Address all correspondence to A.J. Riggs. Email<EMAIL_ADDRESS>
# Flight mask designs of the Roman Space Telescope Coronagraph Instrument

A J Eldorado Riggs  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  0000-0002-0863-6228
Dwight Moody  Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Dr., Pasadena, CA 91109, USA  Jessica Gersh-Range  Princeton
University, Department of Mechanical and Aerospace Engineering, Olden Street,
Princeton, NJ 08544, USA  Dan Sirbu  NASA Ames Research Center, Moffett Field,
Mountain View, CA 94035, USA  Ruslan Belikov  NASA Ames Research Center,
Moffett Field, Mountain View, CA 94035, USA  Eduardo Bendek  Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,
CA 91109, USA  0000-0002-9408-8925 Vanessa P. Bailey  Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,
CA 91109, USA  0000-0002-5407-2806 Kunjithapatham Balasubramanian  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,
Pasadena, CA 91109, USA  Daniel W. Wilson  Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109,
USA  Scott A. Basinger  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  John Debes  Space
Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA
Tyler D. Groff  NASA Goddard Space Flight Center, 8800 Greenbelt Rd,
Greenbelt, MD 20771, USA  0000-0001-5978-3247 N. Jeremy Kasdin  University of
San Francisco, College of Arts and Sciences, 2130 Fulton St, San Francisco, CA
94117, USA  Bertrand Mennesson  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Douglas
M. Moore  Jet Propulsion Laboratory, California Institute of Technology, 4800
Oak Grove Dr., Pasadena, CA 91109, USA  Garreth Ruane  Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,
CA 91109, USA  0000-0003-4769-1665 Erkin Sidick  Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109,
USA  Nicholas Siegler  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  John Trauger  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,
Pasadena, CA 91109, USA  Neil T. Zimmerman  Space Telescope Science Institute,
3700 San Martin Dr, Baltimore, MD 21218, USA

###### Abstract

Over the past two decades, thousands of confirmed exoplanets have been
detected; the next major challenge is to characterize these other worlds and
their stellar systems. Much information on the composition and formation of
exoplanets and circumstellar debris disks can only be achieved via direct
imaging. Direct imaging is challenging because of the small angular
separations ($<1$ arcsec) and high star-to-planet flux ratios (${\sim}10^{9}$
for a Jupiter analog or ${\sim}10^{10}$ for an Earth analog in the visible).
Atmospheric turbulence prohibits reaching such high flux ratios on the ground,
so observations must be made above the Earth’s atmosphere. The Nancy Grace
Roman Space Telescope (Roman), set to launch in the mid-2020s, will be the
first space-based observatory to demonstrate high-contrast imaging with active
wavefront control using its Coronagraph Instrument. The instrument’s main
purpose is to mature the various technologies needed for a future flagship
mission to image and characterize Earth-like exoplanets. These technologies
include two high-actuator-count deformable mirrors, photon-counting detectors,
two complementary wavefront sensing and control loops, and two different
coronagraph types. In this paper, we describe the complete set of flight
coronagraph mask designs and their intended combinations in the Roman
Coronagraph Instrument. There are three types of mask configurations included:
a primary one designed to meet the instrument’s top-level requirement, three
that are supported on a best-effort basis, and several unsupported ones
contributed by the NASA Exoplanet Exploration Program. The unsupported mask
configurations could be commissioned and used if the instrument is approved
for operations after its initial technology demonstration phase.

###### keywords:

coronagraph, masks, deformable mirror, Roman Space Telescope, exoplanet,
circumstellar disk

##  1 INTRODUCTION

Over four thousand exoplanets have been discovered to date, primarily via
indirect methods that detect changes in the flux or Doppler shift of the host
stars. Several of these worlds are even Earth-sized and lie within the
habitable zones of their stars. To determine if any of these or other
exoplanets are habitable, we need spectroscopic measurements of their
atmospheres. Except for rare cases in which the exoplanet transits its host
star, this can only be achieved by directly imaging reflected light from the
exoplanet. The direct imaging of exoplanets is challenging because of the high
star-to-planet flux ratios at small angular separations. For example, the Sun
and Earth viewed from ten parsecs away have a planet-to-star flux ratio of
${\sim}10^{-10}$ in the visible and a maximum separation of just 0.1
arcseconds. Imaging planets with such a high flux ratio will only be possible
with a space-based observatory, avoiding the limitations of atmospheric
turbulence.

In conjunction with a large space telescope for its collecting area and
angular resolution, specialized optics are needed to suppress the bright glare
of the starlight at the exoplanet’s location. One of the leading technologies
for this is the internal coronagraph. A coronagraph is a set of masks and/or
mirrors that blocks or redirects the on-axis starlight and transmits off-axis
sources such as exoplanets and debris disks. Two flagship mission concepts
proposed for the 2020 Astrophysics Decadal Survey, the Habitable Exoplanet
Observatory (HabEx)[1] and the Large Ultraviolet/Optical/Infrared Surveyor
(LUVOIR)[2], would use a coronagraph instrument to image and characterize
Earth-like exoplanets around nearby stars.

High-contrast coronagraphs with active wavefront control have been
successfully demonstrated in laboratory testbeds but have not yet flown in
space. To bridge the the gap between current capabilities and a future Earth-
analog characterization mission, NASA is including the Coronagraph Instrument
as a technology demonstrator on the Nancy Grace Roman Space Telescope (Roman),
a 2.4-meter flagship observatory set to launch in the mid-2020s.[3, 4, 5] Some
of the key technologies that the Roman Coronagraph Instrument will demonstrate
are: high-order wavefront correction with two high-actuator-count deformable
mirrors (DMs),[6, 7, 8, 9, 10, 11] low-order wavefront sensing and control
(LOWFS/C) of pointing jitter and thermo-mechanical deformations of the
observatory,[12, 13, 14] photon-counting detectors,[15] and two different
types of high-contrast coronagraphs.

The two complementary types of high-contrast coronagraphs included in the
Roman Coronagraph Instrument are the Hybrid Lyot Coronagraph (HLC)[16, 17, 18,
7] and the Shaped Pupil Coronagraph (SPC)[19, 20, 21, 22, 23]. The HLC uses
two DMs with large deformations, a phase-and-amplitude modifying focal-plane
occulting spot, and a traditional Lyot stop to destructively interfere light
and create a high-contrast dark hole in the final image. Much of the remaining
starlight diffracts just outside the dark hole, so a field stop is needed with
the HLC on Roman to limit the dynamic range on the detector and prevent
ghosting through the imaging lens. The SPC uses a binary-amplitude apodizer to
reshape the point-spread function (PSF) and then a focal plane mask and Lyot
stop to block most of the rest of the light at later planes. Relatively little
residual starlight remains at the final image plane, so field stops are not
required for the SPCs on Roman in imaging mode, although field stops are
required for slit spectroscopy and recommended for polarimetry).

Because of the challenging pupil obscurations (a large central obscuration and
thick secondary mirror support struts) of the Roman Space Telescope, multiple
coronagraphic mask configurations are needed to enable the desired high-
contrast usage cases of 1) imaging close to the star, 2) performing
spectroscopy close to the star, 3) imaging over a wide field of view (FOV),
and 4) performing polarimetry. The aspirational science cases enabled by these
engineering goals are, respectively, to detect cool gas giant exoplanets and
image inner debris disks, spectrally characterize the imaged exoplanets, image
the outer part of faint debris disks, and measure the polarization of disks.

In this paper, we show all the coronagraphic masks included in the Roman
Coronagraph Instrument and all the designed combinations of them. The designed
mask configurations are described in Section 2. The layouts of the individual
masks on their substrates are shown in Appendices A, B, C, and D. The
available color filters are listed in Appendix E.

