## Optical implementation of iterative fractional Fourier transform algorithm

Optics Express, Vol. 14, Issue 23, pp. 11103-11112 (2006)

http://dx.doi.org/10.1364/OE.14.011103

Acrobat PDF (674 KB)

### Abstract

An optical implementation of iterative fractional Fourier transform algorithm is proposed and demonstrated. In the proposed implementation, the phase-shifting digital holography technique and the phase-type spatial light modulator are adopted for the measurement and the modulation of complex optical fields, respectively. With the devised iterative fractional Fourier transform system, we demonstrate two-dimensional intensity distribution synthesis in the fractional Fourier domain and three-dimensional intensity distribution synthesis simultaneously forming desired intensity distributions at several multi-focal planes.

© 2006 Optical Society of America

## 1. Introduction

7. H. Kim and B. Lee, “Optimal nonmonotonic convergence of the iterative Fourier-transform algorithm,” Opt. Lett. **30**, 296–298 (2005). [CrossRef] [PubMed]

*a*th order FRFT is substituted for its complementary (2-

*a*)th order FRFT.

10. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A **12**, 743–751 (1995). [CrossRef]

14. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. **34**, 7451–7456 (1995). [CrossRef] [PubMed]

18. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**, 1268–1270 (1997). [CrossRef] [PubMed]

## 2. Optical implementation of fractional Fourier transforms

*a*th order two-dimensional fractional Fourier transform is defined by

*m*is an integer. The important properties of the fractional Fourier transform are the associative and the communicative properties as

*a*th order two-dimensional FRFT in optical system is defined by the linear integral transform

*G*(

*x*′,

*y*′)=(1/

*s*)

*Ĝ*(

*x*/

*s*,

*y*/

*s*) into [1/(

*sM*)]exp(

*jπ*(

*x*

^{2}+

*y*

^{2})/(

*λR*))

*Ĝ*

_{a}(

*x*/(

*sM*)),

*y*/(

*sM*)) where

*Ĝ*

_{a}(

*u*,

*v*) is the

*a*th order two-dimensional fractional Fourier transform of

*Ĝ*(

*u*,

*v*).

*a*th order two-dimensional FRFT and its complementary (2-

*a*)th order FRFT. Form Eq. (2d) we see that the overall system is an FRFT with the order of 2 because

*a*+(2-

*a*)=2. Then, from Eq. (2c) we see that this overall system is just inversing the image in transverse coordinates from

*G*(

*x*

_{1},

*y*

_{1}) to

*F*(

*x*

_{3},

*y*

_{3}), which should be true because the overall system is a 4-

*f*imaging system. A diverging spherical wave is incident on the filter denoted by

*G*(

*x*

_{1},

*y*

_{1}) and the optical field on the filter is Fourier-transformed to

*P*(

*x*

_{2},

*y*

_{2}) that is given by

*R*is the radius of the spherical phase, with which the input optical field

*G*(

*x*

_{1},

*y*

_{1}) is wrapped.

*P*(

*x*

_{2},

*y*

_{2}), the extra quadratic phase term exp(-

*jπ*(

*λR*) can be neglected in Eq. (4). Comparing Eqs. (3b) and (4), we can estimate the transform order and the scaling factor of the corresponding FRFT, respectively, as

*a*)th order fractional Fourier transform is similarly analyzed. To construct the (2-

*a*)th order fractional Fourier transform, the converging phase factor exp(

*jπ*(

*λR*′) must be multiplied by the optical field

*P*(

*x*

_{2},

*y*

_{2}), where -

*R*′ is the converging radius of curvature (

*R*′>0). Then the optical field is Fourier-transformed to

*F*(

*x*

_{3},

*y*

_{3}). The second stage is equivalent to the first stage except the sign change of the incident spherical phase factor. Then the transform order of the second stage can be found just by changing

*R*with -

*R*′:

*f*imaging system, i.e., the image

*F*(

*x*

_{3},

*y*

_{3}) is the same as

*G*(

*x*

_{1},

*y*

_{1}) in Fig. 2 except the coordinate inversion. Hence the (2-

*a*)th order FRFT, which is called a complementary transform for the

*a*th order FRFT, can be considered as the inverse transform of the

*a*th order FRFT. We can control the FRFT order

*a*by adjusting the curvature

*R*of the incident spherical field as can be seen in Eq. (5a). With changing the sign of the curvature of the incident spherical wave, we can realize the forward FRFT and its inverse FRFT in the same FRFT optic setup. This is an important factor to be made use of to optically implement the iterative FRFT algorithm.

