Estimation and correction of wavefront aberrations using the self-coherent camera: laboratory results
AAstronomy & Astrophysics manuscript no. articleJMSCC2013 c (cid:13)
ESO 2018November 8, 2018
Estimation and correction of wavefront aberrations using theself-coherent camera: laboratory results
J. Mazoyer , P. Baudoz , R. Galicher , M. Mas , and G. Rousset LESIA, Observatoire de Paris, CNRS, UPMC Paris 6 and Denis Diderot Paris 7, Meudon, France. e-mail: [email protected] Laboratoire d’Astrophysique de Marseille, CNRS, Aix-Marseille Univ., Marseille, FrancePreprint online version: November 8, 2018
Abstract
Context.
Direct imaging of exoplanets requires very high contrast levels, which are obtained using coronagraphs. Butresidual quasi-static aberrations create speckles in the focal plane downstream of the coronagraph which mask theplanet. This problem appears in ground-based instruments as well as in space-based telescopes.
Aims.
An active correction of these wavefront errors using a deformable mirror upstream of the coronagraph is manda-tory, but conventional adaptive optics are limited by differential path aberrations. Dedicated techniques have to beimplemented to measure phase and amplitude errors directly in the science focal plane.
Methods.
First, we propose a method for estimating phase and amplitude aberrations upstream of a coronagraph fromthe speckle complex field in the downstream focal plane. Then, we present the self-coherent camera, which uses thecoherence of light to spatially encode the focal plane speckles and retrieve the associated complex field. This enable usto estimate and compensate in a closed loop for the aberrations upstream of the coronagraph. We conducted numericalsimulations as well as laboratory tests using a four-quadrant phase mask and a 32x32 actuator deformable mirror.
Results.
We demonstrated in the laboratory our capability to achieve a stable closed loop and compensate for phaseand amplitude quasi-static aberrations. We determined the best-suited parameter values to implement our technique.Contrasts better than 10 − between 2 and 12 λ/D and even 3 . − (RMS) between 7 and 11 λ/D were reached inthe focal plane. It seems that the contrast level is mainly limited by amplitude defects created by the surface of thedeformable mirror and by the dynamic of the detector. Conclusions.
These results are promising for a future application to a dedicated space mission for exoplanet characteri-zation. A number of possible improvements have been identified.
Key words. instrumentation: high angular resolution – instrumentation: adaptive optics – techniques: high angularresolution
1. Introduction
Direct imaging is crucial to increase our knowledge of extra-solar planetary systems. On the one hand, it can detectlong-orbit planets that are inaccessible for other methods(transits, radial velocities). On the other hand, it allowsthe full spectroscopic characterization of the surface andatmosphere of exoplanets. In a few favorable cases, directimaging has already enabled the detection of exoplanets(Kalas et al. 2008; Lagrange et al. 2009) and even of plane-tary systems (Marois et al. 2008, 2010). However, the maindifficulties of this method are the high contrast and smallseparation between the star and its planet. Indeed, a con-trast level of 10 − has to be reached within a separationof ∼ . (cid:48)(cid:48) or lower to allow the detection of rocky planets.To reduce the star light in the focal plane of a telescope,several coronagraphs have been developed, such as the four-quadrant phase mask (FQPM) coronograph (Rouan et al.2000), the vortex coronograph (Mawet et al. 2005) and thephase-induced amplitude apodization coronograph (Guyonet al. 2005). However, the performance of these instrumentsis drastically limited by phase and amplitude errors. Indeed,these wavefront aberrations induce stellar speckles in theimage, which are leaks of the star light in the focal planedownstream of the coronagraph. When classical adaptive optics (AO) systems correct for most of the dynamic wave-front errors that are caused by to atmosphere, they use adedicated optical channel for the wavefront sensing. Thus,they cannot detect quasi-static non-common path aberra-tions (NCPA) created in the differential optical paths bythe instrument optics themselves. These NCPA have to becompensated for using dedicated techniques, for ground-based telescopes as well as for space-based instruments.Two strategies have been implemented to overcome thequasi-static speckle limitation. First, one can use differen-tial imaging techniques to calibrate the speckle noise in thefocal plane. Theses methods can use either the spectral sig-nature and polarization state of the planet or differentialrotation in the image (Marois et al. 2004, 2006). Second,even before applying these post-processing techniques, anactive suppression of speckles (Malbet et al. 1995) has tobe implemented to reach very high contrasts. It uses a de-formable mirror (DM) controlled by a specific wavefrontsensor that is immune against NCPA. The techniques de-veloped for this purpose include dedicated instrumental de-signs (Guyon et al. 2009; Wallace et al. 2010), or creatingof known phases on the DM (Bord´e & Traub 2006; Give’onet al. 2007) to estimate the complex speckle field. a r X i v : . [ a s t r o - ph . I M ] S e p . Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC Ground-based instruments that combine these twostrategies are currently being developed, such as SPHERE(Beuzit et al. 2008) and GPI (Macintosh et al. 2008), todetect young Jupiter-like planets with an expected con-trast performance of 10 − at 0 . (cid:48)(cid:48) . Better contrasts mightbe achieved to reach the rocky planet level with instrumentsusing dedicated active correction techniques embedded inspace telescopes (Trauger & Traub 2007).In this context, we study a technique of wavefront sens-ing in the focal plane that allows an active correction in aclosed loop. This paper has two main objectives. First wegive an overview of how the amplitude and phase errors up-stream of a coronagraph can be retrieved from the complexamplitude of the speckle field (Section 2) and how they canbe compensated for using a DM (Section 3). In Section 4,we introduce the self-coherent camera (Baudoz et al. 2006;Galicher et al. 2008). This instrument uses the coherence ofthe stellar light to generate Fizeau fringes in the focal planeand spatially encode the speckles. Using both the aberra-tion estimator and the self-coherent camera (SCC), we areable to correct phase and amplitude aberrations. The sec-ond objective of the paper is a laboratory demonstration ofthe active correction and an experimental parametric studyof the SCC (Section 5).
2. Wavefront estimator in the focal plane of acoronagraph
In this section, we aim to prove that one can retrieve thewavefront upstream of the coronagraph using the measuredcomplex amplitude of the electric field in the focal planedownstream of the coronagraph. We assume in the wholesection that we can measure this complex amplitude with-out error using an undetermined method. We describe onetype of this method (the SCC) in Section 4. In Section 2.1,we express the complex electric field that is associated tothe speckles as a function of the wavefront errors in thepupil upstream of a phase mask coronagraph. From thisexpression, we propose an estimator of the wavefront errorsfrom the speckle electric field (Section 2.2) and analyze itsaccuracy for an FQPM (Section 2.3).
We consider here a model of a phase mask coronagraphusing Fourier optics. Figure 1 (top) presents the principleof a coronagraph. We assume that the star is a spatiallyunresolved monochromatic source centered on the opticalaxis. The stellar light moves through the entrance pupil P .Behind this pupil, the beam is focused on the mask M inthe focal plane, which diffracts the light. Hence, the nonaberrated part of the stellar light is rejected outside of theimaged pupil in the next pupil plane and is stopped by theLyot stop diaphragm L . The aberrated part of the beamgoes through the Lyot stop, producing speckles on the de-tector in the final focal plane (Figure 1, bottom).We note whith α and φ the amplitude and phase aber-rations in the entrance pupil plane and define the complexwavefront aberrations Φ asΦ = φ + iα. (1) Entrance pupil P Focal plane Detector ’ ψ s Lyot stop L mask MCoronagraphicA’ s s ψ A s Star light Final focal planeLyot plane
Figure 1.
Principle of a coronagraph (top). Aberrations inthe entrance pupil plane induce speckles in the focal plane(bottom).
