Qingwei Zeng , Lei Liu * , Kejin Zhang, Shuai Hu, Taichang Gao, Chensi Weng and Ming Chen
College of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101, China; [email protected] (Q.Z.); [email protected] (K.Z.); [email protected] (S.H.); [email protected] (T.G.); [email protected] (C.W.); [email protected] (M.C.) * Correspondence: [email protected]; Tel.: +86-25-8083-0652
Received: 14 August 2019; Accepted: 25 September 2019; Published: 9 October 2019
Abstract: The eﬀects of water vapor ionization on the nonlinear propagation of femtosecond laser pulses with a 248 nm wavelength are numerically investigated in this paper. It is found that ionization of H2O molecules plays a signiﬁcant role in air ionization, which seriously aﬀects the dynamic and energy deposition of ﬁlamentation. The propagation of femtosecond pulses in air with diﬀerent humidity levels are compared. The total number of electrons and total deposited pulse energy increase with the humidity increases. However, they tend to be saturated in high humidity conditions. Results presented here are conducive to characterizing the long-range propagation of ﬁlaments under atmospheric conditions.
Keywords: ultrafast nonlinear optics; atmospheric propagation; self-focusing; multiphoton absorption; water vapor
1. Introduction In the past decade, investigation of the formation of light ﬁlaments in air has become a topic of active experimental and theoretical research due to their various potential atmospheric applications, such as remote sensing of chemical/biological agents  and meteorological elements , high-voltage discharge guiding , air waveguiding , and weather-related applications [5–7]. The widely acceptable physical mechanism that sustains the formation of light ﬁlaments is the result of a dynamic balance between the Kerr self-focusing eﬀect and the plasma defocusing eﬀect . Firstly, the optical Kerr eﬀect focuses the pulse and then the beam intensity experienced obviously increases. When the beam intensity becomes suﬃciently large, it ionizes the medium and creates an electron plasma . Then, the electron plasma defocuses the trail of the pulse and strongly shortens its leading edge around and beyond the nonlinear focus . Many ﬁlamentation-based applications rely on long distance propagation in atmosphere . It is well known that real atmospheric air is composed of various types of gas molecules, which vary in content. In previous theoretical simulation studies, the ionization of N2 and O2 was usually only considered since they are the main components of air gas [12–14]. The ionization of water vapor molecules is often neglected because their concentration is usually too low. However, inﬂuences of water vapor on multiple physical eﬀects during the ﬁlamentation process are expected to be very diﬀerent when a laser pulse wavelength increases . Up to now, only a few studies have considered the inﬂuence of relative humidity on the propagation of femtosecond pulses. Manuilovich et al. 
Appl. Sci. 2019, 9, 4201; doi:10.3390/app9204201 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4201 2 of 10 showed that the spatiotemporal shape of femtosecond pulse at a central wavelength of 800 nm broadens and deforms under high relative humidity conditions. Simulation studies also showed that physical eﬀects like dispersion and linear absorption become more and more important in the formation and propagation dynamics of 10 µm ﬁlaments with the increase of relative humidity . More recently, Shutov et al.  experimentally identiﬁed that the photoionization of water vapor acts as the dominant ionization process in atmospheric air for femtosecond laser pulses at a 248-nm central wavelength and the relevant ionization coeﬃcients were also measured. With increasing air humidity, the plasma channels generated by ultraviolet laser pulses in air become longer and wider . However, the detailed dynamic evolution of UV ﬁlamentation and its energy deposition characteristics in an atmospheric environment with diﬀerent humidity levels has rarely been reported. Allowing for the reported valuable applications of UV laser ﬁlaments in meteorology [19–21], it is also necessary to evaluate how water vapor will aﬀect the transmission of femtosecond lasers in a cloud environment. In this paper, we numerically investigate the inﬂuences of ionization of water vapor on nonlinear propagation of ultraviolet (UV) ﬁlaments. The formation and propagation dynamics of UV ﬁlaments under diﬀerent humidity conditions are studied in detail. The remainder of this paper is organized as follows: the model used to describe the propagation of femtosecond pulse in gas medium is introduced in Section2; in Section3, the evolution of ﬁlament along propagation distance z in di ﬀerent gas media is investigated; and the conclusions are summarized in Section4.