## 3. Optical implementation of iterative fractional Fourier transform algorithm

^{st}beam path with a negative lens for the

*a*th order FRFT, (c) the 2

^{nd}beam path with a positive lens for the (2-

*a*)th order FRFT, (d) the 4-

*f*imaging system for matching the scaling factors between the CCD and the SLM, and (e) LabView-based system control unit.

*µm*. The optical field distribution is measured by the phase-shifting holography technique.

*µm*) is bigger than that of CCD (9

*µm*). However, matching the scaling factors between the CCD and the SLM is necessary to correctly carry out the forward and the inverse FRFTs. The scaling factor mismatch will result in very complicated FRFT relation [5

5. T. Shirai and T. H. Barnes, “Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device,” J. Opt. Soc. Am. A **19**, 369–377 (2002). [CrossRef]

*f*imaging optic setup with magnification feasibility is employed for matching the scaling factors of the SLM and the CCD. By the 4-

*f*imaging optics, the optical field represented on the SLM is adjusted to fit the scale of the CCD.

*a*)th order FRFT. At the following steps, the forward and the successive inverse transforms are conducted iteratively. The beam paths are selected by mechanical shutters to implement the

*a*th order FRFT (1st beam path) and the (2-

*a*)th order FRFT (2nd beam path) alternatively. That is, the

*a*th order FRFT and the (2-

*a*)th order FRFT in Fig. 1 are split into time sequential steps along the 1st and 2nd beam paths.

*a*th order FRFT domain can be attained through this iteration procedure. With additional control of the spherical curvature of the incident spherical wave, we can obtain different intensity distributions at different FRFT domains. Multiplexing several DOE phase profiles enable simultaneous generation of intensity distributions at several FRFT domains with different FRFT orders. In other words, two-dimensional intensity distribution synthesis in the fractional Fourier domain and three-dimensional intensity distribution synthesis simultaneously forming desired intensity distributions at several multi-focal planes can be realized in real time with the devised iterative FRFT system.

*a*th order FRFT domain is illuminated by a plane wave, the diffraction pattern is formed at the defocused plane. In Fig. 6, the transform of the wave wrapped with diverging spherical phase (denoted by dotted lines) to the focal plane is formulated as

*G*(

*x*

_{1},

*y*

_{1}) to a Δ

*d*distance from the focal plane is formulated as

*R*and the defocus Δ

*d*is given by

*R*and -

*R*, we can design DOE phase profiles to generate diffraction images at distinguished image planes. Also, the obtained DOE phase profiles can be multiplexed to form three-dimensional multi-focal intensity distribution at several distinguished planes.

## 4. Experimental results

*R*=∞ (i.e. a plane wave input). Figure 7(b) shows the improvement for the case in which the spherical phase of the radius

*R*=62

*mm*is used. In both cases, the focal length of the Fourier lens is set to

*f*=150

*mm*. The FRFT order of the first case is 1 and the stagnation of the iteration is reached at the 16th iteration stage, where the ratio in soft clipping is set to zero. The FRFT order of the second case is a complex number 0.9746

*j*and the stagnation of the iteration is arrived at the 7th iteration stage, where the ratio in soft clipping is set to zero.

## 5. Conclusion

*f*imaging part for matching the scaling factors of the CCD and the SLM. By adjusting the spherical phase curvature, we can obtain the DOE phase profiles to generate diffraction images at several defocused planes. The experimental feasibility of multi-focal image synthesis using the simple DOE multiplexing method opens the possibility of shaping three-dimensional volumetric intensity distribution synthesis in volumetric region by using a fully optical implementation setup.

