## All-optical axial super resolving imaging using a low-frequency binary-phase mask

Optics Express, Vol. 14, Issue 7, pp. 2631-2643 (2006)

http://dx.doi.org/10.1364/OE.14.002631

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### Abstract

In this paper we present a new approach for obtaining all-optical axial super-resolving imaging by using a non-diffractive binary phase mask inserted at the entrance pupil of an imaging lens. The designed element is tested numerically and experimentally on various practical testing benches and eventually is inserted into the lens of a cellular phone camera.

© 2006 Optical Society of America

## 1. Introduction

14. A. Sauceda and J. Ojeda-Castaneda, “High focal depth with fractional-power wavefronts,” Opt. Lett. **29**, 560–562 (2004). [CrossRef] [PubMed]

15. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. **26**, 875–877 (2001). [CrossRef]

## 2. Theoretical derivation

_{m}determines the severity of the error. The coefficient W

_{m}is also denoted as:

_{o}is the distance between the imaging lens and the object, Z

_{i}the distance between the imaging lens and the sensor and F is the focal length. When imaging condition is fulfilled:

_{n}are binary coefficients equal either to zero or to a certain phase modulation depth: a

_{n}= (0, Δϕ) of the phase only element that we design. Δϕ is the phase depth of modulation. Δx represents the spatial segments of the element. Since we do not want to create a diffractive optical element, i.e. spatial high frequency periodicity (such that there will be no wavelength dependence) we force Δx≫λ. The mathematical formulation for the optimization criteria will be as follows: Compute a phase only element that will provide maximum for the minimal value of the OTF within desired spectral region of interest while the OTF is composed out of two terms. The first is the OTF having strong defocusing deformation with parameter of W

_{m}and the second term is the in-focused OTF, i.e.:

**Δϕ≈π/2**. As previously mentioned the value of Δx is chosen such that it will be much larger than the optical wavelength in order to avoid chromatic distortions and dispersion. In the performed simulations we chose Δx to be 1/8 of the lens aperture.

## 3. Numerical simulations

## 4. Experimental verification

*a*is a coefficient and K(x) the kernel that is used for the sharpening algorithm. The kernel K depends on the point spread function of the optics. The coefficient

*a*is approximately 0.6 and the size of the kernel is 7 by 7 pixels while it is angularly symmetric and separable.

18. T. Q. Pham and L. J. van Vliet, “Separable bilateral filtering for fast video processing,” http://www.qi.tnw.tudelft.nl/~lucas.

_{D}and σ

_{R}are the two standard deviations of the two Gaussian functions.

## 5. Conclusions

## Appendix

_{i}and therefore A(μ) can be approximated as A(0) or for large W

_{m}values causing the argument of the exponent to oscillate rapidly. Next we assume that the phase element carries only one stripe of non-zero phase. When such an element is attached to the aperture one obtains:

_{i}otherwise the approximation of Eq. (A2) will not be valid) one may approximate:

## References and links

1. | W. T. Cathy and E. R. Dowski, “Apparatus and method for extending depth of field in image projection system,” US patent 6069738 (May 2000). |

2. | W. T. Cathy and E. R. Dowski, “Extended depth of field optical systems,” PCT publication WO 99/57599 (November 1999). |

3. | W. T. Cathy, “Extended depth field optics for human vision,” PCT publication WO 03/052492 (June 2003). |

4. | E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. |

5. | J. van der Gracht, E. Dowski, M. Taylor, and D. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. |

6. | C. M. Hammond, “Apparatus and method for reducing imaging errors in imaging systems having an extended depth of field,” US patent 6097856 (August 2000). |

7. | D. Miller and E. Blanko, “System and method for increasing the depth of focus of the human eye,” US patent 6554424 (April 2003). |

8. | N. Atebara and D. Miller, “Masked intraocular lens and method for treating a patient with cataracts,” US patent 4955904 (September 1990). |

9. | J. O. Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasi optimum real and positive transmittance apodizer,” Appl. Opt. |

