Abstract

The interaction of two colored solitons was analyzed in the framework of a particle-like model, derived from a soliton perturbation theory. From “energy” considerations, a soliton capture threshold and the re-coloring of the escaping solitons were derived. The results were compared to the spectral boundaries of a second order soliton as well as to previous reports. The capture of colored solitons was shown to be impractical without additional means. This particle-like model was further generalized to apply also for non-equal intensity colored solitons. Detailed calculations—beyond the particle-like approximation, exhibited additional mechanisms, namely dissipation and friction-like forces, which served as sources for the relaxation of the solitons oscillations within the captured state, thus enhancing the capture phenomenon.

©2004 Optical Society of America

1. Introduction

Colored solitons are possible means for delivery of information in long-haul wavelength division multiplexing (WDM) optical communications. In contrary to the scenario of communications, where minimum interaction between the different colored data channels is desired, optical processing and storage elements may take advantage of large inter-soliton interactions. Soliton storage rings [1,2], designed to function as all-optical memory devices, can be exploited as memory buffers for fast routing and data rate conversion [3]. Previous studies of optical buffers focused mainly on the data integrity issues rather than the read-write (R/W) mechanisms of the stored information. In references [4,5] all-optical erasing/reading schemes were suggested, based on a pulse controlled switch (using a nonlinear optical loop Mirror or Sagnac interferometer) for a monochromatic data stream. In the current paper we propose and analyze nonlinear interactions between colored solitons with the flavor of all-optical R/W mechanisms. The suggested mechanisms, which may be used for all-optical bit reading, are based on shifting the data bit stream into a second color, due to colored control bits. The use of color diversity instead of the polarizing diversity, allows for multiple storage channels, in the spirit of WDM, and furthermore avoiding polarization-maintaining issues.

Soliton capture of two orthogonally polarized solitons was reported to exhibit an intensity threshold as observed using numerical solution of two Non-Linear Schrödinger Equations (NLSEs), one for each soliton, coupled by Cross Phase Modulation (XPM) terms [6–8]. This mechanism was suggested for the realization of all-optical logic gates [9], and for fast header reading [10]. It was studied analytically using the variational approximation [11,12] and by using the perturbation method for the integrable Manakov model [13,14].

The current paper focuses on the interactions of colored solitons – a subject of importance for WDM communications and storage systems. The nature of the interaction of colored soliton is complex: initially – they may be color-wise incoherent, however, the frequency shift mediated by the interaction modifies the coherence degree during the interaction interval – especially in the capture process. This necessitates a strict examination of a proper NLSE model to describe the scenario [6,15,16].

For capture of polarized solitons to occur in a birefringent fiber, the solitons must acquire a certain carrier difference so that the chromatic dispersion compensates for their modal dispersion. On the contrary, when initially colored solitons are captured, their carrier frequencies become the same. Therefore, the two cases (colored and polarized solitons) differ by the initial group velocity difference source, although in both cases the solitons frequency carrier is the varying parameter.

Closed form analysis of colored solitons interaction based on the variational approximation of XPM coupled NLSEs was performed and validated using simulation of XPM coupled NLSEs [17]. As discussed above this model has to be examined cautiously and therefore our closed form expressions were validated versus simulation of the full-fledged NLSE, as well as with simulation of the XPM coupled NLSEs, pointing out some differences.

Moreover, the previous analysis of colored soliton interaction [17] was based on normalized equations. Therefore it is impossible to learn directly from the result whether the capture threshold can be maximized changing either of the fiber or the solitons parameters. In this paper the un-normalized capture threshold closed-form expression is derived, which reveals that the capture is unattainable for practical distinguishable colored solitons. However the particle-like model derived here suggests that capture enhancement is possible applying friction-like mechanism, such as synchronized modulator.

In order to model the colored soliton interaction we choose to apply the soliton perturbation method [2,18–20]. This linearized method enables convenient incorporation of different perturbation sources simultaneously. This will prove beneficial here for incorporation additional source terms to introduce a more comprehensive (coherent) modeling and in future work [21], for incorporation of a modulator as a perturbation source. It should be emphasized that the perturbation method was not used in the past, for the case of colored solitons interaction – for the strong interaction regime. First the validity of perturbation method in this regime where the solitons’ parameters are considerably modified due to the interaction has to be validated. Past studies analyzing the colored soliton interaction using perturbation method [18] dealt with weakly interacting passing by colored solitons, for which the solitons chirp is smaller than the actual frequency difference and can be referred as slight perturbation to initial central frequency. In the capture case, and even if the perturbation scheme can be validated, the cumulative chirp is higher than the frequency difference, which requires a modification of the perturbation method which is performed here. For practical applications – we should explore also the interaction nature for colored solitons with (at least slightly) unequal intensities. Applying our analysis for the unequal intensity solitons, we derived closed-form expressions of capture and re-coloring for the first time, to the best of our knowledge.

The solitons perturbation approximation is thus briefly introduced, in section 2, including the modifications to describe the central wavelength shift (re-coloring). In section 3, the perturbation method is applied to the XPM coupled NLSEs, resulting in the equation of motion for the soliton parameters, from which particle-like equations of motion were extracted. Energy representation was used to extract thresholds and parameters, in section 4. The validity and the breaking of the particle-like model were examined in section 5. In section 6, the particle-like model is modified to apply for non-equal intensity colored soliton and the model predictions are verified using the direct simulation.

2. The perturbation approximation with colored solitons

The propagation of a pulse envelop in an optical fiber is described by the NLSE which includes the second order dispersion and the nonlinear third order susceptibility [20,22]:

zu=j{12β"2T+δu2}u,

where u is a slowly varying amplitude with exp(jβ0z-jωT) carrier, β0 is the mode propagation constant and ω is the carrier angular frequency, z is the propagation distance along the fiber, T is a z dependent time coordinate (traveling with the carrier), β” is the group velocity dispersion coefficient, and δ is the Kerr coefficient.

