Trapping and controlling the dispersive wave within a solitonic well

Zhixiang Deng; Xiquan Fu; Jun Liu; Chujun Zhao; Shuangchun Wen

doi:10.1364/OE.24.010302

1. Introduction

Various aspects of ultrashort pulses dynamics in nonlinear media, such as the formation of solitons, dispersive wave (DW) generation, supercontinuum generation, and interaction of solitons with DW have been the issue of fundamental studies and technological applications in photonics [1–5]. One salient topic of research in this area has been the nonlinear interaction of dispersive wave with solitons, which have recently attracted widespread interests [6–9]. Manipulating soliton with dispersive wave based on optical pulse interaction mechanism can mimic the physics at an event horizon [10,11]. This analogue takes place when a weak dispersive wave cannot penetrate an intense soliton, in spite of propagating at a different velocity [11–14]. In the time-domain, the underlying mechanisms have been described in terms of a soliton-induced refractive index barrier that changes the group velocity of the dispersive wave [13]. In the frequency-domain, this description can be equally understood to arise from the generation of new frequencies based on the four-wave mixing (FWM) phase-matching condition [6,15]. Experimentally, several studies have also addressed the physical description of fibre-optic event horizons in terms of cascaded FWM between discrete single-frequency fields [10].

Studies of soliton-DW dynamics have recently gone a step forward by taking into account the DW’s interaction with two identical, time-separated solitons [16–20]. DWs temporally located in a zone between the twin-solitons interact with one soliton after being bounced off the other. This process can be repeated multiple times, leading to the formation of solitonic cavity behaving like two reflective mirrors for the DWs [16–19]. When the intensity of DWs increases to a considerable degree, the twin-solitons are attracted to each other, to cause collision or even fusion [16,18–20]. It has been demonstrated that this phenomenon is responsible for the emergence of multiple soliton knot patterns during complicated supercontinuum generation process [17]. Recently, such a solitonic cavity has been successfully realized experimentally [18]. Besides, the DW’s interaction with two dark solitons can lead to the formation of the “convex mirror” cavity with mutually excluding [19]. A “concave mirror” solitonic cavity created by two second-order solitons with external dispersive wave trapped between them was also studied [20]. In previous papers, two solitons considered generally have the same group velocity. However, to create a solitonic cavity, the same velocity is not necessary for twin-solitons. So far, to the best of our knowledge, the DWs’ interaction with two bright solitons having slightly mismatched group-velocities has not been investigated yet.

With respect to the soliton-soliton collisions dominated in the supercontinuum generation process, it has been widely studied in the past decade [21–23]. Generally, the soliton-soliton collision can lead to the DW generation. Recently, the DW induced by soliton-soliton collisions in long pulse supercontinuum generation in a photonic crystal fiber with two zero-dispersion wavelengths (ZDW) have been investigated, and its peak power exhibits extreme-value “rogue” characteristics with one order of magnitude greater than that derived from single-soliton generation [21]. Experimentally, the evidence of collision-induced dispersive wave (CDW) in fiber with two ZDW has been reported [22]. By tuning the initial delay between twin-solitons, the occurring soliton-solitons collision can be controlled, which will lead to the generation of CDW [22,23]. In addition, it is necessary to find alternative and efficient way to control both the soliton-soliton collisions and the generated CDW.

In this paper, we explored the possibility to create a solitonic well made from the two initial solitons moving away from each other to trap the incident DW numerically. Besides, we demonstrated that both the collision point of twin-solitons and the intensity of CDW can be controlled by adjusting either the peak power or initial temporal width of the incident DW.

2. Theoretical model

The dynamics of the dimensionless amplitude $A (Z, T)$ in the anomalous dispersion regime, including the third-order dispersion (TOD) and the Raman effect, can be governed by a generalized nonlinear Schrödinger equation, as follows:

i \partial_{Z} A + \frac{1}{2} \partial_{T}^{2} A - i δ_{3} \partial_{T}^{3} A + (1 - θ) | A |^{2} A + θ A \int_{- \infty}^{+ \infty} R (t^{'}) | A (T - t^{'}) |^{2} d t^{'} = 0,

where

Z

and

T

denote the rescaled dimensionless propagation distance and retarded time, respectively, and they satisfy the following relation:

Z = z / Z_{0}, T = t / T_{0},

and

Z_{0} = T_{0}^{2} / | β_{2} | \equiv 1.

