Abstract

An explicit analytical solution for the asymmetric attenuation of optical pulses by self-produced free carriers in silicon waveguides is derived. It allows us to quantify the pulse distortion and to calculate explicitly the free-carrier density and the nonlinear phase shifts caused by the Kerr effect and by free-carrier refraction. We show that omitting two-photon absorption (TPA) as a cause of attenuation and accounting only for free-carrier absorption (FCA) as done in the derivation appropriately models the pulse propagation in short or highly lossy silicon-based waveguides such as plasmonic waveguides with particular use for high-energy input pulses. Moreover, this formulation is also aimed at serving as a tool in discussing the role of FCA in its competition with TPA when used for continuum generation or pulse compression in low-loss silicon waveguides. We show that sech-shaped intensity pulses maintain their shape independently of the intensity or pulse width and self-induced FCA may act as an ideal limiter on them. Pulse propagation under self-induced free-carrier absorption exhibits some features of superluminal propagation such as fast or even backward travelling. We find that input pulses need to have a sufficiently steep front slope to be compressible at all and illustrate this with the FCA-induced pulse broadening for Lorentzian-shaped input pulses.

© 2012 OSA

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2012

2011

2010

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

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

2009

2008

J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express16, 1280–1299 (2008).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express16, 1300–1320 (2008).
[CrossRef] [PubMed]

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett.93, 091114 (2008).
[CrossRef]

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

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun.281, 5889–5893 (2008).
[CrossRef]

2007

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett.90, 211105 (2007).
[CrossRef]

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys.40, R249–R271 (2007).
[CrossRef]

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007).
[CrossRef]

L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett.32, 2031–2033 (2007).
[CrossRef] [PubMed]

N. Suzuki, “FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides,” J. Lightwave Technol.25, 2495–2501 (2007).
[CrossRef]

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

2006

2004

Abedin, K. S.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Afshar, S. V.

Agrawal, G. P.

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Heidelberg, 2005).
[CrossRef]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Heidelberg, 2005).
[CrossRef]

Badding, J.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Baets, R.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Beausoleil, R. G.

Bhadra, S. K.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun.281, 5889–5893 (2008).
[CrossRef]

Bogaerts, W.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Boyd, R. W.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074–1077 (2009).
[CrossRef] [PubMed]

Boyraz,

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

Boyraz, O.

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007).
[CrossRef]

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

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007.
[CrossRef]

Brinkmeyer, E.

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 Meeting, San Diego, California, April 11–13. Joint IPRA/NPIS Oral Session: Frontiers in Nanophotonics (paper JWA3), (2005).

Brouckaert, J.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

Chen, X.

Chen, X. G.

Chou, C. Y.

Claps, R.

Cohen, O.

Dadap, J. I.

Day, T.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

De Vos, K.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

Dekker, R.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys.40, R249–R271 (2007).
[CrossRef]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μ femtosecond pulses,” Opt. Express14, 8336–8346 (2006).
[CrossRef] [PubMed]

DeVore, P. T. S.

P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012).
[CrossRef]

Dimitropoulos, D.

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

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express12, 2774–2780 (2004).
[CrossRef] [PubMed]

Dissanayake, C.

Driessen, A.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys.40, R249–R271 (2007).
[CrossRef]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μ femtosecond pulses,” Opt. Express14, 8336–8346 (2006).
[CrossRef] [PubMed]

Dulkeith, E.

Dumon, P.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Först, M.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys.40, R249–R271 (2007).
[CrossRef]

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μ femtosecond pulses,” Opt. Express14, 8336–8346 (2006).
[CrossRef] [PubMed]

Foster, M. A.

Gaeta, A. L.

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074–1077 (2009).
[CrossRef] [PubMed]

Gradstein, I. S.

I. S. Gradstein and I. M. Ryshik, Table of Integrals, Series and Products (Academic Press, Boston, 1994) p. 104.

Green, W. M. J.

Hak, D.

