Transcript Slide 1

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Introduction to
Displacement
Measuring
Interferometry
Information in this document is subject to change without
notice. Portions of this document describe patented systems
and methods and does not imply a license to practice
patented technologies. No liability is assumed with respect to
the use of the information contained in this
documentation. No part of this document may be reproduced
or transmitted in any form or by any means, electronic or
mechanical, for any purpose, without the express written
permission of Zygo Corporation.
© 2008 Zygo Corporation. All rights reserved.
2
What is this presentation about? Who is it for?
• Restricted to interferometric measurement
of displacement
– Does not cover form, surface roughness
• Fundamentals
• Intended for an audience with a minimal
background in displacement interferometry
– Only knowledge of basic physics is assumed
3
Outline
• Displacement measurement
• Basics of Displacement Measuring
Interferometers (DMIs)
• Common interferometer configurations
• Introduction to uncertainty sources
• Specialized interferometer configurations
• Some application examples
• Summary
4
Some terms that are used throughout this presentation
DMI
Displacement Measuring Interferometer
OPL
Optical Path Length
OPD Optical Path Difference
f
Split frequency
ppm
Parts per million = multiplier of 1 X 10-6
ppb
Parts per billion = multiplier of 1 X 10-9
5
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Displacement
Measurement
What does ‘displacement’ mean in this context?
• Denotes a change in position
– How far something has moved
• Implies a
– Start point and an end point
– Relative motion
• Distinguished from ‘distance’
– Absolute separation between two points
• Displacement measurement tools can
establish distance indirectly
7
Distance and displacement are two different things!
Target
Retroreflector
Two-frequency
laser
Cannot
measure
distance from
beamsplitter!
8
Can measure
displacement
of target
Another example of the distinction between distance &
displacement
Transparent artifact whose
length needs to be
determined
• Direct measurement of length is not possible with
DMI
• Indirect measurements are possible by measuring
displacement of a probing mechanism
9
Consequences of relative nature of measurement
• If the beams of a DMI are broken and signal
is lost, system loses track of target position
• When beam is reestablished, system starts
counting from current position of target
• System has no knowledge of the new
position relative to the beamsplitter
• Critical that beam not be interrupted!
10
‘Absolute’ interferometers exist
• Absolute measurements can be performed
interferometrically
• Based on different working principle
– Multi-wavelength
– Frequency sweeping
• Tutorial restricted to displacement
measuring interferometers
11
Many methods exist for the high-precision measurement of
displacement
• Displacement
•
interferometers
• Encoders
• Capacitance gages
• Electronic indicators
(LVDT, LVDI, etc.)
• Ultrasonic
•
12
Optical probes
– Triangulation
– Chromatic
aberration
– Confocal
– Interferometric
– Fiber optic
And many others…
One way to compare these methods is based on range &
resolution
10-6
Resolution (m)
10-7
Encoders
10-8
10-9
10-10
Interferometers
10-5
10-4
10-3
10-2
10-1
13
Max. Range (m)
100
101
102
DMIs and encoders are unique
• Most displacement measuring devices have
a relatively fixed resolution/range ratio
– Gain one at the cost of the other
• DMIs and encoders do not suffer from this
trade-off
– Same resolution regardless of range
• Encoders are limited in range by maximum
length that can be manufactured
• Encoders often suffer from location conflicts
14
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Physics of Optical
Interference
Light waves are represented by sinusoids
A
Phase (1)
Phase difference
(=2- 1)
B
Amplitude
Phase (2)
Wavelength ()
• Electric field of an electromagnetic (EM) disturbance
can be represented as a sinusoid
• Three parameters completely define the wave
– Amplitude (Typically of the electric or E field)
– Frequency () or wavelength (λ)
– Phase (phase difference more significant than absolute
phase)
16
Equations of interference are derived by the Principle of
Superposition
Two light waves E1 & E2 of amplitude E10 & E20 with a
phase difference  between them
E1  E10 sin(t )
E2  E20 sin(t   )
Resultant wave E is given by sum of E1& E2
E  E1  E2  E10 sin(t )  E20 sin(t   )
irradiance of resultant wave I  E2
I  ( E1  E2 )2
T
 ( E10 sin(t )  E20 sin(t   )) 2
I  I1  I 2  2 I1I 2 cos( )
17
In the most general case the interfering waves have unequal
irradiances
I 
I1  I 2
 2 I1 I 2 cos( )
Sum of irradiances
Interference term
Fundamental equation of Interference
Resultant irradiance is the sum of the irradiances of the
two waves combined with modulation due to
interference term
Irradiance maxima
I  I1  I 2  2
Irradiance minima
I  I1  I 2 -2 I1 I 2
I1 I 2
when   2 m
when    m
m  1, 3, 5...
m  0, 1, 2,...
18
Interference of waves of unequal irradiance (amplitude)
does not produce complete cancellation
A+B
A
B
In phase
Constructive Interference
A
A+B
B
Out of phase
Destructive Interference
19
The special case of equal irradiance is of practical interest
I 
I1  I 2
 2 I1 I 2 cos( )
Sum of irradiances
Interference term
Fundamental equation of Interference
For the special case I1= I2 = I0 above equation reduces to
 
I  4 I 0 cos  
 2
2
Irradiance maxima
Irradiance minima
I 0
when    m
I  4I0
when   2 m
m  1, 3, 5...
m  0, 1, 2,...
20
Waves of equal irradiance can produce complete
cancellation
A
A+B
B
In phase
Constructive Interference
A
A+B
B
Out of phase
Destructive Interference
21
Optical Path Difference (OPD) determines the phase
difference between two waves
I 
I1  I 2
 2 I1 I 2 cos( )
Sum of irradiances
Interference term
Fundamental equation of Interference
Phase difference () is a function of the optical path difference
Optical path difference (OPD) is the difference in optical path length as
distinct from the physical path length
Optical path length (OPL) is defined as
OPL  n * l
where n = refractive index of the medium of propagation
l = physical path length traversed by beam
22
Optical Path Difference (OPD) is the difference in OPL
traversed by two waves
A
OPD  ng l  na l
Air (na)

B
Glass (ng)
l1

l
2
OPD

2
ng l  nal 


OPD  na l1  na l2
A
B
Air (na)

