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# Overcoming the effect of pupil distortion in multiconjugate adaptive optics

Marcos A. van Dam  Flat Wavefronts, 21 Lascelles Street, Christchurch 8022,
New Zealand  Yolanda Martín Hernando  Instituto de Astrofísica de Canarias, C/
Vía Láctea s/n 38205, La Laguna, Spain  Miguel Núñez Cagigal  Instituto de
Astrofísica de Canarias, C/ Vía Láctea s/n 38205, La Laguna, Spain  Luzma M.
Montoya  Instituto de Astrofísica de Canarias, C/ Vía Láctea s/n 38205, La
Laguna, Spain

###### Abstract

Multiconjugate adaptive optics (MCAO) systems have the potential to deliver
diffraction-limited images over much larger fields of view than traditional
single conjugate adaptive optics systems. In MCAO, the high altitude
deformable mirrors (DMs) cause a distortion of the pupil plane and lead to a
dynamic misregistration between the DM actuators and the wavefront sensors
(WFSs). The problem is much more acute for solar astronomy than for night-time
observations due to the higher spatial sampling of the WFSs and DMs, and the
fact that the science observations are often made through stronger turbulence
and at lower elevations. The dynamic misregistration limits the quality of the
correction provided by solar MCAO systems.

In this paper, we present PropAO, the first AO simulation tool (to our
knowledge) to model the effect of pupil distortion. It takes advantage of the
Python implementation of the optical propagation library PROPER. PropAO uses
Fresnel propagation to propagate the amplitude and phase of an incoming wave
through the atmosphere and the MCAO system. The resulting wavefront is
analyzed by the WFSs and also used to evaluate the corrected image quality. We
are able to reproduce the problem of pupil distortion and test novel non-
linear reconstruction strategies that take the distortion into account. PropAO
is shown to be an essential tool to study the behavior of the wavefront
reconstruction and control for the European Solar Telescope.

###### keywords:

multiconjugate adaptive optics, solar, wavefront reconstruction, pupil
distortion

##  1 Introduction

Multiconjugate adaptive optics (MCAO) uses two or more deformable mirrors
(DMs) at different conjugate altitudes in order to correct the three-
dimensional structure of turbulence and hence increase the corrected field-of-
view relative to conventional single conjugate adaptive optics (SCAO).

MCAO has enjoyed some success in night time astronomy, with the MAD
demonstrator at the VLT[1] and GeMS[2, 3] on Gemini South paving the way. GeMS
has been routinely producing corrected science images over an
$$85\text{\,}\mathrm{\SIUnitSymbolArcsecond}$\times$85\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$
field since 2013. Solar MCAO has an even longer history,[4, 5] although early
experiments were hampered by problems with loop stability. The 3 DM MCAO
system Clear at the Goode Solar Telescope has provided high-order AO
correction over a
$30\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$times$$30\text{\,}\mathrm{\SIUnitSymbolArcsecond}$
at visible wavelengths since 2016.[6]

The combination of one or more DMs at a very high altitude and the small
subaperture sizes in solar MCAO can lead to an unstable loop in the presence
of moderate to strong seeing. Table 1 records the salient differences between
night-time and solar MCAO that affect the loop stability.

Table 1: Comparison of relevant parameters for the MCAO systems on Gemini South and Gregor.[7] Parameter | Night-time | Solar
---|---|---
Highest DM conjugate altitude | $9\text{\,}\mathrm{km}$ | $25\text{\,}\mathrm{km}$
Subaperture size | $0.50\text{\,}\mathrm{m}$ | $0.10\text{\,}\mathrm{m}$

In addition, solar telescopes typically point at lower elevations than night-
time telescopes, which have the luxury of observing targets when they are
higher in the sky. If the wavefront sensors (WFSs) are conjugate to the ground
(i.e., in the pupil plane), then the propagation between the high altitude DM
and the ground leads to a pupil distortion. A distorted pupil leads to a
change in registration between the WFSs and the DM. If the interactuator
spacing is matched to the size of the subaperture, a pupil distortion of half
a subaperture leads to an unstable loop, because an actuator poke will be
incorrectly reconstructed as originating from a neighboring actuator.

