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Abstract
States that are switchable from single soliton to pulse bunch in a compact semiconductor saturable absorber mirror (SESAM) mode-locked fiber laser with a fundamental repetition rate of 3.2 GHz are experimentally investigated and further studied via simulations. A composite filtering effect comprising an intracavity low-finesse Fabry-Perot (FP) filter, an artificial optical low-pass filter, and a gain filter implements the state switching to pulse bunch. A numerical model is proposed to clarify the mechanism underlying the switching. It reveals that, for pulse interval ∆T > τA (relaxation time of the SESAM) in a pulse bunch, the laser operates in pulse-bound build up. In an inverse mechanism the state returns to single soliton, in which the ∆T is obtained from the free spectral range Ωc of the intracavity FP filter by mechanically controlling the distance between the SESAM and gain fiber. This pulse bunch regime of operation ought to be amenable to a quasi-steady-state treatment. It represents an alternative emergence trait in the temporal domain between a main soliton with strong sidelobes in both sides and a bound soliton pair with weak sub-sidelobes. Another profile of the pulse bunch state is that the side peak amplitude in the autocorrelation trace is more than 50%, which is distinct and larger than that in the conventional bound state regime in fiber lasers. The optical spectra, radio frequency spectra, and frequency chirp are further analyzed. These numerical results agree well with the experimental ones within the variation range of the crucial values of Ωc and enable the explicit understanding of such behavior in SESAM mode-locked high-repetition-rate fiber lasers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
Yi Zhou, Wei Lin, Huihui Cheng, Wenlong Wang, Tian Qiao, Qi Qian, Shanhui Xu, and Zhongmin Yang, "Composite filtering effect in a SESAM mode-locked fiber laser with a 3.2-GHz fundamental repetition rate: switchable states from single soliton to pulse bunch: erratum," Opt. Express 26, 17458-17458 (2018)https://opg.optica.org/oe/abstract.cfm?uri=oe-26-13-17458
1. Introduction
The development of multi-gigahertz pulse repetition rate mode-locked laser sources has been motivated by the introduction of optical frequency combs and their applications, for example, optical communications and the precision calibration of astronomical spectrograms in the search for Earth-like exoplanets [1–4]. Harmonic mode-locking techniques [5] and methods based on pulse repetition multiplication [6,7] are capable of producing pulse trains with extreme pulse repetition frequency values (>100 GHz); however, fundamental mode locking is more favorable for certain practical applications owing to its intrinsic low timing-jitter and amplitude fluctuation. Aside from the temporal soliton synthesis in passive optical microresonators [8,9], saturable absorber (SA) based mode-locking [10] and Kerr lens mode-locking (KLM) [11] are the two most widespread approaches to produce fundamentally mode-locked multi-gigahertz pulses in active laser cavities. Considering these techniques, there are distinct benefits and trade-offs. KLM frequency comb with comb-line spacing as large as 10 GHz has already been implemented [12], while it is difficult to achieve the self-starting operation and critical cavity alignment is required. Nevertheless, SA-based mode-locked lasers can readily generate robust pulse train with gigahertz-level repetition rate [13–16], but suffer from pronounced Q-switching instability (QSI) before the continuous-wave (CW) mode-locking. Mode-locked lasers can be categorized into different groups according to a wide range of adopted laser gain, including Ti:sapphire, rare-earth (RE)-ion doped solid-state gain materials (crystal or glass), semiconductor and active glass fibers. Fiber laser, regarded as a promising choice for realizing ultrafast pulses, can offer high beam quality, reliability and efficient heat dissipation in a compact size. With the development of heavily RE-ion-doped multicomponent glass fibers, the gain coefficients of the fibers have been significantly boosted (e.g., 5.7 dB/cm in 1.0 μm [17], 5.2 dB/cm at 1535 nm [18], and 3.6 dB/cm at 1950 nm [19]), which enable the ultrafast lasers with gigahertz repetition rate. Until now, highest pulse repetition rates in fiber lasers have been reported as 5 GHz at a wavelength of 1.0 μm [20], 19 GHz at a wavelength of 1.5 μm [21] and 1.6 GHz at a wavelength of 2.0 μm [22]. Although various approaches, such as nonlinear polarization evolution (NPE), nonlinear optical loop mirror (NOLM), nonlinear amplifying loop mirror (NALM), and SA are available for the initialization of the pulsing [23,24], the options to build gigahertz mode-locked fiber lasers are limited. In principle, it is owing to the severe Q-switching instability. Based on the stability analysis of the rate equations [25–28] of the oscillator and gain fiber, the critical small-signal gain coefficient gcri (related to the pump power under practical conditions) for CW mode-locking is given by
