Carsten Bree et al- Self-pinching of pulsed laser beams during filamentary propagation

Self-pinching of pulsed laser beamsduring filamentary propagation

Carsten Bree1,2, Ayhan Demircan1, Stefan Skupin3,4, Luc Berge5, andGunter Steinmeyer2

1 Weierstraß-Institut fur Angewandte Analysis und Stochastik (WIAS), Mohrenstraße 39,10117 Berlin, Germany

2 Max-Born-Institut (MBI), Max-Born-Straße 2a, 12489 Berlin, Germany3 Max-Planck-Institut fur Physik komplexer Systeme, 01187 Dresden, Germany

4 Institut fur Festkorpertheorie und -optik, Friedrich-Schiller-Universitat, 07743 Jena,Germany

5 CEA-DAM, DIF, F-91297 Arpajon, France

[email protected]

Abstract: Competing nonlinear optical effects that act on femtosecondlaser pulses propagating in a self-generated light filament may give rise to apronounced radial beam deformation, similar to the z-pinch contraction ofpulsed high-current discharges. This self-generated spatial beam contractionis accompanied by a pulse break-up that can be beneficially exploited foron-axis temporal compression of the pulse. The pinching mechanism there-fore explains the recently observed self-compression and the complicatedspatio-temporal shapes typical for filament propagation experiments.

© 2009 Optical Society of AmericaOCIS codes: 320.5520 (Pulse compression); 320.7110 (Ultrafast nonlinear optics)

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#113266 - $15.00 USD Received 25 Jun 2009; revised 5 Aug 2009; accepted 6 Aug 2009; published 31 Aug 2009

(C) 2009 OSA 14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16429

13. L. Berge and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001).14. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,”

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tosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006).21. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagat-

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U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15,5394–5405 (2007).

Magneto-hydrodynamics (MHD) provides effective mechanisms for increasing the electrondensity within high-current pulsed discharges. In the plasma channel the self-generated mag-netic field may act to radially focus the electron fluence to near-thermonuclear current densi-ties, with the z-pinch [1, 2] being one of the most prominent examples. In contrast, laser pulsecompression [3–5] has traditionally pursued energy concentration along the longitudinal axisrather than radial contraction. In the following we show that inside a light filament, the com-bination of only three effects, namely diffraction, Kerr self-focusing, and plasma-induced self-defocusing, holds for a radial contraction mechanism acting on the photon fluence. In analogyto the z-pinching in MHD, we call this mechanism self-pinching. This phenomenon gives riseto spatio-temporally inhomogeneous configurations of the optical field, implying strong tem-poral variations of the beam radius [6, 7]. In contrast to previous explanations (see, e.g., [7–9])of the self-compression in filaments that indicated a complex interplay of some ten effects, weshow that only the above-mentioned three spatial effects suffice for self-compression.

Propagation of short laser pulses in a filament involves numerous linear and nonlinear opti-cal processes that are typically modeled in the framework of a Nonlinear Schrodinger Equation(NLSE). It is quite remarkable that MHD bears a very similar NLSE for the magnetic field,which may give rise to ionospheric filaments and mechanisms analogous to self-pinching [10].As all these scenarios exhibit a complex interplay of linear and nonlinear processes it is gener-ally difficult to isolate the primary processes leading to the observed phenomena. For the opticalcase, however, one can compute characteristic lengths of the participating processes [12] to sortout group-velocity dispersion, absorption, Kerr-type self-phase modulation and self-steepening,leaving mainly plasma effects and transverse self-focusing and -defocusing as suspected driversbehind the experimentally observed self-compression. Such analysis, in particular neglection ofdispersion, is indicative of vanishing energy flow along the temporal axis of a pulse in the fila-ment. This essentially leaves particle densities and respective fluences as key parameters, simi-lar to the situation in MHD. Let us therefore restrict ourselves to analyzing radial energy flow,for which we use an extended NLSE in cylindrical coordinates (r, t) [13]. This extended NLSEeffectively couples the photon density to the electron density ρ . Compared to the full model



equations [7], we neglect energy flow along the t-axis and dissipation. These effects have beenproven unimportant in gaseous media at low or atmospheric pressure both, in previous theo-retical and experimental studies [6, 7]. Consequently, we employ a reduced model consideringonly Kerr-type self-focusing and plasma defocusing as main dynamic effects during filamentformation in gases:

∂zE =i

2k0r∂rr∂rE +

iω0

cn2|E |2E − iω0

2n0cρcρ(I)E , (1)

ρ(I) = ρnt

(1− exp

(−t∫

−∞

dt ′W [I(t ′)]))

. (2)

Here, z is the propagation variable, t the retarded time, and ω0 is the central laser frequency atλ0 = 2πn0/k0 = 800 nm. n2 is the nonlinear refraction index. Photon densities are described viathe complex optical field envelope E , with I = |E |2. The wavelength-dependent critical plasmadensity is calculated from the Drude model according to ρc ≡ ω2

0 meε0/q2e , where qe and me are

electron charge and mass, respectively, ε0 is the dielectric constant, c the speed of light, andρnt denotes the neutral density. Plasma generation is driven by the ionization rate W [I], whichis suitably described by Perelomov-Popov-Terent’ev (PPT) theory [14]. For our investigations,we use data for argon [7] at atmospheric pressure.

