Centre for Sensors, Instruments and Systems Development

Universitat Politècnica de Catalunya

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Derivation of analytical refraction, propagation and reflection equations for higher order aberrations of wavefronts

 On June 25, Gregor Kornél Esser defended his doctoral thesis titled Derivation of analytical refraction, propagation and reflection equations for higher order aberrations of wavefronts.

The thesis was co-directed by Dr. Josep Arasa from CD6 and Dr. Peter Baumbach from Aalen University of Applied Sciences


Aberrations play a decisive role in optics. They describe the deviation from the perfect mage. Wavefront aberrations are usually described by a power series expansion or by Zernike polynomials. The wavefront aberrations describe the differences in optical path length between  the ideal and the actual wavefront. From literature the calculation of Lower Order Aberrations (LOA) as Power and Astigmatism of a local wavefront after refraction and reflection at a given surface is known. In the case of orthogonal incidence this relation is described by the  “Vergence Equation, and in the case of oblique incidence by the “Coddington Equation”. For Higher Order Aberrations (HOA) equivalent  analytical equations do not exist. The awareness of the role of HOA has significantly increased also in optometry and ophthalmology.

Hitherto for determining HOA, the wavefront in the pupil was calculated by ray-tracing a precise method when a large number of rays are used. In the field of spectacle optics the use of local wavefronts (determined by their local derivatives) to calculate the LOA (Power and Astigmatism) is well established. Wavefront tracing is a very fast semi-analytical method because it is only necessary to calculate the chief ray by numerical ray tracing. The coefficients of the wavefront itself, determined by their local derivatives, are calculated analytically. For calculating the wavefront aberrations of an entire lens, it is necessary to propagate the wavefront from the intersection point of the chief ray at the front surface along the chief ray to the intersection point at the rear surface and further to the vertex sphere or the entrance pupil of the eye. Because the refracting plane (plane of incidence) at the front and rear surface are not congruent, it is also necessary to rotate the coefficients of the wavefront. For LOA the propagation and rotation of the coefficients of the wavefront is known and described by the analytical Transfer equation. In this context, the purpose of this PhD was to extend the analytical Generalized Coddington Equation and the analytical Transfer Equation, which deals with LOA (power and astigmatism), to the case of HOA (e.g. Coma and Spherical Aberration) which are published by Esser et al in J. Opt. Soc. Am. A 27, 218–237 (2010), J. Opt. Soc. Am. A 28, 2442–2553 (2011) and AIEP 171, in press (2012). Therefore, it is now possible for the first time to calculate analytically the wavefront HOA’s of a spectacle lens or in general of an optical system by wavefront tracing.

This new approach has significant advantages with respect to the state of the art methods. First, the analytical nature of the solution yields more detailed insight into the underlying optical process. Second, the dramatic reduction of computational time in comparison to numerical methods opens new possibilities for the solution of practical problems in optics. Although the method is based on local techniques, it yields results which are by no means restricted to small apertures, as it is been shown theoretically as well as in two examples.
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