The coefficient

δ_{3} \equiv β_{3} / (6 | β_{3} | T_{0})

represents the relative strength of the TOD, and

θ

measures the fractional contribution of the delayed Raman response to nonlinear polarization (

θ = 0.18

for fused silica).

R (t)

is the Raman response function, which can be written in the form

R (t) = \frac{τ_{1}^{2} + τ_{2}^{2}}{τ_{1} τ_{2}^{2}} Θ (t) \exp (- \frac{t}{τ_{2}}) \sin (\frac{t}{τ_{2}}),

where

Θ (t)

is Heaviside step function (unity for

t > 0,

and zero for

t < 0.

), and τ₁, τ₂ are both the delay times which were taken to be

τ_{1} = 12.2 f s / T_{0}

and

τ_{2} = 32 f s / T_{0},

respectively. Equation (1) can be solved numerically using the split-step Fourier method, where linear and nonlinear effects are modeled individually in the frequency and in the time domain, respectively. The nonlinear step is propagated in the time-domain, but the convolution between the field intensity and the nonlinear response is evaluated in the frequency domain using the convolution Fourier theorem. Finally, it is worth noting that self-steepening effect and dispersion terms with an order higher than 3 are neglected, owing to the fact that this effect was found to play a negligible role in the following results. With respect to the fundamental Kerr-type scattering process, the Raman term, which can induce a soliton self-frequency shift and thus the group velocity, is only included in simulations displayed below.

3. Results and discussions

The main prerequisite of our scheme is to establish an effective refractive index barrier and initiate the copropagation of the soliton (anomalous dispersion regime) and the dispersive wave (normal dispersion) at an almost identical group velocity. The moving refractive index barrier is created by a soliton via self-phase modulation and will be imposed on the dispersive wave to prevent its passage through the soliton.

3.1 Forming of a solitonic well to trap the incident dispersive wave

To create a solitonic well to trap the incident DW, we need to recapitulate results for the single scattering event of a weak DW on a single soliton. The wavenumber $κ$ , i.e. $κ = - \frac{1}{2} ω^{2} + δ_{3} ω^{3},$ and the relative group delay are shown in Fig. 1. The injected light consists of two pulses launched with different frequencies, namely soliton and DW, $A (0, T) = u_{s o l} (0, T) + u_{D W} (0, T) .$ The fundamental soliton has the initial form $u_{s o l} (0, T) = \sqrt{P_{s}} sech [\sqrt{P_{s}} (T + t_{0})] \exp (- i ω_{s 2} T),$ with the initial relative frequency of $ω_{s 2}$ , and the temporal delay and peak power of corresponding fundamental soliton $t_{0}$ and $P_{s},$ respectively. The incident DW is given by $u_{D W} (0, T) = \sqrt{P_{i n c}} \exp (- T^{2} / 2 w^{2})] \exp (- i ω_{i n c} T) .$ Here $P_{i n c}$ represents the peak power of incident DW, $w$ is the temporal width of DW, and $ω_{i n c}$ is the relative angular frequency of the incident DW. A typical simulated result is presented in Fig. 2. As is shown in Fig. 2(a), the incident DW initially approaches the soliton for vanishing temporal overlap and no interaction occurs. When they begin to overlap, an extended interaction builds up, preventing two pulses from crossing each other. The two pulses may be temporally locked due to their mutual interaction. Such extended interaction considerably changes the properties of both pulses. It is obvious that DW pushes the soliton away from its original trajectory, and the DW is almost completely reflected. Nevertheless, the combined effects of cross-phase modulation and normal dispersion lead to the phenomenon that the DW can split into multiple scattering waves with slightly different velocities [24,25].

Fig. 1 Wavenumber $κ$ and relative group delay of the fiber, exhibiting frequency combinations for initial twin-solitons moving away from each other and an incident DW with slightly group-velocity mismatched.

