# ABYSS I: Targeting strategy for APOGEE & BOSS young star survey in SDSS-V

Marina Kounkel Department of Physics and Astronomy, Vanderbilt University, VU
Station 1807, Nashville, TN 37235, USA<EMAIL_ADDRESS>Eleonora
Zari Max-Planck-Institut für Astronomie, Königstuhl 17 D-69117 Heidelberg,
Germany  Kevin Covey Department of Physics and Astronomy, Western Washington
University, 516 High St, Bellingham, WA 98225, USA  Andrew Tkachenko Institute
of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium  Carlos
Román Zúñiga Universidad Nacional Autónoma de México, Instituto de Astronomía,
AP 106, Ensenada 22800, BC, México  Keivan Stassun Department of Physics and
Astronomy, Vanderbilt University, VU Station 1807, Nashville, TN 37235, USA
Amelia M. Stutz Departamento de Astronomía, Universidad de Concepción,Casilla
160-C, Concepción, Chile  Guy Stringfellow Center for Astrophysics and Space
Astronomy, Department of Astrophysical and Planetary Sciences, University of
Colorado,Boulder,CO, 80309, USA  Alexandre Roman-Lopes Departamento de
Astronomia, Facultad de Ciencias, Universidad de La Serena. Av. Juan Cisternas
1200, La Serena, Chile  Jesús Hernández Universidad Nacional Autónoma de
México, Instituto de Astronomía, AP 106, Ensenada 22800, BC, México  Karla
Peña Ramírez Centro de Astronomía (CITEVA), Universidad de Antofagasta, Av.
Angamos 601, Antofagasta, Chile  Amelia Bayo Instituto de Física y Astronomía,
Universidad de Valparaíso, Chile  European Southern Observatory, Karl-
Schwarzschild-Strasse 2, 85748 Garching bei München, Germany  Jinyoung Serena
Kim Steward Observatory, Department of Astronomy, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721-0065, USA  Lyra Cao Department of
Astronomy, The Ohio State University, Columbus, OH 43210, USA  Scott J. Wolk
Center for Astrophysics — Harvard & Smithsonian 60 Garden St. Cambridge MA,
02138 USA  Juna Kollmeier Canadian Institute for Theoretical Astrophysics,
University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada
Ricardo López-Valdivia Universidad Nacional Autónoma de México, Instituto de
Astronomía, AP 106, Ensenada 22800, BC, México  Bárbara Rojas-Ayala Instituto
de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

###### Abstract

The fifth iteration of the Sloan Digital Sky Survey (SDSS-V) is set to obtain
optical and near-infrared spectra of $\sim$5 million stars of all ages and
masses throughout the Milky Way. As a part of these efforts, APOGEE & BOSS
Young Star Survey (ABYSS) will observe $\sim 10^{5}$ stars with ages $<$30 Myr
that have been selected using a set of homogeneous selection functions that
make use of different tracers of youth. The ABYSS targeting strategy we
describe in this paper is aimed to provide the largest spectroscopic census of
young stars to-date. It consists of 8 different types of selection criteria
that take the position on the HR diagram, infrared excess, variability, as
well as the position in phase space in consideration. The resulting catalog of
$\sim$200,000 sources (of which a half are expected to be observed) provides
representative coverage of the young Galaxy, including both nearby diffuse
associations as well as more distant massive complexes, reaching towards the
inner Galaxy and the Galactic center.

††software: TOPCAT (Taylor, 2005), PyXCSAO, APOGEE Net

##  1 Introduction

The Sloan Digital Sky Survey (SDSS) has obtained spectra of hundreds of
thousands of stars, both within the Milky Way and beyond. Most of these are
evolved stars; in particular red giants have historically been favored for the
observations due to the ability to use them as tracers of stellar populations
at large distances. However, SDSS has observed spectra of thousands of young
stellar objects (YSOs) through its auxiliary programs (Román-Zúñiga et al.,
2022).

During SDSS-III, young stars were targeted by the IN-SYNC program (Cottaar et
al., 2014). The targeting strategy focused on known members of nearby
populations visible from the northern hemisphere and containing large
concentrations of young stars within the field of view of the telescope. The
latter condition was required to fill a large fraction of the available
spectral fibers. These included NGC 1333 (Foster et al., 2015), IC 348
(Cottaar et al., 2015), Orion A molecular cloud (Da Rio et al., 2016), as well
as NGC 2264. In total, spectra of $\sim$3,600 YSOs were taken with the APOGEE
spectrograph. The selection strategy was based on existing catalogs of
members, and thus yielded samples without significant contamination, but
cannot be considered homogeneous nor complete.

SDSS-IV expanded its footprint in the targeting of YSOs and complemented
previous data by observing the Orion Complex (Cottle et al., 2018), the Taurus
Molecular Clouds, Upper Sco, W3/W4/W5 clusters, Cygnus X, Rosette Nebula,
Carina Nebula, more evolved clusters such as the Pleiades and $\alpha$ Per,
and others (for a complete overview of the regions observed see Román-Zúñiga
et al., 2022). With respect to SDSS-III, a greater emphasis was given to
targeting sources in a more homogeneous manner, for instance by using color
cuts or photometric variability to select sources rather than relying on
existing confirmation of their youth. However different selection criteria
were developed for each individual region, since the observations fell under
the auspices of various programs (Beaton et al., 2021; Santana et al., 2021;
Román-Zúñiga et al., 2022). Spectra of $>$30,000 stars were taken across the
plates covering these regions. Since many of these targets were selected prior
to the release of parameters from the Gaia mission, a significant fraction of
them are evolved field stars, with the actual census of young stars within
them being $<$10,000.

With the transition to SDSS-V, several changes have been implemented to the
survey strategy.

  * •

All sky accessibility: the spectrographs utilized by the survey are installed
on two telescopes in the northern and the southern hemisphere.

  * •

Fast instrument reconfiguration: rather than using pre-drilled plates to
position the fibers of the spectrographs, robotic fiber positioners are now
used instead (Pogge et al., 2020). This allows both a greater flexibility in
targeting, and minimizing operational overheads. It is now possible to create
a comprehensive sample of spectra of the stars across the Galaxy, without
necessarily being limited to a particular line of sight or field of view.

  * •

Prioritization: the young star program is no longer considered auxiliary;
rather it is now one of the core programs of the survey.

  * •

Improved selection function: a homogeneous targeting strategy across the
entire sky would improve the subsequent modeling of the sample. We can no
longer treat individual star forming regions independently. Regardless of the
distance to a particular population, or the number of stars within it, a
“simple” selection of all young stars in the Solar Neighborhood is needed.

However, devising an homogeneous targeting strategy is rather arduous, as,
depending on their mass and age, young stars have a great degree of variety in
the observational signatures that could be used to confirm their youth. As
such there is no one single criterion that can uniformly select all young
stars at different evolutionary stages across all masses and distances. Driven
by these exigencies, we are forced to develop more sophisticated target
strategies than previously implemented.

In this paper we provide an overview of the criteria used to target the young
star in SDSS-V, how it has evolved to date, and the general observation
strategy. We also give a brief overview of the data collected during the first
year of operations, which began in 2021.

##  2 ABYSS overview

The APOGEE & BOSS Young Star Survey (ABYSS) is the name of the young star
program in SDSS-V. This program is set to produce optical and near-IR spectra
of $\sim 10^{5}$ young stars across the entire sky. The primary goals for
these data include (but not limited to):

  * •

Characterizing the dynamical, spatial, and temporal structure of individual
star forming regions, how these populations evolve over time.

  * •

Tracing of the global structure traced by the young stars, from the solar
neighborhood as a whole, to the Galactic scales. Characterizing the spiral arm
structure, and the kinematic and dynamic properties of the young Milky Way
disk.

  * •

Examining the connection between young stars and gas from which they have
formed.

  * •

Measuring the fundamental stellar properties of young stars, their comparison
to the state-of-the-art models of stellar structure and evolution. Examining
the role of the environment in which a star is born on these properties.

  * •

Characterizing multiplicity and orbital parameters of young stars across
different populations.

In this section we present the underlying logistics behind ABYSS, including
survey structure, data acquisition strategy, and target selection.

###  2.1 Survey strategy

SDSS-V utilizes two 2.5m telescopes; one is located in the northern hemisphere
at the Apache Point Observatory (APO), and the second one is located in the
southern hemisphere at Las Campanas Observatory (LCO) (Bowen & Vaughan, 1973;
Gunn et al., 2006; Blanton et al., 2017). Both of these telescopes have two
multi-object fiber spectrographs: APOGEE & BOSS. APOGEE covers H band, with
the wavelength of 1.51–1.7 $\mu$m with R$\sim 22,500$ (Wilson et al., 2010;
Majewski et al., 2017; Wilson et al., 2019). BOSS is an optical spectrograph,
covering wavelength range of $\sim$3600–10400 Å, with $R\sim$5000 (Smee et
al., 2013). A total of 300 APOGEE fibers and 500 BOSS fibers can be placed
simultaneously in 3∘ field of view at APO, and 2∘ at LCO, with the fiber
diameter of $\sim$2” and $\sim$1.3” respectively between these observatories.

With the new capability of SDSS to rapidly reconfigure the fiber placement due
to the robotic fiber positioners (Sayres et al., 2021), and the ambitious
goals of a comprehensive survey obtaining spectra across entire sky (Kollmeier
et al., 2017), the exposure time on all fields is set at 15 minutes.

SDSS-V divides its efforts into three mappers: Milky Way Mapper (MWM), Black
Hole Mapper (BHM), and Local Volume Mapper (LVM). All the programs aiming to
obtain stellar spectra, including ABYSS, are a part of MWM. Each program
within each mapper can define one or several “cartons”: a subset of stars
selected by a particular set of criteria that share the instrument
configuration, requirements on cadence, and number of epochs.

ABYSS has defined 8 cartons of stars to be observed with APOGEE, of which 5
cartons are considered to be optically bright to also be observed with BOSS.
These definitions are described in Section 2.2. A significant fraction of
targeted sources ($\sim$66%) are observed by both instruments. This offers
several advantages: precise sub-km s-1 radial velocities from high resolution
APOGEE spectra, versus various lines that can clearly inform on stellar youth
(such as Li I and H$\alpha$) that can be accessed with BOSS. The two
instruments have different faint limits set by the program: $H<13$ mag for
APOGEE and $G_{RP}<15.5$ mag for BOSS. The faint limit is set at the typical
magnitude that would reach signal-to-noise ratio of 30 in a coadded spectrum
of requisite number of 15 minute exposures, which was deemed sufficiently high
to extract the fundamental stellar parameters from the spectra.

These limits apply for all of the defined cartons. As some of the YSOs are too
faint to be detected in the optical regime due to extinction, some of the
cartons can only be observed with APOGEE. Individual stars can also meet the
faint limit in optical, or in infrared, but not both, as such, $\sim$20% of
the stars for either spectrograph are unique, with the remaining $\sim$80%
being targeted by both instruments.

In total for ABYSS targets, a “complete” set of observations requires 3 APOGEE
and 3111For fainter optical targets, 4 BOSS epochs are requested for sources
with 14.76$<G_{RP}<$15.075 mag, 5 epochs for 15.075$<G_{RP}<$15.29 mag, and 6
epochs for 15.29$<G_{RP}<$15.5 mag. BOSS epochs for a given object, brightness
limits permitting. Such a number of exposures is needed to confirm
multiplicity or variability within a spectrum. No firm constraints on the
cadence of observations have been imposed.

###  2.2 Carton definitions

Figure 1: Spatial distribution of sources in galactic coordinates for each of
the cartons.

In this subsection we present 8 independent definitions for cartons that were
used to target sources for observations as part of the ABYSS program (Figure
1). These cartons rely on various criteria, such as infrared excess, position
on the HR diagram, photometric variability, and membership of moving
groups/clusters/etc.

The initial set of observations conducted during the first 6 months of
operations, still using plug plates (see Section 3) relied on the V0 version
of the targeting. With the instrument upgraded to robotic positioners, the
targeting criteria were updated to V0.5. Transition to V1 will occur in 2023.
Data release 18 makes available V0.5 version of the targeting, which is the
focus of this paper. However, for the sake of the historical record, the full
evolution of the selection is described.

####  2.2.1 Disk

Figure 2: Criteria for the Disk and Embedded cartons, showing color-color
diagrams of disk-bearing dusty YSOs from Megeath et al. (2012) in Orion A & B
molecular clouds, used as a reference to constrain the targeting selection.
Sources in yellow are those with $G<18.5$, $\pi>0.3$ mas, used as a template
to map the Disk carton; the relevant color cuts to select this carton are
shown as red solid lines. Sources in blue are optically faint, representative
of the Embedded carton; the color cuts are shown in red dashed lines. Bottom
right panel shows the data for sources obtained as part of ABYSS observations
with APOGEE, separating the sources into likely YSOs and the contaminating red
giants through $\log g$$>$3.2 cut. The black line shows the color cut
introduced for the Embedded carton in the V1 targeting, as prior to this it
had significant contamination.