10. | J. O. Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. |

11. | E. Ben-Eliezer, Z. Zalevsky, E. Marom, N. Konforti, and D. Mendlovic, “All optical extended depth of field imaging system,” PCT publication WO 03/076984 (September 2003). |

12. | E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A: Pure Appl. Opt. |

13. | E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. |

14. | A. Sauceda and J. Ojeda-Castaneda, “High focal depth with fractional-power wavefronts,” Opt. Lett. |

15. | W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. |

16. | Z. Zalevsky, “Optical method and system for extended depth of focus,” US patent application 10/97494 (August 2004). |

17. | J. W. Goodman, |

18. | T. Q. Pham and L. J. van Vliet, “Separable bilateral filtering for fast video processing,” http://www.qi.tnw.tudelft.nl/~lucas. |

**OCIS Codes**

(110.4850) Imaging systems : Optical transfer functions

(170.1630) Medical optics and biotechnology : Coded aperture imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 3, 2006

Revised Manuscript: February 15, 2006

Manuscript Accepted: March 21, 2006

Published: April 3, 2006

**Virtual Issues**

Vol. 1, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Zeev Zalevsky, Amir Shemer, Alexander Zlotnik, Eyal Ben Eliezer, and Emanuel Marom, "All-optical axial super resolving imaging using a low-frequency binary-phase mask," Opt. Express **14**, 2631-2643 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2631

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### References

- W. T. Cathy and E. R Dowski, "Apparatus and method for extending depth of field in image projection system," US patent 6069738 (May 2000).
- W. T. Cathy and E. R Dowski, "Extended depth of field optical systems," PCT publication WO 99/57599 (November 1999).
- W. T. Cathy, "Extended depth field optics for human vision," PCT publication WO 03/052492 (June 2003).
- E. R Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
- J. van der Gracht, E. Dowski, M. Taylor, D. Deaver, "Broadband behavior of an optical-digital focus-invariant system," Opt. Lett. 21, 919-921 (1996). [CrossRef] [PubMed]
- C. M. Hammond, "Apparatus and method for reducing imaging errors in imaging systems having an extended depth of field," US patent 6097856 (August 2000).
- D. Miller and E. Blanko, "System and method for increasing the depth of focus of the human eye," US patent 6554424 (April 2003).
- N. Atebara and D. Miller, "Masked intraocular lens and method for treating a patient with cataracts," US patent 4955904 (September 1990).
- J. O. Castaneda, E. Tepichin and A. Diaz, "Arbitrary high focal depth with a quasi optimum real and positive transmittance apodizer," Appl. Opt. 28, 2666-2669 (1989). [CrossRef]
- J. O. Castaneda and L. R. Berriel-Valdos, "Zone plate for arbitrary high focal depth," Appl. Opt. 29, 994-997 (1990). [CrossRef]
- E. Ben-Eliezer, Z. Zalevsky, E. Marom, N. Konforti and D. Mendlovic, "All optical extended depth of field imaging system," PCT publication WO 03/076984 (September 2003).
- E. Ben-Eliezer, Z. Zalevsky, E. Marom and N. Konforti, "All-optical extended depth of field imaging system," J. Opt. A: Pure Appl. Opt. 5, S164-S169 (2003). [CrossRef]
- E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, "Experimental realization of an imaging system with an extended depth of field," Appl. Opt. 44, 2792-2798 (2005). [CrossRef] [PubMed]
- A. Sauceda and J. Ojeda-Castaneda, "High focal depth with fractional-power wavefronts," Opt. Lett. 29, 560-562 (2004). [CrossRef] [PubMed]
- W. Chi and N. George, "Electronic imaging using a logarithmic asphere," Opt. Lett. 26, 875-877 (2001). [CrossRef]
- Z. Zalevsky, "Optical method and system for extended depth of focus," US patent application 10/97494 (August 2004).
- J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York 1996) pp.126-151.
- T. Q. Pham and L. J. van Vliet, "Separable bilateral filtering for fast video processing," http://www.qi.tnw.tudelft.nl/~lucas.

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