where Lc and TR are the cavity length and round-trip time, respectively. qns is the net unsaturable loss (including the non-saturable loss of the SA and other linear loss), and q0 and Esat,A are the modulation depth and saturation energy of the SA. Esat,G is the gain saturation energy while TG is the lifetime of the RE-ion. The deviation relies on the limit Ep >> Esat,A, in which Ep represents the pulse energy. It can be seen from Eq. (1) that the parameter χA, defined by the product of the modulation depth q0 and saturation energy Esat,A, is a key factor to determine the gcri. Plotting the relative critical gain values (gcri matrix divided by its minimum) against different unsaturable losses qns and modulation depths q0 as shown in Fig. 1, reveals a noticeable broadening of the QSI regime with the growing qns and q0 (only up to 10%). Regarding the artificial SAs (ASAs), in most cases, e.g., NPE, NOLM, NALM, they exhibit considerable linear loss and modulation depth. Despite the efforts to enhance the characteristics of these ASAs [29–32], repetition rates of the relevant fiber lasers are typically no more than 1 GHz. By contrast, SAs with controllable modulation depth and reasonably low insert loss can be ideal choices to address the problems introduced by QSI. So far, A. Martinez and S. Yamashita have successfully built GHz fundamental mode-locked fiber lasers by using graphene [33] and carbon nanotubes (CNTs) [34]. However, CNTs, graphene and topological insulator-based SAs are prone to thermal damage at low pump power due to the mixed polymer materials.
Fig. 1 Relative critical gain values versus different unsaturable losses and modulation depths. Values of the adopted parameters can be found in Ref [28].
Among several types of SAs, semiconductor saturable absorber mirrors (SESAMs) are known for their highly-designable trait, reliability, and a certain level of thermal damage resistance as well. They enable self-starting soliton mode-locking with multi-gigahertz repetition rate both in solid-state and fiber lasers [35,36], hence they are regarded as an excellent additive intensity modulator. In most ultra-short fiber oscillators, SESAM is either employed after the optical lenses [37–39] or directly attached to the polished gain fiber [40]. The latter construction retains the compactness of the fiber laser and can be further stabilized by a temperature controller. Despite the unique advantages of the second scheme, damage, induced by the high peak powers in the case of QSI, frequently occurs because of the small mode-size (e.g., < 10 μm in diameter) on the SESAM. The method to circumvent the damage is always a crucial concern [41]. Recently, periodically poled lithium niobate crystal acting as a self-defocusing element is utilized in a bulky structure to suppress the damage induced by Q-switching [42]. Here, as a preliminary attempt, we enlarge the impacted spot size by introducing an air gap between the gain fiber and SESAM without additional beam collimation. The loss induced by the air gap is proved to be tolerable for a SESAM mode-locked fiber laser in gigahertz regime. Interestingly, a switchable state from single soliton to pulse bunch was observed in the experiment. Consider a low-finesse Fabry-Perot (FP) interferometer created by the facet of the fiber and SESAM's front, a latent FP filtering effect might be the trigger. Hence, understanding the underlying mechanism of this switchable state is helpful for us to engineer the pulse operation on purpose, and it is always fascinating to explore an exotic pulse dynamics in a typical nonlinear system.
In this paper, we report the switchable status from single soliton to pulse bunch in a SESAM mode-locked fiber laser with a 3.2 GHz fundamental repetition frequency. In the linear laser resonator, a tunable air gap (~300 μm distance) between the gain fiber and SESAM is intentionally generated, thus inducing an FP subcavity. The weak FP filtering effect facilitates the formation of the pulse bunch, and by adjusting the gap, the transition from single pulse to pulse bunch is observed experimentally. Based on the non-collimated, angled beam propagation and dispersion of the semiconductor SA, another underlying filtering effect is revealed. It contributes to the pulse spectral shaping, together with the gain profile of the Er3+/Yb3+ co-doped gain fiber. As indicated by numerical simulation, once the pulse separation is less than the relaxation time of the SESAM, the pulse bunch eventually evolve to the single-pulse-operation. In the transition process, pulses within the bunch attract each other due to the absorption dynamics and results in a longer soliton. Regarding the mutual interaction between the filtering effects and gain shaping, such fiber laser oscillator can be treated as a nonlinear system characterized by the Swift-Hohenberg equation (SHE).
