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Symmetry Breaking and Harmonic Generation in Metasurfaces and 2-Dimensional Materials
Jared S. Ginsberg
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
under the Executive Committee
of the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2021
© 2021
Jared S. Ginsberg
All Rights Reserved
Abstract
Symmetry Breaking and Harmonic Generation in Metasurfaces and 2-Dimensional Materials
Jared S. Ginsberg
A strong argument can be made that physics is, at its core, the study of symmetries.
Nonlinear optics is certainly no exception, with an enormous number of distinct processes each
depending in its own way on the underlying symmetries of the physical system, the light, or of
nature itself. Restricting ourselves to optical harmonic generation, we will explore three unique
physical systems as well as three symmetries. In each case, the controlled breaking of that
symmetry will lead to optical enhancements, novel nonlinearities, or deep physical insights.
Beginning with silicon metasurfaces, we will explore the effects of even and odd spatial
symmetries in optical systems. The periodic breaking of this symmetry will lead us to the highly
engineerable physics of bound states in the continuum. By studying the harmonic emission from
an atomic gas in the volume surrounding the metasurface, we will come to understand that
significant nonlinear optical enhancements can be engineered with any linewidth and at any
wavelength.
In the context of the two-dimensional material hexagonal boron nitride, we will
investigate and break crystal inversion symmetries. Using an intense laser tuned to the phonon
resonance of hexagonal boron nitride, large amplitude anharmonic ionic motions will provide us
a powerful degree of control over the internal symmetries of the system at an atomic level.
Breaking this symmetry, we measure short-lived even-order nonlinearities that would otherwise
be forbidden in equilibrium. Our observations for second- and third- harmonic generation are
confirmed by time-dependent density functional theory. Those simulations further extend the
understanding of this symmetry-breaking effect to even higher-order processes.
Lastly, single-crystal graphene and graphite provide an ideal platform through which to
explore time-reversal symmetry. Chiral photons, or optical beams with ellipticity and
handedness, are well known to break time-reversal symmetry. While applying high-power, chiral
light to graphene, the breaking of time-reversal lifts a degeneracy of the K and K’ valleys in the
momentum space Brillouin zone. Lifting this degeneracy, we unveil underlying spatial symmetry
properties of graphene in odd-order third- and fifth- harmonic generation which should otherwise
be unobservable. We also show experimentally, for the first time, that valley polarization and
population can be extracted using our technique.
i
Table of Contents
List of Figures ................................................................................................................................ iv
Acknowledgments......................................................................................................................... vii
Dedication ...................................................................................................................................... ix
Introduction ..................................................................................................................................... 1
Chapter 1: Low- and High- Order Harmonic Generation ............................................................... 4
Section 1.1: The Perturbative Regime of Harmonic Generation ................................................ 4
Section 1.2: The Semi-Classical Recollision Model for High-Order Harmonics ....................... 7
Step 1: Ionization ................................................................................................................... 8
Step 2: Acceleration ................................................................................................................ 9
Step 3: Recollision ................................................................................................................ 11
Section 1.3: Extension of the Semi-classical Model to Solids.................................................. 12
Chapter 2: Enhanced Harmonic Generation in Gases Using an All-Dielectric Metasurface ....... 15
Section 2.1: Quasi- Bound States in the Continuum................................................................. 16
Fundamental Principles:........................................................................................................ 16
Symmetry- Protected BICs: .................................................................................................. 18
Generalized Asymmetry Parameter and Quality Factor: ...................................................... 19
Section 2.2: Dimerized High- Contrast Gratings ...................................................................... 22
Dimerization and Spatial Mode Control: .............................................................................. 22
Final Device Design:............................................................................................................. 27
Section 2.3: Enhanced Nonlinearities in Gases ........................................................................ 28
ii
Wavelength Dependence: ..................................................................................................... 28
Polarization Dependence: ..................................................................................................... 30
Pressure Dependence: ........................................................................................................... 32
Section 2.4: Final Remarks ....................................................................................................... 33
Chapter 3: Nonlinear Phononics ................................................................................................... 37
Section 3.1: The Charged Harmonic Oscillator Model and Linear Optical Responses............ 37
Section 3.2: Lattice Anharmonicity .......................................................................................... 40
Chapter 4: Enhanced Harmonics and Phonon-Induced Symmetry Breaking in HBN ................. 44
Section 4.1: The Hyperbolic Phonon-Polaritons of Hexagonal Boron Nitride ........................ 45
Section 4.2: Frequency Dependent Third Harmonic Enhancement From Phonons ................. 52
Section 4.3: Time- Resolved Even-Order Nonlinearity in hBN ............................................... 59
Section 4.4: Polarization Dependence of Second Harmonic Generation ................................. 63
Section 4.5: Outlook on the Future of Nonlinear Phononics in hBN ....................................... 66
Chapter 5: Graphene Harmonic Generation and Time- Reversal Symmetry Breaking................ 68
Section 5.1: An Introduction to Graphene ................................................................................ 69
Electronic Properties ............................................................................................................. 69
Optical Properties.................................................................................................................. 72
Section 5.2: Elliptically Polarized Excitation of Graphene ...................................................... 75
Section 5.3: A Brief Review of the Role of Point-Groups of Symmetry in Harmonic
Generation from Solids ............................................................................................................. 82
Section 5.4: Time- Reversal Symmetry Breaking and Valley Polarization Effects ................. 86
Experimental Setup ............................................................................................................... 86
iii
Measurements ....................................................................................................................... 87
Theoretical Explanation of Emergent Flower Symmetry ..................................................... 89
Conclusions ................................................................................................................................... 96
References ..................................................................................................................................... 99
iv
List of Figures
1.1.1: Potential energy landscapes of anharmonic potentials…………….……..………….……...6
1.2.1: Trajectories of electrons vs tunneling times…………………………..……………..………8
1.2.2: Ponderomotive kinetic energy as a function of creation phase…………………….……….9
1.2.3: Lack of even harmonics in a centrosymmetric medium……………………………..……..10
1.3.1: Three-step high harmonic model for solids………………………………………….…….12
2.1.1: Depiction of bound state in the continuum………………………………………….……..17
2.1.2: Quality factor of BICs vs asymmetry parameter….…………………………………..……21
2.2.1: SEM image of dimerized high contrast grating in SOI……………………………….…...23
2.2.2: Band diagrams of HCG and DHCG…………………………………………………..……24
2.2.3: FDTD simulation of BIC mode in DHCG…………………………………………….…...25
2.2.4: Simulated and measured unpolarized reflectance from DHCGs……………………….….26
2.3.1: Wavelength dependence of THG on and off DHCG……………………………………....29
2.3.2: Polarization dependence of THG on DHCGs……………………………………………...30
2.3.3: Argon pressure dependence of THG and 5HG from DHCG…………………...………….32
2.4.1: Simulations of field and reflectance from 2D DHCG proposed design…………………. ..34
3.1.1: Optical and acoustic phonons in a harmonic potential……………………………………..38
3.1.2: Field and phonon response to harmonic potential……………………………………….....39
3.2.1: Field and phonon response with FT to anharmonic potential………………………………41
4.1.1: Toy model dispersion relations deriving Reststrahlen band……………………………….46
4.1.2: Image of hBN flake on silicon substrate…………………………………………………...47
4.1.3: Cartoon of hBN lattice with TO phonon changes also shown……………….…………….48
4.1.4: Phonon band diagram of hBN……………………………………………………………...49
v
4.1.5: Real permittivity of hBN both in plane and in z direction………………………………....50
4.1.6: Reflectance measurement on hBN at upper Reststrahlen band…………………………....51
4.2.1: Simulated atomic motion of B and N under phonon resonant excitation………………….52
4.2.2: Wavelength and power dependence of THG on resonance with hBN phonon………….....54
4.2.3: Experimental setup for THG in hBN……………………………………………………....55
4.2.4: Simulated HHG spectra for TO and LO phonon pumping…………………………………56
4.2.5: Pump intensity dependence simulations of resonant HHG in hBN………………………..57
4.3.1: TDDFT simulation of even order harmonics when pumping phonons……………………58
4.3.2: Experimental setup for pump-probe SHG in hBN…………………………………………59
4.3.3: Measured SHG time delay from hbn pump probe plus FT………………………………..61
4.3.4: Pump and probe power dependence of hBN SHG…………………………………………62
4.4.1: Theoretical and measured polarization dependence of hBN SHG…………………………63
4.4.2: Linecuts for polarization dependent SHG in hBN…………………………………………64
4.4.3: Simulated polarization dependence of SHG from hBN……………………………………65
5.1.1: Calculated electronic band dispersion of graphene in 2D…………………………………70
5.1.2: Dirac cones of graphene with various filling levels/ Fermi energies……………………….71
5.1.3: Real and imaginary parts of optical conductivity of graphene vs Fermi level…………….73
5.2.1: Reproduction of ellipticity dependence of harmonics in graphene………………………..74
5.2.2: Gating mechanism experimental design for graphene harmonics…………………………77
5.2.3: Experimental setup for gated graphene HHG..……………………………………………78
5.2.4: Gate dependent and ellipticity dependent THG and 5HG of graphene……………………79
5.2.5: Parallel and perpendicular components of THG in gated graphene…………….…………80
5.3.1: Comparison of CVD grown and exfoliated graphene crystals…………………………….81
vi
5.3.2: Theoretical polarization dependence of copolarized SHG in TMDs………………………83
5.3.3: Theoretical polarization dependence of copolarized THG in graphene……………………84
5.4.1: Experimental measurements of orientation dependent 5HG in graphite…………………..86
5.4.2: Flower pattern modulation depth vs pump ellipticity……………………………………..87
5.4.3: Orientation dependent 5HG for a thicker graphite flake…………………………………..88
5.4.4: Orientation dependent 5HG for a range of pump powers…………………………………88
5.5.1: Theory of 5HG from graphene as the sum of K and K’ valley contributions……..………90
5.5.2: Rotation of SHG flower pattern due to valley polarization……………………………..…91
5.5.3: Emergence of harmonic modulations resulting from valley polarization…………………92
5.5.4: 5HG orientation dependence (theoretical) for different valley populations……………….93
vii
Acknowledgments
There is not one project presented within this dissertation that I could have completed
without extensive help from my colleagues, collaborators, friends, and family. So many people
have contributed to this work, directly and indirectly. I hope they all enjoyed working with me as
much as I have with them.
First my advisor, Professor Alex Gaeta, who from the very beginning gave me the
freedom to forge my own research path, always trusting that we’d arrive at a place that was
exciting for us and impactful for others. I had not imagined before joining his group that a
graduate student could be given the resources and responsibility to drive their own research so
early on. It made for a truly unforgettable experience.
Everything I know about laboratory techniques, setups, lasers, the whole lot of it, I either
learned from or learned with Gauri Patwardhan. I think back to my first year in the office,
completely clueless and often times totally terrified, and I am so grateful to have had a mentor
like Gauri around to build my confidence in the lab. There’s nobody I’d rather flip an OPA
upside down with.
Our Ultrafast subgroup has been a tight-knit group of only a few people these past six
years. I sincerely hope that I’ve been as useful of a mentor to Cecilia Chen as Gauri was to me.
I’ve never worked with someone so willing to explore new techniques, new devices, or whacky
ideas as Cecilia. In working with her I’ve probably had more fun and unexpected days in the lab
than with anyone else. I can feel confident that the future of our little team is very safe with her.
I would be the first to admit that a new colleague showing up and flipping everything we
know and do on its head can be a painful experience. The opposite was true when Mehdi Jadidi
viii
came to Alex’s group. Mehdi took my research in directions that I may have never been exposed
to otherwise. I think it’s fair to say that our projects together have been the most rewarding of all.
Those of us who work with incredible devices and samples often take for granted our
collaborators who live in cleanrooms for our benefit. From Prof. Nanfang Yu’s group I would
especially like to thank Adam Overvig, who saw the potential in one of my earliest ideas and
worked with me to see it through to the very end. From James Hone’s group I’d like to mention
Sang Hoon Chae, who was unreasonably patient with me each time a laser annihilated one of his
beautiful samples.
Jin Zhang and Angel Rubio must be the most patient collaborators I have ever known.
Aside from his vital role as the primary theorist for my favorite paper, Jin went on all manner of
wild goose-chases looking for fascinating physics with me these past two years. I hope that it
was as worthwhile for their team as it was for me, and I hope to keep the collaboration going for
some time.
As proud as I am to have completed my degree at Columbia University, I am far prouder
to have done the work in such an incredible group. Yoshi, Yun, Bok, Chaitali, Chaitanya, Jae,
Miri, Yair, Andrew, Ricky, MJ, Alessandro, you’ve all contributed to a work environment that
I’ve truly loved every minute of.
Lastly, my friends and my family. I don’t know how I ended up so lucky to be
surrounded by people who support my every decision. Thank you all so much for supporting me
these last 6 years.
ix
Dedication
To my family.
1
Introduction
Strong field physics and attosecond science comprise a many decades long journey built
on first understanding, and then applying, phenomena of electron tunneling and recombination.
Beginning with noble gas experiments in the earliest days and only just being applied to solid-
state systems some ten years ago, strong field physics is now explored at the frontier of every
conceivable material system. From nanostructured arrays to the most exotic two-dimensional
materials, attosecond science now provides as much fundamental physics about the systems
themselves as those systems add to our ability to tailor beams of light.
One of the pillars of ultrafast optics, and the common thread that will link the various
explorations in this dissertation, is harmonic generation. In atomic systems, the ionization and
recollision of electrons and their parent ions by intense near- and mid-infrared lasers has led to
truly profound technological advancements. These culminated in the ability to generate
unbelievable optical bandwidths covering as many as hundreds of octaves and reaching energies
in the x-ray regime. These broad pulses allow scientists to perform some of the fastest
measurements possible, with time-resolutions of a few 10-18 seconds, and all with table-top laser
systems.
Though it may be many years before solid-state systems reach the performance abilities
of their atomic counterparts, interest in harmonic generation from solids is at an all-time high.
The large conversion efficiencies made possible by the densely packed lattices of solids was just
the first of what is now a plethora of advantages. The ability to (1) manufacture resonances, (2)
explore band-structures, and vibrational properties of matter, or (3) study symmetries and
generate previously impossible optical polarizations, makes solids a clear favorite platform for
future research.
2
The goal of this thesis is therefore to provide some of the earliest demonstrations of each
of these three avenues of study in solid-state devices. In doing so, I hope to prove to the reader
that solid and hybrid solid-gas systems are the future of ultrafast nonlinear optics and harmonic
generation. For each of the three areas I’ve listed above, we will need to utilize a unique material
system, specifically tailored to our goals. As such, this dissertation will be broken up into the
following main sections:
• The first chapter is where we will establish foundational knowledge of harmonic
generation and nonlinear optics. We’ll begin with the well-known perturbative regime
and its application to lower-order processes before exploring in more detail the semi-
classical theory of higher-order harmonic generation in gases. A beautiful analogy
between the atomic process and what takes place in bulk solids will cover the full scope
of systems for the remainder of the chapters.
• In Chapter 2 we will dive into our first specific system and symmetry, a hybrid solid-gas
platform for nonlinear optics that aims to apply the biggest advantages of nano-structured
devices to the well-established technique of gas-phase high-harmonic generation. In
doing so, we will take a detour into the physics of bound states in the continuum a
powerful concept in which spatial symmetry breaking can be leveraged to design
semiconductor Metasurfaces at any operating wavelength and with any quality factor.
Our unique insight will be to add spatial control to the long list of advantages we get
from BICs.
• Chapters 3 and 4 introduce an entirely new type of material resonance, one which is
much more rare in nature, yet every bit as applicable to our pursuit of advancing
nonlinear optics. Here we will be focusing on phonon-polaritonics in the van Der Waals
3
material hexagonal-boron nitride. Chapter 3 will establish the theoretical basis for
nonlinear phononics, building up from a harmonic oscillator model straight through to a
robust theory of lattice anharmonicity. Chapter 4 then applies that lattice anharmonicity
to boron nitride in two ways. First, we show enhanced nonlinearities due to phonon
coupling. Second, strong optical pumping is utilized to break the inherent inversion
symmetry of a material, enabling entirely new nonlinearities.
• Finally, we will touch on some of the most curious topics to have emerged in harmonic-
generation related experiments in the last 3 years, all relating to graphene and its thicker
cousin graphite. As a material without a bandgap, graphene exists in a unique space,
seemingly having optical properties of a semi-metal at certain times and those of an
atomic system at others. We will come to understand the role of graphene’s fermi level
and how it relates to this fascinating dichotomy. After that, we will explore ways in
which graphene’s unique relationship to time-reversal symmetry allows us to determine
material characteristics that should be entirely impossible to measure optically in a
system with its rotational symmetry properties.
4
Chapter 1: Low- and High- Order Harmonic Generation
Section 1.1: The Perturbative Regime of Harmonic Generation
Around the time of the completion of this dissertation, the first demonstration of optical
harmonic generation by Franken, et. al [1] will be celebrating its 60th anniversary. Using a ruby
laser emitting at 694 nm and quartz as a nonlinear medium, Franken and his team observed
second harmonic generation (SHG) in what we now understand to be the perturbative (or weak-
field) nonlinear regime. The years since this pioneering experiment have seen the field of
ultrafast nonlinear optics and harmonic generation extended far beyond the weak regimes and
simple bulk systems. And yet now, decades later, some of the most interesting harmonic
generation experiments are aimed at bending and manipulating foundational concepts of
harmonic generation. So that is where we will begin.
Linear optics supposes that the polarization in a material, induced by an applied electric
field 𝐸(𝑡), is given by 𝑃(𝑡) = 𝜖0𝜒(1)𝐸(𝑡) for a linear susceptibility 𝜒(1), which acts as a
material dependent proportionality constant. We allow for the possibility of nonlinear optical
interactions by generalizing this relationship to:
𝑷(𝒕) = 𝝐𝟎[𝝌(𝟏)𝑬(𝒕) + 𝝌(𝟐)𝑬𝟐(𝒕) + 𝝌(𝟑)𝑬𝟑(𝒕) + ⋯ ] ( 1)
which has been expanded as a power series in the electric field and now includes higher order
(nonlinear) susceptibilities, 𝜒(𝑛). Supposing now that the applied field is monochromatic with
frequency 𝜔0, amplitude 𝐸0, and of the form 𝐸(𝑡) = 𝐸0𝑒−𝑖𝜔0𝑡 + 𝑐. 𝑐, it is clear how the
nonlinear polarization of Equation 1 leads to the creation of new frequency components:
𝑷(𝒕) = 𝝐𝟎[𝝌(𝟏)𝑬𝟎𝒆𝟎
−𝒊𝝎𝟎𝒕 + 𝝌(𝟐)𝑬𝟎𝟐𝒆𝟎
−𝟐𝒊𝝎𝟎𝒕 + 𝝌(𝟑)𝑬𝟎𝟑𝒆𝟎
−𝟑𝒊𝝎𝟎𝒕 +⋯] ( 2)
This highly simplified expression fails to consider the possibility of multiple frequencies, but
nonetheless illustrates some important features. First, the integer multiples of the applied
5
frequency 𝜔0 are the harmonics we will be concerned with in this thesis. Second, we can read off
the field scaling of the different harmonic components, observing that the nth polarization order
scales as 𝐸0𝑛, and by extension the measured power of the nth harmonic scales as 𝐸0
2𝑛. The
perturbative definition of the nonlinear polarization given in Equation 1 is precisely why these
field dependencies constitute the “perturbative regime” of harmonic generation. Deviations from
that scaling, such as those we will see ourselves in Chapter 2, are considered to be the domain of
“nonperturbative” high-harmonic generation (HHG) which will be described in detail in the
following Section 1.2.
