An analytic expression for coronagraphic imaging through turbulence. Application to on-sky coronagraphic phase diversity

0.82 MB·

by  Olivier Herscovici-Schiller

MNRAS000,1–6(2017) Preprint31January2017 CompiledusingMNRASLATEXstylefilev3.0
An analytic expression for coronagraphic imaging through
turbulence. Application to on-sky coronagraphic phase
diversity.
Olivier Herscovici-Schiller1(cid:63), Laurent M. Mugnier1 and Jean-Fran¸cois Sauvage1,2
1ONERA – The French Aerospace Lab, 92322 Chaˆtillon, France
2Aix-Marseille Universit´e, CNRS, Laboratoire d’Astrophysique de Marseille UMR 7326, 13388 Marseille, France
7
1
AcceptedXXX.ReceivedYYY;inoriginalformZZZ
0
2

n ABSTRACT
a The ultimate performance of coronagraphic high contrast exoplanet imaging systems
J
such as SPHERE or GPI is limited by quasi-static aberrations. These aberrations

0 produce speckles that can be mistaken for planets in the image. In order to design in-
3 struments,correctquasi-staticaberrationsoranalysedata,theexpressionofthepoint

spreadfunctionofacoronagraphicinstrumentinthepresenceofresidualturbulenceis
]
M most useful. Here we derive an analytic expression for this point spread function that
is an extension to coronagraphic imaging of Roddier’s expression for imaging through
I turbulence. We give a physical interpretation of its structure, we validate it by nu-
.
h mericalsimulationsandweshowthatitiscomputationallyefficient.Finally,weincor-
p porate this imaging model into a coronagraphic phase diversity method (COFFEE)
-
and validate by simulations that it allows wave-front reconstruction in the presence
o
r ofresidualturbulence.Thepreliminaryresults,whichgiveasub-nanometricprecision
t in the case of a SPHERE-like system, strongly suggest that quasi-static aberrations
s
a could be calibrated during observations by this method.
[
Key words: techniques: high angular resolution – turbulence – instrumentation:

1 adaptive optics – methods: analytical – methods: data analysis – techniques: image
v
processing
3
3
6
8
0 1 INTRODUCTION shapeandpropertiesofthecoronagraphicpointspreadfunc-
.
1 tion have been explored, e.g. in Perrin et al. (2003); Soum-
Direct detection of exoplanets is limited by the presence of
0 meretal.(2007b),andananalyticapproachhaspreviously
speckles on scientific images. These speckles are induced by
7 been developed in Sauvage et al. (2010). However, Sauvage
quasi-static aberrations in the optical system that are not
1 etal.(2010)wasbasedonthehypothesisofaperfectcoron-
: corrected by the adaptive optics loop. On SPHERE on the agraph,whichdoesnotexist,andyieldsunacceptableerrors
v VLT, the quasi-static aberrations can be calibrated on an
i when used for coronagraphic phase diversity, as shown by
X internalsourcethankstoCOFFEE(seePauletal.(2014)), Paul et al. (2013b).
anextensionofphasediversity(seeGonsalves(1982)forthe
ar principle of phase diversity, and Mugnier et al. (2006) for a The purpose of this letter is firstly topresent ageneral
review)tocoronagraphicimaging.However,thequasi-static analytic expression for coronagraphic imaging through tur-
aberrations evolve during the observations, so it would be bulence – which is an extension to the coronagraphic case
useful to calibrate them during the observations in order ofthewell-knownexpressionbyRoddier(1981)forimaging
to reach the ultimate performance of the instrument. Since throughturbulence–andsecondlytoshowthatitcanbein-
COFFEEusesamodeloftheinstrument,ananalyticexpres- tegrated into COFFEE to measure quasi-static aberrations
sionofthelongexposurepointspreadfunctionofacorona- with nanometric precision during observations.
graphicinstrumentinthepresenceoftheresidualturbulence Moreover, the expression that we derive in this letter
of an adaptive optics system is needed. couldbeofgreatuseforatleasttwootherapplicationsthan
To the best of our knowledge, no such general analytic wave-front sensing. One is the design, simulation and opti-
expression has been previously published, even though the misationofcoronagraphicsystems.Theotheristheprocess-
ing of high contrast images based on an imaging model, as
proposed by Ygouf et al. (2013), which is essential for the
(cid:63) E-mail:[email protected](OHS) detection and characterisation of exoplanets.
