## Linear phase retrieval with a single far-field image based on Zernike polynomials

Optics Express, Vol. 17, Issue 17, pp. 15257-15263 (2009)

http://dx.doi.org/10.1364/OE.17.015257

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### Abstract

Wavefront aberrations can be represented accurately by a number of Zernike polynomials. We develop a method to retrieve small-phase aberrations from a single far-field image with a Zernike modal-based approach. The difference between a single measured image with aberration and a calibrated image with inherent aberration is used in the calculation process. In this paper, the principle of linear phase retrieval is introduced in a vector–matrix format, which is a kind of linear calculation and is suitable for real-time calculation. The results of numerical simulations on atmosphere-disturbed phase aberrations show that the proposed Zernike modal-based linear phase retrieval method works well when the rms of phase error is less than 1 rad, and it is valid in a noise condition when the signal-to-noise ratio (SNR)>3.

© 2009 Optical Society of America

## 1. Introduction

*et al*. found that in a small-phase condition, the odd and even parts of a phase aberration can be obtained with a simple linear calculation method. The difference between a single measured image with aberration and a calibrated image with inherent aberration was used in the calculation process. But it was impractical due to the point-to-point calculations. Its calculative quantity was huge and it was too sensitive to noise [7

7. M. Li, X.-Y. Li, and W.-H. Jiang. “Small-phase retrieval with a single far-field image,” Opt. Express **16**, 8190–8197 (2008). [PubMed]

**=**

*a***·**

*R***·**

*m**is the vector of Zernike polynomials,*

**a****is the linear matrix, and**

*R***is the vector of the measure parameter. This linear phase reconstruction is favorable for actual application in wavefront sensing.**

*m**et al*. can also realize the Zernike modal phase retrieval with a single far-field image. Though some of the literature [9

9. X.-Y. Li, M. Li, B. Chen, and W.-H. Jiang “A kind of novel linear phase retrieval wavefront sensor and its application in close-loop adaptive optics system,” in Proc. Sixth International Workshop on Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Imperial College Press, 2007), pp. 212–218.

## 2. Theory

7. M. Li, X.-Y. Li, and W.-H. Jiang. “Small-phase retrieval with a single far-field image,” Opt. Express **16**, 8190–8197 (2008). [PubMed]

*B*of the imaging system itself. This calibrated far-field image intensity

*P*serves as the standard.

*W*. We break

*W*into two parts:

*W*and

_{e}*W*are the even and odd parts of

_{o}*W*, respectively. It is based on the theory of odd–even function decompositions. Any function f(x,y) can be decomposed uniquely into the sum of an even and an odd function:

**,**

*f***, and**

*f*_{o}**are the vector formats of functions**

*f*_{e}*f*(

*x*,

*y*),

*f*(

_{o}*x*,

*y*), and

*f*(

_{e}*x*,

*y*); and

**J**is the transform matrix to make the x-direction and the y-direction flip over a matrix.

_{xy}*B*=

*B*

_{e}+

*B*. It is proved in [7

_{o}7. M. Li, X.-Y. Li, and W.-H. Jiang. “Small-phase retrieval with a single far-field image,” Opt. Express **16**, 8190–8197 (2008). [PubMed]

*B*≫

_{e}*B*or

_{o}*B*=0 and

_{o}*W*and

*B*are small, the odd part of disturbed phase

*W*is proportional to the difference of odd part of intensities

*P*and

_{Bo}*P*:

_{o}*P*and

_{Be}*P*,

_{e}*P*is the calibrated far-field image intensity,

*P*=

*P*+

_{e}*P*, and

_{o}*P*is the far-field intensity with inherent system aberration and measuring aberrations

_{B}*P*=

_{B}*P*+

_{Be}*P*.

_{Bo}**̂**

*w**and*

_{o}**̂**

*w**are the vector formats of*

_{e}*W*̂

*and*

_{o}*W*̂

*.*

_{e}

*R**is the response matrix between phase aberration*

_{o}**̂**

*w**and intensity*

_{o}

*Δ*

_{o}*, which is the vector format of Δ*

^{o}

*I*^{o}; Δ

**I**^{o}=

*P*-

_{Bo}*P*; and

_{o}*is the response matrix between phase aberration*

**R**_{e}*w*̂

*and intensity*

_{e}*, which is the vector format of Δ*

**Δ**_{e}*I*, Δ

_{e}*I*=

_{e}*P*-

_{Be}*P*.

_{e}**R**is the linear retrieval matrix between phase aberration

*̂ and image difference*

**w****,**

*Δ**=*

**Δ***+*

**Δ**_{o}*. Therefore, there is an inverse linear relationship between them:*

**Δ**_{e}

*R*^{+}is the pseudo inverse of matrix

**and describes the linear matrix between them as well.**

*R*9. X.-Y. Li, M. Li, B. Chen, and W.-H. Jiang “A kind of novel linear phase retrieval wavefront sensor and its application in close-loop adaptive optics system,” in Proc. Sixth International Workshop on Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Imperial College Press, 2007), pp. 212–218.

