1. Introduction

The specification of modern multifocal intraocular and contact lenses is usually given as their power profile in diopters along a meridian, with the implication that the lens can be modeled by a series of concentric zones whose centers of curvature lie on a straight line. This constraint has been removed by recent design and manufacturing developments [1–3], adding a variable that can substantially impact the resulting retinal images. It is explored here as a feature of a lens with a single bifocal annulus, serving as an exemplar of the more general case of lenses with circular symmetry whose surface evolute does not coincide with the axis. Considering the situation in one meridian of the surface is then wholly representative.

The illustrative example allows a direct comparison of two different regimens: either the annulus has additional refractive power, or it has imposed on it a radial tilt, i.e., a direction difference of propagation of the beam with respect to that of the main lens. The proof-of-concept analysis, intended to serve as foundation of empirical and clinical studies, proceeds in terms of wavefronts. Assuming the lens to have neutralized the eye’s refractive anomalies, emerging from the eye’s exit pupil there will be a spherical wavefront whose center of curvature lies on the axis in the plane of the retina (Fig. 1). Adding plus power to the annulus will take the form of replacing the wavefront in this zone by one of higher curvature, and hence shifting its center of curvature forward but still on the axis. The off-axis strategy, on the other hand, introduces a radial tilt to the wavefront in the annular zone, shifting its direction of propagation so that the meridional centers of curvature form a ring around the axis.

 

Fig. 1. Meridional section showing schematically the light rays from the annular bifocal zone converging from the eye’s exit pupil to the retina in (right) a typical on- axis condition in which both the main lens portions and the annular zone with its plus ADD have the centers of their wavefront located on the axis but in different planes, and (left) an off-axis condition in which the rays from the annulus share their focal plane with that of the main lens but are deflected inwards so that their meridional centers of curvature form a circle surrounding the axis. Above each panel is a cross-sectional outline, not to scale, of the wavefront from a target at infinity as modified by the optical structure in the annular zone. On the right it becomes, for the width of the annulus, part of a plus power Add spherical wavefront (dotted arc). On the left, the off-axis condition, the annulus portion of the plane wave front is merely tilted (see also Fig. 4).

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This study is devoted to the comparison of the retinal image quality under various viewing conditions.

2. Optical formulation

To start with, an idealized formulation is adopted to clarify the basic layout. A lens corrects the eye’s refractive error so that a target at infinity – an impingent plane wavefront – is focused on the retina. In the absence of accommodation, aberrations and inhomogeneities and for monochromatic light and a round pupil, the retinal image arrived at by diffraction calculation would then be the familiar Airy disk. When the target is moved towards the eye, the focus shifts back causing increasing degradation of the retinal image.

The intent of multifocal lenses is to mitigate this degradation by zones with a plus power addition, the Add, to assist the focusing of close-by target. This causes a hybrid situation: when the target is at infinity, the light passing through the main portion of the lens is focused on the retina, but it is diluted in part by the light from the annulus, necessarily defocused. The opposite holds for close-up targets.

Remaining for a moment in the domain of the incoming plane wave, for purposes of diffraction calculations, this can be regarded as substituting, in an annular zone, a band of a spherical wavefront with its center on the axis and in a position governed by the Add power, F. If the inner and outer radii of the annulus are r₁ and r₂, then e_F(h) deviations from the main portion’s wavefront, applicable only within the annulus bounded by r₁ and r₂, is given [4, Eq. (6)-35], as a function of radial distance in the pupil h, by

It is at this juncture that an off-axis lens differs. Its annulus is constructed to impose a radial tilt on the wavefront, directing the beam either inwards or outwards. If t is the angle of tilt in radians, then e_T(h), the deviation along the meridians within the bounds of the annulus, is given by

The particular configuration gives to the annular ring the properties of an axicon, whose multifocal potential has been the subject of earlier studies [5,6].

If the target is moved from infinity towards the observer to a position 1/D meters, i.e., a target vergence D meters is introduced, all parts of the wavefront will have imposed on them the additional deviations associated with a superimposed spherical wavefront with its center at 1/D meters on the object side, i.e.,

For an observation distance 1/D, annulus power F diopters and annulus tilt t radians, the total wavefront deviation as a function of h, distance along a meridian, is therefore given by

e(h) = t (h – r₁) + h² F/2 - h² D/2 within the bounds of the annulus, i.e., r₂> h > r₁

and

As written, in these equations D and F are in diopters and all distance in meters, expressing the specific multifocal structure of the lens as deviations from planarity in the impinging object-sided wavefront which is then handed over and optically processed by the refractive components of the lens and eye to generate the retinal image.

