Abstract

The nonlinear process of stimulated Raman scattering is important for silicon photonics as it enables optical amplification and lasing. To understand the dynamics of silicon Raman amplifiers (SRAs), a numerical approach is generally employed, even though it provides little insight into the contribution of different SRA parameters to the signal amplification process. In this paper, we solve the coupled pump-signal equations analytically under realistic conditions, and derive an exact formula for the envelope of a signal pulse when picosecond optical pulses are amplified inside a SRA pumped by a continuous-wave laser beam. Our solution is valid for an arbitrary pulse shape and fully accounts for the Raman gain-dispersion effects, including temporal broadening and group-velocity reduction (a slow-light effect). It can be applied to any pumping scenario and leads to a simple analytic expression for the maximum optical delay produced by the Raman dispersion in a unidirectionally pumped SRA. We employ our analytical formulation to study the evolution of optical pulses with Gaussian, exponential, and Lorentzian shapes. The ability of a Gaussian pulse to maintain its shape through the amplifier makes it possible to realize soliton-like propagation of chirped Gaussian pulses in SRAs. We obtain analytical expressions for the required linear chirp and temporal width of a soliton-like pulse in terms of the net signal gain and the Raman-dispersion parameter. Our results are useful for optimizing the performance of SRAs and for engineering controllable signal delays.

© 2010 Optical Society of America

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    [PubMed]

2010 (3)

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Optimization of Raman amplification in silicon waveguides with finite facet reflectivities,” IEEE J. Sel. Top. Quantum Electron. 16, 226–233 (2010).
[CrossRef]

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]

I. D. Rukhlenko, I. Udagedara, M. Premaratne, and G. P. Agrawal, “Effect of free carriers on pump-to-signal noise transfer in silicon Raman amplifiers,” Opt. Lett. 35, 2343–2345 (2010).
[CrossRef] [PubMed]

2009 (6)

2008 (2)

2007 (6)

2006 (8)

Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14, 2317–2322 (2006).
[CrossRef] [PubMed]

L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006).
[CrossRef] [PubMed]

E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
[CrossRef] [PubMed]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. F¨orst, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
[CrossRef] [PubMed]

M. Krause, H. Renner, and E. Brinkmeyer, “Efficient Raman lasing in tapered silicon waveguides,” Spectroscopy 21, 26–32 (2006).

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[CrossRef]

2005 (5)

2004 (7)

2003 (3)

2002 (2)

R. Claps, D. Dimitropoulos, and B. Jalali, “Stimulated Raman scattering in silicon waveguides,” Electron. Lett. 38, 1352–1354 (2002).
[CrossRef]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of Raman emission in silicon waveguides at 1.54 μm,” Opt. Express 10, 1305–1313 (2002).
[PubMed]

Agrawal, G. P.

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Optimization of Raman amplification in silicon waveguides with finite facet reflectivities,” IEEE J. Sel. Top. Quantum Electron. 16, 226–233 (2010).
[CrossRef]

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]

I. D. Rukhlenko, I. Udagedara, M. Premaratne, and G. P. Agrawal, “Effect of free carriers on pump-to-signal noise transfer in silicon Raman amplifiers,” Opt. Lett. 35, 2343–2345 (2010).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2009).
[CrossRef]

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Raman amplification of optical pulses in silicon waveguides: Effects of finite gain bandwidth, pulse width, and chirp,” J. Opt. Soc. Am. B 26, 17–25 (2009).
[CrossRef]

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34, 476–478 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Continuous-wave Raman amplification in silicon waveguides: Beyond the undepleted pump approximation,” Opt. Lett. 34, 536–538 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express 17, 5807–5814 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
[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]

L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
[CrossRef] [PubMed]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007).
[CrossRef] [PubMed]

L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006).
[CrossRef] [PubMed]

Baets, R.

Benton, C.

Bhadra, S. K.

Boyd, R. W.

Boyraz, O.

Brinkmeyer, E.

M. Krause, H. Renner, and E. Brinkmeyer, “Efficient Raman lasing in tapered silicon waveguides,” Spectroscopy 21, 26–32 (2006).

