N. Akozbek et al- Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation

8/3/2019 N. Akozbek et al- Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation an

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Propagation dynamics of ultra-short high-power laser pulses

in air: supercontinuum generation and transverse ring

formation

N. AKO ZBEK, C. M. BOWDEN

U.S. Army Aviation and Missile Command, AMRDEC, ATTN:

AMSAM-RD-WS-ST, Redstone Arsenal, Alabama 35898-5000, USA

and S. L. CHIN

De partement de Physique, de Ge nie Physique et dOptique & Centre

dOptique, Phtonique et Laser, Universite Laval, Quebec City,

Quebec, G1K 7P4, Canada

(Received 21 March 2001; revision received 28 June 2001 )

Abstract. Numerical and semi-analytical results of the propagation of high-power ultra-short near IR laser pulses propagating in ionizing air are presented.

1. Introduction

The formation of long laments in air by the use of high-power femtosecond

laser pulses has been the subject of interest both experimentally and theoretically

for the past several years [113]. The long-range propagation of these pulses

depends on the shortness of the pulse duration in which there is not enough time

for optical breakdown to occur. In other words the pulses must be shorter than the

characteristic time for collision in air (one picosecond) to avoid cascade ioniza-tion. For example, if one used a laser pulse with the same input power but longer

pulse length, then before the self-focusing threshold is reached the leading edge ofthe pulse produces strong plasma and the rest of the pulse becomes strongly

absorbed and scattered. Thus these pulses would only create a small plasma

volume. In that case, no propagation takes place and long-range propagation and

supercontinuum generation does not occur. When the pulse duration is short

enough to avoid breakdown of the medium, the pulse creates only low-density

plasma, which stops the self-focusing process and provides further balancing

between self-focusing and defocusing of the laser beam, which creates a self-

induced light lament in air. In this light channel the laser pulse can maintain its

power density and temporal integrity over long distances, in which a white-light

continuum is generated. The creation of a white-light source at a remote distance

gives the possibility of detecting and identifying simultaneously dierent atmos-

pheric and spectral components using a LIDAR type of conguration [4]. How far

these pulses can propagate in the atmosphere is still an open research topic and

depends strongly on the initial laser parameters, and initial and boundary con-

ditions. However, an experiment performed at the University of Jena in Germany

claims that a 2.2 TW and 120 fs laser pulse propagated at least 12 km in the

Journal of Modern Optics ISSN 09500340 print/ISSN 13623044 online # 2002 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/09500340110090396

journal of modern optics, 2002, vol. 49, no. 3/4, 475486
http://www.tandf.co.uk/journals


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atmosphere through the detection of the backscattered white-light [4]. The

detailed characteristic of the lament is not known in their experiment but they

do observe the generation of the white-light continuum along the lament.

In this paper the propagation phenomenon of such pulses propagating in air is

studied. It is shown that the long-range propagation is due to a dynamic competing

eect between self-focusing and plasma defocusing which alternates along thepropagation channel. The overall dynamics of these pulses is complicated owing to

the strong reshaping of the pulse both spatially and temporally. Theoretical studies

so far have been concentrated on the so-called slowly varying envelope approx-

imation (SVEA) [2,3,5]. The proposed model goes beyond this approximation to

show that the SVEA breaks down and does not correctly describe the white-light

continuum generation observed in air [11]. The paper also discusses transverse

ring formation of a focused near IR laser pulse propagating in air [12]. All results

are in qualitative agreement with the experiment.

2. Propagation model

Consider the propagation of a linearly polarized laser pulse in ionizing air with

a wavelength centred at 0 800 nm. The electric eld is assumed to be given byEr; z; tAr; z; teikxi!t c:c. The underlying propagation equation for the envel-ope function Ar; z; t, derived from the Maxwells equations, can be written in theretarded coordinate frame as follows [12]:

i@

@z

1

2kr2?

k00

2

@2

@2 n2k0jAj

2 2e2Ne

kmec2

A

i

!

@

@

1

2kr2?A n2k0jAj

2A

2e2NeA

kmec2

0:

1a

The electron density Ner; z; is generated via multiphoton ionization andobtained from

@Ne@

N0RjAj2: 1b

Here, k k0n0, where n0 1 is the linear index of refraction andn2 4 10

19 cm2 W1 is the nonlinear index of refraction of air. The coecient,

k00 0:2 fs2 cm1 (at 0 800 nm) describes group-velocity dispersion in air. The

inclusion of the dispersion due the plasma is more dicult, since each of part of the

pulse sees a dierent electron density and changes along the propagation distance.

