Measurements of Ultra Strong
Magnetic fields in Laser Produced
Plasmas
by
Amrutha Gopal
This thesis is submitted in partial fulfilment of the requirements for
the degree of Doctor of Philosophy of the University of London and
for the Diploma of Membership of the Imperial College
Department of Physics
Imperial College
Prince Consort Road
London SW7 2BZ
2004
Mammy and Achan
2
Abstract
This thesis discusses experiments to measure the ultra large magnetic fields generated in the laboratory when a high power laser pulse is focussed onto solid. Two
novel techniques to measure these fields have been developed. The experiments were
carried out on the femtosecond ASTRA and the picosecond VULCAN laser systems at the Rutherford Appleton Laboratory at intensities ranging from 1018 − 1020
W/cm2 . A brief overview of the laser systems and laser diagnostics is included.
Theoretical and computational calculations predict the existence of magnetic fields
of 1 GGauss strength at laser intensities of 1021 W/cm2 . However, in previous studies the diagnostic techniques used limited the measured maximum field to ∼ 30
MGauss. The two new techniques used in this thesis are called the cut-off method,
which is based on the detection of the cut-off of the extraordinary component of
the self generated harmonics of the the laser, and harmonic polarimetry using the
Cotton-Mouton effect, based on the measurement of the depolarisation of the harmonics propagating through the magnetised plasma. With the cut-off method the
maximum magnetic field measured was 340 ± 50 M Gauss at laser intensities of 1020
W/cm2 . A similar field was measured using the Cotton-Mouton method.
The temporal evolution and the spatial distribution of the magnetic field were also
studied. The temporal measurements show that the magnetic field increases almost
linearly with the laser intensity. The spatial distribution measurements show that
there is an asymmetry in the field when the laser is incident at an angle on the
target. The thesis also describes a new method for making comprehensive spatial
distribution measurements of the magnetic field. The experimental results were
3
compared with computer simulations using the OSIRIS 2D3V PIC code.
4
Contents
Dedication
2
Abstract
3
Table of Contents
5
List of Figures
8
List of Tables
13
Units
14
Acknowledgments
16
1 Introduction
18
1.1
An overview of previous work . . . . . . . . . . . . . . . . . . . . . . 20
1.2
Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4
The role of the author . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Laser-matter interaction and magnetic field generation
25
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2
Plasma parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1
Plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2
Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.3
Ponderomotive force . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4
Density Scalelength . . . . . . . . . . . . . . . . . . . . . . . . 30
5
2.3
2.4
2.5
Absorption mechanisms
. . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1
Inverse bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 31
2.3.2
Resonance absorption . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.3
Vacuum heating . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.4
Hole boring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.5
Filamentation and channeling . . . . . . . . . . . . . . . . . . 35
Harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1
Harmonic generation from solids . . . . . . . . . . . . . . . . . 36
2.4.2
Theoretical calculations . . . . . . . . . . . . . . . . . . . . . 37
2.4.3
Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Generation of MegaGauss dc magnetic fields . . . . . . . . . . . . . . 41
2.5.1
∇ne × ∇Te mechanism (Thermo electric mechanism) . . . . . 43
2.5.2
Current of fast electrons . . . . . . . . . . . . . . . . . . . . . 45
2.5.3
Spatial and temporal variation of the incident laser pulse (ponderomotive force term) . . . . . . . . . . . . . . . . . . . . . . 46
3
2.5.4
B fields due to resonance absorption . . . . . . . . . . . . . . 49
2.5.5
∇ne × ∇Te field near filaments, composition jumps, shocks . . 50
2.5.6
B field due to thermal instabilities . . . . . . . . . . . . . . . . 50
2.5.7
B field due to Weibel instabilities . . . . . . . . . . . . . . . . 51
Laser systems and diagnostics
52
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2
Chirped Pulse Amplification (CPA) . . . . . . . . . . . . . . . . . . 52
3.2.1
VULCAN laser system . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1.1
Target Area West . . . . . . . . . . . . . . . . . . . 56
3.2.1.2
The laser diagnostics . . . . . . . . . . . . . . . . . . 56
3.2.1.3
Pulse length measurement- Autocorrelation . . . . . 58
3.2.1.4
Focal spot measurements . . . . . . . . . . . . . . . 59
3.2.1.5
The equivalent plane monitor . . . . . . . . . . . . . 60
3.2.1.6
Penumbral imaging . . . . . . . . . . . . . . . . . . . 60
3.2.1.7
Energy measurements . . . . . . . . . . . . . . . . . 61
6
3.2.2
ASTRA laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Magnetic field measurements using the Cut-off method
63
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1
Faraday rotation (propagation parallel to the magnetic field,
k k B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2
Cotton-Mouton effect (propagation perpendicular to the magnetic field, k ⊥ B) . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.3
Electromagnetic wave propagation in plasma . . . . . . . . . . 67
4.2.3.1
Propagation parallel to the magnetic field (k k B0 ) . 69
4.2.3.2
Propagation perpendicular to the magnetic field (k ⊥
B0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.3.3
4.3
Cut-offs and Resonances . . . . . . . . . . . . . . . . 70
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1
Polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1.1
Calibration of polarimeters . . . . . . . . . . . . . . 78
4.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Magnetic field measurements - using Stokes vector analysis
87
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2
The Cotton-Mouton effect and the Polarimetric technique
5.2.1
5.3
Configurations of polarisers and retarders in the polarimeter . 93
Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1
5.3.2
The Vulcan laser . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1.1
Analytical calculation of Stokes vectors . . . . . . . . 94
5.3.1.2
Calculation of plasma transition matrix . . . . . . . 97
The Astra laser . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.2.1
5.4
. . . . . . 88
Results . . . . . . . . . . . . . . . . . . . . . . . . . 104
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4.1
Simulation set up . . . . . . . . . . . . . . . . . . . . . . . . . 106
7
5.4.2
5.5
Spatial asymmetry measurements . . . . . . . . . . . . . . . . . . . . 109
5.5.1
5.6
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 106
The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Harmonic Ellipsometry - A new technique to plot the angular distribution of the magnetic field . . . . . . . . . . . . . . . . . . . . . . . 112
5.7
5.6.1
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 113
5.6.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Time resolved measurements of the self-generated magnetic field
using laser harmonics
117
6.1
The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 Conclusions and Discussions
131
7.1
Summary of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2
Consequences of large B fields in laser-matter interaction . . . . . . . 134
7.3
Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8 Appendix I
8.1
The cold plasma dispersion relation . . . . . . . . . . . . . . . . . . . 138
8.1.1
8.2
138
Dielectric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 139
CUT- OFF - Mathematical derivation . . . . . . . . . . . . . . . . . . 141
9 Appendix II
9.1
144
General representation of an electromagnetic wave . . . . . . . . . . . 144
9.1.1
Horizontally or vertically linear polarised light . . . . . . . . . 145
9.2
Stokes vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.3
Muller Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Bibliography
163
8
List of Figures
2.1
Dispersion ω(k) for a transverse electromagnetic wave propagating in
cold plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2
a. radial ponderomotive force b. longitudinal ponderomotive force . . 30
2.3
Schematic representation of plasma produced by an intense laser pulse
interaction with solid target showing main laser plasma interactions. . 32
2.4
Resonance absorption for an obliquely incident p-polarised laser . . . 33
2.5
Moving mirror model, the electrons undergo excursions across the
plasma vacuum boundary. The trajectory of a free electron in a plane
polarised wave is figure of eight . . . . . . . . . . . . . . . . . . . . . 38
2.6
Electric dipole formation at the plasma vacuum boundary . . . . . . 39
2.7
Electron orbit for s-polaised and p-polarised light respectively. . . . . 40
2.8
Computer simulation studies showing the generation of self-generated
dc magnetic fields due to various mechanisms at normal incidence for
an intensity 1020 W/cm2 using the Osiris 2 1/2 PIC code at a time
162.7(1/ω0 ) The dotted red lines shows the density profile changing
from 0 to 10nc . A - magnetic field generated due to thermoelectric
term, B - field due to ponderomotive force, C - due to fast electron
current and instabilities . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9
DC magnetic field generation due to thermo electric term . . . . . . . 44
2.10 B field generation due to fast electron current. The return current
generated to balance the charge neutrality creates an azimuthal magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1
An illustration of the CPA technique . . . . . . . . . . . . . . . . . . 53
9
3.2
Different stages of Vulcan CPA laser system . . . . . . . . . . . . . . 54
3.3
Schematic layout of VULCAN laser bay
3.4
Schematic layout of Target Area West
3.5
Frequency doubling in an autocorrelator . . . . . . . . . . . . . . . . 58
3.6
Focal spot measurements using equivalent plane monitor . . . . . . . 60
3.7
Schematic layout of ASTRA laser system . . . . . . . . . . . . . . . . 61
4.1
Graphical representation of linearly polarised light . . . . . . . . . . . 65
4.2
Wave propagation perpendicular to an external magnetic field . . . . 67
4.3
Dispersion relation for extraordinary wave plotted on a refractive invφ2
dex or 2 - frequency scale. Hatched regions are regions of non propc
agation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4
CMA diagram showing the phase velocity surfaces for different wave
. . . . . . . . . . . . . . . . 55
. . . . . . . . . . . . . . . . . 57
solutions of dispersion relation perpendicular propagation to the magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5
Cut-off magnetic field plotted for various harmonics of 1.053µ radiation against electron density . . . . . . . . . . . . . . . . . . . . . . . 74
4.6
Schematic layout of the interaction chamber . . . . . . . . . . . . . . 75
4.7
Schematic layout of 4ω polarimeter . . . . . . . . . . . . . . . . . . . 76
4.8
A typical low energy shot shows that only p - polarised harmonics
are produced at low intensities . . . . . . . . . . . . . . . . . . . . . . 79
4.9
Typical data shots with 2ω polarimeter showing (a) cut-off at high
energy and (b) no cut-off with a low energy shot. . . . . . . . . . . . 79
4.10 An example of the cut-off data from the 3ω polarimeter (351 nm).
(a) low intensity shot showing all polarisations. (b) p-component has
vanished (cut-off). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.11 Cut-off data from 4ω polarimeter (264nm). (a) low intensity shot
showing all polarisations. (b) p-component has vanished (cut-off) . . 80
4.12 A typical 5ω polarimeter (210 nm) data where no extinction of ppolarisation is observed. . . . . . . . . . . . . . . . . . . . . . . . . . 81
10
4.13 A high intensity shot showing cut-off of all lower order optical harmonics below 5ω at the same intensity. . . . . . . . . . . . . . . . . . 82
4.14 x -wave cut-off for 3rd , 4th harmonics. The 5th harmonic does not show
any cut-off at the same intensity, the y-axis is the ratio of x-wave over
o-wave on a logarithmic scale. . . . . . . . . . . . . . . . . . . . . . . 83
5.1
Representation of an elliptically polarised wave traveling in the z
direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2
The Poincaré sphere. A useful way to represent the polarisation of
light in a three dimensional vector space. . . . . . . . . . . . . . . . . 90
5.3
Initially the radiation is linearly polarised at an angle b to the x-axis
5.4
An example of 4ω polarimeter raw data showing all channels to mea-
92
sure the Stokes vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5
An example of a low energy raw data showing only p-polarised light . 95
5.6
A calibration shot at 70J with no polarisers and a quarter wave plate.
The s -component is missing as there is no polariser in the beam path.
The four spots show all four channels of the polarimeter
5.7
. . . . . . . 95
Examples of 2ω and 3ω polarimeter data showing data shots and
calibration shots at the same intensities. . . . . . . . . . . . . . . . . 96
5.8
Typical data from the 4ω polarimeter . . . . . . . . . . . . . . . . . . 97
5.9
Second harmonic probe images (shadowgraphy) showing the plasma
expansion performed with the Vulcan laser (λ = 1µm) . . . . . . . . . 101
5.10 Estimated strength of magnetic field using Stokes vector analysis for
various harmonics of the Vulcan laser plotted on an intensity scale. . 101
5.11 Schematic experimental layout . . . . . . . . . . . . . . . . . . . . . . 102
5.12 Typical data shots for 3rd (266nm) at maximum intensities. . . . . . . 103
5.13 Magnetic field measured using Stokes vectors plotted against intensity
for third harmonic (264 nm) of the Astra laser. . . . . . . . . . . . . 104
5.14 Schematic of the 45o simulation geometry. The density profile is
shown in the right hand side. The value is multiplied by critical
density for 1µ laser (i.e., 1.1 × 1021 cm−3 ) . . . . . . . . . . . . . . . 105
11
5.15 Simulation results showing generation / evolution of dc magnetic field
at different times with an intensity 1020 W/cm2 . The scale shown is
a relative scale and the actual value of the magnetic field is a factor
of me ωp e−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.16 The electron and ion density at maximum B field. The actual value
of the density is the right hand scale multiplied by the critical density
(1.1 × 1021 cm−3 ). The line-out is taken at the point of laser incidence.108
5.17 Intensity dependance of self generated magnetic field studied using
three different methods a. Theoretical calculation of field from ponderomotive force mechanism, b. Experiment, c. Osiris PIC simulation 108
5.18 The schematic setup of spatial asymmetry measurements . . . . . . . 110
5.19 The estimated magnetic field strength . . . . . . . . . . . . . . . . . . 112
5.20 The layout of the experiment using ellipsoidal mirrors as collection
optics for self generated harmonics . . . . . . . . . . . . . . . . . . . 113
5.21 A sample raw data of third harmonic(266 nm). Each circle (dotted
black lines) represent different cone angle angles of harmonic emission.
The right hand figure shows the specifications of the ellipsoidal mirror.114
5.22 The intensity distribution of third harmonic (266 nm) at various solid
angles. Each color shows the distribution of harmonics at different
theta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.1
The experimental layout for time resolved measurements . . . . . . . 118
6.2
Operating principle of a streak camera [1] . . . . . . . . . . . . . . . 120
6.3
The main interaction beam. The blue line is the raw data and the
dotted red line is a smoothed fitted Gaussian curve . . . . . . . . . . 121
6.4
Top figure is a typical raw data at the highest intensity (∼ 9 ×
1018 W/cm2 ). Figure below shows the P and S polarisation components plotted on a time vs. intensity scale . . . . . . . . . . . . . . . 122
6.5
Plot of p and s harmonics at an intensity 9 × 1018 W/cm2 . The
red line indicates the p- polarisation and the blue line indicates the
s-harmonics. The dotted green line shows the ratio of s/p. . . . . . . 123
12
6.6
Plot of p and s harmonics for intensities 9 × 1018 W/cm2 (A) and
∼ 1 × 1018 W/cm2 (B) . The blue line indicates the p- polarisation
and the red line indicates the s-harmonics. The dotted green line
shows the ratio of ’s/p.’
6.7
. . . . . . . . . . . . . . . . . . . . . . . . . 124
B field measured using Stokes vectors in the earlier chapter with a
short pulse beam (blue line). The respective s/p ratio is plotted for
the same intensity (red line). . . . . . . . . . . . . . . . . . . . . . . . 126
6.8
Osiris PIC simulations showing the evolution of the laser intensity
(red line) and the self-generated magnetic field (blue line) . . . . . . . 126
6.9
Osiris PIC simulations showing the evolution of dc magnetic field is
plotted against laser periods(time) at different laser intensities . . . . 127
8.1
Variation of phase velocity near cut-off region . . . . . . . . . . . . . 142
8.2
Physical sketch of cut-off and resonance
9.1
Polarisation ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2
Poincaré sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.3
Optical field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.4
Co-ordinate system for wave propagation . . . . . . . . . . . . . . . . 160
13
. . . . . . . . . . . . . . . . 142
List of Tables
5.1
Calculation of Stokes vectors from sample data . . . . . . . . . . . . . 99
14
Units
Here are a list of the most used units in this thesis.
Electron density
ne - cm−3
Intensity
I - W cm−2
Magnetic field
Irradiance
Electric field
B - Gauss
Iλ2 - W cm−2 µm2
E - V /cm
Plasma frequency ωp - radians/sec
Temperature
T - eV
15
Acknowledgments
Many people have given me support, encouragement and help throughout the completion of this work; I would like to thank them all.
My supervisor Prof. Karl Krushelnick for his support, funding and advice.
Bucker Dangor for his patience and constant encouragement, and also teaching me
plasma physics.
I owe a big thank you to Michael Tatarakis, big brother, for all the inspiration,
experimental and theoretical expertise he provided me for the completion of this
thesis.
Zulfikar Najmudin, for your invaluable knowledge on plasma physics and Apple computers. I also take this opportunity to thank Mubaraka and you for helping me to
settle down in London.
Our previous head of the group, Prof. Malcom Haines for accepting me as a PhD
student and providing the financial support.
Peter Norreys, you have been very supportive and encouraging during the experiments at RAL.
Matt Zepf for all valuable discussions on Physics and teaching me many experimental techniques.
Roger Evans for helping me with the osiris simulations.
Mingsheng Wei, Ulrich Wagner and Kevin Cassou for spending many late nights at
RAL during the experiments. Farhat Beg and Eugene Clark for their cooperation
and entertainment during the experiments.
I must also thank the RAL staff, Dave Neely, Rob Clarke, Margarette Notley, Rob
Heathcote, Peta Foster, Darren Neville, Pete Brummit, Laser staff of the Astra and
16
Vulcan laser facilities and the Target prep staff.
I would also like to thank my other colleagues, Barney, Stuart and Alex, with whom
I shared the office, for helping me with many physics related and other problems.
John Pasley, for lending me the laptop when I needed it the most and also for being
a good friend and colleague.
I also appreciate Andrew Knight, Diana Moore, Fran Adams and Alison McCann
for their help with many official matters.
I need to thank also my innumerable friends, making me realise that the cultural
and language barriers do not make any hindrance in understanding human values.
Mingi Kam and Sadaf, Silvio and Koti, thank you for always being there for me.
Friends at Vic league, for making my stay in London memorable.
Pieter, words are not enough to say thank you. Dank u voor uw liefde en steun. My
family, mammy and achan for making me what I am today. Ashechi and Annan for
being there for me through thick and thin. Sunilchettan, Dhanya, Puja, Punya and
little Nandana for all your love and support.
17
Chapter 1
Introduction
Everyday experience suggests that most of the matter in the universe exists in three
forms: solid, liquid and gas. In actual fact, approximately 99% of the universe is
thought to be made up of plasma. Plasma is a quasi-neutral state of charged and
neutral particles where the motion is governed both by hydrodynamics and electromagnetic forces. All of the stars including the sun are thought to be entirely
composed of plasma.
The source of energy for the stars is nuclear fusion. Nuclear fusion is the process
of combining two light nuclei into a heavy nucleus. However, in order to combine
the nuclei it is necessary to overcome the repulsive nuclear potential. In the sun
the hydrogen nuclei combine to form helium at temperatures of tens of millions of
degrees Celsius (> keV ). It is of great importance to find ways to release this energy
in the laboratory as an alternative source of energy as reserves of conventional fossil
sources of energy are diminishing rapidly.
The most suitable fusion reaction is that between Deuterium and Tritium (D-T):
D + T → He4 + n + ∆E(17.65M eV )
(1.1)
For nuclear fusion to be an alternative energy source, it is necessary to heat the
mixture of D-T to temperatures required for fusion reactions to commence, and to
confine it long enough so that more energy is released than that used for plasma
18
heating and confinement. In order to achieve breakeven the Lawson criterion need
to be fulfilled. The criterion is that the product of the number density n and the
energy confinement time τ should be greater than 1020 s/m3 .
There are various ways which have been proposed to produce controlled thermonuclear fusion in the laboratory. Magnetic confinement (MCF) was the first method
proposed for commercial energy production using nuclear fusion. In the MCF approach it is intended that D-T plasma of density 1014 cm−3 is confined at a temperature of 10 keV by a powerful magnetic field for tens of seconds [2]. The inertial
confinement (ICF) approach has the same breakeven condition, but the density is
increased by many orders of magnitude required, and thus decreases the confinement
time. The basic idea is that if the density is high enough, the material can react
before the plasma expands significantly [3]. In ICF a spherical mass of the D-T
fuel is heated by a laser by depositing energy at the outside of the sphere. The hot
plasma, which ablates outwards, produces compression waves propagating inwards
due to momentum conservation. The invention of high power lasers has paved the
way for this technology [4, 5]. Present state-of-the-art lasers can deposit energies of
Mega joules in a pulse with several nanoseconds. Also the invention of the chirped
pulse amplification technique (discussed in detail in section 3.2) helped to concentrate hundreds of joules of energy into a sub picosecond pulse. This development
has revolutionised the field of high intensity laser - plasma interaction.
The coupling of high power laser light with matter is an extremely rich topic with
numerous other applications to x-ray lasers, particle accelerators etc. During such
interactions, various processes lead to the coupling of laser light to plasma and also
various other phenomena occur, such as harmonic generation [6,7], multi MegaGauss
magnetic field generation [8–13] and energetic charged particle generation occur. A
detailed discussion of these processes is provided in chapter 2.
The large magnetic fields are of great significance in laser fusion experiments, as
they can affect the energy flow from the light absorption zone to the ablation re19
gion. i.e., the thermal conductivity will drop with an increase in the ratio of electron
mean free path to the Larmor radius. It is also possible that the strength of these
fields can become comparable to those that may exist in many astronomical bodies.
Hence creating models of such astronomical systems in the laboratory may open up
new avenues of research in astrophysics.
1.1
An overview of previous work
The first observation of self generated magnetic fields in a plasma produced in optical breakdown in air was carried out by Korobkin and Serov in 1966 [14]. Later in
1972, Stamper et al. [15] recorded the first measurement of self generated magnetic
fields in laser produced high density plasmas. The studies on solid and spherical targets [14,16–22] were carried out by external physical probes. At first, measurements
were carried out using magnetic probes made of coils connected to a fast oscillograph. The mobile magnetic probes were placed near the irradiated target and the
∂B
signal obtained was proportional to ∂t . The spatial and temporal resolution of the
measurements was dependent on the coil size, the inductance of the coil and the
frequency bandwidth of the recording oscillograph. The maximum field measured
was 1 M G. These results were based up on the measurements in the low density region and extrapolating the result to the high density coronal region. Measurements
based on contact techniques have many limitations, and could not explore the high
density region where the temperature is hundreds of electron volts.
The development of optical methods made measurements more reliable. These were
based on measuring the rotation of the polarisation of linearly polarised radiation
propagating through a magnetised plasma or measuring the splitting of spectral
lines emitted by magnetised plasma. The first method is called the Faraday rotation method and the latter is called the Zeeman effect. For Faraday rotation
measurements, an external probe beam, usually a linearly polarised second harmonic of the incident laser is sent across the expanding plasma [23–26]. The first
measurement of magnetic fields using Faraday rotation was carried out by Stamper
20
et al. [27]. The maximum recorded field using Faraday rotation measurements is
∼ 10M G [28]. The measured rotation angle of the plane polarised radiation after
R
going through the magnetised plasma is proportional to ne Bdl (a discussion of the
theory of Faraday rotation measurements is given in section 4.2.1.). This method,
however, has several limitations: first, to determine the magnetic field it is necessary to measure the electron density simultaneously. This is normally done using
interferometry, shadowgraphy or Schlieren photography. Secondly, the probe beam
propagating through the plasma can also experience refraction, birefringence and
depolarisation due to inhomogeneous magnetic fields and density gradients. Third,
the highest density region is opaque to external optical probes and this is where the
largest fields are predicted to exist.
The measurements using Zeeman splitting of spectral lines has advantages over
Faraday rotation measurements, as they do not depend explicitly on electron density. However, at high temperatures and densities Stark and Doppler shifting of
spectral lines become important and can make Zeeman splitting difficult to observe.
The highest field measured using this technique was 0.5 M G [29].
The use of protons has facilitated the measurement of magnetic fields as high as
30 M G [30]. These measurements give an estimate of the strength of the magnetic
field existing inside a solid density target which are created due to the fast electron
current. However, the analytical and computational calculations based on various magnetic field generating mechanisms estimate that a field of several hundred
MegaGauss [8,9,15] should be present. This indicates that there is a requirement to
develop new diagnostic methods that are capable of making realistic measurements
of the self generated magnetic fields in these ultra-high field regions.
1.2
Scope of the thesis
This thesis discusses the measurement of large magnetic fields generated during ultra
short laser solid interactions. Two new techniques using the self generated harmonics
21
of the incident laser are developed in this thesis. When a high power laser interacts
with matter harmonics of the incident laser are generated. Harmonics are electromagnetic emissions at multiples of the incident laser beam frequency (the generation
mechanism is explained in section 2.5). It is also known that the harmonics are generated at the critical density surface at the same time as the magnetic fields are
generated. The polarisation of electromagnetic radiation changes as it propagates
through a magnetised medium. Hence it would be ideal to study the polarisation
properties of the self generated harmonics before and after going through the magnetised plasma, as this would give information about the propagating medium. Plasma
in the presence of a magnetic field can act as a birefringent medium. Therefore, the
harmonics propagating out of the plasma with their propagation vectors perpendicular to the magnetic field experience an induced birefringence.
Two new techniques have been developed for the measurement of magnetic fields
during short pulse laser plasma interactions. The first technique is called the cutoff method where the p-component (the electric field vector is perpendicular to the
magnetic field) of the self generated harmonic decreases with increased magnetic
field (or laser intensity) and experience cut-off. This is a simple direct measure of
the magnetic field. The second approach is called the harmonic polarimetry method
using the Cotton-Mouton effect. This technique can be used when no cut-offs of the
self generated harmonics are observable. The highest recorded magnetic field in a
laboratory is measured using these techniques [31, 32].
1.3
Thesis outline
The thesis is divided into chapters as follows:
Chapter 2 discusses the basic theory of laser matter interactions, followed by a
detailed discussion of the dominant mechanisms of magnetic field generation relevant to this thesis.
22
Chapter 3 describes the laser systems which have been used to perform the experiments.
Chapter 4 explains the theory of the cut-off method and describes the measurements made using the Vulcan laser system. This includes the detection of the highest
magnetic fields ever recorded in the laboratory [32].
Chapter 5 details the self generated magnetic field measurements made using the
harmonic polarimetry method using the Cotton-Mouton effect. The theory is discussed in the first section followed by a description of the experiments carried out
using the Vulcan and Astra laser systems. The experimental results are compared
with computer simulations carried out using the Osiris particle in cell code (PIC).
The simulation results at oblique incidence show non-uniformities in the spatial distribution of the magnetic field. This is subsequently studied using the Vulcan laser.
In the final section a new diagnostic technique is discussed that allows for the mapping of the magnetic field using harmonic polarimetry.
