Conventional Soliton and Noise-Like Pulse Generated in an Er-Doped Fiber Laser with Carbon Nanotube Saturable Absorbers

applied sciences

Article Conventional Soliton and Noise-Like Pulse Generated in an Er-Doped Fiber Laser with Carbon Nanotube Saturable Absorbers

Zikai Dong 1, Jinrong Tian 1, Runlai Li 2 , Youshuo Cui 1, Wenhai Zhang 3 and Yanrong Song 1,* 1 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China; [email protected] (Z.D.); [email protected] (J.T.); [email protected] (Y.C.) 2 Department of Chemistry, National University of Singapore, Singapore 637551, Singapore; [email protected] 3 College of Environment and Energy Engineering, Beijing University of Technology, Beijing 100124, China; [email protected] * Correspondence: [email protected]

Received: 6 July 2020; Accepted: 27 July 2020; Published: 11 August 2020

Featured Application: Multi-functional ultrafast laser sources, conventional soliton and noise-like pulse laser system, nanomaterial saturable absorption eﬀect test system.

Abstract: Conventional soliton (CS) and noise-like pulse (NLP) are two different kinds of pulse regimes in ultrafast fiber lasers, which have many intense applications. In this article, we experimentally demonstrate that the pulse regime of an Er-doped fiber laser could be converted between conventional soliton and noise-like pulse by using fast response saturable absorbers (SA) made from different layers of single-wall carbon nanotubes (CNT). For the monolayer (ML) single-wall CNT-SA, CS with pulse duration of 439 fs at 1560 nm is achieved while for the bilayer (BL) single-wall CNT, NLP at 1560 nm with a 1.75 ps spike and a 98 ps pedestal is obtained. The transition mechanism from CS to NLP is investigated by analyzing the optical characteristics of ML and BL single-wall CNT. The further theoretical simulation illustrates that CNT-SA enables the switching between CS and NLP in anomalous dispersion regime in Er-doped fiber lasers.

Keywords: ﬁber laser; nonlinear interaction; nanomaterials; conventional soliton; noise-like pulse

1. Introduction Passive mode-locked laser generating ultrashort pulse (USP) is now exceedingly essential for diverse applications in fundamental science, material processing, and medical treatment [1,2]. Nanomaterial saturable absorber (SA) based fiber laser has developed into a potential technology as nanomaterials exhibit excellent optical properties, including broadband optical response, fast relaxation, and high nonlinearity [3,4]. Till now, several types of pulse shapes have been demonstrated, such as conventional soliton (CS), dissipative soliton (DS), and noise-like pulse (NLP) [5–12]. Among them, CS with a hyperbolic secant temporal profile generated in the anomalous dispersion regime is practically useful in optical soliton communications [13]. NLP with sub-pulses in different phases and amplitudes generated in normal or anomalous dispersion regime has potential applications in optical coherence imaging and micro-machining process [14]. DS with linear chirp and high pulse energy that is compressible into high peak power and narrow pulse duration is in demand in nonlinear optics [15]. Since applications may not only require a particular type of USP but also their combinations in one period, switching the operation states of USP from each other in a single fiber laser is critical. From a fundamental point of view, USP can be regulated by the nonlinear absorption of SA, namely,

Appl. Sci. 2020, 10, 5536; doi:10.3390/app10165536 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5536 2 of 12 the saturable absorption effect (SAE) and reverse absorption effect (RAE) arisen from the third-order susceptibility under the incident light field [16–19]. Their differences lie in the reduction of absorption at high intensities for SAE whereas RAE increases. SAE is also a process of filling near the stages of the conduction and valance bands which is cataloged as accumulative nonlinearity [20]. In contrast, RAE is an instantaneous nonlinearity (e.g., two-photon absorption effect) that is always triggered in relatively high intensity compared to SAE [21]. The correlation between SA properties and USP behaviors has been revealed in several studies. For instance, H. Xu et al. reported the reduction of USP pulse duration with increasing SA modulation depth, indicating the stabilization and regulation of SAE on the USP behaviors [18]. Q. Wang et al. achieved the transformation between DS and NLP in a CNT-SA fiber laser operating in normal dispersion regime, owing to the trigger of RAE in higher pump power and the restriction on further USP peak power increasing [22]. The switch between DS and DS resonance using RAE is also possible in the normal dispersion regime. These indicate a nanomaterial-based SA with RAE in relatively low intensity can switch the USP operation states in a fiber laser oscillator. Recent advance on nanoscience has offered a wide range of selection for such saturable absorber, from the classical graphene [20], carbon nanotubes (CNT) [23], topological insulators [24], black phosphorus [25], transition-metal dichalcogenides [26,27], to more recent MXene and bismuthene [28,29]. Among these, CNT possesses a two-photon absorption (TPA) effect to respond to RAE in high peak power conditions, as well as a substantial nonlinear scattering loss from its quasi-one-dimensional cylindrical nanostructure. Based on both nonlinear effects, CNT is highly competitive to introduce RAE into one fiber laser system with relatively low peak power [30,31]. Introducing RAE through CNT-SA into a fiber laser is practical to transform the USP operation states between CS and NLP that has never been explored. To properly integrate CNT-SA in a fiber laser, side polishing and tapered fiber integration methods are both utilized in the evanescent field, which decays exponentially to interaction distances [32–34]. Comparatively, the SA integration method with free-standing polymers is more suitable since it acts directly on the laser and enhances the RAE in relative low peak power conditions [35]. In this article, we demonstrated the production of conventional soliton and noise-like pulse with monolayer and bilayer CNT-SAs in an Er-doped fiber laser in anomalous dispersion regime, for the first time. The saturable absorption effect and reverse absorption effect enabled the transformation of USP operation states. Further numerical simulations verified the essential correlation between nonlinear behaviors and operation state transformations.

