Current address: ]Department of Physics, Hamilton College, Clinton, NY 13323,
USA

# The loop-zag resonator: A loop-gap resonator design for improved sensitivity
in electron-spin resonance experiments

Brendan C. Sheehan  Department of Physics and Astronomy, Amherst College,
Amherst, MA 01002, USA  Department of Physics, University of Massachusetts
Amherst, Amherst, MA 01003, USA  Guanchu Chen  Department of Physics and
Astronomy, Amherst College, Amherst, MA 01002, USA  Department of Physics,
University of Massachusetts Amherst, Amherst, MA 01003, USA  Sai Chauhan
Department of Physics and Astronomy, Amherst College, Amherst, MA 01002, USA
Rilla McKeegan  Department of Physics and Astronomy, Amherst College, Amherst,
MA 01002, USA  Francisca Abdo Arias  Department of Physics and Astronomy,
Amherst College, Amherst, MA 01002, USA  William Henshon  Department of
Physics and Astronomy, Amherst College, Amherst, MA 01002, USA  Charles A.
Collett  [  Department of Physics and Astronomy, Amherst College, Amherst, MA
01002, USA  Jonathan R. Friedman  Department of Physics and Astronomy, Amherst
College, Amherst, MA 01002, USA  Department of Physics, University of
Massachusetts Amherst, Amherst, MA 01003, USA

###### Abstract

We present a novel design of loop-gap resonator, the loop-zag resonator, for
sub-X-band electron-spin resonance spectroscopy. The loop-zag design can
achieve improved coupling to small-sample spin systems through the improvement
of sample filling factor and RF $B_{1}$ field. By introducing “zags” to the
resonator’s gap path, the capacitance is increased, accommodating a smaller
loop size and thereby a larger filling factor to maintain the requisite
resonant frequency. We present experimental spectra on five different
resonators, each with approximately the same resonant frequency of $\sim 2.9$
GHz, showing that an increase in the number of zags and reduction in loop size
gives rise to higher sensitivity. Finite-element simulations of these
resonators provide estimates of the improved filling factors obtained through
the addition of zags. The frequency range over which this loop-zag design is
practical enables a breadth of future applications in microwave engineering,
including ESR and ESR-like quantum information microwave techniques.

##  I Introduction

Electron spin resonance (ESR) spectroscopy has long served as a valuable tool
across many fields of science and industry; today, several commercial ESR
spectrometer models are available on the market. Additionally, many research
groups across the world take advantage of nontraditional or home-built ESR
hardware. In every ESR spectrometer a method of coupling microwave radiation
to the sample is necessary; many spectrometers, including commercial models,
use resonant microwave cavities Mett  _et al._ (2001); Anderson  _et al._
(2002). Other spectrometers take advantage of coplanar waveguide (CPW)
resonators Malissa  _et al._ (2013), microstrip resonators Johansson  _et al._
(1974); Cebulka and Del Barco (2019), or loop-gap resonators (LGRs) Hardy and
Whitehead (1981); Froncisz and Hyde (1982); Wood  _et al._ (1984).

Commercial spectrometers are typically built to operate in a narrow frequency
range, often in X-band. At frequencies below X-band, however, cavity
resonators often become impractical because increasing radiation wavelength
requires cavity sizes that are impractically large. CPW and LGR designs offer
the possibility of usage at bands below X-band. CPW designs allow frequency
tunability and the use of higher harmonics to work with a wide frequency
range, but they often demonstrate poor RF field homogeneity across the sample
space Joshi  _et al._ (2016). Specialized designs can be used to concentrate
the rf magnetic field and enable high filling factor, but do not provide a
particularly homogeneous field over the volume of the sample space Abhyankar
_et al._ (2020); Choi and Englund (2023); Ranjan  _et al._ (2020). LGRs, on
the other hand, provide a high quality factor ($Q\sim 2000$) and good RF field
homogeneity, high filling factors, frequency tunability via dielectric
insertion into the gap of the resonator, and flexibility of geometry Twig  _et
al._ (2010); Blank  _et al._ (2017); Rinard and Eaton (2005).

The geometry of the LGR is described by the name: each LGR consists of a loop
and a gap. LGRs can be described by an equivalent RLC circuit; the loop
functions as the inductor and the gap functions as the capacitor. A sample
placed within the loop experiences the resonator’s RF magnetic field $B_{1}$,
allowing the sample’s spectral properties to be probed. With careful
adjustment of the position of an antenna, critical coupling with a $Q\sim
2000$ is typically achieved. Because the inductance $L$ and capacitance $C$
depend on the geometry of the loop and gap, respectively, the resonant
frequency $\omega=(LC)^{-1/2}$ can be tuned by adjusting the thickness of the
resonator, the width and length of the gap, and/or the radius of the loop.
Furthermore, insertion of a dielectric material into the gap provides an
additional method of tuning the capacitance and therefore the resonant
frequency.

The filling factor $\eta$ can be written as

$\eta=\frac{\int_{\text{sample}}|B_{1}|^{2}dV}{\int_{\text{mode}}|B_{1}|^{2}dV};$
(1)

this describes the ratio of energy felt by the sample to the total energy in
the resonant mode, typically the volume of a shielded space containing the
resonator. The continuous wave ESR signal size $\nu$ is related to the filling
factor by $\nu=\chi Q_{L}\eta P_{0}^{1/2}$, where $\chi$ is the RF
susceptibility of the sample, $Q_{L}$ is the loaded $Q$, and $P_{0}$ is the
incident power Feher (1957); Hyde  _et al._ (1989). The sample response in the
nonsaturating regime therefore increases linearly with $\eta$.

