Abstract

A deep insight into the inherent anisotropic optical properties of silicon is required to improve the performance of silicon-waveguide-based photonic devices. It may also lead to novel device concepts and substantially extend the capabilities of silicon photonics in the future. In this paper, for the first time to the best of our knowledge, we present a three-dimensional finite-difference time-domain (FDTD) method for modeling optical phenomena in silicon waveguides, which takes into account fully the anisotropy of the third-order electronic and Raman susceptibilities. We show that, under certain realistic conditions that prevent generation of the longitudinal optical field inside the waveguide, this model is considerably simplified and can be represented by a computationally efficient algorithm, suitable for numerical analysis of complex polarization effects. To demonstrate the versatility of our model, we study polarization dependence for several nonlinear effects, including self-phase modulation, cross-phase modulation, and stimulated Raman scattering. Our FDTD model provides a basis for a full-blown numerical simulator that is restricted neither by the single-mode assumption nor by the slowly varying envelope approximation.

© 2010 Optical Society of America

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2010 (4)

2009 (5)

S. Afshar and T.M. Monro, "A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity," Opt. Express 17, 2298-2318 (2009).
[CrossRef]

C. Dissanayake, I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Raman-mediated nonlinear interactions in silicon waveguides: Copropagating and counterpropagating pulses," IEEE Photonics Technol. Lett. 21, 1372-1374 (2009).
[CrossRef]

R. M. Osgood, Jr., N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, "Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersionengineered silicon nanophotonic wires," Adv. Opt. Photonics 1, 162-235 (2009).
[CrossRef]

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]

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]

2008 (3)

B. Jalali, "Can silicon change photonics?" Phys. Status Solidi(a) 2, 213-224 (2008).
[CrossRef]

F. L. Teixeira, "Time-domain finite-difference and finite-element methods for Maxwell equations in complex media," IEEE Trans. Antennas Propag. 56, 2150-2166 (2008).
[CrossRef]

H. K. Tsang and Y. Liu, "Nonlinear optical properties of silicon waveguides," Semicond. Sci. Technol.  23, 064007(1-9) (2008).
[CrossRef]

2007 (8)

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

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]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
[CrossRef]

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

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

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, "Supercontinuum generation in silicon photonic wires," Opt. Express 15, 15242-15249 (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]

2006 (18)

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, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

C. Manolatou and M. Lipson, "All-optical silicon modulators based on carrier injection by two-photon absorption," J. Lightwave Technol. 24, 1433-1439 (2006).
[CrossRef]

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]

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]

I.-W. Hsieh, X. Cheng, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, "Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides," Opt. Express 14, 12380-12387 (2006).
[CrossRef] [PubMed]

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

Y.-H Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, "Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides," Opt. Express 14, 11721-11726 (2006).
[CrossRef] [PubMed]

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

S. E. Thompson and S. Parthasarathy, "Moore’s law: The future of Si microelectronics," Mater. Today 9, 20-25 (2006).
[CrossRef]

B. Jalali and S. Fathpour, "Silicon Photonics," J. Lightwave Technol. 24, 4600-4615 (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]

Q. Xu and M. Lipson, "Carrier-induced optical bistability in silicon ring resonators," Opt. Lett. 31, 341-343 (2006).
[CrossRef] [PubMed]

L. X. Dou and A. R. Sebak, "3D FDTD method for arbitrary anisotropic materials," Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[CrossRef]

N. C. Panoiu, X. Chen, and R. M. Osgood, Jr., "Modulation instability in silicon photonic nanowires," Opt. Lett. 31, 3609-3611 (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. B 39, 2737-2746 (2006).
[CrossRef]

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

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]

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical amplification and lasing by stimulated Raman scattering in silicon waveguides," J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

2005 (8)

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

R. Espinola, J. Dadap, R. Osgood, Jr., S. McNab, and Y. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 8336-8346 (2005).
[CrossRef]

