Methods for investigating optical characteristics of bifocal diffractive-refractive intraocular lenses

$Page 1: Methods for investigating optical characteristics of bifocal diffractive-refractive intraocular lenses$
Methods for Investigating Optical Characteristics of Bifocal

Diffractive-Refractive Intraocular Lenses

G. A. Lenkova

Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, Novosibirsk

E-mail: [email protected]

Received 23 June 2006

Abstract—Peculiarities of designing and determining the refraction (lens power) and the diffraction

efficiency of bifocal hybrid (diffractive-refractive) intraocular lenses (IOL)—implants—are considered.

Methods for designing the refractive IOL for near and far vision are analyzed and compared by means of

measuring the focal lengths in the air and medium. It is shown that in investigating the hybrid lenses it is

more convenient to estimate the conformity of the profile height with the designed value from the

relationship of intensities of the 1st and 2d diffraction orders in the air and of the zero and 1st orders in the

medium. Admissible errors of measurements and deviations of the hybrid IOL parameters are

determined and used for calculating the refraction and diffraction efficiency.

DOI: 10.3103/S8756699007030090

INTRODUCTION

Bifocal and multifocal intraocular lenses (IOL) that enable to see without glasses at any distance to ob-

jects find more and more application in eyesight recovery. The IOLs can act as refractive, diffractive, or

diffractive-refractive (hybrid) lenses. The latter are of particular interest because they enable to correct cor-

nea and spherical component aberrations, vary the relationship of intensities in images of near and far ob-

jects, decrease the lens thickness, etc. [1].

In IOL manufacturing, it is necessary to control that the optical characteristics should correspond to the

design parameters. International ISO/DIS 11979-2 Standard [2] and Russian R 52038-2003 State Standard

[3] (hereafter SS] are available for usual refractive IOLs. They set the optical characteristic requirements and

recommend methods for testing the accuracy of manufactured IOLs. Methods recommended by the SS are

only partially applicable to control of the diffractive-refractive IOLs because of their specificity. This paper

is aimed at considering possible methods for controlling the hybrid IOL parameters and at estimating

admissible measurement errors.

CALCULATING THE DIFFRACTIVE-REFRACTIVE IOL REFRACTIONS

The focal power (refraction) of the simplest two-component system which the refractive IOL refers to

can be represented as [4]

� � � � ��

�

��

�

��

n f d n n nR R

d n n

1 1 2 1 2 2 2 1

1 2

2 11 1( )

( ),

2

2 1 2n R R

(1)

where � � �, ,1 2

and are the focal powers of lenses or components (lens surfaces), d is the distance between

the lenses or lens components, n1

and n2

are the refractive indices of the environment and the lens material,

respectively, and R R1 2

and are the curvature radii of the lens surfaces. Hereinafter we assume that f is the

back focal distance (usually denoted as f � in the geometric optics) and that the focal distances and the

curvature radii are given in millimeters and the focal powers, in diopters. We should note that the IOL

262

ISSN 8756-6990, Optoelectronics, Instrumentation and Data Processing, 2007, Vol. 43, No. 3, pp. 262–273. © Allerton Press Inc., 2007.

Original Russian Text © G.A. Lenkova, 2007, published in Avtometriya, 2007, Vol. 43, No. 3, pp. 85–99.

OPTICAL INFORMATION TECHNOLOGIES,

ELEMENTS, AND SYSTEMS

refraction mark is given for the conditions inside the eye, i.e., in the medium, hence in (1), f is the focal

distance in the medium.

The bifocal diffractive-refractive (hybrid) IOLs are convex-plane or equiconvex lenses with a diffraction

structure formed on the plane surface or on one of the spherical surfaces. Without loss of generality we will

consider only IOLs having a convex-plane refractive part with a diffraction structure on the plane surface.

The form of the structure is similar to Fresnel zone plates. The zone plate does not work in the zero diffrac-

tion order and IOL forms images of far objects on the retina as a usual refractive lens designed for infinity. In

the �1st order (hereafter the order number is given without the sign “+”), the diffractive component produces

a positive additional focal power and the IOL becomes able to project near objects on the retina. Thus, the bi-

focal IOLs have two focal powers, one of them ( )�0,aq

for far vision (the zero order) and the other ( )�1,aq

for near vision (the 1st order).

The focal power for the far vision, based on (1), for R R1

� and R2

� � has the form

�0,aq aq 0,aq IOL aq

� � �n f n n R( ) , (2)

where f0,aq

is the focal distance of the eye aqueous medium for far vision, R is the curvature radius of the

spherical surface, and naq

and nIOL

are the refractive indices of the environment and the lens material,

respectively. The focal power of IOL for near vision �1,aq

is produced by the refractive part and the

diffraction structure. Based on (1) and taking into account that the lens thickness d is much smaller than the

curvature radius (usually d ~ 1 mm, R1

and R2

10~ mm, in our case, R2

� �) it can be represented as

� � �1,aq aq 1,aq 0,aq

� � �n fd, (3)

where f1,aq

is the IOL focal distance in the eye medium for near vision, �d

is the diffractive component

focal power that is calculated for the given near vision distance and the chosen eye model [5].

