## Feasibility of symmetry-based speckle noise reduction for faint companion detection

Optics Express, Vol. 15, Issue 8, pp. 4705-4710 (2007)

http://dx.doi.org/10.1364/OE.15.004705

Acrobat PDF (244 KB)

### Abstract

Great interest has been focused on the problem of detecting faint companions, possibly including extrasolar planets, very close to other stars. A promising approach involves coupling high-correction adaptive optics to coronagraphs, for which many new and innovative designs have emerged. Detection of faint companions will require suppression of noise due to fluctuating speckles from the remnant fraction of stellar light not adaptively controlled. At high correction, the speckle halo takes on distinct spatial symmetries that may allow partial speckle noise reduction through relatively simple post-processing that rejects one spatial symmetry in the image. This paper quantitatively examines potential companion-detection sensitivity improvements that might be expected, and shows that realistic operational parameters will allow them to be realized.

© 2007 Optical Society of America

## 1. Introduction

1. J. J. Lissauer, “Extrasolar planets,” Nature **419**, 355 (2002). [CrossRef] [PubMed]

^{-10}the signal of its parent star. A classical technique for suppressing light from the primary is coronagraphy [2

2. 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 (2001). [CrossRef]

## 2. Symmetry properties of speckles at high correction

### 2.1 Phase and amplitude speckle symmetries

4. 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 (2000). [CrossRef]

*Ā*|

^{2}is the point-spread function (PSF), much brighter than all the other terms. Several speckle terms containing more factors of the small quantities α or

*ϕ*have been omitted from Eq. (1) because they are generally fainter at high correction. Since A,

*ϕ*, α are real, their Fourier transforms are Hermitian, causing each term to have definite spatial symmetry. For the regimes to be considered in this paper, the brightest phase speckles come from the term linear in

*i A*̅

^{*}

*Aϕ*̅}, which is spatially antisymmetric, and the quadratic term |

*Aϕ*̅|

^{2}, which is spatially symmetric. (The 4th term in Eq. (1), -Re{

*Ā*

^{*}

*Aϕ*

^{2}̅}, is primarily a PSF correction plus true speckles that are fainter than those from the quadratic term [8

8. E. E. Bloemhof, “Static point-spread function correction dominating higher-order speckle terms at high adaptive correction,” Opt. Lett. **29**, 2333 (2004). [CrossRef] [PubMed]

*α*̅|

^{2}and 2Re{

*Ā*̅

^{*}

*α*̅}, each of which is spatially symmetric. The last two amplitude-speckle terms in Eq. (1

1. J. J. Lissauer, “Extrasolar planets,” Nature **419**, 355 (2002). [CrossRef] [PubMed]

*ϕ*and so are fainter than the linear phase speckles also retained here. (Later, in §3, simple image processing will remove the symmetric speckle species and leave the antisymmetric behind; the new speckle contrast floor will then be limited by linear phase speckles, the brightest spatially antisymmetric species.)

### 2.2 Speckle intensity estimates

*Ā*. So they reduce diffraction-ring brightness locally, keeping total intensity non-negative. Although the total linear-term power in the halo vanishes, speckles fluctuate locally and cause speckle noise or contribute to the speckle-limited imaging contrast. Owing to the amplification by

*Ā*, linear-term speckles will tend to be brightest in the inner halo; being of lowest order in

*Aϕ*̅, they will always dominate in the limit of high adaptive correction (|

*ϕ*̅| small). The PSF amplitude

*Ā*for a round unobscured aperture is the familiar Airy amplitude. The “speckle amplitude”

*ϕ*̅ and “intensity speckle amplitude”

*λ*/a [9

9. E. E. Bloemhof, “Suppression of speckle noise by speckle pinning in adaptive optics,” Astrophys. J. **582**, L59 (2003). [CrossRef]

_{α}is the analogous spatial frequency characterizing intensity variations on the pupil. Typical intensities of individual speckles may be estimated to within a factor of ~2 by combining Eq. (1) and Eq. (2). Speckle noise is then directly proportional to speckle intensity [10

10. R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. **111**, 587 (1999). [CrossRef]

### 2.3 Attenuation of linear-term speckles by coronagraph

*ϕ*̅ or in

*α*̅ are suppressed, while quadratic-term speckles are little affected. The linear terms 2Re{

