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Abstract
Noise-like pulses (NLP) are extremely sought after in low-coherence tomography and supercontinuum, etc. Here, we propose an effective method to form the NLP in the all-normal-dispersion (ANDi) fiber laser by weakening the spectrum filtering for the first time. Numerical explorations are performed in detail and demonstrate that the NLP can be originated from the clustering behavior of amplified random sub-pulses led by the saturable absorber. By simulating the pulse-pattern distribution in the two-dimensional parameter space, it is further found that this kind of NLP pattern is widely distributed where are with a weak spectrum filtering. Since, the weaker the filtering, the harder the dissipative system to achieve balance, which helps to avoid the evolution from the pulse cluster to other coherent states and supports the NLP stability. To prove the feasibility experimentally, we built an ANDi fiber laser based on the nonlinear polarization rotating (NPR). The built laser can operate at the stable dissipative soliton (DS) state with a spectrum filter, which also with a 45 nm wavelength tuning performance. Replacing the filter, only NLP with a 40.2 ps pedestal and 237 fs spike can be attained. The experiments agree well with the numerical predictions. This exploration significantly broadens the design possibilities for ultrafast lasers, making them much more accessible to produce desired pulse patterns.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, the ANDi fiber laser plays a dominant role in achieving high power and energy ultrafast pulse [1,2]. Various pulse states can be formed in ANDi fiber lasers, such as dissipative soliton (DS) [1], dissipative soliton resonance (DSR) [3], and noise-like pulse (NLP) [4], multiple pulsing [5], etc. The formation of DS is a composite balance involving spectrum filtering, nonlinearity, dispersion, and gain [2], and DS with single pulse energies over 20 nJ could be attained [6]. For DSR, thanks to wave-breaking free, the DSR laser is a powerful source of realizing high pulse energy [7]. NLPs have gained recent research interest due to their low temporal and spatial coherence [8,9], and on propagation through dispersive fiber NLPs do not broaden significantly [10]. NLPs are widely used in low-coherence spectrum interferometry and supercontinuum where low coherence, broad-spectrum, and high peak power are required [11–14]. Moreover, NLPs contain very complex fine structures and dynamics providing a benchmark to study extreme events like optical rogue waves [15].
Driven by these distinctive properties and miscellaneous applications, many efforts have been carried out to produce the NLP pattern by analyzing the influence of laser elements. It is widely known that the ANDi cavity mainly consists of a saturable absorber (SA), fibers, spectrum filter, output coupler, etc. The saturable absorber is a key element to modulation pulse, which may exhibit both saturable absorption and reverse saturable absorption properties [16–20]. Reports reveal that NLPs are attributed to the amplitude modulation of a single pulse introduced by reverse saturable absorption [21]. They fixed other parameters and increase pump power to strengthen the amplitude modulation, the DS will evolve to NLP. Besides, in NPR laser, adjusting the polarization controller may change the SA dynamic, which may also strengthen the modulation. In Ref. [22], Smirnov et al. observed a transient lasing regime between DS and NLP by adjusting the polarization controllers. In their experiments, implicit spectral filtering is performed by the effective Lyot filter action of SA. The critical saturable power (CSP) is the power at which the SA feedback is transformed from saturable absorption to reverse saturable absorption. It should be emphasized that the CSP will also introduce a peak power limiting effect, which has an important impact on the formation of DSR [18,20]. For the equivalent saturable absorber, such as NPR and nonlinear optical loop mirror, the CSP is related to the fiber length [18]. In Ref. [23], Mei et al. increased the loop length (∼250 m) to form a very low CSP and a strong peak power limiting to achieve DSR. In our next exploration, the fiber length is only a few meters meaning that the CSP is very high, which could explain why DSR disappears. The role of fiber always focuses on dispersion and nonlinearity effects [24]. The excessive accumulation of the nonlinear phase shift will lead to pulse splitting, and various pulse patterns could be observed [25]. In Ref. [24], Li et al. have proposed that shortening the single-mode fiber after the gain will make the pulse a high probability of entering the reverse-saturable absorption, resulting in the NLP pattern.
