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Abstract
Fundamental repetition rates of 3.1 GHz, 7.0 GHz, and 12.5 GHz in passively modelocked Yb-doped fiber lasers are demonstrated. To the best of our knowledge, the fundamental repetition rate of 12.5 GHz is the highest value for 1.0 μm mode-locked fiber lasers. The mode-locked oscillator has a peak wavelength of 1047.5 nm and a pulse duration of 1.9 ps. The repetition rate signal has a signal-to-noise ratio of 57 dB. The peak wavelength of mode-locked spectra gradually makes a blue-shift and the modelocked threshold power increases with an increase in pulse repetition rate. Furthermore, in contrast to most of the all-normal-dispersion mode-locked fiber lasers, the present linear resonator (e.g., length < 1 cm) allows the buildup of gain-guided soliton without any filter effect. To unveil the underlying pulse shaping mechanism, a combined model comprising dynamic rate equations and the generalized nonlinear Schrödinger equation is established. Surprisingly, an essential gain filtering effect, which is contributed by a 26-nm gain bandwidth, is revealed and can verify the gain-guided pulse dynamics. Moreover, the pulse build-up in temporal and frequency domain, like spectral evolution and gain bandwidths, is numerically carried out in detail.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ultrafast lasers with high repetition rate (>10 GHz) are of interest for a variety of applications in nonlinear bio-optical imaging, material processing, light communication system, and laser frequency combs for astronomical observations [1–4]. For nonlinear bioimaging techniques such as two-photon fluorescence excitation, higher pulse repetition rate with constant pulse energy can improve the number of pixels for certain distance and hence increase the image frame rate [5,6]. Recently, an optical transmission system based on a 10-GHz laser enabled a data transmission rate up to 661 Tbit s−1, which is faster than the total Internet traffic today [7]. Within the progress in radial-velocity technique, the laser frequency comb-based detection of Earth-mass extra-solar planets requires laser repetition rate of 10-20 GHz range to achieve the spectrograph calibration precision of cm∙s−1 level [8,9]. Unlike harmonic mode-locking techniques [10], modelocked lasers with fundamental cavity repetition rate (a single pulse circulating in the laser resonator) are more reliable for high-precision photonic applications owing to their high spectral purity and stability. Over the past decade, modelocked lasers with fundamental repetition rates of >10 GHz have been realized based on laser gain mediums such as solid-state glasses/crystals [11–13], semiconductor disks [14–16], waveguides [17,18], and fibers [19,20]. 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. Ultrafast fiber lasers with multi-GHz repetition rate have been studying both in theories and experiments [21–32]. Martinez et al. [20] have successfully presented a passively modelocked Er3+/Yb3+ co-doped fiber laser with a repetition rate of 19.45 GHz, which is the highest fundamental repetition rate at 1.5 μm wavelength range. However, as for Yb:fiber based oscillators that have positive group velocity dispersion (GVD) at the wavelength of ~1.0 μm, a fundamental repetition rate of >10 GHz has not yet been reported.
First proposed by Bélanger and associates in 1989 [33], gain-guided pulse refers to that a finite-bandwidth gain medium supports soliton propagation in positive dispersion regime. Despite the fact that action of gain bandwidth has been recognized in the soliton formation for long time [34,35], until 2006 gain-guided soliton in an Er-doped fiber ring laser made of purely positive GVD fibers was experimentally observed by Zhao et al. [36,37]. In analogy to a real spectral filter, the gain with finite-bandwidth can play a crucial role in pulse compression during the soliton propagation [38]. As for the gain-guided soliton dynamics in laser cavity, the scenario can become more intricate because effects of both the gain filtering and the self-amplitude modulation from the saturable absorber contribute to the pulse shaping process [39]. Hitherto, previous work on gain-guided soliton were concentrated on mode-locked Er-doped fiber lasers with a relatively narrow gain bandwidth of approximately 40 nm [36–40], but the gain-guided soliton in Yb-doped fiber lasers has been rarely reported. To the best of our knowledge, there are only several similar reported works, which, however, employed an artificial spectral filter within laser cavities. Dissipative solitons were realized by incorporating a section of polarization-maintaining (PM) fiber [41], a loop mirror as a spectral filter component [42], or an absorber mirror with deep modulation depth [43]. The sources have demonstrated the modelocked operation in all-normal-dispersion Yb-doped fiber lasers, but the filter components of the pulse-shaping do not arise in the gain medium. Moreover, the characteristics of gain filter (e.g. gain bandwidth) in these works were not revealed.
