Slow saturable absorption for optimal operation in a soliton comb laser

: In this article, we study how the choice of parameters of a slow saturable absorber (SSA) affects the stable operation of a soliton fiber comb laser. We show that a shorter recovery time for the SSA does not always lead to shorter modelocked pulses. Instead, increasing the cavity gain plays a critical role in generating stable modelocked pulses with higher energy and shorter durations. We find that more stable, shorter, and more energetic output pulses can be achieved with lower saturation energies of the SSA and/or higher anomalous dispersion within the cavity.


Introduction
In recent years, optical frequency combs have become widely used in both research and the industry and in both military and civil fields [1][2][3][4][5][6]. A sizable fraction of current frequency comb sources are based on passively modelocked fiber lasers [7][8][9][10][11][12][13]. Among such modelocked lasers, millimeter-scale semiconductor saturable absorber mirrors (SESAMs) are widely used as a key intracavity component to achieve stable operation [8,[13][14][15][16]. To design modelocked lasers and comb sources for particular applications, the selection of a proper SESAM is vital. The SESAM parameters such as the saturation fluence, the recovery time, as well as the unsaturated loss (or the flat loss) must be carefully specified to obtain the optimal output from the laser cavity. In particular, it is useful to understand where in the system parameter space stable stationary pulse solutions exist, and how the pulse profile changes as parameters change.
SESAMs are slow saturable absorbers (SSA), in which "slow" implies a recovery time T A that is longer than the optical pulse duration in the frequency comb source. The typical absorption recovery time of a GaAs-based SESAM is usually between 0.5 ps and 30 ps [17][18][19], which is considerably longer than the duration of the generated modelocked pulse (∼100s fs), although shorter than the round trip times (∼10 ns). Other bulk-structured materials that have similar slow absorption recovery times include carbon nanotubes [20][21][22], graphene [23], and few-layer molybdenum diselenide [24]. Currently, the recovery time of bulk-structured saturable absorbers ranges from 0.1 ps to hundreds of picoseconds [25].
Multiple studies have described the dependence of the stationary modelocked pulse profile as the energy balance varies in modelocked lasers using slow saturable absorbers. Paschotta and Keller [26] derived an analytical estimate of the pulse duration using the SSA's absorption depth and saturation energy. In addition, they found that the pulses become unstable when the recovery time becomes sufficiently large. In a later study, it was demonstrated that the stable pulse profiles are consistent with the soliton area theorem [27]. Kärtner, et. al [28] showed that as the small-signal gain increases, the pulse profiles become similar to the profiles of the modelocked pulses that are generated by lasers that use a fast saturable absorber, while the pulse duration and time-bandwidth product remains almost unchanged. However, there has not been a careful study of the effect that the SSA's recovery time and saturation energy have on the profile of the output pulses.
Fast saturable absorbers (FSAs), like those based on nonlinear polarization rotation (NPR) [9][10][11], are a commonly used alternative to SSAs. These saturable absorbers, which rely on interferemetric effects on the scale of the optical wavelength and the Kerr effect, are effectively instantaneous. While maintaining interferemetric stability is challenging, these lasers are remarkably stable when that can be accomplished. Hence, there exists a popular assumption [26,29,30] that SSAs with shorter recovery times will produce shorter modelocked pulses and thus broader bandwidth output combs. However, using the evolutionary simulation method [31][32][33], Lee et al. recently showed that the pulse duration and thus the bandwidth remain almost unchanged for soliton lasers as the recovery time decreases [25]. Lee et al.'s results appear to contradict the notion that a shorter recovery time is always preferred, and this assumption requires a closer look. More generally, it is important for the design and optimization of future SSA-based soliton lasers to provide a more thorough description of where in the experimentally-adjustable parameter space stable pulses exist and to determine the variation of their profiles within the stable parameter regions.
