Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses.

LABORATORY SCIENCE

Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses Sowmya Ravikumar, PhD, Arthur Bradley, PhD, Larry N. Thibos, PhD

PURPOSE: To evaluate the impact of target distance on polychromatic image quality in a virtual model eye implanted with hybrid refractive–diffractive intraocular lenses (IOLs). SETTING: School of Optometry, Indiana University, Bloomington, Indiana, USA. DESIGN: Experimental study. METHODS: A pseudophakic model eye was constructed by incorporating a phase-delay map for a diffractive optical element into a reduced eye model incorporating ocular chromatic aberration, pupil apodization, and higher-order monochromatic aberrations. The diffractive element was a monofocal IOL with a C3.2 diopter (D) diffractive power or 2 types of bifocal IOLs (nonapodized or apodized) with a C2.92 D addition (add) power. Polychromatic point-spread functions and image quality for white and monochromatic light were quantified for a series of target vergences, wavelengths, and pupil diameters using modulation transfer functions and image-quality metrics. RESULTS: Ocular longitudinal chromatic aberration was largely corrected by the monofocal design and by both bifocal designs for near targets. In the bifocal design, add power and the ratio of distance:near image quality changed significantly with wavelength and pupil size. Also, image quality for distance was better with the apodized design. CONCLUSIONS: Achromatization by the diffractive IOL provided significant improvement in polychromatic retinal image quality. Along with apodization and higher-order aberrations, it can significantly affect the near–distance balance provided by a diffractive multifocal IOL. Financial Disclosure: No author has a financial or proprietary interest in any material or method mentioned. J Cataract Refract Surg 2014; 40:1192–1204 Q 2014 ASCRS and ESCRS

Diffractive optical elements1 have been included in monofocal2 and multifocal3,4 intraocular lens (IOL)

Submitted: June 13, 2013. Final revision submitted: November 5, 2013. Accepted: November 7, 2013. From the School of Optometry, University of California Berkeley (Ravikumar), Berkeley, California, and the School of Optometry, Indiana University (Bradley, Thibos), Bloomington, Indiana, USA. Supported in part by funding from Alcon Research Laboratories, Fort Worth, Texas, and by the National Institutes of Health (grant RO1 EY005109; Dr. Thibos), Bethesda, Maryland, USA. Corresponding author: Sowmya Ravikumar, PhD, School of Optometry Minor Hall, University of California Berkeley, Berkeley, California 94720, USA. E-mail: [email protected].

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Q 2014 ASCRS and ESCRS Published by Elsevier Inc.

designs and in multifocal contact lenses.5 Diffractive IOLs use hybrid refractive–diffractive designs that include a low-power diffractive element on the front (eg, Restor, Alcon Laboratories, Inc.) or back surface of the IOL (eg, Tecnis ZM900, Abbott Medical Optics, Inc.).2,6 Diffractive monofocal designs use a 2p phase wrapping that produces a single high-efficiency image and have longitudinal chromatic aberration of a sign opposite to that in the human eye. This IOL can, in theory, produce an approximate chromatic aberration correction of the human eye and improved polychromatic image quality.2,7,8 Diffractive bifocals typically use p phase wrapping in the diffractive optical element to produce 2 low-efficiency images (approximately 40% each) instead of a single high-efficiency image. The extra power in the 1st-order image lends the additional (add) power required to focus near targets.6,9 0886-3350/$ - see front matter http://dx.doi.org/10.1016/j.jcrs.2013.11.035

LABORATORY SCIENCE: IMAGE QUALITY WITH DIFFRACTIVE MULTIFOCAL IOLS

Computational modeling9 experiments with model eyes and in vivo optical studies of pseudophakic eyes10 confirm the bifocality of these p phase wrapped diffractive IOLs. Psychophysical studies of pseudophakic eyes with diffractive bifocal IOLs found high visual acuities at distance and at near.10 Near and distance peaks in contrast sensitivity are also seen in eyes with bifocal diffractive IOLs.11 Postsurgical clinical reports from patients with multifocal diffractive IOLs indicate a high prevalence of spectacle independence and patient satisfaction, suggesting these IOLs are a valuable alternative to monofocal designs.10,11 Theory predicts that add power and the amount of light forming the distance and near images will vary with wavelength in pseudophakic eyes with diffractive bifocal IOLs.1 Also because the achromatizing effect of the diffractive IOL will be restricted to the near image formed by a diffractive bifocal,2,12 the balance between near images and distance images should be different in polychromatic light compared with the design wavelength. Finally, because 1 commercial diffractive IOL uses an apodized diffractive optical element (Restor) and the human eye also exhibits pupil apodization, we would expect the near–distance balance to vary with pupil size.6 Therefore, to develop a more clinically realistic model of ophthalmic diffractive optical element performance, we constructed a polychromatic pseudophakic model eye incorporating the following 3 diffractive–refractive IOL designs: monofocal, nonapodized bifocal, and apodized bifocal. Metrics of image quality were used to evaluate the impact of spectral composition and pupil size on bifocal add power and through-focus imaging characteristics. Analysis showed the dependence of IOL add power and the relative distance–near image efficiency on source spectra as well as pupil size. By incorporating the Stiles-Crawford effect and corneal aberrations, we simulated the polychromatic retinal image quality expected in pseudophakic eyes fitted with diffractive IOLs. The results highlight the importance of using polychromatic analysis with appropriate target vergence and pupil size when determining performance of multifocal ophthalmic lenses that use diffractive optical elements.

