Journal of Refractive Surgery

The articles prior to January 2013 are part of the back file collection and are not available with a current paid subscription. To access the article, you may purchase it or purchase the complete back file collection here

Analysis of the Optical Performance of Presbyopia Treatments With the Defocus Transfer Function

Jim Schwiegerling, PhD

Abstract

ABSTRACT

PURPOSE: A variety of techniques for treating presbyopia are available. However, these techniques are often disparate in the mechanisms used to cope with the lack of accommodation. Multizone-refractive, diffractive, and apodized intraocular lens technologies are currently being used. This article evaluates the optical performance of these markedly different technologies in a simple and comparable manner.

METHODS: The Defocus Transfer Function is a mathematical technique for illustrating the Optical Transfer Function for all levels of defocus. By calculating the Defocus Transfer Function, the simulated performance of presbyopia correction technologies on distance, intermediate, and near vision can be evaluated simultaneously. Simulations of an opaque annular ring, a zonal refractive lens, a full-aperture diffractive lens, and an apodized diffractive lens are analyzed at pupil sizes of 3, 4, and 6 mm with this technique.

RESULTS: The opaque annul us begins to act like a Fresnel amplitude zone plate for small pupils. The diffractive lens technologies had sharp optical performance at two distinct foci with a shift in performance from balanced to distance-biased for the apodized diffractive lens. The zonal refractive lens exhibited a multifocal effect. However, the optical performance of this technology fell below that of the diffractive lenses.

CONCLUSIONS: The Defocus Transfer Function is a useful tool for analyzing presbyopia treatments. Different strategies for creating simultaneous vision or extended depth of field are easily compared. It may also be possible to reverse this technique to create an ideal pupil function that meets a desired multifocal performance specification. [J Refract Surg. 2007;23:965-971.]

Abstract

ABSTRACT

PURPOSE: A variety of techniques for treating presbyopia are available. However, these techniques are often disparate in the mechanisms used to cope with the lack of accommodation. Multizone-refractive, diffractive, and apodized intraocular lens technologies are currently being used. This article evaluates the optical performance of these markedly different technologies in a simple and comparable manner.

METHODS: The Defocus Transfer Function is a mathematical technique for illustrating the Optical Transfer Function for all levels of defocus. By calculating the Defocus Transfer Function, the simulated performance of presbyopia correction technologies on distance, intermediate, and near vision can be evaluated simultaneously. Simulations of an opaque annular ring, a zonal refractive lens, a full-aperture diffractive lens, and an apodized diffractive lens are analyzed at pupil sizes of 3, 4, and 6 mm with this technique.

RESULTS: The opaque annul us begins to act like a Fresnel amplitude zone plate for small pupils. The diffractive lens technologies had sharp optical performance at two distinct foci with a shift in performance from balanced to distance-biased for the apodized diffractive lens. The zonal refractive lens exhibited a multifocal effect. However, the optical performance of this technology fell below that of the diffractive lenses.

CONCLUSIONS: The Defocus Transfer Function is a useful tool for analyzing presbyopia treatments. Different strategies for creating simultaneous vision or extended depth of field are easily compared. It may also be possible to reverse this technique to create an ideal pupil function that meets a desired multifocal performance specification. [J Refract Surg. 2007;23:965-971.]

Presbyopia is the gradual loss of accommodation with age. Without accommodation, visual performance of the eye can be optimized for one distance but the performance quickly degrades for other object vergences due to defocus. A variety of treatments exist for presbyopia, including bifocal and progressive addition spectacle lenses, multifocal contact and intraocular lenses, surgical treatment, and accommodating intraocular lenses. In addition, monovision, where one eye is corrected with a single- vision lens for distance and the other eye is corrected for near vision, is also a choice.

Currently, surgical treatments and accommodating intraocular lenses have had limited success. Therefore, patients who wish to be spectacle free are left with options of monovision or multifocal contact or intraocular lenses. These techniques work in slightly different manners. In monovision, each eye is focused at a different distance. The brain then melds the inputs from the two eyes to provide simultaneous in-focus and out-of-focus images for interpretation. With multifocal contact and intraocular lenses, each eye is given simultaneous in-focus and out-of-focus images, and again the brain needs to properly interpret these images. It is these latter types of elements that will be explored in this article.

