## Closed loop, DM diversity-based, wavefront correction algorithm for high contrast imaging systems

Optics Express, Vol. 15, Issue 19, pp. 12338-12343 (2007)

http://dx.doi.org/10.1364/OE.15.012338

Acrobat PDF (117 KB)

### Abstract

High contrast imaging from space relies on coronagraphs to limit diffraction and a wavefront control systems to compensate for imperfections in both the telescope optics and the coronagraph. The extreme contrast required (up to 10^{-10} for terrestrial planets) puts severe requirements on the wavefront control system, as the achievable contrast is limited by the quality of the wavefront. This paper presents a general closed loop correction algorithm for high contrast imaging coronagraphs by minimizing the energy in a predefined region in the image where terrestrial planets could be found. The estimation part of the algorithm reconstructs the complex field in the image plane using phase diversity caused by the deformable mirror. This method has been shown to achieve faster and better correction than classical speckle nulling.

© 2007 Optical Society of America

## 1. Introduction

^{-10}), to image extrasolar planets from space [1]. In this note we describe an algorithm using a single DM for estimating and correcting static wavefront error in a coronagraphic imaging system. Our approach works across the full spectrum of high-performance coronagraphs.

2. R.K. Tyson*Introduction to Adaptive Optics*. SPIE Press, 2000. [CrossRef]

3. L.A. Poyneer and B. Macintosh “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. **A 21(5)**, 810–819 (2004). [CrossRef]

5. J. T. Trauger, C. Burrows, and B. Gordon et al, “Coronagraph contrast demonstration with the high-contrast imagaing testbed,” Proc. SPIE **5487**, 1330–1336 (2004). [CrossRef]

8. F. Malbet, J.W. Yu, and M Shao, “High dynamic range imaging using a deformable mirror for space coronography”, PASP , **107**, pp. 386 (1995). [CrossRef]

## 2. Energy Minimization: a closed-loop correction algorithm for high contrast imaging systems

### 2.1. The correction stage

*C*between the electric field at the deformable mirror (DM) plane,

*E*

_{0}, and the electric field at the science camera plane,

*E*, where the measurements will take place,

_{f}9. 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(2)**, 1147–1161, (2003). [CrossRef]

*C*{

*E*

_{0}}=

*𝓕*{

*SE*

_{0}}, where

*𝓕*represents the Fourier transform and

*S*represents the shaped pupil function (the DM is assumed to be at a conjugate plane to the shaped pupil). Similarly, for the band-limited Lyot coronagraph [10],

*C*{

*E*

_{0}}=(

*𝓕*{

*SE*

_{0}}

*M*)⊗

*𝓕*{

*L*}, where

*S*is the entrance pupil of the coronagraph,

*M*is the image plane mask,

*L*is the Lyot pupil and ⊗ represents the convolution (the DM is assumed to be at the same plane as the entrance pupil). Since all real optical systems have aberrations induced by errors in the optics, the input field in Eq. 1 can be modeled,

*A*is the un-aberrated, ideal, electric field,

*α*and

*β*represent the amplitude and phase aberrations, respectively, and ψ represents the phase difference caused by the deformation of the DM surface.

*e*

^{α}^{+iβ}-1 was defined for expansion purposes and the cross terms between ψ and Φ is assumed to be negligible.

*𝓔*, is given by the inner product of the electric field,

*𝓔*=〈

*E*,

_{f}*E*〉, or,

_{f}*f*,

*g*〉=∫∬

*f**

*gd*ξ

*dη*and the asterisks represents a complex conjugate.

*influence function model*,

*N*×

_{dm}*N*actuators,

_{dm}*a*

_{k}_{,l}is the

*kl*

*coefficient (actuator command), and*

^{th}*f*

_{k}_{,l}is the DM’s influence function, centered at the location of the kl th actuator. The influence function is defined as the surface profile of the DM when one actuator is commanded (note that this model assumes no coupling between actuators and that superposition holds).

*k*and

*l*,

*C*. The criteria for minimum energy in Eq. 6 can be written in matrix form as

*G*=ℜ{〈

_{r,q}*C*{

*Af*},

_{r}*C*{

*Af*}〉} and

_{q}*H*=-ℑ{〈

_{r}*C*{

*Af*},

_{r}*C*{

*Ae*

^{α+iβ}}〉} and

*r*,

*q*=1,2, …,

*N*

^{2}

*. Therefore, in order to find the coefficients for the DM configuration that minimizes the energy in the dark zone, we need to estimate*

_{dm}*C*{

*Ae*

^{α+iβ}} which is essentially the complex valued electric field in the science camera plane of an uncorrected, aberrated system.

### 2.2. The reconstruction stage

*, following similar steps as the ones leading to equation 3, the electric field in the final science camera plane is given by*

_{k}*C*{

*A*ΦΨ

*} is negligible.*

_{k}*, is therefore approximately given by,*

_{k}*I*

_{0}, be taken with

*ψ*=0. The intensity of light in the image plane is then

*. Combining equations 9 and 10, for each image taken gives,*

_{k}*C*{

*Ae*

^{α+iβ}} via the following matrix equation,

*m*and

*n*, such that

## 3. Experimental results

*k*=1, 2), so that the total number of images per iteration is 3. Each DM configuration was a sum of cosine ripples, designed to create a grid of overlapping PSFs in the image plane that completely covers the dark zone. The two DM configurations were the same except they were in quadrature phase (in which case they satisfy Eq. 14). Figure 1 shows the experimental results.