MATERIALS AND METHODS Optical Theory of Diffractive Lenses When light enters an isolated diffracting element, constructive interference distributes the light into its many diffracting orders starting from zero order, G1st order, G2nd order, and so on. In the case of a diffracting optical element designed to act as a focusing lens (structured like an annular diffraction grating), the diffracted orders of images are lined up along the optical axis of the lens system.9 The zero order of

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diffraction consists of light that passes through the diffractive optical element unaffected, and with positive diffraction orders, the lens exhibits increasing optical power. In the absence of any other focusing element in the optical path, the zero order of diffraction will form a far-field diffraction pattern. In the presence of another refracting surface, such as the combination of the nondiffractive surface of the IOL and the curvature of the diffractive surface, the zero-order diffracted image is brought closer than infinity. In doing so, the rest of the positive diffraction orders are also brought closer than infinity. The dioptric distance between the zeroorder and 1st-order image is determined by the ring geometry of the diffractive optical element and the wavelength of light.1 The relative distribution of light in the diffractive orders is determined by the phase delays introduced by each ring of the diffractive optical element.1 In the case of a monofocal hybrid IOL, the diffracting element directs nearly all the incoming light to its 1st order of diffraction, forming a single focal point. In this case, the power of the IOL is the sum of the refractive power and the diffractive power.2 This combined power of the hybrid IOL is usually designed to focus the image of a distant object on the retina (thereby rendering the pseudophakic eye emmetropic), and the diffractive optical element contribution is intended to correct for the eye's longitudinal chromatic aberration.2,7 In the case of a simple diffractive bifocal IOL, part of the incoming light (w40%) forms the zero order of diffraction and approximately 40% of the light forms the 1st diffraction order. Therefore, the distance image is determined by the refracting optics alone, whereas the near optics is the sum of the refractive power and the 1st-order power of the diffractive optical element. The 1st-order power of the diffractive optical element is chosen so near objects at a typical working distance of about 30 to 50 cm are focused on the retina. Irrespective of the target distance, both the zero and 1st diffractive orders of a diffractive bifocal will simultaneously contribute to the retinal image. That is, for a distant object, the zero-order optics will create a focused retinal image while the 1st-order optics will generate a defocused image due to the excess power provided by the 1st order of the diffractive lens. Conversely, for targets at the appropriate near location, the 1st-order optics will create a focused image while the zero-order image will be defocused. The multifocality of this IOL, therefore, arises from the fact that nearly half the light from distant objects and near objects will be well focused. The power of a diffractive optical element is given by equation 1.1 P Z 2ml rm2 (1) where P is the diffractive optical element power in diopters (D), rm indicates the radius of the mth zone in meters, and l is the wavelength of light in meters. For instance, if a monofocal IOL is designed to have a power of 4.0 D for a wavelength of 550 nm, the radius of the first zone will be r1 Z 0.524 mm, the radius of the second zone will be r2 Z 0.742 mm, and so on.3 As equation 1 shows, for a given ring geometry the diffractive power varies in direct proportion to wavelength. Thus, the longitudinal chromatic aberration of a diffractive optical element lens has a sign opposite to that of a refracting lens. Also, in contrast to a standard refracting lens, the longitudinal chromatic aberration of a diffractive element does not depend on refractive index changes and will be much larger than that of a

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Figure 1. Change in 1st-order optical power from the design wavelength (550 nm) for 3 different diffractive optical elements: solid line (3.2 D), dashed line (4.0 D), and dotted line (2.0 D). Solid line with filled diamonds shows change in the eye's optical power from 550 nm as a function of wavelength.

refracting lens of the same power.8 A graphic of equation 1 (Figure 1) shows that a C3.2 D diffractive optical element is effective at correcting the ocular longitudinal chromatic aberration across the central part of the spectrum, slightly overcorrecting at longer wavelengths while undercorrecting at shorter wavelengths. In addition to the wavelength dependency of the focusing properties of diffractive lenses, the imaging efficiency of a diffractive optical element also varies across the visible spectrum.13 The efficiency h of the diffractive lens is given by equation 2.1 h Z sin c2 ða NÞ

(2)

where a is the amplitude of phase step/2p, the sinc function is defined as sin(px)/px, and N is the diffraction order. Factor a in equation 2 is defined by Buralli et al.1 as the fraction of 2p phase delay introduced for arbitrary wavelength l. They considered the case of a diffractive optical element made from a dispersive material with refractive index nD that varies with wavelength. When the lens is in air, aZ

lo nD ðlÞ 1 l nD ðlo Þ 1

(3)

where lo is the design wavelength (ie, the wavelength at which the diffractive optical element generates a 2p phase step). In the case of an IOL incorporated into a pseudophakic model, the surrounding medium is the aqueous humor of the eye with refractive index nA that also varies with wavelength. The general formula for factor a is given by equation 4. aZ

lo ½nD ðlÞ nA ðlÞ l ½nD ðlo Þ nA ðlo Þ

(4)

Based on data for refractive index of common IOL materials,8 we assume that the Indiana eye model that describes the phakic eye's longitudinal chromatic aberration is well representative of the refractive longitudinal chromatic aberration of a pseudophakic eye. Because diffractive optical element efficiency (equation 2) depends on the amplitude of the phase step introduced in the diffractive optical element (equation 3), it will be altered by the physical depth of the echelettes engraved in the lens

Figure 2. Diffractive optical element efficiency as a function of wavelength for a monofocal 1st-order image (solid line with open squares), bifocal zero-order image (dashed line), and bifocal 1st-order image (solid line). Efficiency was calculated using Buralli's formula described in equation 2. The secondary abscissa also shows the corresponding fraction of the phase delay to that at the design wavelength. In the monofocal case, the phase step at 550 nm is 2p and in the bifocal case, it is p.

and by wavelength. These 2 effects can be seen in Figure 2, which shows efficiency as a function of wavelength. For a given height of the echelette, a lens designed to have a 2p phase step at 550 nm will perform as a monofocal at that wavelength, whereas a lens designed to have a p phase step at 550 nm will perform as a bifocal at that wavelength. Because phase step changes as a function of the ratio between design and incident light wavelength, the relative efficiency of 1st-order and zero-order diffraction images also changes as a function of wavelength (Figure 2). The wavelength dependency of both diffractive optical element power (Figure 1) and image efficiency (Figure 2) predicts that its optical performance with naturally occurring polychromatic objects will differ from its performance at the monochromatic design wavelength. Indeed, polychromatic modulation transfer functions (MTFs) for hybrid diffractive–refractive multifocal IOLs2,8 differ significantly from monochromatic MTFs.