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Fig. 2 Density maps of the temporal evolution of the interaction of a single DW packet with a single soliton. The incident DW appears to reflect off the soliton and break up into multiple scattering waves during the collision. Top row: for different temporal separation between them $t_{0}$ : (a) 4, (b) 5, (c) 6, (d) 7, but $ω_{s 2}$ is fixed to 1.5. It shows that one can achieve even greater numbers of scattering waves by increasing $t_{0}$ . Bottom row: for the different initial relative carrier frequency of soliton $ω_{s 2}$ : (e) 0.3, (f) 0.7, (g) 1, (h) 2, but $t_{0}$ is fixed to 5. It displays that the scattering waves became more sparse with the increasing initial relative carrier frequency $ω_{s 2}$ . Here, the parameters are $δ_{3} = - 1, P_{i n c} = 5, P_{s} = 25, ω_{i n c} = - 38.2$ and $w = 0.22.$

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A very interesting scenario occurs when either the temporal separation between soliton and incident DW or the initial group-velocity of soliton is changed. We have conducted a series of numerical simulations. As can be seen from Figs. 2(a)-2(d), one can achieve even greater numbers of scattering waves by increasing the temporal separation $t_{0}$ between soliton and incident DW. A simple relation can be found that the number of scattering waves is increased by 2 if the temporal separation is increased by 1 unit. We now wonder what happens when the initial group-velocity of soliton is changed, i.e., changing scattering angles. The different group-velocity of soliton arises from the different initial carrier frequency. When changing the initial carrier frequency of soliton $ω_{s 2}$ , it is obvious that the number of scattering waves has not changed but the larger the initial relative carrier frequency $ω_{s 2}$ of soliton, the more sparse the scattering waves can be achieved [Figs. 2(e)-2(h)].

Therefore, both the temporal separation between soliton and incident DW and the initial group-velocity of soliton have effect on dynamics of interaction of soliton with weak DW. The temporal separation between them changes the number of scattering waves, while the initial group-velocity of soliton affects the sparseness of scattering wave.

An even more interesting phenomenon arises when we consider an input consisting of two well separated solitons and DW launched in between them, such as

\begin{matrix} A (0, T) = \sqrt{P_{s}} sech [\sqrt{P_{s}} (T - t_{0})] \exp (- i ω_{s 1} T) + \sqrt{P_{s}} sech [\sqrt{P_{s}} (T + t_{0})] \exp (- i ω_{s 2} T) \\ + \sqrt{P_{i n c}} \exp (- T^{2} / 2 w^{2})] \exp (- i ω_{i n c} T), \end{matrix}

where

t_{0}

is the absolute value of time delay, and the sign “+/−” represents whether the soliton is ahead or behind of DW, respectively;

ω_{s 1}, ω_{s 2}

and

ω_{i n c}

denote the relative carrier frequencies of the twin-solitons and of the DW, respectively, with a selection of an exemplary frequency combination as shown in Fig. 1. However, the soliton group velocity is proportional to its frequency because the wavenumber in the soliton frequency range is quasi-parabolic. By choosing the initial soliton frequencies, they move away from each other during the propagation. We started with a relatively weak DW with amplitude of

\sqrt{P_{i n c}} = 1

reflecting off the soliton with

\sqrt{P_{s}} = 5

. A typical simulation result is presented in Figs. 3(a) and 3(b). As shown in Fig. 3(a), it can be seen that the dispersive wave do not strongly affect the motion of the solitons. In contrast, the weak DW is reflected at the trail edge of the soliton, owing to the soliton-induced refractive index barrier, and then drifts apart at a smaller group velocity. Simultaneously, we find that the frequencies of DW scattered on the first (leading) and second (trailing) solitons get closer, manifesting as the spectrum narrows, which can be indeed seen in Fig. 3(b). As the two fundamental solitons move at slightly different velocities, they separate from each other and the spectrum is modulated at a frequency of the inverse of twin-solitons separation, resulting in the spectral fringes in Fig. 3(b).

