STARK BROADENING APPROACH FOR - Université Laval

JENS BERNHARDT

STARK BROADENING APPROACH FOR MEASURING THE PLASMA DENSITY INSIDE A

FILAMENT INDUCED BY A FEMTOSECOND LASER PULSE IN A GAS OR GAS MIXTURE

Thèse présentée à la Faculté des études supérieures de l'Université Laval

dans le cadre du programme de doctorat en physique pour l'obtention du grade de Philosophiae Doctor (Ph.D.)

DÉPARTEMENT DE PHYSIQUE, GÉNIE PHYSIQUE ET D'OPTIQUE FACULTÉ DES SCIENCES ET DE GÉNIE

UNIVERSITÉ LAVAL QUÉBEC

2009

@Jens Bernhardt , 2009

Résumé

Suite à la propagation d 'une impulsion laser femtoseconde intense dans un milieu

gazeux, l'impulsion laser s'effondre sur elle-même et forme des filaments de lumière.

Ces filaments sont induits par un équilibre dynamique entre l'autofocalisation par effet

Kerr et la défocalisa:tion due au plasma « auto généré ». L'équilibre de ces deux effets

aboutit au phénomène universel. de la « saturation de l'intensité ». L'intensité saturée

est assez grande pour ioniser ou dissocier les différentes espèces de gaz par l'ionisation

multiphotonique j tunnel (MPlj TI) , produisant la fluorescence 'propre'.

Le phénomène de la filamentation est riche en concepts et applications. Ceci inclut la

détection et l'identification des gaz polluants, le contrôle de la foudre ou d 'une décharge

ou, finalement, la génération d'impulsions puissantes de peu de cycles.

L'objectif de cette thèse était. de développer un outil spectroscopique qui peut être

utilisé pour mesurer la densité de plasma à l'intérieur du filament. La connaissance de

ce paramètre clé est importante pour la caractérisation de la saturation de l'intensité

du processus de filamentation. Ce défi pourrait être relevé en développant une nouvelle

approche basée sur l'élargissement Stark de lignes atomiques du spectre d 'émission du

filament.

Cette thèse. traite des critères de l'applicabilité et de la validité de l'approche de

Résumé 111

l'élargissement Stark. L'approche de l'élargissement Stark est d'abord illustrée par la «

spectroscopie de plasma induite par filament» (FIBS) du plomb métallique. On montre

pourquoi la technique FIBS est avantageuse comparée à la spectroscopie conventionnelle

de plasma induite par laser nanoseconde (ns-LIBS). Ensuite, la preuve de l'applicabilité

de la méthode de l'élargissement Stark à un milieu gazeux (utilisant l'argon comme

exemple) est fournie. Elle s'avère. utile pour mesurer la densité de plasma à l'intérieur

du filament dans l'air ambient dans différentes conditions de propagation. De plus , la

saturation d 'intensité du processus de filamentation dans l'hélium est confirmée. Ceci,

en particulier, est effectué en mesurant les densités de plasma en fonction de l'énergie

et de la pression, respectivement.

La réalisation des objectifs ci-dessus serait profitable pour obtenir une meilleure

compréhension de la physique fondamentale et développer les applications mentionées.

Court résumé

Dans cette thèse, on présente le développement d 'une nouvelle approche qui peut être

utilisée pour mesurer la densité de plasma à l'intérieur d 'un filament induit par une im

pulsion de laser femtoseconde dans les gaz. Cette approche est basée sur l'élargissement

Stark de lignes atomiques de fluorescence du filament. L'approche de l'élargissement

Stark est d 'abord présentée par la « spectroscopie de plasma induite par filament» du

plomb métallique et est ensuite appliquée au filament dans l'argon, l'air et l'hélium.

Abstract

As a consequence of the propagation of an intense femtosecond laser pulse propagating

in a gaseous medium, the laser pulse collapses and forms light filaments. These filaments

are induced by a dynamic equilibrium between Kerr self-focussing and defocussing by

the self-generated plasma. The equalisation of these two effects results in the universal

phenomenon of "intensity clamping". The clamped intensity is high enough to ionize or

dissociate the different gas species by pure multiphoton/ tunnelling ionization (MPI/ TI) ,

pr,oducing 'clean' fluorescence.

The phenomenon of filamentation is ri ch in both concepts and applications. This in

cludes the detection and identification of gaseous pollutants, discharge/ lightning control

or, ultimately, the generation of powerful few-cycle pulses.

The objective of this thesis was to develop a spectroscopic tool that can be used to

measure the plasma density inside the filament. The knowledge of this key parameter is

important for the characterisation of the intensity clamping of the filamentation process.

It was proposed that this challenge could be realised by developing a new approach,

based on the Stark broadening of the atomic lines that can be observed in the filament

spectra.

This thesis discusses the criteria of the applicationj validity of the Stark broadening

Abstract VI

approach. The Stark broadening approach is first illustrated by the "filament-induced

breakdown spectroscopy" (FIBS) of metallic lead. It .is shown why the FIBS technique is

advantageous compared with conventional, nanosecond laser-induced breakdown spec

troscopy (ns-LIBS). Next , the proof-of-principle of the application of the Stark ·broa

dening approach to a gaseous medium (using argon as an example) is provided. This

proves useful to measure the plasma density inside the filament in the ambient air under

different propagation conditions. Moreover, the intensity clamping of the filamentation

process in helium is confirmed. This, in particular, is performed by measuring the

plasma densities as a function of energy and pressure, respectively.

The implementation of the above objectives would be beneficial for a better under

standing of the underlying physics and developing the mentioned applications.

Acknowledgements

First of aIl , 1 would like to thank my advisor, Prof. See Leang Chin for his direction, help

and support . He has not only guided me but also given me the freedom to pursue my

research topic. Through him, 1 have learned a lot about the physics and the scient ific

world. 1 would also like to thank his wife, Mrs. May Chin, who has warmly welcomed

me and is kind.

1 am indebted to Prof. Roland Sauerbrey for his invaluable support and advice ever

sin ce 1 have been his student in J ena.

1 am grateful to Prof. Jie Zhang, who has provided me with the unique opportunity

to work in his group in Beijing.

1 appreciate my piano studies withProf. Leonid Chizhik in Weimar.

1 owe special thanks to Dr. Weiwei Liu, Dr. Huailiang Xu and Dr. Francis Théberge.

They have helped and taught me so much. 1 am glad that 1 could work together with

them.

1 appreciate that 1 have had the opportunity to interact with the other students

and postdocs of Prof. Chin 's group, Jean-François Daigle, Dr. Guillaume Méjean,

Patrick Tremblay Simard, Dr. Quan Sun, Yanping Chen, Claude Marceau, Feng Liang,

Acknowledgements VUl

Dr. Zhen-Dong Sun, Ali Azarm, Yousef Karilali and Mehdi Sharifi.

l am fortunate to have collaborated with the people from the Defence department

(DRDC Valcartier) led by Dr. Jacques Dubois.

It is my pleasure to express my sincere grat itude to aIl the staffs in the physics

department and the Centre d 'Optique, Photonique et Laser. EspeciaIly, l would like to

acknowledge the 'technical assistance of Mr. Mario Martin.

l wish to thank Ms. Zhao Jing for aIl her love and support. l also wish to thank my

parents, my mother Chamaipon and my father Helge, and my sister Vip a for their care

and love. It is to them, l wish to dedicate my thesis.

FinaIly, l acknowledge the Government of Canada Awards -from the Canadian Bu

reau for International Education (Bureau canadien de l'éducation internationale) , as

respresented by Ms. Julita Palka, as well as Prof. Chin 'sfinancial support.

Copyright statement

J'autorise le Professeur See Leang Chin à utiliser cette thèse en totalité ou en partie.

Contents

Résumé .

Court résumé

Abstract

Acknow ledgements

Copyright statement

Contents

1 Introduction

2 Basic concepts

2.1 Brief description of filamentation

2.1.1 Definition of a filament . .

2.1.2 Slice-by-slice self-focussing .

2.1.3 Kerr self-focussing ....

2.1.4 Cri tical power for self-focussing

2.1.5 Self-focussing distance

2.1.6 Generation of plasma .

2.1.7 Defocussing by plasma

2.1.8 Intensity clamping ..

ii

IV

V

vii

ix

x

1

6

7

8

8

9

10

Il

12

14

15

Contents

2.2 Short vs. long pulse gas breakdown

2.3 Fluorescence "and plasma spectroscopy

2.3.1 Plasma characterisation

2.3.2 Concept of local thermal equilibrium

2.3.3 Line radiation. .... ..

2.3.4 Mechanisms of line broadening. .

2.3.5 Self-absorption of spectral lines

2.4 Overview of applications

3 High-power laser system

3.1 Femtosecond oscillator

3.2 One-kilohertz repetition rate CPA system.

3.3 Ten-Hertz high-power amplifiers . . . .

4 Filament-induced breakdown spectroscopy

4.1 Introduction... . ..

4.2 Experimental setup .

4.3 Resultsand discussion

4.4 Conclusions . . .

5 Proof-of-principle in argon

5.1 Introduction .....

5.2 Experimental setup

5.3 Results and discussion

5.4 Conclusions . . .

6 Stark broadening analysis of filament in air

6.1 Introduction.

6.2 Experiment

Xl

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24

28

29

33

34

35

39

41

42

43

45

55

56

56

58

59

65

66

67

68

Contents

6.3 Resul ts and discussion

6.4 Conclusion........

7 Critical power of helium

7.1 Introduction.

7.2

7.3

7.4

Experiment

Resul ts and discussion ·

Concl usion . . . . . . .

8 Pressure independence of intensity clamping

8.1 Introduction....

8.2 Analytical analysis

8.3 Experiments and results

8.4 Conclusion........

Xll

70

79

80

81

82

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86

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89

89

91

95

9 Intensity clamping of high-power filaments under 'naturaI' conditions 96

9.1 Introduction

9.2 Experiment


9.4 Conclusion.. . .

10 Conclusion

Bibliography

96

97

99

110

111

114

Chapter 1

Introduction

When a high-power (2: 10 GW) , ultra-short CS 1 ps) laser pulse propagates in a gaseous

medium (pressure: rv Torr) , filaments appear. The laser pulse is self-guided (and con

fined) in these filaments, which consist of weak plasma. The plasma is produced from

multiphoton/ tunnelling ionization (MPI/ TI) of the gas atoms or molecules residing in

side the filament. The filamentation process is a consequence of the dynamic equilibrium

between Kerr self-focussing and defocussing by the self-generated plasma, w hich leads

to the phenomenon that the laser intensity inside the filament is clamped ("intensity

clamping") [1- 4]. The critical (clamped) intensity is high enough (rv 5 X 1013 W /cm2,

in air [1,2]) to ionize or dissociate the gas species into fragments. This produces cha

racteristic fluorescence that is practically free from plasma continuum ("clean fluores

cence") [5,6]. The clean fluorescence can be observed, because the filament plasma

is generated by MPI/ TI of the gas atoms or molecules only, which are uni-molecular

processes, without the occurrence of inverse Bremsstrahlung and cascade (avalanche)

ionization [7].

Chapter 1. Introduction 2

These properties of filamentation have inspired new, exciting applications such as the

detection and identification of gaseous pollutants [6,8- 12], discharge j lightning control

[13- 18] or, ultimately, the generation of powerful few-cycle pulses [19- 24].

The aforementioned phenomena and applications are related to the filament plasma.

Therefore, it is the objective of this work to develop a spectroscopic tool that can be

used to measure the plasma density (in units of cm-3 ) inside the filament. Why is this

important? What insight can this give us? The answer is the following: the plasma

density is a key parameter of the filamentation process. As mentioned above , the fi

lamentation process results from the balance between Kerr self-focussing and plasma

defocussing. The Kerr contribution is determined by the nonlinear index of refraction ,

while the contribution of plasma defocussing is determined by the plasma density gra

dient (i.e., the number of free electrons per cm-3). The filameritation process occurs

only when the laser power exceeds the critical power for self-focussing. Note that the

critical power is related to the nonlinear index of refraction. This condition determines

whether intensity clamping and Kerr self-focussing set in.

In principle, the above information can be obtained, provided that the plasma den

sity is known. As l will show later, the plasma density can be used to infer the critical

power (or, equivalently, the nonlinear index of refraction). Moreover, the plasma den

sity (when compared to the neutral gas density) is related to the ionization degree and

the clamped intensity. In the case offilamentation, the plasma density must be well

below the brea~down limit (rv 1019 cm-\ for gases at 1 atm). Otherwise, losses by the

absorption of plasma due to inverse Bremsstrahlung would be significantly high.

It is noteworthy to point out that the measurement of the plasma density inside the

filament is difficult. This is because of the low value (rv 1014 to 1017 cm-3 [15 ,25- 30]).


In air, for example, the ratio I"'V Ne/Ne is ,:S 10-4 , where Ne is the plasma density and

Ne I"'V 1021 cm -3 the critical density (laser wavelength: À == ~OO nm). This leads to only

a small phase shift !3.cp I"'V J Ne dl/Ne. As a result, interferometric measurement of the

plasma density becomes problematic. (Note that optical interferometry is the preferred

technique to measure the plasma density with high precision.) In the literature, the

value of the plasma density differs by several orders of magnitude; one can assume that

large uncertainties are involved.

In this thesis, l will present a new approach for measuring the plasma density inside

the filament. This approach is based on the Stark broadening of the atomic lines, which

can be observed in the filament spectra. However, the Stark broadening approach has

never been applied to the case of the filament before. There~ore, it will be shown that

the Stark broadening approach is valid, despite of the low plasma density. The Stark

broadening approach will be illustrated by the important cases of air and helium. From

the experimental point of view, different conditions must be fulfilled: the observation

of weIl contrasted atomic lines and a high enough spectral resolution.

To this end, to avoid misunderstanding, we note that our definition of a filament

is the narrow column (about 100 {lm in diameter [31,32]) where there is a plasma

generated [33]. It is also pointed out that the plasma densities determined from the

spectral measurement represent average values integrated over the filament core. Thus,

the intensity which is derived from the plasma density is also an average intensity over

the very narrow plasma column.

To introduce the Stark broadening approach, the spectroscopie study of metallic

lead was carried out [34]. Using longer (ns) pulses, the breakdown spectroscopy of solid

targets was studied before. However, the drawback of this technique ("laser induced


breakdown spectroscopy") is the strong continuum background, especially in the early

evolution of the plasma. It was anticipated that the detection sensitivity (i.e. , the

signal-to-noise ratio) could be improved by the filament. This assumption was shown

to be true, based on the Stark broadening of the atomic line Pb l 373.99 nm.

The creative idea was to apply the Stark broadening approach to the filament in

a gas or gas mixture. The proof-of-principle was first illustrated by the example of

argon [35]. The reason for this choice was because argon is important for producing

few-cycle pulses, carried out in gas cells. Moreover, the Stark broadening of the atomic

line Ar l 696.54 nm, which was used for the measurement of the plasma density, is weIl

referenced in the literature. With this preparatory knowledge, it was possible to carry

out the following experiments:

The first experiment was dedicated to the Stark broadening analysis of the fila

ment in air [30]. This case represents filamentation in a (molecular) gas mixture, where

plasma is generated by MPI/ TI. The spectrally-resoived atomic oxygen triplet 0 l

777.4 nm was observed in the filament spectra. Using the Stark broadening approach,

the plasma density was measured by varying the laser power, beam diameter and fo

cussing condition.

The second experiment was dedicated to the measurement of the critical power of

helium [36]. This atomic gas (plasma generation by pure TI) is of interest for few-cycle

pulse or high-order harmonics generation. However, it is important to know the critical

power, indicating the threshold for the filamentation process. Therefore, for the first

time, the critical power of helium was measured, using the Stark broadening of the

atomic line He l 587.56 nID.


The problem of measuring the plasma density inside the filament in a gas or gas mix

ture is relevant to many applications involving filamentation. Thus, the Stark broaden

ing approach could be very beneficial. The next chapter discusses the physical principles

that underlie the experiments.

Chapter 2

Basic concepts

Phenomena of filarrientation are many and complex [32, 37- 39]. This chapter briefly

reviews the basic concepts involved. Section 2.1 introduces the full picture of filamen

tation , È>ased on the "moving focus model" ("slice-by-slice self-focussing") [40,41]. This

is followed by a quick derivation of the underlying processes, leading to the concept of

intensity clamping.

Section 2.2 contrasts the scenario of short (fs) with long (ns) pulse gas breakdown.

Section 2.3 presents the principles of fluorescence and plasma spectroscopy. This in

cludes the concept of ·local thermal equilibrium. Section 2.3 also discusses mechanisms

of Hne broadening (esp. , via the Stark effect) , shapes and widths.

Finally, this chapter ends with an overview of applications. It is fascinating to realize

that filamentation extends the frontiers of nonlinear optics [42].

Chapter 2. Basic concepts 7

2.1 Brief description of filarnentation

In its most general description , filamentation is described by the following scenario:

• Because of the dependence of the refractive index on the (transverse) intensity

profile l l (r) of the pulse (Eq. 2.2) , Kerr self-focussing occurs. The dependence on

the (longitudinal) profile l (z) leads to self-phase modulation (SPM) , which causes

the pulse spectrum to broaden with propagation. The Kerr/ plasma (Eq. 2.2/ 2.9)

induced instantaneous frequency is w (t) rv -won2z /cxdI (t) /dt or rv NO{5K Z /WO X

I K (t) , respectively [44]; Wo is th~ central frequency and l (t) the pulse envelope.

The former describes a linear Stokes (red) shift in the neutral, while the latter t .

corresponds to a nonlinear anti-Stokes (blue) shift in the plasma. SPM is the

origin of white-light (supercontinuum) .

• The generation of free plasma electrons induced by MPI/ TI counteracts Kerr

self-focussing.

For this reason, the essential physics can -be understood while keeping in mind the

dynamic equilibrium between these two nonlinear phenomena.

1 For a Gaussian pulse, the electric field is given by

E (r, z, t) Eowo/.w (z ) x exp (_r2 /w2 (z )) exp (-ikr2 /2R (z ))

x exp ( - ((t - z /c) /7')2) exp (i (wt - kz + cp (z ))) , (2.1)

- _ - 1/ 2 where Eo is the field amplitude, w (z ) = Wo (1 + (z / zO)2) the beam diameter, Wo the waist width, Zo

the Rayleigh length (Sec. 2.1.1) , k the wave number, R (z ) = z (1 + (zo / z ) 2 ) the wavefront curvature, c

the speed of light , 7 = v2ln 27' the pulse duration at full width half m-aximum (FWHM) of intensity, w the laser frequency and cp (z ) = tan- 1 (z / zo) the Gouy phase (e.g., [43]). Thus, l (r , z = 0) I"'V

exp (-2r2 /wô) and 1 (z, t = 0) l"'Vexp ( _2Z2 / (C7')2).


2.1.1 Definition of a filament

There are, basically, two definitions, which are more or less restrictive [32,37]:

1. In [37], a "filament" is defined as a dynamic structure, which is self-guided over

distances much larger than the Rayleigh length. In Gaussian optics, the Rayleigh

length Zo == 1rW5 / À (À is the laser wavelength and Wo the beam waist) is defined as

the distance along the propagation direction z , where the beam radius increases

by a factor of y'2. Ionization is not a necessary conditions.

2. In the moving focus model [40, 41], the term "filament" denotes the plasma col

umn left behind by the series of self-foci from different intensity / power slices of

the front part of the pulse (Fig. 2.1 a). This definition explicitly presumes the

occurrence of ionization and generation of plasma. Defocussing by plasma acts as

a counteracting mechanism of Kerr self-focussing.

In this thesis, the presence of weak plasma inside the filament was confirmed. Therefore,

filamentation will be described by the latter model. Table 2.1 lists typical properties of

a filament in air.

2.1.2 Slice-by-slice self-focussing

This section describes the moving focus model (slice-by-slice self-focussing) [40,41] (after

[45]). l t is valid for short pulses (rv 100 fs) propagating freely (or weakly focussed) in air

(or other gases or gas mixtures). The pulse is considered to be divided into neighbouring

Chapter 2. Basic concepts

Diameter Critical power Plasma density Clamped intensity

rv 100/Lm rv 10 GW rv 1016 cm-3 rv 5 X 1013 W /cm2

Table 2.1: Typical properties of a filament in air.

9

intensity / power slices (Fig. 2.1 a). The minimum 'thickness ' ds of each slice is rv cTo,

where To is the laser period. Only slices whose power is above the critical power (Eq. 2.3)

will self-focus. It is assumed that aIl slices are independent .of each other.

The most powerful slice, i.e., the central one (Fig. 2.1 a, darkest red) , will self

focus at the short est distance (Eq. 2.4). As it self-focusses, its intensity will increase.

Simultaneously, the plasma density rises strongly, because MPI/ TI are highly nonlinear

pro cesses (Eq.2.6). This, eventually, leads to intensity clamping (Eq. 2.11) in the self

focus of the central slice (Fig. 2.1 a).

The leading slices will self-focus at a farther distance (Eq. 2.4) , because their power

is lower (Fig. 2.1 a). The process of intensity clamping, however, is identical for aIl (i.e. ,

the central and leading) slices. The trailing slices will be diffracted from the plasma

left behind by the front .part of the pulse (self-steepening [46,47]). Finally, the series of

self-foci (plasma column) constitutes the filament (Fig. 2.1 a).

2.1.3 Kerr self-focussing

The refractive index n (r) due to the optical Kerr effect is given by

(2.2)


0.8

0.6

0.4

0.2

~z

(a) (b)

Figure 2.1: a) Slice-by-slice self-focussing in the moving focus model [40, 41]. Typical (Gaussian) intensity profile le (z, t == 0) rv exp (-C'Z2); .c': arbitrary const. , z is the distance in the propagation direction (highest intensity: darkest red). The thickness ds

of each slice (vertical bars) must be equal (or larger) than cTo, w here To is the laser period ds 2: cTo· b) Self-focussing distance Zf (Eq. 2.4) normalized by diffraction length ka2 as a function of laser power (P S 10 Pc).

where no and n2 (rv 3 X 10-19 cm2/W, in air [48]) are the linear and nonlinear refractive

indices, respectively (e.g., [49]). The transverse intensity profile 1 (r) is characterised

by an intensity gradient 81/ 8r < 0 (r == 0 to (0). Thus, the refractive index decreases ,

8n/8r < 0, while the phase velocity Vph (r) rv l/n (r) increases, 8Vph/8r > 0, away

from the optical axis. Consequently, this results in Kerr self-focussing (Fig. 2.2) [50].

