Optimizing Fourier-Filtering WFS to reach sensitivity close to the fundamental limit V.Chambouleyrona O. Fauvarquec C. Plantetd J-F. Sauvageba N. Levraudab M. Ciss eab B.
2025-04-29
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Optimizing Fourier-Filtering WFS to reach sensitivity close
to the fundamental limit
V.Chambouleyrona, O. Fauvarquec, C. Plantetd, J-F. Sauvageb,a, N. Levrauda,b, M. Ciss´ea,b, B.
Neichela, and T. Fuscob,a
aAix Marseille Univ, CNRS, CNES, LAM, Marseille, France
bDOTA, ONERA, Universit´e Paris Saclay, F-91123 Palaiseau, France
cIFREMER, Laboratoire Detection, Capteurs et Mesures (LDCM), Centre Bretagne, ZI de la
Pointe du Diable, CS 10070, 29280, Plouzane, France
dINAF - Osservatorio Astronomico di Arcetri
ABSTRACT
To reach the full potential of the new generation of ground based telescopes, an extremely fine adjustment
of the phase is required. Wavefront control and correction before detection has therefore become one of the
cornerstones of instruments to achieve targeted performance, especially for high-contrast imaging. A crucial
feature of accurate wavefront control leans on the wavefront sensor (WFS). We present a strategy to design new
Fourier-Filtering WFS that encode the phase close from the fundamental photon efficiency limit. This strategy
seems promising as it generates highly sensitive sensors suited for different pupil shape configurations.
Keywords: Adaptive optics, wavefront sensing, Fourier-filtering wavefront sensors, noise propagation
1. INTRODUCTION
Adaptive optics (AO) is a technique allowing to compensate for the impact of atmospheric turbulence on tele-
scopes that has become essential for a large number of astrophysical applications. Motivated in particular by
the hunt for exoplanets,1AO systems pushing the limits of performance are currently being developed, called
extreme AO system (XAO). These systems relies on high-order deformable mirrors with fast real time compu-
tation. The fundamental limit of such instruments is based on the quality of the measurements provided by the
optical device at the heart of the AO system: the wavefront sensor (WFS). One key aspect driving the XAO
instruments is the WFS sensitivity, that can limit the number of controlled modes and the speed of the loop.
The Fourier Filtering WFS (FFWFS) represents a wide class of sensors of particular interest thanks to their
superior sensitivity. From a general point of view, a FFWFS consists of a phase mask located in an intermediate
focal plane which performs an optical Fourier filtering. This filtering operation allows the conversion of phase
information at the entrance pupil into amplitude at the pupil plane, where a quadratic sensor is used to record
the signal.2–4The goal of this paper is to optimise, in the sense of the highest possible sensitivity, the shape of
the critical optical component of a FFWFS: its focal plane filter device. To that end, we first present in section
2 a noise propagation model for FFWFS allowing to define quantitatively what we call their sensitivities. In
section 3, we present a new way to generate highly sensitive masks thanks to a numerical optimization of their
shape.
2. A NOISE PROPAGATION MODEL FOR ALL FFWFS
A new noise propagation model was developed for all kind of FFWFS.5This model, that was derived in the
small phase approximation, allows to link the estimation error σon a given mode due to noise propagation with
the number of photons Nph available for the measurement. This relationship is described through the following
formula:
Futher author information: vincent.chambouleyron@lam.fr
arXiv:2210.12744v1 [astro-ph.IM] 23 Oct 2022
σ2
φi=Nsap ×σ2
ron
s2(φi)×N2
ph
+1
s2
γ(φi)×Nph
(1)
Where φiis the considered mode, Nsap is the number of sampling points in the pupil, σ2
RON the detector
RON, and finally sand sγare the so-called sensitivities. One can spot two part in this formula: the left-hand
side is describing the RON noise propagation (it actually apply to any kind of uniform noise on the detector),
and the right hand side corresponds to the photon noise propagation. For a given mode, the two sensitivity
terms can be found using:
•The corresponding signal of the mode in the interaction matrix: δI(φi).
•The reference intensities I0, i.e the intensities corresponding to the reference wavefront (we assume a flat
wavefront for this study).
s(φi) = pNsap ×
δI(φi)
2and sγ(φi) =
δI(φi)
√I0
2(2)
These two sensitivities have no reason to be equal. Therefore, when comparing two FFWFS, it is required
to compare both sensitivities in order to have a fair comparison. In the rest of this paper, we will call ssimply
“sensitivity” because it actually corresponds to the sensitivity for any uniform noise on the detector (but the
main one for our application is the RON) and we will call sγthe “sensitivity to photon noise”. These two
sensitivities are bounded,6,7and they can’t reach value above 2. This value of 2 set an upper fundamental limit
for both sensitivities.
