and above which the width of the continuum increases with the band-gap. This is ...... One can see that positive focusing brings the self-focus closer to the medium entrance, ..... propagation axis during seIf-focusing and white-light continuum generation in ... After the beam splitter the beam is focused with lens fi (focal length.
T H E PROPAGATION O F P O W E R F U L ULTRASHORT LASER PULSES IN TRANSPARENT MEDIA: SELF-FOCUSING, CONTINUUM GENERATION AND CONICAL EMISSION
Thèse présentée à la Faculté des études supérieures
de l'université Laval pour l'obtention du grade de Philosophiar Doctor (Ph.D.)
Département de physique Faculté des sciences et de génie Université Laval
Québec
Novembre 1997
@André Brodeur, 1997
1+1
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Le déveIoppement récent des lasers ferntoseconde de haute puissance à créé un besoi d'explorer à nouveau certains aspects fondamentaux de la propagation des impulsior laser. Cette thèse est une étude expérimentale des phénomènes d e propagation laser d m les milieux condensés trmspaxents et dans l'air. Les expériences ont été réalisées avec un chaîne laser titane-saphir produisant des impulsions femtoseconde puissantes situées dan l'infra-rouge rapproché.
L a propagation dans les milieux condensés transparents peut mener à l'autc focalisation du faisceau et à la génération d'un continuum spectral de lumière blanche Une technique est présentée qui permet de mesurer le profil du faisceau, le spectre e l'énergie de l'impulsion à différentes étapes pendant l'auto-focalisation et la génération d continuum. Les observations démontrent que la génération de continuum est dédenché par 17aut~focaIisation et dépend fortement de la largeur de la bande interdite du mi
lieu. Un seuil est trouvé sous lequel il n'y a pas de génération de continuum et au-dessu duquel la largeur du continuum augmente avec la largeur de la bande interdite. Pour 1 première fois un paramètre prédismt la largeur du continuum dans les milieux condensé est présenté. L'auto-modulat ion de phase stimulée par la génération d'électrons librc
par excitation multiphotonique est proposée comme principal mécanisme de génératioi de continuum. La divergence anomalique et l'émission conique associée au continuum es aussi explorée.
La propagation dans l'air peut. mener à un régime de propagation filamentaire initi, par l'auto-focalisation et soutenu sur une distance de plusieurs dizaines de mètres.
LI
filament est habituellement accompagné par une émission conique ayant l'apparence d'ui
arc-en-ciel circulaire. Des mesures expérimentales du filament révèlent que le filament s( termine à une distance égale à la longueur de diffraction du faisceau, indépendemmen de la puissance crête initiale. Une explication du filament est fournie dans le contexte di niodèle du foyer mobile d'auto-focalisation modifié par l'ionisation de l'air. Dans la mêml
Abstract Recent developments in the technology of powerful ultrafast lasers have created a need to revisit some fundamental aspects of laser pulse propagation. This thesis investigates phenomena arising in the propagation of powerful near-infrared femtosecond laser pulses in transparent condensed media and in air. Experiments were performed using a titaniumsapphire laser chain. Propagation in transparent condensed media can lead to self-focusing of the beam and a transformation of the pulse spectrum into a continuum covering the entire visible spectrai range (white light). A new technique is presented that allows monitoring of the bearn profile, pulse spectrurn and pulse energy at various stages of propagation during self-focusing and continuum generation. The observations show that continuum generation is triggered by self-focusing and rcveal a strong dependence on the band-gap of the medium. A band-gap threshold is found below which there is no continuum generation and above which the width of the continuum increases with the band-gap. This is the first report of a parameter predicting the width of the continuum in condensed media. Self-phase modulation enhanced by free-electron generation due to multiphoton excitation
is proposed as the primary mechanism of continuum generation. The anomalous beam divergence and the conical emission associated with the continuum are also investigated. Propagation in air can give rise to a filamentary propagation mode initiated by selffocusing and sustained for several tens of meters. The filament is usually accompanied by rainbow-Iike conical emission. Experimental measurements of the filament energy reveal that the filament ends at a distance equal to the diffraction length of the beam (- 100 m in Our experiment), independently of the initial peak power. An explanation for the filament is provided in terms of the moving-focus mode1 of self-focusing modified
by ionization of the air. Along the same lines a new mechanism is proposed t o explain the conical emission in terms of laser-plasma interactions. The measiired filament and conical
Avant-propos Ces travaux ont été réalisés dans le Laboratoire de lasers ultra-rapides et intenses du Centre d'optique, photonique et laser de l'université Laval. Ce laboratoire de haute performance est le fruit du dynamisme et de l'engagement du Professeur See Leang Chin. Je suis reconnaissant envers le Professeur Chin, qui a dirigé mes travaux et m'a fait entrer de plein pied dans le monde de la recherche scientifique.
Au cours de mon doctorat j'ai eu la chance de cotoyer le Dr Fedor Ilkov, qui a été pour moi un modèle de rigueur, de patience et de générosité. Je le remercie de m'avoir initié
au travail de laboratoire. Je suis reconnaissant envers le Dr Steve Augst, pour la justesse et La pertinence de ses enseignements. Je remercie Simon Lagacé, dont la compétence et
le professionalisme m'ont été d'une valeur inestimable.
J e tiens égaiement à remercier mes collègues de laboratoire, ainsi que le personnel de soutien du COPL et du département de physique. En particulier, j'aimerais remercier Samad Talebpour pour de nombreuses discussions enrichissantes. Finalement, je remercie le Conseil de recherche en sciences naturelles et en génie ainsi que le Professeur See Leang Chin pour leur support financier.
Table of Contents
Résumé Résumé court Abstract
v
Avant-propos
vii
Table of Contents
i2
List of Figures xvii
List of Tables Chapter 1 : Introduction
1
Chapter 2 : Survey
6
2.1
Nonlinear opt ics
.................................
6
2.1.1
Nonlinear polarization
.........................
S
2.1.2
The nonlinear index of refraction . . . . . . . . . . . . . . . . . . .
9
2.1.3
Optical nonlinearity in the Lorentz-Drude mode1 .
..........
10
Table of Contents
xii
.-
5.4.3
Numerical simulations . . . . . . . . . . .
5.4.4
Universabty of conical ernission . . . . . . . .
+
. . . . . . . . . . - 129 . . . . . . . . . . . 134
Chapter 6 : Conclusion
138
References
140
Appendix A: Characterization of pulsed laser bearns
147
4 . ReIations between field strength, intensity, power, Buence and energy . . . 147
-4.2 Pulse shape and beam shape .
..................- -
A.3 Beam divergence . . . . . . . . . . . . . . . . . . . . . . . A.4 Spectrum of a Gaussian pulse
. . - . . 149
. . . . . . . . 150
. . . . . .. . . ... . . . ... .... . ..
151
Appendix B: The intensity-dependent index of refraction
152
Appendix C: Numerical code
153
Appendix D: Color pictures
161
List of Figures
..............................
2.1
Anharmonic potential
2.2
Transition rate in Keldysh theory of multiphoton excitation for condensed matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Self-focusing distance for a laser pulse
.....................
Il
14 20
2.4 Self-focusing in the moving-focus model: Intensity spikes in the pulse distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Instantaneous frequency in self-phase modulation . . . . . . . . . . . . . . 25
2.6
Self-phase modulation of a laser pulse . . . . . . . . . . . . . . . . . . . . . 27
3.1
Front end of the Ti:Sapphire laser chain . . . . . . . . . . . . . . . . . . . 31
3.2
Terawatt amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Transmission of BG1S and RG850 filters . . . . . . . . . . . . . . . . . . . 39
4.3
Effect of focusing lem displacement . . . . . . . . . . . . . . . . . . . . . . 41
4.4
Alignment of the imaging lens
4.5
Beam diameter at the geornetrical focus vs power . . . . . . . . . . . . . . 45
4.6
TypicaI white-light continuum spectra generated in water. UV-grade fused silica and NaCl
.........................
35
42
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
List of Figures
r --
.-
.
4.26 Experimental setup to measure the divergence of the continuum
......
S
4.27 Sketch representing the self-focused beam in the vicinity of the output
surface of the medium
.............................
4.25 Divergence of the continuum beam generated in UV-grade Eused silica 4.29 Divergence of the continuum beam generated in CaF2 4.30 Sketch of the continuum divergence
Si;
. . . 8::
............