2.1.4 Critical power for self-focussing

In the case of a non-paraxial Gaussian beam, the critical power required for Kerr self-

focussing to overcome linear diffraction is [51 r

Pc = 3.77,\2 87rnOn2

(2.3)

Equation 2.3 can be derived by solving the nonli!lear Schrodinger equation au / az ==

i/2k x \7iU +iw/c x n2/U/2U == 0, where U ls the field envelope [38]. In the paraxial


Intensity profile

(/)

Refractive index profile

(il)

n(O»n(r)

Phase velocity

(iil)

z

Plane wavefront

(iv)

Converging wavefront

Il

Figure 2.2.: Principle of Kerr self-focussing. (i) Typical (Gaussian) intensity profile le (r , Z == 0) rv exp (-c'r 2

); c': arbitrary const., r (0 to (0) is the distance away from the optical axis. The (ii) refractive index n (r) follows the (i) intensity profile (Eq. 2.2) , while the (iii) phase velo city Vph (r) increases, BVph/ Br > 0, away from the optical axis (r == 0). Consequently, the (iv) initial plane wavefront will (v) converge, which results in Kerr self-focussing [50].

approximation, the critical power is P~ == À2/27rn2nO [52]. For other beam profiles , the

coefficient in Eq. 2.3 is different from 3.77. In the case of a Townes mode , for example,

it is 3.72 [53].

The critical power (in W, energy per unit time) does not depend on the laser inten

sity, Pc =1- Pc (1). For a typical laser wavelength of À == 800 nm, the critical power in air

is about rv 3 GW (Eq.2.3). Recently, Liu et al. measured a value of rv 10 GW [54].

2.1.5 Self-focussing distance

The self-focusing distance (or nonlinear focallength) is given by Marburger's equation

0.367ka2

Zf== ~~~~~~~~~~~

J(p/ Pc - 0.852)2 - 0.0219 ' (2.4)

where a is the beam radius (at lie level ofintensity) and P the laser power (Eq. 2.3) [51] . .

In the case of external focussing by a lens with a focal length j , the combined self-


focussing distance zf is given by the lens transformation formula zf == Zf f 1 ( Zf + f).

Figure 2.1 b shows the self-focussing distance Zf normalized to the diffraction length

ka2 as a function of laser power up to P == 10 Pc.

2.1.6 Generation of plasma

The history of multiphoton/ tunnelling ionization (MPI/ TI) is outlined in [7]. MPI/ TI

sets in at laser intensities2 in the range of rv 1013 to 1014 W 1 cm2 . In MPI, the electron

can be ionized from its bound state to the free continuum state. In this case, it simul-

. taneously absorbs n photons (n == 0, 1, 2, 3, ... ) of energy nw to reach the ionization

energy Eo == nlïw. When the laser intensity l becomes comparable to the atomic inten-

sity la, the Coulomb potential V (r) rv -l/r will be distorted by the strong laser field

E (Fig. 2.3 b). Then, there is sorne probability that the electron can 'tunnel' through

the Coulomb barrier [7]. Figure 2.3 shows a schematic of both processes.

In 1965, Keldysh introduced a parameter IK == J Eoi CPP to distinguish between the

regimes of MPI and TI [55]. CPP == e2 E 2 14mew2 is the "ponderomotive energy" 3 of the

laser field E, e the elementary charge and me the electron mass . [57,58]. Typically,

for low (/K » l) j high (/K « 1) intensities, ionization is induced by MPlj TI [55]. In

practical units, the Keldysh parameter can be written as

IK ~ 2.3 X 106 Eo[eV] (2.5)

2The strength of the Coulomb field that binds the electron is Ea = e/41rcoa~ (atomic field strength) , where aB = 0.5 A is the Bohr radius. This corresponds to the atomic intensity of la rv 4 X 1016 W /cm2 .

3The ponderomotive energy is the quiver energy of the oscillating electron in the E and B field of the laser. In the classical case, the speed of the electron v is much slower than the speed of light c, v « c. Thus, the influence of the B field is negligible [56].


V(rl o

(a)

V(r)

o

(b)

13

Figure 2.3: Principle of multiphotonj tunnelling ionization (MPlj TI). (a) In MPI, the electron simultaneously absorbs n photons of energy !iw to reach the ionizat ion energy Eo = n!iw. b) In TI, the Coulomb potential V (r ) rv -l/r (r is the position of the electron) is distorted by the strong laser field E as V' (r) rv -1/ r - c'Er (c' : arbitrary const.). In this case, there is sorne probability that the electron can tunnel through the Coulomb barrier (arrow).

where 1 (in W /cm2) is the laser intensity. The ponderomotive energy reads cjJp [eV] ~

9.33 x 10-141 [W /cm2] À [JLm2]. For the filament , the Keldysh parameter is close to

one, Î K ~ 1 (E~2 rv 12.1 eV, À = 0.8JLm and l e rv 5 x 1013 W/cm2, in air [1 , 2]).

The ponderomotive energy is cjJp rv 3 eV. To ionize the oxygen (02 ) species , about 8

photons of energy !iw ~ 1.6 eV are required. Î K ~ 1 means that ionization takes place

in a transition regime between MPI and TI.

In MPI/ TI, the generation of plasma is governed by

Ne = No J R(I) dt, (2.6)

where No (in cm-3) is the neutral density and R (1) (in S-l) the ionization rate [2 , 59].

The ionization rate sc ales as R (1) rv O"KIK , where O"K (in S-l cm2K W- K ) is the cross

section and K the effective order of ionization (K ~ 8, in air [2 ,59]). Therefore, MPI/ TI

are highly nonlinear processes.


2.1.7 Defocussing by plasma

In the Drude model, the refractive index of plasma reads 4

(2.7)

where wp == (e2 Ne/come) 1/2 is the plasma frequency, Ne == cOmew2 / e2 the critical density

and co the vacuum permittivity (e.g., [57]). The plasma frequency is a characteristics

of the oscillation of the electron plasma and scales as rv N;/2. For the filament plasma

(Ne rv 1016 cm-3), the plasma frequency is rv 1012 S-l.

The critical density Ne is the density where the plasma frequency equals the laser

frequency, wp ~ w. For Ne ~ Ne the plasma is ·referred as underdense or overdense,

respectively. Pulse propagation is only possible in an underdense plasma (Ne < Ne).

In an overdense plasma (Ne> Ne), the refractive index becomes imaginary (Eq.2.7) ,

leading to an opaque plasma. In this case, the pulse evanescently penetrates into

the plasma, while its amplitude decreases by a factor of 1/ e in the distance d1/ e

c/wp (1 - w2 /w;) -1/2 [60]. In practical units, the critical density can be written as

(2.8)

Thus, the critical density (rv 1.7 X 1021 cm-3 ) is two orders of magnitude higher than

the neutral density (rv 1019 cm-3 , for gases at 1 atm).

4Equation 2.7 can be derived by inserting the classical dispersion relation of plasma w2 = w; + k2c2

into the refractive index n = ck/w.


Because of Ne « Ne, Eq. 2.7 cau be approximated by

(2.9)

Consequently, plasma contributes negatively to the index of refraction (n :s 1). This

leads to defocussing by the self-generated plasma, which counteracts Kerr self-focussing.

If there was no counteracting mechanism, the beam would collapse and the laser inten-

sity grow infinitely.

2.1.8 Intensity clamping

The intensity clamping of the filamentation process occurs when Kerr self-focussing

(Eq.2.2) dynamically balances defocussing by plasma (Eq. 2.9) -[1- 4]

~nK (neutral) ~ ~n (plasma) . (2.10)

In the balance betweendiffraction, Kerr and ionization responses, fundamental relations

can be derived, yielding the clamped intensity le, the plasma density Ne and the radius

Te of the filament [38]

(2.11)

Note that le, Pe (Eq.2.3) and re are related by le rv Pe/1fr;. Note also that no , k and

Ne are known; no depends on the gas properties (no ~ 1), while k and Ne (Eq.2.8) are

functions of À only.

l


This leads to the conclusion that intensity clamping is essentially determined by the

key parameters Ne, n2 (or, equivalently, Pe (Eq. 2.3)) and le.

2.2 Short vs. long pulse gas breakdown

The scenario of long (ns) pulse optical breakdown in gases is described in [7]. It is ,

essentially, a three-step process (after [7]):

1. MPI/ TI pro duces a few free plasma electrons with low kinetic energy provided

by the front part of the pulse.

2. The free plasma electrons can be further accelerated by the back part of the pulse

in an inverse Bremsstrahlung pro cess ("free-free transition "). In this process , the

electron absorbs energy from the laser (photon) field, while eolliding with a heavier

particle (atom, molecule or ion). Th~ heavy particle is to satisfy conservation of

momentum during the process.

3. In the inverse Bremsstrahlung process, the free plasma electrons can acquire ki

netic energies high enough to induce collisional ionization. This pro duces more

and more electrons, collisional absorption and, eventually, cascade (avalanche)

ionization will occur. The latter leads to complete breakdown of the gas (i.e. , the

gas is fully ionized).

The electro~-electron collision time te (in s) is given by

(2.12)


where.T (in K) is the plasma temperature and InA the "Coulomb logarithm"5 [60,61].

Alternatively, the electron-electron collision frequency (" 90° scattering rate" [62])

is Vee rv l/te or ~ 4 x 10-12NelnA/T3/2 (in S-l) , where InA == In(67rNeÀ~) , ÀD ~

(

~ ) 1/2 ~ ~ 7.4 X 103 T / Ne (in m) is the Debye length, T == kBT (in eV) the thermal energy

and kB (in eV /K) the Boltzmann constant6 . [The frequencies of ion-ion collisions Vii ,

electrons scàttering from ions Vei or ions scattering from electrons Vie are related to Vee

by Vei rv Vee , Vii rv (me/mi)1/ 2 Vee or Vie rv (me/mi) Vee , where mi is ' the ion mass [64].

These pro cesses are, thus, slower.] The free-free absorption coefficient scales as [60,61]

(2.13)

For the filament plasma (Ne rv 1016 cm-3 , T ;S 1 eV [30,65]), however, the collision

time is long, te ~ 1 ps (Eq.2.12), compared with the pulse duration (fs). Moreover,

because of the low plasma density (rv 1016 cm-3 ), the free-free absorption coefficient ~i

(Eq. 2.13) is small.

For this reason, collision al ionization/ absorption and therefore losses by inverse

Bremsstrahlung are negligible. As a result, the filament density is not saturated

(Ne « No) and the plasma 'cold' (T;S leV) [30,65].

5The Coulomb logarithm ln A is a slowly varying function of Ne and T [60,61]. 6In practical units, the Boltzmann constant is kB ~ 8.617 X 10-5 eV /K [63].


2.3 Fluorescence and plasllla spectroscopy

Cooper provides a concise definition of the term "plasma spectroscopy" [61]: "( ... ) it

(plasma spectroscopy) can be said to be the study of elec~romagnetic radiation emitted

from ionized media. ( ... ) However, in contrast to conventional spectroscopy, where one

is mainly interested in the atomic structure of an isolated atom, the radiation from a

plasma depends, not only on the properties of the isolated radiating species, but also

on the properties of the plasma in the immediate environment of the radiator. ( ... )"

The spectral distribution of the radiation emitted from the plasma is referred as

"fluorescence spectrum". A discrete spectrum is generated by radiative transitions of

the atom (ion or molecule) between two bound states (upper and lower atomic levels).

Besides, the emitting atom is perturbed by the plasma (electrons) via the Stark effect.

This results in a broadening of the energy levels (Stark broadening).

The analysis of the Stark width and intensities of fluorescence lines provides valuable

information about the properties of the plasma itself, i.e., the plasma density Ne and

temperature T.

2.3.1 Plasma characterisation

A "plasma" is a sufficiently ionized gas that constitutes a 'fourth state of matter ' [62 ,66].

ldeally, this presumes the following concepts (after [62,64])

• Th ermal energy (cf. Sec. 2.3.2). The (averaged) kinetic energy (Tkin ) of the plasma


electrons is associated with the thermal energy t = 2/3 X (Tkin ) . The thermal

energy is related to the plasma temperature by t = kBT. Their "thermal speed"

(

_ ) 1/2 is Vt = T/me .

• Quasi-neutrality. The number densities (i.e. , the numbers of particles per unit

volume) of ions Ni and electrons Ne are approximately equal, Ni f"'..I Ne. This ,

however, is only valid over scale lengths L much larger than the Debye length,

ÀD / L « 1. The Debye length À D corresponds to the distance VtTp that a free

electron of thermal velocity Vt can travel within a plasma period Tp f"'..I l/wp . In

contrast, there are also non-rieutral plasmas (esp., charged particle such as e- or

p+ beams).

• Debye shielding. Inserting a "test" particle with charge qt into a neutral plasma

will result in a shielding (screening) of its Coulomb potential as cp (r) = qt/41féor x

exp ( -r / ÀD ), where the Debye length ÀD = (éokBT / e2 Ne) 1/2 is the scale length of

the screening region ("sheath"). The Debye sphere is the volume 4/3 X1f À~ outside

of which the plasma is quasi-neutral. For r » ÀD' the potential is completely

screened, cp (r » ÀD ) ~ o.

• Collective behavior. The plasma electrons oscillate collectively. Besides, they are

quasi-free (i.e., not bound, but coupled via the Coulomb fields with the positive ion

background). Their fast oscillation frequency corresponds to the plasma frequency

. wp . Collective behavior is only relevant over time scales T much longer than the

plasma period, Tp/T « 1. Its time scale is given by the inverse of the collision

frequency, te f"'..I l/vee , while its scale length is determined by the "mean-free-path"

The detailed behavior of a plasma is complex [62,64]. However, its state can be uniquely

described by a small set of parameters (Tab. 2.2) [64]. The fundamental parameters are


rv ps

Table 2.2: Typical plasma parameters of a filament in air. Fundamental parameters - - 1/2 1/ 2 Ne and T (T = kBT) and subsidiary parameters ÀD rv (T j Ne) , Wp rv Ne and

t rv v-1 rv T 3/ 2 jN e ee e·

Ne and T (or, equivalently, T) , while the subsidiary parameters (ÀD , wp , t e, Àm , Vt, etc.)

can be inferred from Ne and T [64]. In comparison with a gas, the plasma conductivity

ae rv N eT- 1/

2 is high (ae rv 73 mhojm) [65]. Table 2.2 lists typical plasmaparameters

of a filament in air.

2.3.2 Concept of local th~rmal equilibrium

The concept of local thermal equilibrium (LTE) is powerful (after [61,67]). LTE in a

plasma is defined as a state in which the populatIon density Nk of the · upper level k of

the transition of the (atomic) species S follows a Boltzmann distribution

(2.14)

where U (T) . is the partition function 7 and gk and Ek are the statistical weight and

excitation energy of the upper level k of the species S, respectively [61,67]. Moreover,

opposed to a system in complete thermal equilibrium (TE) , the radiation is not neces

sarily thermal [69]. However, the population densities Nk (Eq.2.14) are the same as in

TE.

7In statistical thermodynamics, partition functi'ons can be used to derive thermodynamic variables such as the free energy F or entropy S. The partition function U (T) of different species is tabulated in the literature (e.g., in [68]).


It is noteworthy to point out that ' TE can be approached in stellar interiors but

never in laboratory plasmas [69]. However, the concept of TE is the basis for the

derivation of the transition probabilities Akh Bki and Bik (Einstein coefficients) for

radiative proc~sses (e.g. , [70]). In TE, the radiation is thermal, i.e. , the spectral energy

density (i.e. , energy per unit volume per unit frequency) is according to Planck's law,

P (Vki , T) == 81rhv~ijc3 x (exp (hVkijkBT) - 1)-\ where h is the Planck constant and lIki

the frequency of the photon of the transition [71].

Especially, to achieve LTE, the plasma density must be high enough so that radia

tive pro cesses (i.e. , transitions between bound states) become negligible compared with

collision al ones (i.e. , electron impact excitations j deexcitations) [61 ,69]. The process by

which LTE is obtained is the thermalisation of the velocity distribution functions, le ad

ing to Maxwellian profiles8 , esp., via electron-electron9 collisions [64]. · In cold plasmas,

these pro cesses are faster (te rv T3/2 , Eq. 2.12).

The criterion for LTE reads [67, 73]

- 12 ~ 3 Ne 2 Ne == 1.6 x 10 yT(~E) , (2 .16)

where ~E (in eV) the difference in the energy levels of the transitions and Ne (in cm- 3 )

the critical density for thermal equilibrium. Note that Ne is not identical to the critical

density Ne (Eq.2.8) as defined in Sec. 2.1.7. If Eq. 2.16 holds , the state of the plasma

can be described by the fundamental parameters Ne and T (Sec. 2.3.1) [61 ,67].

8For a Maxwellian velocity distribution, the number of electrons dN (v) (per unit volume) with velocities between v and v + dv is [72]

me m eV 2 ( )3/2 ( 2)

dN (v ) = N e47f 27fkBT exp - 2kBT v dv . (2.15)

9The electrons are the most efficient particles to induce collision transitions [73J


In this thesis, the condition for LTE (Eq.2.16) is always fulfilled. Therefore , the

spectroscopie analysis is expected to be valid (cf. Tab. 2.3).

2.3.3 Line radiation

When a transition from the upper level k to the lower level i occurs10, the energy of

the emitted photon (spectralli~e) is given by

(2.17)

where Àki is the wavelength of the photon and Ei is the energy of the lower level i [75].

Table 2.4 provides the conversion factors for different energy units [63,74]. The total

power (per unit volume per unit solid angle) radiated in the spectral line of frequency

. (2.18)

where Aki is the transition probability (in S-l) from the upper level k to the lower level

i and hVki the photon energy [61,75]. Thus, in the optically thin case Il, the (absol~te)

line intensity (i.e. , the energy flux per solid angle) reads

(2.20)

lOIn atomic spectroscopy, k and i are the princip le quantum numhers. Besides, the selection rules have to he fulfilled: usuaIly, for one-electron systems ~l = ±1 , ~j = 0, ±1; for many-electron systems ~J = 0, ±1 (0 ~ 0 forbidden); in pure LS coupling, also ~L = 0, ±1 (0 ~ 0 forbidden), ~S = 0 [74] .

llOpticaIly thin means that the optical depth dTv = - x (v, x) dx , where X (v, x ) is the absorption coefficient (per unit length) and dx the optical path, is small ,

x (v, x) ls » 1. (2.19)

In this case, the absorption of radiation is negligihle and aIl the radiation escapes from the plasma [61].


Spectroscopie condition Criteria for confirmation

No self-absorption Good fi tting by Lorentzian (Voigt) profile

LTE Ne ~ Ne (Eq.2.16)

Spectral resolution Voigt profile (ai ~ ~À~/2)

No other broadening mechanisms ~ÀP/2 ' ~Ài ~ ~À~/2

Table 2.3: Confirmation of the validity of the Stark broadening approach. LTE: local thermal equilibrium. Ne is the plasma density, Ne the critical density for thermal equilibrium (Eq. 2.16), ~À~/2 the Stark width and ai the instr:umental resolution of

the spectrometer; ~ÀP/2 and ~Ài are the Doppler and ion broadening line widths , ~espectively.

where ls is the source depth of the plasma [61,75]. In LTE, the population density of

the upper level k follows a Boltzmann distribution (Eq. 2.14), thus

(2.21 )

In particular, the analysis of atomic lines can be used for different purposes [67, 76]:

• Because the (photon) energy of the transition ~E is characteristic of the emitting

atom, the central wavelength Àki uniquely identifies the particle. Of course, this

is only true if the level structure (energy diagram) of the atomic species is known.

It should also be noted that the wavelength can be shifted due to the Stark effect .

• In LTE, the plasma temperature T can be measured by the ratio 12 /11 of the

intensities of spectrallines Il == Il (VI) and 12 == 12 (V2),

(2.22)

In addition, the full width at half maximum (FWHM) of the Stark broadening

line profile ~À~/2 is characteristic of the plasma density Ne (Eq.2.27).


Unit J cm-1 Hz eV

1J 1 5.034 x 1022 1.509 X 1033 6.241 X 1018

1 cm-1 1.986 x 10-23 1 2.997 X 1010 1.239 X 10- 4

1 Hz 6.626 X 10-34 3.335 X 10- 11 1 4.135 X 10- 15

leV 1.602 x 10-19 8.065 X 103 2.417 X 1014 1

Table 2.4: Conversion factors for energy equivalents [63 ,74]. AIso, 1 eV is equivalent to "-J 1.160 x 104 K [75].

This thesis makes use of fluorêscence and plasma spectroscopy to realize the objectives

as described in Ch. 1.

2.3.4 Mechanisms of line broadening

Because of.the finite spectral resolution (Ti of the spectrometer and different mechanisms

of line broadening, observed (discrete) lin es have a spectral distribution l (v) of the emit

ted intensity around the central frequency Vki [75, 77]. The full width at half maximum

(FWHM) ("half-width") of l (v) is defined as the frequency interval ~V1/2 == IV2 - VII ,

where VI and V2 are the frequencies for which l (VI) == l (V2) == l (va) /2 (Fig. 2;4 a).

Due to À == c/v and, thus, ~À == -C/V2~V, the relative half-widths for frequency v

and wavelength À are the same I~v/vl == I~À/ÀI [77]. The relevant mechanisms ofline

broadening are (w == 27r V ) :

• N atural line broadening arises because of the finite lifetime of the excited states

and is characterised by a Lorentzian profile

(

2)-1 1 w -wo IL(w)==- 1+( ) ,

1rWN WN (2.23)


I(vo) ----- ----

L'1v1/2 1 (v 0)/2 - - - - - - - - - - -1

(a)

1

1

1

25

(b) (c)

Figure 2.4: a) Intensity profile. b) Comparison betweèn Lorentzian IL (v) (red) , Gaussian IG (v) (blue) and Voigt profile Iv (v) (black) of equal half-width (FWHM:

- +00 - . rv 3.60114); Iv (v) is normalized as J-oo Iv (~)/) d (~À') == 1. For large arguments , IG (v) approaches zero much faster than IL (v) [77,78]. c) Self-absorption; p == 0 (black), Pl > 0 (blue), P2 > Pl (red); P is the absorption parameter as defined in Sec. 2.3.5. In the case of strong self-absorption (red) , the line can reveal a central minimum at Vki

(dip) .

where Wo is the central frequency and WN is the half-width at half maximum [61].

