Development of Nonlinear Optics-based Reduced
Electric Field Diagnostics

A THESIS SUBMITTED TO THE FACULTY OF THE UNIVERSITY
OF MINNESOTA BY

Grayson LaCombe

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE

Advisor: Prof. Marien Simeni Simeni

May, 2025



© Grayson LaCombe 2025



Acknowledgments
First and foremost, I would like to thank my advisor Dr. Marien Simeni for his men-

torship throughout the completion of my Masters degree. Without him, none of this work
would be possible, and has pushed me both academically and as a scientist in the lab.

Next, I would like to express my sincere gratitude to Prof. Peter Bruggeman (De-
partment of Mechanical Engineering, University of Minnesota) and Prof. Renee Frontiera
(Department of Chemistry, University of Minnesota) for serving on my thesis defense com-
mittee and for their valuable feedback and support.

I want to thank my undergraduate advisor Dr. Todd Zimmerman (University of Wisconsin-
Stout) who helped kindle my interest in physics and first introduced me to nonlinear optics.

I am also grateful to Dr. Jianan Wang, Dr. Jérémy Rouxel (Argonne National Labora-
tory), and Dr. Kraig Frederickson (Naval Undersea Warfare Center) for their collaborative
efforts and contributions to this work.

Special thanks go to Dr. Donald Bethune, Prof. Yuen-Ron Shen (University of Califor-
nia, Berkeley), and Thomas Cameron (Department of Mechanical Engineering, University
of Minnesota) for their insightful discussions, which greatly enriched this research.

I would also like to acknowledge the members of the Ultrafast Laser Plasma Diagnos-
tics Laboratory (ULPDL) for their assistance and camaraderie in the lab.

Finally, I gratefully acknowledge the financial support provided by the U.S. National
Science Foundation (Award No. PHY-2308946), the Air Force Office of Scientific Re-
search (Award No. FA9550-23-1-0745), the College of Science and Engineering, and the
Department of Mechanical Engineering at the University of Minnesota.

i



Abstract

This thesis presents the development and application of advanced nonlinear optical di-

agnostics for electric field and number density measurements in gaseous plasmas. Lever-

aging Electric Field Induced Second Harmonic Generation (E-FISH), a highly sensitive

diagnostic setup was designed to achieve sub-1 V/cm electric field detection in atmo-

spheric pressure air with picosecond temporal resolution. This represents a 2 to 3-order-

of-magnitude improvement over conventional E-FISH approaches. A passive homodyne

detection mechanism, arising from the interference between E-FISH signals and stray

surface-generated second harmonic signals, was analytically modeled and experimentally

validated. This mechanism also enabled polarity sensitivity, a key advancement for di-

agnosing electric field reversals in plasma discharges. Additionally, the thesis reports the

experimental observation of non-resonant, second-order sum-frequency generation (SFG)

in five different gases. The observed SFG signals demonstrate quadratic scaling with gas

pressure and insensitivity to applied electric fields. Together, these diagnostics offer new

capabilities for spatially and temporally resolved mapping of electric fields and gas densi-

ties in reactive flows and plasma environments, advancing our ability to probe and model

transient plasma phenomena.

ii



Contents

List of Figures iv

List of Tables vii

1 Introduction 1

2 Sub-1 V/cm E-FISH in Atmospheric Pressure Air 9
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Non-resonant Sum Frequency Generation in the Gas Phase 30
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 Supplementary Information . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7.1 Photomultiplier signal traces . . . . . . . . . . . . . . . . . . . . . 41
3.7.2 Third-harmonic generation signal . . . . . . . . . . . . . . . . . . 42
3.7.3 Discussion on SFG signal and polarization . . . . . . . . . . . . . 43

4 Conclusion 48

Bibliography 51

iii



List of Figures

1.1 Relative ion and electron densities mapped across a plasma reactor. In
the bulk plasma, the ion and electron density are equal, creating a neu-
trally charged region. Near the edge of the plasma, known as the plasma
sheath, the ion and electron densities diverge creating large gradients in
electrical potential. Figure reproduced with permission from [1] license
6039380342154. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The energy loss fraction or air, expresses what fraction of the total energy
is used to drive different reactions for a given reduced electric field. As the
reduced electric field increases more energy is available to enable higher
energy transitions, which then become the dominant form of energy loss in
the gas. Tr, translational excitation. rot, rotation excitation. v, vibrational
excitation. el, electrical excitation. dis, atomic dissociation. Ion, ioniza-
tion. a+b, O2 transition from a to b. Figure reproduced with permission
from [2] license 1615621-1. . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Summary of plasma diagnostics techniques. Emission, Imaging, Scatter-
ing, Fluorescence, and Absorption are all optical diagnostic techniques that
can probe many different properties of plasmas. Figure reproduced with
permission from [3] license 1614612-1. . . . . . . . . . . . . . . . . . . . 6

1.4 (a) Sum Frequency Generation (SFG), a 3-wave-mixing process, depicted
in a medium that lacks symmetry. Two photons, ω1 and ω2, interact with the
medium to produce a new photon, ω3. (b) The energy level diagram depict-
ing SFG, ω1, excites the medium to a virtual state and excites the medium
again to a higher energy virtual state. The medium transitions from the
higher energy virtual state back down to the initial state emitting a single
photon, ω3, with the same energy as the combined energy of the initial pho-
tons. Figure reproduced with permission from [4] license 6039381472801. . 8

2.1 Schematic of the E-FISH experimental setup. The different voltage power
supplies are not shown. The electrodes are 3.5 cm long along the laser
beam propagation direction. . . . . . . . . . . . . . . . . . . . . . . . . . 12

iv



2.2 PMT waveforms from 27,000 laser shots averaged E-FISH tests using a 1
cm electrode gap and 5 mJ laser pulse energy. (a) PMT waveforms taken
with DC-applied voltages ranging from 0 to -19 V to the top electrode
resulting in a positive electric field. (b) PMT waveforms from a) with the 0
V/cm background signal subtracted highlighting the increase in signal from
the applied electric field. (c) Diagram of the electrode set up for positive
electric fields. (d) PMT waveforms taken with DC-applied voltages ranging
from 0 to -30 V to the bottom electrode resulting in a negative electric
field. (e) PMT waveforms from c) with the 0 V/cm background signal
subtracted highlighting the change in signal from the applied electric field.
(f) Diagram of the electrode set up for negative electric fields. . . . . . . . . 15

2.3 Time-integrated DC E-FISH signals for different applied E-Fields. The
electrode gap distance is 10 mm with a width of 35.3 mm. (a) Negative
electric fields. (b) Positive electric fields. . . . . . . . . . . . . . . . . . . . 17

2.4 400 ps time-resolved E-FISH data for a 10 MHz low voltage sine wave
applied to the electrode assembly. The electrode gap distance is 3 mm. (a)
Time-resolved E-Field and corresponding PMT signals. (b)

√
Imeas plotted

VS. Time together with |E| VS. Time. . . . . . . . . . . . . . . . . . . . . 19
2.5 Measured PMT signal as a function of applied E-Field together with a

parabola fit of the data. The laser pulse energy is 5 mJ. . . . . . . . . . . . 20
2.6 Comparison between simulations and experimental results. (a) Time-integrated

PMT signal VS Applied E-Field. (b) Inferred
√
PMT , |E| VS Time. . . . . 25

2.7 Single laser pulse simulations for the case where the AC electric field is
maximum (ϕAC=0), generating maximum EEFISH and Imeas. (a) Incident
laser E-field. (b) Incident laser intensity. (c) Applied AC E-field. (d) In-
duced E-FISH E-field. (e) Generated homodyne signal intensity. . . . . . . 27

2.8 Single laser pulse simulations for the cases where the applied E-field is
null (left) and -10 V/cm (right). (a) Incident laser E-field. (b) Incident laser
intensity. (c) Applied E-field for ϕAC=π

2
. (d) Induced E-FISH E-field for

ϕAC=π
2
. (e) Generated homodyne signal intensity for ϕAC=π

2
. (f) Incident

laser E-field. (g) Incident laser intensity. (h) Applied E-field for ϕAC=3π
5

.
(i) Induced E-FISH E-field for ϕAC=3π

5
. (j) Generated homodyne signal

intensity for ϕAC=3π
5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Schematic of the experimental setup. . . . . . . . . . . . . . . . . . . . . . 34
3.2 (a) SFG signal in room air at 355 nm as a function of the combined power

of the 1064 and 532 nm input beams, (b) Scaled intensity of the SFG signal
at 355 nm for 5 different gases as a function of the gas pressure with a
power of 2 fitting for each gas. . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Polarization state of the TWM signal at 355 nm in room air. A sinusoidal
wave is fitted for each combination. . . . . . . . . . . . . . . . . . . . . . 38

v



3.4 SFG signal with and without externally applied E-Field. The difference is
shown in the subplot. The 1064 and 532 incident beams are both vertically
polarized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Sample PMT waveforms at different pressures in N2. . . . . . . . . . . . . 41
3.6 THG in room air polarization as a function of pressure . . . . . . . . . . . 43
3.7 Measured polarization state of THG in room air generated from a vertically-

polarized 1064 nm input beam. . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8 Polar plot representing the polarization of the 355 nm signal for V1064-

V532 and for H1064-H532 combinations. . . . . . . . . . . . . . . . . . . 44
3.9 (a) Experimental and modeling results for 355 nm polarization as a function

of 532 nm polarization. All polarization angles are taken with respect to
the polarization of the 1064 nm beam. The fitting parameter b varies in an
agreement area from 1.6 to 2, best fitting result is obtained for b = 1.74.
(b) Schematic illustration of our SFG process hypothesis. . . . . . . . . . . 46

vi



List of Tables

3.1 Relative nonlinear hyperpolarizabilities of five atomic and molecular gases
[5, 6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vii



Chapter 1

Introduction

Plasmas are ionized gases. As a result of the ionization, plasmas typically contain free

electrons, ions, and electrically neutral species [7]. The degree of ionization, which is es-

sentially the ratio of the number density of free electrons in the plasma by the total gas num-

ber density, is one of the multiple metrics commonly employed to classify types of plasma

[8]. Around us, low-ionization degree plasmas such as flames or auroras coexist with fully-

ionized plasmas such as the interior of stars. Beyond their ionization degrees, plasmas can

also be categorized by the pressure (or collisionality) of the background gaseous environ-

ment. On one end, low-pressure plasmas encompass for example the so-called Hall-effect

thruster plasmas [9], often leveraged for electric propulsion of satellites or even magnetron

sputtering plasmas, which are commonly deployed for thin film deposition in semiconduc-

tor processing [10]. On the other end, high-pressure plasmas include atmospheric pres-

sure transient discharges such as plasma jets or metal-to-metal (pin-to-pin, more precisely)

discharges with applications reaching the biomedical (wound healing) and combustion (re-

duction of ignition delay for combustible mixtures) fields, respectively [11]. This broad

categorization of plasmas therefore leads to an equally broad set of applications. Interest-

ingly, the extremely high temperature and pressure plasmas found in the interior of stars

make up 99.9% of the visible universe. However, the conditions needed to sustain these

plasmas make them practically unattainable in laboratory and industrial settings. While

1



plasmas in the interior of stars are driven by fusion reactions together with gravitational

force confinement, in laboratory and industrial settings, plasmas are conversely typically

electrically driven through electrically-powered metal electrodes.

