Femtosecond Pulse Trains through Dual-Pumping of Optical Fibers: Role of Third-Order Dispersion
FFemtosecond Pulse Trains through Dual-Pumping of Optical Fibers:Role of Third-Order Dispersion
Aku Antikainen [email protected] Institute of Optics, University of Rochester, Rochester, New York 14627, USA
Govind P. Agrawal
The Institute of Optics, University of Rochester, Rochester, New York 14627, USALaboratory for Laser Energetics, 250 East River Rd, Rochester, NY 14623, USA
Abstract:
Generation of high-repetition-rate, femtosecond, soliton pulse trains through dual-wavelength pumpingof a dispersion-decreasing fiber is studied numerically. The achievable shortest pulse width is found to be limited bythird-order dispersion that has a significant effect on the pulse-compression dynamics. The output wavelength is redshifted because of intrapulse Raman scattering and depends heavily on third-order dispersion, whose positive valueslead to the most red shifted solitons ( >
25% of the input pump center wavelength). The proposed scheme allowsthe generation of ultrashort pulse trains at tunable high repetition rates with a wide range of output wavelengths andpulse durations through dispersion engineering. The resulting frequency combs extend over a wide bandwidth witha tunable spacing between the comb lines.
1. Introduction
Soliton compression in active fibers [1, 2] anddispersion-decreasing fibers (DDFs) [3–8] is a well-understood effect [9, 10]. It is known that the character-istics of compressed solitons can be controlled by tailor-ing fiber dispersion. In the extreme, this process can leadto the generation of intense solitons with durations ofonly a few optical cycles [11]. Moreover, due to the ten-dency of pulses of suitable peak powers to re-adjust in-side the fiber to form solitons, soliton compression doesnot necessarily have to be seeded by solitonic pulses.Indeed, pulses of Gaussian or other shapes can be usedas input to the DDF. When the input is in the form of acontinuous wave (CW) or a long pulse, modulation in-stability (MI) can lead to the spontaneous generation offundamental solitons that can then be compressed.An effective way to enhance soliton formation froma CW input is to seed the MI process through ampli-tude modulation [12]. Induced MI in a fiber with con-stant (anomalous) dispersion leads to compression oflow-amplitude temporal modulations, eventually result-ing in a train of solitons. The formation of solitons, aswell as their further compression, manifests as spec-tral broadening in the frequency domain that is use-ful for supercontinuum generation [13–15]. However,pulse compression is not the only spectral broadening mechanism in the dual- or multi-pumping case [16].Amplitude modulation can lead to significantly broaderspectra in the normal dispersion regime as well dueto enhanced self-phase modulation and optical wave-breaking [17–19], and temporal reflections of dispersivewaves off nonlinear waves can extend the spectrum tothe blue side [20–22]. Modulation can be done eitherthrough direct amplitude-modulation of a single CWlaser [8] or through dual-wavelength pumping using twoCW lasers [23–26].The frequency of direct amplitude modulation is lim-ited by electronics since amplitude modulators cannotoperate effectively beyond 40 GHz. Dual-wavelengthpumping suffers from no such constraints and is onlylimited by the availability of CW lasers of suitablewavelengths. Moreover, modulation depths depend onlyon the relative powers of the two pumps. Therefore,ultrahigh repetition rate (800 GHz in this study) pulsetrains can be generated through dual-pumping of an op-tical fiber. Such pulse trains have a variety of applica-tions ranging from controlling the motion of molecules[27] to generating plasma waves [28] and terahertz radi-ation [29]. The objective of this paper is two-fold. First,we show that the lower limit for pulse duration in dual-pump soliton train generation is determined by higher-order dispersion; in the absence of third-order dispersion1 a r X i v : . [ phy s i c s . op ti c s ] A p r TOD) the pulses could be compressed down to a fewcycles in duration in a suitable DDF. Second, we showthat the sign and magnitude of the TOD plays a cru-cial role in determining the wavelength of the generatedtrain of solitons and that the output pulses can be redshifted by more than 25% from the initial pump centerwavelength. Sub-100 fs soliton trains can thus be gen-erated at a wide range of wavelengths by properly engi-neering the dispersion profile of the fiber. Our findingsshould help in designing fiber-based, high-repetition-rate, femtosecond-pulse sources and wide-band, opti-cal frequency combs with a tunable spacing between itscomb lines.
