Ultrafast Infrared Nano-Imaging of Strongly Coupled
Phonon and Carrier Dynamics in Quantum Materials
bySamuel C. Johnson
Bachelor of Science, Ohio University, 2015

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021

Committee Members:
Prof. Markus B. Raschke
Prof. Henry Kapteyn
Prof. Daniel S. Dessau
Prof. Ana Maria Rey
Prof. David JonasProf. Rafael Piestun

iiJohnson, Samuel C. (Ph.D., Physics)
Ultrafast Infrared Nano-Imaging of Strongly Coupled Phonon and Carrier Dynamics in Quantum
Materials
Thesis directed by Prof. Markus B. Raschke

Infrared (IR) vibrational scattering scanning near-field optical microscopy (s-SNOM) has
advanced to become a powerful nano-imaging and -spectroscopy technique to probe molecular andlattice vibrations, low-energy electronic excitations and correlations, and related collective surface
plasmon, phonon, or other polaritonic resonances. In combination with scanning probe microscopy,
near-field infrared nano-spectroscopy and -imaging enables the study of complex heterogeneous materials with simultaneous nanoscale spatial resolution and quantum state spectroscopic specificity.
IR s-SNOM has also been extended to studying dynamics in the time domain, where ultrafast vibrational and electronic spectroscopy unravels mechanisms underlying functionality in quantum materials. Femtosecond-to-picosecond dynamics are convolved with multiscale spatial heterogeneities,
ranging from microscopic defects to the macroscopic domain level. This technique opens the door
to elementary processes and interactions in functional materials with full spatio-temporal-spectral
resolution.
In this thesis, I will describe my work in advancing s-SNOM through the design, testing,and implementation of ultrafast hyperspectral imaging modalities on both molecular and quantum
systems.
Following the introduction, I will summarize the fundamentals of s-SNOM as advancements
detailed here are based on these starting points. Then, I will detail three distinct method advancements made to improve the image quality, acquisition speed, and information content. By utilizing
knowledge of the resonance being probed, the bandwidth of the excitation source, and the desired
spectral resolution, I can transition from the stationary frame to the rotating frame to reduce the
Nyquist maximum frequency cut-off. With the necessary maximum frequency reduced, fewer data

iii
points are needed along the reference arm axis compared to conventional hyperspectral imaging, enabling faster data acquisition of a higher resolution image over a higher field of view with automatic
drift correction. I further have theoretically demonstrated that prior knowledge about the limited
frequency range of the optical source in a nano-FTIR measurement and the fact that samples are
typically composed of only a few different molecular constituents allows for faster detection algorithms. Specifically, compressive sensing and matrix completion techniques can be used to reduce
the necessary data points from the Nyquist limit by a factor of 10. Further, this technique can beapplied to active systems for adaptive algorithms that use previous measurements to inform future
measurements during acquisition. Finally, I will detail advances made in ultrafast pump-probe
nano-imaging and -spectroscopy, where pump modulation with selective interferometric detection
isolates the difference between the excited and ground state. This advancement enables higher
signal to noise ratios for low excited state contrast systems, especially necessary for the case of low
repetition rate, far-from-equilibrium pump excitation parameter regimes.
Next, I will discuss my work performing spectroscopy with broad band infrared (IR) lightsources, such as synchrotron radiation, fs broadband lasers, and ps narrowband lasers. This varietyof light sources proved extremely useful for a variety of material systems. Synchrotron infrarednano-spectroscopy (SINS) has been used as a consistent diagnostic tool for molecular systems,
like the self assembled monolayer 4-NTP, hydrocarbon and mineral systems, like oil filled pores inshale, and quantum systems, like strongly coupled heterostructures composed of multi-quantumwells and gold antennas. The advancements and discoveries here represent the successes of applying
infrared spectroscopy to a variety of systems to discern inter-molecular coupling, spatially resolvedhydrocarbon mapping, and phase resolved hybridization controlled weak to strong coupling.
Lastly, I will present ultrafast nano-FTIR on 2D transition metal dichalcogenide (TMD) systems, interfacial energy transfer, lattice expansion and phonon softening, and spatially inhomogeneous electron dynamics. Here, I performed visible pump, infrared probe nano-imaging spectrscopy,
where I pumped the interband electronic state in WSe2 , producing free carriers that thermalize
through phonon emission within the TMD and then dissipate heat through the substrate. We

iv
perform ultrafast nano-FTIR on a phonon mode of the substrate as a means of monitoring local
temperature. We find that the electron population in the TMD relaxation is also heterogeneous as
an increase in carriers from a hot phonon bottleneck leads to an Auger recombination and stronger
bright state population for faster excited state photo-relaxation.
I will close with a short summary of my work, and an outlook to future experimental efforts.

Acknowledgements

I have and this work has been supported by many different people:
I of course would like to thank my advisor, Prof. Markus Raschke for his guidance throughout
my research carreer in his laboratory. The laboratory space, equipment, and life lessons he supplied
me with shaped the scientist I am today.
Much of my work was in collaboration with other professors, students, and scientists. This
work would have been impossible without them. I would like to specifically call out Hans Bechtel,
who was the beamline scientist at the Advanced Light Source (ALS) in Berkeley. He supplied advice
and support on numerous projects and is responsible for my success with everything relating to
SINS in addition to helpful conversations about scientific research at large.
I also would like to specifically thank Eric Muller and Omar Khatib, who taught me visible
and infrared ultrafast non-linear optics, near-field spectroscopy, and scientific research at a large
research institution. Jun Nishida was also invaluable for his method development, spectroscopy
expertise, and words of wisdom.
Finally, I want to thank my partner, Elizabeth, for supporting me through the challenge thatis obtaining a PhD.

Contents

Chapter
1 Introduction

1

2 Scattering Scanning Near-Field Optical Microscopy

4

2.1

Introduction to near-field microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Tapping Mode Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.3

Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.4

Optical Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4.1

Continuous Wave Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4.2

Optical Parametric Amplification . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.3

Difference Frequency Generation . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.4

Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5

2.6

2.7

Near-field Optical Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1

Point Dipole Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.2

Finite Dipole Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Signal Isolation and Artifact Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.1

Field Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.2

Sideband and Tandem Demodulation . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.3

Fourier Transform Spectroscopy

2.6.4

Artifacts in s-SNOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii
3 Method Development

28

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2

Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1

Rotating frame parameter space . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2

Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.3

Sample preparation

3.2.4

Shifting frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.5

Proof of principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.6

Imaging protein matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.7

Drift and artifact compensation . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.8

Robust imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.9

Alternative chemical mapping approaches . . . . . . . . . . . . . . . . . . . . 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.10 Spatial resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.11 Spectral Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.12 Theoretical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.13 Applying rotating frame for chemical imaging . . . . . . . . . . . . . . . . . . 46
3.3

3.4

Compressed Sensing and Matrix Completion . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1

Conventional s-SNOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.2

Methods for improving acquisition speed . . . . . . . . . . . . . . . . . . . . . 49

3.3.3

Spatio-spectral imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.4

Compressive sensing and matrix completion . . . . . . . . . . . . . . . . . . . 52

3.3.5

Adaptive sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.6

Reconstruction algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.7

Comparison to original measurements . . . . . . . . . . . . . . . . . . . . . . 55

3.3.8

Further improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Heterodyne Pump Probe s-SNOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

viii
3.4.1

Ultrafast phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.2

Ultrafast near-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.3

Experimental implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.4

Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4.5

Ultrafast nano-imaging and -spectroscopy of Perovskite . . . . . . . . . . . . 75

3.4.6

Comparison to other approaches . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4.7

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Strongly Coupled Multi-Quantum-Well Antenna Heterostructures

84

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2

Strongly coupled polaritonic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3

4.4

4.2.1

Device design and fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.2

Nonlinear optical switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.3

Strong coupling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.4

MQW device characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Nonlinear reflectivity ensemble demonstration . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1

Ultrafast optical switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3.2

Performance limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.3.3

Ensemble performance summary . . . . . . . . . . . . . . . . . . . . . . . . . 104

Near-field quantum phase and hybridization control . . . . . . . . . . . . . . . . . . 105

5 Ultrafast visible pump infrared probe of WSe2 on SiO2

111

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2

Transition metal dichalcogenides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3

Nano-imaging dark excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4

Nano-thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

ix
6 Conclusion and Outlook
6.1

6.2

6.3

130

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.1.1

Multi quantum well strong coupling . . . . . . . . . . . . . . . . . . . . . . . 130

6.1.2

Tungsten diselenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.1

Method Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2.2

Scientific progress on quantum materials . . . . . . . . . . . . . . . . . . . . . 132

Final thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bibliography134

Figures

Figure1.1

Characteristic lengths and time scales for condensed matter phenomena . . . . . . .

2

2.1

Combining optical and mechanical channels . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Broadband light from a synchrotron source . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

Coupling of probe and image dipole at small distances . . . . . . . . . . . . . . . . . 15

2.4

Comparison of a typical s-SNOM probe-sample configuration with the finite-dipole
and the point-dipole model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5

s-SNOM Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6

Near-Field Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1

Rotating Frame s-SNOM (R-sSNOM) . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2

Rotating frame R-sSNOM imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3

Comparison of Imaging Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4

Larger field of view R-sSNOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5

R-sSNOM and conventional nano-FTIR spatial resolution . . . . . . . . . . . . . . . 40

3.6

Method of Accessing Rotation Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7

Smart s-SNOM schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.8

Application of smart s-SNOM to different materials systems (biological, molecular,
and quantum), different light sources (Laser, ALS synchrotron) and different spatial
scanning (single point, line and 2D scan) . . . . . . . . . . . . . . . . . . . . . . . . . 58

xi
3.9

Illustration of the effect of law rank assumption on the reconstruction error. . . . . . 60

3.10 Extraction of physically relevant information from resonance peaks at different reduction factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.11 Error as a function of parameter settings for two different samples: PTFE and Amide. 63
3.12 Heterodyne pump-probe nano-imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.13 Ultrafast pulse characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.14 Retrieval of transient nanoscale response function . . . . . . . . . . . . . . . . . . . . 72
3.15 Modeling parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.16 Polaron-cation coupling in hybrid organic-inorganic perovskite FAMACs . . . . . . . 75
3.17 Transient vibrational nano-spectroscopy of FAMACs perovskite . . . . . . . . . . . . 77
3.18 Relationship between polaron-cation coupling constant and transient vibrational
peak shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.19 Comparison of the excitation modulated and unmodulated detection . . . . . . . . . 78
3.20 Transient vibrational nano-spectroscopy of a lead halide perovskite . . . . . . . . . . 79
3.21 Fano-type interference induced by carrier background4.1. . . . . . . . . . . . . . . . . 80

Ultrafast, highly nonlinear metasurface with saturable absorption and optical power
limiting functionalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2

Linear response of the polaritonic metasurface . . . . . . . . . . . . . . . . . . . . . . 87

4.3

Effect of ultrastrong coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4

Maxwell-Bloch model fits to FTIR spectra . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5

Model parameters from the fitting procedure . . . . . . . . . . . . . . . . . . . . . . 97

4.6

Experimental nonlinear tuning of the coupling coefficient . . . . . . . . . . . . . . . . 98

4.7

Transient all-optical control of metasurface reflection . . . . . . . . . . . . . . . . . . 100

4.8

Limits to metasurface absorption contrast . . . . . . . . . . . . . . . . . . . . . . . . 104

4.9

Comparison of 2 versus 3 oscillator coupling . . . . . . . . . . . . . . . . . . . . . . . 106

4.10 Near-field Michelson interferometer for nano-FTIR and pseudo-heterodyne . . . . . . 107

xii
4.11 Phase resolved single antenna strong coupling . . . . . . . . . . . . . . . . . . . . . . 108
4.12 Phase resolved single frequency imaging between strong and weak coupling regimes . 109
4.13 Single fluence coupling strength tuning through tip positioning . . . . . . . . . . . . 110
5.1

Ultrafast heterodyne pump-probe dynamics on Au . . . . . . . . . . . . . . . . . . . 114

5.2

Ultrafast heterodyne pump-probe dynamics of bulk WSe2 on SiO2 . . . . . . . . . . 114

5.3

Ultrafast heterodyne pump-probe dynamics of monolayer WSe2 on SiO2 . . . . . . . 115

5.4

Ultrafast heterodyne pump-probe imaging bulk and monolayer WSe2 on SiO2 . . . . 116

5.5

Ultrafast heterodyne pump-probe nano-imaging lattice dynamics . . . . . . . . . . . 121

5.6

SiO2 phonon resonance shift as ultrafast temperature sensor . . . . . . . . . . . . . . 123

5.7

Local variations in temperature and thermal boundary conductance . . . . . . . . . 125

5.8

Molecular dynamics modeling of SiO2 phonon . . . . . . . . . . . . . . . . . . . . . . 127

Chapter 1

Introduction

Infrared (IR) spectroscopy is routinely used for chemical identification and structural determination.1 It is a minimally invasive and label-free technique that has applications in disciplines
ranging from biology, chemistry, and condensed matter physics to space sciences. IR spectroscopy
can be used to probe the Drude conductivity of metals and 2D quantum materials like graphene aswell as low energy bandgaps, plasmons, and intersubband transitions.2–4 In combination with conventional microscopy, infrared spectroscopy provides for chemical analysis and imaging at diffraction
limited length scales.5, 6 The highest spatial resolution that IR microscopy has conventionally beenable to achieve is ∼2-20 µm at 3-20 µm wavelengths.7
Near-field scanning optical microscopy (NSOM) was first demonstrated in the 1980s, yet it
had limited applicability, as the fiber throughput typically constrained sensitivity and minimum
resolution to be ∼100 nm.8–10 The development of tip-enhanced scattering near-field microscopy
increased the applicability of deep sub-diffraction limited imaging.
s-SNOM has enabled the study of fundamental interactions at the mesoscopic scale, but itlacked ultrafast temporal resolution until pump-probe spectroscopy was applied to the near-field.
Two color time resolved measurements, where the signal from the pump alone can be removed with
an optical filter, are now common and have been applied to quantum and molecular materials.11–13
The combination of nm resolution with infrared frequencies at ultrafast time scales is a powerful
technique to explore phenomena ranging from electron-electron interactions to vibrational strong
coupling to surface plasmon polaritons (Figure 1.1).

2Dimensionality
Edges, Defects, Domains, 3D
Granularity
Confinement Interfaces

0D Quantum

Timescale

ns

ps

fs

Structural Domain Formation
Surface Phonons Lattice
Vibrational Coupling/Dynamics
Vibrational Phase Coexistence
Excitons
Exciton Diffusion
CT States
Surface Plasmons
Electronic Dephasing

Electronice-e Interactionsnm

μm

Lengthmm Scale

Figure 1.1: Characteristic lengths and time scales for condensed matter phenomena | Condensed
matter phenomena of fundamental particle interactions – caused by lattice or electronic degrees of
freedom. Figure adapted from a design by O. Khatib.

In Chapter 2, I will introduce the fundamentals of s-SNOM. To begin, I summarize the basics
of tapping mode AFM and the diffraction limit from a wave equation perspective. I then transition
to discussing demodulation techniques and asymmetric heterodyne spectroscopy. Following this,I discuss light sources ranging from CW and fs pulsed lasers to synchrotron radiation. Finally, I
conclude with a description of two tip-sample interaction models that serve as the basis for newly
developed pump-probe modeling.
I then delve into near-field imaging and spectroscopy method advancements (Chapter 3).
I have improved image acquisition speed by moving from the stationary frame into the rotating
frame. Transforming frames lowers the Nyquist cut-off frequency, which reduces the necessary
number of data points. I also discuss the collaborative effort of improving hyperspectral imaging
by using prior knowledge for both compressive sensing and matrix completion techniques. I conclude
with advancements made to ultrafast near-field spectroscopy and imaging, where pump-modulation
together with heterodyne detection improves excited state contrast.
I then detail far-field spectroscopy, degenerate pump-probe dynamics, and near-field imaging
and control of multi-quantum wells that are strongly coupled to infrared resonant antennas in

3
Chapter 4. I utilized control through polarization, fluence, and tip localization to actively tune
coupling strength.
Finally, I discuss transition metal dichalcogenides (TMDs) under far-from-equilibrium pumpconditions. In Chapter 5, I use ultrafast heterodyne pump-probe spectroscopy to measure carrier
behavior in WSe2 . I build on this work by measuring the phonon softening in the SiO2 substrate. I
use this temperature dependent phonon mode both to measure thermal dissipation at the atomic interface and to measure spatial heterogeneity in substrate surface temperature. Thermal dissipationfrom WSe2 to the substrate affects excited state electron dynamics through intra- and inter-valley
coupling.
I conclude this thesis with a summary of the work conducted over the course of my PhD andan outlook for future directions in Chapter 6.

Chapter 2

Scattering Scanning Near-Field Optical Microscopy

2.1

Introduction to near-field microscopy
In this chapter, I detail the background of near-field imaging. Specifically, I cover atomic

force microscopes (AFMs) with their modes of operation and the tip-sample interactions. I follow this with the diffraction limit that conventionally obscures nanoscale heterogeneity. Then, I
discuss optical sources for spectroscopy including continuous wave and pumped lasers as well assynchrotron radiation and near-field optical interaction models that are used to quantify nano-IR
spectra. Finally, I conclude with signal isolation and artifact rejection for near-field data that arenecessary for analyzing data.

2.2

Tapping Mode Atomic Force Microscopy
Atomic force microscopy serves as the basis for several specialized techniques, among thems-SNOM.14, 15 In 1985, Binnig and his coworkers built upon the STM by developing the AFM to
study general samples, not just conductive ones.16 Since then, the AFM has been refined andimproved. A metallized silicon tip is used to probe the sample. The conductive tip acts as an
optical antenna in the near-field region around the tip’s apex.17 The sample is scanned below thetip so that the tip remains in the focus of the IR light. AFM feedback is maintained throughtapping amplitude. The tip is driven sinusoidally with a typical tapping amplitude between 40 and
80 nm.18 The cantilever stiffness defines the resonant tapping frequency, which is between 200 and
300 kHz for the experiments described in this thesis. When the tip approaches the sample, the5resonant frequency red-shifts and tapping amplitude decreases. As the tip makes contact with thesample, the tapping amplitude sharply drops. To maintain good contact, the tapping amplitude is
set to 80% of the free space value.
As an extension to tapping mode AFM, peak-force AFM can be used to extract mechanicalproperties about a sample’s surface (Figure 2.1a). Previous studies19 have correlated optical signals
from a s-SNOM experiment (Figure 2.1b) with mechanical channels from an AFM (Figure 2.1c).

Figure 2.1: Combining optical and mechanical channels | a. Peak force mechanical channels as the
tip approaches and withdraws from the sample surface. b. Asymmetric Michelson interferometerfor s-SNOM measurements. c. Combining mechanical and optical channels for correlation andanti-correlation analysis. (Figure from19 )I also contributed to this multidimensional imaging by applying different light sources to
s-SNOM measurements and correlating those results with mechanical properties from peak force
AFM.206

2.3

Diffraction Limit
Conventionally, imaging systems have a minimal resolvable feature ultimately given by the

wavelength of light. In order to study physical phenomena on their natural nm length scales, weneed to first understand the limit to resolution. Historically, Ernst Abbe21 was the first to describe
the minimal distance d that two objects need to be away from each other to be resolvable
d =

λ
,
2 NA

(2.1)

where λ is the wavelength of the light and N A is the numerical aperture. When visible light andmodern objective lenses are used, this separation distance is typically approximated as 250 nm, as
500 nm light is in the center of the visible spectrum and objectives with NA of ∼1 are common.22
While this is still too large of a sampling area to resolve nanoscopic fundamental phenomena, thesituation is more than an order of magnitude worse for the infrared. Ignoring the limitations of
IR objectives and assuming N A = 1, 10 µm light can only be focused down to 5 µm, which isclearly insufficient to study nanoscale phenomena at these optical frequencies.7 Therefore, we must
understand what gives rise to this limit.
To better understand and quantify the diffraction limit, we consider the case of imaging
an object through a circular aperture, as this is a simple imaging system that can be used tounderstand the underlying physics. From time reversal symmetry, we can consider the projectionof monochromatic light from an aperture to a screen as an analogous problem.23 We use the
Huygens-Fresnel principle to integrate over the aperture all light and propagate that light using
the far-field traveling wave solutions to the wave equation,24 such that
Z
U∝

2π

′

′

A(x′ , y ′ )e−i λz (x x+y y) dx′ dy ′(2.2)

aperture

where U is the complex field amplitude, A is the complex amplitude of the aperture, λ is the
wavelength of light, z is the propagation axis through the aperture, and x’ and y’ are the location

7of the aperture. In polar coordinates, we make the substitution
x′ = ρ′ cos(θ′ )y ′ = ρ′ sin(θ′ )(2.3)
x = ρcos(θ)
y = ρsin(θ),
which transforms our previous equation to
Z

2π

′

′

A(ρ′ )e−i λz ρρ cos(θ−θ ) ρ′ dρ′ dθ′ ,

U∝

(2.4)

aperture

where I have assumed angular symmetry of the aperture and used the cosine difference formula.
Further, the integral equation for the zeroth order Bessel function can be substituted for
2π
J0 ( ρρ′ ) =
λz

Z 2π

2π

′

′

e−i λz ρρ cos(θ−θ ) dθ′ ,(2.5)

2π ′ ′ ′ρρ )ρ dρ .
λz

(2.6)

0

to compactify the complex field amplitude toZ

A(ρ′ )J0 (U∝
aperture

In the case of a circular aperture, the limits of integration for dρ′ are from 0 to the radius of
the aperture. Using the recurrence property of Bessel functions and the fundamental theorem ofcalculus, we can evaluate this integral
U ∝ J1 (

πRρ
),
λz

(2.7)

which has its first zero at an argument of 3.8317. This combined withρ= tan(θ)z(2.8)

yieldstan(θ) =3.8317 λ
,
π D

(2.9)

which, upon the small angle approximation and simplifying constants, is the Rayleigh criterion forthe diffraction limit.25 While this is only one description for the resolution of an imaging system, thefundamental physical insight gained with this example generalizes to other, more complex systems.

8
In order to elucidate the underlying resolution limitation, I give a practical example. A 1
cm diameter objective, with a focal length of 1 cm, will be able to resolve two objects if they are
610 nm apart. Near the diffraction limit, tan(θ) ∼ θ = r/d, where r is the distance between two
objects and d is the distance from the aperture to the objects. Typical high quality objectives will
have an aperture to focal length ratio of ∼2, which means objects can typically be resolved if they
are separated by a distance larger than λ/2. Therefore, we need an imaging mechanism that can
go beyond the diffraction limit.
One class of approaches to go beyond the diffraction limit is super resolution techniques.
Specifically, stimulated emission depletion (STED) microscopy,26, 27 photo-activated localization
microscopy (PALM),28 and stochastic optical reconstruction microscopy (STORM)29 have beenused in biological systems to image large complex structures over wide fields of view. However,most super resolution techniques require chemical labeling, which is invasive and not generally
applicable. Further, these techniques only image the system indirectly and do not offer accessto fundamental physical phenomena on the nanoscale. Therefore, molecular vibrations, latticephonons, and intersubband transitions, which all lie in the mid infrared (5 µm − 20 µm), are notaccessible. With our example given above, and the r/d ratio of most good infrared objectives(r/d ∼ 1), resolution is conventionally limited to ∼10 µm, which is much larger than the naturallength scale of mesoscopic physical phenomena.30–32 Therefore, we seek to image in the near-field,
where the diffraction limit does not apply.

2.4

Optical Sources
In this section, I will briefly go over the different light sources used throughout the work

presented in this thesis. The implementation of each of these light sources poses slight variations tothe standard s-SNOM setup, and some imaging modalities are only possible through certain lightsources.

9
2.4.1

Continuous Wave Laser

I begin with continuous wave (CW) lasers, which are the simplest in this section. In CW
operation, the laser is continuously pumped, and, for the applications listed below, a single resonantmode to be probed is selected. CW lasers in s-SNOM are used both for the IR probe and for visible
guide lasers to aid in alignment. The IR probe is often either a CW quantum cascade laser (QCL)
or gas laser. The visible guide lasers are typically either a gas laser (Helium Neon, He-Ne), or,more common recently, simple diode lasers.
(1) Gas lasers used here utilize a gain medium that is electrically pumped inside a simple tube
cavity, where one gaseous species (nitrogen in the case of a CO2 laser and Helium in thecase of a He-Ne laser) is stimulated by the electrical pumping, decays, and transfers energy
to another gaseous species CO2 for an appropriately named CO2 laser and Ne for the He-Ne
laser.33 This generates the necessary population inversion for lasing to occur. Because thelasing medium is a gas, the output light remains coherent over longer time frames than solidstate lasers and therefore leads to exceptionally narrow linewidths. These light sources are
exceptional for their stability and long coherence lengths but are limited as they are notable to be used for spectroscopy.34(2) Solid state lasers, like QCLs and diode lasers, are still CW yet exhibit notably shorter
coherence lengths.35 Diode lasers use electron hole recombination and the resulting photon
emission to produce light with energy dependent on the bandgap, whereas QCLs operate in a qualitatively distinct manner. The tuning range of diode lasers is limited and iscontrollable only through temperature and doping levels, which alter the band diagramof the material. In contrast to this, QCLs string together a series of quantum wells, thesize confinement of which modifies the band structure to produce subbands.36 The energy
difference between these subbands is considerably smaller than the bandgap. Therefore,
QCLs are useful in producing IR light compared to the typical visible light in diode lasers.37
The intersuband transitions used in QCLs occur within the conduction band as opposed

10
to between conduction and valence bands. A bias is then applied to the series of quantum wells so that the photon produced in one intersubband transition then stimulates a
subsequent well. This process continues down the chip of quantum wells, producing highpowered output at relatively long wavelengths.38 Again, similar to the slight adjustments
possible to the bandgap energy in diode lasers with temperature, QCLs are tuned throughlattice expansion and contraction through temperature variations. As the QCL expands,
the quantum well widens and the bound states lower in energy. During contraction, thequantum well narrows and the bound states rise in energy. While this type of tuning istypically quite limited, broader tuning, which is common of modern QCLs, can be achieved
through single wavelength stabilization. This approach uses an external cavity, where agrating (tunable through rotation) is used as the output coupler, selectively decreasing thegain threshold for a desired frequency.39 The shorter coherence lengths of these CW lasers
can make heterodyne techniques more difficult, but their ability to be tuned and theirresulting applicability to spectroscopy makes them useful candidates for chemical imaging.
The CW nature of gas, diode, and quantum cascade lasers implies a very narrow linewidthand correspondingly, through its Fourier relationship, a long coherence time. The narrow line-widthand single frequency nature means CW lasers application to s-SNOM is the easiest of the sources
listed here and is primarily used for pre-characterization and quick measurements where spectral
frequency mapping is not required. The most common implementation of a CW laser in s-SNOM iswith a CO2 laser at approximately 10.6 µm for Drude mapping in 2D materials like, e.g., graphene.3

2.4.2

Optical Parametric Amplification

I now discuss pulsed light sources, whose broad bandwidth enables Fourier transform spectroscopy. While more complex than single wavelength sources, optical parametric amplification,either in optical parametric amplifiers (OPAs) or optical parametric oscillators (OPOs), offers awide tuning range through nonlinear phase matching pulsed output.40 When a crystal lacks inver-

11sion symmetry, the crystal allows for a collection of non-linear processes to occur, namely frequency
doubling, sum frequency generation (SFG), and difference frequency generation (DFG).41 This χ(2)process (optical parametric amplification) uses a signal beam as a seed together with a higher frequency pump beam to convert photons of the pump into more photons of the signal beam. An idler
beam is also generated and has energy equal to the difference in energy between the pump andseed.42 This is most often carried out in the non-degenerate case, where the signal output is higher
in energy than the idler. In the degenerate case, the seed is tuned to half the energy of the pumpbeam, and the resulting signal and idler pulses have equal energy. For efficient conversion to occur,
the optical phases of the output waves must match the crystal structure. The crystal is rotated forphase matching until maximum output power is achieved.43 The accessible bandwidth is limitedby phase matching and typically, the thicker the nonlinear crystal, the more strongly the phasematching requirement is enforced, resulting in a narrow output spectrum. When the nonlinear
process occurs in more material (since the crystal is thicker), higher powers are achievable.44 OPAs
and OPOs are primarily attractive because there is a lack of suitable ultrafast light sources in thenear and mid IR.
The underlying process behind parametric amplification is an optical non-linear conversion.
For the nonlinear process to occur, high peak strength electric fields must be employed, typically necessitating an ultrafast pulsed laser upstream of the amplification process.44 Ultrafast mode-lockednear-IR (Ytterbium based lasers, for example) or visible (titanium sapphire lasers, for example)
pump lasers are split into two paths. One path is for the optical parametric amplification pumpand the other is for white light generation in another non-linear crystal (YAG, for example). The
resulting white light is then co-focused with the pump into a non-linear crystal, and phase matchingis used to select the amplified portion of the white light spectrum.45 OPAs and OPOs operate on
the same underlying principle, but an OPO places the parametric amplification within a cavity fora higher output power at the expense of a slightly longer and spectrally narrower pulse.46 Withsufficient tuning and choice of non-linear crystal, the idler can be tuned to ∼4-5 µm and be directly
used in s-SNOM. Difference frequency generation is used to get further into the IR.

