We present a new accommodative intraocular lens based on a two-element varifocal Alvarez lens. The intraocular lens consists of (1) an anterior element combining a spherical lens for refractive power with a cubic surface for the varifocal effect, and (2) a posterior element with a cubic surface only. The focal length of the IOL lens changes when the superimposed refractive elements shift in opposite directions in a plane perpendicular to the optical axis. The ciliary muscle will drive the accommodation by a natural process of contraction and relaxation. Results of ray-tracing simulations of the model eye with the two-element intraocular lens are presented for on-axis and off-axis vision. The configuration of the lens is optimized to reduce refractive errors as well as effects of misalignment. A prototype with a clear aperture of ~5.7 mm is manufactured and evaluated in air with a Shack-Hartmann wave-front sensor. It provides an accommodation range of ~4 dioptres in the eye at a ~0.75-mm lateral displacement of the optical elements. The experimentally measured on-axis optical performance of the IOL lens agrees with the theoretically predicted performance.
©2006 Optical Society of America
Replacing the lens of the human eye with an artificial intraocular lens (IOL) in cataract surgery or for other medical reasons restores clear vision, but results in a significant loss of accommodation, i.e. ability of the eye to focus on near objects [1–3]. Restoration of accommodation by using variable focal-power intraocular lenses is of vital importance in modern ophthalmic practice [4–6].
In the young, phakic eye (with the natural crystalline lens present) the accommodation is accomplished by contraction of the ciliary body which releases tension on the zonular fibers holding the natural crystalline lens in dynamic suspension [2, 7–9]. The anterior surface of the crystalline lens moves forward whereas the posterior surface moves slightly backward . Due to its natural elasticity the lens reverts to a lens of stronger curvature with a higher focal power. When the eye is focused on infinity, the ciliary muscle is relaxed and regains a larger diameter, the lens is flattened and its focal power drops to a lower value [7–9]. For a young individual the accommodation range of a phakic eye may exceed 10 diopters (D) [1, 11]. This range is reduced significantly with age and most persons become presbyopic due to lens hardening and a resulting loss in accommodative power of the eye at ~35 years.
Cataract, possibly leading to complete blindness, generally develops in persons over 60 years old. Cataracts are optical defects of the natural lens which can increase scattering of light and light absorption, both effects leading to a clouded and increasingly opaque natural lens. To restore vision, an eye surgeon can replace the natural lens by an artificial lens with the optical power adjusted for sharp distant vision (~6 m). However, after surgery the accommodative power of the eye is largely lost. Some pseudo-accommodation, or limited accommodation, can occur in an IOL implanted eye [8, 12, 13]. This accommodation is not based on deformation of the lens itself but on other factors such as movement of the artificial IOL along the anterior-posterior axis by a distance of 0.25-1 mm [10, 12] due to contraction of the ciliary body and possibly some lengthening of the optical path of the eye due to ciliary muscle contraction. Pseudo-accommodation can add 0.3-1.9 D  of accommodative power.
The desired focal power of the standard, non-accommodative IOL is determined before surgery with aim to correct the eye for emmetropia, i.e. having the sharp image on the retina conjugated with a distant target in an unaccommodated eye. Generally, this requires an IOL in the +15-25 D range. Persons with IOL implanted eyes need additional optical correction, (e. g., progressive spectacles), to extend the range of vision.
To restore accommodation of the human eye several configurations of accommodative IOLs have been proposed [4, 14]. These designs concern mainly fixed multifocal IOLs [5, 15, 16] and axial-traveling monofocal IOLs [5, 6, 14].
The multifocal IOLs have several foci which produce superimposed images on the retina. The brain is supposed to interpret the images as separate, selecting the one relating to the object of interest, and suppressing the other [15–16]. A single-piece IOL with a pinhole aperture can be also attributed to the multifocal implants as it provides high depth of focus . Multifocal IOLs are largely similar to standard single-focus IOLs with regard to surgical procedures. In comparison with monofocal IOLs, however, multifocal IOLs cause a reduced contrast sensitivity [12, 15, 16] and tend to produce pronounced halos, flare and glare [12, 17].
Accommodating foldable IOLs with optical elements traveling along the optical axis of the eye have been reported in [5, 6], and references therein. The optical elements, or one of them, travel in the longitudinal direction back and forth due to contraction and relaxation of the ciliary muscle and, likely, due to increased pressure generated by the posterior part of the eye . Most single-element movable IOLs have flexible haptics with a spring action positioned in the capsular bag of the eye [6, 19]. However, it is not clear yet whether the eye can provide sufficient forward distance to generate satisfactory accommodation. Theoretically, a 1-mm traveling of a single-element monofocal IOL of +19 D results in an accommodation of ~1.2 D [5, 6]. In practice, a mean accommodation of 1.2 D for a 0.63-mm mean change in anterior chamber depth was obtained in clinical tests . Stachs et al. reported a mean accommodative amplitude of 0.44 D for a mean forward shift of 0.13 mm measured with an ultrasound biomicroscope . Other studies confirmed these data by concluding that single-element lenses result in ~0.5 D accommodation . Despite extensive studies with movable IOLs, the clinical results are not unequivocal , ranging from no accommodative effect  to some degree of accommodation [20–22, 24, 25].
