 (2)

2011 (2)

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

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]

2010 (1)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4(8), 535–544 (2010).
[CrossRef]

2009 (2)

2008 (2)

2007 (2)

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(19), 191104 (2007).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15(25), 16604–16644 (2007).
[CrossRef] [PubMed]

2006 (4)

E. Dulkeith, F. N. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express14(9), 3853–3863 (2006).
[CrossRef] [PubMed]

E. Dulkeith, Y. A. Vlasov, X. G. Chen, N. C. Panoiu, and R. M. Osgood., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express14(12), 5524–5534 (2006).
[CrossRef] [PubMed]

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

H. Garcia and R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz–Keldysh effect in indirect gap semiconductors,” J. Phys. At. Mol. Opt. Phys.39(12), 2737–2746 (2006).
[CrossRef]

2005 (3)

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (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. Express13(12), 4629–4637 (2005).
[CrossRef] [PubMed]

2004 (4)

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express12(5), 829–834 (2004).
[CrossRef] [PubMed]

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express12(21), 5269–5273 (2004).
[CrossRef] [PubMed]

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron.10(5), 1149–1153 (2004).
[CrossRef]

J. P. Douglas, “Si/SiGe heterostructures: from material and physics to devices and circuits,” Semicond. Sci. Technol.19(10), R75–R108 (2004).
[CrossRef]

2002 (2)

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

R. Claps, D. Dimitropoulos, Y. Han, and B. Jalali, “Observation of Raman emission in silicon waveguides at 1.54 µm,” Opt. Express10(22), 1305–1313 (2002).
[CrossRef] [PubMed]

1996 (1)

1993 (1)

G. Tittelbach, B. Richter, and W. Karthe, “Comparison of three transmission methods for integrated optical waveguide propagation loss measurement,” Pure Appl. Opt.2(6), 683–700 (1993).
[CrossRef]

1992 (1)

1991 (1)

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

1989 (2)

K. W. DeLong, K. B. Rochford, and G. I. Stegeman, “Effect of two-photon absorption on all-optical guided-wave devices,” Appl. Phys. Lett.55(18), 1823–1825 (1989).
[CrossRef]

V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett.14(20), 1140–1142 (1989).
[CrossRef] [PubMed]

1987 (1)

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23(7), 1205–1210 (1987).
[CrossRef]

Agrawal, G. P.

Alloatti, L.

Andreadakis, N. C.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Andrejco, M. J.

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

Baets, R.

Batagelj, B.

B. Batagelj, “Conversion efficiency of fiber wavelength converter based on degenerate FWM,” in 2nd International Conference on Transparent Optical Networks (ICTON), (2000), pp. 179–182.
[CrossRef]

Bhat, R.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Biaggio, I.

Bogatscher, S.

Bogris, A.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Boskovic, A.

Boyraz, O.

Braun, R.

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23(7), 1205–1210 (1987).
[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(19), 191104 (2007).
[CrossRef]

Brun, M.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Cassan, E.

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Chaisakul, P.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Chen, X. G.

Chernikov, S. V.

Chrastina, D.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Claps, R.

Cohen, O.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Coudevylle, J. R.

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Coudevylle, J.-R.

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

Delong, K. W.

V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett.14(20), 1140–1142 (1989).
[CrossRef] [PubMed]

K. W. DeLong, K. B. Rochford, and G. I. Stegeman, “Effect of two-photon absorption on all-optical guided-wave devices,” Appl. Phys. Lett.55(18), 1823–1825 (1989).
[CrossRef]

Diederich, F.

Dimitropoulos, D.

Douglas, J. P.

J. P. Douglas, “Si/SiGe heterostructures: from material and physics to devices and circuits,” Semicond. Sci. Technol.19(10), R75–R108 (2004).
[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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

Dulkeith, E.

Dumon, P.

Edmond, S.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Ettabib, M. A.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Fang, A.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Foster, M. A.

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

Freude, W.

Frigerio, J.

Fukuda, H.

Gaeta, A. L.

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

Garcia, H.

H. Garcia and R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz–Keldysh effect in indirect gap semiconductors,” J. Phys. At. Mol. Opt. Phys.39(12), 2737–2746 (2006).
[CrossRef]

Grant, R. S.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Green, W. M. J.

Gruner-Nielsen, L.

Hak, D.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Hammani, K.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Han, Y.

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[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]

Humlícek, J.

Ikonen, E.

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

Indukuri, T.

Isella, G.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Itabashi, S.

Jalali, B.

Johnson, T. J.

Jones, L.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Jones, R.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Kalyanaraman, R.

