## Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics

Optics Express, Vol. 14, Issue 17, pp. 7499-7514 (2006)

http://dx.doi.org/10.1364/OE.14.007499

Acrobat PDF (641 KB)

### Abstract

High-contrast imaging with adaptive optics (AO) for planet detection requires a sophisticated AO control system to provide the best possible performance. We evaluate the performance improvements in terms of residual error and point-spread function intensity provided by optimal Fourier control using detailed end-to-end simulation. Intensity, however, is not the final measure of system performance. We explore image contrast through analysis and simulation results, showing that speckles caused by atmospheric errors behave very differently in a temporal fashion from speckles caused by wavefront sensor noise errors.

© 2006 Optical Society of America

## 1. Introduction

1. J. Schneider, *The Extrasolar Planets Encyclopaedia*, Tech. rep. (CNRS-LUTH, Paris Observatory, 2006). http://vo.obspm.fr/exoplanetes/encyclo/catalog.php.

2. 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]

*d*≈ 20 cm) and high update rates (1000 – 2000 Hz) and hence still be photon-starved, particularly since many of the stars with the brightest planets are often distant and faint. As a result, these systems must employ optimal controllers to achieve the best possible performance from their available photons. We have proposed a system using an optimal Fourier controller (OFC) [3

3. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A **22**, 1515–1526 (2005). [CrossRef]

4. 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**, 702–712 (2003). [CrossRef]

## 2. System specifications

### 2.1. Wavefront control

3. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A **22**, 1515–1526 (2005). [CrossRef]

5. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A **19**, 2100–2111 (2002). [CrossRef]

*k*and

*l*range from -

*N*/2 + 1 to

*N*/2, for reconstruction done on a

*N*×

*N*grid.

3. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A **22**, 1515–1526 (2005). [CrossRef]

4. 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**, 702–712 (2003). [CrossRef]

*k*,

*l*] in the Fourier reconstructor scatters light in the PSF to a location

*λk*/

*D*,

*λl*/

*D*, where

*λ*is the wavelength of light in the science leg and

*D*is the aperture diameter. This means that optimizing the Fourier modes in this form of wavefront control optimizes specific locations in the PSF. This is very useful for high-contrast imaging where we are not concerned with a single parameter like Strehl ratio, but rather with the structure of the PSF. Second, the Fourier modes are eigenfunctions of linear, shift-invariant systems (LSI). Since the AO system components are LSI, this allows us to represent each element in frequency space as a simple complex gain. That means that the control system breaks into independent loops, one for each spatial frequency pair [

*k*,

*l*]. Temporal analysis of the control system can then be done for each mode following established techniques[7]. Figure 1 shows the refined block diagram for the temporal control of a modal coefficient. We now control the cosine and sine modes at a specific spatial frequency jointly as a complex coefficient. This makes all the modal time series in the diagram complex-valued. Note that this means the complex time series is directly used in the modal gain optimization.

*g*

_{wx}and

*g*

_{wy}and DM gain

*g*

_{d}, the filter is termed custom. This is in contrast to theoretical filters based on analytic models of the elements, which we have used previously. These theoretical filters perform well but do not produce unity gain through the system. In our simulations we have generated such a custom filter and have used it.

*s*[

*t*] is sent on the control law. We use a leaky-integrator control law of the form

*c*is just less than one and the gain

*g*is bounded from 0 to 1 by stability. After integration, the modal signal is sent to the DM. Because the modes are Fourier modes, the DM influence function simply scales each mode by a gain

*g*

_{d}(see Ref. [3

**22**, 1515–1526 (2005). [CrossRef]

### 2.2. Simulation

*D*= 8 meter class telescope. The subaperture diameter is

*d*=

*D*/44, and the Fourier reconstruction uses a

*N*= 48 grid for reconstruction and filtering. The block diagram of Fig. 1 describes the simulation’s behavior.

9. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” in *Amplitude and Intensity Spatial Interferometry II*, J. B. Breckinridge, ed., Proc. SPIE **1237** (1994) pp. 372–383. [CrossRef]

_{wfs}/

*d*in the focal plane before the lenslet array, where λ

_{wfs}is the WFS wavelength and

*d*is the subaperture size in the pupil. The spatial filter removes spatial frequency components in the phase that are beyond the ability of the system to measure or correct. As a result, it removes the aliasing error term and enables a dark region of improved correction inside the controllable band of the AO system.

*I*, the signal-to-noise ratio (SNR) of the WFS measurement can be calculated using standard methods [10]. In our specific case, the WFS SNR is exactly

*F*is the frame rate of the AO system.

## 3. Performance of the gain optimizer

*ε*[

*t*]. Assuming that the signal

*m*[

*t*] and the noises

*n*

_{x}[

*t*],

*n*

_{y}[

*t*] are wide-sense stationary (WSS) random processes, we can easily analyze how the control system modifies them using temporal PSDs and transfer functions. We will further assume for clarity of notation that the random processes are also zero-mean.

*m*[

*t*] has temporal PSD

*P*

_{m}(

*ω*), then the residual error

*ε*[

*t*] due to uncorrected atmosphere has temporal PSD

*P*

_{m}(

*ω*), the residual error variance will generally decrease as the gain is increased.

*ε*[

*t*] due to WFS noise has temporal PSD

*g*> 0.85 in this case), the error variance will become very large due to the large overshoot in the transfer function.

### 3.1. AO simulation parameters

*r*

_{0}of 24.6 cm and moves in frozen flow at speed 2.3 m/s along and angle -124 degrees from the x-axis. Layer 2 is weaker, with

*r*

_{0}of 34.1 cm, but with a higher wind speed of 16.5 m/s at an angle of -95 degrees. This gives a total

*r*

_{0}of 18.7 cm. The dominant wind direction is clearly visible in the AO-corrected PSFs.

*I*from 4 to 9.

### 3.2. Results: Necessary range of gains

**22**, 1515–1526 (2005). [CrossRef]

*k*,

*l*] = [0,0] in the middle. All the filters exhibit the same fundamental structure: a butterfly pattern oriented with the wings just slightly off the y-axis. This is due to the direction of the wind in the atmospheric model. In this butterfly region the atmospheric power is much higher, leading to higher optimal gains.

*I*= 9 here) the optimal gains are simply scaled as

*I*changes.

### 3.3. Results: Improvement in PSF intensity by region

*λ*/(2

*d*), except the core, than the uniform-gain PSFs. The in-band error is the error in spatial frequencies out to frequency 1/(2

*d*), set by Nyquist. We will examine this for both signal and noise error reduction through two animated figures.

*I*= 5, 2 kHz case. Before gain optimization, the modal gains are uniformly 0.3. The apodized PSF is shown on log-scale on a color range from intensity of 1e-6 to 1e-2. (Intensity 1 is the peak of a diffraction limited PSF.) This is a moderate SNR case and the temporal error shows very clearly as a butterfly pattern. The speckles in this area are still well-defined, because only one second has passed. The dominant wind layer is moving at 2.3 m/s and the aperture is 8 m in diameter, so little averaging has occurred. Because error due to the atmosphere is dominant, the gain optimization increases the modal gains, up to a value of 0.6, in the spatial frequency range corresponding to this butterfly shape. These higher gains result in less temporal error. As the animated figure shows in a smooth dissolve between PSFs, the intensity of the atmospheric speckles is greatly reduced, though the overall structure remains the same. In the prior case, the median in-band rms error is 24.0 nm. The gain optimization reduces this to 18.6 nm. This reduction by 15.3 nm rms is concentrated in a specific region. In this butterfly pattern, the intensity of the PSF is cut in half.