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[CrossRef] [PubMed]

Chen, X.

Chou, C. Y.

Claps, R.

Cohen, O.

Dadap, J.

Dadap, J. I.

De La Rue, R. M.

Dekker, R.

Dimitropoulos, D.

Ding, W.

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]

Dissanayake, C.

Driessen, A.

Dulkeith, E.

Dumon, P.

Espinola, R.

F¨orst, M.

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Pannicia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Fang, A. W.

Fauchet, P. M.

Foster, M. A.

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]

Gnan, M.

Gorbach, A. V.

Green, W. M.

Hak, D.

Han, Y.

Houshmand, B.

Hsieh, I.-W.

Indukuri, T.

Jalali, B.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

O. Boyraz, and B. Jalali, “Demonstration of directly modulated silicon Raman laser,” Opt. Express 13, 796–800 (2005).
[CrossRef] [PubMed]

O. Boyraz, and B. Jalali, “Demonstration of 11 dB fiber-to-fiber gain in a silicon Raman amplifier,” IEICE Electron. Express 1, 429–434 (2004).
[CrossRef]

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004).
[CrossRef] [PubMed]

O. Boyraz, and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
[CrossRef] [PubMed]

D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of Raman effect in silicon-on insulator waveguides,” Opt. Lett. 28, 1954–1956 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of Raman emission in silicon waveguides at 1.54 μm,” Opt. Express 10, 1305–1313 (2002).
[PubMed]

R. Claps, D. Dimitropoulos, and B. Jalali, “Stimulated Raman scattering in silicon waveguides,” Electron. Lett. 38, 1352–1354 (2002).
[CrossRef]

Jones, R.

Kawanishi, T.

Knight, J. C.

Krause, M.

M. Krause, H. Renner, and E. Brinkmeyer, “Efficient Raman lasing in tapered silicon waveguides,” Spectroscopy 21, 26–32 (2006).

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[CrossRef] [PubMed]

Leonardis, F. D.

V. M. N. Passaro, and F. D. Leonardis, “Solitons in SOI optical waveguides,” Adv. Stud. Theor. Phys. 2, 769–785 (2008).

Liang, T.

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, 1149–1153 (2004).
[CrossRef]

Lin, Q.

Lipson, M.

Liu, A.

Liu, X.

McNab, S. J.

Moormann, C.

Niehusmann, J.

Nunes, L.

Okawachi, Y.

Osgood, R.

Osgood, R. M.

Painter, O. J.

Paniccia, M.

Paniccia, M. J.

Pannicia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Pannicia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Panoiu, N. C.

Passaro, V. M. N.

V. M. N. Passaro, and F. D. Leonardis, “Solitons in SOI optical waveguides,” Adv. Stud. Theor. Phys. 2, 769–785 (2008).

Piredda, G.

Premaratne, M.

Priem, G.

Qian, F.

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]

Raghunathan, V.

Renner, H.

M. Krause, H. Renner, and E. Brinkmeyer, “Efficient Raman lasing in tapered silicon waveguides,” Spectroscopy 21, 26–32 (2006).

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[CrossRef] [PubMed]

Rong, H.

Roy, S.

Rukhlenko, I. D.

Sakamoto, T.

Sasagawa, K.

Sharping, J. E.

Skryabin, D. V.

Soref, R. A.

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

Sorrel, M.

Thourhout, D. V.

Tien, E. K.

Tsang, H.

Tsang, H. 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, 1149–1153 (2004).
[CrossRef]

Tsuchiya, M.

Udagedara, I.

Vlasov, Y. A.

Wadsworth, W. J.

Wahlbrink, T.

Xia, F.

Xu, Q.

Yin, L.

Yuksek, N. S.

Zhang, J.

