The mutual capture of two colored solitons is enhanced by a modulator, to a level which enables its practical exploitation, e.g., for a read- write mechanism in a soliton buffer. The enhanced capture was analyzed using closed form particle-like soliton perturbation, and verified by numerical simulations. Optimal modulator frequency and modulation depth are obtained. This mutual capture can be utilized for all-optical soliton logic and memory.
©2004 Optical Society of America
As communications evolves towards higher rates, all-optic devices may become a necessity. Routing of data packets in nodes of the communications network requires short term memory buffers. All-optical storage rings are preferable, avoiding the optics-to-electronics conversions, thus enabling light-speed functionality. These devices may also be utilized to convert data rates in the network nodes .
The mutual capture of colored solitons (i.e. having a relatively shifted carrier wavelength) was suggested in  for exploitation as a bit selective all-optic read-write mechanism. Schematically (Fig. 1), a solitons based data stream, is stored in a fiber storage ring [3–5] and can be selectively read out. This is accomplished by capturing the data bits of interest by control solitons, having a different color, and filtering out the captured pair (having the average carrier frequency of the control and data solitons). The capture of colored soliton was predicted and analyzed using several methods [2, 6–11]. The capture is possible for colored solitons having initial frequency difference smaller than a certain value, termed as a capture threshold. However, only recently it was noted  that a basic boundary exists which makes the exploitation of this effect to be very implausible. The capture threshold (frequency difference between interacting solitons) was found to be only about 40% larger than the soliton spectral full width at half maximum, independently on fiber or soliton parameters. A necessary condition for capture is thus a very substantial spectral overlap of the interacting solitons.
In Ref.  the interaction is modeled using a particle-like analogy, based on a modified soliton perturbation method. Each soliton constitutes a perturbation source for the other, leading to a mutual attraction (due to interaction incoherency). The model relates the solitons temporal peak location, central frequency and peak amplitude to the respective particle-like displacement, velocity and mass. The capture threshold is reached when the particle-like kinetic energy, associated with the initial frequency difference, is equal to the particle-like potential energy, associated with the initial temporal centers displacement. This particle-like analogy suggests that introduction of a damping mechanism may enhance the capture threshold. Moreover it will hopefully damp the post capture oscillations of the solitons pair, making it more favorable for possible implementations.
Here we show that the introduction of a modulator introduces a damping source into the process which results in enhancement of the capturing. The particle-like description assumes that the colored solitons interact incoherently, while actually, the coherency of the interaction varies in the process as the inter-solitons frequency difference is modified by the interaction – between a small to a substantial value. This necessitates a validation of the results by a full-fledged Non Linear Schrödinger Equations (NLSE)  simulation. Related work, approximated the interaction of communications-like solitons with modulators for the retiming of the solitons . In this reference – the model was too complex to be solved for the simultaneous interaction of the two solitons and of the modulator, thus it is not adequate for modeling the capture process.
Here, the modulator is first analyzed using the perturbation model. It is shown to introduce a damping source, and its parameters are optimized. Subsequently, the model predictions are verified with simulation results. Finally the impact of a “real” (lumped) modulator versus its distributed model is discussed.
2. Particle-like description
The interaction of the two colored solitons, having the same intensity, is described using a particle-like model, resulting in a mechanical like dynamics of the soliton center frequency and center time [2,13]:
where z denotes the propagation distance, β” the group velocity dispersion coefficient, δ the Kerr constant. W, τ, p and θ stand for the solitons peak amplitude, temporal center, central frequency and phase. ε is the first-order soliton bandwidth to peak-amplitude ratio (ε=(δ/|β”|)1/2). The coordinate system is symmetrical in respect to the initial conditions: τ1=-τ2=τ, p1=-p2=p. The LHS of Eq. (1a) is an equivalent force term for a particle-like model, which asserts an incoherent interaction. Even though the frequency difference varies along capture process, and hence the interaction coherency does vary too, the particle-like model describes reasonably well this process . This may be explained by the fact that whenever the two captured oscillating solitons are either temporally overlapping they are spectrally remote and vice versa. The incoherency justifies the use of a pair of NLSE coupled by a cross-phase modulation (XPM) term.
