1. Introduction

Solitons are self-reinforcing wave packets that maintain their shape and velocity during propagation due to the balance between nonlinear and dispersive effects. Since first observed in 1834, they have initiated remarkable mathematical theories and provided new insight into many physical problems [1]. Though solitons were originally defined as localized solutions of conservative nonlinear systems, this concept has been recently extended to dissipative systems. The extension is achieved by incorporating nonlinear dynamics and nonequilibrium physics into the standard soliton theory. In order to distinguish them from classical solitons, a novel term ’dissipative soliton’ was coined to describe localized structures stabilized through an energy or matter exchange with the environment in presence of nonlinearity and dispersion. With dissipative effects taken into account, these new solitons do not necessarily have a unified form anymore and could show marvelous variety and complexity, which has become a fascinating field of research [2,3].

Dissipative solitons appear in a wide range of optical, mechanical, thermal, chemical and biological systems [4–8]. Here we focus on passively mode-locked lasers, which are not only the workhorse of ultrafast optics but also proved to be excellent model systems for studying physics of dissipative solitons [9,10]. A dissipative soliton exists in a mode-locked laser as an ultrashort laser pulse. The envelope and phase of the pulse can be determined by a complex Ginzburg-Landau equation [11], which also describes various physical phenomena other than mode-locking of lasers, such as second-order phase transitions, superconductivity, superfluidity, and Bose-Einstein condensation [12]. From a perspective of nonlinear dynamics, this dissipative soliton is a fixed point of the complex Ginzburg-Landau equation governed system, and the mode-locking transition can be regarded as a heteroclinic orbit towards the fixed point [13]. The fixed point can also transform into a limit circle or a strange attractor in specific regions of the parameter space, resulting in even more complicated orbits [14,15].

The huge variety of orbits in phase space implies the presence of highly diverse and complicated intracavity pulse dynamics during the transition to mode-locking. Early studies of these dynamics, starting in the 1990s, focused on addressing self-starting conditions of mode-locked lasers [16,17]. Pulse-forming dynamics during mode-locking were experimentally studied by capturing the evolution of the laser power with a fast photodiode. Besides, transient optical spectra and autocorrelation traces were acquired by scanning trigger delays in multiple measurements [18,19]. In the meantime, the concept of dissipative solitons was introduced into optical systems [20,21]. More recently, the advent of novel experimental techniques, such as time-stretch dispersive Fourier transform (TS-DFT) and time-lens, enables real-time tracking of evolving optical waveforms at high frame rates [22–26]. This advance reveals colorful details of pulse-forming dynamics during mode-locking and inspires enthusiasm for studying these phenomena in the framework of dissipative solitons [27–32].

The significance of these studies is twofold. On the one end, the observation and interpretation of the intracavity pulse dynamics during the buildup of mode-locking are essential for developing better-performance ultrafast lasers. For example, a deeper understanding of these dynamics in the framework of dissipative solitons has stimulated innovative laser cavity designs that allow for much higher pulse energy and better stability [33,34]. On the other end, the relative simplicity of implementation makes mode-locked lasers ideal playgrounds for studying nonlinear dynamics that can also arise in other physical processes with theoretical similarity. Many unusual dynamics of dissipative solitons have been revealed in these powerful playgrounds, including soliton pulsations [14,35], soliton explosions [15,36], soliton rain [37,38], soliton molecules [39,40], and noise-like pulse multiscale instabilities [41,42].

Despite the great achievements of studies on dissipative solitons in mode-locked lasers, most of them are restricted to qualitative regimes. More quantitative studies are often hindered by the difficulty to quantify dissipative effects in a mode-locked laser, which can be regarded as the net gain of the laser. The net gain depends on instantaneous laser powers and can change on the scale of femtoseconds according to the pulse shape. This power-dependent, thus also time-dependent, gain amplifies the center of a pulse but suppresses the tails of it, resulting in generating ultrashort mode-locked pulses [43]. Though some analytical and numerical methods have been proposed to estimate the net gain according to optical properties of components constituting the laser cavity [44–46], direct measurement is not yet achievable for femtosecond mode-locked lasers. Indeed, the difficulty in quantifying dissipative effects is a common problem that can be also encountered in many other dissipative systems.

