APODIZED PUPIL LYOT CORONAGRAPHS FOR ARBITRARY APERTURES. II. THEORETICAL PROPERTIES AND APPLICATION TO EXTREMELY LARGE TELESCOPES

In this section, we follow the notations of Aime et al. (2002) and Soummer et al. (2003a) to recall the general formalism of coronagraphy with apodized pupils. The general layout of an APLC, given in Figure 1, is similar to a classical Lyot coronagraph with an additional upstream pupil apodization. The setup consists of an ensemble of apodizers, masks, and stops in four successive planes A, B, C, D, where A is the entrance aperture, B is the focal plane with the occulting mask, C is an image of the entrance aperture where a pupil mask called Lyot stop is placed, and D is the final image plane. We will consider the usual approximations of paraxial optics, and that the optical layout is properly designed to cancel the quadratic phase terms associated with the propagation, so that a Fourier transform exists between each plane.

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The telescope aperture function with the position vector $\textbf {r}$ is denoted by $P(\textbf {r})$ (index function equal to 1 inside the aperture $\mathcal {P}$), and $\Phi (\textbf {r})$ is the apodizer transmission (1 without apodization). A hard-edged mask of transmission $1-M(\textbf {r})$ is placed in the focal plane. M is the index function that describes the mask shape $\mathcal {M}$, equal to 1 inside the coronagraphic mask and 0 outside (Ferrari et al. 2007). In plane C, the Lyot stop consists of a pupil mask, which can be eventually slightly undersized for optimization, but we will not consider this possibility in this paper. The field amplitude in the four successive planes are as follows:

$$\begin{eqnarray} &&\Psi _A(\textbf {r})=P(\textbf {r}) \Phi (\textbf {r}), \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&\Psi _B(\textbf {r})= \widehat{\Psi }_A(\textbf {r})(1-M(\textbf (r))), \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\Psi _C(\textbf {r})= \Psi _A(\textbf {r})-\big (\Psi _A(\textbf {r})\ast \widehat{M}(\textbf {r})\big)P(\textbf {r}), \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&\Psi _D(\textbf {r})= \widehat{ \Psi }_A(\textbf {r}) - \big (\widehat{\Psi }_A(\textbf {r}) M(\textbf {r})\big) \ast \widehat{P}(\textbf {r}). \end{eqnarray} \tag{ 4 }$$

In these equations, $\widehat{ }$ denotes the Fourier transform, and we assume that the focal lengths of the successive optical systems are identical (if not, an appropriate change of variables leads to the same result). Also, we re-orientate the axis in the opposite direction and omit the −1 proportionality factor for better readability. We consider the monochromatic case in this paper to analyze the properties of the solutions for arbitrary apertures. The chromatic formalism was given in previous papers and achromatization was studied by Soummer et al. (2003b) and Aime (2005b).

Lyot (or Roddier coronagaphs) can be well understood by the analysis of the Lyot plane amplitude (Equation (3)), which can be interpreted as a two-wave interference between the pupil amplitude and the wave diffracted by the mask. In the case of a classical Lyot coronagraph, the subtraction of these two terms is not efficient (Figure 2, left panel). One can deduce two complementary coronagraphic approaches from the analysis of this figure.

• 1.  
  A perfect subtraction of the two terms is obtained if the wave diffracted by the mask is equal to the pupil amplitude, at least in a reduced area of the pupil. This case corresponds to the band-limited mask coronagraph (Kuchner & Traub 2002).
• 2.  
  The APLC approach is to apodize the pupil transmission so that the amplitude profile of the wave diffracted by the mask corresponds to the shape of the pupil transmission shape. Optimal apodization functions exist for this problem and are given by the prolate spheroidal functions.

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The intensity in the final focal plane is not invariant by translation, but we will use the terminology of coronagraphic PSF according to the general usage in the community.

According to Equation (3) and Figure 2, the condition for a total extinction of the starlight is obtained if the wave diffracted by the mask (the convolution integral) is equal to the entrance pupil amplitude (the apodization function):

$$\begin{equation} \Psi _A(\textbf {r})=\big (\Psi _A(\textbf {r})\ast \widehat{M}(\textbf {r})\big)P(\textbf {r}). \end{equation} \tag{ 5 }$$

Solutions to Equation (5) can be found if $\Phi (\textbf {r})$ is the eigenfunction of the convolution integral. This problem is well known in signal processing and is equivalent to a generalization of the uncertainty principle (Papoulis 1968), or to the search of band-limited functions having the maximum encircled energy within a given spatial domain. A total extinction solution exists for a Roddier coronagraph with a prolate apodization (Ferrari et al. 2007). Unfortunately, a total extinction is not achievable with an APLC, but thank to the properties of prolate functions discussed below, it is possible to approach this perfect solution with optimal properties.