QL
. . . . . . . . . . . . . . . . . . . . . . Y(
4.3 1 Energy contour plots in the angle-vs-wavelength plane in the visible range
for CE generated in UV-grade fused silica
. . . . . . . . . . . . . . . . . . 85
4.32 Dependence of conical emission angles on wavelength
............
4.33 Continuum spectra at pulse durations 140 fs. 500 fs and 5 ps
S5
. . . . . . . . 91
4.34 Dependence on external focusing of the continuum spectrurn generated in
water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.35 Beam cleaning by self-focusing
.........................
4.36 Cleaning of a beam strongly diffracted by a metal mesh 4.37 Setup to image bearn-breakup in water
94
. . . . . . . . . . . 95
. . . . . . . . . . . . . . . . . . . . 9i
4.38 Sketch of the spatial distribution of filaments in water
. . . . . . . . . . . . 98
4.39 Spatial distribution of a beam undergoing severe filameotation . . . . . . . 99
4.40 Critical power vs ce11 length . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.1
Experimentai setup for the rneasurement of filament energy
.........
109
5.2 Average total pulse energy vs distance . . . . . . . . . . . . . . . . . . . . 110
5.3
Ratio of filament energy to average total energy vs distance
.........
111
5.4 Sketch of the filament arising from the moving-focus dynamics . . . . . . . 112
List of Figures
xv
5.5 Visualkation of the moving focus . . . . . . . . . . . . . . . . . . . . . . . 111 5.6
SpaticAemporal distribution of the pulse intensity . . . . . . . . . . . . . . 12(
5.7
Pulse intensity and corresponding nonlinear index . . . . . . . . . . . . . . 121
5.8
Position of maximum intensity vs time in the moving-focus mode1 . . . . . 12:
5.9
Filament energy vs distance in the simulation . . . . . . . . . . . . . . . . 1%
5.10 Experimental setup to measure the conical emission . . . . . . . . . . . . . 1% 5.11 Meaçured conical emission angles
. . . . . . . . . . . . . . . . . . . . . . . 1%
5-12 Intensity distribution and spatio-temporal spectrum in the filament . . . . 131
5.13 Measured and simulated conical emission
. . . . . . . . . . . . . . . . . . . 13:
5.14 Wavefronts at various wavelengths in the filament
. . . . . . . . . . . . . . 132
5-15 Distribution of free electroas in the simulated filament
...........
135
.5 .16 Transverse distribution of electrons in the filament . . . . . . . . . . . . . . 136
D.l White-light continuum beam generated in water at input peak power around threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
D.2 White-light continuum beam generated in water at input peak power just above threshoId . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
D.3 White-light continuum beam generated in water at input peak power of a few thresholds
..................................
164
D.4 White-light continuum beam generated in water a t input peak power well above threshold
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
D.5 Multiple filaments in water
...........................
D.6 Conical emission arising from the propagation of a 110 m of air
.100-GW beam
165
t hrough
...................................
166
List of Tables
................... 6
4.1
Self-focal characteristics in various media
4
Threshold power for self-focusing and associated nonlinear index of refraction in various media
..............................
xvii
10
Chapter 1 Introduction
The propagation of a powerful laser pulse in a medium can lead to light-matter interaction that modify the laser pulse itself. This fundamental aspect of laser optics bas bee thoroughly investigated in the past. However, recent developrnents in the technolog of powerful ultrafast lasers* have created a need to revisit the issue. In this thesis w investigate the transformations undergone by powedul ultrashort laser pulses in a variet of transparent solids and liquids and in air. This fundamental problem is relevant to an:
application involving the propagation of powerful ultrashort laser pulses. In our daily experience of light-matter interaction, light is produced by the sui
and various kinds of lamps. W t h cuch light the strength of the optica! electric field i
restricted to relatively low levels and the propagation of a light beam is described withii the framework of "Iinearn optics [1,2]. In this Iinear regime the bound electrons in
;
medium are weakly driven by the optical electric field and their response is harmonic. II macroscopic terms, the interaction of light with the driven electrons is expressed by th1 medium's index of refraction [1,2]. The laws of optics applying to weak opticd electric fields do not hold for the stronl electric fields achieved with powerful ultrafast 1asers.t The electric field of a powerful lase 'Laser pulses in the femtoçecond (fs, I W L 5sec) range with peak power up to the terawatt (TW, 10' Watt) range can now be generated. tThe field strength at the focus of a 1 gigawatt (GW, 109 Watt) laser pulse can easily reach
-
10
Chapter I : Introduction
2 -
-
pulse
c m
drive the bound electrons so strongly that their response is not harmonie. T h e
propagation of the laser pulse must then be described within the framework of "nonlinear" optics [3-71.
In this regime appears a wealth of new phenornena which depend on the
strength of the optical electric field. In our study of the propagation of powerful ultrashort laser pulses we are rnainly concerned wit h two nonlinear effects:
(i) Nonlinear index of refraction, In the nonlinear regime the index of refraction de pends on the intensi ty of the optical electric field. For a pulsed laser bearn the nonlinear index of refraction results in self-focusing [S-111 and self-phase modulation [IO-121. In the case of self-focusing, the intense laser beam causes the medium to act as a lens which in turn causes the beam to contract with propagation. In the case of self-phase modulation, the rapid rise and fa11 of the index of refraction dong t h e intense laser pulse causes the pulse's spectrum to broaden with propagation. ( i i ) Generation of lree electrons.
An extreme case of nonlinear optical response
occurs when the strength of the optical field is comparable to the strength of the atomic Coulomb field that binds an electron. The electron can then be freed by multiphoton excitation from a bound state to a free (or quasi-free) state. [13-171 The free electrons appearing in the path of a laser pulse contribute negatively to the index of refraction.
They therefore act as a negative Iens, which counteracts self-focusing, and they induce a phase modulation.
In this thesis we experimentally explore the interplay between these mechanisms in order to explain phenornena observed in the propagation of powerful ultrashort laser
pulses in transparent condensed media and in air. The next chapters are organized as follows: V/cm, while the field strength of unfocused Sun light is of the order of
-
10 V/crn.
Chapter 1: Introduction
3 -
C h a p t e r 2: S u r v e y
A brief survey of the present state of knowledge in the fields pertaining to this thesis is presented. We glance at the basic principles underlying nonlinear optics. Particular attention is paid to the multiphoton excitation of electrons and to the nonlinear index of refraction, which leads to self-focusing and self-phase modulation of a pulsed laser beam.
C h a p t e r 3: T h e l a s e r chah The ultrafast laser chain used in the experiments is presented. It is based on titaniumdoped sapphire (Ti:Sapphire) as a gain medium [18.19] and uses the technology of chirpedpulse amplification [20,21]. It consists primarily of an oscillatort pulse stretcher, regenerative amplifier: two-stage amplifier and pulse cornpressor. The out put beam consists alternatively of 140-fs pulses with peak power up to 3
G W (10-Hz or 1-kHz repetition
rate) or 250-fs pulses with peak power up t o 1.2 TW (IO-Hz repetition rate). The central wavelength is in the near infrared, at 800 nm.
Chapter 4: White-light continuum g e n e r a t i o n a n d self-focusing i n condensed media :\ powerful ultrashort laser pulse propagating in a transparent medium can be trans-
formed into a white-light continuum covering the entire visible range [22-361. Despite the continuum's widespread use [23], its generation mechanism remains far from being well understood. For instance, it is unlcnown why the continuum generated in water, a widely used medium, is among the broadest observed. Self-focusing is a rnechanism known to pIay an important part in continuum generation in extended media [29,31,32]. The purpose of this chapter is to provide new insight into continuum generation and self-focusing in extended condensed media using femtosecond TizSapphire laser pulses.
A new technique is presented that allows monitoring of the beam profile, pulse spectrum and pulse energy at various stages of propagation during seif-focusing and conti-
Chapter 1: Introduction
4
nuum generation. The observations confirm that continuum generation is triggered by self-focusing and reveai a strong dependence on the medium's band-gap. -4 band-gap threshold is found below which the medium cannot generate a continuum and above which the continuum width increases with the band-gap. This is, to our knowledge, the first report of a parameter predicting the width of the continuum in condensed media. The observations contradict self-phase modulation theory, the mechanism generally accepted as responsible for continuum generation. Enhancement of self-phase modulation by free electrons generated by MPE is proposed as the primary rnechanism of continuum generation. The anomdous bearn divergence 1291 and the colorful conical emission [22,26,29,30,33] associated with the continuum are investigated.