One way to increase $\eta$ is by maximizing the amount of sample in the loop,
but there are many cases in which obtaining larger samples is impractical or
impossible. In such cases, an improved resonator design that increases filling
factor by more effectively focusing the $B_{1}$ field onto the sample can be
used. One method of achieving such an effect is to reduce the loop radius to
improve the filling factor; this results in a smaller resonator inductance
that may shift the resonant frequency to an undesirably high range. Thus,
there are contexts in which a traditional LGR cannot properly provide the
desired resonant frequency while furnishing a high filling factor and a strong
RF magnetic field. In this paper we introduce an LGR design that addresses
this conundrum: the loop-zag resonator (LZR) Friedman (2021). The LZR design
takes advantage of similar physics as the LGR; however, the gap can be
continuously elongated through the use of “zags” in the gap path: the gap can
wind back and forth across the plane of the resonator, producing in an
interdigitated capacitor. The resulting increased capacitance allows one to
maintain the desired resonant frequency while working with a small loop
radius. The smaller loop yields both a stronger RF magnetic field for a given
input RF power, and an increased filling factor.

Figure 1: Loop-gap resonator (LGR) and four-zag loop-zag resonator (LZR)
designs are shown in panels (a) and (b), respectively. The colormap in each
figure shows the strength of the RF ($B_{1}$) field in each resonator, for an
RF input power of 1 mW (0.0 dBm). Note each colormap has a different scale;
the LZR shows a higher maximum field in the loop than the LGR. The black
framed outline in the LGR gap in (a) shows placement of dielectric material
(rutile). Insets show focus on loops of each resonator; shaded black lines
cutting diagonally across each loop in the inset provide references for which
RF field strength is shown in (c): RF field as a function of distance from the
left edge of the loop. Traces shown in (c) go through the center of the loop.
(d) and (e) show comparisons of simulated and measured response of the LGR and
four-zag LZR as a function of detuning, respectively. Measured $Q$ and
resonant frequencies are given in blue (d) and red (e), while simulated $Q$
and frequencies are given in black for both panels. All simulations were
produced using Ansys Electronics Desktop Suite, release 2022 R1.

The LZR design could permit strong coupling between the a small number of
spins and the resonator, potentially opening up a number of novel applications
in quantum information, such as using single spins as qubits or using spin
systems as a form of quantum memory Grezes  _et al._ (2015); Wu  _et al._
(2010). “Strong coupling” of the radiation to a collection of $N$ spins
requires the vacuum Rabi frequency
$\Omega^{N}_{VR}=\sqrt{N}g\mu_{B}B^{(1)}_{1}/\hbar$ to be greater than the
relaxation times of the resonator and the spins Kubo  _et al._ (2010);
Schuster  _et al._ (2010); Chiorescu  _et al._ (2010). Here $B^{(1)}_{1}$ is
the RF field of a single photon, $g$ is the Landé $g$-factor, and $\mu_{B}$ is
the Bohr magneton. In practice, this requirement results in
$\Omega^{N}_{VR}/2\pi>\sim 1-10$ MHz when working with solid-state spins.
Unlike cavity and CPW resonator designs, the LGR/LZR are lumped-element
devices (to a first approximation) and therefore the field distribution is not
constrained by the Rayleigh limit, allowing for miniaturization of the loop
and concentration of the field within the loop volume. If the RF field does
not substantially leak outside of the loop, $B^{(1)}_{1}$ scales with loop
volume $V$ as $V^{-1/2}$ while the number of spins coupled to the loop scales
as $N\propto V$, so by scaling the loop down to the nanoscale one can in
principle couple to a small number of spins or a single spin while remaining
in the strong coupling regime. The LZR design thus enables an important step
toward the single-spin regime through minimization of the loop size.

##  II Design

For this study, we designed six resonators using the Ansys Electronics Desktop
Suite and fabricated five of the designs out of oxygen-free high thermal
conductivity (OFHC) copper using a CNC milling machine – see Table 1. One
resonator, the five-zag LZR, had a gap width/loop radius that was too small to
be machined with the available equipment. Two of our resonators are shown in
Fig. 1. The standard LGR design, shown in Fig. 1(a), features a long gap with
a gap width of 0.02 in, a gap length of 0.42 in, and a loop radius of 0.03 in.
The thickness of the resonator is 0.02 in. Furthermore, the use of a
dielectric brings the simulated resonant frequency in Fig. 1(a) from 4.38 GHz
to 2.89 GHz. The dielectric is represented by a black-edged block with
diagonal hashes filling the length of the gap. The four-zag LZR (Fig. 1(b))
has similar features; the gap width is 0.021 in. and resonator thickness
remains 0.02 in. The loop radius has been reduced to 0.0105 in, making the
loop a semicircle that terminates the gap. The increased gap length of the LZR
allows for a smaller loop while yielding a resonant frequency of 2.92 GHz,
similar to the LGR design with dielectric.

The insets in Figs. 1(a) and 1(b) zoom in on each resonator’s loop to show the
simulated $B_{1}$ field strength distribution in the loop. To perform field
simulations, input power was set to 0.0 dBm. While quality factor was not
fixed, it was relatively consistent ($\sim 2000$) among the five designs
(Table 1).

Each inset shows a black line across the diameter of the loop at the midpoint
of the resonator thickness. Fig. 1(c) shows the $B_{1}$ field strength
calculated along each of these lines. The LGR’s large loop size causes a
decrease in $B_{1}$ field strength at the center of the loop (shown in blue);
the sample can therefore either be placed at the center of the loop (to take
advantage of the more homogeneous $B_{1}$ field) or near the edge (to maximize
the field strength). The LZR, however, allows the sample to experience a
large, relatively homogeneous $B_{1}$ field throughout the loop (Fig. 1(c),
shown in red). The $B_{1}$ field strength in the center of the four-zag LZR
loop is $\sim 65\%$ larger than the field strength in the center of the LGR
loop. Averaging over the loop volume, we find the mean field strength is 26%
larger in the four-zag LZR compared to the LGR.

