Dynamics Simulation of Self-Mode-Locking in a Semiconductor Disk Laser Using Delay Differential Equations

: Self-mode-locked semiconductor disk lasers possess compact resonant cavity and stable construction. These devices have a wide application prospect because of their picosecond to sub-picosecond pulse width, excellent beam quality and tailorable emission wavelength. In this paper, dynamics simulations of self-mode-locking in a semiconductor disk laser are performed by using delay differential equations for the ﬁrst time. The corresponding conditions of different modality of mode-locking, including Q-switched mode-locking, continuous-wave mode-locking and harmonic mode-locking are calculated, and their dynamics evolution processes are presented. We also analyze the characteristics of the three different mode-locking modalities and summarize their overall dynamics evolution tendency. This kind of numerical simulation and analysis provides an understanding of the dynamics process of self-mode-locking, and may be referenced for related experiments.


Introduction
Semiconductor disk lasers (SDLs) are perfectly suitable for the generation of ultrashort pulses with pulse duration from picosecond to sub-picosecond and a pulse repetition rate at a gigahertz magnitude because of their intrinsic time characteristics of carriers [1][2][3].Modelocked SDLs combine key features such as good beam quality, moderate output power, ultrashort pulse duration, steady pulse amplitude and tailorable emission wavelength, while maintaining their construction simplicity and mechanism stability [4,5].They are of great interest for different applications, including biophotonics [6], spectroscopy [7] and laser cooling and trapping [8].
Since the first passively mode-locked SDL was reported [9], performances of such devices have been greatly improved.The maximum average output power has been increased to 6.4 W [10], the minimum pulse width has been decreased to 60 fs [11], and the repetition rate directly arisen from an oscillator has been pushed to 101.2 GHz [12].Many groups have also done excellent research on the mechanism of passively mode-locking (ML) in a SDL, e.g., the effect of dispersion [13] and the nonequilibrium carrier dynamics [14].Those passively mode-locked solid-state and fiber lasers can produce some wavelengths for specific utilizations, use saturate absorbers made of various new materials, and obtain high single-pulse energy by reducing the repetition rate [15][16][17][18].In contrast, mode-locked and Q-switched semiconductor disk lasers have more flexible and designable emission wavelengths, higher average output power and pulse repetition rate.
In addition to the passive ML based on a semiconductor saturable absorption mirror (SESAM), researchers have found another method for ML in SDLs: self-mode-locking (SML) [19,20].Some experiments of SML require inserting a slit or another Kerr medium Photonics 2022, 9, 859 2 of 17 into a laser resonator [21,22], while other reports of SML do not require placing any additional elements into the cavity [23][24][25].
Currently, it is widely accepted that SML in an SDL originates from the Kerr effect in a semiconductor gain medium [26,27].The Kerr effect means that the refractive index of medium changes nonlinearly with the electric intensity, but linearly with the light intensity of an incident laser, i.e., n = n 0 + n 2 I.Where n 0 is the refractive index without light, I represents the light intensity, and n 2 is the nonlinear refractive index.In solid-state lasers, the refractive index of a crystal depends on the real part of complex dielectric function.Because the response time of polarization in crystal can be almost ignored, the Kerr-lens (caused by the change of refractive index) and the hard (or soft) aperture in the laser cavity constitute a "fast" saturable absorber, which can start the SML and produce a pulse width of femtosecond magnitude [28].
For semiconductor lasers, the situation is slightly different.The nonlinear refractive index of semiconductor gain medium is caused by the change of carrier concentration.As the inter-and intra-band relaxation time of carriers are in the order of nanoseconds and picoseconds [29], the recovery time of the saturable absorber formed by the Kerr-lens and an aperture in a semiconductor laser is not as fast as that in a solid-state laser.In the numerical simulations in this paper, we will consider the Kerr-lens along with an aperture in self-mode-locked SDL as a "slow" saturable absorber, i.e., a saturable absorber with a recovery time similar to a SESAM.
There have been many reports on the numerical analysis of the physical mechanism and dynamic process of SESAM mode-locked SDLs.A fully microscopic model was used to investigate the transverse mode instabilities and the ultrafast nonequilibrium carrier dynamics of electron-hole excitations in a semiconductor quantum-well gain and absorber [14,30,31].The rate equation method was employed to disclose the soliton-like pulse-shaping mechanism and harmonically ML [13,32].An extended master equation approach based on partial differential equations was introduced to describe the complex pulse train and the influence of group velocity dispersion on the stability boundary [33].
The delay differential equations (DDEs), which have been successfully used to describe various dynamics systems, were also utilized to investigate a passively mode-locked SDL.DDEs have been employed to verify the fundamental and soliton bound states in a low repetition rate mode-locked SDL [34], to depict the pulse cluster dynamics [35], and to explain the dispersive instability and ML behavior in an SDL with different cavity configurations [36,37].
As a new method for ML in SDLs, SML has not yet been numerically simulated.In this paper, the DDEs, a powerful mathematical tool suitable for describing dynamics systems, is used to conduct a comprehensive simulation and analysis of the SML in SDLs.This work mainly includes the calculation of the dynamics process of three different ML regions: the Q-switched ML, the continuous-wave (CW) ML and the harmonic ML.A related analysis of condition and the corresponding performance of the above three ML regions are also presented.This kind of numerical simulation and analysis provides some understanding as to the dynamics process of SML and may be referenced for related experiments.

