“Attosecondelectrondynamicsin complexmolecularsystems” · “Attosecondelectrondynamicsin...

Politecnico di Milano

DEPARTMENT OF PHYSICS

“Attosecond electron dynamics incomplex molecular systems”

Supervisor: PhD Thesis by:

Prof. Mauro Nisoli Andrea Trabattoni

The Chair of the Doctoral Program: Mat.: 785033

Prof. Paola Taroni

Doctoral Program in Physics, XVII cycle (2014)

“... Il linguaggio dell’Universo ... ”

Pino Mascolo

CONTENTS

Introduction 5

1 Attosecond physics 13

1.1 Isolated attosecond pulses . . . . . . . . . . . . . . . . . . . . . 15

1.1.1 Gating methods . . . . . . . . . . . . . . . . . . . . . . 17

1.1.2 Characterization techniques . . . . . . . . . . . . . . . . 21

1.1.3 FROG-CRAB retrieval . . . . . . . . . . . . . . . . . . . 25

1.2 Attoseconds for molecular physics . . . . . . . . . . . . . . . . . 26

1.2.1 From H2 to multielectron diatomic molecules . . . . . . 26

1.2.2 Charge motion in complex molecular systems . . . . . . 30

2 Experimental setup 43

2.1 Attosecond beamline . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Detection systems . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.1 Electron time-of-flight spectrometer . . . . . . . . . . . 46

2.2.2 KEIRAlite mass spectrometer . . . . . . . . . . . . . . . 49

2.2.3 Velocity Map Imaging (VMI) Spectrometer . . . . . . . 50

2.3 VIS/NIR pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3

2.3.1 Hollow-core fiber spectral broadening . . . . . . . . . . 57

2.3.2 Temporal characterization . . . . . . . . . . . . . . . . . 59

2.3.3 CEP stability . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4 Isolated attosecond pulses . . . . . . . . . . . . . . . . . . . . . 69

2.4.1 Tunability in XUV generation . . . . . . . . . . . . . . . 69

2.4.2 Temporal characterization . . . . . . . . . . . . . . . . . 72

3 Quantum interference in N2 molecules dissociation 87

3.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.1.1 N+2 states and pump-probe pulses . . . . . . . . . . . . 89

3.1.2 Retrieval of 3D momentum distribution . . . . . . . . . 91

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.2.1 N+ KER spectrum as a function of time delay . . . . . 95

3.2.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . 101

3.2.3 Quantum interference along the N+2 PECs . . . . . . . . 103

4 Charge migration in the amino acid Phenylalanine 113

4.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.1.1 Molecular target . . . . . . . . . . . . . . . . . . . . . . 114

4.1.2 HHG spectra and VIS/NIR pulses . . . . . . . . . . . . 116

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.2.1 Molecular fragments . . . . . . . . . . . . . . . . . . . . 118

4.2.2 XUV-pump VIS/NIR-probe scans . . . . . . . . . . . . 120

4.2.3 Theoretical calculations . . . . . . . . . . . . . . . . . . 128

4.2.4 Observation of charge migration . . . . . . . . . . . . . 131

Conclusions and future perspectives 141

Appendix 145

List of publications 151

Acknowledgements 157

INTRODUCTION

In 1981 Zewail and coworkers published a pioneering work on quantum coher-

ence effects in the vibrational states of anthracene [1], paving the way for the

study of ultrafast dynamical processes in isolated molecules. In the same years

the laser sources were experiencing a dramatic development thanks to the ap-

pearence of the first subpicosecond dye lasers (1974) [2] and, few years later,

the achievement of pulses with a duration down to 6 fs (1987) [3]. The great

results on both sides converged in the development of ultrafast spectroscopy

and femtochemistry [4], providing an “ultrahigh-speed photography” at the

atomic and molecular level. Nowadays this research field is really well estab-

lished and gives a direct access to dynamical processes of great importance in

physics, chemistry and biology [5].

From quantum mechanics we know that femtosecond temporal scale is intrin-

sically related to the nuclear motion, for this reason a typical experiment with

femtosecond resolution is able to investigate in real time the evolution of a

reaction, the breaking of a chemical bond, the fragmentation of a complex

system after the perturbation of the initial quantum state, down to the vibra-

tional oscillation in diatomic molecules (the ground state vibrational period

of H2 is in the order of 10 fs). Electron dynamics occurs on a faster temporal

5

Introduction

scale ranging from a few fs down to a few hundreds as (1 as = 10−18 s), for this

reason in order to track the electronic motion in matter shorter light pulses

are required.

In 1987 and 1988 two indipendent experiments were able to produce coherent

extreme ultraviolet (XUV) radiation by exploiting the interaction between

a strong IR laser field and the atoms of a rare gas [6, 7]. The result was a

series of odd harmonics of the fundamental wavelength, corresponding to a

train of subfemtosecond bursts. Only few years later this process was fully

underestood and called High Harmonic Generation (HHG) [8–11]. Since then,

great effort was made to investigate more in detail the HHG process, until the

first experimental demonstration of attosecond pulses generation, performed

in 2001 by Paul and coworkers, who were able to generate a train of 250 as

pulses [12]. During the same year a single attosecond pulse with a time dura-

tion of 650 as was successfully isolated from a train of attosecond pulses [13].

These results paved the way for the birth of attosecond physics.

In the last two decades a strong effort was made to characterize attosecond

sources and to apply this technology to the investigation of ultrafast elec-

tronic dynamics in matter. The main problem the community has to face is

the low intensity of attosecond sources, since the convertion efficiency of HHG

process is quite low (in the order of 10−6), resulting in XUV energies usually

in the range between hundreds of picojoules up to few nanojoules. This level

of energy, and correspondent intensity, is tipically too low for initiating non

linear processes in matter, thus for performing attosecond-pump attosecond-

probe experiments (for high energy XUV sources and recent results concerning

XUV-pump XUV probe experiments, see for example [14, 15]). For this rea-

son the common solution is to combine the XUV pulses with a VIS/NIR laser

field, with an attosecond-pump femtosecond-probe configuration. This setup

can still preserve a temporal resolution in the attosecond timescale and in the

last years gave important results in investigating ultafast electron dynamics

in atoms, and recently even in simple molecules [16–21]. Despite these posi-

tive results, attosecond physics still didn’t show the capability of investigating

6

Introduction

complex systems, for example biomolecules, where ultrafast electron dynamics

are expected to play a fundamental role in many biological processes such as

catalysis, respiration, DNA damage by ionizing radiation and photosynthesis.

I dedicated my PhD activity to the ELYCHE (ELectron-scale dYnamics in

CHEmistry) project, in the high energy attosecond lab of Politecnico di Mi-

lano - department of Physics, with the aim of applying attosecond tools to

the investigation of purely electronic dynamics in complex systems, such as

multielectron molecules, nanoparticles and molecules of biological interest.

In the first part of my PhD we installed and developed the attosecond-pump

VIS/NIR probe beamline in the ELYCHE laboratory. A strong effort was

made to compress the laser pulses down to 4 fs of duration and use them for

the generation of isolated attosecond pulses (IAP).

In the last two years of my PhD I applied these tools in some important

pump-probe experiments. First we investigated ultrafast relaxation process

in multielectron diatomic molecules. We concentrated on the N2 molecule,

that is the most abundant species in the Earth’s atmosphere, with the goal

of understanding the interaction of molecular nitrogen with extreme ultra-

violet (XUV) radiation, that is of crucial importance to completely disclose

the atmospheric radiative-transfer processes. By performing a Velocity Map

Imaging (VMI) experiment, we were able to disclose the ultrafast dissociative

mechanisms leading to the production of N atoms by XUV photoionization,

and to observe a predissociation quantum interference between the electronic

states of the molecular cation. We also managed to extract information about

the slope and shape of nitrogen (in particular N+2 ions) potential curves, a sort

of “real-time mapping” of molecular electronic states.

Then we tried to push our invstigation to more complex systems with the aim

of studying ultrafast electronic dynamics in molecules of biological interest.

We perfomed a mass spectrometry experiment on Phenylalanine (one of the

essential amino acids) and analyzed the temporal evolution of molecular frag-

mentation after XUV ionization. We were able to measure a charge oscillation

7

Introduction

in the yield of immonium dication fragment, providing for the first time an

experimental demonstration of charge migration in a biological molecule.

The thesis is organized as follows:

• The first chapter treats some background concepts concerning attosec-

ond technology applied to molecular physics. Particular attention is

dedicated to the description of Polarization Gating (PG) technique for

the generation of isolated attosecond pulses and the temporal charac-

terization of such pulses. Then the applications of attosecond tools to

the investigation of molecular systems are introduced.

• The second chapter is dedicated to the experimental setup we developed.

Particular attention is devoted to the generation and characterization of

isolated attosecond pulses and VIS/NIR probe pulses. Concerning the

characterization of the probe pulses, we carefully investigated the tem-

poral duration and the carrier-envelope phase (CEP) stability. Then we

performed Attosecond Streak Camera (see the section 1.1.2) experiments

to fully characterize the temporal structure of attosecond pulses.

• The third chapter describes the investigation of ultrafast electronic dy-

namics in N2 molecules. The experiment of photoionization is presented.

Numerical simulations are introduced and results analyzed, discussing

the possibility of experimentally mapping N+2 electronic states.

• The fourth chapter reports about the measurement of charge migra-

tion in the aminoacid Phenylalanine. The experimental setup is first

presented. After that the experimental results are discussed according

with the numerical simulations.

• The conclusion consists in a summary of the principal results described

in the thesis and a discussion upon the possible future perspectives.

8

BIBLIOGRAPHY

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dephasing in isolated large molecules cooled by supersonic jet expansion

and excited by picosecond pulses: Anthracene”, J. Chem. Phys. 75, 5958

(1981);

[2] C. V. Shank and E. P. Ippen, “Subpicosecond kilowatt pulses from a mode-

locked cw dye laser”, Appl. Phys. Lett. 24, 373 (1974);

[3] Fork, R. L.; Brito Cruz, C. H.; Becker, P. C.; Shank, C. V., “Compression

of optical pulses to six femtoseconds by using cubic phase compensation”,

Optics Letters, Vol. 12 Issue 7, pp.483-485 (1987);

[4] Peter M. Felker and Ahmed H. Zewail, “Purely rotational coherence effect

and timeresolved subDoppler spectroscopy of large molecules. I. Theoreti-

cal”, J. Chem. Phys. 86, 2460 (1987);

[5] Ahmed H. Zewail, “Femtochemistry: Atomic-Scale Dynamics of the

Chemical Bond”, J. Phys. Chem. A 104, 5660-5694 (2000);

[6] A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIn-

tyre, K. Boyer, and C. K. Rhodes, “Studies of multiphoton production of

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pp. 595-601 (1987);

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[14] Eiji Takahashi, Yasuo Nabekawa, Tatsuya Otsuka, Minoru Obara, and

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[15] P. Tzallas, E. Skantzakis, L. A. A. Nikolopoulos, G. D. Tsakiris

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11


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12

CHAPTER 1

ATTOSECOND PHYSICS

The motion of an electron or a nucleus inside a molecule can be simply seen as

a dynamical process which occurs with a precise time-dependent probability

distribution. There exist three requirements in order to study the motion of

such a system [5]. First, we need to clock the dynamics by defining the zero-

time, with a precision that should approach the resolution of the experiment.

Second, the dynamics must be synchronized, since a huge amount of events

are typically recorded and create the statistics necessary to map and solve

the process. Third, coherence must be induced in the molecule in order to

obtain localization.

A pump-probe experiment is able to satisfy each of these requirements. The

pump pulse excites the molecule coherently creating a quantum wavepacket

with electronic and nuclear degrees of freedom and initiating the dynamics.

The probe pulse provides the shutter speed for freezing the molecular motion,

defining the clock. By varying the delay between the pump and the probe

pulses a series of “snapshots” of the dynamics are obtained.

In order to solve a time-dependent process, it is crucial to define the temporal

13

Attosecond physics

resolution of the experiment. This is not a trivial task and doesn’t come simply

from a careful characterization of pump and probe pulses. Paradoxically, the

resolution of an ultrafast pump-probe experiment on a molecular target is

estimated by taking into account the physical process under study. With

the aim of investigating ultrafast dynamics of nuclei and electrons inside a

molecule, we know that the motion of these elementary particles is governed

by the laws of quantum mechanics. The characteristic timescale can be found

by considering a system with two eigenstates φ1, φ2 with correspondent energy

eigenvalues E1, E2. The wavefunction ψ of the system is described by:

|ψ(t)〉 = a1e− i

~E1t |φ1〉+ a2e

− i~E2t |φ2〉 (1.1)

where a1, a2 are amplitudes that describe the coherent superposition of the

two eigenstates. This analysis is absolutely general but the two states can

be thought as two rotational, vibrational or electronic states of a molecule.

The time evolution of the wavefunction |ψ(t)〉 results from the difference in

the evolution of the amplitudes of the two states, in particular in the value of

phase. This time-dependence can be probed by a measurement of the inter-

ference of the two states. The operator A that represents such a measurement

is described in the basis of the eigenvectors of the hamiltonian in the form:

A =

[

c1 c12

c∗12 c2

]

(1.2)

The expectation value of A gives the measurement of the interference between

φ1 and φ2:

〈ψ| A |ψ〉 = c1 | a1 |2 +c2 | a2 |2 +2R(c12a∗1a2)cos

∆E

~t+

−2I(c12a∗1a2)sin

∆E

~t

(1.3)

with ∆E = E2 − E1. When c12 6= 0 and a1, a2 6= 0, Eq. 1.3 displays an

oscillatory dynamics with a period of T = h/∆E. This motion can be seen

as the result of the beating between the two waves corresponding to the two

14

Isolated attosecond pulses

eigenstates. The beating can be observed when there is a non-zero amplitude

in both eigenstates and the eigenstates of A are different from the eigenstates

of the hamiltonian that describe the system. From this simple analysis it can

be deduced, for example, that rotations of a molecule occur on picosecond (1

ps = 10−12 s) timescales. Chemical reactions are usually thought as processes

in which it is the atomic motion in molecules to drive the transformation

from the initial to the final state. The fastest chemical reactions evolve on

femtosecond timescales and can be studied in time-resolved experiments using

femtosecond laser pulses [5]. An example of femtosecond nuclear motion is the

vibration of a molecule. The fastest molecular vibration in nature is that of

a H+2 molecular ion. The energy splitting between the two lowest vibrational

levels is 270 meV, that corresponds to a vibrational period of 15.2 fs. As shown

in the next sections, even purely electronic dynamics inside a molecular system

can strongly affect the temporal evolution of the system from the initial to

the final state. Energy splittings between electronic states are typically on

the order of ≃ 1 eV, which corresponds to an electronic motion occuring in

the attosecond timescale. Therefore attosecond pulses are required to capture

electronic motions inside a molecular system.

1.1 Isolated attosecond pulses

At the end of the 80’s, two indipendent experiments were able to produce

coherent extreme ultraviolet (XUV) radiation by exploiting the interaction

between a ultraintense IR laser field (1013 − 1014 W/cm2) and the atoms of a

rare gas target [6,7]. The XUV spectrum so obtained displayed unprecedented

characteristics beyond the well known perturbative regime: after a significant

intensity drop in the lowest harmonics, the spectrum was characterized by a

plateau of odd harmonics of the fundamental wavelength, with approximately

constant intensity. The plateau then ended at a sharp cut-off beyond which

no harmonic emission was found. Furthermore, the photon spectrum that

results from this process can extend well above the ionization potential (Ip)

15


of the atom. It was found that the XUV cutoff satisfies a universal rule and

lies at approximately Ip + 3Up for all wavelengths, targets and driving laser

intensities [9] (Up is the ponderomotive energy).

After the initial experimental results, numerous theoretical efforts led to a

deep understanding of this new non-perturbative process, that was dubbed

High Harmonic Generation (HHG). Quantum calculations, based on the time-

dependent Schrodinger equation for a single electron interacting with the fun-

damental laser field, reproduced the principal observations of HHG experi-

ments. Such calculations have helped to determine, for example, the cutoff

law for HHG [8]. Explanations were also found in terms of a classical model

for the interaction between the electrons and the generating field [9–11].

A few years after the first experimental demonstration of HHG, several pro-

posals suggested the generation of attosecond pulses based on HHG [22–24],

since the XUV bandwidth produced by HHG is large enough to support such

short pulses. Although it was long expected that HHG could produce at-

tosecond pulses, it has taken almost a decade before it could be confirmed

experimentally [12], with the first observation by Paul and coworkers of a

train of attosecond pulses. Nowadays HHG is a really well enstablished tech-

nique, and great achievements were reached in the generation of attosecond

pulses in this way. In particular, in 2001 Hentschel et al. demonstrated the

generation of isolated attosecond pulses (IAP) [13], paving the frontier for the

temporal resolution in ultrafast pump-probe experiments. This result was of

great importance, since for many applications it is crucial to reduce the pulse

train to a single attosecond pulse. This is particularly important, for exam-

ple, in order to investigate the sub-femtosecond evolution of single coherent

wavepacket by using the pump-probe technique. Several approaches for effi-

ciently generating IAPs have been proposed and demonstrated in the years.

All of them, anyway, are based on the ability of properly confining the HHG

process, in order to create a gate in the driving field and select a single burst

from the train of attosecond pulses.

16


1.1.1 Gating methods

Isolated attosecond pulses produced by HHG have been experimentally demon-

strated exploiting a number of experimental techniques; these include spectral

selection of half-cycle cutoffs [25, 27, 29] as in amplitude gating [30, 31] and

ionization gating [18,24,28,32], temporal gating methods such as polarization

gating (PG) [24, 25] and double optical gating (DOG) [26–28], and spatio-

temporal gating with the attosecond lighthouse effect [29].

It is worth noting that, in general, the generation of broadband isolated at-

tosecond pulses requires a series of challenging technological tools. One of

these is to have CEP(Carrier-Envelope phase)-stable driving pulses. By defin-

ing the electric field of the generating pulses as:

E(t) = A(t)ei(w0t+ϕ) (1.4)

(where A(t) is the complex envelope of the electric field and w0 is the carrier

frequency of the pulse), the term ϕ represents a temporal offset between the

maximum of the envelope (assumed at t = 0) and the maximum of the carrier

wave of frequency w0. The term ϕ is usually indicated as Carrier-Envelope

phase (CEP). Nowadays several methods exist to lock the CEP value in the

laser pulses, as described in chapter 2. The generation of isolated attosecond

pulses, in general, requires a “light switch”, a method that can effectively

turn on the HHG process during only a single half cycle of the driving electric

field, with the carrier-envelope phase (CEP) of the driving laser tuned so that

only electron trajectories originating from a single ionization event produce

attosecond pulse generation. To date, the most efficient light switches are

based on the polarization gating technique.

This method is based on the fact that the generation of attosecond pulses

is really sensible to the driving laser field polarization, as explained by the

semiclassical recollision model [11]. Indeed the HHG efficiency strongly drops

when the driving electric field is elliptically or circularly polarized [30, 31].

The efficiency of attosecond pulse generation decreases by ≃ 50% when the

ellipticity is increased from 0 to only 0.1, and by about an order of magni-

17


tude for an ellipticity of 0.2. Therefore, a proper manipulation of the driving

laser polarization, producing linear polarization only in a subcycle temporal

window of the pulse, allows for the selection of only a single attosecond burst.

This is possible by using a system of two birefringent (quartz) plates, a first

thick retardant plate and a zero-order quarterwave plate [24, 32–34]. This

system ensures a reliable modulation of the laser ellipticity and a 100% trans-

mission of the incoming pulses.

The polarization gating setup is described in Fig. 1.1: the incoming pulse,

Figure 1.1: A schematic diagram of Polarization Gating (PG) method. The inco-

ming pulse, that is linearly polarized with an angle α = 45◦ with respect to the

neutral axis of the first quartz plate, is projected on the two axes of the plate, pro-

ducing a pair of cross-polarized twin pulses. Transmission through a second plate,

that is a zero-order quarterwave plate with the neutral axis at β = 0◦ (“narrow

gate”) with respect to the original laser polarization, changes the circular polariza-

tion into linear polarization, and conversely, producing a temporal linearly-polarized

gate close to the principal cycle of the driving field. Another configuration exists,

where α = 45◦ and β = 45◦, and it is dubbed “large gate”.

that is linearly polarized with an angle α = 45◦ with respect to the neutral

axis of the first quartz plate, is projected on the two axes of the plate, produc-

18


ing a pair of cross-polarized twin pulses. Since each pulse propagates inside

the plate with a different group velocity (due to the difference in the refrac-

tive index), a delay δ between the two is created, resulting in an alteration

of the original overall polarization. The overlap between the two projections,

indeed, produces circular polarization, whereas the tails are still linearly po-

larized. The delay depends on the thickness of the plate and the refractive

indexes at the carrier frequency of the pulses. Transmission trough the zero-

order quarterwave plate (with the neutral axis at β = 0◦ with respect to the

original laser polarization) changes the circular polarization in linear polariza-

tion, and conversely, producing a temporal linearly-polarized gate close to the

principal cycle of the driving field. This configuration, in which the neutral

axes of the two plates are oriented, respectively, with α = 45◦ and β = 0◦ re-

spect to the polarization of the incoming laser pulses, is called “narrow gate”.

