© ESO, 2013

1. Introduction

Exoplanet imaging is one of the main challenges in today’s astronomy. A direct observation of these planets can provide information on both the chemical composition of their atmospheres and their temperatures. Such observations have recently been made possible (Kalas et al. 2008; Marois et al. 2008; Lagrange et al. 2009), but only thanks to their high mass or their wide apparent distance from their host star.

Being able to image an object as faint as an extra-solar planet very close to its parent star requires the use of extreme AO (XAO) systems coupled to a high-contrast imaging technique, such as coronagraphy. Instruments dedicated to exoplanet imaging using these two techniques (SPHERE on the VLT, Beuzit et al. 2007; GPI on Gemini North, Macintosh et al. 2008) are currently being integrated. The performance of such systems is limited by residual speckles on the detector. These speckles, which originate in quasi-static non common path aberrations (NCPA), strongly decrease the extinction provided by the coronagraph and can be difficult to distinguish from an exoplanet. To achieve the ultimate system performance, these aberrations must be measured and compensated for. The current-generation instruments, SPHERE and GPI, respectively rely on phase diversity (Gonsalvez 1982) and an interferometry approach (Wallace et al. 2010) to compensate for these NCPA.

Several techniques dedicated to high-contrast imaging system optimization have been proposed for future systems. Some of them rely on a dedicated wave-front sensing hardware (Guyon et al. 2009), others use scientific focal plane data assuming small aberrations. Speckle nulling iterative techniques (Bordé & Traub 2006; Give’on et al. 2007) estimate the electric field in the detector plane using at least three images. The technique proposed by Baudoz et al. (2006) relies on a modification of the imaging system, but requires only one image. These techniques aim at minimizing the energy in a chosen area (“dark hole”), leading to a contrast optimization on the detector (Trauger et al. 2010; Baudoz et al. 2012) in a closed loop process.

We have recently proposed a focal-plane wave-front sensor, COFFEE (Sauvage et al. 2012), which is an extension of conventional phase diversity (Mugnier et al. 2006) to a coronagraphic system. Since COFFEE uses focal-plane images, it is possible to characterize the whole bench without any differential aberration. This method requires only two focal-plane images to estimate the aberrations upstream of the coronagraph without any modification of the coronagraphic imaging system or assuming small aberrations. COFFEE’s principle and its application to the apodized Roddier & Roddier phase mask (ARPM) are described in Sect. 2. In Sect. 3, we evaluate the quality of NCPA estimation by realistic simulations. In Sect. 4, we present the experimental results from the laboratory demonstration of COFFEE on an in-house adaptive optics bench (BOA) with an ARPM. Section 5 concludes the paper.

2. COFFEE: principle

2.1. Extension of phase diversity to coronagraphic images

Figure 1 describes the coronagraphic imaging scheme considered in this paper. We consider four successive planes denoted A (circular entrance pupil of diameter D_u), B (coronagraphic focal plane), C (Lyot Stop), and D (detector plane). The optical aberrations are considered as static and introduced in the pupil planes A and C. The coronagraphic device is composed of a focal plane mask located in plane B and a Lyot Stop situated in plane C. No particular assumption is made on the pupil shape or intensity. Thus, the description of COFFEE is compatible with several coronagraphic devices. COFFEE uses two images, ${{i}}_{\mathrm{c}}^{\mathrm{f}}$ and ${{i}}_{\mathrm{c}}^{\mathrm{d}}$, recorded on the detector (plane D in Fig. 1) that, as in phase diversity, differ from a known aberration, φ_div, to estimate aberrations both upstream (φ_u) and downstream (φ_d) of the coronagraph.

Fig. 1 Coronagraphic imaging instrument: principle.

Considering the calibration of the instrument with an unresolved object, we use the following imaging model: $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}{{i}}_{\mathrm{c}}^{\mathrm{foc}}\mathrm{=}\mathit{\alpha }\mathit{.}{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}\mathrm{\left(}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\mathrm{+}{{n}}^{\mathrm{foc}}\mathrm{+}\mathit{\beta }\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}{{i}}_{\mathrm{c}}^{\mathrm{div}}\mathrm{=}\mathit{\alpha }\mathit{.}{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}\mathrm{(}{{\phi }}_{\mathrm{u}}\mathrm{+}{{\phi }}_{\mathrm{div}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{)}\mathrm{+}{{n}}^{\mathrm{div}}\mathrm{+}\mathit{\beta }\end{array}$(1)where α is the incoming flux, h_c the coronagraphic “point spread function” (PSF) of the instrument (i.e. the response of a coronagraphic imaging system to a point source), h_det the known detector PSF, n^foc and n^div are the measurement noises, β is a uniform background (offset), and ⋆ denotes the discrete convolution operation. Such an imaging model can be used for any coronagraphic PSF expression h_c. The measurement noises n^foc and n^div comprise both photon and detector noises. Because calibration is assumed to be performed with high flux levels, we adopt a non-stationary white Gaussian model, which is a good approximation of a mix of photon and detector noises. Its variance is the sum of the photon and detector noise variances: ${{\sigma }}_{\mathrm{n}}^{\mathrm{2}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{=}{{\sigma }}_{\mathrm{ph}}^{\mathrm{2}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{+}{\mathit{\sigma }}_{\det }^{\mathrm{2}}$ (Mugnier et al. 2004), with t the pixel position in the detector plane. The former can be estimated as the image itself thresholded to positive values, and the latter can be calibrated prior to the observations.

We adopt a maximum a posteriori (MAP) approach and estimate the aberrations φ_u and φ_d, the flux α, and the background β that minimize the neg-log-likelihood of the data, potentially penalized by regularization terms on φ_u and φ_d designed to enforce smoothness of the sought phases: $\mathrm{\left(}\mathit{\alpha ̂}\mathit{,}\mathit{\beta ̂}\mathit{,}\widehat{{{\phi }}_{\mathrm{u}}}\mathit{,}\widehat{{{\phi }}_{\mathrm{d}}}\mathrm{\right)}\mathrm{=}\underset{\mathit{\alpha ,\beta ,}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}}{\arg \begin{array}{c}\min \end{array}}\mathit{J}\mathrm{\left(}\mathit{\alpha ,\beta ,}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{\right)}$(2)where $\begin{array}{ccc}\mathit{J}\mathrm{\left(}\mathit{\alpha ,\beta ,}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}{\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}\frac{{{i}}_{\mathrm{c}}^{\mathrm{foc}}\mathrm{-}\mathrm{(}\mathit{\alpha }\mathit{.}{{h}}_{\mathrm{d}}\mathit{\star }{{h}}_{\mathrm{c}}\mathrm{(}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{)}\mathrm{+}\mathit{\beta }\mathrm{)}}{{{\sigma }}_{\mathrm{n}}^{\mathrm{foc}}}\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}}^{\mathrm{2}}\\ & & \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}\frac{{{i}}_{\mathrm{c}}^{\mathrm{div}}\mathrm{-}\mathrm{(}\mathit{\alpha }\mathit{.}{{h}}_{\mathrm{d}}\mathit{\star }{{h}}_{\mathrm{c}}\mathrm{(}{{\phi }}_{\mathrm{u}}\mathrm{+}{{\phi }}_{\mathrm{div}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{)}\mathrm{+}\mathit{\beta }\mathrm{)}}{{{\sigma }}_{\mathrm{n}}^{\mathrm{div}}}\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}}^{\mathrm{2}}\\ & & \mathrm{+}\mathrm{ℛ}\mathrm{\left(}{{\phi }}_{\mathrm{u}}\mathrm{\right)}\mathrm{+}\mathrm{ℛ}\mathrm{\left(}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\end{array}$(3)where ∥x∥² denotes the sum of squared pixel values of map x, ${{\sigma }}_{\mathrm{n}}^{\mathrm{foc}}$, and ${{\sigma }}_{\mathrm{n}}^{\mathrm{div}}$ are the noise standard deviation maps of each image, and ℛ is a regularization metric for the phase.

Any aberration φ is expanded on a basis { Z_k } that is typically either Zernike polynomials or the pixel indicator functions in the corresponding pupil plane: φ =  ∑ _ka_kZ_k where the summation is, in practice, limited to the number of coefficients considered sufficient to correctly describe the aberrations. In this paper, the phase is expanded on a truncated Zernike basis. The impact of using a regularization metric with such a basis is studied later in this paper. In the MAP framework, the regularization metrics ℛ(φ_u) and ℛ(φ_d) are deduced from the assumed a priori statistics of φ_u and φ_d. Assuming these aberrations are zero-mean, Gaussian, and neglecting a priori correlations between Zernike modes, we obtain, for an estimation performed on N Zernike modes: $\mathrm{ℛ}\mathrm{\left(}{{\phi }}_{\mathit{x}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{a}}_{\mathit{x}}^{\mathit{t}}{\mathit{R}}_{{\mathit{a}}_{\mathit{x}}}^{-1}{{a}}_{\mathit{x}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{N}}\frac{{\mathit{a}}_{{\mathit{x}}_{\mathit{k}}}^{\mathrm{2}}}{{\mathit{\sigma }}_{{\mathit{x}}_{\mathit{k}}}^{\mathrm{2}}}\mathrm{,}$(4)where ${\mathit{\sigma }}_{{\mathit{x}}_{\mathit{k}}}^{\mathrm{2}}$ is the assumed phase variance per Zernike mode, R_ak the covariance matrix, and a_x a N element vector containing the estimated Zernike coefficients a_xk. Here x is either u (upstream) or d (downstream).