A dual-element accommodating foldable IOL can consist of either two positive lenses or a combination of a positive and a negative lens [3, 5]. The optical arrangement follows that of traditional refracting telescopes. Generally, a lens with a high focal power travels along the optical axis, resulting in increased accommodative range over the single traveling lens configuration. The refractive error caused by the high focal power lens is subsequently corrected by a second lens. Precise positioning of the lenses in the eye is required because misalignment of the parts will drastically reduce the optical quality of the combination. An accommodation of 2.2 D can be obtained theoretically at a 1 mm forward shift of a +32 D anterior lens followed by a -12 D correcting posterior lens, and 0.5 mm distance between the lenses . Recently a two-element accommodating IOL has shown to provide ~2.75 D of accommodation in clinical trials [6, 14]. Note that 0.3-1.9 D of pseudo-accommodation is likely to contribute to the accommodating effect.
Other designs of accommodating IOLs include mimicking the natural lens by injection of elastic polymers or lens-shaped polymer bag filled with fluid into the capsular bag of the human eye [14, 26]. These designs are now in animal trials of which initial levels of 3-5 D accommodation are reported .
The majority of reported accommodative IOLs are currently in the early stages of development and have not become widely-accepted in ophthalmic practice . Some carry an increased risk of optical distortions of the eye due to large surgical incisions, while others show insufficient contrast over the range of accommodation, reduced peripheral vision, and high sensitivity to misalignments. In most cases, the optical design of the IOLs is not optimized to reduce the influence of monochromatic and chromatic aberrations on the overall performance of the eye, although their correction significantly improves the visual acuity .
Ideally, an accommodating IOL should provide excellent on-axis optical performance of the eye, compared with that of an emmetropic eye, weak dependence of the retinal image quality on viewing field angle and wavelength, and sufficient accommodation for near work, e.g. reading. It should preferably be manufactured from approved IOL materials according to industry standards and allow implantation into the eye using standard surgical procedures. In practice, the IOL has to comply with the EN/ISO standard  and provide an accommodation range of ~4 D [3, 5, 8].
In this paper, we present a novel two-element accommodative lens for intraocular applications [30, 31]. The accommodative IOL is based on a varifocal Alvarez lens . It provides high-contrast imaging over the whole range of accommodation and is suitable for use in the human eye. An accommodative IOL with this optical configuration is presently being evaluated in a pre-clinical stage of development for cataract and presbyopic patients .
2. Dual-element Alvarez-type IOL
Alvarez  invented a two-element varifocal lens which changes its focal power by shift of elements in the plane perpendicular to the optical axis. Such a lens consists of two superimposed phase plates shaped as cubic polynomials. In Cartesian coordinates the sag of each refractive element is given by:
where A is the amplitude of the cubic terms, x is the horizontal meridian, y is the vertical meridian, and z corresponds to the direction of light. The focal length ( F ) of the resulting parabolic lens depends on the elements shift ( ∆x ) in the X-direction, i. e. the elements are shifted by ∆x in opposite directions along the x-axis, and the difference of refractive indices between the lens material (n) and the surrounding medium ( n′ ): F = 1 /[4∆x (n - n′)] . For an accommodative IOL the constants A and ∆x can be determined using the optical parameters of the eye.
The anatomical variation of the focal power of the human crystalline lens ranges from +17 to +26 D with a mean value of ~19 D . The amplitude of accommodation changes from ~15 D in children to ~0.5 D in old age [1, 2, 34]. An emmetropic eye with an accommodation of ~4 D, providing sharp vision from infinity (considered to be 6 m for visual acuity measurements) to 25 cm, is generally considered to be more than adequate to suit the majority of patients [3, 5, 8]. In accommodation, the radius of the ciliary muscle ring varies with an amplitude ∆r 0 = 0.79 mm . Thus, the lateral shift ∆x of each IOL element should not exceed ∆r 0. The pupil of the eye is positioned directly in front of the crystalline lens and its diameter determines the clear aperture of the IOL. Under average illumination conditions, the pupil has a diameter of 3-4 mm . The maximum size of the IOL is determined by the equatorial diameter of the crystalline lens that has a mean value of 8.86 mm . Using these data, and assuming that the varifocal lens is immersed in a medium with a refractive index n 0 = 1.3374 (aqueous of the human eye), we arrive at A = 0.0109 mm-2 and ∆x 0 = 3.56 mm. In this model, a 4-D accommodation can be achieved by shifting the refractive elements in opposite directions by ξ ≅ 0.75 mm relative to the preset value ∆x 0. Here, we set ∆x = ∆x 0+ + ξ.
