Microscopy. Lecture 3: Physical optics of widefield microscopes Herbert Gross. Winter term

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1 Microscopy Lecture 3: Physical optics of widefield microscopes --9 Herbert Gross Winter term

2 Preliminary time schedule No Date Main subject Detailed topics Lecturer 5.. Optical system of a microscope I overview, general setup, binoculars, objective lenses, performance and types of lenses, tube optics Gross.. Optical system of a microscope II Physical optics of widefield microscopes Performance assessment 5.. Fourier optical description Etendue, pupil, telecentricity, confocal systems, illumination setups, Köhler principle, fluorescence systems and TIRF, adjustment of objective lenses Point spread function, high-na-effects, apodization, defocussing, index mismatch, coherence, partial coherent imaging Wave aberrations and Zernikes, Strehl ratio, point resolution, sine condition, optical transfer function, conoscopic observation, isoplantism, straylight and ghost images, thermal degradation, measuring of system quality basic concepts, -point-resolution (Rayleigh, Sparrow), Frequency-based resolution (Abbe), CTF and Born Approximation Gross Gross Gross Heintzmann Methods, DIC Rytov approximation, a comment on holography, Ptychography, DIC Heintzmann Multibeam illumination, Cofocal coherent, Incoherent processes (Fluorescence, Raman), Imaging of scatter OTF for incoherent light, Missing cone problem, imaging of a fluorescent plane, incoherent confocal OTF/PSF Heintzmann Incoherent emission to improve resolution Fluorescence, Structured illumination, Image based identification of experimental parameters, image reconstruction Heintzmann 9.. The quantum world in microscopy Photons, Poisson distribution, squeezed light, antibunching, Ghost imaging Wicker 7.. Deconvolution Building a forward model and inverting it based on statistics Wicker 7.. Nonlinear sample response STED, NLSIM, Rabi the information view Wicker two-photon cross sections, pulsed excitation, propagation of ultrashort pulses, (image 4.. Nonlinear microscopy formation in 3D), nonlinear scattering, SHG/THG - symmetry properties Heisterkamp 3.. Raman-CARS microscopy 4 8. Tissue optics and imaging principle, origin of CARS signale, four wave mixing, phase matching conditions, epi/forward CARS, SRS. Tissue optics, scattering&aberrations, optical clearing,optical tomography, lightsheet/ultramicroscopy Heisterkamp Heisterkamp Optical coherence tomography principle, interferometry, time-domain, frequency domain. Heisterkamp

3 Contents of 3rd Lecture 3. Point spread function. High-NA effects 3. Apodization 4. Defocussing 5. Index mismatch 6. Coherence 7. Partial coherent imaging

4 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 wave image plane object point truncated spherical wave point spread function object plane x =. / NA

5 intensity 3 Physical optics of widefield microscopes Perfect Point Spread Function Circular homogeneous illuminated Aperture: intensity distribution transversal: Airy scale:. D Airy NA axial: sinc n scale R E NA Resolution transversal better than axial: x < z,,8,6,4, vertical lateral, J v Iu I, v I v u / v sin u / 4, I u / 4 Scaled coordinates according to Wolf : axial : u = p z n / NA transversal : v = p x / NA Ref: M. Kempe

6 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 Circular pupil ring pupil, dipol, quadrupole 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 far field conditions near field conditions linear emission/excitation non linear methods

7 Perfect Lateral Point Spread Function: Airy Airy function : Perfect point spread function for several assumptions Distribution of intensity: p r J NA ( ) I r p r NA Normalized transverse coordinate x p ar R akr krsin u' ak sin ' R I(r) D Airy / r Airy diameter: distance between the two zero points, diameter of first dark ring D Airy.976 n' sin u'

8 Perfect Axial Point Spread Function Axial distribution of intensity Corresponds to defocus Normalized axial coordinate p NA z u z 4 Scale for depth of focus: Rayleigh length I(z) z sin sin u / 4 I( z) I Io z u / 4 R E nsin u n NA Zero crossing points: equidistant and symmetric, distance of zeros around image plane 4R E z/p R E z = R E 4R E

9 Defocussed Perfect Point Spread Function Perfect point spread function with defocus Representation with constant energy: extreme large dynamic changes z = -R E z = -R E focus z = +R E z = +R E normalized intensity I max = 5.% I max = 9.8% I max = 4% constant energy

10 Point Spread Function with Aberrations Zernike coefficients c in Spherical aberration Astigmatism Coma Spherical aberration, circular symmetry Astigmatism, split of two azimuths Coma, asymmetric c =. c =.3 c =.5 c =.7 c =.