2. Experimental setup and principle of the composite filtering effect (CFE)
2.1 Experimental setup
The experimental setup for the ultrafast fiber laser operated two states between single soliton and pulse bunch is shown in Fig. 2(a). A 3-cm-long highly Er3+/Yb3+ co-doped fiber (EYDF) with a gain coefficient of 5.2 dB/cm at 1535 nm acts as a gain medium. The gain fiber was pumped by a 974-nm laser diode (LD) with a maximum power of 600 mW through a wavelength division multiplexer (WDM). The EYDF was inserted in a ceramic ferrule with an inner diameter of 125.5 µm and outer diameter of 2.5 mm. The common port of the WDM was spliced to a fiber pigtail dielectric mirror, which was butt-coupled to one end of EYDF. The mirror was fabricated by coating multiple-layer dielectric films (DFs) onto a fiber ferrule using a plasma sputter deposition system, achieving a reflectivity of ~99% at a wavelength of 1550 nm and a reflectivity of ~0.5% at 974 nm. The SESAM (Batop GmbH) used in the experiment, with a modulation depth of 6%, a non-saturable loss of 3%, a relaxation time of 2 ps and a saturation fluence of 50 μJ/cm2, was soldered on a copper heat sink and subsequently mounted on a fixed bracket. In order to realize the state switching, the ceramic ferrule and EYDF assembly was fixed on a 6-axis stage to interact with the SESAM. The relative position between the facets of the EYDF and SESAM could be controlled accurately; thus, a gap between the two surfaces was readily formed and can be finely adjusted with the adjustment precision of 0.5 μm at x-axis. A photograph of the experimental set can also be seen in Fig. 2(b). In addition, a polarization controller (PC) was employed to further enhance the pump coupling efficiency toward the EYDF. In practice, using a PC can improve the signal-to-noise ratio (SNR) of the fundamental repetition rate signal. Owing to the compact linear cavity configuration, the total laser cavity length is approximately 3 cm and it can produce a pulse train with a fundamental repetition rate of 3.2 GHz. The laser was emitted from the signal port of the WDM. The optical spectrum of the laser was measured by using an optical spectrum analyzer (YOKOGAWA AQ6370B), and the pulsed characteristics were detected by using a 12.5-GHz photodetector, a 6-GHz bandwidth digital oscilloscope (Keysight MSOX6004A), a phase noise analyzer (R&S FSWP26), and an autocorrelator.
Fig. 2 (a) Schematic of the 3.2-GHz Er3+/Yb3+ co-doped ultrafast fiber oscillator operated in pulse bunch state. (b) Photograph of the experimental setup.
2.2 Principle of the composite filtering effect
The CFE comprises an intracavity low-finesse Fabry-Perot (FP) filter, an artificial optical low-pass filter, and a gain filter. The first factor of the CFE is related to the FP interferometer introduced by the gap. Despite the fact in the experiments, e.g., the angle of the fiber facet/ SESAM relative to the SESAM/fiber facet (less than 7 degree due to the maximum angular range as shown in Fig. 3) and the angle of incidence (AOI) θ, it is justified to use a standard parallel-surface-model by assuming perpendicularly incident light beam. Hence, the FP filtering is approximately calculated in the form of:
where the reflectivity of the fiber facet is RF = 0.045, the reflectivity of the SESAM R2sat = 0.97 (in the saturated state), and the gap distance lgap is depending on the adjustment of the multi-axis stage. A representative curve for lgap = 303 μm is shown in Fig. 4(c).Fig. 4 Schematic of the principle of the CFE. (a) Net gain window (in purple) created by GF (in red) and HPF (in blue). (b) Experimental demonstration of the tunable mode-locked spectra (in color) stemmed from the versatile gain window, the grey curve represents the simulated power distribution of the resonant modes. (c) Reflectivity curve of the FP subcavity. (d) Plot of the phenomenological filtering function Tcom for the parameter set (αc, Ωc) = (0.0435, 3.11), the dashed line corresponds to the FP reflectivity in (c).