In the following, we search for a field configuration that represents a stationary state inregimes where a Kerr-induced optical collapse is saturated by plasma defocusing. The cor-responding temporal intensity profiles that maintain a balance between competing nonlineareffects in every temporal point are derived from a time-dependent variational approach, withthe following trial function

E =

√P

πR2 exp

[− r2

2R2 + ik0r2∂zR

2R

]. (3)

The quadratic phase guarantees preservation of continuity equations through self-similar sub-stitutions, and the pulse radius R ≡ R(z, t) depends on both the longitudinal and temporal vari-

ables. For conservative systems preserving the power P(t) ≡ 2π∞∫0

I(t)rdr along z, straightfor-

ward algebra provides the virial-type identity [15]

∂ 2z

∞∫0

r3|E |2dr =2

k20

∞∫0

r|∂rE |2dr

−2n2

n0

∞∫0

r|E |4dr− 1

n20ρc

∞∫0

|E |2r2∂rρdr. (4)

Inserting the trial function (3) with R(z, t) = w(z, t)/√

2 being related to the Gaussian spotsize w(z, t), one obtains a dynamical equation governing the evolution of the pulse radius Ralong z [15]. For the derivation of analytical expressions for the plasma term on the r.h.s. ofEq. (4), we approximate the PPT ionization rate by a power law dependence W [I] = σN∗ IN∗

,with parameters N∗ = 6.13 and σN∗ = 1.94×10−74s−1cm2N∗

W−N∗fitted to the PPT rate for the

intensity range of 80TW/cm2. Using I(t) = P(t)/πR2(t), we impose a Gaussian power profileP(t) = Pin exp(−2t2/t2

p ) with duration tp and peak input power Pin as a boundary condition,which results in the following integral equation for steady state solutions,

0 = 1− P(t)Pcr

+ μP2(t)t∫

−∞

dt ′IN∗+1(t ′)

P(t ′)1

(I(t)+N∗I(t ′) P(t)P(t ′) )

2, (5)



Delay (fs)0

-20-40

100%

50%

0Inte

nsity

(b)

0

100

Radi

us (µ

m)

200

300

−45 −30 −15 0 15 300

10

20

30

40

50

60

70

Delay (fs)

Inte

nsity

(TW

/cm

2)

(a)

−t*

Fig. 1. (a) Spectrum of solutions I(t) of Eq. (5). (b) Spatio-temporal representation of thesolution of Eq. (5) marked in red [see (a)], obtained by rotating the line segment generatedby R(t) =

√P(t)/πI(t) around the t−axis. Color corresponds to on-axis intensity.

with Pcr = λ 20 /(2πn0n2) and μ = k2

0N∗σN∗ρnt/πρc.Equation (5) is basically a generalization of a Volterra-Urysohn integral equation [16], with

a kernel depending not only on I(t′) but also on I(t). Using additional simplifying assump-tions, steady-state solutions with soliton-like qualities have been previously discussed [13].Here we solve Eq. (5) without the approximations made in Ref. [13]. Taking into account thatthe integral term of Eq. (5) is strictly positive, it immediately follows that nontrivial solutions

only exist on the temporal interval −t∗ < t < t∗ where P(t) > Pcr, with t∗ = (ln√

Pin/Pcr)1/2

tp.From a physical point of view, Kerr self-focusing can compensate for diffraction only on thisinterval, enabling the existence of a stationary state. For computing a stationary solution I(t)of the integral equation, we use the method presented in [17] which combines a Chebyshevapproximation of the unknown I(t) with a Clenshaw-Curtis quadrature formula [18] for theintegral term. As the laser beam parameters, we choose a ratio Pin/Pcr = 2 and a pulse dura-tion tFWHM =

