Higher-order Kerr improve quantitative modeling of laser filamentation

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Reference

Higher-order Kerr improve quantitative modeling of laser filamentation

PETRARCA, Massimo, et al.

Abstract

We test numerical filamentation models against experimental data about the peak intensity and electron density in laser filaments. We show that the consideration of the higher-order Kerr effect improves the quantitative agreement without the need of adjustable parameters

PETRARCA, Massimo, et al . Higher-order Kerr improve quantitative modeling of laser filamentation. Optics Letters , 2012, vol. 37, no. 20, p. 4347

DOI : 10.1364/OL.37.004347 PMID : 23073458

Available at:

http://archive-ouverte.unige.ch/unige:37084

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Higher-order Kerr improve quantitative modeling of laser filamentation

M. Petrarca,¹Y. Petit,¹S. Henin,¹R. Delagrange,¹P. Béjot,^1,2and J. Kasparian^1,*

1GAP-Biophotonics, Université de Genève, Chemin de Pinchat 22, Geneva 4, Geneva 1211, Switzerland

2Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, BP 47870, Dijon Cedex F-21078, France

*Corresponding author: [email protected] Received July 30, 2012; accepted September 11, 2012;

posted September 13, 2012 (Doc. ID 173436); published October 15, 2012

We test numerical filamentation models against experimental data about the peak intensity and electron density in laser filaments. We show that the consideration of the higher-order Kerr effect improves the quantitative agreement without the need of adjustable parameters. © 2012 Optical Society of America

OCIS codes: 320.2250, 190.3270, 320.7110, 190.5940.

Laser filamentation [1–4] is a propagation regime typical of high-power lasers. It stems from a dynamic balance between self-focusing and self-defocusing nonlinearities of different orders, i.e., different intensity dependences.

Self-focusing by the Kerr effect is well established.

Conversely, the relative contributions to defocusing are still discussed. The two main effects considered in gases are the laser-generated plasma and the saturation of the medium polarizability under strong-field illumina- tion [5–7], empirically described as a power series of the intensity and referred to as higher-order Kerr effect (HOKE). The latter has first been introduced as freely adjustable parameters [8,9], before experimental values were reported [10–12].

Strong efforts have been dedicated to reach a quanti- tative agreement between numerical models of nonlinear pulse propagation and experimental data about filamen- tation [13–16]. In that purpose, a common approach consists of tweaking simulation inputs like the nonlinear index or the ionization rates, and/or the experimental parameters, such as the beam energy, pulse duration, beam diameter or profile, until a satisfactory match with the experiments is achieved. The remaining discrepan- cies are generally attributed, among others, to the diffi- culty of the experimental measurements inside the filaments, and the associated uncertainties. However, the introduction of new techniques, e.g., for measuring the peak intensity from the ratio of nitrogen fluorescence lines [17,18], enhances the precision and renews the challenge for theoretical models to precisely match those data.

Here, we experimentally measure the electron density over a wide range of pulse durations and show that con- sidering the HOKE in numerical simulations [19] allows a good agreement without any parameter adjustment.

Comparing the models with previously published data about the filament peak intensity [18] and electron den- sity [20] confirm this finding.

The filament ionization was characterized as a function of the pulse duration by using 4 mJ laser pulses centered at 800 nm, with 100 Hz repetition rate and 3 cm initial beam diameter. The slightly diverging beam was focused by anf 2.8m lens to form a 20 cm long filament in air.

The pulse duration was varied from 80 fs to 1.6 ps by detuning the grating compressor of the chirped pulse

amplification chain. The charge was measured by propa- gating the filaments between a pair of parallel electrodes (1×1cm; 1 cm spacing) under 1 kV bias: The transient current through the circuit is proportional to the electron density generated by the filaments [21], although with an arbitrary calibration factor (Fig. 1). This charge measurement was double-checked by sonometric measurements [22].

To get information about the role of the HOKE, we per- formed numerical simulations considering a linearly polarized incident electric field at a wavelength λ₀ 800nm with cylindrical symmetry around the propaga- tion axisz. According to the unidirectional propagation pulse equation [23], the scalar envelopeεr; t; z(defined such that jεr; z; tj²Ir; z; t, I being the intensity) evolves in the frame traveling at the pulse velocity according to [24]

∂_zε~i qk²ω−k²_⊥

−k⁰ω

~ ε 1

k²ω−k²_⊥ q

iω²

c² P~NL− ω 2ϵ0c²J~

−α;~ (1)

wherecis the velocity of light in vacuum,ωis the angular frequency, kω nωωc, k⁰ its derivative at ω0 2πc∕λ0, nω is the linear refractive index at the fre- quency ω, k_⊥ is the spatial angular frequency. PNL is the nonlinear polarization, J is the free-charge induced current, andαis the nonlinear losses induced by photo- ionization.f~denotes simultaneous temporal Fourier and spatial Hankel transforms of function f. The nonlinear

Compressor

CPA chain

z= 0 f = 2.8 m

Photodiode (synchronization) 4 mJ, 790 nm

80 fs – 1.6 ps

Filaments +1 kV

Electrodes

Fig. 1. (Color online) Experimental setup.

