Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2023; 7(1): 90-96
Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.90
Copyright © Optical Society of Korea.
Seong-Yeon Lee1, Byeong-Jun Park1, Seong-Hoon Kwon2, Ki-Ju Yee1
Corresponding author: *kyee@cnu.ac.kr, ORCID 0000-0002-1076-2354
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
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 [1–7]. 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
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 [19–24]. 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.
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,
Here,
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.
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]
where
where
While the pulse propagation in the crystal is described by Eq. (1), an abrupt loss at the output coupler can be expressed by
where
Table 1 Parameters used in the numerical simulations
Component | Parameter | Value | Description |
Yb:KGW Crystal | 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 |
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,
In order to simulate the pump power dependence of soliton-like pulse characteristics, the pulse formation was simulated at the total GDD condition of
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
For the case of ideal soliton-like mode-locking, the value of κ
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.
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.
The authors declare no conflicts of interest.
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).
Curr. Opt. Photon. 2023; 7(1): 90-96
Published online February 25, 2023 https://doi.org/10.3807/COPP.2023.7.1.90
Copyright © Optical Society of Korea.
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
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
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 [1–7]. 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
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 [19–24]. 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.
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,
Here,
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.
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]
where
where
While the pulse propagation in the crystal is described by Eq. (1), an abrupt loss at the output coupler can be expressed by
where
Table 1 . Parameters used in the numerical simulations.
Component | Parameter | Value | Description |
Yb:KGW Crystal | 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 |
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,
In order to simulate the pump power dependence of soliton-like pulse characteristics, the pulse formation was simulated at the total GDD condition of
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
For the case of ideal soliton-like mode-locking, the value of κ
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.
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.
The authors declare no conflicts of interest.
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).
Table 1 Parameters used in the numerical simulations
Component | Parameter | Value | Description |
Yb:KGW Crystal | 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 |