Elongation of energy exchange between femtosecond laser pulses via plasma formation in air
Zuoye Liu, Yu Cao, Yanchao Shi, Mingze Sun, Pengji Ding, Zeqin Guo, Bitao Hu
aa r X i v : . [ phy s i c s . op ti c s ] M a y Elongation of energy exchange between femtosecond laser pulses via plasmaformation in air
Zuoye Liu, Yu Cao, Yanchao Shi, Mingze Sun, Pengji Ding, Zeqin Guo, and Bitao Hu a) School of Nuclear Science and Technology, Lanzhou University, 730000, China (Dated: 30 January 2018)
We experimentally demonstrate energy exchange between a delay-tuned femtosecond beam and two delay-fixed ones as they spatiotemporally overlapped and experienced filamentation in air. The energy exchangeprocess in the relative time delay is dramatically elongated up to 40 ps in the presence of plasma grating,indicating that filamentary beams coupling may be an effective method for filament control.PACS numbers: 52.38.Hb, 42.65.Jx, 42.60.HwFemtosecond laser pulse filamentation in air is aninteresting phenomenon , and has intensive applica-tions including few-cycle pulse generation, THz radia-tion, remote-sensing of atmospheric pollution, lightingand discharge triggering. Recently, much attention hasbeen given to the interaction among few noncollinearcrossing filaments. Because of laser field interference, anone- or two- dimensional plasma grating can be formedin the intersection region when the noncollinear filamen-tary pulses overlap in time and space. Energy exchangebetween two filamentary pulses has been demonstratedin several experiments including filamentation in air andliquid methanol , and the formation of plasma grat-ing plays an important role in this coupling process. Forinstance, the traveling plasma grating formed at the in-tersection of two filamentary pulses with slightly differentcentral frequency is responsible for an efficient energy ex-change between the filaments . It has been found thatthe direction of energy transfer depends on the relativetime delay between filamentary pulses, the initial chirp,the laser intensities, the location, the intersecting angleand the relative polarization. However, the underlyingphysical mechanisms of energy exchange in literaturesare different and puzzling, including the impulsive Ra-man nonlinear response of the molecules , the plasma-mediated forward stimulated Raman scattering , thetraveling plasma grating , the configuration akin pro-duced by the crossed filaments to coupled waveguides and the classical two beam coupling model . Bernstein et al obtained a conclusion that no significant affect ofthe filament formation on the energy exchange , but laterY. Liu et al found that the direction of energy exchangereverses due to filament formation when the laser power P = 3 . P cr . Besides, all these works were done by usingtwo filamentatary pulses interacting configuration.In this letter, we experimentally studied the filamen-tary pulse coupling between a delay-tuned beam andtwo delay-fixed beams, keeping in mind underlying phys-ical mechanisms of energy exchange. Our results areconsistent with those of previous works demonstrating a) Author to whom correspondence should be addressed. Electronicmail: [email protected] energy exchange between two filaments in air and liq-uid methanol, when the relative time delay between thedelay-tuned beam and the two delay-fixed ones is variedamong approximately 1 ps. Unexpectedly, we found apreviously undocumented energy exchange with approx-imate 40 ps time-scale in the relative time delay.The interaction of three filaments is schematicallyshown in Fig .1. The laser system we used is a Ti:sapphireamplifier system which is capable of producing 33 fspulses centered at 810 nm, at a repetition rate of 1 kHz.The laser pulses are chirp-free, which can be confirmedby the FROG measurements. The laser beam was splitinto three arms by two successive beam-splitters. Onearm with pulse energy of 0.6 mJ, which was defined asBeam-1, and the other two arms defined as Beam-2 andBeam-3 with pulse energies up to 0.8 mJ and 0.7 mJrespectively, were focused by lens. The minimal transla-tion setup corresponds to a temporal resolution of 16.7fs. The energy of each beam was measured by an energymonitor. The similar details of experimental setup canrefer to our previous work .In the experiment a plasma grating was formed whenthe two delay-fixed pulses without central frequency dif-ference were spatiotemporal overlapped. Because of theplasma density modulation in the plasma grating, thenonlinear refractive index originated from the formationof plasma ∆ n plasma ∼ − ρ ( −→ r , t ) /ρ cr changes periodi-cally, where ρ cr = 1 . × cm − is the critical plasmadensity for 810 nm laser pulse in air . The period ofthis one-dimensional plasma grating can be expressedby Λ = λ/ [2 sin ( θ/ λ and θ = 2 . ◦ are thewavelength of the incident delay-fixed pulses and theircrossing angle respectively. For 2 . ◦ crossing geometryof two delay-fixed beams, the one-dimensional gratingperiod could be estimated as Λ D ≈ . µ m which isclose to our experiment results in Fig. 1(b), and its re-fractive index planes would be approximately orientedalong the bisector of the angle formed by the two delay-fixed beams. With the presence of the delay-tuned beamwith a crossing angle 3 . ◦ with respect to Beam-2, a vol-ume plasma grating was formed when the three beamstempoprally overlapped, as shown in Fig. 1(c). As aresult, its orientation was slightly deviated from that ofone-dimensional plasma grating and tended to the prop- FIG. 1. (Color online) (a) Schematic illustration of three filaments interaction. (b) CCD image of one-dimensional plasmagrating generated by two delay-fixed filaments when the spatiotemporal overlap is fulfilled. (c) CCD image of the volumeplasma grating generated by the delay-tuned filament and two delay-fixed filaments. agation direction of the delay-tuned beam. Since theinvolved laser pulses possess the equal central frequence,their interference field will lead to the formation of therelatively stationary plasma grating rather than a trav-eling one. -0.6 0.0 20 40 60 80 100 120 1400.30.40.50.6
Energy of delay-tuned pulse Energy of delay-fixed pulses
Delay (ps) E n er gy o f d e l ay -t un e d pu l s e ( m J ) E n er gy o f d e l ay -f i x e d pu l s e s ( m J ) FIG. 2. (Color online) The energies of delay-tuned pulse(circle curve) and two delay-fixed ones (square curve) as afunction of the relative time delay. The initial energy of thedelay-tuned pulse is 0.53 mJ, while the initial energy of thetwo delay-fixed pulses is 1.27 mJ.
