Laser beam self-symmetrization in air in the multifilamentation regime

4.4 MB·English

by  Carles Milian

Laser beam self-symmetrization in air in the multifilamentation regime
C. Mili´an,∗ V. Jukna, and A. Couairon
Centre de Physique Th´eorique, CNRS, E´cole Polytechnique, F-91128 Palaiseau, France
A. Houard, B. Forestier, J. Carbonnel, Y. Liu, B. Prade, and A. Mysyrowicz
Laboratoire d’Optique Appliqu´ee, ENSTA ParisTech,
E´cole Polytechnique, CNRS, F-91761 Palaiseau, France
Weshowexperimentalandnumericalevidenceofspontaneousself-symmetrizationoffocusedlaser
beamsexperiencingmulti-filamentationinair. Thesymmetrizationeffectisobservedasthemultiple
filamentsgeneratedpriortofocusapproachthefocalvolume. Thisphenomenonisattributedtothe
nonlinear interactions amongst the different parts of the beam mediated by the optical Kerr effect,
5 which leads to a symmetric redistribution of the wave vectors even when the beam consists of a
1 bundle of many filaments.
0
2
I. INTRODUCTION shown that the beam cleaning process only occurs for
n
the frequency downshifted components of the supercon-
a
J An intense laser pulse propagating in air undergoes tinuumgeneratedduringthefilamentationprocess. They

0 variousselfinducedtransformationsifitspeakpowerex- attributed this effect to the role of self-focusing as a spa-
3 ceedsacriticalvalue[1]P ≈4.7GW.TheopticalKerr tial filter, filtering out the high-order mode components
cr
effectisresponsibleforanincreaseoftherefractiveindex of the beam whereas the fundamental mode is focused,
s] leadingtobeamselffocusing,acumulativeprocessevolv- producing a high quality filament core. Naturally this
c ing towards a collapse. An electron plasma is efficiently process is promoted for redshifted frequencies which are
ti generated via optical field ionization when the beam in- generated in the leading edge of the pulse and undergo
p tensityhasincreasedabove∼1013 W/cm2. Ionizationis mainly self-focusing and diffraction [8]. From these ex-
o
associated with nonlinear absorption of laser energy and amples,itisclearthatevenifbeamself-cleaningreceived
.
s with a decrease of the local refractive index. Both ef- differentinterpretations,thiseffectoccursforbeampow-
c
i fects act against the growth of intensity by self-focusing ers moderately above the critical power.
s
y andeventuallyarrestthecollapse. Theinterplaybetween ForP (cid:29)Pcr,however,abeamrisesmultiplefilaments
h nonlinear effects leads to the formation of a narrow light across the profile, as manifestations of the dynamics fol-
p filamentleavingaplasmachannelinitswake,surrounded lowing local collapse. These filaments are born on beam
[ by a laser energy reservoir. This reservoir maintains an intensity fluctuations [13, 14] and grow due to modula-

energy flux toward the filament core that compensates tioninstability[15,16](MI).Severalfilamentinteractions
1
v fornonlinearabsorption. Thisopticalentitycanliveover manifestedascrossphasemodulation(XPM),fusion,re-
3 several meters and be generated at long distances [2, 3]. pulsion, spiraling, and fission have been reported in this
0 Ionization in the atmosphere by filamentation has been high power regime [13, 17–20]. It has also been shown
9 reported at a distance of 1 km from the laser [4]. When recently that focused multi-filamented beams maintain
7 the reservoir is exhausted and fails to feed the filament a substructure during the focal region and may produce
0
efficiently, a slowly diverging bright channel is observed thickfilamentsassociatedtounusuallyhighintensityand
.
1 and can be detected at several kilometers away from the plasma levels, coined as superfilaments [21]. Up to date,
0 lasersource. Severalteamshavereportedaneffectcalled allstudiesdonewithhighpowerbeamsfocusonthefila-
5
1 beam self-cleaning in the low power regime P ≈ Pcr [5– mentfeatures, ratherthanonthewholebeamdynamics.
