S.L. Chin et al- Transverse ring formation of a focused femtosecond laser pulse propagating in air

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Transverse ring formation of a focused femtosecond laser pulse

propagating in air

S.L. Chin a, N. Akozbek b, A. Proulx a, S. Petit a, C.M. Bowden b,*

a Centre d'Optique, Photonique, et Laser (COPL) and Dept. de Physique, Universite Laval, Que., Qc, Canada G1K 7P4b US Army Aviation and Missile Command Research, Engineering and Development Center, Huntsville, AL 35898-5000, USA

Received 22 September 2000; accepted 20 November 2000

Abstract

We observe the formation of ring patterns of a focused femtosecond near IR laser pulse propagating in air before the

geometrical focal point. These rings are due to the combined eects of self-focusing and defocusing created by the

generated plasma via multiphoton/tunnel ionization of air. Qualitative agreement is found with numerical simulations,

using input conditions similar to those in the experiment. Ó 2001 Elsevier Science B.V. All rights reserved.

Keywords: Nonlinear optics; Self-focusing; Photoionization; Ultrafast processes; Nonlinear phenomena; Waves, Wave propagation

and other interactions

1. Introduction

The formation of ®laments in air by the use of

high-power femtosecond laser pulses has been the

subject of intense interest both experimentally and

theoretically [1±26] for the past several years. Po-

tential practical applications of this phenomenon

include lightning discharge control [1,2] and remote

sensing [12,14]. Besides its possible applications it

is also very interesting from a fundamental non-

linear dynamics point of view. The underlying

physical mechanism, which leads to the formation

of ®laments in air, is due to a dynamic competition

between self-focusing and defocusing created by the

generated plasma [22,23]. However, in general the

full dynamics of these pulses is complicated since

they undergo strong reshaping both temporally and

spatially. Due to the high intensity in the ®lament

incorporating measurement devices directly into

the beam is very dicult. To further elucidate this

phenomenon we use a focused laser pulse and re-

corded damage patterns on a silicate glass plate for

various propagation distances. We observe a com-

plicated ring formation on these plates, which are

attributed to self-focusing, and plasma defocusing.

To the best of our knowledge this is the ®rst ex-

perimental observation of ring formation with near

IR pulses, propagating in air. The damages are

scanned by a DekTak II pro®lometer. This allows

us to get a direct measure of the ablation pro®le,

which gives us a measure of the distributed trans-

verse ¯uence of the laser pulse. This in turn provides

information pertinent to the ®lament, which can be

compared with theoretical predictions. Numerical

simulations using similar laser input parameters

agree qualitatively with the experimental results.

1 February 2001

Optics Communications 188 (2001) 181±186

www.elsevier.com/locate/optcom

* Corresponding author. Fax: +1-256-9557216.

E-mail addresses: [email protected] (N. Akozbek),

[email protected] (C.M. Bowden).

0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.P II: S0 0 3 0 -4 0 1 8 (0 0 )0 1 1 2 9 -9



2. Experiment and results

The laser system consists of a Ti:sapphire os-

cillator followed by a regenerative and two multi-

ple pass Ti:sapphire ampli®ers that can deliver

pulses with energies of up to 100 mJ with a central

wavelength of 800 nm. In this experiment we used

a 350 fs (FWHM) pulse with energy of 85 mJ, and

a 320 fs (FWHM) pulse with energy of 75 mJ. The

beam diameter at 1=e2 of intensity was 1 cm. The

experimental setup is depicted in Fig. 1. The laser

pulse is focused by a lens with a focal length of

150 cm corresponding to an f -number F 150.

At various positions a silicate glass is inserted.

The photograph of the damage at the surface of

the plate was taken with a phase-contrast micro-

scope and the corresponding depth pro®le across

the diameter was taken with a DekTak scan. Fig.

2 shows the beautiful rings burnt into the glass at

a position of 149.5 cm just before the geometrical

focus. We observe in Fig. 2(a) a central damage

zone surrounded by a small unablated region

followed by a rather homogeneous ablated region,

which is then surrounded by a more complex

outer structure. The corresponding depth pro®le

across the damage diameter is shown in Fig. 2(b),

which exhibits indeed a strong central damagecrater followed by a secondary crater. The in-

tensity between the ®rst and secondary crater was

not high enough to create damage; thus the for-

mation of the damage pattern is a direct result

from the ¯uence pro®le of the laser pulse. The

outer ®ne rings do not show up on the pro®le due

to the resolution of the DekTak scanner, but a

third crater is slightly visible giving rise to another

ablated region. We show in Fig. 3, for slightly

dierent input parameters (320 fs (FWHM) and

75 mJ), the evolution of the depth pro®le fordierent propagation distances. Fig. 3(a) and (b)

are the depth pro®les before the geometrical

Fig. 2. Shown (a) is the damage pattern created by the laser pulse (350 fs, 85 mJ) at a position of 149.5 cm (before the geometrical

focus) and (b) the corresponding depth pro®le taken with a DekTek scan across the damage diameter, which is a direct measure of the

¯uence pro®le of the laser pulse.

Fig. 1. Shown is the experimental setup. The laser pulse is fo-

cused by a lens with a focal length of 150 cm. At various po-

sitions (before the geometrical focal point) a glass plate was

placed in front of the beam, creating a damage pattern on thesurface of the plate. The damage was then analyzed with a

phase contrast microscope.

182 S.L. Chin et al. / Optics Communications 188 (2001) 181±186



focal point at 143.5 and 149.5 cm, respectively.