2. Experimental Results Figure1 is the schematic of the passively mode-locked Er-doped fiber laser based on CNT-SA. The laser implements a ring cavity configuration, using Er-doped fiber (Liekki, Er110-4/125, 30 cm, nLIGHT, Vancouver, Wash., USA) as the gain medium. A 980 nm laser diode was used as the pump source through a wavelength division multiplexer (WDM, 980/1550 nm, Connet, Shanghai, China). Laser operation direction was ensured by a polarization-independent isolator (PI-ISO, 1550 nm, Connet, Shanghai, China). Cavity polarization and intra-cavity birefringence were regulated by a polarization controller (PC, Connet, Shanghai, China). Output power was controlled by the 10% intensity fiber output coupler (OC, Connet, Shanghai, China). CNT/PVA hybrid film as a SA was assembled between two fiber ferrules by the sandwiching structure. For CNT-SA preparation, single-wall carbon nanotube powder was purchased from XFNANO, and polymer fabrication method was similar to previously reported [36]. The cavity length and net cavity dispersion were 6.5 m and 0.023 ps2, respectively. − A stable mode-locking was self-started without adjusting the PC when ML CNT-SA inserted into the flange with a pump power of 40 mW. The pulse characteristics are demonstrated in Figure2. The pulse duration is 439 fs with a sech2-pulse assumption as shown in Figure2a. In Figure2b, the spectrum is centered at 1560 nm with a 3-dB bandwidth of ~5.3 nm, and the Kelly sideband indicates it is typical conventional soliton. Pulse trains and radio frequency (RF) spectrum are correspondingly illustrated in Figure2c,d. The pulse period is 32.5 ns and fundamental frequency is 28.7 MHz, both of Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 13

Appl. Sci. 2020, 10, 5536 3 of 12

which depend on the cavity length and the signal-to-noise ratio (SNR) of the RF spectrum is about Appl.73 Sci. dB, 2020 which, 10, x showsFOR PEER the REVIEW mode-locking is very stable. 3 of 13

Figure 1. Schematic diagram of the mode-locked fiber laser based on polymer CNT-SA. Inserted left, the autocorrelation traces of CS with ML CNT-SA and NLP with BL CNT-SA. Inserted right, the corresponding CS and NLP spectra.

A stable mode-locking was self-started without adjusting the PC when ML CNT-SA inserted into the flange with a pump power of 40 mW. The pulse characteristics are demonstrated in Figure 2. The pulse duration is 439 fs with a sech2-pulse assumption as shown in Figure 2a. In Figure 2b, the spectrum is centered at 1560 nm with a 3-dB bandwidth of ~5.3 nm, and the Kelly sideband indicates it is typical conventional soliton. Pulse trains and radio frequency (RF) spectrum are correspondingly illustratedFigure in 1.FigureSchematic 2c,d. diagram The pulse of the period mode-locked is 32.5 ﬁber ns laserand basedfundamental on polymer frequency CNT-SA. Inserted is 28.7 left,MHz, both Figure 1. Schematic diagram of the mode-locked fiber laser based on polymer CNT-SA. Inserted left, of whichthe depend autocorrelation on the cavity traces oflength CS with and MLthe CNT-SAsignal-to-noise and NLP ratio with (SNR) BL CNT-SA. of the RF Inserted spectrum right, is about the autocorrelation traces of CS with ML CNT-SA and NLP with BL CNT-SA. Inserted right, the 73 dB, whichthe corresponding shows the CS mode-locking and NLP spectra. is very stable. corresponding CS and NLP spectra.

Figure 2. Output characteristics of CS using ML CNT-SA with a pump power of 40 mW in experiment. Figure(a) The 2. Output autocorrelation characteristics trace; ( bof) spectrum;CS using ML (c) pulseCNT-SA trains; with (d )a RF pump spectrum, power the of 40 inset mW shows in experiment. the RF (a) spectrumThe autocorrelation with a span oftrace; 180 MHz.(b) spectrum; (c) pulse trains; (d) RF spectrum, the inset shows the RF spectrum with a span of 180 MHz. The pulse width increases from 439 fs to 628 fs with the increase of pump power. However, limitedThe pulse by the width soliton increases area theory, from further 439 increasing fs to 628 thefs with pump the power increase results of in pump the burst power. of following However, limitedpulse by trains the soliton into bound area solitontheory, state. further These increasing CS behaviors the pump associated powerwith results increasing in the burst pump of powerfollowing pulseautocorrelation trains into tracebound (AC) soliton and corresponding state. These spectraCS behaviors evolution associated are presented with in increasing Figure3a,b. pump We notice power autocorrelationthat the two bound trace solitons (AC) and always corresponding exhibit ﬁxed, discretespectra temporal evolution separations, are presented and the in phase Figure diﬀ erence3a,b. We is always about π where the interaction optical spectra own a dip in the center of the spectra, as shown notice that the two bound solitons always exhibit fixed, discrete temporal separations, and the phase in Figure3b3. Figure 2. Output characteristics of CS using ML CNT-SA with a pump power of 40 mW in experiment. (a) The autocorrelation trace; (b) spectrum; (c) pulse trains; (d) RF spectrum, the inset shows the RF spectrum with a span of 180 MHz.

The pulse width increases from 439 fs to 628 fs with the increase of pump power. However, limited by the soliton area theory, further increasing the pump power results in the burst of following pulse trains into bound soliton state. These CS behaviors associated with increasing pump power autocorrelation trace (AC) and corresponding spectra evolution are presented in Figure 3a,b. We notice that the two bound solitons always exhibit fixed, discrete temporal separations, and the phase Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13 Appl. Sci. 2020, 10, 5536 4 of 12 difference is always about π where the interaction optical spectra own a dip in the center of the spectra,Appl. Sci. as 2020 shown, 10, x FORin Figure PEER REVIEW 3b3. 4 of 13

difference is always about π where the interaction optical spectra own a dip in the center of the spectra, as shown in Figure 3b3.