(cid:13)c 2017TheAuthors
L2 O. Herscovici-Schiller, L. M. Mugnier and J.-F. Sauvage
2 CLASSICAL IMAGING THROUGH
TURBULENCE AND INSTANTANEOUS
CORONAGRAPHIC POINT SPREAD
FUNCTION

2.1 Classical imaging through turbulence
Figure 1.Sketchofatelescopeequippedwithacoronagraph
Letusstartwithareminderofthenon-coronagraphiccaseof
imagingthroughturbulence.TheFouriertransformofEqua-
tion(4.15)oftheclassicaltextbyRoddier(1981)showsthat
function of the instrument without turbulence. It writes
the long exposure point spread function h of a telescope
le
through a (possibly residual) turbulent atmosphere is hc(α;ψu,ψd)=(cid:12)(cid:12)F−1(cid:8)ψd×F(cid:2)M×F−1(ψu)(cid:3)(cid:9)(α)(cid:12)(cid:12)2,
(3)
h =h (cid:63)h , (1)
le a s whereαisthevectorofangularcoordinatesofapositionon
where h is the point spread function of the optical system the sky, and M the coronagraphic focal plane mask. This
s
(which accounts for quasi-static aberrations), (cid:63) is the con- equation simply expresses the propagation from successive
volutionoperator,andh istheso-calledatmosphericpoint planes to the next ones and the quadratic detection, illus-
a
spread function, which is the inverse Fourier transform of trated in Figure 1. For notation simplicity we take λ = 1
the atmospheric transfer function h˜ : in the equations, that is to say we use the same coordinate
a
system for the pupil phases and for the transfer functions.
(cid:18) (cid:19)
1
h˜ =exp − D , (2) In the presence of turbulence, the instantaneous point
a 2 φ
spread function of this instrument writes
Dφbeingthephasestructurefunctionofthe(possiblyresid- h (α,t;ψ ,ψ )=h (α;ψ ×eiφt(t),ψ ). (4)
sec u d c u d
ual) atmospheric turbulence.
This expression is appealing because it untangles the Herehsec(α,t,ψu,ψd)(secstandsfor“shortexposurecoron-
influence of the atmospheric turbulence from the influence agraphic”) is the instantaneous point spread function taken
of the quasi-static aberrations on the global point spread atcoordinateαinthedetectorfocalplaneattimetandφt(t)
function. is the instantaneous residual turbulence phase at time t if
We show below that an analogous formula can be de- the telescope is ground-based and equipped with adaptive
rived for coronagraphic systems. optics.
2.2 Instantaneous point spread function of a 3 LONG EXPOSURE CORONAGRAPHIC
telescope equipped with a coronagraph POINT SPREAD FUNCTION
Let us consider an imaging system on a telescope equipped In current high contrast systems such as SPHERE or GPI,
withacoronagraphasdescribedinFigure1.Wewilldenote theexposuretimeislargewithrespecttothetypicalevolu-
byPutheamplitudetransmissioninapupilplanebeforethe tion time of corrected turbulence, so that the signal that is
coronagraph,or“upstreampupil”,φu thephaseaberrations actuallyrecordedbythedetectoristhelongexposurepoint
in this plane, which will be called“upstream aberrations” spread function, which is an average over time of the short
hereafter, and ψu = Pueiφu the complex amplitude in this exposure point spread function:
plane.Similarly,wewilldenotebyP thedownstreamtrans-
d
mission in a pupil plane after the coronagraph, or“down- hlec(α;ψu,ψd)=(cid:104)hsec(α,t;ψu,ψd)(cid:105)t. (5)
streampupil”(alsocalledtheLyotstopplane),φ thephase
d Here h is the long exposure coronagraphic point spread
aberrations in this plane, which will be called“downstream lec
function, and (cid:104)(cid:105) is the averaging over time. Let us denote
aberrations”hereafter, and ψd = Pdeiφd the multiplicative by f∗ the comptlex conjugate of any function f, and by ⊗
term–duetothedownstreamaberrations–thataffectsthe
the bi-dimensional correlation product:
complex amplitude of the wave that has propagated from
(cid:90)(cid:90)
theentranceaperturetothisplane.Forthesakeofsimplic- f ⊗g(r)= f∗(s)g(r+s)ds.
