APODIZED PUPIL LYOT CORONAGRAPHS FOR ARBITRARY APERTURES. V. HYBRID SHAPED PUPIL DESIGNS FOR IMAGING EARTH-LIKE PLANETS WITH FUTURE SPACE OBSERVATORIES

We briefly recall the chromatic formalism of the Lyot-style coronagraph with apodization, following the notations of Aime et al. (2002) and Soummer et al. (2003a, 2003b), for a given wavelength λ within the bandpass ${\rm{\Delta }}\lambda$ centered at the wavelength ${\lambda }_{0}$. The vectors ${\boldsymbol{r}}$ and ${\boldsymbol{\xi }}$ (of modulus r and ξ) denote the two-dimensional position vectors in the pupil and focal planes.

The coronagraph setup involves four successive planes (A, B, C, D) in which A defines the entrance pupil $P={P}_{0}\;{\rm{\Phi }}$ that combines the telescope pupil shape P₀ and the apodization Φ, B sets the location of a hard-edged FPM of diameter m and transmission $1-M$ with $M({\boldsymbol{\xi }})=1$ for $\xi \lt m/2$ and 0 otherwise, C includes the Lyot stop L to filter out the diffraction due to the FPM, and D represents the final image plane. The optical layout of the coronagraph is such that the complex amplitudes of the electric field in two successive planes are related by a Fourier transform (FT). We call $\hat{f}$ the FT of the function f. In the absence of magnification, the electric field amplitude in the four successive planes is written as follows.

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{A}({\boldsymbol{r}},\lambda )=P({\boldsymbol{r}})\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{B}({\boldsymbol{\xi }},\lambda )=\displaystyle \frac{{\lambda }_{0}}{\lambda }{\widehat{{\rm{\Psi }}}}_{A}\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }{\boldsymbol{\xi }},\lambda \right)(1-M({\boldsymbol{\xi }}))\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}{{\rm{\Psi }}}_{C}({\boldsymbol{r}},\lambda ) & = & \left({{\rm{\Psi }}}_{A}({\boldsymbol{r}},\lambda )-{{\rm{\Psi }}}_{A}({\boldsymbol{r}},\lambda )\ast \displaystyle \frac{{\lambda }_{0}}{\lambda }\widehat{M}\right.\left.\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }{\boldsymbol{r}},\lambda \right)\right)L({\boldsymbol{r}})\\ \end{eqnarray} \tag{ 1c }$$

$$\begin{eqnarray}{{\rm{\Psi }}}_{D}({\boldsymbol{\xi }},\lambda ) & = & {\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }\right)}^{2}\left({\widehat{{\rm{\Psi }}}}_{A}\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }{\boldsymbol{\xi }},\lambda \right)(1-M({\boldsymbol{\xi }}))\right)\ast \widehat{L}\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }{\boldsymbol{\xi }},\lambda \right)\quad \\ \end{eqnarray} \tag{ 1d }$$

where ∗ symbolizes the convolution operator. In the absence of FPM, the electric field amplitude in the final image plane is simply given by

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{D}^{0}({\boldsymbol{\xi }},\lambda )={\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }\right)}^{2}{\widehat{{\rm{\Psi }}}}_{A}\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }{\boldsymbol{\xi }},\lambda \right)\ast \widehat{L}\left(\displaystyle \frac{{\lambda }_{0}}{\lambda }{\boldsymbol{\xi }},\lambda \right).\end{eqnarray} \tag{ 2 }$$

Assuming a flat stellar spectrum, the coronagraphic image intensity over the bandpass ${\rm{\Delta }}\lambda$ can be deduced from Equation (1d) by means of an operator ${{ \mathcal I }}_{{\rm{\Delta }}\lambda }$ as

$$\begin{eqnarray}&&{{ \mathcal I }}_{{\rm{\Delta }}\lambda }[{\rm{\Phi }}({\boldsymbol{r}})]({\boldsymbol{\xi }})=\displaystyle \frac{1}{{\rm{\Delta }}\lambda }{\displaystyle \int }_{{\rm{\Lambda }}}{| {{\rm{\Psi }}}_{D}({\boldsymbol{\xi }},\lambda )| }^{2}\;d\lambda ,\end{eqnarray} \tag{ 3 }$$

where Λ defines a set of wavelengths λ such that $| \lambda -{\lambda }_{0}| \lt {\rm{\Delta }}\lambda /2$.