It may be noted parenthetically that the equation, when written in the form

3. Methods

3.1 Optical properties of corrections

Given the stimulus and lens specification, the next step is the retinal point-spread function, from which all other image measures are derived. As an initial step, this will be performed using a model of an unaccommodated, unaberrated eye with homogeneous media, monochromatic light (λ = 555 nm) and round pupil. Later, an attempt is made to relax these restrictions, but examining this idealized situation provides a best-case estimate. Although for large defocus, ray optics provide serviceable estimates, the more rigorous diffraction approach is here adopted because ray optics does not adequately cover hybrid cases of multiple zones and small net defocus. The wavefront deviations, multiplied by 2π/λ, are then factored in as phase errors in the diffraction calculations.

Lenses are circularly symmetrical with the axis of their central zone passing through the center of the pupil of the eye. For purposes of illustrating the computational approach and its results, a typical generic bifocal is chosen with a 3 mm diameter central zone, surrounded by an annular bifocal zone 0.5 mm wide with a 3 D Add, it in turn surrounded by the distance correction to the full pupil diameter, initially set at 5 mm.

3.2 Point-spread functions computed by diffraction theory

A complete characterization of the imagery is afforded by the point-spread function, the retinal image of a point target. In diffraction theory it is derived by the superposition at each retinal point of the summed amplitudes of electro-magnetic disturbance arriving there from all points of the wavefront emerging into the eye’s image space through the exit pupil [7]. When squared, it gives the point-spread function. Of relevance in the current context is the map of relative phase, the deviations across the full pupil from the default situation of the ideal case of the sphere centered on the retina. The computations were carried out in MATLAB and involved digital summation of the contributions at retinal points 0.1 arcmin apart, from pupil locations in 0.01 mm steps, This fine computational mesh assured the recognition of the minute phase differences between on and off-axis conditions.

In a separate, parallel computation, the self-convolution of the pupil aperture amplitude and phase distribution – here needing operation in the domain of complex numbers – gives the optical transfer function for this optical system, showing the demodulation experienced by sinusoidal target patterns as a function of spatial frequency.

3.3 Measures of image quality

The point-spread function provides the full information of image degradation, but its detailed configuration is subject to minor variations in the parameters. It is often helpful to collapse the multifaceted manner in which wavefront deformations leads to degradation of the retinal image into one or two parameters. Just how knotty the problem is can be seen in Thibos et al.’s [8] listing of 31 numerical indexes characterizing point-spread distribution patterns. Two ways of easily visualizing image quality were chosen here: edge- and bar-spread functions. The first depicts the retinal light distribution in the image of a sharp black/white border, arrived at by double integration of the point-spread function. The shallower the slope at the dark/light transition the more blurred the image. Normalized edge-spread, however, fails to make explicit the contrast reduction of the image of small objects, a significant feature of degraded images. This is, however, captured by the light distribution in the image of a narrow bar, the width of a limb of a 20/40 (0.3 log MAR) Snellen letter. In addition, four single-value indexes of image quality are derived: A, steepness (reciprocal of gradient) at the inflexion point at an edge, an indication of resolving capacity; B, area under the optical transfer function, C, proportion of light of a blurred point-spread function that falls in the area covered by an idealized point image, a measure of image compactness [9], and D, contrast of the image of a 2 arcmin wide bar, a measure of the challenge to small target detection when the image is blurred.

3.4 Through-focus analysis

The primary intent of prescribing multifocal lenses is increase in depth-of-focus, that is, to extend the range of object distances for which the retinal image is minimally degraded. Given an object distance for which the eye is fully corrected, of interest is the image quality for other distances, expressed in the reciprocal measure of diopters – the defocus. A common depiction of the performance of visual aids in this regard is the “through-focus” graph [10], a figure showing a measure of image quality as a function of defocus in diopters. This is here realized in two ways. One is by arraying of the bar-spread function (showing the degradation both in contrast and crispness), and the other by graphing the values of the four degradation indexes A-D described above.

4. Results

All the components of the analysis have now been assembled. The diameter of the full pupil, the inner and outer radii and the additional refractive power of the annular multifocal zone, and, for the off-axis condition its tilt have been specified. Calculations, following the standard diffraction formulation are performed for each case and show in each figure: (A) the `wavefront deformation, expressed as the phase distribution of the entering beam, (B) the optical transfer function, (C) the point-spread in cross-sectional profile. (D) and (E), the cross-sectional light distributions in an edge and a narrow bar, respectively, indicating image quality. This computational package, with results displayed in five panels, is generated for a designated object distance.

Figure 2 depicts the full set of data for the baseline condition of targets at infinity and also at 40 cm with a 3D Add 0.5 mm wide annulus on-axis and also an annulus without additional power but in an off-axis configuration, the tilt parameter having been chosen to have the annulus beams intersect in the focus plane of the annulus when it has the Add power in the accompanying on-axis situation, as illustrated in Fig. 1.

