## Iterative linear focal-plane wavefront correction |

JOSA A, Vol. 30, Issue 10, pp. 2002-2011 (2013)

http://dx.doi.org/10.1364/JOSAA.30.002002

Acrobat PDF (520 KB)

### Abstract

We propose an efficient approximation to the nonlinear phase diversity (PD) method for wavefront reconstruction and correction from intensity measurements with potential of being used in real-time applications. The new iterative linear phase diversity (ILPD) method assumes that the residual phase aberration is small and makes use of a first-order Taylor expansion of the point spread function (PSF), which allows for arbitrary (large) diversities in order to optimize the phase retrieval. For static disturbances, at each step, the residual phase aberration is estimated based on one defocused image by solving a linear least squares problem, and compensated for with a deformable mirror. Due to the fact that the linear approximation does not have to be updated with each correction step, the computational complexity of the method is reduced to that of a matrix-vector multiplication. The convergence of the ILPD correction steps has been investigated and numerically verified. The comparative study that we make demonstrates the improved performance in computational time with no decrease in accuracy with respect to existing methods that also linearize the PSF.

© 2013 Optical Society of America

## 1. INTRODUCTION

1. R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A **9**, 1072–1085 (1992). [CrossRef]

2. L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. **141**, 1–76 (2006). [CrossRef]

2. L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. **141**, 1–76 (2006). [CrossRef]

7. F. Martinache, “Kernel phase in Fizeau interferometry,” Astrophys. J. **724**, 464–469 (2010). [CrossRef]

8. S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. **35**, 3036–3038 (2010). [CrossRef]

9. C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. **52**, 1695–1728 (2005). [CrossRef]

5. R. A. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. **26**, 684–685 (2001). [CrossRef]

6. C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE **8447**, 844721 (2012). [CrossRef]

10. I. Mocœur, L. M. Mugnier, and F. Cassaing, “Analytical solution to the phase-diversity problem for real-time wavefront sensing,” Opt. Lett. **34**, 3487–3489 (2009). [CrossRef]

5. R. A. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. **26**, 684–685 (2001). [CrossRef]

11. W. J. Wild, “Linear phase retrieval for wave-front sensing,” Opt. Lett. **23**, 573–575 (1998). [CrossRef]

12. J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A **29**, 2428–2438 (2012). [CrossRef]

6. C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE **8447**, 844721 (2012). [CrossRef]

13. D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramer-rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A **16**, 1005–1015 (1999). [CrossRef]

11. W. J. Wild, “Linear phase retrieval for wave-front sensing,” Opt. Lett. **23**, 573–575 (1998). [CrossRef]

12. J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A **29**, 2428–2438 (2012). [CrossRef]

14. A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. **118**, 1165–1175 (2006). [CrossRef]

8. S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. **35**, 3036–3038 (2010). [CrossRef]

## 2. OPTICAL SYSTEM

### A. Image Formation

### B. Measurement Noise

## 3. APPROXIMATIONS OF THE PSF

12. J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A **29**, 2428–2438 (2012). [CrossRef]

13. D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramer-rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A **16**, 1005–1015 (1999). [CrossRef]

13. D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramer-rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A **16**, 1005–1015 (1999). [CrossRef]

### A. First-Order Approximations

#### 1. Small Total Phase Approximation

**Property 3.1.**

*The linear term of the approximated PSF in Eq. (12) is invariant in the even aberrations.*

**Property 3.1**makes it impossible to neglect the quadratic term of the PSF when the Born approximation is used to formulate an estimation problem. This is also what [15

15. A. Polo, S. F. Pereira, and P. H. Urbach, “Theoretical analysis for best defocus measurement plane for robust phase retrieval,” Opt. Lett. **38**, 812–814 (2013). [CrossRef]

#### 2. Small Aberration Approximation

**Property 3.2.**

*The linear terms of the first-order approximation of the PSF and of the PSF resulting from the first-order Taylor approximation of the GPF are equal.*

**Property 3.3.**

*The approximation in Eq. (14) has the property that for*

*the even modes do not cancel in the linear term.*

**Remark 3.1.**

*Note that this approximation is valid for any diversity. Due to the fact that the linear term is not invariant in the even modes, we can estimate the even and odd modes with just a linear equation, as will be shown in a later section.*

### B. Second-Order Approximations

#### 1. Small Total Phase Approximation

6. C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE **8447**, 844721 (2012). [CrossRef]

**Property 3.4.**

*The expression in Eq. (16) is also obtained when the PSF is approximated using a second-order Taylor expansion around*

#### 2. Small Aberration Approximation

**Property 3.5.**

*The second-order Taylor approximation of the PSF in*

*is more accurate than the PSF obtained from the first-order GPF approximation in*