9]. Moll and Gaeta showed that slightly elliptical beams
: In this work we investigate an effect of the global
v with power close to P tend to increase their roundness
cr structure of powerful (P (cid:29) P ) beam profiles that re-
i andrecover acircularlysymmetricprofilewhentheyun- cr
X mainedunexplored. Wedemonstrateexperimentallyand
dergoself-focusing,justbeforeionization. Thiseffecthas
r by means of numerical simulations that under focusing
a been interpreted as due to beam collapse and reshaping
conditions, powerful beams undergo self-induced sym-
into a universal self-similar spatial profile, the Townes
metrization in the global scale above certain threshold
mode [5, 10], which is intrinsically spatially symmetric.
power, P (cid:29)P , evenifhundredsoffilamentsareborn
Prade et al. [6] measured conical emission for ultraviolet th cr
along propagation. As a consequence, the beam washes
pulses undergoing filamentation in air and reported self-
outallinteractionsanddistortionsitmighthavesuffered
improvement of the beams’s spatial mode quality, sug-
priortothefocus. Howeverthenatureofthisveryrobust
gesting the generation of propagation invariant modes
phenomenon differs from the previously reported beam
called nonlinear X-waves [11, 12]. Liu and Chin have
self-cleaning effect associated with a single filament, in
the sense that a fundamental mode does not necessar-
ily emerge after self-induced symmetrization. We used
∗ [email protected] two indicators to distinguish between beam self-cleaning
2
and self-induced symmetrization: the global beam sym- B. Beam characterization
metrydegreeΦandthebeamqualityfactorM2. Forthe
single filamentation regime, beam self-cleaning reported Foreachrecordedbeamintensityprofile,wecomputed
previously is essentially associated with an improvement the center of mass (CM) on the xy- (transverse) plane
of the beam quality factor, even if the beam symmetry andthenconsideredtheradialintensitytraces(withfixed
degree may improve as well. In contrast, in the case of polar angle φ), I (r) (r = 0 at the CM). We define the
φ
highpowerbeamsconsideredhere,onlyglobalsymmetry degree of symmetry, able to resolve the internal struc-
improves substantially, whereas the beam quality factor ture (intensity fluctuations) of the beam profiles, as the
always remains far from that of a Gaussian beam or the weightedaverageoftheintensityfluctuations: 0≤Φ≤1
Townes mode (M2 ∼ 1). To illustrate this effect, we (see appendix)
performedexperimentsandnumericalsimulationsonthe
nonlinear propagation of heavily distorted input beams
cdheagrraeectearfitzeerdpbayssaindgrathmraotuigchdeincrteeansseityofmthaesikrs.symRmeseutlrtys Φ≡1− 2π12 (cid:90) 2πdφ1(cid:90) φ1dφ2(cid:12)(cid:12)(cid:12)(cid:12)(cid:82)(cid:82)r{{IIφ1((rr))+−IIφ2((rr))}}(cid:12)(cid:12)(cid:12)(cid:12),
0 0 r φ1 φ2
clearly demonstrate that the combination of the Kerr ef- (1)
fectandfocusingiscrucialtoobserveself-symmetrization which measures the similarity in between all pairs of the
of highly asymmetric beams. radial intensity traces, {I (r),I (r)}. Perfect circularly
φ φ(cid:48)
symmetric profiles are characterized by Φ = 1, whereas
Self-symmetrization of powerful (multiterawatt) laser
beams with a large asymmetry are characterized by a
beams may prove useful when it is important to homo-
relatively low symmetry degree.
geneously illuminate a target placed in the focal region.
Regarding the beam quality factor, M2, we use the
This effect also allows us to obtain a relatively uniform
widelyuseddefinitionbyPotemkinetal. [32]forabeam
cylindrical plasma channel around focus which shape is
withcomplexfieldE(x,y)≡A(x,y)exp(iφ)andintensity
independent from the quality of the initial laser beam
I(x,y)≡|E(x,y)|2:
profile. Such property is particularly interesting for ap-
plicationsbasedonlaserfilamentationathighpowersuch
M2 =(cid:0)(cid:104)r2(cid:105)(cid:104)k2(cid:105)−(cid:104)r·k(cid:105)2(cid:1)1/2 (2)
as guiding of electric discharges [22–24] and contact-less
capture of currents [25], control of aerodynamic flows
where
[26], or lasing effect in air [27–31].
