## Optimal pupil apodizations of arbitrary apertures for high-contrast imaging |

Optics Express, Vol. 19, Issue 27, pp. 26796-26809 (2011)

http://dx.doi.org/10.1364/OE.19.026796

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### Abstract

In the context of exoplanet direct detection and characterization, where high-contrast imaging is mandatory, we present fully optimized two-dimensional pupil apodizations for which no specific geometric constraints are put on the pupil plane apodization, apart from the shape of the aperture itself. Masks for circular and segmented apertures are displayed, with and without a central obstruction and spiders. We can now optimize apodizers for any aperture shape, and examples of optimal masks are shown for the Subaru telescope, the Space Infrared telescope for Cosmology and Astrophysics (SPICA) and the James Webb Space Telescope (JWST). Several high-contrast regions are considered with different sizes, positions, shapes and contrasts. It is interesting to note that all the masks that result from these optimizations tend to have a binary transmission profile.

© 2011 OSA

## 1. Introduction

^{−10}in the case of an Earth-like planet observed in the visible), direct detection is a significant challenge. These conditions are however partially relaxed when looking at bigger, younger objects, such as young giant planets. More information can be found in the literature (see, e.g., [2

2. A. Burrows, “A theoretical look at the direct detection of giant planets outside the solar system,” Nature **433**, 261–268 (2005). [CrossRef] [PubMed]

*pupil apodization*creates a high-contrast point-spread function (PSF) by varying the amplitude transmission in the entrance pupil of the telescope (see, e.g., [3]). In fact, Slepian [4

4. D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. **55**, 1110 (1965). [CrossRef]

5. C. Aime, R. Soummer, and A. Ferrari, “Total coronagraphic extinction of rectangular apertures using linear prolate apodizations,” Astron. Astrophys. **389**, 334–344 (2002). [CrossRef]

6. R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J. **618**, L161–L164 (2005). [CrossRef]

7. P. Martinez, C. Dorrer, E. Aller Carpentier, M. Kasper, A. Boccaletti, K. Dohlen, and N. Yaitskova, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. **495**, 363–370 (2009). [CrossRef]

9. R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spiderweb masks for high-contrast imaging,” Astrophys. J. **590**, 593–603 (2003). [CrossRef]

11. R. J. Vanderbei, N. J. Kasdin, and D. N. Spergel, “Checkerboard-mask coronagraphs for high-contrast imaging,” Astrophys. J. **615**, 555–561 (2004). [CrossRef]

13. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R.J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express **15**, 24, 15935–15951 (2007). [CrossRef]

## 2. Formalism of the two dimensional pupil optimization

*A*(

*x,y*): Here, the aperture has diameter

*D*along the

*x*and

*y*axes (the aperture is therefore assumed to be contained within a

*D*×

*D*square), the focal length of the instrument is

*f*, and

*λ*denotes the light’s wavelength.

*x*and

*y*with new variables, for which we use the same notation, that measure length in units of

*D*(formally,

*x*→

*xD*and

*y*→

*yD*). That is,

*x*= 1 means one diameter. Similarly, in the image plane we replace the physical lengths

*u*and

*v*with values that represent radians on the sky (formally,

*u*→

*uλF*/

*D*and

*v*→

*vλF*/

*D*). With this substitution, an image plane variable

*u*represents a physical position of

*uλf*/

*D*on the detector. It also represents an angular position of

*uλ*/

*D*radians on the sky. Throughout the rest of this paper we view image plane variables as measures of

*λ*/

*D*radians on the sky. Making these substitutions, Eq. (1) becomes This two-dimensional Fourier transform can be approximated using discrete sets of points in both the pupil and the image planes: with

*k*indexed on the set {1,...,

*M*} and

_{u}*l*on the set {1,...,

*M*}. We will assume equal sizes along the

_{v}*x*and

*y*axes, as well as along the

*u*and

*v*axes, making

*N*=

_{x}*N*=

_{y}*N*and

*M*=

_{u}*M*=

_{v}*M*. Furthermore, we will consider a uniform partition in the (

*x,y*)-plane, and thus Δ

*x*= Δ

_{i}*y*= 1/

_{i}*N*for all

*i*. When the apodizer can actually be described as an array of pixels, then Eq. (3) is the

*exact*amplitude of the PSF.

