Collision between a dark soliton and a linear wave in an optical fiber

We report an experimental observation of the collision between a linear wave propagating in the anomalous dispersion region of an optical fiber and a dark soliton located in the normal dispersion region. This interaction results in the emission of a new frequency component whose wavelength can be predicted using phase-matching arguments. The measured efficiency of this process shows a strong dependency with the soliton grayness and the linear wave wavelength, and is in a good agreement with theory and numerical simulations.


I. INTRODUCTION
When a bright soliton and a linear wave co-propagate in an optical fiber, four-wave mixing (FWM) interactions can occur and give rise to new frequency components [1][2][3][4]. This nonlinear interaction can be physically seen as a frequency shift of the linear "probe" wave which reflects partially from the bright soliton. To study this interaction, the linear wave can be launched into the fiber directly with the soliton, as in the case of [5,6]. The linear wave can also be emitted directly by the soliton itself and is interpreted as dispersive Cherenkov radiation [7]. In this case, the dispersive wave is emitted by the bright soliton under the action of higher-order dispersion when its spectrum leaks across the zero-dispersion wavelength (ZDW) [8,9]. The soliton then decelerates due to the Raman effect which allows the collision between the emitted dispersive wave and the soliton [4]. Due to its crucial importance in the supercontinuum generation process, this interaction has been the subject of intense research over the last decade [4]. These studies have shown that the FWM process results in the generation of new components at shorter wavelengths, thus explaining the spectral broadening of the supercontinuum towards the blue/UV region [4,10]. This FWM process can also be interpreted in terms of an event horizon in analogy with black holes [7,11,12].
The interaction of bright solitons with linear waves has also been studied in the case of high-order solitons [13], orthogonally polarized solitons [14], or in cavities made by solitons in which multiple collisions can be observed [5,[15][16][17]. The collision of linear waves with dark solitons has been also investigated theoretically and numerically in [18]. In this work, it has been demonstrated that a linear wave propagating on top of a black soliton background may undergo the reflection process described above leading to the frequency shift of the probe wave. However, this process has not been observed experimentally yet with any dark solitons.
In this work, we report the experimental observation of the collision between a dark soliton (black or gray) with a weak linear wave. In agreement with theoretical analysis and numerical simulations, we show that this interaction leads to the generation of a new frequency component whose conversion efficiency varies with the dark soliton grayness as well as the linear wave frequency.

A. Dark Soliton
The evolution of the field with a slowly varying envelope A(z, t) propagating in an optical fiber can be described by the following generalized nonlinear Schrödinger equation (NLSE): where D(i∂ t ) = n≥2 βn n! (i∂ t ) n is the dispersion operator expanded around ω 0 , the carrier frequency. t is the retarded time in the frame travelling at the group velocity V 0 = V 0 (ω 0 ) = β −1 1 of the soliton carrier frequency and γ is the nonlinear parameter. We checked that Raman and self-steepening effects do not play any significant role in the present study, thus they will be neglected.
If we only consider normal second-order dispersion (n = 2 and β 2 > 0), Eq. (1) admits a dark soliton solution A DS (z, t) of the form [19] where T 0 and P 0 are respectively the duration and peak power of the dark soliton. ϕ ∈ [−π/2, π/2] is the grayness parameter of the soliton, ϕ = 0 corresponds to the so-called black soliton and ϕ = 0 to gray solitons. Dark solitons are also associated with a phase jump localized at the center of the dip. A black soliton has an abrupt π phase jump while gray solitons have smaller and smoother phase jumps. Furthermore, it has to be noted that when ϕ < 0 (resp. ϕ > 0) the soliton travels faster (resp. slower) than the carrier frequency.
In presence of higher-order dispersion (n > 2), dark solitons are perturbed and are known to emit radiations [20][21][22] similarly to bright solitons. The dispersive wave emission from dark solitons was unambiguously observed very recently [23,24]. Here, we focus on the interaction occurring when a dark soliton and a linear (dispersive) wave collide in the presence of significant higher-order dispersion. In the following, we limit our investigations to third-order dispersion term β 3 (n = 3) and we neglect higher-order terms.