(cid:90)
P = dxdyI(x,y) (3)
(cid:90)
(cid:104)r2(cid:105)=P−1 dxdy(x2+y2)I(x,y) (4)
(cid:90)
II. EXPERIMENTAL AND NUMERICAL (cid:104)k2(cid:105)=P−1 dxdy[(∇A)2+I(∇φ)2] (5)
PROCEDURES
(cid:90)
(cid:104)r·k(cid:105)=P−1 dxdy[r·∇φ(x,y)]I(x,y) (6)
A. Experiments
andr≡(x,y). Sincethecalculationofthequalityfactor
requires knowledge not only of the beam intensity but
The collimated output 50 fs pulse from a linearly po- alsoofthespatialphase,wecharacterizedbeamsinterms
larizedmulti-TerawattTi:SaCPAlaser(Enstamobile)at of M2 only for the results of our numerical simulations.
λ = 800 nm is weakly focused with an f = 5 m lens in
0
airatatmosphericpressure. Burnpatternsfromthelaser
beamarerecordedoncalibratedpresolarizedKodakpho- C. Simulations
tographic plates at different distances from the focusing
lens. This simple technique reliably detects in a single
Propagation of the monochromatic field envelope
shot the stage of evolution of multiple filaments formed
E(x,y,z) in air (n ≈1), with frequency ω =2πc/λ , is
0 0 0
out of a single laser beam [2]. Plasma channels give rise
modeled by means of a unidirectional beam propagation
tocharacteristic∼50−100µmwidecircularburnsonthe
equation accounting for diffraction, optical Kerr effect,
photographic plate, which are easily distinguished from
multiphoton absorption, plasma absorption, and plasma
the ∼ 1 mm circular dark spots from bright channels
defocusing, respectively [33]:
and the less intense energy reservoir. Similar measure-
ments were performed by intentionally adding chirp to ∂E
the pulse to adjust the peak intensity. The same self- =
∂z
sfoyrmtmheettrriaznastfioonrmeffliemctitoefdm50ulftsippluelfiselasmanendtfsowrapsreo-bchseirrpveedd i (cid:18)∂2 + ∂2(cid:19)E+iω0n |E|2E−1(cid:0)β |E|14+σ[1+iω τ ]ρ(cid:1)E.
2k x2 y2 c 2 2 8 0 c
pulses (see figure captions). This allows us to dismiss 0
(7)
temporal effects as the origin of symmetrization.
3
III. RESULTS
A. Undistorted beams
In the absence of any mask, a beam with power P ∼
550 P undergoes typical multi-filamentation dynamics.
cr
Figure 1 shows the darkening pattern impressed by such
a beam at several distances from the focusing lens on
photographic plates. Already after 2 m of propagation
several ionizing filaments are present. These converg-
ing filaments grow in number and connect into networks
upon further propagation ((cid:38)4 m). Before the geometric
focus, the ionized zone fills a central circular spot of a
few mm in diameter and ∼ 1 m long, where filaments
are in close contact [21]. The number of ionized spots is
significantly reduced to a few which are located near the
center of the beam. Losses incurred by going through
the focus have been measured with a calorimeter placed
before and after the focus. 82% of the initial beam en-
ergy is found in the beam at 3.5 m after the focus. This
rather high value could be connected to the idea that in
certainsituationsthefilamentsareconnectedstrongerto
FIG. 1. Experimental transverse profiles of an undistorted plasma defocusing than to plasma absorption [17]. Be-
focused beam. Images recorded on the photographic papers low we show that independently on the intensity mask
atvariousdistancesforinputpulsesof50fs(notpre-chirped),
used to distort the input beam profile, the beam recov-
carrying 130 mJ each and having a beam width of ∼3.5 cm.
ers after focus the same symmetry degree, hence similar
The lens has its geometrical focus at 5 m. Beam and pulse
roundness, as the one shown in the final stage of Fig. 1.
widths are given at the intensity FWHM and propagation
In contrast, low energy beams (P (cid:28)P ) do not exhibit
distances are marked on top of each image. The 1 cm length cr
any improvement of their symmetry degree along prop-
bar is included for reference.
agation since the final beam profile corresponds to the
diffraction pattern induced by the mask, with identical
symmetry degree as the initial distorted beam.