Much of the nonlinear behavior in a perturbative harmonic generation system can be
captured by an anharmonic oscillator model [2], describing the trajectories of electrons in a
potential well such as those provided by their atoms. An introduction to this technique is
included for the sake of preparing us for Chapter 3, in which the anharmonic behavior will
become the role of the lattice ions of a crystal, as opposed to the relatively freer electrons.
However, for brevity we will truncate this derivation at the point where we can start to extract
some useful conclusions, particularly about symmetries, and from there we will generalize.
Take the position of an electron to be x. Subjected to an applied electric field 𝐸(𝑡) with
frequency 𝜔0 and a potential well with damping constant 𝛾, the equation of motion for the
electron is:
𝒅𝟐𝒙
𝒅𝒕𝟐+ 𝟐𝜸
𝒅𝒙
𝒅𝒕+𝝎𝟎
𝟐𝒙 + 𝒂𝒙𝒏 = −𝒆𝑬(𝒕)
𝒎𝒆. ( 3)
I have deliberately chosen not to specify the exponent of the restoring force proportional to a.
The value of n defines the symmetry of the potential well, with even n leading to non-symmetric
potentials, and odd n restoring that symmetry (Figure 1.1.1). To solve, we expand the position x
6
as a power series, 𝑥 = 𝜖𝑥(1) + 𝜖2𝑥(2) + 𝜖3𝑥(3) +⋯, for small 𝜖. Substituting in and separating
the different perturbation orders, we find:
𝑑2𝑥(1)
𝑑𝑡2+ 2𝛾
𝑑𝑥(1)
𝑑𝑡+ 𝜔0
2𝑥(1) = −𝑒𝐸(𝑡)
𝑚𝑒
{
𝒅𝟐𝒙(𝟐)
𝒅𝒕𝟐+ 𝟐𝜸
𝒅𝒙(𝟐)
𝒅𝒕+𝝎𝟎
𝟐𝒙(𝟐) + 𝒂𝒙(𝟏)𝟐 = 𝟎, 𝒏 𝒆𝒗𝒆𝒏
𝒅𝟐𝒙(𝟐)
𝒅𝒕𝟐+ 𝟐𝜸
𝒅𝒙(𝟐)
𝒅𝒕+𝝎𝟎
𝟐𝒙(𝟐) = 𝟎, 𝒏 𝒐𝒅𝒅 ( 4)
𝑑2𝑥(3)
𝑑𝑡2+ 2𝛾
𝑑𝑥(3)
𝑑𝑡+ 𝜔0
2𝑥(3) + 2𝑎𝑥(1)𝑥(2) = 0
Let’s consider the steady state solutions of the middle pair of equations describing the second-
order in perturbation theory. With all time derivatives neatly set to 0 we are left with:
{ 𝒙(𝟐) = −
𝒂(𝒆𝑬𝟎𝒎)𝟐𝒆−𝟐𝒊𝝎𝟎𝒕
(𝟐𝒊𝝎𝟎𝜸)𝟐 , 𝒏 𝒆𝒗𝒆𝒏
𝒙(𝟐) = 𝟎, 𝒏 𝒐𝒅𝒅
( 5)
Some steps involving the solution of the first order component and its substitution into the next
step of the perturbation have been omitted. Carrying these two expressions forward, one finds
that the second order susceptibility for even n for a monochromatic field is given by 𝜒(2) =
𝑁(𝑒3
𝑚2)𝑎
𝜖0(2𝑖𝜔0𝛾)2[𝜔02−(2𝜔0)2−2𝑖𝜔0𝛾 ]
, whereas for odd n, the value of 𝜒(2) is identically 0. By considering
Figure 1.1.1: The potential energy landscape for anharmonic centrosymmetric media (red) and non-
centrosymmetric media (blue) compared to a perfectly harmonic potential (black). The symmetry of the
material, and hence the symmetry of the potential energy, places selection rules on the allowed harmonic orders.
7
an anharmonic oscillator model of nonlinear optics we have arrived at a key result, namely that
centrosymmetric media (odd n) are forbidden to have even order nonlinearities, while symmetry-
broken media have no such restriction placed on them. In Chapters 3 and 4 this distinction will
become our motivation to design a system in which the inversion symmetry can be changed
optically and at will.
In writing the nonlinear susceptibilities for a single frequency component and in the form
of a proportionality constant, the true tensor nature of the various 𝜒(𝑛) was lost. In Chapter 5 we
will return to extend the above discussion of inversion symmetries to include spatial symmetries
of crystalline solids, giving us a tool to understand the symmetries of emitted light as a function
of the arrangements of atoms.
Section 1.2: The Semi-Classical Recollision Model for High-Order Harmonics
While there are many different theories for describing the emission of higher- order
harmonics (HHG) from atomic systems [3–5], none have been so widely accepted and applied
with such success as the semi-classical model of Paul Corkum [6,7]. As luck would have it, this
most useful explanation also happens to be the simplest, consisting of three distinct steps, each of
which we will explore in some detail. Broadly speaking these steps include: (1) The tunneling of
an electron away from its parent atom, a type of ionization process resulting in a free electron
with little or no kinetic energy. (2) Acceleration of the electron by the electric field of the laser
that released it. (3) Recollision of the electron with its ion, caused by the changing sign of the
electric field, and resulting in the conversion of the electron’s kinetic energy to a high energy
8
photon. Each of these three stages is sensitive to different field and laser parameters, and by
treating each one in detail, we can establish for ourselves some benchmarks and goals that will
be revisited in future sections.
Step 1: Ionization
Of the three stages involved in the model, the first is the one which most heavily relies on
a quantum mechanical description, while providing the least information about the high energy
harmonics that will ultimately result from the full process. However, by considering the
combined atomic potential and applied electric field, we are able to write the ionization rate as:
𝑾𝒅𝒄 =𝑬𝒔𝟎
ℏ|𝑪𝒏∗𝒍∗|
𝟐𝑮𝒍𝒎 (𝟒𝑬𝒔𝟎
ℏ
𝒆𝑬(𝟐𝒎𝒆𝑬𝒔𝟎)
−𝟏𝟐
)
𝟐𝒏−𝒎−𝟏
𝒆−𝟒𝑬𝒔𝟎
ℏ/𝟑𝒆𝑬(𝟐𝒎𝒆𝑬𝒔
𝟎)−𝟏𝟐, ( 6)
where 𝐸𝑠0 and 𝐸𝑠
ℎ are the ionization potential of the atom and the ionization potential of
hydrogen, respectively. 𝐺𝑙𝑚 and 𝐶𝑛∗𝑙∗ are uniquely determined by azimuthal and magnetic
quantum numbers, and E is the amplitude of the applied electric field. That is all to say, that the
Figure 1.2.1: Colors represent a range of one-dimensional electron trajectories for different tunneling times
(phases). The black curve is included as a guide, representing the period of the monochromatic field that
initiates the electron tunneling. The horizontal dotted line represents the atom’s position, illustrating the
range of different recollision times. Some electron trajectories, such as the top brown one, never return to
the atom.
9
probability of ionization 𝑊𝑑𝑐(𝐸(𝑡))𝑑𝑡, within the time interval dt, is purely a function of the
atomic potential, field strength, quantum numbers, and fundamental constants. The key
takeaway then is that the tunneling is a fundamentally probabilistic process, and yet the exact
moment of ionization is significant. As we will see in the following steps, the driving field phase
at the free electron’s creation time will lead to a strongly varying trajectory (Figure 1.2.1) and
final kinetic energy (Figure 1.2.2).
Step 2: Acceleration
We can now formulate the second step in the following simple way: we have an
effectively free electron in vacuum to which we are applying an oscillating electric field
𝐸 = 𝐸0 cos𝜔0𝑡 . The initial position x and momentum 𝑚𝑒𝑥 at time t0 are both 0. Then as a
function of time and carrier frequency, the position of the electron may be expressed as:
𝒙(𝒕) =𝑬𝟎
𝝎𝟎𝟐 [(𝒄𝒐𝒔𝒕 𝝎𝟎𝒕 − 𝒄𝒐𝒔𝝎𝟎𝒕𝟎) + (𝝎𝟎𝒕 − 𝝎𝟎𝒕𝟎)𝒔𝒊𝒏𝝎𝟎𝒕𝟎], ( 7)
Figure 1.2.2: Bottom: the kinetic energy imparted to the electron as a function of the phase of the laser at
the time of tunneling. Above: the laser amplitude at that time (phase). It is found that the maximum kinetic
energy that can be imparted is 3.17 Up, which occurs at a tunneling phase of 18o.
10
and it therefore has a kinetic energy of
𝑬𝒌𝒊𝒏𝒆𝒕𝒊𝒄(𝜽) = 𝟐𝑼𝒑(𝒔𝒊𝒏𝜽 − 𝒔𝒊𝒏𝜽𝒊)𝟐, ( 8)
where we have made two simplifications. First, we have replaced 𝜔𝑡 with the relevant phase 𝜃,
and introduced a new quantity, the ponderomotive energy 𝑈𝑝 =𝑒2𝐸0
2
4𝑚𝑒𝜔2, which represents the
average energy of a free electron in a monochromatic field. There are two key observations to be
made already, which will be revisited again and again in the coming chapters. That is, the
ponderomotive energy is linearly proportional to laser power, and scales quadratically with
wavelength. Given Eq (2) and (3), the final kinetic energy of the electron can be computed
numerically as a function of the driving laser’s phase, as shown in Figure 1.2.2. We find that at a
driving phase of 18o, a maximum kinetic energy of 3.17 times the very same ponderomotive
energy is possible. This maximum energy plus the atom’s potential energy is the highest energy
possible in an HHG process and fundamentally limits the short wavelength cut-off of a high-
harmonic spectrum [8,9].
Figure 1.2.3: (a) When performing harmonic generation with a perfectly symmetry atom such as Argon, only
odd order harmonics can be generated. This is due to the destructive interference of even orders from successive
high harmonic pulses, illustrated in the inset. (b) In atomic systems, elliptically polarized light can prevent the
proper recollision of the electron and ion, see here as a steep monotonic fall-off in harmonic generation power.
11
Step 3: Recollision
Having established that the maximum energy of an emitted photon during the recollision
process is predetermined to be equal to the ionization energy plus the kinetic energy calculated in
the previous section, we can turn our attention to two important subtleties of the HHG recollision
process. The first is that experimental harmonic spectra cannot be explained by a single
tunneling and recollision process. Specifically, it is well established that in HHG experiments
that utilize noble gases, only odd-order harmonics of the driving frequency are observed [10].
We will see this behavior in the following chapter in our own projects utilizing Argon, an
example of which is reproduced in Figure 1.2.3 (a) for a 3 μm pump beam. The full 3 step
process established in this chapter occurs every half-cycle of the driving laser, and to understand
the full picture of experimental observations, we need to consider not one, but a series of
tunneling, acceleration, and recollision processes.
Consider the emission of the nth harmonic in the first recollision process to be a simple
oscillating field 𝐶𝑜𝑠(𝑛𝜔0𝑡), and from the subsequent emission one half-cycle later to be
𝐶𝑜𝑠(𝑛(𝜔0𝑡 + 𝜋)). For combined emission we find that constructive interference occurs for all
odd orders of n, while destructive interference prevents any even-order harmonics from being
experimentally observed. A careful reader will realize that the requirement we’ve placed on the
harmonic orders is not so different from that which we derived for centrosymmetric media in
Section 1.1.1. In fact, the perfect spherical symmetry of a noble gas atom is an equally valid way
of arriving at the same conclusion.
The second important subtlety relates to whether the recollision takes place at all. Up
until now, we have considered the simplest situation of a one dimensional free-electron
trajectory. However, as we will see in later chapters the introduction of ellipticity to the driving
12
field can have profound effects on harmonic emission. In the case of an atomic gas, the
ellipticity causes a deviation of the electron from its straight path [11], increasing the likelihood
that the electron will fail to recollide with its original ion. Figure 1.2.3 (b) quantifies this picture,
showing the reduction of harmonic yield as the pump ellipticity is increased- a result of a failed
recollision step.
Section 1.3: Extension of the Semi-classical Model to Solids
Much of what we’ve learned in the previous section assumed a single electron tunneling
away from a single atom, and therefore feels as though it should fail to describe the interaction of
intense light with a crystalline solid consisting of many neatly arranged atoms. The trick to
making a tidy analogy between the atomic and solid cases is to recognize that the most powerful
description of electrons in a lattice is not a spatial picture, but an energy-momentum one [12,13].
Rather than tracking the trajectory of an electron in real-space, we will understand the 3-step
model of HHG in solids by their propagation along valence and conduction bands in a so-called
band diagram.
Figure 1.3.1: Illustration of the band
diagram for the one-dimensional chain
of atoms with two different masses.
Also shown are the three steps of the
solid HHG process. First, a large
number of fundamental photons excite
an electron from a lower band up to the
conduction band. The electron (hole)
accelerates in the conduction (valence)
band. The acceleration of a charge
contributes optical emission known as
intra-band HHG. Finally the electron
and hole recombine, emitting a high
energy photon in the process.
13
The calculation of a band diagram for a real material can be a substantial undertaking,
requiring powerful computation techniques such as Density Functional Theory (DFT) [14]. We
will instead derive the simplest two-band model of a one-dimensional diatomic chain of atoms,
and use that to illustrate the three steps. Consider two species of atoms with masses m and M,
connected by identical springs of spring constant 𝜆. The equations of motion for the positions
𝑢 of the two species are given by
{ 𝒎�̈�𝟐𝒏 = −𝝀(𝟐𝒖𝟐𝒏 − 𝟐𝟐𝒏−𝟏 − 𝒖𝟐𝒏+𝟏) 𝑴�̈�𝟐𝒏+𝟏 = −𝝀(𝟐𝒖𝟐𝒏+𝟏 − 𝟐𝟐𝒏 − 𝒖𝟐𝒏+𝟐)
. ( 9)
We make the standard ansatz that the solutions for 𝑢 oscillate in space and time according to
𝑢2𝑛 = 𝐴𝑒−𝑖𝜔𝑡−2𝑖𝑘𝑛𝑎 and 𝑢2𝑛+1 = 𝐵𝑒−𝑖𝜔𝑡−2𝑖𝑘𝑛𝑎. Plugging these in and solving for the
frequencies we find two solutions to the equations of motion:
𝝎± =𝝀
𝒎𝑴[𝒎 +𝑴±√(𝒎−𝑴)𝟐 + 𝟒𝒎𝑴 𝑪𝒐𝒔𝟐(𝒌𝒂) ] ( 10)
These two solutions are referred to as the dispersion curves for the one-dimensional diatomic
chain, and together make up the band diagram in Figure 1.3.1. An electron that exists in a lower
energy band, hereafter referred to as the valence band, may be promoted up to the higher energy
collection of states (the conduction band) by the absorption of one or more photons from an
applied laser. This takes the role of the tunneling step in analogy to the atomic system. The
difference now is that the role of an ionization potential is played by the bandgap.
The second step is likewise very similar to the free electron case. Now, a promoted
electron (as well as the hole that it left in the valence band) is accelerated within its new band,
acquiring kinetic energy. An important distinguishing detail here is that the oftentimes
anharmonic shape of the band leads to radiation in this intermediate step. This additional
contribution to the overall observed radiation is referred to as intraband harmonic
generation [13,15,16]. Finally, the electron and hole recombine, emitting a new photon at an
14
energy equal to the bandgap plus the accumulated energy that the electron gained during its
acceleration. This final process is known as interband HHG, and completes the analogy.
With this we have established much of the groundwork necessary to understand the
different examples of low- and high- harmonic generation throughout this thesis. Where
necessary, further theoretical frameworks will be presented to complete each story, such as
bound states in the continuum in Chapter 2 or phonon-polaritons following that. However, the
reader should always keep in mind the laser parameter dependencies, symmetry arguments, and
methods described thus far, as they will inform many of our motivations moving forward.
15
Chapter 2: Enhanced Harmonic Generation in Gases
Using an All-Dielectric Metasurface
In the previous chapter we derived the ponderomotive energy, a relationship between laser
power, wavelength, and the maximum energy photon that can be generated from a gas by HHG.
Though counterintuitive, the key to reaching ever shorter final wavelengths is to begin with the
longest wavelength possible! Next-generation studies of ultrafast atomic phenomena demand
higher repetition rates and shorter pulses with greater bandwidths. Thus, the quadratic
dependence of the maximum HHG bandwidth on pump wavelength [17] provides strong
motivation to push the wavelength of the pump field deeper into the mid-infrared. Strict intensity
requirements of nearly 100 terawatt/cm2 in order to access the non-perturbative regime of HHG
in gases has, however, mostly restricted the field to the use of chirped pulse, multi-pass, and
regeneratively amplified lasers. These large-footprint systems have low repetition rates, high
average power, and are limited to the near-visible regime, hindering progress towards mid-
infrared pumping. This chapter is motivated by a simultaneous pursuit of efficient HHG at longer
wavelengths and lower pump powers.
The lower intensities required to observe HHG from solids [18] has generated significant
interest and led to rapid innovation in the decade since its initial discovery. A promising
direction, and the topic of the current Chapter, has been the merging of nonlinear optics with
metasurfaces to create an efficient platform for HHG [19–24]. We will see how these
nanostructured devices allow for the engineering of far-field emission profiles, selective
wavelength enhancement, and exceptionally strong field-confinement. Sub-wavelength scale
plasmonics (made of metallic antennas) and dielectrics have both been demonstrated for these
16
purposes. While the strong field-confinement permitted by these two types of systems further
reduces the pump power requirements for harmonic generation in solid-state systems, enhancing
intensities within the device material poses its own challenges. Plasmonic devices are severely
hampered by ohmic losses and are susceptible to melting, even at modest intensities [25]. Both
plasmonics and dielectrics must also contend with the reabsorption of harmonics within an
opaque generation material [26], reducing the effective interaction length of the nonlinear
medium to only a few tens of nanometers. For these reasons we will focus on a hybrid approach
in this Chapter, combining the designable aspects of metasurfaces with the robustness of gases.
To do so, spatially selective confinement of pump energy within the regions just outside the
metasurface will be accomplished with bound states in the continuum (BIC), a class of optical
resonance with broadly engineerable mode profiles supported by all-dielectric metasurfaces [27–
30]. A variety of BIC-enhanced light-matter interactions including optical absorption [31], third-
harmonic generation (THG) in solids [32], and Raman scattering [33] were previously reported.
Section 2.1: Quasi- Bound States in the Continuum
Fundamental Principles:
In every discipline of physics from optics to quantum mechanics, the principle of a bound
state arises as a consequence of a potential well with discretized energy states. Look no further
than the quantum mechanical square well, with discrete, bound energy levels given by ℏ2𝑛2
2𝑚𝑎2 for
integer 𝑛, mass 𝑚, and well width 𝑎. In the limit that the potential well becomes finite, these
bound modes exist separate from the surrounding continuum of states, which are capable of
radiating energy out of the system to infinity. Though it may appear as if energy can be confined
within a continuum of states, it is often the case that there is still some coupling through which
17
energy is lost, and these states are referred to as resonances. Only in the limit where the lifetime
of the resonance goes to infinity and its linewidth goes to zero does the resonance truly become
bound. What could it mean then for a state to be bound despite existing in the continuum? Figure
2.1.1 is included as a graphical representation, but only serves to deepen the mystery.
Furthermore, for a mode to be entirely decoupled from all other channels and to have zero
linewidth is to say that it cannot be excited by any external fields, in addition to not radiating into
them. These two questions of existence and coupling will be tackled in turn, but not before a
brief history.
Bound states in the continuum were first proposed as a mathematical curiosity of
quantum mechanics by von Neumann [34], who painstakingly wrote out an infinite potential that
would support an electronic bound state. Though an experimental analog to von Neumann’s first
theoretical approach has never been realized, the universality of the potentials and Schrodinger
equation on which the concepts were formulated have motivated further searches for these
Figure 2.1.1: Various waveforms representing the continuum and bound states. The continuum states outside
of the potential well in any given system will have no spatial localization and tend to be a higher energy. By
comparison a leaky-mode, labeled here as a resonance, will have some spatialization but will ultimately radiate
away. Without any means of radiating, the state is perfectly localized, represented by the black curve.
