We identify a new regime where laser pulses represent the dynamics of rectangular-shaped wave packets (RSWPs) in a passively mode-locked Tm3+-doped fiber laser. In this regime the laser consists of a train of mode-locked pulses underneath a rectangular-shaped envelope. The density of pulses within a RSWP can be as high as 2.8 GHz, which is consistent with cavity fundamental repetition rate. The effects of small-signal gain value, pulse repetition rate, and net dispersion on the RSWP performance are analyzed. These results imply that this new regime particularly favors high-repetition-rate ultrafast lasers. We further reproduce the phenomenon from using numerical simulations and understand such behavior by referring to the nonlinear dynamics.
© 2017 Optical Society of America
The pulse repetition rate of multi-gigahertz mode-locked fiber lasers operating at a wavelength of 2 μm are of great interest for various applications, including optical communications, high speed optical sampling, and high-precision frequency metrology [1,2]. Fundamental mode-locking provides a better spectral purity than that in harmonic mode-locking . In this case, with the presence of a single pulse in the cavity, the pulse-repetition-rate fREP is equal to the free spectral range of the cavity fFSR. Therefore, to increase the pulse repetition-rate to multi-gigahertz frequencies, the corresponding laser resonant cavity should be shortened to a few centimeters [4–7]. In the past, the highest pulse repetition rate achieved was reported at up to 3 GHz  and 19 GHz  by respectively employing a 1-cm Yb-doped fiber and 0.5-cm Er/Yb co-doped fiber to construct the ultrashort laser cavities.
Yb3+-doped, Er3+-doped, and Er3+/Yb3+ co-doped gain fibers that are used to shorten the laser cavity length, are always expected to possess the considerably high gain coefficient among early experiments. Unfortunately, Tm3+-doped fibers (TDFs) have been suffering from restrictions such as high Tm3+-doped concentration and low OH- content of glass fibers [8,9]; thus, it has been difficult to satisfy the gain requirements to achieve the same fundamental pulse-repetition-rate level as reported at 1 or 1.5 µm. Until recently, our group reported a pulse repetition rate of 1.6 GHz at a wavelength of 1960 nm  using a 5.9-cm heavily Tm3+-doped barium gallo-germanate (BGG) glass fiber as the gain medium. However, compared to the results at 1 and 1.5 µm, this value is still low and will restrict some photonics applications that require higher pulse repetition rate. Thus, this is a motivation for further shortening the length of the TDF in the laser resonant cavity. Generally, it will lead to the removal of the continuous wave (CW) mode-locking operation because the total gain is not sufficient to reach the threshold for CW mode-locking. Alternatively, a pulse dynamics regime has emerged in the ultrashort laser cavity.
For typical mode-locked fiber lasers, pulses can exist in various dynamic patterns with different cavity parameters, such as soliton bound states [11,12], harmonic mode-locked states [13,14], soliton rains [15,16], soliton compounds , noise-like pulses , and pulse bursts [19,20]. These pulse patterns allow them to be widely applied in biomedical diagnostics, optical measurement instruments and high-speed optical communication. However, those pulse characteristics are very (or partially) different from the ones of the new dynamics regime. For an example, the pulse repetition rate in pulse bursts, directly produced from mode-locked fiber laser, is not determined by cavity fundamental repetition rate . Moreover, the pulse dynamics in ultrashort (e.g., a few centimeters) cavity ultrafast lasers has hitherto received little attention. What happens in a laser cavity when its small-signal gain coefficient g0 is very low with gradually shortening the laser cavity? The current work answers this question from experiments and simulation.
In this paper, we experimentally and theoretically investigated the dynamics of a rectangular-shaped wave packet (RSWP) in an ultrashort cavity passively mode-locked TDF laser. In the regime, the laser output consists of a train of mode-locked pulses underneath a rectangular-shaped envelope. The density of pulses within a packet can be as high as 2.8 GHz corresponding to a 3-cm Fabry-Pérot (FP) laser cavity. The RSWP operation represents unique optical characteristics, i.e., as the pump power increases, the duration and repetition rate of individual packet simultaneously increase but the pulse repetition remains constant at the cavity fundamental repetition rate. Furthermore, the effects of small-signal gain value, net dispersion on the RSWP performance are analyzed. We reproduce the phenomenon using numerical simulations and understand such behavior by referring to the nonlinear dynamics.
2. Experimental setup
The heavily Tm3+-doped BGG glass fiber used in the experiment were fabricated by the rod-in-tube technique . The Tm3+ doping concentration reaches 4.5 × 1020 ions/cm3, and the gain coefficient of the fiber is 2.3 dB/cm at 1950 nm. The TDF has 8.6/125 µm core/cladding diameters with the numerical aperture of 0.145. The group velocity dispersion of the fiber is estimated to be ~-100 fs2/cm.