Chapter 5: Beam filamentation and conical emission in air The propagation of a collimated powerful ultrashort laser pulse in air can lead to the formation of a light filament whose length can attain several tens of meters [37-421. The light filament is initiated by self-focusing and appears as a mode of propagation
with a drastically contracted bearn diameter. Its enormous length is a most intriguing feature. The filament is usually accompanied by an angular emission of anti-Stokesshifted radiation, or conical emission, which appears on a screen as a concentric rainbow [39,40,42]. This chapter reports a n investigation of these phenomena and proposes new mechanisms for their expianation. Results of filament energy measurements are presented. The most striking observation is the termination of the filament at a position independent of input peak power, at a distance equal to the Rayleigh range of the bearn (- 100 m in the experiment). An expIanation for the observed filament is provided in terms of the moving-focus mode1 of self-focusing [IO] modified by ionization of the air. Measurements of the conical emission angles are then presented, d o n g with a new mechanism of conical ernission arising from laser-plasma interactions. The investigation of these phenomena is supported by numericd simulations of pulse propagation involving ionization of the air [43].
Chapter 1: Introduction
Chapter 6: Conclusion
In this chapter self-focusing, self-phase modulation and free-electron generation by MPE are presented as mechanisms whose interplay accounts for several phenornena observed in the propagation of powerful ultrashort laser pulses in transparent media. The underlying unifying concepts are emphasized and guidelines for future work are presented.
Chapter 2 Survey
In this chapter ive review basic principles of nonlinex optics. Some insight is provided into the nature of nonlinear optical behavior occuring in the propagation of powerful ultrashort laser pulses. Particular attention is given t o the nonlinear refractive index and multiphoton excitation of electrons. The essential features of theories on self-focusing and self-phase modulation are presented.
2.1
Nonlinear optics
The propagation of an optical electric vector E in a medium is governed by the macroscopic Maxwell equations [1,2].' These can be combined to form a wave equation which, in the dipole approximation, is given by
where P is the dipole moment per unit volume, or polarization,
€0
is the vacuum permit-
tivity and c is the velocity of light in vacuum. The polarization vector P is a function of the total electric field E and represents the coupling between the electric field and the medium. 'The optical magnetic field is not considered since its interaction with the medium is negligible at the field strengths considered.
Chapter 2: Snrvey
The total eIectric field E can be represented as a sum of harmonic fields:
) E(w,,,)'. This field is a superposition of the driving field at frequenc where E ( - L J ~= Q ,J
= wo and the fields at various frequencies radiated by the medium via the polarizatio
P ( t ) . The latter can be written
with P(-y,) = P(wn)=. In conventional Iow-intensity optics P(t) is a linear function of E ( t ) (hence the tern "linear optics") [2]. In a non-dissipative and dispersionless medium the polarization i given by
P(t) = €0
: E(t) 7
(2.4
where the second-rank tensor ~ ( ' 1is a constant called linear susceptibili ty.t The fact t ha
P(t) depends linearly un E(t) implies that a linear medium radiates only a t the drivinl frequency ao:
Thris, a single driving frequency in a linear medium yieldr a total fie!d at the sanu frequency. The linear dependence of the polarization on the field allows us to rewrite (2.1) in
2
simpler form, by combining (2.1) and (2.4). In the case of an isotropic medium, the lineai susceptibility is defined as the scalar quantity
and we obtain
tIn a dispersive medium a11 frequency components of E(t) must be treated separately. In a dissipativc
medium f ( ' )is a complex quantity whose real and imaginary parts are associateci respectively with thi radiation of a field by the medium and the absorption of energy by the medium ( i . e . attenuation of thc
field). In this work t h e real part of the susceptibility is imptied, unless stated otherwise. With thesc compIications it is often convenient to work in the frequency domain rather than the tirne domain [4].
Chapter 2: Survey
E
where
is the index of refraction and c/no is the phase velocity of the field E.
2.1.1
Nonlinear polarization
The linear relation (2.4) is only approximate; its validity breaks down at high fieId strengths, where the dependence of the polarization on the field becomes nonlinear. -4 general description of the nonlinear dependence of the polarization P on E can then be expressed as a power series in E [3-71:
In that regime the polarization oscillates at several frequencies, even if there is a single driving frequency. These new frequencies arise in (2.9) from the products of electric fields. In a centrosymmetric medium even powers of E cannot contribute to the polarization. Indeed, a polarization with even powers of E tvould radiate a field that is açymmetrical about the propagation axis. The nonlinear poIarization thus reduces to
In the case of weak nonlinearity, tvhen the excitation field is far from any resonance, the series (2.10) can be truncated. In many circumstances it is sufficient to retain only the third-order term. Considering an isotropic medium and treating the polarization as a scalar. we have
P(~)(w= , ) €0
and
E(wo)
(2.10)
Chapter 2: Survey
where
X(')
and
X(3)
are scalars, and (a&) indicates that the sum is taken over al1 fre
quencies u,,wa, w, leading to the sum (w,+ w p da, 9, w,.
+ w,)
and ail distinct permutations c
The third-order contribution to the polarization thus results from an interac
tion between four waves a t frequencies w,,
wp, w,
+
and (w, wp + LA,). Usually, at leas
one of these waves is the driving field with frequency d o .
2.1.2 The nonlinear index of refraction In the nonlinear regime the third-order susceptibility ,y(3)affects the phase velocity of th1 wave and hence contributes to the index of rekaction. This is the so-cdled Kerr effect [4]
The index of refraction of an isotropic third-order nonlinear medium c m be obtainec by considering the third-order polarization at frequency d a . The third-order poIarizatioi at LJO can be obtained from (2.11) by substituting w, + u a
= wo +wo - d o .
+LA,
Taking inti
account the three possible permutations in (2.11) and the factors f in ( A . l ) and (2.3), w( obtain P(wo) = €0
f
3 $3)
I E ( ~ o ) IE(wo) ~] .
(2.12
Substituting this polarization into the wave equation (2.1) and reducing the Iatter to thc form of (2.6) yields the index of refraction
This index of refraction can be written in terms of the intensity of the optical field
w here
(see Appendix A) and
Chapter 2: Survey
11
Therefore? a medium with third-order noniinearity has
a4
intensity-dependent inde:
of refraction. At optical frequencies wo far from a medium resonance
n2
is generall!
positive, such that the index increases with intensity (2.e. an intense field has a slowe:
phase velocity than a weak field).
Optical nonhearity in the Lorentz-Drude model
2.1.3
The Lorentz-Drude model provides an excellent description of the optical properties
O:
atoms at low field strengths [2]. In this section we look at an extension of the LorentzDrude model to the nonlinear regime. The accuracy of the model is however limitec by the assumption of a single resonance frequency for the atom; a quantum-mechanica
analysis of the nonIinear susceptibility is required for a correct description with rnuItiplc resonance frequencies. Nevert heless, the Lorentz-Drude mode1 is wor t h considering for the physical insight that it provides. The motion of an electron bound to a nucleus and driven by a field at frequency can be described by [4]
where x is the position of the electron, rn is the mass of the electron, factor,
-e
r,
is a damping
is the charge of the electron, Eo is the fieid strength and u ( x ) is the potential
associated with the force that binds the electron to the nudeus.
In the conventional Lorentz-Drude model the electrons in an atom are described as being harmonically bound to their nuclei, implying a quadratic potential U(x).Here an anharmonic term is added to U ( x ) in order to investigate nonlinear behavior. A centrosymmetric medium is considered where the restoring force is symmetric about
xo,
the
electron's rest position relative to its nucleus. This prechdes a potential with odd-power terms. The lowest-order correction term to the quadratic potential must therefore be of the fourth power, so we taise IC;
-O)*
--p-xo)
4 7
Chapter 2: Survey
1
Figure 2.1: Anharmonic potential for the Lorentz-Drude model.
where w, is the resonance frequency and
is the anharmonicity parameter. This potentia
K
is represented as the solid curve in Figure 2.1.
For small displacements about xo the potential(2.18) is effectively a parabola. Hence
for low Eo the electron motion is harmonic. The equation of motion (2.17) can be solvec in this low-field regime by setting il = O in
Z l ) , yieIding -e I
x =
2
U(x)(represented as the dotted line in Figurc
Eo e - ' ~ ~ t
+
C.C.
m (w,2 - w; - illu,,)
where c c . denotes the complex conjugate. The bound electron is thus driven at frequencq ;~o by
the electric field. For an ensemble of N bound electrons this motion yields a t i m e
varying polarization P =
t(l)
1 Eo e-iwot + CL, where the susceptibility
k ( ' ) is given
bv
For
GO
< ua the susceptibility increases with increasing wo,L e . the dispersion is normal.