B. Self induced symmetrization of highly distorted
beams
In order to visualize the self symmetrization effect we
have put different masks on the path of the input beam,
Here n =2×10−19 cm2/W, β =8×10−98 cm13W−7, just before the focusing lens (at 0 m). Examples are
2 8
σ = 5.6×10−20 cm2, and τ ≈ 350 fs. Our (2+1)D shown in Figs. 2 and 3 where a quarter circle (or Pac-
c
modeling does not contain temporal dynamics. Hence, man), square, half-plane, and slit masks are used. Sim-
plasma effects are accounted for in the so-called frozen ilarly to the undistorted case, 89% of the input chirped
time approximation where the explicit standard rate pulse energy was measured 3 m beyond the focus. In-
equations describing electron generation by optical field spection of the burnt papers in Figs. 2 and 3(a) reveals
ionizationandavalancheareusedtogenerateamapping that ionizing filaments get first organized along patterns
oftheelectricfieldpeakintensityI ≡|E|2totheelectron- dictated by the profile of the input beam [13]. Indeed,
plasma density, ρ(I). This calculation is performed for close to the sharp intensity jumps created by the mask,
reference pre-chirped Gaussian pulses with peak inten- large intensity modulations drive filamentation in a de-
sity I with duration given by experimental conditions. terministicmanner. Uponfurtherbeampropagation,the
Theelectrondensity,ρ(I),usedinthepropagationmodel self-symmetrization effect appears around the geometric
Eq. (7) is determined from this mapping, at the tempo- focus symmetrizing the slowly diffracting output beam,
ralcenterofthereferencepulse. Thisprocedureprovides, despite its initial strong distortion. Solid experimental
asshownpreviouslyandbelow,numericalresultsingood evidenceofthesymmetrizationeffectisprovidedbycom-
agreementwithexperiments[20,21,34]. Simulationsare paringtheprofilesinFig. 2recordedat1.5mbeforeand
initialized with 10% intensity and 0.2% phase noise in after the focus (i.e., at 3.5 and 6.5 m).
order to mimic experimental irregularities in the input A detailed comparison between experimental and nu-
beam. merical results is presented in Fig. 3 for the case of the
4
highest input distortion. Here, a 10 mm wide slit trans-
forms the input circular profile into a rectangle with as-
pect ratio ∼ 1/4. Upon propagation the symmetriza-
tion degree increases as the beam approaches and goes
through the focal region at ∼ 5 m. Figure 4(a) shows
symmetrization of the beam along propagation for the
beamprofiles presented in Figs. 3a-b. For acomparison,
two additional numerical results for linear propagation
and for an input intensity I =5 GW/cm2 are also pre-
0
sented. Thesymmetrydegreeisseentobehigherbythe
endofthenonlinearpropagations,reachingitsmaximum
after the focus and holding it for a distance of at least
∼ 2−3 m. The minimum of the symmetry is observed
forthelinearpropagationatthefocalplane. Thisissim-
ply due to the sharp horizontal line (y = 0) apparent in
the beam profiles at this position, z = 5 m (see below,
e.g.,inFig. 6),correspondingtotheFouriertransformof
the slit induced sharp profiles along x (which is a sinc(x)
function at focus). One can easily see that the width of
the Fourier transforms of a Gaussian, ∆κ (at FWHM),
G
FIG. 2. Experimental beam profile evolution under spatial andaslit, ∆κs (fullwidthof√thecetral”sinc” lobe), sat-
distortion of the pre-chirped 700 fs pulses carrying 295 mJ isfy ∆κ /∆κ = w/L ×π/ ln2, where w ≈ 3.5 cm is
s G s
before the masks (P/Pcr ≈102). The different masks screen the FWHM of the input (z = 0) Gaussian profile along
(a) 25 %, (b) 10 %, and (c) 50 % of the input energy. Input y and L = 1 cm the slit width (along x). It therefore
s
beam widths and focusing conditions are those of Fig. 1. follows that whilst L (cid:46) w, the horizontal line appear-
s
Propagationdistancesareindicatedontopofeachcolumnof
ing at focus will be much longer than the spot size along
images.