A first order soliton is a solution of the NLSE [2,23], obtained by the IST (Inverse Scattering Transform) method [24]:

u=Wsech{εW(Tτ)}exp{j(pT+θ)},
zτ=β"p,
zθ=12(δW2+β"p2),

where W, θ, p and τ are the respective soliton peak amplitude, phase, relative frequency (frequency shift from initial carrier), and temporal center for a respective zero relative frequency soliton. The soliton temporal width, ∆T, is related to the soliton amplitude by ∆T=(εW)-1, where ε=(δ/|β”|)1/2. The soliton energy (2/ε)W is proportional to its peak amplitude. The specific calculations in this paper were performed for the fiber parameters β”=-2[ps2/km] δ=1.3[(Watt km)-1] and soliton peak amplitude W=1[Watt1/2].

Following [2, 18–20, 23] we apply the soliton perturbation approximation, in order to examine the evolution of a soliton when a perturbing source (j s(z,T)) is added (to the RHS of the NLSE). The resulting equations of motion for the soliton perturbation parameters are:

zΔW=SW,
zΔτ=Sτ+β"Δp,
zΔp=Sp,
zΔθ=Sθ+δWΔW,

where the source terms in the above equations are projected out by the adjoint perturbation functions (f̱m) given in [2,20]:

SmImdT{fm*s(z,T)exp{j2W2z}}.

m represents the soliton parameters, m∈{W,τ,p,θ}.

The perturbation expressions of soliton parameters in Eq. set (3) are given in its self-system (p=0). Since, colored solitons propagate with different group velocities, a common self-coordinate system does not exist. Therefore the perturbation method, described previously, is modified. Each calculated step is split into phases: solitons perturbation calculation in their self-system (Eq. set (3)) and the propagation of the solitons self-systems itself (Eqs. (2b) and (2c)). The perturbation calculation for each soliton is done in its self-coordinate system, where the other soliton serves as a source for the perturbation. The solitons parameters are updated at the beginning of each step, with the accumulated change due to both perturbation and propagation terms at the previous step. In this way, the solitons parameters are being tracked at an “inertial” system:

zW=SW,
zτ=Sτ+β"p,
zp=Sp,
zθ=Sθ+δW(SWdz)+12(δW2+β"p2).

The second term of Eq. (4d) RHS, is the Kerr phase perturbation due to ∆W.

3. Particle-like colored solitons interaction

The interaction of two different color solitons of the form:

u1=W1sech{εW1(Tτ1)}exp{j(p1T+θ1)},
u2=W2sech{εW2(Tτ2)}exp{j(p2T+θ2)},

is commonly described by two NLSEs, coupled by an XPM term [15,16]:

zu1=j(12β"2T+δ(u12+2u22)),
zu2=j(12β"2T+δ(u22+2u12)).

In Eq. (6) we neglected two terms obtained when substituting the combined solitons field u=u1+u2 into the NLSE (Eq. (1)), namely:

δ(u2)2u1*+δ(u1)2u2*.

These terms contribute time dependent phase and their accumulated contribution is small for a large frequency difference between the solitons. For different colored solitons, the XPM coupled NLSEs is valid (see discussion on validity in section 5) and the perturbation source is thus:

sXPM=2δu22u1.

As will be shown, although in the capture process the frequency difference of the solitons shrinks eventually to zero, the predictions of the perturbation approximation based on the XPM assumption is in a good agreement with simulations.

The sources of parameters evolution equations (Eq. (3c)) obtained using the solitons expressions (Eq. (5)):

SWXPM=0,
SτXPM=0,
SpXPM=2δε2W12W22dT{tanh(εW1(Tτ1))·
·sech2(εW1(Tτ1))sech2(εW2(Tτ2))},
SθXPM=2δεW1W22dT{(1εW1(Tτ1)tanh(εW1(Tτ1)))·
·sech2(εW1(Tτ1))sech2(εW2(Tτ2))},

Due to the incoherent interaction, Sτ=0, SW=0 and Sp is phase independent. τ evolution thus depends only on p. In the case of equal intensity solitons (W1≡W2≡W) this leads to a second order evolution equation for the soliton “center of mass” (CM), similar to that of Newtonian particle displacement. Denoting ξ≡εW(T-τ1) and Dτ≡τ21, and defining an equivalent force (F) as:

F()=(2δβ"εW3)+{tanh(ξ)sech2(ξ)sech2(ξ)},
z(β"p)=F(2τ)=F(2τ),
zτ=β"p,

where we use coordinates system in which the solitons are symmetric in time-position and frequency (τ1≡-τ2≡τ, p1≡-p2≡p). Eq. (9) includes the contributions of the parameters due both the perturbation and evolution of the non-perturbed soliton, according to Eq. (4). From these equations we derive a force equation:

2zτ=F(2τ).

The force F is proportional to the z second derivative of the soliton center position (equivalent acceleration), and is of an attractive nature. For small inter-solitons distance the force is quasi-linear (spring-like) while for a distance exceeding a threshold value the force decays, and vanishes for infinite distance.

Two classes of soliton interactions were resolved. Capture which results with the two fused solitons moving together with a new group velocity. Escape occurs when the two solitons are experiencing a momentary attractive interaction and subsequently move apart. Similar characteristics were obtained using the variational approximation for colored solitons [17] and for two orthogonally polarized solitons they were termed trapping and dragging respectively [7].

