Fresnel diffraction formulae 2.71/2.710 Optics wk11-b-1

spherical wave kernels Fresnel propagation: Huygens picture Thin transparency field at observation plane superposition of spherical wave kernels (Huygens “wavelets”) weighed by the incoming wavefront complicated incoming wave-front field just before the transparency field just after the transparency 2.71/2.710 Optics wk11-b-2

Fresnel propagation: Fourier picture Thin transparency The field immediately past the transparency is “decomposed” into a multitude of plane waves, weighed by the Fourier transform optical axis complex amplitude for this component: complicated incoming wave-front field just before the transparency field just after the transparency 2.71/2.710 Optics wk11-b-3

Fresnel propagation: Fourier picture Thin transparency The field at the observation plane is the superposition of the fields generated by all these plane waves (i.e., the “Fourier components” of the t(x,y)g2(x,y) field); note the quadratic phase delay between different components: 2.71/2.710 Optics wk11-b-4

Fresnel propagation: Fourier picture Thin transparency In the continuous limit, the plane wave superposition becomes an inverse Fourier transform; note that the quantity being transformed (the “operand”) is the product of G2(u,v) times a quadratic phase representing the FT of a Huygens “wavelet” in agreement with the Convolution theorem Inverse FT kernel 2.71/2.710 Optics wk11-b-5

as a linear, shift-invariant system Fresnel diffraction as a linear, shift-invariant system output amplitude impulse response convolution Fourier transform Fourier transform transfer function (≡plane wave spectrum) multiplication 2.71/2.710 Optics wk11-b-6

The 4F system object plane Fourier plane Image plane 2.71/2.710 Optics wk11-b-7

The 4F system object plane Fourier plane Image plane 2.71/2.710 Optics wk11-b-8

The 4F system with FP aperture object plane Fourier plane: aperture-limited Image plane: blurred (i.e. low-pass filtered) 2.71/2.710 Optics wk11-b-9

The 4F system with FP aperture Transfer function: circular aperture Impulse response: Airy function 2.71/2.710 Optics wk11-b-10