##  2 Mask Configurations

###  2.1 Precision Alignment Mechanisms (PAMs)

As shown in the instrument’s optical layout in Fig. 1, the coronagraphic masks
reside in four planes on different Precision Alignment Mechanisms (PAMs),
which are the Shaped Pupil Alignment Mechanism (SPAM), Focal Plane Alignment
Mechanism (FPAM), Lyot Stop Alignment Mechanism (LSAM), and Field Stop
Alignment Mechanism (FSAM). For completeness, in Appendices A, B, C, and D we
show the mask PAMs and the layouts of all the masks on their substrates. Most
of the mask layouts have yet to be fabricated, so the final products may
differ slightly (e.g., extra alignment fiducials might be added) in appearance
from what is shown here.

Figure 1: The optical layout of the Roman Coronagraph Instrument. The
instrument’s beam path begins at the fast steering mirror (FSM) and ends at
the direct imaging detector (labeled DI FPA).

The two other PAMs, located downstream of the coronagraph masks, are the Color
Filter Alignment Mechanism (CFAM) and the Dispersion Polarizer Alignment
Mechanism (DPAM). The CFAM contains four main bandpass filters: Band 1 is a
10.1% bandwidth111Bandwidth is defined here as the full width at half maximum
(FWHM) of the filter transmission profile. centered at 575 nm; Band 2 is a
17.0% bandwidth centered at 660nm; Band 3 is a 16.7% bandwidth at 730nm; and
Band 4 is a 11.4% bandwidth at 825nm. Within each full band, there are also
several narrower-bandwidth engineering filters used for digging the dark
holes. The complete list of CFAM filters and their designed bandpasses is in
Table 7, and their final, manufactured properties will be updated in the list
of instrument parameters at https://roman.ipac.caltech.edu.

The DPAM contains several standalone lenses as well as two polarization
modules (with Wollaston prisms) and two spectroscopy modules (with Amici
prisms).[24] The polarization modules are planned for use only in Bands 1 and
4 but could be used without performance degradation in Bands 2 and 3. One
spectroscopy module is designed for zero optical deviation and a spectral
resolution of 50 at the center wavelength of Band 2, and the other is designed
to have those properties at the center of Band 3. The spectroscopy modules
could be used in Bands 1 and 4 as well but would have nonzero (but minimal)
optical deviation and different spectral resolutions.

###  2.2 Classes of Mask Configurations

Based on their levels of testing and software support, there are three
different classes of mask configurations in the Roman Coronagraph Instrument:
primary, supported, and unsupported.

The primary mask configuration will fulfill the instrument’s single, top-level
threshold requirement and will therefore receive full, system-level testing on
the ground. This top-level requirement is to perform broadband, high-contrast
imaging in Band 1 at a small angular separation. It states:

> The Roman Coronagraph shall be able to measure, with SNR $\geq 5$, the
> brightness of an astrophysical point source located between 6 and 9
> $\lambda$/D from an adjacent star with a VAB magnitude $\leq 5$, with a flux
> ratio $\geq 1\times 10^{-7}$. The bandpass shall have a central wavelength
> $\leq 600$ nm and a bandwidth $\geq 10$%.

To mitigate the programmatic risks associated with demonstrating a high-
contrast coronagraph with active wavefront control in space for the first
time, this threshold requirement is intentionally set to be easier than the
current best estimates of instrument performance, which predict broadband dark
holes with raw contrasts in the $10^{-9}$ to $10^{-8}$ range.222Performance
predictions for the primary mask configuration and two of the supported ones
are available at https://github.com/nasavbailey/DI-flux-ratio-plot and are
updated periodically. Note that Band 1 polarimetry is supported on a best-
effort basis because it is not tied directly to the top-level requirement.

The three supported mask configurations serve to perform spectroscopy in Bands
2 and 3 as well as wide FOV imaging and polarimetry in Band 4. Being part of
the plan but not tied to requirements, these supported mask configurations are
included on a best-effort basis. They are receiving component-level hardware
support and full software support. The main difference is that they will not
receive end-to-end, system-level testing prior to launch.

The NASA Exoplanet Exploration Program (ExEP) is contributing another twenty
unsupported mask configurations. The masks for these are being placed in
otherwise unused space alongside the supported masks. Other than being
installed and aligned, they will not receive any hardware testing or software
support as part of the planned technology demonstration. The unsupported mask
configurations could be commissioned and used if the instrument is approved
for operations after its initial technology demonstration phase.

The unsupported mask configurations are divided into three sub-categories:
high-contrast, low-contrast, and calibration. The unsupported high-contrast
configurations are largely chosen to fill gaps in the covered parameter space
of capabilities; for example, providing narrow FOV imaging to Band 4. The
unsupported low-contrast configurations are all simply Lyot coronagraphs that
can utilize the instrument’s full field of view. Finally, the unsupported
calibration masks are Zernike wavefront sensors (ZWFS) to enable sensitive
pupil-plane wavefront measurements, as is desired for the future mission
concepts HabEx and LUVOIR.[25]

###  2.3 Designed Mask Configurations

The complete set of designed mask configurations in the Roman Coronagraph
Instrument is shown in Table 1. (Other possible mask configurations are
discussed in Section 2.4.) Each row in Table 1 is for a different mask
configuration, and the rows are grouped into the different classes of mask
configurations (primary, supported, and the three types of unsupported). The
columns indicate key optical elements and FOV parameters for each
configuration.