y affecting substantially the symmetry degree (see the
sharp minimum Φ≈0.5 in Fig. 4a). This feature disap-
pears gradually as the input peak intensity is increased
in simulations up to I ≈ 5 GW/cm2, because of the
0
novel spatial frequencies that are generated (see discus-
sion in section V). Also, the photographic paper used in
experimentshasadifferentdefinitioninthelowintensity
parts of the beam and therefore the threshold intensity
at which the horizontal line disappears is expected to be
different than for simulations. Note in the linear prop-
agations for the masked beams (not shown) the beam
profiles at a fixed distance before and after the focus are
almost equal to each other after a rotation of π in the
XˆY plane. Theequalitydoesnotholdexactlyduetothe
input noise and mask induced fluctuations, however the
difference is small at sight. This is the reason why Φ(z)
is (almost) symmetric from the focal plane (see Fig. 4a).
In our experiments, the slit mask is the one screening a
biggest fraction (∼64%) of the input 300 mJ carried by
laser pulses, which was the maximum available energy.
We believe this is why the experimental symmetry de-
gree along propagation (circles in Fig. 4a) starts to drop
abit2−3mafterfocus. Thisfeaturemaybeappreciated
evenvisuallyinFig. 3(a),converselytowhatisshownin
FIG.3. Beamevolutionafterdistortingtheinputbeamwith Fig. 2 for the other masks. Additionally, the symmetry
a 1 cm wide slit mask, screening ∼64 % of the input energy
degree presents a local minimum around focus, which is
∼ 300 mJ. (a) Experimental and (b) simulated beam inten-
a reminiscence of the linear propagation (indeed we see
sities along propagation at selected distances, marked in (a).
similar Φ(z) trends in our monochromatic modeling for
Pulse width is 100 fs (pre-chirped) and energy ∼ 100 mJ:
I (cid:46)10GW/cm2). Still,thesymmetrydegreeisremark-
P/P ≈ 210 (after mask). Simulations are initialized with 0
cr
a peak intensity of ≈ 250 GW/cm2. Position of geometrical ably high during 2-3 m and we expect (see below) that
focus is at 5 m and intensities given by the color bar are in this would improve for higher input pulse energies.
Bels.
Thequantitativeagreementinbetweenourmonochro-
5
1 I=250 GW/cm2 0 Townes
experiment 0 (c)
0.9 simulations I=5 GW/cm2 −1 Gaussian
0 (linear)
0.8 Linear −2) −2
Φ 0.7 g(M10−3 slit mask Linear
o
l
0.6 geometrical Φ −4
(a) I=250 GW/cm2
0
0.5 −5
0 5 10 0 5 10
z (m) z (m)
FIG.4. (a,b)Symmetrydegreeofthebeamprofilesversuspropagationandinputpeakintensity,respectively. (c)Smoothness
alongpropagationforTownesmode(dashed),undistortedfocusedGaussianbeampropagatinginthelinearregime(solid-red),
and heavily distorted beams, corresponding to the linear and I = 250 GW/cm2 cases shown in (a). Results shown here
0
correspond to the slit mask (see also figures 3 and 5).
matic numerical simulations and experimental results in of powerful focused beams is indeed the optical Kerr ef-
Figs. 3 and 4(a) strongly suggests that the spontaneous fect, we made a set of simulations with identical initial
symmetrization shown here is an effect essentially dom- conditions as those in Fig. 5 in which the different non-
inated by spatial dynamics of the beam. We have com- lineartermsinEq. (1)areswitchedon oroff atwill(see
puted the symmetry degree at a fixed distance z = 9 m Fig. 6). Below,weareonlyshowingbeamprofilesatand
(4 m after focus) and varying input power (Fig. 4(b)). afterthefocus(z ≥5)becausethosebeforethefocusare
These measurements reveal an increase of Φ(P/P ) ex- indistinguishable by sight from those in Fig. 3 (b) for all
cr
hibiting saturation: Φ(P/P (cid:38) 40) ≈ 0.95. Such be- cases.