2. Methods To investigate the propagation dynamics of a femtosecond pulse in a gas medium, we modeled the propagation of a laser pulse with the nonlinear Schrödinger equation (NLSE), which governs the evolution of the electric ﬁeld envelope E(x,y,z,t) of the pulse in the reference frame moving at the group velocity vg = ∂ω/∂k : ω0
2 2 (Ks) ∂E i ik00 ∂ E ω ωpe X β = ∆ E + i 0 n E 2E k E s E 2Ks 2E (1) 2 2 0 2 − ∂z 2k0 ⊥ − 2 ∂τ c | | − ω − 2 | | 0 s=N2,O2,H2O where E(x,y,z,t) represents the envelope of the electric ﬁeld along propagation direction z, and τ refers to the retarded time variable, t-z/vg, according to group velocity as the reference system. k0 = 2π/λ0 and ω0 = 2πc/λ0 are the central wave number and the angular frequency of the carrier wave corresponding 2 2 the wavelength λ = 248 nm, respectively. The Laplacian operator ∆2 = ∂ + ∂ denotes the beam 0 ∂x2 ∂y2 ⊥ transverse diﬀraction. The second terms on the right-hand side of Equation (1) account for group 28 2 velocity dispersion (GVD) with a coeﬃcient of k00 = 1.21 10 (s /m). The Kerr eﬀect is related to × − n = 8.0 10 19 cm2/W, which denotes the nonlinear refraction index of air. Plasma defocusing 2 × − q 2 is described by the plasma oscillation frequency, ωpe = qe ρ/meε0 (qe is the electron charge, me is the electron mass, and ρ is the free electron density). The last term in Equation (1) accounts for multiphoton absorption (MPA). The MPA term accounts for energy absorption due to multiphoton ( ) Ks K ionization . The coeﬃcient βs = Ks}ω0ρat,sσ s is related to the multiphoton ionization coeﬃcient, which corresponds to the number of K-photons needed to ionize the corresponding gas molecule. The evolution equation of the electron density ρes is governed by:
(Ks) ! ∂ρ X β ρ s 2Ks σ 2 es = E + ρes E 1 (2) t K U ∂ s s}ω0 | | s | | − ρat,s
22 where σ = 5.1 10− is the cross section for inverse Bremsstrahlung, } = h/2π is Planck constant, Us is × ionization potential of the gas, and ρat,s denotes the initial neutral gas atom density. The coeﬃcients used in Equations (1) and (2) are listed in Table1. Appl. Sci. 2019, 9, 4201 3 of 10
Table 1. Parameters used in the model. We adopted most of the theoretically calculated parameters for K O2 and N2 from Reference . The eﬀective multiphoton ionization cross section σ for H2O molecules is taken from Reference .
Gas Types Symbol (Units) Values  The Measured Values K 3 3 K 1 2K K 40 40 O2 σ (s m /W ) 1.35 10 1.5 10 − × − × − Ui (eV) 12.07 12.07 K 4 3 K 1 2K K 60 41 N2 σ (s m /W ) 3.22 10 2.4 10 − × − × − Ui (eV) 15.58 15.58 K - 3 K 1 2K K 38 H2O σ (s m /W ) - 6.8 10 − × − Ui (eV) - 12.62
The initial pulse investigated in this paper is modeled by a Gaussian beam: s 2 2 2 2Pin x + y t E(x, y, z = 0, t) = exp (3) 2 − 2 − 2 πw0 w0 tp where Pin, w0, and τp are the input peak power, initial pulse radius, and the pulse duration, respectively. A standard split step Crank-Nicholson scheme is used to integrate the linear part of Equation (1) along the propagation axis z and the fourth-order Runge–Kutta method is employed to solve the evolution equation (Equation (2)) .
