W. Liu et al- Multiple refocusing of a femtosecond laser pulse in a dispersive liquid (methanol)

8/3/2019 W. Liu et al- Multiple refocusing of a femtosecond laser pulse in a dispersive liquid (methanol)

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Multiple refocusing of a femtosecond laser pulse in a

dispersive liquid (methanol)

W. Liua,*, S.L. China, O. Kosarevab, I.S. Golubtsovb, V.P. Kandidovb

a Departement de physique, de geenie physique et doptique and Centre doptique, photonique et laser,

Universitee Laval, Que., Canada G1K 7P4b International Laser Center, Physics Department, Moscow State University, Moscow 119992, Russia

Received 30 April 2003; received in revised form 10 July 2003; accepted 14 July 2003

Abstract

In this work multiple refocusing of femtosecond pulse in liquid (methanol) is studied both experimentally and

numerically. We focused a 38 fs laser pulse, which is spatially filtered, into a glass cell containing a very dilute solution

of Coumarin 440 in methanol. Via three photon fluorescence of coumarin, up to 6 refocusing peaks were observed along

the direction of the pulse propagation. This is the first evidence showing a way to control multiple refocusing.

2003 Elsevier B.V. All rights reserved.

PACS: 42.65.Jx; 42.65.Vh; 32.80.Wr

1. Introduction

Propagation of femtosecond laser pulses in bulk

transparent media is the subject of considerable

current interest. The reason for this is the possi-

bility to localize ultrashort radiation inside the

material. The localization may be achieved due toeither geometrical focusing and/or self-focusing. In

the latter case, the result is a continual localization

of a high-intensity spot (zone) of the pulse along

the propagation direction together with a modifi-

cation of its spatial, temporal and spectral prop-

erties. The consequence is what we call a chirped

white light laser pulse [1,2]. It leaves behind a long

weak plasma column popularly called filament

because of its small diameter (of the order of 100

lm in air and several microns in condensed mat-

ter). The processes accompanying this nonlinear-

optical self-transformation are important formany applications. For example, the white light

laser pulse formed in air serves simultaneously as

the white light source and as the tool for delivering

this white light to the point of investigation [17],

which is important for atmospheric remote sens-

ing. The plasma channels, accompanying filamen-

tation, have potential applications for lightning

control [79]. Liquids serve as the compact source

of the white light generation [10]. In transparent

solid dielectrics nonlinear-optical ultrashort pulse

Optics Communications 225 (2003) 193209

www.elsevier.com/locate/optcom

* Corresponding author. Tel.: +1-418-656-2131ext4482; fax:

+1-418-656-2623.

E-mail address: [email protected] (W. Liu).

0030-4018/$ - see front matter 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2003.07.024
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


2/17

self-transformation can be used for writing wave-

guides with prescribed parameters [1115].

In the first experiments on high-power femto-

second laser pulse propagation in air, the fila-mentation produced by 50 GW 250 fs collimated

pulses was considered as self-guided propagation

of the radiation with nearly constant diameter of

the transverse fluence distribution (%100 lm)along the direction of the propagation [16,17].

Further experimental study performed with 38

GW 250 fs collimated pulses in air demonstrated

that the energy contained in the 500 lm pinhole

centered on the beam axis was changing non-

monotonically along the propagation direction

[18]. This effect was entitled refocusing [19].

Theoretically and numerically it was shown that

the energy distribution in the filament is the re-

sult of the complex spatio-temporal transforma-

tion of the radiation in the conditions of Kerr

self-focusing, ionization and plasma-induced de-

focusing, material dispersion and diffraction [19

22].

The following interpretation of the nonmono-

tonic dependence of the filament energy on dis-

tance was obtained in [1,18,19]. The initial

increase in energy in the near-axis part of the

pulse caused by self-focusing is replaced by adecrease in energy due to the plasma-induced

defocusing and multiple ring formation at the

trailing part of the pulse. In the course of defo-

cusing the intensity decreases along the beam axis

and simultaneously increases in the rings sur-

rounding the filament [23,24,32]. The ionization

energy loss is not large and the power in the ring

structure of the pulse exceeds the critical power

for self-focusing in the medium. Therefore the

high-intensity/power rings are again contracted

towards the beam axis and the energy containedin the small pinhole placed on the beam axis

increases. Numerical simulations demonstrate the

following. Refocusing is the result of a strong

energy interchange between the narrow high-in-

tensity near-axis part of a slice (or a thin zone

along the propagation axis) of the pulse and the

whole low-intense periphery surrounding the fil-

ament [22]. The transverse size of this interaction

region is of the order of or larger than the beam

diameter at the laser system output.

In particular, experimental study of the refo-

cusing phenomenon in air and molecular nitro-

gen was performed in [25,26]. Due to the high

intensity in the filament, precise measurements ofthe spatio-temporal intensity distribution of the

pulse in the course of the propagation are diffi-

cult. An alternative way of determining the in-

formation about the intensities inside the

filament is offered by the observation of quan-

tities outside the interaction volume. Since exci-

tation and ionization of atoms and molecules in

intense laser pulses are highly nonlinear pro-

cesses, the measurements of fluorescence spectra

provide us with the information on the intensity

distribution along the filament. In [26] fluores-

cence measurements have been performed after

focusing 1100 mJ 250 fs pulses centered at 800

nm into an interaction chamber filled with N2 at

0.63760 Torr pressure. The fluorescence spectra

were averaged over 100 laser shots. The non-

monotonic dependence of the fluorescence signal

on distance along the filament indicates refo-

cusing of the radiation.

The experiments on refocusing in gases were

performed with multiple laser shots, because of the

difficulties associated with the registration of

the refocusing peaks of the intensity separated bythe distance of the order of 1 [25,26] to 10 [18] m in

a single-shot pulse.

