Speckle temporal stability in XAO coronagraphic images II. Refine model for quasi-static speckle temporal evolution for VLT/SPHERE
P. Martinez, M. Kasper, A. Costille, J.F. Sauvage, K. Dohlen, P. Puget, J.L. Beuzit
aa r X i v : . [ a s t r o - ph . I M ] F e b Astronomy&Astrophysicsmanuscript no. SpeckleStability c (cid:13)
ESO 2018October 9, 2018
Speckle temporal stability in XAO coronagraphic images(Research Note)
II. Refine model for quasi-static speckle temporal evolution for VLT/SPHERE
P. Martinez , M. Kasper , A. Costille , J.F. Sauvage , K. Dohlen , P. Puget , and J.L. Beuzit Laboratoire Lagrange, UMR7293, Universit´e de Nice Sophia-Antipolis, CNRS, Observatoire de la Cˆote d´Azur, Bd. del´Observatoire, 06304 Nice, France European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748, Garching, Germany UJF-Grenoble 1 / CNRS-INSU, Institut de Plan´etologie et d’Astrophysique de Grenoble UMR 5274, Grenoble, F-38041, France Laboratoire d’Astrophysique de Marseille, UMR 7326, CNRS, Universit´e de Provence, 38 rue Fr´ed´eric Joliot-Curie, 13388,Marseille Cedex 13, France O ffi ce National d´Etudes et de Recherches Aerospatiales (ONERA), Optics Department, BP 72, F-92322 Chatillon cedex, FrancePreprint online version: October 9, 2018 ABSTRACT
Context.
Observing sequences have shown that the major noise source limitation in high-contrast imaging is due to the presence ofquasi-static speckles. The timescale on which quasi-static speckles evolve, is determined by various factors, among others mechanicalor thermal deformations.
Aims.
Understanding of these time-variable instrumental speckles, and especially their interaction with other aberrations, referredto as the pinning e ff ect, is paramount for the search of faint stellar companions. The temporal evolution of quasi-static speckles isfor instance required for a quantification of the gain expected when using angular di ff erential imaging (ADI), and to determine theinterval on which speckle nulling techniques must be carried out. Methods.
Following an early analysis of a time series of adaptively corrected, coronagraphic images obtained in a laboratory conditionwith the High-Order Test bench (HOT) at ESO Headquarters, we confirm our results with new measurements carried out with theSPHERE instrument during its final test phase in Europe. The analysis of the residual speckle pattern in both direct and di ff erentialcoronagraphic images enables the characterization of the temporal stability of quasi-static speckles. Data were obtained in a thermallyactively controlled environment reproducing realistic conditions encountered at the telescope. Results.
The temporal evolution of the quasi-static wavefront error exhibits linear power law, which can be used to model quasi-staticspeckle evolution in the context of forthcoming high-contrast imaging instruments, with implications for instrumentation (design,observing strategies, data reduction). Such a model can be used for instance to derive the timescale on which non-common pathaberrations must be sensed and corrected. We found in our data that quasi-static wavefront error increases with ∼ Key words.
Techniques: high angular resolution –Instrumentation: high angular resolution –Telescopes
1. Introduction
Observing sequences have shown that the major noise sourcelimitation in high-contrast imaging is due to the presence of in-strumental speckles, and more precisely to quasi-static speckles(Marois et al. 2003; Boccaletti et al. 2003, 2004; Hinkley et al.2007). Speckle noise originates from wavefront errors caused byvarious independent sources, and evolves on di ff erent timescalespending to their nature. The first class of speckle to overcomecomes from the large, dynamical wavefront error that the atmo-sphere generates, but real-time adaptive optics systems measureand correct it down to fundamental limitations. The fast-varyingspeckle noise floor left uncorrected by such systems would av-erage out over time, as it consists in a random and uncorrelatednoise for which the intensity variance converges to null contribu-tion for an infinitely long exposure. For a non-photon noise lim-ited observing run, speckle noise associated to wavefront aber-rations introduced in the optical train are fundamental to tackle.In this context, instrumental speckles can be divided into twodi ff erent flavors: long-timescale wavefront errors present in the Send o ff print requests to : [email protected] optical train (e.g., optical quality, misalignment errors) that gen-erate static speckles that constitute a deterministic contributionto the noise variance, and slowly-varying instrumental wavefrontaberrations, amplitude