## All-optical axially multi-regional super resolved imaging

Optics Express, Vol. 15, Issue 26, pp. 17912-17921 (2007)

http://dx.doi.org/10.1364/OE.15.017912

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### Abstract

In this paper we present a new approach of all-optical extended depth of focus providing two (or more) discrete ranges of focused imaging for close as well as far ranges. The fact that the extended depth of focus is not continuous allows obtaining improved contrast in the two (or more) axial regions of extended depth of focus. The design is aimed for the cell phone camera applications where dual range extended depth of focus can allow simultaneous reading of business cards at very short distance as well as very high contrasted imaging at far range.

© 2007 Optical Society of America

## 1. Introduction

5. J. van der Gracht, E. Dowski, M. Taylor, and D. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. **21**, 919–921 (1996). [CrossRef] [PubMed]

10. J. O. Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. **29**, 994–997 (1990). [CrossRef]

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. **44**, 2792–2798 (2005). [CrossRef] [PubMed]

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]

17. Z. Zalevsky, A. Shemer, A. Zlotnik, E. Ben-Eliezer, and E. Marom, “All-optical axial super resolving imaging using low-frequency binary-phase mask,” Opt. Express **14**, 2631–2643 (2006). [CrossRef] [PubMed]

17. Z. Zalevsky, A. Shemer, A. Zlotnik, E. Ben-Eliezer, and E. Marom, “All-optical axial super resolving imaging using low-frequency binary-phase mask,” Opt. Express **14**, 2631–2643 (2006). [CrossRef] [PubMed]

17. Z. Zalevsky, A. Shemer, A. Zlotnik, E. Ben-Eliezer, and E. Marom, “All-optical axial super resolving imaging using low-frequency binary-phase mask,” Opt. Express **14**, 2631–2643 (2006). [CrossRef] [PubMed]

**14**, 2631–2643 (2006). [CrossRef] [PubMed]

## 2. Theoretical derivation

**14**, 2631–2643 (2006). [CrossRef] [PubMed]

*a*are binary coefficients equal either to zero or to a certain phase modulation depth:

_{n}*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.

*λ*is the wavelength and

*µ*is the coordinate of the OTF plane.

*P*is the aperture of the lens having coordinates of

*x*(the plane of the CTF) and

*Z*is the distance between the imaging lens and the sensor. Since we do not want to create a diffractive optical element, i.e. spatial high frequencies and periodicity (since we wish to have no wavelength dependence) we force

_{i}*Δx≫λ*. Figure 1(a) shows an imaging system with a phase-element that is attached to the imaging lens, an object at distance

*Z*from the lens and a detection array that is positioned at distance

_{o}*Z*from the imaging lens. The OTF function that is given in Eq. (1) can be used to characterize the optical system that is depicted in Fig. 1(a).

_{i}*W*. The second term has a small defocusing deformation (denoted as

_{m}*βW*in Fig. 1(b) where

_{m}*β≪1*). The coefficient

*W*determines the severity of the defocusing error and is defined as:

_{m}*ψ*is a phase factor represents the severity of out of focus:

*2b*denotes the diameter of the lens,

*λ*is the wavelength,

*Z*is the distance between the imaging lens and the object,

_{o}*Z*the distance between the imaging lens and the sensor and

_{i}*F*is the focal length. When imaging condition is fulfilled,

*ψ*is zeroed, since:

*W*will now determine two new in-focus regions that are positioned before and after the previous plane (see Fig. 1(b)).

_{m}**14**, 2631–2643 (2006). [CrossRef] [PubMed]

**14**, 2631–2643 (2006). [CrossRef] [PubMed]

*Δϕ=π/2*, which stems from the constraint of a continuous focused region. That element was able to cancel the sign inversions of the quadratic phase that was generated in the aperture plane due to defocusing. Correction was obtained by the addition of proper phase to the spatial frequencies where inversion appeared. Although it seems like in order to cancel sign inversions one has to add a phase of

*π*, the best solution in this case was obtained with a phase of

*π/2*. The reason for a phase of

*π/2*was due to the requirement for a continuous focused region. In a continuous focused region solution, one has to cancel the inversions of quadratic phase due to defocusing while not creating new sign inversion to the in-focus plane. Thus, a phase of

*Δϕ=π/2*contributes equally both to the defocused planes as well as the in-focus plane.

