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## Article

Curr. Opt. Photon. 2023; 7(1): 90-96

Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.90

## Numerical Simulation of Soliton-like Pulse Formation in Diode-pumped Yb-doped Solid-state Lasers

Seong-Yeon Lee1, Byeong-Jun Park1, Seong-Hoon Kwon2, Ki-Ju Yee1

1Department of Physics, Chungnam National University, Daejeon 34134, Korea
2Pohang Accelerator Laboratory, Pohang 37673, Korea

Corresponding author: *kyee@cnu.ac.kr, ORCID 0000-0002-1076-2354

Received: October 26, 2022; Revised: December 27, 2022; Accepted: December 28, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

We numerically solve the nonlinear Schrӧdinger equation for pulse propagation in a passively mode-locked Yb:KGW laser. The soliton-like pulse formation as a result of balanced negative group-delay dispersion (GDD) and nonlinear self-phase modulation is analyzed. The cavity design and optical parameters of a previously reported high-power Yb:KGW laser were adopted to compare the simulation results with experimental results. The pulse duration and energy obtained by varying the small-signal gain or GDD reproduce the overall tendency observed in the experiments, demonstrating the reliability and accuracy of the model simulation and the optical parameters.

Keywords: Femtosecond laser, Mode-locking, Soliton-like pulse formation, Yb:KGW laser

OCIS codes: (140.3580) Lasers, solid-state; (140.3615) Lasers, ytterbium; (140.4050) Mode-locked lasers

### I. INTRODUCTION

Ultrashort lasers on the femtosecond scale are widely used in a variety fields such as material processing, nonlinear optical imaging and lithography, medical applications, ultrafast phenomena diagnosis, and so on [17]. Though the laser requirements may differ from application to application, shorter and more intense pulses are generally in demand. The Kerr lens mode-locking of a Ti:sapphire laser demonstrated by Spence et al. [8] can be seen as a milestone in the development of femtosecond solid-state lasers (SSLs). Because of very broad gain, few-cycle pulses down to 5 fs have been achieved with Ti:sapphire lasers [9]. However, Ti:sapphire lasers are generally pumped by comparatively expensive and bulky systems such as frequency-doubled Nd:YVO4 lasers. In contrast, femtosecond Yb-doped SSLs operating at around 1,030 nm can be directly pumped by commercially available diode lasers at around 980 nm and thus have merits in achieving high-power systems with reduced space and cost [1015]. Recently, a diode-pumped Yb:KGW laser was reported to produce femtosecond pulses with an average power of 13 W [16]. In terms of the pulse duration, 17.8 fs pulses were generated from a Yb:CALGO laser [17].

Mode-locked SSLs generally operate in the negative group-delay dispersion (GDD) condition, for which a soliton-like pulse can be formed through the combined effect of the GDD and the self-phase modulation (SPM) [17, 18]. Because the nonlinear SPM depends on the pulse energy while the GDD is a linear parameter, the balancing between the two effects is reached at a specific pulse energy for a predefined pulse shape. By solving the nonlinear Schrӧdinger equation, which governs the pulse propagation in a laser cavity, the soliton-like pulse formation in mode-locked lasers has been studied both theoretically and in combination with experiments. Since the soliton-like pulse formation has been mostly studied for Ti:sapphire lasers, its application to Yb-doped SSLs is rare and worthy to carry out [1924]. We need to note that the beam spot size in the crystal is relatively large for Yb-doped SSLs pumped by diode lasers coupled to multi-mode fibers, and that the nonlinear SPM is weak in comparison to tightly focused Ti:sapphire lasers. Thus, rigorous analysis of the pulse formation in a diode-pumped high-power SSL will aid the interpretation and upgrade of diode-pumped femtosecond laser systems.

In this paper, we report on the numerical simulation of soliton-like pulse formation in a femtosecond Yb:KGW laser that operates in the negative GDD regime and uses a semiconductor saturable absorber (SESAM) for passive mode-locking. The solution to the nonlinear Schrӧdinger equation was found after considering the negative GDD at chirped mirrors, saturable absorption at SESAM, and the dispersion and SPM in gain crystal. To validate the reliability of the model simulation through direct comparison with previous experiments [25, 26], we deliberately chose the optical parameters that closely match those used in the experiment. Relatively good correspondence was obtained between the simulation and experimental results in terms of pulse energy and GDD dependence.

### 2.1. Soliton-like Pulse Formation Model

Soliton-like pulses are formed from the interplay between negative GDD and nonlinear SPM in passively mode-locked SSLs. For the ideal soliton-like laser, pulses with a sech2 temporal profile are generated, where the pulse duration, τs, is given by [20, 21]

τs=3.53β2tκΕp

Here, β2t is the total intracavity GDD, κ is the SPM parameter, and Ep is the pulse energy. According to Eq. (1), the duration of the soliton-like pulse is linearly proportional to the negative GDD and inversely proportional to the SPM coefficient. However, the pulse characteristics deviate from the equation in practical lasers because the GDD and SPM are not evenly distributed throughout the laser cavity. Taking into account the discrete contributions of the GDD and SPM from each optical component used in a Yb:KGW laser, we numerically simulate the pulse evolution as it repeats round-trip propagations in the cavity. For the cavity design, cavity mode, and optical element parameters, we adopt those used in our previous experimental work [25, 26]. There, the gain crystal of 3-mm-thick 3.6% Yb-ion doped Yb:KGW crystal is pumped by a 981 nm laser diode, and the cavity, with a standard X-shaped configuration, includes a 20% transmission output coupler, a SESAM with relaxation time of 800 fs, and Gires-Tournois-interferometer (GTI) mirrors for negative GDD.

The schematic in Fig. 1 shows the optical elements and pulse propagation in the cavity. The cavity is composed of GTI mirrors, a SESAM with slow self-amplitude modulation (SAM), a Yb:KGW crystal, and an output coupling mirror. In one round trip, the pulse encounters six segmental changes as denoted by the numbers in Fig. 1. In the modelling, we include each step in the sequence of propagation and find the soliton state for which the pulse shape and energy is stable upon iterations.