Multifocal lenses basically fall into two categories. The first category is called zonal refractive, where the lens is composed of two or more distinct regions of different powers. The size of these regions is large compared to the wavelength of light and consequently the optical properties are dominated by refraction at the interface of the lens surface. Wesley1 described a zonal refractive contact lens with a central distance power and a surrounding near zone. This basic design has evolved to include more than two zones, strategies for reversing the distance and near zones, and non-rotationally symmetric zones.

The second category of multifocal lenses is diffractive elements. These lenses take advantage of the wave nature of light. They typically consist of concentric zones with sharp steps between the zones. The height of these steps is typically on the order of microns and each step is precisely made to create a relative phase shift in the wavefront between adjacent zones. As these wavefronts propagate towards focus they interfere to create a multifocal effect. The result is two (or more) foci for a single diffractive lens. Full-aperture diffractives have diffractive structures that cover the entire element, whereas apodized diffractives only have diffractive structures in a subregion of the element. The term apodized, in this case, differs from the traditional definition of apodization. This point is discussed further below; both zonal refractive and diffractive lenses will be investigated. Davison and Simpson2 provided an excellent review of these types of lenses and their evolution.

An apodizing filter is another means for providing a presbyopia treatment. Historically, apodizing filters are transmission filters that modify the amplitude of the wavefront passing through the pupil of an optical system. They can be both opaque structures and a filter of reduced or spatially variable transmission. An opaque annulus and its optical properties will be examined below. As mentioned previously, the term apodizing has also been applied to diffractive lenses. In the case of diffractive lenses, the term means a gradual reduction in the size of the diffraction structures, which is slightly different than the historical definition.

Because these presbyopia treatments vary widely in the mechanisms they employ, a general metric of comparing their performance is needed. The Optical Transfer Function (OTF) is a useful means of evaluating the performance of optical systems.3 This function describes the loss in contrast and possible phase shift of sinusoidal patterns as a function of spatial frequency. In cases of multifocal optics, the through-focus OTF (or its modulus) can be evaluated for a given spatial frequency. Lang et al4 applied this technique to evaluate the performance of a multifocal intraocular lens.

An extension of this concept is the Defocus Transfer Function (DTF). FitzGerrell et al5 introduced the DTF as a means of evaluating rotationally symmetric systems with extended depth of field. The DTF is a twodimensional function and lines through the origin of the DTF at various angles are equivalent to the OTF with different object vergences. FitzGerrell et al also suggested that the DTF would be beneficial for analyzing multifocal systems and illustrated an example of a full aperture diffractive. In this article, FitzGerrelFs work is extended to multiple types of currently available presbyopia treatments to understand how the mechanisms achieve treatment and compare the performance of drastically different technologies.

METHODS

Calculation of the Defocus Transfer Function

Note that Eq. (3) is a one-dimensional Fourier transform in the x-variable and a standard integration in the y- variable. Thus, instead of computing a second Fourier transform on the result of the first transform (as occurs in a two-dimensional Fourier transform), we simply integrate the result of the first transform over the second dimension.

Figure 1. A) The Defocus Transfer Function (DTF) for an aberratio ? -free system. A horizontal slice through the DTF corresponds to no defocus, whereas a slice along a line with slope of -2 corresponds to an object vergence of -1.1 diopters for ? = 0.55 ??t?. B) Profiles along the two lines from A) correspond to the Optical Transfer Function (OTF) for object vergences of 0 and -1.1 diopters, respectively.

Figure 1. A) The Defocus Transfer Function (DTF) for an aberratio ? -free system. A horizontal slice through the DTF corresponds to no defocus, whereas a slice along a line with slope of -2 corresponds to an object vergence of -1.1 diopters for ? = 0.55 ??t?. B) Profiles along the two lines from A) correspond to the Optical Transfer Function (OTF) for object vergences of 0 and -1.1 diopters, respectively.