The second factor of an equivalent high-pass filter (HPF, or to say as optical low-pass filter) in the frequency domain is presented due to an inevitable slant of the EYDF relative to the SESAM. For a detailed explanation, a schematic of the laser resonator is presented in Fig. 3 with the part of the gap magnified. Typically, the tilt of the SA is manifested according to the magnified view, which leads to a deviation x1 when the light beam travels back to the facet of the fiber. Consequently, the distance x1 is frequency-dependent for the dispersion of the SA, which satisfies the relation
In the above equation, except for the parameters already defined in Fig. 3, n2 and n2' account for the refractive index of the SA and its first derivative with respect to the frequency w. As can been seen from Eq. (3) the lower frequency results in a larger x1 for n2' > 0 and the increase of the AOI θ can also cause a larger x1. Namely, the components with lower frequencies are more likely to be blocked from coupling back the fiber, thus yielding to the HPF. Aside from the SA layer with the thickness of d, analogous process causing another similar HPF effect should occur within the Bragg mirror (consisting of a multilayer-stack of alternate high and low index films) of the SESAM, which would dramatically aggravates the beam deviation and makes the HPF effect more susceptible to the AOI. In the practical operation of CW mode-locking, we have observed a series of optical spectra with persistent narrowing at the long-wavelength edge only by adding ~0.1° to the θ, as shown in Fig. 4(b). This experimental result gives a strong evidence for the existence of HPF.On the other hand, the nearly invariant short-wavelength edge indicates another inherent pulse shaping mechanism–gain filtering (GF), which is regarded as the third factor responsible for the CFE. The result shown in Fig. 4(b) reveals two points: i) the short-wavelength edge of the resultant spectrum coincides well with those of the measured optical spectra; ii) it literally covers the span of the tunable spectra recorded in experiment. The process of assessing the GF is referred to the Appendix. In consequence, as can be seen in Fig. 4(a), owing to the combined action between the HPF and GF, a net gain window (purple region), controllable by the AOI, is opened.
By summarizing these three aspects, the composite filter is characterized by a FP-induced weak spectral modulation superimposed upon the net gain window constituted by the GF and HPF. To describe it in a concise mathematical formalism, we propose a phenomenological transfer function:
As displayed in the Fig. 4(d), the spectral response Tcom exhibits two distinct maxima of the periodically modulated function RFP, and inherits both edges from the GF and HPF. Referring to the Eq. (4), the two maxima attains at w = -Ωc/2 and w = Ωc/2, respectively; moreover, the contrast between the maximum and central dip is interpreted by the exponent exp(-αc). For a better understanding, we summarize the principle of the CFE incorporating the FP, HPF and GF effect in Fig. 4.3. Experimental results
According to the experiments, careful adjustments on the linear translation and the deflection angle are carried out after an initial rough alignment. By ramping up the pump power to 280 mW, we observe a stable CW mode-locking with the output power of 2.18 mW. The performances of the output pulses are analyzed in both frequency and time domains, and the results are concluded in Fig. 5. As can be seen in Fig. 5(a) that the optical spectrum is centered at 1562.8 nm with a 3-dB bandwidth of 1.18 nm. It corresponds to a transform-limited hyperbolic-secant pulse width of 2.27 ps, indicating that the extracted pulse with a duration of 2.49 ps [sech2-assumed, see Fig. 5(b)] might be slightly chirped. By magnifying on the central part, a spectral fringe with a period of 0.026 nm is distinguishable, which reflects the intrinsic 3.2-GHz longitudinal mode spacing of the FP laser resonator. Interestingly, a dip-type sideband, highlighted by the blue arrow in Fig. 5(a), is visible at 1654.9 nm. Despite multiple possible mechanisms of such exotic structure available, for instance, destructive interference between the dispersive waves and solitons [43], absorption dynamics of the slow saturable absorber [22], we have a distinct hypothesis here that the dip derives from the CFE under certain conditions, which is discussed in detail in the next section. Regarding the oscilloscope trace shown in Fig. 5(c), the pulse repeats itself at every 312.5 ps, revealing the fundamental repetition rate of 3.2 GHz. To be more explicit, the RF spectrum centered at ~3.187 GHz is demonstrated in Fig. 5(d); its 77.2 dB SNR confirms a good short-term stability of the CW mode-locking.
Fig. 5 Experimental results of the single soliton operation. (a) Optical spectrum. (b) Autocorrelation trace. (c) Oscilloscope trace. (d) RF spectrum. The inset of (a) is a magnification of the top peak of the optical spectrum.
Interestingly, we find that a further translation of the x-axis away from the SESAM can remarkably raise the sub-peak (next to the dip) and simultaneously affects the pulse in time. Likewise, measurements are taken at a pump power of 284 mW when the optical spectrum develops a dual-peak-structure by continuous x-axis adjustment; the results are given in Fig. 6. Firstly, we have to mention that the mode-locking still maintains its fundamental repetition rate, which is corroborated via the 0.026 nm modulation period of the optical spectrum [see the inset of Fig. 6(a)], 312.5 ps timing interval between the neighboring pulses [see Fig. 6(c)], and 3.187 GHz prime comb of the RF spectrum [see Fig. 6(d)].