For a varifocal Alvarez lens with a diameter of 6 mm, the relief depth of each element reaches ~1.02 mm, resulting in a bulky system with a total thickness of more than 2 mm. The IOL thickness can be reduced by substituting the average tilts from the shape functions defined by Eq. (1). The shape of the optical element shifted by ∆x 0 can be expressed as:
After subtracting the tilt contribution .Ax∆ from Eq. (2) and neglecting the last term which does not depend on coordinates, the profile of the shifted element becomes a combination of an Alvarez surface and a paraboloid. The parabolic term provides the focal power of the IOL while SA (x, y) allows for accommodation.
One may suppose that the parabolic shape is not optimal for intraocular implants since all optical surfaces in the human eye are highly curved aspherics. Actually, experimental studies of ocular refraction show that the cornea, the most powerful optical element of the eye with a focal power of +38-48 D , has a prolate asphericity of both the anterior and posterior surface . In turn, the anterior and posterior surfaces of the natural crystalline lens can be described with hyperbolic and parabolic profiles, respectively, see  and references therein. In mathematical terms the asphericity of the eye can be described in even-order polynomial terms r 2, r 4, etc. [36, 37] or by using a conicoid as a shape function. In the latter case, a conic parameter characterizes the degree of asphericity [35, 38]. Typical values of aspheric parameters for the refractive surfaces of the human eye are summarized in Table 1, the data are compiled from [36–39].
With these data, we can expect the general expression for the optimized profile of an IOL should also include aspheric terms:
where, ; R is the radius of curvature; k is the conic parameter that specifies the type of conicoid; an is the (2n+2)-th order polynomial coefficient, in most cases n ≤ 2 . Note that the simultaneous use of the conic constant and polynomial series in Eq. (3) is redundant. Actually, expansion of the quadric part of S(x,y) in power series of r yields even-order polynomial terms.
Implementation of an accommodative IOL with the shape of refractive elements specified by Eq. (3), in principle, allows reducing the monochromatic aberrations for a certain distance of vision by an appropriate choice of aspheric parameters. But the quality of vision at other distances may degrade as a result of lateral move of the elements. Apart from defocus, the accommodative shift of the aspherics may lead to higher-order aberrations of the IOL. A possible solution to this problem is to use an additional refractive element fixed in the eye to correct the high-order monochromatic aberrations. The additional element can also include a spherical surface to provide the required focal power in the IOL. In the present work, however, this configuration is not considered.
The values given in Table 1 for the crystalline kens can be substituted into Eq. (3) as a zero-order approximation for the aspherics in the IOL shape. Further optimization of the IOL profile requires exact ray-tracing analysis of the model eye with the accommodative intraocular implant.
3. Ray-tracing analysis of Alvarez-type refractive surfaces
Virtual ray-tracing is based on geometrical principles and consists of two steps: (1) propagation of a ray from one surface to the next in sequence along the optical axis Z, and (2) the refraction of an incoming ray at a surface .
For the Alvarez surface defined by Eq. (3) the propagation analysis is performed by geometrical considerations of lines (ray paths in an isotropic and homogeneous medium with the refractive index n 1) intersecting the surface z = S(x,y).
The intersection point M(x,y,z) of the incoming ray a with the surface z = S(x,y), and its optical path t can be found by iteration :
where N is the total number of iterations; ai is the i-th component of a . The starting point M0(x 0,y 0,z 0) of the ray a belongs to the previous optical surface. It specifies the zero approximation for the optical path t (0) in Eq. (4).
Let b be the unit vector of the refracted ray that intersects the surface z = S(x, y) at the point M(x,y,z) and q is the unit normal at M. The Cartesian components of b can be determined using Snell’s law of refraction that relates the tangential components of a and b :
where the subscript i = x,y,z denotes the respective projections; the values n 1, n 2 specify the refractive indices in front and behind the interface; a ∥ = (aq) is the projection of the incident ray on the surface normal. Eq. (4) requires a ∥ < 0, otherwise the incident ray a misses the surface. For the refractive surface z = S(x,y) defined by Eq. (3), the normal unit vector q at the point M can be determined as follows:
or in terms of Cartesian components:
being the normalization factor, and . The Cartesian components of the refracted ray b can be finally calculated by substituting Eqs. (8) into Eq. (6). Note that bz must be a positive number as the light propagates along the optical axis (in positive Z-direction).