The distorting effect of the high-altitude DMs on the exit pupil in MCAO was
first noticed by von der Lühe while observing rapid intensity fluctuations in
the MCAO corrected image plane at the VTT.[8] Later, it was realized that the
ever changing shape of the high-altitude DM that continuously changes the
registration between the pupil DM and the WFS could potentially pose a
challenge to MCAO control. In a 2010 paper, Schmidt et al. demonstrated the
dynamic misregistration using ray tracing and experimentally. Figure 1
illustrates the pupil distortion and the associated dynamic misregistration of
the the Gregor MCAO system by poking the central actuator of the high altitude
DM.[9]

Figure 1: Photographic illustration of pupil distortion: the same grid pattern
is shown with the DMs flat and with a $400\text{\,}\mathrm{nm}$ poke of the
central actuator on high-altitude DM (left). The region corresponds to an area
spanning approximately $3\times 3$ actuators marked with a cross (right).[9]

In the second generation of solar MCAO systems at Gregor and Clear at the
Goode Solar Telescope, it was a goal to mitigate the misregistration issue. In
the Gregor system, a high-order on-axis WFS was placed between the pupil DM
and the high-altitude DMs, and a low-order multi-directional WFS was placed
behind all DMs. The same scheme was available in Clear as one of six different
WFS and DM concepts implemented. This control strategy is stable and produced
AO corrected images.[11, 10, 6]

In Section 2, we compare existing AO simulation and describe how they
propagate the wavefront. To our knowledge, there is no existing end-to-end
Monte-Carlo simulation tool that models solar MCAO and reproduces this effect.
We considered three different ways to model the pupil distortion:

  * •

Fresnel propagation

  * •

Ray tracing (using the start point and the gradient of the ray to determine
the end point)

  * •

The irradiance transport equation along with the associated wavefront
transport equation[12]

The quickest way to create a new simulation tool with the required properties
was to adapt PROPER,[13] an optical propagation code based on Fresnel
propagation (Section 3).

In this paper, we develop PropAO, the first AO simulation tool (to our
knowledge) to model the effect of pupil distortion. PropAO takes advantage of
the optical propagation library PROPER and uses Fresnel propagation to
propagate the amplitude and phase of an incoming wave through the atmosphere
and the MCAO system. The resulting wavefront is analyzed by the WFSs and also
used to evaluate the corrected image quality. We are able to reproduce the
problem of pupil distortion and test novel non-linear reconstruction
strategies that take the distortion into account and produce well-corrected
wide field images.

Existing AO systems use linear wavefront reconstruction techniques, where the
estimates wavefront is a linear combination of the measured wavefront
derivatives. In Section 4, we propose new reconstructors to combat the pupil
distortion problem. Simulation results using a toy MCAO problem are presented
in Section 5 and conclusions are drawn in Section 6.

The motivation for this work is the design of the $4.2\text{\,}\mathrm{m}$
diameter European Solar Telescope.[14, 15] The current design has a very
ambitious
$40\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$\times$$40\text{\,}\mathrm{\SIUnitSymbolArcsecond}$
AO-corrected science field of view. In order to deliver near diffraction-
limited performance at visible wavelengths over such a wide field of view, an
MCAO system with three to five DMs is under consideration. The ability to
model the pupil distortion is essential to the success of the EST project. The
PropAO tool and the results of this work are also applicable to other solar
and night-time astronomy MCAO systems.

##  2 Existing simulation codes

In this section, we describe publicly available and well-documented end-to-end
simulation codes. We evaluate them on their suitability to model the pupil
distortion of a solar MCAO system and find that none of these codes can be
used to model the pupil distortion.

###  2.1 DASP

The Durham Adaptive optics Simulation Platform (DASP) is an open-source
configurable AO simulation environment written in Python.[16, 17] DASP offers
support for tomographic reconstructors and pseudo-open loop control, so it is
well suited for MCAO simulations. The turbulence propagation is performed
either by simple integration of the wavefront along the line of sight or using
Fresnel propagation. Only the former option is currently available for
propagation through the DM to the WFSs. Implementing this using Fresnel
propagation, while not a lot of work, is not in the pipeline.[18]

###  2.2 YAO

A full-featured end-to-end Monte-Carlo simulation tool, known as yorick
adaptive optics (YAO) and written by Francois Rigaut, is used extensively in
the AO community.[19, 20] The wavefront propagation is performed by summing
the wavefront through each atmospheric layer or DM surface in the direction of
each WFS or science target. Fresnel propagation is not implemented. The pupil
distortions are not modeled.