A schematic of the experimental setup is shown in Fig. 1. A piece of 3 cm TDF as a gain medium was pumped by a 793 nm/250 mW laser diode (LD) through a wavelength division multiplexer (WDM). The TDF was glued in a ceramic ferrule with an inner diameter of 125 µm, two end facets of which were then perpendicularly polished. One end of the TDF was butt-coupled to a fiber-type dielectric mirror, which was spliced to a common port of the WDM. This mirror was fabricated by directly coating multiple-layer SiO2/Ta2O5 dielectric films onto a fiber ferrule using a plasma sputter deposition system. A photograph of the mirror is displayed in the left inset of Fig. 1. The other end of the TDF was connected to a semiconductor saturable absorber mirror (SESAM), which is capable of being sandwiched between the TDF and fiber ferrule because of the compact size. The right inset of Fig. 1 displays the SESAM placed onto the end of a fiber ferrule. The laser at ~2 µm wavelength is output from the signal port of the WDM. A polarization controller (PC) is utilized for optimization of the performance of RSWP operation. In the miniature FP oscillator, the length of the 3 cm TDF represents the entire cavity length, indicating that the fundamental pulse repetition rate is approximately 2.8 GHz. Besides that, the length of TDF is shortened to 3 cm, and the parameters of the mirror, SESAM, and associated measurement equipment, can be found in another work .
3. Results and discussion
The RSWP operation starts at a pump power of 40 mW. The representative waveforms of the RSWP in the temporal domain are illustrated in Fig. 2. It can be seen that, the wave packets are uniformly distributed with an interval of 35.56 µs. Though the envelope of the pulse is periodic and seems similar to the Q-switched mode-locking operation, but they are different in performance and mechanism. It is apparent that, the rising/trailing edges for the RSWP operation are considerably sharp. The duration of a packet is about 4 μs. In order to characterize the details within a packet, a portion of the waveform is magnified in Fig. 2(b), indicated by a white arrow in Fig. 2(a). The pulse train of the oscilloscope trace revealing the laser pulses with a repetition rate as high as 2.8 GHz, indicates that the oscillator operated at a fundamental pulse-repetition-rate. For conventional Q-switched mode locking, which comprises a train of mode-locked pulses under a Q-switched envelope, the pulse energy appreciably changes over several cavity round-trips. By comparison to the case of the RSWP, the pulse energy and pulse shape remain constant over many cavity round-trips within a packet, whose performance acts as a combination of a stable CW mode-locked laser and an optical chopper, making it underpin photonics research in fields such as fluorescence lifetime measurement.
The evolutions of the individual packet profiles in the temporal domain with the increase in pump power are presented in Fig. 3. With the increase in pump power from 40 to 102 mW, the duration of the RSWP is observed to gradually increase from 4.1 to 12.8 µs. With regards to the round-trip time of the cavity of 0.357 ns, it indicates that one packet consists of 11480 pulses for the duration of 4.1 µs and 35840 pulses for the duration of 12.8 µs. This feature is distinct from the behavior of the Q-switched mode-locking operation, in which the duration of the Q-switched envelope decreases with the pump power [22,23]. Moreover, as the pump power increases from 40 to 102 mW, the repetition rate of the packet correspondingly increases from 28 to 59 kHz. The simultaneous increase in duration and repetition rate of the packet suggests that the RSWP operation is a stable regime but tends toward CW mode-locking. In the process, however, the repetition rate of the pulses within a packet always remains constant at 2.8 GHz.
Figure 4(a) illustrates the optical spectrum of the RSWP operation in the experiment when the pump power is 69 mW. The RSWP operating in the wavelength range of 1900-1990 nm can be efficiently initiated which mostly covers the emission band of the 3F4 to 3H6 level of the Tm3+ ions. Moreover, a sharp peak at a wavelength of 1946 nm with a 3-dB bandwidth of 0.7 nm is present in the broad spectrum measured with a spectrometer resolution of 0.05 nm. The measurement shows a mass of absorption peaks because of the water absorption. The radio-frequency (RF) spectrum of the RSWP operation with a 2.8 GHz repetition rate is displayed in Fig. 4(b). It represents the unique RF profile for the RSWP because a wide pedestal occurs under the fundamental frequency of 2.8 GHz. The signal-to-noise ratio and signal-to-pedestal ratio are 43.82 and 22.61dB, respectively. In practice, the RSWP operation can remain stable for hours even in the presence of small perturbations such as pump power changes, small linear polarizations, and temperature drifts.
In order to understand such behavior of the RSWP operation, we attempt to reproduce the unique characteristics using numerical simulations and understand the operation by referring to the nonlinear dynamics. The lumped model of the RSWP operation in the compact laser cavity is established in terms of the following equations [24,25]:Equations (1) and (2) account for the light field propagating in the TDF that includes of the gain dynamics, while the latter two represent the saturable absorber (treated as a fast type here) and mirror (dielectric film), respectively. In Eq. (1), ui (i = 1, …, N) is the light envelope of the ith roundtrip and z represents the propagation distance along the cavity. Regarding the repetition rate of 2.8 GHz for the linear configuration, the cavity length is 2Lc = 6.3 cm and the roundtrip time is Tr = 357 ps, where Lc represents the length of the gain fiber. T and τ = i·Tr are the fast and slow time variables, respectively. β2 = −10 ps2 km−1 and γ = 0.8 W−1 km−1 are the second-order dispersion and nonlinearity parameter of the TDF, respectively. The gain coefficient g characterized by a gain bandwidth Ω = 20 THz, small-signal gain coefficient g0, saturation energy Eg = 1 × 106 J, and lifetime Tg = 1 ms are governed by the rate equation, in which accounts for the pulse energy. The form above can be easily solved by means of the Runge-Kutta algorithm in the temporal domain, where more details of the algorithm and methods for solving them can be found in . The saturable loss q in Eq. (3) that plays a role in intensity discrimination at the position of z = Lc is treated as an instantaneous response to the light field, where the modulation depth q0 = 12%, saturation energy Ea = 50 pJ and relaxation time Ta = 10 ps. The iterative procedure is implemented by Eq. (4). The mirror at z = 2Lc extracts 7% of the total energy from the oscillator.
