# Point spread function reconstruction for SOUL+LUCI LBT data

Matteo Simioni  INAF - Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, Padova, Italy, I-35122  Carmelo Arcidiacono  INAF -
Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, Italy,
I-35122  Roland Wagner  Industrial Mathematics Institute, Johannes Kepler
University Linz, Altenberger Strasse 69, Linz, Austria, 4040  Andrea Grazian
INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova,
Italy, I-35122  Marco Gullieuszik  INAF - Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, Padova, Italy, I-35122  Elisa Portaluri  INAF -
Osservatorio Astronomico d’Abruzzo, Via Mentore Maggini, Teramo, Italy,
I-64100  Benedetta Vulcani  INAF - Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, Padova, Italy, I-35122  Anita Zanella  INAF -
Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, Italy,
I-35122  Guido Agapito  INAF - Osservatorio Astrofisico di Arcetri, Via E.
Fermi 5, Firenze, Italy, I-50125  Richard Davies  MPE - Max-Planck-Institut
für extraterrestrische Physik Giessenbachstrasse 1, Garching, Germany, D-85748
Tapio Helin  LUT University, P.O.Box 20, Lappeenranta, Finland, FI-53851
Fernando Pedichini  INAF - Osservatorio Astronomico di Roma, Via Frascati 33,
Monte Porzio Catone, Italy, I-00078  Roberto Piazzesi  INAF - Osservatorio
Astronomico di Roma, Via Frascati 33, Monte Porzio Catone, Italy, I-00078
Enrico Pinna  INAF - Osservatorio Astrofisico di Arcetri, Via E. Fermi 5,
Firenze, Italy, I-50125  Ronny Ramlau  Industrial Mathematics Institute,
Johannes Kepler University Linz, Altenberger Strasse 69, Linz, Austria, 4040
RICAM - Johann Radon Institute for Computational and Applied Mathematics,
Altenberger Strasse 69, Linz, Austria, 4040  Fabio Rossi  INAF - Osservatorio
Astrofisico di Arcetri, Via E. Fermi 5, Firenze, Italy, I-50125  Aleksi Salo
LUT University, P.O.Box 20, Lappeenranta, Finland, FI-53851

###### Abstract

This paper presents the status of an ongoing project aimed at developing a PSF
reconstruction software for adaptive optics (AO) observations. In particular,
we test for the first time the implementation of pyramid wave-front sensor
data on our algorithms. As a first step in assessing its reliability, we
applied the software to bright, on-axis, point-like sources using two
independent sets of observations, acquired with the single-conjugated AO
upgrade for the Large Binocular Telescope.

Using only telemetry data, we reconstructed the PSF by carefully calibrating
the instrument response. The accuracy of the results has been first evaluated
using the classical metric: specifically, the reconstructed PSFs differ from
the observed ones by less than $2\%$ in Strehl ratio and $4.5\%$ in full-width
at half maximum. Moreover, the recovered encircled energy associated with the
PSF core is accurate at $4\%$ level in the worst case. The accuracy of the
reconstructed PSFs has then been evaluated by considering an idealized
scientific test-case consisting in the measurements of the morphological
parameters of a compact galaxy.

In the future, our project will include the analysis of anisoplanatism, low
SNR regimes, and the application to multi-conjugated AO observations.

###### keywords:

adaptive optics; astronomy; infrared imaging; optical transfer functions;
point spread functions

*Matteo Simioni<EMAIL_ADDRESS>
##  1 Introduction

Next generation workhorse facilities for 8-10m class ground-based telescopes
[e.g., ERIS (Enhanced Resolution Imager and Spectrograph) and MAVIS (MCAO-
Assisted Visible Imager and Spectrograph) at the Very Large Telescope; SHARK-
visible (System for coronagraphy with High order Adaptive optics from R to K
band) and SHARK-near-infrared (NIR) at the Large Binocular Telescope (LBT);
KAPA (Keck All-sky Precision Adaptive-optics) and KPF (Keck Planet Finder) at
the W. M. Keck Observatory] will improve Adaptive Optics (AO) assisted imaging
and spectroscopy, both in the visible and at NIR wavelengths. The great
advances obtained by these AO systems will pave the road for extremely large
telescopes (e.g., the Giant Magellan Telescope, the European Southern
Observatory’s Extremely Large Telescope (ELT) and the Thirty Meter Telescope)
in the future to obtain sharper and deeper images than the ones obtained with
big space telescopes (e.g., Hubble Space Telescope and James Webb Space
Telescope), exploiting all their ground-breaking imaging capabilities.

In many scientific applications exploiting diffraction limit AO
capabilities[1, 2, 3, 4, 5, 6] , a serious challenge experienced during data
analysis is the precise knowledge of the two-dimensional point spread function
(PSF): approaching the diffraction limit of ground-based cornerstone
facilities is not an easy task. A severe drawback in this sense is that the
PSF of an AO system is a very complex entity, which has a high number of
degrees of freedom involved, varies spatially in the field of view, is not
constant in time, and behaves differently at each wavelength. Accurate
knowledge of the PSF is mandatory to perform high-quality photometric,
spectroscopic, and astrometric analysis of astronomical AO data-sets. For
instance, a detailed knowledge of the PSF is an unavoidable ingredient to
carry out transformational science cases e.g. resolved stellar clusters beyond
the Milky Way, stellar populations beyond the Local Group up to the Virgo
cluster, galaxy size evolution, stellar mass growth across cosmic time,
substructures in intermediate-$z$ galaxies, detailed studies of high-redshift
galaxies, and the formation of the smallest galaxies and proto-globular
clusters at high $z$. In addition, a precise reconstruction of the PSF for
astronomical images is fundamental e.g. for: i) quality control of scientific
frames (especially blank extra-galactic fields); ii) flux calibration of
spectroscopic data (since the PSF depends on the wavelength); iii) unbiased
photometry in crowded fields; iv) image restoration (e.g. Richardson-Lucy
deconvolution); v) forward modelling. Currently there is a growing interest in
the development of tools to infer the observed PSF from AO systems. The three
main approaches of PSF estimation are 1) based on focal-plane data (e.g. PSF
classification, PSF fitting/interpolation, PSF marginal estimation and
deconvolution, empirical PSF derivation); 2) based on pupil-plane data [e.g.
PSF simulation, PSF reconstruction (PSF-R)]; 3) hybrid techniques that combine
the use of point sources in the observed scientific (co-added) images, models
of atmospheric turbulence, and instrumental aberrations, hence optimizing the
reconstruction (e.g. adaptive PSF-R, myopic deconvolution, calibrated PSF-R).
All these methods have been extensively described in Ref. 7. Focal-plane based
methods by default include the non common path aberrations (NCPA) components,
which means that imperfections in the optics of the instruments are
automatically included in the reconstruction. But they are also sensitive to
the image noise, and they can be considered representative of a specific
positions in the field and/or for a particular wavelength. Moreover, they
require the presence of at least one non-saturated point-like source in the
imaged field of view. Pupil-plane based methods, instead, refer to techniques
aimed at deriving the post-AO PSF only from telemetry control loop data, e.g.
time series of wavefront sensor (WFS) measurements or applied deformable
mirror commands and associated calibration, without any recourse to focal
plane data. NCPA or segment differential piston should be taken into account
separately, following dedicated independent calibrations. Testing the result
of pupil-plane methods can only be carried out via comparison of PSFs in the
science data.