18
elusive bound states. In the decades following, many such systems have been found, ranging
from entirely accidental bound states due to continuous parameter tuning [35,36] to
separability [37,38]. Perhaps the simplest such system, and the one we will describe in the next
section, is the bound state in the continuum protected by symmetries.
Symmetry- Protected BICs:
Our work in this chapter will primarily focus on one specific avenue for arriving at a
bound state in the continuum, and that is by exploiting symmetry. Those readers who have
experience in fiber optics or some mathematics may already be aware of the following fact: that
the modes belonging to two different symmetry classes naturally decouple from each other. This
can be understood as a trivial consequence of the mode overlap efficiency 𝜂, defined generally as
𝜼 =| ∫ 𝑬𝟏
∗𝑬𝟐𝒅𝑨|
∫|𝑬𝟏|𝟐𝒅𝑨 ∫|𝑬𝟐|
𝟐𝒅𝑨 ( 11)
for modes described by electric fields 𝐸1and 𝐸2 covering an area 𝑑𝐴. If we consider the simplest
case, in which our even mode is described by cos(𝑥), and our odd mode by sin (𝑥) in one
dimension, then clearly the mode coupling vanishes as ∫ sin(𝑥) cos(𝑥) 𝑑𝑥 = 0 over any integer
number of periods of oscillation.
There are as many different approaches to understanding symmetry protected BICs as
there are physical systems to explore. BICs have been realized in air pressure waves [39], water
waves [40], and a bit more recently quantum wires [41]. In the field of photonics there have even
been a number of nice demonstrations from which we are able to extract a key result: periodic
structures provide a natural starting place for symmetry- protected BIC design [42–44]. To be a
little more specific, planar photonic crystals fabricated on semiconductor wafers can be written
in a huge number of different ways. Let’s consider the likely scenario in which a planewave or
gaussian-shaped laser beam is incident on a structured photonic crystal. Then the symmetric
19
nature of the mode necessitates an anti-symmetric device mode in order for the resulting total
system to contain an odd state that is bound in the even-state continuum. This can be
accomplished when the periodic structure follows both time-reversal and inversion symmetry in
the plane. The resulting BIC occurs at one of the high symmetry points in momentum space. This
is an oversimplification of the full design considerations to come in later sections, but before
focusing in on any one specific design, let us tackle the looming question of how to access (and
make us of) an inaccessible resonance.
Generalized Asymmetry Parameter and Quality Factor:
The previous sections have established the principles behind the symmetry-protected
BIC, however it is important to realize that the issues of infinitesimal linewidth and hence zero
coupling make these bound states experimentally unviable. Without coming up with some
scheme that allows for the excitation of such states, we’ll have no hope of using them for any
impactful application. We are actually left with only two options in order to proceed: break one
of the two symmetries in the system- either that of the incident mode or the one you wish to
couple to- in such a way that the linewidth of the resonance becomes experimentally viable. The
applied perturbation, which can come in any number of forms depending on the device design
and BIC scheme, should not go so far as to make the BIC mode lifetime too short, so some care
should be taken.
Returning to our model from before, let’s imagine a planewave (even symmetry)
normally incident on a planar, periodic photonic crystal supporting a mode with odd symmetry.
The overlap integral of the two modes is zero and hence they cannot couple. To break the even
symmetry of the incident beam is actually a very simple task- it can be accomplished by slightly
altering the angle of incidence to be off the normal axis. In practice, this is not the preferred
20
approach, as the quality and wavelength of the resonance will become a sensitive function of the
incident beam, and not of the device design. The second option is considered the best practice-
to break the odd symmetry of the device mode [45] and allow coupling in and out. As we are
interested in periodic structures in this Chapter, and we wish to maintain periodicity in our total
system, the perturbation that we apply must also be periodic. For a series of semiconductor
pillars making up a metasurface, this can include anything from adding material, removing
material, shifting the position of, or tilting the constituent pillars. Regardless of which
perturbation approach is taken, we will now see that the coupling to a leaky mode is universal.
We start with a brief derivation of the inverse lifetime 𝛾𝑟𝑎𝑑 of the mode following [45], because
if we can observe a change from an infinite lifetime to a finite lifetime, we will have succeeded
in our goal. Consider a total resonance electric field 𝐸𝑟 = 𝐸0 + Δ𝐸, where 𝐸0 solves the Maxwell
equations and Δ is a small perturbation.
Each resonant state 𝐸𝑟 with complex frequency 𝜔𝑟 = 𝜔0 −𝑖𝛾
2, must come with a
corresponding complex conjugate state 𝐸𝑟∗. The complementarity of the complex conjugate states
actually gives us an elegant way to solve for the inverse lifetime,
[𝝎𝒓𝟐 −𝝎𝒓
∗𝟐] ∫ 𝒅𝑽 𝝐𝑬𝒓𝑬𝒓∗ − 𝒄𝟐𝑺𝟎∑ 𝒔(
𝑬𝒓𝜹𝑬𝒓∗
𝒅𝒛−
𝑬𝒓∗𝜹𝑬𝒓
𝒅𝒛)𝒔=+𝟏,−𝟏𝑽 ( 12)
Which can be simplified and estimated to first order using perturbation theory to actually be
𝜸
𝒄= 𝑺𝟎∑ 𝑺𝟎(|𝑬𝒓,𝒙|
𝟐+ |𝑬𝒓,𝒚 |
𝟐)𝒔=+𝟏,−𝟏 ( 13)
This sum can be computed with a few tricks that are beyond the scope of this section. To
summarize the result though, we can find that
𝜸
𝒄= |
𝒌𝟎
√𝟐𝑺𝟎[𝒑𝒙 −
𝒎𝒚
𝒄+
𝒊𝒌𝟎
𝟔𝑸𝒛𝒙]|
𝟐
+ |𝒌𝟎
√𝟐𝑺𝟎[𝒑𝒚 +
𝒎𝒙
𝒄+
𝒊𝒌𝟎
𝟔𝑸𝒚𝒛]|
𝟐 ( 14)
21
Where 𝑝 and 𝑚 are the electric and magnetic dipole and 𝑄 are the components of the electric
quadrupole. 𝑘0 is the wavevector and 𝑆0is the surface area of the integration. 𝑐 as always is the
speed of light. Now we can return to considering our periodic perturbations, and observe the
impact that this has on the lifetime we’ve computed. We proceed with the derivation for any
perturbation operation that maintains the mirror symmetry of the unit cell (x) goes to (-x). The
perpendicular components of the total 2D electric field can be described according to how they
transform under this symmetry operation, noting that 𝑥 and 𝑦 components will always be of
opposite parity. Let’s take the case where 𝑥 is the even mode direction and 𝑦 is odd. The second
term in Equation (14) vanishes for this scenario and the inverse lifetime simplifies dramatically
to
𝜸
𝒄=
𝒌𝟎𝟐
𝟐𝑺𝟎|𝒑𝒙|
𝟐. ( 15)
Let’s now transform our language from lifetime to quality factor, defined to be proportional to
that lifetime and resonance frequency according to 𝑄 =𝜔0
𝛾. The key to finishing the derivation
Figure 2.1.2: The generalized asymmetry parameter, a relevant example of which is given in the inset,
universally scales the quality factor according to 𝑸 ∝ 𝜶−𝟐. As the asymmetry parameter approaches zero, the
BIC lifetime goes to infinity.
22
lies in the fact that the dipole moment is proportional to the magnitude of the perturbation, and so
in total, 𝑄 =𝑆0
2𝑘0
1
Δ2|𝑝0|2 , where I have brought back Δ as the perturbation magnitude and 𝑝0 as
the dipole moment amplitude. We have quickly derived an extremely important result, that once
it is made accessible, the quality factor of a BIC scales inversely as the square of the size of the
perturbation.
All that remains is to generalize what we mean when we say “size of the perturbation”.
Let’s define a new perturbation parameter called the asymmetry, which is normalized according
to the device parameters in such a way that it scales from 0 to 1. 0 indicates no perturbation,
while 1 represents the extreme. As an example, if the perturbation is to the length of a lattice
“atom”, then the asymmetry parameter would be defined as the change in length divided by
original length, etc. This generalized result is depicted in Figure 2.1.2.
What we’ve just described is the BIC which is no longer completely bound, and because
of this it is generally referred to as a quasi-bound state in the continuum. The discussions of
bound states and asymmetry parameters have been kept as general as possible so far, but in the
following Section we will finally elaborate on the specific system for our experiments.
Section 2.2: Dimerized High- Contrast Gratings
Dimerization and Spatial Mode Control:
In Chapter 1.3.1 we derived the energy-momentum dispersion relationship for a simple
system of masses joined by identical springs. What we neglected to discuss at the time was that
this toy system, which I claimed to be the simplest, was quite clearly a step up in complexity
compared to a system with all identical masses. By alternating between two different “atoms” of
masses 𝑚 and 𝑀, we replaced the monomer unit cell of the system with a two atom “dimer”. At
23
the time, this was a necessary step to ensure that our model system would have a bandgap. As we
have now seen, the disruption of the periodicity can have more profound effects (such as creating
leaky modes) depending on the perturbed system.
An ideal platform to study these periodic perturbations (and the one we will perform our
experiments on) is a high-contrast grating [46–50]. In this context, “contrast” refers to the
difference in refractive index between the active dielectric medium and the surrounding
environment. Originally, lower contrast periodic gratings grew to prominence because the
lifetime of the guided mode resonance that they supported could be controlled as a function of
the depth of the etch surrounding the grating fingers [51]. High contrast gratings are then the
limit of dielectric gratings as the contrast becomes maximal, that is, the surrounding index
approaches unity. Deep etching solved an issue of scale in dielectric gratings, which is outside
the realm of this Chapter. But to put it simply, shallow corrugation necessitated fabrication of
very large gratings [52].
Figure 2.2.1: SEM of a dimerized high contrast grating used in this work. The effectively infinite extent of the
grating fingers in the y-direction makes this, in essence, a one-dimensional system. The dimerizing perturbation
can clearly be seen to be applied to the gaps between bars as opposed to their widths. The reasons for this choice
will become evident in the next section.
24
Following the recipe developed in the previous section of introducing a periodic
perturbation to a regular array of dielectric bars or resonators, we will now discuss a subset of
high contrast gratings known as dimerized high contrast gratings (DHCGs) pioneered by
collaborators to this work Nanfang Yu and Adam Overvig [46]. In one periodic dimension, let’s
consider each resonator to be an infinitely long silicon bar (sometimes referred to as a grating
finger here) in the y-direction, and periodic in x. Reducing the system to effectively only one
dimension also limits the total number of dimerizing perturbation classes to just two. On one
hand, every other grating finger can have its width adjusted in the x-direction. Alternatively, the
widths can be held constant and every bar can be shifted from its original location along x in
alternating directions. An SEM image of a gap-perturbed dimerized high-contrast grating is
shown in Figure 2.2.1. As discussed above, the change in width or position divided by their
respective starting dimensions specifies our asymmetry parameter, and hence the bandwidth and
Q-factor of the system.
Figure 2.2.2: HCG vs DHCG band structures. (a) The band structure of a 1D HCG. The arrow points to a mode
of interest to us that exists at the edge of the first Brillouin zone (FBZ). This mode is below the light line,
illustrated by the shaded region, making it inaccessible. Inset: a sketch of the corresponding grating. (b)
Dimerization of the HCG folds the FBZ. The arrow points to the same mode as before, now located at k=0. It
can be experimentally accessed.
25
The dimerization of a high-contrast grating alters the physics of the system in ways
beyond the perturbation to the mode symmetry described in Section 2.1. In the momentum space
picture, the doubling of the spatial period of the structure implies a halving in k-space, known as
Brillouin zone folding [53]. This “folds” modes at the edges of the first Brillouin zone (FBZ) to
the gamma point, making them accessible to normal incidence excitation. Such modes had
previously only existed in an unusable state below the light-line. The Brillouin zone folding of a
high contrast grating is illustrated in Figure 2.2.2.
Brillouin zone folding comes with one more inherent advantage: the modes that exist at
the Brillouin zone boundaries are typically much flatter than modes found at the gamma point,
and the slope of the band is directly proportional to the group velocity of light in the mode [54].
Therefore, high degrees of confinement in small devices benefit from the flattest possible bands.
Furthermore, the combination of small slopes and a reduced FBZ means DHCGs can exhibit
sharper spectral features with narrower linewidths than either unperturbed or low-contrast
gratings.
Figure 2.2.3: Finite-Difference Time-Domain simulation of the gap-centered mode originating from a gap-
perturbation to a 1D DHCG. Though the mode extends into the silicon and substrate, the mode volume in the
gaps is more than sufficient to enhance the optical intensity felt by argon atoms by orders of magnitude.
26
What remains is to choose between the width and gap varieties of perturbation. This
choice is ultimately tied to the intended application of the final metasurface, so let’s quickly
review the motivation presented at the start of the Chapter. We are envisioning a system in which
the resonance provided by the DHCG can enhance the electric field strength being used to
perform high-harmonic generation in gases. We must therefore choose a periodic perturbation
which provides the quasi-BIC mode most localized in the gaps between grating fingers. Those
gaps will be the perfect place to flow our atomic sample.
We choose to modulate the spacing (i.e., the gap-perturbation) because this folds the
unique mode depicted in Figure 2.2.3 to the gamma point. Mode and device simulations are
performed with the commercial finite-difference time-domain (FDTD) solver licensed by
Lumerical. The mode is centered in the gap between adjacent grating fingers, where it provides a
uniform two orders of magnitude enhancement in the local field intensity for a modest quality
factor Q = 1500. This mode affords us the ability to inject our gaseous nonlinear medium (e.g.,
Figure 2.2.4: (a) Simulated unpolarized reflectance spectrum of DHCGs with alternating gaps of 190 nm ± δ.
The reflectance peaks at the BIC Fano-resonance design wavelength near 1550 nm. Smaller perturbations
correspond to larger Q-factors and narrower linewidths. (b) FTIR measured reflectance of unpolarized near-
infrared light. Losses lead to the observed variation in reflectance magnitude.
27
Argon atoms) into the gaps. Alternatively, a mode centered purely within the fingers could have
been achieved with a width perturbation and fixed gaps.
Final Device Design:
We designed and fabricated silicon-on-insulator (SOI) DHCGs on a standard wafer with
a 250-nm thick silicon device layer, 1-μm of oxide, and a 500-μm silicon substrate. Figure 2.2.1
shows a top view SEM image of that very device, fabricated by e-beam lithography, which
consists of a series of silicon fingers 270-nm wide with a total grating period of 920-nm. The
large grating period is a result of the dimerization of an unperturbed high contrast grating with a
period of 460-nm. We also fabricated a second grating with approximately twice the width and
spacing, allowing us to test two very different operating wavelengths of 1550 nm and 2900 nm.
By scaling the period and duty cycle, the system can be made to operate at any wavelength and is
limited only by fabrication sensitivity.
We both simulated and experimentally tested a variety of perturbation sizes to the silicon
gratings. Figures 2.2.4 (a) and (b) show the simulated and measured unpolarized reflectance
spectra of a set of DHCGs designed with a central wavelength of 1550 nm and a range of Q-
factors as high as 1000. We also include the symmetry-protected case of unperturbed gap widths
of 190 nm and confirm that no resonance is observed (this is a true BIC, not Quasi). In our
experiments we choose to increase the size of the perturbation in order to accommodate the
bandwidth of our laser pulses, while simultaneously making it easier for the field to couple into
the mode.
Only when the incident laser polarization of the pump is oriented along the finger
direction can the required BIC mode be efficiently excited. The total number of modes in the
28
system is also a function of the grating depth. By restricting the device to 250 nm in height we
thereby reduce the likelihood of a parasitic resonance being located near our gap-centered mode.
Section 2.3: Enhanced Nonlinearities in Gases
Wavelength Dependence:
At last, our ideal metasurface is designed and fabricated, capable of providing field
enhancements throughout the infrared wavelength range and with any Q-factor we desire. The
experimental setup used to test the devices for enhanced harmonic generation is as follows:
An optical parametric amplifier (Light Conversion HE Topas Prime) pumped by an
amplified Titanium-sapphire laser system (Coherent Legend Elite) operating at a 1-kHz
repetition rate with 6 mJ of pulse energy to generate tunable, 60-fs duration signal pulses with
center wavelengths from 1485 nm to 1580 nm. This parametric amplifier output is used to pump
an additional difference frequency generation module for our mid-infrared measurements at 2.9
μm with a pulse duration of 70 fs. The pump-beam polarization is controlled with a broadband
zero-order half wave plate before being focused on the sample by a 5-cm focal length CaF2 lens.
Any unconverted seed light is removed with the appropriate long-pass filters. Reflected
harmonics are collected and measured on a fiber-coupled ultraviolet-visible spectrometer (Ocean
Optics) with a 350 nm to 1100 nm detection range. The devices are mounted in a high-pressure
stainless-steel gas cell with a 1-inch diameter sapphire window. For sufficiently large beam
diameters, the intensity on the sapphire window is low enough that it does not contribute to the
nonlinear signal in our experiments. Argon pressures in excess of 200 psi could be held on the
devices within the cell.
The first thing we wish to test is whether enhanced nonlinearity is restricted to the design
wavelength of our DHCG. The third-harmonic-generation (THG) signals generated on and off
29
the 1550 nm- resonant DHCG are shown in Figure 2.3.1. Off the metasurface structure, the pump
wavelength was swept from 1485 nm to 1580 nm, and the emitted third harmonic signal was
recorded in Figure 2.3.1 (a). We observe the expected behavior of harmonic emission that tracks
the pump wavelength without any additional spectral features. This picture changes when we
repeat the same wavelength scan on the device itself. The harmonic yield dramatically increases,
requiring over an order of magnitude lower power to generate, but within a limited bandwidth.
At pump wavelengths below 1530 nm or above 1580 nm no measurable increase is observed in
the third-harmonic signals plotted in Figure 2.3.1 (b), and only a small signal, far off-resonance
due to the peak of the 1485 nm pump is measured (blue arrow). Compared to this small signal,
the 1550-nm pump beam displays the greatest harmonic yield enhancement of a factor of 45,
implying the greatest overlap with the BIC resonance. This concept is illustrated in Figure 2.3.1
Figure 2.3.1: (a) Normalized third-harmonic signal generated from the bare substrate for pump wavelengths
in the range of 1485 nm to 1580 nm. Pump powers are 10X greater than in (b). (b) Third-harmonic spectra for
the same five pump wavelengths shown in (a), now on the DHCG. The large field enhancement on resonance
pins the third harmonic to 513 nm. The small signal at 495 nm (blue arrow) corresponds to the third-harmonic
signal from the peak of the 1485-nm pump that is generated off-resonance. (c) Schematic of the pump field
and resonance overlap resulting in the harmonics measured in B. The shaded overlap regions contribute to the
nonlinear signal.
30
(c), in which the overlap of the Gaussian pump spectra and Lorentzian BIC resonance determines
efficient harmonic generation. This explains the observed pinning of the THG to roughly 516 nm
and the reduced linewidths of detuned harmonics. The cut-off shape of the short wavelength
harmonics gives the clearest indication of the resonance edge.
