# The Circumgalactic H$\alpha$ Spectrograph (CH$\alpha$S)
I. Design, Engineering, and Early Commissioning

Nicole Melso Department of Astronomy, Columbia University, 550 W. 120th Street
MC 5246, New York, NY 10027, USA  Columbia Astrophysics Laboratory, Columbia
University, 550 W. 120th St. MC 5247, New York, NY 10027, USA  University of
Arizona, Steward Observatory, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
David Schiminovich  Department of Astronomy, Columbia University, 550 W. 120th
Street MC 5246, New York, NY 10027, USA  Columbia Astrophysics Laboratory,
Columbia University, 550 W. 120th St. MC 5247, New York, NY 10027, USA  Brian
Smiley  Columbia Astrophysics Laboratory, Columbia University, 550 W. 120th
St. MC 5247, New York, NY 10027, USA  Hwei Ru Ong  Columbia Astrophysics
Laboratory, Columbia University, 550 W. 120th St. MC 5247, New York, NY 10027,
USA  Bárbara Cruvinel Santiago Department of Physics, Columbia University, 538
W. 120th Street 704 Pupin Hall MC 5255, New York, NY 10027, USA  Columbia
Astrophysics Laboratory, Columbia University, 550 W. 120th St. MC 5247, New
York, NY 10027, USA  Meghna Sitaram Department of Astronomy, Columbia
University, 550 W. 120th Street MC 5246, New York, NY 10027, USA  Columbia
Astrophysics Laboratory, Columbia University, 550 W. 120th St. MC 5247, New
York, NY 10027, USA  Ignacio Cevallos Aleman  Department of Physics, Columbia
University, 538 W. 120th Street 704 Pupin Hall MC 5255, New York, NY 10027,
USA  Columbia Astrophysics Laboratory, Columbia University, 550 W. 120th St.
MC 5247, New York, NY 10027, USA  Sarah Graber Department of Astronomy,
Columbia University, 550 W. 120th Street MC 5246, New York, NY 10027, USA
Columbia Astrophysics Laboratory, Columbia University, 550 W. 120th St. MC
5247, New York, NY 10027, USA  Marisa Murillo Department of Engineering,
Columbia University, 500 W. 120th Street $\\#$510, New York, NY 10027
Columbia Astrophysics Laboratory, Columbia University, 550 W. 120th St. MC
5247, New York, NY 10027, USA  Marni Rosenthal  Department of Physics $\&$
Astronomy, Barnard College, 3009 Broadway, Altschul Hall 504A, New York, NY
10027  Columbia Astrophysics Laboratory, Columbia University, 550 W. 120th St.
MC 5247, New York, NY 10027, USA  Ioana Stelea  Department of Astronomy,
Columbia University, 550 W. 120th Street MC 5246, New York, NY 10027, USA
Columbia Astrophysics Laboratory, Columbia University, 550 W. 120th St. MC
5247, New York, NY 10027, USA

###### Abstract

The Circumgalactic H$\alpha$ Spectrograph (CH$\alpha$S) is a ground-based
optical integral field spectrograph designed to detect ultra-faint extended
emission from diffuse ionized gas in the nearby universe. CH$\alpha$S is
particularly well suited for making a direct detection of tenuous H$\alpha$
emission from the circumgalactic medium (CGM) surrounding low-redshift
galaxies. It efficiently maps large regions of the CGM in a single exposure,
targeting nearby galaxies (d $<35$ Mpc) where the CGM is expected to fill the
field of view. We are commissioning CH$\alpha$S as a facility instrument at
MDM Observatory. CH$\alpha$S is deployed in the focal plane of the Hiltner
2.4-meter telescope, utilizing nearly all of the telescope’s unvignetted focal
plane (10 arcmin) to conduct wide-field spectroscopic imaging. The
catadioptric design provides excellent wide-field imaging performance.
CH$\alpha$S is a pupil-imaging spectrograph employing a microlens array to
divide the field of view into $>60,000$ spectra. CH$\alpha$S achieves an
angular resolution of $[1.3-2.8]$ arcseconds and a resolving power of
R$=[10,000-20,000]$. Accordingly, the spectrograph can resolve structure on
the scale of $1-5$ kpc (at 10 Mpc) and measure velocities down to 15-30 km/s.
CH$\alpha$S intentionally operates over a narrow (30 $\AA$) bandpass; however,
it is configured to adjust the central wavelength and target a broad range of
optical emission lines individually. A high diffraction efficiency VPH grating
ensures high throughput across configurations. CH$\alpha$S maintains a high
grasp and moderate spectral resolution, providing an ideal combination for
mapping discrete, ultra-low surface brightness emission on the order of a few
milli-Raleigh.

instrumentation: spectrographs — techniques: imaging spectroscopy — galaxies:
halos

##  1 Introduction

Where are the atoms in the Universe? This question highlights the last
remaining frontier in our study and census of the evolving distribution of
baryonic matter. The answer remains elusive due to the challenges of detecting
gas at its lowest densities. A true mapping of diffuse emission requires
observations that can reach exceedingly low surface brightnesses. Pioneering
work with Fabry-Perot, deep long-slit, and integral-field unit (IFU)
spectroscopy (e.g. Vogel et al., 1995) has sought to measure faint light from
intergalactic clouds and galaxy halos. This remains very much a field in its
early stages, requiring novel instrumentation that pushes the boundaries of
ultra-low surface brightness spectroscopy.

To meet this challenge, we have designed the Circumgalactic H$\alpha$
Spectrograph (CH$\alpha$S), an IFU spectrograph tailored to measure faint
emission from diffuse gas in the local universe. The primary science goal for
CH$\alpha$S is to observe the circumgalactic medium (CGM) around low-redshift
galaxies. As an integral field spectrograph, CH$\alpha$S can simultaneously
map the mass, distribution, and kinematics of ionized gas undergoing cooling
or recombination. Observations with CH$\alpha$S will address fundamental
questions, investigating how much baryonic mass surrounds nearby spiral
galaxies, how this matter is distributed, and how gas flows between the
circumgalactic medium and the galactic disk.

CH$\alpha$S occupies an important new parameter space in the design of low-
surface brightness integral field spectrographs; it is a fast, ultra-
narrowband, moderate-resolution spectrograph with exceptional grasp.
CH$\alpha$S collects over 60,000 simultaneous spectra with a band-width
optimized to study the relevant dynamics of low-redshift galaxies and their
surrounding gaseous halos. Integral field spectroscopy is well-suited for
faint, diffuse observations because it offers unconstrained pointing, wide-
field spatial sampling, and superior survey speed (Bershady, 2009; Bacon &
Monnet, 2017; Morrissey et al., 2018). CH$\alpha$S is able to compete in
survey speed with IFU spectrographs on 8-10 meter class telescopes at a
fraction of the cost.

Many of the CH$\alpha$S design goals are motivated by the observational and
theoretical properties of ionized gas in the low-redshift CGM. These
properties drive the following scientific requirements.

  1. 1.

Target a broad range of optical emission lines: H$\alpha$ ($\lambda 6563$),
H$\beta$ ($\lambda 4861$), [S II] ($\lambda 6718$), [N II] ($\lambda 6584$),
[O III] ($\lambda 5007$), [O II] ($\lambda 3727$), [O I] ($\lambda 6300$)

  2. 2.

Cover the extended physical size of the CGM ($50-100$ kpc) in the local
universe (d $<35$ Mpc) while resolving individual structures/features on
scales of 0.1-1 kpc at a distance of 10 Mpc.

  3. 3.

Coverage over a continuous systemic velocity range from $\sim(0-2400]$ km/s (z
$<0.008$) with the ability to measure rest frame Doppler kinematics as large
as $\sim\pm 600$ km/s down to a velocity resolution of 15-30 km/s.

  4. 4.

Achieve sensitivity to emission on the order of $\simeq$ 1-10 mR
(approximately $0.1\%-1\%$ the sky background intensity).

Several sources and ionization mechanisms contribute to the ionization of the
CGM (Shull et al., 2012; Haardt & Madau, 2012; Bland-Hawthorn et al., 2017).
Emission radiation traces the recombination of photoionized gas and the
recombination/cooling of collisionally ionized gas and collisionally excited
gas. The namesake emission signature for CH$\alpha$S is the prominent Balmer
H$\alpha$ line; however, CH$\alpha$S targets a broad range of optical emission
lines. Spectral line ratios can be used to map the efficacy of different
ionization mechanisms throughout the galactic halo (Baldwin et al., 1981).

The CGM is a massive gas reservoir extending hundreds of kpc from the disk of
a galaxy out to the viral radius and beyond (Shull, 2014; Borthakur et al.,
2016). In order to cover the extended physical size of the CGM in a single
exposure, CH$\alpha$S is optimized to operate over a wide field of view
($10^{\prime}\times 10^{\prime}$) set by the observatory telescope$-$detector
pairing. The spatial resolution within the CH$\alpha$S FOV is motivated by the
expected gas distribution and the size of the small-scale substructure that
populates the halo. Observations and simulations of the CGM are often in
agreement that cicumgalactic gas distribution is patchy (Borthakur et al.,
2016; Oppenheimer et al., 2018; Lehner et al., 2020) and hosts many sources of
multi-scale (0.01 - 1000 kpc) structure including small clumps/clouds
(Churchill et al., 2003; Stocke et al., 2013; Crighton et al., 2015), radial
gradients (Borthakur et al., 2016), and large scale filaments (Haffner et al.,
1998; Churchill et al., 2012; Joung et al., 2012). CH$\alpha$S is designed to
capture radial/spatial dependencies in the extended gas distribution while
resolving features on the average length-scale of substructure in the CGM.

The CH$\alpha$S spectral window is tuned to detect circumgalactic gas
kinematically associated with low-redshift galaxies ($z<0.008$) up to a
systemic velocity of 2400 km/s. Despite the variety of dynamic components in
the halos of these galaxies (Martin et al., 2012; Rubin et al., 2012; Ho et
al., 2014, 2017; Zheng et al., 2017; Rodríguez del Pino et al., 2019; Zabl et
al., 2019; French & Wakker, 2020; Ho & Martin, 2020), gas within the virial
radius is typically gravitationaly bound with CGM velocities not exceeding the
escape velocity (550 km/s for Milky Way mass galaxies) (Tumlinson et al.,
2017). Accordingly, CH$\alpha$S aims to measure Doppler kinematics as large as
$\pm 600\rm\ km\ s^{-1}$ in a given filter ($\sim 25\ \AA$), encompassing the
majority of gas in the low-redshift CGM of Milky-Way-like galaxies. The
velocity resolution selected for CH$\alpha$S ($15-30\rm\ km\ s^{-1}$) is on
par with typical spectral line widths observed in the CGM (Werk et al., 2013),
which are limited by thermal broadening and non-thermal turbulent motions on
the order of $\lesssim 20-50\rm\ km\ s^{-1}$ (Tumlinson et al. (2017) and
references therein).

