Expected gain in the pyramid wavefront sensor with limited Strehl ratio
INAF−Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy
email: valentina.viotto@oapd.inaf.it
Received: 21 December 2015
Accepted: 30 June 2016
Context. One of the main properties of the pyramid wavefront sensor is that, once the loop is closed, and as the reference star image shrinks on the pyramid pin, the wavefront estimation signaltonoise ratio can considerably improve. This has been shown to translate into a gain in limiting magnitude when compared with the ShackHartmann wavefront sensor, in which the sampling on the wavefront is performed before the light is split into four quadrants, which does not allow the quality of the focused spot to increase. Since this property is strictly related to the size of the reimaged spot on the pyramid pin, the better the wavefront correction, the higher the gain.
Aims. The goal of this paper is to extend the descriptive and analytical computation of this gain that was given in a previous paper, to partial wavefront correction conditions, which are representative for most of the wide field correction adaptive optics systems.
Methods. After focusing on the low Strehl ratio regime, we analyze the minimum spatial sampling required for the wavefront sensor correction to still experience a considerable gain in sensitivity between the pyramid and the ShackHartmann wavefront sensors.
Results. We find that the gain can be described as a function of the sampling in terms of the Fried parameter.
Key words: instrumentation: adaptive optics
© ESO, 2016
1. Introduction
It has been almost twenty years since the pyramid wavefront sensor (PWFS) was first proposed (Ragazzoni 1996) as an alternative to other types of wavefront derivative measuring sensors. The first single star adaptive optics (AO) loop with a PWFS was closed on AdOpt@TNG (Ragazzoni et al. 2000a), and one of the PWFS best results on sky came recently with FLAO (Esposito et al. 2011), the First Light Adaptive Optics system at the 8m Large Binocular Telescope (LBT; Hill & Salinari 2000), which further increased the popularity of the pyramid sensor. This kind of wavefront sensor has also been successfully implemented in the configuration of multi conjugated adaptive optics (MCAO; Beckers 1988), during the MCAO demonstrator (MAD) experiment at the Very Large Telescope (Marchetti et al. 2007; Arcidiacono et al. 2004) and it is the heart of the next Fizeau interferometric MCAOequipped focal station LINCNIRVANA (Herbst et al. 2003) for the LBT, which couples the layeroriented technique (Ragazzoni et al. 2000b) with the multiple field of view concept (Ragazzoni et al. 2002). Moreover, the PWFS has been studied and implemented in other science fields, like ophthalmology (Iglesias et al. 2002; Chamot et al. 2006) or phase microscopy (Iglesias 2011). Nowadays, the PWFS is considered a very powerful and promising tool both for extreme AO and for MCAO for wide corrected fields of view (Kasper et al. 2014). Being the core of very different systems and concepts, it is not, however, always used in the same configuration, so its properties are exploited in different ways. While, in the following, we examine the difference in performance between the PWFS and the ShackHartmann wavefront sensor (SHWFS), it is noticeable that these do not exhaust the whole range of WFSs available in AO and specifically for astronomical purposes. Adopting the distinction between WFSs whose detection is performed in the pupil rather than the object plane, we would like to mention, in the first group, the curvature (Roddier 1981) wavefront sensor. This is, however, sensitive to the second derivative (Laplacian) of the wavefront rather than the first derivative, a common property of PWFS and SHWFS. Various kinds of shearing interferometers (Hardy et al. 1977) have been implemented as well. Another interesting WFS is represented by the Smartt (Smartt & Strong 1974) in which a signal almost proportional to the wavefront departure from flatness is available, when used close to the diffraction limit condition.
2. Approach and assumptions
In Ragazzoni & Farinato (1999), the authors investigated the nonmodulated PWFS sensitivity with respect to an SHWFS in perfect closed loop conditions and carried out an analysis on the gain of the PWFS in terms of guide star limiting magnitude, with respect to the SHWFS. (In this paper, we will refer at this parameter as Δ_{MAG}.) The result comes from the fact that the variance of the PWFS measurements is the same as the SHWFS for the highest sampled spatial frequency, but it decreases to a factor (D/r_{0})^{2} times lower than the SHWFS for low order modes. This applies to the PWFS when it is pushed to its best regime, i.e. when the loop is successfully closed with the highest number of modes allowed by a pupil sampling on scales of the order of r_{0}. The predicted gain was confirmed shortly after using numerical simulations (Esposito & Riccardi 2001) and onsky measurements (Ghedina et al. 2003). The sensitivity of the two mentioned WFSs has also been compared in the case of a modulating pyramid in Verinaud (2004), while a more extended comparison, which includes also other WFS techniques, such as the curvature WFS (Roddier & Roddier 1988), has been presented by Guyon (2010). In this paper, we use the same approach and arguments adopted by Ragazzoni & Farinato in the mentioned paper, but consider the possibility which happens, for example