H. Garcia and R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz–Keldysh effect in indirect gap semiconductors,” J. Phys. At. Mol. Opt. Phys.39(12), 2737–2746 (2006).
[CrossRef]

Kapsalis, A.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Karthe, W.

G. Tittelbach, B. Richter, and W. Karthe, “Comparison of three transmission methods for integrated optical waveguide propagation loss measurement,” Pure Appl. Opt.2(6), 683–700 (1993).
[CrossRef]

Kekoua, M. G.

Khoutsishvili, E. V.

Koos, C.

Koza, M. A.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Labeye, P.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Lamminpaa, A.

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

Le Roux, X.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

LeBlanc, H. P.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Leuthold, J.

Levring, O. A.

Liang, T. K.

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron.10(5), 1149–1153 (2004).
[CrossRef]

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

Lin, Q.

Lipson, M.

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

Liu, A. S.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Liu, Y.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol.23(6), 064007 (2008).
[CrossRef]

Lockwood, D. J.

L. Tsybeskov and D. J. Lockwood, “Silicon-Germanium nanostructures for light emitters and on-chip optical interconnects,” Proc. IEEE97(7), 1284–1303 (2009).
[CrossRef]

Ludvigsen, H.

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

Marris-Morini, D.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Marti, J.

S. Mas, J. Matres, J. Marti, and C. J. Oton, “Accurate chromatic dispersion characterization of photonic integrated circuits,” IEEE Photon. J.4(3), 825–831 (2012).
[CrossRef]

Marttila, P.

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

Mas, S.

S. Mas, J. Matres, J. Marti, and C. J. Oton, “Accurate chromatic dispersion characterization of photonic integrated circuits,” IEEE Photon. J.4(3), 825–831 (2012).
[CrossRef]

Matres, J.

S. Mas, J. Matres, J. Marti, and C. J. Oton, “Accurate chromatic dispersion characterization of photonic integrated circuits,” IEEE Photon. J.4(3), 825–831 (2012).
[CrossRef]

Michael, C. P.

Mizrahi, V.

Nicolaescu, R.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Nicoletti, S.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Niemi, T.

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

Osgood, R. M.

Oton, C. J.

S. Mas, J. Matres, J. Marti, and C. J. Oton, “Accurate chromatic dispersion characterization of photonic integrated circuits,” IEEE Photon. J.4(3), 825–831 (2012).
[CrossRef]

Painter, O. J.

Paniccia, M.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Panoiu, N. C.

Parmigiani, F.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Penty, R. V.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Perahia, R.

Petropoulos, P.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Richter, B.

G. Tittelbach, B. Richter, and W. Karthe, “Comparison of three transmission methods for integrated optical waveguide propagation loss measurement,” Pure Appl. Opt.2(6), 683–700 (1993).
[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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

Rochford, K. B.

K. W. DeLong, K. B. Rochford, and G. I. Stegeman, “Effect of two-photon absorption on all-optical guided-wave devices,” Appl. Phys. Lett.55(18), 1823–1825 (1989).
[CrossRef]

Rong, H. S.

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Röseler, A.

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(19), 191104 (2007).
[CrossRef]

Rouifed, M. S.

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Rouifed, M.-S.

Saifi, M. A.

Schares, L.

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

Scimeca, M. L.

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

Shibata, N.

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23(7), 1205–1210 (1987).
[CrossRef]

Shoji, T.

Sibbett, W.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Soole, J. B. D.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[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]

Stegeman, G. I.

V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett.14(20), 1140–1142 (1989).
[CrossRef] [PubMed]

K. W. DeLong, K. B. Rochford, and G. I. Stegeman, “Effect of two-photon absorption on all-optical guided-wave devices,” Appl. Phys. Lett.55(18), 1823–1825 (1989).
[CrossRef]

Syvridis, D.

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

Takahashi, J.

Takahashi, M.

Taylor, J. R.

Tittelbach, G.

G. Tittelbach, B. Richter, and W. Karthe, “Comparison of three transmission methods for integrated optical waveguide propagation loss measurement,” Pure Appl. Opt.2(6), 683–700 (1993).
[CrossRef]

Tsang, H. K.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol.23(6), 064007 (2008).
[CrossRef]

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron.10(5), 1149–1153 (2004).
[CrossRef]

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Tsuchizawa, T.

Tsybeskov, L.

L. Tsybeskov and D. J. Lockwood, “Silicon-Germanium nanostructures for light emitters and on-chip optical interconnects,” Proc. IEEE97(7), 1284–1303 (2009).
[CrossRef]

Turner, A. C.

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

Vallaitis, T.

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(19), 191104 (2007).
[CrossRef]

Vivien, L.

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

Vlasov, Y. A.

Waarts, R.

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23(7), 1205–1210 (1987).
[CrossRef]

Watanabe, T.

