## Optical wavefront measurement using phase retrieval with transverse translation diversity

Optics Express, Vol. 17, Issue 2, pp. 624-639 (2009)

http://dx.doi.org/10.1364/OE.17.000624

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### Abstract

We demonstrate the use of transverse translation-diverse phase retrieval as a method for the measurement of wavefronts in situations where the detected intensity patterns would be otherwise undersampled. This technique involves using a smaller moving subaperture to produce a number of adequately sampled intensity patterns. The wavefront is then retrieved using an optimization jointly constrained by them. Expressions for the gradient of an error metric with respect to the optimization parameters are given. An experimental arrangement used to measure the transmitted wavefront of a plano-convex singlet using this technique is described. The results of these measurements were repeatable to within approximately *λ*/100 RMS.

© 2009 Optical Society of America

## 1. Introduction

*d*is the detector sample spacing (pixel pitch),

_{u}*λ*is the wavelength, and

*Q*= (

*λ f*/#)/

*d*is a parameter describing the sampling rate [1]. If

_{u}*Q*≥ 2 the intensity pattern is adequately (Nyquist) sampled and if

*Q*≥ 1 the field is adequately sampled. If a detector with a 5 μm sample spacing is used with 632.8 nm illumination and the field is adequately sampled, the f-number must be greater than 7.9 (or the NA less than 0.063). This limits the use of most phase retrieval algorithms to high

*f*/# wavefronts [2

2. J. R. Fienup, “Phase Retrieval for Undersampled Broadband Images,” J. Opt. Soc. Am . A, **16**, 1831–1839 (1999). [CrossRef]

5. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–78 (2008). [CrossRef] [PubMed]

6. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett . **93**, 023903 (2004). [CrossRef] [PubMed]

## 2. Nonlinear optimization over multiple transverse translations of a subaperture

5. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–78 (2008). [CrossRef] [PubMed]

5. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–78 (2008). [CrossRef] [PubMed]

*n*

^{th}position as

*h*(

*x, y*) is the field impinging on the subaperture,

*a*(

*x, y*) is the complex amplitude transmittance of the subaperture, and the subaperture is translated by (

*x*) for the

_{n}, y_{n}*n*

^{th}position. Depending on the propagation method used, the phase of

*h*(

*x, y*) may be the deviation from a plane or spherical wavefront. This can equivalently be written as

*h*(

*x, y*) and

*a*(

*x, y*) do not need to be the same, with the larger being truncated to the size of the smaller during the calculation of

*g*(

_{n}*x, y*). Usually the unknown field,

*h*(

*x, y*), will be the larger array. Subpixel shifted versions of

*h*(

*x, y*) or

*a*(

*x, y*) are calculated making use of the discrete Fourier transform shift theorem. Since a field

*h*(

*x, y*) is usually band-limited, this sinc-like interpolation can be more accurate for it than for an aperture function

*a*(

*x, y*), thereby making it advantageous to model the field as translating.

*g*(

_{n}*x, y*) using a generalized Fourier optics propagation,

**P**[ ] is a linear propagation operator, which may be as simple as a Fourier transform, used in Ref. [5

**16**, 7264–78 (2008). [CrossRef] [PubMed]

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

*z*away. This propagator can be written as

*DFT*[ ] and

*IDFT*[ ] represent the forward and inverse discrete Fourier transform,

*z*is the distance to the paraxial image plane, ∆

*z*is the small defocus distance,

*r*and

*s*are spatial frequency indices,

*u*′ and

*v*′ are coordinates indices in the paraxial image plane, and constant factors are ignored. The relation between the sample spacings (

*d*) in the input plane and (

_{x},d_{y}*d*) in the focal plane and in defocused measurement planes is

_{u},d_{v}*a*(

*x,y*),

*g*(

_{n}*x,y*), and

*G*(

_{n}*u,v*) by arrays of size

*N*×

*N*, and

*h*(

*x, y*) by an array of size

*M*×

*M*. Usually

*M*is larger than

*N*.