After building up the model of the RSWP with a fundamental pulse repetition rate of 2.8 GHz, the evolution of the initial stationary pulse solution (achieved from the fixed point of the gain rate equation) can be traced by increasing the value of g0. Figure 5 presents a sequence of such curves labeled for different values of g0. This suggests a transition to stable CW mode-locking for g0t = 665; namely, the width of the square envelope generally broadens when g0 approaches g0t. This tendency in the experiments can be explained by Fig. 3, that is, the duration of the RSWPs gradually increases with the pump power from 40 to 102 mW. Furthermore, it can be seen that the numerical results of the red trace in Fig. 3, corresponding to g0 = 480, 550, and 620, adequately fits the experimental packet duration.
We further try to understand this phenomenon by comparing with the formation of the heteroclinic orbit and referring to the nonlinear dynamics [22,27]. The states of the CW and CW mode-locking, considered as two saddles before g0 reaches the turning point g0t, constitute a loop analogous to the limit cycle. For the state, the initial pulse solution in a RSWP decays to the CW operation in a specific trajectory owing to the decrease in gain level g(τ). Moreover, the CW operation between the two adjacent packets is not a stable solution, which results in pulsing because of the increase in g(τ). The nonlinear dynamics is referred to as the process of the RSWP operation. In particular, the spike on the top of the spectrum, illustrated in Fig. 4(a), is likely to be the evidence of the existence of the CW component. Hence, as indicated by the theoretical approach, it is implied that the gain of the rare earth -doped fiber is insufficient when the RSWP phenomenon is detected.
Figure 6 shows the minimum values of the small-signal gain coefficient g0 required to attain CW mode-locking operation with different fundamental pulse repetition rates. The value g0 gradually increases from 200 to 665 with the increase in pulse repetition rate from 1.59 to 2.80 GHz. Meanwhile, it should be noted that the value g0 of more than 200, exploited to realize CW mode-locking with the pulse repetition rate of 1.59 GHz, is consistent with the reported experimental results in the 5.9-cm laser cavity . These data imply that with a higher pulse repetition rate, the RSWP will occur more easily because of a lower value of g0 when the length of the laser cavity is shortened. This is why we emphasize that this new regime of RSWPs particularly favors high-repetition-rate ultrafast lasers.
As the net dispersion of the laser cavity is different for different fiber manufacturing processes and cavity mirrors, we have further calculated the temporal features as a function of net dispersion at −1230, −630, −30 fs2, respectively. As shown in Fig. 7, the temporal operation gradually evolves from the RSWP to the CW mode-locking while the net dispersion increases from −1230 to −30 fs2. Throughout these calculations, all other parameters except net dispersion were kept constant, and the value of −630 fs2 particularly matches the experiment described above. It can be concluded that in the 2 µm wavelength range, the RSWP operation occurs more easily in the dispersion-management-free anomalous dispersion regime than in the stretched-pulse regime. This is highly essential because it will be beneficial to constrict the RSWP operation in high-repetition-rate ultrafast lasers by cavity dispersion management to achieve stable CW mode-locking.
In conclusion, a novel regime of RSWPs operated in a TDF laser with a pulse repetition rate of up to 2.8 GHz has been demonstrated both theoretically and experimentally. In this regime, the laser output consists of a train of mode-locked pulses underneath a rectangular-shaped envelope. The simultaneous increase in packet duration and repetition rate indicates that the RSWP operation is a stable regime. The effects of small-signal gain value, pulse repetition rate, and net dispersion on the RSWP performance are analyzed. The results of numerical simulations performed using the lumped model exhibit good quantitative agreement with the experiments supporting this conclusion. The nonlinear dynamics referring to g0 reaching the turning point can explain the unique performances. The RSWP operation can be maintained to attain long-term stability and reliable repeatability. The investigation of the dynamics of RSWPs here is helpful to understand the nonlinear gain dynamics in high-repetition-rate ultrafast lasers.
China Postdoctoral Science Foundation (2016M602462); China National Funds for Distinguished Young Scientists (61325024); National Key Research and Development Program of China (2016YFB0402204); High-level Personnel Special Support Program of Guangdong Province (2014TX01C087); Fundamental Research Funds for the Central Universities (2017BQ110); Science and Technology Project of Guangdong (2015B090926010).
We thank Qi Qian for providing the platform for numerical calculation and Yizhong Huang for discussion on laser pulse performance.
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