R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, "Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode," IEEE Photon. Technol. Lett. 17, 1968-1970 (2005).
[CrossRef]

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

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, "An all-silicon Raman laser," Nature 433, 292-294 (2005).
[CrossRef] [PubMed]

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, "Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides," Electron. Lett. 41, 320-321 (2005).
[CrossRef]

Q. Xu, V. R. Almeida, and M. Lipson, "Demonstration of high Raman gain in a submicrometer-size silicon-oninsulator waveguide," Opt. Lett. 30, 35-37 (2005).
[CrossRef] [PubMed]

2004 (6)

2003 (4)

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

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, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Anti-Stokes Raman conversion in silicon waveguides," Opt. Express 11, 2862-2872 (2003).
[CrossRef] [PubMed]

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

2002 (2)

H. Iwai and S. Ohmi, "Silicon integrated circuit technology from past to future," Microelectron. Reliab. 42, 465-491 (2002).
[CrossRef]

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

1999 (1)

M. Schulz, "The end of the road for silicon?" Nature 399, 729-730 (1999).
[CrossRef]

1998 (1)

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-oninsulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

1997 (1)

R. M. Joseph and A. Taflove, "FDTD Maxwell’s equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
[CrossRef]

1993 (1)

J. Schneider and S. Hudson, "A finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

1992 (1)

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell’s equations," IEEE J. Quantum Electron. 28, 2416-2422 (1992).
[CrossRef]

1990 (1)

E. B. Graham and R. E. Raab, "Light propagation in cubic and other anisotropic crystals," Proc. R. Soc. London, Ser. A 430, 593-614 (1990).
[CrossRef]

1984 (1)

1983 (1)

R. Holland, "Finite-difference solution ofMaxwell’s equations in generalized nonorthogonal coordinated," IEEE Trans. Nucl. Sci. NS-30, 4589-4591 (1983).
[CrossRef]

1980 (1)

H. H. Li, "Refractive index of silicon and germanium and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data 9, 561-658 (1980).
[CrossRef]

1971 (1)

J. Pastrnak and K. Vedam, "Optical anisotropy of silicon single crystals," Phys. Rev. B 3, 2567-2571 (1971).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1965 (1)

P. H. Wendland and M. Chester, "Electric field effects on indirect optical transitions in silicon," Phys. Rev. 140, A1384-A1390 (1965).
[CrossRef]

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, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Afshar, S.

Agrawal, G. P.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Analytical study of optical bistability in silicon ring resonators," Opt. Lett. 35, 55-57 (2010).
[CrossRef] [PubMed]

B. A. Daniel and G. P. Agrawal, "Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects," J. Opt. Soc. Am. B 27, 956-965 (2010).
[CrossRef]

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Yu. S. Kivshar, "Polarization rotation in silicon waveguides: Analytical modeling and applications," IEEE Photon. J. 2, 423-435 (2010).
[CrossRef]

C. Dissanayake, I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Raman-mediated nonlinear interactions in silicon waveguides: Copropagating and counterpropagating pulses," IEEE Photonics Technol. Lett. 21, 1372-1374 (2009).
[CrossRef]

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]

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]

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]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
[CrossRef]

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

Alex, V.

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

Almeida, V. R.

Baets, 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, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

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]

Bogaerts, W.

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, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Boyd, R. W.

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
[CrossRef]

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]

Boyraz, ¨O.

Boyraz, O.

Carter, G. M.

R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, "Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode," IEEE Photon. Technol. Lett. 17, 1968-1970 (2005).
[CrossRef]

Cheben, P.

Chen, X.

Cheng, X.

Chester, M.

P. H. Wendland and M. Chester, "Electric field effects on indirect optical transitions in silicon," Phys. Rev. 140, A1384-A1390 (1965).
[CrossRef]

Chou, C.-Y.

Chu, T.

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

Claps, R.

Cohen, O.

Coppinger, F.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-oninsulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Dadap, J.

