## Fast computation of Lyot-style coronagraph propagation

Optics Express, Vol. 15, Issue 24, pp. 15935-15951 (2007)

http://dx.doi.org/10.1364/OE.15.015935

Acrobat PDF (509 KB)

### Abstract

We present a new method for numerical propagation through Lyot-style coronagraphs using finite occulting masks. Standard methods for coronagraphic simulations involve Fast Fourier Transforms (FFT) of very large arrays, and computing power is an issue for the design and tolerancing of coronagraphs on segmented Extremely Large Telescopes (ELT) in order to handle both the speed and memory requirements. Our method combines a semi-analytical approach with non-FFT based Fourier transform algorithms. It enables both fast and memory-efficient computations without introducing any additional approximations. Typical speed improvements based on computation costs are of about ten to fifty for propagations from pupil to Lyot plane, with thirty to sixty times less memory needed. Our method makes it possible to perform numerical coronagraphic studies even in the case of ELTs using a contemporary commercial laptop computer, or any standard commercial workstation computer.

© 2007 Optical Society of America

## 1. Introduction

1. 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,” in *Advances in Adaptive Optics II. Edited by*Ellerbroek Brent L., *Bonaccini Calia, Domenico. Proceedings of the SPIE*, Volume 6272, *pp. (2006)*. (2006).

2. J. L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, T. Fusco, P. Puget, S. Udry, R. Gratton, H. M. Schmid, M. Feldt, and M. Kasper, and The Vlt-Pf Consortium, “A ”Planet Finder” instrument for the VLT,” in *Tenth Anniversary of 51 Peg-b: Status of and prospects for hot Jupiter studies*,
L. Arnold, F. Bouchy, and C. Moutou, eds., pp. 353–355 (2006).

3. M. Tamura, K. Hodapp, H. Takami, L. Abe, H. Suto, O. Guyon, S. Jacobson, R. Kandori, J.-I. Morino, N. Murakami, V. Stahlberger, R. Suzuki, A. Tavrov, H. Yamada, J. Nishikawa, N. Ukita, J. Hashimoto, H. Izumiura, M. Hayashi, T. Nakajima, and T. Nishimura, “Concept and science of HiCIAO: high contrast instrument for the Subaru next generation adaptive optics,” in *Ground-based and Airborne Instrumentation for Astronomy. Edited by*
McLean
Ian S. and
Masanori
Iye. *Proceedings of the SPIE*, Volume 6269, *pp. 62690V* (2006)., vol. 6269 of *Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference* (2006).

4. B. Macintosh, M. Troy, R. Doyon, J. Graham, K. Baker, B. Bauman, C. Marois, D. Palmer, D. Phillion, L. Poyneer, I. Crossfield, P. Dumont, B. M. Levine, M. Shao, G. Serabyn, C. Shelton, G. Vasisht, J. K. Wallace, J.-F. Lavigne, P. Valee, N. Rowlands, K. Tam, and D. Hackett, “Extreme adaptive optics for the Thirty Meter Telescope,” in *Advances in Adaptive Optics II. Edited by*
Ellerbroek
Brent L., *Bonaccini Calia, Domenico. Proceedings of the SPIE*, Volume 6272, *pp. 62720N* (2006)., vol. 6272 of *Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference* (2006).

5. L. Close“Extrasolar Planet Imaging with the Giant Magellan Telescope,” in *Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007*. University of California, Berkeley, CA, USA. *Edited by*
Paul
Kalas and P. Kalas, ed. (2007).

6. C. Verinaud, M. Kasper, J.-L. Beuzit, N. Yaitskova, V. Korkiakoski, K. Dohlen, P. Baudoz, T. Fusco, L. Mugnier, and N. Thatte “EPICS Performance Evaluation through Analytical and Numerical Modeling,” in *Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century. June 04 - 08, 2007*. University of California, Berkeley, CA, USA. *Edited by*
Kalas
Paul and P. Kalas ed. (2007).

