Propagation of Ultrashort Pulses in a Nonlinear Medium
Cao Long V. 1, Thuan Bui Dinh 1, Goldstein Piotr P. 2, Hung Nguyen Viet 2, Son Doan Hoai 3
1Institute of Physics, University of Zielona Góra,
ul. Prof. Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: vanlongcao@yahoo.com
2Soltan Institute for Nuclear Studies,
Hoża 69, 00-681 Warszawa, Poland
3Vinh University, 182 Le Duan str., Vinh, Nghe An, Vietnam
Received:
Received: 29 June 2010; accepted: 13 September 2010
DOI: 10.12921/cmst.2010.SI.02.77-96
OAI: oai:lib.psnc.pl:706
Abstract:
In this paper, using a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [8] we study in the specific case of Kerr media. An obtained ultrashort pulse propagation equation which is called Generalized Nonlinear Schrödinger Equation usually has a very complicated form and looking for its solutions is usually a “mission impossible”. Theoretical methods to solve this equation are effective only for some special cases. As an example we describe the method of a developed elliptic Jacobi function expansion. Several numerical methods of finding approximate solutions are simultaneously used. We focus mainly on the following methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon: the collapse of solitons.
Key words:
Generalized Nonlinear Schrödinger Equation, Kerr media, solitons, ultrashort pulses
References:
[1] R.W. Boyd, Nonlinear Optics. Academic Press Inc., 2003.
[2] G.P. Agrawal, Nonlinear Fiber Optics. Academic, San Diego, 2003.
[3] U. Bandelow, A. Demircan, M. Kesting, Simulation of Pulse Propagation in Nonlinear Optical Fibers. WIAS, 2003.
[4] Cao Long Van, Dinh Xuan Khoa, Marek Trippenbach, Introduction to Nonlinear Optics. Vinh 2003.
[5] Cao Long Van, P.P. Goldstein, A Concise Course in Nonlinear Partial Differential Equations. University of Zielona Gora 2008.
[6] V. Cao Long, P.P. Goldstein, M. Trippenbach, Acta Phys. Polonica A105, 437 (2004).
[7] V. Cao Long, P.P. Goldstein, S. Vu Ngoc, Acta Phys. Polonica A106, 843 (2004).
[8] V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan, Propagation Technique for Ultrashort Pulses. I. Computational Methods in Science and Technology 14 (1), 5 (2008).
[9] V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan, Propagation Technique for Ultrashort Pulses. II, Computational Methods in Science and Technology 14 (1), 13 (2008).
[10] V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan, Propagation Technique for Ultrashort Pulses. III. Computational Methods in Science and Technology 14 (1), 21 (2008).
[11] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego, 2003.
[12] Yang Qin, Dai Chao-qin, Zhang Jie-fang, Higher-Order Effects Induced Optical Solitons in Fiber. International Journal of Theoretical Physics 44, 1117 (2005).
[13] Cao Long Van, M. Trippenbach, Dinh Xuan Khoa, Nguyen Viet Hung, Phan Xuan Anh, Conference on Theoretical Physics, Sam Son, Vietnam, 12-14 August 2003; Journal of Science, Vinh University 1A, 50 (2003).
[14] Cao Long Van, M. Trippenbach, New Optical Solitary waves in the higher order nonlinear Schrödinger Equation. Scientific Biuletin 126, Technical University of Zielona Gora, Poland 2001, p. 19 (Cao Long Van ed.).
[15] R.S. Tasgal, Y.B. Band, private communication, 2003.
[16] P.L. Francois, J. Opt. Soc. Am. B 8, 276-293 (1991).
[17] J.H.B. Nijhof, H.A. Ferwerda, B.J. Hoenders, Pure Appl. Opt. 4, 199-218 (1994).
[18] R.H. Stolen, W.J. Tomlinson, J. Opt. Soc. Am. B 6, 565- 573 (1992).
[19] R.H. Stolen, J.P. Gordon, W.J. Tomlinson, H.A. Haus, J. Opt. Soc. Am. B 6, 1159-1166 (1989).
