## Extension ratio of depth of field by wavefront coding method

Optics Express, Vol. 16, Issue 17, pp. 13364-13371 (2008)

http://dx.doi.org/10.1364/OE.16.013364

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### Abstract

An analysis of the depth of field (DOF) of the wavefront coding imaging system with a cubic phase mask is presented. A necessary condition on the base of that MTF of wavefront coding system is defocus-independent is described. Then the extension ratio of the DOF relative to that of traditional optical system is calculated. And the conclusion is also verified by the simulation results.

© 2008 Optical Society of America

## 1. Introduction

1. E. R. Dowski and W. T. Cathey, “Extended depth of field through Wavefront Coding,” Appl. Opt. **34**, 1859–1866 (1995). [CrossRef] [PubMed]

## 2. A theoretical analysis of extension ratio of the DOF for wavefront coding imaging system

### 2.1 The basic principles of wavefront coding system

*x*and

*y*are normalized coordinates.

9. E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE **3779**, 137–145 (1999). [CrossRef]

### 2.2 The DOF of Traditional Optical System

_{i}is the distance from the pupil plane of the lens to the image plane; Δ is the distance from actual image plane to the image plane.

_{20}is

### 2.3 The DOF of Wavefront Coding System

*f*(

*x*) in the exponential function, the point satisfying

_{0}, then

*f*(

*x*) in Taylor series at point x

_{0}includes only two terms because the three and above all derivatives are zeros, it is simplified as

_{0}) is 1 as long as x is in the overlapping region of the optical pupil, otherwise the value is 0, so g(x) could be replaced by g(x

_{0}) if x is in the overlapping region of the optical pupil. Substituting Eq. (12) into Eq. (8) yields the following expression:

### 2.4 The Extension Ratio of the DOF for Wavefront Coding Imaging System

4. S. Bradburn, E. R. Dowski, and W. T. Cathey, “Realizations of Focus Invariance in Optical-Digital Systems with Wavefront Coding,” Appl. Opt. **36**, 9157–9166(1997). [CrossRef]

## 3. The Simulation of Extension Ratio of the DOF

## 4. Conclusion

## Acknowledgments

## References and links

1. | E. R. Dowski and W. T. Cathey, “Extended depth of field through Wavefront Coding,” Appl. Opt. |

2. | S. C. Tucker, W. T Cathey, and E. R. Dowski, “Extended depth of field and aberration control for inexpensive digital microscope systems,” Opt. Express |

3. | H. Wach, E. R. Dowski, and W. T. Cathey, “Control of Chromatic Focal Shift through Wavefront Coding,” Appl. Opt. |

4. | S. Bradburn, E. R. Dowski, and W. T. Cathey, “Realizations of Focus Invariance in Optical-Digital Systems with Wavefront Coding,” Appl. Opt. |

5. | D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional Tomography using a Cubic-Phase Plate Extended Depth-of-Field System,” Opt. Lett. |

6. | R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. |

7. | R. Plemmons, M. Horvath, E. Leonhardt, P. Pauca, S. Prasad, S. Robinson, H. Setty, T. Torgersen, J. Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveir, “Computational Imaging Systems for Iris Recognition,” Proc. SPIE |

8. | P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. |

9. | E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE |

10. | J. W. Goodman, |

**OCIS Codes**

(110.1758) Imaging systems : Computational imaging

(110.7348) Imaging systems : Wavefront encoding

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 1, 2008

Revised Manuscript: July 17, 2008

Manuscript Accepted: August 10, 2008

Published: August 15, 2008

**Citation**

Chao Pan, Jiabi Chen, Rongfu Zhang, and Sonlin Zhuang, "Extension ratio of depth of field by wavefront coding method," Opt. Express **16**, 13364-13371 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13364

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### References

- E. R. Dowski and W. T. Cathey, "Extended depth of field through Wavefront Coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
- S. C. Tucker, W. T. Cathey, and E. R. Dowski, "Extended depth of field and aberration control for inexpensive digital microscope systems," Opt. Express 4, 467-474 (1999). [CrossRef] [PubMed]
- H. Wach, E. R. Dowski, and W. T. Cathey, "Control of Chromatic Focal Shift through Wavefront Coding," Appl. Opt. 37, 5359-5367 (1998). [CrossRef]
- S. Bradburn, E. R. Dowski, and W. T. Cathey, "Realizations of Focus Invariance in Optical-Digital Systems with Wavefront Coding," Appl. Opt. 36, 9157-9166(1997). [CrossRef]
- D. L. Marks, R. A. Stack, and D. J. Brady, "Three-dimensional Tomography using a Cubic-Phase Plate Extended Depth-of-Field System," Opt. Lett. 24, 253-255 (1999). [CrossRef]
- R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, "Extending the imaging volume for biometric iris recognition," Appl. Opt. 44, 701-712 (2005). [CrossRef] [PubMed]
- R. Plemmons, M. Horvath, E. Leonhardt, P. Pauca, S. Prasad, S. Robinson, H. Setty, T. Torgersen, J. Gracht, E. Dowski, R. Narayanswamy, and P. E. X. Silveir, "Computational Imaging Systems for Iris Recognition," Proc. SPIE 5559 346-357.
- P. E. X. Silveira and R. Narayanswamy, "Signal-to-noise analysis of task-based imaging systems with defocus," Appl. Opt. 45, 2924-2934 (2006). [CrossRef] [PubMed]
- E. R. Dowski and G. E. Johnson, "Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems," Proc. SPIE 3779, 137-145 (1999). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 3rd Ed. (New York: Roberts and Company Publishers Inc., 2005), p128.

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