7. O. Guyon, J. R. P. Angel, C. Bowers, J. Burge, A. Burrows, J. Codona, T. Greene, M. Iye, J. Kasting, H. Martin, D. W. McCarthy Jr., V. Meadows, M. Meyer, E. A. Pluzhnik, N. Sleep, T. Spears, M. Tamura, D. Tenerelli, R. Vanderbei, B. Woodgate, R. A. Woodruff, and N. J. Woolf “Telescope to observe planetary systems (TOPS): a high throughput 1.2-m visible telescope with a small inner working angle,” in *Space Telescopes and Instrumentation I: Optical, Infrared, and Millimeter*. *Edited by*Mather John C., MacEwen Howard A., and de Graauw Mattheus W. M.*Proceedings of the SPIE*, Volume 6265, *pp. 62651R* (2006)., vol. 6265 of *Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference* (2006).

8. J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature **446**, 771–773 (2007). [CrossRef] [PubMed]

9. A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, “Ground-based Coronagraphy with High-order Adaptive Optics,” Astrophys. J. **552**, 397–408 (2001). astro-ph/0103012. [CrossRef]

10. C. Aime and R. Soummer, “The Usefulness and Limits of Coronagraphy in the Presence of Pinned Speckles,” Astrophys. J. **612**, L85–L88 (2004). [CrossRef]

12. O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical Limits on Extrasolar Terrestrial Planet Detection with Coronagraphs,” Astrophys. J. **167**, 81–99 (2006). arXiv:astro-ph/0608506. [CrossRef]

17. R. Soummer, “Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures,” Astrophys. J. **618**, L161–L164 (2005). [CrossRef]

19. F. Roddier and C. Roddier, “Stellar Coronagraph with Phase Mask,” PASP **109**, 815–820 (1997). [CrossRef]

21. D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The Four-Quadrant Phase-Mask Coronagraph. I. Principle,” PASP **112**, 1479–1486 (2000). [CrossRef]

24. G. Foo, D. M. Palacios, and G. A. Swartzlander Jr. “Optical vortex coronagraph,” Opt. Lett. **30**, 3308–3310 (2005). [CrossRef]

*D*), is not sufficient here. Indeed, the small occulting mask has a typical size of about 5

*λ*/

*D*, and would not be well represented by a disk ten pixels in diameter. This is even more problematic for phase masks of size ~

*λ*/

*D*[19

19. F. Roddier and C. Roddier, “Stellar Coronagraph with Phase Mask,” PASP **109**, 815–820 (1997). [CrossRef]

27. A. Ferrari, R. Soummer, and C. Aime, “An introduction to stellar coronagraphy,” Comptes Rendus Physique **8**, 277–287 (2007). arXiv:astro-ph/0703655. [CrossRef]

28. D. H. Bailey and P. N. Swarztrauber, “The Fractional Fourier Transform and Applications,” SIAM Review **33(3)**, 389–404 (1991). [CrossRef]

30. J. O. Smith, *Mathematics of the Discrete Fourier Transform (DFT)* (W3K Publishing, http://www.w3k.org/books/, 2007).

## 2. Lyot-type Coronagraphs

*et al.*[15] and Soummer

*et al.*[16, 18]. The general layout is given in Fig.1. The setup consists of an ensemble of apodizers, masks and stops in four successive planes

*A*,

*B*,

*C*,

*D*, respectively, where

*A*is the entrance aperture,

*B*is the focal plane with the occulting mask,

*C*is an image of the entrance aperture where a pupil mask called Lyot stop is placed and

*D*is the final image plane. We will consider the usual approximations of paraxial optics [25], and that the optical layout is properly designed to cancel the quadratic phase terms associated with Fresnel propagation, so that a FT relationship exists between two successive planes. The tele-

**r**=(

*x*,

*y*) is denoted by P(

**r**) (index function equal to 1 inside the aperture 𝒫). This aperture can be apodized by a function Φ(r). Note that these functions do not have to be radial. A mask of transmission 1-ε

*M*(

**r**) is placed in the focal plane.

*M*is the index function that describes the mask shape

*𝓜*, equal to 1 inside the coronagraphic mask and 0 outside.

