Accurate simulation and modeling of the human eye can be tricky. Entire volumes have been written on the subject, and it is a subject that continues to spur new developments. In this study, we will create model of a human eye in ZEMAX using the Liou & Brennan 1997 eye model. This is a fairly up-to-date and comprehensive model of the eye. It accounts for many realistic factors that other models do not, such as an offset pupil, a curved retina surface, an inward-pointing eyeball, and a crystalline lens with two different gradient refractive index profiles (one for the front half and a second profile for the rear half).

The prescription for this eye model (along with a wealth of other extremely important and interesting information) can be found in the book titled “Optics of the Human Eye,” by David A. Atchison and George Smith (2000).

After successfully generating this eye model in ZEMAX, we will use it to design a free-form progressive eyeglass lens.

For other approaches, see the article ZEMAX Models of the Human Eye, by Rod Watkins.

We’ll begin by setting up the human eye model. You can open the attached file “Human_Eye_Model.ZMX” to skip typing in all the surfaces by hand. The file is included in the zip archive that can be downloaded va the link on the last page of this article.

If you’re going to type the file in by hand instead of loading the attached file, you’ll want to first put ZEMAX into Sequential Mode, and then set the System|General|Units Lens Units to “Millimeters”. Next you’ll want to set the Wavelengths (found in the System section) to “F, d, C (Visible)” as shown below:



Next, go to System|General|Aperture and set the Aperture Type to “Float By Stop Size” and then go to System|General|Glass Catalogs and add the catalog “MISC” to your Glass Catalogs. Set just one Field, of Type “Angle(Deg)” with an X-Field value of 5:



Now insert 3 surfaces before the STOP and insert another 3 surfaces after the STOP. Below is a step-by-step guide to setting up all the surfaces, one at a time.

Surface 0
This surface is not actually labeled Surface 0 in the ZEMAX Lens Data Editor, it’s labeled “OBJ” and it’s the object surface. Below are the settings for Surface 0 (note that any settings not mentioned here should be left with their default values):

Surf:Type = Standard
Comment = Object
Radius = Infinity
Thickness = 1.00E+009

Surface 1
The first surface (after the Object) is just a dummy plane, and we use it to make our layout drawings easier to understand. Below are the settings for Surface 1:

Surf:Type = Standard
Comment = Input Beam
Radius = Infinity
Thickness = 50.0

(Note that the actual value of the thickness of this surface is not important: feel free to change it as necessary to make your layout drawings look just right)
Since we’re not actually interested in seeing this surface in the layout drawings (we only want to see the rays after they pass through the surface), let’s right click the Surf:Type cell for this surface, then click the Draw tab, and then check the Do Not Draw This Surface box.

Surface 2
This is the outer cornea surface. Below are the settings for Surface 2:

Surf:Type = Standard
Comment = Cornea
Radius = 7.77
Thickness = 0.55
Glass = Model; 1.376, 50.23
Semi-Diameter = 5.00
Conic = -0.18

Note: to set these glass parameters you will need to right-click the Glass cell, select “Model” as the Solve Type from the drop down list, and then type in the values like this:



Surface 3
This is the interface between the cornea and the aqueous humor. Below are the settings for Surface 3:

Surf:Type = Standard
Comment = Aqueous
Radius = 6.4
Thickness = 3.16
Glass = Model; 1.336, 50.23
Semi-Diameter = 5.00
Conic = -0.60

Surface 4

This surface is not actually labeled Surface 4 in the ZEMAX Lens Data Editor, it’s labeled “STO” and it’s the aperture stop of the system. This is our eye model’s pupil plane. Below are the settings for Surface 4:

Surf:Type = Standard
Comment = Pupil
Radius = Infinity
Thickness = 0.00
Glass = Model; 1.336, 50.23
Semi-Diameter = 1.25

In order to model the offset pupil location, we’ll need to decenter this surface. Right click the Surf:Type cell for this surface, and then click the Tilt/Decenter tab. Set this surface to have an X-decenter value of -0.5mm and then for After Surface select “Reverse This Surface,” as shown below:



Surface 5

This is the anterior (front) portion of our model’s crystalline lens. Below are the settings for Surface 5:

Surf:Type = Gradient 3
Comment = Lens-front
Radius = 12.4
Thickness = 1.59
Semi-Diameter = 5.00
n0 = 1.368
Nr2 = -1.978E-003
Nz1 = 0.049057
Nz2 = -0.015427

Surface 6

This is the posterior (rear) portion of our model’s crystalline lens. Below are the settings for Surface 6:

Surf:Type = Gradient 3
Comment = Lens-back
Radius = Infinity
Thickness = 2.43
Semi-Diameter = 5.00
n0 = 1.407
Nr2 = -1.978E-003
Nz2 = -6.605E-003

Surface 7

This is the rear surface of the crystalline lens (that is, it is the interface between the crystalline lens and the vitreous body of the eye). Below are the settings for

Surface 7:
Surf:Type = Standard
Comment = Vitreous
Radius = -8.1
Thickness = 16.23883
Glass = Model; 1.336, 50.23
Semi-Diameter = 5.00
Conic = 0.96

Surface 8
T
his surface is not actually labeled Surface 8 in the ZEMAX Lens Data Editor, it’s labeled “IMA” and it’s the image surface. This is the retina of our model. Below are the settings for Surface 8:

Surf:Type = Standard
Comment = Retina
Radius = -12.0
Semi-Diameter = 5.00

A quick check of the system’s 3D Layout (where First Surface is set to Surface 2, instead of the default Suface 0) shows that something is not quite right:



{Note: I like to make ZEMAX draw graphics as white lines on a black background. This is set under File...Preferences...Graphics:


There are other choices available...experiment!}

Now back to the layout drawing. The top and bottom marginal rays are getting cut off by the pupil. This can’t be right, especially since we’ve chosen our system’s Aperture Type to be “Float By Stop Size”: the top and bottom marginal rays should, by definition, successfully pass though the pupil. Something is amiss with our pupil. When our pupil is suspect, we always check the Pupil Aberration Fan (this is an analysis tool that can be found under Analysis | Fans | Pupil Aberration). Our system’s pupil aberration fan looks like this:




The pupil aberration fan is telling us that there is 40% pupil aberration in the negative X-direction. This is caused by the fact that ZEMAX is aiming rays at the (undecentered) paraxial entrance pupil. Remember that we decentered the pupil 0.5mm – which is 40% of the pupil semi-diameter – in the negative x-direction.

Luckily, ZEMAX has an easy fix for this problem: Ray Aiming. Go to System|General and then click the Ray Aiming tab. Now set Ray Aiming to Paraxial, as shown below:



Click the OK button and now we see that the Pupil Aberrations are eliminated:



And the 3D Layout shows the top and bottom marginal rays successfully passing through the decentered pupil:




This is the Liou & Brennan (1997) eye model. At this point, your Lens Data Editor should look like this:




Change the Settings on the 3D Layout so that Rotation Z = 90, and set it so that the First Surface is Surface 1 (input beam) and you’ll see the top-down view of the model, complete with offset pupil and off-axis field (I’ve added some annotations to the drawing below to label the various parts of the model):

To analyze this lens, let’s first look at the spot diagram:



Next we’ll check the FFT MTF:



Note that I’ve chosen a Max Frequency setting of 100 lp/mm. This corresponds to a single-spot resolution of 5 um, which corresponds closely to the resolution required for 20/20 Snellen Visual Acuity. You can see that this model should produce a well-resolved image up to 100 lp/mm, and therefore is a good model of a 20/20 human eye.

Finally, in order to get a good feel for how well this “patient” can see, we’ll check the Diffraction Image Analysis. The diffraction image analysis feature is similar to the geometric image analysis feature, except the complex system Optical Transfer Function (OTF) is used to compute the image appearance. This method accounts for the finite pass band and other diffraction related effects real optical systems have on image formation.

For this, let’s set it up to image the letter “F” at a size that corresponds with 20/20 Snellen Acuity. Go to Analysis|Image Analysis|Diffraction Image Analysis, and then click Settings and set the values as shown below:



Here is the resulting Diffraction Image Analysis diagram:

At this point, with a good human eye model set up in ZEMAX, it’s possible to add external elements to the design. For example, if you have a measured cornea-shape map, you could replace the cornea surface with a “Grid Sag” surface and import the data into ZEMAX. Or if you had a specially-designed intraocular lens (IOL) to model, you could replace the two gradient surfaces with the IOL prescription.