Another configuration exists, where α = 45◦ and β = 45◦, and it is dubbed

“large gate”.

Concerning the narrow gate configuration, it is possible to find a relation be-

tween the duration τg of the polarization gate, the duration T of the driving

field and the delay δ accumulated by the two projections of the incoming

pulses inside the retardant plate. Let us consider a Gaussian driving pulse,

linearly polarized along z, with a temporal duration T (FWMH):

Ez(t) = EA(t)cos(w0t+ ψ)

EA(t) = E0e(−(t/σ)2)

σ = T/√

2ln(2)

(1.5)

After the first plate, with the ordinary and extraordinary axes along the x

and y directions, the driving pulse can be written as:

Ey(t) =√22 E

−A cos(w0t− π

2 + ϕ)

Ex(t) =√22 E

+A cos(w0t+ ϕ)

(1.6)

19


where:

E−A = EA(t− δ/2)

E+A = EA(t+ δ/2)

(1.7)

After the second plate (the zero-order quarter waveplate) with the ordinary

and the extraordinary axes along the z and w directions, the field can be

written as:

Ez(t) =12E

−A cos(w0t− π + φ) + 1

2E+Acos(w0t− π

2 + φ)

Ew(t) =12E

−A cos(w0t− π

2 + φ) + 12E

+Acos(w0t+ φ)

(1.8)

By using trigonometric identities and changing the reference frame from (z,w)

to (x, y), it is possible to obtain the following expression:

Ey(t) =12 [E

−A + E+

A ]sen(w0t+ φ− π4 )

Ex(t) =12 [E

−A − E+

A ]cos(w0t+ φ− π4 )

(1.9)

We can see that in the center of the pulse (i.e. around t = 0) the x component

of the field is zero, i.e. the field is almost linearly polarized along the y axis.

The ellipticity ǫ is defined as:

ǫ =E−

A − E+A

E−A + E+

A

=1− exp(− 2δt

σ2 )

1 + exp(− 2δtσ2 )

(1.10)

This parameter gives information about the polarization of the driving field: if

ǫ = 0 the pulse is linearly polarized, if ǫ = 1 the pulse is circularly polarized.

In order to investigate the polarization gate, we are interested only in the

temporal range where the field is almost linearly polarized, i.e. | −2δt/σ2 |<<1, which yields:

ǫ ≈ δt

σ2(1.11)

Therefore the temporal gate is:

τg =ǫthrT

2

ln(2)δ(1.12)

20


Usually the threshold ellipticity ǫthr is defined as the ellipticity value for which

the HHG efficiency is decreased to the 50% of the maximum value (obtained

for ǫ = 0). Considering that τg < T0/2 (where T0 is the optical cycle of

the driving field), it is possible to extract a good estimation also about the

duration of generating pulses. According with [34], T < 2.5 T0. Therefore

few-cycle (CEP-stabilized) driving pulses are required in order to confine the

HHG process in a temporal polarization gate and efficiently generate isolated

attosecond pulses.

1.1.2 Characterization techniques

The generation of isolated attosecond pulses requires methods for accurate

temporal characterization of attosecond XUV fields, a problem that has long

obstacled the development of attosecond science. Nevertheless, during the

past years, a number of proposals has been demonstrated [12,13,30,35–38] to

characterize the temporal structure of attosecond pulses [39, 40].

Today, femtosecond metrology tools allow for the full characterization of utral-

short laser fields in the visible or near-infrared range. To do this, it is necessary

to use at least one time-nonstationary filter [41], e.g. an amplitude gate, and

usually the pulse to be measured itself is turned into a time-nonstationary

filter, by means of a nonlinear effect. However, in principle nonlinear ef-

fects are absolutely not required for ultrashort pulse characterization. Be-

cause attosecond light pulses usually have available low intensities, the use

of nonlinear effects for their characterization is still challenging. Therefore,

generally attosecond measurement methods rely on a different approach. In

particular, attosecond XUV fields can efficiently ionize atoms by single-photon

absorption. This ionization generates an attosecond electron wavepacket in

the continuum, which is a replica of the attosecond field when it is far from

any resonance. The phase and amplitude of the XUV field are transferred

to the photoelectron wavepacket. A characterization of this wavepacket, that

is analogue to the characterization of ultrashort light pulses, gives a direct

information on the temporal structure of the XUV attosecond field. To this

21


end, near-infrared or visible laser fields constitute ideal time-nonstationary

filters, by acting as ultrafast phase modulators on the electron wavepacket.

The first step for characterizing the attosecond pulses is to write a relation

between the XUV radiation and the correspondent electron wavepacket. Let’s

consider the ionization of an atom by an attosecond field alone (in the sin-

gle active electron approximation). By applying the first-order perturbation

theory, at times large enough for the attosecond field to vanish, the transi-

tion amplitude av from the ground state to the final continuum state |v〉 withmomentum v is given by:

av = −i∫ +∞

−∞dtdvEXUV (t)exp[i(W + Ip)t] (1.13)

where W = v2/2 is the energy of the final continuum state (in atomic units).

EXUV is the XUV electric field, dv is the dipole matrix element for the tran-

sition from the ground state to the final state |v〉 and Ip is the ionization

potential of the atom. Eq. 1.13 directly connects the attosecond field and the

correspondent electron wavepacket generated after ionization.

The two spectra might differ in phase because of a possible phase dependence

of dv on v, which can be expected to occur, for example, near resonances in

the continuum. If this phase dependence is negligible (or known) the spectral

phase of the attosecond pulse can be directly deduced from the spectral phase

of the electron wavepacket. Under these conditions, the electron wavepacket

can be treated as a perfect replica of the attosecond field.

By considering also the presence of the VIS/NIR field, acting as an ultrafast

phase modulator, strong field approximation (SFA) [8] can be used in addi-

tion to the single active electron approximation. It consists of neglecting the

effect of the ionic potential on the electron motion after ionization. However,

when the photoionization is induced by an XUV field of central frequency wX ,

the requirement wX >> Ip already ensures that the ionic potential can be

neglected for continuum states. Within these approximations, at times large

enough for the attosecond field and the laser field to vanish, the transition

amplitude av(τ) (where τ is the delay between XUV and VIS/NIR fields) to

22


the final continuum state |v〉 with momentum v can be written as:

av(τ) = −i∫ +∞

−∞dtdp(t)EXUV (t− τ)exp[i(Ipt−

∫ +∞

t

dt′p2(t′)/2)] (1.14)

p(t) = v−A(t) is the instantaneous momentum of the free electron in the laser

field, A(t) being the vector potential of this field in the Coulomb gauge, such

that EL(t) = −∂A/∂t. The exponential in the integral of Eq.1.14 accounts

for the phase of the ionization/phase-modulation process, which is the sum

of the phase Ipt accumulated in the fundamental state until time t, and of

the phase subsequently accumulated in the continuum. Within SFA, this last

term is the Volkov phase, i.e. the integral of the instantaneous energy of

a free electron in the laser field, p2(t′)/2, from the ionization time t to the

observation time. By rewriting Eq. 1.14 as:

av(τ) = −i∫ +∞

−∞dte[iφ(t)]dp(t)EXUV (t− τ)e[i(W+Ip)t] (1.15)

φ(t) = −∫ +∞

t

dt′(v ·A(t′) +A2(t′)/2) (1.16)

it is evident that the main effect induced by the VIS/NIR field is a temporal

phase modulation φ(t) on the electron wavepacket generated in the contin-

uum by the XUV field. Because of the scalar product v · A in Eq. 1.16,

the phase modulation of photoelectrons needs to be defined for a given di-

rection [13, 30, 35]. There exist many experimental implementations of this

principle. One of these is the Attosecond Streak Camera method [35]. It is

a photoionization experiment in which an isolated attosecond pulse ionizes a

gas target (usually consisting of a rare gas jet). The photoelectrons are col-

lected by an electron time-of-flight spectrometer, that is able to measure the

kinetic energy accomulated by the charge particles. A VIS/NIR pulse, usu-

ally linearly polarized, is able to drive the photoelectrons in the continuum

after photoionization and thus to be used as phase modulator. In this case

the VIS/NIR pulse is also dubbed streaking pulse. In the Fig. 1.2, a typi-

cal streaking effect by the driving phase-modulator field on the photoelectron

23


Figure 1.2: A typical streaking spectrogram. The streaking pulse drives the pho-

electron spectrum (vertical axes), according with the value of the delay between the

attosecond pulse and the streaking pulse.

spectrum is reported.

In the case of a linearly polarized streaking field, and in the slow-varying

envelope approximation [42], the phase term of Eq. 1.16 can be written as:

φ(t) = φ1(t) + φ2(t) + φ3(t),

φ1(t) = −∫ ∞

t

dtUp(t),

φ2(t) = ([8WUp(t)]1/2/wL)cosθcoswlt,

φ3(t) = −(Up(t)/2wL)sin(2wLt)

(1.17)

Up(t) = E20(t)/4w

2L is the ponderomotive potential of the electron in the laser

field at time t (E0(t) is amplitude of the laser field). The observation angle θ

is defined as the angle between v and the laser polarization direction. φ1(t)

varies slowly in time, i.e. on the time-scale of the laser pulse envelope, while

24


φ2(t) and φ3(t) oscillate at the laser field frequency and its second harmonic,

respectively. We emphasize that around θ = 0, the amplitude ∆φ of the phase

modulation can be large even at moderate laser intensities, due to the W 1/2

factor in φ2(t).

It is important to observe that the streaking pulse has to be fast enough

to work as a proper non-stationary filter, therefore the maximum value of

| ∂φ/∂t | should be a significant fraction of that of the unknown attosecond

field, which will typically range from a few eV to several tens of eV. This can

be easily calculated from Eq. 1.17.

1.1.3 FROG-CRAB retrieval

Among the variety of methods for the retrieval of the temporal structure of

isolated attosecond pulses, such as Attosecond Spider [39] or phase retrieval

by omega oscillation filtering (PROOF) [43], Frequency Resolved Optical Ga-

ting for Complete Reconstruction of Attosecond Bursts (FROG-CRAB, or

simply CRAB) is a very well established technique and offers the advantage

of accurately recostructing arbitrary attosecond fields [44].

FROG-CRAB is inspired from frequency-resolved optical gating (FROG), a

widely used technique for the full characterization of visible pulses [45]. In a

FROG measurement the pulse to be characterized is decomposed in temporal

slices due to a temporal gate G(t), and then the spectrum of each slice is

recorded. This provides a two-dimensional set of data, called a spectrogram

or FROG trace, given by:

S(w, τ) =|∫ ∞

−∞dtG(t)E(t − τ)eiwt |2 (1.18)

where E(t) is the field of the unknown pulse and τ is the delay between the

gate and the pulse. The gate may either be a known function of the pulse, as

in most implementations of FROG, or an unrelated, and possibly unknown,

function (blind FROG) [46]. Various iterative algorithms can then be used

to extract both E(t) and G(t) from S(w, τ). The principle of FROG-CRAB

25

Attoseconds for molecular physics

can be derived by comparing the Eq. 1.15 and 1.16 with Eq. 1.18, in partic-

ular the spectrum S(w, τ) with the photoelectron spectrum | a(v, τ) |2. This

comparison shows that, by scanning the delay τ , the dressing laser field can

be used as a temporal phase gate G(t) = eiφ(t) [47] for FROG measurements

on photoelectron wave packets generated by attosecond fields.

The full characterization of the wave packets provides all the information on

the temporal structure of the XUV attosecond pulses as well as the unknown

gate fields, as for a blind-FROG measurement. Therefore the iterative Prin-

cipal Component Generalized Projections Algorithm (CPGPA) [46, 48, 49],

developed for the optical blind FROG, can be used to retrieve the amplitude

and phase of the pulses (see Appendix A).

1.2 Attoseconds for molecular physics

Isolated attosecond pulses hold great potential for time-resolved measure-

ments on unprecedented timescales. The first attosecond experiments on

atomic systems studied continuum electron dynamics and made use of the

principle of the Attosecond Streak Camera, described above. This method,

indeed, can be used not only for the characterization of attosecond pulses, but

also to investigate the photoionization processes occuring in the gas target.

These measurements are named Streaking Spectroscopy experiments, and are

particularly suitable to study photoionization time delays [16, 18].

Applying attosecond science to molecular physics, instead, the observables

change, moving towards charge localization, molecular dissociation, or charge

transfer processes [21, 52–54].

1.2.1 From H2 to multielectron diatomic molecules

For H2 (or D2) single ionization, one of the simplest molecular excitation

processes, the ionizing pulse usually removes the outermost electron of the

highest occupied molecular orbital (HOMO) from the neutral molecule and

launches a wave packet on the potential curves of the cation (many alternative

26


to direct ionization exist, such as the population of an autoionization state

of the neutral molecule). After ionization many processes can occur in the

molecule. For example, if the populated curve is dissociative (2pσ for hy-

drogen), the wavepacket proceeds monotonically outward, producing, at large

internuclear distance R, one charged and one neutral fragment. On the other

hand, if the potential energy curve (PEC) of the parent has a potential well

in the Franck-Condon (FC) region (1sσ), the wave packet will oscillate in the

well.

In the first attosecond pumpprobe experiment on the hydrogen molecule, an

isolated attosecond pulse was used to excite neutral H2 molecule, both ioniz-

ing and exciting the molecule [21]. Then a probe pulse, that is a co-polarized

few-cycle NIR pulse interacted with the excited molecule or molecular ion,

driving the localization of the one remaining bound electron in the H + H+

dissociative ionization channel that was monitored. The momentum-matched

neutral H atom and charged H+ moved in opposite directions. Depending

on the XUV-NIR time delay and the kinetic energy of the detected H+ frag-

ment ion, this fragment ion moved preferentially upward or downward along

the XUV/NIR polarization axis, resulting in a preferential localization of the

single remaining bound electron downward or upward.

In the case of multielectron molecules, even diatomic, the interpretation

of any excitation/ionization process in a pump-probe experiment becomes

very complex. First of all, expecially in the case of broadband pump pulses,

a huge amount of charge states of the parent molecular ion can be created

and, for each of these, wave packets are launched on many different PECs,

simultaneously and coherently. After that the molecule begins to stretch and

the wave-packet propagates along the PECs. If the distance in energy among

nuclear and electronic states are comparable, the overall molecular dynamics

is strongly affected by the coupling between electronic and nuclear degrees of

freedom, causing the breakdown of the Born-Hoppeneimer approximation [55]

(this is true also for H2 molecule). In this perspective any adiabatic picture

27


is not valid anymore, and procedures of diabatization on the electronic states

have to be considered in the theoretical calculations. The broadband excita-

tion, furthermore, can open many channels for the evolution of the molecular

dynamics, leading for example to the dissociation of the molecule, or the pop-

ulation of autoionizing states, or even to Coulomb explosion. On the other

hand the presence of the probe pulse gives more than a “real-time picture”

of the molecular dynamics, since it can actively alterate the propagation of

the wavepackets along the PECs and affect the overall dynamics itself. One

of the principal effects induced by the probe is the coupling between different

states with an energy gap that is favorable for one-to-few photon transitions,

and this can induce population transfers before fragmentation.

The action of the probe pulse on the potential curves can be easily defined and

investigated by introducing the Floquet picture [56], that in general describes

a quantum system interacting with a coherent electromagnetic radiation. By

defining the interacting Hamiltonian as:

H =∑

i

1

2mi[pi −

qicA(ri, t)]

2 + V (ri) (1.19)

where A is the vector potential of the electromagnetic radiation, the Floquet

theorem states that any solution ψ(r, t) of the Schrodinger equation written

using this Hamiltonian (that has the same time-periodicity of A) can be

expressed in the form:

ψ(r, t) = e(−iEt/~)φ(ri, t) (1.20)

where φ has the same periodicity of H and A. Therefore it can be written as

a Fourier expantion:

ψ(r, t) = e(−iEt/~)∞∑

n=−∞e(inwt)φn(ri) (1.21)

where w is the frequency of the electromagnetic radiation. The integer n refers

to the number of photons absorbed (or emitted) by the system. Coming back

to the molecular potential curves, the result of this periodic absorbtion process

28


is a periodic deflection of the PECs and the appearance of possible crossings,

as reported in Fig. 1.3. This can cause the coupling between different elec-

Figure 1.3: Dressed potential energy curves. (a) Unperturbated configuration. (b)

One-photon absortion induces a dressing process in the state 2, resulting in a crossing

with state 1. This can produce population transfer between state 1 and state 2.

tronic states (during the propagation of the wavepacket along the potential

curves) and consequent population transfers, that can ultimately affect the

dynamics of the parent molecular ion.

The broadband excitation/ionization by XUV radiation and the conseguent

interaction of the molecule (molecular ion) with the probe pulse can ultimately

lead to important information about real-time dynamics of the system after

the initial perturbation. The insight provided by this approach, in principle,

could even provide a real-time mapping of the quantum paths the excited

molecule (molecular ion) goes trough before fragmentation, giving an exper-

imental signature of the “real” potential energy curves of the system (see

chapter 3). In particular the possibility of directly accessing different chan-

nels of molecular dissociation allows one the control over the overall process of

fragmentation, that is a fundamental issue in light-matter interaction physics.

29


1.2.2 Charge motion in complex molecular systems

A chemical rearrangement occurs when atoms in a molecule change their spe-

cific positions. For this reason the time scale in chemistry is traditionally

considered as the time scale for the dynamics of atoms. Actually, many ex-

periments recently demonstrated that the breakdown of Born-Oppenheimer

approximation is really common in complex molecules. For example, it is

known that the photostability of DNA bases is based on photoprotection

mechanisms, involving ultrafast relaxation around conical intersections [57].

Also in this case, anyway, it is still the nuclear motion to drive the dynamics.

This type of electron dynamics mediated by nuclear rearrangement is usually

called charge transfer.

In 1995 and 1996 Weinkauf and coworkers performed some pioneering exper-

iments in which they induced the photoionization of a chromophore (trypto-

phan, phenylalanine or tyrosine) on the C-terminal end of a peptide chain [58,

59], and observed that the fragmentation pattern was dominated by species

related to the N-terminal end whenever this amino acid (glycine or leucine)

had a lower ionization potential than the chromophore. This observation sug-

gested the presence of a purely electronic charge transfer from the N-terminal

to the chromophore named charge migration. A new idea was rising, i.e. the

creation of an electronic wavepacket could bypass the rearrangement of the

nuclei and then be used to control chemical reactivity. Subsequent theoret-

ical works demonstrated that the prompt ionization of large molecules may

produce ultrafast charge migration along the molecular skeleton, which can

precede any nuclear rearrangement [60–63](Fig. 1.4). This purely electronic

dynamics, evolving on an attosecond or few-femtosecond temporal scale, can

determine the subsequent relaxation of the molecule. This behavior play a

crucial role in many biological and chemical processes, such as vision, photo-

syntesis and radation damage of biomolecules.

According to Cederbaum and Zobeley [62], this charge migration may rely on

the existence of a coherent superposition of hole states, leading to coherent

hole propagation within the molecular ion. In this case, the ionization of the

30


(a) (b)

Figure 1.4: (a) Migration of hole charge in glycine. At t = 0 the the charge is

located on the left-hand side of the molecular skeleton. The charge migrates to

the right-hand side of the molecule. At t = 3.5 fs the charge is mainly located on

the right-hand side of the molecule. Figure adapted from [61]. (b) Snapshots of

the densities of the hole as a function of time for a hole created on the HOMO of

TrpLeu3. The hole migrates to the N end in 0.75 fs and it returns to the origin after

1.5 fs. Figure adapted from [60].

molecule must involve a mechanism that coherently produces a superposition

of cationic states. In the independent-electron picture, the ultrafast removal

of an electron from an orbital creates a stationary hole that can be assigned

to a well-defined peak in the photoelectron spectrum (Koopmans’ theorem).