Figure 3. Function of experimental CS behavior with pump power. (a1)–(a3) CS pulse broadening and Figure 3. Function of experimental CS behavior with pump power. (a1)–(a3) CS pulse broadening and breaking-upbreaking-up with with increasing increasing pump pump power power from from 40 40 to to 200200 mW; ( (b1b1)–()–(b3b3) )the the corresponding corresponding spectra. spectra. Figure 3. Function of experimental CS behavior with pump power. (a1)–(a3) CS pulse broadening and In orderIn breaking-uporder to confirm to confirm with the increasing feasibilitythe feasibility pump to power switchto switch from pulse 40puls to operation200e operation mW; (b1 states)–( statesb3) the by by corresponding significantlysignificantly spectra. changing changing the the SA parameters,SA parameters, we assembled we assembled the bilayer the bi CNT-SAlayer CNT-SA in the in flange the flange without without other other modifications. modifications. Typical Typical pulse durationpulse and durationIn optical order toand spectrum confirm optical the generatedspectrum feasibility generated byto switch BL CNT-SA puls by eBL operation areCNT-SA demonstrated states are bydemonstrated significantly in Figure changingin4a,b. Figure Intriguingly, the 4a,b. the spectrumIntriguingly,SA parameters, has the a triangle spectrum we assembled shape has a the triangle and bilayer the shape KellyCNT-SA and sideband inthe the Kelly flange is sideband not without observed. is other not observed. modifications. The measured The measuredTypical AC trace pulse duration and optical spectrum generated by BL CNT-SA are demonstrated in Figure 4a,b. changesAC fromtrace sechchanges2-shape from to sech NLP2-shape shape to whoseNLP shape AC tracewhose has AC a uniquetrace has characteristic a unique characteristic of a narrow of a spike narrowIntriguingly, spike on the top spectrum of the broadhas a triangle pedestal. shape The and NLP the spike Kelly widthssideband remain is not observed.nearly unchanged The measured at 1.75 on top ofAC the trace broad changes pedestal. from Thesech2 NLP-shape spike to NLP widths shape remain whose nearlyAC trace unchanged has a unique at characteristic 1.75 ps with of increasing a ps with increasing pump power (inset of Figure 4c) while NLP pedestals broaden from 15 ps to 98 pump powernarrow (inset spike ofon Figuretop of the4c) broad while pedestal. NLP pedestals The NLP spike broaden widths from remain 15 ps nearly to 98 unchanged ps. The NLPat 1.75 spectra ps. The NLP spectra are centered at 1531 nm as a triangle shape and the 3-dB bandwidth gradually are centeredps with at increasing 1531 nm aspump a triangle power (inset shape of andFigure the 4c) 3-dB while bandwidth NLP pedestals gradually broaden broadensfrom 15 ps withto 98 higher broadens with higher power in Figure 4d. The above results prove the ability to transform the pulse ps. The NLP spectra are centered at 1531 nm as a triangle shape and the 3-dB bandwidth gradually poweroperation in Figure states4d. Thein an above Er-doped results fiber prove laser by the changing ability toCNT-SA transform parameters. the pulse operation states in an Er-dopedbroadens fiber laser with byhigher changing power in CNT-SA Figure 4d. parameters. The above results prove the ability to transform the pulse operation states in an Er-doped fiber laser by changing CNT-SA parameters.

Figure 4. Function of experimental NLP characteristics and NLP behavior with pump power. (a) Typical FigureFigure 4. 4.Function Function of of experimental experimental NLPNLP characteristicscharacteristics andand NLPNLP behaviorbehavior with with pump pump power. power. (a ) (a) NLP ACTypicalTypical trace NLP at NLP the AC AC pump trace trace powerat at the the pump ofpump 280 power power mW, inserted ofof 280280 mW, shows insertedinserted the NLP showsshows spike the the NLP area;NLP spike (spikeb) thearea; area; corresponding (b )( bthe) the spectrum;correspondingcorresponding (c) NLP behavior spectrum; spectrum; with (c ()c )NLP increasingNLP behavior behavior pump withwith powerincreasing from pumppump 160 power mWpower tofrom from 400 160 mW,160 mW mW inserted to to400 400 mW, shows mW, the NLP spikeinsertedinserted area; shows shows (d) thethe the correspondingNLP NLP spike spike area; area; spectra( d(d)) thethe corresponding evolutioncorresponding with spectraspectra zoomed-in evolution evolution NLP with with spectra. zoomed-in zoomed-in NLP NLP spectra.spectra. We first performed a balanced intensity scan twin-detected method (I-scan) to distinguish the correlation between CNT-SA parameters and output characteristics, where the CS generated, abovementioned, was set as a seed laser and injected into an amplifier. Detailed configurations are demonstrated in Figure5. The amplified laser was set as a test source of the I-scan technique to acquire the same intracavity power scale of the CNT-SA mode-locked fiber laser generated. The pulse duration associated with optical spectrum of the amplifier was kept identical in the ML CNT-SA mode-locked Appl. Sci. 2020, 10, 5536 5 of 12

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13 fiber laser. Transmittance curves functioned with peak power using ML and BL CNT-SA are shown in Figure6 and fittedWe first by performed the saturable a balanced absorption intensity formula: scan twin-detected method (I-scan) to distinguish the correlation between CNT-SA parameters and output characteristics, where the CS generated, abovementioned, was set as a seed laser and injected into an amplifier. Detailed configurations are T(I) = 1 α exp( I/Isat) αns β I (1) demonstrated in Figure 5. The amplified− ·laser was− set as− a test −source· of the I-scan technique to acquire the same intracavity power scale of the CNT-SA mode-locked fiber laser generated. The pulse where T representsduration associated the transmission, with optical spectrumαns is the of the non-saturable amplifier was kept loss identical coefficient, in the αMLis CNT-SA the modulation depth, β ismode-locked reverse absorption fiber laser. Transmittance coefficient, curvesIsat is fu thenctioned saturable with peak intensity, power using and MLI andis the BL CNT- incident laser intensity. BySA are fitting shown the in Figure experimental 6 and fitted data, by the thesaturable estimated absorption ML formula: CNT-SA modulation depth is about TI()=⋅ 1-αα exp(- II )- -β ⋅ I 3.1%, Isat is 50 kW, and αns is 41.5%, respectively. Thesat estimated ns BL CNT-SA modulation(1) depth is 4 1 about 4.7%,whereIsat isT represents 38 kW, β theis transmission, 2.68 10 αWns is the, and non-saturableαns is 61%, loss respectively.coefficient, α is Thesethe modulation results indicate × − − different modulationdepth, β is reverse depths absorption and linear coefficient, losses I insat is the the ML saturable and BLintensity, CNT-SA and configurations.I is the incident laser Specifically, BL CNT-SAintensity. owns the By abilityfitting the to introduceexperimental reverse data, the absorption estimated ML coe CNT-SAfficient modu into thelation laser depth cavity, is about whereas ML 3.1%, Isat is 50 kW, and αns is 41.5%, respectively. The estimated BL CNT-SA modulation depth is about CNT-SA does not. The reasons can be concluded that single-wall CNT has large scattering loss than 4.7%, Isat is 38 kW, β is 2.68×10−4 W−1, and αns is 61%, respectively. These results indicate different other nanomaterials,modulation depths thus and enhancing linear losses the in the nonlinear ML and BL scatteringCNT-SA configurations. loss with Specifically, more CNTs. BL CNT- Meanwhile, the two-photonSA owns absorption the ability to e ffintroduceect of single-wall reverse absorpti CNTon coefficient will also into excite the reverselaser cavity, absorption whereas ML coe fficient. Both are consideredCNT-SA does as not. primary The reasons factors can leading be concluded to the that transformation single-wall CNT has from large CS scattering to NLP loss with than BL CNT-SA other nanomaterials, thus enhancing the nonlinear scattering loss with more CNTs. Meanwhile, the setup. To betwo-photon honest, absorption polyvinyl effect alcohol of single-wall (PVA) CNT has will a relatively also excite lowerreverse thermalabsorption threshold coefficient. Both compared to CNT nanomaterial,are considered high as primary peak power factors leading pulses to may the transformation degrade the from PVA CS polymer to NLP with hence BL CNT-SA decreasing the laser transmittancesetup. To be ratio. honest, These polyvinyl mechanisms alcohol (PVA) of has CNT a relatively polymer lower matrix thermal may threshold give an compared inspection to of why ML-CNT andCNT BL-CNT nanomaterial, could high respond peak power to di pulsesfferent may transmittance degrade the PVA curves polymer in experiments. hence decreasing To the decrease the laser transmittance ratio. These mechanisms of CNT polymer matrix may give an inspection of why possibilityML-CNT of PVA degradation,and BL-CNT could long respond time CSto different and NLP tran outputsmittance stabilities curves in experiments. were tested To in decrease experimental as shown in Figurethe possibility7a,b. of PVA degradation, long time CS and NLP output stabilities were tested in experimental as shown in Figure 7a,b.