Delay Differential Equations for Passively ML
The dynamics of ML can be well explained by Haus master equations [38].This analytical approach is widely used because of its insight into the underlying mechanism of ML.However, the dynamics of some mode-locked lasers (e.g., fiber lasers and semiconductor lasers) can no longer be described satisfactorily by the master equation because it is based on small changes of gain and losses in a laser pulse shape within one round-trip.In comparison, DDEs, which have been successfully used to describe various dynamics systems, can effectively reproduce the dynamics while circumventing the approximation of small gain and loss.Here we use the DDEs model proposed by A. G. Vladimirov et al. for passively mode-locked semiconductor lasers [39,40].This model avoids the approximation of small gain and loss per cavity round-trip, and is closer to the actual condition of semiconductor laser devices.
The DDEs can be expressed as: (2) where A is the electric field amplitude of the laser.γ is the bandwidth of the spectral filter.κ describes the total nonresonant linear intensity loss per cavity round-trip.α g and α q are linewidth enhancement factors for the gain medium and saturable absorber, respectively.g and q indicate time dependent saturable gain and absorption.g 0 and q 0 denote the initial value of g and q. γ g and γ q are relaxation rate in the gain and absorber, and s is the ratio of saturation energy of the gain and absorber.It should be noted that the delay parameter τ is equal to the cavity round-trip time, and in this work, all times in the above equations are normalized by the cavity round-trip time.