This is dubbed a one-hole (1h) configuration. When electron correlation con-

tributes to the molecular dynamics (for a molecule containing N electrons),

the wavefunction resulting from the removal of one electron by photoioniza-

tion (thus representing N-1 electrons) is not an eigenstate of the cation, and it

31


corresponds to a non-stationary state that can be described as a coherent su-

perposition of several cationic eigenstates. Several cases can be distinguished.

First, situations exist where only 1h configurations play a role in the dynam-

ics. This configuration is called “hole mixing” and can produce oscillations in

the population of “Koopmans-like” hole states [63]. Another situation occurs

when second-order (or even higher order) configurations become important

in the dynamics, for example when the hole formation by single-electron re-

moval is accompanied by the generation of a second hole by a shake-up process

(2h1p). The result is a photoelectron spectrum displaying a satellite peak,

in addition to the peak corresponding to the main 1h configuration. Also in

this case the coherent superposition of states leads to charge migration (that

is, oscillations in the population of the two holes in the 2h1p configuration).

Higher-order processes are also possible when many hole configurations play

a role in the dynamics. In this case, the molecular orbital picture breaks

down, and the main 1h character is completely redistributed over many possi-

ble configurations; consequently, there is no main 1h line in the photoelectron

spectrum. The coherent preparation of many cationic states again drives

the hole density to rearrange in an ultrafast timescale. In fact, the possible

existence of a “universal” timescale of about 50 as has been suggested [64],

corresponding to the contribution of an infinite number of hole states to the

ionization mechanism. Hole migration has been theoretically investigated for

several systems. Hennig and coworkers studied N-methylacetamide and found

ultrafast hole motion from one end of the linear molecule to the other within

a few femtoseconds [65]. A similar effect was found for DNA bases (glycine),

with different isomers showing different timescales for charge propagation [66].

Another important example was given by Remacle and Levine, who studied

hole migration in small peptides [60]. It is important to observe that the

occurrence of hole migration requires a number of conditions. In order to pre-

pare a non-stationary cationic state, the electron should be removed quickly

enough to avoid rearrangement of the electronic density during the ionization

process. This non-adiabatic or “sudden” ionization allows the (N-1)-electron

32


state to be written as a superposition of cationic eigenstates. This means that

the photoelectron should be ejected with a sufficiently high velocity (for in-

stance, by using a high-energy photon). Furthermore, in order to have charge

migration through the molecular structure, at least one of the eigenstates

formed in the photoionization process must be delocalized. In addition, the

timescale of hole migration must be short compared to that of diabatic vi-

bronic couplings, which lead to a loss of coherence. Finally, to be observable

in a time-resolved measurement, the dynamics must be triggered by a suffi-

ciently short coherent event.

For all these reasons attosecond pulses are excellent candidates in order to

solve this type of phenomena, since they could provide the necessary tempo-

ral resolution to track charge migration in real time.

33

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42

CHAPTER 2

EXPERIMENTAL SETUP

2.1 Attosecond beamline

The laser installed in the ELYCHE laboratory is the FEMTOPOWER PRO

V CEP, a commercial system developed by Femtolasers. It consists of a Ti:Sa

oscillator (Femtosource Raimbow) and a 2-stage Ti:Sapphire multi-millijoule

(mJ) amplifier. It provides IR pulses (at the wavelength of ≃ 800 nm) with

an energy of 6 mJ, 25 fs of duration and a repetition rate of 1 KHz. The laser

pulses are Carrier-Envelope Phase (CEP) stabilized [1], thanks to two differ-

ent stabilization stages. The first one (Menlo Systems XPS800), consisting

of a f0 − f interferometer, is installed just outside the oscillator cavity and

it’s able to phase lock the offset frequency at a 1/4 of the oscillator repetition

frequency. The second CEP-stabilization stadium (Menlo Systems APS800)

is a f−2f interferometer installed at the end of the laser system, and compen-

sates for the slow CEP drift introduced by the 2-stage amplifier. The 2-stage

CEP stabilization system provides a single-shot phase stability with an RMS

of ≃ 190 mrad. Few-femtosecond pulses were required by our experiments, so

43

Attosecond beamline

(a)

(b)

Figure 2.1: A schematic view of the attosecond-pump femtosecond-probe beamline.

In the interaction area several detection systems have been installed, alternatively:

Velocity Map Imaging spectrometer, TOF spectrometer, KEIRA mass spectrometer.

The beamline is split in the figure for a better view.

the hollow-core fiber compressor [2] was installed at the output of the laser

system and used to compress the pulse duration from 25 fs down to 4 fs. The

laser beam is focused into the hollow fiber made of silica by a 1750 mm fo-

44

Attosecond beamline

cal length mirror, as described in section 2.3.1. After the fiber another focal

mirror collimates the radiation into a chirped-mirror compressor made of 11

dielectric multilayer mirrors with a 450 nm - 970 nm bandwidth acceptance.

Fig. 2.1 shows a scketch of the experimental setup after the chirped-mirror

compressor.

An ultrabroad band beamplitter divides the pulses entering the pump-probe

interferometer. 70% of the incoming energy is used for the attosecond-pump

arm, 30% for the probe, in order to maximize the photon flux of attosecond

pulses. On the pump arm, the VIS/NIR pulses pass through the two plates

for the Polarization Gating (PG) [3, 4] (see section 1.1.1) and a pair of glass

wedges, that are used to finely tune the spectral dispersion of the pulses. A

50 cm focal length mirror focuses the beam into the vacuum chamber where

high harmonics generation occurs. Here the gas cell for HHG is mounted

on a XYZ motorized translational stage, that allows one to properly align

the cell with respect to the laser focus position. In order to select the short

trajectories in HHG, the cell was positioned few millimeters after the laser

focus position [5–7]. The XUV radiation enters the recombination chamber

where metallic foils are used to filter out the fundamental radiation and the

low-order harmonics, and then reaches the central hole of a drilled mirror,

where recombination between pump and probe pulses occurs.

On the probe arm, the VIS/NIR pulses pass through a pair of wedges (also

in this case they were used to optimize the pulse compression by controlling

the dispersion) and a coarse delay line mounted on a micrometric translation

stage. It was used to roughly approach the temporal overlap between the

pump and the probe. A 75 cm focal length mirror focuses the laser beam into

the recombination chamber. After the focus the beam inpinges the pump-

probe delay line mounted on a piezoelectric translation stage by Piezosys-

tems Jena. The resolution of the translation stage is 1 nm, corresponding, in

the temporal domain, to a delay resolution of ≃ 3 as. The VIS/NIR probe

pulses recombine with attosecond pump pulses on the drilled mirror and both

reach a 80cm focal length toroidal mirror. This is a grazing incidence optics,

45

Detection systems

coated of gold, that is able to reflect ≃ 85% of XUV incoming radiation (for

s-polarization), and all the VIS/NIR radiation, and focalize the beams into

the interaction chamber. Here several detection systems were used, accord-

ing with the experiments. An electron time-of-flight spectrometer was used

to characterize the attosecond pulses by streaking measurements and to per-

formed straking spectroscopy experiments (see section 2.2.1). We also used

a Velocity Map Imaging (VMI) spectrometer to investigate ultrafast electron

dynamics in diatomic molecules (see section 2.2.3) and a mass spectrome-

ter to study charge transfer processes in complex biological molecules [8](see

the KEIRAlite spectrometer described in section 2.2.2). After the interaction

area, a XUV spectrometer is installed to monitor the spectral properties of

attosecond pulses. The XUV spectrometer consists of a toroidal mirror and a

grating. The toroidal mirror has a 800 mm entrance arm and two exit arms,

700 mm (horizontal plane) and 1519 mm (vertical plane). The grating, in-

stead, that is positioned at 1050 mm after the toroidal mirror, has a 350 mm

entrance arm and a 469 mm exit arm (in the horizontal plane, no focaliza-

tion on the vertical plane). 469 mm after the grating, a MCP+phosphorus

screen+CCD camera detection system is able to record the XUV spectra

at different integration time values, down to (sampled) single-shot measure-

ments.

2.2 Detection systems

2.2.1 Electron time-of-flight spectrometer

An electron time-of-flight (TOF) spectrometer (Kaesdorf ETF 10) was in-

stalled in the interaction area for streaking experiments (see section 1.1.2),

to collect photoelectrons produced by the interaction between a gas target

and the attosecond pulses. The spectrometer is optimized for a distance of

3 mm between the ionization zone and the entrance cone, so we aligned the

TOF spectrometer in order to have the laser focus centered at this height with

respect to the cone. The gas source consists of a stainless steel nozzle with

46

Detection systems

(a)

(b)

Figure 2.2: (a) HHG pattern usually used to calibrate the TOF spectrometer, by

linking the peaks of photoelectrons (green points in (b)) with the correspondent

XUV harmonic order. (b) Calibration fit functions. The green points represent

the peaks of photoelectrons in the time-of-flight axes. Orange line: parabolic fit of

low-tof (high energy) peaks; blue line: parabolic fit of all photoelectron peaks; red

line: fit function from Eq. 2.2.

47

Detection systems

a internal diameter of 500 µm. The position of the nozzle can be controlled

by a XYZ translation stage mounted outside the vacuum chamber. After the

interaction between the gas target and the pump-probe pulses, the phoelec-

trons pass through the entrance aperture of the cone of the TOF and the

electron beam is collimated using a lens module. If the size of the ionization

zone is smaller than 200 µm the acceptance angle of the spectrometer is 45◦

(full angle). To lower the effect of the earth magnetic field which may pen-

etrate through the back into the spectrometer the electrons are accelerated

to 2000 eV before hitting the detector. The detector is a two-stage channel

plate (Photonis Chevron MCP) with a diameter of 40 mm and a pore size of

5 µm. The electronic signal coming out from the MCP detector is amplified

by a preamplifier stage and digitilized by a Time-to-Digital Converter (TDC)

card (Roentdek TDC8HP), with a time-of-flight resolution of 250 ps.

To calibrate the TOF spectrometer (i. e., to convert the time-of-flight axis

in an kinetic energy axis), a known pattern of sharp XUV harmonics (Fig.

2.2 (a)) was generated and used to photoionize the gas target atoms in the

TOF spectrometer. Then the peaks of photoelectrons were linked to the XUV

harmonic order, and the energy axes was obtained by applying the following

fit function:

Ek = A(t− t0)−2 +B(t− t0)

2 + C (2.1)

where t0 is the ionization time. In particular, in the approximation of a

constant electric field, the kinetic energy acquired by the electron after the

photoionization can be written as a function of the time of flight t as:

Ek =md2

2(t− t0)2+dEe

m+E2e2(t− t0)

2

8m(2.2)

Since the quadratic term in Eq. 2.2 usually dominates over the other contri-

butions, it is possible to calibrate the TOF-energy axes even by applying only

a parabolic fit function (orange and blue curves in Fig. 2.2 (b)). Nevertheless

this calibration is able to properly reproduce only a limited spectral range, as

evident in the picture.

The fine characteristics of TOF electron spectra are dependent not only on

48

Detection systems

the properties of the incident XUV radiation, but also on the parameters of

the TOF spetrometer itself, such as the position of the nozzle and the vol-

tages applied to the electron beam, by the drift tube, the lens module and the

MCP detector itself. Therefore a new calibration is needed every time these

parameters are alterated.

2.2.2 KEIRAlite mass spectrometer

Fig. 2.3 reports a schematic diagram of KEIRAlite [2]. The spectrometer

was constructed and developed in Queen’s University, and it was designed to

have an ion source in-built to one of the electrodes, removing the necessity

to couple an ion source to the spectrometer and the additional complexity

associated with this.

KEIRAlite can be operated in two distinct modes; either as a straight time

of light device or as an ion trap. It is composed of a number of cylindrically

symmetric electrodes which are supported on four ceramic rods. Five elec-

trodes at either end of the device, labelled in figure 3.5 as E1 to E5, create

the electrostatic mirroring regions when the appropriate voltages are applied,

and a pick-up assembly (P1, P2, and P3) in the centre is used for detection of

ions in trapping mode. The device measures 256 mm from sample electrode

E1 to the end electrode. The distance between E1 and E2, which form the

interaction region, is 5 mm. During the experiments the spectrometer was

operated in the time-of-flight mode. In this operating mode, voltages are ap-

plied only to the electrodes at one end, the sample end. Referring to Fig. 2.3,

the molecular source is embedded in electrode 1 of the sample end, which is

effectively a repeller electrode. This is known as the sample electrode, and

is normally held at a voltage of 5 kV. Electrodes 2 and 3, mirror high and

low respectively, are connected by a glass cylinder with a resistive coating

(Photonis) which maintains a uniform decrease in potential between the the

two ends. Applying a voltage of around 4.7 kV to electrode 2 and maintaining

electrode 3 at ground gives a smooth potential decrease. Given that electrode

3 is held at ground, electrode 4 acts as the central electrode of an einzel lens

49

Detection systems

Figure 2.3: A schematic diagram showing the electrodes, E1 - E5, which form the

electrostatic mirror regions, and P1 - P3, the pick-up assembly of KEIRAlite.

which is completed by electrode 5 which is also maintained at ground. The

lensing voltage on electrode 4 was adjusted according to other experimental

parameters, but the value was typically close to 0 V. With no other voltages

applied, the remainder of the device acts as a field free drift region. An ad-

ditional flight tube of 0.5 m in length was added to the vacuum chamber,

creating a longer field-free region and increasing the mass resolution of the

spectrometer. At the end of the flight tube a channeltron electron multiplier

(supplied by a voltage of 2.3 kV) works as detection system adn collects the

charge molecular fragments.

2.2.3 Velocity Map Imaging (VMI) Spectrometer

In the experiment on Nitrogen molecules (see chapter 3) we collected the yield

of N+ ions in a Velocity Map Imaging (VMI) spectrometer. In general, in an

experiment of time-resolved photoionization the information about the phys-

ical processes occuring in the target is contained both in the photoelectrons

and the positive charge particles created by the interaction with the laser

field. The simplest way to access this information is a time of flight (TOF)

measurement. In this type of experiment, the charged particles are acceler-

50

Detection systems

ated into a drift-tube by an electric field towards a charge detector, and the

time needed by the particles to reach the detector is recorded. One of the

main drawbacks of this technique is that no information about the angular

distribution of the charged particles can be obtained. Such an additional in-

formation can be measured by using a different kind of spectrometer with a

two-dimensional (2D) position sensitive detector. In this way both the final

position and time of arrival of the charged particles can be recorded, thus

leading to the possibility to reconstruct the initial momentum distribution of

the photofragments.

In 1987 some pioneering imaging experiments were performed by combining

a channel plate detector with a phosphorescent screen and a digital-image

processing device [12]. In 1997 Eppink and Parker proposed a new method

to measure the kinetic energy spatial distribution of charged particles called

velocity-map imaging [13]. In a VMI spectrometer a static extraction electric

field is used to accelerate ions or electrons onto a position-sensitive detector,

for example a MCP multiplier and a phosphor screen, then a CCD cam-

era is used to record images of the phosphors. The time required to reach

the detector and the final position on the detector itself provide an indirect

measurement of the three spatial components of initial velocity of the charge

particle. The velocity distribution can be retrieved by using a deconvolution

algorithm.

Let’s consider a reference xyz frame as the one introduced in Fig. 2.4. The

z-axis corresponds to the laser beam propagation axis. In order to recon-

struct, univocally, the all three dimensional distribution of the momentum

of a charge particle from a two-dimensional image, some symmetry assump-

tions have to be done. All the known retrieval algorithms are based on the

assumption that the starting velocity distribution has cylindrical symmetry,

having the symmetry-axis in a plane parallel to the plane of the detector.

This forces the light polarization axis to be the y-axis in our reference frame.

Lets consider a charged particle, with mass m and charge q, created (e.g. by

photoemission or photodissociation) in the interaction area. The particle is

51

Detection systems

Figure 2.4: Illustration of the coordinate system used to describe the 3D velocity

distribution in the interaction area and the 2D projection in the detection plane.

then accelerated by an extraction field along the TOF tube (oriented as x)

towards the position-sensitive detector. Particles sharing the same modulus

of the initial velocity ( i.e having a spherical distribution of their momenta),

are projected within a circle of maximum radius R on the detector plane. The

value of R coincides with the hit position of a particle with initial velocity

~v3D perpendicular to x and it is equal to R = v3D · t, where t is the particle

time of flight.

If a static potential Vs exists between the interaction region and the detector,

the charged particle create by photoionization will be accelerated acquiring

a final velocity (x-component)√

vx = 2qVs/m. If the acceleration region is

small compared to the total time-of-flight length L, t can be written as:

t ∼= L/vx = L

√

m

2qVs(2.3)

giving a maximum radius R of the 2D angular distribution image:

R ∼= L

√

T0qVs

(2.4)

52

Detection systems

where T0 = 12mv

23D is the initial kinetic energy of the particle. Particles with

the same charge, mass and x-component of the initial velocity but different

absolute value of v3D, will hit the detector at a different positions. Conversely,

particles with the same V3D,yz component but different v3D,x, will arrive onto

the detector at different times.

Fig. 2.5 reports a scheme of the VMI used in the experiment. The accelera-

tion zone is characterized by a curved-lines electric field, created by applying

a voltage to a couple of electrods, called Repeller (R) and Extractor (E),

with respect to the Ground (G) reference electrod. The interaction area is

located between the repeller and the extractor. The two electrods form an

electrostatic lens which focuses the particles from different origins, but with

the same velocity, to one spot on the detector. If VE is the voltage applied

to the extractor, and VR the repeller voltage, the focusing condition (for the

imaging of the energy-dependent particle distribution) is guaranteed by a fix

value of their ratio f = VE/VR. The precise value of f is determined by the

particular geometry adopted such to combine the mass-focusing (ions with the

same mass arrive together on the detector) with the velocity-focusing (ions

with the same initial velocity share the final impinging position) requirements.

Moreover, in this configuration, the maximum radius R is observed to scale

as:

R ∼= NL

√

T0qVs

(2.5)

where N is magnification factor. As the final position of electrons/ions on the

detector is almost exclusively determined by their initial velocity this tech-

nique was called “velocity map imaging”.

In a common gas-phase VMI experiment, where a supersonic expansion of

the target into the vacuum is used, the typical resolution is ∆E/E ≃ 1%,

with ≃ 103 counts/shot (depending on the size of the interaction region)

and a rather well defined initial velocity of the atoms/molecules. The main

drawback of this approach is the low density of the target. To overcome this

problem, Ghafur et. al. [14], proposed to integrate the gas injection system in

53

Detection systems

Figure 2.5: Schematic of the the VMI spectrometer. The electrostatic lens is made of

a repeller electrode (R), an extractor (E) and a ground plate (G). The electrods are

isolated from the housing by ceramic isolators. The laser beam propagates between

the repeller and extractor electrodes.

the repeller electrode. The gas flows through a laser drilled capillary 200−µmlong and with a typical diameter down to 50 µm. In order to increase the

target density in the interaction area, the conventional flat repeller electrode

54

Detection systems

is replaced by a conical one. The upper part of the repeller is composed by

a flat disk with a diameter of 1mm and the laser focus is aligned in front of

the capillary. In this way the gas density can be increased by 2-3 orders of

magnitude, still maintaining an energy resolution of ∆E/E ≃ 2%.

In order to further increase the gas density, a 1 KHz-pulsed valve (triggered

with the laser pulses) is used in our setup to introduce the gas in the spec-

trometer [15]. A series bimorph piezo (PXE5) is clamped between two metal

sheets placed on an housing in order to support the piezo and ensure that the

floating ground potential is equal to the repeller voltage. The repeller disk

with the 50− µm diameter nozzle is mounted as a cap on the valve housing.

A vespel pillar, ending with a viton plate (0.5-mm diameter) is glued on the

piezo cantilever in order to close the repeller nozzle. When a pulse of tension

is applied to the valve, the piezo bends away from the repeller disk and the

valve opens. The relative position between the repeller disk and the valve

housing is adjusted by a wave spring and a vespel adjustment ring. The pulse

of gas has a temporal duration of ≃ 20 µs. In this way, by using the pulse

valve, it is possible to keep the same average gas density in the chamber while

the local gas density in the interaction region is increased by 1 or 2 orders of

magnitude.