In condensed matter there is a possibility to

register refocusing peaks in a single-shot pulse

since the filament length varies from several

hundred microns to several centimetres depending

on the experimental conditions [1114,27,28]. In

fused silica such measurements were done in [29],

where 120 fs 800 nm pulses were focused in a

sample using the objectives with different nu-

merical apertures. The number of refocusingpeaks in the filament registered in a single-shot

pulse was increasing from two to three with in-

creasing pulse energy from 0.25 to 1 lJ and the

intermediate value of the numerical aperture of

the objective is equal to 0.4. With tighter focusing

into the sample no refocusing peaks were ob-

served. Theoretical interpretation of the results is

based on the semianalytical analysis of the beam

radius change with propagation distance in the

medium with instantaneous Kerr response and

194 W. Liu et al. / Optics Communications 225 (2003) 193209


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multiphoton ionization. Time dependence during

the pulse is included parametrically in the ana-

lytical solution for the beam radius. However,

filamentation of a femtosecond pulse is a strongly

nonstationary process and the dynamics of mul-

tiple peak formation along the filament can be

explained only on the basis of time-dependent

numerical simulations of the pulse propagation inthe conditions of diffraction, material dispersion

and nonlinear-optical interaction of the pulse

with the medium. Such simulations of the pulse

focused in condensed media confirm nonmono-

tonic distribution of fluence along the filament

[27,30].

More quantitative investigation of the multiple

peak formation and the dynamics is important for

understanding the supercontinuum generation, the

uniformity of the plasma channel created in the

medium and, hence, the quality of waveguideswritten in transparent solids.

In this paper we present the first study of the

dynamics of multiple refocusing in a single-shot

pulse in liquid and show a new way of making the

filamentation inside the liquid visible to eyes.

We dissolve some dye inside the liquid. Due to the

multiphoton fluorescence emission and the fact

that fluorescence intensity is directly related to the

laser pulse intensity, the intensity distribution

along the filament could be traced by studying the

corresponding fluorescence distribution. Via this

method 1 we observed multiple refocusing in

methanol dissolved with Coumarin 440 (Exciton).

Successive formation of refocusing peaks is pre-

sented for a wide range of input pulse energies and

analyzed numerically on the basis of the self-con-

sistent nonstationary model of the pulse propa-

gation in the conditions of multiphoton andavalanche ionization, self-focusing, material dis-

persion and diffraction.

2. Experimental setup

The experimental setup is schematically shown

in Fig. 1. Our CPA femtosecond laser system

(builded by Spectra Physics) consists of an oscil-

lator (Maitai, 28 fs and 80 MHz) and stretcher,

followed by a regenerative amplifier (Spitfire),which works at 1 kHz repetition rate. The ampli-

fied long pulse is finally compressed to 38 fs

(FWHM) (measured by positive light single-shot

autocorrelator). The output beam after the

Fig. 1. Experimental setup for the focusing of the laser pulse into the cell with methanol.

1 This idea was first proposed and used by H. Schroeder of

the Max Planck Institute for Quantum Optics in Garching,

Germany. He used a two-photon fluorescence dye. (Private

communication to S.L. Chin, 2002).

W. Liu et al. / Optics Communications 225 (2003) 193209 195


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compressor has a maximum energy of 640 lJ/

pulse. The central wavelength k 810 nm and thebeam diameter is 5 mm at 1=e2. A half wave plate

is located before the compressor and used tochange the output power. Our laser power fluctu-

ation is less than 3%. After the compressor, the

reflection from a beam splitter (R : T 1:5.6) pas-ses through two irises. The aperture diameter of

iris 1 is 0.7 mm. At a distance of 90 cm from this

iris 1, the iris 2 is set to filter out the high-order

diffractions. As a result of the spatial filtering a

beam with smooth transverse fluence distribution

(i.e., no hot spots) corresponding to the Airy

pattern is formed after the second iris. Between

those two irises, a shutter with 1 ms exposure time,

corresponding to our laser repetition rate, is inset

to select only one laser pulse during each opening.

The shutter works in a single-shot mode. After the

second iris, a lens, whose focal length is 33 cm,

focuses the laser pulse into a glass cell. The dis-

tances from the second iris to the lens and from the

lens to the cell entrance window are 2.45 and 40

cm, respectively. The glass cell is 10 cm long and 2

cm in diameter. Both the entrance and exit win-

dows are 1 mm thick. The cell contains methanol,

which has been dissolved with Coumarin 440. The

concentration of Coumarin 440 is 0.13% in g/g.The advantage to use Coumarin 440 is that its

fluorescence is excited via three photon absorption

at our laser wavelength. As a result, the interfer-

ence from the background light is less as compared

to two photon absorption-induced fluorescence.

Coumarin 440s fluorescence wavelength ranges

from 419 to 469 nm. It is visible to the eyes and can

be recorded by a CCD camera. So from the side of

the sample cell, a camera lens is used to image the

central axis of the cell on to a Cohu 4810 CCD

camera. The CCD dimension is 8.8 mm 6.6 mm.Up to 8 cm of the cells longitudinal axis can beimaged onto the CCD. In addition, a power meter

is put behind the beam splitter to calibrate the

energy going into the cell. We carefully checked

that in the case of pure methanol, no fluorescence

signal could be registered by our CCD, whether

the shutter is working at single-shot mode or

continuously opened. Therefore, we can conclude,

that the CCD image was purely due to three-

photon fluorescence from Coumarin 440.

3. Experimental results

Figs. 2(a)(e) show the dependence of the

fluorescence signal on distance along the cell filledwith the dilute methanol solution of coumarin.

