APODIZED PUPIL LYOT CORONAGRAPHS FOR ARBITRARY APERTURES. IV. REDUCED INNER WORKING ANGLE AND INCREASED ROBUSTNESS TO LOW-ORDER ABERRATIONS

The APLC is a Lyot-style coronagraph which involves four planes (A, B, C, D) and combines an entrance pupil apodization in plane A, a downstream FPM in plane B, and a Lyot stop in the relayed pupil plane C, to form the coronagraphic image of an unresolved, on-axis bright star on a detector located in the re-imaged focal plane D, see Figure 1. We recall the formalism of the APLC and establish the expression of an operator $\mathcal {C}_{\lambda }$ that linearly relates the entrance pupil function P and the coronagraphic field at a given wavelength λ within the bandpass Δλ centered at central wavelength λ₀.

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In the following, we consider axisymmetric pupils and masks, allowing us to use radial functions to describe the formalism, with no loss of generality, see, e.g., Carlotti et al. (2011). The optical layout of the coronagraph is such that the complex amplitudes of the electric field in two successive planes are related by a Hankel transform. We call $\widehat{f}$ the Hankel transform of a function f:

$$\begin{equation} \widehat{f}(\xi)=\int _{0}^{+\infty } f(r)\,J_0(2\pi \,r\xi)\,2\pi \,r\,dr, \end{equation} \tag{ 1 }$$

where J₀ denotes the zero-order Bessel function of the first kind and ξ represents the focal plane coordinate expressed in units of angular resolution (λ₀/D).

A telescope aperture with central obstruction of respective diameters D and d forms the coronagraph entrance pupil of the system. Assuming the presence of an apodizer in the same plane, the entrance pupil function P is defined as non null for d/2 < r < D/2. The electric field Ψ_A in the entrance pupil plane A of the system is simply given by

$$\begin{equation} \Psi _A(r, \lambda)=P(r)\,. \end{equation} \tag{ 2 }$$

A FPM of angular diameter m is introduced in the following focal plane B. Its transmission is given by 1 − M with M(ξ) = 1 for ξ < m/2 and 0 otherwise. At the operating wavelength λ, the electric field Ψ_B in the following focal plane is given by

$$\begin{equation} \Psi _B(\xi, \lambda)=\frac{\lambda _0}{\lambda } \widehat{\Psi }_A\left(\frac{\lambda _0}{\lambda }\xi, \lambda \right) [1-M(\xi)]\,. \end{equation} \tag{ 3 }$$

By developing $\widehat{\Psi }_A(\xi \lambda _0/\lambda, \lambda)$ in the second term, we obtain

$$\begin{eqnarray} \Psi _B(\xi, \lambda) &=&\frac{\lambda _0}{\lambda }\left[\widehat{P}\left(\frac{\lambda _0}{\lambda }\xi \right)\right. \nonumber\\ &&\left. -M(\xi)\!\int _{0}^{+\infty }\! P(u)\,J_0 \left(2\pi \,u\frac{\lambda _0}{\lambda }\xi \right)\,2\pi \,u\,du\!\right].\quad \end{eqnarray} \tag{ 4 }$$

The complex amplitude Ψ_C in the relayed pupil plane C is obtained by Hankel transform of the previous expression followed by Lyot stop filtering:

$$\begin{equation} \Psi _C(r, \lambda)=\frac{\lambda _0}{\lambda }\widehat{\Psi }_B\left(\frac{\lambda _0}{\lambda }r, \lambda \right) L(r), \end{equation} \tag{ 5 }$$

where L is the Lyot stop function defined as L(r) = 1 for d_S/2 < r < D_S/2 and 0 otherwise, where d_S and D_S denote the sizes of the inner and outer edge of the aperture stop, with d_S  ⩾  d and D_S  ⩽  D. The complex amplitude Ψ_C can be rewritten as