Image Quality of Intraocular Lenses The image formation characteristics of a diffractive IOL, especially a bifocal IOL, have been described as a combination of refraction, interference, and diffraction.9 The amplitude point-spread function (PSF) for such an IOL may be computed as the Fourier transform of the pupil function associated with the phase-delay map of the diffractive optical element at the pupil plane.9 The mathematic form of this phase map is given in section below. To perform similar calculations when the diffractive optical element is embedded in the eye as an IOL, a mathematical model of the wavefront aberrations of the pseudophakic eye (see schematic model) was constructed. The numerical representation of the diffractive optical element requires a high level of digital sampling. Pilot studies showed that 512 samples 512 samples were sufficient for the pupil sizes used in this study. (See sampled kinoform wavefronts in Figure 3, A.)



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The following 3 measures of image quality were quantified: (1) the diffraction normalized metric light-in-the-bucket, (2) a measure of absolute image quality based on the non-normalized visual Strehl ratio from the optical transfer function (VSOTF), and (3) the MTF.14 The light-in-thebucket metric was chosen because of its ability to capture the property of IOL efficiency as well as image blur due to aberrations. Light-in-the-bucket quantifies the total amount of light in the central core of the PSF relative to that in a monofocal diffraction-limited PSF for the same wavelength and pupil size. Image quality was assessed for a range of target distances by incrementally adjusting the defocus term in the wavefront error maps. All modeling and IOL profile constructions were performed using Matlab software (Mathworks, Inc.). Detailed descriptions of each stage of this modeling process are given below.

Optical Model of Diffractive Intraocular Lenses Three hybrid IOL designs were modeled; that is, monofocal, bifocal, and apodized bifocal. The diffractive component of these IOLs was modeled as a finite thickness kinoform lens based on the formula for phase function 4(r) as described in equation 5.1 r2 ; rm !r!rmþ1 4ðrÞ Z a2p m (5) 2lo f

Figure 3. A: The 256 numerical phase samples across the half diameter of a 2.0 D bifocal (550 nm design wavelength). Diffractive optical element sampled at 512 locations across pupil diameter 5 mm (green line), 3.5 mm (black dots), and 2.0 mm (black circles). B: Phase profiles of 3 types of diffractive IOLs, each with identical 1st-order power (C4.0 D), are shown as a function of distance from the IOL center as follows: monofocal (green dotted line), nonapodized bifocal (dark green line), and the apodized bifocal IOL (light green line). C: Phase delay maps produced by a 2.5 mm diameter bifocal diffractive optical element designed to have C4.0 D of power at 550 nm. Phase maps are shown for 3 wavelengths as follows: 400 nm (blue line), 550 nm (green line), and 700 nm (red line).

For the monofocal diffractive optical element, factor a (see equation 4) is modeled using l Z lo Z 550 nm. For the bifocal IOL, factor a is modeled using l Z 2lo Z 550 nm. In equation 5, m refers to the ring number, r refers to the radius of the mth ring, and f refers to the focal power of the diffractive optical element. In the case of a bifocal, a Z 0.5, and in the case of a monofocal, a Z 1 at the design wavelength of 550 nm. The apodized bifocal of a commercially available Restor IOL was modeled using parameters specified in a published patent.3 All IOLs had a design wavelength of 550 nm. Figure 3, B, shows a comparison of their phase maps at the design wavelength. The term apodized in this context refers to damping the phase portion of the pupil function rather than the conventional damping of amplitude. Ocular apodization (ie, Stiles-Crawford effect) was included by using a Gaussian weighting to the pupil amplitude function. For any given design wavelength, desired power determines the required ring radius of a kinoform lens (equation 1). Ring locations can also be varied by adjusting the phase step included in the central core.3 Each ring in the Restor IOL is closer to the center than those in the nonapodized IOL because it uses a small central core with the reduced amplitude phase step.

Schematic Model of the Pseudophakic Eye The wavefront aberrations of the pseudophakic eye for a given wavelength were computed as the sum of the following 3 components: stimulus vergence, refractive elements (cornea and refractive portion of the hybrid IOL), and the diffractive surface of the IOL. Wtotal Z Wstimulus þ Wrefraction þ Wdiffraction

(6)

The map of Wstimulus is the Zernike polynomial for defocus due to target vergence. The map of Wrefraction was constructed


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Airy Radius1 Z

1:22l ðf aÞ d1

(8)

where a is the distance from the IOL plane to the corneal apex. For the 2 Airy patterns to have the same size, it is required that f f a Z d2 d1

Figure 4. Pseudophakic model eye (a Z distance from the IOL plane to the corneal apex; C Z corneal plane that forms a tangent to the corneal apex; Cf Z posterior focal point of the cornea; d1 Z radius of the aperture at IOL plane; d2 Z radius of the equivalent diffractive optical element at the corneal plane; f Z posterior focal length of the cornea; IOL Z IOL plane; n' Z refractive index of the aqueous humor; r1 Z radius of the first zone of the diffractive optical element; r2 Z radius of the first zone of the equivalent diffractive optical element at the corneal plane; z-axis Z optical axis). The inset counterclockwise-rotated schematic of Airy disk (normalized intensity pattern at the posterior focal plane), is shown adjacent to Cf. The arrow points to the first zero of the Airy diffraction pattern.