We examine the escape process by setting the initial position difference to be large (τ1→-∞ for p2>p1). The frequency (p) and the position (τ) of the solitons are calculated from Eq. (9) and depicted in Fig. 1. The only parameter modified by the escape process is τ, where the soliton is experiencing a momentary frequency shift, while W and θ perturbations are identically zero. Note that the mutual attraction causes the solitons to move apart. Assuming that the solitons displacement due to cross-perturbation is negligible regarding the different color dispersion (τ=β“p0z), integrating Eq. (10) twice in respect to z gives accumulated shift each soliton gains due to their interaction:

Δτ(z:+)=εWp02,

where p0 denotes the soliton initial central frequency. Similar results for WDM inter-soliton interaction have been reported [18,25]. The accuracy of Eq. (11) improves as the initial frequency difference increases, according to the postulation made (τ=β“p0z).

 figure: Fig. 1.

Fig. 1. Perturbation calculation of escaping colored solitons. p0=0.145×2π, τ0=5. (a) Center (τ) of escaping soliton (blue) vs. the center of non-perturbed one (black). (b) Solitons carrier (red) vs. center accumulated perturbation (i.e. difference of the curves depicted at figure 1(a)) (blue).

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4. Colored solitons capture

For equal intensity, temporally overlapping (Dτ→0) solitons, the value of the integral (in Eq. (9a)), in its central linear interval (order of (εW)-1), is about (8/15)Dτ and the force can be linearly approximated by FL:

2zτ(1615δ2W4)FL=2FLτ.

The initial velocity of the equivalent particles is the frequency difference of the solitons times the dispersion coefficient. The solution of Eq. (12) describes a harmonic motion of the solitons around their average group velocities. The frequency of the oscillations is proportional to the square root of the force.

The kinetic energy related to the equivalent particle-like velocity is:

Ek=12(β")2p2.

The potential energy Ep= -∫dτ{F(-2τ)} related to the solitons center difference is:

Ep=2(β"εW)21(2εWτ)coth{2εWτ}sinh2{2εWτ}.

The potential energy is minimal for τ=0, i.e. when the solitons overlap. For an increasing values of p0 (for τ0=0), an initial frequency threshold exists, under which the soliton is captured. The soliton minimum potential energy, calculated for τ→0 is:

Epmin=23(β"εW)2.

To resolve the conditions for capture and to derive the capture threshold, the interaction is presented in the parametric plane Ep-Ek (Fig. 2). The potential energy value is defined as negative, and the kinetic energy as positive. Due to energy conservation the curves slope should be identically -1. Capture occurs only if p gains a zero value or equivalently Ek=0. Hence, the line |Ek|=|Ep| divides the energy plane into capture and escape regions, and the initial energy conditions determine completely the capture characteristic. The trajectories having initial conditions “under” this threshold line |Ek|=|Ep|, reach the point Ek=0, thus the soliton is captured. Dynamical examination of the curves (as the pulse propagates) shows that an escape trajectory starts “above” the threshold line and evolves towards the y-axis (which is analog to τ→∞). The capture trajectory maintains a periodic orbit – bouncing back and forth from the initial position to the x-axis.

 figure: Fig. 2.

Fig. 2. The soliton trace in the energy plane. The red and blue curves are for p0=-0.137×2π and -0.16×2π respectively with τ0=0.

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This elucidates a capture criterion, namely the relative absolute value of initial potential and kinetic energy. |Ep0|>Ek0 will result in a capture process (where the zero denotes initial value of the energy). The minimal value of Ep0 is obtained at τ0=0 and is determined by W, β” and δ. For a given scenario (W, β”, δ) there is an upper limit of the initial kinetic energy that allows capture. It is when the maximum potential energy equals the kinetic energy:

p0TH=43εW.

Equation (16) degenerates to the normalized result obtained by M. Karlsson et. al. using the variational approach [17]. However Eq. (16) gives much more transparency to the dependence on soliton intensity and fiber constants. Since the soliton bandwidth (FWHMp) is proportional to the same term FWHMp≈1.68(εW), the ratio (2p0 TH)/(FWHMp)≈1.37 is a constant. This elucidated immediately the fact that in order to be captured, the solitons should overlap substantially in spectrum regardless of fiber parameters and solitons intensity. This makes the possible exploitation of this effect to be very implausible, unless other means to broaden the capture regime will be applied [21].

5. Beyond the particle-like results

The perturbation, tracks the interactions between solitonic like entities, while the interaction with other types of nonlinear waves – e.g. dispersive waves is disregarded. To explore where the particle-like approach may fail, we performed direct calculations of the solitons interactions by the Split Step Fourier Transform method [15], using the XPM coupled NLSEs. As illustrated in Fig. 3, we can distinguish between three interaction classes.

 figure: Fig. 3.

Fig. 3. Simulation (XPM coupled NLSEs) of equal intensity colored solitons interaction. The intensity envelope of one soliton as the two solitons propagate simultaneously in the fiber for: (a) (1MB) Escape (p0=-0.20×2π), (b) (1MB) Intermediate (p0=-0.15×2π) and (c) (0.83MB) Capture (p0=-0.07×2π). τ0=0.

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The capture and escape were observed and discussed using the perturbation approximation and the additional class is an intermediate process. Soliton escape is shown in Fig. 3(a), for initially overlapping solitons (Dτ=0). The coordinates system is such that the solitons position and frequency are symmetric. The simulation consists of two coupled equations, one for each soliton. In Fig. 3 the temporal amplitude distribution for one of the solitons is depicted. For the other soliton, the amplitude is symmetric in respect to the temporal axis. z-axis is normalized by the soliton period (z0). Fig. 3(c) illustrates the solitons’ mutual capture with a final averaged group velocity (the velocity of the coordinate system). In Fig. 3(b), the intermediate case is depicted, exhibiting soliton breaking: a portion of the soliton is escaping, and the remaining portion is captured by the second soliton. The emerging solitons are re-colored, as evident from the trajectory slope change in the T-z plane. This captured portion of soliton was observed numerically for orthogonally polarized solitons and colored solitons [7,17] and termed as a moving “shadow” of the escaping soliton.