In this example, we will be designing a Progressive Addition Lens (PAL) and we’ll start by adding an eyeglass lens to the front of our model. We’ll optimize this PAL to provide good imaging for near, mid-range, and far objects.

To ensure a realistic modeling of the eye’s movement as the “patient” moves her eye up and down, we will want to keep the eyeglass in place, and have the eye model rotate about its center. That is, we’ll have to place a coordinate break at the center of the eyeball, and then have the entire eye model rotate about this point. To do this, we’ll first place the eyeglass lens into the Lens Data Editor, and then we’ll move forward an appropriate amount of Thickness to get to the center of the eye. Then we’ll put in a Coordinate Break surface, and then we’ll move backwards to the desired location of the eye model’s first surface, the cornea. This might seem a little confusing at first, but it’s really quite simple, as you’ll see.

First, insert three surfaces between the Input Beam (Surface 1) and the Cornea (Surface 2). These three surfaces will represent the front and rear surfaces of the eyeglass lens, and a coordinate break to tilt the eye. You’ll notice that I’ve chosen to place the eyeglass lens 15mm from the eye.

We’re going to place the Coordinate Break surface at the center of the eyeball (which I estimate to be approximately 13mm from the front cornea surface). This way, the eye will rotate about its center, in a way that mimics the actual motion of a human eye. Below is a guide to setting up these three new surfaces.

You’ll see below that the thickness from the back of the eyeglass lens to the coordinate break (at the eyeball’s center) is 28mm (that’s 15mm from eyeglass to cornea plus 13mm from cornea to center of eyeball). Then, after the coordinate break, there is a thickness of negative 13mm, to get back to the cornea surface.

Surface 2
This is the front surface of the eyeglass lens.
Surf:Type = Even Asphere
Comment = glasses-front
Radius = 100.0
Thickness = 3.0
Glass = POLYCARB
Semi-Diameter = 20.0

Surface 3 
This is the rear surface of the eyeglass lens.
Surf:Type = Extended Polynomial
Comment = glasses-back
Radius = 100.0
Thickness = 28.0
Semi-Diameter = 20.0

Surface 4
This is the coordinate break located at the center of the eyeball.
Surf:Type = Coordinate Break
Comment = center of eye
Thickness = -13.0

Now let’s add a couple of new configurations. Open the Multi-Configuration Editor (MCE), and hit Ctrl-Shift-Insert twice. We’ll need two extra Multi-Configuration Operands, so now hit Insert twice. There should now be nine empty cells in the MCE. We’re going to make the first Configuration represent the eye looking straight ahead through the lens, at an object located far away. The second Configuration will represent the eye looking slightly down through the lens, at a mid-range object. Finally, the third Configuration will represent the eye looking far down, at a very near object. Changing the object distance is handled with a THIC operand (changing the thickness of Surface 0) and changing the eye’s up-down angle is handled by changing the Tilt About X (which is Parameter 3) of Surface 4 (the Coordinate Break surface). Below is a table showing how the MCE should be filled in:



We’ve now got an eyeglass lens and an eyeball, and we’ve set the eyeball so that it rotates in a realistic manner. There’s one more minor step that will help make our layout plots easier to understand: we need to set the Global Coordinate Reference Surface to one that is before the Coordinate Break in the Lens Data Editor (LDE). The idea here is that after our Coordinate Break (surface 4) the rest of the surfaces in the LDE will be tilted some amount, relative to the surfaces before the Coordinate Break.

When we make layout plots, we would like to have the plots showing the eyeglass lens staying in place, while the eyeball rotates (instead of having the eyeball stay in place while the eyeglass moves around the eye). To ensure that this happens, we go to System|General and click on the Misc. tab. Then set the Global Coordinate Reference Surface to a value of 4 or lower (representing a surface before the Coordinate Break takes place in the LDE).

A quick check of the 3D Layout (with First Surface set to a value of 1; Rotation X, Rotation Y, and Rotation Z values set to 0; Configuration set to All; and Offset Y set to -50) shows that something is not quite right:



The problem is that as our eye model rotates down, it’s not really “looking” downward. This is because the Field value is remaining the same as the eye rotates, and what we want is for the Field to change so that the focused beam always hits the exact same part of the retina (this focal point on the retina is called the fovea centralis). So what we really want to keep constant is the image height, not the field angle. To do this, we’re going to change our Field Type from “Angle(Deg)” to “Real Image Height.”

