Issue 
A&A
Volume 545, September 2012



Article Number  A151  
Number of page(s)  32  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201219614  
Published online  24 September 2012 
Instantaneous phase retrieval with the vector vortex coronagraph
Theoretical and optical implementation
^{1} 60 rue des bergers, 75015 Paris, France
email: riaud.pierre@gmail.com
^{2} Université de Liège, 17 Allée du 6 Août, 4000 Sart Tilman, Belgium
^{3} European Southern Observatory, Alonso de Cordóva 3107, Vitacura, Santiago, Chile
email: dmawet@eso.org
^{4} NASAJet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
^{5} TechspaceAero, route de Liers 121, 4041 Milmort, Belgium
email: arnaud.magette@techspaceaero.be
Received: 16 May 2012
Accepted: 19 July 2012
Coronagraphs are used as highcontrast imaging tools. However, it is wellknown that the achievable contrast is primarily limited by wavefront aberrations in the optical train. Various kinds of devices and methods have been proposed to correct and calibrate these errors and, hence, improve the efficiency of coronagraphs. Here, we present an innovative idea that allows instantaneous measuring of the phase and the amplitude of residual stellar speckles in coronagraphic images. The technique is based on the unique polarization properties of the vector vortex coronagraph, which serves as a new type of phase diversity (POAM diversity), as well an extension of the NijboerZernike theory of aberrations. We also propose and discuss a simple practical optical implementation of the technique, which only requires polarization splitting at the backend of any existing vector vortex coronagraph systems.
Key words: instrumentation: high angular resolution / instrumentation: adaptive optics / methods: numerical / techniques: polarimetric
© ESO, 2012
1. Introduction
The direct detection of exoplanets is limited by both the contrast and the angular separation between the planet and its host star. For example, in the visible, Earthlike exoplanets would be 10^{10} times fainter than their host stars and located within a fraction of an arcsecond even for nearby systems. Many coronagraphs have been proposed to suppress the starlight diffracted light, but all of them are limited by the imperfections of the entrance wavefront: residual speckle patterns are the dominant source of noise in the highcontrast imaging. Even when using highorder or extreme adaptive optics (XAO), the performance of coronagraphs is still limited by the phase and the amplitude knowledge used for the darkhole generation, and maintenance while acquiring science data (Bordé & Traub 2006; Give’on et al. 2007). We propose to measure the speckle phase and amplitude simultaneously with the science integration using the polarization properties of the vectorial vortex coronagraph (VVC), Mawet et al. (2005), Mawet et al. (2009). For that, we split the output pupil field into its two orthogonal circular polarization components with a simple circular polarization splitter (see Sect. 6). In this paper, we analytically demonstrate, using the polar NijboerZernike diffraction theory of light (Magette 2010), that the polarized VVC coronagraphic images present sufficient diversity to retrieve the phase and the amplitude information of the wavefront at the telescope entrance pupil. The main advantage of the proposed optical implementation and phase retrieval scheme is its simplicity and quasiinstantaneous nature. It allows minimizing the noncommon path wavefront errors in the optical system. It also presents a high transmission coefficient, higher than 90%.
The retrieved wavefront complex amplitude of the telescope pupil can be used directly as a synthetic reference image with image subtraction or as a dynamic speckles calibration system using adaptive optics corrections (amplitude and phase).
The paper is organized as follows: first, we briefly present the NijboerZernike diffraction theory for low and high levels of aberrations in Sect. 2. The extension of the NZ theory for the vortex coronagraph is presented in Sect. 3, followed by a detailed presentation of the phase and amplitude retrieval procedure in Sect. 4. The overall system architecture adopted for our speckles calibration system is presented in Sect. 6. Finally, the phase retrieval accuracy under “endtoend” numerical simulations is presented in Sect. 7.
2. The NijboerZernike theory
The NijboerZernike (NZ) theory emerged from the work of Nijboer (1943) on the diffraction theory of aberrations in polar coordinates. Nijboer first introduced the relation between the diffraction equation and the optical aberrations expressed in the form of Zernike polynomials. However, the complexity of the equations forced him to limit his theory to small aberrations, typically smaller than one wave (Nijboer 1947). Janssen (2002) later completed Nijboer’s theory by using an explicit Bessel series representation for the diffraction integral (highorder Hankel transforms). He also proposed a convenient way to numerically compute the expressions that involve these Bessel series (the functions, discussed below). This extended NZ theory allows one to quickly and analytically compute the intensity point spread function (PSF) of any complex system with circular pupil from its known aberrations.
2.1. Fraunhofer diffraction integral of an aberrated pupil in polar coordinates
The expression of the complex amplitude in the image plane U(r,φ) as a function of the pupil complex amplitude P(ρ,θ) can be calculated by the wellknown Fraunhofer diffraction integral, expressed here in polar coordinates: (1)The integration limits are defined by the pupil function 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π. The pupil function itself can be expressed in terms of Zernike polynomials using classical Zernike coefficients α, or the generalized coefficients β, defined as follows (Magette 2010): (2)The Zernike polynomials are defined as usual: (3)where is a radial polynomial defined as (4)The radial polynomials are even or odd in ρ depending on the n,m values. Note that the polynomials corresponding to m = 0 are treated as even polynomials since cos(0) = 1. Wavefront surface development with Zernike polynomials, and the full expression of the C_{z}(n,m,s) coefficients is detailed in Appendix A.
2.2. General Zernike coefficients
Under the small aberrations assumption, the simplified expression of the pupil aberrations is generally sufficient to describe the entrance wavefront, and simply connects the and coefficients by identification of the terms in Eqs. (2). The and coefficients capture both phase and amplitude aberrations.
The use of general Zernike coefficients ^{1} is preferable since they corresponds to the proper aberration basis of the NZ theory. However, the physical interpretation of the coefficients is not as easy as for the usual Zernike coefficients. Here, for the sake of completeness and accuracy, we chose to use the generalized Zernike coefficients . Subsequently, the Fraunhofer diffraction integral given in the Eq. (1) can be seen as a linear system: (5) is the image plane complex amplitude corresponding to the Zernike polynomial (n,m).
For the sake of simplicity, we split the coefficients into two categories: for even Zernike polynomials (cos), and for odd Zernike polynomials (sin), (6)Detailed analytical expressions of the complex amplitudes are given in Appendix C.
3. Extension of the NZ theory for the vector vortex coronagraph
3.1. The vector vortex voronagraph
The VVC is a transparent focal plane phasemask that creates two opposite phase screw dislocations Exp, with l_{p} the topological charge or the photon orbital angular momentum (POAM, Poynting 1909; Yao & Padgett 2011), and φ the azimuthal coordinate. When the phase singularity is centered on the PSF, it redirects the starlight outside the pupil where it can be blocked by a Lyot stop (Mawet et al. 2005).
3.2. Field expression at the coronagraph
The effect of a VVC with a topological charge l_{p} on the complex amplitude of the PSF can be written as (7)where l_{p} = 2 and l_{p} = −2 correspond to the righthanded and lefthanded POAM in circular polarization vector basis, respectively.
Using the expression for U(r,φ) previously defined, the complex amplitude in the coronagraph image plane, U_{v}(r,φ,l_{p}) becomes: (8)The C_{m}(φ) and S_{m}(φ) functions are defined as r,φ are the polar coordinates in the the coronagraphic image plane, and l_{p}, Sign(l_{p}) are the VVC topological charge and the chirality of the modulation, respectively.
The functions are defined for cosine and sinus modes as follows (see Appendices D and E): (12)The impact of the on the intensity distribution is to azimuthally modulate the residual aberrations. Note that the are normalized in intensity (see Appendix F). Let us now extract the dominant term (n = N, m = 0) from the sum and rewrite Eq. (7) as (13)where the ~ symbol means that the term (n = N, m = 0) is absent from the sum. is larger than the other . This separation allows us to virtually create a dominant linear term with respect to the optical aberrations. In other words, as we will see in the following section, the extraction from the sum yields the coupled modal functions , present in the linear term of the final intensity expression, which have the key property .
3.3. Expression of the coronagraphic intensity
The action of the VVC is to redirect the field amplitude outside of the relayed pupil, conjugate to the entrance pupil. The coronagraphic suppression of starlight is then obtained by inserting a diaphragm smaller than the pupil diameter into this pupil plane, called the “Lyot stop”. The coronagraphic intensity in the camera plane downstream from the Lyot stop plane is (see Appendix M) (14)It is composed of three different terms:


: a linear function of inner products between the coefficients and the term.