Determination of Parameters
In the subsequent numerical simulations, we try to make parameters consistent with reported or commonly used experimental data as much as possible, so that the numerical simulation results have broader suitability.We choose a linear resonator composed of a distributed Bragg reflector (DBR) at the bottom of the gain chip and an external flat-concave end mirror with a 150 mm curvature radius.The high-reflection coated end mirror has a reflectivity of R 1 = 99.9% at the laser wavelength, and the reflectivity of the bottom DBR for the laser wavelength is about R 2 = 99%.In the DDEs (1)-(3), κ is defined as the fraction of power remaining in the cavity after each round-trip.So, we have κ = √ R 1 R 2 = 0.9945.According to the existing research results, the linewidth enhancement factors α g and α q will slightly change the mode-locked region and pulse width [39][40][41], which is also obvious in theory.There is no real saturable absorber for a self-mode-locked semiconductor disk laser (SDL) discussed in this paper.The "equivalent saturable absorber" utilized in a SDL to start mode-locking is composed of the Kerr lens in the gain medium and the soft aperture provided by the pump spot.In this sense, the gain is also the saturable absorber.Therefore, we approximately assume that α g is equal to α q , thus ignoring their influence on the calculation results, i.e., both α g and α q are chosen to be zero in the simulation [39][40][41].For a generally used SDL with emission wavelength at 1 µm waveband, the active region consists of InGaAs multiple quantum wells, which is similar to that in reference [35], so the value of gain bandwidth can be selected as 10 nm and the corresponding bandwidth of the spectral filter is γ = 200.
Since the radius of curvature of the output coupler is provided to be 150 mm, for a frequently used value of 200 µm diameter of the pump spot, the cavity length meeting the match-condition of the laser and pump spot on the gain chip is about 145 mm.This results in an approximate cavity round-trip time of τ r ~1 ns.Considering that the typical value of the gain recovery time can be chosen as 10 ns, and it should be normalized by τ r as mentioned before, the relaxation rate of gain medium will be γ g = 0.1.As for the value of g 0 , it is determined by referring to the literature [39][40][41].
The parameter selection of a saturable absorber is a little more complicated, because there is no real saturable absorber in a self-mode-locked SDL, but an equal saturable absorber composed of an equivalent Kerr-lens and a soft aperture.With a provided pump power (i.e., a given focal length of the Kerr-lens), the primary mean to change the modulation depth of a saturable absorber is to change the size of the soft aperture.That is, to change the size of overlapping area between the pump and laser spot.To obtain the above modulation, we suppose a pump power of 10 W and a pulse peak power of 50 kW, which correspond to 250 W intracavity circulating power and 250 mW average output power when the transmittance of output coupler is 0.1%, the pulse width is 5 ps and the pulse repetition rate is 1 GHz) [19,20,26,42].First, the nonlinear refractive index n 2 of gain medium under different pump spot sizes is calculated according to the data provided by reference [43].The results are shown by the green dotted line in Figure 1 (the coordinate axis of n 2 is not shown).Then, based on the focal length formula of the Kerr-lens f = ω 2 /(4πn 2 I p L) (where ω is the radium of light spot, I p is the peak light intensity and L is the length of Kerr medium), the focal length of the equivalent Kerr-lens is computed and plotted as the blue solid line in Figure 1.Finally, by using the light field transmission formula and the split-step Fourier method, the amplitude of the light field is calculated when it completes a round-trip in the cavity, and the difference of round-trip loss experienced by the laser beam with (pulse operation) and without (free running) the Kerr-lens is carried out, as shown by the red solid line in Figure 1.
The parameter selection of a saturable absorber is a little more complicated, because there is no real saturable absorber in a self-mode-locked SDL, but an equal saturable absorber composed of an equivalent Kerr-lens and a soft aperture.With a provided pump power (i.e., a given focal length of the Kerr-lens), the primary mean to change the modulation depth of a saturable absorber is to change the size of the soft aperture.That is, to change the size of overlapping area between the pump and laser spot.To obtain the above modulation, we suppose a pump power of 10 W and a pulse peak power of 50 kW, which correspond to 250 W intracavity circulating power and 250 mW average output power when the transmittance of output coupler is 0.1%, the pulse width is 5 ps and the pulse repetition rate is 1 GHz) [19,20,26,42].First, the nonlinear refractive index n2 of gain medium under different pump spot sizes is calculated according to the data provided by reference [43].The results are shown by the green dotted line in Figure 1 (the coordinate axis of n2 is not shown).Then, based on the focal length formula of the Kerr-lens is the length of Kerr medium), the focal length of the equivalent Kerr-lens is computed and plotted as the blue solid line in Figure 1.Finally, by using the light field transmission formula and the split-step Fourier method, the amplitude of the light field is calculated when it completes a round-trip in the cavity, and the difference of round-trip loss experienced by the laser beam with (pulse operation) and without (free running) the Kerr-lens is carried out, as shown by the red solid line in Figure 1.It can be seen from Figure 1 that the nonlinear reflective index n2 increases monotonously with decreasing pump spot size.Its value changes from negative to positive, and it is zero when the pump spot diameter is about 150 μm.Accordingly, when the value of n2 is negative, the absolute value of focal length increases with decreasing pump spot size, but the absolute value of focal length drops with falling pump spot size when n2 is positive.Obviously, the focal length of the Kerr-lens is infinite when n2 equals zero.
