Physical Optics. Lecture 4: Quality criteria and resolution Herbert Gross.

Transcription

1 Physical Optics Lecture 4: Quality criteria and resolution Herbert Gross

2 Physical Optics: Content No Date Subject Ref Detailed Content Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation, interferometry Diffraction G Slit, grating, diffraction integral, diffraction in optical systems, point spread function, aberrations Fourier optics G Plane wave expansion, resolution, image formation, transfer function, phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point G resolution resolution, criteria, contrast, axial resolution, CTF Photon optics K Energy, momentum, time-energy uncertainty, photon statistics, fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Coherence K Temporal and spatial coherence, Young setup, propagation of coherence, speckle, OCT-principle Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence, components Laser K Atomic transitions, principle, resonators, modes, laser types, Q-switch, pulses, power Nonlinear optics K Basics of nonlinear optics, optical susceptibility, nd and 3rd order effects, CARS microscopy, photon imaging PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping, confocal PSF Scattering L Introduction, surface scattering in systems, volume scattering models, calculation schemes, tissue models, Mie Scattering Gaussian beams G Basic description, propagation through optical systems, aberrations Generalized beams G Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy beams, applications in superresolution microscopy Miscellaneous G Coatings, diffractive optics, fibers K = Kempe G = Gross L = Lu

3 3 Contents Introduction Geometrical aberrations Wave aberrations Rayleigh and Marechal criteria Strehl ratio and other PSF criteria -point resolution Resolution and Contrast Edge criteria Laser beam quality Axial resolution Coherent transfer function

4 Light Properties and Quality Criteria Basic properties of light spectral monochromatic polychromatic linear polarized unpolarized polarization coherent incoherent coherence geometry collimated divergent/convergent

5 5 Performance Criteria - Overview Geometrical optical criteria: 1. Aberrations. Spot diagrams 3. Uniformity illumination irradiance Wave front: 1. Zernike or other coefficients. PV- and rms-value Point spread function: 1. Strehl ratio. Diameters, nd order moments, curtosis, threshold-width Resolution and contrast 1. -point-resolution. Contrast, line resolution, modulation depth 3. Edge image gradient Other: 1. Encircled energy. Fidelity, correlation, sharpness, structural content 3. M

6 6 Performance Criteria Overview Representations Quantitative numbers Applications Advantages Limitations Problems Longitudinal aberrations scaling on Rayleigh unit spherical aberration astigmatism axial chromatical field curvature not useful in the field not defined for afocal Geometrical model Transverse aberration curves Spot diagrams scaling on Airy diameter rms pv scaling on Airy diameters camera lenses simple direct analysis possible 1 curve per field point to be re-defined for afocal any illustrative analysis complicated Wave aberrations rms pv Zernike decomposition any direct measurable scaling on wavelength all orders separated only one wavelength only one field point normalization radius of Zernikes Diffraction model Point spread function Modulation transfer function Strehl ratio scaling on Airy diameter Hopkins number microscopy astronomy diffraction limited camera lenses lithography projection lenses direct relation to resolution easy white light formulation direct analysis possible easy white light formulation computational problems for large aberrations computational problems for large aberrations analysis complicated

7 7 Specifications 1. Functional specification Related to application, valid for the complete system. Plays the major role of a basic agreement between customer and development. Here two levels must be distinguished: 1.1 Basic data 1. Image quality. Manufacturing specification Declaration of several points of the engineering approach to clearify the technological and business goals.1 Business specification to ensure an economic development. Engineering and technology specifications, especially interface data 3. Component specifications Warranaty of reacing the theoretical goals in practice. Main concept of, all complicated relationships and dependencies of subsystems and components. 4. Assembly specification All parts of assembly and interfaces between mechanical and optical components Signal and image processing software, electronical and digital recording systems

