Generation of Mid‐Infrared Noise‐Like Pulses from a Polarization‐Maintaining Fluoride Fiber Oscillator

Noise‐like pulses (NLPs) are becoming increasingly attractive for a variety of applications such as supercontinuum generation, materials processing, and low‐coherence spectral interferometry. Related research in the near‐infrared region is extensive, yet few studies have been reported in the mid‐infrared (MIR) region. Herein, a systematic investigation of MIR NLPs is made for the first time. An approach is presented by exploiting a polarization‐maintaining fluoride fiber in the mode‐locked oscillator to cause a polarization‐dependent delay between the orthogonal‐polarized components of intracavity circulating pulses, thus generating a series of ultrashort pulses which will eventually form a stable wave packet, namely, NLP. Numerical simulations based on the extended coupled nonlinear Schrödinger equations predict the generation of NLPs, and reveal key aspects of the pulse evolution. Experiments yield linearly polarized NLPs at 2.8 μm with a maximum average power of 498 mW and a spike width of 4.3 ps, corresponding to a pulse energy of 12 nJ. The experimental results are in good agreement with the simulation results. This work constitutes a major step toward the development of MIR ultrafast fiber lasers with NLPs output.

Noise-like pulses (NLPs) are becoming increasingly attractive for a variety of applications such as supercontinuum generation, materials processing, and low-coherence spectral interferometry. Related research in the near-infrared region is extensive, yet few studies have been reported in the mid-infrared (MIR) region. Herein, a systematic investigation of MIR NLPs is made for the first time. An approach is presented by exploiting a polarization-maintaining fluoride fiber in the mode-locked oscillator to cause a polarization-dependent delay between the orthogonal-polarized components of intracavity circulating pulses, thus generating a series of ultrashort pulses which will eventually form a stable wave packet, namely, NLP. Numerical simulations based on the extended coupled nonlinear Schrödinger equations predict the generation of NLPs, and reveal key aspects of the pulse evolution. Experiments yield linearly polarized NLPs at 2.8 μm with a maximum average power of 498 mW and a spike width of 4.3 ps, corresponding to a pulse energy of 12 nJ. The experimental results are in good agreement with the simulation results. This work constitutes a major step toward the development of MIR ultrafast fiber lasers with NLPs output. femtosecond) spike located on the top of a broad (picosecond to nanosecond) pedestal. Because of the broad temporal width, NLPs can operate in the high-pulse-energy region and allow a further pulse energy scaling through fiber amplification without any pulse shaping, [19] rendering them attractive for applications such as supercontinuum generation, materials processing, and low-coherence spectral interferometry. [20][21][22][23] Moreover, the complex dynamics and intrastructure of NLPs make them a good platform for studying extreme events like optical rogue wave. [24] NLPs were first reported by Horowitz et al. in a mode-locked erbium-doped silica fiber laser, of which the generation was attributed to the fiber birefringence combined with a nonlinear transmission element and the gain response of the amplifier. [18] Since then, NLPs have been demonstrated to be one of the intrinsic mode-locking states in silica fiber lasers regardless of cavity configurations or cavity dispersion regimes. [18][19][20][21][22][23][24][25][26][27][28][29] For MLFFLs, however, few studies have been reported. The oscillator in ref. [30] was thought to operate in partial NLP mode-locking, yet no further discussion was given.
In the anomalous dispersion regime, Tang et al. found that the formation of NLPs could be caused by the combined effect of soliton collapse and positive cavity feedback. [25] In MLFFLs, the positive cavity feedback can be provided by nonlinear polarization rotation, whereas the threshold of soliton collapse is high due to the low nonlinear coefficient and large anomalous dispersion of fluoride fibers in the MIR range. Elongation of the fluoride fiber laser cavity to hundreds of meters or increasing the pump power to an ultrahigh level is the most straightforward way of inducing soliton collapse, despite the difficulty in practical experiments. Alternatively, inspired by Horowitz et al., [18] the fiber birefringence can be exploited to generate MIR NLPs in fluoride fiber lasers. Fortunately, thanks to the rapid development of fluoride fiber technology, polarization-maintaining (PM) fluoride fibers (Le Verre Fluoré, France) with moderate birefringence are now commercially available, [31][32][33][34] which can be used to provide uniform birefringence instead of bending the conventional fragile fluoride fibers.