4. Model eye with the accommodative IOL
In optical simulations we use the wide-angle model eye described by Pomerantzeff et al. [36, 37]. This model accounts for aspherics in terms of even-order polynomials in r [ Eq. (3) and Table 1]. The conic parameter k is assumed to be zero. Other parameters of the model eye are summarized in Table 2. The listed values are based on in vivo measurements and agree with the data reported by Le Grand et al.  and Koijman .
Figure 2 shows the schematic of the model eye with a two-element varifocal Alvarez-type element. The configuration of the accommodative implant is depicted in the inset. The arrows show the positive directions of the optical elements shift ∆x resulting in the increase of the IOL focal power.
We suppose the implant with an aperture D = 6 mm is placed in the capsular bag exactly behind the pupil of the eye, i. e. the distance from the anterior cornea to the IOL is 3.6 mm. During accommodation, the optical elements move in opposite directions at a distance of 2∆x≤1.5 mm.
Simplified models of the human eye are widely used in engineering and scientific applications [36–38, 42, 43]. Assumption that the retinal region of maximum neural resolution (fovea) is centered on the optical axis of the eye (α = 0° ) is a commonly used approximation for IOL design [5, 6]. The temporal displacement of the fovea by the angle α (Fig. 2), however, is more consistent with the anatomical structure of the human eye and should be accounted for when assessing the visual performance. This is important for IOL applications because α significantly affects the magnitude of off-axis aberrations on the fovea . For modeling, we used a value α = 5° [34, 45].
The optical parameters of the IOL have been optimized to keep the image of a point object focused on the retina over a 4-D accommodation. The ray-tracing simulations take into account that the line-of-sight makes an angle α with the optical axis of the eye, as shown in Fig. 2. We assume that the line of sight (fixation axis) and visual axes coincide . According to the definition adopted by VSIA taskforce members , the line of sight passes through the center of the eye’s entrance (E) and exit (E') pupils and is equivalent to the path of the foveal chief ray (Bennett and Rabbetts defined this axis as the visual axis ). Optimization of the IOL has been performed by minimizing the RMS spot size on the fovea with respect to the chief ray at different distances ( L ) from the eye to the object of regard.
In order to account for the effects of chromatic aberrations in the model eye, the refractive indices of the ocular media and IOL in the visible spectrum have been approximated by interpolation formulae obtained by fitting the experimental data.
For the implant made of copolymer 2-hydroxyethylmethacrylate/ methylmethacrylate (HEMA/MMA) with UV blocker AEHB and containing 26% of water, the dependence of the refractive index versus wavelength λ (measured in microns) was modeled using the Schott interpolation expression:
Expansion coefficients in Eqs. 9 and 10 for the copolymer, cornea, aqueous and vitreous are summarized in Table 3. In particular, with these parameters the refractive index of the HEMA/MMA copolymer becomes n = 1.45997 at λ = 0.55 μm, which is near the peak (0.555 μm) of the photopic curve .
5. Simulation results
Ray-tracing analysis of the optical elements of the accommodative IOL and optimization of their shape for maximum performance of the model eye was carried out using Zemax optical design software (Focus Software, Inc.). The parameters of the model eye are listed in Table 2. Eqs. (9) and (10) together with Table 3 specify dispersion of the ocular media. Eqs. (4–6) were used for ray-tracing of the refractive surfaces defined by Eq. (3). We implemented these equations as a set of external functions in a compiled library file linked into Zemax dynamically.
The clear aperture D = 6 mm of the accommodative IOL allows modeling of the eye with the pupil diameter up to ~5 mm, representing vision under low illumination. The mechanical design of the IOL requires the flat inner surfaces of the optical elements to be separated by a distance of d = 0.5 mm (Fig. 2). This distance can be further optimized along with other parameters of the IOL. However, the distance should probably not be less than ~0.1 mm to prevent adhesion of the flat optical surfaces.
For a pupil diameter of 3 mm, considered as an average value at normal illumination, and fixed d = 0.5 mm, the ray-tracing analysis of the model eye results in the following shapes of the anterior ( S 1 ) and posterior ( S 2 ) IOL parts:
Where h 1 = 1 mm and h 2 = 0.35 mm are the thicknesses of the optical elements at their centers; A 1 = 0.012 mm and A 2 = 0.0148 mm-2 are the corresponding amplitudes of Alvarez terms; R = 6.89 mm is the radius of curvature; a 1 = -7×10-5 mm-3, a 2 = 1×10-5 mm-5 are the aspheric coefficients. Figure 3 illustrates the surfaces of the IOL optical elements, as given by Eqs. (11) and (12) with the obtained parameters.