from the sum of higher-order aberrations (HOAs) of the cornea and/or IOL (expressed as Zernike polynomials). The map for Wdiffraction represents the diffractive optical element and was computed by the method described in detail above. Although Wstimulus is invariant with wavelength, both Wdiffraction and Wrefraction vary with wavelength because of diffractive aberration and refractive chromatic aberration, respectively. From the total aberration map, traditional image-plane descriptors of optical quality, such as the PSF and optical transfer function, were computed. Because the model eye is optimally focused for distance, Wrefraction has no spherical defocus component at 550 nm. For the wavefronts to be integrated as in equation 6, they must lie in the same optical plane. Wstimulus and Wrefraction are defined at the corneal plane. However, the diffractive optical element sits at the IOL plane. The equivalent diffractive optical element at the corneal plane was computed; this element is defined as the Wdiffraction that produces the same diffraction pattern on the retina as a physical diffractive optical element in the IOL plane. This equivalency was derived by scaling the phase profile in the (x, y) plane of the pupil without changing the phase values. The equivalent diffraction pattern argument is established most easily for the simplest case of a circular aperture. First, consider the case of diffraction by a circular aperture present at the corneal plane (C) of the reduced eye as shown in Figure 4. The focal power of the cornea brings the far-field diffraction pattern to focus at its posterior focus Cf, which was treated as the observation plane in this study. In an aberration-free system, the result is an Airy pattern with a radius of the first zero equal to Airy Radius2 Z

1:22lf d2

(7)

where d2 is the aperture diameter, l is wavelength, and f is the focal length of the refracting surface. Similarly, an aperture of diameter d1 placed at the IOL plane would generate an Airy pattern with radius given by equation 8 as follows:

(9)

where a is positive when the effective aperture is moved toward the image plane and negative when the effective aperture is moved toward the object plane. Thus the ratio of equivalent aperture diameters is d2 f Z d1 f a

(10)

Because Fraunhofer's diffraction pattern is described by the Fourier transform kernel for both the diffractive optical element and a simple aperture,9 the size of the diffraction patterns produced by the 2 diffractive optical elements will be equal if the phase profile in the IOL plane is stretched so that the ratio r2/r1 is the same as d2/d1 in equation 10. By assuming an average corneal power of 420 D15 and a Z 4.6 mm16 and that the aqueous humor refractive index is 1.336,17 we can calculate the ratio of d2/d1 as a numerical constant for this specific pseudophakic model eye. This is similar to vertex distance calculation, accounting for the refractive index of the aqueous humor. 0

n =K d2 cornea Z Z 1:17 d1 0 n =K a

(11)

cornea

We conclude that a diffractive optical element in the corneal plane will have the same effect as a diffractive optical element in the IOL plane provided the diameters of the kinoform rings are increased by the factor 1.17. The power of the diffractive optical element at the IOL plane is given by equation 12. Fiol Z 2l=r2 1

(12)

Substituting r2 Z 1.17r1, the power of an equivalent diffractive optical element at the corneal plane is given by Fc Z

2l 2l Z r22 ð1:17r1 Þ2

(13)

An IOL that is 2.92 D in power at the corneal plane will, therefore, be the equivalent of a diffractive optical element of 4.0 D at the IOL plane (2.92 Z 4/1.172). Previous authors have attributed an effective add power of 3.2 D at the spectacle plane for a 4.0 D diffractive optical element.4,18 When a value of 3.5 mm is adopted for a in equation 11, the effective power of a 4.0 D IOL is 3.2 D at the corneal plane. Monochromatic wavefront aberrations from the cornea19 as well as from target vergence were added to the optical path difference generated by the diffractive optical element. After cataract surgery, corneas have slightly greater HOAs20 than preoperative controls. Therefore, the current study adopted the Zernike aberration values for the cornea obtained in a study by Wang et al.19 that measured the corneal aberration in 228 eyes of 134 subjects. The population mean absolute values for individual modes were adopted and



assigned the sign of the population mean. This should have the effect of slightly increasing the total aberrations in the individual eye while not retaining the correlation across individual modes. A wavefront was constructed using this aberration vector and added as described in equation 6 to the target vergence as well as the diffractive optical element wavefront. Including target vergence enables computation of image quality as a function of viewing distance, which is critical for bifocal designs. Through-focus calculations were typically performed in 0.20 D steps and fitted at finer dioptric intervals of 0.05 D using piecewise cubic spline interpolation (Mathworks, Inc.).

Evaluating Chromatic Aberration of a Diffractive Intraocular Lens and Polychromatic Image Quality The polychromatic PSFs of the pseudophakic model eye were calculated by adopting a methodology of summing monochromatic PSFs weighted by the product of the radiance spectrum of a light source and the human photopic spectral sensitivity function.21 Each monochromatic PSF includes the effects of chromatic blur produced by ocular longitudinal chromatic aberration typical of phakic eyes. The longitudinal chromatic aberration dictated by the Indiana eye chromatic model21 is assumed to be a reasonable estimate for that in the pseudophakic eyes, although it is a slight overestimation for some pseudophakic eyes and an underestimation for others depending on the Abbe number of the IOL material and refractive power.8 Wtotal is used to calculate the phase component of the pupil function. The phase components for each wavelength (lo) were therefore scaled by the ratio l (design)/lo (Figure 3, C). For example, if the optical path distance corresponds to a p phase wrap for 550 nm, it creates 2p phase delay at 225 nm, and 0.786p at 700 nm (Figure 3, C). Wavelength-dependent changes in efficiency and power are then revealed through the Fourier transformation of these pupil functions.22 A 100% efficient diffraction-limited diffractive optical element will have 83.8% of the light within Airy's disk23 and a normalized light-in-the-bucket of 1. The IOL material dispersion also influences diffractive optical element efficiency (equations 2 and 4). Computations applying equation 2, using published Abbe numbers8 for acrylic SA60AT and acrylic Tecnis Z9003 showed that on average, material dispersion contributed to a 7.0% difference in efficiency across the visible spectrum and 4.4% across the central visually significant portion of the spectrum (450 nm to 650 nm). In comparison, rescaling phase delay alone contributes a 72.0% difference in efficiency across the visible spectrum. Because the impact of chromatic dispersion on IOL efficiency is minor, subsequent calculations to calculate efficiency were performed using a linear model of alpha as follows: aZl=lo . Polychromatic image quality was computed with diffractive chromatic aberration alone and in combination with the longitudinal chromatic aberration of the Indiana eye model.21 In addition to the standard effects of pupil size on diffraction and on chromatic and monochromatic aberrations, the apodized bifocal produces a pupil-size-dependent balance between distance and near images due to the radially varying efficiency of the diffractive optical element. The zeroorder distance image is generated by the full aperture with increasing efficiency as the distance from the center increases (reverse apodization effect), whereas the 1st-order near