*NM*, where

*N*and

*M*are respectively the number of “pixels” in the pupil and image plane. A basic 2D FT leads to a complexity of

*N*

^{2}

*M*

^{2}; for a typical desktop computer, the maximum value for

*NM*is roughly 4500, which puts serious limits on how large

*N*and

*M*can be. For example, if

*M*is chosen to be 30, which is about the minimum one can get away with in the image plane, then

*N*can be at most 150. Consequently, the smallest detail in a pupil mask of 1 cm in diameter would have a width of ≈ 33

*μm*. This is too low a resolution to produce sufficient contrast, as explained in more detail in Section 3.

13. R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R.J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express **15**, 24, 15935–15951 (2007). [CrossRef]

*N*

^{2}

*M*+

*NM*

^{2}if the 2D Fourier transform is made in two steps, first along one axis, and then along the other one, using as inputs the results of the first computation in the second 1D FT, This two-step method dramatically decreases the complexity by reducing the number of computations at the expense of increased memory allocation (the intermediate matrix

*E*has to be stored during this process).

_{temp}*μm*length. Note that, in this particular case, we gain a factor 1000

^{2}× 50

^{2}/(1000

^{2}× 50 + 1000 × 50

^{2}) ≈ 48 in efficiency over the classical, brute-force, computation of the 2D FT.

^{−c}is the targeted contrast. The electric field in the image,

*E*(

*u*,

_{k}*v*), is subject to this constraint only in a specific discovery space Δ

_{l}*(defined in particular by an IWA and an OWA). The apodization,*

_{F}*A*(

*x*,

_{i}*y*), is also defined on a specific domain, Δ

_{j}*, the initial aperture shape.*

_{P}*λ*

_{0}, they can be used in broadband. We can define two new quantities: IWA

_{Δλ}and OWA

_{Δλ}, the effective IWA and OWA, functions of the bandwidth Δ

*λ*. IWA

_{Δλ}is the IWA for the largest wavelength, and OWA

_{Δλ}is the OWA for the shortest wavelength. The contrast level that was specified in the optimization problem is maintained in the region that these two angles define over the broadband. We can write that: The increase of the bandwidth decreases the extension of the high-contrast zone:

16. B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Veran, “The Gemini Planet Imager,” Proc. SPIE **6272**, 62720L (2006). [CrossRef]

17. K. Dohlen, J.-L. Beuzit, M. Feldt, D. Mouillet, P. Puget, J. Antichi, A. Baruffolo, P. Baudoz, A. Berton, A. Boccaletti, M. Carbillet, J. Charton, R. Claudi, M. Downing, C. Fabron, P. Feautrier, E. Fedrigo, T. Fusco, J.-L. Gach, R. Gratton, N. Hubin, M. Kasper, M. Langlois, A. Longmore, C. Moutou, C. Petit, J. Pragt, P. Rabou, G. Rousset, M. Saisse, H.-M. Schmid, E. Stadler, D. Stamm, M. Turatto, R. Waters, and F. Wildi, “SPHERE: A planet finder instrument for the VLT,” Proc. SPIE **6269**, 62690Q (2006). [CrossRef]

*λ*/

*D*lead to an IWA

_{Δλ}of ≈ 3.3

*λ*/

*D*, and an OWA

_{Δλ}of ≈ 13.5

*λ*/

*D*. The effective angular extension of the high-contrast zone is 85% of the original one.

