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Optical Metrology and Sensing

Lecture 1: Introduction

2017-10-19

Herbert Gross

Winter term 2017

2

Preliminary Schedule

No Date Subject Detailed Content

1 19.10. Introduction Introduction, optical measurements, shape measurements, errors,

definition of the meter, sampling theorem

2 26.10. Wave optics Basics, polarization, wave aberrations, PSF, OTF

3 02.11. Sensors Introduction, basic properties, CCDs, filtering, noise

4 09.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry

5 16.11. Interferometry I Introduction, interference, types of interferometers, miscellaneous

6 23.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods

7 30.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods

8 07.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,

Scheimpflug setup

9 14.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology

10 21.12. Holography Introduction, holographic interferometry, applications, miscellaneous

11 11.01. Measurement of basic

system properties Bssic properties, knife edge, slit scan, MTF measurement

12 18.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy

13 25.01. Metrology of aspheres

and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms

14 01.02. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous

15 08.02. 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 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

Ref: W. Osten

tanx y a

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

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, 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

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 physical 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

without back influence limited resolution

surface related interaction with surface

fast material dependent

flexibel and integrabel

high lateral resolution

13

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

14

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

15

Dimension Classification

x

Ref: R. Kowarschik

16

Shape Measurement

Micro mechanical part depth map

Ref: W. Osten

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,...)

Ref: W. Osten

roughnessfigurewaviness

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

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/

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

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. 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

20

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

21

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

22

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

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

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

24

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

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

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

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°, 101.33 kPa, carbonic acid 0.03 volume percent

Disadvantages: 1. wavelength in air: = 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 = 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

27

Basics - Sampling

Point detector

Ref: R. Kowarschik

28

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

30

Sampling PSF Calculation

PSF calculation by FFT:

- coupling of coordinates in Pupil and image

- zero-guard band necessary to get enough significant points in the PSF

Ref: B. Stone Proc SPIE

Gaussian profile in the spatial domain

Fourier transform

Sampling theorem

N: number of discrete points

D: size of calculation domain

Zero padding with large D/w:

finer pixels in frequency space

31

FFT-Sampling of a Gaussian Profile

2

2

)( xw

x

exf

2

2

222

)( vx w

v

x

vw

x ewewvF

DvvN

D

NvxND

1,, max

x

vw

w

1

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nx = 25

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nv = 25

w = 0.1 w = 0.0353

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nx = 71

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nv = 9

w = 0.01

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nx = 7

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nv = 91

Periodic spectra must be separated

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