[1]Simone Lionetti

1] Lucerne University of Applied Sciences and Arts, Suurstoffi 4, Rotkreuz,
CH-6343, ZG, Switzerland 2] Federal Office of Meteorology and Climatology
MeteoSwiss, Chemin de l’Aérologie 1, Payerne, CH-1530, VD, Switzerland 3]
École Polytechnique Fédérale de Lausanne, Station 2, Lausanne, CH-1015, VD,
Switzerland 4] University of Basel, Hegenheimermattweg 167b, Allschwil,
CH-4123, BS, Switzerland 5] Swisens AG, Meierhofstrasse 5a, Emmen, CH-6032,
LU, Switzerland

# Self-Supervised and Few-Shot Learning for Robust Bioaerosol Monitoring

Adrian Willi  Pascal Baumann  Sophie Erb  Fabian Gröger  Yanick Zeder  *
<EMAIL_ADDRESS>[  [  [  [  [

###### Abstract

Real-time bioaerosol monitoring is improving the quality of life for people
affected by allergies, but it often relies on deep-learning models which pose
challenges for widespread adoption. These models are typically trained in a
supervised fashion and require considerable effort to produce large amounts of
annotated data, an effort that must be repeated for new particles,
geographical regions, or measurement systems. In this work, we show that self-
supervised learning and few-shot learning can be combined to classify
holographic images of bioaerosol particles using a large collection of
unlabelled data and only a few examples for each particle type. We first
demonstrate that self-supervision on pictures of unidentified particles from
ambient air measurements enhances identification even when labelled data is
abundant. Most importantly, it greatly improves few-shot classification when
only a handful of labelled images are available. Our findings suggest that
real-time bioaerosol monitoring workflows can be substantially optimized, and
the effort required to adapt models for different situations considerably
reduced.

###### keywords:

Pollen, holographic imaging, self-supervised learning, few-shot learning

##  1 Introduction

Real-time aerosol monitoring provides key information for public health,
particularly for managing allergies and asthma [1]. Pollen is the most common
cause of respiratory allergies in European countries, making pollen monitoring
key to diagnosing, managing, and treating symptoms. Although Hirst-type
impactors [2] have been providing reliable airborne pollen data for decades,
they have limitations such as low sampling rates (10 l/min), processing
delays, and labor-intensive operations.

Recent advances in laser and artificial intelligence technologies address some
of these limitations and have lead to the development of new automatic
bioaerosol monitoring instruments [3]. The SwisensPoleno is one such new
system based on airflow cytometry [4]. It measures particles through
holographic imaging, laser-induced fluorescence, light scattering and
polarisation. These measurements are then used to classify airborne particles
using deep-learning models. To date, this has been carried out in a setting
which requires substantial amounts of labelled data. The annotation of this
data needs to be performed by experts and is labor-intensive, and thus often
constitutes a bottleneck. Another limitation of this approach is that models
developed for a certain geographical region can perform poorly in other
regions where atmosphere composition is different. To classify new types of
bioaerosol not included in the training data it is typically necessary to re-
train models. Current research, including European efforts like the EUMETNET
AutoPollen Programme [5] and the SYLVA Horizon Europe project [6], are
developing models and networks at the European level to address this issue.

In this paper, we focus on the identification of ambient particles based on
holographic images acquired by the SwisensPoleno. Starting with a general-
purpose deep-learning model trained on ImageNet with supervision, we refine it
on 20 million unlabelled images of airborne particles with a Self-Supervised
Learning method called SimCLR. Using pollen as a case study, we demonstrate
that, when a small number of labelled examples are available, the combination
of self-supervised training with Few-Shot Learning is greatly beneficial for
particle identification. We also show that model refinement with self-
supervision leads to enhanced robustness to variations in data acquisition
settings.

##  2 Methods

This study leverages deep learning to process automatically captured
holographic images of ambient particles. Generalized and robust
representations of unlabelled images are constructed using Self-Supervised
Learning (SSL). Subsequently, these representations are combined with Few-Shot
Learning (FSL) to improve classification performance using only a minimal set
of labelled samples.

###  2.1 Data

The SwisensPoleno instrument [4] takes in-flight holographic images of
particles as they flow through the measurement system. Passing particles with
sizes between 5 and 200 µm trigger two cameras, which are perpendicular to
each other and to the particle flow. Their raw holographic images are
reconstructed in focus, centered, and cropped to 200$\times$200 pixels grey-
scale pictures where one pixel corresponds to 0.595 µm.

