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Metrology and Sensing
Lecture 1: Introduction
2016-10-xx
Herbert Gross
Winter term 2016
2
Preliminary Schedule
No Date Subject Detailed Content
1 18.10. Introduction Introduction, optical measurements, shape measurements, errors,
definition of the meter, sampling theorem
2 19.10. Wave optics (ACP) Basics, polarization, wave aberrations, PSF, OTF
3 01.11. Sensors Introduction, basic properties, CCDs, filtering, noise
4 08.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry
5 09.11. Interferometry I (ACP) Introduction, interference, types of interferometers, miscellaneous
6 22.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods
7 29.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods
8 06.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,
Scheimpflug setup
9 13.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology
10 20.12. Holography Introduction, holographic interferometry, applications, miscellaneous
11 03.01. Measurement of basic
system properties Bssic properties, knife edge, slit scan, MTF measurement
12 10.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy
13 17.01. Metrology of aspheres
and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms
14 24.01. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous
15 31.01. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method
3
Outline
Introduction
Optical measurements
Shape measurement
Errors of measurements
Definition of the meter
Sampling theorem
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 exhibit 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
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 maximum and minimum values, the system is able to detect
True value:
Value of the signal, if the system would be perfect.
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
Ref: R. Kowarschik
0x x x
ox
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 meachanical means and uncertainties are therefore not affecting the result
The scale shoud follow the the movements in measurement
If a tilt a is obtained and y is the Abbe offset, the error is of the range
Ref: W. Osten
tanx y a
7
Optical Methods
Generation of structures for shape measurement:
1. projection
2. interference
Optical methods:
1. fringe projection
2. Moire technique
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
Wavelength Ranges
Ref: W. Osten
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
Optical Measuring Instrument
Characterization of measuring device:
1. Test piece / specimen / object scanning / sensing
2. Measurement signal (material measure, standrad, 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
Classification of Optical Metrology
Ref: W. Osten
Measuring properties coordinates heights distances 3D shapes roughness
changes in shape shifts expansions strain
deviations material data internal external
Measuring principles model geometrical wave optical
light field coherent incoherent
dimension 1D - point 2D - line 3D / 2,5D - surface
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
Ref: R. Kowarschik
advantages disadvantages
contactless indirect
wihtout back influence limited resolution
surface related interaction with surface
fast material dependent
flexibel and integrabel
high lateral resolution
13
Basic Methods
Basic principles:
1. coherent
2. incoherent
Projection: evaluation of contour lines
Moire: usage of 2 sources
Structured detector: usage of 2 wavelength
Triangulation methods
Ref: R. Kowarschik
14
Method Overview
Ref: R. Kowarschik
Shape acquisition techniques
Contact
Non-destructive Destructive
CMM Jointed arms Slicing
Non-contact
Reflective Transmissive
Non-optical
Optical
Industrial CT
Microwave radar Sonar
Passive
Active
Stereo Shading Silhouettes Texture Motion
Shape from X
Imaging radar
Triangulation
Interferometry
(Coded) Structured light
Moire Holography
Stereo
15
Coherence
Observability of interference and coherence
Ref: R. Kowarschik
16
Coherence
Coherence: capability to interfere
Spatial coherence:
- defined by size of light source
- measurement procedure: Young interferometer
Temporal coherence:
- finite wave train,
axial length of coherence lc
- finite bandwidth l
- no interference for long
path differences
- Measurement procedure:
Michelson interferometer
- typical values: table
Ref: R. Kowarschik
17
Measurement Quantities
Interferometric fringes
Ref: R. Kowarschik
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
18
Dimension Classification
x
Ref: R. Kowarschik
19
Shape Measurement
Micro mechanical part depth map
Ref: W. Osten
20
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,...)