R. Espinola, J. Dadap, R. Osgood, Jr., S. McNab, and Y. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 8336-8346 (2005).
[CrossRef]

Dadap, J. I.

Dalacu, D.

Daniel, B. A.

Dekker, R.

Dimitropoulos, D.

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.

I. D. Rukhlenko, C. Dissanayake, and M. Premaratne, "Visualization of electromagnetic-wave polarization evolution using the Poincar’e sphere," Opt. Lett. 35, 2221-2223 (2010).
[CrossRef] [PubMed]

C. Dissanayake, I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Raman-mediated nonlinear interactions in silicon waveguides: Copropagating and counterpropagating pulses," IEEE Photonics Technol. Lett. 21, 1372-1374 (2009).
[CrossRef]

Donecker, J.

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

Dou, L. X.

L. X. Dou and A. R. Sebak, "3D FDTD method for arbitrary anisotropic materials," Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[CrossRef]

Driessen, A.

Dulkeith, E.

R. M. Osgood, Jr., N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, "Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersionengineered silicon nanophotonic wires," Adv. Opt. Photonics 1, 162-235 (2009).
[CrossRef]

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]

Dumon, P.

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, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

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]

Eggleton, B. J.

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, "Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides," Electron. Lett. 41, 320-321 (2005).
[CrossRef]

Espinola, R.

R. Espinola, J. Dadap, R. Osgood, Jr., S. McNab, and Y. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 8336-8346 (2005).
[CrossRef]

F¨orst, M.

Fang, A.

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, "An all-silicon Raman laser," Nature 433, 292-294 (2005).
[CrossRef] [PubMed]

Fathpour, S.

Fauchet, P. M.

Foster, M. A.

Fu, L.

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, "Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides," Electron. Lett. 41, 320-321 (2005).
[CrossRef]

Fukuda, H.

Gaeta, A. L.

Garanovich, I. L.

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Yu. S. Kivshar, "Polarization rotation in silicon waveguides: Analytical modeling and applications," IEEE Photon. J. 2, 423-435 (2010).
[CrossRef]

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. B 39, 2737-2746 (2006).
[CrossRef]

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

Goorjian, P. M.

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell’s equations," IEEE J. Quantum Electron. 28, 2416-2422 (1992).
[CrossRef]

Graham, E. B.

E. B. Graham and R. E. Raab, "Light propagation in cubic and other anisotropic crystals," Proc. R. Soc. London, Ser. A 430, 593-614 (1990).
[CrossRef]

Green, W. M.

Green, W. M. J.

R. M. Osgood, Jr., N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, "Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersionengineered silicon nanophotonic wires," Adv. Opt. Photonics 1, 162-235 (2009).
[CrossRef]

Hagness, S. C.

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell’s equations," IEEE J. Quantum Electron. 28, 2416-2422 (1992).
[CrossRef]

Hak, D.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical amplification and lasing by stimulated Raman scattering in silicon waveguides," J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, "An all-silicon Raman laser," Nature 433, 292-294 (2005).
[CrossRef] [PubMed]

Han, Y.

Holland, R.

R. Holland, "Finite-difference solution ofMaxwell’s equations in generalized nonorthogonal coordinated," IEEE Trans. Nucl. Sci. NS-30, 4589-4591 (1983).
[CrossRef]

Horton, T. U.

R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, "Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode," IEEE Photon. Technol. Lett. 17, 1968-1970 (2005).
[CrossRef]

Hsieh, I.

R. M. Osgood, Jr., N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, "Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersionengineered silicon nanophotonic wires," Adv. Opt. Photonics 1, 162-235 (2009).
[CrossRef]

Hsieh, I.-W.

Hudson, S.

J. Schneider and S. Hudson, "A finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

Itabashi, S.-I.

Iwai, H.