The finite (mean) lifetime Tk == 1/ Aki of the atom in the upper level k is associated

with an effective spread in energy Ô.E, given by Heisenberg's uncertainty principle,

Ô.E ~ Iï/Tk. This corresponds to a broadening of ~w == ~E /Iï ~ l/Tk. In

laboratory plasmas, naturalline broadening is usually negligible [61].

• Doppler broadening is due to the thermal motion of the emitting atome The

observed frequency of the emitted radiation w' == Wo (1 ±VA/ c) is alter~d (w'

increases or decreases wh en the atom moves closer or away from the observer) by

the Doppler effect as ~w / Wo == vAl c; VA is the velocity component of the emitting

atom along the line-of-sight [61]. Wh en the atoms follow a Maxwellian velocity

distribution (Eq. 2.15,where me is to be replaced by the mass of the atom mA),

the Doppler-broadened spectral line is characterised by a Gaussian profile

1 ((W - WO)2) l G (w) = {3Jrl/2 exp - {32 . (2.24)

where {32 == 2kBTw6/mAc2 and WD == (ln 2)1/2WO (2kBT /mAc2)1/2 is' the half half

width [61]. The Doppler width WD is linear proportional to the frequency rv Wo


and scales as rv (T / mA) 1/2. Thus, WD is largest for the Lyman ex line of hydrogen

Ha (MH == 1 amu) [77].

In practical units , the FWHM of the Doppler-broadening line width ~ÀP/2 can

be written as

(2.25)

where MA (in amu) is the atom mass [79] .

• Stark broadening arises from the effect of charged particles ("perturbers") (i.e. ,

electrons or ions) on the emitting atom ("radiator"). The interaction between the

radiator and the perturber can be approximated by a dipole interaction V (t) ==

-d. Ê, where dis the dipole moment of the radiator and Ê the electric field induced

by the perturber at the centre of the radiator [61]. The atomic energy levels are

perturbed (Stark broadened) via the linear, ~v rv E, or quadratic, ~v rv ,E2,

Stark effectl2. Note that "detailed calculations of Stark broadening are extremely,

complicated ( ... ) and have been done in detail only for a féw atoms" [69].

The (standard) theory and a rigorous treatment of Stark broadening can be found

in [61,67]. Basically, there are two different approaches. Stark broadening can be

calculated on the basis of the impact or quasi-static approximation [61 , 67]:

- In the impact approximation, it is assumed that the radiator is occasionally

perturbed by (electron) collisions. This leads to a Lorentzian profile. The

Lorentzian profile can be derived by Fourier analysis of the oscillations of an

equivalent (damped) classical oscillator [61 , 77].

- In the quasi-static approximation, the motion of the heavy ions is completely

neglected and the perturbation is given by a statistical distribution of the

electric fields at the radiator. Usually, in calculations on Stark broadening,

12In quantum mechanics, the Stark effect is the shift of spectral Hnes induced by electric fields .


the impact approximation is used for electron broadening, while the quasi-

static approximation is used for ion broadening [61 ,67].

The Stark broadening line width (FWHM) of hydrogen (H) is given by

(2.26)

where Œl / 2 is, the half-width parameter [79]. In the case of hydrogen, the Stark

effect is linear, ~v rv E. The Stark width scales as rv N;/3 and is large [79].

The Stark width (in nm) due to the quadratic Stark effect , ~v rv E 2 , is given by

(2.27)

where Wl/2 (in nm) is the ele~tron impact parameter [67]. From Eq. 2.27 follows

that the Stark width ~À~/2 is linearly proportion al to the plasma density rv Ne.

The electron impact parameter Wl/2 is · tabulated in the literature (esp. , in [67,80])

and shows a weak dependence on the plasma temperature T [67].

The correction for ion Stark broadening (in nm) is given by

(2.28)

where A is the static ion broadening parameter and ND ~ 1.72 X 109T 3/ 2 N e-l

/2

the particle number in the Debye sphere (cf. Sec. 2.3.1) [67].

If two (or more) mechanisms of line broadening become significant (including the spec

trometer's instrumental broadening), the observed line profile must be 'deconvoluted '.

Consider two different line profiles, Il (~À) and 12 (~À) , then , the observed profile


j (~À) is given by the convolution integral [67]

(2.29)

In the case of a Gaussian profile 1 G (e.g. , via Doppler broadening) and a Lorentzian

profile IL (esp. , via Stark broadening) , the convolution is referred as a Voigt profile Iv.

The Voigt profile Iv (Fig. 2.4 b) is frequently used in spectroscopy [78].

In this thesis , the Doppler width ~À~/2 and the ion broadening line width ~Ài are

always negligible compared with the Stark width ~~/2. Besides, the Voigt profile Iv is

used when ~~/2 is close to the instrumental resolution ai (cf. Tab. 2.3).

2.3.5 Self-absorption of spectral lines

In the case of radiative transfer of the emitted intensity 1 (v) through an optically thick

(cf. Eq.2.19), inhomogeneous plasma with a strong temperature gradient, 8TI8x » 1

(i.e. , the inner layers are much hotter than the outer ones), the observed line inten

sity l (v) can reveal a central minimum at Vki (self-absorption) (Fig. 2.4 c) [61]. [The

equation of radiative transfer is dl (v, x) IdTv == 1 (v, x) - Sv (x), where 1 (v, x) is

the emitted intensity, Sv (x) == J (v, x) lx (v, x) the "source function" and J (v; x) the

emission coefficient [61].]

In particular, l (v) can be expressed by

l (v) rv 1 (v) exp ( -pl (v) II (Vki)) , (2.30)

where p rv BikI (Vki) J~s Ni (x) dx is the absorption parameter, Bik (per unit time per


unit energy density) the Einstein coefficient for absorption, l (v) the originalline profile

and J~s Ni (x) dx the population density · Ni of the lower level i of the (emitting) species

S, integrated over the source depth of the plasma ls [81]. Thus, self-absorption is largest

for transitions to the ground state (fi rv Ni) and at the line centre Vki (Fig. 2.4 c).

In this thesis , the good fitting with Voigt (or Lorentzian) profiles always indicates

the absence of self-absorption (cf. Tab. 2.3). Only for sorne transitions of Pb l , a decrease

of the line intensity ('dip ') was observed. However, these lines were not used for the

Stark broadening analysis.

2.4 Overview of applications

Potential applications of the above phenomena of filamentation include [32,37- 39]:

• Detection and identification of gaseous pollutants [6,8- 12]. Gravel et al. recorded

the fluorescence spectra of different greenhouse gases 13, in particular, those of

the halocarbons PCFs CF 4 and C2F6 and HCFC-124 (2-chloro-l,I,I,2-tetrafluoro

ethane) and the hydrocarbon methane (CH4 )14 [6]. For C2F6 in air, the 3a con-

centration sensitivity ((T is the standard deviation of the background level) was

measured to be C3a (C2F6) ;S 500ppm (volume/ volume) [6].

In [11], the backscattered fluorescence of CH4 was recorded in a Lidar (Light

13The "Kyoto Protocol" explicitly includes the "greenhouse gases" carbon dioxide (C02), methane (CH4 ), nitrous oxide (N20), hydrochlorofluorocarbons (HCFCs), perfluorocarbons (PCFs) and sulphur oxide (SF6) [82].

141n organic chemistry, hydrocarbons consist of hydrogen (H) and carbon (C) atoms, halocarbons consist of carbon and halogen (fluorine (F), chlorine (Cl), bromine (Br) or iodine (1)) atoms. Perfluorocarbons (PCFs) are derived from hydrocarbons, where aH the H atoms are replaced with F atoms, while hydrochlorofluorocarbons (HCFCs) are derived from halocarbones , where not aH the H are replaced with Cl or F atoms.


Detection and Ranging) configuration. Opposed to atmospheric sensing using

long (ns) pulses, short (fs) pulses can produce a high intensity (rv 5 X 1013 W /cm2,

in air [1,2]) at a remote distance [83]. (In the linear (ns) regime, however, the

beam diameter d increases linearly with the focal distance j, d rv j. Thus, the

intensity decreases as l rv 1/ d2 rv 1/ j2 .) For CH4 in air, the 30" concent ration

s,ensitivity was determined to be C3a (CH4 ) rv 40 ppm (volume/ volume) , while the

detection distance was estimated to be in the km range [11].

In [12], the fluorescence spectra of methane (CH4 ) with C3a (CH4 ) rv 1 ppm and

acetylene (C2 H2 ) with C3a (C2H2 ) 'rv 300 ppb in air were simultaneously detected

and identified. In particular, the species and concentrations of an arbitrary ('un-

known') mixture of CH4 and C2H2 were obtained [12].

In 2007, the method for remote sensing of pollutant molecules using nonlinear

fluorescence spectroscopy was patented [84].

A complementary technique is based on the white-light (supercontinuum) com

bined with linear absorption spectroscopy [9]. The supercontinuum ranges from

rv 230 nm in the UV [22,85,86] to rv 4.5 Mm in the IR [47,87], which covers absorp-. . .

tion bands of atmospheric pollutants such as CH4 , CO2 , NOx , VOCs ("volatile

organic compounds"15) or H2 0 [39] .

• Discharge/ lightning control [13-18]. In 1971 , Koopman and Wilkerson proposed

to use (ns) pulses to ionize the air, induce a conducting path (plasma channel) and

trigger lightning [89]. This approach, however, was limited because of significant

losses inherent to the long (ns) pulse gas breakdown (Sec. 2.2).

In contrast, the use of (fs) filamentation is advantageous. In 2002, Rodriguez

et al. have shown that filaments could trigger and guide high-voltage (~ 2 MV)

15 According to a European Union directive, VOC means "any organic compound having an initial boiling point less than or equal to 250°C measured at a standard pressure of 101.3 kPa." The emission of VOCs into the air, esp. , from paints, varnishes and vehicle refinishing products, are problemat ic [88].


discharges over a distance of ;S 3.8 m. The breakdown voltage was typically

reduced by rv 32% [17] .

• Generation of powerful few-cycle pulses [19- 24]. Recently, pulse compression for

producing few-cycle pulses 16 in gases has emerged as a powerful technique (see ,

e.g. , [91] and references therein). The mechanism of pulse compression can be .

explained by SPM of the central slice of the pulse spatially confined inside the

core of the filament , while the leading and trailing slices are diffracted from the

plasma [92,93].

In 2004, Hauri et al. used two gas cells filled with argon at different pressures 17

of 840 and 700 mbar, respectively. In the experiment, 1 kHz, 43 fs pulses with

energies of rv 840 j1J were loosely focussed inside the first cell. After using chirped

mirrors for dispersion compensation, the compressed pulses (rv 10.5 fs , rv 790 j1J)

were further compressed (rv 5.7fs, rv 380 j1J) inside the second cell [19].

In 2006, Stibenz et al. realized a ' pure (i.e., without requiring any dispersion

compensation) self-compression scheme inside a single cell. In this scheme, 1 kHz,

45 fs pulses with energies of ;S 5 mJ were focussed (f == 1.5 m) into a 50 kPa

argon-filled cell. The compressed pulses were characterised by a pulse duration of

;S 10 fs and a strong spectral blueshift. (The blueshift increased with increasing

pressure, which was identified as being typical of optical nonlinearity induced by

fre~ electrons.) [23].

Alternatively, Théberge et al. developed a highly-efficient (conversion efficiency:

;S 25% in air, ;S 40% in argon) source for . generating tunable, few-cycle (rv 12 fs,

in air) pulses in the visible (VIS) spectrum via four-wave mixing (4WM). In this

scheme, the frequency W4WM of the 4WM pulse was determined by the process

16The single-cycle limit is given by the period of one laser oscillation To = >"0/ c ('V 2.7 fs (>"0 = 800 nm) [90].

17The conversion factors for typical pressure units are: 1 atm ~ 1.01325 x 105 Pa, 1 mbar ~ 1.0 x 102 Pa, 1 Torr ~ 1.333224 x 102 Pa [94].


W4WM == 2WNIR - WIR, where WNIR and WIR are the frequencies of co~propagating

near-infrared (NIR) pump (~ 2 mJ) and infrared (IR) seed (~ J.1J) pulses.

The generated 4WM pulse was stable (intensity fluctuation: ~ 0.1 %), of high

spatial mode quality (M2 < 1.01)18 and could be tuned in the spectral range of

475 and 650nm [24].

18In Gaussian optics, the M 2 parameter is defined as M 2 = ()wo1f / À, where () is the divergence angle and À./1f the beam parameter product of a diffraction-limited Gaussian beam. The M 2 parameter is always larger than one, M 2 > 1.

Chapter 3

High-power laser system

Our high-power laser system is based on the principle of chirped pulse amplification

(CPA) (Fig. 3.1) [95, 96]. The purpose of CPA is to efficiently amplify the low-energy

('" nJ) , ultra-short ('" fs) pulses provided by the oscillator, while avoiding the "gain

saturation" 1:

1. The oscillator pulses are temporally stretched (factor: '" 104 ) in a pulse stretcher,

thus reducing their peak intensity as l '" l/T. This allows for a higher gain

(Eq. 3.1) during amplification and prevents damage of the amplifier optics.

2. The stretched pulses are amplified in a regenerative amplifier and multi-pass am-

plifiers.

1 In the basic (hyperbolic) model , the gain of an amplifier medium is given by

9 (1) = go / (1 + 1/18), (3.1)

where go is the small-signal gain, l the intensity and 18 the saturation intensity [97]. 18 is the intensity at which the gain g (1) is reduced by a factor of two, 9 (1 = 18) = go/2. Equation 3.1 shows that t he gain reduces for high intensities , l » 18 [97].

Chapter 3. High-power laser system 34

Compressor

Regenerative Multi-pass Oscillator Stretcher amplifier amplifiers

Stretcher Compressor

Figure 3.1: Principle of chirped pulse amplification (CPA) [95 ,96]. First , the oscillator pulses (rv nJ, rv fs) are temporally stretched (factor: rv 104 ) in a grating stretcher. Then, the stretched pulses are amplified in a regenerative amplifier and multi-pass amplifiers, respectively. Finally, the amplified (rv mJ) pulses are re-compressed in a grating compressor (peak power: rv TW).

3. The amplified pulses are re-compressed in a pulse compressor. However, because

of "gain narrowing" 2, the original pulse duration cannot be fully achieved.

The pulse parameters of the high-power laser system are listed in Table 3.1.

3.1 Ferntosecond oscillator

The ultra-short (rv fs) pulses are produced with a mode-Iocked Ti:sapphire oscillator

(Tsunami). The gain medium of the oscillator is a Ti:sapphire (Ti3+:Ab03) crystal ,

providing a large (rv 650 to 1100 nm) spectral gain (bandwidth: rv 200 THz) [97, 98].

This large bandwidth allows for the generation of few-cycle pulses down to about 5 fs

[99]. Moreover, because of the high thermal conductivity and high saturation fluence

2In an amplifier, gain narrowing, i.e. , the narrowing of the spectrum 1 (v) of the pulse, arises , because the central part of 1 (v) is amplified more than the wings [97].


Central wavelength Pulse energy Pulse duration Peak power

rv 800nm ;S 80rnJ ;S 50 fs ;S2TW

Table 3.1: Pulse parameters of the high-power laser system (Fig. 3.2 and 3.5). Three output pulses are available: (i) 1 kHz (;S 2 mJ, rv 42 fs) and 10 Hz pulses with (ii) ;S 0.2 TW and (iii) ;S 2 TW, respectively.

(rv 1 J / cm2) of the Ti:sapphire crystal, it is optimum for high-power amplifiers operating

at high (up to rv kHz) repetition rates [97].

The oscillator crystal is pumped (pump. power: 4.2 W) by the second harmonie

(wavelength: 532 nm) of a yttrium-vanadate (Nd:YV04 ) continuous-wave laser (Mil

lenia). The oscillator generates the femtosecond pulse duration through passive Kerr

lens mode-Iocking [97, 100]. To enable mode-Iocking at the oscillator start-up, an

acousto-optic modulator (AOM) is used. Two pairs of prisms (Pl, P2; P3 , P4) are

implemented to compensate for the positive group velocity dispersion (GVD) , pre

dominantly induced by the Ti:sapphire crystal and the AOM. The oscillator pulses

(repetition rate: 74 MHz) have a central wavelength of rv 800 nm, a spectral width

of rv 40 nm (FWHM), an energy of rv 6 nJ and a transform-limited pulse dutation of

rv 30fs (FWHM).

3.2 One-kilohertz repetition rate CPA system

Figure 3.2 shows the layout of the 1 kHz repetition rate CPA system (Spectra Physics ,

Spitfire). It consists of a Faraday isolator, a grating stretcher, a regenerative amplifier,

a pulse picker, a two-pass amplifier and a grating compressor. The seed pulses of the

oscillator pass a Faraday isolator. This device (Fig. 3.3) consists of a Faraday rotator

(FR), two polarisers (PLI , PL2) and a half-wave plate (À/2):

Chapter 3. High-power laser system

Evolution 30

~ 1-10 l

! ~ :~±~ :

Q5 .c. (.)

~ êi5 ---------

ar ~ 1 C") 1 Il::: ro 1 ~ 1 1

1 1: - -+- - - - - - _ 1- -7 1 . 1 CD

~ 1 ([ 1 Q) -(/)

N I o

Cs (/) (/) Q)

a. Il E

1 Il 8 l ___________ I- __ _

~

36

CI)

Figure 3.2: One-kilohertz repetition rate CPA system (Spectra Physics , Spitfire). The following abbreviations are used: AOM: acousto-optic modulator, BS: beam splitter, CM: concave mirror, FR: Faraday rotator, G: grating, À.j2: half-wave plate , À.j 4: quarter-wave plate, M: mirror, .p: prism, PL: polariser, PR: periscope, S: spectrometer, Ti:Sa: Ti:sapphire crystal.


PL 1 FR '),)2 PL2

Figure 3.3: Principle of the Faraday isolator. The Faraday isolator protects the oscillator against back-reflected pulses from the l kHz system.

• From the oscillator to the l kHz system: The first polariser (PL l) is aligned 0°

relative to the polarisation (vertical) of the incoming pulses. The Faraday rotator

(FR) rotates (rotation: -45°) the polarisation to -45°. The half-wave plate (À/2)

rotates (rotation: -45°) the polarisation to -90°. The second polariser (PL2) is

aligned 0° relative to the polarisation (horizontal) of the Qutgoing pulses. Thus,

the pulses can pass in this direction.

• From the l kHz system to the oscillator: The second polariser (PL2) is aligned

0° relative to the polarisation (horizontal) of the incoming pulses. The half-wave

plate (À/2) rotates (rotation: +45°) the polarisation to -45°. The Faraday rotator

(FR) rotates (rotation: -45°) the polarisation to -90°. The first polariser (PLI)

is aligned 90b relative to the polarisation (horizontal) of the outgoing pulses. Thus,

the pulses cannot pass in this direction.

The purpose of the Faraday isolator is to prevent the pulses from the l kHz system to

be reflected back, since these could interfere with the Kerr-Iens mode-Iocking of the

oscillator. Inside the grating stretcher, the seed pulses are stretched to a pulse duration

of rv 200 ps (positively chirped) (Fig. 3.4). The stretched pulses are injected into the

regenerative amplifier by activating the first Pockels cell (PCI). The activation time

of PCI is set after the double-pass of the stretched pulses through the quarter-wave

plate (À/4) during the first round-trip. Now, the seed pulses are trapped inside the


E

L

t=O

Negative chirp

t=O '

Positive chirp

38

Figure 3.4: Chirped pulse. For a positively / negatively (+ / -) chirped pulse, the instantaneous frequency w± (t) == dcP±/dt == Wo ± at, where a > 0, increases/ decreases linear ly wi th time [97].

regenerative cavity and amplified until the gain ' saturation of the Ti:sapphire crystal

' is reached. The Ti:sapphire crystal of the regenerative amplifier is pumped (pump

power: rv 8 W) by the second harmonic (527 nm) of a neodymium-doped yttrium

lithium-fiuoride '(Nd:YLF) laser (Evolution-30), operating at a repetition rate of 1 kHz.

The total power of the Nd:YLF laser is rv 17 W. By activating the second Pockels

cell (PC2), 1 kHz amplified (rv 1.6 mJ) pulses are extracted and refiected out from

the regenerative cavity by the polariser PL4. After the regenerative amplifier, a pulse

picker (Fig. 3.2) is installed to select 10 Hz seed pulses for the 10 Hz system (Fig. 3.5).

The pulse picker consists of two polarisers (PL5, PL6), a half-wave plate ()"/2) and

a Pockels cell (PC3). PC3 is activated at a repetition rate of 10 Hz, producing both

1 kHz (vertically polarised) and 10 Hz (hQrizontally polarised) pulses. The 1 kHz pulses

are sent to the 1 kHz two-pass amplifier, while the 10 Hz pulses are refiected out by

the polariser PL6 and sent to the 10 Hz system (Fig. 3.5). The Nd:YLF pump power

for the Ti:sapphire crystal of the 1 kHz two-pass amplifier is rv 8.6 W. After the 1 kHz

two-pass amplifier, the 1 kHz amplified (now, rv 2.8 mJ) pulses are sent to the grating

compressor. Here, the positive chirp of the ampli,fied pulses is compensated by the

negative chirp introduced by the grating compressor. The 1 kHz compressed pulses

(Tab. 3.1) are characterised by a central wavelength of rv 800 nm, an energy of ;S 2 mJ

and a transform-limited pulse duration of rv 42 fs (FWHM).


3.3 Ten-Hertz high-power arnplifiers

Figure 3.5 shows the layout of the 10 Hz high-power amplifiers. The 10 Hz seed is

divided by a 50/ 50 beam splitter into two seeds, which .are directed to the 10 Hz two

and four-pass amplifiers , respectively. The Ti:sapphire crystals of the 10 Hz high-power

amplifiers are pumped by the second harmonics (532 nm) of two neodymium-doped

yttrium-aluminum-garnet (Nd:YAG) lasers (Quanta-Ray) , operating at a repetition

rate of 10 Hz. The Nd:YAG pump pulses (pulse duration: rv 15 ns) have energies of

rv 1200 and 910 mJ, respectively. The pump pulses of the first Nd:YAG laser (Quanta

Ray 1) is divided into two portions. The first portion (rv 500 mJ) provides the pump

pulses for the 10 Hz two-pass amplifier, while the second portion, together with the

pump pulses of the second Nd:YAG laser (Quanta-Ray 2), provides the pump pulses

for the four-pass amplifier. The Ti:sapphire crystal of the four-pass amplifier is pumped

from both sides in a "butterfly" configuration. For higher pumping efficiency, optical

relays (4f-systems) are installed between the output of the Nd:YAG pump lasers and

the high-power-amplifier crystals. Optical breakdown is prevented by guiding the pump

pulses through vacuum tubes made of glass. The 10 Hz compressed pulses (Tab.3.1)

are characterised by a central wavelength of rv 800 nm, a sub-50 fs pulse duration and

powers of ;S 0.2 and ;S 2TW, respectively.


