Combined action of the bound-electron nonlinearity and the tunnel-ionization current in low-order harmonic generation in noble gases
aa r X i v : . [ phy s i c s . op ti c s ] N ov Combined action of the bound-electronnonlinearity and the tunnel-ionizationcurrent in low-order harmonicgeneration in noble gases
Usman Sapaev, Anton Husakou and Joachim Herrmann ∗ Max-Born-Institute for Nonlinear optics and Fast Spectroscopy, Max-Born-Str. 2a, BerlinD-12489, Germany ∗ [email protected] We study numerically low-order harmonic generation in noble gases pumped by intense fem-tosecond laser pulses in the tunneling ionization regime. We analyze the influence of the phase-mismatching on this process, caused by the generated plasma, and study in dependence on thepump intensity the origin of harmonic generation arising either from the bound-electron non-linearity or the tunnel-ionization current. It is shown that in argon the optimum pump intensityof about 100 TW/cm leads to the maximum efficiency, where the main contribution to low-order harmonics originates from the bound-electron third and fifth order susceptibilities, whilefor intensities higher than 300 TW/cm the tunnel-ionization current plays the dominant role.Besides, we predict that VUV pulses at 133 nm can be generated with relatively high efficiencyof about 1 . × − by 400 nm pump pulses. References and links
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1. Introduction
Low-order harmonic generation (LOHG) in gases pumped by ultrashort near-IR laser pulses isan important technique to generate ultraviolet (UV) and vacuum ultraviolet (VUV) femtosec-ond pulses for a wide variety of applications, in particular, for time-resolved spectroscopy ofmany molecules, clusters or biological specimens and for material characterization [1]. The useof noble gases as nonlinear medium instead of solid-state crystals is a preferable way to avoidstrong dispersion, bandwidth limitations, low damage thresholds and strong absorption below200 nm. In particular, by using different gases UV and VUV pulses with a duration down to 11fs have been generated by third [2,3] and fifth harmonic [4] conversion. Similar to other nonlin-ear frequency conversion processes, the efficiency of LOHG in gases is usually relatively lowin practice. This is mainly caused by two factors: first, low conversion results from relativelymall values of the third (and higher) order susceptibilities compared to crystalline media. Asecond problem is the realization of phase matching, which can be partially solved for variousfrequency transformation processes, e.g., by using the anomalous dispersion of hollow-corefibers [5–7], modulated hollow-core waveguides [8], a modulated third order nonlinearity byultrasound [9] or by noncollinear four-wave mixing [10, 11].In the intensity range below the ionization threshold the efficiency of frequency transforma-tion increases with increasing pump intensity. However, as soon as the intensity rises abovethe ionization threshold, different additional processes play a role leading to a more complexdynamics. On the one hand the effective third-order nonlinearity decreases, since c +( ) of anionized gas is lower than c ( ) of a corresponding neutral gas [12]. On the other hand harmonicsof the fundamental frequency emerge due to the ionization of the atoms and the interaction ofthe freed electrons with the intense pump field. The majority of studies of harmonic generationhave been performed for relatively high orders of harmonics much in excess of the ioniza-tion potential which is well described by the three-step process of ionization, acceleration inthe continuum and recombination with the parent ion (see e.g. [13]). Much less studied is anadditional, physically different, mechanism of optical harmonic generation. In the regime oftunneling ionization the density of ionized electrons shows extremely fast, nearly stepwise in-creases at every half-cycle of the laser field. This stepwise modulation of the tunnel ionizationcurrent induces optical harmonic generation [14], which arises in the first stage of ionizationand not in the final recombination stage in the three-step model. As shown in [15], the emissionof the lowest harmonics up to about 9 are accounted for with the tunnel ionization current whilehigher orders are attributed with the recombination process. Further theoretical studies of thisprocess has been published in [16, 17]. Note that the generation of THz pulses by two-colorfemtosecond pulses is also intrinsically connected to the optically induced step-wise increaseof the plasma density due to tunneling ionization [18–20].To date only few direct experimental observations of harmonic generation or frequencymixing due to the modulation of the tunnel ionization current has been reported [21,22]. On theother hand the generation of third harmonics with efficiencies up to the range of 10 − in a noblegas with pump intensities significantly larger than necessary for ionization has been reportedin [2, 23]. Detections of this type of nonlinear response in the regime with intensites above theionization threshold requires better understanding of the complex dynamics and insight into thecompetition of harmonic generation originating from atomic or ionic susceptibilities of boundstates and the tunneling ionization current. The present paper is devoted to a theoretical study ofthis issue. Since here only low orders up to 7 are considered we neglect the recombination pro-cess and consider only the first stage of ionization. At least, up to our knowledge the combinedoccurrence of the two mechanisms by the bound-electron third (and higher) order nonlinearityand the tunnel-ionization current were not studied before.