The detection system in the spectrometer consists of a MCP-assembly (MCP

+ phosphor screen) and a CCD camera. The MCP-assembly consists of a

stack of a 75-mm diameter dual MCP detector and a phosphor screen (F2226-

24P136, Hamamatsu) and it is designed in order to operate in two different

ways: DC-mode or gating-mode. In the DC-mode the MCP and phosphors

are supplied with a maximum voltages of 2 kV and 5 kV, respectively. This

mode is usually used to collect and solve the angular distribution of photo-

electrons momentum, or to record time-of-flight spectra of electrons/ions (by

measuring the current signal on the back of the phosphor screen). For several

applications, it is important to isolate particular fragments to be detected.

This cannot be done in the DC-mode since it requires to confine the acquisi-

tion in a precise and finite temporal window (centered at the time-of-flight of

55

VIS/NIR pulses

the fragment to be detected), i.e. to operate in a gating-mode. In this case,

the MPC-out and the phosphors are kept at 1 kV and 4 kV respectively. The

gate is realized by supplying the MCP-out with a supplemenary 1-kV pulse

of finite time width (typically around ≃ 150 − 200 ns) added over the 1 kV

pedestal. In order to reach the maximum gain supported by the detector, the

phosphors and the MCP are coupled by a capacitor which keeps the voltage

difference between the two fixed to a value of 3 kV. In this way, when the gate

pulse is applied to the MCP-out, the phosphor-voltage rises up to 5 kV. The

minimum gate duration that can be applied to the MCP-assembly is deter-

mined by the RC time of the electronic circuit and it is ∼ 150 ns (rising edge

∼ 40 ns).

The CCD camera device (Pike f145b, ALLIED) has a 1000× 1000 pixels de-

tector and works in a full read-out mode with a repetition rate of 15 frames

per second. The camera is controlled by a Labview program which stores the

experimental data.

2.3 VIS/NIR pulses

In order to produce isolated attosecond pulses by exploiting the polaritazion

gating technique, few-cycles ultraintense generating pulses are required. Fur-

thermore, in an attosecond-pump femtosecond-probe measurement, the du-

ration of the probe pulses can be crucial to determine the resolution of the

experiment. In particular the investigation of ultrafast electronic processes

in molecules, that are expected to occur in the temporal scale from hundreds

of attosecond up to few femtoseconds, may require ultrashort probe pulses in

order to resolve the dynamics under study.

For this reason we made a strong effort in order to produce ultrashort high-

energy VIS/NIR pulses. The result was ultrabroad VIS/NIR pulses (the spec-

trum is reported in Fig. 2.7) with a duration down to 4 fs and an energy of

about 2.7 mJ.

56

VIS/NIR pulses

2.3.1 Hollow-core fiber spectral broadening

High-energy 4-fs VIS/NIR pulses were obtained by using the hollow fiber tech-

nique, a well enstablished compression method that is particularly suitable for

high energy laser pulses [2, 9, 10]. This technique is based on pulse spectral

broadening by self-phase modulation in a hollow cylindrical fiber made of

fused silica and filled with nobles gases at high pressure. Wave propagation

along hollow fibers occurs by grazing incidence reflections at the dielectric

inner surface. Since the losses caused by these reflections greatly discrimi-

nate against higher order modes, only the fundamental mode, with large and

scalable size, will be transmitted through a sufficiently long fiber. For fused

silica hollow fibers the fundamental mode is the EH11 hybrid mode, whose

intensity profile, as a function of the radial coordinate, can be written as:

I(r) = I0J20 (2.405r/a) (2.6)

where I0 is the peak intensity, J0 is the zero-order Bessel function and a is the

capillary radius [11]. The possibility given by hollow fiber of selecting a single

guiding mode with a large diameter present an extremely high advantage for

the propagation of high energy pulses. Furthermore the control over the gas

pressure and the possibility of using different rare gases allow for a fine tuning

of the spectral properties of the guided pulses.

Figure 2.6: Hollow-core fiber in pressure gradient (PG) setup. The entrance of the

fiber (section A) is mantained in vacuum (rough vacuum of the order of 1 mbar).

We tested several configurations for the hollow-core fiber, in terms of lentgh

and core diameter, and the best results have been obtained with a 1 m long

57

VIS/NIR pulses

fiber with 320 µm core diameter. The IR pulses at the output of the laser

system were focalized into the capillar using a mirror with a focal length of

1750 mm, producing a focus diameter of 200 µm (≃ 65% of core diameter)

a the entrance of the fiber. Under these conditions we were able to obtain

almost 4 mJ of output energy with the capillar mantained in vacuum. The

spectral broadening was then obtained by filling the fiber with 1 bar of Helium.

The main side effect of inserting the gas into the capillar was a strong drop

in the throughput, down to 1.7 mJ, due to moltiphoton ionization of the

gas target and optical filamentation occuring at the entrance of the fiber.

While these effects are usually almost negligible for femtosecond pulses with

an energy of 1 or 2 mJ, the massive ionization of the gas target by multi-mJ

laser pulses strongly affects the focusing properties of the pulses and the fiber

throughput as a consequence. In order to avoid this drawback and optimize

the throughput efficiency, it is crucial to mantaining the entrance of the fiber

at a good level of vacuum, even in the precence of gas inside the fiber. The

solution to this problem is the pressure gradient configuration [12], reported

in Fig. 2.6. The entrance of the capillar (section A) is isolated from the rest of

the system and mantained in vacuum. The fiber is filled with the gas for the

spectral broadening; the gas flows through the fiber (section B), towards the

entrance, and the hollow core itself works as a differential pumping element.

Under these conditions, with the fiber filled with 1 bar of He, we measured

a pressure at the entrance of the fiber of 1 mbar (three orders of magnitude

in ∆p) and a maximum throughput of 2.7 mJ, with spectrum corresponding

to a bandwidth limited pulse duration of 3.9 fs, reported in Fig. 2.7. The

troughput achieved with the pressure gradient configuration is much higher

than the one obtained with the static cell setup, but still shows a significant

drop compared with the troughput in vacuum, most probably due to a residual

multi-photon ionization process occuring in the fiber.

58

VIS/NIR pulses

Figure 2.7: IR spectrum obtained after the spectral broadening performed by self-

phase modulation inside the hollow-core fiber. The sectrum width extends from 500

nm up to 950 nm, corresponding to a bandwidth limited pulse duration of 3.9 fs.

2.3.2 Temporal characterization

With the spectral broadening produced by the laser propagation along the

hollow-core fiber we were able to produce pulses with a fourier transfrom li-

mited duration close to 4fs. The syncronization of the spectral components of

the pulses was performed by an ultraboroadband (450 nm - 970 nm) chirped-

mirror compressor (Mosaic OS, Femtolasers).

The full temporal characterization of the VIS/NIR pulses was performed

by exploiting the WIZZLER device, developed and patented by Fastlite, and

based on the measurement technique named “Self-Referenced Spectral Inter-

ferometry” (SRSI) [13, 14]. The principle is simple: from the pulse to be

measured, a replica is created on the perpendicular polarization and delayed,

59

VIS/NIR pulses

Figure 2.8: Schematic design of SRSI Wizzler device.

using a birefringent plate cut at θ = 90◦ and slightly rotated compared to

the input polarization. The main pulse is used to generate a reference pulse

with a broader spectrum and a flatter spectral phase, but with the same car-

rier frequency, via Cross-Polarized Wave Generation (XPW) [15, 16]. XPW

generates a linearly polarized wave, orthogonal to the polarization of a high-

intensity linearly polarized input wave. Within the slowly-varying envelope,

the undepleted regime and the thin crystal approximations [17], The XPW

temporal amplitude is linked to the input temporal amplitude by the following

formula:

EXPW (t) ∝| EIN (t) |2 ·EIN (t) (2.7)

As can be seen, the XPW effect acts like a temporal filter: a XPW generated

pulse is a replica of the initial pulse, filtered by its own temporal intensity.

Thus, it is expected to be shorter in time, i.e. to have a broader spectrum and

a flatter spectral phase than the input pulse. This is true, according to [18],

when the pulse is close enough to the Fourier transform limit, i.e. when the

spectral phase is flat enough for the pulse to be filtered by the XPW effect.

It is worth noting that XPW generation is more sensitive to the chirp than

to higher orders of the spectral phase, and that its efficiency vanished for

pulses chirped to above 2 times their FTL pulse duration. In particular, a

60

VIS/NIR pulses

5 fs (FWHM) Fourier Transform limited (FTL) pulse can be measured if its

quadratic chirp is lower than about 15 fs2. Therefore, due to the strong de-

pendence of XPW efficiency on the pulse chirp, the comparison of the input

pulse spectrum with the reference pulse spectrum already provides an indica-

tion of the measurement validity.

Since reference pulse is created on the perpendicular polarization, a polarizer

Figure 2.9: Spectral amplitudes of XPW pulse (grey line) and pulse replica (black

line) and spectral phase of input pulse (blue line).

can transmit the XPW pulse and the replica to a spectrometer which records

the interference signal. Fourier-Transform Spectral Interferometry (FTSI)

treatment is then applied to this interferogram, and both spectral phase and

spectral amplitude of the input pulse can be extracted by an iterative algo-

rithm (See Appendix B).

We installed the WIZZLER device in our experimental setup, extracting a

portion of the beam by a beamsplitter after the chirped-mirror compressor.

Since the amount of chirp is crucial for the goodness of the SRSI measure-

ment, two wedges with a small aperture angle (2.7◦) were used to finely tune

the dispersion. Then portion of the beam was aligned in the WIZZLER device

61

VIS/NIR pulses

Figure 2.10: Retrieved temporal intensity profile of VIS/NIR pulses in linear scale

and on log scale (inset).

(Fig. 2.8) and the replica of the pulse was created in a 1.5 mm thick calcite

plate (replica generator). The dispersion was adjusted in order to maximize

the XPW generation in a LiF crystal (XPW crystal), resulting in the spec-

trum reported in Fig. 2.9 (grey line). Then an output polarizer selected the

replica and the XPW pulse and the interference spectrogram was recorded

in a broadband spectrometer. A known amount of additional dispersion is

introduced by the XPW crystal and it had to be considered in the retrieval

algorithm.

The black line in Fig. 2.9 represents the spectrum of the pulse replica, and

the blue line the spectral phase of the input pulse. A feedback loop between

the WIZZLER and the DAZZLER (an Acousto-Optic Programmable Disper-

sive Filter that works as an ultrafast pulse shaper inside the laser system)

was also installed, in order to finely tune the laser dispersion and optimize

the duration of the pulses. The amplitude and phase of the VIS/NIR pulses

were retrieved via a FTSI algorithm, and the best result is reported in Fig.

62

VIS/NIR pulses

2.10 4-fs (FWMH) pulses (3.9-fs Fourier transfom limited) were measured in

a temporal window of ±500 fs and with 40 db dynamics.

2.3.3 CEP stability

A characterization of crucial importance for our VIS/NIR pulses concerned

Carrier-Envelope Phase (CEP) stability. The phase stability ensured by the

laser system (single-shot rms of 190 mrad after the amplification stages, moni-

tored upon several hours) doesn’t garantuee a good level of phase stabilization

for an attosecond pump-probe experiment. Indeed, while the attosecond-

pump femtoseond-probe beamline is mantained in vacuum and allows for a

interaction-free propagation of the laser pulses, the hollow-core fiber compres-

sor may be origin of a CEP additional noise, due to the laser interacting with

the gas inside the fiber. However, several attosecond experiments carried out

with systems consisting of CEP-stabilized CPAs and hollow-core fiber com-

pressors (HFC) demonstrate that CEP stability is basically preserved during

compression. Furthermore, several direct measurements of the phase stability

of the pulses emerging from fiber compression have been carried out [18–20].

Yet these measurements detect the overall CEP drift accumulating throughout

the entire system including the HFC. For this reason we developed a simple

and robust method for measuring the nonlinear CEP changes emerging only

within the HFC [21]. With the insight provided by this measurement, one

can devise HFC compression schemes with improved inherent CEP stability.

If the demands on CEP stability over long periods of time are high [22], the

measurement might provide accurate feedback for active stabilization of the

CEP shift introduced by the HFC. Finally, the measurement provides valu-

able design guidelines for the energy upscaling of CEP-stable HFCs.

The easiest way of determining the phase shift experienced by an optical sig-

nal is via linear interference with a reference replica that propagates under

precisely known conditions. In principle, this could be applied to measure

the phase shift introduced by a HFC by splitting a small fraction of the pulse

energy at the fiber entrance, propagating it in a vacuum over a distance equal

63

VIS/NIR pulses

to the path length of the fiber and overlapping it with the main pulse at the

exit of the waveguide. The experimental implementation of this scheme would

be challenging, however, since timing jitter and spatial mode mismatch would

affect the accuracy of an interferometer with such long and complex arms.

We instead implemented this measurement in a collinear setup. In our ap-

proach, dubbed in situ single-shot interferometry (ISI), a small, copropagating

replica of each pulse that is coupled into the fiber is generated by means of a

thin birefringent plate, as for the temporal characterization described above.

The orientation of the plate in the plane perpendicular to the beam deter-

mines the amount of energy of the pulse replica, while the delay between main

pulse and replica is determined by the thickness of the plate and the optical

properties of the crystal. For a linearly polarized pulse, the replica will also

be linearly polarized, with an electric field vector oscillating perpendicularly

to the polarization direction of the main pulse. The main pulse experiences

self-phase modulation (SPM) and spectral broadening in the gas-filled hollow

fiber, whereas the replica will propagate linearly, provided that its intensity

is sufficiently low. This can be easily verified by measuring the spectrum in

Figure 2.11: Experimental setup. Before the HFC, the beam passes a calcite plate

oriented such as to produce a weak, orthogonally polarized post-pulse. Spectral

interference of the two is recorded after the HFC. Two in-line f-to-2f interferom-

eters are used to measure the CEP before and after the fiber, allowing separate

measurement of the phase changes.

64

VIS/NIR pulses

the polarization direction of the replica and comparing it to the spectrum

recorded at the entrance of the fiber. Since the Kerr-effect-induced refractive

index changes are highly adiabatic, they will have no effect on the replica

as long as it does not overlap in time with the main pulse. Phase shifts in

the replica caused by refractive index changes due to the generation of free

electrons through ionization by the main pulse can be excluded by using a

pre-pulse replica. As described before, ionization obstructs the throughput,

and the use of a post-pulse configuration therefore allows a comparison indi-

cating the magnitude of ionization caused by the main pulse. Linear spectral

interference between the main pulse and its replica is detected by means of

a polarizer and a spectrograph at the output of the HFC. Fourier analysis of

the single-shot spectral interferogram yields the phase difference between the

main pulse and the reference replica accumulated along the common beam

path.

The layout of the experimental setup is reported in Fig. 2.11. The IR pulses

coming from the laser passed a 1.5 mm thick calcite plate oriented such as to

produce a weak (roughly 60 µ J), orthogonally polarized replica. A temporal

separation of 900 fs between the two pulses was determined experimentally in

good agreement with the value expected from the difference in group delay be-

tween the optical axes. The birefringent plate was placed as far away from the

fiber entrance as its clear aperture allows and does not cause significant SPM.

The pulses were subsequently coupled into the hollow fiber, that was operated

both in the static pressure (SP) and pressure gradient (PG) regime [12], in

order to have a comparison of the CEP noise produced by the two configura-

tions. The addition of the calcite plate at the entrance of the fiber reduced

the transmission of the HFC by about 10%. After recollimation by a mirror

the pulses are sampled by a beam splitter and coupled into a spectrometer.

A polarizer placed before the device is adjusted for maximum contrast of the

resulting interference fringes. The spectrometer is triggered such as to record

interference patterns of single amplifier shots at an acquisition rate of 500

Hz. The remaining fiber output was compressed using the chirped-mirrors

65

VIS/NIR pulses

compressor. Furthermore, a supplementary in-line f-to-2f interferometer was

placed after the dispersive mirror compressor to independently measure the

CEP. While the f-2f interferometer before the HCF (APS800, IF1 in Fig. 2.11)

uses a 1 mm sapphire plate to broaden the spectrum to an optical octave, in

the second f-to-2f interferometer (IF2), the spectral bandwidth of the pulses

is sufficiently large to obviate the need for additional broadening. This de-

vice, in addition, uses an all-reflective (dispersion-free) beampath up to the

frequency-doubling stage. This proved essential in the measurement, as dis-

persive elements between the mirror compressor and the frequency doubling

crystal increase the time delay between the interfering spectral components.

The correspondingly reduced fringe period leads to unreliable CEP extraction

as its value approaches the spectrometer resolution. The beam path after the

fiber exit between IF2 and the ISI detection was covered for protection against

air currents. Both f-to-2f devices were triggered to measure single-shot inter-

ferograms, at a repetition rate of 158 Hz before and 335 Hz after the HFC.

As in the ISI technique, phase information in the conventional f-to-2f CEP

detection was extracted from the spectrograms by use of Fourier analysis.

In order to assess the noise floor of the ISI technique, we first performed a

measurement with the fiber under vacuum, resulting in a CEP noise contri-

bution from the HFC of 6 mrad rms over 105 shots (200 s) at the full input

energy of 6 mJ, most likely limited by the signal-to-noise ratio of the spec-

trometer. Applying a constant backing pressure of approximately 1 bar of

Helium in the pressure-gradient HFC setup, the output pulses had an energy

of 2.4 mJ and spectra corresponding to the bandwidth limited pulse dura-

tion of 3.9 fs, the same condition used for the temporal characterizion of the

VIS/NIR pulses. Under these conditions, the rms phase noise detected with

the ISI method increased to 40 mrad for the same measurement time. Moni-

toring the mean phase shift emerging in the HFC with the ISI method while

slightly varying the input pulse energy, we determined a coupling constant

of 1.19 rad/mJ, or 71 mrad/% (see Fig. 2.12). This is comparable to the

value of 128 mrad/% obtained for a different setup in [18]. In order to allow a

66

VIS/NIR pulses

Figure 2.12: Mean phase change and phase jitter emerging in the HFC. Red circles:

mean phase extracted by ISI from 105 consecutive spectra each. Red triangles: rms

CEP jitter extracted from the same dataset. Blue squares: rms CEP jitter calculated

from shot-to-shot energy fluctuation measured simultaneously. The leftmost data

point was omitted in the fit, its lower value probably due to decreased gas pressure

in the fiber.

consistency check, the pulse-to-pulse energy fluctuations were simultaneously

measured with a photodiode in front of the HFC. The phase jitter calculated

with the previously determined coupling constant agreed with the value indi-

cated by ISI to within 10% (see Fig. 2.12). The slight overestimation can be

explained by the low energy fluctuation of the amplifier of about 0.5% rms,

corresponding to voltage fluctuations close to the resolution of the sampling

oscilloscope. For changes in backing gas pressure, we determined a coupling

constant on the order of 100 mrad% relative pressure change was determined.

Due to the low precision of the pressure measurement available, this value

provides but a coarse estimate. However, considering the leakage rates and

67

VIS/NIR pulses

backing pressures usually encountered in HFC systems, we can safely assume

that pressure-related CEP changes are limited to slow, monotonous drifts. We

found that phase values retrieved from the ISI measurement are independent

of which pulse (main or probe) passes the fiber first, showing that long-life

(> 1 ps) plasma-related effects do not affect the measurement.

If the phase noise emerging in the HFC and detected with the ISI technique

was uncorrelated to the phase noise of the input pulses, the standard devia-

tions of the two noise contributions would add geometrically:

σtot =√

σ2amp + σ2

hfc (2.8)

Considering that the overall standard deviation of the CEP encountered in

our amplifier is typically about 190 mrad as measured by IF1, an additional

uncorrelated contribution of 30 − 50 mrad would result in negligible extra

noise. In the case of direct correlation, however, the standard deviations

would add up arithmetically.

In order to characterize the correlation of the CEP noise added in the HFC

process, we performed synchronous measurements of the CEP using both ISI

and the two conventional f-to-2f interferometers IF1 and IF2. We operated

the HFC in PG configuration at identical conditions as in the experiments

mentioned above. During the 5 min measurement, the f-to-2f interferometers

IF1 and IF2 detected CEP rms errors of 188 and 233 mrad before and after

the fiber, respectively. In the same time window, the CEP error recorded by

ISI amounted to 40 mrad rms. The difference of 45 mrad between the rms

phase fluctuations detected by IF2 and IF1 is very close to the value detected

with the ISI method, indicating that the phase noise emerging in the HFC

adds arithmetically and is strongly correlated to the phase noise of the input

pulses. In other words, the CEP noise measured before compression and the

additional jitter arising in the HFC likely share the same origin, which is

intensity fluctuation.