A straightforward way to measure the power-dependent net gain of a mode-locked laser is to measure the shot-to-shot change of the instantaneous power across the ultrashort intracavity pulse. However, this measurement is technically challenging for picosecond pulses and still unachievable for femtosecond pulses. In this paper, we demonstrate a method for measuring the power-dependent net gain of a femtosecond mode-locked laser without the necessity of a femtosecond-scale temporal resolution. The key is to find a period of evolution in which the pulse keeps a fixed form. Then, the dependence of the net gain on the instantaneous laser power can be deduced from the evolution of the pulse energy. In practice, we acquire the energy evolution of the pulse that seeds mode-locking via TS-DFT and fit it with a reduced mode-locking model derived from the soliton perturbation theory [47–49]. The good fitting accuracy confirms the existence of a perturbed soliton in the mode-locking transition of a realistic mode-locked laser. This fact as well as the ability to obtain values of the power-dependent net gain leads to a deeper insight into mode-locking. It should be noted that the method proposed here can be generalized to quantify dissipative effects in other complex Ginzburg-Landau equation governed or even more complicated systems.

2. Principles of power-dependent net gain measurement

The complex cubic-quintic Ginzburg-Landau Eq. (CCQGLE), which describes passive mode-locking as well as many other phenomena, can be expressed as [50]

For a soliton mode-locked laser, the evolving pulse can be viewed as a perturbed soliton when it evolves into states that are close to the mode-locked state. According to the soliton perturbation theory, a simple ordinary differential equation (ODE) describing the pulse energy evolution can be derived from Eq. (1) (see Supplement 1, Section 1):

Once obtaining the pulse energy change per roundtrip as a function of the pulse energy, we will be able to find values of $\alpha$, $\gamma$ and $\gamma _2$ by fitting it with Eq. (3). However, the evolution of a pulse into a mode-locked pulse is always accompanied by the evolution of background pulses, as will be shown later. Besides, it is also unknown when the evolving pulse can be viewed as a perturbed soliton. In order to tackle these problems, it is necessary to track the evolution of the intracavity optical waveforms during mode-locking in real time, which can be realized by TS-DFT.

3. Methods

The experimental setup for the TS-DFT measurement is shown in Fig. 1(a). A Ti:sapphire oscillator (KMLABS Collegiate) using an output coupler with a transmission efficiency of 10$\%$ is employed. Before mode-locking, the output laser power is 370 mW. After mode-locking, it outputs ultrashort laser pulses with an average power of 450 mW and a 3-dB bandwidth of 32 nm at a repetition rate of 92 MHz. The laser beam produced by the oscillator is split into two. One impinges on a 2-GHz-bandwidth photodiode (PD1, Thorlabs DET025A) directly, and the other is stretched by a 1.26-km-long fiber with a group velocity dispersion of $-125$ ps/(nm$\cdot$km) and then detected by a 26-GHz-bandwidth photodiode (PD2, New Focus $\#$1024). Finally, electric signals from PD1 and PD2 are simultaneously recorded by a 4-GHz-bandwidth oscilloscope (Tektronix DPO70404C) running at 25 Gsamples/s.

 

Fig. 1. Real-time characterization of the mode-locking transition of a Ti:sapphire oscillator. (a) Experimental setup for the TS-DFT measurement. OI, optical isolator; NDF, neutral density filter; PD, photodiode. (b) TS-DFT result. The data depict the timing of the picosecond fluctuations before mode-locking and the spectral evolution of the dominant pulse. The rapidly oscillating peak is marked by an arrow.

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In order to suppress the nonlinear effects in the fiber, the pulses are pre-chirped by a 50-mm-long SF57 glass block before being coupled to the fiber. Besides, the laser power is properly attenuated to assure both photodiodes work in linear regions. Before the TS-DFT measurement, the spectrum of the mode-locked laser transmitting through the fiber was checked, and no distortions caused by high-order dispersion or nonlinear effects were found (see Supplement 1, Section 2).