## 3. Circular unobstructed aperture

### 3.1. Case of a circular symmetric high contrast region

9. R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spiderweb masks for high-contrast imaging,” Astrophys. J. **590**, 593–603 (2003). [CrossRef]

^{−6}inside a ring with an IWA of 3

*λ*/

*D*and an OWA of 15

*λ*/

*D*, as can be seen in Fig. 2. A comparison of the same region is shown; the size of the arrays has a clear influence on the quality of the optimal transmission. In particular, the value of the transmission tends to converge to either 0 or 1. The number of pixels for which the transmission adopts a value between 0 and 1 decreases greatly when the size of the array is increased; the low resolution pupil has 1.8% of its transmitted intensity between 0.1 and 0.9, while less than 0.2% of the high resolution pupil has this property.

^{−6}can be seen inside the high contrast ring when the pupil is too poorly sampled (N=200). The same does not occur with the higher resolution (N=1000). Apart from one exception, discussed next, all of the PSFs that we show in the rest of the paper are computed for the binary masks obtained by rounding all pixels either to zero or to one.

9. R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spiderweb masks for high-contrast imaging,” Astrophys. J. **590**, 593–603 (2003). [CrossRef]

*λ*/

*D*) and the same contrast (10

^{−10}). The original concentric ring mask and the new one have similar throughputs (23.5% instead of 25%) but the number of rings is not the same in the two cases (5 instead of 6). As seen in Fig. 4, the mask (designed with N=1000, M=100), only provides a mean contrast of 2 × 10

^{−9}when its transmission values have been rounded. Less than 0.12% of the pixels have an apodization value between 0.1 and 0.9, but the targeted high contrast level makes this small ratio significant enough to change the effective contrast by more than an order of magnitude. We should, however, point out that the 1D mask was designed using a two-step process. In the first step, a linear programming problem was solved to find a discretized approximate solution to the problem. For the second step, the on-off and off-on thresholds from the approximate solution were used as input to a 1D nonconvex nonlinear optimization problem that hones these thresholds to highly precise values. We have not adapted this second stage to the current 2D design process. We are currently working on such an enhancement.

### 3.2. Case of two symmetric dark holes

## 4. Circular pupil with a central obstruction and spiders

### 4.1. SPICA

19. K. Enya, T. Kotani, K. Haze, K. Aono, T. Nakagawa, H. Matsuhara, H. Kataza, T. Wada, M. Kawada, K. Fujiwara, M. Mita, S. Takeuchi, K. Komatsu, S. Sakai, H. Uchida, S. Mitani, T. Yamawaki, T. Miyata, S. Sako, T. Nakamura, K. Asano, T. Yamashita, N. Narita, T. Matsuo, M. Tamura, J. Nishikawa, E. Kokubo, Y. Hayano, S. Oya, M. Fukagawa, H. Shibai, N. Baba, N. Murakami, Y. Itoh, M. Honda, B. Okamoto, S. Ida, M. Takami, L. Abe, O. Guyon, P. Bierden, and T. Yamamuro, “The SPICA coronagraphic instrument (SCI) for the study of exoplanets,” Adv. Space Res. **48**, 323–333 (2011). [CrossRef]

*λ*/

*D*and its OWA is 12

*λ*/

*D*. The discovery zone for which it is designed is similar to the one created by the first mask that appears in [19

19. K. Enya, T. Kotani, K. Haze, K. Aono, T. Nakagawa, H. Matsuhara, H. Kataza, T. Wada, M. Kawada, K. Fujiwara, M. Mita, S. Takeuchi, K. Komatsu, S. Sakai, H. Uchida, S. Mitani, T. Yamawaki, T. Miyata, S. Sako, T. Nakamura, K. Asano, T. Yamashita, N. Narita, T. Matsuo, M. Tamura, J. Nishikawa, E. Kokubo, Y. Hayano, S. Oya, M. Fukagawa, H. Shibai, N. Baba, N. Murakami, Y. Itoh, M. Honda, B. Okamoto, S. Ida, M. Takami, L. Abe, O. Guyon, P. Bierden, and T. Yamamuro, “The SPICA coronagraphic instrument (SCI) for the study of exoplanets,” Adv. Space Res. **48**, 323–333 (2011). [CrossRef]

### 4.2. The Subaru telescope

^{−5}at 3

*λ*/

*D*for a throughput of 24%.

