INTERACTION OF INTENSE LASER FIELDS WITH
CARBON NANOTUBES: HIGH-ORDER HARMONIC
GENERATION
By
Nader Daneshfar
Supervisor
Heydar Khosravi
Advisor
Ali Bahari
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN PHYSICS(OPTICS )
AT
RAZI UNIVERSITY
DEPARTMENT OF PHYSICS
2009
c Copyright by Nader Daneshfar, 2009
RAZI UNIVERSITY
DEPARTMENT OF PHYSICS
The undersigned hereby certify that they have read and recommend
to the Faculty of Science for acceptance a thesis entitled “Interaction Of
Intense Laser Fields With Carbon Nanotubes: High-Order Harmonic
Generation” by Nader Daneshfar in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics (optics and nano-optics).
Dated: 2009
B
RAZI UNIVERSITY
DEPARTMENT OF PHYSICS
Date: 2009
Author:
Nader Daneshfar
Title:
Interaction Of Intense Laser Fields With Carbon
Nanotubes: High-Order Harmonic Generation
Department:
Degree: Ph.D.
Convocation: September
Year: 2009
Signature of Author: Nader Daneshfar
C
Dedicated To:
my Wife, Faranak
and
my Son, Aryan
D
Contents
Contents
E
List of Figures
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Abstract
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Acknowledgements
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1 Introduction
1.1 Introduction . . . . . . . . . . . . .
1.2 Motivation and Organization of this
1.2.1 Motivation . . . . . . . . . .
1.2.2 Organization of this work .
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2 High harmonic generation: gaseous, molecules and solid
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 HHG in gases . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Single-atom response (three-step model) . . . . . .
2.2.2 Lewenstein Model of HHG . . . . . . . . . . . . . .
2.3 HHG in molecules . . . . . . . . . . . . . . . . . . . . . . .
2.4 HHG in solid surfaces . . . . . . . . . . . . . . . . . . . . .
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3 Carbon Nanotubes
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2 Molecular structure of CNTs . . . . . . . . . . . . . .
3.2.1 Bonding mechanisms . . . . . . . . . . . . . .
3.3 Single-Walled Carbon Nanotubes . . . . . . . . . . .
3.3.1 Symmetry of Single-walled Carbon Nanotubes
3.4 CNTs as a source of coherent x-ray pulse . . . . . . .
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4 High harmonic generation by carbon nanotubes
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2 Theoretical background . . . . . . . . . . . . . . . . .
4.2.1 Kinetic Equations for Electrons . . . . . . . .
4.2.2 The electron dispersion relation for CNTs . .
4.2.3 Axial Surface Current Density in a Nanotube
4.3 Initial and boundary conditions . . . . . . . . . . . .
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4.4
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HHG by carbon nanotubes
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6 Conclusions
6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
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Appendix
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A Surface current density
76
B The density matrix
78
C The energy dispersion relation of graphite
80
Bibliography
83
4.5
Theoretical study of the HHG by CNTs . . . . . . . . .
4.4.1 Numerical results . . . . . . . . . . . . . . . . . .
HHG by CNTs in bichromatic laser field . . . . . . . . .
4.5.1 HHG in intense laser fields . . . . . . . . . . . . .
4.5.2 The effect of laser intensity on the HHG spectrum
5 Effect of static magnetic field
5.1 Introduction . . . . . . . . .
5.2 Theory . . . . . . . . . . . .
5.3 Numerical results . . . . . .
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List of Figures
1.1
Geometry of second-harmonic generation [4]. . . . . . . . . . . . . . . . . . .
4
1.2
Experimental setup for the detection of SHG light [5]. . . . . . . . . . . . . .
4
1.3
Quantum mechanical description of optical second-harmonic generation [5]. .
4
1.4
Wavelength/photon-energy ranges of HHG, VUV, XUV, soft X-ray and hard
X-ray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
7
Typical setup for the measurement of high-order harmonics. (a) Measurement
with a thin filter to discriminate against stray light of the laser fundamental
(typically a 100 nm Al filter). (b) Measurement without filter, but with a
beam block to avoid damage of the spectrometer due to direct illumination
with the strong laser fundamental [22]. . . . . . . . . . . . . . . . . . . . . .
