11institutetext:  School of Physics and Astronomy, Cardiff University, Queenâs
Buildings, The Parade, Cardiff, CF24 3AA, UK 22institutetext: LLR (Laboratoire
Leprince-Ringuet), CNRS, Ãcole Polytechnique, Institut Polytechnique de Paris,
Palaiseau, France 33institutetext: AIM, CEA, CNRS, Université Paris-Saclay,
Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France
44institutetext: Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53,
avenue des Martyrs, 38000 Grenoble, France 55institutetext: Institut
d’Astrophysique Spatiale (IAS), CNRS, Université Paris Sud, Orsay, France
66institutetext: Institut Néel, CNRS, Université Grenoble Alpes, France
77institutetext: Institut de RadioAstronomie Millimétrique (IRAM), Grenoble,
France 88institutetext: Aix Marseille Univ, CNRS, CNES, LAM (Laboratoire
d’Astrophysique de Marseille), Marseille, France 99institutetext: Dipartimento
di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma,
Italy 1010institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble,
France 1111institutetext: Centro de Astrobiología (CSIC-INTA), Torrejón de
Ardoz, 28850 Madrid, Spain 1212institutetext: Instituto de Radioastronomía
Milimétrica (IRAM), Granada, Spain 1313institutetext: LERMA, Observatoire de
Paris, PSL Research University, CNRS, Sorbonne Université, UPMC, 75014 Paris,
France 1414institutetext: School of Earth and Space Exploration and Department
of Physics, Arizona State University, Tempe, AZ 85287, USA 1515institutetext:
Univ. Toulouse, CNRS, IRAP, 9 Av. du colonel Roche, BP 44346, 31028, Toulouse,
France 1616institutetext: Laboratoire de Physique de lâÉcole Normale
Supérieure, ENS, PSL Research University, CNRS, Sorbonne Université,
Université de Paris, 75005 Paris, France 1717institutetext: Department of
Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street,
Philadelphia, PA, 19104, USA 1818institutetext: Institut dâAstrophysique de
Paris, Sorbonne Université, CNRS (UMR7095), 75014 Paris, France
1919institutetext: Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2020institutetext: Caltech, Pasadena, CA 91125, USA

# Galactic star formation with NIKA2 (GASTON):
Filament convergence and its link to star formation

N. Peretto Galactic star formation with NIKA2 (GASTON): Filament convergence
and its link to star formationGalactic star formation with NIKA2 (GASTON):
Filament convergence and its link to star formation
<EMAIL_ADDRESS>R. Adam Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation P. Ade Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation H. Ajeddig
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation P. André Galactic star formation
with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation E. Artis Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation H. Aussel Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation A. Bacmann
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation A. Beelen Galactic star formation
with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation A. Benoît Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation S. Berta Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation L. Bing Galactic
star formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation O. Bourrion Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation M. Calvo Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation A. Catalano
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation M. De Petris Galactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation F.-X. Désert Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation S. Doyle Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation E. F. C.
Driessen Galactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formationGalactic star formation with NIKA2 (GASTON):
Filament convergence and its link to star formation A. Gomez Galactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation J. Goupy Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation F. Kéruzoré Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation C. Kramer
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation B. Ladjelate Galactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation G. Lagache Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation S. Leclercq Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation J.-F. Lestrade
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation J.-F. Macías-Pérez Galactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation A. Maury Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation P. Mauskopf Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation F. Mayet Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation A. Monfardini Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation M. Muñoz-
Echeverría Galactic star formation with NIKA2 (GASTON): Filament convergence
and its link to star formationGalactic star formation with NIKA2 (GASTON):
Filament convergence and its link to star formation L. Perotto Galactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation G. Pisano Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation N. Ponthieu Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation V. Revéret
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation A. Rigby Galactic star formation
with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation I. Ristorcelli Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation A. Ritacco Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation C. Romero Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation H. Roussel Galactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formationGalactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formation F. Ruppin
Galactic star formation with NIKA2 (GASTON): Filament convergence and its link
to star formationGalactic star formation with NIKA2 (GASTON): Filament
convergence and its link to star formation K. Schuster Galactic star formation
with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation S. Shu Galactic star formation with NIKA2 (GASTON):
Filament convergence and its link to star formationGalactic star formation
with NIKA2 (GASTON): Filament convergence and its link to star formation A.
Sievers Galactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formationGalactic star formation with NIKA2 (GASTON):
Filament convergence and its link to star formation C. Tucker Galactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formationGalactic star formation with NIKA2 (GASTON): Filament convergence and
its link to star formation R. Zylka Galactic star formation with NIKA2
(GASTON): Filament convergence and its link to star formationGalactic star
formation with NIKA2 (GASTON): Filament convergence and its link to star
formation

###### Abstract

In the past decade filaments have been recognised as a major structural
element of the interstellar medium, the densest of these filaments hosting the
formation of most stars. In some star-forming molecular clouds converging
networks of filaments, also known as hub filament systems, can be found. These
hubs are believed to be preferentially associated to massive star formation.
As of today, there are no metrics that allow the systematic quantification of
a filament network convergence. Here, we used the IRAM 30m NIKA2 observations
of the Galactic plane from the GASTON large programme to systematically
identify filaments and produce a filament convergence parameter map. We use
such a map to show that: i. hub filaments represent a small fraction of the
global filament population; ii. hubs host, in proportion, more massive and
more luminous compact sources that non-hubs; iii. hub-hosting clumps are more
evolved that non-hubs; iv. no discontinuities are observed in the properties
of compact sources as a function of convergence parameter. We propose that the
rapid global collapse of clumps is responsible for (re)organising filament
networks into hubs and, in parallel, enhancing the mass growth of compact
sources.

