Abstract

For the first time, an unconditionally stable finite-difference time-domain (FDTD) method for 3-D simulation of dispersive nonlinear media is presented. By applying a new adopted alternating-direction implicit (ADI) time-splitting scheme and the auxiliary differential equation (ADE) technique, the time-step in the FDTD simulations can be increased much beyond the Courant-Friedrichs-Lewy (CFL) stability limit. Thus, in comparison to the classical nonlinear FDTD method, the computational time for the proposed approach is decreased significantly while maintaining a reasonable level of accuracy. Numerical examples are presented to demonstrate the validity, stability, accuracy and computational efficiency of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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  49. J. H. Greene and A. Taflove, “Scattering of spatial optical solitons by subwavelength air holes,” IEEE Microw. Wirel. Components Lett. 17, 760–762 (2007).
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2017 (1)

S.-M. Sadrpour, V. Nayyeri, M. Soleimani, and O. M. Ramahi, “A new efficient unconditionally stable finite-difference time-domain solution of the wave equation,” IEEE Transactions on Antennas Propag. 65, 3114–3121 (2017).
[Crossref]

2015 (1)

B. Salski, T. Karpisz, and R. Buczynski, “Electromagnetic modeling of third-order nonlinearities in photonic crystal fibers using a vector two-dimensional FDTD algorithm,” J. Light. Technol. 33, 2905–2912 (2015).
[Crossref]

2011 (4)

D. Li and C. D. Sarris, “Time-domain modeling of nonlinear optical structures with extended stability FDTD schemes,” J. Light. Technol. 29, 1003–1010 (2011).
[Crossref]

I. S. Maksymov, A. A. Sukhorukov, A. V. Lavrinenko, and Y. S. Kivshar, “Comparative study of FDTD-adopted numerical algorithms for Kerr nonlinearities,” IEEE Antennas Wirel. Propag. Lett. 10, 143–146 (2011).
[Crossref]

V. Nayyeri, M. Soleimani, J. R. Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method,” IEEE Transactions on Antennas Propag. 59, 2268–2279 (2011).
[Crossref]

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 9 (2011).
[Crossref]

2010 (1)

2009 (2)

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (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]

2007 (2)

J. H. Greene and A. Taflove, “Scattering of spatial optical solitons by subwavelength air holes,” IEEE Microw. Wirel. Components Lett. 17, 760–762 (2007).
[Crossref]

E. L. Tan, “Unconditionally stable LOD–FDTD method for 3-D Maxwell’s equations,” IEEE Microw. Wirel. Components Lett. 17, 85–87 (2007).
[Crossref]

2006 (4)

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Light. Technol. 24, 1433 (2006).
[Crossref]

J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
[Crossref] [PubMed]

C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
[Crossref]

G. Sun and C. W. Trueman, “Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 54, 2275–2284 (2006).
[Crossref]

2005 (4)

J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett. 41, 1046–1047 (2005).
[Crossref]

S. Nakamura, N. Takasawa, and Y. Koyamada, “Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12-fs laser pulse propagation in a silica fiber,” J. Light. Technol. 23, 855 (2005).
[Crossref]

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

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Light. Technol. 23, 2094 (2005).
[Crossref]

2004 (1)

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[Crossref]

2003 (3)

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]

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158, 485–505 (2003).
[Crossref]

E. P. Kosmidou and T. D. Tsiboukis, “An unconditionally stable ADI-FDTD algorithm for nonlinear materials,” Opt. Quantum Electron 32, 931–946 (2003).
[Crossref]

2002 (1)

S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
[Crossref]

2000 (2)

D. Sullivan, J. Liu, and M. Kuzyk, “Three-dimensional optical pulse simulation using the FDTD method,” IEEE Transactions on Microw. Theory Tech. 48, 1127–1133 (2000).
[Crossref]

F. Zhen, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 48, 1550–1558 (2000).
[Crossref]

1999 (2)

F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guid. Wave Lett. 9, 441–443 (1999).
[Crossref]

T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Transactions on Microw. Theory Tech. 47, 2003–2007 (1999).
[Crossref]