This behavior is observed in media with atomic resonances that are driven by optical fields.
At high field strengths the electron is strongly driven and c m reach the part of
the potential curve that departs from a parabola. The behavior of the electron is then
Chapter 2: Survey
1
anharrnonic. In particular, its osciiiatory motion h a components at odd harmonics of wo In the approximation wa - w; WO
> ï, wo, the susceptibility at
the fundamental frequenc
is given by
One can see that for a driving frequency wo the nonlinearity decreases with increasin resonance frequency
~
3
~Since .
the ionization potential of an atom is proportional t
w,,one can deduce from this mode1 that
x ( 3 ) is
small for atoms with large ionizatio
potential. Similarly, the condensed media with Large bard-gap can be expected t o be th least nonlinear.
2.1.4
Generation of free electrons
Let us consider the situation where the strength Eo of the driving field is comparabl to the strength E,, of the Coulomb field that binds the electron.5 The laser field the1 drives the electron with a force that rivals the atomic Coulomb force, and the electroi can be excited into a free state. This process, called ionization in the case of gases has been traditionally divided into muitiphoton and tunneling regimes [13-171. In t hl rnultiphoton regime the excitation of the electron can best be described as a multiphotoi excitation (MPE) from a bound state to a free state via several virtual excited states, b: absorption of severai photons. En the tunneling regime. excitation can best be describec as a tunneling of the electron through the potentid banier of the binding electric field
which is periodically lowered by the optical field. In the present work the experimenta parameters pertain to the rnultiphoton regime.
In this work, when considering field strengths Eo approaching E,, we consider MPI processes rather than high-order contributions of the susceptibility to the index of ce fraction (i.e. we consider processes associated with the imaginary part of the high-orde: -
-
-
-
-
t ~ e r the e potential asymrnetrical about x = +O, even harrnonics would appear. §Viecan estimate that Est is of the order of e/4ncoai = 5 x 10" V/m, where -e is the charge of th1 electron and
cl0
= 0.53 .i is the
Bohr radius. At I -- 1oL3W/crn"
EO is about 1% of Est.
Chapter 2: Survey
1
susceptibility rather than the real part). We do so because at field strengths very close to E,,. since at
Eo
-
One c m argue that
x(5)
becornes important on1
X(3)
is of the order of 1/E:
Eat the third- and first-order contributions to the polarization shodd E
comparable. Similarly,
X(5) is
of the order of l/E:, [4]. This means that even when
about a third of E,, the contribution of lesç than the contribution of
X(5)
to the index of refraction is still
-
Eo
10 timc
X(3).
Multiphoton excitation in condensed matter
The Keldysh theory of multiphoton excitation in condensed rnatter [1.5] calculates pei turbatively the transition rate of an electron from an initiai state in the valence band
c
the material to a final state in the conduction band, where it is essentially free [44] an oscillates in the laser field. In that transition the electron gains an energy in excess
c
the band-gap energy Egapbetween the valence and conduction bands. In terrns of t h Keldysh parameter
the transition rate W, which has units of cm-3 sec-', is given by
xexp where
/
K(*)-B(L) -NT
/
dG7 \
Chapter 2: Survey
14
Figure 2.2: Transition rate given by the Keldysh theory of multiphoton excitation in condensed matter.
and
Figure 2.2 shows t h e transition rate as a function of field intensity for various values of the band-gap. T h e range of band-gaps and intensities is such that y
> 1, so the regime is
multiphoton. In this regime the general dependence of W on the intensity I is of t h e type 9 The W cc. IN. The exponent N thus appears as the slope of t h e curves of Figure 9,.-.
kinks in the curves correspond t o channel-closing due to Stark shifting of the conduction band [14].
Chapter 2: Survey
1
Collisional excitation in condensed m a t t e r In condensed media the generation of free electrons can d s o occur via a collisional prc cess which is made possible by the high density of atoms. -4 seed electron (generated b ionization of a shallow trap or impurity) is heated by the laser field until it has sufficien energy to excite another electron into the conduction band by collision. These two elet trons then go on and excite two more electrons, and so on. In this avalanche process th free-electron density Ne grows exponentially [45]:
where
lVo
is the number of seed electrons and a (E) is the transition rate.
The generation of free electrons by avalanche requires lower intensities than MPE, bu it requires time. Until the development of ultrafast lasers this process was the dominan mechanism in condensed media. Bowever, MPE can be dominant with femtosecond pulse: This will be further discussed in Section 2.3.1.
M u l t i p h o t o n excitation in gases The rate of free-electron generation in a gas by multiphoton ionization (MPI) is calculatei in lieldysh's seminal paper 11.51. T h e Kelrlysh theory caIciiIates the transition rate of ai
electron from an initiaily bound state to a free state where the electron oscillates in t h laser field ( a nonperturbative Volkov final state). T h e resulting rate formula has a forn very similar to that of (2.13). Many refinements have been introduced in the theory of MPI of gases [14]. Importan
contributions were made by Faisal and Reiss, leading t o the so-caIled KFR theory [46,47] In this theory the calculations are performed in the radiation gauge instead of the Couloml gauge, leading to analytical results simpler than those of Keldysh. A further developmen to KFR theory was made by Szoke [48,49], who superimposes on the final Volkov statc a constant potential equal to the ionization potential. In the Szoke mode1 the ionizatior rate for a given species is ubtained by substituting its ionization potential into the ratc
Chapter 2: Survey
1
equation, The Sz6ke mode1 is used in the numericd simulations of Chapter 5.
Propagation of a pulsed laser beam in the linea: regime In this section we lay the basic concepts of the propagation of a pulsed laser beam i. the linear regime. When describing a pulsed laser bearn, the term "beam" refers to th transverse distribution of the electric field, with dependence o d y on the radial coordinat r when radial symetry of the beam is assumed. The term "pulse" refers to the longitudina
distribution of the pulse, with dependence on the Longitudinal coordinate z and on tim
t . More details are provided in Appendix A. The electric field of a pulsed laser beam can generally be described by
E(r. ;, t ) = f E,,
(T, Z ,
t ) ei(k'-l*~t)
+ $ Eà ( r ,L, 1 ) e - i ( k = - ~ ~ t )
(2.29
where Eo ( r ,2 , t ) is the cornplex amphtude of the field. uo is the central angular frequencl and k = n o ( w )w / c is the wavenumber. The r-dependence of the envelope Eo (r.2, t expresses the beam's intensity distribution, while its time-dependence describes the pulsc shape and implies a continuous distribution of frequencies around wo. The phase velocity of the wave components of the pulse is given by
and the velocity of the pulse envelope ( L e . the wave packet) is the so-called group velocity
The group velocity usually depends on w , since no is a function of
W.
Loosely speaking
this implies that the various frequency cornponents of the pulse propagate with differen,
group velocities. Due t o this groupvelocity dispersion (GVD) the pulse envelope change: with propagation. GVD is important when v, changes significantly over the width of tht pulse spectrum.
Chapter 2: Survey
1
When GVD is neglected (i-e. in the approximation no(w)= no) we have
It is then useful to defme the retarded time
which allows us to write the field simply as
E ( r ,T ) = 5I Ea (r,T ) e - i w ~ T+
Et ( r ,r ) e ' ~ ' .
(2.34
The absence of GVD also implies that the pulse envelope does not change with propagâ tion. A Gaussian pulse envelope is then given by
where
TO
is the pulse duration at FWHM of power. Similady, a hyperbolic secant
can be written Eo(r, T) = Eo(r)sech(r/r;)
.
Ultrashort pulses In pulsed laser technology, hi& powers are achieved by maximizing the pulse energy anc
minimizing the pulse duration. Today's lasers can generate pulses in the femtosecunc range. Such pulses have broad spectra, since the pulse duration is inversely proportionai to the pulse's bandwidth (see Appendix A). For instance, a 100-fs pulse bandwidth, corresponding to
-
haç
a
-
10 nrr
4% of the visible spectrum. With large bandwidths GVC
can become important and cause pulse stretching with distance. For example, a 100-fs pulse with central wavelength 800 nm propagating in water will double its duration withir:
-
40 cm. In air, this will occur within
-
1 km.
Chapter 2: Survey
1'
-
- -
Propagation of a pulsed laser beam in the non.