Design | Resonant Frequency | Quality Factor | Loop Radius | Gap Length | Gap Width | Mean Field in Loop
---|---|---|---|---|---|---
LGR | 2888 MHz | 1400 | 0.03 in | 0.42 in | 0.02 in | 1.061 G
One-zag LZR | 2887 MHz | 2200 | 0.01 in | 1.16 in | 0.02 in | 1.295 G
Two-zag LZR | 2956 MHz | 2300 | 0.0225 in | 1.25 in | 0.021 in | 0.914 G
Three-zag LZR | 2976 MHz | 2900 | 0.015 in | 1.49 in | 0.021 in | 1.136 G
Four-zag LZR | 2921 MHz | 3700 | 0.0105 in | 1.68 in | 0.021 in | 1.339 G
Five-zag LZR | 2959 MHz | 3100 | 0.0025 in | 1.165 in | 0.005 in | 3.972 G
Table 1: Parameter values for each resonator in simulations performed for
Figs. 1 and 3. The resonant frequency of each resonator could be tuned by
changing the loop radius, the gap length, and the gap width. The resonator
thickness was 0.02" for all resonators. The input power was set identically in
simulation for each, to 0.0 dBm. The antenna position was adjusted to achieve
a coupling of -20 dB, undercoupled, for each design; the quality factor was
determined when this condition was met. The LGR’s quality factor is diminished
due to the inclusion of the rutile dielectric with loss tangent
$\delta=0.001$.

To calculate both the $B_{1}$ field strength and the filling factor, several
parameters need to be held constant for the sake of comparison between
resonator designs. The filling factor is referenced to the total magnetic
field energy in the system
$E_{\text{B}}=\frac{1}{2\mu_{0}}\int_{\text{mode}}B_{1}^{2}dV=\frac{P_{0}Q}{2\omega}$,
where $P_{0}$ is the incident power and $\omega$ is the resonant frequency.
Parameters for each resonator simulation are shown in Table 1.

Figures 1(d) and (e) show a comparison of the simulated and measured
resonances. Coupling was set to $-20$ dB in our experiments since working with
a slightly undercoupled resonance appears to improve sensitivity. The same
undercoupling condition was implemented in the simulations. Each panel notes
the $Q$ value measured for each resonator. We measured the LGR $Q$ value to be
approximately half that of the LZR $Q$ due to the use of the dielectric
material (rutile); in simulation we determined the dielectric loss tangent for
the rutile to be $\delta=.001$ via comparison with the measured resonance.

The introduction of an interdigitated capacitor via the addition of zags
increases the capacitance by increasing the length of the gap. In addition to
a strong RF field in the loop, Fig. 1(b) shows some RF field extending into
parts of the gap, indicating some parasitic inductance of the capacitor. To
quantify the enhanced sensitivity of the LZR while accounting for the
extraneous RF field outside the loop, we calculate the filling factor $\eta$
as per Eq. 1, where the volume of the mode is considered to be the volume of
the entire shielded space in which the resonator sits, and the volume of the
sample is taken to be a small spherical volume of radius 0.0025" near the
center of the loop that approximates the actual volume of the sample used in
experiments. The results of these calculations are shown in Fig. 3, overlaid
with the results of sample measurement discussed below.

##  III Results

To compare the sensitivity of each resonator, we measured an ESR standard
sample, 2,2-Diphenyl-1-picrylhydrazyl (DPPH). DPPH (C18H12N5O6) is an organic
radical with a $g=2.0036$ ESR signal coming from an unpaired electron in each
molecule in the ensemble. It is commercially available as a powder sample and
has been used as an ESR standard for decades Slifkin and Suleiman (1975);
Yordanov and Christova (1994). Inhomogeneous broadening results in a full
width at half maximum (FWHM) typically measured between 1 and 5 Oe at ESR
frequencies below Q-band. These characteristics make it an good continuous
wave (cw) ESR standard, but its short relaxation times are such that pulsed
ESR techniques cannot be used to extract a Hahn echo response.

We employed a heterodyne cw technique with a homebuilt ESR spectrometer built
from commercially available microwave components and a Tabor SE5082
programmable arbitrary waveform generator (see Appendix for details). We
experimentally examined four LZR designs, each with a measured resonant
frequency between 2.89 GHz and 3.05 GHz. A typical measured quality factor for
all four LZR designs was $Q\sim 2500$ at 10 K. The LGR has a resonant
frequency of 4.38 GHz and $Q\sim 2500$ at 10 K. Similar to the simulations
performed for Fig. 1, we inserted a piece of rutile, with dielectric constant
of 86, into the gap of the LGR to bring its resonant frequency down to 2.95
GHz (very close to the simulated resonant frequency of 2.89 GHz). The large
dielectric loss tangent of rutile pushed the LGR $Q$ to $Q\sim 1100$ at 10 K.
The sample and resonator were placed in an OFHC copper shield at the bottom of
a custom-designed ESR probe Joshi  _et al._ (2020). To avoid power broadening
and saturation, the ESR cw power was reduced until the FWHM of the measured
signal peak no longer varied.

We used the same sample of DPPH to characterize each resonator. By encasing
the DPPH powder in a thin layer of vacuum grease, the sample could be
transferred between resonators with no loss of powder. We estimated the sample
to contain $\sim 7\times 10^{15}$ spins within the resonator’s loop Williams
(1967).

Figure 2: CW ESR measurement of DPPH at the resonant frequency of each
resonator at 10 K. For the LGR and the one-zag LZR, the cw power applied to
the sample was $-45$ dBm. For the two, three, and four-zag resonators, the cw
power applied was $-55$ dBm. Signal size increased as the number of zags
increased, with the largest signal from the four-zag resonator. The inset
shows a semilog scale of the same data.