Unlike the refractive component the focal power �d

of the diffractive component remains constant re-

gardless of the medium because it is determined only by the law of changing the radii of diffraction structure

zones. In fact, when the refractive index naq

of the environment changes, the focal distance fd

of the

diffractive component also changes, the change being proportional ( )., ,

f n fd daq aq air

� Therefore, �d

is al-

ways equal to the focal power in the air:

�d d d

n f f� �aq aq air, ,

,1 (4)

where f fd d, ,aq air

and are the focal distances, and n fdaq aq,

and 1 fd , air

(in the numerator, the air

refractive index nair

� 1) are the focal powers in the medium and air, respectively.

For manufacturing IOL with the given refraction the curvature radii of its surfaces are chosen from a

standard set of radii with regard to the refractive indices and formulas (1) and (2). The refraction of the ob-

tained lens not always corresponds to the design value because of the total and local deviations of the curva-

ture radii and the real values of the refractive indices. It is known that there are no methods and devices for

direct refraction measuring, hence, the real refraction is calculated from the measured values of lens parame-

ters, which are involved in (1) and (2). In so doing, measuring the focal distances we obtain general informa-

tion on the IOL refraction whereas while measuring the refractive indices, thickness, and curvature radii of

the lens we obtain concrete information on the correspondence between the real and design parameters.

METHODS FOR MEASURING REFRACTIONS OF DIFFRACTIVE-REFRACTIVE IOLS

The back vertex refraction of usual refractive IOLs can be determined according to the SS by three alter-

native methods. The first method is calculating the refraction from formula (1), based on the measured re-

fractive indices, thickness, and curvature radii of the lens. In the second method, the back vertex refraction

�aq

is calculated by measuring in the air the IOL back focal distance from the formula

� �aq air

� Q , (5)

where �air

is the IOL refraction in the air, which is related to the paraxial focal distance in the air ( )fair

as

�air air

� 1 f ; (6)

OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 43 No. 3 2007

METHODS FOR INVESTIGATING OPTICAL CHARACTERISTICS 263

Q is the coefficient of conversion of the IOL refraction in the air to refraction in the medium; it is calculated

from the formula

Q � � �aq, nom air, nom

. (7)

Herein, �aq, nom

and �air, nom

are the IOL refractions in the conditions similar to the work inside the eye (in

situ) and in the air, both of them were calculated from (1) using the nominal parameters of the lens.

Paper [6] analyzed the relationship between the IOL optical characteristics in the air, aqueous liquid, and

cell. It was shown that for thin lenses like most present-day IOLs the refraction conversion coefficient de-

pends only on the refractive indices and can be written as

Q n n n� � �( ) ( ).IOL aq IOL

1 (8)

Substituting (6) and (8) into (5) yields

�aq IOL aq IOL air

� � �( ) ( )( ),n n n f1 1 (9)

where fair

is found by measuring or calculated from the formula

f R nair IOL

� �( ).1 (10)

In the third method, the IOL back vertex refraction is found by measuring the object enlargement. The

IOL focal distance ( )f is calculated from the formula

f F h h� ( ) ,target image

(11)

where htarget

and himage

are the linear sizes of the scale target and its image and F is the focal distance of the

collimator objective. Then we calculate �aq

from (2), or from (9) if f is measured in the air, assuming

f fair

� .

When determining the refraction by the second or third methods, the controlled IOL is placed behind the

collimator, the linear scale or test-object being at the focus. Then the focal distance (the distance from the

lens to the scale image) is measured by a microscope in the second method and the image size in the third.

The second method is most convenient and frequent because it does not require additional equipment for

measuring the curvature radii, linear size of the scale, and focal distance of the collimator objective. How-

ever, application of this method for testing bifocal diffractive-refractive IOLs in the form recommended by

the SS is impossible. The SS determines testing methods only for refractive IOLs.

Let us consider how we can change the second method for calculating the refraction �aq

, as applied to

the hybrid IOLs. It is known [1] that the microstructure of the hybrid IOL diffractive component is fabricated

as a sawtooth profile whose height is chosen such that the light intensities in the zero (far vision) and 1st

(near vision) diffraction orders in the medium of the eye are close or equal, and negligible in higher orders.

In the air, the phase delay on the structure increases and the light is primarily directed to the 1st and 2d dif-

fraction orders, the zero order being practically absent. Hence, while measuring in the air, it is impossible to

calculate the hybrid IOL refraction for the far vision from formula (9), and for the near vision we have to

consider the difference of refraction transformation for the diffractive and refractive components (formulas

(4) and (9)).

