We demonstrate that the optical event horizon can provide an effective technique to actively control the propagation properties of a dark soliton with another weak probe wave. Careful power adjustment of the probe wave enables the black soliton converted into a gray one with varying grayness through the nonlinear interaction, corresponding to a nearly adiabatic variation of the soliton’s speed. The sign of the phase angle for the newly formed gray soliton at optical event horizon is significantly dependent on the frequency of the launched probe wave. Linear-stability analysis of dark solitons under the perturbation of a weak probe wave is performed to clarify the intrinsic mechanism of the nonlinear interaction. The probe wave manipulated collisional dynamics between both dark solitons are investigated as an analogue of the combined white-hole and black-hole horizons which provides some insights into exploring the transition between integrable and non-integrable systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Light-by-light controlling, as we know, is considered the cornerstone of optical signal processing application such as future optical transistor-like devices [1,2]. One of the effective approaches to realize this is by applying the well-known cross-phase-modulation (XPM) effect arising from Kerr-nonlinear media [3–5]. With suitable conditions one may observe strong interaction between an intense optical soliton pulse and a weak probe pulse that results in an optical analogue of the gravitational phenomenon known as event horizons . This analogue occurs in typical occasions where the weaker probe wave cannot penetrate through the optical soliton pulse during their collision process. The underlying physics was described in the time domain in terms of a soliton-induced refractive index barrier (Kerr effect) that changes the velocity of the probe wave [7–9]. The fundamental mechanism can also be interpreted in the frequency-domain  since the velocity variation of the probe wave is directly resulting from the generation of new spectral components [11–15]. It is worth mentioning that the frequency conversion process is reversible . The optical event horizons created by nonlinear collision between pulses in an optical fiber have attracted considerable attention in recent years [17–21]. Researchers found that the essential mechanism for new frequency generation at optical event horizon can be applied as a potentially effective method for highly coherent supercontinuum generation without soliton fissions and modulation instability [17, 18]. Moreover, it has been numerically [8, 12, 19] and experimentally  shown that a weak DW can be produced and trapped within a solitonic cavity formed by two solitons at optical event horizon. To date, the reported results involving optical event horizon focus mainly on the active control of fundamental [1–5] or higher-order bright solitons [12, 21] by a weak probe wave and the scattering of a weak probe wave from solitons [7, 15–18, 20]. However, to the best of our knowledge, the tunable propagation properties of dark solitons, the basic physics behind which is quite fundamental and different to that involving the bright solitons, have not been systematically explored with optical event horizon so far.
Optical dark solitons, existing only in the regime of normal group-velocity dispersion (GVD), exhibit a local notch over their uniform intensity background with a phase slip across it, and travel with constant velocity and unchangeable profile . Dark solitons were predicted  and experimentally  observed as early as several decades ago. Since then they had gained considerable attention, and have been studied in various branches of physics [22, 25, 26], especially in the field of nonlinear optics [27–32]. The dark soliton dynamics in optical fibers under the perturbation of higher-order dispersive and nonlinear effects have been investigated numerically [27–29] and experimentally [30, 31] where emission of DWs can be observed with their frequency and amplitude largely dependent on the soliton grayness. Recently, researchers reported that the short and long wavelength wings of the fiber supercontinuum spectrum can be shaped by the collision and interaction of the dark and bright solitons . This dark pulse collision phenomenon continues to be a topic of contemporary research.
Here, we show numerically the nonlinear interaction between dark solitons and weak probe waves at optical event horizon and analyze the mechanism behind through the linear-stability analysis of dark solitons under the probe wave perturbation, which can provide innovative perspectives into the light-by-light control. We demonstrate that the dynamics of optical dark solitons can be manipulated by a weak probe wave carefully adjusted at the input in terms of peak power and frequency detuning. Besides, it is shown that the weak probe wave can induce a dissipative effect upon the dark soliton in terms of amplitude decay and oscillated tail generation, and the “catastrophic collapse” of the dark soliton is associated with an exponential growth of a perturbation induced by cross-phase-modulation. We also explore the possibility of manipulating of the nonlinear collision between both dark solitons by the probe waves with varied peak powers.