Air (na)
l2

23
2
OPD

2
nal1  nal2 


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Basics of
Displacement
Measuring
Interferometry
(DMI)
All displacement measuring interferometers have these
basic components
Reference beam
S
o
u
r
c
e
Recombine
Split
Detector
Meas. beam

Target motion
Phase shifted
beam
25
The Michelson interferometer – a simple interferometer
Fixed Mirror
Movable
Mirror
Monochromatic
Light Source
Beamsplitter
λ/4
Observed
intensity at the
detector
Phase
Measurement
Electronics
26
λ/4

The desired displacement d is related to raw phase output 
Assuming that the medium the interferometer is
operating in has refractive index n and the vacuum
wavelength of the light source is vac, the wavelength
in the medium of operation  is given by

vac
n
Also, 2 radians of phase corresponds to a path
length change of , phase change  corresponds to
a path length change d given by
vac
d

2  2 n
27
Three additional pieces of information are required to
extract displacement d from the raw phase output 
 1  vac
d  

 2  2 n
• Vacuum wavelength
• Refractive index of medium
• Additional scale factor which depends on
interferometer configuration (½ in this
case) must be taken into account
28
What does a DMI really measure?
Displacement ?
DMIs infer displacement from changes in
optical path length (OPL) differences between
measurement and reference arms
Indeed, it is possible to build a ‘displacement’
interferometer with no moving parts!
29
Phase changes in either reference or measurement beams
contribute to measured displacement
Reference
Reference motion
Surface deviation
S
o
u
r
c
e
Index changes
Recombine
Split
Detector
Target
Target motion
Surface deviation
Index changes
30

Phase shifted
beam
Measured displacement is a function of OPL changes in
both measurement and reference arms
Reference
Mirror
OPD  nR lR  nM lM
Movable
lR
Mirror
nR

nM
d meas
vac
 OPD 

4  nR lR  lR nR

vac   nM lM  lM nM 
Laser
Beamsplitter
4
lM
nR
nR
nM

lR 
lR  lM 
lM
nM
nM
nM
31
Good displacement metrology requires careful
consideration of spurious terms
d meas
nR
nR
nM

lR 
lR 
lM lM
nM
nM
nM
Desired true
Sources of uncertainty
displacement
• lm is the desired term
• All other terms are sources of uncertainty
in measurement
• Assumption that the reference arm is
‘fixed’ should be evaluated carefully!
32
Another way to think of these effects is in terms of the
metrology loop
Reference
Mirror
Metrology
Loop
Movable
Mirror
Laser
Beamsplitter
Metrology loop is an imaginary closed contour
that passes through all components of the
system that influence the measurement result
33
Changes in the metrology loop affect the measurement
• Changes in index in measurement and
reference arm
• Change in beamsplitter (BS) ‘position’
– Expansion of mounts
– Index changes in BS
• Changes in target and reference mirrors
– Changes in surface shape (mounting, thermal)
– Surface figure related changes
34
DMIs have several advantages over other methods
• Eliminate Abbé offsets
– Measure directly at point of interest
•
•
•
•
•
•
High resolution (< 0.5 nm)
High velocity (> 5 m/sec)
Long range capability (> 10 meters)
Measure multiple degrees of freedom
Non-contact
Directly traceable to the unit of length
35
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Heterodyne
Interferometers
Commercial systems come in two major ‘flavors’
• Homodyne or single frequency
• Heterodyne or dual frequency
– ZYGO DMI
• Significant differences in
– Light source
– Detection electronics
• Focus of this tutorial is heterodyne systems
37
A short detour into homodyne systems
• Homodyne interferometers use a single
frequency laser
• Based on measurement of intensity
variation at detector
38
Homodyne systems use specialized detectors & electronics
• Provision for power normalization to
mitigate sensitivity to spurious intensity
variations
• Quadrature outputs to provide direction
information
• Low noise electronics to compensate for
operation near DC (in presence of large 1/f
noise)
39
A simple homodyne interferometer system
Reference
Retroreflector
Target
Retroreflector
Single
frequency laser
Detector
BS
• Basic system simply counts changes in
intensity at detector
• No direction sense
• Sensitive to variations in intensity of source
and changes in ambient light level
• Inefficient use of light from source
40
A more robust commercial implementation
Reference
Retroreflector
Target
Retroreflector
Single
frequency laser
Polarization
sensitive
detector
Polarization
sensitive
detector
Polarization
sensitive
detector
Special
optic
PBS
• Special optic produces a rotating plane of
polarization depending on the OPD
• Detectors are polarization sensitive
• Multiple detectors produce quadrature
outputs and power normalization functionality
41
Heterodyne interferometers are based on the principle of
heterodyning
• Heterodyne receivers used in radios
• Also known as AC interferometers
• Example of frequency shifting the signal into
a more favorable part of the spectrum
– Avoid 1/f noise at low frequencies
– Eliminate sensitivity to low frequency intensity
variations of light source
– Enable use of sophisticated phase
measurement techniques instead of intensity
42
Heterodyne systems extract displacement by making phase
measurements
• Phase or frequency measurement
– Equivalent methods
– Direct measurement of frequency change
(phase) using Doppler shifted signal
• Requires changes to the hardware
– Two frequency laser source
43
Heterodyne systems are typically polarization encoded
• Polarization encoding requires special
components but confers many advantages
– More efficient use of light
– More flexibility in routing of beams through
interferometer
– Potential for varied interferometer
configurations
– More measurement axes from a given source
44
What is polarization in this context?
• Polarization state of an electromagnetic
disturbance defines the direction that the
electric field is pointing
• Polarization states encountered in
heterodyne DMI systems
– Linear (horizontal and vertical)
– Circular
• Left handed (LCP)
• Right handed (RCP)
45
We will make a slight detour to take a look
at the various polarization states and some
of the polarization components. I will be
using software developed at the University
of Mississippi for optics education and
hereby acknowledge the WebTOP project.
Software is available as a free download at
http://webtop.msstate.edu/
46
Special components are required to manipulate the
polarization state and include polarizers…
Polarization
plane of
polarizer
Linearly
polarized
Light
Unpolarized
Light
Linear
polarizer
47
• Basic element that
converts unpolarized
light to linearly
polarized light
• Azimuthal orientation
determines orientation
of output polarization
state
• Known as an analyzer
when used to
determine state of
polarization
beamsplitters…
Non-polarizing
Beamsplitter (NPBS)
Polarization Beamsplitter
(PBS)
Splits incoming beam
regardless of polarization
Splits incoming beam based
on polarization state
48
… and quarter-wave plates
Vertical pol.
Horizontal
pol.
Fast axis
45
45
Right circ.
pol. polarized
light
Left circ. pol.
polarized light
Vertical pol.  RCP
Horizontal pol.  LCP
Linearly polarized light is turned into circularly polarized
light by passage through a quarter-wave plate with its
fast axis at 45 to the incoming polarization state.
49
Polarizers are also used to combine orthogonal pol. states
Vertical pol.
Polarization
plane of
polarizer
45
Horizontal
pol.
Components
of vertical and
horizontal pol.
50
• Incoming orthogonal
polarization states do
not interfere
• Polarizer at 45 to
both states produces
a component of each
state along
polarization plane
• Interference can now
occur
Another detour to examine the behavior of
the polarization components discussed in
the preceding slides.
51
What does a heterodyne system look like? How is it
different from a homodyne system?
Reference
Retroreflector
f2
f1
Optical
fiber
f1
f2
Two frequency
laser
Fiber optic
Pickup f - (f ± f )
2
1
1
Target
Retroreflector
PBS
(f1 ±  f1)
Measurement signal
Digital
Position Data
Reference signal
Phase Interpolator
52
What happens when waves of two frequencies interfere?
Consider two light waves EM & ER of equal amplitude E0
and frequencies f1 and f2 in the reference and
measurement arms of the interferometer respectively with
a phase difference  between them
EM  E0 sin(2 f1t   )
ER  E0 sin(2 f 2t )
Interference between these waves produces a sinusoidal
intensity variation with a difference frequency equal to the
difference between the two frequencies
I  I1  I 2  2 I1 I 2 cos(2ft   )
f  f1  f 2
53
What are the implications of this result?
I  I1  I 2  2 I1 I 2 cos(2ft   )
f  f1  f 2
• For a constant phase  intensity I varies
with a frequency f (split frequency)
• Operating point of the system has been
translated from 0 Hz (DC) to the split
frequency
• If f =0, eq. reduces to homodyne case
54
Outputs of the two kinds of interferometers are different
Detector
Reference
Retroreflector
Target
Retroreflector
Spectrum
Analyzer
Translation of
operating point
0 Hz
0 Hz
HOMODYNE
HETERODYNE
55
Split
Frequency
Why do we observe this?
For a target moving with velocity v, phase change  is given by

