Effect of the eye pupil diameter and the phase shift in the diffraction structure on bifocal properties of diffractive-refractive intraocular lenses

$Page 1: Effect of the eye pupil diameter and the phase shift in the diffraction structure on bifocal properties of diffractive-refractive intraocular lenses$
ISSN 8756-6990, Optoelectronics, Instrumentation and Data Processing, 2010, Vol. 46, No. 3, pp. 264–273. c© Allerton Press, Inc., 2010.Original Russian Text c© G.A. Lenkova, 2010, published in Avtometriya, 2010, Vol. 46, No. 3, pp. 74–85.

OPTICAL INFORMATIONTECHNOLOGIES

Effect of the Eye Pupil Diameter and the Phase Shift

in the Diffraction Structure on Bifocal Properties

of Diffractive–Refractive Intraocular Lenses

G. A. Lenkova

Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences,pr. Akademika Koptyuga 1, Novosibirsk, 630090 Russia

E-mail: [email protected]

Received January 19, 2010

Abstract—As the eye pupil decreases, the bifocal properties of the intraocular lens (IOL), which isan artificial eye lens, can be violated. To avoid it, a phase shift is introduced into the structure ofthe IOL diffractive component. Results of an analytical study of the light intensity distribution on theaxis of a bifocal IOL for different values of the eye pupil and phase shift in the diffractive componentwith a focal power of 4.2 D (diopters) are presented. Because of broadening and asymmetry of theintensity distribution function arising at small pupil diameters, the foci of orders 0 (far vision) and +1(near vision) of the bifocal IOL are demonstrated to affect each other: they come closer or farther,become separated or overlapped. The total intensity distribution may be continuous or have a dipwhose magnitude depends on the pupil diameter and phase shift. It is noted that the optimal phaseshift at which separation of the foci (bifocality) begins from the pupil diameter of ∼0.9 mm and is notviolated with further pupil expansion is 180◦ (π radian).

DOI: 10.3103/S875669901003009X

Key words: intraocular lens, artificial eye lens, diffractive–refractive lens, bifocal lens.

INTRODUCTION

The diffractive–refractive intraocular lens (IOL) or the artificial eye lens is a usual refractive lens (re-fractive component) with an annular diffractive microstructure (diffractive component), which is similar tothe Fresnel zone plate, being formed on one of the refractive surfaces [1, 2]. Depending on the height andshape of the diffractive structure, the incident light is predominantly directed to two or three orders ofdiffraction of equal intensity (40.5 or 28.8%) [3], and the refractive component together with the diffractivecomponent forms a bifocal or trifocal diffractive–refractive lens. If the height of this structure corresponds toa half-wavelength phase delay (λ/2), then diffraction orders 0 and +1 are formed in the case of a saw-tooth(kinoform) shape of the structure similar to the Fresnel phase lens, while orders −1 and +1 are formed in thecase of a binary (rectangular) shape. In the latter case, the light beam is separated into three orders (−1,0, and +1) if the phase delay equals 0.32λ. In the case of a saw-tooth structure, the diffractive componentdoes not affect the refractive power of the refractive component in order 0, and the IOL forms images ofdistant objects on the retina, whereas an additional optical power is generated on order +1, and the IOLbecomes capable of projecting closely located objects onto the retina. In the case of a binary structure,distant objects are formed in order −1, and closely located objects are displayed in order 0 or in orders 0and +1, depending on the structure height.

In contrast to usual artificial lenses, diffractive–refractive IOLs allow the patient to see distant and closelylocated objects without spectacles. The ability to see objects at different distances from the eye (pseudo-accommodation), however, can be violated under conditions of bright lighting where the pupil diameterdecreases. This feature is related to the diffractive component structure. Depending on the profile shape(saw-tooth or binary), the optical power of the diffractive component is ∼4.2 [1, 2] or ∼2.1 D (diopters).With such an optical power, the size of the central zone diameter is rather large: ∼1.0 or ∼1.4 mm [4]. The

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EFFECT OF THE EYE PUPIL DIAMETER AND THE PHASE SHIFT 265

Fig. 1. Optical scheme of the eye model: (1) the cornea; (2) the di-aphragm imitating the eye pupil; (3) the MIOL-Akkord lens.

