Principles & Practice of Light Microscopy 2

Transcription

1 Principles & Practice of Light Microscopy 2 Resolution Aberrations The Point Spread Function The Optical Transfer Function Spatial frequencies Edited by: Zvi Kam, Weizmann For Advance Light Microscopy course

2 RESOLUTION

3 Aperture and Resolution Objective Tube lens Intermediate image plane Diffraction spot on image plane = Point Spread Function Sample Back focal plane aperture

4 Aperture and Resolution Objective Tube lens Intermediate image plane Diffraction spot on image plane = Point Spread Function Sample Back focal plane aperture

5 Aperture and Resolution Objective Tube lens Intermediate image plane Diffraction spot on image plane = Point Spread Function Sample Back focal plane aperture

6 Aperture and Resolution Objective Tube lens Intermediate image plane Diffraction spot on image plane = Point Spread Function Sample α Back focal plane aperture Image resolution improves with Numerical Aperture (NA) NA = n sin(α) where: α = light gathering angle n = refractive index of sample

7 Back focal plane Resolution Ernst Abbe s argument (1873) Consider a striped sample a diffraction grating Objective lens Sample Condenser d! β Diffracted beams" d sin(β) = λ Smaller d" larger β Light source Consider first a point light source If β > α, only one spot makes it through no interference no image formed Resolution (smallest resolvable d):" dmin = λsample/sin(α) = λ/n sin(α) = λ/na

8 (Abbe s argument, continued) Now consider oblique illumination (an off-axis source point): One spot hopelessly lost, but two spots get through interference image formed! d! βout βin d [sin(βin) + sin(βout) ] = λ Two spots get through if" βout < α and βin < α. Resolution (smallest resolvable d) with incoherent illumination (all possible illumination directions): dmin = λ/(naobj + NAcondenser )" λ/2 NA if NAcondenser NAobj ( Filling the back focal plane ) "

9 Filling the back focal plane In trans-illumination microscopy, to get maximum resolution, the illumination must fill the back focal plane Back focal plane Objective Condenser α objective α condenser For the highest resolution, we need to have! α condenser α objective!! NA condenser NA objective" " with oil immersion objectives, we need an oil immersion condenser! Light source

10 Grade of correction The Condenser Tasks: Illuminate at all angles < α objective Concentrate light on the field of view for all objectives to be used NA Problem: Low mag objectives have large FOV, High mag objectives have large α (With 2X and 100x objectives we need (100/2) 2 = 2500 times more light than any objective uses!) Solutions: Compromise Exchangable condensers, #swing-out front lenses,

11 Aperture, Resolution & Contrast Can adjust the condenser NA with the aperture iris Imaging path Intermed. image Tube lens Illumination path Back aperture Objective Sample Condenser lens Aperture iris Field lens Field iris Collector Light source Q:#Don t we always want #it full open?? A:#No Why? Tradeoff: #resolution vs. contrast

12 Spatial frequencies & the Optical Transfer Function (OTF) 1" (Image" contrast)" OTF(k)! Resolution limit:" k max = 2 NA / λ k" (Spatial frequency," periods/meter)" Observed image Object

13 Resolution & Contrast vs. Illumination aperture Pupil appearance Contrast" NA condenser 0 NA condenser NA obj ( Coherent illumination ) (= Full aperture, incoherent illumination ) Resolution" Resolution limit:" k max = (NA objective + NA condenser )/ λ Increasing the illumination aperture increases resolution but decreases contrast

14 Definitions of Resolution 1" OTF(k)! Cutoff frequency! k max = 2 NA / λ As the OTF cutoff frequency 1/k max = 0.5 λ /NA k " As the Full Width at Half Max (FWHM) of the PSF FWHM λ /NA As the diameter of the Airy disk (first dark ring of the PSF) = Rayleigh criterion Airy disk diameter 0.61 λ /NA

15 Contrast-resolution link"

16 Rayliegh criterion: zero background I 2 / I 1 ~ 70% d=0.61λ / NA NA=1.4 d=0.22 µm!

17 What is the complex optics for?!! The practice of microscopy,! or:! The fight for contrast.

18 Aberrations They are the enemy

19 Aberrations: wave description Chromatic aberrations Longitudinal chr. Ab. Lateral chr. Ab. Curvature of field Distortion Wavefront aberrations Spherical aberration Astigmatism Coma

20 Wavefront Aberrations Ideal wavefront in the pupil Aberrated wavefront in the pupil

21 Wavefront Aberrations (piston) (tilt) Astigmatism Defocus Coma Trefoil Spherical ab. Secondary coma Secondary spherical ab.

22 PSF Aberrations (piston) (tilt) Astigmatism Defocus Coma Trefoil Spherical ab. Secondary coma Secondary spherical ab.

23 2D PSF for different defocus Z=+2µm" The 3D Point Spread Function (PSF) The image of a point object 3D PSF z" 2 Calculated Measured Z=0" y" Z=-2µm" x" x"

24 Spherical Aberration Point spread function 0 1 wave p p of primary spherical ab. z" x"

25 Spherical Aberration Point spread functions Ideal 1 wave of spherical ab z" x" x"

26 Spherical Aberration Point spread functions Ideal 1 wave of spherical ab y" x" x"

27 Sources of Spherical Aberration Design compromises Manufacturing tolerances Immersion fluid index error Temperature variation Cover slip thickness (high-na objectives except oil immersion) Correction collar setting Sample refractive index mismatch

28 Index Mismatch & Spherical Aberration objective n 1 Immersion fluid Cover glass n 2 Sample Focus at cover slip Focus into sample Spherical aberration unless n 2 = n 1