8 8 Gaussian Moment Spot Spot pattern with transverse aberrations x j and y j 1. centroid xs 1 x j ys 1 y j N N. nd order moment M 3. diameter Generalized: Rays with weighting factor g j : corresponds to apodization M G j x x y y r 1 N G j S j S j D M G Worst case estimation: size of surrounding rectangle D x =x max, D y = y max j 1 r g x x y y N G j j j S j S

9 9 Spot Diagram Variation of field and color Scaling of size: 1. Airy diameter (small circle). nd moment circle (larger circle, scales with wavelength) 3. surrounding rectangle 486 nm 546 nm 656 nm axis field zone full field

10 10 Best Focal Plane Geometrical spot diagram Depends on wavelength and field position Best compromise: not trivial axis field zone field z 1 = -100 m z = -50 m z 3 = 0 z 4 = +50 m z 5 = +100 m

11 11 Wave Aberration Definition of the peak valley value Reference sphere corresponds to perfect imaging Rms-value is more relevant for performance evaluation pv-value of wave aberration wave aberration image plane phase front exit aperture reference sphere

12 1 Wave Aberrations Mean root square of wave front error W rms Normalization: size of pupil area A ExP W dxdy W Worst case / peak-valley wave front error pv 1 A x, y W x, y Generalized for apodized pupils (non-uniform illumination) ExP W x, y W x y W max W, W rms 1 A max p p min p p p p mean ( w) x, y W x, y W x y I ( w) ExP p p p p mean p, ExP p p p dx p dx dy p p dy p

13 13 Criteria of Rayleigh and Marechal Rayleigh criterion: 1. maximum of wave aberration: W pv < l/4. beginning of destructive interference of partial waves 3. limit for being diffraction limited (definition) 4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil 5. different limiting values for aberration shapes and definitions (Seidel, Zernike,...) Marechal criterion: 1. Rayleigh crierion corresponds to W rms < l/14 in case of defocus Rayleigh W rms l 19 l l 14. generalization of W rms < l/14 for all shapes of wave fronts 3. corresponds to Strehl ratio D s < 0.80 (in case of defocus) 4. more useful as PV-criterion of Rayleigh

14 14 Rayleigh Criterion The Rayleigh criterion W PV l 4 gives individual maximum aberrations coefficients, depends on the form of the wave a) optimal constructive interference b) reduced constructive interference due to phase aberrations c) reduced effect of phase error by apodization and lower energetic weighting d) start of destructive interference for 90 or l/4 phase aberration begin of negative z-component Examples: aberration type coefficient defocus Seidel a defocus Zernike c spherical aberration Seidel a spherical aberration Zernike c astigmatism Seidel a 0. 5 astigmatism Zernike c coma Seidel a coma Zernike c

15 Strehl Ratio Important citerion for diffraction limited systems: Strehl ratio (Strehl definition) Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity ) ( real) iw ( x, y) I PSF 0,0 A( x, y) e dxdy DS ( ideal DS I PSF 0,0 A( x, y) dxdy D S takes values between D S = 1 is perfect I( x ) 1 Critical in use: the complete information is reduced to only one number The criterion is useful for 'good' systems with values D s > 0.5 distribution broadened peak reduced Strehl ratio ideal, without aberrations real with aberrations r

16 16 Psf with Aberrations Psf for some low oder Zernike coefficients The coefficients are changed between c j = l The peak intensities are renormalized trefoil coma 5. order astigmatism 5. order spherical 5. order c = 0.0 c = 0.1 c = 0. c = 0.3 c = 0.4 c = 0.5 c = 0.7 coma astigmatism spherical defocus

17 Point Spread Function for Coma Aberration The PSF can be observed very sensitive In a well constructed system, 5-7 diffraction rings are observalble with by visual inspection In the case of coma, the asymmetry of the pattern is in particular sensitive The 1st diffraction ring is influenced very sensitive, a Zernike coefficient of l/30 can be detected c 31 = 0.03 l c 31 = 0.06 l c 31 = 0.09 l c 31 = 0.15 l