1998 (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Transactions on Antennas Propag. 46, 334–340 (1998).
[Crossref]

1997 (2)

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Transactions on Antennas Propag. 45, 364–374 (1997).
[Crossref]

P. M. Goorjian and Y. Silberberg, “Numerical simulations of light bullets using the full-vector time-dependent nonlinear Maxwell equations,” JOSA B 14, 3253–3260 (1997).
[Crossref]

1996 (2)

C. V. Hile and W. L. Kath, “Numerical solutions of Maxwell’s equations for nonlinear-optical pulse propagation,” JOSA B 13, 1135–1145 (1996).
[Crossref]

D. M. Sullivan, “Z-transform theory and the FDTD method,” IEEE Transactions on Antennas Propag. 44, 28–34 (1996).
[Crossref]

1995 (1)

D. M. Sullivan, “Nonlinear FDTD formulations using Z-transforms,” IEEE Transactions on Microw. Theory Tech. 43, 676–682 (1995).
[Crossref]

1994 (1)

R. W. Ziolkowski and J. B. Judkins, “Nonlinear finite-difference time-domain modeling of linear and nonlinear corrugated waveguides,” JOSA B 11, 1565–1575 (1994).
[Crossref]

1993 (2)

R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” JOSA B 10, 186–198 (1993).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Applications of the nonlinear finite difference time domain (NL-FDTD) method to pulse propagation in nonlinear media: Self-focusing and linear-nonlinear interfaces,” Radio Sci. 28,901–911 (1993).
[Crossref]

1992 (1)

1991 (1)

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas Propag. 14, 302–307 (1966).
[Crossref]

1928 (1)

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen differenzengleichungen der mathematischen physik,” Math. Annalen 100, 32–74 (1928).
[Crossref]

Adibi, A.

C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
[Crossref]

Agrawal, G. P.

Al-Mudhaffar, H. M.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).
[Crossref]

Alsunaidi, M. A.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).
[Crossref]

Boor, C. De

S. D. Conte and C. De Boor, Elementary Numerical Analysis: an Algorithmic Approach (SIAM, 2017), vol. 78, pp. 153–156.

Buczynski, R.

B. Salski, T. Karpisz, and R. Buczynski, “Electromagnetic modeling of third-order nonlinearities in photonic crystal fibers using a vector two-dimensional FDTD algorithm,” J. Light. Technol. 33, 2905–2912 (2015).
[Crossref]

Chen, Z.

F. Zhen, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 48, 1550–1558 (2000).
[Crossref]

F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guid. Wave Lett. 9, 441–443 (1999).
[Crossref]

Claps, R.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Light. Technol. 23, 2094 (2005).
[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]

Conte, S. D.

S. D. Conte and C. De Boor, Elementary Numerical Analysis: an Algorithmic Approach (SIAM, 2017), vol. 78, pp. 153–156.

Corporation, O.

O. Corporation, OptiFDTD Technical Background and Tutorials (Optiwave Systems Inc., ON, Canada, 2005).

Courant, R.

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen differenzengleichungen der mathematischen physik,” Math. Annalen 100, 32–74 (1928).
[Crossref]

Dehmollaian, M.

V. Nayyeri, M. Soleimani, J. R. Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method,” IEEE Transactions on Antennas Propag. 59, 2268–2279 (2011).
[Crossref]

Dimitropoulos, D.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Light. Technol. 23, 2094 (2005).
[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]

Dissanayake, C. M.

Fauchet, P. M.

Fornberg, B.

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158, 485–505 (2003).
[Crossref]

Freude, W.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[Crossref]

Friedrichs, K.

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen differenzengleichungen der mathematischen physik,” Math. Annalen 100, 32–74 (1928).
[Crossref]

Fujii, M.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[Crossref]

Fukuda, H.

Goorjian, P. M.

P. M. Goorjian and Y. Silberberg, “Numerical simulations of light bullets using the full-vector time-dependent nonlinear Maxwell equations,” JOSA B 14, 3253–3260 (1997).
[Crossref]

P. M. Goorjian and A. Taflove, “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992).
[Crossref]

Greene, J. H.