N >. cu

c::: 1

cu ë cu :::J a

2-pass amplifier

L-

a> ~ a. E cu CI) CI)

cu c..

1 ~

Figure 3.5: Ten-Hertz high-power amplifiers.

40

Chapter 4

Filament-ind uced breakdown

spectroscopy

This chapter introduces the Stark broadening approach by presenting the filament

induced breakdo~n spectroscopy of metallic lead. Th~ gathered experience will be useful .

for the realisation of the main objective of this thesis as outlined in the introduction

(Ch. 1).

The results presented in this chapter are based on the following paper: H. L. Xu,

J. Bernhardt , P. Mathieu, G. Roy, and S. L. Chin, "Understanding the advantage of

remote femtosecond laser-induced breakdown spectroscopy of metallic targets ," J. Appl.

Phys., vol. 101, no. 3, p. 033124, Feb. 2007.

Chapter 4. Filament-induced breakdown spectroscopy 42

4.1 Introduction

Laser-induced breakdown spectroscopy (LIBS) is a powerful tool for rapid, on-line and

multielemental material analysis without sample preparation (see, e.g. , Ref. " [101]). It

is based on the emission spectroscopy of materials ablated into a small-scale plasma

by a tightly focussed laser pulse, usually from a Q-switched nanosecond laser system

(ns-LIBS). In recent years , LIBS using femtosecond laser pulses (fs-LIBS) , has attracted

much attention because of its promising properties including lower ablation threshold ,

low "continuum emission, higher sensitivity and improved detection precision [102 , 103].

However, most fs-LIBS experiments have been performed with tightly focussed laser

beams only at short distances.

Since 2004, filament-induced breakdown spectroscopy (FIBS), a special configu

ration of fs-LIBS, has been developed for remote elemental analysis of metallic as

well as biological samples [104,105]. The FIBS scheme is based on the filamenta

tion phenomenon ind uced by nonlinear propagation of femtosecond laser pulses in air

(Sec. 2.1) [32,37-39]. Since filamentation can overcome the diffraction limit of conven

tional ns-LIBS to deliver the high laser intensity at a long distance, it is natural to

assume that FIBS is a more suitable technique for remote, contact-free pro cess control

and multielemental analysis. In order to justify this assumption, this chapter analyses

the physical characteristics of FIBS. In particular, we study the filament-induced plasma

from a metallic target using spectroscopic methods, including the electron number den

sity and the plasma temperature [106]. The former determines the signal strength and

the latter the continuum contamination. Metallic lead, a highly toxic substance that

may cause severe disease [107], is selected for spectroscopic analysis. In this work , we

have determined the electron number density of filament-induced lead plasmas using the

Stark line broadening and the plasma temperature using a Boltzmann plot. We have


demonstrated that the continuum emission of the filament-induced plasma is mainly as

sociated with the filamentation phenomena itself by means of spectral measurements in

different filament positions. Plasma recombination white light is practically unmeasur

able. Compared to ns-LIBS, the measured high electron-density (rv 8 X 1017 cm- 3 ) and

low plasma temperature (rv 6700 K), together with the controllable continuum back

ground, give rise to a high signal-to-noise ratio even in the early stage of the plasma,

making this technique of FIBS more appealing. In addition, by measuring the back

scattered fluorescence signal in a light detection and ranging (LIDAR) configuration, we

estimate the single-shot detection limit to be in the kilometer range, showing promising

potential for remote applications of this FIBS technique . .


The experimental setup used for the FIBS measurements is schematically illustrated in

Fig. 4.1. The Ti:sapphire femtosecond laser system depicted in Ch. 3 was used to create

a filament-induced plasma on the surface of a lead sample. The 10 Hz pulse train,

extracted from the 1 kHz output of the regenerative amplifier (Fig. 3.2) , was further

amplified in the high-power two-pass amplifier (Fig. 3.5). A portable compressor was

used to short en the pul~e duration, measured with a single-shot autocorrelator (SSA

Positive Light), to about 45 fs. The energy of the pulse was controlled by a half

wave plate ("\/2) and a polariser (PL 7) 10cated before the amplifier (Fig. 3.5) , and

could be varied from 0.5 - 12mJ. In our .experiment, the energy was fixed to 5mJ.

The compressed pulse spectrum was centered at 800 nm with a bandwidth of 23 nm

(FWHM) , and the laser beam had a radius of 2.7mm (at l/e level of intensity). As

shown in Fig. 4.1 , two dielectric mirrors (Ml, M2: diameter d == 25.4 mm) with high


1 : 1 Side-image

/,,: ro:::~~~~::;"_OOO) " ,',' 1 "

1 JI: ,,'-

6m,/,': , // / 4m 0/'//' Fiber

, 1 1 ~

", 1 "

'-------,--·,--------'-//--1-- - -r---' L 1 800nm, 45fs, 10Hz Fiber " / ,: ~+_'__-----4t------... --------.... M 1

"" 1 Il. ,

" 1\ 1 '" \ 1 ," \ , t f=5m '" \ 1 ", \'

M3 ~,'y \,: f=1.5m~ Femtosecond Ti:sapphire laser

LI DAR mirror

Figure 4.1: Experimental setup. Side imaging and LIDAR detection of a filamentinduced lead plasma.

reflectivity at around 800 nm were used to reflect the beam. The laser pulses were

focussed in ai.r using a fused silica lens (LI: f == 5 m, thickness of 6.3 mm) , which was

selected because the intensity produced inside the filament core, in this case, is similar to

that generated by femtosecond pulses propagating freely in space [29]. The lead sample

was fixed on a rotating stage to provide new, unprocessed material for successive laser

shots. The sample under study was placed 4 m away from LI and perpendicular to the

laser beam. The distance between LI and M2 was about 50 cm.

For the measurements of the electron density and the plasma temperature, the

fluorescence emission from the plasma plume was 1:1 side-imaged by a biconvex quartz

lens (L2: f == 5 cm) onto an optical fiber, which was coupled to a 0.5 m spectrometer

(Acton Research Corp., SpectraPro-500i), equipped with a 2400 grooves/mm grating.

The spectrometer has been calibrated in the spectral range of 250 - 700 nm using a

tungsten lamp. The width of the entrance slit of the spectrometer was always set to

50 J1m. The instrumental broadening of the spectral Jines was measured, using a He-Ne

laser, to be 0.055 nm. The dispersed fluorescence ,vas detected by a gated intensified

charged cou pIed device (ICCD, Princeton instruments Pi-Max 512), which can permit

Chapter 4. Filamen.t-induced breakdown spectroscopy 45

time-resolved spectral measurements with nanosecond precision~ For remote detection ,

the fluorescence signal was .collected in a LIDAR configuration, by a concave alumin~m

mirror (M3: f == 1.5 m, diameter of 30 cm) and focussed onto the optical fiber. The

distance between the collection mirror M3 and the sample was about 6 m, limited by the

size of our laboratory. In this case, a 1200 groovesjmm (blazed wavelength at 500 nm)

was used and the spectral resolution was about 0.4 nm with 100 J-lm entrance slit width.


The main parameters that affect the fluorescence emission from a laser-produced plasma

are the electron density and the plasma temperature (Sec. 2.3). In this work, the elec

tron density of the filament-induced plasma was determined from Stark broadening .

(Sec. 2.3.4), which is one of the most frequently used spectroscopic methods to deter

mine the electron density with reasonable accuracy. Figures 4.2 (a) and 4.2 (b) show

the emission spectra of the filament-ind-uced lead plasma in the region of 350 to 380 nm,

measured using agate width of 20 ns for both cases and delay times of 20 and 240 ns ,

respectively, revealing noticeable line broadening. The spectral lin es have been assigned

to the atomic Pb l transitions according to the energy levels by Wood and Andrew [108]

(see Fig. 4.3). This measured line broadening profile results from different broadening

and shift mechanisms such as Stark broadening and Doppler broadening (Sec. 2.3.4) [67].

It is assumed that the major mechanism contributing to the line broadening is the Stark

broadening and other contributions can be ignored [106, 109, 110]. In our case, this as

sumption is reasonable, because the estimated contribution due to Doppler broadening

is negligible. For example, accorqing to Eq; 40 in Ref. [79], Doppler broadening is , for

the spectral line at 373.99 nm (7 s 3 P2 -t 6p2 1 D2), only 0.002 nm, which is much nar-


120000 (a)

11 80000 ·ë ::J

.ri 40000

~ ~ 0 • Ci) c: Q)

:Ë 20000 (b) ëij c: Cl 10000 Ci)

0

350 355 360 365 370 375 380

Wavelength (nm)

Figure 4.2: Spectra of filament-induced lead plasma with delay times of (a) 20 ilS and (b) 240 ilS, respectively.

49439.62 (i) 357.27nm 48188.63

(ii) 363.96nm

(iv) (i) (iii) 368.35nm

E 35287.2 (iv) 373.99nm 34959.9

~ >-e> ID c w

21457.7

(iii) (ii)

7819.2

Figure 4.3: Partial energy diagram of Pb land corresponding transition wavelengths.


rower than the experimental ones (0.3 - 2.2 nm). It can be seen in Fig. 4.2 that there

are two dips in the spectral lin es at 363.96 and 368.35 'nm, respectively, which origi

nate from self-absorption (Sec. 2.3.5) of Pb atoms populating the lower level, 6p2 3 I!l '

with an energy of 7819.2 cm- l . In addition, a very small dip can 'be observed in the

spectral line at 373.99 nm, showing much weaker self-absorption. This is because the

lower level of this transition has a high energy of 21457.79 cm - 1 corn pared to the 6p2 3 Pl

level (7819.2 cm -1). Therefore, the spectralline at 373.99 nm has be~n selected for the

estimation of the electron density at different delay times.

The FWHM of the Stark broadened Pb l lin es (L1ÀPb r) related to the electron density

is given by the expression (Eqs. 2.27 and 2.28) [67]

( 4.1)

where Ne is the electron density, WPbI the electron impact para~eter, A pbI the ion

broadening parameter, and ND the number of particles in the Debye sphere (Sec. 2.3.1) ,

which can be estimated by ND ~ 1.72 x 109T;';Ne-l

/2 [67]. The first term in Eq.4.1

refers to broadening due to the electron impact, whereas the second term provides the

ion broadening correction. The values of WPbI == 0.014 nm and ApbI == 0.0082 nm were

taken from Fishman et al. [111] for , the spectral Pb l line at 373.99 nm. Since the

contribution of the nonhydrogenic ion Stark broadening is normally very smaIl, with a

contribution estimated to be about 4% (TpbI == 6500 K and Ne == 1018 cm-3, see below) ,

it can be neglected and Eq. 4.1 thus becomes

(4.2)

The spectral line at 373.99 nm can be fitted quite weIl to a Lorentzian line profile ,


which is typical for a Stark broadened line, as illustrated in Fig. 4.4. This spectrum

was obtained using agate width of 20 ns and a delay time of 40 ns. The dashed line

represents experimental data and the sol id lines represent the Lorentzian fits. Since the

instrumental broadening linewidth has been measured to be 0.055 nm using a He-Ne

laser, the FWHM (~ÀPb r) of the Stark broadening can be corrected by subtracting the

instrumental contribution from the ·measured FWHM of the line. Using Eq.4.2 , the

electron density of Ne ~ 8 X 1017 cm - 3 was obtained for a 20 ns time delay with respect

to the laser pulse arriving on the target and a 20 ns gate width. This result is similar to

those produced in ns-LIBS (rv 1018 cm-3), typically with the energy of > 50 mJ, which

is much larger than that (5 mJ) used in this work (see, e.g. , Ref. [112]). By setting the

gate width to 20 ns and changing the time delay, we obtained the temporal behaviour of

the electron density, as shown il! Fig. 4.5. The last point was obtained at a delay time

of 220 ns with a linewidth 5.5 times larger than the instrumental broadening in order

to ensure the estimation reliability. The solid line in Fig. 4.5 corresponds to a least

square fit of the experimental data to an exponential decay, showing the behaviour of

the electron density decay.

To determine the plasma temperature, an analysis of the line intensity versus the

energy· of the upper level of the transition was carried out, since the population density

of the excited electronic state can be described by the well-known Boltzmann distribu

tion. The line intensity related to the plasma temperature (Tpbr ) can be expressed as

(Eq.2.21) [67]

(4.3)

where Iki is t~e intensity for the transition from the upper level k to the lower level i,

U (T) the partition function of the species, No the number density of the species, and

Ek and gk the excitation energy and statistical weight of the upper level k, respectively.


100000

80000

~ ·c ::1 60000 .e ~ ~ 40000 • Ci) c: ~ ..s

20000

0 350 355 360 365 370 375 380

Wavelength (nm)

Figure 4.4: Spectrum of the filament-induced"lead plasma with a delay of 40 ns (dotted) and multipeak Lorentzian fitting (solid lines). The Stark broadening profile of Pb l at 373.99 nmwas used for the determination of the electron density.

Taking the log of Eq. 4.3 yield~'

" ln( Àkilki) ex: gkAki

(4.4)

Ek, the plasma temperature TpbI can be obtained. Besides, a necessary condition has

to be satisfied, that is, the assumption of the validity of the local thermal equilibrium

(LTE), which is given by the following formula (Eq. 2.16) adapted to the lead plasma

(4.5)

where Npb l is the lower limit of the electron density, E (in eV) the energy difference

and TpbI (in K) the plasma temperature. Under our experimental conditions, using the

78 1 Pl ---+ 6p2 1 D2 transition at 357.27nm, with an energy difference of ~E == 3.47 eV,

and substituting the highest temperature of 6794 K obtained for a delay time of 20 ns

(see below) , NpbI was determined to be about 5.5 x 1015 cm-3. This is mueh smaller


10

-C")

8 'E u

r---0 :s 6 ~ .u; c: Q) 4 "'0 c: e ~ 2 Lü

0 0 50 100 150 200 250

Delay time (ns)

Figure 4.5: Electron densities of the lead plasma as a function of delay time.

than the values (Ne == 8 X 1017 - 8 X 1016 cm-3 ) deduced over the time scale under

study.

The plasma temperature was estimated from the emission intensity of three lead

(Pb 1) lines, as listed in Tab. 4.1, together with the transition channels , the statistical

weights of the upper states and the corresponding transition probabilities. These three

lines were selected because of several reasons: (1) they are reasonably intense to provide

a high signal-to-noise ratio, (2) the transition probability values of the lines are available

in the literature and (3) the differences in the energies of the upper levels are large

enough to prpvide better precision. It should be pointed out that the spectral line

at 368.35 nm in Fig. 4.4 has not been selected because it is related tb two overlapping

transitions of atorhic Pb l at 367.15 and 368.35 nm [108], which cannot be distinguished

in the early stage of the plasma emission. It can also be noted in Fig. 4.4 that there is a

dip in the profile of the spectralline at 363 nm, which is due to self-absorption. However,

because of the Stark shift, the spectral line can ' still be used in the early stage of the

plasma. Figure 4.6 (a) shows, as an example, the Boltzmann plots for a t ime delay of


À(nm) Ek ( cm-1) Ei(cm- 1) Transitions gk Aki (106 S-l)

373.99 48188.630 2145"7.798 7s 3p~ ---+ 6p2 1 D2 5 53.3

363.96 35287.224 7819.263 7s 2p~ ---+ 6p23P1 3 30.8

357.27 49439.616 21457.798 7s 1 P~ ---+ 6p2 1 D2 3 95.3

Table 4.1: The spectral lead lines used for calculating the plasma temperature of the filament-induced breakdown plasma.

40 ns, giving rise to a temperature of 6421 ± 65 K according to th~ slope obtained from

the linear fitting. This vàlue i~ two times smaller than that obtained in a typical ns-LIBS

experiment on lead (rv 13000 K, averaged without any temporal resolution) , using an

ArF excimer laser at 193 nm [113]. As delay times increase, the Stark shifts will decrease

and eventually the dip will be observed in the centre of the profile, dramatically affecting

the temperature measurements. Therefore, we only deduced the plasma temperatures

for the early stage emission of the plasma, with delay time less than 100 ns [Fig. 4.6(b)].

Because of the short pulses and the relatively low plasma temperature, the con-

tinuum emission associated with the plasma itself is much weaker when compared to

conventional ns-LIBS [110,114], which generates a high density hotter plasma. Specifi-

cally, the continuum emission arises from free-free or free-bound transitions that result

from collisions between electrons and ambient gas species and electron-ion recombina-

tion in the plasma. However, in the filament-induced breakdown spectra, another source

that can contribute to the continuum emission is the white light produced during the

filamentation pro cess in air , as shown in Fig. 4.7. These spectra were recorded with

the sample located at (a) 4.8m, (b) 4.2m, (c) 3.2m and (d) 2.8m away'from the focal

lens (the filament started at a fixed distance around 2.5 m away from the focal lens)

and a time delay of -3 ns with regard to the laser arriving time on the target. The

fluorescence signal was detected using the LIDAR configuration (see Fig. 4.1). It can

be seen from Fig. 4.7 that the white light continuum was much weaker when the sample

Chapter 4. Filament-induced breakdown spectroscopy

1x10-11

:;{ Cl

~ 1x10-12

c:

1x10-13

Delay time=40ns T=6421K

35000 40000 45000 50000

Energy (cm -1 )

(a)

8000

g 7000

e .a i 6000

~5000

1 4000 w

~ ~ W ~ 100 1~

Delaytime (ns)

(b)

52

·Figure 4.6: (a) Typical Boltzmann plot obtained with a delay time of 40ns and (b) plasma temperature as a functioh of delay time.

was placed in the front part [Fig. 4. 7( d)), close to the starting point of the filament.

This can be explained by the fact that the white light spectral width is linearly propor

tional to the propagation distance [115]. During the filamentation process, self-phase

modulation (SPM) occurs inside the filament, leading to a strong spectral broadening

~f the femtosecond pulse from the ultraviolet (UV) to the IR (supercontinuum) [4,116].

Therefore, when the filament hits the sample surface, the white light in the filament

would be scattered back to the detection system. This backscattered white light, how

ever, can be easily eliminated by·controlling weIl the interaction position of the filament

with the sample. This property of FIBS is very favourable for practical, remote ap-

plications, because the detection window can be opened much earlier than in classical

ns-LIBS, making even nongating technique feasible.

In order to get more insight into the application of FIBS technique, it will be help-

fuI to estimate the ultimate detection distance based on the data obtained in our ex-

periment. Because of intensity clamping, the single filament-induced FIBS signal is

expected to be independent of the propagation distance of the laser pulse before the

filament hits the sample. As shown in Fig. 4.8, we measured a single-shot spectrum by

using a moderate ·ICCD gain (g == 200, while the maximum gain is 250). The distance

between the sample and the collecting mirror was 6 m. The intensity of the strongest


160000 160000 140000 140000

4.2m 120000 4.8m 120000 100000 100000 80000 80000 60000 60000

~ 40000 40000 :J

S 20000 20000 Z' 0 0 .u; c: 200 300 400 500 600 700 200 300 400 500 600 700 ID ~

160000 160000 cu c: 140000 3.2m 140000 2.8 m 0) ü5 . 120000 120000

100000 100000 80000 80000 60000 60000 40000 40000 20000 20000

0 0 200 300 400 500 600 700 200 300 400 500 600 700

Wavelength (nm)

Figure 4.7: Spectra recorded with a LIDAR configuration. The sample is located at (a) 4.8 m, (b) 4.2 m, (c) 3.2 m and (d) 2.8 m away from the focallens. For aIl cases, the gate widths are 2 J.1S and the delay times are -3 ns (with regard to the laser arriving time on the target).

spectral line at 405.78 nm (7 s 2 Pl -+ 6p2 3 P2) was as high as 36850 ICCD counts. If we

used the maximum gain (g == 250), the intensity could reach 105580 counts, according

to the gain curve of our ICCD. Therefore, from the LIDAR equation l ex: 1/ R 2, where l

is the signal intensity and R the distance between the sample and the detector [117], the

detection limit of 3a will be reached at about 325 m for this single-shot spectrum; here

a is the standard deviation of the background noise level. If we assume that the collec-

tion and detection system could be improved, e.g., using a larger telescope or replacing

the ICCD by a photomultiplier tube (PMT), and if multifilamentation occurs (over 30

filaments have been observed across the beam profile when a practical terawatt laser

system was used) [104], the signal could be enhanced by a factor of 10. The single-shot

detection limit of 3a would be reached at a distance of about 1 km, as shown in Fig. 4.9.


40000

2' 35000

"ë 30000 ::J

.ri ~ 25000

(/)

20000 ë ::J 0

15000 u 0 Ü 10000 ~

5000

0

200 300 400

Wavelength (nm)

Figure 4.8: Single-shot spectrum of the lead plasma obtained with a 20 ns delay t ime, 2 ILS ' gate width, 1200 grooves/mm grating and a gain of 200.

Ci) 108

ë :::J 107

0 ()

0 106

Ü 105 If the signal is enhanced ü

~ bya factor of 1O, ~. 104

"Ci) 1 km for 30 limit c 103

Q) \ Ë 102

"U Q) 10' ca 0, Q)

10° 325m for 30 limit ~ 10"'

0 200 400 600 800 1000 1200

Distance (m)

Figure 4.9: Extrapolation of the single-shot detection limit.


4.4 Conclusions

By studying the fluorescence emission spectra of a filament-induced plasma from metal

lic lead using the Stark broadening line profile of the transition 7 s 3 P2 ---7 6p2 1 D2 ' the

electron density of the plasma was determined to be of the order of 1017 cm- 3 . This

is similar to that obtained in ns-LIBS. The plasma temperature of around 6700 Kwas

obtained for the early stage of the plasma by means of Boltzmann plot. Besides, we

verified that the continuum emission originates principally from the white light due to

filamentation in ambient air. This white light emission can be significantly reduced by

moving the starting point of the filament with respect to the target surface, showing

the feasibility of using nongating technique. High electron density means that the sig

nal is high; low plasma temperature indicates a low background blackbody continuum.