In order to compare the pressure-gradient HFC scheme with the conventional

SP HFC in terms of phase noise, the vacuum pump was turned off, and the

chamber was statically filled with helium. With the same input pulse energy of

68


6 mJ, the pressure was adjusted such that the output spectrum corresponded

to the same bandwidth-limited pulse duration as in the PG measurement.

Coupling instabilities due to strong nonlinearities in helium before the fiber

entrance resulted in a decreased output energy of 1.7 mJ, as expected, and in

increased phase noise. The ISI measurement indicated an accumulated phase

fluctuation of 66 mrad over a time interval of 4 min (versus 40 mrad in the

PG configuration).

2.4 Isolated attosecond pulses

The investigation of ultrafast electron dynamics in molecules requires a high

level of tunability in the spectral properties of attosecond-pump pulses. In-

deed the ability of selectively exciting the target in a certain energy regime

allows one to have a deep insight in the processes occuring in the molecule

and balances the intrinsic lack of spectral resolution of broadband attosecond

pulses. Therefore we made a strong effort in order to develope a robust system

that is able to produce isolated attosecond pulses in different spectral ranges.

Then we carefully characterized the attosecond pulses in the temporal domain

by photoionization streaking esperiments.

2.4.1 Tunability in XUV generation

The pulses used for the XUV generation have a duration of 4 fs, as described

before, and an energy of ≃ 1 mJ, after splitting in the pump-probe inter-

ferometer. Before entering in the HHG chamber, the beam passes through

the two quartz plates used for the polarization gating (PG) technique [3, 4].

The first one is a 192 µm-thick single-layer birefringent plate, with the prin-

cipal axis oriented at 45◦ with respect to the polarization of incoming pulses.

The second one, instead, is a quarterwave plate designed for a spectral range

between 500 and 1000 nm. After the PG plates, a pair of wedges (with an

aperture angle of 2.7◦) allow one for a fine control of dispersion, that is crucial

to access a good condition for polarization gating.

69


We investigated different configurations of focalization and interaction length

for HHG, with the aim of optimize the generation efficiency and the quality

of XUV radiation. We explored both the regime of loose and tight focus-

ing [23, 24], starting from a soft loose-focusing condition (1.5 m focal length)

and then moving to soft tight-focusing, down to a 40 cm focal length. Ac-

cording with the focalization, different lengths for the HHG gas cell were used

and tested, from 20 mm down to 3 mm long cells. The investigation was con-

ducted by producing isolated attosecond pulses (IAP) by HHG in Krypton.

The best results were obtained by focusing the generating pulses with a 50

cm focal length mirror into a 3 mm gas cell. Under these conditions we were

able to generate isolated attosecond pulses with an energy of more than 1 nJ,

in the spectral range between 17 and 40 eV (an Al foil was used to filter out

the low-order harmonics and the fundamental radiation), as reported in Fig.

2.13 (a).

We produced IAPs also in other spectral ranges, by perfoming HHG in Ar-

gon and Xenon. No changes were applied to focusing condition and gas cell

length, only the amount of gas was tuned in order to approach the saturation

regime. Fig. 2.13 (b) and (c) display the tipical XUV spectra obtained under

these conditions for the different targets. As evident in the figures, we were

able to tune the XUV cutoff over a range of about 20 eV, from a value 30 eV

in the case of Xenon up to 45− 50 eV in the case of Argon (we were able to

push the cutoff up to almost 60 eV). Due to the difference in the ionization

potential, the energy of XUV radiation spanned a range between 800 pJ (for

Argon) up to 1.5 nJ (for Xenon). To further improve the level of spectral

tunability, we installed different metallic foils in the attosecond beamline just

after XUV generation, in order to selectively filter the XUV spectra. We de-

veloped a highpass-lowpass setup, by inserting, alternatively, an Aluminum

filter or an Indium filter (the correspondent trasmission curves are plotted in

Fig. 2.14). The Al foil transmits the high energy region of the XUV spectra,

filtering out all the components below 15 eV. Indium, instead, has a strong

peak in transmittion at ≃ 15 eV and cuts higher energy components, apart

70


(a)

(b)

(c)

Figure 2.13: XUV spectra for the generation of isolated attosecond pulses, in the

case of HHG in Krypton (a), Argon (b), Xenon (c).

71


Figure 2.14: Trasmission curves of Al (blue line) and In (red line) 100-nm thick foils.

The green spectrum was generated by HHG in Xenon with the In low-pass filter.

from a little window centered at 22 eV.

The tunability in XUV generation is a crucial achievement in the perspective

of attosecond experiments on a molecular target. As it will be described be-

low (see chapters 3 and 4), the possibility of tuning the spectral range of

attosecond-pump pulses allows one to understand the energy states of the

molecule (or molecular ion, or even dication) involved in the ultrafast pro-

cesses observed in the measurement and have a deep insight in the physical

phenomena under study.

2.4.2 Temporal characterization

The generation of continuum XUV spectra usually is not enough to draw con-

clusions about the production of isolated attosecond pulses, for this reason it is

necessary to carefully characterize the temporal structure of the XUV pulses.

In this approach, before performing a complete temporal characterization of

the attosecond bursts, it is useful to check the quality of the polarization ga-

72


ting applied to the VIS/NIR pulses. To do this, a pair of wedges installed

after the PG plates and mounted on a motorized stage was used to remotely

scan the XUV spectrum dependence over the CEP of the VIS/NIR genera-

ting pulses [25]. A typical result is shown in Fig. 2.15 a strong periodicity of

Figure 2.15: CEP scan of XUV spectrum generated in the Polarization Gating (PG)

setup. A strong periodicity of continuum-to-modulated XUV spectrum is evident,

with a period of π in the CEP. For a CEP value equal to k π2

the gate selects an

isolated attosecond burst, resulting in a continuum spectrum; For a CEP equal to

kπ, two attosecond pulses enter in the gate and the interference between the two

produces a modulated spectrum.

continuum-to-modulated XUV spectrum is evident, with a period of π in the

CEP. This is compatible with a proper operation of polarization gating: for a

CEP equal to k π2 the gate selects only a single attosecond burst, resulting in

73


a continuum spectrum; instead, for a CEP equal to kπ, two attosecond pulses

enter in the gate and the interference between the two produces a modulated

spectrum. This CEP scan analysis allows for a feasible guess about the num-

ber of pulses produced in the high harmonics generation, but can’t be used

as a characterization of attosecond pulses.

In order to precisely measure the temporal duration of isolated attosecond

pulses photoionization streaking experiments are required [27, 29]. Therefore

we installed an electron time-of-flight spectrometer (see section 2.2.1) and per-

formed streaking measurements by using the 4-fs VIS/NIR pulses as phase

modulator.

The pressure of the gas target inside the TOF spectrometer was chosen in

a way to maximize the signal/noise ratio, but without inducing any spatial

charge effects, that could introduce strong correlation between photoelectrons

and affect the physical information extracted from the measurements. After

installing a pressure gauge close to MCP detector, we set the gas flow in order

to read a pressure of ≃ 1 · 10−6 mbar (over a background of ≃ 1 · 10−8 mbar),

that is a value compatible with the vacuum requirements of the detector and

allows for the generation of photoelectron TOF spectra that are perfect replica

of XUV spectra, without alterations coming from spatial charge effects.

We characterized all the XUV spectra presented in Fig. 2.13. We used also

different gases on target to produce photoelectrons. Fig. 2.16 shows a strea-

king spectrogram obtained by generating isolated attosecond pulses in Kryp-

ton and using Argon on target. The phase modulation of the photoelectron

spectrum is evident, apart from a kinetic energy region close to 11 eV (red

dashed line in Fig. 2.16), corresponding to an excitation energy close to 26.75

eV; the region displays a flat response to the phase modulation. This is due

to the excitation of an autoionization channel in Argon [28], for this reason

the wavepacket created by XUV photoionization at this energy accumulates

a large phase with respect to the other energetical contributions (due to the

lifetime of the autoionization state), resulting in a flat response in the delay

window reported in the picture.

74


Figure 2.16: Streaking trace obtained by generating isolated attosecond pulses in

Krypton and using Argon on target. The red dashed line corresponds to an autoion-

ization channel of Argon.

For this reason we characterized the duration of attosecond pulses by using

Nitrogen, that does not displays resonances in this energy region. Fig. 2.17

(a) shows a straking trace obtained by using Nitrogen on target.

We implemented the FROG-CRAB method, in combination with the CPGPA

algorithm (see section 1.1.3), in a MATLAB-based software to retrieve the am-

plitude and the phase of the attosecond pulses. From the trace reported in

Fig. 2.17 we retrieved a pulse duration (in intensity profile) of ≃ 290 as with a

phase depicted in Fig 2.18. This result was obtained by running the algorithm

over 10000 iteations, with a error function value of 1 · 10−2, evaluated as the

root mean square error per element of the trace.

75


(a)

(b)

Figure 2.17: (a) Experimental streaking trace obtained by generating attosecond

pulses in krypton and using Nitrogen on target. (b) FROG-CRAB retrieved strea-

king trace.

The asymmetry in the trace of Fig. 2.17, i.e. the difference in signal between

the rising edge and the decay slope in every cycle of the streaking field is

due to attochirp, resulting in the non-flat retrieved phase reported in Fig.

76


Figure 2.18: Intensity profile (blue line) and phase (red dashed line) of isolated

attosecond pulse retrieved from the trace of Fig. 2.17 (a).

2.18. The HHG process intrinsically induces a chirp (positive chirp for short

trajectories [24]) on the emitted attosecond pulse, as high-energy photons are

emitted slightly after those with lower energy. This causes the group delay

to linearly increase with increasing photon energy. It is known [30–32] that

metallic filters are able to compensate the positive chirp by introducing a neg-

ative group delay dispersion (GDD). In order to compensate the attochirp,

we installed a 250 µm-thick Al filter. The resulting streaking trace and the

correspondent retrieval of the attosecond pulse are reported in Fig. 2.19. We

retrieved a temporal duration of 260 as, as reported in Fig. 2.20. It is impor-

tant to underline that the streaking trace itself gives a double check about the

duration of the VIS/NIR pulses. The phase gate, indeed, is mapped by the

attosecond pulses as a function of pump-probe delay, giving a nice image of

the vector potential A(t) of the few-femtoseconds probe pulses. To precisely

extract the shape of the field, we calculated the center of gravity of the trace,

that represents a good estimation of A(t). After applying a finite-difference

derivative to the vector potential, we obtained the temporal evolution of the

77


Figure 2.19: Streaking trace obtained by compensating the positive attochirp with

a 250 µm-thick Al filter.

Figure 2.20: Intensity profile (blue line) and phase (red dashed line) of isolated

attosecond pulse retrieved from the trace of Fig. 2.19.

78


electric field of the VIS/NIR pulses, resulting in an intensity profile of 4 fs,

that is in a pretty good agreement with the measurement performed via SRSI

technique.

79

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85

CHAPTER 3

QUANTUM INTERFERENCE IN N2 MOLECULES

DISSOCIATION

In 2002 the real-time observation of the femtosecond Auger decay in krypton

was the first application of attosecond science to atomic physics [1]. This

demonstration was followed by other important experimental results, such as

the measurement of temporal delays of the order of a few tens of attoseconds

in the photoemission of electrons from different atomic orbitals of neon [2] and

argon [3]. Attosecond techniques have been applied in the field of ultrafast

solid state physics, with the measurement of delays in electron photoemission

from crystalline solids [4] and the investigation of the ultrafast field-induced

insulator-to-conductor state transition in a dielectric [5].

In the last few years, attosecond pulses have also been used to measure ul-

trafast electronic processes in simple molecules [6]. Sub-femtosecond electron

localization after attosecond excitation has been observed in H2 and D2 mo-

lecules [7], and control of photo-ionization of D2 and O2 molecules has been

achieved by using attosecond pulse trains (APTs) [8, 9]. More recently an

APT, in combination with two near-infrared fields, was employed to coher-

87

Experiment

ently excite and control the outcome of a simple chemical reaction in a D2

molecule [10]. The direct access to the ultrafast relaxation dynamics of the

excited states in multielectron diatomic molecules is experimentally difficult

to achieve and the theoretical analysis of the physical problem is particularly

challenging for a multi electronic system. On the other hand, the investigation

of ultrafast dynamics in these systems, such as N2, O2 or CO2, is of crucial

importance, for example, for the full understanding of the radiative-transfer

mechanisms activated in the Earth’s atmosphere.

Molecular nitrogen, in particular, is the most abundant species in the atmo-

sphere. The extreme ultraviolet (XUV) spectral region of the solar radiation

is mostly attenuated by the presence of N2 in the upper atmosphere. Absorp-

tion of XUV radiation inevitably leads to ionization and dissociation of the

molecule via non-adiabatic relaxation of highly-excited electronic states. In

this context, understanding the ultrafast dissociative mechanisms leading to

the production of N atoms is of crucial importance to completely disclose the

atmospheric radiative-transfer processes.

3.1 Experiment

In order to study ultrafast electron dynamics in diatomic multielectron mo-

lecules we performed XUV-VIS/NIR pump-probe experiments on Nitrogen

(N2) molecules in gas phase. The epxeriment was performed in the ELYCHE

laboratory - Physics Department (Politecnico di Milano). The group of Prof.

Fernando Martin (Universitad Autonoma de Madrid) performed the theoret-

ical simulations in order to give an interpretation of the experimental data.

The measurement consisted in the photoionization of N2 molecules by iso-

lated attosecond pulses and the subsequent interaction with an ultrashort

(4-fs) VIS/NIR probe pulse. We collected the N+ ions, produced by molec-

ular fragmentation, in a Velocity Map Imaging (VMI) spectrometer and we

investigated the temporal evolution of N+ ions yield as a function of the delay

between pump and probe pulses. The goal of the measurements was to ex-

88

Experiment

tract information about ultrafast predissociation processes by analyzing the

dynamics of N+ ions producted by molecular dissociation itself, and demon-

strate the possibility of a controlling the overall dynamics, in particular by

linking different quantum dissociation paths.

3.1.1 N+2 states and pump-probe pulses

Figure 3.1: Σu,g N+2 PECs and low-energy N2

2+ PECs. The ionization energy for

the cation and dication are reported.

In Fig. 3.1 some electronic states of the parent ion (N+2 ) and dication (N2+

2 )

of nitrogen are reported, with the correspondent ionization potential (only Σ

symmetry states of N+2 are presented in the picture, since the contribution

from Πg,u is negligible in this energy range, as reported in [11]). As evident

in the figure, the low energy states of the N+2 ion (C2Σ+

u and F2Σ+g ), apart

89

Experiment

(a)

(b)

Figure 3.2: XUV spectra used in the experiment. The first spectrum (a) was pro-

duced by high harmonic generation in Argon; the most of the signal is centered

between 35 and 40 eV, with a cutoff extending up to 55 eV. The second spectrum

(b) was obtained by high harmonic generation in Xenon; it is peaked at about 25

eV, with a cutoff at around 40 eV.

90

Experiment

from the ground state X2Σ+g , require a vertical excitation in the energy range

between 25 eV and 30 eV. Instead, in order to access highly excited states of

the N+2 ion, an energy up to 45 eV is necessary. Even more energy is required

to excite the low energy states of the dication.

Therefore, for the experiment, we produced isolated attosecond pulses (pump

pulses) by exploring principally two spectral regions. The first spectrum (Fig.

3.2 (a)) was produced by high harmonic generation in Argon; the most of the

signal is centered at 40 eV, with a cutoff extending up to 55 eV. This spectrum

was used to ionize the nitrogen molecule and populate a large amount of highly

excited states of N+2 , also producing a small population of N2+

2 . The second

spectrum (Fig. 3.2 (b)), instead, was obtained by high harmonic generation

in Xenon; it is peaked at about 25 eV, with a cutoff at around 40 eV and

it was used to ionize the N2 molecule and populate the low energy states of

the cation. In both cases the presence of the Al foil filtered out low energy

components.

4-fs VIS/NIR CEP-stable pulses, with a peak intensity of 8 ·1012 W/cm2 were

used as probe pulses.

3.1.2 Retrieval of 3D momentum distribution

The raw experimental data acquired by the VMI spectrometer are 2D velo-

city map images. By using a retrieval algorithm it is possible to reconstruct

the original three-dimensional momentum distribution of the charged parti-

cles created in the photoionization/fragmentation processes.

The retrieval method is based on the Abel inversion of the two-dimensional

images. Indeed, if the initial 3D momentum distribution has cylindrical (or

spherical) symmetry, with the axis in the plane of the detector, the 3D velo-

city distribution can be retrieved from the 2D-projected images by means of

an inverse Abel transform.

In a commonly used inversion technique each line perpendicular to the sym-

metry axis in the image is treated as an independent 1D Able transform (as

in Fig. 3.3) [16]. Due to the symmetry, one slice Si(x, z) of the initial distri-

91

Experiment

Figure 3.3: 2D projection of a 3D angular distribution as a superposition of 1D-

slicing Abel transformations.

bution can be written as a mono-dimensional function fi(√x2 + z2) = fi(ρ).

The Abel transformation of Si(x, z) is represented by a 1D function Gi(R)

and the two 1D functions are related by:

Gi(R) =∫∞R

fi(ρ)rho√ρ2−R2

dρ

fi(ρ) =1π

∫∞ρ

dGi(R)/dR√R2−ρ2

dR(3.1)

Therefore, by applying an inverse Abel transformation to Gi(R) it is possible

to retrieve the corresponding fi(ρ). The cut through of the 3D momentum

distribution is then obtained by the collection of the fi(ρ) functions. Un-

fortunately, the singularity and the derivative in the integrands of Eq. 3.1

make the numerical inversion nontrivial. Different method exist to perform

this inversion procedure, such as the onion peeling method or the iterative

inversion technique [17,18]. Anyway these methods suffer for being time con-

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Experiment

siming and introducing an appreciable level or noise. Nevertheless, an inver-

sion method exists that is computationally faster and intrinsically connected

to the underlying physical process. It exploits the fact that 3D momentum

distributions of charged particles measured in a VMI spectrometer are the

result of photon-matter interaction. Indeed, it is known that in this case

the angular distribution consists of a superposition of Legendre polynomials,

where the highest-order polynomial involved is determined by the order of the

multi-photon ionization process [19, 20]. The 3D momentum distribution to

be retrieved, P3D, can be written as:

P3D(v3D, θ3D, φ3D) =∑

l

al(v3D)pl(cos(θ3D)) (3.2)

where θ3D and φ3D are the spherical coordinates of a reference system oriented

as in Fig. 2.4, v3D is the initial velocity of the charged particle and pl is the

l-th order Legendre polynomial. By applying the normalization condition∫

VP3DdV = 1, together with Eq. 3.2, it is possible to write P3D as:

P3D(v3D, θ3D) = 2πv23D∑

l

al(v3D)pl(cos(θ3D))sin(θ3D)) (3.3)

or equally as:

P3D(v3D, cos(θ3D)) = 2πv23D∑

l

al(v3D)pl(cos(θ3D)) (3.4)

The point irradiated on the detector defines a projected v2D that is directly

connected with the yz components of v3D:

v3D,y = v2D · cos(θ2D)

v3D,z = v2D · sin(θ2D)(3.5)

where θ2D is the angle with respect to the y-axis and takes a value in the

interval (0, 2π). The 2D-projection of the three-dimensional distribution can

be written as:

P2D(v2D, θ2D) =∑

l

bl(v2D)pl(cos(θ2D)) (3.6)

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Experiment

with:

P2D(v2D, θ2D |π0 ) = P2D(v2D, θ2D |2ππ ) (3.7)

due to the cylindrical symmetry. The keypoint of the retrieval method is to

build a connection between the two representations {al(v3D)} and {bl(v2D)}.For a fixed energy, a matrix M exists for which the al and bl coefficients can

be written as:

b = M · a (3.8)

The matrix M is absolutely general, it doesn’t depend on the particular pro-

jected velocity map, but only on the number of velocity points and the max-

imum order of polynomial adopted. Once M and its inverse M−1 are calcu-

lated, the array a, that completely defines P3D, is given by:

a = M−1 · b (3.9)

The M matrix elements are calculated in the following way. By defining θ′

the angle between v3D and the x -axis, v2D can be written as:

v2D = v3Dsin(θ′) (3.10)

The incremental flux detected on an area around the point (v2D, θ2D) is thus

given by:

δS2D = P2D(v2D, θ2D)δv2Dv2Dδθ2D (3.11)

The correspondent emitted flux is:

δS3D = P3D(v3D, θ3D, φ3D)v3Dδθ′v2Dδθ2D (3.12)

putting δS2D = δS3D:

P2D(v2D, θ2D) = v3Dδθ′

δv2DP3D(v3D, θ3D, φ3D) (3.13)

which defines the matrix M. Once M−1 has been determined, the inversion

of experimentally obtained images can be readily performed by fitting the

experimental angular distributions for each value of v2D to a superposition

94

Results

of Legendre polynomials, in the form of Eq. 3.6. The retrieval of the 3D

momentum distribution then immediately follows by applying Eq. 3.9.

