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Multidisciplinary Ophthalmic Imaging  |   April 2016
Polychromatic Image Performance of Diffractive Bifocal Intraocular Lenses: Longitudinal Chromatic Aberration and Energy Efficiency
Author Affiliations & Notes
  • María S. Millán
    Departament d'Òptica i Optometria Universitat Politècnica de Catalunya, BARCELONATECH, Terrassa, Spain
  • Fidel Vega
    Departament d'Òptica i Optometria Universitat Politècnica de Catalunya, BARCELONATECH, Terrassa, Spain
  • Inmaculada Ríos-López
    Departament d'Òptica i Optometria Universitat Politècnica de Catalunya, BARCELONATECH, Terrassa, Spain
  • Correspondence: María S. Millán, Departament d'Òptica i Optometria, Universitat Politècnica de Catalunya, BARCELONATECH, Violinista Vellsolà 37, 08222 Terrassa, Spain; millan@oo.upc.edu
Investigative Ophthalmology & Visual Science April 2016, Vol.57, 2021-2028. doi:10.1167/iovs.15-18861
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      María S. Millán, Fidel Vega, Inmaculada Ríos-López; Polychromatic Image Performance of Diffractive Bifocal Intraocular Lenses: Longitudinal Chromatic Aberration and Energy Efficiency. Invest. Ophthalmol. Vis. Sci. 2016;57(4):2021-2028. doi: 10.1167/iovs.15-18861.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: The study evaluated—theoretically and experimentally—the longitudinal chromatic aberration (LCA) and through-focus energy efficiency (TF-EE) of diffractive–refractive bifocal intraocular lenses (2f-IOLs).

Methods: Four aspheric 2f-IOLs (Tecnis +4.00 diopter [D] ZMA00, +2.75 D ZKB00, and AcrySof +4.0 D SN6AD3, +2.5 D SV25T0) of same base power 30 D, but different design, additional (add) power, and different material, were tested in vitro in terms of TF-EE when illuminated by 3 red (λR = 625 nm), green (λG = 530 nm), and blue (λB = 455 nm) lights. The LCA affecting the distance and near foci was derived theoretically and measured experimentally from the contributions of the IOLs' refractive and diffractive powers. Longitudinal chromatic aberration was evaluated in a pseudophakic schematic eye.

Results: The distance focus of all 2f-IOLs showed lower energy efficiency (EE) for the blue than for the red light. AcrySof IOLs showed the largest amount of positive LCA in the distance focus that, combined with corneal LCA, would increase the resulting distance LCA in a pseudophakic eye. The near focus of all 2f-IOLs showed higher EE for the blue than for the red light. Better compensation for the LCA of a pseudophakic eye at near focus is obtained with Tecnis than with AcrySof 2f-IOLs.

Conclusions: The energy distribution between the foci of diffractive 2f-IOLs depends on the lens design, the illumination wavelength, and to a lesser extent, the add power. In distance vision, 2f-IOLs' refractive base power increases the positive LCA of prior ocular media, and the resulting LCA may even surpass the natural LCA of human eye. In near vision, however, the achromatizing effect of diffractive 2f-IOLs may compensate, in part, the natural eye's LCA.