Figure 5. SchematicFigure 5. Schematic of I-scan of I-scan system. system. Seed Seed laser laser pulse pulse ty typepe is isthe the CS, CS,inserted inserted shows showsthe AC trace the ACand trace and corresponding spectrum of the CS laser; amplifier is used to increase the output CS power to reach corresponding spectrum of the CS laser; ampliﬁer is used to increase the output CS power to reach the the intra-cavity power level, insert shows the amplifier spectra evolution; I-scan setup is used to detect intra-cavitythe power saturable level, absorption insert of shows CNT-SA. the ampliﬁer spectra evolution; I-scan setup is used to detect the saturable absorptionAppl. Sci. 2020, 10 of, x FOR CNT-SA. PEER REVIEW 6 of 13

Figure 6. NonlinearFigure 6. Nonlinear saturable saturable absorption absorption of ofML MLCNT-SA CNT-SA and BL CNT-SA and using BL homemade CNT-SA I-scan using homemade system. I-scan system.

Figure 7. (a,b) Long-term stabilities of CS and NLP every 1 h sampling over 6 h.

3. Simulation Results From above experimental data, we conclude a saturable absorber with lower modulation depth and no reverse absorption enables the generation of CS, where a saturable absorber with larger loss and reverse absorption leads to NLP. This provides the tunability of pulse-type in one laser system by modifying the SA parameters. A simulation was then conducted to explain the influence of the SA parameters on laser performance. The simplified numerical model of cavity setup is shown in Figure 8 with a piece of Er- doped gain fiber (EDF), three segments of single-mode fiber (SMF1, SMF2, and SMF3), an optical coupler (OC), a wavelength division multiplexer (WDM), and a saturable absorber (SA). The cavity parameters in simulation are estimated from experimental data which were shown in Table 1, including the same cavity length, relative position of each device, and the same net cavity dispersion. Saturable coefficient and reverse absorption coefficient are described by the transmittance function in Equation (1). To simulate the transformation of pulse operation states in the fiber laser, coherently coupled nonlinear Schrödinger equation (NLSE) including dispersion, nonlinearity, gain, and loss was used as the master equation to describe the pulse propagation in the fiber system (Equations (2) and (3)). Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 13

Figure 6. Nonlinear saturable absorption of ML CNT-SA and BL CNT-SA using homemade I-scan Appl. Sci. 2020, 10, 5536 6 of 12 system.

Figure 7. (a,b) Long-term stabilities of CS and NLP every 1 h sampling over 6 h. Figure 7. (a,b) Long-term stabilities of CS and NLP every 1 h sampling over 6 h. 3. Simulation Results 3. Simulation Results From above experimental data, we conclude a saturable absorber with lower modulation depth and noFrom reverse above absorption experimental enables data, the we generation conclude ofa saturable CS, where absorber a saturable with absorber lower modulation with larger depth loss andand reverse no reverse absorption absorption leads enables to NLP. the This generation provides theof CS, tunability where a of saturable pulse-type absorber in one laserwith systemlarger loss by modifyingand reverse the absorption SA parameters. leads to NLP. This provides the tunability of pulse-type in one laser system by modifyingA simulation the SA was parameters. then conducted to explain the influence of the SA parameters on laser performance.A simulation The simplifiedwas then numericalconducted modelto explain of cavity the influence setup is shown of the inSA Figure parameters8 with aon piece laser ofperformance. Er-doped gain The fiber simplified (EDF), three numerical segments model of single-modeof cavity setup fiber is (SMF1,shown in SMF2, Figure and 8 SMF3),with a piece an optical of Er- couplerdoped (OC),gain fiber a wavelength (EDF), three division segments multiplexer of single-mode (WDM), fiber and a(SMF1, saturable SMF2, absorber and SMF3), (SA). The an cavityoptical parameterscoupler (OC), in simulationa wavelength are division estimated multiplexer from experimental (WDM), and data a saturable which wereabsorber shown (SA). in The Table cavity1, includingparameters the in same simulation cavity length, are estimated relative position from ex ofperimental each device, data and which the same were net shown cavity dispersion.in Table 1, Saturableincluding coe theffi samecient cavity and reverse length, absorption relative position coefficient of each are describeddevice, and by the the same transmittance net cavity function dispersion. in EquationSaturable (1). coefficient To simulate and thereverse transformation absorption coeffi of pulsecient operation are described states by in the fibertransmittance laser, coherently function coupledin Equation nonlinear (1). To Schrödinger simulate the equationtransformation (NLSE) of including pulse operation dispersion, states nonlinearity, in the fiber laser, gain, coherently and loss wascoupled used asnonlinear the master Schrödinger equation toequation describe (NLSE) the pulse including propagation dispersion, in the nonlinearity, fiber system (Equationsgain, and loss (2) andwas (3)). used as the master equation to describe the pulse propagation in the fiber system (Equations (2) and (3)). ∂A i ∂2A g g ∂2A = β + iγ A 2A + A + (2) 2 2 2 2 ∂z −2 ∂t | | 2 2Ωg ∂t