4 n
vac
vt
for a double pass interferometer. Substituting for  in
expression for intensity and rearranging results in


I  I1  I 2  2 I1 I 2 cos  2



 

 
2n
v t
 f 
vac

 
Frequency shift  


 frequency shift is proportional to velocity.
56
Direction and magnitude of frequency shift contain
information


I  I1  I 2  2 I1 I 2 cos  2



 

 
2n
v t
 f 
vac

 
Frequency shift  


• Magnitude of frequency shift is
proportional to the velocity
• Direction of frequency shift depends on
direction of motion
• Direction is encoded in sign of shift
57
Changes in frequency and phase are related
t
  2 
0
2n
vac
vdt 
4 n
vac
t
 vdt 
0
4 n
vac
d
• Frequency shift gives us velocity
• Phase is the integral of the frequency,
which corresponds to displacement
• Integral of frequency shift produces
expression for displacement
58
Notation is required for interferometer ray diagrams
Polarization
symbol
l= p pol.
= s pol.
Circular pol.
• Multiple attributes of
the beam need to be
represented
• Beam direction
Arrow head:
direction of
propagation
– Arrow direction
• Polarization state
f 1 ±  f1
Notation:
Base or altered
frequency
– Pol. symbol behind
arrow
Arrow color:
base
frequency
Green = f1
Red = f2
• Frequency (f1 or f2 or
altered frequency)
– Arrow color & notation
59
Reference for phase detection is optically generated in
newer systems…
External Optical
reference
or
Internal Optical
reference
Two frequency
Laser
Reference
Retroreflector
f1 - f2
f1
f2
f1
NPBS
Target
Retroreflector
f2
f2 - (f1 ± f1)
Fiber optic
Pickup
Optical
fiber
f2
PBS
f1 ± f1
(f1 ±  f1)
Measurement signal
Digital
Position Data
Reference signal (optical)
Phase Interpolator
60
… and electrically generated in older systems
Reference
Retroreflector
f2
f1
f1
f2
Two frequency
Laser
f2
f2 - (f1 ± f1)
Fiber optic
Pickup
Optical
fiber
Target
Retroreflector
PBS
f1 ± f1
(f1 ±  f1)
Measurement signal
Digital
Position Data
Reference signal (electrical)
Phase Interpolator
61
Goal is to determine the phase shift between reference &
measurement signals
Measurement
(Doppler shifted)
Reference
(Fixed split
frequency)

62
Phase difference is determined by measurement electronics
Linear Interferometer
Target
Two frequency laser
f1 , f 2
FOP (Lens w/polarizer)
ZMI 4004
Oscilloscope
Measurement Signal
fmeas = f2 - (f1 ± f1)
ZMI phase measurement
electronics convert phase
shift into displacement
fref = f1 – f2
Reference
Reference Signal
63
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Heterodyne DMI
System
Components
Components of a Heterodyne DMI System
• Laser source
• Beam directing
optics
• Interferometers
• Measurement
electronics
• Target
65
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Laser Source
Laser source fulfills multiple requirements
• Produce coherent radiation at a fixed
wavelength
– High stability
• Generate two overlapping beams
– Linearly polarized
– Orthogonal
– At two slightly different frequencies
• Production of desired polarization state and
stabilization may be coupled
67
Laser wavelength establishes the unit of length
vac
d