This leads to a high signal-to-noise ratio. Furthermore, the sup'ercontinuum is con

trollable by controlling the position of the target. These qualities make FIBS a highly

attractive technique for remote sensing metallic targets. With reasonable laser energy

and improved detection system, the single-shot FIBS detection limit can extend to the

kilometer range, which appears promising for the r~mote detection of environmental

' contamination, caused by toxicants such as lead, and instant, contact-free control in

metallic processing.

Chapter 5

Proof-of-principle in argon

This chapter shows how the Stark broadening approach cou Id be applied to the filament

in a gaseous medium. This is not only important to illustrate the proof-of-principle but

also provides the basis for the experiments in air and helium (Chs.6-8).

The results presented in this chapter are based on the following paper: W. Liu, J.

Bernhardt, F. Théberge, S.L. Chin, M. Châteauneuf, and J. Dubois , "Spectroscopie

characterization of femtosecond laser filament in argon gas," J. Appl. Phys., vol. 102,

no. 3, p. 033111 , Aug. 2007.

5.1 Introduction

Recently, producing energetic few-cycle pulses through self-compression during the fem

tosecond laser filamentation process has becoine an attractive subject [19,23, 118- 120].

Chapter 5. Proof-of-principle in argon 57

Applying this new method, the output energy limit associated with optical damage

could be avoided by using gases as the interaction media. It is advantageous compared

with the conventional hollow fiber technique [121]. Furthermore, the complexity of th~

experimental setup has been remarkably reduced as compared to the alternative means

of a non-collinear optical parametric chirped pulse amplifier NOpePA [122]. Thus, a

lot of expectation has been raised on this novel approach to pro duce few-cycle pulses.

On the other hand, we have noticed that noble gases , such as argon, have been very

often employed in such filamentational self-compression type experiments. The most

recent experiment, in which tunable ultra-short pulses have been generated through

a four-wave mixing pro cess during the filamentation in gases, also implies that noble

gases could lead to higher conversion efficiency [24]. Therefore, the knowledge about

the characteristics of the filament (especially, the plasma density) plays an essential role

for optimizing the out-going self-compressed pulse. A relatively convenient method of

studying the filament will help to establish the bridge between the empirical experience

and theoretical analysis for the sake of perfecting the experimental performance. Such

a technique will als.o be welcome in the other cases like the filamentation in air (Ch. 6).

In this chapter, we report a spectroscopic approach to acquire the electron density

distribution in the filament inside an argon gas cell sin ce it is the key information that

one most wants to know. The relationship between the electron density and the Stark

broadening due ta electron impact of the atomic argon lines is (Eq.2.27)

(5.1)

where ~ÀArI indicates the full width at half maximum (FWHM) of the electron impact

broadening induced line width, Ne is the electron density and wAr l the electron impact

parameter.


Therefore , with prior knowledge (cf. Ch. 6) àf the value of wAr l , one cou Id calculate

the electron density from the experimentally obtained Stark width.


The experimental setup is shown in Fig. 5.1. The output pulses from the commercial

1 kHz Ti:sapphire CPA laser system (Spectra Physics , Spitfire) (Fig. 3.2) had a dura

tion of 42fs (FWHM) as weIl as a diameter of 5mm (1/ e2 ). The pulse energy was

2 mJ / pulse at 1 kHz repetition rate. A 1 m focal length lens focussed the laser pulse

into a transparent Plexiglas pipe 10 cm in diameter and 1.5 m "in length, which was filled

with appr:oximately 1 atm pressure of argon gas. The distance between the gas ceIl's

input window and the lens was 10 cm. The filamentation phenomenon took place inside

the gas cell and a filament was created before the geometrical focus. In order to record

the argon fluorescence spectra emitted from the filament , a fused silica lens (f == 2.5 cm,

diameter == 2.5 cm) was used to image the filament onto one end of a fiber bundle. The

magnification was roughly 1:1. Right after the imaging lens, a 2.5 cm diameter dielectric

mirror made of fused silica with high r~flectivity around 800 nm at 0° incident angle

was inserted to filter out the pump laser spectrum. The light collected by the fiber was

then coupled into a spectrometer (Acton Research Corp. , SpectraPro-500i) , which was

equipped with an intensified CCD (ICCD, Princeton Instruments Pi-Max 512) camera

as the detector. The grating with a density of 2400 grooves/mm was chosen in this

experiment. The entrance slit opening was 100/-Lrri. The corresponding resolution was

about 0.03 nm, verified by a He-Ne laser spectrum. Furthermore, the ICCD gate width

was set to be 10 ilS with zero delay to the laser pulse arriving time. The reason for

such a configuration is that after verifying the temporal evolution of the Ar l 696.54 nm

Chapter 5. Proof-of-principle in argon

800 nm, 42 fs, 2 mJ Plexiglas tube filled with argon gas

~------;-:-----LI: f= 100 cm ~ -- L2: f= 2.5 cm

l '-. :' \. (0 = 2.5 cm)

mirror (fused silica) )

0° 800 nm dielectric

Spectrometer " Optical fiber

Figure 5.1: Illustration of the experimental setup (argon fluorescence).

59

argon atomic line, which was selected to be the major spectral line inspected in the

later discussions , we found that its decay time at 1/ e level was longer than 60 ilS. Thus ,

the electron density measured during the first 10 ilS would be a good approximation

to the transient one right after the laser pulse has gone, which is of central interest to

us. In order to ensure satisfying signal-to-noise ratio, the acquired spectra were accu-

mulated for 20000 shots and averaged ten times. The spectral response of the whole

detection system, consisting of the Plexiglas pipe, imaging lens , 800 nm mirror, fiber ,

spectrometer; and ICCD camera, was measured by a calibrated tungsten lamp. Hence,

aIl the spectra were carefully calibrated.


The black squares in Fig. 5.2 represent a typical example spectrum recorded by our

detection system at a distance of 99 cm from the focussing lens. Note that the plot

shows no evident continuum background, which is one of the unique features of the

filamentation phenomenon and consistent with the "clean fluorescence" observations

[5,6,123].

Chapter 5~ Proof-of-principle in argon

80

--- 60 .$!3 'ë ::J

~ 40 ro c: 0>

ü5 20

-.- rv1easured spectrum - Voigt profile fitting

696.0 696.4

Wavelength (nm)

60

696.8 697.2

Figure 5.2: Black squares: measured Ar l 696.54 nm line; red line: corresponding Voigt fitting.

As mentioned in Sec. 2.3.4, the electron impact parameter present in Eq. 5.1 has a

weak temperature dependence [67]. Hence , in order to ensure the precision of our final

results , we decided to first measure the electron temperature. Assuming tHe dominance

of the collision (electron impact) process (Sec. 2.3.2) , the filament plasma is in local

thermal equilibrium (LTE). In the case of the filament in argon, the criterion of LTE

reads (Eq. 2.16) [67]

- 12;;:r;-- 3 Ne 2 NArI == 1.6 X 10 V TArI (~E) . (5.2)

where TArI (in K) is the plasma temperature, E (in eV) is the difference in the energy

levels of the transitions and NArI (in cm-3) the critical density, as defined in [67].

Inserting ~E '== 1. 78 eV and a reasonable estimation of TAr I == 10000 K, the critical

density N Ar I is calculated to be about 1015 cm -3. This value lies more than two orders

of magnitude below the plasma density one would expect from a theoretical study [124].

This justifies the condition ofLTE (Eq. 5.2). As a consequence, the electron temperature

can be determined by the weIl known Boltzmann method from relative lin~ intensities,

provided that their transition probabilities (Aki ) from a given excitation state are known.


The populations of the excited states follow a Boltzmann distribution. Thus, their

relative intensity (Iki ) results from Eq.2.21 [67]

(5.3)

where Àki; Akh and gk are the wavelength, the transition probability and the statis-

tical weight for the upper level, respectively;Ek is the excited level energy, TAr I the

excitation temperature, U (T) the partition function and N (T) the atomic density.

To perform the Boltzmann plot, the atomic line Ar l 430.01 nm was used together

with the 696.54 nm line. These two lines were selected, because they are well-isolated,

having a sufficient signal-to-noise ratio. Moreover, the energy difference of their up-

per levels is large enough to provide reasonable accuracy. The Ar l 696.54 nm and

430.01 nm lin es were assigned to the transitions 3s23p5 (2 Pf/2 ) 4s-3s23p5 (

2 P~/2) 4p

and 3s23p5 (2 Pf/2) 4s-3s23p5 (

2 Pf/2) 5p, respectively (Tab.5.1) (Fig.5.3) [1251. Ac

cordingly, the slope -l/kB T ArI of the Boltzmann plot , ln (ÀkiIki/gkAki) vs. Ek (Eq.5.3) ,

yields the plasma temperature. By repeating this procedure, the plasma temperature

was computed along the propagation axis and is presented in Fig. 5.4 (black squares).

The averaged temperature is about 5800 K while the peak value is less than 6200 K.

The variation between the average and peak temperatures is less than 6.5%. Since the

electron impact parameter is not temperature sensitive (it varies less than 30% in the

range of 5000 to 10000 K based on the data listed in Ref. [67]), the value of WArI at

5000 Kwas used for our calculation. The value is 0.00409 nm [67]. On the other hand, .

based on the folfowing relationship (Eq.2.25) [79]

(5.4)

(MArI is the mass of the argon atom in atomic mass units) , the temperature shown in

Fig. 5.4 corresponds to a Doppler broadened line width of 0.006 nm at 696.54 nm. Such


À (nm)

696.54nm 107496.417 93143.760 3 6.39 X 106

430.01 nm 116999.326 93750.598 5 3.77 X 105

Table 5.1: Spectral data of Ar l '696.54nm and Ar l 430.01 nm [125].

a broadening is roughly 10% of the total measured line width in our experiment and

has to he considered in the next discussion. '

It is weIl known that Stark broadening due to electron impact is characterised by a

Lorentzian line profile (L), w hile the instrumental hroadening of the spectrometer and

Doppler broadening both follow Gaussian profile (G) (Sec. 2.3.4) [67]. The convolution

of Lorentzian and Gaussian profiles gives rise to the Voigt profile (V) (Seç.2.3.4) of the

measured spectralline shape in our experiment, which can he expressed as (Eq.2.29)

VAr! (.\) = 1:000 (.\') L (.\ - .\') d.\' , (5.5)

where L (À) == ,j [,2 + (À - ÀO)2J, G (À) == exp [-2 (À - ÀO)2 ja2J, LlÀArI == 2, and a

is known in our experiment since the spectrometer resolutio.n and Doppler broadening

were both determined. Indeed, it has been c.onfirmed by our experimental observation

that the measured Ar l 696.54 nm line profile could be fitted rather weIl by a Voigt

shape as shown in Fig. 5.2 (red line). The good fitting of the Ar l 696.54 nm line also

hints at the negligible self-absorption effect (Sec. 2.3.5). Note that there is a weak peak

appearing to the right edge of the main 696.54 nrn peak. The origin of this peak is

not clear. We assume it cornes from sorne unknown irnpurities contained in our gas

bottle. However, it would not influence our fitting process , via which the width of the

Lorentzian LlÀAr l, required for calculating Ne in Eq. 5.1, could be deduced.

Having both WAr 1 and LlÀAr l, the electron density in the filament was calculated


120000

115000

E 110000

~ >. 105000 0> L-Q) c

430.01nm 696.54nm w 100000

95000 ,,..

90000

Figure 5.3: Partial energy diagram of Ar 1 and corresponding transition wavelengths [125].

according to Eq.5.1. The results are summarized in Fig. 5.4 as the red circles curve.

Beyond the range between 93 and 107 cm, a plasma was observed but the signal-to-

noise ratio was too poor to infer the plasma density. From the plasma density it can be

observed that the filament plasma density starts to increase from the distance of 93 cm.

It reaches a plateau at 97 cm. The plasma density remains at a level of 5.5 x 1016 cm- 3 for

6 cm and begins to decline again from 103 cm. The plateau of the plasma density arises

due to the weIl described intensity clamping phenomenon (Sec. 2.1.8) that occurred

during the filamentation pro cess [2- 4]. Note that the finally obtained electron density

again confirms that LTE holds in our case [Eq.5.2].

Furthermore, we notice the clear fluctuation of the filament electron density be-

yond the pla~eau region. We propose that it could be caused by the instability of

the longitudinal position of the filament induced by the input laser energy fluctuation.

ln the experiment, besides Doppler broadening, instrumental broadening, and Stark

broadening induced by electron impact, the spectral lines emitted from argon plasma

are subjected to the other broadening mechanisms, mainly natural broadening and

Stark broadening due to ion impact (Sec. 2.3.4) [67]. For the Ar 1 696.54 nm transition,

the natural broadening is generally negligible [126]. AIso, the contribution from ion

Chapter 5. Proof-of-principle in argon

6 --Temperature --.- Density 8000

(")

E 5 <oU

~ 4

~

~ 3 c ID "'0 2 cu E ~ 1 a: o~~~~~~~~~~~~~

7500

rot 7000 3

"'0 CD

6500 §. c CD

6000 .3

5500

92 94 96 98 100 102 104 106 108

Distance (cm)

64

Figure 5.4: Measured longitudinal distribution of plasma temperature and density inside filament. Black squares: electron temperature; red circles: electron density.

impact broadening (Eq.2.28) [67],

(5.6)

is less than 2% in the current work according to our estimation TArI == 6000 K and

Ne == 1016 cm-3 , which can therefore be neglected. Here, AArI == 0.040 nm is the static

ion broadening parameter [67] and ND is the particle number in the Debye sphere given

by ND ~ 1.72 x 109T!~~Ne-1/2 .

. It is noteworthy to point out that, though Doppler broadening was taken into ac-

count in our experiment, it has a min or contribution to the overall line broadening.

Either for larger electron impact parameter or for heavier atom, Doppler broadening

could be reasonably neglected (cf. Ch. 6) [30], making the analysis straightforward.


5.4 Conclusions

In summary, we carried out the measurements of the filament plasma density and tem

perature in a noble gas. The measurement is based on the fact that the characteristics

of plasma, such as density and temperature, are reflected in the correlative atomic flu

orescence emission properties. Especially, the plasma density is simply proportional to

line width broadening due to the Stark effect of the electron impact. Our measurements

elucidate that the filament plasma density in 1 atm pressure argon gas is in the order

of 1016 cm- 3 , and the maximum value reaches 5.5 x 1016 cm-3 . Moreover, the mean

temperature of the plasma inside the filament column is about 5800 K while the peak

value is less than 6200 K.

This provides a useful tool to conveniently diagnose the properties of the filament

created by an intense femtosecond laser pulse in agas cell. The spectroscopie method

we have developed in this chapter can now be used for the other gases (Chs. 6-8).

Chapter 6

Stark broadening analysis of filament

• • ln aIr

The study of filamentation of intense femtosecond laser pulses in air is essential for

practical applications in atmospheric sensing of pollutantsj chemical-biological agents

and in lightning or electrical discharge control (Ch. 2.4). This , especially,. addresses

issues of environmental health, public, security and defence. \

The results presented in this chapter are based on the following paper: J. Bernhardt,

W. Liu, F. Théberge, H.L. Xu, J.F. Daigle, M. Châteauneuf, J. Dubois, and S.L. Chin,

"Spectroscopic analysis of femtosecond laser plasma filament in air," Opt. Commun.,

vol. 281 , no. 5, pp. 1268-1274, Mar. 2008.

Chapter 6. Stark broadening analysis of filament in air 67

6.1 Introduction

In air, the laser intensity inside a plasma filament has a value of about 5 x 1013 W / cm2

("intensity clamping") [1- 4]. 'This intensity is high enough to ionize or dissociate the

air molecules , leading to the observation of the 'clean' fluorescence [5 , 6] (cf. C·h. 4).

Moreover, in the filament core, when performing the four-wave mixing process (Ch. 2.4) ,

a new pulse with very low intensity fluctuation of about 0.1 %, combined with high

spatial mode quality, can be achieved [24]. The plasma filament is generally surrounded

by an energy reservoir ("photon bath fi) [127,128] that can perpetuate filamentation ,

even under adverse atmospheric conditions [129]. Moreover, filament persistence has

been observed at far distances of a few kilometers in the atmosphere [83]. All the above

mentioned phenomena and applications involve the weak plasma column inside the

filament. Therefore, it is essential to precisely characterize the filament plasma density

distribution in air.

In the literature [15,26,29,65, 130, 131], the most favoured method for this purpose,

optical interferometry [25;27], is limited in sensitivity, while plasma conductivity [15 ,65]

is limited in precision (for a review, see, e.g., [73]). Furthermore, quantitative shad

owgraphy [132] and longitudinal diffraction [133] require extensive processing of the

shadowgrams or interferograms obtained, while sonographic probe [134] and nitrogen

fluorescence [29] rely on calibration by an independent method. The measured value

of the plasma density differs by six orders of magnitude, ranging from 1012 [15] to

3 x 1018 cm-3 [130]. This discrepancy, however, has been most recently resolved by

Théberge et al., via the measurement of the nitrogen fluorescence calibrated by longi

tudinal diffraction [29].

However, it would be nice to have a more convenient and reliable measurement


technique. This challenge has been realized with a Stark broadening analysis of the

atomic oxygen fluorescence emission. In fact, earlier spectroscopie study has reported

the molecular nitrogen (i.e. , Nt: B2~~ ~ X2~t , N2: C3IIu ~ B3IIg [5,123]) but not

the atomic oxygen fluorescence emission from the filament. However, in this work, the

characteristic Stark broadened atomic oxygen triplet centered at 777.4 nm has been

observed and utilized. This has been possible because the criteria for the detection of

the triplet have been satisfied: distinguishable from the background, well-isolated and

spectrally resolved.

6.2 Experirnent

The experimental setup is shown in Fig. 6.1. The 1 kHz Ti:Sapphire CPA laser system

(F~g. 3.2) was used to create a plasma column in air. A pulse duration of about 42 fs

at full width at half maximUIn (FWHM) was measured using a second-order single

shot autocorrelator (SSA, Positive Light). The pulse spectrum was characterised by a

central wavelength of 805 nm and a bandwidth of 23 nm (FWHM). The radius of the

laser beam was about 2.5 mm (1/ e2 level of intensity).

In the experiment, 1 kHz laser pulses with different energies in the range of 100 f-LJ

to 2 mJ were focussed (LI, f == 30, 50, 100 and 150 cm) in the ambient air (see Fig. 6.1).

The beam radius was varied from 2.5 to 15 mm using different beam expanders located

before the focussing lens. This series of experiments was carried out under single fila

ment condition, which was verified by recording the beam pattern with a burn paper.

A systematic study in the case of multiple filaments is beyond the scope of this thesis.

Chapter 6. Stark broadening analysis of filament in air

LI: f=30, 50, , 'L2"f-25cm

100, 150 cm ~\,,/'? (0'= ~5 ~m)

[§J j optical fiber Spectrometer

69

Figure 6.1: Fluorescence spectroscopy of a femtosecond laser plasma filament in air.

For the measurement of the plasma density, the fluorescence emission from the

plasma column was side imaged by a bi-convex fused silica lens (L2: f == 2.5 cm,

magnification: rv 1) onto an optical fiber, which could be scanned along the optical

axis. The section of the plasma column, covered by the field of view of the fiber ,

was about 1 mm x 1 mm. The optical fiber was cou pIed to a 0.5-m spectrometer

(Acton Research Corp., SpectraPro-500i)., whose two interchangeable gratings were

used: the 300 grooves/mm grating for the time-resolved fluorescence detection and the

2400 grooves/mm grating for the measurement of the Stark broadening data (i.e. , ~Àl/2

and the electron impact parameter of the 0 l (777.4 nm) triplet w~Xf). Using a helium

neon laser spectrum, the instrumental resolution with the 2400 grooves/mm grating

was measured to be 6À rv 0.03 nm (slit width: 100 f.-lm). The dispersed fluorescence

was recorded with a gated intensified charge coupled device (ICCD, Princeton Instru

ments Pi-Max 512). The temporal gating allowed for a time-resolved detection with

nanosecond precision (minimum gate width: rv 5 ns).

For the measurement of the (i) plasma temperature and (ii) w~xf (detailed procedure

described later), a modified setup was used. 1 kHz laser pu~ses , with the maximum

energy of 2 mJ, were focused (LI, f == 100 m) into a 1 m long plexiglas tube , which was

filled \vith (i) argon gas at almost atmospheric pressure (730 Torr argon, 30 Torr air) or

(ii) air, using a 25 % admixture of argon (75 % air, 25 % argon, total pressure: rv 1 atm).


In the case (ii), only when measuring the Ar l 696.54 nm line in the admixture, a long

pass filter (> 600 nm) was placed after L2 to eliminate the second-order lines of the

strong moleçular nitrogen fluorescence; in addition, a dielectric, fused silica mirror with

a reflectivity of 99.9% for wavelengths of (800 ± 50) nm at 0° incidence angle was placed

before L2 to suppress the scattered laser light.

The spectral response of the spectrometer was calibrated in the range of 250 to

700 nm using a tungsten lamp. The recorded spectra presented here were corrected

for background and the spectral response of the de te ct or é (À i ). Data acquisition was

performed by averaging over 104 or more laser shots to provide a sufficient signal-to

noise ratio. If not noted otherwise, aIl the presented results were ob.tained at agate

width integration time of 20 ns.


Figure 6.2 shows a typical spectrum of the atomic oxygen line emission in the spectral

range of 776.5 to 778.5 nm. This spectrum was measured at agate width of 20 ns and a

delay time of 0 ns with respect to the laser arrivaI. The 0 l triplet centered at 777.4 nm

can be clearly identified. The spectral lines at 777.19, 777.42 and 777.54 nm can be

assigned to the atomic transitions 2s22p3 (48°) 3s - 2s22p3 (48°) 3p, where Jk == 3, 2 and

1, respectively [125]. The separation of the outer lines of the 0 l (777.4 nm) triplet

is 0.35 nm. The individual lines could be resolved by the 2400 grooves/mm grating.

However, when using a focallength shorter than ;S 30 cm, the line width increases , so

that neighbouring lines merge together. The intensity ratio of the individual peaks is

close to 7:5:3, which is given by the ratio of the statistica) weights of the upper levels '


1 ()()()()()

80000

.. o~ .... ~~~~~ .. --~

776.5 777.0 777.5 778.0 778.5 Wavelength (nm)

Figure 6.2: 0 l (777.4nm) triplet. Spectralline of the oxygen triplet (black, squares) fitted by a Voigt profile (red, circles). The spectrum is taken in pure air. The energy is 2 mJ and f 1# == 200.

of the transitions [135] . .

