Using diffractive surfaces to model intraocular lenses

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Intraocular lenses (IOLs) are medical devices that are surgically implanted into the eyes of patients suffering from conditions such as cataracts or myopia. Essentially a plastic lens, IOLs may be used to replace the natural lens in the eye when the lens has become cloudy causing a decrease in vision. This article will demonstrate how users can use the Binary 2 surface to model an intraocular lens.  

Authored By James E. Hernandez

download example (.zip)

Introduction

Intraocular lenses (IOLs) may be used to replace the natural lens in the eye when the lens has become cloudy, causing a decrease in vision.  The clouding of the lens is referred to as a cataract, and the IOL that replaces the lens in this case is termed a pseudophakic intraocular lens. IOLs may also be used to correct the vision of individuals with myopic, hyperopic, or astigmatic eyes.

Years of research and development have resulted in a multitude of IOL designs. OpticStudio provides designers with an excellent tool to model and analyze the performance of most types of IOLs. Here, the IOL is placed over the natural lens, such that its optical power is altered; this type of IOL is termed a phakic intraocular lens.

Diffractive intraocular lenses

Refractive, monofocal IOLs provide patients with corrected vision for either distance vision or near vision. The issue with this design, however, is the inability to correct for both simultaneously. For this reason, modern IOL designs have provided an alternative approach: multifocal IOLs. These lenses provide more than one optical power, allowing recipients to focus at multiple ranges. Bifocal IOLs are a popular iteration of the multifocal design, and feature two optical powers, one for near vision and the other for far vision. In this article, we will focus our attention on bifocal IOL designs.

Bifocal IOLs function by utilizing the diffractive nature of light. As a reminder, when different light waves propagate and coincide at one location, they interfere. This interference may be destructive or constructive (or a partial variation of either), and is a function of the optical path lengths of the individual light waves. For an optical path difference of half a wavelength, for instance, the interference between two coinciding waves will be completely destructive and result in 0 intensity. Bifocal IOL designs utilize these concepts strategically. Lenses are designed with a base optical power, and feature concentric annular zones on at least one surface. Light passing through these annular zones interferes in a controlled fashion with optical “steps” between zones. Below is a sketch of a bifocal IOL showing an aspheric surface and annular zones on the lens.  (The size of the rings is exaggerated.)

To understand how the diffractive IOL works, consider the sketch below showing a lens with evenly spaced rings. In every other ring, there is a step that adds 1/2 wave to the optical path length of a ray passing through it. This creates two lenses that focus in the same location but are exactly 1/2 wave out of phase with one another.  Destructive interference occurs at that focus, and there is no intensity at that point.  For example, at point A in the sketch below, the phase difference between two rays that pass through the outer two rings might be exactly 1/2 wave, and destructive interference would occur.  Further out along the axis, though, at point B, the pathlength difference between the outer two rings might be 0, and constructive interference can occur.  The multiple foci correspond to different diffraction orders for the lens.

The height of each step can be adjusted to produce foci at the desired locations.  For an IOL design, the heights and shapes of the steps are chosen to create two foci on the retina of the eye, one for the near point of the eye and one for objects far away.  Both foci are always present in the eye. When an object is held at the viewer's near point, one image is in focus, and the other focus produces a defocused image on the retina.  The brain learns to ignore the defocused image and concentrate on the in-focus image.

Other foci from the diffractive IOL are also present in the eye.  The exact surface shape of each ring in the IOL can be chosen to maximize the intensity in the two orders that are used, and minimize the intensity in the other orders.  OpticStudio, though, changes the phases of the rays but does not directly model the surface shape of each ring. OpticStudio assumes that each order contains 100% of the incident intensity.  In order to calculate the intensity actually present in each diffraction order for the lens, other modeling software must be used.  

Modelling in OpticStudio

In this example, we will utilize the Binary 2 surface to model a bifocal IOL. The Binary 2 surface is a diffractive surface where the phase added to each ray varies as a rotationally symmetric polynomial. Phase is delayed or advanced via the following expression:

where the coefficients A_i are in units of radians.

Here, N is the number of polynomial coefficients in the series, M is the diffraction order, p is the normalized radial aperture coordinate with coefficients represented by the A terms. To begin this design, assume that an IOL with the following parameters is to be modelled. A pupil diameter of 3.5 mm will be used, to approximate the iris size of the human eye in moderate illumination conditions.

To begin, a simplified model of the human eye will need to be established. Instead of doing so from scratch, we will utilize a human eye sample file. This sample file is included in the article attachments for the following article: OpticStudio models of the human eye. Open up the file Eye_Retinal Image.zar. The Lens Data Editor and layout for this file are shown below. Note that the pupil diameter is as desired, so only changes to the “Lens” will need to be made.

This model of the eye takes into account prominent elements of the eye: the corneal surface, pupil, crystalline lens, retina, as well as the aqueous and vitreous humors. Note that the crystalline lens here is modelled using two standard surfaces. Using the Binary 2, this lens element can be swapped with a diffractive, bifocal IOL.

Surface 6 refers to the front surface of the crystalline lens. Since the requirements of this design indicate that the front surface of the IOL must feature diffractive power, the surface type should be switched from Standard to Binary 2. In doing so, a multitude of new parameters will appear in the LDE for this surface: diffractive order, aspheric coefficients, and the maximum number of terms and normalization radius for the Binary 2 phase expansion.

Since two diffractive orders must be modelled, one corresponding to far vision focus and the other for the near vision focus, multiple configurations must be setup. The first configuration will model the 0^th diffractive order of the bifocal IOL, and will require an infinite object distance. The second configuration will model the 1^st diffractive order of the lens, and will require a finite object distance. The window below shows how we can setup the multi-configuration editor.

