Issue 
A&A
Volume 612, April 2018



Article Number  A80  
Number of page(s)  13  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201732512  
Published online  30 April 2018 
Theoretical performance of serrated external occulters for solar coronagraphy
Application to ASPIICS
^{1}
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS UMR 7293,
Lagrange, Parc Valrose,
06108
Nice Cedex 2, France
email: raphael.rougeot@esa.int
^{2}
European Space Research and Technology Center, European Space Agency,
Keplerlaan 1,
2201
Noordwijk, The Netherlands
^{3}
Thales Alenia Space,
5 allée des Gabians,
06150
Cannes, France
Received:
19
December
2017
Accepted:
17
January
2018
Context. This study is made in the context of the future solar coronagraph ASPIICS of the ESA formationflying mission Proba3.
Aims. In the context of solar coronagraphy, we provide a comparative study of the theoretical performance of serrated (or toothed) external occulters by varying the number and size of the teeth, which we compare to the sharpedged and apodized disks. The tooth height is small (a few centimeters), to avoid hindering the observation of the solar corona near the limb. We first analyze the diffraction pattern produced by such occulters. In a second step, we compute the umbra profile by integration over the Sun.
Methods. We explored a few methods to compute the diffraction pattern. Two of them were implemented. The first is based on 2D fast Fourier transformation (FFT) routines and a multiplication by the Fresnel filter of the form exp(−iπλzu^{2}). Simple rules were derived and discussed to set the sampling conditions. The Maggi–Rubinowicz representation is then proposed as an alternative method, and is proven to be very efficient for this study.
Results. Serrated occulters tend to create a twolevel intensity pattern, the inner being the darker, which perfectly matches a previously reported geometrical prediction. The diffraction in this central region is lower by two to four orders of magnitude when compared to the sharpedged disk. The achieved umbra level at the center ranges from 10^{−4} to below 10^{−7}, depending on the geometry of the teeth.
Conclusions. Our study shows that serrated occulters can achieve a high rejection and can almost reach the performance of the apodized disk when very many teeth are used. We prove that shaped occulters must be preferred to simple disks in solar and stellar coronagraphy.
Key words: Sun: corona / instrumentation: high angular resolution / methods: numerical
© ESO 2018
1 Introduction
The first experiment using an external occultation was made by Evans (1948) to measure the brightness of the sky near the Sun. On a highly reduced scale, this experiment simulates the phenomenon of a lunar eclipse and has been the precursor of a series of coronagraphs using an external occulter in space, as was well described by Koutchmy (1988) in his review of spaceborne coronagraphy. The principle of these experiments appears to be much simpler than the classical Lyot coronagraph (Lyot 1939), but characterizing the straylight involves Fresnel diffraction, the effects of which are very complex to model accurately.
Very soon after the first experiments in the 1969s, it was considered that a single perfect circular disk was not the best external occulter to observe the solar corona. Following the concept of apodization in optics as formalized by Jacquinot & RoizenDossier (1964), apodized or shaped occulters were imagined. They are intended to provide a shadow much darker than the disk. Multiple disks or serrated occulters were thus tried out, as described in Newkirk & Bohlin (1965) or Purcell & Koomen (1962), for example. The intuitive reasoning of the latter authors to use a toothed disk was that it this disk would diffract the incoming light waves in a direction perpendicular to the edge. Therefore, a serrated mask made of successive teeth was thought to reject the light outside a central part, creating a strong darkness. Conversely, a simple disk would reject the light toward the center, resulting in an inferior good darkness. Based on this reasoning, Boivin (1978) deduced by geometrical considerations that such a serrated occulter would produce a dark inner region whose radius, which we name the Boivin radius, depends on the number and the size of the teeth.
In stellar coronagraphy, it is planned to use shaped occulters for the direct detection of exoplanets, and they have been extensively investigated, which led to numerous publications in the past decade. Typically, the occulter is about 30 m in diameter, located at ~80 000 km from a ~4 m telescope. The contrast to be achieved is 10^{−10}. It has been proven that an optimized apodization function for occulter provides the best rejection performance. However, it is known that ideal radial apodization remains an ideal that cannot be manufactured with the required precision. As an alternative, Cash (2006) and Vanderbei et al. (2007) proposed and investigated the use of a petalshaped occulter, which is an improved version of the serrated occulter made of petals of more elaborated shape compared to the mere triangular tooth envisaged in previous solar studies. Moreover, the adopted point of view is rather different from the geometrical reasoning discussed above for the solar case. Here, the petalized occulter is seen as a discrete substitute of the ideal radially apodized occulter, as described in Vanderbei et al. (2007).
Furthermore, the goals and therefore the geometry of solar and stellar coronagraphy are quite different. The relative closeness of observation from the bright source is not as severe for the exoplanet as it is for the solar corona, and the extended light source that is the Sun is not comparable to a stellar pointsource. This implies that the diffraction pattern must be known to a large extent, and not simply in a very central part. We recall that the solar coronagraph ASPIICS is made of an occulter with a 1.42 m diameter positioned 144 m in front of a telescope that in turn has a 5 cm aperture (Lamy et al. 2010; Renotte et al. 2015). This Lyot coronagraph will fly on the future ESA formationflying mission Proba3, and is split between two spacecraft. Related modeling of Fresnel diffraction for the sharpedged occulting disk has previously been studied in Aime (2013) and Rougeot et al. (2017), where the diffracted wavefront is propagated through the hybrid Lyot coronagraph.
Only a few studies about serrated occulters for solar coronagraphy can be found in the literature. Fort et al. (1978) illustrated the gain in straylight of a polished toothed circular disk by experimental measurements, which supports the early suggestion of Purcell and Koomen. In particular, they highlight the required quality of the edge of the occulter, and associated manufacture difficulties. In a more recent study, related to the ASPIICS solar coronagraph, Verroi et al. (2008) developed a diffraction modeling based on Fresnel–Kirchhoff theory and a clever splitting of the plane of the occulter in Fresnel zones. They computed the diffraction intensity for a toothed external occulter and reported a remarkable reduction of the amount of diffracted light at the entrance aperture. However, they considered only a few teeth, about 20, and the shape is different from the classic straight line that we investigate in this paper. Our results are therefore not comparable to their results. Finally, Landini et al. (2016) presented an experimental study of the rejection of various external occulters, based on a reduced scale test bench.