3. Results and Discussion
We ﬁrst considered incident laser pulses with w0 = 1 mm, τp = 90 fs, and Pin = 10Pcr. The total 25 3 atmospheric air molecule density ρat was assumed to be 2.7 10 /m . For simplicity, dry air was × assumed to only consist of N2 and O2 at a ratio of 0.78 to 0.21. For wet air, we imposed a composition of 78.0% N2, 21.0% O2, and 1.0% H2O (gas). Figure1 presents the on-axis intensity and on-axis electron density along the propagation axis. The solid blue and green lines represent the results for pulses in dry and wet air conditions, respectively. K The multiphoton ionization cross section σ for N2,O2, and H2O are taken from the measured results by Shutov , as listed in Table1. The black dash line represents the simulated results for dry air K when the cross section σ for N2 and O2 were obtained from the theoretical formula by Couairon and Mysyrowicz . As shown in Figure1a, the beam intensity in both dry air and wet air ﬁrst increased rapidly at about z = 2.0 m, which implies that the water vapor had no inﬂuence on the self-focusing position (deﬁned as the distance between the initial transmission position of the beam and the position where the on-axis light intensity increases sharply). Almost simultaneously, the on-axis electron density also obviously increases, as shown in Figure1b. This is in accord with the characteristics of intense a femtosecond laser pulse at its ﬁrst propagation stage, as identiﬁed by Bergé et al. . During the ﬁrst stage, the beam intensity increases due to the optical Kerr eﬀect focusing and the electron density being excited when the beam intensity becomes suﬃciently large. Subsequently, the generated electron density has a threshold-like response, which limits the further increase of peak intensity inside the ﬁlament by defocusing the beam in turn . When a balance of these two processes is achieved, the laser intensity inside the ﬁlament gradually stabilizes. This special phenomenon is usually named intensity clamping . This light guide coupling with a signiﬁcant electron density level is identiﬁed as the second stage by Bergé et al. . Based on the values of the peak intensity, it can be clearly seen from Figure1a that the clamping intensity in dry air is much higher than that in wet air. The clamping intensity of UV ﬁlaments decreased more obviously in wet air conditions compared with the results of Appl. Sci. 2019, 9, 4201 4 of 10
Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 10 mid-infrared lasers ﬁlaments . The peak intensity is maintained for several meters and is coupled withseveral a signiﬁcant meters and electron is coupled density with level a significant (Figure1b). electron The laser density pulse tendslevel ( toFigure form 1 longerb). The ﬁlaments laser pulse in wettends air. to Due form to ionizationlonger filament of waters invapor, wet air. the Due on-axis to ionization electron density of water of UV vapor, pulses the in on wet-axis air iselectron much greaterdensity thanof UV that pulses in dry in air, wet which air is can much be clearly greater seen than in that Figure in 1dryb. At air, a largerwhich z,can the be pulse clearly ultimately seen in diFigureﬀracts 1 asb. ρAtdecreases a larger to z, zero. the In pulse addition, ultimately the simulation diffractsresults as ρ decreases from the measured to zero. MPAIn addition, parameters the aresimulation very close results to those from from the measured the theoretical MPA calculation parameters results are very in dryclose air to (as those shown from with the blacktheoretical dash lines).calculation results in dry air (as shown with black dash lines).
Figure 1.1. Evolution of the on-axison-axis (a) intensity, ((b)) electronelectron densitydensity asas functionsfunctions of zz forfor aa UVUV pulsepulse propagatingpropagating in air.air. The blue and green solidsolid lines represent the simulatedsimulated results in dry and wet air KK conditions,conditions, respectively. respectively. The cross section section σσ forfor NN22,,O O2,, and H2OO is obtained from the measured measured resultsresults byby ShutovShutov [ 17].. The The dry dry air air is is assumed assumed to to only only consist consist of Nof2 Nand2 and O2 Oin2 ain ratio a ratio of 0.78 of 0.78 to 0.21. to 0.21. For wetFor wet air, weair, imposewe impose a composition a composition of 78.0%of 78.0% N2 ,N 21.0%2, 21.0% O2 O, and2, and 1.0% 1.0% H H2O2O (gas). (gas).The The blackblack dashdash lineline K representsrepresents thethe simulatedsimulated resultsresults ofof drydry airair whenwhen itsits crosscross section sectionσ σK forfor NN22 andand OO22 are obtained fromfrom thethe theoreticaltheoretical formulaformula inin ReferenceReference [ 9].. TheThe input input beam beam is is a a Gaussian Gaussian shape shape with with P inPin= =10P 10Pcrcr,, ww00 == 1 mm, andand ττpp = 9090 fs. fs.