The initial frequency difference and initial temporal pulse center shift are associated with particle-like kinetic and potential energy . Introduction of a damping mechanism (equivalent to a mechanical friction) into Eq. (1), is expected to reduce the frequency variations subsequent to the soliton capture process. Moreover, it introduces an energy loss, which enables the capture of solitons with initial kinetic energy higher than the initial potential energy, i.e. the initial frequency shift difference threshold may be enhanced. In order to obtain a damped second order equation out of Eq. (1), a perturbation source (Sτ) is introduced into Eq. (1b), to yield:
This damping source can be generated by a modulator synchronized with the average group velocity, as is shown later. The source terms in the evolution equations of the solitons parameters, such as Sτ, are projected out of the NLSE perturbation source, using the adjoint perturbation functions (fm) given in .
The harmonic modulator transmission function is:
where M, ωm and tm are the depth, frequency, and temporal center of the modulator respectively and t is the z dependent time coordinate (traveling with the carrier). The NLSE perturbation source is obtained from the transmission function:
u stands for the slowly varying amplitude of the soliton. The modulator is uniformly distributed along the fiber, which is a good approximation, disregarding lumped periodic effects such as Kelly’s sidebands [15,16], since the modulator period (zm) is smaller than the soliton period (z0). Furthermore, the actual modulator period should be negligible in terms of the colored solitons walk-off due to dispersion. This dictates an accumulated solitons temporal shift along the modulator period (2β”pzm) smaller than soliton effective width (~10/εW):
For β”=-2(ps2/Km), δ=1.3(1/Watt Km), W=1(Watt1/2), zm=10(m), the soliton frequency difference should be smaller than 600(THz), whereas we discuss a much smaller difference of the order of 1(THz). A detailed comparison to lumped modulator is given at the end.
The τ-perturbation source (Sτ) introduced by the modulator into Eq. (1b) is:
Thus the modulator serves as the required damping source. The NLSE perturbation source is imaginary, thus the modulator does not introduce a frequency perturbation source into Eq. (1a).
Defining ξ=εW(t-τ) and Δτm=tm-τ and using Eq. (2) and the perturbation source of Eq. (5) the damping coefficient (V) is:
V is positive when the soliton peak coincides with the modulator transmission peak (|τ-tm|<π/(2ωm)) and hence, the modulator attracts the soliton to this peak. Deviation of the soliton center generates asymmetric Kerr refractive index, pulling the soliton back to the transmission peak (tm). The damping coefficient (V) is enhanced with the modulation depth, and has a maximal value for an optimal modulation bandwidth. f V(ωm) vanishes both for zero and infinity ωm, with a maximum for ωm~1.6εW. As ωm is raised, the variations of the solitons around tm, in terms of the modulator temporal period, become larger and cos(ωm Δτm) may accept negative values. Consequently the ωm value, for optimal V is lower than fv optimal value (~1.6εW), and is similar of the soliton bandwidth (εW).
At this value, both the capture threshold as well as the decay of the post-capture oscillations are enhanced. The optimal modulator bandwidth (Figs. 2(a), 2(b)) depends on the soliton bandwidth (εW) but not on the initial frequency difference (p0). These results were obtained from direct calculations of the modified particle-like equations (Eq. (1),(2),(5)). The value of ωm in Fig. 2(a) is 0.9(THz) - similar to soliton bandwidth (εW~0.8(THz), ωm~1.1εW). In Fig. 2(b), the soliton bandwidths (εW) are smaller, resulting in the decrease of optimal modulator frequency at the same ratio.
3. Simulation results
The perturbation analysis was based on two assumptions: the full NLSE can be replaced by two XPM coupled NLSEs and the solitons retain their nature while interacting. To verify the results we simulated the full NLSE by the Split Step Fourier propagation method .