2.3
linear regime in this section we consider the effect of medium nonlinearity on the propagation of
i
pulsed laser bearn. For a laser frequency far from medium resonances the dominan noniinear mechanism is the Kerr effect, which is discussed in Sections 2.1.2 and 2.1.3 In the femtosecond regime the Kerr effect is due m a i d y to a distortion of the electror cloud [50]. The time response of this mechanism is of the order of an electronic orbita period,
-
10-l6 sec [SOI, which is much faster than the pulse duration. Intramolecula
vibrations can also contribute to the ultrafast Kerr effect in liquids, since their responst is in the femtosecond range. For instance, the period of the stretching vibration of water is
-
.-
1 x 10'4--Hz symmetric
10 fs [51]. Vibrations in solids are about ten time!
slower and therefore less important in the femtosecond regime. For instance, the perioc of the
-
I x IOL3-HZvibration in fused silica is
-
100 fs [52].
Two important effects associated with the Kerr nonlinearity are self-focusing and self-phase modulation. Since al1 media possess an electronic nonlinearity, these effects can occur in any transparent medium; they are therefore important in any application involving the propagation of powerful ultrashort pulses. Peak powers in the megawatf
range are required for self-focusing and self-phase modulation to be important in typica; condensed media. Higher powers are reqtiired in atmospheric air, typically in the gigawatt range.
Self-focusing is a well-known phenomenon arising in the propagation of powerful lasa beams in extended transparent media with an intensity-dependent index of refraction [S-111.
Extensive work was done on this subject in the late 1960's and early 1970's.
Thorough reviews can be found in Refs. [10,ll].
in a medium with index n = no+n2ï a beam with a Gaussian intensity profile inducec
Ii
Chapter 2: Survey
a Gaussian index profile. The optical path thus h a a maximum on the propagation axii
and decreases r a d i d y , much like in a lem. T h e beam therefore contracts radially witl propagation.
Catastrophic self-focusing occurs a t bearn powers exceeding the critica
power [IO] Pcrit
=
3.77x: Qxnona '
(2.37'
where Xo is the laser wavelength. The beam then collapses t o a singularity at a distanct
Pol
where P is the beam power, k = 27r/Xo is t h e wavenumber and a0 is the input beam radiuz a t the l / e level of intensity. This distance is measured from t h e waist of t h e input bearr
a t the entrance of t h e nonlinear medium. The dependence of zr on the Rayleigh rang€ ka: (see Appendix A ) is explicit. FVhen
P = Pcntself-focusing
is balanced by diffractioc
and the beam self-focuses at infinity.
The moving-focus mode1
The above description of self-focusing was developped for the case of a continuous-wavc
( CW) laser bearn. In the case of a laser pulse, the beam power varies in t ime. In the slotvly-
iwying-enidope approximation, which is r d i d for pulsés as short as roughly 10 optical
cycles ( 4 0 fs a t 300 n m ) [6] and in the approximation of a dispersionless medium, the
laser pulse c m be viewed as consisting of a longitudinal stack of infinitely thin transverse
slices that propagate independently from each other a t the group velocity of the pulse.
Each slice obeys t h e time-independent theory of self-focusing and behaves according t o its
own power. Given a Gaussian beam profile, t h e slices with power exceeding the critical
power Pcntundergo a collapse at the distance given by (2.38). Since the position of the
self-focus depends on power, the slices with supercritical power will collapse a t various
positions depending on their power (see Figure 2.3). Note t h a t slices a t t h e front and
back of the pulse having the sarne supercritical power wiIl form their self-foci a t the same position but at times corresponding to their position within t h e pulse.
Chapter 2: Survey
2
Figure 2.3: The self-focusing distance zf is different for the various Iongitudinal dices of the puIse with supercritical power.
These dynamics can be visuahed in the sketch of Figure 2.4. When the pulse is mucl shorter t han the self-focusing distance. the first slice to self-focus is the most powerfu slice at the center of the pulse.q This implies that an intensity spike develops at the cente of the pulse { h
e
L i in Figure 2.4 j. Witli f u r t h propagation this central spike splits ir
two, as less powerful slices self-focus and the central slice defocuses (time t z in Figure 2.4)
The separation between these two spikes increases with propagation, as slices with eveI less power self-focus further almg the axis (tirne t s in Figure 2.4). One can visualize thai
a pulse with supercritical peak poiver forms a continuous distribution of self-foci along
the propagation axiç, which can be perceived as a filament under poor time resolutior [lO,ll]. a ~ i t hlong pulses the first slice to self-focus is at the front of the pulse. However, the central slia
has a shorter self-focal distance since it is the most powerful. Therefore, the self-focus moves backward! in the medium between the time when the first slice self-focuses and the time when the central slici self-focuses
[IO].
Chapter 2: Survey
2
Figure 2.4: Self-focusing in the moving-focus model. As the pulse selffocuses two intensity spikes deveiop on asis, one at the front of the puise and one at the back of the pulse.
Limit ing mechanism The transverse diameter of a self-focus is expected to be limited to a finite size by mecha nisms cornpeting with the Kerr nonlinearity. Nonparaxial calculations show that diffrac
tion should stop the collapse at a self-focal diarneter of about one optical wavelengtl
i53j. However, very high intensities are achieved before this lower lirnit is attained anc the approximation of a strictly-Kerr nonlinearity ( L e . third-order nonlinearity) does no
hold. An important mechanisrn which Limits self-focusing of nano- and pico-second pulse: in condensed rnatter is the generation of free electrons by collisional avalanche [45]( s e Section 2.1.4).
The generated free electrons induce a negative change to the index
O
refract ion, given in the Dmde approximation by (in electrostatic units) [54]
where Ne is the electron density, v is the electron collision frequency and frequency- When Ne reaches
-.
10" cm-3 - 10" cm-3 the change of index
is the lase: 7221
due tc
the Kerr effect is cancelled by the free-electron index An, and self-focusing stops [45]
Chapter 2: Survey
22
For shorter pulses, MPE and collisionai avalanche can both generate free electrons (see Section 2.1.4). The relative importance of these mechanisrns depends on pulse duration. Recent numerical simulations on the laser-induced breakdown of water have shown that avalanche ionization is negligible for pulse durations less than
-
40 fs.
MPE then accomts
for al1 the generated free electrons [55,56]. For longer pulses, both mechanisms are active.
Bearn breakup The self-focusing behavior discussed so far is of the wholebeam type, where the entire bearn contracts towards a self-focus. At higher powers srnall-scale self-focusing can occur, where the beam breaks up into several transverse ceils distributed on the beam profile. Each of these cells then self-focuses independently. This occurs at powers larger than
-
10PCnt (depending on the magnitude of the initial inhornogeneities of the beam profile).
Each ce11 then contains a few
Pcnt [57,58]. In general, the self-focusing distance is shorter
in small-scaie self-focusing than in whole-beam self-focusing [58].
External focusing In most experimentd investigations of self-focusing an external lens is used to force selffocusing within the Iength of the medium. Self-focusing is then reinforced by the external lens, such that the self-focus appears before the position of the geometrical focus. The
self-Çocusing distance z; associated with self-focusing coupled with a lens with focal length
f is obtained by applying the lens transformation to zr from (3.38) [59]:
One can see that positive focusing brings the self-focus closer to the medium entrance, such that z;
5 j. On the other had, negative focusing pushes the self-focus farther dong
the propagation axis and increases the critical power to [IO]
Chapter 2: Survey
2.3.2
2;
Self-phase modulation
Self-phase modulation (SPM) [12,10111]arises €rom the temporal distribution of nonfineai phase in a laser pulse, in much the same way that self-focusing a i s e s from the spatia distribution of nonlinear phase in a laser hem.
A pulse with intensity distribution I(r,r ) propagating over a distance L in a mediun with index n = no + n21will accumulate a nonlinea phase distribution
1
L
d N L ( r )=
L'O
n2ï(z!T)-dz C
,
(2.42;
yielding a total phase
+O N L ( ~ )
d ( r )=
distributed dong the pulse. The nonlinear phase results in a broader spectrum. given bj
A more intuitive understanding of SPM can be achieved by considering the instant aneous frequency
With linear propagation in2 = O), w e simply have w;,,, = i ~ owhile with nonlinear propagation winst varies throughout the pulse. The spectrum of the self-phase modulated pulse then covers frequencies ranging from the maximum Stokes extent anti-Stokes extent 4 w Y M , which are given by
=
(A&) min
and
to the maximum
Chapter 2: Survey
One sees that ~
C 1
u
?
