Filamentation in air with ultrashort mid-infrared pulses

We theoretically investigate filamentation of ultrashort laser pulses in air in the mid-infrared regime under conditions in which the group-velocity dispersion (GVD) is anomalous. When a high-power, ultrashort mid-infrared laser beam centered at 3.1-μm forms a filament, a spatial solitary wave is stabilized by the plasma formation and propagates several times its diffraction length. Compared with temporal self-compression in gases due to plasma formation and pulse splitting in the normal-GVD regime, the minimum achievable pulse duration (∼ 70 fs) is limited by the bandwidth of the anomalous-GVD region in air. For the relatively high powers, multiple pulse splitting due to the plasma effect and shock formation is observed, which is similar to that which occurs in solids. Our simulations show that the energy reservoir also plays a critical role for longer propagation of the air filament in the anomalous-GVD regime. © 2011 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (320.7110) Ultrafast nonlinear optics. References and links 1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou,“Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20, 73–75 (1995), http://www.opticsinfobase.org/ol/abstract. cfm?URI=ol-20-1-73. 2. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. 21, 62–65 (1996), http://www. opticsinfobase.org/ol/abstract.cfm?URI=ol-21-1-62. 3. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22, 304–306 (1997), http://www.opticsinfobase. org/ol/abstract.cfm?URI=ol-22-5-304. 4. J. R. Peñano, P. Sprangle, B. Hafizi, A. Ting, D. F. Gordon, and C. A. Kapetanakos, “Propagation of ultra-short, intense laser pulses in air,” Phys. Plasmas 11, 2865 (2004). 5. A. Ting, I. Alexeev, D. Gordon, R. Fischer, D. Kaganovich, T. Jones, E. Briscoe, J. Peñano, R. Hubbard, and P. Sprangle, “Measurements of intense femtosecond laser pulse propagation in air,” Phys. Plasmas 12, 056705 (2005). 6. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in opticalmedia: physics, applications, and new challenges,” Can. J. Phys. 83, 863–905 (2005). 7. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). 8. L. Bergé, S Skupin, R Nuter, J Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70, 1633 (2007). 9. J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16, 466–493 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI= oe-16-1-466. #141064 $15.00 USD Received 12 Jan 2011; revised 4 Apr 2011; accepted 16 Apr 2011; published 26 Apr 2011 (C) 2011 OSA 9 May 2011 / Vol. 19, No. 10 / OPTICS EXPRESS 9118 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 04 APR 2011 2. REPORT TYPE 3. DATES COVERED 00-00-2011 to 00-00-2011 4. TITLE AND SUBTITLE Filamentation in air with ultrashort mid-infrared pulses 5a. CONTRACT NUMBER

In this Letter, we present the first simulation results for air filamentation and spatial solitarywave formation in the anomalous-GVD regime of air.When a high-power (> 100-GW), ultrashort pulse undergoes self-focusing due to the Kerr nonlinearity, multi-photon absorption (MPA) and plasma formation halt beam collapse.As a result, a spatial solitary wave is formed and stabilized during the filamentation process, and its shape can be maintained for several diffraction lengths.Although spectral broadening induced by phase modulation occurs, the relatively narrow bandwidth of the anomalous-GVD regime (approximately 200-nm) near 3-μm inhibits formation of a temporal solitary wave, which contrasts to the generation of few-cycle optical pulses predicted for solids in the broadband anomalous-GVD region [23,24] and to pulse self-compression down to few-cycles which occurs via plasma formation and/or pulse splitting in gases for the normal-GVD regime [31,32,[39][40][41][42][43][44].