latter circumstance can be the reason for bifocal vision violations. As the eye pupil is contracted, the numberof open zones of the diffractive component decreases and, therefore, the efficiency of bifocal functioning alsodecreases. In addition, the IOL may completely turn from a bifocal to a monofocal lens as the eye pupildiameter approaches the central zone size. To maintain bifocality and increase the efficiency, the centralzone diameter is reduced by introducing a phase shift to the diffractive component structure [2, 5]. As aresult, all zones are shifted toward the center, the number of the zones in the pupil field increases, the areaof the central zone decreases, but the areas of other (annular) zones remain identical. The inequality of thezone areas with a small pupil size can lead to intensity redistribution, alter the focal distance in diffractionorders, and, therefore, change the formation of images of the distant and close objects. Determining theconditions at which the bifocal IOL functioning is not violated is extremely important in choosing thediffractive component structure.

The objective of the present work is an analytical study of bifocal properties of the diffractive–refractiveIOL, namely, the changes in the focal distances (or the IOL optical power) and diffractive efficiency (ratioof intensities in the foci or diffraction orders) in near and far vision as functions of the eye pupil diameterand phase shift in the diffractive component structure; another task is to determine the optimal phase shift.

FORMULATION OF THE PROBLEM

The study was performed for IOLs whose parameters are similar to those of the MIOL-Akkord bifo-cal diffractive–refractive intraocular lens developed at the Institute of Automation and Electrometry of theSiberian Branch of the Russian Academy of Sciences (Novosibirsk, Russia) with assistance of the NovosibirskBranch of the Fedorov Intersectoral Research and Technology Complex Federal State Institution “Eye Mi-crosurgery” and the Reper-NN Research and Production Enterprise (Nizhnii Novgorod, Russia) [2]. The lensmaterial is oligocarbonatemetacrystale (elastic acryl) with the refractive index of 1.505 for λD = 0.5893 μm.The refractive component has a plane–convex shape. The diffractive structure is formed on the flat sur-face turned toward the retina. The structure has a saw-tooth profile typical for the majority of bifocalIOL structures [1, 5].

The optical scheme of the eye model [6] used to calculate the optical characteristics and analyze thebifocal IOL properties is shown in Fig. 1 and includes the cornea (1), the diaphragm imitating the eyepupil (2), and the MIOL-Akkord lens (3). The retina is not shown in the figure, and the lens is shownschematically. The real height of the diffraction structure of the MIOL-Akkord lens is ∼1.6 μm, which isapproximately 500 times smaller than the lens thickness, and the number of zones (∼9) on the 3-mm pupildiameter is greater than that in the figure. A light beam formed by the cornea and converging at the point Aat a distance f from the lens is incident onto the IOL. Passing through the refractive component, the beamis separated on the diffraction structure into two parts. One of them passes through the structure withoutchanging its direction and is focused at the point F0, which corresponds to the diffraction order 0, whereasthe second part experiences additional refraction in order +1 and is focused at the point F+1.

OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 46 No. 3 2010

266 LENKOVA

The design wavelength is chosen to be λ = 0.5461 μm recommended in [7] to control the IOL quality. Theoptical power (refraction) of the examined IOL corresponds to the mean value of refraction of implanted IOLs.The curvature radius of the spherical surface (R = 8.24 mm) is chosen from a standard series of radii; theIOL thickness is d = 0.7 mm. The refractive indices n1 = 1.3377 for the aqueous humor and n2 = 1.5061 forthe IOL material, the optical power Φd = 4.2 D, and the focal distance fd = 318.96 mm of the diffractivecomponent correspond to λ = 0.5461 μm. The design values of IOL refraction for the far and near visionare Φ0 = 20.55 and Φ+1 = 24.75 D.

Corrections in the structure of the diffractive component of the MIOL-Akkord lens were ignored in thepresent study. These corrections are used to compensate for aberrations of the cornea, vitreous body, andrefractive component, but they do not affect the bifocal properties of the IOL. The bifocal properties arealso independent of the distance to the object; therefore, the analysis is performed with eye illumination bya parallel beam to simplify the calculations. The cornea is shown by the dashed line in Fig. 1 because in thecase considered it serves only to form a converging bunch of beams. The distances f0 and f+1 are actuallythe rear focal distances of the eye model [6] with the IOL. For the above-mentioned IOL parameters, wehave f0 = 20 mm and f+1 = 18.82 mm.