Historically, the YSOs that have been easiest to identify are those that have
large infrared excess due to the presence of a protoplanetary disk,
particularly in the mid-IR regime. In the last few decades, telescopes such as
Spitzer and WISE have particularly expanded the census of dusty YSOs. In
particular, WISE, due to being an all-sky survey, is particularly informative
for targeting. Several studies have used WISE to search for YSOs (e.g. Koenig
& Leisawitz, 2014; Kang et al., 2017; Marton et al., 2016), but they either
focused only on a specific star-forming region, or they had a large degree of
contamination across the entire sky.

As at this stage in the survey, the goal is to create a census of sources that
should be targeted for follow-up observations (rather than explicit
classification of YSOs into evolutionary stages), thus we use simple color
cuts in WISE photometry for this carton. To minimize a selection of very
distant, highly extincted field stars (which are the main source of
contamination), we impose a parallax cut as well - this implicitly requires
all of the identified sources to be bright enough in the optical regime to be
detected and have reliable astrometry with Gaia. We select sources satisfying:

  * •

$W1-W2>0.25$ mag

  * •

$W2-W3>0.5$ mag

  * •

$W3-W4>1.5$ mag

  * •

$\pi>0.3$ mas.

These cuts have been evaluated against known dusty YSOs in the Orion Molecular
Clouds (Megeath et al., 2012), and they are shown in Figure 2.

ALLWISE photometry was used for the selection. Although there have been recent
re-reductions, such as unWISE or neoWISE, their improvements are primarily in
W1 and W2 band photometry, W3 and W4 mostly remain as is. Longer wavelength
bands lack the sensitivity of shorter wavelength bands, furthermore, they have
not been observed for as long due to WISE running out of cryogenic coolant
needed to suppress telescope emission at these wavelengths. Nonetheless, W3
and W4 bands are critical for reliably identifying dusty disk-bearing stars.
As such, by requiring these bands and using merged photometry, any
improvements in W1 or W2 have negligible effect on the selection.

In the V0 version of the targeting, these cuts formed the basis of YSO_S1
carton. In version V0.5, the carton was renamed and split into YSO_Disk_APOGEE
and YSO_Disk_BOSS, containing 28,832 and 37,478 stars respectively. In version
V1, Gaia DR2 astrometry was upgraded to EDR3.

####  2.2.2 Embedded

Some disk-bearing sources are too faint to have reliable Gaia parallaxes. This
is usually the case for Class I protostars that are still embedded in their
natal envelopes, Class II YSOs that have edge-on disks, or for the sources
that are more distant and thus have more extinction along the line of sight.
Without a distance estimate, it can be difficult to separate bona-fide YSOs
from distant and heavily extincted red giants. As such, more stringent color
cuts, not just on ALLWISE photometry but also 2MASS, are required.

  * •

$G>18.5$ mag, or undetected

  * •

$J-H>$1 mag, $H-K>$0.5 mag

  * •

$W1-W2>0.5$ mag, $W2-W3>1$ mag, $W3-W4>1.5$ mag

  * •

$W3-W4>0.8\times(W1-W2)+1.1$ mag

These cuts are shown in Figure 2.

In V0 version of targeting, the carton was referred to as YSO_S2; in V0.5 it
was renamed as YSO_Embedded_APOGEE, containing 11,086 stars. Following the
first year of operations, the carton was re-examined; approximately half of
the observed sources were red giants, as shown in Fig. 2 (bottom right panel).
Therefore in V1 we added an additional cut

  * •

$H-K>0.65\times(J-H)-0.25$ mag

to minimize the contamination by evolved stars and further restrict the
selection to the parameter space that is most commonly inhabited by
spectroscopically confirmed YSOs, limiting the sample to 5,455 stars.

####  2.2.3 Nebula

Figure 3: Spatial distribution (in Galactic coordinates) of the selected
sources in Disk, Embedded, and Nebula cartons toward the Orion Nebula Cluster.
Note that the sources in the Nebula carton fill in a gap in the other two
cartons.

The selection criteria of disk-bearing and embedded sources rely on WISE bands
$W3$ and $W4$. $W3$ and $W4$ however become less effective in regions of high
nebulosity, as they saturate, producing gaps in the coverage. To fill them, we
used shorter wavelength data, using a set of criteria that is tuned to
autonomously find such nebulous regions:

  * •

If W4 is not reported, $W2-W3>4$ mag

  * •

If W3 and W4 are not reported, $J-H>1.1$ mag

This preferentially selects sources found in gaps of the previous cartons in
discrete regions on the sky, such as, e.g., in the center of the Orion Nebula
(Figure 3).

Additionally, some of the sources that are selected by these cuts are found
off of the Galactic plane and/or away from known star forming regions. This
creates an excess of targets in a narrow line following the scanning law of
WISE telescope, as such, these sources appear to be suspect. Thus, we also
required $b<5^{\circ}$ and a combination of $b>-5^{\circ}$ or $l>180^{\circ}$,
to exclude this contamination.

This carton was introduced in V0 as YSO_S2.5. In V0.5 it was renamed as
YSO_Nebula_APOGEE, containing 1,112 stars.

####  2.2.4 CMZ

The inner Galaxy, including the central molecular zone (CMZ), has been
surveyed with Spitzer as a part of GLIMPSE and MIPSGAL programs (Churchwell et
al., 2009; Carey et al., 2009; Gutermuth & Heyer, 2015). These data offer a
substantial improvement on sensitivity and resolution in comparison to WISE,
therefore they are advantageous in targeting stars outside of the Solar
Neighborhood.

Using the properties of massive YSOs towards the Galactic center identified by
An et al. (2011), we select a sample of candidates with

  * •

$[8.0]-[24]>2.5$ mag, i.e., very red sources, using photometry from Gutermuth
& Heyer (2015).

  * •

$\pi<0.2$ mas or not detected/measured, to ensure the sources are distant.

In V0 of targeting, YSO_CMZ carton also imposed a spatial limit of
$358<l<2^{\circ}$ and $-1<b<1^{\circ}$ to focus solely on the central
molecular zone. In V0.5, the carton was renamed to YSO_CMZ_APOGEE and removed
the spatial restriction, allowing the sources from the entire MIPSGAL
footprint. This enables to identify YSO candidates across the inner Galaxy,
containing 13,170 stars.

####  2.2.5 Variable

Figure 4: Criteria for the selection of stars in the Variable carton. Top
panels show variability distribution in the sample of stars toward the Orion
Complex, with known members highlighted in blue, and field stars shown in
yellow. The red lines show the cuts to the sample based on the correlation of
variability on the order of unity described in the text. Bottom panel shows
the full sample of stars within 500 pc that meets the minimum variability and
variability correlation cuts (See Section 2.2.5), showing the sources that
have been selected as YSO candidates in blue, and the other variable stars in
yellow.

On average, YSOs tend to be more variable than main sequence stars (e.g.,
Kounkel et al., 2022b). Part of the reason for this variability is the
presence of stronger magnetic fields that leads to more prominent star spots
with a larger filling factor. Also the presence of protoplanetary disks that
would occult the photosphere, and accretion events can lead to an increase in
brightness.

To estimate variability $V_{x}$, we use multi-epoch photometry by Gaia.
Following Belokurov et al. (2017), we define the photometric variability in a
given filter $x$ as:

$V_{x}=\sqrt{\mathrm{phot\\_x\\_n\\_obs}}/\mathrm{phot\\_x\\_mean\\_flux\\_over\\_error},$

where $x$ corresponds to the Gaia $G$, $G_{BP}$, or $G_{RP}$ bands,
$\mathrm{phot\\_x\\_n\\_obs}$ is the number of observations which contributed
to the photometry in a given band,
$\mathrm{phot\\_x\\_mean\\_flux\\_over\\_error}$ is the mean flux in a given
band divided by its error. For strongly variable sources, the photometric
uncertainty is comparable to the amplitude of variability.

Using the list of members of the Orion Complex from Kounkel et al. (2020) as a
representative sample of young stars, we develop a set of criteria based on
variability that allows to most cleanly preserve a large fraction of members
while rejecting field stars within the volume of space surrounding Orion. This
set of criteria was later applied to the entire sky and further modified to
preserve the morphology of nearby star forming regions and minimize
contamination, resulting in the following:

  * •

$V_{G}>0.02$, $V_{BP}>0.02$, $V_{RP}>0.02$. The reference YSOs tend to be more
variable than the field stars.

  * •

$V_{G}^{0.75}<V_{BP}<V_{G}$, $0.75V_{G}<V_{RP}<V_{G}^{0.95}$. Correlation of
variability in different bandpasses on the order of unity appears to be a
strong indicator of YSOs. On the other hand, while YSOs with different
correlation in variability do exist, it is difficult to reliably separate them
from the field stars, as such many young variable stars may be excluded from
the selection. The slopes of these power laws were determined from examining
Figure 4.

  * •

$G_{BP}-G_{RP}>1.3$. Hot stars do not have convective atmospheres, and thus
they generally don’t have spotted photospheres. While this specific cut is
redder than the convective limit, it also minimizes the extreme contamination
from the red giants. Variability among hotter stars may often be an indicator
of other processes, such as, e.g., eclipsing binaries, which is not an
indicator of youth.

  * •

$M_{BP}>5\log_{10}{V_{BP}}+11$ \- to preferentially exclude the evolved
subgiants, despite some overlap in the parameter space with bona-fide YSOs.

  * •

$2.5(G_{BP}-G_{RP})-1<M_{G}<2.5(G_{BP}-G_{RP})+2.5$. Previous criteria
preferentially select YSOs, but it still includes some strongly variable main
sequence stars. This selection confines the parameter space on the HR diagram
to minimize contamination.

  * •

$\pi>0.3$ mas, $G<18.5$, $H<13$, to limit the selection to the sources to
brighter stars with reliable parallaxes. Note that $H$ band cut is applied
both to APOGEE and to BOSS sample, as sources with $H>13$ & $G_{RP}<15.5$
preferentially trace out more distant stars that seem to be more strongly
contaminated that the sources within 1 kpc. As such, BOSS variable sample is a
subset of the APOGEE variable sample, with the faint limit in place.

These cuts are shown in Figure 4.

The carton based on this selection has been introduced in V0 as YSO_S3. In
V0.5 it has been renamed and split into YSO_Variable_APOGEE and
YSO_Variable_BOSS, containing 52,691 and 47,758 stars respectively. In V1, the
selection has been upgraded from Gaia DR2 data onto Gaia EDR3.

We note that Gaia DR3 has produced a catalog of young star candidates based on
their variability (Marton et al., 2022). The selection presented here was
originally derived prior to the availability of these data. Out of 79,375
sources presented in Gaia DR3, our selection has 3,963 stars in common with
this carton, and 16,474 stars across all of the cartons. Of the remaining
20,225 candidates in Gaia DR3 that would meet our faint limit, as much as a
half appear to be contamination from highly reddened distant main sequence and
red giant stars, but, in future versions of the targeting definition, with
some refinement it may be possible to take advantage of this catalog. On the
other hand, our current selection does not extend to as faint magnitudes (and,
indeed, applying the same criteria to fainter stars does appear to
significantly increase contamination), but it does appear to have greater
sensitivity to the populations with ages of up to a few 10s of Myr.

####  2.2.6 Cluster

Most of the selection criteria devised in this work preferentially target low-
mass YSOs, still in the pre-main sequence (PMS) phase of stellar evolution.
Young late B, A, & F stars reach the main sequence quickly and become
difficult to separate from field stars using conventional photometric
selection criteria. To identify them, it is however possible to take advantage
of the fact that young stars generally form in large associations, typically
with hundreds or thousands other members. Young populations tend to be
dynamically cold, with velocity dispersion of less than a few km s-1. Thus, it
is possible to find young moving groups by performing clustering analysis in
position and velocity phase space. As a selection of likely members using this
method does not have a dependence on the spectral type, it makes possible to
include young B, A, and F stars alongside later type stars.

Kounkel et al. (2020) have applied hierarchical clustering on Gaia DR2 data
within 3 kpc of the Sun. The initial data selection consisted of $\pi>0.2$
mas, $-30<b<30^{\circ}$, $v_{\alpha,\delta}^{lsr}<60$ km s-1, as well as
additional cuts based on astrometric and photometric quality. The clustering
was performed with HDBSCAN (Campello et al., 2013) in several slices in
distance and then stitched together. In total more than 8,000 moving groups
were identified consisting of $\sim$1 million stars. The ages of the
identified moving groups were estimated through an isochrone fitting using a
neural net Auriga (Kounkel et al., 2020).

We selected all sources in the moving groups with an age $\log
t\mathrm{(Myr)}<7.5$. The resulting subset forms the basis of YSO_Cluster
carton, introduced in V0 version of targeting, and split into
YSO_Cluster_APOGEE and YSO_Cluster_BOSS in V0.5, containing 45,461 and 59,065
stars respectively.

Older populations, with $\log t\mathrm{(Myr)}>7.5$ are being considered by the
survey as a part of an Open Fiber Program, but only as targets of opportunity,
with a single epoch obtained with either BOSS or APOGEE, and are not included
among the core programs of the survey.

Figure 5: Distribution of distances toward the stars in the optically bright
cartons.  Figure 6: A Venn diagram showing relative sizes of all the various
cartons in the ABYSS program, as well as the largest overlap in targets
between them. Note that due to a large number of cartons, overlap between some
of the cartons cannot be shown.  Figure 7: Distribution of ABYSS sources that
have been observed with SDSS through 2021. Stars in yellow have been observed
with APOGEE prior to the beginning of SDSS-V. Sources in blue are those that
have been observed as a part of SDSS-V plate program, typically with both
APOGEE and BOSS fibers assigned in a given field.