J. H. Greene and A. Taflove, “Scattering of spatial optical solitons by subwavelength air holes,” IEEE Microw. Wirel. Components Lett. 17, 760–762 (2007).
[Crossref]

J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
[Crossref] [PubMed]

Hagness, S. C.

Han, Y.

Hile, C. V.

C. V. Hile and W. L. Kath, “Numerical solutions of Maxwell’s equations for nonlinear-optical pulse propagation,” JOSA B 13, 1135–1145 (1996).
[Crossref]

Hon, N. K.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 9 (2011).
[Crossref]

Itabashi, S.

Jafarpour, A.

C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
[Crossref]

Jalali, B.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 9 (2011).
[Crossref]

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Light. Technol. 23, 2094 (2005).
[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]

Joseph, R. M.

Judkins, J. B.

R. W. Ziolkowski and J. B. Judkins, “Nonlinear finite-difference time-domain modeling of linear and nonlinear corrugated waveguides,” JOSA B 11, 1565–1575 (1994).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” JOSA B 10, 186–198 (1993).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Applications of the nonlinear finite difference time domain (NL-FDTD) method to pulse propagation in nonlinear media: Self-focusing and linear-nonlinear interfaces,” Radio Sci. 28,901–911 (1993).
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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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Karpisz, T.

B. Salski, T. Karpisz, and R. Buczynski, “Electromagnetic modeling of third-order nonlinearities in photonic crystal fibers using a vector two-dimensional FDTD algorithm,” J. Light. Technol. 33, 2905–2912 (2015).
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I. S. Maksymov, A. A. Sukhorukov, A. V. Lavrinenko, and Y. S. Kivshar, “Comparative study of FDTD-adopted numerical algorithms for Kerr nonlinearities,” IEEE Antennas Wirel. Propag. Lett. 10, 143–146 (2011).
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E. P. Kosmidou and T. D. Tsiboukis, “An unconditionally stable ADI-FDTD algorithm for nonlinear materials,” Opt. Quantum Electron 32, 931–946 (2003).
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Koyamada, Y.

S. Nakamura, N. Takasawa, and Y. Koyamada, “Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12-fs laser pulse propagation in a silica fiber,” J. Light. Technol. 23, 855 (2005).
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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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D. Sullivan, J. Liu, and M. Kuzyk, “Three-dimensional optical pulse simulation using the FDTD method,” IEEE Transactions on Microw. Theory Tech. 48, 1127–1133 (2000).
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I. S. Maksymov, A. A. Sukhorukov, A. V. Lavrinenko, and Y. S. Kivshar, “Comparative study of FDTD-adopted numerical algorithms for Kerr nonlinearities,” IEEE Antennas Wirel. Propag. Lett. 10, 143–146 (2011).
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J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158, 485–505 (2003).
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C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
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C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Light. Technol. 24, 1433 (2006).
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D. Sullivan, J. Liu, and M. Kuzyk, “Three-dimensional optical pulse simulation using the FDTD method,” IEEE Transactions on Microw. Theory Tech. 48, 1127–1133 (2000).
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Maksymov, I. S.

I. S. Maksymov, A. A. Sukhorukov, A. V. Lavrinenko, and Y. S. Kivshar, “Comparative study of FDTD-adopted numerical algorithms for Kerr nonlinearities,” IEEE Antennas Wirel. Propag. Lett. 10, 143–146 (2011).
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C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Light. Technol. 24, 1433 (2006).
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M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).
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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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V. Nayyeri, M. Soleimani, J. R. Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method,” IEEE Transactions on Antennas Propag. 59, 2268–2279 (2011).
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C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett. 41, 1046–1047 (2005).
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A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Transactions on Antennas Propag. 46, 334–340 (1998).
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S. Nakamura, N. Takasawa, and Y. Koyamada, “Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12-fs laser pulse propagation in a silica fiber,” J. Light. Technol. 23, 855 (2005).
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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett. 41, 1046–1047 (2005).
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T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Transactions on Microw. Theory Tech. 47, 2003–2007 (1999).
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S.-M. Sadrpour, V. Nayyeri, M. Soleimani, and O. M. Ramahi, “A new efficient unconditionally stable finite-difference time-domain solution of the wave equation,” IEEE Transactions on Antennas Propag. 65, 3114–3121 (2017).
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V. Nayyeri, M. Soleimani, J. R. Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method,” IEEE Transactions on Antennas Propag. 59, 2268–2279 (2011).
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S.-M. Sadrpour, V. Nayyeri, and M. Soleimani, “A new 2d unconditionally stable finite-difference time-domain algorithm based on the crank-nicolson scheme,” in 2016 IEEE International Conference on Computational Electromagnetics (ICCEM), (IEEE, 2016), pp. 55–57.