The complex amplitude of the star in the entrance pupilplane ψ (cid:48) S can be written as ψ (cid:48) S ( ξ , λ ) = ψ P ( ξ ) exp ( i Φ( ξ )) , (2)where ψ is the mean amplitude of the field over the pupilP, ξ the coordinate in the pupil plane and λ the wavelength.We assume that the aberrations are small and defined inthe pupil P ( P Φ = Φ), thus ψ (cid:48) S ( ξ , λ ) ψ (cid:39) P ( ξ ) + i Φ( ξ ) . (3)The complex amplitude of the electric field A (cid:48) S behind thecoronagraphic mask M in the first focal plane is A (cid:48) S = F [ ψ (cid:48) S ] M, (4)where F is the Fourier transform. Using Equation 3, we canwrite the electric field F − ( A (cid:48) S ) before the Lyot stop F − [ A (cid:48) S ] ψ = P ∗ F − [ M ] + i Φ ∗ F − [ M ] , (5)where ∗ is the convolution product. We call Φ M the aber-rated part of the field after the coronagraph:Φ M = φ M + iα M = Φ ∗ F − [ M ] . (6)After the Lyot stop L , the electric field ψ S is ψ S ψ = ( P ∗ F − [ M ]) .L + i Φ M .L. (7)We assume a coronagraph for which the non aberrated partof the electric field is null inside the imaged pupil. Thisproperty of the perfect coronagraph (Cavarroc et al. 2006)has also been demonstrated analytically for several phasecoronagraphs such as FQPM coronagraphs (Abe et al.2003) and vortex coronagraphs (Mawet et al. 2005). Theremaining part Φ L of the normalized electric field after theLyot stop readsΦ L = φ L + iα L = Φ M .L = (Φ ∗ F − [ M ]) .L. (8)
2. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 2.
Simulations of an aberrated phase in the en-trance pupil plane ( φ ), and the real part of the field in thenext pupil plane before ( φ M ) or after ( φ L ) the Lyot pupil.We also show the estimate ( φ est ) and the difference between φ est and φ . L = P in this case. In the final focal plane, the complex amplitude A S is A S = ψ F [ i Φ L ] ,A S = ψ F [ i (Φ ∗ F − [ M ]) .L ] ,A S = iψ ( F [Φ] .M ) ∗ F [ L ] . (9)This complex amplitude is directly related to the wavefrontaberrations in the entrance pupil. If one can measure A S , wecan invert Equation 9 and retrieve the complex wavefronterrors Φ in the entrance pupil. We still assume in this section that an undefined methodprovides access to A S . Using this complex amplitude A S asthe measurement, we therefore propose the following esti-mator Φ est for the wavefront:Φ est = i F − (cid:20) A S M ψ (cid:21) .P. (10)This estimator can be used for any phase mask coronograph(for which M is nonzero over the full focal plane). To justifythe pertinence of this estimator, we can re-write it using thevariables of our model. Using Equation 9, in a noise-freemeasurement case, this estimator readsΦ est = (cid:20) ((Φ ∗ F − [ M ]) .L ) ∗ F − (cid:20) M (cid:21)(cid:21) .P. (11)Theoretically, if no Lyot stop is applied ( L = 1),Equation 11 becomes Φ est = P Φ = Φ. We propose thisestimator based on the assumption that most of the infor-mation about the aberrations is not diffracted outside of theimaged pupil by the coronagraphic mask. Therefore, usingthis assumption, we intuit that for L (cid:39) P , we still haveΦ est (cid:39) Φ. This assumption is verified by the simulation inSection 2.3, and by the experiment described in Section 5.For a symmetrical phase mask such as the FQPM, either F − [ M ] and F − (cid:2) M (cid:3) are real. Thus, in the estimator wecan separate the real ( φ est ) and imaginary part ( α est ) ofthe estimator in Equation 11: (cid:26) φ est = (cid:2) (( φ ∗ F − [ M ]) .L ) ∗ F − (cid:2) M (cid:3)(cid:3) .Pα est = (cid:2) (( α ∗ F − [ M ]) .L ) ∗ F − (cid:2) M (cid:3)(cid:3) .P. (12)This relation ensures that within the limits of our model,this estimator independently provides estimates of thephase and amplitude aberrations. In this section we test the accuracy of the estimation φ est for a phase aberration φ and no amplitude aberrations( α = 0). In the following numerical simulations, we as-sumed an FQPM coronagraph. It induces a phase shift of π in two quadrants with respect to the two others quadrants.We simulated FQPM coronagraphs in this paper using themethod described in Mas et al. (2012). This coronagraph iscompletely insensitive to some aberrations, for instance toone of the astigmatism aberrations (Galicher 2009; Galicheret al. 2010). Because these aberrations introduce no aberra-tion inside the Lyot pupil, we are unable to estimate them.We assumed an initial phase with aberrations of 30 nm rootmean square (RMS) over the pupil at λ = 635 nm , with apower spectral density (PSD) in f − , where f is the spatialfrequency.In these simulations, we studied two cases. First, weused a Lyot pupil of the same diameter as the entrancepupil ( L = P ). Then, we studied the case of a reducedLyot ( D L < D P , where D L and D P are the diameters ofthe Lyot and entrance pupil, respectively). L = P Figure 2 shows the effect of phase-only aberrations φ in dif-ferent planes of the coronagraph. Starting from the left, werepresent the initial phase φ , the real part of the amplitudedue to the aberrations φ M ( φ L ) before the Lyot stop (afterthe Lyot stop), derived from Equation 6 (Equation 8) forphase-only aberrations. The last two images are the esti-mator φ est and the difference between the estimate and theentrance phase aberrations ( φ − φ est ).The estimate φ est is very close to the initial phase φ .For initial phase aberrations of 30 nm RMS, the difference φ − φ est presents a level of 10 nm RMS in the entire pupil.The vertical and horizontal structures in this difference aredue to the cut-off by the Lyot stop of the light diffractedby the FQPM (the light removed between φ M and φ L ),which leads to an imperfect estimate of the defects on thepupil edges. Aberrations to which the FQPM coronagraphis not sensitive (such as astigmatism) are also present inthis difference.Assuming a perfect DM, we can directly subtract φ est from φ in the entrance pupil. Then, we can estimate theresidual error once again, and iterate the process. The aber-rations in the Lyot pupil φ L converge toward zero (0.2 nmin ten iterations). This is important because these aberra-tions are directly linked to the speckle intensity in the focalplane downstream of the coronagraph. However, the differ-ence φ − φ est does not converge toward zero in the entrancepupil. The fact that φ L converges toward zero proves thatthe residual phase is only composed of aberrations unseenby the FQPM. D L < D P In a more realistic case, we aim to remove all the lightdiffracted by the coronagraphic mask, even for unavoidablemisalignments of the Lyot stop. For this reason, the Lyotstop is often chosen to be slightly smaller than the imagedpupil. We consider here a Lyot stop pupil L whose di-ameter is D L = 95% D P . In a first part, we show belowthat phase defects at the edge of the entrance pupil can be
3. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 3.