Fig. 6 Experimental results of the multi-pulsing state. (a) Optical spectrum. (b) Autocorrelation trace. (c) Oscilloscope trace. (d) RF spectrum. The inset of (a) is a magnification of the top peak of the optical spectrum, the curve in grey represents the spectrum of single soliton; the red dashed curve is the reconstructed autocorrelation trace via a tightly bound soliton pair.
Figure 6(a) shows symmetric maxima, divided by a central dip at 1564.9 nm, with the separation of 3.9 nm, whose structure is akin to that of the out-of-phase soliton molecule. In addition, we compared the spectrum with the one shown in Fig. 5(a); the identical position of the dips supports the hypothesis associated with the CFE to a certain extent. The autocorrelation trace in Fig. 6(b) is a typical pattern corresponding to the multi-pulsing with a temporal separation of 2.04 ps, which is consistent with above obtained 3.9 nm separation in frequency domain. An unexpected growth of the side peaks’ amplitude (SPA) is recognized, making the intensity ratio of the three peaks deviate from the normal 1:2:1 [see the black dashed line in Fig. 6(b)]. We attempt to retrieve this from a tightly bound pulse pair [44]; however, the fitting [red dashed curve in Fig. 6(b)] is not quite satisfying because the actual trace exhibits a more significant increase in side peaks and a raise in the pedestal. This mismatch almost rules out the case of bound state (BS). Figure 6(d) shows two kinds of symmetrical sidebands in the RF spectrum: the broad and blurred one centers at a 6.16 MHz frequency offset while the other with a contrast ratio > 20 dB located at ~12 MHz apart from the main peak. The less bright sidelobes can primarily be attributed to the relaxation oscillations [45,46]; whereas, the latter type, in most cases, can be regarded as evidence for periodic oscillations of certain quantities with respect to the pulse dynamics [47], e.g., variations of the pulse separation and phase difference in the circumstance of a vibrating soliton pair (VSP). Here, the oscillation frequencies ~12 MHz [12.012 and 12.156 MHz to be more specific, emphasized by the arrow in Fig. 6(d)] are much higher than the values reported before (e.g., 630 Hz) [47]. Yet, if we interpret the frequency ~12 MHz as the number of round-trips, the number of cavity round-trips of ~250 in this work are in the same order of magnitude compared to the ones in other laser systems [48,49].
Refer to the trait of the VSP, the current intensity ratio of the three peaks of the auto-correlation trace is confusing, as the vibrating process would reduce and not increase the SPA [47,50]. Hence, a numerical simulation was performed to obtain an unambiguous explanation.
4. Numerical simulation and discussion
The calculation is based on a lumped model that accounts for the non-instantaneous response of SA and CFE. During the iterative procedure, an equivalent ring configuration is established to substitute for the linear cavity scheme with the inclusion of the standing-wave effect; that is, EYDF with the length of Lc, SESAM, another segment of EYDF with identical length, and the DFs are handled in sequence (more details can be found in [28]). Correspondingly, the system of equations comprises a generalized nonlinear Schrödinger equation, rate equation for the SESAM, and phenomenological transfer function for the CFE:
The three equations in Eq. (5) characterize the gain fiber, SESAM (with the consideration of the CFE), and the DF (output port), respectively. Among them, ui(z,t) is the electric field amplitude for the i-th round-trip, in which z and t represent the propagation distance and the retarded time, respectively. The dispersion and nonlinearity coefficient of the fiber are denoted by β2 and γ, respectively. g is the saturable gain expressed by g = g0/(1 + ||u||2/Esat,G), where g0 and Esat,G account for the small signal gain coefficient and gain saturation energy, respectively. The reflectivity of the DF is R1 = 0.99. With regard to the equation for the time dependent absorbance q, τA is the relaxation time, q0 is the modulation depth and Esat,A is the saturation energy. Moreover, the dispersion of the SESAM induced by the Gires-Tournois interferometer cannot be neglected here for β2,sat = −1200 fs2 [51] and saturated reflectivity of R2sat = 0.97. It is worth noting that the decaying dynamics of the SESAM (SAM-1550-9-2ps, Batop GmbH) exhibits a multi-scale feature at an intense fluence [52], e.g., a fast intraband relaxation with a lifetime of ~80 fs arises at the laser fluence of 458 μJ/cm2. We assess the intra-cavity pulse fluence just in case and verify that the relation τA = 2 ps maintained at the fluence