Microscope Imaging. Colin Sheppard Nano- Physics Department Italian Ins:tute of Technology (IIT) Genoa, Italy

Transcription

1 Microscope Imaging Colin Sheppard Nano- Physics Department Italian Ins:tute of Technology (IIT) Genoa, Italy

2 Objec:ve lens Op:cal microscope Numerical aperture (n sin α) Air / oil immersion / water immersion Corrected for cover slip (No. 1 1 / 2 = 0.17mm) or not Corrected for infinity or not e.g 100X 1.4NA Oil 0.17/ Eyepiece Illumina:on system Condenser Aperture stop (diaphragm) Field stop

3 Aberra:on func:on In general we can expand Φ as a power series in r and also as series cos nθ: Φ = n,m a mn r m cos nϕ = a 00 + a 20 r 2 + a 40 r a 11 r cosϕ a 22 r 2 cos2ϕ + spherical aberration coma astigmatism focus term (cancels in focal plane) other aberrations m+n has to be even

4 Spherical aberra:on Spherical aberration, W 40 ρ 4 ρ = r / a (16 wavelengths) Marginal Plane of least confusion Paraxial (Gaussian) Paraxial Focus Marginal Focus Born & Wolf

5 Coma Coma, W 31 ρ 3 cos φ Born & Wolf

6 As:gma:sm, 2.7λ Astigmatism, W 22 ρ 2 cos 2φ Focal line Central plane Focal line Born & Wolf

7 Cri:cal illumina:on (source focused on to object) Oversimplified diagram! (Aperture stop in plane of condenser) Each point on object illuminated by a point on the source

8 Köhler illumina:on Each point on source illuminates whole object Each point on object illuminated by whole source

9 Köhler illumina:on f p f p f o f o source imaged onto aperture stop

10 Effec:ve source (incoherent)

11 Imaging in a microscope In a microscope, the spa:al coherence of the illumina:on depends on the aperture of the condenser lens rela:ve to that of the objec:ve. A small condenser aperture gives coherent illumina:on, whereas a very large aperture gives incoherent illumina:on. For intermediate values, we introduce the coherence ra:o S = 0, coherent S = 1, full, complete or matched S large, incoherent Born and Wolf

12 Rayleigh criterion Resolved Not resolved Bradbury, An introduction to the optical microscope

13 Rayleigh two point resolu:on 2 points are just resolved if the second point is placed on the first dark ring of the first. Separation is r 0 = 0.61 λ / (n sin α) Or half-separation is v 0 = 1.92 Then the ratio of the intensity midway to that at the points is 0.735

14 Two- point resolu:on

15 Resolu:on depends on coherence S = 0, coherent illumination S = 1, full or complete illumination S, incoherent illumination Fluorescence behaves as incoherent imaging Coherence parameter S = Resolution depends on coherence

16 Images of two points v 0 = 2.0 is close to the Rayleigh resolution for the incoherent case

17 Two- point resolu:on It is found that the resolu:on depends on the value of S. According to the generalized Rayleigh criterion, the points are just resolved when the ra:o of the intensity at the centre to that at the points is The normalized distance between two points when they are just resolved reaches a minimum when In practical microscopes we usually use Köhler illumination, L(S) which has the advantage that the illumination is more uniform, but the details of the image formation are identical. Born and Wolf coherent S

18 Perfect imaging Object iφ (x,y ) t(x, y) = a(x, y)e Perfect image is modulus (amplitude), real is phase, real No phase information in perfect image

19 Image forma:on (coherent) Add amplitudes of different parts of object. e.g. 2 points:

20 Coherent imaging

21 object 1st harmonic constant 3rd harmonic Fourier series for periodic func:on sum of first three terms = 2π/k 1/m

22 Fourier transforming property of a lens U(x) F{U(x)} f f Position, slope Slope, position

23 Abbe theory (coherent imaging)

24 Introduce object spectrum Abbe theory

25 Coherent transfer func:on For a partially coherent system C(m, n; p, q) does not separate

26 Incoherent imaging

27 2- D transfer func:ons

28 Image of a straight edge 1/3 S = 0.32 (nearly coherent) S = 1 (full illumination) Watrasiewicz, Optica Acta 12, 391 (1965)

29 Straight edge Image depends on coherence Slope is greater for small S, so greater precision for measurement - S = 0, slope =1/π = S = 1, slope = S, slope = Intensity at edge is - S = 0, 1/4 - S = 1, 1/3 - S, 1/2 Important for measuring (edge appears to be at 1/2) Fringes for small S