Figure 1.Schematic of a linear laser cavity with a Yb:KGW gain crystal, negative GDD GTI mirrors, SESAM with slow SAM, and output coupler. The Yb:KGW crystal also provides positive GDD, SPM, and fast SAM. GDD, group-delay dispersion; GTI, Gires-Tournois-interferometer; SESAM, semiconductor saturable absorber; SAM, self-amplitude modulation.

The Yb:KGW crystal provides the gain, SPM, positive GDD, and fast SAM. The pulse propagation through the crystal corresponding to steps 3 and 5 can be expressed by [27, 28]

az,tz=g1+1Ωg22t2az,t+i2β2c2t2+16β3c3t3az,t+iγa2az,t+ηa2az,t

where a(z, t) is the normalized envelope field such that |a|2 is the power, g and Ωg are the saturated gain and the gain bandwidth, respectively, β2c and β3c are the group velocity dispersion and third-order dispersion of the crystal, respectively, γ is the SPM parameter, and η is the SAM parameter provided by the Kerr effect in the gain crystal. We approximate the saturated gain as follows [27]:

g=g01+EP/PgTI

where g0 is the small-signal gain, EP the integrated pulse energy defined by EPt=t |a t'|2dt', TI the time duration between pulses at the crystal, and Pg is the saturation power given by Pg = Aeff × ℏω/στ with the effective mode area in the crystal Aeff, the emission cross section σ, and the upper state lifetime τ. The SPM coefficient is given by γ = 2πn2/λAeff, with n2 being the nonlinear refractive index of the crystal. In the experimental work [25, 26], positive power enhancement was observed with the mode-locking due to the onset of Kerr lensing at the crystal. To implement this effect in the simulation, we add the fast SAM parameter of the crystal, η, which is proportional to the SPM parameter and is assumed to be 0.01 × γ.

While the pulse propagation in the crystal is described by Eq. (1), an abrupt loss at the output coupler can be expressed by a(z+, t) = rOC × a(z, t), with z+ (z) indicating the position after (before) the output coupler with the reflectivity rOC. The role of GDD provided by GTI mirrors, β2m, can be easily implemented in the Fourier domain of a(z, t) by multiplying exp[iβ2m (ωω0 )2] to a(z, ω). The effect of the SESAM, which can be regarded as a slow saturable absorber, is implemented by multiplying exp[−q(t)] to a(z, t), where q(t) satisfies Eq. (4):

qt=qq0TAa2EAq

where q0 is the saturable absorption loss, and TA and EA are the relaxation time and saturation energy of the SESAM, respectively. Here, EA = Isat × ASESAM is obtained from the saturation intensity Isat and the effective beam area at the SESAM, ASESAM. The parameters used in the simulation are listed in Table 1. As the initial pulse, we use a sech2 pulse with a pulse duration of 1 ps.

Parameters used in the numerical simulations

 Component Parameter Value Description Yb:KGW Crystal l 3 mm Length σ 3 × 10−20 cm2 Emission Cross Section τ 0.36 ms Upper State Lifetime n2 20 × 10−16 cm2/W Nonlinear Refractive Index Ωg 35.7 THz Gain Bandwidth β2c 178 fs2/mm Group-velocity Dispersion β3c 236 fs3/mm Third-order Dispersion g0 Variable Small-signal Gain Aeff 2.0 × 10−4 cm2 Effective Mode Area TI 8.35 ns Inter-pulse Time GTI Mirrors β2m −550 × n fs2 GDD at GTI Mirrors (n: No. of Bounces) SESAM q0 0.0132 Saturable Absorption Loss TA 800 fs Relaxation Time Isat 120 mJ/cm2 Saturation Intensity ASESAM 6.0 × 10−4 cm2 Mode Area at SESAM Output Coupler rOC 0.894 Output Coupler Reflectivity

### 2.2. Simulation Results and Comparison with Experiment

To get an insight into the soliton-like pulse formation under different conditions, we carried out numerical simulations tuning the small-signal gain, the round trip GDD, and the SPM parameter. In a previous study [25], we experimentally demonstrated mode-locked pulses from a Yb:KGW oscillator in which the negative GDD was provided by using GTI mirrors. Including the positive GDD in the Yb:KGW crystal, the total GDD, β2t, was −5,532 fs2 at the condition of six GTI mirror bounces. Figures 2(a) and 2(b) show how the pulse energy and duration are stabilized with the round trips of the laser cavity under the small-signal gain condition corresponding to g0l = 0.5 with l being the crystal length and for the total GDD of β2t = −5,532 fs2. The pulse duration exhibits damped oscillations with the number of round trips, which is stabilized after about 600 round trips. This oscillatory behavior at early times originates from the nonlinear nature of the model equation, Eq. (2). Figure 2(c) shows both the initially assumed temporal profile and the one finally reached after 2,000 round trips, while Fig. 2(d) shows the corresponding laser spectra. We need to mention that the soliton-like pulse is stabilized faster as the larger value of the SAM parameter of the gain crystal is assumed.

Figure 2.Pulse evolution with round trips. (a) Pulse energy and (b) pulse duration as a function of the number of round trips simulated for the small-signal gain corresponding to g0l = 0.5 and total GDD of β2t = −5,532 fs2. (c) The initially prepared and final stabilized temporal pulse profiles and (d) laser spectra. GDD, group-delay dispersion.

In order to simulate the pump power dependence of soliton-like pulse characteristics, the pulse formation was simulated at the total GDD condition of β2t = −5,532 fs2 by varying the value of g0l. As shown in Fig. 3, the pulse energy increases more or less exponentially with the small-signal gain. In addition, as the small-signal gain increases, the pulse duration is shortened. According to Eq. (1), the product of the pulse energy and pulse duration is constant in ideal soliton-like lasers. The simulated product of the pulse energy and pulse duration, plotted with a dotted line in Fig. 3, matches the soliton model for g0l below 0.5 well but increases for g0l beyond 0.5. We think that this tendency at high g0l can result from such effects of limited saturated absorption at SESAM, the pulse broadening due to the finite gain bandwidth, and the discrete arrangement of the GDD and SPM in each optical element.