Adv. Stud. Theor. Phys. (1)

V. M. N. Passaro, and F. D. Leonardis, “Solitons in SOI optical waveguides,” Adv. Stud. Theor. Phys. 2, 769–785 (2008).

Appl. Phys. Lett. (1)

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

Electron. Lett. (1)

R. Claps, D. Dimitropoulos, and B. Jalali, “Stimulated Raman scattering in silicon waveguides,” Electron. Lett. 38, 1352–1354 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (5)

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Optimization of Raman amplification in silicon waveguides with finite facet reflectivities,” IEEE J. Sel. Top. Quantum Electron. 16, 226–233 (2010).
[CrossRef]

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]

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, 1149–1153 (2004).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

IEICE Electron. Express (1)

O. Boyraz, and B. Jalali, “Demonstration of 11 dB fiber-to-fiber gain in a silicon Raman amplifier,” IEICE Electron. Express 1, 429–434 (2004).
[CrossRef]

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

Nature (1)

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Pannicia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[CrossRef] [PubMed]

Opt. Express (22)

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004).
[CrossRef] [PubMed]

R. Espinola, J. Dadap, R. Osgood, S. J. McNab, and Y. A. Vlasov, “Raman amplification in ultrasmall siliconon-insulator wire waveguides,” Opt. Express 12, 3713–3718 (2004).
[CrossRef] [PubMed]

A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004).
[CrossRef] [PubMed]

O. Boyraz, and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
[CrossRef] [PubMed]

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[CrossRef] [PubMed]

R. Jones, H. Rong, A. Liu, A. W. Fang, M. J. Paniccia, D. Hak, and O. Cohen, “Net continuous-wave optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 13, 519–525 (2005).
[CrossRef] [PubMed]

O. Boyraz, and B. Jalali, “Demonstration of directly modulated silicon Raman laser,” Opt. Express 13, 796–800 (2005).
[CrossRef] [PubMed]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716–1723 (2005).
[CrossRef] [PubMed]

T. Liang, L. Nunes, T. Sakamoto, K. Sasagawa, T. Kawanishi, M. Tsuchiya, G. Priem, D. V. Thourhout, P. Dumon, R. Baets, and H. Tsang, “Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides,” Opt. Express 13, 7298–7303 (2005).
[CrossRef] [PubMed]

Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14, 2317–2322 (2006).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of Raman emission in silicon waveguides at 1.54 μm,” Opt. Express 10, 1305–1313 (2002).
[PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003).
[CrossRef] [PubMed]

E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14, 5524–5534 (2006).
[CrossRef] [PubMed]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. F¨orst, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
[CrossRef] [PubMed]

I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M. Osgood, Jr., “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 1135–1146 (2007).
[CrossRef] [PubMed]

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

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007).
[CrossRef] [PubMed]

I.-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C. Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, and R. M. Osgood, Jr., “Supercontinuum generation in silicon photonic wires,” Opt. Express 15, 15242–15249 (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]

W. Ding, C. Benton, A. V. Gorbach, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. Gnan, M. Sorrel, and R. M. De La Rue, “Solitons and spectral broadening in long silicon-on-insulator photonic wires,” Opt. Express 16, 3310–3319 (2008).
[CrossRef] [PubMed]

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express 17, 5807–5814 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
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Opt. Lett. (7)

Spectroscopy (1)

M. Krause, H. Renner, and E. Brinkmeyer, “Efficient Raman lasing in tapered silicon waveguides,” Spectroscopy 21, 26–32 (2006).

Other (7)

H. Renner, M. Krause, and E. Brinkmeyer, “Maximal gain and optimal taper design for Raman amplifiers in silicon-on-insulator waveguides,” in Integrated Photonics Research and Applications Topical Meetings (IPRA, 2005), paper JWA3.

M. Krause, H. Renner, and E. Brinkmeyer, “Efficiency increase of silicon-on-insulator Raman lasers by reduction of free-carrier absorption in tapered waveguides,” in Conference on Lasers and Electro-Optics (CLEO 2005), paper CThB1.

A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, (Chapman & Hall/CRC, Boca Raton, 2002).