Simulations of an aberrated phase with a local-ized default in the entrance pupil plane ( φ ), and in the nextpupil plane ( φ M , φ L ). We represent the entrance pupil sizeby a dark ring around φ L . We show the estimate ( φ est ) andthe difference between φ est and φ in the last two images. D L = 95% D P in this case. partially retrieved, then we study the convergence of theestimator in this case.As in Figure 2, Figure 3 corresponds to the simulation ofthe consecutive steps of the model ( φ , φ M before Lyot stops, φ L after Lyot stop, then estimated phase φ est and differ-ence with initial phase). We added a small localized phasedefault, indicated by the black arrow, inside the entrancepupil P , but outside of the Lyot stop L (Figure 3, left).Around φ L , the complex amplitude after the Lyot stop, wedrew a circle corresponding to the entrance pupil, slightlylarger than L . For an FQPM, the additional defect isdiffracted in the Lyot stop plane ( φ M ). After applying theLyot stop of 95% ( φ L ), most of the default disappears, butwe can still see its signature. As the estimator φ est decon-volves by the phase mask, it partially retrieves the default,as seen in the estimate (indicated by the black arrow). Inthe error ( φ − φ est ), we notice a remarkable cross issued fromthis defect, which is due to the information lost during thefiltering by the Lyot stop.The wavefront estimation is limited when compared tothe D L = D P case. Because of the light filtered by theLyot stop, some information about the wavefront aberra-tions close to the border of the entrance pupil is inevitablylost. Due to these unseen aberrations, φ − φ est does notconverge toward zero. However, the residual aberrations inthe Lyot pupil φ L still converge toward zero, practically asquick as in the L = P case.By this first rough analysis, we see that we can estimatephase aberrations upstream of a coronagraph using thecomplex amplitude A S of the speckle field and Equation 10.The same conclusion can be drawn for amplitude aberra-tions and, because the estimation is linear, for a complexentrance wavefront. In the next section, we demonstratethat one can compensate for the wavefront errors in theentrance pupil.
3. Entrance pupil wavefront correction
In this section, we use the estimator Φ est to numericallysimulate the correction of phase and amplitude aberrationsin a closed loop. In the loop, we can remove constant factorsin the estimator, which can be adjusted with a gain g Φ est = gi F − (cid:20) A S M (cid:21) .P. (13)We still assume that we have a perfect sensor that measuresthe complex amplitude A S in the focal plane downstreamof the coronagraph. We used a deformable mirror (DM) of NxN actuators upstream of the coronagraph, in the en-trance pupil plane. We started with phase-only correction.We explain how to correct for the effects of phase and am-plitude aberrations with only one DM in Section 3.3.We define the correction iterative loop by the expressionof the residual phase φ j +1 at iteration j + 1: φ j +1 = φ − φ j +1 DM = φ − [ φ jDM + N − (cid:88) i =0 k j +1 i f i ] , (14)where φ jDM is the shape of the DM at iteration j , k j +1 i isthe incremental command of the DM actuator i at iteration j + 1, and f i is the DM influence function, i.e., the WFdeformation when only poking the actuator i . Note that φ j is the phase to be estimated at iteration j +1. The objectiveis now to determine the command vector { k j +1 i } from thephase estimator φ j +1 est . To derivate the command vector { k j +1 i } , we minimize thedistance between the measurements and the measurementsthat accounts for the parameters to be estimated. Ideally,we would like to find the { k i } minimizing the distance d j { k i } between the residual phase and the DM shape: d j { k i } = (cid:107) φ j − N − (cid:88) i =0 k j +1 i f i (cid:107) . (15)As presented in the previous section, a possible estimatorof φ j is given by Equation 13, allowing us to compute Φ j +1 est from A jS (directly related to φ j ). So we minimize d j { k i } = (cid:107)(cid:60) (cid:104) Φ j +1 est (cid:105) − D { k j +1 i }(cid:107) , (16)where D is the interaction matrix because we are using alinear model. This matrix is calibrated off-line directly us-ing the wavefront sensor (Boyer et al. 1990). As in the con-ventional least-squares approach, we derived the pseudo in-verse of D , denoted D † , by the singular value decomposition(SVD) method. Therefore, the command vector solution ofEquation 16 is given by { k j +1 i } = D † (cid:60) (cid:104) Φ j +1 est (cid:105) . (17)Equation 17 is applicable for different estimators (only D † changes). To create the interaction matrix, we poked oneby one the actuators while the others remain flat, as shownin Figure 4. Each estimated phase vector obtained hencegives the column of the interaction matrix correspondingto the moved actuator. The influence function (which wesimulated as a Gaussian function) is at the left, the esti-mator given by Equation 13 at the right. At the center, wealso plot another estimator of the wavefront that does notinclude the deconvolution by the coronagraph mask. It isdefined by Φ est, = gi F − [ A S ] .P. (18)The chosen estimator applied to the influence function mustbe as spatially localized as possible: we have to filter thenoise and it is far more efficient if the relevant information
4. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 4.
Simulations of the influence function f i in thepupil plane (left), and the effect using the two different es-timators: we deconvolve by the mask ( Φ est , right) or not( Φ est, , center) is gathered around one point. For this reason, it is prefer-able to use Φ est (Figure 4, right) instead of Φ est, (Figure 4,center). We notice in Figure 4 that even after deconvolutionby the FQPM, the estimate (right) shows a negative crosscentered on the poked actuator, whereas it is not presentin the initial phase (left). This artifact is generated to thetransitions of the quadrants, which diffract the light outsideof the Lyot stop. This cross can be a problem for two rea-sons. First, it is difficult to properly retrieve it in a noisyimage. Then, because it enhances the cross-talk betweenthe actuator estimates, it may lead to unstable corrections(see Section 4.4).For these reasons, we chose to create a synthetic inter-action matrix, i.e. , use a slightly different model for theestimation. One interest of interaction matrices is to calibrate the mis-registration between the DM and the wavefront sensor whenconsidering a complex optical system. It also allows us tocalibrate the shape and magnitude of each actuator re-sponse. Since the cross in Figure 4 (right) is 40 times lessintense than the poke actuator in the center, it may be onlypartially retrieved in noisy images, which may lead to anunstable loop. To avoid this problem, we decided to builda synthetic interaction matrix based on the measured posi-tion and shape of each actuator. Because we considered aniterative measurement and correction loop, we finally dis-carded the magnitude calibration of each actuator, whichlead to a slight increase of the required iteration number forconvergence. We decided to only adjust the mis-registrationand estimation shape of the actuator set in the output pupilon the measured interaction matrix, as seen by the sensor.Out of the NxN actuators of the square array in the DM,we chose to limit the response adjustment on the 12x12actuators centered on the entrance pupil. These actuatorswere alternately pushed and pulled using known electricalvoltages and we recorded A S for both positions. Assumingthe complex amplitude A S is a linear function of the wave-front errors in the entrance pupil plane and that other aber-rations of the optical path remain unchanged between twoconsecutive movements (the same pushed and pulled actu-ator), the difference between these two movements leads tothe estimate ˆ f i of the influence function of a single pokedactuator using Equation 13. For each estimate, we adjusteda Gaussian function defined by its width and position in theoutput pupil. From these 144 Gaussian fits, we can build theactuator grid as observed in the plane, where aberrations are estimated and determine the inter-actuator distance ineach direction and the orientation of this grid. We also de-termined the median width of the adjusted Gaussian func-tions and computed a synthetic Gaussian function, whichwas translated onto the adjusted actuator grid to create anew set of NxN synthetic estimates ˆ f isynth .From these synthetic estimates, we built the syntheticinteraction matrix D synth . Some of the actuators are out-side of the entrance pupil, and their impact inside the pupilis negligible. We excluded these actuators from D synth . Forany wavefront estimate, the distance to be minimized isnow d j { k i } = (cid:107)(cid:60) (cid:104) Φ j +1 est (cid:105) − D synth { k j +1 i }(cid:107) . (19)The solution is given by the pseudo-inverse of the interac-tion matrix D synth using the SVD method. As explained in Bord´e & Traub (2006), a complex wavefrontΦ = φ + iα can be corrected for on half of the focal planewith only one DM. The idea is to apply a real phase on theDM to correct for the phase and amplitude on half of thefocal plane. Because the Fourier transform of a Hermitianfunction is real, we define A hermiS as ∀ x ∈ R × R + , A hermiS ( x ) = A S ( x ) ∀ x ∈ R × R − , A hermiS ( x ) = A ∗ S ( − x ) , (20)where A ∗ S is the complex conjugated of A S and introduce itinto the estimator of Equation 13. The resulting estimatedwavefront is real, which allows its correction with only oneDM.Now we have a solution to correct for the wavefrontaberrations upstream of the coronagraph when the com-plex amplitude in focal plane is known. We introduce inSection 4 a technique to measure A S : the self-coherent cam-era.