in MCAO or in open loop configuration, that the pyramid is not working under perfect illumination conditions, meaning that the correction of the wavefront is only partial. The computations and results are mostly focused on the pupil size regime of the next generation of large telescopes, such as the European Extremely Large Telescope (Gilmozzi & Spyromilio 2007), the Giant Magellan Telescope (Johns 2008), and the Thirty Meter Telescope (Szeto et al. 2008), generically referred to as extremely large telescopes (ELTs). Here we intend to consider a PWFS and an SHWFS, working with the same main parameters, with the goal of obtaining a diffraction limited correction. In particular, unless differently specified and independent from the kind of WFS considered, in this paper we always assume the pupil as being sampled at r_{0} scale. Moreover, the following discussion, analytical computations, and simulations only apply to AO systems using unresolved sources. Extended references that can be resolved by the combination of WFS and telescope, such as asteroids (Ribak & Rigaut 1994), extragalactic nuclei (Le Louarn et al. 1998), laser guide stars (Foy & Labeyrie 1985), and even, for ELTs, some natural guide stars (Weiner et al. 2000), are not treated here. For the variety of cases that we consider in this work, in open loop, the spots focusing on the wavefront derivative sensitive element of the two systems (i.e. the pyramid pin, for the PWFS, and the detector, for the SHWFS) share approximately the same size, which is related to the Fried parameter of the atmosphere. Once the loop is closed, however, in a geometrical approximation, the correction performed on the wavefront has no impact on the SHWFS performance, while it allows the spot on the pyramid pin to shrink by a factor of ~D/r_{0}, increasing the sensor signaltonoise ratio (S/N). In Ragazzoni & Farinato (1999), the authors analytically compute the effect of this spotshrinking on the error that is associated with the tiptilt measurement. Then they use the Heisenberg uncertainty principle to extend the same argument to the error of the PWFS, with respect to the SHWFS variance, for a Zernike polynomial of qth radial order. While the same conclusions can be attained through Fourier optics, the Heisenberg principle is applied to the combined uncertainties on the position over the pupil and the momentum associated with the incoming direction, along an axis lying on the pupil plane. Diffraction limited performances are classically obtained through the same argument. The resulting dependence of the measurement variance on the considered radial order is (1)where q is the Zernike radial order and Q = D/r_{0} is the maximum radial order that can be corrected by a system with an aperture D (assuming sampling down to r_{0} scale), when r_{0} is the Fried parameter. To quantify the actual noise for the SH wavefront sensor, the result given in Rigaut & Gendron (1992) was considered as a gauging point. They derived a general expression for the noise propagation coefficients of each Zernike mode in the case of an SHWFS, to be applied to the noise which affects the slope measurement of each subaperture. This result, in which spatial and temporal errors are neglected, leads to a global reconstruction noise given by (2)where is the photonic noise error. Even if this analytical expression is not directly used in the following, here it is reported to show how the SHWFS global reconstruction noise is proportional to the photon noise, enabling us to gauge the PWFS noise to that of the SHWFS, assuming the same reconstructor. The prediction given in 1999 was confirmed with high confidence for low radial orders (up to Q = 7) with a laboratory experiment called Pyramir (Peter et al. 2010). We contacted the authors and were informed that their data were obtained in a laboratory and that the Strehl ratio (SR) on the pyramid pin was of the order of 90−95% (priv. comm., 2012). From a course inspection of Fig. 8 of Peter et al. (2010), a level of confidence better than 5% is estimated. The lower propagation coefficients, for the PWFS in closed loop, translate into a gain Δ_{MAG}> 2.5 mag, which can be achieved with respect to a SHWFS, for the generation of ELTs in the visible and IR bands (Fig. 1, reconstructed from the results in Ragazzoni & Farinato 1999). A few words are required to explain the expression “gain in magnitude”. In the framework of this paper, this type of gain is computed using the ratio of the noise coefficients for the PWFS vs. conventional wavefront sensing approach. This gain will only affect the area of the plot SR vs. magnitude, where the SR is not saturated because of the dominance of other errors, especially the fitting one. In the intermediate and low SR regime, under this type of assumption, a curve that is rigidly shifted towards higher magnitudes should be experienced. It is clear, however, that for extremely low SRs, whose limit is dictated by the proximity to the minimum SR defined in the next section, the assumptions behind the calculation are not confirmed. For completeness, we must say that a similar result has been differently quantified by Verinaud (2004) with another model, which includes the filtering effect of the subapertures, resulting in a maximum Δ_{MAG} that is smaller by about 0.5 mag than the one shown in Fig. 1.
Fig. 1 PWFS vs. SHWFS magnitude gain at different maximum radial orders. Data taken from Ragazzoni & Farinato (1999). 