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image is generated by the apodized diffractive optical element, which has approximately 40% efficiency at its center that declines to zero 1.87 mm from the IOL center. Assessment of the effect of pupil size on absolute image quality for white light was performed by modifying the VSOTF calculation by removing normalization to the diffraction limit. This metric assesses contrast at all visible spatial frequencies, weighted by the neural sensitivity function, and therefore gives a measure of image quality that is independent of pupil diffraction. The pseudophakic model included a Restor apodized diffractive bifocal IOL (Figure 3, B) and corneal monochromatic aberrations. Image quality was calculated as a function of target vergence for 3 pupil diameters designed to span the range of pupil sizes experienced by older eyes at typical light levels (3.00 mm, 3.75 mm, and 4.50 mm). For each of type of diffractive optical element, throughfocus monochromatic image quality and polychromatic image quality with and without incorporating refractive ocular longitudinal chromatic aberration were examined. Also assessed was the impact of monochromatic HOAs with the apodized IOL design. All powers, vergences, and wavefronts are specified at the corneal plane.

RESULTS Image Quality with a Monofocal Diffractive Optical Element For a pseudophakic model eye rendered emmetropic at 550 nm by a C3.2 D monofocal diffractive optical element but with no refractive monochromatic or chromatic aberrations, light-in-the-bucket was maximized at the design wavelength (550 nm) with a stimulus vergence of 0.0 D (Figure 5, A). As expected based on the power and efficiency calculations (Figure 1 and Figure 2, respectively), the monofocal diffractive optical element was less efficient at shorter and longer wavelengths and had increased power at longer wavelengths and decreased power at shorter wavelengths (Figure 5). For a spectrally uniform white-light source, peak image quality was below that of the individual wavelengths (w80% of diffraction limited) due to diffractive chromatic aberration. Polychromatic optimal refraction shifted by 0.05 D in the myopic direction relative to the design wavelength refraction. This shift was due to the higher efficiency for longer wavelengths and also because visual sensitivity peaks at 555 nm and not at the design wavelength of 550 nm. The C3.2 D monofocal design was 1.16 D more powerful at 650 nm than at 450 nm, which approximately corrected the longitudinal chromatic aberration in the typical eye, which was approximately 1.23 D more hyperopic at 650 nm than at 450 nm. When the refractive chromatic aberration of the pseudophakic model eye was incorporated (Figure 5, B), the diffractive optical element slightly undercorrected the eye at shorter wavelengths and overcorrected it at longer wavelengths. Thus, both ends of the spectrum were slightly more myopic relative to the middle of


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the spectrum. When pupil diameter was 2.5 mm, in the absence of monochromatic aberrations, this imperfect longitudinal chromatic aberration correction can produce an almost diffraction-limited white-light MTF (Figure 5, C). Image Quality with a Nonapodized Bifocal Diffractive Optical Element

Figure 5. A: Through-focus image quality for an emmetropic pseudophakic model eye with monofocal hybrid IOL, quantified using the PSF metric, light-in-the-bucket, (normalized to diffraction-limited case) as a function of target vergence. The eye model in this case only incorporates diffractive optical element chromatic aberration. The diffractive optical element 1st-order IOL power is C3.2 D at the design wavelength 550 nm and has a 2.5 mm diameter. Through-focus image quality is shown for 4 light sources; that is, 550 nm (green), 400 nm (blue), 700 nm (red), and a uniform flat white (black). B: Image quality as a function of defocus for the same IOL and model parameters as described in A, with the inclusion of ocular refractive longitudinal chromatic aberration into the model. Source spectrum and associated color coding is the same as A. C: Best-focus MTFs at design wavelength (green line), white-light MTF without ocular longitudinal chromatic aberration (black continuous line), white-light MTF in the presence of ocular longitudinal chromatic aberration (black dashed line with dots), and whitelight MTF without incorporation of the diffractive optical element but with refractive ocular chromatic aberration (black dotted line).

Using the same approach, through-focus image quality (light-in-the-bucket metric) was examined for a pseudophakic model of a nonapodized bifocal diffractive optical element (Figure 6) with a 2.92 D add power (C4.00 D at the IOL plane), 3.75 mm pupil diameter, and no monochromatic aberrations. Two imagequality peaks occur at 0 D and 2.92 D at 550 nm. Although theoretically the efficiency of the bifocal diffractive optical element at its design wavelength is 40.5% at both the zero-order and 1st-order images, the value of the image quality metric light-in-the-bucket was slightly higher than 40.5% because the image of a distant object consists of the in-focus zero-order diffraction image as well as some of the light from the out-of-focus 1st-order image and negligible light from other orders. In the absence of ocular longitudinal chromatic aberration (Figure 6, A), maximum image quality for near objects was better at the blue end of the spectrum (due to higher efficiency of 1st-order image at short wavelengths; Figure 2), and image quality for distant objects was better at the red end of the spectrum (due to higher efficiency of the zero-order image; Figure 2). The zero-order images of distant targets did not have diffractive chromatic aberration, whereas the 1st-order image of near targets had approximately 1.00 D of diffractive chromatic aberration between 450 nm and 650 nm. As a consequence, white-light image quality for the near image dropped to 23.61%. White-light image quality for a distant target was marginally better (by approximately 0.72%) than at the design wavelength. This can be attributed to the fact that longer wavelengths have higher efficiency than 550 nm for in-focus zero-order diffraction. Through-focus analysis was repeated for the C2.92 D bifocal, incorporating ocular longitudinal chromatic aberration (Figure 6, B). Ocular longitudinal chromatic aberration was undercorrected at shorter wavelengths and marginally overcorrected at longer wavelengths by the diffractive optical element near add power. In white light, the image quality for a near target benefited from the compensatory effects of diffractive longitudinal chromatic aberration, producing white-light light-in-the-bucket of 36.4% compared with 26.17% for the distance image. Optimum refraction with white light was shifted by approximately 0.05 D in the myopic direction for a