Polarization Dependence:
We also examine the polarization dependence of our fabricated near-IR metasurfaces. As
described in an earlier Section, the grating mode under study is excited by TE polarized light
parallel to the long direction of the grating fingers. We hereafter refer to this polarization
direction as 0o. The theoretical dependence of the peak reflectance from the grating is studied in
a series of FDTD simulations and plotted as a blue line in Figure 2.3.2 (b). For clarity, we
Figure 2.3.2: (a) (A) Normalized third-harmonic spectra of a 1550 nm pump, corresponding to the red points
in B. Also included is the signal generated off the device, which is unmeasurable. 1 atmosphere of Argon
pressure was flowed over the sample. Contributions to the third harmonic from both gas and silicon are
present. (B) Simulated peak reflectance of the ideal grating (blue line), and integrated experimental third
harmonic yield (red dots), as a function of the pump polarization. 0o is the polarization parallel to the grating
fingers. The red dashed line indicates 𝑪𝒐𝒔 𝟔𝜽, corresponding to the expected scaling of perturbative THG.
The mismatch between the red theoretical curve and experimental dots indicates that the measured THG is
instead following a non-perturbative scaling typical of higher pump intensities.
31
subtract any contribution to the reflectance due to etalon effects of the thin device and oxide
layers, hence the reflectance has a minimum of zero. As the polarization is rotated by an angle 𝜃
beginning from 0o, the peak power reflectance due to the BIC follows precisely the expected
cos2 𝜃 behavior. The component of the field oriented along 90o therefore makes no contribution,
and the cos2 𝜃 behavior is a result of the projection of the pump beam onto the 0o axis. We
compare to this reflectance curve the integrated (measured) THG yield of 1550-nm pulses on
resonance, shown as red dots in Figure 2.3.2 (b), corresponding to the spectra in Figure 2.3.2 (a).
The experimental data follow a similar trend, but with some important differences. Were it a
perturbative third-order nonlinearity with intensity 𝐼 ∝ cos2 𝜃, a THG efficiency proportional to
cos6 𝜃 may be expected, given by the red line in (b). The measured harmonics follow neither the
exact cos2 𝜃 behavior of the resonance or the perturbative cos6 𝜃 dependence.
That observed saturation of the intensity dependence of the Nth order harmonic below 𝐼𝑁
is a signature of non-perturbative harmonic generation. Thus, we conclude that in these
experiments the generated signal corresponds to the non-perturbative regime with more than 32
times larger emission at 0o than at 80o. Within the nonperturbative regime for THG pumped in
the mid-IR, a conversion efficiency upper-bound of about 10-7 is found for low-pressure Argon
gas [55]. Fifth harmonic conversion efficiencies saturate about one order of magnitude lower at
10-8. While the intensity enhancement provided by the DHCGs demonstrated here does not raise
this efficiency limit, it does successfully lower the input power at which the maximum
conversion efficiency is reached. At 90o polarization, the harmonic yield converges back to the
signal off-resonance.
32
Pressure Dependence:
A final, crucial experimental check should be performed when working in a hybrid solid-
gas system. We need to be certain that the measured harmonics originate from the gas sample
and not the substrate or silicon device. To do so we also characterize the performance using the
scaled-up mid-infrared devices, allowing us to explore the third and fifth harmonics, both of
which fall within our detection range for a 2.9 μm pump wavelength. Figure 2.3.3 shows the
third- and fifth-harmonic spectra of these structures for a range of Argon gas pressures. For both
harmonic orders, the harmonic yields scale quadratically with gas pressure (see (b)) confirming
that the emission is due to the atoms in the gap regions. At a pump intensity of 2 x 1012 W/cm2,
this emission occurs at field strengths where efficient harmonic generation in Argon is not
expected to take place. In the absence of Argon, we still observe a non-zero third- and fifth-
harmonic signal, which we attribute to the silicon finger surfaces. This is further supported by
Figure 2.3.3: (a) Third and fifth harmonics of a 2.9 μm pump beam on a 2.9 μm resonant DHCG. Argon
pressures are varied from 100 psi to 200 psi in a high-pressure gas cell. The fifth harmonic has been scaled up
for visibility. (b) Integrated harmonic yields from the spectra in (A), matched by color. Pressure dependence
of both harmonic orders are fit to the square of the pressure added to a pressure independent contribution
that we assume is due to the silicon fingers. The fifth harmonic is similarly scaled up.
33
the expected overlap of the mode and grating predicted in our simulation (Figure 2.2.3). At 200
psi of Argon pressure, the harmonic yields from Argon significantly exceed that of the device
material.
Section 2.4: Final Remarks
Over the course of this Chapter, we have developed an understanding of quasi-bound
states in the continuum as well as a scheme for utilizing them for enhancing harmonic generation
in argon atoms. We have observed a greater than 10-fold enhancement of the third and fifth
harmonic yields in the presence of a BIC- supporting DHCG, with the yield enhancement for
higher-order harmonics being greater due to the higher order dependence on pump intensity. We
have evidence based on the polarization dependence of THG depicted in Figure 2.3.2 that we
have achieved the strong-field, non-perturbative regime, as the integrated THG does not follow a
cos6 𝜃 dependence on the polarization angle. From this observation we note that the hybrid gas-
DHCG technique lowers the required pump input intensity to saturate the THG conversion
efficiency at about 10-7. Furthermore, we have determined that for pressures between 100 psi and
200 psi, the contributions to our total signal from enhanced harmonics within the grating material
and Argon are of the same order of magnitude. With this we confirm the mode profile produced
by our FDTD simulations which predicts a gap-centered mode with leakage into the silicon
fingers.
Further improvement of our hybrid metasurface-gas harmonic scheme requires the ability
to restrict the strong field-confinement mode to only the gas-filled region. Ideally, this would be
accomplished while maintaining relatively large mode areas and all of the tunable wavelength
and quality factor properties afforded to us by the DHCG platform. We propose a device in
Figure 2.4.1, based on a 2-dimensional realization of a DHCG, that meets all of these stated
34
requirements. Figure (a) shows a representation of the device, which consists of silicon dimers
containing one circular and one elliptical nanopillar. The perturbation of the unit cell is to the
ellipse minor axis. For a target resonance wavelength in the mid-infrared, the disk major axes
should be 500 nm, and the grating period should be 2.1 μm. Figure (b) shows the BIC mode of
interest in the system, chosen at a wavelength of 3.15 μm. In this system, the strongest field
enhancement no longer spatially overlaps with the silicon. This offers tremendous flexibility to
the proposed system in two forms. First, the contribution to the harmonic signal from any non-
gas sources would be greatly diminished. Second, the quality-factors could be raised to their
fabrication limits, greatly increasing the local powers experienced by the atomic targets without
increasing the likelihood of device damage. Quality-factor tuning of the 2D DHCG observes the
Figure 2.4.1: (a) Proposed 2D DHCG. The height of the grating is 250 nm, disk radii 500 nm, and the period is
2.1 μm. The dimerizing perturbation is to every other disk’s minor axis. (a) FDTD simulation showing the
relative power enhancement in the region marked by a red square in (a) (top view). The strong field-
confinement mode is entirely contained in the Argon gas region, with no leakage into the walls of the grating
or substrate. (c) Simulated polarized reflectance spectrum of 2D DHCGs with alternating minor axes of 500
nm - δ for four magnitudes of the perturbation as well as the unperturbed case. The reflectance peaks at the
BIC Fano-resonance wavelength near 3.15 μm. Smaller perturbations correspond to larger Q-factors and
narrower linewidths.
35
required 𝛼−2 dependence, but now the lower-Q peaks in the reflectance curve tend toward the
short wavelength side as in Figure (c). Smaller ellipse minor axes corresponding to larger
perturbations result in less silicon in each unit cell, and hence a lower effective index, which
leads to a blueshift in the reflectance peak.
The potential for greatly amplified field strengths within gas-filled gaps gives renewed
relevance to high-harmonic generation from gases in the presence of metasurfaces. Since a
critical factor in the generation of high energy harmonics is, among other things, a sufficiently
large intensity, the approach we have developed in this Chapter can be extended readily to
higher-order harmonic generation. The possibility to tailor mode profiles to exclude leakage into
the grating, and therefore eliminate the damage that has so far limited plasmonic and dielectric
devices alike, deserves further study. Using a transparent nonlinear medium such as Argon
extends the interaction length of the harmonic generation process well beyond the 10’s of
nanometers available in either metal or silicon-based sources, while dramatically reducing
harmonic absorption when compared with solid-state media. The 250-nm tall gratings used in
our experiments, for example, are tall enough to allow greater coherent buildup of the harmonic
signal, while being sufficiently subwavelength in extent to sidestep phase-matching restrictions.
Lower power requirements open up the possibility to use higher repetition rate sources to initiate
HHG in gases and have the potential to make smaller footprint table-top attosecond sources more
viable. The ease with which Q-factors can be adjusted in the fabrication process means the
DHCG devices shown here can be made to match the bandwidth of a given pump laser.
Alternatively, the resonance linewidth can be chosen to select for the desired linewidth of the
generated harmonics. Moreover, the control provided by DHCGs can be further extended to
simultaneously include both wavefront and polarization shaping.
36
Looking to the future, our hybrid gas-DHCG scheme could be applied to the entire suite
of nonlinear optical studies being conducted in atomic systems. Beyond the gas phase, two-
dimensional materials are emerging as a promising nonlinear optical platform moving
forward [56–58] and will become the focus in the next few Chapters. The single- to few- atom
thickness of graphene and transition metal dichalcogenides limits their nonlinear interaction
length, making them prime candidates for integration with field-confining metasurfaces.
Furthermore, the all-dielectric metasurface is a leading platform for meta-lenses, capable of
providing the same level of phase-control and integrability with 2D materials as seen in previous
plasmonic nonlinear metalenses [59]. This is due to the high degree of engineerability of modes
provided by all-dielectric DHCGs, but they also bring the added benefit of eliminating potential
field hotspots in sensitive 2D materials, especially at higher harmonic orders and greater field
intensities.
37
Chapter 3: Nonlinear Phononics
Just as quickly as the development of metasurfaces led to their immediate application for
nonlinear optical enhancements, the discovery of two-dimensional materials has led to intense
interest in their electronic and nonlinear optical properties. We will see in this chapter and the
next that lower orders of harmonic generation can be used to gain valuable insights into the
behavior of special two-dimensional materials when excited by intense resonant radiation. We’ll
be especially focused on hexagonal boron nitride (hBN), a thin insulating material with one of
the highest energy infrared-active phonons known. After describing phonons in general, we will
develop a straightforward but robust theory of the nonlinear driving of phonons, referring back to
the harmonic oscillator picture of Chapter 1. One of the major consequences we will observe
from this theory is the opportunity to control the structural characteristics of hBN. Only then in
Chapter 4 will we go on to show that the nonlinear driving of the phonons leads to enhanced
odd-order harmonic generation on resonance, as well as induced even-order harmonic generation
which should, as we’ll soon understand, be forbidden.
Section 3.1: The Charged Harmonic Oscillator Model and Linear Optical Responses
In Chapter 1 we considered a model of electrons oscillating in an anharmonic potential
and from there derived the even and odd order harmonics based on symmetry. Returning to the
same picture, let’s now consider a variation of the situation, where the electron motion can be
neglected and instead the entire atom moves in a harmonic potential. This situation describes the
excitation of a phonon, a collective vibrational mode of all of the different sites in a lattice. An
illustration of two flavors of phonon in a 1-dimensional chain of atoms is given in Figure 3.1.1.
38
Alternating atoms with masses 𝑚 and 𝑀, joined by springs, move as either a longitudinal wave,
in which the two species can move together, or a compressional wave where different species
move oppositely. The former is called an acoustic phonon and the latter is known as an optical
phonon. As we will see, optical phonons are the only type that couple directly to incident light,
which will be extremely important for our later experiments.
Consider an atom in a lattice subject to small displacements 𝑟𝐼𝑅 due to infrared radiation.
The formalism of the harmonic oscillator tells us then that the atom must feel a restoring force
𝑭 = −𝝎𝒑𝒉𝟐 𝒓𝑰𝑹 ( 16)
for a potential that is quadratic in the position and frequency of the phonon 𝜔𝑝ℎ. Assuming the
infrared radiation drives the phonon according to the Coulomb force, then we arrive at an
equation of motion for the optical phonon amplitude under excitation by a field 𝐸:
�̈�𝑰𝑹 +𝝎𝒑𝒉𝟐 𝒓𝑰𝑹 −
𝒁
𝒎𝒆𝑬. ( 17)
Figure 3.1.1: The two types of phonon modes illustrated with a toy model of two species of masses joined by
identical springs. The springs provide a restoring force that leads to a harmonic potential, sketched on the
central atom. (a) Acoustic phonons, often drawn as longitudinal waves, do not directly couple to light and are
often seen when atoms of different species (charge) move together. (b) Optical phonons on the other hand
couple to free space radiation and are distinguished by species moving oppositely, which induces a changing
dipole.
39
Here 𝑚𝑒 is not the electron mass but the effective mass of the atom, and Z, the Born effective
charge, replaces the charge of the electron in the Coulomb force. A driven oscillator without any
channel through which to lose energy is not a particularly realistic system. The final ingredient
we will need to include recalls the type of resonance lifetime 𝛾 we explored in the context of the
bound state in the continuum. Here the lifetime will be determined by a damping constant Γ
which is inversely proportional to the lifetime. To enforce an exponential saturation on our
equations of motion we make one final addition in the form of:
�̈�𝑰𝑹 + 𝚪𝒓 𝑰𝑹 +𝝎𝒑𝒉𝟐 𝒓𝑰𝑹 −
𝒁
𝒎𝒆𝑬. ( 18)
Oscillatory electric fields with amplitude 𝐸0 therefore lead to solutions for 𝑟𝐼𝑅 with a time
and frequency dependence 𝑟𝐼𝑅 = 𝑟0𝑒−𝑖𝜔𝑡 for phonon oscillations in a damped, driven, harmonic
potential. Combining this with our equation of motion leads us to the final definition of the
phonon trajectory:
𝒓𝑰𝑹(𝒕) =𝒁𝑬𝟎𝒆
−𝒊𝝎𝒕
𝝁(𝝎𝒑𝒉𝟐 −𝝎𝟐−𝒊𝚪𝝎)
. ( 19)
Figure 3.1.2: An electric field (top) applied to a material with a phonon resonance near the electric field
frequency will oscillate and damp similar to the depiction shown on the bottom.
40
It’s clear looking at Equation (19) that the maximum amplitude of the phonon 𝑟𝐼𝑅 occurs when
the driving frequency 𝜔 approaches the phonon frequency. This is the resonance condition of a
phonon, and as we will see once we dive deeper into hexagonal Boron Nitride, the resonance
wavelength is a function of atomic masses, binding energies, as well as other factors. The
response of the phonon amplitude for an applied few-cycle pulsed electric field is illustrated in
Figure 3.1.2. The illustrated lifetime is taken to be just a few times that of the pulse duration. In
general however, the lifetime can be anywhere from a few hundred femtoseconds to many
picoseconds, depending on the system, temperature, and other experimental parameters.
Section 3.2: Lattice Anharmonicity
Short optical pulses such as the one illustrated in Figure 3.1.2 tend to be accompanied by
extreme intensities, often in the regime of terawatts per square centimeter when properly focused
down to the diffraction limit. Intensities this high (on pico- or nanosecond timescales) cannot
have their dynamics fully captured by a linear model. Once again, we are presented with a need
for anharmonicity in our model, and just as we came to powerful conclusions about electronic
harmonic generation in Chapter 1, so too will anharmonicity lead us down some novel and
interesting paths in the context of phonons.
We once again approach this problem by redefining the energy potential to include terms
that are higher-order in position. In general, there is no limit to the higher-orders that can
contribute to the potential energy, and it takes the form:
𝑈 =1
2𝜔𝑝ℎ2 𝑟𝐼𝑅
2 + ∑𝑎𝑛𝑟𝐼𝑅𝑛
∞
𝑛=3
+ ∑ 𝑏𝑚𝑟𝐼𝑅𝑛
∞
𝑛,𝑚=1
𝐸𝑚
(20)
41
The first summation represents all of the purely ionic contributions to the potential, and the
second summation couples the electron and phonon nonlinearities with their own set of constants
𝑏𝑚.
Let’s now focus on two different conclusions that can be drawn from the various terms of
the potential energy: (1) The generation of pure ionic high harmonics, and (2) the structural
changes induced in the lattice by the motion of the ions. First, let’s investigate the terms that are
Figure 3.2.1: (a) Top: pump electric field, unchanged from the previous section. Bottom: the response of the
ions to the electric field when the equations of motion now include higher order terms in the displacement.
This leads to anharmonicity and the generation of new frequency components. (b) Fourier Transform of the
phonon amplitudes of (a), showing new frequency components at integer multiples of the driving frequency.
42
purely phononic in nature [60]. The equation of motion considering only the potential due to the
first series is given by:
�̈�𝐼𝑅 + 2𝛾𝑟 𝐼𝑅 + 𝜔𝑝ℎ2 𝑟𝐼𝑅 + ∑ 𝑎𝑛𝑟𝐼𝑅
𝑛−1
∞
𝑛=3
−𝑍𝐸 = 0
(21)
This series of terms represent the ionic equivalent of high harmonics (just as in Chapter 1), with
each term 𝑟𝑛 representing the 𝑛𝑡ℎ harmonic. As we will see in the following Chapter, under
resonant conditions the harmonics generated as a result of ionic motion can be many times larger
than the more standard electronic nonlinear contributions. Figure 3.2.1 generalizes the phonon
amplitudes of Figure 3.1.2 to include the higher order terms of the new equation of motion. In
Figure 3.1.2 (b), a Fourier transform of the atomic motion shows the generation of the new
frequency components resulting from the atomic anharmonicity.
Now we turn to the effect that the motion of the atoms has on the structure and symmetry
of the lattice. We can subdivide this study further into two sections: the coherent motion of the
phonons as an oscillating electric field is applied, and the rectification of the phonons or creation
of a new, temporary equilibrium position. The coherent motion of the phonons follows trivially
from the driving frequency 𝜔, observed in Equation (19) and in the phonon behaviors of both
Figures 3.1.2 ad 3.2.1.
To understand the rectification effects in the lattice positions, we turn our focus to the
second summation in Equation (20). When considering just the lower term proportional to 𝑟𝐼𝑅2 ,
the equation of motion takes the form
�̈�𝑹 + 𝟐𝜸𝒓 𝑹 +𝝎𝑹𝟐𝒓𝑹 = 𝒃 𝒓𝑰𝑹
𝟐 ( 22)
Where our attention has turned to the position of a Raman mode, specifically. The quadratic term
in the restoring force causes a non-oscillating force along the direction of 𝑟𝑅. This rectified
43
displacement, combined with the coherent oscillations, form the basis for the symmetry control
we will exert on hexagonal boron nitride in the next Chapter.
With these derivations completed, we now have both the historically relevant low- and
high- harmonic generation formalisms as well as a robust model of anharmonicity in lattice ions
based on an anharmonic potential. In theory, we understand how resonant driving of atoms can
lead to purely phononic nonlinearities and changes to the positions of lattice sites (and hence
changes to the crystal symmetries). However, in practice the material under study will determine
how different harmonics present themselves when strongly driven. Chapter 4 will now apply
these concepts to hexagonal Boron Nitride.
44
Chapter 4: Enhanced Harmonics and
Phonon-Induced Symmetry Breaking in HBN
Optical nonlinearities in solids reveal information about both the in-plane rotational and
out-of-plane inversion symmetries of a crystal. In the van der Waals material hexagonal boron
nitride (hBN) both these symmetries and the linear vibrational properties have led to the rich
physics of mid-infrared phonon-polaritons. However, the role of strong electron-phonon
nonlinearities which we began to derive in the previous Chapter requires further study. In this
Chapter, the hybrid phonon-polariton modes will be derived from first principles, and then the
specific case of hexagonal Boron Nitride will be addressed. Narrowing in on the specific
phonons and Reststrahlen bands of hBN, we will motivate the current tremendous interest in
hBN as a 2D photonics platform.