Ohtani, M.

S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
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Russer, P.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
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Sadrpour, S.-M.

S.-M. Sadrpour, V. Nayyeri, M. Soleimani, and O. M. Ramahi, “A new efficient unconditionally stable finite-difference time-domain solution of the wave equation,” IEEE Transactions on Antennas Propag. 65, 3114–3121 (2017).
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S.-M. Sadrpour, V. Nayyeri, and M. Soleimani, “A new 2d unconditionally stable finite-difference time-domain algorithm based on the crank-nicolson scheme,” in 2016 IEEE International Conference on Computational Electromagnetics (ICCEM), (IEEE, 2016), pp. 55–57.

Sakagami, I.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
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Salski, B.

B. Salski, T. Karpisz, and R. Buczynski, “Electromagnetic modeling of third-order nonlinearities in photonic crystal fibers using a vector two-dimensional FDTD algorithm,” J. Light. Technol. 33, 2905–2912 (2015).
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Sarris, C. D.

D. Li and C. D. Sarris, “Time-domain modeling of nonlinear optical structures with extended stability FDTD schemes,” J. Light. Technol. 29, 1003–1010 (2011).
[Crossref]

Shibayama, J.

J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett. 41, 1046–1047 (2005).
[Crossref]

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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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S.-M. Sadrpour, V. Nayyeri, M. Soleimani, and O. M. Ramahi, “A new efficient unconditionally stable finite-difference time-domain solution of the wave equation,” IEEE Transactions on Antennas Propag. 65, 3114–3121 (2017).
[Crossref]

V. Nayyeri, M. Soleimani, J. R. Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method,” IEEE Transactions on Antennas Propag. 59, 2268–2279 (2011).
[Crossref]

S.-M. Sadrpour, V. Nayyeri, and M. Soleimani, “A new 2d unconditionally stable finite-difference time-domain algorithm based on the crank-nicolson scheme,” in 2016 IEEE International Conference on Computational Electromagnetics (ICCEM), (IEEE, 2016), pp. 55–57.

Soltani, M.

C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
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S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
[Crossref]

Soref, R.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 9 (2011).
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I. S. Maksymov, A. A. Sukhorukov, A. V. Lavrinenko, and Y. S. Kivshar, “Comparative study of FDTD-adopted numerical algorithms for Kerr nonlinearities,” IEEE Antennas Wirel. Propag. Lett. 10, 143–146 (2011).
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D. Sullivan, J. Liu, and M. Kuzyk, “Three-dimensional optical pulse simulation using the FDTD method,” IEEE Transactions on Microw. Theory Tech. 48, 1127–1133 (2000).
[Crossref]

Sullivan, D. M.

D. M. Sullivan, “Z-transform theory and the FDTD method,” IEEE Transactions on Antennas Propag. 44, 28–34 (1996).
[Crossref]

D. M. Sullivan, “Nonlinear FDTD formulations using Z-transforms,” IEEE Transactions on Microw. Theory Tech. 43, 676–682 (1995).
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G. Sun and C. W. Trueman, “Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 54, 2275–2284 (2006).
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Tahara, M.

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
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Takahashi, J.

Takahashi, M.

Takasawa, N.