DIFFRACTIVE STRUCTURE

The radii r0 and rk and the areas S0 and Sk of the central and annular zones of the diffractive componentstructure were calculated by formulas that follow from the relation (k + a0)λ = a1r

2k:

r0 =√

a0λ/a1, (1)

rk =√

(k + a0)λ/a1 =√

2fd(k + a0)λ/n1, (2)

S0 = πr20 = πa0λ/a1, (3)

Sk = πΔr2k = πλ/a1 = const. (4)

Here, k and rk are the zone number and radius (k = 0 corresponds to the central zone and k = 1, 2, . . . arethe numbers of the annular zones), λ is the design wavelength in vacuum, a0 is the phase shift expressedin fractions of λ, a1 = n1/2fd = Φd/2000 is the coefficient that depends on the refractive index of theaqueous humor (n1) and on the focal distance (fd in millimeters) or the optical power (Φd) of the diffractivecomponent, and Δrk = rk−rk−1. For the above-given values of n1, fd, and Φd, we have a1 = 0.002097 mm−1.Equation (2) with a0 = 1 predicts that r0 = 0.51 mm and rk = 0.72 (k = 1), 0.88 (k = 2), 1.02 (k = 3), 1.14(k = 4), 1.25 mm (k = 5), etc.

Formulas (1) and (2) yield only the zone boundaries, and the height of the structure is specified on thebasis of the desired diffraction efficiency. It follows from Eqs. (3) and (4) that the area of the central zone(S0) varies in proportion to a0, and the areas of the annular zones (Sk) are equal to a constant quantitydepending on a1, i.e., on the optical power of the diffractive component fd.

At a0 = 1, the right side of Eq. (2) corresponds to a usual diffractive lens whose typical feature is theequality of the areas of the central and annular zones. In this case, however, the central zone diameter hasa rather large size, which is ∼1 mm for a diffractive component with Φd = 4.2 D. For eye pupil diametersof the same order, the IOL becomes monofocal. To maintain bifocality, the zones are brought closer to thecenter by decreasing a0.

Figure 2 shows possible types of diffractive structures for a0 = 1.0, 0.5, 0.25, and 0.125, which correspondsto the phase shifts (in radians) ϕ = 2π, π, π/2, and π/4. The 2π shift actually equals the zero shift, but itwas introduced for clearer comparisons.

As a0 is diminished, the radii (r0) and areas (S0) of the central zones decrease and correspond in ourcase to r0 = 0.51, 0.36, 0.26, and 0.18 mm and to S0 = 0.82, 0.41, 0.21, and 0.10 mm2. The areas of theannular zones remain equal to the area of the central zone at a0 = 1. The height of the structure at itscenter decreases (see Fig. 2), and the number of zones increases, but no more than by one. The shape of thecentral zone and the pupil diameter affect the intensity distributions along and across the optical axis and,therefore, the ratio of intensities in the foci of diffraction orders 0 and +1.



Fig. 2. Types of diffractive structures for a0 = 1.0 (a), 0.5 (b), 0.25 (c),and 0.125 (d).

EFFECT OF THE PUPIL DIAMETER ON THE INTENSITY DISTRIBUTIONALONG THE OPTICAL AXIS OF THE EYE

The eye pupil diameter affects primarily the total number of the working zones. If there are few zones,the intensity distribution function is broadened, and partial overlapping of the orders becomes possible. Inaddition, the shape of the intensity distribution along the axis changes depending on the pupil diameter,because the diffractive optical power of the pupil (Φr) acting as a diaphragm affects the foci positions.

The effect of the diaphragm is manifested as follows. When a parallel light beam passes through thediaphragm, some part of light becomes diffracted in the form of converging and diverging beams correspond-ing to negative (ahead of the diaphragm) and positive (behind the diaphragm) diffraction foci (maximumsof intensity). The distances between the foci and the diaphragm are

fr = ±r2n/Nλ, (5)

where N are the Fresnel numbers whose odd integer values N = 1, 3, 5, . . . correspond to intensity maximums.The most intense maximums with N = 1 forming the optical power of the pupil

Φr = n/fr = λ/r2, (6)

where r is the pupil radius, are particularly noticeable. For instance, if r = 0.10, 0.25, 0.50, 0.75, 1.0, and1.5 mm, then Φr = 54.60, 8.74, 2.18, 0.97, 0.55, and 0.24 D, which corresponds to the diffractive focaldistances fr = 24.5, 153.1, 612.4, 1377.9, 2449.6, and 5511.5 mm. It can be noted that the effect of thediffractive optical power of the pupil is clearly manifested at r < 1 mm with Φr > 0.55 D (fr < 2449.6 mm).

