Ray-Tracing

OKULIX is a program package which calculates single rays exactly. The visual impression of extended objects (e.g. Landolt’s rings) can be simulated by the superposition of many rays. Diffraction from the pupil aperture is taken into account additionally ³. Exactly in this context means, that the refraction of rays at each optical surface is calculated using Snell’s law. For a single ray passing multiple surfaces the calculation cannot be performed by analytical formulae, because otherways so-called "transcendental equations" occur which are unsolvable for principal mathematical reasons. Instead of an analytical calculation the problem therefore has to be solved by numerical methods requiring a computer. In former times, when computers where not available, approximations have been used under which Gaussian optics is best known ². In this approximation, the sine in Snell’s law is substituted by the arc: sin(α)≈α This, however, is sufficiently accurate only for very small angles, i.e., close to the optical axis. Gaussian optics therefore is also called "paraxial optics". In contrast, the calculation error inside OKULIX is the same for all distances to the optical axis (residual error ≤ 0.001dpt). In all OKULIX calculations the refraction error is minimized. The definition of refraction- and wavefront errors is shown in fig.11.

Figure 11: Refraction- and Wavefront Errors

A schematic cross-section of the eye is shown with lens L and cornea C, the optical axis from central retina R to the intersection point F with the off-axis ray r. Wavefront differences are calculated as differences between the optical path lengths of (R,L1,L2,C1,C2, F) and (R, ˜L1, ˜L2, ˜C1, ˜C2, F). The optical path lengths are the sums over the products of the geometrical lengths and the corresponding refractive indices. The difference between the spherical wave W starting from F and the plane wave P along r has to be added to the path length of r if F is not at infinite distance. The meridional refraction error of r is by definition the reciprocal of the distance from C2 to F. The deviation of the ray perpendicular to the graphics plane is described by the azimuthal refraction component. All optical path lengths are calculated in three dimensions. In case of a decentered IOL the bended ray through central retina and central cornea is calculated first and then used as reference instead of the optical axis.