2. Pulse Propagation Model
To simulate propagation of electromagnetic wavesin optical fibers we use the generalized nonlinearSchr¨odinger equation (GNLSE) [13, 30]. In a referenceframe moving at the envelope group velocity, the equa-tion for the electric-field envelope A ( z , t ) can be writtenas ∂ A ∂ z + α A − ∑ n (cid:62) i n + n ! β n ∂ n A ∂ T n = i γ (cid:18) + i τ shock ∂∂ T (cid:19) · (cid:18) A ( z , T ) (cid:90) ∞ − ∞ R ( T (cid:48) ) (cid:12)(cid:12) A ( z , T − T (cid:48) ) (cid:12)(cid:12) dT (cid:48) (cid:19) , (1)where T is the retarded time given by T = t − z / v g and v g is the group velocity at the input central wave-length. The left side of Eq. (1) includes linear effectswith α corresponding to losses and the β ’s being thedifferent-order dispersion coefficients of the fiber at thecentral input frequency. The right side models the non-linearities through the response function R ( T ) that in-cludes the Kerr contribution that is assumed instanta-neous [13] and the delayed Raman contribution that ismodeled through the experimental Raman-gain profileof silica fibers [31]. Note that, contrary to a commonmisconception, the GNLSE for fibers does not assume aslowly-varying envelope for the electric field in the timedomain and is in fact valid down to the few- and evensingle-cycle regime, as long as the wavelengths in ques-tion are far from material resonances so that the slowly-evolving-wave approximation | ∂ z E | (cid:28) β | E | is satisfied,as shown by Brabec and Krausz [32]. The validity of themodel down to the few-cycle regime requires the inclu-sion of the shock term characterized by the time scale τ shock [30].We solve Eq. (1) numerically with an input corre-sponding to launching two CW pumps simultaneously with a frequency difference ∆ ω : A ( , T ) = √ P e i ( ω + ∆ ω / ) t + √ P e i ( ω − ∆ ω / ) t (2)where ω is the central frequency and P and P arethe input powers of the two pumps. In our simulations,the group-velocity dispersion (GVD) parameter β in-creases linearly from its initial negative (anomalous)value of −
10 ps / km over the entire fiber length thatvaries in the range 100 - 200 m. The final values of β range from −
10 to +
10 ps / km, the former correspond-ing to a constant-dispersion fiber. The TOD parameter β is kept constant for each fiber with values rangingfrom − . . / km. It should be noted that usu-ally DDFs are manufactured by tapering the fibers, inwhich case the taper can induce significant losses andthus impose limitations on pulse compression [7]. How-ever, dispersion can also be modified through doping,allowing for the losses to be curbed. The nonlinear pa-rameter in our simulations is γ = . / ( Wm ) , andthe pump powers are taken to be equal with P = P = ω corresponds to a wave-length of 1060 nm. The shock time τ shock is taken to be1 / ω . Throughout this paper the frequency separation is ∆ ω / ( π ) =
800 GHz (3 nm) but we have also verifiedthat frequency separations of 600 GHz and 1000 GHzyielded qualitatively similar results.