##  1 Introduction

The existence of interstellar filaments has been known for more than four
decades SE1979 . Their ubiquity within the interstellar medium, however, has
only been recognised with the launch in 2009 of the Herschel mission, and its
mapping of the sky in the far-infrared andre2010 ; molinari2010 . Since then
filaments are front and centre of a large fraction of star formation research,
and numerous breakthroughs have been made. It has been shown, for instance,
that the majority of prestellar cores, and, by extension, the majority of
stars form as the result of the fragmentation of individual 0.1pc-wide self-
gravitating filaments (e.g. arzoumanian2011, ; polychroni2013, ). Massive
star-forming cores, on the other hand, seem to preferentially find themselves
at the centre of converging networks of filaments, or hubs (e.g. myers2009, ;
peretto2013, ). This raises a number of questions on the physical relationship
between filament convergence and the evolution of star-forming clouds. It has
been proposed that hub formation is a two step process, with the formation of
individual filaments first and then the collision of filament pairs in a
second step kumar2020 . One key aspect of such scenario is the existence of
two distinct formation path for individual and hub filaments. In the study
presented here, we use the very sensitive 1.15mm continuum data of the GASTON
large programme on the IRAM 30m to determine the time evolution of compact
millimetre sources as a function of the local filament network convergence.

##  2 GASTON observations and catalogue

GASTON is a 200h guaranteed-time large programme on the IRAM 30m telescope
(PI: N. Peretto). About a third of the allocated time has been used to map a
$\sim 2$ deg2 region of the Galactic plane around $\ell=24^{\rm{o}}$ at 1.15mm
and 2mm using the NIKA2 instrument adam2018 ; perotto2020 . In the present
study, we only make use of the 1.15mm image which has an angular resolution of
$12^{\prime\prime}$, a pixel size of $3^{\prime\prime}$, and a rms sensitivity
of 3.45 mJy/beam (see Fig. 1), achieved with only half of the allocated time.
In addition, we also make use of the GASTON catalogue of 1.15mm compact source
presented in rigby2021 . In that catalogue, the distances and infrared
brightness of all $\sim 1400$ sources are given. The latter provides an
indication on how bright the source parent clump is at 8$\mu$m, and is used
here as a time evolution tracer rigby2021 ; watkinsprep . Finally, all GASTON
sources have been cross-correlated with the Hi-GAL source catalogue
molinari2016 .

##  3 Quantifying filament convergence and hub definition

Figure 1: (left): GASTON 1.15mm image of the $\ell=24^{\rm{o}}$ region (colour
scale). The identified filament skeleton is overlaid on top as grey solid
lines. (right): Map of the convergence parameter $f_{c}$. Same as in the left
hand side panel, the skeleton is overlaid on top of the image.

Even though the study of hub filament systems has become, in the past 10
years, relatively widespread, there is no clear definition of what a hub is.
For the first time, we propose to quantify the convergence of filaments in
star-forming regions through the construction of a new convergence parameter,
$f_{c}$. In order to do this, we first need to build the skeleton of the
filament network within the GASTON 1.15mm image of the $\ell=24^{\rm{o}}$
region. For that purpose, we use the same second derivative method as
presented in (e.g. orkisz2019, ). Skeletons are obtained by first using a
local thresholding on the eigenvalue images, selecting only regions that have
negative eigenvalues lower than -3 times the local standard deviation. The
python skeletonize function is then used to thin each region to a 1-pixel
width skeleton. We set a lower limit on the length of individual filaments to
twice the beam size ($\sim 24^{\prime\prime}$ \- 0.6pc at 5.2 kpc, the median
cloud distance in the $\ell=24^{\rm{o}}$ field). Figure 1 shows the resulting
skeleton overlaid on top of the GASTON 1.15mm image. A total of $\sim 2600$
filaments are identified in the $\ell=24^{\rm{o}}$ field. Compared to
published catalogues of Galactic plane filaments, the GASTON filament density
is larger, however those studies were typically geared towards the detection
of larger-scale filaments (e.g. schisano2020, ; li2016, ). Since hubs have
been in the past mostly visually identified in mid-infrared extinction, a
visual check between our GASTON skeleton and the Spitzer 8$\mu$m image of the
field is performed (see Fig. 2). The agreement is excellent and shows the
power of our GASTON 1.15mm image in identifying small scale filamentary
structures.

Next we want to construct a new metric allowing to quantify how hub-like a
network of filaments really is. This is the first ever attempt to provide an
objective definition to what a hub is. For that purpose we define the
following convergence parameter $f_{c}$:

$f_{c}(x,y)=N_{\rm{fil}}\frac{\sum_{i=1}^{N_{\rm{pix}}}\cos(\Delta\theta)}{C_{n}}$
(1)

where $(x,y)$ is the pixel coordinate in the image, $N_{\rm{fil}}$ is the
number of unique filaments that enter the search radius $r_{s}$,
$N_{\rm{pix}}$ is the total number of skeleton pixels that are within the
search radius, $\Delta\theta$ is the angle difference between the direction
defined by the pixel $(x,y)$ and the skeleton pixel $i$, and the tangential
direction of the skeleton at pixel $i$. Finally, $C_{n}$ is a normalisation
constant such that a hub of six perfectly radial parsec-long connected
filaments has a $f_{c}$ value of 1. The presence of the $N_{\rm{fil}}$ is
needed to ensure that, for a fixed value of $N_{\rm{pix}}$, networks of
converging filaments have larger $f_{c}$ values than single filaments. This
definition of the convergence parameter implies that high $f_{c}$ value
positions exhibit a high level of filament convergence within a radius
$r_{s}$, while low$f_{c}$ value implies the opposite. In this study we chose
$r_{s}=39^{\prime\prime}$, which corresponds to $\sim 1$ pc at the cloud
median distance of 5.2 kpc. Figure 1(right) shows the resulting convergence
map obtained from the underlying filament skeleton where high convergence
spots can clearly be seen.