Configuration Class Coro. Type Configuration Name Band No. SPAM Pos’n FPAM
Pos’ns LSAM Pos’n FSAM Positions Az FOV (deg) IWA ($\lambda$/D) IWA (mas) OWA
($\lambda$/D) OWA (mas) Primary HLC Narrow-FOV 1 1 4-C2R1:9, 4-C3R1:8,
4-C4R7:9 3 R1C1, (R1C3), (R6C6), R5C3, R5C4 360 3.0 150.6 9.7 486.8 Supported
SPC Bowtie 2 4 2-R3C2, 2-R4C2, 2-R4C4 5 none, R2C5, R3C1, R3C2, R6C5 130 3.0
172.8 9.1 524.2 SPC Bowtie 3 4 6-R2C2, 6-R4C3, 6-R6C1 5 none, R1C2, R3C1,
R3C2, R4C2, R4C6 130 3.0 191.2 9.1 579.8 SPC Wide-FOV 4 2 6-R5C1, 6-R8C1,
6-R8C3 4 none, R1C5, R6C1, R6C2 360 5.9 424.9 20.1 1447.4 Unsupported (High
Contrast) SPC Rotated bowtie 2 3 2-R3C1, 2-R3C3, 2-R4C3 1 R4C3, R4C4 130 3.0
172.8 9.1 524.2 SPC Rotated bowtie 3 3 6-R2C3:4, 6-R4C2 1 R2C2, R4C2 130 3.0
191.2 9.1 579.8 SPC Wide-FOV 1 2 2-R1C1:2, 2-R2C1 4 none, R1C5, R6C1, R6C2 360
5.9 296.1 20.1 1008.8 SPC Multi-star 1 5 2-R1C1:2, 2-R2C1 4 R4C1 360 5.9 296.1
20.1 1008.8 SPC Multi-star 4 5 6-R5C1, 6-R8C1, 6-R8C3 4 R1C4, (R4C1) 360 5.9
424.9 20.1 1447.4 HLC Narrow-FOV 2 1 4-C4R1:6, 4-C5R1:8 3 R3C3, R2C3 360 3.0
172.8 9.7 558.8 HLC Narrow-FOV 3 1 3-R5C1:3, 3-R6C1:3, 3-R7C1:2, 3-R8C1 3
R3C4, R4C5, R5C1, R5C2 360 3.0 191.2 9.7 618.1 HLC Narrow-FOV 4 1 3-R2C1:3,
3-R3C1:3, 3-R4C1:3 3 R3C5 360 3.0 216.0 9.1 655.3 Unsupported (Low Contrast)
Lyot 0.281” occulter 1, 2 1 4-C2R1:9, 4-C3R1:8, 4-C4R7-9 3, 4 open, any 360 -
140.5 - 3600 Lyot 0.322” occulter 1, 2 1 4-C4R1:6, 4-C5R1:8 3, 4 open, any 360
- 161.1 - 3600 Lyot 0.803” occulter 1, 2 1 4-C1R3:4 3, 4 open, any 360 - 401.5
- 3600 Lyot 0.921” occulter 1, 2 1 4-C1R2 3, 4 open, any 360 - 460.7 - 3600
Lyot 1.807” occulter 1, 2 1 4-C1R1 3, 4 open, any 360 - 903.4 - 3600 Lyot
0.357” occulter 2,3,4 1 3-R5C1:3, 3-R6C1:3, 3-R7C1:2, 3-R8C1 3, 4 open, any
360 - 178.6 - 3600 Lyot 0.403” occulter 2,3,4 1 3-R2C1:3, 3-R3C1:3, 3-R4C1:3
3, 4 open, any 360 - 201.6 - 3600 Lyot 1.020” occulter 2,3,4 1 3-R1C3 3, 4
open, any 360 - 509.8 - 3600 Lyot 1.152” occulter 2,3,4 1 3-R1C1 3, 4 open,
any 360 - 576.1 - 3600 Lyot 2.294” occulter 2,3,4 1 3-R1C2 3, 4 open, any 360
- 1147.1 - 3600 Unsupported (Calibration) ZWFS transmissive and reflective 1 1
4-C7R2:8 2 open N/A N/A N/A N/A N/A ZWFS reflective only N/A 1 2-R2C2:4,
6-R4C1, 6-R6C2, 6-R8C2 N/A N/A N/A N/A N/A N/A N/A

Table 1: List of designed mask configurations in the Roman Coronagraph
Instrument. The unsupported mask configurations could be commissioned and used
only if the instrument is approved for operations after its initial technology
demonstration phase. Other combinations of masks are possible but most would
have worse contrast, throughput, and/or FOV compared to the configurations
listed above. Imaging, spectroscopy, and polarimetry are technically possible
in all four bandpasses; however, polarimetry is officially supported only in
Bands 1 and 4, and spectroscopy is officially supported only in Bands 2 and 3.
Spectroscopy and polarimetry cannot be done concurrently. Mask PAM positions
reference the labeled mask array layouts shown in the Appendices. As an
example for FPAM, 4-C2R1:9 means column 2 and rows 1 to 9 in overall FPAM
position 4. For the SPC Multi-star, the FOV is set by the field stop opening
but can be placed anywhere within the listed annular FOV.

Column 4 of Table 1 specifies the bandpasses in which each mask configuration
can work. Only the full bandpasses are listed, but each mask configuration
also works in all the subbands of the full bands listed; the filter bandwidths
are given in Table 7. All of the high-contrast mask configurations work only
in their designed bandpass since they require high-order wavefront correction.
The unsupported, low-contrast Lyot coronagraphs do not need wavefront
correction and are thus spectrally limited only by the anti-reflective (AR)
coatings used on the glass focal plane mask (FPM) substrates. The unsupported
Zernike wavefront sensor (ZWFS) mask configurations work in reflection with
the low-order wavefront sensor (LOWFS), which has its own 128nm-wide filter
centered at 575nm. The ZWFS that also works in transmission is designed to
work in Band 1 only. It may work in other bands, but that has not been
studied.

Columns 5-8 of Table 1 specify PAM mask positions that combine to form that
mask configuration. There is only one FPM design per mask configuration; the
multiple FPAM locations listed per row are all copies of that given FPM
design, which provides redundancy against manufacturing defects and/or dust
contamination over the course of the mission. The HLC occulters–when combined
with flat DMs–can be used as part of the simple Lyot coronagraphs with inner
working angles $<0.5$ arcsec. There are only one SPAM mask and one LSAM mask
per mode. The low-contrast, traditional Lyot coronagraphs can be used with
either Lyot stop that has struts like the telesope pupil (Figs. 11(c) and
11(b)), but most users would probably prefer the one in LSAM position 4
because it provides higher throughput and a sharper PSF at the expense of some
contrast.

There are usually several field stops that can work with each mask
configuration because the field stops are not used in a diffractive manner to
help create the dark hole. Only the field stops designed specifically for each
mask configuration are listed in each row of column 8; other desirable
pairings are left for the reader to determine based on the field stop
specifications provided in Table 6. Of the 31 field stops at the FSAM, 27 are
unique. The positions of the four that are duplicates are shown in parentheses
in Table 1. All the slit-shaped field stops are intended for use with the
spectroscopy modules. The two largest circular field stops (having
1.9-arcsecond and 3.5-arcsecond radii) are primarily intended for use with the
Wollaston prisms to eliminate crosstalk between polarization states in the
event that the astrophysical source or detectable instrumental scattered light
is present at large separations. (The field centers of the two polarization
state images are separated by 7.5” on the detector).

The FOV is specified by the azimuthal coverage, inner working angle (IWA), and
outer working angle (OWA). All configurations have 360-degree azimuthal FOV
except for the bowtie SPCs, which each have $2\times 65^{\circ}$ coverage. The
IWA and OWA are, respectively, the angular separations at which the off-axis
throughput ramps up or ramps down to half the maximum value. With the
unsupported, traditional Lyot coronagraphs, the IWA column instead lists the
occulting spot radius and the OWA column gives the instrument’s full,
unvignetted FOV in imaging mode. For the multi-star imaging mask
configurations,[26] the field of view is limited to the $9{\times}9$
$\lambda$/D field stop, but that field stop can be placed anywhere within the
annular field of view listed in Table 1.

Figure 2 provides a visual representation of the DM surfaces, masks, and
stellar PSFs corresponding to all the high-contrast mask configurations listed
in Table 1. All the focal plane masks, field stops, and PSFs are shown to
scale relative to each other (in terms of $\lambda/D$). All pupil masks and DM
surfaces are also shown to scale relative to each other (in terms of beam
diameter).

Figure 2: Masks in all the designed high-contrast mask configurations of the
Roman Coronagraph Instrument. In general, many field stops could be used with
each mask config, but only the nominal field stop for each config is shown
above. The DM surfaces shown are in addition to what is needed to correct
aberrations in the optical system. The multi-star imaging dark holes are shown
for a single star only, as any two-star geometry would be case dependent. All
the focal plane masks, field stops, and PSFs are shown to scale relative to
each other (in terms of $\lambda/D$). All pupil masks and DM surfaces are
shown to scale relative to each other (in terms of beam diameter).

Looking at the combined capabilities of the primary, supported, and
unsupported high-contrast mask configurations, much of the possible parameter
space is covered with respect to bandpass, field of view, and spectroscopy and
polarimetry. The HLC configurations enable imaging and polarimetry at small
separations in all four bandpasses. The SPC wide FOV allows those out to
larger separations in Bands 1 and 4. For spectroscopy in Bands 2 and 3, the
SPC bowtie and SPC rotated bowtie mask configurations cover approximately two-
thirds of the available azimuthal FOV at any given time. The Band 2 and 3 HLCs
could also be used for spectroscopy, but the SPC bowties are preferred since
they are more robust to the observatory jitter and wavefront drift expected
over long integration times.