cr
havior vividly manifests the nonlinear (intensity depen-
dent)natureofthespontaneoussymmetrization. Indeed,
a systematic numerical study reveals that the Kerr ef-
fect plays a major role in self-symmetrization (see sec- When the only nonlinear effect is MPA (n , σ = 0,
2
tion IV below). We also characterized the beam quality Fig. 6(a)),propagationdoesnotdiffersubstantiallyfrom
via beam quality factor [32, 35]. A Gaussian beam at the one observed in the linear regime, except for the big
waist is characterized by M2 = 1 and is considered as a absorptioninducedacrosstheprofilethatleadtotheap-
reference for perfect quality. Figure 4(c) shows that the pearance of vertical fringes. Linear propagation would
highly distorted beams always present a beam quality 3 lead to the observation of the diffraction pattern of the
to 5 orders of magnitude worse than that of a Gaussian slit. Nonlinear absorption plays the role of a distributed
beam at focus or the Townes mode. Therefore, even if stopper, leading to modulations in the diffraction pat-
there is a partial improvement of M2 with propagation terninaneffectsimilartotheAragospoteffect. Nonlin-
distance, the main effect is self-symmetrization. Beam ear absorption participates to the fringe formation since
self-cleaning occurs in the sense of an improvement of it enhances the self-healing process of a beam that is
the beam quality factor but the latter effect is not so ef- knowntoreshapeGaussianbeamsintoBessel-likebeams
ficient as for beam self-cleaning obtained in the context [36–38]. Addition of plasma absorption and defocusing
of lower power beams [5, 6, 8], where P/Pcr (cid:38)1. (σ (cid:54)= 0, Fig. 6(b)) has a strong impact in the profile
after focus: generation of plasma leads to refractive in-
dex jumps along the sharp intensity edges and the de-
IV. NUMERICAL ANALYSIS OF KEY focusinginducesstrongscatteringtowardsthedirections
PHYSICAL EFFECTS IN BEAM perpendicular to those edges (light propagates towards
SYMMETRIZATION
the higher index regions). As a consequence of this, the
scatteringoflightafterfocusoccurspredominantlyatan
A. Impact of the different nonlinear effects angle π/2 from the long axis of the slit. It is only when
theKerreffectisswitchedonthatsymmetrizationdegree
We consider below the case of an input beam with a improves substantially, as shown in Fig. 6(c) (β = 0)
8
peak intensity of I = 50 GW/cm2 under the distortion and Fig. 5. Note from Fig. 6(d) that with Kerr effect
0
of the slit mask. This situation corresponds to the mini- symmetry is higher when both MPA and plasma effects
mum peak intensity needed to observe a high symmetry are accounted for, presumably due to the fact that MPA
degree (see Fig. 4(b)) and only a few filaments may be andplasmatendtoinducescatteringalongperpendicular
observed on the output beam profile, note there are only directions(atleastinthecaseoftheslitmask). Remark-
fourfilamentsinFig. 5(markedbythesquares). Inorder ably, the symmetrization effects persists even for beams
toshowthatthemainresponsibleforself-symmetrization experiencing a large multifilamentation (see Fig. 3).
6
4 m 8 m 5 m (focus) 8 m
1 cm 1 cm
(a)
5 m
2 mm 1 cm
(b)
2 mm
FIG.5. Beamsymmtrizationinthelowfilamentlimit. Loga-
rithmicscaleplotsofbeamIntensityprofilesatvariousprop-
agation distances (see labels) for a beam focused with a lens
of5moffocallengthandperturbedwitha10×35mm2 slit
mask (as in Figs. 3 and 4). Input peak intensity I = 50 (c)
0
GW/cm2. All nonlinear effects of Eq. 1 are switched on (n ,
2
β , σ(cid:54)=0).
