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Abstract
A single-shot experimental method is proposed to study non-repetitive evolutions of high order solitons. In our experiments, high order solitons are prepared in the building up process of a soliton fiber laser, and the order of high order soliton is controlled via changing the parameters of the laser. The evolution of high order soliton is recorded by the single-shot spectral measurements-time stretch dispersive Fourier transform. A 4th order soliton evolution under perturbations of gain saturation and saturable loss is studied, showing how a leading pulse wins the competition against the tailing one. Our work provides a controllable technique to study the high order solitons evolutions, which can be applied in the research of ultrafast laser amplifications and supercontinuum generations.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical fibers, with low loss and homogeneously small cross-section, provide a both flexible and controllable experimental platform for resolving nonlinear dynamics, including rogue waves and high order solitons [1,2]. High order soliton (HOS), a periodic solution to the nonlinear Schr$\ddot {\textrm o}$dinger equation, is important for soliton mode locking, soliton compression, supercontinuum generation and possible high-speed communication [3–7]. The study of nonlinear dynamics in fibers, including HOSs, is normally achieved either by simulations or by monitoring the output from a length of passive fiber experimentally while scanning the input pulse energy or other parameters [4,8]. Only the output rather than evolution can be analyzed with spectral measurements and autocorrelation in experiments.
The evolution of HOS along the propagation distance is important for understanding the mechanism and for finding new kinds of nonlinear effects. The evolution is difficult to obtain with the abovementioned methods, but can be resolved with mode-locked lasers [9–11]. In previous experiments, HOS were formed and observed in a colliding-pulse mode-locked dye laser, and the laser was tuned cautiously to generate a temporally periodic pattern with a period of $\sim$1000 roundtrips corresponding to the HOS period. The evolutions of the 3rd order soliton and the asymmetric 2nd order soliton were studied with synchronized autocorrelation and spectral measurements [10,11]. However, there is no technique for measuring non-repetitive HOS evolutions under other perturbations, which are unavoidable in ultrafast and nonlinear applications, for example, soliton compression and supercontinuum generation [4–6].
Here, we propose and implement a single-shot experimental method to study non-repetitive evolutions of high order solitons by preparing and controlling them in the building up process of soliton mode-locked fiber lasers. The HOS evolution is recorded by the single-shot spectral measurements-time stretch dispersive Fourier transform [12]. High order solitons are prepared in a soliton laser, and the soliton orders are controlled via the saturable power of the applied saturable absorber in the laser. The spectral evolution of the 4th order soliton is studied including the impacts from gain saturation and saturable loss.
2. Laser states in the building up process of soliton mode-locked lasers
The nonlinear effects, including soliton effects, contribute to diverse laser states during the building up process of mode-locked lasers, which has been intensively studied since its invention. People divide the starting dynamics into three phases as Q-switching, relaxation oscillation, and final mode locking, analyzing both the direct laser output and the integrated second-harmonic generation signal with the output [13]. Recently, a single-shot spectral measurement method, time stretch dispersive Fourier transform, is widely applied to study the starting mechanisms of different lasers, such as soliton laser, dissipative soliton laser, and Mamyshev oscillator [14–17]. With this method, beating dynamics between the relaxation oscillation and final mode locking are further observed and analyzed in a Ti:sapphire laser, which suggests that there is a bound state in the building up process [14].
For a soliton mode-locked laser, gain, saturable loss, group velocity dispersion (GVD) and self-phase modulation (SPM) are important. The soliton-like pulse, generated from the laser under stable operation, can be much shorter than the recovery time of the saturable absorber (SA) since the time delayed saturable loss only initializes the mode locking rather than shapes the pulse [18]. The soliton effect, referring to the balance between GVD and SPM, shapes the pulse and maintains it as a nearly fundamental soliton in general applications. Besides, the pulse energies in the building up process are higher than those of fundamental soliton states in stable operation, which makes it possible for people to investigate high order solitons. Therefore, one can study high order solitons in the building up process of a soliton mode-locked laser.
We introduce a road map model to describe the laser states during the building up process, shown in Fig. 1. In this model, the peak power $P_0$ and pulse duration $T_0$ are chosen to represent the laser state as $(T_0,P_0)$. Two regions represent conditions that different effects is stronger enough for several roundtrips (RTs) in Fig. 1. The soliton-effective region can be defined as
Fig. 1. The road map for the building up process of soliton lasers. The zero gain line includes the gain saturation and saturable loss effects and represents the region where net gain is equal to zero, shown with purple dash-dot line. The soliton effective region, defined by Eq. (1), is drawn with green area inside the black thick-lined triangle. The SA effective region, defined by Eq. (2), is shown with the gray rectangle. The evolution direction of the laser states is along the zero gain line in the building up process of soliton mode-locked lasers, presented with an arrow. The interaction $(T_1,P_1)$ which represents the bound state (the high order soliton) is marked with the filled red dot.