Ref: W. Osten
roughnessfigurewaviness
21
PSD Ranges
Typical impact of spatial frequency
ranges on PSF
Low frequencies:
loss of resolution
classical Zernike range
High frequencies:
Loss of contrast
statistical
Large angle scattering
Mif spatial frequencies:
complicated, often structured
fals light distributions
log A2
Four
low spatial
frequency
figure errormid
frequency
range micro roughness
1/l
oscillation of the
polishing machine,
turning ripple
10/D1/D 50/D
larger deviations in K-
correlation approach
ideal
PSF
loss of
resolution
loss of
contrast
large
angle
scattering
special
effects
often
regular
22
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. experimenter / 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:
2. statistical errors:
Ref: R. Kowarschik
dzz
fdy
y
fdx
x
fdf
2
2
2
2
2
2
dzz
fdy
y
fdx
x
fu
23
Measurement Errors
Scattering of values by repeating the measurements
Distribution of errors:
Repeatability, width 6s
Expected value:
average for large number of repeated measurements
Variance:
Standard deviation
root mean square (rms):
Higher order moments:
1. Skewness, kurtosis
2. Peakedness
Ref: W. Osten
true
value
systematic
deviation
distribution of real
measured values
6s
1
1lim
N
jN j
x xN
2
2
1
1 N
j
j
x xN
s
2
1
1 N
j
j
x xN
s
3
1
1 N
j
j
K x xN
4
1
1 N
j
j
P x xN
24
Distribution of Statistical Errors
Gaussian or Normal Distribution:
Within interval s are 68.27 % measured values (statist. certainty: 68.27 %)
Within interval 2s are 95.45 % measured values (statist. certainty: 95.45 %)
Within interval 3s are 99.73 % measured values (statist. certainty: 99.73 %)
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
68.27 %
0
0.2
0.4
0.6
0.8
1
ss 2s 3s2s3s
2xp e
25
Distribution of Statistical Errors
Gaussian or Normal distribution
Idealized model function for purely statistical
influences
Standardized formulation
Inversion: error function:
Probability, that the variable t
lies within the intervall -z...+z
(interval of confidence, integral)
Examples: p = 0.683 for z=s
p = 0.5 gives interval z = 0.6745 s
Ref: R. Kowarschik
2
221
, ,2
x x
G x x e ss s
x xt
s
2
2
0
2( )
2
z t
p erf z e dt
26
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
Confidence range of the mean
Example: K = 1: confidence +-s
a = 0.3174
Histogram of values for N repeated
measurements:
Number Nj of results inside the same
interval
Ref: R. Kowarschik
N
ss
C KN
s
Nj
xx
27
Linear Trend
Trend of measurement data as
a function of a variable x
Calculation of slope (LSQ fit)
Absolute value / constant
Special aspects:
weighting of point inversely to error bars
Ref: R. Kowarschik
y
x
i iy m x b
2
i i
i
i
i
y x x
mx x
b y m x
x
28
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
Uncertainty of the prototype:
1. external conditions: T = ±0.001° I/I = ± 10-8
2. measurement procedure
- engraved lines
- illumination, cross section, contamination I/I = ± 10-7
3. Instability
Total uncertainty: ± 10-7 < I/I < ± 10-6
Problems with the prototype: unique sample, arbitrary, seconfdary standards
Ref: R. Kowarschik
29
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°, 101.33 kPa, carbonic acid 0.03 volume percent
Disadvantages: 1. wavelength in air: l = 643.84696 nm ± 10-7
2. Cd emission is not monochromatic
3. Michelson usd a lamp
4. bad reproducibility
5. insensitive SEM's
1906 Benoit, measurement repeated with Fabry-Perot
1960 11th GCPM, new standard is Kr wavelength
1 m = 1 650 763.73 times the vacuum wavelength of the transition
2p10 ---> 5d5 of 36Kr, wavelength is l = 605.8 nm
Advantages: 1. vacuum
2. no hyperfine structure of transition
3. no instruction for the generation of the radiation
Uncertainty: l/l = ± 10-8 ... ± 4 10-9
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
30
Basics - Sampling
Point detector
Ref: R. Kowarschik
31
Basics - Sampling
Detector of finite Size
Ref: R. Kowarschik
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
Sampling Theorem
dxexFvf
x
xvi
max
0
2)()(
Nvx
1
max
max 12
xN
vv
xvNy
122 max
Periodic spectra must be separtated
Overlapp of spectra:
- aliasing
- pseudo pattern and Moire generated
Sampling Theorem
original
spectrum
f()
ny
- ny
2ny
4ny
-4ny -2
ny0
replicas replicas
F
F'
convolution
overlap
Necessary sampling in spatial domain to separate spectra in frequency domain
comb function creates periodicity
Sampling Theorem
f(x)
xsampling comb
spatial grid
x x
F()
x
spectra
2max
x > 2max
f(x)
xsampling comb
fine structures
not resolved
spatial grid
x x
F()
x
spectra x < 2
max
undersampling
2max
Discrete ring pattern
Circular aliasing patterns in outer region
Aliasing Errors
Digital discrete signal in spatial domain
comp function as sampling
Signal band-limited
finite extend in spatial domain
Back-transform
sampling corresponds to convolution
with sinc-function
Ideal reconstructor:
sinc function
Sampling of Bandlimited Signals
x
xcombxFxF )()(
~
maxmax
)()(~
)(~~
x
xrect
x
xcombxF
x
xrectxFxF
x
x
x
x
xx
xcombxFxF
sin1
)(~
)(
)()(~
)( xRxFxF
xc
x
xxR ny
ny
ny
sin
sin)(
Sampling of Bandlimited Signals
original
signal
discretized
signal
reconstructed
signal
x
x
x
sinc-function
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