 

Fig. 2. (Left) Retinal Imaging in unaccommodated eye for target at infinity viewed with bifocal lens (dashed blue lines) with a 3D Add on-axis annulus and, (dotted red lines) an annulus with a tilt and no Add power. Relevant geometrical constructs for these conditions are depicted in Fig. 1. Thin solid black lines in all panels show the values of the particular variable for the zero power, single vision case with a clear 5 mm pupil aperture. A. Phase map of wavefront deformation, showing the planar center portion and the outlying zone conjugate to infinity, and the annular zone with 3D power, resp equivalent tilt. B. Optical transfer function, i.e., autocorrelation of A. C. Cross-section through the point-spread function. D. Light distribution in the retinal image of a sharp edge. E Light distribution in the cross-section of the retinal image of a long bright bar, 2 arcmin wide. (Right) Same data for 40 cm object distance. Image quality for targets at infinity are degraded minimally compared with the ideal case, and equally whether the annulus has a power Add or merely a tilt.

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Inspection of the figure leads to the insight that in the case examined here there is full equivalence between adding dioptric power in the annulus or giving it tilt.

Figure 3, top, shows a through-focus display of the spread of light in a narrow bar in a single vision lens and in the two classes, on and off axis, as a target shifts through a range of object distances. The through-focus graphs, Fig. 3, bottom, extracts four single-valued parameters of image degradation emphasizing, each to a differing extent, resolution deficit or outlying light spread, which manifests itself in haloes and ghosts. In spite of their differences, all four convey a good sense of the image quality for the various target vergences, although the bar-spread function (top) provides at a glance a broader account.

 

Fig. 3. Through-focus depiction of image quality in three related configurations: (left) single vision lens, 5 mm pupil diameter; (middle) bifocal lens; the 0.5 mm wide annulus has no power addition but incorporates an equivalent tilt of the wavefront; (right) bifocal lens; the 0.5 mm wide annulus has 3D added power. Top: the light distributions in the image of a 2 arcmin bright bar on a dark background for a range of viewing distances between -1D and +4D, (between virtual 1 meters and 25 cm in front of the observer) arrayed vertically. Bottom: Conventional through-focus graph for four single-value indexes of image quality. Solid lines: steepness of edge gradients, a measure of resolving capacity; Dashed line: area under the optical transfer function. Dotted line: “Light in Bucket” index, proportion of actual point-spread within the area of the diffraction limited ideal. Dot-dash line: contrast of the image of a 2 arcmin-wide bright bar. Improved depth of focus is almost identical in the two ways of configuring the annulus.

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If the intent of the zonal annulus bifocal lens is to aid near vision while at the same time minimizing image degradation for distance targets, then its purpose has been fulfilled. And judicious selection of, equivalently, the Add power or the degree of tilt permits optimization for the individual demands of pupil dimensions and observation distance.

4. Discussion

4.1 Essence of the on/off-axis structural difference

The essence of the on/off-axis structural difference in its impact on the retinal imagery is revealed by a more detailed examination of the phase map in the pupil in the annular zone of the bifocal lens. Figure 4 shows the phase maps, and the bar-spread functions derived from it, for three related optical configurations of the annulus: (a) 3 D addition, on-axis; (b) no power change but a tilt that aligns the normal with the normal in condition (a), and (c), a 5D addition but with tilt that aligns the normal with those in the other two configuration. That all three configurations, spanning a 5D range of power additions, yield almost identical retinal images shows the significant variable to be not the dioptric power in the annular zone but the general orientation of its normal. Raytracing, Fig. 1, lays out the situation and also points to the limitations.

 

Fig. 4. Illustration showing that the overall inclination of the wavefront segment in the bifocal annulus and not its refractive power controls the secondary focus image quality. Top wavefront detail in the region of the annulus in meridional section. Bottom, light spread in a 2 arcmin wide bright bar on a dark background at the secondary focus level (3D in Fig. 4). Three conditions: Dotted line Annulus has no added power and is merely titlted, solid line, on-axis annulus with 3D power add, dashed line 5D power Add but with the tilt needed to align beam withy that in the other two conditions. Substantial overlap of the lines in all three conditions show that for a 0.5 mm annulus width the controlling variable for image quality is the overall tilt and not the curvature.

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4.2 Annulus width

As the width of the annulus increases, so does the associated width of the blur patch, i.e., the retinal intercept in defocus. Analyzed by the more rigorous diffraction theory, the effect of annulus width, for otherwise identical conditions, is shown in Fig. 5. The wider the bifocal annulus, the more disseminated its influence in the domain of defocus: there is lower image sharpening but in a more extended range of object distances.

