Toward Space-like Photometric Precision from the Ground with Beam-shaping Diffusers

Transparency variations include shifting cloud cover. The exact variations will depend on the weather and the observing site, and as such, this source of noise is particularly difficult to estimate for any given observation. This is generally minimized through differential photometry, where the core assumption is that the transparency variations affect the target and reference stars equally. Transparency variations are further minimized by performing observations from a good observing site at high elevation. Efforts have been made to estimate the impact of this effect for some observing sites (e.g., Mann et al. 2011) using different atmospheric models, demonstrating that the median noise due to transparency variations is typically smaller than Poisson and scintillation noise for photometric nights (Mann et al. 2011). For our observations, we assume that for a clear photometric night at a good observing site, this error source is typically much smaller than the expected Poisson and scintillation noise.

A related issue to transparency variations is variable molecular absorption. Commonly used broadband filters, such as the SDSS $u^{\prime} g^{\prime} r^{\prime} i^{\prime} z^{\prime}$ (Fukugita et al. 1996) and UBVRI Johnson-Cousins filters (Bessell 1990), each operate over a wide bandpass and include a number of molecular absorption lines (with water, oxygen, and ozone being the primary absorbers in the optical). For water, the depth of these lines is strongly dependent on the water column at the time of observation, which is dependent on the exact weather conditions and the airmass of the target being observed. This effect is minimized by observing in a bandpass not contaminated by such lines.

Differential extinction is of two types: first order and second order. First-order differential extinction is caused by the variation in the airmass difference of the target and reference star throughout the observation, resulting in a relative brightness change. Mann et al. (2011) estimated that the magnitude of this effect at Maunakea can be on the order of $\sim {10}^{-4}\mbox{--}{10}^{-3}$, depending on the observing conditions and passband being used. However, being a systematic trend correlated with airmass, Mann et al. (2011) mention that this effect can generally be detrended out at high precision if the extinction variation is minimal. Second-order differential extinction is caused by the target and the reference star not being of the same spectral type. Stars with different spectral types will vary differently with extinction throughout the observation. This color effect is smaller in redder passbands, with stars of later spectral type and narrower bandwidth filters, and can further be minimized by a judicious choice of reference stars of the same or similar spectral types.

Both molecular absorption and extinction can be minimized by observing in a red-optical bandpass filter with little to no molecular absorption lines (Mann et al. 2011). We explored the parameter space of commercially available filters and converged on an off-the-shelf filter from Semrock (part number: 857/30), operating in a red passband between 842 and 872 nm.¹⁸ Figure 1 shows the transmission curve of this filter, along with the typical molecular absorption bands calculated by TERRASPEC (Bender et al. 2012) around this region. As this filter is centered at the red end of the optical spectrum, the recorded photometric signal will be less sensitive to variable Rayleigh scattering.

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Thus, with an informed choice of bandpass filter in the red-optical, minimally contaminated by molecular absorption lines, we assume scintillation and photon noise to be the dominant error sources.

For bright nearby stars, the photometric precision is often not limited by photon noise or by background sky counts, but rather by intensity fluctuations—or scintillation—produced by Earth's atmosphere (Osborn et al. 2015). Scintillation is caused by the spatial intensity fluctuations crossing the pupil boundary, and the timescale is determined by the wind speed of the turbulent layer (Young 1967; Dravins et al. 1998; Osborn et al. 2011).

The expected scintillation noise for a given star is described by Young (1967) and Dravins et al. (1998) in units of relative flux, with the following approximation:

$$\begin{eqnarray}&&{\sigma }_{s}=0.09{D}^{-\tfrac{2}{3}}{\chi }^{1.75}{(2{t}_{\mathrm{int}})}^{-\tfrac{1}{2}}{e}^{\tfrac{-h}{{h}_{0}}},\end{eqnarray} \tag{ 5 }$$

where D is the diameter of the telescope in centimeters, χ is the airmass of the observation, t_int is the exposure time in seconds, h is the altitude of the telescope in meters, and ${h}_{0}\simeq 8000\,{\rm{m}}$ is the atmospheric scale height. The constant 0.09 factor in front has units of ${\mathrm{cm}}^{2/3}\,{{\rm{s}}}^{1/2}$, giving the scintillation error in units of relative flux. This equation is approximate and highly reliant on the site and the strength and direction of winds in the upper atmosphere, and the exponent above the airmass term can range from 1.5 to 2.0, depending on the wind direction (Southworth et al. 2009; Osborn et al. 2011). However, for exposures longer than 1 s (long exposure regime for scintillation), the wind profile tends to average out (Kornilov 2012; Osborn et al. 2015). Additionally, it has been suggested by Osborn et al. (2015) that the median value of scintillation is a factor of 1.5 higher than suggested by Equation 5. In the case of differential photometry, the strength of scintillation depends on the number of uncorrelated reference stars n_E (Kornilov 2012). The degree of correlation depends on the angular separations of stars from each other (Kornilov 2012), where 20'' is generally the radius within which they are correlated. Combining these two terms and assuming our target and reference stars are uncorrelated, we have the following equation for the scintillation for differential photometry,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{scint}}=1.5{\sigma }_{s}\sqrt{1+1/{n}_{E}},\end{eqnarray} \tag{ 6 }$$

assuming n_E uncorrelated reference stars. This illustrates the advantages of using multiple reference stars for precision photometry. The total error including scintillation is then

$$\begin{eqnarray}&&{\sigma }_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{rel}\,\mathrm{flux}}^{2}+{\sigma }_{\mathrm{scint}}^{2}},\end{eqnarray} \tag{ 7 }$$

assuming that the other errors, e.g., from transparency variations and differential extinction, are minimal.