√2ln2tp ≈ 100 fs, leading to t∗ ≈ 50 fs. The spectrum of solutions thus obtained

is depicted in Fig. 1(a). As 1−P(t)/Pcr vanishes at the boundaries, there exists a continuumof multiple roots. All solutions show a strongly asymmetric temporal shape, with an intenseleading subpulse localized at t = −t∗ [13] and a minimum [dashed line in Fig. 1(a)] localizednear zero delay, followed by a region of rapid intensity increase, suggesting singular behav-ior of the solutions. Filamentation is known to proceed from a dynamical balance betweenthe Kerr and plasma responses, and a steady-state solution cannot strictly be reached by thephysical system. Nevertheless, Eq. (5) provides deep insight into the configuration that thepulse profile tends to achieve in the filamentary regime. The structure of the emerging solutions[Fig. 1(b)] indeed indicates the formation of two areas of high on-axis intensity being separatedby an approximately 20 fs wide defocused zone of strongly reduced intensity. While similardouble-peaked on-axis intensities have already been observed in numerical simulations and ex-periments [6, 21–24] many authors considered a parasitic dispersive break-up in bulk media oroptical fibers. Despite its superficial similarity, however, such a break-up cannot be exploitedfor the compression of isolated femtosecond pulses as will be done below. Interestingly, weobserve a comparable dynamical behavior as reported for condensed media, where temporalbreak-up around zero-delay and the subsequent emergence of nonlinear X-waves occurs. TheseX-waves were recently proposed to constitute attractors of the filament dynamics [19, 20].

For a deeper substantiation of our analytical model, we perform direct numerical simulationsusing the reduced radially symmetric evolution model Eq. (1) for the envelope of the opticalfield. The incident field is modeled as a Gaussian in space and time with w0 := w(z = 0, t) =2.5mm and identical peak input power and pulse duration as used for the solutions of Eq. (5).The field is focused into the medium with an f = 1.5m lens. The result of these simulations can



Fig. 2. (a) Evolution of the on-axis temporal intensity profile along z for the reduced numer-ical model governed by Eq. (1). (b) Same for the simulation of the full model equations [7].

be considered as prototypical for the pulse shaping effect inside filaments. These simulationsalso demonstrate that spatial effects alone already suffice for filamentary self-compression. Asthe evolution of the on-axis temporal intensity profile in Fig. 2(a) reveals, filamentary com-pression always undergoes two distinct phases. Initially, while z approaches the nonlinear focus(z = 1.4−1.5 m), a dominant leading peak is observed. When the trailing part of the pulse refo-cuses in the efficiently ionized zone (ρmax ≈ 5×1016cm−3) a double-spiked structure emerges.This transient double pulse structure confirms the pulse break-up predicted from the analysis ofEq. (5), see Figs. 3(a) and (c), and is compatible with the stationary shapes detailed in Fig. 1(a).Subsequently, only one of the emerging peaks survives and experiences further pulse shapingin the filamentary channel.

At z = 1.7 m the leading subpulse has already been reduced to a fraction of its original on-axis intensity. This effective attenuation of the leading pulse isolates the trailing pulse that nowexhibits a duration tFWHM = 27 fs. The combined split-up and isolation during the first phasetherefore already provides an about fourfold compression of the 100 fs input pulses. In thesubsequent weakly ionized zone of the filament (z > 1.6m), the surviving, trailing subpulse isthen subject to additional temporal compression. At z = 2.5 m our simulations indicate pulsesas short as tFWHM = 13 fs [Fig. 3(b)], which agrees favorably with the experimental resultsin Ref [6]. In contrast to the plasma-mediated self-compression in the strongly ionized zone,compression in the second zone is solely driven by the Kerr nonlinearity (ρ < 1013cm−3).With time slices of higher optical powers being able to compensate diffraction by Kerr self-focusing, these portions of the pulse diffract less rapidly than time slices with less optical power.Compared to a linear optical diffraction-ruled scenario, the nonlinear optical effects thereforelead to the formation of a characteristic pinch. This emerging spatial structure is depicted inFig. 3(d). This clearly sets the self-pinching mechanism apart from numerous previous reportson seemingly similar dynamics.

For an analytic description of temporal compression during further filamentary propaga-tion, we use the dynamical equation for the time dependent beam radius derived from Eqs. (3)and (4) [15], yet neglecting the plasma term. With the initial conditions R(z = z0, t) = R0

and ∂zR(z = z0, t) ≡ 0 the resulting problem is analytically solvable, and we find R(z, t) =

R0

√1+[(z− z0)/k0R2

0]2(1−P(t)/Pcr). This equation models the evolution of the plasma-

free filamentary channel from z > 1.6 m, assuming P(t) ≤ Pcr. The profile P(t) representsthe power contained in the filament core region only. For simplicity, we assume here P(t) =Pcr exp(−2t2/t2

p), R0 = 100 μm and tp = 23 fs. This corresponds to the duration of the pulseat z0 = 1.7 m shown in Fig. 3(a). Resulting characteristic spatio-temporal shapes are shown in