B. Phase-matching
The interactions between linear waves and bright solitons has been theoretically studied in [1][2][3][4]25]. In these works, it has been shown that the FWM interaction between linear waves and bright solitons leads to the generation of new spectral components. This process follows a phase matching relation, involving soliton and linear wave properties, that can be derived by a perturbative approach [9]. In a similar way, Oresnikov et al. [18] used this perturbative method to derive the phase matching condition in the specific case of a linear wave interacting with a black soliton. In the present study, we generalize this approach to take into account the grayness of the dark soliton interacting with a linear wave. A similar phase marching condition has been obtained for the radiation emitted by shock waves [26][27][28]. In the following, the linear wave is externally launched at the fiber input and therefore acts as a probe. It will therefore be termed probe wave, following the terminology used in the literature.
The dispersion relation of linear waves propagating over a background is (see Appendix A 1) where Ω = ω − ω 0 and ω is the frequency of the linear wave. Three main contributions are involved in this relation : the dark soliton (first term), in which β 1sol = √ γβ 2 P 0 sin(ϕ) is the soliton group velocity, the dispersion of linear waves (second and third terms) and a nonlinear contribution k N L = γP 0 due to the background of the dark soliton (last term). The resonance condition leading to the new frequency generation is verified when the wave vectors of the probe and generated waves are equal, which leads to the following equation where Ω p = ω p −ω 0 and ω p is the absolute frequency of the probe wave. The real solution of Eq. (4) gives the frequency of the generated wave. Note that in the case of a black soliton (ϕ = 0 and therefore β 1sol = 0), the expression is the same than for a bright soliton [2]. However, when ϕ = 0, the soliton carries an extra momentum whose effect is to change the frequency of the generated wave. Equation (4) also shows that the collision process depends on the fiber properties (β 2 , β 3 and γ) as well as the probe wave frequency (ω p ). In the following, the frequencies predicted by Eq.
(4) will be compared with the experimental and simulation results.

C. Conversion efficiency
In order to fully describe the collision process, we focus now on the amplitude of the generated wave. If we refer to the case of bright solitons [7,25], this amplitude can be calculated by considering the soliton as a potential. In this case, the probe and generated signal waves are the transmitted and reflected waves, respectively. However, in the case of quasi-continuous linear wave, only a small part of the probe interacts with the soliton. The duration of this interaction can be approximated as T in = L f iber /(∆V sp ), where ∆V sp is the group velocity mismatch between the dark soliton and probe wave. The energy of the probe interacting with the soliton can be expressed in the time domain as E in = P p × T in , where P p is the probe wave peak power.
Consequently, the energy of the generated wave is taken from the reflected part of the probe wave interacting with the soliton : E g = R × E in , where R is the reflection coefficient corresponding to the soliton potential (see appendix A 2). However, the experimental peak power of the probe wave is generally not exactly known. Therefore, we express the energy of the generated wave in the normalized form [25]: As expected, when the group velocity mismatch between the soliton and the probe wave becomes large, the energy of the generated wave tends to zero. However, as we can see, the generated wave energy mainly depends on the soliton grayness and the probe wave frequency. Therefore, the impact of those parameters will be studied experimentally in what follows.

III. SIMULATION
To illustrate the collision process between a dark soliton and a probe wave, we performed numerical simulations based on Eq. (1). Thus, we take the example of a gray soliton of grayness ϕ = π/8, duration T 0 = 600 fs and peak power P 0 = 0.94 W colliding with a weak Gaussian probe wave A p (t) of the following form: where P p = 0.02 × P 0 = 18.9 mW. The full-width at half maximum (FWHM) duration of the probe pulse (T F W HM = 1.665 T p ) is equal to 2 ps. The delay between the probe wave and the soliton is set to τ = 3 ps. Figures 1(a) and (b) represents respectively the temporal and spectral evolution of the collision process in a 3.15 km long dispersion shifted fiber (DSF) whose ZDW is located at 1549.5 nm. The soliton and probe wavelengths are respectively λ 0 = 1542 nm and λ p = 1554 nm, corresponding to β 2 = 0.68 ps 2 /km and β 2 = −0.405 ps 2 /km, respectively. The nonlinear parameter γ is equal to 2 W −1 .km −1 and the third order dispersion β 3 is equal to 0.115 ps 3 /km. Note that the temporal dynamics in Fig. 1 (a) represents the field intensity minus the soliton background power ||A(z, t)| 2 − P 0 |, for the sake of clarity.
As can be seen in Fig. 1(a), the probe wave and the soliton collide around 1 km of propagation. After the collision, a part of the probe is transmitted and the other one is reflected. In the spectral domain, one can see that the probe and the transmitted wave wavelengths are equal. Around 2 km of propagation a new spectral component associated to the reflected wave is emitted at 1561 nm. This generated wave can be attributed to the FWM interaction between the dark soliton and the probe wave occurring at their collision and resulting in the frequency shift of the reflected wave. The theoretical wavelength of the generated wave can be calculated from the phase matching condition given by Eq. (4) and is drawn in red dotted line in Fig. 1(b). It corresponds to 1561 nm, which is in very good agreement with the simulation and therefore confirms the validity of the previous analysis.

