## Nonlinear optimization algorithm for retrieving the full complex pupil function

Optics Express, Vol. 14, Issue 2, pp. 474-486 (2006)

http://dx.doi.org/10.1364/OPEX.14.000474

Acrobat PDF (825 KB)

### Abstract

Techniques for retrieving the phase of an optical field typically depend on assumptions about the amplitude of the field in a desired plane, usually a pupil plane. We describe an approach that makes no such assumptions and is capable of retrieving both the amplitude and phase in the desired plane. Intensity measurements in two or more planes are used by a nonlinear optimization algorithm to retrieve the phase in the measurement planes. The complex field (amplitude and phase) in the desired plane is then computed by simple propagation. We show simulation results and examine the convergence of the algorithm.

© 2006 Optical Society of America

## 1. Introduction

1. J. R. Fienup, “Phase Retrieval Algorithms: a Comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

4. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble Space Telescope Characterized by Using Phase-retrieval Algorithms,” Appl. Opt. **32**, 1747–1767 (1993). [CrossRef] [PubMed]

5. D. L. Misell, “A Method for the Solution of the Phase Problem in Electron Microscopy,” J. Phys. D: Appl. Phys. **6**, L6–L9 (1973). [CrossRef]

6. C. Roddier and F. Roddier, “Combined Approach to Hubble Space Telescope Wave-Front Distortion Analysis,” Appl. Opt. **32**, 2992–3008 (1993). [CrossRef] [PubMed]

6. C. Roddier and F. Roddier, “Combined Approach to Hubble Space Telescope Wave-Front Distortion Analysis,” Appl. Opt. **32**, 2992–3008 (1993). [CrossRef] [PubMed]

7. S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. **41**, 2095–2102 (2002). [CrossRef] [PubMed]

7. S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. **41**, 2095–2102 (2002). [CrossRef] [PubMed]

## 2. Description of algorithm

### 2.1. Multiple plane nonlinear optimization

*a priori*knowledge of the pupil. Instead, we initially employ one of the measured amplitudes as the known amplitude distribution,

*I*(

_{j}*r*,

*s*) is the measured intensity in the

*j*

^{th}measurement plane,

*F*(

_{j}*r*,

*s*) is the complex field in this plane, and

*r*and

*s*are pixel indices of the sampled field and intensity distribution. We refer to the plane where we are using the intensity measurement in this way as the “master” plane. An estimate of the phase

*θ*(

_{j}*r*,

*s*), which is initially a guess, is combined with the known amplitudes to form a numerical field,

9. Joseph W. Goodman, *Introduction to Fourier Optics 2 ^{nd} Ed*. (McGraw-Hill, 1996), Chap. 3. [PubMed]

*k*

^{th}plane, propagated a distance Δ

*z*from the

_{kj}*j*

^{th}plane, can be written as

*DFT*{} and

*IDFT*{} are the discrete Fourier transform and its inverse, respectively, both over arrays of size

*M*×

*N*, which are computed using a fast Fourier transform (FFT). The quantities

*d*and

_{r}*d*represent the spacings between these samples of the field in plane

_{s}*j*, and

*t*and

*u*are indices in the Fourier domain. The nomenclature

*G*(

_{kj}*r*,

*s*) means the field in the

*k*

^{th}plane obtained by propagating the field in the

*j*

^{th}plane. We can write this propagation as a linear operator,

*j*

^{th}plane is the master plane, and

*k*indexes over the slave planes.

*W*(

_{k}*r*,

*s*) is a weighting function that is used, for example, to discard the effect of bad pixels in the detector array or emphasize the contribution of certain regions of the pattern. It may be different for each plane.

*E*. It is important for the algorithm to take advantage of the gradient of the objective function with respect to each of the phase values, but not require second derivatives. This is because, for

_{j}*M*×

*N*phase values there will be

*M*

^{2}×

*N*

^{2}second derivatives, which can easily form an array larger than can be straightforwardly stored in a 32-bit computer. The algorithm we use that meets these requirements is the conjugate gradient algorithm [10]. In our situation, the gradient can be calculated very efficiently using a relatively simple analytic expression,

3. J. R. Fienup, “Phase-retrieval Algorithms for a Complicated Optical System,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

*G*(

_{jk}^{w}*r*,

*s*) is given by

*K*measurement planes, these expressions allow us to compute the derivatives of the objective function with respect to the phase value at

*MN*points using just 2(

*K*- 1) propagation operations, which can be efficiently implemented using the FFT. This technique is vastly more efficient than using a finite difference technique to compute the partial derivatives individually. Optimization over the coefficients of a basis-set (such as Zernike polynomials) expansion of the phase may also be done [3

3. J. R. Fienup, “Phase-retrieval Algorithms for a Complicated Optical System,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

### 2.2. Algorithm iteration details

*g*

_{0}(

*m*,

*n*), is propagated to a focal plane using standard Fresnel diffraction,

**P**,

_{0j}*d*=

_{r}*d*and

_{p}*d*=

_{s}*d*.