11 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

12 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(x p,y p ), 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

13 Point Spread Function with Apodization Apodisation of the pupil: I(w). Homogeneous. Gaussian 3. Bessel Psf in focus: different decrease of the focal Airy Bessel Gauss intensity for larger radii. Encircled energy: same behaviour E(w) w Airy Bessel Gauss 3 4 w

14 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 E z 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 c 4 alone

15 Axial Magnification at High NA Paraxial case: Relation between lateral and axial magnification m z z' z n' m n Systems with sine condition fulfilled: pupil must have spherical shape Transfer between entrance and exit pupil m y p y' p entrance pupil n sin m n ' sin ' y p z sin exit pupil y p y' p ' z z'

16 High-NA Focusing Transfer from entrance to exit pupil in high-na:. Geometrical effect due to projection (photometry): apodization with A A s sin u s 4 NA n Tilt of field vector components A s r A r s r cos y E r E E x E dy y dy/cosu y y' e Ey e y s E r R s x u u y' x' entrance pupil exit pupil R image plane

17 High-NA Focusing Total apodization corresponds astigmatism Example calculations A x lin ( r, ) A ( r, ) s r s s r 4 r cos NA =.5 NA =.8 NA =.9 NA =.97 A(r).5 y x r

18 High-NA-Systems Systems with high numerical aperture: - distortion of geometry - nonlinear relationship between aperture angle and pupil height r p r p f g ( ) f g( ) This distortion corresponds to an apodization g( ) d g( ) a( ) sin d The function a() depends on the system correction Helmholtzcondition Paraboloidal r p Lagrange condition Herschel condition Sine condition f z

19 Vectorial Diffraction for high-na Vectorial representation of the diffraction integral according to Richards/Wolf E( r', z') E E E x y z E e iu 4sin / i I I cos i I sin I cos Auxiliary integrals o I( r', z) cos sin ( cos) J kr' sin e o I( r', z') cos sin J kr' sin e ikz'cos ikz 'cos d d General: axial and cross components of polarization o I( r', z') cos sin ( cos) J kr' sin e ikz 'cos d

20 Vectorial Diffraction for high-na Point spread function components I x, I y and I z Strong differences in the profiles dependent on defocussing Large differences in size (here: normalized) Example: initial polarization: x E x E y E z

21 High NA and Vectorial Diffraction Relative size of vectorial effects as a function of the numerical aperture Characteristic size of errors: I / I o error axial lateral axial lateral NA

22 High-NA-Systeme Vectorial diffraction integral in Fraunhofer representation ip z' n' sin u' x p y p E r F P x y e ' p, p n' sin u' x y n' sin n' sin u' y u' y y n' sin u' x y n' sin p p p p p p p u' y p All components x,y,z must be considered In particular due to the large bending angle an axial component E Z occurs

23 High-NA Point Spread Function Comparison of Psf intensity profiles for different models as afunction of defocussing error of model NA =.98 low-na paraxial scalar high-na vectorial NA paraxial high-na scalar high-na vectorial

24 Depth of Focus: Diffraction Consideration Normalized axial intensity for uniform pupil amplitude I( u) I sin u u r Decrease of intensity onto 8%: z.493 nsin u diff R u focal plane depth of focus Scaling measure: Rayleigh length - geometrical optical definition depth of focus: R E R u n' sin u' n' NA beam caustic.8 I(z) z - Gaussian beams: similar formula R u n' p o intensity at r = -R u +R u z

25 Microscope Objective Lens: Defocusing Defocusing in high-na lenses Geometrical: change of NA.5 sin u eff f tan uo f d f d tan uo starting planes pupil intra focal focal extra focal.4 d / f = -. d d.3 d / f = d / f = -.5 d / f =.5 d / f = -.3 d / f =.3 d / f = -. d / f =. d / f =. sinu o Relative change factor sin u sin u eff o -.4 d / f =