The diversity of plasma properties is one of the greatest strengths they possess as a tool

in both research and industry. As a simple example, an electrical discharge generated in

room air has the potential to not only excite N2 and O2 molecules but can also dissociate and

ionize these species, leading hypothetically to a very rich mixture of neutral and charged

species and radicals (N2, O2, N+
2 , N, O, NO, N+, N2(A, B, C), N2(ν), O2(ν), OH, O3,..) [3].

The ability to predominantly generate some species is intuitively tied to parameters such as

the input energy, gas density, gas composition, and input excitation waveform (AC vs DC

vs Radiofrequency vs Nanosecond pulsed vs Microwave).

The diversity in plasmas hence also arises from the complexity of the plasma dynamics

and chemistry, as varying the pressure, gas composition, reactor geometry, and the energy

applied will all greatly influence the properties of the generated plasma. For example, as

the pressure changes, the applications of plasmas vary greatly. At low pressures, plasmas

can be used for high energy efficiency etching and electric space propulsion due to the low

probability of intermolecular collisions. As the pressure rises, more complex interparti-

cle chemistry is unlocked, enabling processes like nanoparticle synthesis as the interaction

pathways can still be easily controlled. When operating at atmospheric pressure, the reac-

tion pathways grow exponentially, increasing complexity beyond simple models. But at the

same time, atmospheric pressure plasmas enable the generation of highly reactive oxygen-

nitrogen species (RONS). This enables plasmas to be used in combustion applications, as

well as for water [12] and gas treatment.

Throughout this diverse set of applications, understanding the driving forces of dynamic

plasma systems becomes critical to building and validating the models to predict the func-

tionality of each plasma. Despite the broad types of plasmas, the vast majority of plasmas

2



deployed for common industrial applications are driven by external electromagnetic fields,

whether through AC and DC high-voltage sources, or inductively coupled radio-frequency

sources. Measuring and mapping these effects has proved to be integral to understanding

plasmas as dynamic systems.

In sub-breakdown, ”Laplacian”, environments. the behavior of electromagnetic fields is

to a certain degree straightforward, since the behavior can be captured by solving Maxwell

equations across the domain of the system. In plasmas, the complexity is greatly increased

as ions and electrons are driven in opposite directions by the fields, leading to ambipolar

diffusion [8]. This results in a phenomenon known as quasi-electroneutrality in the bulk of

the plasma, where the relative number of positive and negative charges in a region roughly

cancel each other out, and the majority of the electrical potential is concentrated on the

edge of the plasma in the sheath, depicted in Figure 1.1.

Figure 1.1: Relative ion and electron densities mapped across a plasma reactor. In the bulk
plasma, the ion and electron density are equal, creating a neutrally charged region. Near the edge
of the plasma, known as the plasma sheath, the ion and electron densities diverge creating large

gradients in electrical potential. Figure reproduced with permission from [1] license
6039380342154.

3



The reduced electric field (E/N) is a property used to summarize these different plasma

conditions by expressing the ratio of electrical potential to the gas density. As electrons

have a mass at least 1800 times smaller than that of the surrounding gas atoms/molecules,

they respond quicker to external electric fields, building up kinetic energy measured in

electron volts (eV) where 1 eV is equivalent to the kinetic energy gained by an electron

accelerated in a 1 V/cm electric field over a 1 cm distance (hence an electric potential

difference of 1 volt). Kinetic energy from electrons is subsequently transferred to the larger

gas species through elastic and inelastic collisions, leading to three potential outcomes:

increased thermal energy, excitation (electronic, vibrational, or rotational), or ionization.

The type of outcome is strongly dependent on the energy of the electron, as each type of

interaction requires different threshold energy magnitudes.

As the electric field is the driving force for the kinetic energy of electrons, knowing the

local electric field strength gives direct knowledge of the forces accelerating the electrons

(E). This knowledge in combination with the gas density (N) can be used to find properties

like the expected collision frequency and electron mean free path and most importantly the

electron energy distribution function (EEDF). With this information the allowed reaction

pathways and their probabilities can be calculated as shown in Figure 1.2.

Attempting to make direct measurements of the reduced electric field leads to several

complications. First, electrically driven discharges can be highly transient. This is espe-

cially the case when driven by ns pulsed or radio-frequency electric fields. The time scales

of the dynamics of interest can range from tens of picoseconds to hundreds of nanosec-

onds. In many conditions where the plasma showcases a large mean free path, a device like

a Langmuir probe can be inserted into the plasma and the current produced by the insertion

of the probe in the plasma can be used to infer these properties. But as the plasma size

decreases or spatial variations increase in scale, diagnostics like probes will begin to intro-

duce disturbances into the plasma, changing what is being probed/measured [3]. An ideal

4



Figure 1.2: The energy loss fraction or air, expresses what fraction of the total energy is used to
drive different reactions for a given reduced electric field. As the reduced electric field increases
more energy is available to enable higher energy transitions, which then become the dominant

form of energy loss in the gas. Tr, translational excitation. rot, rotation excitation. v, vibrational
excitation. el, electrical excitation. dis, atomic dissociation. Ion, ionization. a+b, O2 transition

from a to b. Figure reproduced with permission from [2] license 1615621-1.

diagnostic technique would be able to perform the same measurements without perturbing

the plasma properties with high spatio-temporal resolution.

Optical diagnostics fill this role well, as these techniques either leverage the light nat-

urally emitted by the plasma or monitor how the plasma affects an external light source,

minimizing the disturbance to the plasma [3]. This is not to say all optical diagnostics are

created equal, each subset has its advantages. Imaging and spectral profiling of plasma

emission are the only optical methods that cannot perturb the plasma, as these methods

measure the light that is naturally produced. However, the signals generated through emis-

sion are uncontrolled, any species that do not naturally emit light cannot be measured, lim-

iting diagnostic capabilities. Diagnostics based on scattering and fluorescence can probe

gas densities as well as electron and gas temperatures with high accuracy. Fluorescence

methods, for instance, often have limited time resolution due to decay time scales and

camera speeds.

5



Figure 1.3: Summary of plasma diagnostics techniques. Emission, Imaging, Scattering,
Fluorescence, and Absorption are all optical diagnostic techniques that can probe many different

properties of plasmas. Figure reproduced with permission from [3] license 1614612-1.

Nonlinear laser diagnostics enable the combination of high spatial-temporal resolution

that is required through coherent generation and the instantaneous timescale of nonlinear

processes.

The nonlinear optical process important for this thesis is optical wave-mixing. In non-

linear optical wave-mixing, the incoming photon’s energies can be mixed to produce a new

photon with the energy of the incident photons. The simplest example of optical wave-

mixing is optical second harmonic generation, here two photons with frequency ω interact

with a nonlinear crystal to produce a single new photon with frequency 2ω. This is often

expressed in terms of wavelength, in this case, 2 photons of wavelength λ produce a new

photon with wavelength ½λ [4]. In the case of second harmonic generation, there are two

incident photons, making it a 2nd order nonlinear effect, and one photon is generated for a

total of 3 photons in the interaction, making it a 3 wave-mixing process, as demonstrated

in Figure 1.4. The probability of a nonlinear process occurring in any material is governed

by the nonlinear susceptibility χ(n) and the intensity of the light as described by Equation

6



1.1.

I∑ω ∝ χ(2)Iω1Iω2 + χ(3)Iω1Iω2Iω3 + · · ·+ χ(n)Iω1 ∗ · · · ∗ Iωn (1.1)

Where the even terms of the nonlinear susceptibility are only nonzero where there is a

break in symmetry like in a nonlinear crystal where the crystal structure is inherently non-

symmetric implying 3 wave-mixing can only occur with a break in symmetry is present

[4, 13]. The odd terms can always be non-zero, meaning a process like 4 wave-mixing

can always occur. These processes happen at an almost instantaneous timescale as the

photons interact with the molecules in a medium, implying the generated signal must have

been generated on the time scale of the laser pulse, with the prevalence of picosecond and

femtosecond pulsed lasers, the limiting factor for time resolution becomes the detectors and

computers used to process data. The produced photons are also generated in phase with the

original photons and must follow the conservation of momentum implying that if the light

source used to produce the photons is a laser beam, the generated photons will also behave

as a laser beam under the same focusing conditions. Maintaining the focal conditions

enables the spatial resolution of these diagnostics to be on the same spatial scale of the laser

beam which can easily be on the micron scale. All while allowing the generated signal to

remain coherent and be propagated away from the plasma without losing information about

intensity, polarization, or phase.

A 4 wave-mixing process can be leveraged to generate second harmonic signals in the

presence of an external electric field. The external electric fields act as the third input wave

with a frequency ω = 0. Enabling what is known as Electric Field Induced Second Har-

monic generation (E-FISH), whereby controlling the power of the incident laser the gen-

erated signal is directly proportional to the magnitude of the external electrical field. First

introduced to the low-temperature plasma community in 2017 [14], E-FISH has become

7



Figure 1.4: (a) Sum Frequency Generation (SFG), a 3-wave-mixing process, depicted in a medium
that lacks symmetry. Two photons, ω1 and ω2, interact with the medium to produce a new photon,
ω3. (b) The energy level diagram depicting SFG, ω1, excites the medium to a virtual state and

excites the medium again to a higher energy virtual state. The medium transitions from the higher
energy virtual state back down to the initial state emitting a single photon, ω3, with the same

energy as the combined energy of the initial photons. Figure reproduced with permission from [4]
license 6039381472801.

one of the standard methods for in-situ electric field measurements [15], but the systems

currently used to measure these conditions often only utilize the most basic version of E-

FISH where only intensity is measured. The methods presented in Chapter 2 demonstrate

the improved capabilities of E-FISH as a diagnostic when intensity, phase, and polarization

information are measured.

