Effects of Fresnel number in focusing and imaging

The effects of a finite value for Fresnel number in focusing systems are discussed. For an aperture illuminated by a convergent spherical wave, a shift of the maximum intensity towards the aperture results, associated with a coordinate scaling with regions further from the aperture stretched. There is also a dependence on the numerical aperture of the system, the focal shift decreasing with apertures. For high apertures, the focal field can be expressed in terms of a scaled Debye integral. An alternative geometry is that of a focusing system with an aperture stop at an arbitrary position. If the stop is situated in the front focal plane of the lens, the amplitude in the back focal plane is given by the Fourier transform of the aperture amplitude, and the effective Fresnel number is infinite. For positions further from the lens it is negative, so that the maximum in intensity is shifted further from the lens, and the scaling is such that regions closer to the lens are stretched. A high aperture system can be modeled using the concept of the equivalent refractive locus. The field in the front focal plane is transformed into an angular spectrum of plane waves with an appropriate apodization term, so that if the aperture stop is in the front focal plane the effective Fresnel number is infinite.