The spectrum filter has three main impacts on ANDi cavity: firstly, the filter can modulate the spectrum; secondly, the filter will introduce a spectrum-related loss; lastly, the filter could balance the positive-chirp, and narrow the pulse width [1,26]. The role of spectrum filter on pulse dynamics has been widely explored, especially in the framework of the cubic-Ginzburg-Landau equation or pulse-tracing model [18,26–31]. Numerical research demonstrated that spectral filtering has an essential impact on the formation of DS, DSR, and multi-pulsing state [1,2,18,32]. In Ref. [26], Xu et al. numerically demonstrated that strong filtering can also modulate the single pulse amplitude to result in the NLP. Overall, the NLP is generally attained by strengthening the amplitude modulation of a single pulse, and this process is sensitive to polarization controller settings or pumps power, leading to high randomness and low self-starting performance [21,22,33]. A natural question is: is there any other mechanism to attain the NLP in ANDi laser?
Enlightened by some recent experiments [34,35], in this work, we introduce an effective mechanism to form NLP in the ANDi fiber laser. Numerical explorations are performed in detail and demonstrate that the NLP can also be originated from the clustering behavior of random sub-pulses directly. The saturable absorber compresses and stabilizes the amplified sub-pulses to form a dynamic stable pulse cluster (NLP). By simulating the pulse-pattern distribution in the two-dimensional parameter space, it is found that this kind of NLP pattern is widely distributed where are with a weak spectrum filtering. The weaker the filtering, the harder the dissipative system to achieve balance, which helps to avoid the evolution from the pulse cluster to other coherent states and supports the NLP stability. To prove the feasibility experimentally, we built an ANDi laser based on the NPR. With the insertion of a spectrum filter, the laser operates at a stable DS state with a duration of 3.84 ps and maximum pulse energy of 2.2 nJ. The DS laser is also with a 45 nm wavelength tuning performance. Replacing the filter, the self-started NLP with a 40.2 ps pedestal and 237 fs spike can be easily achieved, and the measured envelope efficiency and pulse energy are around 37% and 2.5 nJ. Whether ergodically adjusting the waveplate or the pump power, only the NLP pattern can be achieved. The experiments are in good agreement with the numerical predictions and have further verified the method’s controllability, stability, and self-starting performance.
2. Theoretical analysis and simulation
2.1 Theory model
For explaining the pulse dynamics, we erected a theoretical model based on the pulse-tracing. The simulation cavity is consistent with the following experiment, which mainly consists of 0.3 m Yb3+ doped fibers (YDF), 4 m single-mode fiber (SMF), a Gaussian bandpass filter, an NPR mode locker, and a 35% output coupler. The pulse evolution in fiber is described by the generalized nonlinear Schrödinger equation [29,36]. The nonlinear coefficients are set to $4.7\,{W^{ - 1}}k{m^{ - 1}}$ for SMF and $5\,{W^{ - 1}}k{m^{ - 1}}$ for YDF; $\textrm{T}$ he second-order dispersion are set to $26.2\,f{s^2}/mm$ for YDF and $24.7\,f{s^2}/mm$ for SMF. Higher-order dispersion and nonlinear effects are ignored. The formula $\,g = {g_{avg}} \times g(\omega )$ can calculate the gain coefficient with a gain spectrum g(ω) in the Lorentzian profile. The g(ω) is with a center wavelength (${\lambda _0}$) of 1030 nm and gain bandwidth of 50 nm. Considering the gain saturation, there are:
Where ${g_0}$ is the small-signal gain at the central wavelength, which is set to $10\,{m^{ - 1}}$ for YDF and $0\,{m^{ - 1}}$ for SMF; Esat represents the gain saturation energy, which is relative to pump power [37], setting Esat = 9.5 nJ.The NPR mode-locker exhibits a sinusoidal shape transmissivity verse instantaneous power, the transmissivity can be written as [21,38,39]:
Fig. 1. (a) NPR transmission (blue line) and the product of instantaneous power and transmission (red line); (b) calculated spectrum filtering curves.