Another unsupported capability in the instrument is multi-star imaging at high
contrast. The apodizer for multi-star imaging (at SPAM position 5) is a near-
duplicate of the SPC Wide FOV apodizer (at SPAM position 2) except that it
contains a square grid of dots to act as a mild diffraction grating. This
creates a grid of bright satellite spots in the focal plane that can be used
for super-Nyquist wavefront control (SNWC),[27] which is a critical component
of multi-star wavefront control (MSWC).[28] Bendek et al.[26] (these
proceedings) describe in detail the design and usage cases of the Band 1 and
Band 4 multi-star imaging mask configurations in the Roman Coronagraph
Instrument.

###  2.4 Other Possible Mask Configurations

Other than the designed mask configurations listed in Section 2.3, there are
many other possible combinations of masks and filters. Some may serve a useful
purpose while others are highly discouraged due to poor performance. Either
way, all of these other possible mask configurations are unsupported.

####  2.4.1 Unusable Mask Configurations

In most instances, arbitrary mask combinations would perform worse in most or
all cases compared to the mask configurations already specified.

All the focal plane masks are on glass substrates with AR coatings, which
restrict the bandpasses in which they can be used. Inside the intended
bandpasses the reflectivity of each substrate surface is $<$0.5%, but outside
it is $>$10%. Substrates in FPAM positions 2 and 4 can be used in Bands 1 and
2, and substrates in FPAM positions 3 and 6 can be used in Bands 2, 3, and 4.

The SPMs were optimized to make dark holes in their specified bandpasses and
fields of view, so decent high-contrast dark holes are not possible out of
band or with a different FPM. Because the SPMs increase the brightness of
speckles outside the dark hole, one would be better off using the SPAM flat
mirror instead of any SPM when performing low-contrast imaging over the
instrument’s full field of view.

####  2.4.2 Other, Viable Mask Configurations

As mentioned before, the field stops are not diffractive masks (unlike the
SPMs, FPMs, and Lyot stops), so there are many field stops that could work for
a given combination of other masks. For any high-contrast HLC cases, the dark
hole must be dug in a region at least as large as the field stop to prevent
bright speckles from reaching the detector (which could damage the detector
and would cause ghosting). Because there are 27 unique field stop shapes, we
will not describe all the possible field stop usages.

The only known high-contrast mask configurations not already listed would be
formed by swapping out one HLC’s occulter for another. (The previously
mentioned limitation from FPM substrate AR coatings still applies, though.)
This is possible because the DMs generate most of the contrast for the HLC,
and the DM surfaces are changeable on orbit. A slightly smaller IWA could be
obtained (at the expense of worse contrast) in Band 4 by using the Band 3 HLC
occulter instead. Similarly, the Band 1 HLC occulter could be used in Band 2.
In all other cases, though, using a different HLC occulter (or one of the
simple Lyot occulters) would result in a larger IWA (but possibly better
contrast). The SPC FPMs could also technically be used with DMs instead of an
SPM (similar to the HLCs), but we do not believe these hybrid combinations
would work as well as the existing mask configurations.

##  3 Summary and Current Status

In this paper, we have shown all the coronagraph masks that will be included
in the Roman Coronagraph Instrument. We listed and described all the designed
mask and color filter combinations. There are four high-contrast mask
configurations included as part of the mission’s technology demonstration.
These configurations accomplish imaging and polarimetry at small angular
separations in Band 1, imaging and polarimetry at larger angular separations
in Band 4, and imaging and spectroscopy in Bands 2 and 3. The NASA Exoplanet
Exploration Program is also contributing a number of unsupported mask
configurations that could be used after the initial technology demonstration
phase of the mission. These include eight more high-contrast mask
configurations to enable multi-star imaging and to complement the FOV and
wavelength coverage of the supported mask configurations; several more
occulters for low-contrast Lyot coronagraphy; and a Zernike wavefront sensor
for high-sensitivity, pupil-plane wavefront estimation.

These masks are all being fabricated by JPL’s Microdevices Laboratory.[29, 30,
31] The masks for the primary mask configuration are being fabricated first
and are nearly completed. The remaining masks will be manufactured later in
2021.

##  Appendix A Shaped Pupil Masks

The Shaped Pupil Alignment Mechanism (SPAM), shown in Fig. 3, carries four
different shaped pupil masks (SPMs) for the SPCs as well as a standard flat
mirror used for the HLCs and calibrations. The HLC flat mirror is in position
1, the SPC wide FOV SPM is in position 2, the SPC rotated bowtie SPM is in
position 3, the SPC bowtie SPM is in position 4, and the SPC multi-star
imaging SPM is in position 5. A face-on view of the SPM array is provided in
Fig. 4.

Figure 3: The Shaped Pupil Alignment Mechanism (SPAM) in the Roman Coronagraph
Instrument. The five nominal positions are labeled.  Figure 4: The array of
shaped pupil masks in the Roman Coronagraph. All four masks are manufactured
together on a 44mm-wide by 47mm-high by 4mm-thick silicon wafer. The black
regions on the substrate front face are black silicon, and the gray areas are
unprotected aluminum.[29, 30, 31] The small circles in between masks are used
for calibration and are not illuminated when the beam is centered on the SPMs.

##  Appendix B Focal Plane Masks

The Focal Plane Alignment Mechanism (FPAM), shown in Fig. 5, has the most
masks of any PAM. FPAM positions 2, 3, 4, and 6 each contain a 23mm-diameter,
fused silica substrate carrying many FPMs. Position 2 has FPMs for SPCs in
Bands 1 and 2, as shown in Fig. 6 and listed in Table 2. Position 6 has FPMs
for SPCs in Bands 3 and 4, as shown in Fig. 7 and listed in Table 3. Position
4 has FPMs for HLCs in Bands 1 and 2 as well as ZWFS dimples, as shown in Fig.
8 and listed in Table 4. Position 3 has FPMs for HLCs in Bands 3 and 4, as
shown in Fig. 9 and listed in Table 5. Several copies of each FPM design are
included in case of fabrication defects or dust contamination. The IWA and OWA
for each FPM are given in Table 1.

Figure 5: The Focal Plane mask Alignment Mechanism (FPAM) in the Roman
Coronagraph Instrument. The mechanism plate has four 23mm-diameter, fused
silica substrates carrying the focal plane masks, located at positions 2, 3,
4, and 6. Positions 1 and 7 have neutral-density (ND) filters, and position 5
has a through-hole.

Figure 6: Mask layout at FPAM position 2, primarily for SPC FPMs in Bands 1 and 2. Note that the vertical line in the annular FPMs is a plotting artifact and is not present in the actual masks. Mask Name | Band No. | Array Locations
---|---|---
SPC Wide-FOV FPM | 1 | R1C1, R1C2, R2C1
SPC Bowtie | 2 | R3C2, R4C2, R4C4
SPC Rotated Bowtie | 2 | R3C1, R3C3, R4C3
⌀=0.5$\lambda_{1}$/D Pinhole | 1 | R4C1, R4C5, R5C1, R5C2, R5C3, R5C4
Reflective-only ZWFS Pimple | (LOWFS) | R2C2, R2C3, R2C4
Table 2: Mask names and locations on the substrate at FPAM position 2. There
are several copies of each mask design. Pinholes are used only during ground
calibration. The LOWFS has its own, fixed filter in a separate beam path.