: a term quadratic in the coefficients and cos/sin cross terms.
Note that in practice, the Lyot stop is always slightly undersized compared to the pupil, which in the present NZ theory simply yields a normalization of the radial Zernike polynomials with the diaphragm size d_{c} < 1. The general coefficients then become a function of (d_{c}): (see Sect. 5.2).
4. NZ phase retrieval theory
The NZ phase retrieval method is based on the projection of the measured PSF on the basis of template modes, which leads to a system of decoupled linear equations. We first assume linearity: (15)However, note that the quadratic term will be accounted for later on by a recursive corrector approach (see Sect. 4.3).
In practice, its implementation is a twostep process. On one hand, it requires projecting the measured PSF on the basis of radial template modes by means of a polar Fourier transform: (16)where I_{meas}(r,φ,l_{p}) is the measured coronagraphic image, which depends on l_{p}, the topological charge.
On the other hand, the same projection is performed analytically on the final intensity expression Eq. (14): (17)where I_{c}(r,φ,l_{p}) is the analytical expression of the final coronagraphic image for the topological charge l_{p}.
Comparing both the measured PSF decomposition and the analytically modeled one leads to the formation of decoupled systems of linear equations.
4.1. Modal analysis of the analytical intensity equation
The analytical expression of the linear term f^{(1)} of the coronagraphic intensity is (18)ℜ and ℑ are the real and imaginary parts, respectively. The complete demonstration is detailed in Appendix M.
To facilitate the analytical computation of Eq. (17) using Eq. (18), we introduce the following functions: (21)The Ψ and χ functions correspond to the phase and to the amplitude aberration templates, respectively. The full analytical computation presented in Appendix N demonstrates that the measured intensity in the polar Fourier plane is a linear combination of the phase and amplitude aberration templates and .
We now multiply this result with and , which produces the “inner” products of the aberration templates, physically corresponding to the autocorrelation between the phase and amplitude aberration templates: (22)It can easily be demonstrated that these inner products have the following properties: (23)Using the inner products, it is now possible to build a linear system of equations, (24)where G is a Gram matrix, and is defined with all possible inner products coefficients. is the vector containing the unknown coefficients , and r is a vector with the polar Fourier transform image analysis coefficients. Note that after the proper integration of the inner products the G matrix includes all three Pancharatnam topological charges − l_{p},0, + l_{p}.
Fig. 1 Schematic view of the NZ vortex phaseretrieval process. It consists of using the real input coronagraphic images, corresponding to l_{p} = 0, ± 2 (the l_{p} = ± 2 images are provided by polarization splitting, while the unpolarized image (l_{p} = 0) is directly given by the sum of the two polarized images) projected on aberrations templates, and finding the unknown coefficients by resolving a system of linear equations. Left: the three images needed for the full aberration analysis. Center: linear systems to retrieve the and the f^{(1)} terms. Right: the predictorcorrector approach to evaluate the quadratic correction . 
The r vector must be defined for three different cases: radial modes (m = 0), purely cosine modes, and purely sine modes. For radial modes, we use the cosine description. The end of Appendix N details the analytical expressions for all coefficients used in the matrix G and the two vectors u and r.
4.2. The f^{(1)} term retrieval: solving a linear system of equations
Introducting the inner product (see Eq. (21)) allows us to calculate the linear term of the intensity equations (see Eq. (18)). The vectorial vortex complex amplitude retrieval is sensitive to the amplitude and the phase effects in the image. In practice, and to accelerate the convergence of the algorithm, we must bind the real part of the parameters (amplitude effect) to the most probable range of values. Indeed, amplitude and phase effects in the final intensity images are completely indistinguishable and potentially, many solutions exist. However, if we restrict () to a physical solution, which corresponds to small amplitude errors, the algorithm converges quickly.
4.2.1. The (d_{c}) coefficient
The first term (piston) is equal to zero in the ideal case (Mawet et al. 2005). For a singledish telescope, the piston phase is a gauge invariant and we chose 0 for both the imaginary parts of and , , the defocus term, is generally small, and can be easily minimized in practice. The first dominant purely radial mode is thus the spherical aberration (). The first set of unknown coefficients to retrieve, , corresponds to the main aberration. When the images are projected onto the radial mode template basis , all coefficients will be compared to this term. As we developed in Eqs. (8), the C_{m}(φ) and S_{m}(φ) functions show a great variability for different m modes. Each case must then be processed separately (see the NZ retrieval diagram in Fig. 1).
4.2.2. Resolution of the system for m = 0
If m = 0, the vector of unknown coefficients u can be written as (25)where n_{max} is the maximum number of modes. The first system (u_{1}) allows determining the coefficient. Previously, we fixed the piston term but in the practical reallife case (Fresnel diffraction and finite coronagraphic mask), this coefficient is not zero. This piston term corresponds to the mask limitation in the considered optical system (chromaticity, Fnumber, small Lyot dot in the center, Lyot stop in the pupil plane, etc.) and must be calibrated on the optical system before astronomical use. In that case, the complex value of the coefficient can be fixed in the diffraction model and the coefficient required for the retrieval is known thanks to the resolution of the first equation with , and .
4.2.3. Resolution of the system for 0 < m < n
If 0 < m < n, the vector of unknown coefficients u given by Eq. (24) can be written as (26)
4.2.4. Resolution of the system for m = n
In the case where m = n, the vector of unknown coefficients u has to be calculated separately, and the vector can be written as: (27)The complete demonstrations for the coefficients are detailed in Appendix N.
4.3. The quadratic correction f^{(2)}: the predictorcorrector approach
Keeping only the linear term of the final coronagraphic intensity allows us to simplify the retrieval process using linear algebra only. This approximation allows determining the coefficients with a precision of a few percents. However, in a coronagraphic system, coupling between phase and amplitude appears due to the crossed terms in “sin / cos”. These terms are also quadratic and are taken into account in the predictorcorrector analysis.
In this section, we introduce an improvement that allows us to account for the quadratic terms. When a coronagraphic device is used, is attenuated and the purely quadratic phase terms and the amplitudephase coupled terms are of the same order of magnitude. They should thus be taken into account. Hereafter, we present a simple way of doing this using a predictorcorrector approach. This technique proceeds as follows: (28)The retrieval approach presented previously is based on the following simplification: I_{c} ≈ f^{(1)}.
Solving this equation leads to a first approximation of , which corresponds to an image such that .
is then defined based on approximating the exact quadratic term.
After several iterations, tends to if, and only if, f^{(2)} < f^{(1)}.
The predictorcorrector approach as in the classical NZ phase retrieval (Magette 2010) leads to a better estimation of the coefficients.
4.4. Phase amplification
An optical vortex phasemask “amplifies” the small input pupil phase error () on the output coronagraphic pupil. As already suggested in Sect. 4, the first term . This is why in the coronagraphic image plane all wavefront errors in the entrance pupil plane are enhanced by the phasemask coronagraphic device. Indeed, in the proposed phaseretrieval analysis, we compare all Zernike polynomials with the first nonzero term in the linear system equations to avoid numerical singularities. A good estimation of the term remains mandatory to allow a correct wavefront retrieval.
4.5. The coefficient
The coefficient remains a free variable, and it is naturally obtained with the value of the nulling factor.
Indeed, the nulling coefficient ϵ_{c} can be defined as the sum of all residual aberrations in the coronagraphic pupil plane, (29)We calibrate the nulling factor with an instrumental PSF of the optical system without any coronagraphic device (the l_{p} = 0 term). In our NZ development, the sum of the two circular polarization images can be used as l_{p} = 0 image. This composite image can be seen as the instrumental PSF from which the perfect Airy pattern J_{1}(r)/r^{2} has been subtracted.
4.6. Optimum number of images for phase retrieval
In the classical phase diversity algorithm (Gonsalves 1982; Blanc et al. 2003), the estimation of the entrance aberrations from the sole focused image does not ensure the uniqueness of the solution. Indeed, two different aberrations can produce the same PSF in the image plane. At least, two images with known phase variations are needed to remove this indetermination. If the phase diversity is performed in the pupil plane as in Roddier & Roddier (1993), the result is the same: two images of defocused pupil are needed. The indetermination is fully removed if the two images are π phaseshifted. This is the case of phase retrieval using classical NZ theory (with focus variation ± f), and in this modified NZ theory for the VVC device (with POAM modulation ± l_{p}).
In coronagraphy, another indetermination appears: the residual speckle pattern changes in intensity with the entrance Strehl ratio due to the induced variation of the coronagraph rejection factor, but not in overall morphology. Therefore, the absolute values of need to be retrieved using a good estimation of , (see the previous section). For that, a third image, such as a simple PSF is the key to ensure a true phase retrieval. Moreover, for space telescopes, the PSF allows taking “telescope breathing” (equivalent to Strehl variations) effects into account.
5. Phase retrieval in the presence of practical limitations
In addition to phase and amplitude wavefront errors, starlight suppression is also limited by intrinsic properties of the optical system and features of the coronagraphic device: for instance, the telescope central obscuration (see Appendix H.2) and support structures (see Appendix L). In the following, we review the impact of these characteristics on the phase retrieval process.
5.1. Annular pupil
For an onaxis telescope with a central obscuration ϵ (ϵ < 1), the full aberration retrieval can be obtained directly by replacing the C_{z}(n,m,s) Zernike coefficients (see Eq. (4)) with the C_{z}(n,m,s,ϵ) using the normalization described in Mahajan (1981a,b), (30)Even though the complex amplitude pupil retrieval process presented in this paper is fully applicable (see Appendix B) to onaxis telescopes, for the sake of simplicity, the ϵ coefficient is omitted.
5.2. Impact of the Lyot stop diameter
The main effect of the diaphragm d_{c} (Lyot stop) in the coronagraphic pupil plane (Lyot plane) is to reduce the coherent term of the diffraction by a factor ϵ_{c}. The second effect of the Lyot stop is to rescale the Zernike radial polynomial as (31)We derived a recurrence formula to calculate the in Appendix I, yielding, with n = m,m + 2,...: where and . The coefficients are simply normalized to the size of the diaphragm in the Lyot plane.
5.3. Lyot stop optimization
The Lyot plane field expression is presented in Appendix D under the assumption of Fraunhofer diffraction, and in Appendix H under the Fresnel propagation assumption, respectively. The Fresnel number is generally larger than 100, where d_{c} is the radius of the Lyot stop and z is the propagation distance. The Fresnel case thus only shows a slight blurring effect of the pseudoZernikes polynomials given in the Fraunhofer approximation. Therefore, the Fraunhofer diffraction is sufficient to describe coronagraphic aberration residuals in the pupil plane. As far as the pupil edge effect is concerned, the Lyot stop has to cover enough residual rings inside the pupil plane to ensure a proper nulling ratio (see Appendix K).
5.4. The imperfect vortex device
The VVC manufacturing imperfections lead to the following error terms:

The phaseshift error with respect to π (chromaticity).