Chapter 6 considers the temporal evolution of the magnetic field. An experiment was carried out using a long pulse (∼ 8 ps) CPA beam and the self generated
third harmonic was used as the diagnostic. A fast optical streak camera with ∼ 1ps
resolution was used as the detector. The experimental results are discussed in detail
followed by PIC simulations and analytical study of the evolution of the magnetic
field with time. The evolution of the magnetic field with laser intensity is in agreement with the simulation results and is also the theory proposed by Sudan [9].
Chapter 7 summarises the thesis and discusses possible future research directions.
Appendix I gives the derivation of the plasma refractive index.
Appendix II gives the derivation of Stokes vectors which are used for the har23
monic polarimetry.
1.4
The role of the author
The results presented in chapter 4 and the first part of chapter 5 were part of a
single experiment performed on the Vulcan picosecond beam. This was carried out
along with Ulrich Wagner and Dr. Michael Tatarakis. The author was involved in
setting up the polarimeters, data acquisition and target alignment. The analysis of
the data was performed solely by the author.
The spatial and temporal asymmetry measurements were carried out on the Vulcan
laser facility and the author was part of a team of several people who ran various
diagnostics simultaneously. The author was involved in the planning, and was fully
responsible for all aspects of target area operation and day-to-day running including
shot selection, set-up, and laser alignment and focussing. The polarimeters for the
spatial measurements were set up by the author. The set up of the streak camera for
time resolved measurements was assisted by Dr. Matt Zepf. The data acquisition
and analysis were also the sole responsibility of the author.
The author was responsible for the planning and implementation of the Astra experiments. The author was solely responsible for the day-to-day running of the target
area. The data acquisition and analysis were carried out by the author.
The results presented in this thesis are compared with the particle-in-cell simulations using the OSIRIS code developed by UCLA. The code was implemented and
run for different experimental conditions with the assistance of Roger Evans.
24
Chapter 2
Laser-matter interaction and
magnetic field generation
2.1
Introduction
In this chapter a brief overview of the physical mechanisms of high power laser
matter interaction is discussed. The theory of ultra large magnetic field generation
is given in the last section. Advances in high power laser technology allow the present
state-of-the -art lasers to deliver kilo joules of energy in pulses of picosecond regime.
These laser pulses can be focussed down to a diffraction limited spot sizes and an
focussed intensity of 1021 W/cm2 can be achieved. The electric field of these lasers
is strong enough to overcome the electrostatic Coulomb potential. For hydrogen
atoms the electric field at a distance of the Bohr radius is given as
Ehydrogen =
1 e
4πo a20
(2.1)
2
~
18
−2
where the Bohr radius a0 = mc
,
2 = 0.51Å. For laser intensities higher than 10 W cm
the electric field is of the order of 1012 V cm−1 , which is many times higher than the
atomic Coulomb field and can easily ionise the atom. The large electric fields associated with the laser pulse ionises the material in first few optical cycles of the
laser pulse and a plasma is formed rapidly. The free electrons in the plasma are
driven by the laser field to relativistic quiver energies at intensities higher than 1018
W/cm2 . The quiver energy is the energy acquired by the electron oscillating in a
25
laser field. At relativistic intensities the quiver energy becomes comparable to the
rest mass energy and will alter the physics of the interaction. The laser intensity can
be expressed in terms of quiver momentum or the normalised laser vector potential
ao .
posc
me c
(2.2)
Iλ2µm
1.37 × 1018
(2.3)
ao ≡
ao ≡
eEo
=
me ωc
s
where posc = me vosc is the non relativistic quiver momentum of the electron in the
laser electric field with an amplitude Eo , I is the laser intensity in W/cm2 and λµm
is the wavelength is microns. Depending on whether the value of ao > 1 or ao < 1
the intensity is above or below the relativistic limit. The relativistic factor γ can be
calculated as
γ=q
1
1−
(2.4)
p2osc
γ 2 m2e c2
i.e.,
γ=q
Solving for γ gives, γ =
p
1
1−
(2.5)
a2o
γ2
1 + a2o for circularly polarised light and γ =
q
1+
a2o
2
for
linearly polarised light.
2.2
2.2.1
Plasma parameters
Plasma frequency
The electron plasma frequency is the natural frequency of the plasma and gives
the fundamental time scale in plasma physics. This quantity represents the natural
frequency of electrons when they are displaced from their equilibrium position in
the plasma. The electrostatic field due to the charge separation gives rise to a
restoring force on the electrons and they oscillate past the equilibrium position. At
low temperatures and small amplitudes the electron plasma frequency is given by,
ωp2 =
ne2
o me
26
(2.6)
where n, e, me are the electron density, charge and mass respectively. o is the
permittivity of free space. It can be simplified as,
√
ωp = 5.64 × 104 ne
radians/sec
(2.7)
where ne is in cm−3 .
2.2.2
Dispersion relation
The dispersion relation determines the dispersion characteristics of an electromagnetic wave in ω − k space. Many properties of the medium determine the dispersion
relation such as plasma density, wave frequency, temperature and magnetic field in
the plasma. A simplified form is shown in figure 2.1. From Maxwell’s equations the
ω
ω=ck
ωp
no wave propagation
c
0
k
Figure 2.1: Dispersion ω(k) for a transverse electromagnetic wave propagating in cold plasma
dispersion relation for an electromagnetic wave propagating through cold plasma [33]
is given by
ω 2 = ωp2 + k 2 c2
(2.8)
or the index of refraction n
kc
n=
=
ω
ωp2
1− 2
ω
27
1/2
(2.9)
where k is the wave number.
Therefore, for an electromagnetic wave to propagate through plasma, ω > ωp . When
= 0 and phase velocity vφ = ωk = ∞, the wave
ω = ωp the group velocity vg = ∂ω
∂k
will be reflected at the critical density, nc .
o m e ω 2
nc =
= 1.1 × 1021
2
e
1µm
λL
2
cm−3
(2.10)
where λL is the wavelength. A cut-off of the incident wave occurs when the tangent
ck
to the dispersion curve is horizontal. i.e., refractive index n = ω being zero.
In solid plasmas, beyond the critical density surface i.e. ωp > ω , the k vector is
imaginary and the wave decays evanescently. Beyond this surface the amplitude
of the electric field decreases exponentially over a characteristic distance called the
skin depth δ.
Skin effect
The skin depth is the distance over which an electromagnetic wave can penetrate
beyond the critical density surface. Inside the skin depth the electric field of the
laser can accelerate the electrons and the energy is dissipated to plasma via collisions
with ions [34]. The skin depth, δ is given as,
δ=
c
ωpe
(2.11)
The skin effect is more effective at relatively low intensities (I ≤ 1016 W/cm2 ),
where the electron mean free path is much lower than the skin depth. Therefore,
the distance over which a thermal electron penetrates is much less than the mean
free path. As the intensity increases the electron mean free path can become much
larger than the skin depth and the electron energy is directly coupled to the plasma.
This process is called the anomalous skin effect [35]. The critical density surface is
the point where most laser energy absorption usually takes place.
28
At relativistic intensities γ > 1 (Iλ2 > 1018 W cm−2 µm2 ) the plasma frequency will
decrease due to the relativistic increase in electron mass. Therefore, the modified
dispersion relation will be
ωp2
ω =k c +
γ
2
2 2
and hence the relativistic refractive index will become
1/2 1/2
ωp2
ne
n= 1−
= 1−
γω 2
γnc
(2.12)
(2.13)
and the electromagnetic radiation will be able to propagate to higher densities before
being reflected.
2.2.3
Ponderomotive force
The ponderomotive force arises from the intensity gradient in the laser pulse. The
equation of motion of an electron in an electromagnetic field is
F=m
dv
= e (E + v × B)
dt
(2.14)
where E = E0 cos(ωt − kx)ẑ, v is the electron oscillatory velocity or quiver velocity.
The v × B term is usually negligible for low intensities.
In short pulse lasers these intensity gradients are high and quite significant. The
electrons oscillating in such a field quiver faster in the high intensity region than
the low intensity region in the opposite phase of the pulse. Therefore, there is a net
movement in the high intensity region of the pulse. The ponderomotive force is then
the net movement due to the Lorentz force experienced by an electron in a spatially
varying electric field and is given by
F = e {E (x, y, z : r1 = 0) + r1 · Er1 =0 + v × B}
(2.15)
where r1 is the first order displacement of the electron ignoring the v × B term.
The first term gives the force due to the electric field. The second and third terms
are second order non-linear force terms. The time averaged non-linear force can be
calculated as
hFN L i = − e2 1 + a2 /4mω 2 E 2 (x, y, z)
29
(2.16)
i.e. hFN L i = − {e2 (1 + a2 ) /4mω 2 } E 2 ∝ −UP ponderomotive energy. The gradient
of the ponderomotive energy is called the ponderomotive force and it arises due to
two types of gradients.
1. radial ponderomotive force (figure 2.2a) arises due to variation of intensity in the
radial direction such that the electrons are pushed radially outwards.
2. longitudinal ponderomotive force : Since the intensity is also varying in time the
electrons will experience a ponderomotive force in the beam propagation direction.
This gives rise to a longitudinal ponderomotive force (figure 2.2b), which affects
the behaviour of relativistically self-guided beams, and can lead to the longitudinal
modulation of the laser beam at the plasma frequency. At relativistic intensities
r
Fp α − ∇I
I
F pα −
∂I
∂t
I
z=ct
Figure 2.2: a. radial ponderomotive force b. longitudinal ponderomotive
force
a ≥ 1, the electrons oscillate relativistically and the electron quiver velocity is given
eE
as v = γmω , where γ is the Lorentz factor.
2.2.4
Density Scalelength
The interaction of an intense laser pulse with solid material produces an ablated
coronal plasma that expands rapidly at the sound speed. In short pulse experiments the main pulse is accompanied by a pedestal pulse. i.e., before the arrival of
30
the main pulse a pre-plasma is formed. The density scale length is determined by
the amount of pre-plasma formed.
1 dn
The plasma density scale length is given by, L−1 = n dx , where n is the electron denq
kB (ZTe +Ti )
sity. For a freely expanding plasma, L ∼ cs τ and cs =
≈ 107 m/s, is
mi
the ion acoustic speed and τ is the time from plasma formation. For a prepulse free
short pulse interaction there is not enough time for the coronal plasma to expand
which leads to a sharp density gradient for the main pulse. The major mechanisms
for laser energy coupling to the plasma are discussed below.
2.3
Absorption mechanisms
The efficient transfer of laser energy to the plasma is important in laser fusion
experiments. The extent of energy transferred to the plasma depends on many
parameters such as absorption processes, parametric instabilities, density scalelength
etc. There are many mechanisms through which the laser energy is coupled to the
plasma. Collisional absorption or inverse bremsstrahlung [36–38] is dominant at
intensities (Iλ2 ) ≤ 1015 W cm−2 µm2 . When the laser intensity is not high and the
coronal temperature is not too high electron-ion collisions become dominant. Above
intensities 1014 W/cm2 , most of the absorption depends on Iλ2 , making thus the
absorption phenomena depend on laser wavelength. At higher intensities (Iλ2 >
1018 W cm−2 µm2 ), the electron motion becomes relativistic, resonance absorption,
vacuum heating and the anomalous skin effect become dominant.
2.3.1
Inverse bremsstrahlung
As the name indicates inverse bremsstrahlung is the reverse of bremsstrahlung. During bremsstrahlung a photon is emitted when an electron is decelerated by the
Coulomb field of the nucleus. In collisional absorption or inverse bremsstrahlung,
an electron acquires energy from the electric field of the laser during a collision with
an ion. The electrons oscillating in the electric field of the laser loses energy to the
31
Figure 2.3: Schematic representation of plasma produced by an intense laser
pulse interaction with solid target showing main laser plasma
interactions.
stationary ion via collisions. Including the collision term in the equation of motion
of the electron in an electromagnetic wave (E = Re {E0 eiωt }) gives us
me
dv
+ me υei v = −eE
dt
(2.17)
where, v is the electron velocity and υei is the electron-ion collision frequency. Solving the above equation gives
eE
v=
me ω
i−
1+
υei
ω
υei 2
ω
!
(2.18)
The current generated in the plasma J = ne ev
2
∴J =
i−
ne E
me ω
1+
υei
ω
υei 2
ω
!
(2.19)
The rate at which the energy is absorbed due to collisions will be 2εq υei , where εq
e2 E02
is the quiver energy
. The absorption coefficient is given as [39, 40]
4me ω 2
−7
kIB = 3.1 × 10
Zn2e lnΛIB Te−3/2 ω −2
32
ωp2
1− 2
ω
1/2
cm−1
(2.20)
where lnΛIB is the Coulomb logarithm [35] and Z is the charge of the ions. At low
intensities collisional absorption is effective and is in good agreement with experiments. At intensities greater than 1015 W/cm2 collisions become ineffective during
the interaction [41].
2.3.2
Resonance absorption
Resonance absorption occurs when a linearly polarized electromagnetic wave is incident obliquely on a plasma density gradient and resonantly excites plasma waves
at the critical surface [42]. The energy is transferred to the plasma by collisional
or collisionless damping of the plasma waves. Consider an electromagnetic wave
incident on the plasma density gradient at an angle θ which is specularly reflected
by the density gradient. At oblique incidence, p polarised light has components
of electric field perpendicular and co-planar to the plane of incidence (figure 2.4).
Light is reflected at a density much lower than the critical density. For s-polarised
Figure 2.4: Resonance absorption for an obliquely incident p-polarised laser
laser beams, which have no ⊥ component, the wave electric field gives rise to electron oscillations in y-direction along which the density is uniform. In the case of
a p-polarised wave, the electric field at the turning point is in the x-direction and
33
causes electrons to oscillate across the non-uniform density region and the wave is
no longer purely electromagnetic. Combining Snells law and the dispersion relation
one can calculate the turning point for the electromagnetic wave. For s-polarised
light, reflection takes place where the density is below the critical density given by
the following equation.
ne (θ) = ncr cos2 θ
(2.21)
where θ is the angle of incidence. From the figure 2.5 it is clear that for an incident
p-polarisation the electric field vector has a component in the plane of k along the
density gradient. Therefore, the electric field will be able to tunnel through the
plasma and couple energy to Langmuir waves [41, 42]. The plasma wave grows
and [38] the interaction between the resonant electrons and Langmuir waves excited
in this way is called resonance absorption. If the electric field vector is perpendicular
(out of plane) the density gradient at the turning point there will not be a component
of E along the density gradient hence no plasma waves. Also, the extent of resonance
absorption depends on the angle of incidence. If θ is too large then the evanescent
electrostatic wave beyond the turning point needs to tunnel too far and the plasma
wave is not driven efficiently. On the other hand if θ is too small and the component
of E along the density gradient at the turning point is small, then the excitation of
plasma waves is much less. The optimum angle for absorption [37] is given as,
1/3
c
sin θ =
(2.22)
ωL L
2.3.3
Vacuum heating
The concept of resonance absorption is dominant in short density scalelength plasmas. In short pulse experiments the coronal plasma does not have much time for
expansion before the main pulse arrives. For sharp density gradients, the electron
v
oscillation amplitude becomes comparable to the scale length ( osc
ω > L), hence they
are directly heated by the laser electric field. The electrons undergo large oscillations
across the plasma vacuum interface and energy is absorbed to the plasma [43, 44].
Computer simulation studies using a particle in cell code showed that a magnetic
field is generated from the average current of electrons drawn from the overdense
34
plasma [43]. A magnetic field is generated without having a temperature gradient.
These magnetic fields saturate the absorption mechanism due to deflection of electron orbits by the v × B force. The saturation effect can be partially overcome by
using two incident beams at ±45 angles. For finite density gradients, the transition
between resonance absorption and vacuum heating depends on the value of Iλ2 and
the scale length [45]. At very high intensities vacuum heating is much larger than
resonance absorption (as the field can penetrate only to the skin depth).
2.3.4
Hole boring
The spatial and temporal variation of the laser pulse can create density gradients
in the plasma by expelling electrons from the axis of the beam thereby focussing
the beam. This is called ponderomotive self- focussing. The extent of steepening
depends on the ratio of the light pressure and thermal pressure of the plasma. At
extreme laser intensities the light pressure is much higher than the thermal pressure
of the plasma and pushes back the critical density surface. This process is called
hole boring [8, 46]. The extent of hole boring can be measured from the Doppler
shifting of the harmonics [47]. Hole boring can lead to shifting harmonic frequencies
from their central frequencies.
∆υh
v
=
υh
c
(2.23)
I
Calculations done by Wilks et al., [48] have shown that c = nM v 2 ,
where, M is the mass of the ion. q
nc Zme Iλ2
v
Hole boring velocity,
=
c
ne mi 2.74×1018
2.3.5
Filamentation and channeling
Filamentation is the process by which the intensity modulation in the original laser
pulse is amplified as it propagates through plasma [49, 50]. The high intensity regions of the beam push the plasma aside and creates low density thereby, a high
index of refraction in the high intensity region. As the wavefront propagates perpendicular to the energy flow, creates more focussing effect. If the electrons are
completely expelled from the focal spot then this process is called cavitation [51,52].
35
So this focussing effect amplifies the non-uniformities initially present in the beam.
Computer simulations and experimental studies have shown this phenomenon. [53]
When a high power short pulse laser interacts with matter various phenomena
occur, such as, generation of energetic particles, x-rays, harmonic generation and
self-generated magnetic fields. Self-generated harmonics and magnetic fields are of
interest in this thesis, hence they will be discussed in detail.
2.4
Harmonic generation
Harmonics generated in the plasma are an important diagnostic tool to understand
many physical processes that can occur during laser matter interactions. This section discusses the processes involved in the generation of harmonics during laser
solid interactions. Harmonics are the higher multiples of the fundamental incident
frequency. The process of harmonic generation dates back to 1961 when Frank et
al., [53] observed the second harmonic of a ruby laser pulse using the non-linear
properties of quartz crystals. The conversion efficiency falls off dramatically for
higher order therefore, use of crystals is restricted to the wavelength regime where
transparent crystals are available. Hence, use of other non-linear media such as
plasma is essential. Experimental observation of harmonic emission at wavelengths
down to 2.3 nm has been reported using a 800 nm Ti-Sapphire laser focussed onto
noble gases [54]. The wavelength range between 2.3 nm and 4.4 nm is called the
water window and has many potential applications in biology (for biological imaging) since carbon is strongly absorbing and oxygen molecules are transparent in this
wavelength range. Harmonic generation from gases is due to the highly non-linear
electronic response to the laser electric field [55–57].
2.4.1
Harmonic generation from solids
HHG from solids was first observed by Burnett et al., [58] in 1977 using a 2ns,
10.6µ m CO2 laser at a focussed intensity of 1014 W/cm2 . They observed harmonics
up to 11th order. Later work carried out by Carman et al., [59] observed the 46th
36
harmonic at an intensity of 1016 W/cm2 . These harmonics are generated due to the
anharmonicity in electron motion across the density gradient produced by the laser
pulse. Theoretical and computational work carried out by Gibbon et al., [60, 61]
explained that short pulse lasers (<1ps) operating at intensities 1018 − 1019 W/cm2 ,
in the visible or IR region could produce harmonics extending into the water window region. In experiments by measurements done by Norreys et al., [6], they
observed harmonics up to 75th order using the Vulcan laser CPA beam at intensities
∼ 1019 W/cm2 . It was also shown that harmonics were produced isotropically and
that the efficiency scales as Iλ2 . The main mechanism of harmonic generation can
be simply described using the moving mirror model. The moving mirror model was
initially proposed by Wilks et al. [8], using PIC simulations and later detailed studies were done by Bulanov et al., [62] and Litchers et al., [63, 64]. No cut-offs were
observed in the harmonic orders while using laser-solid interactions as compared to
the cut-offs observed in gas harmonics.
During the interaction, the head or tail of the incident laser pulse generates a plasma
surface before the arrival of the main pulse. The electrons generated undergo excursions across the vacuum - plasma boundary by the laser electric field and the
plasma restoring force. If the density gradient is smaller than the amplitude of the
electron oscillation then the motion becomes strongly anharmonic. The electron
moving across the vacuum-plasma boundary experiences different forces of restoration as it passes through different local densities at different points of the orbit. This
is responsible for the nonlinear response of the electrons and subsequent generation
of higher harmonics.
2.4.2
Theoretical calculations
The moving mirror model proposed by Bulanov et al., [62] and later modified by
Lichters et al., [63] and von der Linde et al., [65] gave a satisfactory explanation
of harmonics generation during laser -solid interactions. A detailed derivation of
the oscillating critical density surface is given by von der Linde [7]. Consider an
37
Reflected
laser
E
k
Incident
laser
B
Skin depth
Density
Figure 2.5: Moving mirror model, the electrons undergo excursions across
the plasma vacuum boundary. The trajectory of a free electron
in a plane polarised wave is figure of eight
electromagnetic wave with a frequency ω and wave vector k interacting with critical
density surface. As discussed earlier most of the laser light gets reflected from the
critical density surface. The electromagnetic forces of the laser pulls electrons out
of plasma and accelerates them back periodically across the boundary. The motion
of the ions can be neglected, as the laser pulse duration is much shorter than ion
oscillations. There is no modification to the electron density profile as well. Consider
the oscillation of the critical density surface as a mirror oscillating at a frequency ω.
The vibration of this mirror (critical density) surface produces phase modulations
and the spectrum of the reflected light exhibits sidebands of multiple modulation
frequency, which is the incident laser frequency.
The oscillation of the surface can be written in the form
u = uo sin(ωm t − kk x + φ)
(2.24)
The phase shift (φ) of the reflected wave resulting from the sinusoidal displacement
of a reflecting surface is
φ(t) =
2ωu0
c
cos θ sin ωm t
(2.25)
where, θ is the angle of incidence and ωm is the modulation frequency (incident laser
frequency).
38
Figure 2.6: Electric dipole formation at the plasma vacuum boundary
The incident wave can be written as
x
z
sin θ − cos θ)]
(2.26)
c
c
ω
where the incident angle θ is chosen in such a way that cm sin θ = kk . The reflected
wave can be represented as
Einc = Eo exp[−iωm (t −
Eref l = G(t −
x
z
sin θ + cos θ)
c
c
(2.27)
assuming the condition that the total electric field vanishes at the oscillating surface,
i.e.,Einc + Eref l = 0.
χ
2ω uo
x
Making substitutions, τ = ξ+ 2 sin(2ωm ξ), χ = m
c cos θ, ξ = t− c sin θ.
G(τ ) = Eo exp[−i(ωm τ − χ sin(2ωm ξ))]
(2.28)
χ
where ξ = ξ(τ ) is the solution of τ = ξ+ 2 sin(2ωm ξ) The frequency spectrum of the
incident wave can be obtained by taking the Fourier transform of equation 2.28.
The oscillating periodic boundary also acts as a source of harmonics. When the
electrons are pulled out of the periodic boundary to vacuum there is a change in
the charge density at the critical density surface. Oscillation of the surface due to
the laser electric force leads to charge separation spatially and an electric dipole is
39
formed at the boundary, considering ions as immobile. For an obliquely incident s
-polarised light the dipole sheet is oscillating at 2ω. i.e., the dipole sheet produces
p- polarised second harmonic in the specular direction. If the dipole sheet is driven
at sufficiently high velocity then p- polarised higher order even harmonics are generated. A more detailed theoretical calculation is given in [65]
The orbital motion of electrons and polarisation dependence on harmonic
generation - The electrons follow figure of eight motion in a high intensity electromagnetic field as shown in figure 2.7. The trajectory of the electron motion can be
Figure 2.7: Electron orbit for s-polaised and p-polarised light respectively.
obtained by solving the equation of motion
F = mr̈ = e (Ey + v × Bz )
(2.29)
Along with planar motion there is also motion parallel to the critical density surface
eEo
in the direction of wave vector. The maximum height of each lobe is
. For weak
mc2
fields the transverse component dominates over the longitudinal component. The
single electron motion trajectory can be used to describe the collective motion of
electrons. Hence, the dependence of harmonic generation on angle of incidence and
polarisation [7, 66, 67] can be calculated.
2.4.3
Selection rules
P-polarised light - for p-polarised incident light, the electron oscillates in the plane
of incidence. Therefore the electron trajectory is in the same plane as well. i.e.,
40
the normal component of the electron motion oscillates with a frequency ω of the
driving laser thus producing p -polarised odd and even harmonics.
S-polarisation : for s-polarised light at normal incidence the electric field direction is in the plane parallel to the plasma-vacuum interface or perpendicular to the
plane of incidence. Therefore, electrons move in a plane perpendicular to the plane
of incidence twice during a laser cycle and only the longitudinal component of the
electron motion contributes. Therefore the periodic motion is driven by ωm = 2ω
and only s-polarised odd harmonics are produced.
For obliquely incident p-polarised light p-polarised odd and even harmonics are produced. In order to have very efficient generation of harmonics using this mechanism
the contrast ratio of the laser should be high. The work presented in chapters IV,
V and VI have used self-generated solid harmonics as the main diagnostic tool.
2.5
Generation of MegaGauss dc magnetic fields
The generation of dc (by dc meaning that, the quantity is averaged over the fast laser
frequency time scale) magnetic fields in laser-produced plasmas has been a field of active research for more than three decades. Previous studies [8,10,15,27,28,63,68–85]
have shown that fields of the order of several megaGauss are produced during such
interaction. Theoretical and computational studies have predicted that these fields
are of the order of 1 GigaGauss at intensities > 1020 W/cm2 [9]. There are various
mechanisms capable of producing large and small-scale magnetic fields in plasmas.
Figure 2.8 shows a computer simulation of magnetic field distribution for a normal
incident (1µm) laser pulse using the Osiris particle in cell code.
The hot electrons produced during high power laser matter interaction are the
principal source of magnetic generation. The energetic electrons leaving the target generates an electric field which in turn accelerates the ions. In order to keep
charge neutrality there will be a return current and a net current flow which in
turn generates magnetic fields. The growing magnetic field induces a back electric
41
Figure 2.8: Computer simulation studies showing the generation of selfgenerated dc magnetic fields due to various mechanisms at normal incidence for an intensity 1020 W/cm2 using the Osiris 2
1/2 PIC code at a time 162.7(1/ω0 ) The dotted red lines shows
the density profile changing from 0 to 10nc . A - magnetic field
generated due to thermoelectric term, B - field due to ponderomotive force, C - due to fast electron current and instabilities
42
field to oppose the current flow. The magnetic field generated can be calculated by
combining Faradays’ law and the generalised Ohm’s law
∂B
= −∇ × E
∂t
J × B qe × B ∇Pe
∇T
me ∂J
+
E=− v×B−
+ 5
+β
− ηJ −
ne e
ne e
e
ne e2 ∂t
p
2 e
(2.30)
(2.31)
where v is the electron velocity, J is the plasma current density, ∇Pe is the pressure
gradient term, Pe = ne Te . ne is the electron density, Te is the electron temperature,
e charge of the electron, η plasma resistivity [86]. Considering only the major source
terms the we get,
∂B
B
1
= ∇ × (v × B) − ∇ × J ×
+
∇Te × ∇ne − ∇ × (ηJ)
∂t
ne e
ne e
(2.32)
The above equation contains source terms (second and third) as well as dissipative(fourth) and convective loss (first) terms. Each of the major source terms will
be discussed in detail.
2.5.1
∇ne × ∇Te mechanism (Thermo electric mechanism)
Non parallel density and temperature gradients in a hot collisional plasma can give
rise to magnetic field. Biermannn proposed the original concept in the context of
rotating stars in 1950. However, the concept was first used in plasma physics by
Stamper et al., [87] in 1971. In the case of an expanding plasma, the electron pressure
gives rise to charge separation thereby an electric field which in turn accelerate the
ions
i.e.,
ene E = −∇pe
(2.33)
pe = ne kB Te
(2.34)
From Maxwell’s equations,
∂B
∂t
1
∂B
∇×
∇pe =
ne e
∂t
∇×E=−
43
(2.35)
(2.36)
Figure 2.9: DC magnetic field generation due to thermo electric term
or
kB
∂B
∇ne × ∇Te =
ne e
∂t
(2.37)
In the expanding plasma the density gradient is towards the laser E field direction at
the point of interaction (target normal) and the temperature gradient is in the radial
direction. Thus a dc magnetic field is produced when there is an angle between the
temperature and density gradient, and the generated B field is toroidal. The self
