Questions-and-Answers-for-Fourier-Optics-傅里叶光学问题解答

2014QuestionsandAnswersforFourierOpticsByLiPei2014QuestionsandAnswersforFourierOptics1.WhatistheFouriertransformofatwo-dimensionalfunctiong(x,y)?A:2(){(,)}(,)(,)XYjfxfyXYgxyGffgxyedxdyF2.ThepropertiesoftheFouriertransform.A:①Linearitytheorem:②Similaritytheorem③Shifttheorem④Rayleigh’stheorem⑤Convolutiontheorem3.Whatistheangularspectrumofplanewaves?Howcanwedeterminetheangularspectrumofthelightfield?A:Acrosstheplanez=0,acomplex-exponentialfunctionexp[2()]XYjfxfyrepresentsaplanewavepropagatingwithdirectioncosinesforaspecificfrequencies(,)XYff22;;1()(XYX)Yffff.Theangularspectrumofthedisturbanceacrossthezplaneisgivenby(,,)Uxyz(,;)(,,)exp[2()]AzUxyzjxydxdy4.Howcanwecalculatethepropagationoftheangularspectrumofthedisturbancefromoneplanetoanother?A:FromtheHelmholtsequation,thepropagationoftheangularspectrumoverdistancezfor221canbegivenby222(,;)(,;0)exp(1)AzAjzwhere(,;0)Aistheangularspectrumofthedisturbance.(,,0)Uxy5.Whatistheeffectofadiffractingapertureontheangularspectrumofthedisturbanceincidentonit?A:Consideringtheamplitudetransmittancefunctionoftheaperture,the(,)Atxy-1-2014QuestionsandAnswersforFourierOpticsByLiPeiangularspectrumofthetransmittedfieldcanbedescribedbytheconvolutionoftheangularspectrumoftheincidentfield(Ai)withtheangularspectrumoftheaperture(T).Thatis(,)(,)(,)tiAATwhere(,)(,)exp[2()]ATtxyjxydxdy).6.WhatarethegeneralpropertiesoftheOTF?A:①;②(0,0)1H*(,)(,XYXYfffHHf;③(,)(0,0)XYffHH7.Fromthewaveopticspointofview,whataretheeffectsofathinlensonthelightincidentonitssurface?A:Athinlens,actingasaphasetransformation,simplydelaysanincidentwavefrontbyanamountproportionaltothethicknessofthelensateachpoint.Thephasetransformationisdeterminedby22exp[()]2kjxyf.8.WhatistherelationshipbetweentheamplitudetransferfunctionHofanopticalsystemanditspupilfunctionPunderthecoherentillumination?A:(,)(,)XYiXiYHffPzfzf,whereziisthedistancefromtheexitpupiltotheimageplane.9.Whatisthephasorofthemonochromaticscalarlightfieldgivenby,cos2uPtAPtt,whereA(P)andaretheamplitudeandphase,respectively,ofthewaveatpositionP,whilePistheopticalfrequency.A:Thephasorisgivenby()()exp[()]UPAPjP.Itdescribesthespatialdependenceofboththeamplitudeandphaseonthespatialposition.10.Whatisthetransferfunctionofthewavepropagationphenomenoninfreespace?A:22221exp21()()(,)0XYXYXYzjffffHffotherwiseNOTE:Whycanthepropagationphenomenonberegardedasalinearspatialfilter?Answer:Thepropagationphenomenonactsasalinearspace-invariantsystemwitha-2-2014QuestionsandAnswersforFourierOpticsByLiPeitransferfunction22221exp21()()(,)0XYXYXYzjffffHffotherwiseTheisnonzeroonlyfor(,)XYHff221XYff,showingthepropagationphenomenonasalinearspatialfilterwithafinitebandwidthcharacterizedbyacircularregionofradius1infrequencyplane.11.IntheFraunhoferdiffractionregion,pleasegivetheexpressionforthelightfieldandexplainitintermsofFouriertransform.A:①222()()2(,)(,)kjkzjxyjxyzzeUxyeUeddjz②Itcanbeseenthatifwelet,XYxyffzz,theexpressionabovewillbeexactlyaFouriertransform.12.Whycanwemodelgeneralopticalimagingsystemsintermsoftheexitpupilorentrancepupil?A:Becausethediffractioneffectofanyopticalsystemcanberepresentedbythediffractionoftheexitpupil.13.Forthepolychromaticlightfield,howcanwerepresentitintermsofthephasor?(Anotherquestion:Howcanwedescribethedistributionofthepolychromaticlightfield)A:Forthepolychromaticlightfield,uPt,wesuppressthepositivefrequencycomponentsanddoublethenegativecomponents,sowehavethefollowingexpression:002[(,u2()2_(,)(,))]jvtjvvtjvtuPtuPvedvPvedve,wherevisthemeanfrequency.Consideringthecomplexexponentialfactor2jvteslowlyvaringwhen1vv(,)UPt,weuseatime-varyingphasortodescribethedistributionofthepolychromaticlightfield,namely2jvt(,)uPt(,UPt)eg.14.Whatistheeffectofthediffractionphenomenonontheimagegeneratedbyanopticalimagingsystem?A:Theobject-imagerelationshipis,where(,v)h(,v)(,v)iUuuUu-3-2014QuestionsandAnswersforFourierOpticsByLiPei1(,v)(,)gouUuUvMMMisthegeometrical-opticspredictionoftheimage,andisthepoint-spreadfunctionintroducedbydiffraction.22(,v)(,)exp[2()]huPzxzyjuxvydxdy15.TheFouriertransformpropertiesofapositivelens.A:222()exp[(1)()]2(,)(,)(,)juvffAkdAjuvddffUuvtPuveddjfff①Inputplacedagainstthelens.Wedo0,,dxy;andPcanbeneglectedintermsofthesmallerphysicalextentoftheinputthanthelensaperture.②Inputplacedinfrontofthelens,justtheexpressionabove.16.Underwhatconditionisalinearimagingsystemspace-invariant(orequivalently,isoplanatic)?A:2222(,;,)(,)hxyhxy22(,;,)hxy.Thefunctionformoftheimpulseresponsehisinvariant.Anddependsonlyonthespatialrelativedistances2()xand2()y.17.WhatisthewaveequationortheHelmholtzequationthatsatisfiesbythecomplexamplitudeofanymonochromaticopticaldisturbance?A:,whereUisthephasorofthelightfield.22()kU018.WhatarethepropertiesofthesecondarysourcespredictedbyfirstRayleigh–Sommerfeldsolution?A:①I