P. Piot, PHYS 630 – Fall 2008Fraunhofer vs Fresnel approximation• Consider an incoming wave f(x,y) that propagates in free space (d)• Now assume the function is confined to• We also consider d to be large enough so that issmall• ThenP. Piot, PHYS 630 – Fall 2008Diffraction• Consider an incoming wave f(x,y)• The wave is intercepted by an aperture with transmission p(x,y)also called “pupil function”• Then propagates a distance d• The final complex amplitude of the wave is g(x,y)f(x,y)dg(x,y)p(x,y)P. Piot, PHYS 630 – Fall 2008Fraunhofer diffraction• In Fraunhofer diffraction, the complex wave amplitude downstream ofthe aperture is computed using the Fraunhofer approximation• This is valid if the Fresnel number is <<1• Consider an incoming plane wave• Downstream of the aperture with transmission p(x,y) we have• And after a drift of length dwhereP. Piot, PHYS 630 – Fall 2008Fraunhofer diffraction: rectangular aperture I• Consider an incoming wave intercepted by a rectangular aperture ofsize Dx and Dy. What is the intensity of the diffraction pattern?• We have• Using the previous slidesP. Piot, PHYS 630 – Fall 2008Fraunhofer diffraction: rectangular aperture II• We finally obtainDx=DyDx=2DyxyxyP. Piot, PHYS 630 – Fall 2008Fraunhofer diffraction: circular aperture I• Now we take a circular aperture of radius a• The Fourier transform of the transmission function is• Here we change to cylindrical coordinates because of the cylindricalsymmetry. IntroducingP. Piot, PHYS 630 – Fall 2008Fraunhofer diffraction: circular aperture II• We can write• So finallyAnd the diffraction pattern intensity isP. Piot, PHYS 630 – Fall 2008Fraunhofer diffraction: circular aperture III• Diffraction pattern intensityrrJ1(r)/r[J1(r)/r]2xyP. Piot, PHYS 630 – Fall 2008Fresnel diffraction I• In Fresnel diffraction, the complex wave amplitude downstream ofthe aperture is computed using the Fresnel approximation• The intensity is given byP. Piot, PHYS 630 – Fall 2008Fresnel diffraction II• Written in a normalized coordinate system• This is the convolution of the transmission function p(X,Y) of theconsidered aperture with the functionxcos(πx2)sin(πx2)size ~ a/[λ d]1/2P. Piot, PHYS 630 – Fall 2008Fresnel diffraction III• In the equationThe result of this convolution is governed by the Fresnel numberNF=a2/(λd)• If NF is large the convolution is going to yield a function similar top(X,Y).• In the limit NF→∞, ray optics is applicable (λ → 0) and the pattern isthe shadow of the aperture• In the opposite limit Fresnel diffraction pattern converge to theFraunhofer pattern.size ~ a/[λ d]1/2P. Piot, PHYS 630 – Fall 2008Fresnel diffraction: slit aperture• Consider a slit infinitely long in the y-direction then• From we need to compute g(X)P. Piot, PHYS 630 – Fall 2008Fresnel diffraction: slit aperture II• Fresnel patterns for different Fresnel numberNf=10Nf=1Nf=0.5Nf=0.1P. Piot, PHYS 630 – Fall 2008Summary• In the order of increasing distance from the aperture, diffractionpattern is• A shadow of the aperture.• A Fresnel diffraction pattern, which is a convolution ot the“normalized” aperture function with exp[-iπ(X2+Y2)].• A Fraunhofer diffraction pattern, which is the squared-absolute value of the Fourier transform of the aperturefunction. The far field has an angular divergence proportionalto λ/D where D is the diameter of the