^{−8}, although a larger IWA of 5

*λ*/

*D*is necessary for the throughput to remain comparable (35% of the light is transmitted).

## 5. Asymmetric pupil with a segmented mirror

*π*/2 if necessary), but the second FT must be completed without it. This results in an array twice as large as with the other pupils (512 × 1024).

*E*(

*u,v*)| ≤ 10

^{−2.5}

*E*(0,0) is replaced with a pair of constraints, one for the real part and one for the imaginary part:

*λ*/

*D*. A smaller IWA (4

*λ*/

*D*) could be obtained with a similar throughput by changing the dark hole to a hexagonal ring, with internal and external contours rotated by

*π*/2 with respect to each other. The resulting mask and PSF are shown in Fig. 9. Several other geometric aspects still remain to be tested.

## 6. Final remarks

20. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. **404**, 379–387 (2003). [CrossRef]

21. D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph. I. principle,” Publ. Astron. Soc. Pac. **112**, 1479–1486 (2000). [CrossRef]

22. D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. **633**, 1191–1200 (2005). [CrossRef]

23. J. Lozi, F. Martinache, and O. Guyon, “Phase-induced amplitude apodization on centrally obscured pupils: design and first laboratory demonstration for the Subaru telescope pupil,” Publ. Astron. Soc. Pac. **121**, 1232–1244 (2009). [CrossRef]

24. R. Galicher, P. Baudoz, and J. Baudrand, “Multi-stage four-quadrant phase mask: achromatic coronagraph for space-based and ground-based telescopes,” Astron. Astrophys. **530**, id.A43 (2011). [CrossRef]

25. D. Mawet, E. Serabyn, W. J. Kent, and L. Pueyo, “Improved high-contrast imaging with on-axis telescopes using a multistage vortex coronagraph,” Opt. Lett. **36**, 1506–1508 (2011). [CrossRef] [PubMed]

^{−10}were tested in the laboratory, and contrasts smaller than 7 × 10

^{−8}were achieved in laser light [19

19. K. Enya, T. Kotani, K. Haze, K. Aono, T. Nakagawa, H. Matsuhara, H. Kataza, T. Wada, M. Kawada, K. Fujiwara, M. Mita, S. Takeuchi, K. Komatsu, S. Sakai, H. Uchida, S. Mitani, T. Yamawaki, T. Miyata, S. Sako, T. Nakamura, K. Asano, T. Yamashita, N. Narita, T. Matsuo, M. Tamura, J. Nishikawa, E. Kokubo, Y. Hayano, S. Oya, M. Fukagawa, H. Shibai, N. Baba, N. Murakami, Y. Itoh, M. Honda, B. Okamoto, S. Ida, M. Takami, L. Abe, O. Guyon, P. Bierden, and T. Yamamuro, “The SPICA coronagraphic instrument (SCI) for the study of exoplanets,” Adv. Space Res. **48**, 323–333 (2011). [CrossRef]

^{−7}for a central wavelength

*λ*

_{0}= 650 nm (Δ

*λ*= 8 nm) to 2.6 × 10

^{−6}for

*λ*

_{0}= 850 nm (Δ

*λ*= 55 nm). In both experiments the wavefront was not actively corrected.