4 n
• Interferometer measures phase
• Wavelength is required to convert phase
difference to displacement
• Uncertainty in wavelength produces an
uncertainty in displacement
68
Laser source is frequency stabilized to control wavelength
• Uncertainty in absolute wavelength is less
critical is most applications, stability is more
critical
• Scale factor is ‘calibrated out’ in some
applications
• Wavelength measured in some applications
• Wavelength stability 1-10 ppb
• Production of ‘red’ light around 633 nm
guarantees a level of traceability
69
Lasers consist of a resonator and a gain medium
Resonator mirrors
2L
k 
k
Gain Medium
Output
k-1
k
• Radiation gains energy from the gain
medium as it oscillates in resonator
• Each medium has a gain curve that
defines the wavelength range of laser
• Actual wavelength is determined by
resonator length
70
Different techniques are used for stabilization
k-1
k
Vary resonator length
k-1
k
• Consider case where two
modes (wavelengths) are under
gain curve
• Two modes are orthogonal and
linearly polarized
• Matching the intensity of the two
modes provides feedback for
stabilization
• Varying tube length provides the
tuning mechanism
71
Laser is stabilized by changing tube length
Heater
power
supply
Controller
-
Detectors
s
p
Laser output
Beamsplitter
Heater
• Laser tube length determines wavelength
• Length is changed by varying temperature
• Stabilized by balancing intensity of two
adjacent modes
72
Two frequencies are commonly generated by two methods
Axial magnetic field
Circular linear
polarizations
f 1= f + f
f 2= f - f
Zeeman effect
Orthogonal
linear
polarizations
f+f f-f
Laser
tube
AOM
f 1= f + f
f 2= f
f
f
73
Accoustooptic
modulator
Two methods have some differences
• Zeeman method
• AOM
• Small difference
frequency (max. of ~
4MHz)
• Variation in the split
frequency from one
laser to the next
• Low laser output
power
74
– Bragg cell
– RF drive at split
frequency
– Greater split
frequency (~20MHz)
– Small variation
between lasers due
to crystal oscillator
Polarization states of laser output are critical
Nominally  to
base plane
Base plane
• Heterodyne source
Nominally  to
produces two orthogonal
base plane
linear polarizations at two
slightly different
frequencies (wavelengths)
• Polarizations are
Base plane
nominally perpendicular &
parallel to laser head base
• States must be linearly
polarized to prevent
mixing
75
Polarization states have slightly different frequencies
f1
f2
Base plane
• Two polarizations differ in
frequency by 20MHz
• For ZYGO systems f1 > f2
• f1 corresponds to
polarization  to base
• f2 corresponds to
polarization // base
• Corresponding
wavelengths are slightly
different
76
Split frequency determines the maximum target velocity
f
• Maximum velocity is
limited by permissible
drop in frequency
• Max. permissible drop
corresponds to a
Doppler shift that
drives signal
frequency to zero
• Usually limited to
some fraction of f

Stationary target
max
f

Target at max. velocity
77
Split frequency sets an upper bound on the target velocity
The maximum frequency shift max is some fraction k of f
 max  k f
The maximum velocity vmax is related to max by
vmax 
 max
N

