Numerical investigation of GHz repetition rate fundamentally mode-locked all-fiber lasers

GHz repetition rate fundamentally mode-locked lasers have attracted great interest for a variety of scientific and practical applications. A passively mode-locked laser in all-fiber format has the advantages of high stability, maintenance-free operation, super compactness, and reliability. In this paper, we present numerical investigation on passive mode-locking of all-fiber lasers operating at repetition rates of 1-20 GHz. Our calculations show that the reflectivity of the output coupler, the small signal gain of the doped fiber, the total net cavity dispersion, and the modulation depth of the saturable absorber are the key parameters for producing stable fundamentally mode-locked pulses at GHz repetition rates in very short allfiber linear cavities. The instabilities of GHz repetition rate fundamentally mode-locked allfiber lasers with different parameters were calculated and analyzed. Compared to a regular MHz repetition rate mode-locked all-fiber laser, the pump power range for the mode-locking of a GHz repetition rate all-fiber laser is much larger due to the several orders of magnitude lower accumulated nonlinearity in the fiber cavity. The presented numerical study provides valuable guidance for the design and development of highly stable mode-locked all-fiber lasers operating at GHz repetition rates. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Mode-locked laser sources capable of producing ultrashort pulses with femtosecond (fs) to picosecond (ps) durations have enormous impacts on many disciplines of science and technology. A growing number of scientific and practical applications rely on mode-locked lasers with high peak power sufficient to accomplish the tasks on an ultrashort timescale without inducing any other side effects [1]. Mode-locked lasers fundamentally operating at gigahertz (GHz) repetition rates exhibit large mode spacings and high signal-to-noise ratios, and have shown their advantages in a variety of applications including ultra-high speed communication, spectroscopy, precision optical sampling, material processing, and biology [2][3][4][5][6][7][8][9][10][11][12][13]. For instance, a GHz repetition rate mode-locked laser frequency comb allowing to access and manipulate of each individual comb line with high accuracy can be used for optical arbitrary waveform generation, high-speed analog-to-digital conversion and highresolution spectroscopy [8]. A mode-locked laser source operating at a repetition rate greater than 10 GHz is an ideal broadband calibration standard that can be used to search the earthlike exoplanets [9][10][11][12][13].
However, mode-locked solid-state lasers usually lack long-term stability and reliability due to their free-space configurations, which mainly limits their use in many applications. Modelocked semiconductor lasers can be very compact and robust and have been used to generate ultrashort pulses at very high repetition rates (> 10 GHz) for fast data transmitters in optical fiber communications and all-optical signal processing. However, their pulse qualities are generally poor due to the additional satellite pulses, timing jitters, and chirped phases [23,24]. Fiber lasers having the advantages of low manufacturing and maintaining cost, excellent beam quality, and outstanding heat dissipation capability, have become excellent laser platforms for GHz repetition rate mode-locked laser sources with high reliability and good pulse quality [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].
A fiber laser can be mode-locked either actively with an optical modulator or passively with a saturable absorber (SA), technique of nonlinear polarization evolution, or technique of four-wave-mixing [19][20][21][22]. Fundamental mode-locking of a fiber laser with an electro-optic modulator usually operates at MHz repetition rate constrained by the cavity length [18]. Actively mode-locked pulses at GHz repetition rates can be achieved with harmonic modelocking of a fiber laser [19]. However, the pulse width of an actively mode-locked fiber laser is usually long and the pulses usually do not have constant energies and coherent mutual phases. Moreover, actively mode-locked fiber lasers require external electronic control devices and high-speed modulators which make the laser source very complicated and expensive. Compared to actively mode-locked fiber lasers, passively mode-locked fiber lasers can generate much shorter pulses with high mutual phase coherence and even GHz repetition rate fundamentally mode-locked pulses can be obtained with very simple and