2. Propagation model
To gain physical insights into the optical event horizon dynamics, we employ the symmetric Fourier split-step method to solve the non-integrable nonlinear Schrödinger equation (NLSE) governing the pulse propagation in dispersive nonlinear media, as following [8,10]:33]. Apart from the dominant terms of Kerr and normal GVD that are required to form dark solitons, we also include the third-order dispersion term in the full model in order to obtain the regime of optical event horizon. It has been found that the higher-order dispersion with n>3 and self-steepening (SS) term do not play any significant role [5, 33].
The input light consists of two pulses launched with different frequencies namely the dark pulse and the probe wave. The overall electric field of the input light is determined as following:22]Fig. 1. The blue curves illustrated in Fig. 1 is given under the limiting case of where Eq. (4) reduces toFig. 1(b). The time-dependent phase-shift represents an effective frequency shift, resulting in a slightly different propagation velocity from that at the carrier frequency used to define our retarded reference frame.
3. Results and discussions
3.1 Propagation dynamics of dark solitons manipulated with a probe wave
The nonlinear collision dynamics between a soliton pulse and a probe wave can create a group-velocity-led optical event horizon for specific probe frequencies, which can provide the potential technique to manipulate the dark soliton by the weak probe wave. The main prerequisite for the group-velocity-led optical event horizon is to establish an effective refractive index barrier and make the group velocities reasonably close for both the soliton and the probe wave. In Fig. 2, we show a typical cubic dispersion relation characteristic of a step-index single-mode fiber in the following numerical simulations, as well as the corresponding retarded-frame group-velocity. Here,, where is the fiber propagation constant, andis the group velocity at the soliton frequency ; vs is the group velocity of the dark soliton (highlighted as a horizontal line). The green (yellow)-shaded area in the frequency region represents the fiber-optical analogue of black (white)-hole horizon in which the probe appears to be visibly reflected on the trailing (leading) edge of soliton, with the resulting frequency up- (down-) converted. The frequency that is group velocity matched to the dark soliton represents the transition point between white- and black-hole horizons.
Figure 3 summarizes the obtained modeling results based on the NLSE describing the collision between a dark soliton and a weak probe wave with varying carrier frequencies. At the input, the dark soliton is aligned to temporally precede or trail the probe wave to realize the collision process according to their relative frequencies and the resulting different traveling velocities [See Fig. 2(b)]. Figure 3(a) shows that the probe wave is elastically scattered and partially transmitted through an effective refractive index barrier induced by the dark soliton. The dynamical process corresponds to the analogue of white-hole horizon. An important feature here is that the probe wave pushes the dark soliton from its original trajectory, which is attributed to the weak but non-vanishing repulsive force between the soliton and the probe wave benefiting from the typical “anti-particle” character of dark solitons at optical event horizon. The steering of trajectory for the dark soliton takes place in the direction opposite to that of the reflected wave, which is fundamentally contrary to what is the case involving with a bright soliton. It can be seen from Fig. 3(a) that the collision results in an acceleration of the dark soliton with a corresponding 0.20 ps temporal shift at the output relative to its initial soliton center (denoted by red dotted lines). This evolution process has recently been referred as diametrically driven self-acceleration . Based on the energy-speed relation for a dark soliton , the increase of the soliton speed is necessarily associated with the energy decay for the soliton. Figure 3(c) shows the intensity profile of the dark soliton before and after the collision, and indicates that the quasi-stationary black soliton () is transformed into an accelerated gray soliton with. Thus, it can be shown that the optical event horizon can directly result in the dissipative effect of the dark soliton.
Researchers found that swapping the frequencies of the probe and reflected waves simply changes the dynamics of the optical event horizon . If the original frequency of the probe wave is shifted to the frequency of the reflected wave in case of an analogue of the white-hole horizon, the dark soliton initially trails the probe wave temporally by 3.5 ps for the realization of the interaction between them. Here, the time-reversal symmetry for the black- and white-holes horizons can be invoked: the process that occurs at the black hole horizon is simply the time-reverse of a corresponding process at the white hole horizon. We can see from Fig. 3(b) that the reflection process can cause a deceleration of the dark soliton. The output soliton profile in Fig. 3(d) shows the time delay and dark soliton decay in contrast to its initial temporal profile (denoted by red curve). These observations suggest that the black soliton dissipates energy and is transformed into a decelerated gray soliton with. The soliton-probe collision dynamics discussed above demonstrate that we can use the frequency of the probe wave as an effectively controllable variable for all-optical manipulation of dark soliton.