Under the perturbation and pulse shortening induced by the SA, the laser starts from noise with low peak power and large pulse duration and evolves to the pulse with high peak power and short duration. The laser states evolving from noise to mode locking will be along to the zero gain line due to the gain saturation, and the direction of the pathway is presented with an arrow in Fig. 1. A purple dash-dot line in Fig. 1 represents the line where net gain is equal to zero,
where $g, E_{sat}, l_0, A_0, L$ are the gain, the saturation energy of the gain medium, the linear loss, the modulation depth of the SA and the gain fiber length, respectively. $\epsilon =\int {|U|^2\textrm {d}t}/(P_0T_0)$ is a parameter related to the pulse shape $|U|^2$, and $\epsilon = 2$ for hyperbolic secant pulse. The precise path for the building up of the mode-locked pulse depends on specific initial conditions and outside perturbations.3. Controlling the soliton order
A bound state in the building up process of a soliton laser can be further utilized to study the HOS dynamics, and the soliton order can be controlled by setting the self amplitude modulation parameters. For a soliton fiber laser, a bound state, induced by the intense soliton effects, has been found in the building up process of soliton fiber lasers [3,15]. This bound state, high order soliton state, shows up when the laser state first enters the soliton effective region shown in Fig. 1. This start point can be considered as the intersection $(T_1,P_1)$ between the boundary of soliton effective region, namely, the boundary of the first inequality in Eq. (1), and the zero gain line, determined by Eq. (3). Therefore, the duration of this bound state is
and the peak power $P_1$ is the solution to Eq. (3) while $T_0=T_1$.This intersection represents when the soliton effects starts to dominate during $\sim$100 RTs in the building up process of the mode locking. As the pulse duration decreases in the building up process, the soliton effects gradually dominate while the dispersion length decreases. When it is close to the effective length of the cavity, the soliton effects trap the laser pulse into a bound state as a start point for soliton-like evolution. Under different laser parameters, the soliton order of this state will be different.
A high order soliton with normalized electrical field envelope $u(x=0,\tau )$, propagating through lossless fiber with only GVD and SPM, can be described with nonlinear Schr$\ddot {\textrm o}$dinger equation [19] as
where $x,\tau$ are the propagation distance and the retarded time, respectively. The initial condition is chosen as where $N$ is the order of the high order soliton.Of this intersection $(T_1,P_1)$, the soliton order, $N=\gamma P_1^{}T_1^2/|\beta _2|$, varies when the laser parameters change. For a general set of parameters for soliton mode-locked laser, the soliton order decreases from 3.4 to close to 1.6 as the saturation power increases from 5 W to 60 W, shown in Fig. 2. Therefore, the theoretical model suggests a possibility to control the soliton order via setting parameters of a laser, and we prove it experimentally (see Section 5).
Fig. 2. The soliton order can be controlled by changing the SA saturation power of the laser, and the model prediction is shown as blue curve. The corresponding soliton order range for three experiments under their parameters are presented with black vertical lines.
4. Experimental setup
In a soliton fiber laser, the GVD and SPM effects distribute along the cavity length, making the soliton state an attractor of the system. Therefore, a soliton fiber laser, suitable for investigating the soliton dynamics, is utilized here to form and control a high order soliton in experiments, shown in Fig. 3. We use polarization maintaining fibers and fast-axis polarization mode blocked elements to eliminate polarization related effects which may leads to chaotic dynamics [20]. The soliton laser is mode-locked by a slow saturable absorber. This semiconductor saturable absorber mirror (SESAM, BATOP) is used to initialize the mode locking and to stabilize the soliton against the noise. A 0.4 m length of Erbium doped fiber (EDF) is used for laser gain in the cavity, with a 0.4 m length of EDF with normal GVD for slightly dispersion compensation. And the pulse energies can be increased with the piece of EDF with normal dispersion while the soliton dynamics holds with short laser cavity length. Notably, this piece of EDF with normal GVD is not necessary for the HOS studies, and an all-anomalous-GVD cavity is also suitable. The other fibers are all polarization maintaining single mode fiber with total length of 1.7 m. A power-splitting coupler are used for laser outputting. The net group delay dispersion (GDD) as $\rm -83800~fs^2$ and the Kelly side bands shown in the spectrum in Fig. 3 implies the soliton pulse shaping mechanism. The averaged GVD is as $\rm -17~fs^2/mm$. The detailed parameters are listed in Table 1.
Fig. 3. The soliton mode-locked laser and the time stretch dispersive Fourier transform measurement setup. SESAM, semiconductor saturable absorber mirror; EDF, Erbium-doped fiber; WDM, wavelength division multiplexing coupler; LD, pump laser diode; OC, 20:80 optical coupler; M, mirror; SMF, single-mode fiber; PD1, the 1st photodiode with resolution of ${\rm 12}~{\rm ps}$; PD2, the 2nd photodiode with resolution of ${\rm 140}~{\rm ps}$.