Although the case of soliton’s breakdown is not expected to be correctly described by a perturbative analysis, the results obtained by the perturbation, are following surprisingly well the simulation prediction, as evident from Fig. 4. It can be explained by the fact that although the two original solitons break down – the four fractures generate again two well behaved solitons. The solitons re-coloring (RC) is calculated based on the fact that the total initial particle-like energy is transformed into kinetic energy when the solitons are far apart:

RC=pfp0=143(εW)2p02,

where index f stands for final.

 figure: Fig. 4.

Fig. 4. Solitons re-coloring vs. initial frequency (τ0=0).

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Since EKf = EK0 + EP0 and RC2 = EKf / EK0, the trajectory of constant RC over energy plane is a straight line:

Ek0=11RC2Ep0.

Equation (18) provides an analytical measure of the borders of the capture, intermediate and escape zones over the energy plane. Accounting for over 90% of solitons re-coloring as escaping ones, the intermediate zone lies between EK0 = EP0 (RC=0, capture threshold) and EK0 ~ -5.26EP0 (RC=90%). In figure 5, Eq. (18) is compared to XPM simulations. Better agreement is observed for higher RC values (i.e. RC=90%).

 figure: Fig. 5.

Fig. 5. Trajectories of re-coloring in the energy plane.

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In Fig. 6 the split step simulation (using the XPM coupled NLSE) is compared with the perturbation approximation. An additional effective damping factor is observed in the split step simulation results. This damping relaxes the harmonic motion. A possible explanation for this distinction is the secondary dispersive waves emanated from the solitons during their continuous encounter, (visible in Fig. 3(c)), which produces an energy transfer mechanism to non soliton entities [26].

 figure: Fig. 6.

Fig. 6. Temporal amplitude calculated by XPM simulations for one of the captured solitons (contour) and the soliton center calculated using Eq. (9) (bold curve). τ0=0, θ1= θ2=0, p0=-0.07×2π.

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To understand if coherent effects are important, especially when the re-colored solitons becomes very close in central frequency and thus highly coherent, we compare the above results to those obtained by numerical solution of the full fledged NLSE – without applying the XPM assumption. In Figs. 7(a) and 7(b) the (combined) solitons amplitude propagation and spectral evolution are respectively depicted. The oscillations amplitudes as well as the perturbations zero crossings are similar to the split step simulation peaks, corresponding to the solitons encountering at the z-axis. A slight reduction in the oscillations amplitude is observed in the simulation.

 figure: Fig. 7.

Fig. 7. Comparison of solitons capture using full fledged NLSE simulation (contours) and the perturbation calculations (bold curves) curves: (a) soliton center, (b) center frequency. p0=0.05×2π, θ12=0, τ0=0.

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Comparing the threshold of the capture frequency predicted by the three different calculation methods (for W=1, θ12=0, τ0=0) we obtained: ±0.147×2π by the XPM perturbation calculations, ±0.139×2π by the split-step simulation based on XPM coupled NLSE, and ±0.193×2π by the full fledged NLSE. There is a slight decrease in threshold frequency of the XPM based simulation relative to the perturbation estimation, while the capture threshold of the full fledged NLSE is increased relative to the XPM based approximation. This suggests the existence of yet an additional damping mechanism, not encountered in the XPM coupled NLSEs approximation. The source of this damping can be traced analytically to the coherent cross terms of Eq. (6c) as an additional perturbation source, resulting in Sτ source within Eq. (4b). Formally it implies the emergence of a friction coefficient (V) in the dynamics of the perturbed soliton center (Eq. (12)) in the form:

2zτ=2FLτVzτ.

The perturbation method can thus be applied also when including the coherent terms, neglected in the framework of XPM assumption. This is described in details elsewhere [27] to give a better agreement with full-fledged simulations including solitons phase difference sensitivity. It should be noted that in the refined perturbation model (which includes the coherent terms) the source terms exist for all four soliton parameters, forming a 4’th order dynamics, instead of the Newtonian particle-like system described here. In Fig. 5 the re-coloring closed-form expression (Eq. (17)) is compared to the full-fledged simulations for initial phase difference of Dθ= π/2 and 0. From the standpoint of re-coloring, the particle-like model fits the simulation results, with higher inaccuracy in the capture region. The case of Dθ = π/2 gave a better match, presumably due to an initial complex amplitude orthogonality (where the temporal overlap is complete).

The slight decrease in threshold frequency of the XPM based simulation relative to the perturbation estimation, is due to the generation of dispersive waves causing an effective decrease in soliton amplitude (W), and therefore in its frequency threshold (from Eq. (16)). The increase in capture threshold of the full fledged NLSE value relative to the XPM based approximation, is caused by the additional “friction” mechanism (associated with V in Eq. (19)) that enables particles (solitons) with higher initial kinetic energy (initial frequency) to loose mechanical energy and cross the |Ep|=|Ek| line (In Fig. 2), thus to become captured. The results also indicate that the XPM approximation is a major cause for inaccuracy.

 figure: Fig. 8.

Fig. 8. A comparison of a second order soliton spectrum at its temporal propagation peaks and the combined spectrum of two co-centered colored solitons at capture threshold. The two colored solitons parameters: Dτ=0, W=1, Dp=2×0.193×2π, Dθ=0.

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The capture process of the two frequency shifted solitons, described in this paper, can be partially viewed as the generation of a second order soliton, with the group velocity that is the average value of the two first order solitons. Simulations showed that the first stage of the capture is the amplitude reshaping via emission of dispersive waves, Fig. 3(c). As depicted at Fig. 8, the evolved second order soliton have two spectral peaks at the peak intensity points (related to the complete overlap of the two interacting solitons). This spectrum resembles that of a co-centered two colored solitons at the capture threshold. The CM of the spectral lobes of the second order soliton are at ±0.22×2π, while for the two colored solitons are at ±0.20×2π. Nevertheless, we should emphasize that our observations show that the captured solitons oscillation might decay rapidly or indefinitely depending on the frequency difference. Those observations can be related to the analysis made by N. C. Panoiu et. el. [28], which predicts the number of solitons that emerge from a capture process.