Building off the nonlinear laser system used in Chapter 2, while investigating other

types of nonlinear interaction Chapter 3 explores the generation of a 2nd order, 3 wave-

mixing, process that otherwise should not occur in the gas phase [4, 13]. Sum Frequency

Generation (SFG) is the mixing of two photons with different energies to generate a third

photon with the sum of these energies, making it a 3-photon process and otherwise forbid-

den in a symmetric medium, but this process was observed in Chapter 3 and demonstrated a

clear dependence on density showing potential for development into a new density diagnos-

tic. Utilizing the combination of these techniques would enable spatial-temporal mapping

of the reduced electric field with unprecedented precision.

8



Chapter 2

Sub-1 V/cm E-FISH in Atmospheric

Pressure Air

2.1 Abstract

We report on the development of a highly sensitive Electric Field Induced Second Har-

monic (E-FISH) generation diagnostic setup capable of measuring electric field strengths

as low as 1 V/cm at the picosecond time scale under atmospheric pressure conditions. This

unprecedented sensitivity is achieved through passive homodyne detection, which utilizes

stray signals generated by an optical component in the beam path. Our detection limit rep-

resents a 2-3 orders of magnitude improvement over previous reports (100 – 1000 V/cm)

in the literature. Additionally, we demonstrate sensitivity to the polarity of the electric

field. Experimental results are corroborated by simulations of the 400 ps time-resolved ho-

modyne process, offering deeper insights into the enhanced detection capabilities and the

system’s ability to resolve the field sign.

A modified version of this chapter is published as [16] Grayson LaCombe, Jianan Wang, Kraig Fred-
erickson, and Marien Simeni Simeni. Sub-1 v/cm e-FISH-based picosecond electric field measurements in
atmospheric pressure air. 34(2):025001, 2025-02. Publisher: IOP Publishing.

9



2.2 Introduction

In electrical gas discharges, the input electrical energy is preferentially transferred to

kinetic energy of free electrons. This preference arises from the significantly lower mass

of electrons compared to other gas particles. In detail, free electrons gain kinetic energy

when accelerated by an externally applied electric field typically resulting from an applied

voltage difference on a pair of metal electrodes. Electrons subsequently generate complex

chemistry by transferring a portion of their energy through collisions with the surrounding

gas particles. Phenomena such as gas heating, rotational excitation of molecules within

the gas, vibrational excitation of those molecules, electronic excitation of both atomic

and molecular species, as well as dissociation and ionization all occur as consequences of

these collisional energy transfers initiated by electrons. As highlighted in the 2022 Plasma

Roadmap, the field of Low-Temperature Plasma science and technology heavily relies on

our capability to harness, engineer, and control these complex energy transfers toward very

diverse applications [17].

A crucial parameter influencing the aforementioned energy transfers and electron kinet-

ics is the reduced electric field, denoted as E/N, where E represents the magnitude of the

electric field within the plasma and N is the total gas number density. This parameter intu-

itively accounts for scaling the effects of accelerating electric fields by the number density

of the available collisional partners [7]. The accurate measurement of the electric field

magnitude, particularly in high-pressure conditions, becomes imperative due to the expo-

nential dependence of rate coefficients for electron impact-driven processes on E/N [8, 11].

Furthermore, sub-nanosecond resolved E-field magnitude measurements are often needed

under high-pressure conditions because of the very transient electric field dynamics when

plasmas are generated using excitation voltages featuring fast nanosecond rise times [18].

The Electric Field Induced Second Harmonic (E-FISH) generation diagnostic [14] has re-

10



cently gained a lot of attention for electric field measurements in high-pressure electrical

discharges. The impressive potential of this technique was put on full display through

the large variety of measurements. We can list for instance measurements in plasma-

enhanced flames [19, 20, 21], in plasma jets [22, 23, 24], in volumetric ionization waves

[25, 26, 27, 28, 29, 30], in corona discharges [31, 32, 33] and in surface ionization waves

[34, 35]. Very recent works related to this technique have focused either on the understand-

ing of the fundamental question of the coherent growth of the signal under tight focusing

conditions [36], on leveraging pulse-burst lasers to achieve single acquisition measure-

ments [37] or on improving the spatial resolution of this diagnostic along the laser beam

path [38, 39, 40]. Interestingly, regarding the latter aspect, research groups at Ohio State

University and Sandia National Laboratories have reported a factor of 2 increase of the

spatial resolution in the propagation direction of the laser beam to about 500 µm when

deploying a 1◦ crossed-probe beam strategy [38, 39]. However, this gain came with the

drawback of a decrease in the measured signal by over two orders of magnitude. Achieving

a micrometer spatial resolution is key when probing high-pressure filamentary discharges

featuring typical diameters around 50-100 µm.

In this context, we report on the development of a very sensitive Electric Field Induced Sec-

ond Harmonic (E-FISH) generation diagnostic setup. This system is capable of measuring

electric field magnitudes as low as 1 V/cm in room air and at the picosecond timescale.

This advancement represents an improvement by over two orders of magnitude compared

to most E-FISH systems encountered in the literature, where reported detection limits are

typically around 100 V/cm – 1 kV/cm [14, 34]. This enhanced capability is especially

important when characterizing electric field reversals in plasma discharges [41, 42]. Ac-

curately characterizing electric field reversals requires an E-FISH diagnostic capable of

measuring low-magnitude electric fields while also being sensitive to the E-field polarity.

However, the standard E-FISH approach cannot meet this polarity sensitivity requirement,

11



as the measured signal is proportional to the square of the electric field [14]. {Beyond elec-

tric field reversals, the ability to measure low-magnitude electric fields is critical for un-

derstanding numerous key phenomena across plasma physics. These include: (1) surface

charges decay on dielectric surfaces in the afterglow of low to high-pressure discharges

[41]; (2) space charge dynamics in the afterglow of low to high-pressure discharges, espe-

cially in inert gases [43]; (3) electron energy partition in atmospheric pressure discharges

under low E/N conditions [8, 38]; (4) trapped charged-particles dynamics in the sheath of

low to intermediate pressure dusty plasmas [44]; (5) charged-species dynamics in Hall-

effect thrusters [45]; (6) physics of electrical double layers [46]; (7) edge localized modes

at the periphery of tokamak plasmas [47]; and (8) ion-ion plasmas in electronegative gases

[48]. Through a comparative analysis with standard E-FISH systems, we examine neces-

sary upgrades and propose pathways for further development.

2.3 Experimental Setup

Figure 2.1: Schematic of the E-FISH experimental setup. The different voltage power supplies are
not shown. The electrodes are 3.5 cm long along the laser beam propagation direction.

Figure 2.1 depicts a schematic of the experimental setup. The vertically-polarized fun-

12



damental output of a mode-locked, diode-pumped picosecond Nd:YAG laser (EKSPLA,

PL2231-50, 30 ps, 30 mJ maximum output at 1064 nm) operating at 50 Hz is focused be-

tween a pair of parallel plate copper electrodes using 45° incidence angle silver-protected

mirrors (ThorLabs, PF10-03-P01) and an AR-coated 1-m focal length BK-7 focusing lens

(Lambda Research Optics, VAR2-PCX-25.4B-1000-1064). Two colored-glass long-pass

filters (ThorLabs, FGL850M, with a cut-on wavelength of 850 nm) are used to filter out

any stray second harmonic (SH) signal emanating from the interaction of the picosecond

laser beam with the silver mirror surfaces as well as with the focusing lens (”Lens 1”).

The copper electrodes are identical and separated by an adjustable gap distance. These

electrodes can be powered by a DC high-voltage power supply (Spellman, SL10PN150)

or by the voltage output of a digital delay generator (Stanford Research Systems, DG645).

Following the approach pioneered by Chng et al. [36], we use triangular-shaped electrodes

such that we can vary the effective length of the interaction region between the focused

Gaussian laser beam and the region of space where the E-field is applied. Although the

optimum electrode length could be directly derived from the knowledge of the Rayleigh

range of the Gaussian beam, placing the triangular-shaped electrode assembly on a 3-axis

translational stage allows for better control over the strength of the generated second har-

monic signal. Throughout the manuscript, we used an electrode length of 3.5 cm, which

was found to be optimized based on E-FISH signal intensity testing. Because of the in-

teraction between the 1064 nm beam and the DC-applied electric field, a co-propagating

E-FISH signal at 532 nm is generated. A Nd:YAG laser harmonic separator dichroic mirror

(Lambda Research Optics, HHS-2506U-R532/T1064-45) reflecting 532 nm and transmit-

ting 1064 nm is placed downstream of the electrode assembly. This results in a significant

reduction of the 1064 nm beam intensity. A 1-ns rise time silicon photodiode (ThorLabs,

DET10A2) is placed in the transmitted direction after the dichroic mirror. This photodi-

ode is used for monitoring the pulse-to-pulse fluctuations of the incident 1064 nm beam

13



intensity and for timing purposes. The photodiode is preceded by a neutral density fil-

ter wheel to avoid saturation effects. Next, a 1-inch diameter AR-coated, 100-cm focal

length lens (Lambda Research Optics, VAR2-PCX-25.4B-1000-532) collimates both the

E-FISH signal at 532 nm and the remnant 1064 nm beam. A 2.5-cm CaF2 equilateral dis-

persion prism (Thorlabs, PS863) then spatially separated the two different wavelengths.

A polarizing cube beamsplitter (Lambda Research Optics, BPB-25.4SF2-550) is then em-

ployed to only select for vertically polarized light. A 10-cm focal length AR-coated N-BK7

plano-convex lens (Thorlabs, LA1509-A) is then leveraged to focus the E-FISH signal onto

the active area of a very sensitive photomultiplier tube (PMT) at 532 nm (Hamamatsu,

H7422PA-40). This PMT is powered by a dedicated low-noise power supply (Hamamatsu,

C8137-02), and the signal from the PMT is amplified through a pre-amplifier (Hamamatsu,

C11184). Note that the PMT is preceded by a hard-coated bandpass filter (center: 532

nm, width: 10 nm, ThorLabs, FLH532-10). The E-FISH, photodiode, and voltage wave-

forms are monitored using a 1-GHz bandwidth, 5 GSa/s sampling rate digital oscilloscope

(LeCroy, WaveSufer4101HD).