*i A*̅

^{*}

*Aϕ*̅} and 2Re{

*A*̅

^{*}

*α*̅} contain a factor of the PSF amplitude,

*Ā*, which the coronagraph naturally suppresses in suppressing the on-axis PSF, |

*Ā*|

^{2}. The action of the coronagraph at any given position in the focal plane may be described as reducing the PSF from |

*Ā*|

^{2}to a new contrast level C=|

*Ā*|

^{2}/R, with the suppression parameter R>1. Then the Airy amplitude

*Ā*and each linear speckle term will be reduced [11

11. E. E. Bloemhof, “Remnant speckles in a highly corrected coronagraph,” Astrophys. J. **610**, L69 (2004). [CrossRef]

12. C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. **612**, L85 (2004). [CrossRef]

*antisymmetric*speckle noise variance fraction f, which is proportional to the antisymmetric fraction of the squared speckle intensity [10

10. R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. **111**, 587 (1999). [CrossRef]

*symmetric*noise variance fraction f’ given by

## 3. Potential gains from symmetry-based speckle noise reduction

*λ*/100 rms wavefront, or S~0.996) with spatial frequency content D/a~10. Coronagraph performance is assumed high, suppressing the PSF to contrast ~10

^{-7}on the 4

^{th}Airy ring [13

13. A. Chakraborty, L. A. Thompson, and M. Rogosky, “10^{-7} contrast ratio at 4.5*λ*/D: New results obtained in laboratory experiments using nanofabricated coronagraph and multi-Gaussian shaped pupil masks,” Opt. Express **13**, 2394 (2005). [CrossRef] [PubMed]

^{th}Airy ring, the pre-coronagraph PSF amplitude is

*Ā*~2.8×10

^{-2}. The speckle amplitude is

*Aϕ*̅~10

^{-2}anywhere in the halo. Assuming pupil-plane intensity variations of 2.6% [14

14. F. Shi, A. E. Lowman, D. C. Moody, A. F. Niessner, and J. T. Trauger, “Wavefront amplitude variation of TPF’s high-contrast imaging testbed: modeling and experiment,” Proc. SPIE **5905**, 59051L-1 (2005). [CrossRef]

*α*̅~2×10

^{-3}anywhere in the halo. So quadratic-term phase speckles |

*Aϕ*̅|

^{2}dominate, with intensity (contrast) ~1.2×10

^{-4}, while linear-term phase speckles 2Re{

*i*

*Ā*

^{*}

*Aϕ*̅} have intensity ~5×10

^{-7}near the 4

^{th}Airy ring. (Note that the PSF amplitude has been reduced by √R to capture the effect of the coronagraph, as discussed in §2.3. Also, each linear-term intensity has been reduced an order of magnitude from the small-pixel, monochromatic values of Eq. (2) to account for partial cancellation over an octave of spectral bandwidth on

*λ*/D pixels.) Quadratic amplitude speckles |

*α*̅|

^{2}will have intensity ~5×10

^{-6}anywhere in the halo, while linear amplitude speckles 2Re{

*Ā*

^{*}

*α*̅} will have intensity ~10

^{-7}near the 4

^{th}Airy ring.

_{n}~1.2×10

^{-4}, set by the bright quadratic phase speckles. Antisymmetrizing images through simple image processing will reject all the speckle species listed in the previous paragraph except linear phase speckles, which will then limit the contrast to C

_{n}'~5×10

^{-7}, an improvement by a factor of about 240. Source signal is halved when one symmetry component is rejected, so a companion intensity C

_{s}=2C

_{n}'~10

^{-6}would just match the final speckle contrast level C

_{n}'. Hence, search sensitivity has been enhanced by about two orders of magnitude by using simple image processing, but no sophisticated optics, to reject the bright symmetric component of speckle noise.

^{*}the rate of detected photons from the primary, C

_{n}the noise contrast set by initial speckle intensity, and C

_{s}the contrast describing the signal to be detected. (The two subtracted frames both have shot-noise fluctuations.) In the visible,

*λ*=400–800 nm, r

^{*}~3×10

^{8}photons/s are detected from a m=5 G star by a ~20m

^{2}telescope having overall efficiency 5% [15

15. A. Quirrenbach, “Coronagraphic methods for the detection of terrestrial planets,” conclusions from a workshop held at Leiden Univ., Feb. 2–6, 2004; preprint at (http://arXiv.org/astro-ph/0502254) (2005).