There is one plot in the panel (a) and two plots in

each panel (b)(e). On the upper plots (with black

background) one can see the fluence of fluores-

cence signal registered on the CCD and its change

along the propagation direction from the left to

the right. The simulation results are presented in

the lower plot (see Section 4 for details). In the

experiment, the white light threshold was found to

be 0.42 lJ and we were able to record a single-shot

fluorescence signal starting from the energy of 0.54

lJ (all energy values are related to the pulse after

the first iris). From that point we increased the

input energy until 2.15 lJ (Figs. 2, panels (a)(d),

upper plots). The width of each image corresponds

to 8 cm and the left-hand edge of the image is at

1.6 cm from the left cell window. The long gray

vertical line in Fig. 2 indicates the geometrical

focus position, and it was determined by the peak

fluorescence intensity on the CCD with the shutter

continuously open at 0.3 lJ input energy, which is

well below the white light threshold (i.e., geomet-

rical focusing dominates the propagation insidethe cell). At this energy the intensity in the geo-

metrical focus is not high enough to cause the

fluorescence of coumarin in a single-shot pulse.

As we noted earlier, the signal recorded by the

CCD was mainly due to three-photon fluorescence

from Coumarin 440. Hence, the brightness of the

image at a certain distance inside the cell is pro-

portional to the cube of the laser radiation intensity

integrated over the pulse time and the transverse

coordinate at this propagation distance. Thus, the

CCD image characterizes the distribution of thelaser pulse peak intensity along the cell. In Fig. 2(a)

the CCD image consists of a single weakly pro-

nounced maxima slightly shifted from the geo-

metrical focus towards the cell entrance window.

The offset from the geometrical focus is associated

with self-focusing of the pulse in methanol. As

shown in Fig. 2(b), with increasing input pulse

energy the peak moves forward to the entrance

window and the intensity of the signal increases.

With further input pulse energy increase, the



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growth of the intensity saturates and a second peak

appears on the CCD image (Fig. 2(c)). Both peaks

move towards the entrance window with the first

one keeping the same intensity. The intensity of thesecond peak grows up to the level of the first peak

intensity and stops. Then the third peak appears

and develops in the similar way as the first and the

second maxima. In Fig. 2(d), four peaks are shown.

The maximum intensity is nearly the same for each

subsequent peak. Up to 6 consecutive peaks are

observed in Fig. 2(e). The last peak appears after

the geometrical focus (see upper plot).

To analyze the behavior of the fluorescence

peaks, we have plotted the position of each ex-

perimentally observed peak (number 16, 1 being

the first appeared peak) relative to the cell entrance

window as a function of input pulse energy (see the

curves marked by open symbols in Fig. 3). Each

maximum approaches the entrance window and

the spacing between the maxima decreases with

increasing input pulse energy.

Fig. 2. (ae) Upper plots: fluorescence signal registered by the

CCD camera in the experiment. (be) Lower plots: the simu-

lated fluorescence signal Fz given by Eq. (9) from Section 4 ofthe paper. For each particular energy in the range 0.742.15 lJ

the signal Fz is normalized to its maximum value Fmax. Inputpulse energy is (a) 0.54 lJ, (b) 0.74 lJ, (c) 1 lJ, (d) 1.6 lJ, (e)

2.15 lJ. The first peak position in the simulations is adjusted to

the first peak position in the experiment independently for each

particular energy. Vertical line indicates the geometrical focus

position in both experiment and simulations.

Fig. 3. Refocusing peak position inside the cell as the function

of input pulse energy. No adjustment in the simulated and ex-

perimentally obtained peak positions is performed. Peaks are

numbered from the cell entrance window (see the corresponding

number near each experimental peak position). First peak: open

squares experiment, filled squares simulations; second peak:

open circles experiment, filled circles simulations; third

peak: open up triangles experiment, filled up triangles

simulations; fourth peak: open down triangles experiment,

filled down triangles simulations; fifth peak: open diamonds

experiment, filled diamonds simulations; sixth peak: cross ( ) experiment, cross (+) simulations.



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4. Numerical simulations

Our theoretical model for the nonlinear prop-

agation in methanol dissolved with coumarin isbased on the nonlinear envelope equation [31]

coupled to the equation for the free electron den-

sity. In addition, we take into account the three-

photon absorption of coumarin dissolved in

methanol. Assuming the propagation along the z-

axis with the group velocity vg, the equation for the

light field envelope Er;z; t is

2ikoE

oz

1

vg

oE

ot

1

i

x

o

ot1D

?E

kk00

x

o2E

ot2

2k2

n01

i

x

o

otDn

k

1

i

x

o

ot

Dnp

E ikaE; 1

where k 2pn0=k0 is a wavenumber, x is the lasercentral frequency. The first term on the right-hand

side describes diffraction; the second term de-

scribes the group velocity dispersion. In the third

term we take into account the nonlinearity of the

medium and the last term is responsible for

the energy losses due to electron transitions to the

conduction band in methanol and due to three-

photon excitation of coumarin. Linear absorptionwas considered to be negligible in comparison with

the nonlinear losses. The operators 1 i=xo=ot describe self-steepening of the pulse in thecourse of propagation.

The nonlinearity consists of the instantaneous

Kerr and the plasma contribution. The Kerr con-

tribution Dnk is given by

Dnk 1

2n2effjEj

2; 2

where n2eff is defined through the experimentallyobtained critical power for self-focusing in the

solution: n2eff 3:77k2=8pn0Pcrit % 10

16 cm2/W.

In turn, the critical power Pcrit 8 MW was ob-tained using the fact that Pcrit is close to the white

light appearance power [19,35]. The latter we

found from the input pulse energy of 0.42 lJ

measured after the first iris (see Section 3 of the

paper). When calculating the effective critical

power for self-focusing we took into account

%16% loss of energy on the second iris as well as

%12% loss on the focusing lens and the cell en-trance window since there is no antireflection

coating on them. Thus, the input pulse energy is

decreased to 0.31 lJ. Introduction of the effectivevalue of the nonlinear coefficient n2 allows us to

take into account noninstantaneous nonlinear re-

sponse of the solution [33]. Note that characteristic

response time of the Kerr nonlinearity in methanol

is approximately 100 fs [34]. Therefore, on a 30 fs

time scale mainly the electronic nonlinear response

is important.