$$\begin{eqnarray} \Psi _C(r, \lambda) &=&\left[P(r)\right. \nonumber\\ &&\left. - \left(\frac{\lambda _0}{\lambda }\right)^2\!\int _0^{+\infty }\! P(u)\,K_M \left(\frac{\lambda _0}{\lambda }u, \frac{\lambda _0}{\lambda }r \right)\,u\,du \right] L(r),\nonumber\\ && \end{eqnarray} \tag{ 6 }$$

where the second term describes the wave diffracted by the FPM. This term is then expressed using the Kernel convolution K_M with

$$\begin{equation} K_M(u, r)=(2\pi)^2\int _0^{m/2} J_0\left(2\pi \,\eta \,u\right)\,J_0\left(2\pi \,\eta \,r\right)\,\eta \,d\eta \,. \end{equation} \tag{ 7 }$$

The coronagraphic electric field Ψ_D is finally obtained with a scaled Hankel transform of the Lyot plane field:

$$\begin{equation} \Psi _D(\xi, \lambda)=\frac{\lambda _0}{\lambda } \widehat{\Psi }_C\left(\frac{\lambda _0}{\lambda }\xi, \lambda \right)\,. \end{equation} \tag{ 8 }$$

Developing this equation makes appear a relation between the electric field Ψ_D and the entrance pupil function P that can be underlined by means of an operator $\mathcal {C}_\lambda$ as follows:

$$\begin{eqnarray} \mathcal {C}_\lambda [P(r)](\xi) &=&\Psi _D(\xi, \lambda)\nonumber\\ &=& \left(\frac{\lambda _0}{\lambda } \right) \left[\!\int _{d_S/2}^{D_2/2}P(r)\,J_0 \left(2\pi \,r\frac{\lambda _0}{\lambda }\xi \right)\,2\pi \,r\,dr\right. \nonumber\\ &&\left. -\left(\frac{\lambda _0}{\lambda }\right)^2\! \int _{0}^{m/2}\! \widehat{P}\left(\!\frac{\lambda _0}{\lambda }\eta \!\right) K_S\left(\!\frac{\lambda _0}{\lambda }\xi, \frac{\lambda _0}{\lambda }\eta \!\right)\eta \,d\eta \!\right],\nonumber\\ && \end{eqnarray} \tag{ 9 }$$

where the behavior of the Lyot stop on the wave diffracted by the FPM is described by the second term, expressed using the Kernel convolution K_S:

$$\begin{equation} K_S(\xi, \eta)=(2\pi)^2\int _{d_S/2}^{D_S/2} J_0\left(2\pi \,r\,\xi \right)\,J_0\left(2\pi \,r\,\eta \right)\,r\,dr\,. \end{equation} \tag{ 10 }$$

The Kernel convolutions K_M and K_S reflect the filtering effect induced by the FPM and the Lyot stop. An analytical closed form for these Kernels is given in Soummer et al. (2003a).

Pueyo & Norman (2013) have previously derived an expression for $\mathcal {C}_{\lambda }$ (see Equation (2) of their paper¹) at λ₀ and in the absence of Lyot stop undersizing (D_S = D and d_S = d). Equation (9) constitutes an extension of their expression since our operator $\mathcal {C}_{\lambda }$ enables Lyot stop undersizing and expresses the coronagraphic electric field at any wavelength λ within the spectral bandpass Δλ centered at λ₀.

The coronagraphic intensity over the bandpass Δλ can be deduced from Equation (9) by means of a second operator $\mathcal {I}_{\Delta \lambda }$ as

$$\begin{equation} \mathcal {I}_{\Delta \lambda }[P(r)](\xi)=\frac{1}{\Delta \lambda }\int _{\Lambda } \left[\mathcal {C}_{\lambda }[P(r)](\xi)\right]^2\,d\lambda , \end{equation} \tag{ 11 }$$

where Λ defines a set of wavelengths λ such that |λ − λ₀| < Δλ/2.