We present here a theoretical and numerical study of diffraction for serrated occulters, using the parameters of the longbase solar coronagraph ASPIICS. We aim to analyze the rejection performance of serrated occulters for solar coronagraphy by varying the number and the size of the teeth, and to compare a sharpedged and an apodized disk. We report that a diffraction computation like this remains a very delicate operation, for instance, for the sampling constraints, and several models have been investigated and implemented for this problem. The penumbra profile is obtained by a mere convolution with the solar image. In particular, we give the trend of the integrated illumination over the 5 cm aperture of the ASPIICS telescope with respect to the number and the size of the teeth.
The paper is organized as follows. Section 1 contains this introduction. In Sect. 2 we introduce the different models we investigated to compute diffraction, and the related numerical implementation. In Sect. 3 we analyze the diffraction pattern obtained for serrated occulters. We show that our results are in good agreement with the intuitive reasoning of Boivin (1978). The computation of the penumbra profile is addressed and discussed in Sect. 4. Conclusions are given in Sect. 5.
2 Diffraction by an external occulter
2.1 Fresnel integral
The Fresnel diffraction of a thin external occulter that is assumed to be a flat structure lying in a plane can be obtained using classical Fourier optics, as described in Goodman (2005), for example. This theory is based on paraxial optics, which approximates a sphere by a paraboloid. This defines the Fresnel regime of diffraction, which has proved to be sufficiently accurate for optical astronomy.
In the following, we set a Cartesian coordinate system (x, y, z). We associate a set of polar coordinates (r, θ) to the transversal plane (x, y). The occulter is set in the plane z = 0. In the Fourier optics formalism, all waves propagate in the z positive direction, and amplitudes are described in the transversal planes (x, y) for different z values. For example, the propagation of a monochromatic plane wave of unit amplitude (corresponding to rays parallel to the zaxis) is represented in this scalar theory of light by the phase term exp(2iπz∕λ), where λ is the wavelength of the light and z is the covered distance from the origin. A tilted plane wave propagating with small angles (α, β) with regard to the zaxis is affected, in addition to this, by a phase term exp(−2iπ(αx + βy)∕λ). The tilted wave is still considered as propagating toward the z direction in this paraxial optics theory, so that the interaction of coherent waves of different angles (α, β) can be considered. As an illustration, two such coherent waves of different angles produces Young fringes.
More generally, let Ψ_{0}(x, y) be the complex amplitude of a wave in plane z = 0 (including the occulter). Neglecting the irrelevant constant phase term exp (2iπz∕λ) that is related to the propagation of a plane wave, the complex amplitude Ψ _{z} (x, y) of the wave at the distance z can be obtained in the direct space by applying a Fresnel freespace propagation: (1)
The symbol ⊛ stands for the convolution, and φ_{z}(x, y) is the parabolic approximation of the spherical wave originating from a radiant pointsource at a distance z. Equation (1) is the socalled Fourier–Fresnel transformation of the function Ψ _{0} (x, y), which describes the Huygens–Fresnel principle. The number N_{F} of Fresnel zones corresponds to the number of times this phase term increases in π, and in our case, N_{F} = R^{2}∕λz, with R the radius of the occulter. For ASPIICS, N_{F} ~ 6400 at λ = 550 nm, while it would be on the order of 15 for an exoplanet experiment. The expanded form of Eq. (1) shows the diffracted wave as a scaled Fourier transformation of the wave Ψ _{0} (x, y) that is affected by the quadratic phase term φ_{z}(x, y). Since Eq. (1) is a convolution, Ψ_{z}(x, y) can be written as follows: (2)
where and denote the 2D direct and inverse Fourier transformations from the domain (x, y) to frequency domain (u, v). The Fresnel filter is given by (3)
It is worth emphasizing that the formulation in Eq. (2) is valid for any occulter shape.
We recall the case of a perfectly circular sharpedged occulter of radius R. For an impinging plane wave of unit amplitude, coming from the onaxis pointsource at infinity, Eq. (1) can be expressed as a radial function: (4)
J_{0}(.) is the Bessel function of the first kind. This Hankel transformation consists of the exact analytic expression of Fresnel diffraction in that particular case. At this stage, a radial apodization of the occulter disk can be accounted for with the formulation given in Eq. (5). We refer to the work of Aime (2013) for a more extensive study and for the alternative use of the Lommel series. In the following, we consider the sharpedged occulting disk as a reference case in order to validate the other models we investigated.
2.2 Theoretical approach of Vanderbei et al.
Although stellar coronagraphy consists of a quite different problematic compared to solar coronagraphy, the diffraction that occurs follows the same Fresnel regime, and the equations used to describe the waves are similar. From the Fresnel theory, Vanderbei et al. (2007) investigated an analytical formulation to compute Fresnel diffraction for a petalshaped occulter. Based on the procedure of Jacobi–Anger expansion (Vanderbei et al. 2003), Ψ _{z} is written as the sum of Hankel transformations with highorders Bessel functions.
We tried to apply this approach to our solar case, the toothed disk being a more elementary version of the petalshaped disk. We define the serrated occulter by its number of teeth, denoted as N_{t}, its inner radius R, and the tooth height Δ. Using the notations of our study, Eq. (9) of Vanderbei et al. (2007) becomes (5)
where is the Fresnel diffraction wave produced by the smoothed apodized occulter, the substitute of which is the serrated occulter. The related radial transmission function is 1 − f(r), as it appears in Eq. (5). This can be computed by a pseudoanalytical integration of Eq. (5), for instance. We recall that φ_{z}(r) is given in Eq. (1). For a serrated occulter, the transmission is fully opaque in the central part, so that the factor sin (jπf(ρ)) is zero when ρ ≤ R. Thus, the integration boundaries in Eq. (5) can be reduced to [R, R+ Δ].
The functions become exponentially small near the center of the shadow for large N_{t} when j > 0. As the inner working angle is relatively small for exoplanet detection, the formulation in Eq. (5) is appropriate for computation, as only the first term dominates and the sum is neglected (Vanderbei et al. 2003). However, in our solar case, the field is much larger and the number of teeth usually on the order of several hundreds. We can no longer neglect the highorder Bessel functions, which renders the computation very complex and a delicate operation. Numerical implementation did not provide any substantial results because we were unable to make the sum converge. This approach was therefore discarded for our study.