Mode-locking of the oscillator is initiated by shaking one of the intracavity prisms, and the electric signal induced by the unstretched laser pulses triggers the oscilloscope. Since the temporal shape of an extensively stretched laser pulse highly resembles its own spectral shape, shot-to-shot spectra in the laser cavity can be captured by recording the electric signal of the extensively stretched laser pulses with an oscilloscope [25,51].

The waveform data saved by the oscilloscope in the H5 format are then imported to MATLAB for processing. The data processing starts with segmenting the one-dimensional waveform amplitude array into many subarrays, each of which was recorded within a duration exactly equal to the period of the mode-locked pulse train. Whereafter, all the subarrays are transposed and then rejointed in the original order to form a two-dimensional array, as visualized in Fig. 1(b). The wavelength axis of the plot is acquired through a calibration procedure, in which the mode-locked spectrum measured by TS-DFT is fitted to that measured by a conventional spectrometer according to a linear frequency-to-time mapping relation (see Supplement 1, Section 3).

4. Results

4.1 Analyses of the perturbed soliton observed by TS-DFT

If looking closely at Fig. 1(b), one can find the mode-locking buildup of the Ti:sapphire oscillator involves three types of wave packets: a dominant pulse that broadens its spectrum smoothly and evolves into a mode-locked pulse, an auxiliary picosecond pulse with fixed wavelength and bandwidth which oscillates rapidly due to its interference with the dominant femtosecond pulse [24], and a noise-like background consisting of numerous picosecond pulses which gradually die out (see Fig. S3 in Supplement 1 and Visualization 1 for more details). During the evolution, the background pulses are barely influenced by nonlinearities and dispersion due to low intensity and narrow bandwidth, so they all hold an invariable shape and evolve in the same pattern.

In order to realize decoupling between evolutionary dynamics of the dominant pulse and the background pulses, the sum of a smoothly broadening peak, a rapidly oscillating peak and a gradually damping background is employed to fit the TS-DFT result, as illustrated in Fig. 2(a). More specifically, the broadening peak is described by $A_1$sech$^2[(\omega -C)/B]$, the oscillating peak is described by $A_2e^{-(\omega /B_0)^2}$, the background is described by $A_3f_{bg}(\omega )$, and $f(\omega )$ is the sum of these three parts. Here, $\omega$ is the optical frequency, and $f_{bg}(\omega )$ is given by the TS-DFT envelope measured exactly before the appearance of the broadening peak. The expression of the background necessitates that all the peaks contained in the background maintain invariant shapes and decay at the same rate in the following evolution. This necessity can be well satisfied because the picosecond fluctuations constituting the background are beyond the temporal resolution of the TS-DFT system and barely altered by nonlinearities and dispersion. Finally, the values of parameters $A_1$, $B$, $C$, $A_2$, $B_0$ and $A_3$ are acquired by fitting $f(\omega )$ to the TS-DFT envelope measured at each roundtrip.

 

Fig. 2. Decomposition of the TS-DFT result. (a) Illustration of the decomposition scheme. (b) Coefficients of determination ($R^2$) at different roundtrips. (c) Dominant pulse energy and total intracavity energy evolving with the roundtrips. (d) Evolution of the background intensity. The ensemble evolution is obtained through the decomposition process, and the intensity evolution of the four strongest background peaks is also presented. The linear fit (red dashed line) indicates a constant linear net gain of $-0.0001$. (e) Amplitude of the rapidly oscillating peak evolving with the roundtrips. (f) Intensity of the beating picosecond pulse and the beat frequency evolving with the roundtrips.

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The fitting accuracy is good, as characterized by the coefficients of determination $R^2$ shown in Fig. 2(b). The energy evolution of the dominant pulse is shown in Fig. 2(c). The evolving pulse energy acquired by detecting the unstretched pulses and the total intracavity energy acquired by integrating single-shot spectra are presented for comparison. Their deviations from the evolution of the dominant pulse extracted from the TS-DFT result are mainly caused by the inclusion of the disturbing background energy. The evolution of the background intensity versus roundtrips is shown in Fig. 2(d). All the background pulses show almost the same evolution, in which their damping rates keep constant in the initial stage and then increase rapidly after the dominant pulse begins its growth.

