## Depth of field extension with spherical optics

Optics Express, Vol. 16, Issue 17, pp. 12995-13004 (2008)

http://dx.doi.org/10.1364/OE.16.012995

Acrobat PDF (264 KB)

### Abstract

The introduction of spherical aberration in a lens design can be used to extend the depth of field while preserving resolution up to half the maximum diffraction-limited spatial frequency. Two low-power microscope objectives are shown that achieve an extension of ±0.88 λ in terms of wavefront error, which is shown to be comparable to alternative techniques but without the use of special phase elements. The lens performance is azimuth-independent and achromatic over the visible range.

© 2008 Optical Society of America

## 1. Introduction

1. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. **50**, 749–754 (1960). [CrossRef]

2. S. Bagheri and B. Javidi, “Extension of depth of field using amplitude and phase modulation of the pupil function,” Opt. Lett. **33**, 757–759 (2008). [CrossRef] [PubMed]

^{3–53. S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709–2721 (2004). [CrossRef] [PubMed] }special aspheric surfaces,

^{66. N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A: Pure Appl. Opt. 5, S157–S163 (2003). [CrossRef] }utilizing birefringence,

^{77. S. Sanyal and A. Ghosh, “High focal depth with quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000). [CrossRef] ,88. Z. Zalevsky and S. Ben-Yaish, “Extended depth of focus imaging with birefringent plate,” Opt. Express 15, 7204–7210 (2007), http://www.opticsexpress.org/abstract.cfm?uri=oe-15-12-7202. [CrossRef] }or utilizing longitudinal chromatic aberration.

^{99. G. Frédéric, “Advances in camera phone picture quality,” Photonics Spectra, Nov. 2007, p. 50 (no archival literature references found)}

- The Modulation Transfer Function (MTF) should stay above zero (or above a minimum value that depends on the system noise level) throughout the range of frequencies of interest. This condition assures that no loss of information occurs through the pre-blurring mechanism.
- The Optical Transfer Function (OTF) should be invariant through the extended range of focus. This condition ensures that a single filter can be used to reconstruct all focal planes simultaneously. Any variation of the OTF will ultimately produce image artifacts when a single filter is used. Since complete invariance is impossible, the significance of the artifacts will depend on the application.
- The OTF should ideally be the same through field, again to allow a single reconstruction filter. This condition can be relaxed at the expense of significant computational complexity, by using different filters across the field of view. Thus in principle, it is not a fundamental requirement but a convenience.
- The OTF should be rotationally symmetric if image recovery is to be invariant with orientation. This condition becomes significant in finely sampled, high-resolution systems.

^{66. N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A: Pure Appl. Opt. 5, S157–S163 (2003). [CrossRef] }demonstrated a tenfold DOF extension with a specially designed “logarithmic asphere”. Their system had 23 µm pixel size and operated at about F/4. Thus the detector Nyquist frequency (21.7 c/mm) was only about 1/20

^{th}of the MTF diffraction limit (~450 c/mm) for that aperture. From the present viewpoint, this is a very coarse frequency. For the systems examined here, the pixel is smaller than the diffraction-limited spot size; thus spatial frequencies up to or above one-half the diffraction limit are of interest. Evidently, the way of quantifying DOF extension must include the spatial frequency range relative to the diffraction limit if it is to be made in a detector-independent way.

^{22. S. Bagheri and B. Javidi, “Extension of depth of field using amplitude and phase modulation of the pupil function,” Opt. Lett. 33, 757–759 (2008). [CrossRef] [PubMed] }made a comparison between amplitude, phase, and mixed modulation for DOF extension and showed that pure amplitude modulation is optimum for DOF extension in high resolution applications because it preserves information up to the diffraction limit. Crucially, the amplitude modulation produces a rotationally symmetric MTF, whereas the cubic phase term produces high MTF only along the sagittal and tangential orientations.

^{1313. P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 46, 2924–2934 (2006). [CrossRef] }For a circularly obscured pupil with a ratio of obscured to total diameter equal to

*δ*, the extended DOF range through amplitude modulation was given as

*W*

_{20}=±0.5(1+

*δ*

^{2}) in number of wavelengths. Thus a DOF extension of ±0.8 λ requires a ~75% linear obscuration.

