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  • Vol. 3, Iss. 10 — Sep. 22, 2008
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Depth of field extension with spherical optics

Pantazis Mouroulis  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12995-13004 (2008)
http://dx.doi.org/10.1364/OE.16.012995


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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

Every technique has its own advantages, but in seeking an expanded depth of field for a wide field, low-power microscope system, a different approach was found beneficial. The approach follows the usual principle of pre-blurring the image in a controlled way that produces an extended DOF, and using digital restoration to recover contrast. However, it utilizes no special elements and instead allows the lens aberrations to perform the required pre-blurring function.

Successful image reconstruction from an extended DOF system requires certain conditions to be met. These are:

  1. 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.
  2. 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.
  3. 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.
  4. 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.

Claims of tenfold or greater DOF extension have been made in the literature. However, the method of calculating the extension is not uniform. Also, quantitative image reconstruction criteria are typically lacking, substituted by visual comparison of a single sample image. George and Chi6

6. 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 1/20th 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.

More recently, Bagheri and Javidi2

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]

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.13

13. 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.

Spherical aberration (SA) can be used to provide DOF extension. Charman and Whitefoot14

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 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 Harvey15

15. 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

Two low-power microscope systems are considered in this study. Both have a linear magnification of ~ -4x. The field of view has a 2.7 mm radius and the wavelength range is the visible band. The two systems differ in the object-space numerical aperture (0.167 and 0.08 respectively), allowing the exploration of two different resolution regimes. A suitable detector array is the Kodak KAF-8300 monochrome array, with approximately 3500×2500 pixels of 5.4 µm size, although the conclusions do not depend on the exact array specifications. The particular model is the closest suitable candidate for this high-resolution, wide field application and the field size was chosen to match its semi-diagonal. Color is to be provided with separate R, G, B illuminants. However, it is expected that all wavelengths will have the same range of focus and similar MTFs.

Two diffraction-limited optical designs that satisfy the above specifications are shown in Fig. 1. For both designs, the stop is placed at the rear surface. These designs avoid cemented interfaces, consistent with ultimate space applications.

Fig. 1. Two diffraction-limited low-power objectives, with NA=0.167 (top) and 0.08 (bottom) respectively. The image semi-diagonal shown is ~11 mm in both cases.

The optimum starting point for introducing spherical aberration in the desired manner has been found to be a nearly diffraction-limited, achromatic design. Once that design has been achieved, a different merit function is constructed that comprises operands giving the MTF values at various frequencies. These operands are used to keep the MTF above a certain level at each frequency and also to equalize the MTF across configurations (focus positions). This procedure is similar to that described by Sherif et al.3

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. 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®).

For a rotationally symmetric system, the only aberration that satisfies all the above conditions is spherical. Thus the optimization is forced to converge to a unique solution. Indeed an alternative way of optimizing without using the MTF is to demand that the system wavefront error should contain only primary and secondary SA terms in specified amounts. Faster convergence was found using the MTF, but this conclusion depends on the specifics of the optimization routine. An additional advantage of spherical aberration is that the phase transfer function (PTF) remains identically zero when the MTF is greater than zero.16

16. 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.

The systems resulting from this second optimization step are shown in Fig. 2. These have been obtained using the systems of Fig. 1 as starting points. For the slower system, it was not found necessary to change the optical glasses during re-optimization. For the faster system, it was found beneficial to change the glass of the last element.

Fig. 2. Re-optimized objectives with spherical aberration introduced. The Gaussian specifications are the same as those of the systems of Fig. 1. The spherical aberration is evident as ray aberration at the image plane.

Table 1 gives the prescription of the simpler objective, which, as explained below, suffices to verify all our conclusions. The object distance of 5.2 mm represents the middle of the extended DOF range.

3. Design assessment and DOF extension

δW20=0.5(NA)2δz
(1)

for a system in air.17

17. 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.

In terms of the Point Spread Function (PSF), the critical characteristic of these SA systems is that they retain a strong central lobe, which corresponds to the band of transmitted frequencies in the MTF. The PSF cross-sections for the three focus positions are shown in Fig. 5.