In coronagraphic terms, the solution of this problem corresponds to the pupil apodization that produces the most concentrated starlight behind a given focal plane mask (and thus blocked). This formal problem has been solved analytically for rectangular and circular unobstructed apertures (Aime et al. 2001, 2002; Soummer et al. 2003a), using linear and circular prolate functions (Slepian & Pollak 1961; Frieden 1971; Papoulis 1981). Formal solutions exist in the generalized case for any shape of aperture (Slepian 1964, 1965; Papoulis 1968), and such solutions find a direct application in coronagraphy. In Appendix A, we provide a sketch of the proof of the existence of this set of functions for an arbitrary geometry and review the properties that make them appealing for coronagraphs with central obstructions, segments, or spiders. Using a normalized prolate function for entrance pupil apodization $\Phi (\textbf {r})=\varphi _0(\textbf {r})/\max _{\textbf {r}}(\varphi _0(\textbf {r}))$, and writing as Λ₀ the associated eigenvalue, we obtain a new set of equations for the four planes:

$$\begin{eqnarray} \Psi _A(\textbf {r})&=&P(\textbf {r}) \Phi _0(\textbf {r}), \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \Psi _B(\textbf {r})&=& \widehat{\Psi }_A(\textbf {r})(1-M(\textbf {r})), \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \Psi _C(\textbf {r})&=& (1-\Lambda _0)\Phi _0(\textbf {r})P(\textbf {r}), \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \Psi _D(\textbf {r})&=& (1-\Lambda _0)\widehat{ \Psi }_A(\textbf {r}), \end{eqnarray} \tag{ 9 }$$

If the focal plane mask $\mathcal {M}$ is symmetric, the Fourier transform of its index function, $\widehat{M}$, is real and symmetric and the eigenfunctions are real. If the mask is not symmetric, its Fourier transform is Hermitian and the eigenvalues Λ_n are real, but the eigenfunctions are complex. In the following, we will only consider symmetric focal plane masks (e.g., circular), since there is no particular reason to choose an asymmetric mask in most common cases, and the symmetry of the mask guarantees a real apodizer.

One of the possible criteria to evaluate the performance of a coronagraph is the residual energy inside the Lyot stop (here identical to the pupil), normalized to the apodizer throughput. This criterion finds a very simple expression as a function of the eigenvalue:

$$\begin{equation} e=\frac{\int _\mathcal {P}|\Psi _C(\textbf {r})|^2 d\textbf {r}}{\int _\mathcal {P} |\Psi _A(\textbf {r})|^2 d\textbf {r}}=(1-\Lambda _0)^2. \end{equation} \tag{ 10 }$$

As discussed by Ferrari et al. (2007), important analytical simplifications are obtained if the mask domain $\mathcal {M}$ is a scaled version of the pupil domain $\mathcal {P}$. More generally, prolate solutions exist in any case, and we will study in Section 3.2 the impact of the choice of the mask geometry on the overall performance.

It is remarkable that the Lyot stop amplitude is proportional to the entrance pupil amplitude, inside the aperture area $\mathcal {P}$, even in the presence of secondary mirror support structures: all the discontinuities in the pupil appear in negative, with no light bleeding into the pupil as illustrated in the Lyot plane in Figure 3. This property derives straightforwardly from the eigenvalue problem and we can recognize the prolate apodization, identical inside the aperture in both planes to a scaling factor. The apodization function can be exactly recovered to the eigenvalue attenuation factor by multiplying the Lyot stop amplitude by the pupil function. In the case illustrated in Figure 3, the apodizer has been calculated for the exact geometry and although the apodizer looks rotationally symmetric to the eye, it is slightly different from the apodizer that would have been calculated with the same mask and no spider vanes in the pupil. In the case where the support structures are not included in the definition of the apodizer, residual light is diffracted in the Lyot plane and has to be mitigated by the Lyot stop optimization (Sivaramakrishnan & Lloyd 2005). This might be preferable for practical reasons to use a rotationally symmetric apodizer in the case of pupil rotation. Small features like segmentation behave in a similar way. However, in the cases of small gaps between segments, it is not beneficial to include the segmentation in the calculation of the apodizer and the diffracted light can be optimized in the Lyot plane with a reticulated Lyot stop (Sivaramakrishnan & Yaitskova 2005).