In this work, gain-guided solitons in all-normal-dispersion Yb-doped fiber lasers with fundamental cavity repetition rates of 3.1 GHz, 7.0 GHz, and 12.5 GHz are demonstrated. During scaling process of the pulse repetition rate, the peak wavelength of mode-locked spectra gradually blue-shifted and the mode-locked threshold power increased. In the past, gain profile of Lorentzian line with an estimated gain bandwidth in the parabolic-gain approximation was generally considered in theoretical calculation [44–46]. In order to obtain exact value of gain bandwidth, a combined numerical model constituting of dynamic rate equations and the generalized nonlinear Schrödinger equation is proposed. An essential gain filtering effect contributed by a 26-nm finite-bandwidth is revealed and it confirms the gain-guided pulse dynamics. Moreover, the model gives more temporal and spectral evolution details in pulse shaping.
2. Experimental setup and results
2.1 Experimental setup
The passively mode-locked Yb-doped fiber lasers with fundamental repetition rate reaching 12.5 GHz can be implemented in an all-fiber Fabry-Perot (FP) laser cavity, which is shown in the photograph of Fig. 1(a). The robust laser cavity consisted of a section of Yb:glass fiber (YGF), a fiber-type dielectric mirror, and a commercially-available semiconductor saturable absorber mirror (SESAM). As a key component for pulsed behavior with GHz fundamental repetition rate, the gain fiber used in this work is a piece of heavily Yb3+-doped phosphate glass fiber with an Yb3+ doping concentration of 15.2% [47]. The YGF was secured with epoxy inside a ceramic ferrule with an inner diameter of 125 μm, and both end facets were perpendicularly polished. A ceramic ferrule, coated by the dielectric films that have a reflectivity of 99% at the lasing wavelength and a transmission of 60% at the pumping wavelength, was pigtail spliced to the common port of a wavelength-division multiplexer (WDM). One end of the YGF was butt-coupled to the dielectric mirror, and the opposite end of the YGF was affixed to the SESAM. The SESAM has a modulation depth of 5%, and a saturated fluence of 40 μJ/cm2 at the wavelength of 1040 nm for the self-started mode-locking operation; to be more specific, it possesses 3% non-saturable loss, 1 ps recovery time, and 23 fs2 second-order dispersion at the signal wavelength. The pump light was provided by a 976-nm laser diode, coupled into the laser oscillator through the pump port of the WDM. The entire cavity lengths were 30.5 mm, 13.8 mm and 7.6 mm for the fundamental repetition rates of 3.1 GHz, 7.0 GHz, and 12.5 GHz, respectively. Furthermore, the whole laser cavity was temperature controlled by a thermo electric cooler to guarantee the long-term stable mode-locking operation.
Fig. 1 (a) The photograph of the Fabry-Perot fiber laser cavity, and the inset shows spectral reflectance of the DF and SESAM (b) Schematic diagram of the laser cavity for numerical simulation. DF, dielectric films; SESAM, semiconductor saturable absorber mirror; GF, gain fiber.
2.2 Results
The power thresholds for self-started mode-locking were measured to be 105 mW, 111 mW, and 148 mW for the GF length of 30.5 mm, 13.8 mm, and 7.6 mm, respectively. The modelocked threshold gradually increased with a decreasing in GF length, which is attributed to the decrease of net gain that can be compensated by a stronger pump intensity [16,22]. The measured optical spectra of the passively mode-locked oscillators in the scaling process are summarized in the first row of Fig. 2. All the optical spectra were recorded by using an optical spectrum analyzer (YOKOGAWA AQ6370B) with a resolution of 0.02 nm. The peak wavelengths are 1060.7 nm, 1050.3 nm, and 1047.5 nm and spectral bandwidths at full-width half maximum (FWHM) are 1.02 nm, 1.0 nm and 0.9 nm, for the GF length of 30.5 mm, 13.8 mm, and 7.6 mm, respectively. The peak wavelength of mode-locked spectra gradually blue-shifted with an increase in the pulse repetition rate, which could be ascribed to the reduced intracavity influence at higher repetition rate. As the mode-locked operation results from a balance of dynamic gain and saturable absorber loss, the reduced intracavity influence contributes the lasing wavelength shifts to short-wavelength side where a higher emission cross section will compensate the increasing saturable loss. In particular, the longitudinal modes in the mode-locked spectrum of 12.5-GHz laser can be resolved, as shown in Fig. 2(c). The longitudinal mode spacing is 0.06 nm, which matches the cavity length of 7.6 mm. In addition, in contrast to conventional soliton possessing regular optical spectrum, the output optical spectra in high-repetition-rate oscillators have an asymmetric shape, which results from the gain-induced pulse shaping process and will be explained in the theoretical part.