Healy, N.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Hsieh, I. W.

Hsieh, I.-W.

Ippen, E. P.

Jalali, B.

P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012).
[CrossRef]

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett.93, 091114 (2008).
[CrossRef]

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

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express12, 2774–2780 (2004).
[CrossRef] [PubMed]

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

Kamke, E.

E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen: Gewöhnliche Differentialgleichungen I. (B. G. Teubner, Stuttgart, 1983), p. 298, Eq. (1.34).

Kärtner, F. X.

Khilo, A.

Knights, A. P.

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
[CrossRef]

Koonath, P.

P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012).
[CrossRef]

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett.93, 091114 (2008).
[CrossRef]

Krause, M.

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 Meeting, San Diego, California, April 11–13. Joint IPRA/NPIS Oral Session: Frontiers in Nanophotonics (paper JWA3), (2005).

H. Renner and M. Krause, “Maximal total gain of non-tapered silicon-on-insulator Raman amplifiers,” in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, OSA Technical Digest Series(CD) (Optical Society of America, 2006), paper OMD2.

Liang, T. K.

T. K. Liang and H. K. Tsang; “Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett.84, 2745–2747 (2004).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

T. K. Liang and H. K. Tsang, “Pulsed-pumped silicon-on-insulator waveguide Raman amplifier,” in Proceedings of International Conference on Group IV Photonics, 29 Sept.–1 Oct. 2004, paper WA4.

Lin, Q.

Lipson, M.

Liu, A.

Liu, X.

Liu, X. P.

Liu, Y.

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

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett.90, 211105 (2007).
[CrossRef]

McNab, S. J.

Mehta, P.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Miyazaki, T.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Monro, T. M.

Moormann, C.

Motamedi, A. R.

Nejadmalayeri, A. H.

Niehusmann, J.

Nunes, L. R.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Osgood, R. M.

Painter, O. J.

Paniccia, M.

Panoiu, N. C.

Peacock, A.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Premaratne, M.

Qian, F.

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007).
[CrossRef]

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007.
[CrossRef]

Raghunathan, V.

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

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express12, 2774–2780 (2004).
[CrossRef] [PubMed]

Reed, G. T.

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
[CrossRef]

Renner, H.

H. Renner, “Upper limit for the amplifiable Stokes power in saturated silicon waveguide Raman amplifiers,” in 7th International Conference on Group IV Photonics (GFP), Beijing, China, 1–3 Sept. 2010, paper P1.15.

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 Meeting, San Diego, California, April 11–13. Joint IPRA/NPIS Oral Session: Frontiers in Nanophotonics (paper JWA3), (2005).

H. Renner and M. Krause, “Maximal total gain of non-tapered silicon-on-insulator Raman amplifiers,” in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, OSA Technical Digest Series(CD) (Optical Society of America, 2006), paper OMD2.

Rong, H.

Ropers, C.

P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012).
[CrossRef]

Roy, S.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun.281, 5889–5893 (2008).
[CrossRef]

Rukhlenko, I. D.

Ryshik, I. M.

I. S. Gradstein and I. M. Ryshik, Table of Integrals, Series and Products (Academic Press, Boston, 1994) p. 104.

Sazio, P.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Sekaric, L.

Selvaraja, S. K.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

Slavik, R.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Solli, D. R.

P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012).
[CrossRef]

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett.93, 091114 (2008).
[CrossRef]

Sparks, J.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Suzuki, N.

Tien, E. K.

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007).
[CrossRef]

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007.
[CrossRef]

Tsang, H. K.

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

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett.90, 211105 (2007).
[CrossRef]

T. K. Liang and H. K. Tsang; “Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett.84, 2745–2747 (2004).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

T. K. Liang and H. K. Tsang, “Pulsed-pumped silicon-on-insulator waveguide Raman amplifier,” in Proceedings of International Conference on Group IV Photonics, 29 Sept.–1 Oct. 2004, paper WA4.

Tsuchiya, M.