Figure 6.3 shows the temporal evolution of the 0 l (777.4 nm) triplet. The sequence

has been recorded with the 300 grooves/mm grating and is therefore not spectrally re-

solved. The individu al curves correspond to the fluorescence signal at different delays in

the range from 651 to 696 ns (step size: 5 ns). The laser pulse arriving time corresponds

to a time delay of 666 ns, which has been set as a zero reference. The fluorescence

lifetime of the 0 l triplet is less than I"'V 5 ns. It is important to note that the tempo

ral resolution of this estimate was limited by the minimum gate width (I"'V 5 ns) of the

spectrometer's detector. This fluorescence lifetime is typical for a high density plasma,

where collision al (electron impact) processes, contrary to radiative ones, dominate the

decay mechanism of the excited atoms (Ch. 2.3.2) [61].

Because of the dominance of the collisional (electron impact) pro cesses (Sec. 2.3.2) ,

the plasma filament is in local thermal equilibrium (LTE). The criterion of LTE, adapted

to the filament in air, reads (Eq.2.16) [67,73]

- 12 ~ 3 Ne 2: NOl == 1.6 X 10 V TOI (~E) , (6.1)


7000

6000

5000

4000 :j

.ri ~ 3000 CI)

2000

1000

0 740 760 780

À (nm) 800

---651 ns 656 ns 661 ns

--T- 666 ns ---+- 671ns -.-. 676 ns

-+- 681 ns 686 ns 691 ns 696 ns

820

72

Figure 6.3: Temporal evolution of the 0 1 (777.4nm) triplet. The sequence is recorded in pure air. The energy is 2 mJ and f / # == 200. The 300 grooves/mm grating is used .

. The time delay of 666 ns, which corresponds to the laser pulse arriving time, is set as a zero reference. The minimum detector gate width is rv 5 ns.

where TOI (in K) is the plasma temperature, ~E (in eV) the difference in the energy

levels of the transitions and NOl (in cm-3 ) the critical density for thermal equilibrium,

as defined in [67]. In the case of the 0 1 (777.4 nm) triplet, the energy difference

of the atomic transitions 2s22p3 (4So) 3s - 2s22p3 (4So) 3p, where J k == 3, 2 and 1, is

~E ~ 1.59 eV. Therefore, inserting the measured plasma temperature of TOI rv 5800 K,

the critical density is calculated to be NOl rv 1014 cm-3 . This value lies two orders of

magnitude below the plasma density obtained in this work (Ne rv 1016 cm- 3). This

justifies the condition of LTE (Eq.6.1).

The measured line profile of the 0 1 (777.4 nm) triplet originates from different

broadening mechanisms such as Stark broadening and Doppler broadening (Sec. 2.3.4)

[79]. Here, Stark broadening prevails, resulting from the perturbation of the excited

levels of the radiating oxygen atoms due to collisions mainly with the plasma electrons,

whereas other broadening mechanisms are negligible [67,136]. In particular, the contri-


. bution of Doppler broadening (Eq.40 in [79]) is calculated to be less than rv 0.01 nm.

This contribution can be neglected compared to the FWHM of the experimental line

(from 0.08 to 0.2 nm). The plasma density inside the filament in air, associated with

electron impact, is proportional to the FWHM of the Stark broadening line width

(6.2)

where Ne is the plasma density and w~xf the electron impact parameter. The correction

for non-hydrogenic ion Stark broadening is given by (Eq.2.28)

ion Ne 3 -1/3 exp

(

. ) 5/4 ( ' ) ~'\OI == 3.5AoI 1016 1 - '4 ND WOI , (6.3)

where Ao 1 is the static ion broadening parameter and ND the particle number in the

Debye sphere, given by ND == 1.72 X 109T~~2 Ne-1

/2 [67]. Inserting w~xf == 0.0166nm,

AOI == 0.035 nm [67] and the plasma parameters deduced in this work (Ne rv 1016 cm-3

and TOI rv 5800 K), ion broadening is only :s 0.01 nm and can therefore be neglected.

As discussed in Sec. 2.3.4, Stark broadening due to electron impact is characterized

by a Lorentzian line profile [67,79], while instrumental broadening follows a Gaussian

profile (~ec. 2.3.4) [67]. For the 0 l (777.4 nm) triplet, because the instrumental con

tribution (6'\ rv 0.03 nm) ranges from 15 to rv 40 % compared to the experimental

line widths, the Stark broadening line width (~,\o 1) cannot be simply corrected by

subtracting 6'\ from the FWHM of the experimentalline (cf. Ch. 4). Instead, the 0 l

(777.4 nm) triplet (Fig. 6.2, black squares) can be fitted quite weIl by a convolution of

a Gaussian (G) and a Lorentzian profile (L), i.e., a symmetrical Voigtprofile (Fig. 6.2,

red circles) (Eq.2.29) [67]

(6.4)


Here, the FHWM of the Gaussian is given by the instrumental broadening (6'\ rv

0.03 nm, fixed) , and the FWHM of the Lorentzian is extracted from a three-peak Voigt

fitting to yield the Stark broadening line width (~'\o r). The good fitting of the 0 l

(777.4 nm) triplet also indicates the negligible effect of self-absorption (Sec. 2.3.5).

The plasma temperature is determined, using the modified setup, from the corrected

line intensities (Iki) of the atomic argon lines Ar l 696.54 nm and 430.01 nm (Tab.5.1)

(Fig. 5.3) (cf. Ch. 5) [125]. Since the first ionization potentials of argon ( 15.8 eV) and

oxygen (12.1 eV) are similar, we assume that the plasma temperature in argon (experi

mental section, case i) does not significantly differ from that in pure air. In addition,

the precision of the plasma temperature has only litt le influence on the absolute value

of the plasma density. BasicaIly, the electron impact parameter is not sensitive to

the plasma temperature: for Ar l (696.54nm), in particular, the variation is less than

;S 40 % in the range of 2500 to 10000 K [67]. The use of argon gas may therefore be

justified.

In LTE, the plasma temperature evaluated from Eq. 5.3 is rv 5819 K (aerr == ±275 K).

This value is obtained by averaging over the whole length of the filament (Fig. 5.4, black

squares). Along the plasma column, no significant change (;S 6.5 %) of the plasma

temperature is observed (Fig. 5.4, black squares) [35]. In the following analysis , because

of the temperature insensitivity of the electron impact parameter (Ch. 2.3.4) , we chose

Tor rv 5000 K. This plasma temperature of rv 5 X 103 K is relatively low compared to

that in a typical nanosecond laser induced air plasma, i.e., 2 to 3 X 104 K, measured

within;S 100 ns after the laser arrivaI [135]. Therefore, the plasma blackbody continuum

is relatively low as weIl [5,123].

As mentioned before, the value of w~xf is to be experimentally determined. For

Chapter 6. Stark broadening ap.alysis of filament in air 75

this purpose, the plasma density was registered in air (75 %), with a 25 % admixture

of argon (experimental section, case ii), at three different points around the foeus (dis

tances: 0, ±1 cm). In order to obtain the same effective plasma density (see Fig. 6.4) ,

w~xf requires correction by a factor of I"'..J 6.7, i.e., from the o~e calculated by Griem,

woa~c(5000K) == 0.00248nm [67], to w~Xf(5000K) == 0.0166nm (20%). The estimated,

overall error, ~w~xf == ±20 %, includes experimental errors and the uncertainty of

I"'..J 15 %.in the electron impact parameter of Ar l (696.54nm) [126,137- 140]. In order to

provide higher accuracy, w~xf was measured at the highest density of I"'..J 5 x 1016 cm- 3,

where the line width was largest.

As a reference for the above analysis, the most investigated, recommended Ar l

(696.54 nm) spectral line is used, extensively reviewed in [126, 137-140]. As in the

case of the 0 l (777.4 nm) triplet, the Ar l (696.54 nm) fluorescence line is fitted by a

Voigt profile (Fig. 5.2, red line) [35]. As discussed in Ch.5, the contributions due to

ion CS 2%) (Eq.5.6) and Doppler broadening, I"'..J 10% (Eq. 5.4) , ~re again negligible

[35]. The value of the electron impact parameter of the Ar l (696.54 nm) spectralline,

WArI (5000 K) '== 0.00409 nm, is taken from Griem [67]. Since the above procedure was

carried out under a fixed laser plasma condition (T, Ne: constant), a slight change of

the plasma properties, induced by the 25 % argon admixture,would not significantly

affect the precision of w~f.

The divergence in w~xf and wût exceeds the limit of error (~w~Xf). In an earlier

work, similar behaviour has been observed by Baronets and Bykova [141]. Their ex

periment was carried out under similar plasma condition: Ne I"'..J 1015 to 1016 cm-,-3 and

T I"'..J 7000 to 9000 K. The authors concluded then (and referred to Zhivotov et al.,

"Diagnostics of nonequilibrium, chemically active plasmas", Moscow (1985) [in Rus

sian]) that an unknown mechanism of line broadening of the oxygen atom could be


-.-N --+- N e, ArI e, Ol

35 .-------.-------. 8 30 u

1.0

0 25 ~

.g 20 00 ~ Q)

15 "'C c'\S

8 10 00 c'\S

p:: 5 .-------.-------.

-1 0

Distance / cm

Figure 6.4: Determination of the electron impact parameter (w~Xf) of the 0 1 (777.4 nm) triplet. The gas concentration is 75 % air, admixed with 25 % argon (total pressure: rv 1 atm). The energy is 2 mJ and 1/# == 200. Vnder a fixed laser plasma condition (T, Ne: constant), the plasma density is registered at three different points around the focus (distances: 0, ±1 cm). The plasma density is evaluated from the 0 1 (777.4nm) spectral lines (red, circles) and the Ar l (696.54 nm) spectrallines (black, squares) , respectively. Here, the electron impact parameters for Stark broadening are taken from [67].

involved due to its high chemical activity [141]. In addition, because oxygen belongs

to one of the elements with a more complex structure, this may complicate theoretical

calculations [73].

The longitudinal plasma density distribution, evaluated by inserting ~Ào land

w~f (5000 K) == 0.0166 nm into Eq.6.2, is in the order of rv 1016 cm-3 (Fig. 6.5). This

showsgood agreement with previous measurements of single filament [29]. The plasma

density was recorded over a range of about 7 cm. At a distance of more than about 5 cm

away from the focal region (1 == 100 cm) in both directions, the signal-to-noise ratio

was poor, and no data were taken. It has to be mentioned that the dynamic range of ·

the measurement was limited by the accumulation time "On the ICCD chip, as weIl as

the low sensitivity of the 2400 grooves/mm grating in this wavelength range. As can be

seen in Fig. 6.5, the plasma column is stable within a region of about 4 cm in the case of

the 1 == 100 cm lens. Farther away from the stable region, the plasma density decreases


2.5 M

S 2.4 u

'; 2.2

~ 2.1

.~ 1.9 (])

"'d cIj 1.8 S ~ 1.6

0: 1.5 +------,--...,.----,---.-----,--..,---~_____,

96 98 100 102 104 Distance / cm

77

Figure 6.5: Longitudinal plasma density distribution of femtosecond laser plasma filament (pure air). Theenergyis 2mJ and thebeamradius 2.5mm (lje2 Ievelofintensity). The distance 100 cm corresponds to the geometrical focus of the lens (f = 100 cm).

rapidly. Here, the distance of rv 100 cm corresponds to the geometrical focus , of the

f = 100 cm lens. This behaviour can be explained by intensity clamping: plasma gen

eration balances self-focussing, which leads to a limited, stabilized peak intensity [2- 4].

The observed characteristics of self-siabilization is fundamental to the filamentation

process, as pointed out by Chin et al. [42].

The effective plasma density is shown in Fig. 6.6 as a function of laser energy under

different focussing conditions. In the case of the shorter focal lengths (f = 30 and

50 cm), increasing the input laser energy, the plasma density increases rapidly until

the slope tends towards a constant at high laser energy (Fig. 6.6). This latter effect

is again due to intensity clamping, which leads to saturation of the laser intensity

and therefore the plasma density [2- 4]. The characteristic energy of about rv' 500 f..LJ,

indicated by the change of the slope in the plasma density, corresponds to a power of

about 10 GW, which is the critical power for self-focussing measured in air recently [54].

, The steep rise below thecritical power results from the highly nonlinear dependence

of the plasma density on the laser intensity before intensity clamping occurs [29]. This

can be 0 bserved for aIl the focal lengths (f = 30 and 50 cm). In the case of the

f = 100 and 150 cm lens, the signal-to-noise Tatio below the critical power was not


5.2 - --f=30cm '-II- f= 50 cm ,..., • f= 100 cm -. f= 150 cm '8

4.5 /.-. . ....--.-.-. u 'C) • / . ---... / 0 • / •

3.7 / • • • • . f' • • •

• rJJ • • • ~ 3.0 j •• (])

"0 ~ - • 8 • • • rJJ 2.2 •• . - ~ - . • • • • • •• ~ • • • • E: • • •

1.5 0 500 1000 1500 2000

Energy / ~J

Figure 6.6: Plasma density as a function of laser energy under different focussing conditions (pure air). The beam radius is 2.5 mm (1/e2 level of intensity).

sufficient. Furthermore, the use of shorter focallengths was problematic because of the

merging together of the individuallines of the 0 l (777.4 nm) triplet. The use of longer

focal lengt hs was limited to constraints of our laboratory. The average plasma density

increases with shorter focal length (Fig. 6.6). This agrees with earlier measurements

and has been discussed in detail by Théberge et al. [29]. The maximum plasma density

is about 5 x 1016 cm-3 for the f == 30 cm lens. An increase in the diameter of the

beam radius, introduced by a beam expander installed before LI (f == 100 cm) , leads

to a linear increase of the plasma density (Fig. 6.7). The plasma density . in the order

of l'V 1016 cm -3 , together with the relatively low plasma temperature of l'V 5 x 103 K,

give rise to a good signal-to-noise ratio of the 0 l (777.4 nm) signal [34]. However, this

plasma density is two orders of magnitude below that measured earlier with nanosecond

l~ser pulses (l'V 1018 cm-3 [135]). Assuming a number density of 2.5 x 1019 cm- 3 in air

(1 atm) , the plasma filament has a relatively low ionization degree of l'V 10- 3.


4.5

E 3.7 u

-0

'0 ..........

Z' 'Vi c 3.0 Q.)

"0 m E VI m a:

2.2

----Ll:f=100cm /

/ -/~-

5 10 15 20 25 . 30

Diameter /mm

Figure 6.7: Plasma density as a function of laser energy under different beam diameter conditions (pure air). The beam diameter is varied from 5 to 30 mm (1/ e2 level of intensity) and f == 100 cm.

6.4 Conclusion

In summary, we have carried out a spectroscopie analysis of a filament generated by

a femtosecond laser pulse in air. As opposed to earlier work [5,123], the spectrally

resolved Stark broadened atomic oxygen triplet centered at 777.4 nm has been observed.

U nexpectedly, the electron impact Stark broadening parameter of the triplet has been

measured to be larger than the theoretical value by Griem [67] by a factor 6.7. However,

the experimental value 0.0166 nm leads to plasma densities that are not different from

those most recently reported by Théberge et al., using a different measurement method

[29]. This makes the Stark broadening approach reliable. Moreover, the validity of the

Stark broadening analysis has been verified. Therefore, this much simpler technique

could become a standard method to measure the filament plasma density distribution

in air under different propagation conditions.

Chapter 7

Critical power of helium

This chapter describes how the critical power for self-focussing of a femtosecond laser

pulse in helium was measured by ·using the moving focus method. In addition, the plots

of the electron densities versus energy and pressure were also used to determine the

critical . power . of helium, based on the intensity clamping of the filamentation process.

The value agrees weIl with the one by the moving focus method.

The results presented in this chapter are based on the following paper: J. Bernhardt ,

P. Simard, W. Liu, H. Xu, F. Théberge , A. Azarm, J. Daigle, and S. Chin, "Critical

power for self-focussing of a femtosecond laser pulse in helium," Opt. Commun. , vol.

281, no. 8, pp. 2248-2251 , Apr. 2008.

Chapter 7. Critical power of helium 81

7.1 Introduction

Recently, there has been important progress in producing few-cycle pulses via filamen

tation of femtosecond laser pulses in gases (Ch. 2.4) [19 ,23,24,118,119]. Couairon et al.

have proposed that one could make use of the filamentation pro cess in a noble gas with

a pressure gradient to produce compressed pulses down to one optical" cycle [142]. The

perhaps most striking results are those obtained by Stibenz et al. , who have reported

highly efficient (conversion efficiency: ;S 85%) pulse compression in argon gas down

to less than ;S 10 fs with a peak power of more than ~ 100 GW (Ch.2.4) [23]. It is

important to note · that this has been achieved without any dispersion compensat ion or

pressure gradients. However, Skupin et al. 's theoretical analysis has revealed that t his

compression scheme is only optimum up to about 5 times the critical power [92].

Alternatively, a Japanese research group has used an argon-fi lIed hollow fiber with

a pressure gradient to produce sub-10 fs pulses with an energy as high as rv 5 mJ [143].

Moreover, N urhuda et al. have proposed a highly efficient compression scheme (conver

sion ratio: ;S 88%) using 100 mJ, 40 fs pulses · in a helium-filled (;S 1 Torr) multi-pass

cell (length: rv 6 m, mirror radii: each rv 3.1 m) [144] .

. In 2006, Painter et al. measured the spatial evolution of a 5 mJ, 30 fs pulse in a

helium-filled (rv 80 Torr) gas cell [145]. The authors concluded ·the "direct observation

of laser filamentation in high-order harmonie generation". However, the input power

of rv 160 GW was well below the critical power rv 2.4 TW for a helium pressure of

80 Torr inferred from the literature value of the nonlinear refractive index nMt (1 atm) =

3.5 x 10- 21 cm2/W [48]. Thus, it has been suggested that the Kerr nonlinearity of

. helium should be re-examined. However, it was later conceded by the same authors

that their results could also be interpreted using the above value nMt [146, 147].


Therefore, it is important to carry out a direct measurement of the critical power

of helium. In this work, we have experimentally measured the critical power of helium

using the moving focus method [54]. AIso, the critical power has been obtained,from the

. electron densities , based on the intensity clamping process, which have been measured

as a function of energy and pressure.

7.2 Experilllent

The experiments were performed using 42 fs pulses (repetition rate: 10 Hz) with a center

wavelength of about 800nm (Fig. 3.5). The pulses were focused (f = 100 cm) into a

gas chamber filled with pure helium gas whose pressure cou Id be varied from 50 Torr

to 1 atm. The energy could be varied in the range of 1 - 63 rnJ. The pulses ·created

a plasma filament inside the charnber. The fluorescence was c611ected from the side

by imaging the length of the filament onto the entrance slit of a spectrometer (Act on

Research Corp., Spectra Pro-500i) (Fig. 7.1). The images of the filament were recorded

in the spectrometer's imaging mode (accumulations: 20) by using the zero-order grating

reflection with the slit widely opened. The presence of a single filament was verified.

The spectra were taken with the 1200 grooves/mm grating. The spectral resolution

of this grating was about 0.4 nm (slit width: rv 100/Lm) (Ch.4) [34]. The dispersed

fluorescence was detected with a gated intensified CCD (ICCD, Princeton Instruments

Pi-Max 512). The ICCD gate width was set to 20ns. The detection window was

opened with zero delay after the laser-plasma interaction. The instrumental response

was calibrated in the range of 250 - 800 nm using a tungsten lampe

Chapter 7. Critical power of helium

f=100cm

BOOnm <63mJ -42fs

83

Gas cell filled with helium gas

Figure 7.1: Illustration of the experimental setup used for the measurement of the critical power of helium.


Fig. 7.2 shows a log-linear plot of the peak position of the fluorescence signal as a func-

tion of energy in the range of 3 - 45 mJ. The pressure was fixed at 1 atm. The peak

positions were retrieved from Gaussian fittings of the on-axis fluorescence distributions

versus distance in units of ICCD chip pixels [54]. Note that smaller pixel values cor-

respond to distances closer to the focussing lens. It can be seen in Fig. 7.2 that the

peak position moves closer towards the focussing lens as the energy is increased. This

behaviour has been explained before [54] as a consequence of the power dependence of

the self-focussing distance above the critical power [51].

The plasma effect that shifts the center of gravit y of the plasma in the linear focal

region is known in long pulse laser-induced breakdown (see, e.g., [135]). This latter shift

is due to the high density plasma created in the linear focal region that partially blocks

the transmission of the laser pulse through the plasma. This is true in both the long

pulse regime where the plasma is generated principally through collision al ionization

and the short pulse regime (current case) where the plasma is due to pure tunnelling

ionization. Note that the plasma density in this case is very high, more than 1017 cm- 3

(see below).


500

450

~ 400 :§: c: o ~ 350 &. ~ tU

~ 300

250

84

-11 .25 mJ "'. ... 10

Energy (mJ)

Figure 7.2: Critical power of helium. The peak position of the fluorescence signal is plotted versus energy in a log-linear scale. The vertical axis is in units of ICCD chip pixels. Smaller pixel values correspond to distances closer to the focussing lens. The pressure is fixed at 1 atm. The critical energy is determined from the crossing point of two linear fits (red lines) through the data (black squares). The value of rv II.25 mJ corresponds to a critical power of Pcexp (1 atm) rv 268 GW (pulse duration: rv 42 fs).

The critical energy is determined ·from the crossing point of two linear fits (Fig. 7.2,

red lines) through the data (Fig. 7.2, black squares) [54]. The value of rv II.25 mJ

(Fig. 7.2) corresponds to a critical power of Pcexp (1 atm) rv 268 GW (pulse duration:

rv 42 fs). The relative errors in the measured energy (;S 18%) and pulse duration

(;S 12%) give an uncertainty of ;S 30% in the critical power. Marburger 's equation

(Eq.2.3) adapted to heliurrt reads [51]

(7.1)

where À is the central wavelength, no the linear and n~e the nonlinear index of refraction.

Inserting À == 800 nm, n~e == 1 and Pc == Pcexp (1 atm) (±80 GW) into Eq.7.1 yields a

nonlinear refractive index of n~xp (1 atm) rv 3.6 X 10-21 cm2 jW (±1.1 x 10-21 cm2 jW).

This value is essentially equal to the one cited above [48].

