## Wavefront sensorless adaptive optics: a general model-based approach |

Optics Express, Vol. 19, Issue 1, pp. 371-379 (2011)

http://dx.doi.org/10.1364/OE.19.000371

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### Abstract

Wavefront sensorless adaptive optics (AO) systems have been widely studied in recent years. To reach optimum results, such systems require an efficient correction method. In this paper, a general model-based correction method for a wavefront sensorless AO system is presented. The general model-based approach is set up based on a relationship wherein the second moments (SM) of the wavefront gradients are approximately proportionate to the FWHM of the far-field intensity distribution. The general model-based method is capable of taking various common sets of functions as predetermined bias functions and correcting the aberrations by using fewer photodetector measurements. Numerical simulations of AO corrections of various random aberrations are performed. The results show that the Strehl ratio is improved from 0.07 to about 0.90, with only N + 1 photodetector measurement for the AO correction system using N aberration modes as the predetermined bias functions.

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## 1. Introduction

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. **32**(1), 5–7 (2007). [CrossRef]

5. M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express **14**(4), 1339–1352 (2006). [CrossRef] [PubMed]

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. **32**(1), 5–7 (2007). [CrossRef]

5. M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express **14**(4), 1339–1352 (2006). [CrossRef] [PubMed]

## 2. The general model-based method

### 2.1 The wavefront sensorless AO correction system

4. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. **32**(1), 5–7 (2007). [CrossRef]

*x*’ and

*y*’ are the rectangular coordinates in the input plane of the photodetector; I

_{0}is proportional to the incident light power; A is the amplitude of the input wavefront and is set to be uniform in this paper;

*λ*is the wavelength of the input wavefront;

*z*is the focal length of the positive lens; and

*j*is the imaginary unit.

*Φ*and

*Ψ*can be represented by a series of M orthonormal functions and N orthonormal functions, respectively. In most cases, the number M is greater than N. The orthonormal functions are called modes in the following text, each mode denoted by Fi(x,y):

*Φ*of the input wavefront can be denoted by vector V, whose elements are the coefficients of

*can be represented by vectors U and Z, respectively. The elements of U and Z are*

_{R}### 2.2 Relationship between the second moment (SM) of the aberration gradients and the FWHM of far-field intensity distribution

_{0}is the slop of the trend line, which is determined by the detector radius R.

**32**(1), 5–7 (2007). [CrossRef]

**32**(1), 5–7 (2007). [CrossRef]

### 2.3 Modeling and analysis

*Φ*’s gradients in the

*x*and

*y*axes are [9]where

*x*and

*y*axes, respectively, and

*x*and

*y*axes, respectively.

_{i}, the N orthonormal mode

_{i,0}between the SM of the gradients of aberration

*Φ*and that after adding a predetermined bias function

*F*

_{i}(

*x*,

*y*) are added sequentially by the DM, we will obtain N equations

_{0}and MDS

_{i}are the corresponding MDS of input wavefronts

*Ф*(

*x*,

*y*) and

*Ф*(

*x*,

*y*) +

*α*

*F*(

_{i}*x*,

*y*), respectively. That is,

*α*, and the correction error comes only from Eq. (8). Furthermore, it is known that Eq. (8) is set up based on geometric optics and is satisfied with all kinds of aberrations as long as a suitable detector radius R is selected.

## 3. Numerical simulations and result analysis

_{i}. The 18 Zernike modes and 18 L–Z modes are taken sequentially to be the biases of the model-based method and added by the DM to the input wavefront.

^{−1}are shown in Fig. 5(a) and Fig. 5(b) respectively.

_{i}are calculated. The detector radius of far-field intensity

*R*equals

*i*= 1,2…,18.

_{0}is the intensity distribution when no aberrations are present.

^{−1}, S

_{m}, and MDS

_{0}are known in advance and do not need recalculation or measurement for each correction, the total detector measurements for one aberration correction would be 19. One aberration correction is considered as one iteration in the following results.

*α*) on correction results is under consideration. We take the L–Z modes and Zernike functions as the predetermined bias functions and change the value of coefficient

*α*. The curves of such iterations are show in Fig. 7 . As we have expected in the conclusions of Section 2, our method is insensitive to the choice of bias values.

## 4. Conclusions

## Acknowledgments

## References and links

1. | B. Wang and M. J. Booth, “Optimum deformable mirror modes for sensorless adaptive optics,” Opt. Commun. |

2. | M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A |

3. | P. Piatrou and M. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. |

4. | M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. |

5. | M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express |

6. | M. Born and E. Wolf, |

7. | J. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A |

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

9. | W. J. Hardy, |

**OCIS Codes**

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

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

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: August 17, 2010

Revised Manuscript: October 11, 2010

Manuscript Accepted: November 24, 2010

Published: December 24, 2010

**Citation**

Huang Linhai and Changhui Rao, "Wavefront sensorless adaptive optics: a general model-based approach," Opt. Express **19**, 371-379 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-371

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### References

- B. Wang and M. J. Booth, “Optimum deformable mirror modes for sensorless adaptive optics,” Opt. Commun. 282(23), 4467–4474 (2009). [CrossRef]
- M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17(8), 1440–1453 (2000). [CrossRef]
- P. Piatrou and M. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. 46(27), 6831–6842 (2007). [CrossRef] [PubMed]
- M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]
- M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1983).
- J. Braat, “Polynomial expansion of severely aberrated wave fronts,” J. Opt. Soc. Am. A 4(4), 643–650 (1987). [CrossRef]
- R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]
- W. J. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford Univ. Press, 1998).

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