## Acknowledgment

## References and links

1. | J. Turunen and F. Wyrowski, |

2. | B. Kress and P. Meyrueis, |

3. | H. Kim, K. Choi, and B. Lee, “Diffractive optic synthesis and analysis of light fields and recent applications,” Jpn. J. Appl. Phys. |

4. | V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, |

5. | T. Shirai and T. H. Barnes, “Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device,” J. Opt. Soc. Am. A |

6. | J. Hahn, H. Kim, K. Choi, and B. Lee, “Real-time digital holographic beam-shaping system with a genetic feedback tuning loop,” Appl. Opt. |

7. | H. Kim and B. Lee, “Optimal nonmonotonic convergence of the iterative Fourier-transform algorithm,” Opt. Lett. |

8. | H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A |

9. | K. Choi, H. Kim, and B. Lee, “Synthetic phase holograms for auto-stereoscopic image display using a modified IFTA,” Opt. Express |

10. | H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A |

11. | A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. |

12. | A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. |

13. | H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, |

14. | D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. |

15. | T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A |

16. | P. Andrés, W. D. Furlan, G. Saavedra, and A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A |

17. | E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, and P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. |

18. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(090.1760) Holography : Computer holography

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: September 5, 2006

Revised Manuscript: October 31, 2006

Manuscript Accepted: October 31, 2006

Published: November 13, 2006

**Citation**

Joonku Hahn, Hwi Kim, and Byoungho Lee, "Optical implementation of iterative fractional Fourier transform algorithm," Opt. Express **14**, 11103-11112 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11103

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### References

- J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).
- B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).
- H. Kim, K. Choi, and B. Lee, "Diffractive optic synthesis and analysis of light fields and recent applications," Jpn. J. Appl. Phys. 45, 6555-6575 (2006). [CrossRef]
- V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).
- T. Shirai and T. H. Barnes, "Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device," J. Opt. Soc. Am. A 19, 369-377 (2002). [CrossRef]
- J. Hahn, H. Kim, K. Choi, and B. Lee, "Real-time digital holographic beam-shaping system with a genetic feedback tuning loop," Appl. Opt. 45, 915-924 (2006). [CrossRef] [PubMed]
- H. Kim and B. Lee, "Optimal nonmonotonic convergence of the iterative Fourier-transform algorithm," Opt. Lett. 30, 296-298 (2005). [CrossRef] [PubMed]
- H. Kim, B. Yang, and B. Lee, "Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements," J. Opt. Soc. Am. A 12, 2353-2365 (2004). [CrossRef]
- K. Choi, H. Kim, and B. Lee, "Synthetic phase holograms for auto-stereoscopic image display using a modified IFTA," Opt. Express 12, 2454-2462 (2004). [CrossRef] [PubMed]
- H. M. Ozaktas and D. Mendlovic, "Fractional Fourier Optics," J. Opt. Soc. Am. A 12, 743-751 (1995). [CrossRef]
- A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions," Opt. Commun. 120, 134-138 (1995). [CrossRef]
- A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters," Appl. Opt. 37, 2130-2141 (1998). [CrossRef]
- H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).
- D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, "Anamorphic fractional Fourier transform: optical implementation and applications," Appl. Opt. 34, 7451-7456 (1995). [CrossRef] [PubMed]
- T. Kim and T-C. Poon, " Three-dimensional matching by use of phase-only holographic information and the Wigner distribution," J. Opt. Soc. Am. A 12, 2520-2528 (2000). [CrossRef]
- P. Andrés, W. D. Furlan, G. Saavedra, and A. W. Lohmann, "Variable fractional Fourier processor: a simple implementation," J. Opt. Soc. Am. A 14, 853-858 (1997). [CrossRef]
- E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, and P. Andrés, "White-light optical implementation of the fractional Fourier transform with adjustable order control," Appl. Opt. 39, 238-245 (2000). [CrossRef]
- I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]

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