The plasma contribution Dnp is given by:

Dnp 1

2

x2p

x

2

m2

c i

mc

x

x2p

x

2

m2

c!: 3

Here x2p 4pe2Ne=m is the plasma frequency

and Ner;z; t is the electron density in the con-duction band of methanol, mc Naverc is theelectron collision frequency, expressed through the

density of neutrals Na, root-mean-square electron

velocity ve and the electron collision cross-section

rc. The velocity ve is proportional to the square

root of the laser intensity. The equation for the

electron density Ne is [36,37]:

dNedt

RjEj2Na Ne miNe bN2e : 4

The avalanche ionization frequency is mi 1=Wge

2jEj2=2mx2 m2c mc, where Wg is the band

gap energy of methanol. The optical-field-induced

ionization rate RjEj2 depends on the light inten-

sity and is calculated according to [38], where the

order of the multiphoton transition to the con-

duction band of methanol is M 5 (band gap inmethanol is Wg 6:2 eV). The radiative electronrecombination coefficient is given by [36]:

b 8:75 1027

T9=2Ne cm

3=s; 5

where T is the electron temperature in the laser-

produced plasma in electron Volts.

The excitation of coumarin molecules occurs

through the three-photon absorption of the laser

radiation with the central frequency x, corre-

sponding to the wavelength k 810 nm. Thedensity of the excited molecules NCe is calculated

from the equation:



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dNCe

dt NC NCe

rKIK

KhxK; 6

where NC is the density of coumarin in the ground

state, K 3 is the order of the multiphoton pro-cess, rK is the three-photon absorption cross-

section, I cn0=8pjEj2

is the light field intensity.

Knowing the expressions for the free electron

density and the density of the excited molecules of

coumarin, we can calculate the nonlinear absorp-

tion coefficient a on the right-hand side of Eq. (1):

a NCrKIK1 hx

1K

Mhx RI Na Ne miNe=I; 7

where the first term on the right-hand side of Eq.(7) describes the absorption due to the three-pho-

ton excitation of coumarin molecules and the

second term describes the absorption due to the

transition of electrons to the conduction band in

methanol and due to avalanche electron forma-

tion.

The light field distribution before the first iris

corresponds to the one used in the experiment:

Er; s;z zin E0 expr2=2a20

exps2=2s20; 8

where s0 24 fs (corresponding to 38 fs FWHM)and a0 1:7 mm (corresponding to the diameter5 mm at 1=e2 intensity level), s tz=vg is thetime in retarded coordinate system, zin is the co-

ordinate of the first iris in Fig. 1. In the simula-

tions we reproduced the geometry of the optical

system delivering the pulse into the cell with

methanol. This system (Fig. 1) includes iris 1, iris

2 and the focusing lens. The transmission of the

pulse through the iris 1 is modelled by cutting the

light field given by Eq. (8) at r 0:35 mm andsmoothing the transverse intensity distribution bya supergaussian function for all r> 0:35 mm. Thepulse propagation between the first and the sec-

ond iris for the distance of 90 cm was simulated

according to the theory of linear diffraction (i.e.,

the only term considered on the right-hand side of

the Eq. (1) was D?E). On the second iris we re-

moved the higher order diffraction maxima by

cutting the beam at the first local minimum

(r% 1:2 mm) and smoothing the intensity by a

supergaussian function for all r> 1:2 mm. Afterthe second iris the pulse was focused into the cell

with methanol. Again, the propagation for a

distance of 40 cm between the lens and the en-trance window to the cell with methanol was

simulated assuming the linear diffraction regime.

The beam radius on the entrance window was

aw 135 lm at 1=e intensity level. To verify themodelled beam shape introduced by the first iris

we compared the simulated and experimentally

obtained beam diameter at a distance of 99 cm

after the first iris without the focusing lens. In

both experiment and simulations the beam di-

ameter at 1=e intensity level was 1.3 mm. Notethat in order to obtain the geometrical focus at a

distance %8 cm inside the cell, the geometricalfocusing distance of the lens in the experiment

had to be chosen 35 cm instead of 33 cm.

We note that in the course of the simulations of

propagation between the first iris and the metha-

nol we did not take into account the group velocity

dispersion and self-focusing in air, the focusing

lens, and the cell entrance window. Indeed, the

dispersion length in glass is approximately 2 cm

for a 40 fs pulse if the coefficient k00x % 3:6 1028

s2/cm [39] was used, while the entrance window is 1

mm thick and the lens is 2 mm thick. The disper-sion length in air is of the order of 50 m (compare

with less than 2 m of propagation in air in our

experiment). The self-focusing distance of the

beam with the radius aw 135 lm in the cell en-trance window is 1.3 cm (compare with 1 mm thick

entrance window) for the input pulse energy of 2

lJ and the critical power for self-focusing in glass 2

MW. The self-focusing distance inside the lens is

much larger than 1 cm since the beam radius on

the lens is approximately 1 mm.

The nonlinear propagation in methanol dis-solved with coumarin started from the cell en-

trance window. The pulse energy after the first iris

ranges from 0.5 to 2.5 lJ. The following parame-

ters of the medium were considered: the density of

neutral molecules of methanol Na 1:49 1022

cm3, the electron collision cross-section rc 1015 cm2 [36], the group velocity dispersion coef-

ficient of methanol, obtained by extrapolating the

data on the refractive index dependence on wave-

length presented in [40], is k00x 0:8 1028 s2/cm.



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To take into account the coumarin dissolved in

methanol, we calculated the density of coumarin

molecules in the ground state to be NC 3:5 1018

cm3

. Since the three-photon absorption cross-section of Coumarin 440 is not, to our knowledge,

published in the literature, we used the typical

value of three-photon absorption cross-section for

dyes rK 1082 cm6 s2/ph2.We calculated the three-photon fluorescence

yield according to:

Fz 2p

Z10

rdr

Z11

NC NCer; s

rKIK r; s

Khx

K1ds; 9

where K 3 and Ir; s is the light field intensity inthe cell. The dependence of the fluorescence signal

on distance is shown in Fig. 2, where the simula-

tion results are presented on the lower plots of

each panel (b)(e). Note, that for each energy va-

lue on plots (b)(e) the first peak position in the

simulations is independently adjusted to the first

peak position in the experiment in order to stress

the relative but not the absolute peak position.

