**Purpose.**:
To determine the energy distribution between the distance and near images formed in a model eye by spherical and aspheric apodized diffractive multifocal intraocular lenses (IOLs).

**Methods.**:
The IOL was inserted in a model eye with an artificial cornea with positive spherical aberration (SA) similar to that of the human cornea. The energy of the distance and near images, as a function of the pupil size, was experimentally obtained by image analysis. The level of SA on the IOL, which is pupil-size–dependent, was determined from simulations. The influence of the SA was deduced from results obtained in monofocal IOLs and by comparison of the experimentally obtained energy efficiency to theoretical results based solely on the diffractive profile of the IOL.

**Results.**:
In contrast with theoretical predictions, the energy efficiency of the distance image strongly decreased for large pupils, because of the high level of SA in the IOL. The decrease was smaller in the apodized diffractive multifocal lens with aspheric design. As for the near image, since the diffractive zone responsible for the formation of this image was the same in the spherical and aspheric lenses and the apertures involved were small (and so the level of SA), the results turned out to be similar for both designs.

**Conclusions.**:
For large pupils, the energy efficiency of the distance image is strongly affected by the level of SA, although aspheric IOLs perform slightly better than their counterparts with a spherical design. For small pupils, there are no differences between the spherical and aspheric IOLs.

^{ 1 }The principle of operation of most diffractive IOLs is to use the base lens curvature and the zero (

*m*= 0) and first (

*m*= 1) diffraction orders to achieve two focal points, sometimes referred to as optical powers, simultaneously along the optical axis.

^{ 2,3 }The power corresponding to the 0-order diffraction is used to image distant objects, and the other is used for near vision. This approach to providing the pseudophakic eye with the ability to see at different distances has an inherent drawback. The focused retinal image, provided by one of the lens powers, is always overlaid by an out-of-focus image from the second lens power. This side effect may be visually disturbing, depending on the distance along the optical axis between the two images and their relative energy distribution. As a consequence, the contrast sensitivity in eyes implanted with a DMIOL may be worse than in eyes implanted with a monofocal IOL,

^{ 4 }and unwanted optical effects such as glare and/or halos are often reported by patients with DMIOLs in mesopic and escotopic conditions.

^{ 5 }For these reasons, it is interesting to characterize, both theoretically

^{ 6 }and experimentally, the optical performance of these IOLs in terms of the energy distribution between the distance and near images and its variation with the pupil diameter. This approach

^{ 7 }has rarely been used, in comparison with more common metrics such as the modulation transfer function (MTF) or the point spread function (PSF).

^{ 8,9 }

^{ 10,11 }or in clinical studies,

^{ 12,13 }the conditions and the degree to which the aspheric design might be advantageous versus a conventional spherical IOL, particularly when some studies have shown little or no benefit of aspheric IOLs in patients with small pupils.

^{ 14,15 }

^{ 16,17 }Next, the energy distribution was experimentally determined from the analysis of the near and distance images obtained with the IOL inserted in a model eye that agrees with the one proposed by the International Organization for Standardization (ISO),

^{ 18 }except for the artificial cornea. Instead of the aberration-free cornea proposed in the ISO standard, our model eye has an artificial cornea that provides a realistic level of SA on the IOL

^{ 19 }that is similar to the values found in human corneas.

^{ 20,21 }The differences between the theoretical and experimental results allow us to determine the influence of the level of SA on the energy balance between distance and near images, particularly in eyes with large pupils. In addition, they permit us to establish the ocular conditions for which the aspheric design may be advantageous versus the spherical one.

^{ 20,21 }This compensation, expressed in terms of the c[4,0] Zernike coefficient, is as much as 0.20 μm for a 6-mm pupil.

^{ 22 }

*r*= 0) and the add power, is given by the equation

^{ 23 }: where

*i*is the zone number (

*i*= 1–12), λ is the design wavelength, and

*D*

_{add}is the add power (in diopters). For λ of 550 nm and

*D*

_{add}of +4 D, the radius of the central disc (

*r*

_{1}) is 0.375 mm, whereas the outer diffractive ring (

*i*= 12) has a radius (

*r*

_{12}) of 1.8 mm (or 3.6 mm of diameter as stated above).