In this study, we use both unlabelled and labelled data. The unlabelled data
consists of about 20 million holographic images of ambient airborne particles
measured by a SwisensPoleno located on the rooftop of the MeteoSwiss building
in Payerne, Switzerland, throughout one year. The labelled data were obtained
from specific measurements during which pollen collected directly from known
plants were aerosolized. These data were then manually filtered and cleaned as
described in previous work [7]. Eleven different plant taxa were considered
(Alnus glutinosa, Carpinus betulus, Cupressus sempervirens, Cynosurus
cristatus, Fagus sylvatica, Fraxinus excelsior, Picea abies, Populus sp.,
Quercus robur, Taxus baccata and Ulmus glabra). The labelled data were
obtained from two SwisensPoleno systems denoted P4 and P5. Although the large
unlabelled collection comes from P5, there is no overlap between the labelled
and unlabelled data.

###  2.2 Machine Learning

Figure 1:  Simplified illustration of the SimCLR framework using holographic
images of airborne particles from SwisensPoleno. Every input image in a batch
is augmented, resulting in different views of the same particle. These views
are then all passed through the same deep neural network encoder to obtain
vector representations. Finally, representations from the same input image are
attracted while the ones from different input images are repelled. This
produces semantic representations without the need for labelled data.

SSL offers a way to build models without labelled data. In particular SimCLR
[8], the method used in this work, belongs to the family of contrastive SSL
approaches. These methods learn vector representations of images by generating
multiple augmented versions for each original image through augmentations like
rotation and clipping, and using a deep neural network encoder to map the
augmented images into vectors. The encoder is trained by pulling together
these vectors if they come from the same original image and repelling them
otherwise, as illustrated in fig. 1. We apply SimCLR to SwisensPoleno
holographic images and initialize it with a general-purpose image encoder pre-
trained on ImageNet in a supervised way, as this performs better compared to
random initialization [9].

To classify new data with a limited number of labelled images, we combine SSL
with FSL. First, we test a linear classifier, where we combine image
representations with one-vs-all logistic regression. Second, we learn a
prototype in representation space for each taxon by minimizing cross-entropy
after assigning samples to prototypes with cosine similarity. The latter
approach was proven to perform consistently better than the linear classifier
and on par with more sophisticated techniques across many FSL scenarios [10].

We compare the model trained on ImageNet to the one that has been specialized
for holographic aerosol images with SimCLR, always using the same
EfficientNet-B0 architecture [11]. We measure performance using balanced
accuracy as our dataset does not reflect atmospheric composition.

##  3 Results

Table 1:  Balanced accuracy for the classification of 11 pollen taxa from the features of the ImageNet+SimCLR model and ImageNet, using a linear classifier (one-vs-all logistic regression) trained on a labelled P5 training dataset and evaluated on the labelled P5 and P4 test datasets.  | Balanced Accuracy Score
---|---
Method | P5 test dataset | P4 test dataset
ImageNet | 89.9 % | 75.5 %
ImageNet+SimCLR | 90.4 % | 79.0 %
|  |

We first consider the case where the model is trained on all available
labelled data. We use 400 to 6’600 pollen grains per taxon for training and
evaluate previously learned representations with one-vs-all logistic
regression. Results are presented in table 1. When evaluating the model
trained on data from the same instrument P5, we observe that specializing the
supervised ImageNet features on SwisensPoleno data with SimCLR marginally
improves the balanced accuracy (+0.5%) compared to ImageNet alone. However,
the gain is larger (+3.5%) when evaluation is carried out on data from the
other instrument P4. In these conditions, the ImageNet performance drops to
75.5% while that of ImageNet+SimCLR only drops to 79%.

Figure 2:  Balanced accuracy for pollen classification as a function of the
number of available labelled images per taxon, evaluated on data from the same
SwisensPoleno (P5) used for SSL (left), and on data from another SwisensPoleno
(P4) (right). Results are shown for the linear classifier (top) and prototype
learning (bottom) on top of features from ImageNet (teal) and ImageNet+SimCLR
(purple). Colored bands represent the uncertainty due to the random choice of
the labelled images.

Next, we address situations where training data are scarce. Figure 2 shows the
performance of different methods as a function of the number of labelled
images available per taxon. When evaluation is performed on test data from the
same measurement system as the training data, i.e. from P5 (left column),
ImageNet+SimCLR outperforms ImageNet with only one image per taxon. As the
amount of labelled data increases, ImageNet gains an edge over
ImageNet+SimCLR, and differences among models become smaller. If less than 10
images per taxon are available, the best results are obtained by combining
ImageNet+SimCLR with prototype learning, while the ImageNet model does not
profit from this FSL technique. When the evaluation is performed on test data
from a different measurement system compared to the training data, i.e. P4
(right column), ImageNet+SimCLR outperforms ImageNet throughout. Moreover, the
synergy of ImageNet+SimCLR and prototype learning yields significantly better
results compared to the other three alternatives. When a single example is
available, this combination achieves 50% balanced accuracy while all other
methods are around 40%. With only 5 or 10 labelled examples, one can already
expect a balanced accuracy around 65% and 70% respectively.

