Nonlinear evolution of a pulse with a linear frequency...

5. Conclusions

[15] Propagation of a weakly nonlinear short pulse with a linear frequency modulation in a graded-index waveguide with a weak longitudinal irregularity has been investigated on the basis of a nonlinear wave equation. Depending on the ratio between the spectral width of the pulse and the frequency modulation depth, the pulses were classified as chirped and strongly chirped. The work was devoted to the detailed investigation of propagation of chirped pulses, that is, the pulses whose modulation depth is much less than the spectral width.

[16] The model nonlinear wave equation was solved by using an asymptotic procedure, whose small parameter was the order of magnitude of the pulse amplitude. The longitudinal irregularity of the waveguide had the scale of the order of a square of the small parameter, and the magnitude of the phase modulation of the chirped pulse was proportional to the third power of this parameter. The use of ansatz (3) and (4) for the solution of the nonlinear wave equation allows one to detach in a natural manner the linear Sturm-Liouville problem (5), whose eigenvalue defines the propagation constant of the high-frequency carrier as a function of the longitudinal coordinate, and its eigenfunction describes evolution of the transverse distribution of the pulse field as it propagates in the graded-index waveguide.

[17] Propagation of a pulse is a nonlinear process characterized by three velocities. A fast process, i.e., propagation of a high-frequency carrier, is modulated by the envelope whose evolution has two scales and is formed by the evolution of the envelope phase with a medium velocity and a slow amplitude variation. The pulse envelope is described by (9) which in a special case can be reduced to the equation of the Painlevé class (10). Numerical solution of (10) has shown that a chirped pulse has a steeper leading edge as compared with a soliton pulse in the absence of frequency modulation, and an oscillating, but decaying, tail of the envelope is formed at the trailing edge.

[18] In the process of a successive realization of the asymptotic procedure, an important relationship (7) and (8) between the phase of the high-frequency carrier, envelope phase, and modulation coefficient of the pulse has been established. The physical consequence of this relationship is that the chirp cannot be specified in an arbitrary manner in a graded-index waveguide channel; it must be tailored to the parameters of the transverse and longitudinal irregularities of the waveguide. The result obtained in our work explains this fact and provides a suitable tool for exploitation of chirped pulses.