As the photon and scintillation errors tend to be the largest sources of noise in ground-based photometry, it is instructive to look at the ratio of the two, ${\sigma }_{\mathrm{scint}}/{\sigma }_{\mathrm{phot}}$, to see when each dominates. To do this, we adapt the calculation and methodology similar to those described by Osborn et al. (2015), showing the dependence of this ratio on the target star magnitude and telescope diameter plane (Figure 2). The pure photon noise in normalized relative flux units, ${\sigma }_{\mathrm{phot}}=1/N$, is calculated using Equation (1) (see also Equation (4)), assuming no sky background and a perfect detector (no read or dark noise) on a telescope with 100% throughput, and the scintillation error is calculated using Equation (6), assuming an airmass of 1.0, an altitude of 2700 m, and one reference star. For other observational parameters, the results must be scaled accordingly. The solid black line in Figure 2 shows where the scintillation and shot noise errors are equal. Therefore, stars below this curve (${\sigma }_{\mathrm{scint}}/{\sigma }_{\mathrm{phot}}\lt 1$) are scintillation limited, and stars above this curve (${\sigma }_{\mathrm{scint}}/{\sigma }_{\mathrm{phot}}\gt 1$) are photon limited. The dotted curve shows where the scintillation error is an order of magnitude larger than the photon noise (${\sigma }_{\mathrm{scint}}/{\sigma }_{\mathrm{phot}}=10$). The dependence of this ratio with diameter is ~${D}^{1/3}$, so the ratio will increase modestly with telescope aperture, but we stress that both error terms, scintillation and photon noise, decrease with telescope aperture as $\sim {D}^{-2/3}$ and ∼${D}^{-1}$, respectively. Due to this modest dependence on telescope diameter, as mentioned by Osborn et al. (2015), we can say that stars brighter than a V-band mag of ∼13 will be scintillation-limited across different telescopes.¹⁹

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Ground-glass diffusers are the simplest of the four diffuser types discussed here. These diffusers are generally produced by sand-blasting glass using various grit sizes to create small randomized surface features. As the surface features are randomized, ground-glass diffusers offer little control over their diffusing characteristics, resulting in limited control on angular divergence. As such, these diffusers are only capable of producing Gaussian intensity profiles. Although commercially available at low cost, these diffusers have low optical transmission efficiencies and diffuse light at large angles. Ground-glass diffusers with opal coatings can be made to achieve close to Lambertian diffusion.

Holographic diffusers rely on the holographic recording of a speckle pattern on the diffuser substrate. This speckle pattern creates pseudo-random semi-periodic surface structures that can be controlled in a statistical sense, offering precise control over the angular distribution of the output light. These types of diffusers are available from many vendors, including Edmund Optics and, notably, Luminit LLC, which offers holographic Light Shaping Diffusers®, which can be made to have very high transmission efficiencies of over 92%.²⁰ However, as the surface structures are only controlled in a statistical sense, this limits the angular diffusion patterns to be either circular or elliptical, and only offers Gaussian-like intensity profiles (Sales et al. 2004). This is suboptimal for high-precision aperture photometry, as the Gaussian profile has broad and extensive wings, spreading out the signal outside the photometric aperture. Furthermore, Gaussian-shaped PSFs have significant slopes across the full PSF except in the very center, making them more subject to guiding errors and changes in seeing.

Diffractive diffusers are based on fabricating a phase mask for a single central wavelength and can be made to have efficiencies between 80% up to 90%–95%.²¹ However, the output pattern is highly sensitive to the wavelength of light used, and as such, these diffusers are largely limited to monochromatic applications in laser systems, but could, however, have possible applications in narrow-band astronomical studies. Due to their high sensitivity with wavelength, we did not study these types of diffusers further for our broadband photometry applications.

Unlike the other types of diffusers that only offer statistical control of the surface features, diffusers that precisely control the shape, size, and location of its surface features in a deterministic manner have the highest degree of control over their output. Such diffusers, capable of molding the output to a desired intensity profile and light distribution pattern, are now commercially available. We worked closely with RPC Photonics in Rochester, NY, to test and design the diffusers used in this paper. These Engineered Diffusers^TMoffer precise beam-control capabilities and utility for many applications.

Engineered Diffusers^TM (Morris & Sales 2006) are composed of individually manipulated unit cells or microlenslets (Figure 4). By precisely controlling the design and manufacturing processes, a surface can be engineered to produce a desired intensity profile and light distribution pattern for a given input beam. To ensure that the diffuser output is stable toward varying beam input, the size, shape, and location of the microlenslets are varied according to a pre-defined probability distribution chosen to implement the desired beam-shaping functions (Sales et al. 2004). Additionally, this microlens distribution can be carefully designed to avoid discontinuities and minimize scattered light and diffraction artifacts from the output. In this manner, Engineered Diffusers^TM retain the best properties of both random and deterministic diffusers.

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The diffusers we used are manufactured by RPC Photonics using a proprietary laser writing process (Figure 4). The process starts by making a master, consisting of a substrate coated with a thick layer of photoresist. A focused UV laser beam is scanned across the surface, and by modulating the intensity of the laser beam, different exposures can be achieved, which breaks down the photoresist. By developing out the exposed areas, a deterministically structured surface with controlled size, shape, and depth is produced (Figure 4(b)). The master can then be used to produce submasters and replicas in different materials, including fused silica, silicon, or on polymer layers on top of glass, allowing these diffusers to span a wide application wavelength range from 193 nm to 10.6 μm (Sales et al. 2004).

Although the details of the exact design and manufacture of these structures are proprietary to RPC Photonics, a few general design rules are noteworthy.

