MIT 2.71/2.710 Optics11/14/05 wk11-a-1Coherent imaging as a linear, shift-invariant systemThin transparency()yxt ,()yxg ,1()),( ,),( 12yxtyxgyxg==(≡plane wave spectrum)()vuG ,2impulse responsetransfer function()),(),(, 23yxhyxgyxg∗==′′Fourier transform),(),(),( 23vuHvuGvuG==output amplitudeconvolutionmultiplicationFourier transformilluminationtransfer function H(sx,sy): aka pupil functionMIT 2.71/2.710 Optics11/14/05 wk11-a-2Spatial filtering with the 4F system()yxg ,in()yxHfyfxG′′′′×⎟⎟⎠⎞⎜⎜⎝⎛′′′′,,11inλλ()⎟⎟⎠⎞⎜⎜⎝⎛′−′−∗ yffxffhg2121in, object planetransparencyFourier planetransparencyImage planeobserved field1f1f2f2f⎟⎟⎠⎞⎜⎜⎝⎛′′′′11in,fyfxGλλ()yxH′′′′,field arrivingat Fourier planemonochromaticcoherent on-axisilluminationfield departingfrom Fourier planeℑℑFouriertransformFouriertransformx′′x′xMIT 2.71/2.710 Optics11/14/05 wk11-a-3Single-lens imaging systemImpulse response (PSF)spatialspatial“LSI” system“LSI” systemgin(x,y) gout(x’,y’)Ideal PSF: ()()()myymxxyxyxh−′−′=′′δδ,;,Diffraction--limitedPSF: ()⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−′′+⎟⎠⎞⎜⎝⎛−′′=′′22jinc,;,sysysxsxRyxyxhλMIT 2.71/2.710 Optics11/14/05 wk11-a-4Imaging with incoherent lightMIT 2.71/2.710 Optics11/14/05 wk11-a-5Two types of incoherenced1d2Michelson interferometer Young interferometer1r1r2rspatialspatialincoherenceincoherencetemporaltemporalincoherenceincoherencepointsourcematchedpathspoly-chromatic light(=multi-color, broadband)mono-chromatic light(= single color, narrowband)MIT 2.71/2.710 Optics11/14/05 wk11-a-6Two types of incoherenced1d21r1r2rspatialspatialincoherenceincoherencetemporaltemporalincoherenceincoherencepointsourcematchedpathswaves from unequal pathswaves from unequal pathsdo not interferedo not interferewaves with equal pathswaves with equal pathsbut from different pointsbut from different pointson the wavefronton the wavefrontdo not interferedo not interfereMIT 2.71/2.710 Optics11/14/05 wk11-a-7Coherent vs incoherent beamsMutually coherent: superposition field amplitudeamplitudeis described by sum of complex amplitudessum of complex amplitudes1e11φiaa =2e22φiaa =2212212121eeaaaIaaaaaii+==+=+=φφMutually incoherent: superposition field intensityintensityis described by sum of intensitiessum of intensities1I21III +=2I(the phases of the individual beams vary randomly with respect to each other; hence,we would need statistical formulation todescribe them properly — statistical optics)MIT 2.71/2.710 Optics11/14/05 wk11-a-8Imaging with spatially incoherent light2f2fx′′x′x1f1fsimple object: two point sourcesnarrowband, mutually incoherent(input field is spatially incoherentspatially incoherent)MIT 2.71/2.710 Optics11/14/05 wk11-a-9Imaging with spatially incoherent light2f2fx′′x′x1f1fincoherent: adding in intensity ⇒()( )()2020xxhxxhxI +′+−′=′2x0MIT 2.71/2.710 Optics11/14/05 wk11-a-10Imaging with spatially incoherent light2f2fx′′x′x1f1fGeneralizing:thin transparency withsp. incoherentsp. incoherent illumination()()()xxxhxIxI d 2−′=′∫()xIintensity at the outputof the imaging systemMIT 2.71/2.710 Optics11/14/05 wk11-a-11Incoherent imaging as a linear, shift-invariant systemThin transparency()yxt ,()yxI ,1()),( ,),( 12yxtyxIyxI==incoherentimpulse response()223),(),(, yxhyxIyxI∗==′′output intensityconvolutionilluminationIncoherent imaging is linear in intensitywith incoherent impulse response (iPSF)where h(x,y) is the coherent impulse response (cPSF)()2),(,~yxhyxh =MIT 2.71/2.710 Optics11/14/05 wk11-a-12Incoherent imaging as a linear, shift-invariant systemThin transparency()yxt ,()yxI ,1()),( ,),( 12yxtyxIyxI==(≡plane wave spectrum)()vuI ,ˆ2incoherentimpulse responsetransfer function()223),(),(, yxhyxIyxI∗==′′Fourier transform),(~),(ˆ),(ˆ 23vuHvuIvuI==output intensityconvolutionmultiplicationFourier transformilluminationtransfer function of incoherent system:()yxssH ,~optical transfer function (OTF)MIT 2.71/2.710 Optics11/14/05 wk11-a-13The Optical Transfer Function()(){}()( )()∫∫∫∫′′′′′′−′−′′′=ℑ≡vuvuHvuvvuuHvuHyxhvuHdd,dd ,, 1 tonormalized , ,~2*2umax–umax12umax–2umaxreal(H)()H~real1MIT 2.71/2.710 Optics11/14/05 wk11-a-14some terminology ...Amplitude transfer function(coherent)Optical Transfer Function (OTF)(incoherent)Modulation Transfer Function (MTF)()vuH ,()vuH ,~()vuH ,~MIT 2.71/2.710 Optics11/14/05 wk11-a-15MTF of circular aperturephysical aperturefilter shape (MTF)f1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/14/05 wk11-a-16MTF of rectangular aperturephysical aperturefilter shape (MTF)f1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/14/05 wk11-a-17Incoherent low–pass filteringMTFIntensity @ image planef1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/14/05 wk11-a-18Incoherent low–pass filteringMTFIntensity @ image planef1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/14/05 wk11-a-19Incoherent low–pass filteringMTFIntensity @ image planef1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/14/05 wk11-a-20Diffraction-limited vs aberrated MTF2umax–2umax()H~real1ideal thin lens,ideal thin lens,finite aperturefinite aperturerealistic lens,realistic lens,finite aperturefinite aperture& aberrations& aberrationsMIT 2.71/2.710 Optics11/14/05 wk11-a-21Imaging with polychromatic light()()()yxyyxxhyxIyxI dd ;,;,;,2000λλλ−′−′=′′∫∫Monochromatic, spatially incoherent responseat wavelength λ0:Polychromatic (temporally and spatially incoherent) response:()()()( )020000d dd ;,;, d ;,,λλλλλ∫∫∫∫−′−′=′′=′′yxyyxxhyxIyxIyxIMIT 2.71/2.710 Optics11/14/05 wk11-a-22Comments on coherent vs incoherent• Incoherent generally gives better image quality:– no ringing artifacts– no speckle– higher bandwidth (even though higher frequencies are attenuated because of the MTF roll-off)• However, incoherent imaging is insensitive to phase objects• Polychromatic imaging introduces further blurring due to chromatic aberration (dependence of the MTF on