###  2.3 OOMAO

OOMAO is a Matlab based AO simulation tool that is highly configurable. The
wavefront propagation is performed in the same way as YAO, so the pupil
distortion is not modeled, either.

###  2.4 CAOS

The Software Package CAOS is a subset of the CAOS Problem-Solving Environment
and consists of an IDL based AO simulations tool.[21, 22] It uses linear
wavefront propagation to calculate the wavefronts. It was claimed in 2005 that
“a Fresnel module for atmospheric turbulence has been written and will be part
of a forthcoming release.”[24] This claim is still true today! Work on Fresnel
propagation through turbulence was paused, has recently restarted but is not
yet ready for release.[23] It is possible to also implement Fresnel
propagation between DMs at different conjugate altitudes, but this
functionality is not planned.

###  2.5 CEO

CEO (an acronym for Cuda Engined Optics) is a simulation tool written for the
Giant Magellan Telescope.[25, 26] This open-source software applies ray
tracing in CUDA and has a Python interface. It has the capability to apply ray
tracing through the atmosphere and the telescope, but not through an AO
system. This capability would be needed to model the pupil distortions.

Figure 2 shows how the wavefront propagation leads to variations in wavefront
and amplitude using ray tracing in CEO.

Figure 2: Phase before propagation (left) and phase (center) and amplitude
(right) after propagation.

##  3 PropAO: an extension to the PROPER optical propagation tool

The PROPER optical propagation code, available in IDL, Matlab and Python, is a
library of routines for optical propagation.[13, 27] PROPER uses full Fresnel
propagation at every step and is suitable for modeling the complex amplitude
of a wave propagating through the atmosphere and an MCAO system. There is
support for pupil stops, DMs, and point-spread function (PSF) generation. An
example of Fresnel propagation using PROPER is shown in Figure 3, where a
Kolmogorov phase screen is propagated $20\text{\,}\mathrm{km}$ in broadband
light over a the range of $400\text{\,}\mathrm{nm}$ to
$800\text{\,}\mathrm{nm}$ nm. The amplitude and the phase evolve as the wave
propagates.

Figure 3: Initial amplitude and phase (top) and final amplitude and phase
(bottom) following a $20\text{\,}\mathrm{km}$ propagation using the PROPER
code.

In this section, we show how we extend the functionality of PROPER into an
end-to-end simulation tool. PropAO uses the Python 3 version of the PROPER
optical propagation tool as a starting point and adds new features missing in
PROPER, such as atmospheric turbulence and WFSs, in order to create an end-to-
end MCAO simulation tool. As will become evident, the development process
consisted a tool that mimicked the behavior of YAO as closely as possible in
order to quickly and efficiently debug and verify the new code using a simple
single conjugate AO system. Once we were confident in the results, the
simulation was extended to an MCAO with multiple DMs and WFSs.

###  3.1 Atmospheric turbulence

PROPER has the capability of inserting amplitude or phase screens anywhere in
the optical train, but no function to create Kolmogorov phase screens.
Atmospheric phase screens were generated using YAO to avoid writing code
unnecessarily, and to make it easier to compare the results between the two
codes. The optical propagation begins at the top of the atmosphere, where the
first atmospheric phase screen is inserted. The complex amplitude is
propagated to the next atmospheric phase screen, where the phase corresponding
to the second atmospheric phase screen is added. This procedure is repeated
until we reach the top of the telescope.

###  3.2 Telescope

The telescope is modeled as an ideal telescope. The complex amplitude of the
incoming wave is multiplied by a circular function representing the exit pupil
of the telescope by calling a PROPER function. In the future, we could include
a realistic optical model of the telescope.

###  3.3 Deformable mirrors

The first DM is conjugate to the pupil, so no propagation from the telescope
exit pupil is required. The complex amplitude is multiplied by the phase
induced by the DM and the wave is propagated to the first altitude DM. This
process was repeated for all DMs. Finally, the wave is progated back to the
pupil plane, where the wavefront sensors (WFSs) and the science instruments
are situated.

The functions in PROPER used to mode DMs were initially used. However, two
limitations drove us to write our own DM module: flexibility and speed. The DM
influence functions in PROPER are hard coded to a particular function that is
loaded as a FITS file. We wanted to be able to specify the same influence
functions as YAO initially, and then to be able to use the influence functions
corresponding to a real DM. The DM implementation in PROPER was also
prohibitively slow.