Fig. 2 Experimental optical spectra and autocorrelation traces: Optical opectra of oscillators with repetition rate of 3.1 GHz (a) with a peak wavelength of 1060.7 nm, 7.0 GHz (b) with a peak wavelength of 1050.3 nm, and 12.5 GHz (c) with a peak wavelength of 1047.5 nm. The inset in the right of (c) is the optical spectrum at a span of 0.2 nm, indicating a longitudinal mode spacing of approximately 0.06 nm, as expected for a laser operating at a repetition rate of 12.5 GHz. Measured autocorrelation trace of (d) 3.1-GHz laser showing a pulse duration of 3.2 ps, (e) 7.0-GHz laser showing a pulse duration of 3.2 ps, and (f) 12.5-GHz laser showing pulse duration of 1.9 ps by fitting with a hyperbolic secant pulse profile.
Furthermore, measured autocorrelation traces of the oscillators are shown in the second row of Fig. 2. Because the pulse peak power is very low for the multi-GHz repetition rate oscillators, the average output powers were amplified for measuring autocorrelation in experiment by an Yb-doped fiber amplifier. With sech2-pulse shape assumed, the pulse durations are 3.2 ps, 3.2 ps, and 1.9 ps for the GF lengths of 30.5 mm, 13.8 mm, and 7.6 mm, respectively.
The pulse characteristics were measured by a radio-frequency (RF) spectrum analyzer (Rohde&Schwarz FSWP26, 26.5-GHz bandwidth), a real-time oscilloscope (Keysight DSOV204V, 20-GHz bandwidth) together with a photodetector with a bandwidth of 25-GHz. As illustrated in Figs. 3(a)-3(c), the RF spectra are plotted; particularly, the panel (c) portrays a frequency peak centered at 12.5374 GHz in a scanning range from 12.53710 to 12.53780 GHz. The signal-to-noise ratio (SNR) of 57.1 dB implies a mode-locked operation free of Q-switching instability. Likewise, high-SNR performances of the mode-locking with repetition rates of 3.1175 and 7.0213 GHz (relevant to the fiber lengths of 30.5 and 13.8 mm) are demonstrated by examining the RF spectra in panels (a) and (b). That is, high-quality modelocked pulse trains, with fundamental repetition rate up to 10 GHz, are readily generated through the simple FP configuration. In Figs. 3(d)-3(f), the pulse trains registered by the oscilloscope reveal time periods of ~313 ps, 143 ps, and 79 ps which correspond to the GF lengths of 30.5 mm, 13.8 mm, and 7.6 mm, respectively. Temporal waveforms measured by an oscilloscope had not been shown in previous work, that reported passively modelocked fiber lasers with fundamental repetition rate of over 10 GHz [19,20]. Here, for the first time to the best of our knowledge, the measured temporal waveforms of continuous-wave (CW) mode locked pulses are given for pulse repetition rate higher than 10 GHz.
Fig. 3 RF spectra and waveform measurements: Spectra of photodiode signal for laser cavity lengths of 30.5 mm (a), 13.8 mm (b) and 7.6 mm (c) acauired with an RF spectrum analyzer. Laser waveforms for laser cavity lengths of 30.5 mm (d), 13.8 mm (d) and 7.6 mm (f) measured using an oscilloscope and a photodiode having bandwidths of 25 GHz and 12.5 GHz, respectively, and the intervals between intensity peaks are ~313 ps, ~143 ps and ~79 ps.
Figure 4(a) shows the average output power as a function of the launched pump power for the oscillator with GF length of 13.8 mm. When the launched pump power is lower than 110 mW, the mode-beating and Q-switching instabilities prevail. Further gain ramping facilitates the transition into CW mode-locking, simultaneously, the corresponding slope efficiency drops slightly, by a factor of 0.18. Moreover, it is crucial to apply thermal management to the laser cavity, because of the temperature increase attributed by the heat shed from the saturable absorption of the SESAM as well as the heat accumulation on the YGF. In experiment, the temperature of the packaged laser resonator was kept at 20°C for the wavelength stabilized operation. A record of successive optical spectra acquired with a sampling interval of 40 second was used to evaluate the long-term stability of the laser emission. In Fig. 4(b), the ensemble of the spectra, manifested as the false color map, indicates that the modelocked operation can be well sustained for 12 hours without obvious wavelength drifting.