In general, we observe significant performance drops when evaluation is
performed in a different setting compared to training, even if holographic
images exhibit very little variations among measurement systems. These drops
are markedly reduced in the case of ImageNet+SimCLR with prototype learning,
highlighting the robustness of distance-based classification with self-
supervised features. By contrast, purely supervised models are more prone to
take shortcuts for classification, which might explain why they generalise
poorly when conditions change [12].

##  4 Discussion and conclusion

ImageNet+SimCLR offers more robust performance when models are evaluated under
different conditions compared to training. This is already an advantage for
pollen monitoring networks. However, the combination of SSL with FSL can
produce flexible identification models, and therefore has an even greater
appeal for networks that cover a large geographical region. Although SSL
requires additional training compared to off-the-shelf models, this only needs
to be done once and taxa can be specified a posteriori. As long as the initial
SSL training dataset is broad enough, the model learns features to distinguish
arbitrary particles, and good performance is expected with FSL for any new
taxon. The significant potential benefit for the development of new pollen
identification models is that the effort of labelling pollen data may be
reduced to a handful of particles, if performance remains within acceptable
limits. This opens the door to faster development of models across climatic
regions.

In this study, we showed that self-supervision can effectively leverage large
volumes of unlabelled data to improve the robustness of deep-learning models
for pollen grain identification, not only against geographical variation but
also against changes in data collection conditions. The approach makes data
preparation more feasible and produces flexible models, since other bioaerosol
types can be classified with FSL as long as the SSL training dataset is
sufficiently diverse. Results could be improved further by training the model
on data which covers different collection conditions, including several
instruments and different geographical regions. Similarly, incorporating data
from other sensors could enhance identification accuracy. Future work should
include testing other SSL methods, especially non-contrastive ones [13], and
merging current research efforts to develop a foundation model handling a
large set of bioaerosol.

## Declarations

### Funding

The data used in this work was provided by the Federal Office of Meteorology
and Climatology MeteoSwiss. PB, FG, and SL gratefully acknowledge the support
of the Swiss Innovation Agency Innosuisse through preliminary study voucher
59965.1 INNO-ICT. SE was funded by the Swiss National Science Foundation
(IZCOZ0_198117).

### Competing interests

YZ is an employee of Swisens AG, and SL discloses support from Swisens AG
covering the supervision for AW’s master thesis as foreseen by the regulations
of the Lucerne University of Applied Sciences and Arts. All investigations
were carried out in compliance with good scientific practices and the declared
relationships have no effect on the results presented.

### Availability of data and materials

The data presented in this paper were kindly provided by the Federal Office of
Meteorology and Climatology MeteoSwiss, and cannot be shared publicly.

### Code availability

The code used in this work is experimental and tailored to the SwisensPoleno.
Considering that it cannot be used without the specific data to reproduce
results nor for applications with other devices, there is little benefit
making it open-source.

### Authors’ contributions

SL, YZ, and FG planned the work. SE provided the data. AW, PB, FG, and SL
contributed to the analysis. AW, SE, SL, and YZ wrote the manuscript, and FG
and PB revised it.

## References

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# An alternative form of the fundamental plasma emission through the
coalescence of Z-mode waves with whistlers

Sulan Ni  Institute of Space Sciences, Shandong University, Weihai, Shandong
264209, People’s Republic of China  Center for Integrated Research on Space
Science, Astronomy, and Physics, Institute of Frontier and Interdisciplinary
Science, Shandong University, Qingdao, Shandong, 266237, People’s Republic of
China  Yao Chen<EMAIL_ADDRESS>Institute of Space Sciences, Shandong
University, Weihai, Shandong 264209, People’s Republic of China  Center for
Integrated Research on Space Science, Astronomy, and Physics, Institute of
Frontier and Interdisciplinary Science, Shandong University, Qingdao,
Shandong, 266237, People’s Republic of China  Chuanyang Li  Institute of Space
Sciences, Shandong University, Weihai, Shandong 264209, People’s Republic of
China  Center for Integrated Research on Space Science, Astronomy, and
Physics, Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao, Shandong, 266237, People’s Republic of China  Jicheng Sun
Department of Physics, Auburn University, Auburn, AL, 36849, USA  Hao Ning
Institute of Space Sciences, Shandong University, Weihai, Shandong 264209,
People’s Republic of China  Center for Integrated Research on Space Science,
Astronomy, and Physics, Institute of Frontier and Interdisciplinary Science,
Shandong University, Qingdao, Shandong, 266237, People’s Republic of China
Zilong Zhang  Institute of Space Sciences, Shandong University, Weihai,
Shandong 264209, People’s Republic of China  Center for Integrated Research on
Space Science, Astronomy, and Physics, Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao, Shandong, 266237,
People’s Republic of China