generated magnetic field due to this mechanism can be up to several MegaGauss.
For a finite focal spot the density gradient points into the target whereas at the
outer edge the temperature gradient points radially inward. Since the intensity
distribution at the centre of the focal spot is uniform compared to the edges the
magnetic field at the centre will be small as there will be no temperature gradient,
since the laser energy is flat at the centre.
44
Figure 2.10: B field generation due to fast electron current. The return
current generated to balance the charge neutrality creates an
azimuthal magnetic field
2.5.2
Current of fast electrons
Hot electrons generated due to resonance absorption are accelerated in the direction of the density gradient which in turn produces an inward flow of cold electron
current, hence producing a toroidal magnetic field. This mechanism is called the
fountain effect [87]. For an infinite plane geometry no magnetic field generated as the
net current is zero. For small focal spots the finite geometry of the plasma becomes
important. The orientation of the magnetic field generated by this mechanism is
same as that of thermoelectric effect. However, the field can also be asymmetrical
depending on the angle of incidence and the absorption mechanism for ejecting electrons [88].
A simple estimate of the magnetic field is generated due to the electron transport
inside the target can be calculated [89] as follows. The electric field generated during
the interaction is.
E = ηjb
45
(2.38)
where, jb is the return current and η is the resistivity . Applying the quasineutrality
condition, the total current j = jb + jf ast , where jf ast is the current due to hot
electrons. Therefore,
η
∇×B
µo
∂B
η
= ∇ × ηjf ast − ∇ × ∇ × B
∂t
µo
E = −ηjf ast +
(2.39)
(2.40)
Using the equation [90] for plasma resistivity the magnitude of magnetic field growth
due to fast electron current can be calculated as,
∂B
∼
∂t
fabs
0.4
keV
Tb
23 Z ln Λ
20
I
5 × 1019
23 6µm
rs
(2.41)
where, fabs is the fraction of the laser energy absorbed into the fast electrons, ln Λ
is the Coulomb logarithm, I is the laser intensity and Tb is the temperature of the
background electrons and rs is the laser focal spot size.
∂B
∼ 32M G/ps
∂t
(2.42)
Clark et al. [30]. have recorded the magnetic field generated inside the target by
the electron current using high energy protons. The maximum field observed was
∼ 30 M G. Computer simulation studies carried out by Davies [91, 92] by solving
the Fokker-Plank equation for fast electrons observed a line average field of 15 − 20
M G with a peak field up to 30 M G.
Large scale (several MegaGauss) fields can restrict the electron flow across B fields
mv
when the Larmor radius ΓL = eBe is less than collisional mean free path λmf p
under these circumstances the step size for a random walk for the electrons become
ΓL instead of λmf p .
2.5.3
Spatial and temporal variation of the incident laser
pulse (ponderomotive force term)
This theory was originally proposed by Sudan [9], to explain the generation of large
scale magnetic fields generated (several hundred megaGauss) when an ultra intense
short pulse interacts with solid density plasma. This field arises from the dc current
46
generated by the temporal and spatial non-uniformities of the ponderomotive force
exerted by the laser pulse on the plasma electrons. Magnetic field generated due to
this mechanism is comparable to the oscillating laser magnetic field. Also the duration of the magnetic field is same as that of the main pulse. A detailed analytical
solution is given in [9].
The plasma is considered as singly ionised. When
ωt <
mi
me
1/2 ω
ωe
eA
me c2
−1
(2.43)
the ion motion can be neglected. ω is the laser frequency and ωp is plasma frequency.
A is the magnitude of laser vector potential. me and mi are the electron mass and
ion mass respectively. The electron momentum equation gives,
P = γme ve −
where γ = 1 −
v2
c2
−1/2
eA
c
(2.44)
is the relativistic factor, A is the vector potential and ve is
the electron fluid velocity. Taking the curl of the momentum gives
∂ (∇ × P)
= ∇ × (ve × ∇ × P)
∂t
(2.45)
Initially electrons have zero momentum and hence, ∇ × P = 0 i.e., ∇ × P is going to
be zero all the time. Also the electron fluid is still at t = 0 before the arrival of the
main laser pulse then P = 0 at t = 0 everywhere an ∇ × P = 0 everywhere. Also
if the electron fluid is still initially then P = 0 at t = 0 everywhere and thus P = ∇ψ.
Maxwell’s equations and the relativistic cold electron fluid equations can be
written as
∂γme ve
∂A
2
= e ∇φ +
− ve × (∇ × A)
∂t
∂t
(2.46)
∂ne
+ ∇ · (ne ve ) = 0
∂t
(2.47)
2 ∇2 φ = e (ne − ni )
(2.48)
∇·A=0
(2.49)
47
2
∂2A
∂φ
∇ A− 2 −∇
∂t
∂t
2
= ne eve
(2.50)
ω2
2 = 2 and φ and A are electrostatic and vector potentials respectively.
ωe
Normalising all the variables to dimensionless form.
φ=
ne
ni
eA
rω
eφ
, ne = , ni =
,A =
, t = tω, r =
2
2
me c
mc
c
n
n
where n is the background ion density.
For short pulse laser solid interaction the ion density gradient distance is taken
to be shorter than the laser wavelength. For laser pulse incident normally on the
plasma vacuum interface with linear or circular polarisation the skin depth is of the
order of laser wavelength. i.e. the density gradient is higher in the propagation
direction than the transverse direction (determined by the spot size). Also the laser
pulse amplitude varies on a time scale slower than the laser period. Expanding the
equations 2.46- 2.50 spatially and temporally,
1
z
a⊥ = [A⊥ ( , x⊥ , 2 t) exp−it +..] + A0⊥ z/, x⊥ , 2 t
2
1 2
az = Az z/, x⊥ , 2 t exp−it + .. + 3 A0z z/, x⊥ , 2 t
2
1 φ = φo z/, x⊥ , 2 t + φ1 z/, x⊥ , 2 t exp−it + ...
2
1 2
ψ = ψo z/, x⊥ , t + ψ1 z/, x⊥ , 2 t exp−it + ..
2
1 2
n = 1 + no z/, x⊥ , t + n1 z/, x⊥ , 2 t exp−it + ..
2
(2.51)
(2.52)
(2.53)
(2.54)
(2.55)
The quantities with subscript and superscript zero are time averaged over a laser
period and the laser frequency is normalised to unity. Substituting the above expansions (equation 2.51-2.55) in equations 2.46- 2.50,
12
1
2
γ0 = 1 + |A⊥ |
= 1 + φ0
2
∂ 2 φ0
= n0
∂ζ 2
∂ 2 A⊥
= (1 + n0 ) A⊥ /γ0
∂ζ 2
where ζ =
(2.56)
(2.57)
(2.58)
z
Equation 2.58 has the solution,
√ √ 1
1
−1
tanh
sinh
A⊥ / 2 = exp−ζ tanh[ sinh−1 A⊥ / 2 ]
4
4
48
(2.59)
Here A⊥ is the resultant amplitude of the incident and reflected laser pulse. The
density n0 and potential φ0 can be obtained from 2.56 -2.57. Applying the boundary conditions, we can obtain the linear equations for average dc magnetic field
generation,
3
Bdc = ∇ × A⊥ + ẑAz
dc
= θ̂
∂Ar
∂ζ
,
(2.60)
dc
From the above solution it is clear that the dc B-field generated is comparable to
the laser magnetic field. The magnetic field will be maximum at z = 0 and will fall
off rapidly with z and depends on A⊥ of laser. Also it vanishes on axis at r = 0.
Intensities of the order of 1020 W/cm2 the predicted magnetic field is 109 G. However, experimental measurements are yet to prove it. The direction of the B field is
azimuthal and can be determined by the electron flow in the z direction.
Apart from large scale dc magnetic fields there are also small scale magnetic fields
produced due to other mechanisms that are explained below.
2.5.4
B fields due to resonance absorption
Theoretical and computational studies [68, 93–95] have suggested the generation of
MegaGauss magnetic fields due to resonance absorption of incident laser light [96].
The electric field can be written as ,
E=−
1
∇ · PR
ene
(2.61)
where, PR is the electromagnetic stress tensor [93]. The growth of magnetic field
can be written as,
∂B
1
= −∇ × E = ∇ ×
∇ · PR
∂t
ene
(2.62)
A steady state B field can arise from balancing the momentum imparted to the
high frequency plasma by high frequency fields with momentum convected out of
the resonance region or with momentum dissipated by local drag forces. In the
presence of absorption ∇ × (∇ · PR ) does not vanish and can give rise to dc magnetic
field. Magnetic fields of the order of several MegaGauss can be produced by this
mechanism. The field is localised near the critical density surface over a fraction of
49
vacuum laser wavelength hence, it does not play a major role in inhibiting the heat
flow towards higher densities.
2.5.5
∇ne × ∇Te field near filaments, composition jumps,
shocks
Initial intensity modulations in the incident laser pulse become amplified as it propagates through the plasma towards the critical density surface [23]. If the intensity
modulation affects only a small portion of the beam, then it is called filamentation.
Temperature and density gradient of small scale are associated with these filaments
and can produce magnetic fields.
Also, strong density gradients can occur near composition jumps or shock waves
and can produce ∇ne × ∇Te magnetic fields. Experimental observation of magnetic
field due to these mechanisms is reported in Raven et al., [26].
2.5.6
B field due to thermal instabilities
Dependence of magnetic field on the electron thermal conductivity can create a
thermal instability that can generate dc magnetic fields [69]. The perturbation in
the plasma temperature varies across the density gradient and creates small scale
magnetic field due to ∇ne × ∇Te mechanism. The dependence of B field on electron
thermal conductivity produces heat flow such that the original temperature perturbation is enhanced and this acts as a feedback mechanism for instability. However,
this effect requires larger instability wavelength compared to the collisional mean
free path. The wavelength of magnetic perturbations giving maximum growth (when
Z 1) is given by
λmax ' 4
c
ωp λmf p
12
(Ln LT )1/2
(2.63)
where λmf p is the mean free path length. Magnetic field generated due to this
thermal instability mechanism is significant in high Z plasmas. Ln and LT are the
scalelength of the of the density and temperature.
50
2.5.7
B field due to Weibel instabilities
Electromagnetic instabilities driven by the inward heat flow can also produce small
scale magnetic fields in plasma. The electromagnetic instabilities are generated in
the plasma by temperature gradients. The anisotropy induced by the density and
temperature gradients and classical collisions can give rise to further anisotropy in
the electron distribution function and this can make low frequency electromagnetic
modes with wave vectors orthogonal to the flux flow unstable. The fields generated
due to this mechanism are oscillating, but can appear quasi-static if the electron
transit time is less than the oscillation period. The magnitude of magnetic perturbation is calculated in detail in [97] and is given by,
1/4
δB
ne
≈
2
106 Gauss
1021 cm−3
Te 1µm
1keV Ln
1/2
(2.64)
The requirement for the instability to occur is λinst < λmf p < Ln .
The above mechanisms are capable of generating magnetic fields of the order of
several kiloGauss to GigaGauss with present high power lasers. With several MegankT
Gauss magnetic fields produced, resulting in the reduction in the β = 2
,
(B /2µo )
they may affect the hydrodynamics of laser plasma interaction. The theory predicted by Sudan (discussed in section 2.5.3) predicts magnetic fields of the order of
a megaGauss generated near the critical density region. Also the Larmor radius will
be small compared to the scale length and the geometry of the field will affect the
interaction. Previous efforts to measure the huge fields were limited to few MegaGauss due to the limitations of the technique employed. In the coming chapters two
novel measurement techniques will be presented along with the first experimental
results.
51
Chapter 3
Laser systems and diagnostics
3.1
Introduction
This chapter describes the laser systems used to carry out the experiments described
in this thesis. The experiments were performed at the VULCAN and ASTRA laser
facilities at the Rutherford Appleton Laboratory, UK. The diagnostics to characterise the laser pulses are discussed in the latter part.
The invention of lasers has led to numerous discoveries in physics. Over the past
decade the output power of short pulse lasers has increased many thousand times.
Present high power lasers in operation can deliver pulses in few hundred femtosecond duration with kilojoule energies. This was made possible by the use of the
Chirped Pulse Amplification (CPA) technique [98]. Most of the tabletop ultra short
pulse lasers today use this technology, which allow them to deliver pulses with multiterawatt powers at high repetition rates.
3.2
Chirped Pulse Amplification (CPA)
The introduction of Chirped Pulse Amplification technique has been a giant leap in
the development of high power short pulse lasers. In conventional master oscillator
power amplifier laser systems (MOPA), a prototype of the laser pulse is amplified
using a chain of optical amplifiers. At high intensities this leads to nonlinear prob52
lems such as self-focussing, filamentation, self phase modulation etc. Using the
chirped pulse amplification method, the pulse is temporally stretched using a pair
of gratings. The technique is to stretch the low power short pulse using a positively
dispersive medium, i.e., by relative delay of the different frequency components, by
for example using the different path lengths of a pair of parallel gratings (figure
3.1). This is called chirping. The stretched pulse is then amplified as in a MOPA
Figure 3.1: An illustration of the CPA technique
at low enough intensity to gain energy efficiently without destroying the amplifying
medium. The amplified pulse is then transmitted through a negative dispersion
medium, which compresses the pulse to the original short pulse but at much higher
power. The distance between the gratings has to be adjusted correctly to achieve
the shortest pulse. Both laser systems mentioned in this thesis use this technique
to achieve higher powers.
3.2.1
VULCAN laser system
Vulcan is a versatile, state of the art, high power neodymium: glass laser, which can
produce pulses of several nanosecond to sub picosecond duration [99]. It can deliver
upto 2.5 KJ in eight beams into three target areas depending on user requirements.
The normal operating wavelength is 1054nm. A detailed layout is given in figure
3.3. Except for beam 8 and beam 7 all of the other beams can only be operated
53
Figure 3.2: Different stages of Vulcan CPA laser system
in the long pulse mode. Beam 8 is the main short pulse interaction beam, and can
deliver 100 TW, sub pico second pulses. Beam 7 can also be compressed, but has
its final compression gratings placed in air which makes the final pulse duration
greater than a picosecond. It is generally used for second harmonic probing using
a frequency doubling crystal. These two Terawatt beam lines use CPA techniques
to produce a multi Terawatt beams into Target Area West and a petawatt beam
to the newly constructed PetaWatt Target Area. Experiments described in chapter
IV and V were performed in Target Area West. The VULCAN laser system can
be divided into three stages as shown in figure 3.2. At the front end is a Tsunami
oscillator, which produces ultra short pulses (5nJ/pulse, 80MHz, 120fs), followed
by a stretcher (80Mhz, 2.5nJ/pulse, 2.4ns) to chirp the beam. An optical gating
system is used to select a single 2.5nJ pulse from stage I which is injected to the
second amplification stage. Here the stretched beam is amplified using a series of
pre-amplifiers to pulse energy of 0.5J (300 ps). The pulse is then sent through the
VULCAN main amplifier chain. In the final stage the pulse from the amplifier output (120J, 300ps) is recompressed to ∼ 1ps, 120J, which is the maximum energy
available on target. The maximum energy is limited by the damage threshold of the
compressor gratings. The efficiency of the compressor grating system is 70% which
makes the maximum energy on target ∼85J. The maximum intensity achieved at
focus is 1 × 1020 W/cm2 . Focusing is done using an f/3 off- axis parabolic mirror.
54
Figure 3.3: Schematic layout of VULCAN laser bay
55
A microscope objective-CCD camera combination is used to optimise the position
of the parabola to get an optimal focal spot. The CCD camera is then connected to
the video monitor can help to optimise the parabola. After optimising, the parabola
can be driven along the focal axis to get the smallest focal spot as possible. A 10
micron diameter glass fibre is placed at the centre of the chamber. The parabola
is driven along the focal axis so that the fibre obscures all of the transmitted light.
This defines the best focal position and all the diagnostics are aligned with reference to it. For solid target experiments a new target is replaced for every shot. The
maximum repetition rate is a shot every 20 minutes.
3.2.1.1
Target Area West
Target Area West is designed for short pulse high intensity laser matter experiments.
The CPA beam is three times diffraction limited and can be focused to 10 micron
spot diameter using suitable optics. In our experiments we used an f/3 off-axis
parabola and the smallest focal spot achievable was 10 microns diameter. A leak
(< 1%) from the final turning mirror of beam 8 is taken for diagnostic purpose. The
layout of TAW is shown in figure 3.4 below.
3.2.1.2
The laser diagnostics
It is important to know the laser intensity on target for proper interpretation of
experimental results. The intensity can be calculated as
Energy
Intensity = pulselength×f ocalspot W/cm2 . For ultra short pulse experiments the
measurement of contrast ratio is very important as it can affect the physics of interaction. The pre-pulse present can be of sufficient intensity to ionise the target prior
to the arrival of the main pulse and could change the physics of interaction. The size
and structure of the focal spot plays a crucial role as it affects the intensity of the
pulse. Also any anomalies in the shape of the laser pulse could lead to the creation
of instabilities like filamentation. Hence it is important to measure the shot to shot
laser pulse parameters. Both VULCAN and ASTRA laser systems use a comprehensive suite of laser diagnostics including spectral bandwidth, pulse length, beam
energy, focal spot size and contrast ratio measurements. Spectral bandwidth mea56
Figure 3.4: Schematic layout of Target Area West
57
surements play a very important role in CPA laser systems because any variation in
the pulse can be enhanced while it passes through the system. The beam pointing
monitor ensures that the stretched beam from the laser bay hits the compressor
gratings at the correct angle, thus ensuring that the final pulselength is constant
throughout the experiment. The timing slide adjusts the timing between the main
interaction beam (B8) and the probe beam (B7).
3.2.1.3
Pulse length measurement- Autocorrelation
Shot to shot pulse length measurements are done using second order autocorrelation.
Approximately 1% of the beam is taken as a leak from the Vulcan laser system for
diagnostic purposes. It is important to measure the pulse duration for every shot,
even though the grating alignment is constant, as there may be slight variation in the
oscillator output, as well as that due to gain pulling of the amplifier chain. The pulse
Figure 3.5: Frequency doubling in an autocorrelator
58
duration can vary between 0.7 ps− 1ps. The leakage beam is split into two using a
50% reflectivity mirror and then recombined at an angle inside a frequency doubling
crystal. The pulses are overlapped temporally and spatially. At each overlapping
point second harmonic is generated and its intensity is proportional to the product
of local intensity in each path. The direction of propagation is parallel to the bisector cross over angle. The measurement of intensity as a function of relative delay
between the beams gives the autocorrelation of the pulse envelope. A schematic is
shown in figure 3.5. The pulse duration can be obtained from the FWHM of the
autocorrelation function by assuming the pulse shape to be Gaussian. However, the
second order autocorrelation does not give a measure of the pulse contrast ratio,
because it can not distinguish between a pre-pulse and a post-pulse. Therefore, a
scanning third order autocorrelator is needed for contrast ratio measurements. Basically it is a sum frequency mixing process. Initially a second harmonic of intensity
function I 2 (t)is produced of an optical signal with intensity function I(t). A later
process generates a response by mixing two arbitrary intensity functions I(t) and
I 2 (t). The third order autocorrelation function is given as
R
I (t − τ ) × I 2 (t)dt
A3 (τ ) = R
I(t) × I 2 (t)dt
(3.1)
where, τ is the delay parameter. The function A3 (t) is asymmetric and it is possible
to differentiate the temporal structure before and after the main pulse. The third
order autocorrelator is not a permanent diagnostic as it is a time consuming diagnostic for the low repetition rate VULCAN pulses. However, previous measurements
have indicated that the pre-pulse on Vulcan is < 10−6 × the main pulse [99]
3.2.1.4
Focal spot measurements
The alignment of the focusing optics inside the interaction chamber is done using
a microscope objective and a CCD camera. The parabola is then optimised to get
maximum focused intensity on target. For a Gaussian beam the diffraction limited
spot size is 1.22f λ , where f is the focal number of the focusing optics and λ is
the wavelength. The measured focal spot size has been measured to be 3 − 5 times
the diffraction limited spot size due to wavefront distortions during amplification.
59
The focal spot size is measured using an equivalent plane monitor or by penumbral
imaging. An experimental measurement is given in figure 3.6.
Figure 3.6: Focal spot measurements using equivalent plane monitor
3.2.1.5
The equivalent plane monitor
A long focal length lens (f=10m) is used to measure the beam quality at a plane
equivalent to the focus at the centre of the chamber. The low intensity leakage beam
taken from the final turning mirror inside the chamber ensures that the beam has
not been distorted by B-integral or spatial distortion. Therefore, any deterioration
in the beam quality will be noticeable. Use of a long focal length lens gives a large
image on the detector thereby giving better resolution. The only factor that may
affect the measurement is the alignment of the focussing optics inside the chamber.
This is overcome by initially aligning both systems to give a best focus with a very
low power (high quality) alignment beam.
3.2.1.6
Penumbral imaging
Penumbral imaging gives an alternative measure of the focal spot size. The soft xrays emitted by the target are imaged using a pinhole onto a CCD. Unlike a pinhole
camera the information is contained in the wings of the image (penumbra or half
shadow of the image). Here we assume that during a short pulse interaction soft
x-rays are presumably emitted in the full width half maximum of the focal spot and
that the lateral energy transport is negligible. The magnification of the system is
60
the ratio of the distances camera-pinhole and pinhole-source and thereby one can
calculate the source size.
3.2.1.7
Energy measurements
Energy measurements are made by use of a calorimeter measuring the leakage beam
from a mirror in the laser bay area. This calorimeter is calibrated relative to a
calorimeter placed inside the chamber for full power shots.
3.2.2
ASTRA laser
Astra is a high power ultra short pulse Titanium-sapphire laser facility. It provides
∼ 50fs pulses of 800nm wavelength with a maximum energy 250mJ into 2 target
areas [100]. The standard Master Oscillator-Power Amplifier (MOPA) configuration
Figure 3.7: Schematic layout of ASTRA laser system
is used along with the chirped pulse amplification (CPA) technique to achieve very
short pulses of TW power. The oscillator generates short pulses of 20fs duration
with 5 nJ energy. The short pulse is stretched temporally to 530ps. Prior to sending
it to the main power amplifier chain the pulse is amplified using a pre-amplifier to
1mJ energy. The first power amplifier amplifies the spatially filtered beam from the
pre-amplifier to 200 mJ. Half of the beam is directed to target area I and the rest
is sent to the second power amplifier for further amplification and can produce up
to 1.5J. The pulse is then compressed and sent to Target Area 2 at a frequency
of up to 10 Hz. Due to technical limitations (primarily poor beam quality and
grating degradation) the energy on target has been kept below 250mJ. A schematic
61
lay out of Astra laser system is shown in figure 3.7. The magnetic field mapping
measurements were carried out in target area 2 of the Astra laser system. Both
VULCAN and ASTRA uses similar laser diagnostic techniques.
62
Chapter 4
Magnetic field measurements
using the Cut-off method
4.1
Introduction
The Cut-off method is one of the novel techniques developed in this thesis for the
measurement of self generated magnetic fields during high power laser interaction
with solid targets. In this chapter the first measurements of the magnetic field
strength using the cut-off method [31] are presented. The self-generated harmonics of the incident laser are used as the diagnostic probe. The cut- off method is
based on the effect of a magnetised plasma on an electromagnetic wave propagating
through it.
It was previously postulated that large magnetic fields are generated during high
intensity laser matter interactions and are localised near the critical density region
which is opaque to most of the external probing methods [28,101]. The use of external material probes can perturb the plasma equilibrium. Hence, the electromagnetic
waves are an excellent probe to understand the internal processes happening during
high power laser matter interaction provided their intensity is not too high. However, external optical probing with visible or ultra violet probe beams is limited due
to :
63
i) refraction effects at higher densities and large density gradients
ii) strong self emission at visible wavelengths.
Also the small spatial (10µm) and temporal (< 1ps) scale of the plasma produced
during short pulse laser matter interaction make the use of conventional methods
difficult.
Therefore self-generated harmonics are a useful tool since :
a) they are generated at the critical density surface and propagate out of the dense
region isotropically [67, 102].
b) they are generated about at the same time as the magnetic field.
c) and are linearly polarised with the same polarisation as the p-polarised incident
laser.
In previous experiments up to the 75th harmonic has been observed with a conversion efficiency of 10−6 [102]. The polarisation properties of these self-generated
harmonics are explained in chapter 2 and have been verified experimentally with
low energy (< 1 J) shots. In order to understand the cut-off method it is necessary
to be familiar with the behaviour of electromagnetic wave propagation in plasma.
4.2
Theory
Electromagnetic radiation propagating through a plasma in the presence of a magnetic field will experience optical activity (Faraday rotation) or birefringence (the
Cotton-Mouton effect) depending on the direction of propagation of the electromagnetic wave with respect to the magnetic field B.
4.2.1
Faraday rotation (propagation parallel to the magnetic field, k k B)
Linearly polarised light can be represented as a combination of left circularly polarised light and right circularly polarised light with different phase velocities as
64
shown in figure 4.1 and can be represented mathematically by two waves of refrac-
Figure 4.1: Graphical representation of linearly polarised light
tive index n− and n+ as right handed cicularly polarised and left handed circularly
polarised waves propagating in the z direction.
ω
(n+ z − ct)]
c
ω
Ey = E0 sin[ (n+ z − ct)]
c
Ex = E0 cos[
ω
(n− z − ct)]
c
ω
Ey = −E0 sin[ (n− z − ct)]
c
Ex = E0 cos[
(4.1)
(4.2)
(4.3)
(4.4)
When they propagate through a magnetised medium, each wave will travel with a
different phase velocity. The wave amplitude at any position can be measured by
superimposing the waves at that point.
n+ + n− ω
4φ
4φ
Ez = E0 exp i
z cos
, sin
2
c
2
2
(4.5)
the phase difference due to the difference in the refractive indices of the waves is,
4φ = (n+ − n− )
ω
z
c
(4.6)
hence the polarisation angle is
α=
1
ω
4φ
= (n+ − n− ) z
2
2
c
65
(4.7)
this means that for radiation traveling parallel (k k B0 ) the wave remains linearly
polarised but the polarisation is rotated by an angle and this rotation angle increases
with distance travelled by the wave along the direction of magnetic field. This is
called Faraday rotation. The total rotation angle can be obtained by taking the
integral over the path length [103]. Using the dispersion relation discussed later in
this chapter the angle of rotation can be written as,
Z
4φ
e
ne B0 · dl
α=
=
12
2
2me c
nc 1 − nnec
(4.8)
where, ne , me , nc and dl are the electron density, electron mass, critical density,
and the optical path length respectively. For Faraday rotation measurements in
laser plasma experiments a probe beam (frequency multiple of the main interaction
beam) is typically sent across the plasma expansion direction. However, the region
over which measurements can be made is limited by the refraction of the beam due
to steep density gradients.
4.2.2
Cotton-Mouton effect (propagation perpendicular to
the magnetic field, k ⊥ B)
When radiation propagates perpendicular to the magnetic field (k ⊥ B0 ), the electric field vector can be parallel or perpendicular to the magnetic field and can give
rise to two kinds of waves. The incident ray is split into an ordinary wave (o-wave)
(E k B0 ) and an extra -ordinary wave (x-wave) (E ⊥ B0 ) depending on the direction of orientation of the electric field vector with respect to the magnetic field. The
o - wave has a polarisation vector parallel to the magnetic field and for the x-wave
the polarisation vector is perpendicular to the magnetic field. The o-waves travel
slower than the x-waves. i.e., the index of refraction of an o-wave is higher than
that of x-wave. The difference between the index of refraction between the o-wave
and x-wave gives the degree of birefringence of the medium.
66
Figure 4.2: Wave propagation perpendicular to an external magnetic field
4.2.3
Electromagnetic wave propagation in plasma
The dispersion relation of an electromagnetic wave propagating in a magnetised
plasma is obtained from Maxwell’s equations,
∂B
,
∇×E=−
∂t
∂E
∇ × B = µ0 j + 0
∂t
(4.9)
(4.10)
We know that any periodic motion can be decomposed into sinusoidal oscillations
of frequencies ω using Fourier analysis. Therefore, the Fourier components of the
electric and magnetic fields can be written as,
E = E0 e−i(ωt−k·r)
(4.11)
B = B0 e−i(ωt−k·r)
(4.12)
where, ω is the wave frequency and k is the wave vector. Linearising equations 4.9
and 4.10 gives
ik × E = iωB
67
(4.13)
ik × B = −iω0 µ0 K · E
(4.14)
where, K is the dielectric tensor (a detailed derivation is given in Appendix I ).
kc
Eliminating B from equation 4.14 and substituting for the wave vector n = ω ,
n × (n × E) + K · E = 0
(4.15)
where the refractive index vector n has the same direction as the wave vector k. k is
in the x-z plane and the external magnetic field B0 in the z - direction and θ is the
angle between the propagation vector k and the z-axis, then equation 4.15 becomes



