MIT 2.71/2.710 Optics11/09/05 wk10-b-1Today• Coherent image formation– space domain description: impulse response– spatial frequency domain description: coherent transfer functionImaging with coherent lightMIT 2.71/2.710 Optics11/09/05 wk10-b-2Coherent imaging with a telescopeMIT 2.71/2.710 Optics11/09/05 wk10-b-3The 4F system1f1f2f2fFourier transformrelationshipFourier transformrelationshipMIT 2.71/2.710 Optics11/09/05 wk10-b-4The 4F system1f1f2f2f(){}{}( )yxgyxg−−=ℑℑ,,Theorem:MIT 2.71/2.710 Optics11/09/05 wk10-b-5The 4F system1f1f2f2f()yxg ,1⎟⎟⎠⎞⎜⎜⎝⎛′′′′111,fyfxGλλ⎟⎟⎠⎞⎜⎜⎝⎛′−′− yffxffg21211,object planeFourier planeImage planeMIT 2.71/2.710 Optics11/09/05 wk10-b-6The 4F system1f1f2f2f()yxg ,1⎟⎟⎠⎞⎜⎜⎝⎛′′′′111,fyfxGλλ⎟⎟⎠⎞⎜⎜⎝⎛′−′− yffxffg21211,object planeFourier planeImage plane()yxssG ,1θxλθλθyyxxsssinsin==MIT 2.71/2.710 Optics11/09/05 wk10-b-7The 4F system with FP aperture1f1f2f2f()yxg ,1⎟⎠⎞⎜⎝⎛′′×⎟⎟⎠⎞⎜⎜⎝⎛′′′′RrfyfxG circ,111λλ()⎟⎟⎠⎞⎜⎜⎝⎛′−′−∗ yffxffhg21211, object planeFourier plane: aperture-limited Image plane: blurred(i.e. low-pass filtered)()vuG ,1θxMIT 2.71/2.710 Optics11/09/05 wk10-b-8Impulse response & transfer functionA point source at theinput plane ... ... results not in a point imagebut in a diffraction patternh(x’,y’) Point source at the origin ↔ delta function δ(x,y)h(x’,y’) is the inpulse response of the systemMore commonly, h(x’,y’) is called theCoherent Point Spread Function (Coherent PSF)MIT 2.71/2.710 Optics11/09/05 wk10-b-9Coherent 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(u ,v): aka pupil functionMIT 2.71/2.710 Optics11/09/05 wk10-b-10Transfer function & impulse response of circular apertureTransfer function:circular apertureImpulse response:Airy function⎟⎠⎞⎜⎝⎛′′Rrcirc⎟⎟⎠⎞⎜⎜⎝⎛′2jincfRrλ2RRf222.1λMIT 2.71/2.710 Optics11/09/05 wk10-b-11Coherent imaging as a linear, shift-invariant systemThin transparency()yxt ,()yxg ,1(≡plane wave spectrum)impulse responsetransfer function⎟⎟⎠⎞⎜⎜⎝⎛′=2jinc),(fRryxhλ()),(),(, 23yxhyxgyxg∗==′′()⎟⎟⎠⎞⎜⎜⎝⎛+=RvufvuH22circ,λFourier transformoutput amplitudeconvolutionmultiplicationFourier transformillumination()),( ,),( 12yxtyxgyxg==()vuG ,2),(),(),( 23vuHvuGvuG==Example: 4F system with circularcircular aperture @ Fourier planeMIT 2.71/2.710 Optics11/09/05 wk10-b-12Transfer function & impulse response of rectangular apertureaaf22λ⎟⎠⎞⎜⎝⎛′′⎟⎠⎞⎜⎝⎛′′byaxrectrect⎟⎟⎠⎞⎜⎜⎝⎛′⎟⎟⎠⎞⎜⎜⎝⎛′22sincsincfbyfaxλλMIT 2.71/2.710 Optics11/09/05 wk10-b-13Coherent imaging as a linear, shift-invariant systemThin transparency()yxt ,()yxg ,1(≡plane wave spectrum)impulse responsetransfer function⎟⎟⎠⎞⎜⎜⎝⎛′⎟⎟⎠⎞⎜⎜⎝⎛′=22sincsinc),(fbyfaxyxhλλ()),(),(, 23yxhyxgyxg∗==′′()⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛=bfvafuvuHλλrectrect,Fourier transformoutput amplitudeconvolutionmultiplicationFourier transformillumination()),( ,),( 