ity,wewanttokeepthesameorientationoftheimagesfrom
one focal plane to the following, and the same pupil image
In developed form, the long exposure coronagraphic
orientation from one pupil plane to another. For this rea-
pointspreadfunctionwrites,thankstoWiener-Khintchine’s
son, we rotate the axis of the downstream plane by 180Aˆˇr
theorem:
with respect to the upstream plane, and similarly we ro-
tate the orientation of the focal mask plane by 180Aˆˇr with hlec(α;ψu,ψd)=
(cid:68) (cid:16)(cid:110) (cid:104) (cid:16) (cid:17)(cid:105)(cid:111)
respect to the final focal plane. This axis change is taken F−1 ψ ×F M×F−1 ψ eiφt(t)
d u
intoaccountmathematicallybychoosinganinverseFourier
(cid:110) (cid:104) (cid:16) (cid:17)(cid:105)(cid:111)(cid:17) (cid:69)
transform (F−1) instead of a forward one (F) to model the ⊗ ψ ×F M×F−1 ψ eiφt(t) (α) .
d u
propagationoflightfromapupiltoafocalplane.Thiscon- t
(6)
vention allows for the image orientation to always remain
the same. Of course, our main result (Equation (11)) is in- Tosomewhatsimplifycalculations,wewillconsiderthe
dependent of this orientation convention. opticaltransferfunctioninsteadofthepointspreadfunction.
Let us denote by h the coronagraphic point spread Let h˜ = F(h ) be the Fourier transform of the point
c lec lec
MNRAS000,1–6(2017)
Coronagraphic imaging through turbulence L3
spreadfunction.Theintegraloverspacecommuteswiththe Finally,thelongexposurecoronagraphicpointspreadfunc-
averaging over time, so tion reads
h˜lec(r;ψu,ψd)= hlec(α;ψu,ψd,Dφ)=
(cid:68)(cid:110)ψd×F(cid:104)M×F−1(cid:16)ψueiφt(t)(cid:17)(cid:105)(cid:111) (cid:90)(cid:90) ha(α(cid:48);Dφ)(cid:12)(cid:12)(cid:12)F−1(cid:110)ψd.F(cid:104)M.F−1(cid:16)ψuei2πα(cid:48)·Id(cid:17)(cid:105)(cid:111)(α)(cid:12)(cid:12)(cid:12)2 dα(cid:48).
⊗(cid:110)ψ ×F(cid:104)M×F−1(cid:16)ψ eiφt(t)(cid:17)(cid:105)(cid:111)(r)(cid:69) . (10)
d u
t
This expression gives the long exposure coronagraphic
Inordertoexpresstheaverageovertime,wemustfullyde- point spread function as a function of three determinis-
velopthisexpression.Weintroducethepupil-planevariables tic parameters, namely: upstream aberrations, downstream
r1,r2andr3,thefocal-planevariablesα1andα2.Aftersome aberrations, and residual turbulence-induced phase struc-
manipulations, the fully developed expression reads ture function.