We can determine the refractions �0,aq

and �1,aq

for the far and near visions, respectively, by measure-

ments in the air in two ways. The first way is using the design value of the diffractive component focal power

�d

and the measured focal distance in the air in the 1st diffraction order ( ).f1,air

Then the formulas have the

form

� �0,aq 1,air

� �Q fd

( ),1 (12)

� � � �1,aq 0,aq 1,air

� � � � �d d

Q f Q( ) .1 (13)


264 LENKOVA

The other way is calculating the refractions from the measured values of focal distance in the air in the 1st

( )f1,air

and 2d ( )f2,air

orders, i.e., without using �d, from the formulas

�0,aq 1,air 2,air

� �Q f f( ),2 1 (14)

� � � �1,aq 0,aq 0,aq 2,air 1,air 2,air

� � � � � � �d

f f Q f( ) ( )1 1 1 � �( ) .1 2Q f1,air

(15)

In (12)–(15), Q is the conversion coefficient that is found from (8).

The third way is measuring the refractions for far and near visions in the medium if IOL is placed in a

cuvette filled with a salt solution (the SS allows distilled water). In this case, we observe the zero and 1st dif-

fraction orders. Paper [6] showed that refractions in the medium �0,aq

and �1,aq

) are equal to those behind

the cuvette �0,cuv

and �1,cuv

):

� �0,aq aq 0,aq 0,cuv 0,cuv

� � �n f f1 ;

(16)

� �1,aq aq 1,aq 1,cuv 1,cuv

� � �n f f1 ,

where f f0,aq 1,aq

and , f f0,cuv 1,cuv

and are the distances from the IOL to the focus in the medium and air

behind the cuvette, respectively. In the real conditions, the thickness of the aqueous layer and cuvette wall

are much smaller than the focal distance, hence after the light beam escapes the cuvette the distance from the

IOL to the focus in the air decreases with the medium refractive index naq

. Formula (16) follows from

relationships f n f0,aq aq 0,cuv

� and f n f1,aq aq 1,cuv

� . We might enter corrections for focus shifts caused

by the action of the medium layer and cuvette wall, but it is unnecessary if the focal distances are measured

by a microscope. In the latter case, the shifts of the images of focus and structure (or the lens vertex) occur

simultaneously in the same direction and by the same value.

The peculiarity of the third method is that �aq

is calculated simply as the inverse value of fcuv

expressed

in meters without entering the conversion coefficient. Thus, in this case, the measurement accuracy of the re-

fractive indices does not affect the result of measuring the refraction.

Contrary to the third method the refraction measurements according to the SS are carried out in the air, al-

though we showed above that measuring the IOL refraction using a cuvette has some advantages. The SS

recommends to place IOL in the eye model, i.e., in a cuvette filled with a salt solution in our case, only for in-

vestigating the image quality, that is, for measuring the resolution and the modulation transfer function.

The fourth and simpler method for refraction measurement, which is not considered in the SS, is as fol-

lows. A parallel laser beam illuminates the IOL, and the focal distance is measured behind the cuvette or in

the air as the difference between indications of the microscope longitudinal scale when it is targeted at the

lens surface and at the focused image of the point source (Fig. 1). This method has advantages over the de-

scribed methods. Firstly, it is not required to calibrate the scale. Secondly, the form of intensity distribution

in the focal plane (point spread function (PSF)) evidences either the presence or absence of spherical, astig-



Fig. 1. A system for measuring focal distances: 1, laser; 2, collimator; 3 and 4, deflecting mirrors; 5, cuvette filled with

distilled water and IOL; 6, microscope; 7, eye.

matic, or other aberrations. In addition, the PSF Fourier transform that is a modulation transfer function

(MTF) can show how the IOL transfers different spatial frequencies.

We should note that measurements in the air can cause problems because the IOL diffractive component

is calculated for the medium in such a manner that to correct the cornea and refractive component aberra-

tions in the 1st diffraction order for a standard wavelength. In the air, the correction conditions are violated

and spherical aberrations appear; they are most evident in the 2d diffraction order. Moreover, when measur-

ing in the white light we observe that the focal power of the IOL diffractive component depends on the wave-

length. In the 1st order, the chromatic aberrations of the diffractive and refractive components are mutually

compensated whereas in the 2d order, in the presence of spherical aberration they appreciably decrease the

image contrast and practically prevent measuring the focal distances in this order. Therefore, investigating

in the air in the white light the second method for calculating the refractions from (14) and (15), based on

measuring the focal distances in the 1st ( )f1,air

and 2d ( )f2,air

orders, i.e., without using �d

, can lead to er-

rors. In the monochromatic light, the aberrations in the 2d order are smaller and the second method is

possible.