Table 1. Design prescription for SA objective #2

table-icon
View This Table
Fig. 3. MTF curves for the blue (486 nm) wavelength, at the distant extreme of focus for the two systems of Fig. 2. It can be seen that the curves show very little variation with field or orientation and are similar for the two systems after scaling the abscissa.

4. Image recovery

In this section, a quick visual appreciation of the extended DOF using a single image is provided. It should be evident that this is merely a visualization aid of little quantitative value. All the quantitative information is contained in the MTF curves of Fig. 4. Because the MTF curves of the two systems are basically the same except for the scaling of the frequency axis, the image recovery will also be the same but will correspond to different spatial frequencies. The images shown correspond to the NA=0.08 system sampled by a detector with 5.4 µm pixel size, as discussed in Sec.2. In this case, the Nyquist frequency of the detector is higher than the diffraction limit at a middle wavelength of 550 nm (~93 c/mm vs. 73 c/mm, for image space NA=0.02). Thus this is a very finely sampled system. The comparison is made with a real rather than ideal diffraction-limited system of the same specifications (the one shown in Fig. 1), which has a small amount of residual aberration (Strehl ratio >0.9). In Fig. 6, the object is the letter F and the ideal (geometric) height of the image is 50 µm. As expected, even the in-focus diffraction-limited image appears degraded because of the high frequencies involved. The fundamental frequency corresponding to the distance of the two horizontal limbs is ~40 c/mm, which is over half the maximum, diffraction-limited frequency.

Fig. 4. Axial MTF curves for the NA=0.167 system at three through focus positions and two wavelengths. The green wavelength shows an intermediate behavior between the two extremes. Also, the full-field MTFs are very similar to the corresponding axial ones.
Fig. 5. Point spread functions for three focus positions and 586 nm wavelength, for the second SA system of NA=0.08.

Since the location of an object within the DOF range is unknown in principle, the processed images are obtained by using a single blurring function corresponding to the mean of the three PSFs. A maximum-likelihood criterion is used, which makes no a priori assumptions about the form of the image or the noise level. It is probable that alternative image processing techniques can result in a better reconstruction. However their value would have to be proven in a general way, or alternatively be shown to be well-suited to a specific application.

Fig. 6. Through focus images for the diffraction-limited system, and the SA system (raw and processed). The DOF range is 300 µm, shown on the right. The box size is approximately 200 µm. All images are normalized to a maximum intensity of one. The bottom two images show the ideal geometric image without aberrations or diffraction, and the same image re-sampled to the detector pitch.

5. Comparison with other techniques

As has already been pointed out,2

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]

,13

13. 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.

Fig. 7. MTF curves of a wavefront-coded system corresponding to the NA=0.167 lens for the center of the field only. The graph on the left shows three curves, one for each focus position, corresponding to the x (or y) direction of the phase function. The graph on the right shows curves corresponding to the directions at ±45°, labeled as T and S. Notice that the inherent asymmetry of the phase function means that different MTF curves are obtained along the two ±45° orientations even for the on-axis position.

Next, we compare with an amplitude modulated (obscured) system. Note that acceptable image reconstruction has been achieved for the SA systems with a defocus of ±0.88 λ. Using the formula provided in ref. 2

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]

, this corresponds to a linear obscuration of δ=0.87. If this obscuration is applied to the SA system with NA=0.167, then the lens becomes equivalent to an unobscured one of NA=0.082 in terms of collecting aperture. This is practically the same as the second, slower SA system examined here. Welford1 observed that the obscured system basically achieves the same DOF as an unobscured one of the same collecting aperture, while retaining the resolution of the larger aperture. The difference in the present case is that the introduction of spherical aberration produces additional increase in DOF, albeit at further expense of resolution. Thus the obscured system produces the DOF of the first SA system while retaining its full resolution, but having the equivalent light collection of the second SA system. Or, we may say that the second SA system produces approximately 8× greater DOF than the obscured system but at one-fourth the resolution. Therefore, this investigation opens up the complete trade space for the designer, between resolution, DOF extension, and light collection. It should, however, be noted that this comparison is complicated by the fact that a highly obscured system produces rather low MTF values, such as shown in Fig. 8. In practice, it was found that a more modest obscuration of δ=~0.81 produced a higher MTF while retaining the resolution over the same extended DOF of ±35 µm.