• 1.  
  Designing and fabricating diffusers with larger angles of diffusion is easier than those smaller angles, if a top-hat like shape with a steep fall-off is desired.
• 2.  
  Polymer diffuser patterns bonded to glass substrates (like those we tested at the APO 3.5 m) are less expensive since the laser writing process can be used to make and replicate them. Pure fused silica diffusers (like those we tested at Palomar) are made with a process similar to a reactive ion etch and are more time consuming and expensive.
• 3.  
  Discontinuities in the surface structures can lead to additional scattered light. This can be mitigated in the design process if the requirements are well stated.
• 4.  
  Systematic periodic errors or shifts in the laser writing process can result in the diffuser pattern being grating-like, diffracting light at very low efficiencies into non-zero orders. This was discovered with tests on the ARC 3.5 m (see Section 7.2). This can also be mitigated in future designs (RPC Photonics 2017, private communication).

To verify the operation of diffusers, we tested various diffusers on-sky using the Penn State PlaneWave 24 inch telescope. The telescope was installed in 2014 at Davey Lab Observatory in University Park, PA, at an altitude of 360 m above sea level. The telescope has an Apogee/Andor Aspen CG 42 camera, using a CCD42-10 2048 × 2048 pixel chip from E2V with 13.5 micron pixels. This results in a plate scale of ∼077 pixel^–1and an FOV of 24' × 24'. The telescope is equipped with a dual filter wheel (AFW50-10S dual filter wheel) capable of housing 20 2'' × 2'' filters. The diffuser was placed in the filter wheel closer to the camera, so that the diffuser was 50 mm away from the detector. Using a dual filter wheel allowed us to easily perform diffuser-assisted observations in different filters.

55 Cnc—As an illustration of our observations with Penn State's CDK 24 inch telescope, we discuss our out-of-transit observations of 55 Cnc using a 20 off-the-shelf diffuser on this telescope. 55 Cnc is a nearby bright (V = 5.95 mag) G8 V binary star (its companion 55 Cnc B is an M4.5V dwarf). We used 53 Cnc, a nearby bright M3III star (V = 6.23), ∼45 away from our target, as our main reference star. We observed the system in Johnson I, as both stars are well matched in brightness in that filter, and to minimize the impact from the Moon brightness, which was at ∼88%. The observations were done on 2016 March 27, from 04:00 UT to 06:30 UT. The target was setting, and starting at airmass 1.16, ending at airmass 1.9. The conditions were good, with few to no clouds. The 20 diffuser allowed us to spread the PSF over ∼120''. Without the diffuser, the detector saturated almost instantly.

The exposure time was 120 s, with 11 s of dead time between exposures. This allowed us to collect >10⁸ counts in the target and reference stars. We took 20 dome flats and 20 dark frames, and median-combined them using AstroImageJ (Collins et al. 2017) to create master dark and flat frame images. We used AstroImageJ for the photometric reduction. The aperture setting that gave the smallest residuals was 100, 150, and 200 pixels for the aperture radius, and the inner and outer annuli, respectively. To arrive at the final light curve, we detrended the raw data with airmass, a straight line, and x and y pixel centroid coordinates using the detrend function in AstroImageJ.

In 2016 September, we installed a custom Engineered Diffuser^TM as part of the Astrophysical Research Consortium Telescope Imaging Camera (ARCTIC) on the ARC 3.5 m Telescope at Apache Point Observatory. ARCTIC uses a back-illuminated STA4150LN BI 4096 × 4096 pixel CCD with 15 micron pixels. This gives an unbinned plate scale of 0114 pixel^–1and a field of view of 75 × 75. The detector has four amplifiers and can be read out using one amplifier (lower left) or using all amplifiers simultaneously dividing the frame into four quadrants.

Figure 7(a) shows an image of the final diffuser at ARCTIC, along with the dedicated holder and rotator we designed. The holder is capable of sliding the diffuser in and out of the telescope beam and rotating the diffuser during observations with a pneumatic motor that moves with the diffuser holder assembly. The holder places the diffuser in front of the ARCTIC six-position filter wheel and 200 mm away from the detector plane, creating a top-hat PSF with ∼9'' FWHM. Figure 7(b) shows a schematic diagram of the diffuser location in ARCTIC, where the rays are traced with no diffuser in the beam path. The inset shows a footprint diagram of the beam at the diffuser location: an annulus due to the central obstruction of the telescope. These parameters—the size of the beam and the distance from the detector plane—were the key parameters in the optimization process of the diffuser.

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For some of our on-sky observations with ARCTIC and the diffuser, we used an off-the-shelf narrow-band filter from Semrock operating in a band with minimal water absorption centered at 857 nm (Figure 1). However, the filter was only 2'' × 2'' in size, truncating the FOV from 8' × 8' to about 180'' × 180''. Therefore, our on-sky tests of the diffuser in this filter were limited to bright, closely separated targets. We did not observe any noticeable fringing effects in this filter with ARCTIC, which is commonly seen with redder filters such as SDSS ${z}^{{\prime} }$.

16 Cygni—Using ARCTIC with a diffuser, we observed 16 Cygni, a nearby bright G-dwarf binary star, where 16 Cyg A and B have V magnitudes of 5.95 and 6.2, respectively, and a separation of 395. The observations were performed on 2016 September 20 from 2 am to 6 am UT. The target rose during the observations, starting at airmass 1.07, peaking at 1.05 at the meridian, and ended at 1.30 at the end of the observations. Moon illumination was ∼85%. The night was not photometric, with variable seeing $\gt 2^{\prime\prime}$ FWHM and intermittent clouds. Nevertheless, due to the flux-homogenizing properties of the diffuser, the PSF remained stable throughout the night. To maximize observing efficiency and minimize scintillation noise, the detector was configured to use one amplifier in fast readout mode, reading out only a subframe of 783 × 813 pixels encompassing 16 Cyg A and the reference star 16 Cyg B. This resulted in a short readout time of 5 s. Binning was set to 1 × 1 due to the brightness of the target, resulting in an exposure time of 16 s to reach ∼40,000 peak counts per pixel for 16 Cyg A, and thus a total observing cadence of 21 s and duty cycle of 76%.

