Optical Metrology and Sensing

Transcription

1 Optical Metrology and Sensing Lecture 1: Introduction Herbert Gross Winter term

2 2 Preliminary Schedule No Date Subject Detailed Content Introduction Introduction, optical measurements, shape measurements, errors, definition of the meter, sampling theorem Wave optics (ACP) Basics, polarization, wave aberrations, PSF, OTF Sensors Introduction, basic properties, CCDs, filtering, noise Fringe projection Moire principle, illumination coding, fringe projection, deflectometry Interferometry I (ACP) Introduction, interference, types of interferometers, miscellaneous Interferometry II Eamples, interferogram interpretation, fringe evaluation methods Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight, Scheimpflug setup Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology Holography Introduction, holographic interferometry, applications, miscellaneous Measurement of basic system properties Bssic properties, knife edge, slit scan, MTF measurement Phase retrieval Introduction, algorithms, practical aspects, accuracy Metrology of aspheres and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method

3 3 Outline Introduction Optical measurements Shape measurement Errors of measurements Definition of the meter Sampling theorem

4 4 General Terms of Measurement Accuracy: In situations where we believe that the measured value is close to the true value, we say that the measured value is accurate (qualitative) Precision: When values obtained by repeated measurements of a particular quantity ehibit little variability, we say that those values are precise (qualitative) Reproducibility: Ability for different users to get the same reading when measuring a specific sample. Repeatability: How capable a gage is of providing the same reading for a single user when measuring a specific sample. Ref: R. Kowarschik

5 5 General Terms of Measurement Resolution: Smallest amount of input signal change the instrument can detect reliably. Reasons for limited resolution: diffraction, noise, hysteresis, discretization. Typically it corresponds to half of the sampling rate. Sensitivity: Smallest signal the instrument can measure. Reproducible change of output signal for changes of the measured property Tolerance/dynamic range: Limiting maimum and minimum values, the system is able to detect True value: Value of the signal, if the system would be perfect. o If this is known for a special case, the system can be calibrated (corrected for systematic errors) Measurement error: Difference between measure value and true value 0 Ref: R. Kowarschik

6 6 Abbe Comparator Principle Basic idea: the measured property and the scale of measurement should aligned Avoid the influence of tilt and bending on the result Errors due to mechanical means and uncertainties are therefore not affecting the result The scale should follow the the movements in measurement If a tilt a is obtained and y is the Abbe offset, the error is of the range y tana Ref: W. Osten

7 7 Optical Methods Generation of structures for shape measurement: 1. projection (incoherent imaging of a grid) 2. interference (mostly coherent) Optical methods: 1. fringe projection (contour lines) 2. Moire technique (2 sources used) 3. holographic contouring 4. speckle contouring 5. photogrammetry Shape measurement for quality control applications 1. digitization of prototypes 2. replacement of mechanical systems Ref: R. Kowarschik

8 8 Wavelength Ranges Ref: W. Osten

9 9 Scales and Dynamic Range 10 orders of magnitude for geometrical measurements: AFM white light holographic pattern projection SNOM confocal speckle Ref: W. Osten

10 10 Optical Measuring Instrument Characterization of measuring device: 1. Test piece / specimen / object scanning / sensing 2. Measurement signal (material measure, standard, etalon) 3. Amplification of the signal 4. Indication of the measured value If one of the first three aspects is performed out optically: optical measuring instrument Methods based on the wave nature of light: 1. Diffraction 2. Interference (coherent): Interferometer Holography Speckle techniques Laser based measurements Ref: R. Kowarschik

11 11 Classification of Optical Metrology Measuring properties coordinates heights distances 3D shapes roughness changes in shape deviations shifts epansions strain material data internal eternal Measuring principles physical model geometrical wave optical light field dimension coherent incoherent 1D - point 2D - line 3D / 2,5D - surface Ref: W. Osten

