1. Introduction

Various strategies have been employed to address presbyopia, including the incorporation of presbyopia-correcting intraocular lenses (IOLs) during cataract surgery. The continuous advancement of these premium IOLs has been fueled by the growing demand for optimal visual quality at all distances. Diffractive multifocal IOLs enhance visual quality across multiple discrete foci but may lead to undesired photic phenomena such as halos, glare, and starbursts [1,2]. In recent times, refractive extended depth of focus (EDoF) IOLs have gained prominence for delivering continuous optical quality from far to intermediate distances while mitigating photic phenomena [1,3–6]. These EDoF IOLs have garnered attention, particularly as intermediate vision has become vital due to the prevalence of digital electronics and display devices. Typically, refractive EDoF designs are based on relatively smooth wavefront profiles, effectively minimizing the significant degradation of image quality between foci and reducing dysphotopsia [1,7].

The proprietary nature of EDoF designs limits our access to detailed optical profiles of EDoF IOLs. The optical performance of EDoF IOLs is commonly assessed by measuring Zernike (modal) spherical aberrations (SAs) using commercially available Shack–Hartmann wavefront sensors (SHWFS) [8–12]. Spot displacements measured with SHWFSs are typically subjected to modal wavefront reconstruction, where Zernike polynomials are fitted to the wavefront slope data [13]. This modal method tends to smooth irregularities caused by abrupt local slope changes in the wavefront profiles. In contrast, zonal wavefront reconstruction can retain most features of the slope data, allowing the reconstructed wavefront to avoid the loss of profile details caused by Zernike fitting. The zonal reconstruction algorithm may be a suitable method to capture detailed local slope changes in the IOL profiles. Most commercial ocular SHWFSs [14] do not provide the zonal wavefront profile and can indicate up to 6^th- or 8^th-order Zernike coefficients. Meanwhile, commercial IOL metrology devices based on SHWFSs [10,11,15] can present the zonal wavefront as well as up to 10^th-order Zernike coefficients. However, the zonal wavefronts of EDoF IOLs have not been meticulously described. The objective of the present study was to assess the reliability of the zonal method in representing IOL profiles and compare it with the Zernike-based modal method up to the 16^th order using a high-resolution SHWFS (26.6 µm). The findings of this study can impact clinical practice by enhancing our understanding of the optical characteristics of EDoF IOLs.

2. Materials and methods

2.1 High-resolution SHWFS

The sampling resolution of a SHWFS is constrained by the lenslet size, as higher spatial frequency features in a wavefront are averaged within each lenslet and cannot be reliably measured. While the most straightforward approach to increasing sampling resolution is to reduce the lenslet size, the pitch of commercially available lenslet arrays is limited by manufacturability. Instead, we opted for magnifying the pupil aperture using a 4-f telescope. Figure 1 illustrates the high-resolution SHWFS designed with 5× pupil magnification, a 1-inch diameter lenslet array (Adaptive Optics Associates 0133S, pitch: 133 µm), and a large CMOS sensor (24 by 36 mm, Canon EOS RP, full-frame mirrorless camera), resulting in a 26.6 µm sampling resolution. A fiber-coupled handheld laser source (HLS635, Thorlabs, Newton, NJ, USA) served as the light source and was transmitted through the IOLs, which were submerged in a wet chamber filled with balanced salt solution (BSS). Each IOL functioned as a collimator, and the absolute IOL powers were neither measured nor corrected. The spot array pattern captured by the sensor comprised 150 spots across the diameter of the 4 mm optical zone and 17,433 spots within the entire area. The intensity-weighted centroiding method was adopted for the centroiding algorithm [16].

 

Fig. 1. Schematic of the high-resolution Shack–Hartmann wavefront sensor. BSS, balanced salt solution; IOL, intraocular lens.

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To address system aberration and straylight (background noise), the wet cell containing the IOL was substituted with an achromatic doublet (50 mm focal length, 20D, matching the IOLs), and 20 spot array patterns were acquired, averaged, and then subtracted from each measurement. To mitigate photon and readout noise, as well as intensity variations induced by the light source, each IOL measurement was repeated 20 times and subsequently averaged. Upon inserting the IOLs into the wet chamber, occurrences of IOL glistening and scatter resulting from multiple microvacuoles were inevitable. The light scatter induced by IOL glistening was validated using the Mie theory [17], and it might have influenced our high-spatial-frequency wavefront measurements. In an effort to minimize the impact of IOL glistening, we endeavored to reduce the duration of IOL exposure to water [18,19] and maintained a constant temperature environment [20,21] (room temperature under an air conditioner).