The region in the center of the mask may present a deviation from the overall largescale pattern called the region of disorientation (this defect largely depends on the technological approach chosen to manufacture the vortex device).

The finite size of the mask in the coronagraphic image plane.
These terms can be taken into account in the coronagraphic NZ retrieval by a simple Fourier simulation to estimate the term (see Appendices J and L for more details).
6. Practical implementation
In this section, we present the instrumental concept we propose to instantaneously acquire the phaseamplitude information about the residual wavefront errors. We propose to use a simple Wollaston device coupled to a pair of achromatic quarterwave plates to separate the coronagraphic images into left and right circular polarizations, allowing for a simple optical implementation that minimizes noncommon path errors. Figure 2 illustrates the optical implementation downstream of the Lyot stop.
Fig. 2 Polarization splitter analysis: possible implementation of the polarization splitter system used after the Lyot stop. An achromatic quarterwave plate (quartz/MgF_{2}) and a Wollaston in MgF_{2} combined here with a simple achromatic doublet. This simple optical implementation allows us to image the two residual coronagraphic images in the left and right circular polarization basis on the detector. The two images presented in this figure in logarithmic gray scale show the two polychromatic PSF (650–900 nm) obtained by this simple scheme for the two circular polarization images. Note that the PSF shows a small chromatic smearing residual on the xaxis (Strehl = 96.5%) due to the substantial wavelength bandpass. This polychromatic smearing is detailed in Appendix P. 
6.1. Tolerancing of the optical design
To show the validity of our approach, we considered potential sources of disturbances one by one. First of all, the entrance pupil introduces its own timeevolving phase defects due to polishing errors and timedependent thermal effects. The latter affect loworder Zernikes and will be simulated with a power spectral density (PSD) of f^{2}. The coronagraphic device possesses its own limitation (manufacturing defect and chromatism, see Sect. 5.4), which need to be taken into account as well.
Thick optical devices such as quarterwave plates, Wollaston and the lens/mirrors are the sources of four types of secondorder aberrations:

Polishing errors: represented by a PSD with wavefront error (WFE) < λ/50 rms for low spatial frequencies, and < 0.2 nm rms in the roughness scale (see optical specifications in Riaud et al. 2003).

Noncommon path aberration: due to the small beam separation angle, it is expected that only roughness will play a significant role; we chose to represent it with two phase screens of 0.2 nm rms each.

Light scattering in thick materials, ghost features: scattering limitation and ghost features of the proposed calibration analysis system are calculated directly with the scattering function (see the Appendix O) and the Zemax software.

Residual polarization ellipticity due to nonideal retardation over the entire wavelength range: the quarterwave plates were optimized Riaud (2003), and we obtained a plate thickness of 0.24104 mm and 0.30792 mm at 20°C for the MgF_{2} and the quartz material, respectively. These thicknesses must be controlled to 1 μm with a Babinet compensator and the plate temperature must be controlled to ±0.5°C. The residual phase error becomes σ^{2} = 0.021 radian for the entire spectral bandpass.
The numerical simulations show that the scattering in the glass material is about 10^{6}−10^{7} of the entrance residual coronagraphic starlight, which is not a problem even for Earthlike planet detection provided that the first coronagraphic stage is efficient enough to remove most of the starlight (ϵ_{c} < 10^{4}).
7. Endtoend modeling
This section presents thorough numerical simulations. The goal of these simulations is to show that the entrance pupil wavefront phase can be retrieved even in the presence of reallife optical imperfections, and to which level of accuracy it can be retrieved. Usually, we model a coronagraphic instrument using FFTbased optical propagation; three FFT are needed to fully simulate a coronagraphic image. Here, we used our NZ modal decomposition of the entrance aberrations seen through the VVC to directly construct the final image. Before going into the details of the endtoend modeling, we tested our analytical method vs. a FFTbased propagation prescription to show their equivalence.
7.1. Analytical vs. Fourier propagation
To study the accuracy of the modal decomposition with the analytical functions, we performed numerical simulations with FFT and the direct expression of modal functions. The entrance pupil is a perfect unobscured circular pupil affected by wavefront (polishing) errors described by a set of Zernikes up to n = 860 with a weighting of 1/(n + 1)^{2}, leading to an initial Strehl ratio of 95%. Note that the FFTs must be performed with large arrays (2K × 2K) or (4K × 4K) to minimize aliasing effects. Supersampling is also used in the coronagraphic plane. All simulations are preformed for the following two cases: the β_{n,m} representation in the pupil plane () and the α_{n,m} classical representation ( ). The NijboerZernike uses the first pupil representation and is not limited to high Strehl ratio (see Appendix A for a full analysis). Figure 3 compares the analytical NZ functions under the two pupil representations. Note that FFT simulations always show a highfrequency residual noise. Moreover, the FFT propagation through the VVC acts as a highpass filter, artificially minimizing the residual coronagraphic stellar flux in the final image near the center. The NZ analytical functions do not present these problems: they are very fast and more accurate. For a complete comparison between the semianalytical and pure numerical method, see Appendix G.
7.2. Summary of simulation parameters
We present realistic polychromatic simulations incorporating the various defects detailed above.

Using the first 860 Zernikes (40 complete modes), the input Strehl ratio is set to 95% @ 650 nm.

An imperfect phasemask: where the stransmittance is 97%, the ptransmittance is 98% and the local phase retardance is π ± Δφ, while following a quadratic law (see Eq. (P.1) in Appendix P.2).

The quarterwave plates have an absolute phase dependence equal to (λ in μm): φq_{λ} = 0.3442 + 2.94887λ − 1.74126λ^{2}.

Common and noncommon path errors are λ/71 rms @ 650 nm both (the total is λ/50 rms at 650 nm).

Polishing error: DSP f^{2} and noncommon path error.

Polychromatic speckle smearing due to residual Wollaston chromatism given by the Zemax model.