*I*= 8, 2 kHz case, where WFS noise is the dominant error source. The PSFs are shown on the same color scale as in Fig. 3. With uniform 0.3 gains, the PSF’s controllable region is dominated by a smooth, symmetric halo of WFS noise. It is smooth because 2000 realizations of WFS (1 s at 2 kHz) have been averaged out. This default gain of 0.3 is too high. The gain optimization lowers all gains into the range 0.05 to 0.25. The PSF after optimization has a visible butterfly pattern from the temporal error. There is still some noise halo, but at a greatly reduced level. The median in-band rms error before gain optimization is 69.6 nm rms. After gain optimization it is 58.4, a reduction of 37.9 nm. There is a slight reduction in the butterfly area, but the biggest gain is in the orthogonal noise-dominated region along the x-axis. In this region the PSF intensity is reduced by up to 80%; the PSF intensity is just one-fifth of what it had been. Because the optimization minimizes the residual error power, it maximizes the Strehl. In the signal-dominated case, the already high Strehl improves from 96.1% to 96.3%. In the noise-dominated case, the Strehl ratio improves from 91.0% to 92.8%. These small increases in Strehl disguise the significant reduction in intensity in the dark hole, emphasizing that for high-contrast imaging, the Strehl ratio is too simple a metric.

### 3.4. Benefits of gain optimization

## 4. Image contrast and speckles

^{-5}times less than the PSF peak value. It is located in the upper right region at 0.6” separation. The smoothness of the long exposure makes the target planet much more clearly detectable.

11. J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” in *Techniques and Instrumentation for Detection of Exoplanets*, D. R. Coulter, ed., Proc. SPIE **5170** (2003) pp. 25–37. [CrossRef]

12. B. L. Ellerbroek, “Linear systems modeling of adaptive optics in the spatial-frequency domain,” J. Opt. Soc. Am. A **22**, 310–322 (2005). [CrossRef]

*et al*. [13

13. L. Jolissaint, J.-P. Véran, and R. Conan, “Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach,” J. Opt. Soc. Am. A **23**, 382–394 (2006). [CrossRef]

14. C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. **612**, L85–L88 (2004). [CrossRef]

15. M. P. Fitzgerald and J. R. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. **637**, 541–547 (2006). [CrossRef]

### 4.1. AO simulation parameters

*I*= 6, 2 kHz case and also for the same case but with no WFS noise, so that the atmospheric behavior could be examined directly. One 5.12 second exposure with no atmosphere and only WFS noise was also run.

### 4.2. Image behavior as exposure time increases

*i*[

*t*]. The exact mean and autocovariance of this time series will be discussed later, but it is worth noting now that this random process is not zero-mean. A long exposure image will be the average of many of these consecutive values to preserve normalization of intensity. This long-exposure value for

*T*consecutive frames will be

*i*[

*t*] is a WSS random process, the same techniques as used in Section 3 can be applied to determine the PSD and variance of

*p*

_{N}[

*t*]. Given a temporal PSD for the instantaneous PSF intensity

*P*

_{i}(

*ω*), the temporal PSD for a long-exposure is

*m*

_{i}is the mean of

*i*[

*t*].

*P*

_{i}(

*ω*) is constant and this integral is easily determined to be 1/

*T*, by Parseval’s theorem. This gives us the averaging out with exposure time that we would expect. If

*P*

_{i}(

*ω*) is not white, we need to consider how the integral behaves as the exposure time increases. When

*T*= 2, the sinc-squared function is very broad. As

*T*increases, the lowpass filter rapidly narrows and suppresses the high temporal frequencies. Once

*P*

_{i}(

*ω*) is relatively uniform over the width of the lowpass filter, the integral behaves exactly as in the white noise case: the variance drops off as 1/

*T*. The level of the variance is set by the power at the lowest temporal frequencies of

*P*

_{i}(

*ω*). We can identify the point at which the intensity variance drops off to one-half of the instantaneous exposure level. This is our analytic basis for the speckle lifetime, a concept that was introduced by Racine

*et al*. [16

16. 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**, 587–594 (1999). [CrossRef]