The new implementation for the DMs is as follows. First, the user can define
an actuator influence function or use one of the pre-programmed ones. The
options include the sinc interpolator, the bilinear intepolator, an
approximation to the measured influence function of a Xinetics actuator, and
the same function that is programmed in YAO. The two-dimensional actuator
commands are convolved with the influence function. The convolution takes
place as a multiplication in the Fourier domain using FFTs.

###  3.4 Wavefront sensors

There are no WFSs in PROPER. Shack-Hartmann WFSs were implemented in the
following manner. The complex amplitude at the pupil plane is subdivided by
the lenslet array and each subaperture is propagated onto the pupil plane to
form an image. The centroid of the image on each subaperture is computed and
recorded.

###  3.5 Wavefront reconstruction and control

The main purpose of the tool is to investigate wavefront reconstruction and
control strategies. The tool supports closed-loop and open-loop (where the
WFSs do not see the DMs) modes as well as pseudo open-loop control, as
described in Section 4.

###  3.6 Science instrument

The science instrument is modeled as an imager which takes an image of a point
source and uses this image to estimate the Strehl ratio at the science
wavelength(s). There is the option of including a pupil stop in the science
camera.

###  3.7 Use of GPUs with the CuPY library

Following the completion of the first operational version PropAO, it was
evident from modeling a $1\text{\,}\mathrm{m}$ diameter telescope that the
simulations could not be scaled to a $4.2\text{\,}\mathrm{m}$ diameter
telescope like the European Solar Telescope due to the computational load.
Since most of the computations consist of FFTs and matrix multiplications,
PropAO is an ideal candidate for the use of Graphical Processing Units (GPUs).
The CuPY library is a Python library that uses CUDA acceleration for the heavy
numerical computations.[28] It is almost a like-to-like replacement for the
NumPY library, which is used in PROPER. By making a small number of changes
(fewer than ten different lines of code), we were able to convert the
computations in PropAO from CPU to GPU. The simulation time using an off-the-
shelf laptop (Dell Precision 7710 with an NVIDIA Quadro M4000M GPU on a
laptop) decreased by a factor of 100! As an example, a 3 DM system on a
$4\text{\,}\mathrm{m}$ telescope with 5 $32\times 32$ subaperture WFSs runs at
2.5 iterations/second.

##  4 Wavefront reconstruction and control

In this section, we describe a number of different approaches to the problem
of wavefront reconstruction and control, which are subsequently evaluated
using PropAO simulations.

###  4.1 Closed-loop control with regularized least-squares reconstructor

The standard closed-loop reconstructor is what is used on almost all AO
systems. An interaction matrix is created by poking the actuators one by one.
The reconstructor is computed as a regularized inverse of the interaction
matrix, either using a singular value decomposition, a regularized least-
squares inversion or a modal basis set. We use the regularized least-squares
reconstructor, $R$:

$R=(H^{T}C_{nn}^{-1}H+\alpha C_{\phi}^{-1}+P)^{-1}H^{T}C_{nn}^{-1}$ (1)

where $H$ is the interaction matrix, $P$ is a matrix that penalizes piston for
each DM, $C_{nn}$ is the noise covariance matrix, $C_{\phi}$ is the covariance
matrix of the uncorrected turbulence and $\alpha$ is a regularization
constant. Figure 4 plots the interaction matrix, covariance matrix and piston
penalization matrix for a 3 DM, 5 WFS MCAO system. The noise covariance matrix
is simply a diagonal matrix.

Figure 4: Interaction matrix (left), covariance matrix (center) and piston
penalization matrix (rigth) for a 3 DM, 5 WFS system.

The wavefront residual, $u$, is given by the matrix multiplication of the
reconstructor, $R$, and the centroids, $s$:

$u=Rs$ (2)

A leaky integrator with leak $l$=0.01 loop gain $k$=0.6 is used to update the
actuator commands, $a$ at time $n$:

$a[n]=(1-l)a[n-1]+ku[n].$ (3)

This reconstruction and control approach can lead to instabilities because the
high altitude DMs will cause a shift in the registration between the actuators
of the lower altitude DM and the WFS lenslets.

