Design and Correction of optical Systems

Transcription

1 1 Design and Correction of optical Systems Part 11: Performance criteria Summer term 01 Herbert Gross

2 Oeriew 1. Basics Materials Components Paraial optics Properties of optical systems Photometry Geometrical aberrations Wae optical aberrations Fourier optical image formation Performance criteria Performance criteria Correction of aberrations Correction of aberrations Optical system classification

3 3 Contents 11.1 Modulation transfer function 11. Contrast s resolution 11.3 Special criteria 11.4 Field dependence of aberrations 11.5 Best focussing 11.6 Measurement of image performance 11.7 Miscellaneous

4 4 Contrast The MTF-alue corresponds to the intensity contrast of an imaged sin grating Visibility Ima Imin V I I The maimum alue of the intensity is not identical to the contrast alue since the minimal alue is finite too Concrete alues: ma min I 1 peak decreased slope decreased object I ma I I ma V image minima increased I min

5 MTF of a Perfect System Aberration free circular pupil: Reference frequency Cut-off frequency: Analytical representation Separation of the comple OTF function into: - absolute alue: modulation transfer MTF - phase alue: phase transfer function PTF sinu' f a o ' sin 0 u n f na G arccos H MTF,,, y PTF ih y MTF y OTF e H H 5

6 MTF and Contrast Object Contrast Object spectrum Image spectrum Image cos 0 a c I obj c a a c a c a c a c I I I I V min ma min ma ˆ a a c I F I obj obj H I I MTF obj ima cos 0 0 ˆ ˆ ˆ ' H a c e H a e H a H c H a H a H c F H I F I F I MTF i MTF i MTF MTF MTF MTF MTF MTF obj ima ima 6

7 7 Real MTF Real MTF of system with residual aberrations: 1. contrast decreases with defocus. higher spatial frequencies hae 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 ma = 0.05 ma = 0.1 ma = 0. ma = 0.3 ma = 0.4 ma = 0.5 ma = 0.6 ma = 0.7 ma = 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 z in R U

8 8 MTF-Cure sagittal / tangential Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the azimuthal orientation of the object structure Generally, two MTF cures are considered g MTF tangential plane y 1 ideal 0.5 sagittal tangential sagittal tangential / ma arbitrary rotated tangential sagittal sagittal plane

9 9 Resolution Estimation with Test Charts Measurement of resolution with test charts: bar pattern of different sizes two different orientations calibrated size/spatial frequency

10 10 Test: Siemens Star Determination of resolution and contrast with Siemens star test chart: Central segments b/w Growing spatial frequency towards the center Gray ring zones: contrast zero Calibrating spatial feature size by radial diameter Nested gray rings with finite contrast in between: contrast reersal, pseudo resolution

11 11 Resolution Test Chart: Siemens Star a. original b. good system c. defocus d. spherical e. astigmatism f. coma

12 1 Hopkins Factor Resolution/contrast criterion: Ratio of contrasts with/without aberrations for one selected spatial frequency g MTF real gmtf ideal g MTF Real systems: Choice of seeral application releant frequencies e.g. photographic lens: 10 Lp/mm, 0 Lp/mm, 40 Lp/mm

13 13 MTF-Area-Criteriom Consideration of the complete area under the MTF cure in teh releant interal of spatial frequencies A MTFa K 1 H MTF d In anisotropic systems: olume under MTF-surface Quite good correlation with isual perception for isual systems

14 14 Contrast s Resolution Balance between contrast and resolution: not triial Optimum depends on application Receier: minimum contrast cure seres as real reference Most detector needs higher contrast to resole high frequencies CSF: contrast sensitiity function g MTF 1 : high contrast threshold contrast b : 1 is better threshold contrast a : is better :high resolution

15 15 Contrast and Resolution High frequent structures : contrast reduced Low frequent structures: resolution reduced contrast brillant blurred sharp milky resolution

16 16 Contrast and Resolution Contrast s contrast as a function of spatial frequency Typical: contrast reduced for increasing frequency Compromise between resolution and isibilty is not triial and depends on application

17 17 Resolution: Loss of Information Blurred imaging: - limiting case - information etractable Blurred imaging: - information is lost - what s the time?

18 18 Contrast / Resolution of Real Images Degradation due to 1. loss of contrast. loss of resolution

19 19 ESF, PSF and ESF-Gradient Typical behaior of intensity of an edge image for residual aberrations The width of the distribution roughly corresponds to the diameter of the PSF Deriatie of the edge spread function: edge position at peak location edge spread function point spread function deriation of edge spread function y'

20 0 Structural Content Structural content: Contrast capacity of an image S I I image image, y d dy, y d dy g g otf otf,, y y d d d d y y Equal Strehl ratio D S = 0.8 but different structural content: g MTF ideal c 0 = 0.19 c 40 = c 60 = 0.0 c 80 =

21 1 Aberrations of a single lens Single plane-cone lens, BK7, f = 100 mm, = 500 nm y Spot as a function of field position Coma orientation towards the ais

22 Variation of Performance with Field Position PSF as a function of the field height = 0 = 0% = 40% = 60 % = 80 % = 100 % Orientation of coma in the field y

23 3 Variation of Performance with Field Position Strehl ratio: 1. Photo objectie lens: strong dependence. Microscope objectie lens: weak dependence D S photo objectie D S micro objectie nm 468 nm 587 nm nm nm 587 nm field field