_{s}~10

^{-6}with SNR=7 from an initial speckle contrast C

_{n}~1.2×10

^{-4}, would require integrating for just ~20 seconds.

## 4. Practical feasibility of symmetry-based speckle noise suppression

_{sp}~FWHM/(2√(ln2))~0.62

*λ*/D. The minimum possible uncertainty in the single-axis centroid location of a Gaussian spot of size σ

_{sp}when only photon noise is considered is [16

16. S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. **371**, 323 (2006). [CrossRef]

_{ph}= C

_{n}r

^{*}t is the number of photons per speckle, C

_{n}the initial contrast, and r

^{*}the rate of detected photons from the primary star, as in §3. (Broadband radial speckle streaks are located transversely to roughly this accuracy.) Locating the center between symmetric speckle pairs improves the accuracy by √2, and combining measurements from all speckle pairs in the halo gives a further improvement by √(N

_{sp}/2), where N

_{sp}≈0.342(D/a)

^{2}is the number of speckles in a halo characterized by actuator density (D/a) [17

17. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. **19**, 281 (1981). [CrossRef]

_{0}Δ/σ

_{sp}, where I

_{0}=C

_{n}is the intensity of each Gaussian speckle. The resulting post-processing contrast is

^{-8}assuming t~100 s for the illustrative space-borne observatory (with initial speckle contrast floor C

_{n}~1.2×10

^{-4}) and the representative primary star (r

^{*}~3×10

^{8}photons/s) of §3. This value is small compared to the final noise contrast limited by linear-term speckles, C

_{n}'~5×10

^{-7}, so frame registration accuracy is adequate to realize this speckle contrast improvement. This is an important check on realistic speckle suppression: past studies have assumed perfect image antisymmetrization [18

18. A. Sivaramakrishnan, P. E. Hodge, R B. Makidon, M. D. Perrin, J. P. Lloyd, E. E. Bloemhof, and B. R. Oppenheimer, “The adaptive optics point-spread function at moderate and high Strehl ratios,” Proc. SPIE **4860**, 161 (2003). [CrossRef]

## 5. Conclusions

_{n}. A

*λ*/160 correction (S=0.9985), D/a=100, and R=7800 on the 4

^{th}Airy ring imply C

_{n}~5×10

^{-7}and a final contrast C

_{s}=2C

_{n}'~6×10

^{-8}, limited by the antisymmetric linear-term phase speckles that survive image processing (Fig. 2). Integration times to overcome speckle and PSF shot noise are still quite short, t~25 seconds.

15. A. Quirrenbach, “Coronagraphic methods for the detection of terrestrial planets,” conclusions from a workshop held at Leiden Univ., Feb. 2–6, 2004; preprint at (http://arXiv.org/astro-ph/0502254) (2005).

19. J. T. Trauger, C. Burrows, B. Gordon, J. J. Green, A. E. Lowman, D. Moody, A. F. Niessner, F. Shi, and D. Wilson, “Coronagraph contrast demonstrations with the high-contrast imaging testbed,” Proc. SPIE **5487**, 1330 (2004). [CrossRef]

^{-10}contrast needed for terrestrial planet detection, while adding little mission complexity to that needed for nulling.

## Acknowledgments

## References and links

1. | J. J. Lissauer, “Extrasolar planets,” Nature |

2. | A. Sivaramakrishnan, C. D. Koresko, R. B. Makidon, T. Berkefeld, and M. J. Kuchner, “Ground-based coronagraphy with high-order adaptive optics,” Astrophys. J. |

3. | J. W. Hardy, |

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

5. | E. E. Bloemhof, R. G. Dekany, M Troy, and B. R. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. |

6. | A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. |

7. | M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. |

8. | E. E. Bloemhof, “Static point-spread function correction dominating higher-order speckle terms at high adaptive correction,” Opt. Lett. |

9. | E. E. Bloemhof, “Suppression of speckle noise by speckle pinning in adaptive optics,” Astrophys. J. |

10. | R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. |

11. | E. E. Bloemhof, “Remnant speckles in a highly corrected coronagraph,” Astrophys. J. |

12. | C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. |

13. | A. Chakraborty, L. A. Thompson, and M. Rogosky, “10 |

14. | F. Shi, A. E. Lowman, D. C. Moody, A. F. Niessner, and J. T. Trauger, “Wavefront amplitude variation of TPF’s high-contrast imaging testbed: modeling and experiment,” Proc. SPIE |