*L*is the index function of the Lyot stop. We recall that both the Lyot coronagraph (opaque mask) and the Roddier coronagraph (

*π*phase mask) can be described by this common formalism, with

*ε*=1 for Lyot and

*ε*=2 for Roddier using that

*e*

*=-1, as detailed in [15, 27*

^{iπ}27. A. Ferrari, R. Soummer, and C. Aime, “An introduction to stellar coronagraphy,” Comptes Rendus Physique **8**, 277–287 (2007). arXiv:astro-ph/0703655. [CrossRef]

31. C. Aime, “Principle of an Achromatic Prolate Apodized Lyot Coronagraph,” PASP117, 1012-+ (2005). [CrossRef]

## 3. Classical numerical propagation and FFTs

*x*

*=(*

_{n}*n*-

_{γ}*N*

*/2)δ*

_{A}*x*,

*n*∊[0,

_{γ}*N*

*-1], corresponding to the sampling points of*

_{A}*f*(

*x*), and

*u*

*=(*

_{k}*k*-

*N*

*/2)δ*

_{B}*u*,

*k*∊[0,

*N*

*-1] are the sampled values of the Fourier transform.*

_{B}*N*

*is the number of pixels along the pupil diameter*

_{A}*D*, and

*N*

*is the number of pixels in the focal plane. Note that under the Riemann sum approximation, independent sampling grids can be chosen in both domains. However, when one chooses the same size*

_{B}*N*=

_{γ}*N*

*=*

_{A}*N*

*for both arrays, and the integration steps as:*

_{B}*N*is 5

*N*log

_{2}

*N*. It is generally assumed that this is also an approximation for the number of operations in any complex FFT [32

32. M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proceedings of the IEEE93(2), 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”. [CrossRef]

*u*=

*λ*/(

_{γ}*), as illustrated on Fig. 2. Under this condition, the occulting mask of*

_{D}*m*resolution elements is therefore sampled with

*γ*

*m*pixels. Because stellar coronagraphs use very small masks (typically 4~5λ/

*D*for an APLC, and λ/

*D*for a PM or DZPM), decent sampling of these masks (at the very least a few tens of pixels) imposes large zero padding factors

*γ*. A general consensus for coronagraphic calculation is to use

*γ*=6 or

*γ*=8. The memory and computational speed problems associated with classical coronographic algorithms stem from the requirement of finely sampling the image plane mask. This can lead to prohibitively large padding factors for some applications which can benefit for particularly fine sampling, such as tip-tilt tolerancing [33

33. J. P. Lloyd and A. Sivaramakrishnan, “Tip-Tilt Error in Lyot Coronagraphs,” Astrophys. J. **621**, 1153–1158 (2005). arXiv:astro-ph/0503661. [CrossRef]

34. A. Sivaramakrishnan and J. P. Lloyd, “Spiders in Lyot Coronagraphs,” Astrophys. J. **633**, 528–533 (2005). arXiv:astro-ph/0506564. [CrossRef]

19. F. Roddier and C. Roddier, “Stellar Coronagraph with Phase Mask,” PASP **109**, 815–820 (1997). [CrossRef]

## 4. Semi-analytical coronagraphic propagations

### 4.1. Principle

*A*(

**r**)] is truncated by the occulting spot

*M*(

**r**), and that the second FT is truncated by the Lyot Stop

*L*(

**r**). This means that we are only interested in the knowledge of the FTs

*inside*limited areas, viz., the limited occulting mask area, and the limited Lyot stop area. We can thus completely circumvent the sampling problem by restricting the information of the FTs to these two zones: the semi-analytical approach consists of computing these limited-area FTs numerically, and subtracting the result from the pupil complex amplitude, according Eq. 9.

28. D. H. Bailey and P. N. Swarztrauber, “The Fractional Fourier Transform and Applications,” SIAM Review **33(3)**, 389–404 (1991). [CrossRef]

30. J. O. Smith, *Mathematics of the Discrete Fourier Transform (DFT)* (W3K Publishing, http://www.w3k.org/books/, 2007).