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Turner, A. C.

Turner, M. D.

Usechak, N.

R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys.40, R249–R271 (2007).
[CrossRef]

Van Thourhout, D.

W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
[CrossRef]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “Nonlinear self-distortion of picosecond optical pulses in silicon wire waveguides,” in Conference on Lasers and Electro-Optics (CLEO) 2006, 21–26 May 2006, Long Beach, paper JThC44.

Vlasov, Y. A.

Wahlbrink, T.

Watts, R.

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

Willner, A. E.

Xia, F. N.

Yan, Y.

Yin, L.

Yue, Y.

Yuksek, N. S.

E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007).
[CrossRef]

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007.
[CrossRef]

Zhang, L.

Adv. Opt. Photon.

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P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett.93, 091114 (2008).
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P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” Appl. Phys. Lett.100, 101111 (2012).
[CrossRef]

T. K. Liang and H. K. Tsang; “Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett.84, 2745–2747 (2004).
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E. K. Tien, F. Qian, N. S. Yuksek, and O. Boyraz, “Influence of nonlinear loss competition on pulse compression and nonlinear optics in silicon,” Appl. Phys. Lett.91, 201115 (2007).
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Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett.90, 211105 (2007).
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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).
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I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron.16, 200–215 (2010).
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W. Bogaerts, S. K. Selvaraja, P. Dumon, J. Brouckaert, K. De Vos, D. Van Thourhout, and R. Baets, “Silicon-on-insulator spectral filters fabricated with CMOS technology,” IEEE J. Sel. Top. Quantum Electron.16, 33–44 (2010).
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B. Jalali, V. Raghunathan, D. Dimitropoulos, and Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron.12, 412–421 (2006).

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R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys.40, R249–R271 (2007).
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Opt. Commun.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Femtosecond pulse propagation in silicon waveguides: variational approach and its advantages,” Opt. Commun.281, 5889–5893 (2008).
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Opt. Express

S. V. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express17, 2298–2318 (2009).
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M. D. Turner, T. M. Monro, and S. V. Afshar, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering,” Opt. Express17, 11565–11581 (2009).
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I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express19, 206–217 (2011).
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L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express20, 1685–1690 (2012).
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A. R. Motamedi, A. H. Nejadmalayeri, A. Khilo, F. X. Kärtner, and E. P. Ippen, “Ultrafast nonlinear optical studies of silicon nanowaveguides,” Opt. Express20, 4085–4101 (2012).
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H. Renner and M. Krause, “Maximal total gain of non-tapered silicon-on-insulator Raman amplifiers,” in Optical Amplifiers and Their Applications/Coherent Optical Technologies and Applications, OSA Technical Digest Series(CD) (Optical Society of America, 2006), paper OMD2.

H. Renner, “Upper limit for the amplifiable Stokes power in saturated silicon waveguide Raman amplifiers,” in 7th International Conference on Group IV Photonics (GFP), Beijing, China, 1–3 Sept. 2010, paper P1.15.

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I. S. Gradstein and I. M. Ryshik, Table of Integrals, Series and Products (Academic Press, Boston, 1994) p. 104.

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Effect of TPA and FCA interplay on pulse compression in silicon,” in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society 2007, Lake Buena Vista, Fl, paper ThY2, 2007.
[CrossRef]

P. Mehta, N. Healy, R. Slavik, R. Watts, J. Sparks, T. Day, P. Sazio, J. Badding, and A. Peacock, “Nonlinearities in silicon optical fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) OFC 2011, Los Angeles, (Optical Society of America, 2011), paper OThS3.

T. K. Liang and H. K. Tsang, “Pulsed-pumped silicon-on-insulator waveguide Raman amplifier,” in Proceedings of International Conference on Group IV Photonics, 29 Sept.–1 Oct. 2004, paper WA4.