Using a SESAM-based fiber comb laser as an example, we show in detail how changing the recovery time T A affects the region of stable operation. We stress that our goal is not to carry out a quantitative performance comparison between an FSA and an SSA. Such a comparison would not only be difficult, but it would also be unfair because of the fundamental difference in their physics, as we will show in Sec. 3. In this article, our goal is to provide general guidance for selecting an SSA for soliton comb lasers. We achieve this goal by studying the regions of stable modelocking operation for the SSA parameters. We show that although the SESAM's absorption recovers faster as T A decrease, which helps to stabilize modelocking, the absorption also significantly weakens, which works against stable modelocking. Hence, the long-held assumption [26,29,30] that a lower T A is always preferred should be revised. Plus, careful parameter selection for the SSA is needed for optimal modelocking performance. We then find that in order to obtain more energetic and broadband stable modelocked pulses, the key is to achieve larger cavity gain. In order to maximize the cavity gain, the SESAM's recovery time T A must be carefully selected to be around 0.5 ps. We also find that the modelocked pulse profile can be further optimized by using lower values of the SESAM saturation energy and larger negative group delay dispersion.
In Sec. 2, we present the computational model that we use for conducting our parameter studies. In Sec. 3, we show the fundamental dynamical structure of the SESAM-based modelocking system in terms of the SESAM recovery time T A , as well as the unsaturated cavity gain g 0 . We also obtain an optimal output pulse when the recovery time T A = 0.43 ps, which is close to the limit of commercially available SESAMs at this time. In Sec. 4, we further show that a lower saturation energy of the SESAM and/or a larger anomalous dispersion can help stabilize the modelocking and thus enable the generation of more energetic and larger bandwidth output combs. We conclude this work in Sec. 5.

SESAM-based soliton fiber comb laser
Our study is based on a reported highly stable fiber comb laser with a repetition frequency of 200 MHz [8]. We use an averaged model [34,35] and dynamical methods [36] to carry out the parameter study. The averaged model [35,37] that we use is a variant of the Haus modelocking equation [38]. The evolution of the wave envelope is described by where T R is the round trip time, T is the slow time of propagation, t is the retarded fast time, u(t, T) is the slowly varying field envelope, and each operator on u on the right-hand side of Eq. (1) is averaged over one round trip. The time t s is the time shift in the fast time t of the pulse centroid the variable ϕ is the phase change, g(|u|) is the saturated gain, ω g is the gain bandwidth, l is the linear loss in the cavity, β ′′ is the group-delay dispersion, γ is the Kerr coefficient, ρ is the saturable absorption coefficient, and f sa (t) is the saturable absorption function. We assume that the recovery time of the gain is significantly longer than the round trip time, so that the gain g(|u|) may be written as where g 0 is the unsaturated gain, w(|u|) is the intra-cavity pulse energy, and P sat is the saturation power of the doped fiber. Here, we model f sa (t) using a two-level slow saturable absorption model where n is the ratio of the lower level population to the total population of the SESAM [37], w A is the saturation energy of the SESAM, and we assume that the response time T A ≪ T R . The parameters that we use are listed Table 1. The variable l corresponds to a total cavity loss of 22%, which includes both the output coupling ratio and the insertion losses among the cavity components. The saturable coefficient ρ corresponds to a SESAM modulation depth of about 10%. From the reported saturation fluence Φ sat and the effective area A eff,SESAM , the saturation energy is w A = Φ sat A eff,SESAM = 50 µJ/cm 2 × 314µm 2 = 157 pJ [19] (Here, we correct the units of Φ sat and A eff,SESAM on page 2363 in [35]). In Fig. 1, we show the change in normalized absorption n(t) of the SESAM as T A and w A vary, in which n(t) = 1 indicates that the absorption is unsaturated. The incoming pulse profile is chosen to be similar to that of the experimental pulse in [8]. In general, we observe that the absorption n(t) decreases as the incoming pulse appears, and recovers gradually to the unsaturated level, n(t) = 1, in the wake of the pulse. By comparison, in Fig. 1, we also show the profile of a fast saturable absorber (FSA) based on the equation where P sat is the saturation power [39]. Here, we set P sat = w A /T R , where w A = 157 pJ, to be convenient for illustrations. In practice, w A is an SSA parameter and has no connection to the FSA parameter P sat . In Fig. 1, we show that as T A decreases, with w A = 157 pJ, the absorption recovery of the SESAM occurs more quickly. Hence, the SESAM recovers more quickly, which can potentially help stabilize the modelocked pulse by suppressing the wake