Very few direct detections of CGM emission have been made in the low-redshift
regime beyond the Milky Way halo (Bland-Hawthorn et al., 1997; Fumagalli et
al., 2017). One of the main obstacles impeding a direct detection of
warm/warm-hot ionized gas in the CGM is the diffuse nature of the gas
($10^{-6}<n_{H}<10^{-2}$) (Tumlinson et al., 2017), producing an extremely
tenuous emission signal on the order of $\sim 1-10$ mR (Corlies &
Schiminovich, 2016; Zhang et al., 2016). This signal is easily overpowered by
sky background or detector noise. CH$\alpha$S is designed to detect emission
with a sensitivity of to $0.1\%-1\%$ of the sky-background intensity
111Dependent on the filter bandpass ($\rm I_{sky}=2\ R\AA^{-1})$ (Leinert et
al., 1998).

CH$\alpha$S is a dedicated instrument for the Hiltner 2.4-meter telescope at
MDM Observatory on Kitt Peak, bringing full-field spectral imaging
capabilities to the observatory. The MDM facilities are owned and operated by
a consortium consisting of Dartmouth College, Columbia University, Ohio State
University, Ohio University, and the University of Michigan. CH$\alpha$S will
be a permanent facility instrument at MDM Observatory, available to the
consortium. This is the debut paper for the Circumgalactic H$\alpha$
Spectrograph. We present the optical design in Section 2, the mechanical
design in Section 2.5, the expected performance in Section 3, and the early
commissioning results in Section 4. We conclude with a discussion of the
spectrograph optimization in Section 5 and a summary of key points from this
publication in Section 6.

##  2 Optomechanical Design

The CH$\alpha$S optomechanical design is summarized in Figure 1 and Figure 4.
While the CH$\alpha$S performance and diffraction efficiency were originally
optimized for H$\alpha$ emission, reduced chromaticity in the catadioptric
design and the narrow instrument bandpass have allowed us extend the
operational wavelength to nearly the full optical range. The CH$\alpha$S
design triumphs in its configuration flexibility and in the cost-savings
afforded by the incorporation of many commercially available optics.

Figure 1: Zemax ray-trace of the CH$\alpha$S design. In this figure, we split
the design into three subsets: the relay optics, the entrance optics, and the
spectrograph optics. The total sequential optical path through all three is:
(relay optics) telescope focal plane, flat pick-off $\\#1$, concave spherical
$\\#1$, convex spherical, concave spherical $\\#2$, (entrance optics) flat
pick-off $\\#2$, microlens array, (spectrograph optics) collimator field
corrector bundle, collimator primary, collimator asphere, grating, camera
asphere, camera flat, camera primary, camera field corrector, detector.

###  2.1 Focal Plane Optics

CH$\alpha$S sits in the focal plane of the MDM Hiltner 2.4-m telescope. It
attaches to the telescope Multi-Instrument System (MIS), utilizing the finder
module (with calibration lamps) and guider module. CH$\alpha$S has a separate
shutter installed near the focal plane of the telescope 222Uniblitz CS65
shutter and D880C driver, which is triggered by the detector exposure commands
through the MDM 4K detector electronics box. A system of relay optics is used
to extend and re-image the telescope focal plane down into the entrance of the
spectrograph with enough clearance to avoid collision between the MIS and the
CH$\alpha$S collimator. The Offner relay design used in CH$\alpha$S can be
seen in Figure 1. It consists of two identical concave spherical mirrors, one
convex spherical mirror, and two flat pick-off mirrors, which in total relay
the focal plane by $\sim$1.5 m with a nearly one-to-one mapping ($\rm
m=-1.001265$). All of the optics are coated with a UV Enhanced Protective
Silver High Reflective Coating ($>97\%\ @\ 400-1000$ nm) applied by Teledyne
Acton Optics. The final flat in the relay is positioned to redirect the re-
imaged focal plane into the spectrograph entrance optics.

###  2.2 Microlens Array

The entrance to the spectrograph is a microlens array, which sits directly in
the re-imaged focal plane exiting the relay system. The lenslets themselves
re-image the pupil of the telescope and act as a focal reducer, converting the
F/7.5 telescope input to an F/2.5 output that can be fed into the fast
collimator design. We have purchased two custom mirolens arrays from Advanced
Microoptics Systems (a$\mu$s): one array with 250 $\mu$m pitch lenslets and
one array with 125 $\mu$m pitch lenslets. Hereafter, these are referred to as
the ML250 array and ML125 array. The ML250 and ML125 arrays have $>60,000$ and
$>240,0000$ individual lenslets respectively, creating a powerful integral
field spectrograph with high spatial and spectral resolution.

We present the CH$\alpha$S microlens array parameters in Table 1. Each array
is a $68\times 68$ mm clear aperture Fused Silica optic composed of hexagonal
plano-convex lenslets, covering nearly the full field of view of the Hiltner
2.4-m telescope. An anti-reflection (AR) coating is applied to the planar side
of each array with a reflectance of $<1\%$ between 400 nm and 900 nm. On the
convex side of the array, a black chromium mask is applied between the
lenslets, filling in the gaps where the lenslets intersect and creating a
circular aperture around each hexagonal lenslet profile. The fused silica and
black chromium mask on this side has a $<4\%-5\%$ reflectance between 400 nm
and 900 nm. This mask is used to stop down the lenslet aperture slightly and
to keep light from scattering at the lenselet interstices. An additional field
mask can be installed next to the microlens array to block a specific pattern
of lenslets, masking bright regions and maximizing the spectral filling factor
at the detector.

As seen in the non-sequential ray trace in Figure 2, each plano-convex array
can be flipped 180 degrees in the optical path without altering the imaging
capabilities. In both configurations the focal plane of the telescope is
incident on the side of the array with optical power. Both microlens arrays
were designed with F/2.5 lenslets. They are slightly slower than the F/2.2
collimator in order to reduce the amount of light lost off the edges of the
grating. For plano-convex lenslets, the effective focal length is independent
of the array thickness, and each array is 2 mm thick to avoid deformation.
When calculating the effective focal length, each lenslet can be approximated
as a thick lens re-imaging the telescope pupil via refraction. We incorporate
aberration effects and calculate the nominal best focus position of the
lenslet array via non-sequential ray trace. This positioning can be fine-tuned
using the motorized collimator focus and telescope focus.

CH$\alpha$S is a pupil imaging spectrograph. It uses a microlens array to
sample the focal plane, as was originally done in the design of the TIGER IFS
(Bacon et al., 2001; Courtes, 1982). The CH$\alpha$S microlens array covers
the full field of view, and each lenslet bins the focused field positions from
a small patch of sky. The exit pupil of each lenslet is a demagnified image of
the telescope pupil (Courtes, 1982), referred to as a micropupil. The
microlens array creates a grid of micropupils that form the focal plane of the
collimator and the entrance to the spectrograph (Figure 2). Since the
telescope pupil (not the field) is re-imaged at each lenslet, imaging
information within each spatial resolution element is not preserved and the
PSF is independent of spatial field variation (Bacon, 1995). Without the
diffraction grating, these micropupils are directly re-imaged onto the
detector with demagnification through the collimator/camera optics (Figure 2).
With the grating in place, the space between micropupils can be filled with
the dispersed spectra. A camera aligned with the dispersion direction images
the spectra on the detector. Overlap between adjacent spectra restricts the
spectral bandpass, and a narrowband filter is used to shorten the spectra.
Slightly rotating the lenslets with respect to the dispersion direction allows
us to lengthens the vertical distance to the next micropupil/spectrum (Bacon
et al., 2001). With a slight rotation of approximately 5-6 degrees, each
spectrum now grazes past its immediate neighbors, improving spectral packing
and extending the allowable bandpass.

Figure 2: The top panels show non-sequential Zemax ray-trace showing a section of the CH$\alpha$S plano-convex microlens array in two orientations: (left) convex side facing towards the telescope focal plane and (right) convex side facing away from the telescope focal plane. The array can be flipped 180 degrees in the optical path without altering the imaging capability. Orientation 1 re-images the telescope pupil inside the array. Orientation 2 re-images the telescope pupil outside the array, allowing us to access it and add additional masking. The bottom panels show non-sequential ray trace imaging at the microlens array and at the detector. The bottom left panel shows the geometric pupil of the telescope in the telescope focal plane. The plano-convex lenslets re-image the telescope pupil, resulting in the micro-pupils shown in the bottom middle panel. This re-imaging/demagnification adds space between lenslets for the spectra to disperse. At the detector (bottom right), the spectra extend vertically and do not overlap for many resolution elements.  Table 1: Microlens array parameters as manufactured by Advanced Microoptics Systems (A$\mu$S). Effective focal lengths are calculated using the thick lens approximation at a central wavelength of $\lambda=0.6583$ nm for low-redshift H$\alpha$. Both arrays are made of Fused Silica (n = 1.4563 @ 0.6583 nm). Pitch | Aperture | Radius | Eff Focal | F/$\\#$ | Thickness
---|---|---|---|---|---
(mm) | (mm) | (mm-1) | (mm) |  | (mm)
0.25 | 0.225 $\pm$ 0.001 | 0.257$\pm 3\%$ | 0.563 | 2.5 | 2.0
0.125 | 0.113$\pm$ 0.001 | 0.129 $\pm 3\%$ | 0.283 | 2.5 | 2.0
Table 2: Optical performance of spectrograph convolved with lenslet pupil imaging. Pitch | RMS Radius | GEO Radius | FWHM | sigma
---|---|---|---|---
| Zemax | Zemax | Measured | Measured
(mm) | ($\mu$m) | ($\mu$m) | ($\mu$m) | ($\mu$m)
0.25 | 19.0 | 27.3 | 57 | 24
0.125 | 8.9 | 13.5 | 30.5 | 13

###  2.3 Collimator $\&$ Camera

CH$\alpha$S has a Schmidt collimator and a Schmidt camera. This design
achieves excellent wide-field optical performance while utilizing many
commercially available components and leveraging the comparably low-cost
manufacturing of optical spheres. The diverging beam from from the microlens
array is collimated by a Celestron 36 cm Rowe-Ackermann Schimdt Astrograph
(RASA). Among commercially available telescopes, the RASA has a large aperture
(356 mm), and a fast (f/2.2) wide-field ($4.3^{\circ}$ FOV) design that
combined result in an impressively high étendue. The optical performance is a
$<6.3$ micron RMS spot size across the full FOV and spanning the full optical
bandpass.

The CH$\alpha$S camera is a custom folding Schmidt design consisting of an
aspheric corrector plate, a folding flat mirror, and a spherical primary
mirror. The asphere at the camera entrance is the mass-produced Celestron 28
cm RASA corrector plate 333the same as the C12 Schmidt-Cassegrain corrector
plate. The folding flat is a custom optic manufactured by Sydor, and the
primary is a custom sphere manufactured by Optimax. The folding Schmidt design
places the detector behind the flat outside of the optical path, significantly
reducing obscuration. The flat has a cut machined out of the center to
accommodate the return beam and pass light through to the detector. The
CH$\alpha$S camera design achieves an excellent optical performance, producing
RMS spot sizes of $<10$ microns over a $10\times 10$ arcmin FOV and a wide
range of central wavelengths spanning $[400-900]$ nm.

