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.

It is important to realise that there is no perfect optical model of the eye that is best for every purpose. An appropriate model is one that gives valid results for a particular project, and a more sophisticated model is not necessarily better. 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 optimisation 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 radius 5.55mm, a refractive index of 1.333 and an axial length of 22.22mm.

Users of the models included here should feel free to change the parameters or modify the models in any way if it seems to be appropriate for a particular purpose.

There are two common uses of eye models- the first in which the retina 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 in which 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. These files are included in the zip file which you can download from the last page of this article. 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, above.

The Eye_Retinal Object model, above.

Also included is a model of an eye accommodated to 250mm (four dioptres 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 posterior chamber so the axial length of the lens increases, the diameter of the lens decreases slightly, 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_Accommodated model, above

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 have a significant effect. (For example, the axial length is 24.0mm, the retinal radius is 11.0mm and the anterior lens surface is spherical with a radius of 6.0mm.) 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. This surface has been measured in real eyes to be more or less hyperboloidal, and the model eye shows that this is a critical factor in off-axis aberration control. The model eye posterior lens surface has been flattened slightly less than actually occurs to compensate for the lower refractive index towards the equator and this is partly offset by choosing the refractive index of the homogeneous lens close to the real eye maximum core index, which slightly increases flattening of the surfaces.

This homogeneous lens has the advantage of greatly reducing the time for optimisation and for NSC ray tracing and is adequate for most purposes. However in some cases, such as where the optical system of the crystalline lens itself is being explored, it is essential to use a gradient index model. The Knowledge Base article How to Model the Human Eye in ZEMAX describes how to do this.

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 focussed onto the retina, such as in laser treatment of diabetic retinopathy, and in other cases light is focussed 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 ZEMAX 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.

Note 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 retina of interest, and the pixel size may need to be increased.

The Eye_NSC model, above.

The glass catalog files EYE.AGF and EYE.BGF provided here must be copied to the ZEMAX glass catalog folder to use these eye models. The default folder is c:\Program Files\ ZEMAX\ Glasscat.

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, often 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 spectrum, and the Nd values are not rounded.

If the wavelength range needs to be extended into the UV or IR, it is useful to note that the ZEMAX 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.

ZEMAX has many tools to make eye models more useful by customising them for particular applications. 

1. Layout: 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 Lens Data Editor by right clicking the Surface Type and opening the Draw tab.
   
   In the sequential models given 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 the dummy surface in the anterior chamber is distracting, this surface can be removed and the posterior hemisphere edges drawn to connect with the lens edge. It is possible to also draw the outer surface of the sclera to connect to the front surface of the cornea, to look more like a real eye, but this produces an additional dummy surface in the anterior chamber. In the non-sequential model these dummy surfaces are removed so the sclera is realistically represented, but the corneal edges cannot be turned off.
   
   
2. Wavelengths: A very useful ZEMAX 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 the Wavelength Data Editor and click Select.
   
   
   
   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 very similar to the predictions of these model eyes.
   
3. 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. On the other hand, when the retina is the image surface the relative visual acuity (where the acuity at the point of highest visual acuity, the fovea, is 1.0) falls to 0.5 at 2.5°, 0.2 at 10°, 0.1 at 20° and 0.025 at the periphery. (The fovea is actually normally located slightly away from the optical axis, but for most eye model purposes it can be considered to be on axis.)
   
   Choosing incorrect weightings when optimising a system can give quite invalid results. Field angle weightings are set in the Field Data Editor.
   
4. Image quality: 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, ZEMAX has some very powerful additional tools:
    
   a. ZEMAX menu Analysis/ Image 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 retinal image size is 0.024mm. The Eye_Retinal Image model and Geometrical Image Analysis/ Settings/ Image Size shows the significant variation in image quality with wavelength due to longitudinal chromatic aberration. (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 or limiting MTF frequency in a model eye to visual acuity.)
   
   b. ZEMAX menu Analysis/ Image 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 open Geometric Bitmap Analysis/ Input ALEX200.BMP and select Field Size 1.0, X and Y Pixels 100, X and Y Pixel Size 0.005). This can be a very useful way of estimating differences between what a person will actually see when changes are made to an optical system, although like Geometrical Image Analysis it is hard to make quantitative judgements based on a single image.
   
   
   
   
    
5. 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 ZEMAX menu General/ Ray Aiming. Users are encouraged to read the manual to understand the implications of ray aiming. (I have used the term “pupil” in this article and in the models 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.)
   
6. Other useful ZEMAX tools:
   a. Toroidal surfaces. Most real eyes have astigmatism due to the cornea being curved more steeply vertically than horizontally. This can be modelled in the Lens Data Editor/ Surface Type/ Toroidal.
   b. Eye rotation, surface tilt and decentration: These effects can be modelled using co-ordinate breaks as described in the Knowledge Base article “How to Model the Human Eye in ZEMAX”. In some cases where the eye rotates by a large angle to look into an optical system it is important to realise that there is no fixed centre 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 centre 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 centre. However, it is simplest in the model eyes here to locate the co-ordinate break to rotate the eye at the geometric centre of the retinal globe (in these models that is 13mm behind the cornea and on axis) and we have not found a case where that has given significant errors.
   c. Biocular Analysis: ZEMAX can analyse the field of view for up to four configurations in a system where two eyes are looking through the same optical system. The manual describes how to use this tool.
   d. 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. ZEMAX tolerancing offers a powerful way of investigating this and matching measured distributions with theoretical ones.

There are many uses for optical models of the eye, and no single model is best for every application. Often a very simple model will quickly give the answer needed, and a complex model often gives no more valid results than a simple one.

ZEMAX has many powerful tools for creating and using eye models, and time spent investigating these tools can be very rewarding.