Although we will consider the AcrySof ReSTOR IOL (Alcon Ltd.) to be the reference lens in this work, the method outlined herein can be applied to any DMIOL. The AcrySof ReSTOR has an optical zone of 6-mm diameter with a hybrid diffractive–refractive design on its anterior surface (
Fig. 1). The diffractive region covers the central 3.6 mm of the lens and is formed by 12 zones (a central disc and 11 concentric diffractive rings) that divert light simultaneously into distance and near powers. The outer region of the lens to the 6-mm edge is purely refractive and sends light to the distance power exclusively. The anterior IOL surface may be either spherical or aspheric, the latter introducing negative SA to compensate for the natural positive SA of human corneas.
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
The relationship between the radii of the diffractive zones, measured from the optical axis (
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).
The most distinct characteristic of the design of the AcrySof ReSTOR IOL is that the height (
hi) of the diffractive steps at the zone boundaries gradually decreases from the center to the periphery, a design described as
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
ho is the maximum height at the optical axis (
r = 0). By setting
ho at 1.3 μm,
17equation 3 produces a continuous reduction of the step height up to approximately
h11= 0.2 μm and
h12 = 0 at the outer ring, in good agreement with the reported features of this type of IOL.
16, 17
The diffractive profile of the lens acts as an optical interface between the IOL material (refractive index
nIOL = 1.55) and the aqueous medium (refractive index
naqueous = 1.336), in which the lens is immersed. Because of the refractive index difference
nIOL −
naqueous, 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 (
ai) in a diffractive ring is
with
hi calculated by
equation 3. The apodization of the diffractive profile of the DMIOLs implies that the value of α
i progressively decreases from α
0 = 0.51 (i.e., nearly half wave) at the first diffractive zone to α
i = 0.06 at the outer diffractive ring. This fact has important implications on how the light is distributed between the
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.
If α
i were constant for all the rings, the throughput efficiency (
TE) of the
m = 0 and
m = 1 diffraction orders would be given by
3,24:
However, since the value of α
i varies with the radius, the
TE for each α
i (
TEm−0.1i) has to be weighted by a factor that corresponds to the
ith 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
cte is a proportionality constant, and
Ai is the area of the
ith diffractive ring. It is worth emphasizing that, when the IOL operates only with the first diffractive zone (i.e., α
i = α
0), there is a nearly equal diffraction throughput efficiency for the distance (
TEm=00 = 0.38) and near (
TEm=10 = 0.43) powers. The progressive reduction of the phase shift of the waves α
i as they pass through the outer diffractive rings implies that
TEm=0i >
TEm=1i, 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).
In the case of the purely refractive region of the ADMIOL, the light goes exclusively to the distance power (i.e.,
TEm=0refractive = 1), and therefore the energy is simply:
where
Arefractive is the area of the illuminated refractive region of the ADMIOL. Then, in the case where the size
AIOL 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.,
AIOL = ∑
i=112Ai +
Arefractive), 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
IIOLtotal is the total energy transmitted through the IOL-pupil. This energy is proportional to the IOL-pupil area
AIOL, if any loss of energy caused by scattering in the diffractive steps
25 is neglected:
Incidentally,
equation 15 implies a quadratic dependence of
IIOLtotal with the IOL-pupil diameter, a well known fact in photography where the image illumination scales quadratically with the diaphragm diameter.
26
We calculated the energy efficiencies according to
equations 13 and
14, as a function of the IOL-pupil diameter of the AcrySof ReSTOR lens. The results, plotted in
Figure 2, are in excellent agreement with those reported elsewhere
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