To verify the reliability of the value obtained from the moving focus method , we


have measured the electron densities of helium plasma as a function of energy and

pressure. The electron densities have been derived from Stark broadening of the atomic

line He l 587.56nm (ls2p 3 po-1s3d 3D) (Fig. 8.1) [125, 148]. As known from Chs.5 and

6, the Stark broadening approach [67] applied to plasma filaments in gaseous media has

proved reliable before (Tab.2.3) ( [30] and references therein).

Fig. 7.3 shows the electron density as a function of energy in the range of 1- 63 mJ.

The pressure is fixed at rv 320 Torr. It can be seen in Fig. 7.3 that the electron density

increases until the change of slope tends towards a constant above a critical energy

of around rv 25 mJ (Fig. 7.3, linear fit). This behaviour is characteristic of intensity

clamping (Sec. 2.1.8) , which has been discussed in detâil in [2,3,29, 30]. The nonlinear

index of refraction is linearly proportional to the pressure (Eq. 8.5). Thus, the critical

power is inversely proportional to the pressure

1 1 Pc ex: - ex: -

n2 P (7.2)

Using Eq.7.2 , the critical point (rv 25 mJ, 320 Torr) corresponds to a critical power '

of p~xp (1 atm) rv 251 GW (pulse duration: rv 42 fs), which is in agreement with the

measurement obtained in Fig. 7.2.

Fig. 7.4 shows the electron density as a function of pressure in the range of 50 -

760 Torr. The energy is fixed at rv 35 mJ. The linear dependence on pressure above

a critical pressure of around rv 250 Torr (Fig. 7.4, linear fit) can again be explained by

intensity clamping (Ch. 8) [149]. In 'short, the electron density (Ne) is a function of the

intensity (1) according to Eq. 2.6 adapted to helium (cf. Ch. 8)

(7.3)


Q) z

6

5

2

, -..-----_---. \1 __ -- ' J

/-/ -/

i ! -25 mJ,

-o 10 20 ' 30 40 50 60 70

Energy (mJ)

86

Figure 7.3: Intensity clarnping: electron density versus energy. The data (black squares) shows the electron density as a function of energy in the range of 1 - 63 rnJ. The pressure is fixed at rv 320 Torr. The electron density increases until the change of slope tends towards a constant above a critical energy of around rv 25 mJ (Iinear fit). This

, behaviour is characteristic of intensity clarnping (see text). The critical point ( rv 25 rnJ , rv 320 Torr) corresponds to a critical power of Peer~f (1 atm) rv 251 GW (pulse duration: rv 42 fs). There fs no saturation by depletion (ionization degree: rv 10-2

).

where NHe is the neutral density and RHe the ( tunnelling) ionization rate. The critical

(clamped) intensity is independent of pressure le i= le (p) (Ch.8) [38,45,149, 150].

Thus, the electron density is linearly proportional to pressure above the critical pressure

(Eq. 7.3). The critical point (rv 35 mJ, rv 250 Torr) corresponds to a critical power of

p~xp (1 atm) rv 274 GW (pulse duration: rv 42 fs). Note that the electron densities are

in the order of rv 1017 cm-3 (Figs. 7.3 and 7.4). This corresponds to an ionization degree

of rv 10-2, which is weIl below the depleti<?n lirnit [151].

7.4 Conclusion

In conclusion, the criticalpower of helium has been rneasured to be Peexp (1 atm)" rv

268 GW (±80 GW) using the moving foeus method. This value corresponds to a non-

Chapter 7. Critical power of heliùm 87

7 -;:;7 - • . -6

• '1- 5

E 0 ,...

/ ~~4

CD z3 / .

/ -250 Torr 2

•

0 100 200 300 400 500 600 700 800

Pressure (Torr)

Figure 7.4: Intensity clamping: electron density versus pressure. The data (black squares) shows the electron density as a function of pressure in the range of 50 to 760 Torr. The energy is fixed at rv 35 mJ. The electron density shows a linear dependence on pressure above a critical pressure' of around rv 250 Torr (linear fit). This behaviour can be explained by intensity clamping (see text). The critical point ( rv 35 mJ, rv 250 Torr) corresponds to a critical power of pcer~f (1 atm) rv 274 GW (pulse duration: rv 42 fs). There is no saturation by depletion (ionization degree: rv 10-2

).

linear refractive index of n~xp (1 atm) rv 3.6 X 10-21 cm2/W (±1.1 x 10-21 cm2/W) , which

agrees weIl with the theoretical value n~t (1 atm) = 3.5 x 10-21 cm2/W in Ref. [48]. The

critical power has also been measured using the plots of the electron densities versus

energy and pressure, which give the same value (within the error bar) as the one by the

moving focus method.

Chapter 8

Pressure independence of intensity

clamping

This chapter shows that because of the dynamic equilibrium between Kerr self-focussing

and plasma induced defocussing and the inexistence of collision al ionization the critical

intensity during femtosecond laser filamentation in air (or other gases) is independent

of pressure. An an?lytical analysis is given which is justified by a direct experimental

verification.

The results presented in this chapter are based on the following paper:

• J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, "Pressure independence of

intensity clamping du ring filamentation: theory and experiment ," Appl. Phys. B ,

vol. 91, no. 1, pp. 4548, Apr. 2008.

Chapter 8. Pressure independence of intensity clamping 89

8.1 ' Introduction

The universal process of intensity clamping (Sec. 2.1.8) has been reported in Ref. [2- 4].

Since then, much work was also performed in the long range propagation of intense

femtosecond laser pulses in air [83, 152]. Because of the possibility of generating a

filament in the atmosphere for up to many km in altitude [83] where the atmospheric

pressure is significantly reduced , it is natural to ask what the clamped intensity is at

high altitude [38, 129, 150, 153]. It turns out that the clamped intensity is independent

of pressure because the only mechanism that controls this clamping phenomenon is

MPI/ TI [7] of the gas molecules, which are uni-molecular processes. No collisional

pro cess could be involved with such a short time scale. This chapter gives a physical

account of this phenomenon [149, 154].

8.2 Analytical analysis

The fundamental relation of the balance between Kerr self-focussing and plasma defo

cussing is the equalisation of the nonlinear indices due to Kerr effect and due to the

plasma as given by the following equation (Eq.2.10)

~nK (neutral) ~ ~n (plasma) . (8.1)

or

(8.2)


where t:,.n is the index change, n2 the Kerr nonlinear index of refraction, l the intensity,

Ne the electron density induced by tunrtelling ~~nization of air molecules and Ne the

critical plasma density as defined in Sec. 2.1.7. Detailed theoretical work ( [155] and

references therein) has revealed that the equilibrium between Kerr self-focussing and

plasma induced defocussing of the laser light is of a complex dynamic nature. However,

the essential physics and the relevant orders of magnitude are governed by Eq.8.2. In

the following, we treat las~r propagation in air as an example. Essentially, identical

arguments apply to any gas or gas mixture. In air, the electron density Ne cornes from

MPI/ TI of both the oxygen and nitrogen molecules through the following rate equation.

dNe (1) dt

where RA, NA (A == N 2 or O2) denote the rate of ionization and the density of molecule '

A respectively. Integrating this equation, we obtain

Ne (1) = Nair [0.78 J RN2 (1) dt + 0.21 J R02 (1) dt] , (8.4)

where the integrals inside the bracket are independent of the air density Nair . This

approximation is valid as long as the electron density Ne stays weIl below the neutral

air density Nair (Ne « Nair ) , which isjustified for usual experimental condition where

Ne has been measured to be in the range of Ne ~ 1014 - 1017 cm-3 [15, 25- 30], which

is small compared to Naïr ~ 2.5 X 1019 cm-3 at atmospheric pressure. The nonlinear


index of refraction n2 is proportional to the air density expressed as

. (8.5)

where ~ is a proportionality constant that depends on the detailed electronic structure

. of the material , but is virtually density independent for gases of atmospheric pressure,

and has only a weak density dependence for solids. Putting Eqs.8.4 and 8.5 into 8.2,

we obtain

KNairI::::; (1/2Nc) Nair [0.78 J RN2 (1) dt + 0.21 J R02 (1) dt] . (8.6)

Eliminating Naïr from thè two sides of Eq. 8.6, we get

1::::; (l/2Nc) [0.78 J RN2 (1) dt + 0.21 J R02 (1) dt]. (8.7)

The solution of Eq. 8.7 yields the critical intensity I~ïr ~ 5 X 1013 W / cm2 for atmospheric

air [2]. The ionization rates Rare uni-molecular processes and are independent of

pressure. Thus, the critical intensity is independent of pressure. The reason for this

surprising behaviour lies in the equilibrium between Kerr effect and ionization as weIl as

the fact that the pulse is so short that no collisional ionization effect is involved in the

propagation process. Similar arguments would lead to only a weak density dependence

of the critical intensity in solids.

8.3 ExperÎlllents and results

So far , the theory tells us that the critical (clamped) intensity in air or other gases is

independent of pressure. It is important that this theoretical understanding be verified


by a direct experiment. As a case study, we chose the noble gas helium (cf. Ch. 7) , which

is very 'clean'. It is mono-atomic and there is no Raman effect nor any complication

such as molecular association and fragmentation. AIso, the Kerr nonlinearity of helium

is two orders of magnitude smaller (n~e rv 10-21 cm2 jW [36,48]) than that of air ( n~ir rv

10-19 cm2 jW). Therefore, in helium the critical intensity would be much higher (IJIe rv

7.8 X 1014 W jcm2 [156]).

In the experiment , 10 Hz, 42 fs pulses (central wavelength: rv 800 nm) (Fig. 3.5) with

the energy of 35 mJ were weakly focused (f == 100 cm) into agas chamber filled with

pure helium whose pressure could be varied up to 1 atm. The pulse propagated through

helium and created a plasma-filament. The helium fluorescence was collected from the

side by imaging the length of the filament onto the entrance slit of a spectrometer

(Acton Research Corp. , Spectra Pro-500i) (Fig. 7.1). The presence of a single filament

was verified by observing the fluorescence in the spectrometer's imaging mode by using

the zero order of the 1200 groovesjmm grating with the slit fully opened. The spectral

resolution of the 1200 groovesjmm grating was about 0.4 nm (slit width: 100 /-Lm) . (Ch. 4)

[34]. The spectrometer was equipped with a gated intensified CCD (ICCD, Princeton

Instruments Pi-Max 512). The ICCD gate width was set to 20ns and the detection

window was opened with zero delay after the laser-plasma interaction. The spectra

were accumulated 20 times. The detector's response was calibrated in the range of ·250

to 800 nm using a tungsten lampe

As a spectral signature, we chose again the atomic line He l 587.56 nm (ls2p 3 p o_

ls3d 3 D) (Fig. 8. i) [125, 148]. This line is the consequence of tunnelling ionization

followed by the recombination of He+ and an electron. Note that this signature is very

'clean ' . In particular, the dynarriic range with respect to the 30- standard deviation of

the background noise level is larger than rv 103 (20 accumulations, moderate ICCD gain


of 125, where the maximum gain is 250). The clean fluorescence is unique to "filament

induced plasma spectroscopy" [5,6], which is another consequence (besides the pressure

independence of intensity clamping) of the fact that the filament plasma is generated

by tunnelling ionization only without the occurrence of inverse Bremsstrahlung and

cascade (avalanche) ionization [7].

Since the independence of the critical intensity on pressure (Eq. 8.7) is under the ap-

proximation Ne « Nair of Eq. 8.4, we justify this approximation first in our experiment.

From the Stark broadening of the He l 587.56 nm Hne, the maximum electron density is

estimated to be about 7 x 1017 cm-3 . This leads to an ionization degree of about 10- 2 ,

justifying the above approximation. The critical intensity derived from the ionization

curve1 of He+ has been inferred to be t~e f"'V 7.8 X 1014 W jcm2 [156]. Note that the

contribution of He+ prevails over that of He++ in the relevant intensity range [151].

Fig. 7.4 shows the electron density versus pressure curve. The electron densities

derived from Stark broadening .of the He l 587.56 nm line (Fig. 8.1) were measured for

different pressures in the range of 50-760 Torr. The input power wasfixed at f"'V 833 GW

(42 fs at 35 mJ). The electron density increases linearly with pressure (Fig. 7.4, linear

fit) beyond a (critical) pressure of about f"'V 250 Torr. It is important to note that there

is no depletion of neutrals in the ionization zone (ionization degree: 10-2). The linear

behaviour beyond 250 Torr indicates intensity clamping which is expected from Eq. 8.4

adapted to He gas

(8.8)

lit is acknowledged that M. Sharifi provided the intensity calibrated ionis.ation curves of He+ and Ar+ relevant to Chs. 8 and 9 of this thesis.

Chapter 8. Pressure independence of intensity clamping

~

€ ~ ~ ëii c: .s .ç: (ij c: Cl

ü5

1200000

1000000

800000

600000

400000

200000

o~~ .. ~~~~~~~~ .. ~~ 576 578 580 582 584 586 588 590 592 594 596 598

Wavelength (nm)

94

Figure 8.1: He l 587.56 nm spectral line. The data show the Stark broadened He l 587.56 nm liiles recorded at pressures of r-v 150 Torr (black circles) and r-v 760 Torr (black squares) , respectively, fitted by Lorentzian profiles (red curves). The pulse duration and energy are the same .as in Fig. 7.4. The good fitting of the He l 587.56 nm line also indicates the absence of self-absorption.

where NHe and RHe are the corresponding values for helium. Eq.8.8 shows that the

electron density scales linearly with pressure if the integral is independent of pressure.

Since the integral is a function of the intensity (1) , it means that the intensity should

be constant as the pressure changes. To recap the experimental observation, at the

beginning at low pressures, the critical power for self-focussing is still too high and only

linear geometrical focussing gives rise to the fluorescence. At higher pressures starting

at around 250 Torr, the critical power decreases to a value equal to the input power.

Self-focussing and filamentation sets in at this pressure. From this pressure on, the

intensity no longer changes with pressure.

Note that we are still in a power regime where relativistic effects are negligible. In

fact , the critical power for Kerr self-focussing is given by Marburger's equation [51]

(8.9)


where À is the central laser wavelength and no the linear and n2 the nonlinear index of

refraction. However, relativistic self-focussing sets in at a mtich higher critical power,

which is given by [157]

p~el ~ 16.2 ~c GW e

(8.10)

Inserting the maximum electron density of r-v 7 x 1017 cm -3 yields p{el r-v 40 TW. This

is about two orders of magnitude higher than pckerr I"'J 102 GW (putting n~e I"'J 3.6 x

10-21 cm2 jW [36,48] in .Eq. 8.9); Therefore, relativistic self-focussing is negligible.

8.4 Conclusion

In conclusion, we have shown both theoretically and experimentally that the critical

intensity in air or other gases is independent of pressure. A consequence of this phe-

nomenon in vertical atmospheric propagation is that the filament size (diameter) will

become larger and larger as the altitude increases because of the following reason. Since

the critical power for self-focussing Pc is inversely proportional to n2 (Eq.8.9) and since

n2 is proportional to the air density (Eq. 8;5) , the critical power for self-focussing in-

creases as the pressure at higher altitude decreases. Hence, to obtain self-focussing at

higher altitude, the input peak power of ~he pulse has to increase. However, since the

intensity is clamped at the value at sea level, the higher peak power of the pulse has to

be contained inside a region with a larger diameter than that at Bea level.

Chapter 9

Intensity clamping of high-power

filaments under 'natural' conditions

In this chapter, we study the universal pro cess of intensity clamping under 'natural '

(i.e. , external focussing without propagation control) conditions at high (P » Pc)

peak powers. In air and argon, the average intensities were determined to be [Air rv

6.4 X 1013 W/cm2 (f == 20cm) and [Ar rv 1.2, 1.3 and 1.7 x 1014 W/cm2 (f == 100, 60

and 20 cm) , respectively.

9.1 Introduction

Nowadays, the intensity clamping (Sec. 2.1.8) of the filamentation . process is weIl un

derstood [2- 4] (see Ch.8). In particular, the knowledge of the clamped intensity is

particularly important for practical applications (Ch.2.4) [32, 37- 39]. However, the

Chapter 9. Intensity clamping of high-power filaments under 'natural' conditions 97

direct measurement of this key parameter, especially in the high-power (P » Pc) fil

amentation regime, has not been possible [32,37- 39]. This is because any probe that

was exposed to the high laser intensities inside the filament would be destroyed.

Therefore, l will describe in the following how the intensities (1) can be derived

nevertheless, approximately. The principle is based on the Stark broadening approach,

together with the knowledge of the ionisation curves of the gas species.

The method presented in this chapter was introduced in the following paper:

J. Bernhardt, W. Liu, F. Théberge, H.L. Xu, J.F. Daigle, M. Châteauneuf, J.

Dubois, and S.L. Chin, "Spectroscopie analysis of femtosecond laser plasma filament in

air," Opt. Commun., vol. 281, no. 5, pp. 1268-1274, Mar. 2008.

9.2 Experilllent

The experiment consists of two (i, ii) parts: ' 10 Hz pulses with a central wavelength of

about 800 nm were focussed with (i) a f== 20 cm lens into the ambient air or (ii)different

(f == 100, 60 and 20 cm) focallength lenses into a transparent Plexiglas tube (diameter:

f"'..I 5 cm, length: 1.5 m) filled with pure argon at f"'..I 1 atm. In the case (i), the pulse

duration was f"'..I 47fs at full width at half maximum (FWHM), while in the case (ii) , it

was f"'..I 100 fs (FWHM) negatively chirped. In both cases (i, ii), the pulse energy could

be varied up to about 70 mJ.

The fluorescence emissionfrom the plasma filament was side imaged onto the en-


trance slit of a spectrometer (Acton Research Corp., Spectra Pro-500i). In order to

increase the collection efficiency, the slit was arranged to be parallel to the filament

column. Two interchangeable gratings of the spectrometer were used: In .the case (i) ,

the 1200 groov~s/mm grating was used for the measurement of the Nt signal at 391 nm

(B2Et ~X2Et [5,123]), while the 2400 grooves/mm grating was used for the spectrally

resolved detection of the characteristic Stark broadened, atomic 0 1 (777.4 nm) triplet

(Ch.6) [30]. In the case (ii) , the 1200 or 2400 grooves/mm gratings were used for

. the measurement of the Stark broadened, atomic Ar 1 696.54 nm line (Ch. 5) [35] for

the shorter (f == 20 cm) or longer (f == 100 and 60 cm) focal lengths , respectively.

The instrumental resolutions of the 1200 or 2400 grooves/mm gratings , respectively,

were rv 0.4 or 0.03 nm (slit width: rv 100/-lm) [34,35]. The dispersed fluorescence was

recorded using a gated intensified CCD (ICCD, Princeton Instruments Pi-Max 512).

The temporal gating of the ICCD allowed for a time-resolved detection with nanosecond

precision (minimum gate width: rv 5 ns).

In the case (i), the Nt (391 nm) signal was recorded over 500 laser shots at a fixed

ICCD gain of 200 (maximum gain: 250), while the 0 1 (777.4nm) triplet was recorded

over rv 5 X 104 laser shots by setting the ICCD gain at 250 or 200 for pulse energies

in the range of 5 - 35 or 40 - 70mJ, respectively. In the case (ii), the Ar 1 696.54nm

line was recorded over 100 laser shots by setting the ICCD gain at 235 or 100 for the

f == 100 or 60 cm lenses, respectively, while the ICCD gain was fixed at 200 for the

f == 20 cm lens.

In the case (ii), the full spectra (wavelength range: 300 - 900 nm) were recorded

over 10 laser shots by setting the ICCD gain at 120 for the f == 100 and 20 cm lenses

or 125 for the f == 60 cm lens, respectively. In both cases (i, ii) , the spectrallines were

recorded at a fixed ICCD gate width of 10 ns. The detection window was opened at


Ons delay with respect to the laser-plasma interaction.

The experimental setup for the measurement of the ionisation curve 1 of Ar+ is

described elsewhere [158,159]. In short , 10 Hz, rv 45 fs (FWHM) pulses were focussed

(f == 100 cm) into a high va~uum (background pressure: rv 10-9 Torr) interaction cham

ber, and the Ar+ ions were detected in a time-of-flight (TO F) mass spectrometer. The

gas pressure was scanned in the range of rv 10-8 to 10-5 Torr.


The plasma densities of the high-power filaments in air (Nt- ir ) can be derived from

Stark broadening of the 0 1 (777.4 nm) triplets in analogue to Ch. 6 [30]. Fig. 9.1 shows

typical spectra of the 0 1 (777.4nm) triplet (Fig. 9.1 , blue ~ circles) in the wavelength

range of ab.out 777 - 777.7nm, fitted by three-peak (Fig. 9.1, black, dashed) Voigt

profiles (Eq.6.4) (Fig. 9.1, red, lines). The good fitting indicates the absence of self

absorption (Sec. 2.3.5). Note that at low energies (~ 5 rnJ), the signal-to-noise ratio

was poor, ànd no data were taken.

By using the ionisation parameters 2 from the S-Matrix theory by A. Becker (valid up

to rv 1014 W/crn2) , the plasma (electron) density (flt-ir

) can be related to the intensity

(1) via

(9.1)

1 It is acknowledged that M. Sharifi provided the intensity calibrated ionisation curves of He+ and Ar+ relevant to Chs.8 and 9 of this thesis.

2F. Théberge. Private communication. 2008.

Chapter 9. Intensity clamping of high-power filaments under 'natural ' conditions 100

160000 (a)

~ o 150000

..0

~ 140000 ~ °ê 130000 Q)

~ 120000 co c

02> 110000 Cf)

777.0 777.1 777.2 777.3 777.4 777.5 777.6 777.7

200000 (b)

~ 180000

-e ~ 160000

~ °ê 140000 Q)

Ë 120000 ro c Cl 100000

ü5

Wavelength (nm)

, , . ,,,, - , ., . /' " '" " . " '" , ' ~ -.,,; --

--==- -- - - -

777.0 777.1 777.2 777.3 777.4 777.5 777.6 777.7

Wavelength (nm)

Figure 9.1: 0 1 (777.4nm) triplets of high-power (P » pcAir) filaments in air. Typical spectra of the 0 1 (777.4nm) triplet (blue, circles) fitted by three-peak (black, dashed) Voigt profiles (red, lines). The two different spectra were recorded at pulse energies of (a) ~ 10 mJ (gain: 250) and (b) 70 mJ (gain 200) , respectively.The pulse duration was ~ 47.fs (FWHM) and the 'focallength f == 20cm. The good fitting indicates the absence of self-absorption.