26 Microscope Objective Lens: Defocusing Real systems: 5 m defocussing in the object focussed 5 m defocussing. Macroscopic change of ray path. Pupil shrinks down macroscopic changes in the ray path 3. Vignetting surface index changes, sharp bending of curve.9.8 NA front surface.6.4 rear stop tube lens z in Ru

27 Microscope Objective Lens: Defocussing Change of performance for variation of the object distance Strehl increased for shorter object distance Strehl.95.9 long object distance nominal magnification short object distance magnification m

28 Microscope Objective Lens: Index Mismatch Objective lens with immersion 3 materials : Immersion (I), cover glass (C) and sample (S) Refraction law: NA n I cos n I C cos C n S cos S Problems by index mismatches with sample points deep inside immersion first lens cover glass probe medium enlarged picture of the ray caustic Strong spherical marginal focus aberrations for high-na n M paraxial focus n CG

29 Microscope Objective Lens: Index Mismatch Calculation of spherical aberration: comparison of optical path length with ideal case ( object on back side of glass) immersion liquid n I defocussing with mismatch focussed on underside of cover glass I cover glass n C C t C t I t C t S sample medium n S t S S a d I I Spherical aberration: d I y y s C d C y d C d S S W t S n S n n I S ( NA/ n S ) / s Vanishing effect for n I =n S n I n C n S immersion liquid cover glass sample medium Independent from cover glass

30 Microscope Objective Lens: Defocusing Aberrations:. Zernike coefficients change approximately linear. Psf strongly disturbed z in Ru d= Ru d=3 Ru d=6 Ru d=9 Ru -3 3 r in D airy d= Ru d=5 Ru d=8 Ru d= Ru.5 c j in.4.3. c 4 d=4 Ru d=7 Ru d=3 Ru d=33 Ru. -. c 8 c d=36 Ru d=39 Ru d=4 Ru d=45 Ru R U z in [mm]

31 Microscope objective Lens: Defocusing Peak height of Psf : decreasing I max No longer symmetry between aberrations for object - image reversal.8.6 c j in 4.4 3,5 c. 3,5 solid lines : image side dashed lines : object side c t in Ru c 6 c 8,5,5,,4,6,8, NA

32 Microscope Objective Lens: Index Mismatch Change of aberration with depth t s, index ratio and aperture angle Psf strongly perturbed z in Ru 5 d= Ru d=4 Ru d=8 Ru d= Ru -3 3 r in D airy d=6 Ru d= Ru d=4 Ru d=8 Ru W / t S n C =. n S =.3 d=3 Ru d=36 Ru d=4 Ru d=44 Ru n C =.5 n S =.4 n C =.5 n S =.33 n C =. n S =.7 n C =.4 n S =.4 n C =.4 n S = S in d=48 Ru d=5 Ru d=56 Ru d=6 Ru

33 Microscope Objective Lens: Index Mismatch Point spread function:. Lateral : only scaling. Axial : strong decrease with ripple.8 I(r) a =...6 Ru.6 I(z).4.8 a =...6 Ru r in D airy z in Ru

34 Point Spread Function for Index Mismatch Focussing into an index-mismatched sample Strong degradation of the point spread function with increasing depth I(r,z).5 3 r in r airy z in R u

35 Coherence in Optics Statistical effect in wave optic: start phase of radiating light sources are only partially coupled Partial coherence: no rigid coupling of the phase by superposition of waves Constructive interference perturbed, contrast reduced Mathematical description: Averagedcorrelation between the field E at different locations and times: Coherence function G Reduction of coherence:. Separation of wave trains with finite spectral bandwidth. Optical path differences for extended source areas 3. Time averaging by moved components Limiting cases:. Coherence: rigid phase coupling, quasi monochromatic, wave trains of infinite length. Incoherence: no correlation, light source with independent radiating point like molecules