Figure 2 shows spectra taken of the DPPH powder sample in each resonator at 10
K. To quantify the signal, we integrate the observed ESR peak with respect to
magnetic field. Integrated signal strength from DPPH in the four-zag LZR
(green) was 5.4 times larger than the integrated signal strength from the LGR
(blue) at comparable frequencies. The measured FWHM for DPPH from each
resonator ranged from was 1.63 Oe (the one-zag resonator, red) to 0.52 Oe (the
two-zag resonator, black), in agreement with previously established values
Chaudhuri and Mahendiran (2020). The inset of Fig. 2 shows a semilogarithmic
scale of the same data, making the background noise of the spectra more
visible.

Figure 3: Blue shows the integrated signal strength, integrated with respect
to magnetic field from Fig. 2, as a function of loop radius; red shows the
calculated filling factor. The red dashed curve is a guide to the eye. Boxes
show the length of the gap for each resonator design. The LGR and four-zag LZR
(shown in Figs. 1(a) and 1(b), respectively) have gap lengths of 0.42 in. and
1.68 in, respectively. For most resonators, the integrated signal size scales
with the filling factor, indicating that the minimization of the loop radius
allows for a better field concentration on the sample and therefore a stronger
sample response. All simulations were produced using Ansys Electronics Desktop
Suite, release 2022 R1.

Figure 3 shows the calculated filling factor (red) as a function of loop
radius for the six resonator designs. This shows a clear increase in filling
factor as the loop radius is reduced. The largest calculated filling factor
was found for the five-zag LZR, which has the smallest loop radius. While that
design was not tested, the signal from the five experimental resonators (blue)
from DPPH shows a similar trend to the calculations. Introduction of zags in
the resonator gap affords a smaller loop radius and therefore improved signal
size.

##  IV Discussion

In quantum information, one generally needs to control and measure the quantum
state of individual qubits. Many qubit experiments, such as those using
superconducting qubits, are done at microwave frequencies of a few GHz, often
employing 1D (stripline, CPW) or 3D (cavity) geometries. Such resonators work
well with superconducting qubits, which have large transition dipole moments.
Potential nanoscale qubits such as phosphorus Morton  _et al._ (2008) or
bismuth Morton and Bertet (2018) impurities in silicon, or molecular
nanomagnets Shiddiq  _et al._ (2016), however, have smaller moments and ESR
signals are found in bulk but rarely on the single-spin level. To see single-
spin ESR requires specialized techniques like STM Rugar  _et al._ (2004), ODMR
Grotz  _et al._ (2011), or EDMR Harneit  _et al._ (2007). The LZR approach
potentially allows for miniaturization to the scale of single qubits, both
superconducting and even individual spin-based systems, and would afford
addressability of single qubits. In the strong coupling regime, photon states
could be “parked” in the qubits and later retrieved. Many experiments on
superconducting qubits are done near zero field as a field would disrupt
superconductivity. For such applications, an LZR could be fabricated out of
superconducting metals such as niobium or aluminum instead of the Cu used here
for ESR. A superconducting LZR would have a significantly higher $Q$ than one
made of copper.

For our five-zag design, we estimate a single-photon magnetic field of
$B^{(1)}_{1}=43$ pT. This is comparable to or larger than the field strengths
found in mode-engineered cavity designs Choi and Englund (2023). This
demonstrates the primary utility of the LZR design: it affords high RF field
strengths, which translates into a large filling factor in ESR experiments,
has large $Q$, and, importantly maintains a good field homogeneity over the
working volume of the resonator. By increasing the number of zags and
incorporating the use of a low-loss dielectric, the resonator capacitance can
be enhanced further, potentially allowing the loop size to be reduced
substantially. This, in turn, would result in larger RF magnetic fields in the
loop, a greater filling factor, and the ability to measure, manipulate and
control systems containing small numbers of spins.

In conclusion, we have presented a novel loop-gap resonator design for the
purpose of coupling to smaller samples while preserving the desired resonant
frequency. This loop-zag resonator design takes advantage of increased
capacitance to allow for a smaller loop with a more concentrated $B_{1}$ field
coupling to the sample, resulting in a larger filling factor. Characterization
of an ESR standard sample shows increased signal with smaller loop radius and
larger number of zags. This LZR design represents an important step toward the
ability to couple to and address systems with a limited number of spins.

###### Acknowledgements.

We thank G. Joshi and D. S. Hall for useful conversations and advice, and J.
Kubasek for assistance with the machining of the resonators. Support for this
work was provided by the National Science Foundation under grant nos.
DMR-1708692 and DMR-2207624.

*

##  Appendix A Appendix

The home-built spectrometer designed to perform the experiments presented in
this paper was developed from commercially available microwave components and
a Tabor Electronics SE5082 arbitrary waveform generator (AWG). A block diagram
depicting the circuit is shown in Fig. 4. Channel 1 of the AWG produces the cw
ESR signal, which is sent to the sample via a coaxial line fed into ESR probe
Joshi  _et al._ (2020). The custom-built probe is built for direct-detection
reflection spectroscopy; the resonator coupling can be tuned in situ. For all
experiments in this paper, the microwave radiation is slightly undercoupled
(to -20 dB loss below coaxial background at resonance). The probe is
integrated into a Quantum Design Physical Property Measurement System (PPMS)
cryostat with a base temperature of 1.8 K. A circulator sends the return
signal to a microwave mixer, which mixes the signal with a reference from
channel 2 of the AWG. The reference is offset from the ESR frequency by 1 MHz.
The 1 MHz down-converted output of the mixer is measured with a Tektronix
mixed-domain oscilloscope and read by software for analysis. A virtual lock-in
technique is implemented with software, whereby the signal is demodulated,
filtered with a low-pass second-order Butterworth filter, and integrated.