To illustrate this discussion, Fig. 1 plots a cut of the diffraction intensity of a circular apodized occulter. The transmission evolves linearly from 0 at R = 710 mm to 1 at R + Δ, with Δ = 10 mm. The computation is based on the Hankel transformation as given in Eq. (5). The umbra level is much darker than the simple disk, by a gain of about six orders of magnitude.
Fig. 1 Red: Fresnel diffraction intensity of the apodized occulter. The transmission evolves linearly from 0 at 710 mm radius to 1 at 720 mm. Black: sharpedged disk as a reference. 
2.3 Numerical 2D FFT technique
The Fresnel diffraction Eq. (2) can be implemented as a brute force numerical integration. We computed the 2D fast Fourier transformation (FFT) of an array of K × K sampling the external occulter, then multiplied this quantity with the Fresnel filter exp (−iπλz(u^{2} + v^{2})), and finally returned to the real plane taking the inverse Fourier transformation. We used regular FFT algorithms, which made the spatial sampling σ the same onthe initial and final arrays.
With K × K points and a sampling σ, the occulter is padded in a square finite region L × L, with L = K × σ. We have an unusual padding here, since the occulter of transmission 0 is padded in a field of 1. Now, we must address the sampling problem for Eq. (2).
A first point concerns the occulter itself. Our reasoning is based on the case of the sharpedged disk. The Fresnel diffraction figure at a distance z is the wellknown Arago spot. At the center, its amplitude is equivalent to J_{0} (2πRr∕λz), as shown in Aime (2013). We now consider the reverse Fourier transformation of this function. Mathematically, it is a circle of radius R∕λz. In the Fourier space, the maximum spatial frequency is given by 1∕2σ. This quantity must be larger than R∕λz, otherwise the circle overlaps and thereby aliasing effects occur. We derive thus a first condition: (6)
With ASPIICS parameters, we find σ < 55.9 μm
A second consideration comes from the 1or0 occulter, which provides thus a discontinuity in transmission. In the sense of a Fourier analysis, such an object is not band limited. The Shannon–Nyquist criterion is not applicable here. To mitigate this problem, we imposed that the spatial extent L has to be much larger than the diameter of the occulter, in order to perform 1padding and avoid too strong aliasing effects when computing the Fourier transformation, (7)
where the relation ≫ must be understood as larger than a factor 2. We highlight that this condition is less critical when considering an apodized occulter because the sharpness of the edge is smoothed.
The third sampling problem concerns the Fresnel filter of the form exp (−iπλzu^{2}). It is a function showing extremely fast variation when u increases, and therefore it is very difficult to sample. To avoid sampling problems, the approach of Mas et al. (1999) might be followed in their otherwise very interesting study of Fresnel diffraction, and a frequency cutoff might be assumed, which means setting the spectrum to zero after some given frequency. We tested applying a lowpass filter to the Fresnel spectrum. The diffraction pattern was smoothed, but the computed result was not satisfactory in the case of the simple disk: the Arago spot was degraded. We therefore chose not to follow their approach. The alternative would be to neglect the sampling condition, but then strong aliasing effects appear, which makes any subsequent calculation irrelevant because it propagates the wave inside the coronagraph. We report below the condition that must be imposed to perform a satisfactory sampling on the basis of the Fresnel filter.
After the FFT routine is applied, the sampling step in the Fourier plane is 1∕(K × σ), and the highest frequencies in u and v are u_{c} = v_{c} = ±1∕(2σ). We assumed for convenience that a shift is applied, and that the zero of the FFT transformation is at the middle of the array, at the point (K∕2 + 1, K∕2 + 1). We considered the u direction of the axes for simplicity, and let k be a point of the array in the region u > 0. We recall that k = 0 at K∕2 + 1 of the array. The corresponding frequency at point k is u_{k} = k∕(Kσ) = u_{c}k∕(2K). The value of the Fresnel filter at this point is thus (8)
Again, it is difficult to use Shannon’s theorem directly for this problem, since the Fresnel filter is not band limited. However, a suitable criterion is that the phase difference of this quadratic filter does not exceed π between two successive points. This gives the value of k such that for k + 1 we have an increase in phase no greater than π. We obtain (9)
This variation of π is required to occur at the edge of the array, that is, for k = K∕2, and we derive the following relation: (10)
It is interesting to note that when we inject Eq. (10) into Eq. (6), we find L = K × σ < 2R, which simply means that the array must encompass the occulter. In practice, K is very large, and we used the highest value that did not saturate the computer memory. In the present study, K = 156 000, and the sampling σ is derived from Eq. (10). We verified that Eqs. (6) and (7) were satisfied and that the computation was accurate enough. Table 1 gives several possible sets of parameters.
Rougeot et al. (2017) propagated the diffracted wave further through the coronagraph and discussed another sampling condition. The reasoning was to have the same number of points padding the diameter of the telescope and padding the image of the external occulter formed by the objectives. In the hybrid case of ASPIICS, we recall that the Lyot mask (or internal occulter) is set at this particular image plane, instead of at the exact focal plane where the solar image is located, as for the classical Lyot coronagraph. If the propagation of the diffracted waves through the coronagraph is to be computed, this additional sampling condition of Rougeot et al. (2017; their Eq. (13)) needs be taken into account. We leave this for future work.
As a reference test case, we considered the circular sharpedged occulter, in order to validate the 2D FFT technique. Figure 2 plots a mere central cut of the 2D diffraction pattern, which we compare to the reference (Eq. (5)) as given in Fig. 2 of Rougeot et al. (2017). We used an array of size K = 156 000 and a spatial sampling of σ = 22.6 μm. Panel a shows the full pattern extended in logarithmic scale; panel b shows a zoom in the central spot in linear scale, whose peak intensity is 1 for r = 0, as expected; panel c shows the transition zone between shadow and light. As expected, the numerical FFT results present small noisy fluctuations. The FFT results are also very similar to the reference, despite the low sampling of 22.6 μm; this is clearly visible in panel b. As our main objective is to compute the penumbra profile by integration over the solar disk, which has a smooth pattern, this is not a critical concern. Finally, we may investigate the effect of deviating from the sampling condition (10). This is reported in Appendix A.