Figure 3.Simulated pulse energy (solid line) and duration (dashed line) of soliton-like pulses as a function of g0l at a total negative GDD of β2t = −5,532 fs2. The dotted line is the product of the pulse energy and pulse duration. GDD, group-delay dispersion.

Next, we discuss on the effect of the GDD on the pulse characteristics. Figure 4 shows the pulse duration and energy as a function of total round-trip GDD for a fixed small-signal gain condition of g0l= 0.5 at three different SPM parameters of 0.9κ0, 1.0κ0, and 1.1κ0, where κ0 = 2γl is the estimated SPM parameter. It is found that the pulse energy does not change much either with the round-trip GDD or with the SPM parameter. However, the pulse duration changes nearly linearly to the amount of negative GDD for all SPM parameters. This behavior is consistent with Eq. (1) for ideal soliton-like lasers. Regarding the dependence on the SPM parameter, the shorter pulse formation at larger SPM is also consistent with the ideal soliton-like laser model. Though the pulse duration is monotonically shortened with a smaller value of GDD in the calculated region, this trend is limited by the pulse breaking and the finite gain bandwidth.

Figure 4.Simulated pulse energy and duration of soliton-like pulses as a function of total round-trip GDD at a fixed small-signal gain of g0l = 0.50 and three different SPM parameters of 0.9κ0, 1.0κ0, and 1.1κ0, where κ0 = 2γl is the estimated SPM parameter. GDD, group-delay dispersion; SPM, self-phase modulation.

For the case of ideal soliton-like mode-locking, the value of κEpτs / |β2t| is invariant at 3.53. In Fig. 5, we plot the numerically obtained values of κEpτs / |β2t| as a function of negative GDD for the three SPM parameters. Because the pulse energy Ep just before the output coupler drops to 0.8 Ep after the output coupler, the effective pulse energy was assumed to be 0.9 Ep. The simulated values are slightly larger than 3.53, exhibiting more deviations at small negative GDDs. It is our opinion that this GDD dependence is related with the finite gain bandwidth, in that the pulse broadening due to the gain bandwidth becomes severer as the pulse duration is shortened at small negative GDDs.

Figure 5.Numerically obtained values of κEpτs/|β2t| as a function of negative GDD for three SPM parameters, 0.9κ0, 1.0κ0, and 1.1κ0 (κ0 = 2γl). The dashed line indicates the constant value expected from an ideal soliton-like laser. GDD, group-delay dispersion; SPM, self-phase modulation.

We now compare the results of the numerical simulations with a previous experimental work [25]. In the previous work, we reported on a SESAM mode-locked Yb:KGW laser producing femtosecond pulses at around 1,028 nm at a repetition rate of 60 MHz with an average power of around 6 W. As the pulse energy and duration were measured while changing the pump power, the pulse duration was inversely proportional to the pulse energy, as is plotted with symbols in Fig. 6(a). We need to note that in the experiment, multiple pulses of double and triple pulses per round trip were generated as the pump power was increased. For those cases, the pulse energy was assumed to be evenly distributed between pulses. The comparison in Fig. 6(a) indicates that the numerical simulation explains the experimentally acquired pulse duration versus energy curve well. In another experimental work [26], we characterized the mode-locked pulses while varying the negative GDD by changing the number of GTI mirror bounces in the cavity. There, the pump power was adjusted to the point of maximum single pulse per round trip operation. In Fig. 6(b), the experimental results with different GDDs are plotted along with the simulated pulse duration versus energy curves under the same GDD conditions as the experiment. The overall correspondence between the experiment and the simulation is relatively good, with both exhibiting shorter pulses for small negative GDD. But the slight discrepancies are possibly due to the combined effects of the inhomogeneous beam profile, uncertainties in the adopted optical parameters, and so on. These comparisons, showing that the numerical simulation explains the experimental results of the pulse energy and GDD dependence well, demonstrate that the modelling procedure and the parameters in the simulation are reliable, and thus this work can be extended to the analysis of other diode-pumped mode-locked SSLs.

Figure 6.Comparison with experimental results. (a) Comparison of the numerical simulation (dashed line) with experimental results (symbols) from [25] for the pulse duration versus energy at the GDD condition of β2t = −5,532 fs2. (b) Comparison of the simulation with experimental results (symbols) from [26] obtained at four different GDDs. GDD, group-delay dispersion.

### III. CONCLUSIONS

We numerically simulated the pulse formation in a mode-locked Yb:KGW laser with a deliberate consideration of each optical process occurring in the cavity, including the nonlinear SPM and optical gain in the Yb:KGW crystal, the GDD in dispersive mirrors, and the SAM in the SESAM. The simulation shows that the pulse duration is inversely proportional to the pulse energy and is almost linearly proportional to the negative GDD, closely following the ideal soliton-like lasing model. The good correspondence with previous experimental results obtained with the corresponding laser parameters supports the accuracy and reliability of the numerical method, which may be improved if the parameters in the simulation are estimated more rigorously. We expect that the numerical method in this study will be fruitful in interpreting and predicting the performance of other diode-pumped SSLs.

### DISCLOSURES

The authors declare no conflicts of interest.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

National Research Foundation of Korea (NRF-2020 R1A2C1008368, NRF-2020R1A6A1A03047771).