(cid:90)(cid:90) (cid:90)(cid:90) (cid:90)(cid:90)
h˜ (r;ψ ,ψ )= ψ∗(r )ψ (r+r ) ei2πr1·α1
lec u d d 1 d 1
×e−i2π(r1+r)·α2M∗(α )M(α ) 4 PHYSICAL INTERPRETATION
1 2
(cid:90)(cid:90) (cid:90)(cid:90)
× e−i2πr2·α1ei2πr3·α2ψ∗(r )ψ (r ) In order to interpret Equation (10) physically, we can re-
u 2 u 3
write it in the more compact following form:
(cid:68) (cid:69)
× ei[φt(r3,t)−φt(r2,t)] dr dr dα dα dr .
t 3 2 2 1 1 hlec(α;ψu,ψd,Dφ)=
Following Roddier (1981), we assume that turbulence is an (cid:90)(cid:90) h (α(cid:48);D )h (cid:16)α;ψ ei2πα(cid:48)·Id,ψ (cid:17) dα(cid:48), (11)
a φ c u d
ergodic stationary process (which is a very reasonable as-
sumptioninthecaseofresidualturbulenceafteranextreme
(cid:16) (cid:17)
adaptive optics system), so we write whereh α;ψ ei2πα(cid:48)·Id,ψ isthecoronagraphicpoint
c u d
(cid:68)ei[φt(r3,t)−φt(r2,t)](cid:69) =e−21Dφ(r3−r2), (7) slepnrceea,dbfuutnwctiitohnaotfilttheα(cid:48)inasdtdreudmteontthineuthpestraebasmencaebeorfrattuirobnus-,
t
thatistosaywiththelightcomingfromthestarbeingtilted
whereD istheturbulentphasestructurefunction,defined
φ by an angle α(cid:48).
by
A physical interpretation of the atmospheric point
spread function defined in Eq. (2) can shed some light on
D (r)=(cid:68)ei[φt(r(cid:48),t)−φt(r+r(cid:48),t)](cid:69) . (8) the expression of Eq. (11). Recall that
φ
t
in EIqnuaotriodner(8t)o, bwee atabklee ttoheseinpvaerrasteeFvoauriraiebrletsrarn2sfaonrmd ro3f ha (cid:44)F−1(cid:20)exp(cid:18)−12Dφ(cid:19)(cid:21)=F−1(cid:104)(cid:68)ei[φt(r+r(cid:48),t)−φt(r(cid:48),t)](cid:69)t(cid:105).
e−21Dφ.Wedenoteα(cid:48) theconjugatevariableofr3−r2,and If we note ψt(r,t) = exp(iφt(r,t)) the contribution to
h istheatmosphericpointspreadfunctiondefinedbyEqua- theelectricfieldatpositionrandtimetbytheatmospheric
a
tion(2).Wewillcomebacktothemeaningofh inSection4. turbulence, we can re-write h as :
a a
We obtain: h =F−1(cid:2)(cid:10)ψ∗(r(cid:48),t)ψ (r+r(cid:48),t)(cid:11) (cid:3) (12)
h˜ (r;ψ ,ψ ,D )= a t t t
lec u d φ
Assumingstationarinessandergodicity,werecognizethein-
(cid:90)(cid:90) (cid:90)(cid:90) (cid:90)(cid:90)
ha(α(cid:48);Dφ) ψd∗(r1) ei2πr1·α1M∗(α1) verseFouriertransformoftheauto-correlationofψt.Hence
we can identify h as the energy spectral density of the
(cid:90)(cid:90) a
× e−i2πr2·α1ψu∗(r2)e−i2πr2·α(cid:48) dr2 dα1 turbulence-induced complex field. In other words, ha(α(cid:48))
gives the fraction of the energy of light that is diffracted
(cid:90)(cid:90) in each direction α(cid:48).
×ψ (r+r ) e−i2π(r1+r)·α2M(α )
d 1 2 Finally, Equation (11) can be interpreted as an inco-
(cid:90)(cid:90) herent plane wave decomposition: the long exposure coro-
× ei2πr3·α2ψ (r )ei2πr3·α(cid:48) dr dα dr dα(cid:48).
u 3 3 2 1 nagraphic point spread function h is a weighted sum of
lec
coronagraphic point spread functions h , without any tur-
c
In a more compact form, using Id to denote the identity bulence,butwithanupstreamtiltα(cid:48).Theweightonanyof
function, this equation can be written:
thesetiltedpointspreadfunctionsisthefractionofthelight
h˜ (r;ψ ,ψ ,D )= energy ha(α(cid:48)) that the atmosphere scatters in the direction
lec u d φ of the corresponding tilt α(cid:48).