~
is given by the maximum rate of increase (decrease) i
the nonlinear phase dong the puise.
In simple SPM theory [4] the pulse intensity I ( r ) does not vary with propagatio~ The accumulated nonlinear phase is then given by
The noniinear phase is thus proportional to the pulse intensity. The case of a Gaussia pulse is represented in Figure 2.5. One can see that the instantaneous frequency is Stoke: shifted (ie. red-shifted) at the front of the pulse and anti-Stokes-shifted (ie. blue-shiftec at the back of the pulse. This effect c m be visualized in Figure 2.6, where the nonlinea phase (a), electric field amplitude ( b ) and spectrum (c) of an 8-fs input pulse are compare with their counterparts (af, bf, cf) after propagation through a Kerr medium with positiv n 2 . One can see that the nonlinear phase causes the wave to -pile up" in the back c
the pulse. Figure 2.6~'shows a modulation in the broadened spectrum. This modulatio is due to the fact that there are always two points d o n g the pulse that have the sarn instantaneous frequency, as can be seen in Figure 2.5. One should note that the simple picture of (3.48) does not hold completely when sell focusing of the pulse is also taken into account. since I ( z , r ) changes with propagatioi distance z . There is unfortunately no theory of coupled SPM and self-focusing. This i a serious limitation for the understanding of pulse propagation, since both effects aris
from the Kerr nonlinearity and t herefore often coexist-11 The SPM theory discussed so far arises from the Kerr nonlinearity. We will see
il
Chapter 3 that the generation of free electrons by the laser pulse can contribute to SPM The nonlinear index change is then a combination of the Kerr index and the free-electroi index; the latter, being negative, contributes to enhance anti-Stokes SPM. l11n fact, self-focusing can be visualized as a spatial SPM.
2
Chapter 2: Survey
Figure 2.5: Nonlinear phase and instantaneous frequency of a Gaussian laser pulse after propagation through a Kerr medium.
2.3.3
Self-focusing and self-phase modulation in the ultrafasi regime
When considering self-focusing of ultrashort pulses medium dispersion becomes important
Recent theoretical work has investigated the connection betwecn self-focusing, GVD anc
SPM [36,60-631.
During self-focusing, as the ultrashort pulse broadens spectrally bj
SPM its frequency cornponents suffer a temporal rearrangement due to GVD. This cause:
a reduction of peak power during self-focusing, such that a higher input power is requirec
to reach catastrophic self-focusing. Therefore, Pcntis higher for ultrashort puIses than f o
long pulses [60,61,63]. The temporal rearrangement of the frequency cornponents due t c
GVD during self-focusing could also result in a temporal splitting of the pulse [62]. An
experimental observation of pulse splitting has been reported in Ref. [36J. However, thir
observation could be the result of moving-focus dynamics and free-electron generation in
the medium, giving rise to intensity spikes a t the front and back of the pulse. This picturc
Chapter 2: Survey
21
is discussed in Chapters 4 and 5.
Self-phase modulation
SPM in the ultrafast regime should be quite strong, since the Stokes (anti-Stokes) broadening is inversely proportional to the pulse rise time (decay time). A rigorous theory 01
SPM
developed in Ref. [2S] to provide a precise description of strong SPM. In thi.
theory an approximation is made to the wave equation 2.1 that is less severe than the slowly-varying-envelope approximation made in simple SPM theory [4,6]. The result gives an asymmetrical SPM broadening, stronger on the anti-Stokes side than on the Stokes side. In the words of the authors of Ref. ['2S], this asymmetry is due to "the reaction ol the phase modulation to the frequency shift in time". For the hyperbolic secant kput pulse defined in (2.36), the broadening is given by
where Q = 2n21 L/cT& For Q
630 nm. ( c ) Stokes part. The intense wavelength components around the laser wavelength are suppressed with a RG850 coIor filter which eliminates A < 850 nm. Al1 spectra are corrected For filter, CCD a n d grat i n g responses.
48
Chapter 4: Whitelight continuum generation and self-focusing in condensed media
49
Figure 4.7: Anti-Stokes part of the continuum generated in LiF at P =
wavelengths would be absorbed by the imaging Lens f2. The sample is placed in front of the spectrograph slit and the beam is focused directly into the sample. The continuum exhibits a very high shot-to-shot stability. This can be seen in Figure
4.8a, which shows ten spectra measured in Ca& at P = 1.1Pz. One can see that the curves overlap. These shots are consecutive in a series of shots selected to have the same input energy within f1%. The standard deviation of the signal is 5.4% at X = 550 nrn
and 4.5% at A = 450 nm (calculated with a sample of 25 shots). The uncertainty in the rneasurement of Aw+ lies more in the measurement of parameters characterizing the laser pulse (e.g. input peak power and pulse duration) than in the measurement of Aw+ itself. This is illustrated in Figure 4.S(b), which shows continua measured in water at
P = 1 . 1 ~ with ~ ' the same measured input pulse characteristics but on different days, separated by a few months. The main uncertainties are in the pulse duration (f10%)
and the determination of the ratio PIP;' (f3%). The input beam profile can also Vary since it depends on the alignment in the spatial filter. Based on statistical observations we assign a typical uncertainty of
f10 nrn in the rneasurement of A#+. This translates
into an uncertainty of f1000 cm-' for a broadening Aw+ --. 20000 cm-' and f500 cm-'
Chapter 4: White-light continuum generation and self-focusing in condensed media
for a broadening Au+
-
5
10000 cm-'.
Let us consider the evolution of the beam profile, pulse spectrurn and pulse energ during continuum white-light generation. The appearance of the continuum during prop agation in water is shown in Figure 4.9, where the anti-Stokes part of the pulse spectrun is given for various positions z relative to the geometrical focus zo (zo is the position wherl d is smallest in the linear regime). The continuum appears over a distance of
.v
200 pm
The strong signal around X = 400 nm is due to the fact that the beam's image on the sli is smailer at X = 400 nm than at longer wavelengths, such that a larger fraction of th, beam at X = 400 nm goes through the slit than at longer wavelengths. The reason fo the smdler beam size at X = 400 nm at the position where the continuum is generated i: discussed in Section 4.5. The bump around 400 nm decreases at positions further alon( the z-axis. This c m be seen as the dotted line in Figure 4.9. which gives the spectrun at 1 mm after the geometricd focus. The anti-Stokes extent is slightly smaler at tha,
position than a t the earlier positions, where the continuum is generated. There is n< furt her decrease of the anti-S tokes widt h wit h further propagation.
The evolution with propagation distance in water of the beam width d, anti-Stoke! broadening Aw+ and pulse energy is shown in Figure 4.10.
One can see in the toi
graph that the beam contracts with propagation (left to right in the figure) and reache: a minimum of about 10 pm at z
diameter up to z t
.v
20.
-. r o -
-
20 -
500 pm- The beam stays contracted at thai
300 Fm and diverges slowly to a diameter of about 11 p m ai
It then diverges more rapidly with further propagation. Increasing the input peaE
power leads to a contraction that starts earlier in propagation but still finishes at z The divergence after
t
-
zo
> 20 is roughly the sarne at al1 powers.
We interpret the region of contracted propagation from z
-
zo - 500 p m to z
-
20 2
a filament arising from the moving-focus dynamics: the most powerful sljce of the pulsc self-focuses around z
-
20 -500
pm and the slice with critical power self-focuses at z
-
20
The region between these two points is a continuous distribution of self-foci associated
with intermediate powers. The observation of a filament that starts earlier d o n g th€ t-axis with higher power is consistent with the moving-focus model, as is the fact that
Chapter 4: White-light continuum generation and selfifocusing in condensed media
Figure 4.8: Stability and reproducibility of continuum spectra: (a) ten spectra measured in CaF2 a t P = l.lP$. Al1 the shots have the same input energy within f1%. The curves overlap perfectly- Note that the anti-Stokes extent is limited by absorption in the imaging lens (lens f2 in the setup of Figure 4.1). (b) spectra measured in water at P = 1.1~;' with the same input puIse characteristics but on different days, separated by a few months.
51
Chapter 4: White-light continuum generation and selfifocusing in condecsed media
Figure 4.9: Evolution of the anti-Stokes part of the spectrum with p r o p ' . is the position agation distance in water at P = 1 . 1 ~ ~zo of the geometricd focus, measured with precision i900 Fm. There is no signai detected at - 550 prn.