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

B. Jalali, O. Boyraz, V. Raghunathan, D. Dimitropoulos, and P. Koonath, “Silicon Raman amplifiers, lasers and their applications,” in Active and Passive Optical Components for WDM Communications V, A. K. Dutta, Y. Ohishi, N. K. Dutta, and J. Moerk, Eds., Proc. SPIE 6014, 21–26 (2005).

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[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Nonlinear reduction factor (NRF) as a function of input power for typical silicon waveguides compared to optical fibers. As the free-carrier lifetime is reduced, NRF increases and approaches the limit determined by TPA (top curve). The right scale shows the enhancement of SRS efficiency in silicon waveguides over silica fibers, obtained by multiplying NRF with the ratio of the Raman gain coefficients for silicon and silica (taken to be 2000).

Fig. 2.
Fig. 2.

Effective length of bidirectionally pumped, 1-cm-long SRA (left panel) and integral intensity (left and right panels) for different contributions of the forward pumping. The green curve on the left panel shows, for comparison, the case of a SRA without nonlinear losses. In the calculations, we used λp = 1434 nm, αp = 1 dB/cm, and τc = 1 ns.

Fig. 3.
Fig. 3.

Evolution of maximum signal gain, G = exp[ψ′(L)], in two bidirectionally pumped SRAs for different contributions of forward pumping. The parameters’ values are: λp = 1.434 µm, λs = 1.55 µm, αp = αs = 1 dB/cm, βp = βs = 0.5 cm/GW, and τc = 1 ns.

Fig. 4.
Fig. 4.

Evolution of three 10-ps pulses with different temporal envelopes (left panel) and relative energies of the pulses as a function of pump intensity for different values of τ 0 (right panel); W 0 = 1 fJ and I 0 = 1 GW/cm2 in the left panel; η = 100% and τc = 1 ns for both panels. Other parameters are given in the text.

Fig. 5.
Fig. 5.

Soliton-like evolution of a Gaussian pulse in (a) forward- and (b) backward-pumped, 1-cm-long SRA. (c) Input FWHM of the soliton-like pulse for 0.2-, 1-, and 5-cm-long SRAs. (d) Evolution of pulse energy as the amount of forward pumping is varied from 100% to zero. I 0 = 1 GW/cm2 in panels (a), (b), and (d); W 0 = 1 fJ and τc = 1 ns for all panels. Other parameters are given in the text.

Equations (29)