MEASURING PARAXIAL FOCAL DISTANCE

The IOL focal power is calculated by geometric optics formulas (2), based on the paraxial focal distan-

ces f . For investigating the IOL quality of manufacturing the focal distance is measured as a distance from

the least-spreading spot to the back lens surface. To agree the measurement results with the calculation, we

have to enter corrections for the distance from the IOL surface to the principal back plane of IOL ( )�f SH1

� �

and from the paraxial focal point to the “best-focusing” point ( ),� �f S2

2� � i.e., to the plane where the mini-

mal spreading point is observed. The first correction is known from the geometric optics [7]:

� f SH

f n n d

n R1

� � �

�( )

.IOL aq

IOL

(17)

For a thin convex-plane lens for f n R n n� �aq IOL aq

( ) formula (17) is transformed into

�f S d n nH1

� � � ( ).aq IOL

(18)

For example, if d � 0.9 mm, then in the medium(naq

� 1.336 and nIOL

� 1.505), � f SH1

� � � 0.8 mm whereas

in the air ( ,n naq air

� � 1 nIOL

� 1.505), � f SH1

� � � 0.6 mm.

The second correction for IOL as a convex-plane lens can be calculated in a first approximation by (2.3)

from [6], taking into account that the minimal spreading spot is located half the value of longitudinal spheri-

cal aberration �S� (in [6], � �S fy

� � ) from the paraxial focus toward the lens:

� �f Sn n

n n

D

f

n n

n n

D

R2

3 2

2

2 3 2 2

22 2

4 1 4

2 2

4 1 4� � �

� �

�

��

�( ) ( ), (19)

where n n n�2 1

(n1

and n2

are the refractive indices of the environment and lens material), D and f are the

diameter and focal distance of the lens, and R is the curvature radius of the spherical surface. In the

conditions of the medium (n n1

� �aq

1.336) and for n n2

� �IOL

1.505 we obtain n � 1.1265 and

� �f S D R2

22 1 56 4� � � . . In the air ( )n n

11� �

airfor the same n

2we have n � 1.505 and

� �f S D R2

22 0 29 4� � � . . For example, for R � 8.24 mm ( f

aq� 65.14 mm and f

air� 16.32 mm) and

D � 4 5. mm (3 mm) for the medium � �f S2

2� � � 0.96 mm (0.43 mm) and for the air

� �f S2

2� � � 0.18 mm (0.08 mm), i.e., is about 5 times less. If the spherical aberration for the near vision

( )f1

is corrected at the expense of the diffractive component, then the second correction �f2

is calculated

only for f0

(far vision).

The calculation of �f2

from analytical formula (19) is convenient because it does not require special soft-

ware, nevertheless it is sufficiently accurate. The results of calculations by formula (19) and the ray-tracing

method for the medium differ by no more than 0.05 mm


266 LENKOVA

CORRECTION FOR FOCUS SHIFT DUE TO AQUEOUS LAYER AND CUVETTE WALL

In measuring the IOL characteristics in conditions similar to the work inside the eye, the lens is placed

into a cuvette filled with a salt solution (or distilled water), the cuvette simulates the eye model. In this case,

the measured distance from the lens to the focus behind the cuvette ( )fcuv.meas

is represented as

f n n f n d n n d dcuv.meas air aq cuv aq aq air cuv cuv a

� � � �( )( ) ( )q cuv

� d , (20)

where n naq cuv

, , and nair

are the refractive indices of the medium, wall, and air, respectively; daq

and dcuv

are

the thicknesses of the aqueous layer and wall; fcuv

is the focal distance behind the cuvette, assuming that the

thicknesses of the aqueous layer and wall are negligible. We can determine from (20) that

f f fcuv cuv.meas

� � �3, (21)

where �f3

is the focus shift caused by the aqueous layer and wall. For nair

� 1, from (20) and (21) we obtain

� f f f d n n d n n3

1 1� � � � � �cuv.meas cuv aq aq aq cuv cuv cuv

( ) ( ) . (22)

If the focal distance is measured by a microscope, the correction is not entered because the visible dis-

tance from the structure to the microscope changes simultaneously by the same value.

ERRORS IN DETERMINING REFRACTIONS OF DIFFRACTIVE-REFRACTIVE IOL

As was shown, the IOL refraction is determined by calculating by parameters or by measured focal dis-

tances. For evaluating how the discrepancy between the manufactured IOL parameters and the given values,

and also the accuracy of measuring the parameters and focal distances affect, the accuracy of determining

the refraction, we calculate the total differentials ��0,aq

and ��1,aq

from (2), (3), (5), (8), (9), and

(12)–(16):

�� 0,aq 1,aq IOL aq IOL aq

� � � � �( ) ( ) [( ) ] ,1 12

R n R n n n R R (2a), (3a)

�� 0,aq 0,air 0,air 0,air

� �Q Q f2

, (5a)

� � �Q n n n n n� � � � �[( ) ( ) ] [ ( )] ,aq IOL IOL IOL aq

1 1 1 12

(8a)

�� 0,aq 0,air 0,air

� � �2

Q f [ ( ) ( ) ] [ ( )]� � � �0,air aq IOL IOL 0,air IOL aq

n n n n n� � � �1 1 12

, (9a)