12 12 Optical Methods Requirements on measurement: 1. high density of measurement points, spatial resolution 2. high velocity 3. contactless 4. absolute 3D coordinates Pros and cons of optical measuring techniques advantages contactless without back influence surface related fast fleibel and integrabel high lateral resolution disadvantages indirect limited resolution interaction with surface material dependent Ref: R. Kowarschik

13 13 Method Overview Shape acquisition techniques Contact Non-contact Non-destructive Destructive Reflective Transmissive CMM Jointed arms Slicing Non-optical Industrial CT Microwave radar Sonar Optical Passive Active Imaging radar Stereo Shape from X Triangulation Motion Stereo Shading Silhouettes Teture Interferometry (Coded) Structured light Moire Holography Ref: R. Kowarschik

14 14 Measurement Quantities Interferometric fringes Primary measured Derived quantity Applications fringe position phase difference length standard refractometry length compensation phase variation interference microscopy optical testing fringe visibility spectrum of source spectral profiles spatial distribution at source stellar diameter full intensity distribution spectrum of source interference spectroscopy Fourier spectroscopy spatial distribution at source optical transfer function radio astronomy Ref: R. Kowarschik

15 15 Dimension Classification Ref: R. Kowarschik

16 16 Shape Measurement Micro mechanical part depth map Ref: W. Osten

17 17 Surface Deviations Typical three different ranges according to power spectral density: 1. figure: long range, overall shape 2. waviness: machine oscillations, errors in production 3. roughness: Short term deviations due to manufacturing interaction (grinding, polish,...) waviness figure roughness Ref: W. Osten

18 18 PSD Ranges Typical impact of spatial frequency ranges on PSF Low frequencies: loss of resolution classical Zernike range High frequencies: Loss of contrast statistical log A 2 Four larger deviations in K- correlation approach oscillation of the polishing machine, turning ripple Large angle scattering Mif spatial frequencies: complicated, often structured fals light distributions low spatial frequency figure error mid frequency range 1/D loss of 10/D 50/D resolution special effects often regular micro roughness loss of contrast large angle scattering 1/ ideal PSF

19 19 Measurement Errors Measurement results: Result of measurement = measured value ± uncertainty Selection of error types: 1. material measures 2. mechanical 'failures' of the system 3. distortion of Abbe comparator principle 4. environmental influences 5. eperimenter / observer Systematic and random errors: Systematic errors: correction of the measured value possible (calibration). Can be reproduced and are constant in amount and sign. Random errors and systematic errors with unknown sign: uncertainty of measurement Propagation of errors: 1. systematic errors: df f d f y dy f z dz 2. statistical errors: u 2 f f y 2 f z 2 2 d dy dz 2 2 Ref: R. Kowarschik

20 20 Measurement Errors Scattering of values by repeating the measurements Distribution of errors: Repeatability, width 6s Epected value: average for large number of repeated measurements 1 lim N j N N j1 true value distribution of real measured values Variance: 2 s 1 N j N 2 j1 systematic deviation 6s Standard deviation root mean square (rms): s 1 N j N 2 j1 Higher order moments: 1. Skewness, kurtosis 1 N j 3 K N j 1 2. Peakedness 1 N j 4 P N j 1 Ref: W. Osten

21 21 Distribution of Statistical Errors Gaussian or Normal Distribution: 2 p e Within interval s are % measured values (statist. certainty: %) Within interval 2s are % measured values (statist. certainty: %) Within interval 3s are % measured values (statist. certainty: %) % 0 3s 2s s s 2s 3s For a given statistical certainty the corresponding range is called ± c s confidence interval (CI) The true value lies within the confidence interval for a given statistical certainty if there are no systematic errors Ref: R. Kowarschik

22 22 Distribution of Statistical Errors Gaussian or Normal distribution Idealized model function for purely statistical influences Standardized formulation 2 2 2s G,, s e 1 2 s Inversion: error function: Probability, that the variable t lies within the intervall -z...+z (interval of confidence, integral) Eamples: p = for z=s p = 0.5 gives interval z = s t s t 2 2 p erf ( z) e dt 2 z 0 2 Ref: R. Kowarschik