The oscillation caused by the interference between the dominant pulse and the auxiliary picosecond pulse, depicted by Fig. 2(e) and 2(f), also provides useful information. The amplitude of the oscillating peak is proportional to that of the picosecond pulse, and the beat frequency is proportional to the peak power of the dominant pulse [24]. With the peak power and energy of the dominant pulse known, its duration can be inferred assuming a fixed sech$^2$ pulse shape. Therefore, we can calculate the root mean square (RMS) time-bandwidth product (TBP) of the dominant pulse evolving with the roundtrips, as shown in Fig. 3(a). One may find some values of the RMS TBP are smaller than the theoretical limit, which is 1/3 for a sech$^2$ pulse. This is because the extraction of the transient beat frequency is not precise in the low-frequency region.

 

Fig. 3. Extraction of the power-dependent net gain. (a) RMS TBP of the dominant pulse evolving with the roundtrips. The oscillating peak is too weak to provide information for calculating RMS TBPs after 688 roundtrips. (b) Dependence of $R^2$ on the center roundtrip number of the fitting range that covers 100 roundtrips. (c) Dependence of $dE/dx$ on $E$. Blue dots denote the experimental data, and the red line is the fitting result. (d) Dependence of the single-roundtrip amplitude net gain on the instantaneous laser power.

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Now one can ascertain when the dominant pulse can be viewed as a perturbed soliton. The evolution of the RMS TPB shows the chirp is nonnegligible, indicating the dominant pulse cannot be treated as a perturbed soliton, at least before 688 roundtrips. It is also necessary to confirm the dominant pulse can be regarded as a chirp-free fundamental soliton in the mode-locked state. This confirmation has been done by measuring the spectrum and autocorrelation trace of the mode-locked pulse (see Supplement 1, Section 5).

Another approach to evaluate when the pulse can be viewed as a perturbed soliton is to check when its energy evolution can be described by Eq. (3). Pulse energy evolution within 100 roundtrips is fitted with Eq. (3). The dependence of the fitting accuracy on the center roundtrip number of the fitting range is shown in Fig. 3(b). The best fitting accuracy is achieved at 784 roundtrips with a coefficient of determination $R^2$ exceeding 0.95, indicating the dominant pulse has evolved into a perturbed soliton at that moment. The fitting accuracy degrades after 784 roundtrips, because the noise in the experimental data overwhelms characteristics of the slow pulse energy evolution in the final stage of mode-locking. Additionally, the dominant pulse undergoes an obvious redshift before the steady state is reached (see Fig. 1(b)). This redshift can be taken as an indicator of how close the system is to the steady state, which assists in the determination of when the dominant pulse can be treated as a perturbed soliton.

4.2 Extraction of the power-dependent net gain

The fitting result obtained in the optimal fitting range is shown in Fig. 3(c). The mode-locked peak power $P_0$ is measured to be 4.0 MW. Then it can be derived from the fitting result that the linear net gain $\alpha$, the SAM coefficient $\gamma$ and the second-order SAM coefficient $\gamma _2$ are $-0.016\pm 0.007$, $0.034\pm 0.011$ MW$^{-1}$ and $-0.014\pm 0.004$ MW$^{-2}$, respectively, at a confidence level of 95%. The measured SAM coefficient is of the same order of magnitude as that estimated by theory (see Supplement 1, Section 6). Ultimately, the dependence of the single-roundtrip net gain on the instantaneous laser power nearby the mode-locked state is obtained, as shown in Fig. 3(d). It shows that the net gain of the Ti:sapphire oscillator reaches the maximum at an instantaneous laser power of 1.2 MW and declines at higher powers. The reason for the decline could be that the laser beam is overfocused by the Kerr lens (see Supplement 1, Section 7), or that the lumped laser propagation is reduced to a distributed process in the model [52].

4.3 Comparison with simulation results

We also compare the simulated pulse evolution with the TS-DFT data. In the simulations, the CCQGLE as described by Eq. (1) is solved numerically with a fourth-order Runge-Kutta method [53]. The SPM coefficient $\delta$ and the single-roundtrip GDD $\psi _2$ are set to 6.7 MW$^{-1}$ and $-2000$ fs$^2$, respectively, according to theoretical calculations (see Supplement 1, Section 8). Values of other system parameters are obtained from the measurement above.