S. Nakamura, N. Takasawa, and Y. Koyamada, “Comparison between finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation and experimental results for slightly chirped 12-fs laser pulse propagation in a silica fiber,” J. Light. Technol. 23, 855 (2005).
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E. L. Tan, “Unconditionally stable LOD–FDTD method for 3-D Maxwell’s equations,” IEEE Microw. Wirel. Components Lett. 17, 85–87 (2007).
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G. Sun and C. W. Trueman, “Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 54, 2275–2284 (2006).
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E. P. Kosmidou and T. D. Tsiboukis, “An unconditionally stable ADI-FDTD algorithm for nonlinear materials,” Opt. Quantum Electron 32, 931–946 (2003).
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Watanabe, T.

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C. M. Reinke, A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani, A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain method for the simulation of anisotropic, χ(2), and χ(3) optical effects,” J. Light. Technol. 24, 624–634 (2006).
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Yamashita, M.

S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
[Crossref]

Yamauchi, J.

J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett. 41, 1046–1047 (2005).
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K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas Propag. 14, 302–307 (1966).
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Yin, L.

York, R. A.

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Transactions on Antennas Propag. 46, 334–340 (1998).
[Crossref]

Yoshida, N.

S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
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F. Zhen, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 48, 1550–1558 (2000).
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F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guid. Wave Lett. 9, 441–443 (1999).
[Crossref]

Zhen, F.

F. Zhen, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 48, 1550–1558 (2000).
[Crossref]

Zheng, F.

F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guid. Wave Lett. 9, 441–443 (1999).
[Crossref]

Ziolkowski, R. W.

R. W. Ziolkowski and J. B. Judkins, “Nonlinear finite-difference time-domain modeling of linear and nonlinear corrugated waveguides,” JOSA B 11, 1565–1575 (1994).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Applications of the nonlinear finite difference time domain (NL-FDTD) method to pulse propagation in nonlinear media: Self-focusing and linear-nonlinear interfaces,” Radio Sci. 28,901–911 (1993).
[Crossref]

R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” JOSA B 10, 186–198 (1993).
[Crossref]

Electron. Lett. (1)

J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett. 41, 1046–1047 (2005).
[Crossref]

IEEE Antennas Wirel. Propag. Lett. (1)

I. S. Maksymov, A. A. Sukhorukov, A. V. Lavrinenko, and Y. S. Kivshar, “Comparative study of FDTD-adopted numerical algorithms for Kerr nonlinearities,” IEEE Antennas Wirel. Propag. Lett. 10, 143–146 (2011).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2-D Kerr and Raman nonlinear dispersive media,” IEEE J. Quantum Electron. 40, 175–182 (2004).
[Crossref]

IEEE Microw. Guid. Wave Lett. (1)

F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guid. Wave Lett. 9, 441–443 (1999).
[Crossref]

IEEE Microw. Wirel. Components Lett. (2)

E. L. Tan, “Unconditionally stable LOD–FDTD method for 3-D Maxwell’s equations,” IEEE Microw. Wirel. Components Lett. 17, 85–87 (2007).
[Crossref]

J. H. Greene and A. Taflove, “Scattering of spatial optical solitons by subwavelength air holes,” IEEE Microw. Wirel. Components Lett. 17, 760–762 (2007).
[Crossref]

IEEE Photonics Technol. Lett. (2)

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).
[Crossref]

S. Nakamura, Y. Koyamada, N. Yoshida, N. Karasawa, H. Sone, M. Ohtani, Y. Mizuta, R. Morita, H. Shigekawa, and M. Yamashita, “Finite-difference time-domain calculation with all parameters of Sellmeier’s fitting equation for 12-fs laser pulse propagation in a silica fiber,” IEEE Photonics Technol. Lett. 14, 480–482 (2002).
[Crossref]

IEEE Transactions on Antennas Propag. (6)

R. M. Joseph and A. Taflove, “FDTD Maxwell’s equations models for nonlinear electrodynamics and optics,” IEEE Transactions on Antennas Propag. 45, 364–374 (1997).
[Crossref]

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Transactions on Antennas Propag. 46, 334–340 (1998).
[Crossref]

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas Propag. 14, 302–307 (1966).
[Crossref]

D. M. Sullivan, “Z-transform theory and the FDTD method,” IEEE Transactions on Antennas Propag. 44, 28–34 (1996).
[Crossref]