In calculations, we assumed the anterior element provides a focal power of +17 D of the accommodative IOL at ∆x = 0 and corrects the aberrations due to the corneal asphericity for the eye focused at a distance L = 6 m. Under these conditions an accommodation range of 4 D is obtained with a shift of 2∆x ≅ 1.52 mm. Pupil apodization (the Stiles-Crawford effect ) was not taken into account because this phenomenon has a retinal basis [42, 47].
To estimate the optical performance of the model eye with the accommodative IOL, sine-wave polychromatic modulation transfer functions (MTFs) were calculated for a 3-mm pupil. Figure 4(a) shows the average of sagittal and tangential MTFs for the model eye focused at L = 25 cm and L = 6 m, depicted as a green line with triangles and a blue line with open squares, respectively. In both figures the results obtained in the literature for both model and real eyes are added for comparison. These polychromatic MTFs were simulated by using wavelengths of 0.480, 0.546, and 0.644 μm, which are Fraunhofer F', e, C'-lines covering the visible spectrum. The weight factors accounting for the photopic spectral sensitivity function  of the retina were 0.15, 1, and 0.15, correspondingly. As seen from the plots, the calculated polychromatic MTFs reasonably agree with the published data on polychromatic MTF of the nonparaxial model eye (4-mm pupil) described by Liou and Brennan  and monochromatic MTFs measured by Artal and Navarro in young emmetropic eyes (3-mm pupil) . A decrease in visual contrast compared to the results of Artal and Navarro may be due to ocular chromatic aberrations . For the model eye of Liou and Brennan , the difference in polychromatic MTS can be caused by a difference in pupil diameters. Thus, the accommodative IOL provides the foveal image quality of the model eye in an accommodation range of 4 D close to that of an emmetropic eye.
Figure 4(b) shows the longitudinal chromatic aberration (LCA) centered at 0.589 μm of the model eye with the accommodative IOL (solid line) and its comparison with the published data [35, 42, 44]. Some discrepancy in LCA values is likely to be related to different dispersion formulae used for calculations of ocular media.
The combined effects of longitudinal and transverse chromatic aberrations on polychromatic MTFs which arise for off-axis fields of view in the horizontal and vertical planes are presented in Fig. 5(a) and (b), respectively. In simulations the eye was focused at L = 6 m (∆x = 0). Angles αx and αY specify the field of view. We set αx > 0 for nasal visual fields (see Fig. 2) and α32 > 0 for fields above the horizontal plane, i. e. lower cornea in image space.
Computations performed for the horizontal fields of view exhibit a strong dependence of image contrast on the eccentricity direction. For the field αX = -5° (red line with diamonds) the visual contrast is much larger than for the line-of-sight (αX = 0°), whereas at αX = 5° (blue line with open squares) the contrast falls off rapidly as the spatial frequency increases. Figure 6 presents the simulated (with polychromatic light) images of the Snellen acuity test on the retina for the central and peripheral fields of view. The angular size of letters on the bottom line is 5′ (minutes of arc). Their vision corresponds to a visual acuity of 20/20.
Figure 5(b) shows the polychromatic MTFs for the vertical fields of view. The vertical eccentricity causes symmetrical change in the polychromatic MTFs of the model eye containing the varifocal IOL but the image contrast decreases rapidly with αY . To the contrary, the optical system of the human eye follows a wide-angle lens design and has a nearly symmetric MTF varying only gradually with eccentricity, although the imaging quality in the fovea is far from the diffraction limit . An obvious way to improve the off-axis performance of the eye with the varifocal IOL is to tilt the implant about Y-axis in the direction of the fovea (see inset in Fig. 2). However, it should be noted that this approach will require a very precise and, intentionally, tilted positioning of the IOL in the capsular bag of the eye.
Accommodation of the IOL introduced in this paper leads to the lateral shift (along the horizontal meridian) of the focal spot on the retina due to asymmetric shapes of the IOL refractive surfaces determined by Eq. (11) and (12). An accommodation of 4 D results in a focal spot displacement of ~0.19 mm. This is equivalent to a ~0.65° change from the central (foveal) field of view. We are studying the feasibility of the IOL design and implantation to achieve the optimized foveal vision for the whole range of accommodation.
6. Compliance with EN/ISO-11979-2 rules and positioning tolerances of the IOL
The EN/ISO standard  specifies the minimum acceptable value of contrast for an IOL implanted model eye. In particular, it requires a monochromatic (at λ = 0.546 μm) MTF to be above 0.43 at a spatial frequency of 100 cycles/mm for a 3-mm aperture. If the IOL design allows only smaller values, the measured MTF at 100 cycles/mm should be above 70% of the theoretically possible value and should always exceed 0.28.