*h*) of the diffractive steps at the zone boundaries gradually decreases from the center to the periphery, a design described as

_{i}*apodized*by the manufacturer (Fig. 1). An apodization factor is given by Lee et al.

^{23}: for which the step height reduction is given by where

*h*

_{o}is the maximum height at the optical axis (

*r*= 0). By setting

*h*

_{o}at 1.3 μm,

^{17}equation 3 produces a continuous reduction of the step height up to approximately

*h*

_{11}= 0.2 μm and

*h*

_{12}= 0 at the outer ring, in good agreement with the reported features of this type of IOL.

^{16, 17}

*n*

_{IOL}= 1.55) and the aqueous medium (refractive index

*n*

_{aqueous}= 1.336), in which the lens is immersed. Because of the refractive index difference

*n*

_{IOL}−

*n*

_{aqueous}, the light waves passing through different parts of the diffractive profile are phase shifted by different amounts. The induced phase shift (Φ) is proportional to the diffractive profile height (

*h*): Equation 4 is commonly expressed as a phase shift in wavelengths units as The maximum phase shift (

*a*) in a diffractive ring is with

_{i}*h*calculated by equation 3. The apodization of the diffractive profile of the DMIOLs implies that the value of α

_{i}*progressively decreases from α*

_{i}_{0}= 0.51 (i.e., nearly half wave) at the first diffractive zone to α

*= 0.06 at the outer diffractive ring. This fact has important implications on how the light is distributed between the*

_{i}*m*= 0 (distance power) and

*m*= 1 (near power) diffraction orders as a function of the pupil aperture or equivalently as a function of the number of diffractive rings that are illuminated and take part in the distribution of the light between the

*m*= 0 and

*m*= 1 orders.

*were constant for all the rings, the throughput efficiency (*

_{i}*TE*) of the

*m*= 0 and

*m*= 1 diffraction orders would be given by

^{3,24}: However, since the value of α

*varies with the radius, the*

_{i}*TE*for each α

*(*

_{i}*TE*

_{m−0.1}

^{i}) has to be weighted by a factor that corresponds to the

*i*th diffractive ring area. Therefore, the energy that the diffractive part of the IOL would divert from an incident plane wave into the

*m*= 0 and

*m*= 1 diffraction orders is calculated by means of linear combinations of the weighted contributions of all the rings: and where

*c*

_{te}is a proportionality constant, and

*A*is the area of the

^{i}*i*th diffractive ring. It is worth emphasizing that, when the IOL operates only with the first diffractive zone (i.e., α

*= α*

_{i}_{0}), there is a nearly equal diffraction throughput efficiency for the distance (

*TE*

_{m=0}

^{0}= 0.38) and near (

*TE*

_{m=1}

^{0}= 0.43) powers. The progressive reduction of the phase shift of the waves α

*as they pass through the outer diffractive rings implies that*

_{i}*TE*

_{m=0}

^{i}>

*TE*

_{m=1}

^{i}, and according to equations 8 and 9, the energy sent to the distance power (

*m*= 0 order) is favored at the expense of the near power (

*m*= 1 order).

*TE*

_{m=0}

^{refractive}= 1), and therefore the energy is simply: where

*A*

^{refractive}is the area of the illuminated refractive region of the ADMIOL. Then, in the case where the size

*A*

_{IOL}of the illuminated region of the IOL (referred to hereinafter as IOL-pupil diameter) would encompass the whole diffractive zone plus a fraction of the refractive one (i.e.,

*A*

_{IOL}= ∑

_{i=1}

^{12}

*A*+

^{i}*A*

^{refractive}), the amounts of energy sent to either the distance and near powers would be, respectively: and which can be expressed in terms of energy efficiency as and where

*I*

_{IOL}

^{total}is the total energy transmitted through the IOL-pupil. This energy is proportional to the IOL-pupil area

*A*

_{IOL}, if any loss of energy caused by scattering in the diffractive steps

^{25}is neglected: Incidentally, equation 15 implies a quadratic dependence of

*I*

_{IOL}

^{total}with the IOL-pupil diameter, a well known fact in photography where the image illumination scales quadratically with the diaphragm diameter.