 R 2   A dt   g = g0 exp | |  (3) − Esat 

where A denotes the slow varying amplitude of the pulse envelope, z is the propagation distance, t is the pulse local time. β2 and γ are group velocity dispersion (GVD) and fiber nonlinear coefficient. g, g0, and Ωg denote gain function, small-signal gain coefficient, and gain bandwidth in Equations (2) Appl. Sci. 2020and, (3), 10, respectively.x FOR PEER SuchREVIEW simulation can provide insights on the evolution of laser performance with 7 of 13 SA parameters and explain the experimental pulse operation states transformation.

Figure 8. Simpliﬁed numerical simulation model. Figure 8. Simplified numerical simulation model.

∂∂22 ∂ Ai=−βγ A +2 + g + gA 2 222iAA A (2) ∂∂zt222 Ω∂ t g

2 A dt gg=−exp 0  (3) Esat 

Table 1. Constant and variable parameters in the simulation.

WDM EDF SMF OC SA

𝛼 = β2 = 4.5 ps2/km β2 = 15.54 ps2/km β2 = −12.2 ps2/km 10% output 15% Γ = 1.1 × 10−3 Γ = 3.0 × 10−3 Γ = 3.0 × 10− 3 i.e., T = 90% 𝛼 = * m−1W−1 m−1W−1 m−1W−1 LWDM = 0.5 m LEDF = 0.5 m LSMF1 = 0.5 m Isat = * Ωg = 40 nm LSMF2 = 0.5 m Β = * g0 = * LSMF3 = 4.5 m Psat,g = 2.3 W * represents variable parameters. where A denotes the slow varying amplitude of the pulse envelope, z is the propagation distance, t is the pulse local time. β2 and γ are group velocity dispersion (GVD) and fiber nonlinear coefficient. g, g0, and Ωg denote gain function, small-signal gain coefficient, and gain bandwidth in equation (2) and (3), respectively. Such simulation can provide insights on the evolution of laser performance with SA parameters and explain the experimental pulse operation states transformation. It should be mentioned that the positions of Kelly sidebands in the optical spectrum reflect the cavity dispersion, despite that the positions of these soliton sidebands could be slightly affected by the amount of chirp acquired by the pulse [37,38]. The total cavity dispersion is estimated by the following Equation (4) with experimentally observed parameters, including the central wavelength, the first-order Kelly sideband separation, Δλ1 = 4.0 nm, and the soliton pulse width t = 439 fs,

λ 2 Δ=±⋅λλ2N − 0 N N 0 0.0787 (4) cDL ()ct 2 where N is an integer, D is the average cavity dispersion parameter, L is the fiber length. The estimated total cavity dispersion is 0.03 ps nm−1. The total cavity dispersion comprises the fiber dispersion and the material dispersion from ML and BL CNT-SA. Thereafter, the material dispersion from CNT polymer could be roughly deduced as:

DML CNT = −0.3 D (5)

2 −1 Variable parameters including β2, g0, 𝛼, Isat, and β were set as −12.2 ps /km, 15 m , 0.15, 50 W, and 0 W−1 to simulate the CS formation process, while the initial pulse condition was set as 10 ps chirp-free pulse and the ML CNT-SA parameters were used. Stable CS can be generated in the Appl. Sci. 2020, 10, 5536 7 of 12

Table 1. Constant and variable parameters in the simulation.

WDM EDF SMF OC SA 2 2 2 β2 = 4.5 ps /km β2 = 15.54 ps /km β2 = 12.2 ps /km 10% output αns = 15% 3 1 1 3 1 1 − 3 1 1 Γ = 1.1 10 m W Γ = 3.0 10 m W Γ = 3.0 10 m W i.e., T = 90% αs = * × − − − × − − − × − − − LWDM = 0.5 m LEDF = 0.5 m LSMF1 = 0.5 m Isat = * Ωg = 40 nm LSMF2 = 0.5 m B = * g0 = *LSMF3 = 4.5 m Psat,g = 2.3 W * represents variable parameters.

It should be mentioned that the positions of Kelly sidebands in the optical spectrum reflect the cavity dispersion, despite that the positions of these soliton sidebands could be slightly affected by the amount of chirp acquired by the pulse [37,38]. The total cavity dispersion is estimated by the following Equation (4) with experimentally observed parameters, including the central wavelength, the first-order Kelly sideband separation, ∆λ1 = 4.0 nm, and the soliton pulse width t = 439 fs, s 2N λ 2 ∆λN = N λ0 0.0787 0 (4) ± · cDL − (ct)2 where N is an integer, D is the average cavity dispersion parameter, L is the fiber length. The estimated 1 total cavity dispersion is 0.03 ps nm− . The total cavity dispersion comprises the fiber dispersion and the material dispersion from ML and BL CNT-SA. Thereafter, the material dispersion from CNT polymer could be roughly deduced as:

D = 0.3 D (5) ML CNT − 2 1 Variable parameters including β2, g0, αs,Isat, and β were set as 12.2 ps /km, 15 m− , 0.15, 1 − 50 W, and 0 W− to simulate the CS formation process, while the initial pulse condition was set as 10Appl. ps Sci. chirp-free 2020, 10, x FOR pulse PEER and REVIEW the ML CNT-SA parameters were used. Stable CS can be generated8 of in13 the simulation in Figure9a. The time domain evolution, AC trace and spectrum of the generated simulation in Figure 9a. The time domain evolution, AC trace and spectrum of the generated CS are1 CS are shown in Figure9a1–a3, respectively. Increasing the value of gain coe ﬃcient g0 to 20 m− −1 shownresults in Figure splitting 9a1–a3, of single respectively. pulse split Incr intoeasing two the (so-called value of two-bound gain coefficient soliton g0 state)to 20 m in Figureresults9 b.in Thesplitting corresponding of single pulsepulse information split into includingtwo (so-called the evolution two-bound map, soliton AC trace, state) and in optical Figure spectrum 9b. The of correspondingtwo-bound soliton pulse are information presented in including Figure9b1–b3, the evol respectively.ution map, These AC simulatedtrace, and results optical indicated spectrum that of two-boundSA with saturable soliton absorption are presented eﬀect in could Figure induce 9b1–b3, the respectively. formation of CSThese pulse. simulated results indicated that SA with saturable absorption effect could induce the formation of CS pulse.