*and*Lyot stop), we actually replace fast calculations with very large arrays by slow calculations with very small arrays. Note that for

*γ*=8, only 40×40 values of the FT need to be calculated in the first focal plane for a 5

*λ*/

*D*mask, independently of the size of the pupil n pixels.

*without*zero padding as a

*N*

*×*

_{A}*N*

*array, where*

_{A}*N*

*is the diameter of the pupil in pixels.*

_{A}*limited*to the occulting mask as a

*N*

*×*

_{B}*N*

*array, with e.g.*

_{B}*N*

*=40 for a 5*

_{B}*λ*/

*D*mask with

*γ*=8.

*limited*to the pupil (

*N*

*×*

_{A}*N*

*array).*

_{A}*N*

*/2+1 (or*

_{A}*N*

*/2+1), and that the focal plane mask can be defined using the gray pixel approximation, as for the classical FFT method. These computation steps are illustrated in Fig.3 and compared to the classical FFT approach. The figure shows the actual arrays that are calculated for both method: note that the semi-analytical method does not use zero-padding in the pupil plane and calculates the focal plane amplitude only*

_{B}*inside*the occulting mask, instead of

*outside*in the case of the classical FFT method.

9. A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, “Ground-based Coronagraphy with High-order Adaptive Optics,” Astrophys. J. **552**, 397–408 (2001). astro-ph/0103012. [CrossRef]

*D*=8

*m*), the secondary mirror support structures are about 1

*cm*wide. Understanding the effects of spiders on coronagraphic performance is important since they appear bright in the Lyot plane and musk be masked out by the Lyot stop [34

34. A. Sivaramakrishnan and J. P. Lloyd, “Spiders in Lyot Coronagraphs,” Astrophys. J. **633**, 528–533 (2005). arXiv:astro-ph/0506564. [CrossRef]

*N*

*=3200 pixels are needed in the pupil. Standard*

_{A}*γ*=8 padding would require 25600×25600 FFTs. The semi analytical method enables this calculation in a few seconds on a standard desktop computer (Table 2).

*outside*the focal plane mask. In fact, we do not lose any information because the method is based on the analytical formulation of Eq. 9 which is equivalent to Eq. 3.

## 4.2. Matrix direct Fourier transform

*m*expressed in resolution elements units (

*λ*/

*D*), we choose the sampling step in plane B such that:

*du*=

*m*/

*N*

*. We compute the Riemann sum directly using a matrix formulation of Eq.6:*

_{B}*x*

*=*

_{k}*y*

*=(*

_{k}*k*-

*N*

*/2)×1/*

_{A}*N*

*and*

_{A}*u*

*=*

_{l}*v*

*=(*

_{l}*l*-

*N*

*/2)×*

_{A}*m*/

*N*

*, for*

_{A}*k*=[0, …,

*N*

*-1] and l=[0, …,*

_{A}*N*

*-1].*

_{B}*m*/(

*N*

_{A}*N*

*) imposes the conservation of energy according to the Parseval theorem [26]: the energy in the limited-area FT is a fraction of the total energy of the FT, corresponding to the limited area, which was calculated.*

_{B}*N*

*/2+1.*

_{B}*du*in the focal plane is obtained at the expense of partial information on the FT. Note that in the case of the

*MFT*,

*γ*still corresponds to the sampling of the FT (number of pixels per resolution element), but not to the zero-padding coefficient. The semi-analytical formalism frees the computation from the rigid sampling constraints of FFT methods. This provides increased flexibility in the computation, since the number of pixels in the pupil and focal planes can be chosen independently.

*E*

_{1},

*f*,

*E*

_{2}are:

*N*

*×*

_{B}*N*

*,*

_{A}*N*

*×*

_{A}*N*

*, and*

_{A}*N*

*×*

_{B}*N*

*. We consider the case of a complex FT for generality. Each element of the result in a matrix product is the result of*

_{A}*N*

*multiplications and*

_{A}*N*

*-1 additions. While complex addition requires 2 flops (floating point operations), complex multiplications require 6 flops. Since the two matrices*

_{A}*E*

_{1}and

*E*

_{2}can be calculated beforehand, in a similar fashion that FFT plans are generated, the total number of operations for the product

*f*.