PSF-R is a versatile and flexible approach, that can also be the initial step
for hybrid or adaptive methods. The PSF-R technique is the only viable
possibility when no other solutions are possible, due to several reasons: e.g.
there are no suitable point sources in the science frames, the target is faint
and complex, as in dense and crowded fields (i.e. the core of globular
clusters, where only the PSF cores are detected for the majority of the
stars), the bright stars are saturated, the integration times are relatively
short, observations have been carried out in poor atmospheric conditions or
faint guide stars have been used. Indeed, for both single conjugated adaptive
optics (SCAO[8]) and multi conjugated adaptive optics (MCAO[9, 10]) observing
modes, the AO reference star can be out of the instrumental field of view.
Moreover, in laser assisted [11] MCAO mode, only faint tip-tilt stars are
usually present. PSF-R is particularly useful in spectroscopic mode, where no
PSF is available in the focal plane, or in non-sidereal tracking mode, where
the stars in the field of view may be blurred along one axis and so cannot be
used as a PSF reference, or where the Solar System targets (used also as AO
reference) can be spatially extended.

The PSF-R approach is particularly convenient for the extra-galactic targets
where the field of view is typically void of bright point sources, necessary
to characterize the PSF of the science data using other methods. As an
example, simulations of stellar densities with TRILEGAL [12] code indicate
that, in a typical extra-galactic field, one can expect on average $\sim 3$
stars per square arcmin in the range Ks=16-25 Vega mag. Considering the
typical field of view of AO assisted instruments (of the order of $1\,{\rm
arcmin}^{2}$), the statistical probability of having at least one available
star is less than 35% to 45%. The PSF-R remains the unique way forward.

In this paper, we present the first results of our ongoing project aimed at
obtaining a PSF-R tool, i.e. the estimation of the PSF using only AO control-
loop telemetry, following the seminal work of Ref. 13. The adopted PSF-R
method is described in Ref. 14 and involves several approximations to
reconstruct the PSF from the residual phase power spectrum density. Such
algorithm was previously tested on simulated data only. Therefore, the present
work is the first application of the method to real telescope data and is
meant to validate the approach. For this reason, we limited the present
analysis to high signal-to-noise ratio (SNR), on-axis observations to reduce
the degree of freedom of the problem and avoid degeneracies in multi-parameter
space (e.g. when taking into account wider AO reference SNR range, off-axis
angles, seeing values). We investigate the performance of the selected
software with archival, SCAO data from the Large Binocular Telescope
(LBT[15]). Specifically, the data consist of two distinct set of observations
of bright, on-axis point sources, preliminary described in Ref. 16. They both
have been taken as part of the Single conjugated adaptive Optics Upgrade for
LBT (SOUL[17]) validation operations, using LBT Utility Camera in the Infrared
(LUCI[18]). Moreover, to quantify and compare the level of accuracy obtained,
we present an idealized scientific case in the area of applicability of the
PSF-R, showing the gain offered by this technique to the analysis of the
photometry and morphology of a high-$z$ galaxy.

The paper is organized as follows: the PSF-R algorithm is described in Section
2; the data are presented in Section 3; in Section 4 the performance of the
PSF-R algorithm is discussed evaluating also the impact on the measure of the
morphological parameters of compact galaxies, and Section 5 is dedicated to
the summary and conclusions. Finally, an interesting relation linking the
encircled energy (EE), the full width at half maximum (FWHM), and Strehl ratio
(SR) of the observed PSF is discussed in Appendix A.

##  2 PSF-R method

In this section, we describe the method used to reconstruct the on-axis PSF
from AO telemetry data, the one described in Ref. 14, which is by itself an
update of the classical algorithm developed by Ref. 13. As detailed in Ref.
14, the algorithm is highly flexible with respect to the WFS, and we performed
minor tuning to adapt it to SOUL+LUCI. Specifically, to our knowledge this is
the first successful implementation of a PSF-R algorithm to a pyramid WFS AO
system. This is especially relevant also in sight of the upcoming next
generation AO instruments that will make wide use of this kind of WFS. The
core idea of this approach is to compute the Optical Transfer Function (OTF),
the Fourier transform of the PSF derived from the pupil mask. This is
advantageous because the OTF can subsequently be factorized into several
independent components (dependent on wavelength and line-of-sight), which can
be either calculated from the wavefront residual phase or using tabulated
values derived from appropriate calibrations.

The OTF can be defined as:

$OTF_{AO+NCPA}=OTF_{AO}\cdot K_{NCPA}$ (1)

where $OTF_{AO}$ is the post-AO OTF describing the effects of the AO corrected
atmospheric turbulence and $K_{NCPA}$ is the non-AO filter describing the
effects due to the imperfection of the optical system.

Following the mathematical analysis of Refs. 13 and 14, $OTF_{AO}$ is
described as:

$OTF_{AO}=OTF_{tel}\cdot\exp(-\frac{1}{2}D)$ (2)

where $D$ is the structure function of the residual incoming phase (i.e. a
time average of the spatial correlation[14]) and thus wavelength dependent,
while $OTF_{tel}$ is the OTF of the telescope computed from the pupil mask.
$D$ is split into two components that are independent as they describe the
part corrected by the AO system, $D_{\|}$ (lower order structure function),
and the part perpendicular to the AO corrected one, $D_{\perp}$ (higher order
structure function). From these two structure functions, the associated
filters $K_{\|}$ and $K_{\perp}$ can be computed as

$K_{x}=\exp(-\frac{1}{2}D_{x})$ (3)

with $x\in\\{\|,\perp\\}$. This gives:

$OTF_{AO}=OTF_{tel}\cdot K_{\|}\cdot K_{\perp}.$ (4)

Therefore, if we are able to measure the $D$ components from telemetry data,
we can go back to the structure function and determine the OTF (and
consequently the PSF) of the system.

To compute $D_{\|}$, one needs the residual phase in the space of the mirror
modes. However, we only have an estimate for it, namely the reconstructed
residual phase computed from the WFS data, which is perturbed by noise and
aliasing effects. Therefore, since the assumptions from Refs. 13 and 14 of
fast frame rate and a least squares reconstructor are fulfilled, we can use
the same decomposition as in Ref. 13 to decompose $D_{\|}$ further as

$D_{\|}=D_{REC}-D_{n}+D_{al}$ (5)

where $D_{REC}$ is the structure function computed from the reconstructed
residual incoming phase $\phi_{REC,t}$, $D_{n}$ is the noise structure
function and $D_{al}$ is the aliasing structure function. It is worth noting
that this decomposition is valid independently of the type of WFS. As detailed
in Ref. 14, the adopted method takes as input the reconstructed residual
phases $\phi_{REC,t}$, which are recovered from the applied updates of the
deformable mirror, to compute $D_{REC}$.