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3. Moving away from the high SR regime
3.1. Strehl ratio effect on tiptilt determination error
The overall gain in magnitude that results from this prediction can approximate the actual PWFS Δ_{MAG} only assuming noiseless measurements or, in other words (at least in this context), considering the full correction achievable in the brightend regime. To estimate the effect of a partial correction onto the pyramid pin (in the diffractiontoseeing limit regime), we introduce the approximation that the shape of the reimaged star is obtained superposing

a seeinglimited spot, with a total flux of(1−S) × n^{∗} photons

a diffractionlimited spot, produced by S × n^{∗} photons,
where n^{∗} is the total number of collected photons and S the SR on the pyramid pin. On first analysis, the two superposed spots have been approximated with cylinders (Fig. 2). The definitions above implicitly assume that the central core of the PSF is clearly distinguishable from its surrounding halo. One can estimate the minimum SR, for which this assumption is verified: since the PSF comprises N_{s} ~ (D/r_{0})^{2} speckles, a central peak one order of magnitude brighter than the halo could be obtained only for SR >S_{min} = 10 /N_{s}. While, for an ELTlike telescope, this is a very weak constraint, which leads to numbers usually well below S_{min} = 1%; for 8mclass telescopes a SR of the order of 10% is instead required.
Fig. 2 Partial correction spot shape assumption. Topleft: Gaussian spot, bottomright: cylindrical spot (scales are different). 

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Figure 3 shows the PWFS tiptilt signal response, as a function of the actual shift of the spot, for different values of the SR. We recall here that, for this kind of WFS, a steeper slope in the signal vs. shift plane describes a higher sensitivity. The latter can be reached either when a higher SR is delivered to the PWFS, or, for a given SR, when the D/r_{0} parameter increases. In this last case, the signal response experiences a reduction of its linear range, which is included between the vertical dashed lines in Fig. 3.
Fig. 3 PWFS tilt signal slope in case of different SR achieved on the pyramid pin (a low value for Q = D/r0 = 4 has been used for the purpose of clarity). 