compact setups. A numbers of GHz repetition rate fundamentally mode-locked fiber lasers have been achieved with centimeter or even sub-cm long gain fibers [23][24][25][26][27][28][29][30][31][32][33][34]. However, some of these lasers were developed with free-space cavities, which have not taken advantages of compactness and robustness of fiber lasers [23][24][25][26][27][28]. The mode-locked lasers developed in all-fiber configurations have shown outstanding stability and reliability [29][30][31][32][33][34]. A 12 GHz repetition rate fundamentally mode-locked all-fiber laser (FML-AFL) was demonstrated with an 8-mm long highly Er 3+ /Yb 3+ co-doped phosphate fiber [34]. Using a 4.75 mm highly Er 3+ /Yb 3+ codoped phosphosilicate fiber, Martinez and Yamashita demonstrated an FML-AFL operating with a fundamental repetition rate as high as 19.45 GHz [32]. However, the output stability of this laser is poor due to the thermal effect of the unabsorbed pump on the saturable absorber. Nevertheless, it is still possible to develop a stable 20 GHz repetition rate FML-AFL with low noise and time-jitter using current highly doped phosphate fiber technology, where a dopant concentration of Yb 3+ up to 12 wt. % has been achieved without concentration quenching effects [35]. GHz repetition rate pulses can also be obtained from harmonic mode-locking of a long-cavity fiber laser with a fundamental repetition rate of MHz [36][37][38][39], in which multiple pulses are formed within a round trip due to the nonlinear effects [37]. Most recently, a modelocked fiber laser operating at the 3436 th harmonic of the fundamental repetition rate was developed and pulses with a repetition rate of 14.5GHz were obtained [38]. Harmonic modelocked fiber lasers with tunable repetition rate at GHz ranges have also been achieved with four-wave mixing mechanism and different frequency multiplication techniques [19][20][21][22]. However, harmonic mode-locking of a fiber laser is usually much less stable than fundamental mode-locking and is also harder to control for practical applications [39]. Therefore, GHz repetition rate FML-AFLs are preferred for the applications mentioned above. However, compared to conventional MHz repetition rate FML-AFLs, it is more challenging to develop stable GHz repetition rate FML-AFLs because the peak power of the laser in the cavity is much lower and there is much less room for dispersion management and instability suppression in a cm-long or sub-cm-long fiber cavity. Therefore, thorough understanding of the mode-locking of cm-long or sub-cm-long fiber lasers with various parameters is very necessary for us to design and develop GHz repetition rate FML-AFLs.

Numerical sim provide guide
Numerical solving the n lasers with va These theoret repetition rate nonlinear dyn analyses and which the pea lasers and th instability of levels was nu easily varied (MD) of the s the output cou performance the influence AFLs is very FML-AFLs w on the GHz dispersion, th different valu AFLs are give

Theoretic
The basic co Figure 1 where, A is the slowly varying amplitude of the pulse envelop, z is the light propagation coordinate, t is the time-delay parameter, 2 β is the second-order dispersion, γ is the ( ) 3 χ nonlinearity parameter, Ω and g represent the gain bandwidth and the gain coefficient of the highly doped gain fiber, respectively, and g is written as [61] where, g 0 and E sat are the small-signal gain and saturation energy of the highly doped gain fiber, respectively. Because the gain fiber length of a GHz repetition rate mode-locked AFL is comparable to the gain medium length of a solid-state laser, we assume that g 0 is constant along the gain fiber. It should be noted that both the peak power and the fiber cavity length of a GHz repetition rate fiber are at least two orders of magnitude smaller than those of a MHz repetition rate fiber laser. Therefore, the split-step Fourier method can provide enough accuracy for our simulation. The reflectivity of the CFBG or the fiber-optic dichroic mirror is given by where, R p is the peak reflectivity of the CFBG or the dielectric thin film output coupler, ω is the optical frequency of the laser, 0 ω and ω Δ are the center frequency and the 3-dB bandwidth of the CFBG or the dielectric thin film output coupler respectively, D 2 is the second-order dispersion of the CFBG and is zero for the dielectric thin film output coupler. The nonlinear absorption dynamics of the SA can be described as [62], where, E A and A τ are the saturation energy and the recovery time of the SA, respectively. It is noted that A τ of the SA for GHz repetition rate FML-AFLs should be much