We note that the CH$\alpha$S camera design is shorter than a classical Schmidt
telescope; the distance from the sphere to the corrector plate is less than
the radius of curvature. While certain optimizations of the shortened Schmidt
design can achieve abberation correction with a singlet self-achromatic444The
chromatic focal shift at the front and back surfaces of the lens are equal and
opposite fused silica field corrector (Wynne, 1977), the CH$\alpha$S field
corrector has a small focal shift on the order of a micron over the full
optical bandpass [400- 900 nm]. The self-achromatic shortened Schmidt
optimization is constrained by the distance between the asphere and the
sphere, which for CH$\alpha$S is partially restricted by the folding geometry.
Moving the flat closer to the primary places it further upstream in the return
beam, requiring a larger cutout that introduces additional obscuration. In the
CH$\alpha$S camera design we minimize this obscuration, as the axial
achromatic shift within each narrowband (3 nm) observation is negligible. The
obscuration from the flat is less than the entrance optics obscuration, which
is approximately 30$\%$ of the stop aperture.

###  2.4 VPH Grating

The CH$\alpha$S diffraction element is a volume phase holographic (VPH)
grating mounted on a rotational stage between the collimator and camera. Many
instruments swap between a collection of VPH gratings in order to optimize the
diffraction efficiency in different wavelength regimes. In contrast,
CH$\alpha$S operates over a wide range of central wavelengths $[400-900]$ nm
with exceptional efficiency using only a single VPH grating. The peak
diffraction efficiency is shifted to the desired central wavelength by simply
rotating the grating to adjust the angle of incidence (Barden et al., 2000;
Baldry et al., 2004). While having a wide field of view and a broad spectral
range usually poses a challenge for VPH gratings, coupling CH$\alpha$S with a
narrow bandpass filter allows us to operate away from the grating’s central
optimization with incredibly high throughput.

The CH$\alpha$S VPH grating was manufactured by Wasatch Photonics with a
fringe frequency of $\nu$ = 1200 lines/mm. This contributes to the moderate
resolving power (R $\sim 10,000-20,000$) and sets a collimator-camera angle
that ranges from $\sim[35^{\circ}-65^{\circ}]$ depending on selected
wavelength. The grating is a large-format optic [340 mm x 290 mm x 20 mm] with
an elliptical clear aperture. It is capped using fused silica substrates, each
polished to excellent surface quality (mean $\rm PV=1.197/RMS=0.21$2 over a 98
mm aperture) and coated with a broadband ($400-1000$ nm) anti-reflective
coating ($<2\%$ reflectance for Bragg incidence at 658 nm). Grating parameters
are summarized in Table 3.

VPH gratings diffract light using a thin layer of gelatin with the index of
refraction modulated to form fringes. The diffraction efficiency is moderated
by Bragg reflection; light is coherently diffracted with high efficiency when
scattering in the gelatin volume creates constructive interference (Baldry et
al., 2004). The Bragg condition 555Parameters in the gelatin have subscript 2
($\alpha_{2}$, $\beta_{2}$, $n_{2}$, $\Delta n_{2}$). Parameters satisfying
the Bragg condition have a subscript b ($\alpha_{2b}$, $\lambda_{b}$).
(Equation 1) is dependent on the fringe frequency in the grating ($\nu_{2}$),
the wavelength of light ($\lambda$), and the angle of incidence in the grating
gelatin with respect to the fringe structure ($\alpha_{2b}$). Therefore, the
diffraction efficiency at a given wavelength can be maximized by selecting an
angle of incidence that satisfies the Bragg condition.

$m\nu_{2}\lambda_{2b}=2n_{2}\sin\alpha_{2b}$ (1)

Table 3: The grating and camera angles for key wavelengths in the three different channels (blue, red, and infrared). The Bragg angle is measured inside the gelatin relative to the grating fringes. The fringe slant is $\phi=4.5^{\circ}$ The angle of incidence and angle of diffraction are measured in air relative to grating normal. The camera angle is the sum of the angle of incidence and angle of diffraction measured relative to the optical axis of the collimator.  Wavelength | Bragg | Incidence | Diffraction | Camera | Anamorphic | Angular Dispersion | Linear Dispersion aa 30 $\rm\mu m$ per pixel in 2x2 detector binning
---|---|---|---|---|---|---|---
|  | $+\phi$ | $-\phi$ | $+\phi$ | $-\phi$ |  | $+\phi$ | $-\phi$ | $+\phi$ | $-\phi$ | $+\phi$ | $-\phi$
490 nm | 12.6∘ | 23.4∘ | 11.0∘ | -11.0∘ | -23.4∘ | 34.4∘ | 0.94∘ | 1.07∘ | $\rm 25^{\prime\prime}\AA^{-1}$ | $\rm 27^{\prime\prime}\AA^{-1}$ | $\rm 72\ \mu m\ \AA^{-1}$ | $\rm 77\ \mu m\ \AA^{-1}$
658 nm | 17.0∘ | 29.7∘ | 17.0 ∘ | -17.1∘ | -29.8∘ | 46.8∘ | 0.91∘ | 1.10∘ | $\rm 26^{\prime\prime}\AA^{-1}$ | $\rm 28^{\prime\prime}\AA^{-1}$ | $\rm 74\ \mu m\ \AA^{-1}$ | $\rm 81\ \mu m\ \AA^{-1}$
658 nm bb high dispersion mode using alternate VPH grating with a fringe frequency of 2173 lines $\rm mm^{-1}$ and a fringe slant of 2.5∘ | 32.0∘ | 49.9∘ | 41.7∘ | -41.7∘ | -49.9∘ | 91.6∘ | 0.86∘ | 1.16∘ | $\rm 33^{\prime\prime}\AA^{-1}$ | $\rm 38^{\prime}\AA^{-1}$ | $\rm 95\ \mu m\ \AA^{-1}$ | $\rm 110\ \mu m\ \AA^{-1}$
900 nm | 23.6∘ | 39.5∘ | 26.2∘ | -26.4∘ | -39.7∘ | 65.9∘ | 0.86∘ | 1.17∘ | $\rm 28^{\prime\prime}\AA^{-1}$ | $\rm 32^{\prime\prime}\AA^{-1}$ | $\rm 79\ \mu m\ \AA^{-1}$ | $\rm 92\ \mu m\ \AA^{-1}$
Figure 3:  The CH$\alpha$S VPH grating diffraction efficiency (average
polarization) as a function of wavelength at a fixed angle of incidence. The
solid lines are rigorous coupled wave analysis (RCWA) carried out at Bragg
incidence for central wavelengths of 490 nm (blue), 660 nm (green), and 900 nm
(red). The same RCWA analysis is done for off-axis incidence angles with a
Bragg deviation of $\pm 2.5$ degrees creating the angular Bragg envelope shown
by the shaded regions. The dotted lines are the average lossless Kogelnik
efficiency predictions again performed at Bragg incidence for 490 nm (blue),
660 nm (green) and 900 nm (red). The plus symbols are the manufacturer
measured first order diffraction efficiencies at three distinct laser
wavelengths (488 nm, 650 nm, 904 nm) taken using a 98 mm aperture and averaged
for measurements across the grating. The RCWA analysis and measured data
products in this plot were supplied by Wasatch Phonics.

Figure 3 shows the predicted (average polarization) CH$\alpha$S diffraction
efficiency. The solid lines are rigorous coupled wave analysis (RCWA) carried
out at Bragg incidence for central wavelengths of 490 nm (blue), 660 nm
(green), and 900 nm (red). We note that over the narrow CH$\alpha$S bandpass
($\sim 3$ nm) there is very little change in the spectral efficiency response.
On axis, CH$\alpha$S is always operating close to the Bragg wavelength, and
the performance is not particularly dependent on the shape of the spectral
efficiency response (spectral Bragg envelope) Kogelnik (1969). Instead,
CH$\alpha$S is more sensitive to deviations from the Bragg angle of incidence.
Off-axis field positions covering the 10 arcmin CH$\alpha$S FOV undergo
angular magnification through the fast, wide-field collimator optics. In the
spectral direction, this angular magnification results in a $\pm 2.5$ degree
spread in off-Bragg incidence angles encountered by the grating, which shifts
the Bragg wavelength by $\pm 70$ nm at a central wavelength $\lambda=660$ nm
(well outside the $\sim 3$ nm filter bandpass) (Robertson et al., 2000). This
can be seen in Figure 3 where RCWA analysis is done at the same central
wavelengths for off-axis incidence angles with a Bragg deviation of $\pm 2.5$
degrees, creating the efficiency response shown by the shaded regions. The
off-axis diffraction efficiency at the central wavelength decreases according
to the profile of the angular Bragg envelope. Within this envelope the grating
still maintains high throughput and the large off-Bragg acceptance angle
allows us to use a fast collimator with high off-axis angular magnification
over a wide field of view.

VPH gratings have a number of additional tuning parameters inherent to the
gelatin layer including the gelatin thickness (d), the average index of
refraction in the gelatin ($\rm n_{2}$), and the index of refraction
modulation amplitude ($\rm\Delta n_{2}$) that creates the fringe pattern. In
the CH$\alpha$S grating we select a thin gelatin depth ($\rm d_{eff}=3\mu m$)
and a high index of refraction modulation amplitude ($\Delta n=0.11$) in order
to maximize the angular bandpass ($\Delta\alpha\propto\frac{1}{\nu_{g}d}$) and
spectral bandpass
($\frac{\Delta\lambda}{\lambda}\propto\frac{\cot\alpha_{B}}{\nu_{g}d}$) while
still ensuring that the diffraction efficiency peaks around a central
wavelength of 660 nm (Equation 2) (Barden et al., 2000).

$\Delta n_{2}d\approx\frac{\lambda}{2}$ (2)

The importance of the tuning parameters on diffraction efficiency is
particularly interesting for an instrument like CH$\alpha$S, as we intend to
use the grating far from its original optimization. A related feature in
Figure 3 is the offset between the peak wavelength (sometimes called the blaze
wavelength) and the Bragg wavelength. This occurs when we rotate the grating
to operate away from the overall optimization at 660 nm. For example, the
solid blue line represents RCWA efficiency predictions at Bragg incidence for
a central wavelength of 490 nm, but the curve actually peaks at a blaze
wavelength of 530 nm. This is a feature at Bragg incidence and is not an off-
axis shift in the Bragg wavelength. To center the efficiency response on 490
nm at Bragg incidence, the ideal $\Delta n_{2}$ should be $\sim 0.08$
(Equation 2). The fringe structure in the CH$\alpha$S grating is over-
modulated (high $\Delta n_{2}$) for bluer wavelengths and under-modulated for
redder wavelengths, manifesting as a shift in the efficiency response and an
offset between the blaze peak and the Bragg wavelength (Wasatch private
correspondence). If CH$\alpha$S had a wider bandpass and the spectral
efficiency response was a more important factor in the overall diffraction
efficiency, we could rotate the grating to a Bragg angle corresponding to a
blaze wavelength peak at 490 nm. However, with the narrow instrument bandpass,
we achieve a much smaller spread in the angular efficiency response by
operating close to Bragg incidence and accepting an offset between the Bragg
wavelength and blaze wavelength. Looking at the shaded off-Bragg angular
efficiency response for 490 nm and 900 nm, the angular efficiency curves
converge at the Bragg wavelength (where the deviation from off-Bragg incidence
angle is zero). For fixed wavelength very close to the Bragg wavelength, the
efficiency response as a function of field angle (in the spectral direction)
is minimized and symmetric. Operating at the blaze wavelength would cause an
asymmetric efficiency response at opposite field angles with a larger spread
in the relative efficiency across the FOV.