4. Self-coherent camera: a complex amplitudesensor in focal plane
The self-coherent camera (SCC) is an instrument that al-lows complex electric field estimations in the focal plane.
Figure 5 (top) is a schematic representation of the SCCcombined with a focal phase mask coronagraph and a DM.We added a small pupil R , called reference pupil, in theLyot stop plane of a classical coronagraph. R selects partof the stellar light that is diffracted by the focal corona-graphic mask. The two beams are recombined in the focalplane, forming Fizeau fringes, which spatially modulate thespeckles. In the following, we call the SCC image the imageof the encoded speckles (Figure 5, bottom). In this section,we briefly demonstrate that this spatial modulation allowsus to retrieve the complex amplitude A S . A more completedescription of the instrument can be found in Baudoz et al.(2006), Galicher et al. (2008, 2010).
5. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC A’ ψ A R ATelescopeDeformablemirror Entrance Lyot Coronagraphic light
S S ’ S S ψ Focal plane Image channelchannel (R)Pupil plane Focal planemask (M) stop (L)Reference Detectorpupil (P) Star light Scattered
Figure 5.
Principle of the SCC combined with a corona-graph and a DM (top). A small hole is added in the Lyotstop plane to create a reference channel. In the final focalplane (bottom), the SCC image is formed by speckles en-coded with Fizeau fringes
The electric field ψ in the pupil plane (Equation 7) afterthe modified Lyot stop is ψ ( ξ , λ ) ψ = (cid:2) ( P ( ξ ) + i Φ( ξ , λ )) ∗ F − [ M ]( ξ ) (cid:3) . ( L ( ξ ) + R ( ξ ) ∗ δ ( ξ − ξ )) , (21)where ξ is the separation between the two pupils in theLyot stop, and δ is the Kronecker delta. ψ can also be writ-ten as ψ ( ξ , λ ) = ψ S ( ξ , λ ) + ψ R ( ξ , λ ) ∗ δ ( ξ − ξ ) , (22)where ψ S is the complex amplitude in the Lyot stop, de-fined in Equation 8, and ψ R is the complex amplitude in thereference pupil. We denote with A R its Fourier transform,the complex amplitude in the focal plane, of the light is-sued from the reference pupil. In monochromatic light, theintensity I = |F [ ψ ] | recorded on the detector in the finalfocal plane can then be written as I ( x ) = | A S ( x ) | + | A R ( x ) | + A ∗ S ( x ) A R ( x ) exp (cid:16) − iπ x . ξ λ (cid:17) + A S ( x ) A ∗ R ( x ) exp (cid:16) iπ x . ξ λ (cid:17) , (23)where A ∗ is the conjugate of A and x the coordinate in thefocal plane. The two first terms are the intensities issuedfrom Lyot and reference pupils, and provide access only tothe square modulus of the complex amplitudes. The twocorrelation terms that create the fringes directly depend on A S and A R .When an off-axis source (planet) is in the field of view,its light is not diffracted by the coronagraphic mask. Thus,it does not go through the reference pupil. Because thelights of the off-axis and in-axis sources are not coherent,the off-axis light amplitude in the focal plane does not ap-pear in the correlation terms ( i.e. , its image is not fringed). In this section, we demonstrate that we can use the SCCimage to estimate the complex amplitude of the specklefield. We first apply a numerical inverse Fourier transformto the recorded SCC image (Equation 23), F − [ I ]( u ) = F − [ I S + I R ] + F − [ A ∗ S A R ] ∗ δ (cid:16) u − ξ λ (cid:17) + F − [ A S A ∗ R ] ∗ δ (cid:16) u + ξ λ (cid:17) , (24)where I S = | A S | and I R = | A R | are the intensities of thespeckles and reference pupil, and u is the coordinate in theFourier plane. F (I ) −1 −
F (I ) ξ Figure 6.
Correlation peaks in the Fourier transform ofthe focal plane. The inverse Fourier transform of I S + I R is circled in blue. The inverse Fourier transform of I − = A S A ∗ R is circled in red. F − [ I ] is composed of three peaks centered at u =[ − ξ /λ, , + ξ /λ ] (Figure 6). We denote with D L the di-ameter of the Lyot pupil and with D L /γ the diameterof the reference pupil ( γ > F − [ I S + I R ]. Its radius is D L because we assume γ > F − [ I − ] and F − [ I + ]hereafter) have a radius ( D L + D L /γ ) /
2. Thus the threepeaks do not overlap only if (Galicher et al. 2010) || ξ || > D L (cid:18) γ (cid:19) , (25)which puts a condition on the smallest pupil separation.The lateral peaks are conjugated and contain informationonly on the complex amplitude of the stellar speckles thatare spatially modulated on the detector. When we shift oneof these lateral peaks to the center of the correlation plane( u = ), its expression can be derived from Equation 24: F − [ I − ] = F − [ A S A ∗ R ] . (26)Assuming γ (cid:29)
1, we can consider that the complexamplitude in the reference pupil is uniform and that A ∗ R is the complex amplitude of an Airy pattern. Therefore,
6. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC TF −1 ψ M Ax 1lateral peakSelectionof the Shift of the lateralpeak3 correlation peaks SCC imageEstimated phase xPReal part TF −1 Imaginarypart φ Estimated amplitude α TF I R − Figure 7.
Steps followed to estimate the phase and ampli-tude from SCC images. knowing A R , we can to retrieve the complex amplitude A S in the focal plane using the SCC (where A ∗ R is not zero): A S = I − A ∗ R . (27) Equation 13 shows how to estimate the wavefront upstreamof a coronagraph is estimated using the complex ampli-tude of the speckle field in the focal plane. CombiningEquations 10 and 27, we have an estimator of the wave-front aberrations Φ as a function of I − :Φ est = (cid:20) i F − (cid:20) I − A ∗ R ψ M (cid:21)(cid:21) .P. (28)This estimator is only limited in frequency by the sizeof the reference pupil. Indeed, where the reference flux isnull, the speckles are not fringed and their estimate can-not be achieved. Small reference pupils produce large pointspread functions ( i.e. , with a first dark ring at large sepa-ration) and allow estimating A S in a large area of the focalplane. The influence of the reference pupil size is detailedin Section 5.5.Figure 7 summarizes the steps followed to estimate thephase and amplitude aberrations with the SCC. From thefringed focal plane, we used a Fourier transform to retrieve I − , from which we deduced the complex amplitude of thespeckle field A S . Using the estimator, we measured thephase and amplitude aberrations. We can use this wavefront estimator to control a DM andcorrect for the speckle field in the focal plane as explainedin Section 3. The DM has a finite number of degrees offreedom and thus can only correct for the focal plane in alimited zone. If the reference pupil is small enough ( γ (cid:29) | A R | is uniform over thecorrection zone ( A ∗ R (cid:39) A ). We discuss this assumption inSection 5.5. Under this assumption, Equation 28 becomesΦ est (cid:39) i F − (cid:20) I − A ψ M (cid:21) .P = gi F − (cid:20) I − M (cid:21) .P. (29) Figure 8.