The most important parameter, the difference of round-trip loss with and without the Kerr-lens, first goes up with decreasing pump spot, reaches the maximum of 0.7% when the pump spot is about 230 μm, and then comes down with decreasing pump spot.Theoretically, the greater the loss difference, the easier it is to start ML.Therefore, under conditions given in Figure 1, the optimal value of the pump spot should be about 230 μm.The existence of maximum value of loss difference is understandable.On the one hand, a smaller pump spot (i.e., a smaller soft aperture) is favorable for a focused beam because it can introduce more loss to an ordinary beam and provide larger loss difference between It can be seen from Figure 1 that the nonlinear reflective index n 2 increases monotonously with decreasing pump spot size.Its value changes from negative to positive, and it is zero when the pump spot diameter is about 150 µm.Accordingly, when the value of n 2 is negative, the absolute value of focal length increases with decreasing pump spot size, but the absolute value of focal length drops with falling pump spot size when n 2 is positive.Obviously, the focal length of the Kerr-lens is infinite when n 2 equals zero.
The most important parameter, the difference of round-trip loss with and without the Kerr-lens, first goes up with decreasing pump spot, reaches the maximum of 0.7% when the pump spot is about 230 µm, and then comes down with decreasing pump spot.Theoretically, the greater the loss difference, the easier it is to start ML.Therefore, under conditions given in Figure 1, the optimal value of the pump spot should be about 230 µm.The existence of maximum value of loss difference is understandable.On the one hand, a smaller pump spot (i.e., a smaller soft aperture) is favorable for a focused beam because it can introduce more loss to an ordinary beam and provide larger loss difference between the case with and without the Kerr-lens.On the other hand, a smaller pump spot also means a higher pump power density, which will result in a smaller n 2 and thus a longer focal length of the Kerr-lens (because when the pump power density is increased, n 2 will change from negative to positive, and its absolute value will first decrease and then increase [42]).This will raise the loss of focused beam, thus weakening the increase trend of loss difference.So, as can be seen from Figure 1 that the difference of round-trip loss with and without Kerr-lens has a peak value at 230 µm pump spot, whose pump density corresponds to a n 2 about −1.0 × 10 −16 m 2 /W.Further reducing of the pump spot will lead to a gradual decrease of loss difference.When the pump spot is reduced to about 150 µm, n 2 is zero, the focal length of the Kerr-lens is infinite, and the corresponding loss difference is reduced to zero.
The calculated round-trip loss difference of the beam with and without the Kerr-lens in Figure 1 is introduced by the equivalent saturable absorber, and can be regarded as the modulation depth of an actual SESAM.In this sense, parameter q 0 in DDEs for a self-mode-locked SDL can be determined by using the above method.
Obviously, SML in a SDL depends on the equivalent saturable absorber, which relies on the Kerr-lens, i.e., the nonlinear refractive index n 2 .In view of the fact that the nonlinear refractive index n 2 in the semiconductor gain medium is essentially caused by the carrier density, and its recovery time is totally the same as in a SESAM, whose typical value of recovery time is about 10 ps, therefore the recovery time of the equivalent saturable absorber is chosen to be 10 ps.After being normalized by the cavity round-trip time tr of 1 ns, the value of relaxation rate of absorber is γ q = 100.
Parameter s is defined as s = E sat,g /E sat,q , and the saturation energy of gain is selected to be E sat,g = 50 nJ (corresponding to a spot radius in the order of 100 µm) [13].Since the equivalent saturable absorber consists of a Kerr-lens and a soft aperture, the parameter E sat,q in this paper is estimated using the critical power of Kerr effect, which can be expressed as P cr = π(0.61) 2  λ 2 /(8n 0 n 2 ) [43,44].For λ = 1000 nm, GaAs material system n 0 = 3.5, 45], and pulse width of 5 ps, the above P cr corresponds to a saturation energy of about 2 nJ, so the estimated value of the saturation parameter is s = 25.
All parameters used in simulations below are summarized and listed in Table 1.Linewidth enhancement factors for gain medium and the saturable absorber are not included as mentioned before.Parameter g 0 can be changed with various pump intensity.q 0 can also be changed with different pump intensity, because the pump intensity will change the nonlinear refractive index of gain medium, thus changing the focal length of the Kerr-lens and having an influence on the value of q 0 .In addition, q 0 will be obviously affected by the radium of soft aperture.Except g 0 and q 0 , other parameters remain a fixed value in numerical simulation.