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With the aforementioned assumptions, we can derive analytically the dependence of the PWFS tiptilt error on the SR value, for small tiptilt signals, as (3)For the sake of completeness, we add a further consideration: the cylindrical spots approximation clearly does not reproduce the actual shape of the spot on the pyramid pin, but it tends to underestimate the sensitivity of the PWFS. If the superposition of two Gaussian spots is considered, in fact, the measurement error to be associated to the tiptilt determination is reduced for very small aberrations. Figure 4 shows that the reducing factor that needs to be applied is of the order of 10 for low values of S. This correction is neglected in the following. This is in fact an underestimation leading to a safer establishment of the predicted gain (i.e. the gain is likely to be lower bounded by the figures found here). However, numerical experiments show that the difference reported in Fig. 4 gets smaller for much lower D/r_{0} and it is, moreover, confined to the solely tiptilt term.
Fig. 4 Correction to be applied (solid line) if Gaussianshaped spots (dotted line) are considered instead of cylindrical spots (dashed line). An EELTlike telescope in r_{0} = 0.2 m conditions has been assumed (D/r_{0} ≈ 200). 

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3.2. Strehl ratio effect extended to higher orders
Since in Eq. (3) the PWFS variance is still proportional to the SHWFS one, we can again introduce the argument of the Heisenberg uncertainty principle. Since the sensitivity of the sensor is inversely proportional to the second power of the focal spot linear dimension, we can extrapolate the dependence of the propagation coefficients for the other modes from the tiptilt one as (4)Figure 5 shows the error propagation coefficients for both the SHWFS and the PWFS in the noiseless case (black lines, the same data presented in Ragazzoni & Farinato 1999), compared to the same errors for different SRs achieved on the pyramid pin in closed loop. The difference between the couple of curves representing the SH and the pyramid WFSs decreases as the performance in terms of SR tends to zero. This is expected since, for low SR values, the spot on the pyramid pin gets more similar to the one the system sees in open loop.
Fig. 5 Solid line: noise propagation coefficients vs. the radial order q for the SHWFS case; dashed lines: the same for the PWFS case. Colors represent the effect of different SR achieved on the pyramid pin in closed loop. The xaxis maximum limit represents the D/r_{0} parameter for the selected case (EELTlike telescope in r_{0} = 0.2 m conditions). 

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The resulting Δ_{MAG}, which is now dependent on the SR, is shown in Fig. 6, for different values of Q = D/r_{0}. As we can see, there is a limit on the maximum achievable SR value, which is related to the maximum corrected radial order. As per Noll (1976), in fact, the residual root mean square error, after the correction of the first J (with J> 10) Zernike modes of a wavefront perturbed according to Kolmogorov statistics, is (5)whose corresponding limit SR is given by the Marechal’s approximation as (6)In the realm of parameters considered in this work, the limit is close to 70%.
Fig. 6 Gain in limiting magnitude (Δ_{MAG}) for different Q = D/r_{0}, as a function of the achieved SR on the pyramid pin. The dashed line shows the limit in the maximum SR that can be obtained for each Qvalue. 

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For a wavefront sensor working in the visible band and installed at an ELTlike telescope, D/r_{0} is usually above 200, which means that a relatively low SR, about 5%, can still be large enough to take advantage of the gain in sensitivity of a PWFS over a SHWFS, since a Δ_{MAG} ~ 1 mag is expected. For lower SR, however, the curves presented in Fig. 6 dramatically drop to Δ_{MAG} = 0.
4. Discussion: which is the gain for a low SR regime?
As already mentioned, the PWFS is not only used as a tool to reach very high resolution in classical AO, but it can also be implemented in other AO techniques, such as MCAO, ground layer AO (Rigaut 2002), multiobjects AO (Hammer et al. 2002), or in the recently proposed global MCAO (Ragazzoni et al. 2010). In all these cases, the PWFS might operate in open or partially open loop and sense the turbulence at a shorter wavelength with respect to those for which the loop parameters are optimized, which leads to very low SRs on the pyramid pin. In this framework, it is important to evaluate which is the gain that can be expected for a system working at low SR by design. Figure 7 shows the expected Δ_{MAG} for an EELTlike telescope, with a Fried parameter taken from the standard ESO Paranal atmosphere (r_{0} = 0.129 m at 500 nm), focusing on the low SR regime. The gain is here computed for different wavelengths, showing the loss in the Δ_{MAG} value experienced by the system when moving to larger wavelengths. We note that, since a r_{0} spatial sampling at the level of the pupil telescope is still assumed, for shorter wavelengths a higher sampling is automatically considered. Equation (6) has been used to derive SR.
Fig. 7 Gain in limiting magnitude Δ_{MAG} for different sensing wavelengths, for an EELTlike telescope (40 m diameter) with a standard ESO Paranal atmosphere. 