12
2.4.3

Difference Frequency Generation

The tunability of both the signal and idler from parametric amplification can be used together
with difference frequency generation to provide a tunable broadband infrared source. As mentioned
above, crystals that lack inversion symmetry can also host difference frequency generation (DFG).41
Here, two beams, focused into such a crystal, are converted into a third beam with energy equal tothe difference of energies of the input beams. Again, as before, phase matching must be implemented
for efficient difference frequency generation.47 For s-SNOM, typical mid-IR broadband generationis performed by using a pump laser in the near-IR in an OPA or OPO. The resulting signal andidler beams are then mixed in DFG to produce mid- to far-IR light. For further IR generation, the
OPA/OPO is simply tuned closer to degeneracy, such that the difference in energy between thesignal and idler is smaller and the resulting light after DFG is lower in energy. The typical main
limit to this process is optical absorption in the non-linear crystal. A common crystal choice is
GaSe, which exhibits a strong absorption at 18 µm, limiting the IR output through DFG to 550
cm-1 .48 DFG is therefore a key useful step in generating tunable broadband mid-IR light.
This concludes the section on pulses light sources for the mid infrared. Their broadband
nature makes them ideal instruments for Fourier transform spectroscopy and their ultrafast pulses
make them indispensable for ultrafast time resolved measurements.

2.4.4

Synchrotron Radiation

The final light source of note for s-SNOM (or SINS) is the synchrotron. My experience withsynchrotron infrared nanospectroscopy (SINS) is primarily at the advanced light source (ALS)in Berkeley, CA, so most of my discussion about synchrotrons will be focused on this example.
Synchrotrons use the tangentially (to the path of motion) emitted electromagnetic radiation from
relativistic electrons that traverse along curved trajectories (Figure 2.2A).
The etendue of the small diameter electron beam source makes the output light diffraction
limited and spatially coherent with approximately 100-1000 times the spectral irradiance (Figure

13

Figure 2.2: Broadband light from a synchrotron source | A. Broadband synchrotron radiation is
produced when relativistic electrons are accelerated by a constant magnetic field. B. Calculated
spectral flux (red) and spectral irradiance (blue) for bend magnet radiation at the ALS, assuming
a collection angle of 17 (V) mrad x 69 (H) mrad (solid line), which is standard for beamlines 2.4
and 5.4, and 80 (V) mrad x 69 (H) mrad (dotted lines), which enables the maximum extraction of
the infrared light to 20 cm-1. The spectral flux was calculated with the SRW package using theaccelerator parameters of the ALS and the spectral irradiance was calculated assuming a diffraction
limited spot size for NA=0.4.49

2.2B) of an incoherent thermal source.50, 51 The electrons used in the synchrotron are packaged into
bunches and have a repetition rate in the hundreds of MHz (500 MHz at the ALS), forming a total
beam current of 0.5 A.49 The resulting infrared radiation is dispersed in time, with a pulse lengthof, at the ALS, 60 ps. The large mass and careful beam pointing of synchrotrons provides for astable light source. The high repetition rate further enables excellent near-field signal optimizationand signal to noise ratios over long periods of time.

14
Although previous synchrotron technology was limited to decay mode, where small losses
over time lead to a reduction in beam current and resulting output radiation, modern advances
have enabled top-off mode,52 where additional electrons are continuously injected into the storage
ring.53 Near-field signal in decay mode would continuously vary with discrete jumps 2-3 times a
day when the ring would be refilled. Top-off mode now enables stable near-field measurements overa period of hours and days, limited only by the stability of the AFM and the hold time of thedetector.
This concludes the section on infrared light sources that can be used for IR s-SNOM. While
this list is incomplete, it represents both the light sources used in the studies throughout the restof the thesis as well as a good overview of common light sources for infrared spectroscopy.

2.5

Near-field Optical Interaction
In this section, I will highlight the two most important models used to understand the tip

sample interaction during s-SNOM. These two models are called out both for being analytical andfor giving a simple intuitive understanding of the extracted quantitites and input parameters forunderstanding s-SNOM signals and spectra. Additional simulation and numerical models54 havebeen developed but are only marginally more quantitatively accurate55 at the expense of a simple
physical understanding.

2.5.1

Point Dipole Model

The simplest of the two analytical models is the point dipole model. In this model, themetalized AFM tip is reduced to a simple sphere with a known dielectric function ϵtip (ω).17 The
dielectric function of metalized AFM tips is typically flat in the infrared, but, for quantitativemodeling, known functions of gold, platinum-iridium, or platinum-silicide can be used. The polarizability of the spherical tip is given by
αtip = 4πr3 ϵ0ϵtip − 1,
ϵtip + 2(2.10)

15
where r is the radius of the spherical tip, ϵtip the dielectric function of the tip, and ϵ0 the vacuum
permittivity.

Figure 2.3: Coupling of probe dipole p and image dipole p′ at small distances z = r − a; the applied
electric field E is assumed a. perpendicular and b. parallel to the sample surface. (Figure from17 )αtip modifies the incident electric field, localizing it to interact with the sample.17 The
back scattered light is further affected by the interaction between the tip and sample. The light
induces a dipole on the tip, which then forms an image dipole in the substrate pim
tip = ptip β, where(ω)−1
β = ϵϵsam. ϵsam is the desired outcome of the measurement (Figure 2.3). The combination ofsam (ω)+1

the tip polarizability and sample dielectric function combines in the point dipole model
P DM
=
αef
f

αtip
α

β

1 − 16πϵ0tip(r+z)3,(2.11)

where z is the tip sample distance. Therefore, the near-field electric field is simply
Escat = (1 + rP )2 αef f E0 ,(2.12)

where rP , comes from the far-field Fresnel reflection of the incident light. A key point that will berevisited later is that near-field imaging and spectroscopy will always contain artifacts and sample
specific responses that can easily convolve sample morphology and composition with the desirednear-field signal. For instance, even in this non-interfered near-field electric field, the far-fieldinteraction with the sample is directly part of the measured near-field quantity. To understandthis, imagine that a sample was coated in a monolayer with a metal carbonyl except for the area
directly under the tip. The tip illumination path would still contain the metal carbonyl monolayer,

16effectively absorbing the incident light before it even reaches the tip. The squaring of the reflection
coefficient term comes from the incident beam shaping and back-scatter beam shaping.56 Whiles-polarized light should couple equally well to a spherical tip as p-polarized light, the true elongatednature of the tip preferentially selects for p-polarized light (Figure 2.3). This particular model isindeed quite simple and gives the correct physical intuition, yet it is quantitatively lacking.

2.5.2

Finite Dipole Model

To better address the quantitative tip sample interaction, the spherical tip approximation isperturbed to an elongated ellipsoid.

Figure 2.4: Comparison of a typical s-SNOM probe-sample configuration with the finite-dipole andthe point-dipole model. a. The probing tip is typically an elongated structure illuminated by plain
waves from the side. b. In the finite-dipole model the tip is approximated by a spheroid in a
uniform electric field E0. c. In the point dipole model the tip is first reduced to a small spherein an uniform electric field E0. The sphere is then further reduced to a point dipole located in its
center. The scattered field is finally calculated from a dipolar near-field coupling with the sample.
(Figure from56 )17This ellipsoid is more similar to the tip’s actual shape (Figure 2.4) and therefore models theeffective polarizability
2L
R
F DM
2
R + ln 4eL

αef
=
R
L
f
ln 4L
e2


2 +


  4L 
β g − R+z
ln 4z+3RL  2L  
3R+4z
ln 4L
−
β
g
−
ln 2z+R
R
4L(2.13)

much better.56, 57 Here, L and R are the half long (L = 600 nm is typical) and short axes (or radiusof curvature) of the ellipsoid, respectively, and g is the constant ratio between a point charge andthe charge on the tip with a value of g = exp (i0.06) empirically used.57
The near-field interaction predicted by both the point dipole model (PDM) and the finite
dipole model (FDM) causes an apparent red shift in center frequency compared to the far-field andnatural energy of a resonant transition. This red shift scales with resonance strength, and therefore
statistical correlations between resonance amplitude and center frequency can be complicated bythis relationship. A similar effect occurs when the phase of a resonant complex optical response isanalyzed, where the increased resonant strength blue shifts the resonance, broadens the linewidth,
and turns an absorptive feature into a dispersive one. While phase is preferred for analysis, since
it is entirely independent of artificial amplitude scaling within the amplitude response, using theimaginary component in its place addresses this issue.
Numerical models beyond the PDM or FDM have also been developed, especially for strongly
resonant systems, like those of plasmonic particles. Specific modeling of a tip’s exact shape, possible
shadowing effects, far-field interference, sample morphology are also being developed.54, 58, 59 Typically, these effects are ignored as it is assumed that self-referencing to a gold or other non-resonant
substrate will normalize out these second order effects.
This concludes the section on near-field optical interaction models. We found that the PDM
was sufficient for gaining a qualitative physical understanding of the tip-sample interaction. While
the PDM is qualitatively useful, we also discussed the FDM, which is more quantitatively accurate
at the expense of additional model complexity.

18

2.6

Signal Isolation and Artifact Rejection
In this section, I will discuss the sources and classification of artifacts. In order to avoid

artifacts in s-SNOM data, or its interpretation, a fundamental understanding of near-field signals
is needed.

2.6.1

Field Detection

To begin, I will discuss the detection, demodulation, and processing of near-field data startingwith the detected electric fields. The current most common implementation of s-SNOM is withan asymmetric Michelson interferometer. Here, I will discuss the fields and detection involvedin this general case and select subsets from this to address the different imaging modalities usedin this thesis. A light source (narrowband or broadband) is sent into an asymmetric Michelsoninterferometer, and its beamsplitter divides the beam into a reference arm and a tip/sample arm(Fig. 2.5). Light from the beamsplitter in the sample arm is focused with an OAP onto the AFM

Light Source

BS

Reference

MCT
OAP

AFM tip

Figure 2.5: s-SNOM Experimental Setup | An s-SNOM experiment. A light source is directed toan AFM tip through an off axis parabolic, which collects the backscattered light and directs it to
the detector. Interference with a reference arm enables spectral and phase resolution.

19
tip, which localizes the field and backscatters the near-field to the OAP, which collimates the lightand directs it through the beamsplitter onto the detector OAP, which also focuses the light into thedetector. In this process, there are three fields of interest, 1) the reference field E ref , 2) the nearfield E NF , and 3) the background field E FF , which overwhelms the near-field from the tip-sample
area.60
Different detectors are used depending on the frequencies being studied. The most common
detector for mid-infrared frequencies is a photovoltaic mercury cadmium telluride (MCT) detector.
If additional bandwidth is needed, photoconductive detectors offer a wider detection tuning range
and higher D* at the expense of non-linearities in its power-to-signal conversion and increased noise.
Near-IR experiments often require different detectors, as MCTs typically drop off in sensitivity
starting at 5 µm and are quite insensitive at less than 2 µm. Detectors with other active elements,
such as InGaAs, InSb, and PbSb, are used in place of MCTs. Far-IR measurements are typically
performed with bolometers and doped Ge detectors.61–63 Currently, the main limitation withthese detectors is their speed, which slows down for detection further into the IR. The typicaldemodulation frequency in near-field measurements is ∼500 kHz, but this can be reduced to ∼150
kHz by using an AFM tip with a lower resonant frequency. At THz frequencies, electro optica
sampling (EOS) is used, where the electric field (as opposed to the intensity) is directly detectedusing an ultrafast gating pulse.64, 65 However, the experimental complexity in performing EOS andthe high signal-to-noise ratio (SNR) of MCTs usually restricts EOS to far-IR/THz measurements,
where MCTs are not sensitive and alternative bandgap materials are not available. Since the work
contained in this thesis uses an MCT, we will restrict the rest of the description to this detection
modality.
I will now discuss near-field extraction. The MCT detects the intensity of light and is therefore
referred to as a square law detector.60 Therefore, the detected signal is20

IDet (x, t, νe) = |EN F (t, νe) + EBkg(t,eν ) + ERef (x, t, νe)|2= |EN F (t, νe)|2 + |EBkg (t, νe)|2 + |ERef (x, t, νe)|2 + 2 Re (EBkg (t, νe)∗ ERef (x, t, νe))
+ 2 Re (EN F (t, νe)∗ EBkg (t, νe)) + 2 Re (EN F (t, νe)∗ ERef (x, t, νe)) .
(2.14)
Here, the detected signal is dependent on the spectrum of the light (eν ), the tip position during its
oscillation (t), and the relative optical path length between the tip arm and reference arm (x). In
this expanded format, the background squared and reference-times-background terms serve no use
and are simply far-field background terms. To isolate only the terms that contain the near-fieldsignal, we use the time dependence of the near-field signal on the tip tapping motion to demodulate
the detected intensity at the tapping frequency. The near-field signal depends nonlinearly on the
tip sample distance, which is being modulated in a sinusoidal motion. The nonlinear dependenceof the signal, based on a periodic function, means the Fourier transform of the near-field signal willcontain signal at all of the harmonics of the tip tapping frequency. The first harmonic contains
a significant far-field contribution, but the second harmonic is typically sufficient for near-field
isolation.17, 66–68
In order to demodulate the signal, a lock-in amplifier is used. The tip tapping frequency fromfrom the AFM photodiode is analyzed within the lock-in amplifier to extract the tapping frequency,such that a synthetic waveform can be generated. For second harmonic near-field detection, theisolated frequency is doubled before the waveform is made. This waveform is then multiplied by thesignal from the detector and the resulting product is integrated over the time constant, which sets
the bandwidth of the detected signal in frequency space. Longer time constants have a narrower
detected bandwidth about the demodulated frequency. This process then results in a detected
intensity of

IDet (x, t, νe) = 2Re(EN F (t, νe)∗ EBkg (t, νe) + 2Re(EN F (t, νe)∗ ERef (x, t, νe),(2.15)

A

B
Tip Motion
Near-Field Signal

Amplitude

Near-Field Signal Tip Height

21

Time

Frequency

Figure 2.6: Near-Field Signal | A The tip sinusoidally approaches and withdraws from the sample,
which increases and decreases the near-field signal from electric field surface enhancement. The
near-field signal is nonlinearly dependent on the tip sample distance and therefore yields a comb
like Fourier transform. B The Fourier transform of the near-field signal, which shows the higher
harmonics resulting from the nonlinear tip sample distance dependence.

where the E NF 2 term was dropped as it is negligibly small. The first term is referred to as selfhomodyne and represents the signal that would be detected if the reference arm were blocked. Whileuseful for alignment, the uncontrolled phase between the near-field and anything/everything in the
focal area makes it effectively useless. The signal is not spectrally resolved, and constructive ordestructive interference at certain wavelengths is dependent on the sample morphology and relative
position of the tip to a scattering center. The second term enables spectral and phase resolution.
Therefore, we wish to extract this heterodyne amplified near-field signal.
In order to isolate the near-field heterodyne signal, three common techniques are used. First,
the reference arm is simply scanned across the zero phase difference, resulting in an interferogram.
The DC offset of the interferogram is the unwanted self-homodyne term, but the AC component
is extracted by subtracting a line of best fit (to account for slow drift) through the interferogram.
This interferogram is asymmetric, resulting from the asymmetric (sample in one arm) nature of theinterferometer. While symmetric interferometers use phase correction algorithms to improve the22
SNR of their spectra by imposing a zero phase response of the interferogram’s Fourier transform, theasymmetric nature here enables full complex valued spectra to be obtained. The second technique
is to modulate the reference arm with an optical chopper and detect at the chopper frequency
sideband about the second harmonic of the tip tapping frequency. This was previously developed
for single wavelength sources to perform chopper-demodulated two phase homodyne, but it has
been extended for use in rotating frame s-SNOM. In the rotating frame, its implementation can beunderstood as chopper demodulated multi-step two phase homodyne, where spectral resolution froma Fourier transform can still be acquired. The final common technique is pseudoheterodyne, where
the reference mirror is sinusoidally driven. Similar to chopper-demodulated two phase homodyne,
sideband detection is performed. Here, instead of one sideband being detected at two different
reference arm phases, two sidebands are detected (the first and second harmonics of the reference
mirror frequency) to simultaneously acquire the in and out of phase signals (real and imaginary).60
The division of imaginary and real components into the first and second sidebands respectively
requires a specific amplitude for the driven mirror, which is wavelength dependent. Because of this,a broadband source is not generally compatible with pseudoheterodyne, as the real and imaginary
components from frequencies outside of the center frequency mix between the two sidebands. For
this reason, pseudoheterodyne is typically restricted to CW laser sources, spectrally focused lasers,or sufficiently filtered broadband light sources. Any of these three techniques extract the near-field
heterodyne signal.
In this section, I have detailed how the near-field heterodyne signal can be extracted from asquare law detector using a lock-in amplifier.

2.6.2

Sideband and Tandem Demodulation

At this point, sideband detection has been mentioned in the context of heterodyne discrimination, but it is also used in the ultrafast studies mentioned later in this thesis. In this section, I will
discuss different approaches to isolating signals from the combination of two distinct frequencies.
Sideband demodulation requires two input frequencies and their chosen harmonics to deter-

23
mine where the signal will be demodulated in frequency space. Both cascaded or tandem demodulation and synchronized parallel demodulation can be used to detect sideband signals. Tandem
demodulation occurs in two steps. First, the carrier signal (second harmonic of the tip tapping
frequency for s-SNOM) is demodulated with a small time constant so that the center frequency andsidebands of interest are all contained within the output of this first stage. This process shifts thecarrier frequency to 0. A second demodulation stage is used where the sideband signal is discriminated from the center frequency by demodulating at the modulated frequency (see Chapter 3).
While tandem demodulation is intuitive, the repeated ADC and DAC add additional noise. Somelock-in amplifiers have an internal setting that enables the direct output of the demodulated signal
for the second stage, avoiding this noise source. A further issue with tandem demodulation is the
bandwidth passed after the first stage. If a sufficiently large enough bandwidth isn’t used, the low
pass filter during the first stage demodulation will partially reject the sideband signal, lowering the
SNR. Tandem demodulation also has the drawback that if the two signals are out of phase witheach other, no amplitude can be extracted, as the demodulation is symmetric about the new zerofrequency. When the sidebands are in phase with each other and the amplitudes of each sideband
are the same, then tandem demodulation is sufficient. On the other hand, synchronized parallel
demodulation instead calculates the sideband frequencies and performs single stage demodulationon the signal directly. In this modality, the amplitudes and phases of the sidebands can be independently measured. The lock-in amplifiers typically used in commercial s-SNOM systems utilize
synchronized parallel demodulation.

2.6.3

Fourier Transform Spectroscopy

In this section, I will discuss the relationship between the detected interferogram and spectrum. The reason why the Fourier transform of an interferogram yields the spectrum is not immediately obvious, so here we derive the relationship between the two bases.
The detector integrates the electric fields over times that are effectively infinite when compared to the optical cycle. Therefore, the intensity on the detector is24

Z
I(x) =

x
∗α(t)Einc (t)Einc
t−
dt,
c

(2.16)

but

∗
Einc



x

x
t−
= Einc− t = Einc (t′ − t)cc(2.17)

with t′ = x/c. Therefore,Z′

EN F (t)Eref (t′ − t)dt.

I(t ) =

(2.18)Taking the Fourier transform of both sides givesZ′

I(t )e

it′ ω

′

Z Z

dt =

′

EN F (t)Eref (t′ − t)eit ω dtdt′ ,(2.19)

which can be simplified by adding a new variable s = t′ − t, such that ds = dt′ toZ′

I(t′ )eit ω dt′ =Z Z

EN F (t)Eref (s)eiω(s+t) dtds.

(2.20)

We can then separateZ′

I(t )e

it′ ω

′

Z

dt =

EN F (t)e

iωt

Z
dt

Eref (s)eiωs ds,(2.21)

which is identified asZ′

I(t′ )eit ω dt′ = EN F (ω)Eref (ω).

(2.22)

By taking the inverse Fourier transform

Z Z′

I(t )ewith the identity

it′ ω −it′′ ω

e

′

′′

dt dt =Z′′

EN F (ω)Eref (ω)e−it ω dt′′

(2.23)

25

Z Z

it′ ω −it′′ ω′

I(t )ee′′′dt dt =

Z

I(t′ )δ(t − t′ )dt′ = I(t′ )(2.24)

finally gives us

Z′

I(t ) =′EN F (ω)Eref (ω)e−it ω dt′ ,(2.25)

which gives us, with a couple of changes of units,Z
I(x) =EN F (eν )Eref (eν )e−i2πxeν deν.

(2.26)

It is now obvious that the detected interferogram I(x) is simply the Fourier transform of I(eν)Z
I(x) =

I(eν )e−i2πxeν deν.

(2.27)As previously alluded to, referencing the tip sample interaction to something that is spectrally flatis done in the frequency domain by division, such thatI(eν ) = EN F (eν )Eref (eν)(2.28)and so that

I(e
ν) =

αsample Einc (eν )Eref (eν)= sN F (eν )eiϕN F (eν ) ,αgold Einc (eν )Eref (eν)(2.29)

where sN F (eν ) and ϕN F (eν ) are the amplitude and phase spectra, respectively.

2.6.4

Artifacts in s-SNOMIn this section, I will discuss the source and types of artifacts in s-SNOM. In addition to
the artifacts mentioned above, I want to take a brief moment to discuss spectral and image interpretation. Spectrally, I have demonstrated how the uncontrolled phase between the tip scatterednear-field and far-field background is a DC offset in interferogram space, which is subtracted before

26
taking the Fourier transform to yield interpretable spectra. However, through the Fresnel coefficient in pump-modulated pump-probe spectroscopy, a Fano type interference can occur, where
the self homodyne signal can still lead to artifacts spectrally.69 This means that, in ground statemeasurements, care must be taken to observe heterogeneity on scales that are smaller than the
wavelength of light, and broad spectral features resulting from preshaping interference should be
ignored. Similar considerations for ultrafast spectroscopy must be taken and will be discussed in
the ultrafast method section of this thesis (Chapter 3).
In addition to interference from the scattering type nature of s-SNOM, topographic features
on a sample lead to artifacts in both spectra and images. Spectral artifacts result from the nonuniform near-field interaction based on a sample’s morphology. Simply the strength of the nearfield interaction can vary spatially from either local topographic height changes, the gradient, orthe second order topographic structure. For example, if the tip is on the back side (relative to the
input light) of a tall sharp cliff, then no far-field light can couple to it, and the signal drops, causing
the interpreted spectrum to be shifted as well. A more common example is when the tip is placed
over a small indent on the surface. Here, the surface enhancement is typically increased and again
shifts the spectrum. In terms of imaging, the changes in signal level will easily be manifest as well.
My preferred approach to analyzing these sort of data is to look at the phase response. This isbecause it is the least sensitive to amplitude changes, as the real and imaginary components are
the product of the amplitude with the respective trigonometric functions of phase. Therefore, caremust be taken when comparing amplitudes of signals, especially on rough samples.

2.7

Conclusion
In this section, I have detailed how the core components of a s-SNOM measurement are oper-

ated and considerations in its typical use. I then discussed the diffraction limit and the fundamental
physics leading to the constraint on measuring physical phenomena of interest at nanoscopic scales.
I detailed the optical sources that will be used throughout the work addressed in the rest of thisthesis and detailed the near-field optical interaction with two models that are frequently used to27analyze s-SNOM data. I concluded with a fundamental discussion of the source of the near-fieldsignal, its isolation, useful modalities, and spectral meaning and finished with sources of artifacts
that must be considered when analyzing data.

Chapter 3

Method Development

3.1

Introduction
In this chapter, I will discuss different approaches to s-SNOM that I have developed during

my PhD. The main advancements that I have made were in developing rotating frame, compressedsensing and matrix completion, and far from equilibrium excitation ultrafast pump-probe extensions
to s-SNOM. The rotating frame was developed by me and this portion of the thesis is adaptedfrom a publication.70 The compressed sensing and matrix completion was work in collaboration
with Simon Labouesse, where we combined my expertise in s-SNOM with his computer science
expertise to apply techniques common in information theory to our specific collaboration and isalso adapted from a previous publication.71 The ultrafast method development, being the mostinvolved project experimentally, involved input from a long line of students and post-docs fromfirst demonstrations to a robust imaging tool and technique. This project was started by Omar
Khatib and Sven Dönges, continued by Jun Nishida, who is the first author on the publication
after which this section is adapted, and myself. All individuals were directly involved in the theory
and development, with Nishida and myself performing experiments once the technique had been
demonstrated.69

3.2

Rotating Frame
To begin, I will discuss imaging in the rotating frame, where data acquisition speed is in-creased by a factor of ∼10.

29
While IR vibrational scattering scanning near-field optical microscopy (s-SNOM) has successfully been applied to applications ranging from biological to quantum materials, the acquisitionof full spatiospectral image sets continue to be hindered by the sheer length of these higher dimensional data sets. As the technique continues to advance, additional data sets from time resolved
measurements to tomography enabled by z dimension modulation and sensitivity will further exacerbate this problem. While non-Nyquist approaches have been proposed and will be discussed
in detail in the following section, we demonstrate here a novel approach, where an increase inacquisition speed is achieved by changing basis from the stationary frame to the rotating frame
at the lowest frequency in the IR carrier envelope. Specifically, we demonstrate here an increasein acquisition speed by a factor of 10 in a real direct comparative example. I stress the contrastof this example to theoretical proof of principle or post processing attempts by noting the actual
experimental demonstration of this approach. To do this, we use prior knowledge about the lightsource and resonances that will be probed. The example demonstrated here is applying R-sSNOM(rotating-frame s-SNOM) to a mollusk shell, where we image ultrathin protein sheets within theprismatic layer. We achieve high voxel density in our hyperspectral image to resolve sparsely distributed molecules within a large field of view. While we applied this approach to a biologicalsample to perform molecular vibrational imaging, this approach is general and is beneficial for any
system that has sparsely located features of interest and can further be applied to other higher
dimensional nanoimaging modalities.
Chemical imaging in a wide range of molecular and quantum systems has been performed
using IR s-SNOM.72–76 Specific to biological systems, which host structural features from proteinsat nanometer scales to cytoskeletons at multiple micron length scales, the active regions determining
the biological function and utility are frequently sparsely distributed.77–81 Here, we define sparsely
distributed as being separated by distances far greater than the size of the feature of interest

lseparation≫ 1.
lf eature

(3.1)

30
Therefore, a hyperspectral imaging method with a high field of view and spatial resolution over awide spectral range is desired. As data sets grow from dense (high resolution) voxels over large
dimensions (field of view and spectral range), associated problems to the long data sets like driftof the sample, light source, and detector further compound as well.82 During this time, if thebiological system is still active, the sample being imaged can of course change as well. Beyond
the simple annoyance of longer measurement times, these problems actively restrict future extensions for measurements. These constraints have limited s-SNOM’s applicability, particularly forbiomaterials.
Biological and chemical systems frequently need high pixel density imaging over wide fields ofview on composite samples. As a representative example, Mollusk shells, with their extended shellstructure, are largely composed of calcite and aragonite,83 two different polymorphs of CaCO3 , buthave a small amount of organic material thought to direct the growth of the shell interspersed in
the prismatic region. The organic material is typically less than 5% by weight and, in addition
to growth control, modifies the crystal habit and type of polymorph that the CaCO3 takes. The
sparsity of its inclusion among the vast biomineral though has remained difficult to locate andstudy. Macroscopic ensemble average techniques measuring de-calcified shells has determined theorganic fraction is at least partially made of proteins.84–87 Proteins and polysaccharides have been
extracted,88–91 yet nano-scopic characterization in situ remains elusive.
In an effort to address these questions and similar challenges in other similarly complicatedsystems, compressive sensing has been applied to broadband s-SNOM through synchrotron based
SINS.92 In this approach, prior knowledge about the data, specifically in what basis the data issparse (taken to be largely filled with zeroes), is used to reconstruct a sub-sampled data set. Thisapproach enables sub-Nyquist sampling and achieves similar resolution and image fidelity to thefully sampled data set. Compressed sensing yields a probabilistic reconstruction to obtain a full
data set from an incomplete one with far fewer actual measured data points. Similarly, nuclear
magnetic resonance (NMR) has used rotating frame techniques where the spectroscopic carrier
frequency is shifted out of the stationary frame, reducing the number of necessary data points to31collect a spectrum.93, 94 For a given desired spectroscopic resolution, faster detection is enabled byshifting the carrier frequency to zero and reducing the necessary Nyquist cut-off frequency. The
rotating frame technique has been extended to time-domain optical spectroscopy in the far-field toimprove acquisition speed and sensitivity.95–99
We have demonstrated rotating frame detection as an extension of s-SNOM in the application
to chemically sensing and resolving proteins within the prismatic region of an oyster shell. The
heterogeneous and sparse distribution of protein within the large field of view dominated by anextended calcite matrix was spectroscopically imaged by mapping the protein amide I IR response.

3.2.1

Rotating frame parameter space

Rotating frame is best applied in conjunction with an IR source of narrow bandwidth, so thatthe frequency shift is maximized. We choose to shift the zero-point frequency to the lowest energyof the laser spectrum. Mathematically, this can be understood as changing co-ordinates from ωto ω-ω 0 , which allows for significantly shorter spectral acquisition times to achieve higher spatial
data densities. Specifically, our direct comparison between R-sSNOM and conventional nano-FTIR
indicates an imaging speed increase of a factor of 60. This faster imaging enables higher spatial
resolution, limited by the radius of the AFM tip and near-field localization, over much wider fieldsof view conventionally accessible. For maximum speed enhancement, broad resonances (where thespectral resolution doesn’t need to be high) and narrow bandwidth light sources should be used.