To characterize the optical performance of the accommodative IOL according to the EN/ISO standard we employed the model eye described in Tables 1 and 2 with the assumption that the line-of-sight matches the optical axis (α = 0°). The latter complies with the description of the model eye given in the EN/ISO standard for measuring the MTF. In the geometry of the model eye a spatial frequency of 100 cycles/mm corresponds to ~29.75 cycles/deg.
Figure 7(a) shows on-axis monochromatic MTFs (average of sagittal and tangential components) of the model eye with the accommodative IOL optimized for the best foveal vision at α = 5° [Eq. (11), (12)] which is consistent with the structure of the human eye. In this case, the MTF curves obtained for the eye focused at L = 6 m (green line with open circles) and L = 25 cm (red line with diamonds) virtually coincide.
As can be expected, the optimal profiles of the IOL optical elements obtained at α = 5° differ from those determined in the geometry of the EN/ISO standard, i. e. at α = 0°. Actually, the optimal IOL parameters in this geometry are: A 1 = 0.012 mm-2, A 2 = 0.0143 mm-2, R = 6.93 mm, a 1 = 1.1×10-4 mm-3, a 2 = 9×10-6 mm-5. Figure 7(b) shows that, with these parameters, the two-element IOL provides near-diffraction-limited performance for the eye focused at L = 6 m (green line with open circles) but a lower performance at L = 25 cm (red line with diamonds).
Ray-tracing calculations for the axially symmetric model eye (α = 0°) indicate a strong dependence of the MTF values on the aspheric parameters a 1 and a 2. These aspherics can be optimized to achieve near-diffraction-limited image quality for a particular distance of accommodation L , as it demonstrated in Fig. 7(b) for L = 6 m (green line with open circles). At the same time, the monochromatic MTF of the model eye with the fovea shifted temporally by α = 5° from the optical axis is less sensitive to the IOL aspheric parameters used in Eq. (11) and almost does not change with L . The on-axis monochromatic MTFs for this case are depicted in Fig. 7(a).
The positioning tolerances for the accommodative IOL as a unit can be estimated within the requirements of the EN/ISO standard. Angular and lateral misalignments of the IOL are considered in the geometry depicted in Fig. 8. The implant is assumed to translate along X-and Y-axes or rotate about them with respect to the apex of the front optical element.
Figure 9 shows the dependences of the on-axis monochromatic MTF (average of sagittal and tangential) at 100 cycles/mm on angular misalignments βX and βY of the IOL about X-and Y-axes, correspondingly.
Rotation about the X-axis primarily causes astigmatism, whereas rotation about the Y-axis produces mainly defocus. Blue solid curves in Fig. 9 correspond to the model eye containing the IOL with aspherics and optimized for the best foveal vision at α = 5°, as specified by Eqs. 11 and 12. Note that the MTF reaches a value of 0.43 only for βY > 0.32°. Neglecting aspheric terms in this model leads to an increase of the on-axis MTF (red curves with squares). For comparison, green curves with open circles represent the on-axis MTFs of the model eye with the IOL optimized for on-axis vision (α = 0°), i. e. in the geometry of the EN/ISO standard.
Figure 10 illustrates the effect of lateral displacements δx and δy on the on-axis monochromatic MTF for the same cases. The shift in the X-direction generates defocus and astigmatism, while the Y-axis shift results mainly in astigmatism. From the MTF plots in Figs. 9 and 10 it can be concluded that the IOL optimized for the model eye, taking into account the angle α between the optical axis and the line-of-sight, is not optimal when the model eye in the EN/ISO standard geometry is used.
According to Figs. 9 and 10, the permissible tolerances for the accommodative IOL satisfying the EN/ISO standard criterion MTF > 0.28 are: | βX |≤2.1°, -1.2° ≤ βY ≤ 4.7°, -0.2 mm ≤ δx ≤ 0.7mm , and | δy |≤ 0.3 mm . A threshold MTF value (at 100 cycles/mm) of 0.43 can be achieved with the IOL for on-axis vision when optimized without aspherics for α = 5°.
A possible way to extend the range of permissible misalignments is to reduce the distance d between the IOL optical elements (Fig. 8). For example, when d = 0.1 mm the condition MTF > 0.28 yields: | βX |≤3.7°, - 8.6° ≤ βY ≤ 6.4°, -0.4mm≤δx≤1mm, and | δy |≤ 0.5 mm. In this case, an MTF value at 100 cycles/mm exceeds 0.43 within the following tolerances: | βX |≤ 0.8°, - 0.9° ≤ βY ≤ 4°, -0.02mm ≤ δx ≤ 0.3mm, and | δy |≤ 0.1mm . We are currently working on novel IOL materials which will allow very close distances between the optical elements without causing adhesion of the surfaces in the eye.