3. Theoretical Limits on Pulse Width
Fundamental solitons are solutions of the GNLSE inthe absence of third- and higher-order dispersion, op-tical shock effects, and delayed nonlinearities. Whenthese effects are present, they manifest as perturbationsto an ideal soliton. Third-order dispersion (TOD) gov-erned by β introduces spectral and temporal asymmetryand forces the soliton to shed radiation in the form of adispersive wave. Shock effects produce self-steepening,again making the soliton asymmetric. Delayed nonlin-earities lead to the well-known phenomenon of solitonself-frequency shift (SSFS) through intrapulse Ramanscattering, causing the soliton to red shift in the spectraldomain. Nevertheless, the robust solitonic nature of thepulse remains. Solitons are robust to the extent that anypulse of suitable shape and energy in the anomalous dis-persion regime of a nonlinear fiber will reshape itself tobecome one [13]. In the context of a beating dual-pumpsignal the sinusoidal oscillations at the beat frequencybecome compressed and evolve to form solitons if theirduration and energy roughly matches those of a funda-mental soliton. If the fiber is long enough, the beatingintensity pattern eventually evolves to become a train of2quidistant solitons. By changing the dispersion alongthe length of a DDF, the solitons can be compressed fur-ther in the temporal domain [3].The dispersion parameters β n in Eq. (1) can be eas-ily tailored through proper design of the refractive in-dex profile, which in the case of photonic crystal fibersmeans appropriately choosing the size and spacing ofthe air holes surrounding the core. The only limita-tions regarding the structure of silica-based photoniccrystal fibers are associated with manufacturing preci-sion. In general, different photonic crystal fiber struc-tures would also lead to different nonlinear coefficientsfor the fibers. However, since it is the relative strength ofdispersion and nonlinearity that determines the propaga-tion of light, we assume here that the nonlinear parame-ter is constant while β changes linearly along the fiber.We also note that ultra-flat highly anomalous dispersionprofiles can be achieved over a wide wavelength rangewith novel designs [33]. The TOD and other higher-order dispersion terms play a relatively minor role forsuch fibers.To understand the dynamics of a dual-wavelength sig-nal inside a DDF, we first neglect the TOD and otherhigher-order dispersion terms so that the effects of a lon-gitudinally varying β can be identified clearly. Figure 1shows the evolution of a dual-pump signal when β in-creases linearly from −
10 to 0 ps / km over 100 m. Thepower of both pumps is 1 W. The two traces on top showchanges in the pulse width and peak powers over the100 m length of the fiber. The initial sinusoidal patterngradually reshapes into a train of solitons whose widthdecreases and peak power increases continuously untilthe numerical model itself breaks down. The spectrumof the resulting pulse train is in the form of a frequencycomb whose bandwidth is inversely related to the widthof solitons and exceeds 100 THz.The compression dynamics in Figure 1 have interest-ing features. The initial sinusoidal pattern with a periodof 1.25 ps evolves into a pulse train within the first 10 msuch that individual pulses are about 200 fs wide (fullwidth at half maximum or FWHM). These soliton-likepulses then broaden with further propagation before be-ing compressed a second time. This process repeats afew times but the pulse duration keeps a downward trendwhile exhibiting transient oscillations. The evolution ofthe solitons is affected by two mechanisms. First, vary-ing fiber dispersion forces them to compress. Second, atthe same time, their speed is reduced as their spectrumred shifts because of SSFS (leading to bending of thetrajectories in Figure 1). The individual solitons growin intensity because of the increasing β , but also be- Fig. 1. The temporal (middle) and spectral (bottom) evo-lution of a dual-pump signal over 100 meters of a DDFwith β increasing from −
10 ps / km to 0 ps / km. Thegray intensity scales are logarithmic. The top two tracesshow the duration (thick blue) and peak power (thin red)of the forming pulses as a function of distance. The ver-tical black dashed lines indicate the distance at whichthe soliton width has been reduced to three optical cy-cles.cause they feed off the darker regions (energy in the low-intensity parts) when they shift in time and overlap tem-porally with them. This mode of energy transfer to thesolitons is evident in Fig. 1, where the regions betweenthe neighboring solitons become darker as the solitonsslow down and pass through these regions. This energytransfer perturbs the solitons, causing their widths andpeak powers to oscillate around their respective trends(decreasing duration, increasing peak power).The GNLSE model given in Eq. (1) accurately de-scribes pulse propagation down to the single-cycleregime [30, 32], and in this study the three-cycle point isused as the cutoff for the validity of the GNLSE model.The distance at which the solitons in Fig. 1 have com-pressed to three optical cycles in duration (about 10 fs) isapproximately 98 meters, and this has been indicated bythe vertical dashed lines in Fig. 1. The important take-3way from Fig. 1 is that the initial beating intensity pat-tern with a period of 1.25 ps (corresponding to 800 GHz)could ideally be reshaped into a train of solitons that areonly three optical cycles long. The input FWHM of thecosine-shaped pulses is 625 fs, implying that the com-pression factor is larger than 50.The simulation shown in Fig. 1 includes all the rele-vant effects that would be present in reality, with the ex-ception of higher-order dispersion and losses. The fiberwas assumed to be lossless and to have perfectly flat dis-persion (constant β ) over all wavelengths at any givenpoint of the fiber. Therefore, Fig. 1 represents the bestcase scenario in terms of how short the solitons can be-come: Under ideal conditions pulse durations of threeoptical cycles or less could be achieved. Several dif-ferent effects might prevent such drastic compressionin practice, but the extent of compression is not lim-ited by GVD, intrapulse Raman scattering, or opticalshock effects. Losses would cause the peak power P ofthe forming solitons to be smaller, which in turn wouldlead to larger soliton durations T such that the solitoncondition of γ P T / β ( z ) = β =
0, and the soli-ton condition can only be satisfied for infinitely nar-row solitons no matter what the peak power might be.Compensation for losses through decreasing dispersion(increasing β ) to keep the soliton duration unchangedupon propagation in lossy fibers has been demonstratedin the past [34]. Decreasing dispersion even faster thanin Fig. 1 would be required to compensate for any pos-sible fiber losses. However, TOD could be expected tochange the compression dynamics more drastically thanlosses because it affects solitons in at least three differ-ent ways: It leads to dispersive-wave emission, it asym-metrically distorts the shape of a soliton, and it makes β frequency-dependent.