###### Abstract

Plasma emission (PE), i.e., electromagnetic radiation at the plasma frequency
and its second harmonic, is a general process occurring in both astrophysical
and laboratory plasmas. The prevailing theory presents a multi-stage process
attributed to the resonant coupling of beam-excited Langmuir waves with ion-
acoustic waves. Here we examine another possibility of the fundamental PE
induced by the resonant coupling of Z-mode and whistler (W) waves. Earlier
studies have been controversial in the plausibility and significance of such
process in plasmas. In this study we show that the matching condition of three
wave resonant interaction (Z+W $\rightarrow$ O) can be satisfied over a wide
regime of parameters based on the magnetoionic theory, demonstrate the
occurrence of such process and further evaluate the rate of energy conversion
from the pumped Z or W mode to the fundamental O mode with particle-in-cell
(PIC) simulations of wave pumping. The study presents an alternative form of
the fundamental PE, which could possibly play a role in various astrophysical
and laboratory scenarios with both Z and W modes readily excited through the
electron cyclotron maser instability.

Plasma emission (PE) is loosely defined as electromagnetic radiations at the
plasma frequency and its second harmonic, corresponding to the fundamental (F)
and harmonic (H) emissions. The process accounts for coherent radiations
observed from a wide range of astrophysical and laboratory plasmas, including
but not limited to (1) the solar atmosphere and the interplanetary space, such
as type-III and II radio bursts McLean & Labrum (1985); Dulk et al. (1984),
(2) the planetary electron foreshocks such as those of the Jupiter and the
Earth Etcheto & Faucheux (1984); Moses et al. (1984); Cairns (1987), (3) the
outer heliosphere Kurth et al. (1984), (4) flare stars Kuijpers (1985); Loeb
et al. (2014), and (5) laboratory experiments Intrator et al. (1984); Whelan &
Stenzel (1985).

The prevailing theory for PE presents a multi-stage nonlinear process,
starting from the kinetic bump-in-tail instability driven by beams of
energetic electrons, scattering of the excited Langmuir waves by ion-acoustic
waves to account for the F radiation, and coalescence of forward- and
backward-propagating Langmuir waves to yield the H radiation. The framework of
PE was originally suggested by Ginzburg and Zheleznyakov Ginzburg &
Zhelezniakov (1958), being developed and improved substantially over the
decades Melrose (1980); Cairns (1987); Robinson et al. (1994). Latest efforts
are mainly along three fronts: (1) developing the electromagnetic weak-
turbulence theory to describe the nonlinear evolution of the beam-plasma
instability Ziebell et al. (2014, 2015); Lee et al. (2019), (2) verifying the
complete multi-step nonlinear PE process driven by single- or counter-
propagating beam using the full electromagnetic PIC simulation Ganse et al.
(2012); Thurgood et al. (2015); Che et al. (2017); Henri et al. (2019), (3)
simulating major characteristics of various types of radio bursts observed in
space, such as the solar type-II and -III bursts, by incorporating the reduced
PE theory into large-scale models Schmidt and Cairns. (2012, 2012); Ratcliffe
and Kontar (2014, 2014); Reid and Ratcliffe (2014).

For other possible forms of PE, in a recent study Ni et al. Ni et al. (2020)
suggested that the F emission could be a result of the resonant coalescence of
magnetoionic Z and whistler (W) modes, and the H emission be a result of the
coalescence of upper hybrid (UH) waves, in weakly magnetized plasmas. These
modes are excited by energetic electrons of the Dory-Guest-Harris (DGH Dory et
al. (1965)) distribution through electron cyclotron maser instability (ECMI),
in a parameter regime with a large ratio of plasma frequency to electron gyro-
frequency ($\omega_{pe}/\Omega_{ce}$ (=10)). The DGH distribution is of the
double-sided loss cone type, usually applied for energetic electrons trapped
within a magnetic structure. The essential nature of ECMI is identical to the
alternative type of coherent emission in plasmas with a small value of
$\omega_{pe}/\Omega_{ce}$ ($<<1$), known as the electron cyclotron maser
emission (ECME Twiss (1958); Wu & Lee (1979)).

Plasma radiation involving whistlers was first suggested by Chiu Chiu (1970)
and Chin Chin (1972) through their coalescence with Langmuir waves. Melrose
Melrose (1975) rejected the idea since (1) for thermal Langmuir waves, as
originally proposed by Chiu, the coupling cannot explain the high brightness
of coherent radiation, (2) for beam-excited Langmuir waves the wave number
($k$) is approximately $\omega_{pe}/v_{b}$ ($>>\omega_{pe}/c$) too large to
meet the conditions of coupling with whistlers whose $k$ is usually less than
$\omega_{pe}/c$, while the F emission of O mode usually has a small $k$
($<<\omega_{pe}/c$), where $v_{b}$ ($<<c$) represents the beam velocity. He
also concluded that the only wave around the plasma frequency that could
coalesce with whistlers is the superluminal Z mode that is electromagnetic
with a phase speed close to or larger than $c$ and a short-enough wave number.