36 Coherence Function Coherence function: Correlation x of statistical fields (complex) * r, r, ) E( r, t ) E ( r, t) ( for identical locations : intensity r r r ( r, r) I( r) t x E(x ) E(x ) x x z normalized: degree of coherence ( ) ( r, r, ) ( r, r, ) I( r ) I( r ) In interferometric setup, the amount of describes the visibility V Distinction:. spatial coherence, path length differences and transverse distance of points. time-related coherence due to spectral bandwidth and finite length of wave trains

37 Double Slit Experiment of Young First realization: change of slit distance D Second realization: change of coherence parameter s of the source light source screen with slits detector z V D x z x D

38 Double Slit Experiment of Young Partial coherent illumination of a double pinhole/double slit Variation of the size of the source by coherence parameter s Decreasing contrast with growing s Example: pinhole diameter D ph = D airy / distance of pinholes D = 4D airy s = s =.5 s =.5 s =.3 s =.35 s =.4

39 Spatial Coherence Area of coherence / transverse coherence length: Non-vanishing correlation at two points with distance L c : Correlation of phase due to common area on source observation area P L c ( r, r ) Radiation out of a coherence cell of extension L c guarantees finite contrast domain of coherence r P O r The lateral coherence length changes during propagation: starting plane receiving plane spatial coherence grows with increasing propagation distance common area

40 Spatial Coherence Incoherent source with diameter D = a Receiver plane indistance z Cone of observation / a Source is coherent in the distance z a / Transverse length of coherence (zeros of -function) L c z a

41 Partial Coherent Imaging Complete description of an optical system:. Light source. Illumination system, amplitude response h ill 3. Transmission object 4. Observation / imaging system with amplitude response h obs illumination field x s, y s object plane x p, y p image plane x i, y i source illumination system observation system pupil P I s h ill I i I o h obs sensor

42 Coherence Parameter Finite size of source : aperture cone with angle u ill Observation system: aperture angle u obs Definition of coherence parameter s: Ratio of numerical apertures Limiting cases: coherent s = u ill << u obs s sin u sin u ill obs incoherent s = u ill >> u obs source object x o, y o lens image x i, y i u ill u obs illumination observation

43 Coherence Parameter Heuristic explanation of the coherence parameter in a system:. coherent: Psf of illumination large in relation to the observation coherent illumination extended source small stop of condenser condenser object objective lens Psf of observation inside psf of illumination. incoherent: Psf of illumination small in comparison to the observation incoherent illumination extended source large stop of condenser Psf of observation contains several illumination psfs

44 Partial Coherence Simulation of partial coherent illumination: Finite size of light source Corresponding finite size of illuminated area in aperture plane Every point in this area is considered to emit independent (incoherent) Off-axis point in aperture plane generates an inclined plane wave in the object Angular spectrum illumination of the object describes partial coherence Estimated sampling of illumination points: Ls s arcsin f c y s.6 n sin aperture plane condenser f c object plane source extension L s s angular spectrum of object illumination y s size of coherence cell

45 Fourier Model of Image Formation Superposition of sourve into coherent beams of several source points Coherent propagation of source point contributions Incoherent summation of all point contributions in the image plane Finite size of source: object illuminated by different directions illumination aperture condenser object plane objective lens pupil tube lens image plane source point S x a f c fo f t x o x p x i chief ray object mask A(x o ) diffracted light pupil mask P(x p ) direct light aberrations x o,w(x o ) E a (x a ) FFT E o (x o ) CTF c (x o ) A(x o ) E o (x o ) FFT E p (x p ) E' p (x p ) CTF o (x p ) P(x p ) CTF t (x t ) FFT E i (x i )

46 Partial Coherent Imaging Image intensity:. Correlation of two points in the object. Integration over all points in the incoherent light source * * * xi Isxs hobsxi, xo, xs hobsxi, xo, xs h ill xo, xs hillxo, xs Oxo O xo dxo dxo dxs Ii light source object plane pupil image y s y o y p y i x s x o x p xi z o (x o,x o ) ' o (x o,x o ) P(x p ) h ill (x L,x ) h obs (x,x') I s (x s ) O(x I i (x i ) ) Simplification: thin object, transmission does not change for moderate inclination angles I i * * xi oxo, xo hobsxi, xo hobsxi, xo O xo O xo dxo dxo