*π*, since now we wish to cancel only the sign inversions of the quadratic defocusing phase in the two regions that are located before and after the original in-focus plane. In this approach, the previously in-focus plane will now be distorted. The fact that two non-continuous regions are required, improves the contrast performance in each of the two regions in comparison to the case of Ref. [17

**14**, 2631–2643 (2006). [CrossRef] [PubMed]

*H(µx,Z0)*is the OTF that primarily depends on object’s distance

*Z*and spatial frequency

_{o}*µ*(without a loss of generality, a one dimensional case is considered). In this example, the considered spatial frequencies are those being smaller than maximal spatial frequency of the photo detecting surface

_{x}*µ*; the region of interest is dual and constructed of separate axial regions of interest

_{d}*R*and

_{1}*R*. The OTF is also dependent on the parameters of the optical system and phase mask; varying these parameters allows optimizing the indicator of Eq. (5).

_{2}*K(µ*is a weight function corresponding to utility of each spatial frequency. Higher spatial frequencies may be assigned lower utility.

_{x})*R*and

_{1}*R*may be decomposed into different spatial frequencies ranges. Then indicator of Eq. (5) may take a form:

_{2}*K(µ*is a utility function dependent both on region of interest and spatial frequency.

_{x},Z_{o})*E*and

_{1}*E*are boundaries (edges) of the near region and

_{2}*K*and

_{1}*K*are weights assigned to these boundaries. The corresponding far region in the first order of approximation is symmetrical to the near region, relatively to the in-focus plane of the lens (the plane with zero defocusing). Here, symmetry is based on defocusing and not the length of the regions. Optimization of Eq. (9) is similar to Eq. (8), but requires less computation. The value maximized in Eq. (9) is composed of two terms with different degrees of defocusing and phase factors.

_{2}*is an in-focus plane of the lens, for the selected PDA distance (i.e. distance between the lensing section and the PDA). From imaged space, the magnitude of geometrical defocusing is the largest either at the left edge of near region or at the right edge of far region. At the right edge of the near region and at the left edge of the far region the geometrical defocusing is the smallest. It should be noted that this smallest defocusing is not zero, as it would be in an EDOF application, but*

**P**_{i}*βW*, where

_{m}*β*is a number between zero and one. This is due to the effect of the phase mask. The effect of phase mask thus relates to overextending depth of focus.

*N*focused axial regions is to be generated the optimization expression takes the form of:

*Δx*was close to 1/8 of the lens aperture. The numerical optimization of the mathematical condition of Eq. (9) was done using Zemax. The parameter

*β*was chosen such that the obtained focusing ranges will be 12.5cm–25cm and 50cm to infinity for camera with F-number of 3 and focal length of 4.8mm. Numerical simulations of the new design reveal the dual region behavior of the designed element.

**14**, 2631–2643 (2006). [CrossRef] [PubMed]

*K*and

_{1}*K*of Eq. (9), in order to obtain an unbalanced response. Examples of two different asymmetric systems are shown below. Figure 3(a) shows the through focus MTF of a system having better near field response compared to the far field response. Figure 3(b) shows the through focus MTF of a system having better far field response compared to the near field response. The simulation of the through focus MTF was performed for spatial frequency of 60 cycles/mm.

_{2}## 3. Experimental results

## 4. Conclusions

**14**, 2631–2643 (2006). [CrossRef] [PubMed]

## 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 (May2000). |

2. | W. T. Cathy and E. R Dowski, “Extended depth of field optical systems,” PCT publication WO 99/57599 (November1999). |

3. | W. T. Cathy, “Extended depth field optics for human vision,” PCT publication WO 03/052492 (June2003). |

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 (August2000). |

7. | D. Miller and E. Blanko, “System and method for increasing the depth of focus of the human eye,” US patent 6554424 (April2003). |

8. | N. Atebara and D. Miller, “Masked intraocular lens and method for treating a patient with cataracts,” US patent 4955904 (September1990). |

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 (September2003). |

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 (August2004). |

17. | Z. Zalevsky, A. Shemer, A. Zlotnik, E. Ben-Eliezer, and E. Marom, “All-optical axial super resolving imaging using low-frequency binary-phase mask,” Opt. Express |

**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: October 11, 2007

Revised Manuscript: December 10, 2007

Manuscript Accepted: December 13, 2007

Published: December 17, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Ido Raveh and Zeev Zalevsky, "All-optical axially multi-regional super resolved imaging," Opt. Express **15**, 17912-17921 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-17912

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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 and 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).
- Z. Zalevsky, A. Shemer, A. Zlotnik, E. Ben-Eliezer and E. Marom, "All-optical axial super resolving imaging using low-frequency binary-phase mask," Opt. Express 14, 2631-2643 (2006). [CrossRef] [PubMed]

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