30 Par:ally coherent image forma:on Propagate mutual intensity through the system: Image intensity source pupil C(m,n;p,q) = transmission cross-coefficient (TCC) m, p are both spatial frequencies in x direction n, q are both spatial frequencies in y direction System and object separated. Although Hopkins propagated mutual intensity, he did not give mutual intensity of the image. Proc. R. Soc. Lond. A 217, 408 (1953)

31 Imaging in a par:ally- coherent microscope For non-periodic objects, replace sums by integrals: C = transmission cross-coefficient (TCC) object spectrum Conventional microscope: condenser objective Confocal microscope:

32 Generaliza:on of coherent imaging For partially coherent, C(m 1, n 1 ; m 2, n 2 ) does not separate

33 C(m, 0; p, 0) as area of overlap of three circles (conven:onal system) objective condenser

34 Transmission cross coefficient (TCC) C S = 1 S is coherence ratio (NA cond /NA obj ) J. Modern Optics 57, (2010)

35 S = 0 coherent C(m; p) (conven:onal) S = 1 S NB m, p are spatial frequencies both in the x direction full, complete, or matched illumination incoherent

36 Introduce central and difference coordinates Introduce central and difference coordinates Δm TCC Transmission cross-coefficient Area of overlap of source and 2 displaced pupils

37 Transmission cross coefficient (TCC) C C(m, Δm) Δm S = 1 S is coherence ratio (NA cond /NA obj ) J. Modern Optics 57, (2010)

38 WOTF C(m;0) and PGTF C(m;m) for conven:onal microscope Partially coherent imaging is complicated, but it becomes simpler for two special cases: Weak object (neglect interference of scattered light with scattered light) Slowly varying phase gradient C(m;0) C (m) 0.6 S = 0 C(m;m) C (m) 0.6 S = S = S = m m Weak object transfer function (WOTF) Phase gradient transfer function (PGTF) S is coherence ratio (NA cond /NA obj )

39 Weak object complex Weak object Spectrum B is skew-hermitian if b is imaginary

40 Weak object transfer func:on (WOTF) Weak object (b is complex) For even C: Weak object transfer function (WOTF) Phase imaged by imaginary part of C

41 Weak phase object An Hermi:an transfer func:on does not give contrast from a weak phase object Make pupil either o complex o asymmetric

42 Defocus Earliest method of phase contrast Like Zernike, based on changing the phase of the signal Only works for a weak object Contrast opposite for different defocus direc:ons Rela:ve condenser aperture S cannot be too large For a coherent system, S = 0, arg[p (ρ)] = uρ 2 /2, so arg[c (m)] = um 2 /2

43 Defocus WOTF, S = 0.01 (nearly coherent) Like cos or sin (ul 2 /2) l is radial spatial frequency, l = (m 2 +n 2 ) 1/2 Sheppard CJR Defocused transfer function for a partially coherent microscope, and application to phase retrieval J. Opt. Soc. Am. A, 21, (2004)

44 WOTF, S = 0.5 Real Imaginary Sheppard CJR Defocused transfer function for a partially coherent microscope, and application to phase retrieval J. Opt. Soc. Am. A, 21, (2004)

45 WOTF, S = 0.99 Real Imaginary (very weak) Sheppard CJR Defocused transfer function for a partially coherent microscope, and application to phase retrieval J. Opt. Soc. Am. A, 21, (2004)

46 Small defocus: analy:c expression Sheppard CJR Defocused transfer function for a partially coherent microscope, and application to phase retrieval J. Opt. Soc. Am. A, 21, (2004)

47 I(Δu) I( Δu) gives phase contrast image Parabolic for small l Sheppard CJR Defocused transfer function for a partially coherent microscope, and application to phase retrieval J. Opt. Soc. Am. A, 21, (2004)

48 Small defocus, inverse Laplacian Phase restored up to l = 1 S Sheppard CJR Defocused transfer function for a partially coherent microscope, and application to phase retrieval J. Opt. Soc. Am. A, 21, (2004)

49 Phase measurement using WOTF Kou SS, Waller L, Barbastathis G, Marquet P, Depeursinge C, Sheppard CJR Quantitative phase restoration by direct inversion using the optical transfer function, Opt. Lett. 36, (2011).