6. Particle-like model for non-equal-intensity solitons

We enhance the particle-like model to mitigate the unequal intensity colored solitons. As in the degenerated case, Sτ=0, SW=0 and the source is phase independent. Thus solitons dynamics is described by Eq. (20):

z(β"pk)=2(δWkW3k)2+{tanh(εWkξ)sech2(εWkξ)sech2(εW3k(ξ(τ3kτk)))},
zτk=β"pk;k∈1,2.

The solitons having different intensities, break the problem symmetry, thus the soliton evolution dynamics must consists of four dependent parameters system [τ1, p1, τ2, p2].

To complete the particle-like analogy, we examine the mass definition applicability. Using the conventional mass (m) definition we extract the mass ratio by:

m2m1t2r1t2r2=+{tanh{εW1ξ}sech2{εW1ξ}sech2{εW2(ξ)}},+{tanh{εW2ξ}sech2{εW2ξ}sech2{εW1(ξ)}},=W2W1.

Therefore the peak amplitudes (W) play a role of mass in the particle-like model, defining the reference mass as W=1[Watt1/2]:

mk=Wk

Using this notation the particle-like dynamics (Eq. (20)) can be written as:

z(β"pk)=Fmk,
F=2β"δε(W1W2)2+{tanh(ξ)sech2(ξ)sech2(W2W1ξεW2)}.

Concluding the mechanical analogy: in the equivalent particle-like model the displacement (r), velocity (v), force (F), mass (m), time (t) and place (r) coordinates play the role of τ, β”p, β”Sp, W, z, T accordingly.

Since we have established particle-like model the capture threshold is calculated in the CM system. The model predicts the solitons are captured to the CM coordinate (rcm = (m1r1+ m2r2)/(m1+ m2)). The solitons dynamics around rcm is described using equivalent particle having the reduced mass μ = (m1m2)/(m1+m2), and the difference coordinate (r≡r2- r1):

μ2tr=F.

Capture occurs if the reduced mass velocity vanishes along the track (∂tr =0). The initial kinetic energy is:

EK=12μ(tr)2=2W1W2W1+W2(β"p0)2,

where the solitons initial velocity is ±p0. The potential energy is derived from the force:

Ep=2β"δε(W1W2)2Δτ+dr{+{tanh(ξ)sech2(ξ)sech2(W2W1ξεW2r)}}.

The capture threshold is reached for lowest potential energy, where the solitons are co-centered (i.e. r=0):

Epmin=2β"δεW12W2+{tanh(ξ)sech2(ξ)tanh(W2W1ξ)}.

An extended comparison is brought elsewhere [27]. It is found that the perturbation predicts correctly the results for small intensity ratios. Using Taylor expansion for Eq. (26b) around W2/W1=1, the capture threshold is calculated:

p0TH=13ε(W1+W2);W2W1.

According to Eq. (27):

poTH(W1,W2)=12{poTH(W1,W1)+poTH(W2,W2)}.

Thus, for W1=W2 Eq. (27) converges to Eq. (16). The re-coloring predicted for the solitons:

Dp=(Dp0)243ε2(W1+W2)2;W1W2,

where Dp denotes solitons carrier frequencies difference.

The closed form expressions for re-coloring are compared to the full fledged simulation in Fig. 9. From the standpoint of re-coloring, the particle-like model seems to fit the simulation, with increased inaccuracy in the capture region. The better match is obtained for Dθ = π/2, perhaps since the initial solitons are orthogonal (i.e. over the complex variables plane).

 figure: Fig. 9.

Fig. 9. Re-coloring (pf/ p0 vs. p0) within particle-like model predictions compared to full fledged simulation. W2=1.5, W1=1, τ0=0.

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7. Conclusions

The soliton perturbation method was modified to model colored solitons. We introduced a particle-like model describing the interactions of two colored solitons. The utilization of soliton perturbation method has proven to be valid and beneficial to resolve and analyze the underlying mechanisms and to synthesize better “systems,” e.g., when particle-like “damping” source is pursuit in order to enhance the capture threshold. The perturbation method became “handy” also for explaining the differences of the particle-like model from simulations, in terms of additional perturbation source introduced into the NLSE to represent the coherent solitons interaction terms. Thus we found that modeling the colored solitons interaction in terms of mutual perturbations gives a clear understanding of the interaction, and enables more easily comparison to other analysis tools.

The solitons capture process entails a color shifting of data bit using a control bit in a different color. This interaction may be exploited as a selective all optical read mechanism e.g. for soliton memory rings. The soliton capture has an oscillatory nature, suggesting that introducing a friction equivalent mechanism may assist enhancing the convergence.

The frequency difference capture threshold was found to be only about 40% higher than the soliton spectral FWHM, with no dependence on fiber or soliton parameters. Thus a necessary condition for capture is that the solitons will overlap “considerably” in spectrum. The mechanical analogy suggests that in order to reach a capture of two colored solitons with higher frequency shifts, an external particle-like damping mechanism should be introduced. Closed form expressions for the re-coloring of the emerging solitons and the boundaries of the intermediate region are derived. Those results are validated simulating the full-fledged NLSE as well as the XPM coupled NLSEs. Since coherency of the colored solitons interaction varies within the process, the validation with the full-fledged NLSE is necessary.

The success of this perturbation approach with no coherent interaction terms, even for capture processes, can be explained by the fact that whenever the two oscillating captured solitons are temporally overlapping (Ep=0) – they have the largest frequency difference (incoherent) while at the maximum oscillation amplitude (Ek=0), where their frequency difference is zero, they are quiet far apart temporally.