24 4 Specifications 1. Functional specification Related to application, alid for the complete system. Plays the major role of a basic agreement between customer and deelopment. Here two leels must be distinguished: 1.1 Basic data 1. Image quality. Manufacturing specification Declaration of seeral points of the engineering approach to clearify the technological and business goals.1 Business specification to ensure an economic deelopment. 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

25 5 Hartmann Shack Waefront Sensor Lenslet array diides the waefront into subapertures Eery lenslet generates a simgle spot in the focal plane The aeraged local tilt produces a transerse offset of the spot center Integration of the deriatie matri deliers the wae front W,y array detector waefront D array h f D meas u spot offset D sub refractie inde n

26 Spot Pattern of a HS - WFS Aberrations produce a distorted spot pattern Calibration of the setup for intrinsic residual errors Problem: correspondence of the spots to the subapertures a spherical aberration b coma c trefoil aberration

27 7 Hartmann Shack Waefront Sensor Typical setup for component testing detector test surface fiber illumination lenslet array collimator beamsplitter telescope for adjustment of the diameter Lenslet array a b subaperture point spread function -dimensional lenslet array

28 8 Array Signal Lenslet array ideal signal array of phase spot pattern cross section of the spot pattern Real signal: 1. discretization. quantization 3. noise original discretized and quantized with noise

29 9 Test by Newton Fringes Reference surface and test surface with nearly the same radii Interference in the air gap Reference flat or cured possible Corresponds to Fizeau setup with contact to detector Broad application in simple optical shop test Radii of fringes beamsplitter r m mr illumination test surface path difference reference surface here: flat

30 30 Testing with Fizeau Interferometer Long common path, quite insensitie setup Autocollimating Fizeau surface quite near to test surface, short caity length Imaging of test surface on detector Straylight stop to bloc unwanted light Cured test surface: auiliary objectie lens aplanatic, double path Highest accuracy

31 31 Testing with Twyman-Green Interferometer Short common path, sensible setup Two different operation modes for reflection or transmission collimated laser beam Always factor of between detected wae and component under test beam splitter objectie lens stop 1. mode: lens tested in transmission auiliary mirror for autocollimation. mode: surface tested in reflection auiliary lens to generate conergent beam detector

32 3 Interferograms of Primary Aberrations Spherical aberration 1 Astigmatism 1 Coma Defocussing in

33 33 Knife Edge Method Moing a knife edge perpendicular through the beam cross section knife edge moement Relationship between power transmission and intensity: Abel transform for circular symmetry beam y P I r r dr r d Eample: geometrical spot with spherical aberration before caustic zone rays below near paraial focus z

34 34 Slit-Scan-Method Method ery similar to moing knife edge Integration of slit length must be inerted: - inerse Radon transform - corresponds to tomographic methods moing slit d beam y

35 35 Phase Retrieal Principle of phase retrieal for metrology: intensity ariation during z-propagation due to phase errors due to the wae equation, the phase information can be recoered from a z-stack ill posed inerse problem easy measurement, algorithms complicated Austrittspupille Bildebene System z z Wellenfläche W,y : lateral y 3D-Intensitäts-Stack I,y,z

36 36 Aufbau für Systemprüfung Possible illumination setups: 1. Pinhole - coherence critical - low power - deconolution for finite size. Epi illumination - only symmetrical aberrations - problems with finite field position 3. Via calibrated objectie lens

37 37 PSF Phase Retrieal Phase retrieal method Image z-stack Correlation of images Phase in pupil

38 38 MTF-Measurement by Edge Spread Function Measurement of an edge image Ealuating the deriatie: Line spread function Fourier transform: optical transfer function I H LSF OTF d ' I ESF ' d ' s Fˆ I ' LSF incoherent illumination I in edge object diffracted light ' detector plane I' geometrical image of the edge d I' d'

39 39 MTF-Measurement by Imaging Gratings Setup: Imaging of a grating diffusor grating object lens under test slit sensor lamp Possible realizations: 1. Density type grating, the sine wae is modelled by gray leels. Area type gratings, the sine wae is modelled by geometrical sine-shaped structures Area coded sine grating:

40 40 Ghost Images Ghost image in photographic lenses: Refle film / surface Ref: K. Uhlendorf, D. Gängler

41 41 Straylight and Ghost Images Different reasons Various distributions a b

42 Summary of Important Topics 4 MTF criteria, Hopkins factor, area criterion Test charts for qualitatie fast ealuations Optimization contrast s resolution: not triial, depends on application Special criteria for edges, information content Problem in practice: field dependence of aberrations Many possible focussing criteria Different types of specifications usual Measurement of system performance ia waefront sensor Shack-Hartmann Quite accurate wae front measurement: interferometry PSF phase retrieal: robust and easy to measure method, algorithms complicated MTF measurement comple: sin-pattern, spatial frequency analysis of edge images Real world problems: ghosts and straylight

43 43 Outlook Net lecture: Part 1 Correction of aberrations 1 Date: Wednesday, Contents: 1.1 Symmetry principle 1. Lens bending 1.3 Correcting spherical aberration 1.4 Coma, stop position 1.5 Astigmatism 1.6 Field flattening 1.7 Chromatical correction 1.8 Higher order aberrations