Our First Trial with a Raytracing Program for the Eye.


Sven-Göran Pettersson

Raytracing


Table of Contents

Summary
Optical characteristics of the human eye
Accommodation
The Resolution of the Eye
Raytracing
Some Examples from the Program

CERTEC Report 2:97
"Widesight" (Bodil Jönsson, Jörgen Gustafsson)

Appendix 1. PEER - Documentation on the Internet (Lars Philipson, Bodil Jönsson)
Appendix 2. RAY TRACING - A List of References (Lars Åke Svensson)





Summary

Raytracing has proved to be a very useful device for simulating and constructing optical systems. Similarly, it is quite clear that this excellent device can be of great significance to the most important of optical systems, that is, to the eye. With better and more precise models of the eye, a simulation of this kind can provide valuable information from which optometrists, doctors and lens designers will benefit. Simulations at small angles of incidence are particularly important in cases where the patient has loss of vision in the visual center.


Optical characteristics of the human eye

There are a number of different models of the eye, which at varying levels of complexity describe its optical characteristics. In the simplest model, all surfaces are spherical and the eye lens is a single lens with a constant refraction index. A more accurate representation may be provided by the Gullstrand eye as set out in tables 1 and 2. In this model, the eye lens consists of an inner nucleus with a high refractive index, which is surrounded by two areas with a lower refractive index. A more complex model is afforded by the Kooijman eye, in which the two surfaces of the cornea as well as that of the lens are aspherical. This model is described in table 3.

Measurements taken of the topography of the cornea show that in reality it consists of areas with different refractive powers such that neither a spherical nor and aspherical model can describe it accurately. One approximation is to regard it as consisting of a number spherical zones with different refractive powers.

The refractive system of the cornea has several parts. First, there is the transition between air and the tearfilm (43.6 D), then between tearfilm and the cornea (5.3 D) and finally the transition to the aqueous humor (-5.8 D). If one disregards the tearfilm, the transition from air to the cornea gives 48.9 D in total. This figure corresponds to a radius of 7.69 mm according to the Gullstrand eye.

Deviations from the normal eye causing defective vision are usually due to the fact that the distance between the retina and the lens is too large (myopia) or too small (hyperopia), but may also be due to deviations in the cornea or in the shape of the lens.


Table 1: The Gullstrand eye focused on the near point:

  Position Radius Refractive
  mm mm index after surface
Cornea 0 7.7 1.376
  0.5 6.8 1.336
Eye lens 3.2 5.33 1.385
 

3.8725

2.655

1.406

 

6.5725

-2.655

1.385

  7.2 -5.33 1.336
Retina 24.0 -11.5  



Table 2: The Gullstrand eye focused in infinity:

  Position Radius Refractive
  mm mm index after surface
Cornea 0 7.7 1.376
  0.5 6.8 1.336
Eye lens 3.6 10.0 1.385*
  4.146 7.911 1.406
  6.565 -5.76 1.385
  7.2 -6.0 1.336
Retina 24.0 -11.5  

* 1.386 according to Physiology of Vision


Table 3: The Kooijman eye

Surface Anterior cornea Posterior cornea Anterior lens Posterior lens

Radius

7.8

6.5

10.2

-6.0

Conic constant

-0.25

-0.25

-3.06

-1.0

Shape Ellipsoid

Ellipsoid

Hyperboloid

Paraboloid

Thickness (mm)

0.55

3.05

4.0

16.6

Index: 486.1 nm

1.3807

1.3422

1.42625

1.3407

Index: 587.6 nm

1.3771

1.3374

1.42

1.336

Index: 656.3 nm

1.37406

1.3354

1.4175

1.3341




Accommodation

Neither the Gullstrand eye nor the Kooijman eye provides any information about what the eye looks like when it is focused for vision at distances other than the near point or with a relaxed eye. The Tübingen group has designed a linear accommodation model. In this model, all parameters that have different values in the two cases set out in tables 1 and 2 are modified by a l parameter. This parameter has the value 1 for a relaxed eye and the value 0 when the eye is focusing on the near point. Accordingly, a radius R can be described with the following formula:



in which Racc, R¥ , Rn are the radii for the accommodated position, for a relaxed eye and for an eye focused on the near point. The Tübingen group makes an adjustment of l for the best image at an arbitrary distance from the object. According to the Gaussian theory of paraxial rays, the focus on the retina within is 0.1 mm with the same parameter values. The reason for the deviation is that the group uses a wide object.