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Figure 6. A: Through-focus image quality for an emmetropic pseudophakic model eye with a bifocal hybrid IOL, quantified using the PSF metric, light-in-the-bucket (normalized to the diffraction-limited case), as a function of target vergence. The eye model in this case only incorporates diffractive optical element chromatic aberration. Through-focus image quality for a diffractive bifocal IOL with an add power of C2.92 D with a 3.75 mm aperture designed at 550 nm. Image quality was quantified using the metric light-in-the-bucket for 3 wavelengths; that is, 550 nm (green), 450 nm (blue), and 650 nm (red) and for a flat-white spectrum (black line for near images and black dots for distance images). B: Through-focus image quality for the IOL and eye model in A. Calculations performed for a 3.75 mm diameter pupil, including chromatic aberration for the diffractive optical element as well as the eye. Color coding of the source spectrum is similar to A except the white-light image of near and distant targets is represented by a black continuous line. C: The MTFs at best focus for the IOL and model in A. Bifocal diffractive optical element MTF at design wavelength of 550 nm (green line) is the same for distance and near targets; the best focus white-light MTFs for near targets (black dashed–dotted line) and for distant targets (black dotted line) are also shown. For comparison, the white-light MTF for a model pseudophakic eye without diffractive chromatic aberration and only ocular refractive longitudinal chromatic aberration (solid black line) and a white light diffraction-limited (dashed black line) MTF is shown. D: The MTFs at near and distance foci for the same IOL and eye model. The monochromatic MTFs for distant and near targets at the design wavelength of 550 nm are identical (green line); the polychromatic white-light best focus MTFs for distant and near targets are plotted as black dotted lines and black dashed–dotted lines, respectively. White-light MTFs for a pseudophakic eye without diffractive chromatic aberration and only ocular longitudinal chromatic aberration with (solid black line) and without ocular longitudinal chromatic aberration (black dashed line) are shown for comparison.

target at near and C0.09 D toward the hyperopic side for a target at distance. This made the add power closer to 3.06 D for white light as opposed to 2.92 D at the design wavelength. The MTFs for the theoretical nonapodized bifocal were calculated at near and distance best focus with and without ocular longitudinal chromatic aberration (Figure 6, C and D). All MTFs were calculated with 3.75 mm pupil diameter. For comparison, the whitelight MTF for a monofocal model eye with ocular

longitudinal chromatic aberration and diffractionlimited MTF are shown. In all bifocal cases, image modulation was reduced to approximately 40% of that in the monofocal case because of the simultaneous presence of the second defocused image plus other unfocused orders. When considering the longitudinal chromatic aberration of the IOL alone (Figure 6, C), the best-focus MTF at the design wavelength was identical for both distant and near targets. Because IOL diffractive


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optical element does not suffer from chromatic aberration for the zero-order diffraction, white-light image quality for distant targets was very close to the monochromatic MTF at the design wavelength (compare the black dotted line with the green line in Figure 6, C). However, the polychromatic MTF for a near target (black dashed–dotted line) had diffractive chromatic aberration of approximately 1.72 D across the visible spectrum, causing an additional 50% demodulation at middle and higher frequencies. When incorporating the diffractive optical element as well as the ocular longitudinal chromatic aberration (Figure 6, D), contrast of near targets in white light improved, approaching the monochromatic MTF at the design wavelength. Conversely, the polychromatic image of distant targets lost contrast due to ocular longitudinal chromatic aberration. At high spatial frequencies, the white-light bifocal MTF for near targets approached that obtained with a monofocal model with ocular longitudinal chromatic aberration (Figure 6, D). Image Quality with Apodized Diffractive Bifocal Intraocular Lens The commercially available apodized bifocal Restor IOL has a phase profile similar to the bifocal diffractive optical element IOL analyzed in the previous section; however, the phase steps successively decrease (from a maximum of p to a minimum of 0) with increasing radial distance from the IOL center (Figure 3, B). The diffractive optical element part of the IOL is 3.2 mm diameter, which covers an effective aperture of 3.744 mm diameter at the corneal plane. At this aperture size, the IOL efficiency was biased toward the zero-order diffraction image. At the design wavelength of 550 nm, the light-in-thebucket for the image of a focused distance target and near target was 57.85% and 28.27%, respectively (Figure 7, A). At shorter wavelengths (450 nm), this imbalance due to apodization was offset because at this wavelength all steps are 550/450 times larger in terms of wavelength, biasing efficiency toward the 1st-order image. The reverse was true for longer wavelengths, at which efficiency was more biased toward the zero-order than at 550 nm. As with the nonapodized bifocal (Figure 6, A), in the absence of ocular refractive longitudinal chromatic aberration, the 1st-order image suffered from diffractive chromatic aberration, further exaggerating the differences between the polychromatic images of distance and near targets; light-in-the-bucket with white light was 58.0% and 16.21% at the distance focus and near focus, respectively (Figure 7, A). The longitudinal chromatic aberration correction effect of the apodized bifocal IOL (Figure 7, B) was very