We then investigate both theoretically and experimentally the rich interplay of phonon
anharmonicity and symmetry in phonon-polariton mediated nonlinear optics. We show that large
enhancements (>30×) of third-harmonic generation occur for incident femtosecond pulses that
satisfy the previously derived resonance condition for the hBN transverse optical phonons in the
mid-infrared. In addition, we predict and observe large transient sub-picosecond duration
second-harmonic signals during resonant excitation, which in equilibrium is forbidden by
symmetry. This surprising result indicates that instantaneous crystal inversion symmetry
breaking can be optically induced and controlled via phonon interactions by both the power and
polarization of the pump laser.
45
Section 4.1: The Hyperbolic Phonon-Polaritons of Hexagonal Boron Nitride
Chapter 3 established a general formalism for nonlinear phononics that we will soon be
able to apply to hBN specifically. But before we can understand the nonlinear phononic
properties of hBN, a baseline knowledge of its linear optical and vibrational properties will help
motivate why hBN holds so much promise as a future nonlinear optical platform. Recall the
solution to the equations of motion of a phonon represented by a charged harmonic oscillator in
the previous chapter. For a bulk material we can now define the polarization in the lattice to be
equal to:
𝑷(𝒕) = 𝑵𝒁𝒁𝑬𝟎𝒆
−𝒊𝝎𝒕
𝝁(𝝎𝒑𝒉𝟐 −𝝎𝟐−𝒊𝚪𝝎)
, ( 23)
For the density of unit cells N. Substituting this expression into the definition of the dielectric
function 𝜖 = 1 + 𝑃/𝜖𝑜𝐸, we find that the dielectric function given by the presence of the phonon
resonance is
𝝐(𝝎) = 𝟏 +
𝑵𝒁𝟐
𝝐𝟎𝝁
𝝎𝒑𝒉𝟐 −𝝎𝟐−𝒊𝚪𝝎
. ( 24)
It’s convenient to define a new frequency, commonly referred to as the plasma frequency 𝜔𝑝𝑙, to
be the square root of the numerator. The dielectric function combined with Maxwell’s equations
is sufficient to fully understand the propagation of light in the medium, as well as how the
photons and phonons interact with each other. Substituting the dielectric function into 𝐷𝑖𝑣 𝐸 =
𝜖
𝑐2𝛿2𝐸
𝛿𝑡2 for a plane wave, we find the familiar dispersion relationship relating the momentum 𝑘
and frequencies 𝜔: 𝑘2 =𝜔2
𝑐2𝜖. Following the full substitutions, we find the total dispersion
relationship
46
𝝎 = √(𝝎𝑳𝑶
𝟐 𝝐∞+𝒄𝟐𝒌𝟐)± √(𝝎𝑳𝑶𝟐 𝝐∞+𝒄𝟐𝒌𝟐)
𝟐−𝟒𝝐∞𝒄𝟐𝒌𝟐𝝎𝑻𝑶
𝟐
𝟐𝝐∞ , ( 25)
where ϵ∞ is standard notation for a constant dielectric background function. Notice that the
previously named ωphonon now appears as two separate frequencies ωTO and ωLO, for transverse
optical and longitudinal optical, respectively. As depicted in Figure 4.1.1, at the point where the
photon and phonon dispersion curves meet, an anti-crossing emerges in the band structure, and
the crystal hosts new hybrid modes called phonon-polaritons [61]. The primary takeaway from
this avoided crossing is that as the momentum 𝑘 approaches 0 (the Gamma point), the lower
energy TO phonon branch behaves more likes a photon, whereas at high momentum, the
behavior switches and the LO phonon takes the role of the photon-like behavior. Furthermore,
as the momentum goes to zero, the TO phonon energy becomes infinitesimal, while the LO
Figure 4.1.1: Phonon Polariton dispersion branches. The TO phonon energy converges to 0 at small momenta,
whereas the LO branch approaches photon-like behavior (represented by the red dashed line) at large values
of k. Between the two branches is the Reststrahlen band, which plays the role of the bandgap. It is the
frequency range in which photons cannot propagate into the material.
47
phonon converges to a finite value. This is significant for our upcoming experiments. It suggests
to us that at normal incidence, the LO phonon branch is more accessible at finite frequencies.
Lastly, the two phonon branches are separated by a well-defined energy gap known as the
Reststrahlen band. Within this range, photons cannot propagate through the material and will
instead be strongly reflected.
At long last we are in a position to apply the tools we have developed to our system of
hexagonal Boron Nitride. This work will focus solely on high-quality, exfoliated flakes of hBN
such as those in figure 4.1.2. To piece together a proper understanding of hBN, we need to
describe (1) the (microscopic) equilibrium structural and symmetry properties, (2) the phononic
band structure of hBN and the particular phonons of interest, (3) the Reststrahlen bands (upper
and lower) of hBN, and briefly (4) the natural hyperbolicity of the hBN phonons.
hBN has an energetically favorable AA’ stacked lattice in equilibrium, with alternating
boron and nitrogen atoms sitting one on top of the other. Figure 4.1.3 shows the hexagonal
arrangement. Not labeled is the experimentally measured b-n bond length of 1.445 Å. hBN
Figure 4.1.2: Microscope image of hexagonal Boron Nitride flakes. The color of the flake can be used as a
means of approximating the thickness of the flake. Atomic force microscopy should be used for more precise
thickness measurements. Flakes typically range in thickness from a few tens of nanometers to a few hundred.
Exfoliated hBN flakes can have diameters of 10s to 100s of microns.
48
belongs to a class of crystals called van der Waals materials, named after the type of weak
interlayer bonding common to such other two-dimensional materials as graphene and transition-
metal dichalcogenides. Having inversion symmetry in its equilibrium shape, hBN is not expected
to have any even-order nonlinearities, just as we derived in Chapter 1. The few exceptions,
particularly at interfaces or as a result of symmetry being broken by strain, have been
experimentally observed and quantified [62–64]. These nonlinearities are present only in the
thinnest few-layer samples with an even number of hBN layers, and decrease substantially with
hBN flake thickness.
hBN, like many real materials, has a band structure that cannot be calculated by any toy
model. To pick out the phonon bands of hBN we must turn to Density Functional Theory (DFT),
capable of computing the band dispersions from first principles [65]. Figure 4.1.4 shows both
the monolayer and bulk phonon dispersions, and labels each branch according to the phonon it
represents. For the experiments in this Chapter, which occur at normal incidence only, we are
restricted to the use of those phonons that have finite energy at the Γ point, which represents 𝑘 =
Figure 4.1.3: Various atomic arrangements for the hBN hexagonal lattice in equilibrium, IR-active phonon
excitation, and Raman-active phonon excitation. The equilibrium arrangement of hBN displays inversion
symmetry due to its AA’ stacking. IR-Active phonons such as the TO(E1u) induce a temporary breaking of
this inversion symmetry.
49
0. There are only two such bunches of phonons, occurring at slightly greater than 100 meV and
200 meV. These are the out-of-plane and in-plane optical phonons, respectively. The middle
panel of Figure 4.1.3 illustrates the atomic motion corresponding to the higher energy (in-plane)
transverse optical phonon. At a resonance wavelength of 7.3 𝜇𝑚, the TO (E1u) mode can be
accessed by our tabletop laser sources. The exceptionally light constituent atoms of the lattice,
with atomic numbers 5 and 6, are the reason for the high energy, mid-infrared phonon mode.
Furthermore, the fact that the oscillating modes of these phonons (depicted in Figure 4.1.3) cause
a rapidly changing dipole moment in the material is what allows these phonons to couple to
radiation, hence the Optical label.
Figure 4.1.4: Phonon dispersion (band) diagram of monolayer (red) and bulk (black) hBN. Adapted from
Mandelli, et. al. The various phonons are labeled according to whether they are Optical or Acoustic, and
Longitudial, Transverse, or in the Z direction. For normal incidence experiments, only the portions of the
bands around 𝚪 i.e. 𝒌 = 𝟎 are accessed. At over 200 meV, the optical phonons of hBN have one of the highest
known energies, making them easily accessible to our laser system.
50
At the beginning of the Chapter, we found that the anti-crossing that occurs between the
phonon dispersion bands and the photon dispersion light-line leads to a Reststrahlen band, a
region akin to a bandgap where light is unable to propagate. Each of the two sets of optical
phonon bands computed for Figure 4.1.4 also lead to their own Reststrahlen bands of
significantly higher reflectivity. Furthermore, these phonons in hBN have one more curious
property of naturally occurring hyperbolicity. Hyperbolicity in this context is the presence of
both positive and negative principle components in the material’s permittivity tensor [66]. Both
the hyperbolicity and Reststrahlen bands can be observed in a plot of the permittivity, defined
by:
𝜺(𝝎) = 𝜺∞(𝟏 +𝝎𝑳𝑶𝟐 −𝝎𝑻𝑶
𝟐
𝝎𝑻𝑶𝟐 −𝝎𝟐−𝒊𝝎𝜸
) ( 26)
Figure 4.1.5: Real parts of the permittivity along the z direction (into the crystal) and the transverse direction
(in-plae). The previously derived phonon branches correspond to strong peaks in the real permittivity. In the
regions where one component is positive and the other negative, the dispersion is hyperbolic. These regions
correspond to the Reststrahlen bands, shaded in gray.
51
where the high energy TO and LO transverse frequencies are 1370 and 1614 𝑐𝑚−1 and the lower
energy out-of-plane phonons have TO and LO frequencies of 760 and 825 𝑐𝑚−1, respectively.
The real parts of the permittivity and the corresponding Reststrahlen bands for the two phonons
are plotted in Figure 4.1.5. In Figure 4.1.6, we also see a clear example of the linear optical
effects of the Reststrahlen region: sharply enhanced reflectivity.
The hyperbolic permittivity is crucially important for applications demanding tight
confinement [67], and is one of the driving motivations for developing hBN as a nonlinear optics
platform. This is because the hyperbolic polaritons allow for propagation of the otherwise
forbidden energies in the Reststrahlen bands. Sub-diffraction confinement [67], negative
refraction [68], and hyperlensing [69,70] are all applications made possible by the strong
confinement. In the coming sections we will develop our own application, making use of the
tight optical confinement to generate large phonon amplitudes along the lines of Chapter 3. This
will lead to enhanced and novel nonlinear optical effects.
Figure 4.1.6: Enhanced reflectivity at the upper Reststrahlen frequency range of hexagonal boron nitride.
52
Section 4.2: Frequency Dependent Third Harmonic Enhancement From Phonons
We are now in a position to properly understand the nonlinear driving of the hBN
phonons, beginning from a simulated model of the anharmonic oscillations, straight through to
the phonon-resonant experiments. We first characterize theoretically the ionic displacements in
bulk hBN under resonant excitation with 25-fs FWHM pulses by performing time dependent
density-functional theory (TDDFT) simulated atomic oscillations spanning 200-fs, or roughly 8
times the theoretical pulse duration (see Figure 4.2.1). For a modest input intensity of 7 x 1010
W/cm2, we estimate that the phonon amplitude is 1% of the equilibrium lattice constant. While
the period of the lattice oscillation is 25-fs, which is consistent with the expected phonon
frequency, the relaxation time, described back in Chapter 3 cannot be theoretically determined
due to a lack of dissipative pathways in TDDFT. The amplitudes of atomic motion are plotted as
a function of pump intensity in Figure 4.2.1 (b). The displacements predicted by TDDFT
Figure 4.2.1: (a) Simulated atomic displacements of boron and nitrogen ions in TO (E1u) excited hBN. A 25-fs
FWHM, 1 x 1012 W/cm2 pulse excites the lattice dynamics. The TDDFT simulations do not include any
damping terms through which to estimate the relaxation time. (b) Peak amplitude of atomic displacements as
a function of pump intensity, fit to I1/2 with a small linear-in-intensity correction. Displacements nearing 5%
of the equilibrium lattice constant are achievable before the onset of damage.
53
calculations are fit by I1/2 with deviations appearing at large intensities and reach nearly 5% of
the equilibrium lattice constant (2.5 Å) [71] at 10 TW/cm2. The deviations strongly indicate the
presence of anharmonicity emerging in the simulation, and as we will now see, the expected
harmonic frequencies can be observed in the computed spectrum. The time-dependent electronic
current is extracted, and from this we generate the theoretical harmonic spectra employed
throughout this work. This process is repeated for wavelengths below, at, and above 7.3 μm, and
the integrated theoretical THG yields are plotted in Figure 4.2.2 (a) as green dots.
To perform the TDDFT simulations, the time evolution of the wave functions and the
evaluation of the time-dependent electronic current were computed by propagating the Kohn-
Sham equations in real space and real time, as implemented in the open source Octopus
code [14,72], in the adiabatic LDA [73] (the findings and trends discussed in the present work
are robust with different functionals) and with semi-periodic boundary conditions [72]. All
calculations were performed using fully relativistic Hartwigsen, Goedecker, and Hutter (HGH)
pseudopotentials [74]. The real-space cell was sampled with a grid spacing of 0.4 bohr and the
Brillouin zone was sampled with a 42 × 42 × 21 k-point grid, which yielded highly converged
results. The boron nitride bond length is taken here as the experimental value of 1.445 Å. The
laser was treated in the dipole approximation using the velocity gauge (that implies that we
impose the induced vector field to be time dependent but homogeneous in space), and we used a
sin-square pulse envelope. In all of our calculations, we used a carrier-envelope phase of f =
0 [75]. The full harmonic spectrum is computed directly from the total electronic current j(r, t)
as:
𝑯𝑯𝑮(𝝎) = | 𝑭𝑻(𝜹
𝜹𝒕∫𝒅𝟑𝒓 𝒋(𝒓, 𝒕))|
𝟐, ( 27)
54
where FT denotes the Fourier transform. The atomic vibrations of phonon modes are prepared
with the following two methods: (i) the time-evolution from a distorted atomic configuration
along the phonon modes of 1% of the bulk hBN lattice. (ii) application of pump laser pulses with
the same frequencies and polarizations of phonon modes. Our calculations confirm the two
methods are equivalent in the simulations of high-harmonic generation. TDDFT simulations
were performed primarily by members of the Rubio Group, Max Planck Institute.
Turning to the experimental side of things, we show the integrated and normalized THG
amplitudes for a range of pump wavelengths from 3 μm to 9.5 μm in Figure 4.2.2 as blue dots,
which are in excellent agreement with the calculations discussed in Figure 4.2.1 and plotted as
Figure 4.2.2: (a) Normalized third-harmonic generation yields of 120-fs pulses as a function of pump
wavelengths throughout the mid-IR. THG yields are below the noise level for all wavelengths less than 6 µm or
greater than 9 µm. Within a roughly 1 µm bandwidth of the phonon-polariton resonance, a THG
enhancement of 30x is observed. The black line is a Lorentzian fit to the data (blue dots) with full-width at
half-maximum of 500 nm. The green dots were obtained by integrating the third-harmonic signal in TDDFT
simulations and show excellent agreement with experiments. (b) Normalized intensity dependence of THG
pumped on-resonance at 7.3 µm and measured at 2.43 µm. The data is a close fit to I3, indicating that the
nonlinear process is scaling perturbatively.
55
green dots in Figure 4.2.2. The third-harmonic exhibits a strong peak for pump wavelengths near
the TO phonon resonance at λ = 7.3 µm. As this wavelength is far from any electronic or
excitonic resonances, the enhancement must be phononic in nature, described by the theory of
Chapter 3. We fit the data to a Lorentzian and extract a resonance full-width at half-maximum of
500 nm. THG yields are below the noise level for all λpump less than 6 µm or greater than 9 µm,
compared to that of the resonant signal which yields at least a 30-fold increase, and thus the
phononic enhancement of the THG coefficient at the phonon-polariton wavelength is
significantly greater than the purely-electronic component in this regime. In Figure 4.2.2 (b) we
plot the measured intensity dependence of the THG signal for λpump = 7.3 µm. The fit to a cubic
function indicates that the measured nonlinearity is third-order and that the scaling is
perturbative, even at high intensities [76]. We note that a similar effect has been observed in the
phononic second-harmonic generation of LiNbO3, which also remained in the perturbative
regime at higher-than-expected intensities. Also, we propose that subwavelength structures that
support confined phonon-polaritons [67,77] can drastically enhance the phonon-induced
nonlinearity in the same way that they dramatically enhance electronic nonlinearities.
Figure 4.2.3: Experimental setup for THG experiments. Detection is performed with PbS and MCT detectors,
a lock-in amplifier, and boxcar-averaging.
56
As previously mentioned, we performed the nonlinear experiments on high-quality hBN
flakes with thicknesses of 10 to 50 nm and typical sizes of tens of microns. The flakes are
exfoliated onto a CaF2 substrate, chosen for its high transparency in both the visible and mid-
infrared and its relatively small nonlinearity. Sample fabrication was performed primarily by
members of the Hone Group, Columbia University. For our long-wave infrared pump pulses we
utilize an optical parametric amplifier (OPA, Light Conversion HE Topas Prime) pumped by an
amplified Titanium-sapphire laser system (Coherent Legend Elite) operating at a 1-kHz
repetition rate with 6 mJ of pulse energy. The OPA produces 60-fs duration signal and idler
pulses with center wavelengths in the near-IR. The parametric amplifier output is then used to
seed an additional difference frequency generation (DFG) module for all mid-infrared
measurements from λpump = 3 to 10 μm with pulse durations ranging from 70- to 120-fs. Pulse
intensities are consistently set below the hBN damage threshold, which we estimate to be 50
TW/cm2. For THG experiments the pump is focused onto an hBN flake using a 2-cm focal
Figure 4.2.4: (a) HHG spectrum for the pump laser parallel with TO (E1u) mode. (b) HHG spectrum for the
pump laser parallel with LO (E1u) mode. Here, we employ an in-plane electric field with a wavelength of 𝝀 =
800 nm and an intensity of I = 1012 W /cm2, and a pulse duration of 25-fs full width at half maximum.
57
length CaF2 lens, and the emitted THG signal is collected in a transmission geometry by an
identical lens. After the residual pump beam is rejected by a short-pass filter, the remaining THG
is measured on a PbS detector for harmonic wavelengths λTHG below 1.7 μm, and on a liquid
nitrogen-cooled MCT detector for all λTHG greater than 2 μm. Figure 4.2.3 illustrates the premise
of the THG setup.
We also performed TDDFT simulations of the wavelength dependence of a higher-order
harmonic (HHG) spectra of bulk hBN. See Figure 4.2.4 for two different pump lasers with a
wavelength of λpump = 7.3 µm (polarized parallel to the TO mode) and λpump = 6.2 µm (polarized
parallel to the LO mode). Changing the wavelength and polarization of the pump laser can lead
to the excitation of different phonon modes and lead to significant modulation of the HHG
spectra. Excitation of either the TO or LO mode leads to noticeable modifications of the high-
harmonic spectra, with the TO (E1u) enhancement being one order of magnitude greater than that
caused by LO excitation. Furthermore, more intense laser pulses can introduce larger atomic
Figure 4.2.5: HHG spectra for different pump laser intensities. Here, the polarizations of pump and probe
laser are parallel with TO (E1u) mode (pump laser with a wavelength of 𝝀 = 7300 nm). For the probe laser, we
use an in-plane driving electric field with a wavelength of 𝝀= 800 nm and an intensity of I = 1012 W/cm2, and a
pulse duration of 25-fs full width at half maximum.
58
displacements and lead to larger nonlinearity. As seen from a pump intensity of 2.5 x 1010
W/cm2, the high-harmonic yields can be increased in a wide energy regime, and the high
harmonic generation plateau is enhanced (Figure 4.2.5), which is attributed to the increased
atomic movement and enhanced nonlinearity.