Pueyo & Norman (2013) developed APLC solutions to generate coronagraphic star image with dark hole $\mathcal {D}$ in monochromatic light observations, relying on the linear relation between the coronagraphic electric field and the apodization at λ₀ in Equation (9). Optimized at a single wavelength, these coronagraphic designs do not form a star image with broadband dark region in white light observations.

Our goal is to optimize the entrance pupil apodization for an APLC to reach a given contrast C over a specific area $\mathcal {D}$ in a broadband coronagraphic image. Equation (11) is a quadratic form between the coronagraphic broadband intensity $\mathcal {I}_{\Delta \lambda }$ and the entrance pupil apodizer P, which makes the search of the optimal apodization difficult. To circumvent this problem and develop novel APLC broadband solutions, we propose a multi-wavelength approach based on Equation (9) which relates the entrance pupil apodization and the coronagraphic electric field at a given wavelength.

The parameters for the optimization of a classical APLC for a given telescope geometry are: mask size, bandpass, Lyot stop geometry, wavelength at which the eigenfunction is defined, and IWA. Quantities such as the contrast and throughput can be derived independently, and the overall APLC design can be selected to reach a certain level of contrast and throughput. However these important quantities are not part of the optimization problem.

Here our approach is to define a numerical optimization problem, including all the classical APLC parameters, plus the throughput, contrast, and outer working angle (OWA). The goal is to find the apodizer transmission profile for the APLC that reaches a contrast target C in the broadband coronagraphic PSF within the search area $\mathcal {D}$, as a function of other parameters (Lyot stop geometry, bandpass, mask size). Among the solutions that can meet these criteria, the most desirable solution is the one that offers the best transmission to provide the highest off-axis throughput.

Since the operator $\mathcal {C}_{\lambda }$ is linear at a given λ, the optimal pupil apodization for the APLC can be found by solving the following linear optimization problem:

$$\begin{eqnarray} & \max _{\lbrace pk\rbrace }[P(r)] \quad {\rm for} \quad d/2<r<D/2 \quad {\rm under\; the\; constraints{:}} \nonumber\\ \end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray} & \left|\mathcal {C}_{\lambda }[P(r)](\xi) \right| \,{<}\, \sqrt{10^{-C}} \quad {\rm for} \quad \left\lbrace\begin{array}{@{}l} \rho _i \,{<}\, \xi \,{<}\, \rho _o \\ |\lambda -\lambda _0| \,{<}\, \Delta \lambda /2 \end{array}\right. \end{eqnarray} \tag{ 12b }$$

$$\begin{eqnarray} & \max _r[P(r)] =1 \end{eqnarray} \tag{ 12c }$$

$$\begin{eqnarray} & \left|\frac{d}{dr}[P(r)] \right| \,{<}\, b. \end{eqnarray} \tag{ 12d }$$

We explain the meaning of theses equations in the following.

(12a) The apodizer throughput is a quadratic function of the field amplitude and maximizing it requires the solution of a nonlinear optimization problem (as described in Vanderbei et al. 2003b and recalled in Pueyo & Norman 2013). In the context of our linear optimization problem, we use as an approximation a linear constraint of the amplitude at each pupil point as a criterion in an attempt to maximize the apodization throughput.

(12b) The absolute value of the coronagraphic electric field is constrained at all the wavelengths over the bandpass to achieve a contrast C over the area $\mathcal {D}$ in broadband light. This region is defined between its inner and outer edges, respectively, ρ_i and ρ_o.

(12c) The maximum value of the apodization function is set to one to exclude the null function in the resolution of our problem.

(12d) We introduce a constraint on the absolute value of the derivative of P with respect to the radial coordinate to limit the oscillations within the function P and avoid bang-bang solutions that make the apodizer manufacturing very challenging. The constant b is defined as the derivative limit.

We solve this problem by using a linear programming solver (Vanderbei 2009) and in the following, we rely on the routine provided with the Mathematica software (Wolfram Research Inc. 2012).