A simple example of an aberration-free pupil illustrates the technique. Figure IA shows the DTF for a circular pupil free of wavefront error. A horizontal line (ie, zero slope) through the DTF corresponds to the case where D = O. The values of the DTF along this line are the same as the OTF for a diffraction-limited optical system. A slice through the DTF origin along a line with a slope of ?2 implies that D = ? 1.1 diopters based on Eq.(5) and a wavelength of λ = 0.55 µm. Again, the values of the DTF along this line are the same as the OTF for an aberration-free system focused at infinity, but with an object vergence of ?1.1 diopters. Plots of these slices are shown in Figure IB.

Because a presbyopic eye does not accommodate when the viewing distance of targets changes, we can interpret the slope of radial lines through the DTF as target vergence. Thus, when displayed pictorially, as in Figure 1, the brightness of the DTF along a radial line indicates the amount of retinal contrast produced at various spatial frequencies for targets with a viewing distance that produces a degree of defocus related to the slope of the radial line.

DESCRIPTION OF PRESBYOPIA TREATMENTS

Four types of presbyopia treatments are simulated with the DTF technique. Specifically, an opaque annulus, a full-aperture diffractive lens, an apodized diffractive lens, and a zonal refractive lens are modeled. These models are similar to current presbyopia treatments, but it should be emphasized that exact prescriptions of these lenses are proprietary to their respective manufacturers so an exact reproduction is impossible. Figure 2 shows the amplitude and phase profiles of each of the lens types. To perform these simulations, the Pupil Function first needs to be constructed. The Pupil Function is a complex function where the amplitude is given by the transmission of each optical element and the phase is given by the profile along the optical surface (eg, the diffractive step heights or the shape of the annular refractive rings). Figure 2 thus describes the amplitude and phase functions used to simulate each presbyopia treatment type. Specific details about each design are given below.

Figure 2. Amplitude and phase profiles of the different simulated presbyopia treatments. A) Annular ring with an amplitude of unity in its transmissive regions and an amplitude of zero in its opaque area. The annular ring also has constant zero phase across its aperture. B) Full-aperture diffractive lens with an amplitude of unity across its entire aperture and a phase profile corresponding to a typical Fresnel zone plate. C) Apodized diffractive lens with a unit amplitude (100% transmission) across its aperture and a phase profile similar to a Fresnel zone plate although the height of each diffractive step gradually reduces and blends into a purely refractive outer region. D) Zonal refractive lens with unit amplitude across its entire aperture and a phase profile corresponding to alternating distance and near zones. The plateaus of the phase regions correspond to the distance zones and the curved slopes correspond to the near zones.

Figure 2. Amplitude and phase profiles of the different simulated presbyopia treatments. A) Annular ring with an amplitude of unity in its transmissive regions and an amplitude of zero in its opaque area. The annular ring also has constant zero phase across its aperture. B) Full-aperture diffractive lens with an amplitude of unity across its entire aperture and a phase profile corresponding to a typical Fresnel zone plate. C) Apodized diffractive lens with a unit amplitude (100% transmission) across its aperture and a phase profile similar to a Fresnel zone plate although the height of each diffractive step gradually reduces and blends into a purely refractive outer region. D) Zonal refractive lens with unit amplitude across its entire aperture and a phase profile corresponding to alternating distance and near zones. The plateaus of the phase regions correspond to the distance zones and the curved slopes correspond to the near zones.

Acufocus (Irvine, Calif) is currently conducting trials of an annular implant placed in the cornea. The implant has microholes to promote corneal health. This type of implant is modeled here as a 0.5-mm wide opaque annular mask with a 1.5 -mm diameter opening in its center. The holes were not modeled.

The Re S tor lens (Alcon Laboratories Ine, Ft Worth, Tex) is an apodized diffractive intraocular lens. It is modeled in a similar fashion as the full-aperture diffractive, but the step heights are systematically reduced over the central 3.5 mm of the lens and no diffractive features appear outside of this region.