Singular values, normalized to their respec-tive maximum, issued from the inversion of the interac-tion matrices D , obtained using the two estimators Φ est (red,dotted) and Φ est, (black, solid) and the synthetic ma-trix (blue, dashed) for γ = 40 . As described in Section 3, we removed the constant termsin the estimation and put them into the gain g. FromΦ est , we created a synthetic matrix, as explained inSection 3.2. Similarly, the other estimator Φ est, introducedin Equation 18, becomesΦ est, = gi F − [ I − ] .P. (30)Using the interaction matrices deduced from these estima-tors (Φ est , Φ est, ) and the synthetic one, we studied the cor-rection loop. We simulated a DM with 27 actuators acrossthe entrance pupil. To build these matrices, we only se-lected the actuators with a high influence in the pupil (633actuators were selected for this number of actuators in thepupil). Lyot stop and entrance pupil have the same radius,and we chose γ = 40 for the reference pupil size.In Figure 8, we plot the singular values (SV), normalizedto their highest values, derived from the inversion of thematrices D obtained using the estimators Φ est and Φ est, and of the interaction matrix built from ˆ f isynth . As alreadyunderlined, the cross in Φ est or Φ est, (Figure 4, centerand right) correlates the estimates of different actuatorsand therefore leads to lower SV (up to five times lowerfor the lowest SV). When inverted in D † , low SV lead tohigher values (in absolute values) and amplify the noise inEquation 17. Applied to noisy data, such D † matrices maylead to an unstable correction. Even in a noise-free case,simulations of the correction with the three methods andthe same number of actuators used (633) showed that onlythe synthetic matrix leads to a stable correction. Between the Lyot stop and the detector, the beam is splitinto two paths (image and reference), which encounter dif-ferent areas in the optics. Thus, differential aberrations ex-ist Galicher et al. (2010). However, because the reference
7. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC pupil is small ( γ (cid:29) Given an OPD d op , we can define phase difference φ op =2 πd op /λ . This phase difference modifies the I − originallydefined in Equation 26: I − = A S A ∗ R exp( iφ op ) . (31)The phase and amplitude estimate φ est,op and α est,op canbe expressed as a function of the estimates made withoutan OPD ( d op = 0): φ est,op + iα est,op = ( φ est + iα est ) exp( iφ op ) , (32)and thus (cid:26) φ est,op = φ est cos( φ op ) − α est sin( φ op ) α est,op = φ est sin( φ op ) + α est cos( φ op ) . (33)Hence, even phase-only aberrations (such as the move-ments of the DM) have an influence on the estimated am-plitude ( i.e. , the imaginary part of the estimator Φ est ) fora nonzero OPD. In this section, we make two assumptions.First, that the DM is perfect and we can correct for any de-sired phase in the pupil plane. Second, that the only errorin the estimator is due to the OPD: if d op = 0, the estimatorretrieves the exact phase and amplitude (Φ est = φ + iα ).We started the loop with a phase φ and an amplitude α . After j iterations the phase in the pupil plane φ j isthe difference between the previous phase φ j − , and theestimate of this previous phase φ j − est,op . Under the previousassumptions, we have α est = α and φ est = φ j − , and φ j = φ j − − φ j − est,op = φ j − (1 − cos( φ op ))+ α sin( φ op ) . (34)Because the OPD biases the estimation, the correction in-troduces an error at each iteration. This sequence convergesif | − cos( φ op ) | <
1. This assumption ( − π/ < φ op < π/ φ end satisfies theequation φ end = φ end (1 − cos( φ op )) + α sin( φ op ) φ end = α tan( φ op ) . (35)Therefore, for a nonzero OPD and a phase-only correction,the SCC correction converges, but the errors on the finalphase depend on the uncorrected amplitude aberrations α .To estimate the OPD effect on the level of the focalplane intensity, we considered the complex amplitude inthe focal plane as a linear function of the phase and ampli-tude aberrations in the entrance pupil plane. We can thusevaluate the energy in the focal plane as a linear function of | φ | + | α | . Without an OPD, a perfect phase-only correc-tion would leave a level of speckles only dependent on theentrance amplitude aberrations | α | . With an OPD, thislevel is slightly higher: | α | (1 + tan( φ op ) ). For a realisticphase difference of 0 . φ op (cid:54) = 0, numerical sim-ulations as well as tests on an optical bench show that thecorrection is unstable: at each iteration, we raised the phaseaberrations by trying to correct for the amplitude aberra-tions and vice versa . Thus, we need an estimate of the OPDto stabilize the correction. In the construction of the synthetic matrix, (Section 3.2),we studied the difference of two SCC images produced bywavefronts that only differ by a movement of an actuator.Because the DM is in the pupil plane, the estimator ap-plied to this difference is real for of an OPD equal to zero.However, for a nonzero OPD d op and using Equation 31with α = 0, we deduce Φ iest,op = Φ iest (sin( d op ) + i cos( d op )).For each of the 12x12 actuators used to build the syntheticmatrix, the arctangent of the ratio of the imaginary part onthe real part of Φ iest,op leads to an estimate of the OPD. Dueto the noise in the image, small differences in the OPD esti-mate can appear from one actuator to another. Calculatingthe median of the estimated OPDs, we obtain the measuredphase difference φ mesop . We modified I − accounting for thisOPD and our estimator (Equation 29) becomesΦ est,op = gi F − (cid:20) I − exp( − iφ mesop ) M (cid:21) .P. (36)We use this new estimator from now on.The OPD variations during the correction are a prob-lem that has to be carefully considered for a telescope ap-plication. In the current installation (bench under a hood,room temperature stabilized) these variations are muchslower than the time of a correction loop. Moreover, onecan change the value of φ op directly during the correction tocompensate for slight changes. However, in an operationalinstrument, this problem will be taken into account by de-sign to comply with the stability requirements (Macintoshet al. 2008).
5. Correction in the focal plane using theself-coherent camera: laboratory performance
We tested the SCC on a laboratory bench at theObservatoire de Paris. A thorough description of this opti-cal bench is given in Mas et al. (2010). We briefly presentthe main components used in the experiments of the cur-rent paper:1. A quasi-monochromatic laser diode emitting at 635nm.2. A tip-tilt mirror built at LESIA, used to center the beamon the coronagraphic mask (Mas et al. 2012). The tip-tilt mirror can also be used in the closed-loop as anoff-load for the DM.3. A Boston Micromachines DM of 32x32 actuators on asquare array. Each actuator has a size of 300 µm . Wecurrently use an entrance pupil of 8.1mm and thus 27actuators across the pupil.4. An FQPM optimized for 635nm.
8. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 9.
Dark holes recorded on the laboratory bench forcorrection with two different sizes of square mask S q : K S q =20 . λ/D L (left) and K S q = 24 . λ/D L (center). The darkhole recorded on the laboratory bench for a correction inphase and amplitude with a square mask of size K S q =24 . λ/D L (right). These images use a different intensityscale but the same space scale