####  2.2.7 PMS

There have been several dedicated studies that focused on identifying pre-main
sequence stars using an HR diagram generated through Gaia photometry and
astrometry. We incorporate the resulting catalogs from two such works.

Zari et al. (2018) have selected low-mass pre-main sequence stars using Gaia
DR2 data within 500 pc that are found above (and therefore are younger than)
the 20 Myr PARSEC isochrone (Marigo et al., 2017) and fainter than $M_{G}>4$
mag. The photometry has been first extinction corrected, excluding sources
with $A_{G}<0.92$ mag, to avoid reddened field stars. Furthermore, additional
cuts have been made to select stars with low parallax errors
($\sigma_{\pi}/\pi<20$%), to limit the sample to disc stars (total tangential
velocity $\sqrt{v_{l}^{2}+v_{b}^{2}}<40$ km s-1), and to produce a ”clean” HR
diagram as suggested by Lindegren et al. (2018, Appendix C).

This selection is effective for nearby populations, but, at larger distances
the sample becomes strongly contaminated by field stars, both due to an
imperfect match of real photometry to the isochrones, and due to imperfections
in the extinction correction. An alternative approach was considered by
McBride et al. (2021), using a neural network Sagitta. It was trained on Gaia
and 2MASS photometry of stars in young populations from Kounkel et al. (2020)
to autonomously identify low-mass pre-main sequence stars, automatically
adjusting color-magnitude threshold based on age and distance of a star. The
constructed sample extended up to $\pi>0.2$ mas. It consisted of sources with
classification probability $>70$%, and it also had several data quality
criteria, such as $\sigma_{\pi}/\pi<10$% or $\sigma_{\pi}<$0.1 mas, precision
in Gaia photometry in all bands $<10$%, recommended cuts based on the
photometric excess noise, as well as Gaia RUWE$<$1.4.

These two catalogs form the basis of YSO_PMS_APOGEE and YSO_PMS_BOSS cartons
that were first introduced in V0.5 version of targeting, containing 76,332 and
73,213 stars respectively. Initially, both catalogs used Gaia DR2 data; in V1
the catalog from McBride et al. (2021) was upgraded to EDR3 version which (due
to magnitude limits) did not change substantially, except for a minor
improvements in sensitivity to more distant PMS stars.

####  2.2.8 OB

Most of the YSO cartons (with exception of Cluster and CMZ) focus exclusively
on low-mass YSOs, as they are most distinct from field stars. Intermediate and
massive stars, on the other hand, reach the main sequence very quickly, and
thus become difficult to differentiate.

However, as OB stars have short lifetimes, they would always be young. Thus,
to fill the gap in targeting, we selected sources based on examining the
placement of known OB stars from Maíz Apellániz et al. (2013).

  * •

$-0.2<(G_{BP}-G_{RP})<1.1$

  * •

$M_{G}<1.6(G_{BP}-G_{RP})-2.2$

  * •

$G<18$ mag, $\pi>0.3$ mas

In V0, this selection formed the basis of YSO_OB carton, which was split into
YSO_OB_APOGEE and YSO_OB_BOSS, both containing 8,670 stars. However, as all of
the selected stars are very bright (typically $G<12$ mag), they currently
cannot be observed with BOSS without offsetting their positions, due to the
saturation limit of the instrument.

In V0.5, these cartons were rendered obsolete, as all of the targets that are
a part of YSO_OB are a perfect subset of OBA_CORE program within SDSS-V (Zari
et al., 2021), thus, they do not require duplication of efforts from multiple
programs.

###  2.3 Sample summary

The map showing the spatial distribution of all stars in all cartons is shown
in Figure 1. Unsurprisingly, almost all sources are found along the Galactic
plane and Gould’s belt. The angular scale height of the disk between the
cartons strongly depends on the typical distance of the stars within it. Of
the optically bright sources, YSOs in the vicinity of the Solar Neighborhood
dominate the PMS carton, while sources found beyond $>$1 kpc dominate the OB
carton (Figure 5).

In total, V0.5 targeting sample consists of 202,726 sources to be observed
with APOGEE, and 196,188 sources to be observed with BOSS. There is some
overlap between the cartons, but, in general, most of the sources in each
carton are unique, sensitive to distinct tracers of youth (Figure 6).

##  3 First year data

Figure 8: Spectroscopic parameters extracted by APOGEE Net from the APOGEE
spectra for new and archival ABYSS targets, highlighting the typical
temperature range encompassed by each carton, as well as a typical rate of
contamination. Note that most sources occupy the parameter space expected by
the pre-main sequence stars, with only minor contamination from the red giants
in most cartons.

In this section we describe the data that have been obtained in the course of
the first few months of SDSS-V operations as well as the available archival
data for the ABYSS targets. We give an overview of the data processing
pipelines that are currently available to process these data, and their
fidelity.

###  3.1 Planning of observations

Although SDSS-V began its operations in 2021, it did not immediately reach its
full capability. As the transition to the robotic fiber positioning system
(FPS) in both hemispheres required significant instrument upgrades, initial
observations at APO were carried out with the SDSS-IV plug-in plate system
(Wilson et al., 2019) during the first six months of operations. As young
stars can often be found inside compact clusters, and the plug plates can pack
the fibers closer together than the robot positioners, such a configuration
was deemed advantageous for ABYSS. Thus, several plates were commissioned to
target fields with the highest density of sources that had not been targeted
in previous iterations of the survey. Additionally, YSO targets were included
in the 2021 plates led by other programs.

In the previous iterations of the survey, BOSS was primarily used in dark time
to look at faint targets. As such, to avoid saturation for a 15-minute
exposure, the nominal bright limit for BOSS is $G>13$ mag, whereas for APOGEE
is $H>7$ mag. In the near future, such limits will be overcome through
offsetting the fiber position from the position of a star. Meanwhile, for the
first year plate operations it was decided that ABYSS targets would be split:
sources brighter than $G_{RP}<13$ mag would be observed with APOGEE, and
fainter stars would be observed with BOSS. At the moment, the split is sub-
optimal, due to a resulting segregation in mass. However, once instrumental
limitations are lifted, the survey is expected to obtain a complementary set
of observations to compensate this problem.

The SDSS-V plate program has obtained spectra for 2,462 ABYSS targets with
APOGEE, and 4,854 targets with BOSS. These data can be supplemented by 7,884
sources for which archival APOGEE spectra exists as part of DR17 (Abdurro’uf
et al., 2022). These include young stars targeted explicitly by SDSS III & IV,
and those that have been serendipitously targeted by other programs (Figure
7). In these data, 193 sources have both APOGEE and BOSS spectra.

###  3.2 APOGEE

Due to the extensive history of observations, various pipelines have been
developed to measure stellar parameters (such as $T_{eff}$ and $\log g$) from
APOGEE spectra (Cottaar et al., 2014; Olney et al., 2020; Sprague et al.,
2022). The latest iteration of these pipelines, APOGEE Net, can extract
parameters for all stars with $T_{eff}$$>3000$ K in a self-consistent manner,
and for PMS stars its $\log g$ is sensitive to age. Thus, using the latest
iteration of the APOGEE Net pipeline, we qualitatively evaluate how prone is
each individual carton to field contamination from red giants (Figure 8). Full
quantitative assessment that also considers contamination from the older main
sequence stars will be presented in future papers in this series.

Optically bright cartons (PMS, Variable, Cluster, Disk, and OB) have only
minimal contamination from red giants ($\sim$4%) relative to the total number
of sources. Approximately half of the sources in the Embedded carton are red
giants. The bulk of this contamination can be reduced in the future through
the 2MASS color cut introduced in V1 version of targeting (Figure 2). The
contamination rate for other two optically faint cartons, CMZ and Nebula, is
difficult to evaluate at the moment due to the limited size of the samples, as
existing SDSS-V observations do not cover the region of space where the bulk
of these targets reside. Similarly, archival DR17 observations impose their
own targeting selections that may favor a particular type of sources, as such
they may not necessarily be representative of a full set of sources for these
cartons. E.g., as during SDSS-III and -IV, the primary targets for the survey
were stars that have photometry consistent with being red giants, which may
make the red giant contaminants to be over-represented in the sample observed
prior to SDSS-V.

Different cartons favor different temperature ranges. The PMS and Variable
cartons typically select sub-solar systems: they are designed for low-mass
stars that are convective and are slow to reach the main sequence. Disk and
Embedded cartons also favor low-mass stars: the Embedded carton is likely to
have fewer massive stars than the Disk carton, as higher mass stars can remain
optically bright through a higher degree of extinction. Nonetheless, low-mass
stars have longer disk lifetimes (e.g., Bayo et al., 2012; Ribas et al.,
2015). Sources in the Cluster carton are not based on stellar properties of
individual stars, but rather on their membership, as such stars of all masses
are represented in it. Finally, sources in OB carton favor sources with
$T_{eff}$$>$10,000 K. In comparison to massive stars in the Cluster carton,
sources in the OB carton tend to skew towards hotter $T_{eff}$ and lower $\log
g$ (on average by $\sim 0.1$ dex).

Typically the lowest $T_{eff}$ observed in the DR17 data is lower than is
currently available for SDSS-V data. As mentioned previously, this is due to
preferentially observing brighter stars with APOGEE due to the saturation
limit in BOSS.

###  3.3 BOSS

Figure 9: An example of a BOSS spectrum of a young star. The inlay shows the
Li I 6707.7 Å line.

Figure 10: Velocity structure observed towards a field observed by ABYSS
towards Cam OB1 association. Three kinematically coherent groups are located
towards it, all found in a similar location on the sky, and at a similar
distance. These groups are distinguishable in the proper motion space (left),
and in RV space (right, same colors). The same velocity structure is recovered
both by APOGEE (thick unshaded curve) and BOSS (shaded curve).

YSOs have a long history of being observed with APOGEE. On the other hand,
fewer than 40 ABYSS targets have archival BOSS observations, most of them
concentrated near 25 Ori (Suárez et al., 2017). LAMOST (Large Sky Area Multi-
Object Fiber Spectroscopic Telescope) low resolution spectroscopy is
comparable to BOSS, both in wavelength coverage and spectral resolution. In
LAMOST DR7, spectra are available for 13,452 ABYSS targets, including several
prominent star forming regions. Despite this, to date only a few studies of
YSOs utilized LAMOST spectra (e.g., Liu et al., 2021, Hernandez, J. et al. in
prep).

As such, with the exclusion of radial velocities (RVs), currently there are no
pipelines capable of extracting reliable stellar parameters from either BOSS
or LAMOST spectra of low-mass YSOs. There are however existing efforts to
rectify this, both by measuring reliable $T_{eff}$ & $\log g$ (e.g., Sizemore,
L. et al. in prep), as well as characterizing a number of features found in
the spectral range of the instrument that could be used as indicators of youth
(such as Li I, or various emission lines for H or Ca; Saad, S., et al. in
prep; Figure 9).

At the moment, BOSS spectra are processed by pyXCSAO (Kounkel, 2022), which is
a Python implementation of IRAF RVSAO package (Tonry & Davis, 1979; Kurtz &
Mink, 1998). The spectra are cross-correlated against synthetic PHOENIX
templates (Husser et al., 2013). Subgrid solutions are derived for parameters
such as $T_{eff}$ and $\log g$ using the quality of the fit. However, there
are significant systematics that affect the quality of the derived parameters
for the young stars in particular, not dissimilar to what was observed in the
original APOGEE YSO pipeline (Cottaar et al., 2014). RVs are measured from the
best fitting template for stars with $T_{eff}$$>$3,500 K. RVs for cooler stars
are determined from best fitting 3,500 K template, as they otherwise appear to
be systematically redshifted relative to the rest velocity of other stars in
the same star forming region. This has also been seen in the APOGEE data,
(e.g., Kounkel et al., 2019), due to a systematic issue in synthetic spectra
of cool stars.

APOGEE can achieve sub-km s-1 precision in its measured RVs. BOSS, being a
lower resolution instrument, can produce RV precision of only $\sim$4–5 km s-1
for spectra of low-mass stars with high signal-to-noise. Nonetheless, both
instruments have been vetted to ensure consistent performance and a lack of a
zero-point offset between them (Figure 10). Note that due to a dearth of
sources observed both with APOGEE and with BOSS so far, a direct comparison of
RVs of specific stars (as opposed to populations as a whole) is not currently
possible.

We defer a more detailed analysis of the BOSS spectra and the parameters to
subsequent publications in the series.

##  4 Gaia DR3 comparison

[h]

Figure 11: A comparison of RVs for young stars between APOGEE and Gaia DR3.
Sources have been selected from Kounkel et al. (2019) for which at least 3
APOGEE epochs have been obtained to confirm their RV stability, excluding any
of the spectroscopic binaries. The typical precision in RV is $<$1 km s-1 for
APOGEE, and $\sim$6 km s-1 for Gaia.

Figure 12: A comparison of APOGEE-derived $T_{eff}$ & $\log g$ for the ABYSS
objects, vs those in Gaia DR3.