[20] C. Headley III, G.P. Agrawal, J. Opt. Soc. Am. B 13, 2170- 2177 (1996).
[21] G.M. Muslu, H.A. Erbay, Mathematics and Computers in Simulation 67, 581-595 (2005).
[22] M.S. Ozyaici, J. Optoelectronics and Advanced Materials 6, 71-76 (2004).
[23] G.P. Agrawal Opt. Lett. 15, 224-226 (1990).
[24] G.H. Shiraz, P. Shum, N. Nagata, IEEE J. Quantum Electron. 31, 190-200 (1995).
[25] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 77 – The Art of Scientific Computing. Cambridge University Press (1992).
[26] J.D. Hoffman, Numerical Methods for Engineers and Scientists. Marcel Dekker (2001).
[27] L. Debnath, Nonlinear partial differential equations for scientists and engineers. Birkhauser (1997).
[28] T. Hohage, F. Schmidt, On the Numerical Solution of Nonlinear Schrödinger Type Equations in Fiber Optics. Berlin (2002).
[29] M. Gedalin, T.C. Scott, Y.B. Band, Phys. Rev. Lett. 78, 448-451 (1997).
[30] A.L. Maimistov, A.M. Basharov, Nonlinear optical waves. Kluwer Academic (1999).
[31] S.A. El-Wakil, M.A. Abdou, A. Elhanbaly, Phys. Lett. A 353, 40 (2006).
[32] D. Anderson, Phys. Rev. A 27, 3135 (1983); D. Anderson et al., J. Opt. Soc. Am. B5, 207 (1988).
[33] B.A. Malomed, in Progress in Optics 43, ed. by E. Wolf, Elsevier Science B. V., Chapter 2, p. 69. (2002).
[34] Nguyen Viet Hung, Doctoral Thesis, to be published.
[35] Cao Long Van, W. Leoński, Nguyen Viet Hung, to be published.
[36] A. Donley et al., Nature 412, 295 (2001).
[37] L. Berge, Phys. Rep. 303, 259 (1998).
[38] Cao Long Van, Dinh Xuan Khoa, Nguyen Viet Hung, M. Trippenbach, Reduction of 2D problem of Bose – Einstein condensate trapped in a harmonic potential to the 1D problem. Paper presented on the 32th National Theoretical Symposium, Da Nang, Vietnam 2008.
[39] T. Bui Dinh, V. Cao Long, B. Nguyen Huy, S. Vu Ngoc, to be published.
In this paper, using a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [8] we study in the specific case of Kerr media. An obtained ultrashort pulse propagation equation which is called Generalized Nonlinear Schrödinger Equation usually has a very complicated form and looking for its solutions is usually a “mission impossible”. Theoretical methods to solve this equation are effective only for some special cases. As an example we describe the method of a developed elliptic Jacobi function expansion. Several numerical methods of finding approximate solutions are simultaneously used. We focus mainly on the following methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon: the collapse of solitons.
Key words:
Generalized Nonlinear Schrödinger Equation, Kerr media, solitons, ultrashort pulses
References:
[1] R.W. Boyd, Nonlinear Optics. Academic Press Inc., 2003.
[2] G.P. Agrawal, Nonlinear Fiber Optics. Academic, San Diego, 2003.
[3] U. Bandelow, A. Demircan, M. Kesting, Simulation of Pulse Propagation in Nonlinear Optical Fibers. WIAS, 2003.
[4] Cao Long Van, Dinh Xuan Khoa, Marek Trippenbach, Introduction to Nonlinear Optics. Vinh 2003.
[5] Cao Long Van, P.P. Goldstein, A Concise Course in Nonlinear Partial Differential Equations. University of Zielona Gora 2008.
[6] V. Cao Long, P.P. Goldstein, M. Trippenbach, Acta Phys. Polonica A105, 437 (2004).
[7] V. Cao Long, P.P. Goldstein, S. Vu Ngoc, Acta Phys. Polonica A106, 843 (2004).
[8] V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan, Propagation Technique for Ultrashort Pulses. I. Computational Methods in Science and Technology 14 (1), 5 (2008).
