April 15, 2005 / Vol. 30, No. 8 / OPTICS LETTERS
917
Chirp-induced dynamics of femtosecond filaments
in air
Rachel Nuter, Stefan Skupin,* and Luc Bergé
Département de Physique Théorique et Appliquée, Commissariat à l’Energie Atomique, Direction des Applications
Militaires—Île de France, B.P. 12, 91680 Bruyères-le-Châtel, France
Received October 14, 2004
We investigate the influence of a chirped phase on femtosecond pulses propagating in air. Pulses with an
initially negative chirp are temporally compressed by compensation with group-velocity dispersion. We demonstrate that this property, combined with plasma defocusing, can be used to trigger filamentation at different foci, increase self-guiding ranges, or even shorten pulse duration. © 2005 Optical Society of America
OCIS codes: 190.5530, 190.7110, 050.1590, 280.3640.
The nonlinear propagation of intense femtosecond
pulses in air has attracted broad interest in the past
decade.1–3 Among their various properties, ultrashort
high-power laser pulses self-focus under the Kerr response of air until their intensity exceeds the ionization threshold. The optical collapse is then arrested
by plasma generation, which stabilizes the beam into
a narrow light channel called a femtosecond filament.
These filaments are able to travel over several
meters in the form of light tubes that are
100– 200 ␮m in diameter and have ⬃1 mJ of energy.
They result from a dynamic equilibrium between optical focusing and plasma defocusing, accompanied
by strong distortions in the temporal pulse profile.
Nonlinearities driven by the Kerr effect lead to selfphase modulation (SPM) and consequently produce
very large spectral broadening from the UV to the
mid-IR. From this, many practical applications have
been suggested, such as lidar remote sensing and
lightning protection.2
Previous technological perspectives often required
the filamentation process to be monitored, which can
be done through chirping techniques.4 Varying the
distance between the compressor gratings of chirpedpulse amplified laser systems makes it possible to
modify the pulse duration. Pulse chirping introduces
a quadratic temporal dependence in the beam phase
at the laser exit. Combined with group-velocity
dispersion (GVD) in air, negatively chirped pulses
with FWHM duration T0 and phase in the form
exp关−2i共ln 2兲Ct2 / T02兴 共C ⬍ 0兲 undergo a compression
in time achieved by minimum temporal extent Tmin
= T0 / 共1 + C2兲1/2 at propagation distance zmin = 关兩C兩 / 共1
+ C2兲兴T02 / 共4k⬙ ln 2兲, where k⬙ denotes the GVD coefficient. Recently,5 chirp-induced compression factors
were measured at distance zmin. The duration
reached at zmin appeared larger at increasing energies, owing to nonlinear effects. Simulations based on
a nonlinear model5,6 accounting for SPM restored
this property, except that the computed values of
pulse duration exceeded those reported experimentally.
In spite of these studies, we are still missing a
clear understanding of the role of a chirped phase in
self-channeling of femtosecond pulses in air and its
potential applications. In this Letter we numerically
0146-9592/05/080917-3/$15.00
investigate the effect of pulse chirping on plasma
generation and self-guiding lengths. We show that
negative chirps help in maintaining the laser beam
envelope focused over long scales. Linear temporal
compression, moreover, superimposes with plasma
defocusing, which opens new trends for handling
pulse shortening.
We assume cylindrical symmetry of the slowly
varying electric field envelope E共r , t , z兲 described in a
reference frame moving with the group velocity of the
pulse by3,7
⳵E
⳵z
i
=
2k0
2
ⵜ⬜
E+
ik0n2
冎
2
再
⫻兩E共t⬘兲兩 dt⬘ E −
2
W共兩E兩兲
−
2
␳atUi
E
兩E兩2
兩E兩2 +
ik⬙ ⳵2E
2 ⳵t
2
1
␶k
−
冕
冋
t
exp −
−⬁
冉
ik0
2␳c
+
␴
2
冊
共t − t⬘兲
␶k
册
␳E
共1兲
,
where k0 = ␻0 / c is the central wave number, with ␻0
being the carrier frequency; k⬙ = 0.2 fs2 / cm; and the
Kerr response of the medium with nonlinear coefficient n2 = 4 ⫻ 10−19 cm2 / W involves a delayed component with relaxation time ␶k = 70 fs. ␳ is the electron
density, and ␴ = 5.44⫻ 10−20 cm2 is the cross section
for the inverse bremsstrahlung related to avalanche
ionization. W共兩E兩兲 is the ionization rate of dioxygene
molecules described by the model of Perelomov et al.8
with ionization potential Ui = 12.1 eV. The quantity
␳c = 1.8⫻ 1021 cm−3 is the critical plasma density, and
␳at = 5.4⫻ 1018 cm−3 is the neutral density. The generation of free electrons is governed by3
⳵
⳵t
␳ = W共兩E兩兲␳at +
␴
Ui
␳兩E兩2 .