V. Nayyeri, M. Soleimani, J. R. Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method,” IEEE Transactions on Antennas Propag. 59, 2268–2279 (2011).
[Crossref]

S.-M. Sadrpour, V. Nayyeri, M. Soleimani, and O. M. Ramahi, “A new efficient unconditionally stable finite-difference time-domain solution of the wave equation,” IEEE Transactions on Antennas Propag. 65, 3114–3121 (2017).
[Crossref]

IEEE Transactions on Microw. Theory Tech. (5)

T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Transactions on Microw. Theory Tech. 47, 2003–2007 (1999).
[Crossref]

F. Zhen, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 48, 1550–1558 (2000).
[Crossref]

G. Sun and C. W. Trueman, “Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method,” IEEE Transactions on Microw. Theory Tech. 54, 2275–2284 (2006).
[Crossref]

D. M. Sullivan, “Nonlinear FDTD formulations using Z-transforms,” IEEE Transactions on Microw. Theory Tech. 43, 676–682 (1995).
[Crossref]

D. Sullivan, J. Liu, and M. Kuzyk, “Three-dimensional optical pulse simulation using the FDTD method,” IEEE Transactions on Microw. Theory Tech. 48, 1127–1133 (2000).
[Crossref]

J. Appl. Phys. (1)

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 9 (2011).
[Crossref]

J. Comput. Appl. Math. (1)

J. Lee and B. Fornberg, “A split step approach for the 3-D Maxwell’s equations,” J. Comput. Appl. Math. 158, 485–505 (2003).
[Crossref]

J. Light. Technol. (6)

B. Salski, T. Karpisz, and R. Buczynski, “Electromagnetic modeling of third-order nonlinearities in photonic crystal fibers using a vector two-dimensional FDTD algorithm,” J. Light. Technol. 33, 2905–2912 (2015).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 A nonlinear slab in a TEM waveguide made of PEC and PMC walls.
Fig. 2
Fig. 2 Recorded values of the transmitted Ex through the 3-D nonlinear slab.
Fig. 3
Fig. 3 Normalized power intensity of the transmitted Ex through the 3-D nonlinear slab.
Fig. 4
Fig. 4 A nonlinear silicon waveguide configuration.
Fig. 5
Fig. 5 Recorded values of Ex at grid (1750,70).
Fig. 6
Fig. 6 Power intensity distribution over the silicon waveguide.
Fig. 7
Fig. 7 Scattering of spatial optical solitons by subwavelength air hole with a size of 150 nm × 250 nm whose center is located at x = 5 μm and y = 0 μm. The figure indicates |E| in the computational domain.

Tables (1)

Tables Icon

Table 1 Run Times of Solving the 3-D Nonlinear Slab

Equations (53)