3.2.2

Experimental implementation

The experimental setup is only slightly perturbed from conventional nano-FTIR, where achopper is included in the reference arm and a piezo mounted mirror is placed on a longer axis
translation stage (Figure 3.1a). The light source used in our demonstration is the tunable output
from difference frequency generation (DFG) between the signal and idler output beams from an
optical parametric oscillator (Levante OPO, APE) using a AgGaS2 crystal (Harmonixx DFG, APE)
that is pumped with a Yb:KGW femtosecond laser (1034 nm, pulse duration 90 fs, 6 W, 75.7

32
MHz repetition rate, Flint, Light Conversion). While the output is tunable, we only use one center
frequency for this experiment and direct 18 mW of 1680 cm-1 into a s-SNOM instrument (nanoIR2s prototype, Anasys Instruments). An off axis parabolic mirror (OAP, NA=0.45, Reflected Focal
Length= 25.4 mm) focuses the light to the apex of a gold coated AFM tip (160AC-GG OPUS,µmash). We use intermittent contact mode to modulate the near-field signal for lock-in extraction
later. The back scattered light is collected with the same OAP and heterodyne amplified withreflected light from the reference arm to be detected by a HgCdTe (MCT KLD-0.5-J1/DC/11,
Kolmar Technologies) detector. In conventional nano-FTIR, the Fourier transform of the resulting
asymmetric interferogram yields the complex valued, phase resolved optical response of the tip
sample interaction with spatial resolution given by field confinement and primarily the radius of
the tip apex.
To extract the near-field, the MCT signal is demodulated at the second-harmonic of the tip
tapping frequency ω tip using a lock-in amplifier (HF2LI, Zurich Instruments), as described above.
When performing R-sSNOM, a chopper in the reference arm is used and the sideband 2ω tip + ω refof the near-field signal is collected to eliminate far-field background self homodyne, where ω ref isthe chopping frequency. We follow established procedure by normalizing the spectrum to a goldreference, whose flat optical response leaves the particular focus, light source, and tip parameters
available to be normalized out of the measurement. Since we are in the water absorption window, we
eliminate artifacts from water vapor by purging the entire instrument with dry air. In addition to
the reference mirror position being controllable with a piezo, the macroscopic longer range position
was controlled with a linear direct drive translation stage (Aerotech, ANT95-50-L-MP).

3.2.3

Sample preparation

A widely studied model mollusk (Crassostrea gigas) has been characterized by its prismatic
layer, which is a distinctive microstructure made of calcite columns that are closely packed and are
more than 10s of microns tall.100 In the cross plane (the dimension measured here), the columns are∼10 µm across with ∼1µm thick gaps made of thin polygonal sheaths of intercrystalline organic

c
ωtip
BS
ωref
Chopper

Piezo Mirror

Delay Stage
ωvib

ωvib

Frequency (a.u.)
Ieven(ω-ω0)x
R (ω0)I2(ω)

Iodd(ω-ω0)

I1(ω)

d 150
100
50
0
-50
-100
-150
0

Laser
nano-FTIR
nanoNL-FTIR

INF (µV)y ωrot

b

Γvib

ωrot

e

33
150
100
50
0
-50
-100
-150
0

f
~ (a.u.)
A(ν)

Laser

Resonance
Laser

AFM

Amplitude (a.u.)

MCT

RNL-SNOM
R-SNOMINF (µV)a

50 100 150 200 250 300
Delay (µm)

1.0
0.8
0.6
0.4
R -SNOM

50 100 150 200 250 300
Delay (µm)

NL
0.2
R-SNOM01550 1600 1650 1700 1750
Frequency (cm-1)

Figure 3.1: Rotating Frame s-SNOM (R-sSNOM) | a, Experimental design with broadband lightsource and asymmetric Michelson Interferometer with necessary modifications for rotating frames-SNOM. b, Schematic of the rotating frame reducing the frequency of the basis waveforms (redand orange) to generate new waveforms (blue and purple). The same information is encoded, butwith the beat frequency between the rotation and basis. c, Resonant IR feature within narrowlaser spectral bandwidth. d, Experimental interferogram (red) with a factor of 10 sub-sampleddatapoints (blue). e, Low frequency interferogram generated by rotating red experimental data by
1550 cm-1 (red) and the similarly rotated sub-sampled points (blue). f, Frequency comparison of
stationary and rotating frame spectral response.70

material. The c-axis of these columns are homogeneously aligned to within 20 degrees and are
perpendicular to the polished surface.101–103 To prepare the samples for IR imaging, the polishing
step was followed by a short ∼3 second 0.1 M HCl etch, re-exposing the organic sheath.

3.2.4

Shifting frequencies

While a faster acquisition method, the rotating frame uses the lowered Nyquist cut-off frequency to reduce the number of data points necessary to resolve a spectral feature. The rotating
frame approach is based on the linewidth Γvib , center frequency ω vib of the vibrational resonanceto be probed (Fig. 3.1c), and rotation frequency ω rot where ω rot is less than ω vib and is typicallyω vib − Γvib or smaller. This concept is illustrated in Figure 3.1b, where the original basis vectors(red and orange) that span the rotation space are collected conventionally with the exception of

34
larger reference arm step sizes (3.1a). The computational aspect of rotating frame can be thoughtof as a complex apodization of the acquired datasets (green spiral). The newly visualized (dark andlight blue) have an apparently lower frequency but span the same space. This can also be thoughtof as spinning about a bloch sphere, where the vibrational resonance precesses with frequency ω vib
around the precession axis at the rotation frequency ω rot . Going to the rotating frame subtracts off
this rotation frequency ω vib -ω 0 so that the near-field signal interferogram only needs to be collectedin steps of 1/Γvib up to the distance set by the desired spectral resolution. Typically, the necessary
step size is based on the maximum frequency you want to be able to recover, but with the rotating frame, it now depends on ω vib Γvib and ω rot while still being Nyquist limited. The example
given here pushes these boundaries to the extreme to best illustrate the speed increase that can beachieved. Moving forward, to avoid aliasing and to ensure good quality spectra, a lower rotation
frequency should be selected and a larger Γ should be assumed.

3.2.5

Proof of principle

To demonstrate the applicability of this technique, we simulate the process by sub-sampling
a conventionally acquired nano-FTIR dataset. A standard nano-FTIR interferogram collected in
the stationary frame is shown in figure 3.1d (red). A ∼30x sub-sampled dataset is shown alongin blue and in the stationary frame seems to not be a good representative of the full dataset,upon transitioning to the rotating frame, both interferograms are in clear close resemblance (Fig.
3.1e). Upon Fourier transformation and carrier frequency inverse shifting, we observe good spectral
agreement between the R-sSNOM and Nqyist limited R-sSNOM spectra shown in 3.1f, red and blue
respectively. In this specific example, chemical imaging is increased by 60 fold while maintaining
voxel integration time so that the signal to noise ratio is the same between the two methods.

3.2.6

Imaging protein matrix

With the proof-of-concept pre-calculation achieved, R-sSNOM is demonstrated as a full hyperspectral imaging method to demonstrate the reduction in sampling needed when using the35rotating frame. To put rotating frame into action, the full field of view is imaged at each reference
arm delay point with spacing described above. To perform the rotation, we need two basis vectors(in and out of phase) to span the rotated space, which we achieve by performing chopper demodulated 2-phase homodyne imaging at each selected reference arm delay using previously described
procedures.81 To spectroscopically resolve voxels, we repeat the two-phase homodyne imaging at
each reference arm, stack the images, correct for spatial drift, dropped pixels, and Fourier transform the interferogram collected by isolating single spatial positions within the hyperspectral data
cube, and using the rotating frame algorithm described above. We use this technique to image
4000 voxels with 100 nm spatial resolution, 20 cm-1 spectral resolution, and 200 cm-1 bandwidth.
To directly compare conventional nano-FTIR to R-sSNOM imaging, we map the protein
locations within an oyster shell to the prismatic region. The spatial distribution and extent ofproteins within the prismatic region is unknown and the hierarchy of scales being imaged here isshown schematically in Figure 3.2a.
By centering the laser at the peak of the amide I response in this sample (determined byconventional nano-FTIR spectroscopy) at 1680 cm-1 , we maximize signal overlap for the spectral
response, while maintaining maximum rotation for the largest speed increase. A 4 µm by 10 µm
field of view was selected as it covered two organic sheets. To image this region with 100 nm by
100 nm spatial resolution in R-sSNOM, we use 40 by 100 pixels (Fig. 3.2b). This is in contrast
to the conventional stationary frame nano-FTIR experiment (Fig. 3.2c), where a much moreconservative 10 by 30 pixel array was used to yield ∼10 times fewer spatial points to match thesame acquisition time (to maintain SNR comparability) with identical spectral resolution. Halfway
through the conventional scan, an AFM image was collected in order to characterize sample drift.
In the conventional image, the entirety of the ∼1 µm gap between calcite columns was mappedby only 3 voxels. The spectra in this measurement need to be locally referenced against a calcite(protein free) spectrum as the large spectral contribution of calcite interferes with the detection of
the amide I response. The corresponding full spatio-spectral R-sSNOM phase data set is shown infigure 3.2d. A single example spectrum of the protein from the organic region is further shown in36aprismatic
layer

organic sheath

130 nm
10 µm

calcite
prism

0 nm
1 cm
C. gigas shellc
Rotating Frame Stationary Frame
2 µm

1.0
Normalized Near-field Intensity (a.u)

2 µm

0.0

25

1720

20

1700
1680

15

(deg)

b

1660

10

1640
80

1620

Pix 30
el 20
nu 10
mb
er

40
20

Pix

60
r
mb e
el nu

5

nano-FTIR
R-sSNOM

0(cm-1)

Figure 3.2: Rotating frame R-sSNOM imaging | a, The prismatic layer (middle schematic, thickness
exaggerated) surrounded by organic sheaths (right). b, Near-field scattering amplitude heterodyne
amplified at zero phase difference between tip and reference arms in the rotating frame and c,stationary frame. d, Spatio-spectral phase image of same region. e, Representative spectrum oflocation indicated in panel d showing typical amide I response of the organic sheath.70

figure 3.2e.
We now use the R-sSNOM amide I phase (Fig. 3.3b) with the AFM topography (Fig. 3.3a)
to locate and map where the protein content is through correlation analysis and identifying 5
distinct regions based on the calcite and amide with the relative surface height (Fig. 3.3c). The
etching process, which yields higher organic features relative to calcite, compares well the locationof proteins in the prismatic region, yet R-sSNOM indicates that the extent of the protein is asmaller fraction of this relieved area. The distribution of amide phase and height data points canbe seen as a correlation plot in figure 3.3d.

37

b Φ°

(nm)
2 µm

2 µm

25

400

20

300

15

200

10

100
0

5
0

c

d
2 µm

Amide

MatrixΦ° (1680 cm-1)

a

Amide

40
20
0
-20
-40
-60

Calcite
0

200

400

600
800
Height (nm)

1000

1200

Figure 3.3: Comparison of Imaging Methods | a, Topography from AFM. b, Strength of near-field
phase response at 1680 cm-1 . c, Correlation comparison of sample topography and amide phase
response. Points of high value indicate high correlation, and points of low value indicate a deviation
between topography and the local Amide population (see color-bar). d, Correlation plot of heightand phase dependence between panels a and b (see color-bar).70

3.2.7

Drift and artifact compensation

The sample, in both R-sSNOM and conventional nano-FTIR, drifts during the acquisition ofdata. The magnitude of drift is given by the thermal instability of the AFM stage and is typicallymore than at least 1 nm/min. In conventional nano-FTIR, drift is typically accounted for by taking
one or more reference images throughout the spectra collection process and using a fiducial marker
on the sample as a tracking point to then correct for drift. However, because the data in R-sSNOM
is collected through imaging with a stationary reference arm, rather than scanning the reference
arm with a stationary sample, we simultaneously collect the height data channel. We then use thischannel, with a cross correlation analysis in Fourier space, by using the first mirror step position
as the reference point to then shift and stack the additional images on top. To calculate the shift,
we search for the pixel with the highest cross correlation and generate a shift vector. We repeat
this for every reference arm position and create a dictionary to stack the X and Y data channels aswell as the larger translation stage steps. Once the images are shifted and stacked on top of each38
other, we crop all of the images so that only the field of view contained within every page in thestack remains.
During the data collection process, occasional data points are lost based on the pixel quality
for every mirror position resulting from instabilities in the laser output, chopper frequency, AFM tip
frequency and their relations to each other during the demodulation process. Inevitably, occasional
instabilities lead to a loss of reasonable value for at least a pixel, resulting in an unusually highvalue. The value typically skyrockets based on a step in the demodulation process losing lock on an
input demodulation frequency, searching for the frequency at lower values, resulting in an increasein signal from 1/f noise. Fortunately, these errors are easy to catch and therefore are correctedby implementing a multistep algorithm. The implementation of 2 phase homodyne results in a
duplicity of scans for each X and Y value as the AFM is raster scanned in a trace and retrace
direction for both X and Y. Therefore, when a pixel is found to be lost in one channel, if thesame spatial location exists in the Y channel, we substitute the redundant point. We perform
this for both trace and retrace and average the resulting two images to increase our SNR. Next,
if pixels remain where the value is still above a threshold, we interpolate the dropped pixel by
averaging the surrounding 8 pixels. Finally, for the few remaining pixels that are clearly off by eye,a conservative threshold is imposed and capped with the maximum value. Finally, we repeat thescan for subsequent thresholds until the image converges.
While the data in conventional s-SNOM is collected in a different order (string wise versus
page wise) than R-sSNOM, the interferograms are processed in the same way. First, the data vector
is zero-padded to a length that is a power of 2, increasing the length of the data by almost a factorof 4, effectively interpolating the spectra with a sinc function. The interferogram is then apodized
with a Blackman-Haris function to increase the SNR of the spectrum. The data is then shifted to
put the center-burst in the middle and Fourier transformed with Matlab’s FFT algorithm. The
spectrum is normalized with a reference spectrum (calcite as mentioned above) by dividing thesample spectrum with the reference spectrum. The spectra are complex, so effectively, amplitude
is divided and phase is subtracted.

39
To increase data fidelity (artifact rejection – see chapter 2 of this thesis), we modify theconventional s-SNOM setup by inserting a chopper into the reference arm, so that sideband demodulation can be used to directly isolate the heterodyne near-field signal. We give the choppera frequency ΩREF and choose chopper blades so that the beam size is comparable to the blade
size in an effort to best sinusoidally modulate the optical signal. Secondly, a piezo actuated mirror
mount is added to the reference arm to have fast, reproducible, stable, and bidirectional accuracyof our mirror position at mesoscale distances to retrieve both the X and Y images to span therotation basis. We use this in conjunction with the longer-range linear stage, which yields spectral
resolution.

3.2.8

Robust imaging

We apply this concept to an image (Fig. 3.4) with many more voxels of data and map thesurface position in the x and y dimensions and the height along the z dimension while coloring thesurface with the relative strength of the phase response at 1680 cm-1 , where the amide responseis greatest. This image is in good agreement with the preferential etching performed to relatively
raise the organic material.

Height (nm)

50
200
0

25

-200
0

0

m
Sa
ple

2000

4000

X
(n

2000

m

4000

)

0

Sa

le
mp

Y

6000
)
(n m

Figure 3.4: Larger field of view R-sSNOM | Additional rotating frame data set acquired in theprismatic region of an Oyster shell showing a high protein concentration in a V-shaped region as
determined by the Amide I response.70

40
3.2.9

Alternative chemical mapping approaches

Previous work has shown spectroscopy performed using tunable single-wavelength sources.32
In these experiments, either self-homodyne or white light imaging can be performed. Unfortunately,
the signal is still not background free, and the spectra obtained are difficult to normalize andinterpret due to the frequency dependent power and stability of the light source. For proper hyperspectral imaging, a broadband source should be used in conjunction with a technique that allows
for frequency dependent imaging to occur as well as a method for being background contamination
free.

2 μm

25
20

b

15

2 μm

10
5
0

c

12
8
4
0

16
12
8
4
0

ΦSF ( d e g )ΦRF ( d e g )30Phase (deg)Amide I (SF) Amide I (RF)a-600 -300 0 300 600
Position (nm)Figure 3.5: R-sSNOM and conventional nano-FTIR spatial resolution | a, Rotating frame phase
response at 1680 cm-1 . b, Stationary frame phase response at 1680 cm-1 . c, Feature edge comparison
between rotating and stationary frame Amide I phase response at regions indicated in panels a andb.70

Most spectral mapping in s-SNOM has followed similar variants to conventional techniques.32, 78
The typical pattern goes as using the topographical channel to determine a region of interest, thena series of tip sample positions are identified where the tip is moved, reference arm swept, and41
spectra collected before moving onto the next point in the list. This creates the 3D data cube,
where a spectrum is associated with each spatial position on the sample. While this is intuitive,
there is quite a bit of ”dead” time and does not scale to higher dimensions well. Further, as mentioned above, this method requires the AFM tip sample locations to be selected ahead of timeand is assumed to remain valid at later times, an assumption often fraught with problems fromthermal drift. Moving into the rotating frame, combined with techniques made for CW lasers,104
the data collection is more targeted, improves with dimensional scaling, and automatically corrects
for sample drift.

3.2.10Spatial resolution

To directly compare the difference in achievable spatial resolution and how it is important forthe demonstrated comparison to conventional s-SNOM, we show line-cuts of the amide slice images
in figure 3.5 and see the slopes at the edge of a feature of interest are markedly different. This
represents a non-trivial improvement on previous hyper-spectral imaging, where the fundamental
resolution is given by the AFM tip radius, but is practicably limited by the acquisition time andassociated problems.

3.2.11

Spectral Artifacts

The spectrum shown above comparing R-sSNOM and conventional nano-FTIR are quite
similar but do exhibit obvious differences, with the phase reconstruction showing larger discrepancies. This results from the inversion symmetry associated with the phase of the Fourier transform.
Briefly, in the rotating frame collected at the Nyquist frequency cut-off limit, this symmetry necessitates a turning point near 1750 cm-1 , which, because of how close this example is to the limit,
contaminates the absolute spectral position and shape of the extracted amide I feature.
The spectral deviation between the conventional and rotating frame datasets is large here fromthe choice of measuring at the Nyquist limit – by choosing a larger effective bandwidth (decreasing
step size) for the rotation, or choosing less broadband light sources and narrower resonances, this42
effect can be minimized. Here, the resonance is quite broad (comparable to the laser bandwidth)and our choice was sufficient for mapping the protein distribution. The shift of the apparent peak
position by ∼10 cm-1 does not affect the identification and mapping in this system. Amide I peak
positions also vary in center frequency and the variation within biological samples is frequently
larger than the deviation observed here.

3.2.12

Theoretical formulation

We now consider the mathematical formulation for imaging in the rotating frame in directcomparison to conventional nano-FTIR. We then compare the two-phase homodyne implementation
with a pseudoheterodyne approach. The near-field heterodyne amplified intensity in both the x
and y demodulated channels in frequency space is given by equation 3.2
N STAT −1

I x (k) =X−2πikx̃n (t)e N
I 2H (x̃n (t))

(3.2)

n=0

Here, NSTAT is the number of data points collected along the interferogram, k is the output
frequency of the transform, x̃ is the position of the interferometer arm (where n is the sampling
number — position index).70 The complex valued response is normalized to a background reference
signal and normalized via division. The Nyquist sampling theorem provides the smallest value that
NSTAT can take, which, for our light source centered at the amide I resonance and a desired spectral
resolution of 20 cm-1 , is 181. In the rotating frame though, the relative optical frequency is thevalue of interest, and is shown in equation 3.3
N ROT −1

I x (k − k 0 ) =X−2πi(k−k0 )(x̃n (t)+ϕ̃)e N
I 2H+Ω (x̃n (t)),

(3.3)

n=0

where we have included the subtracted frequency k0 , a mirror position offset ϕ̃ (controlled
by the reference mirror piezo) and Ω the chopper frequency. The minimum value of NROT is still
given by the Nyquist sampling theorem, where x̃n − x̃n-1 determines the maximum cut-off frequency,
which we choose to be approximately 3 ∗ Γvib . Therefore, the minimum value NROT can take (for

43a resolution of 20 cm-1 ) is 21 — a reduction by almost an order of magnitude. While the minimum
sampling number is lowered by approximately an order of magnitude. The typical implementationof s-SNOM though wouldn’t be able to take advantage of this reduction as the reference arm mirror
would still need to be scanned through the now redundant space. Therefore, the surface scanningtechnique for x and y extraction is borrowed from previously established two phase homodyne.
Because the implementation of rotating frame, for maximum acquisition speed enhancement,
requires surface imaging at one reference arm mirror position, the standard DC subtraction (throughline fitting mentioned in chapter 2) cannot be performed. In its place, we perform sideband detection
off of the second harmonic near-field signal with the chopper frequency added to the reference
arm. Two phase chopper demodulated homodyne has previously been used81 for single wavelength
sources, where the amplitude and phase could be extracted from the collected real and imaginary
channels collected at the two phase positions. Image math then made the computation of amplitude
and phase possible for imaging. We use this technique with a broadband light source by scanning
the reference arm between image acquisition.
Figure 3.6a and b illustrates the possible experimental sideband detected data. The basis
vectors behave normally (comparable to the standard s-SNOM approach) and slowly rotate toproduce interferograms in the rotating frame. Pseudoheterodyne, while faster, is more difficult toimplement with a broadband light source. The difficulty here is the normalization or amplitude
drive condition that is necessary to set the real and imaginary components between the first andsecond sidebands. While technically possible, the real-life implementation of this approach is very
sensitive to drift and not a practical way to implement R-sSNOM.105 In pseudoheterodyne, thedetected signal is given by

1.0

Real
Imaginary

0.5
0.0
-0.5
-1.0
0

50
100
Delay (μm)

150

1.0

1st SB
2nd SBRot Real
Rot Imaginary

50

100

150

Rot 1st SB
Rot 2nd SB

Intensity (a.u.)

1.0

0.5
0.0
-0.5
-1.0
-100

0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

d

c

Normalized Intensity (a.u.)

44bRotated Intensity (a.u.)

Normalized Intensity (a.u.)

a

0.5
0.0
-0.5
-1.0

100

-50
0
50
Delay (μm)

-100

-50
0
50
Delay (μm)

100Figure 3.6: Method of Accessing Rotation Basis | a, Experimental data of two typical interferograms as two basis vectors to span rotation space collected using the fine delay mirror control. b,
Rotated interferograms of experimental data in panel b. c, Resonance-less model data generated
by computing pseudo-heterodyne data collected by demodulating at the second harmonic of thecantilever frequency. d, Rotating frame applied to interferogram basis vectors acquired throughpseudo-heterodyne.70

Z ∞I(ω) =−2σ 2 ϵ2 Z ω
2dte−iωt e c2dω 0 e−iω0 t

−∞

ω1

−2σ 2

−i2ω 0 ϵe c e c2
[J 0 (2ω 0 ξ) + 2(ϵξsin(mt+ϕm )+ξ 2 sin2 (mt+ϕm ))∞Xαeff (ω 0 , t)∗(3.4)(J 2n (2ω 0 ξ)cos(2n(mt + ϕm ))+n=1

iJ 2n-1 (2ω 0 ξ)sin((2n − 1)(mt + ϕm )))]The nontrivial interplay between laser specification, specifically the center frequency ω 0 andimportantly bandwidth σ, and detection parameters like oscillation amplitude ξ and phase offset
ϕm has on the extracted near-field response αeff makes teasing out the wavelength specific real and45
imaginary optical signals practicably impossible.
Specifically, the mirror DC position, center frequency of light source, and bandwidth cause thereal and imaginary near-field response to mix as the mirror moves away from the ZPD. The sensitive
behavior is shown in figure 3.6c. The high frequency oscillations of the signal in the rotating frame
is not recoverable with larger mirror step sizes and we therefore implement multi-step chopperdemodulated two-phase homodyne for complex valued basis and self homodyne subtraction.
To eliminate the background term in pseudoheterodyne, and to simultaneously extract thereal and imaginary optical response, sidebands are generated via the Jacobi-Anger expansion, where
the argument of the electric field is linearly dependent on the sinusoidal motion of the reference
mirror. These sidebands are symmetric about the second harmonic of the tip tapping frequency (orany higher order near-field harmonic) with signal strength dependent on the Jacobi-Anger expansion
coefficients. In the particular case of a single wavelength source, pairs of neighboring sidebands (1and 2, 3 and 4, etc), with appropriate drive strength, can be matched to have the same maximum
amplitude, where signal in one channel is exchanged with the other through a phase offset. Thisis implemented experimentally by selecting a light source and oscillation frequency and tuning thedrive strength, while adjusting phase, to the point where the maximum achievable values are thesame for both sidebands.
We see that in addition to the difficulty of scaling the waveforms generated by pseudoheterodyne (and it being phase dependent — impossible to do in general for a broadband source),
there is a complicated secondary envelope function for the two sidebands. Again, while it is theoretically possible to deconvolve this mixing, the experimental nature of crucially determining theseparameters with stability on the few to 10s of hours timescale is simply not practical. Additionally, assuming the deconvolution could be done, the Jacobi-Anger expansion splits the near-fieldsignal into an infinite series of sidebands, diluting the signal to channels that are discarded anddecreasing the relative signal to noise compared to a method that creates sidebands of finite (orfaster converging) order. From these considerations the choice for background free imaging between
pseudo-heterodyne and chopper demodulation should clearly be the latter.70

46
3.2.13

Applying rotating frame for chemical imaging

The outer organic coating, periostracum, and inner chalky and foliated layers encapsulate theprismatic layer. The prismatic and inner layers are primarily made of calcite. Sampling portionsof the shell was done with a rotary tool (Dremal) with a diamond-impregnated disc to create an
oblique surface relative to the columns. To chemically map the protein distribution, the spatial
resolution needs to be high as the protein deposits are nanoscopic, but the spectral resolution
can be comparatively low. Here is the perfect application for R-sSNOM where these tradeoffs areleveraged.
In the rotating frame, the total scan length is set based on the desired spectral resolution
and the step size is set based on the Nyquist cut-off frequency, which, to first order, is givenby the bandwidth of the light source. The larger step sizes enabled by the reduced maximum
necessary cut-off frequency enables faster data collection, which can be leveraged by the scanning
method detailed above. This simultaneously, during the stacking process, enables passive driftcompensation. Additionally, the static axis of the measurement is shifted from the sample to thereference arm, where the reference mirror only needs to be stable on the ∼µm scale instead of the∼nm scale, enabling faster and more stable hyper-spectral chemical imaging.
Here, we use the same integration time per spatial point per reference mirror position andstill see a speed increase by a factor of ∼60 fold. To enable drift correction in a conventionals-SNOM experiment the sample must be imaged, conservatively, every 50 data points, which for a
100 by 100 pixel image (10,000 spatial positions), the re-imaging time alone (regardless of the timeneeded to actually collect spectra) is more than an order of magnitude more than the complete
data acquisition time for R-sSNOM. Here, for the conventional approach, we select 20 cm-1 spectral
resolution and scan the reference mirror at 10 µm/s, taking 25 seconds per spatial position. To
complete 300 spectra, the total imaging time is 2.3 hours, whereas the same field of view is measured
with R-sSNOM with the same spectral resolution, but 4,000 spatial positions in slightly less time.
Therefore, R-sSNOM has more than an order of magnitude in improvement in the magnitude

47of acquired data points for the same scanning parameters, yielding a drift corrected and higher
spatially resolved image to see more structural detail.
Similar to the issue of being right at the Nyquist limit mentioned above, the first few spectral
datapoints following rotation can be affected by the dual of negative frequency components fromfrequency folding. We therefore choose to rotate this dataset at 1550 cm-1 , which, with our selected
step size, gives a maximum recoverable frequency of 1750 cm-1 .
The capabilities of R-sSNOM are well matched to the needs of a nanometer convoluted
organic-mineral interfaces sparsely distributed sample. The discrepancy between the feature lengthscale and the distribution length scale indicates the necessity of high-resolution large field of view
imaging. In this example, we see a strong improvement in image quality, speed, and resolution witha generic approach that is ideally matched for chemical imaging in diffuse systems.