共2兲
Equations (1) and (2) are numerically solved by considering input Gaussian profiles: E共r , t , z = 0兲
= 共2Pin / ␲w02兲1/2exp关−共r2 / w02兲 − 共t2 / tp2兲 − 共iCt2 / tp2兲兴, where
w0 is the beam waist, Pin is the peak power, and tp
= T0 / 共2 ln 2兲1/2 is the 1 / e2 half-width duration. We
simulate beams with wavelength ␭0 = 800 nm, lead© 2005 Optical Society of America
918
OPTICS LETTERS / Vol. 30, No. 8 / April 15, 2005
ing to critical power for self-focusing of Pcr = ␭02 / 2␲n2
⯝ 2.54 GW.
To start with, we reframe the chirping techniques
in their original physical context: chirped-pulse amplified laser sources can deliver pulses with tunable
durations by moving the compressor gratings. At
equal spectral content, chirping the pulse amounts to
affecting the FWHM duration as T0 = T0C=0共1 + C2兲1/2
C=0
and to varying Pin such as Pin = Pin
/ 共1 + C2兲1/2.6 Let
us thus compare the atmospheric propagation of an
unchirped laser pulse characterized by T0 = 70 fs, w0
= 3 mm, input energy of Ein ⯝ 1.9 mJ, and power ratio
of Pin / Pcr = 10, with its chirped counterpart, setting
C ⯝ ± 1 to reach T0 ⯝ 100 fs.
Figure 1 represents the beam radius, i.e., the width
+⬁ 2
兩E兩 dt兲 as a
at half-maximum of the fluence 共F ⬅ 兰−⬁
function of propagation distance z. The beam first
self-focuses, then forms a channel resulting from the
dynamic balance between Kerr self-focusing and
plasma defocusing. For C = 0 these processes lead to a
spatial spot remaining in a focused state over ⌬z
⯝ 27 m (dashed–dotted curve). The dotted and solid
curves correspond to the radius of chirped fields with
C = 1.02 and C = −1.02, respectively. The self-guiding
length diminishes for a positive chirp 共⌬z = 18 m兲 but
increases for a negative chirp 共⌬z = 40 m兲. In that
case a last compression event occurs at z ⯝ 47 m.
Early propagation stages, moreover, reveal that the
first foci of chirped pulses are displaced further than
for an unchirped beam. This displacement is mainly
due to the fact that temporal broadening induces a
decrease in the input power.
To get a deeper insight into the phase influence, we
now keep both the power ratio and the input duration
constant for three chirp values: C = 0 , ± 20, while
w0 = 0.9 mm. Preliminary computations (not shown
here) indicate that, for subcritical powers, nonlinear
processes such as Kerr focusing and plasma generation can occur with C ⬍ 0. Temporal compression
makes the field intensity increase so much that the
nonlinear refractive index 共⬇n2I兲 becomes dominant
over spatial diffraction. For higher power, Fig. 2 summarizes the chirp action when Pin is well above Pcr.
The dashed–dotted curve 共C = 0兲 shows standard selfguiding [Fig. 2(a)] with peak electron densities reaching 1016 – 1017 cm−3 [Fig. 2(d)]. As expected, a positive
Fig. 1. Beam radius versus z 艌 5 m for C = 0 (dashed–
dotted curve), C = 1.02 (dotted curve), and C = −1.02 (solid
curve).
Fig. 2. (a), (c) Pulse radii and enlarged portions and (b),
(d) peak intensities and related electron densities for C = 0
(dashed–dotted curve), C = 20 (dotted curve), C = −20 (solid
curve), Pin / Pcr = 5, and T0 = 100 fs 共兩⌬zc / zc兩 ⯝ 5 % 兲.
Table 1. Position of the Last Compression Point
Linear regime
Pin / Pcr = 1.35
Pin / Pcr = 5
C = −20
C = −16
C = −12
9.0 m
9.8 m
8.8 m
11.2 m
12.6 m
11.6 m
14.9 m
17.0 m
18.5 m
chirp favors beam divergence, whereas a negative
chirp reinforces on-axis compression (solid curve).
For C ⬍ 0, Fig. 2(b) exhibits localized spikes in the
maximal intensity, the last of which corresponds to a
final residual compression. This latest peak keeps
the pulse shape localized over a longer propagation
range [see Fig. 2(a)]. Refocalization occurs around z
= 9 – 10 m, which belongs to the range z ⬃ zmin, at
which GVD and chirp linearly compensate. In connection with these behaviors, displacements of the focus, zc, vary with the sign of C, inducing back–forth
motions of zc for C ⬍ 0 or C ⬎ 0, respectively [Fig.