The final parameter in the CH$\alpha$S VPH grating design is the fringe slant,
similar to the blaze angle in a ruled grating. The CHaS grating has a
manufactured fringe slant of $\pm 4.5^{\circ}$. Depending on how the grating
is installed in the optical path, the fringe slant can be either positive or
negative, although the sign of the slant will affect the spectral resolution
(see Table 3 and discussion in Section 3.2).

In summary, the CH$\alpha$S VPH grating boasts a high diffraction efficiency
of $>70\%$ for narrowband observations over the full field of view spanning
central wavelengths from [490 - 900] nm. In addition, the grating performance
is particularly superb at 660 nm near the H$\alpha$ line where we achieve a
full-field diffraction efficiency of $>90\%$. The plus markers in Figure 3 are
the manufacturer measured first order diffraction efficiencies at three
distinct laser wavelengths (488 nm, 650 nm, 904 nm) averaged across the
grating aperture. These confirm the predicted on-axis efficiency close to the
central wavelength.

###  2.5 Mechanical Design

Part of the intent behind CH$\alpha$S, and the motivation for incorporating
many commercial optics, is to develop a design that can be replicated or
scaled for future applications. To make this feasible, we also require a
mechanical design that is relatively modular, simple to construct, and cost
effective. Here we detail the commercial and custom aspects of the CH$\alpha$S
mechanical design.

Figure 4: The CH$\alpha$S optomechanical design modeled in SolidWorks. The top
view is the East side of the spectrograph and the bottom view is the West
side. CH$\alpha$S is primarily composed of two trusses. The main truss houses
the relay, focal plane, and collimator optics. The camera truss houses the
custom camera optics. The majority of the mechanical frame is assembled from
prefabricated joint connector kits. The main truss is built from the 80-20
T-slot Aluminum Building System and the camera truss employs the Rock West
Composites CARBONNect carbon fiber system. The camera and grating rotate about
the labeled pivot axis. The relay optics are mounted on the side of the main
truss.

CH$\alpha$S is primarily composed of two trusses connected at a pivot point.
The main truss houses the focal plane optics, relay system, collimator optics,
and grating. It is designed to minimize flexure by keeping the center of
gravity close to the telescope focal plane. The camera truss houses the custom
camera optics. The majority of the mechanical frame is assembled from
prefabricated joint connector kits. The main truss is built from the 80-20
T-slot Aluminum Building System and the camera truss employs the Rock West
Composites CARBONNect carbon fiber system. Both the main truss and the camera
truss are enclosed with black anodized sheet metal. A custom fabric bellows
manufactured by Nabell connects the main truss enclosure to the camera truss
enclosure, completing the light-tight envelope around the instrument optical
path.

The main truss and the camera truss are connected at a pivot point shown in
Figure 4. This is the mounting location for the grating and the axis around
which the camera and grating rotate independently. The camera drive rotation
stage is a Harmonic Drive FHA25-US250-160-BL with a fail-safe brake and rotary
position sensor. This motor was chosen to satisfy our mechanical requirements;
namely, the ability to support the camera load with a 260 ft-lb max torque and
zero backlash precision. The grating angle is controlled separately from the
camera angle using a Newport rotation stage URS100BPP. This motor meets our
$20^{\prime\prime}$ bi-directional repeatability requirement, allowing us to
position the grating angle to within one resolution element on the detector
(two resolution elements with the ML125 array) (See Table 3). In order to
minimize torque on the Camera Drive, the center of mass of the camera truss is
kept as close as possible to the drive axis. The current operation employs
counterweight arms, which are attached to the camera truss whenever the drive
is engaged. Once the camera is at the proper angle, these counterweight arms
are detached replaced with rigid bracing arms that maintain position without
the significant added mass of the counterweight.

We have designed custom mounts for the entrance optics, grating, and camera
optics. An overview of the custom mount designs can be seen in Figure 5. Many
of these optics are retained using a spring-loaded mounting systems in order
to avoid transferring mount deformation to the optic or placing unwanted
stress on the glass substrates. This design also allows us to remove the
optics from their mounts without the need to break any epoxy bonds. Both the
microlens array and grating are mounted such that they can be reversed in the
optical path. In cases where the optics can be removed, reference points are
used to ensure repeatable positioning. All of the optics mounts have built in
tolerance for shimming. Due to the difference in the coefficient of thermal
expansion (CTE) between the aluminum mounts and carbon fiber camera frame,
both of the camera mirror mounts have accordion-like relief slots cut out to
allow for differential compression and expansion during natural thermal
cycling. The collimator optics are left installed in the Celestron optical
tube, which is attached to our main truss using its built-in dovetail rails.
All smaller mirrors in the relay/focal plane are bonded directly to small
aluminum mounting plates with epoxy.

Figure 5: We have designed custom mounts for the entrance optics, grating, and
camera optics. Many of these optics are retained using spring-loaded or
clamped mounting systems so that they can be removed from their mounts without
the need to break any epoxy bonds. Here we show an overview of the custom
mount designs in SolidWorks. Zoom in regions are exploded views of retention
mechanisms.

##  3 Theoretical Performance

We discuss the expected instrument performance assessed in the design stage
via computer modeling and analytic calculations. These performance metrics map
to the scientific requirements presented in Section 1. We have been working to
confirm the expected performance through laboratory testing, calibration
measurements, and engineering/commissioning time at the observatory (Section
4).

###  3.1 Point Spread Function

The point spread function of CH$\alpha$S depends on the imaging properties of
three subsystems: the telescope, the relay, and the spectrograph. The angular
resolution of these subsystems are summarized in Table 7. The spectrograph
imaging performance further depends on the microlens array, collimator optics,
diffraction grating, and camera optics. In this section, the theoretical
optical performance and PSF of the spectrograph are calculated analytically as
well as modeled with Zemax sequential and non-sequential ray tracing.

We begin by reviewing the imaging properties of the spectrograph without the
microlens array. It is important that the underlying PSF be significantly
smaller than the object size at the entrance, or the re-imaged telescope pupil
spots generated by the microlens array; this assures that the imaging
performance is primarily set by the properties of the microlens array and not
degraded by the spectrograph optics. We have designed CH$\alpha$S so that the
spectrograph optics have an underlying PSF with a geometric radius of $\rm<10\
\mu m$ in all three channels across the full field of view.

With the microlens array inserted at the entrance to the spectrograph, each
lenslet generates a pupil image. The pupil image is demagnified by the focal
reduction in the lenslet,

$w^{\prime}=\frac{F_{len}}{F_{tel}}w$ (3)

further demagnified by the camera-to-collimator focal ratio,

$w^{\prime\prime}=\frac{f_{cam}}{f_{col}}w^{\prime}=\frac{f_{cam}}{f_{col}}\frac{F_{len}}{F_{tel}}w$
(4)

and convolved with the spectrograph PSF (Figure 6) before being imaged on the
detector. Using Equation 3 and the clear aperture of the masked lenslets
(Table 1), each pupil image has a radius of $37.5\ \mu m$ with the ML250 array
and $19\ \mu m$ with the ML125 array. Assuming both the pupil image and the
$10\ \mu m$ spectrograph PSF are Gaussian, the blurred radius adds in
quadrature and we expect geometric spot radii of approximately $30\ \mu m$ and
$17.5\ \mu m$ for the ML250 and ML125 arrays respectively. Using Equation 4 to
account for demagnification through the spectrograph, we calculate that each
pupil image on the detector should have a geometric radius of approximately
$28\ \mu m$ for the ML250 array and $14\ \mu m$ for the ML125 array.

Zemax modeling of spectrograph PSF without the microlens array is shown by the
(unconvolved) spot diagrams in the upper panel of Figure 6. We see that the
Schmidt collimator and camera design achieves our desired wide-field imaging
performance of $\rm<10\ \mu m$ in all three channels on and off axis. Modeling
the lenslet and spectrograph together in Zemax at a wavelength of 658 nm, we
generate the convolved pupil images shown in the bottom panel of Figure 6.
From these models we measure an expected geometric radius of 27 microns (19
microns RMS radius) for the ML250 array. The ML125 array model produces a 13.5
micron geometric radius (9 micron RMS radius). These values are on par with
the analytic estimates above. We summarize these models in Table 2 alongside
the optical performance measurements made during early commissioning
(described in Section 4).

Figure 6:  Zemax ray trace showing the optical performance of the spectrograph
(telescope and relay not included). The top panels are the unconvolved PSF
(without the microlens array) in each of the three wavelength configurations.
The bottom panels are the convolved pupil images (with the microlens array)
for each lenslet pitch at a central wavelength of 658 nm.

###  3.2 Spectral Resolution

CH$\alpha$S targets individual emission lines within the optical range of
400-900 nm (Section 1), intentionally limiting the instrument operational
bandpass to 3 nm using a narrowband filter. Each lenslet is dispersed into a
short spectrum with limited overlap. The dispersion at the detector is
expected to be $\rm 0.4\ \AA\ pix^{-1}$ at 658 nm (See Table 3), and each
ML250 array spectral resolution element has a FWHM of about $45\ \mu m$ or 1.5
pixels. Accordingly, CH$\alpha$S has a spectral resolution of approximately
$\rm 0.6\ \AA$ corresponding to a velocity resolution of 27 $\rm km\ s^{-1}$.
The ML125 array with smaller pitch lenslets cuts the size of each spectral
resolution element approximately in half, which allows us to distinguish
emission features down to a velocity resolution of 14 $\rm km\ s^{-1}$.

Many aspects of the CH$\alpha$S design factor into the spectroscopic
performance of the instrument. As seen in Equation 5, the spectral resolution
($\delta\lambda$) depends on the lenslet pitch ($w$), the lenslet focal
reduction ($F_{lens}/F_{tel}$), demagnification through the collimator and
camera optics ($f_{cam}/f_{col}$), anamorphic demagnification from the grating
($r$) (Schweizer, 1979), and the angular dispersion ($d\beta/d\lambda$).

$\delta\lambda=w^{\prime}r\frac{f_{cam}}{f_{col}}\frac{d\lambda}{dl}=\frac{wr}{f_{col}}\frac{F_{lens}}{F_{tel}}\frac{d\lambda}{d\beta}$
(5)

We isolate terms where the diffraction grating contributes to spectral
resolution in Equation 5. The product of the logarithmic angular dispersion
($\frac{d\beta}{d\log{\lambda}}$) and the anamorphic factor ($\frac{1}{r}$) is
often used as a resolution merit function for the diffraction grating
(Bershady, 2009). As mentioned in Section 2.4, fine tuning the grating fringe
slant directly influences the spectral resolution by inversely changing the
anamorphic demagnification and the angular dispersion. A negative (clockwise)
fringe slant increases the angle of diffraction at Bragg incidence, improving
the angular dispersion. A positive fringe slant (counterclockwise) provides
greater anamorphic demagnification at the cost of lowering the angular
dispersion. There are instances where we may benefit from a specific fringe
orientation, balancing these two factors, and the CH$\alpha$S VPH grating can
be installed with the fringe slant in either a positive or negative
orientation.