18 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

19 19 Quality Criteria for Point Spread Function Criteria for measuring the degradation of the point spread function: a) Strehl ratio b) Standard deviation c) Light in the bucket d) Equivalent width SR / D s STDEV LIB EW e) Second moment f) Threshold width g) Correlation width h) Width enclosed area SM FWHM Ref P=50% WEA CW

20 0 Point Resolution According to Abbe Transverse resolution of an image: - Detection of object details / fine structures - basic formula of Abbe x k Fundamental dependence of the resolution from: 1. wavelength. numerical aperture angle 3. refractive index 4. prefactor, depends on geometry, coherence, polarization, illumination,... l n sin Basic possibilities to increase resolution: 1. shorter wavelength (DUV lithography). higher aperture angle (expensive, 75 in microscopy) 3. higher index (immersion) 4. special polarization, optimal partial coherence,... Assumptions for the validity of the formula: 1. no evanescent waves (no near field effects). no non-linear effects (-photon)

21 1 Lateral and Axial Resolution Intensity distributions lateral axial Ref: U. Kubitschek

22 Incoherent -point Resolution Criterions Visual resolution limit: Good contrast visibility V = 6 % : 0.83 l x nsinu D Airy Total resolution: Coincidence of neighbouring zero points of the Airy distributions: V = 1 x D Airy 1. l nsinu Extremly conservative criterion Contrast limit: V = 0 : Intensity I = 1 between peaks 0.51 l x nsinu D Airy

23 Incoherent -Point Resolution: Rayleigh Criterion Rayleigh criterion for -point resolution Maximum of Psf coincides with zeros of neighbouring Psf x 1 D Airy 0.61l n sinu Contrast: V = 0.15 Decrease of intensity between peaks I = I I(x) 1 sum of PSF PSF 1 PSF x / r airy

24 4 Incoherent -Point-Resolution: Sparrow Criterion Criterion of Sparrow: vanishing derivative in the center between two point intensity distribution, corresponds to vanishing contrast d I( x) d x x0 0 Modified formula x Sparrow l D n sin u x Usually needs a priory information Applicable also for non-airy distributions Used in astronomy Rayleigh Airy I(x) x / r airy

25 -Point Resolution Distance of two neighboring object points Distance x scales with l / sinu Different resolution criteria for visibility / contrast V x = 1.l/ sinu total V = 1 x = 0.68l/ sinu visual V = 0.6 x = 0.61l/ sinu Rayleigh V = 0.15 x = 0.474l/ sinu Sparrow V = 0

26 6 -Point Resolution Intensity distributions below 10 % for points with different x (scaled on Airy) x =.0 x = 1. x = 1.0 x = 0.83 x = 0.61 x = x = x = 0.5

27 Incoherent Resolution: Dependence on NA Microscopical resolution as a function of the numerical aperture

28 Image Contrast Image processing: contrast enhancement Ref: T. Sievers

29 Contrast and Resolution original straylight 15% straylight 30% straylight 50% original 56 x 56 blurr 3 pixel blurr 6 pixel blurr 9 pixel

30 Microscope Resolution with Immersion Imaging of a Chromium mask with 15 nm pitch Imaging without / with water immersion Enhancement of resolution and contrast 150x/0.9 air 00x/1. water immersion Lens (Leica) Ref: W. Osten

31 31 Resolution Test Chart: Siemens Star a. original b. good system c. defocus d. spherical e. astigmatism f. coma

32 3 Hopkins Factor Resolution/contrast criterion: Ratio of contrasts with/without aberrations for one selected spatial frequency g MTF ( real) gmtf ( v) ( v) ( ideal) g ( v) MTF 1 g MTF Real systems: Choice of several application relevant frequencies e.g. photographic lens: 10 Lp/mm, 0 Lp/mm, 40 Lp/mm 0.5 real ideal g MTF ideal real g MTF 0 n