A diffractive bifocal intraocular lens (2f-IOL) typically uses a hybrid diffractive-refractive design that consists of a high power refractive base lens (similar to that considered for a monofocal design) and a low additional (add) power diffractive profile engraved on either the front (e.g., AcrySof ReSTOR; Alcon Laboratories, Inc., Duluth, GA, USA) or the back surface of the lens (e.g., Tecnis; Abbott Medical Optics, Inc., Santa Ana, CA, USA). This diffractive profile mainly utilizes the zero and the first diffraction orders to produce two images: The zero-order energy directs to the distance image formed by the high-power refractive base lens, whereas the first-order energy directs to the near image formed by the combined base plus add powers. Many IOL designs also include compensation for the positive spherical aberration of the cornea to some extent.1 
Under white light illumination, the distance image formed by a 2f-IOL is affected by the longitudinal chromatic aberration (LCA) of the refractive part of the hybrid lens, which depends on its base power and the dispersive properties of its optical material (i.e., through the variation of the refractive index with wavelength). This aberration is of the same sign as that in the human eye. The LCA of the near image, however, has the additional contribution of a term of opposite sign—due to the diffractive element—that can be relevant, depending on the add-power value. After implanting a 2f-IOL in the human eye, the final LCA of the distance and near images resulted from the further contributions of other components of the eye (cornea, aqueous and vitreous humors). In addition to LCA, the energy distribution between the distance and near images of diffractive 2f-IOLs also shows a strong dependency on wavelength.2,3 
In this work, we studied and measured the chromatic properties of 4 aspheric diffractive 2f-IOLs (2 nonapodized and 2 apodized of different add power) using an experimental setup arranged on an optical bench. Two monofocal (1f-IOL) counterparts (Tecnis model ZA9003, from Abbott Medical Optics, and AcrySof model SN60WF, from Alcon) of the nonapodized and apodized 2f-IOLs, with similar base power, aspheric design, and material, also have been included as references in our study. Three light-emitting diodes (LEDs) with emission in the blue (B), green (G), and red (R) spectral bands have sequentially illuminated the setup. More specifically, we measured through-focus energy efficiency (TF-EE) of the IOLs with the R, G, and B lights and LCA in both the near (LCAN) and the distance (LCAD) foci of the lenses experimentally obtained from the previous through-focus analysis. Mathematical expressions to calculate the LCA in both foci have been also derived. 
To isolate the LCA and other effects produced by a 2f-IOL in the distance and near foci, the IOL was introduced in a wet cell. By doing so, all the refractive and diffractive power was exclusively due to the IOL. The optical setup was similar to that used by the authors in related works and described in detail elsewhere,4,5 except for the artificial cornea, which was removed from the setup in this experiment. Once the TF-EE and LCA measurements were taken, we additionally set the IOL virtually in a pseudophakic schematic eye (Le Grand eye),6 which provided the average refractive characteristics of other ocular media (cornea, aqueous and vitreous humors) and computed closer estimates for the LCA in the human eye. 
Related research has been extensively developed and reported using different IOL designs and model eyes. It is worth mentioning some closely related works (see Refs. 2, 7, 8); their analyses and numerical simulations have provided a valuable background to our work. However, some of their assumptions meant their methods were not fully applicable to our study on diffractive 2f-IOLs of different material, add power, and design (for more details, see the Discussion section). Thus, we have considered the refractive characteristics of the hybrid IOL (i.e., refractive power and dispersion) as contributions separate from those of the cornea and ocular humors. Our polychromatic analysis and experimental through-focus results show the IOLs' own contribution to the chromatic aberration in the far as well as in the near focus of a pseudophakic schematic eye implanted with such a lens. These results allow us to further compare the experimental chromatic performance of the studied aspheric IOLs (4 diffractive bifocals and 2 reference monofocals) in terms of TF-EE and LCA, something that is more commonly treated just theoretically or by numerical simulation.2,8 
Materials and Methods
Intraocular Lenses
Six IOLs were used in the experimental work of this study. Their technical specifications are listed in Table 1. AcrySof ReSTOR9 2f-IOLs have an apodized diffractive design, that is, with steps of decreasing height from center to periphery. The diffractive zone covers the central part (see Table 1) of the anterior aspheric surface and is surrounded by a peripheral ring that is purely refractive. This design aims at directing a higher amount of the incoming energy to the distance focus with enlarged pupils and, therefore, reducing more effectively the formation of halos in distance vision.10 Tecnis11 2f-IOLs have a posterior spherical surface with a nonapodized diffractive design that fully covers its aperture. This design is intended to produce a balanced distribution of energy between the distance and the near foci independently of the pupil diameter. 
Table 1
 
Technical Specifications of the 2f-IOLs and Reference 1f-IOLs Under Study as Provided by the Manufacturer9,11
Table 1
 