Finally, the Array lens (AMO) is an example of a zonal refractive lens. This lens was modeled with alternating powers of 0 and 3.50 diopters. The 0 diopter central zone has a diameter of 2.1 mm, the next zone of 3.50 diopters goes from a diameter of 2.1 to 3.4 mm. The next 0 diopter zone goes from a diameter of 3.4 to 3.9 mm. The next 3.50 diopter zone goes from 3.9 to 4.6 mm. Outside of 4.6 mm, the power is set to 0 diopters. These zones and powers are consistent with drawings of the Array lens, but again the model only represents an approximation to the real profile.

Pupil sizes of 3, 4, and 6 mm were examined. The effect of the pupil is introduced by setting the amplitude, as shown in Figure 2, to zero outside the diameter of the pupil.

RESULTS

Figure 3 shows the simulations of the four different presbyopia correction techniques for the three pupil sizes examined. The plots in this figure provide a method for visualizing the OTF or the amount of contrast that is passed to the retinal image. The plots allow all different object distances to be viewed simultaneously so that a global performance of the multifocal effect can be seen. The colors in the plots illustrate how well contrast is passed for a given spatial frequency. Warmer colors mean higher OTF and consequently higher contrast. Black means little or no contrast is delivered to the retina.

Figure 3. Defocus Transfer Functions for the different presbyopia treatments and pupil sizes. The columns represent 3-, 4-, and 6-mm pupil diameters. The rows represent the annular ring, the full-aperture diffractive lens, the apodized diffractive lens, and the zonal refractive lens, respectively. The red lines correspond to a spatial frequency of 30 cycles per degree. The colored lines represent the modulation transfer function values for a given object vergence. Warmer colors represent higher retinal contrast for a given object vergence and spatial frequency. The horizontal lines represent zero diopter object vergences or distant objects. The downward sloping lines in the second, third, and fourth rows represent an object vergence of roughly 3.50 to 4.00 diopters.

Figure 3. Defocus Transfer Functions for the different presbyopia treatments and pupil sizes. The columns represent 3-, 4-, and 6-mm pupil diameters. The rows represent the annular ring, the full-aperture diffractive lens, the apodized diffractive lens, and the zonal refractive lens, respectively. The red lines correspond to a spatial frequency of 30 cycles per degree. The colored lines represent the modulation transfer function values for a given object vergence. Warmer colors represent higher retinal contrast for a given object vergence and spatial frequency. The horizontal lines represent zero diopter object vergences or distant objects. The downward sloping lines in the second, third, and fourth rows represent an object vergence of roughly 3.50 to 4.00 diopters.

Consider the plots to be similar to an analog clock with the center of the clock face located on the left hand side of the plot at the point ? = 0.0. The "hands" of the clock represent regions of high contrast performance for the respective presbyopia treatment. The orientation of the hands tell which object vergences have this high performance. If one hand is horizontal (ie, the 3-o'clock position), then the treatment has good performance for an object of zero vergence or, in other words, it performs well for distance. In several of the plots, especially the diffractive lenses and the zonal refractive lens, the other hand slopes downward (somewhere between 4 and 5 o'clock on the scales used here). This orientation corresponds to object vergences of 3.50 to 4.00 diopters. The presence of this second hand suggests high-contrast performance for near objects.

In several cases, the hands of the clock split into multiple branches. This means that performance is split or spread over a range of object vergences, suggesting a depth of focus. In most cases, the splitting causes a decrease in the intensity of the colors, meaning the increased depth of focus is traded against loss of contrast. The specifics of the presbyopia design simulated here are described below.

The annular aperture is shown in the first row. For the 3 -mm pupil, light enters the eye through both the central aperture and around the edges of the annulus. In this case, the annulus begins to act like a Fresnel amplitude zone plate and consequently introduces a multifocal effect that produces radial spokes or lobes in the DTF display. Gaps between spokes correspond to target vergences that produce low-contrast retinal images. The spoke's slope indicates the target vergence needed to produce a high-contrast retinal image. The lobes seen for this pupil size correspond to an add power of approximately 0.85 diopters relative to distance vision. Increasing the pupil to 4 to 6 mm dramatically reduces the add effect and the optical performance only remains primarily for distance vision.