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Figures (7)

Fig. 1
Fig. 1

(a) Pulse intensity I(z,t) at waveguide input z = 0 (dotted red curve) and after z = 1, 10, 20 and 50mm (black solid curves) propagation distance in a silicon waveguide with α = 1dB/cm for a Gaussian input pulse [2t0 = 0.2ns in Eq. (43)] with input peak intensity Î0 = 3W/μm2. The green dash-dotted curve indicates the theoretical temporal peak intensity of an undistorted pulse at z = vt in the presence of linear attenuation exp(−αz) only. The center times of the pulses at positions z = 1, 10, 20 and 50 mm are 0.01, 0.1, 0.2 and 0.5 ns, respectively. (b) Same as (a) in a reference time frame tz/v for better visualization of asymmetric pulse attenuation. (c) Non-linear phase shifts ϕKerr (dashed red curve), ϕFCR (dotted green curve), ϕ = ϕKerr + ϕFCR (solid black curve), and normalized free-carrier density N′ = 10−15cm3 × N with N = γR (blue dash-dotted curve) for the pulse after z = 50mm propagation distance. (d) Total non-linear phase shifts ϕ = ϕKerr + ϕFCR for the pulses shown in (b). (e)–(h) Same as (a)–(d), respectively, for Î0 = 30W/μm2.

Fig. 2
Fig. 2

Interpretation of omitting TPA. See explanation in the main text of Section 5.2.

Fig. 3
Fig. 3

Same as Fig. 1 but for a rectangular input pulse [2t0 = 0.2ns in Eq. (44)].

Fig. 4
Fig. 4

Same as Fig. 1 but for a Lorentzian input pulse [2t0 = 0.2ns in Eq. (45)].

Fig. 5
Fig. 5

Same as Fig. 1 but for a sech-shaped input pulse [2t0 = 0.2ns in Eq. (46)].

Fig. 6
Fig. 6

(a) Peaking time tpeak(z) of sech pulses along the propagation distance z for input peak intensities 0, 1, and 2 times the threshold intensity Iinv of Eq. (55) and a pulse duration of 2t0 = 0.2ns. (b) Q(z)/Q0 for sech pulses as a function of z for different input peak intensities and 2t0 = 0.2ns. (c) Upper limiting peak intensity Ilim(z) (black solid curves) for sech pulses of different durations 2t0, and z-dependent peak intensity Î0 exp [−M(z) −αz] (dashed red curves) of the travelling sech pulses for an input peak intensity Î0 = 10 W/μm2 and the given pulse lengths.

Fig. 7
Fig. 7

Pulse compressibility: Normalized pulse intensity after 10 mm propagation distance (solid black curves), where arrows indicate the increase of the input peak intensities Î0 from 5 to 50 to 500 W/μm2. Dashed red curves indicate the input intensity for (a) Gaussian, (b) rectangular, (c) sech, and (d) Lorentzian input pulse.

Equations (72)