Technical Specifications of the 2f-IOLs and Reference 1f-IOLs Under Study as Provided by the Manufacturer9,11
All the IOL lenses considered in this study have an aspheric surface, designed to compensate to some extent for the natural positive spherical aberration (SA) of the human cornea. Table 1 shows the SA values in terms of the c[4,0] Zernike coefficient for a 6-mm pupil diameter. 
The tested IOLs are made of dispersive optical materials that show different refractive indexes and Abbe values (Table 1). AcrySof IOLs have higher refractive index and show greater dispersion (lower Abbe value) than Tecnis IOLs, but on the other hand, they allow the design of thinner lenses, which is an advantageous property in the surgical practice because thinner lenses require smaller incisions. 
In addition to the standard UV light (<400 nm) filtering exhibited by all the tested IOLs, AcrySof IOLs have a blue-light filtering chromophore that reduces the transmittance of blue light wavelengths (400–475 nm), giving them a yellowish appearance. 
Longitudinal Chromatic Aberration of a Refractive–Diffractive Thin Lens
Longitudinal chromatic aberration is given by the variation in the focal length f or, equivalently, in the optical power P with the illuminating wavelength λ. This variation is caused by the dispersive nature of optical materials, whose refractive index n also exhibits wavelength dependency. The Abbe value (V), also called Abbe number, is widely used to characterize the dispersion of an optical material. It is defined as V = [n(λd) – 1] / [n(λF) − n(λC)], where λF = 486 nm, λd = 588 nm, and λC = 656 nm. The refractive power of a thin lens of refractive index nL(λ) immersed in a medium of refractive index nA(λ) is given by  where K is a geometrical constant of the thin lens involving its front r1 and back r2 radii (K = 1 / r1 − 1 / r2). .We recall that although the aqueous and vitreous humors are different media, their refractive indexes are close, and that is why, in a first approach and for the sake of simplicity, they are considered to have similar value in many studies. We do the same in this work by considering the IOL immersed in a medium with refractive index nA(λ). Since the refractive index n(λ) of optical materials typically decreases when the wavelength increases, Equation 1 implies that a thin lens shows lower refractive power for longer wavelengths (i.e., PR[λC] < PR[λF]). The LCA in the focal plane of the thin lens can be estimated from the variation of the refractive power ΔPR corresponding to F and C wavelengths, that is, ΔPR{FdC} = PR(λF) − PR(λC). Taking into account Equation 1 and the definition of the Abbe value, the following expression for ΔPR can be obtained:    
Equation 2 is the LCA corresponding to the single focus of a 1f-IOL intended for distance vision exclusively. In the case of a diffractive 2f-IOL whose refractive–diffractive design produces 2 main foci at the zero and first diffraction orders, Equation 2 corresponds to the LCA in the zero-order focus or, in other terms, at the distance focus    
Note that the power variation of Equation 2 depends linearly on the design power of the refractive base lens. It also depends on the wavelength change through the refraction indexes, which is a variation relatively slower than the one produced in a diffractive lens, as we will see next. 
Let us denote by PDa(λ0) the diffractive add power for the first diffraction order at the design wavelength λ0. This power in diopters (D) is given by PDa(λ0) = 2mλ0 / Display FormulaImage not available , where rm indicates the radius of the mth zone in meters and the design wavelength λ0 is expressed in meters too.12 The variation of wavelength entails a variation in the diffractive add power given by    
This variation depends linearly on both the wavelength change Δλ and the design add power PDa of the diffractive part. Equation 4 involves a much faster variation of the diffractive add power with wavelength than the refractive base power (Equation 2). The negative sign in Equation 4 accounts for a variation in opposite direction: The diffractive add power is then higher for longer wavelengths (i.e., PDa[λC] > PDa[λF]). 
At the near focus, the power of the bifocal lens PN becomes the addition of both the refractive base power and the diffractive add power, that is PN = (PR + PDa), and so does its variation with the wavelength ΔPN(λ) = ΔPR(λ) + ΔPDa(λ). In equivalent terms,    
For example, assuming λ0λd and Δλ = λCλF, the total chromatic aberration at the near focus (LCAN) would be LCAN = LCAD – ([λCλF] / λd)PDa(λd), where LCAD would be computed using Equations 2 and 3. As mentioned, the chromatic aberration produced by a diffractive component (Equation 4) increases rapidly with wavelength and is opposite the chromatic aberration produced by a refractive element (Equation 2). Equation 5 suggests that a potential compensation of the LCA, totally or in part, can occur in the near focus of a diffractive 2f-IOL. This is not possible in the distance focus, for which only the refractive component of the lens contributes (Equations 2 and 3). So far, we have analyzed how the LCA affects the distance and the near foci of a single diffractive 2f-IOL. To further illustrate how it contributes to the whole chromatic aberration of the human eye in the distance and near vision, the dispersive characteristics of other ocular media need to be considered. This step forward will lead us, as discussed in the Results section, to use a pseudophakic schematic eye to simulate the LCA that would eventually affect the retinal image. 