Fig. 3 (a, b) The radiation spectral narrowing trapped between two separating solitons for the peak power of incident DW P_inc = 1. The leading and trailing solitons have normalized peak power P_s = 25, and frequencies ω_s1 = 0, ω_s2 = 1.5, respectively. Temporally, the leading (first) soliton is initially 5 units ahead of the incident DW while the trailing (second) soliton trails the incident DW by 5 units at the input of fiber. The incident dispersive wave is taken with frequency ω_inc = –38.2. Due to their different group velocity, twin-solitons separate during the course of the propagation. (c, d) The intense dispersive wave field trapped in a solitonic well formed by the twin-solitons moving away from each other. The parameters are the same as in (a, b) except that the peak power of incident DW is set to P_inc = 4.0. The evolutions are in the temporal (a, c) and frequency-domains (b, d) representations.

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When the amplitude of DWs is increased twofold ( $P_{i n c} = 4$ ), the interaction of the DW with the twin-solitons with slightly mismatched group-velocity becomes more complicated with much richer dynamics. The intense DW will split into numerous small pulses that are almost entirely trapped within a solitonic well formed by two solitons and filled with the whole solitonic well. The temporal propagation dynamic is shown in Fig. 3(c) in a reference frame moves at the group-velocity at the initial frequency of the second (trailing) soliton. The DW, travelling faster than the first (leading) soliton, will approach the solitons until it reaches the trailing edge of the soliton, owing to the mismatched group-velocity. At this time, the dispersive pulse begins to interact with the leading soliton. This process can be further explained as follows. Firstly, the soliton deviates from the original trajectory, and drifts apart with a smaller group velocity, maintaining stable propagation as a fundamental soliton. Secondly, due to strong nonlinear refractive index change, the DW is almost completely reflected after the first collision process, and the nonlinear interaction persists only between the DW and the trailing edge of the leading soliton. The DW is almost completely reflected and splitted into multiple scattering waves. Multiple scattering waves then go toward the trailing soliton, and the second collision process occurs quickly. However, this process is different from the first collision. The trailing soliton and the first scattering waves are accelerated by the second collision. On the one hand, the trailing soliton bends toward the dispersive wave, shortening the delay between the two solitons. In addition, the second scattering waves (second collision come out) and first scattering wave (first collision come out, but the second collision does not occur due to smaller relative velocity) are mutual superposition, leading to the DW breaking again within a solitonic well. This is remarkably different from the case with periodical bouncing of an incident DW, as the solitonic cage formed by the same initial frequency of twin-solitons [18].

As the propagation distance increases, similar process occurs again. However, as a result of multiple reflections of the trapped DWs from the bounding solitons that act as mirrors in the solitonic well, the solitons experience strong bending of their trajectories and eventually collide. Eventually, the intense DW will be broken up into numerous small low-amplitude pulses that are almost entirely trapped within the solitonic well.

From Fig. 3(c), it can be seen that the “soliton mirrors” degrade, allowing the trapped DW eventually to escape from the solitonic well boundary at certain distance before the two solitons collision. The reason is that the reflection process is accompanied by energy transfer between the pulses [11]. While any change of the trapped DW pulse parameters that can be induced to manipulate the soliton pulse is constrained by the requirement to enable the reflection process at optical event horizon. With increasing DW energy the efficiency of the reflection process might be reduced [26,27].

In the spectral domain, unlike a back-and-forth wavelength conversion of the probe to an idler wave in the case of the solitonic cage [18], the spectrum of the trapped DW can first shrink, and then expand in the course of the propagation [see Fig. 3(d)]. The reason is that the resonant scattering condition between the leading soliton and the incident DW differs from that between the trailing soliton and the incident DW, owing to the various twin-solitons central frequency. Figures 3(c) and 3(d) show that for relatively low power of the incident DW, the solitons in fact collide in fiber, but both the temporal and the spectra show no evidence of any new DW structure.