The minimization of metric J(α,β,φ_u,φ_d) of Eq. (3)is performed by means of a limited memory variable metric (BFGS) method (Press et al. 2007; Thiébaut 2002), which is a fast quasi-Newton type minimization method. It uses both gradients $\frac{\mathit{\partial J}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}}$ and $\frac{\mathit{\partial J}}{\mathit{\partial }{{\phi }}_{\mathrm{d}}}$. Flux α and offset β are analytically obtained using gradients $\frac{\mathit{\partial J}}{\mathit{\partial \alpha }}$ and $\frac{\mathit{\partial J}}{\mathit{\partial \beta }}$ (implementation details, including gradient expressions, can be found in Appendix A).

Sauvage et al. (2012) established that a suitable diversity phase φ_div for COFFEE was a mix of defocus and astigmatism: ${{\phi }}_{\mathrm{div}}\mathrm{=}{\mathit{a}}_{\mathrm{4}}^{\mathrm{div}}{{Z}}_{\mathrm{4}}\mathrm{+}{\mathit{a}}_{\mathrm{5}}^{\mathrm{div}}{{Z}}_{\mathrm{5}}$ with ${\mathit{a}}_{\mathrm{4}}^{\mathrm{div}}\mathrm{=}{\mathit{a}}_{\mathrm{5}}^{\mathrm{div}}\mathrm{=}\mathrm{80}\mathrm{nmrms}$, introduced upstream of the coronagraph. We therefore use this diversity phase in the following.

2.2. Coronagraphic imaging model

The imaging model used by COFFEE in the criterion minimization (Eq. (3)) requires a coronagraphic PSF expression. In this paper, we use the analytical coronagraphic imaging model developed by Sauvage et al. (2010), whose formalism is developed in this section, where r is the pupil plane position vector, r its modulus, and γ the focal plane position vector. The entrance pupil function P_u(r) is such that: ${{P}}_{\mathrm{u}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{=}{\Pi }\left(\frac{\mathrm{2}\mathit{r}}{{\mathit{D}}_{\mathrm{u}}}\right){\Phi }\mathrm{\left(}{r}\mathrm{\right)}$(5)with ${\Pi }{}^{\mathrm{\right(}}\frac{\mathrm{2}\mathit{r}}{{\mathit{D}}_{\mathrm{u}}}^{\mathrm{\left)}}\mathrm{=}\mathrm{1}$ for $\mathit{r}\mathrm{\le }\frac{{\mathit{D}}_{\mathrm{u}}}{\mathrm{2}}$, pupil entrance diameter, 0 otherwise, and Φ is an apodization function. In this paper, we consider that the impact of amplitude aberrations is negligible, which is a reasonable assumption for a ground-based, high-contrast imaging system such as SPHERE. Considering only static aberrations (no residual turbulent aberrations), the electric field Ψ_A in the entrance pupil plane can be written as ${{\Psi }}_{\mathrm{A}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{=}{{P}}_{\mathrm{u}}\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{e}}^{\mathit{j}{{\phi }}_{\mathrm{u}}\mathrm{\left(}{r}\mathrm{\right)}}\mathit{.}$(6)The field amplitude Ψ_B(γ) in plane B can be calculated, following Sauvage et al. (2010), using the analytical coronagraphic imaging model (which is called “perfect coronagraph model” hereafter): ${{\Psi }}_{\mathrm{B}}\mathrm{\left(}{\gamma }\mathrm{\right)}\mathrm{=}{\mathrm{FT}}^{-1}\mathrm{\left(}{{\Psi }}_{\mathrm{A}}\mathrm{\right(}{r}\mathrm{\left)}\mathrm{\right)}\mathrm{-}{\mathit{\eta }}_{\mathrm{0}}{\mathrm{FT}}^{-1}\mathrm{\left(}{{P}}_{\mathrm{u}}\mathrm{\right(}{r}\mathrm{\left)}\mathrm{\right)}\mathrm{,}$(7)where η₀ is the scalar that minimizes the outcoming energy from focal plane B, whose analytical value is given by ${\mathit{\eta }}_{\mathrm{0}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{𝒩}}{”}_{\mathrm{S}}{{\Psi }}_{\mathrm{A}}^{\mathrm{\ast }}\mathrm{\left(}{r}\mathrm{\right)}{{P}}_{\mathrm{u}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{,}$(8)where $\mathrm{𝒩}\mathrm{=}{”}_{\mathrm{S}}{{P}}_{\mathrm{u}}^{\mathrm{\ast }}\mathrm{\left(}{r}\mathrm{\right)}{{P}}_{\mathrm{u}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathit{.}$(9)It is worthy mentioning that η₀ is the exact definition of the instantaneous Strehl ratio given by Born & Wolf (1989). One can notice that η₀ = 1 when there is no aberration upstream of the coronagraph (φ_u(r) = 0), so that Ψ_B = 0 in such a case. No aberration in the entrance pupil leads to no outcoming energy from plane B, and thus to a perfect extinction in the detector plane D.

Propagating the wave from plane B (Eq. (7)) to plane D, we can write the electric field Ψ_D(γ) in the detector plane: $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}{{\Psi }}_{\mathit{D}}\mathrm{\left(}{\gamma }\mathrm{\right)}\mathrm{=}\hspace{0.17em}{\mathrm{FT}}^{-1}\mathrm{\{}{{P}}_{\mathrm{d}}\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{e}}^{\mathit{j}\mathrm{\left(}{{\phi }}_{\mathrm{u}}\mathrm{\right(}{r}\mathrm{)}\mathrm{+}{{\phi }}_{\mathrm{d}}\mathrm{(}{r}\mathrm{\left)}\mathrm{\right)}}\mathrm{\}}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathrm{-}{\mathit{\eta }}_{\mathrm{0}}{\mathrm{FT}}^{-1}\mathrm{\{}{{P}}_{\mathrm{d}}\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{e}}^{\mathit{j}{{\phi }}_{\mathrm{d}}\mathrm{\left(}{r}\mathrm{\right)}}\mathrm{\}}\mathrm{,}\end{array}$(10)where P_d(r) is the Lyot stop pupil function: ${{P}}_{\mathrm{d}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{=}{\Pi }{}^{\mathrm{\right(}}\frac{\mathrm{2}\mathit{r}}{{\mathit{D}}_{\mathrm{d}}}^{\mathrm{\left)}}{{P}}_{\mathrm{u}}\mathrm{\left(}{r}\mathrm{\right)}$, with D_d the Lyot stop pupil diameter (D_d ≤ D_u). For the sake of simplicity, we omit the spatial variables r and γ in the following. The coronagraphic PSF of the instrument, denoted by h_c, is the square modulus of Ψ_D: $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}{{h}}_{\mathrm{c}}\mathrm{\left(}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\mathrm{=}|{\mathrm{FT}}^{-1}\mathrm{\left(}{{P}}_{\mathrm{d}}{\mathrm{e}}^{\mathit{j}\mathrm{(}{{\phi }}_{\mathrm{u}}\mathrm{+}{{\phi }}_{\mathrm{d}}\mathrm{)}}\mathrm{\right)}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathrm{-}{\mathit{\eta }}_{\mathrm{0}}{\mathrm{FT}}^{-1}\mathrm{\left(}{{P}}_{\mathrm{d}}{\mathrm{e}}^{\mathit{j}{{\phi }}_{\mathrm{d}}}\mathrm{\right)}{|}^{\mathrm{2}}\mathit{.}\end{array}$(11)In this paper, this expression of the coronagraphic PSF is the one used by COFFEE for estimating φ_u and φ_d; i.e., Eq. (11)is inserted into the imaging model (Eq. (1)) used in the criterion minimization described in Eq. (3).

As described by Sauvage et al. (2010), this model, which analytically describes the impact of a coronagraph in an imaging system, considers that the coronagraph removes the projection of the incoming electric field on an Airy pattern, represented by the parameter η₀ (Eq. (8)). Since it does not assume small aberrations, it can be used for any wave-front error upstream of the coronagraph. The quality of the fit of this analytical imaging model with the ARPM coronagraph is discussed later in this paper (Sect. 3.5).