3.3

Compressed Sensing and Matrix Completion
I will now discuss compressed sensing (CS) and matrix completion (MC) as the next approachto improving data acquisition speed. While the rotating frame is Nyquist limited, CS and MS
operate below the Nyquist limit. Similarly to the rotating frame, prior knowledge about the sample
and light source is used in CS and MC to reduce the total number of data points. This work was
performed in collaboration.71

3.3.1

Conventional s-SNOM

While s-SNOM has been successful in providing for spectroscopic imaging of molecular andquantum materials with sub-diffraction imaging at the few nanometer scale, conventional datacollection approaches are too slow to simultaneously capture highly resolved images over large fieldsof view. Weak resonances and light sources with limited spectral irradiance further exacerbate thisissue as longer integration times are required per voxel within the multi-dimensional spatio-spectral
dataset. As additional dimensions are added to s-SNOM imaging, like a delay axis for pump-probe
ultrafast time resolved measurements, the time it takes to collect a complete dataset without

48
sacrificing resolution along at least one axis is prohibitively long, requiring up to days. Sampling
exactly at the Nyquist limit is only possible when the signal to noise ratio (SNR) is sufficiently high.
However, an adaptive sampling method with compressed sampling and matrix completion enables
a significant reduction in the necessary sampling rate, while maintaining data fidelity for chemical
imaging and physical interpretation. We demonstrate this reduction in sampling rate through theuse of complete datasets that were experimentally collected of a wide range of material systems,
ranging from biological, to molecular, to quantum. We use these experimental datasets as theground-truth and then apply deep under-sampling with recovery algorithms to reconstruct the full
dataset. This approach demonstrated below is more than one order of magnitude faster compared
to conventional s-SNOM and is 6 times smaller than the Nyquist limit, enabling a 30-fold reduction
when compared to standard (300 nm stage sampling) methods. Compressed sensing and matrix
completion with an adaptive sampling algorithm is fully generalizable and allows for high spatial
resolution imaging over large fields of view.
Measuring vibrational and phonon resonances in molecular to quantum materials at thenanometer regime can be achieved with s-SNOM.14, 15, 106–109 s-SNOM has been advanced to elucidate fundamental nanoscopic interactions that combine together, creating emergent phenomena
at the meso- and macro-scopic scales. The combination of fundamental length scales together withthe length scales where emergent phenomena occur require an imaging technique that simultaneously resolves nanometer features over micron distances. Conventional s-SNOM (I primarily meannano-FTIR here, figure 3.7) is performed spatially across a sample (two dimensions: x,y) with spectral resolution, enabled by Fourier transform of an interferogram (one dimension: mirror position).
While different imaging techniques exist to retrieve spectral information, the use of broadband IR
light sources, like thermal sources, synchrotrons, and pulsed lasers, is ideal for measuring multiple
vibrational modes, with the typical trade-off of brilliance for bandwidth. Low repetition lasers arefurther difficult to use as a sample’s exposure to them can cause the sample to degrade or become
damaged – reducing the total fluence available, which further limits SNR. The SNR ultimately
limits the acquisition rate of s-SNOM, which is compounded for larger multidimensional datasets.

49
In order to simultaneously image multiple vibrational resonances throughout the mid-IR, the illumination source must be ultra-broad or broadly tunable from ∼400 cm−1 to ∼5000 cm−1 .110–114
Hyper-spectral imaging time is inversely related to the near-field SNR and can ultimately be days
to weeks long.

3.3.2

Methods for improving acquisition speed

As mentioned above, alternative methods, like rotating frame, have been introduced; yet,datasets acquired using those approaches continue to have redundancy in the collected data.70, 115
Previous compressed sensing has shown nano-FTIR acquisition time reduction by ∼ 70%, where thestandard spectral sparsity in the Fourier domain was utilized.116 However, s-SNOM datasets also
exhibit spatial redundancy, where additional previous work on spatio-spectral nano-FTIR datasets
found reductions of 89 % possible.117
Image reduction techniques, while intensively used for other hyperspectral images, have seen
limited application for s-SNOM.118–120 Such examples of hyperspectral problems include the famous
movie recommendation problem, system identification in control theory, and global positioning
sensors.121–124 In these approaches, matrix completion has relied on the small number of individual
constituents present within a dataset.125, 126 We can apply this chemical species being measured
in s-SNOM. We wish to reduce the necessary acquired data without losing physical relevance ofthe reconstructed data by using an adaptive sampling algorithm specific to the nature of s-SNOM
measurements and by using matrix completion and compressed sensing with prior knowledge about,first, the light source, second, spectral sparsity in the Fourier domain, and third, the relatively small
number of constituent chemical (or generally resonant) species.
We design a general reconstruction algorithm by optimizing both each penalty enforcing priorknowledge individually and the combination of optimization penalties on concert for each s-SNOM
dataset. With these considerations, we are able to reconstruct s-SNOM hyperspectral images with
only 3.4% of the original fully sampled data while maintaining data fidelity for physical interpretation. We combine this reconstruction algorithm with an adaptive sampling algorithm that selects

50
mirror positions for each tip sample location. This extends random sampling, which is a universally
accepted strategy for compressed sensing and matrix completion.127, 128 This sampling strategy uses
an estimate of the normalized envelope of the interferogram center burst and free induction decay(FID) based on the reconstruction of measurements at previous tip-sample locations. This emphasis on the center burst and FID preferentially selects for the highest SNR data points and most
relevant data points for Fourier analysis (figure 3.7B).

3.3.3

Spatio-spectral imaging

The fully sampled real experimental datasets that we use to study the performance of oursmart s-SNOM approach are biological, quantum, and molecular materials.72 We then sub-sample
these ground truth measurements with adaptive sampling and apply the reconstruction algorithm
to recover the rest of the data using prior knowledge about the sample and light source.
The high resolution over large field of view requirements for s-SNOM datasets are well suited
to the strengths of compressed sampling (CS) and matrix completion (MC). In conventional nanoFITR, a series of AFM tip-sample locations are selected on a sample, typically in a regularly spaced
rectangular grid. The sample is then moved under the AFM tip to the spatial position of interest
and an interferogram is acquired by scanning a reference mirror. Each point in an interferogram isthe data point from the optical interaction for one optical delay between the sample and reference
arms. Experimentally, this is accomplished by scanning the reference arm at a constant speed withdata points collected at regularly spaced delays with a trigger from the stage or reference arm
position sensor.129 Most of the data collected in an interferogram is redundant and therefore only asmall fraction of the total interferogram datapoints are needed, enabling us to only use some of themirror positions that are conventionally acquired. We use sample and light source prior knowledge
to extract the rest of the datapoints.
Features of interest in infrared spectroscopy are generally categorized into two sets: resonant
and non-resonant. Resonant features are typically thought of either single Lorentzian responses, ora Gaussian sum of Lorentzians. Non-resonant features of interest in the mid- to far-IR are typically51

Figure 3.7: Smart s-SNOM schematic | (A) light source laser L or synchrotron S, beam splitterBS, scanning mirror M on reference arm. Sample on scanning stage under the tip T, point detector
Mercury Cadmium Telluride or MCT. Hyperspectral images can be acquired by scanning the mirror
and the sample. In case of a 2D grid scan or of a 1D line scan of the sample the final output isrespectively a 3D or 2D hyperspectral image. If the sample is not moved but only the mirror
is scanned the final output is a 1D spectrum. B) Diagram of smart s-SNOM. Our contribution:
smart sampling system, provides a list of positions of the scanning mirrors for each sample point
and reconstruction algorithm, to estimates the value of the missing data and hence recreates afully sampled hyperspectral image X. At each new spatial position of the sample an intermediate
reconstruction is calculated in order to evaluate an averaged envelope H(X) of the interferograms.
H(X) is normalized and used as a random distribution to select the mirror positions for the next
sample position.71

represented by the broad Drude response. Here, we restrict our analysis and application to spectrally significant resonant responses coming from either molecular vibrations or lattice phonons.
In either case, the identity and environment dictate the frequency of the resonance. Each inter-52
ferogram, taken at a selected tip-sample location, represents the optical response from a collectionof species, which, upon Fourier transform, gives a sum of the vibrational spectra of the underlying species. Resonances in spectra are sparse, meaning only the spectral peak frequencies yield anon-zero signal. Their resonant nature further limits the bandwidth of non-zero values. Using theconcepts of compressed sampling, the total number of necessary mirror positions in nano-FTIR
can be reduced,116, 117 where established acquisition and reconstruction techniques are used.130–132
The underlying principles of compressed sensing though requires 1) sparsity of the data in a domain distinct from the acquisition domain (how else could sampling be increased other than simply
measuring where the signal is in the original domain) and 2) mathematical incoherence (each interferogram data point depends on the corresponding sparse domain data point differently) of thesensing matrix. The sensing matrix here is the combination of random sampling and the Fouriertransform, which has previously been shown to be incoherent.133 With these conditions satisfied,
CS can be used on in the spectral (interferogram) dimension.

3.3.4

Compressive sensing and matrix completion

A further, and stronger for pulse laser sources, sparsity comes from the limited bandwidthof the light source relative to the entire domain of the Fourier transform. s-SNOM in the rotating
frame has already demonstrated how this prior knowledge can be used to greatly reduce the numberof necessary data points in the mirror dimension, but applied this knowledge in a fundamentally
distinct way from CS.70 At frequencies where the light source fluence is negligible, the spectrum
values are effectively zero. Therefore, the spectrum can be truncated around the known frequency
range of interest, increasing the compression factor and computation speed.
In addition to the sparsity that is used for CS, we know the number of individual elements
giving rise to a resonant response is limited, which implies the measurement matrix will have a small
rank. This assumption becomes stronger for larger data sets where more spatial points are collected.
With a linear mixing model (where individual spectra add together with variable amplitudes), therank of the measurement matrix is smaller than the number of chemical species contained within

53
the sample at the measured locations. One caveat here though, is the spectral shifting that occurs
from environmental effects, which effectively then are new chemical species, increasing the rank of
the measurement matrix. The hyper-spectral image data collected with s-SNOM are consistent
with a generative model, which are still approximately low rank.134
MC works by completing matrices with a penalty related to the low rank assumption.125, 126
To implement MC, the s-SNOM data set, which is 3 dimensional, is flattened into a 2 dimensional
form, where the interferogram axis is along rows and each column is a tip-sample location. The
data is sub-sampled here along rows, which correspond to the mirror positions and the remaining
data can be used to recover the original data since the matrix has a low rank.

3.3.5

Adaptive sampling

The full field of view can be recovered from a smaller dataset by using redundancies intrinsic
to the s-SNOM data. However, s-SNOM data has the most variation (and therefore least amountof sparseness) near the center-burst (maximum interference between sample and reference arms).
This can additionally be used to improve the compression factor by weighting the probabilityof data sampling to the envelope of the interferogram, emphasizing the center-burst and FID.
While the qualitative nature of the interferogram envelopes is consistent, depending on the lightsource and sample, the range of interferogram envelopes can be quite large. We therefore propose
using a periodically updated envelope function that uses prior measurements to inform future
measurements. To do this, the first tip-sample location uses a flat distribution so that random
points along the interferogram axis are selected without bias. For this first spatial point, we use
a conservative Nyquist number of randomly sampled points with the sparsity assumption (CS) to
recover the full interferogram. We then move to the next tip-sample location and use the previously
recovered interferogram to build the envelope function that will serve as the basis for the probability
distribution for the next random sampling. We continue doing this for each next spatial point, where
the average of the all previous interferograms and continually reduce the sampling rate so that the
final compression factor is achieved.

54This is implemented by moving the scanning mirror (figure 3.7) to the positions chosen
by the smart s-SNOM and adaptive sampling algorithm. We apply this algorithm to the fully
sampled data, simulating its implementation in practice. We then compare the recovered resultswith the ground truth through a mean square error calculation to determine the accuracy of thereconstruction. A more physically minded approach is to compare the resonant center frequencies ofinterest to determine data fidelity. We enforce a quadratic norm criterion that must be minimized
while simultaneously adding a sparsity and low rank penalization. We therefore have a convex
optimization criterion with two tunable parameters λ1 and λ∗ based on sample type and lightsource. This criterion is minimized using a previously established algorithm.135

3.3.6

Reconstruction algorithm

The generalized forward-backward algorithm that minimizes the quadratic norm, sparsity,and low rank penalizations enables the reduced number of necessary data points, while maintaininge bymeaningful physical information. We represent the entire dataset in a sparse matrix form X
truncating an applied Fourier transform along the interferogram axis (rows). An optional wavelet
transform along the columns can be used to enforce spatial sparsity.

X = W X F,

(3.5)

where W and F are matrices performing a 2D wavelet transform and a truncated 1D Fouriertransform respectively. This model is the Kronecker compressive sensing model.136 Φ is an n by
n orthonormal matrix that represents the linear transform between the sparse domain and the√measurement domain. The incoherence is measured by µ = n maxi,j |Φij |.137 The sparsity andincoherence conditions required by CS are thus satisfied here.136
We define the measured samples as y, such that

y= S



W† ⊗ F



⃗
wX

(3.6)

55
with the Kronecker product ⊗ and the sampling matrix S composed of 0 and a single 1 per line at
the selected sample positions.
In general, s-SNOM is applied to a wide range of sample types, each with distinct spectralfeatures that require independent tuning of our penalization parameters (figure 3.8). The sample
types considered here are biological, molecular, and quantum. The light source used to measureeach sample also affects the parameter tuning and we therefore use ultra-broadband synchrotron
light (from the Advanced Light Source – ALS – in Berkeley) and broadband pulsed laser sources.
The main contribution to the sampling strategy comes from the bandwidth of the light source,
where the signal from broadband synchrotron light means the interferogram domain is alreadyapproximately sparse. Therefore, we naturally anticipate better compression performance on datacollected with a pulsed laser.

3.3.7

Comparison to original measurements

We now define a reduction factor R as the ratio between the number of data points for the
fully sampled dataset and the number of datapoints needed by smart s-SNOM in order to quantify
the compression. While this is technically representative of the compression, ground truth measurements (normal conventional s-SNOM measurements) over-sample the maximum Nyquist cutoff
frequency (smaller mirror steps) and also over-sample the spectral domain resolution (sample beyond optical interaction between sample and reference arm). Therefore, a more honest comparison
is between the minimally sampled acquisition rate (Nyquist sampling) only over the range of opticalinterference and the smart s-SNOM sampling rate, which we define as the compression factor CF.
While this is more honest, the experimental implementation of Nyquist sampling leads to a significant drop in the quality of the collected spectrum (which is why it isn’t done in the first place),
which is an unfair comparison to the smart s-SNOM results. The experimental measurements
performed here are previously described for both synchrotron IR nano-spectroscopy (SINS) andlaser based nano-FTIR.72 For the most robust interpretation of the application of smart s-SNOM,
we test against these two types of lights sources and 3 materials representing biological, molecular,

56
and quantum systems.
A SINS measurement on 400 nm thick γ-globulin, referenced to Si is shown in figure 3.8A.72
The difficulty of measuring multiple resonant features with a low brilliance light source is highlighted
with this example. Figure 3.8A shows the characteristic amide I,II, and III resonances of secondary
protein structure present in γ-globulin. The entirety of this dataset is a single tip-sample location,
meaning spatial redundancy and the low rank assumptions cannot be implemented. To compress
this data, we use the sparsity of the spectrum and the known bandwidth of the light source, yet
are unable to achieve a compression factor greater than 1, with a reduction factor of 17.
In contrast, figure 3.8B shows a nano-FTIR dataset measuring oriented PTFE with a Au
reference. While the complex valued spectra are typically preferred to be viewed in terms ofamplitude and phase, when the oscillator strength is appreciably strong, yielding a phase shift
more than ∼30 %, real and imaginary spectra are preferred. The strong oscillators in PTFE, withphase amplitudes significantly over ∼30 % necessitate plotting real and imaginary values as theapproximation between phase and imaginary through the small angle relation is no longer valid.
This dataset, which includes multiple tip-sample locations, is compatible with the low rank prior,which is combined with the sparsity assumption to yield a compression factor of 4, indicating
smart s-SNOM’s capabilities for PTFE samples. The effect of compressed sensing and matrix
completion individually are demonstrated in figure 3.9 and the extracted PTFE peak extraction
under compression strength is shown in figure 3.10.
We now transition from strong molecular vibrations for chemical identification to a molecularvibration that is used to perform nano-crystallography through spectral analysis with the example of a metal carbonyl compound (2,3,7,8,12,13,17,18-Octaethyl-21H,23H-porphine ruthenium(II)carbonyl). The physical analysis performed on the spectra acquired with nano-FTIR is based on
the splitting of ω0 into ω− and ω+ as the metal carbonyl crystalizes (figure 3.8C). The ω− and ω+resonances are close, complicating slight frequency differences in the full dataset that must be extracted from the sub-sampled dataset, yet a reduction factor of 4 is still achievable, indicating thatsmart s-SNOM faithfully reconstructs spectra for detailed analysis and can therefore be extended

57
beyond simple chemical mapping.
The final example shown here represents a low brilliance light source again (ALS synchrotron
radiation) to image an SiO2 phonon mode. In contrast with the γ-globulin example described above,
this dataset has multiple tip-sample locations in the form of a linescan. The spatial redundancy
here enabled a much more significant reduction factor of 30 (compression factor of 6.5), even though
the bandwidth of synchrotron radiation is extremely broad and the previous attempt at spectral
sparsity did not yield significant results. The broad bandwidth, and therefore fast center-burst and
FID highlights the importance of the adaptive sampling strategy, where we preferentially selectdata points where the interferogram has the most variation (figure 3.8D).
We now compare spectra in Fig. 3.8B the evenly sub-sampled and reconstructed datasetwith the adaptive sub-sampled and reconstructed dataset, where we see an obvious improvementin the reconstruction fidelity. The broad range of material responses (with varying resonances)
together with the light source used for each measurement in Fig. 3.8 shows how the envelope
sampling strategy, while important, necessitates periodic updating to ensure the correct sampling
distribution is implemented. The selected datapoints are shown in Fig. 3.8D (red crosses), where
the majority are in the center-burst and FID regions. Seeing as the rest of the interferogram is flat(and therefore offers little in terms of non-redundant data), the selection is minimized. The single
tip-sample location dataset used in Fig. 3.8A uses a flat distribution as previously discussed.
One metric for evaluation between the ground truth and reconstructed datasets is the relative
mean square error (RMSE). The RMSE increases as the compression is increased and the example
of PTFE with CF of 1, 10, 20, and 30, the RMSE is shown in Fig. 3.9. If only one spatial point isused, the RMSE is larger than if a full spatio-spectral dataset is used as additional redundancy fromthe spatial domain can be used to help reconstruct individual spectra. The RMSE for the PTFE
dataset sub-selected to a single spatial point for the CF from Fig. 3.9 are 0.019, 0.0197, 0.440,
and 0.794. As expected, the optimal parameter nuclear norm penalty is 0 as there is only one
spatial point. Removing the ℓ1 norm penalization results in an expected RMSE increase to 0.0185,
1.42, 1.53, 1.6 respectively. This improved reconstruction with agrees with previously reported

58

Figure 3.8: Application of smart s-SNOM to different materials systems (biological, molecular, andquantum), different light sources (Laser, ALS synchrotron) and different spatial scanning (single
point, line and 2D scan) | Fully sampled datasets acquired experimentally (Blue curves). Mirror
positions selected by our adaptive sampling strategy (Red crosses). Reconstructed spectrum in
amplitude / phase or |A(ν)|/Φ(ν) and real / imaginary part or Re (A(ν)) /Im (A(ν)) (Red andBlack dotted curves respectively). For the PTFE sample, reconstruction from uniformly sampledmeasurements without the use of adaptive sampling (Green dotted curves). The reduction factor(R) is respectively 17, 30, 4, 30. The compression factor (CF) is respectively 1, 4, 1.6, 6.5. The
regularization parameter called Nuc is respectively 0, 5, 0.75, 0.005. The regularization parameterℓ1 is respectively 0, 10−6 , 0.01, 4 ∗ 10−5 .71

results.116, 117
For the case when more than one tip-sample location is used though, the low rank assumption
can be imposed, and MC can be effective. The crossover when enough tip-sample locations areused to make MC useful is when the number of distinct chemical species is fewer than the number

59of spatial locations. To emphasize this turning point, we compare the RMSE of reconstructions
as a function of additional data points for a variety of CFs. The RMSE decreases with increasing
spatial points (Fig. 3.9A). When used in combination with MC, we see that comparable RMSE canbe achieved at higher CF from CS. The reconstruction of a series of linescans of different lengths
and RF is shown in figure 3.9B, indicating MC’s effectiveness with strong molecular oscillators.
We further note that 6 spatial points is sufficient to demonstrate an advantage when using MC for
PTFE, implying the number of distinct chemical species is quite small indeed (Fig. 3.9C).
In contrast to the holistic measure of RMSE, the extraction of physically relevant parameters
for a resonance, like peak position, linewidth, and amplitude is sufficient. Quite often in nanoFTIR, one or a combination of these parameters is interpreted as physically meaningful to makeconclusions about underlying physical phenomena at the 10s of nanometer scale, or less like inmolecular rulers. Continuing with the PTFE example, the peak position of the symmetric (atwν1 = 1168cm−1 ) and antisymmetric (at wν2 = 1241cm−1 ) modes are of interest for coupling
analysis. The ability to extract these values at different reduction factors is demonstrated in figure
3.10A, where the color map indicates the resonance parameter and the x and y dimensions are thelocation across the sample surface at increasing RF (1, 10, 30). Parameters extracted from theground truth measurement are shown for comparison on top. The estimate of peak positions seems
to be faithful at RF ≤ 100, but the standard deviation increases dramatically with RF. R = 30
yields reconstructions that are still sufficient for physically meaningful interpretation. A final effectof the algorithm employed here is a denoising effect, where noise is rejected by the algorithm basedon redundancies within the data, as noise is evenly distributed in the Fourier domain, where thesparsity assumption is enforced. Further, noise is inherently not of low rank and is therefore further
minimized by MC.
The choices made here in designing our algorithm for smart s-SNOM build on previous
advances within the broad field of hyper-spectral image signal processing. Below, we motivate thechoices made both in and out of favor as well as how our approach has its limitations.

60

Figure 3.9: Illustration of the effect of law rank assumption on the reconstruction error. | We use
a PTFE 2D grid scan to emulate an acquisition of different sized line scans with 10, 20, and 40
pixels (see blue line corresponding to the image of panel B). A) Plots of relative mean square error
as a function of the number of spatial points for different reduction factors. The largest quality
improvement occurs between 1 and 6 spatial points. This can be different for other samples with ahigher diversity of chemical species. B) For each number of points used, different reduction factors
are emulated: 10, 20 and 30. The ground truth is plotted in blue and the reconstructions in red.
At a given reduction factor the visual quality of the reconstruction improves with the number ofspatial points sampled. C) Illustration of the principle of matrix completion. Each color red, green,
and blue corresponds to one chemical species with a specific spectrum. Those images are separable
in space (x, y) and frequency (ν) and are therefore considered rank one images. In the case of alinear mixing model, the final hyper spectral image is a sum of a few (rank one) of these images if
the number of chemical species is small in the sample. This explains the small rank assumption on
hyperspectral images. (Taken from collaboration with71 )61Figure 3.10: Extraction of physically relevant information from resonance peaks at different reduction factors. | A) Characterization of the two spectral peaks of the PTFE sample located at
ν1 = 1168 cm−1 and ν2 = 1241 cm−1 ). Each stack corresponds to 3 images obtained from reconstruction at different reduction factors (from top to bottom respectively 1, 10 and 30). Each peakis characterized by its estimated position ν ′ , amplitude |A(ν ′ )|, phase Φ(ν ′ ) and full width at half
maximum F W HM . B) Plot of the relative mean square error between reconstruction and groundtruth in red. Plot of the localization of the two peaks in blue as a function of the compression
factor (top axis) and to the reduction factor (bottom axis). The standard deviation of the peaklocalization increases with compression factor, leading to potential physical misinterpretation of
the reconstruction for high reduction or compression factors. (Taken from collaboration with71 )3.3.8

Further improvements

We start by considering the spatial redundancy within s-SNOM measurements, where most
samples of interest are composed of chemical species organized in finite domains, rather than being
randomly distributed. Finite domain images are known to be sparse in the wavelet138 and 2D

62Curvelet domains with the additional constraint of smooth boundaries.139 The addition of eitherof these transforms would naturally be implemented in the CS application of smart s-SNOM.140
Combining CS in the spectral domain with these spatial transforms combines to be an example of
CS referred to as Kronecker CS.136 In a rare example of a sample with very sparsely located chemical species, imaging in the spatial domain (as is conventional), neither the wavelet nor curvelet
transform would yield a sparse image as the source domain is already sparse. While the implementation of these additional transforms could, in principle, yield further CF and RF, the additionaltransform and reconstruction for each mirror position limits its applicability as both an additional
assumption is made about the sample is made and the computational time cost grows substantially. We therefore only recommend this combination if the sample is known well enough and thecomputational power is sufficiently high.
A common assumption within the field of information theory and reconstruction algorithms
is the positivity criterion. While the detected optical signal is necessarily positive, the total signal isdemodulated at higher harmonics of the tip tapping frequency to isolate the near-field contribution
based on the non-linear dependence of the near-field signal with tip-sample distance. Since eachtip-sample location measurement comes from this demodulation, positivity can not be enforced.
The CF and RF presented here represent the amount of data that needs to be collected forcomparison to the ground truth, but a practical implementation of how to scan the reference mirror
quickly to these points, while maintaining SNR has thus far not been addressed. Specifically, acomparable tip-sample list with intermediate reference arm scanning can be used if the reference
arm is selectively accelerated to minimize time between smart s-SNOM selected points of interest.
An alternative is to optimize the total tip-sample-reference arm position list for maximum speed.
While the practical implementation has not yet been demonstrated, the significant reduction innecessary data points shows promise for future implementation.
We next comment on the MC enforcement of low rank, where this prior knowledge or assumption reduces the number of necessary data points to complete missing entries in a matrix.
The standard technique is to apply the nuclear norm ∥ · ∥∗ penalization,128 which is effectively the

63
l1 norm of the matrix’s singular values. We minimize the convex criterion to allow for both the
CS and MS penalizations and use a convex criterion to minimize, enabling us easily to incorporate
penalizations used in CS and MC:

J(wX) = F (wX) + G(wX)
F (wX) = ∥ S W † ⊗ F vect(wX) − y ∥2

(3.7)

G(wX) = λ1 ∥ wX ∥1 + λ∗ ∥ wX ∥∗

(3.9)

(3.8)

where we tune both λ1 , λ∗ for the ∥ · ∥1 and l1 penalizations, similar to previous work.141 The
sparsity and low rank of the reconstructed dataset is enforced using the ℓ1 and ℓ∗ norm, but otherpenalizations can be applied.
PTFE

Amide

Figure 3.11: Error as a function of parameter settings for two different samples: PTFE and Amide.
| Light sources are a laser and the synchrotron, respectively, to illustrate the estimator behavior
for two different spectral sparsities. Colors represent reconstruction errors, spatial coordinatescorrespond to (λ1 , λ∗ ) parameter settings. Color saturates to pure yellow for error values bigger
than the error obtained with λ1 = 0 and λ∗ = 0. This way region where errors are reduced are more
visible. The red circle indicates optimal settings, we observe that region around optimal settingsgives similar errors. We observe that tuning the parameters is easy and does not necessarily need
to be optimal to obtain relevant reconstructions. (Taken from collaboration with71 )However, introducing additional norms requires additional parameter tuning, which may or
may not be helpful in overall CF/RF while maintaining data fidelity. For instance, in our example

64
of 2 parameter tuning, we can balance the strength of sparsity and low rank through λ1 and λ∗ .
These effect of these parameters is shown though the RMSE in figure 3.11, with blue showing
smaller reconstruction error than when no penalization is used.

3.3.9

Conclusion

We have demonstrated success in applying CS and MS to s-SNOM datasets. We implemented
prior knowledge about the light source and sample characteristics along with an adaptive samplingstrategy to reduce the number of necessary data points to be collected in a s-SNOM experiment.
We have demonstrated impressive compression and reduction factors, improving on other advancements in improving s-SNOM data acquisition rates. We note that the performance demonstrated
here improves with larger datasets, both in spatial extent (more spatial positions) and in spectral
resolution (interferogram extent) as more redundancies are present. We further predict the additionof additional data axes, like time resolution for ultra-fast time resolved s-SNOM, likely dramatically further increases smart s-SNOM’s performance. Smart s-SNOM therefore enables nanoscalemeasurements over large fields of view and can be extended to higher dimensional datasets.

3.4

Heterodyne Pump Probe s-SNOM
Finally, in this section, I will detail advancements made in near-field pump-probe ultrafast

spectroscopy and imaging. Ultrafast carrier dynamics have been accessed through infrared nanoimaging in semiconductor, correlated-electron, and polaritonic materials. The contrast obtained inconventional approaches has remained too low for important weak and long-lived excitations whenhigh repetition rate lasers are used for excitation in systems like, e.g., low-conductivity carriers,
lattice phonons, and molecular vibrations. In an effort to improve the contrast, we demonstrate
pump-modulated ultrafast excitation for infrared nano-imaging and -spectroscopy to fully andquantitatively characterize electron and vibration dynamics in space, time, and frequency. We
use transient vibrational nano-FTIR to spatially resolve the excited-state polaron-cation couplingin lead halide perovskites to characterize the full dielectric response, on which the photovoltaic

65
performance depends. This demonstrates the power of heterodyne pump-probe nano-imaging toimage elementary processes in quantum and molecular materials. This work was performed incollaboration.69

3.4.1

Ultrafast phenomena

The unique optical and electronic properties of functional materials can be applied to applications including technologies based on quantum phase transitions in Mott transistors,142, 143
photovoltaic carrier transport with polarons in lead halide perovskites,144, 145 singlet fission driven
by coherent phonons and vibrations,146–149 and light-harvesting complexes based on electronic energy transfer.150 The interplay of elementary electronic, vibrational, and phononic quantum statesgive rise to these emergent properties. Ultrafast spectroscopy is able to isolate the individual responses of mode-coupling and competing interactions by exciting the system through one degree of
freedom and probing the dynamic response that is normally convolved in static spectroscopy.151–154
New photoinduced quantum states are formed when the system is strongly perturbed and allow for
the ultrafast control of polaronic, metallic, and superconductive charge carrier functionaliteis.155–157
Extreme nonlinear optical phenomena from strong and ultra-short laser fields enable applications
like high harmonic generation158, 159 and light-field petahertz electronics.160, 161
Naturally, systems based on these sort of materials will exhibit spatial heterogeneity fromlattice defects, grain boundaries, non-uniform doping, and strain.30, 162–166 Therefore, a variety oftime resolved nano-imaging techniques have been previously developed, like, e.g., ultrafast transmission microscopy (TEM),167, 168 X-ray microscopy,169 and electron emission microscopy (PEEM).170
However, these approaches are not sensitive to the low-energy electron lattice coupling that dictate
material and device behavior.