S − n2 cos2 θ −iD n2 cos θ sin θ
Ex






2
(4.16)



iD
S−n
0
E =0

 y 
n2 cos θ sin θ
0
P − n2 sin2 θ
Ez
where,
ωp2
S =1− 2
ω − ωc2
ωc ωp2
D=
ω (ω 2 − ωc2 )
ωp2
P =1− 2
ω
(4.17)
(4.18)
(4.19)
In order to have a nontrivial solution, the determinant of coefficients of should
vanish. This condition gives the cold plasma relation [104],
An4 − Bn2 + C = 0
(4.20)
where,
A = S sin2 θ + P cos2 θ
(4.21)
B = RL sin2 θ + P S(1 + cos2 θ)
(4.22)
C = P RL
(4.23)
RL = S 2 − D2
(4.24)
where,
1
(R + L)
2
1
D ≡ (R − L)
2
S≡
68
(4.25)
(4.26)
The solution of the above biquadratic dispersion relation (equation 4.20)is a quadratic
in n2 with two roots,
2
n =
√
B 2 − 4AC
2A
(4.27)
P (n2 − R) (n2 − L)
(Sn2 − RL) (n2 − P )
(4.28)
B±
or in terms of angle
tan2 θ = −
This is the general condition for the propagation of electromagnetic radiation in the
magnetised plasma.
n2 → ∞
General Resonance Condition,
i.e.,
tan2 θ = −
General Cut-off Condition,
P
S
(4.29)
n2 → 0
i.e.,
C = P RL = 0
4.2.3.1
(4.30)
Propagation parallel to the magnetic field (k k B0 )
In the case, (θ = 0) the numerator of equation 4.26 will be equal to zero. Therefore,
ωp2
either
P =1− 2 =0
→
plasma oscillations, (4.31)
ω
ωp2
2
or n = R = 1 −
→ wave with right handed circular polarisation,(4.32)
ω (ω + ωc )
ωp2
→ wave with lef t handed circular polarisation (4.33)
or n2 = L = 1 −
ω (ω − ωc )
4.2.3.2
Propagation perpendicular to the magnetic field (k ⊥ B0 )
Here (θ = 900 ) the denominator of equation 4.26 is equal to zero. Which implies,
ωp2
ω 2 − ωp2
RL
2
n =
=1− 2 2
→ extraordinary wave E ⊥ Bo (4.34)
S
ω ω − ωp2 − ωc2
and
n2 = P = 1 −
ωp2
ω2
→
ordinary E k Bo wave (4.35)
In this thesis we are interested primarly in the propagation of electromagnetic radiation perpendicular to the magnetic field. The solution for perpendicular propagation
69
gives rise to two types of waves, one is called the ordinary wave and the other is
called the extraordinary wave. The refractive index of the ordinary wave (polarisation parallel to the external magnetic field) is independent of the magnetic field.
However, this is not the case with the extraordinary wave (polarisation perpendicular to the external magnetic field) and it gives rise to two interesting phenomena
as explained below.
4.2.3.3
Cut-offs and Resonances
For certain plasma parameters, n2 goes to infinity or zero. i.e., a transition occurs
from a region of propagation to non propagation. When n2 approaches ∞ the wave
is absorbed and when n2 goes to zero the wave is reflected. The first case is called a
resonance and the latter is called a cut-off .
For perpendicular propagation we have seen that there are two solutions for the
dispersion relation.
i)
n2o
Ordinary wave
ωp2
= 1− 2
ω
As discussed earlier this wave has no dependence on the magnetic field and the
electric field vector is parallel to the magnetic field. The wave experiences a cut-off
at ω = ωp (critical density) and there is no resonance observed.
ii)
Extraordinary wave
ωp2 (ω 2 −ωp2 )
n2e = 1− 2 2 2 2
ω (ω −ωp −ωc )
The dispersion relation of the extraordinary wave depends on the external magnetic field. That means that the refractive index of an extraordinary wave reaches
two extreme values exhibiting resonances and cutoffs.
70
Resonance: At resonance, n2e → ∞ i.e, the wave vector k approaches zero which
implies that for any finite value of wave frequency, ω → ωh . Therefore the resonance occurs at point in the plasma where,
ωh2 = ωp2 + ωc2 = ω 2
(4.36)
At resonance the extraordinary wave loses its electromagnetic nature and becomes
an electrostatic oscillation (upper hybrid wave) [104].
Cut-off: At cut-off, n2 = 0. i.e, k
→
0. Therefore, the dispersion relation
becomes,
ωp2
ω 2 − ωp2
1= 2 2
ω ω − ωp2 − ωc2
(4.37)
ω 2 = ωp2 ± ωωc
(4.38)
Simplifying,
The two roots of the above quadratic equation give two cut-off frequencies which
are called the left hand(ωL ) and right hand(ωR ) cut-off frequencies.
1
1
ωR = [ωc + (ωc2 + 4ωp2 ) 2 ]
2
1
1
ωL = [−ωc + (ωc2 + 4ωp2 ) 2 ]
2
(4.39)
(4.40)
During cut-off the wave propagation is limited by infinite phase velocity and zero
group velocity. While at resonance the wave energy is transferred to plasma particles. i.e., infinite group velocity and zero phase velocity. A representation of the
cut-offs and resonances is plotted on the dispersion diagram which is shown in figure
4.3.
From the figure it is clear that there are two regions of propagation for an extraordinary wave separated by a region of non propagation. Keeping the density
constant and decreasing ω we can see that the phase velocity approaches the velocity of light as the wave travels. When the phase velocity becomes infinite the wave
approaches the right hand cut-off ω = ωR beyond which the wave will not propagate since the refractive index becomes imaginary. At ω = ωh the wave reaches
71
Figure 4.3: Dispersion relation for extraordinary wave plotted on a refracvφ2
tive index or 2 - frequency scale. Hatched regions are regions
c
of non propagation.
resonance as the phase velocity is zero. The wave travels further until at ω = ωL ,
the left hand cut-off is reached. Between ω = ωL and ω = ωh the wave travels with
a phase velocity higher than c depending on whether ω is greater than ωp . It is also
clear that ωR > ωh > ωL . The right hand cut - off and upper hybrid resonance for
4ω (λ = 1µ) is plotted on a magnetic field vs density scale is shown in figure 4.5.
In our experiments we have observed only the right hand cut -off. This is because
the frequency of the harmonic is above the plasma frequency therefore, the cut-off
observed lie on the right hand side of the diagram.
From the dispersion relation of the extraordinary wave it is clear that during short
pulse laser -plasma interactions the refractive index is only dependent on the electron
density and magnetic field strength (using the cold plasma approximation). Even
though the plasma parameters vary over time, but can be assumed as a uniform
medium locally. If we know the electron density the magnetic field strength can
be easily calculated during cut-off and resonances. This is the technique we used
for our magnetic field measurements. During high power laser matter interaction
harmonics of the incident laser are generated at the critical density as discussed in
72
Figure 4.4: CMA diagram showing the phase velocity surfaces for different
wave solutions of dispersion relation perpendicular propagation
to the magnetic field.
detail in chapter 2. The polarisation of these harmonics are determined by the selection rules. These harmonics are generated at the same time as the magnetic field
and while they propagate through the magnetised plasma they experience cut-offs
and resonances.
The solutions of the dispersion relation for electromagnetic wave propagation through
magnetised medium is illustrated in the CMA (Clemmow-Mullaly-Allis) diagram
[103–105]. CMA diagram is a plot of normalised magnetic field vs. density. The
magnetic field increases in the vertical direction and the plasma density increases
in the horizontal direction. Each region in the plot shows which waves are present
along with the variation of phase velocity with angle. The diagram is divided into
areas between cut-offs and resonances which separates regions of propagation and
ωp2
ω
non-propagation. Only the region between ωc = 1and 2 = 1 (i.e., the bottom left
ω
hand area) is of interest in this thesis. Between the right hand cut-off (R cutoff)
and the upper hybrid resonance there is no propagation. The right hand cut- off
magnetic field strength for various Vulcan(CPA, 1.053µ) harmonics are plotted in
73
figure 4.5
Figure 4.5: Cut-off magnetic field plotted for various harmonics of 1.053µ
radiation against electron density
4.3
Experiment
The experiments were carried out using the CPA arm of the Vulcan laser. An experimental layout of the interaction chamber is shown in figure 4.6. The 1.06 micron
Vulcan CPA beam was focussed onto 10mm × 10mm polished glass targets using
a f /3 off -axis parabolic mirror. The alignment of the target was done using the
method described in chapter three. The main beam was focussed at an angle of 45
degrees with respect to the target normal and was p- polarised. i.e., the electric field
of the laser is in the plane of incidence. The measured focal spot size was 10µm
in diameter and the pulse duration was ∼ 1ps. Shots were taken over a range of
intensities from 1 × 1018 W/cm2 to 1 × 1020 W/cm2 . The self-generated harmonics
were collected using an f /8 off-axis parabolic mirror placed at a distance equal to its
focal length so that the relayed beam is collimated. The collimated harmonics were
taken out of the chamber using an UV enhanced aluminium mirror through a quartz
74
Figure 4.6: Schematic layout of the interaction chamber
75
window which transmits wavelengths down to ∼ 190 nm. The collected harmonics
are then sent through a set of three polarimeters aligned outside the chamber as
shown in figure 4.6.
Along with these polarimeters, there were other diagnostics to measure plasma temperature using x-ray pin-hole cameras, external transverse second harmonic probes
for shadowgraphy. Passive stacks were used for electron measurements and ions
were measured using Thomson parabola spectrometers. An XUV spectrometer was
used for measuring the self generated magnetic field with high harmonic polarimetry [106]. The probe was a frequency doubled short pulse beam.
4.3.1
Polarimeter
The polarimeter was designed to measure cut-offs and the Stokes vectors of an electromagnetic beam propagating through it. The schematic layout of a 4ω polarimeter
Figure 4.7: Schematic layout of 4ω polarimeter
is shown in figure 4.7. The collimated harmonics coming out of the chamber were
directed to the polarimeters using two reflective neutral density filters used as beam
splitters. A 0.7 reflective neutral density filter has been used to reflect 80 % of the
76
harmonics to the 4ω polarimeter. The transmitted beam is split into two using a
0.3 reflective neutral density filter and the transmitted beam was sent to the 3ω
polarimeter and the reflected beam to the 2ω polarimeter. Each polarimeter had
four channels in order to measure the three reduced Stokes parameters described in
chapter five. The beam entering the polarimeter was referenced with respect to the
input radiation using two apertures. The size of the beam was adjusted to 5mm
because the polariser cubes were of 8mm × 8mm. In each polarimeter setup, two
pinholes were used at the entrance and end of the channel to produce a reference for
the beam. Narrow band interference(∼ 25nm) filters were used at the entrance of
each channel to select only the particular wavelength of interest. Reflective neutral
density filters of 0.2, 0.3 and 0.4 values and a UV enhanced aluminium mirror were
used to reflect the beam to each arm so that there was an equal amount of light in
each channel.
The 4ω polarimeter used Rochon polarisers with an extinction ratio of 10−4 . The
first channel was called the reference channel with no polarisers and thus it gives the
absolute intensity. The second channel had a Rochon polariser with a split angle
2.50 for cut-off measurements as well as for obtaining the second Stokes parameter
S1 . The third channel measured S2 using a Rochon polariser set at 450 . The fourth
channel consisted of a quarter wave plate with its axis set at 450 , followed by a
polariser for transmitting only p-polarised light This channel measures S3 . All of
the beams were focused to a 16 bit high dynamic range charge-coupled-device arrays
using 1 inch fused silica lenses of 20 cm focal length.
The 3ω and 2ω polarimeters used sheet polarisers (where only one polarisation is
transmitted and the other polarisation is absorbed) and glass polarisers respectively.
In the 3ω channel two sheet polarisers were used simultaneously to get an extinction
ratio of 10−3 . Fused silica lenses of 20 cm focal length were used for focussing the
beam into CCD arrays in the third harmonic channel while BK7 lenses were used
for the second harmonic channel. Special measures were taken to keep all angles
close to normal, so that any depolarisation effects due to optics were reduced. Ad77
ditional neutral density filters were placed at the entrance of each channel to avoid
any saturation of the CCD. All of the optics used for third and fourth harmonic
measurements were UV enhanced and non-polarising. Each of the polarimeters was
light proof, and background shots were taken before each shot to ensure that there
was no stray light. Calibration shots were taken without polarisers to find the absolute intensity of light going through each channel and to check the alignment. We
have selected the 2nd , 3rd , 4th and 5th harmonics of Vulcan CPA beam (1053 nm).
Use of higher harmonics was not possible using this setup as they do not propagate
through air. The 4ω polarimeter was converted into 5ω polarimeter by changing the
interference filter.
4.3.1.1
Calibration of polarimeters
Calibration of the optics and the entire polarimeter set-up was carried out using a
low power 10 Hz Nd:YAG which can generate 3 wavelengths 2nd (527nm), 3rd (351nm)
and 4th (264nm) harmonics using various frequency conversion crystals. A polariser
was used to generate a p-polarised beam whose electric field vector was aligned
parallel to that of the main interaction beam. It was then sent through each polarimeter. The polarisation of the radiation before and after going through each
optical component was measured to make sure that there was no depolarisation due
to optical components.
4.4
Results
Recapitulating the earlier discussion, for propagation vectors perpendicular to the
magnetic field (Bz = 0), the dispersion relation can be written as described earlier,
ωp2
ω2
(4.41)
ωp2 ω 2 − ωp2
=1− 2 2
ω ω − ωp2 − ωc2
(4.42)
n2o = 1 −
n2e
From the above equation it is evident that x − waves can experience cut-offs and
resonances depending on the magnetic field strength and the plasma electron density. Cut-offs occur when refractive index becomes equal to zero where it is reflected
78
and a resonance when the index approaches infinity where it is absorbed and the
energy is converted into upper hybrid oscillations as explained earlier. The ordinary
wave which has an electric field vector parallel to the magnetic field is unaffected
by the magnetic field (since E × Bo = 0). For a cut-off to occur the magnetic field
generated has to be high enough to reflect the extraordinary wave of particular harmonics. This is exactly what we observed in our experiments for the high intensity
shots. Furthermore, at very low intensities, where the magnetic field is negligible
the harmonics are p-polarised as shown in figure 4.8.
Now that we have experi-
Figure 4.8: A typical low energy shot shows that only p - polarised harmonics are produced at low intensities
Figure 4.9: Typical data shots with 2ω polarimeter showing (a) cut-off at
high energy and (b) no cut-off with a low energy shot.
mentally shown that at low energies only p-polarised harmonics are generated we
79
can measure what happens to x-wave (p-polarisation) of different optical harmonics
at higher intensities.
Figure 4.10: An example of the cut-off data from the 3ω polarimeter (351
nm). (a) low intensity shot showing all polarisations. (b) pcomponent has vanished (cut-off).
Figure 4.11: Cut-off data from 4ω polarimeter (264nm). (a) low intensity
shot showing all polarisations. (b) p-component has vanished
(cut-off)
The figure 4.9 shows typical data shots on a 2ω (527 nm) polarimeter where the
figure 4.9 a the x-wave (p - polarisation) is vanished indicating a clear cut-off compared to a less intense shot 4.9 b.
The figure 4.10 shows a cut-off (a) and no cut-off (b) of extraordinary wave of
80
third harmonics (351)nm at different laser energies.
The figure 4.11 shows shots with 4ω polarimeter. The figure 4.11 a is when no
cut-off is observed. Both s (ordinary wave) and p (extraordinary wave) components
from the same polariser are present. The figure 4.11b shows where no p - polarisation is present at a higher intensity. i.e, the extraordinary wave (x-wave) has
experienced cut-off.
Figure 4.12: A typical 5ω polarimeter (210 nm) data where no extinction
of p-polarisation is observed.
The figure 4.12 shows that at the same intensity there is no extinction of extraordinary wave (p- polarisation) component of the fifth harmonic.
The figure 4.13 summarises the observation of cut-offs for all lower order optical
harmonics. The second, third and fifth harmonic results are from the same shot
and the fourth harmonic result is at a similar intensity. From this figure it is clear
that second, third and fourth harmonics are cut-off at a certain intensity while the
fifth harmonic does not show any cut-off. This indicates that a minimum magnetic
field exists at these intensities which is high enough to reflect the second, third and
fourth harmonic while the fifth harmonic still transmits through. These shots are
repeatable. The strength of the magnetic field can be obtained from the magnetic
field vs. density plot (figure 4.5 ). So we have shown that at higher intensities the
x-wave (p-polarisation) of optical harmonics below 5ω experience a cut-off.
This shows that the minimum magnetic field which exists is enough to observe cut81
Figure 4.13: A high intensity shot showing cut-off of all lower order optical
harmonics below 5ω at the same intensity.
82
off of 4ω harmonic and the maximum is below the cut-off strength of 5ω harmonic.
From figure 4.5 the cut-offs of third, fourth and fifth harmonics are 220, 340, 460
MegaGauss respectively. Therefore, the strength of the magnetic field is below 460
and above 340 MG.
Figure 4.14 shows the ratio of extraordinary wave over ordinary wave plotted against
Figure 4.14: x -wave cut-off for 3rd , 4th harmonics. The 5th harmonic does
not show any cut-off at the same intensity, the y-axis is the
ratio of x-wave over o-wave on a logarithmic scale.
intensity. The ratio was calculated by integrating the area of the image over pixels
and deducting the average background from it. It is evident that as intensity increases the ratio decreases and at an intensity 8 × 1019 W/cm2 there is a sharp dip
in the ratio showing cut-off. However, at this intensity we can see that the 5ω ratio
does not change. In order to make a realistic calculation of cut- off magnetic field
the following assumptions need to be taken.
The maximum intensity at which cut-offs are observed is 9 × 1019 W/cm2 . i.e.,
at this intensity the relativistically corrected critical density is ∼ 6.4 × 1021 cm−3 .
At an intensity ∼ 1020 W/cm2 , the electron density is taken to be the relativistically
83
corrected. The relativistic factor γ for a circularly polarised light is given by
γ=
e2 Iλ2
1+ 2
2π 0 m2e c5
21
(4.43)
which can be simplified as
γ'
Iλ2
1+
1.38 × 1018 W cm−2 µm2
12
(4.44)
For linearly polarised light γ is an oscillating parameter and the second term on
the RHS of above equation is divided by 2. The value of γ at intensities where cutoffs were observed is 5.8. Therefore, in our experiments, the relativistically corrected
density is 6.4×1021 cm−3 . In order to calculated the minimum magnetic field present
to observe the cut-off the following conditions are possible .
• If harmonics are generated at relativistic intensities at the relativistically corrected critical density and they propagate through a region which is affected
by relativistic effects where the magnetic field exist too then the fourth harmonic x-wave cut-off strength for magnetic field is 500 MG. i.e., we need to
consider the relativistic density. (ne = γnc ).
• On the other hand if the harmonics are generated at the relativistically corrected critical density and they propagate through a magnetized plasma which
is not affected by relativistic corrections then the minimum magnetic field required to have a fourth harmonic cut-off is ∼ 250M G.
• In the case of a Gaussian focal spot, the laser intensity varies across the focal
spot hence the critical density varies from outer edge of the focal spot to
the centre of the spot. Therefore, harmonics will be generated in gradient
densities over an extended region. This is happening in our experiments as
our measured focal spot is Gaussian. Hence, if we assume that some of the
harmonics we measured are generated at a density which is not relativistically
corrected then the minimum magnetic field present is ∼ 340M G to experience
a fourth harmonic x- wave cut-off (figure 4.5). Hence the highest self-generated
magnetic field measured in our experiment is 340 ± 50M G using the cut-off
84
method. Where the error bar is obtained considering the background level and
the difference between the cut-off levels for different frequencies.
Of the above mentioned possibilities the last condition is the most likely in our
experiment and therefore, the minimum magnetic field required to induce fourth
harmonic x-wave cut-off is 340 M G.
4.5
Summary
The magnetic field measured using the cut-off mechanism is the highest measurement of self generated magnetic field in laboratory plasmas. The third harmonic
cut-off was observed at an intensity > 8 × 1019 W/cm2 and the fourth harmonic
cut-off was observed at intensities > 9 × 1019 W/cm2 . The highest magnetic field
measured was high enough to experience fourth harmonic x-wave cut-off and but
still too low to observe fifth harmonic cut-off. The cut-off results are reproducible
at higher intensities.
The Cut-off method is an efficient and simple way to measure self generated magnetic fields in laser produced plasmas. The method is very straightforward and gives
the exact measure of minimum peak magnetic field generated at higher intensities.
The only diagnostic tool needed for the measurement is the self-generated harmonics
which are produced during laser matter interaction. The self generated harmonics
are an excellent diagnostic as they are generated about the same time the magnetic
field is generated and they do not perturb the medium unlike external probes.
As explained in chapter 2, the source term for magnetic field generation is due
to the spatial gradient and temporal variation of the ponderomotive force. This
field is localised near the critical density surface and is larger than the field gener3
ated by the thermoelectric source term. The polarisation of the 2 harmonic was
3
not changed during shots at different intensities. The 2 harmonics are generated
predominantly at the quarter critical density surface. Hence it is clear that the huge
magnetic fields we are measuring is localised between the critical density surface and
85
quarter critical density surface.
At intensities > 1020 W/cm2 stronger magnetic fields may be produced and can
be measured by observing the cut-off of XUV harmonics using XUV harmonic polarimetry. The resonance can not be observed using this technique. The resonance
occurs at higher magnetic fields while the x- waves have already experienced cut-offs
at a lower magnetic field. The main drawback of this technique is that it can not be
used at lower intensities where the strength of the magnetic field is not high enough
to reflect (cut-off) the harmonics and there is a need to develop another diagnostic
to measure the magnetic fields. Other possible sources of errors in our measurement
may be due to the shot to shot variation in intensity and beam quality.
86
Chapter 5
Magnetic field measurements using Stokes vector analysis
5.1
Introduction
The self generated harmonics of the plasma are a very powerful tool for studying
various laser-plasma phenomena. In the previous chapter it was shown that the
self- generated harmonics propagating through a magnetised plasma can be used to
measure the magnetic field generated during laser matter interaction. The cutoff
method discussed in the previous chapter is an effective diagnostic only at intensities
higher than 1019 W/cm2 . At lower intensities the extraordinary wave of the propagating self generated harmonics do not experience any cutoff. Hence it is necessary to
develop an alternative technique to measure the magnetic field strength. Harmonic
polarimetry using Cotton-Mouton effect is an effective method at lower intensities.
The evolution equation for the polarisation state of the self generated harmonics
while propagating through a magnetised plasma is the basis of harmonic polarimetry. This chapter describes in detail about the theory and the measurements of self
generated magnetic field using the Cotton-Mouton effect measurements [107–109].
A short discussion of the theory of electromagnetic wave propagation through a
magnetised medium is given, followed by the theory of harmonic polarimetry in
87
section two. The third section discusses the measurement of the self generated magnetic field with Vulcan and Astra laser systems using this technique. Section four
1
describes simulations using the OSIRIS, 2 2 D PIC code. The experimental results
are compared with computer simulation results. Section five deals with measuring
the spatial distribution of the magnetic field using Vulcan laser system. Section
six talks about a new diagnostic technique developed for the mapping of magnetic
field. Finally, the advantages and drawbacks of the harmonic polarimetry method
are discussed.
5.2
The Cotton-Mouton effect and the Polarimetric technique
Recollecting the discussion in the previous chapter, when a plane polarised electromagnetic wave propagating perpendicular to the magnetic field the wave experiences
an induced ellipticity. This is called the Cotton-Mouton effect. For any direction of
propagation the incident ray is split into an ordinary wave (o-wave) and an extra
ordinary wave (x-wave) with different phase velocities. The x - wave has polarisation vector perpendicular to the optic axis (magnetic field) and for the o-wave the
polarisation vector is along the optic axis (magnetic field).
The polarisation state of an electromagnetic wave can be represented using two
parameters called χ and ψ as shown in figure 5.1. ψ is the angle between the
direction of major axis and OX and χ is the ellipticity.
tan χ = ±
b
a
a >> b
(5.1)
a and b are the semi-major and semi-minor axis of the ellipse. Since these waves have
c c
different phase velocities ( µ , µ ) a phase difference is induced as they propagate
1
2
through an anisotropic media
Z
ω
δϕ = (µ1 − µ2 ) dl
(5.2)
c
88
Y
Y``
X``
a
b
ψ
X
2b
O
2a
Figure 5.1: Representation of an elliptically polarised wave traveling in the
z direction
for a path length L,
δϕ =
ω
Lµc
c
(5.3)
where µc = µ1 − µ2 , µ1 and µ2 are the refractive indices of slow and fast charecteristic waves which are obtained from the Appleton-Hartree formula. Therefore, the
resultant polarisation of the wave after propagating though the magnetised plasma
will depend on ϕ.
A simple way to represent the evolution of polarisation is using the Poincaré sphere
[110]. The Poincaré sphere is a sphere of unit radius with latitude and longitude 2χ
and 2ψ respectively, where each state of polarisation can be represented by a point
P on the surface of the sphere. In figure 5.2 the point P can be represented in terms
of cartesian co-ordinates,
s1 = cos 2χ cos 2ψ
(5.4)
s2 = cos 2χ sin 2ψ
(5.5)
s3 = sin 2χ
(5.6)
89
Figure 5.2: The Poincaré sphere. A useful way to represent the polarisation
of light in a three dimensional vector space.
where, s = (s1 , s2 , s3 ) = OP. s1 , s2 , s3 are the reduced Stokes vectors. The Stokes
vectors are a set of parameters which can be used to express the optical parameters
of an electromagnetic wave in terms of intensity. A detailed derivation is given in
Appendix II.
Therefore, the evolution of the polarisation after propagating through the non
absorbing, anisotropic magnetised plasma can be represented by a point on the
Poincaré sphere by a rotation equal to the phase shift (ϕ) about an axis passing
through the points representing the characteristic polarisations. We assume that
the plasma is uniform locally and that there is no refraction of the transmited radiation due to density gradients. Hence the evolution equation can be written as,
ds(z)
= Ω × s(z)
dz
(5.7)
d∆ϕ
ω
= nc
dz
c
(5.8)
|Ω| =
where Ω has the direction of fast characteristic polarisation sc2 and nc = no − ne . no
and ne are the refractive indices of ordinary and extraordinary waves. The refractive
90
indices for ordinary and extra ordinary waves are derived in the previous chapter.
ωp2
ω2
(5.9)
ωp2 ω 2 − ωp2
=1− 2 2
ω ω − ωp2 − ωc2
(5.10)
n2o = 1 −
n2e
Hence,
Ω=
ω
nc sc2
c
(5.11)
The polarisation equation can be solved by assuming that the plasma is uniform
locally and the characteristic waves have constant refractive index locally (dz),
Z L
ω
∆ϕ =
nc dz
(5.12)
0 c
ds
(5.13)
= Ω(z) × s(z)
dz
ω
where, Ω(z) = c nc (z)sc2 (z). The approximate solution is obtained by assuming
R
|Ω| dz << 1,
Z
z
Ω(z 0 )dz 0
s(z) = s0 − s0 ×
(5.14)
0
s0 is the initial polarisation at z = 0. i.e., we obtain the final polarisation of the
electromagnetic radiation in terms of initial polarisation and a function Ω(z) which
is a property of the medium. Therefore, knowing the initial and final polarisation
states of the electromagnetic wave propagating through the plasma can give the
transition matrix Ω(z). Thus we can estimate the strength of the self-generated
magnetic field which induced the birefringence.
The diagnostic tool we employed for this purpose was the self generated harmonics
of the incident laser. For the measurements the harmonics were directed from the
plasma to the detector using a pair of optical components and through a polarimeter.
Therefore it is necessary to include the depolarising effects of optical components
during polarimetric measurements. Each optical component has a characteristic
Muller matrix M [110, 111]. The Stokes vector of the initial harmonic (linearly
p-polarised) is Sin and the Stokes parameters of the harmonics after propagating
through the plasma is Sout . They are connected by the equation
Sout (z) = M · Sin (0)
91
(5.15)
where, M is called the plasma transition matrix. If there are more than one optical
component then the resultant Muller matrix is
M = M1 · M2 · M3 · ··
(5.16)
In our experiment the collection optics and the polarimetry setup were positioned
before the detector (16 bit CCD camera). Therefore, M is the resultant transition
matrix due to plasma and other optical components in the beam.
M = MP · Mpol · Mqwp
(5.17)
Figure 5.3: Initially the radiation is linearly polarised at an angle b to the
x-axis
dS(z)
= Ω(z) × S(z)
dz
(5.18)
ω
where Ω = c (η1 − η2 ) and η1 and η2 are refractive indices of ordinary and extra
ordinary waves. Therefore, E(0) = (cos β, sin β). The plasma matrix M is


1
0
0




M =  0 cos(Ωz) sin(Ωz) 
(5.19)


0 sin(Ωz) cos(Ωz)
92
Ω can be obtained from the Appleton-Hartree equation [107]
2
e2 ωp2
B sin 2β
Ω=
(µ1 + µ2 )m2 c3 ω 3 A
1−Γ
ωp2
e 2
where Γ = 2 , A = 1− mωc
ω
(5.20)
B2
1−Γ .
The average angle at which harmonics enter the magnetic field can be estimated
from the solid angle of the collection optics. In our experiment it was estimated
to be (15o ± 5o ). Using equations 5.14 and 5.19 the plasma transition matrix can
be calculated. This is compared with the theoretical value for the corresponding
wavelength at different magnetic fields.
5.2.1
Configurations of polarisers and retarders in the polarimeter
From the plasma evolution equation we know that the final polarisation of the wave
propagating through the plasma is related to the initial polarisation by the plasma
transition matrix. Therefore, by measuring the plasma transition matrix it is possible to determine the plasma parameters. In order to measure the reduced Stokes
parameters an analyser (polariser) and a retarder (quarter wave plate) combination
is used. Let γ be the angle set by the axis of the retarder with respect to the initial
polarisation and θ be the angle set by the polariser axis with respect to the initial
polarisation. Using the Muller matrix for a standard retarder and analyser (given
in appendix 2) the plasma transition matrix can be calculated. s0 is the absolute
intensity of radiation falling on the detector in the absence of any retarder and analyser. In the absence of the retarder the detected intensity (ID ) is given by
ID (θ) =
1
1
(s0 + cos 2θs1 + sin 2θs2 ) = I0 (1 + cos 2θs1 + sin 2θs2 )
2
2
(5.21)
where I0 is the intensity (≡ s0 ) entering the analyser. When θ = 0 and γ = π/4
1
ID (0) = I0 (1 + s1 )
2
93
(5.22)
and
1
ID (π/4) = I0 (1 + s2 )
2
(5.23)
When the retarder is present
1
ID (γ, θ) = I0 (1 + cos(2θ − 2γ)(cos 2γs1 + sin 2γs2 ) + sin(2θ − 2γ)s3 )
2
(5.24)
when θ = γ + π/4
1
ID (γ, γ + π/4) = I0 (1 + s3 )
2
(5.25)
This is how the polariser and analyser angles on the polarimeter were aligned.
5.3
Experiments and Results
The experiments were carried out using the Vulcan and Astra laser systems. The
results from three independent experiments are discussed here.
5.3.1
The Vulcan laser
The experimental setup for Stokes vector calculation is the same as the cutoff method
explained in the previous chapter (figure 4.6). For the Stokes vector calculation all
four channels of the polarimeters (figure 4.7) were used instead of the single channel
in the case of cutoff measurements. Shots were taken over a range of intensities.
5.3.1.1
Analytical calculation of Stokes vectors
The figure 5.4 shows all the channels of the fourth harmonic polarimeter. The refer-
Figure 5.4: An example of 4ω polarimeter raw data showing all channels to
measure the Stokes vectors.
94
ence channel (Iref ) measures the absolute intensity of light entering the polarimeter.
Ip and Is are the intensities of p - polarised and s - polarised harmonics using the
same polariser (in this chapter we are not concerned about the s- polarisation). I45
is the intensity of the linear polarisation at 450 . Iqwp+p can be used to measure the
final Stokes vector S3 .
Figure 5.5: An example of a low energy raw data showing only p-polarised
light
At very low intensities (Energy < 1J) the harmonics are p-polarised as shown in
figure 5.5.
Calibration shots were taken without any polarisers and quarter wave
Figure 5.6: A calibration shot at 70J with no polarisers and a quarter wave
plate. The s -component is missing as there is no polariser in
the beam path. The four spots show all four channels of the
polarimeter
plate to measure the absolute intensity of light going through each channel of the
polarimeter ( figure 5.6). In the case of 2ω and 3ω polarimeter there are only four
spots (figure 5.7). i.e., no s- polarisation is present. This is because the polarisers
we have in these polarimeters transmit only one polarisation and suppress the other
component. As described in the previous chapter the 4ω polarimeter was converted
95
Figure 5.7: Examples of 2ω and 3ω polarimeter data showing data shots
and calibration shots at the same intensities.
96
into a 5ω polarimeter by changing the interference filter. Neutral density filters of
appropriate values were used to avoid saturation of pixels.
We have seen in chapter four that at intensities below 8 × 1019 W/cm2 no cut-offs
are observed. Hence at these intensities the plasma transition matrix has to be
measured to make an estimate of the self-generated magnetic field.
Figure 5.8 shows a sample raw data from 4ω polarimeter at 70J. Figure 5.8 a is a
data shot where no cut- off is observed. Figure 5.8 b is a calibration shot at almost
the same intensity. Absolute intensity on each channel is calculated by integrating
Figure 5.8: Typical data from the 4ω polarimeter
the image over the area (pixels) and subtracting the average background.
5.3.1.2
Calculation of plasma transition matrix
In order to calculate the magnetic field the following procedure is used.
• The initial Stokes vectors are measured using low energy shots.
• The final Stokes vectors S1 , S2 , S3 are calculated from the experimental data
from high energy shots.
97
• Using the evolution equation Ωexperimental is found.
• The theoretical value of Ωtheoretical is calculated from the Appleton-Hartree
formula (derived in the previous chapter) using iterative method for different
values of magnetic field.
• Compare Ωexperimental and Ωtheoretical to find the magnetic field generated during each shot.
Step I - Calculation of Ω theoretically
For 4th (264nm) harmonic:
ωc = 1.758 × 107 B rad/sec
√
ωp = 5.64 × 104 ne , where ne = 2.4 × 1021 cm−3 at an intensity 1 × 1019 W/cm2
ω4ω = 7.14 × 1015 rad/sec
ωp2
= 1− 2 = 0.8608
ω
ωp2
ωp2
1− 2
ω2
ω
2
N2 = 1− 2
ωp ωc2
1− 2 − 2
N12
ω
N22 = 1−
ω
0.1198
0.8608−6.08×10−18 B 2
e 2 1 B2
A = 1− m
ω2
ω2
1− p2
ω
i.e., A = 1 − 7.129 × 10−18 B 2
ωp2
ωp2
e 2
B 2 sin 2b
Ω = mc
.
where
Γ
=
and b is the angle at which
3 A 1−Γ
ω2
(N1 +N2 )cω4ω
harmonics enters the magnetic field.
98
Step II - Calculation of Stokes parameters from the experiment.
The table 5.1 below gives the integrated values over pixels for each polarisation
Ip
α
of the data shown in figure 5.8. for example S1 is calculated as follows. I
= 2
ref
Shot No
160706 - (70J)
Ip
Iref
I45
Iref
Iqwp+p
Iref
0.4839 0.3471
0.2644
180702 - calibration (73J) 0.7478 0.5927
0.4471
S1
S2
S3
0.29412 0.1713 0.1825
Table 5.1: Calculation of Stokes vectors from sample data
(1 + S1 )
where α = I
Ipcal
ref cal
i.e., Ipcal and Iref cal are the intensity of the p-polarised and reference beam respectively from a calibration shot. Taking into consideration the extinction ratios of
polarisers the previous equation becomes,
Ip
α
=
Iref
2 ((τmax + τmin ) + (τmax − τmin )S1 )
where τmax and τmin are the maximum and minimum transmission of the polarisers.
For 4ω polarimeter (τmax + τmin ) ' 1 and (τmax − τmin ) ' 1.
For 3ω polarimeter(τmax + τmin ) ' 0.3819 and (τmax − τmin ) ' 0.3489.
Similarly S2 and S3 are calculated.
Initially

 thebeam is linearlypolarised, i.e., S0 = (cos b, sin b) [107].
S
cos 2χ cos 2ψ
 1  


 

 S2  =  cos 2χ sin 2ψ 

 