Figure2 2 shows shows the the number number of of electrons electrons ionized ionized from from N N22,O, O22,, andand HH22O. The number of electrons R zrz R rmax 2 R 2π waswas calculatedcalculated bybyN Ne(z) z= dzdz max drdr dd r,φρ ( ,r z, τ rmax sin , z ). rThesin φsolid. The lines solid represent lines represent the results the results from e 0 00 0 0 max from the measured MPA parameters under wet conditions. It clearly shows that the number of the measured MPA parameters under wet conditions. It clearly shows that the number of electrons electrons increased substantially in wet air compared with dry air, which demonstrates the remarkable increased substantially in wet air compared with dry air, which demonstrates the remarkable contribution of H2O molecules to the air ionization. The MPA process is a crucial mechanism for contribution of H2O molecules to the air ionization. The MPA process is a crucial mechanism for the the pulse’s energy loss . As shown in Figure2b, the pulse energy decreases rapidly during the pulse’s energy loss . As shown in Figure 2b, the pulse energy decreases rapidly during the ﬁlamentation stage. The ﬁlamentation tends to deposit more energy in wet air conditions. filamentation stage. The filamentation tends to deposit more energy in wet air conditions. To investigate the underlying physics behind the ﬁlamentation for the two cases in the spatial To investigate the underlying physics behind the filamentation for the two cases in the spatial regime, we plot the ﬂuence distribution as a function of the propagation distance, which is shown in regime, we plot the fluence distribution as a function of the propagation distance, which is shown in R + 2 Figure3. The ﬂuence distribution is deﬁned as F(x, y, z) = ∞ E(x, 2y, t) dt. The several highlighted Figure 3. The fluence distribution is defined as F x,,,, y z −∞ E x y t dt . The several highlighted core core areas in Figure3 clearly indicate defocusing-refocusing cycles in the ﬁlaments [ 12]. There are mainlyareas in three Figure cycles 3 clearly in dry indicate air and defocusing wet air. The-refocusing longer bright cycles core in the in wet filaments air further . indicatesThere are that mainly the laserthree pulsecycles forms in dry longer air and ﬁlaments wet air. inThe wet longer air. Compared bright core with in we ﬁlamentationt air further inindicates dry air, that the radiusthe laser of ﬁlamentpulse forms gets longer slightly filament widers andin wet the air. intensity Compared of ﬂuence with filamentation becomes lower in dry inwet air, the air. radius These of results filament are consistentgets slightly with wider the and experimental the intensity results of fluence reported becomes by Shutov lower et in al. wet . air. These results are consistent with the experimental results reported by Shutov et al. . The evolution of the radius and intensity of the filament can also be clearly seen from the transverse energy flux distributions, as shown in Figure 4. The energy is concentrated into the central area with a slightly higher peak fluence in dry air (Figure 4a) than in wet air (Figure 4d) at the initial position of the filament. During the intensity clamping, an obviously higher peak fluence takes place in dry air when propagation distance increases (Figure 4b,e). However, at the end of filament formation, the peak fluence in dry air becomes smaller than in wet air (Figure 4c,f). Relatively, the radius of filament is wider but its intensity is lower in wet air when compared with in dry air. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 10 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 10 Figure 5a–d details the temporal distortions of the pulse at four different propagation distances. DuringFigure the self5a–-dfocusing details the stage, temporal the pulse distortions keeps of its the initial pulse Gaussian at four different shape and propagation undergoes distances. only an inDuringcrease theof its self peak-focusing intensity stage, (Figure the pulse5a). The keeps peak its value initial in Gaussian dry air is shape much andhigher undergoes than in wet only air. an Obviousincrease ofmulti its -peakpeaked intensity structures (Figure and 5a). pulse The broadening peak value can in drybe observedair is much during higher the than filamentation in wet air. process,Obvious as multi noted-peaked in Figure structures 5b,c. The and trailing pulse pulse broadening and leadi canng bepeak observed in temporal during profile the filamentation is created by plasmaprocess, defocusing as noted in [10,22,23] Figure 5b. The,c. The symmetric trailing pulsespikes and of theleadi pulseng peak in time in temporal are associated profile with is created the GVD by plasma. The defocusing GVD can further [10,22,23] broaden. The symmetric the temporal spikes width of the pulse. As shown in time in are in associatedFigure 5b, c,with the thenumber GVD of splits. The in GVD temporal can further profile broaden changes the little temporal in both width types  of .air As gas. shown The in degree in Figure of compression 5b,c, the number of a femtosecondof splits in temporal pulse depends profile onchanges air humidity. little in Theboth broadening types of air of gas. laser The pulses degree in dry of compressionair is much more of a obviousfemtosecond than inpulse wet depends air. In the on end air ofhumidity. filamentation, The broadening the pulse propagated of laser pulses in wet in dry air airbroadens is much and more its peakAppl.obvious Sci.intensity2019 than, 9, in 4201exceed wet air.s that In thein dry end air of (filamentation,Figure 5d). the pulse propagated in wet air broadens and5 of its 10 peak intensity exceeds that in dry air (Figure 5d).