The entrance pupil apodization function can be represented in a number of ways, including directly sampled as independent field amplitude pixel values, or decomposed on a modal basis. Here we use a weighted sum of radial Bessel modes as a pupil representation, and we adopt an expression for the entrance pupil function in the form:

$$\begin{equation} P(r)=\sum _{k=0}^{N_{\rm modes}} p_k\,J_0\left(\frac{r}{\alpha _k}\right), \end{equation} \tag{ 13 }$$

where α_k represents the kth zero of the zero-order Bessel function of first kind J₀. Bessel functions of higher order could be used to decompose P as proposed by Pueyo & Norman (2013); we realized tests with different orders and found similar results for the first few orders. With this modal representation, the numerical optimization seeks the coefficients p_k, instead of the pixel values themselves, which helps to reduce the dimensionality of the problem and accelerates the computation. As a sanity check, we have verified that both pixel-based or modal-based representations of the apodizers both lead to very similar results.

Multi-spectral constraint on the coronagraphic electric field is used in the optimization problem to find solutions producing broadband PSF dark zones with the specified contrast. The problem is solved here by using a number of five wavelengths for the spectral constraints in 10% and 20% bandpasses. We performed some tests with different number of wavelengths that are equally spatially sampled one to another within the bandpass for our optimization. Similar results are found for five wavelengths and beyond. We have also computed the coronagraphic electric field at wavelengths that are unconstrained in the optimization procedure and we have verified that the coronagraph contrast at the wavelengths within the specified bandpass meets our problem constraints.

Before investigating novel designs, we first ran some tests to show the ability of our method to recover the APLC designs using the classical prolate apodization as defined by Soummer et al. (2011b). Our broadband optimization is made under the following assumptions.

We consider a circular aperture with 20% central obscuration (expressed in pupil radius ratio) and a FPM of radius m/2 = 3.31 λ₀/D, corresponding to the relative dimensions of our reflective mask installed in the High Contrast Imager for Complex Aperture Telescope (HiCAT) testbed (N'Diaye et al. 2013a, 2014) and inherited from the Lyot project (Oppenheimer et al. 2004). The standard method to generate APLC prolate apodizations involves a Lyot stop with the same shape as the entrance pupil (Aime et al. 2002; Soummer et al. 2003a) and therefore, no oversizing of the Lyot stop obstruction is introduced in our optimization. The prolate apodization for the mask size and the aperture geometry mentioned above gives a ∼10⁻⁸ contrast at 5.5 λ₀/D. We set a contrast target C = 8 over a 20% bandpass within the search area $\mathcal {D}$ defined by ρ_i = 5.5 λ₀/D and ρ_o = 48 λ₀/D to look for the prolate apodization solution. A large number of modes N_modes = 48 is chosen to control the spatial frequencies in the electric field up to the outer edge of $\mathcal {D}$, since the generation of the prolate apodization does not account on any specific search area delimitations. Finally, since the prolate apodization does not present large amplitude fluctuations, we set the constraint of the derivative limit to a small value b = 5.

Figure 2 shows the apodization and the resulting coronagraphic intensity profile for the linear programming solution and the prolate apodization calculated at the wavelength λ₁  =  1.05 λ₀ for optimal broadband extinction, following Soummer et al. (2011b). This first example confirms the ability of our linear optimization method to achieve the prolate apodization solution previously found by Soummer et al. (2011b). Also, it is important to note that for such linear programming optimizer, the existence of a solution is not guaranteed for a given set of parameters, as opposed to the case of a classical APLC where a prolate apodizer (eigenfunction) always exists for a given geometry and combination of mask size and wavelength.

