foueier optics課件c4 fresnel and fraunhofer diffraction

1、chapter 4fresnel and fraunhofer diffractionreview of physical optics_1lfirst rayleigh-sommerfeld diffraction in rectangular coordinates 11222expcos,expcos, cosjkre pe pdsrjkre pdsrzrxyzrn rreview of physical optics_2lfresnel diffraction2222222112812bbbrxyzxyzz review of physical optics_322,exp2,jkze

2、jkh x yxyj zzu x yuh lconvolution form of fresnel diffractionreview of physical optics_422222,exp,2jkjkzzejku x yxyuej zz flft form of fresnel diffractionpositive vs. negative phasesltime dependence is of the form exp(-j2t)lif we move in space in such a way as to intercept portions of a wavefield th

3、at were emitted later in time, the phasor will have advanced in the clockwise direction, and therefore the phase must become more negativephase sign of spherical waveldivergent spherical wavelconvergent spherical wave222expjxyz222expjxyzearlierearlierlaterlaterphase sign of plane wavelpositive angle

4、lnegative angleexp2jxexp2jxxzkxzk-earlierlaterearlierlaterldistanceearlierpositivetransfer function of fresnel diffractionlft the impulse response function of fresnel diffraction h(x,y) gets2222,exp2exp (421)jkzxyjkzxyejkhffh x yxyj zzejz ffffcomparison of strict diffraction and fresnel diffractionl

5、recall transfer function of propagation through free space 222221exp1,0, otherwise (42xyxyxyjzffffhff 0)lthe expression (4-21) is clearly an approximation to the more general transfer function (4-20).fresnel diffraction between confocal spherical surfaceslconfocal spherical surfacesltwo spheres are

6、said to be confocal if the center of each lies on the surface of the otherparaxial approximationapproximative formulalexpression of fresnel diffraction about planesbecomesfourier transform formlaside from constant multipliers and scale factors, the field observed on the right-hand spherical cap is t

7、he fourier transform of the field on the left-hand spherical cap.lkirchhoff rather than rayleigh-sommerfeld analysis remains valid for diffraction between two spherical caps.review of physical optics_5lfraunhofer diffraction22,exp,2, jkzxyejku x yxyuj zzfxzfyz ffraunhofer diffraction fourier transfo

8、rm laside from multiplicative phase factors preceding the integral, fraunhofer diffraction is simply the fourier transform of the aperture distributionexamples of fraunhofer diffraction_1lrectangular aperturellet the apertures are illuminated by a unit amplitude, normally incident plane wave, then t

9、he light field behind the aperture u(,)= ta(,)fraunhofer diffraction of rectangle aperture2222224,expsinc 2sinc 224,sinc2sinc2jkzxyxxyyxxxyyw w ejku x yxyw fw fj zzw wi x yu x yw fw fz22,exp,2,4sinc 2sinc 2xyjkzfxzfyzxyxxyyejku x yxyuj zzuw ww fw f ffplot of fraunhofer diffraction_1lthe width of the

10、 main lobe isthe first zero2=1z21z2xxxxw fxwxw plot of fraunhofer diffraction_2examples of fraunhofer diffraction_2lcircular aperturelq is a radius coordinate in the plane of the aperture with radius wlr is the radius coordinate in the observation plane 222exp2xyjkzrzffejkru ru qj zzbfraunhofer diff

11、raction of circular aperturewhere a=w22/ 212(/ )( )/jkrzjkzj kwr zau reej zkwr z22rxyairy patternlthis intensity distribution is referred to as the airy pattern. the width of the central lobe is given by=2z1.2221.22 zkwr zwrrdwplot of fraunhofer diffraction_1plot of fraunhofer diffraction_2examples

12、of fraunhofer diffraction_3lthin sinusoidal amplitude gratingcalculation of eq. 4-33diffraction of sinusoidal amplitude gratingplot of fraunhofer diffractionlthe central diffraction pattern is called the zero order of the fraunhofer pattern, while the two side patterns are called the first orders.di

13、ffraction efficiencyldefinition: the fraction of the incident optical power that appears in a single diffraction orderlfor the sinusoidal amplitude grating, we have three des with values ofexamples of fraunhofer diffraction_4lthin sinusoidal phase grating4-38calculation of eq. 4-382220,22 sincsinc2j

14、kxyjkzzqqau x yeej zmwwyjxqfzzzdiffraction of sinusoidal phase gratingif f01/w, the intensity has approximate form diffraction efficiency of sinusoidal phase grating ldiffraction efficiency of the q-th order of this grating isl+1 or -1 order may have diffraction efficiency up to 33.8%, far greater t

15、han for the case of a sinusoidal amplitude gratingfresnel diffraction by a sinusoidal amplitude gratinglobject is periodic structureldiffraction lies within the region of fresnel rather than fraunhoferzzxyspecial case of periodic objectlthe grating is modeled as a transmitting structure with amplitu

16、de transmittancelthere are two approaches to calculating the fields behind the gratinglconvolution form of the fresnel diffraction equationlfourier transformanalysis in frequency domainltransfer function of wave propagation in free spacelfourier transform of the amplitude transmittanceft of the fres

17、nel diffraction fieldlnow the above transfer function has value unity at the origin, and when evaluated at frequencies ( fx, fy) = (1/l, 0) yieldslfourier transform of the field becomesfresnel diffraction fieldlinverse transforming this spectrum we find the field at distance z from the grating to be

18、 given by222122,12coscoscos4zxxi x ymmlllinteresting interpretations_1suppose that the distance z behind the grating satisfies2222, or znlnzlthen the intensity observed at this distance behind the grating islit is an exact image of the grating without a lenstalbot imageinteresting interpretations_2suppose distance z behind the grating satisfies222121, or nlznzlthen the intensity observed at this distance behind the grating islthis is also an image of the grating, but this time with a spatial phase shift phase rever