The analysis of the re-imaged pupil plane offers insight into the coronagraph behavior. According to Equation (1c), perfect starlight suppression is obtained if the direct wave (entrance pupil amplitude) and the wave diffracted by the FPM (convolution integral) matches within the Lyot stop. Different approaches have been considered to obtain a perfect subtraction between these two waves. A first approach consists of modifying the geometry of the mask to obtain perfect subtraction within a reduced area of the pupil. Such a strategy corresponds to the band-limited mask coronagraph (Kuchner & Traub 2002). We follow a second approach seeking for a modification of the entrance pupil amplitude by means of the apodization Φ to match the two waves within the re-imaged pupil (Aime et al. 2002; Soummer et al. 2003a).

The existence of a perfect extinction solution for the Roddier phase mask coronagraph (Roddier & Roddier 1997) in monochromatic light was numerically derived by Guyon & Roddier (2000) and analytically shown by Aime et al. (2002) and Soummer et al. (2003a). For the APLC, there is no solution for complete starlight suppression, but Aime et al. (2002) and Soummer et al. (2003a) analytically show that prolate apodization solutions nearly achieve complete starlight removal with circular and rectangular unobstructed apertures. These solutions have been generalized to the obstructed circular aperture (Soummer 2005) and the arbitrary non-circular aperture in the context of Extremely Large Telescopes (ELTs, Martinez et al. 2007; Soummer et al. 2009). These solutions for opaque masks have also been extended to the complex amplitude FPMs for small IWA coronagraphy (Guyon et al. 2010a, 2014).

In the case of opaque FPM, Soummer et al. (2011) studied the chromatic properties of APLC using prolate apodizers and found the existence of quasi-achromatic solutions over large bandpasses. They derive a continuous family of prolate apodizations corresponding to the telescope geometry and select the optimal set of apodization and Lyot stop geometry using contrast in the final image plane within the controllable adaptive optics area as an optimization metric to obtain the broadband APLC designs that are currently implemented in P1640 and GPI (Hinkley et al. 2011; Macintosh et al. 2014).

The raw performance of the current implementations are, however, limited by the available degrees of freedom (apodization and Lyot stop geometry) in the optimization approach. We recently found new functioning modes for the APLC by proposing an alternative approach based on the SP-like optimization to design a Lyot-style coronagraph with improved performance in terms of contrast and inner working angle (Pueyo & Norman 2013; N'Diaye et al. 2015b). In N'Diaye et al. (2015b), we propose broadband solutions for centrally obstructed circular apertures, using 1D pupil optimization. We here show the existence of solutions for arbitrary apertures with pupil features such as segmentation and spiders by exploiting our approach to 2D pupil.

We define a numerical optimization problem with a Lyot-style coronagraph that includes the aperture, FPM, and Lyot Stop geometries, and constraints on the coronagraphic intensity in a given focal plane region (dark zone) and we look for apodization solutions for the APLC. Among these solutions, we seek the apodization with the best throughput in an attempt to maximize the number of photons from companions around the observed star during observing time. Following the classical SP-type optimization problems (Vanderbei et al. 2003; Kasdin et al. 2003; Carlotti et al. 2011), the optimal pupil apodization for APLC can be found by solving the following optimization problem.

$$\begin{eqnarray}&&\mathrm{max}\left[{\displaystyle \int }_{{P}_{0}}{\rm{\Phi }}({\boldsymbol{r}})d{\boldsymbol{r}}\right]\quad \mathrm{under}\ \mathrm{the}\ \mathrm{contraints}{\rm{:}}\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{| {{\rm{\Psi }}}_{D}({\boldsymbol{\xi }},\lambda )| }^{2}\lt {10}^{-C}{| {{\rm{\Psi }}}_{D}^{0}(0,{\lambda }_{0})| }^{2}\\ &&{\rm{for}}\quad \left\{\begin{array}{c}{\rho }_{i}\lt \xi \lt {\rho }_{o}\\ | \lambda -{\lambda }_{0}| \lt {\rm{\Delta }}\lambda /2\end{array}\right.,\end{eqnarray} \tag{ 4b }$$

$$\begin{eqnarray}&&0\leqslant {\rm{\Phi }}({\boldsymbol{r}})\leqslant 1,\end{eqnarray} \tag{ 4c }$$

where the significance of each line is further explained below.

(4a) The apodizer throughput is a quadratic function of the field amplitude. Assuming positive, real-valued apodization amplitude (see Equation (4c)), we use this linear function of the field amplitude as a proxy to maximize the apodizer throughput over the area defined by the initial aperture shape P₀.