3.2 Manipulating CDW generation within a solitonic well

Numerical spectral signatures of development of a new strong band at low frequency side can be seen as the power of the incident DW increases. At a power of 5.3, Fig. 4(b) shows a strong spectral peak around $ω = - 50$ which is clearly reproduced in the spectral evolution. Simulations of the time-domain evolution allow us to easily identify this DW as generated from the soliton collision occurring at a normalized propagation distance of 6.05 as can be seen from Fig. 4(a). Therefore, such DW has been referred to as CDW. The CDW is emitted at a frequency determined by the standard phase-matching condition for DW generation using the peak power of the soliton superposition field attained during the collision and center frequency taken as the mean frequency of the nonlinear superposition. No dispersive wave is emitted by individual solitons before or after the collision. This is due to the fact that the individual soliton wavelengths before and after the collision are too far from the zero dispersion point to realize DW generation by phase-matching condition, while during the collision it is precisely the increase in bandwidth and peak power that allows the phase-matching condition for DW generation to be satisfied.

Fig. 4 (a, b) The dispersive wave generation induced by twin-solitons collision with the incident DW field $P_{i n c} = 5.3.$ The other parameters are the same as that used in Fig. 3. The white arrows indicate CDW. CDW represents the soliton collision-induced dispersive wave. (c, d) The same as in (a, b) other than the Raman effect is included. The Raman response parameters are $θ = 0.18, τ_{1} = 0.061$ and $τ_{2} = 0.16.$

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In Figs. 4(c) and 4(d) we revisit the simulation shown in Figs. 4(a) and 4(b) by including the Raman scattering effect in the propagation equation. We can again see how the twin-solitons collision is governed by the incident DW and subsequent emission of a CDW. In contrast with the absence of Raman scattering, it can be seen that the existence of Raman scattering effect can result in very unusual CDW dynamics. Firstly, the intensity of CDW is strongly enhanced, which is due to the fact that the frequency difference lies within the bandwidth of the Raman gain curve of a silica fiber. Secondly, the frequency detuning from CDW to the center frequency of two solitons decreases. The phenomenon can be explained as follows: with soliton self-frequency shift induced by the stimulated Raman scattering, individual solitons shift towards ZDW, leading to reduction of the detuning from the center frequency of two solitons to ZDW. However, the frequency detuning from CDW to the center frequency of two solitons is three times that from the center frequency of two solitons to ZDW under the phase-matching condition.

In order to fully understand the formation of a solitonic well to trap incident DW and control the CDW generation, the dependence of the position where twin-solitons collision occur and the intensity of CDW on the peak power and temporal width of incident DW has been studied. By changing the power of the incident DW, the position where the solitons eventually collide can be controlled. The initial pulse width of incident DW is fixed to be 0.22. This is because the effective interaction force induced by the incident DW, which can act between the twin-solitons, is proportional to the intensity of the incident DW, and in turn the greater the power of incident DW, the shorter the distance where solitons collision occurs required. It shows that the relationship between the position where twin-solitons collision occur, $Z_{c},$ and the power of the incident DW, $P_{i n c},$ follows a nearly linear function with negative slope, as can be seen from the red filled circles line in Fig. 5(a). The position where the collision occurs will also affect the peak power of CDW. The blue filled circles line in Fig. 5(a) shows the relationship between the power of CDW and the power of the incident DW. It can be seen that the peak power of CDW increases as the incident DW peak power $P_{i n c}$ increases for $P_{i n c} < 5.1,$ while the peak power of CDW decreased slightly, and gradually become saturated within the range $P_{i n c} > 5.1.$

Fig. 5 $Z_{c}$ and the power of collision-induced dispersive wave, $P_{C D W},$ as a function of (a) peak power and (b) temporal width of the incident wave. The red filled circles lines correspond to the collision point of twin-solitons. The peak power of dispersive wave induced by twin-solitons collision is shown by the blue filled lines.

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The position at which the solitons eventually collide and the intensity of CDW as a function of the initial temporal width of incident DW are calculated and shown in Fig. 5(b) by the red and blue filled circles lines, respectively. The peak power of incident DW is fixed to be 5.3. Clearly, the distance at which the solitons eventually collide $Z_{c}$ decreased significantly with the increased temporal width of incident DW, as shown by red filled circles line in Fig. 5(b). The concave shows that mathematically, the inverse dependence of the position where twin-solitons collision occur, $Z_{c},$ on the temporal width of incident DW $w .$ The relationship between $Z_{c}$ and $P_{i n c}$ follows a nearly linear function with negative slope, clearly indicating that the energy of incident DW strongly affected the dynamics of CDW generation within a solitonic well. As can be seen from blue filled circles line in Fig. 5(b), the intensity of CDW increases greatly with increased temporal width from 0.16 to 0.22, but it slightly fluctuates and eventually become stable when the temporal width is higher than 0.22. We can draw the conclusion that both changing the peak power and temporal width of incident DW are two suitable means to control CDW generation within a solitonic well.

