Microscopy Lecture 3: Physical optics of widefield microscopes 2012-10-29 Herbert Gross Winter term 2012 www.iap.uni-jena.de 3 Physical optics of widefield microscopes 2 Preliminary time schedule No Date Main subject Detailed topics Lecturer overview, general setup, binoculars, objective lenses, performance and types of lenses, 1 15.10. Optical system of a microscope I tube optics Gross Etendue, pupil, telecentricity, confocal systems, illumination setups, Köhler principle, 2 22.10. Optical system of a microscope II fluorescence systems and TIRF, adjustment of objective lenses Gross Physical optics of widefield Point spread function, high-NA-effects, apodization, defocussing, index mismatch, 3 29.10. Gross microscopes coherence, partial coherent imaging Wave aberrations and Zernikes, Strehl ratio, point resolution, sine condition, optical 4 05.11. Performance assessment transfer function, conoscopic observation, isoplantism, straylight and ghost images, Gross thermal degradation, measuring of system quality basic concepts, 2-point-resolution (Rayleigh, Sparrow), Frequency-based resolution (Abbe), 5 12.11. Fourier optical description CTF and Born Approximation Heintzmann 6 19.11. Methods, DIC Rytov approximation, a comment on holography, Ptychography, DIC Heintzmann Multibeam illumination, Cofocal coherent, Incoherent processes (Fluorescence, Raman), 7 26.11. Imaging of scatter OTF for incoherent light, Missing cone problem, imaging of a fluorescent plane, incoherent Heintzmann confocal OTF/PSF Fluorescence, Structured illumination, Image based identification of experimental 8 03.12. Incoherent emission to improve resolution parameters, image reconstruction Heintzmann 9 10.12. The quantum world in microscopy Photons, Poisson distribution, squeezed light, antibunching, Ghost imaging Wicker 10 17.12. Deconvolution Building a forward model and inverting it based on statistics Wicker 11 07.01. Nonlinear sample response STED, NLSIM, Rabi the information view Wicker two-photon cross sections, pulsed excitation, propagation of ultrashort pulses, (image 12 14.01. Nonlinear microscopy formation in 3D), nonlinear scattering, SHG/THG - symmetry properties Heisterkamp principle, origin of CARS signale, four wave mixing, phase matching conditions, epi/forward 13 21.01. Raman-CARS microscopy CARS, SRS. Heisterkamp Tissue optics, scattering&aberrations, optical clearing,Optical tomography, light- 14 28.01 Tissue optics and imaging sheet/ultramicroscopy Heisterkamp 15 04.02. Optical coherence tomography principle, interferometry, time-domain, frequency domain. Heisterkamp 3 Physical optics of widefield microscopes 3 Contents of 3rd Lecture 1. Point spread function 2. High-NA effects 3. Apodization 4. Defocussing 5. Index mismatch 6. Coherence 7. Partial coherent imaging 3 Physical optics of widefield microscopes Diffraction at the System Aperture . Self luminous points: emission of spherical waves . Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave results in a finite angle light cone . In the image space: uncomplete constructive interference of partial waves, the image point is spreaded . The optical systems works as a low pass filter spherical image wave plane truncated object spherical point wave point spread function object plane x = 1.22 / NA 3 Physical optics of widefield microscopes Perfect Point Spread Function Circular homogeneous illuminated Aperture: intensity distribution 1,0 . transversal: Airy vertical 1.22 scale: D 0,8 lateral Airy NA . axial: sinc n 0,6 scale R E NA2 intensity 0,4 . Resolution transversal better than axial: x < z 0,2 0,0 -25 -20 -15 -10 -5 0 5 10 15 20 25 u / v 2 2 2J1v sinu / 4 I0,v I0 Iu,0 I0 v u / 4 Scaled coordinates according to Wolf : axial : u = 2 p z n / NA2 transversal : v = 2 p x / NA Ref: M. Kempe 3 Physical optics of widefield microscopes Abbe Resolution and Assumptions . Abbe resolution with scaling of /NA: assumptions for this estimation and possible changes . A resolution beyond the Abbe limit is only possible with violating of certain assumptions Assumption Resolution enhancement 1 Circular pupil ring pupil, dipol, quadrupole 2 Perfect correction complex pupil masks 3 homogeneous illumination dipol, quadrupole 4 Illumination incoherent partial coherent illumination 5 no polarization special radiale polarization 6 Scalar approximation 7 stationary in time scanning, moving gratings 8 quasi monochromatic 9 circular symmetry oblique illumination 10 far field conditions near field conditions 11 linear emission/excitation non linear methods 3 Physical optics of widefield microscopes Perfect Lateral Point Spread Function: Airy . Airy function : I(r) Perfect point spread function for 1 several assumptions 0.9 0.8 . Distribution of intensity: 0.7 