*G*(

_{n}*u,v*) to the field magnitudes |

*F*(

_{n}*u,v*) = √

*I*(

_{n}*u,v*), where

*I*(

_{n}*u,v*) is the measured intensity pattern, using a squared-difference error metric

*q*is the number of subaperture positions and

*W*(

_{n}*u,v*) are weighting terms that allow us to ignore regions of poor signal to noise ratio or known bad detector pixels.

*h*(

*x, y*),

*a*(

*x, y*) and the subaperture translations (

*x*). For this purpose we typically use gradient search algorithms, e.g., the conjugate gradient search [14]. These algorithms require the gradient of the error metric with respect to the optimization parameters. The large number of free parameters in this problem makes calculation of the gradient using finite difference methods prohibitively expensive, so we use computationally efficient analytic expressions for the gradients. Depending on the particular application and

_{n},y_{n}*a priori*knowledge of the system, different parameterizations of

*h*(

*x, y*) and

*a*(

*x, y*) may be advantageous. The gradients for a number of parameterizations are given in the Appendix.

## 3. Experiment

### 3.1. Experimental Arrangement

### 3.2. Measurement of Subaperture to Camera Distance

*s*is the pinhole separation, and ν is the fringe frequency.

15. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient Subpixel Image Registration Algorithms,” Opt. Lett . **33**, 156–158 (2008). [CrossRef] [PubMed]

*L*, to a resolution of about 5 μm.

### 3.3. Results

*Q*= 0.748. The 9.85 mm diameter subaperture results in a

*Q*= 2.53, which is oversampled for both intensity and field. To cover the area of the composite aperture with significant subaperture overlap, we measured intensity patterns at 43 different subaperture positions illustrated in Fig. 6. A point in the composite aperture could be sampled by as few as one (near the edge) or as many as eight subaperture measurements. Example intensity patterns produced by various subaperture positions are shown in the middle column of Fig. 7, and the corresponding subaperture position is indicated in the left column. For processing in our algorithm, each intensity pattern was centered in a 256 × 256 array of data. This resulted in a subaperture plane sample spacing of 97.4 μm.

*h*(

*x,y*) using Zernike polynomial coefficients up to 8

^{th}order, assuming a uniform amplitude distribution and using the gradient expression in Eq. (22). At the same time we optimized tip, tilt, and focus for each subaperture,

*a*(

_{n}*x, y*), assuming the amplitude distribution shown in Fig. 4, to correct for focus errors and small shifts of the detector, using the gradient expression in Eq. (36). We also optimized for the subaperture positions, using the expressions in Eqs. (47) and (48). This optimization ran for 50 iterations.

*h*(

*x, y*), making use of the gradient in Eq. (19). This captures high frequency variations of the phase not well expressed by Zernike polynomials.

*h*(

*x, y*) at each point in the composite aperture using Eq. (29), at which point we do not rely on the assumption that the amplitude of

*h*(

*x, y*) is uniform.

*h*(

*x, y*) at each pixel in the same way that we did in the previous optimization. The resulting estimate of the field in the composite pupil is shown in Fig. 8. The phase is quite smooth, exhibiting mostly spherical aberration. The amplitude is quite uniform, but does contain artifacts at the edges of the subapertures. It is interesting that amplitude variations due to digs on the lens surface were well reconstructed on the left side of the subaperture. These appear as amplitude variations because the large slope of the dig scatters light away from the detector array. The digs were visible to the naked eye. The reconstructed intensity patterns in the measurement plane are shown in the right column of Fig. 7. The agreement between the measured and retrieved intensity patterns is excellent.

## 4. Conclusion

## Appendix: Analytic gradient calculations for different parameterizations

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

*α*can be written as

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

^{†}[ ] is the inverse operation to 𝗣[ ]. Defining

*h*(

*x, y*) , we employ a change of coordinates for (

*x, y*) and write

*h*(

*x, y*) in terms of its amplitude and phase, we have

*x*’,

*y*’) is given by

*h*(

*x, y*) using a set of polynomial basis functions, such as the Zernike polynomials, with coefficients

*c*,

_{k}^{h}*h*(

*x, y*) as the real and imaginary parts (complex value) of the field at each pixel,