Figure 2: (left): Histogram of the $f_{c}$ values presented in Fig. 1.
(right): zoom-ins of 4 different sources exhibiting the range of $f_{c}$
values with their peak value indicated on the image. For each of the four
pairs of images, we display the convergence map on the left and the
corresponding Spitzer 8$\mu$m image on the right hand side. The skeletons are
overlaid on top of each panel, and $f_{c}$ contours from 0.1 to 0.5 by step of
0.1 are also displayed on the convergence images. For context, the search
radius $r_{s}$ is about $0.01^{\rm{o}}$.

Figure 2(left) shows the histogram of $f_{c}$ values presented in Fig. 1,
ranging from 0 to 0.75. One can see that the distribution can be described by
a single power law, with no obvious breaks. Therefore the segmentation of
clumps between hubs and non-hubs is somehow arbitrary. However one can still
investigate what $f_{c}$ values typical hub filament systems have in order to
separate hubs from non-hubs. Figure 2 shows four sources with increasing peak
$f_{c}$ values and increasing filament complexity, the last two of which have
peak $f_{c}$ values of 0.26 and 0.50 and exhibit a typical hub morphology. We
therefore decide to define hubs as connected groups of pixels with $f_{c}\geq
0.2$. By doing so, we identify 63 hubs and we find that only 5% of skeleton
pixels are located within hubs. Hub filament systems are rare objects.

##  4 Source properties as a function of convergence parameter

Figure 3: (1st row): Histogram of distances for the GASTON 1.15mm compact
sources identified by rigby2021 . Each panel corresponds to a range of $f_{c}$
values indicated in each panel. The vertical dashed lines show the median
values. (2nd row): same as top row but for the masses of the compact sources.
(3rd row): same as top row but for the infrared darkness parameter. (4th row):
Same as top row but for the 70$\mu$m-based bolometric luminosities. The
circular symbols indicate the number of sources with no Hi-GAL 70$\mu$m
sources, i.e. sources for which the bolometric luminosity could not be
computed.

Now that we have established a working definition of a hub, we can investigate
the properties of compact GASTON sources as a function of the type of clump
they find themselves into, and more generally, as a function of their
convergence parameter. For that purpose we use the catalogue of compact
sources identified by rigby2021  and split the $\sim 1400$ compact 1.15mm
GASTON sources in four $f_{c}$ bins: $f_{c}<0.1$, $0.1<f_{c}<0.2$,
$0.2<f_{c}<0.3$, and $f_{c}>0.3$, in which there are 1186, 170, 56 and 19
compact sources, respectively. Based on this splitting, we can already see
that only 8% of GASTON compact sources are embedded within a hub.

First, we want to make sure that there is no distance bias when splitting out
source sample into $f_{c}$ bins. Figure 3(1st row) shows the distance
histograms for each $f_{c}$ bin. The vertical dash line shows the median
value. One can see that there is nearly no change whether in the shape of the
distribution or in the median distance values, all at about 5.2 kpc. Then we
look at the distribution of the source masses. Figure 3 (2nd row) clearly
shows that there is an increase in source mass as the convergence parameter
increases. In fact the median mass for the $f_{c}<0.1$ bin is at 135 M⊙ while
that of the $f_{c}>0.3$ bin is at 1350 M⊙, an order of magnitude larger. This
shows that a larger fraction of compact sources associated with hubs are
massive, although massive compact sources are also found in low $f_{c}$
regions.

Next, we investigate how the infrared-bright fraction ($f_{\rm{IRB}}$) of the
GASTON sources evolves as a function of the convergence parameter. The
brightness parameter is a measurement of how bright a cloud is as 8$\mu$m. For
instance, infrared dark clouds that do not host any star formation activity
yet will have an infrared brightness parameter of 0. The first of the four
clumps shown in Fig. 2 is a good example of such pristine structure. On the
other hand a cloud which is bright at 8$\mu$m, such as the last one of the
four clumps presented in Fig. 2, will have an infrared brightness close to 1.
Infrared brightness is believed to be a good tracer of clump evolution
rigby2021 ; watkinsprep . In Fig.3(3rd row) we show the histograms of infrared
brightness for the same four convergence parameter bins. There, we see a
continuous shift from clumps that are mostly infrared dark at low $f_{c}$
values, to clumps that are mostly infrared bright at large $f_{c}$ values. One
can interpret this in two ways: hubs are the late stages of clump evolution in
which filament convergence increases with time; infrared dark hubs have a very
short lifetime and become very quickly infrared bright via a strong star
formation activity. Note that these two scenarios are not mutually
incompatible. A similar picture can be drawn from Fig. 3(4th row) where the
distribution of bolometric luminosities are presented. Here, the luminosities
are being computed from the 70$\mu$m flux of internal protostellar sources. On
that figure we can clearly see that hubs host the formation of more luminous
sources, including all sources with $L_{\rm{bol}}\leq 10^{5}$L⊙, a result that
was already found by kumar2020 .

##  5 Conclusion

In this study, we have provided, for the first time, a quantitative definition
of what a hub is, and shown that they in fact represent a small fraction of
the filament population. We have also shown that hubs have a higher proportion
of massive, luminous, and more evolved compact sources within them. Also, in
all distributions of properties we have investigated, we do not observe any
discontinuity as a function of the convergence parameter but rather continuous
and smooth transitions from single filaments to highly convergent filament
networks. This is a clear indication that hubs do not represent a different
population of filament with their specific formation scenario. Instead we
propose that it is the rapid global collapse of clumps that is responsible for
(re)organising filament networks into hubs and, in parallel, for leading to
the mass growth of compact sources.