Figure 4: Block diagram showing the spectrometer used for direct-detection
ESR. Channel 1 of the AWG produced the ESR working frequency while Channel 2
produced the reference modulated from the working frequency by 1 MHz. All
components in the circuit, aside from the resonators, are commercially
available.

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# Determination of Plasma Properties in Liquid Jets Through Time-Resolved
Experiments on Third Harmonic Reflection Dynamics

S. Hilal  Laboratory of Femtosecond and Femtotechnologies, ITMO University,
Kadetskaya line., St. Petersburg, 197101, Russia<EMAIL_ADDRESS>M.
Melnik  Laboratory of Femtosecond and Femtotechnologies, ITMO University,
Kadetskaya line., St. Petersburg, 197101, Russia  A. Ismagilov  Laboratory of
Femtosecond and Femtotechnologies, ITMO University, Kadetskaya line., St.
Petersburg, 197101, Russia  A. Tsypkin  Laboratory of Femtosecond and
Femtotechnologies, ITMO University, Kadetskaya line., St. Petersburg, 197101,
Russia  S. Kozlov  Laboratory of Femtosecond and Femtotechnologies, ITMO
University, Kadetskaya line., St. Petersburg, 197101, Russia

###### Abstract

The study of plasma in liquid jets represents a significant area of research
encompassing plasma science, dynamics, and properties. This paper presents
experimental studies on plasma formation processes in liquid jets of water,
ethanol, and isopropyl based on the dynamics of the third harmonic reflection
from the induced plasma surface. The characteristics of plasma formation and
its properties concerning the angle of incidence of pump radiation on the jet
are described. Through time-resolved experiments, and theoretical estimations
using Keldysh theory, plasma properties, including reflectivity, density, and
frequency for three different media: water, ethanol, and isopropyl are
evaluated. Among the studied liquid jets, the isopropyl one demonstrates the
highest values of the characteristics mentioned. These findings hold
significant potential for advancing our understanding of plasma-based
radiation sources e.g. terahertz generation.

††journal: opticajournal

##  1 Introduction

Jets of liquids and gases are of great practical importance for various
applications. Previously, they were used to create ultrashort laser systems
[1]. At the moment, they are widely used in X-ray [2] and terahertz (THz) [3,
4, 5] generation systems, as well as in laser-induced breakdown spectroscopy
(LIBS) systems [6, 7]. They are unique in the fact that, unlike stationary
systems (cells or films), the jets constantly renew the area of interaction.

The effects associated with a breakdown in liquids and the study of plasma
formation have been theoretically and experimentally analyzed in various works
[8, 9, 10]. Plasma formation in liquids occurs at intensities lower than for
gases, on the order of $10^{12}$ $W/cm^{2}$. Recently [4] it was investigated
how material parameters like linear absorption, molecular density and
ionization energy impact the optical-to-THz conversion efficiency during THz
generation via plasma formation in the medium. Furthermore, experiments
revealed how the double-pulse excitation technique improves the conversion
efficiency of THz generation up to 0.1% in the $\alpha$-pinene liquid jet
[11]. Given the fundamental role that plasma formation in liquid jets plays in
THz generation, further research in this area is necessary to enhance the
efficiency of THz generation.

Recently [5] the features of the behavior of plasma optical properties under
different conditions of the experiment for flat jets of several liquids have
been studied. An original technique for measuring plasma dynamics by means of
the method of third harmonic (TH) reflection from induced plasma was proposed,
which makes it possible to evaluate plasma properties. Nevertheless, in real
systems, it is necessary to conduct research and identify certain patterns of
plasma formation with respect to the nonlinear properties of the liquids used.
Knowledge about the dynamics of plasma formation in liquid jets is necessary
for a more detailed understanding of the generation of THz radiation [12, 11].

The first part of this work is aimed at studying the regularities of plasma
properties with a change in the angle of incidence on the jet with respect to
the normal. In the second part, we are implementing an experimental study on
the plasma formation in isopropyl, ethanol, and water liquid jets based on the
dynamics of TH reflection from the induced plasma. The results demonstrate the
patterns of plasma formation when using various liquids which can help to
choose medium for specific task. Theoretical analysis explains all the
obtained experimental regularities.

To study the plasma induced in liquid jets, an experimental setup (Fig. 1a)
was assembled. The method presented in this paper is a time-resolved approach
utilizing the double pump technique based on [5]. As a result, the focusing
system receives two femtosecond pulses at the input, shifted by a time delay
between them $\Delta\tau$. The essence of the method is that the first pulse,
forms a preionized medium, preparing it to reflect the tail of the same pulse
[5]. It creates the conditions for the second pulse, shifted by time delay
$\Delta\tau$, to interact with the ionized region, increasing plasma surface
reflectivity. The second pulse is a delayed copy of the first one so it
carries the same energy and acts as a probe. When focusing both pulses onto
the liquid jet’s surface, the plasma induces in the liquid jet and TH
generation occurs in the ionized air along the path of the laser pulses due to
the high intensity[5]. When the incident radiation frequency is close to the
plasma frequency, the plasma reflects the radiation. Therefore, employing the
pump-probe method allows us to observe the reflection not only of the first
harmonic but also of the TH and measure the reflection dynamics with an
accuracy of up to tens of fs [5]. Notably, the studied TH signals are observed
in two positions: one is in front of the liquid jet’s surface at an angle not
aligned with the pump pulses, and the second position is behind the liquid
jet, aligning with the path of the pump pulses. However, no TH radiation was
observed in the opposing direction relative to the detected TH signal in front
of the liquid jet’s surface. This confirms that the TH radiation is generated
in the ionized air along the path of the pump pulses and reflected from the
plasma formed in the liquid jet. This enables us to study the dynamics of TH
reflection from the plasma formed in the liquid jet, ensuring that the
reflected TH is uncorrelated with the properties of the liquid jet medium due
to its reflection primarily from the induced plasma rather than direct
interaction with the liquid jet. It is important to note that our study does
not encompass the TH radiation originating from the second position located
behind the liquid jet. This omission arises from the fact that the TH
radiation detected in this position is not a result of reflection from the
plasma formed within the liquid jet. Consequently, it does not contribute to
our understanding of the plasma dynamics under investigation.