In this article, by introducing a birefringence-induced polarization-dependent delay between the orthogonal-polarized components of intracavity circulating pulses, we have theoretically and experimentally demonstrated the generation of linearly polarized NLPs at 2.8 μm from an all-PM fluoride fiber modelocked oscillator. Numerical simulations are carried out first based on the extended coupled nonlinear Schrödinger equations, by which the formation of NLPs is predicted and analyzed. According to the simulations, under appropriate pump power and waveplate angles, a maximum average power of 498 mW with a spike width of 4.3 ps is experimentally obtained, corresponding to a pulse energy of 12 nJ. Our work offers a systematic investigation of MIR NLPs for the first time, paving the way for the development of MIR NLP fiber lasers.

Experimental Setup
The schematic of the experimental setup is depicted in Figure 1. A 4 m-length 7 mol% Er 3þ -doped PM fluoride fiber (Le Verre Fluoré), which has a core diameter of 15 μm (NA = 0.12) surrounded by a truncated 260 μm diameter inner cladding (NA = 0.4), is used as the gain medium. The inset of Figure 1 shows its zoomed cross section. The PM behavior is originated from the two air gaps, characterized by a beat length of 4.13 cm. [31] Both the input and output fiber ends are cleaved at an angle of 8°to suppress parasitic lasing. The gain fiber is pumped by a 970 nm laser diode with a multimode pigtail of 105 μm core diameter (NA = 0.22) through a set of lenses consisting of a plano-convex lens L1 ( f = 15 mm) and an aspheric lens L2 ( f = 20 mm). Linearly polarized output pulses are obtained from the output port of the polarization beam splitter (PBS), while the output coupling ratio can be adjusted by rotating the half-wave plate (HWP1) in front of the PBS. The modelocking element is a commercial semiconductor saturable absorber (BATOP GmbH) placed between a pair of ZnSe lenses (L4, L5, f = 6 mm). As stated by the manufacturer, the saturable absorber (SA) is designed to operate at 2.8 μm and has a modulation depth of 6%, a nonsaturable loss of 4%, a saturation fluence of 300 μJ cm À2 , and a relaxation time of 10 ps. The polarizationdependent optical isolator (ISO) ensures a unidirectional light operation. The second half-wave plate (HWP2) is used to change the polarization direction of light before coupling into the PM fiber, thus achieving a misalignment between the fast axis of the fiber and the polarization direction of light. The misalignment allows the light to propagate along both the fast and slow axes, thus introducing a polarization-dependent delay (PDD).
The pulse train is monitored by a MIR photodetector (VIGO System, PCI-9, bandwidth: 250 MHz) connected with a digital oscilloscope (Tektronix, MDO4104C, bandwidth: 1 GHz). The spectrum is measured by an optical spectrum analyzer (Yokogawa, AQ6376) with a resolution of 0.1 nm. The signalto-noise ratio (SNR) is analyzed by a radio frequency (RF) spectrum analyzer (Rohde & Schwarz, FSWP8). The pulse duration is characterized by a commercial intensity autocorrelator (Femtochrome, FR-103 XL, scan range: %185 ps).

Numerical Simulations
In order to develop a better understanding of the NLP buildup process in the cavity, numerical simulations are conducted first. www.advancedsciencenews.com www.adpr-journal.com The nonlinear pulse propagation in the erbium-doped PM fluoride fiber can be described by the extended coupled nonlinear Schrödinger equations [35,36] ∂u ∂z ¼ Ài where u and v are the complex pulse envelopes along the two orthogonal polarization axes of the fiber. Δβ = 2π/L B is related to the fiber birefringence, where L B represents the beat length. 2δ = λ/(cL B ) is the inverse group velocity difference. [36] β m is the mth order dispersion (β 2 = À83 fs 2 mm À1 , β 3 = 476 fs 3 mm À1 ). γ = n 2 ω 0 /(cA eff ), where n 2 = 2.1 Â 10 À20 m 2 W À1 is the nonlinear refractive index, c is the speed of light in vacuum, ω 0 is the carrier angular frequency at the central wavelength of 2.8 μm, and A eff is the effective mode area and is calculated from the V parameter. [37] The gain coefficient g is given in the frequency domain by [17] g where g 0 is the small-signal gain, E p ¼ ∫ juj 2 þ jvj 2 ð Þ dt is the total intracavity pulse energy, E sat is the gain saturation energy, and Δω = 120 nm is the gain bandwidth. Changing the pump power is equivalent to the changing of gain parameters related to either the saturation energy E sat or the small-signal gain g 0 . Here, we choose g 0 as the control parameter, while E sat is fixed at 1.15 nJ. The transmission function T(t) of the SA is calculated by where α ns is the nonsaturable loss and q(t) is the time-dependent absorption loss, which can be described by the following differential equation [38] where τ relax is the relaxation time, q 0 is the modulation depth, and E SA is the saturable energy of the SA. In the simulation, the PBS, HWP1, and HWP2 are represented by their Jones matrixes and operate onto the light vector u v ! to perform the modulation.
