There have been dozens of eye models published over more than 150 years, from very simple “reduced” eyes consisting of a single refracting surface to very complex models with more than 4,000 refracting surfaces. This article presents several sequential and non-sequential models of the human eye in OpticStudio format using only glass catalog data.

The OpticStudio models described below are included in the zip file following the article, which you can download. See the section Glass Catalog below before use. The models are based on particular wavelength ranges and weightings, field angles and field angle weightings and pupil sizes and you should feel free to modify them if it is more appropriate for a particular purpose.

Authored By Rod Watkins - Director of Strategic Development Optometry and Vision Science, Flinders University

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Introduction

Optical models of the eye are used to design instruments to look into the eye (for example to check the uniformity of illumination of a fundus camera), to design instruments that the eye looks through (including some properties of ophthalmic lenses, contact lenses and intraocular lenses), and to investigate the optical system of the eye itself (including the effects on retinal image formation of eye pathology such as corneal scarring and cataracts). This article presents several simple models of the human eye for use in any of the above applications. 

Simplifying the eye

There have been literally dozens of eye models published over more than 150 years, from very simple “reduced” eyes consisting of a single refracting surface to very complex models with more than 4,000 refracting surfaces. Some models have a gradient index crystalline lens, some represent the gradient index with two or more homogeneous shells, and some have a homogeneous lens.

There is no ideal optical model of the eye that is best for every purpose, and a more complex model does not necessarily represent all eyes, or any particular eye, more accurately. There is no point, for example, in using a model that includes a gradient index crystalline lens if that gives no more valid information than a homogeneous lens but slows the computing time significantly during optimization or during calculations on an NSC model with a large number of rays. Often paraxial calculations at a single wavelength are all that are needed, and these can be carried out using a very simple model with spherical surfaces. A common “reduced” eye used for paraxial calculations has a single refracting surface of power 60 diopters and a refractive index of 4/3. It therefore has a surface radius of 5.55mm and an axial length of 22.22mm. This model is particularly useful for calculating retinal image size. Since the nodal point is 5.55mm from the surface, the image size (h in the diagram below) of an object whose position and size or field angle are known can be calculated using simple geometry by projecting the ray along a distance of 16.67mm. This paraxial model is accurate to within a few percent for field angles as large as 10 degrees.

Sequential models

There are two common uses of Sequential eye models - one where the fundus of the eye is being viewed by an external optical system such as an ophthalmoscope or a fundus camera, so the retina is the object surface, and the other where the eye is looking out through an optical system such as a spectacle lens or a visual instrument and so the retina is the image surface.

Models that we have found useful in a wide variety of applications are included here as files Eye_Retinal Image.zmx and Eye_Retinal Object.zmx. Although these models have the same optical system they have considerable differences in the data editors, as described below. The session files are also included. The Eye_Retinal Image model is shown here:

Since the use of this model often concerns visual performance, the model uses photopic weighted wavelengths, field angles of 0, 10 and 20 degrees weighted 1.0, 0.2 and 0.1 respectively to represent the relative visual acuity at those angles, and a 4mm diameter pupil.

Shown here is the Eye_Retinal Object model:

In this model the fundus is treated as a physical object. The model uses F, d, and C wavelengths weighted 0.1, 0.4 and 1 respectively to represent the spectral reflectance of the fundus, equally weighted field angles of 0, 10 and 20 degrees and a 4mm diameter iris aperture. The image space is afocal.

Also included is a model of an eye accommodated to 250mm (four diopters of accommodation referred to the cornea), which is sometimes useful.  The file is Eye_Accommodated.zmx. On accommodation the lens poles move forward into the anterior chamber and backwards into the vitreous chamber, so the axial length of the lens increases, the diameter of the lens decreases, and the surfaces change shape. Most accommodation occurs by an increase in curvature and forward movement of the anterior surface of the lens.

The Eye_Accomodation model uses the same wavelengths, field angles and pupil size as the Eye_Retinal Image model. Note however that this model has been used also to demonstrate the ability of OpticStudio in Sequential Mode to draw the scleral surfaces as hyperhemispheres (see OpticStudio Tools below). This avoids the dummy surface of the above models in the anterior chamber and gives a more realistic diagram of the eye, but the hyperhemispheres introduce ambiguities in ray tracing. If the model is to be used for ray tracing, these surfaces may need to be replaced by the two hemispheres of the previous models.

The values of the various parameters in these models have been taken from a large number of references, and I have not listed the sources here. The parameter values have generally been rounded off for simplicity when this has been found to not be significant. (For example, the axial length is 24.0mm, the retinal radius is 11.0mm and the anterior surface of the relaxed lens is spherical with a radius of 10.0mm.) The models do closely represent an average of measurements on real eyes, with the exception of the use of a homogeneous crystalline lens. The actual gradient index of a real lens is replaced in these models by a small change in the conic factor of the posterior surface. (The model eye posterior lens surface has been flattened slightly less than actually occurs to substitute for the lower refractive index towards the equator.) This surface has been measured in real eyes to be more or less hyperboloidal and is a critical factor in off-axis aberration control.