Dispersion relations of Z mode and whistlers can be approximated by the
magnetoionic theory Hartree (1931); Appleton & Ratcliffe (1932). Using
simplified expressions, Melrose Melrose (1975) deduced that the Z-W resonant
coupling can occur only under very restrictive condition and therefore PE
involving whistlers is unlikely to be important, at least not in coronal
plasmas. Despite this negative statement, some later studies still assumed the
involvement of whistlers to generate the fundamental emission. For example,
Takakura Takakura (1981) and Mann et al. Mann et al. (1987) suggested the
coalescence of whistlers and plasma waves which are scattered by thermal ions
to reach a small wave number, to account for the F emission of Type-III solar
radio bursts. Whistlers are also used to explain formations of fine spectral
structures of Type-IV solar radio bursts through rapid diffusion of energetic
electrons (e.g., Chernov (1976, 2006)). Recently, Ni et al. Ni et al. (2020)
reported the apparent occurrence of the three-wave interaction
(Z+W$\rightarrow$O) according to PIC simulations. Yet, plausibility and
significance of the Z-W-O interaction in plasma radiation process have not
been carefully examined, and direct solution of the matching conditions and
proof of the occurrence of coalescence are not available.

In this Letter, we first present solutions of the matching conditions of
three-wave resonant coupling (Z+W$\rightarrow$O) using the dispersion
relations given by the canonical magnetoionic theory, which describes the
propagation of plasma waves in cold magnetized electron plasmas. The relations
can be found in many textbooks of plasmas and will not be repeated here. The
matching conditions can be written as

$\omega_{Z}+\omega_{W}=\omega_{O},\ \ \ \ \vec{k}_{Z}+\vec{k}_{W}=\vec{k}_{O}$
(1)

where subscripts represent corresponding wave modes. Then we demonstrate the
occurrence of the three-wave coupling process by pumping waves of Z and W
modes, either separately or together, into the PIC system that is simulated
with the open-source Vector PIC (VPIC) code provided by the Los Alamos
National Labs and run on the supercomputers of Beijing Super Cloud Computing
Center (BSCC). It is a fully-kinetic electromagnetic and relativistic PIC code
Bowers et al. (2008a, b, 2009), run in two spatial dimensions ($x,z$) with
three velocity components using periodic boundary conditions. Note that the
main purpose of the present study is to verify whether the matching conditions
can be met by solving them directly and confirm the occurrence of the matching
process using PIC simulations, rather than simulating the complete ECMI-plasma
emission process, which has been done in Ni et al. Ni et al. (2020) and Li et
al. Li et al. (2021)

To solve the resonance conditions for the specific case with
$\omega_{pe}/\Omega_{ce}=10$, we vary the propagation angle ($\theta_{kB}$)
for Z and W modes from 0∘ to 360∘. For each angle we obtain the corresponding
cutoff- and resonance- frequencies which are then used to limit the ranges of
possible frequency ($\omega$) of the two modes. We put 500 grid points in each
range. By solving the magnetoionic dispersion relations, we obtain values of
$k$ corresponding to each combination of $\omega$ and $\theta_{kB}$ for Z and
W modes, ending up with $500^{4}$ sets of [$\omega$, $k$] combinations of wave
solutions. The units of $\omega$ and $k$ are taken to be $\Omega_{ce}$ and
$\Omega_{ce}/c$, respectively. The parameters are then summed up according to
Equation (1), if the sums meet the magnetoionic O-mode dispersion relation
with an accuracy of $<10^{-4}$, then we save the parameter set as a resonance
solution. In total, we obtain 14285 such solutions. We have tried many other
cases with $\omega_{pe}/\Omega_{ce}$ ranging from 3 to 15, getting similar
results.