*f*

_{k}(

*v*

_{x1}-

*v*

_{x2}) +

*f*

_{l}(

*v*

_{y1}-

*v*

_{y2})∣. This temporal frequency is a function of the spatial frequency of interest (given by

*f*

_{k},

*f*

_{l}in the pupil plane) and the x- and y-components of the velocity vectors of the two layers (

*v*

_{x1},

*v*

_{y1}) and (

*v*

_{x2},

*v*

_{y2}). The low temporal-frequency component is set by the changing content of the phase screen as it moves across the pupil. This in turn is controlled by the Kolmogorov profile. In this case of PSF variation due to phase errors, the atmosphere-induced speckle lifetime is implicitly independent of

*r*

_{0}, unlike Racine’s calculation. In our regime,

*r*

_{0}is simply a scaling factor on the power of the atmospheric phase aberration and does not affect the structure of the temporal PSDs. As shown in previous work [17

17. B. Macintosh, L. A. Poyneer, A. Sivaramakrishnan, and C. Marois, “Speckle lifetimes in high-contrast adaptive optics,” in *Astronomical Adaptive Optics Systems and Applications II*, R. K. Tyson and M. Llyod-Hart, eds., Proc. SPIE **5903** (2005) p. 59030J. [CrossRef]

*D*/

*v*, where

*v*is the wind speed.

*T*behavior. The WFS noise-induced intensity follows the 1/

*T*behavior for the range of exposure times.

*T*trend-lines for comparison. This exhibits the same trends as shown as in Fig. 6(b), though the data points were produced with a completely different method.

### 4.3. Intensity as a function of residual modes

4. 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**, 702–712 (2003). [CrossRef]

*a*(

*x*,

*y*)exp[

*jϕ*(

*x*,

*y*)], where both

*a*and

*ϕ*are functions of two spatial variables in the pupil plane, the PSF in the image plane is

*A*(

*X*,

*Y*) is not necessary. Second, the spatial variables will be dropped unless necessary for clarity. This makes the PSF

*ε*[

*t*] represents the residual error power. The variance and PSD of the residual error are dependent on the signal and noise inputs and the behavior of the control system, as detailed in Section 3. The modal coefficient

*ε*[

*t*] produces the real-valued phase aberration

*ε*

_{cos}[

*t*] is just the real part of

*ε*[

*t*] and

*ε*

_{sin}[

*t*] is the imaginary part of

*ε*[

*t*]. The Fourier transform of this phase is

*λ*/

*D*terms from the variables for clarity, the intensity at the prime scattering location [

*k*,

*l*] is simplified to the diffraction term plus the pinned speckle and power-spectrum term

2. 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]

*ε*, namely

*ε*

_{cos}and

*ε*

_{sin}. The variance of the error is the sum of two parts

*E*[

*cs*]. This correlation is nonzero in the case of the atmosphere-induced residual error. The mean intensity is just diffraction plus the power level of the residual error

*ε*

_{1}+

*ε*

_{2}, the resulting intensity variance is not simply the sum

## 5. Gain optimizer and contrast

*I*= 6, 2 kHz operation. This case has a mix of gains being raised (up to 0.5 in the butterfly) and lowered (down to 0.15 orthogonal to the butterfly). A 2 s exposure PSF was accumulated from the 100 Hz short exposure series to generate an estimate of the PSF intensity. The same 100 Hz short exposure series was used as described above to generate an estimate of the image contrast in the 2 s exposure, using the temporal-averaging contrast method.

## 6. Conclusion

*T*drop-off in intensity variance as a function of exposure time for even short exposures, the atmospheric speckles have a much longer decorrelation time. These speckles can dominate the contrast, even when error power is low, as shown in Section 5. As a consequence, the error-minimizing metric of OFC does not optimize contrast.