0

20

40

60

80

I(r=

0) (T

W/c

m2

)

0

1

2

3

4

5

Delay (fs)−100 0 100

r (mm

)

−0.5

0

0.5

Delay (fs)−100 0 100

(b)(a)

(c) (d)

Delay (fs)

−100

0

100

0

100%

50%

On-axis intensity

Fig. 3. (a) Pulse sequence illustrating the two-stage self-compression mechanism. Shownare the on-axis intensity profiles for z = 1.5 m (solid line), z = 1.55 m (dashed line) and z =1.7 m (dashed-dotted line). (b) Self-compressed few-cycle pulse at z = 2.5 m. (c) Spatio-temporal characteristics of the double-spiked structure at z = 1.55 m. (d) Same for the few-cycle pulse at z = 2.5 m.

Fig. 4, clearly revealing the presence of self-pinching in this Kerr-dominated stage of propaga-tion and the dominant role it plays for on-axis temporal compression.

So far our analysis has completely neglected dispersion, self-steepening and losses. To con-vince ourselves that dissipation and temporal coupling between time slices have only a mod-ifying effect on the discussed self-compression scenario, we pursued full simulations of thefilament propagation, including few-cycle corrections and space-time focusing [7]. As shownin Fig. 2(b), minor parameter adjustment, setting w0 = 3.5 mm and leaving all other laser beamparameters the same value, suffices to see pulse self-compression within the full model equa-tions. Now self-steepening provides a much more effective compression mechanism in the trail-ing part. However, the comparison of Fig. 2(a) with (b) also reveals that the dynamical behaviorchanges only slightly upon inclusion of temporal effects. Clearly, the same two-stage compres-sion mechanism is observed as in the reduced model. We therefore conclude that the pulsebreak-up dynamics in the efficiently ionized zone is already inherent to the reduced dynamicalsystem governed by Eq. (1). Rather than relying on the interplay of self-phase modulation anddispersion as in traditional laser pulse compression, filament self-compression is essentially aspatial effect, conveyed by the interplay of Kerr self-focusing and plasma self-defocusing. Thisdominance of spatial effects favorably agrees with the spatial replenishment model of Mlejneket al. [21]. However, our model indicates previously undiscussed consequences on the tempo-ral pulse structure on axis of the filament, leading to the emergence of the pinch-like structure[Fig. 3(d)] that restricts effective self-compression to the spatial center of the filament [7, 25].Our analysis confirms the existence of a leading subpulse, in the wake of which the short self-compressed pulse is actually shaped during the first stage of filamentary propagation. Thisleading structure gives rise to a pronounced temporal asymmetry of self-compressed pulses,which is confirmed in experiments [7].

While qualitatively similar break-up processes have frequently been observed in laser fila-



(a)

(c) (d)

(b)

-250

25Delay (fs)

00.2

0.4

r(m

m)

0

50%

100%

On-axis intensity

Fig. 4. Sequence of pulses illustrating temporal self-compression due to Kerr-induced spa-tial self-pinching in the variational model corresponding to (a) z = 1.7 m, (b) z = 1.9 m, (c)z = 2.1 m and (d) z = 2.3 m.

ments, our analysis identifies the spatially induced temporal break-up as a first step for efficienton-axis compression of an isolated pulse. In our case, the leading break-up portion is eventu-ally observed to diffract out and to reduce its intensity, while the trailing pulse can maintain itspeak intensity. A subsequent stage, dominated by diffraction and Kerr nonlinearity, serves tofurther compress the emerging isolated pulse, and may give rise to almost tenfold on-axis pulsecompression. The main driver behind this complex scenario is a dynamic interplay between ra-dial effects, namely diffraction, Kerr-type self-focusing, and, exclusively close to the geometricfocus, plasma defocusing. The dominance of spatial effects clearly indicates the unavoidabil-ity of a pronounced spatio-temporal pinch structure of self-compressed pulses. The frequentlyobserved pedestals in this method are identified as remainders of the suppressed leading pulsefrom the original split-up. Our analysis also indicates that lower pulse energies < 1 mJ requiringmore nonlinear gases or higher pressures will see an increased influence of dispersive coupling,which can eventually render pulse self-compression difficult to achieve. Higher energies, how-ever, may not see such limitation, opening a perspective for future improvement of few-cyclepulse self-compression schemes.

Financial support by the Deutsche Forschungsgemeinschaft, grants DE 1209/1-1 and STE762/7-1, is gratefully acknowledged. We acknowledge support by the GENCI project No

x2009106003.



Carsten Bree et al- Self-pinching of pulsed laser beams during filamentary propagation

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