The incidence of a converging light beam onto the diaphragm is equivalent to mounting a refractive lensahead of or behind the diaphragm in a parallel beam. The lens projects the images of the diffraction maxi-mums (diaphragm foci) and minimums, which alter the shape of the intensity distribution of the main beam,into the region of focusing of the beam that passes through the diaphragm without deflection. The distancesbetween the images of the maximums and the lens can be easily found by the formulas of geometrical optics

S′+ = ffr/(fr + f); S′

− = ffr/(fr − f), (7)

where f is the focal distance of the lens, fr is the absolute value calculated by Eq. (5), and S′+ and S′

− arethe distances of the images of the positive and negative foci of the diaphragm. The images S′

+ are locatedcloser to the lens than its focus, and the images S′− are located farther from the lens than its focus.

The data calculated by Eq. (7) for f = 20 mm are summarized in table (in the first column, the numberin brackets is the number of zones of the diffractive component with this radius). For instance, for r =0.1 mm, the image of the negative focus of the diaphragm (S′−, column 5) is projected at a large distance(S′

− = 109 mm) from the lens and exerts practically no effect on the intensity distribution along the axis.The image of the positive focus (S′

+, column 4) is projected closer than the lens focus (S′+ = 11.01 mm).

The light directed to this image interacts with the main beam; as a result, the total intensity maximum


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Table

1 2 3 4 5 6 7 8 9 10

r,mm

±fr,mm

Φr ,D

S′+,

mmS′−,

mmzmax, mm

(f = 20 mm)zmax, mm

(a1)r/20,rad

r/20,deg

(Δz)D ,mm

0.1 24.50 54.6 11.01 109.00 13.30 12.82 0.005 0.29 32.66

0.2 97.98 13.65 16.61 25.13 19.08 18.52 0.01 0.57 8.16

0.3 220.46 6.07 18.34 22.00 19.80 19.20 0.015 0.86 3.63

0.4 391.93 3.41 19.03 21.08 19.94 19.34 0.02 1.15 2.04

0.5(∼1 zone)

612.39 2.18 19.37 20.68 19.98 19.36 0.025 1.43 1.31

0.6 881.84 1.52 19.56 20.46 19.98 19.90 0.03 1.72 0.91

0.7(∼2 zones)

1200.28 1.11 19.67 20.34 20.00 19.86 0.035 2.01 0.67

0.8 1567.71 0.85 19.75 20.26 20.00 19.98 0.04 2.29 0.51

0.9(∼3 zones)

1984.14 0.67 19.80 20.20 20.00 19.92 0.045 2.58 0.40

1.0(∼4 zones)

2449.55 0.55 19.84 20.16 20.00 19.98 0.05 2.86 0.33

(zmax = 13.30 mm) does not coincide with the lens focus (f = 20 mm), but becomes shifted toward the lens.As the pupil radius increases, the values of S′

+ and S′− approach the focal distance of the lens; at r ∼ 1 mm,

the difference between these values and the focal distance is smaller than 1%.The overall pattern of the intensity distribution (U2

D and U2K) along the axis can be obtained analytically

using the Debye or Kirchhoff diffraction approximation [8]

U2D =

(kr2

2f2

)2( sin(u/4)u/4

)2

, (8)

U2K =

( kr2

2zf

)2( sin(u′/4)u′/4

)2

, (9)

where UD and UK are the amplitudes of the intensity distribution on the axis as functions of the distance zfrom the lens, k = 2πn/λ is the wavenumber, u = k(r/f)2(z − f), and u′ = k(r/f)2(z − f)(f/z) = u(f/z).

The Debye approximation (8) is valid only for large diaphragm diameters. In this case, the principal max-imum is observed at the point zmax = f , and the minimums (zmin) are located approximately symmetricallywith respect to this point at the distances

(Δz)D = ±(f − zmin) = ±(2λ/n)(f/r)2, (10)

where (Δz)D is actually the longitudinal half-width of the intensity distribution. Column 10 in table yieldsthe calculated values of (Δz)D, which show that it is only at r > 0.5 that the longitudinal half-width issmaller than the distance between the foci f0 = 20 mm and f+1 = 18.82 mm, which is equal to 1.18 nm.