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This proportionality property between the entrance aperture and the Lyot plane enables the possibility of multiple-stage APLCs (Aime & Soummer 2004a). After the Lyot stop, the field amplitude is proportional to the entrance pupil apodization, and can be used as the entrance pupil of a second APLC. For the second coronagraphic stage, the apodization is therefore produced by the output of the first stage, and no additional apodizing element is needed. In principle, several successive stages can be used. Each coronagraphic stage provides the same rejection factor of the starlight. An interesting aspect of this approach is that there is only one physical apodizer involved, at the entrance of the system. Therefore, the throughput is not reduced by successive stages, and the total residual energy after k coronagraphic stages is simply:

$$\begin{equation} e_k=(1-\Lambda _0)^{2 k}. \end{equation} \tag{ 11 }$$

Multiple-stage APLCs require that the Lyot stop field amplitude be proportional to a prolate function. In broadband, with a gray apodizer and a hard-edge mask, this is only verified at a single wavelength and multistage APLCs produce a limited improvement compared to the ideal case, even using a numerical optimization of the mask size. A perfect achromatic solution can be obtained with an apodizer producing the requisite chromatic apodization and the result is close to that of an achromatic APLC (Aime 2005a). Interferometric methods can be used to produce such an apodizer (El Azhari et al. 2006; Carlotti et al. 2007). A simple method for partial achromatization is to manufacture an absorbing apodizer where the material is itself chromatic, and approximates the required wavelength variation (Soummer et al. 2006).

As shown in Section 2.2, the geometry of $\mathcal {P}$ and $\mathcal {M}$ defines an eigenvalue problem, with the optimum solution (Λ₀, φ₀) for coronagraphy. Once the geometry of the mask has been chosen, a scale parameter for its size remains as a free parameter. This parameter corresponds to the so-called prolateness parameter c (Frieden 1971). For rectangular and circular geometries, the eigenvalue Λ₀ can be controlled by the size of the focal plane mask (Aime et al. 2002; Soummer et al. 2003a), and is a strictly increasing function of the mask size that approaches one asymptotically. APLCs take advantage of the particular shape of this function, which saturates quickly, and thus enables very high rejections with values of Λ₀ very close to 1, for relatively small masks (typically a few resolution elements). This monotonic behavior of the eigenvalue with the mask size is less obvious in the case of arbitrary geometries. However, it is still true and we detail the mathematical demonstration of this property in Appendix B. The immediate consequence is that the residual energy (Equation (10)) is monotonically decreasing with the mask size.

Prolate spheroidal function exists for any geometry, and we calculate the eigenfunctions using a numerical algorithm presented in Figure 4. This algorithm was initially proposed by Guyon & Roddier (2000) to calculate optimal apodizations for the Roddier phase mask. We recently discovered that this method is also known in the field of signal processing and laser optics (Papoulis 1968; Schulz 2004) and based on classical tools of functional analysis (Riesz & Sz.-Nagy 1990, p. 491).

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The convergence of the algorithm can be demonstrated using the fact that eigenfunctions φ_n form a complete orthogonal basis over the aperture $\mathcal {P}$. We start the algorithm with an initial estimate of the entrance pupil apodization, Ψ⁽⁰⁾_A, which can be decomposed over the prolate basis:

$$\begin{equation} \Psi _A ^{(0)}=\alpha ^{(0)} \left(\varphi _0+\sum _{n=1}^{\infty } \gamma _n \varphi _n\right) P, \end{equation} \tag{ 12 }$$

where α⁽⁰⁾ is a coefficient that corresponds to the projection of Ψ⁽⁰⁾_A onto φ₀. Superscripts denote the iteration number. We consider the pupil itself P for our initial estimate, as it insures to have a nonzero projection onto the first eigenfunction φ₀. This pupil complex amplitude is propagated to the Lyot plane, and we obtain

$$\begin{eqnarray} \Psi _C ^{(0)}=&&\alpha ^{(0)} \varphi _0 P-\alpha ^{(0)} \Lambda _0 \varphi _0 \nonumber \\ &&+ \alpha ^{(0)} \left(\sum _{n=1}^{\infty } \gamma _n \varphi _n\right)P-\alpha ^{(0)} \sum _{n=1}^{\infty } \gamma _n \Lambda _n \varphi _n. \end{eqnarray} \tag{ 13 }$$