Through-Focus Energy Efficiency of a Refractive–Diffractive Bifocal Lens
The image formation of a diffractive 2f-IOL has been described as a combination of 2 images: the focused image produced by one of the IOL powers surrounded by a blur (halo) that corresponds mainly to the overlying on-axis out-of-focus image produced by the other power. For an insight concerning a geometrical explanation of such compound image, the mathematical description of its characteristics, halo formulae, and experimental intensity profiles obtained for a variety of multifocal IOLs, the interested reader is referred to other studies (see Refs. 13 and 14). 
The TF-EE is the measure of image quality we used in this work to test the diffractive 2f-IOLs with the R, G, B lights. It is worth remarking that the energy efficiency (EE) values of the distance and near foci are straightforwardly obtained from dense sampling of the through-focus measurement, particularly in the axial neighborhood of such focal planes. The same test was repeated for the reference 1f-IOLs, but referred only to their single (distance) focus. 
A method to measure the EE in the image space has been reported in detail for 1 or more wavelengths.3,4 It assumes a pinhole object at infinity and a charge-coupled device (CCD) camera with linear response for digital image acquisition. The method first applies an edge-detection algorithm to segment the central core of the pinhole image at the focus plane of the lens (either the distance or the near focus in a 2f-IOL) and quantifies the amount of light intensity in the core (Icore) relative to the intensity in the full image that comprises the core and the background (Itotal = Icore + Ibackground). The ratio η = Icore / Itotal is easy to compute in the experimental practice and approaches the so-called light-in-the-bucket metric15 used (see Ref. 2) to quantify the polychromatic image quality of 1f- and 2f-IOLs by numerical simulation. For a through-focus analysis, the core contour determined in the best focus plane is applied unchanged to the defocus images obtained by axial scanning of neighbor planes. In 2f-IOLs, this axial scanning stretches to cover the distance and the near foci, thus allowing us to plot the TF-EE between them. 
In this work, a TF-EE curve is experimentally obtained for each IOL under study with a fixed pupil size and for every R, G, and B light. Measurements were taken in the IOL image space, with the origin of vergences (0 D) set at the distance image for the G light (530 ± 15 nm, the closest to the design wavelength of 546 ± 10 nm, with full width at half maximum [FWHM] not greater than 20 nm).16 The through-focus maximum intervals covered 7 D for 2f-IOLs and 4 D for 1f-IOLs in 0.2-D steps, and LCA experimental values were obtained from the power difference between the extreme EE peaks (usually the R and B peaks, but not necessarily) at each focus plane (i.e., LCAD and LCAN for 2f-IOLs and just LCA for 1f-IOLs). 
Experimental Setup
The experimental work of this study has been done using the setup sketched in Figure 1. This setup is similar to others containing an ISO eye model16 in an optical bench already described in detail and used in former works,4 except for the artificial cornea, which has been removed in this study. Three R, G, and B LEDs (Thorlabs, Inc., Newton, NJ, USA)17 with nominal wavelengths (NW) and FWHM, detailed in Table 2, were sequentially used to illuminate the setup. A 200-μm pinhole test object was placed at the front focal plane of a collimating lens of 200-mm focal length. The collimated beam illuminated the wet cell where the IOL was inserted, and thus either 1 or 2 aerial images of the pinhole object were formed behind the wet cell by the 1f- or the 2f-IOL, respectively. A diaphragm, placed in front of the wet cell and used as entrance pupil, limited the IOL aperture to 3.5-mm diameter throughout the experience. The amounts of negative SA of all the tested IOLs were limited accordingly. Behind the wet cell, an infinite corrected microscope mounted in a translation holder focused the aerial image of interest and magnified it onto a monochrome 8-bit CCD camera used for digital image acquisition. The set of microscope and camera could be moved along the bench axis to locate the best focal planes for each IOL and observation distance, with a spatial resolution of ±1 μm. The microscope objective (4× Olympus Plan Achromat; Olympus, Wells Research, Inc., West Covina, CA, USA) had diffraction-limited performance through the visible spectrum and was specifically designed for high-quality imaging applications. For every IOL and wavelength, the intensity of the LED source and the time integration of image acquisition were adjusted to obtain a linear response of the camera in the intensity range of the aerial images (from distance to near images) with no saturation of the camera sensor. 
Figure 1
 