5. A Lyot stop with a diameter of 8mm for an entrancepupil of 8.1mm (98 .
7% filtering) and in the same plane,reference pupils of variable diameters: 0.3mm ( γ =26 . γ = 22 . γ = 20) and 0.5mm( γ = 16) and 0.8mm ( γ = 10).6. A CCD camera of 400x400 pixels with a readout noiseof 16 electrons/pixel and a full well capacity of 13,000electrons/pixel.We used the Labview software to control the bench andthe DM and applied the closed-loop correction at 20 Hz. Owing to the limited number of actuators on the DM, onlyspatial frequencies lower than the DM cut-off can be cor-rected for. For a given diameter D L of the Lyot pupil, thehighest frequency attainable for a NxN actuators DM (Nactuators across the pupil diameter) is N λ/ (2 D L ) in oneof the principal directions of the mirror and √ N λ/ (2 D L )in the diagonal. The largest correction zone, called darkhole ( DH ) in Malbet et al. (1995) is the zone DH max =[ − N λ/ (2 D P ) , N λ/ (2 D P )] × [ − N λ/ (2 D P ) , N λ/ (2 D P )] inthe image plane. During the numerical process of the SCCimage (Figure 7), we can decide to reduce the correction toa smaller zone than the one allowed by the number of actu-ators of the DM. This can be implemented in the SCC cor-rection by multiplying I − by a square mask S q . ModifyingEquation 36, the estimation becomesΦ est = gi F − (cid:20) S q .I − exp( − iφ mesop ) M (cid:21) .P, (37)where S q equals 1 on a square area of K S q λ/D L x K S q λ/D L in the center of the image and 0 everywhere else.Using an SCC with a reference pupil of 0.5 mm ( γ = 16),we applied Equation 37 to estimate the upstream wave-front. We used a square zone to restrain the correction zoneto 24 . λ/D L to optimize the correction of the DM. We builta synthetic interaction matrix as described in Section 3.2.The pseudo inverse of D synth was used to control the DMin a closed loop using Equation 14. The correction loop wasclosed at 20 Hz for the laboratory conditions and ran for anumber of iterations large enough ( j >
10) for the DM toconverge to a stable shape. We recorded focal plane imagesduring the control loop. The typical result obtained on the optical bench for this reference and square zone sizes andfor phase-only correction is shown in Figure 9 (center). Wealso show an image of a DH obtained with a correction witha square zone of size K S q = 20 . λ/D L (Figure 9, left). Aspecific study of the size of the correction zone is made inSection 5.4. In Figure 9, dark zones represent low intensi-ties. The eight bright peaks at the edges are caused by highspatial frequencies due to the print-through of the actua-tors on the DM surface. These peaks are uncorrectable bynature, but probably do not strongly alter the correctionbecause they are situated at more than 20 λ/D L from thecenter.As explained in Section 3.1, the correction of phaseand amplitude with only one DM is possible by replac-ing A S by A hermiS in Equation 13. With Equation 20, wesimilarly define the hermitian function I hermi − from I − .Using Equation 27 and the assumption that | A ∗ R | is anAiry pattern, a phase and amplitude correction is thereforepossible by replacing I − by I hermi − in Equation 37. Thiscorrection allows one to go deeper in contrast but limitsthe largest possible dark hole to half of the focal plane: DH + max = [0 , N λ/ (2 D L )] × [ − N λ/ (2 D L ) , N λ/ (2 D L )]. Onthis half plane, we can also choose to reduce the correc-tion to a smaller zone. A resulting dark hole is presented inFigure 9 (right) for K S q = 20 . λ/D L . In this section, we present contrast results obtained on thelaboratory bench for phase-only correction and for ampli-tude and phase correction. We used a reference pupil of 0.5mm ( γ = 16) to estimate the upstream wavefront and asquare zone of size K S q = 24 . λ/D L to optimize the cor-rection of the DM. The speckles near the FQPM transitions are brighter thanthose in other parts of the DH . Moreover, the contrast inthese region is not relevant, because the image of a planetlocated on a transition would be distorted and stronglyattenuated. Therefore, for phase-only correction, we choseto measure the radial profile of the SCC image only on thepoints (x,y) which verify (cid:26) x ∈ [ − λ/D L ; − λ/D L ] ∪ [1 λ/D L ; 20 λ/D L ] y ∈ [ − λ/D L ; − λ/D L ] ∪ [1 λ/D L ; 20 λ/D L ] . (38)We calculated the profiles by normalizing the intensitiesby the highest value of the PSF measured through theLyot pupil and without coronagraphic mask. In practice,we moved the source away from the coronagraph transi-tions to measure this PSF. In the following figures, thedistances to the center are measured in λ/D L . Figure 10shows the radial profile of the azimuthal standard devia-tion of the intensities obtained in phase-only correction inthe focal plane zone described in Equation 38. The detec-tion level reaches a contrast level of 10 − between 6 and12 λ/D L and 3 . − at 11 λ/D L . As shown in Figure 9(center), speckles are still present in the dark area. Sincewe only corrected for the phase, we can suspect amplitudeeffects.To estimate the amplitude aberration level, we recordedthe pupil illumination on the optical bench without coron-
9. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 10.
Radial profiles of the azimuthal standard de-viation (in RMS) of the intensities in the focal plane typ-ically obtained with this method for phase-only correction,for simulation (blue dashed line) and laboratory bench re-sult (red solid line), for γ = 16 and a square zone of size K S q = 24 . λ/D L . We also plot in this graph the simulationof the focal plane obtained using the amplitude aberrationsrecorded and no phase aberrations (black dash-dotted line). Figure 11.
Pupil illumination recorded on the laboratorybench. agraph, shown in Figure 11. The amplitude defect level isestimated to be about 10% RMS in intensity. The periodof the actuator pitch clearly appears in this pupil image.Due to vignetting effects by the focal coronagraphic mask,these high-frequency structures of the DM surface createillumination variations across the pupil. The first effect ofthese high-frequency aberrations are bright speckles out-side the corrected zone (mostly on the eight bright peaks).The second effect is more critical for our purpose. Becausethe level of high-frequency amplitude errors varies acrossthe pupil, it creates low-frequency amplitude aberrations,which induce bright speckles in the center of the correctionzone.To compare the level of the recorded speckles with theone expected using amplitude and phase errors, we simu-lated the expected focal plane image. We used the ampli-tude aberrations deduced from the intensity measurementon the laboratory bench (Figure 11). From these amplitudeaberrations, we first simulated the the focal plane without phase errors (just amplitude errors). The profile of this fo-cal plane is plotted in Figure 10 with a black dot-dashedline. We then simulated a phase-only correction, assuminginitial phase aberrations of 16 nm RMS over the pupil, anda power spectral density (PSD) in f − where f is the spa-tial frequency. These simulation results (blue dashed line)are compared to the experimental measurement (red line)in Figure 10. The level and shape of the two curves are verysimilar. They show the same structure around 27 λ/D L , dueto the eight bright peaks created by amplitude aberrations.These curves inside the DH match the simulation of thefocal plane without amplitude aberrations. It seems that inphase-only correction, we corrected all phase aberrationsand that we are only limited by amplitude errors. The simulation without amplitude errors (only phase aber-rations) shows that a contrast level of 10 − can be reached,as previously shown in Galicher et al. (2010). Since the am-plitude errors set the limits of our phase-only corrections,we aim to correct both phase and amplitude at the sametime. However, with only one DM, the corrected zone issmaller by half, as shown in Figure 9 (right). Therefore,the radial profile measurement zone becomes (cid:26) x ∈ [1 λ/D L ; 20 λ/D L ] y ∈ [ − λ/D L ; − λ/D L ] ∪ [1 λ/D L ; 20 λ/D L ] . (39)The results for this correction are plotted in Figure 12as a dashed blue line for the simulations and as a red line forthe laboratory bench results. When correcting for the phaseand amplitude aberrations, we obtain contrasts better than10 − between 2 λ/D L and 12 λ/D L , and better than 3 . − between 7 λ/D L and 11 λ/D L . This is an improvement com-pared to the phase-only correction. The simulated profilesmatch the laboratory results from 0 to 8 λ/D L and outsideof the DH .Between 8 and 12 λ/D L , the experimental correctionshows a plateau at 3 . − , while the simulation correc-tion goes deeper. This plateau is a distinctive feature ofa limitation caused by the low dynamic range of the de-tector (our CCD camera has a full well capacity of 13,000electrons/pixels for a readout noise of 16 electrons/pixels).This is confirmed by the last images of the loop which showspeckle levels below the readout noise between 8 and 12 λ/D L : the speckles beyond the readout noise are not visi-ble and thus beyond correction. However, this problem canbe solved by using a detector with a better dynamic range.The number of incoming photons from the observedsource is a critical problem of any speckle-correction tech-nique: the speckles can only be corrected for to a certainlevel of contrast if the source is bright enough for them tobe detected above photon and detector noise at these levels.Although we can correct in a closed loop at 20 Hz in thelaboratory, the correction rate in a real telescope observa-tion will be limited by the shortest exposure time necessary.This shortest exposure time depends on several parameterssuch as stellar magnitude, observational wavelengths, tele-scope diameter, or dynamic range of the camera.The contrast level in the numerical simulation is limitedto 10 − . This is due to the high-amplitude defects (10% inintensity) introduced by the DM in the pupil. Indeed, thebright speckles of the uncorrected half-area diffract their
10. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 12.