of ~130 μJ/cm2 in accordance with the experiments. The transfer function Tcom divided by R2sat derives because of the term (R2sat-q)1/2. Other parameters utilized in the simulation are: β2 = −100 fs2/cm, γ = 1.3 W−1km−1, g0 = 115 m−1, Esat,G = 2 pJ (variable), Ωg = 3π THz, Lc = 3 cm, q0 = 6%, and Esat,A = 12 pJ. The evolution of the electric field u starts from a Gaussian noise.Guided by the practical separation of the spectral maxima, e.g., 3.9 nm (related to ~2π × 0.5 in frequency) shown in Fig. 6(a), we use (αc, Ωc) = (0.0387, 3.11) to duplicate the multi-pulsing operation in the calculation, and subsequently gather the results from both the time and frequency domains in Fig. 7. Although the optical spectrum with two main peaks are not very equal in intensity, the numerically obtained separation of 3.8 nm agrees well with the value of 3.9 nm obtained in experiment as expected [relevant spectral evolution is shown at the right side of the Fig. 7(a)]. Notably, the simulated autocorrelation curve reproduces the two most puzzling traits of the experimental trace, that is, the abnormal intensity ratio of the three peaks with SPA > 0.5 as well as the extension of the pedestal. Figure 7(c) shows a periodic temporal evolution in which the pulse bunch switches back and forth between the single-soliton-shape [labelled 1 in Fig. 7(c)] with considerable sidelobes and the bound-state-structure [labelled 2 in Fig. 7(c)] with a minor pedestal. Both profiles enable the SPA > 0.5 [see the red and green curves in Fig. 7(b)] owing to the merit of the previously mentioned components of the sidelobes in time, and a reduction of SPA presents in the averaged autocorrelation trace owing to the internal pulsation of the soliton. To quantitatively characterize the oscillating feature, we performed the Fourier transformation on the vector of the pulse energy [extracted from an ensemble of 7000 iterative simulations, a standard fast Fourier transformation (FFT) algorithm is implemented as seen in Fig. 7(d)]; as a result, a peak with a 9.5 MHz frequency offset, shown in Fig. 7(e), which is reasonably close to the 12 MHz sideband measured by the signal analyzer [see in Fig. 6(d)].
Fig. 7 Numerical simulation of the pulse bunch. (a) Optical spectrum, and the right inset in (a) shows the optical spectral evolution of the pulse bunch. (b) Autocorrelation traces, the blue line represents the averaged trace calculated from an ensemble of 6000 simulations, the red and green curves are related to the single-solitonshape (labelled 1) and bound state structure (labelled 2), respectively. (c) Temporal evolution of the pulse, with pulse profiles relevant to 1 and 2 are demonstrated in time domain. (d, e) Plot of a vector of pulse energy in (d) and the corresponding function achieved from the Fourier transformation.
We attempted to emulate the x-axis translation in the experiments by varying the value of Ωc; for example, another set of (αc, Ωc) = (0.0387, 4) is applied, representing the decrease of the gap distance lgap from the previous 303.05 (Ωc = 3.11) to 235.62 μm (Ωc = 4). As can be seen in Fig. 8, it is numerically found that a stable single-soliton solution forms in this case, which is in agreement with the experimental observation. As shown by Fig. 8(a), the transition from the pulse bunch to single soliton gives rise to an abrupt degradation of one of the maxima in the frequency domain, with a slight shift of ~1.17 nm in the position of the dip occurring simultaneously. Indicated by the evolved spectra [see the inset of Fig. 8(a), the initial condition is a light field for Ωc = 3.11], the dip-type sideband is originated from the central minimum attributed to the CFE, and further reshaped by the effect of the noninstantaneous SA response. Thus, the previous hypothesis is partially admired, and the specific discussions of the noninstantaneous absorption of the SA is given in the following. The temporal profile of the pulse in Fig. 8(c) manifests a pulsewidth of 2.52 ps, whose lineshape perfers the Gaussian-type (red dots) to hyperbolic-secant [blue dots in Fig. 8(c)]. Furthermore, the conversion factor from the pulse duration to the autocorrelation trace width of 3.61 ps [see Fig. 8(b)] is ~0.7, suggesting that the lineshape is closer to be the Gaussian-type. Thus, the sech2-assumption need to be reconsidered when estimating the pulsewidth in experiment. Correspondingly, the time-bandwidth product of ~0.395 reveals a nearly transform-limited soliton instead of a slightly chirped one; this viewpoint can be well verified by the chirp profile shown in Fig. 8(c). Ultimately, we demonstrate a stable running of the single soliton, as shown Fig. 8(d).