Therefore, we can see that the simulated fluores-

cence distribution along the cell is in agreementwith the experimental data. Both in the experiment

and in the simulations we observe multiple peak

formation with increasing input pulse energy. For

a given value of energy the number of peaks in the

simulations corresponds to the one in the experi-

ment and increases from 1 to 6 for the energies

from 0.54 to 2.15 lJ (Figs. 2(a)(e)). Peaks are

moving towards the cell entrance window and the

spacing between them gradually decreases. Good

quantitative agreement can be noticed for the

spacing between the simulated and experimentallyobtained first and second peaks (Figs. 2(c)(e)).

The third peak in the simulations is slightly behind

in Fig. 2(d) but catches up with the experimentally

obtained third peak in Fig. 2(e). The experimental

and simulated positions of the fourth and the fifth

peaks in Figs. 2(d) and (e) correspond to each

other. On the experimental plot of Fig. 2(e) we can

see just an indication of the sixth peak near the

geometrical focus, while in the simulations this

peak is pronounced more clearly and shifted

towards the entrance window. One possible reason

for the discrepancy in the sixth peak position may

be in the value of the group velocity dispersion

coefficient. The effect of material dispersion onmultiple peak formation and positioning increases

with the increase in the peak number (see Section 6

of this paper).

In order to analyze the dynamics of intensity

peaks formed with increasing input pulse energy,

we plotted the distance between the cell entrance

window and the position of each local fluorescence

maximum numbered from 1 to 6, starting from the

cell entrance window, as the function of energy

(Fig. 3). In contrast to the plots shown in Fig. 2,

no adjustment in the first peak position has been

made. Both in the experiment and in the simula-

tions the peak distance from the cell entrance

window decreases monotonically as the pulse en-

ergy increases and each subsequent peak is formed

further in the propagation direction. Simulta-

neously, the spacing between the neighboring

peaks decreases. In the simulations the first and

the second fluorescence maxima are closer to the

cell entrance window than the first and the second

maxima in the experiment, while the third, the

fourth and the fifth maxima positions are in good

agreement with the experiment. The positions ofthe first peak and, to a large extent, of the second

peak are defined by the self-focusing of the beam

entering the cell. The self-focusing distance inside

the cell is very sensitive to the beam radius aw on

the cell entrance window. A 20% increase in the

value ofaw 135 lm leads to the increase in dis-tance between the cell entrance window and the

first fluorescence peak by %0.6 cm. This shift valueis of the order of the discrepancy between the

simulated and experimentally obtained values of

the peak positions (compare the curves markedwith open squares and filled squares in Fig. 3). A

20% change in aw may be due to the local fluctu-

ations in the phase front of the initial beam en-

tering the first iris. The distances of the higher

order peak (>2) formation are mainly defined bythe refocusing process. Therefore, the simulated

peak positions are in agreement with the experi-

mental ones, except for the sixth peak. The de-

layed formation of the sixth peak was commented

earlier.



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5. Multiple refocusing phenomenon

The physical mechanisms that cause multiple

refocusing in our experiment are self-focusing,material dispersion, ionization of methanol

through 5-photon and then avalanche process as

well as the three-photon absorption by coumarin

molecules. Initially, the pulse contraction in

both space and time is due to the self-focusing

of each slice of the pulse whose peak power is

higher than the critical power for self-focusing.

As the peak intensity reaches %1013 W/cm2 boththree-photon absorption in coumarin and free

electron generation in methanol stop self-focus-

ing and lead to the energy flow from the near-

axis region to the peripheral region of the

transverse beam section. That means no more

self-focusing. Simultaneously the narrow tempo-

ral peak formed by a group of self-focused slices

is broadened temporally due to the material

dispersion.

As the result, all the slices in the pulse become

wide enough both spatially and temporally. Dis-

persive broadening slows down. The intensity is

low and cannot cause any significant absorption

and ionization. However, if these slices still have

enough power, self-focusing would start all overagain resulting in spatial contraction as well as

temporal steepening. After some distance of

propagation, self-focusing is stopped again by the

other nonlinear effects. This cycle repeats as long

as there are slices in the pulse whose peak power is

higher than the threshold power for self-focusing

in methanol. As a result, a sequence of maxima in

the peak intensity and fluence distribution is cre-

ated along the pulse propagation axis.

The phenomenon of multiple refocusing is the

manifestation of the interchange of energybetween a wide background of the radiation and

the narrow near-axis region with r< 10 lm. Asthe demonstration of this effect, we present the

dependence of the pulse energy contained in dif-

ferent regions of the transverse beam section as a

function of the propagation distance (Fig. 4). The

input pulse energy is 1.6 lJ. The considered re-

gions are shown in Fig. 4(a). They include: the

circular region with the radius 10 lm centered on

the beam axis and two ring-shaped regions: the

first one with the inner radius 10 lm and the

outer radius 60 lm and the peripheral region with

the inner radius 60 lm and the outer radius 590

lm. On the cell entrance window (z 1:6 cm onthe plots) 0.5% of the total input pulse energy W0is in the central region (Fig. 4(b)), 14% is in the

first ring-shaped region (Fig. 4(c)) and 85.5% is in

the peripheral region (Fig. 4(d)). The distribu-

tions of fluorescence signal F and the total pulse

energy W along the cell are shown in Figs. 4(e)

and (f), respectively.