The discussion above highlights that two solitons, between which the DWs are trapped, have initially different frequencies but the initial frequency of twin-solitons are fixed. In order to thoroughly understand the effect of initial soliton frequencies on the formation of a solitonic well and the generation of CDW within it, we have investigated the dynamics of the formation of a solitonic well within which the DW are trapped for a wide range of different initial soliton frequencies. The distance at which the solitons eventually collide due to the attraction and the peak power of CDW as a function of the initial temporal width of incident DW have been calculated and are shown in Fig. 6 by the red and blue filled circles lines, respectively. The distance at which the solitons eventually collide increased significantly with the increased initial frequencies difference, as can be seen from the red filled circles lines in Fig. 6. This implies that it is no longer possible to form twin-solitons collision in practice when two solitons initial frequencies difference is large. As the initial soliton frequencies difference is less than 1.8, the intensity of CDW induced by twin-solitons with group-velocity mismatched collision become slight decreased, while the decrease of intensity of CDW become remarkable when the initial soliton frequencies difference is larger than 1.8 as shown by the blue filled circles lines in Fig. 6.

Fig. 6 $Z_{c}$ and the peak power of CDW, $P_{C D W},$ versus the difference between initial carrier frequencies of the twins-soliton, $ω_{s 2} - ω_{s 1} .$ Red filled circles lines correspond to the position where the collision of twin-solitons occur. The peak power of CDW is shown by the blue filled lines. The peak power and temporal width of incident DW are fixed to 5.3 and 0.22, respectively.

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With the parameters used in the numerical simulations, the solitonic well to trap and control the dispersive wave can be designed and implemented in experiment. For example, with ultrafast soliton laser with peak power 500 W, temporal width 40 fs, carrier frequency offset of leading soliton 1.2 THz, time delay 1 ps, weak dispersive wave power 20 W, temporal width 45 fs, nonlinear coefficient γ = 0.05 W⁻¹m⁻¹, dispersion parameters $β_{2} = - 40 p s^{2} / km$ , $β_{3} = - 4.8 p s^{3} / km$ , fiber length 8 m, we can obtain the normalized parameters T₀≈0.2, δ₃≈–0.01, P_inc = 1, w = 0.22 and w_s2 = 1.5, and thus the numerical results in Figs. 2(a) and 2(b) can be realized.

4. Conclusions

In conclusion, we have studied the effect of mutual interactions between soliton and DW and the possibility of to create a solitonic well consisted of the two initial solitons moving away from each other to trap the incident DW. Unlike the case of the solitonic cage formed by the group velocity-matched twin-solitons, the incident DW breaks up into small pulses via multiple scattering process that are trapped almost entirely within the solitonic well. However, most of these trapped small pulses eventually escaped from the solitonic well, owing to the degradation of “soliton mirrors” at certain distance before twin-solitons collision. In this process, the corresponding spectrum of the trapped DW can be first narrowed, and then expanded with the propagation distance. In addition, the twin-solitons collision can result in the new DW generation. It has been shown that arranging the peak power or the duration of the incident DW are two suitable means to control the position of the twin-solitons collision and CDW generation within a solitonic well. Moreover, the formation dynamics of a solitonic well within which the DW are trapped for a wide range of different initial soliton frequencies have been investigated. The numerical results may enrich our understanding for the complicated interactions of solitons with DWs.

Acknowledgments

This work was partially supported by the National 973 Program of China (Grant No.2012CB315701), the National Natural Science Fund Foundation of China (Grant Nos. 61475102, 11574079).

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Trapping and controlling the dispersive wave within a solitonic well

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

3.1 Forming of a solitonic well to trap the incident dispersive wave

3.2 Manipulating CDW generation within a solitonic well

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (3)

Optics Express