Optics of Intraocular Lenses (IOLs)

I. IOL OverviewII. Monofocal IOLsIII. Accommodating IOLsIV. Multifocal IOLs

I. IOL Overview

Primary (optical) functions of the crystalline lens:1. Provide additional refractive power (~ 20 D) to

supplement cornea (~ 40 D) for distance vision.2. Provide accommodation for near vision (~ 4 D).3. Provide aberrations that are opposite in sign to those of the

cornea in order to reduce total aberrations of eye.

When it becomes necessary to remove the crystalline lens, an intraocular lens (IOL) replacement is (hopefully) designed to provide as many of these same functions as possible.

Additionally, the use of phakic IOLs is being developed to supplement the optics of the cornea and crystalline lens as a means to correct vision rather than laser surgery. Since the crystalline lens is not removed, its functions do not necessarily need to be replaced by the IOL.

Three primary classes of IOLs:1. Monofocal – have a single, fixed power to provide correct distance vision2. Multifocal – have a more than one power to provide

correct distance and near vision3. Accommodating – have continuously tunable power that

mimics the accommodative ability of the crystalline lens

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In this discussion we will consider the optics of several types of IOLs that are currently in use or being developed.

II. Monofocal IOL’s

Optical features:

1. Can be relatively simple (single thin lens)

2. Many feature aspheric surfaces to reduce aberrations (particularly spherical) introduced by cornea. Eventually, wavefront techniques could be used to customize the shape to the individual patient. Current techniques simply use data from a study of “typical patients” to take into account average spherical aberration of the typical cornea.

3. One interesting phakic IOL under development (Vision Membrane) utilizes diffractive optics. More on this later…

III. Accommodating IOLs

Optical features:

1. Single or multiple thin lenses used to effectively make a “zoom lens” within the eye.

2. Aspheric surfaces could be used (although not currently as far as I can tell) to help reduce the aberrations introduced by cornea.

How it works (optically): Thin lens model eye (mostly)

Suppose we replace the crystalline lens with a single IOL to make an emmetropic eye that can be modeled as shown:

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Let’s prove that this eye really is emmetropic. How? Locate the image of an object at optical infinity – the image should form on the retina.

1. Find image produced by cornea:

V = U + P, but U = 0 V = P = +40 DReduced vergence V = n/vso, v = 1.33/40 v = 3.3 cm (note that this is behind the retina)

2. Image due to cornea acts as image for IOL. Find image produced by IOL.

V = U + PReduced vergence U = n/u where u = 3.3 cm – 6 mm = 2.7 cmso V = 1.33/0.02725 + 20 = 69.3 Dso, v = 1.33/69.3 v = 19 mm

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cornea (SSRI)

IOL retina

Pc = 40 D PIOL = 20 Dn = 1.33n = 1

19 mm6 mm

image location for cornea

3.3 cm

To gain some further intution, let’s think of the cornea and IOL together as a single thick lens. Recall equivalent power of a thick lens:

Pe = P1 + P2 (d/n) P1 P2

d is the distance between the first and second elements

n is the refractive index of the material between first and second elements

Question: Consider the cornea and IOL together acting as a single thick lens. In what direction would the IOL need to move in order to increase the overall power of the eye (i.e. accommodate)?

A. toward the corneaB. toward the retina

Find equivalent power of emmetropic eye above as thick lens:

Pe = Pc + PIOL – (d/n)PcPIOL

Pe = 40 + 20 – (0.006/1.33)(40)(20) = 56.4 D

Now find equivalent power of same eye if IOL is moved 1 mm in the direction determined in the question above:

Pe = Pc + PIOL – (d/n)PcPIOL

Pe = 40 + 20 – (0.005/1.33)(40)(20) = 57.0 D

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equivalent power of system

power of first lens or surface

power of second lens or surface

correction term for vergence transfer through the thickness of the lens

Note that the thick lens equivalent power calculations show that the overall power of the eye has increased as the IOL moves. However, quantitative calculations about near point location are difficult to do using this information because the principal points and nodal points would have to be located for the eye and these change as the IOL moves.

Question: Suppose we want the near point of the eye to be 40 cm in front of the cornea. How far does the IOL need to move in order to provide the appropriate amount of accommodation?

