**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 (*λ*_{R} = 625 nm), green (*λ*_{G} = 530 nm), and blue (*λ*_{B} = 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.

^{1}

^{2,3}

*) and the distance (LCA*

_{N}*) 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.*

_{D}^{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.

^{2,8}

^{9}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}Tecnis

^{11}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**

*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*(

*λ*) – 1] / [

_{d}*n*(

*λ*) −

_{F}*n*(

*λ*)], where

_{C}*λ*= 486 nm,

_{F}*λ*= 588 nm, and

_{d}*λ*= 656 nm. The refractive power of a thin lens of refractive index

_{C}*n*(

_{L}*λ*) immersed in a medium of refractive index

*n*(

_{A}*λ*) is given by where

*K*is a geometrical constant of the thin lens involving its front

*r*

_{1}and back

*r*

_{2}radii (

*K*= 1 /

*r*

_{1}− 1 /

*r*

_{2}). .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

*n*(

_{A}*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.,

*P*[

_{R}*λ*] <

_{C}*P*[

_{R}*λ*]). The LCA in the focal plane of the thin lens can be estimated from the variation of the refractive power Δ

_{F}*P*corresponding to

_{R}*F*and

*C*wavelengths, that is, Δ

*P*{FdC} =

_{R}*P*(

_{R}*λ*) −

_{F}*P*(

_{R}*λ*). Taking into account Equation 1 and the definition of the Abbe value, the following expression for Δ

_{C}*P*can be obtained:

_{R}*P*

_{Da}(

*λ*

_{0}) the diffractive add power for the first diffraction order at the design wavelength

*λ*

_{0}. This power in diopters (D) is given by P

_{Da}(

*λ*

_{0}) = 2mλ

_{0}/

*r*indicates the radius of the

_{m}*m*

^{th}zone in meters and the design wavelength

*λ*

_{0}is expressed in meters too.

^{12}The variation of wavelength entails a variation in the diffractive add power given by

*λ*and the design add power

*P*

_{Da}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.,

*P*

_{Da}[

*λ*] >

_{C}*P*

_{Da}[

*λ*]).

_{F}*λ*

_{0}≈

*λ*and Δ

_{d}*λ*=

*λ*−

_{C}*λ*, the total chromatic aberration at the near focus (LCA

_{F}*) would be LCA*

_{N}*= LCA*

_{N}*– ([*

_{D}*λ*−

_{C}*λ*] /

_{F}*λ*)

_{d}*P*

_{Da}(

*λ*), where LCA

_{d}*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.*

_{D}^{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 (

*I*

_{core}) relative to the intensity in the full image that comprises the core and the background (

*I*

_{total}=

*I*

_{core}+

*I*

_{background}). The ratio

*η*=

*I*

_{core}/

*I*

_{total}is easy to compute in the experimental practice and approaches the so-called light-in-the-bucket metric

^{15}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.

^{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., LCA

*and LCA*

_{D}*for 2f-IOLs and just LCA for 1f-IOLs).*

_{N}^{16}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**

**Figure 1**

**Table 2**

^{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*=

*I*

_{core}/

*I*

_{total}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,

**Figure 2**

**Figure 2**

^{10}

^{2,8}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.

**Table 3**

- 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 LCA
, LCA_{D}, and LCA labels in Fig. 2)._{N} - By numerical estimation. We have used Equation 2 to compute the LCA
= Δ_{D}*P*{FdC} with the data of refractive index, Abbe value, and IOL base power contained in Table 1. To calculate Δ_{R}*P*_{Da}{*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 LCAwith Equation 5, we have used the experimental measures of LCA_{N}= Δ_{D}*P*{RGB} obtained with the RGB lights instead of the computed values of LCA_{R}= Δ_{D}*P*{FdC}._{R}

^{6,20}along with the chromatic dispersions of the ocular media (cornea and aqueous humor) determined with Cauchy's equation

^{21}and the coefficients provided by Atchison and Smith

^{22}(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.

*P*{FdC} predict constant positive LCA

_{R}*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 Δ*

_{D}*P*{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 {

_{R}*λ*,

_{F}*λ*,

_{d}*λ*} 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 LCA

_{C}*value of every IOL combines with the positive LCA of the cornea in Le Grand pseudophakic schematic eye to produce greater aberration ([LCA*

_{D}*]*

_{D}_{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 [LCA

*]*

_{D}_{Eye}reaching 2.87 D (see values of [LCA

*]*

_{D}_{Eye}in Table 3).

*P*

_{Da}{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 Δ

*P*

_{Da}proves its compensating effect on the aberration introduced by Δ

*P*{RGB} when both terms are totaled in the estimation of the LCA

_{R}*(Equations 3–5). The results obtained in this case are qualitatively different for the Tecnis and AcrySof 2f-IOLs (see third right column in Table 3). While LCA*

_{N}*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), LCA*

_{N}*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 LCA*

_{N}*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), LCA*

_{N}*results in a negative value (LCA*

_{N}*[ZKB00] = −0.20 D). A larger amount of negative LCA is predicted for the ZMA00 Tecnis IOL, with +4 D add power (LCA*

_{N}*[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), LCA*

_{N}*results in a positive value (LCA*

_{N}*[SN6AD3] = 0.41 D). A larger amount of positive LCA is yet predicted for the SV25T0 AcrySof IOL, with +2.5 D add power (LCA*

_{N}*[SV25T0] = 0.82 D). The experimental results of LCA*

_{N}*, 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.*

_{N}*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 ([LCA*

_{N}*]*

_{N}_{Eye}) (first right column of Table 3). In all 2f-IOLs, it satisfies [LCA

*]*

_{N}_{Eye}< [LCA

*]*

_{D}_{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 [LCA

*]*

_{N}_{Eye}values within the natural refractive error of 1.3 D for the same spectral range, AcrySof IOLs exceed this figure.

^{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 [LCA

*]*

_{D}_{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 [LCA

*]*

_{D}_{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.

**M.S. Millán**, None;

**F. Vega**, None;

**I. Ríos-López**, None

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