2.1
10
Scheme of the interaction between a focused laser pulse and a gas jet( z axis
originates at laser focus). When the intense, short-pulsed laser of frequency
ω is sent into a gas jet of noble atoms (such as He, Ne, Ar), high harmonic
fields whose frequencies are odd-integer multiples of ω are generated. . . . .
G
14
2.2
Illustration of the three-step model for HHG: 1. Tunnel ionization of the electron, 2. Acceleration in the laser electric field, 3. Recombination and emission
of a high-energy photon. The energy of the emitted photon depends on the
ionization potential of the atom and on the kinetic energy of the electron upon
its return to its parent ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
18
Typical high harmonic spectrum; This figure illustrates the basic components
of the harmonic spectrum, that it consists of an initial falloff (perturbative
regime) at low orders, the plateau for intermediate orders, and the cut-off at
the highest orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.4
Theoretical prediction of single atom high harmonics [51]. . . . . . . . . . . .
19
2.5
Harmonic spectra of two groups of molecules [52, 53]. . . . . . . . . . . . . .
21
2.6
Diagram of an experimental setup generating of high harmonics in the thin
plasma layer produced by laser impact at the surface of a solid material [52].
3.1
24
Basic hexagonal bonding structure for one graphite layer ; carbon nuclei shown
as filled circles, out-of plane π-bonds represented as delocalized (dotted line),
and σ-bonds connect the C nuclei in-plane. . . . . . . . . . . . . . . . . . . .
28
3.2
Rolling up of a graphene sheet to make a SWCNT . . . . . . . . . . . . . . .
30
3.3
The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. a) The shaded area shows the part of the sheet to be wrapped
and the black arrow identifies the direction of wrapping. (b) Possible vectors specified by a pair of (n1 , n2 ) integers for general nanotubes. The blue
circles correspond to metallic nanotubes and the red circles correspond to
semiconducting nanotubes [93]. . . . . . . . . . . . . . . . . . . . . . . . . .
H
30
3.4
CNTs with different chiralities: armchair, zigzag and chiral nanotube. There
are two kinds of achiral nanotubes called “zigzag nanotubes” and “armchair
nanotubes”. The remaining nanotubes are called “chiral nanotubes” [99]. . .
4.1
31
Configuration of the first Brillouin zone for (a) Graphene, (b) Armchair CNTs
with substituting; px −→ pϕ , py −→ pz and (c) Zigzag CNTs with substituting; px −→ pz , py −→ pϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
The electric field of a sequence of n = 10 pulses. The time parameters of the
pulse train are τp = 10−13 s and Tq = 10−12 s [106]. . . . . . . . . . . . . . . .
4.3
44
51
The HHG power spectra of the induced current in (9, 0) zigzag SWCNT
as function harmonic order corresponding to the laser intensity I = 2 ×
1011 W cm−2 with µ = 0 [106]. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
52
The HHG power spectra of the induced current in (9, 0) zigzag SWCNT
as function harmonic order corresponding to the laser intensity I = 2 ×
1011 W cm−2 with µ 6= 0 [106]. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
52
The HHG power spectra of the induced current in (10, 10) armchair SWCNT
as function harmonic order corresponding to the laser intensity I = 2 ×
1011 W cm−2 with µ = 0 [106]. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
53
The HHG power spectra of the induced current in (10, 10) armchair SWCNT
as function harmonic order corresponding to the laser intensity I = 2 ×
1011 W cm−2 with µ 6= 0 [106]. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
53
HHG spectra the nonlinear current density in zigzag SWCNT as a function of
the harmonic order for the applied laser fields with photon energy ~ω = 1.8
eV and different field intensities: (a) for MLF with I0 = 1 × 1012 W/cm2 [116].
I
58
4.8
HHG spectra the nonlinear current density in zigzag SWCNT as a function of
the harmonic order for the applied laser fields with photon energy ~ω = 1.8
eV and different field intensities: (b) for BLF with β = 1 and ϕ = 0 having
different intensities I2 = I1 = I0 [116]. . . . . . . . . . . . . . . . . . . . . . .