This homogeneous lens has the advantage of greatly reducing the time for optimization and for NSC ray tracing and is adequate for most purposes. However, in some cases, such as where the optical properties of the crystalline lens itself are being explored, it is essential to use a gradient index model. The Knowledgebase article "How to model the human eye in OpticStudio" describes how to do this.

Non-Sequential models

Many ophthalmic instruments direct light into the eye and it is useful to be able to model the efficiency of the lighting delivery system, the uniformity of light distribution on the retina and so on. In some cases, light is focused onto the retina, such as in laser treatment of diabetic retinopathy, and in other cases light is focused onto the pupil so that it illuminates a wide field, such as in indirect ophthalmoscopy. The same NSC model can be used for both these situations, with different source geometry.

The optical media of real eyes are often not completely transparent, and non-Sequential modelling in OpticStudio also provides powerful tools to investigate the effects on vision of a wide range of pathological and physiological changes in real eyes. By adding absorption, scattering and inclusions it is possible to model the effects on vision of such things as corneal scarring, cataracts, vitreous floaters and foreign bodies. It is also possible to look at light scattering from the edges of corneal or intraocular lenses.

The Non-Sequential eye model included here is Eye_NSC.zmx. It uses the same glass catalog as the Sequential models. The first object in the Non-Sequential Component Editor is a reference point located at the geometrical center of the globe of the eye. The eye can be translated or rotated by changing the parameters of this reference point. The NSC Shaded Model in the file is given a brightness of 90% and opacity of 50% to allow the internal structure to be seen (see NSC Shaded Model...Settings):

This model uses F, d, and C equally weighted wavelengths and a 6mm diameter stop to represent a moderately dilated pupil. The default retinal detector subtends about 50 degrees edge to edge at the pupil for wide field illumination of the fundus. The pixel size of the model may be considerably larger than the image of a point object, so the Detector Viewer light distribution might show the pixel size rather than the image size. If point imaging is of interest the pixel size will need to be reduced (and possibly the wavelength range and pupil size also reduced). Note also that the number of pixels in the retinal detector can have a significant effect on computing time. The maximum aperture of the detector should not be too much larger than the area of the fundus of interest.

In the Eye_Binocular.zmx model (below), the interpupillary distance (PD) and convergence angle of this model can be set using the parameters of the null object “Reference Point 1”.  Axial source rays have been added to represent the lines of sight projected to an object surface. (In real eyes, the line of sight is normally about 4° nasal to the optical axis in object space - “angle alpha” - but in this model the two are colinear.) This model can be useful, for example, to track the lines of sight through a binocular instrument with a fixed convergence angle.

Glass catalog

The glass catalog EYE.AGF included in the zip file must be copied to the OpticStudio glass catalog folder to use these eye models. To find the folder location go to Setup Ribbon...Project Preferences...Folders...Glass. After copying, hit F4 on the keyboard to open the Materials Catalog frame and verify that OpticStudio can see the file.

The glass catalog has been constructed from published measurements of the refractive indices of the optical media of real eyes. This has generally been available for a limited number of wavelengths, usually F, D and C. For this reason, the Conrady formula has been used, with the consequence that the wavelength range is limited to the visible and near infrared spectrum, and the Nd and Vd values are not rounded.

If the wavelength range needs to be extended into the UV or IR, it is useful to note that the OpticStudio stock glass catalog MISC contains data for seawater using the Schott formula for wavelengths from 0.334 to 2.325 microns. Since both the aqueous and vitreous humors of the eye have compositions similar to saline, it might be reasonable to assume that while the refractive indices are different, the dispersions can be inferred from that of seawater.

OpticStudio tools

OpticStudio has many tools to make eye models more useful by customizing them for particular applications. 

Layout plots

Because of the steep curves of some surfaces and the fact that in a real eye the edges of the sequential surfaces are not actually connected, the layout is often clearer and a better representation of a real eye if the edges are not drawn. However, in some applications it is necessary to turn on the edges. This is controlled in the sequential Lens Data Editor...Surface Properties...Draw Tab, or in the Non-Sequential Component Editor...Object Properties...Draw Tab.

In the Sequential models here, some edges are drawn while others are not. The anterior hemisphere of the retina is drawn as a separate surface between the cornea and the pupil so that the eye is represented as a complete retinal globe. If this dummy surface in the anterior chamber is distracting it can be removed and the posterior hemisphere edges drawn to connect with the lens edges. In the Eye_Accommodated.zmx model the retina has been forced into a hyperhemisphere by using an object cone angle that creates ambiguity (click System...General...Aperture Tab) and the outer surface of the sclera has also been added. This is a useful layout technique to draw a more realistic eye, but the ambiguity in the sequence of surfaces means that ray tracing is generally not possible. To use this model optically the hyperhemispheres must often be deleted and replaced by the two hemispheres of the other Sequential models.

In the Non-Sequential models there is no difficulty in including objects inside other objects, so there is no ambiguity in using hyperhemispheres to represent the sclera. The layout method for the hyperhemispherical NSC surfaces is simple; the surface apertures are given negative values.

