TY - JOUR
AU - Véran,, J-P
AB - ABSTRACT Adaptive optics (AO) restore the angular resolution of ground-based telescopes, but at the cost of delivering a time- and space-varying point spread function (PSF) with a complex shape. PSF knowledge is crucial for breaking existing limits on the measured accuracy of photometry and astrometry in science observations. In this paper, we concentrate our analyses of the anisoplanatism signature only on to the PSF. For large-field observations (20 arcmin) with single-conjugated AO, PSFs are strongly elongated due to anisoplanatism that manifests itself as three different terms for laser guide star (LGS) systems: angular, focal and tilt anisoplanatism. First, we propose a generalized model that relies on a point-wise decomposition of the phase and encompasses the non-stationarity of LGS systems. We demonstrate that it is more accurate and less computationally demanding than existing models: it agrees with end-to-end physical-optics simulations to within 0.1 per cent of PSF measurables, such as the Strehl ratio, FWHM and the fraction of variance unexplained (FVU). Secondly, we study off-axis PSF modelling with respect to the |$C_n^2(h)$| profile (heights and fractional weights). For 10-mclass telescopes, PSF morphology is estimated at the 1  per cent level as long as we model the atmosphere with at least seven layers, whose heights and weights are known with precisions of 200 m and 10 per cent, respectively. As a verification test, we used the Canada’s National Research Council – Herzberg NFIRAOS Optical Simulator (HeNOS) testbed data, featuring four lasers. We highlight the capability of retrieving off-axis PSF characteristics within 10 per cent of the FVU, which complies with the expected range from the sensitivity analysis. Our new off-axis PSF modelling method lays the groundwork for testing on-sky in the near future. atmospheric effects, instrumentation: adaptive optics, methods: analytical, methods: data analysis 1 INTRODUCTION Ultimately, the scientific exploitation of astronomical observations depends on post-processing techniques, commonly called data reduction. Currently, scientific images are commonly processed using standard packages, such as starfinder (Diolaiti et al. 2000), sextractor (Bertin & Arnouts 1996) and daophot (Stetson 1987). Nonetheless, past studies (Fritz et al. 2010; Yelda et al. 2010) have quantified an astrometric error breakdown on the Galactic Centre (GC), which showed the astrometric accuracy of the point spread function (PSF) model to be at  the 50–60  per cent level. Photometric measurements are also affected by mistaken knowledge of the PSF (Sheehy, McCrady & Graham 2006; Schödel 2010). However, in their recent analysis, Ascenso et al. (2015) have revealed that the accuracy of photometric measurements can be improved from 0.2 to 0.02 mag by providing starfinder with the exact PSF. Ground-based astronomy is improved by adaptive optics (AO), which allows us to restore the angular resolution to nearly the diffraction limit. However, AO produces a PSF that varies across space and time and does not match standard parametric models, such as Gaussian or Moffat functions. Also, it might be difficult to retrieve the AO PSF shape from focal-plane observations when observing crowded fields suffering from large source confusion (Ghez et al. 2008; Schödel 2010; Lu et al. 2013; Turri et al. 2017) or deep cosmological fields where no PSF is observed (Falomo et al. 2008; Schramm & Silverman 2013). This situation calls for an alternative approach such as PSF reconstruction (PSF-R; Véran et al. 1997; Gilles et al. 2012). PSF-R is a data-processing approach that delivers the PSF from AO control loop data, without any priors on its shape. The reliability of PSF-R has been demonstrated on-sky multiple times (Véran et al. 1997; Flicker 2008; Jolissaint, Ragland & Wizinowich 2015; Martin et al. 2016b). The PSF is reconstructed as a convolution of independent patterns that characterize different physical limitations of AO systems (Véran et al. 1997; Gilles et al. 2012). In this paper, we focus on the specific step of PSF extrapolation that relies on anisoplanatism (Fried 1982) and accounts for PSF spatial variations across the field. The AO system measures the incoming distorted wave front in the particular direction of the guide star; wave fronts that propagate along another direction and do not cross the exact same turbulence are partially compensated. This introduces an anisoplanatism error that grows with the angular separation from the guide star and the seeing of altitude layers. AO systems can also be guided using a laser guide star (LGS; Foy & Labeyrie 1985), which are focused in the range  80–100 km. When the LGS beam propagates as a spherical wave downwards to the pupil, it crosses only a portion of the turbulence above the telescope. The phase aberration in the science target location in the field is not fully corrected; it includes an additional term in the residual phase that stands as the cone effect or focal anisoplanatism. In addition, by probing the wave fronts using LGSs, we do not gain access to the wave-front angle of arrival because of the round trip made by the light from the telescope to the sodium (Na) layer. The PSF position in the focal plane is stabilized by measuring the wave-front angle of arrival using a natural guide star (NGS) whose location might be different from the scientific target. This angular separation yields an anisoplanatism effect on tip-tilt modes only, called tip-tilt anisoplanatism or anisokinetism (Winick & Marquis 1988; Fried 1996; Ellerbroek & Rigaut 2001; Flicker, Rigaut & Ellerbroek 2003; Correia et al. 2011). Reconstructing the off-axis PSF for LGS-based systems requires a model of the angular, focal and tilt anisoplanatism plus the relevant input parameter on which the model depends, which is the vertical distribution of turbulence known as the |$C_n^2(h)$| profile. Such models exist in the literature, but are either not laser-compliant (Tyler 1994; Rigaut, Véran & Lai 1998; Jolissaint 2010) or not numerically efficient (Fusco et al. 2000; Britton 2006; Flicker 2008), particularly for future AO systems on the next 40-mclass telescopes with a large number of degrees of freedom. The |$C_n^2(h)$| profile is accessible from internal methods handling the AO telemetry of multiple guide stars (Guesalaga et al. 2016; Martin et al. 2016a; Ono et al. 2017), from external profilers (Wilson 2002; Tokovinin & Kornilov 2007; Osborn et al. 2013; Osborn 2015) or, more recently, from meso-scale models (Masciadri et al. 2017). For single guide star AO systems, the |$C_n^2(h)$| profile cannot be measured using AO telemetry, and this calls for external profilers in order to model the off-axis PSF. However, Ono et al. (2017), for example, have highlighted discrepancies on the retrieved profile by comparing the outputs of external and internal profiling techniques. Consequently, using an external |$C_n^2(h)$| profile along a likely different line of sight from the observations might degrade the accuracy of the PSF modelling. Besides, the prediction of the performance of future AO-based instruments also relies on the |$C_n^2(h)$| profile; we must constrain the accuracy of the |$C_n^2(h)$| profile that is required to retrieve AO-corrected PSFs. In this paper, we propose an investigation of how the PSF is degraded regarding the accuracy of the |$C_n^2(h)$| profile. Yet, such an analysis is compromised if it relies on on-sky observations, and the real |$C_n^2(h)$| profile cannot be perfectly known; downstream results would be contaminated by anisoplanatism model errors, such as the discrete number of layers or the exact LGS height that is not perfectly identified for instance. As an alternative, we have applied this approach to the Herzberg NFIRAOS Optical Simulator (HeNOS) multi-conjugate AO (MCAO) testbed, designed to be the demonstrator for NFIRAOS (Conan et al. 2010). In this paper, we describe the theoretical background that our near-future work will be based on, which will be particularly focused on the application of the PSF characterization to observations of crowded fields, such as the GC. This paper is organized as follows. In Section 2, we derive a generalized anisoplanatism model that combines both focal and tip-tilt anisoplanatism occurring in an accurate and fast implementation. In Section 3, the model is compared to physical-optics simulations on a 10-mtelescope. We consider morphological PSF metrics, such as the Strehl ratio, FWHM and the fraction of variance unexplained (FVU), as well as science metrics (photometry and astrometry for tight binaries). In Section 4, we present how these criteria are sensitive to the accuracy of the |$C_n^2(h)$| profile, including the number of layers and the precision of the weights and heights. We validate the meaning of this approach using observations on the HeNOS test bench in Section 5. We conclude in Section 6. 