Q-Switched ML
In a mode-locked SDL, although the carrier relaxation time of nanosecond order can effectively suppress the Q-switching tendency, if the gain of laser is not sufficient, or, the modulation depth of saturable absorber is relatively large, it will still cause undersaturation of the absorber and result in Q-switching.
Particularly, in a self-mode-locked SDL which is initiated by the combined action of a Kerr-lens and a soft aperture, not only the change of pump intensity will lead to the change of focal length of Kerr-lens, the size of soft aperture is also very sensitive to adjustment of the laser cavity.Both may stimulate the Q-switching tendency.
We numerically solve Equations ( 1)-(3) using the famous Runge-Kutta method, and the simulated Q-switched ML with q 0 = 2.0 and various g 0 are plotted in Figure 2. The horizontal axis is time (normalized by the cavity round-trip time t r ), and the longitudinal axis is amplitude of light field (arbitrary unit).Figure 2a-e show the Q-switching envelope of mode-locked pulses in a longer time region, and (f-j) are their corresponding pulse train in a shorter time scale.
horizontal axis is time (normalized by the cavity round-trip time tr), and the longitudinal axis is amplitude of light field (arbitrary unit).Figure 2a-e show the Q-switching envelope of mode-locked pulses in a longer time region, and (f-j) are their corresponding pulse train in a shorter time scale.In Figure 2, with gradually increased gain parameter g 0 , the transition from Q-switched ML to CW ML is very clear.The period T Q (in t r ) of amplitude envelope and the relative modulation depth (in percentage, defined as the difference of maximum and minimum value of amplitude envelope divided by the maximum value) with different g 0 are shown in Figure 3. Obviously, both the period and the modulation depth decrease with increased g 0 .It can be expected from Figure 3 that when g 0 reaches a critical point, T Q and modulation depth will both fall to zero; that is, the ML will transition from Q-switched ML to CW ML.This point is clearly verified in Figure 2e,j  In Figure 2, with gradually increased gain parameter g0, the transition f switched ML to CW ML is very clear.The period TQ (in tr) of amplitude envelope relative modulation depth (in percentage, defined as the difference of maximum a imum value of amplitude envelope divided by the maximum value) with differe shown in Figure 3. Obviously, both the period and the modulation depth decre increased g0.It can be expected from Figure 3 that when g0 reaches a critical point modulation depth will both fall to zero; that is, the ML will transition from Q-s ML to CW ML.This point is clearly verified in Figure 2e,j, in which g0 = 3.3, the mod depth of Q-switch envelope decreases sharply to a negligible value after about 200 trips, and the amplitude of pulse train does not change any longer.Both inadequate pump intensity and excessive absorption will result in unde tion of the absorber.They are mainly responsible for the appearance of Q-switch Figure 4a shows the undersaturated absorption, the step-by-step saturated gain, amplitude-modulated pulse train in Q-switched ML.Within an envelope in Figur absorber cannot be fully saturated in a single absorption because of the insufficie energy in the cavity due to the insufficient gain.The gain is not fully saturated changes step by step.Therefore, the net gain windows formed by the combined a absorption and gain saturation also change gradually.Thus, the amplitudes of est pulses are modulated by an envelope, i.e., Q-switch.Figure 4b shows the formatio pulses in corresponding net gain windows in detail.It should be noted that thei tudes are not equal.Both inadequate pump intensity and excessive absorption will result in undersaturation of the absorber.They are mainly responsible for the appearance of Q-switch in ML. Figure 4a shows the undersaturated absorption, the step-by-step saturated gain, and the amplitude-modulated pulse train in Q-switched ML.Within an envelope in Figure 4a, the absorber cannot be fully saturated in a single absorption because of the insufficient pulse energy in the cavity due to the insufficient gain.The gain is not fully saturated too, but changes step by step.Therefore, the net gain windows formed by the combined action of absorption and gain saturation also change gradually.Thus, the amplitudes of established pulses are modulated by an envelope, i.e., Q-switch.Figure 4b shows the formation of two pulses in corresponding net gain windows in detail.It should be noted that their amplitudes are not equal.