Table 1: Structure functions computed for the PSF-R. A description on how they are obtained is also provided. |  Name |  Description
---|---|---
$D_{\perp}$ |  Wavefront higher order structure function |  Computed using the corresponding seeing $r_{0}$ and pupil mask via a statistical model of the atmospheric turbulence (von Karman, Kolmogorov).
$D_{REC}$ |  Residual structure function |  Computed from the reconstructed residual incoming phase $\phi_{REC,t}$, for time frames $t$ and related to the direction of the NGS at wavelength $\lambda$. The residual incoming phase is in turn obtained directly from the telemetry. For the pyramid WFS, it has been computed as $\phi_{REC,t}=s\,C\,g\,M$ where $s$ is the WFS measurement, the so-called slopes of the pyramid WFS; $C$ is the control matrix; $g$ is the modal control loop gain vector of the pyramid WFS and $M$ is the description of the mirror modes in terms of phase (a product of the instrument calibtrations).
$D_{n}$ |  Noise structure function |  Computed from the noise model. The control matrix, which already incorporate WFS-specific effects, has been used to propagate from WFS level to phase level.
$D_{al}$ |  Aliasing structure function |  A model is used to propagate simulated higher order parts of the phases through the WFS model and then compute the response of the AO control algorithm for the corresponding seeing $r_{0}$, the used AO control algorithm, and pupil mask. To retrieve a statistical average, this procedure is rerun for many simulated phases before all the computed aliasing phases are used in the structure function computation.
$D_{\|}$ |  Wavefront lower order structure function |  Computed from $D_{REC}$, $D_{n}$, and $D_{al}$ following Eq. 5.

Any WFS-specific issues have to be solved already when setting up the AO
system control algorithm. In the case of SOUL the optical gain is measured
through a low-amplitude sinusoidal modulation of a controlled mirror mode.
Thus, the SOUL slopes telemetry already include the correction for the optical
gain. Therefore, the used PSF-R algorithm is flexible enough to account for
specific characteristics of the pyramid WFS without any major modification. We
refer the interested reader to a more detailed description of the various
structure functions provided in Table 1.

The last term needed for the computation of the post-AO $OTF$ ($OTF_{AO}$) is
$OTF_{tel}$, which is computed from the calibrated pupil mask and then
multiplied by the aforementioned filters following Equation (4).

$K_{NCPA}$ in Eq. (1) is computed from the calibrated phase map and depends on
the specific instrument configuration and telescope pointing. It therefore
varies in the field.

The final PSF is then retrieved by transforming the ${OTF}_{AO+NCPA}$ via
inverse Fourier transform into the ${PSF}_{REC}$, which is also integrated in
wavelength over the pass-band of the selected filter.

We note that focusing only on on-axis point source, the anisoplanatic
component can be neglected, thus reducing the degrees of freedom in the PSF-R.
Moreover, considering that during the observations described in this work the
loop was sampled at frequencies $>500$ Hz, and we expect that non-negligible
physical vibrations occur at much slower variations, we can assume that the
SOUL+LUCI system already accounted for all these variations. Therefore we do
not need to account for any additional term related to vibrations, as it is
already present in the used data.

##  3 Data

As the main goal of this work is to apply the selected algorithm for PSF-R to
real AO data, we used SCAO near infrared images and associated synchronous
telemetry acquired with the SOUL system mounted on the LBT and feeding the
LUCI near-infrared camera. This, in turn, is equipped with three optical zoom
configurations delivering the same number of distinct resolutions (see Ref. 19
for a detailed description of the instrument set-up). We use the diffraction
limited mode, called N30, a $2048\times 2048\,{\rm px}$ configuration with
$30"\times 30"$ field of view ($14.95\,{\rm mas/px}$).

Table 2: Log of the SOUL+LUCI@LBT observations | DAYTIME | NIGHTTIME
---|---|---
PROG. ID | 1183150 | 1286078
Target | illuminated fiber | HD$\,873$
RA [hh mm ss] | N.A. | $00\,13\,13.123$
DEC [${}^{\circ}\,{}^{\prime}\,{}^{\prime\prime}$] | N.A. | $+20\,45\,9.479$
R [mag] | $10.0$ | $8.57$
Date [yyyy/mm/dd] | 2019/03/29 | 2019/11/09
Filter | H | FeII
NDIT$\times$DIT [s] | $90\times 1$ | $20\times 0.313$
AIRMASS | N.A. | $1.0248$

We used archive data, consisting of 2 independent datasets. The first set
refers to observations of an artificial source (i.e. an illuminated fiber -
hereafter ”daytime” dataset) with controlled atmospheric condition [20] . The
second dataset consists of observations of a natural point source (i.e. a
bright star - hereafter ”nighttime” dataset). The log of the observations is
presented in Table 2.

###  3.1 Observed PSFs

The LUCI data reduction procedure is identical for both datasets. First, a
bad-pixel map is obtained from the dark frames and a background image is
generated averaging the sky frames. Individual science frames have been
stacked with a sigma-clipped mean to increase the final SNR; we have then
subtracted the background image from the stacked science one. Finally, we
masked the bad pixels and estimated their values by linear interpolation of
the neighboring ones. We corrected the obtained images for the residual
background and cropped them, selecting a region of $\sim 8\,{\rm arcsec^{2}}$
($190{\rm px}\times 190{\rm px}$), centered at the point source. Finally, we
normalized them to their total flux. These high SNR images will be referred in
the following as “observed PSFs”.

###  3.2 Reconstructed PSFs

In the following, we provide additional technical details on the application
of the adopted PSF-R method in the specific case of SOUL+LUCI data. We only
used the telemetry data associated with the scientific observations to
reconstruct the PSFs.

The telemetry data provided by the SOUL team include: AO WFS slopes data
history, control matrix, interaction matrix, gain vector, and the pupil
definition. In particular, we obtained pupil images taken with LUCI in the
”pupil image” mode used for the alignment of the instrument. This imaging mode
unfortunately does not include the additional pupil obstruction exclusive of
the N30 camera. We derived them from the camera mechanical drawing, to obtain
the pupil shape for daytime dataset. For nighttime, additional contributions
have been taken into account in defining the pupil shape. In this
configuration, in fact, LUCI sees also the secondary mirror mounting
(including the LBT swing arm) shadow occurring in the pupil mask. While the
N30 camera system is fixed with the LUCI detector, the swing arm projection
depends on the orientation of the LUCI camera that is continuously updating to
follow the apparent rotation of the sky. The derived pupil shape is used as a
binary mask assuming negligible modulation of the amplitude (optical system
transmission) over the aperture. Using the LUCI images we verify the
pixelscale value which results $14.95\,{\rm mas/px}$ for both the H and FeII
filters.

Figure 1: NCPA mean residuals, after AO correction, for daytime (left) and
nighttime (right). The pixel values are in ${\rm nm}$.

In all the considered observations, NCPA between WFS optical train and the
LUCI camera are compensated by the AO system making the pyramid loop closing
on non-zero reference slopes. This correction, however, leaves residual
depending on the pyramid optical gain [21] and un-calibrated reciprocal
flexures of the WFS with respect to LUCI. These residuals may count up to
$100\,{\rm nm}$ [19] of wavefront error. Lacking proper calibrations, we
measured these residuals by analyzing the observed PSF: this method, applied
on long exposure PSF, has low sensitivity. In the present case we observed
residuals (tip-tilt free) accounting for about $40\,{\rm nm}$ in both daytime
and nighttime. The measured phase maps for NCPA are shown in Figure 1. As a
consequence, the small residual values also indicate that the pyramid WFS gain
measures registered in the telemetry data are accurate. Even if negligible,
these estimated wavefront errors were also applied in the PSF-R algorithm.