The recent third Gaia data release (Gaia DR3, Gaia Collaboration et al., 2022)
has made available not just the astrometric parameters that have been present
in Gaia DR2 or EDR3, but also RVs for 30 million stars derived with its on-
board spectrograph (Katz et al., 2022), as well as stellar parameters such as
$T_{eff}$ and $\log g$ for 5.5 million stars (Fouesneau et al., 2022). This
census includes a number of ABYSS-targeted objects.

Unfortunately, these parameters were optimized to produce accurate solutions
for “typical” stars. YSOs do not fall into this category, due to a number of
unique spectral features they exhibit from both accretion and activity,
especially near Ca II triplet at $\sim$8500 Å, which is tightly encompassed in
the spectral range of Gaia RVS spectrograph (Recio-Blanco et al., 2022). Thus,
caution has to be utilized in interpreting the available parameters in this
data release for the young stars.

In particular, Kounkel et al. (2022a) identified issues in Gaia RVs for YSOs:
1) they are not precise, as they have typical uncertainties of 5–10 km s-1
which is significantly worse than the instrumental limit at the typical
signal-to-noise of these observations. 2) they are not accurate as when
evaluated against high resolution RVs from existing APOGEE observations of
spectroscopically stable young stars, they typically show a scatter $>5\sigma$
(Figure 11). This issue persists for more than 100 Myr, as a large scatter in
RVs among low-mass stars is detected in a (comparatively) older cluster such
as the Pleiades.

Similarly, we evaluate $T_{eff}$ and $\log g$ of the ABYSS stars that Gaia has
observed, and compare them against the derived parameters from APOGEE (Figure
12). We find that the parameter space occupied by cool pre-main sequence stars
($T_{eff}$$<$5000 K, low $\log g$) is currently not well sampled by the
pipelines employed by the Gaia consortium. The sources that occupy this
parameter space are often recognized to have lower $\log g$ than what is
commonly found in main sequence stars, but they are pushed towards hotter
$T_{eff}$, which results in placing them towards the red giant branch. No
correlation is found between the reported $\log g$ values for low mass stars
between these two data sets, however. Further, the Gaia-derived $\log g$ are
not sensitive to stellar ages, unlike those from APOGEE (e.g., Olney et al.,
2020; Kounkel et al., 2022c). Similar caution should be given to other derived
parameters produced by Gaia for these stars, such as, for instance ages.

We stress that this issue is specific to young stars, most prominently on the
low-mass end, and should not affect more evolved main sequence stars that are
older than a few 100s Myr. In future releases, the Gaia spectra may be
reprocessed with a pipeline that is better tuned to YSOs, both within the
collaboration or through efforts using the publicly released spectra. However,
as Gaia DR4 is not expected until 2026 at the earliest, by which point ABYSS
is expected to approach completion. As such, ABYSS will be the first
comprehensive spectroscopic census of young stars across the entire sky.

##  5 Summary

We present the selection strategy of young stars across the entire sky
targeted by SDSS-V. The target catalogues are released publicly as a part of
data release 18. In total, this selection has resulted in a sample of
$\sim$200,000 sources, down to the limiting magnitude of $H<13$ mag or
$G_{RP}<15.5$ mag, encompassing both diffuse associations as well as compact
massive complexes of young stars across a range of distances, from the solar
neighbourhood to the inner Galaxy. In future years, either optical or near-
infrared spectra (or both) are expected to be available for most of these
sources.

The selection strategy for ABYSS is complex, and relies on a variety of
different tracers of youth, in an attempt to create as complete, as
homogeneous, and as clean sample as possible, across all masses and ages
younger than $\sim$30 Myr. A preliminary examination of the data shows that
the vast majority of the sources observed to-date exhibit spectroscopic
signatures of youth, although some fraction of contamination from more evolved
sources (main sequence or red giants) is present across all cartons. Future
studies employing larger data sets will precisely quantify the contamination
level and allow to select cleaner samples. The development of dedicated
pipelines to derive accurate stellar parameters is underway.

The previous iterations of SDSS have produced a spectroscopic census of young
stars across several selected star forming regions. These data have been
instrumental in understanding the star formation history and the three-
dimensional kinematics of star forming regions as a whole (e.g., Foster et
al., 2015; Da Rio et al., 2016; Galli et al., 2019; Kounkel et al., 2022c), as
well as properties of individual stars, such as accretion (Campbell et al.,
2022), multiplicity (Kounkel et al., 2019), evolutionary properties (e.g.,
Serna et al., 2021; Cao et al., 2022), and stellar parameters (e.g., Roman-
Lopes et al., 2019; Ramírez-Preciado et al., 2020). Similarly to SDSS, other
spectroscopic surveys have targeted nearby star forming regions, such as GALAH
(Kos et al., 2021), or with Gaia-ESO (e.g., Sacco et al., 2015; Bouvier et
al., 2016). Taken together however, these efforts amounted to only
$\sim$10,000 young stars across a very limited footprint on the sky that
generally could not encompass even a given extended population in full. ABYSS
will expand the total spectroscopic census of young stars by more than an
order of magnitude, without the stringent spatial restrictions that were
necessary in the past and it will substantially increase our ability to probe
the recent epoch of star formation in the Galaxy as a whole. Finally, while
sophisticated, the homogeneous selection function described in this study is a
significant improvement on previous efforts that were specifically tailored to
individual star forming regions (Román-Zúñiga et al., 2022), as it enables a
more direct comparison between them.

In the next few years this survey will be complemented by the 4MOST Survey of
Young Stars (4SYS), and we expect future Gaia data releases to improve the
processing of the RVS spectra of young stars. However, ABYSS is the first
major spectroscopic survey to focus on young stars across the sky and to
provide accurate stellar parameters for large, statistical samples. With
yearly data releases (DR19 onward) yielding spectra and stellar parameters, it
is our hope that these data will be of value to the community.

AS gratefully acknowledges support by the Fondecyt Regular (project code
1220610), and ANID BASAL projects ACE210002 and FB210003. C.R-Z acknowledges
support from projects CONACYT CB2018 A1S-9754, Mexico and UNAM DGAPA PAPIIT
IN112620, Mexico. The research leading to these results has (partially)
received funding from the KU Leuven Research Council (grant C16/18/005:
PARADISE) and from the BELgian federal Science Policy Office (BELSPO) through
PRODEX grant PLATO. K.P.R. acknowledges support from ANID FONDECYT Iniciación
11201161. A.B. acknowledges partial funding by the Deutsche
Forschungsgemeinschaft Excellence Strategy - EXC 2094 - 390783311 and the ANID
BASAL project FB210003. RLV acknowledges support from CONACYT through a
postdoctoral fellowship within the program ‘Estancias Posdoctorales por
México’. B.R-A acknowledges funding support from FONDECYT Iniciación grant
11181295 and ANID Basal project FB210003. Funding for the Sloan Digital Sky
Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-
Simons Foundation, the National Science Foundation, and the Participating
Institutions. SDSS acknowledges support and resources from the Center for
High-Performance Computing at the University of Utah. The SDSS web site is
www.sdss5.org. SDSS is managed by the Astrophysical Research Consortium for
the Participating Institutions of the SDSS Collaboration, including the
Carnegie Institution for Science, Chilean National Time Allocation Committee
(CNTAC) ratified researchers, the Gotham Participation Group, Harvard
University, Heidelberg University, The Johns Hopkins University, L’Ecole
polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik
Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-
Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University,
National Astronomical Observatories of China (NAOC), New Mexico State
University, The Ohio State University, Pennsylvania State University,
Smithsonian Astrophysical Observatory, Space Telescope Science Institute
(STScI), the Stellar Astrophysics Participation Group, Universidad Nacional
Autónoma de México, University of Arizona, University of Colorado Boulder,
University of Illinois at Urbana-Champaign, University of Toronto, University
of Utah, University of Virginia, Yale University, and Yunnan University. This
work has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## References