The content of this Section has served primarily as confirmation of the anharmonic model
of Chapter 3. By upgrading from a ball-and-spring model to the far more sophisticated
techniques of Density Functional theory analysis, we have gained further insight. This includes
the extension of the nonlinear enhancement to much higher harmonic orders, but also has the
potential to greatly expand our understanding of the combined electron-phonon coupling effects
in the intense strongly-coupled light-matter regime. Regardless, we come away from the third
harmonic experiment understanding that when driven on resonance, the THG due to lattice
anharmonicity can be far greater than the electron processes of decades past.
Figure 4.3.1: TDDFT simulations show the emergence of even-order nonlinearity during resonant excitation of
the TO phonon mode.
59
Section 4.3: Time- Resolved Even-Order Nonlinearity in hBN
The generation of optical harmonics in solids provides a window into the optical
susceptibility, band-structure, and underlying symmetries of crystals, each of which can
dramatically affect the nonlinear frequency-conversion process [2,16,78]. Symmetries, more so
than any other factor, dictate the allowed higher-order processes in a given nonlinear
system [79]. Following the discussions of lattice control outlined earlier in this work, hBN
presents us with a fantastic opportunity to explore specific instances of lattice motion away from
equilibrium.
Multilayer hBN has inversion (and 6-fold rotational) symmetry due to the natural 2H
stacking of its van der Waals structure [80]. As briefly described in Section 4.1, any contribution
to SHG in few- to many- layer hBN is therefore restricted only to the broken inversion symmetry
cases of interfaces and an odd number of layers and is inherently weak [64]. By conducting
ultrafast pump-probe experiments we show that the crystal inversion symmetry can instead be
Figure 4.3.2: Experimental setup for pump-probe SHG experiments. The time-delay is controlled by a
mechanical delay stage with sub-1 µm step size. The pump and probe are both focused onto the sample with a
reflective objective with 0.5 numerical aperture. Detection is performed with a silicon photomultiplier tube
and lock-in amplifier.
60
controllably broken by excitation of the IR-active TO (E1u) phonon, which leads to a finite
second-order susceptibility. For large amplitudes of the laser-driven lattice deformation, our
simulations reveal the emergence of an ultrafast, transient SHG signal from a secondary 800 nm
laser pulse, as shown in Figure 4.3.1 We distinguish this induced SHG from any nearby odd-
order processes by performing the calculations in the following two different ways: (i) with a 7.3
µm pump field, and (ii) using the time-evolution of an equivalently distorted lattice and no
resonant photons. In both cases the signal at harmonic order 2 emerges, confirming the principal
role of the broken symmetry. Furthermore, simulations confirm that in the case of the Raman-
active hBN TO (E2g) phonon which preserves inversion symmetry, even-order nonlinearity
cannot be observed.
For our SHG measurements we modify the experimental setup to a pump-probe scheme
with the addition of a 792 nm, 45-fs pulse from the same amplified Ti-Sapphire laser, shown in
Figure 4.3.2. A variable time-delay separates the 7.3 µm pump pulse that we use to excite the
phonon, from the near-IR probe which produces the SHG signal at 396 nm. The intensity of both
pulses is maintained at or below the (TW/cm2) range, which is below the hBN damage threshold.
The time delay between pulses is controlled by a mechanical delay line with sub-1-µm step size.
The polarization of the pump and probe beams are independently rotated with zero-order half-
wave plates and wire-grid polarizers before the pulses are combined on a beamsplitter. The
collinear pump and probe are focused onto an hBN flake using a reflective objective (NA = 0.5),
which ensures the same focal plane for the two beams with very different wavelengths. The SHG
signal produced by the 792 nm pulse can be collected either in the reflection geometry by the
61
reflective objective, or in transmission by a CaF2 lens. The signal is then directed through a
bandpass filter with a 10-nm bandwidth to reject the residual 792 nm and 7.3 µm light, as well as
any unwanted χ(3) signals, before detection on a fast photomultiplier tube (PMT) and lock-in
amplifier.
The experimentally measured SHG signal at 396 nm is presented in Figure 4.3.3 as a
function of the time delay between 792 nm and 7.3 µm pulses. The probe pulse from an
amplified Titanium-Sapphire laser and the pump pulse from a mid-infrared OPA and difference
frequency generation module are scanned in time by a mechanical delay line. The powers and
relative polarizations are set with filters and half wave plates (HWP), and the two beams are then
combined on a beamsplitter before being focused onto the sample by a reflective objective.
Figure 4.3.3: (a) Time-resolved SHG yield (normalized) of the 792 nm probe pulse. While the pumps are
temporally overlapped, an ultrafast second-order nonlinearity is measured. The inversion symmetry is
restored following a 200-fs time constant, or about twice the pulse duration. The appearance of wings in the
time-delay scan is a result of a non-perfectly Gaussian pulse, a result of strong atmospheric absorption. (b)
Zoom in of the region outlined by the dotted box in (a). (c) Fourier Transform of the time-series measurement.
The peak in the spectrum at 7.3 𝝁𝒎 confirms that the phonon is at least partly coherent with the field.
62
When the probe pulse precedes the pump pulse, no SHG is measured, indicating that the
interface SHG and odd layer-number contributions are below the noise floor. The time-resolved
SHG then displays a finite signal at the zero-time delay, when the probe pulse’s arrival coincides
with the strong excitation of the hBN phonon-polariton. The SHG signal relaxes back to zero
with a time constant of approximately 120 fs, which is approximately twice the pump pulse
duration. When pumped far off-resonance, no SHG is measured. The presence of fast oscillations
in the measured SHG on top of another transient but finite signal is indicative of a combination
of effects. Whereas the fast fs-timescale oscillations appear to be coherent with the excited
phonons (based on the observation that the Fourier transform of the time-resolved measurement
peaks at the pump wavelength), the constant pedestal suggests a second, rectified signal. This
can occur, for example, when the coupling of the IR-active phonon to a Raman-active mode
creates a new equilibrium position of the lattice atoms [81].
In Figure 4.3.4 (a) and (b) we show the dependence of the SHG yield on the intensity of
the probe and pump, respectively. A quadratic dependence of the SHG intensity on the probe
power is observed, as expected for a second-order nonlinear process. The linear scaling of the
Figure 4.3.4: (a) Dependence of measured SHG yield on probe power. (b) Dependence of measured SHG on
pump power. The SHG yield increases linearly with the phonon driving intensity.
63
SHG signal with respect to the mid-IR intensity in (b) implies a direct dependence of the
transient SHG process on the ionic displacements. These dependences match those reported for
SrTiO3, in which an SHG yield in low temperature experiments was found to steadily increase
over hours of total pump exposure and persist for hours after, with ps-scale modulations that also
increase in frequency with total exposure time [81]. Another similar effect with a very long
response time was observed in the naturally inversion-symmetry-broken van der Waals material
WTe2 [82]. However, in this case the effect of a shear strain and lattice displacement was to
eliminate the naturally present SHG, which is the opposite of our observed effect.
Section 4.4: Polarization Dependence of Second Harmonic Generation
In Section 4.3 we treated crystals of hBN as though they were perfectly uniform, making
no remarks on the rotational anisotropy of its crystal structure. This obviously couldn’t be further
Figure 4.4.1: Theoretical (left) and experimental (right) polarization dependent SHG. White dashed lines on
either plot represent the 60o rotational symmetry of a hexagonal lattice. The functional form of the
polarization dependence is given in the inset on the left. Here, 𝜽 and 𝝓 are the angles of the pump and probe
with respect to the ZZ axis, respectively.
64
from the truth, evidenced by our earlier discussion of the hexagonal structure of the hBN lattice.
Common nomenclature when discussing the high-symmetry directions of any hexagonal lattice
is to define two perpendicular axes, the so-called zigzag (ZZ) and armchair (AC) directions.
Looking at Figure 4.1.3, we would say that the y-direction is the ZZ axis, whereas the x-direction
must be AC. Obviously, these two axes periodically repeat. A 120o rotation of the ZZ axis will
return another ZZ axis, for example. The harmonic oscillator model used to discuss the THG of
ions on a theoretical basis did not leave room for a rotationally anisotropic restoring force. In a
real crystal of hBN, the AC and ZZ directions will feel different restoring forces, and hence will
have a substantial impact on the symmetries of measured harmonic emission. Next, we
establish the dependence of the ultrafast SHG on the orientation of the pump and probe
Figure 4.4.2: Linecuts from the full
polarization plot. Left (red linecut) shows the
typical behavior when the pump is polarized
along the AC direction. Right (blue linecut) is
when the pump is polarized along the ZZ
direction. No matter the pump polarization,
yields are largest along ZZ axes that are at
least partly excited by the pump. The yield
does not go identically to 0 along the AC axes,
perhaps because the slightly higher energy,
perpendicular phonon is weakly excited by the
tail of our pump spectrum.
65
polarizations with respect to the crystal high-symmetry axes. Figure 4.4.1 gives the total
normalized SHG yield for 360o rotation of both pulses (180o rotation is measured and the data is
then mirrored). We observe a polarization behavior unique from either the inherent 6-fold χ(2) or
isotropic χ(3) symmetries of purely electronic hBN nonlinearities [83]. Specifically, the emission
closely follows the functional form,
𝑺𝑯𝑮(𝜽,𝝓) = [𝜶 𝑪𝒐𝒔(𝟑𝜽)𝟐 + 𝜷 𝑺𝒊𝒏(𝟑𝜽)𝟐]𝑪𝒐𝒔𝟐(𝜽 − 𝝓), ( 28)
where θ and ϕ are the angles of the pump and probe relative to the zigzag (ZZ) axis of the
crystal, respectively, and α and β determine the relative strengths of the emission along the ZZ
and Armchair (AC) axes, respectively. This form was phenomenologically derived, based on the
following key observations: each of the AC and ZZ high-symmetry directions contributes a 6-
fold symmetric flower pattern to the SHG, just as one expects in a hexagonal lattice that
naturally lacks inversion symmetry such as [64]. The relative strengths of these two contributions
Figure 4.4.3: TDDFT computed high-harmonic generation spectra for pump and probe pulses co-polarized
along the ZZ (blue) and AC (red) directions. The TO(E1u) phonon present along the ZZ axes leads to the
greatest even order nonlinearity.
66
are left as fit parameter, to be compared with TDDFT theory later. Next, the SHG yield ought to
be greatest when the pump and probe are co-polarized, and could be expected to vanish when
cross-polarized. This leads to the final cos2(𝜃 − 𝜙) term at the end of the expression.
SHG yields peak only along ZZ axes that are being resonantly driven with a phonon-
polariton. This is most clearly visible in the linecuts of the probe polarization dependence for
pump fields aligned parallel to the AC and ZZ axes, given in Fig. 4.4.2. Even when the pump
excitation is aligned with an AC axis, the two adjacent ZZ oriented TO(E1u) phonons break the
inversion symmetry, and we observe SHG, whereas the ZZ axis at exactly 90o from that
excitation shows no emission. From Figure 4.4.2 we determine that the phonon-mediated SHG is
at least 3 times greater parallel to ZZ than AC directions. This is supported by time-dependent
density functional theory (TDDFT) simulations in Figure 4.4.3 which identifies even-order
nonlinearity along both symmetry axes, though much greater for the TO(E1u) phonon than the
relative π phase LO(E1u).
Section 4.5: Outlook on the Future of Nonlinear Phononics in hBN
Over the course of this Chapter we have extended the light-matter interactions confined by
the hyperbolic nature of the hBN phonon dispersion to a strongly nonlinear regime. In doing so,
we demonstrated that the large electron-phonon coupling leads to a nearly two order of magnitude
enhancement of SHG and THG. Efficient coupling of light to hBN phonon-polaritons at normal
incidence places stringent requirements on the allowed optical excitation wavelength. For the free-
space wavevector k = 0, the required photon wavelength of 7.3 µm is fixed, independent of flake
thickness [84]. With the energy-momentum conditions met, we have shown that efficient coupling
of optical energy into the hBN lattice leads to anharmonic driving of the atoms. Phonon amplitudes
can reach a few percent of the equilibrium lattice constant long before the onset of laser-induced
67
damage according to TDDFT simulations. The anharmonicity of the ionic motion leads to a novel,
enhanced nonlinearity, for which hBN is the most attractive platform in the mid-IR due to its
relatively light constituent atoms. Saturation of the THG yield below its perturbative cubic scaling
was not observed and is more likely to occur closer to the onset of sample damage. The extension
of nonlinear phonon enhancement to higher-order nonlinearities and to two-color high-harmonic
generation techniques will likely advance our understanding of electron-phonon coupling and
nonlinearly-driven lattices.
The experiments of this chapter clearly establish SHG as a sensitive probe for ultrafast
symmetry monitoring and eventually control in hBN, complementing earlier demonstrations on
broken-inversion-symmetric transition metal dichalcogenides, while reducing the relaxation time
scale by more than two orders of magnitude. It’s worth noting that the observed transient broken-
inversion symmetry and atomic displacement can also be interpreted as a strong photo-induced
strain field when the phonon mode is resonantly excited [85]. The promise of optically-
controllable strain in hBN lends itself to numerous other applications, from the tuning of
photoluminescence [86] to optical control of van der Waals heterostructures, for which hBN is a
common encapsulating material.
68
Chapter 5: Graphene Harmonic Generation and
Time- Reversal Symmetry Breaking
Graphene, a single-atom layer of carbon atoms arranged in a hexagonal lattice, has been
one of, if not the most hotly studied materials on the planet for 15 years running. First exfoliated
from its bulk form graphite by Geim and Novoselov [87] (who received the 2010 Nobel prize in
physics for their efforts), graphene has now earned its place as one of the most impressive two-
dimensional materials for its electronic [88], optical [89,90], mechanical [91], and thermal [92]
properties. To cover even a fraction of these would take (and surely has taken) many, many
dissertations. Therefore, our discussion of graphene will begin with an introduction only of its
more crucial electronic properties regarding its band structure and lack of a band gap, before
immediately moving on to its optical properties.
From there we will discuss an interesting discrepancy in the early literature on graphene
high harmonic generation, in which two seemingly identical experiments found completely
opposite optical behaviors. On the one hand, graphene appeared to behave like a solid as one
might expect [93]. And yet in the other experiment, graphene displayed all of the properties of
atomic (gaseous) harmonic generation that was characterized by the three-step model all the way
back in Chapter 1 [94]. The resolution of that discrepancy will take us down an extremely novel
path, where we will not only come to understand how both of these papers were in fact correct,
but graphene (the incredible system that it is) can be smoothly tuned from one of the two regimes
to the other.
Taking advantage of the incredible fabrication groups available to us at Columbia
university, we will then take a step beyond either of the two previous demonstrations. By
utilizing large single-crystals of exfoliated graphite, we will gain valuable insights into the
69
rotational and symmetrical behaviors of graphene in the context of elliptically pumped harmonic
generation. In a beautiful demonstration of time-reversal symmetry breaking caused by a chiral
optical field, we will be able to use third and fifth harmonic generation in graphene for two final
experiments: (i) to reveal spatial symmetries that should be entirely unmeasurable in harmonic
experiments and (ii) to probe a population imbalance of electrons in the so-called “valleys” of
the band structure. Though theoretically predicted (many times over), no experimental
measurements of valley polarization in carbon materials have been demonstrated prior to our
own experiments.
Section 5.1: An Introduction to Graphene
Electronic Properties
Like hBN, graphene is a material with a hexagonal lattice that belongs to the class of
easily-exfoliated van der Waals bonded crystals. Having only one species of atom (Carbon) and
a bond length of 14.2 Angstroms [95], graphene is one degree simpler in its structure than hBN.
For that reason, and because it will reveal one of the most crucial properties graphene offers, we
will go through the motions of estimating the graphene band structure. Rather than using either a
1D toy model or the far more complex Density Functional Theory approaches, for graphene we
will turn to the tight binding method for computing the band dispersions. Those readers with
some training in quantum mechanics will be familiar with the concept of the Bloch wave
functions, of which we will need two (𝜓𝐴 𝑎𝑛𝑑 𝜓𝐵), for two sublattices of graphene. We will
ultimately be interested in the total wave function
𝚿(𝐫) = 𝜶𝝍𝑨 + 𝜷𝝍𝑩 ( 29)
70
Consider N total atoms, with those in the A sublattice having positions 𝑟𝐴 and those in the B
sublattice having positions 𝑟𝐵. Then the two Bloch wave functions in terms of a sum of real
atomic orbitals 𝜙 are defined standardly as
𝝍𝑨(𝒓) =𝟏
√𝑵∑ 𝝓𝑨(𝒓 − 𝒓𝑨) 𝒆
𝒊 𝒌 𝒓𝑨𝒓𝑨 ( 30)
𝝍𝑩(𝒓) =𝟏
√𝑵∑ 𝝓𝑩(𝒓 − 𝒓𝑩) 𝒆
𝒊 𝒌 𝑩𝒓𝑩 ( 31)
Bloch’s theorem imposes a periodicity requirement on the total wavefunction:
𝚿𝐣,𝐧 (𝒓 + 𝑹𝒋) = 𝒆𝒊 𝒌 𝑹𝒋𝛙𝐀,𝐣 (𝒓) = 𝟏
√𝑵 𝒆𝒊 𝒌 𝑹∑𝝓(𝒓 − (𝒓𝒋,𝒏 − 𝑹)) 𝒆𝒊 𝒌(𝒓𝒋,𝒏−𝑹𝒋) ( 32)
Where the new subscript j indicates a Bravais lattice vector. The Bloch wavefunction must
satisfy the Schrodinger Equation to continue the derivation, where the total Hamiltonian is given
by the combinations of the two sublattices. In matrix form
Figure 5.1.1: Tight Binding Model estimation of the band energy dispersion of graphene, seen from above. The
6-fold symmetry of the lattice is reflected in its dispersion. Furthermore, some of the high-symmetry points
have been labeled, including the 𝚪 point at the Brillouin zone center and the surround degenerate K and K’
points. At the K and K’ points, the band gap between conduction and valence goes to zero, and the dispersion
is linear.
71
𝑯 = (< 𝝍𝑨(𝒓)|�̂�|𝝍𝑨
∗ (𝒓) > < 𝝍𝑨(𝒓)|�̂�|𝝍𝑩∗ (𝒓) >
< 𝝍𝑩(𝒓)|�̂�|𝝍𝑨∗ (𝒓) > < 𝝍𝑩(𝒓)|�̂�|𝝍𝑩
∗ (𝒓) >) ( 33)
and the change has been made to the bra-ket notation of Dirac for ease of viewing. We can
recognize that for a properly normalized wave function, the diagonal terms are identically 1. The
eigenvalues can be found by the solution of the characteristic equation of what remains:
det[𝐻𝑘 − 𝑆𝑘 𝐸𝑘] = 0. Here, the k subscript hints at the fact that our ultimate dispersion
equation will be found in terms of the wavevectors once again. 𝐸𝑘 are the corresponding energies
and 𝑆𝑘 are the wavefunction inner products of Equation (33). Two solutions can be found for the
energies now.
𝑬(𝒌) = ± 𝒉 √𝟏 + 𝟒𝐜𝐨𝐬 (𝟑
𝟐𝒌𝒙𝒂) 𝐜𝐨𝐬 (
√𝟑
𝟐𝒌𝒚𝒂) + 𝟒𝐜𝐨𝐬𝟐 (
𝟑
𝟐𝒌𝒙𝒂) ( 34)
Where h is the energy needed to hop to the nearest neighbor lattice site. The dispersion relation
Equation (34) is plotted top-down in Figure 5.1.1. Believe it or not, we’ve already stumbled onto
Figure 5.1.2: The three possible doping states of the graphene Dirac point. In the center, the undoped
“intrinsic” graphene has a Fermi level that is exactly at the Dirac point. If electrons are added to the system,
that Fermi level rises as in the n – doped case, and when charge is removed, the left-hand p – doped situation
occurs. The Fermi energy is defined in both doping scenarios as the energy difference from the Dirac point i.e.
the charge neutral point.