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E z + 1 v E t + i β 2 2 2 E t 2 = α 2 E β r + i β i 2 | E | 2 E σ r + i σ i 2 N E ,
N ( z , t ) = γ t I 2 ( z , t ) G ( t t ) d t ,
τ eff = 0 G ( t ) d t ,
G ( t ) = exp ( t / τ exp )
I z + 1 v I t α I σ r N I ,
ϕ z + 1 v ϕ t = β i 2 I σ i 2 N ,
N ( z , t ) { γ R ( z , t ) , t t P ( z ) + Δ t , γ Q ( z ) G [ t t P ( z ) ] , t > t P ( z ) + Δ t .
R ( z , t ) = t I 2 ( z , t ) d t , Q ( z ) = R ( z , t ) = = + I 2 ( z , t ) d t .
1 2 R ˙ z + 1 2 v R ˙ t = α R ˙ γ σ r R R ˙
1 2 R z + 1 2 v R t = α R γ σ r 2 R 2 + K ( z )
x ˜ = t + z / v , y ˜ = t z / v
1 v R x ˜ = α R γ σ r 2 R 2 .
R = 2 α C ( y ˜ ) exp ( 2 α v x ˜ ) γ σ r
R 0 ( t ) = R ( z = 0 , t ) = t I 0 2 ( t ) d t
R ( z , t ) = R 0 ( t z / v ) exp ( 2 α z ) 1 + γ σ r L eff ( 2 ) ( z ) R 0 ( t z / v )
L eff ( m ) ( z ) = [ 1 exp ( m α z ) ] / ( m α )
Q ( z ) = Q 0 exp ( 2 α z ) 1 + γ σ r L eff ( 2 ) ( z ) Q 0 exp ( 2 α z ) γ σ r L eff ( 2 ) ( z ) ,
I ( z , t ) = I 0 ( t z / v ) exp ( α z ) 1 + γ σ r L eff ( 2 ) ( z ) R 0 ( t z / v ) .
1 v ϕ x ˜ = β i 4 I σ i γ 4 R .
ϕ ( z , t ) = ϕ 0 ( t z / v ) + ϕ Kerr ( z , t ) + ϕ FCR ( z , t ) .
ϕ Kerr ( z , t ) = β i I 0 ( t z / v ) 2 α η R 0 ( t z / v ) [ 1 + η R 0 ( t z / v ) ] × arctanh { α L eff ( 1 ) ( z ) η R 0 ( t z / v ) [ 1 + η R 0 ( t z / v ) ] 1 + α L eff ( 1 ) ( z ) η R 0 ( t z / v ) } .
ϕ Kerr ( z , t ) = β i I 0 ( t z / v ) L eff ( 1 ) ( z ) / 2 ,
ϕ Kerr ( z , t ) = β i 2 exp ( α z ) L eff ( 1 ) ( z ) I ( z , t ) h { u [ R 0 ( t z / v ) ] } g [ R 0 ( t z / v ) ] .
h ( u ) = arctanh ( u ) u , u ( R 0 ) = α L eff ( 1 ) ( z ) η R 0 ( 1 + η R 0 ) 1 + α L eff ( 1 ) ( z ) η R 0 , g ( R 0 ) = 1 + 2 α L eff ( 2 ) ( z ) η R 0 1 + α L eff ( 1 ) ( z ) η R 0 .
ϕ FCR ( z , t ) = σ i 2 σ r × ln { 1 + γ σ r L eff ( 2 ) ( z ) R 0 ( t z / v ) }
Δ ω FCR ( z , t ) = σ i 2 × γ L eff ( 2 ) ( z ) I 0 2 ( t z / v ) 1 + γ σ r L eff ( 2 ) ( z ) R 0 ( t z / v )
= σ i γ 2 × exp ( + 2 α z ) L eff ( 2 ) ( z ) I 2 ( z , t ) [ 1 + γ σ r L eff ( 2 ) ( z ) R 0 ( t z / v ) ] ,
f 0 ( ξ ) = f 0 ( t / t 0 ) = I 0 ( t ) / I ^ 0