0≤s≤2
0≤sγ≤2(3)
The Zernike WFS (ZWFS) is often considered as the most sensitive WFS.3In a previous study, we used the
sensitivity metrics introduced previously to assess performance of ZWFS class.8It was shown that the classical
ZWFS (dot diameter p= 1.06 λ/D and phase depth δ=π/2) is actually not the most sensitive sensor (s= 0.9
and sγ= 1.2), and that its sensitivity can be improved by increasing the phase-dot diameter (we introduced the
Z2WFS: p= 2 λ/D and δ=π/2). The aim of this paper is to find even more sensitive configurations for which
sand sγare reaching values even closer from the maximum of 2. We will focus only on the sensitivity in the
framework of the linear regime. Hence, we will not consider the dynamic behaviour of the generated FFWFS in
this study.
3. A NEW TECHNIQUE TO CREATE HIGHLY SENSITIVE MASKS
3.1 Principle of the technique: convolutional model and numerical optimization
The convolutional model is a model which describes any FFWFS measurement as a convolutional operation
described by its impulse response and its associated transfer function. This model, introduced in,9gives an
interesting relationship between the shape of the filtering mask and the RON sensitivity map:
s≈p|TF|2? P SF (4)
where P SF corresponds to the Point-Spread Function in the plane of the filtering mask, and TF is the
transfer function of the considered FFWFS. This transfer function can be written as:
TF = 2Im(m ? m ×P SF ) (5)
where mis the WFS filtering mask. Hence, we have a 2D sensitivity function swhich takes as argument two
maps:
•P SF which corresponds to the Point-Spread Function at the filtering mask plane.
•m: the WFS filtering mask.
For a given pupil shape (and therefore a given PSF), we suggest here to numerically optimise the 2D sensitivity
map through Eq. 4and Eq. 5. This sensitivity map therefore depends only on the mask m:s=s(m). We call
starget the targeted sensitivity. We aim at inverting the problem thanks to numerical optimizations : given a
targeted sensitivity, it is possible to compute the corresponding mask. This technique is illustrated in a schematic
way figure 1.
Formula
vincent.chambouleyron
August 2021
1 Introduction
m|x,y “ei|x,y
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹m!q|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ipp
mq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
m|x,y “ei|x,y
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ipp
mq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
m|x,y “ei|x,y
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ipp
mq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
m|x,y “ei|x,y
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ipp
mq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
m|x,y “ei|x,y
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ipp
mq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyMASK
uvTARGET
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyMASK
uvTARGET
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyMASK
uvTARGET
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyMASK
uvTARGET
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyMASK
uvTARGET
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyUNKOWN mask
uvTARGET sensitivity
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyUNKOWN mask
uvTARGET sensitivity
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyUNKOWN mask
uvTARGET sensitivity
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyUNKOWN mask
non-linear inversion
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Formula
vincent.chambouleyron
August 2021
1 Introduction
xyMASK
uvTARGET
RELATION
!|u,v “PSF
|u,v fixed
s|u,v «a|TF|2‹PSF|u,v (1)
TF|u,v “2Impm‹mˆPSFq|u,v (2)
s|u,v “2
densit´e de probabilit´e de res
Ipq
BI
B|res
i
Ippmq
Ipq“D.“Dcalib.T.(3)
Ipq“Dcalib.G.(4)
Ipq“IR‹(5)
IR (6)
1
Figure 1. Mask optimization. The target sensitivity map is chosen to be maximal on a given spatial frequency range
(u, v).
The least-square criterion is used for the optimization, and the scoring function F(function to be minimized)
is written:
F(m) = ||starget −s(m)||2(6)
Fis a function which goes from Rkin Rwhere kis the number of parameters on which the mask is optimized.
It is therefore possible to implement non-linear numerical optimization methods in order to find a minimum of
this score function. However, it is not obvious that this function is convex: it can thus exhibit several local
minima, and it will consequently be difficult to be ensured to have reached the global minimum at the end of an
optimization process. To carry out this numerical optimization, we use the lsqnonlin function of MATLAB which
is a non-linear least squares problem solver using the Levenberg-Marquardt algorithm. This algorithm is based
on classical gradient descent methods. The goal of this study is to build a mask with an optimal sensitivity, so
the target sensitivity map is chosen to reach a value of 2 at each point in the frequency space: starget = 2. Our
score function thus becomes:
F(m) = ||2−s(m)||2(7)
An important point that must be emphasized: the optimization will be done here on the sensitivity with
respect to a uniform noise (like RON), because it is the one for which the convolutional model gives us an explicit
formula. We will show later that this study could be easily applied to photon noise sensitivity.
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OptimizingFourier-FilteringWFStoreachsensitivityclosetothefundamentallimitV.Chambouleyrona,O.Fauvarquec,C.Plantetd,J-F.Sauvageb,a,N.Levrauda,b,M.Cissea,b,B.Neichela,andT.Fuscob,aaAixMarseilleUniv,CNRS,CNES,LAM,Marseille,FrancebDOTA,ONERA,UniversiteParisSaclay,F-91123Palaiseau,FrancecIFREMER,Labora...
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