4.9
59
HHG spectra the nonlinear current density in zigzag SWCNT as a function of
the harmonic order for the applied laser fields with photon energy ~ω = 1.8
eV and different field intensities: (c) for BLF with β = 1 and ϕ = 0 having
different intensities I2 = 5I1 = 5 × 1012 W/cm2 [116]. . . . . . . . . . . . . . .
59
4.10 HHG spectra in a (9,0) zigzag CNT in MLF with different intensities and
photon energy ~ω = 1.8 eV: I = 0.8I0 (triangles); I = 8I0 (squares) [116]. . . .
60
4.11 HHG spectra in metallic zigzag (a) CNTs for different laser field intensities
and ϕ = 0 with photon energy ~ω = 1.8 eV are plotted, for monochromatic
with I1 = I0 (squares) and bichromatic laser fields with the two components
with frequencies ω1 = ω and ω2 = 2ω having intensities I2 = I1 = I0 (circles)
and I2 = 5I1 = 5I0 (triangles) [116]. . . . . . . . . . . . . . . . . . . . . . . .
61
4.12 HHG spectra in metallic armchair (b) CNTs for different laser field intensities
and ϕ = 0 with photon energy ~ω = 1.8 eV are plotted, for monochromatic
with I1 = I0 (squares) and bichromatic laser fields with the two components
with frequencies ω1 = ω and ω2 = 2ω having intensities I2 = I1 = I0 (circles)
and I2 = 5I1 = 5I0 (triangles) [116]. . . . . . . . . . . . . . . . . . . . . . . .
5.1
62
Harmonic generation spectra by a metallic SWCNT in intense laser and static
magnetic fields for different values of the magnetic field corresponding to (a)
B= 5T [The inset shows HHG spectrum for zero magnetic field, B = 0] [161].
J
68
5.2
Harmonic generation spectra by a metallic SWCNT in intense laser and static
magnetic fields for different values of the magnetic field corresponding to (b)
B= 10T [161]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
69
Harmonic generation spectra by a metallic SWCNT in intense laser and static
magnetic fields for different values of the magnetic field corresponding to (c)
B= 20T [161]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
69
Harmonic generation spectra by a metallic SWCNT in intense laser and static
magnetic fields for different values of the magnetic field corresponding to
(d)B=40T [161]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
C.1 The unit sell of 2D graphite (honeycomb lattice), where a1 and a2 are the
unit vectors in the real space. The unit cell is shown as the dashed line and
containing two carbon atoms: 1 and 2. . . . . . . . . . . . . . . . . . . . . .
K
81
Abstract
The main topic addressed in this thesis is the interaction of intense laser fields with
the π electrons of single-walled carbon nanotubes. In this work we have presented a consistent quantum mechanical approach to nonlinear optics of carbon nanotubes and studied
the nonlinear interaction of the isolated nanotube with intense laser pulses. We consider the
nonlinear motion of π electrons in metallic carbon nanotubes ((9,0) zigzag and (10,10) armchair) driven by intense fields and the induced current density spectrum has been analyzed.
The effect of variation intensity of the applied laser fields on electron current density and
high-order harmonic generation is investigated. Also, we consider a metallic single-walled
nanotube under the influence of the combination of laser and static magnetic fields. Highharmonic generation in the presence of a transverse magnetic field is studied theoretically.
We apply a nonperturbative approach to metallic nanotubes and describe their linear and
nonlinear optical response to intense laser and magnetic fields by using a quantum mechanical method based on the single-electron approximation. The analysis utilizes the quantum
kinetic equations for π-electrons with both intra-band and inter-band transitions (note that
our theory ignores the role of σ electrons that come to play where the high-order harmonic
frequencies ωn = nω become comparable with the frequencies of corresponding transitions).
Nonperturbative approach using numerical solution of the quantum kinetic equations in the
time domain has been developed and the density of the axial electric current in nanotubes
has been calculated.