(4b) The coronagraphic image intensity is constrained at all the wavelengths over the bandpass to achieve a contrast of ${10}^{C}$ over the area ${ \mathcal D }$ in broadband light. This region is defined between its inner and outer edges, respectively, ${\rho }_{i}$ and ${\rho }_{o}$.

(4c) The transmission of the amplitude apodization is real, positive, and normalized.

N'Diaye et al. (2015b) set constraints on the apodizer derivative to obtain continuous solutions with limited oscillations within the pupil apodization and avoid bang-bang solutions. The problem was written in one dimension with linear constraints, allowing resolution with a linear programming solver (Vanderbei 2009).

The 2D problem does not include constraints on the apodizer derivative, thus enabling binary apodization solutions using SP manufacturing technologies. While the 2D optimization problem can also be expressed linearly as the 1D problem, we find that this problem in AMPL language (Fourer et al. 1990) with the solver Gurobi (Gurobi Optimization 2015) is handled slightly more effectively using the image intensity instead of the field as shown in Equation (4b) described above. This somewhat counter-intuitive behavior is likely to be solely due to the technical implementation of these various solvers.

The computation of the coronagraphic electric field for the APLC requires three successive FTs between the electric field amplitude in the four successive planes of the optical layout. We adopt the semi-analytical approach from Soummer et al. (2007) for fast Lyot-style coronagraph computation with simultaneous highly sampled pupils and FPMs. The semi-analytical and FFT approaches are strictly equivalent in the absence of field-of-view limitations in the FPM plane. Further error budget studies should provide the acceptable size of the mirror for a reflective FPM to obtain a compatible 10¹⁰ contrast level. We also express the FTs involved in the coronagraphic electric field calculation using a discrete Riemann sum in Cartesian coordinates. Following Vanderbei (2012), these 2D FTs are performed in two steps with a separation of the variables to reduce the problem complexity. Carlotti et al. (2011) successfully applied this strategy to design SPs in a two-plane configuration that simply involves a single FT for the coronagraphic electric field computation.

We here seek solutions based on an apodizer and FPM with real amplitude transmission (no phase terms). Further solutions with complex amplitude apodizers and FPMs could also be investigated based on the optimization problem set above without loss of generality, but this study is out of the scope of this paper.

To illustrate these new APLC/SP hybrid solutions, we present a design optimization for a 37-hexagonal-segment telescope aperture with central obstruction and spiders, such as one of the envisioned designs for the ATLAST mission concept (Postman et al. 2012) or the geometry corresponding to existing ground-based telescopes W.M. Keck, or Gran Telescopio Canarias. In the following, all of the pupil geometry parameters (obstruction, spiders, gaps) are expressed in aperture diameter ratio.

We assume an inscribed circular aperture over the 37 segments, with 20% central obstruction, 1% spiders, and 0.2% segment gaps, see Figure 1 top left. For the coronagraph design, we consider an FPM of radius $m/2=4\;{\lambda }_{0}/D$. The contrast target is set to 10¹⁰ in a focal plane region ${ \mathcal D }$ delimited by inner and outer edge radii of ${\rho }_{i}=3.5$ and ${\rho }_{o}=$ 20.0 ${\lambda }_{0}/D$. Following the approach developed in N'Diaye et al. (2015b), we set the dark zone inner edge with a smaller radius than the FPM. Such resulting designs do not allow observations of companions at separations smaller than the projected FPM angular size, but the coronagraphic solution will present a reduced sensitivity to low-order aberrations, such as telescope jitter and thermal or mechanical focus drift, as well as a reduced sensitivity to the stellar diameter.

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We finally consider a circular Lyot stop with the same outer size as the entrance aperture, with 40% central obstruction and 2% spiders, corresponding to an oversizing factor of two with respect to the entrance pupil characteristics, but without reticula for the segment gaps (no obstruction for these aperture discontinuities), see Figure 1, middle left. To achieve a broadband APLC design, we run an optimization with constraints at three wavelengths that are centered around ${\lambda }_{0}$ and equally spatially sampled one to another over a 10% bandwidth. Note that while only three wavelengths are used in this case for the optimizer, the broadband profiles illustrated in the figures include 11 simulated wavelengths. The parameters are summarized in Table 1.