For small diaphragms, the Kirchhoff approximation should be used. Figure 3 shows the intensity dis-tributions calculated by Eq. (9), which reveal asymmetry at r < 0.5 mm (Figs. 3a–3c). The intensity ismodulated and rapidly decreases in the direction toward the lens and smoothly decreases in the oppositedirection, with the principal maximum zmax being shifted toward the lens. At r > 0.5 mm (Fig. 3, d),the distribution becomes almost symmetric, and the Debye approximation can be used beginning from thisvalue of r.

To determine zmax in the domain where the Kirchhoff approximation is valid, we equate the derivativedUK/dz to zero and obtain

tan (u′/4) = (u′/4)(f/z). (11)

Equation (11) has no analytical solutions. The values z = zmax satisfying this equation can be determined bya graphical or an iterative method. Column 6 in table gives the values of zmax as functions of r, which werecalculated for f = 20 mm; the values of the relative aperture are listed in columns 8 and 9. As the diaphragm



Fig. 3. Intensity distributions along the z axis for different diaphragm radii: (a) r = 0.1 mm andzmax = 13.30 mm; (b) r = 0.15 mm and zmax = 17.4 mm; (c) r = 0.25 mm and zmax = 19.6 mm;(d) r = 0.5 mm and zmax = 20 mm.

radius is increased, the value of zmax approaches f . For the angular aperture of 0.86◦ (r = 0.3 mm), we havezmax = 19.8 mm, which is smaller than f by 1%. At r > 0.5 mm, the intensity maximum almost coincideswith the focus (zmax = f). If a diffractive component is placed behind the lens (a1 = 0.002097 mm−1), thenzmax (order 0, see column 7 in table) decreases at the same values of r (for instance, zmax = 19.2 at r = 0.3);at r > 0.5 mm, zmax oscillates within 0.7–0.1% around f . The fact that the value of zmax is not constant isexplained by the influence of diffraction of order +1.

INTENSITY DISTRIBUTION ALONG THE AXISOF THE DIFFRACTIVE–REFRACTIVE LENS

As a result of broadening and asymmetry of the intensity distribution functions, the foci of orders 0 and+1 of the diffractive–refractive IOL can diverge, converge, or coincide, and the total distribution can becontinuous or have a dip whose magnitude depends on the pupil diameter and phase shift in the diffractivestructure.

For diaphragm diameters smaller than 1 mm (r ≤ 0.5 mm) and without the phase shift (a0 = 1 or 0),there is no separation into the beams of orders 0 and +1, because the diaphragm radius is comparable withthe central zone radius (r0 = 0.51 mm). Separation in diffraction orders begins at diameters greater than1 mm (r > 0.5 mm), when more than one zone is open. In the interval r = 0.5−0.8 mm where the half-width(Δz)D = 1.31− 0.5 mm (see column 10 in table) of each order is commensurable with the distance betweenthe foci (f0 − f+1 = 1.18 mm), however, the orders do affect each other. If the phase shift is introduced, thecentral zone diameter decreases, and separation into orders begins already at r < 0.5 mm.

As an example, Figs. 4a–4d show the total intensity distributions at r = 0.5 mm for different values ofthe phase shift a0. The plots were constructed with the use of the Microsoft Excel system. The data forthese plots were calculated by a program developed by P. S. Zav’yalov. It follows from the figure that theintensity distribution functions in the foci of orders 0 and +1 almost coincide at a0 = 1.0 (a) and 0.125 (d);at a0 = 0.5 (b), they are separated (100% dip); finally, at a0 = 0.25 (c), they partly coincide (50% dip). Thevalues of the focal distances (f0 and f+1) change in this case. The caption gives the values of f0 and f+1,and the deviations of the optical powers ΔΦ0 and ΔΦ+1 from their nominal values are indicated in brackets.These deviations are identical for the IOL and for the eye model, i.e., for the IOL in a converging beam.