3. Performance assessment by numerical simulation

The aim of this section is to quantify the impact of each error source on COFFEE’s aberration estimation. Such a study will show COFFEE’s sensitivity to the classical error sources that limit the phase retrieval in a real system (and thus the final extinction of the coronagraph), which will be of high interest in defining COFFEE’s upgrades. Likewise, it will allow us to estimate the accuracy level expected on our AO bench. In this section, we present the evolution of this reconstruction error with respect to the incoming flux (Sect. 3.1), to the size of the source (Sect. 3.2), to an error made on the assumed diversity phase used in the reconstruction (Sect. 3.3), and to the number of Zernike modes used in the reconstruction (Sect. 3.4). For each error source, coronagraphic images will be computed using the imaging model presented in Eq. (1), using the perfect coronagraph model to calculate the coronagraphic PSF h_c whose expression is given Eq. (11). COFFEE will then perform the phase estimation using these two images. The compatibility of COFFEE with realistic coronagraphic images will be studied as well (Sect. 3.5) by computing coronagraphic images using a realistic coronagraph model and then running COFFEE to estimate the aberrations both upstream and downstream of the coronagraph.

Table 1 gathers the parameters used for these simulations.

The chosen wave-front error (WFE) values upstream and downstream of the coronagraph for these simulations are typical of the aberrations that will be estimated on our AO bench in Sect. 4 (so that experimental results can be compared to the following simulations). Since these simulations are performed with a small number of Zernike modes (36), there is no need of regularization metrics in such simulations.

To simulate realistic aberrations, we have considered that the variance per Zernike mode ${\mathit{\sigma }}_{\mathit{k}}^{\mathrm{2}}$ was decreasing with the radial order n(k) of the considered Zernike mode k (Noll 1976): ${\mathit{\sigma }}_{\mathit{k}}^{\mathrm{2}}\mathrm{\propto }\frac{\mathrm{1}}{\mathit{n}\mathrm{(}\mathit{k}{\mathrm{)}}^{\mathrm{2}}}\mathrm{·}$(12)This corresponds to a decrease in the static aberration spatial spectrum as $\frac{\mathrm{1}}{\mathrm{|}\mathit{\nu }{\mathrm{|}}^{\mathrm{2}}}$, where ν is the spatial frequency, which is a common assumption for mirror fabrication errors. To evaluate COFFEE’s performance, we define the reconstruction error ϵ_x (x stands for u (upstream) or d (downstream)) as $\mathit{ϵ}\mathrm{=}\sqrt{\sum _{\mathit{k}\mathrm{=}\mathrm{2}}^{\mathit{N}\mathrm{-}\mathrm{1}}\mathrm{|}{\mathit{a}}_{\mathit{k}}\mathrm{-}\mathit{â}\mathit{k}{\mathrm{|}}^{\mathrm{2}}}$(13)with a_k the Zernike coefficients (starting with k = 2 corresponding to tilt) used for the simulation, â_k the reconstructed Zernike coefficients, and N the number of Zernike modes. In this section, every reconstruction error value is an average value, computed from ten independent simulated phases.

3.1. Noise propagation

The ultimate limitation of an instrument lies in the amount of noise in the images. In Fig. 2, we present the reconstruction error for the aberrations upstream (φ_u) and downstream (φ_d) of the coronagraph with respect to the total incoming flux. Photon noise and detector noise (σ_det = 6e⁻) are added in the coronagraphic images for simulation.

Fig. 2 Aberrations upstream (φu (WFE = 80 nm), top) and downstream (φd (WFE = 20 nm), bottom) of the coronagraph: reconstruction error (solid red line) as a function of the incoming flux α. For comparison, $\frac{\mathrm{1}}{\mathit{\alpha }}$ (cyan dashed line) and $\frac{\mathrm{1}}{\sqrt{\mathit{\alpha }}}$ (magenta dashed line) theoretical behaviours are plotted for detector noise only and photon noise only (respectively).

The evolution of the reconstruction error presented in Fig. 2 is proportional to (1/α) for the detector noise limited regime (low flux) and to ($\mathrm{1}\mathit{/}\sqrt{\mathit{\alpha }}$) for the photon noise limited regime (high flux). In this figure, it can be seen that for an incoming flux α ≥ 10⁶ photons, the reconstruction error ϵ_u for the phase upstream of the coronagraph is smaller than 1nmrms. Thus, in a calibration process, where high values of flux (≥10⁶ photons) can be easily reached, COFFEE’s performance will not be significantly affected by noise.

It is noteworthy that the results of many similar simulations with various levels of upstream aberrations show that COFFEE’s reconstruction error does not depend on the amplitude of the aberrations upstream of the coronagraph, as long as the diversity phase amplitude is larger than the WFE of the aberrations to be estimated.

3.2. Impact of the source size on the reconstruction error

Our imaging model, presented in Sect. 2.1 (Eq. (1)), assumes an unresolved object. Thus, the presence of a real source with a given spatial extension will have an impact on the phase reconstruction, which is quantified here. We consider here a Gaussian-shaped laser source, emitted from a single-mode fiber. Because of the incoming light coherence, it can be represented as a Gaussian amplitude in the entrance pupil plane (where COFFEE assumes a uniform amplitude). Knowing this, coronagraphic images are simulated by considering a small coherent Gaussian-shaped beam ($\mathit{FWHM}\mathrm{\le }\mathrm{0.5}\frac{\mathit{\lambda }}{\mathit{D}}$) on the coronagraph, and then processed by COFFEE.

Fig. 3 Error reconstructions upstream (red line) and downstream (blue line) of the coronagraph as functions of the size of the source on the coronagraph.

Since the imaging model assumes an unresolved object, both reconstruction errors for the phases upstream and downstream of the coronagraph increase with the FWHM of the coherent object, as showed in Fig. 3, but remains low: for an FWHM smaller than $\frac{\mathit{\lambda }}{\mathrm{3}\mathit{D}}$, the reconstruction error is indeed sub-nanometric. The size of the laser source will thus definitely not be a limitation for COFFEE: if this error is not negligible in the total error budget, it is possible to include it in the imaging model used by COFFEE (Eq. (1)) as a non-uniform (Gaussian) entrance pupil function P_u(r).

3.3. Sensitivity to a diversity phase error

The diversity phase ${{\phi }}_{\mathrm{div}}\mathrm{=}{\mathit{a}}_{\mathrm{4}}^{\mathrm{div}}{{Z}}_{\mathrm{4}}\mathrm{+}{\mathit{a}}_{\mathrm{5}}^{\mathrm{div}}{{Z}}_{\mathrm{5}}$ has been defined in Sect. 2.1. This phase φ_div is one of the inputs that COFFEE needs in order to perform phase retrieval, so it must be calibrated as accurately as possible. To optimize the use of COFFEE, the impact of an error on such a calibration is studied. In this section, we consider that the diversity phase used to create the diversity image is not perfectly known. The coronagraphic simulated diversity image is computed with a diversity phase ${{\phi }}_{\mathrm{div}}^{\mathrm{\prime }}\mathrm{=}{{\phi }}_{\mathrm{div}}\mathrm{+}{{\phi }}_{\mathrm{err}}$, with φ_err a randomly generated phase of given rms value, and COFFEE’s phase reconstruction is done considering that the diversity phase is equal to φ_div. In Fig. 4, we see that the reconstruction error increases linearly with the calibration error on the diversity phase, with a slope of 0.5. Thus, the requirement on the calibration precision for the diversity phase is typically the precision wanted for the aberration measurement.

Fig. 4 Error reconstructions upstream (solid red line) and downstream (solid blue line) of the coronagraph as functions of the error on the diversity phase.

3.4. Impact of aliasing

The phase estimation is performed here on a truncated Zernike basis. In real images (recorded from a bench), some speckles will originate in high-order aberrations. These aberrations, which cannot be fitted by the truncated Zernike basis, will have an impact on the phase estimation, called aliasing error hereafter. Thus, it is necessary to study this aliasing error as a function of the number of Zernike modes used in the phase reconstruction. Here, we generate a phase on a large number of Zernike modes, and compute the corresponding images using the perfect coronagraph model. Aberrations both upstream and downstream of the coronagraph are then estimated by COFFEE using an increasing number of Zernike modes. Since one of the aims of this simulation is to determine the size of the truncated Zernike basis to be used with experimental data recorded on an in-house bench, the noise level in the simulated images corresponds to the one we have on this bench. The total incoming flux is 5 × 10⁶ photons, and the detector noise is σ_det = 1e⁻ per pixel. Parameters used for this simulation are gathered in Table 2. This simulation has been done with and without a regularization metric, so that we can demonstrate the relevance of this metric on phase estimation.

Fig. 5 Error reconstructions upstream (top) and downstream (bottom) of the coronagraph as functions of the number of reconstructed Zernike modes, with a regularization metric (solid blue line) and without (solid red line).