*E*

_{2}is therefore:

*N*

_{A}*N*

*(8*

_{B}*N*

*-2)⋍8*

_{A}*N*

^{2}

_{A}*N*

*, assuming*

_{B}*N*

*large enough. The number of flops involved in the first complex MFT (pupil to focal) is therefore:*

_{A}*N*, it is interesting to note that the number of operations for the matrix method is proportional to

*N*

^{3}and not

*N*

^{4}as for a direct calculation of the DFT. This is because the two successive matrix products take advantage of some redundancy in intermediate calculations. The FFT is as expected more efficient in this case, and the relative number of operations is:

## 4.3. Fast fractional Fourier transform

*x*

*=*

_{n}*n*δ

*x*-(

*N*

*δ*

_{A}*x*)/2,

*n*∊[0,

*N*

*-1] with δ*

_{A}*x*=

*D*/(

*N*

*) and*

_{A}*u*

*=*

_{k}*k*δ

*u*-(

*N*

*δ*

_{B}*u*)/2,

*k*∊[0,

*N*

*-1] with δ*

_{B}*u*=1/(

*γD*) and

*N*

*=*

_{B}*mγ*, where

*m*is the size of the image plane mask in resolution element units (

*λ*/

*D*). Note that here again

*γ*does not correspond to an oversize of the pupil but directly to the number of pixels per unit of angular resolution in the image plane. The Riemann sum can be re-written as:

*N*

*. We direct the reader to the aforementioned references for details on these computation schemes. In order to use fast circular convolution algorithms, the incoming sequence needs to be zero-padded by factor of two, and the computational cost corresponds then to three FFTs of size 2*

_{A}*N*

*. When the field in the image plane is computed up to the Nyquist limit, the computational cost of the method is 20*

_{A}*N*

^{2}

*log*

_{A}_{2}(

*N*

^{2}

*). When the size of the final array is*

_{A}*N*

*≠*

_{B}*N*

*, partial convolution algorithms can be used, and they reduce the computational cost to 20*

_{A}*N*

^{2}

*log*

_{A}_{2}(

*N*

^{2}

*) [28*

_{B}28. D. H. Bailey and P. N. Swarztrauber, “The Fractional Fourier Transform and Applications,” SIAM Review **33(3)**, 389–404 (1991). [CrossRef]

*N*

*large) the fractional FT method has a lower cost than the MFT and the potential gain is of the order of*

_{B}*N*

*/log*

_{B}_{2}(

*N*

*).*

_{B}## 5. Performance comparison and practical applications

### 5.1. Computation costs for coronagraphic propagation

*γ*. The array sizes are therefore (

*γNA*)

^{2}, where

*N*

*is the number of pixels across the pupil diameter and the number of flops involved in each FFT is 10(*

_{A}*γNA*)

^{2}log

_{2}(

*γNA*). The total number of operations for a propagation from pupil to Lyot plane is:

*γm*)

^{2}flops), which can be neglected in most cases of interest.

*N*

*=*

_{B}*γm*where

*m*is the mask size in resolution elements (we remind here that typically

*m*=5 for a Lyot coronagraph and

*m*=1 for a Roddier or DZPM). The number of operations for the first MFT (Eq. 14). The second MFT transforms the

*N*

*×*

_{B}*N*

*array into a*

_{B}*N*

*×*

_{A}*N*

*array and its cost is*

_{A}*n*(

*MFT*)=8(

*N*

^{2}

_{A}*N*

*+*

_{B}*N*

_{A}*N*

^{2}

*)-2*

_{B}*N*

_{A}*N*

*-2*

_{B}*N*

^{2}

*. The propagation from pupil to Lyot plane includes the two MFTs, the multiplication by the focal plane mask (6*