Figure 2 shows the energies of the delay-tuned pulseand two delay-fixed ones as a function of their rel-ative time delay. The positive delay corresponds tothe translation-stage delay-tuned beam delayed withrespect to the two delay-fixed beams. There aretwo distinct regimes, which is consistent with reportedobservations . The two delay-fixed beams transferenergy to the delay-tuned one during the negative de-lay, while the delay-tuned pulse transfers energy to thetwo delay-fixed ones as the relative time delay is posi-tive. It is similar to the observation by Y. Liu et al andclearly different from that by Bernstein et al in whichthe trailing pulse obtains energy from the leading pulse.Note that the energy reduction of the delay-tuned pulse islarger than that transfered to the two delay-fixed pulses during the pulses overlap time because of the nonlinearmultiphoton absorption. The energy exchange efficiencydecreases gradually with the increasing of the relativetime delay. With the positive time delay larger than 0.3ps, the energy loss can be neglected because there is nostrong coupling between the delay-tuned pulse and thetwo delay-fixed ones due to the decay of temporal over-lap. Until the time delay of ∼
40 ps, the energies ofpulses almost recover to their initial levels. That is tosay, the energy exchange in the domain of relative timedelay has been elongated to the order of magnitude of40 ps. This result was unexpected because no reportedwork has mentioned the process of energy exchange asa function of time delay beyond 3.0 ps. For instance,the duration of energy exchange in the relative time de-lay could be estimated as 400 fs in Ref. [9], and furtherY. Liu et al observed this time region of energy exchangewas about 2.0 ps when two femtosecond laser pulses wereinvolved .Similar to previous work , the ratio of energy exchangebetween the delay-tuned pulse and the two delay-fixedones can be defined as S = ( E tuned − E fixed ) − ( E tuned − E fixed ) E tuned + E fixed , (1)where E tuned and E fixed represent the delay-tuned en-ergy and energy of two delay-fixed pulses, E tuned and E fixed are the initial delay-tuned energy and total en-ergy of two delay-fixed pulses, respectively. As shown inFig. 3, the maximum energy exchange ratio can reach 5%when the time delay is negative, and 23% when the timedelay is positive. When the delay-tuned energy grad-ually approach that of the two delay-fixed pulses, thetwo delay-fixed pulses transfer more energy to the delay-tuned one before the zero time delay, and gain less energyfrom the delay-tuned one after the zero time delay. Fig-ure 4 shows the energy exchange ratio S as a functionof the relative time delay when the delay-tuned beamorthogonally crossed the two delay-fixed beams. The di-rection of energy exchange reversed compared to that ofsmall-angle crossing configuration, and there was no long-time dependence on the relative time delay. In this case, Delay (ps)0 20 40 60 80 100 120 140 E n er gy E x c h a n g e R a t i o S -0.25-0.2-0.15-0.1-0.0500.050.1 (a) Delay (ps)-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 E n er gy E x c h a n g e R a t i o S -0.25-0.2-0.15-0.1-0.0500.050.1 (b) FIG. 3. (Color online) (a) The energy exchange ratio S asa function of the relative time delay. The red solid (greendashed, blue dash-doted, violet dotted) line represents thebest fit by using the formula S ( τ ) ( S ( τ ), S ( τ ), S ( τ ) + S ( τ )). (b) The detailed energy exchange in the relative timedelay from − . ps . the interaction region was dramatically reduced, leadingto a reduction of plasma density. Thus, the flow of energytransfer changed and the elongation of energy exchangein the delay domain vanished. Delay (ps)-1 -0.5 0 0.5 1 E n er gy E x c h a n g e R a t i o S -0.08-0.06-0.04-0.0200.020.040.06 FIG. 4. (Color online) The energy exchange ratio S as afunction of the relative time delay in the orthogonal crossingconfiguration. The red solid (green dashed, blue dash-doted)line represents the best fit by using the formula S ( τ ) ( S ( τ ), S ( τ ) + S ( τ )). Because the electron density within plasma grating hasbeen dramatically enhanced by using three filaments in-teraction, which can be confirmed by the significant en- hancement of fluorescence intensity at the intersectingregion, the plasma grating