Experimental setup. The inset shows the LCA affecting the distance and near foci behind a diffractive 2f-IOL.
Figure 1
 
Experimental setup. The inset shows the LCA affecting the distance and near foci behind a diffractive 2f-IOL.
Table 2
 
Data of the Red (R), Green (G), and Blue (B) LEDs Used to Illuminate the Experimental Setup
Table 2
 
Data of the Red (R), Green (G), and Blue (B) LEDs Used to Illuminate the Experimental Setup
In a closely related work,2 the authors estimated by numerical simulation an imaging quality metric, namely the light-in-the-bucket metric, because of its ability to capture the property of IOL efficiency as well as image blur. They computed through-focus light-in-the-bucket curves to simulate the EE and energy distribution of several IOLs under R, G, B lights. As stated by the authors, “this metric quantifies the total amount of light in the central core of the point spread function (PSF) relative to that in a monofocal diffraction-limited PSF for the same wavelength and pupil size.”2 To implement this metric in practice, the ideal point source is substituted by a pinhole of certain size. We have used a 200-μm pinhole test object, which allows us to have enough energy in the image space to develop the whole experiment, and have computed the ratio η = Icore / Itotal according to the aforementioned procedure.4 A misclassification error occurs in each focal plane due to the central part of the out-of-focus image, which overlays the bucket, thus contributing to the energy efficiency of the in-focus image. This misclassification error can be roughly approximated by the ratio of areas,    
The area of the bucket is the diffracted image of the pinhole, and the size of the out-of-focus image area is calculated using basic geometry. For the 2f-IOLs of our study, the highest error corresponds to the lens with the lowest add power (+2.5 D) and is approximately 3% in both the near and the distance focal planes. 
Results
Through-Focus Energy Efficiency
Figure 2 shows the experimental TF-EE measured for all the studied 1f- and 2f-IOLs with the 3 R, G, B lights, using the implementation of the light-in-the-bucket metric described in previous sections. Energy efficiency values have been normalized to unity for each wavelength and lens. 
Figure 2
 
Experimental measurement of EE, through-focus energy distribution, and LCA in the tested 2f-IOLs: (a) ZMA00 (+4.0D), (b) SN6AD3 (+4.0D), (c) ZKB00 (+2.75D), (d) SV25T0 (+2.5D), and 1f-IOLs: (e) ZA9003, (f) SN60WF.
Figure 2
 
Experimental measurement of EE, through-focus energy distribution, and LCA in the tested 2f-IOLs: (a) ZMA00 (+4.0D), (b) SN6AD3 (+4.0D), (c) ZKB00 (+2.75D), (d) SV25T0 (+2.5D), and 1f-IOLs: (e) ZA9003, (f) SN60WF.
Bifocal IOLs show a double set of R, G, B EE peaks that correspond, as expected, to the distance and the near foci. Monofocal IOLs have, in turn, a single set of R, G, B EE peaks that correspond to their single focus intended for distance vision. A relative displacement of R, G, B plots for every focus and IOL accounts for the existing LCA, which is discussed below. Figure 2 shows that the distribution of the energy between the distance and the near foci of diffractive 2f-IOLs depends on the IOL design (apodized vs. nonapodized IOLs). The energy deflected and focused on the zero and first diffraction orders (distance and near focus, respectively) is balanced when the distance and near EE peaks reach similar height. For the pupil size of 3.5 mm used in our experiment, the distance and near EE peaks are more balanced with nonapodized than with apodized 2f-IOLs. For instance, with the ZMA00 (+4 D) lens (Fig. 2a), we obtain 0.36 distance and 0.37 near EE peak values for the G light, whereas with SN6AD3 (+4 D) lens (Fig. 2b), we obtain 0.47 distance and 0.23 near EE peak values for the same G light. The latter class (apodized AcrySof ReSTOR 2f-IOLs) shows clear energy predominance of the distance focus over the near focus, which is in agreement with the apodized design of these lenses aimed at preventing deleterious glare and halos in distance vision, particularly in dimmer mesopic illumination.10 
Additionally, 2f-IOLs with lower add power (higher add f/#), such as ZKB00 +2.75D and SV25T0 +2.5D, are slightly less efficient in the near focus—with all 3 R, G, B wavelengths—than the respective lenses of the same brand with higher add power (lower add f/#), such as ZMA00 +4 D and SN6AD3 +4 D. For instance, in the near focus of the SV25T0 (+2.5 D) lens (Fig. 2d), the {R, G, B} EE peaks reach the values {0.12, 0.16, 0.20}, whereas in the same focus of SN6AD3 (+4 D) lens (Fig. 2b), they reach the values {0.13, 0.23, 0.26}. 
More importantly, the distribution of energy between the foci also depends on the illumination wavelength as already described2,8 based on the dependence of the diffraction efficiency with wavelength.18,19 Our experimental results in Figure 2 confirm the basics of those predicted by Ravikumar et al.2 using numerical simulations, but they also disclose important differential features between the IOLs under study. Thus, in the near focus of all tested 2f-IOLs, our experiments coincide with theirs in obtaining higher (lower) EE for the blue (red) light than for the design wavelength (546 nm, close to our 530 nm light). Conversely, the opposite effect occurs in the distance focus, for which lower (higher) EE for the blue (red) light than for the design wavelength can be acknowledged. 
Longitudinal Chromatic Aberration
Table 3 shows the values of LCA in the distance and near foci of the set of diffractive 2f-IOLs, as well as in the single focus of the 1f-IOLs (the latter intended exclusively for distance vision) obtained in two different ways: 
Table 3
 