Photon noise, readout noise (6 e^{−}), fullwell capacity of 10^{5}e^{−}, residual flat of 1% rms.
These inputs are commented on in Appendix P, where we also present the full sets of images.
Fig. 3 Numerical simulation illustrating the principle of NijboerZernike retrieval applied on the vortex coronagraph. (Up to down), numerical simulation for each circular polarization with l_{p} = 2 and l_{p} = −2 respectively using the first 860 Zernike polynomials (40 complete modes) and a Lyot stop of 99%. The Strehl ratio of the PSF before coronagraphic filtering is 95%. The Lyot stop remove the strong diffraction value in ρ = 1 but allows us to show all images without scaling the coefficients (). (Left to right), final monochromatic (λ = 650 nm) coronagraphic image obtained with the sum of analytical functions, the FFT of the direct sum of Zernike polynomials in the pupil plane, and finally, the classical phase function obtained by . Due to the difference of basis, the final simulation with the classical phase function presents minor discrepancies in the speckle background (I_{c} < 10^{7}, I_{c} is the coronagraphic final intensity). The image scales are not linear . 
7.3. The accuracy functions
To quantify the quality of the proposed modal decomposition based on the NZ theory, we define the following function: (32)This metric is applicable for all cases (monochromatic and polychromatic). But for a proper knowledge of the modal decomposition quality under adaptive optics correction (Krist et al. 2011), we can define the Err(σ_{A}) function (33)This function is applicable if we know the amplitude function of the coronagraphic residual. Indeed, this metric is only applicable in the monochromatic case. Numerical simulations for all Zernike modes for the monochromatic case are performed in Appendix G. This procedure allows us to estimate the robustness of our method in the two metrics. Table 1 presents the numerical accuracy of the NZ theory in the two presented metrics.
7.3.1. Polychromatic numerical simulations
Phasemask coronagraphs are inherently chromatic, and so is the pure physical propagation process (see Fig. 4). The NZ phase retrieval must take that effect into account. A full set of simulated images is shown in Appendix P. We scanned all physical parameters (spectral bandpass and phaseshift error of the mask), including photon and detector noises below.
Fig. 4 Polychromatic coronagraphic simulation for l_{p} = 2 with Δλ = 250 nm of spectral bandpass and a phase error of Δφ = ± 0.001 radian for the vortex mask. We also include defects of our optical implementation. Left: without the speckle smearing. Right: with the Wollaston speckle smearing. The chromatic effect on the speckles is small in the two cases, the contrast remains high. The brightness scale is the same between the two images and is not linear. 
7.3.2. RESULTS without photon noise
Here we test the maximum likelihood ( minimization) of the VVC coronagraphic images with the sum of monochromatic modal functions (see Appendix M). The first set of simulations is given without photon and readout noises. This process allows us to determine the global NZ phase retrieval behavior with respect to the wavelength bandpass and the phaseshift error on the coronagraphic device. Owing to the larger bandpass, we tuned the sampling of the NZ images. Numerical simulations are presented in Fig. 5, which shows the value variation as a function of the wavelength bandpass and the maximum phaseshift error of the vortex phasemask (10^{1}/10^{2}/10^{3} radian).
Fig. 5 values as a function of the wavelength bandpass and a phasemask with a 10^{1}/10^{2}/10^{3} radian of phaseshift error. The black lines correspond to for the in: : retrieval, the gray lines are for the in: : retrieval: the solid lines are for the “p” modes and the dashed lines are for the “m” modes. The x and y scales are common between the three images. The values for of 10^{2} rad for the mask error are poorer than in the previous simulations. The 10^{1} rad case presents significant limitations because of a strong peak in the image center (β_{0,0} = 0.04). The dotted line with and shows the chromatic limit of a desired retrieval precision in the “AO metric”. 
We notice that simulations are presented for the two input basis and , but the second pupil phase decomposition is better suited for the NZ analysis. These simulations also show that the bandpass sets a limit of Err(σ_{A}) = 10^{10} on the precision of the input electric field. The main result for our 95% of input Strehl ratio is that the NZ decomposition in the basis must be narrow band (Δλ = 65 nm or R = 10) to obtain the desired precision on the wavefront error. Now, with the photon limited image sets, we are only interested in the in: : modal decomposition.
7.3.3. RESULTS on photonnoise limited images
The previous simulations shows that the monochromatic modal decomposition is good with a small for a Δλ < 100 − 150 nm. In this section, we present noisy simulations using the same monochromatic NZ modal functions to compare results with the previous ones. Figure 6 shows NZ retrieval results for coronagraphic images realistically limited by simulated detector (readout,flat) and photon noises. A satisfying wavefront retrieval can be obtained after averaging at least 100 exposures. The gain for more exposures is only on the high order Zernike modes (n > 20).
To increase the precision of the retrieval with a broad bandwidth, we developed in Appendix E a polychromatic set of functions. These functions take into account the contrast loss in the Bessel J rings due to the broad bandwidth. This effect would increase the robustness of the NZ retrieval. This new feature is added in the NZ algorithm.
Fig. 6 values as a function of the signaltonoise ratio in the input coronagraphic images for a in: : retrieval. We present the effect of the modal decomposition quality with respect to the phaseshift error of the mask. Several realistic cases are presented with 1/10/100/1000 exposures in the final image. A good wavefront retrieval is given after an average image of at least 100 exposures. 
8. Discussion
Thanks to the phase retrieval technique, we have shown in the previous sections that using POAM on the starlight given by the VVC, it is possible to instantaneously measure the phase of residual stellar speckles in coronagraphic images and hence improve the sensitivity of highcontrast telescopes. The main limitations are essentially coming from the limited coronagraphic nulling due to the input Strehl ratio, and the mask chromaticity over broad bandwidths. To investigate the impact of these instrumental limitations, we considered various input Strehl ratios in our simulations and estimated the coronagraphic nulling ratio and the maximum wavelength bandpass that are necessary to provide an accuracy as good as Err(σ_{A}) = 10^{10}. The results of this analysis are shown in Table 2. They were computed for an optimized highcontrast optical workbench for a best VVC (Δφ = 10^{3} radian).
Maximum coronagraphic nulling ratio and wavelength bandpass achievable for various levels of input Strehl ratio.
Table 2 shows that our phase retrieval technique can still be used for a Strehl ratio as low as 95% with a mask chromaticity of 10^{3} < Δφ < 10^{2} radian provided that it is applied on a sufficiently narrow wavelength bandpass (65 nm). In practice, this means that the technique becomes more timeconsuming because it must be applied on more spectral channels for decreasing Strehl ratios. Owing to the mask imperfection, a maximum nulling of 10^{5} on the stellar peak seems to be feasible for an input Strehl ratio of 99.6%. The proposed method can handle various instrumental limitations and is sufficiently flexible to be coupled with the EFC algorithm (Give’on et al. 2007) to minimize the electric field in a desired region of the focal plane.
To ensure optimized wavefront corrections in the entrance pupil if we use an EFC algorithm, we need to use the complete Zernike development in the basis. We can null all present residual speckles in the the final coronagraphic image, not an individual speckle or identifiable feature only. Indeed, our process directly gives the input pupil phase and amplitude by image analysis but this is not an EFC algorithm. On the other hand, we must modify the EFC algorithm with our formalism to increase speedup and obtain a direct global electric field minimization problem on the entrance pupil plane. A previous paper (Riaud 2012) on a new deformable mirror architecture provides all mathematical tools to use this NZ theory in the speckle cancellation process. Finally, we notice that if the corrections are not exactly in the pupil plane, some Fresnel propagations occur and small corrections of the Fraunhofer diffraction are needed (see Appendix H for a complete mathematical demonstration).
9. Conclusion
We demonstrated a full analytical modal decomposition of the effect aberrations on the vectorial vortex phasemask final image using polarization properties. This procedure can be used on a very stable coronagraphic system for detecting terrestrial planets in the visible around nearby stars such as TPFC (and precursors). Indeed, this proposed simple optical implementation allows us to calibrate the residual speckle pattern directly on the final coronagraphic images. Endtoend simulations show that the common and noncommon path errors (λ/50 rms at 650 nm) after the filtered pupil due to the beam separation system are not influencing the final images. Indeed, the level of precision can be as good as ≈0.1% for the retrieved phase and is only limited by the detector and the photon noises. We also presented the effect of the polychromatic coronagraphic images on the maximum likelihood between the Fourier and the modal simulations. A precise ( ≈ 1%) modal decomposition with a wavelength bandpass of Δλ < 140 nm can be obtained with the proposed optical implementation. Finally, the main source of error and performance limitation of the presented modal decomposition is the signaltonoise ratio of the two coronagraphic images and the direct PSF image (l_{p} = 0). A very stable optical system is needed to stack several images ( ≈ 100 see Sect. 7.3.3) and reach the 10^{10} speckle level that opens the way to detect terrestrial planets in the visible with a space telescope.
An option is the direct phase correction with extreme adaptive optics (XAO) to increase the rejection factor, the main limitation will again be the phase retrieval precision.
Details on the method for converting to for large aberrations (Strehl ratio > 50%) are given in Appendix A.
Acknowledgments
This work received the support of the University of Liège. This work was partly carried out at the European Southern Observatory (ESO) site of Vitacura (Santiago, Chile). The authors are grateful to C. Hanot (IAGL), D. Defrère (MPIRA) for manuscript corrections. The authors wish to thank the referee Wesley Traub for his useful comments and corrections. The authors also acknowledge support from the Communauté française de Belgique – Actions de recherche concertées – Académie universitaire WallonieEurope. This idea dates back to 20052006 and the first author is grateful to section 17 and the CNAP French commissions for their outstanding recruitment work.
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Appendix A: Expression of the pupil aberration function
A.1. Zernike decomposition of the wavefront
The general pupil function is where are the classical Zernike polynomials. When the aberration level is low (Strehl ratio > 90%), we can approximate this function by (A.1)The coefficients are then directly related to the classical coefficients. Since the Zernike polynomials form an orthogonal basis, the classical pupil function with can always be fully described by a sum of Zernike functions, (A.2)Note that the Strehl ratio Sr can be calculated directly with the coefficients: (A.3)By decomposing the real and imaginary parts of the pupil into Zernike polynomials, it is thus possible to obtain the real and imaginary parts of the β coefficients. If the coefficients represent the weights of the aberrations of the surface, the orthogonality relation yields (A.4)for the real and imaginary parts of Zernike polynomials.
High spatial frequency variations are difficult to represent with a limited sum of Zernike polynomials. More β coefficients than the number of α coefficients are then required to compute aberrations in the pupil. For coronagraphic imaging, where we use a diaphragm smaller than, or equal to, the pupil radius, the strong variations at the edge of the pupil are masked and the number of β and α coefficients are somewhat equal. The phase function φ_{Π}(ρ,θ) of the pupil in the β basis can be determined by (A.5)Because the arctangent function is defined between ± π, the phase function needs to be unwrapped before it is decomposed, if the level of aberrations is relatively high (Strehl < 80%).
A.2. Radial Zernike functions calculation
(A.6)For a proper mathematical stability, the C_{z}(n,m,s) coefficients of the radial Zernike functions must be calculated recursively. The following radial Zernike functions can be calculated easily: (A.7)The loworder radial Zernike functions to can be calculated with the following recurrence formula: (A.8)
Appendix B: Expression of the Hankel transform H_{m}
B.1. The classical Zernike radial functions
To calculate the amplitude in the image plane in the presence of optical aberrations given by the Zernike polynomials, we need the direct mth order Hankel transform H_{m} for the radial coordinate. We have (B.1)The radial integration between the ρ^{n − 2s + 1} polynomial and the Bessel function J_{m} can be calculated in the following way: (B.2)where is the Gauss hypergeometric function. The full expression of the f_{z}(n,m,s) function becomes (B.3)The Gauss hypergeometric function can be expressed in terms of Bessel J functions: (B.4)After a simple variable change, the f_{z}(n,m,s) function becomes (B.5)where P(m) and Q(m) are two polynomials (see Table B.1). In fact, the summation of two f_{z} functions (f_{z}(n,m,x) + f_{z}(n,m,x + 1)) allows us to simplify the general expression of the mth order Hankel transform. We use the following Bessel J transformation: (B.6)After Bessel J functions simplification, we obtain (B.7)
Values of some P(m)/Q(m) polynomials.
B.2. The annular Zernike radial function
For a telescope with a central obscuration ϵ (ϵ < 1), the aberration function can be obtained directly by replacement of the C_{z}(n,m,s) Zernike coefficients (see Eq. (4)) by the C_{z}(n,m,s,ϵ) using the normalization described in Mahajan (1981a,b). The expression of the f_{z}(n,m,s) must take into account the new normalization system (). The mth order Hankel transform becomes (B.8)During the Bessel J simplification process (see Eqs. (B.5), (B.6)), the presence of the terms does not allows us a complete Bessel J loworder elimination (J_{m + x + 2} < J_{n + 1}). This effect is very small and a simple renormalization by coefficients gives a good approximation of the mth order Hankel transform under annular Zernike decomposition, (B.9)
Appendix C: Expression of the U functions
To calculate the amplitude in the coronagraphic image plane in the presence of optical aberrations , we use the diffraction integral in polar coordinates. Depending on the function parity (cos or sin), the functions given by the Eq. (6) becomes In both cases, the integral of the term sin(mθ) between 0 and 2π vanishes. Moreover, using the definition of the Bessel functions, we have (C.3)Substituting this expression into the two previous equations while using properties of the Bessel functions (Gradshteyn & Ryzhik 1994), and the radial result for the direct mth order Hankel transform H_{m} (see Eq. (B.7)), we finally obtain (C.4)
Appendix D: Expression of the V function (Fraunhofer diffraction)
In the classical NijboerZernike theory without defocus the expression of the V_{n,m} is given directly by (D.1)With the VVC, the coronagraphic pupil expression is given by the inverse l_{p} + m order Hankel transform of the Bessel function included in the , (D.2)With l_{p} even (necessary conditions to obtain the VVC coronagraphic effect), we have Outside the coronagraphic pupil geometric area, it is necessary to change variables in the integrals to ensure a proper convergence: n → l_{p} ± m l_{p} ± m → n + 2 ρ → 1/ρ.
For n = 0,m = 0,l_{p} = 2: (D.8)These last two results correct typos in Eq. (C.7) in the appendix of (Mawet et al. 2005) for n = 0,m = 0.
Fig. D.1 Input classical Zernike . 
Fig. D.2 function in the coronagraphic pupil. 
Fig. D.3 function in the coronagraphic pupil. 
Fig. D.4 U_{v}(r,φ),l_{p} = ± 2 functions in the coronagraphic image plane for the 10 Zernike polynomials (Z_{2} − Z_{11}). Top: l_{p} = + 2, Bottom: l_{p} = −2. The vertical solid line separates the real part from the imaginary part of the two U_{v}(r,φ) functions. 
Note that the pupil integration on the ρ coordinate is applied between 0 and d_{c} because of the presence of the Lyot stop, (D.9)To retrieve the coronagraphic amplitude residual in the final imaging plane, we use the direct l_{p} + m order Hankel transform of the coronagraphic pupil plane, (D.10)We do not simplify with 2.cos(mθ) or 2i.sin(mθ) because when n = m the Zernike polynomial or , (D.11)

