Improving VLT/SPHERE without additional hardware: Comparing quasi-static correction strategies.

Direct imaging of exoplanet is currently performed with large ground-based telescopes equipped with state-of-the-art coronagraph instruments[1, 2, 3] . These instruments provide diffraction-limited point spread functions thanks to their extreme adaptive optics systems (XAO) while the coronagraph aims to attenuate the starlight to reveal the faint companions and/or circumstellar disks. Their capabilities are however limited by post-XAO wavefront residuals that enable starlight leakage through the coronagraph, hence creating bright stellar speckles on the science detector. Advanced post-processing techniques like angular (ADI)[4] , spectral (SDI)[5] , polarimetric (PDI)[6] , and reference-star (RDI)[7] differential imaging are therefore used on raw images to further enhance detection capabilities by mitigating speckle noise in the images. These techniques are particularly efficient to remove speckles originating from instabilities and wavefront errors internal to the instruments like the non-common-path aberrations (NCPAs). But the broadly used ADI technique requires 1-2 h sequences of observations and suffers for self-subtraction of astrophysical sources at small angular separations. Mitigating static and quasi-static speckles using focal plane wavefront sensors is now an important study of research [8, 9, 10, 11] . In this work, we present the first on-sky correction of the NCPAs on SPHERE with pair-wise probing (PWP) and electric field conjugation (EFC), originally developed for high-contrast imaging in stable environments like space. In Sec. 2, we describe the algorithms used throughout this study and how they were implemented in the instrument. In Sec. 3, we demonstrate the algorithms on the internal source to investigate the contrast limitation and apply those solutions while observing on-sky. In Sec. 4, we demonstrate the correction loop on-sky in a half DH. In Sec. 5, we use PWP alone to calibrate the static stellar speckles in post-processing. Finally in Sec. 6, we discuss the combination of the different strategies and best practices to optimize the speckle calibration in accordance with the science objectives.

PWP and EFC algorithms have been extensively described in the literature[12, 13, 14] . Their combination in closed-loop aims to minimize the speckle intensity a region of the science image hence called the dark hole (DH). First, PWP probing estimates the focal plane electric field (E-field) through a temporal modulation of the speckle field, equivalently to conventional phase diversity techniques. PWP requires a high-order deformable mirror (DM) to introduce optical path differences (OPDs) called probes that coherently interfere with the speckle field we wish to correct. In the absence of a specific DM for this purpose, the OPDs are introduced while AO runs in closed-loop by modifying the Shack-Hartmann (SH) reference slopes, assuming the sensor remains linear. Such an assumption might be invalid in the case of a pyramid wavefront sensor planned for the upgrade of both VLT/SPHERE[15] and Gemini/GPI[16] . The probes need to be chosen such that they create a different electric field at every location of the speckle field. As explained in a previous work[17] , the first probe is here the poke of one individual actuator, whose phase-shift is mostly unaffected by the coronagraph masks. The second probe is then the individual poke of a second actuator in the direct neighborhood of the first one. The second actuator is chosen depending on the correction zone as explained in the next paragraph. The algorithm requires each probe to be pushed and pulled with an image recorded via the science detector for each DM shape. PWP therefore requires 4 images per E-field estimation (or per iteration when used in closed-loop with EFC).

EFC is the E-field counterpart of phase conjugation in AO. Knowing the E-field with any focal plane wavefront sensor (e.g. either the Self-Coherent Camera[18] or PWP can be used), we aim to create destructive interference in regions of the focal plane by applying the opposite of the estimated E-field, applying OPDs using the DM in the pupil plane. The technique is powerful when applied in half the field of view (called half DH or HDH) since it enables the correction of speckles originating from phase and amplitude aberrations as well as the residual diffraction pattern from the coronagraph masks[17] . The algorithm inverts a Jacobian that describes the effect of each actuator movement on the E-field in the science detector plane. Here, the Jacobian is model-based using Fourier optics and a simplified compact model of the system. The DH regions are chosen in agreement with the set of probes and depend on the science case: two actuators spread vertically (resp. horizontally) are chosen for top and bottom (resp. left and right) DH corrections. The presented DH here ranges from 100 mas to 650 mas from the optical axis. The inversion of the Jacobian is regularized with a smooth filter (Tikhonov regularization[19]) whose knee is defined at the 400th singular value (when sorted in descending order). We use a global loop control gain of 0.2 to ensure convergence.

Focal plane wavefront sensing is performed in the H3 band ($\lambda_{0}=1667$ nm, $\Delta\lambda=54$ nm) with IRDIS[20] behind an Apodized Pupil Lyot Coronagraph[21] (APLC) whose diameter of the central focal plane mask obstruction is 185 mas. On-sky experiments were performed during the second half of the night of February 15th 2022, starting at 5 a.m. UTC. The seeing averaged $\sim$0.8 arcsec at 500 nm through the night while the atmospheric coherence time remained around 10 ms. These good conditions enabled the use of the small pinhole for spatial filtering in front on the SH WFS [22]. Images with 64 s of exposure time are all acquired on HIP 57013 (RA=11 41 19.79 , $\delta$= -43 05 44.40 , R=5.5, H=5.5).

