Design and Correction of optical Systems Part 10: Performance criteria 1 Summer term 01 Herbert Gross
Overview 1. Basics 01-04-18. Materials 01-04-5 3. Components 01-05-0 4. Paraxial optics 01-05-09 5. Properties of optical systems 01-05-16 6. Photometry 01-05-3 7. Geometrical aberrations 01-05-30 8. Wave optical aberrations 01-06-06 9. Fourier optical image formation 01-06-13 10. Performance criteria 1 01-06-0 11. Performance criteria 01-06-7 1. Correction of aberrations 1 01-07-04 13. Correction of aberrations 01-07-11 14. Optical system classification 01-07-18
3 Contents 10.1 Introduction 10. Geometrical criteria 10.3 Wave aberrations, Rayleigh and Marechal criteria 10.4 PSF criteria: Strehl ratio, moments 10.5 -point-resolution 10.6 Focussing 10.7 Miscellaneous
4 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
5 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 3. diameter j. nd order moment M r 1 N x x y y G j S j S j D M G j y max y r rms Generalized: Rays with weighting factor g j : corresponds to apodization M G 1 r g x x y y N G j j j S j S Worst case estimation: size of surrounding rectangle D x =x max, D y = y max x s,y s x max x
6 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
7 Wave Aberration Definition of the peak valley value pv-value of wave aberration wave aberration image plane phase front exit aperture reference sphere
8 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
9 Typical Variation of Wave Aberrations Microscopic objective lens Changes of rms value of wave aberration with 1. wavelength. field position Common practice: 1. diffraction limited on axis for main part of the spectrum. Requirements relaxed in the outer field region 3. Requirement relaxed at the blue edge of the spectrum W rms [ ] 0.30 0.4 0.18 0.1 field zone Achroplan 40x0.65 field edge 0.06 diffraction limit on axis 0 0.480 0.56 0.644 [ m]
10 Spatial Frequency of Surface Perturbations Power spectral density of the perturbation Three typical frequency ranges, scaled by diameter D log A Four oscillation of the polishing machine limiting line slope m = -1.5...-.5 long range low frequency figure Zernike mid frequency 1/D 1/D 40/D micro roughness 1/
11 Criteria of Rayleigh and Marechal Rayleigh criterion: 1. maximum of wave aberration: W pv < /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 < /14 in case of defocus Rayleigh W rms 19 13.856 14. generalization of W rms < /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
1 PV and W rms -Values PV and W rms values for different definitions and shapes of the aberrated wavefront Due to mixing of lower orders in the definition of the Zernikes, the W rms usually is smaller in comparison to the corresponding Seidel definition
13 Rayleigh Criterion The Rayleigh criterion W PV 4 gives individual maximum aberrations coefficients, depends on the form of the wave Examples: aberration type coefficient defocus Seidel a 0 0. 5 defocus Zernike c 15 0 0. spherical aberration Seidel a 0. 40 5 spherical aberration astigmatism Seidel a 5 Zernike c 40 0. 167 0. astigmatism Zernike c 0. 15 coma Seidel a 0. 31 15 coma Zernike c 0. 31 15
14 Psf with Aberrations Psf for some low oder Zernike coefficients The coefficients are changed between c j = 0...0.7 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
15 Quality Criteria for Point Spread Function Criteria for measuring the degradation of the point spread function: 1. Strehl ratio. width/threshold diameter 3. second moment of intensity distribution 4. area equivalent width 5. correlation with perfect PSF 6.power in the bucket
16 Beam Spot Diameter Definitions Threshold diameter of intensity e.g. 50% (FWhM) nd moment (rms) Energy content at threshold (PiB) Petermann definition Diameter definition with entropie of intensity distribution D e S S D E I spot r thresh I peak r r thresh w Peter S rms 0 0 E E( r) r Ex ( ) P ( x ( r) r dr ln r dr x Ex ( ) P s ) ( y y I( x, y) dx dy s ) I( x, y) dx dy dx I( x)ln I( x) dx
17 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 0...1 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
18 Approximations for the Strehl Ratio Approximation of Marechal: ( useful for D s > 0.5 ) but negative values possible Bi-quadratic approximation Exponential approach D s Wrms 1 4 Wrms D 1 s D s e 4 W rms D S 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 Marechal defocus exponential biquadratic exac t 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 c 0 Computation of the Marechal approximation with the coefficients of Zernike D s 1 N n1 cn0 1 n 1 N n n1 m0 c nm n 1
19 Strehl Ratio Criterion In the case of defocus, the Rayleigh and the Marechal criterion delivers a Strehl ratio of D S 8 0.8106 The criterion D S > 80 % therefore also corresponds to a diffraction limit This value is generalized for all aberration types 0.8 aberration type coefficient Marechal approximated Strehl exact Strehl defocus Seidel a 0. 8 0 5 0. 7944 0. 8106 defocus Zernike c 0 0. 15 0.7944 0.8106 spherical aberration Seidel a 40 0. 5 0.7807 0.8003 spherical aberration Zernike c 40 0. 167 0.7807 0.8003 astigmatism Seidel a 0. 5 0.8458 0.857 astigmatism Zernike c 0. 15 0.897 0.901 coma Seidel a 31 0. 15 0.99 0.960 coma Zernike c 31 0. 15 0.99 0.960