###  3.3 Diffraction limited PSFs

We numerically computed the monochromatic diffraction limited PSF thanks to
the knowledge of the pupil shape. This was obtained as the square modulus of
the Fourier transform of the pupil mask, which was properly zero-padded to
obtain the adequate pixelscale. Finally, different sizes of the zero-padding
were used to take into account the whole passband of the selected LUCI
filters. For both datasets, the final diffraction limited PSFs are composed by
$11$ monochromatic ones, calculated at discrete wavelength intervals over each
passband.

###  3.4 Daytime

Figure 2: Daytime PSFs. Left panel: observed PSF. Central panel: reconstructed
PSF. Both are in arbitrary logarithmic color scales. Right panel: normalized
residuals between observed and reconstructed PSF. The residuals are normalized
pixel-per-pixel with the observed PSF. The color scale is linear and the
colorbar is on the bottom. There are no evident residual structures of the two
PSFs. All the images are $\sim 8\,{\rm arcsec^{2}}$ ($190{\rm px}\times
190{\rm px}$).

For this particular dataset, laboratory-like conditions have been used:
optical turbulence disturbance has been generated and corrected [22] using the
Adaptive Secondary [23] of the LBT, with the telescope and AO system
configured for daytime [20] . Specifically, a seeing of $1.2$ arcsec was
simulated with an average wind speed of $15\,{\rm m/s}$ [19] . The simulated
point source has an equivalent R-band magnitude of $10.0\,{\rm mag}$. Three
sets of $30$ $1$s-images have been obtained in H band ($1.5-1.8\,{\rm\mu m}$,
centered at $1.65\,{\rm\mu m}$), achieving a temporal frequency for the AO
correction of $500\,{\rm Hz}$. The resulting saving rate for the associated
telemetry is $500\,{\rm frames/s}$. In all the science frames the point source
is not saturated, within the linear regime of the detector. The resulting SNR
per pixel for the observed PSF goes from $250$ at peak to a mean value of $10$
at a radial distance of $300$ mas ($20{\rm px}$) from PSF centre and reaching
mean SNR per pixel$=1$ around $0.9$ arcsec ($60$px). The observed PSF for
daytime is presented in the left panel of Figure 2. The measured SR - defined
as the ratio between the peak values of a selected PSF and of the diffraction
limited one - is $0.59$ and the FWHM is $2.95\,{\rm px}$. The central panel of
Figure 2 shows the reconstructed PSF. The match between the two PSFs can be
qualitatively inferred from the right panel of the same figure, where the
normalized residuals are presented. No evident residual PSF structures are
present. A quantitative assessment on the performance of the reconstructed PSF
is presented in the following sections.

###  3.5 Nighttime

Figure 3: Nighttime PSFs. Left panel. Observed PSF. Central panel.
Reconstructed PSF. Right panel. Normalized residuals between observed and
reconstructed PSF. Images size, color codes and scales are the same as in
Figure 2.

The nighttime dataset has been obtained in the narrow-band FeII filter,
centered at $1.65\,{\rm\mu m}$, with FWHM $0.02\,{\rm\mu m}$. The use of a
narrow-band filter reduces the impact of atmospheric dispersion on data: LUCI
is not equipped with an atmospheric dispersion corrector and the associated
blur of the PSF is of the order of few pixels, so we properly take into
account the effect while reconstructing the PSF. The dataset consists of on-
axis observation of a bright star (HD$\,873$, R=$8.57\,{\rm mag}$). The star
has been also used as AO reference, thus obtaining the best correction from
the AO system along the direction of the target. In this case, $20$ images of
$0.313\,{\rm s}$ have been considered. The measured seeing was $1.20$ arcsec
and the wind speed $3.4\,{\rm m/s}$. Using a bright target, the frequency of
AO correction could be raised to $1700\,{\rm Hz}$, with a resulting saving
rate for the associated telemetry of $850\,{\rm frames/s}$ (decimated by a
factor 2). The measured SR is $0.58$ and the FWHM $43.95\,{\rm mas}$
($2.93\,{\rm px}$). Also in this case, all the images of the point source are
not saturated and within the linear regime of the detector. The final SNR per
pixel of the observed PSF is $50$ at the peak, decreasing to a mean value of
$10$ at $300$ mas. For this dataset, a mean SNR per pixel$=1$ is reached
around $1.3$ arcsec ($90$px) from PSF centre. The difference between the SNR
radial trend of the two dataset is due to the different bandwidth of the
filters used for each dataset: a broadband one for daytime and a narrowband
one for nighttime. The variation in environmental conditions during the
observations is also a contributing factor. The observed PSF is shown in the
left panel of Figure 3, the reconstructed one in the central panel. In the
reconstructed PSF it can be noted a feeble ring-like structure, with a radius
of $\sim 0.6\,{\rm arcsec}$ ($\sim 40\,{\rm px}$). This approximately
coincides with the expected location of the control radius, which delimits the
region of the PSF affected by AO correction. This ring-like structure leaves a
trace also in the residuals, shown in the right panel of Figure 3, and it is
due to the method used in the adopted PSF-R algorithm to treat corrected
spatial modes.

##  4 Results and Discussion

Table 3: Properties of the observed and reconstructed PSFs for both datasets. See text for details. Parameter | OBS. | REC. | DIFF. $[\%]$
---|---|---|---
DAYTIME |  |  |
SR | 0.59 | 0.58 | 1.7
FWHM[mas] | 44.10 | 43.80 | 0.7
${\rm EE_{core}}$ | 0.43 | 0.43 | 1.9
${\rm R_{50}}$[mas] | 50.23 | 53.97 | 7.4
$\chi^{2}_{RED}$ |  | 4.63 |
Corr. with OBS. |  | 0.996 |
NIGHTTIME |  |  |
SR | 0.58 | 0.59 | 1.7
FWHM[mas] | 43.80 | 45.75 | 4.4
${\rm EE_{core}}$ | 0.43 | 0.45 | 4.0
${\rm R_{50}}$[mas] | 52.92 | 50.83 | 3.9
$\chi^{2}_{RED}$ |  | 3.27 |
Corr. with OBS. |  | 0.995 |

In this section, we detail the performance of the adopted PSF-R algorithm. A
qualitative analysis of Figures 2 and 3 corroborates the good match between
the observed and reconstructed PSF. In particular, the residuals shown on the
right panels suggest that the core of the PSF is well reproduced. More
quantitatively, a collection of the most relevant properties of the observed
and reconstructed PSFs is provided in Table 3. For each dataset and PSF, we
report the measured SR and FWHM. The EE - defined as the amount of PSF energy
that is contained within a fixed circular aperture, centered on the PSF center
- in the core of the PSF, ${\rm EE_{core}}$, is also listed. The radius of the
aperture, in this case, has been fixed to $40.4$ mas ($2.7\,{\rm px}$): the
expected FWHM of the diffraction limited PSF. ${\rm R_{50}}$ is the aperture
(in mas) that contains $50\%$ of the total light.