Note that the FLTP, THIC, and YFIE operands must be used in order to correctly setup the infinite conjugate and finite conjugate system configurations. Defining “0” when using the FLTP operand denotes angular field, while “1” defines object height. Here, 10 degrees and 20 degrees were selected for the far vision FOV, while 10 mm and 20 mm were the chosen object heights for the near vision configuration. A smaller angular subtense was chosen for near vision in an attempt to accurately reflect vision targets such as text. Finally, 250 mm was chosen as the object distance for near vision. This is based on the assumption that 25 cm is a “normal” near point.  This value may vary depending on the source of information.

Before optimizing the lens in an attempt to determine the required parameters for the IOL (including the base radii for both surfaces, aspheric coefficients, and “phase” coefficients), a few parameters must first be entered manually. The material on Surface 6 should be changed to PMMA to reflect the material requirement imposed by the table above. Parameter 13 on Surface 6, Maximum Term #, refers to the number of terms desired in Equation #1; in other words, Parameter 13 refers to “N”. This will be set to 4, but can be set to a high value if desired. Finally, the normalization radius must be entered. Here, we will set the parameter to be equal to the semi-diameter of the lens; the exact value of the normalization radius is not important so long as it is carried along with the Ai coefiicents.

Having entered these final parameters, optimization of the IOL may begin. The following parameters will be set as variables: radii for Surface 6 and 7, 4^th and 6^th order aspheric coefficients for Surface 6 and 7, and the coefficients for p^2, p^4, p^6, and p^8 on Surface 6. At this point, the LDE should similar to what is shown below. The far vision configuration (1^stconfiguration) is shown.

With everything set up properly, optimization can begin. As with all systems, this becomes an iterative process. The process cycles between optimizing for RMS spot size and RMS wavefront until favorable results are found. In the ideal scenario, the user would begin by optimizing a monochromatic system with a single field, then add wavelengths and fields back in as the results become encouraging. To ensure that color is corrected for, operands for longitudinal aberration can be included in the merit function; the user should be sure to define which configuration any additional operands are to be applied to. Below is an example of what the merit function editor might look like at the final stage of optimization.

For this demonstrative design, optimization yielded the following results. Note that values for all varied parameters now have distinct values. For the layout below, the far configuration is the left, while the near configuration is to the right.

To ensure that the system is meeting all of the initially imposed requirements, we’ll use some tools found under the Analyze tab of OpticStudio’s Sequential mode! First, from the Lens Data Editor above, it is already apparent that requirements on material type, lens dimensions, and surface types have been met. To ensure that our MTF requirement has been met, navigate to Analyze...MTF...FFT MTF. Here, sampling was set at 128 x 128, but may be increased if need be. The results for both configurations are shown below (Far: Top, Near: Bottom).

Graphically, it appears that both requirements, 50% contrast at 10 lp/mm and 50 lp/mm for far vision and near vision respectively, have been met. To be sure, the user can navigate to the text viewer (accessed by clicking the “Text” tab below the MTF plot). Scrolling down to the outermost field, which from the plots above are the closest to not meeting requirements, the following data is found. Again, far vision is on the left, near vision is on the right.

Looking at spatial frequencies of 10 lp/mm on the left and 50 lp/mm on the right, it is apparent that the requirement has been met. If this were not the case, additional optimization runs could be performed with the addition of MTF operands. Please refer to the article How to optimize on MTF for more information on optimizing for MTF.

From the table above, a total aberration RMS value of 2.13 +/- 2. um was imposed. To see if this sample design met these requirements the user can navigate to Analyze...Wavefront...Wavefront Map. In the settings, the sampling rate is adjustable, as is the wavelength and field of interest. Assuming the 2.13 +/- 2 applies to all fields and wavelengths, all combinations may be checked. In this design, all configurations successfully met this requirement. Below is a sample showing the setup necessary to view the data in question. Note the RMS value is in waves, and must be multiplied by the wavelength if specifications use microns as a unit type.

An additional analysis tool that users might find useful during IOL design is OpticStudio’s Extended Scene Analysis. In particular, the Geometric Bitmap Image Analysis feature will be demonstrated. This feature creates an RGB color image based up ray tracing data using an RGB bitmap file as the source. This is useful for many applications, such as modelling extended sources, displaying distortion, or representing the appearance of imaged objects. To use this feature, navigate to Analyze...Extended Scene Analysis...Geometric Bitmap Image Analysis. Below are the results for the far configuration using the 0-degree field position, along with the settings used for the analysis.

On a final note, the reader might point out that this IOL was modeled using a corneal “lens”. In IOL design, a major emphasis must be placed on balancing the aberration contribution from the cornea with that of the IOL. Since this may vary from patient to patient, one may have the desire to more accurately model the wavefront following the corneal surface by using Zernike coefficients obtained from corneal topography measurements. Using a Zernike standard phase in conjunction with a paraxial lens with the same power as the corneal surface, the sample eye model used previously may be edited to better suit the circumstances of the IOL design. Below is an LDE and corresponding layout demonstrating just that. This file may also be found in the article attachments.

Note that here the sample eye model used earlier remains essentially the same. The only difference is that the corneal “lens” from earlier has been replaced with a paraxial surface of the same power.  The thickness with respect to the stop has been adjusted taking into account the location of the principle planes for the lens previously representing the cornea. In this new configuration, the wavefront following the (now paraxial) corneal surface will be aberrated as dictated by the Zernike coefficents placed on the Zernike Standard Phase surface. These coefficients may be entered directly in the Lens Data Editor or imported under surface properties, with the following format.

n

norm_rad

Z1

Z2

Z3

...

The multi-configuration editor should be checked before proceeding to optimize/build the IOL, depending on whether corneal measurements were made for more than one pupil size.

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