In this adopted model, there exist three kinds of spectrum filters (hard or soft): a bandwidth tunable first-order Gaussian filter [29], a gain filter introduced by the gain medium, and a sinusoidal filter introduced by NPR [33]. The nonlinear coefficient is related to wavelength, NPR can be both used as a mode locker and a spectral filter to promote pulse evolution. The formula $\gamma = 2\pi {n_2}/\lambda {A_{eff}}$ can calculate the nonlinear coefficient, where ${n_2}$ and ${A_{eff}}$ are the nonlinear-index coefficient and effective mode field area of the fiber, setting ${n_2} = 2.35 \times {10^{ - 22}}{m^2}/W$, ${A_{eff}} = 3.02 \times {10^{ - 11}}\,{m^2},\,\textrm{and}\,P = 1550\,\textrm{W}.$ The calculated filtering curves of Yb3+ gain and the NPR are shown in Fig. 1(b), meaning the weak built-in (soft) filtering. If the bandwidth (Δλ) of the Gaussian filter < 50 nm, it will play a key role to modulate the spectrum. In the following numerical solution, they are all considered at the same time. The total intracavity loss is set at around 10%. The simulation starts from arbitrary Gaussian-windowed white noise after the filter. We employed the split-step Fourier method to solve this model [41], and the time window and sampling points are set to 100 ps and 10,000. The time resolution is 10 fs. Usually, several tens of roundtrips are required for the solution to stabilize, and the total cycle is set to 500.
2.2 Pulse dynamics versus spectrum filtering
To investigate the impact of spectrum filtering on pulse dynamics, we fix other parameters and increase filtering bandwidth Δλ to simulate. Figures 2(a) and 2(b) show the output pulse and spectrum versus Δλ before the NPR mode-locker. When Δλ = 5 nm, the pulse is with a Gaussian envelope, and the “Cat's ear” spectrum proves the forming of DS pulse [2]. The self-phase modulation effect makes the spectrum a sharper edge and large bandwidth of ∼25.7 nm. As Δλ increases from 5 nm to 17.5 nm, the output pulses are always in line with the DS, while the pulse properties have an obvious variation. We also simulated their autocorrelation trace to analyze the pulse state, as shown in Figs. 2(c) and 2(d). To offer a quantitative description, we also simulate the variation of key properties, as presented in Figs. 2(e) and 2(f). As Δλ increases from 5 to 17.5 nm, the pulse width increases from 5.5 to 14.67 ps, and the spectrum width decrease from 25.7 to 21.9 nm, respectively. The pulse energy increased from 5.1 to 18.2 nJ. The peak power increases first and then decreases, and is with a maximum value of 1.36 kW when Δλ = 10 nm. When Δλ is increased, the loss induced by the filter decreases, hence the pulse energy increases. Meanwhile, the pulse-narrowing-ability weakens, therefore the pulse duration goes up, which then results in peak power declining. Since lower peak power causes less self-phase modulation, the spectrum width also declines.
Fig. 2. (a) Pulse and (b) spectrum envelope versus Δλ; autocorrelation trace when (c) Δλ = 10 nm and (d) Δλ = 30 nm; (e) pulse width and spectrum width versus Δλ; (f) peak power and pulse energy versus Δλ.
Further weakening the spectrum filtering, irregular pulse and spectrum profiles can be observed. The pulse exhibits a fine inner structure of many narrow sub-pulses with randomly varying intensity and duration, and the pulse properties have large randomness (Figs. 2(e) and 2(f)). In the case of Δλ = 30 nm (Fig. 2(d)), its intensity autocorrelation trace is two-scaled, consisting of a wide pedestal and a narrow spike of ∼180 fs, proving the formation of NLP. The widths of the pedestal and the spike indicate the width of the wave packet and the average width of the sub-pulses inside the wave packet, respectively. The ratio of the spike to the pedestal is related to the density of the sub-pulse a larger ratio usually means a smaller sub-pulse density [26,42]. With simulation, we find the ratio is positively correlated with the filter bandwidth. The above results indicate that the spectral filtering effect has an impact on the transition of pulse patterns in ANDi fiber laser, and NLPs can be operated under a weak spectrum filtering effect.