###  4.2 Closed-loop control with iterative interaction matrix and regularized
least-squares reconstructor

The interaction matrix is calculated by poking actuators one by one away from
their nominal positions and measuring the centroids. However, the effect of
each actuator on the WFS depends on the actuator commands of the mirrors that
follow it. For example, the effect that the ground-layer DM has on the WFS
centroids depends on the shape of the mid- and high-altitude DMs. In this
control strategy, we calculate the interaction matrix iteratively by poking
actuators referenced to the current DM commands. Each time the interaction
matrix is calculate (once every ten iterations in the simulations presented in
this paper), the reconstructor is recomputed.

###  4.3 Pseudo open-loop control with regularized least-squares reconstructor

The reconstructor in Eq. (1) can be used in pseudo open-loop control. Pseudo-
open loop control uses the open-loop centroids, $s-Ha$ to calculate the
residuals:

$\displaystyle u$ $\displaystyle=$ $\displaystyle R(s+Ha)-a$ $\displaystyle u$
$\displaystyle=$ $\displaystyle Rs+({RH}-I)a$ (4)

where $I$ is the identity matrix.

###  4.4 Non-linear pseudo open-loop control with regularized least-squares
reconstructor

The pseudo open-loop control described in Section 4.3 relies on a linear
relationship between the actuator commands and the centroids to compute the
open-loop centroids. For strong turbulence, this relationship does not hold
due to the pupil distortion effect. A more accurate way to compute the open-
loop centroids is to propagate a wave from the top of the telescope through to
the WFSs and measure the centroids. This propagation takes place once per
iteration.

###  4.5 Separate control of ground-layer DM and high-altitude DMs

In some solar MCAO systems, a high-order on-axis WFS is used to control the
ground-layer DM, while lower-order off-axis WFSs are used to drive the mid-
and high-altitude DMs.[10] The on-axis WFS does not see the other DMs, so the
registration between the WFS and the ground-layer DM is not affected by the
other DMs. Here, the optical setup is different, not just the control law!

##  5 Simulations

Simulations were run using new simulation tool and compared with the same
simulations run in YAO in order to understand the effects of Fresnel
propagation through the atmosphere and AO system.

###  5.1 Simulation parameters

The simulations were run using a $1\text{\,}\mathrm{m}$ diameter telescope
with no central obscuration. In all cases, there are $8\times 8$ subapertures
across the pupil. Two different atmospheric profiles are used, one with 70% of
the turbulence near the ground (Profile 1) and another one with 90% (Profile
2). The turbulence parameters are tabulated in Table 2.

Table 2: Turbulence profiles used in the simulations. | Elevatiion ($\text{\,}\mathrm{m}$) | 0 | 1000 | 2000 | 4000 | 8000 | 16000
---|---|---|---|---|---|---|---
Profile 1 | Turbulence fraction | 0.55 | 0.15 | 0.05 | 0.06 | 0.07 | 0.12
Profile 2 | Turbulence fraction | 0.63 | 0.25 | 0.02 | 0.02 | 0.03 | 0.05
| Wind speed ($\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | 5.6 | 6.25 | 7.57 | 13.31 | 19.06 | 12.14
| Wind direction ($\text{\,}\mathrm{\SIUnitSymbolDegree}$) | 190 | 270 | 350 | 17 | 29 | 66

The outer scale of turbulence used in the simulations is a few meters, which
is not realistic. This occurs because the phase screens are periodic in order
to avoid issues with discontinuities when propagating with the Fast Fourier
Transform (FFT). The effect of the outer scale on AO-corrected performance is
expected to be small, but needs to be considered.

###  5.2 Seeing-limited simulations

Simulations were run with no correction as a sanity check, and compared with a
YAO simulation using the same parameters. There were 3000 iterations with a
$10\text{\,}\mathrm{ms}$ exposure time for a total integration time of
$30\text{\,}\mathrm{s}$. There is excellent consistency between the YAO and
the PropAO results (Table 3), which is not suprising since the same turbulence
phase screens are used. The on-axis Strehl ratio at $500\text{\,}\mathrm{nm}$
is used as the science metric.