All the observables to be studied in the experiment were extracted from the

3D momentum distribution, by integrating over the radial or angular dimen-

sion. For example, the kinetic energy distribution P (E) can be derived by

considering:

∫

P (E)dE = m

∫

P (v3D = (2E/m)1/2)v3Ddv3D = 1 =

=

∫

P3D(v3D, θ3D, φ3D)v23Dd(cosθ3D)dφ3Ddv3D

(3.14)

Therefore, by exploiting the orthogonality of the Legendre polynomials:

P (E) = Ka0(v3D) (3.15)

where K = 4πv3Dm . The angular distribution of charged particles created by

light-matter interaction, is often described in terms of β-parameters [ref]:

P (E, cosθ) = σ(E)[1 +∑

n≥2

βn(E)pn(cosθ)] (3.16)

where σ(E) is the photoionization cross section. The β-parameters can be

immediately obtained according to:

βn =an(v3D)

a0(v3D)(3.17)

3.2 Results

3.2.1 N+ KER spectrum as a function of time delay

Fig. 3.4 (a) shows the 2D angular distribution of N+ ions collected in the

VMI spectrometer, produced by the interaction with attosecond pulses only,

in both the cases described by Fig. 3.2, respectively. By applying the retrieval

method described above, we calculated the correspondent three dimensional

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Results

(a)

(b)

Figure 3.4: Angular momentum distributions. (a) 2D images of angular distribution

of N+ ions produced in the photoionization process by XUV spectra reported in

Fig. 3.2 (a) and (b) respectively. (b) A cut of the correspondent retrieved three-

dimensional momentum distributions.

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Results

momentum distributions. Fig. 3.4 (b) reports, for each case, a cut of the 3D

distribution. The total N+ yield as a function of the kinetic energy release

(KER) is then obtained by angular integration of the retrieved momentum

distribution. The KER of the N+ ion accumulated along a certain dissociation

pathway is defined as:

KER =EB − ED

2(3.18)

where EB is the vertical transition energy for the initial N+2 state (in the

Figure 3.5: N+ kinetic energy release (interaction only with XUV radiation) ob-

tained by integrating the retrieved 3D momentum distribution within 0◦ ± 20◦

around the laser polarization axis, in the case of high cutoff XUV (HHG in Ar-

gon, blue line) and in the case of low cutoff XUV (HHG in Xenon, red line).

case of single ionization) and ED the dissociation energy. Fig 3.5 presents

the N+ KER spectra in the case of attosecond pulses generated in Argon and

Xenon. As evident in the picture a strong signal in the ion yield emerges at

about 1 eV. This channel, also named F-band, is known from literature and

it corresponds to the F 2Σ+g direct dissociation to the third dissociation limit

L3: N+(3P )+ N(2D) [11, 21]. The F 2Σ+g direct dissociation is a particularly

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Results

favorable channel also through vibronic couplings from highly excited states

of the N+2 ion, for this reason the F-band is much more pronounced in the

case of high XUV cutoff excitation.

Even more interesting, a peak at a kinetic energy release of ≃ 2.5 eV is evident

in the case of high cutoff XUV (blue line), while is almost negligible is the case

of low energy cutoff. Again, this means that this KER region is originated by

the dissociation of highly excited states of the cation, or even from Coulomb

explosion of low-energy states of the dication.

After analyzing the kinetic energy spectra produced by XUV radiation only,

we performed pump-probe scans, by varying the delay between the isolated

attosecond pulses and the VIS/NIR pulses. The result, again specialized

to the two different XUV spectra, is reported in Fig. 3.6, were the kinetic

energy spectra, for each delay, were integrated over an angle of 0◦ ± 20◦ with

respect to the laser polarization (parallel transition to Σu,g states) [21, 22].

The presence of the VIS/NIR pulse alters significantly the KER spectrum

as can be seen in the picture. Concentrating on Fig. 3.6 (a), a clock for

the zero time delay between the XUV pulse and the VIS/NIR pulse can be

identified through the sudden increase of the KER up to 4 eV due to the

two-color excitation process, moreover a clear depletion of the F-band with a

correspondent appearance of a band extending up to 2.5 eV can be observed

at 8 ± 1 fs after zero time delay. In addition, a long dynamics is evident in

picture (a), with a temporal evolution (not shown here) of the order of 100

fs and starting at kinetic energies in the range between 2.5 and 4 eV. This

process is known from literature [23], and comes from the excitation of an

aoutoionization state of the molecular cation. Since the vertical transition for

this channel stays above the double-ionization potential, only the XUV with

the high cutoff is able to access it.

By zooming over the temporal interval 5 − 16 fs in the first pump-probe

map, (decreasing the delay step from 150 as down to 75 as), we were able to

observe the presence of a sub-cycle modulation of the ion yield, as shown in

Fig. 3.7. To investigate this modulation we extracted single energy profiles

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Results

(a)

(b)

Figure 3.6: Time-dependent N+ kinetic energy spectra as a function of the delay

between the XUV pump pulse and the VIS/NIR probe pulse, in the case of high

cutoff XUV (a) and in the case of low cutoff XUV (b).

99

Results

Figure 3.7: (a) Time-dependent N+ kinetic energy spectra acquired within the

pump-probe delay interval 5 − 16 fs. (b) N+ yield integrated in a 0.3-eV-wide

energy band around 0.8 eV (black curve), 1.6 eV (blue curve), 1.9 eV (red curve)

and 2.2 eV (green curve). An arbitrary offset has been added to the curves for better

visualization.

from the map and performed a Fourier analysis. The result of the analysis in

the frequency domain reveals a periodicity of 1± 0.07 fs.

It is important to observe that this fast modulation occurs outside the time-

overlapping region between the XUV and the VIS/NIR pulses, furthermore

the pattern of interference displays a clear tilt in the fringes as a function of

the time delay, that means a phase-energy relation. A sub-cycle modulation of

the N+ ion yield is a clear signature of quantum interference between different

dissociative paths. The fact that the fringes, in the pump-probe map, are

delayed with respect to the zero delay suggests that the interference between

the multiple quantum paths occurs far away from the Franck-Condon region.

This point is of crucial importance because allows to access information about

the propagation of the coherent wavepackets along the PECs of N+2 ion before

dissociation, i.e. a possible experimental insight in the shape and properties

100

Results

of the energy curves. The tilt itself, displayed by the fringes pattern, results

from the propagation of the different components of the wavepackets with a

different velocity along the PEC slope.

3.2.2 Theoretical model

In order to investigate the role of the manifold of electronic excited states

involved in the process observed in the experiment, a theoretical model has

been developed.

The diabatic excited N+2 states were computed ab initio in MOLCAS [24].

At the state of the art, explicitly modeling the ionization dynamics due to

the XUV-pump pulse prior to the VIS/NIR pulse is not possible, therefore

an approximation based on Dyson orbitals was used. Then the effect of the

IR-probe pulse impinging on the ionized molecule at various time delays was

investigated, by time-evolving the initial state in the diabatic states and the

kinetic energy reease distribution of the fragments were obtained.

In the calculations nine electrons were considered in an active space comprised

of the p and 2s atomic orbitals (yielding 308 Configuration State Functions

(CSFs)) for symmetries Ag and Au in the Abelian group D2h (maximum

symmetry that can be used by the available codes). As it proved difficult to

reliably obtain the couplings and dipoles for N+2 in their diabatic representa-

tions from adiabatic calculations [25], a new methodology was used to obtain

the diabatic data directly without having to diabatize previously computed

adiabatic data.

The orbitals were optimized for the first 13 molecular states of each symme-

try. This was found sufficient to describe the evolution of the wave function

in the range of energies made available by the XUV pulse. In order to create

the states inherently diabatic, their CI-vectors were forced to be of the form

ci = (..., 0, 1, 0, ...). Thus, assuming the variation of the CSFs over the dis-

tance of the spacing of grid points to be negligible, the resulting states are

quasi diabatic, only the relaxation of the orbitals with the internuclear dis-

tance was neglected. It is important to note, however, that the diabatic PECs

101

Results

Figure 3.8: The N+2 potential energy curves obtained by the simulation code, cor-

responding to the symmetry 2Ag in their diabatic (a) and adiabatic (b) representa-

tions.

intersect and are therefore not ordered by energy across the grid. Thus, in

order to be sure that the evolution is described adequately across the grid, we

cannot simply propagate in the first 13 states of each symmetry, as these no

longer correspond to the 13 states with minimum energy. Rather it is required

to consider all states of the from ci. This leaves us with the quite large set of

308 states (one for every CSF) per symmetry to propagate in.

Fig. 3.8 shows a subset of the PECs of the relevant symmetries, in their adia-

batic and diabatic representations, respectively. The ionization was simulated

at the minimum of the ground electronic state of the neutral N2 assuming a

square-shaped XUV spectrum, with a bandwidth between 20 and 40 eV. First,

the Dyson orbitals were calculated between the ground electronic state of the

neutral and several electronic states of the cation of symmetry Σg and Σu up

to 40 eV. These Dyson orbitals were constructed by integrating the neutral

and cationic electronic wave function over all the common electrons. Further-

more, in agreement with the experimental configuration (parallel transition),

only the dipole moment in the internuclear axis direction was applied over the

Dyson orbitals.

102

Results

Using the calculated PECs and the estimation of the ionization given by the

Dyson orbitals, the time evolution of the system was studied. To do this, the

time-dependent Schrodinger Equation was solved on a grid using operator

splitting. In the temporal domain, in order to incorporate the effect of the

VIS/NIR pulse on the propagation of the wavepackets created by the XUV

ionization, the attosecond pulse was considered as instantaneous.

3.2.3 Quantum interference along the N+2 PECs

The result of the full calculation is reported in Fig. 3.9, in a delay interval

between approximately 5 fs and 16 fs. The main features observed in the

experiment are clearly present. In particular, we can observe a depletion of

the F-band around 0.9 eV, and this is accompanied by a subcycle oscillatory

modulation of the signal with the same phase-energy relation present in the

experimental data. For better visualizing the sub-cycle dynamics, Fig. 3.10

reports the oscillatory pattern obtained by subtracting, by means of a poly-

nomial fitting curve, the slow envelope dynamics, in the case of experiment

and calculations respectively. The really good agreement is confirmed also

by performing a Fourier analysis on the theoretical calculations. The Fourier

peak, in this case is centered at 0.89 PHz, corresponding to a periodicity of

1.12 fs, to be compared to 0.93 PHz obtained from experimental data (see

Fig. 3.10).

An important observation, both from experiment and theory, comes from

the fact that a clear periodicity in the modulation is visible even if a large

amount of states are populated by the XUV bandwidth. This suggests that

only really few states dominate to the interference process, otherwise many

contributions would completely blur the pattern. This is expected also due to

the presence of the VIS/NIR pulse, that selects and couples only the states

that are favorable to a one-to-few photon transition. In this approach we

build a simplified model, by including in the simulations only four states:

12Σu, F2Σg, 3

2Σg and 42Σu. This small subset of states is able to repro-

duce the main features observed in the experiment and allowed us to better

103

Results

Figure 3.9: (a) Time-dependent N+ kinetic energy spectra calculated within the

pump-probe delay interval 5 − 16 fs. (b) N+ yield integrated in a 0.3-eV-wide

energy band around 0.8 eV (black curve), 1.6 eV (blue curve), 1.9 eV (red curve)

and 2.2 eV (green curve). An arbitrary offset has been added to the curves for better

visualization.

indentify the physical process. The simulation with the four state model is

presented in Fig. 3.11 (a) and it suggests the following interpretation (re-

ported in Fig. 3.11 (b)): the strong depletion of the F-band observed about

8 fs after the zero delay is due to bond softening via two different channels.

The VIS/NIR pulse is able to transfer population from F 2Σg to 42Σu and

from F 2Σg to 12Σu. The measured delay of 8 fs represents the time required

by the wave packet to reach the crossing point between the dressed states (in

the Floquet picture). The fringes, on the other hand, are the result of an

off-resonant two-photon transition from the F 2Σg state to the 32Σg state via

the 42Σu state, which interferes with the initial population of the 32Σg state.

The tilt in the fringes, as introduced before, is due to the dispersion of the

initial wave packet induced by propagation along the PECs: the faster com-

ponents reach the interference region (see Fig. 3.11) first, thus being affected

104

Results

Figure 3.10: Oscillaory pattern obtained by subtracting, by means of a polynomial

fitting curve, the slow envelope dynamics, in the case of experiment (a) and calcula-

tions (b) respectively, around 0.8 eV (black curve), 1.6 eV (blue curve), 1.9 eV (red

curve) and 2.2 eV (green curve). (c) Fourier spectrum of the experiment (red line)

and the simulation (black line), around 2.2 eV.

by the VIS/NIR probe pulse for shorter time delays compared to the slower

components. The dispersion is a function of the slope of the PEC along

which the wavepacket propagates, in particular it contains information about

the relative splope between the two dissociation pathways, especially close to

the Frank-Condon region, where the incline is larger. This hypothesis can

be verified by maintaining the F 2Σg state fixed and artificially altering the

gradient of the 32Σg. Fig. 3.12 shows a set of artificially constructed poten-

tials for the 32Σg together with the corresponding interference patterns. It

is evident in the picture that larger slope leads to an increase in the tilt due

to larger dispersion of the wave packet as well as an overall reduction in the

contrast of the fringes as the interference window becomes narrower; smaller

slope leads to the opposite trend. Taken to their extreme, smaller as well as

larger gradients result in a destruction of the fringes pattern. This allows to

105

Results

(a)

(b)

Figure 3.11: (a) Time-dependent N+ kinetic energy spectra calculated with the

four-state model, within the pump-probe delay interval 5− 16 fs. (b) Interpretation

of the ultrafast pre-dissociation process by means of the four-state model.

106

Results

Figure 3.12: Artificially constructed potentials for the 32Σg together with the cor-

responding interference patterns. The alteration of the gradient in the potential

affects the tilt in the fringes pattern.

exctract information from the experiment about the shape of the molecular

PECs populated by the attosecond pulses, giving a unprecedented insight in

the ultrafast relaxation of the molecule after XUV photoionization.

107

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112

CHAPTER 4

CHARGE MIGRATION IN THE AMINO ACID

PHENYLALANINE

The investigation of ultrafast electronic motions in complex molecular sys-

tems is one of the most attractive challenges of attosecond science [1]. The

possibility of accessing purely electronic dynamics inside a molecule of biolog-

ical interest can disclose the origin of a huge amount of biological processes,

such photosynthesis, catalysis, DNA damage due to ionizing radiation and

respiration.

4.1 Experiment

In order to investigate charge migration in biologically relevant molecules we

performed XUV-pump IR-probe experiments on the amino acid Phenylalanine

and we studied the molecular fragmentation as a function of pump-probe de-

lay. The experiments were performed in the ELYCHE laboratory - Physics

Department (Politecnico di Milano), in collaboration with the group of Prof.

Jason Greenwood (Queen’s University of Belfast) who provided the molecular

113

Experiment

target and the KEIRA mass spectrometer [2], and with the group of Prof.

Fernando Martin (Universitad Autonoma de Madrid) who performed the the-

oretical simulations in order to explain and support the experimental results.

The idea of the measurement is to photoionize the amino acid with isolated

attosecond pulses; this prompt ionization promotes the molecule to a super-

position of cation (and eventually dication) states and, after a certain time,

leads to the fragmentation of the amino acid. The presence of a second pulse

(the probe) can alterate the fragmentation process as a function of the delay

respect with the attosecond pulses. The mass spectrum of the molecular frag-

ments is recorded in a mass spectrometer for every dealy step, and the goal

of the experiment is to access some pre-fragmentation dynamics by analyzing

the delay-dependent mass spectra.

In particular, in the very first femtoseconds after photoionization purely elec-

tronic dynamics are expected to be visible, hopefully giving a direct signature

of charge migration.

4.1.1 Molecular target

Phenylalanine (2-Amino-3-Phenylpropanoic acid) is an essential α-amino acid

with the formula C6H5CH2CH(NH2)COOH (Fig. 4.2). The α-amino acids

consist of a central carbon atom (α carbon) linked to an amine (-NH2) group,

a carboxylic group (-COOH), a hydrogen atom and a side chain (R), which

in the case of phenylalanine is a benzyl group (Fig. 4.1). In the isomeric

form of L-Phenylalanine (Fig. 4.1 (a)) it is used in nature for the biochemical

formation of proteins, while the stereoisomer D-Phenylalanine (Fig. 4.1 (b))

doesn’t partecipate to the proteic biosynthesis, although it is found in proteins

in small amounts.

For the experiments, 99% DL-Phenylalanine (a mixture of D-Phenylalanine

and L-Phenylalanine) was acquired from Sigma-Aldrich and used without fur-

ther purification. It was rubbed directly onto the surface of a 301 stainless

steel foil of 10 − µm thickness and 12−mm diameter and then clamped into

the repeller electrode of the KEIRAlite mass spectrometer (see below). The

114

Experiment

(a) (b)

Figure 4.1: (a) L-isomer of Phenylalanine. In nature it is involved in the biochemical

formation of proteins. (b) D-isomer of Phenylalanine. It doesn’t partecipate to the

proteic biosynthesys. Nevertheless it is present in proteins in small amounts.

Figure 4.2: Molecular structure of the most abundant conformer of the aromatic

amino-acid Phenylalanine. Dark gray spheres represent carbon atoms, light gray

spheres hydrogen atoms, blue sphere nitrogen and red spheres oxygen.

115

Experiment

distance from the foil, and hence the sample, to the focal point of the ionizing

laser pulses was approximately 3 mm. To evaporate the sample, the reverse

side of the foil was irradiated with a CW diode laser operating at a wavelength

of 960 nm, with a spot diameter of 6 mm and power in the range 0.3− 0.4 W.

It is well known that amino acids exist in many conformations, as a result

of their structural flexibility. Typically, the energy barrier to interconversion

between different conformers is small, of the order of a few kcal/mol, so that,

even at room temperature, thermal energy is sufficient to induce conforma-

tional changes. Theoretical investigations have shown that such changes can

affect the charge migration process [3]. In the case of phenylalanine, 37 con-

formers have been found by ab initio calculations [4], with a conformational

distribution that depends on temperature.

Since the experiments were performed without any conformer selection, it was

crucial to estimate the temperature of the molecules at the interaction area.

The temperature was estimated by assuming equilibrium heat conduction

along the foil to the repeller electrode (which acts as a room temperature

heat sink) and that 30% of the incident radiation was absorbed by the foil at

this wavelength [5]. Radiative heat losses were negligible for the calculated

temperatures. For the average laser power used, the temperature at the cen-

tre of the foil was estimated to be 430 K and about 410 K at a radius of 2

mm. As the sample depleted, to increase the evaporation at greater distances

from the center, the diode laser power was increased to maintain a constant

target gas density.

At this level of temperature, only the six most stable conformers of pheny-

lalanine are significantly present [4], with the most abundant configuration

shown in Fig. 4.2. All these six conformers were considered in the theoretical

calculations (see below in the Results).

4.1.2 HHG spectra and VIS/NIR pulses

The experimental setup was prepared in order to have some reliability in the

XUV spectral range, with the aim of selecting different excitation regimes in

116

Experiment

(a)

(b)

Figure 4.3: XUV spectra generated during the experiment, by using an Al filter (a)

and an In filter (b).

117

Results

the molecule as explained in the section 2.4.1. XUV radiation was generated

by HHG in Xenon. We choosed this gas in order to maximize the amount of

XUV signal. Fig. 4.3 shows the two different XUV spectra generated during

the experiment. The first one is obtained by inserting the Al filter. As it is

evident, the cutoff extends up to 30 eV, with the maximum signal centered

at about 25 eV. The second one, instead, is obtained by inserting an Indium

foil in the XUV beam-path. The new XUV spectrum was characterized by

a 3-eV (FWHM) peak centred around 14 − 15 eV, followed by a broad and

weak spectral component extending up to 25 eV.

CEP-stabilized 4-fs VIS/NIR pulses were used as probe pulses, with a peak

intensity of about 5 · 1012 W/cm2.