Table 1. Parameters of the laser
The laser dynamics is measured with single shot spectral measurement–time-stretch dispersive Fourier transform. A spool of single model fiber (FiberHome G652D) is used to disperse the input laser output pulses, with the GDD of $\rm -402~ps^2$. After the dispersion, the temporal waveform has the same profile as the input spectrum. Then the optical signal is recorded continuously by a fast photodiode (Newport 1024, 12 ps rise time) and digitized by an oscilloscope (Tektrnoix, 4 GHz bandwidth). The temporal resolution is 175 ps while the corresponding spectrum resolution is 0.76 nm. At the same time, the system can record 1.25 ms (with the highest resolution). We use the power overshoot and pulse duration suddenly shortening as the trigger events for the time-stretch signal recording. In detail, we first optimize the stable operation and then turn off the pump laserdiode. And we set the trigger events for the time-stretch signal recording on the oscilloscope since they mark the final stable operation. We then turn on the pump laserdiode. The signal is recorded, and we can find the HOS evolution before stable operation.
The soliton order is controlled by changing the saturation power of the SA. We use an imaging system to change the beam size on the SA, with which the saturation power of the SA is changed consequently. The fiber end face and the SA reflector are placed close to the focal point of each lens nearby, as shown in Fig. 3. By changing the ratio of the two focal lengths, the beam size on it, $S_0$ is varied. The radius of the beam can be varied from 2.9 $\mu$m to 12.8 $\mu$m with focal lengths ranging from 4.5 mm to 11 mm. Therefore, we can control the effective saturable power of the saturable absorber, $P_{sat} = S_0\Phi _{sat}/\tau$, where $\Phi _{sat},\tau$ are the saturation fluence and the relaxation time of the SA, respectively. An accuracy of 12.9 $\mu$m of the SESAM placement will lead to 5% variation on the saturation power.
5. Experimental results of different order soliton
As predicted by the model in Section 3, the order of the high order soliton can be controlled by the SA saturation power, and we also verify it experimentally. When the saturable power is set to be 9 W, a 4th order soliton is formed and observed during the building up process of mode locking experimentally, shown in Fig. 4(a). Each vertical line represents a single-shot spectrum for one roundtrip (RT). The spectrum evolves from $\sim$0.8 nm bandwidth (FWHM), then splits symmetrically for nearly 20 RTs, and becomes nonasymmetrical afterwards. Notably, the lasers fall into stable mode-locking after the HOS dynamics, which can be used as an indicator when building and optimizing the laser.
Fig. 4. (a)-(f) Experimental results with saturable power of 9 W, 16 W and 57 W: (a)-(c) the spectral evolution acquired by single shot measurements-time stretch Fourier transform; (d)-(f) the energy growth. (g)-(i) Simulation results of the high order soliton spectral evolution: (g) the 4th (h) the 3rd and (i) the 2nd order soliton. The intensities are normalized to the maximum among itself in (a)-(c) and (g)-(i).
To check the soliton order, we compare the pulse energy with those of the pure high order solitons. The experimental energy evolution is obtained by summarizing the undispersed signal during each roundtrip and calibrating with the stable mode-locked pulse energy, presented in Fig. 4(d). The initial pulse energy is 29 pJ, and it increases to 57 pJ during 25 RTs. The initial 0.8 nm spectral bandwidth corresponds to a transform limited hyperbolic secant pulse duration as $T_1=$1.8 ps (for $sech(t/T_1)$). With the averaged GVD of the laser, $\beta _2=-17~\textrm {fs}^2/\textrm{mm}$, the theoretical pulse energy for the $N$th order soliton can be calculated as
The pulse energies for the 2nd, the 3rd and the 4th order soliton are $\textrm {18.8pJ,42.5~pJ,75.5~pJ}$, respectively, shown as horizontal dashed lines in Fig. 4(d) for comparison. The laser evolves with energy larger than that of the 2nd order soliton and increases to be larger than that of the 3rd order soliton. The large pulse energies in the building up process agree with that of the high order soliton.
The spectral evolution provides another evidence for the order of this high order soliton. The evolution is compared with the propagation of the 4th order soliton through lossless fiber with only GVD and SPM included, described with Eq. (5). The propagation of a 4th order soliton in one soliton period is simulated by fourth-order Runge–Kutta split-step Fourier method [21,22], shown in Fig. 4(g). Blue, red and yellow lines highlight three characteristic spectra at $Z = 0.125z_0,0.25z_0,0.45z_0$, respectively, and we find same structured spectra and mark them with same style lines in Fig. 4(a). Those lined characteristic spectral structures in agreement also identify the 4th order soliton in the building up process of mode locking. The differences between the other spectra are attributed to effects like gain and loss, and we will discuss this in Section 6. Notably, the HOS evolution under effects like gain and loss is only possible to be studied with this single-shot method, which can not be obtained via previous averaging method.