and phase errors, originating from vari-ous causes, among others mechanical or thermal deformations.The latest evolve on a shorter timescale than long-lived aberra-tions, and are the so-called quasi-static speckles.Instrumental speckles average to form a fixed pattern, whichcan be calibrated to a certain extent. A deterministic contribu-tion to the noise variance such as static speckles can easily becalibrated, while using a reference image, time-variable quasi-static noise can be subtracted as well, but its temporal evolu-tion ultimately limit this possibility. In particular it is under-stood that some timescales have a larger impact than others.This is especially true as quasi-static speckles interact with otheraberrations, referred to as the pinning e ff ect , or speckle crossterms. The timescale of quasi-static speckles evolution is es-sential to understand and predict the performance of the newgeneration of instruments such as SPHERE (Beuzit et al. 2008),GPI (Macintosh et al. 2008), HiCIAO (Hodapp et al. 2008), andProject 1640 (Hinkley et al. 2011). The temporal evolution of these quasi-static speckles is in particular needed for the quan-tification of the gain expected with angular di ff erential imaging(ADI, Marois et al. 2006), as well as to determine the timescaleon which speckle nulling techniques should be carried out. Forinstance, a typical hour-long ADI observing sequence provides apartial self-calibration of the residuals after a rotation of ∼ λ/ D at a given angular separation, which generally requires less thanfew minutes (e.g., 5 to 7 mn at 1 ′′ on a 8-m class telescopefor stars near the meridian in H -band), though it depends onwavelength, telescope latitude, and object declination. Residualspeckles with decorrelation times faster than the time neededto obtain the ADI reference image cannot then be removed,while quasi-static speckles associated with larger timescales canlargely be subtracted.In this context, several authors have investigated the decor-relation timescale of quasi-static residuals in the particular con-text of ADI but at moderate 20-40 % Strehl levels (Marois et al.2006; Lafreni`ere et al. 2007; Hinkley et al. 2007), while in a for-mer paper (Martinez et al. 2012, hereafter Paper I), we exploredthe realm of very high Strehl ratios (extreme adaptive optic sys-tems, XAO). In paper I, we analyzed a time series of adap-tively corrected, coronagraphic images recorded in the labora-tory with the High-Order Test bench, a versatile high-contrastimaging, adaptive optics bench developed at ESO. We shownthat quasi-static aberrations exhibit a linear power law with timeand are interacting through the pinning e ff ect with static speck-les. We examined and discussed this e ff ect using the statisticalmodel of the noise variance in high-contrast imaging proposedby Soummer et al. (2007). In particular, we found that quasi-static speckles, fast-evolving on the level of a few angstroms tonanometers over a timescale of few seconds, explained the evo-lution of our sensitivity through amplification of the systemat-ics. It is believed that this e ff ect is a consequence of thermal andmechanical instabilities of the optical bench. The HOT bench isindeed localized in a classical laboratory setting, and was notinitially designed to guarantee stability, nor mechanical stabilityat the level that would be required / expected for an actual high-contrast imaging instrument. Indeed Paper I emphasizes the im-portance of such stability for the next generation of high-contrastinstruments, but the estimates found in this former analysis (am-plitude, and slope of the temporal evolution) could not fairly beconsidered as representative of a realistic situation in order toa ff ect operational aspects or designs of nowadays / future real in-struments.In this paper, we confirm the results presented in Paper I withmore representative measurements. We analyze a time series ofadaptively corrected, coronagraphic images with the SPHEREinstrument in a thermally actively controlled environment repro-ducing realistic conditions encountered at the telescope. In thiscontext, we propose a refine model of quasi-static speckle evo-lution that can be used for forthcoming high-contrast imaginginstrument classes that SPHERE represents.The paper reads as follow: in Sect. 2, we briefly recall theformalism of the statistical model of the noise variance in high-contrast imaging proposed by Soummer et al. (2007) and used inPaper I to discuss our former results, in Sect. 3, the experimentalconditions are described, and in Sect. 4 we analyze and discussthe results. Finally, in Sect. 5, we draw conclusions.