m = 0 − → C_{m}(φ) = 1 S_{m}(φ) = 0
0 < m < n − → C_{m}(φ) = cos(mφ) S_{m}(φ) = sin(mφ)
m = n − → C_{m}(φ) = ^{(}cos(mφ) − iSign(l_{p})sin(mφ)^{)}/2 S_{m}(φ) = ^{(}sin(mφ) + iSign(l_{p})cos(mφ)^{)}/2.
In the final imaging plane, a VVC device of order l_{p} adds a POAM l_{p} on the incoming residual light (Poynting 1909; Allen et al. 1999).
Appendix E: Polychromatic V expression
In the broadband case, we can use a polychromatic set of functions to enable a better complex amplitude aberration retrieval. The main effect is to broaden the monochromatic set of functions and to decrease the contrast of the Bessel J_{n + 1} rings of the coronagraphic images. These two effects can be implemented with a simple Gaussian convolution of the monochromatic functions, (E.1)Using the two following relations in the convolution (E.2)we obtain (E.3)Following Gradshteyn & Ryzhik (1994), p1094, the infinite integral can be calculated analytically, (E.4)where I_{k/2} is the Bessel I function. Note that the two Bessel J functions can be compacted as follows: (E.5)The polychromatic set of functions then becomes (E.6)The Bessel I function decreases quickly with respect to the k variable. Therefore, the discrete summation on the first 20 to 30 k coefficients is sufficient to retrieve the final function to an accuracy of 10^{12}. Note that the β_{n,m} coefficients retrieved in the polychromatic case correspond to the worst attenuation factor at the shortest wavelength.
Appendix F: Energy conservation
Each mode must be normalized by its intensity for a proper retrieval analysis, (F.1)(F.2)Similarly, we obtain in the polychromatic case (F.3)
Appendix G: Analytical vs. Fourier simulations
We performed numerical simulations to compare the robustness of the analytical functions in the monochromatic case to the classical FFT computations. These simulations were performed on 2K × 2K arrays using the “superresolution” method presented in Sect. 7.1. A χ^{2} and Err(σ_{A}) are calculated for each Zernike on a box centered on the coronagraphic image of 256 × 256 pixel square. We assume that each function is optimized in terms of sampling, (G.1)The χ^{2} is normalized to the number of pixels inside an annulus centered on the mode: typically, a χ^{2} < 1 is considered as good. The following Figs. G.1 and G.2 show structuring features as a function of the Zernike mode parity number m.
Fig. G.1 Comparison between Fourier simulations for each Zernike polynomial (2860) and mode in terms of χ^{2}. Left: the direct summation in the pupil (β_{n,m}), right: the classical phase function (e^{iαn,m}). 
Fig. G.2 Comparison between Fourier simulations for each Zernike polynomial (2860) and mode in terms of Err(σ_{A}). Left: the direct summation in the pupil (β_{n,m}), right: the classical phase function (e^{iαn,m}). 
Appendix H: Expression of the pupil function in Fresnel diffraction ()
H.1. Circular pupil
This appendix presents the calculation of the Fresnel diffraction of the coronagraphic pupil . Indeed, due to the uncertainty of the Lyot stop position, the diffracted field must be treated according to the Fresnel diffraction theory, (H.1)where k = 2π/λ, z is the Fresnel distance propagation, and is the Fresnel number.
Inside the pupil plane, we have Outside the pupil plane, we have Inside the pupil plane, we have two terms: the Zernike polynomials, and a pupil edge effect. Here, we make use of Babinet’s principle of complementary screens to express the second integral as a classical Fresnel diffraction between 0 to d_{c}, (H.6)The first integral can be calculated with all Zernike polynomials: (H.7)where is the confluent hypergeometric function.
The second integral only represents a diffraction effect to the J_{lp} order, (H.8)We define the Lommel functions of two variable generally used in the Fresnel diffraction of a circular screen (Born & Wolf 1999): (H.11)We include the diaphragm edge effect due to the Fresnel propagation only in the first term . The coronagraphic pupil with Fresnel propagation thus becomes
Fig. H.1 function in the coronagraphic pupil with an extreme case N = 4 to highlight differences on Zernike polynomials. In the real case N > 1000, and only the must be taken into account in the NZ theory. 
Fig. H.2 function in the coronagraphic pupil with an extreme case N = 4 to highlight significant differences on Zernike polynomials. 
H.2. Annular pupil
A telescope with a central obscuration ϵ must be treated with annular Zernike polynomials, but the central obscuration creates a strong diffraction phenomenon at the edge of the pupil (1 > ρ > ϵ). The effect of the optical aberration (n ≠ 0, m ≠ 0) gives the must be calculated taking into account the diffraction of the central obscuration. The function included in the can thus be written (H.14)

: The Fresnel diffraction of the outer edge of the pupil as calculated previously
: The Fresnel diffraction of the the central obscuration edge inside the pupil , ρ′ > ϵ
: Starlight diffraction on a perfect circular pupil.
Appendix I: Scaled radial Zernike polynomial
ming Dai (2006) gives the Zernike coefficients of the scaled pupil as a function of those of the unscaled pupil, (I.1)This expression is somewhat complicated. In this Appendix, we propose to calculate the scaled coefficients in a simpler way. The main idea is to transform with the coronagraphic diaphragm d_{c} < 1. We need to calculate a new orthogonality relation: (I.2)(I.3)Finally, the scaled can be calculated more easily than with the complicated formula in ming Dai (2006) and we obtain (I.4)Similarly, (I.5)In Eq. (I.3), we commute n and n′(I.6)
Appendix J: Vortex phasemask nulling limitation
This section presents the main mathematical properties of an imperfect vortex device owing to manufacturing defects. In real life, the optical vortex phasemask presents three problems:

The phaseshift error with respect to π (chromaticity).

The region in the center of the mask presents a deviation from the overall largescale pattern, called region of disorientation.