Figure 5 presents the evolution of the reconstruction errors when the number of reconstructed Zernike modes increases. Here, every reconstruction error (Eq. (13)) is calculated on a basis of 350 Zernike modes; thus, the error originates both in high-order aberrations, which are not considered by COFFEE because of the Zernike basis finite size (modelling error), and in the impact of these high-order aberrations on the estimated ones (aliasing). The WFE corresponding to the aberrations that are not estimated by COFFEE (from N to 350, where N varies between 15 and 275 according to Table 2) is called “unmodelled WFE” hereafter.

In the plot of the reconstruction error upstream of the coronagraph evolution (Fig. 5, top), one can see that without a regularization metric, the reconstruction error increases for a large number of Zernike modes. An interpretation of this behaviour is the following: because high-order aberrations have a smaller variance, their associated speckle intensity is lower. Thus, owing to the photon and detector noise in the image, the SNR is smaller for these aberrations. Such behaviour leads to a trade-off between aliasing and noise amplification for the optimal number of Zernike modes (Fig. 5). The best number of Zernike modes is then a function of the aberrations level (WFE) and spectrum, as well as of the level of noise. The use of a regularization metric allows us to avoid this noise amplification (Fig. 5): the reconstruction error roughly reaches a saturation level (rather than growing to very high values). Additionally, the use of regularization reduces the aliasing error, and avoids the need for the difficult and somewhat ad hoc choice of number of Zernike modes for the reconstruction.

According to the results presented in Fig. 5, we have chosen to estimate the aberrations upstream and downstream of the coronagraph on 170 Zernike modes with the regularization metric of Eq. (4).

3.5. Model mismatch

We have already demonstrated that ARPM images are compatible with the perfect coronagraph model and therefore with COFFEE estimation in Sauvage et al. (2012). The Roddier & Roddier phase mask (RRPM; Roddier & Roddier 1997; Guyon et al. 1999) consists in a π phase shifting mask slightly smaller than the Airy disk. Additionally, the use of a circular prolate function as entrance pupil apodization Φ_P (ARPM), proposed by Soummer et al. (2003), leads in a perfect case (no aberrations upstream of the coronagraph) to a total suppression of signal in the detector plane. In the simulations presented hereafter, realistic ARPM coronagraphic images are computed following Soummer et al. (2007) to consider an accurate numerical representation of Lyot-style coronagraphs. Then, we use COFFEE to reconstruct both phases upstream and downstream of the coronagraph. Here, when using the formalism developed in Sect. 2.2, the prolate apodization function Φ_P is included in both simulation and reconstruction imaging models.

Fig. 6 Error reconstruction upstream of the coronagraph with respect to the WFE of the aberration upstream of the coronagraph.

Because the perfect coronagraph model is not exactly identical to an ARPM (although their responses to aberrations is very close), there is a model mismatch in the estimation of aberrations upstream of the coronagraph φ_u, which varies linearly with the WFE of φ_u, as shown in Fig. 6. The model mismatch can thus be quantified as 7.5% of the WFE rms value of φ_u, except for very small WFE (≤1 nm rms), where the variation is non-linear, but remains below 1 nm rms.

Since the variation in this model mismatch varies linearly with the WFE of φ_u, it should not limit the ability to compensate for the aberration upstream of an ARPM using COFFEE as focal plane wave-front sensor (WFS).

4. Laboratory demonstration

In this section we present experimental validations in the coronagraphic phase diversity. These validations are done on the bench BOA, described in Sect. 4.1. Section 4.2 describes a carefully designed method developed to introduce calibrated static aberrations on the AO bench to be measured with COFFEE. The error made on the measurements of aberrations upstream of the coronagraph (NCPA) is quantified in Sect. 4.3. Section 4.4 presents the static aberration measurement performance, and Sect. 4.5 details the procedure for compensating for the measured aberrations.

4.1. Experimental setup

Fig. 7 Adaptive optics testbed schematic representation. Mi: fold mirrors; MPi: parabolic mirrors; Li: lenses (doublets); BS: beam splitter; TTM: Tip-Tilt mirror; DM: deformable mirror; RRPM: coronagraphic focal plane mask; Φ: prolate apodizer; WFS: AO wave-front sensor

Figure 7 shows the design of our in-house bench. The input beam, emitted from a fibered laser source (λ = 635 nm) comes through the prolate apodizer Φ, which is in the entrance pupil plane (P_u). The beam is reflected by the tip-tilt mirror (TTM) and then by the deformable mirror (DM, entrance pupil, D_u = 40 mm, 6 × 6 actuators). The beam-splitter sends a fraction of the beam to the AO wave-front sensor (Shack-Hartmann, 5 × 5 sub-apertures). On the other channel, the light is focused onto a RRPM, whose diameter is d_c = 18.1  μm (angular diameter is $\mathrm{1.06}\frac{\mathit{\lambda }}{{\mathit{D}}_{\mathrm{u}}}$). After going through the Lyot stop plane (P_d, with D_d = 0.99D_u), the beam is focused onto the camera (256 × 256 pixels images with an oversampling of 2.75, detector noise σ_det = 1e⁻). For faster computations, recorded images are re-binned to 128 × 128 pixels images with an oversampling of 1.38.

4.2. Introduction of calibrated aberrations

To evaluate COFFEE’s performance, we introduce calibrated aberrations on the bench using a process described in this section. We consider an aberration phase φ_cal to be introduced on BOA. First, since the phase is represented by the DM with a finite number of actuators (6 × 6), the introduced aberration will not match the aberration φ_cal perfectly, as illustrated in Fig. 8 in the case of a pure spherical aberration.

Fig. 8 Introduction of calibrated aberration on BOA: case of a pure spherical aberration. Left: theoretical wave-front (top) and DM introduced wave-front (bottom). Right: corresponding Zernike modes for the theoretical introduced aberration (solid red line) and the DM introduced aberration (dashed blue line).

Our aim is here to introduce, using the DM, the closest aberration to the aberration φ_cal. We let F be the DM influence matrix (obtained by calibration); any DM introduced aberration φ^DM can be described as a set of actuator voltages u (φ^DM = Fu). We are thus looking for the set u_cal which solves the least-squares problem: ${{u}}_{\mathrm{cal}}\mathrm{=}\underset{{u}}{\arg \begin{array}{c}\min \end{array}}{\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}{F}{u}\mathrm{-}{{\phi }}_{\mathrm{cal}}\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}}^{\mathrm{2}}\mathit{.}$(14)The solution of this problem can be written as ${{u}}_{\mathrm{cal}}\mathrm{=}{T}{{\phi }}_{\mathrm{cal}}\mathrm{,}$(15)with T the generalized inverse of matrix F. Using the interaction matrix D (resulting from calibration), we can compute the corresponding set of slopes s_cal (s_cal = Du_cal), which can then be used to modify the AO loop reference slopes s_ref. Thus, closing the AO loop with the reference slopes s_ref + s_cal, we introduce an aberration ${{\phi }}_{\mathrm{cal}}^{\mathrm{DM}}\mathrm{=}{F}{{u}}_{\mathrm{cal}}\mathrm{=}{F}{T}{{\phi }}_{\mathrm{cal}}$ on the bench, which is the best fit of φ_cal in the least squares sense.

We also have to consider that the bench BOA presents its own unknown static aberrations ${{\phi }}_{\mathrm{u}}^{\mathrm{BOA}}$ and ${{\phi }}_{\mathrm{d}}^{\mathrm{BOA}}$ upstream and downstream of the coronagraph (respectively). Thus, if a calibrated aberration φ_cal is introduced in the entrance pupil, aberrations φ_u upstream of the coronagraph will be ${{\phi }}_{\mathrm{u}}\mathrm{=}{{\phi }}_{\mathrm{cal}}\mathrm{+}{{\phi }}_{\mathrm{u}}^{\mathrm{BOA}}\mathit{.}$(16)To get rid of the unknown aberration ${{\phi }}_{\mathrm{u}}^{\mathrm{BOA}}$, we perform a differential phase estimation:

• 1.

  We introduce the aberration${{\phi }}_{\mathrm{cal}}^{\mathrm{DM}}$ on the bench. A phase $\hat{{\phi }}\begin{array}{c}\mathrm{+}\\ \mathrm{u}\end{array}\mathrm{=}\hat{{\phi }}\begin{array}{c}\mathrm{DM}\\ \mathrm{cal}\end{array}\mathrm{+}\hat{{\phi }}\begin{array}{c}\mathrm{BOA}\\ \mathrm{u}\end{array}$ is estimated using focused and diverse images recorded on the camera.

• 2.

  The opposite aberration $\mathrm{-}{{\phi }}_{\mathrm{cal}}^{\mathrm{DM}}$ is then introduced. A phase $\hat{{\phi }}\begin{array}{c}\mathrm{-}\\ \mathrm{u}\end{array}\mathrm{=}\mathrm{-}\hat{{\phi }}\begin{array}{c}\mathrm{DM}\\ \mathrm{cal}\end{array}\mathrm{+}\hat{{\phi }}\begin{array}{c}\mathrm{BOA}\\ \mathrm{u}\end{array}$ is estimated.