*while the quadratic form remains.*

**Property 3.6.**

*The second-order Taylor approximation of the PSF in*

*has the property that the even modes do not cancel in the linear term of the PSF.*

## 4. ITERATIVE LINEAR PHASE DIVERSITY

**29**, 2428–2438 (2012). [CrossRef]

**Property 4.1.**

*Taking the difference between two images significantly decreases the SNR.*

8. S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. **35**, 3036–3038 (2010). [CrossRef]

### A. Convergence Analysis

**35**, 3036–3038 (2010). [CrossRef]

## 5. SIMULATIONS

**ILPD**and

**LIFT**. We first describe the simulation setup. Second, we give one example of

**ILPD**. Next, we analyze the behavior of both methods using a Monte Carlo simulation by varying the noise level and the rms value of the initial aberration. The computer employed for these simulations is a 2.67 GHz quad-core Intel Core (TM)2 Quad CPU Q8400 with 4.0 GB of RAM.

**LIFT**collects one image at iteration 1 and based on this image performs five steps that lead to a phase estimate, which is then used for the correction. On the other hand,

**ILPD**collects a new image at each of the five steps (image which includes the previous corrections) and performs a correction of the wavefront by a DM after each step. When estimating the wavefront using

**ILPD**, we take one image (per correction step) of the same point-like object at defocus 2 rad. For a fixed diversity, like the one used here, the linear coefficients in Eq. (15) can be computed in advance.

**LIFT**uses only one one image with an astigmatism diversity of

**35**, 3036–3038 (2010). [CrossRef]

**LIFT**can be precomputed.

20. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A **66**, 207–211 (1976). [CrossRef]

**ILPD**in Subsection 5.A. Second, in Subsection 5.B, for the same aberration, in the noiseless case, we show the convergence and the corresponding rate of convergence in terms of residual wavefront error and relative residual wavefront error. Subsequently, in Subsection 5.C, we study the convergence properties in terms of the residual error for

**ILPD**and

**LIFT**as a function of increasing read-out noise SNR, photon count, and wavefront rms.

### A. One Example of Iterative Phase Diversity

**ILPD**is much shorter. Tables 1 and 2 list the residual rms values obtained after each iteration for the two methods. Inspecting the tables, it seems that

**ILPD**is more robust to noise than

**LIFT**, but the error difference between them is not significant. One advantage of

**LIFT**is that it only uses one image, while

**ILPD**uses one image per iteration, but the later method is faster. In order to quantify how much faster, we need to make a Monte Carlo analysis. This is the subject of the next subsection.

### B. Iterative Linear Phase Diversity Without Noise

**LIFT**and for all the samples considered for

**ILPD**at this particular rms value. In the noiseless case,

**ILPD**converges to a residual error

**LIFT**converges to a small bias different from zero, but not as fast. There are also cases when

**LIFT**diverges.

**ILPD**converges to a relative residual error equal to zero and it is independent with respect to different realizations of

**ILPD**is smaller.

**LIFT**to complete five iterations is 10.9978 s on average, while

**ILPD**performs them in 0.0028 s on average. Note that the integration time of the CCD is not included in computing these times. This makes

**ILPD**3927.8 times faster. When we also count the CCD integration time (of approximately 47.3 ms),

**ILPD**is 40 times faster. For a fair comparison with

**LIFT**, we have used here all the pixels in order to compute the estimate at each step, but the computational time for

**ILPD**further decreases when using just a fraction of the pixels.

### C. Error Residue in the Presence of Noise

**ILPD**, the residual error decreases with the increase of the SNR, which is what we expected. For

**LIFT**this behavior is not very visible. One reason is the high value of the rms. In our simulations, we have noticed that for smaller rms values, e.g., 0.5 rms,

**LIFT**starts to show this decrease in bias for increasing rms, which shows that

**LIFT**is more appropriate for small rms values.

**ILPD**has a lower error variance and a lower error mean. Clearly, for

**ILPD**, the error variance would converge to zero for an SNR equal to

**ILPD**has a lower error variance and a lower error mean. Furthermore, it is visible in Fig. 7 that at low photon counts

**LIFT**diverges.

**LIFT**starts to diverge for rms values larger than 0.5 rad, while

**ILPD**still corrects for the aberration. This is due to the fact that with each iteration the aberration becomes smaller and the linear model in Eq. (19) is more and more accurate.