similar to that of the nonapodized bifocal IOL because the chromatic difference in power scaled linearly with the total power of the IOL. Thus, for near targets, the through-focus functions for long, short, and middle wavelength approximately collapsed close to the design wavelength, indicating that in the 1st-order image, ocular longitudinal chromatic aberration was almost fully corrected by the apodized bifocal IOL. Once again, there was slight undercorrection at shorter wavelengths and overcorrection at longer wavelengths. However, the longitudinal chromatic aberration correction was sufficiently complete to produce a polychromatic light-in-the-bucket that was 93.34% of the light-in-the-bucket at the design wavelength. Similar to the effect seen with the nonapodized bifocal diffractive optical element, images of distance targets suffered from ocular longitudinal chromatic aberration that reduced white-light image quality to 61.22% that at the design wavelength. Thus, although the zero-order diffraction image was more efficient than the 1st-order image in the apodized bifocal, this imbalance was moderated for polychromatic near targets because of the achromatization of the 1st-order image. Therefore, the ratio of image quality for a distance target to image quality for a near target was reduced from 2.18 at the 550 nm design wavelength to 1.43 for white light. Image analysis using MTFs (Figure 7, C and D) found the same general trends observed with the light-in-the-bucket. In the absence of ocular longitudinal chromatic aberration (Figure 7, C), the white-light MTF for distant targets was similar to that at the design wavelength because it did not suffer from diffractive longitudinal chromatic aberration. On the contrary, the white-light MTF for near targets was significantly worse than the monochromatic MTF at the design wavelength because of diffractive longitudinal chromatic aberration. When ocular longitudinal chromatic aberration was included (Figure 7, D), the opposite-sign diffractive and refractive longitudinal chromatic aberration was mostly canceled, improving white-light image quality at near (black dashed line) almost to the level of that at the design wavelength. For the image of a distance target, ocular longitudinal chromatic aberration lowered the MTF; however, for all spatial frequencies below 20 cycles per degree (cpd), it remained significantly higher than that for near targets due to apodization. At very high spatial frequencies (O20 cpd), the polychromatic distance MTFs and near MTFs were similar. Effect of Pupil Size and Monochromatic Aberrations on Polychromatic Image Quality In the absence of HOAs (longitudinal chromatic aberration alone; Figure 8, A), the through-focus VSOTF



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Figure 7. A: Through-focus image quality for the pseudophakic eye with an apodized bifocal IOL. Ocular longitudinal chromatic aberration was not included in these calculations. The IOL has 2.92 D effective 1st-order power at the entrance pupil plane and a 3.75 mm aperture and was designed at 550 nm. Color coding for data is same as in Figure 6. B: Through-focus image quality for the same IOL and model with ocular longitudinal chromatic aberration included. Color coding for data is same as in A. C: The MTFs at near and distance best foci for the same IOL and model. The pseudophakic model does not include ocular longitudinal chromatic aberration. The distance (green line) and near (green dashed line) MTFs at the 550 nm design wavelength are compared with the polychromatic white light near MTF (black dashed line) and distance MTF (black continuous line). The white-light MTFs for a pseudophakic model eye without diffractive chromatic aberration and only ocular longitudinal chromatic aberration (black dots) and without ocular longitudinal chromatic aberration (black dotted line) are also shown. D: The MTFs at best focus for the same IOL and model with inclusion of ocular longitudinal chromatic aberration. Color coding same as C.

showed 2 very clear peaks at target vergences of approximately 0.0 D and 3.0 D (distance image and near image). For all 3 pupil sizes, polychromatic image quality was better for distant targets. The ratio of distance to near VSOTF increased from approximately 1.5 to 2.6 as pupil diameter increased from 3.0 to 4.5 mm, which reflects the zero-order dominance of the apodized diffractive optical element. Notably, this increase in the proportion of light in the zero-order image counteracted, in part, the defocus caused by longitudinal chromatic aberration as pupil size increased. That is, although the distance image was not achromatized, in the absence of HOAs its image quality remained high even at large pupil sizes due to the apodized diffractive optical element structure. In the presence of HOAs, distance and near image quality declined with increasing pupil size (Figure 8, B). The impact of HOAs on the near image can be

seen by comparing the near image VSOTF with and without HOAs (Figure 8, A and B). As pupil diameter increased from 3.75 mm to 4.50 mm, the near peak of the VSOTF dropped by a factor of 1.5 without HOAs and 1.7 with HOAs. This can be contrasted to the change in the distance peak of the VSOTF as the pupil diameter increased from 3.75 mm to 4.50 mm. In the absence of HOAs, VSOTF increased by a factor of 1.1 as pupil size increased, but in the presence of HOA, it decreased by a factor of 1.4. The decline in nearimage quality with increasing pupil size in the aberrated eye was due to the combined effect of the smaller proportion of light in the 1st-order image (Figure 8, A) and HOAs. Because the diffractive optical element had a fixed size (3.75 mm), optical quality outside this central region had little additional impact on the near image except to increase efficiency of the zero-order image relative to 1st-order image.


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Figure 8. A: Through-focus VSOTF, non-normalized with respect to the diffraction limit for respective pupil size, to enable direct comparison of raw VSOTF values across pupil sizes. Data are shown for the Restor IOL with a 3.0 D effective power incorporated into the eye model with longitudinal chromatic aberration for white light and the following 3 pupil sizes: 4.50 mm (dotted line), 3.75 mm (line with dots), 3.00 mm (solid line). B: Non-normalized throughfocus VSOTF for the same IOL and model, inclusive of HOAs (see text for details of HOAs). The data key and parameters are the same as in A (VSOTF Z visual Strehl ratio from the optical transfer function).