*a*(

*x,y*) in terms of its amplitude and phase,

*a*(

*x,y*) in terms of the coefficients of basis functions as well. In this situation it is useful to have two sets of coefficients, one set that is constant over all subaperture measurements,

*c*, and one that varies for each translation of the subaperture,

_{k}^{a}*c*[2

_{k,n}^{a}2. J. R. Fienup, “Phase Retrieval for Undersampled Broadband Images,” J. Opt. Soc. Am . A, **16**, 1831–1839 (1999). [CrossRef]

*a*(

*x, y*) at each pixel,

*x*). We are assuming here that the function that is shifted is

_{n},y_{n}*h*(

*x, y*). To compute the derivative

**16**, 7264–78 (2008). [CrossRef] [PubMed]

## References and Links

1. | G. R. Brady and J. R. Fienup, “Range of Phase Retrieval in Optical Metrology,” in Frontiers in Optics 2005 / Laser Science XXI (Optical Society of America, Washington DC, 2005), paper FTuS3. |

2. | J. R. Fienup, “Phase Retrieval for Undersampled Broadband Images,” J. Opt. Soc. Am . A, |

3. | P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, “Extending the range of interferometry through subaperture stitching,” Proc. SPIE |

4. | M. Bray, “Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics],” Proc. SPIE |

5. | M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express |

6. | H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett . |

7. | J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett . |

8. | J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy |

9. | J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett . |

10. | H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy |

11. | M. Guizar-Sicairos and J. R. Fienup, “Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity,” manuscript submitted to Opt. Express . |

12. | J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt . |

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

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

15. | M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient Subpixel Image Registration Algorithms,” Opt. Lett . |

**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:**

Image Processing

**History**

Original Manuscript: November 7, 2008

Revised Manuscript: December 23, 2008

Manuscript Accepted: December 28, 2008

Published: January 7, 2009

**Citation**

Gregory R. Brady, Manuel Guizar-Sicairos, and James R. Fienup, "Optical wavefront measurement using phase retrieval with transverse translation diversity," Opt. Express **17**, 624-639 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-2-624

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### References

- G. R. Brady and J. R. Fienup, "Range of Phase Retrieval in Optical Metrology," in Frontiers in Optics2005 / Laser Science XXI (Optical Society of America, Washington DC, 2005), paper FTuS3.
- J. R. Fienup, "Phase Retrieval for Undersampled Broadband Images," J. Opt. Soc. Am. A 16, 1831-1839 (1999). [CrossRef]
- P. Dumas, J. Fleig, G. Forbes, and P. E. Murphy, "Extending the range of interferometry through subaperture stitching," Proc. SPIE TD02, 134-7 (2003).
- M. Bray, "Stitching interferometer for large optics using a standard interferometer: description of an automated system [for ICF optics]," Proc. SPIE 3047, 911-18 (1997).
- M. Guizar-Sicairos and J. R. Fienup, "Phase retrieval with transverse translation diversity: a nonlinear optimization approach," Opt. Express 16, 7264-78 (2008). [CrossRef] [PubMed]
- H. M. L. Faulkner and J. M. Rodenburg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm," Phys. Rev. Lett. 93, 023903 (2004). [CrossRef] [PubMed]
- J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004). [CrossRef]
- J. M. Rodenburg, A. C. Hurst and A. G. Cullis, "Transmission microscopy without lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007). [CrossRef]
- J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-x-ray lensless imaging of extended objects," Phys. Rev. Lett. 98, 034801 (2007). [CrossRef] [PubMed]
- H. M. L. Faulkner and J. M. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005). [CrossRef] [PubMed]
- M. Guizar-Sicairos and J. R. Fienup, "Measurement of coherent x-ray focused beams by phase retrieval with transverse translation diversity," submitted toOpt. Express.
- J. R. Fienup, "Phase Retrieval Algorithms: A Comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- J. R. Fienup, "Phase-Retrieval Algorithms for a Complicated Optical System," Appl. Opt. 32, 1737-1746 (1993). [CrossRef] [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.
- M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, "Efficient Subpixel Image Registration Algorithms," Opt. Lett. 33, 156-158 (2008). [CrossRef] [PubMed]

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