1. Find image produced by cornea:

V = U + P, where U = 1/-0.4 D V = -2.5 D + 40 D = 37.5 DReduced vergence V = 1.33/37.5 = 3.5 cm

2. Image due to cornea acts as image for IOL. Important: want final image to still be on

retina! Find distance IOL must move to make this happen.

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cornea (SSRI)

IOL retina

Pc = 40 D PIOL = 20 Dn = 1.33n = 1

19 mm

d image location for cornea

3.5 cm

6 mm

V = U + PReduced vergence U = n/u and V = n/vwhere u = 29 + d mm and v = 19 + d mm 1.33/(19 + d) = 1.33/(29 + d) + 20 D

this gives a quadratic equation for d to solveresult d = 2.2 mm

How is the IOL made to move in practice? This is accomplished by attaching the IOL to the ciliary muscle of the eye. This muscle can move the IOL by approximately 1 mm. In the case of a single 20 D IOL lens this would provide a near point of approximately 67 cm (rather than 40 cm in the example above).

Greater accommodation can be achieved by using an IOL that consists of two lenses rather than one. In this case, the lenses are designed to move closer together (or farther apart) as the ciliarly muscle contracts or relaxes.

IV. Multifocal IOLs

Optical features:

1. Single lens consisting of either regions with different curvature or diffractive and refractive regions to provide several focal points (and several images on the retina) at the same time. The brain selectively chooses which image to pay attention to and which to ignore based on the relative brightness of the images and other visual cues.

2. Aspheric surfaces are used in some cases to help reduce the aberrations introduced by cornea.

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Two primary types of multifocal IOLs:1. Refractive – Single lens consisting of regions with different curvature to provide two focal points.2. Diffractive – Single lens consisting of diffractive and refractive regions to provide two focal points

We will consider the optics of these two cases separately.

1. Refractive multifocal IOLs

One particular lens is described here:

http://www.rezoomiol.com/rezoom_multifocal_lens.html

(Note: I am not endorsing this particular product, it is just a convenient example.)

Some features to notice:1. Each region of the lens has a radius of curvature that

corresponds to either a power for distance or near vision.

2. A greater percentage of the total lens area has a power for distance vision.

3. In bright viewing conditions the pupil will be small and only the central distance viewing part of the IOL will be used.

4. In lower light conditions the pupil will be larger which will lead to two images on the retina, one for distance and one for near. The distance image will be brighter since more of the light entering the eye will experience the distance power regions of the lens.

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2. Diffractive multifocal IOLs

Diffraction Grating and Zone Plate Review

A. Path Length Difference and Interference

Idea: The degree to which two mutually coherent waves constructively or destructively interfere at some point in space depends on the relative distance that each wave has traveled in getting to that point.

Consider two light sources that are emitting coherent light waves. You are standing at a location where one source is further from you than the other by a distance x. Therefore, the path length difference (how much farther one wave has traveled than the other) between the two waves is x.

For constructive interference: x = m (m = integer)

For destructive interference: x = m/2 (m = odd integer)

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Δxlight sources

B. Two-slit interference

Constructive interference (bright fringes):

d sin mm = 0,±1, ±2,…

Question: The figure below shows the pattern created by a two slits when illuminated by a red He-Ne laser. If the slits are illuminated by a green laser, which pattern below might results?

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Coherent light source

OPD = d sin()

red

A

B

B. Diffraction Grating

Screen with several hundred (or thousands or tens of thousands) of very closely spaced slits. Since the slit spacing (d) can be very small, the angles to the primary maxima can be very large. The primary maxima locations are still given by the equation:

d sin = m

Why?

They are used to separate and measure wavelengths of light in spectrometers and many other instruments.

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OPD = d sin()

OPD = d sin()The OPD between waves from neighboring slits is identical since spacing between slits is same

m = 1m = -1 m = 2m = -2m = 0

C. What is a zone plate?

A zone plate is a type of circular diffraction grating that uses diffraction (instead of refraction) to focus light. Since a zone plate focuses light, it can be used as a type of lens.