The remaining terms in Equation 5 are optimized to improve the spectral
resolution while maintaining high S/N. Note that the camera design is slightly
faster than the collimator (Table 6), with a shorter focal length ($f_{cam}$)
that de-magnifies the field of view and improves the linear dispersion. The
focal reduction ($F_{len}/F_{tel}$) provided by the microlens array is set to
approximately match the telescope aperture ratio to the collimator aperture
ratio. The lenslet pitch ($w$) is selected to carefully balance the angular
resolution, spectral resolution, and S/N. Decreasing the pitch of the lenslets
in the microlens array decreases the angular extent of each lenslet and
reduces the size of the spectral resolution element (similar to reducing the
slit width in long-slit spectroscopy). This both improves the spatial
resolution and boosts the resolving power. The drawback to decreasing the
lenslet pitch is that it also results in a reduced the S/N from each lenslet.
We discuss this trade-off more in Section 3.3 and Section 5.

###  3.3 Sensitivity

To demonstrate the sensitivity performance, we calculate the predicted minimum
detectable intensity CH$\alpha$S can measure while requiring that we achieve a
S/N of 10.

$\displaystyle\rm\frac{S}{N}$
$\displaystyle=\frac{S}{\sqrt{S+S_{sky}+\sigma_{Dark}+\sigma_{RN}}}$ (6)
$\displaystyle=\frac{I_{\lambda}\epsilon A_{g}\Omega
t}{\rm\sqrt{S+I_{S}(\lambda)\epsilon A_{g}\Omega
t\Delta\lambda+N_{p}Dt+(N_{p}/N_{b}^{2})RN^{2}N_{f}}}$ (7)

The emission signal S in units of photons is the product of the line flux
($I_{\lambda}$), the instrument efficiency ($\epsilon$), the geometric area of
the telescope ($A_{g}$), the angular extent on the sky ($\Omega$), and the
exposure time (t). Noise contribution terms added in quadrature include shot
noise from the emission signal, shot noise from the sky-background emission,
dark current from the detector, and read noise from the detector. Our
sensitivity analysis (Figure 14) includes all of these noise sources. The
design goal is to operate in a sky-background limited regime during dark time,
and Equation 6 then simplifies to:

$\rm\frac{S}{N}=\frac{I_{\lambda}\sqrt{\epsilon A_{g}\Omega
t}}{\sqrt{I_{s}(\lambda)\Delta\lambda}}$ (8)

In Figure 14 we model the CH$\alpha$S $10\sigma$ minimum detectable intensity
as a function of angular scale. This calculation follows Equation 6 assuming
an efficiency of $\epsilon=15\%$. In the sky background limited regime, the
signal-to-noise will scale as $\sqrt{\epsilon}$. We show three integration
times (1 hour, 10 hours, and 100 hours) and assume that each integration is
composed of 900s exposures. Lenslets in each microlens array only cover a few
arcseconds in angular size. While it would be exceedingly difficult to reach
sensitivities on the order of 1-10 mR in each resolution element, part of the
power of using a microlens array is that individual lenslets can be binned
together to measure a fixed solid angle on the sky. We measure larger spatial
scales by combining the signal from many neighboring lenslets. Combining the
signal from multiple lenslets is made easier by the fact that PSF of each
lenslet is an image of the pupil and is invariant to the on-sky spatial
structure in each lenslet. Although we are probing a coarser spatial scale
with this lenslet binning, the measurement maintains a fixed spectral
resolution. The S/N improves with increasing scale and integration time in
accordance with Equation 8.

##  4 On-Sky Performance

Using calibration data and on-sky measurements taken during early
commissioning in May-June 2021, we have compiled a preliminary performance
assessment of CH$\alpha$S on the Hiltner 2.4-m telescope. This is intended as
an early validation of the expected performance.

We summarize the on-sky commissioning targets in Table 4 and Figure 7. Here we
compare the expected emission from each target/galaxy with the telluric
emission from the atmosphere shown in the top panel (Leinert et al., 1998).
Each black horizontal line in the bottom panel is a galaxy centered on its
redshifted H$\alpha$ emission. The length of the line corresponds to the
$W_{20}$ velocity width (Kennicutt et al., 2003). The corresponding grey lines
represent the redshifted NII emission (assuming the same width as H$\alpha$).
The shaded regions show the approximate filter response curves for the Baader
H$\alpha$ (6 nm FWHM) and Astrodon NII (3 nm FWHM) filters. Most of our
primary targets have H$\alpha$ emission that is redshifted into the NII
filter.

As an example of the commissioning data, we show a stack of the Deer Lick
Galaxy NGC 7331 in Figure 8. In many ways this stack resembles a narrowband
image of the galaxy, with an H$\alpha$ image from the SINGS survey provided
for comparison. On close inspection we see the imprint of the lenslet array in
the CH$\alpha$S stack, resembling a pointillism image of the galaxy. This
stack is not just an intensity map of NGC 7731, but also a spectral image
dispersed along the x-axis. This can be seen in the continuum spectrum
dispersed for each star.

The two zoom-in panels in Figure 8 show discrete emission. Depending on the
gas velocity in a given lenslet bin, each emission line may be Doppler shifted
slightly with respect to the systemic velocity. Small wavelength shifts are
dispersed through the grating and the pupil image for each lenslet is slightly
offset from the lenslet center along the spectral direction. These pixel-scale
offsets in the emission line position with respect to the centroid of each
lenslet are measured via comparison with a rest-frame image of a calibration
lamp. This effect can be seen in the discrete disk emission. Note how the
pupil images do not form a perfect grid pattern and instead looks distorted,
reflecting velocity offsets in the gas at this location. The discrete sky
emission comes from the atmosphere at a nearly uniform velocity and therefore
has a constant Doppler shift with respect to the microlens hexagonal grid
pattern. Accordingly, we see the sky emission as a uniform image of the
microlens array.

In addition to the list of galaxy targets in Table 4, we have used CH$\alpha$S
to observe a variety of nebulae during early commissioning. Wide field
spectral imaging with CH$\alpha$S is great for mapping large nebular regions
such as planetary nebulae, emission nebulae and supernova remnants.
CH$\alpha$S maps these bright targets extremely efficiently, allowing us to
tile large regions with only a few exposures per pointing. These targets often
fill the full field of view, and are excellent for instrument calibration. In
Figure 9 we show CH$\alpha$S images of the Dumbbell Nebula. Like many nebulae,
the Dumbbell Nebula hosts interesting velocity structure, and we highlight the
capability of CH$\alpha$S to map this structure in a single 180s exposure with
a velocity resolution of $\sim 30$ km $\rm s^{-1}$. A spectrum is extracted
for each lenslet using the calibration lamp as a weighted template to find the
emission peaks. We produce an RGB image of the CH$\alpha$S spectra colored by
velocity (with steps of 30 km $\rm s^{-1}$) to demonstrate the stunning visual
properties of this data. We plot 1D profiles for each spectrum at the center
showing both single and multi-component emission lines from gas expanding away
from the central white dwarf at velocites of $\pm 30$ km/s. In future runs,
CH$\alpha$S will be capable of collecting similar spectral maps in multiple
optical emission lines, probing shock structure and the strength of various
ionization mechanisms in a collection of a few short exposures.

Table 4: Primary target integration summary from early commissioning May/June 2021. Each integration is composed of 180 second exposures. We require a Moon illumination of $\leq 50\%$ and a Moon separation of $\geq 55^{\circ}$.  Target | Alternate Name | Total Integration
---|---|---
NGC 4631 | Whale Galaxy | 3.0 hrs
NGC 4656 | Whale Friend Galaxy | 2.1 hrs
NGC 7331 | Deer Lick Galaxy | 4.1 hrs
NGC 5906 | Cat Scratch Galaxy | 5.05 hrs
NGC $6946^{{\dagger}}$ | Fireworks Galaxy${\dagger}$${\dagger}$Observed in H$\alpha$ filter | 1.2 hrs
NGC 5033 | Dungeness Galaxy | 1.85 hrs
Figure 7: CH$\alpha$S early commissioning targets and observational planning.
The top spectrum shows the sky background emission lines from the atmosphere.
Each black horizontal line is a galaxy centered on its redshifted H$\alpha$
emission. For galaxies in the SING survey, the length of the line corresponds
to the H$\alpha$ velocity width ($W_{20}$) (Kennicutt et al., 2003). For
galaxies not in the SINGS survey (NGC 5477, NGC 5457, NGC 5907, and NGC 4656),
we use HI velocity width measurements (Davies et al., 1980; Rots, 1980). Note
that the black lines for NGC 5907 and NGC 4656 are full width $\rm\Delta v$
and not $\rm W_{20}$ measurements. The grey lines correspond to the redshifted
NII emission (assuming the same width). Nebulae targets are shown as vertical
lines at the emitted wavelength of H$\alpha$ and NII. The shaded regions
behind the spectrum show the approximate filter response curves for the
H$\alpha$ (6 nm FWHM) and NII (3 nm FWHM) filters. The red vertical dashed
line marks the emission line wavelength present in the Ne reference lamp.
Figure 8: Example data from early commissioning. The left panel is an
H$\alpha$ image from the SINGS survey for comparison (SINGS Team, 2020). The
middle panel is a preliminary stack of NGC 7331 (Deer Lick) data taken on 14
June 2021. This is a 1 hour integration composed of 180 second exposures.
These observations were taken with a Moon illumination of $15\%$ and a Moon
separation of $117^{\circ}$. Sky line emission in the background has not been
subtracted, and can be seen in the upper right panel. The cross-hair artifact
is a calibration mask installed throughout the commissioning run. Figure 9:
Example data from early commissioning. The left panel is an RGB image of the
CH$\alpha$S spectra colored by velocity. This map has a resolution of $\pm 2$
pixels or $\pm 60$ km/s. If we zoom in on the center of the nebula and create
a plot of the 1D spectral profiles (middle panel) we see the expansion of the
nebula. Emission lines in each 1D profile have been fit and colored based on
if they are redshifted or blueshifted. In the rightmost panels we see that the
Dumbbell nebula nearly fills the $10^{\prime}\times 10^{\prime}$ FOV. The
smoothed CH$\alpha$S Data resembles a narrowband H$\alpha$ image of the
nebula. The cross-hair artifact is a calibration mask installed throughout the
commissioning run.

###  4.1 Comissioning Data Analysis

The raw CH$\alpha$S data is a flattened spectral cube collected in a single
exposure without the need to scan in frequency space. In order to build up
long integration times we combine a set of exposures from each night into a
stacked image. The stacking/image registration process uses the cross
correlation to calculate sub-pixel offsets between exposures and align them
appropriately. Cosmic rays are removed from each image in the stack by looking
for Laplacian/sharp edges following the prescription in van Dokkum (2001). We
are working on a global solution for the spatio-spectral instrument response
across the full field of view. Flat field correction, sky background
subtraction, and photometric calibration have not been applied to Figure 8.