2.2 Performance validation for the high-resolution SHWFS

We conducted measurements on the Vivity IOL using a commercially available wavefront sensor (SID4, Phasics, France) to assess the performance of our high-resolution SHWFS. This wavefront sensor employs quadri-wave lateral shearing interferometry (QWLSI), a quantitative phase imaging technique [22] widely used in various applications, including the characterization of IOL phase profiles [23]. The manufacturer indicates a 29.6 µm sampling resolution for this sensor, similar to our SHWFS. In this QWLSI measurement, 1:1 pupil magnification was employed. Figures 2(A) and 2(B) illustrate wavefront maps of the Vivity IOL from the SHWFS and QWLSI, respectively, revealing an overall similarity in wavefront heights. Figure 2(C) provides a comparison of the cross-sectional profiles at the center of the IOL between the two wavefront sensors, demonstrating good agreement, although our SHWFS wavefront map exhibits a relatively rough profile due to centroiding noise.

 

Fig. 2. Wavefront maps of Vivity IOL from (A) the high-resolution Shack–Hartmann wavefront sensor (SHWFS) and (B) quadri-wave lateral shearing interferometry (QWLSI). (C) Comparison in cross-sectional profiles of the wavefront maps from the two wavefront sensors.

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2.3 Wavefront reconstruction: modal vs. zonal reconstruction

In the modal wavefront reconstruction method, Zernike polynomials were fitted to the wavefront slope data. In the zonal method, the measured phase slopes (differences) must be converted into absolute wavefront phases. Specifically, linear least-squares equations were employed for each square array of phase differences, with phase estimates at the corners [24]. For very large arrays, as in our case, the numerical complexity (convergence problem or computation speed) is a critical consideration in subroutines. To alleviate the numerical calculation burden, Southwell introduced a matrix iterative technique [25], while Roddier proposed a novel algorithm based on iterative fast Fourier transforms [26]. The two algorithms yielded comparable performance for our application, and the latter method was employed in the present study due to its higher computation speed.

The reconstructed wavefronts obtained through both the zonal and modal reconstruction methods were analyzed for a 4 mm IOL diameter. The modal method computed Zernike coefficients up to the 16^th order in accordance with the OSA convention [27]. With rotational symmetry being a common feature in each IOL design, the various orders of SAs were the only Zernike coefficients exhibiting significant magnitudes. Since the zonal wavefront furnishes the most wavefront details, it served as a reference for evaluating errors introduced by the modal method. To gauge the extent of spatial information lost with the modal method, the difference wavefront error maps between the zonal and modal methods were scrutinized to compute the root mean squared error (RMSE) for each of the four EDoF IOLs.

2.4 Estimation of through-focus retinal image quality of IOL

To evaluate the through-focus retinal image quality of each intraocular lens (IOL), we computed the through-focus area under the modulation transfer function (areaMTF) over a range of object vergences, from -3D (near) to +1 D (beyond infinity). This calculation was based on the measured zonal wavefronts and various Zernike orders of the SAs without a model cornea. The areaMTF, computed for white light, describes the area under the radially averaged modulation transfer function across spatial frequencies [28,29] ranging from 0 to 60 cycles/degree [30,31]. This range represents the spatial resolution limit of the human fovea [32]. To assess through-focus performance, we also calculated and compared the areaMTF for a monofocal IOL profile with -0.27 μm of 4^th-order SA for a 6 mm diameter.

2.5 Intraocular lenses

We measured four refractive EDoF IOLs (Vivity, Lucidis, Eyhance, and Isopure) with a nominal power of +20.00 D for a 6 mm optical diameter. The AcrySof IQ Vivity DFT015 (Alcon, Fort Worth, TX, USA) comprises two zones: a central zone (approximately 2.2 mm in diameter) and a peripheral ring with different aspheric profiles [33]. The two phase-transition elements—a protruding plateau in the central zone and the peripheral ring—work synergistically to produce a continuously extended depth of focus [34]. Lucidis (Swiss Advanced Vision, Switzerland) involves a central aspheric zone surrounded by a peripheral ring; the central aspheric zone (about 1 mm in diameter) serves as an axicon, producing a Bessel beam that generates multifocality for a continuous depth of focus to a near distance [35]. The Tecnis Eyhance ICB00, developed by Johnson & Johnson Vision in Jacksonville, FL, USA, features a seamlessly graduated power profile extending from the periphery to the center, eliminating the presence of a demarcation line. In contrast, the Isopure IOL from BVI in Waltham, MA, USA, is distinguished by its sleek aspheric surfaces on both sides, incorporating aspheric coefficients up to the 10th order. Notably, Eyhance and Isopure share analogous optical characteristics, presenting a fluid transition in power across the aperture and a central zone with heightened power to enhance intermediate vision. [36–38].

