Pulse Trapping and Harmonic Oscillations in the Presence of a Dark Soliton in Optical Fibers

When the fiber dispersion is normal, there exists a pair of bright and dark solitons with orthogonal polarizations. When the power of the dark soliton (pump) on one polarization is stronger than that of the bright pulse (signal) on the other polarization, we found that the bright pulse is trapped into the dip of the dark soliton, and it undergoes harmonic oscillations if the mean frequency of the bright pulse is lower than a certain critical frequency. Otherwise, it escapes the potential well formed by the dark soliton. Analytical expressions derived for the frequency of harmonic oscillations and the asymptotic value of the trapped energy are in good agreement with numerical experiments. This result could have potential applications for ultra-fast all-optical switching and logic gates.


Pulse Trapping and Harmonic Oscillations in the Presence of a Dark Soliton in Optical Fibers
Mohammad Khajezadeh and Shiva Kumar , Member, IEEE Abstract-When the fiber dispersion is normal, there exists a pair of bright and dark solitons with orthogonal polarizations.When the power of the dark soliton (pump) on one polarization is stronger than that of the bright pulse (signal) on the other polarization, we found that the bright pulse is trapped into the dip of the dark soliton, and it undergoes harmonic oscillations if the mean frequency of the bright pulse is lower than a certain critical frequency.Otherwise, it escapes the potential well formed by the dark soliton.Analytical expressions derived for the frequency of harmonic oscillations and the asymptotic value of the trapped energy are in good agreement with numerical experiments.This result could have potential applications for ultra-fast all-optical switching and logic gates.

I. INTRODUCTION
W HEN fiber dispersion is exactly balanced by nonlinear- ity, optical soliton is formed which propagates without change in shape over long distances [1], [2].The nonlinear Schrodinger equation (NLSE) admits bright [2] and dark soliton [3] solutions in anomalous and normal dispersion regimes, respectively.Nonlinear interaction between optical signals can be divided into two types: (i) the signals that are of the same polarizations but of different center frequencies, and (ii) the signals that are of the same center frequency, but of different polarizations.In the case of (i), Trillo et al. [4] and Kivshar [5] have shown that a pair of bright and dark solitons exist provided the center frequencies of bright and dark solitons correspond to anomalous and normal dispersion regimes, respectively.It is hard to observe such a pair experimentally since the center frequencies should be carefully chosen so that the group velocity of these solitons matches.It was just recently [6] that experimental validation was conducted using a fiber laser with near-zero average cavity dispersion, which showed the separate formation of dark and bright solitons on either side of the zero dispersion point.In the case of (ii), Menyuk [7] showed that bright soliton can exist on each polarization and these solitons trap each other propagating stably over long distances.It has been experimentally reported in many works including [8].In the case of a single-mode fiber having a weak birefringence, Christodoulides has shown the existence of a vector dark-bright soliton pair, i.e. dark soliton on one polarization and bright soliton on the other [9].
By averaging polarization states over the Poincare sphere, Wai et al. have obtained the Manakov equation which describes the evolution of polarization states in single-mode optical fibers [10].In this paper, we consider the dark-bright soliton solution of the Manakov equation.In the context of spatial soliton, Sheppard and Kivshar [11] have shown the existence of a dark-bright soliton pair which is the starting point for our analysis.We consider the case in which the bright signal pulse is weak and the dark soliton is strong.The bright signal pulse of a different center frequency passes through the potential well formed by the dark soliton.If its center frequency is lower than a threshold frequency, it is trapped by the dark soliton; otherwise, it escapes the potential well.Thus, the final position of the bright pulse can be controlled by the dark soliton.These findings could be useful to develop ultra-fast all-optical logic gates and switching circuits.The all-optical logic gates can be divided into two types: (i) spatial and (ii) temporal.In the case of a spatial (temporal) logic gate, the spatial (temporal) position of the signal at the gate output varies depending on the input states.Blair et al. [12] proposed a spatial all-optical NOR gate based on spatial soliton shifts due to soliton collisions.Islam [13] utilized the frequency shift resulting from the collision of two temporal solitons to create a temporal all-optical XOR gate.In another work, Islam et al. [14] developed a temporal all-optical NOR gate utilizing time shifts resulting from soliton dragging.In this study, the time shifts caused by soliton trapping affected the control signal, causing it to be received outside of the expected time window.In this paper, we propose a temporal all-optical AND gate based on the bright pulse trapping due to the presence of a dark soliton in the other polarization.In the absence of the dark soliton, the pulse is not trapped and hence, it will be outside the detection window centered around the dip of the dark soliton.The advantages of temporal soliton-based all-optical logic gates are that (i) they can operate at high rates (>100 Gb/s) due to the possibility of forming solitons with very small pulse widths and (ii) they do not suffer from electronic bottleneck since the inputs and outputs are all optical.