2 SPATIAL PSF EXTRAPOLATION 2.1 Anisoplanatism transfer function We define φϵ(r, t) as the residual wave front delivered by the AO system. At the focal plane downstream of the AO system, the long-exposure optical transfer function (OTF) in the science direction 1 is (Roddier 1981) \begin{equation*} {\rm OTF}_{1}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) =\frac{1}{S} \iint _{\mathcal {\mathcal {P}}} \mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho }) \exp \left[-\frac{1}{2}D_{\phi _\varepsilon }(\boldsymbol{r},\boldsymbol{\rho })\right]{\rm d}\boldsymbol{r}. \end{equation*} (1) Here, |$\mathcal {P}$| is the pupil function, |$\boldsymbol{r}$| and |$\boldsymbol{\rho }$| are the location and separation vectors in the pupil, respectively, S is the area of the telescope aperture that normalizes the PSF to unit energy and |$D_{\phi _\varepsilon }(\boldsymbol{r},\boldsymbol{\rho })$| is the residual phase structure function defined by \begin{equation*} D_{\phi _\varepsilon }(\boldsymbol{r},\boldsymbol{\rho }) = \left\langle \left|\phi _\varepsilon (\boldsymbol{r},t) - \phi _\varepsilon (\boldsymbol{r}+\boldsymbol{\rho },t) \right|^2 \right\rangle _t, \end{equation*} (2) where ⟨x(t)⟩t denotes the temporal average of process x. Under the assumption that the anisoplanatism term is not correlated to other AO error terms, such as servo-lag or aliasing, |$D_{\phi _\varepsilon }(\boldsymbol{r},\boldsymbol{\rho })$| can be split as follows \begin{equation*} D_{\phi _\varepsilon }(\boldsymbol{r},\boldsymbol{\rho }) = D_{0}(\boldsymbol{r},\boldsymbol{\rho }) + D_{\Delta }(\boldsymbol{r},\boldsymbol{\rho }). \end{equation*} (3) Here, |$D_{0}(\boldsymbol{r},\boldsymbol{\rho })$| characterizes the AO residual phase structure function in the AO guide-star direction (NGS or LGS as well), while |$D_{\Delta }(\boldsymbol{r},\boldsymbol{\rho })$| is the structure function of the anisoplanatic phase |$\phi _\Delta (\boldsymbol{r},t)$| defined as \begin{equation*} \phi _\Delta (\boldsymbol{r},t) = \phi _1(\boldsymbol{r},t) - \phi _0(\boldsymbol{r},t), \end{equation*} (4) where φ1 and φ0 refer to the atmospheric phase in direction 1 (science) and direction 0 (guide star). As explicitly mentioned in equation (1), the phase structure is a function of both position and separation in the pupil. For on-axis PSF reconstruction, we handle the residual phase as a stationary process (Véran et al. 1997); we assume that |$D_{0}(\boldsymbol{r},\boldsymbol{\rho })$| is a function of separation only, which allows us to average D0 over the pupil location as \begin{equation*} \bar{D}_{{\rm 0}}(\boldsymbol{\rho }) = \frac{\iint _{\mathcal {P}} \mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho }) D_{0}(\boldsymbol{r},\boldsymbol{\rho }) {\rm d}\boldsymbol{r}}{\iint _{\mathcal {P}}\mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho }) {\rm d}\boldsymbol{r} }. \end{equation*} (5) This allows us to remove the exponential term in equation (1) from the integral, which makes for an easier and more convenient numerical implementation. For the LGS case, Flicker (2008) has pointed out that the stationarity hypothesis is no longer accurate and degrades the PSF model because of the cone effect. We have confirmed that this assumption degrades PSF metrics (introduced in Section 3.1) at the level of 5 per cent, while it is maintained at less than 1 per cent in the NGS case. Because we have the numerical ability to derive the full calculation |$D_{\Delta }(\boldsymbol{r},\boldsymbol{\rho })$|⁠, the formalism we present here does not rely on the stationarity hypothesis. We include in D0 all AO residuals in the guide-star direction and we separate any focal and angular anisoplanatism effect into DΔ, in a manner in which we can still apply the pupil-averaged process on D0. The OTF reduces to the expression \begin{eqnarray*} {\rm OTF}_{1}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) &=&\frac{1}{S} \exp \left[-\frac{1}{2}\bar{D}_{0}(\boldsymbol{\rho })\right] \\ &&\times \iint _{\mathcal {P}} \mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho })\exp \left[-\frac{1}{2}D_{\Delta }(\boldsymbol{r},\boldsymbol{\rho })\right] {\rm d}\boldsymbol{r}. \end{eqnarray*} (6) We introduce |${\rm OTF}_{{\rm DL}}(\boldsymbol{\rho }/\lambda )$| as the diffraction-limit OTF that characterizes the angular frequencies distribution imposed by the pupil shape \begin{equation*} {\rm OTF}_{{\rm DL}}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) = 1/S\iint _{\mathcal {P}} \mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho }){\rm d}\boldsymbol{r}, \end{equation*} (7) which allows us to write equation (6) as \begin{equation*} {\rm OTF}_{1}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) = {\rm OTF}_{0}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) {\rm ATF}_{\Delta }\left(\frac{\boldsymbol{\rho }}{\lambda}\right), \end{equation*} (8) where Note that \begin{equation*} {\rm OTF}_{0}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) = {\rm OTF}_{{\rm DL}}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) \exp \left[-\frac{1}{2}\bar{D}_{0}(\boldsymbol{\rho })\right] . \end{equation*} (9) \begin{equation*} {\rm ATF}_{\Delta }\left(\frac{\boldsymbol{\rho }}{\lambda}\right) = \frac{\iint _{\mathcal {P}} \mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho })\exp [-(1/2) D_{\Delta }(\boldsymbol{r},\boldsymbol{\rho })] \,{\rm d}\boldsymbol{r} }{{\rm OTF}_{{\rm DL}}(\boldsymbol{\rho }/\lambda )} \end{equation*} (10) is the anisoplanatism transfer function (ATF) as introduced by Fusco et al. (2000), which is a convenient formulation to derive the residual OTF anywhere in the field with knowledge of the on-axis OTF. A practical result to extrapolate the PSF anywhere in the field consists of computing OTF2. From equations (8) and (10), we obtain \begin{equation*} {\rm OTF}_{2}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) = {\rm OTF}_{1}\left(\frac{\boldsymbol{\rho }}{\lambda}\right) \frac{{\rm OTF}_{\Delta _2}(\boldsymbol{\rho }/\lambda )}{{\rm OTF}_{\Delta _1}(\boldsymbol{\rho }/\lambda )}, \end{equation*} (11) which introduces an OTF ratio that is the key quantity for deriving the OTF from one direction to any other. The PSF is seamlessly computed from the Fourier transform of the OTF. In summary, in order to have the capability of modelling a PSF anywhere in a field, it is necessary first to have access to a PSF in direction 1 using the source extraction method, parametric modelling or PSF-R, and second to be capable of computing an OTF ratio accurately. The latter is the main focus of following sections. 2.2 Modelling anisoplanatism We now focus on the implementation of the phase structure function |$D_{\Delta }(\boldsymbol{r},\boldsymbol{\rho })$| for computing the ATF in equation (10). The results in the remainder of this paper rely on the calculation of the anisoplanatic covariance |$\mathcal {C}_\Delta (\boldsymbol{r},\boldsymbol{\rho })$|⁠, which is written as \begin{equation*} \mathcal {C}_\Delta (\boldsymbol{r},\boldsymbol{\rho }) = \left\langle \phi _\Delta (\boldsymbol{r},t)\phi ^t_\Delta (\boldsymbol{r}+\boldsymbol{\rho },t) \right\rangle \end{equation*} (12) from which |$D_\Delta (\boldsymbol{r},\boldsymbol{\rho })$| is \begin{equation*} D_\Delta (\boldsymbol{r},\boldsymbol{\rho }) = 2\left[\mathcal {C}_\Delta ({\boldsymbol{r}},0) - \mathcal {C}_\Delta (\boldsymbol{r},\boldsymbol{\rho })\right]. \end{equation*} (13) The problem of modelling anisoplanatism is now a matter of describing how the atmospheric phase is spatially correlated. In general terms, phase is described as a linear combination of nm (orthonormal) |$\mathcal {M}$| modes, yielding \begin{equation*} \begin{aligned} &\phi _1(\boldsymbol{r},t) = \sum _{i=1}^{n_m} a_i(t)\boldsymbol{\times} \mathcal {M}_i(\boldsymbol{r})\\ &\phi _0(\boldsymbol{r},t) = \sum _{i=1}^{n_m} b_i(t)\boldsymbol{\times} \mathcal {M}_i(\boldsymbol{r}), \end{aligned} \end{equation*} (14) where |$\boldsymbol{a}(t)$| and |$\boldsymbol{b}(t)$| are modal coefficients in the science and guide-star directions, respectively. The anisoplanatic phase |$\phi _\Delta (\boldsymbol{r},t)$| becomes \begin{equation*} \phi _\Delta (\boldsymbol{r},t) = \sum _{i=1}^{n_m} \Delta _i(t)\boldsymbol{\times} \mathcal {M}_i(\boldsymbol{r}), \end{equation*} (15) with Δi(t) = [ai(t) − bi(t)]. Combining equations (12) and (15) gives the following expression \begin{equation*} \begin{aligned} \mathcal {C}_{\Delta }(\boldsymbol{r},\boldsymbol{\rho }) = \sum _{i=1,j}^{n_m} \left\langle \Delta _i\Delta _j^t \right\rangle \boldsymbol{\times} \mathcal {M}_i(\boldsymbol{r})\mathcal {M}^t_j(\boldsymbol{r}+ \boldsymbol{\rho }), \end{aligned} \end{equation*} (16) where \begin{equation*} \Delta _i\Delta _j^t = {a_i(t)a^t_j(t)} + {b_i(t)b^t_j(t)} - {a_i(t)b^t_j(t)} - b_j(t)a^t_i(t). \end{equation*} (17) Equation (16) recalls the Uij formalism introduced by Véran et al. (1997) for the computation of the residual phase structure function; it simplifies to the multiplication |$\mathcal {M}_i(\boldsymbol{r})\mathcal {M}_j(\boldsymbol{r}+ \boldsymbol{\rho })$| when deriving the covariance. In addition, anisoplanatism modelling requires the calculation of the correlation of modal coefficients |$\left\langle \Delta _i\Delta _j^t \right\rangle$|⁠. Fusco et al. (2000) has derived the ATF using a Zernike expansion of the phase, based on the spatial correlation of Zernike coefficients provided in Chassat (1989). Although this study has delivered successful results on an 