(cid:90)(cid:90) h (α(cid:48);D )(cid:110)ψ ×F(cid:104)M×F−1(cid:16)ψ ei2πα(cid:48)·Id)(cid:17)(cid:105)(cid:111) Moreover,theexpressionofEquation(11)separatesthe
a φ d u
turbulent part and the coronagraphic part of the problem:
⊗(cid:110)ψd×F(cid:104)M×F−1(cid:16)ψuei2πα(cid:48)·Id)(cid:17)(cid:105)(cid:111)(r)dα(cid:48). ha codesforthecharacteristicsoftheturbulentatmosphere,
whileh codesforthecharacteristicsoftheinstrument.This
(9) c
separationofthepointspreadfunctionintoanatmospheric
Toobtainthepointspreadfunctionbackfromthisopti- pointspreadfunctionandapointspreadfunctionofthein-
caltransferfunction,wejusthavetotaketheinverseFourier strumentisstrikinglysimilartotheonefoundintheRoddier
transform, then apply Wiener-Khintchine’s theorem again. expression for non-coronagraphic imaging.
MNRAS000,1–6(2017)
L4 O. Herscovici-Schiller, L. M. Mugnier and J.-F. Sauvage
5 SPECIAL CASES Thus we can express h :
lec
5.1 Non-turbulent point spread function h (α;ψ ,ψ )(cid:39)
lec u d
As a first particular case, let us assume that there is no e−σφ2 ×(cid:20)hc(α;ψu,ψd)+(cid:90)(cid:90) Sφ(α(cid:48))hc(cid:16)α;ψuei2πα(cid:48),ψd(cid:17) dα(cid:48)(cid:21)
turbulence, so that h is reduced to a Dirac distribution.
a
Indeed, in the case where there is no turbulence, φ is a (16)
t
(cid:68) (cid:69)
constant, so Dφ = [φt(r(cid:48))−φt(r+r(cid:48))]2 r(cid:48) =0. Then, ac- This means that hlec is approximately the non-turbulent
cording to Equation (2), h =δ. Hence, by use of Equation coronagraphicpointspreadfunction,withacorrectiveaddi-
a
(11), tivetermthattakesintoaccountthepowerspectrumdensity
of the turbulent phase, all this dampened by the coherent
hlec(α;ψu,ψd)=hc(α;ψu,ψd), (13) energy e−σφ2.
whichispreciselytheclassicalexpressionforacoronagraphic
point spread function in the absence of turbulence.
6 NUMERICAL VALIDATION AND
EFFICIENCY CONSIDERATIONS
5.2 Non-coronagraphic optical transfer function
6.1 Numerical validation
As a second particular case, if we now consider a non-
coronagraphic instrument, M = 1, so Equation (10) reads To validate our point spread function model, we test it
againstanaverageofshortexposurepointspreadfunctions,
each one of them with a different outcome of the residual
h (α;ψ ,ψ ,D )
lec u d φ turbulence-inducedphase.Eachshortexposurepointspread
=(cid:90)(cid:90) h (α(cid:48);D )(cid:12)(cid:12)F−1(cid:110)ψ ψ ei2πα(cid:48)·Id(cid:111)(α)(cid:12)(cid:12)2 dα(cid:48) function is computed using a matrix Fourier transform as
a φ (cid:12) d u (cid:12)
proposedbySoummeretal.(2007a)inordertoperforman
=ha(Dφ)(cid:63)(cid:12)(cid:12)F−1{ψdψu}(cid:12)(cid:12)2(α), accurate computation of the field on the mask focal plane.