less than the round-trip of the ultra-short fiber cavity. q 0 is the non-saturable loss of the SA. In the simulation, the light circulates in the fiber cavity shown in Fig. 1 and stationary output can be obtained until that when the iteration is converged. The criterion for the converge of the iteration is given by where A(k) is the amplitude of a data point describing the pulse shape of the current loop and A'(k) is that of the previous loop, N is the total number of the element of the matrix over entire time window, and k is the element number. N is 2 18 in our simulation. Therefore, the criterion for a stable output is that the normalized difference between the shapes of two sequent pulses is less than 10 −5 . It should be noted that Q-switched mode-locking cannot be simulated with this model and is considered as an unstable output in our simulation. Q-switched modelocking of a fiber laser can be modeled when the Q-switched operation is combined with the mode-locking operation, which needs to solve the rate equations for specific gain medium and thus is out of the scope of this paper. However, in addition to continuous-wave (CW) modelocking, pulsation mode-locking, multi-pulse mode-locking, and harmonic mode-locking can be simulated with this model. Here, we define a parameter d to evaluate the instability of a mode-locked laser, which can be described as

{ }
max min max max min A max pulse peak 1, pulse peak 2, pulse peak 3, , pulse peak n , n 1000, A min pulse peak 1, pulse peak 2, pulse peak Therefore, d is the variation of the peak powers of a pulse train. Here, we define the fundamentally mode-locked operation of the all-fiber laser as CW mode-locking or highly stable mode-locking when d is smaller than 10 −3 (the deep blue regions in following color maps) and as pulsation mode-locking for d > 10 −3 . To verify the reliability and applicability of the modeling, we simulated a 1 GHz repetition rate FML-AFL at 1034 nm that was developed in our lab. The AFL was developed by splicing a 5-cm 6 wt% Yb 3+ -doped phosphate fiber to a CFBG with a reflectivity of 61% and a 3-dB bandwidth of 5 nm. The dispersion of the CFBG is about 0.37 ps 2 . The other end of the Yb 3+ -doped phosphate fiber was butt-coupled to a SESAM with a saturation fluence of 70 µJ/cm 2 , a modulation depth of 3%, and a linear absorption of 5%. The measured and simulated pulse trains and optical spectra of the laser are shown in Figs. 2(a) and 2(b), respectively. The time window for the simulation was set to be 1 ns and the pulse train was obtained by recording the output of the FML-AFL for many loops when stable mode-locking was formed. Clearly, the simulation results are in a great agreement with the experimental results, verifying the correctness and reliability of the modeling.

Simulation results and discussion
In this section, the simulation results of GHz repetition rate mode-locked AFLs are presented and discussed. The performance of an AFL with a total cavity length of 10 cm, consisting of a 5-cm gain fiber, a 1-cm CFBG, and a 4-cm passive SMF, was first simulated. The gain fiber has a dispersion parameter of −36 ps/nm/km and a nonlinear coefficient of 4 W −1 km −1 . Its gain saturation energy and bandwidth were assumed to be 100 pJ and 20 nm, respectively. The SMF length has a dispersion parameter of −38 ps/nm/km and a nonlinear coefficient of 6 W −1 km −1 . The SA has a saturation energy of 100 pJ, a recovery time of 2 ps, and a non-   Fig. 3(a). at a g 0 of ndamental ate modeate modetition rate Fig. 3(b). becomes % when g 0 e-locking starts as g 0 is greater than 320 m −1 . The pulse train of two-pulse mode-locking at a g 0 of 320 m −1 is shown in Fig. 3(d). Harmonic mode-locking is obtained when g 0 is between 360 m −1 and 430 m −1 and the pulse train of 3rd-order harmonic mode-locking is shown in Fig. 3(e). As g 0 exceeds 430 m −1 , a stable mode-locking state can't be achieved any more. Because FML of an AFL is more interesting than multi-pulse mode-locking for most of practical applications, we will only present the simulation and analysis of CW mode-locking and pulsation modelocking of short-length AFLs in the following sections. In this subsection, the performance of 1-GHz repetition rate FML-AFLs operated at ADregime is numerically studied. The instability of the FML-AFLs as a function of g 0 and MD for different D total was calculated and is presented in Fig. 4. The reflectivity of the CFBG was set as 70%. (e) experimental observation [57], highly stable mode-locking can be only obtained at a small range of pump power (g 0 ) and the stability of an FML-AFL degrades with the increased pump power (g 0 ).