5:
Chapter 4: White-light continuum generation and self-focusing in condensed media
5:
Figure 4.10: Evolution of the FWHM bearn width d , the anti-Stokes broadening Aw+ and the pulse energy with propagation dis' . is the position of the tance in water at P = 1 . 1 ~ ~ zo geornetrical focus, rneasured with a precision of f200 prn.
the end of the filament does not change with power. These concepts will be addressed in
more detail in Chapter 5, in relation to the filaments observed in air. We note that in Figure 4.10 the Rayleigh range (see Appendix -4)is
-
0.5 mm in the converging part of
the propagation (as the beam diameter shrinks from 14 pm to 10 pm) and
-
1 mm in the
diverging part of the propagation (as the bearn expands from 10 pm to 14 pm). These values are smailer than the Rayleigh range of using the measured beam waist d
-
2.7 mm in the linear regime (calculated
27 pm).
One can see in the middle graph of Figure 4.10 that the continuum starts t o appear
when the bearn is contracted into a filament. The full development of the continuum occurs over a distance of
-
200 Pm. The bottom graph of Figure 3.10 shows that the
continuum develops concurrently with an energy loss of a few percent. The anti-Stokes width Au+ of the continuum does not depend strongly on the input peak power. This can be seen in Figure 4.11b, which shows the anti-Stokes wing of
the continua generated at various input peak powers. Al1 these continua have the same
Chapter 4: Whitelight continuum generatioo and self-focusingin condensed media
extent within a range of
-
1500 cm-', i.e. within
-
5
10% of their average extent. Thj
range should be compared to the uncertainty on Aw+, which is about f500 cm-L or &3?4 Unlike the anti-Stokes width, the Stokes width Aw- increases with increasing input powei This can be seen in Figure 4.11a, which shows the Stokes wing and central part of th continua generated in water a t various input peak powers. One can see in this figure tha a modulation develops gradually in the spectrum as the input peak power is increasec
This modulation can be attributed to SPM and is due to the fact that there are aiway two points along the pulse where the rate of change of the nonlinear phase are e q w (see Section 2.3.2). Interestingly, the period of the modulation decreases with increasin, frequency at large detuning (Figure 4.1 lb). This behavior is contrary to the prediction c
SPM and can possibly by attributed to the effect of groupvelocity dispersion 1601.
4.2.3
Media that cannot generate a continuum
The white-light continuum described in the previous section is not observed in al1 tram parent condensed media. In some media, such as SF-11 glass, trichloroethylene, benzene toluene and CS?, the pulse does not transform into a continuum when the peak powe exceeds PfL. The spectrum measured aftet propagation in one of these media is compara tively much narrotver and nearly syrnrnetric. Examples of such spectra at P = 1 . 1 ~arc 2 shown in Figure 4.12. After propagation in these media the beam remains invisible to the eye when pro jected on a screen; at several P,Sh' the central part of the beam acquires at best a din reddish tint. An example of a spectrum observed in benzene a t P
E
1 0 P z is show
in Figure 4.13, along with the continuum observed in water at P = 1.1P$ (shown fo: comparison). There is no sharp threshold for the appearance of this visible light anc there is no visible light at powers below P cx lOP,"hf.The appearance of this dim visibli light in the forward direction at high potver is often accompanied by the appearance
O
a colorful pattern of conical emission surrounding the bearn.11 At the power where thi! [[AS rnentioned
in the beginning of this chapter, although the conical emission can be described as
i
spectral continuum with a wavelength-dependent angular distribution we do not refer to it as a white
Chapter 4: Whitelight continuum generation and self-focusingin condensed media
Figure 4.11: White-light continuum generated in water at various input peak powers: (a) central and Stokes parts of the spectrum and (b) anti-Stokes part of the spectrum.
5:
Chapter 4 : White-light con tin uum genera tion and self-focusing in condensed media
(nm) 860
840 m
1 1500
800
820 .
12000
,
-
,
780 .
12500
,
.
760
13000
740
13500
o (cm-') Figurc 4.12: Spcctra gcnerated in trichloroethylene and in SF-11 g l a s at P = 1 . 1 c . There is no long anti-Stokes wing. Note ttiat there is no conical emission at this power.
5
Chap ter 4: White-light con &inu um generation and self-focusing in condenseù media
5i
Figure 4.13: Visible spectrum in the forward direction observed in benzene at P = 10ei,along with the visible spectrum observed in water at P = The spectrum observed in benzene is weaker, strongly rnodulated and does not extend as far as the spectrum observed in water.
1.1~2.
conical emission appears the spectral broadening is Iarger in the conical emission than in the central part of the beam. The conical emission observed in these media is similar to that observed in the media that can generate a white-light continuum in tbe forward direction. This topic d l bc discussed further in Chapter 5.
4.2.4
Band-gap dependence of continuum generation and selffocusing
Investigation of continuuum generation and self-focusing in several media revealed a dependence on the band-gap of the materiai E,,.
T h e latter is readily obtained from the
absorption spectrum of the medium, which generally shows a sharp absorption edge in light continuum. It becornes apparent that separating conceptuaIIy the two phenornena is advantageous. LVe stress that there is no conical emission at P =
1.1~2 in any of the transparent media.
Chapter 4: White-ligh t continuum generation and self-focusingin condensed media
58
the UV corresponding to the edge of the conduction band. The absorption spectra are obtained from Refs [75-791 and from measurements with a spectrophotometer (for SF-11 glass and CS2). We apply the same definition of E,,
to crystals, Iiquids and arnorphous
solids, although the band structures are not as clearly defined in liquids and arnorphous solids as in crystals. The relevant physical quantity is the energy corresponding to the
UV absorption edge. The most striking band-gap dependence observed in this experiment is that continuum generation occurs only in media with band-gap E,,, larger than the threshold
4.7 eV. Also surprising is a trend of increasing Au+ with increasing EgaP. ObEi:, servation of this trend in several media led us to investigate LiF, the medium with the largest known Egap. Indeed, LiF yieIded the broadest continuum (see Figure 4.7). The dependence of Aw+ on E,,, is shown in Figure 4.14 and in Table 4.1, along with various measured self-focal characteristics. Aw+ is obtained from spectra measured with positioned such that the continuum is generated
-
1 mm below the output surface of
the medium. In Table 4.1 Pth designates the meaçured white-light continuum and
fi
Pz' for media that generate a
~ L h ffor media that do not generate a white-light
continuum.
We recall t hat the P,whl 2 P:;. One can see in Table 4.1 that
PZ
tends to increase with Ep,.
This implies that
the Kerr index n2 decreases with Egap,since the theoretical Pcat is inversely proportional to na (101.*- Therefore, Aw+ increases with decreasing
n1.
This observation contradicts
SPM theory, which predicts an increase of Aw+ with increasing Q. The measurements shown in Table 4.1 also reveal that dminis larger in media with
Egap < E:,
than in media with Ega, > Eg,. Note that although dminis similar in al1
> E g p the measured fluence increases with E,, since the measurements and PC6,increases as n? decreases. are taken a t P = 1.1P$ N 1.1PCnt
media with EkP
"This is consistent with the Lorentz-Drude mode1 of Section 2.1.3, which predicts a decreasing nonIinearity with increasing resonance frequency.
Chapter 4: White-iigh t continuum generation and self-focusing in condensed media
Figure 4.14: Aw+ vs band-gap in various media. a-LiF, b - C S a , c-UVgrade fused silica, d-water, e-D20, f-1-propanol, g-methanol, h-NaCl, i-1,4dioxane, j-chloroform, k-CCLI, 1- C2HC13, mbenzene, n-CS2, +SF-l l glass.
1
Chapter 4: White-light continuum generation and self-focusing in con densed media
Table 4.2:
61
Self-focal characteristics measured in various media at P = 1.1Pth. Fo media with Epp > E:,
= 4.7 eV, Pthis the measured Pzi (which is
and for media with Egap< Eth = 4.7 eV, @P '
Ph
is the measured :P
E
Pt;
dm;, i:
the diarneter (FWHM) of the beam waist f f0.5 pm), Fmax is the maximun fluence and Elos is the energy loss.
Egap is obtained from the absorptior
spectrum of the medium, which generdy exhibits a sharp absorption edgt in the UV corresponding to E,.