We show 7 examples in Figure 1, with wave vectors shown in panel (a) and
frequencies plotted onto the corresponding magnetoionic dispersion curves in
panels (b) and (c). These examples distribute within the range of
[$0^{\circ},90^{\circ}$] for the Z-mode propagation angle. Since the whistlers
experience strong damping when reaching sufficiently-high frequency
($\sim>\Omega_{ce}/2$), we do not consider the possibility of Z and W
coalescence into the X mode which has a cutoff frequency around 10.5
$\Omega_{ce}$. We see that the coalescing Z and W modes distribute within a
relatively large range of $\omega$, $k$, and $\theta_{kB}$. This can be
confirmed from Figure 2, in which we plot the histogram distribution of
solution numbers for the three modes. The $k$ values of both Z and W modes
distribute mainly within a range of [2.5, 9] $\Omega_{ce}/c$, the O mode
distributes mainly within a range of [0, 4] $\Omega_{ce}/c$; the frequencies
of Z and W lie mainly in ranges of [9.8, 10.0] $\Omega_{ce}$ and [0, 0.4]
$\Omega_{ce}$ respectively, and their propagation angles present a preference
of distribution over the parallel to oblique propagation; for the O mode the
frequency range is mainly in the range of [10, 10.4] $\Omega_{ce}$ while its
propagation angle manifests a quasi-isotropic distribution with a weaker
preference over the parallel and anti-parallel propagation. These solutions
show that the three-wave coupling process (Z+W$\rightarrow$O) is quite general
and takes place over a broad range of wave parameters, as seen from the
perspective of resonance conditions.

To demonstrate the actual occurrence of the Z+W$\rightarrow$O coalescence, we
develop a wave-pumping technique. Different wave properties, such as
components of the electro-magnetic fields and the velocity, are related
linearly to each other. The relationships can be easily deduced from the mass
and momentum equations for cold-plasma electrons and the Maxwell equations.
All wave properties vary in space in form of cosine function with a prescribed
initial phase angle ($\sim\cos(\vec{k}\cdot\vec{r}+\psi_{0})$, where
$\vec{r}=x\hat{e}_{x}+z\hat{e}_{z}$). Revising the initial conditions of the
PIC simulation accordingly allows us to pump various wave modes into the
system.

In the PIC simulation, the plasma consists of equal number of thermal
electrons and protons with Maxwellian distribution of $2$ MK, typical
temperatures for solar coronal plasmas; $\omega_{pe}/\Omega_{ce}$ is taken to
be $10$, the background magnetic field $\textbf{B}_{0}=B_{0}\hat{e}_{z}$ is
uniform and along the $z$ direction, the wave vector $\vec{k}$ is in the $xOz$
plane. We employ 1000 macro-particles per species per cell. The simulation
lasts for 2000 $\omega_{pe}^{-1}$ within the spatial domain of
$2048\Delta\times 2048\Delta$, where $\Delta=3.25\lambda_{D}$ and
$\lambda_{D}$ is the Debye length of thermal electrons. The time step is set
in order to ensure the Courant conditions $\Delta t=0.029$ $\omega_{pe}^{-1}$.
A realistic proton-to-electron mass ratio of 1836 is used.

The resonance solution with $\theta_{kB}=45^{\circ}$ of the Z mode (see the
red arrows plotted in Figure 1 for this case and exact values of relevant
parameters in its caption) is selected for further PIC simulation. We pump Z
and W modes with corresponding parameters into the PIC system, either
separately or together, obtaining three cases (referred to as Cases Z, W, and
Z+W). Comparing them, we can tell whether the three-wave resonance indeed
takes place and evaluate the percentage of energy that can be converted from Z
or W to O.

The distribution of the maximum spectral intensity in the $\vec{k}$ space for
the three cases is illustrated in Figure 3, and the $(\omega,k)$ dispersion
curves along $\theta_{kB}=45^{\circ}$ and $224^{\circ}$ are presented in
Figure 4 (and the accompanying movie for other relevant propagation angles)
(Multimedia view) superposed by those of magnetoionic modes. Various modes (Z,
W, and O) can be easily identified. The three sets of arrows in Figure 3 point
to parameters given by the above-selected resonance solution. They also point
right at the center of areas with enhanced intensity of the PIC simulation.
This indicates that the modes have been pumped correctly. The Z mode is
dominated by the electric field fluctuation while the W mode is dominated by
the magnetic field fluctuation, as expected. The obtained O mode is dominated
by the parallel component of electric field ($E_{z}$) since its frequency is
at or only slightly larger than $\omega_{pe}$. Note that the simulated wave
characteristics distribute over a relatively broad angular region of $\sim
20^{\circ}$ (see the accompanying movie), therefore the characteristics of
both Z and O modes at $45^{\circ}$ are almost the same as those at
$47.7^{\circ}$, so in Figure 4 (Multimedia view) only diagrams at $45^{\circ}$
are shown for these two modes.

It should be highlighted that the O mode appears with the expected wave vector
only in Case Z+W, confirming the occurrence of the Z+W$\rightarrow$O
coalescence, and the maximum of the O-mode intensity is centered around the
parameters prescribed by the selected coalescence solution. This presents
unambiguous evidence of the occurrence of the three-wave interaction of
Z+W$\rightarrow$O.

The temporal profiles of energy of the three modes are plotted in Figure 5a
for the three cases, calculated using the Parseval’s theorem. For Z and W
modes, the total electro-magnetic field energy is calculated directly from the
PIC simulation, while for the O mode the total energy is calculated by the
simulated magnetic energy multiplied by the theoretical magnetoionic ratio of
the total to magnetic energy, to avoid interference of nearby Z mode.