• 3.

  The half difference $\hat{{\phi }}\begin{array}{c}\mathrm{DM}\\ \mathrm{cal}\end{array}\mathrm{=}\frac{\hat{{\phi }}\begin{array}{c}\mathrm{+}\\ \mathrm{u}\end{array}\mathrm{-}\hat{{\phi }}\begin{array}{c}\mathrm{-}\\ \mathit{u}\end{array}}{\mathrm{2}}$ is our estimate of φ_cal.

The first use of this process is to calibrate the diversity phase itself. Since this phase will be introduced using the AO system, the actually introduced diversity phase will not exactly match the theoretical mix of defocus and astigmatism. We introduce the aberrations φ_div and − φ_div on the bench using the AO system. These two aberrations are then estimated using classical phase diversity (no coronagraph), with a pure defocus of diversity phase introduced using a flat glass plate of known thickness e in a focused beam.

Such a process gives us an accurate estimation of the diversity phase really introduced on the bench, with an estimated accuracy of 4 nm rms on the introduced aberration. This calibration is then used in COFFEE’s estimations performed on experimental images.

4.3. Performance assessment: error budget

From simulations presented in Sect. 3, we establish an error budget for estimating aberrations upstream of the coronagraph using experimental data:

• ⋄

  Photon and detector noise error: on the BOA bench, the typicalincoming flux isf_BOA = 5 × 10⁶ photons. Knowing that we have photon noise and a detector noise with σ_det = 1e⁻, we can evaluate the noise error: ϵ_noise = 0.9 nm rms.

• ⋄

  The diversity phase φ_div has been calibrated using classical phase diversity, using the process presented in Sect. 4.2. Such an estimation has been performed with an error of 4.0 nm rms (value calculated from an error budget evaluated for a classical phase diversity estimation on the BOA bench. Such accuracy has already been obtained on this bench by Sauvage et al. 2007). According to Sect. 3.3, this error on the diversity phase leads to an error ϵ_model = 2.0 nm rms.

• ⋄

  The source is a coherent Gaussian-shaped beam whose FWHM is $\mathrm{0.27}\frac{\mathit{\lambda }}{\mathit{D}}$ on the coronagraph. According to the simulations of Sect. 3.2, this leads to a reconstruction error: ϵ_obj = 0.7 nm rms.

• ⋄

  Residual turbulent speckles, which originate in uncorrected turbulent aberrations, are not included in the imaging model. To measure the impact of these speckle on the reconstruction, several wave-fronts have been successively recorded using a commercial Shack-Hartmann wave-front sensor. From these acquisitions, we calculate the WFE of the residual turbulent phase: σ_φturb = 1.2 nm rms. This residual turbulence will create speckles on the detector, which will be considered by COFFEE as originating in NCPA. Thus, the residual turbulence error ϵ_turb made by COFFEE is estimated to ϵ_turb = σ_φturb = 1.2 nm rms.

• ⋄

  Aliasing error, which originates in high-order aberrations, has been studied in Sect. 3.4. For a phase upstream of the coronagraph estimated on N = 170 Zernike modes, we have ϵ_aliasing = 18.3 nm rms.

• ⋄

  From simulations, we know that the model mismatch is 7.5% of WFE. For this study, we will not estimate aberrations with a WFE stronger than 80 nm rms. For such a WFE, the model error is ϵ_model = 6.0 nm rms.

As one can see in Table 3, the error budget is mainly driven by the aliasing error. The second most important term is the model mismatch (even though it goes to zero with the WFE).

4.4. Measurement of aberrations upstream of the coronagraph

In this section, we introduce calibrated aberrations on the BOA bench upstream of the coronagraph, and then estimate them with COFFEE in order to evaluate its performance. In the course of this study, we realized that the position of the coronagraphic image on the detector (quantified by the tip-tilt downstream of the coronagraph) is a critical issue. Indeed, it occurred that COFFEE was able to perform phase retrieval only for downstream tip-tilt [a₂,a₃] values within the range [−100nmrms;100nmrms] ($\mathrm{[}\mathrm{-}\frac{\mathit{\lambda }}{\mathrm{6}\mathit{D}}\mathrm{;}\frac{\mathit{\lambda }}{\mathrm{6}\mathit{D}}\mathrm{]}$). To get rid of this constraint, we have developed a method to perform a preliminary estimation of the tip-tilt downstream of the coronagraph. This method, which uses the diversity image, is fully described in Appendix B.

4.4.1. Measurement of tip-tilt upstream of the coronagraph

We present the estimation of a tilt aberration upstream of the coronagraph using COFFEE in this section. Using the AO system, we introduce a tilt aberration by adding a constant value δs_TT to the AO wave-front sensor references slopes s_ref, and then closing the AO loop on the slopes s_ref + δs_TT. To accurately calibrate the introduced tilt, for each position, we first estimate the aberrations using classical phase diversity (no coronagraph). Then, the RRPM is put in the focal plane, and the same operation is repeated: for each position, we record two images, and then estimate the aberrations using COFFEE.

Fig. 9 Estimation of a tilt aberration on BOA: calibration (solid blue line) and COFFEE’s estimation with bound on the tip-tilt downstream of the coronagraph (dashed crossed red line) and without boundaries (dashed diamond green line).

From the upstream tilt reconstruction performed by COFFEE (Fig. 9), we calculate an average reconstruction error: ϵ_tilt = 2.1 nm. Part of this error is due to an error on the estimation of tip-tilt downstream of the coronagraph. An improved estimation has been performed by setting boundaries on the downstream tip-tilt. Its value is evaluated before COFFEE’s estimation using the method described in Appendix B with the diversity coronagraphic image recorded for a tip-tilt upstream the coronagraph value close to 0 nm rms (centered coronagraph). Such an estimation process gives us an estimation of tip-tilt downstream of the coronagraph $\mathrm{\left\{}{\mathit{a}}_{\mathrm{2}}^{\mathrm{do}}\mathit{,}{\mathit{a}}_{\mathrm{3}}^{\mathrm{do}}\mathrm{\right\}}$ with an accuracy of ± 1.5 nm rms. Using this estimation as the starting value for the minimization, and setting bounds of ± 1.5 nm rms on it, we processed the same experimental data. This, in turn, results in a better estimation of tilt upstream of the coronagraph (Fig. 9), with an average error ϵ_tilt = 1.5 nm, which is close to the expected error per Zernike mode given in Sect. 4.3 (ϵ′ = 1.6 nm rms).

4.4.2. NCPA measurements

In this section, we introduce aberrations upstream of the coronagraph. The aberration φ_cal is expanded on the first 15 Zernike modes (which is the largest number of modes we can properly describe with our 6 × 6  DM), and then we estimate these aberrations using COFFEE, following the process described in Sect. 4.2. To take the DM action into account on the introduced phase (illustrated in Fig. 8), aberrations φ_cal are first estimated with classical phase diversity (no phase mask in the coronagraphic focal plane, Sauvage et al. 2007). This estimation gives us a calibration of the introduced aberration, which is then used to evaluate the accuracy of COFFEE’s estimation.

Fig. 10 COFFEE: NCPA estimation of an introduced phase φcal on BOA. Top: for an aberration + φcal, recorded coronagraphic image from the bench (left) and computed image using the reconstructed aberration $\hat{{\phi }}\begin{array}{c}\mathrm{+}\\ \mathrm{u}\end{array}$ (right) (log. scale, same range for both images). Middle: same images for an aberration − φcal introduced and a reconstructed aberration $\hat{{\phi }}\begin{array}{c}\mathrm{-}\\ \mathrm{u}\end{array}$ (log. scale, same range for both images). Bottom: calibrated introduced aberration (left) and COFFEE estimated aberration (right).

At convergence of the reconstruction, a very good match can be observed between the experimental images and the ones computed for the estimated aberrations (Fig. 10, top and middle). This, in turn, results in a very good match between the aberrations measured by COFFEE (Fig. 10, right) and the introduced ones (Fig. 10, left).

From the experimental phase estimation presented in Fig. 10, we compute a reconstruction error between the classical diversity phase calibrated aberration and COFFEE’s estimation: ${\mathit{ϵ}}_{\exp }\mathrm{=}\mathrm{22.5}\mathrm{nmrms}\mathit{.}$(17)One can notice that this error is close to the expected error budget, i.e. that there is a good match between the performance assessment study carried out in Sect. 3 and the experimental results presented in this section.

4.5. Low-order NCPA compensation

Lastly, the ability of COFFEE to compensate for the aberrations upstream of the coronagraph is experimented on BOA. In Sect. 4.4, the aberrations upstream of the coronagraph are expanded on 170 Zernike modes, in order to have the smallest reconstruction error (according to Sect. 3.4).