3.4.2

Ultrafast near-field

The recent implementation of ultrafast scattering scanning near-field optical microscopy (sSNOM) contrasts with these more established techniques by probing the associated low-energy

66
processes directly.11, 13, 105, 171–181 Ultrafast nano-FTIR has,11, 173 in particular, given access tocarrier dynamics in a range of semiconducting,11, 175, 177 polaritonic,174, 176 and correlated-electronmaterials,171, 173 together with electro-optic sampling (EOS),177 to resolve spatial heterogeneity inthe temporal dynamics and spectral differences and evolutions. To date, ultrafast infrared nanoimaging has only probed strong and short-lived collective excitations based on low-energy (highrepetition rate > 10 MHz) laser sources, which naturally yield high SNRs. This of course thoughleaves out the ability to probe long-lived transient states and those that need a strong pump to,e.g., induce a phase transition or probe vibrational coupling in molecular materials. Resolving thefull spatial, temporal, and spectral degrees of freedom has only recently175, 176 been achieved with
low-repetition excitation and therefore necessitates a generalized approach to enhance the contrast
for these limited SNR applications.
Here, heterodyne visible pump infrared probe s-SNOM (HPP IR s-SNOM) has been demonstrated with a low repetition-rate laser to measure space, time, frequency, and phase degrees of
freedom. In this approach, the system is excited with an optical pump and this transient excitedstate is probed with a low-energy pulse to determine the electronic and vibrational response. We
use simultaneous sideband lock-in detection to isolate the induced third-order nano-localized polarization and extract interferograms in the time-domain to nano-image the ultrafast excited state fora variable excitation rate. The scheme of modulating the excitation in HPP IR s-SNOM directly
resolves the transient response and is spectrally resolved such that quantitative modeling can extract the dielectric function of the material in its transient state. A combination of the finite dipole
model and a four-layer reflection model is used.
As a representative example of soft molecular material transient nano-spectroscopy and imaging, we apply this technique to image the polaron-cation coupling in a triple cation perovskitethat controls the photovoltaic response.182, 183 The low-repetition-rate excitation, together withhigh sensitivity detection, enables the study of long-lived vibrational responses in a soft molecularmaterial. Therefore, the approach detailed here can be used to study elementary processes at
the nanoscale that control the optical, photophysical, catalytic, and electronic properties of many

67
functional materials.

Experimental implementation

A

B
Yb:KGW
Osc. + RA

VIS
515 nmlock-in amplifierQȦt ȍM

AOM

sig.

BBO

ref.1 ref.2

g

pump

ELO

¨ENF

Q

D

MCT

sample
probe

Ȧt

OAP
pump

P (3)

2ωt ± ΩMP

T = 10 ps
φ=�

T = 100 ps
T = 250 ps

0t (ps)

0.2

0.4

T
Ȟ
model fit(3)~

¨İ(Ȟ,T, {X,Y})X

λpump = 1030 nm
λprobe = 6.25 µm

φ=0

-0.2

,P>¨ENF]

t

Y

ref.

T
Ȟ

FT

T

pump-probe signal

ȍMMIR
5 - 10 ȝmt

MIR probe∆INF(t,T) (a.u.)

DFG

NIR
1030 nm

5H>¨ENF]

epump delay

T
OPA

C

E

∆INF(φ = 0,T)

∆INF(T) (a.u.)

3.4.3

T

∆INF(φ = �,T)
T

∆ISHPP(T) (×0.7)

0200

RHPP(T)

400 600T (ps)

800 1000

Figure 3.12: Heterodyne pump-probe nano-imaging | A. HPP IR s-SNOM, with the ∼185 fs
optical 1.2/2.4 eV (1030/515 nm) pump excitation, ∼170 fs tunable mid-IR 0.12-0.25 eV (5 - 10 µm)
probe, and interferometric heterodyne near-field detection. AOM - acousto-optic modulator, DFG difference frequency generation, OAP - off-axis parabolic mirror, OPA - optical parametric amplifier,Osc. + RA - oscillator and regenerative amplifier laser system. B. Far-from-equilibrium excitation
followed by mid-infrared probe of the transient low-energy vibrational and electronic response
coupled to the excited state. C. The tip-localized time-domain signal of ∆ENF (ν), from which thepump-induced change in the nano-localized complex dielectric function ∆ϵ̃NF (ν) is retrieved withspatio-temporal-spectral resolution. D. Sideband-demodulated pump-probe interferogram ∆I(t, T )for a Ge reference sample. E. T -dependent pump-probe transients from two different ELO phasevalues ∆I(ϕ = 0, T ) and ∆I(ϕ = π, T ), the derived frequency-averaged heterodyne pump-probe
amplitude relaxation RHPP (T ), and the self-homodyne pump-probe signal relaxation ∆ISHPP (T ).
Inset: tip-enhanced pump excitation in nano-localized probe volume, leading to faster relaxationfrom higher excited carrier density. (Taken from collaboration with69 )The schematics of HPP IR s-SNOM are shown in figure 3.12A-C. Here, a femtosecond
Yb:KGW amplified laser (∼185 fs FWHM pulse at 1030 nm with a ∼1 MHz repetition rate –Pharos, Light Conversion) pumps an optical parametric amplifier, producing, with difference frequency generation (DFG) between the signal and idler pulses, broadband infrared light. The
shortest time intervals that can be resolved result from the cross correlation between the visible

68pump and infrared probe. The spatial resolution for nano-imaging and -spectroscopy is primarily
given by the radius of the tip and is approximately 40 nm. Spectral resolution is achieved byfollowing previously established techniques.73, 112 The atomic force microscope (AFM) used here
(Innova AFM, Bruker) is placed in the sample arm of the asymmetric Michelson interferometer andis operated in intermittent, or tapping, mode, where the tip (ARROW-NCPt, NanoAndMore USA)is driven with a tip tapping frequency ωt . Time resolution T is achieved by spatially delaying thepump relative to the probe with a linear delay stage, which precedes and acousto-optic modulator(AOM) or mechanical chopper that modulates the pump excitation frequency ΩM . An off-axis
parabolic mirror (NA = 0.45) focuses both the pump and probe light onto the tip apex. Figure
3.12B shows the detected excited-state absorption and corresponding ground state bleach by amplifying the near-field signal ENF with the reference field ELO detected with a mid-IR HgCdTe (MCT)
detector. Lock-in (HF2LI, Zurich Instruments) demodulation is performed at nωt (n = 1, 2, 3...)for near-field localization and background discrimination.73, 112
The Yb:KGW amplified laser output (7 µJ) is split with ∼6.5 µJ pumping the OPA, which
generates signal and idler outputs that are then collinearly focused in a GaSe crystal, generating
mid-infrared pulses via difference frequency generation that are tunable from 5-10 µm with a typicalpulse energy of ∼ 40 nJ, which is used as the probe in our pump-probe near-field measurements.
Simultaneously, 0.5 µJ is used either at its fundamental wavelength λpump = 1030 nm or frequency
doubled wavelength λpump = 515 nm, which is generated by second-harmonic generation in a 2
mm-thick BBO crystal (CASTECH). The pulse duration of the pump is characterized by SHG
auto-correlation (FROG) to 185 fs (FWHM intensity) with a 0.1 mm-thick BBO crystal (Figure
3.13A). The pump-probe cross correlation duration of 170 fs is determined by third-order SHG-DFG
XFROG with a 0.3 mm-thick GaSe crystal (Figure 3.13B).
The direct detection of the excited minus ground state response is enabled by excitation
modulation and simultaneous sideband detection at nωt ± ΩM , which builds on previous pump
un-modulated schemes11, 177 that measure the sum of both the ground and excited state responses.
The extraction of the time resolved response in this previous implementation is only possible for

69

λpump
λSHG

2

λpump

515
520

-0.2

0

0.2

∆tpump-pump (ps)~260 fs
0
-0.4

-0.2

0.0

0.2

∆tpump-pump (ps)

0.4

2
λpump

1

λpump

λSHG-DFG (nm)4510

Cross-correlation (a.u.)B
λSHG (nm)Auto-correlation (a.u.)A

λprobeλSHG-DFG

560

570

-0.2

0

0.2

∆tpump-probe (ps)~215 fs
0

-0.4

-0.2

0.0

0.2

∆tpump-probe (ps)0.4

Figure 3.13: Ultrafast pulse characterization | A. SHG intensity auto-correlation (FROG) of the1030-nm pump pulse. B. Third-order SHG-DFG cross-correlation (XFROG) between the 1030-nm
pump pulse and the 6000-nm probe pulse. Insets, left - nonlinear optical pathway for each signal.
Insets, right - frequency-resolved optical gating (FROG) trace.69

systems that have a large excited state contrast. The SNR of this detection scheme is significantly
improved and the spectra that are fit with an appropriate sample and tip interaction model are now
quantitatively accurate, enabling spatial, temporal, and spectral resolution to fully characterize thefull 4D transient material response.
We control the pump-probe delay by directing the pump beam to a 20 cm translation stage(SLLA42, SmarACT) fitted with a retroreflector (UBBR2.5-1S, Newport). As previously mentioned, a mechanical chopper (MC2000B, Thorlabs) or acousto-optic modulator (AOMO 3110-120
442-633 NM, Gooch & Housgo) operating at ΩM ∼ 3 kHz (λpump = 1030 nm) and ΩM ∼ 50 kHz(λpump = 515 nm) respectively. We then collinearly combine the pump with the mid-IR probe andsteer them into an asymmetric Michelson interferometer. One arm is a reference arm containing
mesh filter attenuators and scannable precision translation stage (ANT95-50-L-MP, Aerotech; or
SLC1760, SmarACT) fitted with a reference mirror, generating TTL pulses every 300 nm of motion
to trigger lock-in amplification sampling. The other interferometer arm contains the AFM (Innova,
Bruker) and sample, where p-polarized pulses are focused with an off-axis parabolic mirror (NA= 0.45) onto the apex of a PtIr-coated metallic tip (ARROW-NCPt, NanoAndMore USA). The
tip tapping frequency of ωt ∼ 250 kHz with amplitude zA = 60 − 70 nm periodically enhances the70near-field signal, which then scatters and combines with the reference arm light to be focused intoa HgCdTe detector (KLD-0.1-J1, Kolmar Technologies) after going through a germanium long passfilter to block pump scattered light.
The voltage signal from the MCT is fed into a lock-in amplifier and is demodulated at nωtwith the reference frequency coming from the tip feedback laser quad photo diode. To improve
SNR, we demodulate at nωt + ΩM and nωt − ΩM , which corresponds to the near-field pump-probe
excited state minus the near-field ground state. We simultaneously collect the conventional second
harmonic signal as well so that ground state properties can simultaneously be compared to thetransient response.
Specifically, germanium is often used as a reference sample, where we pump above thebandgap and probe the photocarriers produced by taking an interferogram ∆IHPP (t, T ) at a selected pump-probe timing. The overall spectrally integrated time domain response is determinedby scanning the pump probe delay at the zero phase difference (ZPD) reference arm position andat the π phase difference position and subtracting the resulting time traces, yielding the ultrafast
heterodyne amplified temporally resolved decay (figure 3.12E). In germanium the ultrafast spectral
response is dominated by the Drude response of the free carriers injected upon pump excitation.184
The extremely broad nature of the Drude response in comparison to the probe bandwidth (∼100
cm−1 FWHM) implies the heterodyne transient response can be approximated by the two phaselocations of the reference arm.
The recombination of photoinduced carriers that lead to the HPP signal decay RHPP (T ) (red)is clearly different than the self-homodyne decay (blue) where there is no interference. Specifically,
the self-homodyne decay shows a previously far-field observed initial rise and slow decay,184 whereas
the heterodyne RHPP (T ) decay shows a clearly faster recovery. The convolution of the timedependent far-field phase unresolved background ∆ISHPP (T ) causes this difference.105 The nearfield enhanced pump leads to a higher carrier density generation at the tip location compared to the
far-field background (Figure 3.12E inset). Since RHPP (T ) is background free, the faster relaxationreflects the faster recombination and scattering associated with the higher carrier density. Clearly,

71
interferometric heterodyne detection with pump-modulation and sideband amplification is needed
to quantitatively resolve transient dynamics in the high fluence regime and when the dynamics arepump fluence sensitive.
The spectrally integrated relaxation RHPP (T ) is estimated by recording the dynamic relaxation with reference arm at two phase values, specifically ϕ = 0 and ϕ = π, which works well for
broadband, instantaneous responses, like the Drude response in, e.g., semiconductors. The timing
for the ϕ = 0 (constructive interference near the zero-path difference) is independent of frequency,
but, technically, the π phase offset is frequency dependent. However, the narrow linewidth of thelaser source used yields a π phase shift very close for the lowest and highest frequency components
compared to the center frequency for the first half optical cycle near the ZPD. The error for thisphase offset is only ∼6 % or 0.6 fs in a 10 fs half optical cycle. This discrepancy is below theaccuracy of the stage positioning and is therefore negligible. Thus, the two phase reference arm
offset is sufficient for extracting the near-field spectrally integrated amplitude.

3.4.4

Theoretical framework

The pump-induced change in the complex dielectric function of a material is related to thespectrally resolved near-field response that has been evaluated with the point dipole model177 andlightning rod model.175 In order to directly model the experimental observable related to our
new sideband-demodulated detection scheme for interferogram ∆IHPP (t) based on the underlying
transient dielectric response ∆ϵ̃NF (ν), a simple theoretical extension to the finite dipole model57
and multi layered model (four in this case) is employed (figure 3.14).
To illustrate the application of a vibrational response coupled to photo injected carriers,183
we study a thin 600 nm film on a thick substrate. Here, ground-state and excited-state dielectric
functions ϵ̃(0) (ν) and ∆ϵ̃NF (ν) are assumed. Vibrational resonances at ν gs and ν ex are indicated infigure 3.14B, where the top layer d1 = 100 of the sample is excited by the pump and the remaining
d2 = 500 nm is in the ground state. As previously established, the near-field scatter is calculatedwith and without pump excitation for a sinusoidally modulated tip-sample distance, demodulated

A

B
εair

5

ε~(0) (ν)ε~(e) = ε~(0) + ∆ε~NF
ε~ (0)d1
d2

1.1

Re[ε~(0) ]

4

0
1650

1.0

Im[∆ε~NF ]

1

εsub

1.2

-0.9

~
NF

Re[∆ε ]

Im[ ε~(0)]1700∆ε~ NF (ν)

ENF + ∆ENF72

νgs νex

6

-1.0
-1.1
1800

1750

ν (cm-1)
2
1

non-resonant

INF(t)∆IHPP(t)

∆φ

0

0.5

-1
-0.1

0.0

0.1

0.0
0.05

-0.5
-1.0
-0.5

resonant

0.00
-0.05
1.0

0.0

0.5

1.5

νgs νex
0.175

Re[ENF]
0.5

Im[∆ENF]
Im[ENF]
Re[∆ENF]

0

1.5

1.0

D
ENF(ν) (a.u.)

1.0

2.0

2.5

1650

1700

t (ps)

1750

0.150

-0.125∆ENF(ν) (a.u.)

INF(t), ∆IHPP(t) (norm.)

C-0.1501800ν (cm-1)

Figure 3.14: Retrieval of transient nanoscale response function | A. Theoretical framework toquantitatively relate the measured pump-induced near-field signal change ∆ENF to the transientcomplex dielectric function ∆ϵ̃NF . B. Example ground-state complex dielectric function ϵ̃(0) (ν) andits pump-induced change ∆ϵ̃NF (ν) for a model vibrational response, with the ground-state resonance at ν gs and the frequency-shifted excited-state resonance at ν ex . C. Simulated heterodyne IR
s-SNOM interferograms for the ground state INF (t) and its pump-induced change ∆IHPP (t). D.
ENF (ν) and ∆ENF (ν) in frequency-domain, obtained from Fourier transform and ELO (ν) deconvolution of the time-domain interferograms in C. (Taken from collaboration with69 )at the second harmonic of the tip tapping frequency. The ground state is then subtracted from theexcited state to give the pump-modulated sideband detected result.
The calculated ground-state s-SNOM interferogram is shown in figure 3.14C (INF (t) in black).
a broadband carrier response and transient vibrational resonance compose the pump-induced response ∆ϵ̃NF (ν), giving rise to the measured pump-probe interferogram ∆IHPP (t) (red). The
broadband non-resonant term leads to an instantaneous centerburst, only limited by the probe
pulse duration. This is followed by the vibrational free-induction decay (FID)73 with ∆IHPP (t)phase-shifted by the complex nature of the Drude response.

73
We extend the established finite dipole model57 with a four-layer model (figure 3.15A). Here,the four layers have a frequency dependent complex valued dielectric function, indexed by layer
number ϵi (ν) (i = 0, 1, 2, 3). Specifically, the top and bottom layers are given by the dielectric
response of air and substrate respectively. Layers 2 and 3 are the sample of interest, with layer
2 in the excited state, with depth given by the pump penetration depth and layer 3 remaining in
the ground state. The far-field penetration depth is larger than the near-field localization, so thetip-sample optical interaction is assumed to be only with the excited state. The far-field reflection
coefficient contains both ground and excited state optical signals.

A

B
GS

ε0
Ein
d1
d2

Rt

C

Eout = σNFEin

ES

ε0 = 1

Eprobeε0 = 1Epump

σNF(0)Eprobe = ENF

σEprobe= ENF+∆ENF(e)
NF

Eprobe

z

ε1
ε2

rp

ε3

d

rp (0)

ε1 = ε2 = ε~(0)ε1 = ε~(0) +∆ε~ NF

d1d2

rp (e)

ε3 = εsub

ε2 = ε~(0)ε3 = εsubFigure 3.15: Modeling parameters | A. Parameters defined for modeling the near-field scattering
coefficient σNF , in the ground and excited states, based on the finite dipole and four-layer model. B,(0)(e)
C. Parametrization for computing the ground-state (B, σNF ) and excited-state (C, σNF ) near-field
scattering coefficients. (Taken from collaboration with69 )Here, σNF as a function of z (tip sample distance) is given by

σNF (ν, z) = αeff {ϵ̃1 (ν), z} · [1 + rp (ν)]2 ,(3.10)

where the effective polarizability αeff {ϵ̃1 (ν), z} is calculated with the FDM57 and rp (ν) with thefour-layer model, which is derived from the matrix transfer method185

rp =
where

r01 e−iδ1 [e−iδ2 + r12 r23 eiδ2 ] + eiδ1 [r12 e−iδ2 + r23 eiδ2 ],e−iδ1 [e−iδ2 + r12 r23 eiδ2 ] + r01 eiδ1 [r12 e−iδ2 + r23 eiδ2 ](3.11)

74

ki =

2π p
ϵi − sin2 θ0
λ(3.12)

δi = ki dirij =ki ϵj /ϵi − kj.
ki ϵj /ϵi + kj

We use θ0 = 60◦ for the incident angle and σNF (ν, z) is calculated as the second harmonic of thesinusoidally modulated tip sample distance.
To calculate the sideband HPP signal, the ground-state near-field signal(0)

ENF (ν) = σNF, 2H (ν)Eprobe (ν)(3.13)

is subtracted from the excited-state near-field signal(e)

ENF (ν) + ∆ENF (ν) = σNF, 2H (ν)Eprobe (ν).

(3.14)

After deconvolution with ELO (ν), the Fourier transform of ∆IHPP (t) yield the spectral profiles ofthe photoinduced change ∆ENF (ν) (figure 3.14D red). The ground-state complex dielectric functionϵ̃(0) (ν) is approximated by ENF (ν), as previously noted.57, 73 But, the complex convolution of theground and excited state with compounded Fano-type interference between the excited-state carrier
and vibrational responses compose ∆ENF (ν). We use a Lorentz-Drude model for the vibrational
resonance and broadband carrier response57 to quantitatively retrieve the desired complex dielectric
function ∆ϵ̃NF (ν). Therefore, the fundamental electronic and vibrational dynamics is described by∆ϵ̃NF (ν). Fano interference can in general lead to complicated lineshapes in ∆ENF , especiallywhen narrow and broadband responses interfere. The theoretical framework developed here areuniversally applicable for HPP s-SNOM.
To isolate the pure ground-state response, we subtract the sideband interferogram from theconventional interferogram. Both the ground-state and excited minus ground-state interferograms
are averaged, Fourier transformed, and referenced. We use phase line correction to determine
the ZPD and use the same correction on the pump-probe interferogram so that the phase offset
between ground state and pump-probe interferograms can be determined, yielding phase-locked
spectroscopy that is critical for spectral analysis and fitting.

75
3.4.5

Ultrafast nano-imaging and -spectroscopy of Perovskite

We applied HPP IR s-SNOM to a molecular material exemplified by a lead halide perovskitethat can be seen in figure 3.16. Here, the useful optoelectronic response of lead halide perovskites ischaracterized by the spontaneous formation of long-lived free carriers and their significant diffusion
lengths.186, 187 This unique performance has been speculated to be from the formation of polarons,
where the charge-lattice coupling through electron phonon interactions across many unit cells thatenable long diffusion lengths and coherent carrier transport from the coupling stabilization.144, 145

B

X
X
A
A

MA+ (16%)
Cs+ (5%)BX

I (83%)Br (17%)

B Pb 2+FA
vibration
polaron

B 0.3∆T/T (a.u.)

FA+ (79%)

A

Avib, ∆Avib (norm.)

Atransientvibrational
response

0.2
0.1
0
1650
1.0
0.5

polaronbackg.

1700
Avib, GS

1750
1800
Δν
ΔAvib, PP

0
1700

1720
ν (cm-1)

1740

Figure 3.16: Polaron-cation coupling in hybrid organic-inorganic perovskite FAMACs | A. Polaroncation coupling and associated blue-shift of molecular vibration. B. Far-field transmission visiblepump (2.4 eV) IR-probe spectrum at T = 0.5 ps (top), the ground-state (GS) vibrational absorbance
Avib and transient absorbance ∆Avib (bottom). (Taken from collaboration with69 )Previous work have investigated ground state heterogeneity in lead halide perovskites,188, 189
yet the underlying polaronic heterogeneity and the non-uniform optoelectronic response that leads
to photoluminescence intensity, carrier lifetime, open circuit voltage164, 188–190 across multiple lengthscales remains an unexplained relationship. Previous ultrafast infrared vibrational spectroscopy
has elucidated the cation polaron coupling in perovskites, yet was unable to resolve and explain
spatial heterogeneity182, 191, 192 (Figure 3.16A). Here though, we aim to address this nanoscale heterogeneity through HPP IR s-SNOM (figure 3.17). We first establish the ground- and excitedstate vibrational responses of a thin film of triple cation perovskite FAMACs with far-field visible

76pump infrared probe in a transmission geometry (figure 3.17b). Here, conventional far-field pumpprobe spectroscopy uses a collinearly propagating pump and probe beams that are focused witha f = 10 cm silver-coated off-axis parabolic mirror onto the sample surface. We collect and collimate the transmitted IR probe with an identical OAP and direct it into an interfermoter forspectral characterization upon detection and isolation with a HgCdTe detector and lock-in amplifier repsectively. We demodulate he photoinduced change ∆T at the pump modulation frequencyΩM ∼ 10 kHz and use a pump fluence of ∼50 µJ/cm2 . This sample’s chemical composition is[(FA0.83 MA0.17 )0.95 Cs0.05 ]Pb(I0.83 Br0.17 )3 . Based on the pump fluence, focal spot size, and absorption coefficient, we estimate a peak carrier density of 1019 cm−3 , resulting in a broad backgroundas the carriers couple to the lattice and forms a polaron,183 which couples to the anti-symmetricvibrational stretch mode from the CN group in the formamidinium (FA) cation.191 Figure 3.17b(bottom) shows the excited-state absorption that is blue shifted and stronger compared to theground-state. The stronger absorption results from an enhanced transition dipole moment thatagrees with previous measurements on similar perovskites,182, 191 which together with the blue shift
indicates vibrational coupling to the polaron absorption at a lower resonance frequency, hybridizing
the polaron-vibrational state (Figure 3.17A). Quantitatively, the polaron for this type of perovskite
has previously been observed at 1100 – 1200 cm−1 ,183, 192 which with the ∼ 5 cm−1 blueshift, impliesa ∼ 50 cm−1 coupling strength.
This blue shift is attributed to coupling with a polaron absorption, previously observed near
1200 cm−1183, 192 and describe the coupling with a simple phenomenological Hamiltonian



ν̃
J
 pol.
Ĥ = 
,Jν̃vib.

(3.15)

where ν̃pol. ∼ 1200 cm−1 and ν̃vib. = 1715 cm−1 are the uncoupled polaron absorption and themolecular vibrational energies, respectively, and J is the coupling constant. Diagonalizing yields

ν̃± =ν̃pol. + ν̃vib. ±p

4J 2 + (ν̃pol. − ν̃vib. )2.
2(3.16)

77

B

nano-localized response
far-field ensemble

Re[ENF]∆INF(t) (a.u.)(a.u.)

53Cnm

νgs νex

0

Im[∆ENF]

transientvibrational response

-2
-0.5

0

0.5

Re[∆ENF]

-0.25
-0.50

1750

E
Fano interference

0.04

0.50
0.25

Im[ENF]
1700
1725
ν (cm-1)

1.0

0

D

×10-1

0.6

0.2

0.5

1.0

1.5

t (ps)∆ENF(ν) (a.u.)

ENF(ν) (a.u.)

0.8 Re[ENF]

150

(nm)
80
40
0

Im[∆ENF(ν)] (a.u.)

topo.

2ωt ± ΩM
2

∆ENF, car

0.03
0.02

ΔENF, vib

0.01
0
1700

1720
1740
ν (cm-1)

Im[∆ENF, vib(ν)] (norm.)

A

2

1

0

-10

spot 1
spot 2
spot 3010∆ν (cm-1)

20

Figure 3.17: Transient vibrational nano-spectroscopy of FAMACs perovskite | A. Ground-state IR
s-SNOM Re[ENF ] imaging (top; AFM topography, bottom). B. HPP IR s-SNOM interferogram∆INF (t) acquired at T = 2 ps with instantaneous polaron absorption and long-lived transient
vibrational coherence. C. Phase- and frequency-resolved nano-localized pump-probe ∆ENF (ν) andground-state response ENF (ν). D. Decomposition of Im[∆ENF (ν)] into the transient vibrational(Im[∆ENF, vib (ν)]) and carrier (Im[∆ENF, car (ν)]) contributions with a minor feature from Fanotype interference. E. The nano-localized transient vibrational signal Im[∆ENF, vib (ν)] at different
sample locations shows nanoscale spatial heterogeneity in the polaron-cation coupling. (Taken fromcollaboration with69 )Here, ν̃+ corresponds to the transient vibrational frequency. Figure 3.18 plots the transient
peak shift ∆ = ν̃+ − ν̃vib. with respect to the varying coupling constant J. The observed peak shiftof 5 - 8 cm−1 in the near-field pump-probe measurement corresponds to a polaron-cation coupling
constant of 50 - 70 cm−1 .
The conventional pump-unmodulated detection and pump-modulated detection is comparedin figure 3.19, where the conventional technique exhibits low-frequency noise that obscures the fast
dynamics, which are easily seen in the sideband demodulated approach. Further, the conventional

78transient peak shift Δ (cm-1)

20
15
10
5
0

0

20
40
60
80
100
-1
coupling constant J (cm )Figure 3.18: Relationship between polaron-cation coupling constant and transient vibrational peakshift | The shift in the vibrational peak position observed in the excited state, induced by the
coupling between the polaron absorption and molecular vibration as described in Eq. 3.16 forvarious coupling constant J. (Taken from collaboration with69 )B

VO2 (30 s avg.)
2ωt
2ωt ± ΩM

∆I (norm.)

1.0

0.5

0.0

-0.5

1%
-2.5

0.0

2.5

5.0

perovskite (5 min avg.)
2ωt
2ωt ± ΩM

0.030

Im[∆ENF] (a.u.)

