MIT 2.71/2.710 Optics11/07/05 wk10-a-1The ideal thin lens as a Fourier transform engineMIT 2.71/2.710 Optics11/07/05 wk10-a-2Fresnel diffraction()()().ddexp),(2exp1;,22inoutyxlyyxxiyxglililyxg⎭⎬⎫⎩⎨⎧−′+−′⎭⎬⎫⎩⎨⎧=′′∫∫λπλπλThe diffracted field is the convolution convolution of the transparency with a spherical waveQ: how can we “undo” the convolution optically?ReminderReminderxyarbitrary lx´y´),(outyxg′′()yxg ,incoherentplane-waveilluminationMIT 2.71/2.710 Optics11/07/05 wk10-a-3Fraunhofer diffractionxyl→∞x´y´),(outyxg′′()yxg ,in()yxlyylxxiyxglyxg dd2- exp, );,(inout⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛′+⎟⎠⎞⎜⎝⎛′∝′′∫λλπThe “far-field” (i.e. the diffraction pattern at a largelongitudinal distance l equals the Fourier transformof the original transparencycalculated at spatial frequencieslyflxfyxλλ′=′= Q: is there another optical element who can perform a Fourier transformation without having to go too far (to ∞ ) ?ReminderReminderMIT 2.71/2.710 Optics11/07/05 wk10-a-4The thin lens (geometrical optics)Ray bending is proportionalproportionalto the distanceto the distance from the axisobject at ∞(plane wave)point object at finite distance(spherical wave)f (focal length)MIT 2.71/2.710 Optics11/07/05 wk10-a-5The thin lens (wave optics)incoming wavefronta(x,y)outgoing wavefronta(x,y) t(x,y)eiφ(x,y)(thin transparency approximation)MIT 2.71/2.710 Optics11/07/05 wk10-a-6The thin lens transmission function∆0∆(x,y)()()()() ()()()()⎭⎬⎫⎩⎨⎧+⎟⎟⎠⎞⎜⎜⎝⎛−−−⎭⎬⎫⎩⎨⎧∆≈⎭⎬⎫⎩⎨⎧∆−+∆=⎟⎟⎠⎞⎜⎜⎝⎛−+−∆≈∆⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡+−−+⎟⎟⎠⎞⎜⎜⎝⎛⎥⎦⎤⎢⎣⎡+−−−∆≈∆⎟⎟⎠⎞⎜⎜⎝⎛+−−+⎟⎟⎠⎞⎜⎜⎝⎛+−−−∆=∆21112exp2exp,,122exp,112,211211,1111,22210lens0lens21220222212210222212210yxRRniniyxayxniyxaRRyxyxRyxRRyxRyxRyxRRyxRyxλπλπλπλπMIT 2.71/2.710 Optics11/07/05 wk10-a-7The thin lens transmission function∆0∆(x,y)()()length focal theis 1111 whereexp2exp,21220lens⎟⎟⎠⎞⎜⎜⎝⎛−−≡⎭⎬⎫⎩⎨⎧+−⎭⎬⎫⎩⎨⎧∆≈RRnffyxiniyxaλπλπthis constant-phase term can be omittedMIT 2.71/2.710 Optics11/07/05 wk10-a-8Example: plane wave through lensplane wave: exp{i2πu0x}angle θ0, sp. freq. u0≈ θ0/λ()⎭⎬⎫⎩⎨⎧+−=fyxiyxaλπ22lensexp, lens,MIT 2.71/2.710 Optics11/07/05 wk10-a-9() { }(){}()⎭⎬⎫⎩⎨⎧+−−=⎭⎬⎫⎩⎨⎧+−=++fyfuxifuiyxafyxixuiyxaλλπλπλππ22020220expexp,exp2exp, :lensafter wavefront Example: plane wave through lensfuλ0ignorespherical wave,converging off–axisback focal planefMIT 2.71/2.710 Optics11/07/05 wk10-a-10Example: spherical wave through lens()()() { }⎭⎬⎫⎩⎨⎧+−∆=⎭⎬⎫⎩⎨⎧++⎭⎬⎫⎩⎨⎧=−fyxiniyxafyxxifiyxafλππλπλπ220lens220exp2exp,:functionion transmisslensexp2exp, :) distance propagated (has wavespherical0xspherical wave,diverging off–axisfront focal planefMIT 2.71/2.710 Optics11/07/05 wk10-a-11Example: spherical wave through