^{26}

^{ 16,17 }and show that for small IOL-pupils, the energy was nearly equally divided between distance and near powers but that there was a gradual change in the energy balance to favor the distance power for large pupils. However, it must be emphasized that these calculations predict only the amount of the energy sent to either the distance or near powers for a particular IOL pupil, but they do not ensure that this energy would be properly focused on the respective images, the latter depending, among other factors, on the design of the base lens (aspheric versus spherical) from which the diffractive profile is made. This fact is especially relevant when the apodized diffractive IOL is inserted in any type of model eye with an aberrated cornea,

^{ 27 }the latter meaning that there would be a converging beam impinging on the IOL with a value of SA that depends on the size of the pupil aperture. This scenario may compromise the theoretical distance-dominant behavior of the AcrySof ReSTOR IOL for large pupils, particularly when dealing with IOLs of spherical design that tend to increase the optical aberrations of the eye.

^{ 28 }

*x*,

*y*, and

*z*axis) and rotation (tilt and tip) stages. The pinhole object is imaged by the model eye with the ADMIOL in two planes separated along the optical axis. A 10× infinite corrected microscope mounted in a translation holder is used to select either the distance or the near image and magnify it onto an 8-bit CCD camera. Before any measurement, the IOL is centered without any tilt/tip with the aid of a 6× objective lens coupled to a CCD video camera. To this end, the iris diaphragm is closed as much as possible and the light scattered onto the IOL surface is used to ensure the proper position of the lens (Fig. 4). Further details of the experimental setup can be found elsewhere.

^{ 29 }

Surface | Radius (mm) | Thickness (mm) | Refractive Index |
---|---|---|---|

Object | Infinity | 1 | |

Cornea front | 35.99 | 4.29 | 1.4599 |

Cornea back | −35.99 | 8.6 | 1 |

Window front | Flat | 6 | 1.5185 |

Window back | Flat | 6.25 | 1.336 |

Iris pupil* | Flat | 10 | 1.336 |

Window front | Flat | 6 | 1.5185 |

Window back | Flat | 9.24 | 1 |

Image | — | — | — |

^{ 29 }between the diameters Ψ of the EP and the IOL-pupil: We used commercial optical design software (Zemax Development Corporation, San Diego, CA) to obtain the dependence of the SA of the wavefront that leaves the artificial cornea and impinges on the IOL as a function of the EP diameter. The results, expressed in terms of the values of the Zernike c[4,0] coefficient and plotted in Figure 5, show that for EP diameters up to 7 mm, which actually means an IOL-pupil diameter of ∼3.6 mm (i.e., the entire apodized diffractive zone), the SA contribution was relatively small. For larger pupils, there was a significant increase in the level of SA. Thus, for EP diameters above 9 mm (or IOL-pupil diameter >4.7 mm) the value of c[4,0] was larger than 0.2 μm, which corresponds to the maximum value of SA for which the aspheric Acrysof IOL design is theoretically able to compensate.

^{ 27 }From now on, references to small or large IOL-pupil diameters will be understood to indicate that the diameters are smaller or larger than 3.6 mm, respectively.

*I*

_{pinh}in Fig. 6) surrounded by a more or less defined blurred halo (labeled

*I*

_{backg}in Fig 6). This background corresponds primarily to the overlaying defocused image but, in the case of the distance image (Fig. 6b), it will be shown that there is an additional contribution that strongly depends on the level of SA on the IOL.

*I*

_{pinh}) only and the energy of the total image that comprises the pinhole plus background regions (

*I*

_{total}=

*I*

_{pinh}+

*I*

_{backg}) are obtained by integration of the pixel gray level in the corresponding regions: where

*R*is either the pinhole region or the total image (

*R*= pinh, total),

*n*is a pixel contained in the

*R*region, and

*g*(

*n*) is the pixel gray level. Since the images are blurred because of the background, determining the borders of the region that correspond only to the focused pinhole is not a straightforward matter. An edge-detection algorithm was used to unambiguously define a region of interest (ROI) and remove all the background contribution outside this ROI (Figs. 6c, 6d). Then, the

*I*

_{pinh}is calculated from these filtered images, according to equation 17.