 k f
N
N is an integer that depends on the interferometer
configuration. For a single pass system, N=2 and for a
double-pass system, N=4.
One pass = one back-and-forth trip of the measurement beam
78
Large split frequencies enable high velocities
 max  k f
For k=0.8 and f= 20MHz, max = 16 MHz
vmax 
 max
N
For a single pass interferometer N=2 and a wavelength =
633 nm, vmax  5 m/s
For a double-pass system, N=4, leading to an increase in
resolution but a decrease in vmax to  2.5 m/s
79
Laser heads have several key features
• Two-frequency output
• Larger split frequencies enable higher
velocities
• Two-frequencies are orthogonally polarized
• Polarization states are nominally linear
• Typical power output ~1.3 mW
80
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Beam directing
optics
Beam directing optics split and direct the source beam
• Fold mirrors turn beam
through 90
• NPBS split incoming
beams regardless of
polarization
Input
Output 1
Output 2
– Various split ratios
Non-polarizing beam
splitter (NPBS)
• Orientations of f1 and f2
can change on
passage through
directing optics
• Designed for specific
orientation
Input
Output
Fold mirror
82
Fibers route light from the source & to the detectors
• Polarization maintaining (PM) fibers convey
source beam from a remote laser source to
system
• Multi-mode fibers carry return signals from
the interferometer output to remote
electronics
83
Multi-axis systems require the source beam to be directed
to multiple interferometers
Delivery Module (DM)
PM fiber
Measurement Electronics
Laser
Module (LM)
Interferometers
PM fiber
NPBS
DM
Multi-mode fiber
84
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Optical
Components –
Basic Building
Blocks
Most standard interferometers are composed of some basic
optical components
•
•
•
•
•
Polarization beams splitter (PBS)
Quarter-wave plate
Plane mirror
Retroreflector (corner-cube)
Plane mirrors and retroreflectors can also
be used as targets
86
Polarization beamsplitter is the heart of an interferometer
p at f1
s at f2
s at f2
Polarization Beamsplitter
Splits incoming beam
based on polarization
state
• Creates reference &
measurement beams
p at f1 • Separates input
polarization states
• Polarization states
are called s & p &
defined relative to
plane of incidence
• Pneumonic: p passes
87
Retroreflectors (corner cubes) are insensitive to rotations
Output
beam
Input
beam
Offset
• Output beam parallel to
input regardless of
rotation about nodal point
• Output beam translates
at twice the rate as retro
• Hollow or solid
• Coated retroreflectors if
solid
• Can alter polarization
state
88
A quarter-waveplate is used to rotate polarization states
p
p
/4
45
LCP
s
s
RCP
• Changes the polarization state of a linearly polarized
beam to circular
• Two passes through it result in rotation of linear
polarization state by 90 (from p to s in above example)
89
Plane mirrors are used both in the measurement and
reference arms
Measurement beam
Direction of measurement
Plane mirror
Direction of travel
Surface Figure (flatness) /4 = 158nm
• Allows for translation perpendicular to optical axis
• Surface figure is critical if target translates perpendicular
to direction of measurement
• Deformed mirror produces spurious displacement
• Minimal tilt of the mirror is allowed
90
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Common Linear
Displacement
Interferometer
Configurations
Common interferometer configurations are discussed
• Two common configurations are discussed
in detail
• Emphasis is on understanding inner
workings
• Provide a basis for understanding other
configurations
• Not an exhaustive description of the
multitude of configurations in existence (or
possible)
92
The Michelson interferometer revisited
Fixed Mirror
Movable
Mirror
Monochromatic
light source
Beamsplitter
Detector
93
• Tilt sensitive and can
accommodate very
limited mirror tilt
• Mirror tilt causes
reduced beam
overlap
• Difficult to align and
maintain alignment
• Does not make
efficient use of light
from source
A typical linear displacement measuring system
Reference
Retroreflector
Two-frequency laser
Target
Retroreflector
Optical
fiber
Fiber optic
pickup (FOP)
PBS
Measurement signal
Digital
Position Data
Reference signal
Phase Interpolator
94
Notation is required for interferometer ray diagrams
Polarization
symbol
l= p pol.
= s pol.
Circular pol.
• Multiple attributes of
the beam need to be
represented
• Beam direction
Arrow head:
direction of
propagation
– Arrow direction
• Polarization state
f 1 ±  f1
Notation:
Base or altered
frequency
– Pol. symbol behind
arrow
Arrow color:
base
frequency
Green = f1
Red = f2
• Frequency (f1 or f2 or
altered frequency)
– Arrow color & notation
95
A closer look at the linear displacement interferometer
Reference
Retroreflector
# passes = 1
Scale factor = 1/2
N=2
f2
f1
f2
f1
Target
Retroreflector
f2- (f1 ± f1)
f1
(f1 ±  f1)
f2
• Most common linear
lengths
interferometer
• Retroreflector confers
• Thermally balanced –
immunity to rotation
same glass path
• Easy to align
96
A more compact variation – the single beam interferometer
# passes = 1
Scale factor = 1/2
N=2
Reference
Retroreflector
f2
f1
f2
f2- (f1 ± f1)
/4
f1
(f1 ±  f1)
Target
Retroreflector
97
• Similar to previous
configuration
• Smaller target retro
• More compact optics
• Small beams subject
to loss at retro apex
• Polarization state
varies over beam
Retroreflector based interferometers are unsuitable in some
applications
• Retro interferometers can tolerate very
limited motion  to measurement axis
• Unsuitable for applications where target
moves  to measurement (beam) direction
– X-Y stages
– Straightness measurement
• Plane mirrors are ideally suited
• Many interferometer configurations have
been designed for plane mirror targets
98
Plane mirrors interferometers (PMI) are designed be used
with plane mirrors
Reference
retroreflector
f1
f2
f2
f1
f1
f1 ±  f1
# passes = 2
Scale factor = ¼
N=4
Plane
mirror
f2- (f1 ± 2f1)
4
f 1 ±  f1
f1 ± 2 f1
• Plane mirror target permits translation  to
measurement direction
• Ref. arm is retro-reflected within interferometer
• Two passes through /4 rotates pol. state 90
99
PMI design confers tilt tolerance on interferometer
• Tilt of mirror results in shear of
measurement and reference beams rather
than misalignment
• Double pass interferometer
• Scale factor of ¼ (N=4)
• Not a symmetric design, beams travel
different paths and hence not as stable as
HSPMI (to be discussed)
• Temperature coefficient of ~300 nm/°C
100
High Stability PMI (HSPMI) is based on a symmetric design
Reference
plane
mirror
f2
f2
f2
f2
# passes = 2
Scale factor = ¼
N=4
4
f2
f1
f2
f2
f1
4
f1
f1 ±  f 1
Target
plane
mirror
f1 ±  f1
f2- (f1 ± 2f1)
101
f1 ± 2 f1
Symmetric design results in high stability
• Same resolution as PMI
• Thermally stable design; unlike PMI
reference path and measurement path
traverse the same amount of glass (but not
exactly the same path)
• Tolerates mirror tilt; results in shearing of
beams rather than misalignment
• Temperature coefficient 18 nm/°C
102
Mirror tilt is transformed to beam shear
Returning secondary
(Second-pass beam)
Beams shown
separated for
visualization
To FOP
f1
f2
Outgoing primary
(First-pass beam)
103
Beam overlap & signal strength change as the target mirror
tilts
Near Full Signal
~50%
No signal
• The measurement and reference beams must overlap in
order to provide a signal to the electronics
• As the target mirror rotates, the measurement beam will
shear across the reference beam
– Less overlap = decreased AC signal
104
Observed AC signal decreases with increasing beam shear
Signal Strength (%)
100%
1/e2 Beam
Diameter
80%
3 mm
4.6 mm
7.5 mm
60%
40%
20%
0%
0
1
2
3
4
Beam Shear (mm)
• Loss depends of beam size
• Larger beam, smaller signal loss
105
5
Tilt also contributes a displacement error
Displacement measurement
error (nm)
-10
-30
-50
-70
-50
-40
-30
Target distance
-20
-10
0
10
20
Mirror rotation angle (arcsec)
100 mm
200 mm
106
500 mm
30
40
1000 mm
50
Tilt error is function of distance of target mirror from PBS
• Larger target mirror distance results in
larger error
• Error is symmetrical, i.e., direction of tilt
does not matter
• Error is ~ few nm for typical mirror tilts (~ a
few seconds) characteristic of good quality
stages
107
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Measurement
Electronics
Measurement board measures phase change between
measurement and reference signals
• Converts phase change into digital output
• Electronic resolution up to /1024, i.e.,
subdivides 2 radians of phase into 1024
parts
• With one-pass interferometer:
• System resolution = (/1024)*(1/2) =  /2048  0.31
nm
• Maximum velocity  5 m/s
• Real time data rates >20MHz
109
Electronics can accommodate multiple axes
• Current technologies permit up to 64
channels from one laser source
• Each channel requires 70 nW for operation
• Electronics have low data age uncertainty
– Enable synchronization of axes for coordinated
motions
110
Interferometer output is optical
• Interference requires the two beams to have
the same polarization state
• Orthogonally polarized measurement and
reference beams combine at the exit of the
interferometer
• Orthogonal polarization states are combined
by a polarizer to create interference
• Detectors convert optical output to electrical
signals
111
Output of interferometer is coupled to electronics via fiber
• Numerous
advantages
Fiber
connector
Beam input
Fiber optic pickup (FOP)
– Eliminate heat
– Less cost
– Smaller size
• Consist of
Focusing
lens
Multi-mode
fiber
– Lens
– Polarizer oriented
relative to base
– Connector for multimode fiber
Polarizer at
45 to
incoming
polarization
states
112
Polarization states are combined prior to launch into fiber
• Fiber optic cables commonly used for
transfer of mixed signal to measurement
electronics
• Specialized fibers are needed to maintain
polarization states
• Above requirement is avoided by combining
the polarization states with a polarizer
before launch
113
Phase interpolation electronics convert optical output to
digital data
• Convert optical signals into electrical signals
and digitize them
• Measure the phase difference between a
reference signal and measurement signal
• Output phase change which corresponds to
displacement in units of counts
• Output of board needs to be scaled to
provide displacement in units of length
114
Electronics also provide additional functionality
• Outputs in various formats
• Programmable digital filters
• Provision for synchronization with other
devices
– Clock
– Digital I/O
• Cyclic error correction (CEC)
• Error checking
• Absolute phase
115
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Calculation of
Displacement
Displacement calculation requires additional information
Phase in
counts from
phase meter
Desired
displacement
1  vac
d
Nk n
Integer based
on number of
passes
# of counts/2
of phase
117
Vacuum
wavelength
Refractive
index of
medium
Phase value is obtained from measurement electronics
1  vac
d
Nk n
• Phase output from electronics is in units of
counts
• Electronics outputs the accumulated
phase from user specified zero
118
Phase value from measurement electronics is converted to
phase through a constant
1  vac
d
Nk n
• k is a constant that depends on
measurement electronics
• Represents the number of phase meter
counts/2 radians of phase
• Typical values of k are 512 or 1024
119
Vacuum wavelength is obtained from laser head specs.
1  vac
d
Nk n
• Wavelengths for the two frequencies from
a two-frequency laser are slightly different
• Appropriate wavelength value based on
which frequency is in the measurement
arm
120
Appropriate wavelength must be used to scale data
f1
f2
f2
• Wavelength used for
scaling depends on
which frequency (f1 or
f2) is in the
measurement arm
• Use 1vac & 2vac for f1
& f2 respectively
• Use of incorrect
wavelength results in
displacement error
f1
(f1 ±  f1)
1vac = 632.991501 nm
f2 f1
f1
f2
(f2 ±  f2)
2vac = 632.991528 nm
121
Direction sense is determined by the frequencies in the two
arms
f1
f2
f2
f1
• Disposition of
frequencies in each
arm depends on
– Laser head orientation
– Orientation of beam
directing optics
– PBS orientation
(f1 ±  f1)
Phase readout increases
f2
f1
f1
f2
(f2 ±  f2)
Phase readout decreases
• Interchanging
frequencies reverses
the direction sense
• Can be set in software
122
N is a constant that depends on the interferometer config.
1  vac
d
Nk n
One
pass
N=2
Pass 1
Two
passes
N=4
Pass 2
123
• N depends on the
number of passes of
the measurement
beam
• One pass is one
back-and-forth trip of
the beam
• N = 2 and 4 for linear
& plane mirror
interferometer
respectively
Changes in the refractive index change the wavelength
1  vac
d
Nk n
• Wavelength in the medium of operation 
is equal to vac only in a vacuum (n=1)
• For operation in a medium other than
vacuum, n must be known
• n is usually determined from an analytic
expression
124
Index of air is not a constant & depends on many factors
• Index depends on
– Pressure
– Temperature
– Humidity
– Composition
• Very sensitive to presence of hydrocarbons
• Hydrocarbon content is typically not
factored into analytic expressions
125
Index of air may be calculated using Edlen’s equation
• Relationship between the index and
pressure, temperature, humidity and
wavelength is given Edlen’s equation
• Complex equation
• Pressure, temperature, humidity and
wavelength must be known to calculate
index
• Typically obtained from measurements
126
Index of air may also be calculated from the following
approximation
n  n0 1  KT T  20 C   K P  P  760 mmHg   K H  H  50% RH  