The model for spectral extraction in our science data is based on the location
of monochromatic light through each lenslet, which is determined using either
a calibration lamp (Figure 10) or a low-velocity nebula. These locations are
then shifted, using the dispersion solution, to the expected location of
H$\alpha$ at the systemic velocity of the target galaxy, establishing initial
centers for the analysis of lenslet spectra on the detector. We extract a
spectrum at each of these locations; however, overlap with adjacent lenslets
results in a quasi-periodic multi-peaked spectrum. Using a velocity prior,
typically an HI map, we weight the peak detection in the extracted spectrum by
the expected Doppler shift, limiting it to a small velocity range around the
prior velocity.

Cross-talk between neighboring lenslets is partially mitigated by setting an
appropriate lenslet array rotation given the hexagonal packing geometry, but
there is still spectral overlap to contend with in both the dispersion and
cross-dispersion directions. The amount of overlap is dependent on the
selected bandwidth, the lenslet array pitch and cross-dispersion profile and
the (optimized) lenslet array rotation (See Figure 2). As discussed in Bacon
et al. (2001), there is a significant tradeoff to be considered between
maximizing signal and increasing random or systematic noise due to overlap.
Since CH$\alpha$S is primarily targeting a single emission line, the bandwidth
may be optimized for specific targets using ultra-narrowband filters.
Additionally, the design of CH$\alpha$S allows for masking of the microlens
array that can reduce overlap through loss of sky coverage. For the
commissioning runs, we have not used masking or ultra-narrowband filters and
leave more extensive discussion of the full set of tradeoffs and their impacts
on S/N for future work.

For the commissioning implementation, as seen in Figure 8, narrow/individual
emission lines can be identified. We estimate the spectral overlap between
lenslets using the non-sequential ray trace model. For narrowband $3\rm\AA$
emission (e.g. Doppler broadened spectral lines), we find that $<5\%$ of the
flux in a measured spectrum comes from overlap. Still, confusion may arise
from continuum emission, multiple velocity components, or overlap with the
dispersed sky lines. For continuum emission with a filter bandpass of
$20\rm\AA$, the fractional flux in each spectrum that comes overlap is
approximately $66\%$. This overlap fraction is in agreement with measured
noise from the sky continuum background shown in Figure 12.

###  4.2 Optical Performance

During commissioning calibration lamps were routinely collected for all
filters. The optical commissioning performance can be measured across the
field of view using these calibration lamps, an example of which is shown in
Figure 10. This Ne lamp is taken in the NII filter and is a direct measure of
the spectrograph PSF (without the telescope) in discrete emission at 6598
$\AA$ . We measure an average circular FWHM spot size of 1.8 pixels in the
commissioning lamps or 54 microns. This is similar to the effective PSF in
Figure 10 and is on par with predictions in Section 4.2. There is a slight
astigmatism in the PSF, most prominent at the edges of the field. This is most
likely due to a tilt in the co-planar alignment of the camera focal plane with
the detector CCD (Bacon, 1995).

We apply a World Coordinate System solve to the on-sky commissioning images
using stellar catalogs in each pointing. This solve aligns the stellar catalog
positions with the wavelength of the H$\alpha$ emission line redshifted to the
systemic velocity of the galaxy. This provides the best average alignment
between CH$\alpha$S emission features and narrowband imaging. In both
directions, the telescope plate scale of $\rm 0.34^{\prime\prime}\ pix^{-1}$
is magnified by the ratio of the CH$\alpha$S collimator to camera focal
length. We recover a plate scale at the detector of $\rm 0.457^{\prime\prime}\
pix^{-1}$ in the cross-spectral direction and $\rm 0.421^{\prime\prime}\
pix^{-1}$ in the spectral direction. In the spectral direction, the anamorphic
magnification from the grating (installed with a negative fringe slant)
accounts for the additional factor of 1.09 in the plate scale measurement (See
Table 3).

In order to recover an estimate for the angular spatial resolution, we measure
the average separation between detected lenslets in our commissioning data.
Here we use images with a WCS solve and very little expected velocity
structure. Our model predicts that the on-sky angular resolution for the ML250
array should be $2.83^{\prime\prime}$. We measure an angular resolution of
$2.77^{\prime\prime}\pm 0.06^{\prime\prime}$, consistent with this
expectation.

Figure 10: (left) Calibration Lamp in H$\alpha$ filter with the CH$\alpha$S
mask in front of microlens array. This demonstrates the dispersion along the
spectral axis, where each image of the mask is a line in the Ne lamp. (middle
left) Ne calibration lamp in the NII filter. This is a zoomed-in single
quadrant of the lamp, showing the individual lenslets and the cross-hair mask.
(middle right) The effective point-spread function (EPSF) of the calibration
lamp lenslet pupil images. (right) Stars dispersed in the NII filter in a
single image (at the same telescope focus) hitting different combinations of
lenslets. In the broader H$\alpha$ filter, stars dispersed in the CH$\alpha$S
field show prominent absorption features.

###  4.3 Spectral Performance

We measure the linear spectral dispersion using two independent methods. (1)
Using the broader (6 nm) H$\alpha$ filter, we detect multiple emission lines
from the Ne lamp (Figure 10). The separation between lines at known
wavelengths can be used to solve for the dispersion. (2) Bright M-type stars
dispersed in the commissioning data show spectral absorption features in the
NII filter. These absorption lines can be compared with existing high
resolution spectra to solve for the dispersion. Both methods recover an
operational linear dispersion of $\rm 0.373\pm 0.001\ \AA\ pix^{-1}$. We use
this dispersion measurement and the optical performance assessed in Section
4.2 to estimate the spectral resolution. Given a FWHM of 1.8 pixels and a
dispersion of $\rm 0.373\ \AA\ pix^{-1}$ we achieve a commissioning spectral
resolution of $\rm 0.67\ \AA$ or $\rm 30\ km\ s^{-1}$. The ML125 array results
in spots with a measured FWHM of 1.01 pixels, corresponding to a spectral
resolution of $\rm 0.377\ \AA$ or $\rm 17\ km\ s^{-1}$.

###  4.4 Mechanical Performance

The CH$\alpha$S mechanical design must maintain minimal flexure over long
integration times in order to avoid blurring/degrading the spectral imaging
quality. The mechanical performance can be assessed using two critical flexure
measurements: (1) flexure in the entrance optics and relay system (2) flexure
in the spectrograph including the collimator, camera, and pivot. Entrance
flexure blurs the re-imaging of the telescope focal plane on the surface of
the microlens array and degrades the spatial resolution. Spectrograph flexure
causes the microlens array image to shift around on the detector. It is axis
dependent; a drift along the spectral direction degrades the spectral
resolution, while a drift in in the cross spectral direction degrades the
spatial resolution element. CH$\alpha$S does not have imposed pointing
constraints when attached to the telescope, so it experiences a full range of
gravity vectors during normal operation. As a commissioning goal, we aim to
reduce the acceptable drift due to flexure to $<50\%$ of the spectral imaging
performance. This sets the following requirements using the ML250
configuration:

  1. 1.

Entrance flexure (in the re-imaged telescope focal plane) should not exceed
half a lenslet or $1.3^{\prime\prime}$.

  2. 2.

Spectrograph flexure should not exceed half a spectral resolution element or
30 microns (1 pixel in 2x2 binning).

Early commissioning measurements of the instrument flexure were designed to
probe both entrance flexure and spectrograph flexure. Reference cross-hair
masks were installed on both the filter and microlens array. The telescope was
slewed incrementally in RA (the flexure is much smaller in Dec), and the shift
in both cross-hair images was measured at the detector. We summarize the
results in Figure 11. Discrete points are the measured entrance (blue) and
spectrograph (black) flexure. Each measurement is the magnitude of the drift
calculated by summing flexure along the spectral axis and flexure along the
cross-spectral axis in quadrature. However, we note that the flexure in the
entrance optics is primarily in the spectral direction and the flexure in the
spectrograph optics is primarily in the cross-spectral direction. The shaded
regions represent the expected drift due to flexure using the commissioning
requirements over a range of exposure times spanning from 300 seconds (dotted
line) to 600 seconds (solid line). These measurements along the equatorial
axes and the flexure measured during on-sky observations show a very similar
trend. We find that the current flexure is consistent with our limiting
requirements at an exposure time of around 600s. Early commissioning runs used
a very conservative exposure time of 180s. Exposures can be lengthened to $\rm
180s<t_{exp}<600s$ in subsequent runs without degrading the spectral imaging
beyond commissioning requirements. These numbers become more stringent by an
additional factor of $2$ if we swap out the ML250 microlens array for the
smaller pitch, higher resolution ML125 array. Ultimately, we aim to make the
flexure requirements for CH$\alpha$S a factor of $5$ tighter than the
commissioning requirements. This would reduce the acceptable drift due to
flexure to $<10\%$ of the spectral imaging performance.

Figure 11: Early commissioning flexure measurements. The blue points are the
measured entrance flexure and the black points are the measured spectrograph
flexure. Each measurement is the magnitude of the drift calculated by summing
flexure along the spectral direction and flexure along the cross-spectral
direction in quadrature. The shaded regions represent the expected drift due
to flexure over a range of exposure times spanning from 300 seconds (dotted
line) to 600 seconds (solid line). These are calculated using the
commissioning flexure goals outlined in text.

###  4.5 Sensitivity Performance

The CH$\alpha$S noise budget including the sky background and MDM 4K detector
noise is shown in Figure 12. Here we plot the standard deviation $\sigma_{N}$
of each noise source per lenslet per frame as a function of the exposure time.
This is independent of the total integration time composed of N frames. We use
a CCD binning of $2\times 2$ and a spot size of 60 $\mu$m (corresponding to an
area per lenslet of $\sim 4$ pixels). The expected read noise (black dotted
line) is constant per frame but remains a function of the number of frames per
total integration time. The expected dark noise (dashed black line) increases
with the exposure time per frame. These sources are added in quadrature to
estimate the total detector noise (solid black line). Discrete points are
detector dark data from our 2021 commissioning tests. The MDM 4K detector is a
mosaic of 4 CCDs. Each data point is the average detector noise (bias
subtracted) at the center of each quadrant, and the error bars are the
standard deviation between quadrants. These measurements confirm that the read
noise and dark current are similar to the detector specifications and
demonstrate the light-tight capabilities of our instrument enclosure in the
dome environment. In addition to the detector noise, the sky background noise
can have a range of values covered by the blue shaded regions in Figure 12.
These values assume a sky background intensity of 2 $\rm R\ \AA^{-1}$ (Leinert
et al., 1998). If the microlens array is partially masked and the bandpass is
narrow enough that continuum from the sky spectra does not overlap at all,
then the sky background is the lower boundary of the light blue shaded region.
In this case, CH$\alpha$S is detector-limited by the dark noise in the MDM 4K.
However, even with a narrow bandpass of 30 - 60 $\AA$, the continuum from the
sky background in neighboring lenslets overlaps by a factor of 2-3 (shown by
the darker blue shading). In this nominal configuration without any masked
lenslets, the sky background noise becomes comparable to or slightly exceeds
the dark noise and CH$\alpha$S is sky-background limited. The total noise
(again summed in quadrature) is shown by the shaded gray region over the range
of sky background values. We measure the total noise directly (including the
sky and detector) in regions of blank sky in the commissioning data. This
measurement for a 360s frame is plotted as discrete open circle. It confirms
the model used in our signal-to-noise predictions and shows that we are
currently operating in a sky-background limited regime.

