## Super-Gaussian apodization in ground based telescopes for high contrast coronagraph imaging |

Optics Express, Vol. 21, Issue 10, pp. 12744-12756 (2013)

http://dx.doi.org/10.1364/OE.21.012744

Acrobat PDF (1587 KB)

### Abstract

We introduce the use of Super-Gaussian apodizing functions in the telescope pupil plane and/or the coronagraph Lyot plane to improve the imaging contrast in ground-based coronagraphs. We describe the properties of the Super-Gaussian function, we estimate its second-order moment in the pupil and Fourier planes and we check it as an apodizing function. We then use Super-Gaussian function to apodize the telescope pupil, the coronagraph Lyot plane or both of them. The result is that a proper apodizing masks combination can reduce the exoplanet detection distance up to a 45% with respect to the classic Lyot coronagraph, for moderately aberrated wavefronts. Compared to the prolate spheroidal function the Super-Gaussian apodizing function allows the planet light up to 3 times brighter. An extra help to increase the extinction rate is to perform a frame selection (Lucky Imaging technique). We show that a selection of the 10% best frames will reduce up to a 20% the detection angular distance when using the classic Lyot coronagraph but that the reduction is only around the 5% when using an apodized coronagraph.

© 2013 OSA

## 1. Introduction

2. V. F. C. Vidal F. Canales, P. J. V. Pedro J. Valle, J. E. O. Jose E. Oti, and M. P. C. Manuel P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. **7**(8), 720–723 (2009). [CrossRef]

4. V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express **14**(22), 10393–10402 (2006). [CrossRef] [PubMed]

6. D. Slepian, “Analytic solution for two apodization problems,” J. Opt. Soc. Am. **55**(9), 1110–1115 (1965). [CrossRef]

7. R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J. **618**(2), L161–L164 (2005). [CrossRef]

8. R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J. **590**(1), 593–603 (2003). [CrossRef]

9. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. **404**(1), 379–387 (2003). [CrossRef]

11. O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. **622**(1), 744–758 (2005). [CrossRef]

12. R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J. **729**(2), 144 (2011). [CrossRef]

13. J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. **604**(2), L117–L120 (2004). [CrossRef]

14. D. Mawet., “Review of small-angle coronagraphic techniques in the wake of ground-based second-generation adaptive optics systems,” http://arxiv.org/abs/1207.5481 (2012). [CrossRef]

15. M. A. Cagigas, P. J. Valle, and M. P. Cagigal, “Coronagraphs adapted to atmosphere conditions,” Opt. Express **20**(4), 4574–4582 (2012). [CrossRef] [PubMed]

*D/r*is fulfilled, where

_{0}< 8*D*is the pupil diameter and

*r*the Fried parameter. This condition can be satisfied on visible or infrared bands depending on the telescope size and the degree of compensation achieved by the Adaptive Optics system. We consider that this type of apodization combined to a Lucky-Imaging detection strategy may provide new exciting results.

_{0}*D/r*value. In Section 4 we estimate the effects on the coronagraphic images of using a Super-Gaussian apodizing profile on the telescope entrance pupil and/or on the Lyot stop. Results for the prolate spheroidal apodizing function are also included for comparison. Section 5 analyzes the results obtained combining image selection (Lucky Imaging) and apodized coronagraphy. Finally, Section 6 summarizes the main results of the paper.

_{0}## 2. Super-Gaussian pupil functions

### 2.1 Theoretical description of Super-Gaussian and its Fourier transform

*n*and half-width

*σ*is defined as:where

*r*is the usual radial coordinate. It can be shown that, for a particular width value, the Super-Gaussian function ranges from two limiting cases: It is a Gaussian function for

*n*= 2 and a hard-edge circle for

*SG*(

*n,r*) can be understood as the convolution of a hard-edge circle with a Gaussian function as suggested [16

16. S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. **138**(1-3), 35–39 (1997). [CrossRef]

*n*= 2, 4, 10) along with the hard-edge circle pupil of unit diameter (HEP) (the width of Super-Gaussian functions has been fixed so that all have the value 10

^{−3}at the pupil border). We can see how it evolves from a Gaussian to a hard-edge circle as the order increases. When assuming the Super-Gaussian function as a convolution operation, its Fourier transform will be the product of the individual Fourier transforms:where

*A*(

*w*) is the Fourier transform of the hard-edge circle function (its square module

*|A*(

*w*)

*|*is the well-known Airy pattern) and

^{2}*g*(

*w*) the Gaussian function resulting from Fourier transforming

*G*(

*r*).

