21.21. FourierFourier OpticsOptics OBJECT SPECTRUM Fourier Transformation Time domain Temporal frequency [ ft : 1/sec ] Space domain Spatial frequency [ fX : 1/m ] Fourier transform 과 Optics가 무슨 관련이 있는가? Spatial Frequency 란 무엇인가? Fourier Optics 는 어떤 유용성이 있는가? FourierFourier Transforms Transforms in in space-domain space-domain and and time-domain time-domain Object in Time Object in Space ft() g ( ) eit d Object f ()x g ()keikx dk gftedt() ()it Spectrum g()k f ()xeikx dx 2 f Angular Frequency t k 2 fS wave number 2 k Frequency 11 fS m Temporal Frequency Spatial Frequency FourierFourier TransformsTransforms Fourier Transform pair : One dimension Fourier Transform pair : Two dimensions FraunhoferFraunhofer diffraction diffraction andand FourierFourier TransformTransform Spectrum plane Fraunhofer approximation X 2 X kkX rr00 Spatial angular YY2 frequency kkY rr00 The Fraunhofer pattern EP is the Fourier transform of Es ! Remind!Remind!Diffraction Diffraction underunder paraxialparaxial approx.approx. PropertiesProperties ofof 1D1D FTFT PropertiesProperties ofof 1D1D FTFT -- scaling scaling -- There is an inverse scaling relation between functions and their transforms PropertiesProperties ofof 1D1D FTFT -- translation translation (shifting)(shifting) -- Shifting in the time -domain leads to phase delay in the frequency –domain (no shift in frequency -domain), so FT amplitude is unaltered. PropertiesProperties ofof 1D1D FTFT -- modulation modulation -- Modulation in the time -domain leads to frequency shifting in the frequency-domain PropertiesProperties ofof 1D1D FTFT -- addition addition ofof twotwo shiftsshifts -- Cosine modulation of the amplitude In Fourier optics, this represents the interference for two-slit diffraction PropertiesProperties ofof 1D1D FTFT -- convolution convolution theoremtheorem -- Definition of convolution integral : g(t) h(t) g( )h(t )d Convolution theorem Fg(t) h(t) G( f )H ( f ) Convolution in the time -domain leads to multiplication in the frequency -domain PropertiesProperties ofof 1D1D FTFT -- correlation correlation theoremtheorem -- Definition of correlation integral : g(t) h(t) g( )h( t)d Correlation theorem Fg(t) h(t) G( f )H * ( f ) Fg(t) h(t) g( )h( t)de j2ft dt d g( ) h( t)e j2ft dt d g( ) h(T )e j2fT e j2f dT ; T (τ t) j2f * * g( )e d H (f ) G( f )H ( f ) Correlation in the time -domain leads to multiplication in the frequency -domain Convolution Correlation g(t) h(t) g( )h(t )d g(t)h(t) g()h( t)d g(t) h(t) t t -T/2 T/2 -T/2 T/2 g()h(-t) g() h(-) 0 t V(t) g()h(-) t -T 0 T g()h(t-) V(t) 0 t V(ti) ti : ~ V (ti ) g( )h(ti )d t -T T 0 ti ti PropertiesProperties ofof 1D1D FTFT -- Parseval’s Parseval’s theorem theorem -- Area under the absolute value squared of a function is equal to area under the absolute value squared of its transform SomeSome frequentlyfrequently usedused functionsfunctions SomeSome frequentlyfrequently usedused functionsfunctions OpticsOptics andand 2D2D FourierFourier transformtransform 1 g(,)x y =F G (ffxy , ) G (ff xy , )exp2( j xfyf x y )df xydf Superposition of plane waves g(,,)x y zG (ff , )exp2( j xfyf )df df e jkz xy x y xy g(x,y ;z=z0) OpticalOptical spectrumspectrum analysisanalysis --Fourier Fourier transformationtransformation byby lenslens -- X kk X f Y kk Y f Example of Optical Fourier Transform HowHow cancan aa convexconvex lenslens performperform thethe FTFT E(xi ) E0 P(x ) i xi sini i The phase at P(xi) is leading that at the origin by 2 x 1 jk(sini )xi j xi f0 HowHow cancan aa convexconvex lenslens performperform thethe FTFT fo fo FourierFourier TransformTransform withwith LensesLenses back focal plane Input placed Input placed against lens behind lens Input placed in front of lens 이 경우에 대해 자세히 알아보자 FourierFourier transformingtransforming propertyproperty ofof aa convexconvex lenslens The input placed in front of the lens Input Output U (u,v) Ul(x,y) f kd22 Ajexp 1 u 2 ff 2 U u,,exp U x y j xu y dxdy fl jf f A 2 U u,,exp U x y j xu y dxdy If d = f, fl jf f Exact Fourier transform ! Fourier and Inverse Fourier Transformation by Lenses Ff(,xy f ) Ff(,x fy ) 4-f System and Optical Filtering Example of Spatial Filters Band-pass filtering Example of Spatial Filters Low-pass filtering High-pass filtering Band-pass filtering Example of Spatial Filters Example of Spatial Filters : Correlator System evaluation using the transfer function Fourier transform spectroscopy Fourier transform spectroscopy.