Chapter 4: White-ljgh t con tin u um generation and self-focusing in condensed media
61
Limitation of the self-focal diameter by generatioi of fiee electrons In order to gain insight into the band-gap dependence of the continuum, let us fist consider the mechanism that stops the bearn's catastrophic collapse a t powers exceeding Pcnt(see Section 2.3.1). In the case of picosecond pulses, where self-focal diameters of typicdly 6-10 p m are observed [50], the mechanism generally invoked is avalanche ionization of the medium [45,50]. The free electrons generated in the avalanche induce a negative change to the index of refraction, which is given in the Drude mode1 by (in electrostatic units) 1541
where -I, is the electron density, u is the electron collision frequency and wo is the laser frequency. This negative contribution to the index of refraction can cancel the Kerr index
and thus stop self-focusing. This is expected to occur at N,Stop
-
101'
- 10''
[25,45,80]. Questions regarding the mechanism limiting self-focusing for femtosecond pulses were raised initially by Corkum e t al. [29]. The authors observed that the onset of catastrophic
self-focusing leaves the spatial characteristics of the beam practically unchanged after the bearn's focus. Furthermore, Iit tle energy loss was observed. These observations, which are not expected from avalanche ionization as a lirniting mechanism, triggered theoretical work on the connection between self-focusing, group-velocity dispersion (GVD) and SPM
[60-631. An interesting outcome of this work is that the effective critical power for selffocusing is higher than
Pcnt for ultrashort pulses, because of GVD [60,61,63]. The issues
related to GVD will be discussed in Section 4.8.3. The band-gap dependence of the self-focal diameter observed in this work suggests that free-electron generation by multiphoton excitation (MPE?see Section 2.1.4) is important in the limitation of self-focusing in condensed media wi t h 140-fs, Ti:Sapp hire pulse.
Chapter 4: White-light continuum generation and self-focusing in condensed media
6
Indeed, the band-gap is an important parameter governing MPE. We shail therefore con sider the limitation of the self-focal diameter by MPE.
During self-focusing a sharp intensity spike develops in the laser pulse [10,41,42,60
Assuming that self-focusing is limited by MPE, AnK,, should be cancelled by An, at th1 peak of the spike. From (4.1) and (4.2) in the approximation w;
> v 2 , this
cancellatioi
should occur when
In order to find the intensity iStoP and electron density NZtop that satisfy this relation le
us examine the MPE rate W in the case of water. Using the Keldysh theory of M P E (sel Equation (2.23))we obtain the solid line in Figure 4.15,showing the intensity-dependeno of M.' in water. We must now determine at which point on this curve Equation (4.3) wil be satisfied. One can see that W increases rapidly with intensity; this implies that durin! self-focusing most of the free electrons appear at the very peak of the intensity spike
during the haif-cycle (1.3fs) when the electric field is ma..imum. The MPE rate requirec to generate a free-electron density Ne in one half-cycle is given by
Combining (4.3)and (4.4),we find that the Kerr index is cancelled when the MPE ratt 1s
-
zOm,w~ n21 ~.~[crn-~/fs] 2xe2(l.3fs) ' This relation iç plotted as a dashed line in Figure 4.15.tt The two curves cross at IStoP
-
10L2W/cm2 and WstOP 10L8
-
corresponding to the point rvhere (4.5)is satisfiec
-
in the case of water. Using (4.4) we find that N,'tOP
1018 cm-3? which is similar tc
the density required to stop self-focusing by avalanche ionization [25,45].In other medi: with different n2 and E,
these d u e s would differ; NztOP would be slightly different duc
to a slightly different n2, but Istop C O U be I ~ significantly different since E,,
has a strone
impact on MPE. This point is addressed in more detail below.
Let us see if these results are consistent with the experiment. First, the free-electror density Nstop generated in our experiment in water can be estimated by assuming that t t ~ use e n? = 2 x 10-l6 cm2/W, extracted from the rneasured critical power in water.
Chapter 4: White-light continuum generation and self-focusifig in condensed media
10"
1 013
Intensity (W/crn2) Figure 4.15: Intensity and MPE rate a t which self-focusing stops in water. T h e solid line represents the M P E rate vs intensity in water, calculated from Equation 2.23. T h e dashed line represents t h e M P E rate vs intensity for which the Kerr index a n d plasma index cancel in water, from Equation 4.3 (assuming t h a t the M P E occurs in half a n optical cycle). The point where the two lines cross gives t h e intensity and M P E rate when self-focusing stops.
6
Chapter 4: White-light continuum generation and self-focusing in condensed media
61
MPE accounts for most of the energy loss ( a Ioss of EGp= 7.5 eV per electron in water) We approximate the pIasma volume to a 200-pm-long cylinder with a diameter that cannoi be larger than d = 9.8 pm and cannot be smaller than set by diffraction [53]. We obtain 1 x 10" with our theoreticai estimate of !Vzt"P
-
-
1 Pm, the theoretical lower limi.
< N,"'Op < 1 x 10''
This is consisteni
10'' cm-3 obtained from (4.3-4.5).
Before estirnating the experimental self-focal intensity I,,,, achieved in water somc attention should be paid to what is actually measured in the experiment. Indeed, the
maximum intensity reached in the filament cannot be inferred from the smallest measured diameter dminalong the filament, since dminis the time-integrated beam diameter To fully grasp these concepts one has to refer to the moving-focus model of self-focusing. which is discussed in Section 2.3.1. In this model we view the laser pulse a longitudinal stack of infinitely thin transverse slices that propagate independently from each other at the group velocity. Under conditions of linear focusing, L e . when focusing is due solely tc an external lens and not to seIf-focusing, al1 the slices are focused to the same diameter The maximum intensity I,,
achieved in the focus is then given by
-
Fm,/ro, where Fm,
is the maximum fluence and ro is the input pulse duration (FWHM). When self-focusing
occurs, however, various longitudinal slices of the pulse contract to different diameters according to the ratio of their own potver to P,,,.
This is depicted in sketches at the top
and middle of Figure 4.16, showing respectively three pulse slices and their corresponding transverse intensity distribution. At t h e position with the smallest measiired heam diameter dm;,, a slice can be self-focused to the limiting diameter dlimit( ~ . g slice . a , Figure 3.16), whiIe the rest of the pulse is not focused so tightly (e.g. slices 6, c ) . Furthermore,
while a particular slice with supercritical power might form a self-focus at the position where the measurement is made, other slices with supercritical power might form a selffocus at an earlier or Iater position dong the axis. Those slices would not be contracted down to diimitat the position of measurement. For al1 these reasons the measured fluence distribution (time integral of the intensity distribution) has a diameter dmin larger than the limiting diameter dlimi,.As a result, I,,
has to be greater than Fm,/ro.
Using these concepts we estimate that the self-focal intensity achieved in water at
P = 1.1P;h' is greater than S x 10" W/cm2, using dmi, = 9.8 p m and Fm, = 0.62
Chapter 4: White-light continuum generation and self-focusing in condensed media
Figure 4.16: Sketches of: Top, power distribution of the pulse shown with infinitesimal pulse siices a, b and c. Middle, intensity distributions of slices a, b and c in the focal piane. Bottom, tluence distribution in the focal plane.
6
Chapter 4: Whitelight continuum genera &ionand self-focusing in condenseci media
f
Figure 4.17: Intensity Istop required for M P E rate PV = 1018 cm-3 fs-' as a function of band-gap (solid line). The curves for W = fs-' and W = 1019 cm-3 fs-' are displayed for 10'' cornparison. The top axis shows the band-gap norrnalized ta the laser photon energy.
J/cm2. This value is larger than
-
10L2W/cm2, the value calculated using (4.3-4.5
This reinforces the assumption that MPE does occur in the self-focus.
Our estimates of IstOp and Nestop using (4.:3-3.5)also agree with recent studies of laser
induced breakdown of water [55:56], which have shown that a 100-fs pulse with a peal intensity of 5 x IOL2 W/cm2 can generate a free-electron density iV,
10" cm-3 mainl:
by MPE. It is therefore reasonable to assume that in our experiment significant MPE
occurs in the self-focus, and we assume that in our experiment seIf-focusing is stopped b: free-electron generation.
Now that we have established MPE as a mechanism capable of stopping self-focusing let us consider the band-gap dependence of the measured self-focal diameter
dmin.
Wc
have seen in Section 4 2 . 4 and Table 4.1 that the measured dminis larger in media witl
E,,
< E& than in media with Egap> E;;,. To get some insight into this behavior le
Chapter 4: Whitelight con fin u um generation and self-focusing in condensed media
6:
us consider the band-gap dependence of MPE in Keldysh theory [15,55]. We have seer above that MPE stops self-focusing when the MPE rate reaches W
.v
1Ols cm-3 fs".
Th