  * Abdurro’uf et al. (2022) Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35, doi: 10.3847/1538-4365/ac4414
  * An et al. (2011) An, D., Ramírez, S. V., Sellgren, K., et al. 2011, ApJ, 736, 133, doi: 10.1088/0004-637X/736/2/133
  * Bayo et al. (2012) Bayo, A., Barrado, D., Huélamo, N., et al. 2012, A&A, 547, A80, doi: 10.1051/0004-6361/201219374
  * Beaton et al. (2021) Beaton, R. L., Oelkers, R. J., Hayes, C. R., et al. 2021, AJ, 162, 302, doi: 10.3847/1538-3881/ac260c
  * Belokurov et al. (2017) Belokurov, V., Erkal, D., Deason, A. J., et al. 2017, MNRAS, 466, 4711, doi: 10.1093/mnras/stw3357
  * Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28, doi: 10.3847/1538-3881/aa7567
  * Bouvier et al. (2016) Bouvier, J., Lanzafame, A. C., Venuti, L., et al. 2016, A&A, 590, A78, doi: 10.1051/0004-6361/201628336
  * Bowen & Vaughan (1973) Bowen, I. S., & Vaughan, A. H., J. 1973, Appl. Opt., 12, 1430, doi: 10.1364/AO.12.001430
  * Campbell et al. (2022) Campbell, H., Khilfeh, E., Covey, K. R., et al. 2022, arXiv e-prints, arXiv:2211.06454.  https://arxiv.org/abs/2211.06454
  * Campello et al. (2013) Campello, R. J. G. B., Moulavi, D., & Sander, J. 2013, in Advances in Knowledge Discovery and Data Mining, ed. J. Pei, V. S. Tseng, L. Cao, H. Motoda, & G. Xu (Berlin, Heidelberg: Springer Berlin Heidelberg), 160–172
  * Cao et al. (2022) Cao, L., Pinsonneault, M. H., Hillenbrand, L. A., & Kuhn, M. A. 2022, ApJ, 924, 84, doi: 10.3847/1538-4357/ac307f
  * Carey et al. (2009) Carey, S. J., Noriega-Crespo, A., Mizuno, D. R., et al. 2009, PASP, 121, 76, doi: 10.1086/596581
  * Churchwell et al. (2009) Churchwell, E., Babler, B. L., Meade, M. R., et al. 2009, PASP, 121, 213, doi: 10.1086/597811
  * Cottaar et al. (2014) Cottaar, M., Covey, K. R., Meyer, M. R., et al. 2014, ApJ, 794, 125, doi: 10.1088/0004-637X/794/2/125
  * Cottaar et al. (2015) Cottaar, M., Covey, K. R., Foster, J. B., et al. 2015, ApJ, 807, 27, doi: 10.1088/0004-637X/807/1/27
  * Cottle et al. (2018) Cottle, J., Covey, K. R., Suárez, G., et al. 2018, ApJS, 236, 27, doi: 10.3847/1538-4365/aabada
  * Da Rio et al. (2016) Da Rio, N., Tan, J. C., Covey, K. R., et al. 2016, ApJ, 818, 59, doi: 10.3847/0004-637X/818/1/59
  * Foster et al. (2015) Foster, J. B., Cottaar, M., Covey, K. R., et al. 2015, ApJ, 799, 136, doi: 10.1088/0004-637X/799/2/136
  * Fouesneau et al. (2022) Fouesneau, M., Frémat, Y., Andrae, R., et al. 2022, arXiv e-prints, arXiv:2206.05992.  https://arxiv.org/abs/2206.05992
  * Gaia Collaboration et al. (2022) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2022, arXiv e-prints, arXiv:2208.00211.  https://arxiv.org/abs/2208.00211
  * Galli et al. (2019) Galli, P. A. B., Loinard, L., Bouy, H., et al. 2019, A&A, 630, A137, doi: 10.1051/0004-6361/201935928
  * Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332, doi: 10.1086/500975
  * Gutermuth & Heyer (2015) Gutermuth, R. A., & Heyer, M. 2015, AJ, 149, 64, doi: 10.1088/0004-6256/149/2/64
  * Husser et al. (2013) Husser, T.-O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6, doi: 10.1051/0004-6361/201219058
  * Kang et al. (2017) Kang, S.-J., Kerton, C. R., Choi, M., & Kang, M. 2017, ApJ, 845, 21, doi: 10.3847/1538-4357/aa7da3
  * Katz et al. (2022) Katz, D., Sartoretti, P., Guerrier, A., et al. 2022, arXiv e-prints, arXiv:2206.05902.  https://arxiv.org/abs/2206.05902
  * Koenig & Leisawitz (2014) Koenig, X. P., & Leisawitz, D. T. 2014, ApJ, 791, 131, doi: 10.1088/0004-637X/791/2/131
  * Kollmeier et al. (2017) Kollmeier, J. A., Zasowski, G., Rix, H.-W., et al. 2017, arXiv e-prints, arXiv:1711.03234.  https://arxiv.org/abs/1711.03234
  * Kos et al. (2021) Kos, J., Bland-Hawthorn, J., Buder, S., et al. 2021, MNRAS, 506, 4232, doi: 10.1093/mnras/stab1767
  * Kounkel (2022) Kounkel, M. 2022, PyXCSAO, 0.2, Zenodo, doi: 10.5281/zenodo.6998993
  * Kounkel et al. (2020) Kounkel, M., Covey, K., & Stassun, K. G. 2020, AJ, 160, 279, doi: 10.3847/1538-3881/abc0e6
  * Kounkel et al. (2022a) Kounkel, M., Mcbride, A., Stassun, K. G., & Leigh, N. 2022a, MNRAS, 517, 1946, doi: 10.1093/mnras/stac2829
  * Kounkel et al. (2022b) Kounkel, M., Stassun, K. G., Bouma, L. G., et al. 2022b, AJ, 164, 137, doi: 10.3847/1538-3881/ac866d
  * Kounkel et al. (2022c) Kounkel, M., Stassun, K. G., Covey, K., & Hartmann, L. 2022c, MNRAS, 517, 161, doi: 10.1093/mnras/stac2695
  * Kounkel et al. (2019) Kounkel, M., Covey, K., Moe, M., et al. 2019, AJ, 157, 196, doi: 10.3847/1538-3881/ab13b1
  * Kurtz & Mink (1998) Kurtz, M. J., & Mink, D. J. 1998, PASP, 110, 934, doi: 10.1086/316207
  * Lindegren et al. (2018) Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, A&A, 616, A2, doi: 10.1051/0004-6361/201832727
  * Liu et al. (2021) Liu, J., Fang, M., Tian, H., et al. 2021, ApJS, 254, 20, doi: 10.3847/1538-4365/abf4d1
  * Maíz Apellániz et al. (2013) Maíz Apellániz, J., Sota, A., Morrell, N. I., et al. 2013, in Massive Stars: From alpha to Omega, 198.  https://arxiv.org/abs/1306.6417
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94, doi: 10.3847/1538-3881/aa784d
  * Marigo et al. (2017) Marigo, P., Girardi, L., Bressan, A., et al. 2017, ApJ, 835, 77, doi: 10.3847/1538-4357/835/1/77
  * Marton et al. (2016) Marton, G., Tóth, L. V., Paladini, R., et al. 2016, MNRAS, 458, 3479, doi: 10.1093/mnras/stw398
  * Marton et al. (2022) Marton, G., Ábrahám, P., Rimoldini, L., et al. 2022, arXiv e-prints, arXiv:2206.05796.  https://arxiv.org/abs/2206.05796
  * McBride et al. (2021) McBride, A., Lingg, R., Kounkel, M., Covey, K., & Hutchinson, B. 2021, AJ, 162, 282, doi: 10.3847/1538-3881/ac2432
  * Megeath et al. (2012) Megeath, S. T., Gutermuth, R., Muzerolle, J., et al. 2012, AJ, 144, 192, doi: 10.1088/0004-6256/144/6/192
  * Olney et al. (2020) Olney, R., Kounkel, M., Schillinger, C., et al. 2020, AJ, 159, 182, doi: 10.3847/1538-3881/ab7a97
  * Pogge et al. (2020) Pogge, R. W., Derwent, M. A., O’Brien, T. P., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11447, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1144781, doi: 10.1117/12.2561113
  * Ramírez-Preciado et al. (2020) Ramírez-Preciado, V. G., Roman-Lopes, A., Román-Zúñiga, C. G., et al. 2020, ApJ, 894, 5, doi: 10.3847/1538-4357/ab8127
  * Recio-Blanco et al. (2022) Recio-Blanco, A., de Laverny, P., Palicio, P. A., et al. 2022, arXiv e-prints, arXiv:2206.05541.  https://arxiv.org/abs/2206.05541
  * Ribas et al. (2015) Ribas, Á., Bouy, H., & Merín, B. 2015, A&A, 576, A52, doi: 10.1051/0004-6361/201424846
  * Roman-Lopes et al. (2019) Roman-Lopes, A., Román-Zúñiga, C. G., Tapia, M., et al. 2019, ApJ, 873, 66, doi: 10.3847/1538-4357/ab0305
  * Román-Zúñiga et al. (2022) Román-Zúñiga, C. G., Kounkel, M., Hernández, J., et al. 2022, arXiv e-prints, arXiv:2211.09217.  https://arxiv.org/abs/2211.09217
  * Sacco et al. (2015) Sacco, G. G., Jeffries, R. D., Randich, S., et al. 2015, A&A, 574, L7, doi: 10.1051/0004-6361/201425367
  * Santana et al. (2021) Santana, F. A., Beaton, R. L., Covey, K. R., et al. 2021, AJ, 162, 303, doi: 10.3847/1538-3881/ac2cbc
  * Sayres et al. (2021) Sayres, C., Sánchez-Gallego, J. R., Blanton, M. R., et al. 2021, AJ, 161, 92, doi: 10.3847/1538-3881/abd0f2
  * Serna et al. (2021) Serna, J., Hernandez, J., Kounkel, M., et al. 2021, ApJ, 923, 177, doi: 10.3847/1538-4357/ac300a
  * Smee et al. (2013) Smee, S. A., Gunn, J. E., Uomoto, A., et al. 2013, AJ, 146, 32, doi: 10.1088/0004-6256/146/2/32
  * Sprague et al. (2022) Sprague, D., Culhane, C., Kounkel, M., et al. 2022, AJ, 163, 152, doi: 10.3847/1538-3881/ac4de7
  * Suárez et al. (2017) Suárez, G., Downes, J. J., Román-Zúñiga, C., et al. 2017, AJ, 154, 14, doi: 10.3847/1538-3881/aa733a
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Tonry & Davis (1979) Tonry, J., & Davis, M. 1979, AJ, 84, 1511, doi: 10.1086/112569
  * Wilson et al. (2010) Wilson, J. C., Hearty, F., Skrutskie, M. F., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 77351C, doi: 10.1117/12.856708
  * Wilson et al. (2019) Wilson, J. C., Hearty, F. R., Skrutskie, M. F., et al. 2019, PASP, 131, 055001, doi: 10.1088/1538-3873/ab0075
  * Zari et al. (2018) Zari, E., Hashemi, H., Brown, A. G. A., Jardine, K., & de Zeeuw, P. T. 2018, A&A, 620, A172, doi: 10.1051/0004-6361/201834150
  * Zari et al. (2021) Zari, E., Rix, H. W., Frankel, N., et al. 2021, A&A, 650, A112, doi: 10.1051/0004-6361/202039726


# Space-time structured plasma waves

J.P. Palastro<EMAIL_ADDRESS>University of Rochester, Laboratory for
Laser Energetics, Rochester, New York 14623-1299 USA  K.G. Miller  University
of Rochester, Laboratory for Laser Energetics, Rochester, New York 14623-1299
USA  R.K. Follett  University of Rochester, Laboratory for Laser Energetics,
Rochester, New York 14623-1299 USA  D. Ramsey  University of Rochester,
Laboratory for Laser Energetics, Rochester, New York 14623-1299 USA  K.
Weichman  University of Rochester, Laboratory for Laser Energetics, Rochester,
New York 14623-1299 USA  A.V. Arefiev  Department of Mechanical and Aerospace
Engineering, University of California at San Diego, La Jolla, California
92093, USA  D.H. Froula  University of Rochester, Laboratory for Laser
Energetics, Rochester, New York 14623-1299 USA

###### Abstract

Electrostatic waves play a critical role in nearly every branch of plasma
physics from fusion to advanced accelerators, to astro, solar, and ionospheric
physics. The properties of planar electrostatic waves are fully determined by
the plasma conditions, such as density, temperature, ionization state, or
details of the distribution functions. Here we demonstrate that electrostatic
wave packets structured with space-time correlations can have properties that
are independent of the plasma conditions. For instance, an appropriately
structured electrostatic wave packet can travel at any group velocity, even
backward with respect to its phase fronts, while maintaining a localized
energy density. These linear, propagation-invariant wave packets can be
constructed with or without orbital angular momentum by superposing natural
modes of the plasma and can be ponderomotively excited by space-time
structured laser pulses like the flying focus.

A defining characteristic of plasma is its ability to exhibit collective
motion. This motion often manifests as coordinated oscillations of the
constituent particles, mediated by their mutual electrostatic attraction or
repulsion. The oscillations, or electrostatic waves, play a critical role in
nearly every branch of plasma physics. In fusion, electrostatic waves can be
both a feature, providing a means to measure plasma conditions 1; 2; 3; 4; 5;
6, and an impediment, growing unstably to the point of disrupting plasma
confinement and heating 7; 8; 9; 10; 11; 12; 13; 14. Advanced accelerators
harness electrostatic waves to accelerate electrons to relativistic energies
over short distances, with the ultimate goal of miniaturizing radiation
sources and particle colliders 15; 16; 17; 18; 19; 20; 21; 22; 23; 24. As a
final, naturally occurring example, the mode conversion of electrostatic waves
driven by fast electrons can explain the emission of type III radio bursts
from the solar wind 25; 26; 27.

In each of these systems, the evolution of electrostatic waves impacts
performance, dynamics, or observations. The evolution of planar electrostatic
waves, i.e., waves having a single frequency $\omega$ and wavevector
$\mathbf{k}$, is fully determined by the plasma conditions through the
dispersion relation $\varepsilon(\omega,\mathbf{k})=0$. More specifically, the
phase velocity $\mathbf{v}_{p}=[\omega(\mathbf{k})/k]\mathbf{e}_{\mathbf{k}}$
can depend on the density, temperature, ionization states, or details of the
distribution functions. Physically occurring electrostatic waves exist as
superpositions of plane waves with amplitudes and phases imposed by a driver,
such as an intense laser pulse or charged particle beam. A typical driver
excites the wave packets without introducing correlations in
$(\omega,\mathbf{k})$ space. As a result, the wave packets retain properties
similar to those of a plane wave. However, electrostatic wave packets can also
be driven so that they feature correlations in $(\omega,\mathbf{k})$ space.
With appropriate structuring, these correlations can produce emergent
properties that are independent of the plasma conditions.

The structuring of electromagnetic waves with space-time correlations has
provided new opportunities for laser-based applications and basic science 28;
29; 30; 31; 32; 33; 34; 35. This has motivated the development of optical
techniques for creating structured light, such as propagation-invariant 36;
37; 38; 39; 40; 41; 42, flying focus 43; 44; 30; 45; 46; 47, and arbitrarily-
structured-laser (ASTRL) pulses 48. While these techniques cannot be directly
applied to electrostatic waves, much of the mathematical formalism carries
over: at a fundamental level, all waves evolve according to a wave equation.
Thus, by using an appropriate driver, one can construct electrostatic analogs
to propagation-invariant, flying focus, or ASTRL pulses.

This manuscript introduces the concept of space-time structured plasma waves.
A space-time structured plasma wave (STP) can be constructed, with our without
orbital angular momentum, by superposing natural electrostatic modes of a
plasma with a particular correlation in $(\omega,\mathbf{k})$ space. As an
example, we focus on the special case of a linear, propagation-invariant
electrostatic wave packet with a group velocity that is independent of the
plasma conditions. The excitation of such an STP can be achieved
experimentally by using the ponderomotive force of a structured laser pulse
like a flying focus. STPs offer a new class of collective excitations that may
provide additional control over dynamics such as wave-particle interactions,
particularly in situations where the driver can be structured.

Figure 1 contrasts a conventional, localized plasma wave with an STP. The
conventional plasma wave propagates with a group velocity determined by the
plasma conditions. As the wave propagates, diffraction causes a rapid drop in
the peak energy density. The STP travels at a velocity that is independent of
the plasma conditions and maintains its profile, and peak energy density, over
an extended distance. In this example, the peak energy density travels in the
opposite direction as the phase fronts and the nominal group velocity.

Figure 1: Evolution of the cycle-averaged energy density
$\varepsilon_{0}\langle k_{0}^{2}\phi^{2}\rangle$ for a conventional and
space-time structured plasma wave (STP). The conventional plasma wave (left)
diffracts as it propagates from left to right at a nominal group velocity
$v_{n}$ determined by the plasma conditions. The peak energy density of the
STP (right) travels in the opposite direction as the nominal group velocity
and phase velocity while maintaining a constant spatiotemporal profile. In
both cases, $k_{0}w_{0}=20$. The STP has $v_{g}=-v_{n}$ and $\ell=1$. For the
conventional plasma wave, $Z_{0}=4.5w_{0}$ and $\ell=0$. Space is normalized
by $w_{0}$ and time by $\tau=\omega_{0}w_{0}^{2}/2u^{2}$ [see Table I and Eqs.
(29) and (30)]. The contours have the same normalization, while each
projection is normalized to its maximum.

The formulation of STPs will be presented for pure electrostatic waves in the
absence of external fields. Pure electrostatic plane waves have a wavevector
that is parallel to their electric field $\mathbf{E}$ and have no magnetic
field, i.e., $\mathbf{k}\times\mathbf{E}=0$. These waves are completely
described by their electrostatic potential. The electrostatic potential $\phi$
of a plasma wave packet can be expressed as a superposition of plane waves
constrained by the dispersion relation:

$\phi(\mathbf{x},t)=\int\phi_{0}(\omega,\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{x}-\omega
t)}\delta[\varepsilon(\omega,\mathbf{k})]d\mathbf{k}d\omega,$ (1)

where $\delta$ is the Dirac delta function and the conditions
$\phi_{0}(\omega,\mathbf{k})=\phi_{0}^{*}(-\omega,-\mathbf{k})$ and
$\varepsilon(\omega,\mathbf{k})=\varepsilon^{*}(-\omega,-\mathbf{k})$ ensure
that $\phi$ is real. The constraint imposed by the dispersion relation
collapses one of the integrals in Eq. (1) and is typically used to write the
frequency in terms of the wavevector, i.e., $\omega=\omega(\mathbf{k})$ with
$\varepsilon[\omega(\mathbf{k}),\mathbf{k}]=0$ implied.