*P*is the Zernike number,

*c*is the coefficient, and

_{i}*M*(

_{i}*x*,

*y*) is a description of the Zernike polynomials. Or, in matrix format,

**is the response matrix between**

*D**and Zernike polynomial*

**w***, and*

**c***is the vector format of*

**c***c*. Equation (9) becomes

_{i}*represents the relationship between them. Matrix*

**Z***can be obtained formally according to the parameters of the imaging system. Then we can reconstruct the Zernike coefficients by*

**Z**

**Z**^{+}is the pseudo inverse of matrix

*. The wavefront also can be reconstructed.*

**Z**## 3. Numerical Simulations

*W*(

*x*,

*y*) and the estimated phase

*W*̂(

*x*,

*y*), which are expressed by 65 Zernike polynomials, we define the error wavefront as

*η*, which is the ratio between the rms of the error wavefront

*E*(

*x*,

*y*) and the unknown wavefront

*W*(

*x*,

*y*), is used as one criterion to determine the validity of the phase retrieval method:

*η*<1, the retrieval effect is valid.

*SR*), which is the ratio between the peak intensity of the far-field image produced by the error wavefront

_{e}*E*(

*x*,

*y*) and the maximum intensity of the Airy spot. If the

*SR*is closer to 1, the performance of this method is better.

_{e}**16**, 8190–8197 (2008). [PubMed]

*σ*=0.4 rad and the main type of system aberration is defocus.

*σ*=4 rad. The results are shown in Fig. 3.

*σ*̄≥1 rad, the average residual Strehl ratio

*η*̄>0.6. So we can conclude under the conditions of this paper that the valid dynamic range of this method is approximately

*σ*<1 rad.

*is the peak value of the far-field image without noise, and*

**P**_{I}*σ*is the rms value of noise.

_{n}*σ*̄=0.4 rad whose initial average Strehl ratio is 0.84, different levels of noise are added to the far-field image, and 100 frame simulations are performed. The system aberration is defocus and its rms is

*σ*=4 rad. The retrieval results are presented in Table 1.

*η*̄ increases from 0.331 to more than 0.5. It shows that the Zernike modal phase retrieval method is not sensitive to noise. It is still effective when SNR≥3.

**16**, 8190–8197 (2008). [PubMed]

*σ*<0.7 rad and the SNR>100, it is obvious that the range of applicability is extended by a Zernike modal-based approach. The anti-noise ability especially improved greatly. It is due to the modal method, which is more stable on the border of the aperture, and the singular value of the matrix is smaller. So, it is not sensitive to noise. Certainly the performance of this Zernike modal-based approach relates to the order of Zernike polynomials. With the increase of the order, the retrieval result becomes more exact. But by aiming at the 128×128 pixels image plane, 65 Zernike polynomials are enough to retrieve the phase exactly.

## 4. Conclusions

**16**, 8190–8197 (2008). [PubMed]

## Acknowledgments

## References and links

1. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Eng. |

2. | J. M. Wood, M. A. Fiddy, and R. E. Burge, “Phase retrieval using two intensity measurements in the complex plane,” Opt. Lett. |

3. | J. R. Fienup, “Phase retrieval using a support constraint,” presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va., October 28–30 (1985). |

4. | R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. |

5. | B. Ellerbroek and D. Morrison, “Linear methods in phase retrieval,” Proc. SPIE |

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

7. | M. Li, X.-Y. Li, and W.-H. Jiang. “Small-phase retrieval with a single far-field image,” Opt. Express |

8. | L. Xinyang and J. Wenhan, “Zernike modal wavefront reconstruction error of Hartmann wavefront sensor,” Acta. Optica Sinica |

9. | X.-Y. Li, M. Li, B. Chen, and W.-H. Jiang “A kind of novel linear phase retrieval wavefront sensor and its application in close-loop adaptive optics system,” in Proc. Sixth International Workshop on Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Imperial College Press, 2007), pp. 212–218. |

10. | N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. |

**OCIS Codes**

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

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Image Processing

**History**

Original Manuscript: April 16, 2009

Revised Manuscript: July 20, 2009

Manuscript Accepted: July 23, 2009

Published: August 14, 2009

**Citation**

Min Li and Xin-Yang Li, "Linear phase retrieval with a single far-field image based on Zernike polynomials," Opt. Express **17**, 15257-15263 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15257

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### References

- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Eng. 21, 2758-2769 (1982).
- J. M. Wood, M. A. Fiddy, and R. E. Burge, "Phase retrieval using two intensity measurements in the complex plane," Opt. Lett. 6, 514-516 (1981). [PubMed]
- J. R. Fienup, "Phase retrieval using a support constraint," presented at the Institute of Electrical and Electronics Engineers ASSP Workshop on Multidimensional Digital Signal Processing, Leesburg, Va., October 28-30 (1985).
- R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 66, 961-964 (1976).
- B. Ellerbroek and D. Morrison, "Linear methods in phase retrieval," Proc. SPIE 351, 90-95 (1983).
- R. A. Gonsalves, "Small-phase solution to the phase-retrieval problem," Opt. Lett. 26, 684-685 (2001).
- M. Li, X.-Y. Li, and W.-H. Jiang. "Small-phase retrieval with a single far-field image," Opt. Express 16, 8190-8197 (2008). [PubMed]
- L. Xinyang and J. Wenhan, "Zernike modal wavefront reconstruction error of Hartmann wavefront sensor," Acta.Optica Sinica 22,1236-1240 (2002).
- X.-Y. Li, M. Li, B. Chen, and W.-H. Jiang, "A kind of novel linear phase retrieval wavefront sensor and its application in close-loop adaptive optics system," in Proc. Sixth International Workshop on Adaptive Optics for Industry and Medicine, C. Dainty, ed. (Imperial College Press, 2007), pp. 212-218.
- N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).

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