As previously mentioned, the compensation on BOA is limited to the 15th Zernike mode. Thus, what is required in a closed loop process is the most accurate estimation of 15 Zernike modes rather than an accurate measurement of every estimated Zernike mode. Using a basis of 36 Zernike modes for the reconstruction is sufficient to give an accurate estimation of the first 15 Zernike modes: the aliasing error, which is the most important error source, will mainly degrade the estimation accuracy of the reconstructed high orders (close to Z₃₆).

To demonstrate the ability of COFFEE to be used in a closed loop, we introduce a set of aberrations on the DM by modifying the reference slopes, as described in Sect. 4.2. Then, we use the pseudo-closed loop (PCL) method described in Sauvage et al. (2007). This iterative process has two stages: for the PCL iteration i:

• 1.

  acquisition of the focused${{i}}_{\mathrm{c}}^{\mathit{f}}$ and diverse ${{i}}_{\mathrm{d}}^{\mathit{f}}$ images;

• 2.

  estimation of the aberration $\hat{{\phi }}\begin{array}{c}\mathit{i}\\ \mathrm{u}\end{array}$ upstream of the coronagraph;

• 3.

  computation of the corresponding reference slopes correction $\mathit{\delta }{s}\mathrm{=}\mathit{g}{DT}\hat{{\phi }}\begin{array}{c}\mathit{i}\\ \mathrm{u}\end{array}$, where D and T are the interaction and influence matrices defined in Sect. 4.2 and g is the PCL gain;

• 4.

  the AO loop is closed on the modified reference slopes.

The computation time (step 2) varies from 1 min to 2.5 min, allowing us to compensate for quasi-static aberrations upstream of the coronagraph. This compensation process is limited by the estimation accuracy of the first 15 Zernike modes performed by COFFEE, which corresponds to the error budget established in Sect. 4.3), and by the ability of the DM to reproduce a given wave-front. Indeed, the correction introduced on the bench (step 2 of the PCL process) is the best fit of the estimated phase $\hat{{\phi }}\begin{array}{c}\mathit{i}\\ \mathrm{u}\end{array}$ in the least-square sense (as presented in Sect. 4.2). The difference between the estimated aberration and the actual introduced correction will thus limit the compensation performance of the PCL process. Considering these two limitations, one can compute the variance ${\mathit{\sigma }}_{\mathrm{BOA}}^{\mathrm{2}}$ (for the first 15 Zernike modes) that can be reached on the BOA bench: ${\mathit{\sigma }}_{\mathrm{BOA}}^{\mathrm{2}}\mathrm{=}\mathrm{4.4}\mathrm{\times }{\mathrm{10}}^{-2}{\mathrm{radrms}}^{\mathrm{2}}\mathit{.}$(18)The correction and stabilization of the NCPA variance can be seen in Fig. 11. One can see that the variance of the 15 corrected Zernike modes reaches the expected asymptotic value ${\mathit{\sigma }}_{\mathrm{BOA}}^{\mathrm{2}}$. This result is the very first demonstration of COFFEE’s ability to compensate for aberrations upstream of the coronagraph. A compensation at levels compatible with SPHERE or GPI-like instruments will require using a DM with many more actuators, and working on the reduction of the dominant term of the error budget, which is aliasing.

Fig. 11 PCL on the bench BOA (gPCL = 0.5): variance of the residual static aberrations upstream of the coronagraph for the 36 COFFEE estimated Zernike modes (solid red line) and the 15 corrected modes (solid blue line). The magenta dashed line represents the ultimate performance one can reach according to the error budget detailed in 4.3.

5. Conclusion

In this paper, we have presented a thorough simulation study (Sect. 3) and a first experimental validation (Sect. 4) of the coronagraphic wave-front sensor called COFFEE, which consists mainly in the extension of the phase diversity concept to a coronagraphic imaging system. From the validation and careful performance assessment of COFFEE, we showed that COFFEE is currently limited by the aliasing error, due to high-order aberrations, which are difficult to model with a Zernike basis.

In Sect. 4, we presented a first experimental validation of COFFEE using an ARPM. We introduced calibrated aberrations upstream of the coronagraph (NCPA), using the AO sub-system, and estimated them with COFFEE. The accuracy we obtained on these estimation shows a very good match with our error budget. Lastly, we used COFFEE in an iterative process to perform a preliminary validation of COFFEE’s ability to compensate for the aberrations upstream of the coronagraph.

Several perspectives are currently considered to optimize COFFEE: firstly, in order to minimize the impact of the aliasing error on the phase reconstruction, we plan to perform the phase reconstruction on a pixel-wise map, which is more suitable than a truncated Zernike basis. Secondly, we would like to improve the imaging model, both to make COFFEE work with other coronagraph than the ARPM and to reduce the model error, which is currently the second most important one, even though it goes to zero with the WFE. Two solutions are considered. In the absence of residual turbulence, an accurate imaging model is obtained by propagating the electric field through each plane of the coronagraphic imaging system (Fig. 1) for an arbitrary focal plane coronagraphic mask. Such a method, where no model error needs to be considered, can be used for a laboratory calibration. Alternatively, a more accurate analytical imaging model, which could include a residual turbulent aberration, can be developed. Such a model, which could include a residual turbulent aberration, will ultimately allow us to perform NCPA estimation on images from the sky. These improvements should allow us to estimate and compensate for the aberrations upstream of the coronagraph using COFFEE with a nanometric precision in a closed loop process.

A further perspective is to extend COFFEE to phase and amplitude aberration estimation, in order to create a dark hole region in the coronagraphic image.

Acknowledgments

The authors would like to thank Mamadou N’Diaye, Kjetil Dohlen and Thierry Fusco for stimulating discussions, as well as Marc Ferrari, David Mouillet and Jean-Luc Beuzit for their support, and the Région Provence-Alpes-Côte d’Azur for partial financial support of B.P. scholarship. This work has been partially funded by the European Commission under FP7 Grant Agreement No. 312430 Optical Infrared Coordination Network for Astronomy.

References

Appendix A: Implementation details

COFFEE performs a phase estimation by minimizing a criterion J whose expression is given by Eq. (3). To estimate φ_u and φ_d (expanded on a truncated Zernike basis), we need both gradients $\frac{\mathit{\partial J}}{\mathit{\partial }{{a}}_{\mathrm{u}}}$ and $\frac{\mathit{\partial J}}{\mathit{\partial }{{a}}_{\mathrm{d}}}$, where a_x =  { a_x1,a_x2,...,a_xN } is a vector that contains the Zernike coefficients, for an aberration expanded on N Zernike modes (x is for u (upstream) or d (downstream)).

Let us write the numerical expression of J^foc, using the notations defined in Sect. 2.1: $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}\mathit{J}\mathrm{=}\hspace{0.17em}\frac{\mathrm{1}}{\mathrm{2}}\sum _{\mathit{t}}\left|\frac{{{i}}_{\mathrm{c}}^{\mathrm{foc}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{-}\mathit{\alpha }\mathit{.}{{h}}_{\det }\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{foc}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{-}\mathit{\beta }}{{{\sigma }}_{\mathrm{n}}^{\mathrm{foc}}\mathrm{\left[}\mathit{t}\mathrm{\right]}}\right|2\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\sum _{\mathit{t}}\left|\frac{{{i}}_{\mathrm{c}}^{\mathrm{div}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{-}\mathit{\alpha }\mathit{.}{{h}}_{\det }\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{div}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{-}\mathit{\beta }}{{{\sigma }}_{\mathrm{n}}^{\mathrm{div}}\mathrm{\left[}\mathit{t}\mathrm{\right]}}\right|2\\ \begin{array}{c}\end{array}\mathrm{+}{\mathrm{ℛ}}_{{{\phi }}_{\mathrm{u}}}\mathrm{+}{\mathrm{ℛ}}_{{{\phi }}_{\mathrm{d}}}\\ \begin{array}{c}\end{array}\mathrm{=}\hspace{0.17em}{\mathit{J}}^{\mathrm{foc}}\mathrm{+}{\mathit{J}}^{\mathrm{div}}\mathrm{+}{\mathrm{ℛ}}_{{{\phi }}_{\mathrm{u}}}\mathrm{+}{\mathrm{ℛ}}_{{{\phi }}_{\mathrm{d}}}\mathit{.}\end{array}$(A.1)With t the pixel position in the detector plane. ${{\sigma }}_{\mathrm{n}}^{\mathrm{foc}}$ and ${{\sigma }}_{\mathrm{n}}^{\mathrm{div}}$ are the noise variance maps. Considering the expression of J, we derive J^foc, and then deduce the gradients expressions of J^div using a trivial substitution. Expressions of the regularization terms gradients $\frac{\mathit{\partial }{\mathrm{ℛ}}_{{{\phi }}_{\mathrm{x}}}}{\mathit{\partial }{{a}}_{\mathrm{x}}}$ are given by $\frac{\mathit{\partial }{\mathrm{ℛ}}_{{{\phi }}_{\mathrm{x}}}}{\mathit{\partial }{{a}}_{\mathrm{x}}}\mathrm{=}{\mathit{R}}_{{{a}}_{\mathrm{x}}}^{-1}{{a}}_{\mathrm{x}}\mathit{.}$(A.2)The calculation of gradients $\frac{\mathit{\partial J}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}}$ and $\frac{\mathit{\partial J}}{\mathit{\partial }{{\phi }}_{\mathrm{d}}}$ is done following Mugnier et al. (2001): first, we calculate the gradient of J^f with respect to the PSF h_c: $\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{h}}_{\mathrm{c}}^{\mathrm{foc}}}\mathrm{=}\frac{\mathrm{1}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{foc}}}^{\mathrm{2}}}\mathrm{[}\mathit{\alpha }{{h}}_{\det }{\mathrm{(}\mathit{\alpha }\mathit{.}{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{foc}}\mathrm{-}{{{i}}_{\mathrm{c}}^{\mathrm{foc}}}^{\mathrm{)}}}^{\mathrm{]}}\mathit{.}$(A.3)Then, the calculation consists in derivating the gradient of the PSF h_c with respect to phases φ_u [k] and φ_d [l] at pixels k, l in pupils upstream and downstream of the coronagraph, respectively, and applying the chain rule, as already done in a non-coronagraphic case, e.g. in Thiébaut & Conan (1995). The calculation of both gradients $\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}\mathrm{\left[}\mathit{k}\mathrm{\right]}}$ and $\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{\phi }}_{\mathrm{d}}\mathrm{\left[}\mathit{l}\mathrm{\right]}}$ gives