Aside from the dispersion relation, an additional constraint
$C(\omega,\mathbf{k})$ can be applied by writing

$\phi_{0}(\omega,\mathbf{k})=\bar{\phi}_{0}(\omega,\mathbf{k})\delta[C(\omega,\mathbf{k})].$
(2)

The most general form of an STP uses $C(\omega,\mathbf{k})$ to introduce
correlations in $(\omega,\mathbf{k})$ space. Motivated by propagation
invariant and flying focus laser pulses 36; 37; 33, the constraint is chosen
here to allow for an arbitrary, specified group velocity $v_{g}$:

$C(\omega,\mathbf{k})=\frac{v_{g}}{(\omega_{0}-v_{g}k_{0})}[(\omega-
v_{g}k_{z})^{2}-(\omega_{0}-v_{g}k_{0})^{2}].$ (3)

Upon setting Eq. (3) equal to zero in accordance with the delta function, one
can verify that

$\frac{\partial\omega}{\partial k_{z}}=v_{g}.$ (4)

Substituting Eq. (2) into Eq. (1) and applying the constraint provides the
electrostatic potential of the STP:

$\phi(\mathbf{x}_{\perp},\eta,\xi)=\tfrac{1}{2}e^{ik_{0}\eta}\Phi(\mathbf{x}_{\perp},\xi)+\text{c.c.},$
(5)

where $\eta=z-v_{0}t$, $v_{0}=\omega_{0}/k_{0}$, $\xi=z-v_{g}t$,

$\Phi(\mathbf{x}_{\perp},\xi)=\int\bar{\Phi}(\Omega,\mathbf{k}_{\perp})e^{i\mathbf{k}_{\perp}\cdot\mathbf{x}_{\perp}+i\Omega\xi/v_{g}}\delta(\varepsilon)d\mathbf{k_{\perp}}d\Omega,$
(6)

$\Omega=\omega-\omega_{0}$, and
$\bar{\Phi}(\Omega,\mathbf{k_{\perp})}=\bar{\phi}_{0}(\omega_{0}+\Omega,\mathbf{k}_{\perp},k_{0}+\Omega/v_{g})$.
Equation (5) demonstrates that an electrostatic potential constructed with the
correlation $C(\omega,\mathbf{k})$ has phase fronts that travel at the
velocity $v_{0}$ and an envelope $\Phi$ that travels at the group velocity
$v_{g}$.

Thus far, the formulation has been relatively abstract. To make the concept
more tangible, examples will be presented for a non-relativistic, non-flowing
plasma composed of electrons and a single ion species. The dispersion relation
for such a plasma can be derived using the Vlasov-Poisson system of equations
and is given by

$\varepsilon(\omega,\mathbf{k})=1+\chi_{e}(\omega,\mathbf{k})+\chi_{i}(\omega,\mathbf{k})=0,$
(7)

where

$\chi_{s}(\omega,\mathbf{k})=\frac{\omega_{ps}^{2}}{k^{2}}\int\frac{\mathbf{k}\cdot\nabla_{\mathbf{v}}f_{s}}{\omega-\mathbf{k}\cdot\mathbf{v}}d\mathbf{v}$
(8)

is the susceptibility for species $s$,
$\omega_{ps}=(q_{s}^{2}n_{s}/\varepsilon_{0}m_{s})^{1/2}$ is the plasma
frequency, $n_{s}$ is the density, and $f_{s}=f_{s}(\mathbf{v})$ is the
velocity distribution function. Equation (7) predicts the existence of two
elementary plasma waves: a high-frequency electron plasma wave and a low-
frequency ion-acoustic wave.

The dispersion relation for electron plasma waves can be found in the limit
that the phase velocity is much greater than the electron thermal velocity,
i.e., $v_{p}\gg v_{Te}$, where $v_{Ts}=[\int v_{k}^{2}f_{s}d\mathbf{v}]^{1/2}$
and $v_{k}=\mathbf{e}_{\mathbf{k}}\cdot\mathbf{v}$. In this limit, Eq. (7)
reduces to

$\varepsilon(\omega,\mathbf{k})\approx
1-\frac{\omega_{pe}^{2}}{\omega^{2}}-3k^{2}\lambda_{De}^{2}$ (9)

where $\lambda_{De}=v_{Te}/\omega_{pe}$ is the electron Debye length. The
dispersion relation for ion-acoustic waves can be found in the opposite limit
where the phase velocity is much smaller than the electron thermal velocity,
i.e., $v_{p}\ll v_{Te}$. Here, Eq. (7) reduces to

$\varepsilon(\omega,\mathbf{k})\approx
1+\frac{1}{k^{2}\lambda_{De}^{2}}-\frac{\omega_{pi}^{2}}{\omega^{2}}.$ (10)

In both cases, Landau damping has been neglected. This is a good approximation
when $k\lambda_{De}\lesssim 0.2$ or $v_{p}/v_{Ti}\gg 1$ for electron plasma
and ion-acoustic waves, respectively.

Without the constraint $C(\omega,\mathbf{k})$, the plasma waves travel at a
group velocity determined by the plasma conditions. Solving for the frequency
in Eqs. (9) and (10) yields $\omega(k)=(\varpi^{2}+u^{2}k^{2})^{1/2}$ and the
group velocity

$\frac{\partial\omega}{\partial k_{z}}=\frac{u^{2}k_{z}}{\omega},$ (11)

where the values of $\varpi$ and $u$ for each wave are defined in Table I.
With the constraint $C(\omega,\mathbf{k})$ [Eq. (3)], the relation
$\omega(k)=(\varpi^{2}+u^{2}k^{2})^{1/2}$ still holds, but now the transverse
wavenumber is a function of the longitudinal wavenumber (or frequency), such
that

$\frac{\partial\omega}{\partial
k_{z}}=\frac{u^{2}}{\omega}\left(k_{z}+\frac{1}{2}\frac{\partial
k_{\perp}^{2}}{\partial k_{z}}\right)=v_{g},$ (12)

where $k_{z}=k_{0}+\Omega/v_{g}$ has been used. Thus the velocity $v_{g}$ is
completely independent of the plasma conditions.

Table 1: Variable definitions used for the electron plasma wave (EPW) and ion-acoustic wave (IAW). $c_{s}=\omega_{pi}\lambda_{De}$ is the sound speed.  Variable | EPW | IAW
---|---|---
$\varpi$ | $\omega_{pe}$ | 0
$u$ | $\sqrt{3}v_{Te}$ | $c_{s}$

Using the explicit expressions for $\varepsilon$ from Eqs. (9) and (10), the
delta function enforcing the dispersion relation can be written in the general
form

$\delta(\varepsilon)=\left|\frac{\partial\Omega}{\partial\varepsilon}\right|_{\Omega=\Omega_{n}}\delta[\Omega-\Omega_{n}(k_{\perp})],$
(13)

where

$\displaystyle\Omega_{n}$
$\displaystyle(k_{\perp})=-k_{0}v_{g}v_{0}\left(\frac{v_{g}-v_{n}}{v_{g}^{2}-u^{2}}\right)$
(14)
$\displaystyle+\left[(k_{0}v_{g}v_{0})^{2}\left(\frac{v_{g}-v_{n}}{v_{g}^{2}-u^{2}}\right)^{2}+\left(\frac{v_{g}^{2}u^{2}k_{\perp}^{2}}{v_{g}^{2}-u^{2}}\right)\right]^{1/2}$

and $v_{n}=u^{2}/v_{0}$. In arriving at Eq. (14), the choice was made to set
$\omega_{0}=(\varpi^{2}+u^{2}k_{0}^{2})^{1/2}$. With this choice,
$v_{0}=\omega_{0}/k_{0}$ equals the phase velocity of the plasma wave in the
plane-wave limit $v_{g}\rightarrow 0$, and $v_{n}$ equals the nominal group
velocity in the absence of space-time structuring. Applying Eq. (13) in Eq.
(6) collapses the integral over $\Omega$, leaving only the integral over
$\mathbf{k}_{\perp}$:

$\Phi(\mathbf{x}_{\perp},\xi)=\int\tilde{\Phi}(\mathbf{k}_{\perp})e^{i\mathbf{k_{\perp}}\cdot\mathbf{x}_{\perp}+i\Omega_{n}(k_{\perp})\xi/v_{g}}d\mathbf{k_{\perp}},$
(15)

where
$\tilde{\Phi}(\mathbf{k_{\perp}})=|\partial_{\varepsilon}\Omega|_{\Omega=\Omega_{n}}\bar{\Phi}[\Omega_{n}(k_{\perp}),\mathbf{k}_{\perp}]$.
The function $\tilde{\Phi}(\mathbf{k}_{\perp})$ determines the spatiotemporal
profile of the arbitrary group velocity plasma wave.

Analytic expressions for the spatiotemporal profile can be found in the
“paraxial” approximation, i.e., when the condition

$k_{\perp}^{2}\ll\frac{\>v_{0}(v_{g}-v_{n})^{2}}{v_{n}(v_{g}^{2}-u^{2})}k_{0}^{2}$
(16)

is satisfied. Upon using this condition, $\Omega_{n}$ simplifies to

$\Omega_{n}(k_{\perp})\approx\left(\frac{v_{g}v_{n}}{v_{g}-v_{n}}\right)\frac{k_{\perp}^{2}}{2k_{0}}.$
(17)

With the quadratic dependence of $\Omega_{n}$ on $k_{\perp}$, a natural choice
for $\tilde{\Phi}(\mathbf{k}_{\perp})$ is a superposition of Laguerre-Gaussian
modes, i.e.,

$\tilde{\Phi}(\mathbf{k}_{\perp})=\sum_{p,\ell}\tilde{\Phi}_{p\ell}\kappa^{\ell}L_{p}^{\ell}(\kappa)\mathrm{exp}(-\tfrac{1}{2}\kappa^{2})e^{i\ell\theta_{k}},$
(18)

where $\kappa=k_{\perp}w_{0}/\sqrt{2}$, $w_{0}$ characterizes the transverse
width, $L_{p}^{\ell}$ is a generalized Laguerre polynomial, and $\theta_{k}$
is the azimuth in transverse wavenumber space. The spatiotemporal profile of
the STP is then given by

$\displaystyle\Phi($
$\displaystyle\mathbf{x}_{\perp},\xi)=\sum_{p,\ell}\Phi_{p\ell}\frac{w_{0}}{w}\Bigl{(}\frac{\sqrt{2}r}{w}\Bigr{)}^{\ell}L^{\ell}_{p}\Bigl{(}\frac{2r^{2}}{w^{2}}\Bigr{)}e^{i\ell\theta}$
(19)
$\displaystyle\text{exp}\left[-\Bigl{(}1-i\frac{\xi}{\xi_{0}}\Bigr{)}\frac{r^{2}}{w^{2}}-i(2p+\ell+1)\text{arctan}\frac{\xi}{\xi_{0}}\right],$

where $w(\xi)=w_{0}[1+(\xi/\xi_{0})^{2}]^{1/2}$,

$\xi_{0}=\frac{(v_{g}-v_{n})k_{0}w_{0}^{2}}{2v_{n}},$ (20)

$r=(x^{2}+y^{2})^{1/2}$, $\theta=\mathrm{arctan}(y/x)$ is the azimuth in
configuration space, and constant factors have been absorbed into the
amplitudes $\Phi_{p\ell}$. The profile of the STP advects at the group
velocity $v_{g}$, has a characteristic duration $\xi_{0}/v_{g}$, and can have
any orbital angular momentum value $\ell$.

The analysis so far has demonstrated that an arbitrary group velocity STP can
be constructed theoretically, but has not provided a prescription for how to
do so in practice. Plasma waves can either exist as thermal fluctuations or be
driven by external forces. Thermal fluctuations have no correlations in
$(\omega,\mathbf{k})$ space, and other than having to satisfy
$\varepsilon(\omega,\mathbf{k})=0$, $k_{\perp}$ and $k_{z}$ are completely
independent, i.e., $\partial k_{\perp}/\partial k_{z}=0$. As a result, an STP
must be driven by external forces, such as those exerted by particle beams or
electromagnetic waves. In the presence of an external force
$\mathbf{F}(\mathbf{x},t)$, the potential of a generic electrostatic wave is
given by

$\phi(\mathbf{x},t)=\int\frac{i}{ek}\frac{\chi_{e}(\omega,\mathbf{k})}{\varepsilon(\omega,\mathbf{k})}[\mathbf{e}_{k}\cdot\hat{\mathbf{F}}(\omega,\mathbf{k})]e^{i(\mathbf{k}\cdot\mathbf{x}-\omega
t)}d\mathbf{k}d\omega,$ (21)

where $e$ is the elementary charge. Resonant excitation of an STP requires
that the force $\mathbf{F}(\mathbf{x},t)$ be a function of space and time in
the combinations $\eta=z-v_{0}t$ and $\xi=z-v_{g}t$.