During the formation of the first fluorescence

peak, the background low-intensity light field

converges from the peripheral regions towards

both the central region and the first ring-shaped

region (z< 2 cm). Right before the first peak (z% 2cm), the energy in the central region rapidly in-

creases up to 5.3% due to the flow from the first

ring-shaped region, where the energy decreases

(compare Figs. 4(b) and (c) at z% 2:1 cm). Afterthe first peak the energy in the central region de-

creases till the level 2.8% of the total pulse energy

(z 2:2 cm). Simultaneously, the energy in the firstring-shaped region increases locally. Thus, the en-

ergy flows from the near-axis region with

r< 10 lm to the region with 10< r< 60 lm.

Slightly further in the propagation (z% 2:5 cm) thelocal maximum is attained in the peripheral region

r> 60 lm (Fig. 4(d)) due to the energy flow fromthe ring 10< r< 60 lm. The next cycle of theenergy interchange starts at z% 2:6 cm from theenergy increase in the central region (Fig. 4(b)).

Again, one can see the energy extraction from the

ring 10< r< 60 lm and also from the peripheralregion (compare Figs. 4(b)(d) at z% 2:71 cm).This repeated energy interchange among the three

zones lasts till z% 8 cm. The last weakly pro-

nounced fluorescence peak is formed at z% 7 cmnear the geometrical focus. One may see the ten-dency of the peripheral energy increase in the whole

region of investigation 2< z< 8 cm (Fig. 4(d)).This increase is mainly associated with the strong

divergence of the self-focused peaks with a typical

transverse scale of the order of 10 lm. The dif-

fraction of the beam as a whole starts to contribute

to the peripheral energy increase only after the

geometrical focus z% 8 cm. In contrast, the region10 < r< 60 lm (Fig. 4(c)) is gradually loosing its



10/17

Fig. 4. Interchange of energy between the regions in the transverse beam section. Input pulse energy is 1.6 lJ. (a) Three different energy

regions in the transverse beam section: the near-axis region 0 < r< 10 lm; the first ring-shaped region 10 < r< 60 lm; the peripheralregion 60 < r< 590 lm. The change of energy as the function of propagation distance (b) in the near-axis region 0 < r< 10 lm; (c) in

the first ring-shaped region 10 < r< 60 lm; (d) in the peripheral region 60 < r< 590 lm. The change of (e) the normalized fluo-rescence signal F and (f) the total pulse energy W as the functions of propagation distance. Vertical dashed lines indicate the positions

of fluorescence peaks along the propagation direction.



11/17

energy by transferring it to the peripheral region.

Thus, we can observe the following direction of

the energy flow: initially the near-axis region

(Fig. 4(b)) takes the energy from the ring-shapedregion 10< r< 60 lm (Fig. 4(c)) and, due to thestrong divergence, passes it to the peripheral re-

gion. Therefore, the ring-shaped region looses en-

ergy (Fig. 4(c)), the peripheral region acquires

energy (Fig. 4(d)) and the energy in the near-axis

region between the peaks (see, e.g., 2:9 < z< 3:5 cm; 3:9 < z< 4:7 cm) remains at approxi-mately the same level until the geometrical focus is

reached. After the geometrical focus the transverse

intensity and wave front distortions are so strong,

that at this particular pulse energy of 1.6 lJ no

refocusing peaks can be formed.

The result of this energy interchange is the

multiple refocusing picture in the fluorescence

signal (Fig. 4(e)). Note that the overall decrease in

the input pulse energy W0 is around 10% for the

whole propagation distance (Fig. 4(f)). The ex-

traction of energy has a stepwise behavior and

each subsequent step is associated with the for-

mation of a new peak in the fluorescence distri-

bution in Fig. 4(e). The major part of the absorbed

energy is due to the three-photon process in cou-

marin.The destruction of fluorescence peaks in the re-

peated process of refocusing is not only due to the

spatial divergence of the radiation, but also due to

the strong material dispersion in methanol. Indeed,

the dispersion length in methanol is %5 cm fork00x 0:8 10

28 s2/cm and the pulse duration with

s0 24 fs. However, when the pulse self-focuses,the typical width of the temporal peak is 10 fs and

the corresponding dispersion length is %0.5 cm,which is approximately twice smaller than the

typical distance between the refocusing peaks inFigs. 2 and 3 which is of the order of 0.51 cm.

Thus, the intensity peak, created due to the self-

focusing, is diverged spatially due to the ionization

in methanol and absorption in coumarin and

broadened temporally due to the material disper-

sion. Also in the simulations we observe pulse

splitting into two or three subpulses, which is typ-

ical for the propagation in the medium with self-

focusing and group velocity dispersion. This effect

was theoretically predicted in [41,42] and confirmed

experimentally in [43]. Pulse splitting is demon-

strated in the inset to Fig. 5, where spatio-temporal

intensity distributions are shown in the first refo-

cusing peak (z 2:14 cm) and between the peaks(z 2:58 cm) for the case of propagation of a 1.6 lJpulse (Figs. 2(d) and 5(b)). Note a 10-fold decrease

in the maximum intensity between the peaks in

comparison with the one at the peak. In the peak

two subpulses are clearly pronounced, while

between the maxima, a third subpulse can be

observed.

The phenomenon of multiple refocusing in dis-

persive medium is sensitive to the value of the group

velocity dispersion coefficient k00x. Fig. 5 shows the

distribution of the fluorescence signal as a function

of propagation distance inside the cell for different

values of the coefficient k00x. Input pulse energy is

1.6 lJ for all the panels (a)(d) in Fig. 5. With

increasing dispersion coefficients from (a) to (d) the

temporal broadening is stronger and refocusing is

delayed along the z-axis. The distance between the

neighboring peaks increases while the number of

peaks within the image region (1.69.6 cm) de-

creases. For the range 0.5 10281.2 1028 s2/cmof the group velocity dispersion coefficient change,

the position, the intensity and the width of the

first, the second and the third peaks do not change alot. However, further increase in k00x up to

2.5 1028 s2/cm leads to the decrease in the num-ber of peaks located within the region 0 < z< 9 cmand increase in the spacing between the peaks.