2.3 Evolution dynamics of noise-like pulse
To explain the mechanism of NLP in this case, setting Δλ = 10 nm and 30 nm, we simulated the pulse and spectrum evolution versus roundtrips, as present in Fig. 3. In the case of Δλ = 10 nm, after the 95th cycle, the initial noise field gradually evolves to a stable DS pulse with a Gaussian envelope, and the spectrum begins to have a flat top and sharp edges. However, under the large filter bandwidth (Δλ = 30 nm), the initial field gradually evolves into an amplified pulse cluster after 69 cycles, which with many random sub-pulses and obvious jitters. It should be emphasized that the random sub-pulses always exist, no transition steady state could be observed during its evolution. That is, NLP pulse is not the result of modulating DS, but a clustering behavior of sub-pulse. The clustering behavior includes three main factors: initializing, compressing, and stabilizing. Firstly, the initial noise field is amplified by the gain, and a large number of amplified random sub-pulses will be formed in the cavity. Then, when the peak powers of the sub-pulses are lower than CSP, the saturable absorber will continue to compress the pulse envelope and form a narrow pulse cluster (Figs. 4(a) and 4(b)). After the 69th cycle, as shown in Figs. 4(c) and 4(d), the peak power exceeds the CSP, and the peak power limiting effect will occur and stabilize the pulse cluster. Figures 4(e) and 4(f) show the peak power and energy versus roundtrips. One can observe that the peak power and energy are stabilized at a certain level, which makes the pulse cluster achieve a dynamic balance, i.e., NLP is formed. A random energy fluctuation can be observed during its evolution, which can explain why the measured NLP trains are always with large fluctuations. After the NLP is established, reducing Δλ from 30 nm to 10 nm, the evolution from NLP to DS patterns will be observed (Figs. 3(e) and 3(f)). That is, although the spectrum filtering does not have a direct impact on the NLP evolution, weakening the filtering is important to maintain the stable operation of NLP. Since spectrum filtering plays a key role to balance the positive chirp and nonlinearity to form coherent pulses [1]. The weakened filtering is hard to attain the balance of the dissipative system and helps to avoid the evolution from the pulse cluster to other coherent states, supporting the NLP stability. Combining with the previous report, we can conclude that there is a difference in NLP formation in the ANDi fiber laser. Under strong filtering, NLPs generally originate from amplitude modulated single pulse, and the amplitude modulation induced by reverse saturable absorption or spectrum filter plays the leading role [21,26]. While, under a weak spectrum filtering, the NLP patterns originate from the clustering behavior of the random sub-pulse, and the saturable absorber plays an important role in compressing and stabilizing the NLP. Compared with the method of amplitude modulation, this proposed mechanism does not need a transition state and has better self-starting performance and stability.
Fig. 3. Simulated (a) pulse and (b) spectrum evolution versus roundtrips when Δλ = 10 nm; Simulated (c) pulse and (d) spectrum evolution versus roundtrips when Δλ = 30 nm; Simulated (e) pulse and (f) spectrum evolution versus roundtrips with varied Δλ.
Fig. 4. Pulse envelope in the (a) 67th, (b) 68th, (c) 69th, and (d) 70th cycles; (e) peak power and (f) energy versus roundtrips.
2.4 Formation probability of noise-like pulse
To attain the intuitive pulse-pattern distribution, we employ the double-parameter-scanning and vary both the spectral filter bandwidth (from 5 to 30 nm) and gain saturation energy (from 5 to 10 nJ) to simulate. According to the key properties, peak power, pulse width, spectrum width, and energy, we map the pulse-pattern distribution in the two-dimensional parameter (Δλ–Esat) space. As presented in Fig. 5, it can be observed that there are two obvious different regions in the two-dimensional parameter space. The region with a regular variation of pulse parameter corresponds to the DS state, while the highly random mosaic region corresponds to the NLP state. It can also observe that the DS pattern is distributed in the wide parameter space where are with obvious filtering (< 20 nm bandwidth). The allowable bandwidth increases with the decrease of gain saturation energy. In the DS region, the peak power increases with the gain saturation energy and achieved the highest value when the Δλ=10 nm and Esat=10 nJ. The nonlinear variation of peak power versus filter bandwidth shows that a suitable filter can achieve a higher peak in the experiment. For pulse width and energy, they both increase with gain saturation energy and filter bandwidth. The spectrum is with a wider width under high gain and a strong filtering effect. In the NLP region, the gain saturation energy also has an observed impact on the peak power and energy, while these properties are always with strong randomness along with the filtering bandwidth. One can observe that the NLP pattern is distributed in the wide parameter space where are with a weak filtering effect, which can provide a benchmark for the production of this pattern.