To conclude, the brute force 2D FFT technique seems adequate to compute Fresnel diffraction, provided that the size of the array K is large enough and that the sampling condition Eq. (10) is fulfilled. We tested this method for serrated occulters and describe the results below.
Parameters for the 2D FFT technique. See text for details.
2.4 Maggi–Rubinowicz representation
An alternative approach to computing the diffracted wave behind a screen or an aperture is given by the Maggi–Rubinowicz representation. Instead of the 2D integral of the Fresnel formulation, this method relies on a single boundary integral along the edge of the diffracting object. It must be noted that this modeling is only suitable for the 1or0 mask, so no apodization can be investigated.
The analytical derivation of this approach is well described in Born & Wolf (2006) based on the Kirchhoff integral theorem. We consider an incoming wave Ψ_{0} that is diffracted byan aperture lying on an infinite opaque screen. We denote by the complex amplitude of the wave at a point (x, y, z) downstream of the aperture. It is split between two terms (11)
where consists of the wave as dictated by the geometrical optics, that is, Ψ_{0} in the extent of the opening and 0 otherwise, and is the diffraction disturbance. This last term is written as a curvilinear integral along the boundary of the aperture.
In the framework of stellar coronagraphy, Cady (2012) formulated this boundary diffraction wave integral by introducing a potential vector and applying the Stokes theorem. As an occulter is the complementary of an opening in an opaque screen, the Babinet theorem is applicable. The complex amplitude of the diffracted wave behind the occulter,at the same point (x, y, z), is thus linked to following (12)
Equations (11) and (12) give the expression of the diffraction by the occulter: (13)
We verified that the geometrical derivation of Born & Wolf (2006) and the vectorial representation presented in Cady (2012) are consistent. Considering an impinging wave from the onaxis pointsource at infinity, we have Ψ _{0} = 1. The final expression of the disturbance is given by (14)
where l = (l_{ξ}, l_{η}, 0) is the unit vector in the direction tangent to the edge at any point P = (ξ, η, 0) of , s is the distance between P and the observation point (x, y, z), and dl is the differential curvilinear element of the edge. This mathematical expression can be expanded given a specific occulter edge and related vector l. Because occulters usually have a certain cylindrical periodicity, it is convenient to write Eq. (14) using polar coordinates (ρ, ϕ) in the integrand instead of (ξ, η), and perform the integration on ϕ ∈ [0;2π], either by discretization or analytical integration. In contrast to the FFT technique, this approach does not require a discussion of the sampling. No aliasing effects are present, which is intrinsically linked to the way the FFT works.
The detailed expression of the sharpedged disk is given by Eq. (B.2). The computation was performed by directly sampling the edge of the disk, that is, the angle ϕ, using 100 000 points. Figure 3 plots the diffraction intensity Ψ _{z} (r)^{2}, which we compare to the same reference Eq. (5) that is obtained using NIntegrate of Mathematica. Panels a–c show the same as Fig. 2. The diffraction pattern is very smooth and agrees perfect well with the reference. However, a divergence around the transition between shadow and light at r ~ 710 mm is observed because s(ϕ = 0) tends to z when r ~ R; see Eq. (B.2). In addition to this feature, we considered the Maggi–Rubinowicz representation to compute diffraction for serrated occulters.
An interesting result of the Rubinowicz representation for the simple occulting disk is the value reached by the intensity at the center, the socalled Arago bright spot. From Eq. (B.2), we obtain analytically (15)
With ASPIICS parameters, we find a value of 0.99995, which differsfrom the wellknown unitary spot, as obtained using Eq. (5), for instance. Very surprising at first, this discrepancy comes from the parabolic approximation used in Fresnel diffraction equations, which is not assumed by the Rubinowicz representation. When we expand the square root in Eq. (15) in terms of R^{2}∕z^{2}, we obtain the expected value of 1.
Fig. 2 Red: Fresnel diffraction intensity from the 2D FFT computation Eq. (2). Black shows the Hankel computation Eq. (5) as a reference. The occulter is a sharpedged disk of radius 710 mm set at z = 144.348 m. An array of 156 000 × 156 000 and a sampling σ = 22.6 μm have been used. Panel a: full range, in logarithmic scale. Panel b: zoom in the Arago bright spot in the central region [0, 500 μm], in linear scale. Panel c: transition region between shadow and light around 710 mm, in linear scale. 
3 Diffraction pattern of serrated occulters
We now consider thin serrated occulters in the geometrical configuration of ASPIICS coronagraph. We recall that the disk is defined by its number of teeth, denoted as N_{t}, its inner radius R, and the size of the teeth Δ of a few centimeters. The computational results were obtained with a machine with two 14core Intel Xeon processors and 512 GB of RAM.
3.1 Numerical validation
We first validated the numerical computation for the serrated occulters. In the Rubinowicz formulation Eq. (14), the curvilinear integral along the occulter can be performed either analytically or by discretization. The related expanded expression is given in AppendixB. Near the central point, we have observed that the integrand oscillates very quickly with the integration variable ϕ. This numerical consideration does not encourage the use of analytical integration technique, and we finally chose to compute the integral (Eq. (14)) by sampling the edge. We did not find any simple condition tocorrectly sample the occulter edge in the literature. Therefore, our approach is somewhat empirical.
We first imposed the condition that there is a sampling point at every peak and trough of the teeth. We denote by σ_{R} the spatial sampling along the edge of the occulter, that is, the distance between two successive points for the integration. This is different from the angular sampling dϕ. By reducing σ_{R}, which means increasing the number of points along the edge, we intuitively know that the resulting diffraction curve will be more accurate, but will also require more computational time. We aim to find a condition that gives a reasonable value for σ_{R}. We recall that we used 100 000 points for the disk, which corresponds to σ_{R} = 44.6 μm, and the result was very satisfactory. We focused on the particular 1024tooth occulter, which is the most stringent with respect to the number of teeth. We verified that reducing σ_{R} causes the computed diffraction curves to converge. The average relative difference between the computations using σ_{R} = 10 μm, 5 μm, and 2 μm, with respect to the computation using σ_{R} = 1 μm, are 2.4%, 0.55%, and 0.07%, respectively. Therefore we set as empirical criteria σ_{R} = 2 μm, which seemsreasonable. This value is much lower than the value we used for the disk.