�� 0,aq 1,aq 0,air 1,air 1,air

� � �Q Q f f( ) ,2

(12a), (13a)

�� 0,aq 0,air 1,air 1,air 2,air 2,a

� � �Q Q f f Q f f( ) ( )22 2

ir, (14a)

�� 1,aq 0,air 1,air 1,air 2,ai

� � � � �Q Q f f Q f[( ) ] [( )1 2 12

r 2,air

2] ,� f (15a)

�� 0,aq 0,cuv 0,cuv 0,cuv

� � � ( ) ,12

f f ( )16à�

�� 1,aq 1,cuv 1, cuv 1,cuv

� � � ( ) .12

f f ( )16à ��



In (5a) and (9a), the refraction accuracy for the refractive component of the hybrid IOL is denoted by

��0,aq

.This is the same as ��aq

for a usual refractive IOL based on (5) and (9). We should note that formu-

las for calculating the accuracy of giving and measuring the parameters, (2a), (3a), (12a) and (13a) obtained

by differentiating (2), (3), (12), and (13), respectively, have the same form. The explanation is that �d

is the

constant depending only on the law of changing the zone radii of the diffractive structure and independent of

the environment. Formulas ( )16à� and ( )16à �� were derived from (16) for far and near visions, respectively.

Note, in formulas (5a) and (12a)–(15a) the coefficients at �Q are the same although it follows from (5)

and (12)–(15) that their analytical form must be different. For simplifying the formulas, these and some

other coefficients were changed because in fact

� �0,air 0,air 1,air 1,air 2,air

� � � � �1 1 2 1f f f fd

( ) ( ) ( ).

Let us define formulas (2à)–(16a ��) and the tolerances for concrete initial IOL parameters:R � 8.24 mm,

�d

� 4.24 diopters, naq

� 1.336, and nIOL

�1.505. We calculate in addition the nominal values of the param-

eters that enter the formulas and are derivatives of these parameters: Q � 0.3347 (8), � �0,aq 0,cuv

� � 20.5

diopters (2), (16), � �1,aq 1,cuv

� � 24.74 diopters (3), (16), f0,air

� 16.32 mm (10), �0,air

� 61.25 diopters

(6), f1,air

� 15.27 mm ( ( )),fd1,air 0,air

� �1 � � f2,air

� 14.33 mm ( f2,air

�1 2( )),� �0,air

�d

f0,cuv

�

48.78 mm and f1,cuv

� 40.42 mm (16). Substituting the nominal parameter values into (2à)–(16a ��) in such a

manner that the accuracy of measuring �f and �R is expressed in millimeters and �� in diopters we obtain

�� 0,aq 1,aq IOL aq

� � � �121 36 121 36 2 49. . . ,n n R (2b), (3b)

�� 0,aq 0,air

� �61 25 1 26. . ,Q f (5b)

� � �Q n n� �1 32 1 98. . ,IOL aq

(8b)

�� 0,aq 0,air IOL aq

� � � �1 26 80 78 121 18. . . ,f n n (9b)

�� 0,aq 1,aq 1,air

� � �61 25 1 437. . ,Q f (12b), (13b)


� � �61 25 2 87 1 63. . . ,Q f f (14b)


� � �61 25 1 42 3 24. . . ,Q f f (15b)

�� 0,aq 0,cuv 0,cuv

� � �0 42. ,f ( )16b�

�� 1,aq 1,cuv 1,cuv

� � �0 61. .f (16b )��

Let us assume that we want to have ��0,aq

� 0.1 diopters, then according to (2b), (3b), (5b), and

(12b)–(15b) the curvature radius R and the conversion coefficient Q must be known or measured with the ac-

curacy �R � 0.04 mm and �Q � 0.0016. The required accuracy or admissible deviations in measuring the re-

fractive indices: �naq

�0.0008 ((2b), (8b), and (9b)), �nIOL

� 0.0008 (2b), �nIOL

� 0.0012 ((8b) and (9b)),

and in measuring the focal distances: � f0,air

� 0.08 mm ((5b) and (9b)), � f1,air

� 0.07 mm ((12b), (15b),

� f1,air

� 0.035 mm (14b), � f2,air

� 0.06 mm (14b), � f2,air

� 0.03 mm (15b), � f0,cuv

� 0.24 mm ( ),16b� and

� f1,cuv

� 0.17 mm ( ).16b�� These data show that in measuring the focal distances of IOL in the cuvette, the

admissible deviations are several times greater than in measuring in the air.