This Lyot plane amplitude is then subtracted from the pupil amplitude and limited to the pupil:

$$\begin{eqnarray} &&\hspace{-3.6pc}\Psi ^{(0)}=\big(\Psi _A ^{(0)}-\Psi _C ^{(0)}\big) P \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} &&\hspace{2.9pc}=\alpha ^{(0)} \Lambda _0 \left(\varphi _0+\sum _{n=1}^{\infty } \frac{ \Lambda _n}{\Lambda _0} \gamma _n\varphi _n\right) P. \end{eqnarray} \tag{ 15 }$$

We obtain the next iterate after renormalization:

$$\begin{eqnarray} &&\hspace{-4.9pc}\Psi _A ^{(1)}=\frac{\Psi ^{(0)} }{\max (\Psi ^{(0)})} \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} &&\hspace{3.3pc}=\frac{\left(\varphi _0+\sum _{n=1}^{\infty } (\Lambda _n/\Lambda _0) \gamma _n \varphi _n \right) P}{\max \left(\varphi _0+\sum _{n=1}^{\infty } (\Lambda _n/\Lambda _0)\gamma _n \varphi _n\right)}. \end{eqnarray} \tag{ 17 }$$

This expression can be generalized to the kth iterate after some algebra:

$$\begin{equation} \Psi _A ^{(k)}=\frac{\left(\varphi _0+\sum _{n=1}^{\infty } (\Lambda _n/\Lambda _0) ^k \gamma _n \varphi _n\right) P}{\max \left(\varphi _0+\sum _{n=1}^{\infty } \left(\Lambda _n/\Lambda _0\right) ^k \gamma _n \varphi _n\right)}. \end{equation} \tag{ 18 }$$

Because eigenvalues are ordered as Λ₀ > Λ₁ > Λ₂ > ⋅ ⋅ ⋅ >0 (Slepian 1964), the convergence of the algorithm to the first eigenvalue Φ₀ is always guaranteed. In the perfect case, the ratio $(\Psi _C^{(\infty)}(\textbf {r})/\Psi _A^{(\infty)}(\textbf {r}))P(\textbf {r})$ is constant over $\mathcal {P}$ and equal to 1 − Λ₀. We use the uniformity of this ratio as a stopping criterion for the algorithm, as it gives a proxy for the precision on the eigenvalue. We find satisfactory results when we stop the iterations when the precision on the eigenvalue is 0.1%.

In this section we study the performance and sensitivity of an APLC to central obstruction in a circular aperture, with circular focal plane mask. We first generate a large number of apodizers to sample finely the obstruction and mask size parameter space. Depending on the obstruction and mask size, we identify two morphologies of apodizers: bell shaped or bagel shaped. In Figures 5 and 6, we show the prolate apodizers for a range of obstruction and mask sizes.

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Small obstruction and small masks produce bell-shaped apodizers (bottom left in Figure 5), whereas large obstructions and/or large mask (top right in Figure 5) produce bagel-shaped apodizers. Denoting by D₁ the diameter of the primary aperture and by D₂ the diameter of the obstruction, the existence of these two regimes can be understood by comparing the scales of D₁/α and D₂, where α denotes the mask diameter in resolution elements (Soummer 2005). Indeed, in the convolution product of Equation (3), the extent of the term $\widehat{M}(\textbf {r})$ is proportional to D₁/α. For example, if D₁/α ≪ D₂, the convolution product $\big (\Psi _A(\textbf {r}\big)\ast \widehat{M}(\textbf {r})$ will go to zero at the center of the aperture and the apodizer will be bagel shaped. Conversely, if D₁/α ≫ D₂, the convolution product at the center of the aperture will only have a negligible dip, and the resulting apodizer is bell shaped. Empirically, we consider that the bell/bagel transition is obtained for α ≈ D₁/(2D₂). For example, for a 20%, the transition region between bell-shaped and bagel-shaped apodizer is for a mask size greater than 2.5Λ/D.