modes trailing the modelocked pulse [35]. This operating regime was described in [25]. However, as we show in Fig. 1, when T A decreases while w A is fixed at 157 mW, the SESAM's absorption depth n(t) decreases significantly at the same time, and thus increasingly deviates from the behavior of an FSA, which we also show in Fig. 1. Unsurprisingly, we will later show that the decrease in SESAM's absorption depth works against stable modelocking. This result is unsurprising because in the limit as T A → 0, the SESAM effectively disappears. This behavior is remarkably different from FSAs [39,40], which have large parameter regions in which modelocked operation is stable and the output modelocked pulse energy is only limited by the instability of continuous waves [41]. In Sec. 3, we will also show that the stability of the SESAM-based laser is also limited by continuous waves when T A becomes sufficiently small, but this limit is less important than in the FSA-based laser because the absorption depth in this limit is already small. In Fig. 1, we also show the case T A = 2.0 ps, w A = 20 pJ. Comparing with the case T A = 2.0 ps, w A = 157 pJ, we observe that the absorption depth visibly increases and becomes more comparable to the absorption depth of the FSA. Thus, in order to approximate the operation of an FSA and produce more broadband and energetic modelocked pulses, one would need to reducing both T A and w A simultaneously. Hence, careful selection of the SSA parameters is necessary to achieve optimal performance. In Sec. 3, we will explicitly calculate the impact of changing T A on the SESAM-based laser's stable operating region and in Sec. 4, we will do the same for w A .
Despite the comparison that we show in Fig. 1, we do not directly compare the absorption depth and the subsequent modelocking performance of an FSA and an SSA, as it would not be very meaningful to do so. An SSA is typically a single device so that its saturation energy is well-defined and easily quantifiable. By contrast, an FSA is typically realized by combining several optical components. So, for example, a common FSA will combine nonlinear polarization rotation in a nonlinear fiber with polarization-selection elements [42]. As a consequence, the saturation power P sat is usually a heuristic quantity and thus very difficult to quantify. Moreover, with either an FSA or an SSA, we cannot set the absorption depth a priori. It is a parameter that depends on the energy of the modelocked pulse and must be determined a posteriori.

Optimizating the SESAM recovery time
We first determine the stability of modelocked pulses using the dynamical method [34,36]. This method is a set of computational algorithms that make it possible to determine the precise region in the experimentally-adjustable parameter space where stable pulse solutions exist. A modelocked laser can be viewed as a nonlinear dynamical system, and a modelocked pulse u 0 (t) can be viewed as a stationary solution or an equilibrium of such a system. In the dynamical method, we find the stationary solution u 0 , the linear phase rotation coefficient ϕ, and the time shift t s using Newton's method. The computational root-finding procedure is described in detail in [36]. If any possible perturbation ∆u(t, T) of the stationary pulse u 0 (t, T) grows exponentially, then the modelocked pulse is unstable. The stability can be determined by solving a linear eigenvalue problem where the matrix L can be written as in which L 11 = δF /δu, L 12 = δF /δu * , L 21 = δF * /δu, L 22 = δF * /δu * are functional derivatives [34,43], F (u) is defined by the right-hand side of Eq. (1), and λ k and e k are the kth eigenvalue and eigenfunction of the matrix L. Any perturbation ∆u(t, T) can be expressed as a linear combination of the eigenvectors e k . A stable modelocked pulse u 0 exists only when all the λ k have non-positive real parts. The dynamical method allows us to determine the existence and stability of pulses over a broad parameter range efficiently and unambiguously. By contrast, when using traditional evolutionary methods-for example, the popular split-step method [31][32][33]-it is often uncertain how long it takes for the system to evolve into a stable profile, especially when the system is approaching a stability boundary. Additionally, if multiple stable profiles exist, the final evolution profile would depend on the initial conditions that are chosen. These ambiguities are typically difficult to resolve when using evolutionary methods.
In Fig. 2, we show the stable regions in the parameter space of the unsaturated gain g 0 and the SESAM recovery time T A . For any given pair T A and g 0 , we calculate the stationary modelocked pulse solution u 0 and evaluate the dynamical spectrum, which we show later in Fig. 3. When T A becomes sufficiently small, we discover a continuous region of stable operation (the white-shaded area) that is enclosed by three different boundaries: a lower bound and an upper bound in g 0 due to unstable continuous waves (C s ) and an upper bound in g 0 , S s , due to unstable soliton wake modes.