It consists of concentric circular apertures of varying radii and width: wider near the center and more narrow in the periphery. This is different than an ordinary diffraction grating in which the slits are spaced evenly from one another.

D. How does a zone plate work?

1. Suppose plane waves are incident on a circular aperture. We would like to make point Po (that is downstream a distance b from the aperture) the focal point of the zone plate.

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Divide the aperture into circular “zones” (these are the concentric rings in the zone plate pictured above). The edge of each zone is /2 farther away from Po than the edge of the neighboring zone.

2. Notice that the wave that travels along the axis of the aperture experiences constructive interference with waves from some of the zones (OPD = m, m=integer) and destructive interference with waves from other zones (OPD = m/2, m=odd integer) . If we block the light from the zones that lead to destructive interference and allow through the light from the zones that lead to constructive interference we can produce a net constructive interference at Po. Therefore, the zone plate acts like a lens with a focal length f2 = b.

3. “Dioptric power” of a zone plate

The dioptric power of the zone plate is related to the radii of each zone. In particular:

€

P = λh1

2

where h1 is the radius of the 1st zone and is the design wavelength.

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Question: Is the focal length of a zone plate shorter for red or blue light?

Some important features:

a. Chromatic aberration: the power of the zone plate is proportional to , so different colors focus at different distances from the zone plate. Ordinary refractive lenses have chromatic aberration as well, but much less pronounced than zone plates.

For example: +10 D lens made of glass with an Abbe number of 30 (high dispersion) has CA ~ 0.3 D +10 D zone plate designed for use at 550 nm has CA ~ 5 D!

b. Multiple focal points due to higher order diffraction, which is unlike an ordinary refractive lens which only has one focal point.

c. Light is diffracted both toward and away from the zone plate’s optical axis. Therefore, the plate can be designed as either a positive or negative lens.

E. Zone Plate Images

An interesting comparison of photographs taken with lenses, zone plates, and pinholes can be found at:

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http://www.lensbaby.com/optic-comparison.php

Note the poor quality of the zone plate image (from an optics point of view, not an artistic one!) This fuzziness of the zone plate image is due to chromatic aberration and the fact that a significant portion of the diffracted light is spread out into higher orders than the one making the image.

F. Zone Plates as IOLs

Advantages:

1. No need for curved surfaces so lens can be made flat and more easily rolled up for surgical insertion.

2. Optical properties such as percentage of light going to distance and near vision images, aberrations, and power can be fine tuned by proper design of the zones without the need for aspheric surfaces.

Disadvantages – Optical quality not as good as standard, monofocal lens

G. Diffractive IOL Designs

Here we will consider some issues that can be addressed to improve the optical quality of a zone plate.

(Note: I am not endorsing any of the products shown in this section, these are just to illustrate some specific examples of designs currently in use or being developed.)

1. Image dimness – this is partly due to the fact that a large fraction of the incident light is blocked (by design) by the

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zone plate in order to eliminate the waves that would be destructively interfering at the image.

Solution – Alter the phase of the light from those zones (rather than blocking them) in such a way that they also contribute constructively at the image location. How?

Review - Optical Path Length

Recall that the wavelength of light changes when the light enters a medium other than vacuum because of the index of refraction.

Idea: Since the wavelength of light depends on the index of refraction of the medium it is traveling in and the phase of the light depends on the wavelength and the distance traveled the phase of the light will depend on the index of refraction of the medium it is traveling in as well as the distance traveled.

Define optical path length as the product of the index of refraction of a medium and the distance traveled by a light wave inside that medium.

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Light traveling in vacuum

Same light traveling in glass

x

End at different points along the sine curve even though same distance traveled

Alter thickness of zone plate material so that the optical path length is increased by ½ in these regions + ½

+ ½

2. Chromatic aberration – as described above, this is due to the fact that the process of diffraction inherently separates the wavelengths because constructive interference occurs at different angles for each.

Solution – Design the zone plate so that different diffracted orders of different wavelengths meet at the same location. (Called Multi-Order Diffraction, or MOD).