The finite size of the mask in the coronagraphic image plane.
J.1. The phaseshift error
For manufacturing defects, a VVC M_{V} can be modeled in the following way: where T_{s} and T_{p} are the polarization transmittance for s and p polarization, respectively (depolarization issue), Δφ is the local phase retardance deviation around π with respect to the wavelength. In these conditions, the radial mode templates are modified by the C_{M},γ_{M} terms. If C_{M} = 1,γ_{M} = 0 the output image intensity is completely described by the classical NZ theory (without coronagraph). In the general case, the complete coronagraphic intensity development becomes more complicated and shows several coupled terms such as Generally, the term C_{M} ≈ 0. Moreover, remember that the first step of the wavefront retrieval process only uses the linear terms. A simple correction of the vortex phase e^{ ± ilpφ} by γ_{M} allows us to reject all C_{M} Zernike polynomial terms in the coefficient. The C_{M} term can be seen as no POAM (Poynting 1909; Allen et al. 1999) added by the vortex coronagraph. Unfortunately, a few part of aberrations pass through the coronagraphic device without speckle modulation because of the Pancharatnam topological charge l_{p} effect.
J.2. The central region of disorientation
The finite mask extension is easily taking into account in our NijboerZernike development by cutting highorder rings of the Bessel function in the functions. The central region of disorientation is the most important effect on of the coronagraphic rejection. Indeed, most of the stellar energy to be suppressed will fall on it. The uncertainty for the diameter of the central region of disorientation allows us to use a simplistic hole model. The nulling factor can be calculated with the modified NZ theory and the maximum nulling factor N_{v} is given by (J.3)where s is the radius of the disorientation region expressed in λ/d units.
Covering this central region of disorientation with an opaque mask (small Lyotlike coronagraph) allows us to reach the chromatic limit, but create diffraction effects at the edge of the Lyot stop. In our NZ simulation, we can take the presence of this diffraction into account by changing the function (in the perfect case: ). Indeed, the maximum coronagraphic rejection without optical aberration is given by the term, where the vortex mask extent is [s_{1},s_{2}] , Π(0,1) is the unit box function and is the Gauss hypergeometric function. The function inside the pupil can be approximated by a simple parabolic function: 10^{5}ρ^{2} 0 < ρ < 1 l_{p} = 2 as a simple defocus.
Appendix K: Lyot stop optimization
The diaphragm in the coronagraphic pupil plane must be optimized to obtain a good ratio between the nulling factor and the planet throughput attenuation because of this Lyot stop. The total nulling factor ϵ_{c} can be defined as follows: (K.1)The coronagraphic pupil presents in real life three different diffraction patterns: a smooth ρ^{2} residual light in the pupil owing to the small Lyot coronograph in the center (pseudoapodization), a light modulation inside the pupil owing to optical aberrations, and finally a strong light gradient in the edge of the pupil with diffraction oscillations.
K.1. Residual optical aberrations
The diaphragm optimization in the presence of aberrations can be easily calculated as follows: (K.2)where 1/n^{pw} is the power spectral density of the optical aberrations. We consider two extreme cases: pw = 1 (strong aberration residuals on an adaptive optics for example) and pw = 2 (optical polishing). Results are given in the two following figures (Fig. K.1):
Fig. K.1 Variation of the residual energy (dE_{p}/dx) in the residual optical aberrations as a function of the number of Zernike polynomials used (nz – without the piston term) and the power spectral density 1/n^{pw}. The figure shows two diaphragm values: when the gain is maximum and when it is equal to zero. The figure in 1/n presents strong residual oscillations owing to loworder aberrations. 
K.2. Diffraction oscillations
We need to calculate the function (including ) previously given in Appendix H. In the visible/nearinfrared, these oscillations are generally limited to the region 0.8 < ρ < 1 (see numerical simulations in Riaud & Schneider 2007).
Fig. K.2 Fresnel diffraction of the Lyot stop with an optical vortex coronagraph. We present the real and the imaginary parts of the function for four different Fresnel number N. The last figure shows the residual intensity in the coronagraphic pupil plane when we choose a d_{c} radius of 86% (vertical dotted line). 
To minimize the Fresnel residual diffraction in the coronagraphic pupil plane, it is necessary to increase the Fresnel number N. For highcontrast imaging devoted to the research of telluric planets around nearby stars (10^{10} contrast level), the diaphragm (86%) has to be positioned with ± 50 μm accuracy with respect to the real pupil position (N > 10 000) along the optical axis. To simplify the aberration retrieval process, the only contains pseudoapodization pattern owing to the small Lyot dot in the center of the mask and not the one caused by the propagation of the Lommel function of two variables given by the term. Indeed, these functions are somewhat negligible if we use the proper Lyot stop diameter (86% in the case presented here) and a good diaphragm position.
K.3. Annular pupil
The case of the annular pupil is relatively simple, because all simulations already given in the previous section can be used for a larger diameter of the Lyot stop. Concerning the optimization of the smaller diameter of the Lyot stop, the value is completely governed by the starlight diffraction of the telescope obscuration ( ∝ ρ^{ − lp}). For an obscuration ϵ = 0.14 (Very Large Telescope) an inner radius (d_{in}) of 0.3 and l_{p} = ± 2 allows one to obtain a total nulling factor of ϵ^{2}(ln(d_{in}) − ln(ϵ))^{2} ≈ 1/4500.
Appendix L: Spider diffraction in the Fresnel mode
A telescope with a central obscuration possesses a complex spider to hold the secondary mirror. We present here the Fresnel development of the residual diffraction caused by a thin spider in the coronagraphic pupil with the vortex coronagraph.
First of all, we calculate this effect in the classical cartesian coordinates (x,y) taking into account the vortex term (e^{ilpθ}) and after that, we transform the result into the polar coordinates. We follow the Fresnel notation given in the previous analytical section: ρ → ρ′, θ → θ′, and now x → x′, y → y′. A spider is defined as follows: x is ( ± th/2) the thichness of the spider in radius unit and y is (ϵ to 1) the spider length. The spider is on azimuthal position θ_{n}. Generally, a telescope possesses three or four spiders. For the VLTUT telescopes, we sum in amplitude the diffraction contribution of four different spiders: n = 1,2,3,4 with azimuthal angle of 109 degres. The Fresnel diffraction U_{n}(x′,y′) of one spider becomes (L.1)After some variable changes, we obtain the spider diffraction as a function of the modified Fresnel integrals C(ξ,l_{p}) and S(ξ,l_{p}) (L.2)where These modified Fresnel integrals are relatively complicated in cartesian coordinates. Now, we use the polar coordinates to calculate this diffraction in a more convenient way: (L.5)In this coordinate transform y depends only on ρ and the diffraction integral on ρ′ does not depend on the θ coordinate. The Fresnel integrals become In the coronagraphic pupil plane, we use a Lyotstop for a proper starlight filtering, but we also notice that this diaphragm suppresses the Fresnel diffraction on the ρ′ axis. The spider creates only residual diffraction on the x′ coordinate. The Fresnel oscillations in the edges of the spider depend on the ρ′ and the θ′ coordinates. In these conditions, the modified Fresnel integrals must be simplified in the ArcTan() using exponential algebra for a fixed value of l_{p}: The Fig. L.1 shows numerical simulation results for the vortex and simple Fresnel diffraction regime.
Fig. L.1 Spider Fresnel diffraction calculation for a VLT coronagraphic pupil (ϵ = 0.14) and the diameter of the coronagraphic pupil of 18 mm. The figure shows a comparison between the classical Fresnel diffraction (l_{p} = 0) and the optical propagation including the vortex (l_{p} = 2). The thickness of the spider is 225 μm. 
Appendix M: Intensity in the final coronographic plane
Now, we present the coronagraphic intensity in the final plane using the function definition provided by the Fraunhofer diffraction. Indeed, the Fresnel propagation of aberrations is somewhat complicated and in the high Fresnel number regime (N > 10 000) the image result is very close (err < 10^{5}) to the Fraunhofer diffraction. We use here the Fraunhofer diffraction in the final coronagraphic intensity. The intensity on the final coronagraphic plane is given by the square modulus of the complex amplitude: (M.1)The Lyot stop action is equivalent to a normalization of the radial Zernike polynomials with the diaphragm size d_{c} < 1. This is a homothetic transformation in the ρ coordinate and , (M.2)The coronagraphic intensity consists of a linear part f^{(1)} and a quadratic part f^{(2)} as follows: (M.3)Developing the squared terms yields The are defined as follows
Appendix N: Derivation of
We now define two sets of functions and : (N.1)with ϵ_{m} = 1 for m = 0 and 2 in all other cases.
We have already defined the inner product: (N.2)The aberrated images are composed of several modes in sine and cosine functions. The first step of the image analysis is the decoupling of these modes, which can be performed with a polar Fourier transform F_{p} of the coronagraphic images. It is given by (N.3)In the perfect case without photon and readout noise, the NijboerZernike analysis shows that .
For a linear approximation of the final coronagraphic intensity, (N.4)Now, we can write the polar Fourier transform of the the final coronagraphic images using the Ψ,χ functions. – m = 0 (N.8)– 0 < m < n(N.9)– m = n(N.10)Using the inner products between the and the functions, it is now possible to retrieve coefficients of the linear system of equation as follows: – m = 0 – 0 < m < n– m = n
Appendix O: Scattering function
The physical theory of scattering by small particles is the Mie theory (Bohren & Huffman 1983), but this mathematical development is too complicated to be used with spherical Bessel and infinite sum of Legendre polynomials. The HenyeyGreenstein model (Henyey & Greenstein 1941) is a useful approximation to describe the angular distribution of light scattered by small particles. At the beginning, this model was applied to scattering by interstellar dust clouds in the MilkyWay. It is now used in numerous situations, ranging from the scattering of light in biological tissues to scattering caused by bulk material in an optical system.
In this model, the angular distribution of scattered light is given by The HenyeyGreenstein phase function Sc(θ) is governed by the variation of only one parameter, , and ranges from backscattering through isotropic scattering to forward scattering. The θ and φ variables are the classical angular coordinates. Forward scattering is θ = 0, while for backscattering, θ = π. We see that the ratio of forwardtoback scattering is [(1 + g)/(1 − g)] ^{3}.
For g > 0, forward scattering is dominant, while for g < 0, backscattering predominates. To use the HG function for Monte Carlo models, we need the accumulated distribution: (O.1)The Zemax software uses this approximation for scattering calculation, and we obtain
Fig. O.1 Nonsequential Zemax analysis of the proposed optical implementation after the Lyot stop to split the two circular polarizations. Left: the polychromatic ray tracing (650–900 nm) for one polarization, and right: the detector view of nocoherent intensity distribution in logarithmic scale. With these classical materials (MgF_{2}, quartz, infrasil) the maximum scattered light is 10^{6} of the residual coronagraphic intensity. In this simulation, we chose an antireflexion coating of 0.5% per surface. 
Appendix P: Polychromatic simulations and speckles smearing
We present in this section a complete set of numerical simulations obtained in wide bandpass with an imperfect vortex phasemask. The inputs of these simulations were given in Sect. 7.2:

The input Strehl ratio is 95% @ 650 nm with the first 860 Zernikes (40 complete modes).