Table 4: As in Table 3 but for the diffraction limited case and the best fit to the observed PSF with an analytic profile that is the sum of 2 Moffat functions. Parameter |  OBS |  D.L. |  DIFF. [%] |  Moffat |  DIFF. $[\%]$
---|---|---|---|---|---
DAYTIME |  |  |  |  |
SR |  0.59 |  1.00 |  |  0.61 |  3.4
FWHM[mas] |  44.10 |  40.36 |  8.5 |  44.10 |  0.0
${\rm EE_{core}}$ |  0.43 |  0.62 |  44.1 |  0.44 |  2.3
${\rm R_{50}}$[mas] |  50.23 |  30.20 |  39.9 |  48.14 |  4.2
$\chi^{2}_{RED}$ |  |  10.70 |  |  4.97 |
Corr. with OBS. |  |  0.974 |  |  0.997 |
NIGHTTIME |  |  |  |  |
SR |  0.58 |  1.00 |  |  0.60 |  3.4
FWHM[mas] |  43.80 |  40.07 |  8.5 |  43.80 |  0.0
${\rm EE_{core}}$ |  0.43 |  0.62 |  44.1 |  0.44 |  2.3
${\rm R_{50}}$[mas] |  52.92 |  30.50 |  42.4 |  49.78 |  5.9
$\chi^{2}_{RED}$ |  |  8.27 |  |  6.02 |
Corr. with OBS. |  |  0.982 |  |  0.995 |

Finally, to quantify the match between the observed PSF and the reconstructed
one, two parameters are listed in Table 3. The first is $\chi^{2}_{RED}$ which
is the reduced $\chi^{2}$ between observed and reconstructed PSFs. It has been
obtained using GALFIT [24] on the observed PSF with the reconstructed PSF as
input. The center position of a “psf” model has been set as the only free
parameter for the fit and an input rms image has been created directly from
the science data. The second parameter is the correlation level between the
observed PSF and the reconstructed one. It is measured pixel-per-pixel over
the whole PSF image ($8\,{\rm arcsec^{2}}$) by means of Pearson R index. For
comparison, the same parameter list is also provided in Table 4 for the
diffraction limited case. We also include in the table the results associated
to the best fit of the observed PSFs with an analytic profile, the sum of 2
Moffat functions. The choice is motivated by the fact that fitting a single
Moffat function does not provide a good match on both the core and the wings
of the observed PSF simultaneously (Ref. 25 and references therein).

Figure 4: Radial profile (mean over all angles) of the observed PSF (black)
and the reconstructed PSF (red). Left panel is for daytime, while the right
one is for nighttime. The associated standard deviations are indicated with
shaded regions. For comparison, the radial profile of the diffraction limited
PSF (D. L.) is also shown (blue dot-dashed line). The green profile on bottom
shows the normalized deviation, as a function of distance, of the
reconstructed PSF with respect to the observed one. The associated standard
deviation is represented as a shaded area. We computed this quantity as the
quadratic sum of the standard deviations associated to observed and
reconstructed PSFs.

Table 3 shows that the reconstructed PSFs match extremely well the observed
ones, both for daytime and nighttime data. Indeed, the SR is recovered with an
accuracy of $1.7\%$ in both cases. To further discuss the other parameters, we
compare the radial profile of the observed and reconstructed PSFs. They are
reported in Figure 4 for daytime (left panel) and nighttime (right panel). The
radial profile of the diffraction limited PSF is also provided for comparison.
For completeness, the relative differences between observed and reconstructed
PSF are also reported on the insets on top. In both cases, the radial profile
of the reconstructed PSF nicely follows the observed one, particularly in the
central regions. This is confirmed by the measured FWHM of the reconstructed
PSFs that are accurate at the level of $1\%$ and $4\%$ for daytime and
nighttime, respectively. This translates into differences of the order of
$2\,{\rm mas}$ ($0.13\,{\rm px}$) in the worst case. Even the EEcore in the
core is fairly reproduced, with a precision better than $4\%$ in both cases.
Moving outwards to around $50\,{\rm mas}$ ($3.5\,{\rm px}$), the light
distribution of the reconstructed PSF keeps following the observed one with
errors on ${\rm R_{50}}$ of the order of $7\%$ and $4\%$ for daytime and
nighttime respectively ($0.25\,{\rm px}$ in the worst case). For larger
apertures, small deviations (relative difference $<1$) are present in the
radial profiles of the reconstructed PSFs with respect to the observed ones.
In particular, we note the bump in the radial profile of the nighttime
reconstructed PSF, associated to the ring-like structure visible in Figure 3.
This feature, however, does not seem to affect significantly the match between
observed and reconstructed PSF, as demonstrated by the associated low
$\chi^{2}_{RED}$ and high correlation level values. Consistent values are
found also in the daytime case.

For what concerns the comparison with other PSFs, the diffraction limited PSF
does not perform as well as the reconstructed one. All the values reported in
Table 4 indicate larger deviations than what has been obtained with the
reconstructed PSFs. Furthermore, in the case of the two-Moffat fit, ${\rm
EE_{core}}$ and ${\rm R_{50}}$ are well recovered, as expected, but the
$\chi^{2}_{RED}$ and correlation level values indicate that the reconstructed
PSFs work equally well or even slightly better than the analytic profile.

###  4.1 Impact of the PSF-R on an ideal science case

The aim of this work is to obtain a reliable estimate of the differences
between the reconstructed PSF and the observed one (as listed in Table 3), and
to test the applicability of the reconstruction to an ideal science case. In
this section, we aim to define a general method to assess how differences
between observed and reconstructed PSF translate into errors on the scientific
measurements. In fact, the same information should be used to define
constraints on the reconstructed PSFs to meet the requirements for the
precision on selected science measurements. For this reason, we limited as
much as possible the effects of other factors related to specific observation.
In particular, we envisaged a representative case where the reconstruction of
the PSF is critical to reach the scientific goal and take advantage of the
knowledge of the observed (true) PSF to assess quantitatively the potential
differences that can be revealed as a comparison.

As an example, we consider the case of a distant (e.g. $z\sim 1.5$), bright
and compact (e.g. effective radius comparable to the FWHM of the PSF) galaxy.
We investigate what accuracy on the structural parameters is achieved when
modelling the 2D light profile of the galaxy with GALFIT using our
reconstructed PSF model. For simplicity, we consider a galaxy whose light
distribution can be described by a Sérsic profile. We did not simulate
galactic substructures that, albeit more realistic, would complicate the
interpretation of our results. Similarly, we chose to simulate a bright galaxy
to assess the effects of the faint and outer region of the PSF, which would be
hidden with a lower SNR, but are fundamental because these are the most
noticeable differences between the observed and the reconstructed PSF. The
parameters of the simulated galaxy are: integrated magnitude ${\rm
m_{Vega}}=17.5\,{\rm mag}$; Sérsic index ${\rm n}=3.9$; effective radius
$R_{e}=94\,{\rm mas}$ ($6\,{\rm px}$, corresponding to $\sim 2\,{\rm FWHM}$);
axis ratio of $0.28$. These values, with the exception of the integrated
magnitude, are typical for a red nugget, an early type galaxy, at redshift
$z\sim 1.5$ [26] .