As shown in Fig. 2, when T A = 2 ps, which is the value for a typical SESAM [44] that is used in fiber comb lasers, stable modelocked pulses exists when 0.57<g 0 <7.0. The continuous waves become unstable when g 0 <0.57 (indicated by the curve C s ) while the wake modes become   Fig. 2. Here, the initial conditions are a one-photon-per-mode noise seed. We obtained similar stable/unstable propagation profiles when using pulsed initial conditions. These results suggest that the system does not support multiple stable solutions using the selected sets of system parameters. (d)(e)(f) The dynamical spectra of these points; In all three subplots, the eigenvalue λ a is too negative to be shown in the plotted area. We use "CWS" in (f) to represent the continuous waves spectra, which also appear as the orange lines in (d) and (e). unstable when g 0 >7.0 (indicated by the curve S s ). As an example, at point ①, where T A = 2 ps and g 0 = 6.0, the system supports a stable modelocked pulse. We show the corresponding evolution profile in Fig. 3(a), and the dynamical spectrum in Fig. 3(d). We observe in the dynamical spectra that there are two continuous branches of continuous waves eigenvalues, and four discrete eigenvalues on the real axis, λ a , λ φ , λ t , and λ f that correspond to the perturbations to the amplitude, phase, central time, and central frequency, as suggested by soliton perturabtion theory [45][46][47]. Additionally, there are a pair of complex conjugate eigenvalues λ w+ and λ w− that correspond to the soliton wake modes [34,35]. All eigenvalues have non-negative real parts, which confirms that the stationary modelocked pulse is stable, as suggested by Fig. 3(a). By contrast, at point ②, at which T A = 2 ps and g 0 = 12.0, the stationary pulse is unstable due to the wake modes. The corresponding evolution profile is shown in Figs. 3(b), in which we observe the subsequent construction and destruction of new pulses. The dynamical spectra in Figs. 3(e) show that Re(λ w+, w− )>0, which indicates that the wakes modes are unstable. These results agree well with prior results [35].
Starting from T A = 3 ps, we observe that the stable region first expands as T A decreases, and then shrinks when T A decreases below 0.43 ps. The stable region ultimately disappears when T A ≤ 0.05 ps. By comparison, when T A >3 ps, we observe a similar stability structure as when T A = 3 ps and the stable region continues to exist up to T A = 7 ps. We did not study larger values of T A as they are of little practical interest for frequency comb applications.
In Fig. 2, we observe that when g 0 becomes sufficiently large, a stable modelocked pulse is destablized by different mechanisms that depend on T A : • When T A >0.18 ps, a stable modelocked pulse is destabilized by the wake modes, while the continuous waves remain stable, as shown by the blue region (above line S s ) in Fig. 2. This case was discussed in detail in [35] which includes a cartoon of the pulse evolution.
• By contrast, when T A <0.18 ps, the modelocked pulse is destabilized by continuous waves while the wake modes are stable, as shown by the green triangular region (above line C s , upper) in Fig. 2. Prior studies showed that continuous waves destabilize modelocked FSA-based lasers as cavity gain grows [41]. This result is consistent with ours, as the SESAM behaves similarly to an FSA with decreasing absorption depth as T A decreases, as suggested in Fig. 1.
• Although we can still computationally find stationary solutions as T A → 0, these solutions become unstable because the saturable absorption disappears, as shown in Fig. 1.
In Fig. 3, we show an examples in which g 0 = 8.5 and T A = 0.1 ps and continuous waves are unstable. We show the unstable evolution profile in Fig. 3(c), in which we observe that the continuous waves continues to fluctuate around the propagating pulse. In addition, the pulse amplitude experiences quasi-periodic modulations due to its interference with the unstable continuous waves. The corresponding spectrum in Fig. 3(f), in which we observe that parts of the two branches of the continous spectrum have positive real parts, consistent with [35,36]. Both the wake modes and the continous waves are unstable in the purple region above both lines (C s , upper) and S s , as shown in Fig. 2.