Radial profiles of the azimuthal standard devi-ation (in RMS) of the intensities in the focal plane typicallyobtained with this method for phase and amplitude correc-tion, for simulation (blue dashed line) and laboratory benchresult (red solid line), for γ = 16 and a square zone of size K S q = 24 . λ/D L . light into the corrected half-area. This limit, independent ofthe estimation method (Give’on et al. 2006; Galicher et al.2010), may be lowered by the introduction of a second DMon the optical bench (Pueyo et al. 2010).In the next sections (Section 5.4 and 5.5), we study theinfluence of different parameters on the SCC performance. In this section, we compare the performance for dif-ferent sizes of the square zone S q . Using the modi-fied estimator introduced in Equation 37, and for dif-ferent square zone sizes K S q , we experimentally closedthe loop and recorded images after convergence. In thesetests, we used N = 27 actuators across the pupil di-ameter and γ = 16, with phase-only correction. Asexplained in Section 5.2, for this number of actua-tors, we have DH max = [ − . λ/ (2 D L ) , . λ/ (2 D L )] × [ − . λ/ (2 D L ) , . λ/ (2 D L )] (as D L /D P = 8 / . K S q = ∞ ) and three others: K S q =26 . λ/D L , which is only slightly smaller than size of thelargest DH and two smaller square zones ( K S q = 20 . λ/D L and K S q = 24 . λ/D L ). The images obtained in the last twocases can be seen in Figure 9: K S q = 20 . λ/D L (left) and K S q = 24 . λ/D L (center).Figure 13 presents the radial profiles of the focal planesobtained on the laboratory bench, normalized by the high-est value of the PSF obtained without coronagraphic mask.The red, solid curve shows the result for K S q = ∞ ,without square zone. The blue dotted line represents theresult of a square mask of size K S q = 26 . λ/D L , which isonly slightly smaller than the actual cut-off frequency ofthe DM. In this case, we prevented the correction of speck-les outside of the DH and obtained a great improvementinside the DH (0 to 13.5 λ/D L ) and a small depreciation Figure 13.
Experimental radial profile comparison of darkholes obtained on the test bench without square zone (di-rectly using the estimator described in Equation 36) (red,solid) and with square zones of different side lengths: K S q =26 . λ/D L (blue, dotted), K S q = 24 . λ/D L (green, dashed)and K S q = 20 . λ/D L (black, dot-dashed). These phase-onlycorrections were achieved with a γ = 16 reference pupil. Theintensities are normalized by the highest value of the PSFobtained without coronagraphic mask. outside (13.5 to 15.5 λ/D L ). Using a smaller correction zone( K S q = 24 . λ/D L green dashed line) still improves the cor-rection but to the detriment of the size of the DH (the con-trast starts to rise around 12 λ/D L ). Finally, we see that asmaller square zone ( K S q = 20 . λ/D L , black, dot-dashed)produces a smaller but not shallower DH .Going from K S q = ∞ to K S q = 24 . λ/D L , the contrastin the DH progressively deepens. This is because correct-ing fewer of the highest frequencies with a constant num-ber of actuators, we free degrees of freedom. However, for K S q < . λ/D L , the contrast level does not improve be-cause we reach the level of the speckles created by the am-plitude aberrations. Additional shrinking would only reducethe size of the DH . Thus, the reduction of the correctedzone in the wavefront estimation greatly improves the cor-rection performance (up to a factor 10) with only a smallreduction of the DH size. This effect was described in Bord´e& Traub (2006) using 1D simulations.It is important to note that this improvement does notcome from the phenomenon of aliasing in the estimation(Poyneer & Macintosh 2004). Indeed, only the correction isenhanced by this process, because the estimation remainedunchanged. The wavefront estimation with the SCC is onlylimited in frequency by the size of the reference PSF: wecan estimate speckles as long as the reference flux is notnull, i.e. , as long as the speckles are fringed. In most cases(see next section), the first dark ring of the reference PSFis larger than the correction zone and the frequencies insidethe PSF’s first dark ring are well estimated.
11. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 14.
Experimental radial profiles of the PSFs forreference pupils from γ = 10 to γ = 22 . recorded on theoptical bench. The distance to the center is in λ/D L . Thesereference PSFs are normalized by the highest value of theLyot PSF obtained without coronagraphic mask. The ver-tical line correspond to the frequency cut-off for N = 27 actuators in the entrance pupil ( √ N λ/ (2 D L ) ) In this section, we study the effect of the size of the referencepupil on the performance of the SCC. In the previous sec-tions, we used two assumptions on the size of the referencepupil. First, in Section 4.2, we assumed a reference pupilsmall enough to consider that the influence of the aberra-tions inside such a reference pupil is negligible. Simulationsshowed that even for small γ , the level of aberrations inthe reference pupil is very low and uncorrelated to the levelof aberrations in the entrance pupil. Second, in Section 4,we assumed a reference pupil small enough to consider A ∗ R constant over the correction zone in the focal plane. As pre-viously mentioned, the highest frequency attainable by theDM is √ N λ/ (2 D L ). Using the first assumption, | A ∗ R | is aperfect PSF whose first dark ring is located at 1 . λγ/D L .Thus, A ∗ R is roughly constant over the DH if1 . γ > N/ √ . (40)For N = 27 actuators in the entrance pupil, Equation 40reads γ > .
6. In Figure 14, we plot the radial profiles of | A R | recorded on the optical bench for γ from 10 to 22 . − for γ = 10 to 3 . − for γ =22 . γ = 10 (blue, solid) does notsatisfy Equation 40, and the first ring of its PSF is insidethe correction zone (vertical orange dashed line). We testthis case independently in Section 5.5.2. The other referencepupils are studied in Section 5.5.1. The size of the reference pupil can influence the correctionin two different ways: it changes the signal-to-noise ratio (S/N) on the fringes and modifies the flatness of the refer-ence PSF over the correction zone. We develop these effectsin this order in this section.The S/N on the fringes is critical, because I − can onlybe retrieved with well-contrasted fringes. The S/N is di-rectly related to the reference pupil size. Using Equation 23,we deduce that the peak-to-peak amplitude of the fringes inthe focal plane is 2 | A S || A R | . Thus, if | A S | and | A R | are ex-pressed in photons, and assuming only photon and read-outnoise, the S/N can be written as S/N (cid:39) | A S || A R | (cid:112) | A S | + | A R | + σ cam , (41)where σ cam is the standard deviation of the detector noisein photons. A higher S/N allows a better estimate of thespeckle complex amplitude and thus, a better correctionof the aberrations. One can notice that this S/N can besimplified depending on the relative values of its differentterms. We quickly study the following cases: – if | A S | ≈ | A R | (cid:28) σ cam , S/N →
0. In this case, thecorrection is limited by the dynamic range. – if σ cam (cid:28) | A S | and | A R | (cid:28) | A S | , S/N ∼ | A R | . Initialcase, at the beginning of the correction, when the Lyotpupil is a lot brighter than the reference pupil. The S/Nis only a function of | A R | . – if σ cam (cid:28) | A R | and | A S | (cid:28) | A R | , S/N ∼ | A S | . TheS/N is decreasing with deepening correction. The refer-ence brightness is not important.Equation 41 shows that this S/N is an increasing functionof | A R | , but for deep corrections ( | A S | (cid:28) | A R | ), the impactof the size of the reference is probably very weak.The second effect is due to the assumption of a constantreference PSF over the correction zone. Variations of A ∗ R inthe correction zone distort the wavefront estimation. Thiseffect advocates for small reference pupils (large γ ): a ref-erence pupil of γ = 16 generates an A ∗ R that varies from1 to 0 .