4.2 Results

4.2.1 Molecular fragments

Fig. 4.4 (a) ilustrates the mass spectra of Phenylalanine fragments generated

by the interaction with XUV radiation only. The most important fragments

observed in the mass spectra can be assigned as follows:

• m/q = 165: it is the parent ion (M), indicating the Phenylalanine

molecule singly ionized;

• m/q = 120: Immonium ion, that is Phenylalanine with a loss of the

carboxyl group (M - COOH), arising from charge on the amine group;

• m/q = 91 or 92: Breakage of the Cα - Cβ bond, leaving the side chain

(R), with the possible addition of a hydrogen from the NH2 group (R

+ H), with the charge on the phenyl ring;

• m/q = 74: It is the parent molecule with loss of the side chain, (M - R),

with the charge residing on the amine group.

All these fragments (together with less preminant peaks at m/q = 103,77,65)

are singly ionized. At m/q = 60, instead, it is possible to identify a small

118

Results

(a)

(b)

Figure 4.4: Molecular fragments of Phenylalanine collected by mass spectrometer.

(a) Fragmentation produced by the interaction with XUV-pump radiation only. In

the inset a zoom on the yield of immonium dication (m/q = 60) is reported. (b)

Molecular fragments generated by VIS/NIR-probe pulses.

119

Results

signal correspondent to the doubly charged immonium ion (M-COOH)2+ (

inset in Fig. 4.4 (a)). Concerning VIS/NIR pulses, the intensity (in the order

of 5 · 1012 W/cm2) was set in order to generate only few fragments as in Fig.

4.4 (b). Parent ion yield is predominant, togheter with peaks at m/q = 120,

74. Only a small contribution from the chain R (m/q = 91,92) is visible, while

no signal from immonium dication is present in the spectrum.

4.2.2 XUV-pump VIS/NIR-probe scans

After collecting and analyzing the mas spectra of XUV only and VIS/NIR

only, we performed XUV-pump VIS/NIR-probe scans. For each delay step,

the signal for every peak in the spectra has been integrated and normalized

at the yield of peak m/q = 74 (the most predominant signal in the spectrum),

in order to avoid any affection due to intensity flactuations, and plotted as

a function of the delay between XUV and VIS/NIR. (Negative delays corre-

spond to VIS/NIR coming first).

The fractional yields of nearly all the singly charged fragments vary as a func-

tion of pump-probe delay. However, there is a marked difference when the

XUV beam is filtered with the aluminium foil (Fig. 4.3 (a)), compared to the

Indium foil (Fig. 4.3 (b)). This is demonstrated in Fig. 4.5 for the fragment

m/q = 28 (NH2C). It can be seen that in the case of XUV pulses transmitted

by the aluminium filter (see Fig. 4.5 (a)), the yield of the fragment m/q =

28 increases with pump-probe delay. The fit (red dashed lines in the Fig. 4.5

(a)) used to analyzed this process is of the form:

y = a+ b · (1− e−t/τ0) (4.1)

where a is a constant equal to the initial value of the ion yield of the fragment,

averaged over the negative time delay values, b is the ion yield as τ tends to

infinity, t the time delay between the two pulses and τ0 the time constant

which gives the temporal scale of the ion yield increasing. The result of the

fit gives a time constant τ0 = 80 ± 2 fs. Other fragments also increase with

time constants in the range from 50 fs to 100 fs at the expense of the parent

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Results

(m/q = 165) and immonium ions (m/q = 120) which decrease. By contrast,

in the case of XUV pulses transmitted by the Indium filter (see Fig. 4.5 (b)),

a sudden increase in the yield of the fragment m/q = 28 can be observed with

no subsequent dynamics.

This result suggests that when only a valence electron can be ionized (XUV

pulse filtered with Indium) we see no dynamics in the singly charged ions, but

the presence of hole in the valence shell allows the VIS/IR pulse to be ab-

sorbed. For instance, it is known that absorption by the phenyl chromophore

shifts from the ultraviolet to the green when there is a hole in the highest

occupied molecular orbital [7]. This increases the yield of smaller fragments

when the XUV pulse precedes the VIS/IR pulse with a concomitant reduction

in parent and larger fragment ions. When an inner valence orbital is ionized

(by the XUV pulse filtered with aluminium), there is initially no resonant

absorption of the probe but an internal conversion to a lower electronic state

generates a hole in the valence shell opening up absorption of the probe and

hence further fragmentation. The measured timescale of 50 − 100 fs is com-

patible with an internal conversion mechanism as it is mediated by nuclear

motion on potential energy surfaces.

If we instead concentrate our attention on the temporal evolution of the im-

monium dication yield (Fig. 4.6 (a)), a strong dynamics appears and evolves

on much faster temporal scales compared with the charge transfer process

desribed just above. The figure shows the result on a time scale of ≃ 100

fs, with a delay step of 3 fs. The dynamics has been fitted by the convolu-

tion F (t) of a Gaussian pulse of 4-fs full-width at half maximum (FWHM),

corresponding to probe pulse duration, with the following function R(t):

R(t) = A(e−t/τ1 − e−t/τ2) (4.2)

with τ1 and τ2 describing the rise time and exponential decay time, respec-

tively. The experimental data give τ1 = 10± 2 fs and τ2 = 25± 2 fs.

Since the overall dynamics evolves in such untrafast temporal scale, we were

interested in further investigating this temporal evolution, therefore we de-

creased the delay step down to 500 as. The result is surprising: clear oscilla-

121

Results

(a)

(b)

Figure 4.5: Normalized yield of m/q = 28, as a function of pump-probe delay for

an XUV pump pulse transmitted by an aluminium foil (a) and an Indium foil (b).

In (a) the data has been fitted with the convolution of a Gaussian function with

the function y = a + b(1 − exp(−t/τ0)). In (b) the data has been fitted with the

convolution of a Gaussian function with the step function.

tions of the dication yield appear along the slope of the dynamics envelope,

with an evident periodicity for delay between 10 fs and 40 fs (Fig. 4.6 (b)).

122

Results

Figure 4.6: . (a) Yield of doubly charged immonium ion (mass/charge = 60) as

a function of pump-probe delay, measured with 3-fs temporal steps, in the delay

interval from −30 fs to 80 fs. (b) Decay slope of the dynamics occuring on the

fragment m/q = 60, shown in the inset (delay step of 3 fs). The red line in the

inset is a fitting curve with an exponential rise time of 10 fs and an exponential

relaxation time of 25 fs. The decay slope is sampled with 0.5-fs temporal steps,

within the temporal window 10 − 35 fs. Error bars show the standard error of the

results of four measurements. The red line is the fitting curve given by the sum of

the fitting curve shown in the inset and a sinusoidal function of frequency 0.234 PHz

(4.3-fs period). (c) Difference between the experimental data and the exponential

fitting curve. Red curve is a sinusoidal function of frequency 0.234 PHz.

For larger delays a gradual dephasing of the oscillation was observed. In or-

der to analyze the temporal evolution of this periodic modulation we have

first subtracted the fitting curve F (t) from the experimental data acquired in

the case of 0.5-fs delay-steps and then fitted the resulting curve (in the delay

123

Results

between 10 fs and 40 fs) by using the following function:

S(t) = A1sin(2πν1t+ φ1) (4.3)

The curve has been calculated by using the fitting tool of Matlab R2011a

with a confidence level for the bounds of 95%. The result is shown in Fig.

4.6 (c). The calculated frequency is 0.234 PHz (corresponding to an oscilla-

tion period of 4.3 fs), with lower and upper confidence bounds of 0.229 PHz

and 0.238 PHz, respectively. The total deviation is 0.0069 PHz and the root-

mean-square deviation is 0.0119 PHz. The amplitude and phase of the fitting

curve are A1 = 0.022 and φ1 = 1.75 rad, respectively.

To better understand the nature and the temporal evolution of the oscilla-

tions a time-dependent Fourier analysis has been performed, with the aim

of extracting frequency and time information on the same plot. The sliding

window Fourier transforms have been calculated by using a Gaussian window

function:

g(t− td) = e−(t−td)2/t20 (4.4)

with t0 = 10 fs and peak at td (gate delay time). The result is shown in Fig.

4.7 (a). At short pump-probe delays two frequency components are present,

around 0.14 PHz and 0.3 PHz. A strong and broad peak around 0.24 PHz

forms in about 15 fs and vanishes after about 35 fs, with a spectral width

which slightly increases upon increasing the pump-probe delay, in agreement

with the frequency values obtained from best fitting of the data reported in

Fig. 4.6 (b). For this reason we came back to fit analysis in order to overall

investigate the oscillations from negative delays up to 40 fs after the zero-

delay. The results are reported in Fig. 4.8. At short pump-probe delays (t

< 10 fs), as described above, Fourier analysis shows the presence of two main

frequency components, therefore the experimental data have been fitted by

the sum of two sinusoidal functions:

S(t) = A1sin(2πν1t+ φ1) +A2sin(2πν2t+ φ2) (4.5)

The calculated frequencies are: 0.14 PHz (lower and upper confidence bounds:

0.12 PHz and 0.158 PHz, respectively) and 0.293 PHz (lower and upper con-

124

Results

Figure 4.7: 2D spectrogram calculated for the measured data of Fig. 4.6. The

sliding window Fourier transforms have been calculated by using a Gaussian window

function g(t− td) = e−(t−td)2/t20 , with t0 = 10 fs and peak at td (gate delay time).

fidence bounds: 0.281 PHz and 0.304 PHz, respectively). The total deviation

is 0.0063 PHz and the root-mean-square deviation is 0.0148 PHz. Concern-

ing the delays between 10 fs and 40 fs, also in this case we tried to fit the

experimental results with a sum of two sinusoidal functions, The resulting fre-

quencies are: 0.234 PHz (lower and upper confidence bounds: 0.229 PHz and

0.239 PHz, respectively) and 0.292 PHz (lower and upper confidence bounds:

0.277 PHz and 0.307 PHz, respectively). The total deviation of 0.0059 PHz

and the root-mean-square deviation of 0.0113 PHz are slightly reduced com-

125

Results

Figure 4.8: In the first panel the fitted data for short pump-probe delays (t < 10

fs) are shown. Dots correspond to the difference between the experimental data

and the exponential fitting curve F (t). Error bars show the standard error of the

results of four measurements. Red curve is the corresponding fitting curve calculated

considering the sum of two sinusoidal functions of frequencies 0.14 PHz (A1 = 0.016,

φ1 = 2.88 rad) and 0.293 PHz (A2 = 0.0254, φ2 = 3.62 rad). In the second panel

delay range between 10 fs and 40 fs is shown. The corresponding fitting curve

considered the sum of two sinusoidal functions of frequencies 0.234 PHz (A1 = 0.022,

φ1 = 1.61 rad) and 0.292 PHz (A2 = 0.007, φ2 = 4.88 rad). The third panel shows

the hole delay range.

pared to the single-frequency fitting (see above).

From this fitting and delay-dependent Fourier analysis we can draw some im-

portant preliminary conclusions: first of all the temporal evolution of the ul-

trafast oscillation cannot be related to nuclear dynamics, which usually come

into play on a longer temporal scale, ultimately leading to charge localization

in a particular molecular fragment. Indeed, standard quantum chemistry cal-

126

Results

culations in phenylalanine show that the highest vibrational frequency is 0.11

PHz, which corresponds to a period of 9 fs, associated with X-H stretching

modes, while skeleton vibrations are even slower, so that one can rule out that

the observed beatings are due to vibrational motion. The oscillations observed

are therefore purely electronic, even if, in any case, some influence of the nu-

clear motion cannot be completely excluded, since for example stretching of

the order of a few picometers of carbon bonds can occur in a few femtoseconds,

and this could modify the charge dynamics [8, 9]. The gradual dephasing of

the oscillations along the dication yield dynamics (for large delays) can be

seen itself as a signature of the increasing influence of nuclear motion with

the delay increasing.

It is also important to note that the presence of a clear periodic modulation

confirms that only really few conformers of Phenylalanine are present in the

interaction area (see the assumptions on the target temperature described

before). A mixture of many conformers, indeed, (in general any charge mo-

tion is expected to show a different behavior for each molecular conformation)

would completely blur any periodic modulation by averaging on all possible

contributions.

Another crucial observation, in order to improve our investigation, is that

no fragmentation of the molecules was expected prior to interaction with

the ionizing pulses. In other experiments using a softer ionization method

(near threshold single photon ionization), parent phenylalanine ions consti-

tuted more than 90% of the mass spectrum for a molecular temperature of

423 K [10]. Therefore, we can obtain the important conclusion that the ob-

served ultrafast dynamics on the immonium dication yield has to be actually

attributed to the parent phenylalanine cation and that loss of the carboxyl

group to form the immonium dication occurs only after interaction with the

pump and probe pulses.

127

Results

4.2.3 Theoretical calculations

Theoretical calculations were performed first to calculate the ionization am-

plitudes of neutral Phenylalanine, in order to study the ionization channels

opened up by XUV excitation, then the temporal evolution of the hole den-

sity was studied, without taking into account the interaction of the VIS/NIR

probe pulse.

Since XUV field can be considered as a weak field, time-dependent first-order

perturbation theory was used to evaluate the ionization amplitudes cαǫl at

the end of the pulse t = T :

cαǫl(T ) = −i 〈Ψαl(ǫ, ~r)|~e · ~r |Ψ0(~r)〉∫ T

−∞E(t)ei(Eα)+ǫ−E0)tdt (4.6)

where Ψ0 is the all-electron (hereafter called N-electron) ground state of

phenylalanine with energy E0, Ψαl(ǫ) is the N-electron continuum state that

describes a photoelectron ejected from the α molecular orbital with kinetic

energy ǫ and angular quantum number l (for simplicity in the notation, we

have omitted the m quantum number), Eα is the corresponding cationic en-

ergy and E(t) is the electric field associated with the XUV pulse polarized

along the ~e direction.

This electric field is derived from the experimental measurements (XUV spec-

trum reported in Fig. 4.3 (a). The Ψ0 and Ψαl(ǫ) wave functions have

been evaluated in the framework of the fixed-nuclei approximation by us-

ing the static-exchange density functional theory (DFT) [11–13]. The elec-

tronic Kohn-Sham equations were solved by expanding the wave functions in

a basis of multicentric B-spline functions. The static-exchange DFT method

has been successfully used to study photoionization of a large number of di-

atomic and polyatomic molecules [14–17]. The ionization energies of all open

channels for the most abundant (according to [4]) conformer of phenylalanine

were calculated. These energies are approximately given by the Kohn-Sham

orbital energies resulting from the static-exchange DFT calculations. The cor-

responding photoionization cross sections are shown in Fig. 4.9 as a function

of photon energy. In Fig 4.10, instead, all the states of singly charged pheny-

128

Results

lalanine and of doubly-charged phenylalanine are reported. As it is evident in

this diagram, a number of transitions from highly excited states of the cation

to the lowest states of the dication are possible which involve the absorption of

just a few VIS/NIR photons. The formation of dication becomes instead less

probable for low-energy XUV excitation, since in this case transitions from

cation states to the lowest dication states would require the absorption of

many VIS/NIR photons. This observation was confirmed experimentally, by

Figure 4.9: Photoionization cross sections of the phenylalanine molecule from differ-

ent molecular orbitals provided by the static-exchange DFT method. The frequency

spectrum of the attosecond pulse used in the experiment and in the calculations of

the transition amplitudes leading to the coherent superposition of the one-hole states

is represented by a thick orange curve lying over a shaded area.

varying the photon energy and spectral width of the attosecond pump pulse.

By using the spectrum reported in Fig. 4.3 (b), indeed, doubly charged im-

monium fragments were barely visible, suggesting that the dication formation

129

Results

involves relatively highly excited states of the cation.

After calculating the cation and dication states of Phenylalanine and the

Figure 4.10: Energy level diagram containing all the states of singly charged pheny-

lalanine created by the XUV pulse, all the states of doubly-charged phenylalanine

and those for the system immonium dication + COOH.

correspondent ionization amplitudes in the range of XUV excitation, simula-

tions on the temporal evolution of hole densitiy were performed. The ionic

electronic density is given by ( [18, 19]):

ρion(~r, t) =∑

α

(∑

α′ 6=α

γionα′α′)φ2α(~r)−∑

α,α′ 6=α

γionαα′ei(Eα′−Eα

)tφα(~r)φα′(~r) (4.7)

where φα(~r) is the α molecular orbital and γionαα′ is the reduced density matrix

element defined as:

γionαα′ =∑

l

∫

cαl(ǫ)c∗α′l(ǫ)dǫ (4.8)

130

Results

The hole density is then given as usually by the difference between the elec-

tronic density of the neutral molecule, which does not depend on time, and

the electronic density of the cation:

ρhole(~r, t) = ρneutral(~r)− ρion(~r, t) =

=∑

α

(1 −∑

α′ 6=α

γionαα′)φ2α(~r) +∑

α,α′ 6=α

γionαα′ei(Eα′−Eα)tφα(~r)φα′(~r) (4.9)

where:

ρneutral(~r) =∑

α

φ2α(~r) (4.10)

The approach used in our calculations presents some important differences

compared with the Configuration Interaction (CI) approach used for example

in [20, 21] (see Appendix C), and makes really difficult to take into account

second order configurations; therefore they were not included in our calcu-

lations. For this reason, to be sure about the fidelity of this approach, the

time-propagation method was tested by performing theoretical calculations

for the glycine molecule, which has been extensively studied by using the CI

approach and explicitly including 2h1p states in the time propagation. Since

the results (not shown here) are in excellent agreement with those of Kuleff

et al. in [22], we can safely conclude that 2h1p states do not play a signifi-

cant role in the temporal evolution of the hole density, at least in the range

of photon energies leading to the cationic states relevant for the observed

dynamics.

4.2.4 Observation of charge migration

The temporal evolution of the hole-density has been calculated from imme-

diately after XUV excitation up to a 500 fs delay. Such a large delay is not

physically meaningful, but allows for a good resolution in the frequency do-

main, for any Fourier analysis.

Since in the experiments the molecules were not aligned, we have calculated

the charge dynamics resulting from excitation by pulses with the electric field

131

Results

Figure 4.11: Snapshots of the variation of the hole density with respect to the time-

averaged hole density. The charge appears quite delocalized along the molecule, but

it is clear that the largest variations of this density, showing some periodicitiy, are

observed around the amine group (-NH2).

polarized along three orthogonal directions (shown in Fig. 4.2). The re-

sults were then averaged assuming randomly oriented molecules. For a better

analysis, we have integrated the hole density around selected portions of the

molecule. The result is that few clear beating frequencies appear when the

charge density is integrated around the amine group.. This observation is

supported by plotting some snapshots of the variation of the hole density

with respect to the time-averaged hole density as a function of time, shown

in Fig. 4.11. The hole density appears really delocalized, but it is clear that

the largest variations of this density, showing some periodicity, are observed

around the amine group (-NH2). Fig. 4.12 shows the Fourier power spectra

of the temporal evolution of hole densities integrated on the amine group,

132

Results

furthermore reporting the Kohn-Sham orbitals that are responsible for the

most important beatings. The dominant beatings almost always involve two

orbitals (not reported here) with significant density around the amine group,

and at least one of these orbitals allows for delocalization over the whole

molecule. This is the reason why the hole moves all over the molecule and

the dynamics is better observed in the vicinity of the amine group. The re-

sults shown in Fig. 4.12 regards the most abundant conformer. We have

then analysed the numerical results by using the same sliding-window Fourier

transform procedure applied to the experimental data. Fig. 4.13 shows the

resulting spectrogram in a temporal window up to 45 fs, considering an ex-

perimental temporal resolution of about 3 fs. A dominant peak around 0.25

PHz is visible, which forms in about 15 fs and vanishes after about 35 fs, in

excellent agreement with the results of the Fourier analysis of the experimen-

tal data (see Fig. 4.7). A higher frequency component is visible around 0.36

PHz in the delay intervals below ≃ 15 fs and above ≃ 30 fs. At short de-

lays this component favourably compares with the experimental observation

of the frequency peak around 0.30 PHz in the same window of pump-probe

delays. The good agreement between simulations and experimental results is

rather remarkable in light of the fact that simulations do not take into account

the interaction of the VIS/NIR probe pulse. It is interesting to note that the

beating frequencies have been observed experimentally even though the initial

hole density is highly delocalized. An important result of the simulations is

that the measured beating frequencies originate from charge dynamics around

the amine group. This leads to the conclusion that the periodic modulations

measured in the experiment are mainly related to the absorption of the probe

pulse by the amine group, despite the fact that the VIS/NIR pulse is not

locally absorbed only by this group, but also by other molecular sites.