2. Speckle noise and dynamical range
In Paper I, we examined and discussed our data using the statis-tical model of the noise variance in high-contrast imaging pro- posed by Soummer et al. (2007). Following the same formalismas for Paper I, we briefly recall here the main equations.Soummer et al. (2007) proposed the following analytical ex-pression for the variance of the intensity, including speckle andphoton noise in the presence of static, quasi-static, and fast vary-ing aberrations, in the context of a propagation through a coron-agraph: σ I = N (cid:16) I s + NI s + I c I s + NI c I s + I s I s (cid:17) + σ p , (1)where I denotes the intensity, σ p is the variance of the photonnoise, and N is the ratio of fast-speckle and slow-speckle life-times. The intensity produced by the deterministic part of thewavefront, including static aberrations, is denoted by I c , whilethe I s terms corresponds to the halo produced by random inten-sity variations, i.e. atmospheric ( I s ) and quasi-static contribu-tions ( I s ). In this generalized expression of the variance, severalcontributions can be identified by order of appearance: (1 / ) theatmospheric halo, (2 / ) the quasi-static halo, (3 / ) the atmosphericpinning term, the speckle pinning of the static aberrations by thefast evolving atmospheric speckles, (4 / ) the speckle pinning ofthe static by quasi-static speckles, and finally (5 / ) the specklepinning of the atmospheric speckles by quasi-static speckles.Equation 1 provides useful insights in the understanding of theimpact of quasi-static speckles and their interactions through thepinning phenomenon with other aberrations present in a real in-strument.As for Paper I, we focus our analysis on contribution (4 / ) tothis noise budget in XAO coronagraphic images. In particular,we are interested in the speckle pinning of the static by the quasi-static speckles when no atmospheric contribution is present (dy-namical speckles), i.e., when the contribution of I s to Eq. 1 canbe neglected. Furthermore, our study concerns a situation wherethe photon noise is not limiting, so that the contribution σ p fromthe noise variance can be neglected. In such conditions, Eq. 1can be simplified such that: σ I ≃ (cid:16) I s + I c I s (cid:17) , (2)and the present study focuses on the e ff ect of the cross-term I c I s .Since we can fairly assume that I s ≪ I c , I s can be neglectedexcept in the cross-term, and the noise variance in the raw coro-nagraphic image finally becomes: σ I ≃ I c I s . (3)Ultimately, very deep dynamic range imager, such as SPHERE,or GPI, aim to calibrate static speckles ( I c ) such that I c ≈ I s ,which would largely reduce speckle pinning. Speckle nullingtechniques to correct for remnant quasi-static aberrations wouldbe a must to access deeper contrast level, nonetheless, the tem-poral characteristic of I s remains a key parameter in these cir-cumstances (Eq. 2).From Eq. 3 a breakdown of this pinning e ff ect can be carriedout at the level of di ff erential images. Raw coronagraphic im-ages are dominated by static speckle noise. This means that theinteraction between the quasi-static terms of Eq. 3, being time-dependent, and static terms, assumed time-independent, can bestudied through di ff erential imaging from a time series of rawcoronagraphic images, which simply refers to the di ff erence inintensity between an image recorded at time t +∆ t and the refer-ence image registered at t . In this situation, a similar expressionof the noise variance for the di ff erential images ( σ DI ) can be de-rived as the di ff erence of Eq. 3 evaluated at t + ∆ t , to that of thereference, at t , and reads: σ DI ≃ I c ∆ I s , (4) where ∆ I s represents the quasi-static evolution between the twosuccessive images. Therefore, the quasi-static contribution canbe expressed as: ∆ I s ≃ σ DI I c . (5)A general expression of the speckle intensity (Racine et al.1999) is: I speckle ≈ (1 − S )0 . , (6)where S can be related to the wavefront error φ using Mar´echal’sapproximation (Born & Wolf 1993): S ≈ − πφλ ! , (7)and the contribution from static speckles to the wavefront error( φ s ≪ φ c , and φ s neglected) can be expressed as: φ c ≃ λπ √ × p I c . (8)Similarly, the contribution from quasi-static speckles to thewavefront error in the di ff erential images can be expressed as: ∆ φ s ≃ λ π √ × σ DI √ I c . (9)Using Eqs. 8 and 9, the analysis of both direct and di ff eren-tial images allows to characterize the temporal properties ofstatic and quasi-static aberrations. We note that both Eq. 8and 9 are approximated expressions relying on simple rule ofthumbs / assumptions (e.g., general expression of speckle inten-sity, low phase aberration regime) to provide wavefront estima-tion per Fourier component.