Values of LCA at the Distance and Near Foci of 2f-IOLs and at the Single Focus of 1f-IOLs
Table 3
 
Values of LCA at the Distance and Near Foci of 2f-IOLs and at the Single Focus of 1f-IOLs
  1.  
    From the experimental position of all EE peaks in the optical setup accounting for the distance, near, and single powers of the set of IOLs under the sequential illumination of the R, G, and B lights (see LCAD, LCAN, and LCA labels in Fig. 2).
  2.  
    By numerical estimation. We have used Equation 2 to compute the LCAD = ΔPR{FdC} with the data of refractive index, Abbe value, and IOL base power contained in Table 1. To calculate ΔPDa{λ} (Equation 4), we have used the IOL add power (Table 1) and the R and B NWs of the LED sources utilized in the experimental setup (Table 2). For a closer estimation of LCAN with Equation 5, we have used the experimental measures of LCAD = ΔPR{RGB} obtained with the RGB lights instead of the computed values of LCAD = ΔPR{FdC}.
In addition, to better illustrate the LCA effects produced by a given IOL when it is implanted in a human eye, we have simulated the resultant LCA in a model eye for the specific R, G, B wavelengths used in the experiment. To this end, we have considered Le Grand pseudophakic schematic eye,6,20 along with the chromatic dispersions of the ocular media (cornea and aqueous humor) determined with Cauchy's equation21 and the coefficients provided by Atchison and Smith22 (data contained in Supplementary Table S1). Using the powers measured experimentally (Fig. 2), the LCA was computed in the distance and the near foci of Le Grand pseudoaphakic schematic eye for all the IOLs. 
Regarding the LCA in distance due to the IOL immersed in the wet cell, the estimated values ΔPR{FdC} predict constant positive LCAD for 1f- and 2f-IOLs of the same material and base power (first left column in Table 3). For the AcrySof lenses, they are approximately twice the values of the Tecnis lenses, which is consistent with the higher dispersion of AcrySof material (lower Abbe value). The estimated values are in good agreement with the experimental values ΔPR{RGB} (second left column in Table 3), but somewhat lower. This fact can be explained by the difference between the spectral wavelength ranges used for the numerical estimation {λF, λd, λC} and the illumination of the experimental setup {B, G, R}. Although Δλ = 170 nm in both cases, the extreme wavelengths do not coincide. Particularly in the blue region, the difference between the wavelength F (486 nm), used for numerical estimation, and B (455 nm), actually used to illuminate the setup, tends to increase the experimental LCA, more specifically for those materials with lower Abbe value (greater dispersion) such as AcrySof. The positive LCAD value of every IOL combines with the positive LCA of the cornea in Le Grand pseudophakic schematic eye to produce greater aberration ([LCAD]Eye). In this sense, while the natural LCA of the human eye, in terms of the subjective chromatic difference of refraction, results approximately in 1.3 D for the spectral range considered in this work (455–625 nm),20,23 the distance LCA of the pseudophakic eye with any of the Tecnis IOLs is larger than 1.3 D and becomes even worse in the case of the AcrySof lenses, with values of [LCAD]Eye reaching 2.87 D (see values of [LCAD]Eye in Table 3). 
Regarding the LCA in near vision with diffractive 2f-IOLs, the add power variation ΔPDa{RGB}, which is proportional to the add power, is calculated with Equation 4 (center column of Table 3). In all cases, the negative sign of ΔPDa proves its compensating effect on the aberration introduced by ΔPR{RGB} when both terms are totaled in the estimation of the LCAN (Equations 35). The results obtained in this case are qualitatively different for the Tecnis and AcrySof 2f-IOLs (see third right column in Table 3). While LCAN of Tecnis 2f-IOLs is finally negative (which means a reverse order in the wavelengths forming the near focus with respect to the order in the distance focus), LCAN of AcrySof 2f-IOLs is finally positive (which means the same order in the wavelengths forming the near and the distance foci). The magnitude of LCAN also deserves comment. The relatively low refractive power variation of 2f-Tecnis IOLs with wavelength is quickly surpassed by their diffractive add-power variation, for which, even in the case of the lowest add (ZKB00, add +2.75 D), LCAN results in a negative value (LCAN[ZKB00] = −0.20 D). A larger amount of negative LCA is predicted for the ZMA00 Tecnis IOL, with +4 D add power (LCAN[ZMA00] = −0.49 D). On the contrary, the relatively high refractive power variation of 2f-AcrySof IOLs with wavelength cannot be fully compensated by their diffractive add-power variation, for which, even in the case of the highest add (SN6AD3, add +4 D), LCAN results in a positive value (LCAN[SN6AD3] = 0.41 D). A larger amount of positive LCA is yet predicted for the SV25T0 AcrySof IOL, with +2.5 D add power (LCAN[SV25T0] = 0.82 D). The experimental results of LCAN, derived from the EE peak positions at near focus in Figure 2 and included in the second right column in Table 3, agree with these numerical estimations in both the sign and magnitude. 
The experimental LCAN of every IOL combines with the positive and higher LCA of the cornea to produce the final LCA in the near focus of Le Grand pseudophakic schematic eye ([LCAN]Eye) (first right column of Table 3). In all 2f-IOLs, it satisfies [LCAN]Eye < [LCAD]Eye, that is, there is certain compensation in the LCA of the near focus, but the compensation is accomplished considerably better by Tecnis (with only 0.54 D for ZMA00) than AcrySof 2f-IOLs. To further illustrate the difference, while Tecnis IOLs have [LCAN]Eye values within the natural refractive error of 1.3 D for the same spectral range, AcrySof IOLs exceed this figure. 