## 6. CONCLUSIONS

## APPENDIX A: PROOF OF PROPOSITIONS

*Proof of Property 3.1*. We introduce the short-hand notation

*Proof of Property 3.2*. The OTF is given by and the linear term is equal to Eq. (A3).□

*Proof of Property 3.3*. We show that Eq. (A3) for

*Proof of Property 3.4*. The second-order Taylor approximation of the GPF is Dropping terms of order 3 and higher, the resulting approximated OTF reduces to which is exactly the second-order Taylor approximation.□

*Proof of Property 3.5*. The difference between the approximated PSF following from a first-order approximation of the GPF in Eq. (A2) and the second-order Taylor approximation in Eq. (A9) is given by The addition of the missing term from the first-order GPF approximation results in a residue of order

*Proof of Property 3.6*. Inspecting Eq. (A10), we observe that the linear term is not affected; therefore, Property 3.1 and Property 3.3 still hold for the linear terms of Eq. (A9).□

*Proof of Property 4.1*. The intensity of both signals is positive and subtracting two images decreases the mean signal at each pixel. Recall that we assume that all the camera pixels are mutually independent and that the measurement noise is Gaussian distributed In Eq. (A11),

## REFERENCES

1. | R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A |

2. | L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. |

3. | D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008). |

4. | R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik |

5. | R. A. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. |

6. | C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE |

7. | F. Martinache, “Kernel phase in Fizeau interferometry,” Astrophys. J. |

8. | S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. |

9. | C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. |

10. | I. Mocœur, L. M. Mugnier, and F. Cassaing, “Analytical solution to the phase-diversity problem for real-time wavefront sensing,” Opt. Lett. |

11. | W. J. Wild, “Linear phase retrieval for wave-front sensing,” Opt. Lett. |

12. | J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A |

13. | D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramer-rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A |

14. | A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. |

15. | A. Polo, S. F. Pereira, and P. H. Urbach, “Theoretical analysis for best defocus measurement plane for robust phase retrieval,” Opt. Lett. |

16. | ANSI, “Methods for reporting optical aberrations of eyes,” American National Standard for Ophtalmics ANSI-Z80.28-2004 (2004). |

17. | J. W. Goodman, |

18. | F. Gustafsson, |

19. | G. H. Golub and C. F. Van Loan, |

20. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(100.5070) Image processing : Phase retrieval

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 16, 2013

Revised Manuscript: August 19, 2013

Manuscript Accepted: August 21, 2013

Published: September 13, 2013

**Citation**

C. S. Smith, R. Marinică, A. J. den Dekker, M. Verhaegen, V. Korkiakoski, C. U. Keller, and N. Doelman, "Iterative linear focal-plane wavefront correction," J. Opt. Soc. Am. A **30**, 2002-2011 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-10-2002

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### References

- R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992). [CrossRef]
- L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” Adv. Imaging Electron Phys. 141, 1–76 (2006). [CrossRef]
- D. R. Gerwe, M. M. Johnson, and B. Calef, “Local minima analysis of phase diverse phase retrieval using maximum likelihood,” The Advanced Maui Optical and Space Surveillance Technical Conference, Kihei, Hawaii (2008).
- R. W. Gerchberg and W. O. Saxton, “Phase retrieval by iterated projections,” Optik 35, 237–246 (1972).
- R. A. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. 26, 684–685 (2001). [CrossRef]
- C. U. Keller, V. Korkiakoski, N. Doelman, R. Fraanje, R. Andrei, and M. Verhaegen, “Extremely fast focal-plane wavefront sensing for extreme adaptive optics,” Proc. SPIE 8447, 844721 (2012). [CrossRef]
- F. Martinache, “Kernel phase in Fizeau interferometry,” Astrophys. J. 724, 464–469 (2010). [CrossRef]
- S. Meimon, T. Fusco, and L. M. Mugnier, “Lift: a focal-plane wavefront sensor for real-time low-order sensing on faint sources,” Opt. Lett. 35, 3036–3038 (2010). [CrossRef]
- C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005). [CrossRef]
- I. Mocœur, L. M. Mugnier, and F. Cassaing, “Analytical solution to the phase-diversity problem for real-time wavefront sensing,” Opt. Lett. 34, 3487–3489 (2009). [CrossRef]
- W. J. Wild, “Linear phase retrieval for wave-front sensing,” Opt. Lett. 23, 573–575 (1998). [CrossRef]
- J. Antonello, M. Verhaegen, R. Fraanje, T. van Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29, 2428–2438 (2012). [CrossRef]
- D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramer-rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16, 1005–1015 (1999). [CrossRef]
- A. Tokovinin and S. Heathcote, “Donut: measuring optical aberrations from a single extrafocal image,” Publ. Astron. Soc. Pac. 118, 1165–1175 (2006). [CrossRef]
- A. Polo, S. F. Pereira, and P. H. Urbach, “Theoretical analysis for best defocus measurement plane for robust phase retrieval,” Opt. Lett. 38, 812–814 (2013). [CrossRef]
- ANSI, “Methods for reporting optical aberrations of eyes,” (2004).
- J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).
- F. Gustafsson, Statistical Sensor Fusion (Holmbergs i Malmö AB, 2010).
- G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66, 207–211 (1976). [CrossRef]

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