In comparison to simulations a third mode of interaction between the colored solitons, is revealed, as an intermediate behavior between capture and escape. The significance of comparison to both types of simulations is manifested for the capture threshold values. The capture threshold in the XPM simulations was lower than the analytic model, while for the direct simulation values higher than the particle-like predictions were obtained. The decrease in the capture threshold in XPM simulation may stem from secondary dispersive waves emanated while interacting. Oppositely, a natural damping mechanism appears in the full fledged simulation was the reason for enhanced capture threshold.

The particle-like model was modified for non-equal colored solitons. The solitons peak amplitudes was shown to play a role of the particles mass. Using center of mass notation, the capture threshold and re-coloring are found. Those predictions were verified with the full-fledged simulations. The model seems to give reasonable accuracy for small intensity difference.

References and Links

1. M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992). [CrossRef]  

2. H.A. Haus, “Lecture 11” in Opical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz, ed. (Springer, NY,1999).

3. R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996). [CrossRef]  

4. A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

5. H. Avramopoulos and A. Whitaker, “Addressable fiber-loop memory,” Opt. Lett. 18, 22–24 (1993). [CrossRef]   [PubMed]  

6. C.R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987). [CrossRef]   [PubMed]  

7. M.N. Islam, Ultrafast Fiber Switching Devices and Systems ,(Cambridge, GB, 1992).

8. M.W. Chbat, C.R. Menyuk, I. Glesk, and P.R. Pruncnal, “Interactions of bound multiple soliton in strongly birefringent fibers,” Opt. Lett. 20, 258–260 (1995) [CrossRef]   [PubMed]  

9. M.N. Islam, “Ultrafast all-optical logic gates based on soliton trapping in fibers,” Opt. Lett. 14,1257–1259 (1989) [CrossRef]   [PubMed]  

10. C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992) [CrossRef]  

11. D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993) [CrossRef]   [PubMed]  

12. B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998) [CrossRef]  

13. B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991). [CrossRef]   [PubMed]  

14. M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996) [CrossRef]  

15. G.P. Agrawal, Nonlinear Fiber Optics, 2’nd Ed, (Academic Press, NY, 1995), Chap. 2,7.

16. Y.R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), 40.

17. M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994). [CrossRef]  

18. A. Meccozi, J.D. Moors, H.A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992); L.F. Mollenauer, S.G. Evangelides, and J.P. Gordon, “Wavelength division with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362––367 (1991). [CrossRef]  

19. D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999) [CrossRef]  

20. H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996) [CrossRef]  

21. E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted

22. C.R. Menyuk, “Application of multiple-length-scale method to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999). [CrossRef]  

23. J.P. Gordon, “Interaction forces in optical fibers,” Opt. Lett. 8, 596–598 (1983). [CrossRef]   [PubMed]  

24. S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

25. N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999). [CrossRef]  

26. H. A. Haus, W. S. Wong, and F.I. Khatri, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B 14, 304–313 (1997). [CrossRef]  

27. E. Feigenbaum, “Multi-colored optical storage rings – solitons interaction,” Thesis (EE Department -Technion, Israel, 2003)

28. A. Mecozzi and H. A. Haus, “Effect of filter on soliton interactions in wavelength-division-multiplexing systems,” Opt. Lett. 17, 988–990 (1992). [CrossRef]   [PubMed]  

References

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  1. M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
    [Crossref]
  2. H.A. Haus, “Lecture 11” in Opical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz, ed. (Springer, NY,1999).
  3. R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
    [Crossref]
  4. A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)
  5. H. Avramopoulos and A. Whitaker, “Addressable fiber-loop memory,” Opt. Lett. 18, 22–24 (1993).
    [Crossref] [PubMed]
  6. C.R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
    [Crossref] [PubMed]
  7. M.N. Islam, Ultrafast Fiber Switching Devices and Systems ,(Cambridge, GB, 1992).
  8. M.W. Chbat, C.R. Menyuk, I. Glesk, and P.R. Pruncnal, “Interactions of bound multiple soliton in strongly birefringent fibers,” Opt. Lett. 20, 258–260 (1995)
    [Crossref] [PubMed]
  9. M.N. Islam, “Ultrafast all-optical logic gates based on soliton trapping in fibers,” Opt. Lett. 14,1257–1259 (1989)
    [Crossref] [PubMed]
  10. C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
    [Crossref]
  11. D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
    [Crossref] [PubMed]
  12. B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
    [Crossref]
  13. B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
    [Crossref] [PubMed]
  14. M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
    [Crossref]
  15. G.P. Agrawal, Nonlinear Fiber Optics, 2’nd Ed, (Academic Press, NY, 1995), Chap. 2,7.
  16. Y.R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), 40.
  17. M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
    [Crossref]
  18. A. Meccozi, J.D. Moors, H.A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992); L.F. Mollenauer, S.G. Evangelides, and J.P. Gordon, “Wavelength division with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362––367 (1991).
    [Crossref]
  19. D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
    [Crossref]
  20. H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
    [Crossref]
  21. E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted
  22. C.R. Menyuk, “Application of multiple-length-scale method to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
    [Crossref]
  23. J.P. Gordon, “Interaction forces in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [Crossref] [PubMed]
  24. S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79
  25. N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
    [Crossref]
  26. H. A. Haus, W. S. Wong, and F.I. Khatri, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B 14, 304–313 (1997).
    [Crossref]
  27. E. Feigenbaum, “Multi-colored optical storage rings – solitons interaction,” Thesis (EE Department -Technion, Israel, 2003)
  28. A. Mecozzi and H. A. Haus, “Effect of filter on soliton interactions in wavelength-division-multiplexing systems,” Opt. Lett. 17, 988–990 (1992).
    [Crossref] [PubMed]

1999 (3)

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[Crossref]

C.R. Menyuk, “Application of multiple-length-scale method to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[Crossref]

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

1998 (1)

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[Crossref]

1997 (1)

1996 (3)

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[Crossref]

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[Crossref]

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

1995 (1)

1994 (1)

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[Crossref]

1993 (2)

H. Avramopoulos and A. Whitaker, “Addressable fiber-loop memory,” Opt. Lett. 18, 22–24 (1993).
[Crossref] [PubMed]

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[Crossref] [PubMed]

1992 (4)

1991 (1)

B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[Crossref] [PubMed]

1989 (1)

1987 (1)

1983 (1)

Adams, L.E.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Agarwal, A.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Agrawal, G.P.

