4: Fourier Optics. Lecture 4 Outline. ECE 4606 Undergraduate Optics Lab. Robert R. McLeod, University of Colorado

Transcription

1 Outline 4: Fourier Optics Introduction to Fourier Optics Two dimensional transforms Basic optical laout The coordinate sstem Detailed eample Isotropic low pass Isotropic high pass Low pass in just one dimension Phase contrast imaging Pedrotti 3, Chapter 21: Fourier Optics 45

2 Fourier optics Fourier transform pairs From Goodman, Introduction to Fourier Optics, p

3 Fourier optics Properties of 2D FTs j 2π ( f + f ) (, ) F( f, f ) e Definition f df df F(, ) f (, ) Linearit Scaling Shift Rotation Convolution f (, ) β g( ) ( ξ η) g( ξ η) (, 0 0 ) F( u, v) e j 2π θ { f ( ) } R θ { F( u, v) } R, ( f + f ) dd α f +, α F ( f, f ) + β G( f, f ) a b f, a b F( a f, b f ) f,, dξ dη F ( f, f ) G( f, f ) e j 2π ( f + ) 0 0 f Correlation ( ξ, η) g ( ξ η ) * * f, dξ dη F ( f, f ) G ( f, f ) 2 Parseval s thm f (, ) dd F( f, f ) 2 df df f (, )d Projection slice thm F(,0) Real function f ( ) Real F( f ) F ( ) Even/odd function f ( ) ± f ( ) Real F( ) Real Imaginar Each of these has a direct phsical analog with optics. 47 f f f

4 Fourier optics General spatial filtering 4F processing sstem: Collimate Object FT Filter Inverse FT Output f FT f FT f FT f FT Prepare input Input mask Take FT Filter mask 1 F ( ) f, F ( F, ) λ FT λ F FT (, ) G(, ) λ F FT λ F FT λ F FT λ F FT Take inverse FT ( ξ, η) g( ξ η) f, dξ dη Thus the 2D object f(,) has been filtered with the 2D filter with impulse response g(,). 48

5 Fourier optics Coordinate sstem F θ λ f λ 0 sinθ or sinθ λ 0 θ λ 0 F sinθ λ0 F λ Fλ f 0 So spatial frequenc f is related to coordinate b the scale factor F λ 0 49

6 Fourier optics Simple optical Fourier transforms F 100 mm Laser wavelength λ nm Amplitude cosine, aka diffraction grating λ 200µm 100, µm λ λ 2 200µm 282.8µm Rotate object b 45 o 100, µm 50

7 Eample Low pass, sharp cutoff REAL SPACE FOURIER SPACE Multiplied b Filter cutoff frequenc 1/500 µm -1 Filter cutoff position µm 100 mm Laser wavelength 632 nm µm pinhole All plots show amplitude of E Smoothed, but Gibbs ringing due to sharp filter edges 51

8 Eample Low pass, smooth cutoff REAL SPACE FOURIER SPACE Multiplied b Filter cutoff frequenc 1/500 µm -1 Filter cutoff position µm Edge smoothing 132 µm 100 mm Laser wavelength 632 nm Now just nicel smoothed 52

9 Eample High pass, narrowband REAL SPACE FOURIER SPACE Multiplied b Filter cutoff frequenc 1/1200 µm -1 Filter cutoff position 52.6 µm Edge smoothing 58.2 µm 100 mm Laser wavelength 632 nm µm dot Note sharp edges, darkening of large, uniform areas (~DC) Sharp filter used for clarit 53

10 Eample High pass, wideband REAL SPACE FOURIER SPACE Multiplied b Filter cutoff frequenc 1/300 µm -1 Filter cutoff position µm Edge smoothing 46.6 µm 100 mm Laser wavelength 632 nm Onl edges remain. Almost a line drawing 54

11 Eample Vertical low pass REAL SPACE FOURIER SPACE Multiplied b Filter cutoff frequenc 1/200 µm -1 Filter cutoff position 316 µm Edge smoothing 93.6 µm 100 mm Laser wavelength 632 nm 632 µm horiz. slit Horizontal lines at edges of ees gone Vertical lines above nose remain. 55

12 Eample Horizontal low pass REAL SPACE FOURIER SPACE Multiplied b Filter cutoff frequenc 1/200 µm -1 Filter cutoff position 316 µm Edge smoothing 93.6 µm 100 mm Laser wavelength 632 nm 632 µm vert. slit Horizontal lines at edges of ees remain. Vertical lines above nose gone. 56

13 Eample Simpler object Low-pass Original High-pass Low-pass, different cutoff in & 57

14 Phase contrast Phase contrast REAL SPACE FOURIER SPACE e Ansel jπ ma ( Ansel ) Multiplied b Filter cutoff frequenc 1/5000 µm -1 Filter cutoff position 12.6 µm 100 mm Laser wavelength 632 nm Knife edge Phase has become amplitude. Zernike won the 1953 Nobel in Phsics for this. 58