03 inside a correction zone of 27x27 λ/D L . For thisreference pupil, the fringe intensity is weaker at the edgesof the DH . Therefore, the estimate is less accurate at theselocations.Using simulation tools, where we can change the cameraand photon noise easily, we were able to isolate these twodifferent effects and analyzed their influence on the perfor-mance of the instrument separately. A more detailed studyhas previously been presented in Mazoyer et al. (2012).Here, we experimentally tested the influence of the ref-erence size. We used 27 actuators across the pupil diameterand K S q = 24 . λ/D L with phase-only correction. Figure 15shows the radial profiles of the SCC image in RMS ob-tained on the laboratory bench for different reference pupils( γ = 16 and γ = 22 . γ = 16) is prefer-able, even at the edge of the DH , where the reference PSFfor γ = 16 is fainter than the reference PSF for γ = 22 . | A R | (cid:28) | A S | . Deeper corrections wouldnormally depend less on the size of the reference pupil.When we use the SCC as a planet finder there is anotherimpact to consider: detection is possible only if the planetintensity is higher than the photon noise of the reference
12. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC
Figure 15.
Radial profiles obtained on the laboratory benchfor two different reference pupils ( γ = 16 and γ = 22 . )with N = 27 actuators and K S q = 24 . λ/D L for the size ofthe corrected zone. These contrasts are normalized by thehighest value of the PSF obtained without coronagraphicmask. pupil. This effect advocates for small reference pupils. Atrade-off study of the reference size is needed depending onexpected planet intensity and the actual contrast that canbe achieved. A more complete study of the noise in the SCCestimation is given in Galicher et al. (2010). In this section, we experimentally prove that we can stillachieve a correction inside the DH using a reference pupilthat does not satisfy Equation 40 by modifying the phaseestimator. This correction has previously been simulatedin Galicher et al. (2010). A ∗ R is still considered as the com-plex amplitude of a perfect PSF, but we cannot considerit uniform anymore over the DH . First, the speckles in thefirst dark ring of this PSF are not fringed, because thereference PSF intensity is null at this location. The wave-front errors that produce these speckles are not estimatedand are thus not corrected for. Second, the sign of (cid:60) [ A ∗ R ]and (cid:61) [ A ∗ R ] changes between the first and the second darkring ( i.e. , between 1 .
22 and 2 . λ/D L ). These speckles arefringed and we can estimate the wavefront errors that pro-duce them when we consider the sign change. Hence, whenEquation 40 is not satisfied, instead of A ∗ R constant, weassume | A ∗ R | constant and change the sign of A ∗ R over thecorrection zone. We now estimateΦ est = i F − (cid:20) Sign [ (cid:60) [ A ∗ R ]] .I − exp( − iφ mesop ) M (cid:21) .P, (42)where Sign [ (cid:60) [ A ∗ R ]], is the sign of the real part of A ∗ R . Thisfunction is represented in Figure 16 (center).In practice, to achieve the correction with this referencepupil, we multiplied I − by the mask in Figure 16 (center),where the white zones (the black zones) are constant andequal to 1 ( − Figure 16.
PSF of the 0.8mm reference pupil ( γ = 10 )(right). From this PSF we constructed the sign mask (cen-ter). The white zones are uniform and equal to 1 and theblack zones are equal to -1. Multiplying I − by this mask, thecorrection can be achieved (right) for this reference pupil. find the dark rings of the complex amplitude. We were ableto build the sign of the real part of the complex amplitude.The tests on the optical bench were conducted using the0.8mm reference pupil ( γ = 10) and the process describedin Section 5.3. We used no square zone. The resulting DH is presented in Figure 16 (right). We distinctly see the firstreference ring at 1 . λ/D R . As expected, the speckles onthis ring are not corrected for, because they are not fringed.Nevertheless, apart from this ring, the whole DH is cor-rected. Although correction with a large reference pupil ispossible, the level of speckle suppression is much lower (bet-ter contrast) than with smaller reference pupils (higher γ ),because the speckles of the uncorrected dark ring diffracttheir light into the corrected zone (Galicher et al. 2010;Give’on et al. 2006).We showed in Section 5.5.1 that the SCC used witha reference pupil that obeys Equation 40 shows a betterperformance. However, some cases (many aberrations dueto an unknown initial position of the DM, for example)may require the use of large reference pupils that producehighly contrasted fringes even with very aberrated wave-fronts. The correction can then be initiated by correctingfor low spatial frequencies (usually dominating the wave-front errors). Finally, the large reference is replaced witha smaller reference (which satisfies Equation 40) to correcthigher frequencies and reach better contrast levels.
6. Conclusion
In Section2.1, we used Fourier optics to model the propa-gation of light through a coronagraph. We then proposed amethod for estimating phase and amplitude aberrations inthe entrance pupil from the complex electric field measuredin the focal plane after a four-quadrant phase mask corona-graph. We used this model to correct phase and amplitudeaberrations in a closed loop using a DM in the pupil plane,even for a Lyot pupil smaller than the entrance pupil.We implemented this technique, associated with a self-coherent camera as a focal plane wavefront sensor. We cor-rected for phase and amplitude aberrations in a closed loopwhich led to speckle suppression in the central area of thefocal plane (called dark hole).We tested these methods on a laboratory bench wherewe were able to close the loop and obtain a stable correctionat 20 Hz. When correcting for phase aberrations only , weobtained contrast levels (RMS) better than 10 − between6 and 12 λ/D L and 3 . − at 11 λ/D L . We proved that wecorrected for most phase aberrations in the dark hole and
13. Mazoyer et al.: Estimation & correction of wavefront aberrations using the SCC that the contrast is limited by high amplitude aberrations(10% RMS in intensity) induced by the DM. When cor-recting for the phase and amplitude aberrations using oneDM, we obtained contrast level better than 10 − between2 λ/D L and 12 λ/D L , and better than 3 . − between 7 λ/D L and 11 λ/D L . The simulation performance was lim-ited by the diffraction of the speckles of the uncorrectedarea in the focal plane created by the amplitude defects. Inaddition, in laboratory tests, the contrast is currently lim-ited by the camera dynamics in the aberration estimation.We experimentally proved that a small shrinking of thesize of the correction zone can improve the contrast thecontrast up to a factor 10. We analyzed the influence ofthe reference pupil radius on the performance of the SCCand proved that the reference of γ = 16 (the larger refer-ence pupil possible with a nonzero reference flux inside thecorrection zone) provides the best correction in our case.To enhance the performance of the self-coherent cam-era even more, we plan several improvements. First, onecan directly minimize A S , the speckle complex field mea-sured by the SCC and not the phase estimated in thepupil plane. This approach has started to show good re-sults (Baudoz et al. 2012) for the simultaneous correctionof amplitude and phase. The correction for the amplitudeerrors can probably also be improved by the use of twoDMs. Moreover, solutions are considered to use the SCCwith wider spectral bandwidths. First tests in polychro-matic light have already been conducted and show promis-ing results (Baudoz et al. 2012). A preliminary study ofthese effects has been published (Galicher et al. 2010). Aforthcoming paper will present a new version of the SCCthat will probably overcome the current chromatic limita-tion. Acknowledgements.
J. Mazoyer is grateful to the Centre Nationald’Etudes Spatiales (CNES, Toulouse, France) and Astrium (Toulouse,France) for supporting his PhD fellowship. SCC development is sup-ported by CNES (Toulouse, France).
References