CW ML
The role of the saturable absorber is to promote a mode from noise and stabilize the pulses once they are generated.It has been concluded from previous simulations that both insufficient gain and excessive absorption will cause Q-switching in SML.If CW ML is to be started, two conditions must be met: adequate intracavity circulating power and an aperture with appropriate size.The former can ensure that there is desired noise with enough peak power to generate the necessary Kerr effect, and the latter can provide a certain difference between the round-trip losses of ML and continuous operation, so as to initiate and stabilize ML.
Figure 5a-e show the evolution of calculated CW mode-locked pulses under different g0 when q0 = 0.5.Figures on the right side describe performance of the pulses in a smaller time range.As can be seen from Figure 5, if g0 is relatively small (figure (a)) or large (figure (e)), the time required for building a stable pulse is relatively longer.In all CW ML processes, the pulse amplitude gradually increases while the pulse width also increases with increased g0, as shown in Figure 6.Although the simulated pulse width cannot be strictly equal to that produced from the experiment (in fact, most reported pulse widths in SML are far small than the calculated pulse width here), the trend that pulse width increases with increased g0 in numerical simulation still has certain reference significance for experimental design.

CW ML
The role of the saturable absorber is to promote a mode from noise and stabilize the pulses once they are generated.It has been concluded from previous simulations that both insufficient gain and excessive absorption will cause Q-switching in SML.If CW ML is to be started, two conditions must be met: adequate intracavity circulating power and an aperture with appropriate size.The former can ensure that there is desired noise with enough peak power to generate the necessary Kerr effect, and the latter can provide a certain difference between the round-trip losses of ML and continuous operation, so as to initiate and stabilize ML.
Figure 5a-e show the evolution of calculated CW mode-locked pulses under different g 0 when q 0 = 0.5.Figures on the right side describe performance of the pulses in a smaller time range.As can be seen from Figure 5, if g 0 is relatively small (figure (a)) or large (figure (e)), the time required for building a stable pulse is relatively longer.In all CW ML processes, the pulse amplitude gradually increases while the pulse width also increases with increased g 0 , as shown in Figure 6.Although the simulated pulse width cannot be strictly equal to that produced from the experiment (in fact, most reported pulse widths in SML are far small than the calculated pulse width here), the trend that pulse width increases with increased g 0 in numerical simulation still has certain reference significance for experimental design.We plot the establishment process of pulse in the corresponding net gain window in Figure 7a.Compared with Figure 4b, because the absorber and gain can be fully saturated here, the net gain window is deeper, and the pulse amplitude is higher and uniform.Figure 7b shows that when the gain is too large or the pump is too intense, building-up of a steady pulse train needs to undergo more oscillation.In the second harmonic region marked in Figure 7b, pulses have already been split and the second harmonics have appeared.However, the gain is not sufficient to maintain the second harmonics, which disappears after about 100 intracavity round-trips.Finally, the pulses tend to stable fundamental ML.It can be expected that a robust harmonic ML can be achieved by increasing the pump intensity, as described later.We plot the establishment process of pulse in the corresponding net gain window in Figure 7a.Compared with Figure 4b, because the absorber and gain can be fully saturated here, the net gain window is deeper, and the pulse amplitude is higher and uniform.Figure 7b shows that when the gain is too large or the pump is too intense, building-up of a steady pulse train needs to undergo more oscillation.In the second harmonic region marked in Figure 7b, pulses have already been split and the second harmonics have appeared.However, the gain is not sufficient to maintain the second harmonics, which disappears after about 100 intracavity round-trips.Finally, the pulses tend to stable fundamental ML.It can be expected that a robust harmonic ML can be achieved by increasing the pump intensity, as described later.We plot the establishment process of pulse in the corresponding net gain window in Figure 7a.Compared with Figure 4b, because the absorber and gain can be fully saturated here, the net gain window is deeper, and the pulse amplitude is higher and uniform.Figure 7b shows that when the gain is too large or the pump is too intense, building-up of a steady pulse train needs to undergo more oscillation.In the second harmonic region marked in Figure 7b, pulses have already been split and the second harmonics have appeared.However, the gain is not sufficient to maintain the second harmonics, which disappears after about 100 intracavity round-trips.Finally, the pulses tend to stable fundamental ML.It can be expected that a robust harmonic ML can be achieved by increasing the pump intensity, as described later.