7. Experimental evaluation of the dual-optic IOL
To demonstrate the feasibility of two-element Alvarez intraocular implants, a prototype of the accommodative IOL optimized for on-axis vision (α = 0°), as required by the EN/ISO standard was manufactured and characterized experimentally. The optical elements of the IOL with a clear aperture of ~5.7 mm and an outer diameter of ~11.7 mm were made of HEMA/MMA copolymer by diamond lathing (Procornea Nederland B.V., The Netherlands). The surfaces are defined by Eqs. 11 and 12, respectively, with the following parameters: A 1 = 0.012 mm-2, A 2 = 0.0143 mm-2, R = 6.87 mm, a 1 = 0 mm-3 and a 2 = 0 mm-5. Thus, for experimental characterization we implemented the IOL prototype with a spherical anterior surface.
The imaging quality of the IOL was measured in air without simulating the in vivo environment of the anterior chamber of the eye. In order to evaluate the accommodation range and determine the aberrations produced by the IOL versus the lateral shift ∆x of optical elements, the wave-front of a collimated laser beam was measured after passing through the two-element lens. The wave-front characterization was performed using an imaging system due to high focal power of the accommodative IOL in air (~73 D at ∆x = 0 mm) and large variation of its anterior and posterior surface relief.
Figure 11 shows the experimental setup. A collimated laser beam of a He-Ne laser, operating at λ = 0.633 μm, passes through a 5.4-mm diaphragm D1 (entrance pupil) and illuminates the anterior optical element of the IOL. The anterior and posterior optical elements are mounted on translation stages, which allows precise positioning in the X- and Z-directions. An additional ~0.5-mm diaphragm D2 filtered out the scattered light. After the IOL, the diverging beam is collimated by a 50-mm focal length objective O1 and then its wave-front is measured with a Shack-Hartmann sensor placed in the plane conjugate to the entrance pupil after passing the IOL. The wave-front sensor with a hexagonal 127-subaperture lenslet array and the wave-front analysis software were supplied by OKO Technologies .
The distance between the optical elements of the accommodative lens is set to be d = 1.2 mm in the experiments as their minimum separation is limited to about 1 mm by mechanical clamps. For nearly on-axis imaging, the IOL performance depends only weakly on d. Distances between the optical parts, as depicted in Fig.11, were: L 1 = 14 mm, L 2 = 10 mm, L 3 = 47 mm and L 4 = 47 mm. The RMS phase error was measured to be ~0.21 waves and the peak-to-valley phase error was ~1.16 waves after the IOL initial alignment at ∆x = 0 mm. The corresponding theoretical values obtained in Zemax simulations were ~0.11 and ~0.3 waves.
In the plane of the Shack-Hartmann sensor, the incoming wave-front can be expressed as a Zernike series :
where Zi is the i-th Zernike polynomial, N is the number of decomposition modes (N = 44 for the OKO Technologies software and wave-front sensor), R 0 is the decomposition radius of the wave-front sensor, and r is the transverse coordinate vector, | r |≤ R 0. The ordering of the Zernike modes proposed by Noll  is used in Eq. (13). For the decomposition radius R 0 = 2.245 mm, that is determined by the wave-front sensor aperture and the reconstruction algorithm, the effective diameter of the IOL seen by the Shack-Hartmann sensor is R′0 ≅ 1.23 mm.
Figure 12 shows the change in defocus (a 4) caused by the transverse shift ∆x of the optical elements and the calculated variation of the IOL focal power. The calculation was performed with the paraxial model of the imaging system, which results in the following expression for the variation of the IOL focal length ( ∆F ) with ∆x :
where Fo is the focal length of the objective O1 (Fig. 11). Theoretical dependence of ∆F on ∆x , depicted by the solid curve in Fig. 12(b), was evaluated directly by the use of Eqs. 11 and 12 with the IOL parameters specified above. The refractive index of HEMA/MMA copolymer in air was estimated to be n = 1.503 (for a 26% hydrated copolymer, however, n = 1.46).
Plots in Fig. 13 represent astigmatisms (a 5,a 6), comas (a 7,a 8), trefoils (a 9,a 10) and spherical aberration (a 11) produced by the IOL in the imaging system as functions of ∆x . The solid curves depict the simulation results obtained with Zemax software. Defocus is, as expected, the dominant aberration of the accommodative IOL (Fig. 12a) and astigmatism the main factor limiting image quality (Fig. 13). Some discrepancy between measured and simulated data can be attributed to misalignment of the IOL optical elements. In the experiments, the maximum wave-front RMS error, i. e. deviation of the measured wave-front from the calculated spherical wave-front caused by astigmatism and coma, does not exceed ~0.7 waves at ∆x = 0.75 mm, while the IOL focal power changes by ~40 D. It should be noted that astigmatism can not be corrected for by the eye, but that defocus can likely be corrected by the eye itself by accommodation.
