PH880 Topics in Physics

Transcription

1 PH880 Topics in Physics Modern Optical Imaging (Fall 2010)

2 Monday Fourier Optics Overview of week 3 Transmission function, Diffraction 4f telescopic system PSF, OTF Wednesday Conjugate Plane Bih Bright field microscopy Illumination

3 (Updated) Theorems f ( x ) F ( u ) g G h H 0. Linearity: a f ( x) + b g( x) a F( u) + b G( u) f ( x x ) F( u, v) e i2π u x 0 1. Shift theorem: modulation in F space 0 i2π u x 0 2. Modulation theorem: f ( xe ) Fu ( u) shift in F space 0 3. Scaling theorem: 1 u f ( ax) F( ) a a 4. Convolution theorem: g = f h G = F H ( f h)( x') f ( xhx ) ( ' xdx ) 5. Correlation theorem: * g = f h G = F H ( f h)( x') f ( x) h( x' + x) dx *

4 Convolution * =

5 (updated) Useful Fourier Transform pairs Space domain Fourier domain g( x) = δ ( x) Gu ( ) = 1 g( x) i2 u x e π Gx ( ) = δ ( u u) 0 = 0 g( x) = i x e π 2 Gu ( ) = i u e π 2 g ( x ) cos(2 ) ( ) 1 ( ) ( ) 2 1 Gu ( ) = ( u u0) ( u u0) 2i δ δ + = π u 0 x Gu = δ u u 0 + δ u + u 0 g( x) = sin(2 π u x) 0 g( x) =Δ( x) δ ( x n) Gu ( ) = Δ( u) n= g( x) = rect( x) Gu ( ) = sinc( u) g( r) = circ( r) Gr ( ) = jincr ( ') sin( π u ) π u J (2 π r') 1 r '

6 (updated) Useful Fourier Transform pairs Space domain Fourier domain g( x) = δ ( x) Gu ( ) = 1 Plane wave w/ s.f. u 0 g( x) i2 u x e π Gx ( ) = δ ( u u) 0 = 0 Point source shifted by u 0 (diverging) Spherical wave g( x) = i x e π 2 Gu ( ) = i u e π 2 (converging) spherical wave = π u 0 x Gu = δ u u 0 + δ u + u 0 g ( x ) cos(2 ) g( x) = sin(2 π u x) 0 g( x) =Δ( x) δ ( x n) Gu ( ) = Δ( u) n= g( x) = rect( x) Gu ( ) = sinc( u) ( ) 1 ( ) ( ) 2 1 Gu ( ) = ( u u0) ( u u0) 2i δ δ + g( r) = circ( r) Gr ( ) = jincr ( ') sin( π u ) π u J (2 π r') 1 r '

7 Summary: Diffraction (2D) x X X l l y y' y'' gin ( x, y ) gout ( x ) Fresnel diffraction gout ( x ) Fraunhofer diffraction 2 2 x y i g ( x, y ) G ( u, v) x y ') π + g ( x, y g ( x, x) e λl l l out in (diverging) Spherical wave out in u, l v = = λ λl

8 Thin lens can perform Fourier transform x X y l y'' gin (, xy ) x g ( x, y ) G ( u, v ) x y out in u =, v = λl λl gout ( x ) X y f f y'' gin (, xy ) gout ( x ) g ( x, y ) G ( u, v) x y out in u=, v= λ f λ f

9 Monday Fourier Optics Overview of week 3 Transmission function, Diffraction 4f telescopic system PSF, OTF Wednesday Conjugate Plane Bih Bright field microscopy Illumination

10 Example #2: sinusoidal grating x X f f Λ

11 Example #3: sinusoidal grating + beam block x x x f f f f Λ

12 x Example #4: NA revisited (2D) x x f f Λ f f

13 x Example #5: NA revisited (3D) x x f f 1 f f circ () r circ () r jincj (')( r ) J 1 ') ( ') (2 π r jinc r r ' 1