For Case Z (W), the energy profiles of W (Z) and O modes represent the
corresponding noise levels. The input energy of Z (W) modes are above the
noise levels by about 2-3 orders in magnitude, and the obtained O mode energy
in Case Z+W is above the corresponding noise level by one order in magnitude.
The input energy of Z and W modes is taken to be close to each other, referred
to as $E_{Z,W}^{0}\sim 5.8\times 10^{-6}E_{B_{0}}$ hereinafter, where
$E_{B_{0}}$ represents the total energy of the background magnetic field. Note
that the pumped Z and W modes in all cases presented here should be regarded
as small-amplitude waves since their energy are negligible in comparison to
that of the background magnetic field. With time going on, energies of the
pumped modes present an overall gradual decline, while the O mode energy
increases gradually with time during the early stage of the interaction. After
$t=750$ $\omega^{-1}_{pe}$, the energy reaches an asymptotic value of $\sim
0.5\%E_{Z,W}^{0}$, which gives the energy conversion rate of the PE process,
from the pumped wave modes to the escaping radiation mode, in the present Z+W
case. At this time ($t=750$ $\omega_{pe}^{-1}$), the relative decline of the Z
mode energy in both Case Z and Case Z+W is about $\sim$10%, while that of the
W mode energy in both Case W and Case Z+W is about $\sim$20%. Thus, the
decline of energy due to the conversion to O mode in Case Z+W is minor,
indicating that the decline of Z and W mode energies are mainly due to the
damping of individual mode, rather than their interaction.

To investigate the variation of the conversion rate versus the pumped energy,
we gradually increase the pumped energy of one mode of Z and W while fixing
that of the other mode, taking Case Z+W to be the reference solution. This
yields two sets of parameter studies, referred to as Set Z with Z-mode energy
increasing and Set W with W-mode energy increasing. The results and
corresponding lines of linear and polytropic fittings are plotted in Figure
5b. If the pumped energy is less than $8\times 10^{-4}$ $E_{B0}$ or the
obtained O-mode energy is less than ($0.2-0.3$ $E^{0}_{Z,W}$), then the O-mode
energy ($E_{O}$) increases almost linearly with the input energy of either Z
or W mode, while the O mode energy approaches an asymptotic value ($E_{c}$) of
about 2$\times 10^{-6}E_{B0}$ ($\sim 0.3$ $E^{0}_{Z,W}$) if the pumped energy
gets larger. The linear fitting lines for the two sets (Z and W) are
respectively expressed as

$y=0.0025x+5.37\times 10^{-9},\ \ \ \ y=0.0021x+3.63\times 10^{-8}$

The slope of the fitting lines can be regarded as the energy conversion rate
from the pumped Z (for Set Z) or W (for Set W) mode to O mode for the
respective linear-correlation regime.

These results, in particular the linear correlation found above, provide
additional yet strong support to the occurrence of three-wave interaction with
the O mode being a result of the nonlinear coalescence of Z and W. If further
increasing the input energy above the levels of Sets Z and W, there appears
unexpected interfering signals caused by numerical artifacts and/or nonlinear
effect which make the uncertainty of mode evaluations larger. This is
partially due to the usage of cold-plasma magnetoionic approximation in our
wave-pumping method. Note that since the amount of energy pumped into the
system is rather small, no obvious heating or acceleration of electrons are
observed, and the original Maxwellian distribution remains.

To summarize, the Letter explored an alternative form of the fundamental
plasma emission involving the nonlinear resonant coupling of Z-mode waves and
whistlers. On the basis of cold-plasma magnetoionic theory of plasma waves, we
solved the matching conditions of three-wave interaction of Z mode, whistlers,
and O mode. It was found that the conditions can be satisfied within a quite
general and broad regime of wave parameters. In addition, we developed a wave-
pumping technique using the VPIC code to simulate the nonlinear coupling of Z
and W modes and generation of the O-mode emission. By case and parameter
studies, we presented solid evidence of the occurrence of the process and
investigated the energy conversion rate from the pumped Z-mode waves and
whistlers to the fundamental O-mode emission. It was found that there exists a
critical value which determines how the coalesced O-mode energy varies
according to the input energy of Z and W modes, either linearly below a
certain threshold or asymptotically beyond. The maximum energy that can be
converted to O mode was found to be about 30% of the Z- or W- mode energy,
whichever is smaller.