Our data reduction consisted of standard aperture photometry performed with the AstroImageJ software suite. After experimenting with different aperture radii and annuli, the best rms precision was achieved with an aperture radius of 100 pixels, and inner and outer annulus radii of 100 and 200 pixels, respectively. For calibration, a set of 25 bias frames were used, along with 25 dome flats. Each set was median-combined using AstroImageJ.

16 Cygni A and B were observed by Kepler in both short- and long-cadence mode, and in this paper, we compare our precision achieved with ARCTIC to that achieved with Kepler. There have been previous studies in the literature using Kepler data of 16 Cygni A and B, using short-cadence data for astroseismology (e.g., Metcalfe et al. 2012; Lund et al. 2014) and gyrochronology (Davies et al. 2015), and long-cadence data to study the link between radial velocity variations and photometric flicker (Bastien et al. 2014). 16 Cygni A and B have Kp magnitudes of 5.864 and 6.095, respectively, and Kepler Input Catalog (KIC) IDs of KIC12069424 and KIC12069449, respectively. Due to their brightness, both stars are well above the saturation limit of Kepler, which is ${K}_{p}\sim 11.5$ (Gilliland et al. 2011). Still, excellent precision levels can be achieved for saturated stars in the Kepler data, due to the conservation of charge in the Kepler CCDs, by summing up the counts in the surrounding pixels, commonly yielding precisions down to 40 ppm for stars ∼7 mag (Gilliland et al. 2010). The data are easily retrievable from MAST.²³ Short-cadence data are available for both stars for Quarters 6–15, with the exception of Quarter 13. Quarter 6 has a known problem with its photometric precision as it did not use an optimized aperture in the photometric retrieval, resulting in data of rather poor quality (Lund et al. 2014). Therefore, we focused our comparison to Kepler, by looking at both the short-cadence and long-cadence data from Kepler from Quarters 7–15, excluding Quarter 13.

Transit of WASP-85 A b—WASP-85 A b was discovered by Brown et al. 2014 and observed by K2 in Campaign 1 (EPIC 201862715). The star is a G5 dwarf, with a V magnitude of 11.2 and Kp magnitude of 10.247. WASP-85 A forms a close visual binary (angular separation of 15) with a cooler and dimmer (V = 11.9) K0 dwarf companion, WASP-85 B. With the diffuser PSF FWHM being ∼9'', this close proximity of the two stars causes the stellar PSFs to completely overlap in the diffuser images. Despite the overlapping PSFs, we are able to recover very high photometric precisions, as we discuss in Section 5.2.2.

Our transit observations of WASP-85 A b were performed on January 31 from 08:30 to 13:00 UT. The exposure time was initially set at 7 s. However, the target was rising, starting from airmass 1.20 during the beginning of the observations, peaking at 1.11, and ending at airmass 1.54, and the exposure times were reduced to 6 s after 15 minutes of observations to keep the exposures at ∼30,000 peak counts, well within the linear regime of the detector. This change in exposure time did not result in a visible change in photometric precision. Moon brightness was ∼12%. The filter used was SDSS ${r}^{{\prime} }$. The binning mode was 4 × 4, with the detector read out in quad amplifier and fast readout mode, resulting in a readout time of 2.5 s. We assume a total cadence of 8.5 s, as ∼95% of the images were taken at this cadence, resulting in an observing efficiency of 80% for these observations.

The data reduction was performed using two independent photometry pipelines. First, we used a photometry pipeline being developed by B. Morris et al. (2017, in preparation), which implements principal component analysis (PCA) to find an optimal set of aperture radii, comparison stars, and environmental measurements used for detrending. The aperture radius used was 19 pixels, and the radius of the inner and outer radii were 32 and 55 pixels, respectively. The data were independently reduced using AstroImageJ using the same radii and detrending parameters, giving consistent results. A set of three reference stars within the FOV with median fluxes between 0.2 and 1.0 times that of the target were used in the differential photometry. A set of 22 darks and a set of 20 flats were median-combined to create master flat and dark frames. In 4 × 4 binning at ARCTIC, a relatively high fraction of cosmic rays and charge particle events is observed in the science frames. To reduce the effect of these events on our photometry, we ran the data through the astroscrappy package,²⁴ a cosmic-ray rejection package written in Python, based on the Laplacian-edge cosmic-ray rejection algorithm described by van Dokkum (2001). Using the default parameters in astroscrappy resulted in fewer saturated outliers observed in the light curve.

To compare our best-fit planet parameters with the values reported in the literature, we fit the transit using the Markov Chain Monte Carlo (MCMC) approach described in Section 6.

Transit of TRES-3b—TRES-3b is a hot Jupiter ($R\sim 1.3{R}_{\mathrm{Jup}}$) discovered by O'Donovan et al. (2007), and is in a P = 1.306 days orbit around a G4 V dwarf star with a V magnitude of 12.4. This target has been well-studied in the literature (e.g., Sozzetti et al. 2009; Knutson et al. 2014). The TRES-3 field has a number of similarly bright reference stars close by (within the ARCTIC FOV for our purposes), which is beneficial for high-precision ground-based differential photometry. Choosing to observe this target thus allowed us to further test the limits of the diffuser-assisted photometry technique with ARCTIC, by observing a clear transit signal on a clear night at good airmasses.