*n*= 2 we get that

*sg*(

*w*) is a pure Gaussian as expected. In the other limiting case (

*g*(

*w*) multiplying the Airy pattern as it can be seen in Eq. (2).

### 2.2. Moments analysis

*F*(

*r,θ*) can be evaluated from (

*r*and

*θ*are polar coordinates):So that the normalized second-order moment is given by:If

*f*(

*w,ϑ*) is the Fourier transform of

*F*(

*r,θ*), and

*w*and

*ϑ*are polar coordinates in the Fourier domain, the normalized second-order moment is expressed by:Assuming an optical system where

*F*and

*f*are the pupil and image fields respectively, the numerator in Eq. (5) gives a measure of the image spot size whilst the denominator is the total signal intensity in the image plane that, thanks to the Parseval’s theorem, is the same as the total energy in the apodized pupil. Hence, an efficient apodizing function will provide the smallest possible

*nm*(

_{2}*f*) value, that is, the narrowest function with the maximum energy. At this point it is interesting to recall the uncertainty principle namely:The right part may differ from 1/(2

*π*)

^{2}depending on the normalization used. The equality in Eq. (6) is attained when

*F*is a Gaussian function [17]. Hence, it would be interesting, to analyze how do behave the set of Super-Gaussian functions, evolving from the standard pupil (hard-edge circle) to the more balanced, from the point of view of the uncertainty principle, Gaussian function. However, only the second term of the product shown in Eq. (6) is useful for our purpose.

*sg*(

*w*) to the Fourier transform of

*SG*(

*n,r*) shown in Eq. (1), thanks to the derivative property of the Fourier transform we will have:The zero-order moment is given by:Hence, the expression for

*nm*(

_{2}*sg*) will be:Equation (9) is a measure of the image spot size of an optical system with a Super-Gaussian pupil. Figure 3 shows the evolution of

*nm*(

_{2}*sg*) (squares and left scale) as a function of the Super-Gaussian order. The width of each Super-Gaussian (also shown by circles and right scale in Fig. 3) was calculated so that all functions have the value 10

^{−3}at the unit-diameter circle (this value is the one providing the best contrast as it will be seen below). It can be seen that

*nm*(

_{2}*sg*) has a minimum value for

*n*about 5. This means that the fifth-order Super-Gaussian is the best balanced apodizing function from the point of view of spot size and energy lost in the optical system image plane.

*m*(

_{2}*SG*) can be also calculated:Now, since

*m*(

_{0}*SG*) is the same as

*m*(

_{0}*sg*) as stated in Eq. (8), we can obtain an expression for the uncertainty principle for Super-Gaussian functions:This moment product does not depend on the width

*σ*, like in the case of Gaussian functions, it depends only on the Super-Gaussian order

*n*and tends to 1/(2

*π*)

^{2}when

*n*= 2. Similar results have already been obtained in the context of propagation of Super-Gaussian field distributions [18

18. A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. **24**(9), S1071–S1079 (1992). [CrossRef]

### 2.3. Prolate spheroidal comparison

20. R. Soummer, C. Aime, A. Ferrari, A. Sivaramakrishnan, B. R. Oppenheimer, R. Makidon, and B. Macintosh “Apodized pupil lyot coronagraphs: concepts and application to the gemini planet imager,” in *Direct Imaging of Exoplanets: Science and Techniques*, Proceedings IAU Colloquium No. 200*,**2005*, C. Aime and F. Vakili., eds. (Cambridge University, 2006), pp.367–372.