15. | A. Quirrenbach, “Coronagraphic methods for the detection of terrestrial planets,” conclusions from a workshop held at Leiden Univ., Feb. 2–6, 2004; preprint at (http://arXiv.org/astro-ph/0502254) (2005). |

16. | S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. |

17. | F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. |

18. | A. Sivaramakrishnan, P. E. Hodge, R B. Makidon, M. D. Perrin, J. P. Lloyd, E. E. Bloemhof, and B. R. Oppenheimer, “The adaptive optics point-spread function at moderate and high Strehl ratios,” Proc. SPIE |

19. | J. T. Trauger, C. Burrows, B. Gordon, J. J. Green, A. E. Lowman, D. Moody, A. F. Niessner, F. Shi, and D. Wilson, “Coronagraph contrast demonstrations with the high-contrast imaging testbed,” Proc. SPIE |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(030.6140) Coherence and statistical optics : Speckle

(350.1260) Other areas of optics : Astronomical optics

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: January 22, 2007

Revised Manuscript: March 13, 2007

Manuscript Accepted: March 20, 2007

Published: April 3, 2007

**Citation**

E. E. Bloemhof, "Feasibility of symmetry-based speckle noise reduction for faint companion detection," Opt. Express **15**, 4705-4710 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4705

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### References

- J. J. Lissauer, "Extrasolar planets," Nature 419, 355 (2002). [CrossRef] [PubMed]
- 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 (2001). [CrossRef]
- J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford University Press, New York, 1998).
- 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 (2000). [CrossRef]
- E. E. Bloemhof, R. G. Dekany, M Troy, and B. R. Oppenheimer, "Behavior of remnant speckles in an adaptively corrected imaging system," Astrophys. J. 558, L71 (2001). [CrossRef]
- A. Boccaletti, P. Riaud, and D. Rouan, "Speckle symmetry with high-contrast coronagraphs," Publ. Astron. Soc. Pac. 114, 132 (2002). [CrossRef]
- M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702 (2003). [CrossRef]
- E. E. Bloemhof, "Static point-spread function correction dominating higher-order speckle terms at high adaptive correction," Opt. Lett. 29, 2333 (2004). [CrossRef] [PubMed]
- E. E. Bloemhof, "Suppression of speckle noise by speckle pinning in adaptive optics," Astrophys. J. 582, L59 (2003). [CrossRef]
- R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, "Speckle noise and the detection of faint companions," Publ. Astron. Soc. Pac. 111, 587 (1999). [CrossRef]
- E. E. Bloemhof, "Remnant speckles in a highly corrected coronagraph," Astrophys. J. 610, L69 (2004). [CrossRef]
- C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85 (2004). [CrossRef]
- A. Chakraborty, L. A. Thompson, and M. Rogosky, "10-7 contrast ratio at 4.5λ/D: New results obtained in laboratory experiments using nanofabricated coronagraph and multi-Gaussian shaped pupil masks," Opt. Express 13, 2394 (2005). [CrossRef] [PubMed]
- F. Shi, A. E. Lowman, D. C. Moody, A. F. Niessner, and J. T. Trauger, "Wavefront amplitude variation of TPF's high-contrast imaging testbed: modeling and experiment," Proc. SPIE 5905, 59051L-1 (2005). [CrossRef]
- A. Quirrenbach, "Coronagraphic methods for the detection of terrestrial planets," conclusions from a workshop held at Leiden Univ., Feb. 2-6, 2004; preprint at (http://arXiv.org/astro-ph/0502254) (2005).
- S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, "Comparison of centroid computation algorithms in a Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 371, 323 (2006). [CrossRef]
- F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. 19, 281 (1981). [CrossRef]
- A. Sivaramakrishnan, P. E. Hodge, R. B. Makidon, M. D. Perrin, J. P. Lloyd, E. E. Bloemhof, and B. R. Oppenheimer, "The adaptive optics point-spread function at moderate and high Strehl ratios," Proc. SPIE 4860, 161 (2003). [CrossRef]
- J. T. Trauger, C. Burrows, B. Gordon, J. J. Green, A. E. Lowman, D. Moody, A. F. Niessner, F. Shi, and D. Wilson, "Coronagraph contrast demonstrations with the high-contrast imaging testbed," Proc. SPIE 5487, 1330 (2004). [CrossRef]

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