33 33 Real MTF Real MTF of system with residual aberrations: 1. contrast decreases with defocus. higher spatial frequencies have stronger decrease g MTF (z,f) g MTF z = Zernike coefficients: c 5 = 0.0 c 7 = 0.05 c 8 = 0.03 c 9 = 0.05 nn max = 0.05 nn max = 0.1 nn max = 0. nn max = 0.3 nn max = 0.4 nn max = 0.5 nn max = 0.6 nn max = 0.7 nn max = z = 0.1 R u z = 0. R u z = 0.3 R u z = 1.0 R u z = 0.5 R -0.5 u n z in R U

34 34 Modulation Transfer Function Photographic lenses with different performance Objektiv 1 f/ 3.5 Objektiv Lens 1 f/3.5 Lens MTF [%] bei 10, 0, 40 Lp/mm... tan sag max. MTF Bildhöhe [mm] max. MTF Image height 10 c/mm 0 c/mm 40 c/mm

35 35 Resolution: Loss of Information Blurred imaging: - limiting case - information extractable Blurred imaging: - information is lost - what s the time?

36 Edge and Line Imaging Edge and line imaging: - low contrast for large blurr - central part: high image sharpness Typical anisotropic behavior: sagittal/tangential orientation I LSF line good contrast medium contrast low contrast x I ESF edge x

37 Edge Width Definition of Thomas Practical definition of an edge width according to Thomas Identical integrated areas on left/right side defines position of the edge x x lkb RKb 1 I 0 1 I 0 0 I ( x) I ( x) real ideal I ( x) I ( x) real ideal 0 dx dx I esf 1 Determination of the two rectangles with the same area on both sides 0.75 x 75% Width: average of both retangle widths x Kb x lkb x rkb 0.50 x 5% 0.5 x x left x right x width

38 Quality of Laser Beams: Moments Conventional criteria of imaging systems are nor useful for laser beams: 1. significant apodization. no imaging application 3. status of coherence may be complicated Description of the complex fields by moments of second order: 1. spatial moments of intensity profile second moments describes beam width third moment describes asymmetry. angular moment of the direction distribution second moment describes the divergence Alternative descriptions of impuls: 1. angle u x. spatial frequency n x 3. transverse wavenumer k x Mixed moments: description of twist effects x u x m m x l v x x m E( x, y) E( x, y) m E(, ) k k E(, ) x o dxdy dxdy dd dd

39 Quality of Laser Beams: M Characterizing beam quality M M x l w x x w x x Special case: definition in waist plane M w l x ox x Properties of M : 1. Gaussian beam TEM00: M = 1 Smallest possible value. Paraxial optical systems: M remains constant for propagation 3. Real beams: M > 1 describes the decrease in quality and focussability relative to a gaussian beam Reasons for degradation of beam quality: 1. intensity profile. phase perturbation 3. finite degree of coherence Incoherent mixture of modes: additive composition of M General beams: components and mixed terms x y M 1 M M M xy

40 40 Depth of Focus: Geometrical Spot spreading in focus: diameter Detector spatial resolution D Depth of focus: < D Axial interval of sharpness. calculated by geometrical optics object plane image plane z p p' z geo entrance pupil system exit pupil z' geo

41 41 Perfect Point Spread Function Circular homogeneous illuminated Aperture: intensity distribution transversal: Airy scale: l 1. D Airy NA axial: sinc scale nl R E NA Resolution transversal better than axial: x < z intensity 1,0 0,8 0,6 0,4 0, vertical lateral 0, J1 v Iu 0 I 0, v I v 0 u / v sin u / 4, I u / 4 Scaled coordinates according to Wolf : axial : u = z n / l NA transversal : v = x / l NA 0

42 Depth of Focus: Diffraction Consideration 4 Normalized axial intensity for uniform pupil amplitude I( u) I 0 sin u u Decrease of intensity onto 80%: 1 z diff R u l n sin u 1 focal plane r depth of focus Scaling measure: Rayleigh length - geometrical optical definition depth of focus: 1R E R u l n' sin u' n' l NA beam caustic 0.8 I(z) 1 z - Gaussian beams: similar formula R u l n' o intensity at r = 0 -R u / 0 +R u / z

