Long-range interference effects of soliton reshaping in

Received September 25, 1992; revised manuscript received March 19, 1993
We report the observation of spectral oscillations at the output of an optical fiber after the propagation of a reshaped optical soliton over several soliton periods. The experimental results are in good agreement with the results of an analytical study that permits the calculation of the radiated field associated with a hyperbolicsecant-shaped input pulse, after a few soliton periods of propagation. For a nonsoliton input we show that the spectra of the asymptotic soliton and of the radiated field, which is stripped from the input pulse, remain overlapped during propagation, and the phase difference between the two fields is both distance and frequency dependent. This causes an interference that gives rise to the observed oscillations. Furthermore, a numerical simulation that uses a split-step Fourier-transform technique gives results that agree well with the experimental observations.
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with distance in such a way that after a few soliton periods the output field envelope becomes close to the asymptotic soliton. One can understand this behavior better by considering the soliton and the dispersive background fields as two separate coherent fields. The output spectrum therefore features the interference between them. This is equivalent to the observation of the interference between two sequential coherent pulses. The study of this interference suggests a description of the dispersive field resulting from the soliton reshaping. In particular, for a given spectral component, knowledge of the phase difference between the soliton and the radiated field is of interest. The soliton reshaping in optical fibers-or, equivalently, the asymptotic behavior of the solution to the nonlinear Schr6dinger equation-has been studied analytically in the case of a single-output soliton by Segur.4 The first term of his asymptotic expansion of the dispersive field in the distance of propagation was given in terms of the scattering data associated with the input field. However, the phase of the solution was not determined, because of the use of conservation laws as a basis for the calculations. This rules out the use of this expansion for the study of the spectral modulation. Satsuma and Yajima' determined the exact asymptotic soliton solution of the propagation equation for a specific input consisting of a hyperbolic-secant-shaped pulse envelope. In this case the corresponding radiation can be given only in an integral, intractable form. We study experimentally, analytically, and numerically the effect of the radiated field on the spectrum of the output field. An analysis, based on Gordon's study of the dispersive perturbations of solitons,6 gives a simple expansion of the radiated field in the time domain and permits the calculation of the spectrum of this field. We show that, although the peak intensity of the radiated field decays rapidly with the distance of propagation, its spectral com©9 1993 Optical Society of America
creasing discrepancy between the input field and the
asymptotic soliton and (2) an increasingly dramatic effect on the central part of the spectrum as the distance of propagation increases. As much as the first point seems intuitively acceptable, the second seems surprising. Indeed, the intensity of the radiated field decreases rapidly
1.
INTRODUCTION
It is well known that soliton propagation in optical fibers can be considered the only stable regime of propagation for a given nonzero fiber dispersion and a certain range of pulse widths and optical powers.' 2 Furthermore, the study of the solution to the nonlinear propagation equation in a fiber' shows that an optical pulse with sufficient energy in a fiber will reshape into an optical soliton after its propagation over several soliton periods, a characteristic length that depends on the fiber dispersion and the pulse parameters. Soliton reshaping is accompanied by the radiation of a part of the input energy. Because of group-velocity dispersion, the width of the resulting radiated dispersive field broadens and its peak intensity decays as it propagates in the fiber. If the input pulse is not an exact optical soliton but has sufficient power to sustain an asymptotic soliton behavior, the time signature of the field at the output of the optical fiber, after propagation over a certain number of soliton periods that depends on the input condition, consists essentially of the soliton(s). However, if one examines the spectrum of the output field, one sees that it differs considerably from the spectrum of the asymptotic soliton field. That is, the output spectrum features a modulation of the soliton spectrum. The resulting spectral oscillations were experimentally observed, and a numerical simulation of the propagation shows an identical behavior. The two main characteristics of those oscillations are (1) a modulation depth that increases with in-