Harmonic ML
As mentioned above, when the gain is enlarged to a certain extent, pulse will be split and stabilized, and the CW ML in SML will change to harmonic ML. Figure 8a-d show the second harmonic ML obtained by different permutation of g 0 and q 0 , where the value of g 0 is about 20 times that of q 0 .Figure 8e-h on the right side are their corresponding pulses within ten round-trips.It can be concluded from Figure 8 that the pulse width will decrease with raising g 0 and q 0 , and permutations of larger or smaller g 0 and q 0 all will need more intracavity cycles for harmonic ML to reach a stable state.

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Harmonic ML
As mentioned above, when the gain is enlarged to a certain extent, pulse will be split and stabilized, and the CW ML in SML will change to harmonic ML. Figure 8a-d show the second harmonic ML obtained by different permutation of g0 and q0, where the value of g0 is about 20 times that of q0. Figure 8e-h on the right side are their corresponding pulses within ten round-trips.It can be concluded from Figure 8 that the pulse width will decrease with raising g0 and q0, and permutations of larger or smaller g0 and q0 all will need more intracavity cycles for harmonic ML to reach a stable state.Second harmonic ML obtained by different permutation of g 0 and q 0 .(a) g 0 = 0.5, q 0 = 0.02, (b) g 0 = 1.0, q 0 = 0.05, (c) g 0 = 2.0, q 0 = 0.1, (d) g 0 = 3.0, q 0 = 0.15.(e-h) on the right side are their corresponding pulses within ten round-trips.
Figure 9a,b indicate the emergence and gradual enhancement of the second harmonic in a shorter and longer time range when g 0 = 2 and q 0 = 0.1.The second harmonic becomes steady after about 350 intracavity round-trips and maintains the same amplitude as the fundamental wave.Figure 9c shows the establishment of two intracavity circulating pulses in the corresponding net gain windows formed by the successively saturated absorption and gain.Increasing the values of g0 and q0 and keeping the ratio of g0 to q0 roughly the same when the second harmonic ML is obtained, the third harmonic ML can be achieved as shown in Figure 10.At the beginning of the third harmonic, energy will be transferred among the fundamental wave, the second harmonic and the third harmonic, and their respective amplitudes will be changed alternately.Therefore, compared with the second harmonic ML, the third harmonic ML needs more cavity round-trips to reach stabilization.Increasing the values of g 0 and q 0 and keeping the ratio of g 0 to q 0 roughly the same when the second harmonic ML is obtained, the third harmonic ML can be achieved as shown in Figure 10.At the beginning of the third harmonic, energy will be transferred among the fundamental wave, the second harmonic and the third harmonic, and their respective amplitudes will be changed alternately.Therefore, compared with the second harmonic ML, the third harmonic ML needs more cavity round-trips to reach stabilization.Increasing the values of g0 and q0 and keeping the ratio of g0 to q0 roughly the same when the second harmonic ML is obtained, the third harmonic ML can be achieved as shown in Figure 10.At the beginning of the third harmonic, energy will be transferred among the fundamental wave, the second harmonic and the third harmonic, and their respective amplitudes will be changed alternately.Therefore, compared with the second harmonic ML, the third harmonic ML needs more cavity round-trips to reach stabilization.We summarize the second harmonic ML in Figure 8 and the third harmonic ML in Figure 10 together, plotting it in Figure 12.The x-, y-and z-axis represent g0, q0 and the amplitude of electric field of mode-locked pulse, and the color bar on the right side represents the pulse width.In general, with increased g0 and q0, the pulse width in harmonic We summarize the second harmonic ML in Figure 8 and the third harmonic ML in Figure 10 together, plotting it in Figure 12.The x-, y-and z-axis represent g 0 , q 0 and the amplitude of electric field of mode-locked pulse, and the color bar on the right side represents the pulse width.In general, with increased g 0 and q 0 , the pulse width in harmonic ML decreases, which is contrary to CW ML.The critical boundary of second and third harmonic ML is about g 0 •q 0 = 0.6.When g 0 •q 0 < 0.6, there are two pulses circling in the cavity, and when g 0 •q 0 > 0.6, the third harmonic ML happens.
ML decreases, which is contrary to CW ML.The critical boundary of second and third harmonic ML is about g0•q0 = 0.6.When g0•q0 < 0.6, there are two pulses circling in the cavity, and when g0•q0 > 0.6, the third harmonic ML happens.
Harmonic ML in a self-mode-locked SDL has been realized experimentally [21].ML was initiated by using a variable slit in front of one end mirror.In the fundamental modelocked operation, the slit could be opened completely, while in the harmonic operation, ML was only achieved with the slit partially closed.For the harmonic ML, an increased pump power was required to achieve higher harmonics.The reported experimental results show that when the fundamental ML was achieved, larger saturable absorption and higher pump intensity, which means higher q0 and g0, was needed for harmonic ML, and these results are consistent with our simulations.At last, the theoretical laser threshold (given by g = q − ln(κ)), the calculated Qswitched ML threshold, CW ML threshold and harmonic ML threshold are generalized in Figure 13.For fixed q0, after g0 exceeding the laser threshold, continually increased g0 will result the occurrence of Q-switched ML.CW ML can be obtained by further increasing g0, and the boundary of transition is g0~1.5-2q0.The region of CW ML is significantly larger than that of Q-switch ML.When g0 goes beyond 15-20 q0, CW ML will evolve to harmonic ML.Obviously, the range of harmonic ML is minimal.Second and third harmonic ML under different permutations of g 0 and q 0 .The critical boundary of second and third harmonic ML is about g 0 •q 0 = 0.6.
Harmonic ML in a self-mode-locked SDL has been realized experimentally [21].ML was initiated by using a variable slit in front of one end mirror.In the fundamental modelocked operation, the slit could be opened completely, while in the harmonic operation, ML was only achieved with the slit partially closed.For the harmonic ML, an increased pump power was required to achieve higher harmonics.The reported experimental results show that when the fundamental ML was achieved, larger saturable absorption and higher pump intensity, which means higher q 0 and g 0 , was needed for harmonic ML, and these results are consistent with our simulations.
At last, the theoretical laser threshold (given by g = q − ln(κ)), the calculated Qswitched ML threshold, CW ML threshold and harmonic ML threshold are generalized in Figure 13.For fixed q 0 , after g 0 exceeding the laser threshold, continually increased g 0 will result the occurrence of Q-switched ML.CW ML can be obtained by further increasing g 0 , and the boundary of transition is g 0 ~1.5-2 q 0 .The region of CW ML is significantly larger than that of Q-switch ML.When g 0 goes beyond 15-20 q 0 , CW ML will evolve to harmonic ML.Obviously, the range of harmonic ML is minimal.
Photonics 2022, 9, x FOR PEER REVIEW 14 of 17 ML decreases, which is contrary to CW ML.The critical boundary of second and third harmonic ML is about g0•q0 = 0.6.When g0•q0 < 0.6, there are two pulses circling in the cavity, and when g0•q0 > 0.6, the third harmonic ML happens.Harmonic ML in a self-mode-locked SDL has been realized experimentally [21].ML was initiated by using a variable slit in front of one end mirror.In the fundamental modelocked operation, the slit could be opened completely, while in the harmonic operation, ML was only achieved with the slit partially closed.For the harmonic ML, an increased pump power was required to achieve higher harmonics.The reported experimental results show that when the fundamental ML was achieved, larger saturable absorption and higher pump intensity, which means higher q0 and g0, was needed for harmonic ML, and these results are consistent with our simulations.At last, the theoretical laser threshold (given by g = q − ln(κ)), the calculated Qswitched ML threshold, CW ML threshold and harmonic ML threshold are generalized in Figure 13.For fixed q0, after g0 exceeding the laser threshold, continually increased g0 will result the occurrence of Q-switched ML.CW ML can be obtained by further increasing g0, and the boundary of transition is g0~1.5-2q0.The region of CW ML is significantly larger than that of Q-switch ML.When g0 goes beyond 15-20 q0, CW ML will evolve to harmonic ML.Obviously, the range of harmonic ML is minimal.

Conclusions
In conclusion, we have performed the dynamics simulation of SML in a SDL using DDEs.Occurrence conditions and evolution characteristics of Q-switched ML, CW ML and harmonic ML are calculated and numerically analyzed.It has been found that the relatively small g 0 just above laser threshold is the main reason for the undersaturated process of saturable absorption, which leads to Q-switching.CW ML can be obtained by increasing g 0 continuously, and the boundary of transition is g 0 ~1.5-2 q 0 .When g 0 goes beyond 15-20 q 0 , CW ML will evolve to harmonic ML.The critical boundary of second and third harmonic ML is about g 0 •q 0 = 0.6.This kind of numerical simulation and analysis provides some understanding of the dynamics process of SML, and may be referenced for related experiments.
ω is the radium of light spot, Ip is the peak light intensity and L

Figure 1 .
Figure 1.The calculated nonlinear refractive index n2 of gain medium, focal length f of Kerr-lens, and the difference of round-trip loss with and without Kerr-lens under various pump spot sizes.The pump power and the peak power of noise pulse are supposed to be 10 W and 50 kW, respectively.The nonlinear reflective index n2 is obtained from reference [42].