 T  20 C


n0  Index at  P  760 mm Hg
  1.000271374
 H  50% Relative Humidity 


KT  0.95 ppm/ C
K P  0.36 ppm/mmHg
K H  8.6 ppb/%RH
Index values can also be obtained using the index calculator at
emtoolbox.nist.gov
127
Environmental inputs are typically obtained from a weather
station
• Weather station contains instrumentation to
measure environmental parameters
• Weather station may communicate directly
with measurement system
• Station location is critical
• Should be located close to measurement
beam path in order to sense environmental
factors in the space occupied by
measurement beam
128
Another method of tracking index changes is a wavelength
tracker
• Interferometric arrangement that utilizes
fixed length beam paths of known lengths
• Measurement beam passes through
medium of interest
• Reference beam passes through a vacuum
path
• Measures index changes relative to initial
environmental conditions
• Initial conditions provided by other means
129
Wavelength tracker is a differential interferometer
Air
Spacer
Vacuum
Interferometer
Spacer
Air
L
• Unlike a typical weather station used in
conjunction with Edlen’s equation, tracker
also tracks index changes due to
composition changes
130
Measurement beam is in air while reference is in vacuum
Input beam
Cell
Fiber optic pickup
DPMI
Beam in vacuum
Beam in air
131
Tracker does not measure absolute index, only changes
n  ninitial 
vac
4kL
• L is length of the cell
• As before, appropriate vac must be used
depending on the frequency in the
measurement arm
• Sign of measured phase corresponding to
a change in index also depends on
disposition of frequencies
• Initial index is calculated by other means
132
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Uncertainty
Sources and
Analysis
Interferometric displacement measurements have low
uncertainty compared to other methods
• While uncertainty is low in a relative sense it
is finite and can be estimated
• Number of contributors
• Example of a simple uncertainty analysis to
develop a feel for main sources
• Analysis based on the ISO Guide to
Expression of Uncertainty in Measurement
(GUM)
134
We will consider an uncertainty analysis of a displacement
measurement of a linear stage
Deadpath
Ldeadpath

LAbbé
Stage
Stage base
ds
Measurand is the displacement of the stage as
measured at point indicated at the surface of the
stage and indicated by ds
135
We will include the following sources of uncertainty
•
•
•
•
•
•
Wavelength
Refractive index
Phase meter output
Deadpath error
Abbe′ offset
Cosine error
136
We will neglect the following sources of uncertainty
• Cyclical errors due to
target
mixing
• Beam shear
• Data age uncertainty • Surface figure of
• Thermal expansion
target
– Interferometer
• Index gradients
– Target mount
• Inertia loading
• Air turbulence
• Vibration
• Effect of parasitic
• And a host of others…
motions
• Change in index of
137
We will assume the following uncertainty values &
parameters
S. No.
Parameter
Nominal value
1
Stage travel
250 mm
2
Deadpath
250 mm
3
Abbé offset
100 mm
S. No.
Uncertainty Source
Uncertainty
1
Wavelength
10 ppb
2
Temperature
2 C
3
Pressure
15 mm Hg
4
Humidity
20% RH
5
Pitch amplitude
5 arc-sec
6
Index in deadpath
2 ppm
7
Phase meter
1 LSB
138
Uncertainty in input values contributes to uncertainty in
measured displacement
Vacuum
wavelength
uncertainty
Uncertainty in
measured
displacement
vac
d