Our observations of this target were performed on 2017 March 12 from 08:45 UT and 12:20 UT, where the target rose during the night, from an airmass of 1.90 to 1.04. The Moon was full during these observations (brightness ∼100%), and thus to minimize sky-background noise, we observed the target in SDSS ${i}^{{\prime} }$. We used an exposure time of 30 s with ARCTIC in quad amplifier fast-readout 4 × 4 binning mode, resulting in a readout time of 2.5 s. The total cadence was thus 32.5 s, yielding an observing efficiency of 92%.

Similar to the WASP-85 observations above, the data reduction was performed using AstroImageJ, after cleaning the images of cosmic rays and charged particle events using astroscrappy. We used 13 reference stars with a flux between 4% and 180% of the flux of the target star. After systematically testing a number of different aperture settings in AstroImageJ, the aperture radius that gave the smallest unbinned rms scatter was 19 pixels, and with inner and outer radii of 32 and 50 pixels, respectively. To create the final light curve, we fit the transit guided by the parameters presented in O'Donovan et al. (2007), performing a simultaneous transit fit and detrending using AstroImageJ, using airmass, a straight line, and x-, y-centroid pixel coordinates in AstroImageJ.

Similar to the WASP-85 A b transit, to compare our best-fit TRES-3b planet parameters with the values reported in the literature, we fit our TRES-3b observations with the MCMC approach described in Section 6.

Because an imaging system inherently has a converging beam before the detector, the most straightforward way to incorporate a diffuser is in such a beam. However, some telescope systems have locations with collimated beams, where placing a diffuser would be more optimal. To compare the resulting PSF of a diffuser placed in a converging versus a collimated beam, we modeled off-the-shelf diffusers using Zemax Opticstudio in non-sequential mode. For these simulations, we used Bidirectional Scattering Distribution Function data files available from the RPC Photonics Web site measured for off-the-shelf diffusers illuminated with on-axis input beams.

Figure 8(a) shows an image of the optical model: an imaging system consisting of a point source emitting an F/6.3 beam at 550 nm, a collimating lens (L1), and an identical lens (L2) to reimage the beam on a detector. Using this model, we studied the output at the detector by placing a diffuser in the collimated space before L2, and also in the converging beam after L2.

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Figures 8(b) and (c) show the resulting horizontal cuts on the detector for an off-the-shelf diffuser with opening angles of 025, and 05, respectively. Also shown in the insets are the respective modeled PSFs. For both diffuser opening angles, we see that the PSF shape, size, and speckle structures (high-intensity peaks in the image plane) are similar for a diffuser placed in the collimated and converging beams. However, we observe that the intensity for the diffuser in the converging beam is about 20% higher in both cases. This is due to two reasons. First, in the collimated beam, the diffuser is slightly farther away from the detector, broadening the resulting PSF. Second, with the diffuser in collimated space, the diffuser causes the incident rays on the L2 lens to be slightly diverging rather than collimated, moving the original focus position farther away. Although not specifically shown in Figure 8, this can be minimized by optimizing the focus of L2 after placing the diffuser in the collimated beam. With the diffuser in the collimated beam, an additional precaution will be to ensure that the now-diverging beam does not get clipped at the lens, i.e., that L2 has a large enough clear aperture to accommodate the larger beam footprint.

The diffuser can be rotated during observation to smooth out the speckle pattern observed in diffused PSFs (Figure 9). Being a statistically varied pattern of engineered structures, illuminating different parts of the diffuser will result in slight variations in the resulting output PSF. This PSF variation can be averaged over time by moving the diffuser during an exposure. Therefore, by taking an exposure that is longer than the characteristic time of change in the residual PSF variation, the resulting PSF can be effectively smoothed out.

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In practice, the least design-intensive path to move the diffuser in a beam is to rotate it along an axis parallel to the optical axis. We tested this in the lab: continuously rotating the diffuser at ∼1–2 Hz effectively smoothed the output PSF by allowing the diffuser to complete a few full rotations for exposure times of a few seconds. Figure 9 compares our lab measurements of rotated and non-rotated PSFs for two top-hat off-the-shelf diffusers from RPC Photonics with a $0\buildrel{\circ}\over{.} 25$ and $2\buildrel{\circ}\over{.} 0$ opening angle, respectively. Comparing the two non-rotating diffuser PSFs, we see that the amplitude of the speckles is larger for the $0\buildrel{\circ}\over{.} 25$ diffuser than for the $2\buildrel{\circ}\over{.} 0$ diffuser, or about 40% and 5%–10% of the total intensity, respectively. This results from the fact that it is easier to suppress speckling for larger opening angles.

Although not as good as allowing the diffuser to complete a few revolutions per exposure, we note that excellent PSF smoothing can already be achieved with only ∼180° of rotation during an exposure. Therefore, considering that our shortest exposure times for ARCTIC for high-precision photometry on bright stars are on the order of 1–2 s—with most exposure times for high-precision photometry being longer than 10 s to maintain observing efficiency—the ARCTIC diffuser rotator was designed to produce a smooth rotational speed of at least 2–3 Hz.

Small residual ripples with amplitudes of ∼2%–10% of the total intensity are seen in the rotated PSFs in Figure 9. These ripples are concentric around the diffuser rotational axis. We see that for the $2\buildrel{\circ}\over{.} 0$ diffuser exposure (Figure 9) the rotational axis was more closely centered than in the $0\buildrel{\circ}\over{.} 25$ diffuser exposure. However, as shown in the horizontal cuts in Figure 9(c), these ripples are small in comparison to the spikes before rotating.

We note that the diffusers in Figure 9 are marketed as general top-hat diffusers with a specific opening angle. As such, they are advertised to provide a top-hat PSF over a broad parameter space and were not specifically optimized for this test setup, i.e., to have a flat speckle pattern and steep wings. To find a diffuser more suited to our specific needs, we worked with RPC Photonics to design a custom top-hat diffuser that had better top-hat characteristics.