II. THEORETICAL BACKGROUND
The nonlinear interactions between two polarization components in an optical fiber can be described by Manakov equation [10]: where U x and U y are complex field envelopes on x and y polarization, β 2 and γ are 2nd order dispersion and nonlinear coefficients.Dispersion and nonlinear lengths are defined as 8γP 0 , respectively [15], where T 0 is the pulse width and P 0 is the peak power.Manakov equation is an integrable model owing to the symmetrical XPM-induced coupling between two polarization components.If one sets L D = L NL , then we can normalize (1) by introducing these dimensionless parameters: Where j ∈ {x, y} and A x is the amplitude of the pulse on x polarization.Using these parameters (1) transforms into: These equations are dealt with in [11], [16].One can apply Hirota's method to find dark-dark or dark-bright soliton solutions in their most general form.For simplicity, we focus on a special case of equations ( 14) and ( 15) of [11] which are more relevant to our study: where θ ∈ [0, π/2].The amplitude of the bright soliton is determined by θ.Choosing this parameter close to zero leads to a weak signal on y-polarization.We consider the case of |u x | |u y |.We designate u x and u y as pump and signal respectively.Under the weak signal approximation, cos θ is close to one, and (3b) leads to the linear Schrodinger equation, where the potential function is given by Let Using ( 6) and ( 7), ( 5) becomes where the prime indicates differentiation with respect to τ , k 2 = 2λ − 2 and ν = 2. Equation ( 8) describes the Schrodinger equation with the modified Poschl-Teller potential function [17].
This equation can be solved analytically to find the bounded, and scattering states for u y [18].Poschl-Teller potential has played an important role in gravitational physics where it has been used to model the dynamics of fields propagating around compact objects such as black holes and wormholes.[19] obtained quasinormal modes and frequencies associated with generalized Poschl-Teller potential and late-time evolution under this potential function is analyzed for a Cauchy initial value.It is shown that there are no late-time tails, meaning that the time evolution is always stable for long enough propagation times.For (8), it can be shown that there exists a bounded state corresponding to λ = 0.5 and is equal to: and a continuum of normalized scattering states corresponding to λ ≥ 1 given by: It can also be proved that the potential is reflectionless for , where n is an integer.Our special case with ν = 2 fulfills this requirement for n = 1.Thus any incident wave completely passes the potential well without any reflections.Every solution to (5) can be written as a combination of a bounded term and an integral over scattering components: (11) where and When an arbitrary wave packet is incident, only the bounded term (the first term in (11)) contributes to the trapped energy, a crucial factor for the intended application of this theory in the design of optical gates.Looking at the expression of ψ s in (10) reveals that if a signal is far from the time origin, then ψ s → e ikτ and so c s (k) → F {u y (τ, 0)} where F is the Fourier transform.It follows from Parseval's theorem that in this case, all the energy of u y is carried by scattering modes.In the next Section, we show that it is possible to control and manipulate the weak signal, u y by the strong pump, u x .

III. MOVING BRIGHT PULSE
In this section, we study the interactions between a strong dark soliton in the x-polarization which acts as a background pulse for the weak bright pulse on the y-polarization.The dynamics between moving dark-bright solitons have been extensively studied in numerous works including [11].The differences between our work and the previous works are as follows.(i) [11] considers a fully nonlinear case but its results are applicable only for integrable systems while we employ small signal approximation and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
results are applicable to non-integrable systems where the effects of self-phase modulation (SPM) and cross-phase modulation (XPM) are not equal.For example, it is common to encounter a 2/3 coefficient for XPM in many situations.(ii) In [11], the signal on y polarization has to be a soliton whereas in our case it can be any arbitrary weak pulse.Moreover, it is worth noting that when the power of the signal on y-polarization is large, there can be significant displacements and distortions in the shape of the dark soliton [11].This is particularly undesirable when designing gates that rely on time window filtering.
We introduce a moving bright pulse on y-polarization: where A is the amplitude of the bright pulse which satisfies the small signal approximation, τ 0 is the normalized initial delay, Ω is the normalized frequency shift.The frequency shift Ω translates into the inverse group velocity shift β 2 Ω due to group velocity dispersion.This pulse starts from the right side of the potential well and approaches its center due to this velocity shift.Using quantum mechanics analogy, β 2 Ω may be interpreted as the velocity of a particle with the wave function u y .If the kinetic energy of this particle exceeds the potential energy, the particle (i.e.bright pulse) will escape the potential well formed by the dark soliton.If the kinetic energy is lower than the potential energy, the bright pulse will be trapped into the dip of the dark soliton.The kinetic energy is proportional to (β 2 Ω) 2 .