The main idea for a dark soliton manipulation with a weak probe wave at optical event horizon can be characterized by the horizon-induced temporal shift of the dark soliton. We find that both the time shift and the dissipative effect of the black soliton can be investigated by simply adjusting the peak power of the injected probe wave. The obtained output dark pulse profiles in contrast to the injected dark soliton after interaction with the probe wave at two different peak powers of 2 W and 4 W are shown in Fig. 4. A significant increase of the peak power Pp results in a greatly reduced amplitude of the soliton and increasing time delay after the collision: for Pp = 2 W (which is the case in Fig. 3) the soliton’s amplitude decreases by ~5.9% and the temporal shift of soliton increases to 0.2 ps, while for Pp = 4 W the corresponding values are ~14% and 0.39 ps, respectively. Besides, the uniform background is full of a non-vanishing oscillatory tail with similar amplitude, showing that optical event horizon can cause an energy loss for dark solitons in the form of a continuously growing tail, similar to that of a dark soliton confined in boxlike traps .
In this section, we have demonstrated that the propagation of dark solitons dressed by the optical event horizon is typically dissipative. In the next section we will develop an analytic description to gain deeper insights.
3.2 Linear-stability analysis of dark solitons at optical event horizon
We now turn our attention to explore an interesting question concerning the existence of stationary dark soliton solution of Eq. (1) and analyze the mechanism behind the dark soliton manipulation with a weak probe wave at group-velocity-led optical event horizon. Given that dark solitons that are similar to the bright one also appear like an effective particle, the linear-stability analysis can be analytically studied based on the adiabatic approximation of the perturbation theory.
Traditionally, the one-dimensional NLSE can be used to analyze the relative strengths of physical effects governing the pulse evolution. We introduce a new term to describe the probe wave perturbation. We substitute the following dimensionless variables into Eq. (1) :5, 21]:7, 36]:7, 33]. The terms on the left-hand side of the Eq. (7) are associated with the fundamental dark soliton . The right-hand side is reserved for perturbation on the dark soliton evolution at optical event horizon. Obviously, the balancing between the two terms in the NLSE (the linear term and the nonlinear term) is altered by the presence of the latter, and then the integrability is broken. In this sense, the wave packets studied here are not true solitons. However, without loss of generality we still use this term to refer to the non-dispersing pulses formed by nonlinearity.
To analyze the linear stability of this dark soliton pulse, we find solutions of Eq. (7) in the following form:Eq. (7) and linearizing with respect to, we obtain the following linear-stability eigen-frequency problem:37]. Example of the whole stability spectrum for the stationary dark soliton under the perturbation of a probe wave with (corresponding toW) is shown in Fig. 5(a). It can be observed that the dark solitons are unstable in an apparently dynamic manner as a result of the eigen-frequencies with nonzero imaginary part in the spectrum.
In order to well understand these instabilities and obtain an analytical estimate for the relevant eigen-frequencies, we introduce the Newton’s particle result in . One can show in a straightforward manner that the function38]. Then, the stability of the dark soliton solution presented above is dependent on the sign of the second derivative of. Generally, an instability will be occurring with one imaginary eigenvalue pair for, and with exactly one complex eigenvalue quartet for . The instability is dominated by the eigen-frequency of Eq. (10), which bifurcates from the origin as long as the perturbation is present, in sharp contrast to the situation for the higher-order bright counterparts where an instability occurs only when the magnitude of perturbation exceeds a minimum threshold . We find that the relevant eigen-frequencies are determined by the following quadratic characteristic equation ,Eq. (7), and eigenvalues, which are related to eigen-frequencies through. Assuming that satisfies the equation, we can evaluate explicitly to obtain:Eq. (13) and Eq. (14) yields an analytical prediction for the magnitudes of relevant eigenvalue.
Figure 5(b) shows the pertinent analytical solution (depicted by red solid curves) in contrast to the corresponding numerical results (depicted by blue points). For smaller perturbation parameter, the numerical results are in excellent agreement with the analytical one. The distinct deviation for larger perturbation values is attributed to the assumption that the soliton remains unaffected during its interaction with the probe wave that is not the case even for a weak probe wave. The exponential growth rate of the instability is determined by the magnitude of eigen-frequencies imaginary part, and then the dark soliton at optical event horizon becomes mobilized and weakened through the generation of a continuously growing tail from the right of the primary soliton. The larger the imaginary part of eigen-frequencies is, the more the energy of dark soliton is lost. These results suggest that the fundamental features of the dark soliton propagation can be manipulated by careful power adjustment of the probe wave at optical event horizon. As the perturbationis increasing up to a certain value, the probe wave at optical event horizon may precipitate the “catastrophic collapse” of the dark soliton. As shown in , the interaction between a dark soliton and an intense probe wave causes the disintegration of the dark soliton.