We now compare the 2D FFT technique and the Rubinowicz method for serrated occulters. The FFT routine uses an array K × K = 156 000 × 156 000 and a sampling σ = 22.6 μm. We illustrate the comparison in this paper by using a serrated occulter of 64 teeth of 20 mm. Figure 4plots the cuts of the intensity Ψ_{z}(x, y)^{2} along the xaxis, obtained by both computations, in logarithmic scale. The two curves strongly differ in the central region of the diffraction pattern, but they coincide in an outer region where the intensity is brighter. On the one hand, as we previously identified, the accuracy of the FFT routine strongly depends on the size of the array because the computational memory is limited. Considering that the FFT analysis can be made only for a limited field, we may explain the observed discrepancy by an overlapping of the computed 2D images. On the other hand, the Rubinowicz method has proven to be rather efficient and accurate. For our present study, we legitimately decided to discard the 2D FFT technique, mostly because of our computational resources were limited.
In the following, the diffraction was computed using the Rubinowicz method, with a spatial sampling σ_{R} = 2 μm along the occulter edge. However, the use of the 2D FFT technique remains valid in the region relatively close to the transition from shadow to light.
Fig. 3 Red: Diffraction intensity using the Rubinowicz representation. Black shows the Hankel computation Eq. (5) as a reference. The occulter is a sharpedged disk of radius 710 mm set at z = 144.348 m. Panel a: full range, in logarithmic scale. Panel b: zoom in the Arago bright spot in the central region [0, 500 μm], in linear scale. Panel c: transition region between shadow and light around 710 mm, in linear scale. 
Fig. 4 Horizontal cuts of the diffraction intensity Ψ_{z}(x, y)^{2} for a serrated occulter of 64 teeth of 20 mm, in logarithmic scale. Black shows the computation using the 2D FFT technique. Red depicts the computation using the Rubinowicz method. See text for comments. 
3.2 Analysis of the diffraction pattern
Figure 5 shows the shape of a few occulters, and the modulus and phase of the related complex amplitude Ψ _{z} (x, y) near the edge of the Fresnel pattern of the occulter. The representation is limited to a rectangular zone of 80 mm × 60 mm, and the central position is at 720 mm, which is already outside the projected solar limb of about 671 mm radius. Panel a shows the sharpedged disk, as a reference; panels b and c show serrated occulters defined by N_{t} = 512, Δ = 10 mm and N_{t} = 1024, Δ = 20 mm, respectively.We here study the transition zone near the edge and therefore computed the 2D diffraction patterns using the 2D FFT technique, which was proven to be valid in this region.
Figure 6 plots several cuts of the intensity Ψ _{z} (x, y)^{2} along the xaxis in logarithmic scale for a 64, 512, and 1024tooth disk; see the legends of the plots. The sharpedged disk is plotted behind as a reference. We also show the intensity of the corresponding apodized occulter, whose transmission extends linearly from 0 at R to 1 at R +Δ. Numerically, the Rubinowicz computation contains only very few divergence features, as was observed for the disk. Here, only one or two sampling points are invalid (either infinity or “not a number”), and they were removed. This model remains very satisfactory.
Several comments and observations can be made based on these results. First, we observe that the diffraction patterns produced by serrated occulters present a twolevel dark region, the inner being the darker. The spatial extent of this inner dark central zone strongly varies with the number and the size of the teeth. For instance, it is about 180 mm long for the 64tooth disk with 10 mm teeth, but up to 700 mm for the 1024tooth disk with 30 mm teeth. In general, the larger N_{t} and Δ, the longer the dark central zone.
Starting from the idea of Purcell & Koomen (1962) that the edges of the teeth reject the light outside the central part of the dark shadow of the occulter, Boivin (1978) reported the expected radius of this dark inner zone that is produced by the serrated occulter using geometrical considerations. According to his Eq. (1)a, this radius that we denote can be written as (16) where Λ is the halfapex angle of the teeth, and is half the angular separation of two successive teeth, that is, the angle under which the halftooth is seen from the center of the occulter. The angle Λ is derived from the geometry of the system, as given in Eq. (1b) of Boivin (1978), (17)
Figure 7 plots the Boivin radius function of the number of teeth for different tooth heights. As we see in Fig. 6, the Boivinpredicted radii plotted as vertical lines are in very good concordance with the computed diffraction patterns and with the related observation we made.
The diffraction intensity level can reach about 10^{−8}, with a high number of teeth. This is much better than the sharpedged disk; the gain is from two to four orders of magnitude in the central region. However, the apodized occulter remains the best in terms of rejection performance, but many long teeth allow one to comeclose to this level; see panel d of Fig. 6.
Finally, we investigated the effect of rotating the serrated occulter about its axis by computing the diffraction either along a line with a tooth trough or a line with a peak. This is illustrated in Fig. 8, where we used an occulter of 64 teeth. The diffraction curves are very similar inside Boivin’s central region, and they differ outside of it. The transition between shadow and light is shifted by a distance Δ, as expected.
As an additional analysis of the diffraction intensity, Appendix C treats the small variations with wavelength at the center of the umbra.
Fig. 5 2D representation of the occulter, the modulus , and the phase ∠Ψ_{z}(x, y) of the diffraction. The transition region between light and shadow is a rectangular zone of 80 mm × 60 mm with a central position at 720 mm. From top to bottom: Panel a: circular sharpedged occulter. Panel b: serrated occulter N_{t} = 512 and Δ = 10 mm. Panel c: serrated occulter N_{t} = 1024 and Δ = 20 mm. 
Fig. 6 Red: cuts of the diffraction intensity Ψ_{z}(x, y)^{2} for several serrated occulters in logarithmic scale. The vertical lines shows corresponding Boivin radii . Black: sharpedged disk. Blue: apodized disk. Panel a: N_{t} = 64, Δ = 10 mm. Panel b: N_{t} = 64, Δ = 20 mm. Panel c: N_{t} = 512, Δ = 10 mm. Panel d: N_{t} = 1024, Δ = 30 mm. 