A. Amount of Trapped Energy
A more rigorous approach to analyze this behavior is through examining the projection of the moving pulse onto the bounded state of (8): where c b is normalized by the total energy of the signal given by: The ratio of trapped energy (asymptotic) to the pulse energy is |c b | 2 /E y .Fig. 1 shows this parameter for two different values of pulse delays.As can be seen, as the initial speed (∝ Ω) increases, the amount of trapped energy decreases and after a certain threshold speed, the bright pulse escapes the potential well.Also, the amount of trapped energy is lower when the delay τ 0 is longer since the longer delay leads to a large number of scattering components.

B. Scattering Components of the Moving Pulse
Using (13), the spectrum of scattering components for the moving bright pulse given by ( 14) is calculated.Fig. 2 shows this spectrum for the case τ 0 = 1 and three different values of Ω.As the Ω increases, the spectrum moves towards the right and the peak increases meaning that faster and stronger scattering components will be excited as we increase Ω.In the limit when Ω is large, c b → 0 and we find a closed form expression for c s (k):

C. Numerical Simulations and Discussions
The coupled (3a) and (3b) are solved numerically (without making weak signal approximations) using the split-step Fourier technique [20].The following parameters are used in the simulations throughout this paper unless otherwise specified: A = 0.02 and delay τ 0 = 1.The following initial condition for u x is assumed: and the initial condition for u y is given by (14).To verify the expression in (15), numerical experiments are conducted to Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.5,5].In fact, this interval covers more than 99% of bounded state energy.We measure the ratio of energy within this interval to the total energy of u y .The pulse starts from normalized delay τ 0 = 1 and moves toward the dip of the potential well with the initial speed equal to β 2 Ω.Fig. 3 shows how the amount of trapped energy converges to the limit value given by (15).It may be noted that as Ω increases, the amount of trapped energy decreases consistent with Fig. 1.
Next, we investigate the evolution of weak bright pulse in three different scenarios: Case 1: If Ω is chosen to be large, the bright pulse starts from the right side of the potential well and will escape from it ending up on the other side of the well.This case is shown in Fig. 4 when Ω = 5.In terms of quantum mechanics analogy, large Ω means higher kinetic energy and the particle's kinetic energy exceeds the potential energy.

Case 2:
If Ω is small, the kinetic energy is not large enough to escape the potential barrier established by the dark soliton on the other polarization.In this case, the bright pulse will be trapped by the dark soliton and it undergoes harmonic oscillations as shown in Fig. 5 when Ω = 0.2.Based on (15) and Fig. 1, it can be confirmed that more than 70% of the energy will be trapped if Ω < Ω c , where Ω c is a threshold frequency given by When Ω = 5, it exceeds the threshold frequency and from (15), we find that the ratio of the trapped energy and incident energy is negligibly small (≈ 10 −7 ).Case 3: If the dark soliton is not present in x-polarization, the pulse given in ( 14) moves toward the other side regardless of the value of Ω.This is shown in Fig. 6.It may be observed that the pulse is broadened continuously due to dispersion since A is small.

TABLE I AND GATE USING TEMPORAL PORTS
From these three cases above, it can be inferred that the final location of the bright signal pulse can be controlled by the pump u x , and therefore, it can find applications for all-optical systems, especially in all-optical logic circuits and all-optical switching.For example, if Ω < Ω c and if we assign a temporal port 1 for the final position τ f with |τ f | < τ 0 and a temporal port 2 for |τ f | > τ 0 , the signal pulse would arrive at port 1 or port 2 depending on the presence or absence of the dark soliton, respectively which acts as a control pulse.Since these pulses could be of the order of a few ps, all-optical switching at Tb/s is feasible using the proposed approach.
As for the all-optical logic gate, we assign a bit '1' for the center frequency of the pulse, Ω = Ω 1 < Ω c and a bit '0' for Ω = Ω 0 > Ω c .We also assign a bit '1'/'0' for the presence/absence of the dark soliton and, a bit '1' for the final position |τ f | < τ 0 and a bit '0' for |τ f | > τ 0 .It is easy to verify that this device can be used to realize an AND gate.This layout for the AND gate is summarized in Table I.The first row of Table I can be explained as follows.When the frequency shift of the pulse, Ω is greater than Ω c (input state is '0') and if the dark soliton is absent (state '0'), the bright pulse is not trapped and hence, it will be outside the detection window (output state '0').When Ω < Ω c , (input state '1') and if the dark soliton is present (state '1'), the pulse is trapped (output state '1').Similarly, the second and third rows of Table I can be explained.
If we replace the time t with the transverse distance x, temporal ports can be replaced with spatial ports, and the proposed approach can be used for all-optical switching of spatial signals.