3.3 Dark solitons collision induced by dual-probe waves at optical event horizon
Based on the time-reversal symmetry character for the evolution dynamics presented above, it should be interesting to explore what will happen when the white-hole and black-hole event horizons are combined together. In this situation we consider the input optical fields that are consisted of a pair of well-separated dark solitons and two probe waves located outside of them as following:
The pair of dark solitons isFig. 2 to ensure the time reversal quality for the corresponding evolution process. In this case, we will be able to observe the occurrence of soliton-soliton collision and the tunneling behaviour between two dark solitons. It can be seen from Fig. 6(a) that both dark solitons pass through one another in the course of collision, and then continue to propagate with an unchanged velocity. As can be seen from the intensity evolution in Fig. 6(b) alone, the interaction between both dark solitons seem to be repulsive. However, they are in fact transmitted, acting as a discontinuous jump in position relative to the original trajectory as a result of the collision [See Fig. 6(c)]. The apparent node between both dark solitons, resulting from the interference of the two wave packets, leads to a negligible intensity dip at the center of the collision [39, 40]. The jump is a consequence of the nonlinearity interaction of the system. Since the temporal shift of solitons is significantly dependent on the grayness of the colliding solitons , the collision-induced time shift of solitons can also be manipulated by adjusting the strength of the incident probe waves at optical event horizon.
To fully understand the dynamics of dark soliton-soliton collision under the action of the probe wave, the position where both dark solitons eventually collide and the time shifting for both solitons as a function of the initial peak power of the incident probe wave are calculated and shown in Fig. 7. The initial temporal width of the incident probe waves is fixed to 800 fs. It is clear that the distance at which the collision of both dark solitons occurs decreases significantly with the increasing peak power of incident probe waves. This is attributed to the significant variation of the traveling speed and grayness of the dark solitons induced by the increasing peak power of the probe waves at optical event horizon. Meanwhile, the collision-induced time shifting for both solitons is decreased when the peak power of probe wave is increased, as can be seen from the blue filled circle lines in Fig. 7. We can conclude that adjusting the peak power of the incident probe waves at optical event horizon is one of most effective and suitable means to control the collisional dynamics of dark solitons.
In conclusion, we have shown that the analogy between event horizons and the nonlinear reflection of a weak probe wave onto a dark soliton opens innovative perspectives in the control of light. The optical event horizon is demonstrated to have a typical dissipative effect on dark solitons where the black soliton is decayed into a ‘new’ gray soliton (with lower energy) with non-vanishing oscillating tail, in sharp contrast to the situation where the bright soliton interacts with the dispersive wave. The soliton grayness can be effectively controlled by adjusting the peak power of the probe wave with the optical event horizon, corresponding to the manipulation of the dark soliton’s speed. However, whether the gray soliton is accelerated or decelerated is significantly dependent on the region where the selected frequency of probe wave is located. To explore the nonlinear mechanism behind the dark soliton manipulation with the probe wave, we develop a linear-stability eigenvalue problem by the perturbation theory which provides some insights for the dark soliton statics and dynamics, especially the catastrophic collapse. In the soliton-soliton collision dynamics induced by the probe waves, both dark solitons pass through one another and emerge from the collision unchanged in velocity, with a mutual temporal shift. The position at which both dark solitons eventually collide and the collision-induced time shift of both solitons can be effectively controlled by adjusting the intensity of the probe waves. These results can be proposed as a candidate to realize the photonic transistor-like action and provide further insights into the collisional dynamics of dark solitons in a more controllable manner.
European Horizon 2020-MSCA-IF ( 701493); National Natural Science Foundation of China (NSFC) (61505124, 61505122, and 61606166); Science and Technology Planning Project of Guangdong Province of China (2016KCXTD006); Shenzhen Government Plan of Science and Technology (JCYJ20170302153731920); Natural Science Foundation of SZU (2017022).
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