Discussion
We studied the LCA, analytically and measured experimentally, of 2 types of diffractive 2f-IOLs in their distance and near foci and have estimated their contribution to the LCA of the retinal image by numerical simulation in a pseudophakic schematic eye. For comparison, we have included 2 additional reference 1f-IOLs in our study. The distribution of energy between foci has been experimentally determined for every IOL under 3 R, G, B lights by means of through-focus measurements of the EE. 
Our results, mainly contained in Table 3 and Figure 2, are consistent with other results obtained by numerical simulation in related works.2,7,8 They analyzed numerically the design of hybrid diffractive–refractive IOLs inserted in a polychromatic pseudophakic model eye. For instance, an aspheric monofocal hybrid IOL, proposed by López-Gil and Montés-Micó,7 has the diffractive surface intended to correct for the eye's LCA. In this lens, the diffractive profile directs nearly all the incoming light to its first diffraction order, where diffractive and refractive powers add up and, as a result, a single achromatic focal point is formed for distance object imaging in a modified Navarro et al. model eye.20,24 Castignoles et al.8 paid more attention to the diffractive element of multifocal IOLs, more specifically, to the influence of the phase profile function (binary, parabolic, sinusoidal) on the distribution of EE between the ± 2, ± 1, and 0 diffraction orders. They extended their simulation beyond the design wavelength (λ = 550 nm) to include the effects of chromatism in a simplified eye model consisting of a planar diffractive element against a perfect lens equivalent to the human eye. A step closer to our study can be found in the comprehensive work done by Ravikumar et al.,2 who further included three different hybrid IOLs: monofocal, nonapodized bifocal, and apodized bifocal, in their polychromatic analysis of IOL performance. In their numerical simulations, the authors used a reduced eye model optimally focused for distance objects at the wavelength λ = 550 nm. The total refractive power of the reduced eye, coming from the combination of the cornea and the refractive portion of the hybrid IOL, was treated as corresponding to a single diopter at the cornea plane. An equivalent diffractive optical element at the corneal plane was computed to produce the same diffraction pattern on the retina as the physical diffractive profile virtually inserted in the IOL plane. Although the work reported by Ravikumar et al.2 has provided a valuable basis for ours, some of their simplifications could not be assumed in our study. In particular, they assumed that the LCA of the Indiana eye chromatic model was a reasonable estimate for the LCA in pseudophakic eyes, thus neglecting the distinct contribution of different IOL materials (with different refraction index and dispersive characteristics) to the LCA of both the distance and near foci of the pseudophakic eye. As a consequence, they found in their simulations a uniform [LCAD]Eye ≈ 1.3 D in eyes implanted with either Tecnis or AcrySof lenses. In contrast to their simulated results, the formula we derived and our experimental results (Table 3 and Fig. 2) prove the role that the refractive features of 1f-IOLs and diffractive 2f-IOL materials (i.e., refraction index and Abbe value) play in the LCA of their image focal planes. In this regard, we have shown that the [LCAD]Eye may be significantly larger in the case of IOLs made of highly dispersive material, reaching values close to 3 D in distance vision. Moreover, we have also shown the influence of the refractive features of diffractive 2f-IOL materials on the achromatizing effect at near vision. 
Conclusions
The distribution of energy between the near and distance foci of diffractive 2f-IOLs, as measured experimentally in an optical bench, proves to be different depending on the lens design, the illumination wavelength, and, to a lesser extent, the add power. In all the tested lenses, the experimental results of the TF-EE analysis agree with the characteristics expected from the IOL design. 
In distance vision, diffractive 2f-IOLs (and 1f-IOLs) increase the positive LCA of prior ocular media. The more dispersive the IOL material (lower Abbe value), the greater is the LCA. With the tested IOLs implanted, the LCA of pseudophakic eyes would surpass even the natural chromatic aberration of the human eye. 
In near vision, diffractive 2f-IOLs tend to reduce the amount of LCA. This achromatizing effect varies linearly with the add power and, depending on the IOL material and the amount of refractive LCA produced, may compensate, in part, the LCA of the eye in near vision. This fact benefits near vision in a pseudophakic eye implanted with a diffractive 2f-IOL. 
Acknowledgments
Supported by project DPI2013-43220-R from the Spanish Ministerio de Economía y Competitividad y Fondos FEDER, and a PhD scholarship (IR-L; Ref. BES-2014-070687) from the aforementioned ministry. 
Disclosure: M.S. Millán, None; F. Vega, None; I. Ríos-López, None 
No author has a financial or proprietary interest in any product, method, or material mentioned in this article. 
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Figure 1
 