lens() () ()⎭⎬⎫⎩⎨⎧++⎟⎠⎞⎜⎝⎛+∆=×=−+fxxifxifniyxayxayxaλπλπλπ0200lens22exp,,,lensafter wavefront ignorespherical wave,diverging off–axisfront focal planefx0 angleat ≈θplane wave0xMIT 2.71/2.710 Optics11/07/05 wk10-a-12Diffraction at the back focal planefzthintransparencyg(x,y)thinlensback focal planediffraction patterngf(x”,y”)xx’ x”MIT 2.71/2.710 Optics11/07/05 wk10-a-13Diffraction at the back focal planefzxx’ x”1D calculation1D calculation() ()()() ()() ()()xfxxixgxgfxixgxgxzxxixgxg′⎭⎬⎫⎩⎨⎧′−′′′=′′⎭⎬⎫⎩⎨⎧′−′=′⎭⎬⎫⎩⎨⎧−′=′∫∫+−+−d expexpd exp2lensf2lenslens2lensλπλπλπField before lensField after lensField at back f.p.g(x,y)gf(x”,y”)MIT 2.71/2.710 Optics11/07/05 wk10-a-14Diffraction at the back focal planefzxx’ x”1D calculation1D calculation() ()xfxxixgfzfxixg d 2exp 1exp 2f∫⎭⎬⎫⎩⎨⎧′′−⎭⎬⎫⎩⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−′′=′′λπλπ() () d d2exp, 1exp ,22fyxfyyxxiyxgfzfyxiyxg∫∫⎭⎬⎫⎩⎨⎧′′+′′−⎭⎬⎫⎩⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−′′+′′=′′′′λπλπ2D version2D versiong(x,y)gf(x”,y”)MIT 2.71/2.710 Optics11/07/05 wk10-a-15Diffraction at the back focal planefzxx’ x”() , 1exp , 22f⎟⎟⎠⎞⎜⎜⎝⎛′′′′⎭⎬⎫⎩⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−′′+′′=′′′′∴fyfxGfzfyxiyxgλλλπ() () d d2exp, 1exp ,22fyxfyyxxiyxgfzfyxiyxg∫∫⎭⎬⎫⎩⎨⎧′′+′′−⎭⎬⎫⎩⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−′′+′′=′′′′λπλπsphericalwave-frontg(x,y)gf(x”,y”)Fourier transformof g(x,y)MIT 2.71/2.710 Optics11/07/05 wk10-a-16Fraunhofer diffraction vis-á-vis a lensxyl→∞x´y´),(outyxg′′()yxg ,in()∫∫⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛′+⎟⎠⎞⎜⎝⎛′∝′′yxlyylxx-iyxglyxg dd2 exp, );,(inoutλλπxyfx´y´),(outyxg′′()yxg ,inf()∫∫⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛′+⎟⎟⎠⎞⎜⎜⎝⎛′∝′′yxfyyfxx-iyxgfyxg dd2 exp, );,(inoutλλπMIT 2.71/2.710 Optics11/07/05 wk10-a-17Spherical – plane wave dualityxyfx´y´),(outyxg′′()yxg ,in()∫∫⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎦⎤⎢⎣⎡′⎟⎟⎠⎞⎜⎜⎝⎛−+′⎟⎟⎠⎞⎜⎜⎝⎛−∝′′yxyfyxfxiyxgyxg dd2 exp, ),(inoutλλπpoint source at (x,y)amplitude gin(x,y)⎟⎟⎠⎞⎜⎜⎝⎛−−fyfxλλ, towardsplane wave orientedeach output coordinate(x’,y’) receives ... ... a superposition ...... of plane wavescorresponding topoint sources in the objectMIT 2.71/2.710 Optics11/07/05 wk10-a-18Spherical – plane wave dualityxyfx´y´),(outyxg′′()yxg ,ina plane wave departingfrom the transparencyat angle (θx, θy) has amplitudeequal to the Fourier coefficientat frequency (θx/λ, θy /λ) of gin(x,y)() ()()ffffyxyxθθλλθλλθ,, towards =⎟⎟⎠⎞⎜⎜⎝⎛××produces a spherical wave converging()∫∫⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛′+⎟⎟⎠⎞⎜⎜⎝⎛′∝′′yxfyyfxx-iyxgyxg dd2 exp, ),(inoutλλπeach output coordinate(x’,y’) receives amplitude equalto that of the correspondingFourier componentMIT 2.71/2.710 Optics11/07/05 wk10-a-19Conclusions• When a thin