2.6 Correlation with results of published optical bench studies

We conducted a comparative analysis with recent optical bench studies to verify the reliability of our zonal wavefront measurements. Madrid-Costa et al. [39] utilized the areaMTF with a spatial frequency range of 0–50 cycles/mm and reported a bimodal through-focus curve for Vivity, while Vega et al. [40] presented a unimodal through-focus curve for Eyhance. These studies employed artificial corneas with +0.135 µm (5.15 mm diameter) and +0.175 µm (5 mm diameter) of SAs, respectively. Given that our wavefront measurements were conducted without an artificial cornea, we developed a pseudophakic model eye with a model cornea based on the Liou–Brennan model eye [41] using Zemax OpticStudio raytracing software (Ansys, PA, USA). The anterior corneal surface of the model eye was adjusted to match the corneal SAs reported in each of the previous studies. Both Madrid-Costa et al. and Vega et al. also reported IOL diameters of 3 mm, and 4.5 mm, and we adopted their 3 mm diameter data for comparison. As our zonal and modal wavefronts were obtained with a 4 mm diameter, we truncated them to a 3 mm diameter and calculated the areaMTF from the truncated wavefronts. To ensure consistency in units, we utilized a spatial frequency range from 0–15 cycles/degree for calculating the areaMTF from our measured wavefronts.

3. Results

3.1 Reconstructed wavefronts with zonal and modal methods

Figure 3 illustrates the magnitudes of Zernike SA coefficients ranging from the 4^th to the 16^th order, measured within a 4 mm diameter. All IOLs exhibited negative 4^th-order SA, with Isopure and Lucidis displaying the largest (-0.34 µm) and smallest (-0.07 µm) magnitudes, respectively. Additionally, all IOLs manifested 6^th- and 8^th-order SAs, with Isopure displaying exclusively negative SAs. Vivity and Lucidis exhibited higher-order SAs, extending up to the 16^th order.

 

Fig. 3. Bar graph presenting the magnitude of Zernike spherical aberration coefficients from 4^th- to 16^th-order measured with four refractive intraocular lenses within 4 mm diameter.

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In Fig. 4, the wavefront maps reconstructed by the zonal method (leftmost column) and the modal method, encompassing Zernike coefficients from the 4^th to the 16^th order, are presented within a 4 mm diameter. Difference maps between the zonal and modal wavefronts are provided below each modal wavefront map. All defocuses in each wavefront map were corrected using the least-squares method. Generally, the modal wavefront maps appeared to exhibit similarity to the zonal wavefront maps up to the 10^th-order Zernike coefficients. However, the wavefront difference maps revealed significant disparities, particularly up to the 16^th order, with Vivity and Lucidis. Furthermore, for Eyhance and Isopure, none of the modal wavefront maps showed substantial changes, even up to the 16^th order.

 

Fig. 4. Wavefront maps were generated through both zonal and modal reconstruction methods, covering Zernike coefficients from the 4^th to the 16^th order. These reconstructions were conducted for four refractive intraocular lenses within a 4 mm diameter. Difference maps, illustrating variances between the zonal and modal wavefronts, are presented below each modal wavefront map. Consistent color scales were applied across all maps for uniformity.

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More quantitatively, Fig. 5 illustrates the wavefront differences between the zonal and modal methods, expressed in terms of the root mean squared error (RMSE). Up to the 8^th order, Vivity and Lucidis exhibited larger RMSE compared to Eyhance and Isopure. However, extending the Zernike fit to the 10^th order resulted in a reduction of RMSE for Lucidis, while the negligibly small RMSE for Eyhance and Isopure remained unchanged relative to the measurement noise floor. Even when using up to 16^th-order polynomials for Vivity and Lucidis, the RMSE persisted at levels higher than the measurement noise floor.