Figure 9. Numerical simulation of CS output pulse characteristics as a function of pump power using Figurethe ML 9. CNT-SA Numerical parameters. simulation ( aof) SingleCS output CS pulsepulse evolution;characteristics (b) two-boundas a function soliton of pump state power evolution; using (thea1)–( MLa3 )CNT-SA are the simulated parameters. time (a evolution,) Single CS AC pulse trace evolution; and optical (b) spectrum two-bound of thesoliton CS; (b1state)–( b3evolution;) are the (simulateda1)–(a3) are time the evolution, simulated AC time trace evolution, and optical AC trace spectrum and optical of the two-boundspectrum of soliton, the CS; respectively. (b1)–(b3) are the simulated time evolution, AC trace and optical spectrum of the two-bound soliton, respectively.

To simulate the BL CNT-SA, two primary factors should be considered: the introduction of reverse absorption coefficient (β was set as 0.0015 W−1) and higher value of normal dispersion (−0.3D from Equation (5)). Stable mode-locking NLP pulse evolution generated in the fiber laser is demonstrated in Figure 10, with various gain coefficients (g0 = 70 W−1, 100 W−1, and 150 W−1, respectively). By increasing the gain coefficient from 70 to 150 W−1, the NLP envelope broadens while the NLP spike remains nearly unchanged. These simulation results are in accordance with experimental results, illustrating the critical effect of reverse absorption in the NLP pulse operation state formation.

Figure 10. Numerical simulation of NLP output pulse characteristics as a function of pump power using the BL-CNT SA setup. (a1)–(c1) the simulated envelope evolution of the NLP pulse at the gain

coefficient g0 values of 70, 100, and 150 W−1; (a2)–(a4) the simulated time evolution, AC trace and optical spectrum at g = 70 W−1; (b2)–(b4) g = 100 W−1 and (c2)–(c4) g = 150 W−1, respectively. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 simulation in Figure 9a. The time domain evolution, AC trace and spectrum of the generated CS are shown in Figure 9a1–a3, respectively. Increasing the value of gain coefficient g0 to 20 m−1 results in splitting of single pulse split into two (so-called two-bound soliton state) in Figure 9b. The corresponding pulse information including the evolution map, AC trace, and optical spectrum of two-bound soliton are presented in Figure 9b1–b3, respectively. These simulated results indicated that SA with saturable absorption effect could induce the formation of CS pulse.

Figure 9. Numerical simulation of CS output pulse characteristics as a function of pump power using the ML CNT-SA parameters. (a) Single CS pulse evolution; (b) two-bound soliton state evolution; (a1)–(a3) are the simulated time evolution, AC trace and optical spectrum of the CS; (b1)–(b3) are the Appl. Sci. 2020, 10, 5536 8 of 12 simulated time evolution, AC trace and optical spectrum of the two-bound soliton, respectively.

To simulatesimulate the the BL BL CNT-SA, CNT-SA, two two primary primary factors factor shoulds should be considered: be considered: the introduction the introduction of reverse of reverseabsorption absorption coefficient coefficient (β was set(β was as 0.0015 set as W0.00151) and W−1 higher) and higher value ofvalue normal of normal dispersion dispersion ( 0.3D (− from0.3D − − fromEquation Equation (5)). Stable (5)). mode-lockingStable mode-locking NLP pulse NLP evolution pulse generatedevolution ingenerated the fiber laserin the is demonstratedfiber laser is demonstrated in Figure 10, with various gain coefficients1 (g0 = 701 W−1, 100 W−11, and 150 W−1, in Figure 10, with various gain coefficients (g0 = 70 W− , 100 W− , and 150 W− , respectively). respectively). By increasing the gain coefficient from 701 to 150 W−1, the NLP envelope broadens while By increasing the gain coefficient from 70 to 150 W− , the NLP envelope broadens while the NLP thespike NLP remains spike nearly remains unchanged. nearly Theseunchanged. simulation These results simulation are in accordance results withare in experimental accordance results, with experimentalillustrating the results, critical illustrating effect of reverse the critical absorption effect inof thereverse NLP absorption pulse operation in the state NLP formation. pulse operation state formation.

Figure 10. Numerical simulation of NLP output pulse characteristics as a function of pump power using the BL-CNT SA setup. ( a1)–()–(c1)) the simulated envelope evolution of the NLP pulse at the gain 1 coefficientcoefficient g0 values of 70, 100, and 150150 WW−−1;;( (a2a2)–()–(a4a4)) the the simulated simulated time time evolution, AC trace and 1 1 1 optical spectrum at g = 7070 W W−−1; ;((b2b2)–()–(b4b4) )g g == 100100 W W−1− andand (c2 ()–(c2)–(c4c4) g) = g 150= 150 W− W1, −respectively., respectively. 4. Discussion The evolutions of pulse envelope and chirp, together with the corresponding optical spectra and SA transmittance curves, were carefully studied in Figure 11 to reveal the influence of reverse absorption coefficient on the NLP formation process. Numerical simulation of the NLP formation process in different roundtrips was depicted in four typical stages: (a) pulse amplifying, (b) peak power clamping, (c) pulse instability, and (d) NLP envelope stability. First, in stage a, the initial pulse is amplified in the first 11th roundtrips with a gaussian shape in Figure 11a, where its peak power is not sufficient to trigger the reverse absorption regime. In stage b, the next 12th to 140th roundtrips revolution, the pulse time domain is narrowed down by anomalous dispersive effect and the SA saturable absorption modulation, showing a typical broadened optical spectrum from SPM in Figure 11b3. The enhanced peak power in the SA transmittance curves triggers off the reverse absorption regime, rising the peak power clamping effect. In stage c, further narrowing pulse induces a significant increase in peak power, which induces the pulse time domain instability in Figure 11c1 in return. The SA transmittance curves record tiny pulse generation in pulse envelope, and the parasitic tiny pulse further enhances the single pulse shape instability and bursts into noise like pulse regime. Finally, in stage d, the NLP pulse time domain gradually broadens to decrease high nonlinearity from SPM and reverse absorption effect, as a balance among the SPM, dispersion, saturable absorption, and reverse absorption effects in this laser system. The final evolution state exhibits a rectangle shape pulse with free chirp as shown in Figure 11d1. The above simulation results demonstrate the ability of reverse absorption to switch the pulse state from single pulse to NLP through restricting the peak power infinite increase in the anomalous dispersion regime. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13