L
In the case of high-intense incident pulse, the induced current spectrum has shown to
be a set of narrow discrete lines of the driving field harmonics imposed by a continuous
background, thus demonstrating an essential contribution of direct inter-band transitions of
π-electrons between valence and conduction bands. The component of the current density
induced by the interband transitions is strongly modulated and typically is out-of-phase
to the classical component caused by the intraband transitions. This phenomenon leads
to a destructive interference of the quantum and quasi-classical components of the current
and reduces the effectiveness of high-order harmonic generation in comparison with the
semiclassical model.
M
Acknowledgements
Many people have contributed to this work directly and indirectly. First of all, to express
my deepest gratitude to my wife, Faranak, for her patience and encouragement which made
this work possible that i would never be able to pursue my Ph.D without her unconditional
support and love during these years. This dissertation is dedicated to her and my son Aryan.
I would like to acknowledge and thanks, my thesis supervisor, Dr. Heydar Khosravi for
his guidance and discussions that we had during this research. I would also like to thank Dr.
Ali Bahari, my thesis advisor, for his encouragement and suggestions during this research.
I would like to thank my thesis committee members and head of Department of physics
Dr. A. Rabeie for their suggestions. Finally, I would also like to thank all my friends and
colleagues, N. Ghobadi, M. Roshani, A. Moradi, H. Mousavi, R. Chegel, A. Azizi and B.
Asteenchap for their help.
Nader Daneshfar
Razi University
September 2009
N
Chapter 1
Introduction
1
1.1
Introduction
Many physical systems in various areas such as condensed matter or plasma physics,
biological sciences, or optics, give rise to localized large amplitude excitations having a
relatively long lifetime. Such excitations lead to a host of phenomena referred to as nonlinear
phenomena. Nonlinear optics is the study of the interaction of intense laser light with
matter. It is the study of phenomena that occur as a consequence of the modification
of the optical properties of a material system by the presence of light. Typically, only
laser light is sufficiently intense to modify the optical properties of a material system. The
beginning of the field of nonlinear optics is often taken to be the discovery of second-harmonic
generation (SHG) by Franken et al. [1], shortly after the demonstration of the first working
laser by Maiman in 1960 [2]. After the birth of the laser, the demonstration of nonlinear
optical phenomena revolutionized the science of optics, bringing new understanding to the
interactions of light and matter. Nonlinear optical phenomena are nonlinear in the sense
that they occur when the response of a material system to an applied optical field depends
in a nonlinear manner on the strength of the optical field. For example, SHG occurs as
a result of the part of the atomic response that scales quadratically with the strength of
the applied optical field. Consequently, the intensity of the light generated at the secondharmonic frequency tends to increase as the square of the intensity of the applied laser light
[3, 4]. Interest in this field has grown continuously since its beginnings, and the field of
nonlinear optics now ranges from fundamental studies of the interaction of light with matter
to applications such as laser frequency conversion and optical switching.
In nonlinear optics, electrons in the material act like driven oscillators that respond to the
2
laser electric field. Ordinarily, these electrons remain bound but are driven strongly enough
that the potential that binds an electron to its atomic core is no longer a purely parabolic,
harmonic-oscillator potential. The motion of the electrons themselves becomes anharmonic,
which gives rise to a time-dependent nonlinear polarization that reradiates electromagnetic
waves not only at the driving laser frequency, but also at higher harmonics of the driving
laser field [4].
When the laser was focused onto a quartz crystal, a very small signal at half the original
wavelength was observed amid the strong background radiation of the fundamental. This
engendered the new field of nonlinear optics. Optical second- and third-harmonic generation
provide good probes for the material properties. Even surface optical SHG can be applied
to study the microscopic properties at the material surface with submicrometer spatial resolution by near-field harmonic imaging. In the extreme high field area, optical harmonic
generation up to hundreds of orders provides a coherent beam in the extreme ultraviolet or
soft X-ray region. In other words, nonlinear optics has become an inseparable part of everyday scientific phenomena and is used in many high-technology industries, such as optical
communications [5].