20. R. Soummer, C. Aime, A. Ferrari, A. Sivaramakrishnan, B. R. Oppenheimer, R. Makidon, and B. Macintosh “Apodized pupil lyot coronagraphs: concepts and application to the gemini planet imager,” in *Direct Imaging of Exoplanets: Science and Techniques*, Proceedings IAU Colloquium No. 200*,**2005*, C. Aime and F. Vakili., eds. (Cambridge University, 2006), pp.367–372.

## 3. Apodizing atmospheric aberrated wavefronts

22. M. P. Cagigal and V. F. Canales, “Generalized Fried parameter after adaptive optics partial wave-front compensation,” J. Opt. Soc. Am. A **17**(5), 903–910 (2000). [CrossRef] [PubMed]

*D*) and the atmosphere Fried parameter (

*r*) increases since the wavefront phase variance scales as (

_{0}*D/r*)

_{0}*. Given that we are only able to act over the coherent part of the light, we expect that pupil apodization will be effective only when the coherent peak intensity contributes to the Strehl ratio more than the halo peak intensity. This condition is fulfilled for*

^{5/3}*D/r*7.8 [23]. Hence, we expect that the effect of apodization will decrease when

_{0}<*D/r*increases and it will become almost negligible for

_{0}*D/r*values larger than 7.8.

_{0}*x*and

*y*are the spatial Cartesian coordinates. Within the framework of Fourier Optics the telescope image is obtained by Fourier transform the entrance pupil field,

*SG*(

*n,r*) and the phase screen

*r =*(

*x*)

^{2}+ y^{2}*and*

^{1/2}*w =*(

*u*)

^{2}+ v^{2}*). The resulting image, obtained Fourier transforming the new pupil function, is the product between the result of convolving the function*

^{1/2}*A*(

*w*) with

*ϕ*(

*u,v*) (which describes the degraded PSF composed by coherent peak and speckled halo) and the Gaussian function

*g*(

*w*). It is necessary to state that the function

*A*(

*w*) appearing in Eq. (12) is wider than the PSF of the clear pupil of size

*D*since it corresponds to the Fourier transform of the function

*HE*(

*r*) which is always smaller than

*D*. Although the last term of Eq. (12) is an approximation only valid for slightly aberrated wavefronts, it can be helpful to understand how the Super-Gaussian apodizing function works. We see that the speckled PSF, described by the convolution product

*g*(

*w*)) what produces an efficient noisy tails reduction.

15. M. A. Cagigas, P. J. Valle, and M. P. Cagigal, “Coronagraphs adapted to atmosphere conditions,” Opt. Express **20**(4), 4574–4582 (2012). [CrossRef] [PubMed]

*E*(

*x,y*)

*= exp*(

*iφ*(

*x,y*)), where a constant field amplitude is assumed. This field is Fourier transformed to simulate the telescope outcome. Computer simulations were carried out using the FFT routine implemented in Matlab. To achieve a good spatial sampling and to avoid aliasing effects, 1024x1024 data samples were used. The telescope pupil was simulated with a 128x128 data sampling. We simulated series of

*φ*(

*x,y*) for a number of atmospheric conditions following the standard procedure established by [24

24. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. **29**(10), 1174–1180 (1990). [CrossRef]

*D/r*= 1, 5, and 9. It is clear that images obtained with the Super-Gaussian apodized pupil (b, d, f) show a sharper appearance than the corresponding unapodized ones (a, c, e), although the apodized image for

_{0}*D/r*= 9 remains highly aberrated. This sharpness will produce a reduction on the value of the parameter

_{0}*nm*(

_{2}*sg*) as we will see later in Fig. 6.

*D/r*= 1 to 2 for

_{0}*D/r*= 9.