(a) (b) (a) (b) Figure 2. Evolution of (a) number of electrons ionized from N2, O2, and H2O, and (b) pulse energy as zr 2 Figure 2.2. Evolution of (aa)) numbernumber ofof electronselectrons ionizedionized fromfrom NN22,O, O22,, andandmax HH22O, and (b) pulse energy asas functions of z. The number of electrons is calculated by Ne z dz dr d r, max , z r sin . In R0zzr R 0rmaxmax 02R 2π functionsfunctions ofof z.z. TheThe numbernumber ofof electrons electrons is is calculated calculated by byNNe(z) z= dz drdr ddφρ r,( r, τmax , z r, sinz)r sin . Inφ. e 00 0 0 0 max Figure2a, the blue dash line is the results for O2 gas, the red dash line denotes the results for N2 gas, In Figure2a, the blue dash line is the results for O 2 gas, the red dash line denotes the results for N2 andFigure2 the a, black the blue dash dash line line represents is the results the results for O2 forgas, dry the airred gas.dash The line resu denoteslts are the obtained results for from N2 gas, the gas, and the black dash line represents the results for dry air gas. The results are obtained from the calculatedand the black multiphoton dash line absorption represents (MPA the ) results parameters for dry in Ref air erence gas. The  . resuThe ltssolid are lines obtained in both from figures the calculated multiphoton absorption (MPA) parameters in Reference . The solid lines in both ﬁgures arecalculated the results multiphoton from the measured absorption multiphoton (MPA) parameters absorption in Ref (MPAerence) par ameters. The solid in Ref lineserence in both, figureswhich are the results from the measured multiphoton absorption (MPA) parameters in Reference , which are the results from the measured multiphoton absorption (MPA) parameters in Reference, which are clearly indicated in the legends, respectively.respectively. are clearly indicated in the legends, respectively.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 10 (a) (a)
FigureFigure 3. EvolutionEvolution of of the the fluence ﬂuence distribution distribution along along the the propagation propagation distance. distance. (a) Dry (a) Dry air and air and(b) w (bet) wetair. air.
The evolution of the radius and intensity of the ﬁlament can also be clearly seen from the transverse energy ﬂux distributions, as shown in Figure4. The energy is concentrated into the central area with a slightly higher peak ﬂuence in dry air (Figure4a) than in wet air (Figure4d) at the initial position of the ﬁlament. During the intensity clamping, an obviously higher peak ﬂuence takes place in dry air
(a) (b) (c)
(d) (e) (f)
Figure 4. Fluence distribution F/F0(x,y) at different propagation distance z. The top row, (a) z = 2.0 m, (b) z = 3.5 m, and (c) z = 7.0 m for dry case. The bottom row, (d) z = 2.0 m, (e) z = 2.5 m, and (f) z = 7.0 m for wet case.
(a) (b) Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 10
Appl. Sci. 2019, 9, 4201 6 of 10
whenAppl. Sci. propagation 2019, 9, x FOR distancePEER REVIEW increases (Figure4b,e).(b) However, at the end of ﬁlament formation,6 of the 10 peak ﬂuence in dry air becomes smaller than in wet air (Figure4c,f). Relatively, the radius of ﬁlament Figure 3. Evolution of the fluence distribution along the propagation distance. (a) Dry air and (b) wet is wider but its intensity is lower in wet air when compared with in dry air. air.
(a) (b) (c) (b)
Figure 3. Evolution of the fluence distribution along the propagation distance. (a) Dry air and (b) wet air.