268 LENKOVA

It is clear that the coefficients in the formulas will change depending on the IOL refraction. We analyze

the degree of the change using formulas (2b) and (5b). For obtaining the refraction in the most essential re-

gion �0,aq

� 18–23 diopters (�0,air

� 53.7–68.7 diopters) the curvature radius of the lens spherical surface

has to be varied within R � 9.38–7.34 mm ( %)�12 and the coefficients at �n and �R, within 106.6–136.2

( %)�12 and 1.92–3.14 ( %)�24 . Assume ��0,aq

� 0.1 diopters, then � �n naq IOL

� � 0.0009–0.0007

( %)�12 and �R � 0.05–0.03 mm ( %).�25 For the chosen region of refractions the coefficient at �Q varies

within 53.78–68.72 ( %)�12 and the value of �Q, within 0.0019–0.0015 ( %)�12 . Calculations show that

for the chosen region of refractions the change of the coefficients in (2b)–(16 b )� and the tolerances do not

exceed 12–25%, i.e., change insignificantly. To conclude, we note that for calculating the diffraction with an

accuracy of 0.1 diopters the division value of the device in measuring refractive indices should be no more

than 0.0005, and in measuring the focal distances in the air and behind the cuvette, no more than

0.01–0.05 mm and 0.1 mm, respectively.

Measuring the focal distances, one should pay attention that it is necessary to enter corrections. Other-

wise the calculated refraction value will be greater by about 0.5–1.0 diopters because, depending on the

measurement conditions, these corrections can be about 0.6 mm (in the air) and 1 mm (in the medium). The

foregoing calculations evidence that the correction is important for measurements in the air.

METHODS AND ERRORS IN MEASURING DIFFRACTION EFFICIENCY OF HYBRID IOL

An important characteristic of hybrid IOL, besides the refraction, is diffraction efficiency in the zero and

1st diffraction orders. It is usually determined by the ratio of diffracted beam intensity to incident or trans-

mitted beam intensity. The intensity measurement system is shown in Fig. 2. In the general case, the effi-

ciency depends on the form and depth of the diffraction structure profile and on the reflection, absorption,

and scattering light losses. We assume that the light losses are small and the IOL diffraction structure is a

sawtooth structure. Then the intensity in the zero ( )I0

and nth ( )In

diffraction orders is represented as [8]

I0

2

2

2�

�

��

�

��

sin( ),

max

max

�

�(23)

In

n�

�

�

��

�

��

sin( )

( ),

max

max

�

� �

2

2 2

2

(24)

� � �max max

( ) ,� �2 h n nIOL aq

(25)

where �max

and hmax

are the maximal values of the phase delay and profile height within one zone of the

structure, n is the diffraction order, and � is the wavelength. Plots of the intensities are given in Fig. 3. It is

seen that in the medium with the profile height h � 1.61 �m the intensities of the zero and 1st orders are equal

(40.5% each), whereas in the air the zero order almost disappears. The explanation is that in the air, the phase

delay on the structure becomes greater and the light is redistributed and directed primarily to the 1st and 2d

orders (40.5% each). It is convenient to measure the efficiency in the air as well as the refraction in the 1st

and 2d orders, and then convert the results to the medium.



Fig. 2. A system for measuring diffraction efficiency: 1, cuvette with distilled water and IOL; 2, microobjective; 3,

photodiode with diaphragm; 4, microammeter.

Under experimental conditions while investigating the dependence of the diffraction efficiency on the

profile height hmax

it is problematic to take into account the effect of the light loss, including the higher-order

diffraction loss. Nevertheless, since this disadvantage is inherent in all orders, it can be prevented by analyz-

ing the mutual relationship of intensities in the operating orders. Based on (23)–(25), the relationships of in-

tensities of the zero and 1st �0 1,

), and also 1st and 2d ( ),

�1 2

diffraction orders, depending on the profile

height hmax

, have the form

�

�

0 1

0

1

2

,

max

max

( )

( )� �

� �

�

�

�

�

�

�

�

�

�

�I

I

h n n

h n n

IOL aq

IOL aq

�

h n nmax

( ),

IOL aq�

�

�

�

�

�

�

�

�

�

1

2

(26)

�

�

�1 2

1

2

2

,

max

max

( )

( )� �

� �

� �

�

�

�

�

�

�

�I

I

h n n

h n n

IOL aq

IOL aq �

2

. (27)

At the same time, by measuring or giving �0 1,

and �1 2,

we can find the profile height hmax

. From (26) and

(27) it follows that

h

n nmax

,( )( )

,�

� �

�

�0 1

1IOL aq

(28)

h

n nmax

,

,

( )

( )( )

.�

�

� �

� �

�

1 2

1 2

2

1IOL aq

(29)

Assuming that the intensities of the zero and 1st orders are equal, i.e., �0 1

1,

,� for � � 0.5461 �m

(nIOL

� 1.507, naq

� 1.3377, and n nIOL aq

� � 0.1693) from (28) we obtain hmax

� 1.61 �m and from (27)

�1 2,

� 8.98. This wavelength value is recommended by the Standard. In measuring the efficiency in the air,

the phase delay on the diffraction structure, according to (25), increases and approaches 2�. As a result, the

zero order becomes much smaller than the 1st order (,

�0 1

� 0.11). In these conditions, it is senseless to mea-


270 LENKOVA

(a)