4 n
Refractive
index
uncertainty
Pressure
uncertainty
Humidity
uncertainty
139
Phase meter
uncertainty
Temp
uncertainty
Uncertainty in vacuum wavelength
• Represents the lack of knowledge of the
actual value of the wavelength
• Frequency stabilization of laser guarantees
that wavelength is stable
• Does not guarantee any particular value
• For critical measurements wavelength
should be measured or otherwise
accounted for
140
Bounding uncertainty can be estimated from physics of
HeNe laser
2 X 10-6 m
Laser threshold
k-1
k
• If no other information is available and the
HeNe laser produces red light, then the
wavelength uncertainty is ~ 3 ppm
• Consequence of the fact that the width of
the laser gain curve above the threshold is
~ 2 X 10-6 m
141
Some rules of thumb for index dependence on environment
At
• Temperature T = 20C,
• Pressure P = 760 mmHg and
• Relative humidity H = 50%
1 ppm change in index is caused by:
• T 1C
• P  3 mm Hg
• H  100%RH
142
Change in the index of air can be calculated from the
following approximation
n  KT T  KPP  KH H
KT  0.95 ppm/ C
T  Change in temperature
P  Change in pressure
and
K P  0.36 ppm/mmHg
H  Change in humidity
K H  8.6 ppb/%RH
143
Cosine error results from misalignment of beam and axis of
motion

Direction of motion
Actual displacement da
dm  da cos 
144
Measured displacement is less than actual displacement
• Not significant for typical applications until
misalignment is large
• Misalignment also causes shear of
measurement beam as a function of
displacement
• Beam shear reduces overlap between
measurement and reference beams
resulting in reduction in signal
145
Abbé error results from an offset between measurement
axis and axis of interest
Measured
displacement
dm
Pitch 
Measurement axis
LAbbé
Displacement axis
Stage displacement
ds
d m  d s  LAbbe′ tan 
 d s  LAbbe′ 
146
Abbé principle is a fundamental principle of metrology
• Axis of measurement must pass through the
axis of interest, i.e., the line along which we
wish to measure displacement
• If there is an offset, angular error motions of
the stage couple into the measurement
• Magnitude of uncertainty contributed scales
linearly with offset for a given angular error
147
Deadpath = Length between PBS and target retro at
interferometer zero
Point at which
interferometer
‘zero’ is set
Index variation
n
PBS
Dead path
Ldeadpath
Measuring Path
d deadpath  nLdeadpath
d deadpath  Displacement due to
change in index in deadpath
149
Dead path should be as small as possible
• Minimize errors due to refractive index
variations during a measurement
• Causes changes in separation between
zero point and PBS
• Consequence of fact that any compensation
only applies to displacement from zero
• Deadpath contribution can be minimized by
– Short deadpath
– Minimizing changes in index
150
Deadpath can be minimized by simple strategies
Range of motion
• Addition of a fold
mirror can help move
interferometer to a
more advantageous
position
Deadpath
Deadpath
Fold mirror
151
Uncertainty analysis is based on this model equation
vac  1 
n
ds 

  LAbbe′  2
Ldeadpath

4 n  cos   Abbe Error
n
′
Deadpath Error
Cosine Error
152
Results of uncertainty analysis
Uncertainty Source
Contribution (m)
Pitch error motion
Index
Cosine error
Deadpath
Vacuum wavelength
Phase meter
Combined Std. Uc
1.40
0.83
0.25
0.29
0.00
0.00
1.67
153
Dominant source of uncertainty is a function of setup
• Contribution from the pitch error motion
dominates as a result of the Abbé offset
• This contribution applies to any metrology
technique
• Index contribution dominates contributions
linked to interferometer
– Compensation can make a large difference
• Deadpath contribution is significant and
scales linearly with deadpath
154
Index contributions can be reduced
S. No.
Parameter
1
Uncertainty
Compensated
Uncompensated
Temp uncertainty
0.1C
2C
2
Pressure uncertainty
1 mm Hg
15 mm Hg
3
Humidity uncertainty
5% RH
20% RH
• Measure environment and compensate
• Uncertainty in environmental variables is
replaced with uncertainty in the
measurement of these variables
155
In critical applications, index effects can be reduced further
• Operate system in a helium atmosphere
– Helium has lower index sensitivity to
environmental variables
• Operate system in vacuum
– Consider all systems issues associated with
transition to vacuum
156
Compensation reduces index contribution drastically
Contribution (m)
Uncertainty Source
Pitch error motion
Index
Cosine error
Deadpath
Vacuum wavelength
Phase meter
Combined Std. Uc
Compensated
Uncompensated
1.40
0.06
0.25
0.29
0.00
0.00
1.45
1.40
0.83
0.25
0.29
0.00
0.00
1.67
157
Uncertainty analyses are a tool to identify significant
contributors
• Uncertainties associated with refractive index
typically dominate in uncompensated system
• Setup related contributions can usually be
reduced by careful alignment
– Abbé offset
– Deadpath
– Beam alignment to direction of motion
• Capability of a measurement technique should
be judged in the context of the measurement
uncertainty
158
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Specialized
Interferometer
Configurations
Specialized configurations extend the capability of linear
displacement interferometers
• Utilize the principles of linear displacement
interferometry to make other measurements
– Relational (differential) measurements
– Angle
– Straightness
• Emphasis is on introducing configurations,
not on detailed analysis
160
Column reference interferometers perform a differential
measurement between two parts of a machine
161
Column reference interferometer monitors relative
displacement between stage and lens column
Column
mirror
Column
reference
interferometer
(CRI)
Column
Stage
Stage
mirror
162
Column reference interferometer (CRI)
Retroreflector
/4 waveplates
Compensating
Plugs
PBS
Target and
Reference Mirrors
Fold Mirror
Steering Wedges
•
•
•
•
Performs a differential measurement
Reduces deadpath error
Steering wedges facilitate beam alignment
‘Folded’ HSPMI
163
Differential PMI makes meas. between two plane mirrors
164
DPMI can be configured to make differential displacement
measurement
PBS
Target
Shear plate
/4
/2
Reference
165
Some features of the linear DPMI
• Measurement is differential and permits a
short metrology loop
• Resolution is same as a two-pass PMI
166
…or differential angular measurements
PBS
Target
/4
Shear plate
/2
Reference
167
Some features of the angular DPMI
• Measurement is differential and permits a
short metrology loop
• Range is limited and varies inversely with
distance of target mirror from interferometer
and typically < ±1 
• Sub 0.01 arc-second resolution achievable
168
Routing of beams is complex and three-dimensional
DPMI - Linear
DPMI - Angular
169
Larger angular motions are handled by a dual-retro config.
f2
Beam bender
f2 ± f2
f1
f2