In addition to the detector noise performance, the sensitivity relies on the
full instrument throughput. In Table 7 we outline the expected performance of
the telescope, spectrograph and detector combined, predicting a total
throughput of $\epsilon=15\%-20\%$ without the atmosphere. Early commissioning
throughput testing of the spectrograph optics has been completed using a laser
at $6563\AA$ fed into the system (below the filter) with an F/7.5 input to
match the speed of the telescope. The intensity of this input was measured
separately using a beam splitter to a photometer. We measure a spectrograph
efficiency of approximately $15\%$ not including the telescope, filter, or
atmosphere. We have also made an on-sky measurement using a bright star placed
at the center of the field. We compare with the SDSS r-band magnitude, as this
is the closest to our NII filter. This measurement has been corrected for the
$75\%$ fill factor of the microlens array as the star is detected over
multiple lenslets and about $25\%$ of the flux is blocked by the black
chromimum mask between lenslets. We measure an full-system efficiently of
$8\%$ in current configuration. This temporary factor of 2 loss in throughput
is due a combination of the severely aged telescope coating and the commercial
filter performance. We anticipate improvement of these factors and they will
not have a long-term affect on the system throughput.

Figure 12: CH$\alpha$S noise budget including sky background and MDM 4K
detector noise. We plot the standard deviation $\sigma_{N}$ of each noise
source per lenslet per frame as a function of the exposure time per frame. We
assume a CCD binning of $2\times 2$ and a spot size per lenslet of 60 $\mu$m
(corresponding to an area per lenslet of $\sim 4$ pixels). The expected read
noise (black dotted line) is constant per frame but remains a function of the
number of frames per total integration time. The expected dark noise (dashed
black line) increases with the exposure time per frame. These sources are
added in quadrature to estimate the total detector noise (solid black line).
Discrete points are detector dark data from our 2021 commissioning tests. The
MDM 4K detector is a mosaic of 4 CCDs. Each data point is the average detector
noise (bias subtracted) at the center of each quadrant, and the error bars are
the standard deviation between quadrants. Similarly, the total noise shown as
a discrete open circle is measured by taking the lenslet-to-lenslet standard
deviation in a blank sky frame.

##  5 Discussion

Figure 13: A comparison of effective grasp vs. resolving power between
integral field spectrographs. Symbols and coloring distinguish the telescope
size, instrument commissioning status, and focal plane coupling. When an
efficiency measurement is not available we assume $\epsilon=0.2$. The
CH$\alpha$S design (nominal and high resolution) occupies a unique quadrant of
this parameter space, maintaining a high grasp and moderate resolving power.
Contours are exposure time isochrones showing the grasp/resolution needed for
a 10$\sigma$ detection of diffuse H$\alpha$ emission with an intensity of 3 mR
over a solid angle equal to the full field of view for each instrument.
Shading encompasses instruments that can make this detection within 1 hour
(dark blue), 10 hours (medium blue), and 100 hours (light blue). This
calculation excludes detector noise for comparison across platforms. It is
only valid for diffuse emission, as it assumes the signal is binned over the
full field of view.

We discuss CH$\alpha$S in comparison with other IFU designs and review the
trade-off decisions considered while optimizing the instrument for sky-
background limited observations of diffuse signal. The key advantages of the
CH$\alpha$S design can by summarized by stating that CH$\alpha$S is a large
grasp, high throughput instrument with moderate spectral resolution. While
there are a variety of different instrument merit criteria, one common
parameter space for assessing relative performance between instruments is the
combination of coverage vs. resolution (Bershady, 2009). In Figure 13 we plot
the effective grasp vs. resolving power, comparing CH$\alpha$S with many
excellent existing and planned integral field spectrographs. CH$\alpha$S
occupies a unique quadrant of this plot, maintaining both a high effective
grasp and a moderate resolving power.

Maintaining a high grasp and moderate spectral resolution are compelling
instrument criteria for mapping discrete, ultra-low surface brightness
emission. This combination directly drives the scientific capabilities of
CH$\alpha$S. We demonstrate this concept in Figure 13 by overlaying exposure
time isochrones representing the grasp/resolution needed for a 10$\sigma$
detection of diffuse H$\alpha$ emission with an intensity of 3 mR over a solid
angle equal to the full field of view for each instrument. Shading encompasses
instruments that can make this detection within 1 hour (dark blue), 10 hours
(medium blue), and 100 hours (light blue). The dark blue contour segment
requiring the shortest exposure time resides in the top right corner of the
plot, favoring high grasp/high resolution configurations. The CH$\alpha$S
effective grasp is driven by the instrument’s large field of view
($10^{\prime}\times 10^{\prime})$. While many integral field spectrographs are
paired with $4-8$ meter class telescopes, prioritizing high spatial resolution
over wide-field imaging, CH$\alpha$S is paired with a 2.4-meter telescope
operating over a wide field of view. Maintaining a moderate spectral
resolution allows for the detailed sky background rejection essential for
faint sky-background limited observations; sky background noise at a specific
wavelength is reduced by attaining a smaller spectral resolution element
(Equation 8). This resolution-driven sensitivity gain is only realized for
discrete emission line mapping, where the signal is independent of the
bandpass. In selecting the spectral resolution, the CH$\alpha$S design
carefully balances the spectral vs. spatial information. CH$\alpha$S fits an
exceptional number of total ($\rm spatial\ N_{\Omega}\times\ spectral\ N_{R}$)
resolution elements on its detector, optimizing the instrument’s informational
collecting power (Bershady, 2009). The trade-off between high grasp and high
spectral resolution results in a scale-dependent sensitivity; at the same
grasp, an instrument with higher spectral resolution will have greater
sensitivity on smaller scales. This occurs because the S/N in the sky-
background limited regime relies strongly on the grasp per spectral resolution
element and not solely on achieving unparalleled grasp. We leave for future
work a comparison with alternative spectrograph designs such as Fabry-Pérots
or Interferometers, some of which may achieve a similarly high grasp and
moderate spectral resolution. Because of the complexities of these
instruments, there are trade-off assessments and questions regarding their
backgrounds that need to be included in a more detailed comparison.

Figure 14: The predicted minimum detectable H$\alpha$ intensity (S/N = 10) as
a function of angular scale at three different integration times: 1 hr, 10 hr,
100 hr. Soid lines are calcuated with an efficiency $\eta=15\%$ and dashed
lines are calculated with an efficiency $\eta=6\%$. We compare these
sensitivity limits with the expected emission flux from the ultra-faint
extragalactic UV background (EUVB) and the galactic UV background. We show
that CH$\alpha$S is able to reach these sensitivity limits within 10-100 hours
of integration.

$\rm EM=\frac{\Gamma_{z=0}N_{HI}}{\alpha_{B}(T)}=2.75I_{H\alpha}T_{4}^{0.9}\
cm^{-6}\ pc$ (9)

To demonstrate the scientific capabilities of CH$\alpha$S we compare the
CH$\alpha$S predicted sensitivity limits with emission observed in the local
universe (Figure 14). Observational measurements and upper limits are shown as
discrete symbols. Bright H$\alpha$ features include the M82 Cap (Lehnert et
al., 1999), the Virgo filaments (Kenney et al., 2008), edge-on galaxies
(Christlein et al., 2010), NGC 7793 (Dicaire et al., 2008), UGC 7321 (Adams et
al., 2011), NGC 247 (Hlavacek-Larrondo et al., 2011), the outer disk of M31
(Madsen et al., 2001), the Leo Ring (Donahue et al., 1995), and the Virgo HI
cloud (Weymann et al., 2001). The dashed and solid light grey lines around 100
mR represent H$\alpha$ emission observations and surface brightness
predictions for the Magellanic stream/bridge (Bland-Hawthorn et al., 2007;
Barger et al., 2013). CH$\alpha$S is capable of making most of these
detections on arcminute scales in $\lesssim 1$ hour, and observations of many
of these targets have already been carried out with CH$\alpha$S. Emission from
the circumgalactic medium is much more tenuous, requiring measurements that
push the boundaries of low surface brightness imaging. Faint emission from
ionized hydrogen in the outskirts of galaxies and well into the CGM is a probe
of the local ionizing flux (Sunyaev, 1969). The predominant sources of
ionizing radiation in the CGM is the UV continuum from the galactic disk and
the photo-ionizing extragalactic UV-background (EUVB). Constraints on the low-
redshift (z = 0) EUVB photoionization rate range from $\Gamma\lesssim
2-8\times 10^{-14}\rm\ s^{-1}$ (Adams et al., 2011; Madau & Haardt, 2015;
Fumagalli et al., 2017; Caruso et al., 2019). The H$\alpha$ emission intensity
(quoted in Rayleigh) scales as the emission measure given in Equation 9. We
use this approximation to calculate a rough estimate of the CGM surface
brightness assuming a warm gas temperature $\rm T=10^{4}\ K$, an EUVB
ionization rate $\Gamma=4\times 10^{-14}\rm\ s^{-1}$, and a column density
$\rm N_{HI}\geq 10^{17}\ cm^{-2}$ for case B recombination ($\alpha_{B}$) in
optically thick Lymann Limit systems. With these assumptions, warm ionized gas
far into the CGM should produce an exceedingly faint emission signal on the
order of a few mR. The Zhang et al. (2016) statistical detection of
circumgalactic H$\alpha$ emission achieved by stacking millions of SDSS
sightlines has a similar flux of 3 mR. Higher column density gas or gas
ionized by stronger UV continuum flux from the galactic disk would produce
brighter emission on the order of 10 mR. For example, high velocity clouds in
the halo of the Milky Way ionized by the Galactic disk exhibit H$\alpha$
emission intensities of 30-70 mR (Putman et al., 2003). Following a
calculation similar to Equation 9 and assuming a 50 kpc region, the light blue
shaded parallelogram region shows the predicted CGM emission intensity for a
range of hydrogen densities $\rm log\ n_{H}/cm^{-3}=-4.5\ to\ -3.5$
corresponding to an overdensity of $\delta\sim 100-1000$ with clumping on
scales of $2-25$ kpc. We show that CH$\alpha$S can probe the majority of this
phase space within a few tens of hours. Within 10-100 hours, CH$\alpha$S is
capable of reaching a sensitivity faint enough to place our own constraints on
the EUVB photoionization rate. A 50-100h exposure with CH$\alpha$S would be
the deepest H$\alpha$ image and velocity field ever obtained, reaching a
surface brightness of a few mR on scales of a few arcmin. A CH$\alpha$S
science campaign is currently underway, building up 10-50 hour observations on
a small sample of Fall/Spring galaxy targets.