 

Fig. 5. The wavefront difference between the zonal and modal reconstruction is depicted through the root mean squared error (RMSE) across Zernike coefficients from the 4th to the 16th order, using four refractive intraocular lenses within a 4 mm diameter. The dashed line indicates the measurement noise floor.

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3.2 Through-focus retinal image quality estimation from reconstructed wavefronts

Figure 6 illustrates the simulated through-focus retinal image quality represented by the areaMTF, derived from the wavefronts reconstructed by the zonal and modal methods under white light conditions. Evaluating the through-focus retinal image quality from the zonal method, all three refractive IOLs, except Lucidis, exhibited EDoF characteristics spanning from far to intermediate distances. Lucidis, on the other hand, displayed through-focus performance akin to that of a monofocal IOL. Vivity showcased a distinct bimodal pattern (approximately 0 D for far focus and -1.25 D for intermediate focus), while the other IOLs exhibited single peaks at far focus. As evident in the wavefront maps in Fig. 4, the through-focus retinal image quality from the zonal method markedly differed from that of the modal method with Vivity and Lucidis. This discrepancy indicates a loss in the higher spatial frequency wavefront components with the modal method. Conversely, Eyhance and Isopure demonstrated a correlation between the through-focus retinal image qualities obtained from the zonal and modal methods, as long as the modal wavefronts were reconstructed at the 8^th order and higher.

 

Fig. 6. The estimation of through-focus retinal image quality is depicted by the area under the modulation transfer function (areaMTF), derived from reconstructed wavefronts using both the zonal and modal methods, encompassing Zernike coefficients from the 4th to the 16th order. These measurements were conducted with four refractive intraocular lenses within a 4 mm diameter. The through-focus areaMTF curve from a monofocal intraocular lens profile is illustrated by the red dashed line.

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3.3 Correlation with results of published optical bench studies

Figures 7(A) and 7(B) depict the correlation of through-focus areaMTF curves between our simulations and prior optical bench results for Vivity and Eyhance, respectively. Our simulated through-focus areaMTF curves showed that the zonal wavefronts had a stronger correlation with recent optical bench studies than the modal wavefronts, particularly for Vivity. The zonal wavefront of Vivity displayed a slightly greater discrepancy from the optical bench results, possibly attributed to its more intricate wavefront profiles compared to those of Eyhance.

 

Fig. 7. Correlation between two through-focus areaMTF curves is illustrated, comparing our simulation results obtained from both zonal and modal wavefronts with the measured results from previous optical bench studies for (A) Vivity and (B) Eyhance.

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4. Discussion

The objective of our study was to objectively evaluate the wavefront profiles of refractive EDoF IOLs using a high-resolution SHWFS. By employing zonal wavefront reconstruction, our laboratory-developed wavefront sensor achieved sufficient sampling resolution, enabling the measurement of fine spatial details in advanced IOL designs. Particularly, the refractive EDoF IOLs, notably Vivity and Lucidis, exhibited intricate optical profiles. Modal reconstruction based on Zernike polynomials, even up to the 16^th order, proved inadequate to fully characterize these profiles.

As proprietary medical products, the original wavefront designs of these EDoF IOLs were not publicly disclosed. Consequently, we used the measured zonal wavefront as a surrogate for the unknown reference wavefront. An inherent limitation arises from the inability to resolve wavefronts with spatial frequency components exceeding our sampling frequency, irrespective of the reconstruction algorithm (zonal or modal). Without a priori knowledge of the reference wavefront profile, setting a sampling threshold was impractical. Instead, our focus was maximizing the sampling resolution to capture the majority of the high spatial frequency content in the IOL profile. Another noteworthy limitation is the sensitivity of zonal reconstruction to measurement noise, as depicted in Fig. 2(C).

Presently, most commercially available IOLs feature negative 4^th-order Zernike SA to counterbalance the positive 4^th-order SA in the cornea. Conversely, additional SAs can be introduced to extend the depth of focus, with studies suggesting that combining negative 4th-order SA with opposite-signed 6^th-order SA enhances the depth of focus [42–47]. This observation may explain the prevalence of negative 4^th-order and positive 6^th-order SAs in refractive EDoF IOLs, except for Isopure, which shares the same negative sign. However, it is crucial to note that SA magnitudes are influenced by the pupil diameter used for fitting. Since refractive EDoF profiles typically occur within relatively smaller diameters (1-2 mm) compared to our measurement diameter (4 mm), fitting a flat profile to smaller orders of SAs may introduce larger inherent fitting errors. Additionally, SA magnitudes vary for different orders used in the fitting, implying that calculated SA coefficients may not truly represent the wavefront unless larger orders are incorporated in the fitting process.