4. Discussion The evolutions of pulse envelope and chirp, together with the corresponding optical spectra and SA transmittance curves, were carefully studied in Figure 11 to reveal the influence of reverse absorption coefficient on the NLP formation process. Numerical simulation of the NLP formation process in different roundtrips was depicted in four typical stages: (a) pulse amplifying, (b) peak power clamping, (c) pulse instability, and (d) NLP envelope stability. First, in stage a, the initial pulse is amplified in the first 11th roundtrips with a gaussian shape in Figure 11a, where its peak power is not sufficient to trigger the reverse absorption regime. In stage b, the next 12th to 140th roundtrips revolution, the pulse time domain is narrowed down by anomalous dispersive effect and the SA saturable absorption modulation, showing a typical broadened optical spectrum from SPM in Figure 11b3. The enhanced peak power in the SA transmittance curves triggers off the reverse absorption regime, rising the peak power clamping effect. In stage c, further narrowing pulse induces a significant increase in peak power, which induces the pulse time domain instability in Figure 11c1 in return. The SA transmittance curves record tiny pulse generation in pulse envelope, and the parasitic tiny pulse further enhances the single pulse shape instability and bursts into noise like pulse regime. Finally, in stage d, the NLP pulse time domain gradually broadens to decrease high nonlinearity from SPM and reverse absorption effect, as a balance among the SPM, dispersion, saturable absorption, and reverse absorption effects in this laser system. The final evolution state exhibits a rectangle shape pulse with free chirp as shown in Figure 11d1. The above simulation results demonstrate the ability Appl. Sci. 2020, 10, 5536 9 of 12 of reverse absorption to switch the pulse state from single pulse to NLP through restricting the peak power infinite increase in the anomalous dispersion regime.

Figure 11. Numerical simulation of NLP formation process in different roundtrips with the assistance of Figure 11. Numerical simulation of NLP formation1 process in different roundtrips with the assistance reverse absorption coefficient at 0.0015 W− .(a)–(d) Pulse shape, optical spectrum and SA transmittance −1 curvesof reverse of the absorption typical four-stages coefficient NLP at formation: 0.0015 W pulse. (a amplifier,)–(d) Pulse peak shape, power optical clamping, spectrum pulse instability, and SA andtransmittance NLP envelope curves stability; of the typical (a1)–( four-stagesa3) the pulse NLP amplifier formation: stage; pulse (b1 )–(amplifieb3) ther, peak power clamping, clamping stage;pulse instability, (c1)–(c3) the and pulse NLP instability envelope stage; stability; (d1)–( (a1d3)–() thea3) NLPthe pulse envelope amplifier stability stage; stage. (b1)–(b3) the peak power clamping stage; (c1)–(c3) the pulse instability stage; (d1)–(d3) the NLP envelope stability stage. The NLP envelope characteristics are directly corresponding to the reverse absorption mechanism. To figureThe outNLP the envelope relationship characteristics between NLP are characteristics directly corresponding and reverse absorption to the ereverseffect, the absorption variations ofmechanism. intracavity To NLP figure features out the (including relationship pulse between energy, NLP peak characteristics power, and and pulse reverse duration) absorption are analyzed effect, inthe Figure variations 12. of In intracavity one round NLP intracavity features trip, (including the pulse pulse energy energy, increases peak power, from the and EDF pulse component duration) thenare analyzed decreases in through Figure 12. the In OC one and round SA components. intracavity trip, The remainingthe pulse energy SMF component increases hasfrom no the energy EDF consumption. Under this situation, the peak power largely increases through EDF and monotonically increases through the OC and SA consumption components. It is clear that the SA component with reverse absorption is mainly responsible for the decrease in the intracavity peak power. Compared to the intracavity peak power evolution, the pulse duration evolution is slightly monotonically decreased through the whole position in cavity except for the EDF component. To further reveal their relationships on the NLP pulse characteristics, the comparisons in pulse energy, peak power, and pulse duration with modified relative gain coefficient and reverse absorption coefficient are provided in Figure 12. Increasing 1 1 the gain coefficient from 70 to 150 m− with identical reverse absorption coefficient (0.0015 W− ) leads to linearly increased pulse energies and pulse durations with nearly unchanged peak powers. Increasing 1 the reverse absorption coefficient with identical gain coefficient (100 m− ) produces nearly unchanged pulse energies compared to the variations in modifying the gain coefficients, together with linearly decreased peak powers and monotonically increased pulse durations in Figure 12. These indicate that the reverse absorption can balance the intracavity time domain conventional through decreasing the peak power and broaden the pulse duration. To further explore the noise-like pulse states in various dispersion regimes, the noise-like pulse envelopes associated with the corresponding chirps are recorded in Figure 13a1–c1. In normal dispersion regime (the net cavity dispersion is 5.702 ps2), it is mainly transmitted in the range of pulse envelope though the disorder of the center of the dissipative soliton by newly generated frequency components. Partially disordered pulse envelopes exhibit noise-like pulse features such as the chaotic distribution of chirps and intensity auto-correlation traces with a spike in the center of pulse, as well Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13