Note in Fig. 3(a) the small peak in the center of

the main crater. This would translate as a dip in

the ¯uence pro®le. As we approach the geomet-

rical focal point this center peak disappears as

seen in Fig. 3(b). However, such a dip in the

¯uence reappears beyond the geometrical focal

point shown in Fig. 3(c) at a distance of 157 cm,

but the secondary rings have signi®cantly dimin-

ished. This dynamic behavior of the ¯uence pro-

®le is qualitatively in agreement with numerical

simulations as we discuss in detail in the next

section.

3. Theoretical predictions and discussion

We consider the propagation of an input col-

limated paraxial Gaussian beam focused by a lens:

Ar ; s; z 0 A0 expÀr 2=w20 s

2=s20À ikr 2=2 f ,

where w0 and s0 are the initial beam radius and

pulse width, respectively, f is the focal length

of the lens, and k n0k 0 n0x=c with n0 % 1.

The propagation equation including diraction,

self-focusing and plasma defocusing, for the slowly

varying envelope function Ar ; s; z is given as,

(in the retarded time coordinate frame s t À z =m g )

Fig. 3. The depth pro®le taken across the damage pro®le with the DekTek scan is shown at positions (a) 143.5 cm (b) 149.5 cm and (c)

157 cm. Here the input laser pulse width was 320 fs with an input energy of 75 mJ. Initially, there is a strong central crater followed by a

secondary crater, which indicates the formation of rings. Note the peak in the center part of the damage pro®le in (a) which disappears

in (b) and reappears again in (c). This peak would translate as a dip in the center of the ¯uence pro®le.

S.L. Chin et al. / Optics Communications 188 (2001) 181±186 183



io

o z

1

2k

o2

or 2

1

r

o

or

n2k 0j Aj

2 À2pe2 N e

kmec2 iC

Â Ar ; s; z 0; 1a

N e is the generated electron density via multipho-

ton/tunnel ionization and is governed by

o N e

os N 0 R: 1b

Here, n2 is related to the third order nonlinear

susceptibility of air. N 0 is the number density of

neural air molecules and the ionization rate R 0:2rO2

j Aj2n 0:8rN2

j Aj2m

where rO2, rN2

, n, m are

the cross-sections and eective multiphoton num-bers, respectively for O2 and N2 obtained from ex-

perimentally measured ionization rates [27], (Here

we assumed that air is 20% O2 and 80% N2) and C

describes ionization losses. We have neglected the

eect of dispersion since the ratio of the diraction

length of the focused laser beam to the dispersion

length is much smaller than unity. In addition we

consider an instantaneous Kerr response.

We integrate Eqs. (1a) and (1b) with initial

conditions as close as possible to the experiment.

However, due to the ®xed grid points used in the

numerical scheme a smaller input beam diameter

and focal length of the lens is used such that the f -

number is the same as in the experiment. The peak

input power in the experiment was about 0:2 TW

which is about 35±20 times the critical power for

self-focusing in air (experimental estimation of the

critical power lies between 6 and 10 GW [3,8±11]).

The input parameters considered in the numerical

simulations are taken as, w0 0:2 cm, s0 297 fs,

f 60 cm ( f -number F 150), ( z 0 pw20=k0 is the

diraction length of the collimated input beam),

and P 0 20 P cr, where P cr k20=2pn0n2 is the critical

power for self-focusing in the CW limit, which is

about 3 GW for n2 5 Â 10À19cm2=W [28]. This

critical value is somewhat smaller than estimated by

experiment, which could be due to the fact that thenonlinear response is not purely Kerr-like. In Fig.

4(a) the simulated ¯uence (normalized to the peak

input ¯uence) is plotted as a function of the trans-

verse coordinates x and y (measured in units of the

initial beam radius) before the geometrical focal

point at a position of 59.75 cm. Alternatively, in

Fig. 4(b) the positions of the rings are plotted on a

2-D plot. We clearly see the formation of the rings

of the ¯uence pro®le, which agrees well with the

experimental observations. There is a central part

followed by a secondary ring and further out we see

Fig. 4. Shown in (a) are numerical results for the ¯uence (normalized to the peak input ¯uence) pro®le depicted in 3-D as a function of

the transverse coordinates x and y measured in units of the initial beam radius and (b) is the corresponding ring structure plotted in 2-

D, where only the peaks of the rings are shown. The ¯uence distribution in (a) is in good qualitative agreement with the experimentally

observed damage pro®le shown in Fig. 2(b) in which the ring formation is clearly observed. The outer rings depicted in Fig. 2(a) are

also apparent in (b).






4. Conclusion

In conclusion, we observed the formation of

ring structure during the propagation of a focused

femtosecond laser pulse. These rings are due to the

combined eects of self-focusing, and defocusing

by the plasma. Numerical simulations at least

qualitatively agree with the ring formation before

and after the geometrical focal point. We plan to

investigate more quantitatively the ring formation

by including re¯ections and interactions with the

surface into our present theoretical model.

Acknowledgements

N.A. would like to thank the ®nancial support

from the National Research Council and is grate-

ful to Dr. M. Scalora for helpful discussions.

S.L.C. acknowledges the support of NSERC, the

Department of National Defence Canada via the

Defence Research Establishment Valcartier and le

fonds FCAR and would like to thank Drs. A.

Talebpour, O.G. Kosareva, and V.P. Kandidov

for fruitful discussions. The authors also would

like to thank S. Lagace and J. Yang for their

valuable technical help.

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S.L. Chin et al- Transverse ring formation of a focused femtosecond laser pulse propagating in air