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We assume a circular entrance pupil with a 14% central obstruction ratio and we select the FPM of relative radius m/2 = 3.48 λ₀/D in H band. For the coronagraph design, we set our contrast target to 10⁸ in the dark zone over a 20% bandpass, corresponding to one order of magnitude higher contrast than the specification for the GPI coronagraphs (Macintosh et al. 2008; Soummer et al. 2009b, 2011b). The dark-zone region has an outer bound ρ_o = 20 λ₀/D, corresponding to the maximum spatial frequency that can be controlled with the GPI deformable mirror.

In our simulations, we set a pupil size of N = 300 pixels for camera images with maximum spatial frequency of 50 element resolutions (or cycles/pupil). From this last number, we set N_modes = 48. In the following, we explore the entire parameter space by considering the Lyot stop inner diameter (ID) oversizing factor and the inner radius ρ_i of the search area $\mathcal {D}$ by solving the optimization problem described above. The ID oversizing factor is defined as the ratio between the central obscuration diameters of the Lyot stop and the aperture d_s/d. A similar undersizing factor can also be defined as the ratio of the outer diameters of the Lyot stop and the aperture D_S/D. We ran some tests and did not find any optimum for this parameter, which is in accordance with the results described in Soummer et al. (2011b). We set the constraint on the derivative b to 5 to avoid bang-bang solutions that typically have very low throughput, and to avoid the hard-to-manufacture intermediate situation with a continuous transmission profile but very high oscillatory behavior that is obtained if larger derivative values are tolerated.

Figure 3 represents the throughput of the complete coronagraphic system, including the apodizer and Lyot stop, as a function of the search area inner radius ρ_i and the Lyot stop ID oversizing factor for the GPI mask mentioned above. Two contour plots are displayed, representing the 10% and 20% bandwidths (left and right, respectively). The parameters are listed in Table 1.

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In each plot, two different regimes are observed, delimited by the existence or not of solutions to the optimization problem with our constraints. This limit, found at 3.8 λ₀/D for 20% bandwidth, is pushed down to 2.4 λ₀/D for a 10% bandwidth, underlining that enlargement of the solution space is possible for a given constraint on the apodizer derivative at the cost of a bandwidth decrease. This parameter space exemplifies the very important trade-off between bandwidth and IWA in the optimization of the apodizer.

Relaxing the constraint on the apodizer derivative (larger b) can be used to find solutions for a wide spectral bandpass (>10%) and a small search area inner bound (<4 λ₀/D) but in return, the optimizer apodizer will present large and fast amplitude variations over the pupil; these apodizer configurations have very low throughput, and the fast oscillations are most likely very hard to manufacture, e.g., with microdots technology (Martinez et al. 2009; Sivaramakrishnan et al. 2009).

Assuming an ID = 2 oversizing factor for the Lyot stop obstruction, we seek two solutions for 10% and 20% bandwidths with the smallest inner edge of the dark region and the highest coronagraphic throughput observed in Figure 3 plots, see white cross marks. The parameters of these two solutions are recalled in Table 2.

Since these solutions are based on re-using an existing FPM in GPI (in this case a K-band FPM considered for use at H band), the mask size is not part of this optimization. We find two designs with 39% coronagraph throughput for ρ_i = 3.7 and 4.0 λ₀/D with 10% and 20% bandwidth. The apodizers and their coronagraphic profiles are represented in Figure 4. Our designs provide coronagraphic images with 10⁸ contrast dark regions in white light for two different bandwidths.

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The ρ_i  =  2.4 λ₀/D bound is the shortest inner edge of the dark hole obtained with our 10% bandwidth solution for a mask radius m/2 = 3.48 λ₀/D and a Lyot stop with 28% central obscuration in pupil radius ratio, based on the actual diameter of K-band mask and the current Lyot stop geometry of the GPI instrument. As part of our parameter space study we found that solutions exist with ρ_i (i.e., the inner edge of the dark zone) smaller than the mask radius. In this configuration, the actual IWA of the instrument (i.e., defining the ability to detect an object) is entirely determined by the geometry of the FPM, and not that of the coronagraph dark zone (note that the actual instrumental dark zone also depends on wavefront control and diffraction effects from secondary support structures and/or segment gaps, not discussed here). Producing dark holes with an inner bound smaller than the projected FPM radius is a fundamentally new property of APLCs with numerical optimization of the apodizer. We find that the existence of this new type of solution is also strongly dependent on the entrance pupil central obstruction. These solutions are very interesting in practice with the benefit of being quasi insensitive to tip tilt and more generally low-order aberrations. We thus investigate this aspect in more details below.

