MIT 2.71/2.710 Optics12/12/05 wk15-a-1Today• Defocus• Deconvolution / inverse filtersMIT 2.71/2.710 Optics12/12/05 wk15-a-2DefocusMIT 2.71/2.710 Optics12/12/05 wk15-a-3©©2020ththCentury FoxCentury FoxMIT 2.71/2.710 Optics12/12/05 wk15-a-4Focus in classical imaginginin--focusfocusdefocusdefocusMIT 2.71/2.710 Optics12/12/05 wk15-a-5Focus in classical imaginginin--focusfocusdefocusdefocusMIT 2.71/2.710 Optics12/12/05 wk15-a-6Intensity distribution near the focus of an ideal lens()NA61.0λ×=∆x()2NA2λ=∆z(rotationally symmetric wrt z axis)zMIT 2.71/2.710 Optics12/12/05 wk15-a-7Back to the basics: 4F system 2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanezMIT 2.71/2.710 Optics12/12/05 wk15-a-8Back to the basics: 4F system 2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanezmaxxa≡(NA)1()1max1 NAfx=(NA)2()2max2 NAfx=MIT 2.71/2.710 Optics12/12/05 wk15-a-9Back to the basics: 4F system 2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanez()xg⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ⎟⎟⎠⎞⎜⎜⎝⎛′− xffg21MIT 2.71/2.710 Optics12/12/05 wk15-a-104F system with defocused input2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanez()xgMIT 2.71/2.710 Optics12/12/05 wk15-a-114F system with defocused input2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanez()⎟⎟⎠⎞⎜⎜⎝⎛∆+∗zyxixgλπ22expMIT 2.71/2.710 Optics12/12/05 wk15-a-124F system with defocused input2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanez()⎟⎟⎠⎞⎜⎜⎝⎛∆∗zxixgλπ2exp()⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛′′∆⋅⎟⎟⎠⎞⎜⎜⎝⎛′′211 expfxzifxGλλπλℑMIT 2.71/2.710 Optics12/12/05 wk15-a-134F system with defocused input2f2fx′′x′x1f1fimageimageplaneplaneobjectobjectplaneplaneFourierFourierplaneplanez()⎟⎟⎠⎞⎜⎜⎝⎛∆∗zxixgλπ2exp()()⎥⎦⎤⎢⎣⎡′′∆⋅⎟⎟⎠⎞⎜⎜⎝⎛′′2121expfxzifxGλπλℑMIT 2.71/2.710 Optics12/12/05 wk15-a-14Effect of defocus on the Fourier planemild defocus1fxλ′′()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ()z∆λ2110MIT 2.71/2.710 Optics12/12/05 wk15-a-15Effect of defocus on the Fourier planestrong defocus()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ()z∆λ21101fxλ′′MIT 2.71/2.710 Optics12/12/05 wk15-a-16Effect of defocus on the Fourier planemild defocus()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ()z∆λ21⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ×not too different from ⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ101fxλ′′MIT 2.71/2.710 Optics12/12/05 wk15-a-17Effect of defocus on the Fourier plane10⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ×very different from strong defocus()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ⎟⎟⎠⎞⎜⎜⎝⎛′′1fxGλ1fxλ′′MIT 2.71/2.710 Optics12/12/05 wk15-a-18Depth of field()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπ()z∆λ21101fxλ′′system is defocus-insensitiveas long as frequencies that pass through the systemare confined within this regionMIT 2.71/2.710 Optics12/12/05 wk15-a-19Depth of field()z∆λ2110system is defocus-insensitiveas long as frequencies that pass through the systemare confined within this region()()()21max1max221fxzzfx′′≤∆⇒⇒∆≤′′λλλ1fxλ′′()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπMIT 2.71/2.710 Optics12/12/05 wk15-a-20Depth of field()z∆λ2110system is defocus-insensitiveas long as (∆z) is small enoughthatfrequencies that pass through the systemcan be confinedwithin this region()()()121maxNA221λλλ≤∆⇒⇒∆≤′′zzfx1fxλ′′()()⎥⎦⎤⎢⎣⎡′′∆212cosfxzλπDepth of fieldMIT 2.71/2.710 Optics12/12/05 wk15-a-21Depth of field & Depth of focus2f2fx′′x′x1f1fobjectobjectplaneplaneFourierFourierplaneplanez()()()1max112 NANA2fxz′′=≤∆λ()()()2max222 NANA2fxz′′=≤∆λimageimageplaneplaneDepth of fieldDepth of fieldDepth of focusDepth of focus(NA)1(NA)2MIT 2.71/2.710 Optics12/12/05 wk15-a-22NA trade – offs• high NA– narrow PSF in the lateral direction (PSF width ~1/NA)• sharp lateral features– narrow PSF in longitudinal direction (PSF depth ~1/NA2)• poor depth of field•low NA– broad PSF in the lateral direction (PSF width ~1/NA)• blurred lateral features– broad PSF in longitudinal direction (PSF depth ~1/NA2)• good depth of fieldMIT 2.71/2.710 Optics12/12/05 wk15-a-23Depth of focus: Geometrical Optics viewpoint2f2fx′′x′x1f1fz(NA)1(NA)2defocuseddefocusedimageimageplaneplane∆x∆zmaxxa≡()2NAxz∆=∆From similar triangles:Now require defocused spot ≈ diffraction spot:()2NA61.0λ≈∆xTherefore:()22NA61.0λ≈∆zMIT 2.71/2.710 Optics12/12/05 wk15-a-24Defocus and Deconvolution(Inverse filters)MIT 2.71/2.710 Optics12/12/05 wk15-a-252f2fx′′x′x1f1fDepthDepthOf Of FocusFocusDefocusDefocusworsensworsensawayawayfromfromfocalfocalplaneplane22½½DDobjectobjectzImaging a 2½D objectMIT 2.71/2.710 Optics12/12/05 wk15-a-262f2fx′′x′x1f1fportion of objectportion of objectdefocused by defocused by ΔΔzzz∆zImaging a 2½D objectMIT 2.71/2.710 Optics12/12/05 wk15-a-272f2fx′′x′x1f1f……is equivalent to same portion is equivalent to same portion inin--focusfocusPLUS PLUS …………fictitious quadratic fictitious quadratic phase mask phase mask on the Fourier planeon the Fourier plane()⎭⎬⎫⎩⎨⎧∆′′+′′−21222expfzyxiλπ(applied(appliedlocally locally ))zImaging a 2½D objectMIT 2.71/2.710 Optics12/12/05 wk15-a-28Example focal plane2 (DoF)s4 (DoF)sMIT 2.71/2.710 Optics12/12/05 wk15-a-29Distance between planes ≈ 2 Depths of Fieldleft-most “M” : image blurred by diffraction onlycenter and right-most “M”s : image blurred by diffraction and defocusRaw image (collected by camera – noise-free)MIT 2.71/2.710 Optics12/12/05 wk15-a-30Raw image explanation: convolution“M” convolvedwith standarddiffraction PSF“M” convolvedwith diffraction PSFand defocus“M” convolvedwith diffraction PSFand more defocusMIT 2.71/2.710 Optics12/12/05 wk15-a-31Raw image explanation: Fourier domain{}ndiffractioM"" H×ℑ{}()2DoFM"" defocusndiffractioHH××ℑ{}()4DoFM"" defocusndiffractioHH××ℑMIT 2.71/2.710 Optics12/12/05 wk15-a-32Can