8-m telescope, its application on a 40-mclass telescope is computationally demanding because of the |$\mathcal {M}_i(\boldsymbol{r})\mathcal {M}_j(\boldsymbol{r}+ \boldsymbol{\rho })$| derivations. An alternative computation approach was proposed by Gendron et al. (2006) to tackle the numerical complexity of the Uij technique. However, although focal anisoplanatism could be covered by using the covariance terms introduced by Molodij & Rousset (1997), it would not be as accurate as the NGS case because of the intrinsic stationarity assumption of the Zernike expansion. To decrease the numerical complexity whilst maintaining model accuracy, we can resort to a spatial frequency basis (Rigaut et al. 1998; Jolissaint 2010). Although such a technique is efficient from the numerical computation point of view, statistical independence of Fourier modes assumes underlying stationarity; the long-exposure OTF is derived from the AO residual phase power spectrum density (PSD), although such a description is only accurate for linear and space-invariant systems. In order to generalize it to the LGS case, correlations between the residual phase errors at different frequencies must be included (van Dam et al. 2006; Flicker 2008) due to the cone stretching factor. Besides, Fourier expansion does not comply with an accurate tip-tilt filtering (Sasiela 1994) and we assume an infinite pupil that degrades the PSF model. Such considerations call for alternative approaches such as the point-wise method that is our formulation baseline describedbelow. 2.3 Point-wise calculation To maintain the accuracy of the anisoplanatism model whilst reducing the computational complexity, we focus on a point-wise approach. We follow the technique proposed by Gilles et al. (2012). The phase is discretized over a grid of N × Npixels in the real domain; that is, |$\mathcal {M}_i(\boldsymbol{r})$| functions are turned into Dirac distributions |$\delta _i(\boldsymbol{r}) = \delta (\boldsymbol{r}- \boldsymbol{r}_i)$| that are centred at the ith pixel location. The anisoplanatic covariance takes the following form \begin{eqnarray*} \mathcal {C}_{\Delta }(\boldsymbol{r},\boldsymbol{\rho }) &=& \sum _{i,j=1}^{N}\delta (\boldsymbol{r}-\boldsymbol{r}_i)\delta (\boldsymbol{r}+ \boldsymbol{\rho }- \boldsymbol{r}_i - \boldsymbol{\rho }_j)\nonumber\\ &&\times \,\, \left\langle \phi _\Delta (\boldsymbol{r},t)\phi ^t_\Delta (\boldsymbol{r}+\boldsymbol{\rho },t) \right\rangle , \end{eqnarray*} (18) where |$\delta (\boldsymbol{r}-\boldsymbol{r}_i)\delta (\boldsymbol{r}+ \boldsymbol{\rho }- \boldsymbol{r}_i - \boldsymbol{\rho }_j)$| is 1 only for the couple of pixels located at ri and separated by |$\boldsymbol{\rho }_j$|⁠. Computation of |$\mathcal {C}_{\Delta }(\boldsymbol{r},\boldsymbol{\rho })$| is reduced to the determination of phase covariance at specific separations in the bi-dimensional plane. To include the dependence on both pupil location and separation, we compute the covariance of any two samples leading consequently to N4 values. All of these are concatenated into an N2 × N2 matrix, which defines |$\mathcal {C}_{\Delta }$|⁠. For a given separation, the latter is well described for the Von-Kármán spectrum of turbulence as \begin{eqnarray*} C_\phi (\rho ) &= & \left(\frac{L_{0}}{r_{0}}\right)^{5/3} \frac{\Gamma (11/6)}{2^{5/6} \pi ^{8/3}} \left[\frac{24}{5}\Gamma \left(\frac{6}{5}\right)\right]\nonumber\\ && \times \left(\frac{2\pi \rho }{L_{0}}\right)^{5/6} K_{5/6}\left(\frac{2 \pi \rho }{L_{0}}\right) \end{eqnarray*} (19) where L0 and r0 are the outer scale and Fried’s parameter, respectively, Γ is the ‘gamma’ function and K5/6 is a modified Bessel function of the third order. Derivation of |$\mathcal {C}_{\Delta }$| relies on the numerical implementation of equation (19) that is fed with a vector of separations to estimate the covariance terms in equation (18). The main challenge in the anisoplanatic covariance calculation lies in the proper definition of separations. Consider the phase covariance along two different directions θ1 and θ2 at a single layer located at altitude hl. Define z1 and z2 as the source heights in directions 1 and 2, respectively. Finally, let |$\boldsymbol{r}_1$| and |$\boldsymbol{r}_2$| be the global pixel location vectors in the pupil projected from the atmosphere along straight paths. Fig. 1 provides a schematic view of this sketch. Figure 1. Open in new tabDownload slide Schematic diagram representing how the separation vector |$\boldsymbol{\rho }_l$| is defined between two resolution elements at a given turbulent layer l. Vectors |$\boldsymbol{r}_1$| and |$\boldsymbol{r}_2$| are the coordinates back-projected on the pupil. Figure 1. Open in new tabDownload slide Schematic diagram representing how the separation vector |$\boldsymbol{\rho }_l$| is defined between two resolution elements at a given turbulent layer l. Vectors |$\boldsymbol{r}_1$| and |$\boldsymbol{r}_2$| are the coordinates back-projected on the pupil. From Fig. 1, we note that the separation of the two phase samples has two components. The first component is related to the pupil plane coordinates that are stretched down with respect to cone-related squeeze factor 1 −hl/z1 and 1 − hl/z2. There is also a pupil shift in altitude created by the angular separation that leads to hl(θ1 − θ2). The separation vector ρl(i, j), between aperture points raster index i in direction |${\boldsymbol{\theta }}_1$| and index j in direction |${\boldsymbol{\theta }}_2$| at altitude hl is \begin{equation*} \rho _{l}(i,j) = r_1(i) \left(1 - \frac{h_l}{z_1}\right) - r_2(j)\left(1 - \frac{h_l}{z_2}\right) + h_l(\theta _1 - \theta _2). \end{equation*} (20) We assume the turbulence is discretized along nl statistically independent layers. From equation (20), the element (i, j) of |$\mathcal {C}_{\Delta }$| takes the following expression \begin{eqnarray*}\mathcal {C}_{\Delta }(i,j) &=& \sum _{l=1}^{n_l} f_l\left\{\mathcal {C}_{\phi }\left[\left(1 - \frac{h_l}{z_1}\right)r_1(i) - \left(1 - \frac{h_l}{z_2}\right)r_2(j)+ h_l\theta _1\right]\right.\nonumber\\ &&+\mathcal {C}_{\phi }\left[\left(1 - \frac{h_l}{z_1}\right)r_1(i) - \left(1 - \frac{h_l}{z_2}\right)r_2(j) + h_l\theta _2\right]\nonumber\\ && - \mathcal {C}_{\phi }\left[\left(1 - \frac{h_l}{z_1}\right)r_1(i) - \left(1 - \frac{h_l}{z_2}\right)r_2(j) + h_l\Delta \theta \right]\nonumber\\ && \left.- \mathcal {C}_{\phi }\left[\left(1 - \frac{h_l}{z_1}\right)r_1(j) - \left(1 - \frac{h_l}{z_2}\right)r_2(i) + h_l\Delta \theta \right]\right\},\nonumber\\ \end{eqnarray*} (21) where Δθ = θ1 − θ2. Here, fl is the fractional power of the lth layer that is defined from the |$C_n^2(h_l)$| value as \begin{equation*} f_l = 0.06\lambda ^2r_0^{-5/3}/C_n^2(h_l), \end{equation*} (22) where |$C_n^2(h_l)$| is the |$C_n^2(h)$| profile at altitude hl. This implementation of the ATF is made freely available with the simulator OOMAO (Conan & Correia 2014). Other point-wise approaches have been developed (Tyler 1994; Britton 2006; Flicker 2008), but only the approach of Flicker (2008) is laser-compliant. For the purpose of validation, we have also coded Flicker’s method as an alternative point-wise calculation. Our formulation of anisoplanatism is strictly equivalent to that of Flicker from a mathematical point of view; both rely on a non-stationarity calculation and include focal anisoplanatism. However, we propose a direct derivation of the covariance function (instead of the structure function as done by Flicker) that relies on the implementation of optimal routines within the OOMAO framework, which are compliant with 40-m class telescopes in terms of memory and cpu usage. 2.4 Generalization to laser-based systems In this section, we provide the full expression of anisoplanatic covariance for laser-based systems, which includes angular, focal and tilt terms. We define φs, φl and φn as the atmospheric phase in the science, LGS and NGS directions, respectively. The anisoplanatic phase |$\phi _\Delta$| results from the subtraction of the high-order modes measured from the LGS and the tip-tilt measured on the NGS from the phase in the science direction as \begin{equation*} \phi _\Delta = \phi _{\rm s}- \boldsymbol{{P}}_{\rm TTR}\boldsymbol{\cdot}\phi _{\rm l}- \boldsymbol{{P}}_{\rm TT}\boldsymbol{\cdot}\phi _{\rm n},\end{equation*} (23) where |$\boldsymbol{{\rm P}}_{\rm TTR}(\boldsymbol{r})$| is a spatial filter that removes the angle of arrival from LGS measurements. Conversely, |$\boldsymbol{{\rm P}}_{\rm TT}(\boldsymbol{r})$| is the filter that conserves only this angle of arrival. Modal spatial filters are of the kind \begin{equation*} \boldsymbol{{P}}_{\rm TTR}= \mathcal {I}_d - \boldsymbol{{P}}_{\rm TT} ,\end{equation*} (24) where |$\boldsymbol{{\rm P}}_{\rm TT}= \boldsymbol{T}\boldsymbol{T}^\dagger$| and Tis the vector projecting the phase on to tip or tilt modes. Based on the properties of |$\boldsymbol{{\rm P}}_{\rm TT}$| and |$\boldsymbol{{\rm P}}_{\rm TTR}$|⁠, |$\phi _\Delta$| conforms to \begin{eqnarray*} \phi _\Delta &=& \boldsymbol{{P}}_{\rm TTR}\left(\phi _{\rm s}- \phi _{\rm l}\right) + \boldsymbol{{P}}_{\rm TT}\left(\phi _{\rm s}-\phi _{\rm n}\right)\nonumber\\ & =& \boldsymbol{{P}}_{\rm TTR}\phi _{\Delta _{\rm sl}} + \boldsymbol{{P}}_{\rm