1MIT 2.71/2.710 Optics11/23/05 wk12-b-1ResolutionMIT 2.71/2.710 Optics11/23/05 wk12-b-2The meaning of “resolution”[from the New Merriam-Webster Dictionary, 1989 ed.]:resolve v : 1 to break up into constituent parts: ANALYZE;2 to find an answer to : SOLVE; 3 DETERMINE, DECIDE;4 to make or pass a formal resolutionresolution n : 1 the act or process of resolving 2 the actionof solving, also : SOLUTION; 3 the quality of being resolute :FIRMNESS, DETERMINATION; 4 a formal statementexpressing the opinion, will or, intent of a body of persons2MIT 2.71/2.710 Optics11/23/05 wk12-b-3The two–point resolution problemobject: two point sources,mutually incoherent(e.g. two stars in the night sky;two fluorescent beads in a solution)x′ximagingsystemintensitypatternobserved(e.g. withdigitalcamera)The resolution question[Rayleigh, 1879]: when do we ceaseto be able to resolvethe two point sources (i.e., tell them apart)due to the blurring introduced in the image by the finite (NA)?MIT 2.71/2.710 Optics11/23/05 wk12-b-4Numerical Aperture and Speed (or F–Number)medium ofrefr. index nθθ: half-angle subtended by the imaging system from an axial objectNumerical Aperture(NA) = n sinθSpeed (f/#)=1/2(NA)pronounced f-number, e.g.f/8 means (f/#)=8.Aperture stopthe physical element whichlimits the angle of acceptance of the imaging system3MIT 2.71/2.710 Optics11/23/05 wk12-b-5()()()yxyxgδδ=,in()λπλπrfRrfR′⎟⎟⎠⎞⎜⎜⎝⎛′≡11122J2.,.jincobject planeimpulseFourier planecirc-apertureimage planeobserved field(PSF)1f1f()⎟⎠⎞⎜⎝⎛′′=′′′′RryxH circ,monochromaticcoherent on-axisilluminationℑFouriertransformx′′x′x1f1fradial coordinate@ Fourier plane22yxr′′+′′=′′22yxr′+′=′radial coordinate@ image plane(unit magnification)2RPSF vs NAMIT 2.71/2.710 Optics11/23/05 wk12-b-6Fourier planecirc-apertureimage plane1f1fmonochromaticcoherent on-axisilluminationx′′x′x1f1f()()λπλπλπλπλλrrrfRrfRyfRxfR′⎟⎠⎞⎜⎝⎛′=′⎟⎟⎠⎞⎜⎜⎝⎛′≡⎟⎟⎠⎞⎜⎜⎝⎛′−′−NA2NA2J222J22,2jinc111111()1NAfR≡Numerical Aperture (NA)by definition:NA: angleof acceptancefor on–axispoint object2RPSF vs NA4MIT 2.71/2.710 Optics11/23/05 wk12-b-7DiffractionNA22.1λ=dmain lobewidthMIT 2.71/2.710 Optics11/23/05 wk12-b-8Diffraction-limited resolutionNA22.1λδ≥xRayleigh criterion(incoherent imaging)5MIT 2.71/2.710 Optics11/23/05 wk12-b-9PSF vs NA()NA61.0 @ nullλ=′r()()()λπλπrryxh′⎟⎠⎞⎜⎝⎛′=′′NA2NA2J2,1MIT 2.71/2.710 Optics11/23/05 wk12-b-10()()()λπλπrryxh′⎟⎠⎞⎜⎝⎛′=′′NA2NA2J2,1()NA22.1 width lobeλ=′∆rPSF vs NA6MIT 2.71/2.710 Optics11/23/05 wk12-b-11NA in unit–mag imaging systems1f1fmonochromaticcoherent on-axisilluminationx′′x′x1f1f2R12 fmonochromaticcoherent on-axisilluminationx′′x′x12 f2R()1NAfR≡()12NAfR≡()() ()⎟⎠⎞⎜⎝⎛′=′=′′=λrrhyxh NA2jinc,PSFin both cases,MIT 2.71/2.710 Optics11/23/05 wk12-b-12The incoherent case:()NA61.0 @ nullλ=′r()()2,,~yxhyxh′′=′′()()()21NA2NA2J2,~⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡′⎟⎠⎞⎜⎝⎛′=′′λπλπrryxh7MIT 2.71/2.710 Optics11/23/05 