14 x 4f imaging system and Fourier Optics x x f 1 f 1 f f 2 2 f 1 ( xy, ) f 2 ( x, y ) f 3 ( x, y ) Fourier Transform Relationship x' y' f2( x, y ) = F1, λ f1 λ f1 Fourier Transform Relationship x y f3( x, y ) = F2, λ f2 λ f2 f 1 f 1 f 1 f3( x, y ) = f1 x, y f2 f2 f2 FT( FT( f( x)) = f( x)

15 4f imaging system and Fourier Optics x x x f 1 f 1 f f 2 2 Object Plane f ( xy, ) F Fourier Plane x' λ f, y' λ f 1 1 Image Plane f2 f2 f x, y f1 f1

16 4f imaging system and Fourier Optics x x x f 1 f 1 f f 2 2 Object Plane f ( xy, ) F Fourier Plane x' λ f, y' λ f 1 1 Image Plane f2 f2 f x, y f1 f1

17 Impulse response in imaging system x Object Plane Image Plane x Arbitrary Imaging System A point source at the input plane... does not focused into a point image. But into a diffracted pattern δ ( xy, ) h( x, y )

18 Impulse response in imaging system x Object Plane Image Plane x Arbitrary Imaging System A point source at the input plane... ( xy, ) does not focused into a point image. But into a diffracted pattern δ h( x, y ) Then, any arbitrary wave, f ( xy, ) will have an output as: f h * For LSI (Linear Shift Invariant) system

19 Impulse response in imaging system x Object Plane Image Plane x Arbitrary Imaging System A point source at the input plane... ( xy, ) does not focused into a point image. But into a diffracted pattern δ h( x, y ) (, y ) h x : Impulse response of the imaging system (Coherent) Point Spread Function For a given input f(x,y), output field is f Huv (, ) = FThxy ( (, )) h is called Amplitude Transfer Function

20 4f imaging system and Spatial Filtering x (, ) H x y x Fourier Plane after filter x f 1 f 1 Λ f f 1 1 x Object Plane ( xy, ) Spatial Filter Image Plane δ H( x, y ) h( x, y ) * sign ignored (i.e. image inversion ignored)

21 4f imaging system and Spatial Filtering x x' F λ f y', λ f 1 1 Fourier Plane before filter x x' y' F, H x, y λ f 1 λ f 1 Fourier Plane after filter ( ) x f 1 f 1 Λ f f 1 1 x Image Plane Object Plane Spatial Filter f ( xy, ) H( x, y ) f ( x, y ) h( x, y ) x ATF * PSF * sign ignored (i.e. image inversion ignored)

22 (spatial (p domain) Optical Imaging System (Coherent) Input field output field Convolution f ( xy, ) f ( xy, ) hxy (, ) Point Spread Function hxy (, ) Fourier Transform Fourier Transform (spatial frequency domain) Multiplication F ( uv, ) F ( uv, ) Huv (, ) Amplitude Modulation Function Huv (, )

23 Coherent vs. Incoherent λ 1 i 1 A e φ Coherent waves: Superposition field = sum of field i φ 1 1 f = Ae + Ae iφ 1 1 λ 2 i 1 i 1 φ φ i 2 A 2e φ I = f = Ae 1 + Ae 1 2 λs λs I 1 I 2 (temporally) incoherent waves: Superposition Intensity = sum of Intensities I = I1+ I2 Phases of the individual id wavelength vary randomly w.r.t. each other; need statistical optics to describe it

24 (spatial domain) Optical Imaging System (Incoherent) Input Intensity output intensity Convolution 2 I( x, y) = f( x, y) I( x, y) h ( x, y) Incoherent PSF 2 * h ( xy, ) = h ( xy, ) = h h Fourier Transform Fourier Transform Multiplication Iuv ˆ(, ) = Fuv (, ) Fuv (, ) Iuv ˆ(, ) Huv (, ) Optical Transfer Function Huv (, ) = Huv (, ) Huv (, ) (spatial frequency domain) Huv (, ) : Modulation Transfer Function