This discussion can be generalized for any features in the pupil, and in particular for secondary mirror support structures and segments. For a given feature of characteristic size d, we observe a similar effect where the apodization function is modified around the structure, when the mask size is larger than D/(2d) (in resolution element units). This is illustrated in Figure 14 where we can see that the secondary mirror support structures start to influence the optimal apodization shape for masks sizes greater than 5λ/D (in this figure, the vanes are approximately of width D/10).

In the case of telescope geometries where the secondary mirror support structures are very thin (as for example on Gemini where d ≈ 1 cm), their effect is negligible in the calculation of the optimal apodizer. Sivaramakrishnan & Lloyd (2005) estimated the effects of such spiders on APLC coronagraphic contrast. These very thin support structures only affect the theoretical prolate apodizer for very large focal plane masks (a few ten λ/D). The apodizer would present similar features to that observed in Figure 14. Because of the scientific goals on a 8 m class telescope, such large mask are not considered and spiders are ignored for the calculation of the apodizer. Also, a nonrotationally symmetric apodizer would be much more difficult to manufacture and align, especially if pupil rotation is involved for an Alt-Az telescope.

The first consequence of these two regimes (bell/bagel), already discussed in Soummer (2005), is the impact on the throughput. By throughput we mean here the ratio of the number of companion's photons reaching the final science detector over the total number of companion photons that enter the coronagraph. This definition corresponds to the total throughput and does not take into account the final shape of the image of an off-axis source, whose spatial distribution can affect the instrument S/N. However, as seen in Equation (9) the off-axis response field after an APLC is still a prolate function of order zero, whose eigenvalue is very close to unit; as a consequence, most of the light is concentrated in the core of the PSF and making most of the photons readily available for the detection or characterization of the companion. Thus for the remainder of this paper we will use this definition of throughput. bell-shaped apodizers have a much lower throughput than bagel-shaped apodizers. A representation of the throughput for the parameter space we studied is shown in Figure 7. For line plots as a function of central obstruction, see Figure 2 of Soummer (2005). The transition region between low-throughput bell-shaped and high-throughput bagel-shaped apodizers is visible along the diagonal in the figure where the contour lines increase sharply, and is indicated by an arrow. It is interesting to note that for a given geometry, the throughput is not a monotonic function: after a steep increase in throughput, which corresponds to the bell–bagel transition, the throughput decreases again. This throughput decrease with mask size in the bagel regime is analog to that observed for rectangular and circular apertures where the apodizer strength increases with mask size.

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Several criteria can be considered to study the sensitivity and to optimize APLCs. A first obvious criterion is the integrated residual energy, normalized to the apodizer throughput (Equation (10)). Because the apodizer throughput is not a monotonic function of mask size (e.g., due to the bell–bagel transition), it is not intuitive that the normalized residual energy e monotonicly decreases with mask size. Note that when the mask radius increases, the apodization function changes (possibly significantly, e.g., bell/bagel transition), and so does the PSF. We illustrate the residual energy criterion in Figure 8 for the circular geometry with central obstruction.

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Another interesting criterion is to consider the residual energy outside the mask area in the final focal plane, normalized to the throughput. Because the final focal plane amplitude $\Psi _D(\textbf {r})$ is proportional to the prolate function itself (Equation (9)), we can find a simple expression for this term (Soummer et al. 2003a):

$$\begin{eqnarray} e_{om}&=&\frac{1}{T}\, \left(\int _\mathcal {P}|\Psi _C(\textbf {r})|^2 d\textbf {r} - \int _\mathcal {M}|\Psi _D(\textbf {r})|^2 d\textbf {r}\right) \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} &=& e_{om}=(1-\Lambda _0)^3. \end{eqnarray} \tag{ 20 }$$

This last expression shows that the residual energy outside the mask has a similar behavior to the residual energy criterion, with no specific optimum mask size for a given geometry.

A slightly better criterion is the residual starlight energy in the region of the final focal plane where an off-axis companion can be detected, normalized to the apodizer throughput. For that, we define the inner working angle (IWA), as the minimum distance at which the transmission of an off-axis companion is significantly high. Although the off-axis transmission of a coronagraph is a smooth function of the angular position, a good rule of thumb is to consider the IWA equal to the mask radius augmented by one resolution element (see also, Lloyd & Sivaramakrishnan 2005). Intuitively, if a source is located just outside the mask, it is almost fully transmitted by the coronagraph. This criterion does not produce a significant difference compared to e_om in the perfect monochromatic case and we did not represent the result. However, this is a better criterion for broadband optimization and it was used by Soummer (2005).