2(c)]. Relative deviations of this focus depend on both
the pulse power and the chirp value. With the above
parameters, 兩⌬zc / zc兩, limited to 5% at Pin = 5Pcr, was
found to attain 25% at Pin = 2Pcr. These deviations
agree with the discrepancies observed in computed
filamentation patterns when one omits the temporal
variations in numerical models.7
Table 1 specifies the location of the last peak intensities for various values of the chirp parameter. It
suggests that, even if the temporal profile is strongly
perturbed during plasma generation, the compression stage still persists and keeps the laser beam localized over longer distances close to zmin. This phenomenon holds, provided that the beam diffraction is
not too important, to maintain the light spot localized
before z = zmin.
Finally, Eqs. (1) and (2) were used to reproduce experimental data reported by Alexeev et al.5 Here the
initial pulse duration was 150 fs and the chirp parameter, C ⯝ −3, was chosen to obtain a compressed
pulse of 50 fs at zmin ⬃ 110 m in the linear regime. We
fixed the input beam waist to a moderate value of
w0 = 0.3 cm.
April 15, 2005 / Vol. 30, No. 8 / OPTICS LETTERS
FWHM pulse durations were measured near zmin
at z = 105 m and compared with numerical computations accounting for SPM.5 As noted in Ref. 5, these
computations overestimated the importance of SPM.
Characteristic durations as well as those revealed by
direct integrations of Eqs. (1) and (2) are specified in
Table 2. They reveal a good agreement with the experimental data when the model equations also describe plasma generation, delayed Kerr response,
and energy losses.
Table 2. FWHM Temporal Width for Three Energy
Values
a
Experimental
Our results
1.3 mJ
1.9 mJ
2.5 mJ
55 fs
79 fs
70 fs
50 fs
Multipeak
Multipeak
a
Ref. 5.
919
In Fig. 3(a) the temporal extent plotted for, e.g., input energy Ein = 1.9 mJ, displays evidence that the
maximum compression is not obtained at zmin but instead at the first nonlinear focal point, where free
electrons start to be created. Here plasma defocusing
induces a strong pulse shortening, which arises when
the ionization front depletes the trail of the pulse.9
This phenomenon decreases the pulse duration to
values as small as T ⬇ 3 fs, i.e., close to one optical
cycle at 800 nm. From this figure it is worth noting
that the chirped duration develops sharp shortening
and broadening sequences followed by a relaxation
stage, so that the pulse width resulting at z = 105 m is
already augmented from its smallest value. Similar
evolutions were observed with Ein = 1.3 and 2.5 mJ.
At z = 105 m, only a relaxed duration can be measured, compared with the shrunk narrow leading
peak produced in Fig. 3(b), whose occurrence constitutes the generic signature of plasma defocusing.9
In summary, we have shown that chirped pulses
could be used as efficient tools for monitoring femtosecond filaments and their plasma channels. Negative chirps favor enhancement of the self-channeling
length. Applying this property to lidar experimental
setups,4 chirping the pulse from 100 to 600 fs 共C ⯝
−5.92兲 can retrigger filamentation at zmin reaching
the kilometer range. Combining pulse chirping with
plasma defocusing also results in tunable pulse
shortening, whose characteristic distances could easily be accessed in further experiments.
R. Nuter’s e-mail address is [email protected].
*Permanent address, Institut für Festkörpertheorie und-optik Friedrich-Schiller-Universität Jena,
Max-Wien-Platz 1, 07743 Jena, Germany.
References
Fig. 3. (a) FWHM temporal width for Ein = 1.9 mJ and C
= −3. Filled circles denote the z distances at which the
pulse duration was calculated. Bottom, temporal profiles
for Ein = 1.3 mJ (dashed–dotted curve), Ein = 1.9 mJ and C
= −3 (dotted curve), and Ein = 2.5 mJ (solid curve) at (b) z
⯝ zc (19.5, 12, 10 m, respectively) and (c) z = 105 m.
1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G.
Mourou, Opt. Lett. 20, 73 (1995).
2. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E.
Salmon, H. Wille, R. Bourayou, S. Frey, Y. André, A.
Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste,
Science 301, 61 (2003).
3. M. Mlejnek, E. Wright, and J. Moloney, Opt. Lett. 23,
382 (1998).
4. M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian,
J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel,
U. Laux, A. Hatzes, R. Sauerbrey, L. Wöste, and J.-P.
Wolf, Phys. Rev. E 69, 036607 (2004).
5. I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R.
Peñano, R. Hubbard, and P. Sprangle, Appl. Phys. Lett.
84, 4080 (2004).
6. P. Sprangle, J. Peñano, and B. Hafizi, Phys. Rev. E 66,
046418 (2002).
7. S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean,
J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M.
Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey,
Phys. Rev. E 70, 046602 (2004).
8. A. Perelomov, V. Popov, and M. Terent’ev, Sov. Phys.
JETP 23, 924 (1966).
9. S. Champeaux and L. Bergé, Phys. Rev. E 68, 066603
(2003).