1-GHz repetition rate FML-AFLs operating at AD-regime
It can be seen from Figs. 4(a)-4(e) that the ranges of g 0 and MD for stable FML increase with the increased anomalous dispersion of the fiber cavity. In other words, the conditions for stable 1-GHz repetition rate FML-AFLs is more flexible when the net cavity dispersion is anomalous and has a large value such that the interaction of dispersion and nonlinear effect can result in stable soliton pulses. The deep blue regions shown in Figs. 4(b)-4(e) indicate that highly stable fundamental mode-locked pulses at a repetition rate of 1 GHz can be achieved by choosing a proper MD less than 50% and managing the cavity dispersion properly. When the D total is small and close to zero, the ranges of g 0 and MD for stable mode-locking are very small and the instabilities of the mode-locking are large as shown in Fig. 4(a), which is similar to the observation of MHz repetition rate FML. Therefore, deliberate design of an AFL with a small net anomalous cavity dispersion is very critical to achieve stable FML operation.  Another key parameter determining the stability of an FML-AFL is the R of the output coupler of the cavity. Figure 5 shows the calculated instabilities of 1-GHz repetition rate FML-AFLs with varied R. The net cavity dispersion for this calculation was set as −1 ps 2 . The contour plots of the instabilities of the FML for R of 30%, 50%, 70%, 80%, and 90%, are shown in Figs. 5(a)-5(e), respectively. It is clear that the ranges of MD and g 0 for stable mode-locking reduce with the increased R. When the R is 90%, the ranges of MD and g 0 for stable mode-locking is very small and the instability of the mode-locking is large. This is because the power intensity of the laser inside the cavity increases with the increased reflectivity of the output coupler and the interaction of the dispersion and nonlinear effects is hard to arrive in a stable soliton operation. The deep blue regions shown in Figs. 5(a)-5(d) tell us that highly stable fundamental mode-locked pulses at a repetition rate of 1 GHz can be achieved by choosing a proper SA with a MD < 55% and the ranges of MD and g 0 for stable mode-locking are large when the R of the CFBG or fiber-optic dichroic mirror is 50%.
The pulse width, spectral bandwidth, and time-bandwidth product (TBP) of the stable mode-locked pulses generated by the 1-GHz repetition rate FML-AFLs operating at the ADregime as a function of the g 0 , R, D total , and MD were calculated and are plotted in Fig. 6. Figure 6(a) shows the three parameters of the pulses as a function of the g 0 when the R = 70%, MD = 20%, D total = −0.5 ps 2 . It is clear that the pulse width deceases while the spectral bandwidth increases with the increased g 0 . The TBP decreases with the increased g 0 and arrives in a transform-limited value eventually. It should be noted that this conclusion can be applied only for a specific range of g 0 in which stable fundamental mode-locking can be achieved. Further increase of g 0 will arrive in unstable or multi-pulse mode-locking. Figure  6(b) shows the three parameters of the pulses as a function of the R when g 0 = 100 m −1 , MD = 20%, D total = −0.5 ps 2 . Similar to the case of increasing g 0 , both the pulse width and TBP decrease while the spectral width increases with the increased R. Figure 6(c) shows the three parameters of the pulses as a function of D total when R = 70%, MD = 20%, g 0 = 100 m −1 . The calculation results agree well with the general observation and understanding of mode-locked fiber lasers, i.e., the pulse width decreases while the spectral bandwidth increases as the net cavity dispersion reduces. Sub-ps pulses can be achieved when the cavity dispersion approaches zero by careful dispersion management. The TBP is nearly independent of the net cavity dispersion and has the value close to that of transform-limited. Figure 6(d) shows the three parameters of the pulses as a function of the MD when g 0 = 100 m −1 , R = 70%, D total = −0.5 ps 2 . The pulse width decreases and the spectral bandwidth increases with the increased MD. The TBP of the output pulses increases from the value of transform-limited soliton pulses to more than 0.6, indicating that the pulses are chirped as the MD increases.