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If we consider the possibility of having a PWFS working in the NIR regime, and we want to estimate what could be the actual Δ_{MAG} with respect to an analogous SH system, we can refer to the performances obtained by an existing PWFS (FLAO), whose results also include error sources that we did not take into account in our model. Figure 8 shows the performance confirmed by FLAO at the LBT in Kband and an extrapolation to shorter wavelengths, which was performed assuming a constant corrected wavefront shape. We report this as having an estimation of the SR (which translates into gain in limiting magnitude) that we can expect from a real working system. If we focus on the faintend regime (between 15 and 18.5 mag) and the J band, we see that the SRs which can be achieved are always below 20%. This, however, is a lower limit, since the performance here is measured on the scientific camera, and not directly on the pyramid pin. If we look at the curves in Fig. 7, with these types of SR and at NIR wavelengths, we can still expect a Δ_{MAG} value of up to 1.5 mag, which is a nonnegligible increase.
Fig. 8 FLAO performance in K band (solid line), from Esposito et al. (2011), and the one extrapolated for a shorter wavelength. 

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5. Discussion: dimensioning the system
As just discussed, to take advantage of the gain in sensitivity of the PWFS, a minimum SR on the pyramid pin itself should be reached. Another topic affected by this is the dimensioning of the system also in terms of hardware components, since the required pupil sampling is obviously the driver for the choice of both the detector and the correcting device (i.e. deformable mirror). In this section, we also explore the consequence of different spatial samplings at the level of the pupil on the gain parameter. The effect of different samplings on the SR or the Δ_{MAG} is of course related to the considered wavelength, since the Fried parameter directly depends on it. Figure 9 shows the computed Δ_{MAG} with limited correction in terms of SR on the pyramid pin, as a function of the maximum corrected radial order, for an EELTlike system (a 40 m diameter pupil was considered). The expected SR is directly derived from the pupil sampling under noiseless assumption from high spatial frequency residuals (Noll 1976), and the result is shown for different corrected radial orders and the full considered wavelength range, in Fig. 10. The plot shows the result of our simulation for a set of different wavelengths, with a 0.1 μm separation between each of them in the 0.5−2.2 μm range. With this result, we can find a relation between the pupil sampling in r_{0} units (thus including the wavelength dependence), to expect a certain gain Δ_{MAG}. Even if the precise values depend on the approximation used to estimate the number of actuators to correct a given radial order, we can derive a rule of thumb. In fact, to experience any Δ_{MAG}, one should tune the AO correction parameters to spatially sample the pupil with a pitch lower than 5−6r_{0}. If we want to take advantage of a 1 mag gain, instead, the pitch should be smaller than ~3r_{0}, while a pitch ~r_{0} is required to aim for a gain of 2 mag.
Fig. 9 Gain in magnitude of a PWFS with respect to an SHWFS (Δ_{MAG}) in a limited SR regime, as a function of the maximum radial order that can be corrected by the deformable mirror. Solid lines represent the computed gain for different wavelengths, with a 0.1 μm separation between each of them. The extreme wavelengths considered are displayed in green (λ = 2.2 μm) and red (λ = 0.5 μm), being the correspondent D/r_{0} parameters 52 and 310, respectively. Dashed lines represent the maximum slopes of the curves. 

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Fig. 10 SR as a function of the corrected radial order for different wavelengths. This is directly derived from the pupil sampling under noiseless assumption from high spatial frequency residuals. The extreme wavelengths considered are displayed in green (λ = 2.2 μm) and red (λ = 0.5 μm). 

