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Authored by: Rod Watkins, Zemax consultant
This is a revision of an article first published on June 18, 2007. It includes the following
changes:
• The NSC model has been reconstructed around a reference point at the center
of the globe so the eye can be rotated and translated more easily.
• A binocular NSC model has been added with control over convergence and
interpupillary distance. Source rays have been added so the lines of sight can
be visualized.
• Some changes (described below) have been made to the sequential models.
• Some editorial changes have been made to the text.
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. This article presents several
sequential and non-sequential models of the human eye in Zemax format, with glass
catalog data.
IntroductionOptical 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).
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 dioptres and a refractive index of 4/3. It
therefore has a surface radius of 5.55 mm and an axial length of 22.22 mm. This model
Life Sciences - Zemax Models of the Human Eye
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This white paper presents several
sequential and non-sequential models
of the human eye in Zemax format, with
glass catalog data.
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is particularly useful for calculating retinal image size. Since the nodal point is 5.55 mm
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 16.67 mm. This paraxial model is accurate to within a few
percent for field angles as large as 10 degrees.
The Zemax models described below can be downloaded from the the Knowledge Base
article Zemax Models of the Human Eye. 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 size. You should feel free to modify them if it is
more appropriate for a particular purpose.
Sequential ModelsThere 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 as files
in the Knowledge Base Article Zemax Models of the Human Eye. The files are named
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.
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The Eye_Retinal Image model, above.
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 4 mm
diameter pupil.
The Eye_Retinal Object model, above.
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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 4 mm
diameter iris aperture. The image space is afocal.
Also included is a model of an eye accommodated to 250 mm (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 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_Accommodated model, above.
This 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 Zemax in sequential mode to draw the scleral surfaces as hyperhemispheres
(see Zemax 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 24.0 mm, the retinal radius is 11.0 mm
and the anterior surface of the relaxed lens is spherical with a radius of 10.0 mm.)
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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 Knowledge Base article How to Model the
Human Eye in Zemax describes how to do this.
Non-Sequential ModelsMany 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
modeling 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. The first object in the 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 shaded
model in the file is given a brightness of 70% and opacity of 50% to allow the internal
structure to be seen (see NSC Shaded Model | Settings).
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The Eye_NSC model, above.
This model uses F, d, and C equally weighted wavelengths and a 6 mm 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.
The Eye_Binocular.zmx model, above.
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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 CatalogThe glass catalog EYE.AGF included in the zip file must be copied to the Zemax glass
catalog folder to use these eye models. The default folder is c:\ Program Files\Zemax\
GLASSCAT. (To find the folder location, click on File | Preferences… | Folders. After copying,
select F4 to open the Glass Catalog frame and verify that Zemax 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 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 ToolsZemax has many tools to make eye models more useful by customizing 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 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) 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.
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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.
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 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.
(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 Zemax.)
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.
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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 powerful additional tools.
a. See 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
0.024 mm. Geometric Image Analysis of the Eye_Retinal Image model 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 0.024 mm
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. See Zemax menu Analysis | Image Simulation | 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.
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.
Paraxial ray aiming is usually sufficient, but users are encouraged to read the manual
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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 Zemax Toolsa. Toroidal Surfaces: Most real eyes have astigmatism due to the cornea being curved
more steeply vertically than horizontally. This can be modelled in sequential mode in
the Lens Data Editor | Surface Type | Toroidal. In NSC mode the toric 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
toric lens.
b. 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 15.4 mm behind the anterior
corneal surface and 1.6 mm to the nasal side of the geometric center. However, it is
simplest in the model eyes here to locate the coordinate break to rotate the eye at
the geometric center of the retinal globe (in these models that is 13 mm behind the
anterior corneal surface and on axis) and we have not found a case where that has
given significant errors.
c. 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.
SummaryThere 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.
AttachmentsEye_Retinal Object.ZAR(12.80 KB)
Eye_Retinal Image.ZAR(146.31 KB)
Eye_Binocular_ NSC.ZAR(34.76 KB)
Eye_ NSC.ZAR(21.67 KB)
© 2013 Radiant Zemax LLC. Radiant Zemax, ProMetric, TrueTest and Zemax are trademarks of Radiant Zemax LLC. All other marks are the property of their respective owners.770-9005-01 1/13
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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. This white paper presents several sequential and non-sequential
models of the human eye in Zemax format, with glass catalog data.
RadiantZemax.com
