Intraocular lenses (IOLs) are widely used to restore the eye’s refractive power in case of loss of the eye lens. Patients whose crystalline lens has been removed after surgery need to compensate for its base focal power (~20 diopters [D]) and to ensure the best possible quality of foveal imaging for objects located at finite distances from the eye, usually within a range of vergences spanning about 3.5 D, coping with the lack of accommodation. This goal can be met in a variety of ways, among them, using a monofocal IOL combined with progressive addition spectacles or using bifocal or multifocal IOLs. Most IOLs are designed to have one or several focal regions conveniently distributed within the addition range. However, using elements with a set of discrete foci is not the only possible strategy of design: the use of elements with extended depth of focus (EDOF), that is, elements producing a continuous focal segment spanning the required addition, can also be considered. Axicons, for instance, are circularly symmetric EDOF elements that have been proposed as suitable candidates for presbyopia compensation.^{1}

Rotational symmetry, in turn, is not a necessary constraint when it comes to designing and manufacturing EDOF elements for the presbyopic eye. In the last years, some interest has been devoted to the enhanced depth-of-field imaging performance of nonrotationally symmetric elements like the light sword optical element (LSOE).^{2–5} The refractive power of the LSOE does not vary radially like in the axicons, but angularly, which gives rise to some interesting imaging features, including its robustness against changes in pupil size^{6} and the homogeneity of its angularly averaged spatial frequency response over a sizable fraction of the addition range.

Preliminary studies on the suitability of axicons and LSOEs for presbyopia compensation were based on the analysis of the retinal image quality attainable with these elements. This analysis only takes into account the effects of the optical transfer function (OTF)^{7} of the eye, without regard to perceptual aspects. In this work, we extend the previous work by studying the behavior of the quartic axicon and the LSOE using the visual Strehl ratio (SR) computed in the spatial frequency domain (VSOTF),^{8} an objective metric of optical quality that takes into account not only the OTF but also the effects of the neural contrast sensitivity function and that has been shown to correlate noticeably well with the subjective visual acuity.^{9}

The structure of this article is as follows. In section 2, we describe the dioptric power distribution of four types of IOLs (monofocal, bifocal, quartic axicon, and the LSOE) and briefly analyze their optical behavior in terms of their angularly averaged modulation transfer functions (MTFs). Section 3 deals with the definition and meaning of two visually inspired objective metrics of optical quality, namely, the VSOTF described by Thibos et al.,^{8} with the improvement suggested by Iskander,^{10} and the compound modulation transfer function (CMTF) proposed by Dai.^{11} In section 4, we describe the behavior of these four elements in terms of the VSOTF and the CMTF across the presbyopic addition range. Discussion and conclusions are given in section 5.

#### METHODS

##### Eye Model and IOL Power Maps

To compare the performance of the different IOL designs, we model the aphakic eye as a lens of power 40 D within a homogeneous medium, with photopic pupil radius *R* = 2.2 mm. That radius value was chosen according to the study of the pupil size and age disclosed by Winn et al.^{12} The IOLs under study have to provide the remaining 20 D of base power plus the required addition. The eye is considered to be aberration-free, and the image quality is studied within the foveal region, assuming paraxiality and isoplanaticity (Fig. 1).

When light from an on-axis point object with vergence *PObj* comes in on an aphakic eye fitted with an IOL, the field *u*(*r*, *u*) at the exit pupil plane will be given by:

where (*r*, *&thetas;*) are the polar coordinates of points in the eye pupil plane, circ(*r*/*R*) is the eye’s aperture function, valued 1 for *r* < *R* and zero otherwise, *Pe* = 40 D is the dioptric power of the aphakic eye model, *PIOL* (*r*, *&thetas;*) is the local power distribution of the IOL, and λ is the wavelength.

The four solutions analyzed in this work correspond to the monofocal and bifocal IOLs, the quartic axicon, and the LSOE, all with base power *PBase* = 20 D and the last three with addition *PAdd* = 3.5 D. The *PIOL* (*r*, *&thetas;*) power maps are depicted in Fig. 2. Their analytic forms are given by:

Note that the first three power maps, unlike the LSOE one, are rotationally symmetric so that they have no actual dependence on *θ*. The monofocal lens is characterized by a pure quadratic phase: although many commercially available IOLs include in their designs some amount of spherical aberration,^{13} this is intended to compensate for the corneal spherical aberration, so that its net effect is simply attaining an unaberrated eye. The bifocal lens has been modeled by way of example as the IOL AMO Array S40L with rings of alternate powers of 20 and 23.5 D.^{14} The radii defining these rings are the parameters *r*_{1}, *r*_{2}, *r*_{3}, and *r*_{4} of equation 3, which, in our example, correspond to 1.05, 1.7, 1.95, and 2.3 mm, respectively. The quartic axicon has been described^{1} where its imaging performance was analyzed to assess its suitability for presbyopia compensation. The design and basic optical performance of the LSOE have been studied and described.^{2–6} Unlike the other elements, the LSOE has a transmittance function without symmetry of revolution. This translates into an asymmetry on the MTF of the overall imaging system composed of the aphakic eye + LSOE, whose particular form depends on the object vergence (Fig. 3). To compare the performance of the different elements, a unidimensional version of the MTF was obtained for each case by angularly averaging the radial profiles of the corresponding two-dimensional MTF (Fig. 4).

Figure 3 Image Tools |
Figure 4 Image Tools |

##### Visually Related Optical Quality Metrics

##### The VSOTF and the CMTF

The problem of finding an optical quality metric that correlates well with the subjective visual acuity has been a subject of intense research in the last years.^{8,9,15} The interest in such kind of metrics is manifold. For our purposes, they are useful because they allow to compare quantitatively—at least to a first approximation—the visual performance attainable with different designs of optical aids without resorting to the realization of extensive clinical trials. Some common metrics of optical quality—like the widely used SR—do not correlate well with visual performance, although they can serve for making a first assessment of the suitability of any optical element for eye correction.^{9} This weak correlation is not particularly surprising because these metrics are intended to describe the imaging performance of the optical system of the eye but do not take into account the perceptual aspects tied to brain processing. These aspects can be accounted for—at least partially—by including in the metrics the contribution of the neural contrast sensitivity function CSF_{N}^{16,17} plotted in Fig. 5. Analyzing the correlation of 31 single-value metrics of optical quality to visual acuity (for rms wavefront errors of about λ/2 over 6-mm pupils), Marsack et al.^{9} concluded that the best one (accounting for 81% of the variance in high-contrast logMAR acuity) was the VSOTF. The original definition of this metric showed some limitations and consistency problems pointed out by Iskander,^{10} who proposed an enhanced version given by:

where the dependence of the OTF on the object vergence (*PObj*) is explicitly indicated, OTF_{IOL} is the OTF of the system formed by the IOL and the aphakic eye and OTF_{DL} is the OTF of the diffraction-limited emmetropic eye for a point object located at infinity (*PObj* = 0).

We have also calculated for the four elements of the CMTF proposed by Dai.^{11} The CMTF is a weighted average of the MTF at several frequency values deemed relevant for the visual process, namely, 10, 20, and 30 cycles per degree. These frequencies are indicated by dashed lines in Fig. 4 and are broadly associated with periodic structures detectable with visual acuities of 20/60, 20/30, and 20/20, respectively. Note that the CMTF by Dai^{11} tends to measure the optical performance in the high spatial frequencies (>10 cycles per degree), not including the mid frequency range at which the CFS peaks. In this parameter, the MTF values of each element (MTF_{IOL}) are normalized to the MTF values of the diffraction limited emmetropic eye (MTF_{DL}) for *PObj* = 0. Formally, the CMTF (dependent for each element and spatial frequency on the object vergence) is defined as:

where *N* is the number of spatial frequencies *fn* included in the average (in our case, *N* = 3). Because the MTF_{IOL} of the LSOE does not possess symmetry of revolution, the values used for computing equation 7 are its angular averages in circles located at distances *fn* from the origin of the MTF reference frame (Fig. 6).

For completeness, we computed the values of the classical SR associated with each element as a function of the object’s vergence value defined by:

where PSF_{IOL} (P_{Obj}) is the point spread function (PSF) of the aphakic eye plus IOL, with an object point at a vergence *PObj*, and PSF_{DL} is the PSF of the diffraction-limited emmetropic eye for *PObj* = 0. As we have previously pointed out, the SR correlation with the visual acuity is not especially high (R^{2} = 0.55, by Marsack et al.^{9}). Moreover, a recent study on 24 eyes using a double pass system with infrared illumination evaluated the repeatability of the SR measurements, obtaining an estimated SD equivalent to 15.79%.^{18} It is, however, a common and easily recognizable metric of optical quality, which—complementing the angularly averaged MTFs shown in Fig. 4—provides an additional insight on the optical behavior of the different solutions analyzed in this work.^{19}

#### RESULTS

The optical field given by equation 1 with object’s vergences *PObj* ranging from 0 to 3.5 D was computed for each of the elements with the *PIOL* (*r*, *&thetas;*) given by equations 2 to 5 and λ = 555 nm and subsequently propagated to the eye’s retina using an efficient Fresnel transform–based algorithm.^{20} The eye’s incoherent PSF was obtained as the squared modulus of the resulting retinal field; the OTF was computed as the Fourier transform of that PSF and the MTF as the modulus of the OTF.^{7} From these functions (and the CSF_{N} in Fig. 5, when applicable), the VSOTF, the CMTF, and the SR were calculated.