G.P. Agrawal, Nonlinear Fiber Optics, 2’nd Ed, (Academic Press, NY, 1995), Chap. 2,7.

Anderson, D

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[Crossref]

Arbel, D.

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[Crossref]

Avramopoulos, H.

Barry, R. A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Chabat, M.W.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[Crossref]

Chan, V. W. S.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Chbat, M.W.

Doerr, C.R.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Etrich, C.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

Evangelides, S.G.

Feigenbaum, E.

E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted

E. Feigenbaum, “Multi-colored optical storage rings – solitons interaction,” Thesis (EE Department -Technion, Israel, 2003)

Finn, S.G.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Gill, T.L.

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[Crossref]

Glesk, I.

Gordon, J.P.

Hall, K. L.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Haner, M.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Haus, H. A.

Haus, H.A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[Crossref]

A. Meccozi, J.D. Moors, H.A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992); L.F. Mollenauer, S.G. Evangelides, and J.P. Gordon, “Wavelength division with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362––367 (1991).
[Crossref]

H.A. Haus, “Lecture 11” in Opical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz, ed. (Springer, NY,1999).

Höök, A

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[Crossref]

Ippen, E.P.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Islam, M.N.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[Crossref]

M.N. Islam, “Ultrafast all-optical logic gates based on soliton trapping in fibers,” Opt. Lett. 14,1257–1259 (1989)
[Crossref] [PubMed]

M.N. Islam, Ultrafast Fiber Switching Devices and Systems ,(Cambridge, GB, 1992).

Karlsson, M.

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[Crossref]

Kaup, D.J.

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[Crossref] [PubMed]

Khatri, F.I.

Kintzer, E.S.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

kubota, H.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[Crossref]

Kumar, P.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Lai, Y.

Lederer, F.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

Lisak, M.

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[Crossref]

Mahmood, M.F.

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[Crossref]

Malomed, B.A.

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[Crossref]

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[Crossref] [PubMed]

B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[Crossref] [PubMed]

Manakov, S.V.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Meccozi, A.

Mecozzi, A.

Mel’nikov, I. V.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

Menyuk, C.R.

Mihalache, D.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

Mollenauer, L.F.

Moors, J.D.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

A. Meccozi, J.D. Moors, H.A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992); L.F. Mollenauer, S.G. Evangelides, and J.P. Gordon, “Wavelength division with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362––367 (1991).
[Crossref]

Nakazawa, M.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[Crossref]

Novikov, S.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Orenstein, M.

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[Crossref]

E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted

Panoiu, N. C.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

Pitaevskii, L.P.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Prucnal, P.R.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[Crossref]

Pruncnal, P.R.

Rauschenbach, K.A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Shen, Y.R.

Y.R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), 40.

Soccolich, C.E.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[Crossref]

Su, Y.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Suzuki, K.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[Crossref]

Swanson, E.A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Tasgal, R.S.

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[Crossref]

Wang, L.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Whitaker, A.

Wong, W. S.

Wong, W.S.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[Crossref]

Yamada, E.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[Crossref]

Zachary, W.W.

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[Crossref]

Zakharov, V.E.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Elec. Lett. (1)

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[Crossref]

IEEE J. of Quant. Elec. (1)

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[Crossref]

IEEE Phot. Tech. Lett. (1)

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[Crossref]

J. Eng. Math. (1)

C.R. Menyuk, “Application of multiple-length-scale method to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Selected Areas In Com. (1)

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[Crossref]

Opt. Eng. (1)

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[Crossref]

Opt. Lett. (6)

Phys. Rev. A (2)

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[Crossref] [PubMed]

B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[Crossref] [PubMed]

Phys. Rev. E (2)

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[Crossref]

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[Crossref]

Phys. Scripta (1)

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[Crossref]

Rev. Mod. Phys. (1)

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[Crossref]

Other (8)

E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted

G.P. Agrawal, Nonlinear Fiber Optics, 2’nd Ed, (Academic Press, NY, 1995), Chap. 2,7.

Y.R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), 40.

M.N. Islam, Ultrafast Fiber Switching Devices and Systems ,(Cambridge, GB, 1992).

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

H.A. Haus, “Lecture 11” in Opical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz, ed. (Springer, NY,1999).

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

E. Feigenbaum, “Multi-colored optical storage rings – solitons interaction,” Thesis (EE Department -Technion, Israel, 2003)

Supplementary Material (3)

» Media 1: AVI (1064 KB)     
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» Media 3: AVI (828 KB)     

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Figures (9)