Fig. 7 Boivinpredicted radius as a functionof the number N_{t} for different tooth heights Δ. The horizontal black dotted line at 671 mm corresponds to the projected stenope radius of the Sun for the geometry of ASPIICS. 
Fig. 8 Two cuts of the diffraction intensity Ψ_{z}(x, y)^{2}, one along a peak of a tooth, one along a trough, for a serrated occulter with 64 teeth of 10 mm. Black shows the trough of a tooth on the axis. Red plots the peak of a tooth on the axis. Panel a: full range, in logarithmic scale. Panel b: zoom in the central region [0, 500 μm], in linear scale. Panel c: transition region between shadow and light around 710 mm, in linear scale. 
4 Rejection performance of serrated occulters
4.1 Umbra profile
Aime (2013) computed the penumbra profile I_{z}(x, y) on the plane at z. This consists of the convolving the diffraction intensity Ψ_{z}(x, y)^{2} with the solar disk. Here, the assumption is that the shape of the projected geometrical shadow of the
occulter remains the same for every pointsource describing the solar disk. In other words, we assume that cos 2R_{⊙} is equal to 1, the actual value being 0.9997, (18)
where denotes the 2D projected stenope image of the Sun, including the centertolimb darkening. This function is radial and has a limited extent given by r_{⊙} = z × tanR_{⊙}, where R_{⊙} is the angular radius of the Sun. In the case of ASPIICS, r_{⊙} = 671 mm. We used the same centertolimb variation of the Sun as Rougeot et al. (2017), which was derived from Hamme (1993).
We chose to compute only a radial penumbra profile along the xaxis, even if there is no actual axisymmetry for the serrated occulter. Equation (18) becomes (19)
First of all, we can predict the theoretical rejection performance of serrated external occulters based on the discussion in Sect. 3. If the Boivin radius is large enough with respect to the projected solar radius, that is, , the convolution in Eq. (19) will integrate the solar image on the dark region of the twolevel diffraction intensity. On the other hand, if the Boivin radius is rather small, the solar image overlaps the bright area. The resulting umbra intensity at the center will thus not be low – in a relative sense. In Fig. 7, r_{⊙} is plotted as a horizontal line to facilitate comparison. This deep relation between the Boivin radius and the theoretical umbra may give a first indication about the ideal serrated external occulter. We must now verify this prediction.
We computed the penumbra I_{z}(r) for many serrated occulters up to 200 mm because the pupil is only some few centimeters in diameter, and we are not interested in the far penumbra. We preferred not to compute the full 2D diffraction patterns Ψ_{z}(x, y)^{2}, and chose toreconstruct it. To do so, we performed a linear interpolation using three different computations of the diffraction intensity on the xaxis: a tooth peak on the axis, a toothtrough on the axis, and the intermediate curve. This interpolation method was validated against the sharpedged disk. We retrieved the umbra level of 10^{−3.94} and the same penumbra profile as computed and described in Rougeot et al. (2017).
First, we analyze the effect of the number of teeth, given a fixed tooth height. Figure 9 plots the penumbra for 64, 384, 784 and 1024tooth disks with Δ = 20 mm. The sharpedged and apodized disks are also plotted as a reference. As expected, the more teeth, the deeper the umbra. Surprisingly, the 64tooth disk is slightly worse than the disk. In this particular case, the corresponding Boivin radius is small, and the diffraction intensity is brighter than that from the disk; see panel b of Fig. 6. The improvement from 784 to 1024 teeth is small. In this latter case, the gain is about 2.5 orders of magnitude with respect to the sharpedged disk, whereas the apodized disk remains the best. In the same way, we investigated the effect of the tooth size. Figure 10 plots the penumbra for a 484tooth disk with 10, 20, 30 and 40 mm teeth. Again as expected, the umbra deepens with increasing tooth size. A level of 10^{−7} is achieved with Δ = 40 mm, almost similar to the performance of the apodized occulter.
Fig. 9 Penumbra profile I_{z}(r) for several serrated occulters with 20 mm teeth in logarithmic scale in the range r ≤ 200 mm. Red shows N_{t} = 64. Blue shows N_{t} = 384. Green represents N_{t} = 784. Yellow plots N_{t} = 1024. Black showsthe sharpedged disk. The black dashed line represents the apodized disk. 
Fig. 10 Penumbra profile I_{z}(r) for several serrated occulters N_{t} = 464 teeth in logarithmic scale in the range r ≤ 200 mm. Red shows Δ = 10 mm. Blue represents Δ = 20 mm. Green plots Δ = 30 mm. Yellow depicts Δ = 40 mm. Black showsthe sharpedged disk. The black dashed line shows the apodized disk. 
4.2 Analysis and discussion
In order to quantify the gain in rejection performance, we took as a criterion the total power of diffracted sunlight that would enter the coronagraph. We computed the integrated umbra illumination P_{z} over the entrance aperture of radius R_{p}, which is 25 mm for ASPIICS, which we normalized by the case of the sharpedged disk. This criterion is defined as (20)
Figure 11 shows the color map of the power P_{z} (R_{p}) as a function of the number and the tooth size in logarithmic scale. The white line in the plot gives the pairs (N_{t}, Δ) such that above. For sake of comparison, the apodized disk gives a value .
Our prediction is well verified: serrated occulters with a Boivin radius larger than the projected solar radius perform better than the disk. Few teeth, between 200 and 400, would require long teeth to compensate for this in order to gain in rejection. Even more, occulters with N_{t} < 200 are worse than the disk. Conversely, occulters with 1000 teeth provide a substantial gain of two orders of magnitude, which is a factor 100. This satisfying result proves the actual advantage of shaped occulters, which must be preferred to simple disks.
As a final result, we considered a serrated occulter made of N_{t} = 1080 teeth, which consists of three teeth per angle degree. Such a tooth has base length of 4.13 mm, which seems reasonable. The height was fixed to Δ = 20 mm to avoid hiding too much of the inner solar corona. At the center, the umbra reaches a level I_{z} (0) = 10^{−6.61}, and the rejection gain is P_{z}(R_{p}) = 10^{−2.65}. The comparison with the apodized disk shows that the rejection from this latter is only better by a factor 5. Serrated occulters can therefore match this ideal performance closely.
