Beyond the Fresnel approximation for focused waves

By extension of the transitional operator method developed by Wünsche, the Rayleigh–Sommerfeld and Kirchhoff solutions to the diffraction of a converging spherical (or cylindrical) wave are expressed in terms of a series of derivatives of the field estimate that follows from the Fresnel approximation. This result allows a systematic assessment of the error associated with the paraxial wave model for focused fields and offers simple corrections to this model. In particular, for simple diffracting masks, the Fresnel approximation leads to estimates of the field that have a relative error near focus that is of the order of one on the square of the f-number. The number of significant digits in the field estimate is shown to be doubled by retaining just the first of the series of corrections derived here.