We want the position of our on-axis image to remain unchanged after we make this Field Type switch, so the first thing we’ll do is check to see where the on-axis image is formed. This information is calculated in the Spot Diagram:



You can see that just under the actual spot diagram plot, there is a measurement of the (X, Y) location of the reference point (I’ve chosen a Chief Ray reference for my spot diagram) on the image surface. We see here that the Chief Ray intercepts the image surface at an X value of 1.462, therefore we will use a Real Image Height value of X = 1.462:




With our Field Data set up this way, we will be guaranteed that, irrespective of which configuration we choose, the Chief Ray will intercept the image surface (that is, the retina) at X = 1.462mm, Y = 0.0. This is a good simulation of the fact that no matter where the human eye rotates, the center of vision is always in the same exact spot on the retina (the fovea centralis).

Here is a 3D Layout plot showing the system from a side view for the 3 configurations (Far, Mid, and Near object distances):



The MTF curves and Diffraction Images for our system’s three configurations are shown below in order from Far to Near:



Note that because our model does not include accommodation by the crystalline lens (to adapt to the change from far to near object distances) the system shows very poor performance for mid-range and near objects, as we would expect in a presbyopic patient, for example.

We’re going to be optimizing the eyeglass lens to take on a free-form shape, in an attempt to provide accommodation to the eye as it rotates downward. Our next step is to set variables in the Lens Data Editor (LDE). The only things we’re interested in varying are the shapes of the front and back surfaces of the eyeglass lens.

In the LDE, set the Radius, Conic, and 2nd Order Term through 12th Order Term of Surface 2 (glasses-front) to variables – there should be eight parameters total for this surface that are now allowed to vary. Then in the Extra Data Editor (EDE) set the Max Term # of Surface 3 (glasses-back) to 40 and the Norm Radius of the same surface to 20.0. Now set all forty polynomial coefficient terms for Surface 3 to be variables.

Before we can optimize the shape of the eyeglass lens, we need to set up a Merit Function. In the Merit Function Editor (MFE), go to Tools|Default Merit Function. Set the Optimization Function and Reference values to “RMS,” “Wavefront,” and “Centroid.” Set the Pupil Integration Method to “Gaussian Quadrature” and choose 3 Rings and 6 Arms. Note that the Assume Axial Symmetry box will already be un-checked and we will leave it that way:



Click “OK” and ZEMAX will automatically generate a few thousand optimization operands for us.

Before we start optimizing, we just need to add a couple of edge thickness operands, to keep the edges of the eyeglass lens from becoming too thick or too thin. The eyeglass lens comprises an Extended Polynomial surface type, and so it is not going to remain rotationally symmetric during optimization. Therefore, we must use operands that check the edge thickness at multiple points around the lens. XNEG and XXEG are just such operands. We will add these operands to the merit function to ensure that the edge thickness of the eyeglass lens is between 1 and 8mm thick.

Insert two new operands at the beginning of the Merit Function. Below are the settings to use for these two new operands:

First new operand 
Type = XNEG
Surf1 = 2
Surf2 = 3
Zone = 0
Target = 1
Weight = 1

Second new operand
Type = XXEG
Surf1 = 2
Surf2 = 3
Zone = 0
Target = 6
Weight = 1

At this point, all the hard work of the design is complete. The attached file, “Human_Eye_Model.ZMX,” represents all the above work to this point.

The one thing left to do is to optimize the design by going to Tools|Optimization|Optimization and then clicking the Automatic button.

After some time, ZEMAX will finish optimizing and you’ll be left with an interesting, free-form, Progressive Addition Lens (PAL):



The MTF curves and Diffraction Images for our newly-optimized system’s three configurations are shown below in order from Far to Near:

We used ZEMAX to model a human eye using realistic parameters, including a gradient refractive-index crystalline lens, offset pupil, curved retina, and off-axis field angle. We then provided for a realistic rotation of this model and used this rotating eye model to design a progressive addition lens (PAL).