IV. EXPERIMENTAL SETUP
In order to experimentally study the collision process, we need to generate both a dark pulse (required to excite a dark soliton in the fiber) and a probe pulse, but we also need tp control the soliton grayness as well as the time delay between the two pulses. Several methods have been proposed to generate dark solitons [30][31][32][33][34][35][36][37][38][39]. However, none of them allows an accurate control of the soliton grayness. To do that, we implemented a technique based on phase and spectral shaping using commercial waveshapers (Finishar wavesapher 4000S), inspired of [24]. The corresponding experimental setup, shown in Fig. 2(a), will be used to generate simultaneously a dark pulse A DS (0, t) located on a super-Gaussian background pulse A BG (t) and a Gaussian pulse A p (t) which will act as the probe wave. The total field launched in the fiber can be written where T BG is the duration of the super-Gaussian background pulse and n its order.
In order to synthesize this field, we use an optical parametric oscillator (OPO) pumped by a Ti:Sa laser. It delivers Gaussian pulses of 220 fs FWHM duration and tunable around 1550 nm at 80 MHz. The pulses pass though a variable attenuator consisting of half-wave plates and a polarizer and reaches the first waveshaper WS1. Here, the initial pulse is shaped in amplitude (without shaping the phase jump characteristics of dark solitons and delay between the two pulses). This step is represented in Fig. 2(b) and (c) in which the large OPO spectrum [red curve] is shaped into a dark pulse of duration T 0 on a wide super-Gaussian background of duration T BG and a probe wave of power and duration P p , T p [blue curve]. After WS1, this field is amplified with an erbium doped fiber amplifier (EDFA) and reaches the second waveshaper WS2. At this stage, the odd-symmetry phase profile of the dark pulse is applied and the time delay between the dark pulse and the probe pulse is fixed thanks to a linear phase variation of the probe. Additionally, the use of this second waveshaper allows to filter out the amplified spontaneous emission added during the amplification and to correct spectral distortions due to amplification. For all simulations and experiments below, we use the DSF described in the previous section. The parameters of the dark pulse are set to: λ 0 = 1542 nm (β 2 = 0.6795 ps 2 /km), P 0 = 0.94 W, T 0 = 600 fs, T BG = 40 × T 0 = 24 ps, n = 16.
To summarize, with the experimental setup presented in Fig. 2(a), it is possible to generate a dark pulse located onto a super-Gaussian background pulse with a controlled grayness and wavelength, co-propagating with a linear probe pulse with a controlled delay, duration, wavelength and peak power.

V. IMPACT OF THE DARK SOLITON GRAYNESS
In this section, we study the impact of dark soliton grayness on the collision process. Figure 3(a) represents the simulated fiber output spectrum against the soliton grayness changed from ϕ = −π/3 to ϕ = π/3. The probe wavelength is fixed to a constant value of λ p = 1559 nm (β 2p = −0.852 ps 2 /km) with a FWHM duration of 5.9 ps. Its peak power is fixed to P p = 0.0035 × P 0 = 3.3 mW and the time delay to τ = 6.5 ps. The black solid line represents the fiber ZDW. As can be seen, the generation of the new spectral component in the anomalous dispersion region occurs for grayness values from −π/6 to π/4. Its wavelength increases with the grayness because the carrier wavelength of the dark solitons shifts with grayness. As in the previous section, the theoretical wavelength of the reflected wave can be found from the solution of Eq. (4) for each grayness value. The theoretical evolution of this wavelength is represented in black dashed line in Fig. 3. The agreement between theoretical results and numerical simulation is very good. To observe this dynamics experimentally, the method shown in Fig. 2 is used with the same parameters. The corresponding results are reported in Fig. 3(b) which represents the evolution of the fiber output spectrum as a function of the grayness. In agreement with the simulations results, the generated wave wavelength increases for grayness values from ϕ = −π/8 to ϕ = +π/3, following the same tendency as the theoretical prevision [black dashed line]. However, when the grayness is below ϕ = −π/6, one can notice the absence of emission. The small discrepancy between simulations and experiments can be attributed to experimental uncertainties. Figure 3 also shows that the generated wave observed experimentally is efficiently emitted for a specific range of grayness values. Hereafter, we will focus on the efficiency of the collision process. Grayness parameter φ -π -π -π -π π π π π π  show that the efficiency of the collision process presents a maximum for grayness value around −π/8. When the grayness tends to −π/3, the generated wave energy decreases rapidly to become almost zero before −π/4. The same happens for increasing grayness values, i.e. the generated wave energy decreases and almost vanishes for grayness values larger than π/3.