$\begin{array}{ccc}& & \hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}\mathrm{\left[}\mathit{k}\mathrm{\right]}}\mathrm{=}\hspace{0.17em}\mathrm{2}\mathrm{\Im }\left\{{{\psi }}^{\mathrm{\ast }}\mathrm{\left[}\mathit{k}\mathrm{\right]}\left[\mathrm{FT}\left(\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{h}}_{\mathrm{c}}^{\mathrm{foc}}}\mathrm{(}{\Psi }\mathrm{-}{\mathit{\eta }}_{\mathrm{0}}{{\Psi }}_{\mathrm{d}}\mathrm{)}\right)\right]\right\}\mathrm{\left[}\mathit{k}\mathrm{\right]}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathrm{-}\mathrm{2}\mathrm{\Re }\left(\frac{\mathit{\partial }{\mathit{\eta }}_{\mathrm{0}}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}\mathrm{\left[}\mathit{k}\mathrm{\right]}}\sum _{\mathit{t}}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{f}}}{\mathit{\partial }{{h}}_{\mathrm{c}}^{\mathrm{foc}}}{{\Psi }}^{\mathrm{\ast }}{{\Psi }}_{\mathrm{d}}\right)\\ \begin{array}{c}\end{array}\mathrm{+}\frac{\mathit{\partial }\mathrm{|}{\mathit{\eta }}_{\mathrm{0}}{\mathrm{|}}^{\mathrm{2}}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}\mathrm{\left[}\mathit{k}\mathrm{\right]}}\sum _{\mathit{t}}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{f}}}{\mathit{\partial }{{h}}_{\mathrm{c}}^{\mathrm{foc}}}\mathrm{\left|}{{\Psi }}_{\mathrm{d}}\mathrm{\right|}2\end{array}\\ & & \hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{\phi }}_{\mathrm{d}}\mathrm{\left[}\mathit{l}\mathrm{\right]}}\mathrm{=}\hspace{0.17em}\mathrm{2}\mathrm{\Im }(\mathrm{\left(}{{\psi }}^{\mathrm{\ast }}\mathrm{\right[}\mathit{l}\mathrm{]}\mathrm{-}{\mathit{\eta }}_{\mathrm{0}}^{\mathrm{\ast }}{{\psi }}_{\mathrm{d}}^{\mathrm{\ast }}\mathrm{[}\mathit{l}\mathrm{\left]}\mathrm{\right)}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathrm{\times }\left\{\mathrm{FT}\left[\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{f}}}{\mathit{\partial }{{h}}_{\mathrm{c}}^{\mathrm{foc}}}\mathrm{(}{\Psi }\mathrm{-}{\mathit{\eta }}_{\mathrm{0}}{{\Psi }}_{\mathrm{d}}\mathrm{)}\right]\right\}\mathrm{\left[}\mathit{l}\mathrm{\right]})\mathit{.}\end{array}\end{array}$With ℑ and ℜ the imaginary and real part (respectively), and

$\begin{array}{ccc}& & \frac{\mathit{\partial }{\mathit{\eta }}_{\mathrm{0}}}{\mathit{\partial }{{\phi }}_{\mathrm{u}}}\mathrm{=}\mathit{j}{{P}}_{\mathrm{u}}^{\mathrm{2}}{\mathrm{e}}^{\mathit{j}{{\phi }}_{\mathrm{u}}}\\ & & {\psi }\mathrm{\left(}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\mathrm{=}{{P}}_{\mathrm{d}}{\mathrm{e}}^{\mathit{j}\mathrm{(}{{\phi }}_{\mathrm{u}}\mathrm{+}{{\phi }}_{\mathrm{d}}\mathrm{)}} {\Psi }\mathrm{\left(}{{\phi }}_{\mathrm{u}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\mathrm{=}{\mathrm{FT}}^{-1}\mathrm{\left(}{\psi }\mathrm{\right)}\\ & & {{\psi }}_{\mathrm{d}}\mathrm{\left(}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\mathrm{=}{{P}}_{\mathrm{d}}{\mathrm{e}}^{\mathit{j}{{\phi }}_{\mathrm{d}}} {{\Psi }}_{\mathrm{d}}\mathrm{\left(}{{\phi }}_{\mathrm{d}}\mathrm{\right)}\mathrm{=}{\mathrm{FT}}^{-1}\mathrm{\left(}{{\psi }}_{\mathrm{d}}\mathrm{\right)}\mathit{.}\end{array}$(A.6)Since the phases are expanded on a Zernike basis, we need the gradients of J^foc with respect to the Zernike coefficients a_xi of phase φ_x. These gradients are given by the expression (Mugnier et al. 2001): $\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{\mathit{a}}_{{\mathrm{x}}_{\mathit{i}}}}\mathrm{=}\sum _{\mathit{k}}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{foc}}}{\mathit{\partial }{{\phi }}_{\mathrm{x}}\mathrm{\left[}\mathit{k}\mathrm{\right]}}{\mathit{Z}}_{\mathit{i}}\mathrm{\left[}\mathit{k}\mathrm{\right]}\mathit{.}$(A.7)Flux α and constant background β are also analytically estimated during the minimization, considering that ${\mathit{J}}^{\mathrm{p}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\sum _{\mathit{t}}\left|\frac{\mathrm{-}{{i}}_{\mathrm{c}}^{\mathrm{p}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{+}\mathit{\alpha }\mathit{.}{{h}}_{\det }\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{p}}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{+}\mathit{\beta }}{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}\mathrm{\left[}\mathit{t}\mathrm{\right]}}\right|2$(A.8)where p is for “foc” (focused) or “div” (diverse). For the sake of simplicity, we shall omit the variable t. We have $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{p}}}{\mathit{\partial \alpha }}\mathrm{=}\hspace{0.17em}\mathit{\alpha }\begin{array}{c}\sum \end{array}\frac{\mathrm{(}{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{p}}{\mathrm{)}}^{\mathrm{2}}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}}^{\mathrm{2}}}\mathrm{+}\mathit{\beta }\begin{array}{c}\sum \end{array}\frac{{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{p}}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}}^{\mathrm{2}}}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\mathrm{-}\begin{array}{c}\sum \end{array}\frac{\mathrm{(}{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{p}}\mathrm{)}{{i}}_{\mathrm{c}}^{\mathrm{p}}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}}^{\mathrm{2}}}\\ \begin{array}{c}\end{array}\frac{\mathit{\partial }{\mathit{J}}^{\mathrm{p}}}{\mathit{\partial \beta }}\mathrm{=}\hspace{0.17em}\mathit{\alpha }\begin{array}{c}\sum \end{array}\frac{{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}^{\mathrm{p}}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}}^{\mathrm{2}}}\mathrm{+}\mathit{\beta }\begin{array}{c}\sum \end{array}\frac{\mathrm{1}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}}^{\mathrm{2}}}\mathrm{-}\begin{array}{c}\sum \end{array}\frac{{{i}}_{\mathrm{c}}^{\mathrm{p}}}{{{{\sigma }}_{\mathrm{n}}^{\mathrm{p}}}^{\mathrm{2}}}\mathrm{·}\end{array}$(A.9)Which gives us, in a matricial form: $\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}\left(\begin{array}{c}\\ & \\ & \end{array}\right)\left(\begin{array}{c}\\ \\ \end{array}\right)\mathrm{=}\left(\begin{array}{c}\\ \\ \end{array}\right)\mathit{.}\end{array}$(A.10)A simple matrix inversion gives us the analytical estimation of the flux α and the background β for each iteration.