6. Other factors affecting multiple refocusing

Another factor that affects the distance between

the peaks is the wavefront curvature of the radia-

tion introduced by the optical system deliveringthe beam into the cell (Fig. 1). We modelled the

change in the wavefront curvature by changing the

geometrical focusing distance f of the lens, located

after the second iris. Fig. 6 shows the fluorescence

distribution inside the cell in the case off 33 cm(Fig. 6(a)), f 35 cm (Fig. 6(b)) and f 37 cm(Fig. 6(c)) for the same input pulse energy of 1.6

lJ. The corresponding positions of the geometrical

focus in methanol are z% 4 cm, z% 7:5 cm andz% 12:5 cm from the cell entrance window. By



12/17

following the first four refocusing peaks numbered

from 1 to 4 in Fig. 6, one can see proportional

increase in spacing between the peaks with the

increase in the geometrical focusing distance. At

the same time, the relative intensity and the width

of the peaks remain nearly the same for the chosen

values off. Thus, as can be expected, the influence

of geometrical focusing is qualitatively different

from the influence of material dispersion. The

former equally affects all the peaks, while the latter

is mainly pronounced for the higher order refo-

cusing peaks with the number larger than 2.

Possible way of controlling multiple refocusing

is changing the pulse duration at the laser system

output. To study the effect of the pulse duration on

the multiple refocusing picture we performed the

simulations with several different values of the time

constant s0 in the light field distribution before the

Fig. 5. Simulated fluorescence signal Fz as the function of propagation distance inside the cell for different group velocity dispersioncoefficients in methanol. (a) k00x 0:5 10

28 s2/cm, (b) k00x 0:8 1028 s2/cm, (c) k00x 1:4 10

28 s2/cm, (d) k00x 2:5 1028 s2/cm.

Fluorescence signals in panels (ad) are normalized to the same maximumFmax attained forthe case ofk00x 0:5 10

28 s2/cm. Input pulse

energyis 1.6lJ. Theinset shows spatio-temporal intensitydistributions in the peak (z 2:14cm) and between the peaks (z 2:58cm) for

the case of (b) k00x 0:8 10

28

s2

/cm. Note different intensity scales on the plots for z 2:14 cm and z 2:58 cm. I0 1012

W/cm2

.



13/17

first iris (see Eq. (8)). Figs. 7(a)(d) shows thefluorescence signal inside the cell for different input

pulse duration s0 24 fs (38 fs FWHM, Fig. 7(a)),s0 18 fs (30 fs FWHM, Fig. 7(b)), s0 12 fs (20fs FWHM, Fig. 7(c)), s0 6 fs (15 fs FWHM,Fig. 7(d)). The input pulse energy is the same for

all pulse durations and equal to 1.6 lJ. The fluo-

rescence signal in Fig. 7(a) is the same as in

Fig. 2(d) and is shown for reference.

With decreasing input pulse duration from

s0 24 fs to s0 12 fs the first three refocusing

peaks remain approximately the same in bothmaximum fluence and relative position. However,

the fluorescence signal from the fourth refocusing

peak is the strongest for s0 24 fs, undergoes 1.4times decrease for s0 18 fs and 4 times decreasefor s0 12 fs in comparison with the case ofs0 24 fs. For the shortest input pulse durations0 6 fs the fourth refocusing peak does not showup on the propagation distance of 10 cm. This is

due to the increase in the dispersive broadening

with the pulse duration decrease. This broadening

is demonstrated in Fig. 7(e), where the spatio-temporal intensity distributions are shown for

three different pulse durations s0 6; 12; 18 fs. Theintensity distributions in Fig. 7(e) are plotted for

the z-positions located right behind the maximum

of the fourth fluorescence peak for s0 18 fs ands0 12 fs and in the z-position corresponding tothe maximum of the third fluorescence peak for

s0 6 fs. By measuring the time delay Ds betweenthe intensity maxima of the two subpulses

(Fig. 7(e)) one may see the increase in Ds from

Ds 34 fs for s0 18 fs to Ds 60 fs for s0 6 fs.At the same time, the characteristic temporal scaleobtained in the fluorescence peaks is the same for

all input pulse durations s0 under discussion. The

full width of the characteristic peak is approxi-

mately 10 fs at e1 intensity level and the intensity

% 1013 W/cm2 (see subpulses in Fig. 7(e)). There-fore, the interplay between the Kerr nonlinearity

and free electron generation is similar for the

pulses of the same initial energy and duration s0 in

the range 624 fs, because this interplay depends

Fig. 6. Simulated fluorescence signal Fz as the function of propagation distance inside the cell for different geometrical focusingdistances of the lens: (a) f 33 cm, focus position inside the cell is at z% 4 cm from the cell entrance window, (b) f 35 cm, focusposition inside the cell is at z% 7:5 cm from the cell entrance window, (c) f 38 cm, focus position inside the cell is at z% 12:5 cmfrom the cell entrance window. For all cases (ac) input pulse energy is 1.6 lJ, the group velocity dispersion coefficient is

k00x 0:8 1028 s2/cm, fluorescence signal in each panel is normalized to its own maximum Fmax. Peaks are numbered from 1 to 6 from

the cell entrance window.



14/17

on the characteristic temporal scale and maximum

intensity of subpulses. However, due to the over-

all temporal broadening introduced by material

dispersion, the distance between these sub-

pulses increases. As the result, it is harder for the

pulse to refocus and the number of refocusing

Fig. 7. Simulated fluorescence signal Fz as the function of propagation distance inside the cell for different input pulse duration s0.(a) s0 24 fs, (b) s0 18 fs, (c) s0 12 fs, (d) s0 6 fs. Fluorescence signals in panels (ad) are normalized to the same maximum Fmaxattained in the case of s0 24 fs. Input pulse energy is 1.6 lJ. Fluorescence peaks are numbered from 1 to 4 from the cell entrancewindow. (e) Equal-intensity contours for different input pulse duration s0. For the case of s0 18 fs and s0 12 fs intensity distri-butions are shown after the fourth fluorescence peaks, for the case of s0 6 fs the intensity distribution is shown after the thirdfluorescence peak. For all plots in panel (e) the lowest contour and the interval between the contours corresponds to I 0:5 I0, whereI0 1012 W/cm2.



