General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics
Optics Express, Vol. 14, Issue 18, pp. 8305-8310 (2006)
http://dx.doi.org/10.1364/OE.14.008305
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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
1. Introduction
1 . T. Kashiwa and I. Fukai , “ A treatment by FDTD method of dispersive characteristics associated with electronic polarization ,” Microwave Optics 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 ,” IEEE J. of 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]
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]
2. Electromagnetic wave propagation in dispersive nonlinear materials
3. General vector auxiliary differential equation FDTD method
3.1. Linear Lorentz polarization
3.2. Nonlinear Kerr polarization
3.3. Nonlinear Raman polarization
3.4. Solution for electric field
4. GVADE FDTD simulation of temporal and spatial solitons
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]
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]
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 ,” IEEE J. of Lightwave Technol. 23 , 855 – 863 ( 2005 ). [CrossRef]
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]
5. Conclusion
References and links
1 . | T. Kashiwa and I. Fukai , “ A treatment by FDTD method of dispersive characteristics associated with electronic polarization ,” Microwave Optics 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 ,” IEEE J. of 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] |
OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5650) Nonlinear optics : Raman effect
ToC Category:
Nonlinear Optics
History
Original Manuscript: June 26, 2006
Revised Manuscript: August 22, 2006
Manuscript Accepted: August 24, 2006
Published: September 1, 2006
Citation
Jethro H. Greene and Allen Taflove, "General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics," Opt. Express 14, 8305-8310 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-18-8305
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References
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- A. T. Ryan and G. P. Agrawal, "Spatiotemporal coupling in dispersive nonlinear planar waveguides," J. Opt. Soc. Am. B 12, 2382-2389 (1995). [CrossRef]
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