A key aspect of this new setup resides in the removal of the monochromator, which usually

precedes the PMT. We found that removing the monochromator significantly increases the

E-FISH signal. However, following this removal, the system becomes very sensitive to

stray light. This issue was resolved using multiple irises as well as beam tubes in the beam

path. Another essential aspect of this setup is the use of a very sensitive PMT, featuring

the highest efficiency at 532 nm, among the commercially available devices. Next, the op-

timization of the electrode length and finally the use of a diode-pumped laser featuring a

superior beam spatial profile compared to flashlamp-pumped lasers.

14



Figure 2.2: PMT waveforms from 27,000 laser shots averaged E-FISH tests using a 1 cm electrode
gap and 5 mJ laser pulse energy. (a) PMT waveforms taken with DC-applied voltages ranging

from 0 to -19 V to the top electrode resulting in a positive electric field. (b) PMT waveforms from
a) with the 0 V/cm background signal subtracted highlighting the increase in signal from the
applied electric field. (c) Diagram of the electrode set up for positive electric fields. (d) PMT
waveforms taken with DC-applied voltages ranging from 0 to -30 V to the bottom electrode

resulting in a negative electric field. (e) PMT waveforms from c) with the 0 V/cm background
signal subtracted highlighting the change in signal from the applied electric field. (f) Diagram of

the electrode set up for negative electric fields.

2.4 Results

Figure 2.2 displays averaged E-FISH waveforms taken in room air for sub-breakdown

DC electric fields ranging from -30 V/cm to 19 V/cm between two parallel electrodes with a

gap distance of 1 cm. The laser pulse energy was fixed at 5 mJ. Throughout this manuscript,

the magnitude of the Laplacian electric field at the center of the electrode gap is approxi-

mated by the value given by the ratio of the applied voltage to the gap distance. A negative

DC power supply (Stanford Research Systems, PS370) is used to provide a constant low

negative voltage, which is measured by a voltage probe (Teledyne Lecroy, PP026). The

15



negative DC was initially connected to the top electrode, while the bottom electrode was

grounded. In this configuration, the electric field vector pointing ”up”, is defined as a ”pos-

itive” E-field (see figure 2.2-(c)). Conversely, grounding the top electrode and connecting

the bottom electrode to the negative DC power supply results in an electric field vector

pointing ”down” and hence defined as a ”negative” E-field (see figure 2.2-(f)). The PMT

is operated with a control voltage (to realize the gain) of 0.725 V (corresponding to a gain

value of about 5×105) for the positive E-field experiments, while a control voltage of 0.850

V (corresponding to a gain value of about 1.5 × 106) is leveraged for the negative E-field

measurements. Figures 2.2-(a) and 2.2-(d) show the averaged PMT traces for positive and

negative E-Fields respectively, while figures 2.2-(b) and 2.2-(e) show the amplitude change

(compared to the 0 V/cm case) in the PMT waveform caused by the corresponding applied

electric field. Each PMT trace represents an average of at least 27,000 laser shots due to the

weak nature of the signal. It is important to emphasize that this large number of laser shots

does not constitute a limitation to the applicability of this approach to laboratory plasma

experiments, which typically involve experimental drifts over long periods due to changes

in ambient temperature and pressure. Indeed, leveraging MHz repetition rate femtosecond

lasers would address and resolve such challenges [49]. As depicted in figures 2.2-(a),(d),

for both positive and negative electrode configurations, a background signal is measured in

the absence of an applied E-field. From figure 2.2-(b), when a positive E-Field is applied,

the signal increases with increasing electric field from a peak (background removed) sig-

nal of 0 mV at 0 V/cm to about 83 mV at 19 V/cm. But we see a large difference even

between 0 V/cm (0 mV peak) and 1 V/cm (12 mV peak), which is the same increase in

PMT output voltage seen at 3 kV/cm in some of the previous studies in the literature [34].

Such a noticeable difference (0 mV VS. 12 mV) readily suggests that our system is capable

of measuring sub-1 V/cm DC E-fields. In fact, we expect to be able to measure positive

DC E-fields down to 0.3-0.5 V/cm. It is important to point out that the ≈ 700 mV ”offset”

16



signal measured at 0 V/cm (see figure 2.2-(a)) is due to stray second harmonic generation

(SHG) signal generated at the surface of the dichroic mirror. For negative E-fields, the

signal offset at 0 V/cm (see figure 2.2-(d)) is stronger due to the higher PMT gain used.

Surprisingly, for negative E-fields, figure 2.2-(e) evidences there is initially a decrease of

the signal for applied E-field magnitudes ranging from 0 to 10 V/cm (signal decrease from

0 mV to -20 mV). Subsequently, the signal begins to rise quickly from 10 to 30 V/cm E-

fields, corresponding to an increase in peak signal from -20 mV to 160 mV. Although the

variation in peak PMT voltage allowed us to showcase the detection limit of our system,

these signals do not clarify the relationship between measured E-FISH signals and applied

voltages.

Figure 2.3: Time-integrated DC E-FISH signals for different applied E-Fields. The electrode gap
distance is 10 mm with a width of 35.3 mm. (a) Negative electric fields. (b) Positive electric fields.

By time-integrating PMT traces in figures 2.2-(a) and 2.2-(d), we obtained the results

depicted in figure 2.3. The latter figure corresponds essentially to typical E-FISH calibra-

tion plots relating electric field strengths to measured E-FISH signal intensities. Error bars

in figure 2.3 correspond to 95% confidence intervals evaluated from statistical calculations

over the large number of laser shots collected for each field magnitude. The positive and

negative E-fields trends evidenced in figure 2.2 are consistently observed in figure 2.3. For

17



the positive E-field case (figure 2.3-(b)), the collected signal increases with the increase

of the magnitude of the electric field at the center of the electrode gap while a more com-

plex trend is observed for the negative E-field cases (figure 2.3-(a)). From figure 2.3, for

both positive and negative E-field cases, the relationship between the applied E-Field at the

center of the gap and the corresponding measured E-FISH signal intensity does not follow

the expected quadratic dependence (Imeas ∝ |Eappl|2). This is especially apparent for the

negative E-Field cases (Figure 2.3-(a)), for which the E-FISH signal decays from 0 to −10

V/cm and then increases from -10 to -30 V/cm. This unexpected trend calls for a more

in-depth investigation.

To address these unexpected results, a transient low-voltage waveform featuring polarity-

switching is applied to the electrode assembly. An arbitrary function generator (Tektronix,

AFG1062) is used to produce a sine wave with a frequency of 10 MHz (100 ns tempo-

ral period) and an amplitude from -10.5V to +10.5V. The output of the arbitrary function

generator is connected to the ”top” electrode while the bottom electrode is grounded. The

electrode gap distance was 3 mm leading to an applied sub-breakdown electric field be-

tween -35 and 35 V/cm. A fixed PMT control voltage of 0.850 V was employed (corre-

sponding to a gain value of about 1.5× 106). The time-resolved E-Fields and correspond-

ing time-integrated PMT signals for a single voltage period are shown in figure 2.4-(a).

For this experiment, the temporal jitter of the incident laser pulse with respect to the ap-

plied voltage was ”artificially” increased by realizing a constant temporal drift of the laser

pulse with respect to the AC waveform. Subsequently, for each laser shot, the photodi-

ode, PMT, and voltage probe signals were recorded by the digital oscilloscope and placed

into time bins (see [18] for more details about this procedure). Note that the segmented

memory acquisition mode (”sequence” mode) of the oscilloscope was leveraged for this

operation. Time bins of 400 ps were leveraged for this measurement and about 3000 laser

shots were collected per bin. Remarkably, from figure 2.4-(b), the temporal dynamics of

18



the measured square root of the time-integrated PMT signal do not follow that of the ap-

plied E-field waveform. This comes as a surprise as the latter feature (
√
Imeas ∝ |Eappl|) is

often leveraged for absolute calibration of transient electric field measurements in electrical

discharges (see for instance [19, 20] for the case of AC excitation waveforms). Using the

time-resolved data to plot a ”calibration-like” plot featuring time-integrated PMT signal

VS. electric field at the center of the gap, yields figure 2.5. The depicted trends from figure

2.5 are consistent with those from figure 2.3, with again a decrease of the measured signal

between -35 and -10 V/cm and then a subsequent increase from -10 to 35 V/cm. Based

on the ”basic” E-FISH paradigm (Imeas ∝ |Eappl|2), one would expect figure 2.5 to be a

parabola showcasing no offset (measured signal should be 0 for a 0 V/cm applied electric

field) and the line of equation E-field = 0 should be an axis of symmetry.

Figure 2.4: 400 ps time-resolved E-FISH data for a 10 MHz low voltage sine wave applied to the
electrode assembly. The electrode gap distance is 3 mm. (a) Time-resolved E-Field and
corresponding PMT signals. (b)

√
Imeas plotted VS. Time together with |E| VS. Time.

2.5 Discussion

We hypothesize that understanding the intensity offset as well as the asymmetry of the

plot displayed in figure 2.5 could be achieved when developing a formalism factoring in

19



Figure 2.5: Measured PMT signal as a function of applied E-Field together with a parabola fit of
the data. The laser pulse energy is 5 mJ.

the background second harmonic signal generated by the surface of the dichroic mirror.

Because the surface SHG signal generated on the dichroic mirror is a coherent beam, it

also propagates to the PMT detector. The decrease of the PMT signal at negative applied

voltages suggests the contribution of that stray signal to the overall PMT-measured inten-

sity through an interference-like interaction. In the following, we assume that the stray

signal from the dichroic mirror acts as a local oscillator (LO), interfering with the ”regu-

lar” E-FISH signal produced at the electrode assembly. Such an effect has been previously

reported in the literature [50, 51, 49]. Interestingly, the same phenomena were also recently

observed in some E-FISH experiments but were regarded as undesirable parasitic effects

and subsequently mitigated [52, 42]. Using the complex field notation, we can therefore

write:

Imeas ∝ |Etotal|2, (2.1)

20



where the total complex electric field Etotal at 2ω (532 nm) is the sum of the electric fields

emanating from the surface (ELO

2ω ) and E-FISH (EE−FISH

2ω ) processes, respectively:

Etotal = EE−FISH

2ω + ELO

2ω , (2.2)

where EE−FISH

2ω = EE−FISH
2ω eiϕE−FISH . EE−FISH

2ω is the magnitude of the electric field

of the E-FISH signal with phase ϕE−FISH . Similarly, ELO

2ω = ELO
2ω eiϕLO . ELO

2ω is the

magnitude of the electric field of the coherent background signal with a phase ϕLO. To

simplify the following expressions, we define the relative phase difference between the

local oscillator and E-FISH signals as ϕ = ϕLO −ϕE−FISH and define ϕE−FISH as 0. This

means that equation (2) can be simplified as Etotal = EE−FISH
2ω + ELO

2ω eiϕ.