To produce the mock observations, we make extensive use of SimCADO [27, 28] ,
which we tuned accordingly to simulate SOUL+LUCI. We considered an exposure
time of $54000\,{\rm s}$, subdivided into $180$ frames (DIT) of $300\,{\rm s}$
each. To ensure a statistical significance of the results, we generated $100$
mock observations of the same model galaxy, adding at each time random white
noise.

Figure 5: Left panel. Simulated galaxy. A $3.7\,{\rm arcsec^{2}}$ ($128{\rm
px}\times 128{\rm px})$ window is shown. The nighttime observed PSF is used as
input to generate the simulated observation. Central panel. The best-fit
GALFIT Sérsic model. In this case, the reconstructed PSF is used as the input
PSF for GALFIT. Right panel. Normalized residuals. The color scale is linear,
the colorbar is shown on the bottom. The normalization has been done dividing
the residuals by the simulated observation. It can be noted the excellent
agreement between observation and model, despite the two being obtained using
the observed and reconstructed PSF as input, respectively. The contours in the
left and central panels refer to the same count levels.

The analysis is subdivided in two main parts: the simulation of the
observations and the measurement. The simulated observations have been
generated by convolving a pure Sérsic model with the observed PSF. A zoom-in
of the central $128\times 128\,{\rm px}$ is presented in the left panel of
Figure 5. The measurements have been performed with GALFIT giving different
PSFs as input to the software: i) the observed PSF, for the control measure;
ii) the reconstructed PSF, for the actual measure; iii) the diffraction
limited PSF, for comparison; iv) no PSF at all, again for comparison. The
best-fit GALFIT model (a pure Sérsic profile) associated to the reconstructed
PSF is shown in the central panel of Figure 5. The contours in the central
region of the galaxy mark the same count levels as in the left panel: it can
be noted the good agreement between the simulated galaxy and the model. This
is emphasized by the normalized residuals, presented in the right panel of
Figure 5. No evident residual structures can be detected.

Table 5: Scientific evaluation results. The mean deviation from input values and associated standard deviation are reported for the main morphological parameters. Specifically, they are: the centre position (x and y), the integrated magnitude, effective radius, Sérsic index, axis ratio and position angle of the galaxy. The input galaxy is obtained using the observed PSF, while each column refers to the different PSF used by GALFIT to recover the input parameters. In particular: the control measure has been obtained using the observed PSF also in the fit, while the one labelled ”REC.” has been obtained using the reconstructed PSF. For completeness, we reported also the results obtained using the diffraction limited PSF and those without providing any PSF at all as input to GALFIT (”NONE”). Parameter | OBS. [CONTROL] | REC. | D.L. | NONE
---|---|---|---|---
x[mas] | $0.000\pm 0.164$ | $-0.194\pm 0.179$ | $-0.164\pm 0.164$ | $-0.120\pm 0.284$
y[mas] | $-0.045\pm 0.105$ | $-0.030\pm 0.105$ | $-0.105\pm 0.105$ | $0.000\pm 0.149$
int. mag. [mag] | $-0.002\pm 0.019$ | $-0.009\pm 0.016$ | $0.020\pm 0.034$ | $0.119\pm 0.017$
${\rm R_{e}}$[mas] | $-0.807\pm 3.005$ | $2.497\pm 2.272$ | $41.845\pm 11.168$ | $69.5773\pm 6.055$
n | $-0.105\pm 0.324$ | $-0.841\pm 0.188$ | $0.421\pm 0.486$ | $-1.640\pm 0.087$
b/a | $0.002\pm 0.005$ | $0.042\pm 0.005$ | $0.103\pm 0.006$ | $0.217\pm 0.005$
PA[deg] | $0.095\pm 0.167$ | $-0.621\pm 0.150$ | $-0.896\pm 0.232$ | $-0.737\pm 0.229$
Figure 6: Box plots of the measured deviations from input values for
integrated magnitude, effective radius ${\rm R_{e}}$, Sérsic index ${\rm n}$,
and axis ratio $b/a$ from top to bottom. The control measure has been obtained
using the observed PSF (the same used to produce the simulated observations)
as input for GALFIT. On the contrary, the actual measure (REC.) has been
obtained using the reconstructed PSF as input for GALFIT. For completeness,
the box plots are shown also for the case where the diffraction limited PSF is
given as input to GALFIT and when no PSF at all was provided as input to the
software.

Quantitatively, all the morphological parameters are in good agreement. We
report in Table 5 the mean differences between recovered and input values
along with the associated standard deviations. The table includes the results
also for the control measure, the diffraction limited case, and the case where
no PSF has been provided to GALFIT. The associated box plots are shown in
Figure 6, representing the distribution of the deviations from input values
for the $100$ simulations. The main parameters are shown, namely the
integrated magnitude, the effective radius ${\rm R_{e}}$, the Sérsic index
${\rm n}$, and the axis ratio $b/a$ (from top to bottom). The loss of accuracy
is more noticeable for the case where no PSF at all is provided to GALFIT,
confirming the need of the observed PSF in this particular case.

From Table 5 and Figure 6 it can be noted that the recovered Sérsic index is
somewhat biased towards smaller values. The best-fit model obtained providing
the reconstructed PSF to GALFIT has a $n$ value $2\sigma$ lower than expected.
As a general consideration, although an optimization of the analysis is
possible (but beyond the aim of this study), our two-dimensional fit is
performed successfully and the galaxy can be classified in all the cases as an
early type object (having $2.5<{\rm n}<4$).

To further assess whether the discrepancy between the measured and intrinsic
Sérsic index is due to our PSF reconstruction, we also applied a mask on both
observed and reconstructed PSFs selecting only the pixels inside a given
cutting radius (${\rm R_{CUT}}$). We then performed the same analysis
described above, using the masked observed PSF to simulate the observations
and giving the masked reconstructed one as input to GALFIT to recover the
structural parameters of the galaxy. While gradually increasing ${\rm
R_{CUT}}$, we record the variations in the measured morphological parameters.

Figure 7: Mean deviation from the input value of the measured Sérsic index as
a function of the cutting radius (${\rm R_{CUT}}$) used to mask the external
regions of both oberved and reconstructed PSFs. While the PSFs cores are very
similar, the small differences in the wings of the PSFs translate into a
deviation of the measured Sérsic index from the input value. After an initial
settling to $-0.3$, the deviation resumes a shallow increase for ${\rm
R_{CUT}}\gtrsim 375\,{\rm mas}$.

We find that the only morphological parameter affected is indeed the Sérsic
index. The behaviour of the measured ${\rm n}$ as a function of ${\rm
R_{CUT}}$ is reported in Figure 7. If the central $29\,{\rm px}$ of the PSFs
(${\rm R_{CUT}}=3\,{\rm px}=44.85\,{\rm mas}$) are considered, the differences
between the reconstructed and observed PSFs are negligible. Increasing ${\rm
R_{CUT}}$, the recovered Sérsic index progressively shifts towards lower
values, suggesting that the bias in the measure of this morphological
parameter is due to the non-perfect reconstruction of the wings of the
observed PSF. Specifically, for ${\rm R_{CUT}}>45\,{\rm mas}$, the difference
between measured and input $n$ values steeply increases up to $-0.3$ at ${\rm
R_{CUT}}\lesssim 90\,{\rm mas}$. The recovered $n$ value remains almost
constant up to ${\rm R_{CUT}}\lesssim 0.4\,{\rm arcsec}$, from where a shallow
increase in the deviation from input values is resumed.