F 0 ( ξ ) = ξ f 0 2 ( ξ ) d ξ , W 0 = t 0 I ^ 0 J , J = + f 0 ( ξ ) d ξ ,
α TPA α FCA = β r I ^ 0 f 0 ( t / t 0 ) σ r γ t 0 I ^ 0 2 F 0 ( t / t 0 ) = 1 W 0 × β r J f 0 ( t / t 0 ) σ r γ F 0 ( t / t 0 ) .
f ( z , t ) = I ( z , t ) I ^ 0 exp ( α z ) = f 0 ( ξ z / v t 0 ) 1 + Θ FCA F 0 ( ξ z / v t 0 ) ,
Θ FCA = σ r γ t 0 I ^ 0 2 L eff ( 2 ) ( z )
f ( z , t ) = I ( z , t ) I ^ 0 exp ( α z ) = f 0 ( ξ z / v t 0 ) 1 + Θ TPA f 0 ( ξ z / v t 0 ) ,
Θ TPA = β r I ^ 0 L eff ( 1 ) ( z ) .
Θ TPA = β r I ^ 0 L eff ( 1 ) 1 ,
L eff ( 1 ) = 1 exp ( α z ) α = Θ TPA β r I ^ 0 1 β r I ^ 0 .
Θ FCA Θ TPA = σ r γ t 0 I ^ 0 2 L eff ( 2 ) β r I ^ 0 L eff ( 1 ) 1
L eff ( 1 ) ( z ) = σ r γ t 0 β r 2 Ξ ( α z ) Θ TPA 2 Θ FCA , W 0 = J β r Ξ ( α z ) σ r γ Θ FCA Θ TPA ,
L eff ( 1 ) ( z ) σ r γ t 0 β r 2 Ξ ( α z ) Θ FCA , W 0 J β r Ξ ( α z ) Θ FCA σ r γ .
z = 2 α arctanh ( α σ r γ t 0 2 β r 2 Θ TPA 2 Θ FCA ) , W 0 = J 2 β r 2 Θ FCA + α σ r γ t 0 Θ TPA 2 2 β r σ r γ Θ TPA
z σ r γ t 0 Θ TPA 2 β r 2 Θ FCA σ r γ t 0 β r 2 Θ FCA , W 0 J β r Θ FCA σ r γ Θ TPA J β r Θ FCA σ r γ , for α fixed , z 1 / α .
α 2 β r 2 Θ FCA σ r γ t 0 Θ TPA 2 2 β r 2 Θ FCA σ r γ t 0 , W 0 2 J β r Θ FCA σ r γ Θ TPA 2 J β r Θ FCA σ r γ , for z fixed , α 2 / z ,
I 0 ( t ) = I ^ 0 exp ( t 2 / t 0 2 ) , R 0 ( t ) = π / 8 t 0 I ^ 0 2 [ 1 + erf ( 2 t / t 0 ) ]
t < t 0 : I 0 ( t ) = 0 , R 0 ( t ) = 0 , | t | < t 0 : I 0 ( t ) = I ^ 0 , R 0 ( t ) = ( t + t 0 ) I ^ 0 2 , t > t 0 : I 0 ( t ) = 0 , R 0 ( t ) = 2 t 0 I ^ 0 2 ,
I 0 ( t ) = I ^ 0 1 + ( t / t 0 ) 2 , R 0 ( t ) = t 0 I ^ 0 2 2 [ t 0 t t 0 2 + t 2 + arctan ( t t 0 ) + π 2 ]
I 0 ( t ) = I ^ 0 cosh ( t / t 0 ) , R 0 ( t ) = t 0 I ^ 0 2 [ 1 + tanh ( t / t 0 ) ]
I ( z , t ) = I ^ 0 exp [ M ( z ) α z ] cosh [ M ( z ) + ( t z / v ) / t 0 ] = exp [ M ( z ) α z ] I 0 [ t 0 M ( z ) + t z / v ]
R ( z , t ) = t 0 I ^ 0 2 exp [ 2 M ( z ) 2 α z ] { 1 + tanh [ M ( z ) + ( t z / v ) / t 0 ] } = exp [ 2 M ( z ) 2 α z ] R 0 [ t 0 M ( z ) + t z / v ]
M ( z ) = 1 2 ln [ 1 + 2 γ σ r t 0 I ^ 0 2 L eff ( 2 ) ( z ) ] .
I [ z , t peak ( z ) ] I ^ 0 = exp [ M ( z ) α z ] = exp ( α z ) 1 + 2 γ σ r t 0 I ^ 0 2 L eff ( 2 ) ( z ) .