Fig. 4 (a) Measured (blue dots) and simulated (olive line) variation of the laser average output power for 7 GHz oscillator with the launched pump power (976 nm). (b) False color map of optical spectra of 7 GHz mode-locked oscillator recorded for 12 h, one datum per 40 second. While recording the data, temperature of laser cavity was maintained at 20°C.
3. Combined model and pulse-shaping of gain-guided soliton
3.1 Combined model
By referring to the insights into the DS solutions, it is known that in the normally dispersive mode-locked fiber lasers the filtering effect merits attention. Intriguingly, in our present experiments, the SESAM-based mode-locking at multi-GHz repetition rates is fulfilled in an all-normal-dispersion geometry without employing any bandwidth limiting elements (As seen in inset of Fig. 1(a), both DF and SESAM possess broadband reflection and make negligible contribution to the spectral filtering effect). To dissect the latent pulse shaping mechanism here, we exploit a lumped model [22,25] to resolve the build-up of the mode-locking. The schematic of laser cavity is depicted in Fig. 1(b). The typical gain bandwidth of >80 nm [48] in Yb-fiber laser imparts insufficient spectral dissipation to form a DS (calculation in detail is not presented here, and a similar numerical procedure can be seen in [29]). To reproduce the practical situation in a more explicit approach, we rewrite the nonlinear Schrödinger (NLS) equation to yield a modified set of equations:
ui,SESAM represents the light field ui(Lc,t) after being modulated by the SESAM. The operator L in angular frequency Ω domain, accounting for the dispersion curve of the SESAM, is manifested by a truncated Taylor expansion in the form of L(ω) = β2aω2/2! + β3aω3/3! + β4aω4/4! + β5aω5/5!; consider the angular frequency Ω in THz, the dispersive coefficients are: β2a = 2.4 × 10−5, β3a = 2.18 × 10−6, β4a = 1.32 × 10−8, β5a = 4.35 × 10−9. The RSAM(ω) is the wavelength-dependent low-intensity spectral reflectance of the SESAM (Inset of Fig. 1(a)). The other parameters are summarized in Table 1.
Table 1. Parameters of Mode-Locked Ytterbium Fiber Laser at 7-GHz Repetition Rates
Inspired by the fiber amplifier models comprising dynamic rate equations (DREs) and pulse propagation equations [49–51], the gain response g(ω) can be described by means of standard DREs. In our current circumstance, two conditions meet: i) the pulse width is far shorter than the upper-state lifetime τG, ii) the pulse energy ‖u‖2 is much lower than the gain saturation energy EG = τGEg/TR (~30 μJ here); thus, it is justified to write the DREs in the steady-state approximation [51]:
and the population density of the upper-level state N2 satisfies:Notably, the Ω-dependent distributed gain g(z, Ω) in Eq. (2) is determined by the laser rate Eqs. (3) and (4), and represents in the form ofDefinitions and values of the parameters therein are also concluded in Table 1. Besides, h = 6.626 × 10−34 is the Planck constant and c = 3 × 108 is the speed of light in vacuum. Basically, Eq. (1) evinces the dispersive and nonlinear effects imposed on the complex amplitude profile u(z,t) in t-domain; while, the propagation-rate Eq. (3) characterizes the intensity profile Pp(z) and Ps ± (z,λk) in f-domain ( ± represents forward and backward propagations, respectively).Before the establishment of a combined model for the oscillators with millimeter-scale cavity length, it is critical to relate the field envelope with the signal intensity for reconciling the Eqs. (1) and (3). We designate the time window Tω and size of the time vector N used in the split-step Fourier algorithm. In our case it is preferred to choose TR as the time window Tω, which enables a counterpart of the discrete cavity longitudinal modes with a interval of Δω = 2π/TR. The optical spectrum is divided into N channels with uneven spaces, that is, λk = 2πc/(ωk + ωref) where ωk is the k-th element within the Fourier transform of time vector t; ωref is a reference angular frequency so that each Fourier frequency ωk is assigned to a specific wavelength λk, here ωref corresponds to a reference wavelength λref = 1050 nm. In contrast to the Fourier series û(z,ωk) which describes the individual spectral envelope with the Fourier frequency ωk, the Ps(z,λk) represents the sum of power covering the whole spectral components around λk with a spectral range of Δλ. As a result, it is straightforward to connect the spectral envelope û (z,ωk) with signal power Ps(z,λk) by means of [23],