0.6 2 0.5 2pr 2 J1 NA 0.4 I(r) 0.3 2pr NA 0.2 0.1 0 r . Normalized transverse coordinate 2 4 6 8 10 12 14 16 18 20 DAiry / 2 2par akr x krsin u' ak sin' R R . Airy diameter: distance between the 1.21976 two zero points, DAiry diameter of first dark ring n'sin u' 3 Physical optics of widefield microscopes Perfect Axial Point Spread Function . Axial distribution of intensity 2 2 sinz sin u / 4 Corresponds to defocus I(z) I0 Io z u / 4 . Normalized axial coordinate I(z) 1 p NA2 u z z 0.9 2 4 0.8 . Scale for depth of focus: 0.7 0.6 Rayleigh length 0.5 n 0.4 RE 2 2 nsin u NA 0.3 . Zero crossing points: 0.2 equidistant and symmetric, 0.1 distance of zeros around image 0 -4 -3 -2 -1 0 1 2 3 4 z/p plane 4RE RE z = 2RE 4RE 3 Physical optics of widefield microscopes Defocussed Perfect Point Spread Function . Perfect point spread function with defocus . Representation with constant energy: extreme large dynamic changes z = -2RE z = -1RE focus z = +1RE z = +2RE normalized intensity Imax = 5.1% Imax = 9.8% Imax = 42% constant energy 3 Physical optics of widefield microscopes Point Spread Function with Aberrations Spherical aberration Astigmatism Coma . Zernike coefficients c in . Spherical aberration, c = 0.2 circular symmetry . Astigmatism, split of two azimuths c = 0.3 . Coma, asymmetric c = 0.5 c = 0.7 c = 1.0 3 Physical optics of widefield microscopes Point Spread Function with Spherical Aberration . Intensity distribution I(r,z) for spherical aberration . Asymmetry of intensity around the image plane . Usually no zero points on axis intra focal 3 Physical optics of widefield microscopes Point Spread Function with Apodization . Apodization: Non-uniform illumination of the pupil amplitude . Modified point spread function: Weighting of the elementary wavelets by amplitude distribution A(xp,yp), Redistribution of interference . System with wave aberrations: Complicated weighting of field superposition For edge decrease of apodization: residual aberrations at the edge have decreased weighting and have reduced perturbation . Definition of numerical aperture angle no longer exact possible . For continuous decreased intensity towards the edge: No zeros of the PSF intensity (example: gaussian profile) . Modified definition of Strehl ratio with weighting function necessary 3 Physical optics of widefield microscopes Point Spread Function with Apodization . Apodisation of the pupil: I(w) 1. Homogeneous 1 Airy Bessel 2. Gaussian 0.8 Gauss 3. Bessel 0.6 . Psf in focus: 0.4 different decrease of the focal intensity for larger radii 0.2 . Encircled energy: 0 w -2 -1 0 1 2 3 same behaviour E(w) 1 0.8 0.6 0.4 Airy Bessel 0.2 Gauss 0 w 1 2 3 4 3 Physical optics of widefield microscopes High-NA-Systems General features: . Beam profiles and point spread function complicated . No axial symmetry around image plane . No shift-invariance in z-direction . Vectorial effects: axial components Ez and mixing of lateral x-y-field components, therefore polarization is important linear polarized input field breaks symmetry . Apodization of the pupil, depending on correction . Spherical shape of the pupil: defocussing is not described by Zernike c4 alone 3 Physical optics of widefield microscopes Axial Magnification at High NA Paraxial case: z' n' . Relation between lateral and axial magnification m m2 z z n n sin . Systems with sine condition fulfilled: m pupil must have spherical shape n' sin' y p z sin . Transfer between entrance and exit pupil y m p y' p entrance exit pupil pupil yp y'p ' z z' 3 Physical optics of widefield microscopes High-NA Focusing . Transfer from entrance to exit pupil in high-NA: 1. Geometrical effect due to projection y (photometry): apodization y' dy dy/cosu 1 A A 0 4 2 2 with 1 s r u NA R s sinu n y y y' E e . Tilt of field vector components Ey ey sE s x x' r xE r 2 2 E A0 1 1 s r A u 2 2 2 E 1 1 s r cos2 entrance exit image pupil pupil plane R 3 Physical optics of widefield microscopes High-NA Focusing . Total apodization 2 2 2 2 corresponds astigmatism 1 1 s r 1 1 s r cos2 Axlin (r,) A0 (r,) 2 4 1 s2r2 . Example calculations NA = 0.5 NA = 0.8 NA = 0.9 NA = 0.97 A(r) 2 2 2 2 1.5 1.5 1.5 1.5 y 1 1 1 1 x 0.5 0.5 0.5 0.5 0 0 0 0 r -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 3 Physical optics of widefield microscopes High-NA-Systems . Systems with high numerical aperture: - distortion of geometry r f g ( ) f g ( ) - nonlinear relationship between aperture angle p 1 1 1 2 2 2 and pupil height rp . This distortion corresponds to an apodization Helmholtz- r condition p g() d g() a( ) sin d Paraboloidal Lagrange condition . The function a() depends on the system correction Herschel condition Sine condition z f 3 Physical optics of widefield microscopes Vectorial Diffraction for high-NA .