Table 3: Strehl ratio at $500\text{\,}\mathrm{nm}$ from YAO and PropAO simulations of uncorrected turbulence. |  | Profile 1 | Profile 2
---|---|---|---
| Elev. |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$ |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$
YAO | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.0020 | 0.0061 | 0.0160 | 0.0020 | 0.0062 | 0.0164
PropAO | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.0018 | 0.0063 | 0.0179 | 0.0018 | 0.0063 | 0.0176
YAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.0043 | 0.0145 | 0.0394 | 0.0044 | 0.0146 | 0.0405
PropAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.0042 | 0.0156 | 0.0432 | 0.0043 | 0.0153 | 0.0415
YAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.0067 | 0.0232 | 0.0607 | 0.0063 | 0.0221 | 0.0593
PropAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.0067 | 0.0232 | 0.0607 | 0.0063 | 0.0221 | 0.0593

###  5.3 Single conjugate adaptive optics simulations

A simple single conjugate adaptive optics (SCAO) system was simulated using
both codes. A ground-layer DM with $9\times 9$ actuators across the pupil in
the so-called Fried configuration was used. The WFS exposure time was
$1\text{\,}\mathrm{ms}$, and 1000 iterations were run for a total simulation
time of $1\text{\,}\mathrm{s}$. All of the simulations in this section are
performed using pseudo open-loop control with an optimized value of $\alpha$
in the least-squares inversion of Eq. (1). The turbulence was moved to the
ground in order to avoid scintillation effects, which are not modeled in YAO.
The results are tabulated in Table 4.

Table 4: Strehl ratio at $500\text{\,}\mathrm{nm}$ from YAO and PropAO simulations of an SCAO system with all of the turbulence near the ground. |  | Profile 1 | Profile 2
---|---|---|---
| Elev. |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$ |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$
YAO | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.093 | 0.443 | 0.706 | 0.098 | 0.452 | 0.712
PropAO | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.089 | 0.421 | 0.690 | 0.102 | 0.442 | 0.703
YAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.321 | 0.673 | 0.844 | 0.321 | 0.674 | 0.845
PropAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.305 | 0.657 | 0.832 | 0.325 | 0.672 | 0.840
YAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.435 | 0.745 | 0.882 | 0.437 | 0.747 | 0.882
PropAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.431 | 0.741 | 0.876 | 0.451 | 0.754 | 0.882

The theoretical decrease in Strehl ratio, $S$ due to scintillation is given by
the Maréchal approximation for sufficiently small amplitude vatiations:

$S=\exp[-\sigma_{\chi}^{2}]$ (5)

where the scintillation is calculated as[29]

$\sigma_{\chi}^{2}=0.5631k^{7/6}\sec^{11/6}(\zeta)\int dzC_{n}^{2}(z)z^{5/6}.$
(6)

The symbol $\zeta$ represents the zenith angle, and $k$ is the wave number,
$2\pi/\lambda$.

PropAO simulations were repeated with the turbulence phase screens at their
nominal value, and the decrease in Strehl ratio was compared to the expected
decrease in Strehl due to scintillation (Table 5). In general, both sets of
results show the same trends, but the fit is not perfect. Part of the reason
is that the scintillation also has an effect on the WFS, so the Strehl ratio
is worse than predicted when the scintillation is strong. The results at an
elevation of $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ are omitted because the
Maréchal approximation does not hold.

Table 5: Strehl ratio at $500\text{\,}\mathrm{nm}$ from YAO and PropAO simulations of an SCAO system with all of the turbulence near the ground. |  | Profile 1 | Profile 2
---|---|---|---
| Elev. |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$ |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$
Analytic | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.891 | 0.960 | 0.983 | 0.942 | 0.979 | 0.991
PropAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.790 | 0.951 | 0.994 | 0.898 | 0.987 | 0.998
Analytic | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.941 | 0.978 | 0.991 | 0.969 | 0.989 | 0.995
PropAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.929 | 0.991 | 0.998 | 0.970 | 0.996 | 0.999

###  5.4 Multiconjugate adaptive optics simulations

Simulations were run for a toy 3-DM MCAO system using the same
$1\text{\,}\mathrm{m}$ telescope. The DM parameters are as described in Table
6.