A

0.025
0.020
0.015
0.010

7.5

T (ps)

10.0

νex
1650

1700

1750

ν (cm )
-1

Figure 3.19: Comparison of the excitation modulated and unmodulated detection | A. The T dependent near-field pump-probe signal ∆I demodulated at 2ωt (pump unmodulated) and 2ωt ±ΩM(pump-modulated) channels. B. The spectrally resolved near-field pump-probe signal observed at
T = 2 ps. While the 2ωt channel is dominated by noise, in 2ωt ± ΩM channel the peak associatedwith the transient vibrational resonance is evident.69

pump probe spectrum is dominated by noise, whereas the transient vibrational response is easily
seen with the sideband technique.
The spatial nanoscale heterogeneity of polaron-cation coupling in perovskite explored here

79
with HPP IR s-SNOMis summarized in figure 3.17A, where the instantaneous polaron absorption
for the interferogram collected with T = 2 ps delay arises from the instantaneous carrier (polaron)
response183 and the long lived narrow resonance results from the coupled transient vibrationalmode. We then Fourier transform this interferogram and recover the pump-induced transient
vibrational peak at ν ex in Im[∆ENF (ν)], which is blue-shifted from the ground-state peak position
ν gs in Im[ENF (ν)] by ∼ 5 cm−1 . This agrees with the far-field observed polaron-cation coupling.
Experimentally, we similarly (to the far-field experiment) collinearize the pump and probe
beams, which we then focus to the apex of an AFM tip, located in an enclosure that is purged withnitrogen gas (< 1% O2 ) to suppress photoinduced degradation. We select a 15 by 15 µm area toinvestigate polaron-cation coupling strength heterogeneity by spacing measurements 5 µm apart sothat each new location has not been dosed with pump to ensure measurements are independent ofeach other. We perform conventional heterodyne and sideband nano-FTIR at pump-probe delaysof T = 2 with ∼ 15 cm−1 spectral resolution. We repeat this 30 times and average to improve
SNR. Measurement stability is demonstrated by figure 3.20.

A

scan #1 (0 min)scan #30 (10 min)

0.4

5
~(0)

ε (ν)

100
75

4

νgs

0.2

νex

1

50
-0.5

6

-0.2

0
0.0

0.5

t (ps)

1.0

1.5

2.0

1675

1700

1725

ν (cm-1)

1750∆ε~ NF (ν)

INF(t) (a.u.)

125

B

-0.4

Figure 3.20: Transient vibrational nano-spectroscopy of a lead halide perovskite | A. Single INF interferograms demodulated at 2ωt at the beginning and end of 10 minutes of data collection, acquired
under constant illumination of the visible pump pulse with pump-probe timing delay T = 2 ps.
Both interferograms are not normalized or offset, indicating no sign of sample photo-degradation.
B. Retrieved nano-localized ground (black) and pump-induced (blue) complex dielectric functions
from ENF (ν) and ∆ENF (ν) in Figure 5C based on model fitting. (Taken from collaboration with69 )We take the ground state dielectric response and pump penetration depth (100 nm; based on

80
the absorption coefficient of the perovskite film characterized by UV-Vis absorption spectroscopy)
with the substrate dielectric constant (ϵsub ∼ 1.69) to determine ∆ENF by varying ∆ϵNF (ν) withbroadband responses and narrowband resonances to retrieve the dielectric function ∆ϵNF (ν).
In order to retrieve the transient complex dielectric function ∆ϵ̃NF (ν), we apply and fit thisnear-field spectra to the multi-layer finite dipole model described above (figure 3.14). Specifically,
the background carrier response ∆ϵ̃NF ∼ −0.1 + 0.3i, which agrees with ∆ENF (ν, results from thepolaron absorption. Here, the small carrier background does not give rise to a significant Fanotype interference (Figure 3.17D), meaning the nano-localized transient vibrational response can berecovered by simply subtracting the broad carrier background from Im[∆ENF (ν)].
The narrow-band resonance in ∆ENF (ν) results from both the transient vibrational response
and Fano-type interference. Here, to assess the effect a pure Fano-type interference has on the
dielectric function, we model the ground-state as having a narrow-band resonance and the photoinduced ∆ϵ̃NF, car (ν) results from a broad Drude response (Figure 3.21). The resonant feature canbe seen near the ground-state resonance ν gs .

5ε~(0) (ν)

Im[∆ε

]

1.0
-0.9

1

Re[∆ε~NF, car]Im[ ε~(0)]

-1.0
-1.1

1700

ν (cm-1)

1750

0.150

0.5

Im[∆ENF,car]-0.1250.0
1650

Re[∆ENF,car]

Im[ENF]1700

ν (cm-1)

1750-0.150∆ENF(ν) (a.u.)~
NF, car.

C 0.16

0.175

Re[ENF]∆ε~ NF (ν)

4

1650

νgs

1.1

Re[ε~(0) ]

0

B

1.2

Im[∆ENF(ν)] (a.u.)

νgs

ENF(ν) (a.u.)

A 6

νgs

νex

Im[∆ENF, car]

0.14

Im[∆ENF, vib]
0.02

0.00
1700

1720

1740

ν (cm-1)

Figure 3.21: Fano-type interference induced by carrier background | A. Model input dielectric
functions identical to Figure 2B but without the transient vibrational response. B. ENF (ν) and∆ENF, car (ν) calculated based on the input dielectric functions, with ∆ENF, car (ν) exhibiting Fanotype interference between the vibrational resonance and carrier background. C. The separated
carrier and transient vibrational contributions to Im[∆ENF (ν)] in Figure 2D, main text. (Taken
from collaboration with69 )We obtain the vibrational near-field pump-probe signal Im[∆ENF, vib (ν)] by simply subtract-

81
ing out Im[∆ENF, car (ν)]. This results in a spectrum with good agreement to the transient dielectric
function Im[∆ϵ̃NF ] – therefore, we take the Fano-type interference to be at least approximately additive in ∆ENF . Technically though, the individual contributions can be teased out by careful
fitting of the two independent contributions.
We now perform spectroscopy at different sample locations and observe a varying degree ofvibrational peak shifts, ranging from 5 to 8 cm−1 , indicating spatial heterogeneity in the couplingstrength between the polaron and cation between 50 and 70 cm−1 (figure 3.17E). This heterogeneity
is not resolvable in the far-field and such coupling likely depends on the nanoscale heterogeneity inlattice elasticity.
Nanoscale heterogeneity in the local chemical composition that results in disorder of the lattice and strain has previously been experimentally observed and theoretically predicted.188–190, 193
Here though, the ultrafast transient dynamics explored with HPP IR s-SNOM, the polaron formation, lifetime, and transport that defines the photovoltaic device performance is explored.
A mixture of three different A-site cations of formamidinium, methylammonium and cesium
are used here in a composition of [(FA0.83 MA0.17 )0.95 Cs0.05 ]Pb (I0.83 Br0.17 )3 . A thin film is spincoated on a glass substrate as previously established.189, 194 As we see from this example, excitation
modulation and sideband lock-in detection significantly increase sensitivity to the near-field pumpprobe signal of interest.

3.4.6

Comparison to other approaches

While many studies have recently been using un-modulated pump excitation to probe a rangeof low-energy phenomena in 2D, semiconductor, and other quantum materials,11, 13, 105, 171–177, 179
this conventional approach has limited applicability to detecting strong carrier and plasmon responses, where the high contrast between the excited- and ground-state allows for low sensitivity
detection approaches. Systems with lower excited state contrast have not yet been able to bestudied, including dilute carrier, low-conductivity carrier, vibrational, and phonon systems thatare important to many quantum and soft materials.

82
Far-field methods have historically used pump modulation and lock-in or gated detection todiscriminate the pump-induced response from the typically dominating ground-state response.195, 196
Here, we isolate the excited-state response from the ground state response by borrowing from theseadvances to extract the pure transient nano-localized spectrally resolved response. With this development, HPP IR s-SNOM is now universally applicable to resolve weak and strong perturbations
and excitations and is, in general, compatible with ultra-violet to THz frequencies with appropriate
optics and detection choices.197 We have applied this new technique here to study the transient
vibrational response in a lead halide perovskite to retrieve the phase and spectrally resolved vibrational response that we use as a quantitative marker for polaron-cation coupling strength. This
coupling gives rise to the photovoltaic response in perovskites and is therefore of high importance.
The nano-localized contrast and heterogeneity shown here, together with previous nano-localizedmeasurements, indicate nanoscale lattice strain differences give rise to the fundamental carrier
properties in perovskite solar cells.
From a condensed matter perspective, where electron-phonon coupling is of modern importance and study, nanoscale electronic excitation and vibrational probing is ideal for the study of
2D materials,198–200 hybrid photovoltaics,145, 149, 183 and nanoscale thermal transport.201 Therefore, our new technique is an important step in mapping the inherent disorder in electron-phonon
interactions and controlling nanoscale quantum technologies.164, 176, 202

3.4.7

Outlook

Following the same path as far-field advancements, HPP IR s-SNOM can be further extended
to ultrafast spectroscopy techniques currently available in the far-field like pump re-pump probe
spectroscopy as is performed in 2DIR to probe coherent population transfer at the nanoscale.150, 203, 204
Adding interferometric detection to adiabatic plasmonic nano-focusing for four-wave mixing would
enable 2D electronic spectroscopy at the nanoscale, characterizing free and bound carrier dynamics
in quantum materials.205–207 Therefore, we can use HPP s-SNOM to fully resolve the spatio-spectral
evolution of fundamental interactions in functional materials that define the wide range of appli-

83
cations and properties. We have successfully applied the benefits of far-field ultrafast spectroscopy
to probe local dynamics at the nanoscale.

Chapter 4

Strongly Coupled Multi-Quantum-Well Antenna Heterostructures

4.1

Introduction
In this chapter, I will discuss a polaritonic system based on an intersubband transition ina multi quantum well that strongly couples to an infrared resonant gold antenna. The sample
was prepared by Nishant Nookala, the far-field theory was performed by Sander Mann, and theexperiments and near-field theory were performed by me. This project is divided into two parts,
starting with the far-field, where a reflection spectroscopy measurement was extended into thetime domain with degenerate pump-probe spectroscopy and ending with the near-field, where thestrongly coupled field profile is mapped out, demonstrating phase and coupling strength control.4

4.2

Strongly coupled polaritonic systems
The operation of all-optical photonic devices critically depends on the third-order nonlinear

optical response, like power limiters, saturable absorbers, and switches for signal processing andcomputation. Significant changes in the properties of such materials are needed when high lightintensities are not feasible for the typically limited nonlinear response of optical materials, evenconventional non-linear crystals, where their IR performance is lacking. However, syntheticallyfabricated devices like metasurfaces demonstrate a giant third-order nonlinear response on theorder of 3.4 · 10-13 , which can be used with an infrared subwavelength plasmonic nanocavity tocontrol the coupling rate optically faster than 2 ps.

85
An intersubband polaritonic metasurface could be used as an all-optical modulated and controlled highly-nonlinear saturable absorber and optical power limiter. Further, a large coupling
constant could be applied to cavity quantum electrodynamic (cQED) phenomena and related applications like quantum information systems, such as a controlled phase flip gate where the phaseof light reflected from a cavity depends critically on the state that a strongly coupled emitter isin.208
Metasurfaces with highly nonlinear optical coefficients are desired as a wide range of applications call for them, like phase-matching, four wave mixing, beam switching and steering, opticallimiting, and phase conjugation.209, 210 Achieving large both χ(2) and χ(3) has previously been attempted, where intrinsic metal and dielectric nonlinearities were explored.211–217 Special materials,
like epsilon-near-zero and J-aggregates, have also been studied for their nonlinear behavior.218–223
To date though, the best approach for maximizing χ(2) is the combination of an intersubband
transition from a multi-quantum well (MQW) strongly coupled to a plasmonic resonator, like apatch antenna or nanowire, forming an intersubband polaritonic metasurface.224, 225 The electronic
states within the MQW are engineered to have a large nonlinear response when combined with thephotonic field profiles of the resonant antenna.226 These heterostructures give the, to date, largest
non-linear response in any bulk material. The photonic engineering of the field profiles provide
for a strong electric field that is highly confined and overlapped in the near-field with the MQWtransition dipole moment that hosts the nonlinearity. Previous demonstrations of tailored nonlinear
responses in meta surfaces have been performed from the near- to far-infrared, a spectral region
typically devoid of (or weak) nonlinearities in naturally occurring materials.

4.2.1

Device design and fabrication

We have tailored the device’s design to be temporally fast, highly nonlinear, compact andspectrally tunable based on fabrication and optical control. Specifically, the samples measured
here are grown with molecular beam epitaxy on an InP substrate. 10 nm Ti and 20 nm Pt wetting
layers for wafer metallization were formed with e-beam evaporation. 200 nm of Au was subsequently

86
deposited. and thermocompressively bonded to another similarly metalized InP substrate, forming
a metal backplane. Washing, polishing, and etching was performed to remove the original substrate.
A 300 nm SiNx hard mask layer was deposited on top of 10 nm of Ti and 100 nm of Au. Antennas
were mapped with e-beam lithography and formed by dry etching. The structure is fabricated bypatterning down the InGaAs/AlInAs MQW before adding a thin gold strip, which is the plasmonic
antenna that couples to the quantum well (figure 4.1a,b).227–231 The hard mask was then removed,
yielding the final product. A variety of antenna lengths were fabricated to account for material
specific losses, ensuring at least one antenna length is suitable for near degenerate resonance.
bcAbsorption

a

1.0
0.6
0.4
0.2
0.0

d

Intensity

0.8

0.8

0.9

1.0

1.1

1.2

0.8

0.9

1.0

1.1

1.2

1.0

ΔA

0.5
0.0
-0.5

Figure 4.1: Ultrafast, highly nonlinear metasurface with saturable absorption and optical power
limiting functionalities | a) At low intensities, a nanoantenna is strongly coupled to an intersubband
transition in the underlying MQW substrate (shown in green). As a result, polariton splitting isobserved at the central frequency and the incident radiation is fully reflected. b) At high intensities,
the intersubband transition is saturated, and the antenna resonance is not strongly coupled to it.
The incident pulse is now largely absorbed, leading to a limiting response. c) The low intensity
spectrum shows a clear signature of mode splitting due to strong coupling, while at high intensity
only the antenna resonance is observed due to saturation of the intersubband transition (see Eq.
4.1). d) Change in metasurface absorption between low and high intensity illumination, highlighting
saturable absorption (pink shading) and reverse-saturable absorption (green shading). (Taken fromcollaboration with4 )The heterostructure is imaged with a scanning electron microscope (SEM) and shown infigure 4.2a. Figure 4.2b shows the InGaAs/AlInAs MQW band diagram of a single ”unit cell” that
repeats 26 times over 400 nm. First principle modeling and absorption measurements confirm thetransition between the ground state and first excited state is ν 12 = 40 THz, whereas the transition

87
from the first to second excited state is ν 23 = 25 THz. This significant difference points to thepotential being quite anharmonic, as is expected for an approximately square well potential fromthe nature of the barriers and wells.
b

c
600

1 μm

2

300
200

1

100
0

2

4

6
8
10
Thickness (nm)1214Frequency (THz)

Energy (meV)

Lx

400

45
40
35

0.8

45

0.6

40

0.4

35

0.2

30

30
25
1.2

1.0

1.4
1.6
ly (μm)1.8

25

Absorption

Ly

3

55
50

50

500

ly

d

55

Frequency (THz)aexperimental range
25

30

35

40
45
(THz)

50

55

0.0

Figure 4.2: Linear response of the polaritonic metasurface | a, SEM image of the nonlinear metasurface under consideration, with patch antenna size lx = 0.35 µm and ly = 1.75 µm and unit cellsize Lx = 2.1 µm and Ly = 3.15 µm. b, Band structure of the asymmetric multi-quantum well,highlighting the 1-2 and 2-3 transition, which is purposefully detuned from the 1-2 transition. c,
Absorption spectrum measured with Fourier-transform infrared spectroscopy (FTIR) varying theantenna length ly . A clear avoided crossing with the intersubband transition is observed. The
dashed line indicates the transition frequency as obtained from bulk measurements. The color
scale is shown in d. d, Absorption as calculated from Eq. 4.2 at low intensity, where the plasmon
frequency νp is varied rather than the antenna length. (Taken from collaboration with4 )We tune the antenna length within each patch by varying its length to find the antenna closest
to degeneracy between the local surface plasmon resonance (LSPR) and the MQW intersubband
transition (figure 4.2c). We perform a linear absorption measurement with far-field FTIR, where
the strongly coupled system (two resonant peaks) shifts with antenna length. Here, we plot thefrequency response of the absorption measurement on the y-axis, antenna length on the x-axis andthe absorption (A =I in −I reflect, Iin is the incident intensity and Ireflect is the reflected intensity).
I in

The anticrossing behavior that can be read off in figure 4.2c is highlighted in figure 4.2d, where thedark mode at the intersection of the antenna and MQW resonances of fixed frequency is observed.
The MQW transition dipole moment is out of plane and cannot be driven by the far-field incident
radiation directly and is only driven through its coupling to the plasmonic antenna that does
appreciably couple to the far-field.232–234 Bulk absorption measurements of the MQW resonance88
give the expected resonance frequency, which is shown in figure 4.2d (dashed white line).

4.2.2

Nonlinear optical switching

The ultimate limit of the nonlinear conversion efficiency onsets upon saturation in intersubband polaritonic metasurfaces. Specifically, the population in the ground state of the MQWdecreases to half, which lowers the absorption coefficient of the driving field.235 We use this to ouradvantage here though, where extreme nonlinearity that exceeds perturbative responses reduces
the effective coupling rate between the antenna and MQW (figure 4.1a,b)236, 237Ω2 = Ω2 R (ng − ne )(4.1)

where ng and ne are the ground and excited state occupancies in the MQW and Ω2 R is the vacuum
Rabi frequency.
Being able to tune the coupling coefficient allows for all-optical control, where in one limit,
the low fluence regime, significant energy transfer between the antenna and MQW occurs, whereas,in the high fluence regime, only the antenna is able to be driven further. This can be seen spectrally, where the low fluence regime is characterized by two distinct resonances formed by thestrong coupling, forming upper and lower polaritonic branches. This is contrasted with the highfluence regime, where the saturation of the MQW no longer allows for absorption of radiation andtherefore coupling to the antenna, yielding a single spectral resonance (figure 4.1c,d). Using thisscheme of switching between strong and weak coupling as a nonlinear transition has previously
been demonstrated in microcavities and plasmonic arrays coupled to molecules.209, 220, 223, 237
The strong coupling regime is defined as the coupling rate or splitting exceeding the individual resonance decoherence or linewidths respectively. This is contrasted though with the desire foran ultrafast response, which necessitates reasonably large linewidths and associated high loss rates.
Simultaneously matching both of these conditions requires the coupling frequency to approach thefrequency of resonances, the ultrastrong coupling regime, which has also been previously demonstrated.232, 233, 238, 239

89
We leverage the strong dipole moment of the MQW transition and large linewidth of a plasmonic resonator, from its native Drude damping, to achieve nonlinear switching at low intensities
to be a sensitive saturable absorber at low intensities.

4.2.3

Strong coupling theory

The interaction between the MQW and antenna is described by coupled-mode theory andMaxwell-Bloch model in the rotating wave approximation, where the MQW is simplified to N
two-level systems with coupling rate given by g.240–243

ȧ = (iω c − γ r − γ a − γ s (w + 1))a + iN gq +p
2γ r s+(4.2)

q̇ = (iω q − γ 2 )q − igaw

(4.3)

ẇ = 4g Im{qa∗ } − γ 1 (w + 1)

(4.4)

where ȧ describes the complex plasmon amplitude (normalized to the stored number of photons), q̇
off-diagonal elements of the MQW transition, and ẇ inversion of the population.243 The plasmon
resonance is described by its center frequency ω c and loss rates γ r and γ a , which are the radiation
and absorption losses respectively. The MQW resonance on the other hand has its center frequency
given by ω q and its loss rates summarized by γ 1 and γ 2 , which are the population decay andtotal decoherence respectively. The underlying physics is understood as the amplitude operator isexternally pumped (s+ term), internally coupled (N g term), and intrinsic restoring force (a term).
The operator for the MQW is similarly structured without the external driving field, because thetransition dipole moment forbids far-field excitation directly of the MQW. Therefore, we only drive
the system with s+ (normalized for the number of photons/second). The inversion of the system
is captured by w, which includes the antenna energy transfer as well as the MQW relaxation.
We add an intensity dependent decay rate γ s to account for the additional MQW level beyond

90
the simple two level approximation, which comes into play at higher intensities and excited state
populations. At sufficiently high enough intensities, the electrons are excited into the continuum
and aren’t bound by the wells within the MQW. However, at sufficiently low enough intensities, thesystem of equations reduces to coupled simple harmonic oscillators, with damping given by γ. We
extract relevant parameters through low fluence measurements in the linear absorption regime andapply the relevant (fluence independent) parameters to measurements performed in the non-linear
case (high fluence).
The MQW is best described as N two level systems indexed by |1⟩ and |2⟩. The interaction of
the plasmonic cavity with a single 2 level system is described by the dipole transition between the
ground and first excited states though the coupling of the optical electric field from the plasmonic
antenna cavity. The coupling strength and rate is described by the Rabi coupling rate Ω. We
combine creation and anihilation operators to describe the excitation σ 12 = |2⟩ ⟨1| and decay
σ 21 = |1⟩ ⟨2| of carriers within the MQW.
If we restrict the MQW response to be summarized by a single 2 level system (N = 1), theHamiltonian is

H = H 0 + H int + H pump

(4.5)

where the free evolution is given by H 0 , the driving of the system is given by H pump , and theinteraction between the two level system and the plasmonic antenna is given by H int . Specifically,

H 0 = ℏω c â† â +2X

ℏω i σˆii(4.6)

H pump = dc E + σ̂ + h.c.

(4.7)

i=1

where we have used the cavity resonance ω c and the MQW electron energies ℏω i=1,2 to describe theenergy sum of the plasmonic cavity mode and the carriers in the MQW with the standard number
operator. The number operator counts with creation and annihilation operators. The particle

91
number is not conserved with pump as expected since we are supplying energy through the far-field
driving. We now discuss the interaction Hamiltion, which is responsible for the coupling of the
MQW to the antenna and is given by:

H int = ℏg(σ 12 aˆ† + h.c.)(4.8)

with coupling rate g and creation operator aˆ† and 1-2 coupling via σ 12 .
Approximating the expectation value of the product of two operators by taking the productof the expectation values gives the semiclassical approximation for the cavity field, polarization of
the 1-2 transition and the occupation probabilities of the first and second excited states. These are
given byȧ = (iω c − γ r − γ a − γ s )a + igq + idc E(t)

(4.9)

γ1
− γ dp,1 − γ dp,2 )q − ig(σ 22 − σ 11 )a2(4.10)σ˙11 = γ 12 σ 22 − 2q Im{qa∗ }(4.11)σ˙22 = −γ 12 σ 22 + 2q Im{qa∗ },(4.12)q̇ = (iω 12 −where we recognize (γ dp,1 + γ dp,2 )q in equation 4.10 as γ2∗ , the pure dephasing rate, which, whencombined with

γ1
2 , is the total decoherence rate γ2 .

As mentioned above, we track the inversion

by defining w = σ22 − σ11 . With the appropriate choice of normalization, we define the normal√
ized quantity σ11 + σ22 = 1. We further connect our newly defined idc E(t)as 2γ r s+ that weℏ
defined above with normalization of s+ ∗ s+ as the number of photons per second. Upon inserting
normalization conditions and recognized equalities, we nearly recover equations 4.2,4.3, and 4.4.
We now extend the single MQW picture to N MQW systems. To do this, we combine multiple
MQW resonances through non-coupled interaction (with each other) to the cavity (where they do

92couple). To do this, we generalize our system of equations above and cast into matrix form, where
the diagonal terms are the non-coupled free evolution terms and the first column and first row are
the couplings between the cavity mode a and the MQW resonance. The other off diagonal terms
are set to 0 under the assumption that they do not couple with each other. In principle, each MQWmode can have a slightly different coupling to the cavity, which we will label by indexing gj , withj being the index of MQW two level system up to N such systems.






ig1
ig2
 ȧ  iωc − γc
  
  
 q˙1   iωg1
iω21 − γ2
0
  
  
 q˙  =  iωg
0iω21 − γ21 2   
 ..  ..
..
..
 .  
.
.
.
  
  
iωgN
0
0
qN
˙



√
igN   a   2γr s+       q1   0 
0    

,



+
0
0
q
2

  
    ..  
.
.





.
···  .    
0iω21 − γ2
qN


···
···
···
..

.

···



(4.13)

In contrast to equation 4.4, we must keep track of each two level system inversion given by wj andobeyingẇj = 4g Im(qj a∗ ) − γ1 (wj + 1).

(4.14)

If we assume that all MQWs equally couple to the antenna, we may assign all gj to simply aneffective g – doing so reduces our system of 2N + 1 equations to an effective 3 equation model
whereȧ = (iω c − γ r − γ a )a + iN gq +p
2γ r s+(4.15)q̇ = (iω q − γ 2 )q − igaw

(4.16)ẇ = 4g Im{qa∗ } − γ 1 (w + 1),(4.17)

93
which looks like the familiar derivation prior to extending to N two level systems with the inclusionof N in the equation for ȧ and g taking on the new meaning of an effective coupling constant. In√
the low intensity approximation, we can read off the observed mode splitting as 2ΩR = 2 N g.
Now we will consider the case where the MQW is not simply a two level system. In reality,
the MQW has 3 bound levels and a continuum. At sufficiently high enough intensities, where
there is an apprecible pumping of electrons into the continuum, the electrons will experience a
Drude damping from scattering off other electrons. This Drude damping can be added by theterm γ s (w + 1), where the total loss rate is related to the inversion, which approximately tracks
the number (upon scaling) of continuum electrons. With this insertion, we finally recover the fullsystem of equations that describes the antenna MQW heterostructureȧ = (iω c − γ r − γ a − γ s (w + 1))a + iN gq +p
2γ r s+(4.18)q̇ = (iω q − γ 2 )q − igaw

(4.19)ẇ = 4g Im{qa∗ } − γ 1 (w + 1).

(4.20)

At low intensities, the Drude damping of the continuum MQW electrons can be neglected
and we are left withȧ = (iω c − γ r − γ a )a + iN gq +q̇ = (iω q − γ 2 )q − iga,p2γ r s+(4.21)(4.22)

which upon usings− = s+ −p2γr a(4.23)

94
and
r=

s−
,
s+

(4.24)

we calculater =1+2γr (i∆q − γ2 )N g 2 + (i∆c − γr − γa )(i∆q − γ2 )(4.25)∆q = ω21 − ω(4.26)∆c = ωc − ω.

(4.27)

The final assumption that I would like to address is the rotating wave approximation, whichassumes only the linear terms from the semiclassical derivation above. This is valid in the weak andstrong coupling regimes, but must be considered when the coupling strength reaches the ultrastrong
coupling regime. This is defined as the coupling frequency being one tenth or more of the resonanttransition frequency ΩR ≥ 0.1ω0 . In the experiment performed here, the coupling strength isapproximately 0.13 of the resonant transition frequency, edging on the transition between strong
and ultrastrong coupling. With some accuracy loss of absolute parameters, we choose to continue
employing the rotating wave approximation as the physics are well described, with ”effective”
parameters rather than true parameters. The effect of the rotating wave approximation included
versus without it can be summarized by analyzing the polariton branches as a function of couplingstrength for both cases with and without the rotating wave approximation. In this comparison, thetwo photon cavity mode coupling that occurs near 80 THz is included in its coupling to the MQW.

4.2.4

MQW device characterization

We then apply the theory derived in the previous section to fit the FTIR spectra collected
for various antenna lengths figure 4.4.
Both the unit cell size of the antenna MQW heterostructures and the widths of each antenna
are kept constant. The MQW resonant frequency ω21 , total dephasing rate γ2 , and coupling

95

Figure 4.3: Effect of ultrastrong coupling | In the model we apply the rotating wave approximation,
which ignores quadratic and antiresonant terms. It is known that for very high Rabi frequencies
these terms affect the splitting, as shown here by tracking the resonance frequencies. Here we show
the effect of the antiresonant term. In the experiment described in the main text, ΩR /ω = 0.13,
and the resulting differences in splitting may safely be ignored. (Taken from collaboration with4 )strength Ω are kept constant across all antenna lengths, whereas the antenna resonant frequency
ωc , radiative damping γr , and absorption damping γa are fitting parameters for each array.
Figure 4.5 shows the extracted parameters for each antenna length and the frequency crossing between the MQW and antenna resonant frequency is clear. The antenna lengths for which
the resonant frequencies are closest are those used in the experiments shown here. We use theparameters extracted in this experiment to inform our fitting.