_{0}*nm*(

_{2}*sg*) is a measure of the size of the spot to be blocked by the coronagraphic mask. Consequently the smaller

*nm*

_{2}(

*sg*) value the better contrast. We used Fig. 6 to estimate the order of the Super-Gaussian apodizing function providing the smaller

*nm*(

_{2}*sg*) value and accordingly the better contrast. It can be seen that the optimum Super-Gaussian order evolves from

*n*= 5 to

*n*= 2 as the ratio

*D/r*ranges from

_{0}*D/r*= 1 to 9. Hence, results in Fig. 6 can be used to choose the optimum value of the Super-Gaussian function order as a function of the atmosphere seeing conditions.

_{0}## 4. Apodized coronagraphy

15. M. A. Cagigas, P. J. Valle, and M. P. Cagigal, “Coronagraphs adapted to atmosphere conditions,” Opt. Express **20**(4), 4574–4582 (2012). [CrossRef] [PubMed]

24. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. **29**(10), 1174–1180 (1990). [CrossRef]

*I*(

*r*) is the intensity at the radial coordinate in the final star image,

*I*(

_{s}*0*) is the peak intensity obtained without the coronagraphic mask in the optical train, and

*|M*(

*r*)

*|*is the mask intensity transmission that in our case takes value zero inside the radius mask and unity outside it.

^{2}*D/r*values of 1, 3, 5, and 7, respectively. We see that the introduction of apodizing Super-Gaussian (SGP-HEL and SGP-SGL cases) produces an important improvement with respect to hard-edge pupil case (HEP-HEL) for all

_{0}*D/r*values. Besides, the use of a Super-Gaussian Lyot function (SGP-SGL case) improves the values of hard-edge Lyot stop (SGP-HEL). This improvement is particularly important for angular positions close to the coronagraphic mask border.

_{0}*λ/D*units) at which a companion 10

^{6}times fainter than the parent star can be detected (SNR>5) and the peak companion intensities (right numbers) are normalized to those obtained when entrance pupil and Lyot stop are both hard-edge. Data correspond to entrance pupil telescope: HEP, SGP, and apodized with a prolate spheroidal function (PSP); combined with HEL or SGL. Atmosphere conditions correspond to

*D/r*= 1, 3, 5, and 7.

_{0}*D/r*values. However, when

_{0}*D/r*increases, both combinations provide equivalent distances whilst the combinations HEP-SGL offer the highest peaks intensities. For apodized pupils the introduction of Super-Gaussian Lyot stop does not improve angular detection distance but reduces the companion signal. Hence, for low

_{0}*D/r*values it is better to apodize the pupil whilst for large values to apodize the Lyot stop.

_{0}^{6}(left numbers) and 10

^{7}(right numbers) times fainter than the parent star. Data are normalized to those values obtained when the telescope pupil and the Lyot stop are hard-edge. The main conclusion is that the distance reduction is more important when the companion intensity decreases. In this particular case the relative distance reduction in percent can reach a value of 44%. This tendency has also been confirmed for companions fainter than 10

^{−7}times the parent star. It also can be seen that the general behavior of the detection angular distance reduction as a function of

*D/r*basically does not depend on the type of apodizing function used or even on the plane (pupil or Lyot) where they are applied.

_{0}## 5. Frame selection. Lucky imaging

^{6}times fainter than the parent star appear on the left and those for a ratio of 10

^{7}appear on the right. A general conclusion is that frame selection may reduce the angular detection distance up to a 20 percent when telescope pupil and Lyot stop are hard-edge. However, when the pupil or the Lyot stop has been apodized the improvement falls to values around a 6 percent. Hence, the angular position where a companion can be detected may decrease an additional distance of 1

*λ/D*when using apodizing mask and frame selection.