To numerically solved the combined model self-consistently, the forward/backward signal powers Ps ± (z, λk), governed by the Eqs. (3) and (4), only have to coincide with the ones calculated by the u(z,t) in the master Eq. (2) at both ends of the linear resonator, the same for pump power Pp± (z). The boundary conditions are written by
3.2 Pulse shaping of gain-guided soliton
To obtain a visual understanding of the gain dynamic in the pulse shaping process, net gain spectrum gnet(ω) and averaged inversion level Nave are applied by the relations:
where Nave facilitate us to understand the variation of the gnet(ω) [51]. As illustrated in Fig. 5(a), the averaged inversion level Nave has a decreasing tendency in the nascent stage of evolution, and varies in a much lower speed afterwards. The descending averaged inversion level Nave leads to a red shift of the peak wavelength of the net gain gnet(ω). For the gain bandwidth change in detail, the transient shape of gnet(ω) at the 10th RT and the 90th are extracted out, as shown in Fig. 5(c). It indicates that, aside from the tendency for red shift, gain narrowing from 32 nm to 26 nm (refers to [52] for the estimation of the FWHM) occurs. The gain bandwidth is gradually reduced in pulse shaping process, eventually resulting in 26 nm when the average inversion extent barely changes. The steady gain bandwidth could result from the balance dynamic gain and saturable loss from the SESAM in the normal laser cavity where the red component of the optical spectrum experiences a larger group velocity. Therefore, the soliton in the multi GHz-repetition-rate ANDi MLFL is substantially gain-guided [31].Fig. 5 (a) Averaged population inversion level Nave of the 7-GHz per roundtrip (RT) from 10 to 90 RTs; (b) The corresponding net gain profiles; (c) The net gain line shape at the 10th RT with a full width at half maximum (FWHM) of ~32 nm (blue line), together with line shape at the 90th RT with ~26 nm FWHM (green line).
Regarded as a demonstration in Fig. 6, one can observe a robust evolvement from an initial white noise to a stationary pulse solution within 3500 roundtrips (RTs). In the early stage which the temporal signal is still in less localized structure (typically, the first 100 RTs), a red shift of the intensity profile features the spectral evolution, which can be attributed to the gain filtering effecting. When the gain bandwidth retains an approximate steady value, it acts in conjunction with saturable loss, dispersion, and nonlinearity, to yield the generation of stable gain-guided soliton. Furthermore, we proceed with comparing the simulated and experimental results on power curve in Fig. 4(a) and achieve a satisfying agreement, which evidence the validity of the combined model. However, in the absence of the rate equation accounting for dg/dt (see the derivation of the stability criterion against Q-switching in [53]), such combined approach has not prepared to predict the threshold for mode-locking [i.e., the shaded region in Fig. 4(a)] yet. Despite the lack in forecast the operation status, associating the ideas of the usage of delay differential equation [54,55] may open up a way to improve the present model. We look forward to developing a full physical model of the GHz-repetition-rate MLFL.
Fig. 6 (a) The stimulated temporal evolution of the 7-GHz laser; (b) The corresponding optical spectral evolution.
4. Conclusion
We have obtained gain-guided soliton with fundamental cavity repetition rates of 3.1 GHz, 7.0 GHz, and 12.5 GHz in the all-normal-dispersion Yb-doped fiber lasers. To the best of our knowledge, 12.5 GHz is the highest fundamental repetition rate for modelocked Yb-doped fiber lasers. The peak wavelength of the oscillator is 1047.5 nm and the pulse duration is measured to be 1.9 ps. During the scaling of the repetition rates, the CW mode-locking threshold power increases and the corresponding peak wavelength is gradually shifted to short-wavelength range. Furthermore, in contrast to gain profile of Lorentzian line with an estimated gain bandwidth in the parabolic-gain approximation generally exploited in previous theoretical calculation, a combined model comprising dynamic rate equations and generalized nonlinear Schrödinger equation is proposed and calculated. The model not only provides more details both in temporal and spectral domain, exhibiting accurate results in the evolution process, but also reveals a gain filter effect with 26-nm finite-bandwidth, which further confirms the gain-guided pulses dynamics.
Funding
National Key Research and Development Program of China (2016YFB0402204); the Science and Technology Project of Guangdong (2015B090926010, 2016B090925004, 2017B090911005); Fundamental Research Funds for the Central Universities (2017BQ110).
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