Starting from the original SPHERE calibration, we also closed the correction loop while observing HIP 57013. We applied PWP+EFC in the bottom D-shape DH starting at 135 mas from the star to 650 mas. The exposure time of the probe images was set to 64 s for the sake of averaging the intensity of the turbulence halo below the static speckle intensity. Assuming this halo is stable in time (i.e. the turbulence statistic is in steady state for two probe acquisition in a row), it is then removed in the PWP algorithm when the difference of each pair of diversity images is calculated. In total, each image required 80s with overheads for a total of $\sim$400s per iteration (4 images required for PWP and 1 image to confirm loop convergence). We present the resulting processed images after 4 iterations (i.e. $\sim$30mn) in Fig. 3. The resulting normalized intensity RMS at each iteration is also plotted in Fig. 5 , showing that the static speckle intensity is continuously minimized in the DH and demonstrating robustness of the algorithm. A factor $\sim$2 in performance is gained between 165 mas and 270 mas while we achieve a factor $\sim$3 of improvement between 300 mas and 500 mas. Only a few stellar residuals persist in the science image at small separations after 4 iterations. If this signal is coherent with the starlight, we expect these speckles to be corrected with more PWP+EFC iterations or with a better calibration of the PWP diversity amplitude.

One can also take advantage of the starlight coherence to extract the incoherent light, including any substellar faint companion, from the total intensity image. Indeed, PWP can be used alone to estimate the stellar speckle E-field. The squared absolute value of this E-field is then used as a reference image to be subtracted from the total intensity image. The same technique has been used to highlight the halo of internal turbulence in Fig. 2. The processed image is then called equivalently unmodulated component, incoherent component, or the result of coherent differential imaging (CDI). When applying PWP+EFC, this post-processing technique can be done for free since PWP is already employed at each iteration. We therefore show the result of CDI after iteration 1 of PWP+EFC in Fig. 3. This means one single EFC iteration has already been performed in the bottom DH, followed by one PWP set of acquisitions to get the residual speckle E-field that is used for CDI. The resulting CDI image is cleaned out from modt of the stellar speckles in the regions that are well sensed by PWP (top and bottom regions of the field of view for the chosen probes here) and then enable a speckle correction in almost all the field of view. The resulting normalized intensity RMS before and after CDI in the bottom DH is also plotted in Fig. 6. It demonstrates performance equivalent to the full bottom half DH correction in Sec. 4, but after less than $\sim 15$min of on-sky calibration. The CDI itself improves the performance to a factor up to 3 in the bottom region.

In this work, we have presented three different observing strategies using PWP and EFC. First, minimizing the stellar speckle intensity with a half DH correction performed on the instrument internal calibration unit enables a factor up to 3 in contrast performance. The technique is robust since it is not affected by atmospheric turbulence and it does not overlap with science operations because it uses the AO system during the day. It is also readily adaptable (if anticipated) to any target separation, coronagraph, or spectral bandwidth. However, we showed it has a limited field of view and presented the cause of its performance limitations. The second strategy of observation is to close the loop on-sky while directly observing the target of interest. This method provides the best performance at all angular separation because it is iterative and corrects for wavefront errors along the entire beam, but it relies on good observing conditions and stable AO system and is also limited in field of view. The modulation of the science images with the probes in PWP also prevents a 100% science duty cycle. The third strategy of observation is to use the PWP estimations to apply CDI in post-processing. This increases the field of view for science and provides encouraging performance but requires PWP to be active during science operations.

In the future, these different strategies can be combined, depending on the science case. First, deeper contrast in one particular region of the image can be achieved using a combination of strategies 1 and 2. Strategy 1 would start improving the instrument performance for free and reduce the required number of iteration for strategy 2, hence increasing the time dedicated to science observations. Strategy 3 can also be applied after strategies 1 and 2 by introducing PWP on the already corrected image to perform a final clean up of the DH and potentially observe fainter objects. It would currently require about 5 min after a few on-sky iterations of PWP+EFC for an improvement factor up to 10 in the half DH region. Strategy 3 could also be applied right after strategy 1 and the speckle calibration would require only 5 min of on-sky observation but with potentially degraded results in the DH region. For a broader field of view and for either the study of extended objects or the blind search for planets, strategy 3 could also be applied alone at a regular cadence while the CDI results would be used an inputs for conventional ADI, SDI, PDI or RDI postprocessing techniques. These different calibration scenarios will be investigated in ongoing studies of the optimal observing strategy for VLT/SPHERE.