4.3

Nonlinear reflectivity ensemble demonstration
We now proceed to the measurements that characterize the novel functional operating regimes

for the metasurface for, e.g., optical switching and power limiting through saturable absorption.
To begin, we perform spectroscopy of the MQW metasurface by an absorption measurement at two
different fluences, medium peak intensity (∼70 kW/cm2 ) and high peak intensity (∼700 kW/cm2 ).
We use a tunable ps laser (Carmina, APE) with a 15-20 cm−1 bandwidth generated by spectral

96Figure 4.4: Maxwell-Bloch model fits to FTIR spectra | FTIR spectra (blue) from six different
arrays with the same unit cell size and antenna width, but different antenna length (mentioned as
inset in each graph), were fit using the low-intensity Maxwell-Bloch model (red). The array underconsideration in the main text is the bottom left one. The resulting parameters are shown in figure4.5. (Taken from collaboration with4 )focusing of the difference frequency generation between the signal and idler pulse.
Here, a ps optical parametric oscillator (OPO) followed by difference frequency generation(DFG) generates mid-IR light with ∼1.6 ps (intensity FWHM) pulses at a 40 MHz repetition
rate. We focus this light onto a single array of antenna MQW heterostructures with a BaF2 lensand control the fluence by tuning a wire-grid polarizer pair to adjust input power by turning
the first polarizer relative to the second, maintaining final output polarization. We simultaneously
measure the input and reflected power with 2 mercury-cadmium-telluride (MCT) detectors by using
a ZnSe beam-splitter before the focusing lens. We use this geometry with simultaneous detection
to normalize power fluctuations and we normalize our reflectance results to a Au reference as themid-IR reflectance of Au is spectrally flat.
At the lower power used here, the system is not saturated, meaning the coupling strength is97

Figure 4.5: Model √parameters from the fitting procedure | To reduce degrees of freedom, the total
coupling strength N Ω, the MQW 1 - 2 transition frequency ωq and linewidth γ2 were global
fitting parameters. The other parameters were considered free per each metasurface. (Taken fromcollaboration with4 )still significant and splitting between the two hybridized modes can be observed. However, as thepower is increased 10 fold, the MQW effectively saturates, turns transparent and no longer couples
to the plasmonic antenna cavity, causing the polaritonic splitting to vanish, giving a single peakfrom the bare antenna resonance. While infinitely high power would be required to completely
saturate the MQW (as there is no population inversion in this system), the high power used hereis sufficient to take the system out of the strong coupling regime and into the weak coupling
regime where only one mode is resolvable. Figure 4.6 shows the experimental data compared to
the Maxwell-Bloch model with an effective coupling constant g that incorporates the contributionof N two-level systems (see above).

a

b

0

Pulse fluence (μJ/cm2)
0.2
0.4
0.6

c

0
0.9

Pulse fluence (μJ/cm2)
0.4
0.8

d

98

Pulse fluence (μJ/cm2)
0.4
0.8
1.2

0
0.9

0.9

0.8

0.8

0.8

0.8

0.7

0.7

0.7

FTIR
Low intensity
High intensity

0.2
0.0
30

35

40
45
Frequency (THz)

0.5
0.4

50

0.3

0.6
0.5

0

100
200
300
Ipeak (kW/cm2)

0.3
0

lop
er s
pow

0.5
0.4

0.4
35.5 THz

0.6

low

0.6

Absorption

0.4

Absorption

0.6

Absorption

Absorptione1.0

38 THz200
400
Ipeak (kW/cm2)

600

0.3

40.0 THz
0

200
400
600
2
Ipeak (kW/cm )Figure 4.6: Experimental nonlinear tuning of the coupling coefficient | a, Absorption spectra at low(∼70 kW/cm2 , blue) and high (∼700 kW/cm2 , red) peak intensities, demonstrating the transition
from strong to weak coupling of the polaritonic metasurface. We also show the FTIR absorption
spectrum for comparison. b, Absorption at 35.5 THz (left shaded region in a) as a function ofpeak intensity, demonstrating very large nonlinear absorption at low intensity. Experimental datais shown by circles, the predicted response from the model is shown with the solid line. c, Same asin panel b, but at 38 THz (middle shaded region in a). In this case we observe reversed saturable
absorption (optical limiting) as the normal mode passes by. d, Power dependence at 40 THz (rightshaded region in a). Here we observe maximal optical limiting for this sample. The black line
(3)
shows the slope at low intensity, used to estimate the effective Im(χ(ef f ) ).4

While the spectrum changes between the strong and weak coupling regime for two distinct
powers, the total absorption at a single frequency non-linearly depends on the input intensity. To
observe this, we tune to three frequencies as indicated in figure 4.6a. The fluence dependent responseat these frequencies is summarized by figure 4.6b,c,d for 35.5 THz, 38 THz, and 40 THz respectively.
We choose these frequencies as they represent the ultra-strongly coupled lower polariton branch,
the strongly coupled lower polariton branch (at higher fluences), and the bare antenna resonant
frequency, where absorption is maximized at yet higher fluences. When tuned to the ultra-strongly
coupled polariton branch, increasing the fluence results in a reduction of the mode splitting (as the
MQW begins to saturate), which causes the absorption to decrease as the lower polariton branchis swept (via power) to higher frequencies. At the intermediate tuning frequency, the absorption
increases slightly with intensity as the mode splitting reduces with increasing fluence. Finally, when
tuned to the bare antenna resonance, the absorption strongly increases as the MQW saturates andthe bare antenna resonance is recovered, where we are effectively only driving the antenna LSPR. I

99would like to note though that when the MQW saturates the individual absorption rate decreases(trivially from saturation), yet the combined absorption increases at this frequency as the system
is no longer significantly coupled.
As an additional consideration for the fitting performed here, we incorporate the effect a
Gaussian laser source has on the measurements, where the intensity is not constant across the focal
spot and therefore the response is heterogeneously averaged. We take the Gaussian beam profile
to beI(x, y) = I0 e

x2 +y 22
w0,

(4.28)

where w0 is the spot size (1/e2 ) and I0 is the un-normalized total intensity. To model the intensity
dependent absorption, we must average over each antenna in the array

P
A(I0 ) =x,y A(I(x, y))I(x, y)P

x,y I(x, y)(4.29)

To compare the nonlinearity of this metasurface, we treat the sample as homogeneous andcharacterized by χ(3) as the change in absorption from low to high intensities. We define β = ∆A
2dwhere ∆A is the change in absorption between the two intensity regimes and d is the single passdistance, which is 500 nm for this sample. We therefore find β = 1.6 · 107 cmGW -1 which impliesIm χ(3) = 3.4 · 10-13 m2 V -2 , where we have used the relationshipIm χ(3) =2ϵ0 n2 c2 β.
3ω(4.30)To date, this is the highest third order nonlinearity by 2-3 orders of magnitude.218, 219

4.3.1

Ultrafast optical switching

These results have thus far been from the interaction of a single pulse with the metasurface,
but the saturation can be started with a separate pump pulse as is the case for degenerate pumpprobe measurements performed here. We direct the pump pulse along a separate path and focus it

100
onto the sample surface with an incidence angle of 30 degrees to normal. We use a high intensity
pulse to pump the MQW into saturation and use a low fluence (≤ 50kW cm-2 ) pulse to probe thesystem and not repump the MQW back to saturation at normal incidence.
a

b

pump

probe

0.10
0.08

c

35.5 THz

ExperimentModel

40.0 THz

0.0

0.06

-0.1

0.04
0.02

-0.2

0.00

-0.3

-0.02
-5

0.1

0

5
t (ps)

10

-5

0

5

10

t (ps)

Figure 4.7: Transient all-optical control of metasurface reflection | a, Schematic showing the alloptical control experiment. A pump pulse excites the metasurface at an oblique angle, whilea weaker pulse probes the metasurface response under normal incidence with time difference t.
b, A degenerate pump-probe measurement at 35.5 THz, demonstrating how the reflectivity canbe increased all-optically at ultrafast timescales. The measurements are in excellent agreementwith theoretical predictions, shown in red (with a peak intensity of 12.5 kW/cm2 ). Shaded areas
show measurement uncertainty. c, A similar measurement performed at a different frequency todemonstrate optical limiting (40 THz), showing that reflectivity can also be all-optically reduced,
highlighting the versatility of the metasurface operation. The red line again shows theoreticalpredictions, now for a peak intensity of 30 kW/cm2 .4

We use the same ps OPO/DFG system described above for the ground state measurement,
but split the output mid-IR generated light into a strong pump and weak probe beam and direct
them along different paths. The pump beam is focused onto the sample at an angle, whereas
the probe beam is under normal incidence, just as it was for the ground state measurement. We
move the pump beam focusing lens to optimize spatial overlap between the pump and probe at the
sample surface. We again monitor the probe pulse, modulated at the pump and probe frequencies(to discriminate against the dominating ground state response) with two MCT detectors placed
similarly on either side of a ZnSe beam-splitter. We simultaneously monitor the pump modulation
frequency plus probe modulation frequency as a function of relative pump-probe delay to retrieve
the time dependent reflectivity.

101
Figure 4.7 shows the frequency dependent pump probe absorption traces. In figure 4.7a, thelaser is tuned to 35.5 THz, the ultra-strongly coupled lower polariton branch. As expected fromthe power dependent results in figure 4.6, we observe a decrease in absorption immediately after
pumping as the pump saturates the MQW and reduces the mode splitting, reducing the absorption
at the lower polariton branch. We then tune the laser to 40 THz (figure 4.7b) and observe theopposite behavior, as expected. The time dependent Maxwell-Bloch model agrees well with theinclusion of finite pulse length (1.6 ps). We extract the population decay rate (T1 = 1.73ps) that is
combined into the total dephasing rate for the time independent measurements in figure 4.6. The“rise time” (the time it takes to pump the MQW) is limited by the pulse length, which is limited by
the bandwidth of the laser, which means the nonlinearity onset dynamics are on the 100s fs or lesstime scale. This enables the ultrafast control of the metasurface optical response and perfomancein applications like ultrafast optical switching and power limiting through saturable absorption.

4.3.2

Performance limitations

We now look to find the fundamental limit of reflection contrast with our polaritonic metasurface. The ideal case is the optical limiter fully absorbing when the MQW is saturated and fullyreflecting when the MQW is entirely in its ground state. The losses from Drude damping in theplasmonic antenna and the MQW is the fundamental limit of minimum reflection. The maximum
reflection is limited by the critical coupling, when the metasurface is perfectly absorbing. To begin,
we recall our previous definition of s−s− = 1 −p
2γr a(4.31)and the definition of absorbanceA = 1 − |r|2 = 1 −

s2−
,
s2+

which, in combination with the definition of absorbed (damped) power yields(4.32)

102

A=

Pabs
2(γa |a|2 + γ2 |q|2 )=Pinc|s− |2(4.33)

at high intensities w = 0 and a low intensities w = −1. Therefore, upon degeneracy ω = ω0 ,Alow =

4γ2 γr (γ2 γa + Ω2R )(γ2 (γa + γr ) + Ω2R )2(4.34)

4γr (γa + γs ),(γr + γa + γs )2(4.35)

Ahigh =

but for the case γr = γa + γs , Ahigh = 1 and critical coupling is achieved. This condition applied
to Alow though implies

Alow (γr = γa + γs ) =(Ω2R − γ2 γs )2.
(γ2 (2γa + γs ) + Ω2R )2(4.36)We define the low intensity dephasing loss as Γ2 = Ω2R /γ2 , the relative plasmonic antenna absorption
to losses in the MQW as η = γa /(γa + Γ2 ), and the fraction of Drude losses in the high intensity
limit to the low intensity dephasing losses as fs = γs /Γ2 , we can write a simple expression for theabsolute contrast defined as |∆A| = |Ahigh − Alow | as|∆A| =(1 − fs )2.
1+η
( 1−η + fs )2(4.37)

This gives the fundamental limit from the intrinsic losses in the MQW and plasmonic antenna.
We can explore the non-degenerate frequency as well, where we seek to maximize the absorption difference between the strongly coupled system (low fluence) and the weakly coupled system(high fluence) as defined by Alow − Ahigh , where we naturally will look to the peak position of either
the lower or upper polariton branch. We refer to this as resonantly enhanced saturable absorption.
The absorption contrast here is given by|∆A| =γ˜s (γ˜s − 2γ˜2 ) + 1(2γ˜a + γ˜s )2 + 2γ˜m (2γ˜a + γ˜s ) + 1(4.38)

103withγ˜2 = γ2 /ΩR(4.39)γ˜a = γa /ΩR(4.40)γ˜s = γs /ΩR(4.41)γ˜m = γm /ΩR .

(4.42)

Optical limiting at this critical coupling is shown in Figure 4.8. The first example (figure 4.8a)shows the metasurface with low absorption at low intensities at the critically coupled frequency
at high intensities. The optical contrast here is |∆A| = 0.65 for input parameters η = 0.031
and fs = 0.08 (white circle figure 4.8a contour plot). As can be read off from the contour plot,
minimizing both η and fs will maximize absorption contrast and represents photonic and material
engineering respectively. For example, using different materials with less Drude damping for theantenna and designing a MQW with longer total decoherence times would improve performance atthe expense of ultrafast response times.
At the other end of the spectrum (contrast, not frequency), maximizing saturable absorption
can be done in two ways. The first is to use a critically coupled system at low intensities (stillusing the same plasmonic antenna resonance) that then loses its critical coupling condition at highintensities. This behavior is symmetrically the same as what was demonstrated in figure 4.6. The
second is to shift to either the lower or upper polaritonic branch, where the contrast now depends
on γ˜s , γ˜a , and γ˜2 . We highlight small γ˜2 in figure 4.8b and large γ˜2 in figure 4.8c, where they are
0.1 and 0.75 respectively. Again, as expected, maximum performance requires minimizing γa andγs and, ideally, γ2 – similarly looking to reduce the total decoherence rate of the MQW.

a
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Figure 4.8: Limits to metasurface absorption contrast | a, Saturable and reverse saturable absorption when aligned with the bare resonance frequencies. The top part shows an example of reverse
saturable absorption with critical coupling at high intensity. The bottom part shows the absorption
contrast achieved for combinations of η and fs , where the circle corresponds to the parameters in
the top subplot. b, Similar to a, but now for saturable absorption when aligned with a criticallycoupled normal mode at low intensity, when γ2 /Ω = 0.1. c, Saturable absorption when aligned witha critically coupled normal mode at low intensity, when γ2 /Ω = 0.75. (Taken from collaboration
with4 )

So far, we’ve determined that the design of our MQW antenna heterostructure yields an optical contrast of |∆A| = 0.2 (η = 0.09 and fs = 0.32). This deviates from the observed experimental
results, where a maximum contrast of |∆A| = 0.3 is observed. This results in our system not being
critically coupled from γ2 being smaller than designed, which lowers the absorption at low fluences,increasing the observed optical contrast. We note that the greatest avenue for improved opticallimiting contrast in the future will result from improved MQW designs where high intensity Drude
damping is reduced and improved nanocavity antennas, where losses are similarly minimized.

4.3.3

Ensemble performance summary

Ultrafast optical control of the coupling constant between a plasmonic antenna MQW heterostructure has been demonstrated in the mid-infrared with complete control between the ultrastrong, strong, and weak coupling regimes. This represents the highest nonlinear coefficient to datein the mid infrared spectral regime, where modest optical powers influence large spectral changes.

105
Performance of next generation systems will critically depend on the damping rates of the plasmonic
antenna and MQWs used in the heterostructures. To further study the cavity quantum electrodynamics phenomena associated with these heterstructures, we extend far-field measurements tothe near-field, where the optical driving phase and coupling strength can both be simultaneously
controlled with the use of a metalized atomic force microscope tip.239

4.4

Near-field quantum phase and hybridization control
While the results demonstrated above impressively show macroscopic ultrafast power limit-

ing and optical switching behavior, resolving and controlling the phase and coupling strength forsingle antennas remains elusive. Near-field spectroscopy, with its deep sub-wavelength diffraction
unlimited spatial resolution and complex response sensitivity, enables quantum state phase andhybridization control through tip sample positioning.
With applications from biological to quantum materials, infrared vibrational scattering scanning near-field optical microscopy (s-SNOM) has advanced to become a powerful nano-imaging
and -spectroscopy technique and has previously been demonstrated in its application to infrared
vibrational strong coupling.244 Infrared polaritonic heterostructures based on a multi-quantumwell intersubband transition that are coupled to a gold antenna similarly exhibit novel hybrid statephenomena, including record-high nonlinear optical responses and optical power limiting behavior.
However, the collective response of an ensemble of these structures are limited to static passive
performance depending on fabrication parameters.4 Here, we extend our demonstration of ultrafast optical switching to broadband and ultrafast infrared nano-probe imaging and spectroscopy ofsingle antenna quantum well heterostructures to actively tune quantum-well saturation, couplingstrength, and quantum path interference (figure 4.9) – the tip controls the interference pathways
and relative coupling strength through manipulation of the nano-cavity mode volume.
Optical control is enabled through controlling the saturation level of the MQW, i.e. higher
peak powers saturate the MQW and the coupling strength to the antenna decreases.239 Midinfrared light was produced by narrowband mode difference frequency generation (DFG) between

106

Figure 4.9: Comparison of 2 versus 3 oscillator coupling | left, Far-field (previous section) strong
coupling model, where mid-IR resonant antenna strongly couples to MQW. right Near-field strong
coupling model where the tip position controls phase response through far-field driving.

the signal and idler of a 40 MHz 1032 nm pumped optical parametric oscillator (OPO), which
produces light with a bandwidth of approximately 15 cm−1 and a pulse duration of 1 ps. In farfield ultrafast degenerate pump-probe measurements, a Mach-Zehnder interferometer with one armof variable delay length was used to generate the pump and probe pulses. Reflectivity measurements
were detected in a balanced detection scheme to compensate for small drifts in the laser power.
The ps pulse duration of the narrowband mode enabled bridging the gap between low and highpeak intensities in order to observe both strong coupling in the low peak intensity limit and weakcoupling in the high peak intensity limit.
However, full optical control demonstrated in the near-field (schematic figure 4.10) requires avariety of light sources were used, including low peak intensities provided by a synchrotron like theAdvanced Light Source (ALS), which has a pulse duration of 60 ps, a repetition rate of 500 MHz,and a maximum power of 1 mW and the high peak intensities provided by the broadband modeof the same OPO/DFG combination that has 4 times or more shorter pulse lengths and used in ageometry with 5 times smaller focus spot size. The near-field measurements used these varying
light sources to demonstrate the full optical tuning for hybridized states, phase resolved frequency
imaging, and peak position power dependence.

107

Figure 4.10: Near-field Michelson interferometer for nano-FTIR and pseudo-heterodyne | Lightsource (synchrotron or ps laser) excites tip and antenna that couple to MQW, whose backscattered
light is interferometrically detected with MCT.

Figure 4.11 shows the near-field spectral phase evolution down the antenna. Starting on
the near side of the antenna relative to the incident beam and equivalently the back side of thesurface plot, shows a similar spectrum to what was observed in the far-field measurements in theprevious section. We observe two peaks notably different in magnitude, but through extensive
measurements, this was determined to be dependent on the particular antenna being measured.
The far-field equal weighting between lower and upper polariton branches resulting from ensemble
averaging. As the tip is scanned along the antenna, the spectral peaks begin to shift into dispersivelineshapes by the middle of the antenna and finally turn into two distinct dips on the far end.
Through coupled oscillator modeling, this is understood as a difference in the driving phase of
the MQW through the tip and antenna respectively. Specifically, as the sample is moved under thetip, the tip oscillation phase (and resulting near-field driving phase) obviously remains unchanged.
The antenna that is being moved under the tip though, to first order within the beam spot of
the light source focus also does not change phase. However, the antenna nature of the plasmonicresonance implies the phase of the oscillation in the near-field will be ∼180◦ out of phase from one
end of the antenna to the other. This results in the MQW, which is driven both by the tip and theantenna, to experience drastically different driving fields as the sample is scanned below the tip,

108

Figure 4.11: Phase resolved single antenna strong coupling | Near-field phase spectrum in linescan
along wire, starting (back) with tip at laser incident side, when both tip and antenna are in phase,ending (front) with tip at opposite side, where tip and antenna are 180◦ out of phase.

resulting in the evolution of the shape of the near-field phase spectrum.
We then proceed on to measurements using the same light source as the far-field, the ps
OPO/DFG output from the APE Carmina. These ps pulses can be tuned through fluence with awire-grid polarizer pair to drive the MQW from the low excited state population through saturation.
At low fluences (figure 4.12a), we observe, through frequency resolved pseudo-heterodyne imaging,strong coupling at the frequencies observed with the ALS synchrotron. We monitor the couplingstrength by plotting the difference in phase from one end of the antenna to the other, with resonant
frequencies showing stronger phase differences, owing to the antenna like behavior in the hybridized
state (figure 4.12c). We note here again that the asymmetry of peak amplitude is consistent for thesame antenna measured using synchrotron radiation. However, as we increase the input fluence, we109observe a transition of strong coupling to weak, interestingly, only at one end of the antenna (figure
4.12b). We again plot the antenna strength as a function of laser frequency and only observe onevery broad peak (figure 4.12d).

Figure 4.12: Phase resolved single frequency imaging between strong and weak coupling regimes| a, low peak power (strong coupling) pseudo-heterodyne imaging at tuned frequencies (z axis) –color scale is near-field phase response. b, medium peak power (weak coupling) pseudo-heterodyne
imaging at tuned frequencies (z axis) – color scale is near-field phase response. c Phase difference(antenna strength) across wire as function of driving frequency, showing peak splitting, indicatingstrong coupling. d Phase difference along wire, showing no peak splitting, indicating weak coupling.

We attribute this asymmetry in peak splitting to the relative driving strength at one end of
the antenna relative to the other. FTDT simulations similarly predict higher field concentrations
at the near-side (relative to driving field) compared to the far side. We expect this as the tip
effectively shadows the antenna when positioned at the near-side. This relative signal strength anddriving field strength has also been reproducibly observed both in these samples and other antenna

110
molecule samples.244 We then plot the spectrum along the wire (similar to the synchrotron dataset,with less spectral resolution given the psuedo-heterodyne nature of the measurement – figure 4.13).
This emphasizes the effect of the increased field strength on the near side of the antenna, where
the MQW is driven more strongly (top, red), begins to saturate, causing the coupling strength todecrease. As the tip is moved down the antenna and the driving strength decreases, the MQW andantenna remain strongly coupled (bottom, blue).

Figure 4.13: Single fluence coupling strength tuning through tip positioning | Near-field phase
spectra along the wire, starting on laser incidence side (top), ending on opposite side (bottom),where the transition from weak to strong coupling is observed.

We have shown how the nano-optical tip acts as a coupled antenna resonator and how the
tip-sample nano-cavity enhances far-field coupling to control photon emission, electric field orientation, and nanoscopic field heterogeneity. We further extend the picosecond laser far-field opticalpower limiting and pump-probe characterization to probe and control the ultrafast near-field opticalresponse, where the positioning of the tip, polarization of the incident light, and sample orientation
can be tuned to give quantum state hybridization control and to perform qubit phase rotation
operations.

Chapter 5

Ultrafast visible pump infrared probe of WSe2 on SiO2

5.1

Introduction
In this chapter, I will discuss the application of pump modulated pump-probe ultrafast imag-

ing that I developed in chapter 3 to image heterogeneity in carrier dynamics in WSe2 and local
temperatures as a measure of thermal boundary conductance. This work was primarily designed,
performed, and analyzed by me. Theoretical calculations were performed by collaborator Shiqian
Hu under the guidance of Baowen Li.

5.2

Transition metal dichalcogenides
Transition-metal dichalcogenides (TMDs) have attracted significant attention due to theirunique optical properties that depend on the symmetry and dimensionality for applications ranging from spintronics to atomically thin photodetectors and field effect transistors. The ultrafast
dynamics that govern the excited state populations, spin polarization, and coherence are characterized by fast decaying intravalley bright excitons and long lived dark excitons in W centered
TMDs.
In WSe2 , the valence band and conduction band are split by strong Ising spin-orbit couplingthat is hundreds of meV in the valence band, yet only few meV in the conduction band. The split in
the valence band gives rise to so called A and B excitons with A being the lower energy of the two– for the energies used below, we will ignore contributions from the B band. The splitting of the
conduction band though, for monolayer and few layer samples, gives rise to a k-valley opening from112
the broken inversion symmetry. This k-valley opening is what makes TMDs attractive for spintronic
applications as the two bands are spin polarized. In W centered TMDs, an electron populating thethermal minimum state and a hole populating the A band valence band is called a dark exciton
as the optical transition is spin forbidden and therefore slow ∼300 ps. In contrast, populating thehigher conduction band state instead yields the bright exciton as the optical transition is extremely
fast ∼2 ps.
The spin polarization for the k valley, with time reversal symmetry, determines the spinpolarization for the k’ valley. Here, the minimum energy state still forms the dark exciton withthe k’ A exciton valence band, but also couples to the k valley, forming a third coupled state, theintervalley bright exciton (bright here as the spin polarization flips between the k and k’ valleys).
In order to observe the dark exciton, different experimental approaches have been taken to either
suppress the bright exciton (which normally dominates any dark exciton signal), or to enhancethe dark exciton transition rate. One such method is to perform spectroscopic measurements at
ultracold temperatures. The bright exciton is only populated during above resonance excitation
relaxation and from thermal excitations from the dark exciton – by going to lower temperatures,
this excitation channel can effectively be removed.
In a previously demonstrated cryogenic ultracold measurement, the decay of the dark exciton
was found to have 2 time constants fit by a bi-exponential decay, with the additional decay constantcame from coupling to the intervalley bright exciton. The time constants were ∼30 ps and ∼300
ps. Whenever a dark exciton is thermally populated to the bright exciton state, it rapidly decays
through the spin allowed optical transition, and this time constant can then be ignored for thesetime scales. The coupled state equations are given asdNX (t)= −(γX + γX−D + γX−XK )NX (t) + γX−D ND (t) + γX−XK NXK (t)
dt

(5.1)

dND (t)22= γX−D NX (t) − (γD + γX−D )ND (t) + γ2D−2XK (NX(t) − ND
(t))
K
dt

(5.2)

113

dNXK (t)22= γX−XK NX (t) − (γXK + γX−XK )NXK (t) + γ2D−2XK (ND(t) − NX
(t)),
K
dt

(5.3)

where N is the population of the subscripted state, γ is the transition rate of the subscripted
state, and t is time. Here, the subscripted states are X for the intravalley bright exciton, D for
the intravalley dark exciton, and XK for the intervalley bright exciton. Note the dependence on
the square of the population in the equations for ND and NXK , where the bimolecular process
dominates. The coupling rates γ are in principle temperature dependent, but the above bandgap
excitation and resulting carrier cooling emits phonons that contribute thermal excitations to populate the bright state, which is often neglected in cryogenic measurements. We keep this in mind
as we investigate heterogeneity at the nanoscale in carrier generation and decay. We also make a
final comment that going to cyrogenic temperatures, or turning on and off decay pathways is only
necessary in the visible, where the names dark and bright exciton originate. Here, we choose tosidestep this problem and use an infrared probe as the entire carrier population can be directly
assessed, regardless of its spin states.

5.3

Nano-imaging dark excitons
To begin, we located WSe2 flakes on a Au substrate and on an SiO2 substrate. We then foundsmooth regions in the interior of the flake and performed a time trace on bulk WSe2 on both Au
and SiO2 and on monolayer WSe2 on SiO2 . The pump modulated near-field signal (pump probe
amplitude) was greatest on the Au, indicating optical enhancement from the substrate. We thencollected time traces at different fluences as shown in figure 5.1a and fit the curves to bi-exponential
fits (dashed).
We note that the fast time constant for each of the biexponential fits was fluence independent,
indicating the process is intrinsic to the sample itself and does not result from sample heating orAuger like process. The second time constant though is fluence dependent and the relationship isshown in figure 5.1b, where we see to good agreement, a linear relationship between Tc and fluence.

114

Figure 5.1: Ultrafast heterodyne pump-probe dynamics on Au | a. Time traces at varying fluencesof near-field amplitude signal decay. b. Bi-exponential fit slow time constant value as a functionof fluence

We repeat this measurement for the bulk (figure 5.2) and monolayer (figure 5.3) WSe2 on SiO2 .

Figure 5.2: Ultrafast heterodyne pump-probe dynamics of bulk WSe2 on SiO2 | a. Time traces
at varying fluences of near-field amplitude signal decay. b. Bi-exponential fit slow time constantvalue as a function of fluence

We first note the higher pump fluence that we are able to use on the SiO2 sample as thesubstrate enhancement is not as strong compared to the Au. We also note that the 0nJ energy yintercept is remarkably similar between the Au and the SiO2 , indicating a limit for bulk WSe2 . The
main difference between bulk WSe2 on Au versus on SiO2 is the proportionality constant between
time constant and pump energy.
However, the monolayer case clearly shows different dynamics, where again, the fast time
constant is independent of fluence, but the second time constant again is linearly related to fluence.

115

Figure 5.3: Ultrafast heterodyne pump-probe dynamics of monolayer WSe2 on SiO2 | a. Time
traces at varying fluences of near-field amplitude signal decay. b. Bi-exponential fit slow time
constant value as a function of fluence

The 0 energy y-intercept is lower than the bulk case. This sample has the least sensitivity to pumpfluence and could be the result of optical saturation at the fluences experimentally available. To
better observe the spatially dependent dynamics, we take an ultrafast nano-movie by collecting
images (or frames) at varying time delays as shown in figure 5.4.

116

Figure 5.4: Ultrafast heterodyne pump-probe imaging bulk and monolayer WSe2 on SiO2 | Pumpprobe amplitude R(T, νeint ) at varying pump-probe delays.

From this movie, we observe the spatial extent and heterogeneity of what we previously saw
from simple point time traces shown in figures 5.1a 5.2a 5.3a. Specifically, we observe the monolayer
section decaying faster than the bulk, but also observe a significant edge enhancement.
Using the prior knowledge of cryogenic experiments where the dark state is known to couple
to the intervalley bright state, generating a bi-exponential decay, and the knowledge that the
majority of the carriers will be in the thermal minimum state, the dark exciton, we can use theDrude mapping from figure 5.4 as a nano-map of the population and dynamics of the dark excitonin WSe2 . However, we note that the dark exciton dynamics are dictated by 2 time constants – 1
independent of fluence and 1 dependent. Therefore, to determine if this heterogeneity is simply
an artifact of the pump fluence and whose dynamics are controlled by the local temperature or117
if results from edge effects that affect the carrier density in a more intrinsic nature to the WSe2
itself, we need to determine the thermal timescales. If the fluence dependent time constant is on
the same order as the thermal decay time constant, we know the difference in long term dynamics
results from simple Boltzmann excitations. However, if they differ, then the dark exciton within
the WSe2 flake is indeed intrinsically heterogeneous.