S3
sin 2χ
Initially S3 = 0 as ellipticity χ = 0, linearly polarised. Therefore, Ωz can be calculated from
S1 = S01 ,
S2 = cos Ωz S02 ,
S3 = sin Ωz S02 .
99
Using the calculated values of S1 , S2 and S3 the value of Ωz can be estimated.
This is then compared with the theoretical estimation of Ω. Comparing these two
values the strength of the magnetic field is calculated. The angle at which harmonics
enter the magnetic field is calculated from the solid angle of the collection optics
or from the initial Stokes parameters using low intensity shots. The f-number of
the collection optics is f /6. Therefore, the average angle is thus taken as 15o . Also
it was observed that the 3/2 harmonics generated at the quarter critical density
surface had the same polarisation as the incident laser. Therefore, it is clear that
the magnetic field is localised between critical density surface and quarter density
surface. From computer simulations it was shown that the density decreases exponentially with a scalelength of ∼ 1µm. This is in agreement with measurements of
the plasma density scalelength using shadowgraphy (figure 5.9). Also the following
assumptions are considered.
• The frequency of the radiation is much greater than the particle collision frequency as well as plasma frequency so that the absorption of harmonic radiation by the plasma is negligible and a cold plasma approximation can be
used.
• Depolarisation due to plasma density gradient is negligible as the radiation
propagates mainly along the direction of density gradient [24, 108, 109, 112].
• Also the Bk component is negligible.
The figure 5.10 shows the magnetic field calculated using this method with second
(527nm), third (351nm), fourth (264nm) and fifth (210nm) harmonics of the Vulcan
laser. The strength of the magnetic field measured by different harmonics is different,
as their propagation through a magnetised plasma is dependent on electron density
and the strength of the magnetic field. Hence the lower order harmonics will see
only low fields and the maximum field measured using that particular harmonics
is limited by the cut-off. This is the reason why at the same laser intensity the
strength of the magnetic field observed by the different harmonics are different. The
error bars arise from calculating the uncertainty in angle at which the harmonics
100
Figure 5.9: Second harmonic probe images (shadowgraphy) showing the
plasma expansion performed with the Vulcan laser (λ = 1µm)
Figure 5.10: Estimated strength of magnetic field using Stokes vector analysis for various harmonics of the Vulcan laser plotted on an
intensity scale.
101
enters the magnetic field. Also there may be discrepancy in calculating the scale
length of propagation. In addition there may be depolarisation effects due to plasma
density gradient, however, this is negligible as the harmonics are propagating along
the direction of density gradient. Error bars are also due to the calculation of
background level. Also lower order harmonics can be generated on a larger region
than the higher orders implying that they on average sample smaller magnetic fields.
5.3.2
The Astra laser
The harmonic polarimetry technique was also used to measure the self-generated
magnetic fields using the ultra short pulses (∼ 70f s) of the ASTRA laser system.
Astra is a 10 Hz, 70 fs, 800nm Ti-Sapphire laser can produce a maximum of 250mJ.
Figure 5.11: Schematic experimental layout
Detailed description of the Astra vulcan laser system is given in chapter three.
102
Figure 5.12: Typical data shots for 3rd (266nm) at maximum intensities.
The experimental set-up is described below. The Astra beam was focused to a 10
micron diameter spot using an off-axis parabolic mirror. Targets were made of 5
mm diameter aluminium rods polished at 450 on one end. Retro-reflection technique
were used to align the targets, i.e., the light scattered from the target is reflected
back through the optical chain and is refocussed. The maximum energy available
was 250mJ. Shots were taken at different intensities by reducing the energies to as
little as 1% of maximum using wave plate and polarisers. The harmonics were collected on the specular reflection side using f /6 uv optics. The collimated harmonic
beam from the chamber was directed to the polarimeter set up outside the chamber.
The maximum intensity available was ≈ 5 × 1018 W/cm2 . The experimental layout
is shown in figure 5.11.
The set-up and the polarimeter is the same as discussed earlier. At the entrance
of the polarimeter a narrow band interference filter was placed to choose the cor5
3
rect harmonic wavelength. We have selected the 2nd , 3rd , 2 and 2 harmonics. The
measurements using fourth harmonic were difficult as harmonic conversion efficiency
decreases with increasing harmonic number [102, 113]. The measurements were performed separately for each harmonic. No simultaneous measurements of all harmon103
ics were taken.
5.3.2.1
Results
Calculation of Stokes vectors
Since there were no cut-offs observed it is necessary to measure the Stokes vectors of
the harmonics after propagating through the magnetised plasma to obtain the magnetic field. Calibration shots were taken without polarisers and quarter wave plate,
which give the relative intensity of light going through each channel. The angle at
which the harmonics entering the magnetic field is taken to be ∼ 20o , calculated
from the solid angle of the collection optics. The optical path length is estimated
using PIC simulations. Thus we can calculate the reduced Stokes parameters and
thereby the magnetic field. The plasma transition matrix was calculated using the
same method as described in section 5.3.1.2 . Measurements of the half harmonics
were also carried out and were observed to be polarised similarly to the polarisation
3
of the fundamental during low intensity shots. The measurement of 2 harmonics
Figure 5.13: Magnetic field measured using Stokes vectors plotted against
intensity for third harmonic (264 nm) of the Astra laser.
shows again that the magnetic field is localised close to the critical density surface.
104
The results shown in figure 5.13 confirms that the magnetic field increases linearly
with intensity at lower intensities. Measured magnetic fields are in agreement with
the low intensity Vulcan results. The error bars in the calculation of magnetic field
are the same as discussed in the previous section for measurements with the Vulcan
laser.
5.4
Simulations
Particle-in-cell (PIC) simulations were carried out using the OSIRIS code (developed
by UCLA [114]) to model the experimental results at various intensities. Osiris is
a two and a half dimensional PIC code (2D3V ) with particle collisions neglected.
The simulation space is 2 dimensional but the particle parameters like electric field
(E1 , E2 , E3 ), particle momentum (p1 , p2 , p3 ), magnetic field (B1 , B2 , B3 ) and current (j1 , j2 , j3 ) are 3 dimensional. The boundary conditions are periodic in the x2
Figure 5.14: Schematic of the 45o simulation geometry. The density profile
is shown in the right hand side. The value is multiplied by
critical density for 1µ laser (i.e., 1.1 × 1021 cm−3 )
105
direction and Lindman in the x1 direction. Using Lindman-open-space boundary
the particles are allowed to escape the boundary.
5.4.1
Simulation set up
We have used experimental parameters to do the simulation. The density profile
used matches the experimental conditions. The simulation box was of the order of
c
75×134 ω . The box was split into 840×1480 cells. A linear density profile followed
p
by a a sharp increase in the density at the critical density surface from 8 to 15 times
the critical density as shown in figure 5.14. There were 9 particles per cell. The ions
were immobile and there are no collisions in the code. The laser pulse is incident
at 45o to the plane of target in the x2 plane. The laser was launched at a height
c
c
25 ω . The laser focal spot size was 12.5 × ω . The simulation was performed on a
p
p
24 node beowulf cluster at Imperial college. The simulation was carried out for a
range of intensities. The laser reaches it peak intensity in a couple of laser periods
(∼ 7f s) and remains the same. The values of a0 where 1, 3 and 10 respectively and
the corresponding intensities were 1.37 × 1018 , 1.37 × 1019 and 1.37 × 1020 W/cm2 .
The dc magnetic field is obtained averaging over 4 laser periods.
5.4.2
Simulation Results
The evolution of the magnetic field for various times is shown in figure 5.15. As seen
in the figure there is a positive and negative field of different magnitude compared
to toroidal fields produced at normal incidence (figure 2.8) . The positive field is
more localised compared to the the negative field which is more diffuse, however, the
total magnetic flux is conserved. This is due to the fact that at oblique incidence the
components of the laser ponderomotive force pushes the electrons to the specular reflection side. On the low density side the field is advected by the electrons. The field
lines are stuck in the plasma because this is a collisionless simulation. The positive
polarity of the field is on the high density side and is pinched into a very small region
while the negative polarity field is on the low density side and is diffuse. In figure
5.16 the electron and ion density distribution at a time when the peak magnetic
106
Figure 5.15: Simulation results showing generation / evolution of dc magnetic field at different times with an intensity 1020 W/cm2 .
The scale shown is a relative scale and the actual value of the
magnetic field is a factor of me ωp e−1
107
Figure 5.16: The electron and ion density at maximum B field. The actual
value of the density is the right hand scale multiplied by the
critical density (1.1 × 1021 cm−3 ). The line-out is taken at the
point of laser incidence.
Figure 5.17: Intensity dependance of self generated magnetic field studied
using three different methods a. Theoretical calculation of field
from ponderomotive force mechanism, b. Experiment, c. Osiris
PIC simulation
108
field is shown (figure 5.15). The electron density is higher on the specular reflection side because of the ponderomotive steepening. This result showing asymmetric
field strength was also measured experimentally and is described in the next section.
In figure 5.17 the strength of the magnetic field is plotted against laser intensity
from the experiment, simulation and analytical calculation using the laser ponderomotive mechanism (explained in section 2.5.3). The experimental and simulation
results increase linearly with intensity, which is in agreement with the theory of
magnetic field generation by the laser ponderomotive force. This means that the
magnetic field measured in our experiments was likely generated due to the ponderomotive mechanism and is the dominant magnetic field generation mechanism
during high power laser matter interaction.
5.5
Spatial asymmetry measurements
The computer simulation carried out using an oblique incidence laser beam shows
that there is an asymmetry in the magnetic field topology. The field lines are much
closer together in the specular reflection side. An experiment was carried out on the
Vulcan laser to verify this simulation results.
5.5.1
The Experiment
The Vulcan CPA beam was focussed down to a 10µm spot size with a f /3 parabolic
mirror as illustrated in figure 5.18. The targets were made of polished glass of 5 × 5
mm2 and they were aligned using the same obscuration technique described in the
earlier Vulcan experiment. The harmonics were collected at an angle +70o and −70o
with respect to the target normal direction. A fused silica lens of f /10 was used in
the specular reflection side and a f /6 fused silica lens was used in the laser incident
direction to collect the harmonics. UV enhanced aluminium mirrors were used to
direct the beam to the polarimeters set-up outside the target chamber as depicted
in the figure 5.18.
109
Figure 5.18: The schematic setup of spatial asymmetry measurements
110
The polarimeter set up was the same as described in section 4.3.1 and the angles
of polarisers and quarter wave plates were configured as mentioned in section 5.2.1.
Polarimeter II was the same as used in the previous Vulcan experiment. Polarimeter
I was a newly built system with similar type of Rochon polarisers as polarimeter II.
However the incident area of each polariser was 5 × 5 mm2 . The extinction ratio
of these polarisers was the 10−4 . Calibration of polarimeter I was performed using
the same method as described in section 4.3.1.1. The angles were kept small to
avoid any depolarisation. A 3ω interference filter was placed at the entrance of each
polarimeter. Calibration shots were taken with similar intensities by removing the
polarisers and quarter wave plates. The collection angles for the harmonics were
different on both sides of the target normal and were estimated to be ∼ 10 degrees
and ∼ 20 degrees respectively.
The plasma transition matrix was calculated for every shot and is compared at
any intensity for calculations with different density profile taken from simulation
results shown in figure 5.16. The results shown here are calculated using same density scales on both sides of laser incidence. The calculated plasma transition matrix
shows that the magnetic field will be different on both sides of laser incidence if the
density and path length is taken the same. However, the simulation shows that the
density (figure 5.16) is higher in the laser direction. But the path length will be less
in this case compared to the other side where the field is more diffused. i.e., the flux
remains the same. We have calculated magnetic field with the 0 same0 and 0 different0
values of density and path length. It is clear that the measured field is large in
the direction away from the incident laser direction and is in agreement with the
simulations.
The possible error in the calculation of magnetic field is from the calculation (apart
from those described in the previous sections) of the solid angles and the estimate of
the density scale length. It would be suitable to use the same solid angle collection
optics so that harmonics are collected over same spatial dimension on both sides
of laser incidence. A new technique is described in the next section which will be
111
laser direction (264 nm)
away from laser incident direction (351 nm)
away from laser incident direction (264 nm)
laser direction (351 nm)
400
350
Magnetic field (MG)
300
250
200
150
100
50
0
0
1E+19
2E+19
3E+19
4E+19
5E+19
6E+19
7E+19
8E+19
9E+19
1E+20
Intensity W/cm2
Figure 5.19: The estimated magnetic field strength
useful in plotting the magnetic field over a large solid angle.
5.6
Harmonic Ellipsometry - A new technique to
plot the angular distribution of the magnetic
field
In the previous section magnetic field measurements were carried out by collecting
harmonics at two solid angles on both sides of the target normal. A new technique
was tested for carrying out the measurements over a large solid angle simultaneously. A large solid angle collection optics (ellipsoidal mirror) has been employed
for this purpose. An ellipsoidal mirror has two conjugate foci. Light from one focus passes through the other after reflection. The amount of light collected by the
ellipsoidal mirror is many orders higher than light collected by conventional lens or
spherical mirror and can be used for the spatial distribution of the light emitted.
112
The experimental set up is shown in the figure 5.20.
5.6.1
Experimental setup
Figure 5.20: The layout of the experiment using ellipsoidal mirrors as collection optics for self generated harmonics
The high power Astra beam was focussed onto a polished cylindrical aluminum
targets. The target was placed at the first focal point of the ellipsoidal mirror as
shown in the figure. The focussing of the beam was done using retro-reflection
technique (explanation is given in section 5.3.2). The harmonics collected by the
ellipsoidal mirror was focussed on to the second focal point. The collimated harmonics were sent to the polarimeter set up outside the chamber by using a suitable focal
length lens. The polarimeter consisted of single arm due to technical limitations.
The polariser angle was set up for different polarisations and shots were taken over
a range of intensities.
5.6.2
Results
A sample raw data for normal incidence is shown in the figure 5.21. There is no
harmonic emission at the top, this is because the relayed harmonics were blocked
by the target mount. The emission of harmonics on spherical coordinate plane is
given in figure 5.22. The harmonics emission is maximum near the axis. This is
because for p-polarised beam at normal incidence the electric field is perpendicular
to the target normal(x-axis). The broad angular emission can be due to the rippling
113
Figure 5.21: A sample raw data of third harmonic(266 nm). Each circle
(dotted black lines) represent different cone angle angles of
harmonic emission. The right hand figure shows the specifications of the ellipsoidal mirror.
Figure 5.22: The intensity distribution of third harmonic (266 nm) at various solid angles. Each color shows the distribution of harmonics at different theta.
114
of the critical density surface due to Raleigh Taylor instabilities as the expanding
plasma is pushed back by the ponderomotive force [8] or due to wave collapse mechanisms [115]. Earlier studies with oblique incidence have shown that harmonics are
emission is specular (confined to the cone angle of reflected pulse) [116, 117].
The intensity of harmonics (3ω) emitted at various angles is plotted in figure 5.22.
This technique is a very efficient way to plot the harmonics at different angles.
Due to technical limitations only one channel of the polarimeter was focussed on
to the detector. The calibration shots showed no depolarisation of the beam at
various angles. However, after few shots the reflectivity of the mirror on the specular reflection side was decreased. The use of single arm polarimeter limited the
choice of measuring the polarisations necessary to calculate all the Stokes vectors
simultaneously. Use of better imaging techniques can be used in future.
5.7
Summary
The first measurement of self- generated magnetic fields using the Stokes vector
analysis has been made. Harmonic polarimetry is a useful technique when there is
no cut-off observed. Since the harmonics are self generated and are produced at the
same time as the magnetic field is generated it is a reliable and simple technique.
3
As there is no change in the polarisation of 2 harmonics, this clearly shows that
the magnetic field is localised between the critical and the quarter critical density
surface. Also the strength of the magnetic field increases with square root of intensity. This must be due to the fact that the magnetic field measured using Stokes
vectors is the self-generated magnetic field due to the laser ponderomotive force.
This is in agreement with simulations using the Osiris PIC code. The magnetic field
measurements with second and third harmonic saturates at higher intensities as the
harmonics are not able to penetrate to higher densities where higher magnetic field
exists. Magnetic field measurement with Stokes vectors saturates at higher intensities for lower order harmonics and experience cut-off. In order to measure larger
magnetic field higher order harmonics need to be used. The sources of error in our
115
measurements are the uncertainty in the calculation of the angle at which the harmonics entering the magnetic field and the estimation of density scale length. At
higher intensities the relativistic effects on electron and ion motion also need to be
considered. Magnetic field measurements using short pulses are also in agreement
with Vulcan results at lower intensities.
Simulations carried out using the Osiris PIC code are in good agreement with experimental results as well as analytical models using the laser ponderomotive potential.
Asymmetric measurements shows that there is a difference in magnetic field strength
on both sides of laser incidence. However, the total flux remains constant.
The harmonic ellipsometry using the Astra laser is a very useful method for plotting
the distribution of harmonics and thereby mapping the magnetic field. A realistic
spatial distribution of the magnetic field can be done with this technique in the future with a better optical imaging system where each Stokes vector can be measured
simultaneously.
The cut -off method and Cotton-Mouton method can be used for the measurement
of magnetic fields using XUV harmonics [106]. The self generated magnetic field
can play a significant role in the absorption of laser light [118,119]. They can induce
resonance absorption even at normal incidence through upper hybrid oscillations.
They can also give rise to an additional mechanism of second harmonic generation
and studying this second harmonic could explain the size of the magnetic field as
well as the information about the local density gradient [120].
116
Chapter 6
Time resolved measurements of
the self-generated magnetic field
using laser harmonics
The measurements described in the previous chapters have recorded the presence
of ultra strong magnetic fields of the order of several hundred MegaGauss during
high power laser interactions with matter. A quantitative study on the temporal
evolution of these fields is discussed in detail in this chapter. Other studies have
measured the existence of magnetic fields which has ∼ 6ps duration using a 100f s
incident pulse with a maximum intensity ∼ 1016 W/cm2 [121] using an experimental
probe beam. In our experiment the self generated third harmonic of the incident
laser is used as the diagnostic tool. The experiment is described in the first section.
The results are discussed in section two followed by the computational and analytical
calculation of magnetic field evolution. The limitations of these measurements and
future work are discussed in the last section.
6.1
The experiment
The experiment was carried out using the CPA beam of the VULCAN laser. The
short pulse interaction beam was stretched to ∼ 8ps duration so that the temporal
resolution of the optical streak camera was shorter than the incident pulse duration.
117
The schematic of the experimental layout is shown in figure 6.1. A long pulse, ∼ 8
Figure 6.1: The experimental layout for time resolved measurements
ps, beam was focussed onto a glass target using a f /3 parabolic mirror. The targets
were optically polished 1 × 1cm2 glass with a thin coating of Germanium in a cross
pattern. The width of the cross was chosen to be close to the focal spot size. The
alignment was done by observing the shadow of the crosswire in an expanded beam
after it passed through the focus. Best focus was achieved when the beam was totally obscured by the cross wire. The diagnostic set-up consisted of a single channel
polarimeter with a fast optical streak camera as the detector. The harmonics were
collected using a f /10 UV lens. The collimated beam was taken out of the target
118
chamber using UV enhanced aluminium mirrors. At the entrance of the diagnostic set-up an interference filter was placed to choose the appropriate wavelength of
harmonic. The selected harmonic was then sent through a single arm polarimeter
which measures the s and p polarisation. The single arm polarimeter consisted of
a Wollaston prism that splits the beam into s and p polarisations which were then
imaged onto the slit of the Streak camera using a 25 cm UV lens as shown in figure
6.1. A reference arm is necessary for a quantitative measurement of the magnetic
field. However, the width of the front slit of streak camera was not wide enough to
accommodate three beams into the detector so the sum of the p and s could be used
as a reference. The front slit of streak camera was of 25µ wide during the shots. All
the optics were normalised to avoid any depolarisation due to multiple reflections
and the mirrors were set of an angle close to normal to prevent any depolarisation
of the collected harmonics.
Streak cameras are ultra fast light detectors which can measure intensity vs. time
vs. position or wavelength simultaneously. They are highly sensitive such that they
are capable of detecting even single photons. They can handle single event to events
at a repetitive rate of GHz. The dedicated readout system helps the streak images
to be displayed and analyzed in realtime. Figure 6.2 shows the operating principle
of a streak camera. The light to be measured passes through the slit and the optics
form an image of the slit on the photocathode of the streak tube. As shown in
the figure 6.2 two optical pulses which vary in time, space and intensity arrive at
the photo cathode. The incident optical pulse on the photocathode is converted
into a number of electrons proportional to the light intensity. Hence the two optical pulses are converted sequentially into electrons which passes through a pair of
accelerating electrodes and is bombarded against a phosphor screen. The high voltage applied to the sweeping electrodes is synchronous with the incident light so the
electrons generated from the optical pulses arrive the sweeping electrodes exactly
at the same time as the sweeping voltage is applied. This is done using a trigger
control unit which controls the sweep speed using a frequency unit and a delay unit.
We have used a fast optical trigger (beam 9 taken from the main interaction beam
119
sp
sweep circuit
sweep
electrode
streak image
on phosphor screen
lens
time
e
ac
light intensity
trigger signal
slit
photocathode
time
accelerating
mesh
incident light
mcp
phosphor
screen
space
Figure 6.2: Operating principle of a streak camera [1]
in the laser area) for this purpose. The electrons arriving at slightly different times
and at slightly different intensities are deflected at slightly different angles in the
vertical direction and are detected by the micro-channel plate. In a micro-channel
plate(MCP) electrons can be amplified by many orders of magnitude before entering
the phosphor screen where they are converted to light. The image on the phosphor
screen corresponds to different optical pulses with earlier ones on the top and the
horizontal positions corresponds to the horizontal location of the incident light. The
brightness of various phosphor images corresponds to the intensity of corresponding
incident pulses.
A micro-channel plate is an electron multiplier consisting of many thin glass capillaries of diameter typically varying from 10 to20µm. The capillaries are bundled
together to form a disk shaped plate of thickness less than 1mm. The internal walls
of each channels are coated with material to enhance secondary electron production.
The electrons get multiplied as the primary electrons hit the wall.
We used a C6138 Hamamatsu streak camera as the detector. It is a single shot
detector and has a temporal resolution of 200 femtoseconds. Spectral response
ranges between 280 nm to 850 nm. A detailed description of the specification of the