8
B. Impact of diffraction: non focused beams
Thesecondessentialingredientforsymmetrization(to- (d) 1
MPA (a)
getherwithKerr)isthefocusinginducedbythelens. We MPA, plasma (b)
have done extensive numerical modeling with collimated 0.9 MPA, Kerr (c)
all
beams and controlled the density of filaments via the in-
Φ 0.8
put power, P. None of these simulations showed traces
of spontaneous symmetrization in the distances of ∼ 10 0.7
m. Figure 7 shows an example with the slit mask. Here,
inordertodecreasefilamentseparation,theinputpower 0.60 5 10
z (m)
was 4 times larger than for the focused cases shown in
Fig. 3(b) or Fig. 4 with the largest I. The multiple
FIG. 6. Influence of the different nonlinear terms. (a)-(c)
filaments develop along propagation and the phase gra-
Beam profiles at z = 5,8 m from numerical results corre-
dients impinged by the mask make some of the optical
sponding to simulations with identical conditions as in Fig.
energymigrateintotheinitiallyunpopulatedarea. How- 5: (a) n ,σ = 0, (b) n = 0, (c) σ = 0. Color scale is that
2 2
ever, symmetry is seen to remain rather low along the of Fig. 5. (d) Evolution of symmetry degree for (a)-(c) and
10 m, which implies that the self-symmetrizing effect re- Fig. 5. The nonlinear effects that are on in each simulation
ported here is intimately related to the focusing induced are listed in the legend.
interaction amongst the different parts of the beam.
cientlyintheregionswhereintensitygradientsarebigger
V. DISCUSSION: A POSSIBLE EXPLANATION (e.g., sharp edges impinged by the mask on the beam).
FOR THE SYMMETRIZATION EFFECT The transverse wavenumber of these waves satisfies the
proportionality relation κ ∝ ∂ I (∂ denotes spatial
⊥ ⊥ ⊥
In this section we attempt to give a plausible explana- derivativeinsomedirectiononthetransverseplaneXˆY).
tion for the symmetrization effect on the basis on the re- Note here two things: (i) the new spectral component is
sultsreportedanddescribedabove. Firstofallwenotice generated at a sharp edge and travels inwards towards
that results in Sec. IV show that symmetrization occurs the intense part of the beam, and (ii) this wave being
mainly due to the Kerr term and that it is not necessar- generated, implies that other frequencies present at that
ily linked to the presence of many filaments in the beam edge have been annihilated, i.e., the Kerr effect boosts
profile. This strongly suggests that symmetrization oc- energy from the strongly divergent spatial frequencies at
cursasaconsequenceoftheKerr-inducedgenerationand anedgetonewfrequenciestravelingintheoppositedirec-
annihilationofspatialfrequencies(thisisindeedaneces- tionoverthetransverseplane(thisisofcourseequivalent
sarycondition). Alongthislineitiswellknown(seee.g., to the qualitative argument explaining why the Towns
[39], for the description in time domain) that the Kerr mode [10] exists). In the case of the slit mask, the image
term induces generation of spatial frequencies more effi- described above plays a central role along x (horizontal)
7
z=0.5 m z=3 m powerful beams. The absence of symmetrization for col-
limated beams highlights the importance of focusing in
theglobalorganizationintheformofcircularlysymmet-
ric beam. A possible explanation for this effect is also
provided.
z=6 m z=9 m ACKNOWLEDGMENTS
Authors acknowledge financial support from the
French Direction G´en´erale de l’Armement (DGA).
APPENDIX: SYMMETRY DEGREE
FIG. 7. Simulated beam intensity cross sections along prop- First of all, for a given 2D beam profile, we locate the
agation for a collimated cw beam under distortion of the slit
center of intensity (center of mass)
mask. Absenceofbeamsymmetrizationandoffilamentden-
sity increase are evident (compare with Fig. 3(b)). Numbers (cid:82)
(x(cid:126)u +y(cid:126)u )I(x,y)
iGnafiugssuiraenspmualsrekupsreodpaingattheidsdmisotdaenlcinegs aisloonfg1z0.0Tfshaenrdeftehreenicne- (x,y)CM ≡ xy (cid:82)x I(x,yy) (8)
put beam peak intensity is 2.5 TW/cm2. All figures have a xy
cross-section of 4×4 cm2.