f1
f1 ± f1
f2 ± f2
PBS
f1 ± f1
Angular
Retroreflector
170
R
 vac 
  sin 

 2knR 
1
Some features of the angular interferometer
•
•
•
•
Range of ± 10
Resolution < 0.1 arc-second
Insensitive to pure displacement
Commonly used in machine tool metrology
applications
• Used for rotary table calibrations with
appropriate fixturing
• Care required in setup to achieve lowest
uncertainty
171
DMI can be used to measure straightness of an axis
Dihedral
mirror
f1, f2
f1 ± f1

f2 ± f2
Wollaston
prism
 vac    
 
 sin  
 4kn   2 
172
Straightness
error motion

Some features of a straightness interferometer
• Wollaston prism is a birefringent prism that
splits the two polarization states at an angle
• Different sets of optics for short and long
travel ranges
• Dihedral angle  is typically ~1.6 and
~0.16 for short and long travel ranges
respectively
173
Optical probe config. is ideally suited for small targets
Reference
retroreflector
/4 Lens
Target
mirror
PBS
• Single-pass interferometer
• Reference beam reflects off vertex of retro
• Beam routing behavior same as single
beam interferometer
174
Some features of the optical probe configuration
• Ideal when only a small target mirror can be
used
• Range is determined by depth of focus of
lens and can be as large as ± 5 mm (f = 150
mm)
• Focal length determines standoff
• Spot size ~ 100 m
• Signal strength is a strong function of
displacement from focus
175
Fiber fed interferometers have advantages in some
applications
• Remote laser source
• Laser radiation is transported to system via
optical fibers
• Eliminates heat load from laser
• Improves flexibility in terms of optical
‘plumbing’
176
Fiber fed heterodyne interferometers must preserve the
input polarization states
Heterodyne Laser
Source
Remote Laser
Head
Polarization Maintaining (PM) Fiber
Split frequency
generator/polarizer
Beam
expander
Delivery Module
177
Expanded beam
to optics
Other configurations abound
• Literally hundreds of interferometer designs
exist
• Many different approaches to same
measurement problem
• Once the basics are understood, the basic
building blocks can be used to create
numerous custom designs.
178
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Application
Examples
Typical DMI Applications
• Calibration (static)
– Machine tools
– Stage calibration
– X/Y stages
– CTE
• Production (dynamic)
– Semiconductor instruments
– Feedback for diamond turning
– X/Y stage control
180
Use as feedback sensors for motion control is a very
common application
X-Y reticle
stage
Interferometers
X-Y wafer
stage
181
Wafer processing requires measurement of multiple DOF
• Accurate monitoring of X and Y position and
rotation
• Require simultaneous measurement of
multiple degrees of freedom
– Many lithography tools use ~30 axes of DMI per
system
182
Used as feedback for machine tools & measuring machines
Large Optics Diamond Turning Machine (LODTM) at Lawrence Livermore
183
Interferometer system for the LODTM
184
Stage calibration is another common application
Stage Error
500
Position Error (m)
DMI Position (mm)
DMI Calibration Data
250
0
250
500
Commanded Position (mm)
20
15
10
5
0
100
200
300
400
500
Commanded Position (mm)
• Determination of stage errors
• Generation of an error maps
185
Another application involves use as an indicator
HSPMI
Laser out
Reference
mirror
Laser in
Straightedge
Machine stage
186
Machine tool metrology is an example of ‘strap-on’
metrology
• Characterize various
error motions of a
machine tools
• Many different types
of interferometers
• Numerous
accessories
• Measurements made
between tool &
workpiece
Laser Head
Target
DMI
187
Many machine parameters may be evaluated
•
•
•
•
Linear displacement accuracy
Straightness
Angular error motions with exception of roll
Squareness when used with appropriate
accessories
– Optical square
• Dynamic performance can be measured
188
Rotary tables may be calibrated with DMIs
• Variety of configurations depending on
range of angle
• Extremely high angular resolutions (< 0.1
arc-sec)
• All interferometer configurations require
specialized fixturing if calibration is required
over 360
– Hirth coupling based indexer
189
Dilatometry and material stability measurements use DMIs
• Used in setups for measurement of
– Thermal expansion
– Material stability
– Stability of epoxy joints
• Very stable when operated in vacuum
• High resolution critical for sensing small
changes
190
A setup for the measurement of material stability*
• Interferometric
metrology for the
measurement of
material stability
• Interferometers
operate in vacuum
* Patterson, SR., “Interferometric
Measurement of the Dimensional Stability
of Superinvar,” UCRL-53787, LLNL,1988.
191
Other applications
• Actuator calibration
– PZT, electrostrictive, linear motors, capstan
drives, etc.
• Gage calibration
– LVDT
– Capacitance gages
– Encoders
• Vibration analysis
192
Some general comments about DMI applications
• Only a sampling of possible applications
• Numerous other applications possible
• Setup & procedure are critical to good
results
• Minimize geometric errors by design
– Abbé, cosine & deadpath
• Stable environment
• Minimize total measurement time
• Compensate for index change
193
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Summary
DMIs are versatile devices
• Measure at the point of interest
– Eliminate Abbé offsets
•
•
•
•
•
•
High resolution, velocity & low uncertainty
Non-contact
Directly traceable to the unit of length
Many commercial configurations exist
Configured for many geometries
Measure multiple degrees of freedom
simultaneously (64 with one laser head)
195