23 23 Distribution of Statistical Errors Probability, that the value is outside the confidence interval (failure): a = 1-p N measurements: Standard deviation of the mean is reduced to s s N Confidence range of the mean Eample: K = 1: confidence +-s a = CK s N Histogram of values for N repeated measurements: Number N j of results inside the same interval N j Ref: R. Kowarschik

24 24 Linear Trend Trend of measurement data as a function of a variable y m b i i y Calculation of slope (LSQ fit) m i i Absolute value / constant i y 2 b y m i i Special aspects: weighting of point inversely to error bars Ref: R. Kowarschik

25 25 Definition of the Meter History: 1791 French Academy of Sciences: 1 m = one ten-millionth part of the quadrant of the earth's meridian 1875 Treaty of the Meter (Meter convention) General Conference on Weights and Measures (GCPM) International Bureau of Weights and Measures (BIPM) 1889 International prototype final definition 1927 by 7th GCPM conference Ref: R. Kowarschik Uncertainty of the prototype: 1. eternal conditions: T = ±0.001 I/I = ± measurement procedure - engraved lines - illumination, cross section, contamination I/I = ± Instability Total uncertainty: ± 10-7 < I/I < ± 10-6 Problems with the prototype: unique sample, arbitrary, seconfdary standards

26 26 Definition of the Meter 1893 Michelson, 1st measurement of the meter based upon the wavelength red Cadmium line as standard for spectroscopy Conditions: dry air, 15, kpa, carbonic acid 0.03 volume percent Disadvantages: 1. wavelength in air: = nm ± Cd emission is not monochromatic 3. Michelson usd a lamp 4. bad reproducibility 5. insensitive SEM's 1906 Benoit, measurement repeated with Fabry-Perot th GCPM, new standard is Kr wavelength 1 m = times the vacuum wavelength of the transition 2 p > 5 d 5 of 36 Kr, wavelength is = nm Advantages: 1. vacuum 2. no hyperfine structure of transition 3. no instruction for the generation of the radiation Uncertainty: l/l = ± ± Required accuracy of the meter: everyday life: commerce: I/I = ± 10-3 gauge block I/I = ± 10-6 physics: I/I = ± 10-7 Ref: R. Kowarschik

27 27 Basics - Sampling Point detector Ref: R. Kowarschik

28 28 Basics - Sampling Detector of finite Size Ref: R. Kowarschik

29 Sampling Theorem Fourier transform Relation for discrete Fourier transform Frequency sampling depends on spatial sampling Discrete sampling: - periodicity in frequency space, limits bandwidth at Nyquist frequency - 2 points per period necessary to avoid aliasing f ( v) ma 0 1 v N v 2v N F( ) e ma 1 ma 1 2 ma 2v Ny 2 iv d

30 Sampling Theorem Periodic spectra must be separated replicas f() original spectrum replicas -4 ny -2 ny 0 2 ny 4 ny - ny ny Overlapp of spectra: - aliasing - pseudo pattern and Moire generated F F' convolution overlap

31 Sampling Theorem Necessary sampling in spatial domain to separate spectra in frequency domain Comb function creates periodicity f() > 2 ma F() spectra sampling comb spatial grid 2 ma f() < 2 ma undersampling F() spectra sampling comb fine structures not resolved spatial grid 2 ma

32 Aliasing Errors Discrete ring pattern Circular aliasing patterns in outer region

33 Digital discrete signal in spatial domain comp function as sampling Signal band-limited finite etend in spatial domain Back-transform sampling corresponds to convolution with sinc-function Ideal reconstructor: sinc function Sampling of Bandlimited Signals comb F F ) ( ) ( ~ ma ma ) ( ) ( ~ ) ( ~ rect comb F rect F F comb F F sin 1 ) ( ~ ) ( ) ( ) ( ~ ) ( R F F c R ny ny ny sin sin ) (

34 Sampling of Bandlimited Signals original signal discretized signal reconstructed signal sinc-function