camera can be found in [1]. Since the trigger jitter was ± 75ps, the minimum full
screen sweep time was 100 ps.
120
6.2
Results
Figure 6.3: The main interaction beam. The blue line is the raw data and
the dotted red line is a smoothed fitted Gaussian curve
The figure 6.3 shows a plot of the interaction beam. A typical data shot at the
highest intensity is given in figure 6.4. Here note that the highest intensity available
was a factor of 10 times less than the normal Vulcan CPA beam, which has been
used for the experiments described in the previous chapter, as the beam is stretched
out for this experiment. The raw data was analysed by integrating the absolute intensity over time. This is plotted in figure 6.4 as intensity of different polarisations
with time (duration).
Background shots were taken and was reduced from the total intensity. In chapter
four we have measured that the ratio of the p-polarisation over the s-polarisation
decreases as the p-polarisation approaches cut-off at high magnetic fields. However,
in this experiment the intensity was not sufficient to observe cutoffs. The lineout
121
Figure 6.4: Top figure is a typical raw data at the highest intensity (∼
9 × 1018 W/cm2 ). Figure below shows the P and S polarisation
components plotted on a time vs. intensity scale
122
Figure 6.5: Plot of p and s harmonics at an intensity 9 × 1018 W/cm2 . The
red line indicates the p- polarisation and the blue line indicates
the s-harmonics. The dotted green line shows the ratio of s/p.
of the plot shows that at the beginning of the pulse only p-polarised harmonics are
produced and there is insufficient magnetic field to observe birefringence. As intensity increases s-polarised harmonics are produced, as a result of magnetic field. The
p- harmonics peaks as the laser intensity approaches maximum and drops almost
linearly with the incident laser pulse. For the calculation of the magnetic field using
Stokes vectors, all components of Stokes vector are required. This was not possible
in this experimental setup as the slit size of the streak camera was not sufficiently
large to accommodate all the beams without overlapping. Consequently, a complete
determination of the polarisation properties of the light was not possible. However,
the ratio of absolute intensity of the ordinary wave (s-pol) to the extra-ordinary
wave (p-pol) allows a measurement of the initial development of the magnetic field.
Shots were taken at different intensities as well as at short pulse duration. However,
measurements with short pulses are not precise since the resolution of the streak
camera is less than the pulse duration.
Figure 6.6 shows the intensities of s and p harmonics at higher intensity ( 9 × 1018
123
Figure 6.6: Plot of p and s harmonics for intensities 9 × 1018 W/cm2 (A)
and ∼ 1 × 1018 W/cm2 (B) . The blue line indicates the ppolarisation and the red line indicates the s-harmonics. The
dotted green line shows the ratio of ’s/p.’
124
W/cm2 ) and a shot at low laser intensities ( ∼ 1 × 1018 W/cm2 ). In the earlier
chapter it was shown that at lower intensities the value of magnetic field is low. The
ratio of s/p at low intensity (figure 6.6B) is less than 0.1 during the laser rise time.
This agrees with our previous results that the dominant magnetic field mechanism at
high intensity laser matter interaction arise from the spatial and temporal variation
of the laser ponderomotive force. The depolarisation of the harmonics after the peak
intensity could be due to the generation of the third harmonic by other mechanisms.
The magnetic field generated due to the spatial and temporal variation of the ponderomotive force is directly proportional to the laser magnetic field. However, the
s-harmonic peaks few femtoseconds later than the p-polarisation. This could be due
to the fact that there are other mechanisms like Langmuir wave collapse [115], which
give rise to the generation of unpolarised third harmonic emission.
The measurements show that the duration of the harmonic is almost same as the
incident laser pulse. Therefore using the self generated harmonics can only measure
the magnetic field during the laser pulse duration. The field dissipates slowly after
the source is turned off and there is no harmonic generation after the laser pulse
is turned off. Hence an exact measure of magnetic field duration is difficult with
self-generated harmonics.
Also, a comparison has been done with the magnetic field calculated using the
Stokes vectors. This is shown in figure 6.7. The ’s/p’ ratio increases with intensity
and the magnetic field strength.
It is evident that the ’s/p’ ratio increases as the
intensity increases. Also, it is clear that the s-polarisation peaks when the intensity
peaks. At the beginning of the pulse there is no s-polarisation as the magnetic field
is low or negligible at low intensities. From this analysis it is likely that the duration
of the magnetic field is at least as long as that of the incident laser pulse. At high
intensities the s-polarisation signal peaks later than the p-polarisation signal, i.e,
the magnetic field peaks after the incident pulse peaks. In the latter part of the
pulse the ratio changes such that the harmonics are essentially depolarised. This
125
Figure 6.7: B field measured using Stokes vectors in the earlier chapter with
a short pulse beam (blue line). The respective s/p ratio is plotted for the same intensity (red line).
Figure 6.8: Osiris PIC simulations showing the evolution of the laser intensity (red line) and the self-generated magnetic field (blue line)
126
may be because the third harmonic can be generated by other mechanisms which do
not produce polarised radiation (i.e., Langmuir wave turbulence and collapse [115]).
In order to have more understanding, simulations were carried out using a particle in cell code. The code is same as discussed in section 5.4. The simulation was
run in a box of 22µm×40µm. The size of the simulation box is split in to 840×1480
cells in each cell. The density scale was varied from 1 to 10 times critical density.
Figure 6.9: Osiris PIC simulations showing the evolution of dc magnetic
field is plotted against laser periods(time) at different laser intensities
c
Laser was sent at an angle of 45 degree at a height of 25 ω . The focal spot was
p
c
12.5× ω . The laser pulse intensity reaches its peak intensity in few laser periods
p
(∼ 20) and remained the same for ∼ 40 laser periods. After that the laser pulse
was turned off within ∼ 20 laser periods. The simulation ran for longer duration
than the laser duration, to calculate the evolution of magnetic field after the pulse
is turned off.
Simulations were also carried out with a laser intensity 1019 W/cm2 where, the laser
127
was turned off after 70f s. The run time of the simulation was set at different times
with a minimum of 24 laser periods. The magnetic field peaks after few laser periods
and remains the same untill the laser intensity falls. After the laser is turned off
the field decays over a period of time which is almost equivalent to the duration
of the laser pulse. The strength of the magnetic field was calculated for ∼ 140f s.
This is plotted in figure 6.8. The magnetic field strength increases proportionally
with square root of laser intensity and peaks as intensity reaches the maximum.
After the laser is turned off the field decays slowly. A series of runs were carried out
at different laser intensities to calculate the evolution of magnetic field with laser
intensity. The results are shown in figure 6.9 and they agree with the experimental
observations.
The analytical calculation of the magnetic field evolution is performed. The source
terms causing the generation and diffusion of self generated magnetic field can be
obtained from the generalised Ohm’s law,
∂B
1
ck
1
= ∇×(v×B)−
∇×(J×B)+
(∇Te ×∇ne )+
∇×[(∇×B)×B]−∇×ηJ
∂t
ne e
ne e
4πne e
(6.1)
The first term on the right hand side is convection term, the second term is the
electron magnetohydrodynamic source due to Hall effect, the third term is the thermoelectric source term originating from the electron pressure term. The fourth term
contains the curvature term and magnetic pressure term and the fifth term is the
dissipative term.
Simplifying, the evolution equation for the magnetic field can be written as
∂B
η
= ∇ × (v × B) + ∇2 B
∂t
µo
(6.2)
In the earlier measurements we have seen that the magnetic field measured in our
experiments were generated due to the spatial and temporal gradient of the laser
ponderomotive force and is localised near the critical density surface. Therefore, the
other source terms can be neglected. The duration of magnetic field generation due
to ponderomotive source term is the laser duration. Therefore, in order to calculate
the dissipation time the first, fourth and fifth terms need to be examined. Therefore,
128
the dissipation time τ
τ=
µ0 l 2
η
(6.3)
1
where, µ0 is the permitivitty of free space, η is the plasma resistivity η = σ and l is
the length of plasma. The plasma resistivity is given by the Spitzer formula [90]
1
η≈
πe2 m 2
3
(4π0 )2 (KTe ) 2
ln Λ
(6.4)
For laser plasma KTe is of the order of 100eV and ln Λ is around 6.8, therefore
the resistivity approximately 7 × 10−7 Ω − m. In our experiment we performed, the
pulse duration was ∼ 8ps. The magnetic field we measured in our experiments were
localised between the critical and quarter critical density surface. Taking l ∼ 6µm,
the dissipation time is ≈ 45 ps.
6.3
Summary
The temporal evolution of the magnetic field is measured with the self generated
harmonics of the incident laser pulse. The growth of the field is proportional to the
square root of laser intensity. At the beginning of the pulse where the intensity is
low, there is no magnetic field is generated. This is evident in the depolarisation
measurement of the harmonics. As the laser intensity peaks the self generated field
peaks and reaches a maximum value. It is also clear that the p-polarisation peaks
earlier than the s polarised harmonic. This can be explained by the fact that the
magnetic field saturates as the laser pulse peaks. The field decays slowly as the laser
intensity falls. There is an increase in the ratio of s/p, this might be due to other
mechanisms which generates unpolarised third harmonic. The simulation results are
in agreement with the experimental results of generation of magnetic field. In order
to calculate the evolution of magnetic field after the laser pulse is turned off, more
changes like the inclusion of the resistivity term in the code. Also the duration of
the diagnostic tool (the self generated harmonics) is shorter than the laser pulse.
Therefore a measure of the entire evolution of the magnetic field was not possible.
The major limitations of this technique are:
129
1. the wavelength sensitivity of the detector was limited to ∼ 300 nm. Therefore, the use of higher order harmonic for measurements were not possible. Higher
order harmonics can propagate to higher densities where larger magnetic fields are
predicted to exist.
2. The duration of the self-generated harmonics was shorter than the incident pulse
- hence it is difficult to deduce the decay of the magnetic field after the duration of
the incident pulse.
3. The physical dimension of the entrance slit of the streak camera detector is
quite small so it is not possible to have all the Stokes vectors measured to make a
quantitative measure of the magnetic field with time. Accurate measurements are
only possible for low magnetic fields such that B ∝ s/p. Future research is discussed
in the next chapter.
130
Chapter 7
Conclusions and Discussions
7.1
Summary of the thesis
This thesis studied the measurement of large magnetic fields generated during short
pulse high power laser solid interaction in the laboratory. Theoretical and computational studies have predicted the existence of several hundred megaGauss magnetic
fields during such interaction. Previous experimental studies were restricted to low
density regions in the ablated plasma, where the strength of the field is many orders
smaller than the field in the high density region. This was due to the limitations
of the diagnostic used. Two independent methods based on the self generated harmonics generated during the interaction were developed. The first technique is
the cut-off method and the second is the harmonic polarimetry method based on
Stokes vector analysis. The techniques were based on the fact that electromagnetic
radiation propagating perpendicular to the magnetic field experiences a change in
ellipticity (birefringence), the Cotton-Mouton effect. The cut-off method is a simple
and direct measure of magnetic field, where the extraordinary component (x-wave),
whose polarisation is perpendicular to the magnetic field and so does not propagate
and gets reflected. The second method measures the initial and final Stokes vectors
of the harmonics before and after propagating through the plasma.
In Chapter 4, the theory and the experimental setup for the cut-off method were
131
discussed. The lower order optical harmonics of the incident lasers were used as
the diagnostic probe. The cut-off measurements reported in this chapter observed
the cut-off of the x-wave of the second, third and fourth harmonics of the Vulcan
(1.054µm) laser. Cut-offs were observed up to the fourth harmonics. The maximum
magnetic field measured using this technique was calculated to be 340 ± 50 M G. No
assumptions were needed to calculate the peak magnetic field strength.
In Chapter 5, the harmonic polarimetry method using the Stokes vector analysis
was used. Stokes vectors are a set of parameters which describe the state of polarisation of an electromagnetic wave in terms of measurable quantities (intensity).
The depolarisation of the second, third, fourth and fifth harmonic was measured of
the Stokes vectors to give the induced ellipticity and the strength of the magnetic
field. The measured magnetic field were of the order of ∼ 400 ± 50M G. The half
3
harmonics ( 2 ) produced at the quarter critical density surface did not show any
change in polarisation. This shows that the large magnetic field is localised near the
critical density surface. The possible errors in the calculation of magnetic field was
from the estimation of the angle at which the harmonics enters the magnetic field
and also from the calculation of density scale length.
1
The observed field grow with ponderomotive scaling, (Iλ2 ) 2 . This indicates that
the field generated is due to the ponderomotive mechanism proposed by Sudan [9]
. Particle in cell simulation results are also in agreement with the experimental
results. At lower intensities the magnetic field measured using the Vulcan laser is
comparable to the measurements with the Astra laser (which has a shorter pulse).
The theoretical calculations by Sudan predicted that very large magnetic fields will
extend into the plasma only for a collisionless skin depth. However, in the simulations it is seen that these fields exist over a laser spatial extent (several times
the collisionless skin depth) . This could be due to the relativistic motion of the
electron in the large fields generated by the focused laser intensity. The particle in
cell simulation using Osiris 2D3V code at oblique incidence shows that the field is
132
asymmetric and is larger in the specular reflection direction. A possible explanation
for this could be the laser ponderomotive force pushing the electrons so that there is
an accumulation of magnetic flux in a smaller region. This was experimentally studied in a different experiment. The harmonics were collected at two different angles
simultaneously . For similar density scale length calculations, the plasma transition
matrix is the same (i.e., the field is different) on both sides of laser incidence. This
indicates that the total field is conserved. In another experiment performed with the
Astra laser a new technique capable of mapping of magnetic field over a large range
of angles has been proposed and the initial measurements are reported. Using a
large angle collection optics like an ellipsoidal mirror compared to normal spherical
mirror can collect harmonics emission over a large range of angles and therefore,
enable us to calculate the magnetic field strength at different solid angels simultaneously.
In Chapter 6, measurements of the temporal evolution of the magnetic field was
carried out using a longer incident pulse. The self generated third harmonic was
used as the diagnostic probe. The growth of the magnetic field is proportional to
the square root of laser intensity ((Iλ2 )1/2 ). The field saturates as the laser intensity peaks and start to decrease as the laser intensity falls. A measurement of the
absolute value of magnetic field was not possible due to technical limitations. The
evolution of the field is in agreement with Vulcan results as well as the particle in
simulation results.
In summary, the major mechanism of magnetic field generation in high power laser
matter interaction has been identified using two new techniques. The use of self
generated harmonics are superior than external diagnostic techniques(probing) as
the latter can perturb the medium. The cut-off method is a very efficient technique
at intensities higher than 8 × 1019 W/cm2 and does not require any assumptions.
The harmonic polarimetry method using Stokes vector analysis is applicable at lower
intensities. The highest field measured using this technique is limited only by cut-off
of the respective harmonic employed. The lower limit of the measurements is lim133
ited due to the sensitivity of our diagnostics. The experimental results are in good
agreement with the prediction of computer simulations. The temporal evolution and
spatial distribution of self generated magnetic field for 45o laser incidence is carried
out using the self generated harmonics. Another new technique has been demonstrated for the mapping of these ultra huge fields. The magnetic field observed using
the Stokes vector analysis shows that at higher intensities the dominant magnetic
field mechanism is due to the gradient in the ponderomotive force of the incident
laser.
7.2
Consequences of large B fields in laser-matter
interaction
At intensities ∼ 1020 W cm−2 the magnetic field generated is comparable to the oscillating field of the incident laser. The field measured in this thesis is the highest self
generated magnetic field in the laboratory and is more than 10 times larger than the
previously recorded magnetic field in the laboratory and eight orders of magnitude
higher than the earth’s magnetic field. With current high power lasers delivering
1 P W , it will be possible that the strength of the self generated magnetic fields
will begin to approach those required to generate Landau quantization of electron
motion in hydrogen. [122, 123]. The motion of the electron in the orbits may be
affected as the presence of these huge fields will perturb the electron wave function
and the electron will follow cylindrical orbits rather than spherical orbits. As the
intensity of laser systems is increased further, these magnetic fields may begin to
affect fundamental parameters of the plasma such as the equation of state and the
radiative spectral opacities in the plasma. Also the magnitude of these huge fields
are comparable to that may exist in many astronomical bodies (neutron stars and
white dwarfs) [124] and will be possible to test astrophysical models in the laboratory.
The presence of these huge magnetic field can affect the plasma expansion. The
134
evolution of the magnetic field depends mainly on the convection term and the resistive diffusion term. We know from the measurements carried out that the strength
of the magnetic field is approximately 200 M G at 9 × 1018 W/cm2 intensity (refer.
figure 5.10). This will result in a very small plasma beta . The plasma beta is the
ratio of thermal pressure to magnetic presssure, and can be calculated as
β=
nkT
B 2 /2µo
(7.1)
Taking experimental conditions,
β=
1.1 × 1027 · 103 · 1.6 × 10−19 · 8 · π · 10−7
(2 × 104 )2
β ≈ 10−4
(7.2)
(7.3)
This means that at higher intensities where the magnetic field is several hundred
megaGauss the magnetic pressure term dominates. When magnetic field dominates the Lundquist number [125] define the flow of magnetic energy over a distance. i.e., the Alfven speed (vA = B/(µo ρ)1/2 ) dominates over the accoustic speed
(cs = (KTe /M )1/2 ) . When the Lundquist number is high (i.e., when β is very
small) magnetic diffusion dominates.
The diffusion of the magnetic field lines (or the annihilation of the magnetic field)
can be calculated from the induction equation (equation 6.2) ,
τ = µo L2 /η
(7.4)
where L is the scale length of the spatial variation of the magnetic field. τ is the
characteristic time for magnetic field to penetrate into plasma. Ohmic heating can
be caused by the induced current due to the flow of field lines in the plasma. The
energy lost per unit volume in a time τ is ηj 2 τ , where j is the induced current. Thus
the dissipated energy can be calculated from Maxwell’s equation with displacement
term neglected,
ηj 2 τ = B 2 /µo
which is twice the magnetic pressure.
135
(7.5)
In the presence of high magnetic fields the ωc τ (τ is the collision time) will be
an important quantity in confining the plasma. At high magnetic fields ωc τ 1 ,
the diffusion of the plasma across the field lines will be decreased. In these circumstances
• the step length will be Larmor radius and not λm . The diffusion can be
decreased or increased by varying the magnetic field [33]
• For diffusion parallel to B - the diffusion coefficient is inversely proportional to
ν. In diffusion perpendicular to B the diffusion coefficient is proportional to ν.
For parallel diffusion the diffusion coefficient is proportional to1/(m)1/2 and
for perpendicular diffusion, the diffusion coefficient is proportional to (m)1/2 .
In parallel diffusion electrons move faster because their mass is small. In perpendicular diffusion electrons escape slowly because of smaller Larmor radius.
D ∼ λ2m τ and D⊥ ∼
2
γL
τ
The study of growth and decay of the magnetic field can give an insight into the
energy absorption and diffusion of hot plasma. Magnetic fields is important for confinement of plasma in laboratories and in astronomical bodies. At higher intensities
the magnetic pressure can be higher than the thermal pressure of the plasma and
may be important in laser fusion [126] experiments. Use of ultra short pulses for
the generation of ultrashort magnetic fields can be used in magnetic data storage
technologies [127] and also in reversal of magnetisation in thin films [128]. The
self -generated magnetic field can be sufficiently high to inhibit the fast electron
transport in laser plasmas [76, 77, 80, 129, 130].
7.3
Future Research
The self -generated XUV harmonics can be used to observe cut-off, which will be
able to give a direct measure of magnetic fields at higher intensities. Stokes vector
analysis of XUV harmonics can be used where cut-off method can not be employed.
Also external XUV radiation can be used to probe the solid density plasma. These
radiation can be generated using a high power laser solid interaction or laser gas
136
cell interaction. These radiations will be able to propagate to solid densities (for
λ ∼ 105nm, tenth harmonic of Vulcan laser) and will be able to measure the field inside the target due to fast electron current and Weibel instability. By using suitable
time delay it is possible to measure the evolution of the magnetic field at different
times, sending the xuv radiation perpendicular to the target and also at an angle
to the target to find the Cotton-Mouton effect as well as Faraday rotation. Higher
order gas harmonics can be generated very efficiently using ultra short pulse lasers.
Another technique is using a glass target and sending an externally generated harmonic for Faraday rotation measurements which is a straightforward method to
measure the magnetic field inside the target.
Many beam interactions may be useful to study magnetic reconnection experiments
in the laboratory. The reconnection of the magnetic field lines in the blow off region (generated by the thermo electric mechanism) can be studied using an external
probe beam in the optical wavelength region. For higher density regions radiation
in the xuv spectral region can be used.
Future experiments will be conducted in the 1 P W laser, which will take us to
a new regime of laser plasma interaction where the electron motion will be highly
relativistic. The focussed intensity will be high enough for multiple ionisation of
the target material leading to increase in plasma density and thereby affecting the
physics of interaction.
The generation of ultra short magnetic pulse has many application in the development of future magnetic data storage technology [127]. In this thesis we have
demonstrated that magnetic pulses of picosecond regime can be developed in the
laboratory using table top lasers.
137
Chapter 8
Appendix I
8.1
The cold plasma dispersion relation
The cold-plasma dispersion relation was first obtained by Appleton and later derived
extensively by Hartree in 1932 [131] and is known as the Appleton-Hartree dispersion
relation for cold plasma. The derivation is given below.
The dispersion relation of an electromagnetic wave propagating in a magnetised
plasma is obtained from Maxwell’s equations,
∂B
∇×E=−
,
∂t
∂E
∇ × B = µ0 j + 0
∂t
Rearranging the above equations by eliminating the magnetic field term
∂
∂E
∇ × (∇ × E) +
µ 0 j + 0 µ0
=0
∂t
∂t
(8.1)
(8.2)
(8.3)
Taking the Fourier analysis of the fields and current
k × (k × E) + iω(µo σ · E − 0 µ0 iωE) = 0
ω2
k × (k × E) + K · E = 0
c2
1
where o µo = 2 and 1 is the unit dyadic and K is the dielectric tensor.
c
i
K= 1+
σ
ω0
σ is the conductivity. The dielectric tensor is obtained as follows.
138
(8.4)
(8.5)
(8.6)
8.1.1
Dielectric tensor
The equation of motion for a single particle species j with a charge qj in an electromagnetic field is,
dvj
= qj (E + vj × B)
dt
j is the current density. The total current
X
j=
nj qj vj
(8.7)
mj
(8.8)
j
nj is the number density of the species j. Assuming plasma to be uniform and
homogeneous in space and time and taking the Fourier transform of the electric
(E = E0 e−i(ωt−kr) ) and magnetic field (B = B0 e−i(ωt−kr) ) equations and substituting
them in the equation of motion (equation 8.7) we get v which is purely harmonic
(∝ e−iωt ). Hence,
−iωmj vj = qj (E + vj × B)
(8.9)
If the second order terms are neglected and the wave amplitudes are adequately low
enough that linear approximation is valid. The velocity components in terms of the
electric field are
vxj
vyj
iωcj Ey
Ex −
ω
ω2
−iqj
1
iωcj Ex
=
+ Ey
2
mj ω 1 − ωcj
ω
−iqj
1
=
2
mj ω 1 − ωcj
(8.10)
(8.11)
ω2
vzj =
iqj
Ez
mj ω
(8.12)
|qj |B
where, ωcj ≡ m is the electron cyclotron frequency.
j
Using the cold plasma approximation the current density is due to electron motion
is
j=
X
qj nj vj = σ · E
(8.13)
j
where σ is the conductivity tensor and can be written as,