5.4

Nano-thermometry
Thermal transport across interfaces is of high technological relevance in, e.g., semiconducting

nanoelectronics, 2D material nanocomposites, and energy transmission and converting devices. For
materials with little phonon spectral overlap, like semiconductors on silicon with a thermal oxide, at
length scales smaller than the phonon mean free path, the energy transfer resistance is dominatedby the interface. However, the specifics of these interfacial processes are yet poorly understood as
conventional spectroscopic techniques are limited in spatial resolution, interfacial sensitivity, or are
too slow to resolve the ultrafast dynamics during cooling and associated relaxation processes. Here,ultrafast s-SNOM is implemented, with its evanescent signal spatially confined to the nanoscale
to selectively resolve ultrafast dynamics at the interface and discriminated from the conventionallybulk dominated response. At the example for a monolayer to bulk transition metal dichalcogenide(TMD) on SiO2 , pump modulated pump-probe nano-spectroscopy and -imaging is performed tomeasure interfacial thermal transport on few ps timescales through substrate phonon softening.
Combined with molecular dynamic simulations, the relationships between maximum temperature,
TMD layer number, and thermal boundary conductance (TBC) is quantified. The smaller TBC
value of 46.7 MW/m2 K for the monolayer WSe2 /SiO2 compared to the larger TBC value of 66.7
MW/m2 K for the bulk WSe2 /SiO2 results from finite size effects of monolayer WSe2 compared tothe phonon mean free path. The interfacial phonon scattering associated with higher temperatures
further limits its thermal conductivity compared to bulk WSe2 . The application of this approachallows for optimizing interfacial thermal management.
Higher temperatures from poor thermal transport across interfaces reduces the lifespan and118
performance for a range of technological applications, namely semiconducting nano-electronics,245–247
2D material nanocomposites,248–250 and energy transmission and conservation devices. Heat dissipation in these applications are governed by phonon emission and phonon transport.251–253 The
dynamics of phonon emission depend on non-radiative electronic relaxation and their dissipation
depends on thermal transport. This is exacerbated between materials with little phonon spectral
overlap, like semiconductors on silicon with a thermal oxide, at scales smaller than the phononmean free path, where heat dissipation is primarily limited by the Kapitza resistance.254–256
One approach to address thermal management in devices is to model the effects of substrate engineering for increased thermal management by including additional dissipation pathways,248, 250, 257 engineering surface roughness258, 259 ), or by nanostructuring the interface to match
the phonon mean free path.255, 260 Atomically thin field effect transistors, based on a transition
metal dichalcogenide (TMD) gate,261, 262 can bridge the divide between modern semiconducting
nano-electronics and 2D material nanocomposites. Here, fundamental exciton and carrier dynamics are governed by interfacial thermal transport between the TMD and the substrate throughthermal excitations that couple bright and dark states.263–265 Understanding interfacial thermaltransport, informed by direct ultrafast interfacial measurements, would thus illuminate charge carrier dynamics and coupling in 2D semiconductors.
Understanding phonon and carrier dynamics and interactions and technological innovation
for better thermal management design has previously been explored through explicit temperature
mapping but has remained complicated by the nanoscale spatial dimensions involved,266, 267 theeffect of the measurement on the system itself,268 the thermal contact between the sensor anddevice,269 and the optical properties of the device.270
Many experimental techniques have been developed in an attempt to address these shortcomings. Thermal conductivity, diffusivity, and specific heat in low dimensional systems can beinvestigated with the 3ω technique.271 However, the signal is still convolved with bulk contributions and further lack nanoscale spatial resolution. Recently developed techniques, like Raman
thermometry-based approaches, which have demonstrated success272–274 over a wide temperature

119
range and like conventional thermoreflectance (TDTR), which through sophisticated modeling andanalysis can extract interfacial resistance, still lack the necessary spatial and temporal resolution
necessary to study ultrafast thermal dynamics at the nanoscale.
Luminescent thermometers, applied in atomic, molecular, and semiconducting systems, utilize temperature dependent emission lines to spectrally resolve the thermometers temperature,
but are unable to spatially resolve nanoscale heterogeneity and lack thermal contact to the region of interest. Point contact thermoelectric cantilever probes are too slow and require complex
modeling of the impact the thermocouple/thermistor has on the local temperature by opening an
additional dissipation pathway and the thermal contact to the device in question. While ultrafast
thermoreflectance in principle offers a mechanism to study thermal dynamics of devices, it remains
insensitive to nanoscale heterogeneity. No technique offers simultaneous interfacial to bulk context
with nanometer and ultrafast spatio-temporal resolution.
Here, we implement ultrafast heterodyne pump-probe s-SNOM in a new pump-modulated
modality to image ultrafast interfacial transport dynamics.69 The near-field spatial confinement
provides the desired few-nm spatial localization necessary to discriminate the interfacial response
from the bulk. At the example of a monolayer to bulk TMD, we pump the interband electronic stateof WSe2 and quantitatively explore extreme temperatures in excess of 2000 K at ps timescales bymeasuring the SiO2 substrate phonon mode that softens with temperature to measure interfacial
thermal transport. These measurements, combined with molecular dynamics (MD) simulations,determine the monolayer WSe2 /SiO2 TBC 47 MW/m2 KW/K and the bulk WSe2 /SiO2 TBC 67
MW/m2 K. The results show that interfacial phonon scattering associated with the higher temperatures in monolayer WSe2 and its finite size effects both limit its thermal conductivity compared
to bulk WSe2 . Therefore, the effect of thermal excitation coupling between electronic states insemiconductors increases as dimension is reduced.
Samples of WSe2 /SiO2 were prepared by exfoliation as previously established.275 Briefly, bulk
WSe2 (2D Semiconductors) was transferred to Si with a 300 nm oxide layer (MTI Corporation) withadhesive tape (ProTapes Nitto SPV224 PVC Vinyl Surface Protection Specialty Tape from Ama-

120zon). The samples were pre-characterized by far-field micro-photoluminescence (PL) spectroscopy
with a HeNe (10 µW) focused onto the sample with a homemade microscope and the scattered light
was focused into a spectrometer (f = 500 mm, SpectraPro 500i, Princeton Instruments) with a thermoelectrically cooled, electron-multiplied, charge-coupled device (CCD, ProEM+: 1600 eXcelon3,
Princeton Instruments).
Ground state near-field optical amplitude R(T, ν̃) and phase ϕ(T, ν̃) response of SiO2 andWSe2 are measured using a customized near-field s-SNOM nano-scope (Anasys Instruments/Bruker,
nanoIR2-s prototype) (Fig. 5.5a). Femtosecond mid-IR light is generated by difference frequency
generation (DFG) between the signal of one OPA (Light Conversion, Orpheus-1) and the idler ofa second OPA (Light Conversion, Orpheus-F) that are both pumped by a 220 fs 1 MHz 1030 nm
pump (Light conversion, Carbide-40). The 1200 cm−1 output pulses are directed into an asymmetric Michelson interferometer with one arm focusing the IR light with an off axis parabolic mirror(OAP) onto a metalized atomic force microscope (AFM) tip (ARROW-NCPt, NanoAndMore USA)apex (Fig. 5.5b). The back scattered near-field is then heterodyne amplified and collected with amercury cadmium telluride (MCT) detector (Kolmar, KLD-0.1), whose signal is lock-in amplified(HF2LI, Zurich Instruments) at the second harmonic of the tip tapping frequency for near-field
isolation as previously established.69, 173, 258
We then apply ultrafast broadband nano-imaging and -spectroscopy to measure the ps dynamics of the SiO2 phonon shift, following previously established procedures for far-from-equilibrium
pump excitations.69 Here, we use the 670 nm 180 fs signal output from one OPA (Light Conversion
Orpheus-F) as the pump and delay it relative to the probe pulse with a retroreflector on a linear delay stage (SLLA42-270-S, SmarAct). To isolate the ultrafast response, we employ pump modulation
with a chopper (500 Hz, Thorlabs) and near-field sideband detection (nωt + Ωp ). As illustrated
in figure 5.5c, the photo-excited WSe2 emits phonons through its thermalization, which then interfacially couples to the SiO2 substrate, rapidly heating the lattice and causing the SiO2 phonon tosoften. These softened phonons are experimentally observed through pump-probe near-field phase
spectra showing the characteristic phonon dip red-shifting.

121

Figure 5.5: Ultrafast heterodyne pump-probe nano-imaging lattice dynamics | a. HPP IR s-SNOM,
with the 180 fs pump pulse (670 nm), 220 fs probe pulse (8.33 µm), and interferometric near-field
detection. DFG – difference frequency generation, BS – beam splitter, OAP – off axis parabolic,
MCT – mercury cadmium telluride detector. b. Tip localization of pump pulse, which excites
electrons in the WSe2 , and the probe pulse, which measuring the Drude response in the WSe2 andexcited state phonon absorption in SiO2 . c. Top: Ground state WSe2 , with room temperaturelattice spacing and corresponding phonon resonance. Bottom: far-from-equilibrium excitation andassociated electron-lattice thermalization through phonon emission, heating the SiO2 substrate,
expanding the lattice and red-shifting the corresponding phonon absorption. d. SiO2 , Monolayer
WSe2 , and 10 layer WSe2 density of states indicating phonon overlap at low frequencies and not
at the probed SiO2 mid-IR resonance.

The density of states (DOS) is calculated by taking the Fourier-transform of the normalized

122
velocity auto-correlation functions (Cor(t)). The Cor(t) is given as* n+Xmi vi (t) · vi (t)Cor(t) = * i=1
n
X

+,

(5.4)

mi vi (0) · vi (0)i=1

where t is the correlation time, n is the number of atoms, m is the atomic mass, v is the velocity
vector of the atom, and the brackets indicate the ensemble average. Here, the thermal effect on
the DOS is achieved by changing the simulated temperature (affecting the velocity of the atoms)
in molecular dynamics (Fig. 5.5d).
We perform the non-equilibrium molecular dynamics (NEMD) simulation to study the thermal boundary conductance at the interface between SiO2 and multi-layers WSe2 . The thermalboundary conductance G is calculated as, G = J/∆T , where ∆T and J are the temperature difference and the heat flux across the interface, respectively.
We first measure multilayer WSe2 on SiO2 by pumping the interband electronic transition in
WSe2 , followed by infrared (IR) probe on the SiO2 phonon at variable pump fluences. Figure 5.6a(blue) shows the near-field spectral phase response ϕ(T=3 ps, ν̃) of the SiO2 phonon with peak atν 0 = 1175 cm−1 under pump fluence 50 W/cm2 . Then phonon resonance frequency is red-shiftedfrom its ground state value ν 0 = 1203 cm−1 . As the fluence is increased (Fig. 5.6a dark red = 180
W/cm2 , red = 240 W/cm2 ), the phonon frequency continues to red-shift (Fig. 5.6a dark red: ν 0 =1162 cm−1 , red: ν 0 = 1159 cm−1 ). The spectra were fit with Lorentzian resonances in a 4-layer
finite dipole model (FDM), where the 4 layers are air, WSe2 , interfacially excited substrate, andground-state substrate. The WSe2 layer exhibits a broad, non-resonant Drude response that, to firstorder at these fluence values and sample parameters, inverts the spectral phase of the SiO2 phonon
absorption.69 The substrate phonon resonance exhibits a red shifted SiO2 phonon absorption fromlattice expansion and a ground state bleach. Figure 5.6b shows the extracted center frequency as afunction of fluence. The linear relationship between fluence and phonon shift (Fig. 5.6b) matches
well with the linear trend expected from theoretical molecular dynamics simulation (Fig. 5.6binset). (For additional fluence dependent data, see SI.) At the fluence of 240 W/cm2 , we measure

123
a local temperature of 1200 K.

Figure 5.6: SiO2 phonon resonance shift as ultrafast temperature sensor | a. Power dependent ultrafast spectra (T = 3 ps) showing characteristic phonon absorption center frequency red-shift withincreasing fluence. b. Model extracted center frequency as function of fluence. Inset: Lorentzianline shape fit to modeled density of states for varying temperatures (Fig. 5.8c), indicating good
linear agreement with experimental power dependence. c. Time dependent spectra at identicalfluences (120 W/cm2 ), indicating a blue shift with increasing time as the system cools back toroom temperature (center frequency 1200 cm−1 ). d. Center frequencies of SiO2 phonon extracted
through HPP s-SNOM model indicating the asymptotic approach from maximally red-shifted to1200 cm−1 . Green trace: cross correlation between pump and probe. Gray dashed line: room
temperature ν 0 . Inset: propagating thermally excited SiO2 within the nearfield.

In order to address the cooling dynamics, we measure the spectro-temporal evolution of thephase ϕ(ν̃) for a pump fluence of 120 W/cm2 (Figure 5.6c red). The cooling dynamics indicate

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a continuous SiO2 phonon blue-shift from initial pump-induced resonant frequency of ν 0 = 1166
cm−1 as shown in figure 5.6d (visible-IR cross correlation Fig. 5.6d green) at varying pump-probe
delays. (Additional spectra were collected at varying pump-probe delays; see SI for details.69 )We observe a mono-exponential exponential decay with time constant τ = 13 ps (Fig. 5.6d black
dashed line) that asymptotes back to the ground state center frequency (Fig. 5.6d grey dashedline).
In order to observe local temperature differences across the sample, an image of the maximum
near-field amplitude R(T = 3 ps, ν̃ integrated ) from the Drude response of the system was collected
and is shown in Figure 5.7a (top). Here, the Drude response is a good measure for the relative
carrier density that is spatially varying across the sample surface. We identify a strained monolayer
region next to the bulk WSe2 through AFM scanning (Fig. 5.7a bottom) combined with microphotoluminescence spectroscopy (Fig. 5.7b, λ0 = 770 nm, black dashed line) at location 2 (darkblue) in Figure 5.7a (unstrained ML WSe2 center frequency Fig. 5.7b gray dashed line). The imageof R(T = 3 ps, ν̃ integrated ) shows a homogeneously high carrier density in the center of both the ML
and bulk WSe2 , with a slightly decreased carrier density at their boundary. The boundary between
the WSe2 and the bare SiO2 substrate shows a significant increase in carrier density compared tothe center regions. To understand the higher carrier density, we perform nano-FTIR for the same
pump-probe delay of 3 ps as shown in Figure 5.7c.
Spectra were collected at different sample positions of 1) center of the bulk WSe2 (Fig. 5.7c(1) blue), 2) the center of the ML WSe2 (Fig. 5.7c (2) dark blue), 3) the edge of the bulk WSe2 (Fig.
5.7c (3) dark red), and 4) the edge of the ML WSe2 (Fig. 5.7c (4) red). From the degree of red-shift
at each location, we observe both that the center of the ML WSe2 reaches higher temperatures
than the center of the bulk WSe2 and that the edge of both the ML and bulk WSe2 reaches
higher temperatures than the center of either region. This can be intuitively understood as fewer
dissipation pathways are available for the WSe2 to cool in plane and therefore heat builds up at theedge as illustrated in figure 5.7d. This also explains the increased carrier density observed in the
Drude response (Fig. 5.7a). The increased local temperature, corresponding to a higher excited125

Figure 5.7: Local variations in temperature and thermal boundary conductance | a. Top: Pump demodulated integrated near-field amplitude. Bottom: Topographic image of bulk WSe2 , monolayer
WSe2 (dashed green) and substrate SiO2 (dashed white). b. Far-field photoluminescence indicatingstrained monolayer in A. Gray dashed line: unstrained ML center frequency. Black dashed line:
strained ML center Frequency. c. Ultrafast (T = 3 ps) SiO2 phonon absorptions at different locations in the sample (indicated in A). In increasing temperature, center of bulk WSe2 , center ofmonolayer WSe2 , edge of bulk WSe2 , and edge of monolayer WSe2 . d. Schematic comparison ofthermal dissipation at the center of a sample vs. the edge of the sample indicating the intralayer
thermal transport in WSe2 strongly affects local dissipation. e. Molecular dynamic simulation ofthermal boundary conductance as a function of WSe2 layer number, indicating phonon reflection
and saturation once the mean free path is significantly smaller than the WSe2 layer thickness.

phonon density, acting as a hot phonon bottleneck, restricting the carrier’s ability to thermalize
through phonon emission, thus causing an increase in carrier density at this pump-probe delay. We

126further observe that the ML WSe2 red-shifts more and therefore reaches higher temperatures thanthe bulk WSe2 . The difference in ML versus bulk WSe2 temperature is a result of Beer’s law withthe mass of the system increasing faster than the absorbed energy. Molecular dynamic simulations
predict a sharp increase in TBC for ML compared to bulk (Fig. 5.7e) as a finite size effect until
the WSe2 layer thickness exceeds the phonon mean free path.
To better understand these experimental results, we perform molecular dynamic simulations
to model this system. The relationship between temperature and thermal boundary conductanceis shown in figure 5.8a. We observe an inverse linear dependence on the TBC with temperature.
Figure 5.8b shows the effect this has on the cooling dynamics, where hotter systems recover more
slowly than their cooler counterparts. While the higher energy SiO2 phonon shifts with temperature(Figure 5.8c), the low energy phonon modes participating in heat transfer are not temperaturedependent, indicating the modification of the density of states overlap is not responsible for thedifference in the TBC. While our ultrafast nano-thermometry method is sensitive to the surface
volume, we don’t directly measure just the surface layer of SiO2 atoms. The relative temperatureand depth dependent dynamics are shown in Figure 5.8d. Here, we note the trivial observation thatthe surface reaches higher temperatures faster than deeper locations in the SiO2 , but specifically callout the sensitivity of the rise time on depth, where the top 2 nm reaches the maximum temperaturein ∼1 ps and the SiO2 16 nm from the surface takes ∼50 ps (Figure 5.8d inset).
In the following, we discuss the mechanism behind the temperature dependent TBC. Simple
Newtonian cooling predicts faster heat transfer at higher WSe2 temperatures. However, increased
phonon densities at higher temperatures lead to phonon-phonon scattering. We quantified thisrelationship with molecular dynamic simulations, finding the intrinsic Umklapp phonon scattering,which has been observed at the GaN/SiC interface,276 determines the TBC temperature relationship. This is directly observed in the cooling traces simulated for different starting temperatures.
However, the location studied here exhibited a layer dependence as well, where we see both higher
temperatures and thinner WSe2 in the monolayer lead to slower cooling compared to the bulk.
We next discuss the processes that occur within the WSe2 that upon pumping generate the127

Figure 5.8: Molecular dynamics modeling of SiO2 phonon | a. Molecular dynamic simulation ofthermal boundary conductance between WSe2 and SiO2 for different interfacial temperatures indicating monotonically decreasing relationship with temperature. b. Molecular dynamics simulationof SiO2 cooling over time for different starting WSe2 temperatures. Inset: zoom in on first 50 ps
rise and fall of temperature. c. Molecular dynamic simulation of density of states for SiO2 at varying temperatures. Increasing temperature corresponds with red-shift of density of states peak. d.
Molecular dynamics of temperature change in SiO2 at increasing depths, spaced by 2 nm startingat the WSe2 /SiO2 interface (red). Inset: zoom in on the first 50 ps rise in temperature.

thermal load to be dissipated by the SiO2 . The non-resonant, above bandgap excitation initially
produces a population of hot electrons that then rapidly thermalize into excitons and carriers. This
thermalization rapidly heats the WSe2 lattice through phonon emission. The electron dynamics in
WSe2 are then dominated by continued thermalization and radiative emission of bright excitons,
followed by the long lived electronic dark excitonic state.277 The hot WSe2 lattice then thermally

128
interfacially heats the substrate, with temperature increase on the order of ps to 10s of ps dependingon the distance from the surface, with the surface layer reaching maximum temperature the fastest.
In the case of WSe2 on SiO2 , the higher energy phonons from the large bandgap of SiO2 limits theinterfacial energy transfer to primarily occur at the lowest phonon frequency modes as that is where
the DOS overlap is greatest. While the higher energy SiO2 phonons are temperature dependent,
this change in the density of states doesn’t affect the TBC as these modes don’t participate ininterfacial energy transfer.
The observed 10s of ps decay time constant and simultaneous lack of rise time observation
together with the theoretical depth dependent result indicate extreme surface sensitivity. As previously mentioned, photoinjected carriers screen the substrate phonon mode, limiting our temporal
resolution to dynamics that occur 3 ps after excitation. This screening further limits the evanescent near-field propagation from the tip, enhancing our surface specificity. Our pump modulated
detection scheme is most sensitive to the largest deviation from the ground state. Therefore, thesurface, whose temperature deviates the most from the ground state and therefore has the largest
phonon shift, dominates the weaker signals from the bulk of the substrate, further localizing ournear-field signal to the near surface region. The depth dependent molecular dynamic simulations
further predict a gradient in the rise time of the substrate temperature along the surface normal
axis within the first 10s of nm. However, the Drude screening in the first 3 ps completely obscuresthe substrate phonon temperature rise and the following dynamics are exclusively cooling, indicating our surface technique is specific to the first 1 nm and is largely insensitive to even the top 10
nm of the SiO2 substrate.
Previous techniques have demonstrated great success in the wide temperature ranges accessible with Raman thermometry and the surface sensitive time domain thermoreflectance, but thelack of intrinsic surface sensitivity (rather than simply sample selection) limits their applicability
to a wider class of material systems. Specifically, other work also looking at TMD to SiO2 to Si
systems, performing Raman thermometry, modeled a similar system, yet were insensitive to the
TMD SiO2 interface and instead used indirect measurements of the underlying Si.278, 279 Differences

129
in the extracted TBC value increasing with increasing temperature are explained as a positive correlation between temperature and phonon specific heat together with neglecting the temperaturedependent absorption of MoS2 . However, our modeling and dynamics illustrate the importance ofphonon-phonon scattering at the interface. The higher temperatures explored here further suggestthat Umklapp scattering continues to dominate the interfacial thermal resistance at few nm lengthand thousands K temperature scales.
Alternative previous approaches, using thermocouple cantilever probes similarly probe nanoscopic
volumes, but required complex numerical simulations of the tip sample interaction and lacked temporal resolution. Here, we directly observe the substrate phonon frequency shift as a measureof temperature and therefore interfacial thermal transport. Similarly, ultrafast thermo-reflectancespectroscopy has been used to measure the ultrafast response of molecular systems, but lacks spatial
resolution. Our technique uses the combination of nano-probes for signal localization and ultrafast
spectroscopy to resolve the ultrafast nanoscopic thermalization and interfacial thermal transport.
Our approach is generalizable to other substrate systems whose phonons exhibit a temperature dependence. In the example of WSe2 on SiO2 , we’ve imaged heterogeneity in the spatial distributionof temperature, which will affect the dark exciton state lifetimes. The decay of these long lived
states are dominated by coupling between the intravalley dark and bright excitons.280 Therefore,
the electronic relaxation is also expected to be spatially heterogeneous.
In conclusion, ultrafast s-SNOM has been implemented to selectively resolve ultrafast dynamics at the interface of WSe2 and SiO2 . Pump modulated pump-probe nano-spectroscopy and-imaging has been performed to measure interfacial thermal transport on the few ps timescale
through substrate phonon softening. This has been combined with molecular dynamic simulations
to quantify the relationships between maximum temperature, TMD layer number, and thermalboundary conductance (TBC). The lower TBC of the monolayer results from finite size effects
of monolayer WSe2 compared to the phonon mean free path. The interfacial phonon scattering
associated with higher temperatures further limited its thermal conductivity compared to bulk
WSe2 .

Chapter 6

Conclusion and Outlook

6.1

Summary
In summary, this thesis covered the two main branches of research I performed. The first ismethod development of scanning optical microscopy and the second is its application to the studyof electron dynamics in quantum materials. In this final chapter, I will take a look to the future toassess possible extensions, developments, and new applications.

6.1.1

Multi quantum well strong coupling

In summary, my studies (Chapter 4) on MQWs first demonstrated the macroscopic opticalpower limiting and switching behavior enabled by multi-quantum wells (MQWs) that are strongly
coupled to plasmonic antennas. I noted the strong non-linearity of the reflection coefficient uponincident peak fluence through the saturation of the MQW and resulting decreased coupling strength
in the heterostructure. This reduced coupling strength collapsed the hybridized states to thebare antenna resonance, which behaved classically under high fluences. I then extended this work
to the near-field, where I demonstrated tip-sample nano-cavity enhancement. I used tip-sample
positioning to control both the hybridized state phase and coupling strength through near-field
illumination. This new avenue of control could enable qubit phase rotation operations in future
quantum information technology applications.

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6.1.2

Tungsten diselenide

I demonstrated the power of nanoscale ultrafast spectroscopy (Chapter 5. Previous to thismeasurement, temperatures and dynamics were expected to be lower and slower. However, the 2000
K measured temperature and 10s of ps dynamics clearly show the interfacial thermal transport, atleast directly at the interface, is faster with higher temperatures than previously estimated. I used
a temperature decay time constant to calculate a TBC of 50 MW/m2 K. I further qualitatively explored the relationship between the maximum temperature, TMD layer number, and TBC. I notedthat higher temperatures lead to lower TBC from phonon-phonon scattering and that finite size
effects of monolayer WSe2 limit the interfacial thermal transport. Future applications with atomically thin two dimensional materials will require sophisticated substrate engineering for thermal
management.

6.2

Outlook

6.2.1

Method Development

The state of the art s-SNOM has significantly advanced within just my time in the field from
simple self-homodyne imaging, to ultrafast pump-probe, variable temperature, variable voltage,
photochemistry, THz spectroscopy, and in liquid techniques becoming available and more consistently implemented. However, the increased complexity of these extended measurement techniques
is rapidly lengthening measurement times to the point where a ”full” dataset can not even be
acquired as the dimensionality is simply too high. Below, I will address these issues individually.

Errors in measurements and interpretation

On the most basic level, errors in mea-

surements can easily result from improper isolation of the near-field signal as discussed in depth in
Chapter 2. Self-homodyne imaging and topographic artifacts from simple near-field enhancement(where the tip-sample interaction is not as simple as a tip above a perfectly planar surface) confound measurement interpretation. Demodulation techniques can be used to improve heterodyne

132
near-field signal extraction (Chapter 3).

Data acquisition rate

I see dimensionality of complicated datasets as a major limitationin the continued advancement of s-SNOM. Computational approaches like compressed sensing andmatrix completion will become increasingly necessary for lengthy experiments, like, e.g., cavity
mapping (z), temperature dependent (T ), spectroscopy (t), ultrafast pump-probe spectroscopy(T ), on a gated (V ) quantum well (I) coupled (g) to a resonant antenna. In this experiment, which
is a scientifically reasonable desired direction, the dimensionality of the total dataset is nine. For
reference, the largest dataset shown in this thesis is four and took six months to complete. One
further concern about large datasets is not simply the acquisition time, but is rather the analysis.
Careful spectral analysis is important in the physical interpretation of s-SNOM datasets, but isnotoriously difficult to generalize sufficiently for computer automatization.

6.2.2

Scientific progress on quantum materials

The two projects that I wish to discuss a possible future of are strongly coupled multiquantum-wells and transition metal dichalcogenide (TMD) projects. Both of these systems involve
carrier coupling to a resonance, either strongly coupled to a gold antenna, or interfacially thermallycoupled to an SiO2 substrate.

mid-IR strong coupling

Strong coupling has frequently been studied in both molecularand quantum systems both in the far-field and near-field. However, showing all optical couplingstrength control, phase resolved and excitation interference through tip positioning has yet to be
demonstrated (before the work presented in this thesis). Far-field time resolved measurements using
degenerate pump probe measurements showed ultrafast optical control, but the implementationof this technique in the near-field is significantly more complicated. To measure the ultrafast t1
MQW decay in the near-field, we need to develop degenerate pump-probe nano-FTIR. Changing theinterferometer scheme to, e.g., a modified Mach-Zehnder interferometer with one arm responsible

133
for the pump-probe near-field interaction and the other for spectroscopy could work. This scheme
is still asymmetric such that phase information can be extracted, but also is background free fromthe heterodyne process.

Transition metal dichacogenide electron dynamcis

The work presented here in thisthesis covered the ultrafast interfacial thermal transport between photo-excited WSe2 and an SiO2
substrate. However in the process of measuring the resonant feature from the substrate (SiO2
phonon), we also measured the Drude response from the electron gas within the WSe2 itself. Whatis challenging when using visible spectroscopy (dark states are dark in the visible for a reason) is
easily measured in the infrared (the sum of carrier populated states). Future measurements looking
for the spatial evolution of bound exciton wavefunctions through 1s-2p absorption measurements
in MoS2 are planned. The exciton 2p wavefunction is predicted to extend by ∼10 nm and should
therefore be quite sensitive to defects, boundaries, and edges.

6.3

Final thoughts
s-SNOM has shown itself, through its applications to studying quantum and molecular ma-

terials, to be a power technique, giving rise to many high impact publications. I have demonstrated
significant method advancements, imaging electron dynamics in quantum materials at nanoscopic
volumes. Further, I give an outlook of possible future measurements and the field as a whole tostudy novel meso-scopic physical phenomena.

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