The Resolution of the Eye

Because of the pigmentation of the retina and the optical characteristics of the eye, the resolution of the eye is about 1¢.

This corresponds to a distance of approximately 5 mm on the retina. For a pupil with a diameter D of 2 mm and a focal distance f of 22,89 mm the spot size 2q at the diffraction limit is determined by the following formula:


According to this formula, at an air wavelength of 550 nm and a refractive index of 1.337, the spot diameter will be 11 mm. For larger pupil diameters, the diameter of the spot decreases as long as there are not too many aberrations. The spot diameter is somewhat large in relation to the size of the cones in the macula lutea. The diameter of the cones is between 1,5 m m and 3 mm. It is possible that the constant movement of the eye compensates for this difference and that, consequently, a somewhat better resolution is obtained.



Raytracing

With a general raytracing program as the starting point, a special program has been developed for studying the optical characteristics of the eye. In this program, it is possible to simulate an eye with essentially spherical surfaces. However, a number of spherical areas with different refractive powers can be substituted for the cornea. It can also be made paraboloidal or ellipsoidal in order to provide a continuous shape. The lens is simulated with four surfaces and allowance can be made for the eye's ability to accommodate. Two spherical surfaces can be placed in front of the eye. The retina can be moved forwards or backwards in order for myopia or hyperopia to be studied. Furthermore, light that has an angle of incidence of up to 60° in relation to the optical axis of the eye can be studied.

One can also study parallel, divergent, or convergent beams travelling through the optical system. In these beams, the distribution of light is represented by an even distribution of rays within a given cross-section or a given solid angle. The image on the retina is projected onto a plane perpendicular to a radius from the center of the retina. Moreover, the image on the retina can be enlarged to allow details in the image to be studied. The ray path can also be enlarged.

By virtue of its graphic structure, the program provides clear results and is fairly easy to use. For example, it is relatively easy to change certain parameters, add surfaces, change the radius of the pupil or turn incoming rays. However, the program code is on the verge of being too large and, consequently, it is quite difficult to add new options without removing existing functions. Otherwise, it might be possible to let the computer do its own optimizing, for example. Moreover, the program does not contain any exact measurements of the image spot diameter.



Some Examples from the Program

The first figure shows what the display looks like (Fig. 1).


Fig. 1. The Display

At the bottom of the display, a main menu and a survey of the light rays directed towards the eye are shown. In the top left corner, a cross-section of incoming rays just prior to the cornea can be seen and in the top right corner, a centered image of the rays intersecting with the retina is shown. Further comments with respect to the Figure are set out below. Figs 2 and 3 show examples of accommodation based on the model of the Tübingen group described above.

Fig. 2a. Unaccommodated eye

Fig. 2b. Rays on the retina

Fig. 2a. Unaccommodated eye

Fig. 2b. Rays on the retina

Fig. 3a.  Accommodated eye

Fig. 3b.  Rays on the retina

Fig. 3a. Accommodated eye

Fig. 3b. Rays on the retina



Fig. 4 shows a ray diagram of rays incident at a very small angle. The spot diagram of the retina shows a spot in the order of 300 mm.

Fig. 4 a.  Rays incident at an angle

Fig.4b.Spot diagram

Fig. 4 a. Rays incident at an angle

Fig. 4b. Spot diagram



In order to test a correction with glasses a test was performed with two different cylindrical surfaces with perpendicular axes. The best results are shown below (Fig. 5).

Fig. 5a.  The image on the retina corrected with ylindrical surfaces

Fig.5b.Spotdiagram

Fig. 5a. The image on the retina corrected with ylindrical surfaces

Fig. 5b. Spot diagram



Two surfaces with the radii 27.5 mm (+18D) and - 233 mm(-2.15D) have been employed in the correction. It can be seen from the spot diagram that the image spot has been reduced to approximately 40 mm. However, it should be noted that in this simulation the eye was unaccommodated. In an accommodated eye, the image spot is considerably larger. The reason for this is that the rays hit the peripheral area of the nucleus. In this case, an even better model for the nucleus would be able to provide results that are more reliable.