_{A}*N*

^{2}

*flops) and the subtraction of the result from the pupil (2*

_{B}*N*

^{2}

*flops). The total cost of a semi-analytical propagation from pupil to Lyot plane is therefore:*

_{A}*γ*and the pupil size

*N*

*.*

_{A}## 5.2. The case of direct imaging

*et al.*[36

36. A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, “Amplitude and phase correction for high-contrast imaging using Fourier decomposition,” in *Techniques and Instrumentation for Detection of Exoplanets II. Edited by*
Coulter
Daniel R.*Proceedings of the SPIE*, Volume 5905, *pp.*368–378 (2005).,
D. R. Coulter, ed., pp. 368–378 (2005).

38. 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). URL http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v582n2/56580/56580.web.pdf. [CrossRef]

*γ*. If the PSF is only calculated to a FOV of 20 resolution elements, the MFT provides a gain by a factor of several over the FFT (Fig.8). In some applications where only the center of the PSF is needed, gains comparable to that of the coronagraphic cases can be obtained.

## 5.3. Practical applications and implementation

32. M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proceedings of the IEEE93(2), 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”. [CrossRef]

*N*

*and*

_{A}*N*

*are the number of pixels used in the calculation, in the pupil and focal planes. We use the same number of pixels in the Lyot plane as in the pupil. We show the effect of the sampling improvement from*

_{B}*λ*/4

*D*to

*λ*/8

*D*, obtained by zero-padding the pupil with FFTs. Note that for the semi-analytical method, the number of pixels

*N*

*corresponds to the pupil size itself, and that*

_{A}*N*

*corresponds to the focal plane mask size. For the FFT,*

_{B}*N*

*corresponds to the zero-padded pupil. The timings given in the table correspond to the time for 2 FFTs calculated with FFTW3, neglecting the application of the focal plane mask and zero-padding time. For the semi-analytical method, we give timings obtained with Mathematica as example.*

_{A}*λ*/6

*D*to

*λ*/8

*D*sampling, in order to keep the FFTs manageable. With the semi-analytical method, we can increase the precision significantly, and we typically use a

*λ*/10

*D*or

*λ*/20

*D*sampling for the focal plane mask, including gray pixel approximation. This corresponds to a 40~50 or 80~100 pixel diameter mask for a 4~5 resolution element mask.

*λ*/8

*D*sampling in the focal plane, in order to study the effect of segmentation of an ELT. We show an example of a 6000×6000 pupil array. This corresponds to calculations that are typically made on very large computer clusters using the classical method. This would correspond to 36000×36000 FFTs with

*γ*=6. It is also possible to oversample dramatically the occulting mask. This can be used to study the effect of the actual mask shape, such as ellipticity, edge roughness, fine alignment, or effects of atmospheric differential refraction. We give an extreme example with 1000 pixels in the pupil and a resolution of λ/200

*D*. This calculation would corresponds to 200000×200000 FFTs.

*D*, making the sampling problem even more critical for FFTs. The optimization of DZPM [20], which was obtained using the semi-analytical approach by integrating directly the Hankel transforms, can be studied by direct numerical simulations with our method.

## 6. Conclusion

*N*

*=400 and*

_{A}*γ*=10, and the overall gain with the semi-analytical method is a factor of 15.

*et al.*[36

36. A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, “Amplitude and phase correction for high-contrast imaging using Fourier decomposition,” in *Techniques and Instrumentation for Detection of Exoplanets II. Edited by*
Coulter
Daniel R.*Proceedings of the SPIE*, Volume 5905, *pp.*368–378 (2005).,
D. R. Coulter, ed., pp. 368–378 (2005).