White, I. H.

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

Wong, C. S.

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 mu m wavelength,” Appl. Phys. Lett.80(3), 416–418 (2002).
[CrossRef]

Xia, F. N.

Yamada, K.

Zettler, T.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

K. W. DeLong, K. B. Rochford, and G. I. Stegeman, “Effect of two-photon absorption on all-optical guided-wave devices,” Appl. Phys. Lett.55(18), 1823–1825 (1989).
[CrossRef]

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 mu m wavelength,” Appl. Phys. Lett.80(3), 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 nm,” Appl. Phys. Lett.90(19), 191104 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23(7), 1205–1210 (1987).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron.10(5), 1149–1153 (2004).
[CrossRef]

IEEE Photon. J. (1)

S. Mas, J. Matres, J. Marti, and C. J. Oton, “Accurate chromatic dispersion characterization of photonic integrated circuits,” IEEE Photon. J.4(3), 825–831 (2012).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. Chaisakul, D. Marris-Morini, G. Isella, D. Chrastina, M. S. Rouifed, X. Le Roux, S. Edmond, E. Cassan, J. R. Coudevylle, and L. Vivien, “10-Gb/s Ge/SiGe multiple quantum-well waveguide photodetector,” IEEE Photon. Technol. Lett.23(20), 1430–1432 (2011).
[CrossRef]

J. Appl. Phys. (2)

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

H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Two-photon absorption and self-phase modulation in InGaAsP/InP multi-quantum-well waveguides,” J. Appl. Phys.70(7), 3992–3994 (1991).
[CrossRef]

J. Phys. At. Mol. Opt. Phys. (1)

H. Garcia and R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz–Keldysh effect in indirect gap semiconductors,” J. Phys. At. Mol. Opt. Phys.39(12), 2737–2746 (2006).
[CrossRef]

Nat. Photonics (1)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4(8), 535–544 (2010).
[CrossRef]

Nature (2)

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,” Nature441(7096), 960–963 (2006).
[CrossRef] [PubMed]

H. S. Rong, A. S. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433(7023), 292–294 (2005).
[CrossRef] [PubMed]

Opt. Express (11)

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion of 40 Gbaud PSK signals in a silicon germanium waveguide,” Opt. Express (to be published).

R. Claps, D. Dimitropoulos, Y. Han, and B. Jalali, “Observation of Raman emission in silicon waveguides at 1.54 µm,” Opt. Express10(22), 1305–1313 (2002).
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O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express12(5), 829–834 (2004).
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O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express12(21), 5269–5273 (2004).
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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. Express13(12), 4629–4637 (2005).
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E. Dulkeith, F. N. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express14(9), 3853–3863 (2006).
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E. Dulkeith, Y. A. Vlasov, X. G. Chen, N. C. Panoiu, and R. M. Osgood., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express14(12), 5524–5534 (2006).
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Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15(25), 16604–16644 (2007).
[CrossRef] [PubMed]

Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express16(14), 10596–10610 (2008).
[CrossRef] [PubMed]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon-organic hybrid (SOH) waveguide geometries,” Opt. Express17(20), 17357–17368 (2009).
[CrossRef] [PubMed]

P. Chaisakul, D. Marris-Morini, M.-S. Rouifed, G. Isella, D. Chrastina, J. Frigerio, X. Le Roux, S. Edmond, J.-R. Coudevylle, and L. Vivien, “23 GHz Ge/SiGe multiple quantum well electro-absorption modulator,” Opt. Express20(3), 3219–3224 (2012).
[CrossRef] [PubMed]

Opt. Fiber Technol. (1)

A. Lamminpaa, T. Niemi, E. Ikonen, P. Marttila, and H. Ludvigsen, “Effects of dispersion on nonlinearity measurement of optical fibers,” Opt. Fiber Technol.11(3), 278–285 (2005).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

L. Tsybeskov and D. J. Lockwood, “Silicon-Germanium nanostructures for light emitters and on-chip optical interconnects,” Proc. IEEE97(7), 1284–1303 (2009).
[CrossRef]

Pure Appl. Opt. (1)

G. Tittelbach, B. Richter, and W. Karthe, “Comparison of three transmission methods for integrated optical waveguide propagation loss measurement,” Pure Appl. Opt.2(6), 683–700 (1993).
[CrossRef]

Semicond. Sci. Technol. (2)

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol.23(6), 064007 (2008).
[CrossRef]

J. P. Douglas, “Si/SiGe heterostructures: from material and physics to devices and circuits,” Semicond. Sci. Technol.19(10), R75–R108 (2004).
[CrossRef]

Other (2)

B. Batagelj, “Conversion efficiency of fiber wavelength converter based on degenerate FWM,” in 2nd International Conference on Transparent Optical Networks (ICTON), (2000), pp. 179–182.
[CrossRef]

M. A. Ettabib, K. Hammani, F. Parmigiani, L. Jones, A. Kapsalis, A. Bogris, D. Syvridis, M. Brun, P. Labeye, S. Nicoletti, and P. Petropoulos, “FWM-based wavelength conversion in a silicon germanium waveguide,” in OFC/NFOEC (Anaheim, CA, USA, 2013), p. OTh1C.4.