Another common criterion is based on the contrast at a given angular separation r₀, between an average intensity level at a given radial distance in the final focal plane and the maximum of the PSF p for an off-axis source:

$$\begin{equation} \mathcal {C}(r_0)=\frac{1}{p}\, \frac{1}{2 r_0 \lambda /D} \int _{r_0-\lambda /D}^{r_0+\lambda /D} |\Psi _D(r)|^2 r dr. \end{equation} \tag{ 21 }$$

In Equation (21), we consider an arbitrary average of the contrast over an annulus of width 2λ/D, in order to remove the effects of ringing in the coronagraphic PSF. Note that both the apodizer and the Lyot stop impact the maximum intensity of an off-axis source p and that the actual off-axis transmission of the coronagraph can be included readily as a weighting function in p. We show a few of these contrast curves at an angular separation of r₀ = 6 resolution elements in Figure 9. A contrast criterion is very useful for the optimization of a coronagraph over a particular region of interest, since the width of the annulus can be changed to suit science requirements. We also considered a definition of the contrast based on a one-dimensional average over a radial profile:

$$\begin{equation} \mathcal {C}_2(r_0)=\frac{1}{p}\, \frac{1}{2 \lambda /D} \int _{r_0-\lambda /D}^{r_0+\lambda /D} |\Psi _D(r)|^2 dr. \end{equation} \tag{ 22 }$$

This definition produces similar results, which are not shown here. The criterion of the residual energy outside the mask area was used in Soummer (2005), and a discussion of contrast criteria for APLCs was also done by Martinez et al. (2008). A criterion based on the Airy throughput (Vanderbei et al. 2003) would be interesting to include the consideration of the useful flux of the planet, i.e., which fraction of the total planet flux can be used for detection. With an APLC, the off-axis PSF (planet) is simply the PSF produced by the apodized aperture, assuming that the source is sufficiently outside the focal plane mask area (outside the IWA). By definition, this PSF has maximum encircled energy within a circle equal to the focal plane mask area. The fractional energy of the PSF within this circle is Λ₀, which is very close to 1 for an APLC (typicallyΛ₀ > 0.99). This ensures that most of the planet flux is usable for detection with an APLC. Significant effect on the Airy throughput is observed in the case of shaped pupil apodization, and this criterion should be considered if this method is used to produce the apodizer of an APLC.

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In the simple monochromatic case we presented here, these criteria provide consistent results and illustrate that there is no particular optimum for the mask size of an APLC for a given circular geometry with central obstruction. However, in large band, an optimum can be found in some cases, and these criteria are useful for the optimization and designs for a specific geometry. In particular, it is possible to optimize the mask size according to the chromatic properties of the apodizer (gray or chromatic apodizer; Soummer et al. 2006).

It is interesting to note that in some regions of the parameter space shown in Figures 8 and 9, the contrast (or other criterion) can be somewhat independent of the mask size. This has important consequences in the case of broadband optimization with the existence of quasi-achromatic solutions, which are used in the GPI coronagraph design (Macintosh et al. 2008), and which will be detailed in a future paper.

In the previous section, we only considered the case of a circular mask. As explained in Section 2.2, focal plane masks do not have to be circular, and as long as they are symmetric, the prolate apodizer is real. We study in this section the comparison between circular and noncircular masks for two examples.

For a hexagonal aperture, with similarities to the Keck telescope for example, or more generally for future segmented telescopes, the PSF corresponding to the hexagonal aperture shows some hexagonal symmetry. We study if a coronagraph would benefit from having a hexagonal focal plane mask that may match the PSF better. For that, we calculate the optimal generalized prolate apodization in two cases, with a circular mask and with a hexagonal mask (Figure 10). In this calculation, the hexagonal mask and the circular mask have the same surface. We find that the two prolate apodizers are very similar and the residual energy is slightly better for a circular mask. Corresponding contrast curves normalized to the off-axis PSF maximum are shown in Figure 11. Although we have not studied the entire parameter space for the hexagonal geometry, we conclude that there is no interest to use hexagonal masks in this case. A possible explanation for this behavior is that the apodizer circularizes the PSF. This is illustrated in Figure 12 where we show the PSFs corresponding to the same geometry as shown in Figure 10. The apodized PSF has been partially circularized by the apodizer.