(b)

Fig. 3. Plots of changing intensities in the zero (solid curves), 1st (dashed), and 2d (dash-dotted) diffraction orders: (a) in

the medium and (b) in the air versus profile height hmax for � � 0.5461 �m ((nIOL � 1.507 and naq � 1.3377).

sure �0 1,

. It is more convenient to estimate the correspondence of the profile height to the calculated value by

the relationship of the 1st- and 2d-order intensities (,

�1 2

� 1.04). If a suitable light source is not available, the

efficiency can be measured at other wavelengths, taking into account the dependence of the refractive in-

dices on the wavelength. Table 1 presents calculated values of �0 1,

and �1 2,

for measuring at the wavelengths

� � 0.5461, � � 0.6330 �m (He-Ne laser), and � � 0.5320 �m (solid-state laser) for the same profile height of

the diffraction structure (max

h � 1.61 �m). It is seen that the intensity relationships depends greatly on the

wavelength.

Plots of changing �0 1,

in the medium (26) and �1 2,

in the air (27) for the wavelength � � 0.5461 �m are

depicted in Fig. 4. It is seen that insignificant deviations of hmax

from the calculated value have a stronger ef-

fect on �1 2,

in the air than on �0 1,

in the medium. The relationship of the beam intensities behind the cuvette

remains equal to �0 1,

because it is formed before escaping the cuvette and depends only on the phase profile

height in the medium.

We find analytically the degree of effect of the deviations �hmax

, and �nIOL

and �naq

on the changes of

the relative efficiencies ��0 1,

and ��1 2,

near the calculated value of the profile depth hmax

� 1.61 �m. Thus,

let us differentiate (26) and (27):

� � � ��0 1 0 1, , max max

[ ( ) (� � � � � �a h h n n n n n nIOL IOL aq aq IOL aq

)], (30)

� � � ��1 2 1 2, , max max

[ ( ) (� � � � �a h h n n n n n nIOL IOL aq aq IOL aq

)], (31)

where

ah n n

a

h n n

0 1

0 1

1 2

1 22 2

,

,

max

,

, max

( );

(

��

�

��

IOL aq

IOL aq)

[ ( )]

.

max� � �h n n

IOL aq

2(32)



Table 1

Parameters Medium Air

�,�m nIOL

naq

n nIOL aq

� �0 1,

�1 2,

�0 1,

�1 2,

0.5461 1.5070 1.3377 0.1693 1.00 8.98 0.11 1.04

0.5320 1,5077 1.3387 0.1690 0.91 9.28 0.12 0.75

0.6330 1.5030 1.3348 0.1682 1.79 7.55 0.05 6.65

Fig. 4. Changes of the relative intensities �0 1, in the medium(solid curve) and �1 2, in the air (dash-dotted) plotted versus

profile height hmax for the wavelength � � 0.5461 �m (nIOL � 1.507 and naq � 1.3377).

Table 2 represents calculated values of a a0 1 1 2, ,

and from (30) and (31) for three wavelengths in measur-

ing in the medium and air, and Tables 3 and 4 contain values of the coefficients at � � �h n nmax

, ,IOL aq

, and

tolerances � � �h n nmax

, ,IOL aq

in measuring the relative intensities ��0 1,

in the medium and ��1 2,

in the air

with an accuracy of 0.1.

Table 3 shows that for measuring in the medium, �hmax

~ 0.03–0.04 �m and � �n nIOL aq

� ~ 0.003–0.004

if ��0 1,

� 0.1. In the air (see Table 4) for ��1 2,

� 0.1for the first two wavelengths the values of �hmax

are

approximately two times less and �nIOL

are approximately the same; for � � 0.6330 �m, �hmax

and �nIOL

are 13 and 5 times less than for the medium for ��0 1,

� 0.1.

CONCLUSIONS

We have shown that the methods for calculating the refraction by measuring the IOL focal distance in the

air, recommended by the SS for refractive IOL, are inapplicable to hybrid lenses because the zero diffraction

order that ensures far vision is practically absent in the air. The possibilities and peculiarities of calculating

the refraction for near and far vision by measuring focal distances in the 1st and 2d orders in the air and in the

zero and 1st orders in the medium have been considered. Admissible errors of IOL parameters used for cal-

culating the refraction were analyzed. For calculating the refraction with an accuracy of 0.1 diopters the cur-

vature radius and conversion coefficient Q have to be known or measured with an accuracy of 0.04 mm and

0.0016, respectively, the refractive index accuracy should be equal to ~ . ,0 0005 and the focal distance accu-

racy should be 0.01–0.05 mm in measuring in the air and 0.1 mm behind the cuvette. It has been shown that

in the region of the greatest demand, for refractions (18–23 diopters) the tolerances for the accuracy of mea-


