## Small-phase retrieval with a single far-field image

Optics Express, Vol. 16, Issue 11, pp. 8190-8197 (2008)

http://dx.doi.org/10.1364/OE.16.008190

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### Abstract

A method to retrieve small-phase aberrations from a single far-field image is proposed. It is 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 is used in the calculation process. It is proved that most of the inherent phase aberration of the imaging system must be of an even type, such as defocus, astigmatism, etc., to keep the method working. The results of numerical simulations on atmosphere-disturbed phase aberrations show that the proposed small-phase retrieval method works well when the RMS phase error is less than 1 rad. It is also shown that the method is valid in a noise condition when the SNR>100.

© 2008 Optical Society of America

## 1. Introduction

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

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

## 2. Principal of the small-phase retrieval method

*f*(

*x*,

*y*) can be decomposed uniquely into the sum of an even and an odd function:

*iB*)≈1+

*iB*. Then

*S*≈

*A*(1+

*iB*).

*B*and

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

_{o}*B*, respectively, then

*S*=

*A*+

*iA*·

*B*+

_{e}*iA*·

*B*.

_{o}*Z*=

*A*·

*B*,

_{e}*T*=

*A*·

*B*, then

_{o}*s*=

*a*+

*iz*+

*it*, where

*s*,

*a*,

*z*, and

*t*are the Fourier transforms of

*S*,

*A*,

*Z*, and

*T*, respectively. Based on the Fourier transform theory,

*a*and

*z*are real and even, and

*t*is imaginary and odd. We define

*x*=

*it*such that

*x*is real and odd. Then

*s*becomes

*s*=

*a*+

*iz*+

*x*. So the modulus squared of

*s*is

*P*is the far-field intensity with inherent system aberration. The even and odd parts of

*P*are

*W*and

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

_{o}*W*, respectively. Assuming the RMS phase error of the phase aberration W is small, the term B+W is small, too. Then the complex optical field with aberration W and B may be approximated by

*V*=

*A*·

*W*,

_{e}*Q*=

*A*·

*W*,

_{o}*V*is real and even, and

*Q*is real and odd.

*a*,

*z*,

*t*,

*v*, and

*q*are the Fourier transforms of

*A*,

*Z*,

*T*,

*V*, and

*Q*, respectively. From Fourier-transform theory, we know that

*a*,

*z*, and

*v*are real and even, respectively, and

*t*and

*q*are imaginary and odd. We define

*y*=

*iq*such that

*y*is real and odd. Then

*h*becomes

_{B}*h*is

_{B}*P*is the far-field intensity with aberration. The even and odd parts of

_{B}*P*are

_{B}*W*should be calculated first. From Eq. (14),

_{o}*y*is real and odd,

*Y*, the inverse Fourier-transform of

*y*, is imaginary and odd. Using the previous definitions:

*y*=

*iq*,

*Q*=

*AW*, the odd part of the estimated aberration

_{o}*Ŵ*is obtained by

*W*is calculated secondly. Equation (13) becomes:

_{e}*W*is small, and

*v*and

*y*are the Fourier-transforms of the unknown aberration

*W*’s even part

*W*and odd part

_{e}*W*, respectively, so

_{o}*v*and

*y*are small, too.

*B*≪

_{e}*B*,

_{o}*z*≪

*x*, we can solve this equation for

*v*by neglecting

*z*and the quadratic terms of

*y*, then Eq. (17) becomes:

*v*from Eq. (18).

*B*=0,

*z*=0,

*x*=0, then Eq. (17) becomes:

*v*from Eq. (19) either.

*B*≫

_{e}*B*,

_{o}*z*≫

*x*, we can solve this equation for

*v*by neglecting

*x*and the quadratic terms of

*v*and

*y*, then

*v*is real and even, then

*V*, the inverse Fourier-transform of

*v*, is also real and even. Based on the definition

*V*=

*A*·

*W*, then the even part of the estimated aberration is obtained:

_{e}*Ŵ*can be obtained by

*Ŵ*=

*Ŵ*+

_{e}*Ŵ*.

_{o}*a*and

*z*, we have replaced 1/

*a*with

*a*/(

*a*

^{2}+

*e*) and 1/

*z*with

*z*/(

*z*

^{2}+

*e*), respectively, where

*e*is an appropriately small constant. The divisor A in Eqs. (16) and (21) is replaced by

*A*/(

*A*

^{2}+

*e*) to avoid the zero of A, where

*e*is an appropriate small constant. In each equation, the e may be equivalent or not.

*P*and

_{B}*P*. Here

*P*is fixed and calibrated in advance. It only needs to measure intensity

*P*in real time, and then the estimated phase aberration can be obtained directly.

_{B}## 3. Numerical simulations

7. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. **29**, 1174–1180 (1990). [CrossRef]

*σ*=0.7 rad in this simulation. If the imaging system is not changed, the defocus is fixed and calibrated once to get its intensity profile P. In the simulation, S and

*H*are calculated from

_{B}*S*=

*A*exp(

*iB*) and

*H*=

_{B}*A*exp[

*i*(

*B*+

*W*)], respectively.

*W*(

*x*,

*y*) and the estimated phase

*Ŵ*(

*x*,

*y*), we define the error wavefront as

*η*, the ratio between the RMSs of the error wavefront

*E*(

*x*,

*y*) and the unknown wavefront

*W*(

*x*,

*y*), is used as one criterion to determine the validity of 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 of intensity of the Airy spot. If the

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

_{e}*≥1 rad, the average residual Strehl ratio*σ ¯

*>0.65. So we can conclude under the conditions of this paper that the valid dynamic range of this method is approximately*η ¯

*σ*<1 rad.

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

*σ*is the RMS value of noise.

_{n}*σ*=0.38 rad, whose initial average Strehl ratio is 0.86, different levels of noise are added to the far-field image, and 100 frame simulations are performed. The retrieval results are presented in Table 1.

*SNR*changes from infinity to 100,

*increases from 0.468 to more than 0.75. It shows that noise influences the retrieval effect of the sensor as long as it exists. But the phase retrieval method in this paper is effective enough when SNR>100.*η ¯

*σ*=0.467 rad and the initial Strehl ratio of aberration is 0.812 under the condition of SNR=∞ and SNR=120, respectively, are shown in Fig. 3: (a) is the initial disturbed wavefront; (b) and (c) are the even and odd parts of the initial disturbed wavefront; (d) is the retrieved wavefront with SNR=∞; (e) and (f) are the even and odd parts of the retrieved wavefront with SNR=∞; (g) is the retrieved wavefront with SNR=120; (h) and (i) are the even and odd parts of the retrieved wavefront with SNR=120. When SNR=∞, the residual Strehl ratio

*SR*is 0.963, and the error coefficient

_{e}*η*is 0.471; when SNR=120, the residual Strehl ratio

*SR*is 0.945 and the error coefficient

_{e}*η*is 0.656.

## 4. Conclusions

## 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. | 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: January 3, 2008

Revised Manuscript: March 24, 2008

Manuscript Accepted: April 11, 2008

Published: May 20, 2008

**Citation**

Min Li, Xin-Yang Li, and Wen-Han Jiang, "Small-phase retrieval with a single far-field image," Opt. Express **16**, 8190-8197 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-8190

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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). [CrossRef] [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). [CrossRef]
- 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). [CrossRef]
- N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990). [CrossRef]

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