TT}\phi _{\Delta _{\rm sn}}, \end{eqnarray*} (25) where the first term |$\boldsymbol{{\rm P}}_{\rm TTR}\phi _{\Delta _{\rm sl}}$| relates to the anisoplanatism, both angular and foca, and the second term |$\boldsymbol{{\rm P}}_{\rm TT}\phi _{\Delta _{\rm sn}}$| represents tilt anisoplanatism. From equation (25), the anisoplanatic covariance |$\mathcal {C}_{\Delta }$| in equation (12) becomes \begin{eqnarray*} C_{\Delta }(\boldsymbol{r},\boldsymbol{\rho }) &=& \boldsymbol{{P}}_{\rm TTR}\left\langle \phi _{\Delta _{\rm sl}}(\boldsymbol{r})\phi^{t}_{\Delta_{\rm sl}}(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle \boldsymbol{{P}}_{\rm TTR}^{t}\nonumber\\ && + \boldsymbol{{P}}_{\rm TT}\left\langle \phi_{\Delta_{\rm sn}}(\boldsymbol{r})\phi^{t}_{\Delta_{\rm sn}}(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle \boldsymbol{{P}}_{\rm TT}^{t}\nonumber\\ && + \boldsymbol{{P}}_{\rm TT}\left\langle \phi_{\Delta _{\rm sn}}(\boldsymbol{r})\phi^{t}_{\Delta_{\rm sl}}(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle \boldsymbol{{P}}_{\rm TTR}^{t}\nonumber\\ && + \boldsymbol{{P}}_{\rm TTR}\left\langle \phi_{\Delta _{\rm sl}}(\boldsymbol{r})\phi^{t}_{\Delta_{\rm sn}}(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle \boldsymbol{{P}}_{\rm TT}, \end{eqnarray*} (26) which turns into \begin{eqnarray*} C_{\Delta }(\boldsymbol{r},\boldsymbol{\rho }) & =& \boldsymbol{{P}}_{\rm TTR}\mathcal {C}_{\Delta_{\rm sl}}(\boldsymbol{r},\boldsymbol{\rho })\boldsymbol{{P}}_{\rm TTR}^{t}+ \boldsymbol{{P}}_{\rm TT}\mathcal{C}_{\Delta_{\rm sn}}(\boldsymbol{r},\boldsymbol{\rho })\boldsymbol{{P}}_{\rm TT}^{t} \nonumber\\ && + \mathcal {C}_{{\rm cross}}(\boldsymbol{r},\boldsymbol{\rho }) + \mathcal{C}_{{\rm cross}}^{t}(\boldsymbol{r},\boldsymbol{\rho }), \end{eqnarray*} (27) where covariance terms |$\mathcal {C}_{\Delta _{\rm sl}}$| and |$\mathcal {C}_{\Delta _{\rm sn}}$| are focal-angular and tip-tilt anisoplanatism, respectively. These are derived independently using the formalism introduced in the previous section. Equation (27) introduces cross-correlation on high-order and tip-tilt modes. From equations (26) and (25), this cross-term is writtenas \begin{eqnarray*} \mathcal {C}_{{\rm cross}}(\boldsymbol{r},\boldsymbol{\rho })&=&\boldsymbol{{P}}_{\rm TT}\left\langle [\phi _{\rm s}(\boldsymbol{r}) - \phi _{\rm n}(\boldsymbol{r})][\phi _{\rm s}(\boldsymbol{r}+\boldsymbol{\rho })-\phi _{\rm l}(\boldsymbol{r}+\boldsymbol{\rho })]^t \right\rangle \boldsymbol{{P}}_{\rm TTR}^t\nonumber\\ & =& \boldsymbol{{P}}_{\rm TT}\big[\left\langle \phi _{\rm s}(\boldsymbol{r})\phi _{\rm s}(\boldsymbol{r}+\boldsymbol{\rho })^t \right\rangle + \left\langle \phi _{\rm n}(\boldsymbol{r})\phi _{\rm l}^t(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle \nonumber\\ && - \left\langle \phi _{\rm s}(\boldsymbol{r})\phi _{\rm l}^t(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle + \left\langle \phi _{\rm n}(\boldsymbol{r})\phi _{\rm s}^t(\boldsymbol{r}+\boldsymbol{\rho }) \right\rangle \big] \boldsymbol{{P}}_{\rm TTR}^t,\nonumber \\ \end{eqnarray*} (28) and includes the tip-tilt/high-order mode correlation. These cross-terms are generally neglected in the literature because of their small amplitude when compared with other terms; we propose to confirm this assumption using full physical-optics simulations in Section 3.3. We now introduce the terminology `total anisoplanatism' when including all these cross-terms as in equation (27), compared to the term `split anisoplanatism' that considers |$\mathcal {C}_{{\rm cross}} = 0$| in equation (27). 2.5 Spatial filtering The number of phase samples N must be chosen wisely; large values will make the calculation time-consuming, while small values reduce accuracy. In terms of spatial frequencies, N phase samples allows us to derive the PSF within an N/2 × λ/D-wide two-dimensional (2D) region. Besides, the AO correction band breaks at nact/2 × λ/D, with nact being the linear number of deformable mirror (DM) actuators, where the anisoplanatism occurs actually. As a consequence, it is enough to define |$\mathcal {C}_{\Delta }$| as an nact × nact matrix to represent accurately the anisoplanatism in the AO correction band. However, numerical simulations make it necessary to sample the phase with a resolution given by at least two to three points per r0, which potentially gives N > nact. We solve this issue by multiplying |$\mathcal {C}_{\Delta }$| with a filter matrix |$\mathcal {P}_{\rm DMR}$| that is designed to filter out uncontrolled DM modes. It sets to zero all spatial frequencies greater than nact/2D: \begin{eqnarray*} {\rm ATF}_{\Delta }\left(\frac{\boldsymbol{\rho }}{\lambda}\right) &=& \frac{1}{{\rm OTF}_{{\rm DL}}(\boldsymbol{\rho }/\lambda )}\iint _{\mathcal {P}} \mathcal {P}(\boldsymbol{r})\mathcal {P}(\boldsymbol{r}+\boldsymbol{\rho })\nonumber\\ &&\times \exp \left\{\mathcal {P}_{\rm DMR}\left[\mathcal {C}_{\Delta }(\boldsymbol{r},\boldsymbol{\rho }) - \mathcal {C}_{\Delta }(\boldsymbol{r},0)\right]\mathcal {P}_{\rm DMR}^t\right\} \,{\rm d}\boldsymbol{r} . \end{eqnarray*} (29) Here, |$\mathcal {P}_{\rm DMR}$| is a zonal DM filter of size N × N, calculated from the DM influence function |$\boldsymbol{h}$| of size N2 × nact as \begin{equation*} \mathcal {P}_{\rm DMR}= \mathcal {I}_d - \boldsymbol{h}\boldsymbol{h}^\dagger, \end{equation*} (30) where |$\mathcal {I}_d$| is the N × N identity matrix. This filter removes spatial frequencies above the DM cut-off frequency that are beyond the correction space spanned by the latter. The removed high spatial frequencies contribute to the fitting error, which manifests itself as the PSF halo and is a function of the atmospheric seeing only. 3 ANALYTICAL MODELS VERSUS PHYSICAL-OPTICS SIMULATIONS In this section, we compare anisoplanatism models that already exist in the literature with simulations. In a first step, we aim to check whether the point-wise derivation of anisoplanatism using our general formalism, referred to as OOMAO in the following results, complies with existing models, such as as Zernike (Fusco et al. 2000), Fourier (van Dam et al. 2006) and the model of Flicker (2008). 3.1 Metrics The results presented in the following sections evaluate the PSF model's deviations from simulations. First insights on the PSF are given by morphological scalar values, such as the Strehl ratio and the FWHM. Both parameters mostly refer to AO performance; we might prefer to have a metric that encompasses the entire structure of the PSF, including especially the PSF halo outside the AO correction band, as the FVU. If X is a 2D image and |$\widehat{X}$| is the estimation of this, the FVU is defined as (King 1983) \begin{equation*} {\rm FVU}_X = \frac{\sum _{i,j} [X(i,j) - \widehat{X}(i,j)]^2}{\sum _{i,j} [X(i,j) - \sum _{i,j}X(i,j)]^2}, \end{equation*} (31) where (i, j) are the ith and jth pixels of the image, respectively. The great advantage of such a metric is to yield an overall error on all angular frequencies; all the PSF patterns, such as the PSF core and wings, are included in this calculation. Besides PSF-related metrics, the characterization of off-axis PSFs is motivated by science exploitation, calling for science-based metrics, such as photometry and astrometry. Contrary to PSF-related metrics, science-based metrics must refer to a specific observed object and image processing tools (deconvolution or model-fitting, for instance). We focus on a particular science case of imaging a binary system to measure relative fluxes and astrometry in order to derive the binary’s orbit. In this case, the field typically lacks independent PSF stars and the ability to characterize the PSF is of tremendous value. We define a binary model from a reference off-axis PSF and a set of parameters as \begin{eqnarray*} \mathcal {B}({\rm PSF},\Delta F,\Delta \alpha _x,\Delta \alpha _y)&=&\Delta F [{\rm PSF}(\alpha _x,\alpha _y) \nonumber\\ &&+ {\rm PSF}(\alpha _x + \Delta \alpha _x,\alpha _y+\Delta \alpha _y)], \end{eqnarray*} (32) where (αx, αy) represents the angular separations in the focal plane, ΔF is the flux of the relative stars and (Δαy, Δαy) are the differential angular offsets. We did not include any source of noise and AO residual in the focal plane to determine the real impact of anisoplanatism characterization on our metrics. We define a reference binary model from the simulated off-axis PSF, sampled at λ/4D, with ΔF0 = 1 and a star separation set to |$\Delta \alpha _y^0 = \lambda /D$|⁠. Accuracy on photometry and astrometry is evaluated by minimizing the following criterion \begin{eqnarray*} \varepsilon ^2(\Delta F,\Delta \alpha _x,\Delta \alpha _y) &=&||\mathcal {B}({\rm PSF}_\varepsilon ,\Delta F^0,\Delta \alpha _x^0,\Delta \alpha _y^0) \nonumber\\ &&- \mathcal {B}({\rm P}\widehat{{\rm S}}{\rm F}_\varepsilon ,\Delta F,\Delta \alpha _x,\Delta \alpha _y) ||^2_2,\nonumber\\ \end{eqnarray*} (33) where |$\left|\left|\boldsymbol{x} \right|\right|_2^2$| is the |$\mathcal {L}_2$| norm of the vector |$\boldsymbol{x}$|⁠. We retrieve photometry and astrometry by fitting a synthetic PSF-based binary to the reference model. Because we know the exact binary parameters, we can estimate how we deviate from those regarding the PSF model. Particularly, the photometry error is given by \begin{equation*} \Delta {\rm mag} = -2.5\times \log _{\rm 10}\left(\frac{\widehat{\Delta }F}{\Delta F^0}\right) \end{equation*} (34) while the astrometry error results from \begin{equation*} \Delta \alpha = \sqrt{(\widehat{\Delta }\alpha _x - \Delta \alpha _x^0)^2 + (\widehat{\Delta }\alpha _y -\Delta \alpha _x^0)^2}. \end{equation*} (35) Photometry and astrometry, as defined in last equations, can be derived differently regarding the science case and the image-processing method. However, gathering science-based and PSF-related metrics in our analysis will enhance the overall evaluation of anisoplanatism characterization; we will point out the consequence of PSF errors on the exploitation of science images, which is the real information that matters in the end. Finally, we estimate astrometry accuracy with an infinite signal-to-noise ratio; we consider only the effect of the PSF morphology on this derivation. 