H. Iwai and S. Ohmi, "Silicon integrated circuit technology from past to future," Microelectron. Reliab. 42, 465-491 (2002).
[CrossRef]

Jalali, B.

B. Jalali, "Can silicon change photonics?" Phys. Status Solidi(a) 2, 213-224 (2008).
[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]

B. Jalali and S. Fathpour, "Silicon Photonics," J. Lightwave Technol. 24, 4600-4615 (2006).
[CrossRef]

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

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

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in silicon-on-insulator waveguides," Opt. Express 12, 2774-2780 (2004).
[CrossRef] [PubMed]

¨O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, "All optical switching and continuum generation in silicon waveguides," Opt. Express 12, 4094-4102 (2004).
[CrossRef] [PubMed]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Anti-Stokes Raman conversion in silicon waveguides," Opt. Express 11, 2862-2872 (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, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-oninsulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Jones, R.

A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical amplification and lasing by stimulated Raman scattering in silicon waveguides," J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, "An all-silicon Raman laser," Nature 433, 292-294 (2005).
[CrossRef] [PubMed]

Joseph, R. M.

R. M. Joseph and A. Taflove, "FDTD Maxwell’s equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
[CrossRef]

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell’s equations," IEEE J. Quantum Electron. 28, 2416-2422 (1992).
[CrossRef]

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. B 39, 2737-2746 (2006).
[CrossRef]

Kawanishi, T.

Kivshar, Yu. S.

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Yu. S. Kivshar, "Polarization rotation in silicon waveguides: Analytical modeling and applications," IEEE Photon. J. 2, 423-435 (2010).
[CrossRef]

Koonath, P.

Kuo, Y.-H

Li, H. H.

H. H. Li, "Refractive index of silicon and germanium and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data 9, 561-658 (1980).
[CrossRef]

Liang, T.

Liang, T. K.

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, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

Lin, Q.

Lipson, M.

Littler, I.

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Adv. Opt. Photonics (1)

R. M. Osgood, Jr., N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, "Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersionengineered silicon nanophotonic wires," Adv. Opt. Photonics 1, 162-235 (2009).
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Appl. Opt. (1)

Appl. Phys. Lett (1)

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
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Appl. Phys. Lett. (2)

G. W. Rieger, K. S. Virk, and J. F. Young, "Nonlinear propagation of ultrafast 1.5 μm pulses in high-indexconstrast silicon-on-insulator waveguides," Appl. Phys. Lett. 84, 900-902 (2004).
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M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys. Lett. 82, 2954-2956 (2003).
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Electron. Lett. (2)

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IEEE J. Quantum Electron. (2)

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

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R. A. Soref, "The past, present, and future of silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006).
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IEEE Photon. J. (1)

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Yu. S. Kivshar, "Polarization rotation in silicon waveguides: Analytical modeling and applications," IEEE Photon. J. 2, 423-435 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, "Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode," IEEE Photon. Technol. Lett. 17, 1968-1970 (2005).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

C. Dissanayake, I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Raman-mediated nonlinear interactions in silicon waveguides: Copropagating and counterpropagating pulses," IEEE Photonics Technol. Lett. 21, 1372-1374 (2009).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

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[CrossRef]

J. Schneider and S. Hudson, "A finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

R. M. Joseph and A. Taflove, "FDTD Maxwell’s equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
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Figures (9)

Fig. 1
Fig. 1

Staggered space arrangement (left) and leapfrog time ordering (right) of discretized electromagnetic-field components using a cubic Yee cell.

Fig. 2
Fig. 2

Positions of electric field components in the FDTD grid. The values of components Eβ and Eγ at the node (i, j, k) are calculated by averaging their four values over the nodes (i ± 1/2, j ± 1/2, k) and (i ± 1/2, j, k ± 1/2), respectively.