As shown in Fig. 4(a), the linear net gain resulting from the fit agrees well with the value obtained from the damping of the oscillating picosecond pulse, further strengthening the validity of the fit result. It is also found that the linear net gain does not oscillate along the roundtrips, so the influence of relaxation oscillations can be excluded. Besides, the linear net gain keeps almost unchanged before 560 roundtrips and after 650 roundtrips. The simulations focus on the evolution of the dominant pulse in these regions so that the linear net gain $\alpha$ can be set to a constant. For all the simulations, pulses with sech$^2$ shapes are used as the initial pulses.

 

Fig. 4. Comparison between simulation and experimental results. (a) Evolution of the linear net gain during the mode-locking transition. The initial (blue), midterm (magenta) and final (green) evolution is derived from Fig. 2(d), Fig. 2(f), and the fitting process, respectively. Evolution of (b) the energy and (c) the spectrum of the dominant pulse. The blue solid lines represent the TS-DFT results, the magenta dashed lines represent the calculated results assuming a zero initial chirp, and the green dash-dotted lines represent the calculated results assuming an initial linear chirp of 250 fs$^2$. For the two magenta dashed lines, the left one is calculated at $\alpha =-2.0\times 10^{-4}$ and $\gamma =2.994$ MW$^{-1}$, while the right one is calculated at $\alpha =-0.016$ and $\gamma =0.034$ MW$^{-1}$. The beginning of the optimal fitting range is marked by a cross.

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At the initial stage (from 470 to 560 roundtrips), the energy of the dominant pulse increases slowly, which gradually saturates the gain and thus makes the linear net gain increase at a very low rate. For the simulation in this region, the linear net gain is set to the value measured at the 470th roundtrip, namely $\alpha =-2.0\times 10^{-4}$. Besides, we take the bandwidth and pulse energy measured at the 470th roundtrip as initial conditions, and the initial chirp is set to zero. System parameters are set to be the measured or calculated values except for the SAM coefficient. As shown in Fig. 4(b), only when the SAM coefficient is increased up to 88 times the value nearby the mode-locked state can the energy buildup be as fast as the measured result. If the attenuation of $\alpha$ and the non-zero initial chirp are considered, the energy buildup will be slower and thus an even higher SAM coefficient will be required. Obviously, the dominant pulse cannot achieve such strong modulation by itself, so the dominant pulse is prevailingly modulated by the auxiliary picosecond pulse beating with it. This effect facilitates the starting of mode-locking.

After the initial stage, the dominant pulse grows and broadens rapidly with its chirp varying dramatically (see Fig. 3(a)). Moreover, the linear net gain it experiences increases dramatically until the evolution enters the final stage (after 650 roundtrips), at which the linear net gain can be treated as a constant ($\alpha =-0.016$) and the pulse behaves like a perturbed soliton. The simulation in this region is first implemented using a chirp-free fundamental soliton as the initial pulse with its energy slightly above the threshold, below which mode-locking cannot be achieved anymore. Other parameters are set to be the measured or calculated values. As shown in Fig. 4(b), the simulated evolution deviates from the measured one at first, but is asymptotic to that in the end. The moment when the actual evolution begins to agree well with the simulated one is marked by a cross, since then can the dominant pulse be regarded as a perturbed soliton. We attribute the deviation to the chirp in the initial pulse, and achieve a good reproduction of the measured pulse evolution by assuming an initial linear chirp of 250 fs$^2$, as shown in Fig. 4(b) and 4(c). Because the actual initial pulse shape is slightly different from the sech$^2$ one, some ripples appear in the calculated spectrum, which do not appear in the TS-DFT data.