If an average electron density is dened over the beam then one can dene an

average dispersion coecient for the plasma k00

p !2

p=c!3

0where !

pis the plasma

frequency. We predict an upper average plasma density of about 1016 cm3 for

which, k00

p 0:08 fs2=cm. Therefore we do not expect plasma dispersion to

become signicant. The number density of neutral air molecules is

N0 3 1019 cm3 and R njAj2n is the ionization rate for air (it is assumed

that air is a mixture of 80% nitrogen and 20% oxygen), where n=67 is the eective

order of the multiphoton/tunnel ionization process. Equation (1a) includes eects

such as diraction, group-velocity dispersion, self-focusing, plasma generation,

and higher-order nonlinear terms, which appear as rst derivative terms with

respect to time on the diraction, self-focusing, and plasma generation component.

476 N. Akozbek et al.


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We have neglected the nonlinear plasma eect, which becomes more important in

the relativistic limit. In the absence of these higher-order terms equation (1a) takes

the usual form obtained in the (SVEA).

3. Results and discussion

This section presents numerical results and discusses the underlying physical

mechanisms that lead to the formation of long laments in air. An initially

collimated Gaussian beam of the form Ar; A0er2=w2

02=2

0 is assumed, where

w0 and 0 are the initial beam radius and pulse width respectively, measured at 1=e2

of the intensity. Equation (1) is integrated with w0 0:025 cm, 0 85 fs andP0 6Pcr, where Pcr 3GW is the critical power for self-focusing in air. It isimportant to note that in most experimental conditions the initial beam radius is

much larger than used here for numerical simulations. However, for a larger inputbeam radius the ratio between the initial and self-focused beam radius is quite

large, which then poses a numerical diculty in resolving the self-focused beam on

the xed grid used in the numerical scheme. In this view, a smaller initial beam

radius is used to understand the important physical mechanism of this phenom-

enon thus a quantitative agreement between theory and experiment is not

expected.

3.1. Refocusing phenomenon and lament formation in air

In order to understand some of the main features of the propagation, rstresults are presented in which the higher-order terms are neglected in equation (1),

i.e. SVEA. In addition, group velocity dispersion is also neglected.

In gures 1 (b) and (c) the spatio-temporal intensity distribution is plotted at a

propagation distance (normalized to the diraction length of the collimated input

beam kw20=2) z=0.4 and z=0.8, respectively. As the pulse self-focuses, the peakintensity increases very rapidly until there is enough plasma to stop the focusing

process. The strongest part of the pulse will come to a focus rst followed by other

parts of the pulse. On the other hand, plasma generation is an accumulative

process and each slice experiences a dierent magnitude of plasma defocusing.Thus some of the earlier slices need to reach a higher peak intensity before being

defocused. This time-dependent focusing and defocusing process leads to the

temporal reshaping of the pulse. As seen in gure 1(b) there is a sharp leading edge

with a smoother back component but with further propagation a second pulse

appears at the back of the leading pulse, as seen in gure 1(c).

The main physical mechanism of the lament formation is due to a continuous

competing eect between self-focusing and defocusing. To understand this

competing mechanism further, gure 2 (a) plots the lament energy, dened as

the energy contained in a 150 micron diameter to the total pulse energy as a

function of the propagation distance. Initially due to self-focusing, more energy of

the pulse is channelled into the core region until there is enough plasma generated

to stop the self-focusing process, and the beam starts to defocus. However, the

defocusing is stopped and the pulse refocuses again, which can be seen as the

second peak in the lament energy. This process can repeat itself many times

which is apparent from gure 2(a) as a weak third peak in the lament energy. The

multiple refocusing phenomenon has been observed in experiments [2, 8], and our

results are in good qualitative agreement. Figure 2(b) shows the generated

Propagation dynamics of ultra-short high-power laser pulses 477


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Figure 1. The spatio-temporal intensity distribution of (a) an initially Gaussian pulsepropagating in ionizing air at (b) z 0:4 and (c) z 0:8. The intensity is normalizedto the peak input intensity and the radius and time coordinates are scaled to theinitial beam radius and pulse width, respectively.


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Figure 2. (a) Filament energy and (b) corresponding generated electrons per length as afunction of the propagation distance (normalized to the diraction length). As thepulse self-focuses more energy is channelled into the core region in which thelament energy increases until there is sucient plasma generated so that the pulsestarts to defocus. However, the pulse refocuses again around z 0:55 as seen in thesecond peak in (a). A third refocusing appears around z 0:8. This refocusing isconsistent with the generated electrons (b) in which more electrons are generatedduring the refocusing of the pulse. In (c) a particular intensity slice, I(0,0) is plotted

as a function of the propagation direction.