DISCUSSION The basic optical properties of kinoform lenses, including some of the pupil-dependence and chromatic properties, have been described.1,13,22,24 Also,

the polychromatic imaging characteristics of monofocal diffractive optical elements for ophthalmic use is emerging2,7; however, there are few descriptions of the polychromatic behavior of diffractive bifocal IOLs. Atchison et al.12 observed the longitudinal chromatic aberration of eyes wearing diffractive bifocal contact lenses, and our study used a computational model to describe the longitudinal chromatic aberration and polychromatic imaging of pseudophakic eyes with diffractive bifocal IOLs. By separately including the longitudinal chromatic aberration properties of the IOL, chromatic aberration of the eye, and corneal HOAs, we were able to identify the contributions of each component that determines polychromatic distance and near image quality. Our model of a pseudophakic eye with a diffractive IOL quantified the benefits from cancellation of ocular and diffractive longitudinal chromatic aberration in both apodized and nonapodized bifocal IOL designs and showed that polychromatic images of near targets approach that achieved monochromatically at the design wavelength. Because the distance image (zeroorder diffractive image) does not benefit from this achromatizing effect, image modulation for a distant target illuminated with polychromatic light is reduced by ocular longitudinal chromatic aberration. For example, in the absence of monochromatic aberrations, the polychromatic distance MTF is lower than the monochromatic MTF by a factor of approximately 2 over most of the visible spatial frequency range (Figure 6). This inherent near bias in polychromatic imaging can be counterbalanced by using less than p phase shifts in the bifocal to increase the efficiency of the zero order (distance image). A new form of this strategy decreases the diffractive optical element phase shifts from a peak of p at the pupil center to zero at approximately 1.9 mm from the IOL center.3 This apodized diffractive optical element increasingly shifts the bias toward the distance image as pupil size increases,4 achieving a distance bias with polychromatic light for pupil diameters between 3.0 mm and 4.5 mm. Our polychromatic analysis found that the apodized and nonapodized designs provided distance-biased images and near-biased images, respectively, for white light. Our MTF calculations showing the impact of longitudinal chromatic aberration used a model eye with only longitudinal chromatic aberration, analogous to an eye with its HOAs corrected. However, as HOAs increase, the impact of longitudinal chromatic aberration on the ocular MTF decreases.21 In our study, as the pupil diameter increased, the corneal HOAs had an increasing impact on distance and near image quality, as they do for monofocal optics. The effect of longitudinal chromatic aberration in a regular human eye is offset, to some extent, by the



presence of HOAs.8 As a consequence, HOAs introduce a greater percentage loss in image quality by itself than in the presence of longitudinal chromatic aberration, although this is dependent on the magnitude of the HOAs.21 This protective effect is achieved by the improvement in optics at the edges of the visible spectrum, which counteracts the loss due to HOAs at the center, thereby making image quality more uniform across wavelengths. However, in the hybrid bifocal design, for a 3.75 mm diameter pupil (which includes the full diffractive optical element as well as significant HOAs), image quality loss due to HOAs was slightly greater in the presence of longitudinal chromatic aberration at (factor of w1.7) than in the absence of longitudinal chromatic aberration (factor of 1.5). A key reason could be that the very low efficiency of the IOL at shorter wavelengths for distance makes it impossible for the benefit gained at those wavelengths to contribute to overall polychromatic image quality. This is unlike a refractive model that has equal efficiency across the spectrum. This finding indicates that IOLs that introduce HOA correction could potentially benefit the hybrid pseudophakic eye more than what would be predicted based on results in phakic eye studies. Several clinical studies have assessed postoperative contrast sensitivity and quality of life after implantation of multifocal IOLs.18 A common observation in many such clinical studies, documented in a comprehensive review of different IOL designs,25 is that the overlap of focused and defocused images in simultaneous imaging does not affect high contrast visual acuity but does affect contrast sensitivity. These clinical results are anticipated because reductions in image contrast down to 40% have almost zero impact on visual acuity in this age group,26 whereas reductions in image contrast are directly reflected as reductions in psychophysical contrast sensitivity.27 However, due to apodization, the near image has a contrast lower than 40%; thus, near visual acuity with the apodized diffractive optical element is only slightly worse (20/20 to 20/30) than the better than 20/20 observed at distance in pseudophakic eyes fitted with this type of diffractive optical element.18 However, because the primary goal of the near image is to allow for reading and reading speed is very robust to large reductions in contrast in phakic eyes and in pseudophakic eyes, pseudophakic eyes with diffractive bifocals can achieve maximum reading speeds when viewing typical high-contrast characters.28 In our model, the hybrid bifocal IOL provided significant contrast at both distance and near, consistent with laboratory studies that show a high proportion of patients achieve spectacle independence with this type of IOL.18

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WHAT WAS KNOWN Hybrid refractive–diffractive monofocal IOLs can compensate for the longitudinal chromatic aberration of the eye. Apodized hybrid refractive–diffractive IOLs provide increased efficiency for distance targets for monochromatic light. WHAT THIS PAPER ADDS Diffractive multifocal IOLs selectively corrected longitudinal chromatic aberration for near targets. This correction fundamentally altered distance–near image quality balance. Applying 2 different broad-spectrum scalar image quality metrics showed that pupil-dependent IOL design retains significant distance dominance for a range of pupil sizes. A pupil-independent design tended to have equal or slightly near-dominant balance. These modeling results agree closely with those in clinical studies and therefore provide a way to assess preoperative image quality with realistic optical parameters.

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Variations in manufacturing quality of diffractive multifocal lenses.

Quality of vision through diffractive bifocal intraocular lenses.

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Short-term outcomes of combined implantation of diffractive multifocal intraocular lenses with different addition power.

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Filtering chromatic aberration for wide acceptance angle electrostatic lenses.

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Visual performance with accommodating and multifocal intraocular lenses.

Multifocal toric intraocular lenses versus multifocal intraocular lenses combined with peripheral corneal relaxing incisions to correct moderate astigmatism.

Psychophysical Vision Simulation of Diffractive Bifocal and Trifocal Intraocular Lenses.

Optimizing outcomes of multifocal intraocular lenses.

Comparative visual performance with monofocal and multifocal intraocular lenses.

Optical performance of multifocal intraocular lenses.

Intraocular lens power calculations for multifocal intraocular lenses.

Comparison of visual outcomes and subjective visual quality after bilateral implantation of a diffractive trifocal intraocular lens and blended implantation of apodized diffractive bifocal intraocular lenses.

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