4. Effects of Third Order Dispersion
The first thing to note is that the sign of TOD plays animportant role in the evolution of a short solitons un-dergoing intrapulse Raman scattering. The SSFS causesthe soliton spectrum to red shift, and it is the sign of β that then determines whether the soliton will expe-rience a larger or smaller β as a consequence. Sincesoliton compression is based on increasing β from aninitially negative value through dispersion engineering,any TOD-induced change to β will affect the compres-sion of solitons. The presence of TOD also introducesa spectral region or normal dispersion in which solitonscannot exist but also guarantees the existence of a spec-tral region of anomalous dispersion even when β > β and β deter-mine whether the normal dispersion regime is on the redor the blue side of the soliton. The frequency at whichGVP changes sign is given by ω ZDW = ω − β / β where β and β are evaluated at the central frequency ω . The wavelength corresponding to ω ZDW is the zero-dispersion wavelength (ZDW). When β is a linear func-tion of distance z we have β ( ω ) = β in2 + ( β out2 − β in2 ) zL (3)where L is the length of the fiber and β in2 and β out2 arethe output values of β at ω . Consequently the ZDWbecomes a function of z through ω ZDW = ω − β in2 β − ( β out2 − β in2 ) z β L . (4)To illustrate the effects of TOD in a DDF, Fig. 2shows the evolution in a fiber where β changes from −
10 ps / km to 5 ps / km over 150 meters and where β = − .
03 ps / km. Note that the rate of change of β with z is the same as for the fiber in Fig. 1 and the ZWDcoincides with the pump center wavelength at exactly100 meters just like in Fig. 1.The evolution of the dual-pump shown in Fig. 2 dif-fers from that of Fig. 1. The most noticeable differ-ence between the two cases is that the pulses do notbecome infinitely narrow when β (cid:54) = β (cid:54) = β was negative ( β <
0) and hence the normal disper-4ig. 2. The evolution of a 800 GHz dual-pump signal ina fiber in which β grows from −
10 ps / km to 5 ps / kmalong its 150 m length. Third-order dispersion is β = − .
03 ps / km.sion regime was on the red side of the pump. Solitonshave a tendency to try to stay away from the ZDW andremain in the anomalous regime, which can be seen inthe spectrum of Fig. 2 where the spectral trajectory ofthe soliton bends slightly downwards between 90 m and110 m and the solitons blue shift. The blue shift is al-ways accompanied by significant transfer of energy tothe red side of the ZDW to conserve total energy. Nor-mally solitons, especially short ones, have a tendencyto red shift upon propagation because of intrapulse Ra-man scattering. This raises the question whether hav-ing the ZDW approach the soliton spectrum from theblue side instead would help the solitons remain in theanomalous regime for longer distances. Figure 3 showsthe evolution of a 800 GHz dual-pump in a fiber with β = .
03 ps / km. Other than the fiber length and theTOD, the fiber is similar to the ones in Figs. 1 and 2 andagain the ZDW is at the pump center at 100 m. Note thatthe temporal trace in Fig. 3 is now in the reference frameof the solitons instead of moving at the group velocityat the pump frequency.The evolution of the soliton power and duration is Fig. 3. The evolution of a 800 GHz dual-pump sig-nal in a fiber in which β grows from −
10 ps / km to10 ps / km along its 200 m length. Third-order disper-sion is β = .