has a much larger decay timethan the time scale of pulse overlap. X. Yang et al hasdemonstrated that the plasma grating formed by two 45fs filaments could last for about 30 ps . In addition,L.P. Shi et al has demonstrated that the electron densityin the plasma grating formed by two UV filaments decaysexponentially, and its lifetime is on the order of 100 ps .The plasma grating gradually decays and consequentlydisappears due to two distinct processes , e.g. am-bipolar diffusion and electron recombination. The slowerthe plasma decay process, the larger the imbanlance be-tween the positive and negative portions of the energyexchange. Therefore, we can expect the long-time de-pendence of energy exchange on the relative time delay,corresponding to the lifetime of plasma grating formedby the noncollinear filaments interaction.We use the treatment developed by N. Tang et al tofit the energy exchange ratio, shown as solid line in Fig.3(a) and 3(b). Taking the temporal response function R ( t ) of air as exp ( − t/τ n ), where τ n is the exponentialdecay constant , the energy exchange ratio can be esti-mated as: S ( τ ) = S ( τ ) + S ( τ ) + S ( τ ) + S ( τ ) , (2) S ( τ ) = κ √ πτ p I B xxxx {− Im [ Z + ∞−∞ dtu ∗ ( t − τ ) u ( t ) × Z t −∞ dt ′ R ( t − t ′ ) u ( t ′ − τ ) u ∗ ( t ′ )] } , (3) S ( τ ) = κ √ πτ p I B ′ xxxx {− Z + ∞−∞ dt | u ( t − τ ) | × Z t −∞ dt ′ R ( t − t ′ ) | u ( t ′ ) | } , (4) S ( τ ) = κ √ πτ p I B ′ xxxx {− Re [ Z + ∞−∞ dtu ∗ ( t − τ ) u ∗ ( t ) × Z t −∞ dt ′ R ( t − t ′ ) u ( t ′ − τ ) u ( t ′ )] } , (5) S ( τ ) = κ √ πτ p I B ′ xxxx {− Re [ Z + ∞−∞ dtu ∗ ( t − τ ) u ( t ) × Z t −∞ dt ′ R ( t − t ′ ) u ( t ′ − τ ) u ∗ ( t ′ )] } , (6)where τ p the pulse duration, u ( t ) the time-dependentelectric field, and I = R + ∞−∞ | u ( t ) | dt . S ( τ ) representsthe contribution from beam-coupling, and the terms of S ( τ ), S ( τ ) and S ( τ ) represent the nonlinear absorp-tive contributions. κ = 24 √ πkL eff E p /n w cτ p , where k the wave number for the incident laser wavelength invacuum, L eff the effective propagation length (3 cm), E p the pulse energy, and c the light speed. The Gaus-sian beam radius w at the intersection region was es-timated as 100 µm . B xxxx and B ′ xxxx are the delayedcontributions to the nonlinear refractive effects and thenonlinear absorptive effects respectively. Though theseexpressions are derived from the time-domain theory forpump-probe experiment, it is still valid for the situationinvolving three noncollinear filamentary beams, becausethe Beam-2 and Beam-3 were kept spatiotemporally over-lapping.The red solid lines in Fig. 3(a) and 3(b) were thebest fit curves by using Eq. (2-6) with the relevantpulse duration and energies, and with a fit of B xxxx =2 . × − cm /erg · s , B ′ xxxx = 2 . × − cm /erg · s and τ n = 14 . ps . The third-order nonlinear suscep-tibility χ (3) can be estimated by χ (3) ≈ B xxxx τ n =1 . × − cm /erg , and then the delayed nonlin-ear refractive index coefficient n ,d = 12 π χ (3) /n c =3 . × − cm /W , which is close to the reported valueof ∼ . × − cm /W in literature . The fit inFig. 4 obtained n ,d = 2 . × − cm /W , which wasalso in the same order of magnitude. Despite the com-plicated situation involving three filamentary pulses, themodel mentioned above predict well the temporal evolu-tion of energy exchange for the intersecting filamentarypulses. A detailed explanation will require full knowledgeof the temporal and spatial evolution of the filamentarypulses at the pulse overlap region.In summary, we have demonstrated the elongation ofenergy exchange process in the relative time delay be-tween two delay-fixed pulses and a delay-tuned one. Cor-responding to the gradual decay of plasma density, en-ergy exchange exhibits a long-time dependence on therelative time delay. In the orthogonally crossing con-figuration, the flow direction of energy transfer reversedbecause of the reduction of plasma density. By usinga time-domain model including the nonlinear absorptiveeffects, the best fit was obtained and the delayed non-linear refractive index coefficient was extracted. Theseresults indicate that efficient energy exchange betweenfemtosecond pulses could be ensured and controlled byplasma formation assisted by filaments interaction. 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