Figure 1: a. Experimental scheme for studying the properties of a plasma
formed during filamentation in a liquid jet by measuring the dynamics of
plasma-induced reflection of the probe pulse radiation. The method of pump
sample spectroscopy is used, provided that the first pulse forms a plasma in a
liquid jet, and the second one, as a result of filamentation in front of the
jet, generates TH in addition to the main one, which is also used as probe
radiation. The time delay between pulses is controlled by the Michelson
circuit. The registration of the reflected signal from the induced plasma by
the first pulse is recorded using a spectrometer. b.The dependence of TH
reflection dynamics on the jet rotation angle.

The liquid jet is located on a slider that can move it along the $z$ axis. It
is known that when using a focal lens of $10$ $cm$, the waist length will be
about $2$ $mm$. A photograph of a filament in the air without a jet using the
same experimental conditions is shown in the inset in Fig. 1(a). The jet is
located at the central region of the filament ($z=0$). The radiation reflected
from the plasma in the liquid jet is measured using a spectrometer. Each
measurement is taken 40 times and averaged. The jet has a thickness of 100
$\mu m$ and is rotated for all measurements at a different angle concerning
the incidence of pump radiation. The jet is formed using a nozzle [13]. The
pump radiation is a Gaussian beam generated by a femtosecond laser system with
a central wavelength of 800 $nm$, a duration of 150 fs, FWHM waist diameter is
100 $\mu m$, and pulse energy of 300 $\mu J$ for each pulse in the double pump
pulses. In the case of two-pulse excitation of a liquid jet, the Michelson
interferometer scheme is used and the time delay between pulses varies from 0
to 10 $ps$. The dynamics of the reflection of the TH are measured as a
function of the incident pump radiation angle as shown in Fig. 1(b). These
measurements are performed using a liquid jet of water as the medium. The
maximum reflected TH dynamics are observed within a time delay range of 0-3
ps, and within an angular range of 50-60∘, centered around the Brewster angle
of water at approximately 53∘. Observations show that as the angle of
incidence deviates from this optimal range, whether decreasing or increasing,
the maxima exhibit a faster relaxation and minimal shift in position.

Having chosen the conditions for the appropriate energy and pulse duration as
outlined in [5], we compare the properties of plasma formed in various
liquids. The jet position is fixed at the center of the generated filament
($z=0$) as shown in Fig. 1(a), with a 55° rotation angle relative to the
normal, corresponding to the average Brewster’s angle for the studied liquids
(water, ethanol, and isopropyl). The choice of liquids was due to their very
different nonlinear characteristics. Fig. 2 shows the dynamics of the TH
reflection from the induced plasma as a function of time delay between the
double pump pulses at a fixed pulse energy of 300 $\mu$J and a fixed pulse
duration of 150 fs.

Figure 2: Experimental results of TH (265 nm) density reflected from the
induced plasma in water (black dots), ethanol (red dots), and isopropyl (blue
dots) as a function of the time delay between the double pump pulses. With the
exponential fitting curves of the experimental data describing the plasma
relaxation in water (black line), ethanol (red line), and isopropyl (blue
line), at a fixed energy of 300 $\mu$J and pulse duration of 150 fs.

Considering the interaction between the second pulse and the preionized region
caused by the first pulse, the plasma density, and plasma frequency reach
higher values for shorter time delays between the pump pulses. Consequently,
the plasma frequency initiates at its maximum, diminishing as the time delay
increases. When the TH radiation frequency $\omega_{TH}=7.06\times 10^{15}$ Hz
(256 nm) is much lower than the plasma frequency $\omega_{p}$, the plasma
becomes absorptive, and the wave’s energy transfers to the plasma, leading to
attenuation. In this case, the attenuation of TH signal caused by the
absorption, explains the low reflected TH density starting at zero time delay
until the maximum. As time delay increases, the plasma frequency decreases and
approaches values near $\omega_{TH}$, resulting in higher reflectivity. This
explains the increasing in reflected TH density in Fig. 2 until reaching a
maximum when $\omega_{p}\sim\omega_{TH}$. As the plasma frequency continues to
decrease beyond this point, the reflectivity of the plasma diminishes.
Eventually, when $\omega_{p}$ is smaller than $\omega_{TH}$, the plasma
exhibits a weak response to the TH radiation due to the significantly higher
energy associated with the wave compared to the characteristic energy of
plasma oscillations. Consequently, the reflected density decreases which is
seen in the figure. To gain more insight, the plasma relaxation can be
described by the measured TH reflection from the plasma as a function of the
time delay between the double pump pulses by exponentially fitting the
experimental curves in Fig. 2, passing through the peaks, using the following
exponential equation:

$\omega_{p}(\Delta\tau)=\alpha\cdot
e^{-\left(\Delta\tau-\tau_{0}\right)/\tau_{d}}$ (1)

The parameter $\tau_{d}$ is a fitting parameter that represents the
degradation time, which describes how quickly the plasma frequency is
decreasing and, thus how fast the TH reflection occurs. The discrepancy
between the experimental results for reflected TH density and the lines
describing the plasma’s relaxation behavior, for time delay values ranging
from zero to those corresponding to the maximum peaks of TH densities, is due
to the strong absorption of the plasma when $\omega_{p}>\omega_{TH}$. As a
result, with delays shorter than those corresponding to the maximum, we have
absorption.