Experimental setup. The inset shows the LCA affecting the distance and near foci behind a diffractive 2f-IOL.
Figure 1
 
Experimental setup. The inset shows the LCA affecting the distance and near foci behind a diffractive 2f-IOL.
Figure 2
 
Experimental measurement of EE, through-focus energy distribution, and LCA in the tested 2f-IOLs: (a) ZMA00 (+4.0D), (b) SN6AD3 (+4.0D), (c) ZKB00 (+2.75D), (d) SV25T0 (+2.5D), and 1f-IOLs: (e) ZA9003, (f) SN60WF.
Figure 2
 
Experimental measurement of EE, through-focus energy distribution, and LCA in the tested 2f-IOLs: (a) ZMA00 (+4.0D), (b) SN6AD3 (+4.0D), (c) ZKB00 (+2.75D), (d) SV25T0 (+2.5D), and 1f-IOLs: (e) ZA9003, (f) SN60WF.
Table 1
 
Technical Specifications of the 2f-IOLs and Reference 1f-IOLs Under Study as Provided by the Manufacturer9,11
Table 1
 
Technical Specifications of the 2f-IOLs and Reference 1f-IOLs Under Study as Provided by the Manufacturer9,11
Table 2
 
Data of the Red (R), Green (G), and Blue (B) LEDs Used to Illuminate the Experimental Setup
Table 2
 
Data of the Red (R), Green (G), and Blue (B) LEDs Used to Illuminate the Experimental Setup
Table 3
 
Values of LCA at the Distance and Near Foci of 2f-IOLs and at the Single Focus of 1f-IOLs
Table 3
 
Values of LCA at the Distance and Near Foci of 2f-IOLs and at the Single Focus of 1f-IOLs
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