In Fig. 4, we show the variation of the stable modelocked pulses inside the stable region that we show in Fig. 2. Here, we can confirm that the profile of the calculated stable modelocked pulses largely agree with the soliton area theorem [28,48] E 0 t 0 = 2 | β ′′ | /γ = 12.6 (W · ps 2 ), where E 0 = ∫ T R /2 −T R /2 |u 0 | 2 dt is the pulse energy, and t 0 = FWHM/1.763 in which FWHM is the full-width-half-maximum pulse duration. This energy theorem has significant implications for practical laser designs. An important design goal for comb lasers is to increase the output comb power/energy as well as the bandwidth. Here, Eq. (9) indicates that for SESAM-based soliton fiber comb lasers, a growth in pulse energy is conveniently accompanied by decrease in pulse duration and thus broader bandwidth. In Fig. 4(b), we also observe that for a given T A , the pulse duration decreases, while the pulse energy E 0 increases almost linearly when we increase the unsaturated gain g 0 . By comparison, as shown in Fig. 4(a), both the pulse duration and the energy remain almost unchanged as T A decreases when g 0 is fixed, which is consistent with [25]. Thus, this observation provides an unambiguous answer to the question that was posed at the beginning: A shorter recovery time of the SSA does not necessarily lead to a shorter modelocked pulse. Instead, a higher cavity gain leads to a higher pulse energy and a wider pulse bandwidth. Therefore, to optimize the laser's output profile, one should find a recovery time that can allow the highest unsaturated gain for stable modelocking.
For our parameters, we find that by selecting a SESAM with T A = 0.43 ps, we obtain the largest cavity gain g 0 = 14, which corresponds to a small signal gain of 61 dB. In Fig. 2, we label this optimal point as P. In Fig. 4(c), we show the stable pulse profile at P, as well as a few other points for comparison. We observe that when T A = 0.43 ps, increasing g 0 from 6 to 14 leads to a growth of pulse energy from 59.6 pJ to 142.7 pJ. Simultaneously, the pulse duration decreases from 212 fs to 88 fs.

Further improvements in output energy and bandwidth
Next, we show how the system improves as the SSA's saturable energy w A and the cavity GDD β ′′ vary.
In Sec. 3, we showed that the change in the system's dynamical spectrum and the output pulse's profile as the SSA's recovery time T A varies. In Fig. 2, we show that for a given w A , we can find an upper bound which is a composite curve of (C s , upper) and S s . We then show the optimal pulse profile at P in Fig. 4. In Fig. 5, we show the change of the upper stability boundary and the corresponding pulse profiles as the saturation energy w A changes. In Fig. 5(a), we find that the stability boundary (S s , upper) moves to larger values of g 0 as w A decreases. This result suggests that by lowering the SESAM saturation power, the laser cavity can operate stably with higher gains, leading to more energetic modelocked pulses with a larger bandwidth. We then show the change of the largest unsaturated gain, g 0 , consistent with pulse stability as well as the corresponding pulse profiles in subplots (b) and (c). Starting from the optimal point P in Fig. 2, we observe that the largest possible gain g 0 grows almost linearly as w A decreases, as shown in Fig. 5(b). We take four points along the line of T A = 0.43 ps, marked as points {1}, {2}, {3}, and {4} in Fig. 5(b), and show the corresponding pulse profile in Fig. 5(3). In Fig. 5(c), we observe that as w A decreases and g 0 increases, the stable modelocked pulse continues to become narrower and more energetic. When w A is 20 pJ, the corresponding pulse energy is 238.5 pJ, while the pulse duration is 52.7 fs, which is a 67% increase in pulse energy and a 40% decrease in pulse duration. In Fig. 6, based on the data in Fig. 5, we show how the optimal value of the SSA's recovery time T A varies as the saturation energy w A changes. In the range of parameters that we study, we observe that as w A decreases, the optimal value of T A appears to decrease linearly. This observation will be useful for experimentalists to improve their laser designs. In the future, we expect to study the variation of the optimal T A for a wider range of w A , as well as when other system parameters vary. The black dash-dotted line, which shows a linear trend between w A and the optimal T A , is fitted using the optimal values that are obtained when w A = 20 pJ, 60 pJ, 100 pJ, and 157 pJ, shown in Fig. 5(a).