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electrons along the propagation direction and clearly this agrees with the refocus-

ing discussed in the lament energy description. Whenever the pulse refocuses

more electrons are generated which are seen as peaks in gure 2(b) and their

location agrees well with the peaks in the lament energy depicted in gure 2(a).

Alternatively, one can examine each temporal slice of the intensity as a function of

propagation distance. In gure 2(c) we plot the I(0,0) slice which has the highestpeak power. It will come to a focus rst and the peak intensity increases until

plasma defocusing stops the self-focusing process and it starts to defocus, but it

defocuses until self-focusing takes over again. The next section describes briey

how the balance between self-focusing and defocusing occurs and describes the

refocusing phenomenon using a very simple model.

The refocusing of the pulse channel energy back into the core of the beam,

which is one of the important physical mechanisms of the long-range propagation

and lament formation in air.

3.2. Variational method

As seen in the previous section, the dynamics of these pulses is complex owing

to the reshaping of the laser pulse both temporally and spatially. A variational

method [9] has been developed in which a semi-analytical result can be obtained

without resorting to lengthy computational simulations. In particular, it shows

explicitly how the self-focusing process is stopped by plasma defocusing and thus

provides direct physical insight into an otherwise complicated problem. The

variational method is based on dening a Lagrangian functional L for the systemfrom which the equations of motions can be derived from

L

A

@

@z

@L

@Az

@L

@A 0: 2

An appropriate trial solution for A(r,z,t), with sucient variational parameters, is

then inserted into the Lagrangian and integrated over the transverse coordinates.

In this case the higher-dimensional problem is reduced to a one-dimensional

problem. From the equations of motions a set of rst-order dierential equationscan be derived for the variational parameters with respect to the propagation

direction. Using the simplest propagation equation in which group-velocity

dispersion and higher-order terms are neglected the equations of motion can be

reduced to a single equation for the beam radius a(z,t),

1

2

@a

@z

2Ua 0: 3

Equation (3) describes the motion of a classical particle moving in a potential well

U(a). Here a describes the position of the particle and z acts as ctitious time

variable. The potential is shown in gure 3(a) for the case when the initial peak

power exceeds the threshold power for self-focusing. In the absence of plasma

defocusing the particle is released from rest and since the potential diverges

( ) it approaches to zero, which corresponds to beam collapse, and hasbeen studied widely. On the other hand, in the presence of plasma defocusing,

self-focusing is overcome by the plasma defocusing and the particle comes to rest

at a minimum beam radius, amin. However, it then rolls back, refocusing the beam,

and reaches its initial position. Since no losses are included in the motion, it



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oscillates between two points in the potential creating the self-focusing and

defocusing of the laser beam. This can be seen in gure 3(b) in which the

corresponding beam radius is plotted as function of propagation distance. This

self-focusing and defocusing is in qualitative agreement with numerical results

shown in gure 2(c) in which one particular time slice is plotted as a function of

propagation direction and the competing eect between self-focusing and defocus-

ing agrees well with the potential well description. It is important to note that a


Figure 3. (a) Potential Ua in which the particle moves as a function of a (normalizedto the input beam radius) and (b) the beam radius az as a function of propagationdistance z. The solution corresponds to the case where the particle is released froma 1 at rest. When there is no defocusing eect the potential diverges () and theparticle (*) rolls towards a 0, i.e. beam collapse. However, in the presence ofdefocusing there is signicant change in which the defocusing eect becomesdominant at some position and stops the focusing (). Here the particle (*) willoscillate between two points. This oscillation manifests itself as a alternating self-focusing and defocusing of the beam radius shown in (b).


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quantitative agreement is not expected between the numerical and variational

results since in the trial solution the functional dependence of the amplitude and

phase are xed. Nevertheless, using this method we were able to study the entire

pulse dynamics and obtained good qualitative agreement with experimental results

[8].

3.3. Supercontinuum generation

Having shown the main features of this phenomenon, we now examine theeect of the steepening terms on the pulse dynamics. Figure 4 plots the on-axis

total electron density generated during the propagation of the laser pulse. The

beginning of the lament is dened where the plasma is turned on at z=0.2 and

the end at z=1.3, where the electron density falls o rapidly. In order to

understand the eect of the higher-order terms on the pulse dynamics, in gure

5 the pulse spectrum is shown at z=1.0 in the case of the SVEA ( ) and non-SEVEA ( ). For reference the initial pulse spectrum () is alsopresented. It can be seen that the self-steepening of the pulse causes a much longer


Figure 4. Plotted is the on-axis electron density generated over the entire pulse asfunction of the propagation distance using the non-SVEA.