 

Fig. 5. The influence of bifocal annulus width on the depth-of-focus performance. All four conditions have identical 3D ADD on-axis annuli, centered on a diameter 1.75 mm but differing in width ranging between 0.25 and 1 mm. The wider the annulus, the poorer the visual performance for targets at infinity, but the better the quality and range of near vision. For explanation of the panels, see legend for Fig. 4.

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4.3 Testing validity in more general conditions

So far, the calculations have been carried out for an idealized system of spherical surfaces and monochromatic light. To essay their validity in more realistic situations, a few sample calculations were carried out embracing typical values of chromatic aberration, multi-chromatic light, photopic luminosity as well as Stiles-Crawford effect apodization. To achieve this, all computations were carried out at wavelengths 25 nm apart between 450 and 650 nm, with the requisite chromatic aberration and the luminous efficiency for photopic vision. Summation weighted by the luminous efficiency at each wavelength was in the realm of intensity, i.e., upon squaring of the point-spread amplitude distributions. Because the luminous efficiency in photopic vision decreases so much for light nearing either end of the visible spectrum, the influence of chromatic aberration is small. More significant in the context of this study is that there is virtual coincidence of all data when wavefront tilts are substituted for annulus Add power (Fig. 6). The same applies to variation in the other parameters across a plausible range of values; such perturbations affect the measures of image quality minimally. The problem becomes even less troubling the smaller the overall diameter of the pupil, because narrowing the various optical zones diminishes their contribution to the image. Extension to more unspecific optical deficits like higher order aberrations and optical inhomogeneities is difficult to approach because it requires knowledge of the optical and structural properties of each individual eye. Still, the data shown in the figures paint an adequate picture of the expectation in regular typical eyes because, based on diffraction theory, they represent the absolute best performance of any optical device and as such precede and complement empirical testing and clinical validation.

 

Fig. 6. Point-, edge- and bar-spread functions for 40 cm viewing distance with bifocal annuli as in Fig. 2, right, panels C, D, and E, i.e., with 3D power add annuli and wavefront-tilted annuli. The curves were derived using typical chromatic aberration of the eye, Stiles-Crawford apodization and the photopic luminosity curve. The virtual identity of imaging characteristics between the ‘power Add’ and ‘wavefront-tilt’ annulus properties that is seen in the other figures is retained when extending the computation from monochromatic light to more realistic viewing situations.

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4.4 Measuring lens specification

The imperatives of manufacturing these lenses are not topics in this study, but knowledge of their optical properties is. Failing full disclosure by the manufacturers, the specifications have to be acquired by direct measurements [11,12] and here questions arise about the performance of the test instrumentation. In devices based on the Shack-Hartmann principle, they center on the size and spacing of the sampling lenslets, and on the sensitivity for beam displacements. For example, if the wavefront is sampled at intervals of 1/6 mm, a 0.5 mm wide annulus would have 3 readings along a meridian, and the distinction between the on- and off-axis regimens analyzed above would depend on a direction difference between the beams through the middle and outer sampling locations of 0.7 milliradians. This is beyond the capacity of the standard devices and puts published data on multifocal lenses in perspective. However, the problem has been tackled in a recent study, which also addresses computational issues that arise from gathering wavefront deviation samples into a Zernike polynomial framework [13]. Any additional performance problems relating to surface irregularities are a separate problem.

However, the calculations provided here make it clear that the precision of read-out, or the lack of it, does not impact the evaluation of retinal image quality in a major way. On the other hand, they raise interesting questions about how well lens surfaces have to conform to curvature specifications. For example, how small can the number n be when approximating a sphere by a spherical n-hedron?

4.5 Adaptive changes in wavefront tilt

The main conclusion of this study is that for multifocal interocular or contact lenses for relatively narrow annular zones, the tilt of the wavefront dominates the dioptric power in the influence on image quality at the secondary focus. For narrow-enough annuli, the image quality differs only negligibly from the more demanding curvature needed to strictly focus, rather than merely redirect, the light passing through the annulus.

This recognition opens up an intriguing possibility. When the technology has advanced sufficiently to manufacture lenses whose surface properties can be controlled in situ, it would suffice to modify just the tilt in the annular zone to accord to the patient’s particular near-focus need of the moment. It would amount to a “flappable” annulus, acting optically like a louvre of a venetian blind.

5. Conclusions

Zones in intra-ocular or contact lenses occupying circular rings of the eye’s pupil can be made to increase depth of focus, usually by featuring additional dioptric power. The resulting impact on retinal image quality when viewing close-up targets has been compared with that of merely giving the annulus wavefront a radial tilt without adding curvature. The two are almost indistinguishable. For narrow enough zones, the overall direction of the normal to the zone’s wavefront dominates the within-zone changes generated by the curvature. This finding permits the specification and measurement of these lenses to remain within the precision of the sampling grain of standard Shack-Hartmann testing devices.