### References

1. M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast lasers: Technology and Applications, 1st ed. (CRC Press, 2003).
2. U. Keller, “Ultrafast solid-state laser oscillators: a success story for the last 20 years with no end in sight,” Appl. Phys. B 100, 15-28 (2010).
3. W. Sibbett, A. A. Lagatsky, and C. T. A. Brown, “The development and application of femtosecond laser systems,” Opt. Express 20, 6989-7001 (2012).
4. K. Sugioka and Y. Cheng, “Ultrafast lasers—reliable tools for advanced materials processing,” Light Sci. Appl. 3, e149 (2014).
5. F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2, 932-940 (2005).
6. J. Koch, E. Fadeeva, M. Engelbrecht, C. Ruffert, H. H. Gatzen, A. Ostendorf, and B. N. Chichkov, “Maskless nonlinear lithography with femtosecond laser pulses,” Appl. Phys. A 82, 23-26 (2006).
7. A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015-1047 (2005).
8. D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti: sapphire laser,” Opt. Lett. 16, 42-44 (1991).
9. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411-413 (1999).
10. F. Brunner, G. Spühler, J. A. der Au, L. Krainer, F. Morier-Genoud, R. Paschotta, N. Lichtenstein, S. Weiss, C. Harder, A. A. Lagatsky, A. Abdolvand, N. V. Kuleshov, and U. Keller, “Diode-pumped femtosecond Yb: KGd (WO4)2 laser with 1.1-W average power,” Opt. Lett. 25, 1119-1121 (2000).
11. A.-L. Calendron, K. Wentsch, and M. Lederer, “High power cw and mode-locked oscillators based on Yb: KYW multi-crystal resonators,” Opt. Express 16, 18838-18843 (2008).
12. G. H. Kim, U. Kang, D. Heo, V. E. Yashin, A. V. Kulik, E. G. Sall’, and S. A. Chizhov, “A compact femtosecond generator based on an Yb:KYW crystal with direct laser-diode pumping,” J. Opt. Technol. 77, 225-229 (2010).
13. H. Zhao and A. Major, “Powerful 67 fs Kerr-lens mode-locked prismless Yb:KGW oscillator,” Opt. Express 21, 31846-31851 (2013).
14. B. Zhou, Z. Wei, Y. Zou, Y. Zhang, X. Zhong, G. L. Bourdet, and J. Wang, “High-efficiency diode-pumped femtosecond Yb:YAG ceramic laser,” Opt. Lett. 35, 288-290 (2010).
15. S. Manjooran and A. Major, “Diode-pumped 45 fs Yb:CALGO laser oscillator with 1.7 MW of peak power,” Opt. Lett. 43, 2324-2327 (2018).
16. J. Yang, Z. Wang, J. Song, X. Wang, R. Lü, J. Zhu, and Z. Wei, “Diode-pumped 13 W Yb:KGW femtosecond laser,” Chin. Opt. Lett. 20, 021404 (2022).
17. Y. Wang, X. Su, Y. Xie, F. Gao, S. Kumar, Q. Wang, C. Liu, B. Zhang, B. Zhang, and J. He, “17.8 fs broadband Kerr-lens mode-locked Yb:CALGO oscillator,” Opt. Lett. 46, 1892-1895 (2021).
18. T. Brabec, C. Spielmann, and F. Krausz, “Mode locking in solitary lasers,” Opt. Lett. 16, 1961-1963 (1991).
19. F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097-2122 (1992).
20. C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100-1114 (1994).
21. F. X. Kartner, J. A. der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-What’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
22. M. Lederer, B. Luther-Davies, H. Tan, C. Jagadish, N. Akhmediev, and J. Soto-Crespo, “Multipulse operation of a Ti:sapphire laser mode locked by an ion-implanted semiconductor saturable- laser oscillator with 1.7 MW of peak power,” J. Opt. Soc. Am. B 16, 895-904 (1999).
23. V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
24. Y. H. Cha, J. M. Han, and Y. J. Rhee, “Effect of discrete distribution of dispersion and self-phase modulation in a ∼10-fs Ti:sapphire laser,” Appl. Phys. B 74, s283-s289 (2002).
25. B.-J. Park, J.-Y. Song, S.-Y. Lee, D.-Y. Kim, M. Y. Jeon, and K. J. Yee, “Behaviors of multiple pulsing in high power saturable-absorber mode-locked Yb:KGW laser,” Opt. Commun. 527, 128968 (2023).
26. B.-J. Park, J.-Y. Song, S.-Y. Lee, and K.-J. Yee, “High-power SESAM mode-locked Yb:KGW Laser with different group-velocity dispersions,” Curr. Opt. Photonics 6, 407-412 (2022).
27. H. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975).
28. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156-158 (1984).

### Article

#### Article

Curr. Opt. Photon. 2023; 7(1): 90-96

Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.90

## Numerical Simulation of Soliton-like Pulse Formation in Diode-pumped Yb-doped Solid-state Lasers

Seong-Yeon Lee1, Byeong-Jun Park1, Seong-Hoon Kwon2, Ki-Ju Yee1

1Department of Physics, Chungnam National University, Daejeon 34134, Korea
2Pohang Accelerator Laboratory, Pohang 37673, Korea

Correspondence to:*kyee@cnu.ac.kr, ORCID 0000-0002-1076-2354

Received: October 26, 2022; Revised: December 27, 2022; Accepted: December 28, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We numerically solve the nonlinear Schrӧdinger equation for pulse propagation in a passively mode-locked Yb:KGW laser. The soliton-like pulse formation as a result of balanced negative group-delay dispersion (GDD) and nonlinear self-phase modulation is analyzed. The cavity design and optical parameters of a previously reported high-power Yb:KGW laser were adopted to compare the simulation results with experimental results. The pulse duration and energy obtained by varying the small-signal gain or GDD reproduce the overall tendency observed in the experiments, demonstrating the reliability and accuracy of the model simulation and the optical parameters.

Keywords: Femtosecond laser, Mode-locking, Soliton-like pulse formation, Yb:KGW laser

### I. INTRODUCTION

Ultrashort lasers on the femtosecond scale are widely used in a variety fields such as material processing, nonlinear optical imaging and lithography, medical applications, ultrafast phenomena diagnosis, and so on [17]. Though the laser requirements may differ from application to application, shorter and more intense pulses are generally in demand. The Kerr lens mode-locking of a Ti:sapphire laser demonstrated by Spence et al. [8] can be seen as a milestone in the development of femtosecond solid-state lasers (SSLs). Because of very broad gain, few-cycle pulses down to 5 fs have been achieved with Ti:sapphire lasers [9]. However, Ti:sapphire lasers are generally pumped by comparatively expensive and bulky systems such as frequency-doubled Nd:YVO4 lasers. In contrast, femtosecond Yb-doped SSLs operating at around 1,030 nm can be directly pumped by commercially available diode lasers at around 980 nm and thus have merits in achieving high-power systems with reduced space and cost [1015]. Recently, a diode-pumped Yb:KGW laser was reported to produce femtosecond pulses with an average power of 13 W [16]. In terms of the pulse duration, 17.8 fs pulses were generated from a Yb:CALGO laser [17].