Fig. 3
Fig. 3

Relative orientation of FDTD (α, β, γ) and crystallographic (x, y, z) axes adopted in the paper. For ϑ = π 4 , the FDTD axes α, β, and γ coincide, respectively, with the [110], [1̄10], and [001] crystallographic directions. The inset shows the TM and TE polarizations and an arbitrary linear polarization determined by the angle φ.

Fig. 4
Fig. 4

Three one-dimensional Yee cells corresponding to the simplified FDTD model. The electric and magnetic fields along the propagation direction α are ignored.

Fig. 5
Fig. 5

Input–output characteristics for 100-, 200-, and 500-μm-long silicon waveguides pumped by TM- and TE-polarized, 1.4-ps Gaussian pulses. Other parameter values are given in the text.

Fig. 6
Fig. 6

Efficiency of SPM-induced polarization rotation for different input SOPs of a 1.4-ps Gaussian pulse. The output polarizer is perpendicular to the input polarization state of the pulse. We choose τc = 0.8 ns; other parameters are given in the text.

Fig. 7
Fig. 7

Poincaré-sphere representation of SOP variations along a 1.4-ps Gaussian pulse at the output of a 0.5-mm-long silicon waveguide. Numbers near the traces staring from the equator show φ values for the input SOPs (see Fig. 3); φ 0 ≈ 35°. Traces starting from the poles correspond to circularly polarized input pulses. Here, I 0 = 400 GW/cm2; other parameters are the same as in Fig. 6(a).

Fig. 8
Fig. 8

Switching efficiency of a CW beam for a 350-fs Gaussian pump pulse as a function of (a) input polarization angle φ of the CW beam, (b) waveguide length L, and (c) pump's peak intensity Ip 0. Panel (d) shows switching windows for three different waveguide lengths; dashed curve shows the Gaussian pulse that is used in the definition of transmittance. In panels (b)–(d), φ = π/4. In all panels input CW intensity is 1 GW/cm2 and τ c = 0.8 ns; other parameters are given in the text.

Fig. 9
Fig. 9

Peak intensity of TM- and TE-polarized signals amplified via SRS by TM- and TE-polarized pumps. Both input signal and input pump are assumed to be 1.4-ps Gaussian pulses with peak intensities of 1 and 100 GW/cm2, respectively; carrier frequencies are 200 and 215.6 THz; gR = 76 cm/GW; other parameters are given in the text.

Tables (1)

Tables Icon

Table 1 Values of dμ specifying relative positions of components Vξ inside a four-dimensional unit cell.

Equations (48)