Figure 5 illustrates the changes in the focal distances f+1 and f0 and also the differences in the opticalpowers in orders +1 (Φ+1) and 0 (Φ0) for a0 = 1.0, 0.5, 0.25, and 0.125 corresponding to the phase shifts


270 LENKOVA

Fig. 4. Intensity distributions along the z axis at r = 0.5 mm, a1 = 0.002097, and different values ofthe phase shift a0: (a) a0 = 1 and f0 = f+1 = 19.36 mm (ΔΦ0 = +2.21 D and ΔΦ+1 = −1.98 D);(b) a0 = 0.5, f0 = 20.34 mm (ΔΦ0 = −1.12 D), and f+1 = 18.48 mm (ΔΦ+1 = +1.31 D); (c) a0 = 0.25,f0 = 20.14 mm (ΔΦ0 = −0.47 D), and f+1 = 18.64 mm (ΔΦ+1 = +0.69 D); (d) a0 = 0.125,f0 = f+1 = 19.28 mm (ΔΦ0 = +2.49 D, and ΔΦ+1 = −1.70 D).

(in radians) ϕ = 2π, π, π/2, and π/4 versus the pupil radius r. The changes are particularly noticeable atsmall phase shifts a0: first, the focal distance f+1 decreases and f0 increases, i.e., the optical power increasesin order +1 and decreases in order 0; after that the opposite changes occur.

If there is no phase shift (a0 = 1 or 0, r0 = 0.51 mm), the bifocal properties of the IOL are manifestedalready at the pupil radius r = 0.6 mm (see Fig. 5a). At these parameters, a 50% dip is formed in the totalintensity distribution function. Introduction of the phase shift ϕ = π rad (a0 = 0.5 and r0 = 0.36 mm)induces this 50% dip already at r = 0.43 mm (see Fig. 5b). At r = 0.5 mm (see also Fig. 4b), when twozones [the central zone and the first annular zone (r1 = 0.51 mm)] are open, complete separation of theorders is observed (100% dip). Further expansion of the pupil changes the dip in the intensity distributionwithin 80–100%. With a decrease in the phase shift to ϕ = 0.5 and ϕ = 0.25π rad (a0 = 0.25 and 0.125;r0 = 0.26 and 0.18 mm), separation of the foci of orders 0 and +1 starts already at r = 0.30 and 0.21 mm(see Figs. 5c and 5d). With increasing r, however, the foci in the two last cases substantially diverge at first(to f0 = 21.75 and f+1 = 17.4 mm at a0 = 0.25, i.e., they change by 8.75 and 7.54% with respect to thenominal value, and to f0 = 23.8 and f+1 = 15.66 mm at a0 = 0.125, i.e., by 19.0 and 16.8%), then they startto converge, and finally, at a0 = 0.125, they even coincide in the interval r = 0.43–0.57 mm. In the domainof foci coincidence, the bifocality disappears; therefore, the IOL operates as a monofocal lens. Taking intoaccount that the nominal values of the focal distances of orders 0 and +1 lie within the half-width of thetotal intensity distribution function (see Fig. 4d), we can say that the IOL operates as a multifocal lens ora lens with a large focal depth commensurable with the difference f0 − f+1 = 1.18 mm. Pupil expansion(r > 0.57 mm) leads to foci separation and recovery of bifocality (see Fig. 5d).

Analyzing Fig. 5, we can conclude that the phase shift ϕ = π rad (a0 = 0.5, see Fig. 5b) is the optimalone; in this case, foci separation starts from r < 0.4 mm, the focal distances deviate only insignificantly fromtheir nominal values (within 0.5%), and it is only near r = 0.5 mm that the deviation reaches ∼1.7–1.8%(f0 = 20.34 mm and f+1 = 18.48 mm). For diaphragm diameters greater than 1.2 mm (r > 0.6 mm), fociseparation, i.e., bifocality, is observed in all cases independent of the phase shift.

Figure 6 illustrates the changes in the intensity distribution function along the axis for the phase shiftϕ = π rad (a0 = 0.5) as a function of the pupil radius r. As the radius increases, the width of the distributionin each order decreases; as a result, the mutual effect of the orders becomes less intense, and the locations ofthe maximum values of intensity approach the nominal values of the focal distances. Another reason for thedecrease in the width of the distribution intensity is the interference effect. With increasing radius, more



Fig. 5. Changes in the focal distances f+1 and f0 and also differences in the optical powers Φ+1

and Φ0 versus r for the phase shifts a0 = 1.0 (a), 0.5 (b), 0.25 (c), and 0.125 (d).

zones are open, and the number of the interfering beams, which affects the width of the distribution functionsimilar to multibeam interference, increases.