[9] V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan, Propagation Technique for Ultrashort Pulses. II, Computational Methods in Science and Technology 14 (1), 13 (2008).
[10] V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan, Propagation Technique for Ultrashort Pulses. III. Computational Methods in Science and Technology 14 (1), 21 (2008).
[11] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego, 2003.
[12] Yang Qin, Dai Chao-qin, Zhang Jie-fang, Higher-Order Effects Induced Optical Solitons in Fiber. International Journal of Theoretical Physics 44, 1117 (2005).
[13] Cao Long Van, M. Trippenbach, Dinh Xuan Khoa, Nguyen Viet Hung, Phan Xuan Anh, Conference on Theoretical Physics, Sam Son, Vietnam, 12-14 August 2003; Journal of Science, Vinh University 1A, 50 (2003).
[14] Cao Long Van, M. Trippenbach, New Optical Solitary waves in the higher order nonlinear Schrödinger Equation. Scientific Biuletin 126, Technical University of Zielona Gora, Poland 2001, p. 19 (Cao Long Van ed.).
[15] R.S. Tasgal, Y.B. Band, private communication, 2003.
[16] P.L. Francois, J. Opt. Soc. Am. B 8, 276-293 (1991).
[17] J.H.B. Nijhof, H.A. Ferwerda, B.J. Hoenders, Pure Appl. Opt. 4, 199-218 (1994).
[18] R.H. Stolen, W.J. Tomlinson, J. Opt. Soc. Am. B 6, 565- 573 (1992).
[19] R.H. Stolen, J.P. Gordon, W.J. Tomlinson, H.A. Haus, J. Opt. Soc. Am. B 6, 1159-1166 (1989).
[20] C. Headley III, G.P. Agrawal, J. Opt. Soc. Am. B 13, 2170- 2177 (1996).
[21] G.M. Muslu, H.A. Erbay, Mathematics and Computers in Simulation 67, 581-595 (2005).
[22] M.S. Ozyaici, J. Optoelectronics and Advanced Materials 6, 71-76 (2004).
[23] G.P. Agrawal Opt. Lett. 15, 224-226 (1990).
[24] G.H. Shiraz, P. Shum, N. Nagata, IEEE J. Quantum Electron. 31, 190-200 (1995).
[25] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 77 – The Art of Scientific Computing. Cambridge University Press (1992).
[26] J.D. Hoffman, Numerical Methods for Engineers and Scientists. Marcel Dekker (2001).
[27] L. Debnath, Nonlinear partial differential equations for scientists and engineers. Birkhauser (1997).
[28] T. Hohage, F. Schmidt, On the Numerical Solution of Nonlinear Schrödinger Type Equations in Fiber Optics. Berlin (2002).
[29] M. Gedalin, T.C. Scott, Y.B. Band, Phys. Rev. Lett. 78, 448-451 (1997).
[30] A.L. Maimistov, A.M. Basharov, Nonlinear optical waves. Kluwer Academic (1999).
[31] S.A. El-Wakil, M.A. Abdou, A. Elhanbaly, Phys. Lett. A 353, 40 (2006).
[32] D. Anderson, Phys. Rev. A 27, 3135 (1983); D. Anderson et al., J. Opt. Soc. Am. B5, 207 (1988).
[33] B.A. Malomed, in Progress in Optics 43, ed. by E. Wolf, Elsevier Science B. V., Chapter 2, p. 69. (2002).
[34] Nguyen Viet Hung, Doctoral Thesis, to be published.
[35] Cao Long Van, W. Leoński, Nguyen Viet Hung, to be published.
[36] A. Donley et al., Nature 412, 295 (2001).
[37] L. Berge, Phys. Rep. 303, 259 (1998).
[38] Cao Long Van, Dinh Xuan Khoa, Nguyen Viet Hung, M. Trippenbach, Reduction of 2D problem of Bose – Einstein condensate trapped in a harmonic potential to the 1D problem. Paper presented on the 32th National Theoretical Symposium, Da Nang, Vietnam 2008.
[39] T. Bui Dinh, V. Cao Long, B. Nguyen Huy, S. Vu Ngoc, to be published.