2. Fundamentals
Third harmonic generation (THG) in gases by focused beams for pump intensities below theionization threshold has been studied theoretically already four decades ago [24, 25]. In theregime of tunneling ionization besides the nonlinearity due to bound electron states additionalprocesses come into play which has to be accounted for. In particular, the sub-cycle temporaldynamic of the laser field plays an essential role in the ionization process. Therefore, in thetheoretical description the slowly-varying envelope approximation requires the solution of acomplicated, strongly coupled system of partial differential equations and would result in in-creased numerical errors due to relatively short (down to 8 fs) durations of harmonic pulses.Since backward propagating field components are small we can use the unidirectional pulsepropagation equation for the description of pulse propagation [26]. As will be seen later theffective propagation length is much smaller than the Rayleigh length, therefore we can neglectthe diffraction term. Correspondingly the following basic equation for the electric field of linearpolarized pulses will be used: ¶ z ˆ E ( w ) = ik ( w ) ˆ E ( w ) + i m o w k ( w ) ˆ P NL ( w ) (1)Here ˆ E ( w ) is the Fourier transform of the electric field E ( t ) ; k ( w ) = cn ( w ) / w is the frequency-dependent wavenumber, w is the angular frequency, c is the speed of light and n ( w ) is thefrequency-dependent refractive index of the chosen gas; m o is the vacuum permeability. Thefirst term on the right-hand side of Eq. (1) describes linear dispersion of the gas. The nonlinearpolarization is ˆ P NL ( w ) = ˆ P Bound ( w ) + i ˆ J e ( w ) / w + i ˆ J Loss ( w ) / w with ˆ P Bound ( w ) being the non-linear polarization caused by the bound electron states, ˆ J e ( w ) being the electron current andˆ J Loss ( w ) being the loss term due to photon absorption during ionization. The plasma dynamicsis described by the free electron density r ( t ) , which can be calculated by: ¶ t r ( t ) = W ST ( t )( r at − r ( t )) (2)where r at is the neutral atomic density; W ST ( t ) is the quasistatic tunneling ionization ratefor hydrogenlike atoms [27] W ST ( t ) = w a ( r h ) . ( | E a | / E ( t )) exp ( − r . | E a | / E ( t )) , where E a = m e q / ( pe o ) ¯ h , w a = m e q / ( pe o ) ¯ h and r h = U Ar / U h , U h and U Ar are the ionizationpotentials of hydrogen and argon, correspondingly; P Bound ( t ) = e o c ( ) ( − r ( t ) / r at ) E ( t ) + c +( ) ( r ( t ) / r at ) E ( t ) + c ( ) ( − r ( t ) / r at ) E ( t ) , e o is the vacuum permittivity; m e and q beingthe electron mass and charge, respectively; c ( ) and c ( ) are third and fifth order susceptibil-ities of neutral gas, correspondingly, while c +( ) is that of ionized gas. In the following weconsider nearly collimated beams with diameter corresponding to a Rayleigh length larger thanthe propagation lengths. In addition, for these parameters the pump power is below the self-focusing power. Therefore, we can neglect diffraction in the numerical model. The transversemacroscopic plasma current J e ( t ) is determined by [19]: ¶ t J e ( t ) + n e J e ( t ) = q m e E ( t ) r ( t ) (3)where n e is the electron collision rate (for argon n e ≈ . − ). Finally, the ionization energyloss is determined by J Loss ( t ) = W ST ( t )( r at − r ( t )) U Ar / E ( t ) . A critical condition for an effi-cient frequency transfer to harmonics is the realization of phasematching which for intensitieslarger than the ionization threshold is sensitively influenced by the plasma contribution to therefraction index. The change of the linear refractive index of argon at the maximum