## Acknowledgements

NP and AJR would like to thank the STFC for financial support under the
consolidated grant numbers ST/N000706/1 and ST/S00033X/1, and the Royal
Society for providing computing resources under Research Grant number
RG150741. We would like to thank the IRAM staff for their support during the
campaigns. The NIKA2 dilution cryostat has been designed and built at the
Institut Néel. In particular, we acknowledge the crucial contribution of the
Cryogenics Group, and in particular Gregory Garde, Henri Rodenas, Jean Paul
Leggeri, Philippe Camus. This work has been partially funded by the Foundation
Nanoscience Grenoble and the LabEx FOCUS ANR-11-LABX-0013. This work is
supported by the French National Research Agency under the contracts "MKIDS",
"NIKA" and ANR-15-CE31-0017 and in the framework of the "Investissements
dâavenirâ program (ANR-15-IDEX-02). This work has benefited from the support
of the European Research Council Advanced Grant ORISTARS under the European
Union’s Seventh Framework Programme (Grant Agreement no. 291294). F.R.
acknowledges financial supports provided by NASA through SAO Award Number
SV2-82023 issued by the Chandra X-Ray Observatory Center, which is operated by
the Smithsonian Astrophysical Observatory for and on behalf of NASA under
contract NAS8-03060.

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An Edge-Coupled Magnetostatic Bandpass Filter

Connor Devitt, Graduate Student Member, IEEE, Renyuan Wang, Member, IEEE,
Sudhanshu Tiwari, Member, IEEE, Sunil A. Bhave, Senior Member, IEEE

Manuscript received on XX XX, 2023; revised on XX XX, 2024; accepted on XX XX, 2024. This research was developed with funding from the Air Force Research Laboratory (AFRL) and the Defense Advanced Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. This manuscript is approved for public release; distribution A: distribution unlimited. (Corresponding authors: Connor Devitt, Renyuan Wang)

R.W. invented the device concept, and developed baseline model and device design. C.D. performed simulations on fabrication variations and total filter loss, chip fabrication and characterization, as well as data analysis. Manuscript was prepared by C.D. with inputs from R.W., S.T., and S.A.B.

Connor Devitt (e-mail: devitt@purdue.edu), Sudhanshu Tiwari (e-mail: tiwari40@purdue.edu), and Sunil A. Bhave (e-mail<EMAIL_ADDRESS>are with the OxideMEMS Lab, Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA.
Renyuan Wang is with FAST Labs, BAE Systems, Inc., Nashua,
NH 03060 USA (e-mail: renyuan.wang@baesystems.com).

This paper reports on the design, fabrication, and characterization of an edge-coupled magnetostatic forward volume wave bandpass filter. Using micromachining techniques, the filter is fabricated from a yttrium iron garnet (YIG) film grown on a gadolinium gallium garnet (GGG) substrate with inductive transducers. By adjusting an out-of-plane magnetic field, we demonstrate linear center frequency tuning for a 4th-order filter from 4.5 GHz to 10.1 GHz while retaining a fractional bandwidth of 0.3%, an insertion loss of 6.94 dB, and a -35dB rejection. We characterize the filter nonlinearity in the passband and stopband with IIP3 measurements of -4.85 dBm and 25.84 dBm, respectively. When integrated with a tunable magnetic field, this device is an octave tunable narrowband channel-select filter.

Micromachining, magnetostatic wave (MSW), yttrium iron garnet (YIG), tunable bandpass filter, edge-coupled

§ INTRODUCTION

Coupled micro-electromechanical resonators with high quality factors (Q-factor) and miniaturized footprints have been an attractive technology for integration in wireless communication systems as narrowband channel-select filters. Wang [1] has demonstrated high-order micromechanical bandpass filters using one-dimensional (1D) arrays of mechanically-coupled resonators. However, for higher-order filters, long mechanical coupling beams become impractical due to size constraints [2] while sensitivity due to fabrication variation causes increased insertion loss and passband distortion [3]. The high sensitivity of weak electrostatic or mechanical edge-coupled resonators due to structural asymmetries has been leveraged for sensitive parametric mass sensing applications in [2], but prohibits their use in filters. 2D microresonator arrays have shown some success by utilizing weak coupling in one dimension to achieve pass band shape and strong coupling in the other dimension to reduce effects of fabrication variation, but suffer from high insertion loss [3].
Coupled resonator arrays utilizing magnetostatic waves (MSW) have the potential to overcome the weak coupling and extremely narrow bandwidths achieved by electrostatically coupled micromechanical resonators [4, 5] while introducing a degree of tunability.

(a) Chip microphotograph of multiple MSW bandpass filters and 1-port resonators fabricated on a YIG on GGG chip using YIG micromachining technology [6]. A 4-pole filter (on the left) and a 2-pole filter (on the right) are highlighted in red. A 1-port resonator is highlighted in orange. (b) Rendering of a 4-pole bandpass filter featuring four YIG resonators with gold electrodes conformally deposited over the etched YIG. Magnetic bias is oriented out-of-plane along the $z$-axis.

The magnetostatic wave resonance can be tuned more than an octave in frequency using a static magnetic field ensuring that the filter size does not scale to sub-micrometer dimensions at high-frequencies. Yttrium iron garnet (YIG) is the most widely used material for MSW devices due to its low Gilbert damping ($\alpha = 2.8\times10^{-4}$ for a $\SI{100}{\nano\meter}$ film [7]) and experimentally demonstrated Q-factors exceeding $3000$ [8, 9]. In state of the art YIG sphere filters [10, 11, 12], polished YIG resonators are attached to a thermally conductive rod and manually aligned to non-planar inductive loops acting as transducers. The assembled sphere and loop structures are then coupled through transmission lines similar to the coupling beams in [1] to synthesize a filter. Planar YIG resonators can magnetically couple if they are fabricated in close proximity (Fig. $\ref{chip_cartoon}$), analogous to the electrically coupled mechanical resonators in [2, 13] or coupled electromagnetic resonators. Magnetically coupled YIG filters can be fabricated at scale using micromachining techniques on films grown on a gadolinium gallium garnet (GGG) substrate, allowing for miniaturization and eliminating the need for polishing and meticulous manual alignment.