Fig. 5. (a) Peak power, (b) pulse width, (c) pulse energy, and (d) spectrum width in the spectral filter bandwidth and gain saturation energy parameter space.
3. Experimental setup and results
3.1 Experimental setup
To prove feasibility experimentally, we built an ANDi fiber laser based on the NPR. The schematic of the ANDi fiber oscillator is shown in Fig. 6. 30 cm highly Yb3+ doped fibers (Liekki Yb1200-4/125 1200 dB @ 976 nm) with a calculated second-order dispersion of $26.2\; f{s^2}/mm$ are used as the gain medium. The fiber represented in black is about 4 m HI1060 fiber with dispersion $24.7\; f{s^2}/mm$, i.e., a single-mode fiber at 1 µm. The net cavity dispersion is around 0.11 $p{s^2}$. A 976 nm laser diode with a maximum pump of 350 mW is coupled into the cavity through a wavelength-division multiplexer. The fiber collimators transmit the signal to the spatial path. The quarter-wave plates (QWP1 and QWP2), half-wave plate (HWP1), and polarization splitting prism (PBS1) are served as NPR devices to realize mode-locking. Meanwhile, PBS1 is also employed as the output coupler. The Faraday rotator, half-wave plate (HWP2), and longitudinally placed polarization splitting prism (PBS2) act as an isolator by controlling the polarization, and this part plays the role in protecting the pump.
Fig. 6. Schematic of the ANDi fiber laser. YDF: Yb3+ doped fiber; SMF: single-mode fiber; LD: laser diode; WDM: wavelength-division multiplexer; C1 and C2: fiber collimators; QWP1 and QWP2: quarter-wave plates; HWP1 and HWP2: half-wave plates; PBS1 and PBS2: polarization splitting prisms; FR: Faraday rotator.
3.2. Dissipative soliton pulse
Firstly, we insert 300 lines/mm grating to form a tunable single pass Gaussian filter [43]. Stable coherent pulses can be attained by controlling the mode-locker, and the output performances are measured under a pump power of 350 mW. An ultrafast photodetector (Thorlabs, model FDS025) with a rise time of 47 ps and a Rohde & Schwarz oscilloscope with a bandwidth of 2 GHz are used to characterize the pulse train. The output pulse trains observed in time windows of 6 ms and 200 ns are shown in Fig. 7(a). One can observe that the pulse interval is around 23.5 ns, and the corresponding repetition rate is about 42.5 MHz. It is in line with the calculated value with cavity length. Furthermore, an autocorrelator (Femtochrome, model FR-103WS) with a time resolution ratio of 1 fs is adapted to parse the pulse. As shown in Fig. 7(b), the duration is about 3.84 ps after Gaussian fitting. Figure 7(c) shows the output power versus the pump. The laser starts a continuous-wave (CW) operation when the pump is about 60 mW. The self-started mode-locking (ML) can be obtained as the pump exceeds 180 mW. The output power increases almost linearly with the pump, the maximum output is ∼92.8 mW, and the envelope efficiency is around 32%. It can be further calculated that the maximum single pulse energy is about 2.2 nJ and the peak power is ∼573 W.
Fig. 7. Measured DS parameters: (a) oscilloscope train in the range of 6 ms and 200 ns; (b) autocorrelation trace and its Gaussian fitting (red); (c) measured output power and mode-locked spectrum; (d) wavelength tuning of the mode-locked laser.