(d) (e) (f)
FigureFigure 4.4. Fluence distribution FF/F/F00(x,y) at didifferentﬀerent propagationpropagation distancedistance z.z. The toptop row,row, ((aa)) zz == 2.02.0 m, ((bb)) zz == 3.53.5 m,m, andand ((cc)) zz == 7.0 m forfor drydry case.case. TheThe bottombottom row,row, ( d(d)) z z= =2.0 2.0 m,m, ((ee)) z z= = 2.52.5 m,m, andand ((ff)) zz= = 7.07.0
mm forfor wetwet case.case(a.) (b) (c) Figure5a–d details the temporal distortions of the pulse at four di ﬀerent propagation distances. During the self-focusing stage, the pulse keeps its initial Gaussian shape and undergoes only an increase of its peak intensity (Figure5a). The peak value in dry air is much higher than in wet air. Obvious multi-peaked structures and pulse broadening can be observed during the ﬁlamentation process, as noted in Figure5b,c. The trailing pulse and leading peak in temporal proﬁle is created by plasma defocusing [10,22,23]. The symmetric spikes of the pulse in time are associated with the GVD . The GVD can further broaden the temporal width [14 ]. As shown in in Figure5b,c, the number of splits( ind) temporal proﬁle changes little( ine) both types of air gas. The degree(f) of compression of a femtosecond pulse depends on air humidity. The broadening of laser pulses in dry air is much Figure 4. Fluence distribution F/F0(x,y) at different propagation distance z. The top row, (a) z = 2.0 m, more obvious than in wet air. In the end of ﬁlamentation, the pulse propagated in wet air broadens (b) z = 3.5 m, and (c) z = 7.0(a) m for dry case. The bottom row, (d) z = 2.0 m,( (be)) z = 2.5 m, and (f) z = 7.0 and itsm peakfor wet intensity case. exceeds that in dry air (Figure5d).
Figure 5. Cont. Appl. Sci. 2019, 9, 4201 7 of 10 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 10
Figure 5. TemporalTemporal profile proﬁle as as functions functions of of the the propagation propagation distance distance z. z. (a ()a z) = z 2.0= 2.0 m, m,(b) ( bz )= z 2.75= 2.75 m, ( m,c) (zc )= z4= m,4 mand, and (d) ( dz )= z 5.5= 5.5 m. m.The The blue blue and and green green solid solid lines lines represent represent results results from from the the measured measured MPA MPA
parameters inin drydry and and wet wet air air conditions, conditions, respectively. respectively. The The dry d airry air is assumed is assumed to only to only consist consist of N 2ofand N2 Oand2 in O a2 in ratio a ratio of 0.78 of 0.78 to 0.21.to 0.21. For For wet wet air, air, we we impose impose a compositiona composition of of 78.0% 78.0% N N2,2, 21.0% 21.0% OO22,, andand 1.0% H2O (gas) (gas).. The black dash lineline representsrepresents thethe simulated simulated result result from from the the calculated calculated MPA MPA parameters parameters in Referencein Reference . .
In meteorology, relativerelative humidityhumidity (RH)(RH) isis an important physical measure toto describe the amount of water vapor present in ambient conditions.conditions. We further investigated the nonlinear propagation of UV pulses inin conditionsconditions withwith didifferentﬀerent relative relative humidity. humidity The. The relative relative humidity humidity grows grows as as the the amount amount of waterof wat vaporer vapor increases. increas Byes.deﬁnition, By definition, relative relative humidity humidity can be can expressed be expressed as theratio as the of vapor ratio of pressure vapor (e)pressure to saturation (e) to saturat vaporion pressure vapor (pressureesw(T)): (esw(T)): " # e RHRH= (4(4)) eeTsw(T) sw ppT,,T
The saturation vapor pressure of pure water is only a functionfunction of temperature,temperature, which can be calculated according toto Wexler’sWexler’s formulationsformulations [25] asas follows:follows:
eT0.01exph 2991.2729 22 esw = sw0.01 exp 2991.2729T − − 6017.01286017.0128TT11+ 18.8764385418.87643854 − − 0.0283547210.028354721TTT+ 0.17838301 1010424T2 (5) − × − 0.841504170.84150417 10 1099TTT 33 + 0.444125430.44412543 10 1210 4 12T4 (5) − × − × − +2.8584872.858487ln lnTT] where esw is in hPa and e0 = 6.11 hPa is the saturated vapor pressure at 0 ◦C. T is absolute temperature in K. For moist air, a slight correction must be made to calculate the saturation vapor pressure, which where esw is in hPa and e0 = 6.11 hPa is the saturated vapor pressure at 0 °C . T is absolute temperature is expressed by: in K. For moist air, a slight correction must be made to calculate the saturation vapor pressure, which e0sw(p, T) = f (p) esw(T) (6) is expressed by: × where f (p) is called the enhancement factor and