3. Experimental conditions
SPHERE which stands for Spectro-Polarimetric High-contrastExoplanet REsearch, is a second generation instrument for theVery Large Telescope, aiming at direct detection and spectralcharacterization of extrasolar planets. The instrument is nownearing completion in its final integration stage in Europe, be-fore shipping to Chile. Being in its final test phase, it o ff ers aunique opportunity to carried out research on static and quasi-static speckles. SPHERE is a unique instrument, with first lightin 2013, including a powerful extreme adaptive optics system(SAXO), an infrared di ff erential imaging camera (IRDIS), anintegral field spectrograph (IFS), and a visible di ff erential po-larimeter (ZIMPOL). The time series of adaptively corrected,coronagraphic images that will be discussed through this paperhave been obtained using SPHERE with the IRDIS instrument,so that for the sake of clarity only the systems that have beenused and are relevant will be further described.The SPHERE adaptive optics for exoplanet observation(SAXO) uses a 41 ×
41 actuator deformable mirror (DM) of 180mm diameter with inter-actuator stroke > ± µ m and a maximumstroke > ± µ m, and a 2-axis tip-tilt mirror (TTM) with ± ×
40 sub-aperturesShack-Hartmann sensor equipped with a spatial filter for alias-ing minimization. During the test, no dynamical turbulence waspresent in the system, except the low internal turbulence (opticalelements are installed inside an hermetical enclosure). Hence, SAXO is used to correct for internal turbulence, static aberra-tions, and guarantee image and pupil stability. As image andpupil stability are essential in high-contrast imaging, di ff erentialimage movements due to thermo-mechanical e ff ects are mea-sured in real-time using an auxiliary NIR tip-tilt sensor locatedclose to the coronagraph focus plan, and corrected with a di ff er-ential tip-tilt mirror in the wavefront sensor arm. Similarly, pupilrunout is accounted for and corrected by a pupil tip-tilt mirror atthe entrance of the instrument. The near-IR Strehl ratio is >
95 %.In particular, non-common path aberrations (from the DM to thedetector) are measured o ff -line using a phase diversity algorithmand compensated by reference slopes adjustment. In these con-ditions, static wavefront errors left uncorrected in the system areestimated at the level of ∼ ff ers various coronagraphic possibilities (classi-cal Lyot coronagraphs, apodized Lyot coronagraphs – ALC –,and achromatic four quadrants phase masks). During the exper-iment, a 5.2 λ/ D ALC has been used.The infra-red dual beam imaging and spectroscopy (IRDIS)sub-system includes a spectral range from 950 to 2320 nm andan image scale of 12.25 mas per pixel (Nyquist sampling at950nm). The field of view is greater than 11 ′′ square, with a2kx2k Hawaii-II-RG detector. The main mode of IRDIS is thedual band imaging (DBI), providing images in two neighbor-ing spectral channels with minimized di ff erential aberrations.Ten di ff erent filter couples are defined corresponding to di ff erentspectral features in modeled exoplanet spectra. During the exper-iment the narrow H -band couple filters have been used (centeredaround 1593 nm, and 1667 nm, R = ff erentwavelengths for comparison / confirmation purpose.Finally, the SPHERE instrument is installed on an active tripoddamping system, and fully covered by an hermetic enclosure. The SPHERE instrument must conform to several environmentalspecifications. In particular, it shall operate under a temperaturerange from 5 to 18 ◦ C, while the highest gradient of temperatureat the VLT is -0.9 ◦ C / h and -1.4 ◦ C / h for respectively 80% and95% of the nights. In order to validate that SPHERE is compliantwith such conditions, cold tests have been set up for functionaland performance evaluation at di ff erent ambient temperatureswithin the operational range, and with realistic transient condi-tions reflecting situations encountered at the VLT. The wholeSPHERE instrument has been installed in a cold tent (150 m )and cooled down with an e ffi cient air conditioning system. Thecooling system allows to reduce by ∼ ◦ C the temperature in-side the tent compared to the outside (in the integration hall).Numerous temperature probes have been installed at several crit-ical locations in the instrument to accurately monitored the evo-lution inside / outside the SPHERE enclosure. The observing run consists in recording a time series of AOclose-loop IRDIS coronagraphic images. Each coronagraphicimage of the time series corresponds to a series of 3 s short expo-sure images, averaged out over ∼ ◦ C with a rate of
Fig. 1.