Achieving accurate sky subtraction will require a precise and stable flat-
field and careful control of systematic effects. With respect to flat-field,
since we are combining many hours of images (corresponding to $>$100
exposures), and combining spectra within a field (e.g. $>$100 spectra per ‘sky
bin’), random and spatially uncorrelated flat-field uncertainties will
decrease as sqrt(N). Because of this, flat-field accuracies of 1-10% should be
sufficient and feasible. Other aspects will require careful calibration and
analysis, including bias subtraction, removal of continuum from background
point sources, stray and scattered light, detector ghost images, correction
for distortion, flexure and sky and spectral variability. These may reduce the
accuracy of the sky subtraction and the reliability of the measured signal and
will be addressed in future work that seeks to reach the faintest limits.

##  6 Summary

  1. 1.

CH$\alpha$S is a narrowband (3 nm), moderate resolution (R $\sim 10,000$),
wide-field ($10^{\prime}\times 10^{\prime}$) integral field spectrograph
designed to detect faint optical emission from diffuse gas in the local
universe (z $<0.01$).

  2. 2.

CH$\alpha$S has a catadioptric design optimized to perform over a 400 - 900 nm
operational wavelength range. It is capable of targeting a broad number of
optical emission lines including H$\alpha$, H$\beta$, [SII], [NII], [O III],
[O II], and [O I].

  3. 3.

CH$\alpha$S collects a total of $1.75\times 10^{6}$ resolution elements ($\rm
N_{\Omega}\times N_{R}$) in a single exposure while maintaining a high grasp
and moderate resolving power. This an ideal combination for efficiently
mapping discrete, ultra-low surface brightness emission.

  4. 4.

A 50-100 hour integration with CH$\alpha$S would be the deepest H$\alpha$
image and velocity field ever obtained, probing emission with a surface
brightness of a few milli-Rayleigh while maintaining a S/N of 10.

  5. 5.

CH$\alpha$S is deployed on the Hiltner 2.4-m telescope at MDM Observatory. We
have completed a successful early commissioning at the observatory, and we are
working towards making CH$\alpha$S a facility instrument available to the MDM
consortium. The instrument remains a testbed for new design concepts and a
teaching opportunity for many student-led upgrades.

The Circumgalactic H$\alpha$ Spectrograph is funded by NSF AST-1407652 (ATI).
Graduate student Nicole Melso was supported with an NSF GRFP under grant
number DGE-1644869, and graduate student Bárbara Cruvinel Santiago is
supported by NASA FINESST 80NSSC19K1419. CH$\alpha$S is commissioned at the
MDM Observatory, operated by Dartmouth College, Columbia University, Ohio
State University, Ohio University, and the University of Michigan. We
especially thank the MDM Observatory staff Eric Galayda and Tony Negrete for
their help in commissioning CH$\alpha$S on the Hiltner 2.4-meter as well as
John Thorstensen and Jules Halpern for supportive observatory scheduling. This
work is built on the success of protoCH$\alpha$S, a prototype constructed and
tested by Erika Hamden and Sam Gordon. We thank José Manuel Zorrilla Matilla,
Julia Blue Bird, Derek Wacks, and Katherine Gonglewski for their help at
various stages of this project. The construction of this instrument was done
in collaboration with many industry/manufacturing companies listed throughout
this paper. Special thanks go to Dominic Speer, Elroy Pearson, Misty Johnson,
and Wasatch Photonics for the technical discussions and efficiency modeling
that led to the current VPH grating design. Many thanks to Vladimir Leleko at
Advanced Microoptics Systems (a$\mu$s) for advising us on the design of custom
microlens arrays for this application. The CH$\alpha$S optical design relies
on an NDA with Celestron LLC. We are grateful for their collaboration with
with academic institutions and for the support provided by their engineering
and sales teams. MDM:2.4m

Table 5: CH$\alpha$S Offner relay design  | Surface | Radius | Thickness | Semi-Diameter | Material | Decenter X | Decenter Y | Tilt X | Tilt Y
---|---|---|---|---|---|---|---|---|---
1 | Object | Infinity | 654.627 | 33.0 |  |  |  |  |
2 | Coordinate Break |  |  |  |  | 0.0 | 0.0 | 8.364 | 0.0
3 | Concave Sphere | -609.6 |  | 75.0 | MIRROR |  |  |  |
4 | Coordinate Break |  | -305.927 |  |  | 0.0 | 0.0 | 8.364 | 0.0
5 | Coordinate Break |  |  |  |  | 0.0 | 0.0 | -16.728 | 0.0
6 | Coordinate Break |  |  |  |  | 0.0 | -3.0 | 0.0 | 0.0
7 | Convex Sphere | -310.661 |  | 25.4 | MIRROR |  |  |  |
8 | Coordinate Break |  |  |  |  | 0.0 | 3.0 | 0.0 | 0.0
9 | Coordinate Break |  | 305.927 |  |  | 0.0 | 0.0 | -16.728 | 0.0
10 | Coordinate Break |  |  |  |  | 0.0 | 0.0 | 8.5 | 0.0
11 | Concave Sphere | -609.6 |  | 75.0 | MIRROR |  |  |  |
12 | Coordinate Break |  | -551.031 |  |  | 0.0 | 0.0 | 8.5 | 0.0
Table 6: CH$\alpha$S optical design parameters  Parameter | Value | Unit | Comments
---|---|---|---
Design Requirements |  |  |
Operational wavelength range | 400 - 900 | nm | broad range of optical emission lines
Operational Bandpass | 3 | nm | avoid overlapping spectra
Optical Overview |  |  |
Telescope aperture ratio | $f/7.5$ |  | MDM Observatory Hiltner 2.4m
Offner relay aperture ratio | $f/7.5$ |  |
Microlens array aperture ratio | $f/2.5$ |  |
Collimator aperture ratio | $f/2.2$ |  |  $14"$ Rowe-Ackerman Schmidt Astrograph (RASA 14)
Camera aperture ratio | $f/2.1$ |  |
Collimator focal length | 790 | mm |
Camera focal length | 590 | mm |
Microlens Array |  |  |
Array dimensions | $68\times 68$ | mm |
Clear aperture | $66\times 66$ | mm |
Material | Fused Silica |  | refractive index n = 1.4563 @ 0.6583 nm
Thickness | 2.0 | mm |
Aperture ratio | $f/2.5$ |  | w/ black chromium mask
Lenslet pitch [ML250, ML125] | [0.25, 0.125] | mm |
Lenslet clear aperture [nominal, high-res] | $[0.225,0.113]\pm 0.001$ | mm |
Lenslet radius of curvature [ML250, ML125] | $[0.257,0.129]\pm 3\%$ | mm -1 |
Effective focal length [ML250, ML125] | [0.563, 0.283] | mm | thick lens approx. at $\lambda=0.6583$ nm
VPH Grating |  |  |
Fringe frequency ($\nu$) | 1200 | lines/mm |
Fringe slant/Blaze ($\phi$) | 4.5 | deg | positive or negative
Dimensions | $[340\times 290\times 20]$ | mm | fused silica substrate
Bulk index (n2) | 1.35 |  |
Index modulation amplitude ($\Delta n$) | 0.11 |  |
Effective thickness (d) | 3 |  $\mu$m |
Table 7: CH$\alpha$S performance and sensitivity calculations

Parameter | Value | Unit | Comments
---|---|---|---
| Design | CBE ${}^{\dagger}$${}^{\dagger}$footnotemark:  |  |
Telescope Performance |  |  |  |
Primary clear aperture diameter | 2.32 |  | m | MDM Observatory Hilter 2.4-m
Secondary obscuration diameter | 0.745 |  | m |
Geometric area | 37914 |  | cm2 |
Plate scale | 11.33 |  | $\rm{}^{\prime\prime}/mm$ | Focal Plane of 2.4-m
Imaging Performance |  |  |  |
Field of view | 10 x 10 |  | arcmin |
RMS spot radius | $<10$ |  |  $\mu$m | Monochromatic point source at spectrograph input
Pupil image diameter | [60 (4), 30 (2)] | [57(4), 30 (2)] |  $\mu$m (pixels) | [ML250, ML125]
Spatial Resolution | [$2.83$, $1.42$] | [$2.77$, $1.42$] | arcsec | [ML250, ML125]
Image Plate Scale [x, y] | [0.418, 0.455] | [0.421, 0.457] | $\rm arcsec\ pix^{-1}$ | Commissioning Measurement
Spectroscopic Performance |  |  |  |
Spectral resolution $\Delta\lambda$ | [0.69, 0.34] | [0.67, 0.377] | Å |  $\lambda 656.3$ nm [nominal, high-resolution]
Velocity resolution | 30 (15) | 30(17) | km $\rm s^{-1}$ | [ML250, ML125]
Dispersion | 0.4 | 0.373 | $\rm\AA/pix$ |
Number spectral elements (NR) | 35 |  |  | per bandwidth, overlapping
Number spatial elements (NΩ) | $[5\times 10^{4},2.3\times 10^{5}]$ |  |  | [ML250, ML125]
Detector Performance |  |  |  | MDM Observatory B4K Detector
Detector pixel size [angular] | 15 [0.17] |  |  $\mu$m [arcsec] | unbinned
Detector read noise | 5 | 4.4 | $\rm e^{-}$ |
Detector dark current | 0.01 | 0.007 | $\rm e^{-}\ pix^{-1}\ sec^{-1}$ | At a temperature of -124 ∘C
Detector binning | 2 $\times$ 2 |  | pixels |
Predicted Throughput/Efficiency |  |  |  |
Telescope | 0.77 |  |  | R = 0.88 per surface, aged
Filter | 0.97 |  |  | 3nm Astrodon NII Filter
Relay | 0.86 |  |  |  $R>97\%\times 5$ optics
Mircolens | 0.95 |  |  | R $\approx 4\%$ convex fused-silica, R $\leq 1\%$ planar AR-coated
Filling Factor | 0.75 |  |  |
Collimator/Camera | 0.9/0.9 |  |  |
Grating | 0.9 |  |  | H$\alpha$ Figure 3
Detector QE | 0.85 |  |  | MDM B4K
Spectrograph Vignetting | 0.50 |  |  |
System throughput | 0.19 | 0.08 - 0.15 |  | predicted without atmosphere
H$\alpha$ Sensitivity Calculation |  |  |  |
Sky background intensity | 2 ($1.59\times 10^{5}$) |  | R $\rm\AA^{-1}$ (ph s-1 cm2 sr-1, $\rm\AA^{-1}$) | Leinert et al. (1998)
Exposure time per frame | 625 |  | seconds | stacked to reach full integration time
Sky background noise | 50 |  | e- per lenslet per frame | nominal 250 $\mu$m lenslets w/o overlap
Dark current noise | 100 |  | e- per lenslet per frame | nominal 250 $\mu$m lenslets
Read noise | 100 |  | e- per lenslet per frame | nominal 250 $\mu$m lenslets, 2 pixel binning
10$\sigma$ Detection Limit |  |  | See Figure 14 |
1 hour integration | 109 (8674) |  | mR (ph s-1 cm-2 sr-1) | 30 arcsec angular scale, 900s exposures
10 hour integration | 34 (2706) |  | mR (ph s-1 cm-2 sr-1) | 30 arcsec angular scale, 900s exposures
100 hour integration | 11 (875) |  | mR (ph s-1 cm-2 sr-1) | 30 arcsec angular scale, 900s exposures

††footnotetext: The current best estimate (CBE) is the same as the design goal
unless independently measured or modeled

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