We can write the electric field from the E-FISH process as:

EE−FISH
2ω = |EE−FISH

2ω | ∝ |χ(3)EapplEωEω|, (2.3)

where χ(3) denotes the third-order nonlinear susceptibility of the gas (room air in the

present case), whereas Eω represents the complex notation of the electric field of the inci-

dent 1064 nm laser beam. The SHG electric field from the LO can be written as:

ELO
2ω = |ELO

2ω | ∝ |χ(2)EωEω|, (2.4)

where χ(2) is the second-order nonlinear susceptibility of the dichroic mirror surface.

Inserting equation (2) into equation (1) yields:

Imeas ∝ ILO2ω + IE−FISH
2ω + 2EE−FISH

2ω ELO
2ω cos(ϕ), (2.5)

where IE−FISH
2ω = |EE−FISH

2ω |2 = (EE−FISH
2ω )2 and ILO2ω = |ELO

2ω |2 = (ELO
2ω )2 are the

intensities of the E-FISH and LO signals, respectively.

21



Combining equations (3)-(4) into equation (5) yields:

Imeas ∝ β|Eω|4 + α|Eappl|2|Eω|4 + γ|Eappl||Eω|4 cos(ϕ) (2.6)

where α and β are positive proportionality constants related to the efficiencies of the SHG

and E-FISH processes, respectively. γ = 2
√
αβ.

The first term of equation (6) represents the intensity of the SHG signal emanating

from the interaction of the incident laser beam with the dichroic mirror surface (ILO2ω ). The

second term represents the intensity of the E-FISH process (IE−FISH
2ω ). The third term

finally is ascribed to the interference between the surface SHG and E-FISH signals. It is

responsible for a decrease in the intensity of the measured signal with increasing applied

electric field. Constructive interference occurs when ϕ = 0, i.e. cos(ϕ) = 1. Conversely,

destructive interference is achieved for ϕ = π, i.e. cos(ϕ) = −1.

In equation (6)’s paradigm, when cos(ϕ) < 0, meaning π
2
< ϕ < 3π

2
, it is possible to

observe a decrease in measured intensity Imeas with an increase in the magnitude of the

applied electric field |Eappl|. When writing Y = Imeas and X = |Eappl|, equation (6) can

be identified as a quadratic function of |Eappl|:

Y = AX2 +BX + C, (2.7)

where A = α|Eω|4, B = γ|Eω|4 cos(ϕ), and C = β|Eω|4 are proportionality constant

non-dependent on |Eappl|. To validate this interpretation of our measurements, we fitted the

negative E-fields portion of the data displayed on figure 2.5 using equation (7). A great

quality fit (R2 = 0.995) was obtained and overlaid on top of the negative E-fields experi-

mental results. When using coefficients A, -B, and C from equation (7) for positive E-field

values ranging from 0 to 35 V/cm, we obtained a curve showing an excellent agreement

with the positive E-fields data (overall R2 = 0.991). The fact that the quadratic fit some-

22



what overestimates experimental results for positive E-fields above ≈25 V/cm could be

a hint of a slight saturation of the PMT for the corresponding intensity range. Nonethe-

less, the good match obtained suggests our experimental findings are consistent with an

enhanced E-FISH homodyne detection passively achieved through the interference of the

regular E-FISH signal with the SHG signal produced from a dichroic mirror in the beam

path. It is important to note that ”active” homodyne E-FISH detection of gas-phase electric

fields down to about 500 V/cm has been recently reported [53, 54], with also a demonstra-

tion of a sensitivity to the E-field polarity. In contrast, our setup realized it ”passively”, in

a single beam configuration achieving a detection limit better than 1 V/cm. Furthermore,

the sub-nanosecond temporal resolution (a capability not demonstrated by the aforemen-

tioned previous works) of our setup allows us to unravel the fundamental mechanisms of

this interaction.

Our intuitive interpretation of the data relies on the fact that when the sign of the applied

electric field Eappl flips from positive to negative (or vice-versa), we get a π change in

the phase difference (∆ϕ) between the incident laser beam and the applied electric field:

∆ϕ2 = ∆ϕ1 ± π. Where ∆ϕ1 and ∆ϕ2 are the phase differences before and after the sign

flip (electric field reversal), respectively. As a result, cos(∆ϕ2) = − cos(∆ϕ1). Such an

interpretation, although appropriate in the first order of approximation, does not account

for the wave nature of the incident 1064 nm laser beam featuring a frequency of about

2.82 × 1014 Hz. This means that within a single 30 ps laser pulse, while Eappl is constant,

the electric field of the incident light flips about 1.69 × 104 times (two sign flips per time

period).

In the following, we now extend the previous model to account for the temporal evo-

lution of the different electromagnetic waves at play. Assuming the electric field of the

incident laser beam could be written as a 30 ps Gaussian envelope electromagnetic wave

featuring a 1064 nm wavelength (≈ 3.55 fs time period), we get:

23



Eω = E0,ω g(t) cos(ω0t), (2.8)

where E0,ω is the amplitude of the laser electric field, g(t) = e
−4 ln2 ( t

τp
)2 is the 30 ps

Gaussian temporal envelope (τp = 30 ps), and ω0 = 2π c
λ0

≈ 1.77 × 1015 rad/s is the angular

frequency of the incident laser for λ0 = 1064 nm. Note that τp corresponds to the FWHM

of the incident laser beam electric field temporal profile. Consequently, it is equivalent to

the full width at quarter maximum (FWQM) for the temporal profile of the incident laser

beam intensity.

Similarly, the 10 MHz applied AC electric field can be written as:

Eappl = E0,AC cos(2π fAC t+ ϕAC), (2.9)

where E0,AC = 35 V/cm, is the amplitude of the applied AC electric field and fAC = 10

MHz is the frequency of the AC wave. ϕAC is the phase of the applied AC electric field.

Substituting equations (8) and (9) into equations (1) - (4) yields:

Imeas ∝ |αE0,AC cos(2π fACt+ ϕAC)E
2
0,ωg

2(t) cos2(ω0t) + βE2
0,ωg

2(t) cos2(ω0t)e
iϕ|2

(2.10)

Equation (10) can be simplified into:

Imeas(t) ∝ |αF1(t) + βF2(t)e
iϕ|2, (2.11)

where F1(t) = E0,AC cos(2π fACt+ ϕAC)E
2
0,ωg

2(t) cos2(ω0t)

and F2(t) = E2
0,ωg

2(t) cos2(ω0t).

Equation (11) can be further simplified into:

24



Imeas(t) ∝ |F1(t) + ηF2(t)e
iϕ|2, (2.12)

where η = β
α

. η and ϕ are the only two optimization parameters in our model.

Figure 2.6: Comparison between simulations and experimental results. (a) Time-integrated PMT
signal VS Applied E-Field. (b) Inferred

√
PMT , |E| VS Time.

Now considering a single laser pulse, the time-integrated signal S(Eappl) measured by

the PMT can therefore be written as:

S(Eappl) =

∫ τp

−τp

Imeas(t)dt ∝
∫ τp

−τp

|F1(t) + ηF2(t)e
iϕ|2dt, (2.13)

Based on our model, we should be able to use equation (13) for different values of Eappl

and essentially build a Matlab-based fit matching experimental data presented in figures 2.5

and 2.4-(b). For this purpose, the Matlab least-squares nonlinear curve fitting tool lsqcurve-

fit is deployed. Figure 2.6 depicts the results of our model when compared to experimental

results in terms of time-integrated PMT signal VS. applied E-field. In figure 2.6-(a), we

show that our Matlab simulations are able to capture accurately the intensity offset and

asymmetry discussed previously. Furthermore, figure 2.6-(b) demonstrates that when com-

25



pared with the applied AC wave, the temporal dynamics are also fully reproduced.

The excellent agreement obtained from figure 2.6 allows us to dig more into the details

of the homodyne E-FISH process when selecting a few notable points in time from the AC

wave dynamics. For instance, figure 2.7 focuses on characterizing the interaction process

at the peak applied E-field: for Eappl = +35 V/cm (t = 0 ns or t = 100 ns). Note that the

peak homodyne signal for the AC wave is obtained for the peak applied E-field. Figures

2.7-(a),(b),(c),(d),(e) display the temporal evolution of the incident 30 ps laser electric field,

its intensity, the applied AC electric field, the electric field of the generated ”pure” E-FISH

signal, and the intensity of the measured homodyne signal, respectively.

Figure 2.8 showcases the very same calculations but this time for Eappl= 0,−10 V/cm,

respectively. While Eappl= 0 V/cm is obtained at t ≈ 26 ns, Eappl= −10 V/cm, which

corresponds to the point in time for which the minimum homodyne signal is measured, is

obtained at t ≈ 30 ns. Comparing figures 2.7 and 2.8, on one hand we observe that as

expected, a non-null homodyne E-FISH signal is generated for Eappl= 0 V/cm. The signal

depicted in figure 2.8-(e) corresponds in fact to the intensity of the signal from the LO.

The latter signal is comparable in magnitude but a bit stronger than the minimum measured

signal, which occurred at Eappl= −10 V/cm (see figure 2.8-(j)). On the other hand, the

magnitude of the signal measured for the peak applied field is predicted to be a factor ≈ 2

stronger (see figure 2.7-(e)), which is very consistent with our experimental measurements.

Very interestingly, it is observed that from both figures 2.7 and 2.8, our model predicts a

systematic ”temporal compression” of the incident laser pulse by the E-FISH process. The

FWQM of the generated homodyne signal intensity temporal profiles are about 20 to 22 ps,

instead of 30 ps. Such a phenomenon is typical for SHG processes [55].