40. L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” Journal of the Optical Society of America A, vol. 21, Issue 5, pp.810–81921, 810–819 (2004). [CrossRef]

41. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” Journal of Modern Optics **43**, 289–293 (1996). [CrossRef]

## Acknowledgements

## References and links

1. | 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,” in |

2. | J. L. Beuzit, D. Mouillet, C. Moutou, K. Dohlen, T. Fusco, P. Puget, S. Udry, R. Gratton, H. M. Schmid, M. Feldt, and M. Kasper, and The Vlt-Pf Consortium, “A ”Planet Finder” instrument for the VLT,” in |

3. | M. Tamura, K. Hodapp, H. Takami, L. Abe, H. Suto, O. Guyon, S. Jacobson, R. Kandori, J.-I. Morino, N. Murakami, V. Stahlberger, R. Suzuki, A. Tavrov, H. Yamada, J. Nishikawa, N. Ukita, J. Hashimoto, H. Izumiura, M. Hayashi, T. Nakajima, and T. Nishimura, “Concept and science of HiCIAO: high contrast instrument for the Subaru next generation adaptive optics,” in |

4. | B. Macintosh, M. Troy, R. Doyon, J. Graham, K. Baker, B. Bauman, C. Marois, D. Palmer, D. Phillion, L. Poyneer, I. Crossfield, P. Dumont, B. M. Levine, M. Shao, G. Serabyn, C. Shelton, G. Vasisht, J. K. Wallace, J.-F. Lavigne, P. Valee, N. Rowlands, K. Tam, and D. Hackett, “Extreme adaptive optics for the Thirty Meter Telescope,” in |

5. | L. Close“Extrasolar Planet Imaging with the Giant Magellan Telescope,” in |

6. | C. Verinaud, M. Kasper, J.-L. Beuzit, N. Yaitskova, V. Korkiakoski, K. Dohlen, P. Baudoz, T. Fusco, L. Mugnier, and N. Thatte “EPICS Performance Evaluation through Analytical and Numerical Modeling,” in |

7. | O. Guyon, J. R. P. Angel, C. Bowers, J. Burge, A. Burrows, J. Codona, T. Greene, M. Iye, J. Kasting, H. Martin, D. W. McCarthy Jr., V. Meadows, M. Meyer, E. A. Pluzhnik, N. Sleep, T. Spears, M. Tamura, D. Tenerelli, R. Vanderbei, B. Woodgate, R. A. Woodruff, and N. J. Woolf “Telescope to observe planetary systems (TOPS): a high throughput 1.2-m visible telescope with a small inner working angle,” in |

8. | J. T. Trauger and W. A. Traub, “A laboratory demonstration of the capability to image an Earth-like extrasolar planet,” Nature |

9. | A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, “Ground-based Coronagraphy with High-order Adaptive Optics,” Astrophys. J. |

10. | C. Aime and R. Soummer, “The Usefulness and Limits of Coronagraphy in the Presence of Pinned Speckles,” Astrophys. J. |

11. | C. Cavarroc, A. Boccaletti, P. Baudoz, T. Fusco, and D. Rouan, “Fundamental limitations on Earth-like planet detection with extremely large telescopes,” A&A447, 397–403 (2006). |

12. | O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, “Theoretical Limits on Extrasolar Terrestrial Planet Detection with Coronagraphs,” Astrophys. J. |

13. | R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” ApJ 669, 642–656 (2007). arXiv:0706.1739v1. |

14. | B. Lyot, “Étude de la couronne solaire en dehors des éclipses. Avec 16 figures dans le texte.” Zeitschrift fur Astrophysics5, 73-+ (1932). |

15. | C. Aime, R. Soummer, and A. Ferrari, “Total coronagraphic extinction of rectangular apertures using linear prolate apodizations,” A&A 389, 334–344 (2002). |

16. | R. Soummer, C. Aime, and P. E. Falloon, “Stellar coronagraphy with prolate apodized circular apertures,” A&A397, 1161–1172 (2003). |

17. | R. Soummer, “Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures,” Astrophys. J. |

18. | R. Soummer, L. Pueyo, A. Ferrari, C. Aime, A. Sivaramakrishnan, and N. Yaitskova, “Apodized Pupil Lyot Coronagraphs for Arbitrary Telescope Apertures. II. Application to Extremely Large Telescopes,” submitted to ApJ (2007). |