The intensity distributions in all figures are normalized to the intensity of the maximum in order +1.It should be noted that the intensity on the axis in order +1 is greater than that in order 0, though thediffraction efficiencies at a given height of the diffractive structure should be identical. This is explainedby the fact that the efficiency is understood here as the total amount of light directed to each order ofdiffraction, while the maximum intensities on the axis are compared in these plots. The intensity on theaxis depends on the energy density in the distribution of each order; therefore, it is inversely proportionalto the squared focal distance. The calculations show that the ratio of intensities in the foci in domains withno superposition of the orders corresponds to the squared ratio of the focal distances (f+1/f0)2 = 0.886,independent of the phase shift value.


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Fig. 6. Intensity distributions along the z axis for a0 = 0.5, a1 = 0.002097, and different values of r:(a) r = 0.5 mm, f0 = 20.34 mm (ΔΦ0 = −1.12 D), and f+1 = 18.48 mm (ΔΦ+1 = +1.31 D);(b) r = 0.7 mm, f0 = 20.02 mm (ΔΦ0 = −0.07 D), and f+1 = 18.80 mm (ΔΦ+1 = +0.08 D);(c) r = 1.0 mm, f0 = 20.02 mm (ΔΦ0 = −0.07 D), and f+1 = 18.80 mm (ΔΦ+1 = +0.08 D);(d) r = 1.5 mm, f0 = 20.00 mm (ΔΦ0 = 0 D), and f+1 = 18.82 mm (ΔΦ+1 = 0 D).

All studies were performed for the diffractive component with the optical power of 4.2 D, where thecentral zone diameter in the absence of the phase shift (a0 = 1.0) is ∼1.0 mm. In the diffractive componentwith a lower optical power (2.1 D), the central zone diameter is ∼1.4 mm at a0 = 1.0 and ∼1.0 mm ata0 = 0.5. In the latter case, foci separation independent of the phase shift is observed only from thediaphragm diameter of ∼1.7 mm, which is commensurable with the minimum pupil diameter. Possibly, adecrease in the optical power of the diffractive component will change the optimal phase shift value andrestrict the area of applicability of these components. Further investigations are necessary to determine thedependence of the phase shift on the optical power of the diffractive component more exactly.

CONCLUSIONS

The paper describes an analytical study of the effect of the eye pupil diameter and phase shift in thediffractive component with the optical power of 4.2 D on the light intensity distribution along the axis ofa bifocal diffractive–refractive intraocular lens. It is demonstrated that a decrease in the pupil diametercan violate the bifocal properties of the IOL. The total intensity distribution corresponding to the foci oforders 0 (far vision) and +1 (near vision) can be continuous or have a dip whose magnitude depends on thediaphragm (pupil) diameter and phase shift. An increase in the pupil diameter leads to a decrease in thewidth of the intensity distribution in each order, separation of the orders, and recovery of IOL bifocality.The diffraction efficiency or the ratio of intensities in orders 0 and +1 corresponds to the design value,independent of the phase shift.

After introduction of the phase shift to the IOL diffractive structure, foci separation (bifocality) occursat a smaller pupil diameter. It is shown that the optimal shift is π rad (a0 = 0.5) at which bifocality isobserved beginning from the diameter of ∼0.8 mm (r ∼ 0.4 mm) and further remains non-violated, whilethe focal distances deviate from the nominal values by no more than 0.5%. For diaphragm diameters greaterthan 1.2 mm (r > 0.6 mm), the foci of orders 0 and +1 are separated independent of the phase shift.

From the results of this study, we can conclude that the phase shift or the central zone size for normaleyes with the minimum pupil size of 1.8–2.0 mm is of no principal importance for the diffractive structurewith the optical power of 4.2 D. Nevertheless, the effect of the phase shift on the bifocal properties of the



IOL should be taken into account if the diameter of a real pupil or an artificial diaphragm becomes for somereasons smaller than 1.2 mm, because superposition of the orders becomes possible and the focal distancescan significantly deviate from their nominal values. For diffractive structures with a lower optical power, theuse of the phase shift may be required even for eyes with the normal pupil.

REFERENCES

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Documents

Effect of the eye pupil diameter and the phase shift in the diffraction structure on bifocal properties of diffractive-refractive intraocular lenses