Fig. 8 Numerical simulation of the single soliton. (a) Optical spectrum, the blue and grey curves accounts for the solutions of the soliton and pulse bunch, respectively. The inset depicts the transition from the pulse bunch to soliton. (b) Autocorrelation trace. (c) Temporal profile (in grey) and the frequency chirp (in blue), the Gaussian and sech2 fit to the pulse are represented by the red and blue dots, respectively. (d) Temporal evolution of the pulse.
For the better understanding of the generation of the single-soliton solution, we investigated the transient process as to the transition, as shown in Fig. 9(a). Cases I and II, corresponding to the parameter sets (αc, Ωc) = (0.0387, 3.11) and (αc, Ωc) = (0.0387, 3.8), are taken as examples of pulse bunch and single soliton, respectively. Motivated by the relationship between the BS stabilization and relaxation time of the SA studied in Ref [53,54], we evaluate the time intervals ΔT = 2π/Ωc of both cases, namely, ΔT = 2.04 ps for case I and 1.65 ps for case II. In case I, ΔT is slightly beyond the decay time of τA = 2 ps, and the pulse bunch with a periodic internal motion prevailing; when ΔT is shorter than τA in case II, sub-pulses evolved periodically can no longer be sustained. They begin to attract each other and finally become a single soliton because the SA cannot efficiently respond to adjacent pulses with a time separation of ΔT < τA. Figure 9(b) displays a transient status before converging to the stable single-soliton solution in case II, and one can identify a flat-bottom absorption profile of the SESAM at the leading part of the pulse can be indentified. The comparison with a representative absorption in case I reveals that the response time τA of the SESAM defines a critical time separation ΔTc = τA for the sub-pulses within the pulse bunch. Furthermore, we examine the energy trajectores at different Ωc values (ΔT > 2 ps is guaranteed) when every light field u approaches to its local attractor, as shown in Fig. 9(d). It demonstrates an underlying connection between the convergence rate and the Ωc (or say ΔT, lgap). The increase of Ωc (decrease of ΔT or lgap) promotes the convergence to the single-soliton solution; in physical terms, the single-soliton build-up time is shortened.
Fig. 9 Transition from pulse bunch to single soliton. (a) Transient process as to the transition. The curves of the Tcom’ relevant to case I and II are also shown. (b,c) Pulse shape and the corresponding absorption profile in case (c) I and (b) II. (d) Change in energy as a function of round trip number for different Ωc values.
By increasing the gain saturation energy Esat,G up to 8 pJ in the parameter set (αc, Ωc) = (0.0387, 3.11), a stable pulse bunch can be numerically realized. Under this condition, the temporal evolution of the solution is given in Fig. 10(c), exhibiting the equilibrium with a constant group velocity. The optical spectra of the stable and vibrating pulse bunches are compared in Fig. 10(a), and it can be seen that the modulation superimposed upon the envelope of the optical spectrum has brighter contrast ratio in the case of stable multi-pulsing. We also investigate the pulse profile and its phase information in the time domain [see Fig. 10(b)]. The ~π dephased pulse doublet inside the pulse bunch is akin to the class of double-pulse solutions of the complex quintic SHE in the anomalous dispersion regime [55]. It is not surprising because the master equation of the current physical model accounting for Eq. (5) is approximately in the Swift-Hohenberg form.
Fig. 10 Simulation result of the stable pulse bunch. (a) Optical spectra. The blue and grey curves represent the stable and vibrating pulse bunch, respectively. (b) Temporal shape and relevant phase profile of the pulse. (c) Temporal evolution of the stable pulse bunch.
5. Conclusion and outlook
In conclusion, we successfully implemented a SESAM mode-locked fiber laser with a fundamental repetition rate of 3.2 GHz, where the spot size imposed on the SESAM is expanded in a certain extent, which suppresses the Q-switching induced damage by removing the beam collimating and focusing in the free space between the fiber and SESAM. In this case, a mutual interaction between the FP filtering, HPF and GF, denoted by the notion of CFE, attributes to the additive pulse shaping in the soliton mode-locking. By adjusting the distance of the gap, a switchable state from single soliton to pulse bunch can be experimentally achieved; in addition, it was appropriately reproduced by numerical simulation. Further theoretical investigations disclose that the response time of the SESAM plays an important role in determining the regime of single-soliton and pulse bunch solution. Guided by the principle presented here, we believe that the lasing (single-soliton or pulse burst) is fully controllable and can be employed in potential applications. Last but not least, we have confirmed an excellent long-term stability of the present scheme by letting the fiber laser operate within a month.