Historically, the most important nonlinear optical effect is SHG. The first experiment
by Franken et al. [1] led to the birth of the field of nonlinear optics. Second harmonic
generation (also called frequency doubling) is a nonlinear optical process, in which photons
interacting with a nonlinear material are effectively ”combined” to form new photons with
twice the energy, and therefore twice the frequency and half the wavelength of the initial
photons, which is illustrated schematically in Fig. 1.1. It is a special case of Sum frequency
generation. Also, Fig. 1.2 shows the experimental arrangement and Fig. 1.3 shows the
3
Figure 1.1: Geometry of second-harmonic generation [4].
quantum mechanical description of optical second-harmonic generation.
Figure 1.2: Experimental setup for the detection of SHG light [5].
Figure 1.3: Quantum mechanical description of optical second-harmonic generation [5].
The physical mechanism behind frequency doubling can be understood as follows. Due
to the χ(2) nonlinearity, the fundamental (pump) wave generates a nonlinear polarization
wave which oscillates with twice the fundamental frequency. According to Maxwell’s equations, this nonlinear polarization wave radiates an electromagnetic field with this doubled
frequency. Due to phase-matching issues, the generated second-harmonic field propagates
dominantly in the direction of the nonlinear polarization wave. The latter also interacts with
the fundamental wave, so that the pump wave can be attenuated (pump depletion) when
4
the second-harmonic intensity develops: energy is transferred from the pump wave to the
second-harmonic wave [4, 5].
• Linear optics-Optics of weak light: Light is deflected or delayed but its frequency is
unchanged. In linear optics one assumes that the induced dielectric polarization of a
medium is linearly related to the applied electric field, i.e., P = χE.
• Non-Linear optics-Optics of intense light: We are concerned with the effects that
light itself induces as it propagates through the medium. At high optical intensities
(which corresponds to high electric fields), all media behave in a nonlinear fashion. In
the nonlinear optics the induced nonlinear polarization is related to higher-order powers
of the applied electric field. Thus, the relation between P and E will be nonlinear:
P = χ(1) E + χ(2) E (2) + χ(3) E (3) + ... = P (1) + P (2) + P (3) + ...
where the first term is the usual linear polarization, the second term on the righthand side is responsible for SHG, sum and difference frequency generation, parametric
interactions, etc. while the third term is responsible for third-harmonic generation,
intensity dependent refractive index, self-phase modulation, four-wave mixing, etc.
The quantities χ(2) , χ(3) , ... are higher order susceptibilities giving rise to the nonlinear
terms.
Higher-order nonlinear effects, such as self-phase modulation, four-wave mixing, etc. can
also be routinely observed today. The field of nonlinear optics dealing with such nonlinear interactions is gaining importance due to numerous demonstrated applications in many
diverse areas such as optical fiber communications, all-optical signal processing, realization
of novel sources of optical radiation, etc. Nonlinear optical interactions become prominent
5
when the optical power densities are high and interaction takes place over long lengths.
Laser technology – the ability to generate intense and coherent light with controllable
properties – is one of the most significant achievements of 20th century science. In recent
years, nonlinear optical techniques that convert one frequency of light to another have played
an increasingly pivotal role in laser technology. By using the techniques of nonlinear optics
and ultrashort pulse generation to an extreme limit, one can generate coherent light at
even shorter wavelengths using a process called high-order harmonic generation (HHG). One
of the most strange and interesting phenomenon in optics is harmonic generation. There
are few phenomena such as HHG that have captured the attention and energies of optical
physicists over the past decades, the term given to the production of harmonics of the driving
field when an intense optical field interacts with a gas, plasma, or solid target. The key to
generating high order harmonic photons is the presence of an intense laser field. This field is
usually provided by short pulse, subpicosecond laser because short pulses lead to high peak
intensities and thus intense electric fields. However, for some extreme nonlinear processes,
such as HHG, even the perturbative approach fails. HHG represents a most extreme
version of nonlinear optics.
Rapid progress in the optimization of the HHG process (energy/harmonic/pulse, and spatial and optical characteristics of the output radiation) over the past several years reinforces
the conviction that HHG has matured to the point of being a viable quantum electronic tool
for physical, chemical, medical, and materials research in the vacuum ultraviolet (VUV) to
X-ray regions. HHG provides a powerful source of ultrashort coherent radiation in the extreme ultraviolet (XUV) and soft-x-ray range and has the inherent advantage to be realizable
6