Coronagraphic images recorded at t , t +
10 mn, and t +
100 mn. The Strehl ratio is ∼
95 %. The arbitrary color scale anddynamic range (identical for the three images) were chosen to enhance the contrast for the sake of clarity. The field covered in theimages is ∼ ′′ square. Fig. 2. Di ff erential coronagraphic images. Left: di ff erence of the t +
10 mn image to the reference t . Right: di ff erence of the t +
100 mn image to the reference t . The increase in the strength of residuals can be qualitatively observed here (the intensity rangeis similar in both images).2 ◦ C per hour. The total duration of the experiment was 100 mn.The gradient inside the enclosure was estimated at the level of0.2 ◦ C / h. The data-reduction process corrects for bad pixels andbackground, and normalizes the images with respect to exposuretime and flux. Before subtraction, a fine, sub-pixel correctionof the residual tip-tilt component is performed on the raw im-ages. The analysis has been done on the two H -band filters dataset available and converge to similar results, so that only onedata set is presented here. Depending on the nature of the im-age analyzed, we applied di ff erent metrics. The contrast refersto the ratio of intensity in the raw coronagraphic image, averagedazimuthally at a given angular separation, to the peak intensityof the direct flux. When studying a di ff erential image, implyingthe subtraction of a reference image, the average contrast is nolonger suited. The detectability is used then, which stands for theazimuthal standard deviation measured in a ring of width λ/ D . It quantifies the ability to distinguish a companion at a given an-gular distance.
4. Analysis and interpretation
Figure 1 presents three coronagraphic images extracted from thetime series: the coronagraphic image recorded at t (the refer-ence), t + t + ff erence of the t +
10 mn imageto the reference t (left), and the di ff erence of the t +
100 mnimage to the reference t (right). The corresponding profiles forboth raw and di ff erential images are presented in Fig. 3.Qualitatively, the raw coronagraphic images (Fig. 1) demon-strate starlight attenuation, and exhibit static speckles with lowerintensity in the AO-correction domain (from the second or thirdwing of the coronagraphic image to the rise of the AO cut-o ff fre-quency). A radial trend in speckle intensity is observable in the Fig. 3.
Contrast profiles of a time series of coronagraphic images(top) and detectability (1 σ ) of the di ff erential images (subtrac-tion of the time series of coronagraphic images to the referenceone, bottom)image: speckles closer to the center of the image are brighter.The central part of the coronagraphic image is dominated bydi ff raction residuals. The AO cut-o ff frequency can be readilyseen in the image owing to the slope of intensity in the specklefield at 0.8 ′′ (20 λ / D) from the center, as expected. Outside theinner-domain defined by the AO cut-o ff frequency, the AO sys-tem cannot measure or correct the corresponding spatial frequen-cies. Various bright spots are observable at ∼ λ/ D (twice theAO cut-o ff frequency) and correspond to the inter-actuator pitchspatial frequency.As observed in Fig. 1 and presented in Fig. 3 (top curves),raw coronagraphic images are dominated by the static contri-bution, for which contrast profiles are stable over time at anyangular separation. Raw coronagraphic images are dominatedby static speckle noise, as no evolution in the speckle field isvisually detectable over the three images presented that coverthe temporal duration of the experiment. This is consistent withthe fact that the interaction between the quasi-static terms ofEq. 3, being time-dependent, and static terms, assumed time-independent, can be studied through di ff erential imaging from atime series of raw coronagraphic images, and not directly at thelevel of raw images.The static wavefront error amplitude has been evaluated onthe raw coronagraphic images using Eq. 8, and converges to thevalue of ∼ ff erential images. The profiles presented in Fig. 3 (bottomcurves) clearly indicate that the detectability level degrades withtime. These profiles demonstrate that raw coronagraphic imagesevolve temporally, being less and less correlated with the refer-ence over time, even though such an evolution cannot be readilyseen in raw images. Further, this degradation of the detectabil-ity is e ff ective at all angular separations. This result is compliantwith observations formally reported in Paper I. From the timeseries of di ff erential images, using Eq. 9, we derive the quasi-static wavefront error contribution per Fourier component, ofthe pinning e ff ect at several angular separations. Figure 4 shows Fig. 4.