An imperfect phasemask: where the stransmittance is 97%, the ptransmittance is 98% and the local phase retardance is π ± Δφ with a quadratic law for the phase error of the mask.

Common and noncommon path error are λ/71 rms @ 650 nm (the total is λ/50 rms at 650 nm).

Polishing error: DSP f^{2} with a shift of 8.1% for noncommon path error.

Polychromatic speckle smearing due to residual Wollaston chromatism given by the Zemax model.

Photon noise, readout noise (6 e^{ − }), fullwell capacity of 10^{5}e^{ − }, residual flat of 1% rms.
P.1. Telescope pupil and coronagraphic pupil phase screens
We present the entrance pupil for a perfect circular telescope (offaxis configuration) obtained by the direct summation of the first 860 Zernikes, and also the two phasescreens using a density spectral power function (f^{2}) where f is the spatial frequency in the Fourier domain.
Fig. P.1 Left to right: the telescope entrance pupil phase obtained with the first 40 Zernike modes. The Strehl ratio is 95% @ 650 nm. In the center, the phase screen that simulates highfrequency polishing defaults for the l_{p} = 2 beam. Right, the phase screen for the l_{p} = −2 beam. These two last phase screens are the superposition of two different random simulations (common and noncommon path errors). The last case (l_{p} = −2) is simulated with a shift of the second phase screen (8.1% in the yaxis with respect to the pupil diameter). The sigma of the two phase error screens for l_{p} = 2 and l_{p} = −2 are chosen equal to λ/50 rms @ 650 nm. 
Fig. P.2 Set of numerical coronagraphic simulations with a maximum phaseshift (Δφ_{max}) on the vortex mask of 0.001 radian for each bandpass. We chose five different bandpasses: Δλ = 2 nm (corresponding to a singlemode Laser diode), 100 nm, 150 nm (Rfilter), 200 nm, and finally 250 nm (R to I astronomical filters). These simulations show the antisymmetric smearing (bottom: l_{p} = −2, top: l_{p} = 2 images) due to the Wollaston optical component and also a highcontrast in the residual speckle due to the wavelength filtering by the coronagraphic device. Indeed, only wavelengths with larger phaseshift errors on the phasemask (λ_{max},λ_{min}) have a significant effect on the final coronagraphic images. We recall that for λ_{max} the effect is less than λ_{min} due to a better Strehl ratio. 
Fig. P.3 Same coronagraphic simulations with a maximum phaseshift of 0.01 radian. 
Fig. P.4 Same coronagraphic simulations with a maximum phaseshift of 0.1 radian. A coronagraphic strong peak residual appears in the center of the image (β_{0,0} ≠ 0). 
P.2. Phase error on the vortex coronagraph
The π phaseshift of the vortex coronagraph is generally obtained by index material engineering as engraving a ramp in a material for scalar vortex or in liquid crystal polymer (LCP) using birefringent optical properties, see Mawet et al. (2009) to generate the proper phase helix. The refractive index of the material as a function of the wavelength follows a Sellmeiler law (λ^{2}) (also for the LCP material). This property generates a π phaseshift with a quadratic dependance. In these conditions, the phaseshift error increases strongly at the edges of the chosen wavelength bandpass. To perform our numerical simulations, we chose the following phaseshift wavelength dependence: (P.1)where Δφ_{max} is the maximum phaseshift error in radian (0.001,0.01,0.1), λ_{0} is the centering mask wavelength (where the phaseshift is π) and finally λ_{max},λ_{min} corresponds to the wavelength bandpass edges.
P.3. Wollaston residual dispersion
Our optical implementation presents an artificial polychromatic speckle smearing in the xaxis due to the Wollaston residual dispersion. If we use the Sellmeiler law for the refractive index of the MgF_{2} or the position of the spot diagram in the Zemax software, we can know the shift of the PSF as a function of the wavelength on the two coronagraphic images. The position of the PSF center in λ/d units is given with very good approximation ± 0.2 μm by the following function: (P.2)The position of the PSF is completely antisymmetric in the ± 0.01λ/d level with respect to the sign of the vortex charge l_{p}.
P.4. Coronagraphic vortex numerical simulations set
The Fourier simulation was performed on 2K × 2K arrays with an entrance perfect circular pupil of 255 pixels diameter. The chromatism was taken into account by changing the diameter of the entrance pupil with one pixel steps that correspond to ≈ 7 nm bandpass steps. For the smaller bandpass (2 nm – the Laser diode), the simulation was performed with 4K × 4K arrays and a circular pupil of 511 pixels diameter. The entrance pupil phase was scaled and a vortex coronagraphic mask was created for each wavelength with the proper s and p transmittance and the mask phaseshift error. The diaphragm was chosen to be just smaller the pupil diameter (98%, 2 pixels apart) and scaled with respect the wavelength. We chose this nonoptimized configuration (see Appendix K) to use the β_{n,m} coefficients without a scaling factor. A phase screen error was added to each coronagraphic pupil (l_{p} = ± 2) to take the common on the noncommon path error of the optical train (Wollaston and achromatic doublet) into account. This phase screen includes the shift of the two beams due to the polarization separation (noncommon path error). The maximum shift between the two beams is 8.1% at the output of the final imaging lens doublet, this is the value adopted in our Fourier numerical simulations. Finally, we used Eq. (P.2) to produce the Wollaston chromatic smearing in the coronagraphic image plane by adding a tip in the coronagraphic pupil. These simulations complete the previous results presented in Sect. 7.2, and we show here some images obtained for three maximum phaseshift errors of Δφ_{max} = 0.001,0.01,0.1 radian and five different wavelength bandpasses Δλ = 2,100,150,200,250 nm, also including photon noise with 80 000 e^{ − } on the maximum of coronagraphic image (80% of the fullwell capacity), readout noise (6e^{ − }), and flat noise (1% rms).
Appendix Q: FFT vs. NZ for a “sin” ripple case
Fig. Q.1 modal decomposition for the sin function applied as wavefront error (both amplitude and phase). Up to down, we show the first 180 coefficients for the frequencies f_{x} = 2,4,8. 
We present in this appendix a complete set of numerical simulations obtained for a perfect case with a simple “sin” function in the entrance pupil plane. A sin modulation is proposed for both phase and amplitude. The pupil becomes (Q.1)where A_{Φ}, A_{a} are the coefficients of the phase and amplitude errors applied in the pupil respectively, and f_{x} is the spatial frequency of the sin function. The direct decomposition of the three sin modulations is presented in the following figure: the modal decomposition is not wellsuited for a “simple” sin function as wavefront error. Indeed, the term “simple” can be applied for the classical Fourier transform in cartesian coordinates (only one frequency in the Fourier plane), but not in the Zernike polynomials basis, where it is necessary to use many coefficients. We propose to compare the FFT propagation with VVC and the NZ modal decomposition for these three spatial frequencies.
Fig. Q.2 Set of numerical monochromatic coronagraphic simulations with a vortex mask error of 0.001 radian. The input sin frequency is chosen as f_{x} = 2 (loworder wavefront error). Each graphic is organized as follows: Left to right: l_{p} = + 2 coronagraphic image, l_{p} = −2 coronagraphic image, difference between the two previous images with an amplification of intensity level of 100. Upper: pure amplitude wavefront error (A_{a} = 0.01), middle: pure phase error (A_{Φ} = 0.1λ_{0}), and bottom: both errors, amplitude and phase. The first nine sets of figures left is for the FFT propagation, whereas the nine sets of figures right is for direct modal decomposition. The brightness scales are linear and are the same for the FFT and the NZ images sets. We notice that the amplitude modulation effects on the final coronagraphic images (three upper images) are smaller than the phase modulation error and the brightness scales for the amplitude set of simulations are also smaller to enhance the image contrast. Indeed, for the amplitude error, we chose 10^{7} ≤ I ≤ 10^{5} and for the phase error 10^{5} ≤ I ≤ 2.5 × 10^{4}. The direct comparison between the FFT and NZ shows a very significant similarity in the direct images l_{p} = ± 2 and after substraction of the two images. 
Fig. Q.3 Two sets of numerical simulations FFT vs. NZ with a sinus frequency of f_{x} = 4. 
Fig. Q.4 Two sets of numerical simulations FFT vs. NZ with a sinus frequency of f_{x} = 8 (highorder wavefront error). 
All Tables
Maximum coronagraphic nulling ratio and wavelength bandpass achievable for various levels of input Strehl ratio.
All Figures
Fig. 1 Schematic view of the NZ vortex phaseretrieval process. It consists of using the real input coronagraphic images, corresponding to l_{p} = 0, ± 2 (the l_{p} = ± 2 images are provided by polarization splitting, while the unpolarized image (l_{p} = 0) is directly given by the sum of the two polarized images) projected on aberrations templates, and finding the unknown coefficients by resolving a system of linear equations. Left: the three images needed for the full aberration analysis. Center: linear systems to retrieve the and the f^{(1)} terms. Right: the predictorcorrector approach to evaluate the quadratic correction . 