Figure 1 .
Figure 1.The calculated nonlinear refractive index n 2 of gain medium, focal length f of Kerr-lens, and the difference of round-trip loss with and without Kerr-lens under various pump spot sizes.The pump power and the peak power of noise pulse are supposed to be 10 W and 50 kW, respectively.The nonlinear reflective index n 2 is obtained from reference [42].
, in which g 0 = 3.3, the modulation depth of Q-switch envelope decreases sharply to a negligible value after about 200 round-trips, and the amplitude of pulse train does not change any longer.

Figure 3 .
Figure 3.The period TQ (in tr) of amplitude envelope and the relative modulation depth (in age) in Q-switched ML with q0 = 2.0 and various g0.

Figure 3 .
Figure3.The period T Q (in t r ) of amplitude envelope and the relative modulation depth (in percentage) in Q-switched ML with q 0 = 2.0 and various g 0 .

Figure 4 .
Figure 4. (a) Undersaturated absorption, step by step saturated gain, and corresponding amplitudemodulated pulse train in Q-switched ML.(b) Formation of two pulses in corresponding net gain windows.It should be noted that their amplitudes are not equal.

Figure 4 .
Figure 4. (a) Undersaturated absorption, step by step saturated gain, and corresponding amplitudemodulated pulse train in Q-switched ML.(b) Formation of two pulses in corresponding net gain windows.It should be noted that their amplitudes are not equal.

Figure 7 .
Figure 7. Establishment process of pulse in the corresponding net gain window (a).With q0 = 0.5 and a relatively large g0 = 6.5, second harmonics begin to emerge and then disappears (b).

Figure 6 .
Figure 6.Simulated pulse width and pulse amplitude with q 0 = 0.5 and varied g 0 from 1.0 to 6.5.

Figure 7 .
Figure 7. Establishment process of pulse in the corresponding net gain window (a).With q0 = 0.5 and a relatively large g0 = 6.5, second harmonics begin to emerge and then disappears (b).

Figure 7 .
Figure 7. Establishment process of pulse in the corresponding net gain window (a).With q 0 = 0.5 and a relatively large g 0 = 6.5, second harmonics begin to emerge and then disappears (b).

Figure
Figure 9a,b indicate the emergence and gradual enhancement of the second harmonic in a shorter and longer time range when g0 = 2 and q0 = 0.1.The second harmonic becomes steady after about 350 intracavity round-trips and maintains the same amplitude as the fundamental wave.Figure 9c shows the establishment of two intracavity circulating pulses in the corresponding net gain windows formed by the successively saturated absorption and gain.

Figure
Figure 11a-c describe the occurrence and evolution of the second and third pulse in the cavity when g 0 = 5.0 and q 0 = 0.25.(a) shows that the second pulse is split on the right side of the first pulse before about 200 intracavity cycles.(b) depicts that the third pulse is generated on the left side of the first pulse after about 200 intracavity round-trips.(c) indicates the evolution tendency of amplitudes of the first, second and third pulses.(d) shows the amplitude of three pulses circulating in cavity and the corresponding changes of absorption and gain when the third harmonic ML is stable.

Figure 11 .
Figure 11.The occurrence and evolution of second (a) and third (b) pulse in cavity when g0 = 5.0 and q0 = 0.25.(c) The evolution tendency of amplitudes of the first, second and third pulses.(d) The amplitude of three pulses circulating in cavity and the corresponding changes of absorption and gain when the third harmonic ML is stable.

Figure 11 .
Figure 11.The occurrence and evolution of second (a) and third (b) pulse in cavity when g 0 = 5.0 and q 0 = 0.25.(c) The evolution tendency of amplitudes of the first, second and third pulses.(d) The amplitude of three pulses circulating in cavity and the corresponding changes of absorption and gain when the third harmonic ML is stable.

Figure 12 .
Figure 12. Second and third harmonic ML under different permutations of g0 and q0.The critical boundary of second and third harmonic ML is about g0•q0 = 0.6.

Figure 13 .
Figure 13.Theoretical laser threshold, calculated Q-switched ML threshold, CW ML threshold and harmonic ML threshold.

Figure 12 .
Figure12.Second and third harmonic ML under different permutations of g 0 and q 0 .The critical boundary of second and third harmonic ML is about g 0 •q 0 = 0.6.

Figure 12 .
Figure 12. Second and third harmonic ML under different permutations of g0 and q0.The critical boundary of second and third harmonic ML is about g0•q0 = 0.6.

Figure 13 .
Figure 13.Theoretical laser threshold, calculated Q-switched ML threshold, CW ML threshold and harmonic ML threshold.

Figure 13 .
Figure 13.Theoretical laser threshold, calculated Q-switched ML threshold, CW ML threshold and harmonic ML threshold.