The angles of the polarization relative to the x-axis for the HWP1 and HWP2 are denoted as θ HWP1 and θ HWP2 , respectively. Note that θ HWP2 = 0 indicates that the polarization direction of input light is parallel to the fast axis of the PM fiber. To introduce a PDD between the orthogonal-polarized components of input light, the θ HWP2 value is set to 0.05π. The total cavity loss in the simulation is assumed to be 75%, which is mainly caused by the insertion loss of the SA and ISO, and the coupling loss at the input fiber end. The simulation starts with a weak hyperbolic secant pulse as the initial pulse. Equation (1) is numerically solved by the split-step Fourier method implemented using the fourth-order Runge-Kutta method in the interaction picture. [37,39] A typical steady NLP state is obtained by setting the smallsignal gain coefficient g 0 and the θ HWP1 value as 4 m À1 and 0.15π, respectively, as illustrated in Figure 2. The evolution of the NLPs in the time domain within the first 1000 roundtrips is shown in Figure 2a. It can be observed that the pulse has a lot of fine structures which are different for each roundtrip.  Stable NLPs with a pulse energy of %7.1 nJ are obtained after 300 roundtrips. The output pulse profile and corresponding AC trace at the 1000th roundtrip are shown in Figure 2b,c, respectively. The typical double-scale structure of the AC trace demonstrates that the output pulse is indeed in an NLP regime. The width of the spike is 1.7 ps assuming a hyperbolic-secant pulse shape, larger than the typical value (femtosecond level) of near-infrared NLPs. It can be concluded that MIR NLPs can be generated by such a laser cavity design. The principle of the NLP buildup process is illustrated in Figure 2d. Pulses injected into the PM fiber with an orientation α to the slow axis will split due to the PDD. This process will repeat and generate a series of ultrashort pulses. Eventually, under the combined effect of gain and cavity loss, all the pulses will together form a stable wave packet, namely, NLP. A well-known feature of NLPs is that their wave-packet widths can vary with the pump power and the output coupling ratio. [26,40] To give more insight into the characteristics of MIR NLPs, we conduct a further numerical simulation by changing the small-signal gain coefficient g 0 and θ HWP1 . Figure 3a,b depicts the pulse profiles and corresponding AC traces at increasing small-signal gain coefficient g 0 . As g 0 (i.e., the pump power) increases, the number of internal subpulses increases while the intensities remain barely constant. As a result, the wave packet gets broadened, leading to a wider pedestal of the AC trace. However, a slightly different behavior can be observed in Figure 3c,d when we fixed the pump power at the g 0 value of 4 m À1 and adjust the HWP1. The output pulse energy is calculated and indicated on each pulse profile of Figure 3c. With the increase of output coupling ratio, the output pulse energy increases while the wave-packet width decreases, allowing the intensities to increase significantly. The numerical results are in good agreement with the work on NLPs in ref. [40], implying the same regularity between MIR NLPs and near-infrared NLPs.