Figure 5 represents the contrast throughput as a function of the mask radius m/2 and inner bound of the dark region ρ_i (see list of parameters in Table 3). This plot underlines the existence of coronagraph designs for ρ_i < m/2 (see solutions above diagonal), meaning that these solutions generate coronagraphic image dark hole with an inner bound smaller than the mask radius. To illustrate our remarks, we pick up the design denoted by the cross mark in Figure 5. This corresponds to the solution providing the smallest bound for the dark region, see Figure 6.

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It is important to note that such designs with ρ_i < m/2 do not allow to observe companions at distances shorter than m/2 since these objects see their light totally blocked by the opaque mask in the intermediate focal plane B of the coronagraph layout. However, there are two important advantages to this configuration:

• 1.  
  The IWA in the absence of aberrations is solely determined by the geometric IWA from the FPM, and not by the contrast curve of the APLC design, since the dark zone in the PSF extends within the geometrical mask. Therefore this new class of APLC designs decouples completely the IWA from the coronagraph performance itself. The actual ability to observe a faint companion at the shortest separations in this case is therefore entirely determined by the combination of geometric IWA (due to the diameter of the FPM) and wavefront control.
• 2.  
  Since the core of the PSF is smaller than the projected angular size of the FPM, this design allows for both expansion or displacement of the PSF core up to the mask radius with no impact on the contrast in the effective dark region. These designs are virtually insensitive to jitter and low-order aberration in this range.

A large FPM (m/2 = 4 λ₀/D) is used for the configuration displayed in Figure 6, corresponding to a geometric IWA of ∼4 λ₀/D. As a comparison, we recall the GPI configuration for the H-band mask (m/2 = 2.8 λ₀/D), as described in Soummer et al. (2011b). Perfect optical surfaces, alignments and absence of atmospheric turbulence are assumed for raw performance comparison between both designs. Performance degradation due to wavefront errors are studied in the following section. Pupil strut effects are not considered in our analysis since they can be mitigated in the Lyot plane with an independent optimization of the Lyot stop (Sivaramakrishnan & Lloyd 2005; Sivaramakrishnan & Yaitskova 2005) as it was performed for GPI. Spider vane effects, alignment tolerance and ringing effects from the field stop are considered within such an optimization and diffraction effects from struts hence remain below the 10⁻⁸ raw contrast provided by the optimized circularly symmetric coronagraphs with almost no throughput loss (Soummer et al. 2009b; Sivaramakrishnan et al. 2010). With a larger mask, our design provides a smaller IWA and a higher contrast at small angular separations (between 4  and 10 λ₀/D or equivalently 170 and 425 mas in H-band) than the existing coronagraph in GPI.

Even larger masks (m/2 > 4 λ₀/D) are worth being studied since the apodizer can be implemented with the Phase-induced Amplitude Apodization (PIAA; Guyon 2003; Traub & Vanderbei 2003) using a series of 2 aspherical mirrors to reach small angular separations (<2 λ₀/D). With such an implementation, the apodizer throughput is unity and the angular resolution units is magnified by the field-independent centroid-based angular magnification defined in Pueyo et al. (2011). With an estimated PIAA magnification of 3.0, our mask radius becomes 1.3 λ₀/D in PIAA units, allowing the observation of planets that are very close to their host star.

In the following, we analyze and compare the tip–tilt sensitivity of several APLCs (including one design optimized with ρ_i < m/2).