3.2 NGS case We have simulated H-band atmospheric phase screens using the simulator OOMAO (Conan & Correia 2014) to compare all anisoplanatism models described in the previous section with simulations. These latter include only anisoplanatism patterns into the PSF; we do not account for AO residuals in the guide-star direction, static patterns and science camera parasites such as noise. Our goal is strictly dedicated to anisoplanatism determination and impact on the PSF. We have considered the median |$C_n^2(h)$| profile (Sarazin et al. 2013) at the Paranal Observatory degraded to seven layers with r0= 16 cm. See Section 4 for an explanation of the choice of a profile based on seven layers. Fig. 2 provides a qualitative comparison and illustrates how each approach achieves an accurate model of the ATF within 1 per cent of the full physical-optics simulation. We notice that the Zernike approach becomes less accurate when putting the NGS farther away in the field. We believe that the stationarity assumption causes this effect, as invoked by Fusco et al. (2000), but in such a case, we would observe a similar degradation compared to the Fourier approach that also relies on the same assumption. The main issue here is that the DM spatial filtering is translated into a modal truncation at the radial order n given by 0.3 × (n + 1)/D (Conan 1994). Zernike modes do not match exactly the DM frequency behaviour; such an approximation in the DM cut-off frequency can introduce slight errors on the ATF, especially when the anisoplanatism level is stronger. The Fourier method is the most accurate for the NGS case in the FVU sense, despite the approximation made for an infinite pupil translating into potential edgeeffects. Figure 2. Open in new tabDownload slide Relative errors on the ATF azimuthal average deduced from the difference between simulations and analytical calculations, for an NGS offset by 40 arcmin. Negative values on residuals correspond to the overestimation of analytical calculations. Figure 2. Open in new tabDownload slide Relative errors on the ATF azimuthal average deduced from the difference between simulations and analytical calculations, for an NGS offset by 40 arcmin. Negative values on residuals correspond to the overestimation of analytical calculations. Table 1 provides a quantitative assessment of the ATF compared with simulations and it highlights that each method leads to very similar results. What we see is consistent with Fig. 2: all models match end-to-end simulations very well, within 0.2 per cent of the FVU on the OTF. We also verified that the FVU on the PSF is systematically within one or two orders of magnitude lower than the FVU on the ATF. This is explained by the PSF derivation that results from the Fourier transform of the ATF, which is preliminarily multiplied by the diffraction-limit OTF, given by equation (7), that filtered out high-angular frequencies. The Strehl ratio is estimated within a few per cent, which is definitely below the errors bars obtained on images acquired on-sky, while the FWHM is perfectly well estimated whatever the approach. Science metrics reach a level of mmag (0.1  per cent on photometry) and mas (1  per cent of pixel size), which is definitely several orders of magnitude lower than usual estimations (Ascenso et al. 2015; Turri et al. 2017). Table 1. Relative errors obtained on the considered PSF metrics by comparing simulated and modelled anisoplanatic PSFs. Models referred to are OOMAO (O), Flicker (F), Zernike (Ze) and Fourier (Fo). Photometric errors are given in H-band mmag and astrometry in μ as. Zero values correspond to machine precision. The Strehl ratio and FWHM given in the table header are values extracted from the simulation. NGS location (arcmin) 10 20 40 Strehl ratio (per cent) 54 19 5 FWHM (mas) 80 90 260 Model O F Ze Fo O F Ze Fo O F Ze Fo Relative residual errors (per cent) Strehl ratio 3.8 4.0 1.1 2.3 3.4 3.9 1.7 1.6 2.7 3.3 3.5 0.6 FWHM 0 0 0.1 0.1 0.04 0.04 0.2 0.6 0.4 0.06 1.0 1.5 FVUOTF 0.21 0.23 0.02 0.08 0.15 0.18 0.04 0.05 0.08 0.1 0.08 0.03 Science estimates Δmag 25 29 2 13 22 28 4 6 15 20 20 5 Δα 2 9 37 28 13 49 56 50 50 10 14 70 NGS location (arcmin) 10 20 40 Strehl ratio (per cent) 54 19 5 FWHM (mas) 80 90 260 Model O F Ze Fo O F Ze Fo O F Ze Fo Relative residual errors (per cent) Strehl ratio 3.8 4.0 1.1 2.3 3.4 3.9 1.7 1.6 2.7 3.3 3.5 0.6 FWHM 0 0 0.1 0.1 0.04 0.04 0.2 0.6 0.4 0.06 1.0 1.5 FVUOTF 0.21 0.23 0.02 0.08 0.15 0.18 0.04 0.05 0.08 0.1 0.08 0.03 Science estimates Δmag 25 29 2 13 22 28 4 6 15 20 20 5 Δα 2 9 37 28 13 49 56 50 50 10 14 70 Open in new tab Table 1. Relative errors obtained on the considered PSF metrics by comparing simulated and modelled anisoplanatic PSFs. Models referred to are OOMAO (O), Flicker (F), Zernike (Ze) and Fourier (Fo). Photometric errors are given in H-band mmag and astrometry in μ as. Zero values correspond to machine precision. The Strehl ratio and FWHM given in the table header are values extracted from the simulation. NGS location (arcmin) 10 20 40 Strehl ratio (per cent) 54 19 5 FWHM (mas) 80 90 260 Model O F Ze Fo O F Ze Fo O F Ze Fo Relative residual errors (per cent) Strehl ratio 3.8 4.0 1.1 2.3 3.4 3.9 1.7 1.6 2.7 3.3 3.5 0.6 FWHM 0 0 0.1 0.1 0.04 0.04 0.2 0.6 0.4 0.06 1.0 1.5 FVUOTF 0.21 0.23 0.02 0.08 0.15 0.18 0.04 0.05 0.08 0.1 0.08 0.03 Science estimates Δmag 25 29 2 13 22 28 4 6 15 20 20 5 Δα 2 9 37 28 13 49 56 50 50 10 14 70 NGS location (arcmin) 10 20 40 Strehl ratio (per cent) 54 19 5 FWHM (mas) 80 90 260 Model O F Ze Fo O F Ze Fo O F Ze Fo Relative residual errors (per cent) Strehl ratio 3.8 4.0 1.1 2.3 3.4 3.9 1.7 1.6 2.7 3.3 3.5 0.6 FWHM 0 0 0.1 0.1 0.04 0.04 0.2 0.6 0.4 0.06 1.0 1.5 FVUOTF 0.21 0.23 0.02 0.08 0.15 0.18 0.04 0.05 0.08 0.1 0.08 0.03 Science estimates Δmag 25 29 2 13 22 28 4 6 15 20 20 5 Δα 2 9 37 28 13 49 56 50 50 10 14 70 Open in new tab In conclusion, our point-wise method matches accurately the existing angular anisoplanatism models and simulations in the literature within differences at the 1 per cent level. 3.3 LGS case We now focus on anisoplanatism in LGS-based systems; we have simulated the total anisoplanatism from equation (23), with one LGS and one NGS distributed along an L-shaped asterism while keeping the science on-axis. We have estimated the anisoplanatic covariance given in equation (26) and the resulting ATF using equation (10). Furthermore, we have also derived independently the focal and tilt anisoplanatic covariance matrices |$\mathcal {C}_{\Delta _{sl}}$| and |$\mathcal {C}_{\Delta _{sn}}$| in equation (27), using both simulations and analytical calculations. The goal of the analysis is two-fold: first, to demonstrate that the generalized analytical model matches simulations including all anisoplanatism terms; secondly, to confirm that the high-order/tip-tilt cross-terms appearing in equation (26) are not determinants in the anisoplanatism characterization. Fig. 3 provides a comparison of ATF maps for an LGS offset by 20 arcmin and an NGS offset by 40 arcmin from on-axis in perpendicular directions. It highlights clearly that analytical calculations fit the simulation results very well, to within an accuracy of two percentage points. Also, residual errors are mostly located at the map border that corresponds to pupil edges. As previously, the FVU on the PSF reaches a level that is two orders of magnitude lower than the FVU on the OTF. When extrapolating the on-axis OTF, we multiply these angular frequencies above D/λ to zero. Fig. 4 illustrates cuts of the ATF and residuals along the elongated direction. From Table 2, we can see that analytical calculations reproduce the simulations very well, within 2 per cent in maximal range and 0.1 per cent in the FVU. Figure 3. Open in new tabDownload slide Comparison of ATF maps derived from either simulations or analytical calculations. The figure distinguishes total and split anisoplanatisms, which respectively do and do not include tip-tilt/high-order mode cross-correlation, as discussed in Section 2.4. ATF maps are derived for an LGS and NGS offset by 40 arcmin in perpendicular directions. Figure 3. Open in new tabDownload slide Comparison of ATF maps derived from either simulations or analytical calculations. The figure distinguishes total and split