The prevailing theory of the fundamental PE is the scattering of beam-excited
Langmuir waves by ion-acoustic wave or density inhomogeneities. The process
discussed here is an alternative form of the fundamental PE. For this form to
play a role in plasmas, both Z-mode waves and whistlers should be excited
efficiently. This can be done through the ECMI driven by energetic electrons
with velocity distribution functions of positive gradient along the
perpendicular direction, i.e., with ${\partial f\over\partial v_{\perp}}>0$.
Thus, the complete emission process can be referred to as the ECMI-induced PE.
Examples of such distributions include the well-known loss cone distribution,
and the DGH, ring, ring-beam, and horseshoe distributions. The first example
of the ECMI-induced PE driven by DGH electrons has been presented by Ni et al.
Ni et al. (2020).

Here we briefly list a few types of solar radio bursts as examples, to which
the mechanism may be important, (1) type-IV bursts which are believed to be
generated by energetic electrons trapped within eruptive magnetic structures
(e.g., Benz (2002); Vasanth et al. (2016, 2019)), therefore loss-cone like
distribution shall develop and play a role by exciting both W and Z-mode waves
which then coalesce to yield the F emission; (2) type-I bursts which are
likely related to electrons released into coronal loops by small-scale
reconnections (see Li et al. (2017)) and the form of electron distribution may
be different from a beam type; (3) type-II bursts which have been assumed to
be similar to type-IIIs in emission mechanism, yet doubts remain regarding the
exact form of electron distributions, it is likely that type-IIs are excited
by electrons that are trapped and scattered around the shock, thus for the
same reason as above the mechanism discussed here may be significant Yao Chen
(2013); Chen et al. (2014); Kong et al. (2015). Further investigations are
demanded to clarify the role of the ECMI-induced PE mechanism in various
circumstances of astrophysical and laboratory plasmas.

This study is supported by the National Natural Science Foundation of China
(11790303 (11790300), 11750110424, and 11873036). The authors acknowledge the
Beijing Super Cloud Computing Center (BSCC, URL: http://www.blsc.cn/) for
providing high-performance computing (HPC) resources, and the open-source
Vector-PIC (VPIC) code provided by Los Alamos National Labs (LANL). The
authors acknowledge the helpful discussions with Drs. Xin Tao, Quanming Lu,
and Xinliang Gao at USTC, and Dr. Xiaocan Li at LANL.

##  I Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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Figure 1: Seven examples selected from the solutions of resonance conditions.
(a) Wave vectors ($\vec{k}$) of Z, W, and O, (b-c) frequencies ($\omega$) of
Z, O, and W plotted onto the corresponding magnetoionic dispersion curves
(dashed lines for Z mode, dotted lines for W mode, and solid lines for O
mode), the dispersion curve of X mode along $\theta_{kB}=45^{\circ}$ is also
plotted in panel (b). The selected wave mode for further PIC simulation is
labeled by ‘4’, exact values of ($\omega$, $\vec{k}$, $\theta_{kB}$) for this
solution are (9.959 $\Omega_{ce}$, 6.123 $\Omega_{ce}/c$, 45∘) for Z mode,
(0.119 $\Omega_{ce}$, 4.455 $\Omega_{ce}/c$, 224∘) for W mode, and (10.078
$\Omega_{ce}$, 1.671 $\Omega_{ce}/c$, 47.7∘) for O mode. Figure 2: Histogram
distributions of solutions of the matching condition (see Equation (1)) of
three wave resonant interaction (Z+W $\rightarrow$ O): (a) for wave numbers ,
(b) for propagating angles, and (c-d) for frequencies. Figure 3: Maximum
intensity distributions of the six field components in the $\vec{k}$ space for
Case Z (a), Case W (b), and Case Z+W (c), as shown by the colormap of 20
$log_{10}$ [($E_{x}$, $E_{y}$, $E_{z}$, $B_{x}$, $B_{y}$,
$B_{z}-B_{0}$)/($cB_{0}$)]. All panels use the same colorbar as shown next to
panel (a). Wave vectors of the three modes corresponding to the selected
solution (see solution ‘4’ plotted in Figure 1(a)) are plotted as arrows.
Figure 4: Wave dispersion diagrams of (a) $E_{x}$, $E_{y}$, $E_{z}$ along
$\theta_{kB}=45^{\circ}$ and (b) $B_{x}$, $B_{y}$, $B_{z}$ along
$\theta_{kB}=224^{\circ}$. The corresponding dispersion curves of Z, W, O, and
X modes are also plotted. An accompanying animation is available online
(Multimedia view). Figure 5: (a) Temporal profiles of energy of Z mode
(dashed), W mode (dotted), and O mode (solid) for Case Z (green), Case W
(blue), and Case Z+W (red). The energy is normalized with the total energy of
the background magnetic field ($E_{B_{0}}$), (b) variation of O-mode energy
with the increase of the pumped energy of Z and W modes (red for Set Z and
blue for Set W) with stars representing results of the two sets of parameter
study. Dot-dashed and solid lines represent linear and polytropic fitting
curves of the PIC results, respectively.