Abstract
We consider propagation of a short optical pulse in an optical fiber whose refractive index is strongly dependent on the radial coordinate and is weakly dependent on the longitudinal coordinate with allowance for the possible weak spatial bending of the fiber axis. The three-dimensional nonlinear wave equation modelling the pulse propagation is solved asymptotically with respect to a small parameter specifying the order of magnitude of the pulse amplitude. A relationship between the propagating modes and the eigenvalues and eigenfunctions of the singular Sturm-Liouville problem is established. The pulse propagation is shown to have three scales: the high-frequency carrier is modulated by the envelope which evolves in a two-scale manner and is described by a nonlinear Schrödinger equation whose coefficients depend on the longitudinal coordinate. The transverse distribution of the wave field and the envelope soliton are obtained in terms of elementary functions for several types of transverse and longitudinal inhomogeneities of the fiber. The possibility of controlling the pulse parameters by varying the transverse and longitudinal inhomogeneities of the fiber is pointed out.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
G. P. Agrawal, Nonlinear Fiber Optics [Russian translation], Mir, Moscow (1996).
R. H. Stolen, Proc. IEEE, 68, No. 10, 1232 (1980).
A. Hasegawa and F. Tappert, Appl. Phys. Lett., 23, No. 3. 142 (1973).
W. A. Gambling, H. Matsumura, and C. M. Ragdale, Microwaves, Optics, and Acoustics, 3, No. 6, 239 (1979).
C. Lin, A.Tomita, A. R. Tynes, P. P. Glodis, and D. L. Philen, Electron. Lett., 18, No. 20, 882 (1982).
L. G. Cohen, W. L. Mammel, and S. J. Jang, Electron. Lett., 18, No. 24, 1023 (1982).
M. Jain and N. Tzoar, J. Appl. Phys., 49, No. 9, 4649 (1978).
V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wave Di.raction Problems: The Method of Standard Problems [in Russian], Nauka, Moscow (1972)
V. M. Babich, V. S. Buldyrev, and I. A. Molotkov, Spatio-Temporal Ray Method: Linear ans Nonlinear Waves [in Russian], Leningrad Univ. Press, Leningrad (1985).
G. V. Permitin, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 16, No. 2, 254 (1973).
I. G. Kondrat'yev, G. V. Permitin, and A. I. Smirnov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 23, No. 10, 1195 (1980).
I. A. Molotkov, S. A. Vakulenko, and M. A. Bisyarin, Nonlinear Localized Wave Processes [in Russian], Yanus-K, Moscow (1999).
B. A. Dybrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and Applications [in Russian], Nauka, Moscow (1986).
S. N. Vlasov and S. N. Gurbatov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 19, No. 8, 1149 (1976).
A. Hasegawa, in: J. R. Taylor, ed., Optical Solitons-Theory and Experiment, Cambridge Univ. Press, Cambridge, (1992), p. 1.
I. N. Sisakyan and A. B. Shvartsburg, Dokl. Akad. Nauk SSSR, 269, No. 1, 105 (1983).
N. S. Erokhin and R. Z. Sagdeev, Zh. Éksp. Teor. Fiz., 83, No. 1, 128 (1982).
D. H. Auston, Topics in Applied Physics, Springer, Berlin, Heidelberg, New York (1977), Vol. (18), Ultrashort Light Pulses. Picosecond Techniques and Applications, ed. S. L. Shapiro, p. 123.
E. A. Coddington and N. Levinson, Theory of Ordinary Di.erential Equations, McGraw-Hill, New York (1955).
E. S. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Di.erential Equations, Clarendon Press, Oxford (1962).
M. A. Bisyarin, Vestnik LGU, Ser. Fiz.-Khim., No. 2, 80 (1989).
M. A. Bisyarin, Zap. Nauchn. Sem. LOMI, 173, 42 (1988).
T. I. Lakoba and G. P. Agrawal, J. Opt. Soc. Am. B, 16, No. 9, 1332 (1999).
T. Schäfer, V. Mezentsev, K. H. Spatschek, and S. K. Turitsyn, Proc. Roy. Soc. Lond. A, 457, No. 2006, 273 (2001).
K. Rottwitt, B. Hermann, J. H. Povlsen, and J. N. Elgin, J. Opt. Soc. Am. B, 12, No. 7, 1307 (1955).
A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E, 61, No. 3, 3107 (2000).
E. M. Gromov and V. I. Talanov, Zh. Éksp. Teor. Fiz., 110, No. 1, 137 (1996).
M. A. Bisyarin and I. A. Molotkov, Izv. Rossiisk. Akad. Nauk, Ser. Fiz., 65, No. 6, 876 (2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bisyarin, M.A., Molotkov, I.A. Mode Structure and the Envelope of a Short Pulse in a Graded-Index Optical Fiber with a Longitudinal Inhomogeneity and a Spatial Bending. Radiophysics and Quantum Electronics 45, 471–480 (2002). https://doi.org/10.1023/A:1019968702600
Issue Date:
DOI: https://doi.org/10.1023/A:1019968702600