We seek for a solution where the presence of the control soliton determines the data soliton capture, thus the modulator should have a low enough modulation index - to avoid soliton capture in the absence of a control soliton, as well as to inflict minimal losses to non-captured data bits. On the other hand, the modulator should have the highest allowable modulation index to enhance the mutual capture and damp efficiently the oscillations. Figs. 3(b), 3(c) depict the evolution of the soliton amplitude in the presence of the harmonic modulator, with/without a second colored soliton, respectively. Both solitons were initially overlapping with intensity of 1(Watt1/2). These solitons have a capture threshold of 2p0-TH=2×0.19×2π and therefore for initial frequencies difference 2p0-TH=2×0.20×2π, the solitons escape as depicted in Fig. 3(a). In Fig. 3(b), using the same initial frequencies but with a modulator in the loop, the capture of the data soliton to the control soliton, as well as the post capture oscillations damping are evident. In Fig. 3(c) it is clear that for the same initial conditions when no control soliton is applied, but with a modulator in the loop, the data soliton propagates virtually undisturbed by the modulator. To compensate for the losses of the modulator, a distributed gain of 5% per soliton period (z0) was applied.
In Fig. 4 the capture frequency difference is depicted versus the modulation depth for soliton assisted modulator with/without the second soliton. For a given initial frequency difference, the modulation depth should be larger than the value needed for modulator assisted capture, yet smaller than the capture threshold by the modulator itself. An adverse effect is that the gain required for compensating the modulator loss, increases with the modulation depth, resulting in pulse leakage to the time slots of the next modulation minima. We thus define arbitrarily that the cease of capture is where the peak power of the pulse in the adjacent modulation minima is >10% of that in the main modulation minima. Applying an unharmonic modulator may reduce the leakage and the capture threshold further enhanced.
The use of two XPM coupled NLSE, instead of a single NLSE, enables the distinction of the two solitons envelops and hence the calculation of their center position. We have applied the split step simulation for the two XPM coupled NLSE approximation, in order to evaluate the decay coefficient (α) when the solitons are mutually captured. The maximal oscillations amplitude points are extracted, and exponential curve of the form τ~exp(αz) is fitted.
In Fig. 5 the decay coefficient of the solitons oscillations is depicted versus the modulator period. The optimal modulator frequency is similar to the soliton bandwidth, in accordance with the conclusion from the perturbation calculation (Figs. 2(a),(b)). In the simulations, the damping coefficient was used as the merit parameter for the modulator and not the capture threshold, due to an uncertainty in the determination of the latter (in simulations [2,6] the transition from capture to escape process is gradual).
Modeling a lumped modulator, the results depend on the relative phase between modulator and inter-solitons oscillations. In fig. 6 the decay coefficient of the post capture oscillation is depicted versus this relative phase. The initial inter solitons frequency difference is 0.1×2π (THz), well in the capture regime. Without a modulator the decay coefficient is 0.1763 (1/Km). The oscillations are calculated using the split-step simulation assuming XPM coupled NLSE. Applying a distributed modulator the decay coefficient increases 2.34 times. Losses are compensated by a distributed gain. For lumped modulator with a period of 1.2z0, (~ capture oscillation period), the decay coefficient is depicted versus the M-phase (the relative shift in the modulator location, in units of modulator periods). The maximal decay coefficient was obtained for M-phase=0.5, which is the location where the solitons are most apart. The minimal decay coefficient was for overlapping solitons (M-phase=0). The distributed modulator decay coefficient is about the mean value obtained for the lumped modulators.
To avoid lumped periodic effects (such as Kelly’s sidebands) we are restricted to zm≪z0 which usually coincides with typical system parameters, e.g soliton period of kilometers and storage ring meters long. Since M-phase is a fraction of the modulator period, which is negligible relative to the capture oscillation period, the modulator is effectively distributed. However for larger storage rings, M-phase design may improve the capture process.
A modulator enables the capture of two solitons with a substantial initial frequency difference and damps their post capture oscillations. The results of the perturbation model were verified using simulations of the full-fledged NLSE. An optimal modulation frequency is shown to be similar to the soliton bandwidth. The optimized location of a lumped modulator is where the solitons are maximally apart. This is averaged for a modulator period much smaller than the soliton period. This mutual capture can be utilized for an all-optical reading of soliton data storage devices via control solitons stream.
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