Chapter 9. In tensi t y clamping of high-power filaments under 'natural' conditions 101

where (A == 02 .or N2)

(9.2)

N~2 ==" 0.6 X 1019 cm-3 , 0'02 == 1.8 X 10-83 S-l and m02 == 6.8 or N~2 == 2.4 X 1019 cm-3 ,

O'N2 == 4 X 10-110 S-l and mN2 == 8.7, respectively, are the ionisation parameters of

the N 2 or O2 molecules. Here, j rv f exp ( - (t 1 T) 2) is the (G~ussian) intensity profile ,

T == TFWHMI v2ln 2, TFWHM the pulse duration (FWHM) and fo == 1 X 1013 W Icm2.

In Fig. 9.2, the ionisation curve, f\rt"ir == f\rt"ir (1) (Eq. 9.1), is plotted as a function of

intensity in the range of about 5 -7 x 1013 W Icm2. [Note that fl: ir varies within about

1 order of magnitude (rv 1016 -1017 cm -3) in the relevant intensity range (Fig. 9.2). This

is because of the h~ghly nonlinear dependence of fl;ir on f (Sec. 2.1.6).] The intensities

(fAir) (Fig. 9.3 b) can be inferred from the intersects of the experimental values of N:ir 1 •

with the ionisation curve (Fig. 9.2, arrows), approximately.

Fig. 9.3 shows N:ir (Fig. 9.3 a) and fAir (Fig. 9.3 b) as a function of energy in the

range of about 5 - 70 mJ. It can be seen in Fig. 9.3 that N:ir (Fig. 9.3 a , red, circles)

and fAir (Fig. 9.3 b, red, circles) are independent of energy (Fig. 9.3, blue, lines) over

the whole range of input powers (PI pcAir rv 10 - 150, where pcAir rv 10 GW [54]). This

behaviour can be explained by intensity clamping, which typicaUy sets in at PI pcAir rv 1,

leading to a limited plasma density and therefore peak intensity (Sec. 2.1.8) [2- 4].

The mean values of N:ir and fAir averaged over aU energies are f\!:ir rv 5.3 x

1016 cm-3 (Fig. 9.3 a, blue, line) (40%) or [Air rv 6.4 X 1013 W Icm2 (Fig. 9.3 b, blue, line)

(+0.31 -0.4 x 1013 W Icm2), respectively. The estimated, overall error of fl.N:ir rv ±40%

includes fitting errors and the uncertainty of fl.w~xf rv ±20% in the electron impact

parameter of the 0 l (777.4 nm) triplet (Ch. 6) [30]. The error bars of [Air are due to

the determination of fAir from the ionisation curve, N:ir == N:ir (f) (Fig. 9.2).


8.x1016

6.X1016 NeAir-5.3x1016cm-3 (+/-40%)

- - - - - - - - - - - - - - -~

2.x 1016

5.5x1013

Intensity (W/cm1)

Figure 9.2: Ionisation curve in air. The blue curve, fl:- ir == fl:- ir (f) (Eq.9.1) , shows the plasma density in air versus intensity (f) in the range of about 5 - 7 X 1013 W f cm2

.

The intensities (fAir) can be inferred from the intersects of the experimental values of N:ir with the ionisation curve (arrows). Note that the value of fl: ir is very sensitive to f in the relevant (~ 1014 W fcm 2 ) intensity range.

Ne/1016cm-3 • f = 20cm

10 (a) - Ne - 5.3x1 0 16cm-3

8

6 . . . . 4

2

III 013W /cm2

10 (b)

8

6

4

2

• f= 20cm

- 1 - 6.4x1 013W/cm2

----- le - 4.5x1013W/cm 2

--+---'---'---'---'---'~'----'----'-~-'----'----'-~~~~-'-'--'~--'- E/mJ -+--'----'~~~~~~--'---'---'~~~~-'--'-- E/mJ 10 20 30 40 50 60 70 10 20 30 40 50 60 70

Figure 9 .3: a) Plasma densities (N :ir) and b) intensities ( fAir) of high-power (P » pcAir) filaments in air. The data (red , circles) shows (a) N:ir and (b) fAir as a function of energy in the range of about 5 - 70 mJ. The pulse duration and focal length are the same as in Fig.9.1. It can be seen that (a) N:-ir and (b) fAir are constant (blue, lines) over the who le range of input powers (P f p:ir rv 10 - 150, where p:ir rv 10 GW [54]). The mean values (blue, lines) averaged over aIl energies are R:-ir rv 5.3 X ~016 cm- 3 or [Air rv 6.4 X 1013 W fcm2

, respectively. Note that the calculated value of the clamped intensity in air (Eqs. 9.3 and 9.4) is f:- ir

rv 4.5 X 1013 W fcm 2 (black, dashed).


Following Bergé et al., the clamped intensity in air (f~ir) can be calculated by

(Sec. 2.1.8) [38]

(9.3)

(9.4)

where no == 1 is the linear and n~ir the nonlinear index of refraction , R (f~ir) rv

a02 (fAir )K02 (in S- l) the ionisation rate a02 rv 2 9 X 10-99 s-lcm2KW- K the ionisation K c 'K .

cross section and K 02 == 8 the order of MPI (Tab. 9.1). The nonlinear refractive index

(n~ir) is related to the critical power in air (PcAir) by Marburger's equation (Eq. 2.3) [51]

(9.5)

Inserting T == TFwHM/v'2In2, TFWHM rv 47 X 10-15 S-1, N(/2 == 0.6 x 1019 cm- 3 , n~ir ~

1 x 10-19 cm2/W and solving Eqs.9.3 and 9.4 numerically, yields a clamped intensity

of f~ir rv 4.5 X 1013 W /cm2 (cf. Ref. [1,2]).

Fig. 9.4 shows the peak intensities of the Nt (391 nm) signal, S (Nt) (Fig. 9.4, red ,

circles), as a function of energy in the range of about 1 - 70 mJ. It can be seen that

S (Nt) monotonically increases over the whole range of input powers (up to P / Pc rv

150). The increase of S (Nt) is in contrast to the constant behaviour of N:ir (Fig. 9.3 a)

and fAir (Fig. 9.3 b). This can be attributed to the growing of the number of multiple

filaments or the effective increasing of the plasma volume, while f.I: ir and [Air stay

constant due to intensity clamping.

The plasma densities of the high-power filaments in argon (N:r) can be derived from

Stark broadening of the Ar l 696.54 nm spectrallines in analogue to Ch. 5 [35]. Fig. 9.5

shows typical (f == 100 cm) spectra of the Ar l 696.54 nm line (Fig. 9.5, red , circles) in


À (nm) Pc(GW) aK(s-lcm2KW-K) K Uion(eV)

800/ 0 2 rv 10 2.9 X 10-99 8 12.1

800/ Ar rv 5.5 6.0 x 10-140 Il 15.8

Table 9.1: MPI parameters of air (02 ) [38] and argon (Ar) [160].

Nf Signal (arb.u.)

1.5 x 106

500000

---t"-'----'----'----'--'-'---'---'~'---"--"--'---"-'-_'_'_'__~_L_L_L.--'----'----'--__L.....L.....L_'__'_____'_---L- E/rnJ 10 20 30 40 50 60 70

Figure 9.4: Peak intensities, S (Nt), of the Nt (391 nm) signal (B 2L:t ---+X2L:t [5 , 123]). The data (red, circl~s) shows S (Nt) as a function of energy in the range of about 1 - 70 mJ (up to P / pcAir rv 150). The pulse duration and focal 'length are the same as in Fig. 9.1. It can be seen that S (Nt) monotonically increases with input energy, while Nt"ir (Fig. 9.3 a) and JAir (Fig. 9.3 b) stay constant.

the wavelength range of about 695 - 698.5 nm, fitted by Voigt profiles (Eq. 5.5) (Fig. 9.5,

blue, lines). The good fitting indicates the absence of self-absorption. However, in the

,case of the shorter (f == 20 cm) focal length, there is a larger fluctuation of Nt"r and

JAr in the energy range below rv 35 mJ (Fig. 9.7, green, circles/ line), which is due to a

relatively strong plasma continuum.

Fig. 9.6 shows a log-log plot of the ionisation curve of Ar+ in the intensity range of

about 4 x 1013 to 2 X 1015 W /cm2. The saturation intensity, J:r rv 2.4 X 1014 W /cm2

(ionisation degree: 100%), is determined from the crossing point of two linear fits

(Fig. 9.6, blue, lines) through the data (Fig. 9.6, blue, points). It is well known that the

linear fit in the saturation regioh (Fig. 9.6, J ~ J:r, blue, line) has a slope of 3/2 [161].

The deviation is because of the experimental constraint that the effective focal volume

where significant tunnelling ionisation (TI) occurs becomes larger than the opening of


1.0 (a) :tJ Q)

~ 0.8 E CS

~ 0.6 cu Q) Q. z: 0.4 ïn c: Q)

,5 0.2 ëii c: Cl

ü5 695.5 696.0 696.5 697.0 697.5 698.0 698.5

Wavelength (nm)

1.0 (b) :tJ Q) fi)

~ 0.8 E o ~ 0.6 cu Q) Q. z: 0.4 ïn c:

~ 0.2 ëii c: Cl

ü5 695.5 696.0 696.5 697.0 697.5 698.0 698.5

Wavelength (nm)

. Figure 9.5: Ar l 696.54 nm spectrallines of high-power (P » pcAr) filaments in argon. Typical, peak normalised spectra of the Ar l 696.54 nm line (red, circles) fitted by Voigt profiles (blue, lines). The two different spectra were recorded at pulse ' energies of (a) rv 10mJ and (b) rv '70mJ, respectively. The pulse duration was rv 100fs (FWHM) negatively chirped and the ' focal length 1 = 100 cm. The good fitting indicates the absence of self-absorption.

the mass spectrometer at high laser intensities [162].

The intensities (JAr) (Fig. 9.7 b) can be inferred from the experimentél:l values of Nt.r

(Fig. 9.7 a), which are directly related to the ionisation degrees (ŒAr), together with the

ionisation curve of Ar+ (Fig. 9.6), approxlmately. The ionisation degree is given by

N ArjNAr ŒAr = e 0' (9.6)

where Ntr rv 2.5 X 1019 cm-3 is the neutral density at 1 atm. In particular, for N:r rv

2.6 X 1018 cm-3 (1 = 20 cm) (Fig. 9.7 a, green, line) , the ionisation degree (Eq.9.6) is

rv 10-1 , yielding [Ar rv 1.7 X 1014 Wjcm2 (Fig. 9.7 b, green, line).

Fig. 9.7 shows N:r (Fig. 9.7 a) and JAr (Fig. 9.7 b) as a function of energy in the

range of about 1 - 70mJ under different focussing conditions: 1 = 100cm (Fig. 9.7,

red, circles), 1 = 60 cm (Fig. 9.7, blue, circles) and 1 = 20 cm (Fig. 9.7, green, circles).

It can be se en in Fig. 9.7 that N:r (Fig. 9.7 a, coloured, circles) and JAr (Fig. 9.7 b,

coloured, circles) are in,dependent of energy (Fig. 9.7, solid, lines) over the whole range

of input powers (Pj pcAr rv 2 - 136, where p:r rv 5.5 GW [160]). This behaviour can

Chapter 9. Intensity clamping of high-power filaments under 'natural ' conditions 106

L0910Y (arb. u.)

-1

-2

-3

-4

-5

-6 . . -7 •

Is -2.4x1 014W/cm2

l 14.0 14.5 15.0

Figure 9.6: Ionisation curve of Ar+. [The ionisation curve was measured by M. Sharifi.] Log-log plot of the ion signal (Y, blue, circles) of Ar+ in the intensity range of about 4 x 1013 to 2 X 1015 W /cm2

. The saturation intensity I:r rv 2.4 X 1014 W /cm2 (ionisation degree: 100%) is determined from the crossing point of two linear fits (blue , lines) through the data. The slope of the linear fit in the saturation region (blue, line, l ~ It"r) is 3/2 [161]. The deviation is because of the experimental constraint that the effective focal volume where significant tunnelling ionisation (TI) occurs becomes larger than the opening of the mass spectrometer at high laser intensities [162].

Chapter 9. Intensity clamping of high-power filament? l1nder 'natura1' conditions 107

(a)

400

300 . . 200

100

o 10 20 30 40 50 60 70

1/1014W /cm2

2.5

2.0 (b)

1.5

1.0

0.5

. . . . . . .

o 10 20 30 40 50 60 70

• f= 100cm • f= 60cm • f = 20cm

- N - 4.1x1017cm-3

- Ne

_ 59x1017cm-3

- N:- 2:6x101Bcm-3

• f= 100cm • f = 60cm • f = 20cm

- 1-1.2x1014W/cm2

- 1- 1.3x1 014W/cm2

- 1- 1. 7x1 014W/cm2

---- Ic-1.1x1014W/cm2

--- - Is - 2.4x1014W/cm2

E/mJ

Figure 9.7: a) Plasma densities (N :X) and b) intensities ( JAr) of high-power (P » p cAr)

filaments in argon. The data (coloured, circles) shows (a) Nt-r and (b) JAr as a function of energy in the range of about 1 - 70 mJ under different focussing conditions: 1 = 100 cm (red, circles), 1 = 60 cm (blue, circles) and 1 = 20 cm (green, circles). The pulse duration is the same as in Fig. 9.5. It can be seen that (a) Nt-r and (b) JAr are constant (solid, lines) in the whole range of input powers (PI pcAr . f'-I 2 - 136, where pcAr f'-I 5.5 GW [160]). In the case of the shorter (1 = 20 cm) focallength , there is a larger fluctuation below f'-I 35 mJ (green, circlesj line) , w'hich is due to a relatively strong plasma continuum. The mean values (solid , lines) averaged over aIl energies are Rt-r

f'-I 4.1 x 1017 cm-3 (red, line) , 5.9 x 1017 cm-3 (blue, line) and 2.6 x 1018 cm- 3

(green, line) or JAr f'-I 1.2 (red, line) , 1.3 (blue, line) and 1.7 x 1014 W jcm2 (green, li ne ), respectively. The saturation intensity of the ionisation curve of Ar+ (Fig. 9.6) is J~r f'-I 2.4 X 1014 W Icm2 (magenta, dashed). Note that the literature 'value of the clamped intensity in argon is J~r f'-I 1.1 X 1014 W Icm2 (black, dashed) [160].


again be explained by intensity clamping, which sets in at P / pcAr rv 1 (see above). The

mean values (Fig. 9.7, solid, lines) averaged over aIl energies are N:r rv 4.1 X 1017 cm- 3

(Fig. 9.7 a , red; line) , 5.9 Xl017 cm- 3 (Fig. 9.7 a , blue, line) and 2.6 x 1018 cm- 3 (Fig. 9.7

a, green, line) (40%) or JAr rv 1.2 (Fig. 9.7 b, red, line) , 1.3 (Fig. 9.7 b, blue, line)

and 1.7 x 1014 W/cm2 (Fig. 9.7 b, green, line) (+0.1 / -0.2 x 1014 W/cm2) , respectively.

Here, D.N:r rv ±40% includes fitting errors and the uncertainty of D.wArI rv 30% in the

electron impact parameter of the Ar l (696.54nm) line (Ch. 5) [35]. The error bars of

JAr are due to the determination of JAr from the ionisation curve of Ar+ (Fig. 9.6).

In Fig. 9.7, N:r and JAr inc~eases as the focallength decreases (1 == 100, 60 and

20 cm). The dependence on external focussing can be explained in the following way.

Before reaching the self-focus, ionisation will start at the same intensity where the beam

has a certain diameter with and without a lens. Without the lens, the convergence of

the wavefront with this diameter is less than that with the lens. This is why a higher

plasma density is required to counterbalance the st ronger convergence so as to reach

intensity clamping. Intensity clamping means that the wavefront at the balancing point

should be fiat [45].

Fig. 9.8 shows the full spectra in the wavelength range of about 300 - 900 nm under

different focussing conditions: 1 == 100 cm · (Fig. 9 ~ 8, red , line) , 1 == 60 cm (Fig. 9.8,

blue, line) and 1 == 20 cm (Fig. 9.8, green, line). According to Planck's law, the spectral

intensity of the plasma continuum is given by [71]

2hc2 1 S (À , T) rv y (he ) ,

exp ÀkBT - 1 (9.7)

where h == 6.626 X 10-34 Js is the Planck constant, c == 3 X 108 mis the speed of light ,

k B == 1.38 X 10-23 J /K the Boltzmann constant , À (in nm) the wavelength and T (in K)


Signal/arh u.

1.5

1.0

T 3 ...... - .......

/ T "-1 2- - "

/ T' / L,.

/ /

1 / /

/ /

0.5 /

200 300 400 500 600 700 800 900

Wavelength (nm)

- f= 100cm - f=60cm - f=20cm ----- T

1- 7,OOOK

----- T2

- 9,OOOK ----- T

3 - 10,OOOK

Figure 9.8: Full spectra: f ~ 100 cm (red, line) , f ~ 60 cm (blue, line) and f ~ 20 cm . (green, line) in the wavelength range of about 300 - 900 nm. The spectra were taken

at a pulse duration of t"'V 100 fs and an energy of t"'V 70 mJ. The plasma continua of the three different (coloured, lines) spectra are fitted (black, dashed) by Planck's law for blackbody radiation at temperatures of Tl t"'V 7000 K (f ~ 100 cm) , T2 t"'V 9000 K (f ~ 60 cm) or T3 t"'V 10000 K (f ~ 20 cm), respectively. The spectra are normalised with respect to the maximum of the continuum emission at Tl t"'V 7000 K in the case of the f ~ 100cm lens (Tl , black, dashed). Note that the wavelengths below ;S 380nm are cut by the Plexiglas tube. It can be seen that the plasma temperature is relatively low (;S 1 eV) , which is a consequence of the fact that the filament plasma is generated by tunnelling ionisation (TI) only without the occurrence of inverse Bremsstrahlung and cascade (avalanche) ionization.

the plasma temperature. The plasma continua of the three different (Fig. 9.8 , coloured,

lines) spectra are fitted (Fig. 9.8 , black, dashed) by Planck's law (Eq. 9.7) for blackbody

radiation at temperatures of Tl t"'V 7000 K (f ~ 100 cm), T2 t"'V 9000 K (f ~ 60 cm) or

T3 t"'V 10000 K (f ~ 20 cm), respectively. In order to compare the continua, the spectra

are normalised with respect to the maximum of the continuum emission at Tl t"'V 7000 K

for the f ~ 100 cm lens (Fig. 9.8 ~ Tl, black, dashed).

As can be observed in Fig. 9.8, the plasma temperature (T) increases for shorter

focal lengths. This would be a result of the higher intensities (Fig. 9.7 b, solid , lines).

As discussed in Sec. 2.2, T is relatively low CS 1 eV) , which is a consequence of the

fact that the filament plasma is generated by tunnelling ionisation (TI) only, without


the occùrrence of inverse Bremsstrahlung and cascade (avalanche) ionization. Thus,

the plasma is 'cold', even in the case of strongly (f = 20 cm) focussed , high-power

(P » pcAr) filaments in argon.

9.4 Conclusion

In conclusion, we have determined the avèrage intensities of high-power (P» Pc)

filaments in air ([Air) and argon ([Ar) under natural conditions. Specifically, we have

quantitatively confirmed the intensity clamping phenomenon at 'high power level (from

/"'..J 0.1-1.5TW). This is complementary to a previous study, where intensity clamping

was only precisely studied up to /"'..J 0.2 TW in nitrogen molecular gas [3] :

Chapter 10

Conclusion

In this thesis , we have developed a Stark broadening approach for measuring the plasma

density inside a filament induced by a femtosecond laser pulse in gases. It was shown

that the Stark broadening approach is valid, despite of the low plasma density. In

principle, as opposed to other techniques , the Stark broadening approach can be used

to measure the plasma density (in units of cm-3), without requiring calibration ~y an

independent method.

Unexpectedly, the measured electron impact parameter of the Stark broadened

atomic oxygen triplet 777-.4 nm was larger than the theoretical value by Griem [67]

by a factor 6.7. Using the experimental value w~xf ~ 0.0166 nm, the plasma densi

tie's derived from Stark broadening agree weIl with those most recently obtained from

Théberge et al. 's measurement of the nitrogen fluorescence calibrated by longitudinal

diffraction [29]. However, the Stark broadening approach is much simpler and could be

used to measure the filament density in air under different propagation conditions.

Chapter 10. Conclusion 112

However, it should be pointed out that the 0 l (777.4 nm) triplet is very weak com

pared to the (molecular) nitrogen lines observed in "the filament spectra [163]. Therefore,

the precise characterisation of remote filaments produced with high peak power lasers

under more realistic environmental conditions, e.g., distorted laser beam profile, air

turbulence, etc ... , is still an open challenge [164].

Moreover, the critical power for self-focussing of a femtosecond laser pulse in helium

was measured, based on the Stark broadening of the atomic helium line at 587.56 nm

to be Pcexp (1 atm) ~ 268 GW. Usirig this value, the nonlinear refractive index of helium

was inferred to be n~e (1 atm) rv 3.6 X 10-21 cm2 /W. The value of n~e is essentially

equal to the one calculated by Nibbering et al. [48].

In addition, the existence of the intensity clamping of the filamentation pro cess was

confirmed. It was also shown both analytically and experimentally that the intensity

clamping is independent of pressure. This surprising result has the implication that the

filament size (diameter) will increase as the altitude increases in vertical atmospheric

propagation.

This naturally occuring pro cess was also studied in the high-power (P » Pc) multi

ple filamentation regime. In particular, we have quantitatively confirmed the intensity

clamping phenomenon in air and argon at high power level (from rv 0.1 - 1.5 TW).

This is complementary to a previous study, where intensity clamping was only precisely

studied up to rv 0.2 TW in nitrogen molecular gas [3].

Moreover, the clamped intensity of helium was measured to be about I~xp rv 7.8 X

1014 W Icm2. This value is more than one order of magnitude higher than the clamped

intensity in air (rv 5 X 1013 W /cm2 [1,2]). Therefore the use of this noble gas could be

Chapter 10. Conclusion 113

interesting for scaling few-cycle pulse generation to a higher power-Ievel.

Finally, because the filament spectra are very 'clean', the developed approach could

also be used for atomic spectral line broadening analysis in other gases or gas mixtures

(cf. Ref. [80]).

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