Soliton dynamics with fixed and sliding filters

Inline frequency filtering [1, 2] is a powerful method to extend the limits imposed to soliton transmission by the Gordon-Haus jitter [3] and by the intensity noise of the amplifiers [4]. The use of filters gives many advantages. One is the increase of the bitrate distance product, otherwise limited for pulses of high intensity by the Gordon-Haus effect and for pulses of low intensity by the amplitude noise induced on solitons by the amplified spontaneous emission noise. The window of permitted soliton energies is without filters about 3 dB for 2.5 Gbit transmission over transoceanic distances [4]. With filters, the window significantly opens [5]. Filters of arbitrary strength cannot however be used. It was predicted in Ref. [1] that the strength of filters is limited by the growth of continuous wave radiation close to the filter center frequency. An elegant way to overcome this problem is to move with distance the filter center frequency. The transmission line becomes transparent to the signal and opaque to the noise [6]. Unfortunately, the sliding action has undesired effects on soliton propagation. The solitons with larger bandwidth are trapped more efficient by the filters. This means that the offset of the soliton center frequency from the filter center frequency is larger for solitons having smaller bandwidth, and consequently smaller energy. This converts intensity fluctuations into frequency fluctuations producing, ultimately, timing jitter [7]. This is the reason for the Gordon-Haus jitter observed in the experiment of Mollenauer et al. [8] being twice the value predicted by the Gordon-Haus theory, corrected to include the action of fixed filters. If intensity fluctuations convert into frequency fluctuations, also frequency fluctuations produce intensity fluctuations because of the frequency dependent loss of the filters. This cross coupling is the physical reason for the existence of a maximum allowed sliding rate. Beyond a critical value, the positive feedback established between a proper combination of intensity and frequency fluctuations becomes unstable [6, 7]. The maximum sliding rate is proportional to the cube of the soliton energy. Consequently, for a given value of sliding rate, there is a minimum value of soliton energy for stability. Besides a minimum, there is also a maximum for the allowed soliton energies because, for increasing soliton energy hence for increasing bandwidth, larger excess gain of the amplifier is required to overcome the filter loss. For soliton energy exceeding this maximum, the sliding rate is not sufficient to sweep away the amplified spontaneous emission noise at the filter center frequency and the soliton becomes unstable. This limits the window of the allowed energies to 3 dB for an optimized line at 10 Gbit/s [9].