The second and third rows of Figure 3 show the DTFs for the full-aperture and apodized diffractive lenses, respectively. Two high performance regions appear in these DTFs. One region (the horizontal spoke) corresponds to distance vision and the second region (oblique spoke with negative slope) corresponds to an add power of roughly 4 diopters (4 diopters in the plane of an intraocular lens corresponds to a 3-diopter add in the spectacle plane). The effect of the apodization can be seen by examining these plots as the pupil size increases. For all pupil sizes, the full-aperture diffractive lens splits performance between the distance and near portions, whereas the apodized diffractive lens shifts the energy from the distance to the near portion as the pupil size increases.

Finally, the zonal refractive lens, shown in the bottom row of Figure 3, has two regions of high optical performance corresponding to distance and approximately 3.50 diopters, as designed. The DTFs (and their corresponding OTFs) were markedly lower in both regions when compared to the diffractive elements. This effect can be attributed mainly to the limited width of each of the zones in the zonal refractive-type lens. Small widths restrict the spatial frequency content that can pass through a particular zone, just as a small pupil restricts the high spatial frequency content of an image.

DISCUSSION

A variety of different technologies are used to treat presbyopia. Optical techniques include multifocal diffractive and refractive lenses that provide an add power to aid near vision. In addition, apodization filters that affect the transmission of the pupil function can be used. Because these techniques are so different, it is often difficult to compare their performance. In addition, the mechanism by which they function may not be obvious. That the annular ring acts as a two-zone amplitude zone plate is non-intuitive. The DTF allows for comparisons to be made and gives insight into how these technologies work. This insight, in turn, may allow for optimization of the design to achieve better presbyopia treatment. Furthermore, it may be possible to reverse this process to help design new presbyopic treatments. Conceivably, a DTF with desirable properties could be designed and the algorithm reversed to determine classes of solutions that would give the desired effect. Although at this point the reversal is speculative, it is certainly worth further investigation.

Chromatic effects can also be included. The wavelength of light appears in two places in the preceding analysis. First, the analysis was performed for monochromatic light at 555 nm. In white light, each wavelength would have its own DTF. Longitudinal chromatic aberration is effectively a wavelength-dependent defocus. Therefore, different wavelengths would just act as a rotation of the DTF through various angles. Yang et al6 proposed this type of analysis, but used the 1931 Commission Internationale de l'Eclairage (CIE) tristimulus values to weight the response of the eye to each of the wavelengths. Although the concept is valid, the spectral sensitivities of the L, M, and S cones would be more appropriate as weighting functions for the eye's response. Making this switch will be part of a future analysis.

Finally, wavelength is also important when examining diffractive lenses. The diffraction efficiency changes away from the design wavelength of the element. The chromatic aberration of diffractive elements is also reversed compared to traditional refractive systems. As a result, chromatic aberration could be partially compensated by this class of lenses. This effect will also be the subject of future investigation.

REFERENCES

1. Wesley NK. Bifocal Corneal Contact Lens. United States Patent 3,031,927. May 1, 1962.

2. Davison JA, Simpson MJ. History and development of the apodized diffractive intraocular lens. / Cataract Refract Surg. 2006;32:849-858.

3. Goodman JW. Introduction to Fourier Optics. Palo Alto, Calif: Stanford University Press; 1968.

4. Lang AJ, Lakshminarayanan V, Portney V. Ph?nom?nologie al model for interpreting the clinical significance of the in vitro optical transfer function. / Opt Soc Am A. 1993;10:1600-1610.

5. FitzGerrell AR, Dowski ER, Cathey WT. Defocus transfer function for circularly symmetric pupils. Appi ???.1997;36:57965804.

6. Yang Q, Liu L, Lang H. Defocus transfer function for circularIy symmetric pupils under polychromatic illumination. Proc SPIE. 2005;5896:58960Ll-8.

10.3928/1081-597X-20071101-19

Sign up to receive

Journal E-contents