## Acknowledgments

## References and links

1. | J. Schneider, |

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

3. | L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A |

4. | 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. |

5. | L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A |

6. | E. Gendron and P. Léna, “Astronomical adaptive optics, 1: Modal control optimization,” Astron. Astrophys. |

7. | P.-Y. Madec, “Control Techniques,” in |

8. | A. Tokovinin, “Modeling turbulence profile for GLAO,” Tech. rep. (Gemini Observatory, 2004). |

9. | E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” in |

10. | S. B. Howell, |

11. | J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” in |

12. | B. L. Ellerbroek, “Linear systems modeling of adaptive optics in the spatial-frequency domain,” J. Opt. Soc. Am. A |

13. | L. Jolissaint, J.-P. Véran, and R. Conan, “Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach,” J. Opt. Soc. Am. A |

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

15. | M. P. Fitzgerald and J. R. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. |

16. | 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. |

17. | B. Macintosh, L. A. Poyneer, A. Sivaramakrishnan, and C. Marois, “Speckle lifetimes in high-contrast adaptive optics,” in |

18. | A. V. Oppenheim, R. W. Schafer, and J. R. Buck, |

**OCIS Codes**

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

(350.1260) Other areas of optics : Astronomical optics

**ToC Category:**

Focus Issue: Adaptive Optics

**History**

Original Manuscript: March 16, 2006

Revised Manuscript: May 17, 2006

Manuscript Accepted: May 18, 2006

Published: August 21, 2006

**Citation**

Lisa A. Poyneer and Bruce A. Macintosh, "Optimal Fourier control performance and speckle behavior in high-contrast
imaging with adaptive optics," Opt. Express **14**, 7499-7514 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7499

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### References

- J. Schneider, The Extrasolar Planets Encyclopaedia, Tech. rep. (CNRS-LUTH, Paris Observatory, 2006), http://vo.obspm.fr/exoplanetes/encyclo/catalog.php.
- 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]
- L. A. Poyneer and J.-P. Véran, "Optimal modal Fourier transform wave-front control," J. Opt. Soc. Am. A 22, 1515-1526 (2005). [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, 702-712 (2003). [CrossRef]
- L. A. Poyneer, D. T. Gavel, and J. M. Brase, "Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform," J. Opt. Soc. Am. A 19, 2100-2111 (2002). [CrossRef]
- E. Gendron and P. Léna, "Astronomical adaptive optics, 1: Modal control optimization," Astron. Astrophys. 291, 337-347 (1994).
- P.-Y. Madec, "Control Techniques," in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University Press, 1999) pp. 131-154.
- A. Tokovinin, "Modeling turbulence profile for GLAO," Tech. rep. (Gemini Observatory, 2004).
- E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE 1237 (1994) pp. 372-383. [CrossRef]
- S. B. Howell, Handbook of CCD Astronomy (Cambridge University Press, 2000).
- J. J. Green and S. B. Shaklan, "Optimizing coronagraph designs to minimize their contrast sensitivity to loworder optical aberrations," in Techniques and Instrumentation for Detection of Exoplanets, D. R. Coulter, ed., Proc. SPIE 5170 (2003) pp. 25-37. [CrossRef]
- B. L. Ellerbroek, "Linear systems modeling of adaptive optics in the spatial-frequency domain," J. Opt. Soc. Am. A 22, 310-322 (2005). [CrossRef]
- L. Jolissaint, J.-P. Véran, and R. Conan, "Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach," J. Opt. Soc. Am. A 23, 382-394 (2006). [CrossRef]
- C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004). [CrossRef]
- M. P. Fitzgerald and J. R. Graham, "Speckle statistics in adaptively corrected images," Astrophys. J. 637, 541-547 (2006). [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, 587-594 (1999). [CrossRef]
- B. Macintosh, L. A. Poyneer, A. Sivaramakrishnan, and C. Marois, "Speckle lifetimes in high-contrast adaptive optics," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson andM. Llyod-Hart, eds., Proc. SPIE 5903 (2005) p. 59030J. [CrossRef]
- A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time Signal Processing (Prentice Hall, 1999).

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