Appendix B: Tip-tilt estimation downstream of the coronagraph

The tip-tilt downstream of the coronagraph (which represents the image position on the detector) strongly limits COFFEE’s performance. Indeed, we determine that the phase estimation was accurate when − 100  nmrms ≤ a_i ≤ 100  nmrms, with a_i the Zernike coefficient for tip or tilt (i ∈ {2,3}). Beyond this range, COFFEE is unable to properly estimate both phases φ_u and φ_d. Such a phenomenon strongly limits COFFEE’s performance on a bench, since its utilization requires a restrictive location of the PSF on the detector.

To get rid of this limitation, we have developed a simple and fast method of estimating the tip-tilt downstream of the coronagraph before COFFEE’s estimation, based on the diversity image. This image is created by adding a known aberration ${{\phi }}_{\mathrm{div}}\mathrm{=}{\mathit{a}}_{\mathrm{4}}^{\mathrm{div}}{\mathit{Z}}_{\mathrm{4}}\mathrm{+}{\mathit{a}}_{\mathrm{5}}^{\mathrm{div}}{\mathit{Z}}_{\mathrm{5}}$ (${\mathit{a}}_{\mathrm{4}}^{\mathrm{div}}\mathrm{=}{\mathit{a}}_{\mathrm{5}}^{\mathrm{div}}\mathrm{=}\mathrm{80}$ nm rms) to φ_u. Since the amplitude of this aberration is important (σ_φdiv = 113 nm rms), the speckles we have in the coronagraphic diversity image mainly originate in this diversity aberration. This is illustrated in Fig. B.1, where we show two diversity images: one computed with randomly generated phases φ_u (WFE 30 nm rms), φ_d (WFE 10 nm rms), and another computed with no aberrations other than the diversity ones.

Fig. B.1 Coronagraphic diversity images computed for an aberration φu + φdiv upstream, φd downstream of (left) and the only diversity aberration φdiv (right). The shape of both images is mainly driven by diversity aberration.

As one can see in Fig. B.1, we can clearly identify the aberrations which originate in the diversity φ_div. The principle of our method lies in the research of these well-known aberrations (since we know the phase φ_div we introduce) in the diversity image ${\mathit{i}}_{\mathrm{c}}^{\mathrm{d}}$ by comparing it with a theoretical diversity image ${\mathit{i}}_{{\mathrm{c}}_{\mathit{th}}}^{\mathrm{d}}$, calculated with no other aberrations than the diversity ones: ${{i}}_{{\mathrm{c}}_{\mathit{th}}}^{\mathrm{d}}\mathrm{=}{{h}}_{\det }\mathit{\star }{{h}}_{\mathrm{c}}\mathrm{(}{{\phi }}_{\mathrm{div}}\mathit{,}{{\phi }}_{\mathrm{d}}\mathrm{=}\mathrm{0}\mathrm{)}\mathit{.}$(B.1)The comparison of ${\mathit{i}}_{{\mathrm{c}}_{\mathit{th}}}^{\mathrm{d}}$ with ${\mathit{i}}_{\mathrm{c}}^{\mathrm{d}}$ is performed using the method developed by Gratadour et al. (2005), which consists in minimizing the following criterion J_TT${\mathit{J}}_{\mathrm{TT}}\mathrm{\left(}\mathit{x,y}\mathrm{\right)}\mathrm{=}{\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}\frac{{{i}}_{\mathrm{c}}^{\mathrm{div}}\mathrm{\left(}{\mathit{x}}_{\mathit{o}}\mathit{,}{\mathit{y}}_{\mathit{o}}\mathrm{\right)}\mathrm{-}{{i}}_{{\mathrm{c}}_{\mathit{th}}}^{\mathrm{div}}\mathrm{\left(}{\mathit{x}}_{\mathit{o}}\mathit{,}{\mathit{y}}_{\mathit{o}}\mathrm{\right)}\mathit{\star }{\delta }\mathrm{(}{\mathit{x}}_{\mathit{o}}\mathrm{-}\mathit{x,}{\mathit{y}}_{\mathit{o}}\mathrm{-}\mathit{y}\mathrm{)}}{{{\sigma }}_{\mathrm{n}}^{\mathrm{div}}}\begin{array}{c}\mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\\ \mathrm{\parallel }\end{array}}^{\mathrm{2}}\mathrm{,}$(B.2)

where δ is the dirac function. Minimization of J_TT gives us the shift [x_M,y_M]between both images. It is then possible to calculate the corresponding tip (a₂) and tilt (a₃) downstream of the coronagraph knowing the image sampling s: ${\mathit{a}}_{\mathrm{2}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}\mathit{s}}{\mathit{x}}_{\mathit{M}} {\mathit{a}}_{\mathrm{3}}\mathrm{=}\frac{\mathit{\pi }}{\mathrm{2}\mathit{s}}{\mathit{y}}_{\mathit{M}}\mathrm{·}$(B.3)Finally, these estimated tip-tilt values are given to COFFEE as an input of the minimization, and are used as initial values to begin phase reconstruction. This method performs, on our experimental images, a fast preliminary estimation (~1 s for a 256 × 256 image) of the tip-tilt downstream of the coronagraph with an accuracy of 1.5 nm rms, which is far enough, compared to the level of accuracy (± 100 nm rms) required by COFFEE.

All Tables

All Figures

	Fig. 1 Coronagraphic imaging instrument: principle.
In the text

Fig. 2 Aberrations upstream (φu (WFE = 80 nm), top) and downstream (φd (WFE = 20 nm), bottom) of the coronagraph: reconstruction error (solid red line) as a function of the incoming flux α. For comparison, $\frac{\mathrm{1}}{\mathit{\alpha }}$ (cyan dashed line) and $\frac{\mathrm{1}}{\sqrt{\mathit{\alpha }}}$ (magenta dashed line) theoretical behaviours are plotted for detector noise only and photon noise only (respectively).

In the text

	Fig. 3 Error reconstructions upstream (red line) and downstream (blue line) of the coronagraph as functions of the size of the source on the coronagraph.
In the text

	Fig. 4 Error reconstructions upstream (solid red line) and downstream (solid blue line) of the coronagraph as functions of the error on the diversity phase.
In the text

	Fig. 5 Error reconstructions upstream (top) and downstream (bottom) of the coronagraph as functions of the number of reconstructed Zernike modes, with a regularization metric (solid blue line) and without (solid red line).
In the text

	Fig. 6 Error reconstruction upstream of the coronagraph with respect to the WFE of the aberration upstream of the coronagraph.
In the text

	Fig. 7 Adaptive optics testbed schematic representation. Mi: fold mirrors; MPi: parabolic mirrors; Li: lenses (doublets); BS: beam splitter; TTM: Tip-Tilt mirror; DM: deformable mirror; RRPM: coronagraphic focal plane mask; Φ: prolate apodizer; WFS: AO wave-front sensor
In the text

	Fig. 8 Introduction of calibrated aberration on BOA: case of a pure spherical aberration. Left: theoretical wave-front (top) and DM introduced wave-front (bottom). Right: corresponding Zernike modes for the theoretical introduced aberration (solid red line) and the DM introduced aberration (dashed blue line).
In the text

	Fig. 9 Estimation of a tilt aberration on BOA: calibration (solid blue line) and COFFEE’s estimation with bound on the tip-tilt downstream of the coronagraph (dashed crossed red line) and without boundaries (dashed diamond green line).
In the text

Fig. 10 COFFEE: NCPA estimation of an introduced phase φcal on BOA. Top: for an aberration + φcal, recorded coronagraphic image from the bench (left) and computed image using the reconstructed aberration $\hat{{\phi }}\begin{array}{c}\mathrm{+}\\ \mathrm{u}\end{array}$ (right) (log. scale, same range for both images). Middle: same images for an aberration − φcal introduced and a reconstructed aberration $\hat{{\phi }}\begin{array}{c}\mathrm{-}\\ \mathrm{u}\end{array}$ (log. scale, same range for both images). Bottom: calibrated introduced aberration (left) and COFFEE estimated aberration (right).

In the text

	Fig. 11 PCL on the bench BOA (gPCL = 0.5): variance of the residual static aberrations upstream of the coronagraph for the 36 COFFEE estimated Zernike modes (solid red line) and the 15 corrected modes (solid blue line). The magenta dashed line represents the ultimate performance one can reach according to the error budget detailed in 4.3.
In the text

	Fig. B.1 Coronagraphic diversity images computed for an aberration φu + φdiv upstream, φd downstream of (left) and the only diversity aberration φdiv (right). The shape of both images is mainly driven by diversity aberration.
In the text