IV. FREQUENCY OF OSCILLATIONS
As mentioned in the previous section, when Ω is small, the bright pulse undergoes harmonic oscillations.In this section, our goal is to determine the frequency of these oscillations once the signal has propagated sufficiently far to pass the transient state, i.e. when ξ becomes relatively large.Considering that λ = (k 2 + 2)/2 and using the method of stationary phase, we approximate the integral in (11) for large values of ξ to obtain: This approximation is quite accurate for large values of ξ.The mean position of the bright pulse is given by where E y is the energy of the bright pulse given by ( 16).Considering small signal approximation, the Schrodinger equation in (5) guarantees that the denominator in ( 21) is constant and where Y and φ are amplitude and phase of c b c s (0), respectively, i.e., From ( 22), we see that the mean position of the pulse oscillates with an angular frequency, ω 0 = 1/2 or in real units, it can be expressed as: Fig. 7 examines the accuracy of this approximation.The solid line in this figure shows the mean position of the bright pulse obtained by solving (3) with the initial conditions for u y given by ( 14) and for u x given by ( 18), and with τ 0 = 1 and Ω = 0.The broken line in Fig. 7 shows the mean position calculated analytically using (22).This figure demonstrates that when ξ is sufficiently large, (22) precisely predicts the mean position of the bright pulse.Equation ( 22) shows that the amplitude of these oscillations decay at a rate of 1 √ ξ .This implies a frozen trapped energy at infinity.Equation (24) indicates that ω 0 is directly related to A 2 x and inversely related to L D .Fig. 8 shows an example of how the mean position of the bright pulse moves as it propagates down the fiber.This figure is obtained by solving (1) numerically without making weak signal approximations.This numerical experiment is repeated for various combinations of parameters to confirm our analytical results.The base experiment is conducted using these parameters: L D = L NL = 10 Km (β 2 = 10 ps 2 Km −1 , T 0 = 10 ps, γ = 1 W −1 Km −1 , γ = 8γ 9 ), A x = 1, A = 0.01, delay t 0 = 10 ps (this corresponds to the normalized delay τ 0 = 1) and Ω = 0.Then, for each subsequent experiment, one parameter is changed to illustrate the effect of that parameter on the frequency of oscillations.For each experiment, the computed  frequency of oscillation, ω 0 from the numerical experiment, as well as the value of the proposed formula is tabulated in Table II.From these results, it can be confirmed that the measurements are consistent with (24).For example, when A x is changed to 0.5 (Ex2), ω 0 is decreased by a factor of 4.However, when A is changed to 0.04 (Ex3), ω 0 does not change.Also, when β 2 is changed to 20 ps 2 Km −1 (Ex4), L D is halved and ω 0 is doubled.Finally, Experiments 5 and 6 show that changing γ and Ω doesn't change the frequency of oscillations compared to the base experiment.All these experiments show that the proposed formula for frequency is accurate in all situations.
As the results of this section suggest, the frequency of oscillations does not depend on Ω, the frequency shift of the bright pulse.However, this parameter plays an essential role when one is concerned about the amount of energy trapped in the potential well.It is worth recalling from the previous section that the value of Ω determines whether the signal will be trapped at the end.

V. CONCLUSION
The nonlinear interaction between a weak bright pulse and a strong dark soliton is studied.We found that the weak bright pulse is trapped by the potential well formed by the dark soliton and undergoes harmonic oscillations if the mean frequency of the bright pulse is low; otherwise, it escapes the potential well.Thus the weak bright pulse can be controlled by the strong dark soliton which can find applications in all-optical logic and switching circuits.An analytical expression for the frequency of harmonic oscillation is obtained.

Fig. 1 .
Fig. 1.Ratio of asymptotic trapped energy to the pulse energy for two different values of pulse delay.

Fig. 6 .
Fig. 6.Signal evolution in the absence of dark soliton on the other polarization.Ω = 0.5.

Fig. 8 .
Fig. 8. Oscillations of the mean position of a trapped waveform with respect to propagation distance.

TABLE II FREQUENCY
OF OSCILLATIONS FOR EACH EXPERIMENT