The pulsating characteristic of the multi-pulsing was demonstrated by indirect signature only, i.e. sideband in the RF spectrum, based on time-averaged signal analysis. Real-time diagnostic, e.g., dispersive Fourier transform, is readily realized nowadays, especially in the 1.5 μm wavelength range [56,57]. As to the pulse train with ultra-high fundamental repetition rate, the overlap between the pulse waveforms is critical and requires special attentions. One way to avoid the overlapping is to filter out a slice from the pulse output with a bandwidth of ΔλF, where the condition Δλ F < Tr/β2DFT must be satisfied (where Tr is the round-trip time and β2DFT is the total accumulated dispersion induced by the dispersive fiber, i.e., ΔλF < 1.5 nm for β2DFT = 200 ps/nm). This might be presented in our further work.
Appendix (assessing the gain filtering effect)
We have adopted a posterior that the gain profile can be approximately mapped into the resonant modes in the CW lasing. To acquire an intensity distribution of the longitudinal modes in the laser cavity, the following propagation equations for the pump power Pp(z) and amplified spontaneous emission (ASE) powers Ps±(z,λk) (± represents forward and backward propagations, respectively) are used [58,59]:
The differential equations describe both the pump and ASE powers in spatial coordinate z, where z[0, Lc]. Here, we create only 91 channels for longitudinal modes λk ranging from 1520 to 1610 nm with a step Δλ = 1 nm to roughly estimate the ASE profile, with the pump set to λp = 980 nm. In the steady-state condition, Eq. (6) is further constrained by the relations between the local normalized population densities n1, n2, n3, and n6 in the Er3+/Yb3+ co-doped system, described by,where NEr and NYb are the Er3+ and Yb3+ concentration, respectively. Ni, ni, and Aij are the population density, normalized population density, and relaxation rate of the ith level, where i = 1, 2, 3, 4 indicates the 4I15/2, 4I13/2, 4I11/2, 4I9/2 erbium levels while I = 5, 6 indicates the 2I7/2, 2I5/2 ytterbium levels. Wij is the stimulated transfer rate from level i to j, and σij is the transition cross-section between levels i and j. Cup and Ccr are the upconversion and cross-relaxation coefficient. Γp and Γs are the overlap factors of the pump and ASE, in which Γs is assumed to be independent of the wavelength. The propagation loss of the ASE components is denoted by α, Acore is the core area, h = 6.626 × 10−34 is the Planck constant and c = 3 × 108 is the speed of light in vacuum. By imposing the following boundary conditionsEquations (6) and (7) can be numerically solved. Pp0 is the injected pump power, R1 is the reflectivity of the film, and R2 is the non-saturable reflectivity of the SESAM. Values of the primary parameters used in the numerical calculation are as follows [59]: NEr = 6 × 1025 m−3, NYb = 1.2 × 1026 m−3, A21 = 1.43 × 102 s−1, A32 = 2.8 × 106 s−1, A43 = 7 × 109 s−1, A65 = 1 × 103 s−1, σ13(λp) = 1.68 × 10−25 m2, σ56(λp) = 1.01 × 10−24 m2, σ65(λp) = 1.27 × 10−24 m2, Cup = 2 × 10−24 m3/s, Ccr = 2.1 × 10−22 m3/s, Γp = Γs = 0.9, α = 2.3 m−1, Acore = 7.85 × 10−11 m2, Pp0 = 0.2 W, R1 = 0.99 (assumed to be invariant for the 91 channels), R2(λk) is available on the website of BATOP, and absorption/emission cross-sections data σ12/σ21 is available from the corresponding author on request. Then the mode power distribution is obtained for Pout(λk) = [1-R1(λk)]Ps-(0,λk) to designate the gain profile.Funding
National Key Research and Development Program of China (2016YFB0402204); High-level Personnel Special Support Program of Guangdong Province (2014TX01C087); Science and Technology Project of Guangdong (2015B09 0926010); Fundamental Research Funds for the Central Universities (2017BQ110); China Postdoctoral Science Foundation (2016M602462); National Key Research and Development Program of China (2016YFB0402204); the Science and Technology Project of Guangdong (2016B090925004).
Acknowledgments
We thank Dr. Florian Adler from Tiger Optics, LLC and Prof. Alfred Leitenstorfer from University of Konstanz for providing the relaxation time data of the saturable absorber. We thank Dr. Wolfgang Richter of BATOP, GmbH for offering the reflectance and dispersion data of the SESAM.
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