72
one of the great properties of graphene with this simple derivation. First, the hexagonal 6-fold
symmetry of the lattice is clearly reflected in the symmetries of the bands. In Figure 5.1.1, a few
of the significant high-symmetry points have been labeled, including the Γ point at the center of
the Brillouin zone, and the surrounding degenerate K and K’ points. At the K and K’ points, the
dispersion of the bands becomes completely linear, and the gap between the valence and
conduction bands completely closes. This point, know as the Dirac point, will be the source of all
of our interest in graphene from this point forward.
The reason for this extreme intrigue (for our research at least) is the way in which the
Dirac point interacts with the Fermi level of the graphene. The Fermi level (or Fermi energy) is
the energy level of the highest energy electron in the system. In an undoped graphene sample, the
Fermi level sits precisely at the Dirac point, where the conduction and valence bands meet. If
additional charge is added to the system, the Fermi level rises, a situation known as n – doping.
If the opposite were to occur, the Fermi energy would be lower, and that would be called p -
doping. All three of these are visualized in Figure 5.1.2. With that derivation completed we can
now move on to the optical properties of interest.
Optical Properties
We just showed in the previous Section that graphene lacks a band gap. However, for all
intents and purposes the presence of the easily doped Fermi level near the Dirac point gives us an
effective band gap to control. The Fermi level can be rapidly altered by applying a gate voltage
to graphene, though the exact speed and reproducibility of the doping will depend on the size and
quality of the sample, whether it is encapsulated, and other factors of the substrate and
environment. In order to see how the optical conductivity of the graphene changes as a function
of this doping level, the common theoretical treatment is to imagine graphene as a two-
73
dimensional conductive sheet. The general form of the graphene conductance [96] will not be
fully derived here. Instead, let us consider only the contributions to the optical conductivity due
to electrons that are trying to jump from the valence band up in to the conductance band. This
interband conductivity has been derived previously [97,98] and is given by
𝝈𝒊𝒏𝒕𝒆𝒓𝒃𝒂𝒏𝒅(𝝎) =𝒆𝟐
𝟒ℏ[𝟏
𝟐+
𝟏
𝝅𝐚𝐫𝐜𝐭𝐚𝐧 (
ℏ𝝎−𝟐𝑬𝑭
𝒌𝑩𝑻) −
𝒊
𝟐𝝅𝐥𝐧 (
(ℏ𝝎+𝟐𝑬𝑭𝟐)
𝟐
(ℏ𝝎−𝟐𝑬𝑭𝟐)
𝟐+(𝟐𝒌𝑩𝑻)𝟐
)] ( 35)
Where e is the electron charge, ℏ is as always, the reduced Planck constant, and 𝑘𝐵𝑇 is a
measure of the thermal energy, which at room temperature is much lower than the Fermi Energy.
The real and imaginary parts of the interband conductivity are plotted for a variety of normalized
Fermi Energies in Figure 5.1.3. As we can see, the real part of the conductivity undergoes a rapid
jump at twice the Fermi energy for all Fermi energies. There is a corresponding dip in the
imaginary part of the optical conductivity at these same distinct energies.
To understand these features of the optical conductivity, we refer back to the pictures in
Figure 5.1.2 defining the Fermi level. For optical experiments involving normal incidence
Figure 5.1.3: Real and imaginary parts of the optical conductivity of graphene due to interband electron
transitions. The energies have all been normalized to the Fermi level. There is a clear tendency for both the
real and imaginary parts of the conductivity to have a jump and an absolute minimum at twice the Fermi
energy, respectively.
74
excitation, where no dramatic change in momentum is possible for the electrons, only perfectly
vertical movements of electrons from the valence to conduction band are permitted. The
minimum energy difference that must be covered in order for the transition to occur is clearly
twice the Fermi energy. In situations where the energy of the photon is insufficient to promote
the electron up by 2𝐸𝐹, the interband transition is forbidden, and only intraband motion can be
taken into consideration. As we will soon see, this energy requirement, known as Pauli
Blocking [99], can also have profound effects on nonlinear optical processes.
Figure 5.2.1: (a) Ellipticity dependence of fifth harmonic generation in CVD grown graphene. The total
harmonic yield is shown for both monolayer and few-layer samples. The dashed line represents the expected
behavior of an atom undergoing the 3-step HHG process. The authors concluded based on this observation
that graphene had an atom-like behavior. (b) Ellipticity dependence of fifth harmonic generation in the same
CVD grown graphene (same vendor). Here the signal has been polarization resolved into components parallel
and perpendicular to the pump ellipse major axis. The sum of the two curves is the direct comparison to (a).
The authors claim a very much non-atomic behavior.
75
Section 5.2: Elliptically Polarized Excitation of Graphene
Graphene has recently emerged as an intriguing solid-state material for studies of HHG
with elliptically-polarized pump fields. The introduction of this Chapter alluded to a mystery that
has been swirling around this field ever since the publication of two fascinating works back in
2017. In one paper, graphene was shown to exhibit large enhancements in the harmonic yield for
finite pump ellipticities [93]. The ellipticity of a laser is defined as the ratio of its electric fields
perpendicular and parallel to the major axis of the ellipse. 0 ellipticity defines linearly polarized
light, and 1 corresponds to circular polarization. The magnitude, peak ellipticity, and ellipticity
range of the enhancement all vary with harmonic order. This behavior came to be associated
with the semi-metal type electronic structure of graphene. However, in an alternative experiment
under similar conditions [94], differing only by the pump wavelength, no such ellipticity
dependence from low-order harmonics was observed. Graphene in that case instead displayed
“atom-like” behavior with a monotonically decreasing yield versus pump ellipticity. Going all
the way back to Figure 1.2.3 (b), we derived this exact trend when considering atomic gases! In
order to emphasize the nature of this discrepancy, the relevant plots from either paper are
reproduced in Figure 5.2.1.
It so happens that graphene is not the only material to display large enhancements to its
harmonic yield at finite pump field ellipticities. In silicon and MgO [100,101] and ZnO [102],
the different high harmonic orders can be categorized into those displaying atomic-like behavior
and those with a far more non-trivial dependence on the polarization. The dependence which
primarily distinguishes these two regimes from each other is on the differing roles of the
interband and intraband electron dynamics, discussed at the end of Chapter 1 when making the
analogy between solid-state and atomic HHG. Relying on that same analogy, we can make a
76
guess as to the source of novel ellipticity dependence in solid-state HHG. Seeing as the biggest
difference in the two versions of the 3-step model comes in the form of intraband harmonic
contributions coming from the motion of electrons and holes, this is a natural place to look.
The TDDFT simulations necessary to explore the ellipticity dependence are beyond the
scope of this work. Though they have been performed by other groups in the past, to show all of
the results would require the copying of figures whose creation I was not involved in, and I have
done my best to avoid doing so. Instead, I will briefly summarize the findings:
(1) Lower-order harmonics- having energies up to the band gap energy of the studied
material- are far more likely to originate from interband electron-hole recombination.
Because the recombination process is not sensitive to the band dispersion, it is identical
for polarized pumps of either chirality. These harmonics therefore tend to have an
atomic-like, monotonically decreasing yield with ellipticity.
(2) Harmonics that have energies significantly larger than that of the band gap cannot be the
direct result of interband recombination effects. Instead, these emissions must be the
result of intraband anharmonic motion of the electrons and holes in the acceleration (2nd)
step of the 3-step model. The sensitivity of these frequencies to the specific shape of the
conduction band and its dispersion make the higher-order harmonics a sensitive probe of
the electron trajectory. The electron trajectory is going to depend on the ellipticity of the
pump, and hence a non-trivial dependence is found.
When thought of as a gapless material, monolayer graphene is not expected to show a
transition between intraband and interband dominated behavior with changing photon energies
77
the way a traditional semiconductor will. Graphene should, in principle, have interband
contributions allowed at every possible frequency. Hence the understanding we’ve just
developed for the HHG in solids really should be insufficient to resolve the experimental
discrepancies reported for graphene. In Section 5.1 we have shown that in fact, the tunability of
the graphene Fermi energy and the Pauli blocking behavior associated with it provides us an
effective band gap through which to try to understand the inter- and intra- band contributions, in
essence applying the observation in silicon, ZnO, and MgO to graphene.
We perform our experiments in the following way: Our samples are based on
commercially available monolayer graphene grown by chemical vapor deposition (CVD, the
same vendor as was used in all prior experiments). The CVD graphene is transferred to a
sapphire substrate with metallic source and drain contacts to measure the graphene resistance. In
order to control the Fermi energy of graphene, we use an electrolyte top gate to which a variable
Figure 5.2.2: Sample preparation for elliptically polarized excitation and harmonic generation. CVD graphene
is transferred to a transparent sapphire substrate. Elliptically polarized pump beams incident on the graphene
generate variable polarized third and fifth harmonics. Circular polarization can be achieved. Metal contacts
and an electrolyte gel allow for easy manipulation of the graphene Fermi energy.
78
gate voltage spanning +3V to -5V (limited by electrolyte degradation) can be applied. We pump
the graphene at normal incidence with 3-μm wavelength pulses at a repetition rate of 1 kHz, 80-
fs pulse duration, and 1 μJ pulse energy through a CaF2 lens. The beam is defocused such that
the intensity incident on the graphene is just below the threshold for optical damage. A zero-
order mid-infrared quarter-wave plate allows for control of the pump polarization. The generated
3rd harmonic at 1 μm and 5th harmonic at 600 nm are then collected in a transmission geometry
and analyzed on a visible- near-infrared spectrometer (Ocean Optics QE65 Pro). Neither the
sapphire substrate nor the electrolyte is observed to contribute a nonlinear signal in the
experiment. The experimental principle for the sample is shown in Figure 5.2.2. A block diagram
of the rest of the experimental setup is given in Figure 5.2.3.
The third and fifth harmonic signals generated from the graphene monolayer are plotted
in Figure 5.2.4 for three different gate voltages, each of course corresponding to a different
Figure 5.2.3: Basic experimental setup for generation of graphene harmonics. Femtosecond pulses are
generated by an amplified Ti-sapphire laser. The 795 nm wavelength pulses then pump an OPA and
difference frequency generation system to produce mid-infrared pulses. A quarter wave plate sets the pulse
ellipticity. An applied gate voltage sets the Fermi energy of graphene.
79
Fermi energy. We observe a clear trend towards enhancement at finite ellipticity as the gate
voltage is swept through the full range. For positive gate voltages, a linearly polarized input
produces the highest yield. At the largest negative gate voltage (highest Fermi energy), the 5th
harmonic yield is 22% greater for an ellipticity of 0.35 than for a linearly polarized input.
The shifted yield curve can be attributed to a change in the relative contributions of
harmonics emitted parallel and perpendicular to the major axis of the input ellipse as described
in [93]. A very nice consequence of this also happens to be that for a fixed input ellipticity, the
output ellipticity varies with gating. This result confirms our theory relying on the forbidden
inter-band excitations graphene possesses due to the Pauli-blocking of photons with energy less
than twice its Fermi energy [103]. For a lower Fermi level, the pump photon energies are
sufficiently large compared to twice the Fermi energy, making inter-band excitations possible.
For this case, the “atom-like” behavior due to high pump photon energy in [94] is dominant. As
the Fermi energy is increased to a point where Pauli-blocking of the excitations takes place, the
Figure 5.2.4: Third and fifth harmonics of a 3 𝝁𝒎 pump as a function of the ellipticity of the pump. Colors of
the curves correspond to three different gate voltages ranging from +3V to -5V. In all cases the graphene is n –
doped, and has the largest Fermi energy at the most negative voltage. Every curve is normalized to its own
maximum. In all cases, the peak yield shifts towards finite ellipticities with increasing Fermi energy.
80
enhancement at finite ellipticities observed due to low pump photon energy becomes more
apparent and the “non-atomic” case takes over.
This experimental result resolves the discrepancy between previously reported
measurements and establishes two unique regimes for harmonic generation in graphene. One is
an “atom-like” regime with a monotonic ellipticity dependence for which ℏ𝜔𝑝𝑢𝑚𝑝 > 2𝐸𝐹. The
other is a regime that graphene can be smoothly tuned into requiring ℏ𝜔𝑝𝑢𝑚𝑝 < 2𝐸𝐹 and
displaying rich intraband electron dynamics. Having a system that can cover all variations of this
behavior may find use as a powerful tool both for generating specific polarized harmonics at
will, and for understanding the interplay between inter- and intra- band processes.
The emergence of the finite ellipticity peak in the low photon energy regime reveals a
tunable, perpendicularly polarized harmonic component and we’ll benefit from being able to see
the separation of the two orthogonal components. Figure 5.2.5 resolves the parallel and
perpendicular polarization components of the THG for two extreme cases of graphene gating.
Figure 5.2.5: Polarization resolved THG of graphene for two extreme doping situations. Whereas the parallel
component of THG (co-polarized with the pump ellipse major axis) shows little variation with gating, the
perpendicular component can be greatly enhanced by raising the Fermi energy. At the point where the two
components cross, circularly polarized harmonics are obtained.
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The parallel component always appears to follow an atom-like monotonic trend, whereas the
perpendicular component can be greatly enhanced by raising the Fermi energy. Recent
theoretical progress [104] has drawn the same conclusions, based off of TDDFT simulations of
the various inter- and intra- band harmonic contributions. This simulation work performed by the
Angel Rubio group clearly shows that for a fixed input ellipticity, the perpendicular harmonic
component steadily increases with Fermi energy up to some critical value. The parallel
component shows a relatively minor change.
Finally, looking at Figure 5.2.5, and specifically at the crossing of the two perpendicular
components, we see that circularly polarized harmonics can be obtained from finite elliptical
pumping. This can be particularly useful, as we now have access to a tool for rapidly controlling
the output polarization of harmonics via electrostatic gating of the nonlinear medium.
Figure 5.3.1: (a) Representative example of CVD grown graphene, containing many randomly oriented ~1𝝁𝒎
sized grains of graphene. (b) Single crystal exfoliated graphene flake measuring 10s of microns x 10s of microns.
The exfoliated sample has one single orientation of atoms.
82
Section 5.3: A Brief Review of the Role of Point-Groups of Symmetry in Harmonic
Generation from Solids
All of the experiments that have been conducted on graphene harmonic generation thus
far (including our own) have taken place on samples grown by chemical vapor deposition. Figure
5.3.1 compares the arrangement of the CVD graphene lattice (composed of small, randomly
oriented zones) grown in [105] to that of a single crystal exfoliated sample prepared by
collaborator to this work Sang Hoon Chae of the Hone lab. The difference is stark: CVD grown
graphene consists of very many randomly oriented “grains” of graphene measuring only about
1 𝜇𝑚 in size. By comparison, exfoliated graphene (and graphite) is a single crystal, all oriented
in the same direction, with average sizes on the order of 10s of microns. Consider the typical
HHG experiment in which a laser spot size is easily on the order of 10s to 100s of 𝜇𝑚. It is clear
that when using CVD graphene samples many different orientations of graphene will be
contained within a single spot area. Any information that may be contained within the orientation
of the graphene will be averaged over the whole area, and much information will be lost.
The loss of orientation information is felt particularly hard in the context of harmonic
generation, where the crystal point-group symmetry plays the deciding role in the emission
symmetries of all harmonic orders [2]. Building on the very earliest description in this
dissertation, Section 1.1, we now briefly describe the role of symmetry in low-order even and
odd harmonic generation.
Going a step beyond a simple inversion-symmetry argument, we finally circle back to
give the nonlinear susceptibilities 𝜒(𝑛) of Chapter 1 the full tensor treatment they deserve. In
order to prevent these next few pages from getting completely out of hand, we will make use of
83
the more concise contracted notation for the susceptibilities. In this notation, the second order
susceptibility tensor is a 3 x 6 matrix:
𝒅𝒊𝒍 = [
𝒅𝟏𝟏 𝒅𝟏𝟐 𝒅𝟏𝟑𝒅𝟐𝟏 𝒅𝟐𝟐 𝒅𝟑𝟑𝒅𝟑𝟏 𝒅𝟑𝟐 𝒅𝟑𝟑
𝒅𝟏𝟒 𝒅𝟏𝟓 𝒅𝟏𝟔𝒅𝟐𝟒 𝒅𝟐𝟓 𝒅𝟐𝟔𝒅𝟑𝟒 𝒅𝟑𝟓 𝒅𝟑𝟔
] ( 36)
and anything higher than second order is too large and too high in dimension for me to want to
write out in matrix form. We can immediately begin to make arguments based off of various
symmetries and conditions to reduce the number of unique and independent terms in 𝑑𝑖𝑙. First,
the permutation symmetry known as Kleinman’s symmetry [106] reduces the matrix
representation to
𝒅𝒊𝒍 = [
𝒅𝟏𝟏 𝒅𝟏𝟐 𝒅𝟏𝟑𝒅𝟏𝟔 𝒅𝟐𝟐 𝒅𝟑𝟑𝒅𝟏𝟓 𝒅𝟐𝟒 𝒅𝟑𝟑
𝒅𝟏𝟒 𝒅𝟏𝟓 𝒅𝟏𝟔𝒅𝟐𝟒 𝒅𝟏𝟒 𝒅𝟏𝟐𝒅𝟐𝟑 𝒅𝟏𝟑 𝒅𝟏𝟒
]. ( 37)
From here, it is necessary to consider the specific spatial symmetries of the crystal in order to
further reduce the elements of 𝑑𝑖𝑙 from the 10 independent elements under Kleinman’s symmetry
to as few as 3 independent elements in some cases. We must therefore specify a point symmetry
Figure 5.3.2: Rotational symmetry of second harmonic generation copolarized with pump, for crystals of
symmetry point group D3h. There is no clear comparison for the higher symmetry D6h point group representing
graphene, as the centrosymmetry of the lattice forbids SHG.
84
group before we can take this process any further, and once doing so, we may calculate the
polarization according to
[
𝑷𝒙
𝑷𝒚
𝑷𝒛
] = 𝟑𝝐𝟎 [
𝒅𝟏𝟏 𝒅𝟏𝟐 𝒅𝟏𝟑𝒅𝟏𝟔 𝒅𝟐𝟐 𝒅𝟑𝟑𝒅𝟏𝟓 𝒅𝟐𝟒 𝒅𝟑𝟑
𝒅𝟏𝟒 𝒅𝟏𝟓 𝒅𝟏𝟔𝒅𝟐𝟒 𝒅𝟏𝟒 𝒅𝟏𝟐𝒅𝟐𝟑 𝒅𝟏𝟑 𝒅𝟏𝟒
]
[
𝑬𝒙(𝝎𝟏)𝑬𝒙(𝝎𝟐)𝑬𝒚(𝝎𝟏)𝑬𝒚(𝝎𝟐)
𝑬𝒛(𝝎𝟏)𝑬𝒛(𝝎𝟐)
𝑬𝒚(𝝎𝟏)𝑬𝒛(𝝎𝟐) + 𝑬𝒛(𝝎𝟏)𝑬𝒚(𝝎𝟐)
𝑬𝒙(𝝎𝟏)𝑬𝒛(𝝎𝟐) + 𝑬𝒛(𝝎𝟏)𝑬𝒙(𝝎𝟐)
𝑬𝒙(𝝎𝟏)𝑬𝒚(𝝎𝟐) + 𝑬𝒚(𝝎𝟏)𝑬𝒙(𝝎𝟐)]
( 38)
For this to have any real meaning at all, let’s look at a specific example common to 2D materials
such as monolayer hBN and various 2D transition metal dichalcogenides. These materials often
have point group symmetry D3h, for which every element in 𝑑𝑖𝑙 vanishes with the exception of
yyy, yxx, xxy, and xyx, which are all of the same magnitude (different signs though)! From
there, the calculation of the polarization and greatly simplified, and using equations of an
arbitrarily polarized electric field, with no z-component and x and y components given by
𝐸 cos(𝜃) and 𝐸𝑠𝑖𝑛(𝜃), we find that the rotational symmetry of the second harmonic radiation
(parallel to the pump polarization) emitted from a material of this spatial symmetry group goes
Figure 5.3.3: Rotational symmetry of third harmonic generation copolarized with pump, for crystals of
symmetry point group D6h. There is no symmetry information present in the perfectly isotropic signal.