0 Point Resolution According to Abbe Transverse resolution of an image: - Detection of object details / fine structures - basic formula of Abbe Fundamental dependence of the resolution from: 1. wavelength. numerical aperture angle 3. refractive index 4. prefactor, depends on geometry, coherence, polarization, illumination,... x k nsin 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)
1 Incoherent -Point Resolution : Rayleigh Criterion Rayleigh criterion for -point resolution Maximum of psf coincides with zeros of neighbouring psf x 1 D Airy 0.61 n sin u Contrast: V = 0.15 Decrease of intensity between peaks I = 0.735 I 0
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 0.474 0.385 D nsin u 0.770 x Usually needs a priory information Applicable also for non-airy distributions Used in astronomy Rayleigh Airy I(x) 1 0.8 0.6 0.4 0. 0 -.5 - -1.5-1 -0.5 0 0.5 1 1.5.5 x / r airy
3 Incoherent -point Resolution Criterions Visual resolution limit: Good contrast visibility V = 6 % : 0.83 x 0. 680 n sin u D Airy Total resolution: Coincidence of neighbouring zero points of the Airy distributions: V = 1 x D Airy 1. n sin u Extremly conservative criterion Contrast limit: V = 0 : Intensity I = 1 between peaks 0.51 x 0. 418 n sin u D Airy
4 -Point Resolution Distance of two neighboring object points Distance x scales with / sinu Different resolution criteria for visibility / contrast V x = 1. / sinu total V = 1 x = 0.68 / sinu visual V = 0.6 x = 0.61 / sinu Rayleigh V = 0.15 x = 0.474 / sinu Sparrow V = 0
5 -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 = 0.474 x = 0.388 x = 0.5
6 Incoherent Resolution: Dependence on NA Microscopical resolution as a function of the numerical aperture NA = 0. NA = 0.3 NA = 0.45 NA = 0.9
7 Focussing: Definitions Possible definitions for focussing : 1. Maximum intensity value on axis or on reference ray. Image location with largest contrast 3. Image location with highest spatial frequency resolution (at threshold) 4. Smallest value of the Rms of the wavefront aberration 5. Smallest spot diameter with nd order moment definition 6. Smallest PSF diameter at given threshold value (e.g. 50%, 90%) 7. Largest energy content inside a circle of given diameter No definition of global validity Usefulness depends on application Some hidden assumptions for focussability: Polarization, coherence, spectral content, geometry of pupil, apodization,... Diffraction limits focussability (uncertainty relation) Scaling quantity is D n sin Focussability is a measure of beam quality
8 Depth of Field Comparison of systems with 1. Small depth of field. Large depth of field Ref: M. Seesselberg
9 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
30 Best Image Plane: Geometrical Consideration Highest resolution: - tangent line on transverse aberration curve - compact central core of PSF y max best matching in the centre y Best contrast: - fitted mean straight line on transverse aberration curve - rms-optimum of PSF W rms z 0 D s z 0 Different criteria delivers different best image planes minimal difference y min u y max
31 Depth of Focus: Diffraction Consideration Normalized axial intensity for uniform pupil amplitude / z sin I( z) I0 / z r depth of focus Decrease of intensity onto 80%: 1 Scaling measure: Rayleigh length - geometrical optical definition depth of focus: 1R E R E n' sin 0.493 n sin z diff u - Gaussian beams: R E n' o ' u beam caustic intensity at r = 0 focal plane 1 0.8 I(z) -R u 0 +R u z z
3 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
33 Comparison of Performnace Criteria Comparison of different performance criteria as a function of defocus Residual system aberrations: 1. astigmatism 1. spherical aberration 1 Different results for optimal image plane determination Special problems: Strehl criterion fails due to special constellation for astigmatism
34 Distortion y Conventional definition of distortion V y y Special definition of TV distortion H V TV H Measure of bending of lines y real y ideal H H x Acceptance level strongly depends on kind of objects: 1. geometrical bars/lines: 1% is still critical. biological samples: 10% is not a problem Digital detection with image post processing: un-distorted image can be reconstructed
35 Distortion original 0% keystone barrel and pincushion 5% barrel 10% barrel 15% barrel % pincushion 5% pincushion 10% pincushion
36 Chromatical Difference in Magnification Typical color ringing Critical for blackwhite edges Human eye is very sensitive for these effects Ref: J. Kaltenbach
37 Chromatical Difference in Magnification Color rings are hardly seen due to colored image Lateral shift of colored psf positions Ref: J. Kaltenbach
Real Image with Different Chromatical Aberrations
Summary of Important Topics 39 Spot diagram, moments Wave aberration PV and W rms value important criteria, but only one number Rayleigh criterion PV < /4 Marechal criterion W rsm < /14 (equaivalent with Rayleigh for defocus) Psf criteria: Strehl ratio, diameter, correlation,... -point resolution, Abbe resolution Focussing and depth of focus Lateral color aberration: critical
40 Outlook Next lecture: Part 11 Performance Criteria Date: Wednesday, 01-06-7 Content: 11.1 Modulation transfer function 11. Contrast vs resolution 11.3 Special criteria 11.4 Field dependence of aberrations 11.5 Best focussing 11.6 Measurement of image performance 11.7 Miscellaneous