Direct measurement of the ultrafast charge dynamics in an amino acid, ini-

tiated by attosecond pulses, represents a crucial benchmark for the extension

of attosecond methodology to complex systems. The experiments we per-

133

Results

Figure 4.12: Fourier power spectra of the hole density integrated over the amine

group for the most abundant conformer of phenylalanine for the three polarization

directions defined in Fig. 4.2, and by averaging over all polarization directions

(randomly oriented molecules). The Kohn-Sham orbitals that are responsible for

the most important beatings are reported in the figure.

134

Results

Figure 4.13: 2D spectromgram obtained from sliding-window Fourier analysis ap-

plied to numerical results. A dominant peak around 0.25 PHz is visible, which forms

in about 15 fs and vanishes after about 35 fs, in excellent agreement with the results

of the Fourier analysis of the experimental data.

formed clearly demonstrated that charge fluctuations over large regions of a

complex molecule such as phenylalanine can be induced by attosecond pulses

on a temporal scale much shorter than the vibrational response of the system,

then without a predominant mediation by nuclear motion, at least within a

certain delay range. The delocalization of the charge along the molecular

skeleton make the hole dynamics complicated, but not completely caohotic.

Indeed a periodic oscillation in the amine group can be clearly distinguished,

135

Results

corresponding to few strong peaks in the Fourier power spectra of the tem-

poral evolution of the hole density. This is the first experimental observation

of (purely electronic) charge migration in a complex molecular system. The

result was achieved in spite of the broad bandwidth of the attosecond pulses

and, therefore, their low frequency selectivity, thus showing that attosecond

science offers the possibility to elucidate processes ultimately leading to charge

localization in complex molecules.

136

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139

CONCLUSIONS AND FUTURE PERSPECTIVES

This thesis reports important achievements of attosecond technology applied

to complex molecular systems. The XUV-IR pump-probe beamline devel-

oped in the ELYCHE laboratory, consisting of isolated attosecond pulses in

combination with ultrashort VIS/NIR pulses, provided a robust tool able to

investigate ultrafast dynamics in complex systems, in particular multielec-

tron diatomic molecules, dielectric nanoparticles (not presented here) and

large biological molecules. Concerning multielectron diatomic molecules, our

experimental results, supported by full time-dependent simulations, were able

to solve the ultrafast electron/nuclear relaxation preceding the dissociation of

the nitrogen molecular ion, excited by attosecond XUV pulses. By investigat-

ing the propagation of the coherent wavepackets created by XUV radiation, it

was possible to access the slope and the shape of the “real” potential energy

curves of the cation, with an extremely high temporal resolution. This result

allows one to obtain important information about the light-matter physics in-

side this type of molecules, that is crucial, for example, in the understanding

of the atmospheric photochemical processes.

By moving to even more complex systems, the experimental results presented

here demonstrate for the first time the direct access to electronic dynamics in

141

Conclusions and future perspectives

a biologically relevant molecule, disclosing a charge migration process in the

amino acid Phenylalanine. This is a breaktrough for attosecond science, since

the possiblity of capturing attosecond electronic processes in such a complex

system could pave the way for the rising of attobiology. Purely electronic

dynamics, indeed, play a fundamental role, together with nuclear motions, in

the temporal evolution of a large number of biological processes such as cel-

lular respiration, DNA damage in light-tissue interactions, photosynthesis in

plants, and many other phenomena that are of crucial importance in physics,

biology and chemistry. The combination of isolated attosecond pulses (tu-

nable in the spectral range between 15 and 50 eV) with ultrashort VIS/NIR

probe pulses, made possible to resolve the ultrafast dynamics exposed here.

In perspective, ultrafast dynamics in biologically relevant molecules will be

investigated by developing a new UV-XUV pump-probe beamline, based on

the use of ultrashort UV pulses and XUV isolated attosecond pulses.

Furthermore, while the experimental results obtained in a attosecond-pump

femtosecond-probe experiment are still technologically challenging, the inve-

stigation of ultrafast processes in complex molecular systems has also to face

the state of the art of theoretical simulations. The ultrabroad excitation in-

duced by XUV pulses, indeed, together with the extreme complexity of many-

atoms molecules, makes impossible a complete calculation of a pump-probe

dynamics. The ultrafast processes experimentally observed in Penylalanine,

indeed, were simulated and explained without taking into account the effect

of the VIS/NIR probe pulses. This allowed us to understand the oscillatory

migration of the charge along the molecular skeleton, but, for example, leaves

unknown the origin of the envelope dynamics of the immonium dication.

Apart from the future evolution on the computational side, experimental mea-

surements with a high spectral resolution, still preserving a good temporal

resolution, could compensate the intrinsic broadband nature of attosecond

pulses. For this reason, the investigation of the same target, by merging

attosecond experiments (in XUV-IR or UV-XUV setup) together with mea-

surements performed by using a time-compensated monochromator beamline,

142

Conclusions and future perspectives

could provide new information about ultrafast dynamics in large molecules,

giving an unprecedented insght in complex molecular systems.

143

APPENDIX

A - PCGPA algorithm

The first step of CPGPA algorithm is to calculate a FROG trace (in our case

a FROG-CRAB trace) using a pulse E(t) and a gate G(t). The PCGPA is

started using Gaussian pulses with random phase for the initial guess for E(t)

and G(t) (for example the transform limited value). Suppose that E(t) and

G(t) are sampled at given values of t with a constant spacing of ∆t. Then

E(t) and G(t) can be thought as vectors of length N whose elements sample

E and G at discrete times. The outer product of E and G E ·G is given by:

E1G1 ... E1GN

.... . .

...

ENG1 ... ENGN

(4.11)

The outer product contains all of the points required to construct the time

domain FROG trace because it contains all of the interactions between the

pulse and gate for the discrete delay times. Consequently, a one-to-one map-

ping of the elements of the outer product can transform the outer product

into the time domain of the CRAB trace. This is the keypoint to the PCGP

145

Appendix

algorithm. Because the mapping is one-to-one, it is invertible; transforma-

tions can be made from the outer product form to the time domain CRAB

trace and vice versa. This transformation can be accomplished by rotating

the elements of the rows in the outer product to the left by the row number

minus one. Applying this transformation:

E1G1 E1G2 E1GN−1 E1GN

E2G2 E2G3 E2GN E2G1

...... E3G1 E3G2

......

......

ENGN ENG1 · · · ENGN−1

(4.12)

The τ = 0 column is the first column, where τ is the time delay in increments

of ∆t. The next column is the τ = −1 column where the gate is delayed

relative to the probe by one resolution element ∆t. The gate appears to be

shifted by one resolution element with the first element wrapped around to the

other end of the vector. Column manipulation places the most negative delay

on the left and the most positive on the right. Thus, we obtain a matrix that is

the time domain of the CRAB trace formed by the multiplication of the pulse

and gate functions, i.e. a discrete version of the product E(t)G(t− τ). Then,

by Fourier transforming each column, the Fourier transform of E(t)G(t−τ) isobtained, as a function of t. The next step is to apply an intensity constrain

to the CRAB trace just calculated, by using the magnitude of experimental

CRAB trace that we want to invert. It is easy to imagine an infinite number of

complex images that have the same magnitude as the CRAB trace that has to

be inverted; however,there is only one image that can be formed by the outer

product of a single pair of non trivial vectors that has the same magnitude as

the CRAB trace to be inverted. In order to find the two vectors, the phase of

the CRAB trace has to be determined using a 2-D phase-retrieval algorithm.

The magnitude of calculated CRAB trace is replaced by the square root of

the magnitude of the experimental CRAB trace. The result is converted to

the time-domain trace using an inverse Fourier transform. Next, the time-

146

Appendix

domain trace is converted to the outer product form of Eq. 4.11 by reversing

the steps used to construct the time domain CRAB trace. If the intensity

and phase of the CRAB trace are correct, this matrix is a true outer product

with a rank R = 1, i.e. it would have one and only one non zero eigenvalue

and one right eigenvector and one left eigenvector. The right eigenvector is

the attosecond pulse, spans the range of the outer product matrix. The left

eigenvector, instead, is the gate. If the rank of the outer product is 6= 1,

it may have several right and left eigenvector. In this case the best pair of

vectors E and G to be used in the n+ 1 PCGPA iteration may actually be a

superposition of two or more different but linearly independent eigenvectors.

This configuration of course has more than one solution, and in principle

requires a minimization of the FROG-CRAB trace error to find the correct

superposition. Actually, this minimization is not required. If we decompose

the outer product matrix O into three matrices in the following way:

O = U ×W × V T (4.13)

where U and V T are orthogonal square matrices and W is a square diagonal

matrix, the matrix O is now written as the superposition of outer products

OUV between ”pulse” vectors (columns of U) and ”gate” vectors (rows of

V T ). The values of W determine the relative weights of each outer product

contribution. The outer product with the largest weighting factor, or principal

component, will be chosen for the next iteration of the algorithm, in order to

minimize the function error [50, 51]:

ǫ2 =

N∑

i=1

N∑

j=1

| Oi,j −Oi,jUV |2 (4.14)

The column of U and the row of V T that minimize ǫ are the new pulse E and

the new gate G for the next iteration.

The algorithm is iterated until the FROG-CRAB trace error ǫTOT reaches an

147

Appendix

acceptable minimum:

ǫ2TOT =1

N2

N∑

i=1

N∑

j=1

| S(wi, τj)− Scal(wi, τj) |2 (4.15)

where Scal(wi, τj) is the current iteration of the calculated FROG-CRAB

trace, S(wi, τj) is the measured trace.

B - SRSI retrieval

The spectrogram resulting from the interference between the pulse replica and

the XPW radiation can be written as:

S(w) =| EXPW (w) + E(w)eiwt |2= S0(w) + f(w)eiwt + f∗(w)e−iwt (4.16)

where S0(w) =| EXPW (w) |2 + | E(w)eiwt |2 is the sum of the spec-

tra between the XPW and the input pulse to be measured, and f(w) =

EXPW (w)E∗(w) is the interference part of the two pulses. The spectra of the

two pulses are analytically calculated by the following expressions:

| EXPW (w) |= 1

2(

√

S0(w) + 2 | f(w) |+√

S0(w) − 2 | f(w) |) (4.17)

| E(w) |= 1

2(

√

S0(w) + 2 | f(w) | −√

S0(w) − 2 | f(w) |) (4.18)

The spectral phase is first estimated by the argument of f(w), considering

initially that the XPW pulse spectral phase is null or at least negligible com-

pared to the input pulse spectral phase:

φ(w) = φXPW (w) − arg(f(w)) ≃ −arg(f(w)) (4.19)

From this rough estimation, the input pulse temporal profile and the XPW

pulse can be simulated giving a first estimation of the XPW pulse spectral

phase. This phase is then reintroduced in the expression 4.19 and these steps

148

Appendix

repeated until the phase modification is negligible. From the complete algo-

rithm, the XPW spectrum is analytically calculated directly from the mea-

sured signal and also simulated from the input pulse determination. The in-

put spectrum is also analytically calculated and its spectral phase estimated

through the iterative algorithm as long as it converges.

C - CI vs. Density matrix

In that case of Configuration Interaction approach, the temporal evolution of

the cation density is defined as:

ρi(~r, t) =∑

Ij ,Ij

〈Φi| |Ij〉〈Ik| eiHtρ(~r, 0)e−iHt |Ik〉〈Ik| |Φi〉 (4.20)

that is the free evolution of the densitiy operator, where Φi are the initial

cation state and |Ij〉,|Ik〉 are (N-1)-electron states calculated as the CI expan-

tion:

|I〉 =∑

j

cIjaj |Φ0〉+∑

r,k<l

cIrkla+r akal |Φ0〉+ ... (4.21)

where Φ0 is the Hartree-Fock ground state and the a+r akal |Φ0〉 term is the

two-hole one-particle (2h1p) configuration. In the case of density matrix ap-

proach, instead, the (N-1)-electron |I〉 states are sobstituted by N-electron

(ion + photoelectron) |Φ(t)〉 defined as:

|Φ(t)〉 = c0(t) |Ψ0〉+∑

i

cαiǫilimi(t)a+αiǫilimi

aαi|Ψ0〉 (4.22)

where the coefficient cαiǫilimi(t) is the ionization amplitude defined in eq.

4.2.3. In this way a real picture of the molecular ionization is accessed, by tak-

ing in account the experimental frequency spectrum of the attosecond pulses,

(anyway in the approximation of fixed nuclei) and requires the introduction

of a reduced density matrix (eq. 4.2.3) to describe the temporal evolution of

the hole density in eq. 4.2.3.

149

LIST OF PUBLICATIONS

• L. Belshaw, F. Calegari, M.J. Duy,ff A. Trabattoni, L. Poletto, M. Nisoli,

and J. B. Greenwood, “Observation ofUltrafast Charge Migration

in an Amino Acid”, J. Phys. Chem. Lett. 3,3751 (2012);

We present the first direct measurement of ultrafast charge migration

in a biomolecular building block - the amino acid phenylalanine. Using

an extreme ultraviolet pulse of 1.5 fs duration to ionize molecules isolated

in the gas phase, the location of the resulting hole was probed by a 6 fs

visible/near-infrared pulse. By measuring the yield of a doubly charged

ion as a function of the delay between the two pulses, the positive hole

was observed to migrate to one end of the cation within 30 fs. This pro-

cess is likely to originate from even faster coherent charge oscillations

in the molecule being dephased by bond stretching which eventually lo-

calizes the final position of the charge. This demonstration offers a clear

template for observing and controlling this phenomenon in the future.

• L. Poletto, F. Frassetto, F. Calegari, S. Anumula, A. Trabattoni, M.

Nisoli, “Micro-focusing of attosecond pulses by grazing-incidence

151

List of publications

toroidal mirrors”, Opt. Express 21, 13040(2013);

The design of optical systems for micro-focusing of extreme-ultraviolet

(XUV) attosecond pulses through grazing-incidence toroidal mirrors is

presented. Aim of the proposed configuration is to provide a micro-

focused image through a high demagnification of the XUV source with

the following characteristics: i) almost negligible aberrations; ii) long

exit arm to easily accommodate at the output the experimental setups

required for the applications of the focused attosecond pulses; iii) possi-

bility to have an intermediate region where the XUV beam is collimated,

in order to insert a plane split-mirror for the generation of two XUV

pulse replicas to be used in a XUV-pump/XUV-probe setup. We present

the analytical and numerical study of two optical configurations charac-

terized by two sections based on the use of toroidal mirrors. The first

section provides a demagnified image of the source in an intermediate

focus that is free from defocusing but has a large coma aberration. The

second section consists of a relay mirror that is mounted in Z-shaped

geometry with respect to the previous one, in order to give a stigmatic

image with a coma that is opposite to that provided by the first section.

An example is provided to demonstrate the capability to achieve spot

sizes in the 5− 15 µm range with a demagnification higher than 10 in a

compact envelope.

• L. Poletto, F. Frassetto, S. Anumula, F. Calegari, A. Trabattoni, M.

Nisoli, “Micro-focusing of soft X-ray pulses by grazing-incidence

toroidal mirrors”, Proc. SPIE8778, Advances in X-ray Free-Electron

Lasers II: Instrumentation, 877813 (2013);

The design of optical systems for micro-focusing of extreme-ultraviolet

(XUV) and soft X-ray pulses through grazing incidence toroidal mirrors

is presented. Aim of the configuration here presented is to provide a

micro-focused image through a high demagnification of the source with

152


almost negligible aberrations and a long exit arm to accommodate at the

output a large experimental chamber. We present the analytical and nu-

merical study of two configurations to fulfill these requirements with two

toroidal mirrors. The first mirror provides a demagnified image of the

source in the intermediate plane that is free from defocusing but has a

large coma aberration, the second mirror is mounted in Z-shaped geom-

etry with respect to the previous one, in order to give a stigmatic image

with a coma that is opposite to that provided by the first one. Some

examples are provided to demonstrate the capability to achieve spot sizes

in the 5 − 15 µm range both applied to high-order laser harmonics and

free-electron-laser radiation.

• C. Liu, M. Reduzzi, A. Trabattoni, A. Sunilkumar, A. Dubrouil, F. Cale-

gari, M. Nisoli, and G. Sansone, “Carrier-envelope phase effects of

a single attosecond pulse in two-color photoionization”, Phys.

Rev. Lett. 111, 123901 (2013);

The attosecond streak camera method is usually implemented to char-

acterize the temporal phase and amplitude of isolated attosecond pulses

produced by high-order harmonic generation. This approach, however,

does not provide any information about the carrier-envelope phase of the

attosecond pulses. We demonstrate that the photoelectron spectra gen-

erated by an attosecond waveform and an intense synchronized infrared

field are sensitive to the electric field of the attosecond pulse. The depen-

dence on the carrier-envelope phase of the attosecond pulse is understood

in terms of the coherent superposition of two photoelectron wave pack-

ets. This effect suggests an experimentally feasible method for complete

reconstruction of attosecond waveforms.

• F. Lucking, A. Trabattoni, S. Anumula, G. Sansone, F. Calegari, M.

Nisoli, T. Oksenhendler, and G. Tempea “In situ measurement of

nonlinear carrier-envelope phase changes in hollow fiber com-

pression”, Opt. Lett. 39, 2302-2305 (2014);

153


We demonstrate a simple and robust single-shot interferometric tech-

nique that allows the in situ measurement of intensity-dependent phase

changes experienced by ultrashort laser pulses upon nonlinear propaga-

tion. The technique is applied to the characterization of carrier-envelope

phase noise in hollow fiber compressors both in the pressure gradient and

in the static cell configuration. Measurements performed simultaneously

with conventional f-to-2f interferometers before and after compression

indicate that the noise emerging in the waveguide adds up arithmetically

to the phase noise of the amplifier, thus being strongly correlated to the

phase noise of the pulses coupled into the compressor.

• F. Frassetto, A. Trabattoni, S. Anumula, G. Sansone, F. Calegari, M.

Nisoli, and L. Poletto, “High-throughput beamline for attosecond

pulses based on toroidal mirrors with microfocusing capabili-

ties”, Rev. Sci. Instrum. 85, 103115 (2014);

We have developed a novel attosecond beamline designed for attosecond-

pump/attosecond probe experiments. Microfocusing of the Extreme ul-

traviolet (XUV) radiation is obtained by using a comacompensated op-

tical configuration based on the use of three toroidal mirrors controlled

by a genetic algorithm. Trains of attosecond pulses are generated with

a measured peak intensity of about 3× 1011 W/cm2.

• F. Calegari, D. Ayuso, A.Trabattoni, L. Belshaw, S. De Camillis, S. Anu-

mula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. Greenwood,

F. Martin, M. Nisoli, “Ultrafast Electron Dynamics in Phenylala-

nine Initiated by Attosecond Pulses”, Science 346, 6207, 336-339

(2014);

In the past decade, attosecond technology has opened up the investigation

of ultrafast electronic processes in atoms, simple molecules, and solids.

Here, we report the application of isolated attosecond pulses to prompt

154


ionization of the amino acid phenylalanine and the subsequent detection

of ultrafast dynamics on a sub4.5-femtosecond temporal scale, which is

shorter than the vibrational response of the molecule. The ability to

initiate and observe such electronic dynamics in polyatomic molecules

represents a crucial step forward in attosecond science, which is pro-

gressively moving toward the investigation of more and more complex

systems.

155

ACKNOWLEDGEMENTS

Poche parole, solamente, pennellano la conclusione.

Ringrazio Mauro Nisoli e Francesca Calegari, prima di tutti, perche e prezioso

essere mentori, ma molto raro essere allo stesso tempo amici.

Ringrazio Maurizio per essere stato in questi anni il mio “compagno di giochi”,

e per avermi dimostrato molto piu che “fisica, scacchi e alcol”. Un grazie par-

ticolare per i biscotti.

Un grazie grande come un abbraccio a Teo N., Teo L., Cate, Michele, Cristian,

Salvatore, Giuseppe, Paolo e tutti quelli che mi hanno regalato consigli, risate,

sciate, teatri, stadi, e anche tanta fisica, ma e la parte meno importante.

Un grazie speciale a Patrick, per una chiacchierata all’Edelweiss.

Ringrazio Tea, sperando che la nostra amicizia sia sempre una scambio di fili

di lana.

Per ultimi, salendo in alto dove l’aria e rarefatta e la luce piu forte, la dove

le parole non servono piu, ringrazio i protagonisti della mia fiaba personale:

una farfalla, due gigli bianchi, una roccia, e una stella. Grazie a loro, ogni

giorno, scalo il cielo.

157

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