Table 6: Description of the DMs used in the MCAO simulations. DM number | Altitude | Interactuator spacing | Actuators across pupil
---|---|---|---
0 | $0\text{\,}\mathrm{m}$ | $125\text{\,}\mathrm{mm}$ | $9\times 9$
1 | $5000\text{\,}\mathrm{m}$ | $250\text{\,}\mathrm{mm}$ | $7\times 7$
2 | $12\,000\text{\,}\mathrm{m}$ | $375\text{\,}\mathrm{mm}$ | $6\times 6$

There are five identical WFSs with $8\times 8$ subapertures. The guide stars
are modeled as point sources located at
$[$0\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,$0\text{\,}\mathrm{\SIUnitSymbolArcsecond}$]$,
$[\pm$5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,\pm$5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$]$.
The science field of view is a
$$10\text{\,}\mathrm{\SIUnitSymbolArcsecond}$\times$10\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$
square. The results from the YAO simulations using pseudo open-loop control
are displayed in Table 7. All of these YAO simulations exhibited loop
stability and improved the image quality; this is not surprising, because YAO
does not model the pupil distortion. It is clear that some configurations,
such as observing at an elevation of $20\text{\,}\mathrm{\SIUnitSymbolDegree}$
in anything but the best seeing, are not really suitable for wide-field MCAO
correction.

Table 7: Strehl ratio at 500 nm from MCAO simulations using YAO with psedo open-loop control. | Profile 1 | Profile 2
---|---|---
Elev. |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$ |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$
$20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.005±0.001 | 0.037±0.016 | 0.152±0.058 | 0.008±0.002 | 0.092±0.032 | 0.316±0.065
$45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.088±0.023 | 0.417±0.040 | 0.687±0.028 | 0.189±0.021 | 0.548±0.022 | 0.773±0.013
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.268±0.017 | 0.629±0.015 | 0.820±0.008 | 0.361±0.011 | 0.699±0.008 | 0.858±0.004

These simulations were repeated in PropAO using a variety of reconstruction
strategies, described in Section 4:

  * •

Regularized least-squares reconstructor operating in closed-loop (CL)

  * •

Regularized least-squares reconstructor with an iterative interaction matrix
operating in closed-loop (Iter CL)

  * •

Regularized least-squares reconstructor operating in pseudo open-loop (POLC)

  * •

Regularized least-squares reconstructor with non-linear pseudo open-loop (NL
POLC)

The performance achieved was very sensitive to the level of regularization
(the value of the $\alpha$ parameter) used in the reconstructor, with the
regularization adjusted for each simulation in order to maximize the Strehl
ratio. For strong turbulence cases, a high level of regularization was needed
to maintain loop stability.

Table 8: Strehl ratio at $500\text{\,}\mathrm{nm}$ from MCAO simulations using PropAO with varying reconstruction and control strategies. |  | Profile 1 | Profile 2
---|---|---|---
| Elev. |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$ |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$
CL | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | — | 0.045±0.012 | 0.206±0.050 | — | 0.119±0.034 | 0.385±0.047
Iter CL | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.005±0.001 | 0.046±0.021 | 0.206±0.048 | 0.008±0.002 | 0.119±0.031 | 0.375±0.041
POLC | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | — | 0.048±0.019 | 0.206±0.043 | — | 0.128±0.029 | 0.384±0.038
NL POLC | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | — | 0.046±0.016 | 0.200±0.042 | — | 0.116±0.025 | 0.367±0.038
CL | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.064±0.012 | 0.402±0.035 | 0.663±0.024 | 0.142±9.014 | 0.536±0.021 | 0.754±0.013
Iter CL | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.068±0.015 | 0.388±0.035 | 0.658±0.025 | 0.142±0.010 | 0.520±0.022 | 0.729±0.015
POLC | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.075±0.015 | 0.419±0.035 | 0.675±0.024 | 0.182±0.020 | 0.550±0.021 | 0.763±0.021
NL POLC | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.075±0.017 | 0.397±0.034 | 0.660±0.024 | 0.152±0.017 | 0.523±0.021 | 0.747±0.013
CL | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.226±0.014 | 0.587±0.012 | 0.783±0.007 | 0.267±0.012 | 0.670±0.009 | 0.831±0.006
Iter CL | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.194±0.012 | 0.575±0.012 | 0.784±0.007 | 0.288±0.010 | 0.662±0.008 | 0.816±0.008
POLC | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.249±0.015 | 0.618±0.013 | 0.806±0.007 | 0.358±0.011 | 0.696±0.008 | 0.849±0.005
NL POLC | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.224±0.014 | 0.600±0.013 | 0.797±0.008 | 0.318±0.011 | 0.673±0.008 | 0.838±0.005

The results in Table 8 shows that all of the methods could close the loop
successfully, except in the worst case where the turbulence was strong and the
elevation was low. The key to loop stability was a high level of
regularization, but this limits the attainable atmospheric correction.