Experimental Results and Discussion
According to the numerical simulations, by appropriately setting the pump power and half-wave plates, reliable self-starting modelocking is achieved. Under a pump power of 3.2 W, linearly polarized mode-locked pulses with an average power of 303 mW and a repetition rate of 41.4 MHz are obtained. The NLP regime is identified by the measured AC trace shown in Figure 4a, which exhibits a typical double-scale structure with a narrow spike riding on a broad pedestal. The pedestal is clipped on both edges due to the limited scan range of the autocorrelator. The width of the spike shown in the inset of Figure 4a is 4.3 ps assuming a hyperbolic-secant pulse shape, which is consistent with the simulation result. The measured output spectrum is presented in Figure 4b. Different from the broad smooth spectra of NLPs in previous works, the output spectrum centered at 2.8 μm is somewhat narrow and contains several strong dips because of water vapor absorption in the long free-space propagation. We have also characterized the experimental absorption background using a homemade flat MIR supercontinuum source and present the result in Figure 4b. The spectral dips fit well with the peak experimental absorption lines. Figure 4c shows the measured RF spectrum with an SNR value of 54 dB at the     fundamental frequency. The two wide side lobes apart from the fundamental frequency further confirm the NLP regime, revealing the inherent jittering of the NLPs. Figure 4d shows the mode-locked pulse trains measured in 400 ns and 40 μs time scales. The pulse-to-pulse interval is 24.2 ns, matching well with the cavity roundtrip time. Large pulse-intensity fluctuation over successive round trips can be observed, which is also a typical feature of NLPs. The evolutions of the output average power and pulse energy with the pump power for the stable NLP mode-locking state are depicted in Figure 5a,b, respectively. The mode-locking operation is initiated under a pump power of 1.25 W, beyond which the average power increases linearly with the pump power. A slope efficiency of 10.8% is obtained by linear fitting. The maximum average power of 303 mW can be further enlarged by increasing the pump power, which, however, will pose a risk of damage to the end face of the PM fluoride fiber. Figure 5c, d shows the evolution of the output spectra and the AC traces after noise reduction, respectively. As the pump power is increased, the spectrum experiences a slight redshift probably resulting from the dynamical gain spectrum of the gain fiber, while the AC trace has no obvious change. Elongation of the NLP packet width with pump power predicted by the numerical simulations is expected to be observed by an autocorrelator with a wider scan range. The pedestal-to-peak ratio is preserved, indicating a maintenance of the pulse coherence.
The output performance of the oscillator depends on the output coupling ratio, which can be simply adjusted by rotating the HWP1. To further explore the characteristics of the MIR NLPs, we have tracked the changes of the output NLPs while rotating the HWP1 under a fixed pump power of 3.2 W. An intriguing feature is that the NLP mode-locking state can be sustained even under a large degree of adjustment of the HWP1. Figure 6a,b depicts the output average power and pulse energy as a function of the angular position of the HWP1, respectively. Note that the initial azimuth angle (0°) of the HWP1 corresponds to the position at which we obtain the NLPs described in Figure 5. The output power increases from 113 to 498 mW for the azimuth angles from À10°to 16°, respectively. A maximum pulse energy of 12 nJ is obtained. The saturation of the output power may result from the slowing growth of the output coupling ratio. Figure 6c shows the output spectra at increasing output average power levels. As the output power increases, the spectrum gets broadened symmetrically. We attribute this spectral broadening to the enhanced self-phase modulation effect, which is caused by the increase of intensities of the internal subpulses. As a comparison, the spectrum in Figure 5c does not show any noticeable broadening. This is because the intensities of the internal subpulses remain barely constant when changing the pump power, as theoretically demonstrated above. The experimental phenomena agree well with the numerical simulations. The evolution of the NLP packet width with the output power in Figure 6d cannot be observed either due to the limited scan range of the autocorrelator.

Conclusion
In summary, we have made a systematic investigation of MIR NLPs for the first time. Numerical simulations are carried out www.advancedsciencenews.com www.adpr-journal.com first based on the extended coupled nonlinear Schrödinger equations, and suggest that MIR NLPs can be generated by introducing a birefringence-induced PDD between the orthogonal-polarized components of intracavity circulating pulses. According to the simulations, we experimentally demonstrate the generation of linearly polarized NLPs at 2.8 μm from an all-PM fluoride fiber oscillator mode-locked by a semiconductor saturable absorber. By appropriately setting the pump power and waveplate angles, a maximum average power of 498 mW with a spike width of 4.3 ps is obtained, corresponding to a pulse energy of 12 nJ. Moreover, we have analyzed the characteristics of the MIR NLPs, about which the experimental results agree well with the numerical simulations. Our work not only provides a method for generating NLPs in the MIR, but also sheds new insights into the complex nonlinear behavior of NLPs in MLFFLs, which paves the way for further developments of promising applications in materials processing, supercontinuum generation, and low-coherence spectral interferometry.