03 ps / km. Unlike in Figs. 1 and 2, thetemporal frame of reference is now with respect to thesolitons, as their trajectories would look heavily curvedin the pump frame of reference.similar to that of Fig. 2 but the solitons last longer andthe spectral evolution looks very different. The ZDWis now on the blue side of the solitons and the ZDWapproaching the soliton spectrum greatly enhances thenatural SSFS pushing the soliton spectrum all the wayto 1 . µ m from the initial 1 . µ m. Still, the mov-ing ZDW eventually overtakes the soliton spectrum andin the end the pulses end up in the normal dispersionregime and disperse. The minimum soliton duration is125 fs around 185 m.To understand quantitatively the impact of β , we car-ried out a large number of numerical simulations for dif-ferent DDF designs. Figure 4 shows the color-coded du-ration of solitons (range 0–250 fs) for β values varyingfrom − . . / km along the x axis and differentvalues of β ( L ) at the end of a 200-m-long fiber with β ( ) = −
10 ps / km. In each case, β is kept constantalong the fiber. The four plots shows the soliton widthsat distances of (a) 80, (b) 120, (c) 160, and (d) 200 m.5ig. 4. The mean duration (FWHM, color coded) of theforming solitons after a) 80 m, b) 120 m, c) 160 m, andd) 200 m of propagation as a function of β and the finalvalue of β . The initial value of β at the input end of200-meter-long fiber is −
10 ps / km. The striped areasin the upper left corners are regions where the pulseshave lost their solitonic nature by virtue of having trans-ferred energy to the normal dispersion regime.If the solitons forming from the beating input signalare able to keep up with the gradually changing GVDparameter β , then larger final values of β leads toshorter solitons. The general trend in Fig. 4 is that in-creasing the final value of β makes the output pulsesshorter, which means that solitons are mostly able tokeep up with the longitudinally changing GVD, evenwhen GVD becomes normal near the fiber end. This isalso corroborated by Figs. 1 where pulse duration hasa downward linear trend approaching zero with decay-ing transient oscillations. The transient oscillations dieout by the end of a 200-meter-long fiber when the finalvalue of β is larger than − / km, as seen in Fig. 4.The temporal compression continues even after the os-cillations disappear.The effects of TOD are clearly visible in Fig. 4.Larger values of | β | hinder pulse compression, whereassmaller values lead to shorter pulses at shorter distances.The explanation for this lies in how β effects the β that the soliton experiences and in the Raman effectthat causes the soliton spectrum to red shift throughSSFS with propagation. The TOD parameter is given by β = d β ( ω ) / d ω evaluated at the central frequency ω .Negative values of β thus mean that β decreases withoptical frequency and hence increases with wavelength.SSFS then causes the solitons to experience larger GVDcompared to the initial pump center frequency. Nega- -1 0 1 Time (ps) I n t en s i t y ( W ) -100 0 100 Time (fs) AB Fig. 5. Comparison of pulse trains generated with thesame dual-pump input in two different fibers. FiberA is 100 m long and its GVD increases linearlyfrom −
10 ps / km to 0 over this length with β = .
05 ps / km. Fiber B is 97 m long but its GVD increasesfrom −
10 to − .
725 ps / km with β = − .
05 ps / km.The total input power is 2 W and initial pump separationis 800 GHz. The two traces on the right show the pulsearound T = β together with SSFS imply that β at thesolitons’ central frequency increases even faster than β at the pump center frequency, thus causing the solitonsto compress rapidly. The opposite occurs for positivevalues of β . As seen in Fig. 1, solitons could be com-pressed down the three optical cycles in the absence ofTOD, but in practice pulse compression is limited by it.We note that fibers with β = <
100 fs can be achieved with many differentparameter combinations. Even a 100-meter fiber can belong enough to produce such an ultrashort pulse trainif β of the DDF increases rapidly enough with dis-tance [see Fig. 4(b)]. Both negative and positive valuesof β work, and two different sets of fiber parameterscan lead to very similar-looking pulse trains. Figure 5shows portions of two pulse trains generated using twodifferent fibers with the same input. Both fibers havethe same GVD at the input end but their lengths andfinal values of β are different. Their TOD parametersare equal in magnitude but opposite in sign. The soli-tons generated in each fiber are nearly identical: theirenergies and pulse durations are within 2% of one an-other. The only notable difference is that the pulses inthe fiber with β > Wavelength ( m) -60-40-200 N o r m a li z ed s pe c t r u m ( d B ) A Wavelength ( m) B Fig. 6. Spectra of the two pulse trains shown in Fig. 5 atthe output of fibers A and B.near the trailing end. The differences between the pulsetrains are subtle in the time domain but become quiteevident in the spectral domain to which we turn in thenext section.