To understand this, consider a regular diffraction grating where constructive interference is found from:

d sin mm = 0,±1, ±2,…

If we design the grating properly, we could make it so the 2nd order (m = 2) red line occurs at the same location as the 3rd order (m = 3) blue line. To make this happen we would just need to have:

d sin redand d sin blue

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m = 1m = -1 m = 2m = -2m = 0

Notice that the left side of each of these is the same since the wavelengths are both being diffracted by the same grating and the wavelengths are being sent to the same angle.

What does this mean in practice? We would follow the following procedure:

1. Choose our two wavelengths of interest in such a way that

2red = blue(this puts a restriction on which wavelengths will work)

2. Choose an angle (want) that you want the light to be at on the screen and then manufacture the grating so that:

€

d =2λ red

sin(θ want )

This grating will then send the 2nd order red light and 3rd order blue light to the same location on the screen.

In fact, we could make a grating that sends even more wavelengths to the same location as long as the ratios of each pair of wavelengths are ratios of integers!

A similar procedure can be done for zone plates (since they are just circular diffraction gratings) so that multiple wavelengths all have the same focal point.

MOD diffractive lenses are being developed by Vision Membrane Technologies for use as phakic IOL’s.

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Optics and Photonics News, p. 29, September 2004.

3. Light Optimization – Zone plates have multiple focal points leading to multiple images formed on the retina. The percentage of light found in each of these images may not be optimal for all viewing conditions. In fact, higher order diffraction can lead to image degradation.

Solution – Design the zone plate with varying zone optical thickness (called “apodization”) so that the appropriate amount of light is sent to each image.

Apodized diffractive lenses are offered by Alcon Surgical for use as IOL’s.

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Slowly reduce the optical thickness of the zones. This alters the degree of constructive interference and the amount of light found at each focal point.

+ ½

+ ¼ + 3/8

Journal of Cataract and Refractive Surgery, Vol. 32, p. 854, May 2006.

Figure 1: Small-scale planar computer-generated simulation of diffraction at an apodized diffractive structure overlaid with a sketch of an eye [Davidson & Simpson, 2006]

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QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Figure 2: Aclon ReStor IOL: twelve diffractive rings over the central 3.6 millimeters of the lens optic. The inner most ring has a step height of 1.3 microns with subsequent rings gradually decreasing in height until the outer ring with a height of 0.2 microns. The inner rings are further apart and the distance between rings gradually decreases toward the periphery.

From: http://webeye.ophth.uiowa.edu/eyeforum/tutorials/restor.htm

References

M.P. Keating, Geometric, Physical, and Visual Optics, 2 nd ed. , Butterworth-Heinemann (2002).

G.M. Morris and L.T. Nordan, “The New Focus in Refractive Surgery,” Optics & Photonics News, 27-31, September 2004.

M.J. Simpson, “Diffractive Multifocal Intraocular Lens Image Quality,” Applied Optics, Vol. 31, No. 19, 3621-3626 (1992).

T. Terwee, H. Weeber, M. van der Mooren, and P. Piers, “Visualization of the Retinal Image in an Eye Model With Spherical and Aspheric, Diffractive, and Refractive Multifocal Intraocular Lenses,” Journal of Refractive Surgery, Vol. 24, 223-232 (2008).

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S.S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R.B. Wallace III, “Multifocal Intraocular Lenses,” Ophthalmology Clinics of North America, Vol. 19, 89-105 (2006).

D. Fakalis and M.G. Morris, “Spectral Properties of Multiorder Diffractive Lenses,” Applied Optics, Vol. 34, No. 14, 2462-2466 (1995).

J.A. Davidson and M.J. Simpson, “History and Development of the Apodized Diffractive Intraocular Lens,” Journal of Cataract and Refractive Surgery, Vol. 32, 849-858 (2006).

A. Rana, D. Miller, and P. Magnante, “Understanding the Accommodating Inraocular Lens,” Journal of Cataract and Refractive Surgery, Vol. 29, 2284-2287 (2003).

J.M. Artigas, J.L. Menezo, C. Peris, A. Felipe, and M. Diaz-Llopis, “Image Quality With Multifocal Intraocular Lenses and the Effect of Pupil Size: Comparison of Refractive and Hybrid Refractive-Diffractive Designs,” Journal of Cataract and Refractive Surgery, Vol. 33, 2111-2117 (2007).

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