Figure 10(a) compares the azimuthally averaged PSF of our custom developed diffuser pattern to those of commercial off-the-shelf diffusers. To facilitate the comparison, Figure 10(a) shows azimuthally averaged PSFs normalized to be equal to unity at the HWHM of each PSF. We also show a model Gaussian with no background noise or background light. These data were taken at a fixed distance of 106 mm away from the detector. From Figure 10(a), we see that our customized diffuser pattern provides PSFs with steeper wings than produced by off-the-shelf diffusers. We see that for our customized diffuser, the PSF plateaus at a relatively constant level at ∼2HWHM, indicating that most of the signal for this diffuser will be within ∼2HWHM. The other diffusers fall off less steeply and do not show evidence of a plateau in the range tested.

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Figure 10(b) shows the encircled energy (EE; curve of growth) for the same set of diffusers tested. Due to the size of the detector used, we could only record up to ∼3HWHM of the PSF of the 20 diffuser. Therefore, to compare the EEs, we set EE(3HWHM) = 1 for all of the PSFs and compare how quickly the EE or signal strength grows as a function of HWHM. We see that the customized diffuser pattern is better than the model Gaussian and both off-the-shelf diffusers with a similar opening angle, in terms of steepness of fall-off and EE. In this comparison, the 20 diffuser is observed to have better EE and a flatter top-hat, but still to have some residual power toward the higher angles (fall off of wings could be steeper). As mentioned above, it is easier to fabricate and design diffuser patterns producing closer to ideal top hats for diffusers with larger opening angles. However, for our astrophysical applications, we needed diffusers closer to or smaller then ∼04 to achieve PSFs with an FWHM ∼10' with ARCTIC to minimize the sky-background noise and source overlap effects.

Figure 11 shows a result from some of our early tests with off-the-shelf diffusers on a small telescope. Specifically, Figure 11 shows 2.5 hr of out-of-transit observations of 55 Cnc using Penn State's CDK 24 telescope with a 20 off-the-shelf diffuser from RPC Photonics. Using the diffuser allowed us to spread the PSF over a large number of pixels, 160 pixel FWHM, corresponding to ∼20,000 pixels illuminated per target. Spreading out the PSF over so many pixels allowed us to increase our exposure time to 120 s and giving a final cadence of 131 s (11 s readout time), allowing us to achieve a duty cycle of 90% and gather ∼10⁸ electrons in the target and reference star apertures, respectively. The PSF remained stable throughout the observations. The resulting unbinned precision after detrending with airmass and the x and y pixel positions was 1124 ppm (see Figure 11).

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Figure 11 shows how the 55 Cnc observations bin down with increasing bin sizes (black curve). We calculate the error bars on our photometric precision, using the code described in Cubillos et al. (2017), assuming that for the highest binning sizes, the rms scatter as a function of bin size follows an inverse-gamma distribution. The data bin down in a Gaussian-like manner, indicative of minimal residual systematic effects. In 30 minute bins, the data bin down shows a scatter of ${246}_{-81}^{+176}$ ppm. This precision is similar to the precision needed to successfully observe the transit of 55 Cnc e (McArthur et al. 2004; Dawson & Fabrycky 2010)—having a transit depth of 380 ppm (Winn et al. 2011)—in a single night of observations. Although we note that observing only one transit at this precision would result in a marginal detection of the transit, the transit could be more precisely be characterized by co-adding a few transit observations to further increase the precision. This demonstrates that the addition of such diffusers could significantly improve detection thresholds of transiting planets orbiting bright stars.

Furthermore, Figure 11 compares the relative photometric flux errors, ${\sigma }_{\mathrm{rel}\mathrm{flux}}$ (green curve), to the expected total photometric noise (including scintillation noise), ${\sigma }_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{rel}\ \mathrm{flux}}^{2}+{\sigma }_{\mathrm{scint}}^{2}}$ (blue curve), as calculated using Equations (2), and (7), respectively. For the airmass term in Equation (7), we assumed a fixed airmass equal to the mean airmass of the observations, amounting to χ ∼ 1.4, which matches well with the observed scatter. From Figure 11, we see that the total noise is much larger than the photometric noise by an order of magnitude (${\sigma }_{\mathrm{tot}}\gg {\sigma }_{\mathrm{rel}\mathrm{flux}}$), where the total error is dominated by the scintillation noise. This is expected for a 24 inch (0.6 m) telescope observing such a bright star from Figure 2.

Photometric observations of 16 Cygni—To demonstrate the photometric precision capabilities of diffusers on sky, we show 4 hr of differential photometry of 16 Cyg A, taken during our engineering run in 2016 September (see Figures 12–14). The unbinned raw photometry and unbinned differential photometry (without detrending) are shown in the top and middle panels in Figure 12, respectively. The unbinned undetrended precision is 776 ppm for a 21 s cadence. From the top panel, we see that clouds appeared ∼1 hr after the observations started (JD ∼ 0.63) and toward the end, but those transparency changes are effectively canceled out in the differential photometry in the middle panel. However, we do observe a slow downward linear trend in the differential photometry, along with a bump at around 0.70 JD with a clear correlation with the x centroid coordinate of the target star (bottom panel).

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The diffused PSF remained stable throughout the night. Figure 13 shows an image taken at random from this data set. Speckles are still seen in the diffused PSF, but they are smoothed to a certain extent by seeing variations and clouds passing by. An animation of the observations is available. During these observations, the rotator did not work reliably, as it was difficult to rotate the diffuser bearing due to friction. This caused the diffuser to sometimes rotate and sometimes get stuck. Therefore, although the rotator was formally on, the diffuser was only sporadically rotated during the observations presented. This is seen in the animation: the PSF speckle pattern is sometimes smoothed out. Regardless, the overall size and shape of the diffused PSF is not affected.