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× E = μ 0 H t ,
× H = t [ ε 0 ε r E + P T ] ,
× H = ε 0 ε r E t + J T ,
J T = J L + J NL = P T t = P L t + P NL t
J L = p = 1 m J Lp ,
J ˜ Lp = ε 0 β p ω p 2 ( j ω ω p 2 ω 2 ) E ˜ ,
ω p 2 J Lp + 2 J Lp t 2 = ε 0 β p ω p 2 E t .
J Lp n + 1 2 = α p J Lp n 1 2 J Lp n 3 2 + γ p ( E n E n 1 )
α p = 2 ( ω p Δ t ) 2 , γ p = ε 0 β p ω p 2 Δ t .
J L n + 1 2 = p = 1 m { α p J Lp n 1 2 J Lp n 3 2 + γ p ( E n E n 1 ) } .
P NL ( r , t ) = ε 0 t t t χ ( 3 ) ( t τ 1 , t τ 2 , t τ 3 ) E ( r , τ 1 ) E ( r , τ 2 ) E ( r , τ 3 ) d τ 1 d τ 2 d τ 3 ,
P NL ( t ) = ε 0 χ 0 ( 3 ) E ( t ) t g ( t τ ) | E ( τ ) | 2 d τ
g ( t ) = ( 1 α ) g Raman ( t ) + α δ ( t ) ,
P NL = P R + P K
P R ( t ) = ε 0 ( 1 α ) [ χ 0 ( 3 ) g Raman ( t ) * | E ( t ) | 2 ] E ( t ) ,
P K ( t ) = ε 0 α χ 0 ( 3 ) | E ( t ) | 2 E ( t ) ,
S R ( t ) = ( 1 α ) χ 0 ( 3 ) g Raman ( t ) * | E ( t ) | 2 .
S R ˜ = ( 1 α ) χ 0 ( 3 ) g ˜ Raman ( ω ) | E ˜ | 2 ,
g ˜ Raman ( ω ) = ω R 2 ω R 2 + 2 j ω δ R ω 2 .
( ω R 2 + 2 δ R t + 2 t 2 ) S R = ( 1 α ) χ 0 ( 3 ) ω R 2 | E | 2 .
S R n + 1 = α R S R n + β R S R n 1 + γ R | E n | 2 ,
α R = 2 ω R 2 Δ t 2 δ R Δ t + 1 , β R = δ R Δ t 1 δ R Δ t + 1 , γ R = ( 1 α ) χ 0 ( 3 ) ω R 2 Δ t 2 δ R Δ t + 1 .
J R = P R t = ε 0 t ( S R E ) .
J R n + 1 2 = ε 0 Δ t ( S R n + 1 E n + 1 S R n E n ) .
J K = P K t .
J K n + 1 2 = P K n + 1 P K n Δ t ,
P K n = α ε 0 χ 0 ( 3 ) | E n | 2 E n .
J K n + 1 2 = α ε 0 χ 0 ( 3 ) Δ t ( | E n + 1 | 2 E n + 1 | E n | 2 E n ) .
H x n + 1 H x n = Δ t μ D z E y n + 1 2 Δ t μ D y E z n + 1 2 , H y n + 1 H y n = Δ t μ D x E z n + 1 2 Δ t μ D z E x n + 1 2 , H z n + 1 H z n = Δ t μ D y E x n + 1 2 Δ t μ D x E y n + 1 2 ,
E x n + 1 E x n = Δ t ε 0 ε D y H z n + 1 2 Δ t ε 0 ε D z H y n + 1 2 Δ t ε 0 ε J T n + 1 2 , E y n + 1 E y n = Δ t ε 0 ε D z H x n + 1 2 Δ t ε 0 ε D x H z n + 1 2 Δ t ε 0 ε J T n + 1 2 , E z n + 1 E z n = Δ t ε 0 ε D x H y n + 1 2 Δ t ε 0 ε D y H x n + 1 2 Δ t ε 0 ε J T n + 1 2 ,
D x   f i , j , k = 1 Δ x ( f i + 1 2 , j , k f i 1 2 , j , k )
J T ρ n + 1 2 = J N L ρ n + 1 2 + J L ρ n + 1 2 = J R ρ n + 1 2 + J K ρ n + 1 2 + J L ρ n + 1 2 ,
J T ρ n + 1 2 = ε 0 Δ t S R n + 1 E ρ n + 1 ε 0 Δ t S R n E ρ n + α ε 0 χ 0 ( 3 ) Δ t | E n + 1 | 2 E ρ n + 1 α ε 0 χ 0 ( 3 ) Δ t | E n | 2 E ρ n + J L ρ n + 1 2 .
H x n + 1 2 H x n = Δ t 2 μ D z E y n + 1 2 Δ t 2 μ D y E z n , H y n + 1 2 H y n = Δ t 2 μ D x E z n + 1 2 Δ t 2 μ D z E x n , H z n + 1 2 H z n = Δ t 2 μ D y E x n + 1 2 Δ t 2 μ D x E y n ,