Mode-locked SSLs generally operate in the negative group-delay dispersion (GDD) condition, for which a soliton-like pulse can be formed through the combined effect of the GDD and the self-phase modulation (SPM) [17, 18]. Because the nonlinear SPM depends on the pulse energy while the GDD is a linear parameter, the balancing between the two effects is reached at a specific pulse energy for a predefined pulse shape. By solving the nonlinear Schrӧdinger equation, which governs the pulse propagation in a laser cavity, the soliton-like pulse formation in mode-locked lasers has been studied both theoretically and in combination with experiments. Since the soliton-like pulse formation has been mostly studied for Ti:sapphire lasers, its application to Yb-doped SSLs is rare and worthy to carry out [1924]. We need to note that the beam spot size in the crystal is relatively large for Yb-doped SSLs pumped by diode lasers coupled to multi-mode fibers, and that the nonlinear SPM is weak in comparison to tightly focused Ti:sapphire lasers. Thus, rigorous analysis of the pulse formation in a diode-pumped high-power SSL will aid the interpretation and upgrade of diode-pumped femtosecond laser systems.

In this paper, we report on the numerical simulation of soliton-like pulse formation in a femtosecond Yb:KGW laser that operates in the negative GDD regime and uses a semiconductor saturable absorber (SESAM) for passive mode-locking. The solution to the nonlinear Schrӧdinger equation was found after considering the negative GDD at chirped mirrors, saturable absorption at SESAM, and the dispersion and SPM in gain crystal. To validate the reliability of the model simulation through direct comparison with previous experiments [25, 26], we deliberately chose the optical parameters that closely match those used in the experiment. Relatively good correspondence was obtained between the simulation and experimental results in terms of pulse energy and GDD dependence.

### 2.1. Soliton-like Pulse Formation Model

Soliton-like pulses are formed from the interplay between negative GDD and nonlinear SPM in passively mode-locked SSLs. For the ideal soliton-like laser, pulses with a sech2 temporal profile are generated, where the pulse duration, τs, is given by [20, 21]

$τs=3.53β2tκΕp$

Here, β2t is the total intracavity GDD, κ is the SPM parameter, and Ep is the pulse energy. According to Eq. (1), the duration of the soliton-like pulse is linearly proportional to the negative GDD and inversely proportional to the SPM coefficient. However, the pulse characteristics deviate from the equation in practical lasers because the GDD and SPM are not evenly distributed throughout the laser cavity. Taking into account the discrete contributions of the GDD and SPM from each optical component used in a Yb:KGW laser, we numerically simulate the pulse evolution as it repeats round-trip propagations in the cavity. For the cavity design, cavity mode, and optical element parameters, we adopt those used in our previous experimental work [25, 26]. There, the gain crystal of 3-mm-thick 3.6% Yb-ion doped Yb:KGW crystal is pumped by a 981 nm laser diode, and the cavity, with a standard X-shaped configuration, includes a 20% transmission output coupler, a SESAM with relaxation time of 800 fs, and Gires-Tournois-interferometer (GTI) mirrors for negative GDD.

The schematic in Fig. 1 shows the optical elements and pulse propagation in the cavity. The cavity is composed of GTI mirrors, a SESAM with slow self-amplitude modulation (SAM), a Yb:KGW crystal, and an output coupling mirror. In one round trip, the pulse encounters six segmental changes as denoted by the numbers in Fig. 1. In the modelling, we include each step in the sequence of propagation and find the soliton state for which the pulse shape and energy is stable upon iterations.

Figure 1. Schematic of a linear laser cavity with a Yb:KGW gain crystal, negative GDD GTI mirrors, SESAM with slow SAM, and output coupler. The Yb:KGW crystal also provides positive GDD, SPM, and fast SAM. GDD, group-delay dispersion; GTI, Gires-Tournois-interferometer; SESAM, semiconductor saturable absorber; SAM, self-amplitude modulation.

The Yb:KGW crystal provides the gain, SPM, positive GDD, and fast SAM. The pulse propagation through the crystal corresponding to steps 3 and 5 can be expressed by [27, 28]

$∂az, t∂z=g1+1Ωg2∂2∂t2az, t+−i2β2c∂2∂t2+16β3c∂3∂t3az, t+iγa2az, t+ηa2az, t$

where a(z, t) is the normalized envelope field such that |a|2 is the power, g and Ωg are the saturated gain and the gain bandwidth, respectively, β2c and β3c are the group velocity dispersion and third-order dispersion of the crystal, respectively, γ is the SPM parameter, and η is the SAM parameter provided by the Kerr effect in the gain crystal. We approximate the saturated gain as follows [27]:

$g=g01+EP/PgTI$

where g0 is the small-signal gain, EP the integrated pulse energy defined by $EPt=∫−∞t |a t'|2 dt'$, TI the time duration between pulses at the crystal, and Pg is the saturation power given by Pg = Aeff × ℏω/στ with the effective mode area in the crystal Aeff, the emission cross section σ, and the upper state lifetime τ. The SPM coefficient is given by γ = 2πn2/λAeff, with n2 being the nonlinear refractive index of the crystal. In the experimental work [25, 26], positive power enhancement was observed with the mode-locking due to the onset of Kerr lensing at the crystal. To implement this effect in the simulation, we add the fast SAM parameter of the crystal, η, which is proportional to the SPM parameter and is assumed to be 0.01 × γ.