## References and links

1. | National Research Council, |

2. | A. Burrows, “A theoretical look at the direct detection of giant planets outside the solar system,” Nature |

3. | P. Jacquinot and B. Roizen-Dossier, “Apodisation,” Prog. Optics |

4. | D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. |

5. | C. Aime, R. Soummer, and A. Ferrari, “Total coronagraphic extinction of rectangular apertures using linear prolate apodizations,” Astron. Astrophys. |

6. | R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J. |

7. | P. Martinez, C. Dorrer, E. Aller Carpentier, M. Kasper, A. Boccaletti, K. Dohlen, and N. Yaitskova, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys. |

8. | D. N. Spergel and N. J. Kasdin, “A new pupil for detecting extrasolar planets,” Bull. AAS |

9. | R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spiderweb masks for high-contrast imaging,” Astrophys. J. |

10. | N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J. |

11. | R. J. Vanderbei, N. J. Kasdin, and D. N. Spergel, “Checkerboard-mask coronagraphs for high-contrast imaging,” Astrophys. J. |

12. | R. J. Vanderbei, “Fast Fourier optimization: sparsity matters,” MPC (to be published). |

13. | R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R.J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express |

14. | R. Fourer, D. M. Gay, and B. W. Kernighan, “A modeling language for mathematical programming,” Manage. Sci. |

15. | R. J. Vanderbei, “LOQO: An interior point code for quadratic programming,” Optim. Method. Softw. |

16. | B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Veran, “The Gemini Planet Imager,” Proc. SPIE |

17. | K. Dohlen, J.-L. Beuzit, M. Feldt, D. Mouillet, P. Puget, J. Antichi, A. Baruffolo, P. Baudoz, A. Berton, A. Boccaletti, M. Carbillet, J. Charton, R. Claudi, M. Downing, C. Fabron, P. Feautrier, E. Fedrigo, T. Fusco, J.-L. Gach, R. Gratton, N. Hubin, M. Kasper, M. Langlois, A. Longmore, C. Moutou, C. Petit, J. Pragt, P. Rabou, G. Rousset, M. Saisse, H.-M. Schmid, E. Stadler, D. Stamm, M. Turatto, R. Waters, and F. Wildi, “SPHERE: A planet finder instrument for the VLT,” Proc. SPIE |

18. | K. Enya and L. Abe, “A binary shaped mask coronagraph for a segmented pupil,” Publ. Astron. Soc. Pac. |

19. | K. Enya, T. Kotani, K. Haze, K. Aono, T. Nakagawa, H. Matsuhara, H. Kataza, T. Wada, M. Kawada, K. Fujiwara, M. Mita, S. Takeuchi, K. Komatsu, S. Sakai, H. Uchida, S. Mitani, T. Yamawaki, T. Miyata, S. Sako, T. Nakamura, K. Asano, T. Yamashita, N. Narita, T. Matsuo, M. Tamura, J. Nishikawa, E. Kokubo, Y. Hayano, S. Oya, M. Fukagawa, H. Shibai, N. Baba, N. Murakami, Y. Itoh, M. Honda, B. Okamoto, S. Ida, M. Takami, L. Abe, O. Guyon, P. Bierden, and T. Yamamuro, “The SPICA coronagraphic instrument (SCI) for the study of exoplanets,” Adv. Space Res. |

20. | O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. |

21. | D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The four-quadrant phase-mask coronagraph. I. principle,” Publ. Astron. Soc. Pac. |

22. | D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astrophys. J. |

23. | J. Lozi, F. Martinache, and O. Guyon, “Phase-induced amplitude apodization on centrally obscured pupils: design and first laboratory demonstration for the Subaru telescope pupil,” Publ. Astron. Soc. Pac. |

24. | R. Galicher, P. Baudoz, and J. Baudrand, “Multi-stage four-quadrant phase mask: achromatic coronagraph for space-based and ground-based telescopes,” Astron. Astrophys. |

25. | D. Mawet, E. Serabyn, W. J. Kent, and L. Pueyo, “Improved high-contrast imaging with on-axis telescopes using a multistage vortex coronagraph,” Opt. Lett. |

26. | K. Haze, K. Enya, L. Abe, T. Kotani, T. Nakagawa, T. Sato, and T. Yamamuro, “Multi-color coronagraph experiment in a vacuum testbed with a binary shaped pupil mask,” Publ. Astron. Soc. Jpn. |