4.4 Quantitative characterization of mode-locking initiation and stability

The good agreement between the simulation results and the measured ones confirms the accuracy of the measured net gain. Further, we employ the soliton perturbation theory to analyze soliton mode-locking. The phase portrait of the nonlinear system described by Eq. (3) is shown in Fig. 5(a). Because $E$ has been defined as the pulse energy normalized to the steady-state pulse energy, $dE/dx=0$ should hold at $E=1$. The slight deviation of $E$ at $dE/dx=0$ from 1 is caused by the measuring error of the net gain. As can be seen, mode-locking corresponds to a heteroclinic orbit jointing an unstable fixed point and a stable fixed point in phase space. The stable fixed point represents the mode-locked state, and the unstable fixed point accounts for a threshold pulse energy $E_{th}$, which can be derived by assuming $dE/dx=0$:

Here, $\alpha$ represents the linear net gain nearby the mode-locked state. Despite the linear net gain changes throughout the mode-locking buildup significantly, it keeps almost invariant at the final stage of mode-locking. Accordingly, the threshold given by Eq. (4) measures the difficulty in entering the final stage of mode-locking. Generally, the lower the threshold, the easier mode-locking can be initiated.

 

Fig. 5. Quantitative analyses of mode-locking initiation and stability. (a) Phase portrait of the nonlinear system described by Eq. (3). The unstable fixed point corresponding to a pulse energy of $\zeta _{th}$ and the stable fixed point corresponding to a pulse energy of $\zeta _{ml}$ are marked by a hollow dot and a filled dot, respectively. Dependence of $E_{th}$ and $\partial ^2 E / (\partial x \partial E) |_{E=1}$ on (b) $\xi$ and (c) $\alpha$ with other parameters set to be the measured values. Values of $\xi$ and $\alpha$ measured in the experiment are marked by dashed lines.

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The recovery rate after perturbation at the stable fixed point can be measured by [54]

With the simple equations derived above and the parameters therein already measured, it is possible to characterize mode-locking initiation and stability quantitatively. Dependences of mode-locking threshold and stability on $\xi$ and $\alpha$ are shown in Fig. 5(b) and 5(c), respectively, wherein the performance of the employed Ti:sapphire oscillator is marked by dashed lines. These results indicate that promoting the value of $\xi$ is the key to achieving better soliton mode-locking.

5. Discussion and conclusion

In summary, we have proposed a method for quantifying dissipative effects in systems governed by the complex Ginzburg-Landau equation and demonstrated it in a mode-locked Ti:sapphire laser. The key is to find a period of evolution in which a localized structure exists with a fixed form. The dependence of the energy or matter exchange rate on the amplitude of the structure is then deduced from the evolution of the total amount of energy or matter in the structure. This method can be also applicable to other complex Ginzburg-Landau equation governed systems, such as fiber lasers and thin-disk lasers, and can be extended to even more complicated systems. For a mode-locked laser, the dissipative effects are equivalent to the power-dependent net gain and the dynamics of the localized structure can be traced via TS-DFT. The measurement of the net gain allows for quantification of the energy threshold for mode-locking buildup and the stability of mode-locked states based on the soliton perturbation theory. It is found that the way to achieve better soliton mode-locking is to promote the value of $\xi$, which is related to the linear net gain $\alpha$, the SAM coefficient $\gamma$, and the second-order SAM coefficient $\gamma _2$ (see Eq. (5)).

We have also found that the mode-locking transition of a Ti:sapphire oscillator can be divided into three stages: first, an auxiliary picosecond pulse strongly modulates the dominant pulse and thus facilitates its growth, a critical drive to initiate mode-locking; second, the amplitude, bandwidth and chirp of the dominant pulse vary dramatically, accompanied by decays of background pulses; finally, the dominant pulse squeezes out its chirp and evolves into a perturbed soliton. Unlike the mode-locking of the soliton fiber laser we investigated previously, in which a pulse dominates by winning the competition against all other pulses [31], the dominant pulse comes into being under the assistance of another pulse during the mode-locking of our Ti:sapphire oscillator.

This work will shed light on both laser physics and soliton physics. On the laser end, this work shows that the soliton perturbation theory can greatly simplify the characterization and analyses of mode-locking. Besides, direct measurements of the power-dependent net gain will enable the full use of the well-established mode-locking theory and various real-time optical waveform tracking techniques in the diagnosis and improvement of mode-locked lasers. On the end of soliton physics, this work provides an approach to studying the dynamics of perturbed solitons in lasers quantitatively. Furthermore, the ability to determine parameters in the CCQGLE of mode-locking will pave the way to a totally controllable simulator of dissipative soliton dynamics.