its formulation is given below : esw p, T f p e sw T (6) 1 6 f (p) = 1.0016 0.074p− + 3.15 10− p (7) where f(p) is called the enhancement factor and− its formulation× is given below : f p1.0016 0.074 p16 3.15 10 p Based on Equations (5)–(7), the saturation vapor pressure in moist air is 24.991 hPa at T = 21(7◦C.) The vapor pressure e can then be derived from the ideal gas state equation: Based on Equations (5)–(7), the saturation vapor pressure in moist air is 24.991 hPa at T = 21 °C. The vapor pressure e can then be derived frompV the= nRTideal gas state equation: (8) pV nRT (8) where p is the pressure, V is the volume, and n is the amount of substance of the gas, respectively, which can be expressed as: Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 10
N n (9) N A
where N is the number of gas molecules and NA is the Avogadro constant. As above, we find for cases with 10%, 75%, and 90% relative humidity, the relevant water content is 0.245%, 1.866%, and 2.248%, respectively. As for the case with 1.0% water content analyzed above, the relative humidity was 40%. In order to keep the total atmospheric density constant in all cases ( N2 O 2 H 2 O at ), the components of dry air were reduced proportionally. Appl. Sci.Figure2019, 69, shows 4201 the on-axis intensity and electron density along propagation distance for the8 offive 10 different humidity values. In the simulation, we adopted I0 = 2.5 × 1015 W/m2, τp = 160 fs, and r = 1 mm. We can clearly see that the atmospheric humidity levels do influence the overall filament wheredynamicsp is. theAs pressure,the contentV is of the water volume, vapor and increases,n is the amount the initial of substance filamentation of the gas, position respectively, slightly whichincreases. can The be expressed on-axis intensity as: experienced obviously decreases for 0% humidity to 40% humidity. N The evolution of on-axis intensity in 75% humidityn = condition is very close to the 90% humidity(9) N condition. The differences in evolution of on-axis intensityA along z decrease gradually from low wherehumidityN is to the high number humidity. of gas Analogously molecules, andthe peakNA is of the on- Avogadroaxis electron constant. density Asincreases above, rapidly we ﬁnd from for caseslow humidity with 10%, to 75%,high andhumidity 90% relativeand the humidity,trends of increas the relevanting gradually water content decrease iss 0.245%, as the water 1.866%, content and 2.248%,increase respectively.s. As for the case with 1.0% water content analyzed above, the relative humidity + + = was 40%.Figure In 7 order shows to keepthe total the totalnumber atmospheric of electrons density ionized constant from in H all2O cases and (theρN 2overallρO2 energρH2Oy ofρ theat), thewave components packet as of a dry function air were of propagation reduced proportionally. distance z. As presented in Figure 7a, the number of electronsFigure increases6 shows the with on-axis the water intensity content and electron but the density trend of along increase propagation gradually distance decreases for the as ﬁve the 15 2 diﬀerent humidity values. In the simulation, we adopted I = 2.5 10 W/m , τp = 160 fs, and r = 1 mm. humidity increases. In other words, the density of electr0 ons× tends to saturate in high humidity Wecondition can clearlys. This see saturation that the atmospheric is attributed humidity to the nonlinear levels do energy inﬂuence loss thein the overall multiphoton ﬁlament gas dynamics. ionization As theby contentShutov of water. The vapor differences increases, in thenonlinear initial ﬁlamentation energy loss can position be clearly slightly seen increases. in Figure The 7b. on-axis The intensitydeposited experienced energy of filamentation obviously decreases increases for 0%with humidity the humidity to 40% increases humidity. but The the evolution growth of quantity on-axis intensitytakes on reducing in 75% humidity trends in conditionthe higher is humidity very close conditions. to the 90% humidity condition. The diﬀerences in evolutionWe also ofsimulated on-axis intensitythe femtosecond along z decreasefilamentation gradually with frominitial low peak humidity intensity to of high I0 = humidity.1.0 × 1015 Analogously,W/m2 and I0 = the 5.0 peak × 1015 of W/m on-axis2. We electron find the density conclusions increases are rapidly identical from to those low humidity above and to highthe results humidity are andnot plotted the trends again of. increasing gradually decreases as the water content increases.