October 15, 2012 / Vol. 37, No. 20 / OPTICS LETTERS 4347

0146-9592/12/204347-03$15.00/0 © 2012 Optical Society of America

polarization is evaluated as PNLP

jn_2jjεj^2j ΔnRaman

ε, where the n2j are the jth-order nonlinear refractive indices (n2 1.2×10⁻⁷ cm²∕TW, n4

−1.5×10⁻⁹ cm⁴∕TW², n₆2.1×10⁻¹⁰ cm⁶∕TW³, n₈

−8×10⁻¹² cm⁸∕TW⁴ [10]), and ΔnRaman is the Raman- induced refractive index change evaluated by solving the rotational time-dependent Schrödinger equation of N₂ and O₂ in the weak field regime. Alternatively, the model was truncated to the third-order nonlinearity (i.e., the n2jεj²ε term) to disregard the contribution of the HOKE. The current is evaluated as J~ _m^e2_e ν_eiωρεν~ _e²ω², where e and m_e are the electron charge and mass, respectively,ε0 is the vacuum permit- tivity,νeis the effective electronic collisional frequency, and ρ is the electron density. Finally, αP

iO2;N2

Wijεj²Uiρat;i∕2jεj², ρat;i is the density of molecules of speciesi,Wijεj²is their photoionization probability modeled by the Perelomov, Popov, Terent’ev (PPT) formulation, with ionization potentialU_i.

The propagation dynamics of the electric field is coupled with the electron density ρ, calculated as [2]

∂_tρ X

iO2;N2

W_ijεj²ρat;iσi

Uiρjεj²

−βρ²; (2)

whereβis the electron recombination rate and theσiare the inverse Bremsstrahlung cross sections with speciesi, including avalanche ionization. Note that our model re- lies only on published values [4,10], without adjustable parameter. Simulations of experimental situation relied on the published conditions, without any tweaking.

The measured electron density rose very slowly (less than a factor of 4) when the pulse duration decreased from 1.6 ps to 80 fs, although the peak power increased by a factor of 20 (Fig. 2). This relative stability might seem unexpected since the ionization rate scales with I⁸in oxygen [25,26]. But the free electrons need several picoseconds to tens of picoseconds to recombine: They accumulate more efficiently during longer pulses, partly balancing the lower ionization rates. While both models reproduce this slow variation for pulses below a few hun- dreds of femtoseconds, the full model seems slightly more accurate for shorter pulses.

To better discriminate between the two models, we tested them against previously published quantitative experimental data about the peak intensity [18] and elec- tron density [20] measured in filaments by different groups and techniques, so as to avoid individual artifacts or systematic errors to affect our conclusions.

The peak intensity of filaments generated by focused 42 fs pulses of 1 cm diameter at1∕e²(i.e., 5.9 mm FWHM) measured via the ratio of nitrogen fluorescence lines [18]

is much better reproduced by the full model than by the truncated one (Fig. 3). The latter, in which the HOKE defocusing terms are disregarded, overestimates the intensity by a factor of typically 2–3.

The electron density as measured in [20] provides a much more sensitive variable, because its high-order dependence on intensity magnifies the experiment dy- namics, allowing more reliable comparisons. The inten- sity of the nitrogen fluorescence constitutes a classical measurement of the electron density [27]. Consistent with the differences in the predicted intensities, the full model always yields a 4- to 30-times lower electron den- sity than the truncated one. The match with experimental data [20] is perfect for a parallel beam corresponding to typical filamentation conditions [Fig. 4(a)]. However [Figs.4(b)–4(d)], even the full model tends to overesti- mate the electron density produced by focused beams.

This discrepancy could be due to the recently suggested overestimation [7] of the free electron density by the PPT model [4,25], or to optical aberrations and astigmatism in experiments, which are not included in the model: They tend to decrease the peak intensity in the waist region

0 1 2 3 4 5 6

0 500 1000 1500 2000

Electron density (arb. units)

Pulse duration (fs)

Full model Truncated model Experimental

Fig. 2. (Color online) Effect of pulse duration on the electron density in filaments. Each numerical curve is normalized independently to optimally match the experimental results.

0 40 80 120

0 1 2 3 Intensity (TW/cm2)

Energy (mJ)

Truncated model Full model

(a)f = 100 cm

Experimental [17]

0 40 80 120 160

0 1 2 3 Intensity (TW/cm2)

Energy (mJ)

(b)f = 50 cm

Fig. 3. (Color online) Experimental [18] and simulated peak intensity in filaments generated by 42 fs pulses of 1 cm diameter.

10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷

0 50 100 150

Peak electron density (cm-3)

Peak power (GW)

Experiment [27]

Full model Truncated model (a)f =

0 10 20 30 40 50 Peak power (GW)

(b)f = 380 cm

10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸

0 10 20 30 40 50 Peak electron density (cm-3)

Peak power (GW)

(c)f = 100 cm

0 10 20 30 40 Peak power (GW)

(d)f = 50 cm

Fig. 4. (Color online) Experimental [20] and simulated elec- tron density generated by a 45 fs pulse of initial beam diameter of 8.4 mm (full width at1∕e²).

4348 OPTICS LETTERS / Vol. 37, No. 20 / October 15, 2012

[28] and reduce filament length and strength [29].

Besides, this increasing geometrical constraint explains the decrease of the difference between the two models for tighter focusing. Therefore, the less focused condi- tions are more relevant to test filamentation models.

Our results show that considering the saturation of the medium polarization in the strong-field regime (aka HOKE) improves the quantitative modeling of filamenta- tion, especially for collimated beams for which the geometrical constraint does not interfere with the self-guiding process. Such update of the standard laser filamentation model is easy to implement, based on both experimental measurement [10] and quantum mechani- cal justifications [5–7]. It will in particular impact the determination of the optimal conditions for the potential atmospheric applications [30] of laser filamentation, like rainmaking [31], or lightning control [32].

This work was supported by the European Research Council Advanced Grant “Filatmo” and the Conseil Régional de Bourgogne (FABER program). We thank F. Théberge for his assistance in accessing his experi- mental data reproduced in Fig.4[20].

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