1-GHz repetition rate FML-AFLs operating at ND-regime
In this subsection, the performance of 1-GHz repetition rate FML-AFLs operating in the NDregime are numerically studied. The mode-locking instability of the laser as a function of g 0 and MD for different D total was calculated and is presented in Fig. 7. The R and spectral bandwidth of the CFBG are set to be 70% and 5 nm, respectively. The varying ranges of g 0 and MD are 50 m −1 ~500 m −1 and 1% ~90%, respectively. The contour plots of the instabilities of the FML as a function of g 0 and MD for D total of 0.1 ps 2 , 0.3ps 2 , 0.5ps 2 , 0.8ps 2 , and 1ps 2 , are shown in Figs. 7(a)-7(e), respectively. In agreement with the experimental observation [46], stable mode-locking of the AFLs can usually be achieved in large ranges of g 0 and MD.
The ranges of g 0 and MD for stable FML decrease a bit with the increased D total . The deep blue regions shown in Figs. 7(a)-7(e) indicate that highly stable fundamental mode-locked pulses at a repetition rate of 1 GHz can be achieved at large pump powers (large g 0 ) as the MD is large. When the total net cavity dispersion is 1 ps 2 , the ranges of g 0 and MD for highly stable mode-locking are much smaller than those of the other cases and the instability of the FML is due to the large chirp [54]. Compared to the FMLs at the AD-regime shown in Fig. 4, FMLs operating at the NDregime can be achieved in much larger ranges of g 0 and MD as shown in Fig. 7. This is consistent with the experimental observations [46] that stable fundamental normal dispersion mode-locking can be obtained at a very large range of pump power while anomalous dispersion mode-locking can be achieved only at a small range of pump power. Spectral filtering effect is a key requirement for self-started mode-locking in the NDregime [41]. The mode-locking instability of the AFLs as a function of g 0 and MD was calculated for different spectral bandwidths (BWs) of the CFBG and is presented in Fig. 8. The total net cavity dispersion is set as 1 ps 2 . The contour plots of the instabilities of the FMLs as a function of g 0 and MD for BW of 1 nm, 5 nm, 10 nm, 20 nm, and 30 nm, are calculated and shown in Figs. 8(a)-8(e), respectively. Clearly, stability of the mode-locking decreases with the increased BW. When the BW is small, highly stable mode-locking can be achieved in a large range of g 0 and MD. As the BW exceeds 10 nm, highly stable fundamental mode-locked pulses are available only at some specific cases and the MD of a SA for stable mode-locking needs to be > 35%, as shown in Figs  The pulse width, spectral bandwidth, and TBP of the pulses generated by the 1-GHz repetition rate FML-AFLs operated in the ND regime as a function of g 0 , MD, D total , and BW were calculated and are plotted in Fig. 9. Figure 9(a) shows the three parameters of the modelocked pulses as a function of the g 0 , when the R = 70%, MD = 20%, D total = 0.5 ps 2 , BW = 5 nm. It is consistent with the experimental observations [41] that all the three parameters increase with the increased g 0 and the pulses are highly chirped at high pump power levels. Figure 9(b) shows the three parameters of the mode-locked pulses as a function of the MD when g 0 = 100 m −1 , R = 70%, and D total = 0.5 ps 2 , BW = 5 nm. Both the pulse width and TBP decrease while the spectral width increases with the increased MD. Figure 9(c) shows the three parameters of the mode-locked pulses as a function of the D total when R = 70%, MD = 20%, g 0 = 100 m −1 , BW = 5 nm. The pulse width increases while both the spectral bandwidth and TBP decreases with the decreased D total . Figure 9(d) shows the three parameters as a function of the BW when g 0 = 300 m −1 , MD = 60%, R = 70%, D total = 0.5 ps 2 . Pulse width increases while spectral width decreases with the increased BW of the CFBG. TBP of the pulses, however, doesn't changes much, which is consistent with the results presented in [30], indicating the filtering effect of the CFBG do help for avoiding the break-up of the pulses mode-locked in the ND regime.