A few averages of point spread functions are displayed on
(14)
Figure 2. They are simulated with a Lyot coronagraph of
whichisidenticaltoEquation(1),withhs =(cid:12)(cid:12)F−1{ψdψu}(cid:12)(cid:12)2, diameter 3λ/D, the residual turbulence of a SPHERE-like
that is to say that the aberration is the sum of the up- adaptive optics and upstream and downstream white noise
streamanddownstreamaberrations.Thelongexposureop- phase aberrations of variance 0.1rad2.
tical transfer function is thus Let us define the convergence error as follows. We de-
note by h the long exposure point spread function us-
h˜lec(r;ψu,ψd)=h˜a(r)×(ψdψu⊗ψdψu)(r), (15) ingtheanalelcyticformulaofEquation(11).WedenotehsseN
the average of N short exposures. We define the error as
whichistheclassicalformulabyRoddier(1981)ofthelong
err = max(|h −h |/h ), where the division is
exposure transfer function for imaging through turbulence; N sseN lec lec
takenpixelbypixel.Wehaveperformedasimulationofthe
indeed the autocorrelation of ψ ψ is the optical transfer
d u evolutionof error versus N, the number of shortexposures.
functionoftheinstrumentduetodiffractionandquasi-static
The error tends to zero as N−1/2, which is consistent with
aberrations.
the central limit theorem. The error goes below 10−3 just
after N =107 short exposures.
5.3 Approximation in the case of weak turbulence
Additional insight on Equation (11) can be gained by as- 6.2 Computing cost
suming that the residual turbulence phase after the adap-
Theevolutionoftheconvergenceerrorgivesusaneasycrite-
tive optics is a small perturbation. It is straightforward to
riontoquantifythecomparativecomputingcostsofthelong
see that
exposure coronagraphic point spread function and the em-
1D (r)=σ2 −A (r), piricalaverageofshortexposures.Letustakethecomputing
2 φ φ φ cost of a short exposure point spread function as the unit
where A (r)=(cid:104)φ (r(cid:48))φ (r+r(cid:48))(cid:105) is the autocorrelation of computingcost.Then,ifwewantanerroroflessthan10−3
φ t t r(cid:48) onthepointspreadfunction,approximately107 shortexpo-
the phase, hence
sure point spread functions must be averaged, for a cost of
ha =e−σφ2F−1(cid:104)eAφ(cid:105). 107.Theanalyticformulaforthelongexposurepointspread
function has a total cost of the number of points on which
Then,iftheturbulenceissmall,A issmallcomparedto1, the phase structure function is known. This implies that,
φ
and we can perform a first order Taylor expansion: for square images of 512×512 pixels, our exact formula is
about 38 times less costly to evaluate than an average of
ha(α(cid:48))(cid:39)e−σφ2F−1[1+Aφ](α(cid:48)) short exposures.
ha(α(cid:48))(cid:39)e−σφ2δ(α(cid:48))+e−σφ2Sφ(α(cid:48)) In addition, it should be noted that, since the long ex-
posureisanintegral,andthus,inpractice,asum,itisvery
where S is the power spectrum density of the turbulent easytocomputeitinparallelonseveralprocessors.Wemade
φ
phase φ , defined as the (inverse) Fourier transform of A . all calculations in parallel on 16 cores.
t φ
MNRAS000,1–6(2017)
Coronagraphic imaging through turbulence L5
Figure2.Pointspreadfunctions.Fromlefttoright:shortexposure,averagesof10,100and1000exposures,andanalyticlongexposure
calculatedwithequation11.
Finally,itshouldalsobenotedthatausefulapproxima-
tioncanacceleratethecomputingofthelongexposurepoint
spread function by a great deal. Indeed, when the tilt α(cid:48) is
sufficiently greater than the radius of the focal-plane mask
of the coronagraph, the point spread function can be well
approximated by a shifted non-coronagraphic point spread
function that can be computed once and for all, so the sum
in Equation (11) for the computation of the long exposure Figure3.Left:upstreamaberrationsthatweaimtoreconstruct.
pointspreadfunction mustactually only be computedon a Center left: reconstructed aberration. Center right: difference,
square of typical width 8λ/D. For a Nyquist-rate sampled magnifiedahundredtimes.Right:modulusoftheFouriertrans-
image,thatisonly16×16shortexposurepointspreadfunc- formofthephaseestimationerror
tions, with a maximum relative error of less than 10−3 on
each pixel for SPHERE-like parameters.
