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 Femtosecond Laser Matter Interactions Solid Plasma Solid
Transformations at the Extreme Energy Density 2nd Edition
Eugene G. Gamaly

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 Femtosecond
Laser-Matter Interactions
Solid-Plasma-Solid Transformations
at the Extreme Energy Density

Eugene G. Gamaly
Published by
Jenny Stanford Publishing Pte. Ltd.
Level 34, Centennial Tower
3 Temasek Avenue
Singapore 039190

Email: editorial@jennystanford.com
Web: www.jennystanford.com

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Femtosecond Laser-Matter Interactions: Solid-Plasma-Solid

Transformations at the Extreme Energy Density
Copyright © 2022 Jenny Stanford Publishing Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced
in any form or by any means, electronic or mechanical, including
photocopying, recording or any information storage and retrieval system
now known or to be invented, without written permission from the
publisher.

ISBN 978-981-4877-40-4 (Hardcover)

ISBN 978-1-003-25661-8 (eBook)
To Ksana
 Contents

Preface xvii
List of frequently used symbols xxiii

1. Basics of Laser–Matter Interactions: Light and Matter 1

1.1 Laser Beam 1
1.1.1 Macroscopic Electrodynamics 1
1.1.2 Polarization States 2
1.1.3 Spectral Structure 3
1.1.4 Temporal/Spatial Shape of the Incident
Laser Pulse Intensity 4
1.1.5 Focussing Positions of the Laser Beam
on/in a Target 5
1.1.6 Focussing to Diffraction Limit with the
High NA Lens 6
1.1.7 Diffraction-Free Beams 8
1.2 The Matter 9
1.2.1 Electrons’ Oscillations and Scattering
in the High-Frequency Electromagnetic
Field 10
1.2.2 The Drude Model for the Permittivity of
Simple Plasma in the High-Frequency
Electric Field 11
1.2.2.1 Electrons’ plasma frequency 12
1.2.2.2 Critical density of electrons 13
1.2.3 Absorbed Energy Density 14
1.2.4 Hierarchy of the Laser-Affected Material
Transformations as a Function of Laser
Intensity/Fluence 15
1.2.4.1 Melting 16
viii Contents

1.2.4.2 Ablation 16
1.2.4.3 Atomic field intensity 17
1.2.4.4 The relativistic intensity 18
1.3 Summary 18

2. Interaction with Metals 21

2.1 Interaction of the Plane Wave with a Metal
Layer 22
2.2 Electron and Lattice Temperature, Energy
Balance, Two-Temperature Approximation 25
2.3 Temperature Dependence of the Electronic and
Lattice Heat Capacity 26
2.4 Electrons Relaxation Processes in the
Laser-Affected Metal 28
2.4.1 Electron–Electron Collisions 28
2.4.2 Electron–Phonon Momentum Transfer 29
2.4.3 Electron–Phonon Energy Transfer 31
2.4.4 Electron-to-Ion Momentum and Energy
Transfer 34
2.4.5 Electron-to-Ion Energy Exchange Time 35
2.4.5.1 Non-ideality effects 36
2.4.5.1 Effects of the oscillations in
high-frequency electromagnetic
field on the electrons’ collision
rate 37
2.5 Modification of the Electron Distribution
Function: From the Fermi–Dirac to the
Maxwell–Boltzmann 39
2.6 Electronic Heat Conduction 42
2.7 Summary 46

3. Interaction with Dielectrics 47

3.1 Ionization in the Strong High-Frequency
Electric Field: Electrons Transfer from the
Valence to Conduction Band and to Continuum 48
3.1.1 Tunnelling Ionization Rate in the
Limit g << 1 50
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 Contents
Contents I
ix
ix

