Abstract

The auxiliary differential equation finite-difference time-domain method for modeling electromagnetic wave propagation in dispersive nonlinear materials is applied to problems where the electric field is not constrained to a single vector component. A full-vector Maxwell’s equations solution incorporating multiple-pole linear Lorentz, nonlinear Kerr, and nonlinear Raman polarizations is presented. The application is illustrated by modeling a spatial soliton having two orthogonal electric field components. To the best of our knowledge, the numerical technique presented here is the first to model electromagnetic wave propagation with two or three orthogonal vector components in dispersive nonlinear materials. This technique offers the possibility of modeling sub-wavelength interactions of vector spatial solitons.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Opt. Tech. Lett. 3, 203-205 (1990).
    [CrossRef]
  2. R. M. Joseph, S. C. Hagness, and A. Taflove, "Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses," Opt. Lett. 16, 1412-1414 (1991).
    [CrossRef] [PubMed]
  3. 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]
  4. 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. Lightwave Technol. 23, 855-863 (2005).
    [CrossRef]
  5. 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]
  6. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, 2001).
  7. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, CA, 2003).
  8. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, MA, 2005).
  9. A. T. Ryan and G. P. Agrawal, "Spatiotemporal coupling in dispersive nonlinear planar waveguides," J. Opt. Soc. Am. B 12, 2382-2389 (1995).
    [CrossRef]

2005 (1)

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]

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]

1995 (1)

1991 (1)

1990 (1)

T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Opt. Tech. Lett. 3, 203-205 (1990).
[CrossRef]

Agrawal, G. P.

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]

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]

Fukai, I.

T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Opt. Tech. Lett. 3, 203-205 (1990).
[CrossRef]

Hagness, S. C.

Joseph, R. M.

Karasawa, 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).
[CrossRef]

Kashiwa, T.

T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Opt. Tech. Lett. 3, 203-205 (1990).
[CrossRef]

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. Lightwave Technol. 23, 855-863 (2005).
[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]

Mizuta, Y.

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]

Morita, R.

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]

Nakamura, S.

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. Lightwave Technol. 23, 855-863 (2005).
[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]

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).
[CrossRef]

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).
[CrossRef]

Ryan, A. T.

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).
[CrossRef]

Shigekawa, H.

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]

Sone, H.

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]

Taflove, A.

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).
[CrossRef]

Takasawa, N.

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]

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).
[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 Photonics Technol. Lett. (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]

J. Lightwave Technol. (1)

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

Microwave Opt. Tech. Lett. (1)

T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive characteristics associated with electronic polarization," Microwave Opt. Tech. Lett. 3, 203-205 (1990).
[CrossRef]

Opt. Lett. (1)

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, 2001).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, CA, 2003).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, MA, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

GVADE simulation results for temporal soliton propagation in a dispersive nonlinear material. These results reproduce those of [5].

Fig. 2.
Fig. 2.

GVADE simulation results of a +x-directed higher-order spatial soliton with field components {Ex ,Ey ,Hz } in a material with a three-pole Sellmeier linear dispersion, an instantaneous Kerr nonlinearity, and a dispersive Raman nonlinearity: magnitude of Hz .