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B ( r , t ) t = × E ( r , t ) , D ( r , t ) t = × H ( r , t ) ,
H ( r , t ) = B ( r , t ) μ 0 , E ( r , t ) = D ( r , t ) P ( r , t ) ɛ 0 ,
V ( r , t ) = α ^ V α ( r , t ) + β ^ V β ( r , t ) + γ ^ V γ ( r , t ) ,
V ξ | i + d 1 , j + d 2 , k + d 3 n + d 4 V ξ [ ( i + d 1 ) Δ ξ , ( j + d 2 ) Δ ξ , ( k + d 3 ) Δ ξ , ( n + d 4 ) Δ t ] ,
B α | i 1 / 2 , j 1 / 2 , k 1 / 2 n + 1 / 2 = B α | n 1 / 2 S c ( E γ | i , j + 1 / 2 , k n E γ | i , j 1 / 2 , k n E β | i , j , k + 1 / 2 n + E β | i , j , k 1 / 2 n ) , H α | i 1 / 2 , j 1 / 2 , k 1 / 2 n + 1 / 2 = H α | n 1 / 2 + 1 μ 0 ( B α | n + 1 / 2 B α | n 1 / 2 ) , D α | ijk n + 1 = D α | ijk n + S c ( H γ | i , j + 1 / 2 , k n + 1 / 2 H γ | i , j 1 / 2 , k n + 1 / 2 H β | i , j , k + 1 / 2 n + 1 / 2 + H β | i , j , k 1 / 2 n + 1 / 2 ) ,
E ξ | i + d 1 , j + d 2 , k + d 3 n + 1 = E ξ | n + 1 ɛ 0 ( D ξ | n + 1 D ξ | n P ξ | n + 1 + P ξ | n ) .
P ( r , t ) = i P ( i ) + a P ( a ) .
P ξ ( i ) | i + d 1 , j + d 2 , k + d 3 n + 1 = f ξ ( i ) ( P ξ ( i ) | n , P ξ ( i ) | n 1 , , E ξ | n + 1 , E ξ | n , E ξ | n 1 , ) ,
P ξ ( a ) | i + d 1 , j + d 2 , k + d 3 n + 1 = f ξ ( a ) ( P ξ ( a ) | n , P ξ ( a ) | n 1 , , E α | n + 1 , E β | n + 1 , E γ | n + 1 , E α | n , E β | n ) .
E β | ijk n = 1 4 ( E β | i 1 / 2 , j 1 / 2 , k n + E β | i 1 / 2 , j + 1 / 2 , k n + E β | i + 1 / 2 , j + 1 / 2 , k n + E β | i + 1 / 2 , j 1 / 2 , k n ) , E γ | ijk n = 1 4 ( E γ | i 1 / 2 , j , k 1 / 2 n + E γ | i 1 / 2 , j , k + 1 / 2 n + E γ | i + 1 / 2 , j , k 1 / 2 n + E γ | i + 1 / 2 , j , k + 1 / 2 n ) .
P ˜ κ ( r , ω ) = ɛ 0 χ ˜ ( 1 ) ( ω ) E ˜ κ ( r , ω ) + ɛ 0 ( 2 π ) 2 λ μ ν + d ω 1 + d ω 2 χ ˜ κ λ μ ν ( 3 ) ( ω ; ω 1 , ω 2 , ω 3 ) E ˜ λ ( ω 1 ) E ˜ μ ( ω 2 ) E ˜ ν ( ω 3 ) + ɛ 0 χ ˜ FC ( | E | 4 , ω ) E ˜ κ ( r , ω ) ,
V ˜ ( ω ) = F [ V ( t ) ] + V ( t ) e i ω t d t , V ( t ) = F 1 [ V ˜ ( ω ) ] 1 2 π + V ˜ ( ω ) e i ω t d ω .
χ κ λ μ ν ( 3 ) = klmn a k κ a l λ a m μ a n ν χ klmn ( 3 ) ,
a = ( cos ϑ sin ϑ 0 sin ϑ cos ϑ 0 0 0 1 ) .
ɛ ˜ ( ω ) ɛ ˜ ( ω ) + i ɛ ˜ ( ω ) = 1 + χ ˜ ( 1 ) ( ω ) .
n 2 ( ω ) = 1 + j = 1 s a j ω j 2 ω j 2 ω 2 ,