##  5 Summary and Conclusions

PSF reconstruction is a fundamental tool to analyze the data from AO
instrumentation. It will support state-of-the-art scientific analysis of
imaging and spectroscopic data. It is worth stressing here the fact that
having a number of bright and isolated point sources in the observed field
allows the PSF characterization directly from the science frames. This,
however, can be tricky, in the case of AO observations where the PSF changes
both with distance from the AO reference and in time.

The PSF-R technique allows to obtain the PSF of an astronomical observation
simply from synchronous WFS telemetry data, without any access to the focal
plane data. This is an unavoidable solution when there is no bright and
isolated star in the instrumental FoV, suitable for deriving the observed PSF.

We have attempted here to carry out a PSF reconstruction of SOUL+LUCI LBT
observations of bright, on-axis point sources in order to test the PSF-R code
we are developing. To the best of our knowledge, this is the first published
work to apply the PSF-R technique to real AO data acquired with a pyramid WFS.
We have also carried out the PSF-R of an artificial source (an illuminated
fiber), with optical turbulence disturbance generated and corrected using the
Adaptive Secondary of LBT: these results have been already discussed in a
preliminary stage by Ref. 16. The controlled atmospheric conditions allow to
check that the tool is able to successfully recover the injected
perturbations.

The PSF-R software we have used is able to reproduce the observed FWHM, EE,
and SR with an accuracy of 2-4%, which is comparable to the level of accuracy
reached by hybrid or calibrated techniques (see e.g. PRIME[29, 30]). Moreover,
the reconstructed PSF provides a better fit to the observed data than the
diffraction limited one. This level of agreement points to a successful
approach of the adopted PSF-R algorithm, which matches a technology readiness
level equal to 7, following the international ranking defined by the ISO
16290/2013 standard.

With the present work, we aim at setting the foundations for forthcoming
analysis and refinements of the adopted PSF-R method. An effort is needed in
order to improve the representation of the wings (at $>0.3\,{\rm
arcsec}=20\,{\rm px}$) of the reconstructed PSF. It is worth mentioning that
even when bright stars are present in the FoV, empirical PSF methods would not
be able to provide an accurate description of the PSF wings. In the near
future, we plan focussing on the treatment of off-axis (anisoplanatism)
corrections and testing the PSF-R algorithm in the case of crowded stellar
fields, e.g. in the case of a globular cluster.

This kind of studies paves the way for exploiting AO data reduction for ELT-
size telescope observations of dense stellar fields or individual extra-
galactic objects. The PSF-R software shown here will be an important starting
point for developing a similar software for MICADO, the first-light camera of
ELT, which will adopt a pyramid WFS similar to SOUL in its SCAO observing
mode. In light of the future implementation of the MAORY module on MICADO, our
group is also contributing in extending the algorithm to MCAO systems. It is
worth noting that in the ELT the halo of a star will be a factor of $\sim$5
larger than in LBT, and thus it will have a much lower surface brightness (a
factor of $\sim 25$ fainter). On the other hand, the core of the PSF is a
factor $\sim$5 narrower in diameter. This means that, especially at longer
wavelengths, even for bright stars the halo will be barely detected as a
distinct feature, and that multiple halos will be blended into a faint
distributed background. This indicates that the PSF-R approach will be even
more important in the future for 30-40m class telescopes. Only for stars that
are bright enough to saturate, the outer halo becomes visible, thus the hybrid
or calibrated PSF-R techniques will not be suitable in this case.

A first stage of scientific evaluation of the reconstructed PSFs has been
carried out by simulating a long SOUL+LUCI LBT observation of a compact early
type galaxy (red nugget) at $z\sim 1.5$. The reconstructed PSF allows us to
derive the physical parameters (position, magnitude, half light radius) of the
galaxy within 1 $\sigma$ level with GALFIT, while the Sérsic index is
reproduced with an accuracy of $\sim 2\,\sigma$. The axis ratio $b/a$ and
position angle are reproduced within 5% and 0.6 degree, respectively. We
propose that the possible systematic shift of the Sérsic index could be due to
a non perfect representation of the wings of the reconstructed PSF with
respect to the observed one. In summary, PSF reconstruction from telemetry
data only is strongly required if one needs to obtain precise morphological
measurements, since typical high-z galaxies are usually observed in blank
extra-galactic fields. In this respect, the results of the presented analysis
are extremely encouraging.

Lastly, we discuss, in the appendix, an interesting relation linking EE, the
FWHM, and SR of the observed PSF that can be useful e.g. when one is
interested in computing an instrument sensitivity.

We can conclude that the PSF-R approach presented here is an innovative method
for pyramid WFS.

##  Appendix A A practical relation linking SR, FWHM, and EE

In this appendix we present a useful relation that links the observed EE
($EE_{OBS}$) to the measured SR. Its application is of particular interest
i.e. in the case that, for an AO systems, one is interested in the energy
concentration in the core of the observed PSF knowing only the associated SR
and FWHM with the final goal of computing an instrument sensitivity. The SR is
defined as the ratio of the punctual peak values of the observed PSF and the
diffraction limited one. A typical method for sensitivity computation is to
consider the SR for the calculation of the Encircled Energy within the PSF
core:

${\rm EE_{core,OBS,uncorrected}}\simeq{\rm SR}\cdot{\rm EE_{core,DL}}\,.$ (6)

In case of AO systems performing very well, e.g. SR$>30-50\%$, the core of the
PSF can be assumed as the area within the first minimum of the associated Airy
function (i.e. the diffraction limited PSF). Nevertheless, in these
conditions, wrongly assuming that within this area one can find a SR-
percentage of the flux corresponding to the Airy figure, brings to an
underestimation of the flux within the aperture.

To overcome this problem, a correction can be made in the above equation by
means of a factor that accounts for the actual PSF shape. On a first
approximation, this shape factor corresponds to the combined effect of the
very-low order modes corrected by the AO. The shape of the core is dominated
by the residuals of these few modes. In real cases, tip-tilt residuals
represent the majority of the uncorrected low-order fraction for two main
reasons: i) because Kolmogorov power spectrum puts there $90\%$ of the energy
and ii) because the largest non-turbulence wavefront error derives from
telescope and instrument vibration (that may be as large as the atmospheric
jitter and much faster). Moreover, the correlation time of the atmospheric tip
tilt - proportional to the ratio of wind velocity and telescope diameter -
puts it in the $1-10\,{\rm Hz}$[31] , while the typical vibration power
spectrum has peaks in the $10-100\,{\rm Hz}$ region. When the PSF peak is
similar to the diffraction limited regime, corresponding to relatively high SR
(${\rm SR}>30\%-50\%$), one may correct the equation above by adding the
shape-factor:

$\displaystyle{\rm EE_{core,OBS,corrected}}\simeq{\rm SR}\cdot{\rm
EE_{core,DL}}\cdot\left(\frac{FWHM_{OBS}}{FWHM_{DL}}\right)^{2}$ (7)

and if opto-mechanical vibrations dominate the low-order residuals it can be
restricted to the tip-tilt component

$\displaystyle{\rm EE_{core,OBS,corrected}}\simeq{\rm SR}\cdot{\rm
EE_{core,DL}}\cdot\left(\frac{FWHM_{VIBRATION}^{2}}{FWHM_{DL}^{2}}+1\right)\,,$
(8)

where the $FWHM_{VIBRATION}$ is the standard deviation of the jitter due to
the vibration component. In fact, under the hypothesis that both the residual
vibrations component, uncorrected or introduced by the AO system, and the PSF
core can be approximated by Gaussian functions, the broadening of the
registered PSF is a function of the standard deviation associated to the
vibration component. In particular, it is straightforward to derive that the
peak value of the registered PSF is proportional to $FWHM_{OBS}^{-2}$. It is
also important to note that the ratio of EEs operates as a normalization
factor, correlating with the amount of uncorrected photons outside the control
radius of the AO system, defined as the distance from PSF centre where light
distorted by uncorrected high-order spatial frequencies resumes following a
seeing-limited distribution[32] .

If applied to the nighttime dataset, Equation (6) gives
$EE_{core,OBS,uncorrected}=0.36$, in contrast with the observed value of
$EE_{core,OBS}=0.43$. The shape factor in this case results:

$\left(\frac{FWHM_{OBS}}{FWHM_{DL}}\right)^{2}=\left(\frac{43.80}{40.07}\right)^{2}=1.19$
(9)

Thus, applying Equation (7) it results $EE_{core,OBS,corrected}=0.43$ in
perfect agreement with the observed value. From Equation (8) one can also
compute the expected standard deviation associated to the residual vibration
component. In the case of nighttime dataset, this amount to $7.4\,{\rm mas}$.
For comparison, it results from telemetry that the amount of tip-tilt only in
the WF residuals sum up to $3.9\,{\rm mas}$, indicating that other peak-
broadening are likely occurring (i.e. defocus-astigmatisms, chromatic
dispersion).

Figure 8: Application of Equation (6) (UNCORRECTED) and Equation (7)
(CORRECTED) to mock diffraction limited PSFs, each convolved with a 2D
Gaussian function of varying standard deviations. The relative error in the
resulting EE are shown as a function of the standard deviation of the
convolving 2D Gaussian ($\sigma_{VIBRATION}$).

Equation (7) provides accurate results only under particular conditions. The
major requirement is that the low-order aberration effect needs to be smaller
than the EE aperture. Indeed, the higher is the broadening effect, the lower
is EE inside a fixed aperture, therefore invalidating the normalization of the
PSF. Specifically, the shape of the post-AO PSF inside the control radius
matches as close as possible that of the diffraction limited PSF, but with
intensity hampered by the amount of light outside the control radius
associated to uncorrected modes.

We propose that a simple solution to avoid this issue is to take care in
defining the aperture radius for $EE_{core}$ of the order of $FWHM_{OBS}$. In
other words, we encourage the use of aperture referring to actual, post-AO
light distribution that implicitly takes into account the broadening due to
residual vibrations.

To quantify the limits of applicability of the relation in Equation (7), we
performed a simple simulation. We convolved the diffraction limited PSF,
oversampled by a factor $5$, with a 2D Gaussian functions of varying standard
deviations (from $1$ to $20\,{\rm mas}$). In this case, as the observed PSF is
obtained directly from the diffraction limited one, the lowering of SR values
is only due to the residual vibration effects, which do not affect sensibly
the wings of the PSF. Therefore, the effect of the aperture choice can also be
estimated. We then measured the associated $EE_{core,OBS}$ and $EE_{core,DL}$
inside an aperture of radius $R_{EE,core}=FWHM_{OBS}$. We also measured the SR
directly from the resulting PSF. We then compared the measure $EE_{core,OBS}$
with those computed using Equations (6) and (7). The results are reported in
Figure 8. It can be observed that, after properly defining the aperture,
Equation (7) correctly estimates the observed $EE_{core,OBS}$ within $10\%$
accuracy for a wide range of residual vibrations.

### Acknowledgements

We thank the anonymous referees for their thorough revision that enhanced the
quality of the present work. This work has been partly supported by INAF
through the Math, ASTronomy and Research (MAST&R), a working group for
mathematical methods for high-resolution imaging. Based on observations made
at the Large Binocular Telescope (LBT) at Mt. Graham (Arizona, USA). The LBT
is an international collaboration among institutions in the United States,
Italy and Germany. LBT Corporation partners are: The University of Arizona on
behalf of the Arizona university system; Istituto Nazionale di Astrofisica,
Italy; LBT Beteiligungsgesellschaft, Germany, representing the Max-Planck
Society, the Astrophysical Institute Potsdam, and Heidelberg University; The
Ohio State University, and The Research Corporation, on behalf of The
University of Notre Dame, University of Minnesota and University of Virginia.

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desired performance without ever needing to utilize the full-order environment
after training. Interestingly, by injecting a small amount of Gaussian noise
onto the trajectory before querying the neural network, we do not degrade the
performance at all.

Figure 22: Comparison of the six different data collection strategies and the
performance of the resulting dictionary model deployed to the full-order
environment.

###  F.2 Swing-up

Figure 23 demonstrates the effectiveness of the surrogate policy described in
Section 5.4. While the episode returns are larger for the surrogate
dictionary, it’s important to note that the performance is very comparable;
with the median performance only being about 1% different. In Figure 24, we
highlight a case where we continued to train the swing-up policy using
Algorithm 1 beyond what was presented in Section 5.4. At some point, the
neural network starts to perform worse on the full-order environment, leading
to an unstable control policy near the equilibrium. However, when using this
neural network to create a surrogate dictionary, the resulting dictionary is a
stable, consistent control policy. For both the neural network and the policy,
the returns are significantly diminished compared to what’s shown in Figure
23, and this time the surrogate dictionary’s median performance is about 20%
better.

Figure 23: Swing-Up Policy: Trajectories. The evaluation of 15 slightly
different initial conditions using the surrogate dictionary (blue) and neural
network policies discussed in Section 5.4. Left: Timeseries of the
trajectories and control. Right: Distribution of the total cumulative returns
from the 15 trajectories.

Figure 24: Swing-Up Policy: Neural Network Overfitting. The evaluation of 15
slightly different initial conditions using the surrogate dictionary (blue)
and neural network policies. Top: Timeseries of the trajectories and control.
Bottom Left: Distribution of the total cumulative returns from the 15
trajectories. Bottom Right: The control landscape evaluated on a mesh of
$(\theta,\dot{\theta})$ values. (For ease of visualization, $x=\dot{x}=0$).

###  F.3 Swimmer-v4

As with the swing-up environment, we get a comparable surrogate policy using a
cubic dictionary compared to the original neural network, with only a slight
difference in median performance.

Figure 25: Swimmer-v4 Policy Trajectories. Example of one of 15 slightly
different initial conditions using the surrogate dictionary (blue) and neural
network policies. Bottom: Distribution of the total cumulative returns from
the 15 trajectories.