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6. Conclusions
Defining Δ_{MAG} as the nonmodulated PWFS gain in the reference star limiting magnitude with respect to an SHWFS, in perfect closed loop conditions and with the same sampling, we retrieved the dependence of Δ_{MAG} on the SR that was achieved on the pyramid pin. Since the gain is due to the fact that the reference star image at the entrance of the PWFS shrinks, partial wavefront correction conditions reduce the Δ_{MAG} value with respect to the one expected for a noiseless ideal system. Assuming a spatial sampling down to the Fried parameter, we retrieved Δ_{MAG} as a function of the SR for different D/r_{0}, obtaining the result that, for an ELTclass telescope in the Visible band, a Δ_{MAG} ~ 1 mag is expected for a 5% SR, while a higher SR of about 40% translates into a gain Δ_{MAG} ~ 2 mag. Focusing on the low SR regime, we found that the gain can be described as a function of the spatial sampling in terms of the Fried parameter. As a rule of thumb, we can say that, to obtain a Δ_{MAG} higher than 0 mag, 1 mag, and 2 mag, the spatial sampling of the pupil needs to be denser than 5−6r_{0}, ~3r_{0}, and ~r_{0}, respectively. This result can be taken into account for the selection of system parameters, since this low SR regime is in the realm of most of the wide field correction AO systems. Moreover, it can also apply to the cases in which the WFS is working at a shorter wavelength with respect to the scientific camera, for which the pupil spatial sampling is dimensioned.
Acknowledgments
The authors greatly appreciated the useful review of the anonymous referee.
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All Figures
Fig. 1 PWFS vs. SHWFS magnitude gain at different maximum radial orders. Data taken from Ragazzoni & Farinato (1999). 

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In the text 
Fig. 2 Partial correction spot shape assumption. Topleft: Gaussian spot, bottomright: cylindrical spot (scales are different). 

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In the text 
Fig. 3 PWFS tilt signal slope in case of different SR achieved on the pyramid pin (a low value for Q = D/r0 = 4 has been used for the purpose of clarity). 

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In the text 
Fig. 4 Correction to be applied (solid line) if Gaussianshaped spots (dotted line) are considered instead of cylindrical spots (dashed line). An EELTlike telescope in r_{0} = 0.2 m conditions has been assumed (D/r_{0} ≈ 200). 

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In the text 
Fig. 5 Solid line: noise propagation coefficients vs. the radial order q for the SHWFS case; dashed lines: the same for the PWFS case. Colors represent the effect of different SR achieved on the pyramid pin in closed loop. The xaxis maximum limit represents the D/r_{0} parameter for the selected case (EELTlike telescope in r_{0} = 0.2 m conditions). 

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In the text 
Fig. 6 Gain in limiting magnitude (Δ_{MAG}) for different Q = D/r_{0}, as a function of the achieved SR on the pyramid pin. The dashed line shows the limit in the maximum SR that can be obtained for each Qvalue. 

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In the text 
Fig. 7 Gain in limiting magnitude Δ_{MAG} for different sensing wavelengths, for an EELTlike telescope (40 m diameter) with a standard ESO Paranal atmosphere. 

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In the text 
Fig. 8 FLAO performance in K band (solid line), from Esposito et al. (2011), and the one extrapolated for a shorter wavelength. 

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In the text 
Fig. 9 Gain in magnitude of a PWFS with respect to an SHWFS (Δ_{MAG}) in a limited SR regime, as a function of the maximum radial order that can be corrected by the deformable mirror. Solid lines represent the computed gain for different wavelengths, with a 0.1 μm separation between each of them. The extreme wavelengths considered are displayed in green (λ = 2.2 μm) and red (λ = 0.5 μm), being the correspondent D/r_{0} parameters 52 and 310, respectively. Dashed lines represent the maximum slopes of the curves. 

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In the text 
Fig. 10 SR as a function of the corrected radial order for different wavelengths. This is directly derived from the pupil sampling under noiseless assumption from high spatial frequency residuals. The extreme wavelengths considered are displayed in green (λ = 2.2 μm) and red (λ = 0.5 μm). 

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In the text 