85
as cos2(3𝜃), depicted in Figure 5.3.2. We will be seeing many such patterns over the course of
the rest of this Chapter, so we give the 6-fold rotational structure of Figure 5.3.2 the nickname of
a “flower” pattern.
The higher symmetry D6h group that materials like graphene belong to cannot be
compared to the cos2(3𝜃) shape we’ve shown, since the inversion symmetry of graphene’s
lattice forbids the second order nonlinearity. Therefore, we must go up one order of nonlinearity
to 𝜒(3). The third-order nonlinear susceptibility has 81 terms. To write them all out just to study
a single material would not add much to our understanding beyond the derivation we’ve
performed for the second order. Instead, let’s just jump ahead to the plot of the co-polarized
THG from a centrosymmetric hexagonal crystal such as bulk hBN or graphene. Figure 5.3.3
shows that in fact, the THG in such a system is perfectly isotropic, revealing no information
about the rotational symmetries of the underlying structure.
It is apparent from the above discussion that for lower symmetry crystals, polarization-
resolving harmonics can lead to a wealth of information about the lattice. As it so happens,
orientation-resolved SHG has become one of the most common tools for determining the lattice
orientation of thin samples. In 2D transition metal dichalcogenides (TMDs), polarized SHG is
greatest along the AC crystallographic axis, and due to the 𝐷3ℎ point group symmetry of most
TMDs, it is periodically repeating. SHG is then perfectly applied for measuring orientation with
exceptional resolution [107]. Graphene, a centro-symmetric material in both its monolayer and
few-layer forms, has its SHG and other even order nonlinearities forbidden. Furthermore, as a
member of the 𝐷6ℎ point symmetry group, graphene’s low odd-order (third and fifth) harmonics
are perfectly isotropic as seen in Figure 5.3.3. Knowing this, the results of the next and final
86
experimental section will be very surprising, as we demonstrate the emergence of the 6-fold
rotational symmetry in these lower-order odd nonlinearities in graphene.
Section 5.4: Time- Reversal Symmetry Breaking and Valley Polarization Effects
Experimental Setup
In order to go from the non-orientation dependent measurements of Section 5.2 to a new
setup with better control and higher signal to noise ratio, we make some significant changes to
the experimental design. Our new setup utilizes a much higher 1 MHz repetition rate optical
parametric amplifier (OPA) pumped by a Ytterbium fiber laser (KM Labs/ Thor Labs Y-Fi OPA)
centered at 3 µm wavelength, and with pulse durations of 60 fs. A combination of a half wave
plate (HWP) and quarter wave plate (QWP) set the ellipticity of the pump beam while
maintaining a constant ellipse orientation. Single crystal graphene flakes (including monolayer,
up toa10s of nm thickness) are exfoliated onto a 500 µm thick sapphire substrate, chosen for its
high transparency at the pump wavelength and relatively low nonlinearity. The sample is
Figure 5.4.1: Orientation dependent 5th harmonic generation in few-layer graphite sample at three different
pump ellipticities. Red points and curves show the harmonic component parallel to the pump ellipse major axis.
Blue is measured by an analyzer in the perpendicular direction. Each plot is normalized to its own parallel
component.
87
mounted on a rotating holder so that its orientation with respect to the incident laser can be easily
varied. The mid-IR pump beam is focused onto the graphite flakes with a 10 cm focal-length
CaF2 lens, and the generated third and fifth harmonics are collected in a transmission geometry
by a 5 cm UV-fused silica lens. A visible-wavelength polarization analyzer selects for parallel or
perpendicularly oriented harmonics before the signals are measured on a thermo-electrically
cooled ultraviolet-visible spectrometer (Ocean Optics) with a 350 to 1100 nm detection range.
Peak pump intensities are maintained below the regime of optical damage.
Measurements
The surprising results of the first ever orientation dependent 5th harmonic generation
experiment in multilayer graphene are presented in Figure 5.4.1 for different pump ellipticities.
This few-layer graphite sample, when pumped with perfectly linearly polarized light, has an
isotropic odd-order harmonic process predetermined by its 𝐷6ℎ symmetry group. In addition, no
cross-polarized signal can be measure for 0 ellipticity. However, as the ellipticity of the pump is
increased, a modulation of the uniform harmonic pattern emerges. What’s more, that modulation
Figure 5.4.2: Modulation depth of emergent “flower” symmetry in graphite 5HG vs pump ellipticity. Linear
best-fit also shown.
88
quickly converges to the recognizable 6-fold “flower” symmetry pattern of lower-symmetry
materials of different harmonic orders. The modulation depth as a function of the pump
ellipticity is plotted in Figure 5.4.2, and appears to follow a linear dependence. One further key
observation of Figure 5.4.1 beyond the emergent symmetry is the slight rotation of the flower
Figure 5.4.3: Orientation dependent 5th harmonic generation in few-layer graphite sample at three different
pump ellipticities. The sample used here is thicker than that in Figure 5.4.1. and a greater modulation depth is
observed for the same ellipticities.
Figure 5.4.4: Orientation dependent 5th harmonic generation in few-layer graphite sample at three different
pump powers and a fixed ellipticity of 0.4. The relative size of the blue perpendicular harmonic component
slightly increases.
89
with increasing modulation. This rotation will be a key supporting element when we come to
understand the origin of this rotational periodicity.
To round out the experimental observations before jumping into the theoretical
discussions, a similar ellipticity and orientation sweep on a thicker graphite flake is presented in
Figure 5.4.3, and the pump power dependence is investigated in Figure 5.4.4. The thickness has a
dramatic effect on the modulation depth at a given ellipticity, indicating a strong dependence on
the bandgap, which opens in graphite when compared with gapless graphene. The power
dependence presents itself primarily in the perpendicular component of the 5th harmonic.
Theoretical Explanation of Emergent Flower Symmetry
To properly understand the emergence of the flower symmetry pattern in thin graphite
samples under elliptical excitation, we must briefly describe an emerging field of 2D materials
and optics: that of valleytronics [108–113]. Recall the calculation of the graphene dispersion in
Figure 5.1.1 and Equation (34). At the time, some of the high-symmetry points of the Brillouin
zone were given names, in particular the K and K’ points. In an equilibrium system with time-
reversal symmetry, the K and K’ points are degenerate and equivalent. A look below the surface
reveals, however, that the two classes of points display opposite Berry curvatures [114]. A
proper derivation of the Berry curvature involves taking the curl of the Berry vector potential:
𝑨(𝒌) = 𝒊 < 𝝍|𝛁𝐤|𝝍 > ( 39)
Where 𝜓 recalls the Bloch functions that went into deriving the dispersion relations. Jumping
ahead past the derivation (which is beyond our scope), the Berry curvature which is defined in
momentum space plays the same role that a magnetic field does in real space. Hence the
opposite Berry curvatures of the K and K’ points can be thought of as opposite sign magnetic
fields in momentum space at the two types of points. There is an immediate and powerful
90
consequence of the analogy to magnetic fields: despite being degenerate under normal
conditions, the K and K’ “valleys” (local minima regions in the momentum space) are time-
reversal opposites of each other. This provides a neat way to lift the degeneracy by breaking
time-reversal symmetry!
It may seem as though we have strayed quite far away from our initial goal of
understanding the emergent flower symmetry of graphene 5HG. On the contrary, we are now
armed with a powerful new tool. The 6-fold rotational symmetry of graphene and graphite can be
decomposed into two sublattices in momentum space, with each of the K and K’ lattices
contributing a 3-fold rotationally symmetric trigonal lattice. In total, graphene will still belong to
the D6h symmetry group, and all contributions from the two valleys must sum to the equivalent
properties of D6h, but we have a new direction of attack to work from.
Figure 5.5.1 depicts our proposed theory. Each of the K and K’ sub-lattices in momentum
space display a 3-fold rotational symmetry. The odd-order harmonics have a 6-fold rotational
flower pattern for the 3-fold symmetry groups as seen for example in the experimental work
Figure 5.5.1: Decomposition of the graphene D6h 5HG rotational symmetry into the sum of two lower-dimension
“flower” patterns from the K and K’ valleys. Each of the K and K’ valleys has trigonal symmetry, and they are
related to each other by a 90 degree rotation.
91
done on sapphire and silicon [115]. Due to the natural 90-degree rotational relationship between
the two valleys, their independent contributions are oriented with a 𝜋 phase between them. Just
as the sum of sin2 and cos2 is 1, so too do the contributions of the two sublattice return the
isotropic, unmodulated harmonic emission of Figure 5.3.3.
This is all well and good, but unless we are capable of changing one of these two valley
contributions with respect to the other (in essence lifting the degeneracy) we will not be able to
explain the experimental observations. As previously stated, the well-established method for
lifting the K and K’ degeneracy is by breaking time reversal symmetry. It has been shown [116]
that chiral photons are sufficient to break time reversal symmetry, selectively driving electrons
into either of the two valleys. The now- famous valley hall effect is a consequence of circularly
polarized pumping for example [116]. One valley can be selectively pumped with right-hand
chiral excitation, while the opposite valley requires left-handedness. By driving electrons
selectively into one type of valley, a population imbalance is generated and the uneven
distribution of charge leads to a new polarization, called the valley polarization (VP) that
depends on that population difference.
Figure 5.5.2: The addition of a nonzero valley polarization to the total 𝝌𝟐 leads to a 6-fold symmetric flower that
is slightly rotated with respect to the intrinsic second order nonlinearity. The direction of the tilt is determined
by the phase of the valley polarization term, which is experimentally set by the handedness of the chiral photon
used to excite either the K or K’ valley.
92
Many reports have been published on the role of the valley polarization in allowing
second harmonic generation from graphene [117–119] and TMDs [120,121], though there have
not been any corresponding experiments performed in graphene or few-layer graphite. In the
context of SHG, VP contributes a new susceptibility term 𝜒𝑉𝑃 in addition to 𝜒𝑖𝑛𝑡, the intrinsic
susceptibility. This generalizes the matrix equation in Section 5.3 according to
[𝑷𝒙
𝑷𝒚] ∝ [
𝝌𝑽𝑷 −𝝌𝑽𝑷 −𝝌𝒊𝒏𝒕−𝝌𝒊𝒏𝒕 𝝌𝒊𝒏𝒕 −𝝌𝑽𝑷
] [
𝑬𝒙𝟐
𝑬𝒚𝟐
𝟐𝑬𝒙𝑬𝒚
]. ( 40)
The change that this new term induces on the symmetry of the 6-fold flower pattern for SHG that
was derived above for TMDs and other D3h materials is shown in Figure 5.5.2. As we can see,
the new contribution to the nonlinearity induced by 𝜒𝑉𝑃 is 𝜋 out of phase with the intrinsic
susceptibility. The sum of these two components squared manifests as a small rotation of the
overall measured 6-fold pattern, where the size of the rotation is determined by the size of the VP
contribution, and the direction of the rotation depends on the valley that’s been selectively
excited.
Figure 5.5.3: When each of the K and K’ 5HG contributions are altered with the addition of a VP, they rotate
in opposite directions. The grayed-out shapes show the total and component-wise contributions to the nonlinear
signal before valley polarization is induced. The black, red, and blue curves include VP terms. The 6-fold flower
begins to appear as the opposite rotations modulate the total signal.
93
We now address the question of how to control the degree of valley polarization. [120]
shows that for SHG in MoSe2 (a transition metal dichalcogenide), the degree of pump ellipticity
is directly proportional to the measured rotation of the 6-fold flower, and thus ellipticity is a
natural knob by which to control the degree of valley polarization. Hopefully now all of the
puzzle pieces are beginning to fit into place, and the final steps of our phenomenological
derivation are becoming clear. We will apply the concept of the rotation of the 6-fold flower to
higher-order nonlinearities, and observe how the 5th harmonic evolves.
To each of the spatially mirrored K and K’ harmonic contributions, we add a new term,
their respective valley polarizations, which we will assume for simplicity’s sake are equal in
magnitude and opposite in sign to each other. To write out the 6th rank tensor required to derive
the 5HG polarization components would be a crazy undertaking, so we will instead plot out the
components again in Figure 5.5.3. The K and K’ flowers undergo a slight rotation as a result of
VP, just as in the SHG case experimentally demonstrated in TMDs. The two flowers (which are
time-reversal and spatial mirrors of each other) rotate in opposite directions, bringing their petals
Figure 5.5.4: Total co-polarized 5HG for three different values of the valley population, which is experimentally
controlled by the laser ellipticity. The rotation of the flower is also recovered as a consequence of unequal
contributions from the two valleys.
94
closer together. The result is a modulation of the total, symmetric D6h harmonic emission,
causing it to reproduce the experimental observations of Figure 5.4.1.
The final experimental observations that remain to be accounted for are the dependence
on ellipticity as well as the slight rotation that seems to occur with increasing modulation depth.
Figure 5.5.3 is produced by the total function:
5𝐻𝐺(𝜃) = 𝛼 (cos(3𝜃) +𝜒𝑉𝑃
𝜒𝑖𝑛𝑡sin(3𝜃))
2
+ 𝛽 (sin(3𝜃) +𝜒𝑉𝑃
𝜒𝑖𝑛𝑡cos(3𝜃))
2
( 41)
where 𝛼 and 𝛽 determine the relative contributions from the two valleys and the VP contribution
is some small fraction of the total, determined by the ratio of the susceptibilities. Figure 5.5.4
plots three different values of that ratio as well as the situation 𝛼 > 𝛽. The unequal contributions
follow from the fact that the population of one valley is greater than the other, hence its flower
has a larger yield. The ratio 𝜒𝑉𝑃/𝜒𝑖𝑛𝑡 can be controlled with the laser ellipticity, with a nearly
linear dependence. The Figure confirms the experimentally observed modulation depth as a
function of ellipticity. Furthermore, the inequality of 𝛼 and 𝛽 reproduces the gradual turning of
the flower. Valley polarization effects are therefore shown to explain all of the fascinating
observations.
There are tremendous implications associated with the developments of this last section.
Although emergent SHG has already been proposed as a means of measuring valley polarization
in graphene, no experimental demonstration has been done. This is likely due to the fact that
graphene lacks any inherent SHG, and the contribution from valley effects may be too small to
easily measure. We’ve greatly extended this theory to include higher, odd-order nonlinearities
including fifth harmonic generation, and confirmed that VP makes a significant, easily
measurable change to the harmonic emission. This change is exerted on the rotational profile,
creating modulations as well as rotations of a few degrees.
95
Future efforts on this experiment can go in a number of interesting new directions. On the
one hand, the exact degree of valley polarization needs to be extracted from one of our harmonic
plots. In addition, twisted graphene and graphite samples, TMD homostructures, or
heterostructures containing a combination of materials are likely to have an inherent valley
polarization [122]. Using 5HG and linearly polarized light, the presence of modulations in the
rotational profile may provide a method of extracting the inherent VP with a straightforward all-
optical method.
96
Conclusions
The role of symmetry in nonlinear optics can present itself in any number of ways,
forbidding certain processes while dramatically enhancing others. Control over various
symmetries, then, becomes one of the most useful tools in engineering the next-generation
platforms of optical experiments. Low- and high- order harmonic generation are ideal candidates
for exploring the impact of symmetries on nonlinear processes. The strong scaling of high
harmonics with power greatly emphasizes resonant enhancements and tight confinement due to
symmetry breaking. Spatial symmetries and time-reversal become linked to each other in the
context of harmonic generation, and forbidden processes are deeply linked to material symmetry
groups.
In the first main experimental Chapter of this thesis, spatial symmetries were leveraged to
manufacture resonances call quasi-bound states in the continuum. Perfectly periodic structures
are capable of hosting modes with odd symmetry properties in a continuum of even symmetry
states. Being unable to couple energy in or out of that continuum, the odd symmetry state is
bound and inaccessible. We showed in Chapter 2 that by making a periodic perturbation to the
periodic unit cell, we are able to introduce coupling between the even symmetry incident field
and the mode. The size of the perturbation directly determined the quality factor (life time) of the
new resonance, while the materials and duty cycle selected the wavelength. In our particular
system which consisted of silicon-on-insulator gratings, we took further steps towards resonance
control. Understanding that the periodic perturbation of the grating effectively folds the Brillouin
zone in half, we used that knowledge to select for a mode located in the grating gaps. This
overcame two of the great shortcomings of recent semiconductor-based and plasmonics-based
nonlinear enhancement metasurfaces: damage and signal absorption. We proved the
97
effectiveness of the field confinement within the empty volume by performing third and fifth
harmonic generation on argon gas. The observed wavelength, polarization, and pressure
dependences confirmed all of the design principles of the dimerized high-contrast gratings.
From there, we tackled inversion symmetries in the two-dimensional material hexagonal
boron nitride. Being an inversion-symmetric crystal in equilibrium, hBN is forbidden to show
even-order nonlinearities in most cases. We developed a simple but robust theory of anharmonic
motion of lattice ions based on a perturbed harmonic oscillator model from which we were able
to generate new frequency components from ionic displacements. From there, a more
sophisticated theory based on time-dependent density function theory showed the exact response
of an hBN lattice to incident femtosecond pulses. These pulses, when tuned to the hyperbolic
phonon-polariton frequency of hBN, had two incredible impacts. First, the odd-order harmonics
generated by the driven atoms had yields far exceeding the more standard electronic
contributions. Second, the properties of an IR-active phonon, necessitating the creation or change
of a dipole moment in the material, allowed us to transiently break inversion symmetry. That
symmetry breaking was perturbed by a secondary laser pulse from which we were able to time-
resolve ultra-fast second order nonlinearities. There are many implications of this sub-
picosecond phase change in hBN, the applications of which may greatly impact van der Waals
heterostructures.
The final symmetry explored in this dissertation was time-reversal symmetry. We began
with a motivation to resolve an experimental discrepancy in the literature for elliptically driven
harmonic generation in graphene. By demonstrating that graphene could be put in an atomic or
non-atomic state by controlling the ratio between the pump energy and the Fermi level, we
definitively resolved that open question. In the end, it’s now becoming well known that the novel
98
ellipticity dependence originates from intra-band harmonic generation, and the degree to which
intra-band harmonics contribute can be tuned with Pauli blocking. Following those experiments,
we were presented with an opportunity to extend the field towards orientation sensitive
measurements due to our pristine single-crystal samples. What we found was surprising:
isotropic nonlinearities (with respect to polarization) display periodic modulations when time-
reversal is broken by chiral light. These modulations point us towards two extremely useful
future experiments. On one hand, we are presented with a simple, all-optical means of measuring
graphene’s lattice orientation in a way analogous to second harmonic spectroscopy used in
lower-symmetry 2D materials. Second, we have built out a theory understanding that the
emergence of the modulations in the orientation profile is a direct result of valley polarization.
Valley polarization measurements have been proposed in graphene SHG experiments, but have
not yet been successfully demonstrated. Using much larger nonlinearities such as the intrinsic
third and fifth, we have established a superior all-optical means of probing valley populations in
few-layer graphene samples.
Three seemingly unrelated material systems have been explored in this work, each
demonstrating the ways in which harmonic generation is inseparable from symmetry. In some
cases, it was symmetries that were applied as a way of enhancing the harmonics, while other
times the harmonics were applied as a means of understanding deep physical concepts in the
materials. I hope this work has proved to you that not only is harmonic generation one of the
longstanding pillars of nonlinear optics, but that it has much, much more to offer with each new
cutting edge material or platform to be developed.
99
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