Instability of fundamental mode-locking at higher GHz repetition rates
In this subsection, FML-AFLs operating at higher GHz repetition rates are numerically investigated and studied. Since the cavity length needs to be several centimeters or sub-cm, the all-fiber cavity consists of only a short gain fiber, a DC and a SA, as shown in Fig. 1(b). The parameters of the gain fiber and SA are set the same values as above. The total dispersion of the cavity is calculated by total DC gain SA Because the dispersion of the DC is negligible, that of the gain fiber dispersion is generally hundreds of fs 2 , and that of the SA dispersion is around a few thousands of fs 2 , the D total is set as −5000 fs 2 and + 5000 fs 2 for the ND and AD mode-locking regimes, respectively. The varying ranges of g 0 and MD are 50 ~1000 m −1 and 1% ~90%, respectively. Firstly, the instabilities of FML-AFLs operating in the AD-regime with repetition rates of 5 GHz, 10 GHz, and 20 GHz are calculated and plotted as a function of g 0 and MD by the contour figures shown in Fig. 10. It can be seen from Figs. 10(a)-10(c) that the instability of the FML-AFLs decreases while the ranges of g 0 and MD for the stable FML increase with the increased repetition rate. This is because the peak intensity of the mode-locked pulses becomes smaller as the repetition rate increases. It tells us that stable mode-locking can be obtained at larger ranges of g 0 and MD of the SA when the repetition rate is higher. It is also consistent with the experimental observation [26] that CW fundamental mode-locking can be only achieved with SAs with very small MD. Then the instabilities of FML-AFLs operating in the ND-regime with repetition rates of 5 GHz, 10 GHz, and 20 GHz were calculated and are shown by the contour plots in Fig. 11. It is clearly that the ranges of the g 0 and MD for stable mode-locking decrease with the increased repetition rate. The MD of the SA required for highly stable mode-locking also decreases with the increased repetition rate. Therefore, proper selection of a SA is very critical for developing a highly stable mode-locked AFL operating at 10-20 GHz repetition rate. Fig. 11. The calculated instability as a function g 0 and MD for the FML-AFLs operating in the ND-regime with a repetition rate of (a) 5 GHz, (b) 10 GHz and (c) 20 GHz, D total of 5000 fs 2 , and R of 70%. Figs. 12(a) and 12(b), respectively. It can be seen from Fig. 12(a) that the pulse width increases while the spectral bandwidth decreases with the increased repetition rate, and the TBPs of the FML-AFLs don't change much as the repetition rate increases because they operate in soliton mode-locking. As shown in Fig. 12(b), when the FML-AFLs operate in the ND-regime, the TBP, pulse width, and spectral bandwidth, however, decrease with the increase of the repetition rate.

Conclusion
Performance of GHz repetition rate mode-locked AFLs was numerically investigated and the instability of FMLs in terms of total net cavity dispersion, small-signal gain, spectral bandwidth, and MD of the SA were studied. The simulation results show that, when the net cavity dispersion is negative (anomalous dispersion), highly stable FML can be achieved in small specific ranges of MD of the saturable absorber and the small signal gain of the active fiber and the ranges becomes large as the net cavity dispersion becomes large or the reflectivity of the output coupler becomes small. As the net cavity dispersion is positive (normal-dispersion), highly stable FML can be obtained in large ranges of MD of the saturable absorber and the small signal gain of the active fiber and the stability of the modelocking doesn't change much with the net cavity dispersion. A narrow bandwidth of the CFBG or fiber-optic dichroic mirror is helpful for achieving highly stable mode-locking in the ND regimes. The ranges of modulation depth and small signal gain for stable FML decrease with the increased repetition rate of the FML. The highly stable FML at high GHz repetition rate can be obtained only with small MD of the SA for the mode-locking at both AD and ND regimes. The simulation results are consistent with the reported experimental observations that the MD of the SA and small signal gain ranges of stable mode-locking at normal dispersion are much larger than those at anomalous dispersion. The small signal gain range for a GHz repetition rate FML is much larger than what we usually observed with a MHz repetition rate FML [51] because the accumulated nonlinearity of a GHz repetition rate FML is at least 4 orders of magnitude smaller. The simulation results are very helpful for the design and development of GHz repetition rate FML-AFLs that can be used for a variety of applications. The numerical investigation can also help us to understand the pulsed laser performance in short-cavity all-fiber lasers.