ω
0
1 −i ωcj

2
inj qj
1
 ωcj
σ=
·
1
0
2  i ω
mj ω 1 − ωcj

2
ω2
ω
0
0
1 − ωcj2
139





(8.14)
Substituting the value of σ in equation 8.6 and rearranging equation 8.5
n × (n × E) + K · E = 0
(8.15)
kc
where n = ω . The refractive index vector n has the same direction as the wave
vector k. k is in the x-z plane and the magnetic field B in the z -direction and θ is
the angle between the propagation vector k and z-axis, then


2
2
2
S − n cos θ −iD n cos θ sin θ
E

 x




  Ey
iD
S − n2
0


2
2
2
n cos θ sin θ
0
P − n sin θ
Ez



=0

(8.16)
The nontrivial solution can be obtained when the coefficients of the determinant
vanish.
An4 − Bn2 + C = 0
(8.17)
which is the cold plasma dispersion relation [132]. where,
A = S sin2 θ + P cos2 θ
B = RL sin2 θ + P S(1 + cos2 θ)
C = P RL
RL = S 2 − D2
where,
ωp2
1
(R + L) = 1 − 2
2
ω − ωc2
ωc ωp2
1
D ≡ (R − L) =
2
ω (ω 2 − ωc2 )
ωp2
P =1− 2
ω
S≡
Substituting the value of j into Maxwell’s equation [132], the plasma dielectric tensor
can be obtained.