3.1.1.1 Linear
3.1.1.1 Linear polarization
polarization 50
50
3.1.1.2 Tunnel ionization in the
elliptically polarized electric
field 51
3.1.2 Multi-Photon Ionization Rate 51
3.1.2.1 Linear polarization 51
3.1.2.2 Time dependence of the
ionization degree produced by
the Gaussian laser pulse with
the MPI domination 53
3.1.2.3 Circular polarization 53
3.1.3 Ionization by the Electron Impact 54
3.1.3.1 Transition to the avalanche
regime 57
3.1.4 Distribution Function of Electrons
Transferred to the Conduction Band in
Dielectrics 58
3.2 Electrons' Collision Rates: Momentum and
Energy Transfer 58
3.2.1 Electron-Phonon Collisions 59
3.2.2 Momentum and Energy Transfer
Collisions in a Dielectric Converted to
Plasma 60
3.2.3 Effects of the Electrons' Oscillations in
the HF Field on the Collision Rates 61
3.3 Transient Permittivity in the Laser-Affected
Dielectric 61
3.3.1 Non-Linear Contributions to the
Permittivity at the Low Intensity (I « fat) 62
3.3.2 Kerr-Nonlinearity in Silica: Temperature
Dependence 63
3.3.3 Permittivity in the Intense-Laser-Excited
Dielectric 64
3.3.3.1 Reaching the state ere = 0 65
3.4 Laser Interaction with the Electrically
Inhomogeneous Dielectric 66
x Contents

3.4.1 Normal Incidence 67

3.4.2 Oblique Incidence 68
3.4.2.1 s-Polarization 68
3.4.2.2 p-Polarization 69
3.4.2.3 Band gap modification and
collapse 70
3.5 Energy Density Thresholds for Achieving the
Major Steps in Dielectric-Plasma Transformation 71
3.5.1 Threshold to Achieve the State Where
ere = 0 72
3.5.2 Threshold Fluence for Transferring All
Valence Electrons to the Conduction
Band 73
3.5.3 Ionization Threshold: Transfer
Electrons to the Solid Plasma State 73
3.6 Energy Equations: Two-Temperature
Approximation 73
3.7 Linear and Non-Linear Heat Conduction:
Transition from Heat Transfer in Solid to
That in Plasma 74
3.7.1 Heat Diffusion in the Isotropic Medium 74
3.7.2 Thermal Conduction in Dielectrics:
Energy Carriers are Phonons 75
3.7.3 Electronic Heat Conduction in Plasma 76
3.7.4 Applicability of Diffusion Approximation
for Description the Heat Conduction in
the Laser-Created Plasma 77
3.7.5 Linear and Non-Linear Heat
Propagation from the Focal Region 78
3.7.5.1 Linear heat conduction 79
3.7.5.2 Non-linear heat propagation 79
3.8 Summary 81

4. Non-Destructive Transformations: Formation,

Lifetime and Decay of Unconventional States of
Matter 83
4.1 Generation and Decay of the Coherent Phonons 84
 Contents xi

4.1.1 Melting in Equilibrium: Atomic Vibrations

Become Anharmonic 85
4.1.2 Swift Deposition of the Energy Density
Comparable to the Enthalpy of Fusion
by the Ultra-Short Laser Pulse 88
4.1.2.1 Processes during the pulse:
building the electronic pressure
force 89
4.1.2.2 After the end of the pulse:
electronic heat transfer,
electron–lattice temperature
equilibration, and phonon’s
decay rate 90
4.1.2.3 Phonon’s excitation imprinted
into the transient optical
properties 92
4.1.2.4 Experiments 93
4.2 Fast Transformations in Non-Equilibrium:
Transient Phase States—Neither Solid Nor Melt 95
4.2.1 Road to Melting in Equilibrium:
Sequence of the Catastrophes 96
4.2.1.1 Main features of the melting in
equilibrium 96
4.2.1.2 Critical entropy and critical
temperature 97
4.2.2 Ultra-Fast Material Transformations 99
4.2.2.1 The legitimacy of the entropy
concept for the description of
rapidly excited metal 100
4.2.2.2 Entropy changes during the
quick heating 101
4.2.3 Novel Transient Phase States—Neither
Solids Nor Melts 103
4.2.3.1 Excitation of metals/
semi-metals—Ga, Bi, Al 104
4.3 Mixed Dielectric/Metal Properties in Swiftly
Excited Transparent Dielectric 107
xii Contents