A spectrum analyzer (Yokogawa, AQ6370C) is employed to characterize the spectrum. As also presented in Fig. 7(c), the “Cat's ear” spectrum proves the forming of DS pulse. The central wavelength is around 1030 nm and the 3-dB width is ∼11.3 nm. The reflective grating and the collimator are equivalent to a narrow-band single-pass Gaussian filter with variable central wavelength. After the mode-locking is operated, the mode-locked wavelength can be continuously adjustable from 1010 nm to 1055 nm (45 nm) by adjusting the angle of the grating, as shown in Fig. 7(d). The wavelength tuning performance is mainly limited by the competition between the ASE effect and gain [44].
3.3 Noise-like pulse
To weaken the spectrum filtering effect, we replace the grating with a plane mirror and fix the other parameters. The NLP patterns can be easily obtained, and the measured pulse trains are shown in Fig. 8(a). It can be observed that the pulses have a large random amplitude fluctuation, and the interval is around 23.3 ns, corresponding repetition rate of about 43 MHz. As presented in Fig. 8(b), the autocorrelation trace with a spike shows the typical feature of NLP [45]. The measured width of the pedestal is about 40.2 ps, and the spike is with a width of ∼237 fs (Fig. 8(d)). As presented in Fig. 8(c), the spectrum has an irregular shape, the central wavelength is at around 1030 nm and the 3-dB width is about 5.4 nm. Figure 8(c) also shows the output power versus the pump, the self-started NLP mode-locking can be obtained as the pump exceeds 280 mW, and the maximum output is about 108.2 mW. The calculated envelope efficiency and single pulse energy are around 37% and 2.5 nJ. It should be emphasized that, in this case, only NLP can be attained, whether ergodically adjusting the waveplate or the pump power.
Fig. 8. Measured NLP parameters: (a) oscilloscope train in the range of 12 $\mu $s and 200 ns; (b) autocorrelation trace and its Gaussian fitting (red); (c) measured output power and mode-locked spectrum. (d) autocorrelation trace of the spike.
To further describe the stability, the radio frequency spectrum is also measured by a radio frequency analyzer (Keysight, model N9000B). As shown in Figs. 9, the fundamental repetition rates are 42.5 MHz and 43 MHz for DS and NLP, respectively. Since the grating and plane mirror introduces different optical paths, the repetition rates will have a slight change. The signal-to-noise ratio is higher than 80 dB for DS pulse meaning the mode-locking is very stable. By comparison, the NLP is with a signal-to-noise ratio of 60 dB and a sidelobe. Meanwhile, the higher-order spectrum of NLP also has a more obvious modulation. The NLP mode-locking is not stable, but it is in line with the evolutionary dynamics of NLP (see Fig. 3). The results prove that the NLP pattern can be controllably formed only by weakening the spectrum filtering, which has also verified the self-starting performance and stability. The experiments are in good agreement with the numerical predictions.
Fig. 9. (a) DS radio frequency spectrum: around fundamental repetition rate and within 0–500 MHz; (b) NLP radio frequency spectrum: around fundamental repetition rate and within 0–500 MHz.
4. Conclusion
In conclusion, we have presented a concise mechanism, weakening the spectrum filtering, to improve the formation probability of NLP patterns in ANDi fiber lasers. The NLP pattern can be originated from the clustering behavior of random sub-pulses led by the saturable absorber. The further simulated pulse-pattern distribution shows that this kind of NLP pattern is widely distributed where are with weak filtering. Since, the weaker the filtering, the harder the dissipative system to achieve balance, which helps to avoid the evolution from the pulse cluster to other coherent states and provide the NLP stability. The followed experiments are in good agreement with the numerical predictions and have further verified its controllability, stability, and self-starting performance. This mechanism may be attractive for some practical applications, such as low-coherence spectral interferometry and supercontinuum. In addition, we believe this exploration can also assist in understanding the mechanism of pulse-pattern generation and define a compelling way to manipulate the ANDi cavity parameters in an orderly manner. Our exploration can be regarded as an extension and enrichment of the earlier works on this topic.
Funding
National Natural Science Foundation of China (61975137).
Acknowledgments
The authors would like to acknowledge Long Li and Xinxin Sun for their valuable help in paper writing.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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