*I*

_{total}and

*I*

_{pinh}obtained with the monofocal spherical (SN60AT) and aspheric (SN60WF) IOLs is plotted, as a function of the EP diameter, in Figure 7. The corresponding image energy efficiencies, defined as

*I*

_{pinh}

*/I*

_{total}, are plotted in Figure 8. To make the comparison of results easier,

*I*

_{total}and

*I*

_{pinh}are normalized to the value of

*I*

_{total}obtained with the largest EP diameter (11 mm). The results in Figure 7 show a quadratic dependence of

*I*

_{total}with the EP diameter, in good agreement with the theoretical values predicted by equation 15. More interesting, for both types of IOLs and small EP diameters up to 6 mm (at which the level of SA on the IOL is small), the energy

*I*

_{pinh}was nearly the same as the

*I*

_{total}, which means that the background contribution was negligible in these conditions, no matter the type (spherical or aspheric) of monofocal IOL. As a consequence, high image efficiencies on the order of 80% were obtained with small apertures for both IOLs, as is shown in Figure 8.

*I*

_{pinh}remained constant, even though the measured

*I*

_{total}increased (Fig. 7). This result implies that, when the EP is opened, most of the additional available energy is not sent to the pinhole image but is wasted on the background, and consequently, a dramatic reduction in image efficiency occurs (see Fig. 8). For the aspheric SN60WF, the larger the EP the larger the value of

*I*

_{pinh}, but the increase occurred with a slope smaller than the increase in

*I*

_{total}(Fig. 7), which implies a moderate reduction in its efficiency for larger EP diameters, as is shown in Figure 8.

*I*

_{total}(in the case of both distance and near images) scaled quadratically with the EP diameter.

*I*

_{pinh}measured in the spherical and aspheric multifocal IOLs. It increased slightly for EP diameters up to 6 mm (i.e., up to IOL-pupils of ∼3.6 mm that corresponds to the diffractive zone of the IOL) and then kept a low constant value for larger pupils. Since on the other hand the energy

*I*

_{total}of the near image increased with the EP (see Fig. 9), the energy efficiency

*I*

_{pinh}/

*I*

_{total}strongly decreased for large pupils, as is shown in Figure 10.

*I*

_{pinh}first increased with the EP diameter (up to 7 mm) and then kept a constant value for larger pupils. The results of the aspheric SN6AD3 IOL followed a similar tendency, but the constant value of

*I*

_{pinh}measured for large pupils was slightly higher than the value measured for the spherical SN60D3 IOL.

^{ 15 }that did not find significant differences in contrast sensitivity between aspheric and spherical IOLs for small apertures.

^{ 28 }and, as a consequence, more and more energy goes to the background when the EP diameter increases, which strongly reduces its efficiency (Fig. 8). On the contrary, the aspheric IOL tends to partially counteract the SA of the artificial cornea and manages still to focus a good amount of the additional energy in the pinhole region of the image, although it cannot impede a certain reduction of its energy efficiency by large pupils. The best performance of the aspheric IOLs in terms of energy efficiency for large pupils would be in agreement with clinical findings

^{ 13,30 }that showed that patients with implanted aspheric IOLs have better contrast sensitivity in mesopic conditions (i.e., large pupil diameters) than do those with spherical IOLs.

*I*

_{pinh}in the near image remains constant (Fig. 9), independent of the EP diameter and the particular design (spherical or aspheric) of the IOL. In these conditions, the aspheric design of the SN6AD3 proves to be no advantage, in terms of near image energy efficiency, over the spherical design of the SN60D3. These results seem to be in agreement with those in clinical studies

^{ 14,15 }that have shown that for small pupils there is little or no benefit in using aspheric IOLs.

*Handbook of Optics*; vol 3.