Figure 7: Mask Layout at FPAM position 6, primarily for SPC FPMs in Bands 3 and 4. Note that the vertical line in the annular FPMs is a plotting artifact and is not present in the actual masks. Mask Name | Band No. | Array Locations
---|---|---
SPC Wide-FOV FPM | 4 | R5C1, R8C1, R8C3
SPC Bowtie | 3 | R2C2, R4C3, R6C1
SPC Rotated Bowtie | 3 | R2C3, R2C4, R4C2
⌀=0.5$\lambda_{1}$/D Pinhole | 1 | R2C1, R3C1, R7C1
⌀=0.5$\lambda_{3}$/D Pinhole | 3 | R1C4, R2C5, R6C3, R7C2
⌀=0.5$\lambda_{4}$/D Pinhole | 4 | R1C1, R1C2, R1C3
Reflective-only ZWFS Pimple | (LOWFS) | R4C1, R6C2, R8C2
Table 3: Mask names and locations on the substrate at FPAM position 6. There
are several copies of each mask design. Pinholes are used only during ground
calibration. (Band 1 pinholes are included on the position 6 substrate as a
backup because it is supported whereas the position 2 substrate is
unsupported.)

Figure 8: Mask Layout at FPAM position 4, primarily for HLC FPMs in Bands 1 and 2. The substrate also includes four simple Lyot coronagraph occulters in column 1 and seven phase-shifting dimples for a Zernike wavefront sensor in column 7. The dark yellow region indicates where PMGI resist is left on the substrate; in general it is not desirable as it negates the AR coating, but here is used to make the ZWFS dimples. Mask Name | Band No. | Array Locations
---|---|---
⌀=1.807 arcsec Lyot Occulter | 1, 2 | C1R1
⌀=0.921 arcsec Lyot Occulter | 1, 2 | C1R2
⌀=0.803 arcsec Lyot Occulter | 1, 2 | C1R3:4
Open | 1, 2 | C1R5
HLC Occulter | 1 | C2R1:9, C3R1:8, C4R7:9
HLC Occulter | 2 | C4R1:6, C5R1:8
Transmissive-and-reflective ZWFS Dimple | 1 | C7R2:8
⌀=2$\lambda_{1}$/D Alignment Spots for ZWFS | 1 | C6R1:9, C7R1, C7R9, C8R1:9
Table 4: Mask names and locations on the substrate at FPAM position 4.

Figure 9: Mask layout at FPAM position 3, primarily for HLC FPMs in Bands 3 and 4. The substrate also includes three simple Lyot coronagraph occulters in row 1. All these masks are unsupported. The dark yellow region indicates where PMGI resist is left on the substrate. Mask Name | Band No. | Array Locations
---|---|---
⌀=1.152 arcsec Lyot Occulter | 2, 3, 4 | R1C1
⌀=2.294 arcsec Lyot Occulter | 2, 3, 4 | R1C2
⌀=1.020 arcsec Lyot Occulter | 2, 3, 4 | R1C3
Open | 2, 3, 4 | R1C4
HLC Occulter | 4 | R2C1:3, R3C1:3, R4C1:3
HLC Occulter | 3 | R5C1:3, R6C1:3, R7C1:2, R8C1
Table 5: Mask names and locations on the substrate at FPAM position 3.

##  Appendix C Lyot Stop Masks

The Lyot Stop Alignment Mechanism (LSAM), shown in Fig. 10, carries four
different lyot stops as well as a though hole used for alignment and
calibration. The Lyot stop for the unsupported SPC rotated bowtie is in
position 1. The through hole is in the middle at position 2. The Lyot stop for
all HLCs is in position 3. The Lyot stop for SPC wide FOV and SPC multi-star
imaging is in position 4. The Lyot stop for the supported SPC bowtie is in
position 5. Face-on views of the Lyot stops are provided in Fig. 11.

Figure 10: The Lyot Stop Alignment Mechanism (LSAM) in the Roman Coronagraph
Instrument. The five nominal positions are numbered.

(a) Lyot stop for SPC Rotated Bowtie

(b) Lyot stop for HLC Narrow-FOV

(c) Lyot stop for SPC Wide-FOV and SPC Multi-star

(d) Lyot stop for SPC Bowtie

Figure 11: Face-on views of the four Lyot stop substrates. All features are to
scale. The flat-to-flat width of each piece is 21.00mm. The fiducials (arrows
and rotated F’s) are top-layer etches only, not through-holes.

##  Appendix D Field Stop Masks

The Field Stop Alignment Mechanism (FSAM), shown in Fig. 12, has two open
positions (1 and 3), a neutral density filter in position 2, and a silicon
wafer carrying all the through-hole field stops in position 4. A face-on view
of the substrate with all the field stops is provided in Fig. 13. The
specifications for all the field stops are listed in Table 6.

Figure 12: The Field Stop Alignment Mechanism (FSAM) in the Roman Coronagraph
Instrument. Unlike the other PAMs, the FSAM is installed facing downstream for
packing reasons on the optical bench; thus, the field stop array is mounted
face down. Figure 13: The field stop array layout at FSAM position 4 in the
Roman Coronagraph Instrument. All masks are to scale and are labeled with
their position identifiers. The field stops are through holes, whereas the
fiducials (arrows and lowercase “t” shapes) are top-layer etches only. The
specifications of each field stop are in Table 6.

FSAM Position Primary Use Shape Width ($\mu$m) Width ($\lambda$/D) Width (mas)
Height ($\mu$m) Height ($\lambda$/D) Height (mas) Fillet Radii (micron) 1 Open
- - - - - - - - 2 Neutral density (ND) filter - - - - - - - - 3 Open - - - - -
- - - 4-R1C1 HLC Band 1 circle 464.34 19.4 974 464.34 19.4 974 N/A 4-R1C2 SPC
major-FWHM vertical slit Band 3; tall rect 60.77 2 127 364.84 12 765 30.39
4-R1C3 HLC Band 1 [spare] circle 464.34 19.4 974 464.34 19.4 974 N/A 4-R1C4
Multi-star imaging in Band 4 square 309.07 9 648 309.07 9 648 30 4-R1C5 3.5”
radius circular polarimetry field stop circle 3338.28 N/A 7000 3338.28 N/A
7000 N/A 4-R2C1 Multi-star imaging in Band 1 square 215.41 9 452 215.41 9 452
30 4-R2C2 SPC minor-FWHM rotated slit Band 3; tall rect 30.39 1 64 516.58 17
1083 15.19 4-R2C3 HLC FWHM vertical slit Band 2 rect 30.22 1.1 63 164.84 6 346
15.11 4-R2C4 Long slit #3 rect 171.68 N/A 360 476.9 N/A 1000 30 4-R2C5 SPC
major-FWHM vertical slit Band 2; short rect 54.95 2 115 164.84 6 346 27.47
4-R3C1 SPC $\sim$1st null vertical slit; tall rect 115.72 4 243 347.16 12 728
30 4-R3C2 SPC $\sim$1st null vertical slit; short rect 115.72 4 243 173.58 6
364 30 4-R3C3 HLC Band 2 circle 532.98 19.4 1118 532.98 19.4 1118 N/A 4-R3C4
HLC Band 3 circle 589.51 19.4 1236 589.51 19.4 1236 N/A 4-R3C5 HLC Band 4
circle 666.22 19.4 1397 666.22 19.4 1397 N/A 4-R4C1 Multi-star imaging in Band
4 [spare] square 309.07 9 648 309.07 9 648 30 4-R4C2 SPC minor-FWHM rotated
slit Band 3; short rect 30.39 1 64 182.32 6 382 15.19 4-R4C3 SPC minor-FWHM
rotated slit Band 2; tall rect 27.47 1 58 467.04 17 979 13.74 4-R4C4 SPC
minor-FWHM rotated slit Band 2; short rect 27.47 1 58 164.84 6 346 13.74
4-R4C5 HLC FWHM vertical slit Band 3 rect 33.43 1.1 70 182.32 6 382 16.71
4-R4C6 SPC major-FWHM vertical slit Band 3; short rect 60.77 2 127 182.32 6
382 30.39 4-R5C1 HLC FWHM curved slit Band 3 arc 33.43 1.1 70 N/A $\pm
45^{\circ}$ N/A 16.71 4-R5C2 HLC FWHM curved slit Band 3 (reversed) arc 33.43
1.1 70 N/A $\pm 45^{\circ}$ N/A 16.71 4-R5C3 Left half HLC Band 1 arc 232.17
9.7 487 464.34 19.4 974 30 4-R5C4 Right half HLC Band 1 arc 232.17 9.7 487
464.34 19.4 974 30 4-R5C5 empty / dark N/A 0 0 0 0 0 0 N/A 4-R6C1 1.9” radius
circular polarimetry field stop circle 1812.21 N/A 3800 1812.21 N/A 3800 N/A
4-R6C2 SPC Wide-FOV Halves Bands 1 and 4 puzzle 1024.42 42.8 2148 1469.81 42.8
3082 60 4-R6C3 Long slit #2 rect 143.07 N/A 300 476.9 N/A 1000 30 4-R6C4 Long
slit #1 rect 85.84 N/A 180 476.9 N/A 1000 30 4-R6C5 SPC major-FWHM vertical
slit Band 2; tall rect 54.95 2 115 329.68 12 691 27.47 4-R6C6 HLC Band 1
[spare] circle 464.34 19.4 974 464.34 19.4 974 N/A