of the pulseintensity I ′ (assuming Gaussian pulse shape), owing to the formation of laser plasma with a freeelectron density r ′ and the Kerr nonlinearity, is given by (see e.g., [28, 29]): n ( w , I ′ , r ′ ) = n e ( w , r ′ ) + D n Kerr ( I ′ , r ′ ) + D n Plasma ( w , r ′ ) (4)where n e ( w , r ′ ) = ( n o ( w ) − )( − r ′ / r at ) + D n Kerr ( I ′ , r ′ ) = I ′ [ n ( − r ′ / r at ) + n + r ′ + I ′ n ( − r ′ / r at )] and D n Plasma ( w , r ′ ) = − q r ′ / ( e o m e w ) .The nonlinear susceptibility c ( ) for argon is well known from many independent measure-ments, while only few experimental results exist for the higher-order susceptibilities. In [31,32]coincident experimental date on c ( ) for argon which also agrees (up to sign) with a theoret-ical estimation [30] can be found. On the other hand reported data for c ( ) differ by ordersof magnitudes. Correspondingly, neglecting the weak frequency dependence we assume in thefollowing parameters c ( ) = . × − m /V and c ( ) = − . × − m /V . ρ / ρ a t I (TW/cm ) (a) -1x10 -5 -5 -5 -5
0 100 200 300 400 500 ∆ n K e rr ( I , ρ ) I (TW/cm ) (b)0.9920.9940.9960.99811.002 0 100 200 300 400 500 n ( ω , P , I , ρ ) I (TW/cm ) (c) l c o h ( c m ) I (TW/cm ) (d) Fig. 1. Linear and nonlinear optical parameters of argon in the high-intensity regime (nor-mal pressure P = . × m − ); (b)contributions of the nonlinear refractive index, caused by Kerr nonlinearity, when only n = . × − m / W (solid), n and n = − . × − m / W [31] (dashed), n , n and ionized argon n + = . × − m / W [12] (dotted) are taken into account; (c) changeof the total refractive index of pump (black), third (red), fifth (green) and seventh (blue),harmonics; (d) change of coherent length of third (red), fifth (green) and seventh (blue)harmonics.
Figure 1 shows some linear and nonlinear optical parameters of argon, calculated by usingEq. (4) for a 20 fs (FWHM) pump pulse at 800 nm in dependence on the pump intensity.Figure 1(a) shows the normalized plasma density after the pulse as a function of the pumpintensity at the peak of the pulse. As can be seen full ionization at the trailing edge of the pulseoccurs at around 450 TW/cm . Figure 1(b) shows changes of the Kerr-type nonlinear refractiveindex contribution D n Kerr ( I , r ) taking into account (i) only n of neutral argon (solid curve),(ii) n and n of neutral argon (dashed line) and (iii) n , n and n + of neutral and ionizedargon (dotted line). Figure 1(c) shows the change of the refractive indexes of the fundamentalfrequency (black), third (red), fifth (green) and seventh (blue) harmonics. As can be seen therefractive indexes of the fundamental frequency and third harmonic are decreased down to avalue smaller than unity. For higher intensities, the difference between the refractive indexesof the fundamental and the harmonics becomes larger, i.e., the generation of plasma electronsdecreases the phase-matching length. This can be seen in Fig. 1(d) demonstrating the changeof the coherent lengths ( l coh = p / | D k r + | ) for different harmonics in dependence on the pumpintensity. Here the coherent length strongly decreases above 100 TW/cm for all harmonics.Below we show that this behavior appears also in our full numerical calculations of Eqs. (1)-(3).Neglecting bound electron contributions and the dependence of the pump intensity andplasma density on the propagation coordinate an analytical solution for the electric field of theharmonic has been derived in [14]. If we include the bound-electron contributions, the electriceld for the harmonics with order of 2 r + E r + ( z ) = q A P + d r ( A + A ) + d r A sin ( D k r + z / ) / ( D k r + / ) (5)here A p = − F k o / ( p r ( r + ))( w pg / w o ) (cid:2) exp ( − / x ) + r / ( r + ) exp ( − ( r + ) / x ) (cid:3) E o , A = m o c e o c ( ) E o w o / A = m o c e o c ( ) E o w o / A = m o c e o c ( ) E o w o / d i j is Kro-necker’s symbol; w pg = pr at q / m e is the plasma frequency associated with the initial gasdensity; F = √ p ( w a / w o ) x / exp ( − x / ) , x = E a / E o , E o , w o and k o are peak electric field,central angular frequency and wavenumber of pump, respectively; △ k r + is wave mismatchfor ( r + ) th harmonic. The term A describes the contribution of fifth-order susceptibility c ( ) to the generation of third harmonic. We note that the relative phase of the fields generatedby bound-electrons and the plasma current is p /
2. In the following we compare this solutionwith the numerical solutions of the full model as presented in Eqs. (1)-(3). -10 -8 -6 -4 -2
1 3 5 70.8 0.26 0.16 0.11 I ( a r b . u n i t s ) ω/ω ο λ (µ m)(a)00.00350.0070.01050.014 0 0.2 0.4 0.6 0.8 η z (cm)(b) 1 × -5 × -5 × -5 × -5 × -5
0 0.2 0.4 0.6 0.8 η z (cm)(c)10 -10 -8 -6 -4 -2
0 0.2 0.4 0.6 0.8 η z (cm)(d) -6 × -3 × × × -10 0 10 20 30 4002 × -4 × -4 × -4 × -4 J ( A / m ) ρ / ρ a t Time (fs)(e)-6 × -3 × × × -10 0 10 20 30 402 × -4 × -4 × -4 × -4 J ( A / m ) ρ / ρ a t Time (fs)(e)0
Fig. 2. Numerical and analytical calculations for a 20-fs transform limited pump pulsewith a 100 TW/cm peak intensity at 800 nm: (a) spectrum of the output pulse, calculatednumerically with (red) and without (green) taking c ( ) and c ( ) into account; (b) effi-ciency conversion of third harmonic, calculated numerically (red) and analytically (black)for c ( ) = c ( ) =
0, and c ( ) = c ( ) =
3. Numerical results for 800-nm pump
In this chapter we present numerical solutions of Eqs. (1) to (3) using the split-step method withfast Fourier transformation and the fifth-order Runge-Kutta method for 800-nm pump pulseswith a 100 TW/cm peak intensity and 20-fs (FWHM) duration.he spectra in Fig. 2(a) calculated with (red) and without (green) contribution of c ( ) and c ( ) predict that LOHG is dominated by the bound electron contributions with the third andfifth order susceptibilities, since with c ( ) = c ( ) = c ( ) = c ( ) = c ( ) = c ( ) = thebound-electron contribution is much larger than that of the tunnel ionization current. It shouldbe noted that the maximum efficiency of the third harmonic of about 1.4 % appears at 0 . − and 10 − . In Fig. 2(e) the normalized plasma current and the plasma den-sity are presented. Note the steplike nature of the density profile of free electrons (red curve),which explains the source of the harmonic generation due to the tunnel ionization current. -10 -8 -6 -4 -2
1 3 5 70.8 0.26 0.16 0.11 I ( a r b . u n i t s ) ω/ω ο λ ( µ m)(a) 05.0 × -5 × -4 × -4
0 0.01 0.02 0.03 0.04 η z (cm)(b)5.0 × -5 × -4 × -4
0 0.01 0.02 0.03 0.04 η z (cm)(b)-50-2502550-40 -20 0 20 40 E ( G V / m ) Time (fs)(c) -3 × -2 × -1 × × × × -20 0 20 40 0 0.05 0.1 0.15 0.2 0.25 J ( A / m ) ρ / ρ a t Time (fs)(d)-3 × -2 × -1 × × × × -20 0 20 40 0 0.05 0.1 0.15 0.2 0.25 J ( A / m ) ρ / ρ a t Time (fs)(d)
Fig. 3. Numerical calculations for a 20-fs transform limited pump pulse with 400 TW/cm peak intensity at 800 nm: (a) spectrum of the output pulse, calculated with (red) and without(green) taking c ( ) and c ( ) into account; (b) conversion efficiencies of third (red) and fifthharmonics (black); (c) time profile of the pump at the input (blue) and output (red); (d)normalized density of free electron distribution (red) and plasma current (blue). To study the regime where the tunnel-ionization current is dominant, in Fig. 3 results forLOHG are presented for a 20 fs pulse at 800 nm with an