I ( z , t ) I lim ( z ) cosh [ M ( z ) + ( t z / v ) / t 0 ] , I lim ( z ) = exp ( α z ) 2 γ σ r t 0 L eff ( 2 ) ( z ) ,
t peak ( z ) = z v t 0 M ( z ) .
d t peak ( z ) d z = 1 v peak ( z ) = 1 v γ σ r t 0 2 I ^ 0 2 exp ( 2 α z ) 1 + 2 γ σ r t 0 I ^ 0 2 L eff ( 2 ) ( z ) ,
z inv = 1 2 α ln [ γ σ r t 0 I ^ 0 2 ( 1 + α v t 0 ) α + γ σ r t 0 I ^ 0 2 ] .
W 0 = π t 0 I ^ 0 W inv = π t 0 I inv = π / γ σ r v .
Δ ω FCR ( z , t ) = σ i γ L eff ( 2 ) ( z ) I ^ 0 2 exp [ M ( z ) ] cosh [ M ( z ) ] + cosh [ M ( z ) + 2 ( t z / v ) / t 0 ] .
Δ ω FCR ( max ) ( z ) = σ i σ r t 0 × 1 + 2 γ σ r L eff ( 2 ) ( z ) t 0 I ^ 0 2 1 1 + 2 γ σ r L eff ( 2 ) ( z ) t 0 I ^ 0 2 + 1 ,
t Δ ω FCR ( max ) ( z ) = z v t 0 4 ln [ 1 + 2 γ σ r L eff ( 2 ) ( z ) t 0 I ^ 0 2 ]
= z v t 0 M ( z ) 2 = t peak ( z ) + t 0 M ( z ) 2 ,
Δ ω FCR [ z , t peak ( z ) ] = σ i γ L eff ( 2 ) ( z ) I ^ 0 2 2 [ 1 + σ r γ t 0 I ^ 0 2 L eff ( 2 ) ( z ) ] ,
a FCA ( z ) = 0 z α FCA ( z ) d z , ψ FCR ( z ) = σ i 2 σ r a FCA ( z )
α FCA ( z ) = σ r N [ z , t P ( z ) + T ] = σ r γ Q ( z ) G ( T ) = G ( T ) σ r γ Q 0 exp ( 2 α z ) 1 + γ σ r Q 0 L eff ( 2 ) ( z ) .
a FCA ( L ) = G ( T ) ln [ 1 + σ r γ Q 0 L eff ( 2 ) ( L ) ] .
a FCA ( L ) < a ¯ FCA ( L ) .
Q 0 = t 0 F 0 ( ) I ^ 0 2 < [ 1 + 2 σ r γ L eff ( 2 ) ( L ) τ eff I ¯ 0 2 ] 1 / [ 2 G ( T ) ] 1 σ r γ L eff ( 2 ) ( L )
I ¯ 0 > 1 t 0 F 0 ( ) × 2 [ τ eff t 0 F 0 ( ) ] σ r γ L eff ( 2 ) ( L ) ,
I z + 1 v I t = 2 v I x ˜ = ( α + σ r N ) I β r I 2 , N ( z , t ) = γ t I 2 ( z , t ) d t ,
I ( z , t ) = I ^ 0 f 0 [ ( t z / v ) / t 0 ] exp ( α z ) U ( z , t ) 1 + β r I ^ 0 L ˜ eff ( 1 ) ( z , t ) f 0 [ ( t z / v ) / t 0 ] ,
U ( z , t ) = exp { σ r 0 z N [ ζ , t + ( ζ z ) / v ] d ζ } , N ( z , t ) = γ t I 2 ( z , t ) d t .
L ˜ eff ( 1 ) ( z , t ) = 0 z exp ( α χ ) U ( χ , t ) d χ
I ¯ ( z ) = I ¯ 0 exp ( α z ) 1 + 2 σ r γ τ eff I ¯ 0 2 L eff ( 2 ) ( z ) , α ¯ FCA ( z ) = σ r γ τ eff I ¯ 0 2 exp ( 2 α z ) 1 + 2 σ r γ τ eff I ¯ 0 2 L eff ( 2 ) ( z ) .
a ¯ FCA ( L ) = 0 L α ¯ FCA ( z ) d z = 1 2 ln [ 1 + 2 σ r γ τ eff I ¯ 0 2 L eff ( 2 ) ( L ) ] .

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