Table 6: Field stops specifications. Mask array positions (starting with 4-)
correspond to those shown in Fig. 13. Widths and heights will differ slightly
from the designed values reported here because of fabrication tolerances as
well as observatory and instrument magnification tolerances.

##  Appendix E Color Filters

All of the color filters included in the Roman Coronagraph are listed in Table
7. The four broadband filters, to be used after the dark hole is dug, are
named 1F, 2F, 3F, and 4F. (The F stands for full and is generally omitted for
brevity.) All the other filters are subsets of the four larger bandpasses and
are used for digging the dark holes in separate subbands. There are three
subbands each for Bands 1 and 4, and five each for Bands 2 and 3. Because
Bands 2 and 3 overlap, though, subbands 3A and 3B are also used to dig the
dark hole in Band 2. The narrowest filters, 2C and 3D, are used for
spectroscopy wavelength calibration and line spread function measurements,
rather than wavefront correction. Note that 2C also coincides with the
H$\alpha$ spectral line.

Filter Name | Center Wavelength (nm) | FWHM (nm)
---|---|---
1F | 575 | 58
1A | 555.8 | 19.2
1B | 575 | 19.2
1C | 594.2 | 19.2
2F | 660 | 112
2A | 615 | 22
2B | 638 | 18
2C | 656.3 | 6.6
3F | 730 | 122
3A | 681 | 24
3B | 704 | 24
3C | 727 | 20
3D | 754 | 7.5
3E | 777.5 | 27
3G | 752 | 25
4F | 825 | 94
4A | 792 | 28
4B | 825 | 30
4C | 857 | 30
CLEAR | N/A | N/A
ACQ DARK | N/A | N/A
Table 7: All color filters and their designed wavelength coverage. The as-
built filter properties may differ slightly from these and will be updated in
the list of instrument parameters at https://roman.ipac.caltech.edu when they
are available.

###### Acknowledgements.

This work was performed in part at the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration (NASA).

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# Radiation from a Classical Harmonic Oscillator

Paritosh Verma
Narodowe Centrum Bada$\acute{n}$ Ja̧drowych
Andrzeja Sołtana 7, 05-400 Otwock, Polska
<EMAIL_ADDRESS>
##  1 Abstract

This article presents the calculation of gravitational and electromagnetic
radiation emitted from a classical simple harmonic oscillator (SHO). Here we
show only the selected formulae and apply them to a toy problem without
rigorous derivation. First, we calculate the explicit expressions for the
gravitational waves polarizations and then obtain the power radiated away in
gravitational waves using Brans-Dicke theory (BD). We also calculate the
electromagnetic dipole and quadrupole power emitted by the SHO and compare
them with their gravitational counterparts.

##  2 Introduction

n 1961, Robert H. Dicke and Carl H. Brans proposed a scalar-tensor theory,
known as BD theory [1], to describe gravitation by incorporating Mach’s
principle. The foundation of this theory was built on the previous work of
Pascual Jordan [2] as well as Markus Fierz [3], and sometimes it is also
referred to as Jordan-Fierz-Brans-Dicke theory. In Einstein’s general
relativity (GR), the coupling constant between matter and spacetime is given
by G (the Newtonian constant of gravitation). In BD theory, the coupling
depends also on a parameter $\zeta$ through the relation $G(1-\zeta)$. The BD
parameter $(\zeta)$ is obtained through experiments, and the Cassini [4]
experiment in 2003 has imposed the constraint $\zeta<0.0000125$.

A generic metric theory of gravity can possess up to six polarization states
of gravitational waves (GWs); two tensor states; two vector states and two
scalar states [5]. GR has only two tensor polarizations, whereas BD has three
polarizations. The first two are tensor polarizations similar to GR, and the
third is scalar polarization. In recent years, there has been a decent amount
of work either to test GR using the LIGO-Virgo-Kagra (LVK) data [6], [7] or to
modify gravity to explain cosmological ambiguities like the nature of dark
energy and Hubble tension [8], [9], [10]. There are many promising theories,
but we choose BD for the following reason. BD theory is not too far from GR
because the tensor polarizations differ from the ones in GR only by a tiny
parameter $\zeta$. However, with the current sensitivity of detectors, it
would be too hard to differentiate between the tensor polarizations of BD and
GR. And we don?t want any new theory to deviate much from Einstein?s theory
because it has already passed many tests. So, BD theory appears to be a good
candidate to start with. Recently, we have implemented the BD theory to hunt
for the dipole radiation in the LVK O3 data [11] by adding the D-statistics
developed in [12].

We consider a simple harmonic oscillator moving in the x-direction, as shown
in Figure 1. The spring is ideal and massless. We isolate the system so that
the only force acting on the mass m is the spring force. The point particle
also carries a charge q and the equation of the oscillator is given by

$x_{p}(t)=x_{0}\cos\omega t$ (1)

where $\omega^{2}=\frac{k_{eff}}{m}$ and $k_{eff}$ is the effective spring
constant. Since, we are dealing with a classical SHO, we assume that
$x_{0}\omega<<c$ where $c$ is the speed of light.

This problem is a simple mechanical problem that provides insight into
radiation without any knowledge of astrophysics. Also, we start with the BD
theory and then take a limiting case which leads to GR.

Figure 1: A point particle with mass m and carrying a charge q is connected to
a massless ideal spring. The other end of the spring is connected to a fixed
wall. The particle can oscillate in the x-direction about the mean position O.

##  3 A theoretical minimum on radiation

###  3.1 Gravitational radiation

This subsection briefly presents the formulae for GW polarizations in BD
theory and the power radiated from the system. There are three polarization
states in BD theory; two tensor polarizations and one scalar polarization. In
the field theory language, the tensorial states could be conceived as spin-2
particles, whereas the scalar one as a spin-0 particle, and they all are
massless because gravity is a long-range force. The two tensor polarizations
are transverse to the wave, and they differ from each other by 45 degrees. So
it means that if we rotate one polarization by 45 degrees, we get to the
second polarization. And a rotation of 180 degrees gets back to the same
state. A circle can represent the scalar polarization in the BD theory. This
means that a rotation by an infinitesimally small angle leads to the same
polarization.