anisoplanatisms, which respectively do and do not include tip-tilt/high-order mode cross-correlation, as discussed in Section 2.4. ATF maps are derived for an LGS and NGS offset by 40 arcmin in perpendicular directions. Figure 4. Open in new tabDownload slide Relative errors on ATF azimuthal average deduced from the differences between simulations and analytical calculations for an LGS at 20 arcmin and an NGS at 40 arcmin separations in perpendicular directions. Figure 4. Open in new tabDownload slide Relative errors on ATF azimuthal average deduced from the differences between simulations and analytical calculations for an LGS at 20 arcmin and an NGS at 40 arcmin separations in perpendicular directions. Table 2. Relative errors obtained on the considered PSF metrics by comparing simulated and analytical PSFs while considering split anisoplanatism. Sources were distributed along an L-shaped constellation with the science target on-axis. Photometric errors are given in H-band mmag and astrometry in μas. Zero values correspond to machine precision. The Strehl ratio and FWHM given in the table header are values extracted from the simulation. LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Strehl ratio (per cent) 64 56 41 22 19 15 FWHM (mas) 79 84 97 86 92 108 Relative residual errors  (per cent) Strehl ratio 2.2 2.0 2.0 2.5 2.03 2.2 FWHM 0 0.31 0.35 0 0.23 1.3 FVUOTF 0.07 0.06 0.06 0.10 0.08 0.08 Science estimates Δmag 6.3 3.1 4.8 8.6 1.8 6.0 Δα 7.7 41.4 28.2 67.6 60.5 77.6 LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Strehl ratio (per cent) 64 56 41 22 19 15 FWHM (mas) 79 84 97 86 92 108 Relative residual errors  (per cent) Strehl ratio 2.2 2.0 2.0 2.5 2.03 2.2 FWHM 0 0.31 0.35 0 0.23 1.3 FVUOTF 0.07 0.06 0.06 0.10 0.08 0.08 Science estimates Δmag 6.3 3.1 4.8 8.6 1.8 6.0 Δα 7.7 41.4 28.2 67.6 60.5 77.6 Open in new tab Table 2. Relative errors obtained on the considered PSF metrics by comparing simulated and analytical PSFs while considering split anisoplanatism. Sources were distributed along an L-shaped constellation with the science target on-axis. Photometric errors are given in H-band mmag and astrometry in μas. Zero values correspond to machine precision. The Strehl ratio and FWHM given in the table header are values extracted from the simulation. LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Strehl ratio (per cent) 64 56 41 22 19 15 FWHM (mas) 79 84 97 86 92 108 Relative residual errors  (per cent) Strehl ratio 2.2 2.0 2.0 2.5 2.03 2.2 FWHM 0 0.31 0.35 0 0.23 1.3 FVUOTF 0.07 0.06 0.06 0.10 0.08 0.08 Science estimates Δmag 6.3 3.1 4.8 8.6 1.8 6.0 Δα 7.7 41.4 28.2 67.6 60.5 77.6 LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Strehl ratio (per cent) 64 56 41 22 19 15 FWHM (mas) 79 84 97 86 92 108 Relative residual errors  (per cent) Strehl ratio 2.2 2.0 2.0 2.5 2.03 2.2 FWHM 0 0.31 0.35 0 0.23 1.3 FVUOTF 0.07 0.06 0.06 0.10 0.08 0.08 Science estimates Δmag 6.3 3.1 4.8 8.6 1.8 6.0 Δα 7.7 41.4 28.2 67.6 60.5 77.6 Open in new tab We went through systematic comparisons over different asterism geometries to investigate whether cross-terms are always negligible. Table 3 highlights that the assumption of cross-term independence is affecting the PSF within extremely low differences, lower than the model's deviations from the simulations. As a subproduct of our analysis, we confirm that the LGS-based anisoplanatism is accurately represented by two independent terms, as angular+focal terms and tilt anisoplanatism. Table 3. Relative errors obtained on the considered PSF metrics by comparing the simulated PSFs produced by the total and split anisoplanatism. Sources were distributed along an L-shaped constellation with the science target on-axis. Photometric errors are given in H-band mmag and astrometry in μas. Zero values correspond to machine precision. LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Relative residual errors  (per cent) Strehl ratio 0 3 × 10−2 3 × 10−2 0 2 × 10−2 1 × 10−2 FWHM 0 0.35 0.39 0.22 0.39 1.39 FVUOTF 0 1 × 10−4 5 × 10−4 0 2 × 10−3 4 × 10−3 Science estimates Δmag 0 0.30 0.21 2 × 10−3 1.05 1.5 Δα 0 2.0 16.7 0.05 8.1 36.0 LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Relative residual errors  (per cent) Strehl ratio 0 3 × 10−2 3 × 10−2 0 2 × 10−2 1 × 10−2 FWHM 0 0.35 0.39 0.22 0.39 1.39 FVUOTF 0 1 × 10−4 5 × 10−4 0 2 × 10−3 4 × 10−3 Science estimates Δmag 0 0.30 0.21 2 × 10−3 1.05 1.5 Δα 0 2.0 16.7 0.05 8.1 36.0 Open in new tab Table 3. Relative errors obtained on the considered PSF metrics by comparing the simulated PSFs produced by the total and split anisoplanatism. Sources were distributed along an L-shaped constellation with the science target on-axis. Photometric errors are given in H-band mmag and astrometry in μas. Zero values correspond to machine precision. LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Relative residual errors  (per cent) Strehl ratio 0 3 × 10−2 3 × 10−2 0 2 × 10−2 1 × 10−2 FWHM 0 0.35 0.39 0.22 0.39 1.39 FVUOTF 0 1 × 10−4 5 × 10−4 0 2 × 10−3 4 × 10−3 Science estimates Δmag 0 0.30 0.21 2 × 10−3 1.05 1.5 Δα 0 2.0 16.7 0.05 8.1 36.0 LGS location (arcmin) 0 20 NGS location (arcmin) 0 20 40 0 20 40 Relative residual errors  (per cent) Strehl ratio 0 3 × 10−2 3 × 10−2 0 2 × 10−2 1 × 10−2 FWHM 0 0.35 0.39 0.22 0.39 1.39 FVUOTF 0 1 × 10−4 5 × 10−4 0 2 × 10−3 4 × 10−3 Science estimates Δmag 0 0.30 0.21 2 × 10−3 1.05 1.5 Δα 0 2.0 16.7 0.05 8.1 36.0 Open in new tab 4 OFF-AXIS PSF SENSITIVITY TO |$\boldsymbol{C_n^2(h)}$| PROFILE ACCURACY In this section we quantify how knowledge of the |$C_n^2(h)$| profile affects PSF morphological and science metrics. We split our study into two complementary analyses. First, we introduce a bias on the |$C_n^2(h)$| estimation by binning the number of layers. Secondly, we keep the same number of bins, but we introduce random variations on both heights andweights. 4.1 Impact of number of bins We have initially considered a 35-layer |$C_n^2(h)$| profile (Sarazin et al. 2013) as our reference to compute the PSF at different separations (θ0/2, θ0, 1.5 × θ0 and 2 × θ0) with θ0 = 24.5 arcmin in theH band. We have opted for the mean-weighted compression (Robert et al. 2010) method for reducing the problem from 15 to 2 layers. At each iteration, we retrieve the anisoplanatic PSF to compare to the reference full profile PSF. Other binning options could be considered instead (Saxenhuber et al. 2017). Keeping the angular coherence angle constant across profiles seems to us a sensible choice as we are looking particularly into anisoplanatic effects. Figs 5, 6 and 7 report the FVU and accuracy on the Strehl ratio, FWHM, photometry and astrometry as functions of the number of modelled layers for a median profile at the Paranal Observatory and different off-axis positions in the field (0.5 × θ0, θ0, 1.5 × θ0 and 2 × θ0). Curve envelopes are deduced from quartile profiles (Sarazin et al. 2013). An immediate observation is that 15 layers instead of 35 can be used with little impact on the PSFs. If errors of 1 per cent are allowed, at least seven layers are required. For such a profile, photometry errors are at the level of 3 per cent while astrometry is given at a level of 5 per cent of pixel size in the worst case. External profilers commonly deliver profiles along seven up to nine layers; its suggests they must have a sufficient altitude resolution for anisoplanatism characterisation purpose. Figure 5. Open in new tabDownload slide FVU as a function of the number of reconstructed layers. Figure 5. Open in new tabDownload slide FVU as a function of the number of reconstructed layers. Figure 6. Open in new tabDownload slide Top: H-band long-exposure Strehl ratio. Bottom: FWHM accuracy versus the number of reconstructed layers. Figure 6. Open in new tabDownload slide Top: H-band long-exposure Strehl ratio. Bottom: FWHM accuracy versus the number of reconstructed layers. Figure 7. Open in new tabDownload slide Top: H-band photometry. Bottom: astrometry accuracy versus the number of reconstructed layers. Figure 7. Open in new tabDownload slide Top: H-band photometry. Bottom: astrometry accuracy versus the number of reconstructed layers. This is an important point to consider for anisoplanatism characterization and PSF-R on 10-mclass telescopes: for a field roughly given by θ0, we only need a representation of the |$C_n^2(h)$| over seven layers to model the anisoplanatism signature on to the PSF at the 1 per cent level. 