The results presented in this manuscript represent a significant milestone in enhancing

the sensitivity of the E-FISH diagnostic. While we are aware of recent similar advance-

26



Figure 2.7: Single laser pulse simulations for the case where the AC electric field is maximum
(ϕAC=0), generating maximum EEFISH and Imeas. (a) Incident laser E-field. (b) Incident laser

intensity. (c) Applied AC E-field. (d) Induced E-FISH E-field. (e) Generated homodyne signal
intensity.

ments employing single beam homodyne enhancement (where the interference term in

equations (5) & (6) completely dominates the sum) to achieve improved E-FISH sensitivity

by the group at Polytechnique Montréal [56], further improvements in the detection limit

27



Figure 2.8: Single laser pulse simulations for the cases where the applied E-field is null (left) and
-10 V/cm (right). (a) Incident laser E-field. (b) Incident laser intensity. (c) Applied E-field for
ϕAC=π

2 . (d) Induced E-FISH E-field for ϕAC=π
2 . (e) Generated homodyne signal intensity for

ϕAC=π
2 . (f) Incident laser E-field. (g) Incident laser intensity. (h) Applied E-field for ϕAC=3π

5 . (i)
Induced E-FISH E-field for ϕAC=3π

5 . (j) Generated homodyne signal intensity for ϕAC=3π
5 .

remain possible. Potential avenues include: (1) adopting an all-reflective optics approach

in the collection part of the setup; (2) employing a gated photon-counting photomultiplier

tube as the endpoint detector; (3) developing an ”active” homodyne experimental setup

allowing for a precise adjustment of the phase difference and local oscillator intensity,

to operate within different regimes of the interference effects; and (4) utilizing femtosec-

ond MHz repetition rate lasers to address scenarios requiring large numbers of laser shots

(> 104), particularly in plasma applications. Notably, many of these strategies have been

successfully implemented by Dadap et al [49] but for the purpose of surface electric field

measurements. Our findings demonstrate that this approach is well-suited for gas phase

electric field measurements while preserving the picosecond temporal resolution.

28



2.6 Conclusion

In conclusion, we demonstrated DC and AC E-FISH-based electric field measurements

to magnitudes lower than 1 V/cm for Laplacian field configurations in room air. A 400

ps temporal resolution was achieved for the AC measurements. Sensitivity to the phase

of the electric field is achieved through homodyne detection arising from the unexpected

interference of the E-FISH signal with stray SHG signal from the surface of a dichroic

mirror in the beam path. The high sensitivity of our experimental setup is attributed to

several key factors: (1) homodyne detection; (2) use of a very sensitive PMT under high

gain conditions; (3) meticulous optimization of the E-FISH collection leg optics and align-

ment; (4) precise adjustment of the electrode length for our incident laser beam focusing

conditions; and (5) a high-quality spatial profile of the incident laser beam. Our experi-

mental results were found to be in excellent agreement with analytical simulations of the

E-FISH signal for a homodyne process. Interestingly, the analytical simulations of the E-

FISH signal predicted that the E-FISH process, similarly to SHG processes should result in

a temporal compression of the incident laser pulse. This work opens up new opportunities

for improved single-shot and 2-D measurements under high-pressure conditions. Addi-

tionally, we expect the homodyne approach to also improve the performance of the E-FISH

diagnostic under very low-pressure conditions (0.1-1 Pa), for which the ”standard” E-FISH

approach is not well-suited because of the N2 dependence of the signal on the gas density

N . Between 1 atm (≈ 105 Pa) and 1 Pa, because N decreases by 5 orders of magnitude,

the E-FISH signal is therefore expected to decrease by 10 orders of magnitude (when not

accounting for homodyne enhancement). Finally, we plan to leverage the newly gained

sensitivity for precise characterization of electric field reversals in a variety of discharge

geometries.

29



Chapter 3

Non-resonant Sum Frequency

Generation in the Gas Phase

3.1 Abstract

We report on the experimental observation of non-resonant, second-order optical Sum-

Frequency Generation (SFG) in five different atomic and molecular gases. The measured

signal is attributed to an SFG process by characterizing its intensity scaling and its po-

larization behavior. We show that the electric quadrupole mechanism cannot explain the

observed trends and suggest a mechanism based on symmetry-breaking along the inci-

dent beam path arising from laser-induced species ground state number density gradients.

Our results demonstrate that SFG is about four orders of magnitude stronger than Third

Harmonic Generation (THG) and independent from any externally applied electric fields.

These features make this method suitable for gas number density measurements at the pi-

cosecond timescale in reactive flows and plasmas.

A modified version of this chapter is published as [57] Grayson LaCombe, Jianan Wang, Jeremy
R. Rouxel, and Marien Simeni Simeni. Non-resonant picosecond three-wave mixing in the gas phase.
49(23):6717–6720, 2024-12-01. Publisher: Optica Publishing Group.

30



3.2 Introduction

A classic nonlinear optics textbook demonstration [58, 4, 59] shows that even-order

nonlinear susceptibilities vanish in centrosymmetric media. It stems from the sign flip of

electric dipole moments upon an inversion operation. This property has made even-order

techniques useful for probing interfaces [60, 61, 62] or chiral liquids [63, 64]. However,

it is often considered that techniques such as even-order Sum- or Difference-Frequency-

Generation (SFG and DFG respectively) cannot be used to probe the bulk of centrosym-

metric media. Pioneering works by Bethune et al. [65, 66, 67, 68] have demonstrated that

these considerations do not hold when quadrupolar interactions are considered. By specif-

ically targeting an electric quadrupole transition in one step of the interaction pathway,

they were able to show that resonant second-order SFG (which is equivalent to resonant

three-wave mixing: TWM) in centrosymmetric media can be generated from quadrupolar

coupling. Bethune et al. [69], Kim et al.[70], Okada et al. [71] all have measured the

quadrupolar resonant SFG signals in various bulk alkali metal vapor gases (Na, K, Li),

relying on their high nonlinear hyperpolarizabilities [72, 59].

However, later experiments focusing on non-resonant SHG in Na vapor by Miyazaki

et al. [73] and in Xe by Malcuit et al. [74] did not match the predictions of a signal

with quadrupolar coupling. A model based on static electric fields arising from charge

separation following photoionization was proposed to explain a laser-induced symmetry

breaking. Finally, an alternative explanation based on an induced anisotropy following

spatial variation of the densities of ground state atoms along the focused laser beam prop-

agation direction was proposed by Freeman et al. [75]. In this Letter, we demonstrate that

second-order SFG can be measured off-resonance in various atomic gases (He, Ar, Kr) and

diatomic molecules (O2, N2), and rule out quadrupolar SFG and charge separation-induced

static electric fields as possible mechanisms. This work generalizes previous observations

31



to molecular gases but also to SFG instead of only SHG. SFG in the gas phase carries

information about the total gas number density within the probed volume, making it very

attractive to characterize reacting flows and plasmas.

We first present experimental results demonstrating the observation of off-resonant SFG

processes. Measurements of pressure and intensity scaling of the signal demonstrate its

nature. Polarization characterization rules out the quadrupolar origin of the signal.

3.3 Methods

The experimental setup used to measure the SFG signal is displayed in Fig. 3.1. The

experiments are carried out using a 50 Hz 1064 nm Nd:YAG laser system, which delivers

30 mJ 30 ps pulses. A frequency-doubling crystal is used to generate a 532 nm second

harmonic. An equilateral dispersion prism spatially separates the two beams. Two half-

waveplates and a polarizer are incorporated in each of the beam paths to allow independent

control of the polarization angles and pulse energies. An optical delay line is used to ensure

the pulses are coincident in time at the sample location. The beam paths are recombined

by a dichroic mirror (DM1) and focused with a f = 15 cm N-BK7 AR-coated lens into a

small pressure chamber. Unless otherwise specified, the experiments are conducted using

approximately E1 = 1 mJ at 1064 nm and approximately E2 = 0.1 mJ at 532 nm. Therefore,

the corresponding upper bounds laser intensities at the focus are I1 = 3.2 × 1012 W/cm2

and I2 = 1.2× 1012 W/cm2, respectively. The pressure chamber is equipped with inlet and

outlet Swagelok ports connected to a vacuum pump (Specstar, 9.6 cfm) and to mass flow

controllers allowing the chamber to be filled up with different high-purity gases. A baraton

capacitance manometer (MKS, 628B13TDE1B) is used to monitor the gas pressure within

the chamber. A dichroic mirror (DM2) positioned after the chamber transmits the 1064

and 532 nm beams while reflecting 355 nm. The transmitted pump beams are measured

using a silicon photodiode (PD1) to monitor the intensities of the incident beams as well

32



as for timing purposes. The SFG signal is then collimated with a f = 15 cm fused-silica

plano-convex lens (L2), then dispersed (Pr2) to isolate the 355 nm signal from any remnant

of the 1064 and 532 beams before being focused (L3, f = 10 cm) into a 355 nm narrow

bandpass filter (FLH355-10, Center 355 nm, FWHM 10 nm) and a manual Mini-Chrom

monochromator (Edmund Optics, Model C, slit 300 µm, 1800 grooves/mm, resolution 2.18

nm). The signal is then collected by a photomultiplier tube (PMT, Hamamatsu, C5594-44).

The polarization state of the SFG signal is measured using a halfwave plate and a polarizer

(Pol.3) prior to the PMT.

To mitigate both Third Harmonic Generation (THG) of the fundamental beam and THG

from external sources, the following precautions were taken. A 450 nm hard-coated long-

pass filter (OD¿5) is inserted between the focusing lens and the test chamber entrance

window to remove any stray 355 nm signal produced by any optics ahead of the cham-

ber. The pressure chamber dimensions and focal lengths were matched to ensure the spot

size on the chamber windows was large (∼4 mm) in order to decrease the incident inten-

sity, greatly limiting nonlinear interactions at windows. The incident beams are focused

near the center of the pressure chamber with a beam waist of ∼40 µm. The laser-induced

breakdown threshold for a typical 1064 nm 40 ps laser at 1 atm in Ar is about 1013 W/cm2

[76, 77]. While the pulse intensities used in the setup mitigate laser-produced plasmas

(LPP) occurrence under low-pressure conditions, we cannot fully exclude the possibility of

LPP occurring for some of the high-pressure cases.

3.4 Results

Fig. 3.2 (a) displays the intensity scaling of SFG signals using air in the chamber with

respect to the pulse energies E1 and E2. The measured signal increases linearly with both

E1 (black square and blue triangle symbols) and E2 (red circle and green downward triangle

symbols). The latter observation is consistent with the expected intensity scaling for SFG

33



Figure 3.1: Schematic of the experimental setup.

given by

ISFG ∝ |χ(2)|2I1I2 (3.1)

where ISFG is the intensity of the SFG signal, χ(2) is the second-order nonlinear suscepti-

bility. Furthermore, χ(2) scales linearly with n, where n is the number density of the gas

molecules, and thus ISFG scales quadratically with n. The contribution of THG to the signal

measured at 355 nm is negligible (approximately 4 orders of magnitude weaker, see sup-

plemental material) such that the output at 355 nm is only measured when both 1064 nm

and 532 nm beams are present in the beam path and overlap spatio-temporally for a given

PMT amplification. The linear dependences highlighted in Fig. 3.2 (a) are also observed

in room air when the test chamber is removed from the beam path, further ruling out pos-

sible artifacts generated by the chamber windows. Plotting Fig. 2(a) as a function of the

individual laser pulse energies results in plots further asserting the linear dependence of the

SFG signal with respect to the intensities of the individual 532 and 1064 nm beams (see

figure S6 in the supplemental document). Noteworthy, in our experiments, the conversion

efficiency of the SFG process in air at 1 atm was estimated to be about 10-10.