Not only can we form the 4th order soliton, but we can also generate HOS of other orders by setting the saturation power of the SA properly, predicted by our model shown in Fig. 2. We vary the saturation power of the SA to be 16 W and 57 W in experiment. The build-up spectral evolutions under those conditions are shown in Fig. 4(b)(c), respectively. The energies start from that close to the 2nd order soliton for both evolutions, shown in Fig. 4(e)(f). Under the saturation power of 16 W, the energy increases to be close to the energy of the 3rd order soliton. The propagation of the 3rd and the 2nd order soliton through lossless fiber, with only GVD and SPM included, are shown in Fig. 4(h)(i). Characteristic spectra are also found in the evolution between the experiments and the pure HOS, highlighted by blue, red or yellow lines. The spectral evolution and energy identify that the soliton order increase from 2 to 3 under the saturation power of 16 W. And for the saturation power of 57 W, the spectra are strong modulated because of soliton fission under high order effects.
To compare the soliton order under different saturation powers, we deduce the effective soliton order as
where $E$ is the energy of the experimental resolved pulse, and $E_1$ is obtained from Eq. (7). The soliton order increases from 2.5 to 3.5 with the saturation power of 9 W, from 1.9 to 2.7 with 16 W and from 1.9 to 2.4 with 57W, drawn as vertical lines with end dot in Fig. 2. The soliton orders qualitatively agree with our prediction on the soliton orders to the saturation powers of the SA. Therefore, we demonstrate that one can form the high order soliton and control the soliton order both theoretically and experimentally.6. 4th order soliton under gain saturation and asymmetric saturable loss
After preparing a high order soliton in the building up process of the soliton lasers, one can investigate its dynamics and characteristics under other high order effects, like gain and loss. With a suitable model developed, the dynamics can be resolved and patterns or rules can be found. Here, we study the dynamics of the 4th order soliton under gain saturation and asymmetric saturable loss.
In our experiments, the pulse increases in energy and evolves with asymmetric spectrum because of the gain and the slow absorber like saturable loss. We include those effects in the abovementioned simulation of Eq. (5), as
Fig. 5. Simulation results of the 4th order soliton under the gain and the saturable loss, described by Eq. (9). The parameters for this equation are $G_0=10,E_{sat}=180, L_0=1, A_0=12, E_{SA}=4$, respectively. They are all with normalized unit.
Counterintuitively, a leading pulse wins the competition finally. The SPM and the saturable loss result this phenomenon. Of the pulse highlighted by yellow dashed line in Fig. 5(b), the leading edge is delayed relative to the symmetrical position at the tail, attributed to the effects of the SA. However, of the pulse marked by red line, the tail is stronger than the leading parts. And in the final stages, a weaker pulse is left behind and red-shifted in the spectrum, which is confirmed by the interference shown on negative frequency component.
7. Conclusions
We demonstrate a single-shot experimental method to study evolutions of high order solitons under perturbations of gain, saturable loss and other effects in a mode-locked fiber laser. The order of soliton can be controlled by setting the saturable power of the absorber. A theoretical model is built, which can explain the mechanism behind successfully. In experiments, we successfully generate and measure the 2nd, the 3rd and the 4th order soliton by controlling saturable powers, which is achieved by changing the beam size on the SA. With a 4th order soliton generated by this method, we study its spectral evolution under the perturbation of gain saturation and asymmetric saturable loss. The experimental results of this 4th order soliton are also compared with simulations, which are in good agreement. We find that the saturable loss leads to asymmetric spectral evolution and soliton splitting. Under the interaction between the soliton effects and the saturable loss, a leading pulse wins the competition against the tailing one.
Our method provides a new way to study the high order soliton in ultrafast laser amplifications and supercontinuum generation. The HOS evolution is measured in a single-shot way, and no repetitive measurements are needed. One can study the HOS evolution under other effects with this method. Similar to our method, one can obtain the spectral evolution of ultrashort pulse amplification but in oscillator efficiently, which, in fiber amplifier, can only be obtained by cutting the fiber and measuring the spectrum repeatedly. And by inserting a pulse shaper to control the dispersion precisely [23], one can evaluate the spectral evolution of the supercontinuum generation. The single-shot full-field measurements [24] which can get the intensity and the phase, comprising the time-stretch method and time lens measurement simultaneously, may further broaden the applications like soliton optical communications.
Funding
National Key Research and Development Program of China (2016YFB1102404); The Scientific Research Foundation of National Institute of Metrology, China.
Disclosures
The authors declare no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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