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𝓘 Si = { ( 4 π h ¯ c σ τ β λ α Si ) 1 2 tan 1 [ ( σ τ β λ 4 π h ¯ c α Si ) 1 2 P A Si ] in the weak TPA regime , 1 β ln ( 1 + β α Si P A Si ) in the weak FCA regime ,
1 A ˜ s A ˜ s z i β 1 s ω i β 2 s ω 2 2 = α s 2 ( β s + 2 i γ s ) I p ( z ) ( ξ r 2 + i ξ i ) I p 2 ( z ) + G R ( ω ) I p ( z ) ,
G R ( ω ) = g R 2 ω s ω p Ω R Ω 0 [ γ R γ R i ( ω Ω ps + Ω 0 ) γ R γ R i ( ω Ω ps Ω 0 ) ] ,
A s z + [ β 1 s + 1 2 g ̂ R T R I p ( z ) ] A s t + 1 2 [ i β 2 s + g ̂ R T R 2 I p ( z ) ] 2 A s t 2
= [ α s 2 + ( g ̂ R 2 β s 2 + 2 i γ s ) I p ( z ) ( ξ r 2 + i ξ i ) I p 2 ( z ) ] A s .
± 1 I p ( z ) d I p ( z ) dz = α p β p I p ( z ) ξ p I p 2 ( z ) ,
I p ± ( z ) = I 0 exp [ α p ( z δ ± ) ] 1 + I 0 2 ( ξ p α p ) { 1 exp [ 2 α p ( z δ ± ) ] } ,
L eff ( z ) = 1 I 0 0 z I p ( z ) d z = ± tan 1 [ I p ( 0 ) ξ p α p ] tan 1 [ I p ( z ) ξ p α p ] I 0 α p ξ p .
ζ ( z , t ) = t β 1 s z 1 2 g ̂ R T R 0 z I p ( z ) d z = τ ( z , t ) 1 2 g ̂ R T R I 0 L eff ( z ) ,
A s z + 𝓑 ( z ) 2 2 A s ζ 2 = 𝒢 ( z ) 2 A s ,
A s ( z , ζ ) = a [ χ ( z ) , ζ ] exp [ ψ ( z ) 2 ] ,
χ ( z ) = 0 z 𝓑 ( z ) d z = g ̂ R T R 2 I 0 L eff ( z ) + i β 2 s z ,
ψ ( z ) = 0 z 𝒢 ( z ) dz = α s z + ( g ̂ R 2 β s + 4 i γ s ) I 0 L eff ( z ) ( 1 + 2 i ξ i ξ r ) Q ( z ) ,
Q ( z ) = ξ r 0 z I p 2 ( z ) dz = ω p 2 ω s 2 ( ± ln I p ( 0 ) I p ( z ) α p z ) .
a ( χ , τ ) = 1 2 π χ + f ( τ + u ) exp ( u 2 2 χ ) d u .
W ( z ) = A eff exp [ ψ ( z ) ] + a [ 2 χ ( z ) , τ ] f ( τ ) d τ .
a 1 ( χ , τ ) = ϖ 1 π 1 4 τ 1 τ 1 2 + χ exp ( τ 2 2 ( τ 1 2 + χ ) ) ,
a 2 ( χ , τ ) = ϖ 2 2 2 [ erfc ( χ + 2 τ 2 τ 2 τ 2 2 χ ) e τ ( 2 τ 2 ) + erfc ( χ 2 τ 2 τ 2 τ 2 2 χ ) e τ ( 2 τ 2 ) ] e χ ( 8 τ 2 2 ) ,
a 3 ( χ , τ ) = ϖ 3 τ 3 2 χ [ erfc ( τ 3 + i τ 2 χ ) e ( τ 3 + i τ ) 2 ( 2 χ ) + erfc ( τ 3 i τ 2 χ ) e ( τ 3 i τ ) 2 ( 2 χ ) ] ,
W 1 ( z ) = W 0 τ 1 τ 1 2 + χ exp ψ , W 2 ( z ) = W 0 erfc ( χ 2 τ 2 ) exp ( χ 4 τ 2 2 + ψ ) ,
W 3 ( z ) = W 0 τ 3 2 π χ exp ψ + erfc ( τ 3 + i τ 2 χ ) exp [ ( τ 3 + i τ ) 2 ( 4 χ ) ] τ 2 + τ 3 2 d τ .
I ( z , ζ ) = A s ( z , ζ ) 2 = ϖ 1 2 π τ 1 2 ( τ 1 2 + χ ) 2 + ( χ ) 2 exp ( ( τ 1 2 + χ ) ζ 2 ( τ 1 2 + χ ) 2 ( χ ) 2 + ψ ) ,
ϕ NL ( z , ζ ) = Arg [ A s ( z , ζ ) ] = 𝕾 { 1 2 [ χ ζ 2 ( τ 1 2 + χ ) 2 + ( χ ) 2 tan 1 ( χ τ 1 2 + χ ) + ψ ] } ,
( σ 2 1 ) χ τ 1 2 2 σ χ τ 1 2 = ( 1 + σ 2 ) χ 2 .
τ 1 2 ( G 2 1 ) = ( 1 + σ 2 ) χ .
σ = χ ± χ G χ ± G ,
τ 1 = χ χ ( G 2 ± 2 ( χ χ ) G + 1 G 2 1 ) 1 2 χ coth ψ ,
𝓘 ( χ ) = 1 2 π χ + d q f ( q ) + d N [ f ( q N ) ] * exp ( N 2 2 χ ) + d u exp ( χ u 2 χ 2 N u χ )
= 1 4 π χ + d q f ( q ) + d N [ f ( q N ) ] * exp ( N 2 4 χ ) = + d q f ( q ) [ a ( 2 χ , q ) ] * .

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