However, we recall that our study remains theoretical and numerical. The serrated occulters we have considered here are perfect, in the sense that all teeth are perfectly identical. The actual results would be subject to tolerancing and manufacture defects. As discussed in Fort et al. (1978), the edge must be super polished, and the measured performance can degrade very quickly. The rejection performance we have shown here must be viewed as an upper boundary of what can be reached in reality.
Fig. 11 Integrated penumbra illumination P_{z}(R_{pup}) over the entrance aperture, function of the number of teeth N_{t}, and the teeth height Δ in logarithmic scale. The white line plots the pairs (N_{t}, Δ) such that . 
5 Conclusion
We computed the diffraction patterns from the onaxis pointsource at infinity for various serrated occulters made of very many teeth of a few centimeters each. We showed that serrated occulters tend to create a twolevel intensity, with the inner being the darker. Our results agree perfectly well with the prediction of Boivin (1978). The diffraction in this dark central region is lower by far than the diffraction produced by the sharpedged disk.
We showed that the umbra deepness of serrated external occulters is closely related to the Boivin radius, as predicted. Increasing the number and height of the teeth substantially improves the rejection performance. With 1000 teeth, a serrated occulter can reach a gain in rejection of a factor 100 with respectto the disk, and it closely matches the performance of the apodized disk.
Nevertheless, we underline that our study remains theoretical and numerical. A first question concerns the actual feasibility of manufacturing such serrated occulters with very many teeth. Second, the actual performance is driven by the quality of the overall system. Any defects from tolerancing of the edge will degrade it. However, these defects would be purely mechanical, and it is probably easier to achieve a very good quality for the occulter itself, in comparison to what is required for the optics in the Lyot coronagraph. A comparison with an experimental approach would be of interest, as was reported by Landini et al. (2016).
Finally, we are interested in the complete endtoend performance of the instrument that results from the combination of the external occulter and the hybrid Lyot coronagraph. The diffracted wavefront made by serrated occulters must be propagated through the coronagraph, in order to assess the residual level of sunlight on the final focal plane. Moreover, increasing the tooth size introduces vignetting, which may degrade the inner field of view. This would require a delicate imagereconstruction processing for observations of the corona very close to the solar limb. This study is left for future work.
In addition to the strict consideration of serrated occulters, we investigated diffraction modeling in the context of solar coronagraphy, and studied several numerical methods. The brute force FFT technique has the strong advantage of directly computing the 2D diffraction patterns, but it may unfortunately be limited by available computational resources. As an alternative, the Maggi–Rubinowicz representation has proven to be very powerful for a 1or0 occulter, allowing us to integrate only along the edge. These diffraction models will enable us to investigate other types of occulters and to consider tolerancing and defects of the edge, for instance, in both solar and stellar coronagraphy.
Acknowledgements
We would like to thank R. Flamary and D. Mary for their help in this study. We are grateful to the anonymous referee for their time and suggestions to improve our paper. We also thank the European Space Agency and Thales Alenia Space for supporting this activity.
Appendix A: Deviation of the sampling condition for the FFT technique
In this appendix we explore the deviation from the sampling rules we established in Sect. 2. Here, we take as study case a sharpedged disk with a radius R = 300 mm. We fixed the size of the array to K = 2^{15} = 32 768. Equations (10) and (6) give the relations constraining the bestfit sampling σ_{opt}, (A.1) (A.2) (A.3)
Figure A.1 shows a cut of the diffraction intensity Ψ _{z} (x, y)^{2} in logarithmic scale for various samplings: 0.4, 0.5, 0.8, 1, 2, 3, 4, 5, and 6 times the value σ_{opt}. The curves are shifted by two decades in the vertical axis with respect to each other.
Fig. A.1 Horizontal cuts of the diffraction intensity Ψ_{z}(x, y)^{2} in logarithmic scale, computed with the 2D FFT technique, for a fixed size K = 32 768 and for various samplings σ. The occulteris a sharpedged disk of radius R = 300 mm. The curves are shifted by two decades with respect to each other. From bottom to top: σ = 295.3 μm, 246.1 μm, 196.9 μm, 147.7 μm, 98.4 μm, σ_{opt} = 49.2 μm, 39.4 μm, 24.6 μm, and 19.7 μm. 
First, when σ < σ_{opt}, the Fresnel filter is no longer correctly sampled because the variation of its phase at the edge of the array exceeds π. Related curves are therefore much more noisy. We also note that the spatial field decreases accordingly, and Eq. (A.3) is no longer fulfilled. Second, when σ > σ_{opt}, Eq. (A.2) is not respected. The structure of the Arago spot even disappears quickly as the sampling increases with respect to the nominal value because the required high spatial frequencies to described the disk are not captured. In the end, the best diffraction pattern is correctly obtained for σ_{opt}.
Appendix B: Detailed expression of the Rubinowicz boundary integral
We recall the general expression of the diffraction disturbance given in Eq.(14): (B.1) where l = (l_{ξ}, l_{η}, 0) is the unit vector in the direction tangent to the edge at any point P = (ξ, η, 0) of , s is the distance between P and the observation point (x, y, z), and dl is the differential curvilinear element of the edge.
In the case of the disk, we take advantage of the axisymmetry, and the function Ψ ^{(d)} is written as a radial function. The edge can be parametrized by its polar angle ϕ, such that ξ = Rcosϕ and η = Rsinϕ. The unit tangent vector, counted counterclockwise, at any point P(ϕ) is simply l = (sinϕ, −cosϕ, 0). Thus, the diffraction disturbance becomes (B.2)
The same expansion can be performed for a serrated occulter. We recall that the occulter is defined by its number of teeth N_{t} and the tooth height Δ. Two other important geometrical parameters are the halfapex angle Λ of the teeth, and the half angular separation of two successive teeth, as given in Eq. (17). In polar coordinates, the edge of the serrated occulter is given by (B.3) where mod(a, b) is the rest of the Euclidean division of a by b. Then, by differentiating the expressions of ξ = ρcosϕ and η = ρsinϕ, we obtain the two components l_{ξ}, l_{η} of the tangent vectors, where sign(a) is the function returning the sign of a. Injecting Eqs. (B.4), (B.5) into Eq. (B.1), we derive the analytical integral over ϕ that gives the diffraction disturbance . This integral may be numerically solved using dedicated integration tools.
