Analytical Representation of Optical Surfaces

Gaussian optics only consider the curvature term of an optical surface and a tilt term for prisms or mirrors; these terms constitute the two fundamental surfaces of the 1st-order theory. The correction of the optical aberrations (see next sections)

requires taking into account the next order terms. Investigating the case of stigmatic singlet lenses, Descartes laid down the first analytic theory leading to ovals which are non-spherical surfaces. In the present terminology, non-spherical surfaces are called aspherical surfaces or more briefly aspherics.

Representing an optical surface in a cylindrical coordinate frame z, r,d, we may distinguish between axisymmetric aspherical surfaces for centered systems and non-axisymmetric aspherics for non-centered systems. Their representation is usually given with respect to the tangent plane x,y at the vertex, hence z{r=0, 0} = 0.

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