Average focal length and power of a section of any defined surface

PURPOSE: To provide a method to allow calculation of the average focal length and power of a lens through a specified meridian of any defined surface, not limited to the paraxial approximations. SETTING: University of Liverpool, Liverpool, United Kingdom. METHOD: Functions were derived to model back-vertex focal length and representative power through a meridian containing any defined surface. Average back-vertex focal length was based on the definition of the average of a function, using the angle of incidence as an independent variable. RESULTS: Univariate functions allowed determination of average focal length and power through a section of any defined or topographically measured surface of a known refractive index. These functions incorporated aberrations confined to the section. CONCLUSIONS: The proposed method closely approximates the average focal length, and by inference power, of a section (meridian) of a surface to a single or scalar value. It is not dependent on the paraxial and other nonconstant approximations and includes aberrations confined to that meridian. A generalization of this method to include all orthogonal and oblique meridians is needed before a comparison with measured wavefront values can be made.