χ ˜ κ λ μ ν e ( ω ; ω 1 , ω 2 , ω 3 ) χ ˜ α α α α e ( ω ) κ λ μ ν ,
klmn = ( ρ / 3 ) ( δ kl δ mn + δ km δ ln + δ kn δ lm ) + ( 1 ρ ) δ kl δ lm δ mn ,
α α α α = β β β β = 1 + A ϑ , γ γ γ γ = 1 , α α α β = α α β α = α β α α = β α α α = B ϑ , β β β α = β β α β = β α β β = α β β β = B ϑ , α α β β = α β β α = β β α α = α β α β = β α β α = β α α β = ρ / 3 A ϑ , α α γ γ = α γ γ α = γ γ α α = α γ α γ = γ α γ α = γ α α γ = ρ / 3 , β β γ γ = β γ γ β = γ γ β β = β γ β γ = γ β γ β = γ β β γ = ρ / 3 ,
P κ K ( r , t ) = ɛ 0 ɛ 2 λ μ ν κ λ μ ν E λ ( r , t ) E μ ( r , t ) E ν ( r , t ) = ɛ 0 ɛ 2 S κ K ( r , t ) E κ ( r , t ) ,
S α K = ( 1 + A ϑ ) E α 2 + ( ρ 3 A ϑ ) E β 2 + ρ E γ 2 + 3 B ϑ E α E β B ϑ E β 3 / E α ,
S β K = ( 1 + A ϑ ) E β 2 + ( ρ 3 A ϑ ) E α 2 + ρ E γ 2 3 B ϑ E α E β + B ϑ E α 3 / E β ,
S γ K = ρ E α 2 + ρ E β 2 + E γ 2 .
P ˜ κ TPA ( r , ω ) = ɛ 0 η TPA i ω F [ S κ K ( r , t ) E κ ( r , t ) ] .
χ ~ κ λ μ ν R ( ω ; ω 1 , ω 2 , ω 3 ) = 1 2 [ H ~ ( ω 1 + ω 2 ) κ λ μ ν + H ~ ( ω 2 + ω 3 ) κ ν μ λ ] ,
H ~ ( ω ) = 2 ξ R Ω R Γ R Ω R 2 2 i ω Γ R ω 2 .
klmn = δ km δ ln + δ kn δ lm 2 δ kl δ lm δ mn .
α α α α = β β β β = α α β β = β β α α = C ϑ , α α α β = α α β α = α β α α = α β β β = D ϑ , β β β α = β β α β = β α β β = β α α α = D ϑ , α β α β = β α β α = α β β α = β α α β = 1 C ϑ , α γ α γ = γ α γ α = α γ γ α = γ α α γ = β γ β γ = γ β γ β = β γ γ β = γ β β γ = 1 ,
χ κ λ μ ν R ( t 1 , t 2 , t 3 ) = 1 ( 2 π ) 3 + d ω 1 + d ω 2 + d ω 3 χ κ λ μ ν R ( ω ; ω 1 , ω 2 , ω 3 ) e i ( ω 1 t 1 + ω 2 t 2 + ω 3 t 3 ) = 1 2 [ δ ( t 1 t 2 ) δ ( t 3 ) κ λ μ ν + δ ( t 1 ) δ ( t 2 t 3 ) κ ν μ λ ] H ( t 2 ) .
P κ R ( r , t ) = ɛ 0 λ μ ν t d t 1 t d t 2 t d t 3 χ κ λ μ ν R ( t t 1 , t t 2 , t t 3 ) E λ ( r , t 1 ) E μ ( r , t 2 ) E ν ( r , t 3 ) = ɛ 0 λ μ ν κ λ μ ν E λ ( t ) t H ( t t 1 ) E μ ( t 1 ) E ν ( t 1 ) d t 1 = ɛ 0 λ Ξ κ λ R ( r , t ) E λ ( r , t ) ,
Ξ κ λ R ( r , t ) = t H ( t t 1 ) S κ λ R ( r , t 1 ) d t 1 , S α α R = S β β R = C ϑ ( E α 2 E β 2 ) + 2 D ϑ E α E β , S γ γ R = 0 , S α β R = S β α R = D ϑ ( E α 2 E β 2 ) + 2 ( 1 C ϑ ) E α E β , S κ γ R = S γ κ R = 2 E κ E γ , ( κ = α , β ) .