FigureFigure 6.6. EvolutionEvolution ofof thethe on-axison-axis ( a(a)) intensity intensity and and (b ()b electron) electron density density along along propagation propagation distance distance z in z diinﬀ differenterent humidity humidity conditions. conditions. Red Red solid solid lines: lines 0%: 0% humidity; humidity; purple purple solid solid lines, lines, 10%10% humidity;humidity; blueblue solidsolid lines,lines, 40%40% humidity;humidity; greengreen solidsolid lines,lines, 75%75% humidity;humidity; lightlight greengreen solidsolid lines,lines, 90%90% humidity.humidity.
Figure7 shows the total number of electrons ionized from H 2O and the overall energy of the wave packet as a function of propagation distance z. As presented in Figure7a, the number of electrons increases with the water content but the trend of increase gradually decreases as the humidity increases. In other words, the density of electrons tends to saturate in high humidity conditions. This saturation is attributed to the nonlinear energy loss in the multiphoton gas ionization by Shutov . The diﬀerences in nonlinear energy loss can be clearly seen in Figure7b. The deposited energy of ﬁlamentation increases with the humidity increases but the growth quantity takes on reducing trends in the higher humidity conditions. We also simulated the femtosecond ﬁlamentation with initial peak intensity of I = 1.0 1015 W/m2 0 × and I = 5.0 1015 W/m2. We ﬁnd the conclusions are identical to those above and the results are not 0 × plotted again. Appl. Sci. 2019, 9, 4201 9 of 10 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 10
Figure 7. 7.Evolution Evolution of of(a ) (numbera) number of electrons of electrons ionized ionized from H 2 fromO and H (b2O) pulse and energy (b) pulse along energy propagation along distance.propagation Purple distance. solid Purple lines, 10%solid humidity; lines, 10% blue humidity; solid lines,blue solid 40% humidity;lines, 40% greenhumidity; solid g lines,reen solid 75% humidity;lines, 75% lighthumidity; green light solid g lines,reen solid 90% humidity.lines, 90% humidity.
4. Conclusions In conclusion, wewe havehave presentedpresented a a detailed detailed numerical numerical study study of of the the UV UV ﬁlamentation filamentation processes processes in dryin dry and and humid humid air air conditions. conditions. The The results results show show that that clamping clamping intensity intensity and an electrond electron density density are very are sensitivevery sensitive to the to water the vapor. water vapor.Ionization Ionization of H2O ofmolecules H2O molecules plays a signiﬁcant plays a significant role in the role air ionization. in the air Theionization. ﬁlamentation The filamentation tends to deposit tends higher to deposit energy in higher wet air energy conditions. in wet During air condition the ﬁlamentations. During stage, the thefilamentation temporal proﬁlestage, the is compressed temporal profi narrowerle is compressed in wet air narrower than in dry in air.wet Withair than the in humidity dry air. With increase, the bothhumidity the clamping increase, intensityboth the andclamping electron intensity density and tend electron to increase. density The tend number to increase. of electrons The number and pulse of energyelectrons tend and to pulse be saturated energy tend in the to higher be saturated humidity in the conditions. higher humidity The above conditions. results clearly The above demonstrate results theclearly signiﬁcant demonstrate roles of the photoionization significant roles of water of photoionization vapor in research of on water UV ﬁlamentation. vapor in research on UV filamentation. Author Contributions: Conceptualization, L.L.; methodology, Q.Z.; software, K.Z.; investigation, C.W.; resources, X.X.;Author writing—original Contributions: draft conceptualization preparation, Q.Z.;, L.L.; writing—review methodology, and editing, Q.Z.; software, S.H.; visualization, K.Z.; investigation, M.C.; supervision, C.W.; T.G. resources, X.X.; writing—original draft preparation, Q.Z.; writing—review and editing, S.H.; visualization, M.C.; Funding:supervision,This T.G. work has been supported by the Fund of the National Natural Science Foundation of China (NSFC) (41575024, 41875025). These grants are greatly appreciated. Funding: This work has been supported by the Fund of the National Natural Science Foundation of China Acknowledgments: We acknowledge the help of Jingjing Ju from Shanghai Institute of Optics and ﬁne Mechanics, (NSFC) (41575024, 41875025). These grants are greatly appreciated. Chinese Academy of Sciences. ConﬂictsAcknowledgments: of Interest: WeThe acknowledgeauthors declare the no help conﬂicts of Jingjing of interest. Ju from Shanghai Institute of Optics and fine Mechanics, Chinese Academy of Sciences.
ReferencesConflicts of Interest: The authors declare no conflicts of interest.
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