Preset wavelengths

A very useful OpticStudio tool for eye models is the ability to insert either F, d, C visible spectrum wavelengths or photopic (or scotopic) wavelengths with relative luminosity weightings. The F, d, C wavelengths will often be appropriate when looking at the retina (the Eye Retinal Object model) but the photopic wavelengths will often be appropriate when the eye is looking through an external optical system (the Eye Retinal Image model). Open System Explorer...Wavelengths...Photopic (Bright) and click Select Preset.

When wavelength choice is important it is worth noting that transverse chromatic aberration of the eye is very small, since the second principal plane is close to the aperture stop of the system, but longitudinal chromatic aberration is very marked. Measurements in real eyes of about 2.5 diopters of aberration are similar to the predictions of these model eyes.

Field angle weighting

When looking at the retina, for example with a fundus camera, it is necessary that the image resolution does not fall away too much over quite large field angles of 30° or more, and the field angles will need similar weighting. (Ophthalmic instrument manufacturers quote field angles between the edges of the field rather than from the optical axis to the edge, that is, twice the value of OpticStudio.) On the other hand, when the retina is the image surface the relative visual acuity falls from 1.0 at the fovea to 0.5 at 2.5°, 0.2 at 10°, 0.1 at 20° and 0.025 at the periphery. Choosing incorrect weightings when optimizing a system can give quite invalid results. Field angle weightings are set in the Field Data Editor.

Image quality analyses

When the retina is the object surface, the usual aberration and resolution analysis tools (fans, spot diagrams, MTF etc.) are helpful. However, when considering what an eye is seeing, OpticStudio has some powerful additional tools. 

See the Analyze Ribbon...Extended Scene Analysis...Geometric Image Analysis. A number of library image files are available. Particularly useful are the LETTERF.IMA file and the LINEPAIR.IMA file (see Settings...File), as they can be related directly to visual acuity, but custom image files are also very easy to create. Since normal visual acuity (6/6, 20/20 or 1.0) corresponds to resolution of a five-bar letter such as E that subtends 5 minutes of arc in object space, the reduced eye model gives a retinal image size of 0.024mm. Using the Eye Retinal Image model, Geometric Image Analysis shows the significant variation in image quality with wavelength due to longitudinal chromatic aberration. (Open LETTERF.IMA and enter an image size of the order of 0.024mm and a similar field size.) This is particularly useful when comparing retinal images before and after changes in an optical system, but a good deal of care is needed in drawing conclusions about visual acuity, as processing in the neural pathways from the eye to the brain can have a large effect on the perceived acuity. (Also, for this reason, it is not straight forward to relate grating frequency in LINEPAIR.IMA or limiting MTF frequency in a model eye to visual acuity.)

See the Analyze Ribbon...Extended Scene Analysis...Geometric Bitmap Image Analysis. This allows real scenes to be projected as bitmaps onto the retina. A number of library files are available and custom files can be easily used. For example, in the Eye Retinal Image model, from this menu go to Settings...ALEX200.BMP. Set the pixel size to 2.5 microns (about the size of the foveal cone receptors) and choose a field size and number of rays per pixel to balance the computing time and image quality. (The example in the model places Alex about 8 meters from the eye.) This can then be a very useful way of estimating differences in retinal image quality when changes are made to an optical system.

Ray aiming

The entrance pupil of the eye changes shape and position with field angle, so for calculations at even modest field angles and pupil sizes it may be necessary to turn on Ray Aiming. This is done at System Explorer...Ray Aiming. Paraxial ray aiming is usually sufficient, but users are encouraged to read the Help Files to understand the implications of ray aiming. (I have used the term “pupil” both correctly to mean the entrance pupil of the eye and also incorrectly but in accordance with common practice to mean the physical aperture of the iris. I hope the different meanings are clear from the context.) 

Other useful OpticStudio tools include the following.

• Toroidal surfaces: Most real eyes have astigmatism, commonly due to the cornea being curved more steeply vertically than horizontally. This can be modelled in Sequential Mode in the Lens Data Editor...Surface Properties...Type...Toroidal. In NSC mode the toroidal surfaces can be entered directly. It is possible, for example, to look at the retinal image in both Sequential and Non-Sequential mode, of an astigmatic eye and an off-axis correcting toroidal lens.
• Eye rotation, surface tilt and decentration: These can be included in the Sequential models by using Coordinate Breaks and in the NSC models by changing the coordinate parameters. In some cases where the eye rotates by a large angle to look into an optical system it may be important to realize that there is no fixed center of rotation. As each of the six extraocular muscles become more or less important at different angles of rotation, the eye translates as it rotates. For small angles, the center of rotation has been measured to be on average 15.4mm behind the anterior corneal surface and 1.6mm to the nasal side of the geometric center. However, it is simplest in the model eyes here to locate the co-ordinate break to rotate the eye at the geometric center of the retinal globe (in these models that is 13mm behind the anterior corneal surface and on axis) and we have not found a case where that has given significant errors.
• Tolerancing: Many studies have measured the optical parameters of real eyes and have noted that the distribution of refractive errors that is predicted from the convolution of the individual parameter distributions does not match the measured distribution. OpticStudio tolerancing offers a powerful way of investigating this and matching measured distributions with theoretical ones.

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