§ BANDPASS FILTER DESIGN

The bandpass filter shown in Fig. $\ref{chip_cartoon}$ consists of a number of closely-spaced rectangular YIG magnetostatic forward volume wave (MSFVW) resonators with shorted $\SI{300}{\nano\meter}$-thick gold inductive transducers conformally deposited over the outermost resonators. With an out-of-plane DC magnetic bias, the RF magnetic field from the transducers excite MSFVW modes in the YIG mesa. Forward volume waves are a family of highly dispersive modes in a thin film whose lowest order mode is described by the dispersion relation [14]:

\begin{equation}
\omega^2 = \omega_0\left[\omega_0+\omega_m\left(1 - \frac{1-e^{-k_{mn}t}}{k_{mn}t}\right)\right],
\end{equation}

where $\omega_m = \mu_0\gamma_m M_s$, $\omega_0 =\mu_0\gamma_m H^{eff}_{DC}$, $t$ is the film thickness, $\gamma_m$ is the gyromagnetic ratio, $\mu_0$ is the permeability of free space, $k_{mn}$ is the wave vector, $M_s$ is the saturation magnetization, and $H^{eff}_{DC}$ is the effective DC magnetic field. Considering the limits as $k_{mn}\rightarrow0$ and $k_{mn}\rightarrow\infty$ in the dispersion relation, $\omega$ is restricted within the range [14]:

\begin{equation}
\omega_0 \leq \omega \leq \sqrt{\omega_0\left(\omega_0+\omega_m\right)},
\label{spinwave_manifold}
\end{equation}

denoted as the spin wave manifold. When the planar dimensions of the thin YIG film are bounded, the magnetostatic waves reflects off the edges forming a standing waves with wave vectors approximately given by [15, 16, 17, 18, 19]:

\begin{equation}
k_{mn} = \sqrt{\left(\frac{\pi m}{l}\right)^2 + \left(\frac{\pi n}{w}\right)^2}, \ m,n=1,2,3\dots
\end{equation}

where $l$ and $w$ are the length and width of the cavity respectively. These MSFVW resonances can be further understood through an analogy to Lamb waves in a piezoelectric plate [20, 21, 22, 23]. An oscillating electric field perturbs the polarization of the piezoelectric film generating a stress field and exciting an acoustic wave. In the ferrimagnetic film, an oscillating magnetic field, conversely, perturbs the static magnetization leading to a precession of spins. Both the piezoelectric and magnetostatic cavities support a discrete number modes whose wave vectors depend on the cavity dimensions. Nonlinear dispersion relates the wave vectors to the resonant frequencies for both MSFVW and Lamb waves. For MSFVW, this leads to irregularly spaced modes which all reside in the spin wave manifold.

Unique to MSW, the applied out-of-plane magnetic field shifts the MSFVW dispersion relation in frequency where the tuning rate for the fundamental mode is given by

\begin{equation}
\frac{\partial \omega}{\partial H^{eff}_{DC}} = \mu_0\gamma_m \frac{2\omega_0 + \omega_m\left(1-\frac{1-e^{-k_{mn}t}}{k_{mn}t}\right)}{2\sqrt{\omega_0\left[\omega_0 + \omega_m\left(1-\frac{1-e^{-k_{mn}t}}{k_{mn}t}\right)\right]}},\\
\end{equation}

which simplifies to $\mu_0\gamma_m = \SI{2.8}{\mega\hertz / Oe}$ (for YIG) when $k_{mn}t\ll1$. The magnetostatic scalar potential, $\psi$, decays exponentially outside the YIG mesa [14, 24] so the MSFVW resonance in one YIG mesa may couple to adjacent mesas if there is sufficient overlap in their scalar potentials [25]. Consequently, the spacing of adjacent resonators and their vertical sidewall profile are critical to control the inter-resonator coupling strength. For the 4-pole filter in Fig. $\ref{chip_cartoon}$, the resonator spacings are $s_1=\SI{10}{\micro\meter}$, $s_2=\SI{15}{\micro\meter}$, and $s_3=\SI{10}{\micro\meter}$. Each has length $l=\SI{500}{\micro\meter}$. The outermost resonators have a width $w_{1,4}=\SI{70}{\micro\meter}$ while the inner resonators are slightly narrower at $w_{2,3}=\SI{67}{\micro\meter}$ which was found to marginally improve insertion loss and higher order width mode suppression based on finite element simulation.

[<->,>=stealth] (1,1.75) to [bend left] node[pos=0.5,fill=white] $M_{12}$ ++(3.5,0);
[<->,>=stealth] (4.5,1.75) to [bend left] node[pos=0.5,fill=white] $M_{23}$ ++(3.5,0);
[<->,>=stealth] (8,1.75) to [bend left] node[pos=0.5,fill=white] $M_{34}$ ++(3.5,0);
[<->,>=stealth] (0,4.75) to [bend left] node[pos=0.5,fill=white] $M_{IO}$ ++(12.5,0);

(0,3) to[L=$L_0$] (0,5)
to[R=$R_0$, -o] (0,8)

(0,0) to[R=$R_{m1}$] (0,2)
(1,0) to[L=$L_{m1}$] (1,2)
(2,0) to[C=$C_{m1}$] (2,2)

