## 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

In recent years, a lot of attention has been paid to the problem of extending the depth of focus or field (DOF). Many different techniques have been explored, from varying the pupil intensity transmission,1,2 to adding special phase elements,^{3–5} special aspheric surfaces,^{6} utilizing birefringence,^{7,8} or utilizing longitudinal chromatic aberration.^{9}

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:

- 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.

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 Chi^{6} 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.

Dowski and Cathey^{10} who introduced the term “wavefront coding” used a version of the Hopkins defocus criterion through the ambiguity function introduced in ref 11, which refers to the mid-frequency of a system with rectangular aperture. As given in ref. 11, the Hopkins defocus criterion corresponds to a wavefront defocus *W*
_{20}=λ/6. Order of magnitude improvement was claimed for the wavefront coding system (with typically cubic phase function), which would imply that the extended DOF corresponds to *W*
_{20}≈1.6 λ^{~}. In a related publication,^{12} a 10× extension was claimed based on visual assessment of images, but no details were given regarding the detector pixel size or system F-number, so it is impossible to quantify the claim in a comparative way. However, the authors did note that DOF extension in systems with high resolution remains a challenge.

More recently, Bagheri and Javidi^{2} 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} 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 Whitefoot^{14} 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^{15} 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.

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}, 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} 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.

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

To assess DOF extension we will use the *W*
_{20} wavefront defocus coefficient. A change in focus of *δW*
_{20} is related to the numerical aperture NA and the corresponding (small) axial displacement *δz* via

for a system in air.^{17} 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.

## 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.

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.

The images presented in this section are monochromatic. As Fig. 4 shows, there is little difference between the MTF curves for different wavelengths and for this reason polychromatic image reconstruction would in principle be equally successful. However, for an off-axis field point, there is an additional consideration that is not related to DOF extension. In that case, successful polychromatic image reconstruction at high resolution requires also the diffraction limited SA raw SA processed absence of transverse chromatic aberration. This additional condition may be difficult to satisfy, depending on the design. It therefore may be preferable to synthesize a polychromatic image by acquiring three monochromatic images which are then processed individually.

## 5. Comparison with other techniques

As has already been pointed out,^{2,13} 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.

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, 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.

Comparison may also be made in terms of fabrication tolerances between the three techniques considered thus far. A tolerance analysis has revealed the following. 1) The wavefront-coded system must be built to the same standards as a nominally diffraction-limited system, otherwise the residual wavefront error may combine with that of the phase plate to produce unacceptable MTF curves. On the surface this conclusion appears to contradict those of previous references that suggest relaxed tolerances for the wavefront-coded system, but in fact it is a direct consequence of the higher resolution regime that is of concern here. When the MTF must remain high over an extended frequency range, any small additional amount of aberration can prove detrimental. 2) The obscured system must satisfy even tighter tolerances than a nominally diffraction-limited system, again as a consequence of the low MTF values it produces. 3) The SA system has only slightly reduced tolerance sensitivity over a corresponding diffraction-limited system. Even though looser tolerances might have been expected in this case, there remains the need to minimize off-axis aberrations and allow spherical aberration to be the dominant factor through the entire field.

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

Low-power microscope designs have been presented that achieve a depth of field extension equivalent to ~ ±0.88 λ while retaining resolution up to one-half the diffraction-limited frequency. These can be realized without the help of special phase plates or other elements by simply introducing spherical aberration in a controlled way, using as a starting point a well-corrected design. The technique compares well with previous ones and represents a useful alternative.

## 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.

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