While the pulse propagation in the crystal is described by Eq. (1), an abrupt loss at the output coupler can be expressed by a(z+, t) = rOC × a(z, t), with z+ (z) indicating the position after (before) the output coupler with the reflectivity rOC. The role of GDD provided by GTI mirrors, β2m, can be easily implemented in the Fourier domain of a(z, t) by multiplying exp[iβ2m (ωω0 )2] to a(z, ω). The effect of the SESAM, which can be regarded as a slow saturable absorber, is implemented by multiplying exp[−q(t)] to a(z, t), where q(t) satisfies Eq. (4):

$∂q∂t=−q−q0TA−a2EAq$

where q0 is the saturable absorption loss, and TA and EA are the relaxation time and saturation energy of the SESAM, respectively. Here, EA = Isat × ASESAM is obtained from the saturation intensity Isat and the effective beam area at the SESAM, ASESAM. The parameters used in the simulation are listed in Table 1. As the initial pulse, we use a sech2 pulse with a pulse duration of 1 ps.

Parameters used in the numerical simulations.

 Component Parameter Value Description Yb:KGW Crystal l 3 mm Length σ 3 × 10−20 cm2 Emission Cross Section τ 0.36 ms Upper State Lifetime n2 20 × 10−16 cm2/W Nonlinear Refractive Index Ωg 35.7 THz Gain Bandwidth β2c 178 fs2/mm Group-velocity Dispersion β3c 236 fs3/mm Third-order Dispersion g0 Variable Small-signal Gain Aeff 2.0 × 10−4 cm2 Effective Mode Area TI 8.35 ns Inter-pulse Time GTI Mirrors β2m −550 × n fs2 GDD at GTI Mirrors (n: No. of Bounces) SESAM q0 0.0132 Saturable Absorption Loss TA 800 fs Relaxation Time Isat 120 mJ/cm2 Saturation Intensity ASESAM 6.0 × 10−4 cm2 Mode Area at SESAM Output Coupler rOC 0.894 Output Coupler Reflectivity

### 2.2. Simulation Results and Comparison with Experiment

To get an insight into the soliton-like pulse formation under different conditions, we carried out numerical simulations tuning the small-signal gain, the round trip GDD, and the SPM parameter. In a previous study [25], we experimentally demonstrated mode-locked pulses from a Yb:KGW oscillator in which the negative GDD was provided by using GTI mirrors. Including the positive GDD in the Yb:KGW crystal, the total GDD, β2t, was −5,532 fs2 at the condition of six GTI mirror bounces. Figures 2(a) and 2(b) show how the pulse energy and duration are stabilized with the round trips of the laser cavity under the small-signal gain condition corresponding to g0l = 0.5 with l being the crystal length and for the total GDD of β2t = −5,532 fs2. The pulse duration exhibits damped oscillations with the number of round trips, which is stabilized after about 600 round trips. This oscillatory behavior at early times originates from the nonlinear nature of the model equation, Eq. (2). Figure 2(c) shows both the initially assumed temporal profile and the one finally reached after 2,000 round trips, while Fig. 2(d) shows the corresponding laser spectra. We need to mention that the soliton-like pulse is stabilized faster as the larger value of the SAM parameter of the gain crystal is assumed.

Figure 2. Pulse evolution with round trips. (a) Pulse energy and (b) pulse duration as a function of the number of round trips simulated for the small-signal gain corresponding to g0l = 0.5 and total GDD of β2t = −5,532 fs2. (c) The initially prepared and final stabilized temporal pulse profiles and (d) laser spectra. GDD, group-delay dispersion.

In order to simulate the pump power dependence of soliton-like pulse characteristics, the pulse formation was simulated at the total GDD condition of β2t = −5,532 fs2 by varying the value of g0l. As shown in Fig. 3, the pulse energy increases more or less exponentially with the small-signal gain. In addition, as the small-signal gain increases, the pulse duration is shortened. According to Eq. (1), the product of the pulse energy and pulse duration is constant in ideal soliton-like lasers. The simulated product of the pulse energy and pulse duration, plotted with a dotted line in Fig. 3, matches the soliton model for g0l below 0.5 well but increases for g0l beyond 0.5. We think that this tendency at high g0l can result from such effects of limited saturated absorption at SESAM, the pulse broadening due to the finite gain bandwidth, and the discrete arrangement of the GDD and SPM in each optical element.

Figure 3. Simulated pulse energy (solid line) and duration (dashed line) of soliton-like pulses as a function of g0l at a total negative GDD of β2t = −5,532 fs2. The dotted line is the product of the pulse energy and pulse duration. GDD, group-delay dispersion.

Next, we discuss on the effect of the GDD on the pulse characteristics. Figure 4 shows the pulse duration and energy as a function of total round-trip GDD for a fixed small-signal gain condition of g0l= 0.5 at three different SPM parameters of 0.9κ0, 1.0κ0, and 1.1κ0, where κ0 = 2γl is the estimated SPM parameter. It is found that the pulse energy does not change much either with the round-trip GDD or with the SPM parameter. However, the pulse duration changes nearly linearly to the amount of negative GDD for all SPM parameters. This behavior is consistent with Eq. (1) for ideal soliton-like lasers. Regarding the dependence on the SPM parameter, the shorter pulse formation at larger SPM is also consistent with the ideal soliton-like laser model. Though the pulse duration is monotonically shortened with a smaller value of GDD in the calculated region, this trend is limited by the pulse breaking and the finite gain bandwidth.

Figure 4. Simulated pulse energy and duration of soliton-like pulses as a function of total round-trip GDD at a fixed small-signal gain of g0l = 0.50 and three different SPM parameters of 0.9κ0, 1.0κ0, and 1.1κ0, where κ0 = 2γl is the estimated SPM parameter. GDD, group-delay dispersion; SPM, self-phase modulation.

For the case of ideal soliton-like mode-locking, the value of κEpτs / |β2t| is invariant at 3.53. In Fig. 5, we plot the numerically obtained values of κEpτs / |β2t| as a function of negative GDD for the three SPM parameters. Because the pulse energy Ep just before the output coupler drops to 0.8 Ep after the output coupler, the effective pulse energy was assumed to be 0.9 Ep. The simulated values are slightly larger than 3.53, exhibiting more deviations at small negative GDDs. It is our opinion that this GDD dependence is related with the finite gain bandwidth, in that the pulse broadening due to the gain bandwidth becomes severer as the pulse duration is shortened at small negative GDDs.