VI. IMPACT OF THE PROBE WAVELENGTH
To study the impact of probe frequency on the collision process, the soliton grayness is now fixed to a constant value of ϕ = 0 (black soliton case). The wavelength of the probe is varied from 1552 nm to 1564 nm with the waveshapers, corresponding to β 2 values varying between -1.297 and -0.225 ps 2 /km. Starting with numerical simulations, the peak power of the probe wave is set to P p = 0.005 × P 0 = 4.7 mW and the FWHM duration of the probe to 16 ps with a zero time delay between the soliton and the probe. Figure 5(a) represents the evolution of the simulated output spectrum as a function of the probe frequency. It can be seen that the generated wave wavelength decreases from 1561 nm to 1550 nm when the probe wavelength increases from 1551 nm to 1562 nm. This results is in very good agreement with the solution of the phase matching condition [black dashed line in Fig. 5], obtained from Eq. (4). One can also note the presence of a spectral component on the left of the spectrum which varies from 1532 nm to 1521.4 nm with the probe frequency. This peak follows the red dashed line that corresponds to the FWM process between the soliton background and the probe wave at 2ω 0 − ω p (the second peak located at long wavelength 2ω p − ω 0 being too weak to be seen experimentally). Figure 5(b) shows the experimental results obtained using the same parameters as in the simulation. One can see that the generated wave wavelength follows the same evolution as in the simulation. Therefore, this result is in very good agreement with both the experimental and theoretical results and confirms the previous analysis. Figure 6 shows the theoretical evolution of the energy of the generated wave (red curve), based on Eq. (5). We can see two local maxima, the first one being at 1560.5 nm and the second one at 1553.8 nm. The generated wave energy falls rapidly to zero for values less than 1551 nm and greater than 1563 nm. Furthermore, when the probe wavelength tends to 1557 nm, the group velocity of the probe and the soliton are almost equal, implying a vanishing theoretical energy of the generated wave, as it is evident from Eq. (5). To analyze the evolution of the generated wave energy, we followed the same integration method as before. Figure 6 shows the experimental and numerical energy of the generated wave, in black and green respectively. The agreement with theory is qualitatively good, i.e. two local maxima can also be found. One can note that the energy does not fall to zero around 1557 nm as theoretically predicted. This discrepancy with the theoretical curve can be attributed to the fact that the probe wave is not a continuous wave and therefore has a relatively broad linewidth. Indeed, the spectral width of the probe directly impacts the resolution of the energy measurement. In addition, the spectral width may increase during propagation due to cross-phase modulation with the soliton background. Thus, the zero of the theoretical curve, seen around 1557 nm, cannot be found in this case. However, when the probe wave wavelength increases (> 1562 nm), its group velocity mismatch with the soliton increases too, which strongly reduces the conversion efficiency, as in the theoretical evolution. The same arguments apply for decreasing wavelengths (< 1551 nm), showing that the group velocity mismatch between the dark soliton and the probe wave is the control parameter of the new frequency generation and of its efficiency. Despite the quantitative discrepancies, these results confirm that the collision between a black soliton and a linear wave follows the same phase matching rule as in the previous case where the soliton grayness was

VII. CONCLUSION
To conclude, in this work we made the first experimental observation of the collision between a dark soliton (gray and/or black) and a weak linear wave. By generating both the dark soliton and the linear wave with available commercial waveshapers, it has been possible to control all the parameters of both the dark soliton and the linear wave. It has been demonstrated that the collision process leads to the generation of a new frequency components whose conversion efficiency varies with the dark soliton grayness as well as the linear wave frequency, which are all in good agreement with theoretical and numerical predictions.