Table 1.  Parameters for the Design Shown in Figure 1

Parameters	Value
Contrast C	1010
Dark zone inner edge radius ${\rho }_{i}$	3.5 ${\lambda }_{0}/D$
Dark zone outer edge radius ${\rho }_{o}$	20.0 ${\lambda }_{0}/D$
focal plane mask radius $m/2$	4.0 ${\lambda }_{0}/D$
Aperture	20% central obstruction
	1% aperure size spiders
	0.2% aperture size gaps
Lyot stop	40% central obstruction
	2% aperture size spiders
	no segmentation
Broadband optimization	3 wavelengths within 10% band
Broadband PSF profiles calculations	11 wavelengths within 10% band

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With these parameters, we solve our optimization problem and find an APLC design solution with the SP apodization displayed in Figure 1, top right, leading to a 28% total coronagraph throughput. Our solution produces the relayed pupil image that is shown at ${\lambda }_{0}$ before and after the Lyot stop application in the middle right and bottom left frames. As expected, most of the star diffracted light is found at the aperture discontinuities (Sivaramakrishnan & Lloyd 2005; Sivaramakrishnan & Yaitskova 2005) and constrained within the area blocked by the Lyot stop geometry, allowing an efficient starlight rejection by the Lyot Stop. It is interesting to note that the introduction of a Lyot stop reticula (Sivaramakrishnan & Lloyd 2005; Sivaramakrishnan & Yaitskova 2005) in this specific design is counterproductive in terms of starlight suppression: it results in additional diffracted light within the final image plane and a deterioration of the contrast in the dark region. An APLC solution using a Lyot stop with reticula can be designed with our approach if the geometry of the diaphragm is included in the optimization process.

The broadband coronagraphic image and its corresponding averaged intensity profile are displayed in Figure 1, bottom right, and Figure 2 for designs with three different dark zone outer bound, illustrating the ability of our approach to design APLCs that could produce broadband PSFs with a 10¹⁰ dark region in the presence of an arbitrary aperture.

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Because of numerical limitations for the number of points in the pupil, our APLC/SP optimization produces an apodization for which the values are mostly black or white (37% of the apodization points have a 0% or 100% amplitude transmission level to a 1% precision) but that also contains gray points. We investigated the conversion of these apodization solutions into binary apodizers that can be manufactured using the current state-of-the-art technologies developed for SP apodizers (Balasubramanian et al. 2013).

Our gray designs are obtained with a 600-point diameter. A pupil sampling increase or a full exploration of the coronagraph parameter space (FPM size, Lyot stop geometry, dark zone dimensions) could allow us to find fully black and white designs, but proves computationally challenging. Instead, we investigated new versions of our original gray design, using different methods to convert our gray pixels into black and white, such as simple rounding of the values or a sub-pixelization of the gray design pixels followed by the application of the error diffusion algorithm (EDA; Floyd & Steinberg 1976), a process that has been used for the manufacturing of gray apodizers for GPI, VLT/SPHERE and P1640 to reach ${10}^{6}\mbox{--}{10}^{7}$ raw contrast levels (Dorrer & Zuegel 2007; Martinez et al. 2009; Sivaramakrishnan et al. 2009). At 10¹⁰ raw contrast levels, the same process is expected to be efficient if the performance of the binary design is validated numerically and then experimentally and as long as the pixel size of the apodizer prototype remains significantly larger than the wavelength of observation.

Figure 3 displays the contrast curves for our coronagraph design with a dark zone outer edge ${\rho }_{1}$=10 ${\lambda }_{0}/D$ using the original apodizer from the optimizer and its binarized versions from numerical simulations. A contrast loss of two orders of magnitude is observed by simply rounding the pixel values. The contrast loss can be recovered by using the EDA method and sufficient oversampling. With a factor of 16 for the sub-pixelization, we almost recover the initial 10¹⁰ contrast level reached with our optimized design. With this 9600-pixel diameter apodizer, we translate these values into physical units for manufacturing considerations. We assume the use of black silicon technologies with the current state-of-the-art for the fabrication of a reflective apodizer prototype (Balasubramanian et al. 2013). Assuming a 10 μm pixel size, an apodizer prototype with 96 mm diameter can be manufactured fulfilling the specifications that enable the starlight suppression down to a 10⁻¹⁰ intensity level in visible light with a telescope segmented aperture. This result shows the existence of suitable, broadband diffraction suppression solutions with already manufacturable components (SP apodization, FPM, Lyot stop) for the direct imaging and spectral analysis of Earth twins with large segmented apertures. Further coronagraph designs using binary apodization with a smaller sampling are currently under investigation to reduce the pupil dimensions in the context of a coronagraph instrument.