The chosen case clearly traces reflection dynamics changes with different
liquids. Fitting the experimental curves with Eq. (1) reveals that plasma in
water has a maximum degradation time of $\tau_{d}\sim 1.8$ ps, while ethanol
and isopropyl show similar degradation times of approximately $\tau_{d}\sim
1.2$ ps. This difference can be attributed to their molecular structures and
water’s higher polarity. Ethanol and isopropyl’s lower polarity leads to
faster ionization processes, resulting in quicker changes in plasma frequency
values. It can be seen from Fig. 2 that the maximum reflected TH density is
observed for the liquid isopropyl at time delay $\Delta\tau$ $\sim$ 1.5 ps,
and for ethanol liquid jet at lower time delay $\Delta\tau$ $\sim$ 1.25 ps,
while the maximum reflection is observed for water liquid jet at the shortest
time delay among the other two liquids $\Delta\tau$ $\sim$ 0.8 ps. As the
maximum TH reflected density occurs when $\omega_{p}\sim\omega_{TH}$, the
critical plasma frequency at the peak TH reflection corresponds to the TH
frequency ($\omega_{p}\sim\omega_{TH}=7.06\times 10^{15}Hz$). By introducing
this value of plasma frequency as $\alpha$ in Eq. (1) and associating
degradation times $\tau_{d}$ with the peak position for each liquid, it
becomes possible to estimate the plasma frequency as a function of the time
delay $\Delta\tau$ between the double pump pulses as seen in Fig. 3.

In Fig. 3, the plasma frequency values, obtained from the measured TH
reflection dynamics of the plasma, are presented as a function of the time
delay between the double pump pulses (from $\Delta\tau=0$ to $10$ ps). We
estimated that the plasma frequency decreases rapidly in water liquid jets
(from $1.1\times 10^{16}$ Hz to $0.42\times 10^{14}$ Hz) and in ethanol liquid
jets (from $2.00\times 10^{16}$ Hz to $0.48\times 10^{13}$ Hz), while in
isopropyl liquid jets (from $2.46\times 10^{16}$ Hz to $0.56\times 10^{14}$
Hz). It is now feasible to estimate the variation in the maximum plasma
frequency values among the studied liquid jets at zero time delay, where the
plasma is at its highest frequency. The isopropyl liquid jet shows a maximum
plasma frequency 2.24 times higher than that of water, while ethanol
demonstrates a maximum plasma frequency 1.8 times greater than water. The
effects of plasma formation are known to be the main cause of generating THz
fields [11, 14]. Our findings align closely with the efficiency ratio of THz
generation in these liquids, where the THz generation efficiency of isopropyl
is twice that of water, while ethanol’s efficiency surpasses that of water by
1.25 times [14] confirming the significant role of plasma.

Figure 3: Plasma frequency decreasing as a function of the time delay between
the double pump pulses in the liquid jets of water (green line), ethanol (blue
line), and isopropyl (red line). The red points correspond to the TH frequency
at which the highest reflection occurs.

In order to interpret the experimental data, Keldysh theory is considered [15,
16, 17, 18], the theory of prediction of the multiphoton atomic ionization
regimes under a strong laser field. Given that the complete ionization model
is taken into consideration, with diffusion and recombination mechanisms
ignored since they are not noticeable on the sub-picosecond timescale, then
according to Keldysh theory, the induced plasma electron density is determined
by considering both tunneling and multiphoton ionization phenomena. This
density is calculated by solving the ionization rate equation
$\frac{\partial\rho}{\partial t}=W_{NP}$ where $W_{NP}$ (Eq.(37) in [15]):

$W_{NP}=\frac{2\omega}{9\pi}\cdot\left(\frac{m_{e}\omega}{\gamma_{1}\hbar}\right)^{3/2}\cdot
Q(\gamma,x)\\\ \cdot\exp\left(-\pi\langle
x+1\rangle\cdot\frac{K\left(\gamma_{1}\right)-E\left(\gamma_{1}\right)}{E\left(\gamma_{2}\right)}\right)$
(2)

Where $\rho$ is the plasma electron density, and the plasma frequency is given
as $\omega_{p}=\sqrt{\rho e^{2}/m_{e}\varepsilon_{0}}$, with Keldysh parameter
given by the formula:

$\gamma=\frac{\omega\sqrt{m_{e}\Delta}}{eF}$ (3)

Where $\gamma_{1}={\gamma}/{\sqrt{1+\gamma^{2}}}$ and
$\gamma_{2}={\gamma_{1}}/{\gamma}$, $\omega$ is the angular frequency of the
laser field, $e$ is the electron charge, $m_{e}$ is the electron mass,
$\varepsilon_{0}$ is the vacuum permittivity, $\Delta$ is the material
bandgap, $F$ is the laser electric field amplitude, $\hbar$ is the reduced
Planck constant, $K$ and $E$ are the complete elliptic integrals functions of
the first and second kind respectively, and $Q(\gamma,x)$ is a function
pertains to the structure of the spectrum, which is associated with the
discreteness of the absorbed photon count, given by the corrected formula
[17]:

$Q(\gamma,x)=\left(\sqrt{\frac{\pi}{2K\left(\gamma_{2}\right)}}\right)\sum_{n=0}^{\infty}\left[\exp\left(-\pi
n\cdot\frac{K\left(\gamma_{1}\right)-E\left(\gamma_{1}\right)}{E\left(\gamma_{2}\right)}\right)\right.\\\
\left.\cdot\Phi\left(\sqrt{\frac{\pi^{2}(\langle
x+1\rangle-x+n)}{2K\left(\gamma_{2}\right)E\left(\gamma_{2}\right)}}\right)\right]$
(4)

Where $n$ is the number of absorbed photons in the multiphoton absorption
effect and the function $\Phi$ can be defined by
$\Phi(z)=\int_{0}^{z}e^{y^{2}-z^{2}}dy$ and $x={\Delta^{*}}/{\hbar\omega}$.