peak intensity of 400 TW/cm . Due tothe reduced coherence length the maximum conversion efficiencies are smaller than for the caseof lower pump intensity in Fig. 2. Since the efficiencies are roughly the same independent onthe inclusion of c ( ) and c ( ) in the model, we can conclude that the tunnel-ionization currentis the main LOHG mechanism in this case.Due to the high intensity significant spectral broadening caused by self-phase modulationan be seen. The dependence of the efficiencies on the propagation distance indicates pumpdepletion rather than loss of coherence, since it exhibits no maximum. Here pump depletion,owing to ionization loss, appears mainly in the pulse center, as shown in Fig. 3(c).In order to analyze the roles of the c ( ) , c ( ) and c +( ) nonlinearity and the tunnel-ionizationcurrent in dependence on the applied intensity range, we calculated the contribution to LOHGefficiencies of the two considered nonlinear optical processes in a large range of pump intensi-ties. Figure 4 shows the conversion efficiencies in dependence on the pump intensity up to the7 th harmonic from a 800-nm pump with a 20-fs duration, calculated at the coherent length ofthird harmonic. The contribution of c ( ) and c ( ) dominates up to 300 TW/cm , while after ap-proximately 300 TW/cm the plasma current (red curves) becomes the main source for LOHG.The green curve in Fig. 4(a) shows results, which includes c +( ) of the ionized gas. For highintensities up to 500 TW/cm the efficiency of THG decreases down to the range of 10 − . -9 -6 -3
100 200 300 400 500 η I (TW/cm )(a) × -5 × -5 × -5
100 200 300 400 500 η I (TW/cm )(b) × -7 × -6 × -6
100 200 300 400 500 η I (TW/cm )(c) l c o h ( c m ) l w / o ff ( c m ) I (TW/cm )(d)0.10.20.30.40.50.6 100 200 300 400 500 01020304050 l c o h ( c m ) l w / o ff ( c m ) I (TW/cm )(d)0 0 Fig. 4. Efficiency of LOHG in dependence on the pump intensity calculated for (a) third, (b)fifth and (c) seventh harmonic. Efficiencies were calculated with (blue) and without (red)taking c ( ) and c ( ) into account. Green curve in (a) shows results when c ( ) , c ( ) and c +( ) were taken into account. In (d) the change of the coherent length of third harmonicand the length of its temporal walk-off from the fundamental frequency are shown. It seems to be surprising that even for relatively high intensities from 150 to 300 TW/cm the contribution of the c ( ) and c ( ) process remains in the same order as that of the tunnel-ionization current. This can be explained by the fact, that full ionization only occurs at thetrailing edge of the pulses, while at the leading edge the atoms are not ionized and bound-electron contributions still play a significant role. The length of temporal walk-off between thefundamental and the third harmonic is shown in Fig. 4(d). It is much larger than the coherentlengths, and therefore does not play a significant role during propagation.A general observation arising from the results presented above is that the nonlinear suscep-tibilities c ( ) and c ( ) plays an important role in the formation of LOHG, especially for thethird harmonic. Its contribution is dominant up to a pump intensity of 300 TW/cm for argonat normal pressure. As similar behavior can be expected for other noble gases, although thecorresponding intensities will vary dependent on the properties of the noble gas. The tunnel-ionization current is a main source for LOHG for intensities larger than approximately 300TW/cm , especially for the fifth and seventh harmonics. As noted above, the high-intensityegime of pump can not support highly efficient LOHG because of the the contribution of theionized electrons to the refraction index and the associated increased phase mismatch. Below100 TW/cm , the coherent length is roughly constant, but for larger intensities it shows a sharpdecrease as visible in Fig. 1(d) and Fig. 4(d). This establishes a range around 100 TW/cm asoptimum pump intensity for argon for the generation of the third and fifth harmonic. -10 -8 -6 -4 -2