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Beyond starlight removal, estimating the amount of photons from the nearby objects through the coronagraph is important to determine the observation time required for accurate photometry, astrometry, and spectral analysis of the star companions. We consider the Airy throughput as a metric to determine the fraction of transmitted light from off-axis companions through our coronagraphic system. The Airy throughput is defined as the ratio of the total energy inside the core of the off-axis coronagraphic PSF to the total energy inside the core of the telescope PSF. Since our design makes use of a pupil apodization, the image of an off-axis companion is an apodized PSF for which sharpness is defined by the SP. Throughputs are displayed for our coronagraph designs with three different PSF dark zones in Table 2. For the Airy throughput, we adopt a 0.75 ${\lambda }_{0}/D$ core radius for the photometric aperture to account for the modified light distribution in the PSF in the presence of the SP apodization. This ideal value has been determined after calculating the exo-Earth yield, presented in the next section, as a function of the photometric size aperture to maximize the planet's light signal compared with different sources of noise backgrounds (leaked starlight, exozodi, local zodi, etc.). In the background-limited regime, for an Airy pattern PSF, the ideal photometric aperture radius is ∼0.7 ${\lambda }_{0}$/D, which contains ∼70% of the planet's PSF.

Table 2.  Throughput Estimates for Our Coronagraph Designs in Figure 2 with Different Dark Zone Outer Bounds

Values wrt to Aperture Geometry		Clear			Segmented
${\rho }_{o}$ in ${\lambda }_{0}/D$		10.0	20.0	30.0	10.0	20.0	30.0
Airy throughput in % at	4.0 ${\lambda }_{0}/D$	3.1	2.9	2.7	5.8	5.4	5.1
	4.2 ${\lambda }_{0}/D$	5.1	4.7	4.5	9.5	8.9	8.4
	5.0 ${\lambda }_{0}/D$	8.6	8.1	7.7	16.1	15.2	14.4
	6.0 ${\lambda }_{0}/D$	10.0	9.3	8.8	18.7	17.5	16.5
Max. Airy Throughput in %		10.5	9.8	9.3	19.8	18.4	17.4

Note. Airy throughputs are estimated with a 1.5 ${\lambda }_{0}/D$ size photometric aperture, using 5 pixels per ${\lambda }_{0}/D$.

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Our throughputs have been estimated with a sampling of 5 pixels per resolution element (${\lambda }_{0}/D$). For the design showed in Figure 1, the Airy throughput within the dark region and with respect to the segmented aperture is estimated with a maximum of 18.4% and with about half of it at an angular separation of 4.2 ${\lambda }_{0}/D$, giving the IWA of our design. Table 2 summarizes the throughputs for our coronagraph designs, showing its evolution as a function of different separations.

Further improvements in contrast and IWA are expected with a full exploration of the parameter space for this coronagraph (FPM, Lyot stop geometry, bandwidth), following N'Diaye et al. (2015b), to identify the solution with the best throughput for a given aperture geometry.

Since on-axis resolved stars are identified as one of the major concerns for high-contrast imaging of extrasolar planets with future large telescopes (Guyon et al. 2006), we analyze the impact of the stellar angular size on the performance of our coronagraphs.

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 considering a sum of tip-tilt point sources weighted by $2\;\pi \;\xi$, where ξ here corresponds to the offset position. We can devise the coronagraph contrast sensitivity at 5 ${\lambda }_{0}$/D by averaging the solid-line profiles for the coronagraphic images of tip-tilt sources with weights corresponding to the offset positions, as previously done in N'Diaye et al. (2015b).

Figure 4 shows the contrast performance at 5 ${\lambda }_{0}$/D of the 20 ${\lambda }_{0}/D$-OWA design as a function of the angular size of the observed objects. Our designs are robust enough to stellar angular size to allow for the observation of planets around Sun-like stars located beyond 4.4 pc.

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Following our hybrid concept, and assuming a perfect and stable wavefront, a solar system twin with unresolved companions at 13.5 pc could be observed with a 12 m telescope in 8 hr and characterized with less than a two-day exposure time under the photon-noise limited regime, see Figure 5. In this numerical simulation, we did not account for the motion of the planet and the required post-processing methods to increase the signal-to-noise ratio of planets in such long period observations. Promising strategies for such an effect have recently been proposed to efficiently combine the different frames of the sequence of observation for the planet with orbital motion during the observations (Males et al. 2015). Our design is fully functioning for the detection and characterization of Earth twins around nearby stars.

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