(3.5,0) to[R=$R_{m2}$] (3.5,2)
(4.5,0) to[L=$L_{m2}$] (4.5,2)
(5.5,0) to[C=$C_{m2}$] (5.5,2)

(7,0) to[R=$R_{m3}$] (7,2)
(8,0) to[L=$L_{m3}$] (8,2)
(9,0) to[C=$C_{m3}$] (9,2)

(10.5,0) to[R=$R_{m4}$] (10.5,2)
(11.5,0) to[L=$L_{m4}$] (11.5,2)
(12.5,0) to[C=$C_{m4}$] (12.5,2)

(12.5,3) to[L=$L_0$] (12.5,5)
to[R=$R_0$, -o] (12.5,8)

(0,2) – (0,3)
(1,2) – (1,2.5)
(2,2) – (2,2.5) – (0,2.5)

(0,0) – (0,-0.5)
(1,0) – (1,-0.5)
(2,0) – (2,-0.5) – (0,-0.5)

(12.5,2) – (12.5,3)
(11.5,2) – (11.5,2.5)
(10.5,2) – (10.5,2.5) – (12.5,2.5)

(12.5,0) – (12.5,-0.5)
(11.5,0) – (11.5,-0.5)
(10.5,0) – (10.5,-0.5) – (12.5,-0.5)

(3.5,2) – (3.5,2.5)
(4.5,2) – (4.5,2.5)
(5.5,2) – (5.5,2.5) – (3.5,2.5)

(3.5,0) – (3.5,-0.5)
(4.5,0) – (4.5,-0.5)
(5.5,0) – (5.5,-0.5) – (3.5,-0.5)

(7,2) – (7,2.5)
(8,2) – (8,2.5)
(9,2) – (9,2.5) – (7,2.5)

(7,0) – (7,-0.5)
(8,0) – (8,-0.5)
(9,0) – (9,-0.5) – (7,-0.5)

Lumped circuit model of an edge-coupled 4-pole MSW bandpass filter with electrically short transducers.

Since each resonator length is much shorter than the electromagnetic wavelength over the tuning range, the 4-pole filter can be modeled with the lumped element circuit show in Fig. $\ref{lumped_circuit}$. A Butterworth-Van Dyke circuit [26, 27] is typically used to model an acoustic resonance where the mechanical mode is described by a series R-L-C tank circuit and the transducers introduce a shunt plate capacitance. In the case of magnetic resonators, the transducer introduces a parasitic series inductance and the MSFVW is modeled using a parallel R-L-C tank circuit instead. $L_0$ and $R_0$ represent the parasitic inductance and resistivity of the gold electrodes. $R_{m}$, $C_{m}$, and $L_{m}$ represent the MSFVW resonance of each YIG mesa. $M_{nm}$ is the inter-resonator coupling between adjacent YIG mesas while $M_{IO}$ represents input/output inductive coupling of the gold electrodes setting the out-of-band rejection level. Similar to a mechanical coupling coefficient, an effective coupling from the electrical to magnetostatic domain can be defined by ($\ref{effective_coupling}$) where $f_p$ and $f_s$ represents the magnetic resonance and anti-resonance respectively.

\begin{equation}
k_{eff}^2 = \frac{\pi}{2}\left(\frac{f_p}{f_s}\right)\cot\left(\frac{\pi}{2}\frac{f_p}{f_s}\right)
\label{effective_coupling}
\end{equation}

$k_{eff}^2$ is a function of the resonator design determined by the ratio of $L_m$ to $L_0$. Similar to an acoustic filter, $k_{eff}^2$ sets a bound on the maximum achievable filter bandwidth and impacts the passband ripple. The representative 1-port resonator (highlighted in Fig. $\ref{whole_chip}$) exhibits a measured effective coupling and quality-factor of $k_{eff}^2=1.53\%$ and $Q=2206$ at $\SI{3962}{Oe}$ with frequency response shown in Fig. $\ref{resonator_response}$. From the measured resonator impedance response, $R_0$, $L_0$, $R_m$, $C_m$, and $L_m$ can all be extracted through separate fittings near the magnetostatic resonance and outside the spin wave manifold where the resonator behaves as an inductor. Using the same resonator parameters, the measured filter response excluding spurious modes can be fit to the lumped model in Fig. $\ref{lumped_circuit}$ by tuning $M_{IO}$, $M_{12}$, and $M_{23}$ under the assumption that each resonator has the same resonant frequency and the filter is symmetric. Fig. $\ref{4pole_fig_filter_frequency_response}$ shows the frequency response of the lumped model fitted to the measured S21 for a 4-pole filter along with the extracted model parameters.

Measured frequency response of the 1-port resonator highlighted in Fig. $\ref{whole_chip}$ at $\SI{3962}{Oe}$ showing a $Q=2206$ and $k_{eff}^2=1.53\%$.

§ FABRICATION

Fabrication process of the bandpass filter: (a) $\SI{3}{\micro\meter}$ liquid phase epitaxy (LPE) YIG film grown on $500\si{\micro\meter}$ GGG substrate. (b) $7.8\si{\micro\meter}$ thick photoresist (SPR220-7.0) patterned on YIG film as an etch mask. (c) $3\si{\micro\meter}$ ion mill etch of YIG film at a rate of $\SI{36}{\nano\meter / \minute}$. (d) Photoresist mask is removed and etched YIG is soaked in phosphoric acid at $80 \, ^\circ C$ for $ \SI{20}{\min}$. (e) Bi-layer photoresist (SPR220-7.0 and LOR 3B) mask for a liftoff is patterned onto the etched sample. (f) $\SI{10}{\nano\meter}$ Ti and $\SI{300}{\nano\meter}$ Au is deposited using glancing angle e-beam evaporation followed by a liftoff process.