34



Figure 3.2: (a) SFG signal in room air at 355 nm as a function of the combined power of the 1064
and 532 nm input beams, (b) Scaled intensity of the SFG signal at 355 nm for 5 different gases as a

function of the gas pressure with a power of 2 fitting for each gas.

Figure 3.2 (b) displays the SFG signal strength as a function of the pressure for various

gases considered. Vertically-co-polarized incident beams were used for these measure-

ments to maximize the generated SFG signals (see further). These measurements were

performed between 20 and 1100 Torr in Kr, Ar, O2, N2, He. For all five investigated gases,

the SFG signal increases with the gas pressure. Furthermore, we observe that ISFG
(2) (Kr)

≫ ISFG
(2) (Ar) ∼= ISFG

(2) (O2) ¿ ISFG
(2) (N2) ≫ ISFG

(2) (He). These results are in good agree-

35



ment with the relative magnitudes of the investigated gases’ hyperpolarizabilities [5, 6], see

Table 1. The signals pressure scaling in figure 3.2 (b) can be all fitted with quadratic func-

tions in low-pressure ranges, as expected by a quadratic scaling with the number density n

of gas particles. Departures from the quadratic scaling are observed for larger pressures and

correspond to signal amplitudes in which the PMT’s response is nonlinear (for a specific

PMT gain). Departures from perfect quadratic functions occur over a shorter pressure range

for gases with large nonlinear hyperpolarizabilities due to the quadratic dependence of the

SFG signal on both the hyperpolarizability and gas pressure (number density). The scal-

ing with the number density observed in Fig. 3.2 (b) suggests the possibility of leveraging

non-resonant three-wave mixing to measure number densities at the picosecond timescale

in virtually all gases. We note that the same n2 dependence was also reported by Miyazaki

et al. for the case of non-resonant optical second harmonic generation in Na vapor [73]

whereas a n3 dependence was observed by Malcuit et al in Xe for a 1064 nm laser intensity

of 4×1013 W/cm2 (an order of magnitude larger) [74].
Table 3.1: Relative nonlinear hyperpolarizabilities of five atomic and molecular gases [5, 6]

Species Nonlinear hyperpolarizability [arb. units]
He 1
N2 21
O2 31
Ar 32
Kr 77

3.5 Discussion

We now discuss the polarization of the measured ISFG signal as a function of the in-

cident beam polarizations. The polarization state of the SFG signal was characterized by

rotating the halfwave plate in front of the linear polarizer Pol.3. The x-axis in Fig. 3 cor-

responds to the angle of the signal beam to be transmitted through Pol.3 in the laboratory

reference frame. A 90° angle is equivalent to the polarizing axis of Pol.3 being aligned

36



vertically (in the laboratory reference frame) while a 0° angle is equivalent to Pol.3 be-

ing oriented horizontally. In Fig. 3.3 , it can be observed that when both incident beams

have the same polarization, the emitted SFG has an identical polarization. For example,

for vertically-polarized input beams (V-V: black squares), a vertically-polarized SFG sig-

nal is obtained. Likewise, for horizontally-polarized incident beams (H-H: red triangles), a

horizontally-polarized SFG signal is measured. These observations are incompatible with

both experimental observations and theoretical predictions of resonant quadrupolar cou-

pling origin for the SFG signal [69, 78]. As a quadrupolar SFG signal is only expected

when the incident laser beams are in a noncollinear phase matching arrangement, with

orthogonal polarization components.

When incoming crossed polarization states are used, the polarization state of the SFG

signal is different. For a vertically-polarized I1 and a horizontally-polarized I2, the SFG

signal is polarized vertically (see Fig. 3.3). For a vertically-polarized I1 and a 45◦ polarized

I2, the SFG is linearly polarized at ∼120◦. Additionally, it is observed that some SFG signal

can be generated using orthogonally polarized incident beams. However, the intensity of

the mixed output signal in the latter case is significantly lower than that of the collinearly

polarized beams configuration. In summary, our results demonstrate that the polarization

of the non-resonant SFG signal depends on the polarizations of both incident beams and

suggest that electric quadrupole effects are not responsible for our observations. These

observations also differ from Miyazaki et al. [73, 79], which showcase no dependence of

the intensity of the SHG signal on the incident beam polarization angle for non-resonant

SHG.

In principle, four-wave mixing processes are allowed in bulk centrosymmetric media.

In the present case, the closest 4-wave mixing process would be DC electric field-induced

sum-frequency generation (E-FISFG) [66]. E-FISFG would be therefore a third-order, χ(3)

process while three-wave mixing is described as a χ(2), second-order process. To distin-

37



Figure 3.3: Polarization state of the TWM signal at 355 nm in room air. A sinusoidal wave is fitted
for each combination.

guish between χ(2) and χ(3) processes, we conducted additional experiments applying a

DC external electric field to the probed volume. Although unintended, such measurements

would also readily provide a test of the suitability of the observed SFG process to diag-

nostic environments featuring electric fields such as in transient electrical discharges (due

to charge separation following ionization). In order to perform the DC electric field tests,

we replaced the test chamber with an electrode pair assembly. The electrode assembly was

made of a pair of rounded-edge parallel plate copper electrodes of 20 mm in length, in

the propagation direction of the incident laser beams. The electrodes were 10 mm wide

and separated by a gap distance of 2.75 mm. While the upper electrode was powered by a

DC high-voltage power supply (Spellman, SL10PN150) providing up to 10 kV, the bottom

electrode was grounded. In this configuration, the electrode pair was deployed to generate

”vertical” sub-breakdown DC electric fields in room air. Figure 3.4 shows PMT waveforms

recorded in room air with the incident laser beams propagating through the centerline of the

electrode gap. As previously observed, with no applied voltage to the upper electrode and

38



with the 1064 and 532 nm incident beams vertically polarized, a strong signal was readily

measured by the PMT. Remarkably, the latter PMT signal remained unchanged when a 5.5

kV bias voltage (corresponding approximately to a 20 kV/cm electric field at the center-

line) was applied to the upper electrode. The very same result was obtained when rotating

the polarizations of both of the incident beams by 90°, such that horizontally-polarized

532 and 1064 beams were temporally and spatially overlapped at the middle gap of the

metal electrodes. This result indeed suggests that the SFG process is rather a χ(2) process.

Furthermore, because of the observed invariance to applied electric fields, we anticipate

three-wave mixing to be also suitable for gas-phase number density measurements in plas-

mas.

Figure 3.4: SFG signal with and without externally applied E-Field. The difference is shown in the
subplot. The 1064 and 532 incident beams are both vertically polarized.

Noteworthy, past [80, 81, 82] and recent [83, 84] conflicting observations of second-

order SFG from the bulk of centrosymmetric solid materials have led to a recrudescence

of interest in this topic. However, most of these recent activities interpret the second-

order SFG signal as originating from multipolar interactions, which we ruled out from our

39



polarization-sensitive measurements. In the gas phase, Miyazaki et al. [73, 79] ascribed

their observations of non-resonant SHG in Na vapor to laser-induced multiphoton ioniza-

tion resulting in charge separation and therefore static electric fields generation on the order

of 50 V/cm for 1012 W/cm2 28 ps laser pulses. Since from our measurements, no change

in the measured SFG signal was observed for DC electric fields up to 20 kV/cm in room

air, the static electric field hypothesis from Miyazaki et al. is not consistent with the results

of the sub-breakdown DC electric field tests we conducted. However, Freeman et al. [75]

suggested the observed effects could be qualitatively interpreted as a result of the alter-

ation of isotropic properties of the probed gases following a spatial variation of the ground

state densities of the probed atoms/molecules along the laser beam path. With the latter

spatial density variations due to the combined effects of spatial variation of the laser beam

intensity and multiphoton absorption. While we did not comprehensively test the hypothe-

sis from Freeman et al., our experimental observations make the case for the suitability of

leveraging non-resonant picosecond three-wave mixing for number density measurements

in both flows and plasma environments. Absolute species number density measurements at

the picosecond timescale is a much-needed capability for the validation of plasma chemical

kinetic models involving transient phenomena and short-lived species [85, 86, 87].

3.6 Conclusion

In this Letter, we have presented experimental observations of non-resonant second-

order optical SFG in various atomic and molecular gases. This nonlinear effect was pre-

viously believed to be forbidden in the gas phase. While our polarization-sensitive mea-

surements ruled out electric quadrupole and charge-separation mechanisms, the anisotropy

mechanism from Freeman et al. although consistent with our observations, requires a more

in-depth validation. Across all investigated gases, a quadratic relationship between the

40



SFG signal intensity and the ambient gas pressure is observed. Moreover, the signal is in-

dependent of any externally applied E-Field, indicating its potential for gas number density

measurements on the ps timescale for multiple cases including in low and high-pressure

reactive flow environments and plasmas.

3.7 Supplementary Information

3.7.1 Photomultiplier signal traces

Figure 3.5 displays sample photomultiplier (PMT) waveforms recorded at different

pressures with the test chamber filled with high-purity N2 gas. Each trace was averaged

over 1000 laser pulses. As evidenced by Figure 3.5, the PMT signal increases with the

increase of the N2 pressure between 400 and 1000 Torr. The relationship between gas

pressure and measured SFG signal discussed in the main manuscript was obtained by inte-

grating PMT waveforms from Figure 3.5 from 0 ns to 28 ns.

Figure 3.5: Sample PMT waveforms at different pressures in N2.

41



3.7.2 Third-harmonic generation signal

Because we are mixing the fundamentals of a Nd:YAG laser with its second harmonic,

we have the peculiarity of producing third-harmonic generation at the very same wave-

length as that of the SFG signal. SFG and THG a priori cannot be disentangled and must

be therefore characterized accurately. We will show that THG within our experimental con-

ditions is about many orders of magnitude weaker than SFG, and is therefore negligible.