Figure 5. Numerically obtained values of κEpτs/|β2t| as a function of negative GDD for three SPM parameters, 0.9κ0, 1.0κ0, and 1.1κ0 (κ0 = 2γl). The dashed line indicates the constant value expected from an ideal soliton-like laser. GDD, group-delay dispersion; SPM, self-phase modulation.

We now compare the results of the numerical simulations with a previous experimental work [25]. In the previous work, we reported on a SESAM mode-locked Yb:KGW laser producing femtosecond pulses at around 1,028 nm at a repetition rate of 60 MHz with an average power of around 6 W. As the pulse energy and duration were measured while changing the pump power, the pulse duration was inversely proportional to the pulse energy, as is plotted with symbols in Fig. 6(a). We need to note that in the experiment, multiple pulses of double and triple pulses per round trip were generated as the pump power was increased. For those cases, the pulse energy was assumed to be evenly distributed between pulses. The comparison in Fig. 6(a) indicates that the numerical simulation explains the experimentally acquired pulse duration versus energy curve well. In another experimental work [26], we characterized the mode-locked pulses while varying the negative GDD by changing the number of GTI mirror bounces in the cavity. There, the pump power was adjusted to the point of maximum single pulse per round trip operation. In Fig. 6(b), the experimental results with different GDDs are plotted along with the simulated pulse duration versus energy curves under the same GDD conditions as the experiment. The overall correspondence between the experiment and the simulation is relatively good, with both exhibiting shorter pulses for small negative GDD. But the slight discrepancies are possibly due to the combined effects of the inhomogeneous beam profile, uncertainties in the adopted optical parameters, and so on. These comparisons, showing that the numerical simulation explains the experimental results of the pulse energy and GDD dependence well, demonstrate that the modelling procedure and the parameters in the simulation are reliable, and thus this work can be extended to the analysis of other diode-pumped mode-locked SSLs.

Figure 6. Comparison with experimental results. (a) Comparison of the numerical simulation (dashed line) with experimental results (symbols) from [25] for the pulse duration versus energy at the GDD condition of β2t = −5,532 fs2. (b) Comparison of the simulation with experimental results (symbols) from [26] obtained at four different GDDs. GDD, group-delay dispersion.

### III. CONCLUSIONS

We numerically simulated the pulse formation in a mode-locked Yb:KGW laser with a deliberate consideration of each optical process occurring in the cavity, including the nonlinear SPM and optical gain in the Yb:KGW crystal, the GDD in dispersive mirrors, and the SAM in the SESAM. The simulation shows that the pulse duration is inversely proportional to the pulse energy and is almost linearly proportional to the negative GDD, closely following the ideal soliton-like lasing model. The good correspondence with previous experimental results obtained with the corresponding laser parameters supports the accuracy and reliability of the numerical method, which may be improved if the parameters in the simulation are estimated more rigorously. We expect that the numerical method in this study will be fruitful in interpreting and predicting the performance of other diode-pumped SSLs.

### DISCLOSURES

The authors declare no conflicts of interest.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

### FUNDING

National Research Foundation of Korea (NRF-2020 R1A2C1008368, NRF-2020R1A6A1A03047771).

### Fig 1.

Figure 1.Schematic of a linear laser cavity with a Yb:KGW gain crystal, negative GDD GTI mirrors, SESAM with slow SAM, and output coupler. The Yb:KGW crystal also provides positive GDD, SPM, and fast SAM. GDD, group-delay dispersion; GTI, Gires-Tournois-interferometer; SESAM, semiconductor saturable absorber; SAM, self-amplitude modulation.
Current Optics and Photonics 2023; 7: 90-96https://doi.org/10.3807/COPP.2023.7.1.90

### Fig 2.

Figure 2.Pulse evolution with round trips. (a) Pulse energy and (b) pulse duration as a function of the number of round trips simulated for the small-signal gain corresponding to g0l = 0.5 and total GDD of β2t = −5,532 fs2. (c) The initially prepared and final stabilized temporal pulse profiles and (d) laser spectra. GDD, group-delay dispersion.
Current Optics and Photonics 2023; 7: 90-96https://doi.org/10.3807/COPP.2023.7.1.90

### Fig 3.

Figure 3.Simulated pulse energy (solid line) and duration (dashed line) of soliton-like pulses as a function of g0l at a total negative GDD of β2t = −5,532 fs2. The dotted line is the product of the pulse energy and pulse duration. GDD, group-delay dispersion.
Current Optics and Photonics 2023; 7: 90-96https://doi.org/10.3807/COPP.2023.7.1.90

### Fig 4.

Figure 4.Simulated pulse energy and duration of soliton-like pulses as a function of total round-trip GDD at a fixed small-signal gain of g0l = 0.50 and three different SPM parameters of 0.9κ0, 1.0κ0, and 1.1κ0, where κ0 = 2γl is the estimated SPM parameter. GDD, group-delay dispersion; SPM, self-phase modulation.
Current Optics and Photonics 2023; 7: 90-96https://doi.org/10.3807/COPP.2023.7.1.90

### Fig 5.

Figure 5.Numerically obtained values of κEpτs/|β2t| as a function of negative GDD for three SPM parameters, 0.9κ0, 1.0κ0, and 1.1κ0 (κ0 = 2γl). The dashed line indicates the constant value expected from an ideal soliton-like laser. GDD, group-delay dispersion; SPM, self-phase modulation.
Current Optics and Photonics 2023; 7: 90-96https://doi.org/10.3807/COPP.2023.7.1.90