5. Output Frequency Comb and its Central Wave-length
The output spectrum of any periodic ultrashort pulsetrain generated through dual-pumping is in the form ofa frequency comb whose comb lines are separated bythe initial spacing between the frequencies of the twoinput pumps. Figure 6 shows the spectra correspondingto the two identical-looking pulse trains shown in Fig.5. The spectra resemble mirror images of one anotherbecause of the opposite signs of the TOD parameter β .The soliton part of the spectrum (dominant peak) of fiberA is centered at 1086.1 nm, while that of fiber B is at1067.5 nm, a difference of 18 . .
81 THz). As areminder, the input center wavelength of the two pumpsis at 1060 nm.The central frequency at each point in the fiber is de-termined by several processes. The first one is SSFSwhich causes the solitons to red shift. The second oneis the tendency of solitons to stay away from the ZDWin the spectral domain [35], and a moving ZDW canmanifest as an effective push for the soliton spectrum.Depending on whether this push comes from the redside or the blue side, it can respectively hinder or en-hance the red shift (See Figs. 2 and 3, respectively). For β > ω ZDW > ω , and ω ZDW approaches ω from the blue side, enhancing the red shift and pushingthe solitons further into the red. When β > ω ZDW approaches ω from the red side and SSFS is thus hin-dered. This is the reson the spectrum out of fiber A inFig. 6 is more red shifted than that of fiber B.If β out2 > ω ZDW always surpasses ω no matter howfast or slow its rate of change. The rate of change is pro- portional to 1 / β as seen in Eq. (4), which means thatwhen β is close to zero, ω ZDW changes rapidly with dis-tance z . Based on this argument, it seems likely that soli-tons could be pushed towards even longer wavelengthsby making β smaller while keeping it positive. Figure 7shows the central wavelength of the pulse trains gener-ated through dual-pumping at distances of 80 m, 120 m,160 m, and 200 m under conditions identical to thoseof Fig. 4. The initially forming solitons are wide at firstand, as a result, red shifts of < β out2 for whichadiabatic soliton compression kicks in and makes thesolitons shorter thus enhancing their SSFS. The largestred shifts occur in the regime where β out2 > β is small but positive. The soliton central frequency canbe red shifted by more than 25% to 1 .
35 nm before ω ZDW moves beyond the soliton central frequency anddisperses the solitons.Fig. 7. The central wavelengths λ soliton of the formingsolitons for the parameters used in Fig. 4 after a) 80 m,b) 120 m, c) 160 m, and d) 200 m of propagation. Thestriped regions indicate that the pulses have lost theirsolitonic nature and have dispersed. The upper colorbaris for the top row and the lower one for the bottom row;note the different scales.
6. Conclusions
It was numerically demonstrated that the technique ofdual-wavelength pumping can be used to generate soli-ton pulse trains at ultrahigh-repetition rates (up to 1 THzor more) and that the solitons could be compressed tem-porally inside a dispersion-decreasing fiber down to thefew-cycle regime (pulse widths as short as 10 fs at wave-lengths near 1 µ m). The repetition rate used in this study7as 800 GHz, but since it is set by the frequency sepa-ration of two CW pumps, it can be tuned over a widerange by choosing the input pump wavelengths suit-ably. It was further pointed out that the soliton com-pression is limited by higher-order dispersion with smallvalues of the GVD slope β = d β / d ω leading to short-est pulses. It was also shown that third-order disper-sion is crucial in determining the output wavelength ofthe pulses. We found that small positive values of theGVD slope lead to the largest red shifts and the longestoutput wavelengths. Sub-100 fs solitonic pulses with awavelength anywhere between 1060 nm and 1350 nmcould be achieved in our numerical simulations, mak-ing dual-wavelength pumped optical fibers a versatileplatform for generating femtosecond pulses at high-repetition rates that have a variety of applications rang-ing from biomedical imaging to the manipulation of mo-tion of individual molecules.The spectral features of the generated pulse trainsare also remarkable. Our results clearly show that thedual-pumping scheme is capable of generating fre-quency combs that extend over 50 THz and whose cen-ter frequency is tunable over 60 THz in the vicinity of1150 nm. Moreover, the comb spacing in itself can betuned over a wide range ( ∼ ∼ References
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