^{1414. W. N. Charman and H. Whitefoot, “Pupil diameter and the depth-of-field of the human eye as measured by laser speckle,” Optica Acta 24, 1211–1216 (1977). [CrossRef] }noted an apparent increase in the DOF of the human eye with increased pupil diameter, which could be attributable to increased spherical aberration. Mezouari and Harvey

^{1515. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28, 771–773 (2003). [CrossRef] [PubMed] }examined theoretically spherical aberration terms and compared what they called a quartic filter (containing SA terms) with other types. It is evident however, that spherical aberration can be introduced into a design without the need for a special phase plate. We demonstrate below that the deliberate introduction of spherical aberration into a low power microscope system can provide depth of focus extension and azimuth-independent image quality comparable with other techniques and even be advantageous in certain respects.

## 2. Optical system requirements and design optimization

^{33. S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709–2721 (2004). [CrossRef] [PubMed] }, but with three additional conditions, as follows: 1) For a broadband system, we demand also the equalization of the MTF values across wavelength. If the polychromatic MTF is optimized instead, it is possible that a very low MTF value for one wavelength will be balanced by a high MTF value for another, thus compromising polychromatic image reconstruction. 2) We also demand the equalization of MTF values for the middle and the extreme field, thus ensuring isoplanatic performance, and 3) we demand equalization of the sagittal and tangential responses. (Optimization and computations in this paper have been performed using ZEMAX®).

^{1616. V. N. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, Ch.2, SPIE Press, Bellingham, WA (2001).}Thus it is not necessary to account specifically for the PTF behavior in the merit function. This also shows the importance of starting out with a system that is nearly diffraction-limited through the entire field, otherwise residual off-axis aberrations can cause strong variation of the PTF.

## 3. Design assessment and DOF extension

^{1717. P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design, p. 324, Oxford University Press, New York/Oxford (1997).}This relation can apply equally to the object or image space. As shown below, the depth of field achieved for the NA=0.167 objective is ±35 µm, and for the NA=0.08 objective ±150 µm. In both cases the amount of defocus in terms of

*W*

_{20}is ±0.88 λ for a mid-wavelength of 550 nm. The fact that the defocus is the same in both cases also means that the same amount of spherical aberration is introduced, scaled only by the aperture. Thus the MTF curves of the two systems are practically identical in shape, differing only in the scaling of the frequency axis. Sample MTF curves for the two systems are shown in Fig. 3. In Fig. 4 the MTF curves at the two extremes and the middle of the field range for one system only are shown, with the understanding that the other system behaves very similarly. Also, only the axial field point is shown, since, as can be seen from Fig. 3, there is very little difference between the MTF curves for the edge of the field and the axial ones, and also very little difference between the S and T (as well as intermediate) orientations.

## 4. Image recovery

## 5. Comparison with other techniques

^{22. S. Bagheri and B. Javidi, “Extension of depth of field using amplitude and phase modulation of the pupil function,” Opt. Lett. 33, 757–759 (2008). [CrossRef] [PubMed] ,1313. P. E. X. Silveira and R. Narayanswamy, “Signal-to-noise analysis of task-based imaging systems with defocus,” Appl. Opt. 46, 2924–2934 (2006). [CrossRef] }the wavefront-coded system excels at extending the depth of focus along two orthogonal directions, but its inherent asymmetry means that there is inferior reconstruction along the diagonals. Figure 7 shows the MTF curves of a system optimized for DOF extension with the same specifications as the NA=0.167 system. The “near” and “far” points correspond to ±65 µm, which is almost twice the range of the SA system (±35 µm). The MTF curves that correspond to the two principal directions of the wavefront coded system (normally co-aligned with the detector array orientation) are indistinguishable and show through-focus variation similar to that of the SA system but over a larger range of focus. However, the MTF curves corresponding to the ±45° orientation show considerable degradation. The system would therefore fail to reconstruct detail at that orientation for almost any focus position. This may not be a problem in low resolution systems especially since detector resolution suffers along the diagonal, but it becomes important in highly sampled systems such as considered here.