Simulation and refractive index of air
In our simulations, we use the radially-symmetric nonlinear envelope equation (NEE) in normalized units including diffraction, dispersion, self-focusing with the delayed Raman response, MPA, and plasma de-focusing and absorption, which is given as [23,43,[45][46][47], where ψ is the field normalized by the peak input field amplitude A 0 , ζ = z/L d f is the propagation distance normalized by the diffraction length L d f = n 0 πw 2 0 /λ 0 , n 0 is the refractive index of air, w 0 is the 1/e 2 spot size radius, λ 0 is the central wavelength, ∇ 2 ⊥ is the transverse Laplacian, τ is the retarded time normalized by the 1/e 2 input pulse duration τ p , β n is the n th -order dispersion parameter [48], L nl = c/(ωn 2 I 0 ) is the nonlinear length, n 2 is the nonlinear refractive index, I 0 = cn 0 |A 0 | 2 /2π is the peak input intensity, τ k = 70 fs is the Raman relaxation time, ) is the m-photon absorption length, β (m=31) = 3×10 −384 cm 59 /W 30 is the 31-photon absorption coefficient [7], L pl = 2/(σρ 0 ωτ c ) is the plasma length, σ is the inverse bremsstrahlung cross section, ρ 0 = β (m) I (m−1) 0 τ p /(mhω) is the total electron density that would where α = σ I 0 τ p /(n 2 0 E g ) is the avalanche ionization coefficient, and E g = 12.1 eV the band-gap energy for oxygen.The dispersion parameters at 3.1-μm are calculated using the Taylor expansion formula, which is a function of wavelength λ , temperature T , pressure p, and humidity h [49].Figure 1 shows the calculated GVD for different values of humidity at T = 17.5 • C and p = 101,325 Pa (standard atmospheric pressure).As humidity and temperature (not shown) increase, the absolute magnitude of the GVD and the wavelength range of anomalous GVD decrease slightly, and the peak of the GVD shifts toward longer wavelengths.We assume 10 % humidity (h = 10) for our calculations such that β 2 = -0.53fs 2 /cm, β 3 = 3.02 fs 3 /cm, and higher-order dispersion parameters (n ≥ 4) are all positive.The anomalous-GVD region near 3.1-μm which spans 200-nm is related to the water vapor absorption, and the fitting coefficients used for index calculation are valid between 2.8-μm and 4.2-μm.There exist strong resonance absorption regions between 2.5 -2.8-μm and 4.2 -4.4-μm due to the presence of water vapor and carbon dioxide (CO 2 ) [49-51].The calculated critical power P cr = αλ 2 /(4πn 0 n 2 ), where α = 1.8962 for the input Gaussian beam profile [52], is equal to 66-GW [53].We limit the peak power of the input pulse P ≤ 8P cr to avoid multi-filamentation. with L ds = τ 2 p /β 2 = 306-m.As the peak intensity increases due to self-focusing and anomalous GVD, a low-density plasma is created as shown in Fig. 2(b).At that point, plasma absorption and de-focusing combined with MPA arrest beam collapse so that an air filament with I = 5 × 10 12 W/cm 2 forms and propagates stably about 0.03, 0.05 and 0.06 times the diffraction length of the input beam for P/P cr = 2, 3 and 4. For increasing powers, collapse occurs at shorter distances, and the filament length is extended.According to the calculated beam diameter (FWHM) [Figs.2(c)], the filament maintains its diameter (1.4-mm FWHM), which is 1/10 that of the initial beam and thus a spatial solitary wave is generated during filamentation, propagating for at least 3 times of the diffraction length based on its minimum spot size.As is shown in Fig. 2(d), although the pulse duration initially decreases due to anomalous GVD, it suddenly increases near the peak intensity due to spectral broadening into the normal GVD regime via self-phase modulation and slowly decreases again since the field components at wavelengths in the anomalous GVD regime undergo compression as the pulse propagates.Therefore, compared with calculated few-cycle spatio-temporal solitary waves in the anomalous-GVD regime for solids [23,24], a solitary wave is not generated near 3.1-μm due to the relatively narrow bandwidth of the anomalous-GVD region.

Conclusion
In conclusion, we investigate air filamentation for relatively large diameters in the anomalous-GVD regime centered at 3.1-μm.The mm-sized filament can propagate several times its diffraction length, and the propagation distance increases with the higher laser input power.However, the potential formation of a spatio-temporal solitary wave is inhibited by the narrow bandwidth of the anomalous-GVD regime.Two other wavelength regions below 10-μm with the anomalous-GVD and weak absorption include two 100-nm bandwidth regions centered at 4.7 μm related to CO 2 absorption and at 9.5 μm related to O 3 absorption [49][50][51].Since the high-power, ultrashort mid-infrared laser technology has rapidly progressed in recent years, we expect that the necessary power (> 100-GW) for experimental studies should be available soon [36][37][38].

Fig. 1 .
Fig. 1.Calculated group-velocity dispersion (GVD) for different values of the humidity using the Taylor expansion formula based on Ref. [49] at T = 17.5 • C, p = 101325 Pa (standard atmospheric pressure).

Figure 2 (Fig. 2 .
Figure 2(a) shows a plot of the peak intensity as a function of normalized distance for different input powers.Here we assume the collimated, initial spot size (1/e 2 radius) is 12-mm and the initial pulse duration (FWHM) is 150 fs such that L d f = 146-m approximately matches
be produced by the input laser pulse via multi-photon ionization, τ c is the electron-ion collision time, η = ρ e /ρ 0 is the normalized electron density.The operator (1 + i∂ /ωτ p ∂ τ) accounts for space-time focusing in the diffraction term and self-steepening in the self-focusing term.The plasma is generated by multi-photon ionization and avalanche ionization, and the electron density satisfies the equation, #141064 -$15.00USD Received 12 Jan 2011; revised 4 Apr 2011; accepted 16 Apr 2011; published 26 Apr 2011 (C) 2011 OSA 9 May 2011 / Vol. 19, No. 10 / OPTICS EXPRESS 9121