χ ~ F C ( ω ) = n 0 Δ n FC ( N ) c n 0 2 i ω Δ α FC ( N ) ,
Δ n FC ( N ) = ζ ( ω r / ω 0 ) 2 N , Δ α FC ( N ) = σ ( ω r / ω 0 ) 2 N , ω r = 2 π c / ( 1.55 μ m ) , ζ = 5.3 × 10 27 m 3 , σ = 1.45 × 10 21 m 2 .
N ( r , t ) t = N ( r , t ) τ c + β TPA 2 ω 0 I 2 ( r , t ) ,
ζ TOE = 2 ω κ θ C τ c ρ ( ω 0 ω r ) 2 ,
E ~ κ ( r , ω ) = ( 1 / ɛ 0 ) [ D ~ κ ( ω ) P ~ κ LD ( ω ) P ~ κ K ( ω ) P ~ κ R ( ω ) ] + η TPA i ω F [ S κ K ( r , t ) E κ ( r , t ) ] n 0 Δ n FC E ~ κ ( ω ) + c n 0 i ω α L + Δ α FC 2 E κ ~ ( ω ) ,
E κ | n + 1 E κ | n Δ t = D κ | n + 1 D κ | n ɛ 0 Δ t P κ LD | n + 1 P κ LD | n ɛ 0 Δ t ɛ 2 S κ K | n + 1 E κ | n + 1 S κ K | n E κ | n Δ t + η TPA S κ K | n + 1 E κ | n + 1 + S κ K | n E κ | n 2 λ Ξ κ λ R | n + 1 E λ | n + 1 Ξ κ λ R | n E λ | n Δ t n 0 ( Δ n FC | n + 1 / 2 E κ | n + 1 E κ | n Δ t + E κ | n + 1 + E κ | n 2 Δ n FC | n + 1 Δ n FC | n Δ t ) c n 0 α L + Δ α FC | n + 1 / 2 2 E κ | n + 1 + E κ | n 2 ,
P κ LD | n + 1 = j = 1 s P κ j LD | n + 1 .
2 P κ j LD t 2 + ω j 2 P κ j LD = ɛ 0 a j ω j 2 E κ .
P κ j LD | n + 1 = 4 2 + ω j 2 Δ t 2 P κ j LD | n P κ j LD | n 1 + 2 ω j 2 Δ t 2 2 + ω j 2 Δ t 2 ɛ 0 a j E κ | n .
2 Ξ κ λ R t 2 + 2 Γ R Ξ κ λ R t + Ω R 2 Ξ κ λ R = 2 Γ R Ω R ξ R S κ λ R ,
Ξ κ λ R | n + 1 = 4 2 + 2 Γ R Δ t + Ω R 2 Δ t 2 Ξ κ λ R | n 2 2 Γ R Δ t + Ω R 2 Δ t 2 2 + 2 Γ R Δ t + Ω R 2 Δ t 2 Ξ κ λ R | n 1 + 4 Γ R Ω R Δ t 2 2 + 2 Γ R Δ t + Ω R 2 Δ t 2 ξ R S κ λ R | n .
N | n + 1 / 2 = 2 τ c Δ t 2 τ c + Δ t N | n 1 / 2 + τ c Δ t 2 τ c + Δ t ɛ 0 η TPA 2 ω 0 ( E α 2 | n + E β 2 | n + E γ 2 | n ) 2 .
Δ n FC | n + 1 Δ n FC | n Δ n FC | n + 1 / 2 Δ n FC | n 1 / 2 ,
E κ | n + 1 = κ κ + E κ | n + D κ NL | n + 1 D κ NL | n ɛ 0 κ + ν κ Ξ κ ν R | n + 1 E ν | n + 1 Ξ κ ν R | n E ν | n κ + ,
κ ± = 1 + Ξ κ κ R | n + 1 / 2 ± 1 / 2 + ( ɛ 2 ± η TPA Δ t / 2 ) S κ K | n + 1 / 2 ± 1 / 2 ± n 0 c Δ t 4 ( α L + Δ α FC | n + 1 / 2 ) ± n 0 2 [ ( 1 ± 2 ) Δ n FC | n + 1 / 2 Δ n FC | n 1 / 2 ] .
E κ | n + 1 = κ κ + E κ | n + D κ NL | n + 1 D κ NL | n ɛ 0 κ + Ξ κ ν R | n + 1 E ν | n + 1 Ξ κ ν R | n E ν | n κ + ,
φ 0 = tan 1 1 1 2 sin 2 ( 2 ϑ ) .

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