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As mentioned previously and detailed by Ferrari et al. (2007), analytical derivations are possible if the mask $\mathcal {M}$ is a scaled version of the pupil shape $\mathcal {P}$. We analyze the performance of a pierced mask, compared with the usual circular mask. We calculated the generalized prolate solution in both cases. In Figure 13, we show that the performance of the pierced mask is degraded for a Gemini/VLT-like aperture geometry. Intuitively, this can be explained because the hole at the center of the mask creates a leakage of light where the PSF is the brightest. Also, its diffraction is a uniform background in the Lyot stop since the pinhole at the center of the mask is much smaller than the resolution element.

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The generalized prolate spheroidal solutions are highly relevant for APLCs on future ELT projects. Indeed, these telescopes will not be monolithic but will include necessarily segmentation, with relatively large central obstruction, and large secondary mirror support structures. All these characteristics are challenging for coronagraphy and typically reduce the delivered contrast. We have shown above that the generalized prolate functions enable APLCs to mitigate these adversities. Moreover, because of the ELTs' very high angular resolution, IWA is not a crucial parameter, and they can accommodate coronagraphs that work at a few resolution elements of IWA, as the APLC does.

In this section, we give a few examples of generalized prolate solutions for three possible ELT geometries. The aperture geometries we consider might not reflect the current status of these projects. Our point here is not to make a detailed technical study, but to illustrate the type of theoretical solutions we can consider for arbitrary aperture shapes. We will consider the following.

• 1.  
  The ESO 100 m Overwhelmingly Large Telescope (OWL) (this geometry has been abandoned and now reduced to 42 m). This geometry has a large obstruction, thick support structures, and a large number of segments.
• 2.  
  The Thirty Mirror Telescope (TMT). This geometry has segments, a triangular obstruction, and a complex system of support structures.
• 3.  
  The Giant Magellan Telescope (GMT). This geometry consists of a daisy-type arrangement of seven circular telescopes. There are large gaps between the circular telescopes.

In Figure 14 we show three optimal apodizers for OWL with increasing mask sizes. The smallest mask size leads to a simple bagel-shaped apodizer, where smaller features (spider vanes, and support structures) do not visibly affect the shape of the apodizer. This geometry was also used in Figure 3 for the illustration of the Lyot plane before the application of the Lyot stop. In this case, it would be preferable to consider a rotationally symmetric apodizer and ignore the support structures, as Sivaramakrishnan & Lloyd (2005) do. For mask sizes about 5λ/D, the secondary mirror support structures start to alter the shape of the apodizer in a similar transition to the bell–bagel regimes. This transition makes the apodizer shape more complex and its throughput is also affected by this transition.

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In Figure 15, we show a calculation for the OWL geometry where the segments were included during the computation of the apodizer. Segment gaps are too small compared to the mask size to alter the shape of the optimal apodizer, and they should be ignored in this case. However, they will appear bright in the Lyot plane, and the Lyot stop should be reticulated to remove these bright segment gaps (Sivaramakrishnan & Yaitskova 2005).

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In Figure 16, we show three apodizers for the geometry of GMT. It is interesting to see that solutions exist even for this complex, nonconvex aperture. In some cases the result is somewhat surprising. For example in the right panel, the prolate apodizer almost completely obscures the central aperture.

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In Figure 17, we show three examples for the geometry of TMT. Even in the smallest mask case, it is possible to see that the support structures modify the apodizer, which is not perfectly rotationally symmetric.

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The features of ELT geometry include large secondary obstructions and support structures much larger than for 8–10 m class telescopes, for compactness and mechanical reasons. For example, in the case of TMT, the secondary supports can be as large as 30–50 cm in some designs (Ellerbroek et al. 2006). Compared to the 1 cm Gemini secondary supports, this is an order of magnitude larger (relative to the telescope diameter). This means that secondary supports start to matter more for prolate apodizations when applied to ELT geometries, especially for masks diameters between five and ten resolution elements. A detailed study is needed to determine whether it is better to consider smaller mask sizes with a multiple-stage APLC or larger masks with more complex apodization profiles.