2. S. Bagheri and B. Javidi, “Extension of depth of field using amplitude and phase modulation of the pupil function,” Opt. Lett. **33**, 757–759 (2008). [CrossRef] [PubMed]

## 6. Conclusions

## Acknowledgments

## References and links

1. | W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. |

2. | S. Bagheri and B. Javidi, “Extension of depth of field using amplitude and phase modulation of the pupil function,” Opt. Lett. |

3. | S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. |

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

5. | G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express |

6. | N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A: Pure Appl. Opt. |

7. | S. Sanyal and A. Ghosh, “High focal depth with quasi-bifocus birefringent lens,” Appl. Opt. |

8. | Z. Zalevsky and S. Ben-Yaish, “Extended depth of focus imaging with birefringent plate,” Opt. Express |

9. | G. Frédéric, “Advances in camera phone picture quality,” Photonics Spectra, Nov. 2007, p. 50 (no archival literature references found) |

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

11. | H. Bartelt, J. Ojeda-Castañeda, and E. E. Sicre, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. |

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

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

14. | W. N. Charman and H. Whitefoot, “Pupil diameter and the depth-of-field of the human eye as measured by laser speckle,” Optica Acta |

15. | S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. |

16. | V. N. Mahajan, |

17. | P. Mouroulis and J. Macdonald, |

**OCIS Codes**

(080.3620) Geometric optics : Lens system design

(110.4850) Imaging systems : Optical transfer functions

(110.7348) Imaging systems : Wavefront encoding

**ToC Category:**

Geometric optics

**History**

Original Manuscript: June 4, 2008

Revised Manuscript: August 6, 2008

Manuscript Accepted: August 6, 2008

Published: August 11, 2008

**Virtual Issues**

Vol. 3, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Pantazis Mouroulis, "Depth of field extension with spherical optics," Opt. Express **16**, 12995-13004 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-17-12995

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### References

- W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749-754 (1960). [CrossRef]
- S. Bagheri and B. Javidi, "Extension of depth of field using amplitude and phase modulation of the pupil function," Opt. Lett. 33, 757-759 (2008). [CrossRef] [PubMed]
- S. S. Sherif, W. T. Cathey and E. R. Dowski, "Phase plate to extend the depth of field of incoherent hybrid imaging systems," Appl. Opt. 43, 2709-2721 (2004). [CrossRef] [PubMed]
- 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]
- G. Mikula, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, "Imaging with extended focal depth by means of lenses with radial and angular modulation," Opt. Express 15, 9184-9193 (2007), http://www.opticsexpress.org/abstract.cfm?uri=oe-15-15-9184. [CrossRef] [PubMed]
- N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003). [CrossRef]
- S. Sanyal and A. Ghosh, "High focal depth with quasi-bifocus birefringent lens," Appl. Opt. 39, 2321-2325 (2000). [CrossRef]
- Z. Zalevsky and S. Ben-Yaish, "Extended depth of focus imaging with birefringent plate," Opt. Express 15, 7204-7210 (2007), http://www.opticsexpress.org/abstract.cfm?uri=oe-15-12-7202. [CrossRef]
- G. Frédéric, "Advances in camera phone picture quality," Photonics Spectra, Nov. 2007, p. 50(no archival literature references found)
- E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
- H. Bartelt, J. Ojeda-Castañeda and E. E. Sicre, "Misfocus tolerance seen by simple inspection of the ambiguity function," Appl. Opt. 23, 2693-2696 (1984). [CrossRef] [PubMed]
- S. C. Tucker, W. T. Cathey and E. R. DowskiJr, "Extended depth of field and aberration control for inexpensive digital microscope systems," Opt. Express 4, 467-474 (1999). http://www.opticsexpress.org/abstract.cfm?uri=oe-4-11-467. [CrossRef] [PubMed]
- P. E. X. Silveira and R. Narayanswamy, "Signal-to-noise analysis of task-based imaging systems with defocus," Appl. Opt. 46, 2924-2934 (2006). [CrossRef]
- W. N. Charman and H. Whitefoot, "Pupil diameter and the depth-of-field of the human eye as measured by laser speckle," Optica Acta 24, 1211-1216 (1977). [CrossRef]
- S. Mezouari and A. R. Harvey, "Phase pupil functions for reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003). [CrossRef] [PubMed]
- V. N. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, Ch.2, SPIE Press, Bellingham, WA (2001).
- P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design, p. 324, Oxford University Press, New York/Oxford (1997).

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