Figure 7 shows the contrast performance at 5 λ₀/D of the three APLCs as a function of tip tilt, defocus, astigmatism, coma, trefoil and spherical aberration in each of the six panels. In the tilt case (top left panel), the curves of our optimizer designs present a plateau for pointing errors ranging from 0 to 0.25 λ₀/D (11 mas for an 8 m class telescope in H-band) with an intensity level below 10⁻⁷ for tip–tilt smaller than 0.5 λ₀/D (22 mas) while the profile for the current GPI design exhibits a contrast loss as the star pointing error increases with an intensity level above 10⁻⁷. For the other aberrations, our designs also show an intensity level is one to two order of magnitude smaller than the current GPI design.

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Our designs show a clear improvement both in contrast and in robustness to low-order aberrations compared with the GPI current implementation. This is particularly interesting in the context of GPI for which telescope jitter has been identified as an important issue in the first runs of the instrument (beyond the 3 mas initially specified during the instrument design process; Hartung et al. 2014). More generally, jitter and vibrations are a potential concern for any coronagraphic instrument either on the ground or in space, and optimizing instrument designs with increased robustness to tip/tilt errors is therefore highly valuable.

Our new designs with the optimizer remove the diffraction at smaller angular separations from the star than with the current GPI design (i.e., between 4  and 5 λ₀/D), and increase the raw contrast between 4  and 8 λ₀/D by one order of magnitude or more. However the coronagraph throughput is reduced slightly from 46% with GPI implementation to 36% in our best case.

Our designs constitute promising options for a potential coronagraph upgrade for GPI. In particular, the insertion of the design with m/2 = 3.48 λ₀/D only requires the replacement of the apodization in the filter wheels of the instrument. The insertion of the design with m/2 = 4.0 λ₀/D requires the introduction of a new FPM in addition to replacement of an apodizer, offering an even larger discovery space and a more robust system to low-order aberrations than the design mentioned above.

These results are detailed for the GPI design as an illustration comparison point without loss of generality, and apply to any other coronagraphic configuration.

The study of the robustness to tip–tilt errors can be extended to stellar angular sizes. Indeed, an on-axis resolved star can be seen as a combination of off-axis point sources at all the angular positions ranging up to the stellar angular radius. This is equivalent to consider a sum of tip–tilt point sources weighted with their offset position. We can devise the coronagraph contrast sensitivity at 5 λ₀/D by averaging the solid-line profiles showed in Figure 7 with weights corresponding to the offset positions.

Figure 8 displays the contrast performance at 5 λ₀/D of the three designs as a function of the angular size of the observed objects (dashed curves). As with the pointing errors, our optimizer designs are more robust to stellar angular size than the current GPI implementation.

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Our studies have been mainly focused on designing APLC for an aperture with 14% centrally obstruction in pupil radius ratio to yield a 10⁻⁸ contrast in the search area $\mathcal {D}$, offering promising possible upgrades for GPI and SPHERE, and for the baseline design of the HiCAT testbed coronagraph (N'Diaye et al. 2013a, 2014). We here provide some examples of APLCs for different central obstruction sizes (10% and 20%) to produce 10⁻¹⁰ broadband PSFs dark region, see Figure 9. These designs produce a large dark region inner radius (>5 λ₀/D) but they can be combined with PIAA as the apodization implementation to reach small IWA (e.g., PIAACMC; Guyon et al. 2010a), proving promising options for future exoplanet direct imaging missions.

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Our previous solutions have been given for apertures with obstructions up to 20%. However, telescopes such as WFIRST-AFTA in space or ELTs on the ground will have aperture with very large central obstructions (>30%). We here give an example of solutions for such geometry with a design for 36% central obstruction, see Figure 10. This APLC design produces star images with 10⁻⁹ contrast dark region for a 20% bandpass. Such a solution can also be combined with a PIAA as an apodization implementation to reach small IWA (down to 2 λ₀/D).

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