### Fig 6.

Figure 6.Comparison with experimental results. (a) Comparison of the numerical simulation (dashed line) with experimental results (symbols) from [25] for the pulse duration versus energy at the GDD condition of β2t = −5,532 fs2. (b) Comparison of the simulation with experimental results (symbols) from [26] obtained at four different GDDs. GDD, group-delay dispersion.
Current Optics and Photonics 2023; 7: 90-96https://doi.org/10.3807/COPP.2023.7.1.90

Table 1 Parameters used in the numerical simulations

 Component Parameter Value Description Yb:KGW Crystal l 3 mm Length σ 3 × 10−20 cm2 Emission Cross Section τ 0.36 ms Upper State Lifetime n2 20 × 10−16 cm2/W Nonlinear Refractive Index Ωg 35.7 THz Gain Bandwidth β2c 178 fs2/mm Group-velocity Dispersion β3c 236 fs3/mm Third-order Dispersion g0 Variable Small-signal Gain Aeff 2.0 × 10−4 cm2 Effective Mode Area TI 8.35 ns Inter-pulse Time GTI Mirrors β2m −550 × n fs2 GDD at GTI Mirrors (n: No. of Bounces) SESAM q0 0.0132 Saturable Absorption Loss TA 800 fs Relaxation Time Isat 120 mJ/cm2 Saturation Intensity ASESAM 6.0 × 10−4 cm2 Mode Area at SESAM Output Coupler rOC 0.894 Output Coupler Reflectivity

### References

1. M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast lasers: Technology and Applications, 1st ed. (CRC Press, 2003).
2. U. Keller, “Ultrafast solid-state laser oscillators: a success story for the last 20 years with no end in sight,” Appl. Phys. B 100, 15-28 (2010).
3. W. Sibbett, A. A. Lagatsky, and C. T. A. Brown, “The development and application of femtosecond laser systems,” Opt. Express 20, 6989-7001 (2012).
4. K. Sugioka and Y. Cheng, “Ultrafast lasers—reliable tools for advanced materials processing,” Light Sci. Appl. 3, e149 (2014).
5. F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2, 932-940 (2005).
6. J. Koch, E. Fadeeva, M. Engelbrecht, C. Ruffert, H. H. Gatzen, A. Ostendorf, and B. N. Chichkov, “Maskless nonlinear lithography with femtosecond laser pulses,” Appl. Phys. A 82, 23-26 (2006).
7. A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81, 1015-1047 (2005).
8. D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti: sapphire laser,” Opt. Lett. 16, 42-44 (1991).
9. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411-413 (1999).
10. F. Brunner, G. Spühler, J. A. der Au, L. Krainer, F. Morier-Genoud, R. Paschotta, N. Lichtenstein, S. Weiss, C. Harder, A. A. Lagatsky, A. Abdolvand, N. V. Kuleshov, and U. Keller, “Diode-pumped femtosecond Yb: KGd (WO4)2 laser with 1.1-W average power,” Opt. Lett. 25, 1119-1121 (2000).
11. A.-L. Calendron, K. Wentsch, and M. Lederer, “High power cw and mode-locked oscillators based on Yb: KYW multi-crystal resonators,” Opt. Express 16, 18838-18843 (2008).
12. G. H. Kim, U. Kang, D. Heo, V. E. Yashin, A. V. Kulik, E. G. Sall’, and S. A. Chizhov, “A compact femtosecond generator based on an Yb:KYW crystal with direct laser-diode pumping,” J. Opt. Technol. 77, 225-229 (2010).
13. H. Zhao and A. Major, “Powerful 67 fs Kerr-lens mode-locked prismless Yb:KGW oscillator,” Opt. Express 21, 31846-31851 (2013).
14. B. Zhou, Z. Wei, Y. Zou, Y. Zhang, X. Zhong, G. L. Bourdet, and J. Wang, “High-efficiency diode-pumped femtosecond Yb:YAG ceramic laser,” Opt. Lett. 35, 288-290 (2010).
15. S. Manjooran and A. Major, “Diode-pumped 45 fs Yb:CALGO laser oscillator with 1.7 MW of peak power,” Opt. Lett. 43, 2324-2327 (2018).
16. J. Yang, Z. Wang, J. Song, X. Wang, R. Lü, J. Zhu, and Z. Wei, “Diode-pumped 13 W Yb:KGW femtosecond laser,” Chin. Opt. Lett. 20, 021404 (2022).
17. Y. Wang, X. Su, Y. Xie, F. Gao, S. Kumar, Q. Wang, C. Liu, B. Zhang, B. Zhang, and J. He, “17.8 fs broadband Kerr-lens mode-locked Yb:CALGO oscillator,” Opt. Lett. 46, 1892-1895 (2021).
18. T. Brabec, C. Spielmann, and F. Krausz, “Mode locking in solitary lasers,” Opt. Lett. 16, 1961-1963 (1991).
19. F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097-2122 (1992).
20. C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100-1114 (1994).
21. F. X. Kartner, J. A. der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-What’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
22. M. Lederer, B. Luther-Davies, H. Tan, C. Jagadish, N. Akhmediev, and J. Soto-Crespo, “Multipulse operation of a Ti:sapphire laser mode locked by an ion-implanted semiconductor saturable- laser oscillator with 1.7 MW of peak power,” J. Opt. Soc. Am. B 16, 895-904 (1999).
23. V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
24. Y. H. Cha, J. M. Han, and Y. J. Rhee, “Effect of discrete distribution of dispersion and self-phase modulation in a ∼10-fs Ti:sapphire laser,” Appl. Phys. B 74, s283-s289 (2002).
25. B.-J. Park, J.-Y. Song, S.-Y. Lee, D.-Y. Kim, M. Y. Jeon, and K. J. Yee, “Behaviors of multiple pulsing in high power saturable-absorber mode-locked Yb:KGW laser,” Opt. Commun. 527, 128968 (2023).
26. B.-J. Park, J.-Y. Song, S.-Y. Lee, and K.-J. Yee, “High-power SESAM mode-locked Yb:KGW Laser with different group-velocity dispersions,” Curr. Opt. Photonics 6, 407-412 (2022).
27. H. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975).
28. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156-158 (1984).

Wonshik Choi,
Editor-in-chief