Electromagnetic waves provide a flexible option for driving STPs. Laser
pulses, in particular, can exhibit correlations between two or more degrees of
freedom, including polarization, orbital angular momentum, and spatio-spectral
content, and can interact in geometries ranging from co- to counter-
propagating. When the frequencies of the electromagnetic waves are much
greater than $\omega_{0}$, the disparity of time scales allows for a cycle-
averaging over their periods. The end result is a “ponderomotive guiding
center” equation of motion with the effective force

$\mathbf{F}(\mathbf{x},t)=-\frac{1}{2}m_{e}c^{2}\nabla\langle\mathbb{a}\cdot\mathbb{a}\rangle$
(22)

where $\mathbb{a}(\mathbf{x},t)=e\mathbb{A}(\mathbf{x},t)/m_{e}c$ is the total
normalized vector potential of the electromagnetic waves, satisfying
$|\mathbb{a}|\ll 1$, and $\langle\rangle$ represents a cycle-average.

An STP can be resonantly excited using a superposition of two flying focus
pulses. Flying focus pulses feature an intensity peak that can travel at any
velocity $v_{f}$, while maintaining a near-constant spatiotemporal profile.
The interference of two flying focus pulses with $v_{f}=v_{g}$ and distinct
frequencies and wavenumbers satisfying $\omega_{1}-\omega_{2}=\omega_{0}$ and
$\mathbf{e}_{z}\cdot(\mathbf{k}_{1}-\mathbf{k}_{2})=k_{0}$ produces the
ponderomotive force neccessary to resonantly drive an STP. Specifically, the
superposition

$\mathbb{a}(\mathbf{x},t)=\tfrac{1}{2}\sum_{j\in(1,2)}\bm{a}_{j}(\mathbf{x}_{\perp},\xi)e^{i(k_{j}z-\omega_{j}t)}+\mathrm{c.c.},$
(23)

where $\bm{a}_{j}(\mathbf{x}_{\perp},\xi)$ is the envelope of each pulse,
results in a ponderomotive force term

$\mathbf{F}_{d}(\mathbf{x}_{\perp},\eta,\xi)=-\frac{i}{8}m_{e}c^{2}k_{0}(\bm{a}_{1}\cdot\bm{a}_{2}^{*})e^{ik_{0}\eta}\mathbf{e}_{z}+\mathrm{c.c.}.$
(24)

In writing Eq. (23), it has been assumed that the durations of the flying
focus pulses are much longer than their periods $2\pi/\omega_{j}$.

Without further specification of the $\bm{a}_{j}$, the electrostatic potential
of the driven STP is given by
$\phi(\mathbf{x}_{\perp},\eta,\xi)=\tfrac{1}{2}e^{ik_{0}\eta}\Phi(\mathbf{x}_{\perp},\xi)+\text{c.c.},$
with

$\Phi(\mathbf{x}_{\perp},\xi)=\int
S(\Omega,\mathbf{k}_{\perp})e^{i\mathbf{k}_{\perp}\cdot\mathbf{x}_{\perp}+i\Omega\xi/v_{g}}d\mathbf{k_{\perp}}d\Omega$
(25)

and

$\begin{split}S(\Omega,\mathbf{k}_{\perp})&=\frac{m_{e}c^{2}}{32\pi^{3}e}\frac{\mathrm{s}(v_{0}-v_{g})}{|v_{g}|}\frac{k_{0}\chi_{e}(\omega,\mathbf{k})}{k\varepsilon(\omega,\mathbf{k})}\\\
&\qquad\int(\bm{a}_{1}\cdot\bm{a}_{2}^{*})e^{-i\mathbf{k}_{\perp}\cdot\mathbf{x}_{\perp}-i\Omega\xi/v_{g}}d\mathbf{x_{\perp}}d\xi,\end{split}$
(26)

where $\mathrm{s}$ is the sign function, and $\chi_{e}$, $\varepsilon$, and
$k$, are evaluated at $\omega=\Omega+\omega_{0}$ and
$k_{z}=k_{0}+\Omega/v_{g}$. Thus, the ponderomotive force of the two flying
focus pulses drives an electrostatic potential with phase fronts that travel
at $v_{0}$ and an envelope $\Phi(\mathbf{x}_{\perp},\xi)$ that travels at
$v_{g}$. Note that while the frequency and wavenumber matching conditions,
i.e., $\omega_{1}-\omega_{2}=\omega_{0}$ and
$\mathbf{e}_{z}\cdot(\mathbf{k}_{1}-\mathbf{k}_{2})=k_{0}$, are identical to
those required for stimulated Raman or Brillouin scattering (electron and ion-
acoustic waves, respectively), excitation of an STP does not require
instability.

Equations (24) and (26) provide an exact, linear solution for a driven STP in
the spectral domain. While these solutions demonstrate the salient physics,
they are “monochromatic,” that is, they oscillate in $\eta$ with a single
period $2\pi/k_{0}$. More generally, the potential will be a superposition of
these solutions, such that

$\phi(\mathbf{x}_{\perp},\eta,\xi)=\tfrac{1}{2}e^{ik_{0}\eta}\int\breve{\Phi}(\mathbf{x}_{\perp},k^{\prime},\xi)e^{ik^{\prime}\eta}dk^{\prime}+\text{c.c.},$
(27)

where $k^{\prime}$ represents a wavenumber shift about the central wavenumber
$k_{0}$ and the envelope of the potential
$\Phi(\mathbf{x}_{\perp},\eta,\xi)=\int\breve{\Phi}(\mathbf{x}_{\perp},k^{\prime},\xi)e^{ik^{\prime}\eta}dk^{\prime}$
now depends on $\eta$.

Direct evaluation of Eq. (21) can be challenging. As an alternative, when
$\omega_{0}$ is close to the natural mode frequency of the plasma wave, Eq.
(21) can be recast as the configuration-space wave equation

$\left(\partial^{2}_{t}+\varpi^{2}-u^{2}\nabla^{2}\right)\phi(\mathbf{x},t)=\pm\frac{1}{8}\omega_{0}^{2}(\bm{a}_{1}\cdot\bm{a}_{2}^{*})e^{ik_{0}\eta}+\mathrm{c.c.},$
(28)

where $\phi$ has been normalized by $m_{e}c^{2}/e$, and the top and bottom
signs are taken for electron plasma and ion-acoustic waves, respectively. If
the $\bm{a}_{j}$ are independent of $\eta$ (or approximately so), Eq. (28)
reduces to

$\left[2i\kappa\frac{\partial}{\partial\xi}+\frac{u^{2}-v_{g}^{2}}{u^{2}}\frac{\partial^{2}}{\partial\xi^{2}}+\nabla_{\perp}^{2}\right]\Phi_{s}(\mathbf{x}_{\perp},\xi)=\mp\frac{\omega_{0}^{2}(\bm{a}_{1}\cdot\bm{a}_{2}^{*})}{8u^{2}},$
(29)

where $\kappa=k_{0}(v_{n}-v_{g})/v_{n}$ and the subscript $s$ refers to the
STP. The homogeneous dispersion relation for Eq. (28) is given by Eq. (14) and
the homogeneous, paraxial solutions by Eq. (19). The evolution of
$\Phi_{s}(\mathbf{x}_{\perp},\xi)$ contrasts that of a conventional plasma
wave for which Eq. (28) is often simplified as

$\left[2i\frac{\omega_{0}}{u^{2}}\frac{\partial}{\partial
t}+\frac{\partial^{2}}{\partial\zeta^{2}}+\nabla_{\perp}^{2}\right]\Phi_{c}(\mathbf{x}_{\perp},\zeta,t)=\mp\frac{\omega_{0}^{2}(\bm{a}_{1}\cdot\bm{a}_{2}^{*})}{8u^{2}},$
(30)

where $\zeta=z-v_{n}t$, the subscript $c$ refers to a conventional plasma
wave, and $|\partial_{t}\Phi_{c}|\ll|\omega_{0}\Phi_{c}|$ has been assumed. In
Fig. 1 the homogeneous solutions to Eqs. (29) and (30) are compared for the
initial conditions
$\Phi_{s}(\mathbf{x}_{\perp},0)=\Phi_{0}(\sqrt{2}r/w_{0})\exp{(-r^{2}/w_{0}^{2})e^{i\theta}}$
and
$\Phi_{c}(\mathbf{x}_{\perp},\zeta,0)=\Phi_{0}\exp{(-r^{2}/w_{0}^{2}-\zeta^{2}/Z_{0}^{2})}$,
respectively. The $\ell=1$ mode was chosen for the STP to illustrate its
ability to carry orbital angular momentum.

As a final note, the delta function that enforces the dispersion relation in
Eq. (6) can also be written in terms of the perpendicular wavenumber
$k_{\perp}$:

$\delta(\varepsilon)=\left|\frac{\partial
k_{\perp}}{\partial\varepsilon}\right|_{k_{\perp}=k_{\perp,n}}\delta[k_{\perp}-k_{\perp,n}(\Omega)],$
(31)

where

$\displaystyle k_{\perp,n}(\Omega)$
$\displaystyle=\frac{1}{u}\left[\Omega^{2}\left(1-\frac{u^{2}}{v_{g}^{2}}\right)+2\omega_{0}\Omega\left(1-\frac{u^{2}}{v_{0}v_{g}}\right)\right]^{1/2}.$
(32)

This allows one to write $\tilde{\Phi}$ as a function of $\Omega$ instead of
$\mathbf{k}_{\perp}$ when evaluating $\Phi(\mathbf{x}_{\perp},\xi)$ in Eq.
(15). With this convention, the paraxial approximation is given by
$k_{\perp,n}(\Omega)\approx\frac{1}{u}[2\omega_{0}\Omega(1-\frac{u^{2}}{v_{0}v_{g}})]^{1/2}$.

Space-time structured plasma waves (STPs) exhibit properties that are
independent of the plasma in which they exist. Unlike conventional plasma
waves, which are devoid of correlations in $(\omega,\mathbf{k})$ space and are
therefore constrained by the plasma conditions, STPs are constructed with
correlations that provide control over their evolution. An example of
arbitrary-group-velocity STPs was presented, which was motivated by the
subfield of structured light dedicated to controlling the trajectory of peak
laser intensity, i.e., spatiotemporal pulse shaping. While much of the
analysis from spatiotemporal pulse shaping carries over 38; 49; 50,
unstructured plasma waves are distinct in that their nominal group velocity
can be significantly different than their phase velocity. STPs can be realized
experimentally, with or without orbital angular momentum, by using the
ponderomotive force exerted by two space-time structured laser pulses. More-
advanced correlations may allow for STPs with more-exotic structures, such as
spatiotemporal optical vortices 51; 52. Further work will generalize STPs to
magnetized plasma waves, consider STPs driven by charged particle beams, and
explore whether STPs can provide control over wave-particle interactions,
including linear and nonlinear Landau damping, trapped particle instabilties,
or kinetic inflation 53; 54; 55; 56; 57; 58; 59; 60; 61.

###### Acknowledgements.

The authors would like to thank A. Raymond, K.L. Nguyen, and T.T. Simpson for
discussions. This report was prepared as an account of work sponsored by an
agency of the U.S. Government. Neither the U.S. Government nor any agency
thereof, nor any of their employees, makes any warranty, express or implied,
or assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product, or process
disclosed, or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial product, process, or
service by trade name, trademark, manufacturer, or otherwise does not
necessarily constitute or imply its endorsement, recommendation, or favoring
by the U.S. Government or any agency thereof. The views and opinions of
authors expressed herein do not necessarily state or reflect those of the U.S.
Government or any agency thereof. This material is based upon work supported
by the Office of Fusion Energy Sciences under Award Numbers DE-SC0023423 and
DE-SC00215057, the Department of Energy National Nuclear Security
Administration under Award Number DE-NA0003856, the University of Rochester,
and the New York State Energy Research and Development Authority.