component then decreases through the OC and SA components. The remaining SMF component has no energy consumption. Under this situation, the peak power largely increases through EDF and Appl. Sci. 2020monotonically, 10, 5536 increases through the OC and SA consumption components. It is clear that the SA 10 of 12 component with reverse absorption is mainly responsible for the decrease in the intracavity peak power. Compared to the intracavity peak power evolution, the pulse duration evolution is slightly as the featuremonotonically of up-chirp decreased trend through distribution the whole in position Figure in 13 cavitya1. except Different for the from EDF thecomponent. normal To dispersion regime, highfurther peak reveal power their excites relationships new on pulse the NLP envelope pulse characteristics, in small anomalous the comparisons dispersions in pulse (energy,1.448 ps2) that peak power, and pulse duration with modified relative gain coefficient and reverse absorption− consist of chirp-freecoefficient are pulses provided with in Figure same 12. interval Increasing in the Figure gain coefficient13b1. Pulse from separation 70 to 150 m−1 canwith beidentical achieved with a decreasedreverse reverse absorption absorption coefficient coe ffi(0.0015cient W in−1)a leads larger to anomalouslinearly increased dispersion pulse energies regime and ( pulse21.688 ps2) in − Figure 13c1.durations Compared with nearly to the unchanged small dispersion peak powers. regime, Increasing the the junction reverse ofabsorption two pulses coefficient has nowith interaction −1 meaning thatidentical the separated gain coefficient pulses (100 and m pulse) produces envelopes nearly areunchanged different. pulse The energies simulation compared suggests to the noise-like variations in modifying the gain coefficients, together with linearly decreased peak powers and pulses generatedmonotonically in normal increased dispersion pulse durations regime in Figure are compressible12. These indicate with that the an reverse up-chirp absorption distribution can trend, whereas noise-likebalance the pulses intracavity generated time domain in anomalous conventional regimethrough decreasing are incompressible. the peak power and broaden the pulse duration.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 13 that consist of chirp-free pulses with same interval in Figure 13b1. Pulse separation can be achieved with a decreased reverse absorption coefficient in a larger anomalous dispersion regime (−21.688 ps2) in Figure 13c1. Compared to the small dispersion regime, the junction of two pulses has no interaction Figure 12.Figure 12. Intracavity pulse energy, peak power, and pulse duration as a function of position in cavity. meaning that theIntracavity separated pulse pulses energy, and peakpulse power, envelo andpes pulseare different. duration The as a simulation function of suggests position in noise- cavity. TheThe line line charts charts on on the the rightright show the the influence influence of changing of changing relative relative gain coefficient gain coe andfficient reverse and reverse like pulses generatedabsorption coefficientin normal on pulse dispersion energy, peak regime power, andare pulse compressible duration. with an up-chirp distribution absorption coefficient on pulse energy, peak power, and pulse duration. trend, whereas noise-like pulses generated in anomalous regime are incompressible. To further explore the noise-like pulse states in various dispersion regimes, the noise-like pulse envelopes associated with the corresponding chirps are recorded in Figure 13a1–c1. In normal dispersion regime (the net cavity dispersion is 5.702 ps2), it is mainly transmitted in the range of pulse envelope though the disorder of the center of the dissipative soliton by newly generated frequency components. Partially disordered pulse envelopes exhibit noise-like pulse features such as the chaotic distribution of chirps and intensity auto-correlation traces with a spike in the center of pulse, as well as the feature of up-chirp trend distribution in Figure 13a1. Different from the normal dispersion regime, high peak power excites new pulse envelope in small anomalous dispersions (−1.448 ps2)

Figure 13.13. Typical pulsepulse envelopesenvelopes with thethe corresponding chirps at the net cavity dispersion of (a1) 5.7025.702 psps22,(, (b1)−1.4481.448 ps ps22,, and and ( (c1c1))−21.68821.688 ps ps2.2 .((a2a2)–()–(c2c2) )are are the the corresponding corresponding au auto-correlationto-correlation traces. traces. − − 5. Conclusions This article demonstrated the pulse state transformations between CS and NLP with SA parameters by experimental and simulation results. Changing the SA parameters such as modulation depth and reverse absorption coefficient generates soliton and transforms into noise-like pulses, which are further evidenced in a numerical simulation on the formation process. Such noise-like pulses generated in normal dispersion are compressible, whereas it becomes incompressible in anomalous dispersion regime.

Funding: This research was funded by National Natural Science Foundation of China (61975003); National Key Research and Development Program of China (2017YFB0405200); Natural Science Foundation of Beijing Municipality (4192015).

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Malinauskas, M.; Zukauskas, A.; Hasegawa, S.; Hayasaki, Y.; Mizeikis, V.; Buividas, R.; Juodkazis, S. Ultrafast laser processing of materials: From science to industry. Light Sci. Appl. 2016, 5, e16133. 2. Fermann, M.E.; Hartl, I. Ultrafast fibre lasers. Nat. Photonics 2013, 7, 868–874. 3. Liu, X.; Guo, Q.; Qiu, D.J. Emerging Low-Dimensional Materials for Nonlinear Optics and Ultrafast Photonics. Adv. Mater. 2017, 29, 1605886. 4. Geim, A.K.; Novoselov, K.S. The rise of graphene. Nat. Mater. 2007, 6, 183–191. 5. Kutz, J.N. Mode-Locked Soliton Lasers. SIAM Rev. 2006, 48, 629–678. 6. Grelu, P.; Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photonics 2012, 6, 84–92. Appl. Sci. 2020, 10, 5536 11 of 12

5. Conclusions This article demonstrated the pulse state transformations between CS and NLP with SA parameters by experimental and simulation results. Changing the SA parameters such as modulation depth and reverse absorption coeﬃcient generates soliton and transforms into noise-like pulses, which are further evidenced in a numerical simulation on the formation process. Such noise-like pulses generated in normal dispersion are compressible, whereas it becomes incompressible in anomalous dispersion regime.

Author Contributions: Conceptualization, Z.D. and Y.S.; Data curation, Z.D. and Y.C.; Formal analysis, Z.D.; Funding acquisition, Y.S.; Investigation, Z.D., J.T. and Y.C.; Methodology, Z.D., J.T., R.L. and W.Z.; Project administration, J.T. and Y.S.; Resources, W.Z.; Software, Z.D.; Supervision, Y.S.; Writing—review & editing, J.T., R.L. and W.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China (61975003); National Key Research and Development Program of China (2017YFB0405200); Natural Science Foundation of Beijing Municipality (4192015). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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