−iD 0
S


K =  iD

0
140
S
0



0 

P
(8.18)
where,
2
X ωpj
1
(R + L) = 1 −
2
2
ω 2 − ωcj
j
(8.19)
2
X j ωcj ωpj
1
D ≡ (R − L) =
2
2
ω ω 2 − ωcj
j
(8.20)
S≡
P =1−
ω2
R=S+D =1−
X
j
2
ωpj
j
ω (ω + j ωcj )
L=S−D =1−
X
j
8.2
2
X ωpj
ωp2
ω (ω − j ωcj )
(8.21)
(8.22)
(8.23)
CUT- OFF - Mathematical derivation
The solutions of the dispersion relation breakdown in the immediate vicinity of cutoffs and resonances. The physical process of the cut-off can be explained as follows.
In a magnetised or a slowly varying density plasma as the wave approaches a cut-off
the refractive index tends zero figure 8.1. Near the cut-off region the wavelength
increases infinitely so does the phase velocity. However, the field remains finite and
can be calculated using the differential form of the wave equations assuming the
cut-off point is located near ẑ = 0 (where a > 0). Therefore,
n2 = a(z) + O(z 2 )
(8.24)
In the region where z > 0 the electromagnetic radiation propagate and in the region
where z < 0 the wave will decay evanescently. The wave solution for this physical
condition is given by
d2 E
+ ẑE = 0
d2 ẑ
(8.25)
1
where ẑ = (k02 a) 3 z. The above equation 8.14 is an Airy equation [131, 133] and
ˆ and Bi(−z).
ˆ
the solutions of the above equation are Ai(−z)
The first solution has
asymptotic behavior when z → −∞
1
1
ˆ ∼ √ |ẑ|− 4 exp
Ai(−z)
2 π
141
“
3”
− 23 |ẑ| 2
(8.26)
Figure 8.1: Variation of phase velocity near cut-off region
Figure 8.2: Physical sketch of cut-off and resonance
142
When ẑ → ∞ the equation becomes,
1
1
ˆ ∼ √ ẑ − 4 sin
Ai(−z)
π
2 3 π
ẑ 2 +
3
4
(8.27)
An electromagnetic radiation polarised in the y-direction, traveling towards the cutoff point (z = 0) can be represented as a linear combination of propagating WKB
solutions in terms of reflected wave and transmitted wave,
Z ẑ
Z
− 12
− 12
ndz + Rn exp +ik0
−ik0
Ey (ẑ) = n
0
z
ndz
(8.28)
0
where, the first term on the RHS represents the incident wave, while the second
term represents the reflected wave and R is the coefficient of reflection. Near the
cut-off point the above equation reduces to
Ey (ẑ) =
ko
a
16 1
2 3
2 3
− 14
−
ẑ
−i ẑ 2 + Rẑ 4 exp +i ẑ 2
3
3
(8.29)
Also
1
C
ˆ ' √ ẑ − 4 sin
Ey (ẑ) = CAi(−z)
π
2 3 π
ẑ 2 +
3
4
.
(8.30)
is an equation representing the same region (ẑ is small and positive)comparing the
above two equations can give the value of the reflection coefficient R
R = −i
(8.31)
π
Hence it is clear that at cut-off point there is total reflection with a − 2 phase shift.
143
Chapter 9
Appendix II
9.1
General representation of an electromagnetic
wave
The transverse components of the electric field can be represented as
Ex (z, t) = E0x cos (τ + δx )
(9.1)
Ey (z, t) = E0y cos (τ + δy )
(9.2)
where τ = ωt − kz and the subscripts represents the components in the x and y
direction. E0x and E0y are the maximum amplitudes and δx and δy are phases. As
the field propagates the resultant vector defines the locus of the field at any instant
of time. It can be derived as follows,
Ex
= cos τ cos δx − sin τ sin δx
E0x
Ey
= cos τ cos δy − sin τ sin δy
E0y
(9.3)
(9.4)
therefore,
Ex
Ey
sin δy −
sin δx = cos τ sin (δy − δx )
E0x
E0y
Ex
Ey
cos δy −
cos δx = sin τ sin (δy − δx )
E0x
E0y
(9.5)
(9.6)
which gives,
Ey2
Ex2
Ex Ey
+
−
2
cos δ = sin2 δ
2
2
E0x E0y
E0x E0y
144
(9.7)
Equation 9.7 represents an ellipse and shows that at any point of time the locus of
the optical field is an ellipse which is called polarisation ellipse. Different states of
Y
Y``
X``
a
b
ψ
X
2b
O
2a
Figure 9.1: Polarisation ellipse
polarisation can be obtained by when E0x or E0y is equal to zero or equal and/or
δ = 0 or
9.1.1
π
2
orπ radians.
Horizontally or vertically linear polarised light
When E0y = 0 we have
Ey = 0
(9.8)
Ex = E0x cos (τ + δx )
(9.9)
In this case light is horizontally linearly polarised. When E0y = 0 it is vertically
linear polarised.
When δ = 0 or π when δ = 0 or π, equation 9.7 reduces to
Ey2
Ex2
Ex Ey
+
±2
=0
2
2
E0x E0y
E0x E0y
145
(9.10)
i.e
Ex
Ey
±
E0x E0y
2
=0
(9.11)
which can be written as
Ey = ±
E0y
E0x
Ex
which is the equation of a straight line with slope ±
(9.12)
E0y
E0x
and intercept at origin.
For values of δ = 0 the slop is negative and δ = π the slope is positive. When
E0y = E0x the slope is 1 and the wave is said to be ±450 linearly polarised.
When δ =
π
2
or
3π
.
2
The polaristion ellipse reduces to
Ey2
Ex2
+ 2 =1
2
E0x
E0y
(9.13)
this is the characteristic equation of an ellipse.
When δ =
π
2
or
3π
2
and E0x = E0y = E0 The equation 9.7 reduces to
Ex2 Ey2
+
=1
E02 E02
this is the standard equation of a circle. When δ =
polarised and when δ =
3π
2
(9.14)
π
2
the wave is right circularly
wave is left circularly polarised.
The general expression of the polarisation ellipse is
Ey2
Ex2
Ex Ey
+ 2 −2
cos δ = sin2 δ
2
E0x E0y
E0x E0y
(9.15)
where,δ = δx − δy . If the axes of the ellipse are not parallel to the X-Y axes then
the third term in equation 9.15 would appear. From figure 9.1 OX and OY are the
original axis of the ellipse. When the ellipse is rotated by an angle ψ with respect
to the original axis, hence, the electric field components become
Ex0 = Ex cos ψ + Ey sin ψ
(9.16)
Ey0 = −Ex sin ψ + Ey cos ψ
(9.17)
If 2a and 2b are the major and minor axis respectively, then equation of ellipse can
be written as
Ex0 = a cos (τ + δ 0 )
(9.18)
Ey0 = ±b cos (τ + δ 0 )
(9.19)
146
i.e
Ex02 Ey02
+ 2 =1
a2
b
(9.20)
from equation 9.16 and 9.17 we can obtain,
Ex
= cos (τ + δx )
E0x
Ey
= cos (τ + δy )
E0y
(9.21)
(9.22)
From the above equations we can get
2
2
a2 + b2 = E0x
+ E0y
(9.23)
±ab = E0x E0y sin δ
2E0x E0y cos δ
tan 2ψ =
2
2
E0x
− E0y
(9.24)
let α be the angle of polarisation ellipse 0 ≤ α ≤
tan α =
π
2
(9.25)
then
E0y
E0x
(9.26)
then equation 9.26 becomes
tan 2ψ =
or
2 tan α
cos δ
1 − tan2 α
(9.27)
= tan 2α cos δ
when δ = 0 or π the angle of rotation is ψ = ±α
when δ =
π
2
or
3π
2
the angle of rotation is zero.
The ellipticity of the polarisation ellipse is given as
tan χ =
±b
a
−π
π
≤χ≤
4
4
(9.28)
e.g for linearly polarised light b = 0, therefore, χ = 0, for circularly polarised light
b = a, hence, χ = ± π4 .
Also
±2ab
2E0x E0y
= 2
sin δ = sin 2α sin δ
2
2
2
a +b
E0x + E0y
sin 2χ = sin 2α sin δ
(9.29)
(9.30)
From the above equations it is clear that the polarisation ellipse can be described
either in terms of orientation and ellipticity angles or in terms of major and minor
147
axes of the ellipse along with the phase shift δ.
The use of right handed and left handed polarisation is based on the direction of
the electric field vector clockwise or counterclockwise with respect to the observer
who is looking from the direction from which the light is coming.
9.2
Stokes vectors
However, in reality the light vector traces an ellipse in a plane perpendicular to the
propagation direction at a rate of 1015 times per second. So it is difficult to trace the
ellipse in such a short time duration. Since the polarisation ellipse is the physical
observation of light at a particular instant in time, the above analysis of observing
polarisation ellipse is valid for polarised light only. Therefore, it is necessary to introduce new parameters which can be used to describe measurable quantities. The
four Stokes parameters discovered by Sir George Gabriel Stokes [110] can measure
all quantities needed to describe fully light of any polarisation state. The first parameter gives the total intensity of the optical field. The remaining three parameters
define the polarisation states.
Intensity is the observable quantity which can be measured as square of amplitude which is not observable. Hence if we take the time average of the square of the
unobservables of the polarisation ellipse will give us the observables of the polarisation ellipse.
Consider a pair of plane waves orthogonal to each other at a point in space, at z=0
then,
Ex (t) = E0x (t) cos [ωt + δx (t)]
(9.31)
Ey (t) = E0y (t) cos [ωt + δy (t)]
(9.32)
where each notation has the same definition as explained earlier. At a particular
instant of time the polarisation ellipse will be
Ey2 (t)
Ex2 (t)
2Ex (t)Ey (t)
+
−
cos δ(t) = sin2 δ(t)
2
2
E0x (t) E0y (t) E0x (t)E0y (t)
148
(9.33)
where δ(t) = δy (t) − δx (t). For monochromatic radiation the amplitude and phase
are constant, therefore the above equation reduces to
Ex2 (t) Ey2 (t) 2Ex (t)Ey (t)
+
−
cos δ = sin2 δ
2
2
E0x
E0y
E0x E0y
(9.34)
E0x and E0y and δ are constants and Ex and Ey are dependent on time. Therefore,
the observables of the polarisation ellipse can be obtained by taking the time average
of the observables of the optical field.
hEx2 (t)i hEy2 (t)i 2hEx (t)ihEy (t)i
+
−
cos δ = sin2 δ
2
2
E0x
E0y
E0x E0y
(9.35)
where,
1
hEi (t)ihEj (t)i = lim
T =∞ T
Z
∞
Ei (t)Ej (t)dt
i, j = x, y
(9.36)
0
which gives rise to
2
2
4E0y
hEx2 (t)i + 4E0x
hEy2 (t)i − 8E0x E0y hEx (t)Ey (t)i cos δ = (2E0x E0y sin δ)2 (9.37)
i.e.
1 2
hEx2 (t)i = E0x
2
1 2
2
hEy (t)i = E0y
2
1
hEx (t)Ey (t) = E0x E0y cos δ
2
(9.38)
(9.39)
(9.40)
substituting above eqns in to eqn.36 gives,
2
2
2
2
2E0x
E0y
+ 2E0x
E0y
− (2E0x E0y cos δ)2 = (2E0x E0y sin δ)2
(9.41)
In order to express the above equation in terms of intensity we add and substract
4
4
E0x
+ E0y
from the above equation.
2
2
+ E0y
E0x
2
2
2
− E0x
− E0y
2
− (2E0x E0y cos δ)2 = (2E0x E0y sin δ)2
(9.42)
the quantities inside the parentheses can be written as
2
2
E0x
+ E0y
= S0
2
2
= S1
E0x
− E0y
(9.43)
(2E0x E0y cos δ) = S2
(9.45)
(2E0x E0y sin δ) = S3
(9.46)
149
(9.44)
and
S12 + S22 + S32 = S02
(9.47)
the above quantities are called Stokes parameters and it is evident that all of them
are real and observables. The first parameter S0 is the total intensity of the optical
field. S1 gives the horizontal or vertical linear polarisation, S2 gives the amount of
linear +450 or −450 polarisation and S3 gives the amount of right or left circularly
polarisation contained in the beam. In the case of a partially polarised light the
above parameters are valid for short interval of time as the amplitude and phase
fluctuates slowly. Using the Schwarz’s inequality, eqn. 46 can be written as
S02 ≥ S12 + S22 + S32
(9.48)
Equality sign is applicable for fully polarised light and inquality sign for partially
or unpolarised light. Also the orientation angle of the ellipse can be deduced as
tan 2ψ =
S2
S1
(9.49)
S3
S0
(9.50)
and the ellipticity angle χ is given by
sin 2χ =
and the degree of polarisation P for any state of polarisation is
P =
Ipol
(S 2 + S22 + S32 )2
= 1
Itot
S0
0≤P ≤1
(9.51)
where, Ipol is the intensity of the sum of the polarisation components and Itot is the
total intensity of the beam. P = 0 corresponds to unpolarised light and P = 1
corresponds to partially polarised light.
Each polarisation state can be uniquely represented by a point P on the Poincaré
sphere having a latitude 2χ and longitude 2ψ. Poincaré sphere is a sphere of unit
radius in (s1 , s2 , s3 ) space. Two orthogonal polarisations can be represented by
two diametrically opposite points on the sphere. The change of polarisation due to
interaction with polarising elements can be described using Poincaré sphere.
The Stokes parameters for various polarisations can be expressed as follows
150
Figure 9.2: Poincaré sphere
Linear horizontal polarised light E0y = 0
therefore,
2
S0 = E0x
(9.52)
2
S1 = E0x
(9.53)
S2 = 0
(9.54)
S3 = 0
(9.55)
2
S0 = E0y
(9.56)
2
S1 = −E0y
(9.57)
S2 = 0
(9.58)
S3 = 0
(9.59)
Vertically polarised Light E0x = 0
Linear +450 In this case E0x = E0y = E0 and δ = 00 .
S0 = 2E02
151
(9.60)
S1 = 0
(9.61)
S2 = 2E02
(9.62)
S3 = 0
(9.63)
Linear −450 In this case the amplitudes E0x = E0y = E0 and δ = 00 and phase
difference δ = 0
S0 = 2E02
(9.64)
S1 = 0
(9.65)
S2 = −2E02
(9.66)
S3 = 0
(9.67)
Right circularly polarised light The amplitudes E0x = E0y = E0 and δ = 900
S0 = 2E02
(9.68)
S1 = 0
(9.69)
S2 = 0
(9.70)
S3 = 2E02
(9.71)
Left circularly polarised light For right circularly polarised light the amplitudes are
equal but the phase difference between orthogonal and transverse components is
δ = 2700 , hence the Stokes parameters are
S0 = 2E02
(9.72)
S1 = 0
(9.73)
S2 = 0
(9.74)
S3 = −2E02
(9.75)
The representation of Stokes parameters in a column matrix is called Stokes vector.


S
 0 


 S1 

S=
(9.76)


 S2 


S3
152
which is in terms of observables of the optical field.


2
2
+ E0y
E0x




2
2
 E0x
− E0y 


S=

 2E0x E0y cos δ 


2E0x E0y sin δ
(9.77)
therefore the Stokes vectors for different states of polarisation can be found from
equation 9.76. Stokes vectors for linearly horizontally polarised light is
 
1
 
 
 1 

S = I0 
 
 0 
 
0
similarly of linearly vertically polarised light the Stokes vectors are


1




 −1 


S = I0 

 0 


0
(9.78)
(9.79)
2
where I0 is the total intensity E0i
, where i = xory
For linear 450 and −450 polarised light theStokes vectors are
 
1
 
 
 0 

S = I0 
 
 1 
 
0


1




 0 

S = I0 


 −1 


0
153
(9.80)
(9.81)
respectively. where I0 = 2E02 .
Also for left and right circularly polarised light, the Stokes vectors are


1




 0 


S = I0 

 0 


−1

(9.82)

1
 
 
 0 

S = I0 
 
 0 
 
1
(9.83)
Similarly, the orientation angle ψ and ellipticity χ are given by
tan 2ψ =
sin 2χ =
S2
S1
S3
S0
0≤ψ≤π
(9.84)
π
−π
≤χ≤
4
4
(9.85)
for δ = 00 or 1800 , S3 is 0 Stokes vector becomes


2
2
E + E0y
 0x

 2

2
 E0x − E0y 

S=


 ±E0x E0y 


0
(9.86)
This can be used for the representation of linear polarised light.
If we introduce α as the auxillary orientation angle, then
2
2
S = E0x
+ E0y
= E02
(9.87)
from the figure
E0x = E0 cos α
(9.88)
E0y = E0 sin α
(9.89)
154
E0y
E0
α
E0x
Figure 9.3: Optical field
where, 0 ≤ α ≤ π2 . Therefore, the Stokes vector for a linearly polarised light is


1




 cos 2α 

(9.90)
S = I0 


 sin 2α 


0
where, I0 = E02 . Hence,

1


 cos 2α
S = I0 

 sin 2α cos δ

sin 2α sin δ








The normalised Stokes vector can be obtained by setting I0 = 0


1




 cos 2α 

S=


 sin 2α cos δ 


sin 2α sin δ
155
(9.91)
(9.92)
and orientation angle ψ is
tan 2ψ = tan 2α cos δ
(9.93)
sin 2χ = sin 2α sin δ
(9.94)
ellipticity χ is given by
Equation 9.92 gives the representation of the Stokes vector using auxiliary angle α
and phase difference. When α = 0 the polarisation ellipse is represented using only
the phase shift between the orthogonal amplitudes.


1




 0 


S=

 cos δ 


sin δ
(9.95)
The orientation angle ψ always 450 and the ellipticity angle is
sin 2χ = sin δ
(9.96)
i.e., χ = 2δ . In this case the polarisation ellipse is rotated by 450 from the horizontal
axis and with polarisation state varying from linear polarisation (δ = 0, π) to circular
polarisation (δ = 900 or 2700 ).
In the case of δ = 900 or 2700 , the Stokes vector reduces to


1




 cos 2α 

S=




0


± sin 2α
(9.97)
and the orientation angle ψ is zero and ellipticity angle
sin 2χ = ± sin α
(9.98)
or χ = α2 . i.e., the light will be elliptically polarised. For α = ±900 it is right or left
circularly polarised and for α = 00 or 1800 we get horizontally or vertically polarised
light. The representation of the Stokes vector in terms of S0 , ψ and χ is
S2 = S1 tan 2ψ
(9.99)
S3 = S0 sin 2χ
(9.100)
156
or
S1 = S0 cos 2χ cos 2ψ
(9.101)
S2 = S0 cos 2χ sin 2ψ
(9.102)
S3 = S0 sin 2χ
(9.103)
i.e.,

1


 cos 2χ cos 2ψ
S = S0 

 cos 2χ sin 2ψ

sin 2χ








(9.104)
As explained earlier the Stokes parameters give a direct measurement of the observables of an optical beam. Measurement of the Stokes vector is easier as it is
the intensity formulation of the polarisation state of the optical beam. The optical
beam is sent through a retarder and a polariser. The polarisation state of an optical
beam is changed when it interacts with matter. Hence, the Stokes vectors of the
incident optical beam is a function of the Stokes vectors of emerging optical beam
with a unique quantity called Muller matrix, which is a property of each optical
component. i.e.,

0


m
S
 0   00
 0  
 S1   m10

 
 0 =
 S2   m20

 
0
m30
S3
m01 m02 m03
m11 m12 m13
m21 m22 m23
m31 m32 m33


S
 0 


  S1 




  S2 


S3
(9.105)
or
0
S = M.S
(9.106)
0
where, M is the Muller matrix and S and S are the final and initial Stokes vectors
of the optical beam respectively. The transverse components of electric field of a
plane wave is
Ex (z, t) = E0x cos(ωt − kz + δx )
(9.107)
Ey (z, t) = E0y cos(ωt − kz + δy )
(9.108)
157
In the above equation can be changed by changing E0x or E0y or phase or δx or δy .
The first method is called Cut-off method, where the extra-ordinary wave of the
self-generated harmonics of the incident laser experiences cut-offs and resonances
depending on the strength of the self-generated magnetic field.
Second method is called harmonic polarimetry which uses the amount of depolarisation of the self-generated harmonics when they propagate through a transverse
magnetic field. In order to explain the above two methods it is necessary to explain
what happens when electromagnetic wave propagate through magnetised plasma.
Using third and fourth Maxwell’s equations,
∂
∂E
∇×∇×E=−
µ0 j + 0 µ0
∂t
∂t
(9.109)
where E and B are the electric and magnetic field field vectors and 0 and µ0 are
the permittivity and permeability of free space respectively. jf and ρf are the free
electron current density and free volume charge density respectively.
In order to solve the above equation we need to make few assumptions.
i) plasma is homogeneous in space and time. i.e., the dielectric constant 0 and
electric conductivity are independent of position and time.
ii) current is a linear function of electric field. i.e., any variation in the electric field
E1 and E2 gives rise to a current j1 and j2 respectively, then a variation of E1 + E2
gives rise to j1 + j2 .
iii) Since plasma is an anisotropic medium, the conductivity σ is considered as a
tensor. Hence, the Fourier analysis of current and electric field gives,
Z
j(r, t) = j(k, t)ei(k·r−ωt) d3 kdω
Z
E(r, t) = E(k, t)ei(k·r−ωt) d3 kdω
(9.110)
(9.111)
Each of the above Fourier mode satisfy equation 9.114. The relationship between
evolution of current density and electric field is given by the Ohm’s law,
j(k, ω) = σ(k, ω) · E(k, ω)
Therefore, equation 9.114 can be written as
k × (k × E) = −iω (µ0 σ · E − 0 µ0 iωE)
158
(9.112)
i.e., using the vector identity k × (k × E) = (k · E) E − (k · k)E we get,
ω2
2
kk − k 1 + 2 · E = 0
c
where, 1 is the unit matrix and the dielectric tensor.
i
= 1+
σ
ω0
(9.113)
(9.114)
To derive a non-zero solution for the equation 9.118 we need to set the determinant
of the matrix should be zero. i.e.,
ω2
2
det kk − k 1 + 2 = 0
c
(9.115)
the above equation is called dispersion relation. Using the cold plasma approximation, thermal motion of the electrons and ions are considered negligible Ti = Te = 0.
Hence, the collisions are also negligible. In the presence of an external magnetic
field the equation of motion can be written as,
me
∂v
= −e (E + v × B0 )
∂t
(9.116)
Since we are assuming the fluctuations are only in the linear approximation, we can
ignore the second order term of v × B and v · ∇v. The velocity can be written as
v (r, t) = v(r)e−iωt
(9.117)
if we take the components of equation 9.121,
−me iωvx = −eEx − eB0 vy
(9.118)
−me iωvy = −eEy + eB0 vx
(9.119)
−me iωvz = −eEz
(9.120)
where, B0 is in the z direction. Solving the above equations for vx , vy and vz ,
−ie
Ω
1
Ex − i Ey
vx =
(9.121)
ωme 1 − Ωω22
ω
−ie
1
Ω
i Ex + Ey
vy =
(9.122)
ωme 1 − Ωω22
ω
−ie
vz =
Ez
(9.123)
ωme
159
where,Ω =
eB0
me
is the electron cyclotron frequency. The conductivity tensor σ can
be obtained using the equation j = −ene v

ine e2 1
σ=
me ω 1 − Ωω22




1
−iΩ
ω

0
iΩ
ω
1
0
0
0
1−
Ω2
ω2




(9.124)
The above equation gives the electron conductivity. Ion conductivity can be obtained
in a similar way by substituting for ion mass, charge and density. Therefore, σtot =
σi + σe . Using the conductivity equation, equation 9.119 can be written as


ωp2
iωp2 Ω
1 − ω2 −Ω2 ω(ω2 −Ω2 )
0


2
 −iωp2 Ω

p
=  ω(ω2 −Ω2 ) 1 − ω2ω−Ω
(9.125)

0
2


2
ω
0
0
1 − ωp2
where, ωp is the plasma frequency. Ion contribution can be found by replacing
electron parameters with corresponding ion parameters. It is now easier to solve
the characteristic wave propagating through plasma. Considering the co-ordinate
system as shown in figure 9.3. where, k = k(0, sin θ, cos θ), where, θ is the angle
z
B0
θ
k
y
x
Figure 9.4: Co-ordinate system for wave propagation
160
between k and B0 . So we obtain the solution for equation 9.120 as
ω2
ω2
ω2
2
−k + c2 11
2 12
2 13
c
c
2
ω2
ω2
ω2
2
2
2
−k + k sin θ + c2 22
k sin θ cos θ + c2 23 = 0
c2 21
2
2
2
ω
ω
ω
2
2
2
2
k sin θ cos θ + c2 32 −k + k cos θ + c2 33 c2 31
In the above equation we substitute α =
ωp2
,β
ω2
=
Ω
ω
(9.126)
and the refractive index η =
kc
ω
and solving the determinant equation using the cold plasma approximation where,
is independent of the direction of k we get a quadratic equation for k 2 . thereby,
α (1 − α)
η2 = 1 −
1−α−
1 2
β
2
2
sin θ ±
rh
1 2
β
2
2
sin θ
(9.127)
2
2
+ (1 − α) β 2 cos2 θ
i
The above equation is called Appleton-Hartee formula for refractive index and this
is the formula to explain various types of wave propagation through plasma [134].
In an isotropic plasma non magnetised plasma (B0 = 0), Ω = 0. i.e., 11 = 22 =
33 = = 1 −
ωp2
ω2



=

1−
ωp2
ω2
0
0
1−
0
0
ωp2
ω2
0
1−
0
and the dispersion relation is

2
−k 2 + ωc2 0



0
−k 2 +

0
0

ωp2
ω2





0
ω2
c2
0
−k 2 +
(9.128)
ω2
c2


=0

(9.129)
solution for E transverse and Elongitudinal are given below.
−k 2 +
ω2
=0
c2
ω2
=0
c2
(9.130)
(9.131)
In the longitudinal mode the electric field is parallel to the wave vector. In transverse
mode the dispersion relation is ω 2 = ωp2 + c2 k 2 . The electric and magnetic field
vectors are perpendicular to each other and normal to the wave vector k as well. The
wave will propagate only if the wave frequency is higher than the plasma frequency.
When ω is less than ωp , the propagation vector is imaginary and the wave will decay
evanescently.
161
9.3
Muller Matrix
We have seen that the effect of optical components need to be included in the calculation of Stokes parameters. In our polarimetric measurements we had polarisers
and retarders. The effect due to an optical component can be represented by Muller
matrix. Muller matrices are (4 × 4) matrices [110]. The Muller matrix for an ideal
polariser is given by

1
cos 2θ
sin 2θ
0





2
 1 cos 2θ
cos 2θ
sin 2θ cos 2θ 0 


MP (θ) = 

2
2
 sin 2θ sin 2θ cos 2θ
sin θ
0 


0
0
0
0
(9.132)
where θ is the angle between the polariser axis and the initial polarisation of the
radiation. Similarly the Muller matrix for a retarder with a relative retardation ρ
and an angle γ between the axis and the initial polarisation direction of radiation


1
0
0
0




 0 cos2 2γ + cos ρ sin2 2γ (1 − cos ρ) sin 2γ cos 2γ − sin ρ sin 2γ 

MR (ρ, γ) = 


2
2
 0 (1 − cos ρ) sin 2γ cos 2γ sin 2γ + cos ρ cos 2γ
sin ρ cos 2γ 


0
sin ρ sin 2γ
− sin ρ cos 2γ
cos ρ
A 3 × 3 matrix is obtained by simplifying the above formula. For a quarter wave
plate (ρ = π/2)

cos2 2γ
sin 2γ cos 2γ − sin 2γ


Mr (π/2, γ) =  sin 2γ cos 2γ

sin 2γ
162
2
sin 2γ
− cos 2γ



cos 2γ 

0
(9.133)
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