Figure 14(a) shows the final detrended photometry of 16 Cygni, detrended with a line, airmass, and the x and y centroid coordinates. The unbinned photometry (cadence: 21 s) shows a precision of 494 ppm, while in 30 minute bins, the rms precision bins down to ${62}_{-16}^{+26}$ ppm. The photometric error bars shown in Figure 14(a) are the total errors including scintillation, as calculated from Equation (7). The error bars increase toward the end due to the increasing airmass. Also shown is the calculated photometric noise error (including photon noise, dark, read, and sky-background noise), calculated using AstroImageJ from Equation (2) and a representative scintillation error calculated from Equation (6). The values for these errors in the unbinned photometry are ∼100 ppm and ∼500 ppm, respectively, indicating that these observations are scintillation limited.

To study the effect of residual systematics, we plot the rms scatter of our 16 Cyg A observations as a function of increasing bin size (Figure 14(b)). Also overplotted are the relative photometric flux errors (green curve) as calculated with AstroImageJ (Equation (1)), including photon, dark, read, and sky-background noise, and the expected total photometric noise (including scintillation noise), ${\sigma }_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{rel}\ \mathrm{flux}}^{2}+{\sigma }_{\mathrm{scint}}^{2}}$ (blue curve), as calculated using Equations (5) and (7). For the airmass term in Equation (5), we assumed a fixed airmass equal to the mean airmass of the observations, amounting to $\chi =1.10$, which matches well with the observed scatter. We see that our data (black curve) largely bin down as white noise (blue curve).

In Figure 14(b), we also compare our precision to Kepler. We ran the 6.5 hr combined differential photometric precision (CDPP) metric, as defined in Gilliland et al. (2011) for each of the individual quarters for 16 Cygni A in both long- and short-cadence mode, that were available from MAST. The resulting 6.5 hr CDPP precision is consistently at the 7–9 ppm level across different quarters, giving a 6.5 hr CDPP precision of >22 ppm, except for Quarter 6, which was known to have poor data quality due to the use of a non-optimized photometric aperture (Lund et al. 2014). Scaling this result to 30 minutes (i.e., ${\sigma }_{\mathrm{CDPP},30\mathrm{minutes}}={\sigma }_{\mathrm{CDPP},6.5\mathrm{hr}}\sqrt{6.5\,\mathrm{hr}/0.5\,\mathrm{hr}}$) gives a precision ranging from 26 to 30 ppm in 30 minute bins across the different quarters, with a median precision of 27.4 ppm. Calculating the standard deviation of the long-cadence data (∼30 minute cadence), after throwing out 5σ outliers and running the same two-day Savitsky–Golay high-pass filter as performed in Gilliland et al. (2010) and in the Everest 2.0 pipeline (Luger et al. 2016, 2017), we get a precision of 28–44 ppm with a median value of 31.6 ppm, in 30 minute bins. This way of estimating the precision might be biased toward outliers, but it gives a precision estimate that is roughly consistent with the scaled CDPP precision. We choose to use the better of the two, or 27.4 ppm, as our Kepler precision comparison of 16 Cyg A, and we plot this number in Figure 14(b). Additionally, in Figure 14(b), to graphically compare an equal length segment of Kepler photometry to our 4 hr ground-based photometry, we plot a representative 4 hr segment of the Kepler short-cadence data of 16 Cyg A. We say that this 4 hr segment is "representative" of Kepler photometry on this star, as in 30 minute bins the precision of this 4 hr segment is similar to the 27.4 ppm 30 minute CDPP precision value discussed above. Additionally, to perform a head-to-head comparison with our ground-based differential photometry to Kepler's differential photometry of 16 Cyg A, we calculated the same 30 minute CDPP metric for Quarter 7 on the differential16 Cyg A short-cadence Kepler light curve, using 16 Cyg B as a reference star. In doing this, the resulting 30 minute CDPP metric for the 16 Cyg Quarter 7 differential light curve degraded to 36.3 ppm, effectively adding a factor of $\sqrt{2}$ to the photometric noise. By this metric, our ground-based precision of 16 Cyg A of ${62}_{-16}^{+26}$ ppm in 30 minute bins is a factor of $\lt 2$ from Kepler's differential precision (37.4 ppm), but a factor of ∼2 in the non-differential photometry case (27.4 ppm). We show the exact 4 hr segments of Kepler photometry of 16 Cyg A and B used in Figure 18(a) in Appendix A.

Transit of WASP-85 A b—Figure 15(a) shows the transit of WASP-85 A b as observed with ARCTIC, along with our best-fit transit model after detrending the light curve with the x and y pixel coordinates, airmass, and a line. The unbinned residuals are shown offset for clarity, showing a photometric precision of 1771 ppm. Also shown are the residuals binned to 1 minute bins, with a precision of 689 ppm. Additionally shown in the plot are representative scintillation and photon noise error bars.

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We observe a small brightening—or "bump"—in the middle of the transit. We speculate that this is caused by the planet crossing a starspot, as there is no clear correlation with other observational parameters, suggesting the bump is of astrophysical origin. This is a likely scenario, as WASP-85 A has been observed to have repeated starspot crossing events through detailed analysis of K2 data (Močnik et al. 2016). If the bump in Figure 15(a) is indeed a starspot crossing, this marks the first ground-based detection of a starspot crossing event for this target.