43 43 Depth of Focus Depth of focus depends on numerical aperture 1. Large aperture:. Small aperture: small depth of focus large depth of focus Ref: O. Bimber

44 Depth of focus for Annular Pupil 44 Ring pupil illumination Enlarged depth of focus Lateral resolution constant due to large angle incidence Can not be understood geometrically

45 Depth of focus for Annular Pupil 45 Farfield of a ring pupil: outer radius a a innen radius a i parameter Ring structure increases with Depth of focus increases z nsin Application: Telescope with central obscuration Intensity at focus a a l u 1 i a 1 I I(r) J1( x) J1( x) ( x) 1 x x = 0.01 = 0.5 = 0.35 = 0.50 = 0.70 r

46 Cubic Phase Plate (CCP) 46 Phase Mask with cubic polynomial shape cubic phase PSF MTF P( x) e 0 ix 3 für x 1 sonst Effect of mask: - depth of focus enlarged - Psf broadened, but nearly constant - Deconvolution possible Problems : - variable psf over field size - noise increased - finite chief ray angle - broadband spectrum in VIS - Imageartefacts y-section z x-section

47 Cubic Phase Plate: PSF and OTF 47 Conventional imaging System with cubic phase mask focus defocussed focus defocussed

48 EdoF: CPP for microscopic Imaging 48 Conventional microscopic image Image with phase mask with / without deconvolution Ref: E. Dowski

49 49 Ewald Sphere Assuming an object as grating with period L k obj L Scattering of a wave at the object with - conservation of energy k in k out - conservation of momentum The outgoing k-vector must be on a sphere: Ewald's sphere for possible scattered wave vectors k k k in obj out grating Ewald sphere k obj k out k out k in k obj k in

50 50 CTF in Microscopy Applicattion of Ewalds sphere on the plane wave vectors of: 1. illumination n i. object scattering n o 3. image signal n s Only signal vectors inside the numerical aperture cone are resolvable The principle defines the lateral and the axial resolution This defines the 3D coherent transfer function illumination/pupil object pupil image n i n o n s resolvable range

51 51 McCutchen Formula and Axial Resolution Imaging of a plane wave at a volume object x: minimum value resolution n: maximum interval Uncertainty relation: nx = 1 P(n x ) n z transverse pupil n x n z Radius of the Ewald sphere generalized 3D pupil: red area Transverse resolution due to Abbe l x v R sin n / l NA / n NA x n R l cap light cone n x n z n x axial pupil P(n z ) Axial resolution: - height of the cap of the cone - McCutchen formula 1 z v z n 1 R Rcos n / l 1 n l NA nl NA 1 1 sin n/l Ewald sphere

52 5 3D Transfer Function - Missing Cone Realistic case: finite numerical aperture illumination n i Blue cone: possible incoming wave direction due to illumination cone scattered wave 3D coherent transfer function: limited green area, that fulfills all conditions n s n obs n x n z transfer function object Missing cone: certain range of spatial axial spatial frequencies can not be seen in the image missing cone n x transfer function object n z

53 53 Coherent Transfer Function Definition of the coherent transfer function (CTF): similar as OTF, but FFT of complex field Coherent imaging: complex field in the image plane H CTF ( v) E psf ( r ) e irv dr E ima ( x', y', z') e ikz' 1 Eobj vx, vy, vz HCTF vx, vy, vz e l i xvx yv y ( v z 1 ) M l z dv x dv y dv z Offset 1/l: optical path length in thick object sn r 0.5 Special case of lens with pupil function P 1.0 H with v ( rotsym) CTF v, v z P v vz v z 0 l 1 sin 0 H CTF 0 ns z P(n P(s r ) n z0 s z0

54 54 3D Transfer Function Imaging as 3D scattering phenomen Only special spatial frequencies are allowed due to energy conservation and momentum preservation Green circle: supported spatial frequencies of the transmitted wave vector n z n obj = n s - n i n o-max n/l n i n x n s n obj n i backward forward Ewald sphere