Fig. 1.
Fig. 1. Perturbation calculation of escaping colored solitons. p0=0.145×2π, τ0=5. (a) Center (τ) of escaping soliton (blue) vs. the center of non-perturbed one (black). (b) Solitons carrier (red) vs. center accumulated perturbation (i.e. difference of the curves depicted at figure 1(a)) (blue).
Fig. 2.
Fig. 2. The soliton trace in the energy plane. The red and blue curves are for p0=-0.137×2π and -0.16×2π respectively with τ0=0.
Fig. 3.
Fig. 3. Simulation (XPM coupled NLSEs) of equal intensity colored solitons interaction. The intensity envelope of one soliton as the two solitons propagate simultaneously in the fiber for: (a) (1MB) Escape (p0=-0.20×2π), (b) (1MB) Intermediate (p0=-0.15×2π) and (c) (0.83MB) Capture (p0=-0.07×2π). τ0=0.
Fig. 4.
Fig. 4. Solitons re-coloring vs. initial frequency (τ0=0).
Fig. 5.
Fig. 5. Trajectories of re-coloring in the energy plane.
Fig. 6.
Fig. 6. Temporal amplitude calculated by XPM simulations for one of the captured solitons (contour) and the soliton center calculated using Eq. (9) (bold curve). τ0=0, θ1= θ2=0, p0=-0.07×2π.
Fig. 7.
Fig. 7. Comparison of solitons capture using full fledged NLSE simulation (contours) and the perturbation calculations (bold curves) curves: (a) soliton center, (b) center frequency. p0=0.05×2π, θ12=0, τ0=0.
Fig. 8.
Fig. 8. A comparison of a second order soliton spectrum at its temporal propagation peaks and the combined spectrum of two co-centered colored solitons at capture threshold. The two colored solitons parameters: Dτ=0, W=1, Dp=2×0.193×2π, Dθ=0.
Fig. 9.
Fig. 9. Re-coloring (pf/ p0 vs. p0) within particle-like model predictions compared to full fledged simulation. W2=1.5, W1=1, τ0=0.

Equations (51)

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z u = j { 1 2 β " 2 T + δ u 2 } u ,
u = W sech { ε W ( T τ ) } exp { j ( pT + θ ) } ,
z τ = β " p ,
z θ = 1 2 ( δ W 2 + β " p 2 ) ,
z Δ W = S W ,
z Δ τ = S τ + β " Δ p ,
z Δ p = S p ,
z Δ θ = S θ + δ W ΔW ,
S m Im dT { f m * s ( z , T ) exp { j 2 W 2 z } } .
z W = S W ,
z τ = S τ + β " p ,
z p = S p ,
z θ = S θ + δW ( S W dz ) + 1 2 ( δ W 2 + β " p 2 ) .
u 1 = W 1 sech { ε W 1 ( T τ 1 ) } exp { j ( p 1 T + θ 1 ) } ,
u 2 = W 2 sech { ε W 2 ( T τ 2 ) } exp { j ( p 2 T + θ 2 ) } ,
z u 1 = j ( 1 2 β " 2 T + δ ( u 1 2 + 2 u 2 2 ) ) ,
z u 2 = j ( 1 2 β " 2 T + δ ( u 2 2 + 2 u 1 2 ) ) .
δ ( u 2 ) 2 u 1 * + δ ( u 1 ) 2 u 2 * .
s XPM = 2 δ u 2 2 u 1 .
S W XPM = 0 ,
S τ XPM = 0 ,
S p XPM = 2 δ ε 2 W 1 2 W 2 2 dT { tanh ( ε W 1 ( T τ 1 ) ) ·
· sech 2 ( ε W 1 ( T τ 1 ) ) sech 2 ( ε W 2 ( T τ 2 ) ) } ,
S θ XPM = 2 δ ε W 1 W 2 2 dT { ( 1 ε W 1 ( T τ 1 ) tanh ( ε W 1 ( T τ 1 ) ) ) ·
· sech 2 ( ε W 1 ( T τ 1 ) ) sech 2 ( ε W 2 ( T τ 2 ) ) } ,
F ( ) = ( 2 δ β " ε W 3 ) + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( ξ ) } ,
z ( β " p ) = F ( 2 τ ) = F ( 2 τ ) ,
z τ = β " p ,
2 z τ = F ( 2 τ ) .
Δ τ ( z : + ) = εW p 0 2 ,
2 z τ ( 16 15 δ 2 W 4 ) F L = 2 F L τ .
E k = 1 2 ( β " ) 2 p 2 .
E p = 2 ( β " ε W ) 2 1 ( 2 ε W τ ) coth { 2 ε W τ } sinh 2 { 2 ε W τ } .
E p min = 2 3 ( β " ε W ) 2 .
p 0 TH = 4 3 ε W .
RC = p f p 0 = 1 4 3 ( ε W ) 2 p 0 2 ,
E k 0 = 1 1 RC 2 E p 0 .
2 z τ = 2 F L τ V z τ .
z ( β " p k ) = 2 ( δ W k W 3 k ) 2 + { tanh ( ε W k ξ ) sech 2 ( ε W k ξ ) sech 2 ( ε W 3 k ( ξ ( τ 3 k τ k ) ) ) } ,
z τ k = β " p k ; k∈ 1,2 .
m 2 m 1 t 2 r 1 t 2 r 2 = + { tanh { ε W 1 ξ } sech 2 { ε W 1 ξ } sech 2 { ε W 2 ( ξ ) } } , + { tanh { ε W 2 ξ } sech 2 { ε W 2 ξ } sech 2 { ε W 1 ( ξ ) } } , = W 2 W 1 .
m k = W k
z ( β " p k ) = F m k ,
F = 2 β " δ ε ( W 1 W 2 ) 2 + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( W 2 W 1 ξ ε W 2 ) } .
μ 2 t r = F .
E K = 1 2 μ ( t r ) 2 = 2 W 1 W 2 W 1 + W 2 ( β " p 0 ) 2 ,
E p = 2 β " δ ε ( W 1 W 2 ) 2 Δ τ + dr { + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( W 2 W 1 ξ ε W 2 r ) } } .
E pmin = 2 β " δ ε W 1 2 W 2 + { tanh ( ξ ) sech 2 ( ξ ) tanh ( W 2 W 1 ξ ) } .
p 0 TH = 1 3 ε ( W 1 + W 2 ) ; W 2 W 1 .
p o TH ( W 1 , W 2 ) = 1 2 { p o TH ( W 1 , W 1 ) + p o TH ( W 2 , W 2 ) } .
Dp = ( D p 0 ) 2 4 3 ε 2 ( W 1 + W 2 ) 2 ; W 1 W 2 ,

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