15/17

peaks decreases with decreasing value of pulse

duration s0.

The experimental results discussed in this paper

are based on the fluorescence of coumarin dissolvedin methanol. It is important to be aware, at least in

the simulations, how the addition of the dye with a

high three-photon absorption coefficient can affect

the phenomenon of multiple refocusing in the

conditions of our experiment. Fig. 8(a) shows the

simulated fluorescence signals with consideration

of the nonlinear three-photon absorption in cou-

marin according to Eqs. (6) and (7). In Fig. 8(b) the

quantity given by Eq. (9) is plotted without the

consideration of the effect of coumarin on the pulse

propagation, i.e., the density of coumarin mole-cules NC 0 in Eqs. (6) and (7), but NC 6 0 in Eq.(9) when processing the simulated intensity distri-

bution. Comparing Figs. 8(a) and (b) we see that

addition of coumarin nearly does not affect the

position of the refocusing peaks, which mainly de-

pends on the self-focusing and dispersive properties

of the medium as well as the geometry of the ex-

periment. At the same time, the peaks are wider and

their maximum intensity is smaller with the pres-

ence of coumarin. In addition, the fine structure of

the peaks observed in pure methanol (Fig. 8(b))

disappears. This can be explained by the lower or-

der of the multiphoton process in coumarin (K 3)

in comparison with methanol (M 5). Indeed, inthe case of pure methanol the maximum intensity

achieved in the medium is five times larger than in

the presence of coumarin.

Higher value of the clamped intensity in pure

methanol leads to the peak narrowing. Large free

electron density produced from methanol mole-

cules leads to the strong defocusing and formation

of the subpeaks right after the refocusing peaks

(see, e.g., double-maxima structure in the vicinity of

the first peak in Fig. 8(b): z% 2:07 and %2.13 cm).

Thus, the addition of coumarin changes thedetails of the intensity distribution inside the cell.

However, the overall picture of multiple refocusing

in methanol is independent of the addition of

coumarin if its amount is 0.13% g/g or lower.

7. Conclusions

We performed uniqueexperimental observations

of multiple refocusing phenomena in a single-shot

Fig. 8. Simulated fluorescence signal Fz as the function of propagation distance inside the cell (a) with consideration of the effect ofcoumarin on the pulse propagation; (b) without the consideration of the effect of coumarin on the pulse propagation. The coumarin

density is NC 3:5 1018 cm3 and the input pulse energy is 1.6 lJ. Fluorescence signals in panels (a) and (b) are normalized to the

same maximum Fmax attained without the effect of coumarin on the propagation (case (b)).



16/17

pulse propagating in liquid methanol dissolved with

coumarin. The three-photon fluorescence from

coumarin was an indicator of the intensity distri-

bution along the propagation direction in the cellwith methanol. The number of refocusing peaks

increases from one to six with increasing input pulse

energy from 0.54 to 2.15 lJ. Analysis of the peak

position inside the cell with methanol demonstrates

that each peak approaches the cell entrance window

and the distance between the neighboring peaks

decreases with increasing pulse energy.

Our study shows that multiple refocusing is a

repeated process consisting of several stages. In the

initial stage the radiation is contracted both spa-

tially and temporally due to self-focusing. This

leads to the formation of the first intensity peak

inside the cell. Then three-photon absorption in

coumarin together with multiphoton and ava-

lanche ionization limits the intensity growth and

leads to the spatial divergence of the radiation.

Simultaneously, material dispersion in methanol

splits the peak into subpulses and leads to the

temporal broadening. The next stage, which leads

to the formation of the subsequent peaks, is due to

the fact that in the diverging part of the pulse the

power is higher than the critical power for self-

focusing. Therefore, the contraction starts again,etc. Numerical analysis reveals strong energy in-

terchange between the near-axis and peripheral

parts of the transverse beam section. The simu-

lated refocusing peak positions and shapes are in

agreement with the experimental ones.

Peak positions are very sensitive to the group

velocity dispersion coefficient. With increasing

material dispersion the distance between the peaks

increases, while the number of the peaks decreases.

The change in the geometrical focusing distance of

the lens leads to proportional change of spacingbetween the refocusing peaks.

With decreasing duration and the constant en-

ergy of the input pulse the relative position of the

first two to three refocusing peaks remains ap-

proximately the same, while the fluorescence signal

decreases. For a sub-20 fs pulse duration (the case

of s0 6 fs) the number of refocusing peaks de-creases in comparison with the case of the longer

pulse with the same initial energy. The reason for

this is the stronger effect of material dispersion on

a shorter pulse. Dispersive broadening is revealed

in the increasing distance between the subpulses in

the spatio-temporal intensity distribution.

Numerical simulations show that the additionof coumarin into the methanol changes the details

of the intensity distribution inside the cell. How-

ever, the overall picture of multiple refocusing in

methanol is independent of the additive if its

amount is 0.13% g/g or less.

The study of multiple refocusing widens the

concept of filament as an extended region along

which the intensity is changing weakly. In the

conditions of the group velocity dispersion of the

pulse with peak power much higher than the crit-

ical power for self-focusing, multiple refocusing

takes place and the filament is represented as a

sequence of high-contrast intensity peaks.

Acknowledgements

The experimental work was supported in part

by NSERC, DRDC-Valcartier, Canada Research

Chair, FTRNQ and CIPI. O.G.K., V.P.K. and

I.S.G. acknowledge the support of the Russian

Fund for Basic Research, Grant N 03-02-16939.

The encouragement of previous linkage grantsfrom NATO resulting in the current sustained

collaboration is appreciated by the Canadian and

the Russian partners.

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