at which we locate the origin of the polar coordinates,
(r,φ): r = 0 at (x,y) . Then we consider the radial
CM
intensitytracesatafixedpolarangle,I (r),andthenwe
φ
and not so much along y (vertical) because the Gaussian
compare them by pairs. Given a pair of these traces we
profile is much smoother than the rectangular shape. As
define the local relative difference
a consequence of this, one could say that Kerr acts as
a moderator which kills waves with high |κ | and there-
⊥
foresymmetrizesthespectraldistribution. However,this I (r)−I (r) I (r)−I (r)
D¯(r,φ ,φ )≡ φ1 φ2 = φ1 φ2 ∈[−1,1].
is not enough to explain the effect reported in this pa- 1 2 I (r)+I (r) 2(cid:104)I (r)(cid:105)
per. We believe that the second necessary condition for φ1 φ2 φ1,φ2
(9)
symmetrization is the presence of the external focusing. Since D¯ values are more significant in regions where the
Not only because it enhances intensity levels and the ef-
average intensity, (cid:104)I (r)(cid:105), is locally high, we define
ficiency of the Kerr effect, but also because it tends to φ1,φ2
the weighted average relative difference in between the
localize all the frequency components of the beam into
pair of traces:
a reduced focal region that acts as a quasi-point source
with symmetric spectrum in the transverse plane. This
unavoidably leads to an after-focus beam that exhibits a (cid:12)(cid:82) (cid:104)I(cid:105)D¯(cid:12)
substantiallyhighoverallroundnessandwecantherefore wD¯(φ1,φ2)≡(cid:12)(cid:12)(cid:12) (cid:82)r (cid:104)I(cid:105) (cid:12)(cid:12)(cid:12)∈[0,1], (10)
saythatthebeamself-symmetrized. Apreciseanswerto r
why this effect still happens in the multi filamentation
which nullifies for equal intensity traces and tends to 1
regime (P (cid:38) 150 P ) and what is the role of the tur-
cr for very dissimilar ones. The overall symmetry degree
bulent, by nature, multi filament dynamics [17] requires
is obtained by comparing all possible pairs of intensity
further work and understanding.
traces:
VI. CONCLUSION 1 (cid:90) 2π (cid:90) φ1
Φ≡1− 2π2 dφ1 dφ2wD¯(φ1,φ2)∈[0,1], (11)
0 0
We have demonstrated a self-symmetrization effect in
air occurring for high power laser beams experiencing wherethemeaningofΦisreversedfromthatofwD¯, i.e.,
multifilamentation and the action of an external focus- Φ is good (bad) for values close to 1 (0) and the nor-
ing force. The collective organization process is very malization factor is (cid:82)2πdφ (cid:82)φ1dφ =2π2. Eqns. (9-11)
0 1 0 2
different from self-cleaning and is linked to the Kerr trivially combine to give the symmetry degree presented
induced isotropic wavenumber redistribution leading to in Eq. (1). In the numerical analysis we have used the
beams with improved overall symmetry. Simulations are discreteversionoftheaboveequationtakingintoaccount
in excellent qualitative and quantitative agreement with only a finite number of intensity traces (φ → m,n),
1,2
the observed features of this effect induced by focusing which adopts the form:
8
sume the probability for this to be small). This could be
regardedasadrawbackofthissimilaritymeasureandits
2 M(cid:88)−1 (cid:88)M significance depends in reality on the nature of the ana-
Φ≡1− M[M −1] wD¯(m,n)∈[0,1]. (12) lyzeddata. However, theabsolutevaluebarsinEq. (10)
m=1n=m+1 ensure that little non-zero values of ωD¯ will accumulate
when performing the angular integrals (or sums), wors-
Note that because D¯ can take positive and negative ening the overall symmetry. To finalize, we stress here
values, wD¯ might be in principle nullified not only by that we tried six different ways (not specified here) of
identical intensity traces, but also for those which are obtaining symmetry indices and the one presented here
different but cross each other one or more times. By do- proofed (unlike all others) to give systematically a good
ing so we allow a pair of similar traces to be regarded as qualitative correspondence for all situations with differ-
equal(i.e.,toleranceinthiscaseisbetterthanifabsolute ent masks and input peak intensities. We recall that the
valuebarsareintroducedinEq. (9))butweriskthattwo purpose of Φ is the one of obtaining a global feature of a
very different traces are taken as equal (we therefore as- beam profile with complex structure.
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