The optical setup in Fig. 11 allows measurement of the shape of the wave-front generated by the dioptric (anterior) Alvarez element. Figure 14 shows the measured and calculated wave-fronts excluding tilts and defocus. In the plots, an area of the dioptric optical element of 1.23 mm in diameter is mapped onto a decomposition area of 4.45 mm in diameter. Theoretical plot in Fig. 14(b) was calculated using Eq. (11), in which tilts and spherical term responsible for defocus were excluded. In this case, the expression for the wave-front amplitude (in waves) becomes:
The parameters employed in Eq. (15) for the dioptric optical element in air are specified above. As can be seen from Fig. 14, there is an excellent agreement between the theoretical and experimental results.
We have demonstrated a novel accommodative IOL based on a two-element varifocal cubic lens . The IOL consists of (1) a plano-convex anterior element combining spherical, even-order aspheric and cubic (third-order) terms, and (2) a posterior optical element comprising only third-order terms as a shape function. Such a configuration of the IOL allows a high dioptric power of ~17D (in the eye) and provides a nearly emmetropic on-axis vision over a 4-D accommodation. The IOL parameters have been optimized for the model eye taking into account corneal asphericity (in terms of 4th and 6th order polynomials) aw well as temporal displacement of the fovea with respect to the optical axis. The calculated on-axis MTF of the model eye containing the accommodative IOL agrees with earlier experimental results on the retinal image quality [48, 49] and simulated data obtained with the anatomically accurate model eye . The longitudinal chromatic aberration of the model eye with the novel two-element IOL fits well with the experimental and with calculated results for the human eye [35, 38, 42, 44].
In comparison with a phakic eye which exhibits only a weak dependence of the retinal image contrast on the visual field , off-axis vision of the eye with the accommodative IOL is sensitive to the angular position of the viewed object, especially in the horizontal plane. This effect can, theoretically, be reduced by tilting the IOL horizontally toward the fovea. Alternatively, the peripheral vision can be improved by minimizing the distance d between the optical elements of the IOL.
The positioning tolerances of the accommodative IOL, as a unit, were estimated using the requirements of the EN/ISO standard . It was found that the permissible angular and rotational misalignments were highly dependent on the model eye and the IOL configuration. In particular, for the model eye with the temporally displaced fovea the two-element IOL has to be positioned with a high precision. However, decreasing d reduces this sensitivity significantly.
To the best of our knowledge, the IOL optimization taking into account the displacement of the fovea relative to the optical axis of the eye is performed for the first time. The optical elements of the accommodative IOL have been optimized to achieve the maximum image contrast on the fovea, which is located 5° temporally . In other terms, the RMS size of the focal spot has been minimized with respect to the foveal chief ray. This is in contrast to the EN/ISO-11979-2  standard which requires only high on-axis MTF values for IOLs with the model eye.
Our simulations show that the asphericity of the corneal surfaces of the human eye can be corrected for a particular distance of vision L by the anterior optical element of the IOL that contains aspheric terms. At other distances, the image quality on the retina remains reasonable. The aspheric parameters of the accommodative IOL are optimized for the eye focused at L = 6 m.
The feasibility of a two-element accommodative implant was demonstrated experementally. A prototype of the accommodative IOL with a clear aperture of 5.7 mm optimized for on-axis vision, as required by the EN/ISO standard was manufactured and characterized in air using a Shack-Hartmann wave-front sensor. In the experiments, the maximum RMS deviation of the measured wave-front from the calculated spherical wave-front, caused by astigmatism and coma, did not exceed ~0.7 waves for a shift ∆x = 0.75 mm, whereas the IOL focal power changed by ~40 D (~4 D in the eye).
Further studies on the accommodative IOL with optical configuration similar to that described in this paper are currently underway. Biological effects are of significant practical importance and are being studied in a series of medical tests. The capsular bag can react to the trauma of surgery by opacification, shrinkage and hardening of the capsular bag due to fibrosis. These reactions can negatively affect or even block the functioning of accommodative IOLs with movable parts. Presently we are evaluating the biological effects of various IOL designs described in this paper in series of animal trials to find the optimal IOL configuration for cataract and presbyopic patients .
A mechanical structure of the two-element IOL fitted in the eye and driven by the natural process of contraction and relaxation is shown in Fig. 15. The optical performance of this accommodative IOL in vitro as described in [29, 52] and optimization of its mechanical design are in progress as well as studies in vivo in which the optical and mechanical behavior of the lens is measured in the eye.
The authors thank Dr. Oliver Stachs (University of Rostock, Germany) and Dr. Jos Rozema (University Hospital Antwerp, Belgium) for helpful comments on the manuscript, and Puck Rombach for editing.
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