The difference in polarization can also be understood in the language of
particle physics. Let $\phi$ be the minimum angle by which one polarization
should be rotated to get the same state. The $\phi$ is given by

$\phi=\frac{2\pi^{c}}{s}$ (2)

where s is the spin of the graviton. For scalar polarization, $s=0$, $s=1$ for
vector polarization and $s=2$ corresponds to the tensor polarization. This
formula is valid only for Bosons (integer spin particles) and not for Fermions
(half odd integer spin particles). If we substitute $s=0$ in the above
formula, $\phi$ tends to infinity, which implies that even an infinitesimal
rotation leads to the same polarization. We can call tensor polarizations
$2$-fold symmetric, vector polarizations $1$-fold symmetric and scalar
polarization $\infty$-fold symmetric.

Here, we present only important formulae without any derivations. For a
detailed analysis, one can follow [12], [13].

Assuming that the wave propagates in the $+z$ direction, we have the following
formulae for $h_{+}$ and $h_{\times}$ polarizations

$\displaystyle h_{+}(t)$ $\displaystyle=$
$\displaystyle\frac{G}{rc^{4}}(1-\zeta)(\ddot{Q}^{xx}_{mW}(t^{\prime})-\ddot{Q}^{yy}_{mW}(t^{\prime}))$
$\displaystyle h_{\times}(t)$ $\displaystyle=$
$\displaystyle\frac{2G}{rc^{4}}(1-\zeta)\ddot{Q}^{xy}_{mW}(t^{\prime})$ (3)

where G is the gravitational constant, c is the speed of light, $t^{\prime}$
is the retarded time, r is the distance of the source, $\ddot{Q}^{ij}_{mW}$ is
the second time derivative of the mass quadrupole moment in the wave frame. In
the limit $\zeta\rightarrow 0$ the above expressions for the two polarizations
reduce to the expression in classical general relativity given by Eqs. (1.114)
of [13].

The scalar polarization $(h_{S})$ in BD theory is given by

$h_{S}(t)=\frac{2G}{rc^{2}}\zeta\left[M(t^{\prime})+\frac{1}{c}\dot{D}_{mW}^{z}(t^{\prime})-\frac{1}{2c^{2}}\ddot{Q}^{zz}_{mW}(t^{\prime})\right]$
(4)

where $\dot{D}_{mW}^{i}$ is the first time derivative of the mass dipole
moment in the wave frame and $M$ is the mass monopole moment. In GR, there is
no contribution from the mass monopole moment and dipole moment because of
mass and linear momentum conservation. But monopole and dipole radiation
appear in BD theory because the scalar field in BD theory does not satisfy a
conservation law.

The total power radiated away per unit angle in BD theory can be written as

$\frac{dP_{grav}}{dA}=\frac{c^{3}}{16\pi
G(1-\zeta)}<\dot{h}^{2}_{+}(t)+\dot{h}^{2}_{\times}(t)+\left(\frac{1-\zeta}{\zeta}\right)\dot{h}_{s}^{2}(t)>=\frac{dP^{(T)}}{dA}+\frac{dP^{(S)}}{dA}$
(5)

where
$dA=r^{2}\int_{\rho=0}^{2\pi}\int_{\iota=-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos\iota
d\iota d\rho$ and $<\cdot>$ implies the time average.

$\frac{dP^{(T)}}{dA}$ is the power emitted in the tensor wave

$\frac{dP^{(T)}}{dA}\equiv\frac{c^{3}}{16\pi
G(1-\zeta)}<\dot{h}^{2}_{+}(t)+\dot{h}^{2}_{\times}(t)>$ (6)

and $\frac{dP^{(S)}}{dA}$ is the power emitted in the scalar wave

$\frac{dP^{(S)}}{dA}\equiv\frac{c^{3}}{16\pi\zeta G}<\dot{h}^{2}_{s}(t)>$ (7)

###  3.2 Electromagnetic radiation

An electric charge at rest produces a static electric field, whereas a charge
moving with a uniform velocity gives rise to the magnetic field. Therefore,
only an accelerating charge can generate EM radiation. (Note: A charge moving
with constant velocity in a vacuum can?t produce the radiation, but it can
produce radiation in plasma). So, we can say that there is no monopole
radiation in the EM theory because of the charge conservation. The significant
contribution in the radiation is due to the time-varying dipole moment, and
then there are also quadrupole, octupole moments and so on. We can also
understand the property mentioned above in the language of particle physics. A
photon is a massless particle with spin 1, so it cannot exist in the state of
total angular momentum j = 0, and this implies the absence of the monopole
radiation. The two transverse polarizations (vertical and horizontal) in the
EM theory differ by 90 degrees, meaning that a rotation of 360 degrees takes
us to the same polarization. This can be seen by putting $s=1$ in Eq. (2)

The total power emitted in the electromagnetic radiation is given by

$P_{em}=\frac{2}{3}\frac{K}{c^{3}}<\ddot{D}_{q}^{2}(t^{\prime})>+\frac{1}{20}\frac{K}{c^{5}}<\dddot{Q}_{q}^{ij}(t^{\prime})\dddot{Q}_{q}^{ij}(t^{\prime})>+\frac{2}{3}\frac{K}{c^{3}}<\ddot{\mu^{2}}(t^{\prime})>$
(8)

where $K=\frac{1}{4\pi\epsilon_{0}}$, $D_{q}(t^{\prime})$ is the time-varying
electric dipole moment in the inertial frame, $Q_{q}^{ij}(t^{\prime})$ is the
time-varying electric quadrupole moment in the inertial frame and
$\mu(t^{\prime})$ is the time-varying magnetic dipole moment in the inertial
frame.

##  4 Multipole moments of a SHO

###  4.1 Dipole moment

Let the position of the particle be $x_{p}(t)=x_{0}\cos(\omega t^{\prime})$ at
an instant $t^{\prime}$. The x-component of mass dipole moment in the inertial
frame can be written as

$d(t^{\prime})=mx(t^{\prime})=mx_{0}\cos(\omega t^{\prime})$ (9)

The y and z components vanish because the mass distribution is along the
x-axis. The dipole moment in the vector form is

$D_{m}(t^{\prime})=\begin{bmatrix}d_{m}(t^{\prime})\\\ 0\\\ 0\end{bmatrix}$
(10)

Consider a wave making an angle $\iota$ with the z-axis. The mass dipole
moment in the wave frame can be computed as

$D_{mW}(t^{\prime})=S\cdot D_{m}(t^{\prime})$ (11)

where $S$ is the rotation matrix given by

$S=\begin{bmatrix}\cos\iota&0&-\sin\iota\\\ 0&1&0\\\
\sin\iota&0&\cos\iota\end{bmatrix}$ (12)

The matrix multiplication gives

$D_{mW}(t^{\prime})=\begin{bmatrix}d(t^{\prime})\cos\iota\\\ 0\\\
d(t^{\prime})\sin\iota\end{bmatrix}$ (13)

The analogous charge dipole moment in the inertial frame is

$D_{q}(t^{\prime})=\begin{bmatrix}d_{q}(t^{\prime})\\\ 0\\\
0\end{bmatrix}=\begin{bmatrix}qx_{0}\cos(\omega t^{\prime})\\\ 0\\\
0\end{bmatrix}$ (14)

###  4.2 Quadrupole moment

The linear mass density of a point particle oscillating in one dimension can
be represented by a Dirac delta function as

