## Complex signal recovery from multiple fractional Fourier-transform intensities

Applied Optics, Vol. 44, Issue 23, pp. 4902-4908 (2005)

http://dx.doi.org/10.1364/AO.44.004902

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### Abstract

The problem of recovering a complex signal from the magnitudes of any number of its fractional Fourier transforms at any set of fractional orders is addressed. This problem corresponds to the problem of phase retrieval from the transverse intensity profiles of an optical field at arbitrary locations in an optical system involving arbitrary concatenations of lenses and sections of free space. The dependence of the results on the number of orders, their spread, and the noise is investigated. Generally, increasing the number of orders improves the results, but with diminishing return beyond a certain point. Selecting the measurement planes such that their fractional orders are well separated or spread as much as possible also leads to better results.

© 2005 Optical Society of America

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(100.5070) Image processing : Phase retrieval

**History**

Original Manuscript: September 15, 2004

Revised Manuscript: February 25, 2005

Manuscript Accepted: March 1, 2005

Published: August 10, 2005

**Citation**

M. Günhan Ertosun, Haluk Atli, Haldun M. Ozaktas, and Billur Barshan, "Complex signal recovery from multiple fractional Fourier-transform intensities," Appl. Opt. **44**, 4902-4908 (2005)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-44-23-4902

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### References

- D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993). [CrossRef]
- H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993). [CrossRef]
- A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional order Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993). [CrossRef]
- L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994). [CrossRef]
- H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994). [CrossRef]
- H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994). [CrossRef]
- O. Aytür, H. M. Ozaktas, “Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms,” Opt. Commun. 120, 166–170 (1995). [CrossRef]
- H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995). [CrossRef]
- M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999). [CrossRef]
- İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition,” in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, 2000), pp. I:93–96. [CrossRef]
- H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdağı, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996). [CrossRef]
- H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
- H. Stark, ed., Image Recovery: Theory and Application (Academic, 1987).
- H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002). [CrossRef]
- M. G. Ertosun, H. Atli, H. M. Ozaktas, B. Barshan, “Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence,” Opt. Commun. 244, 61–70 (2005). [CrossRef]
- J. Miao, D. Sayre, H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998). [CrossRef]
- B. Dong, Y. Zhang, B. Gu, G. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997). [CrossRef]
- W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Recursive algorithm for phase retrieval in the fractional Fourier transform domain,” Appl. Opt. 37, 6906–6910 (1998). [CrossRef]
- W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A new method for phase retrieval in the optical system,” Chin. Phys. Lett. 15, 24–26 (1998). [CrossRef]
- W.-X. Cong, N.-X. Chen, B.-Y. Gu, “A recursive method for phase retrieval in Fourier transform domain,” Chin. Sci. Bull. 43, 40–44 (1998). [CrossRef]
- H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]
- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), pp. 231–275.
- G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta Phys. Sin. 30, 410–413 (1981).
- B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981).
- G. Yang, B. Dong, B. Gu, J. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994). [CrossRef] [PubMed]
- Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996). [CrossRef] [PubMed]
- Y. Zhang, B. Dong, B. Gu, G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998). [CrossRef]
- B. Hennelly, J. T. Sheridan, “Fractional Fourier transform-based image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003). [CrossRef]
- M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994). [CrossRef] [PubMed]
- D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995). [CrossRef] [PubMed]
- X. Liu, K.-H. Brenner, “Reconstruction of two-dimensional complex amplitudes from intensity measurements,” Opt. Commun. 225, 19–30 (2003). [CrossRef]
- D. Dragoman, “Redundancy of phase-space distribution functions in complex field recovery problems,” Appl. Opt. 42, 1932–1937 (2003). [CrossRef] [PubMed]
- M. Testorf, “Comment on ‘Redundancy of phase-space distribution functions in complex field recovery problems’,” Appl. Opt. 44, 55–57 (2005). [CrossRef]
- D. Dragoman, “Reply to comment on ‘Redundancy of phase-space distribution functions in complex field recovery problems’,” Appl. Opt. 44, 58–59 (2005). [CrossRef]
- T. Alieva, M. J. Bastiaans, “On fractional Fourier transform moments,” IEEE Signal Process. Lett. 7, 321–323 (2000). [CrossRef]
- T. Alieva, M. J. Bastiaans, L. Stankovic, “Signal reconstruction from two close fractional Fourier power spectra,” IEEE Trans. Signal Process. 51, 112–123 (2003). [CrossRef]
- M. Bastiaans, K. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. A 20, 1046–1049 (2003). [CrossRef]
- R. Rolleston, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986). [CrossRef] [PubMed]
- R. Rolleston, N. George, “Stationary phase approximations in Fresnel-zone magnitude-only reconstructions,” J. Opt. Soc. Am. A 4, 148–153 (1987). [CrossRef]
- D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973). [CrossRef]
- D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D 6, 2200–2216 (1973). [CrossRef]
- D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D 6, 2217–2224 (1973). [CrossRef]
- T. E. Gureyev, A. Pogany, D. M. Paganin, S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004). [CrossRef]
- T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003). [CrossRef]
- W.-X. Cong, N.-X. Chen, B.-Y. Gu, “Phase retrieval in the Fresnel transform system: a recursive algorithm,” J. Opt. Soc. Am. A 16, 1827–1830 (1999). [CrossRef]
- W. Coene, G. Janssen, M. Op. de Beeck, D. Van Dyck, “Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy,” Phys. Rev. Lett. 69, 3743–3746 (1992). [CrossRef] [PubMed]
- K. A. Nugent, T. E. Gureyev, D. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996). [CrossRef] [PubMed]
- L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004). [CrossRef]
- L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001). [CrossRef]
- L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev E 63, 037602-1–037602-4 (2001). [CrossRef]
- V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992). [CrossRef]
- C. Candan, M. Alper Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000). [CrossRef]
- R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990). [CrossRef]

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