SEM of the MSFVW bandpass filter showing vertically etched YIG resonators with an inter-resonator spacing of 10.5 and conformal gold electrodes over the edges of etched YIG.

The fabrication process for the MSW bandpass filter is outlined in Fig $\ref{fig_fab_process}$ where steps (a)-(c) are adapted from [6]. A thick photoresist mask (SPR220-7.0) is patterned onto a $\SI{3}{\micro\meter}$ YIG film grown via liquid phase epitaxially (LPE) on a $\SI{500}{\micro\meter}$ GGG substrate. The YIG film is etched through at a rate of $\SI{36}{\nano\meter / \minute}$ using an optimized ion milling recipe for vertical sidewalls with sufficient intermittent cooling to prevent burning of the photoresist. Optimized lithography for the thick SPR photoresist is crucial to the filter's final performance since the inter-resonator coupling factors, $M_{ij}$, are sensitive to the physical separation of the etched YIG. With this process, we are able to fully etch resonator spacings as narrow as $\SI{3}{\micro\meter}$ setting the maximum achievable $M_{ij}\leq 1.37\%$ based on finite element simulation. After etching, the resist is removed and the sample is soaked in phosphoric acid at $80 \, ^\circ C$ to remove redeposited material. For the gold electrodes, a SPR220-7.0 photoresist mask with a liftoff resist (LOR 3B) bi-layer is patterned for a liftoff process. Using a glancing angle e-beam evaporation, $\SI{300}{\nano\meter}$ of gold and a $\SI{10}{\nano\meter}$ titanium adhesion layer is conformally deposited over the etched YIG resonators. Finally, the sample is soaked in  overnight to complete the liftoff process. Fig. $\ref{fig_fab_sem}$ shows an SEM of a fabricated filter illustrating the vertically etched YIG and conformal gold electrodes. Due to the combination of bi-layer liftoff and angled metal deposition, the gold electrodes are larger than designed and show a tapered thickness.

§ EXPERIMENTAL RESULTS

Measured 4-pole MSW bandpass filter frequency response at different out-of-plane magnetic biases from $\SI{3205}{Oe}$ to $\SI{5303}{Oe}$.

Experimental setup showing the fabricated filter chip resting on the pole of an electromagnet, two GS probes connected to one device under test, and an optical microscope used for probe landing and device alignment.

Frequency responses for the (a) 2-pole and (b) 4-pole filters highlighted in Fig. $\ref{chip_pictures}$ near the passband at $\SI{3864}{Oe}$. (b) also shows a comparison of the fitted lumped element model in Fig. $\ref{lumped_circuit}$ with measured S21. The fitted circuit assumes all four resonators are identical and excludes the spurious pass bands caused by higher order MSFVW modes.

Measured $\SI{3}{\decibel}$ bandwidth and center frequency of the 4-pole MSFVW filter showing a tuning rate of $\SI{2.7}{\mega\hertz / Oe}$.

Filter s-parameters are measured using an Agilent PNA-L N5230A network analyzer with a pair of ground-signal (GS) probes from $\SI{3205}{Oe}$ to $\SI{5303}{Oe}$ corresponding to a center frequency tuning over $\SI{5.6}{\giga\hertz}$ as shown in Fig. $\ref{fig_filter_frequency_response_tuning}$.A stack of $\SI{1}{in}$ diameter, grade N52 Neodymium permanent magnets attached to a three axis stage provides the required out-of-plane magnetic field and a three axis hall effect sensor (THM1176-MF) is used to calibrate the applied field prior to each measurement. A single-pole electromagnet powered by a constant current source provides the required out-of-plane magnetic bias (pictured in Fig. $\ref{measurement_setup}$) and a single-axis Gauss meter is used to map the source current to the applied field. Prior to each measurement, the device under test is aligned to the center of the electromagnet's pole to ensure field uniformity. Fig. $\ref{frequency_response}$ shows the frequency response near the passband for a 2-pole and 4-pole filter at $\SI{3864}{Oe}$. Outside of the spin wave manifold, no magnetostatic waves may propagate so the filter behaves as two coupled inductors. Consequently, the out-of-band rejection is governed by the inductive coupling strength between the input and output electrodes. For the 2-pole filter with an electrode spacing of $\SI{67}{\micro\meter}$, the rejection is  $\SI{-25}{\decibel}$ while for the 4-pole filer with a spacing of $\SI{229}{\micro\meter}$, the rejection is $\SI{-35}{\decibel}$. At $\SI{3864}{Oe}$, the 2-pole filter shows an insertion loss (IL) of $\SI{-3.55}{\decibel}$ and a $\SI{3}{\decibel}$ bandwidth of $\SI{57.0}{\mega\hertz}$ while the 4-pole filter has an IL of $\SI{6.94}{\decibel}$ and a $\SI{3}{\decibel}$ bandwidth of $\SI{17.0}{\mega\hertz}$. From Fig. $\ref{2pole_fig_filter_frequency_response}$ and $\ref{4pole_fig_filter_frequency_response}$, higher-order magnetostatic spurious modes are visible to the right of the passband with a frequency separation of $\SI{29}{\mega\hertz}-\SI{40}{\mega\hertz}$. As described in [15], the current distribution along the transducer length can excite either even or odd ordered modes. Based on the resonator dimensions and electrically short transducer, the frequency spacing of these spurs agree well with odd ordered length modes. Fig. $\ref{fig_filter_tuning}$ shows the linear center frequency tuning at a rate of $\SI{2.7}{\mega\hertz/Oe}$ and the $\SI{3}{\decibel}$-bandwidth over the applied bias for the 4-pole filter. With a well-calibrated bias field, the extrapolated center frequency tuning line should intersect $-\omega_m$ at $\SI{0}{Oe}$. However, the Gauss meter used for the field calibration is thicker than the fabricated chip, so the reported bias is underestimated by approximately $\SI{219}{Oe}$. The filter's tuning was also measured using a neodymium permanent magnet mounted on a 3-axis stage to precisely calibrate the field accounting for any thickness difference between the chip and sensor. In this setup, the extrapolated $\SI{0}{Oe}$ intersection is at $\omega_m= \mu_0\gamma_m \cdot \SI{1751}{Oe}$ which agrees well the saturation magnetization of LPE YIG.