Figure 15(b) shows how the rms scatter of the WASP-85 A b residuals change with increasing bin size. Also shown are the expected ${\sigma }_{\mathrm{rel}\mathrm{flux}}$ (green curve) and total errors ${\sigma }_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{rel}\ \mathrm{flux}}^{2}+{\sigma }_{\mathrm{scint}}^{2}}$ (blue curve), as calculated using Equations (2) and (7), respectively. The scintillation noise was calculated using Equation (6) assuming three reference stars and using the mean airmass of the observations of 1.3. Different from our 16 Cygni observations, where ${\sigma }_{\mathrm{rel}\mathrm{flux}}\ll {\sigma }_{\mathrm{scint}}$, these observations are in the regime where ${\sigma }_{\mathrm{rel}\mathrm{flux}}\sim {\sigma }_{\mathrm{scint}}$. We do observe that the unbinned precision (1771 ppm) we obtained is slightly higher than the expected one (1500 ppm). This might be attributed to some degree to the uncertainty in the estimation of the scintillation noise, which has a strong dependence on the variable wind profile in the upper atmosphere (Osborn et al. 2015). An additional factor helping explain this discrepancy is that the three reference stars are all much fainter than the target star (having fluxes of 3%, 10%, and 3% of the target star, respectively), lowering their S/N when correcting for atmospheric transparency fluctuations. Lastly, some of the correlated noise behavior toward higher bin sizes in Figure 15(b) might be explained by astrophysical noise, including the candidate starspot crossing event.

Similar to Figure 14, we compare our achieved precision on WASP-85 A with the 30 minute CDPP precision achieved by K2 in Figure 15(b). The K2 short-cadence photometry of WASP-85 A from Campaign 1 was retrieved and detrended with the Everest 2.0 pipeline (Luger et al. 2016, 2017). The 6.5 hr CDPP precision of K2 as calculated by Everest is 11.7 ppm, and using a similar scaling to the discussion above to calculate the 30 minute CDPP precision (i.e., ${\sigma }_{\mathrm{CDPP},30\mathrm{minutes}}\,={\sigma }_{\mathrm{CDPP},6.5\mathrm{hr}}\sqrt{6.5\,\mathrm{hr}/0.5\,\mathrm{hr}}$) yields a 30 minute CDPP of 42.3 ppm. Similarly to the discussion above for 16 Cyg A in Figure 14(b), in Figure 15(b) we plot the σ-versus-bin-size curve for a representative 4.5 hr segment of K2 short-cadence data, the same length as our observations with ARCTIC. The specific segment is shown in Figure 18(b) in Appendix A. We see that on this star, our 30 minute ground-based precision (${180}_{-41}^{+66}$ ppm) is a factor of ∼4 of the K2 observations on this star (42.3 ppm).

We note that our precision of ${180}_{-41}^{+66}$ ppm in 30 minute bins for our WASP-85 A b transit observations is worse than the precision for our 16 Cyg A observations. This is due to a few reasons. First, 16 Cyg A and B are both brighter than WASP-85 A and the available reference stars in the WASP-85 A field. This allowed us to suppress the photon noise better for the 16 Cyg A observations, while the WASP-85 A b transit observations are in the regime where the photon noise and scintillation noise are similar. Additionally, 16 Cyg A and B are spectrally well matched, reducing the effects from secondary extinction. Second, for the 16 Cyg A observations, we used a filter (Semrock 857/30) with little-to-no water absorption in the red-optical, minimizing systematics due to molecular absorption and extinction. Furthermore, astrophysical systematics for WASP-85 A, in particular from our candidate starspot crossing event, further add to the systematic noise floor for those observations. Still, even though the precision level achieved for our WASP-85 A transit observations is worse than for our 16 Cygni observations, the precision of ${180}_{-41}^{+66}$ ppm in 30 minutes demonstrates that diffusers are enabling precision observations across a wide magnitude range, even for targets with overlapping PSFs.

Transit of TRES-3b—Figure 16(a) shows our in-transit observations of TRES-3b as observed by ARCTIC with a diffuser, along with our best-fit transit model after detrending the light curve with x and y pixel centroids and airmass. We did not detrend with a straight line as we did for the other observations, as it did not yield a significant improvement in the rms scatter.

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Figure 16(b) shows how the rms scatter of the TRES-3b residuals change with increasing bin size, similar to Figures 11(b), 14(b), and 15(b). Furthermore, Figure 16(b) compares the relative photometric flux errors, ${\sigma }_{\mathrm{rel}\mathrm{flux}}$ (green curve), to the expected total photometric noise (including scintillation noise), ${\sigma }_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{rel}\ \mathrm{flux}}^{2}+{\sigma }_{\mathrm{scint}}^{2}}$ (blue curve), as calculated using Equations (2) and (7), respectively. For the scintillation term in Figure 14, we assumed a fixed airmass equal to the mean airmass of the observations ($\chi =1.20$) and 10 reference stars. For the smallest bins, we see that the scatter bins down roughly as white noise. However, at the largest bins, we see a dip in the precision below the expected Gaussian behavior at bin sizes >10 minutes: the data bin down to ${54}_{-14}^{+23}$ ppm in 30 minute bins (black curve), slightly below the expected photometric precision of 101 ppm (blue curve). The drop in Figure 16(b) is somewhat below the expected photometric noise (green curve in Figure 16(b)), but is overall largely within the 1σ or 2σ error bars. Similar excursions below the expected Gaussian behavior at large bin sizes have been reported in the literature (e.g., Blecic et al. 2013; Cubillos et al. 2013), and Cubillos et al. (2017) demonstrate that those fluctuations are not statistically significant after taking into account the increasingly skewed inverse-Gamma distribution of the bins at large bins sizes. Therefore, we argue that a precision much below the expected photometric noise (blue curve) is likely an overestimate of the actual precision, and we thus conservatively say that our achieved precision in 30 minutes for these observations equals the expected photometric precision of 101 ppm. We discuss this further in Section 6, where we compare our precision with other values reported in the literature.