Fig. 8. Typical MTF curves for an annular aperture system with δ=0.86 and 0.81 linear obscuration ratio. These curves have been obtained using the NA=0.167 diffraction-limited system of Fig. 1. The legend shows the value of δ, and also the location within the DOF. Only the on-axis field point is shown, for a wavelength of 586 nm. Other locations, fields and wavelengths are omitted for clarity, but show similar behavior. The value of the lowermost curve at 80 c/mm is ~0.025.

A comment about the logarithmic asphere design follows. The logarithmic asphere is based on the principle of a smoothly variable focus distance as a function of pupil radius. Seen from the aberration viewpoint, this is exactly the effect of spherical aberration. Indeed, a rotationally symmetric surface can introduce only rotationally symmetric phase terms on axis. Therefore, the wavefront emanating from the logarithmic asphere can be approximated to an arbitrary degree of accuracy through rotationally symmetric Zernike polynomial terms, which represent spherical aberration of various orders. By using a Zernike polynomial term optimization with a variable phase surface, it was found that the third and fifth order (fourth and sixth in pupil coordinates) spherical aberration terms are dominant, with higher orders having negligible contribution. For systems of low to moderate numerical aperture, it is those low order terms that are introduced when the optimization variables are the curvatures and separations of spherical elements instead of the strength of the Zernike polynomial coefficients of a phase surface. Thus it is expected that a logarithmic asphere design, optimized over the same range of focus and other conditions as the present designs, will end up producing a similar amount of spherical aberration and similar performance as the systems shown here.

Finally, it is noted that the design was made to be achromatic, with the DOF range independent of wavelength. This is the exact opposite of the idea of using longitudinal chromatic aberration for DOF extension. There is nothing to prevent the designer from attempting to gain even greater DOF by using chromatic aberration while following the basic design technique described previously. However, the possibility of introducing chromatic artifacts would have to be carefully scrutinized, depending on the application.

6. Conclusions

Acknowledgments

This research has been performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. I wish to thank Sven Geier for generating the processed images and Holly Bender for discussions that led to the idea of removing the special phase elements.

References and links

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.

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]

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

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 15, 9184–9193 (2007) , http://www.opticsexpress.org/abstract.cfm?uri=oe-15-15-9184. [CrossRef] [PubMed]

6.

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

7.

S. Sanyal and A. Ghosh, “High focal depth with quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000). [CrossRef]

8.

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]

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. 34, 1859–1866 (1995). [CrossRef] [PubMed]

11.

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]

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 4, 467–474 (1999). http://www.opticsexpress.org/abstract.cfm?uri=oe-4-11-467. [CrossRef] [PubMed]

13.

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]

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 24, 1211–1216 (1977). [CrossRef]

15.

S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28, 771–773 (2003). [CrossRef] [PubMed]

16.

V. N. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, Ch.2, SPIE Press, Bellingham, WA (2001).

17.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design, p. 324, Oxford University Press, New York/Oxford (1997).

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

  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. 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]
  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. 44, 2792-2798 (2005). [CrossRef] [PubMed]
  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 15, 9184-9193 (2007), http://www.opticsexpress.org/abstract.cfm?uri=oe-15-15-9184. [CrossRef] [PubMed]
  6. N. George and W. Chi, "Extended depth of field using a logarithmic asphere," J. Opt. A: Pure Appl. Opt. 5, S157-S163 (2003). [CrossRef]
  7. S. Sanyal and A. Ghosh, "High focal depth with quasi-bifocus birefringent lens," Appl. Opt. 39, 2321-2325 (2000). [CrossRef]
  8. 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]
  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. 34, 1859-1866 (1995). [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. 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]
  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 24, 1211-1216 (1977). [CrossRef]
  15. S. Mezouari and A. R. Harvey, "Phase pupil functions for reduction of defocus and spherical aberrations," Opt. Lett. 28, 771-773 (2003). [CrossRef] [PubMed]
  16. V. N. Mahajan, Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, Ch.2, SPIE Press, Bellingham, WA (2001).
  17. P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design, p. 324, Oxford University Press, New York/Oxford (1997).

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