**OCIS Codes**

(110.1220) Imaging systems : Apertures

(110.6770) Imaging systems : Telescopes

(220.1230) Optical design and fabrication : Apodization

(350.1260) Other areas of optics : Astronomical optics

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: August 31, 2011

Revised Manuscript: October 8, 2011

Manuscript Accepted: November 12, 2011

Published: December 14, 2011

**Citation**

A. Carlotti, R. Vanderbei, and N. J. Kasdin, "Optimal pupil apodizations of arbitrary apertures for high-contrast imaging," Opt. Express **19**, 26796-26809 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26796

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### References

- National Research Council, New worlds, new horizons in astronomy and astrophysics (National Academies Press, 2010).
- A. Burrows, “A theoretical look at the direct detection of giant planets outside the solar system,” Nature433, 261–268 (2005). [CrossRef] [PubMed]
- P. Jacquinot and B. Roizen-Dossier, “Apodisation,” Prog. Optics3 (1964).
- D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am.55, 1110 (1965). [CrossRef]
- C. Aime, R. Soummer, and A. Ferrari, “Total coronagraphic extinction of rectangular apertures using linear prolate apodizations,” Astron. Astrophys.389, 334–344 (2002). [CrossRef]
- R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J.618, L161–L164 (2005). [CrossRef]
- P. Martinez, C. Dorrer, E. Aller Carpentier, M. Kasper, A. Boccaletti, K. Dohlen, and N. Yaitskova, “Design, analysis, and testing of a microdot apodizer for the apodized pupil Lyot coronagraph,” Astron. Astrophys.495, 363–370 (2009). [CrossRef]
- D. N. Spergel and N. J. Kasdin, “A new pupil for detecting extrasolar planets,” Bull. AAS33, 1431 (2001).
- R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spiderweb masks for high-contrast imaging,” Astrophys. J.590, 593–603 (2003). [CrossRef]
- N. J. Kasdin, R. J. Vanderbei, D. N. Spergel, and M. G. Littman, “Extrasolar planet finding via optimal apodized-pupil and shaped-pupil coronagraphs,” Astrophys. J.582, 1147–1161 (2003). [CrossRef]
- R. J. Vanderbei, N. J. Kasdin, and D. N. Spergel, “Checkerboard-mask coronagraphs for high-contrast imaging,” Astrophys. J.615, 555–561 (2004). [CrossRef]
- R. J. Vanderbei, “Fast Fourier optimization: sparsity matters,” MPC (to be published).
- R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R.J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express15, 24, 15935–15951 (2007). [CrossRef]
- R. Fourer, D. M. Gay, and B. W. Kernighan, “A modeling language for mathematical programming,” Manage. Sci.36, 519–554 (1990). [CrossRef]
- R. J. Vanderbei, “LOQO: An interior point code for quadratic programming,” Optim. Method. Softw.12, 451–484 (1999). [CrossRef]
- B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Veran, “The Gemini Planet Imager,” Proc. SPIE6272, 62720L (2006). [CrossRef]
- K. Dohlen, J.-L. Beuzit, M. Feldt, D. Mouillet, P. Puget, J. Antichi, A. Baruffolo, P. Baudoz, A. Berton, A. Boccaletti, M. Carbillet, J. Charton, R. Claudi, M. Downing, C. Fabron, P. Feautrier, E. Fedrigo, T. Fusco, J.-L. Gach, R. Gratton, N. Hubin, M. Kasper, M. Langlois, A. Longmore, C. Moutou, C. Petit, J. Pragt, P. Rabou, G. Rousset, M. Saisse, H.-M. Schmid, E. Stadler, D. Stamm, M. Turatto, R. Waters, and F. Wildi, “SPHERE: A planet finder instrument for the VLT,” Proc. SPIE6269, 62690Q (2006). [CrossRef]
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