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1、13.FresneldiffractionRemind!DiffractionregimesFresnel-KirchhoffdiffractionformulaEexp(ikr)0E()PF=()θdA0∫∫irλ∑rzObliquityfactor:F()cosθθ==rzEexp(ikr)Ex,y0dd()=ξη∫∫2irλ∑Aperture(ξ,η)Screen(x,y)222rzx=+−+−()()ξyη2222⎡⎤11⎛⎞⎛⎞xy−−ξη()xy−−ξ()η≈+zz⎢⎥1⎜⎟⎜⎟+=++⎢⎥⎣⎦22⎝⎠⎝⎠zz22zz222
2、2⎛⎞xy⎛⎞ξη⎛⎞xyξη=z++++−+⎜⎟⎜⎟⎜⎟⎝⎠22zz⎝⎠22zzzz⎝⎠E0⎡⎤k22Exy(),e=+xp()ikzexp⎢⎥i()xyizλ⎣⎦2z⎡⎤kk22⎡⎤×+∫∫exp⎢⎥ii()ξηξexp⎢⎥−()x+yηddξη∑⎣⎦2zz⎣⎦Exy(),e=+E0xp()ikzexp⎡⎤ik()xy22r⎢⎥izλ⎣⎦2z⎡⎤kk22⎡⎤×+∫∫exp⎢⎥ii()ξηξexp⎢⎥−()x+yηdξdηAperture(ξ,η)Screen(x,y)∑⎣⎦2zz⎣⎦⎡⎤kk22⎡⎤=+Ci∫∫exp⎢⎥()ξηξe
3、xp⎢⎥−i()x+yηdξdη∑⎣⎦2zz⎣⎦Fresneldiffraction⎡⎤kk22⎡⎤E()xy,(=+CU∫∫ξ,ηξ)exp⎢⎥i()ηexp⎢⎥−i()xξ+yηddξη⎣⎦2zz⎣⎦k22⎧j()ξη+⎫Exy(,)U,e2z∝F⎨()ξη⎬⎩⎭Fraunhoferdiffraction⎡⎤kExyCU(),(=−∫∫ξη,)exp⎢⎥ix()ξ+yηddξη⎣⎦z=−CU(,)expξηξ⎡⎤ik()sinθ+ηsinθddξη∫∫⎣⎦ξηExy(,)∝F{U(ξ,η)}Fresnel(near-field)di
4、ffractionThisismostgeneralformofdiffraction–Norestrictionsonopticallayout•near-fielddiffractionk22⎧j()ξη+⎫•curvedwavefrontUxy(,)FU,e2z≈⎨()ξη⎬–Analysissomewhatdifficult⎩⎭Curvedwavefront(parabolicwavelets)ScreenzFresnelDiffractionAccuracyoftheFresnelApproximation3π222z〉〉[(
5、)()x−ξ+y−η]max4λ•AccuracycanbeexpectedformuchshorterdistancesforU(,)ξηξsmooth&slowvaringfunction;x2−=≤D4λz2Dz≥Fresnelapproximation16λInsummary,Fresneldiffractionis…13-7.FresnelDiffractionbySquareApertureFresnelDiffractionfromaslitofwidthD=2a.(a)Shaded2w2aareaisthegeometr
6、icalshadowoftheaperture.ThedashedlineisthewidthoftheFraunhoferdiffractedbeam.(b)Diffractionpatternatfouraxialpositionsmarkedbythearrowsin(a)andcorrespondingtotheFresnelnumbersN=10,F1,0.5,and0.1.Theshadedarearepresentsthegeometricalshadowoftheslit.Thedashedlinesatx=(λrepr
7、esentthe/D)dwidthoftheFraunhoferpatterninthefarfield.Wherethedashedlinescoincidewiththeedgesofthegeometricalshadow,theFresnelnumberN=0.5.F2Naz=/λ:FresnelnumberFjkz∞()e()⎧k[()()22]⎫Ux,y=∫∫Uξ,ηexp⎨jx−ξ+y−η⎬dξdηjλz−∞⎩2z⎭2Δ=vNaz=/λ:FresnelnumberFFresneldiffractionfromawireFr
8、esneldiffractionfromastraightedgeFromHuygens’principletoFresnel-KirchhoffdiffractionHuygens’principleEv