Specifically, the wavefront profiles of Vivity and Lucidis could not be fully decomposed with the limited orders of Zernike SAs. As previously mentioned, Vivity features an elevated plateau in the central zone, exhibiting abrupt slope changes around the edges of the plateau. The central zone's phase shift characteristic includes a protruded plateau (approximately 0.2 mm wide), with a 1 μm increased thickness relative to an adjacent inner portion and a 0.6 μm increased thickness relative to an adjacent outer portion [48]. Mathematically, this specific jump in the wavefront can be represented by an infinite Zernike polynomial series, akin to the Heaviside step function expressed by the sum of an infinite number of sinusoids or a Fourier series. Similarly, Lucidis possesses an axicon-like central zone, and this unsmooth part may also necessitate expression using an infinite Zernike polynomial series. The zonal reconstruction is crucial to describe these sharp edges in the wavefront profiles, as illustrated in Fig. 4.

Meanwhile, there are several other options for commercially available wavefront sensors based on different techniques, such as the pyramidal sensor, interferometric techniques such as QWLSI and the phase-shifting Schlieren technique. The pyramidal sensor (Osiris, CSO, Italy) offers a high sampling resolution, i.e. a lateral resolution of approximately 41 μm for a 9-mm pupil diameter [49,50]; however, it commonly suffers from diffraction effects between different pupil images [51]. The QWLSI (SID4, Phasics, France) provides a 29.6-μm sampling resolution and has achromaticity, i.e., wavelength independency [22]; however, it cannot completely avoid the influence of unwanted parasitic diffraction orders from the two-dimensional cross-phase grating [52]. The phase-shifting Schlieren technique (NIMO TR1504, Lambda-X, Belgium) provides a high sampling resolution (36 μm) [53] and up to 13^th-order Zernike coefficients [54]; however, its precision has been reported to be reduced within the central portion (0.5–1.0 mm diameter) due to the high sensitivity to noise [54,55]. It would be worthwhile to perform comparison studies among these devices with identical optical conditions.

The recent surge in refractive EDoF IOLs as treatment options for presbyopia has been overwhelming. Given the limited disclosure of detailed profiles for these proprietary designs, the capability of our high-resolution SHWFS to reconstruct these intricate profiles can offer crucial clinical insights into how visual performance may vary among individual patients with such IOLs. With advancements in ocular biometry technology, it is now feasible to design individual eye models using biometric data. Once we obtain the wavefront profiles of EDoF IOLs with the model cornea, these profiles can be utilized in conjunction with matrix optics through various software packages to predict their optical performance in individual eyes. Performing this process pre-surgery allows for personalized cataract surgeries [3,56], optimizing clinical outcomes in the future. For instance, an optimal IOL with appropriate SAs can be selected based on corneal asphericity, a factor that may be significantly altered after corneal refractive surgery [57].

One limitation of our study is the absence of an artificial cornea in the optical setup. The objective of our study was to evaluate the intrinsic optical quality of the IOLs rather than estimating the impact of IOL SA on total ocular SA and through-focus MTF after implantation. Consequently, we did not employ a recommended model eye for our measurements. Notably, due to differences in vergence conditions, the IOLs may induce different SA magnitudes from setups using a model cornea. Moreover, to meet the requirement of large wavefront magnification in our setup, in addition to the requirement of a large sensor and lenslet array, the overall size of the system would be unavoidably larger than the conventional one and the measurement sensitivity would be lower due to the magnification.

Our findings underscore that recent refractive EDoF IOLs possess intricate optical profiles not adequately characterized by modal reconstruction algorithms based on relatively low orders of Zernike SAs. We demonstrated that our laboratory-developed high-resolution SHWFS, coupled with zonal wavefront reconstruction, allows for an objective assessment of the high-spatial-frequency wavefront profiles of these advanced IOLs. By utilizing the measured wavefront profiles, it becomes possible to select the appropriate IOL for each individual before surgery, simulating the impact of various factors, including corneal aberrations and potential post-surgery IOL decentration. This approach enables the optimization of clinical outcomes.