Beyond the Fresnel approximation for focused waves

By extension of the transitional operator method developed by Wunsche, the Rayleigh-Sommerfeld and Kirchhoff solutions to the diffraction of a converg ing spherical (or cylindrical) wave are expressed in terms of a series of d erivatives of the field estimate that follows from the Fresnel approximatio n. 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 appro ximation leads to estimates of the field that have a relative error near fo cus that is of the order of one on the square of the f-number. The number o f significant digits in the field estimate is shown to be doubled by retain ing just the first of the series of corrections derived here. (C) 1999 Opti cal Society of America [S0740-3232(99)01508-2].