_{q}*g*(

*m*,

*n*), in the pupil that is closest to the true field,

*f*(

*m*,

*n*), is not yet known. Another approach is to average in the pupil plane the inverse propagated results (either the fields or the amplitudes and phases separately) from all of the measurement planes. We then must choose the appropriate method to average these complex numbers. We examine these results in Sect. 3.3 below.

## 3. Simulation Results

### 3.1. Example results

### 3.2. Algorithm convergence

### 3.3. Effects of pupil plane field averaging

### 3.4. Using results as input to phase-only retrieval algorithm

3. J. R. Fienup, “Phase-retrieval Algorithms for a Complicated Optical System,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgements

## References and Links

1. | J. R. Fienup, “Phase Retrieval Algorithms: a Comparison,” Appl. Opt. |

2. | J.N. Cederquist, J.R. Fienup, C.C. Wackerman, S.R. Robinson, and D. Kryskowski, “Wave-Front Phase Estimation from Fourier Intensity Measurements,” J. Opt. Soc. Am. A |

3. | J. R. Fienup, “Phase-retrieval Algorithms for a Complicated Optical System,” Appl. Opt. |

4. | J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble Space Telescope Characterized by Using Phase-retrieval Algorithms,” Appl. Opt. |

5. | D. L. Misell, “A Method for the Solution of the Phase Problem in Electron Microscopy,” J. Phys. D: Appl. Phys. |

6. | C. Roddier and F. Roddier, “Combined Approach to Hubble Space Telescope Wave-Front Distortion Analysis,” Appl. Opt. |

7. | S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. |

8. | J. H. Seldin and R. G. Paxman, “Joint Estimation of Amplitude and Phase from Phase-Diversity Data,” in “Signal Recovery and Synthesis” Topical Meeting of the Optical Society of America (June, 2005), paper JTuB4. |

9. | Joseph W. Goodman, |

10. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

11. | D. Malacara and S. L. DeVore, “Interferogram Evaluation and Wavefront Fitting” in |

12. | G. R. Brady and J. R. Fienup, “Retrieval of Complex Field Using Nonlinear Optimization”, in “Signal Recovery and Synthesis” Topical Meeting of the Optical Society of America (June, 2005), postdeadline paper JTuC3. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Focus issue: Signal recovery and synthesis

**Citation**

Gregory R. Brady and James R. Fienup, "Nonlinear optimization algorithm for retrieving the full complex pupil function," Opt. Express **14**, 474-486 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-474

Sort: Journal | Reset

### References

- J. R. Fienup, "Phase Retrieval Algorithms: a Comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- J.N. Cederquist, J.R. Fienup, C.C. Wackerman, S.R. Robinson and D. Kryskowski, "Wave-Front Phase Estimation from Fourier Intensity Measurements," J. Opt. Soc. Am. A 6, 1020-1026 (1989). [CrossRef]
- J. R. Fienup, "Phase-retrieval Algorithms for a Complicated Optical System," Appl. Opt. 32, 1737-1746 (1993). [CrossRef] [PubMed]
- J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, "Hubble Space Telescope Characterized by Using Phase-retrieval Algorithms," Appl. Opt. 32, 1747-1767 (1993). [CrossRef] [PubMed]
- D. L. Misell, "A Method for the Solution of the Phase Problem in Electron Microscopy," J. Phys. D: Appl. Phys. 6, L6-L9 (1973). [CrossRef]
- C. Roddier and F. Roddier, "Combined Approach to Hubble Space Telescope Wave-Front Distortion Analysis," Appl. Opt. 32, 2992-3008 (1993). [CrossRef] [PubMed]
- S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, "Sensing wave-front amplitude and phase with phase diversity," Appl. Opt. 41, 2095-2102 (2002). [CrossRef] [PubMed]
- J. H. Seldin and R. G. Paxman, "Joint Estimation of Amplitude and Phase from Phase-Diversity Data," in "Signal Recovery and Synthesis" Topical Meeting of the Optical Society of America (June, 2005), paper JTuB4.
- Joseph W. Goodman, Introduction to Fourier Optics 2nd Ed. (McGraw-Hill, 1996), Chap. 3. [PubMed]
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1986), Chap. 10.
- D. Malacara and S. L. DeVore, "Interferogram Evaluation and Wavefront Fitting" in Optical Shop Testing, D. Malacara, Ed. (Wiley, 1992), Chap. 13.
- G. R. Brady and J. R. Fienup, "Retrieval of Complex Field Using Nonlinear Optimization", in "Signal Recovery and Synthesis" Topical Meeting of the Optical Society of America (June, 2005), postdeadline paper JTuC3.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.