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Figures (9)

Fig. 1
Fig. 1

(a) Scanning Electron Microscopy (SEM) image of a SiGe waveguide, before and after (inset) encapsulation. (b) Sketch of SiGe waveguides before encapsulation.

Fig. 2
Fig. 2

Field of the TM fundamental mode for a waveguide with width 1μm and height 1.4μm

Fig. 3
Fig. 3

Dispersion curves for (a) Ge concentration x = 0.2 and different waveguide width values and, (b) for width (W) equal to 1 μm and different Ge concentrations.

Fig. 4
Fig. 4

(a) Experimental setup for loss measurement. PC = polarization controller, LF = Lensed Fiber, MO = Microscope Objective, BS = Beam Splitter, IRC = Infrared Camera, PM = Power meter. (b) Spectral fringes and associated FP fit for W = 1 µm at x = 0.2 (b1), and at x = 0.3 (b2).

Fig. 5
Fig. 5

(a) Experimental setup for the TPA measurement. VOA = variable optical attenuator (b) Inverse transmission versus coupled input peak power for x = 0.1 (red circles), x = 0.2 (green triangle) and x = 0.3 (blue square). The solid lines correspond to a linear fit.

Fig. 6
Fig. 6

(a) Experimental setup for nonlinearity measurement. EDFA = Erbium Doped Fiber Amplifier, OSA = Optical Spectrum Analyzer. (b) Typical spectrum where the conversion efficiency can be read (c) Typical plots of the Nonlinear Phase shift vs. the signal power for W = 1 µm and either x = 0.1 (red circles) or x = 0.2 (green triangles). The dotted-dashed lines correspond to linear fits.

Fig. 7
Fig. 7

Nonlinear parameter γ as a function of the waveguide width for x = 0.1 (red circles), 0.2 (green triangles) and 0.3 (blue squares). The experimental results (symbols) are compared with numerical simulations in dotted (x = 0.1), dotted-dashed (x = 0.2) and dashed (x = 0.3) lines.

Fig. 8
Fig. 8

Conversion efficiency as a function of wavelength detuning from 1550 nm for (a) W = 1.5 µm and x = 0.1 (red circles) (b) W = 1.0 µm and x = 0.2 (green triangles) and (c) W = 0.6 µm and x = 0.3 (blue squares). The solid lines correspond to numerical simulations. The input power was close to 295 mW.

Fig. 9
Fig. 9

Figure of Merit as a function of the waveguide width for x = 0.1 (red circles), 0.2 (green triangles) and 0.3 (blue squares). The solid lines, corresponding to cubic polynomial fits, are only to guide the eye.

Tables (3)

Tables Icon

Table 1 Effective mode area, effective mode index and confinement factor for all waveguides studied.

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Table 2 Loss in dB/cm for the SiGe waveguides depending on width and concentration in Ge.

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Table 3 TPA values in cm/GW for the SiGe waveguides depending on concentration in Ge.

Equations (9)

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A P z ={ i( β p + β p f ) α p 2 } A P + i( γ p P p +2 γ ps P s +2 γ pi P i ) A P +2i γ pspi A s A i A p * ,
A s z ={ i( β s + β s f ) α s 2 } A s + i( γ s P s +2 γ sp P p +2 γ si P i ) A s +i γ spip A i * A p 2 ,
A i z ={ i( β i + β i f ) α i 2 } A i + i( γ i P i +2 γ ip P p +2 γ is P s ) A i +i γ ipsp A s * A p 2 ,
A eff = [ | E ~ | 2 dxdy ] 2 | E ~ | 4 dxdy ,
Γ xy = W/2 W/2 H/2 H/2 | E( x,y ) | 2 dxdy xy | E( x,y ) | 2 dxdy .
αL=ln( R 1+ P min / P max 1 P min / P max ),
P in P out = 1 T =exp( αL ) β TPA A eff L eff P in +exp( αL ).
φ SPM =2γ L eff P,
[ C E linear ] 1 = J 0 2 ( φ SPM /2 )+ J 1 2 ( φ SPM /2 ) J 1 2 ( φ SPM /2 )+ J 2 2 ( φ SPM /2 ) ,

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