272 LENKOVA

Table 2

�, �m

Medium Air

n nIOL aq

� a0 1,

a1 2,

nIOL

� 1 a0 1,

a1 2,

0.5461 0.1693 4.02 11.92 0.5070 0.44 12.47

0.5320 0.1690 3.74 13.06 0.5077 0.45 9.23

0.6330 0.1682 6.25 7.18 0.5030 0.34 84.58

Table 3

�, �m

Medium ��0 1,

� 0.1)

Coefficients Tolerances

n nIOL aq

� a h0 1, max

a n n0 1,

( )IOL aq

� �hmax

, �m � �n nIOL aq

,

0.5461 0.1693 2.50 23.74 0.040 0.0042

0.5320 0.1690 2.32 22.13 0.043 0.0045

0.6330 0.1682 3.88 37.16 0.026 0.0027

Table 4

�,�m

Air ��1 2,

� 0.1)

Coefficients Tolerances

nIOL

� 1 a h1 2, max

a n1 2

1,

( )IOL

� �hmax

, �m �nIOL

0.5461 0.5070 7.75 24.60 0.013 0.0040

0.5320 0.5077 5.74 18.20 0.017 0.0055

0.6330 0.5030 52.53 168.15 0.002 0.0006

suring or giving the parameters change depending on the refraction value by no more than 12–25%. Com-

parison of the efficiency of hybrid lenses in the air and medium shows that in the air, it is more convenient to

estimate the correspondence of the profile height to the calculated value by the relationship of intensities in

the 1st and 2d orders. For obtaining the relative intensities with a 0.1 accuracy the deviations of the diffrac-

tion structure profile depth and refractive indices from the given values in the medium should be less or

equal to 0.03–0.04 �m and 0.003–0.004 �m, respectively. In the air, the tolerances for these deviations de-

crease several times (2.5–13), depending on the parameters and wavelength. We should note that the effect

of each parameter is estimated separately. The total IOL refraction deviation can exceed 0.1 diopter but ac-

cording to the SS it should not exceed 0.4 diopter.

The obtained relationships and inferences were used in investigating the accuracy of manufacturing

diffractive-refractive IOLs on special test benches and a setup for measuring optical characteristics of eyes

in the IAiE SB RAS (Novosibirsk, Russia). A pilot batch of MIOL-Akkord diffractive-refractive IOLs was

fabricated by IntraOL (Novosibirsk) and Reper-NN (Nizhni Novgorod) enterprises in 2005. The lenses were

based on a diffraction structure matrix developed in the IAiE SB RAS. The MIOL-Akkord lenses have run

clinical trials in the Novosibirsk and other departments of FGU MNTK “Eye Microsurgery”. After implant-

ing, patients have a good far vision and can read without glasses. Results of the trials were sent to the Minis-

try of Health and Social Development for licensing the MIOL-Akkord lenses in medical practice.

ACKNOWLEDGMENTS

This research was supported by the Russian Foundation for Basic Research, grant no. 06-08-00541.

The author is grateful to V. P. Koronkevich for fruitful discussions and A. N. Remennoi for his attention.

REFERENCES

1. V. P. Koronkevich, G. A. Lenkova, I. A. Iskakov, et al., “Bifocal Diffractive-Refractive Intraocular Lens,”

Avtometriya, No. 6, 26 (1997) [Optoelectr., Instrum. Data Process., No. 6, 26 (1997)].

2. ISO/DIS 11979-2, Optics and Optical Instruments—Intraocular Lenses. Pt. 2: Optical Properties and their

Methods, 1996.

3. GOST R 52038-2003 (ISO11979-2-99), Ocular Implants. Intraocular Lenses, Part 2. Optical Properties and

Methods of Trials (Gosstandart, Moscow, Russia) [in Russian].

4. M. I. Apenko and A. S. Dubovik, Applied Optics (Nauka, Moscow, 1982) [in Russian].

5. G. A. Lenkova and M. M. Myznik, “Investigation of Dependence of the Image Quality of a Schematic Eye with a

Hybrid Bifocal Intraocular Lens on the Diameter of the Pupil,” Avtometriya, No. 3, 61 (2002) [Optoelectr.

Instrum. Data Process., No. 3, 50 (2002)].

6. G. A. Lenkova, “Optical Characteristics of Intraocular Lenses in Air, Water, and Cell,” Avtometriya, No. 3, 35

(1997) [Optoelectr. Instrum. Data Process., No. 3, 32 (1997)].

7. B. N. Begunov, Geometric Optics (Mosk. Un., Moscow, 1961) [in Russian].

8. G. A. Lenkova, “The Effect of Phase Profile Depth on Intensity Distribution in Diffraction Orders of a Bifocal

Element,” Avtometriya, No. 5, 16 (1995) [Optoelectr. Instrum. Data Process., No. 5, 15 (1995].



Documents

Methods for investigating optical characteristics of bifocal diffractive-refractive intraocular lenses