## References

  * Behn  _et al._ (1989) R. Behn, D. Dicken, J. Hackmann, S. A. Salito, M. R. Siegrist, P. A. Krug, I. Kjelberg, B. Duval, B. Joye, and A. Pochelon, Phys. Rev. Lett. 62, 2833 (1989).
  * Glenzer  _et al._ (1999) S. H. Glenzer, W. E. Alley, K. G. Estabrook, J. S. De Groot, M. G. Haines, J. H. Hammer, J.-P. Jadaud, B. J. MacGowan, J. D. Moody, W. Rozmus, L. J. Suter, T. L. Weiland, and E. A. Williams, Physics of Plasmas 6, 2117 (1999), https://pubs.aip.org/aip/pop/article-pdf/6/5/2117/12562483/2117_1_online.pdf .
  * Froula  _et al._ (2002) D. H. Froula, L. Divol, and S. H. Glenzer, Phys. Rev. Lett. 88, 105003 (2002).
  * Bindslev  _et al._ (2006) H. Bindslev, S. K. Nielsen, L. Porte, J. A. Hoekzema, S. B. Korsholm, F. Meo, P. K. Michelsen, S. Michelsen, J. W. Oosterbeek, E. L. Tsakadze, E. Westerhof, and P. Woskov (the TEXTOR team), Phys. Rev. Lett. 97, 205005 (2006).
  * Korsholm  _et al._ (2011) S. B. Korsholm, M. Stejner, H. Bindslev, V. Furtula, F. Leipold, F. Meo, P. K. Michelsen, D. Moseev, S. K. Nielsen, M. Salewski, M. de Baar, E. Delabie, M. Kantor, and A. Bürger (TEXTOR Team), Phys. Rev. Lett. 106, 165004 (2011).
  * Hansen  _et al._ (2021) A. M. Hansen, K. L. Nguyen, D. Turnbull, B. J. Albright, R. K. Follett, R. Huff, J. Katz, D. Mastrosimone, A. L. Milder, L. Yin, J. P. Palastro, and D. H. Froula, Phys. Rev. Lett. 126, 075002 (2021).
  * Kruer  _et al._ (1996) W. L. Kruer, S. C. Wilks, B. B. Afeyan, and R. K. Kirkwood, Physics of Plasmas 3, 382 (1996), https://pubs.aip.org/aip/pop/article-pdf/3/1/382/12467611/382_1_online.pdf .
  * Porkolab  _et al._ (1977) M. Porkolab, S. Bernabei, W. M. Hooke, R. W. Motley, and T. Nagashima, Phys. Rev. Lett. 38, 230 (1977).
  * Tang (1978) W. Tang, Nuclear Fusion 18, 1089 (1978).
  * Jenko and Dorland (2001) F. Jenko and W. Dorland, Plasma Physics and Controlled Fusion 43, A141 (2001).
  * Michel  _et al._ (2012) D. T. Michel, A. V. Maximov, R. W. Short, S. X. Hu, J. F. Myatt, W. Seka, A. A. Solodov, B. Yaakobi, and D. H. Froula, Phys. Rev. Lett. 109, 155007 (2012).
  * Proll  _et al._ (2012) J. H. E. Proll, P. Helander, J. W. Connor, and G. G. Plunk, Phys. Rev. Lett. 108, 245002 (2012).
  * Michel  _et al._ (2015) P. Michel, L. Divol, E. L. Dewald, J. L. Milovich, M. Hohenberger, O. S. Jones, L. B. Hopkins, R. L. Berger, W. L. Kruer, and J. D. Moody, Phys. Rev. Lett. 115, 055003 (2015).
  * Montgomery (2016) D. S. Montgomery, Physics of Plasmas 23, 055601 (2016), https://pubs.aip.org/aip/pop/article-pdf/doi/10.1063/1.4946016/15940279/055601_1_online.pdf .
  * Tajima and Dawson (1979) T. Tajima and J. M. Dawson, Physical Review Letters 43, 267 (1979).
  * Lu  _et al._ (2007) W. Lu, M. Tzoufras, C. Joshi, F. S. Tsung, W. B. Mori, J. Vieira, R. A. Fonseca, and L. O. Silva, Physical Review Special Topics - Accelerators and Beams 10, 061301 (2007).
  * Esarey  _et al._ (2009) E. Esarey, C. B. Schroeder, and W. P. Leemans, Reviews of Modern Physics 81, 1229 (2009).
  * Gonsalves  _et al._ (2019) A. Gonsalves, K. Nakamura, J. Daniels, C. Benedetti, C. Pieronek, T. de Raadt, S. Steinke, J. Bin, S. Bulanov, J. van Tilborg, C. Geddes, C. Schroeder, C. Tóth, E. Esarey, K. Swanson, L. Fan-Chiang, G. Bagdasarov, N. Bobrova, V. Gasilov, G. Korn, P. Sasorov, and W. Leemans, Physical Review Letters 122, 084801 (2019).
  * Miao  _et al._ (2022) B. Miao, J. Shrock, L. Feder, R. Hollinger, J. Morrison, R. Nedbailo, A. Picksley, H. Song, S. Wang, J. Rocca, and H. Milchberg, Physical Review X 12, 031038 (2022).
  * Caldwell  _et al._ (2009) A. Caldwell, K. Lotov, A. Pukhov, and F. Simon, Nature Physics 5, 363 (2009).
  * Assmann  _et al._ (2014) R. Assmann, R. Bingham, T. Bohl, C. Bracco, B. Buttenschön, A. Butterworth, A. Caldwell, S. Chattopadhyay, S. Cipiccia, E. Feldbaumer, R. A. Fonseca, B. Goddard, M. Gross, O. Grulke, E. Gschwendtner, J. Holloway, C. Huang, D. Jaroszynski, S. Jolly, P. Kempkes, N. Lopes, K. Lotov, J. Machacek, S. R. Mandry, J. W. McKenzie, M. Meddahi, B. L. Militsyn, N. Moschuering, P. Muggli, Z. Najmudin, T. C. Q. Noakes, P. A. Norreys, E. Öz, A. Pardons, A. Petrenko, A. Pukhov, K. Rieger, O. Reimann, H. Ruhl, E. Shaposhnikova, L. O. Silva, A. Sosedkin, R. Tarkeshian, R. M. G. N. Trines, T. Tückmantel, J. Vieira, H. Vincke, M. Wing, G. Xia, and A. Collaboration), Plasma Physics and Controlled Fusion 56, 084013 (2014).
  * Litos  _et al._ (2014) M. Litos, E. Adli, W. An, C. I. Clarke, C. E. Clayton, S. Corde, J. P. Delahaye, R. J. England, A. S. Fisher, J. Frederico, S. Gessner, S. Z. Green, M. J. Hogan, C. Joshi, W. Lu, K. A. Marsh, W. B. Mori, P. Muggli, N. Vafaei-Najafabadi, D. Walz, G. White, Z. Wu, V. Yakimenko, and G. Yocky, Nature 515, 92 (2014).
  * Gessner  _et al._ (2016) S. Gessner, E. Adli, J. M. Allen, W. An, C. I. Clarke, C. E. Clayton, S. Corde, J. P. Delahaye, J. Frederico, S. Z. Green, C. Hast, M. J. Hogan, C. Joshi, C. A. Lindstrøm, N. Lipkowitz, M. Litos, W. Lu, K. A. Marsh, W. B. Mori, B. O’Shea, N. Vafaei-Najafabadi, D. Walz, V. Yakimenko, and G. Yocky, Nature Communications 7, 11785 (2016).
  * Joshi  _et al._ (2018) C. Joshi, E. Adli, W. An, C. E. Clayton, S. Corde, S. Gessner, M. J. Hogan, M. Litos, W. Lu, K. A. Marsh, W. B. Mori, N. Vafaei-Najafabadi, B. O’shea, X. Xu, G. White, and V. Yakimenko, Plasma Physics and Controlled Fusion 60, 034001 (2018).
  * Lin  _et al._ (1986) R. P. Lin, W. K. Levedahl, W. Lotko, D. A. Gurnett, and F. L. Scarf, Astrophys. J. 308, 954 (1986).
  * Ergun  _et al._ (1998) R. E. Ergun, D. Larson, R. P. Lin, J. P. McFadden, C. W. Carlson, K. A. Anderson, L. Muschietti, M. McCarthy, G. K. Parks, H. Reme, J. M. Bosqued, C. D’Uston, T. R. Sanderson, K. P. Wenzel, M. Kaiser, R. P. Lepping, S. D. Bale, P. Kellogg, and J.-L. Bougeret, The Astrophysical Journal 503, 435 (1998).
  * Reid and Ratcliffe (2014) H. A. S. Reid and H. Ratcliffe, Research in Astronomy and Astrophysics 14, 773 (2014).
  * Howard  _et al._ (2019) A. J. Howard, D. Turnbull, A. S. Davies, P. Franke, D. H. Froula, and J. P. Palastro, Phys. Rev. Lett. 123, 124801 (2019).
  * Debus  _et al._ (2019) A. Debus, R. Pausch, A. Huebl, K. Steiniger, R. Widera, T. E. Cowan, U. Schramm, and M. Bussmann, Physical Review X 9, 031044 (2019).
  * Palastro  _et al._ (2020) J. Palastro, J. Shaw, P. Franke, D. Ramsey, T. Simpson, and D. Froula, Physical Review Letters 124, 134802 (2020).
  * Caizergues  _et al._ (2020) C. Caizergues, S. Smartsev, V. Malka, and C. Thaury, Nature Photonics 14, 475 (2020).
  * Di Piazza (2021) A. Di Piazza, Phys. Rev. A 103, 012215 (2021).
  * Ramsey  _et al._ (2022) D. Ramsey, B. Malaca, A. Di Piazza, M. Formanek, P. Franke, D. H. Froula, M. Pardal, T. T. Simpson, J. Vieira, K. Weichman, and J. P. Palastro, Phys. Rev. E 105, 065201 (2022).
  * Kabacinski  _et al._ (2023) A. Kabacinski, E. Oliva, F. Tissandier, J. Gautier, M. Kozlová, J.-P. Goddet, I. A. Andriyash, C. Thaury, P. Zeitoun, and S. Sebban, Nature Photonics 17, 354 (2023).
  * Simpson  _et al._ (2023) T. T. Simpson, J. J. Pigeon, M. V. Ambat, K. G. Miller, D. Ramsey, K. Weichman, D. H. Froula, and J. P. Palastro, In Review (2023).
  * Longhi (2003) S. Longhi, Phys. Rev. E 68, 066612 (2003).
  * Kondakci and Abouraddy (2017) H. E. Kondakci and A. F. Abouraddy, Nature Photonics 11, 733 (2017).
  * Kondakci and Abouraddy (2019) H. E. Kondakci and A. F. Abouraddy, Nature Communications 10, 929 (2019).
  * Li and Kawanaka (2020a) Z. Li and J. Kawanaka, Communications Physics 3, 211 (2020a).
  * Li and Kawanaka (2020b) Z. Li and J. Kawanaka, Scientific Reports 10, 11481 (2020b).
  * Yessenov  _et al._ (2022) M. Yessenov, J. Free, Z. Chen, E. G. Johnson, M. P. J. Lavery, M. A. Alonso, and A. F. Abouraddy, Nature Communications 13, 4573 (2022).
  * Besieris and Saari (2022) I. M. Besieris and P. Saari, J. Opt. Soc. Am. A 39, 1449 (2022).
  * Sainte-Marie  _et al._ (2017) A. Sainte-Marie, O. Gobert, and F. Quéré, Optica 4, 1298 (2017).
  * Froula  _et al._ (2018) D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, Nature Photonics 12, 262 (2018).
  * Jolly  _et al._ (2020) S. W. Jolly, O. Gobert, A. Jeandet, and F. Quéré, Opt. Express 28, 4888 (2020).
  * Simpson  _et al._ (2022) T. T. Simpson, D. Ramsey, P. Franke, K. Weichman, M. V. Ambat, D. Turnbull, D. H. Froula, and J. P. Palastro, Optics Express 30, 9878 (2022).
  * Ambat  _et al._ (2023) M. V. Ambat, J. L. Shaw, J. J. Pigeon, K. G. Miller, T. T. Simpson, D. H. Froula, and J. P. Palastro, Optics Express (2023), accepted.
  * Pierce  _et al._ (2023) J. R. Pierce, J. P. Palastro, F. Li, B. Malaca, D. Ramsey, J. Vieira, K. Weichman, and W. B. Mori, Physical Review Research 5, 013085 (2023).
  * Palastro  _et al._ (2018) J. P. Palastro, D. Turnbull, S.-W. Bahk, R. K. Follett, J. L. Shaw, D. Haberberger, J. Bromage, and D. H. Froula, Physical Review A 97, 033835 (2018).
  * Ramsey  _et al._ (2023) D. Ramsey, A. D. Piazza, M. Formanek, P. Franke, D. H. Froula, B. Malaca, W. B. Mori, J. R. Pierce, T. T. Simpson, J. Vieira, M. Vranic, K. Weichman, and J. P. Palastro, Physical Review A 107, 013513 (2023).
  * Hancock  _et al._ (2019) S. W. Hancock, S. Zahedpour, A. Goffin, and H. M. Milchberg, Optica 6, 1547 (2019).
  * Hancock  _et al._ (2021) S. W. Hancock, S. Zahedpour, and H. M. Milchberg, Phys. Rev. Lett. 127, 193901 (2021).
  * Bernstein  _et al._ (1957) I. B. Bernstein, J. M. Greene, and M. D. Kruskal, Phys. Rev. 108, 546 (1957).
  * Kruer  _et al._ (1969) W. L. Kruer, J. M. Dawson, and R. N. Sudan, Phys. Rev. Lett. 23, 838 (1969).
  * Morales and O’Neil (1972) G. J. Morales and T. M. O’Neil, Phys. Rev. Lett. 28, 417 (1972).
  * Katsouleas (1986) T. Katsouleas, Phys. Rev. A 33, 2056 (1986).
  * Liu and Tripathi (1986) C. Liu and V. Tripathi, Physics Reports 130, 143 (1986).
  * Manfredi (1997) G. Manfredi, Phys. Rev. Lett. 79, 2815 (1997).
  * Vu  _et al._ (2001) H. X. Vu, D. F. DuBois, and B. Bezzerides, Phys. Rev. Lett. 86, 4306 (2001).
  * Yin  _et al._ (2007) L. Yin, B. J. Albright, K. J. Bowers, W. Daughton, and H. A. Rose, Phys. Rev. Lett. 99, 265004 (2007).
  * Chapman  _et al._ (2012) T. Chapman, S. Hüller, P. E. Masson-Laborde, A. Heron, D. Pesme, and W. Rozmus, Phys. Rev. Lett. 108, 145003 (2012).