29 Index matching

30 Index Mismatch & Spherical Aberration z=0 µm n 1 =1.515 (oil) n 2 =1.44 (Vectashield) z=25 µm z=50 µm

31 Index Mismatch & Axial Scaling n 1 n 2 Mechanical focus step δz m Optical focus step δz o "z o # "z m n 2 n 1 If there is index mismatch, your z pixel size is not what you think

32 Getting Rid of Spherical Aberration Use correct and consistent cover slips Pick a room with stable temperature 20 C Use special immersion iof for 37 C live work Adjust correction collar If no collar, adjust immersion medium index Use an objective that is matched "to the mounting medium: For aqueous samples, use a #water immersion objective For fixed samples viewed with #oil immersion objectives, ideally use a #mounting medium with index For fixed samples in commercial media, #ideally use a glycerol immersion objective

33 Astigmatism Point spread function 1 wave p p of x-y astigmatism y" x"

34 Coma Point spread function 1 wave p p of y coma y" x"

35 Sources of Astigmatism & Coma Off-axis (edges of field of view) All objectives have some vignetting Present in the design You get what you pay for On-axis (center of field of view) Should be none, by symmetry. If they are there, they could be from: manufacturing or assembly tolerances dirt or abuse Misalignment (tilt, off-axis shift of something) bad downstream components (mirrors, dichroics, filters ) Air bubble in the immersion fluid or sample Tilted cover slip (dry and water-immersion high-na lenses)

36 More about Spatial frequencies & the Optical Transfer Function (OTF) For the record only, not for this course Attribute to Mats Gustafsson

37 The response to pure waves is well-defined by the Optical Transfer Function (OTF) 1" (Image" contrast)" OTF(k)! Resolution limit:" k max = 2 NA / λ k" (Spatial frequency," periods/meter)" Observed image Object

38 Think of Images as Sums of Waves or spatial frequency components one wave another wave (2 waves) Position" + = Intensity" (25 waves) (9855 waves) + ( ) = + ( ) =

39 Frequency Space To describe a wave, we need to specify its: Frequency (how many periods/meter?) Direction Amplitude (how strong is it?) Phase (where are the peaks & troughs?) Can describe it by a value at a point Distance from origin Direction from origin Magnitude of value Phase of value complex k y! direction k = (k x, k y ) " k x! period

40 Frequency Space and the Fourier Transform k y! k y! Fourier Transform k x! k x!

41 Properties of the Fourier Transform F (k) = # f (r)e 2"ik r dr Completeness: The Fourier Transform contains all the information" of the original image" Symmetry: The Fourier Transform of the Fourier Transform is the original image Fourier transform

42 The OTF and Imaging the PSF are Fourier transforms of each other True Object convolution PSF Observed Image?? = Fourier Transform OTF =

43 Why do we care? They are everywhere The convolution theorem: If then h(r) = (f g)(r),! h(k) = f(k) g(k)! Convolutions (f g)(r) = f(a) g(r-a) da! So what is a convolution, intuitively? Blurring Drag and stamp A convolution in real space becomes a product in frequency space & vice versa f! g! f g!! = Symmetry: g f = f g" y y y x! x = x

44 The Transfer Function Lives in Frequency Space OTF(k)! Object Observed image k " k y! Observable Region k x!

45 The 2D In-focus Optical Transfer Function (OTF) OTF(k)! OTF(k)! k " k y " k x " (Idealized calculations)

46 The 3D OTF 2D PSF 2D OTF 2D F.T. 3D PSF 3D OTF 3D? F.T.?

47 Values of the 3D OTF k x k z

48 3D Observable Region = OTF support = Region where the OTF is non-zero k y " k y " k z " k z " Why this weird region?

49 In free space Monochromatic light Monochromatic light has only one wavelength λ only one k = 1 / ι support of" A(k)" ky kz The light amplitude A(i)" has Fourier components only on a shell of radius 1 / λ (amplitude)" R=1/! support of" I (k)" (intensity)" k y k x R=2/ λ The intensity I(r) = A(r) 2" " In reciprocal space, this product A*A! becomes a convolution A A of the shell with itself, which is a sphere of twice the radius" " This sphere indicates all frequency components of the light intensity that can ever be non-zero "

50 Consider light from a camera pixel into the sample. What are the Fourier components of this light? Light enters the sample only from those directions allowed by the aperture angle α Only a bowl-shaped segment of the shell makes it through the objective: k y α k z support of A(k) The intensity now becomes a convolution of the bowl with itself, which has a a donut-shaped region of support. This is the OTF support: the set of sample frequencies that can be detected through the microscope k y " k z " support of I (k)

51 So what is the resolution? Radius" = 1 / λ sample = n / λ k x " α K xy max =" 2 n sin(α) / λ = 2 NA / λ" k z " Missing Cone of information K z max =" n (1-cos(α)) / λ"

52 So what is the resolution? Radius" = 1 / λ sample = n / λ k x " α Missing Cone of information k z " K xy max =" 2 n sin(α) / λ = 2 NA / λ" Lowering the NA Degrades the axial resolution faster than the lateral resolution But low axial resolution = long depth of field This is good, if 2D is enough K z max =" n (1-cos(α)) / λ"

53 So what is the resolution? Radius" = 1 / λ sample = n / λ k x " Example: a high-end objective α K xy max =" NA = 1.4" 2 n sin(α) / λ Missing Cone of information k z " = 2 NA / λ" K z max =" n (1-cos(α)) / λ" n=1.515" α = 67.5 " λ = 600 nm" " " Lateral (XY) resolution:" 1/ K xy max = 0.21 µm! " Axial (Z) resolution:" 1/ K z max = 0.64 µm"

54 Nomenclature Optical Transfer Function, OTF Complex value with amplitude and phase Contrast Transfer Function, CTF Modulation Transfer Function, MTF Same thing without the phase information