In the text 
Fig. 2 Polarization splitter analysis: possible implementation of the polarization splitter system used after the Lyot stop. An achromatic quarterwave plate (quartz/MgF_{2}) and a Wollaston in MgF_{2} combined here with a simple achromatic doublet. This simple optical implementation allows us to image the two residual coronagraphic images in the left and right circular polarization basis on the detector. The two images presented in this figure in logarithmic gray scale show the two polychromatic PSF (650–900 nm) obtained by this simple scheme for the two circular polarization images. Note that the PSF shows a small chromatic smearing residual on the xaxis (Strehl = 96.5%) due to the substantial wavelength bandpass. This polychromatic smearing is detailed in Appendix P. 

In the text 
Fig. 3 Numerical simulation illustrating the principle of NijboerZernike retrieval applied on the vortex coronagraph. (Up to down), numerical simulation for each circular polarization with l_{p} = 2 and l_{p} = −2 respectively using the first 860 Zernike polynomials (40 complete modes) and a Lyot stop of 99%. The Strehl ratio of the PSF before coronagraphic filtering is 95%. The Lyot stop remove the strong diffraction value in ρ = 1 but allows us to show all images without scaling the coefficients (). (Left to right), final monochromatic (λ = 650 nm) coronagraphic image obtained with the sum of analytical functions, the FFT of the direct sum of Zernike polynomials in the pupil plane, and finally, the classical phase function obtained by . Due to the difference of basis, the final simulation with the classical phase function presents minor discrepancies in the speckle background (I_{c} < 10^{7}, I_{c} is the coronagraphic final intensity). The image scales are not linear . 

In the text 
Fig. 4 Polychromatic coronagraphic simulation for l_{p} = 2 with Δλ = 250 nm of spectral bandpass and a phase error of Δφ = ± 0.001 radian for the vortex mask. We also include defects of our optical implementation. Left: without the speckle smearing. Right: with the Wollaston speckle smearing. The chromatic effect on the speckles is small in the two cases, the contrast remains high. The brightness scale is the same between the two images and is not linear. 

In the text 
Fig. 5 values as a function of the wavelength bandpass and a phasemask with a 10^{1}/10^{2}/10^{3} radian of phaseshift error. The black lines correspond to for the in: : retrieval, the gray lines are for the in: : retrieval: the solid lines are for the “p” modes and the dashed lines are for the “m” modes. The x and y scales are common between the three images. The values for of 10^{2} rad for the mask error are poorer than in the previous simulations. The 10^{1} rad case presents significant limitations because of a strong peak in the image center (β_{0,0} = 0.04). The dotted line with and shows the chromatic limit of a desired retrieval precision in the “AO metric”. 

In the text 
Fig. 6 values as a function of the signaltonoise ratio in the input coronagraphic images for a in: : retrieval. We present the effect of the modal decomposition quality with respect to the phaseshift error of the mask. Several realistic cases are presented with 1/10/100/1000 exposures in the final image. A good wavefront retrieval is given after an average image of at least 100 exposures. 

In the text 
Fig. D.1 Input classical Zernike . 

In the text 
Fig. D.2 function in the coronagraphic pupil. 

In the text 
Fig. D.3 function in the coronagraphic pupil. 

In the text 
Fig. D.4 U_{v}(r,φ),l_{p} = ± 2 functions in the coronagraphic image plane for the 10 Zernike polynomials (Z_{2} − Z_{11}). Top: l_{p} = + 2, Bottom: l_{p} = −2. The vertical solid line separates the real part from the imaginary part of the two U_{v}(r,φ) functions. 

In the text 
Fig. G.1 Comparison between Fourier simulations for each Zernike polynomial (2860) and mode in terms of χ^{2}. Left: the direct summation in the pupil (β_{n,m}), right: the classical phase function (e^{iαn,m}). 

In the text 
Fig. G.2 Comparison between Fourier simulations for each Zernike polynomial (2860) and mode in terms of Err(σ_{A}). Left: the direct summation in the pupil (β_{n,m}), right: the classical phase function (e^{iαn,m}). 

In the text 
Fig. H.1 function in the coronagraphic pupil with an extreme case N = 4 to highlight differences on Zernike polynomials. In the real case N > 1000, and only the must be taken into account in the NZ theory. 

In the text 
Fig. H.2 function in the coronagraphic pupil with an extreme case N = 4 to highlight significant differences on Zernike polynomials. 

In the text 
Fig. K.1 Variation of the residual energy (dE_{p}/dx) in the residual optical aberrations as a function of the number of Zernike polynomials used (nz – without the piston term) and the power spectral density 1/n^{pw}. The figure shows two diaphragm values: when the gain is maximum and when it is equal to zero. The figure in 1/n presents strong residual oscillations owing to loworder aberrations. 

In the text 
Fig. K.2 Fresnel diffraction of the Lyot stop with an optical vortex coronagraph. We present the real and the imaginary parts of the function for four different Fresnel number N. The last figure shows the residual intensity in the coronagraphic pupil plane when we choose a d_{c} radius of 86% (vertical dotted line). 

In the text 
Fig. L.1 Spider Fresnel diffraction calculation for a VLT coronagraphic pupil (ϵ = 0.14) and the diameter of the coronagraphic pupil of 18 mm. The figure shows a comparison between the classical Fresnel diffraction (l_{p} = 0) and the optical propagation including the vortex (l_{p} = 2). The thickness of the spider is 225 μm. 

In the text 
Fig. O.1 Nonsequential Zemax analysis of the proposed optical implementation after the Lyot stop to split the two circular polarizations. Left: the polychromatic ray tracing (650–900 nm) for one polarization, and right: the detector view of nocoherent intensity distribution in logarithmic scale. With these classical materials (MgF_{2}, quartz, infrasil) the maximum scattered light is 10^{6} of the residual coronagraphic intensity. In this simulation, we chose an antireflexion coating of 0.5% per surface. 

In the text 
Fig. P.1 Left to right: the telescope entrance pupil phase obtained with the first 40 Zernike modes. The Strehl ratio is 95% @ 650 nm. In the center, the phase screen that simulates highfrequency polishing defaults for the l_{p} = 2 beam. Right, the phase screen for the l_{p} = −2 beam. These two last phase screens are the superposition of two different random simulations (common and noncommon path errors). The last case (l_{p} = −2) is simulated with a shift of the second phase screen (8.1% in the yaxis with respect to the pupil diameter). The sigma of the two phase error screens for l_{p} = 2 and l_{p} = −2 are chosen equal to λ/50 rms @ 650 nm. 

In the text 
Fig. P.2 Set of numerical coronagraphic simulations with a maximum phaseshift (Δφ_{max}) on the vortex mask of 0.001 radian for each bandpass. We chose five different bandpasses: Δλ = 2 nm (corresponding to a singlemode Laser diode), 100 nm, 150 nm (Rfilter), 200 nm, and finally 250 nm (R to I astronomical filters). These simulations show the antisymmetric smearing (bottom: l_{p} = −2, top: l_{p} = 2 images) due to the Wollaston optical component and also a highcontrast in the residual speckle due to the wavelength filtering by the coronagraphic device. Indeed, only wavelengths with larger phaseshift errors on the phasemask (λ_{max},λ_{min}) have a significant effect on the final coronagraphic images. We recall that for λ_{max} the effect is less than λ_{min} due to a better Strehl ratio. 

In the text 
Fig. P.3 Same coronagraphic simulations with a maximum phaseshift of 0.01 radian. 

In the text 
Fig. P.4 Same coronagraphic simulations with a maximum phaseshift of 0.1 radian. A coronagraphic strong peak residual appears in the center of the image (β_{0,0} ≠ 0). 

In the text 
Fig. Q.1 modal decomposition for the sin function applied as wavefront error (both amplitude and phase). Up to down, we show the first 180 coefficients for the frequencies f_{x} = 2,4,8. 

In the text 
Fig. Q.2 Set of numerical monochromatic coronagraphic simulations with a vortex mask error of 0.001 radian. The input sin frequency is chosen as f_{x} = 2 (loworder wavefront error). Each graphic is organized as follows: Left to right: l_{p} = + 2 coronagraphic image, l_{p} = −2 coronagraphic image, difference between the two previous images with an amplification of intensity level of 100. Upper: pure amplitude wavefront error (A_{a} = 0.01), middle: pure phase error (A_{Φ} = 0.1λ_{0}), and bottom: both errors, amplitude and phase. The first nine sets of figures left is for the FFT propagation, whereas the nine sets of figures right is for direct modal decomposition. The brightness scales are linear and are the same for the FFT and the NZ images sets. We notice that the amplitude modulation effects on the final coronagraphic images (three upper images) are smaller than the phase modulation error and the brightness scales for the amplitude set of simulations are also smaller to enhance the image contrast. Indeed, for the amplitude error, we chose 10^{7} ≤ I ≤ 10^{5} and for the phase error 10^{5} ≤ I ≤ 2.5 × 10^{4}. The direct comparison between the FFT and NZ shows a very significant similarity in the direct images l_{p} = ± 2 and after substraction of the two images. 

In the text 
Fig. Q.3 Two sets of numerical simulations FFT vs. NZ with a sinus frequency of f_{x} = 4. 

In the text 
Fig. Q.4 Two sets of numerical simulations FFT vs. NZ with a sinus frequency of f_{x} = 8 (highorder wavefront error). 

In the text 
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