$\lambda(t^{\prime},x)=m\delta(x-x(t^{\prime}))$ (15)

and the expression for the symmetric-trace free (STF) quadrupole moment is

$Q_{m}^{ij}(t^{\prime})=\int\left[x^{i}x^{j}-\frac{1}{3}r^{2}\delta_{ij}\right]dm$
(16)

The $Q_{m}^{xx}$ component is

$Q_{m}^{xx}(t^{\prime})=m\int\delta(x-x_{p}(t^{\prime}))dx\left[x_{p}(t^{\prime})^{2}-\frac{1}{3}\left(x_{p}(t^{\prime})^{2}+0+0\right)\delta_{11}\right]$
(17)

or,

$Q_{m}^{xx}(t^{\prime})=\frac{2}{3}m\left[x_{p}(t^{\prime})\right]^{2}=\frac{2}{3}mx_{0}^{2}\cos^{2}(\omega
t^{\prime})$ (18)

Similarly, we can compute other components of the quadrupole moment tensor to
obtain

$Q_{m}^{ij}(t^{\prime})=Q_{m0}\begin{bmatrix}2&0&0\\\ 0&-1&0\\\
0&0&-1\end{bmatrix}$ (19)

where $Q_{m0}$ is defined as

$Q_{m0}(t^{\prime})\equiv\frac{1}{3}mx_{0}^{2}\cos^{2}(\omega t^{\prime})$
(20)

The following orthogonal rotation gives the mass quadrupole moment tensor in
the wave frame

$Q_{mW}(t^{\prime})=S\cdot Q_{m}(t^{\prime})\cdot S^{T}$ (21)

and we get

$Q_{mW}^{ij}(t^{\prime})=Q_{m0}(t^{\prime})\begin{bmatrix}2\cos^{2}\iota-\sin^{2}\iota&0&3\sin\iota\cos\iota\\\
0&-1&0\\\ 3\sin\iota\cos\iota&0&2\sin^{2}\iota-\cos^{2}\iota\end{bmatrix}$
(22)

The analogous charge quadrupole moment tensor in the inertial frame is

$Q_{q}^{ij}(t^{\prime})=Q_{q0}(t^{\prime})\begin{bmatrix}2&0&0\\\ 0&-1&0\\\
0&0&-1\end{bmatrix}$ (23)

where $Q_{q0}(t^{\prime})$ is defined as

$Q_{q0}(t^{\prime})\equiv\frac{1}{3}qx_{0}^{2}\cos^{2}(\omega t^{\prime})$
(24)

##  5 Power radiated

The three GW polarizations are obtained using Eq. (3.1) and Eq. (4). Their
expressions are

$\displaystyle h_{+}(t)$ $\displaystyle=$
$\displaystyle\frac{-2G}{rc^{2}}(1-\zeta)m\frac{v^{2}}{c^{2}}\cos^{2}\iota\cos(2\omega
t^{\prime})$ $\displaystyle h_{\times}(t)$ $\displaystyle=$ $\displaystyle 0$
(25)

$h_{S}(t)=\frac{2G}{rc^{2}}\zeta\left[-m\frac{v}{c}\sin\iota\sin(\omega
t^{\prime})+\frac{m}{3}\frac{v^{2}}{c^{2}}\cos(2\omega
t^{\prime})\left(3\sin^{2}\iota-1\right)\right]$ (26)

In the above expressions, we use $v=x_{0}\omega$ where $v$ is the maximum
speed of the SHO at the mean position. When $\zeta$ tends to zero, we approach
the limit of GR. In this case, the scalar polarization vanishes. Also, there
is no cross-polarization from a classic SHO both in BD theory and GR. We have
ignored the mass monopole term in the scalar polarization because there is no
mass transfer.The radiation due to the dipole moment comes at frequency
$\omega$ whereas, waves due to the quadrupole moment originates at $2\omega$.

We use Eq. (6) and Eq. (7) to obtain the power emitted in tensor wave and
scalar wave respectively. The expressions are

$\displaystyle P^{(T)}$ $\displaystyle=$
$\displaystyle\frac{16}{15}\frac{G(1-\zeta)}{c}m^{2}\omega^{2}\left(\frac{v}{c}\right)^{4}$
$\displaystyle P^{(S)}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\frac{G\zeta}{c}m^{2}\omega^{2}\left(\frac{v}{c}\right)^{2}+\frac{8}{45}\frac{G\zeta}{c}m^{2}\omega^{2}\left(\frac{v}{c}\right)^{4}$
(27)

The total power emitted in the gravitational radiation is
$P_{grav}=P^{(T)}+P^{(S)}$

$P_{grav}=\frac{16}{15}\frac{G}{c}m^{2}\omega^{2}\left(\frac{v}{c}\right)^{4}+\frac{1}{6}\frac{G\zeta}{c}m^{2}\omega^{2}\left(\frac{v}{c}\right)^{2}-\frac{8}{9}\frac{G\zeta}{c}m^{2}\omega^{2}\left(\frac{v}{c}\right)^{4}$
(28)

In this case, we can consider the last two terms as a correction to GR because
of the absence of parameter $\zeta$ in the first term. In practice, we can
ignore the energy emitted due to the quadrupole moment because of the extra
factors of c in the denominator.

For the case of electromagnetic radiation, we directly calculate the total
power emitted using Eq. (8) without finding the fields at the point of
observation. For the dipole part, we get

$<\ddot{D}_{q}^{2}(t^{\prime})>=<q^{2}x_{0}^{2}\omega^{4}\sin^{2}\omega
t^{\prime}>$ (29)

and the quadrupole term gives

$<\dddot{Q}_{q}^{ij}(t^{\prime})\dddot{Q}_{q}^{ij}(t^{\prime})>=6<(\dddot{Q}_{q0})^{2}>$
(30)

We ignore the contribution from the magnetic dipole term because it is
proportional to the angular momentum. A little algebra yields

$P_{em}=\frac{1}{3}\frac{K}{c}q^{2}\omega^{2}\left(\frac{v}{c}\right)^{2}+\frac{4}{15}\frac{K}{c}q^{2}\omega^{2}\left(\frac{v}{c}\right)^{4}$
(31)

To calculate the total power for both gravitational and electromagnetic
radiation, we use the fact that $<\sin^{2}n\omega t>=<\cos^{2}n\omega
t>\approx\frac{1}{2}$ and $<\sin n\omega t>=<\cos n\omega t>\approx 0$.

The total power radiated by the point particle is $P_{grav}+P_{em}$. In the
absence of any radiation, the total energy of the SHO is constant and equal to
$\frac{1}{2}m\omega^{2}x_{0}^{2}$. But radiation takes away energy and
momentum from the system. As a result, the amplitude of oscillations will
decrease, and finally, the particle will come to rest. We can find the rate of
change of amplitude using the equation

$\frac{d}{dt}\left(\frac{1}{2}m\omega^{2}x_{0}^{2}\right)=P_{grav}+P_{em}$
(32)

##  6 Conclusion

The numerical value of K is $8.98\times 10^{9}N\ m^{2}\ C^{-2}$, and that of G
is $6.67\times 10^{-11}N\ m^{2}\ kg^{-2}$. Let’s consider a 1 Coulomb of
charge and 1 kg of mass. It can be easily seen that power emitted in
gravitational radiation is too low compared to electromagnetic radiation. That
is why we can safely ignore gravitational radiation loss from objects in our
vicinity.

## Acknowledgement

I cordially thank Prof. Andrzej Krolak for his valuable guidance and support
during my PhD studies. I also warmly thank Prof. Enrico Barausse for his time
and helpful remarks to enhance this article.

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