in this case is much less than 1 nm RMS, which is a very
encouraging result. Additionally, the usefulness of this re-
construction is even better than what the estimated error
7 APPLICATION: ESTIMATION OF
suggests.Indeed,ifweexaminetheFouriertransformofthe
QUASI-STATIC ABERRATIONS WITH
difference on Figure 3, we notice that the error is mainly
COFFEE IN THE PRESENCE OF RESIDUAL
located on the spatial frequencies that are on the border of
TURBULENCE
thearray,thatistosay,thosethatarenotcorrectedbythe
7.1 Motivation deformablemirror.Quantitatively,theerrorinthecorrected
zoneaccountsforlessthan10%ofthetotalestimationerror
In current systems such as SPHERE, quasi-static aberra- variance.
tions are only corrected during day-time. For an upgrade,
or for future high contrast instruments on extremely large
telescopes, it would be useful to correct them during the
8 CONCLUSION AND PERSPECTIVES
scientificacquisition.Toachievethis,wecombineCOFFEE
(see Paul et al. (2013a), Paul et al. (2014)) with the long To summarize this work, we have derived an analytic ex-
exposurephasediversityproposedbyMugnieretal.(2008). pression to model long exposure image formation for a
coronagraphic telescope through the residual turbulence of
an extreme adaptive optics system. This model is general
7.2 Simulation results
and valid for any coronagraphic mask. It only depends on
We have incorporated the long exposure coronagraphic three deterministic quantities: quasi-static aberrations be-
imaging model developed above into the coronagraphic foreandafterthecoronagraph,andtheresidualphasestruc-
phase diversity method COFFEE. In the following, we per- ture function. Its expression shows, quite nicely, that the
form a preliminary validation of the quasi-static aberration long exposure coronagraphic point spread function is the
estimationwithCOFFEEinthepresenceofresidualturbu- weightedsumofcoronagraphicpointspreadfunctionswith-
lence by means of simulation. The parameters of the sim- out turbulence witha weightdistributiongiven only by the
ulations are the following: the phase map size is 64×64 turbulence-induced residual phase structure function. It is
pixels,theincomingfluxis109 photoelectrons,thestandard computationally effective, allowing a time gain of at least
deviationofthereadoutnoiseis1electron,the(supposedly anorder of magnitudeonstandardsizedimages.Lastly,we
monochromatic) wavelength is 1589 nm. The phase struc- have obtained numerical results that suggest that this long
ture function is typical of the residual turbulence of the ex- exposure model can be applied to the on-sky measurement
treme adaptive optics system of the SPHERE instrument andcorrectionofaberrationsinthecontextofground-based
of the VLT, the coronagraph is a Roddier & Roddier phase direct exoplanet detection.
mask, and there are no downstream aberrations. The up- These encouraging results suggest that we carry on
stream aberration is 50 nm RMS. Figure 3 shows the true working in two directions. On the one hand, we will ex-
upstream aberrations, the estimated upstream aberrations, ploretherobustnessoftheCOFFEEestimationstoanerror
and the difference between the upstream aberrations and on the phase structure function of the residual atmospheric
the estimated ones, magnified a hundred times. The error turbulence. On the other hand, we will demonstrate exper-
MNRAS000,1–6(2017)
L6 O. Herscovici-Schiller, L. M. Mugnier and J.-F. Sauvage
imentally that COFFEE can estimate aberrations through
turbulence, first on a laboratory bench and then on sky.
ACKNOWLEDGEMENTS
The PhD work of O. Herscovici-Schiller is co-funded by
CNESandONERA.WethankJ.-M.LeDuigouforhissup-
port. This work received funding from the E.U. under FP7
Grant Agreement No. 312430 OPTICON, from the CNRS
(D´efi Imag’In), and from ONERA in the framework of the
VASCOresearchproject.WethankDrFaustineCantalloube
for useful discussions. We thank the reviewer for his careful
review and constructive comments.
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