4.2 Impact of the precision of heights and weights We consider the seven-layer equivalent profile at the Paranal Observatory (Sarazin et al. 2013) as the new reference for this section; our purpose is to evaluate how much the precision of both weights and heights of turbulent layers affect the PSF. We denote |$h_l^0$| and |$w_l^0$| as the values of reference of the lth layer heights and weights, respectively. We apply a random variation on each layer, on either its weight wl or height hl. We define for each of those layers a zero-mean Gaussian statistical variable (i.e. ηh and ηw), by setting its standard deviation to unity. We then define σh and σw as p-size vectors such as \begin{eqnarray*} h_l(p,k) &=& h_l^0 + \sigma _h(p)\eta _h(l,k)\nonumber\\ w_l(p,k) &=& w_l^0\left[1 + \sigma _w\eta _w(l,k)\right], \end{eqnarray*} (36) where k refers to the random selection and p to the value of deviation introduced. For height sensitivity, we allow up to 1 km of variability, while weights are detuned by up to 30 per cent. For each iteration k, we define a new seven-layer atmosphere and we obtain the PSF at any separation starting from the PSF on-axis as described earlier. We finally compute metrics as a function of σh and σw, the separation θ and the random selection k. Metrics are averaged out over 1000 realizations providing error bars as well. The choice of 1000 iterations is a compromise between the computation time and the relevance of results. Fig. 8 illustrates FVU errors regarding the accuracy of height and weight. Figs 9, 10, 11 and 12 provide curves of mean values at multiple separations versus the precision of weights and heights given by σp . Curve envelopes are not represented, but have a similar range as in Figs 5, 6 and 7 for a seven-layer profile. Figure 8. Open in new tabDownload slide FVU as a function of height accuracy (top) and weight accuracy (bottom). Figure 8. Open in new tabDownload slide FVU as a function of height accuracy (top) and weight accuracy (bottom). Figure 9. Open in new tabDownload slide Strehl ratio (top) and FWHM (bottom) accuracy versus accuracy on weight estimation in the H band. Figure 9. Open in new tabDownload slide Strehl ratio (top) and FWHM (bottom) accuracy versus accuracy on weight estimation in the H band. Figure 10. Open in new tabDownload slide Strehl ratio (top) and FWHM (bottom) accuracy versus precision on height estimation in the H band. Figure 10. Open in new tabDownload slide Strehl ratio (top) and FWHM (bottom) accuracy versus precision on height estimation in the H band. Figure 11. Open in new tabDownload slide Photometry (top) and astrometry (bottom) accuracy versus precision on weights estimation in the H band. Figure 11. Open in new tabDownload slide Photometry (top) and astrometry (bottom) accuracy versus precision on weights estimation in the H band. Figure 12. Open in new tabDownload slide Photometry (top) and astrometry (bottom) accuracy versus precision on height estimation in the H band. Figure 12. Open in new tabDownload slide Photometry (top) and astrometry (bottom) accuracy versus precision on height estimation in the H band. Globally, PSF metrics deviate monotonically with respect to input precision level, with a speed that grows with the angular separation from the guide star. This is an expected result: the PSF model differs more when introducing more errors on inputs and for stronger anisoplanatism cases. The Strehl ratio is known to follow an |$\exp (\theta _0^{-5/3})$| law while the FWHM is proportional to |$\theta _0^{-5/3}$|⁠. This latter is given by |$r0\{\sum [h^{5/3}C_n^2(h)]^{3/5}\}$|⁠, which makes the FWHM proportional to |$r_0^{-5/3}\{\sum [h^{5/3}C_n^2(h)]\}$| and the Strehl ratio proportional to |$1+ \{r0^{-5/3}\sum [h^{5/3}C_n^2(h)]\}$| for a small amount of variations on inputs. As introduced in Fig. 8, weights scale with r0−5/3; we can see directly why the FWHM and the Strehl ratio are supposed to be linear regarding weight. Regarding the altitude, both the FWHM and Strehl ratio should not be linear in height. However, because of the exponential term involved in the expression for the Strehl ratio, the non-linear regime on the Strehl ratio appears after 1 km of altitude precision, whereas we do not have this mitigation effect of the non-linear component on the FWHM that follows a 5/3 power law on altitude. The Strehl ratio and FWHM are estimated within an accuracy of 1  per cent as long as we ensure a precision of 10 per cent on weights and roughly 200 m on heights. The latter precision means that heights of the retrieved seven-layer profile must be measured within 200 m at 1σ. Astrometry is more critical regarding the weight precision that must be ensured to a level of 7  per cent to obtain an astrometry of 10 per cent of the pixel scale. Also, photometry is only affected by height precision, which must also be within 200 m to obtain a level of 1 per cent for photometry. Such a level of accuracy on |$C_n^2(h)$| weights is accessible from external profilers (Butterley, Wilson & Sarazin 2006), but altitude resolution reaches generally 500 m up to 1 km, leading to an accuracy of  5–10 per cent on PSF estimates, which can be still acceptable for some science cases that are noise-limited, for instance. Fig. 9 highlights that the error introduced on weights does not degrade the PSF linearly with the separation. Errors at 2 × θ0 are lower compared to 1.5 × θ0. There is a physical explanation for this. At such a separation, the phase is largely decorrelated. Equation (18) involves both the phase autocovariance and cross-covariance terms. The latter converge towards zero when the separation goes to infinity. However, these are the terms that carry the sensitivity to the fractional weights of the |$C_n^2(h)$| profile, which means the off-axis PSF is less and less sensitive to weight precision for increasing separations. How do our results translate to Extremely Large Telescope (ELT) scales? Anisoplanatism variance degrades with (θ/θ0)5/3, where θ0 is only an atmosphere-dependent variable. Therefore, it seems to us that the findings ought to be conserved at an ELT scale, so we would need about 10 layers to describe the anisoplanatism. However, our analysis is focused on the anisoplanatism only and it does not include any tomography or multi-laser configuration. Also, other metrics might be more relevant regarding the science case. In conclusion, we have an order of magnitude for the number of layers of about 10 for anisoplanatism characterization only. 4.3 Metrics correlation We have gathered all metrics values from previous analyses and compared them in order to track correlations. We found the following regression relationship: \begin{eqnarray*} \Delta {\rm mag\,\, (mag)} &=& 0.0067 \pm 0.00052 \times \Delta {\rm SR\,\, ({per\, cent})}\nonumber\\ \Delta {\rm ast\,\, (mas)} &=& 0.25 \pm 0.024 \times \Delta {\rm FWHM \,\,({per\,cent})} . \end{eqnarray*} (37) This allows us to match observations accurately, as illustrated in Fig. 13. Figure 13. Open in new tabDownload slide Top: photometry accuracy versus Strehl ratio accuracy. Bottome: astrometry accuracy versus FWHM accuracy. Figure 13. Open in new tabDownload slide Top: photometry accuracy versus Strehl ratio accuracy. Bottome: astrometry accuracy versus FWHM accuracy. We have noticed a quadratic dependence of the FVU on the accuracy of the photometry (and also the Strehl ratio); for accurate photometric measurements (Δ mag < 5 per cent), we have FVU = 5.4± 0.6 × Δmag, whilefor less accurate measurements, we have FVU = 24± 3 × Δmag. We also observed a clear correlation trend between the FVU and astrometry accuracy values, given by FVU = 1.2± 0.2 × Δast, although we observed a large discrepancy of samples around the linear regression. In summary, the FVU is an efficient metric to characterize the PSF; it defines a comprehensive scale value that depends on both PSF-related parameters and key science observables. 5 TOWARDS ELT SCALES: SENSITIVITY ANALYSIS VALIDATED USING HENOS HeNOS is a MCAO test bench designed to be a scaled-down version of NFIRAOS, the first light AO system for the Thirty Metre Telescope (TMT; Rosensteiner et al. 2016; Mieda et al. 2018). We used HeNOS in single-conjugated mode in closing the loop using one of the LGS distributed over a constellation with a side length of 4.5 arcmin2, while the atmosphere is created using three phase screens. A summary of main parameters is given in Table 4. To simulate the expected PSF degradation across the field on the NFIRAOS at the TMT, all altitudes are stretched by a factor of 11. Moreover, at the time we acquired the HeNOS data, the science camera was conjugated at the altitude of the LGSs; LGS beams are propagating along a cone but are arriving in-focus at the entrance to the science camera. Table 4. Summary of HeNOS set up. Parameter Value Asterism side length 4.5 arcmin Source wavelength 670 nm r0 (670 nm) 0.751 θ0 (670 nm) 0.854 arcmin Fractional r0 (74.3,17.4,8.2)  per cent Altitude layer (0.6, 5.2, 16.3) km Source height 98.5  km Telescope diameter 8.13 m DM actuator pitch 0.813 m Parameter Value Asterism side length 4.5 arcmin Source wavelength 670 nm r0 (670 nm) 0.751 θ0 (670 nm) 0.854 arcmin Fractional r0 (74.3,17.4,8.2)  per cent Altitude layer (0.6, 5.2, 16.3) km Source height 98.5  km Telescope diameter 8.13 m DM actuator pitch 0.813 m Open in new tab Table 4. Summary of HeNOS set up. Parameter Value Asterism side length 4.5 arcmin Source wavelength 670 nm r0 (670 nm) 0.751 θ0 (670 nm) 0.854 arcmin Fractional r0 (74.3,17.4,8.2)  per cent Altitude layer (0.6, 5.2, 16.3) km Source height 98.5  km Telescope diameter 8.13 m DM actuator pitch 0.813 m Parameter Value Asterism side length 4.5 arcmin Source wavelength 670 nm r0 (670 nm) 0.751 θ0 (670 nm) 0.854 arcmin Fractional r0 (74.3,17.4,8.2)  per cent Altitude layer (0.6, 5.2, 16.3) km Source height 98.5  km Telescope diameter 8.13 m DM actuator pitch 0.813 m Open in new tab We did acquire closed-loop data when using an LGS in single-conjugated