19. | F. Roddier and C. Roddier, “Stellar Coronagraph with Phase Mask,” PASP |

20. | R. Soummer, K. Dohlen, and C. Aime, “Achromatic dual-zone phase mask stellar coronagraph,” |

21. | D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The Four-Quadrant Phase-Mask Coronagraph. I. Principle,” PASP |

22. | L. Abe, F. Vakili, and A. Boccaletti, “The achromatic phase knife coronagraph,” A&A 374, 1161–1168 (2001). |

23. | M. J. Kuchner and W. A. Traub, “A Coronagraph with a Band-limited Mask for Finding Terrestrial Planets,” ApJ 570, 900–908 (2002). |

24. | G. Foo, D. M. Palacios, and G. A. Swartzlander Jr. “Optical vortex coronagraph,” Opt. Lett. |

25. | J. Goodman, |

26. | R. N. Bracewell, |

27. | A. Ferrari, R. Soummer, and C. Aime, “An introduction to stellar coronagraphy,” Comptes Rendus Physique |

28. | D. H. Bailey and P. N. Swarztrauber, “The Fractional Fourier Transform and Applications,” SIAM Review |

29. | L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The Chirp z-Transform Algorithm and its Application,” Bell Sys. Tech. J. |

30. | J. O. Smith, |

31. | C. Aime, “Principle of an Achromatic Prolate Apodized Lyot Coronagraph,” PASP117, 1012-+ (2005). [CrossRef] |

32. | M. Frigo and S. G. Johnson, “The Design and Implementation of FFTW3,” Proceedings of the IEEE93(2), 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”. [CrossRef] |

33. | J. P. Lloyd and A. Sivaramakrishnan, “Tip-Tilt Error in Lyot Coronagraphs,” Astrophys. J. |

34. | A. Sivaramakrishnan and J. P. Lloyd, “Spiders in Lyot Coronagraphs,” Astrophys. J. |

35. | L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Transactions on Audio and Electroacoustics |

36. | A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, “Amplitude and phase correction for high-contrast imaging using Fourier decomposition,” in |

37. | A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, “On representing and correcting wavefront errors in high-contrast imaging systems,” JOSA A 23 (2006). |

38. | 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. |

39. | |

40. | L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” Journal of the Optical Society of America A, vol. 21, Issue 5, pp.810–81921, 810–819 (2004). [CrossRef] |

41. | R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” Journal of Modern Optics |

**OCIS Codes**

(350.1260) Other areas of optics : Astronomical optics

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

(070.7145) Fourier optics and signal processing : Ultrafast processing

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: August 21, 2007

Revised Manuscript: October 23, 2007

Manuscript Accepted: October 23, 2007

Published: November 16, 2007

**Citation**

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, "Fast computation of Lyot-style coronagraph propagation," Opt. Express **15**, 15935-15951 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15935

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### References

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- O. Guyon, E. A. Pluzhnik, M. J. Kuchner, B. Collins, and S. T. Ridgway, "Theoretical Limits on Extrasolar Terrestrial Planet Detection with Coronagraphs," Astrophys. J. 167, 81-99 (2006). arXiv:astro-ph/0608506. [CrossRef]
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- C. Aime, R. Soummer, and A. Ferrari, "Total coronagraphic extinction of rectangular apertures using linear prolate apodizations," A&A 389, 334-344 (2002).
- R. Soummer, C. Aime, and P. E. Falloon, "Stellar coronagraphy with prolate apodized circular apertures," A&A397, 1161-1172 (2003).
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- M. J. Kuchner and W. A. Traub, "A Coronagraph with a Band-limited Mask for Finding Terrestrial Planets," ApJ 570, 900-908 (2002).
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- A. Give’on, N. J. Kasdin, R. J. Vanderbei, and Y. Avitzour, "On representing and correcting wavefront errors in high-contrast imaging systems," J. Opt. Soc. Am. A 23 (2006).
- 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). URL http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v582n2/56580/56580.web.pdf>. [CrossRef]
- URL http://www.primatelabs.ca/geekbench/>.
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