Another approach to further increase the output energy and bandwidth is to introduce more negative dispersion. In Fig. 7, we show the upper stability bounds in g 0 as we vary the group delay dispersion (GDD) β ′′ while keeping other system parameters fixed. We see that the basic structure is the same as in Fig. 2. When β ′′ becomes more negative, both lines (C s , upper) and S s , as well as the location of the optimal point P, shift to lower values of T A and larger values of g 0 . Fig. 7. The stability limits of the SESAM laser as the cavity GDD β ′′ varies. Here, the lines (C s , upper) and S s , as previously defined in Fig. 2, are the stability limit for the continuous waves and the wake modes, respectively.

Conclusion
In summary, we have studied how the parameters of the slow saturable absorption (SSA) affect the performance of a soliton fiber comb laser. As the recovery time T A decreases from picoseconds to sub-picoseconds, the modelocking can be destablized by different mechanisms. In general, as T A decreases, the absorption recovery occurs more quickly while the absorption depth decreases. No modelocking exists as T A approaches zero as the absorption depth becomes too small to support a modelocked pulse. Saturable absorbers with small non-zero values of T A behave similarly to a fast saturable absorber with a small saturation depth, in which case the modelocking can be destabilized by continuous waves. When T A grows, the wake modes instability appears as cavity gain increases. Wake modes appear in the net gain window that is formed by the slow recovery of the absorption and the even-slower saturable gain, destabilizing the modelocked pulse.
In this article, we described two approaches that can help suppress the wake modes and stabilize the modelocking. Smaller values of saturation energy saturate the absorption more easily, leading to deeper and narrower net gain windows. In addition, large values of negative group-delay dispersion (GDD) helps suppress the wake modes and stabilize the modelocking by sweeping the growing modes out of the gain window [35].
Based on this dynamical stability structure, we show that significantly larger output power and bandwidth can be achieved by selecting a SSA with carefully selected parameters. In particular, the largest cavity gain is the key factor enabling more energetic modelocked pulses with a broader bandwidth. For the laser parameters that we consider here, the cavity gain can be significantly increased when the recovery time is reduced from 2 ps to 0.43 ps, which leads to more than doubling the laser's output energy and bandwidth. This output can be even further improved by introducing a more negative GDD and a lower saturation energy for the SESAM. Previous results [49] have shown that higher output coupling ratios can further increase the pulse energy and bandwidth without sacrificing the noise performance. Combining all these results, we can predict that many SSA-based fiber comb lasers can be optimized to support significantly higher power comb outputs with a broader bandwidth.
In this artile, our optimization goal is to increase the laser's output power and bandwidth by varying the SSA's recovery time T A and saturation energy w A , as well as the group delay dispersion. Here, based on our prior optimization efforts [35,43,49,50], we provide a brief summary to our prior results in Table 2. For future work, we plan to conduct a more exhaustive optimization effort in the laser's adjustable parameters, which might include fiber parameters, the saturation parameters, as well as the cavity designs structures. In addition, we also plan to include lowering the comb lasers' noise, including the carrier envelope phase noise [51] and the timing jitter phase noise [52], as a key optimization goal. Table 2. To optimize the output parameters of a soliton comb laser, the preferred variation of the parameters of the laser cavity.

Parameter
Perferred variation Notes g 0 larger Achieving higher values of g 0 is the key to optimize the soliton comb lasers' output parameters. Larger values of g 0 is achieved by increasing pump power.
β ′′ more negative Larger values of negative GDD can help avoid the most unstable perturbations: the wake modes [35].
w A smaller See an example in Fig. 5. Smaller values of w A can be achieved with smaller saturation fluence.
t out larger Larger output coupling ratios can allow higher values of g 0 without triggering the wake mode instability while increasing the power efficiency [49,50].
T A from 0.2 ps to 0.5 ps In this work, optimal values of T A are from 0.2 ps to 0.5 ps, where smaller values of w A and more negative values of β ′′ leads to smaller optimal values of T A . In general, the optimal value of T A depends on the specific values of w A , β ′′ , and t out .
Here, our work focuses on a generic model for slow saturable absorbers, Eq. (5), without considering the manufacturing details. In practical designs, availability of cavity components can also pose limits on which SESAM to choose. In addition, it has been reported that two-photon absorption can lead to inaccurate evaluations of the SESAM's absorption curves, especially in GaAs-based SESAMs [53,54] with high fluences. While particular model inaccuracy should not affect our results since we focus on SESAMs with lower values of w A , as described in Sec. 4, it might affect other designs. These issues will have to be tackled in future work.