1 2 30.40 0.20 0.13 I ( a r b . u n i t s ) ω/ω ο λ ( µ m)(a) 05.0 × -4 × -3 × -3
0 0.025 0.05 0.075 0.1 η z (cm)(b)10 -10 -8 -6 -4 -2
1 2 30.40 0.20 0.13 I ( a r b . u n i t s ) ω/ω ο λ ( µ m)(c) 05.0 × -4 × -3 × -3 × -3
0 0.01 0.02 0.03 0.04 0.05 η z (cm)(d) Fig. 5. Results of numerical and analytical calculations for 400 nm pump pulses with 100TW/cm [(a),(b)] and 300 TW/cm [(c), (d)]. In (a), (c) the spectrum of the output pulses,calculated with (red) and without (green) taking c ( ) and c ( ) are presented. In (b), (d)the efficiency of third harmonic, calculated numerically (red) and analytically (black) areshown.
4. Numerical results for 400-nm pump
Nowadays, the generation of pump pulses at 400 nm with high energy by second harmonicgeneration in nonlinear crystals from near-IR ones is a standard method. Using THG with thesepump pulses allows frequency conversion with relative high efficiency into the VUV spectralrange at 133 nm. Figure 5 shows the results for such pump pulses with two different peakintensities for 100 TW/cm (a, b) and for 300 TW/cm (c, d) and the same pulse duration of 20fs. The coherent length for the third harmonic is around a 0 .
05 cm ( . ) cm for 100 (300)TW/cm , calculated by Eq. (4) and visible in Fig. 5(b) and 5(d). The tendency visible from Figs.2 and 3 that for a lower intensity THG is caused by the third and fifth order susceptibilities whilefor higher intensities the tunnel ionization current plays the dominant role, is also observed for400 nm as can be seen by comparison of Fig. 5(a) and 5(c). The maximum THG efficiency ofabout 1 . × − for a pump intensity of 100 TW/cm ( ) [Fig. 4(b)] is in the same range as inthe case of a 800 nm pump pulses compare [Fig. 2(b)], but now a spectral transformation to theVUV range at 133 nm is realized. Higher harmonic orders above third, experience high linearloss of in the vacuum ultraviolet region for argon [33] due to the strong absorption band below106 nm. . Conclusions In conclusion, we numerically studied the generation of low-order harmonics in argon in thehigh-intensity regime, when tunneling ionization takes place. The used numerical method isbased on the unidirectional pulse propagation equation combined with the nonlinear responseby the bound-electrons and a model for tunneling ionization and the associated plasma current.We analyzed LOHG in the regime of pump intensities up to 500 TW/cm arising either from thethird- and fifth-order bound-electron nonlinearity or from the tunnel-ionization current. It wasnumerically observed that up to 300 TW/cm the formation of LOHG is caused mainly by thebound-electron nonlinearity, while for higher intensities the tunnel-ionization current plays thedominant role. It was also shown that a high intensity of the pump does not necessary lead toefficient LOHG, rather, due to the reduced coherence length by the plasma contribution to therefraction index an optimum around 100 TW/cm with efficiencies in the range of 3 × − and10 − for third and fifth harmonic generation, respectively, is predicted. Further on, we studiedTHG by intense pump pulses at 400 nm and predicted frequency transformation to the spectralrange of 133 nm with maximum efficiency of about 1 . × − . Acknowledgments
We acknowledge financial support by the German Research Foundation (DFG), project No. He2083 / −−