E x n + 1 2 E x n = Δ t 2 ε 0 ε D y H z n + 1 2 Δ t 2 ε 0 ε D z H y n Δ t ε 0 ε J 1 x n , E y n + 1 2 E y n = Δ t 2 ε 0 ε D z H x n + 1 2 Δ t 2 ε 0 ε D x H z n Δ t ε 0 ε J 1 y n , E z n + 1 2 E z n = Δ t 2 ε 0 ε D x H y n + 1 2 Δ t 2 ε 0 ε D y H x n Δ t ε 0 ε J 1 z n ,
H x n + 1 H x n + 1 2 = Δ t 2 μ D z E y n + 1 2 Δ t 2 μ D y E z n + 1 , H y n + 1 H y n + 1 2 = Δ t 2 μ D x E z n + 1 2 Δ t 2 μ D z E x n + 1 , H z n + 1 H z n + 1 2 = Δ t 2 μ D y E x n + 1 2 Δ t 2 μ D x E y n + 1 ,
E x n + 1 E x n + 1 2 = Δ t 2 ε 0 ε D y H z n + 1 2 Δ t 2 ε 0 ε D z H y n + 1 Δ t ε 0 ε J 2 x n + 1 , E y n + 1 E y n + 1 2 = Δ t 2 ε 0 ε D z H x n + 1 2 Δ t 2 ε 0 ε D x H z n + 1 Δ t ε 0 ε J 2 y n + 1 , E z n + 1 E z n + 1 2 = Δ t 2 ε 0 ε D x H y n + 1 2 Δ t 2 ε 0 ε D y H x n + 1 Δ t ε 0 ε J 2 z n + 1 .
J T ρ n + 1 2 = J 1 ρ n + J 2 ρ n + 1 ,        ( ρ = x , y , z ) .
J 1 ρ n = ε 0 Δ t S R n E ρ n α ε 0 χ 0 ( 3 ) Δ t | E n | 2 E ρ n + J L ρ n + 1 2 ,
J 2 ρ n + 1 = ε 0 Δ t S R n + 1 E ρ n + 1 + α ε 0 χ 0 ( 3 ) Δ t | E n + 1 | 2 E ρ n + 1 .
( 1 a D y 2 ) E x n + 1 2 = A n E x n a D x D y E y n + b D y H z n b D z H y n 2 b J L x n + 1 2 , ( 1 a D z 2 ) E y n + 1 2 = A n E y n a D y D z E z n + b D z H x n b D x H z n 2 b J L y n + 1 2 , ( 1 a D x 2 ) E z n + 1 2 = A n E z n a D z D x E x n + b D x H y n b D y H x n 2 b J L z n + 1 2 ,
( A n + 1 a D z 2 ) E x n + 1 = E x   n + 1 2 a D x D z E z   n + 1 2 + b D y H z   n + 1 2 b D z H y   n + 1 2 , ( A n + 1 a D x 2 ) E y n + 1 = E y   n + 1 2 a D y D x E x   n + 1 2 + b D z H x   n + 1 2 b D x H z   n + 1 2 , ( A n + 1 a D y 2 ) E z n + 1 = E z   n + 1 2 a D z D y E y   n + 1 2 + b D x H y   n + 1 2 b D y H x   n + 1 2 ,
A n + 1 = 1 + c S R n + 1 + d | E n + 1 | 2
a = Δ t 2 4 μ ε 0 ε ,    b = Δ t 2 ε 0 ε ,    c = 1 ε ,    d = α χ 0 ( 3 ) ε .
D x 2   f i , j , k = 1 ( Δ x ) 2 ( f i + 1 , j , k 2 f i , j , k + f i 1 , j , k ) .
| E n + 1 | 2 = ( E x n + 1 ) 2 + ( E y n + 1 ) 2 + ( E z n + 1 ) 2 ,
E n + 1 2 = E n + 1 + E n 2 .
E n + 1 = 2 E n + 1 2 E n
A n + 1 = 1 + c S R n + 1 + d | 2 E n + 1 2 E n | 2 .
ω R   2 = τ 1   2 + τ 2   2 τ 1   2 τ 2   2 ,      δ R = 1 τ 2 .
E x = exp  [ ( t t 0 ) 2 τ 2 ] × ( cos  ( 2 π f 1 t ) + cos  ( 2 π f 2 t ) )
H z = H 0  exp  [ ( t t 0 ) 2 τ 2 ] cos  ( 2 π f 3 t )
H z = H 0  cos  ( 2 π f 0 t )  sech ( y / w )

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