4.4 Inverse Population in Laser-Excited Sapphire

and Silica 109
4.5 Summary 112

5. Ablation of Metals and Dielectrics 113

5.1 Evaporation in Equilibrium 114
5.2 Major Relaxation Processes under Swift Short
Pulse Excitation 116
5.3 Short Pulse Ablation Thresholds 119
5.3.1 Ablation of Metals 119
5.3.1.1 Extremely short pulses of
duration shorter tc,i 119
5.3.1.2 Comparison to the Coulomb
explosions of clusters 123
5.3.1.3 Ultra-short npulses
a r0 wpi < tp < theat
Al 2 1
124
5.3.1.4 Ablation threshold fluence for
metals 126
5.3.2 Ablation of Dielectrics 127
5.4 Dependence of the Ablation Thresholds on the
Laser Wavelength for Metals and Dielectrics 131
5.5 Long Pulse Ablation Thresholds 132
5.6 Ablation Thresholds in the Ambient Gas 133
5.7 Ablation Rate, Depth and Mass Per Single Pulse 136
5.8 Control over the Phase State of the Ablated
Plume 141
5.8.1 Criterion for Complete Atomization of
Ablated Plume 142
5.8.2 Experimental Verification of the Plume’s
Atomization 145
5.9 Multiple Pulse Action: Accumulation Effects 146
5.9.1 Dwell Time: Control over the Number
of Pulses Hitting the Spot 148
5.9.2 Multiple-Pulse Action on Dielectrics:
Energy Accumulation 148
5.9.3 Achieving Thermal Ablation Threshold
with the Low-Energy Multiple Pulses 149
 Contents xiii

5.9.4 Smoothing of the Intensity Distribution

across the Focal Spot 150
5.9.5 Change of the Interaction Mode: Density
Build-Up near the Target Surface 151
5.10 Summary 152

6. Extreme Energy Density Confined inside a Transparent

Crystal—Novel Path for New Materials Creation:
Solid-Plasma-Solid Transformations 155
6.1 Introduction 155
6.2 Interaction of the Gauss Beam Tightly Focused
inside a Transparent Crystal 158
6.2.1 Beam Propagation 159
6.2.2 Focusing of the Gauss Beam with the
High-NA Optics 160
6.2.2.1 Conventional approach to the
high-NA focusing 160
6.2.2.2 Experimental evidence: high-NA
focus produces nano-sharp
intensity distribution 162
6.2.3 Laser Modification of the Dielectric
Permittivity: Conversion to Solid
Density Plasma 166
6.2.3.1 Threshold for Conversion of
Dielectric into Absorbing
Medium (ere = 0) 167
6.2.3.2 Conversion a dielectric to the
solid density plasma 168
6.2.3.3 Ionization after the end of the
pulse 169
6.2.4 Relaxation Processes 170
6.2.4.1 Range for the maximum
temperature/pressure value
in the absorption volume 171
6.2.4.2 Ion’s acceleration by the
gradient of the electronic
pressure 173
xiv Contents