The results are plotted in Fig. 7. As expected, the monofocal and bifocal solutions show higher values of the quality metrics at the precise object vergences for which they are designed and optimized (the 0 D and the 0 and 3.5 D, respectively). Not surprisingly, the quartic axicon and the LSOE show smaller performance at these points (although not significantly smaller in comparison with the bifocal lens). However, they provide very reasonable values for these metrics throughout the whole addition range. The relative decrease in performance of both EDOF elements at the end points of the addition range can easily be overcome by slightly expanding their focal segments in the design step. Interestingly enough, the behavior of the LSOE is noticeably more uniform than that of the axicon, with a smoother dependence on the object’s vergence.

#### DISCUSSION

The results in Fig. 7 support the idea that EDOF elements like the axicons and LSOEs can be interesting options to be used as IOLs to restore the imaging ability of the aphakic eye. Unlike the monofocal or bifocal solutions, their designs seek to obtain a reasonable degree of focusing across the object vergence range, providing a consistent level of vision. This increased uniformity in the response to different object vergences comes at the expense of a slightly smaller performance at certain particular points in comparison with that of the elements specifically designed to image them. This is true in particular of bifocal lenses that have been extensively studied^{21–24} and whose use has been generally rated as satisfactory by many users.

In comparison with the quartic axicon, the LSOE presents two main advantages: its behavior is more uniform against changes in the object vergence and its angular power distribution makes it relatively insensitive to variations in the observer’s pupil size.^{6} The phase discontinuity at *θ* = 2π of the LSOE power map (equation 5) can be avoided by slightly redesigning the *PIOL_LSOE* (*r*, *&thetas;*) transmittance function to ensure a smooth refractive profile without jumps in its height or its derivatives, for example, using a 2π-periodic dependence on *&thetas;* instead of a linear one. The amount of higher order aberrations introduced by the LSOE design and their relative visual significance compared with those introduced by the other elements is a subject requiring further quantitative studies. Additional research is also needed to quantify the effects that small misalignments in IOL placement can have on the final image quality. However, preliminary visual observations through LSOE manufactured by photosculpture of photoresist^{5,25} did not show any relevant amount of stray light.

The VSOTF has proved to be a reliable image plane metric of neuro-optical quality in a wide variety of situations,^{8} for example, the accurate prediction of the changes in visual acuity produced when selected high-order aberrations are introduced to the eye^{9,15} and the prediction of the lens power that maximizes the visual acuity in a thorough-focus experiment.^{9} A word of caution must, however, be made regarding the use of the VSOTF to assess the relative merits of different solutions for presbyopia compensation when nonrotationally symmetric elements are present. In its original form,^{8} the VSOTF definition was equivalent to the visual SR in the image domain evaluated using not the maximum of the visual PSF but its value at the center of the optical axis. Because the LSOE peak is slightly off center, the classical VSOTF will underestimate the performance of this element. The modified VSOTF proposed by Iskander,^{10} equation 6, could in principle be affected by the same bias. Because our purpose is the relative intercomparison of different solutions and the LSOE generally performs better than the axicon, this underestimation of performance would not alter—it would rather reinforce—the general trends shown in our study.

In conclusion, we have studied the performance of two kinds of EDOF elements (quartic axicon and LSOE) using visually oriented (also called neuro-optical) objective quality metrics that take into account the neural contrast sensitivity function. The angularly averaged VSOTF and CMTF of the LSOE outperform those of the quartic axicon in terms of stability against changes in the object vergence. Both kinds of EDOF elements provide better optical quality than bifocals throughout the whole addition range, e at the end points, where they are surpassed (however slightly) by the bifocal.

Augusto Arias Gallego

Center for Research

Instituto Tecnológico Metropolitano

A.A. 54954 Medellín

e-mail: augustoariasgallego@gmail.com

##### ACKNOWLEDGMENTS

*Supported by the Spanish Ministry of Science and Innovation (MICINN) under grant FIS2008-03884 and by the Polish Ministry of Science and Higher Education under grant NN 514149038.* *The authors thank José M. Gómez Ojeda, Laura Remón Martín, and the anonymous reviewers for their useful suggestions.*

*Received January 26, 2012; accepted August 15, 2012.*