## 6. Conclusions

## Acknowledgments

## References and links

1. | F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE |

2. | V. F. C. Vidal F. Canales, P. J. V. Pedro J. Valle, J. E. O. Jose E. Oti, and M. P. C. Manuel P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. |

3. | F. M. Dickey and S. C. Holswade, |

4. | V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express |

5. | P. Jacquinot and B. Roizen-Dossier, “II Apodisation,” in |

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

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

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

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

10. | R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J. |

11. | O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J. |

12. | R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J. |

13. | J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J. |

14. | D. Mawet., “Review of small-angle coronagraphic techniques in the wake of ground-based second-generation adaptive optics systems,” http://arxiv.org/abs/1207.5481 (2012). [CrossRef] |

15. | M. A. Cagigas, P. J. Valle, and M. P. Cagigal, “Coronagraphs adapted to atmosphere conditions,” Opt. Express |

16. | S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. |

17. | H. Weyl, |

18. | A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. |

19. | H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. |

20. | R. Soummer, C. Aime, A. Ferrari, A. Sivaramakrishnan, B. R. Oppenheimer, R. Makidon, and B. Macintosh “Apodized pupil lyot coronagraphs: concepts and application to the gemini planet imager,” in |

21. | T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354. |

22. | M. P. Cagigal and V. F. Canales, “Generalized Fried parameter after adaptive optics partial wave-front compensation,” J. Opt. Soc. Am. A |

23. | J. W. Hardy, |

24. | N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. |

25. | J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J. |

**OCIS Codes**

(100.2980) Image processing : Image enhancement

(110.2970) Imaging systems : Image detection systems

(110.6770) Imaging systems : Telescopes

(350.1260) Other areas of optics : Astronomical optics

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 21, 2012

Revised Manuscript: January 30, 2013

Manuscript Accepted: March 22, 2013

Published: May 16, 2013

**Citation**

Miguel A. Cagigas, Pedro J. Valle, and Manuel P. Cagigal, "Super-Gaussian apodization in ground based telescopes for high contrast coronagraph imaging," Opt. Express **21**, 12744-12756 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12744

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### References

- F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE66(1), 51–83 (1978).
- V. F. C. Vidal F. Canales, P. J. V. Pedro J. Valle, J. E. O. Jose E. Oti, and M. P. C. Manuel P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett.7(8), 720–723 (2009). [CrossRef]
- F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker Inc., 2000).
- V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express14(22), 10393–10402 (2006). [CrossRef] [PubMed]
- P. Jacquinot and B. Roizen-Dossier, “II Apodisation,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1964), Vol. 3, pp. 29–186.
- D. Slepian, “Analytic solution for two apodization problems,” J. Opt. Soc. Am.55(9), 1110–1115 (1965). [CrossRef]
- R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J.618(2), L161–L164 (2005). [CrossRef]
- R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003). [CrossRef]
- O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003). [CrossRef]
- R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J.626(2), 1079–1090 (2005). [CrossRef]
- O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005). [CrossRef]
- R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011). [CrossRef]
- J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J.604(2), L117–L120 (2004). [CrossRef]
- D. Mawet and ., “Review of small-angle coronagraphic techniques in the wake of ground-based second-generation adaptive optics systems,” http://arxiv.org/abs/1207.5481 (2012). [CrossRef]
- M. A. Cagigas, P. J. Valle, and M. P. Cagigal, “Coronagraphs adapted to atmosphere conditions,” Opt. Express20(4), 4574–4582 (2012). [CrossRef] [PubMed]
- S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997). [CrossRef]
- H. Weyl, Theory of Groups and Quantum Mechanics (Dover Publications, 1950).
- A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992). [CrossRef]
- H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J.40, 65–84 (1961).
- R. Soummer, C. Aime, A. Ferrari, A. Sivaramakrishnan, B. R. Oppenheimer, R. Makidon, and B. Macintosh “Apodized pupil lyot coronagraphs: concepts and application to the gemini planet imager,” in Direct Imaging of Exoplanets: Science and Techniques, Proceedings IAU Colloquium No. 200,2005, C. Aime and F. Vakili., eds. (Cambridge University, 2006), pp.367–372.
- T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354.
- M. P. Cagigal and V. F. Canales, “Generalized Fried parameter after adaptive optics partial wave-front compensation,” J. Opt. Soc. Am. A17(5), 903–910 (2000). [CrossRef] [PubMed]
- J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
- N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29(10), 1174–1180 (1990). [CrossRef]
- J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007). [CrossRef]

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