6.2.4.3 Electrons-to-ions energy transfer

by the Coulomb collisions:
Spatial separation of the light
and heavy ions 173
6.2.4.4 Diffusion of ions with the
different masses: The
experimental evidence of the
spatial separation 174
6.2.5 Micro-Explosion in Sapphire Considered
as a Strong Point-Like Explosion 176
6.2.5.1 Interpretation of the
micro-explosion experiments
using the strong point-like
explosion solution 179
6.2.6 Summary of the Conditions after the
Electron–Ion Energy Equilibration;
Shock Wave Generation, Propagation
and Stopping 180
6.2.6.1 Analysis of the experimentally
observed structure on the basis
of the energy and mass
conservation laws 181
6.2.6.2 Evolution of the conditions and
the structure of the laser-
affected zone during expansion 183
6.2.7 Modelling the Hydrodynamics of the
Confined Explosion 185
6.2.8 Relaxation of the Laser-Affected
Material to the Ambient Conditions 187
6.3 Interaction of the Gauss Beam Tightly Focused
on the Surface of an Opaque Solid Buried under
the Transparent Cover 188
6.3.1 Upper Limit for the Energy Density
Directly Delivered to the Opaque
Medium through the Transparent Layer 189
6.3.2 High-Energy-Density Interaction at
the Silicon/Silica Interface 191
6.3.3 Shock Wave Generation in Silica Layer 193
 Contents xv

6.4 Summary of the Gauss Beam–Generated

Confined Micro-Explosions 194
6.5 Interaction of the Bessel Beam Focused inside a
Bulk of the Transparent Crystal 195
6.5.1 Status of the BB–Transparent Crystal
Interactions 195
6.5.2 Non-Diffractive Bessel Beam 196
6.5.3 Creation of the BB by the Circular Slit 197
6.5.4 Formation the BB by Transmitting the
GB through the Axicon 199
6.5.5 BB Interaction with the Transparent
Medium at Low Intensity 201
6.5.6 Intense BB Interaction: Conversion a
Dielectric into Plasma Near the Axis 202
6.5.7 BB Interaction with Plasma 205
6.5.8 Energy Deposition in Plasma to the
End of the Pulse 207
6.5.9 Strong Cylindrical Explosion 208
6.5.10 Experimental Evidence of the High
Energy Density Achieved by the BB
Induced Cylindrical Explosion 210
6.6 Solid-Plasma-Solid Transformation in Confined
Micro-Explosion: New Path for High-Pressure
Material Phase Formations 212
6.6.1 Discovery of Superdense bcc-Al in
Sapphire Irradiated by the Tightly
Focused Gauss Beam 214
6.6.2 Discovery of New Tetragonal
Polymorphs of Silicon 215
6.7 Summary: Confined Micro-Explosion by the
Gauss and Bessel Beams 218

Conclusions and Future Directions 221

Appendices 227
Bibliography 253

Index 267
 Preface

The aim of this book is to present a succinct description of

the non-equilibrium physics of the laser–solid interactions
induced by the action of the ultra-short laser pulse on metals and
dielectrics. The ultra-short pulse stands for the pulses shorter
of the major relaxation times in the laser-excited solids (in
the sub-picosecond time domain). Such a pulse excites only
electrons while the atoms remain at the initial temperature and
at the initial positions. We consider interactions in a broad
range of intensities from the subtle excitation of the coherent
phonons at low intensity up to the creation of the extreme
energy density by the intense short laser focused into sub-micron
volume.
The basic physics of the interaction is the focus of the studies.
Therefore, we present mainly the analytical estimates in a form
allowing to a reader using (or creating) the numerical code on
a clear physical basis. The effects of the laser field on the matter
and the effects of the excited matter on the field are considered
intertwined.
The goal of the book is the study of the interaction of the
intense ultra-short laser pulse focused to the sub-micron volume
inside the transparent dielectric at the laboratory tabletop. The
interaction zone and laser-affected material remain confined
inside the pristine crystal. The absorbed energy density exceeds a
few TPa (1 TPa = 106 J/cm3 = 10 Mbar) that is higher than Young’s
modulus of any existing material. Hence, the micro-explosion
after the end of the pulse creates a void inside an undamaged
crystal, the size of which can be controlled by the proper choice
of the laser and material parameters. It was found that in the
conditions of micro-explosion the unusual high-pressure phase
transformations occurred resulting in the formations of bcc
aluminium and novel phases of silicon.
The phase transitions in the laser-induced confined micro-
explosion take place along the unique transformation path first
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