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

× E = μ 0 H t ,
× H = ε 0 E t + J ,
J Lorentz = p = 1 3 J Lorentz p ,
P ˜ Lorentz p = ε 0 χ ( 1 ) E ˜ = ε 0 β p ω p 2 ω p 2 ω 2 E ˜ ,
P NL r t = ε 0 χ ( 3 ) t t 1 t t 2 t t 3 E r t 1 E r t 2 E r t 3 d t 1 d t 2 d t 3 ,
P NL ( t ) = ε 0 χ 0 ( 3 ) E g ( t t ) E ( t ) 2 dt ,
g ( t ) = αδ ( t ) + ( 1 α ) g Raman ( t ) ,
g Raman ( t ) = ( τ 1 2 + τ 2 2 τ 1 τ 2 2 ) exp ( t τ 2 ) sin ( t τ 1 ) U ( t ) ,
P Kerr ( t ) = ε 0 χ 0 ( 3 ) E αδ ( t t ) E ( t ) 2 dt = α ε 0 χ 0 ( 3 ) E 2 E ,
J Kerr ( t ) = P Kerr t = t α ε 0 χ 0 ( 3 ) E 2 E .
P Raman ( t ) = ε 0 E [ χ Raman ( 3 ) ( t ) * E 2 ] ,
χ Raman ( 3 ) ( t ) = ( 1 α ) χ Raman ( 3 ) ( t ) g Raman ( t ) .
J ˜ Lorentz p = ε 0 β p ω p 2 ( ω p 2 ω 2 ) E ˜ .
ω p 2 J Lorentz p + 2 J Lorentz p t 2 = ε 0 β p ω p 2 E t .
J Lorentz p n + 1 = α p J Lorentz p n J Lorentz p n 1 + γ p E n + 1 E n 1 2 Δ t
α p = 2 ω p 2 ( Δ t ) ; γ p = ε 0 β p ω p 2 ( Δ t ) 2 .
J Lorentz p n + 1 2 = 1 2 [ ( 1 + α p ) J Lorentz p n J Lorentz p n 1 + γ p 2 Δ t ( E n + 1 E n 1 ) ] .
J Kerr n + 1 2 = α ε 0 χ 0 ( 3 ) Δ t { ( E n + 1 ) 2 E n + 1 ( E n ) 2 E n } .
S ( t ) χ Raman ( 3 ) ( t ) * E ( t ) 2
S ( ω ) χ Raman ( 3 ) ( ω ) { E ( t ) 2 } ,
χ Raman ( 3 ) ( ω ) = ( 1 α ) χ 0 ( 3 ) ω Raman 2 ω Raman 2 + 2 δ Raman ω 2 ,
ω Raman τ 1 2 + τ 2 2 τ 1 2 τ 2 2 ; δ Raman = 1 τ 2 .
ω Raman 2 + 2 δ Raman S t + 2 S t 2 = ( 1 α ) χ 0 ( 3 ) ω Raman 2 E 2 .
S n + 1 = [ 2 ω Raman 2 ( Δ t ) 2 δ Raman Δ t + 1 ] S n + [ δ Raman Δ t 1 δ Raman Δ t + 1 ] S n 1
+ [ ( 1 α ) χ 0 ( 3 ) ω Raman 2 ( Δ t ) 2 δ Raman Δ t + 1 ] ( E n ) 2 .
J Raman n + 1 2 = ε 0 Δ t ( E n + 1 S n + 1 E n S n ) .
× H n + 1 2 = ε 0 Δ t ( E n + 1 E n ) + p = 1 3 J Lorentz p n + 1 2 + J Kerr n + 1 2 + J Raman n + 1 2 .
[ X Y ] = × H n + 1 2 + ε 0 Δ t ( E n + 1 E n )
+ 1 2 p = 1 3 [ ( 1 + α p ) J Lorentz p n J Lorentz p n 1 + γ p 2 Δ t ( E n + 1 E n 1 ) ]
+ α ε 0 χ 0 ( 3 ) Δ t { ( E n + 1 ) 2 E n + 1 ( E n ) 2 E n } + ε 0 Δ t ( E n + 1 S n + 1 E n S n )
[ G x g + 1 G y g + 1 ] = [ G x g G y g ] ( J 1 [ X Y ] ) g ,
J 11 = ε 0 Δ t + 1 t ( γ 1 + γ 2 + γ 3 ) + ε 0 Δ t [ α χ 0 ( 3 ) ( 3 G x 2 + G y 2 ) + S n + 1 ] ,
J 12 = 2 ε 0 Δ t α χ 0 ( 3 ) G x G y , J 21 = 2 ε 0 Δ t α χ 0 ( 3 ) G x G y ,
J 22 = ε 0 Δ t + 1 t ( γ 1 + γ 2 + γ 3 ) + ε 0 Δ t [ α χ 0 ( 3 ) ( G x 2 + 3 G y 2 ) + S n + 1 ] ,
H z ( t ) = H 0 sin ( ω c t ) sech ( y w ) ,

Metrics