*D*for the first 17 iterations of energy minimization and the first 50 iteration of speckle nulling, and then switched to 6 to 14 λ/

*D*in the horizontal direction and -2 to 2 λ/

*D*in the vertical. (We use sky angle as image plane coordinates, in units of 1λ/

*D*, which is the angular diffraction limit of a telescope of aperture D and wavelength

*λ*). Figure 1 shows the images taken before correction, after Classical Speckle Nulling correction bottomed out, and after Energy Minimization bottomed out. The optical axis in all images is close to the right edge of the image and the images show the left dark zone out to about 25λ/

*D*, together with a portion of the image plane stop which blocks the core of the star. The colormap is logarithmic, with contrast levels shown in the colorbar to the right. It is evident that in both cases of correction, a dark hole appears with improved contrast.

^{-6}on the Princeton testbed.) It should be noted, however, that classical speckle nulling was observed to be much less sensitive to noise and errors in the model than energy minimization. The limiting factor in our classical speckle nulling experiment is, based on simulations, believed to be the inefficiency of Classical Speckle Nulling to correct for spatially small speckles, which arise at around the 10

^{-6}level. Energy minimization does not suffer from this limitation and therefore achieves a deeper null. However, it also hits a limit at roughly 6×10

^{-7}in our lab. The limiting factor is the haze around 4λ/

*D*that is apparent on the rightmost image in Figure 1. This residual halo behaves as incoherent light, i.e. adding in intensity rather than amplitude (it is not known yet what it actually is) and thus is not eliminated by theDMcorrection. One of the appealing features of the subtraction based estimation formula in Eq. 11 is that any incoherent light common to each measurement is eliminated, resulting in an estimate of the desired coherent portion only. The contrast of this residual coherent light estimate is roughly 10

^{-7}, averaged across the control region. At this level, quantization of the DM voltage signal is believed to be the limiting factor. Simulations further show that with no quantization (and no other non-fundamental limiting factors such as incoherent light) the contrast reaches 10

^{-10}with energy minimization.

## 4. Summary and conclusion

## Acknowledgments

## References and links

1. | W. A. Traub, ed. |

2. | R.K. Tyson |

3. | L.A. Poyneer and B. Macintosh “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. |

4. | 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. |

5. | J. T. Trauger, C. Burrows, and B. Gordon et al, “Coronagraph contrast demonstration with the high-contrast imagaing testbed,” Proc. SPIE |

6. | R. Belikov, A. Give’on, J.T. Trauger, M. Carr, N.J. Kasdin, R.J. Vanderbei, F. Shi, K. Balasubramanian, and A. Kuhnert, “Toward 10 |

7. | P. J. Borde and W. A. Traub, “High-contrast imaging from space: Speckle nulling in a low aberration regime,” Astrophys. |

8. | F. Malbet, J.W. Yu, and M Shao, “High dynamic range imaging using a deformable mirror for space coronography”, PASP , |

9. | 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. |

10. | M. J. Kuchner, J. Crepp, J. Ge, and Astrophys.“Eighth-Order Image Masks for Terrestrial Planet Finding,” |

11. | N.J. Kasdin, R. Belikov, J. Beall, R.J. Vanderbei, M.G. Littman, M. Carr, and A. Give’on, “Shaped pupil coronagraphs for planet finding: optimization, manufacturing, and experimental results,” Proc. SPIE |

**OCIS Codes**

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

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: July 9, 2007

Revised Manuscript: September 6, 2007

Manuscript Accepted: September 9, 2007

Published: September 13, 2007

**Citation**

Amir Give'on, Ruslan Belikov, Stuart Shaklan, and Jeremy Kasdin, "Closed loop, DM diversity-based, wavefront correction algorithm for high
contrast imaging systems," Opt. Express **15**, 12338-12343 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12338

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### References

- W. A. Traub, ed., Proceedings of Coronagraph Workshop 2006. (JPL Publication 07-02).
- R. K. Tyson, Introduction to Adaptive Optics (SPIE Press, 2000). [CrossRef]
- L. A. Poyneer and B. Macintosh, "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21, 810-819 (2004). [CrossRef]
- 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, 1063-1073 (2006).
- J. T. Trauger, C. Burrows, B. Gordon, et al., "Coronagraph contrast demonstration with the high-contrast imagaing testbed," Proc. SPIE 5487, 1330-1336 (2004). [CrossRef]
- R. Belikov, A. Give’on J. T. Trauger, M. Carr, N. J. Kasdin, R. J. Vanderbei, F. Shi, K. Balasubramanian, and A. Kuhnert, "Toward 1010 contrast for terrestrial exoplanet detection: demonstration of wavefront correction in a shapedpupil coronagraph," Proc. SPIE 6265, 626518 (2006).
- P. J. Borde and W. A. Traub, "High-contrast imaging from space: Speckle nulling in a low aberration regime," Astrophys. J. 638, 488-498 (2006).
- F. Malbet, J. W. Yu, and M. Shao, "High dynamic range imaging using a deformable mirror for space coronography," Publ. Astron. Soc. Pac. 107, 386 (1995). [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]
- M. J. Kuchner, J. Crepp, J. Ge "Eighth-order image masks for Terrestrial Planet finding," Astrophys. J. 628, 466-473 (2005).
- N. J. Kasdin, R. Belikov, J. Beall, R. J. Vanderbei, and M. G. Littman, M. Carr, and A. Give’on "Shaped pupil coronagraphs for planet finding: optimization, manufacturing, and experimental results," Proc. SPIE 5905, 128-136 (2005).

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