Figure 5. Power spectrum of the pulse at a propagation distance z 1:0 (in units ofdiraction length), in the case of the SVEA ( ) and non-SVEA ( ).The solid curve is the initial pulse spectrum at z 0. It is apparent that higher-order terms in equation (1) cause a much stronger blue shift when compared to the

SVEA. The spectral intensity is normalized to the peak input spectral intensity.


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blue shift. This is due to the fact that a shock is formed at the back of the pulse

depicted in gure 6. The overall spectral shape and frequency range is in good

qualitative agreement with the supercontinuum generation observed in air [4] and

rare gases [13].Although plasma defocusing contributes to the spectral broadening of the pulse

spectrum, the self-steepening pushes the pulse energy further to the blue and the

development of the shoulder in the spectrum is directly related to the shock

formation at the trailing edge of the pulse [11].

3.4. Transverse ring formation

The intensity of the lament is estimated to be high 1013 W cm2, and in this

case incorporating measurement devices directly into the beam is very dicult. In

order to understand this phenomenon further, a focused laser pulse was used anddamage patterns were recorded on a silica glass plate for various propagation

distances. A complicated ring formation is observed on these plates, which are

attributed to self-focusing, and plasma defocusing. To the best of our knowledge

this is the rst experimental observation of ring formation with near IR pulses,

propagating in air. The damage is scanned by a DekTak II prolometer. This

allows a direct measure of the ablation prole, which gives a measure of the

distributed transverse uence of the laser pulse. This in turn provides information

pertinent to the lament, which can be compared with theoretical predictions.

In this experiment a 350 fs laser pulse with 85 mJ energy was focused

externally by a lens with a focal length of 150 cm. In numerical simulations initial

conditions as close as possible to that of the experiment [12] are used and equation

(1) is integrated using the SVEA. The same f-number, F=150, is used as in the

experiment but an initial beam radius of 0.2 cm and a lens focal length of 60 cm

are used. The input power was taken as 20 times the critical power for self-

focusing in air, which is about 3GW. Figure 7 shows the uence distribution at

dierent propagation distances and the inset of each gure shows the experimental

prole of the damage created at the surface of the glass plate. Before the


Figure 6. The spatio-temporal intensity prole plotted with all higher-order termsincluded at a propagation distance z 1:0. Notice the sharp shock formation at theback of the pulse, which causes a strong blue shifting of the spectrum depicted ingure 5 ( ).


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Figure 7. Fluence (normalized to the peak input uence) plotted as function of theradius at positions (a) 59.5 cm, (b) 60.5 cm, and (c) 68 cm. Note the appearance ofthe dip in the centre of the uence prole which disappears in (b) and reappears

again in (c). Experimental results are shown in the insets.


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geometrical focal point at 59.5 cm shown in gure 7(a) there is a central dip in the

uence prole, which is also observed in the experiment, which shows up as a peak

in the centre of the crater. Around the geometrical focal point, gure 7(b) the dip in

the uence disappears and there is mainly a centre part with an outer ring

structure. This result is also in agreement with the experiment. Well beyond the

geometrical focal point, gure 7(c), it is seen that the outer ring structure hasdiminished but the dip in the uence has reappeared, which is also the case in the

experiment. Thus the numerical uence distribution evolution as a function of

propagation distance using similar experimental input conditions is in good

qualitative agreement with the experiment. To the best of our knowledge such

ring formation has not been observed before with near IR laser pulses propagating

in air.

4. Conclusion

In conclusion, it has been shown that ultra-short near IR laser pulses can

propagate in air owing to the competing eects of self-focusing and plasma

defocusing. The propagation of such laser pulses is complex due to the high

nonlinear interaction between the laser pulse and air molecules. However, a

variational method has been used to describe how the self-focusing and defocusing

alternates along the propagation direction, leading to the formation of a self-

induced lament in air.

The model, which goes beyond the slowly varying envelope approximation,correctly describes the white-light continuum generation in air. It has been shown

that the higher-order correction terms to the propagation equation push the pulse

energy into the blue, creating a shock front at the back of the pulse. It is believed

that the formation of the shock formation has a close connection with the super-

continuum generation in air. It has also been shown that during the propagation of

these pulses a complicated ring structure is formed and a good qualitative

agreement with the experiment was obtained.

Acknowledgments

N.A. is a National Research Council (NRC) associate working at AMCOM and

would like to thank the NRC for the nancial support of this research project.

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N. Akozbek et al- Propagation dynamics of ultra-short high-power laser pulses in air: supercontinuum generation and transverse ring formation