The key medium parameters are the bandgap energy ($\Delta$) and the refractive
index. The medium under the laser field can be described using the effective
ionization potential formula:

$\Delta^{*}=\frac{2\Delta E(\gamma_{2})}{\pi\gamma_{1}}$ (5)

Given that the concept of bandgap energy pertains to solid-state physics, it
is more appropriate to prioritize the consideration of ionization energy
rather than bandgap energy when discussing liquid mediums [19]. The
approximate ionization energy values for Ethanol, Water, and Isopropyl are as
follows: 10.7 eV [20], 11.67 eV [21], and 10.49 eV [22], respectively. With
the provided pump parameters, it becomes feasible to estimate the plasma
density by obtaining $W_{NP}$ and integrating over the pump pulse duration
[23], thereby calculating the plasma frequency in each liquid jet. Keldysh
theory allows us to estimate the plasma parameters in a medium excited by a
single pulse of the double pump pulse technique. Following ionization by the
initial pulse after a time delay $\Delta\tau$, the determination of crucial
medium parameters such as ionization energy and refractive index becomes
unattainable. Consequently, the introduction of a second pulse after time
delay $\Delta\tau$ into the calculation is precluded. However, In order to
compare with the predicted maximum plasma frequency values obtained in Fig. 3,
it is possible to estimate the effect of double pulse excitation when the time
delay between them is zero $\Delta\tau=0$. In this case, the liquid jet is
simultaneously pumped by the two pulses, resulting in the liquid jet surface
receiving double the intensity of the single pulse in the double pump
configuration. Table 1 presents the calculated plasma density and frequency
based on a single pulse and based on double pulses when $\Delta\tau=0$.

Table 1: The calculated values of plasma densities and frequencies formed in the studied liquid jets based on a single pulse and based on double pulses when $\Delta\tau=0$, with the estimated plasma frequency obtained experimentally in Fig. 3 | Single-Pulse | Double-Pulse $\Delta\tau=0$
---|---|---
Liquid Jet | $\begin{array}[]{c}\text{Plasma Density}\\\ (cm^{-3})\end{array}$ | $\begin{array}[]{c}\text{Plasma Frequency}\\\ {(Hz)}\end{array}$ | $\begin{array}[]{c}\text{Plasma Density}\\\ (cm^{-3})\end{array}$ | $\begin{array}[]{c}\text{Plasma Frequency}\\\ {(Hz)}\end{array}$ | $\begin{array}[]{c}\text{Plasma frequency}\\\ \text{from Fig.\leavevmode\nobreak\ \ref{fig3} (Hz)}\end{array}$
Water | $2.99\times 10^{21}$ | $3.08\times 10^{15}$ | $5.57\times 10^{22}$ | $1.33\times 10^{16}$ | $1.1\times 10^{16}$
Ethanol | $1.16\times 10^{22}$ | $6.07\times 10^{15}$ | $1.57\times 10^{23}$ | $2.23\times 10^{16}$ | $2.01\times 10^{16}$
Isopropyl | $1.40\times 10^{22}$ | $6.69\times 10^{15}$ | $1.92\times 10^{23}$ | $2.47\times 10^{16}$ | $2.46\times 10^{16}$

The calculated plasma frequency values based on the single pulse are initially
lower than the TH frequency $\omega_{TH}=7.06\times 10^{15}Hz$ ($256$ $nm$),
indicating that the first pulse alone is insufficient to achieve maximum TH
reflected density. However with introducing the second pulse of the double
pump pulse technique, ionization is enhanced, resulting in plasma frequency
values greater than the TH frequency as evident in the calculations for the
double pulse at $\Delta\tau=0$. This observation aligns with previous studies
[24, 25] that highlight the augmentation of ionization when employing the
double pump pulse technique. On the other hand, the calculated plasma
frequency values based on double pump pulse excitation at $\Delta\tau=0$
closely align with the experimentally estimated values at $\Delta\tau=0$ from
Fig. 3, showing slightly higher values in the theoretical model. This
validates the accuracy of our experimental estimation and explanation.

In conclusion, our study investigated plasma formation in liquid jets,
focusing on TH reflection dynamics from induced plasma. We have examined
experimentally how the jet rotation angle influenced TH reflection, observing
maximum reflection at the Brewster angle for the liquid jet used. For plasma
property evaluation, we have determined the plasma frequency values from time-
resolved experiment on the dynamics of TH reflection. We have compared water,
ethanol, and isopropyl liquid jets. Isopropyl has exhibited the highest plasma
frequency, ethanol lower, and water the lowest, correlating with their
terahertz (THz) generation efficiency [14]. Utilizing Keldysh’s theory, we
have employed a theoretical model that estimated plasma parameters, aligning
well with experimental evaluations in the investigated liquid jets.
Furthermore, we have shown that using the double pump technique enhances the
ionization in liquids resulting in increasing the plasma density and
frequency. Our results regarding the variation of plasma frequency among the
studied liquids are consistent with the efficiency of THz generation in these
liquids, which is significantly influenced by the process of plasma formation
[11, 14]. These findings hold significant potential for the development of
plasma as a THz radiation source.

Funding This work was supported by the Russian Science Foundation (RSF)
(2019-0903).

Disclosures The authors declare no conflicts of interests.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

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