12yxtyxgyxg==Example: 4F system with rectangularrectangular aperture @ Fourier plane()vuG ,2),(),(),( 23vuHvuGvuG==MIT 2.71/2.710 Optics11/09/05 wk10-b-14Aperture–limited spatial filtering1f1f2f2fobject plane:grating generatesone spatial frequencyFourier plane: aperture unlimitedImage plane:grating is imagedwith lateralde-magnification(all orders pass)x′′x′xMIT 2.71/2.710 Optics11/09/05 wk10-b-15x′′x′xAperture–limited spatial filtering1f1f2f2fobject plane:grating generatesone spatial frequencyFourier plane: aperture limitedImage plane:grating is not imaged;only 0thorder(DC component)surviving(some orders cut off)MIT 2.71/2.710 Optics11/09/05 wk10-b-16Spatial frequency clipping() ()[]() () ()()⎥⎦⎤⎢⎣⎡++−+=⇒+=00in0in2121212cos121uuuuuuGxuxgδδδπ() () ()()⎥⎦⎤⎢⎣⎡+′′+−′′+′′=′′− 0101f212121vfxufxxxgλδλδδfield before filterfield after filter() ()xxg′′=′′+δ21f() () ( )2121outout=′⇒= xguuGδfield at output(image plane)field after input transparencyMIT 2.71/2.710 Optics11/09/05 wk10-b-17Effect of spatial filteringOriginal object(sinusoidal spatial variation,i.e. grating)Frequency-filtered image(spatial variation blurred out,only average survives)Fourier plane filterwith circ-apertureMIT 2.71/2.710 Optics11/09/05 wk10-b-18Spatial frequency clippingobject planetransparencyFourier planecirc-aperture1f1fmonochromaticcoherent on-axisilluminationx′′xx′1f1ff1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/09/05 wk10-b-19Space-Fourier coordinate transformationsδx:pixel size∆x:field sizespacedomainspatial frequency domainδfx:frequencyresolutionfx,maxxfxfxx∆==21 21max,δδNyquistrelationships:MIT 2.71/2.710 Optics11/09/05 wk10-b-204F coordinate transformationsδx:pixel size∆x:field sizespacedomainFourierplaneδx”xfxxfx∆=′′=′′2 2maxλδδλNyquistrelationships:x”maxMIT 2.71/2.710 Optics11/09/05 wk10-b-21Spatial frequency clippingobject planetransparencyFourier planecirc-apertureImage planeobserved field1f1fmonochromaticcoherent on-axisilluminationx′′x′x1f1ff1=20cmλ=0.5µmMIT 2.71/2.710 Optics11/09/05 wk10-b-22x′′x′xFormation of the impulse response1f1f2f2fobject plane:pinhole generatesspherical waveFourier plane: circ-aperture limitedImage plane:Fourier transformof aperture,Airy pattern(plane wave is clipped)MIT 2.71/2.710 Optics11/09/05 wk10-b-23Low–pass filtering()()()()1,,inin=⇒= vuGyxyxgδδ()1,f=′′′′−yxgfield before filterfield after filter()⎟⎟⎠⎞⎜⎜⎝⎛′′+′′=′′′′+Ryxyxg22fcirc,()()⎟⎟⎠⎞⎜⎜⎝⎛′+′∝′′⇒⎟⎟⎠⎞⎜⎜⎝⎛+=fyxRyxgRvufvuGλλ22out22outjinc, circ,field at output(image plane)field after input transparency(Airy pattern)ℑMIT 2.71/2.710 Optics11/09/05 wk10-b-24Effect of spatial filteringOriginal object(small pinhole ⇔impulseimpulse,generating spherical wavepast the transparency)Impulse reponse(akapointpoint--spread functionspread function,original point has blurred toan Airy pattern, or