(a) Measured S21 and (b) total loss for 2-pole filters with $\SI{300}{\nano\meter}$ and $\SI{3}{\micro\meter}$ thick gold transducers biased at $\SI{3652}{Oe}$ and $\SI{3660}{Oe}$ respectively. Frequency is plotted relative to the center frequency to account for the slight difference in bias strength.

Considering the total loss ($1-\left\lvert S_{11}\right\rvert^2-\left\lvert S_{21}\right\rvert^2$) for the 2-pole filter biased at $\SI{3652}{Oe}$ and measured far away from all magnetostatic resonances, an average of $43\%$ of the input power is dissipated in the thin $\SI{300}{\nano\meter}$ gold transducers, radiated, or absorbed by the YIG on GGG substrate. Based on finite element simulations, the loss is primarily attributed to the resistance of the gold transducers. A second sample was fabricated with $\SI{3}{\micro\meter}$ electroplated gold to reduce resistive losses. Fig. $\ref{ElectrodeThickness}$ compares the measured insertion loss and total loss for the same 2-pole filter with different gold thicknesses. As expected, the mean out-of-band loss shows significant improvement from $43\%$ to only $12\%$. The average loss within the 3dB bandwidth exhibits a slight improvement dependent on bias, ranging from $0.5\%$ to $14.5\%$. The insertion loss improvement reflects the change in mean in-band loss with a maximum improvement from $\SI{4.43}{\decibel}$ to $\SI{2.92}{\decibel}$ around $\SI{3660}{Oe}$.

Two-tone IIP3 measurement in the passband of a 4-pole filter at $\SI{3652}{Oe}$ bias

Upper Input tone frequencies for IIP3 measurements

Bias Field      Stopband Low     Passband     Stopband High
3652 Oe     $\SI{5.465}{\giga\hertz}$     $\SI{5.799}{\giga\hertz}$     $\SI{6.055}{\giga\hertz}$

Summary of 4-pole filter IIP3

Bias Field      Stopband Low     Passband     Stopband High
3652 Oe     $\geq \SI{37.95}{dBm}$     $\SI{-4.85}{dBm}$     $\SI{25.84}{dBm}$

The linearity of the MSW bandpass filter is evaluated by measuring the input referred $3^{rd}$ order intercept point (IIP3) in the passband as well as the stopband both below and above the passband at a bias of $\SI{3652}{Oe}$. The nonlinearity measurements are performed using two Keysight E8257D signal generators with a frequency separation of $\Delta f = \SI{15}{\mega\hertz}$. The higher of the two tone frequencies for each region are listed in Table $\ref{IIP3_summary_frequency_table}$. A wideband power divider combines the two tones while an Agilent PXA spectrum analyzer measures the resultant spectrum. A two-stage calibration is performed to remove all cable and system losses at every tone frequency and input power level. Far away from the passband, the filter is expected to be linear and no intermodulation products were observed. Based on the maximum output power of the signal generators and noise floor of the spectrum analyzer, the lower stopband IIP3 is estimated to be greater than $\SI{37.95}{dBm}$ at $\SI{3652}{Oe}$. The passband shows the greatest nonlinearity with an IIP3 of $\SI{-4.75}{dBm}$ at $\SI{3652}{Oe}$ as shown in Fig. $\ref{passband_IIP3}$. The measured IIP3 in each frequency region is summarized in Table $\ref{IIP3_summary_table}$.

§ CONCLUSION

Performance Comparison with Other Tunable Bandpass Filters

Reference      Frequency Tuning (GHz)      Insertion Loss (dB)     Bandwidth (MHz)     Rejection (dB)
This work (2-pole)     4.5-10.1     $<6$     29-39     $>25$
This work (4-pole)     4.5-10.1     $<11$     11-17     $>35$
YIG Sphere [10]      2-8     $<5$       $20$      $>50$
Magnetostatic Surface Wave [28]      3.4-11.1     $<5.1$     18-25     $>25$
RF MEMS Tunable Filter [29]     6.5-10     $<5.6$     306-539      $>50$
Evanescent-Mode Cavity [30, 31]      3-6.2     $<4$     15-25     $>50$

In this paper, we have demonstrated a novel edge-coupled highly-tunable magnetostatic bandpass filter using state-of-the-art micromachining fabrication techniques. The designed 2-pole and 4-pole filters have been tuned over an octave from $\SI{4.5}{\giga\hertz}$ to $\SI{10.1}{\giga\hertz}$ showing a consistent passband shape with performance comparable to other state-of-the-art frequency tunable bandpass filters as summarized in Table $\ref{Performance_Comparison}$. We have also characterized the linearity of the filter in three distinct frequency regions. Our micromachining process enables precise control over the YIG mesa shape and spacings to synthesize miniaturized MSW channel-select filters analogous to electromagnetic cavity filter design.

§ DATA AVAILABILITY

The code and data used to produce the plots within this work will be released on the repository Zenodo upon publication.

§ ACKNOWLEDGMENTS

Chip fabrication was performed at the Birck Nanotechnology Center at Purdue. Resonator and filter measurements and characterization were performed at Seng-Liang Wang Hall at Purdue. The Purdue authors would like to thank Dave Lubelski for assistance with the glancing angle metal deposition and Yiyang Feng for discussions on the fabrication recipes.

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