THG signal measurements were taken with the 532 nm beam path blocked. Similarly to the

SFG PMT traces, the THG signal was taken in the sealed chamber filled with N2. Figure

3.6 shows the THG signal as a function of the incident 1064 nm laser beam pulse energy at

three different gas pressures (or gas densities). The THG signal, as a third-order nonlinear

process (a four-wave mixing process) is expected to follow the relation [88, 89]:

ITHG ∝ |χ(3)|2I31064;χ
(3) ∝ n, (3.2)

where χ(3) is the third-order nonlinear susceptibility, I1064 is the intensity of the incident

1064 nm laser beam, and n is the gas number density. Equation (3.2) appears to be con-

sistent with Figure 3.6 since for all three investigated pressures, a cubic dependence of the

THG signal was found as a function of the incident laser intensity. It is important to point

out that the greater THG signal at 201 Torr (black symbols) compared to that at 390 Torr

(red symbols) is due to the fact these measurements were conducted under a different phase-

matching configuration employing a 1-m focal length instead of the 15-cm one reported in

the main manuscript. More details explaining such behavior can be found in LaCombe et

al. [90]. Nonetheless, a cubic dependency was equally observed for the phase-matching

geometry discussed in the main manuscript.

Figure 3.7 depicts the polarization of the THG signal for a vertically-polarized incident

1064 nm beam. As expected it is evidenced that the polarization of the THG signal follows

42



Figure 3.6: THG in room air polarization as a function of pressure

that of the 1064 nm beam. Furthermore, scaling the THG signal intensity consistently with

the SFG signal intensity presented in Figure 3 of the main manuscript yields three and four

orders of magnitude differences with the orthogonally-polarized and collinearly-polarized

cases, respectively. This demonstrates that THG effects within the collected SFG signal

can be essentially ignored.

For the purpose of comparison between THG and SFG signal polarizations, Figure 3.8

shows a polar plot depicting the polarization of the 355 nm SFG signal for V1064-V532 as

well as H1064-H532 combinations.

3.7.3 Discussion on SFG signal and polarization

Black squares in Figure 3.9 (a) depicts the measured polarization angle of the SFG

signal at 355 nm (with respect to that of the 1064 nm beam) as a function of the relative

polarization angle of the incident 532 nm beam with respect to that of the 1064 nm beam,

which is taken as the reference state of polarization. The polarization angle of the 1064 nm

43



Figure 3.7: Measured polarization state of THG in room air generated from a vertically-polarized
1064 nm input beam.

Figure 3.8: Polar plot representing the polarization of the 355 nm signal for V1064-V532 and for
H1064-H532 combinations.

beam is therefore fixed at 0◦ for that figure. For these measurements, the pulse energies of

the incident 1064 and 532 nm beams were made equal, with a common value of 0.12 mJ.

Data was collected by rotating the polarization of the incident 532 nm beam by steps of 10◦

44



from 0◦ to 90◦. The measurement points are overlapped with the results of a phenomeno-

logical model aiming at reproducing our experimental results. The model is described in

great detail as follows.

Using a planar Cartesian coordinate system with unity vectors
−→
i and

−→
j , for the laboratory

reference frame x- and y-axis, respectively, we can write the unity polarization vectors for

the 1064, 532, and 355 coherent radiations as:

−→e 1064 = cos(θ1064)
−→
i + sin(θ1064)

−→
j (3.3)

−→e 532 = cos(θ532)
−→
i + sin(θ532)

−→
j (3.4)

−→e 355 = cos(θ355)
−→
i + sin(θ355)

−→
j (3.5)

where θ1064, θ532, and θ355 are the polarization angles of the 1064, 532, surprisingly, and 355

beams, respectively. Based on our experimental observations, θ355 appears to depend on

both of the incident beam’s polarizations and intensities. We therefore have the following

set of equations:

−→e 355 ∝ (−→ϵ near−IR +−→ϵ green) ∝ (α1
−→e 1064 + α2

−→e green) (3.6)

where α1 = α1(I1064) and α2 = α2(I532) are adjustable parameters dependent on inten-

sities I1064 and I532 of the 1064 and 532 nm beams, respectively. −→ϵ near−IR = α1
−→e 1064

represents the contribution of of the 1064 beam to the polarization of the SFG signal at 355

whereas −→ϵ green = α2
−→e green represents the contribution of the 532 nm beam. In order to

match our experimental observations regarding the polarization of the SFG signal (for in-

stance, V1064-H532 yields a vertically-polarized SFG signal whereas H1064-V532 yields

45



a horizontally-polarized signal at 355 nm), we need −→e green to be written as (see Figure 3.9

(b)):

−→e green = cos(2θ532)
−→
i + sin(2θ532)

−→
j (3.7)

Figure 3.9: (a) Experimental and modeling results for 355 nm polarization as a function of 532 nm
polarization. All polarization angles are taken with respect to the polarization of the 1064 nm
beam. The fitting parameter b varies in an agreement area from 1.6 to 2, best fitting result is

obtained for b = 1.74. (b) Schematic illustration of our SFG process hypothesis.

This summarizes Equation (3.6) into:

−→e 355 ∝ (α1 cos(θ1064) + α2 cos(2θ532))
−→
i + (α1 sin(θ1064) + α2 sin(2θ532))

−→
j (3.8)

Now if we refer to all polarization angles with respect to θ1064, Equation (3.8) yields:

−→
e′ 355 ∝ (α1 + α2 cos(2θ

′
532))

−→
i′ + (α2 sin(2θ

′
532))

−→
j′ (3.9)

where θ′532 is now the polarization angle of the 532 nm beam with respect to the polarization

of the 1064 nm beam.
−→
i′ and

−→
j′ are the unity vectors characterizing this new Cartesian

coordinate system obtained from a θ1064 rotation of the previous one. In that reference

frame, we can hence deduce:

46



θ′355 = arctan

(
α2 sin(2θ

′
532)

α1 + α2 cos(2θ′532)

)
(3.10)

Equation (3.10) can be further simplified by introducing the parameter b:

b =
α1

α2

=
α1(I1064)

α2(I532)
(3.11)

Equation (3.11) in Equation (3.10) yields:

θ′355 = arctan

(
sin(2θ′532)

b+ cos(2θ′532)

)
(3.12)

b hence becomes the unique adjustable parameter of our model. Performing a Matlab-based

least-squares nonlinear fitting (see lsqcurvefit function of Matlab) of our experimental data

using Equation (3.12) yields an optimum value of b = 1.74 for a R2 value of 0.98. In-

terestingly, reasonable fits can also be obtained with b values in the range [1.6; 2.0] with

corresponding R2 values of 0.95 and 0.91 for b = 1.6 and b = 2.0, respectively. Fits for

these two values of the parameter b were added in Figure 3.9 (a) as red lines delimiting

an ”agreement” area. Remarkably, although very simplistic, the present phenomenological

model is able to reproduce our experimental measurements with a surprisingly excellent

agreement.

47



Chapter 4

Conclusion

In this thesis, two advanced optical diagnostics for gas-phase plasmas were presented:

an improved method for electric field measurement using nonlinear optical second har-

monic generation, and a novel approach for gas density measurement utilizing sum fre-

quency generation.

Chapter 2 introduced the Ultrasensitive Electric Field Induced Second Harmonic Gen-

eration (E-FISH) technique, which innovatively combines existing diagnostic methods to

achieve sub-1 V/cm sensitivity, an enhancement of two to three orders of magnitude over

conventional approaches. Notably, this method demonstrated, for the first time, the capa-

bility to directly resolve the polarity of external electric fields using E-FISH at the picosec-

ond timescale. The underlying physics enabling polarity sensitivity was explored in detail,

and key parameters were identified that could lead to further sensitivity improvements and

clearer signal differentiation between positive and negative fields. These developments set

the stage for advancements in signal processing at high electric fields and may facilitate the

transition of E-FISH from high-pressure to low-pressure plasma environments, as well as

potential single-shot measurements.

Chapter 3 detailed the application of Sum Frequency Generation (SFG) for gas density

48



measurements. Despite being a nonlinear optical process traditionally considered forbid-

den in the gas phase, SFG was shown to be viable under specific experimental condi-

tions. Crucially, the SFG signal was found to be independent of external electric fields, en-

abling simultaneous and decoupled measurements of gas density and electric field strength.

Power and polarization dependence tests confirmed the observed signal was of a true sum-

frequency nature, distinguishing it from other nonlinear effects. While further investigation

is needed to fully elucidate the mechanisms permitting SFG in gases, the current results

validate its practical diagnostic potential.

Both the sub-1 V/cm E-FISH and SFG diagnostics were developed using a shared laser

system with common fundamental and harmonic wavelengths, ensuring strong compati-

bility. This paves the way for an integrated diagnostic setup capable of performing si-

multaneous, co-located measurements of electric field and gas density. With appropriate

modifications to the signal collection paths, both SFG and E-FISH signals can be isolated

from each other and residual fundamental light, while probing the same spatial volume.

Such a system would enable direct, non-perturbative, high-resolution measurements of the

local reduced electric field throughout a plasma, offering critical insights into the reaction

kinetics of low-temperature plasmas.

To this end, both of the diagnostic techniques presented have only been implemented

under idealized conditions, featuring sub-breakdown electric fields, spatially uniform num-

ber density of species, and simple mixtures of species not involving ionized nor electron-

ically/vibrationally excited species. Although our approaches enabled direct characteriza-

tion of the non-linear effects driving both processes, it significantly simplified the com-

plexity that exists when a plasma is generated. Due to ionization, attachment, and recombi-

nation effects in actual transient discharges, the electric field profile is expected to change

dynamically throughout the discharge. The generated plasmas typically exhibit spatial gra-

dients of species number density. These effects can potentially mitigate the maximum

49



increase in the signal generated through homodyne E-FISH because phase-matching con-

ditions cannot be fully optimized throughout the entire discharge (for example, showing

changes in sizes and species concentrations). A specific point of interest lies in accounting

for the nonlinear susceptibilities of excited species and ions. Given that the SFG process

demonstrated in Chapter 3 shows a strong dependence on the species’ nonlinear susceptibil-

ities, until the underlying mechanisms that enable this interaction are fully understood, the

effects of the spatiotemporally varying species concentrations that form in plasmas cannot

be necessarily predicted.

Ultimately, these diagnostics still represent a significant advancement in demonstrating

the still underutilized utility of existing diagnostic capabilities and demonstrating previ-

ously unseen nonlinear phenomena. Together, these diagnostics open new avenues for val-

idating plasma kinetic models and deepening our understanding of electron impact-driven

processes in transient plasmas.

50



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