The best control strategy is to use linear pseudo open-loop control. The non-
linear control strategies did not improve the performance across the board and
often made things worse.

###  5.5 Multiconjugate adaptive optics simulations with decoupled DM control

The simulation is run using the same DM configuration is used, but two sets of
WFSs are used:

  * •

a pick-off after the ground-layer DM that feeds an on-axis WFS with 8x8
subapertures across the pupil.

  * •

A second pick-off following the three DMs feeds the four off-axis WFSs, which
have 4x4 subapertures across the pupil.

The correction of the ground-layer DM is decoupled from the high-altitude DMs,
as described in Schmidt et al.[6]

Simulations run using YAO and PropAO show consistent results, with the PropAO
values lower than the YAO results mostly due to scintillation (Table 9). The
Strehl ratios are much lower than those obtained in Section 5.4. The decoupled
DM control strategy has lower performance because the on-axis wavefront is not
the same as the wavefront at the ground. This strategy should not be used.

Table 9: Strehl ratio at $500\text{\,}\mathrm{nm}$ from YAO and PropAO simulations using MCAO with decoupled control. |  | Profile 1 | Profile 2
---|---|---|---
| Elev. |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$ |  $r_{0}=$ $8\text{\,}\mathrm{cm}$ |  $r_{0}=$ $15\text{\,}\mathrm{cm}$ |  $r_{0}=$ $25\text{\,}\mathrm{cm}$
YAO | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.004±0.001 | 0.025±0.006 | 0.101±0.022 | 0.007±0.001 | 0.057±0.012 | 0.230±0.034
PropAO | $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.001±0.000 | 0.018±0.004 | 0.086±0.033 | 0.001±0.000 | 0.038±0.011 | 0.187±0.045
YAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.033±0.019 | 0.260±0.049 | 0.555±0.042 | 0.101±0.022 | 0.436±0.030 | 0.700±0.020
PropAO | $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.019±0.005 | 0.210±0.032 | 0.516±0.033 | 0.063±0.011 | 0.379±0.025 | 0.660±0.019
YAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.097±0.028 | 0.421±0.040 | 0.688±0.027 | 0.217±0.026 | 0.581±0.023 | 0.793±0.013
PropAO | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ | 0.056±0.012 | 0.367±0.029 | 0.660±0.022 | 0.160±0.015 | 0.544±0.022 | 0.770±0.021

##  6 Conclusion

A major stumbling block in attaining diffraction-limited images over a wide
field using MCAO on solar telescopes is the dynamic change in registration
between the pupil DMs and the WFSs. While this phenomenon is well understood,
a simulation tool to study this problem and test potential solutions have not
been available until now.

In this paper, we present a new AO simulation tool, PropAO, based on the
PROPER optical propagation tool which propagates waves from plane to plane
using Fresnel propagation. The Python version of the PROPER tool has been
extended by adding features needed for an end-to-end MCAO simulation and has
been modified to run on GPUs using the CuPY library, considering the large
number of Fresnel propagations required. The simulation tool has been
extensively tested by comparing outputs from simple cases to the open-source
Monte-Carlo simulation tool, YAO, with excellent agreement between the codes.

The effect of the pupil distortion on the registration between the DMs and the
WFSs is readily seen in PropAO simulations. While the ultimate solution to
wavefront reconstruction in the presence of pupil distortion remains elusive,
it can be partially mitigated by using using a heavily regularized least-
squares reconstructor in conjunction with pseudo open-loop control. This leads
to a stable reconstruction for a wide range of observing conditions, and much
better performance than what is attained when decoupling the control of the
pupil DM and the high altitude DM using dedicated WFSs for each one.

PropAO will be used to simulate the MCAO system for the EST and to test
potential implementations of wavefront control. It can also be used for any
MCAO system for solar or night-time astronomy.

## Acknowledgments

This work was supported by the EST Project Office, funded by the Canary Island
Government (file SD 17/01) under a direct grant awarded to EST on ground of
public interest.

We extend our gratitude to Dirk Schmidt (NSO), who has been battling the
dynamic misregistration problem for many years, for his advice and help with
the manuscript. The authors would also like to thank John Krist (JPL) for
advice on the use of PROPER and Rodolphe Conan (GMT) for the suggestion to use
the CuPY library in place of the NumPY library.

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