mode in 2017 July. Two data sets have been acquired with and without phase screens, respectively, in the LGS beams. This second measurement allows us to measure the best PSFs limited by non-common path aberrations (NCPAs) over all directions. See Lamb et al. (2016) about phase diversity/focal plane sharpening methods deployed on HeNOS to calibrate NCPAs. Our purpose is to demonstrate that the anisoplanatism model we have developed allows for a good representation of PSF characteristics within the expected accuracy given by the sensitivity analysis. We derive off-axis PSFs from the PSFs in the guide-star direction by using equation (11). Although off-axis PSFs are largely dominated by anisoplanatism, static aberrations that vary across the field must be taken into account to improve the PSF modelling. We have compensated for on-axis NCPAs and restore off-axis static aberrations for each individual direction by convolving PSFs with the best NCPA-limited PSFs. These PSFs include both NCPA calibration and field static aberrations residual in the off-axis direction. The OTF in the off-axis direction θ is thus yielded by the following calculation \begin{equation*} {\rm OTF}_{}(\theta ) = {\rm OTF}_{}(0) \frac{{\rm OTF}_{{\rm stat}}(\theta )}{{\rm OTF}_{{\rm stat}}(0)} {\rm ATF}_\Delta (\theta ) ,\end{equation*} (38) where OTFstat(θ) is derived from observations without phase screens in direction θ while OTF(0) is the on-axis OTF during AO operation on phase screens. Because all sources are focused at the same height, we only need to consider angular anisoplanatism to extrapolate the on-axis PSF to any other direction. Fig. 14 provides focal-plane images acquired in visible (670 nm) whilst closing the AO loop on LGS 1 (top-left PSF). Anisoplanatism on LGS 2 (bottom-right), 3 (bottom-left) and 4 (top-right) is clearly visible and produces a strong PSF elongation in the guide-star direction. We also illustrate off-axis PSFs modelled using equation (38). At a glance, we are capable of reproducing the good shape and elongation of off-axis PSFs. Table 5 reports a quantification of measured versus estimated PSF characteristics, which shows that we have an accuracy at the 10 per cent level on PSF metrics and reach 5 per cent of the FVUs on all PSFs. Figure 14. Open in new tabDownload slide Top: 670-nmHeNOS PSFs while running the AO loop on LGS 1. Middle: PSFs predicted from on-axis PSF and anisoplanatism. Bottom: residual on the prediction. Figure 14. Open in new tabDownload slide Top: 670-nmHeNOS PSFs while running the AO loop on LGS 1. Middle: PSFs predicted from on-axis PSF and anisoplanatism. Bottom: residual on the prediction. Table 5. PSF characteristics measured on laboratory PSFs compared to predicted values from on-axis PSFs and anisoplanatism. Ensquared energy (EE) is taken at 10λ/D and FVU values are estimated using equation (31) derived using 60 pixels. Bench Predicted LGS 2 LGS 3 LGS 4 LGS 2 LGS 3 LGS 4 Strehl ratio (per cent) 4.1 6.0 5.0 3.2 4.2 4.2 FWHM (mas) 117 90 100 105 84 82 Aspect ratio 1.41 1.27 1.42 1.41 1.34 1.33 EE  (per cent) 53 56 64 65 67 64 FVUPSF (per cent) – – – 5.8 4.1 6.0 Bench Predicted LGS 2 LGS 3 LGS 4 LGS 2 LGS 3 LGS 4 Strehl ratio (per cent) 4.1 6.0 5.0 3.2 4.2 4.2 FWHM (mas) 117 90 100 105 84 82 Aspect ratio 1.41 1.27 1.42 1.41 1.34 1.33 EE  (per cent) 53 56 64 65 67 64 FVUPSF (per cent) – – – 5.8 4.1 6.0 Open in new tab Table 5. PSF characteristics measured on laboratory PSFs compared to predicted values from on-axis PSFs and anisoplanatism. Ensquared energy (EE) is taken at 10λ/D and FVU values are estimated using equation (31) derived using 60 pixels. Bench Predicted LGS 2 LGS 3 LGS 4 LGS 2 LGS 3 LGS 4 Strehl ratio (per cent) 4.1 6.0 5.0 3.2 4.2 4.2 FWHM (mas) 117 90 100 105 84 82 Aspect ratio 1.41 1.27 1.42 1.41 1.34 1.33 EE  (per cent) 53 56 64 65 67 64 FVUPSF (per cent) – – – 5.8 4.1 6.0 Bench Predicted LGS 2 LGS 3 LGS 4 LGS 2 LGS 3 LGS 4 Strehl ratio (per cent) 4.1 6.0 5.0 3.2 4.2 4.2 FWHM (mas) 117 90 100 105 84 82 Aspect ratio 1.41 1.27 1.42 1.41 1.34 1.33 EE  (per cent) 53 56 64 65 67 64 FVUPSF (per cent) – – – 5.8 4.1 6.0 Open in new tab Residuals are mostly a combination of high spatial frequency patterns and a large structure oriented towards the guide-star direction. The first component can be introduced by residual speckles and static aberrations, while the second feature suggests that the anisoplanatism effect is not perfectly well characterized. It echoes the previous discussion on sensitivity: the parameters in Table 4 that we have considered as inputs of our model have been measured with a certain accuracy. Weights are estimated within 10 per cent while heights are retrieved every 1 km. To confirm whether such a level of accuracy can explain the bias on PSF estimates, we have deliberately introduced a random error on |$C_n^2(h)$| following the methodology presented in Section 4. Fig. 15 illustrates how the off-axis PSF FWHMs vary regarding the layer height's precision. At this range of separations (4.5 arcmin compared with 0.854 arcmin for θ0), the anisoplanatism characterization does not depend on the weight precision, as discussed earlier, and this has been confirmed for HeNOS. In Fig. 15, we highlight the fact that the relative error on FWHM, averaged for the three LGSs, reaches zero for an altitude precision of 750 m. From Rosensteiner et al. (2016), the altitude has been measured every 1 km, which complies with our results. This does not mean that we will obtain a better PSF characterization by shifting all layers by this quantity; instead, it confirms that the precision on input parameters translates into accuracy on PSF metrics. In future work, we will investigate the inversion of the problem, in order to retrieve the |$C_n^2(h)$| profile from a collection of observed off-axis PSFs. Figure 15. Open in new tabDownload slide Predicted PSF FWHMs versus the absolute precision in altitude layers. Plots also represent FWHM variations on each individual off-axis PSF as the overall mean FWHM over PSFs. Figure 15. Open in new tabDownload slide Predicted PSF FWHMs versus the absolute precision in altitude layers. Plots also represent FWHM variations on each individual off-axis PSF as the overall mean FWHM over PSFs. 6 CONCLUSIONS We carried out developments to characterize anisoplanatism in LGS (and NGS) systems in order to estimate off-axis PSFs using a point-wise method that is accurate, numerically efficient and can potentially be used with 40-mclass telescopes featuring a large number of degrees of freedom. For a seven-layer median profile at the Paranal Observatory, we have demonstrated that our modelling complies with physical-optics simulations to within 0.1 per cent on the FVU, together with 1 per cent on PSF-related metrics, such as the Strehl ratio and FWHM, with the LGS and NGS at 20 and 40 arcmin off-axis, respectively. The stars' flux and position are conserved at mmag and μas levels, respectively. Finally, the total anisoplanatism with the LGS is accurately split into a focal+angular and a tilt component. With the purpose of determining how knowledge of the |$C_n^2(h)$| profile constrains off-axis PSFs, we have investigated how input parameter errors on the number of layers, its heights and weights translate into PSF errors. First, we have highlighted for 10-mclass telescopes that per cent level of accuracy on the Strehl ratio, FWHM and FVU can be met with seven turbulent layers. On tight binaries, photometry can be retrieved at 3 per cent while astrometry stays within 5 per cent of pixel size. If we admit a 10 per cent accuracy on PSF metrics, |$C_n^2(h)$| must be provided within an accuracy of 500 m on altitude height. These results can be extended to the ELT scale, with the conclusion that we need roughly 10 layers to characterize a PSF at the 1 per cent on a 40-mclass telescope. We have finally validated this methodology on the HeNOS testbed where |$C_n^2(h)$| values and its accuracy are known. By introducing random errors on anisoplanatism model input parameters, we have improved the off-axis PSF FWHM estimation. It demonstrates the precision in those parameters measurements by explaining the bias we have observed on off-axis PSF modelling. Future work will aim at applying our analytical formulation to real on-sky observations of crowded fields at Keck for improving the PSF characterization and the estimation of key science parameters. Moreover, we are also focusing on inverting the problem in order to retrieve the |$C_n^2(h)$| profile from a collection of observed off-axis PSFs. Such a technique is investigated in order to provide an accurate reconstructed PSF for single-conjugated AO systems for which the|$C_n^2(h)$| profile cannot be measured using the AO control loop data. ACKNOWLEDGEMENTS This work was supported by the A*MIDEX project (no. 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TI - Off-axis point spread function characterization in laser guide star adaptive optics systems
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/sty1103
DA - 2018-08-21
UR - https://www.deepdyve.com/lp/oxford-university-press/off-axis-point-spread-function-characterization-in-laser-guide-star-GbofkIjBSv
SP - 4642
VL - 478
IS - 4
DP - DeepDyve
ER -