25 summary (spatial domain) (spatial frequency domain) h( ( xy, ) : (coherent) Point Spread Function (PSF) Huv (, ) : Amplitude Modulation Function (AMF) hxy (, ) = hxy (, ) = hh : (incoherent) PSF 2 * H ( uv, ) = H ( uv, ) H ( uv, ) Optical Transfer Function (OTF) Huv (, ) = Huv (, ) Huv (, ) Modulation Transfer Function (MTF)

26 Spatial filtering x x x f f f f Λ Original pattern Filter at Fourier plane Low pass filtered image (High s.f. are blocked out)

27 Spatial filtering Original pattern Fourier filter image plane * Images are adapted from

28 Original pattern Spatial filtering Fourier filter image plane Blocking DC components ~ blocking non scattered light e.g. dark field microscopy

29 Conjugate planes in 4f telescopic system f f g ( x, y ) g ( x, y ) = 2 f3 f3 Fourier Conjugate planes g 2 ( ', ') g 4 (, ) x x x X f 1 f 1 f f 2 2 f 3 f 3 g 1 ( x, y) g 3 ( x, y ) Imaging forming Conjugate planes f1 f1 g3( x, y ) = g1( x, y) f f 2 2

30 Conjugate plane: example

31 Monday Fourier Optics Overview of week 3 Transmission function, Diffraction 4f telescopic system PSF, OTF Wednesday Conjugate Plane Bih Bright field microscopy Illumination

32 Optical trains in bright field microscopy

33 Imaging formation in microscopy # = a b f 4f imaging syst.

34 Accomodation (focusing) in eyes = z z f = cm cm ~3cm = 25cm 3cm 2.68cm ~25cm ~3cm Comfortable viewing up to 25cm away from the cornea

35 Imaging formation in microscopy # = a b f 4f imaging syst.

36 Conjugate planes in bright field microscopy Aperture or illuminating Conjugate planes Imaging forming Conjugate planes

37 Conjugate planes in bright field microscopy

38 Objective lens

39 Objective lens Manufacturer Tube Lens Focal Length (Millimeters) Parfocal Distance (Millimeters) Thread Type Leica M25 Nikon M25 Olympus RMS Zeiss RMS Parfocal distance f 2 f 2 When you have a tube lens with ihcorrect focal llength, M l =f2/f1=60x

40 Immersion oil NA = nsinθ max No immersion oil (n=1) Oil immersion objective lens (n~1.51, why?) * Water immersion (n=1.33) or glycerol immersion (n=1.473) * Solid immersion lens (n~2 or above, see the reading list)

41 Tungsten halogen Light sources Arc Lamps

42 Köhler Illumination

43 MTF

44 3D PSF 2D PSF for various focal plane (i.e. various z position) 3D PSF z x y x

45 Deconvolution * z x BioTechniques 31: (November 2001)

46 Reading List (wk 3 day 2) 1. Davidson, M. W. and Fellers, T. J. Understanding conjugate planes and Köhler illumination. Nikon MicroscopyU Whitepaper 4 pages (2003). 2. Mansfield, S. M. and Kino, G.S.. Solid immersion microscopy, App. Phys. Lett. 72, 2778 (1998) 3. Agard, D. A. Optical Sectioning Microscopy: cellular architecture in three dimensions. Annual Review of Biophysics and Bioengineering 13: (1984). 4. Sibarita, J. B. Deconvolution microscopy. Advances in Biochemical Engineering and Biotechnology 95: (2005). 5. Carrington, W. A., Lynch, R. M., Moore, E. E. W., Isenberg, G., Fogarty, K. E. and Fay, F.S. Superresolution three dimensional images of fluorescence in cells with minimal light exposure. Science 268: (1995).