Ray tracing through a schematic eye containing second‐order (quadric) surfaces using 4 × 4 matrix notation

Abstract

Ray tracing is used in ophthalmology for evaluation of the optical properties of the eye. We demonstrate an algebraic method for tracing a bundle of rays through the optical system of an eye containing aspheric surfaces. Restricting to second-order surfaces (quadric surfaces) such as ellipsoids, paraboloids or hyperboloids, a surface is described by a 4 x 4 matrix. In this case, the normal vector can be derived analytically and the ray-surface intersection is calculated by solving a quadratic equation. We applied this straightforward matrix-based strategy to the spherical 4-surface Le Grand schematic eye, and the Le Grand eye modified by Kooijman containing four aspheric surfaces. We calculated the spot diagram for the focal plane as well as a pre- and post-focal plane for both model eyes, and found that the optical quality of the aspheric model characterized by the ray scatter in the spot diagram at the focal plane is much better than that of the spherical model. This calculation strategy may be helpful for evaluating the image distortion of decentred or tilted spherical or aspheric artificial intra-ocular lenses.

Customized Aspheric IOL Design by Raytracing through the Eye Containing Quadric Surfaces

“…5,6 If we restrict the system to paraxial optics, the linear optical conditions are determined with simple straightforward mathematical strategies: (1) the vergence from the object plane to the image plane can successively be traced for an optical system containing refractive or reflective surfaces and a homogeneous optical medium at the interspaces between the surfaces, or (2) refractive or reflective surfaces can be characterized by using matrix notation using so-called power matrices: 2 × 2 matrices for the spherical 10,11 or 4 × 4 matrices for the spherocylindrical case. [12][13][14][15][16] Paraxial raytracing yields proper results, if the ray height and the slope angle relative to the optical axis are low.…”

“…2,5,6 For concave surfaces, R in Eq. 2 has to be considered as the negative value of the apical radius of curvature.…”

“…The factor 2 in the linear and mixed terms is used to simplify the equivalent matrix notation. 5 A convex conic surface is a particular case of rotationally symmetric quadric surface with the apex at the origin of the coordinate system with orientation in the z-axis:…”

simEye: computer-based simulation of visual perception under various eye defects using Zernike polynomials

“…10-12͒. Ray tracing with aspheric surfaces can still be analytically calculable: Langenbucher et al, 19 for example, report on an algebraic method for ray tracing through the optical system of an eye with aspheric surfaces. Their method is restricted to secondorder surfaces ͑quadric surfaces͒.…”

Irregularity of the Posterior Corneal Surface After Curved Interface Femtosecond Laser-Assisted Versus Microkeratome-Assisted Descemet Stripping Automated Endothelial Keratoplasty

“…[21][22][23][24][25][26][27] Second-order surfaces have previously been used as a mathematical model of the corneal surface. 28 In the present retrospective clinical study, we compared graft morphologies and visual outcome in patients after curved interface femtosecond laser-assisted EK and microkeratome-assisted EK.…”