Time variability of wavefront error due to quasi-staticspeckles, evaluated at various angular separations (observationaldata).the temporal evolution of φ s at 3, and 10 λ/ D , i.e. the quasi-static wavefront error (rms) as function of time at several an-gular separations. It clearly indicates that φ s is time-dependentand increases with time, justifying the constant degradation ofdetectability observed in Fig. 3 (bottom curves). This is true forall angular separations, and the shorter the separation the higherthe amplitude. Quantitatively, the level of quasi-static wavefronterror that limits the sensitivity in the di ff erential images ( ∼ − to ∼ − , 1 σ ) is found to be in the regime of ∼ ff erence in amplitude is notsignificant , we attempt to generalize the expression of the powerlaw for any angular position in the field, which reads as the fol-lowing approximation: ∆ φ s ( t ) ≃ . + . × t , (10)where t is the time in minutes, and ∆ φ s is expressed in nm rms.In Paper I, the parameters found for the fit were 0.250 forthe value at the origin, and a slope of 0.012. This means thatSPHERE is definitely more stable in term of thermo-mechanicalvariations than the HOT bench, though this was expected (aslope reduced by at least an order of magnitude, since data werenot recorded a transient regime as performed here, but in sta-bilized temperature environment). While scaling the fit parame-ters to model quasi-static temporal evolution observed with theHOT bench to real instrument was highly non-trivial as it sig-nificantly dependents on the instrument environment (tempera-ture or pressure changes, mechanical flexures, etc...), we believethat the estimates found with SPHERE and presented in Eq. 10are representative of the new high-contrast imaging instrumentgeneration that SPHERE represents, and can be used as such.In our data we found that quasi-static wavefront error increaseswith ∼ ff ect is a consequence Fig. 5.
Quasi-static aberrations power spectral density at varioustimescales (using Eq. 5).of thermal and mechanical instabilities (pressure changes, me-chanical flexures...) of the optical bench. This is qualitativelysupported by the fact that outside the AO control radius, a char-acteristic ”butterfly shape” drawn by the speckle pattern in thedi ff erential image at t + mn is observable (Fig. 2, right). Thisindicates that a beam-shift is at work, though it is di ffi cult to ob-tained quantitative information on it. I functions described in Sect. 2 essentially represent powerspectral densities (PSD). While I c symbolizes the PSD of thestatic wavefront, ∆ I s stands for the PSD of the di ff erential aber-rations, which is calculated by Eq. 5. Plotting Eq. 5 gives accessto the quasi-static aberration PSD as function of time (Fig. 5).From the DSPs presented in Fig. 5, we can observe that: (1 / ) atvery low frequencies (from 0.1 to 3 cycles per pupil) the PSD istemporally roughly stable and exhibits a f power law, where f is the spatial frequency, while it must be pointed out that in thisfrequency domain, Eq. 5 might not be entirely valid, (2 / ) at lowfrequencies (from 3 to 8 cycles per pupil) the PSD exhibits a f − power law, (3 / ) at intermediate frequencies (from 8 to 20 cyclesper pupil) the PSD exhibits a f power law, then essentially dom-inated by a white noise, (4 / ) at high frequencies (outside the AOcontrol domain, from 20 to 30 cycles per pupil, noise dominatedat farther spatial frequencies) the power law is again f − . FromFig. 5 it is di ffi cult to extract further unambiguous informations,or initiate preliminary explanations, such as the f − power lawbeing in contradiction with the generally adopted and standardPSD slope of f − for surface optics (static aberrations).
5. Conclusion
In this paper, we confirm the results formerly reported in PaperI with new measurements, and derive a practical and general-ized expression to model quasi-static speckles temporal evolu-tion for any angular position in the field (Eq.10), in the contextof the forthcoming high-contrast planet imagers. Quasi-staticaberrations observed in a time series of extreme adaptive optics-corrected coronagraphic images exhibit a linear power law withtime and are interacting through the pinning e ff ect with staticaberrations. We examine and discuss this e ff ect using the sta-tistical model of the noise variance in high-contrast imagingproposed by Soummer et al. (2007), where the e ff ect of pinningquasi-static to static speckles as described by the expression forthe variance (Eq. 9) is found to reflect the situation in our dataset fairly well. We found that quasi-static wavefront error increases witha rate of ∼ / operationalstrategies for high-contrast imaging class instruments. In addi-tion, Eq. 5 provides a useful insight in the PSD of quasi-staticaberrations in the system, and at first glance to identify whatmoves in the system. The proposed model of quasi-static speck-les temporal evolution can be used to derive timescales for cal-ibration / operational aspects, such as ADI, or non-common pathaberrations correction.The case considered in this present paper represents a staticsystem subject to a thermal gradient. The foreseen impact ofrotating components in the instrument, such as, e.g., the atmo-spheric dispersion compensators (ADCs, not seen by the AOsystem), or the derotator (seen by AO system) will be treatedin a separated study. Acknowledgements.
SPHERE is an instrument designed and built by a consor-tium consisting of IPAG, MPIA, LAM, LESIA, Laboratoire Lagrange, INAF,Observatoire de Gen`eve, ETH, NOVA, ONERA, and ASTRON in collaborationwith ESO.
References