Appendix C: Variation of diffraction intensity with wavelength
In this section, we illustrate the variation in diffraction intensity Ψ _{z} ^{2} with wavelength. Here, the occulter is made of N_{t} = 128 teeth of Δ = 20 mm. We consider different wavelengths λ in white light, the bandpass of ASPIICS. Figure C.1 shows some results to illustrate this: a cut of the diffraction intensity Ψ _{z} (x, y)^{2} in logarithmic scale for λ = 540, 550, 560, and 570 nm.
Fig. C.1 Horizontal cuts near the center of the diffraction intensity Ψ_{z}(x, y)^{2} in logarithmic scale. The occulter is made of N_{t} = 128 teeth of Δ = 20 mm. Panel a: λ = 540 nm. Panel b: λ = 550 nm. Panel c: λ = 560 nm. Panel d: λ = 570 nm. 
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All Tables
All Figures
Fig. 1 Red: Fresnel diffraction intensity of the apodized occulter. The transmission evolves linearly from 0 at 710 mm radius to 1 at 720 mm. Black: sharpedged disk as a reference. 

In the text 
Fig. 2 Red: Fresnel diffraction intensity from the 2D FFT computation Eq. (2). Black shows the Hankel computation Eq. (5) as a reference. The occulter is a sharpedged disk of radius 710 mm set at z = 144.348 m. An array of 156 000 × 156 000 and a sampling σ = 22.6 μm have been used. Panel a: full range, in logarithmic scale. Panel b: zoom in the Arago bright spot in the central region [0, 500 μm], in linear scale. Panel c: transition region between shadow and light around 710 mm, in linear scale. 

In the text 
Fig. 3 Red: Diffraction intensity using the Rubinowicz representation. Black shows the Hankel computation Eq. (5) as a reference. The occulter is a sharpedged disk of radius 710 mm set at z = 144.348 m. Panel a: full range, in logarithmic scale. Panel b: zoom in the Arago bright spot in the central region [0, 500 μm], in linear scale. Panel c: transition region between shadow and light around 710 mm, in linear scale. 

In the text 
Fig. 4 Horizontal cuts of the diffraction intensity Ψ_{z}(x, y)^{2} for a serrated occulter of 64 teeth of 20 mm, in logarithmic scale. Black shows the computation using the 2D FFT technique. Red depicts the computation using the Rubinowicz method. See text for comments. 

In the text 
Fig. 5 2D representation of the occulter, the modulus , and the phase ∠Ψ_{z}(x, y) of the diffraction. The transition region between light and shadow is a rectangular zone of 80 mm × 60 mm with a central position at 720 mm. From top to bottom: Panel a: circular sharpedged occulter. Panel b: serrated occulter N_{t} = 512 and Δ = 10 mm. Panel c: serrated occulter N_{t} = 1024 and Δ = 20 mm. 

In the text 
Fig. 6 Red: cuts of the diffraction intensity Ψ_{z}(x, y)^{2} for several serrated occulters in logarithmic scale. The vertical lines shows corresponding Boivin radii . Black: sharpedged disk. Blue: apodized disk. Panel a: N_{t} = 64, Δ = 10 mm. Panel b: N_{t} = 64, Δ = 20 mm. Panel c: N_{t} = 512, Δ = 10 mm. Panel d: N_{t} = 1024, Δ = 30 mm. 

In the text 
Fig. 7 Boivinpredicted radius as a functionof the number N_{t} for different tooth heights Δ. The horizontal black dotted line at 671 mm corresponds to the projected stenope radius of the Sun for the geometry of ASPIICS. 

In the text 
Fig. 8 Two cuts of the diffraction intensity Ψ_{z}(x, y)^{2}, one along a peak of a tooth, one along a trough, for a serrated occulter with 64 teeth of 10 mm. Black shows the trough of a tooth on the axis. Red plots the peak of a tooth on the axis. Panel a: full range, in logarithmic scale. Panel b: zoom in the central region [0, 500 μm], in linear scale. Panel c: transition region between shadow and light around 710 mm, in linear scale. 

In the text 
Fig. 9 Penumbra profile I_{z}(r) for several serrated occulters with 20 mm teeth in logarithmic scale in the range r ≤ 200 mm. Red shows N_{t} = 64. Blue shows N_{t} = 384. Green represents N_{t} = 784. Yellow plots N_{t} = 1024. Black showsthe sharpedged disk. The black dashed line represents the apodized disk. 

In the text 
Fig. 10 Penumbra profile I_{z}(r) for several serrated occulters N_{t} = 464 teeth in logarithmic scale in the range r ≤ 200 mm. Red shows Δ = 10 mm. Blue represents Δ = 20 mm. Green plots Δ = 30 mm. Yellow depicts Δ = 40 mm. Black showsthe sharpedged disk. The black dashed line shows the apodized disk. 

In the text 
Fig. 11 Integrated penumbra illumination P_{z}(R_{pup}) over the entrance aperture, function of the number of teeth N_{t}, and the teeth height Δ in logarithmic scale. The white line plots the pairs (N_{t}, Δ) such that . 

In the text 
Fig. A.1 Horizontal cuts of the diffraction intensity Ψ_{z}(x, y)^{2} in logarithmic scale, computed with the 2D FFT technique, for a fixed size K = 32 768 and for various samplings σ. The occulteris a sharpedged disk of radius R = 300 mm. The curves are shifted by two decades with respect to each other. From bottom to top: σ = 295.3 μm, 246.1 μm, 196.9 μm, 147.7 μm, 98.4 μm, σ_{opt} = 49.2 μm, 39.4 μm, 24.6 μm, and 19.7 μm. 

In the text 
Fig. C.1 Horizontal cuts near the center of the diffraction intensity Ψ_{z}(x, y)^{2} in logarithmic scale. The occulter is made of N_{t} = 128 teeth of Δ = 20 mm. Panel a: λ = 540 nm. Panel b: λ = 550 nm. Panel c: λ = 560 nm. Panel d: λ = 570 nm. 

In the text 
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