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 CSFN16,17 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, OTFIOL is the OTF of the system formed by the IOL and the aphakic eye and OTFDL 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 Dai11 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 (MTFIOL) are normalized to the MTF values of the diffraction limited emmetropic eye (MTFDL) 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 MTFIOL 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 PSFIOL (PObj) is the point spread function (PSF) of the aphakic eye plus IOL, with an object point at a vergence PObj, and PSFDL 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 (R2 = 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
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 CSFN 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.
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 studied21–24 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 photoresist5,25 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 eye9,15 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
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.
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Keywords:© 2012 American Academy of Optometry
vision; presbyopia; intraocular lens; extended depth of focus