wk12-b-13Resolution in optical systemsx() ()NA61.0NA0.3λλ>=∆r()⎟⎟⎠⎞⎜⎜⎝⎛+′NA5.1~λxh()⎟⎟⎠⎞⎜⎜⎝⎛−′NA5.1~λxhMIT 2.71/2.710 Optics11/23/05 wk12-b-14Resolution in optical systemsx() ()NA61.0NA0.3λλ>=∆r()()⎟⎟⎠⎞⎜⎜⎝⎛−′++⎟⎟⎠⎞⎜⎜⎝⎛+′NA5.1~NA5.1~λλxhxh8MIT 2.71/2.710 Optics11/23/05 wk12-b-15x() ()NA61.0NA4.0λλ<=∆rResolution in optical systems()⎟⎟⎠⎞⎜⎜⎝⎛+′NA2.0~λxh()⎟⎟⎠⎞⎜⎜⎝⎛−′NA2.0~λxhMIT 2.71/2.710 Optics11/23/05 wk12-b-16Resolution in optical systemsx() ()NA61.0NA4.0λλ<=∆r() ()⎟⎟⎠⎞⎜⎜⎝⎛−′+⎟⎟⎠⎞⎜⎜⎝⎛+′NA2.0~NA2.0~λλxhxh9MIT 2.71/2.710 Optics11/23/05 wk12-b-17x()NA61.0λ=∆rResolution in optical systems()⎟⎟⎠⎞⎜⎜⎝⎛+′NA305.0~λxh()⎟⎟⎠⎞⎜⎜⎝⎛−′NA305.0~λxhMIT 2.71/2.710 Optics11/23/05 wk12-b-18Resolution in optical systemsx()NA61.0λ=∆r() ()⎟⎟⎠⎞⎜⎜⎝⎛−′+⎟⎟⎠⎞⎜⎜⎝⎛+′NA305.0~NA305.0~λλxhxh10MIT 2.71/2.710 Optics11/23/05 wk12-b-19Resolution in noisynoisy optical systemsx()NA61.0λ=∆rMIT 2.71/2.710 Optics11/23/05 wk12-b-20x()NA22.1λ=∆r“Safe” resolution in optical systems() ()⎟⎟⎠⎞⎜⎜⎝⎛−′+⎟⎟⎠⎞⎜⎜⎝⎛+′NA61.0~NA61.0~λλxhxh11MIT 2.71/2.710 Optics11/23/05 wk12-b-21Diffraction–limited resolution (safe)Pushy definition:(1/2–lobe spacing)Safe definition:(one–lobe spacing)One–dimensional systems(e.g. slit–like aperture)Two–dimensional systems(rotationally symmetric PSF)()NA22.1λ=′∆r()NA61.0λ=′∆r()NAλ=′∆x()NA5.0λ=′∆xYou will see different authors giving different definitions.Rayleigh in his original paper (1879) noted the issue of noiseand warned that the definition of “just–resolvable” pointsis system– or application –dependentTwo point objects are “just resolvablejust resolvable” (limited by diffraction only)if they are separated by:MIT 2.71/2.710 Optics11/23/05 wk12-b-22Also affecting resolution: aberrationsAll our calculations have assumed “geometrically perfect”systems, i.e. we calculated the wave–optics behavior ofsystems which, in the paraxial geometrical optics approximationwould have imaged a point object onto a perfect point image.The effect of aberrations (calculated with non–paraxial geometricaloptics) is to blur the “geometrically perfect” image; includingthe effects of diffraction causes additional blur.geometrical optics description12MIT 2.71/2.710 Optics11/23/05 wk12-b-23Lens aberrationsbarrelpincushionMIT 2.71/2.710 Optics11/23/05 wk12-b-24Effect of aberrations on resolution2sx,max–2sx,maxH~1“diffractiondiffraction––limitedlimited”(aberration–free) 1D MTF2sx,max–2sx,maxH~11D MTF with aberrationsℑFouriertransformℑFouriertransformdiffraction–limited 1D PSF(sinc2)somethingwiderwave optics picture13MIT 2.71/2.710 Optics11/23/05 wk12-b-25Typical result of optical design(FoV)field of viewof the systemMTF is neardiffraction–limitednear the centerof the fieldMTF degradestowards thefield edgesshiftshiftvariantvariantopticalopticalsystemsystemMIT 2.71/2.710 Optics11/23/05 wk12-b-26The limits of our approximations• Real–life MTFs include aberration effects, whereas our analysis has been