Optical Interferometry
and Industrial Interferometers
- a Tutorial Friedemann Mohr
Pforzheim University of Applied Sciences
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
1
Outline
1 Physical Basics and components
2 Interferometry for path measurement
3 Laser vibrometry for vibration
measurement
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
2
The photoelectric conversion process
W
W

Number of photons impinging: n 
Wph h
Gives total charge:
q  e
n
W
W

Wph h
W
h
 e dW  e


P R P
h dt
h
Current:
i
where
R
e e

  = Responsivity, Sensitivity
h h c
Interferometry ITSS 2007
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3
Detecting a light wave
 1  
S  E  H * Total power:
2
Poynting vector

E

H
Z0
E
A
(where Z0=characteristic impedance of medium) 
S
P


P   S  dx dy
1
2 Z0

A
2
A 2
E dx dy 
E
2 Z0
i  R P R 
H
2
A 2
E  K  E = photo current
2 Z0
2
I  E = intensity or, irradiation
using E  E 0  e
j( t  kz)
 E0  e
j( t  )
i
S

i  K  E0
2
i.e., detection of light is a nonlinear process.
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
A
4
Components I: Lenses, beam transformation
The
telescope
D2 f 2

D1 f 1
f1
f2
Gaussian
beam
transformation
w 
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci

is an invariant

5
i
Components II:
Mirrors and retroreflectors
r
a
'i1
1
b
'r2
2
c
Interferometry ITSS 2007
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6
Components III:
Lossless
beamsplitters
2
2
2
(Special case of 3dB power splitters: E1  E 2 
2
1
2
E 0 whence
2
2
E1  E 2 
(1)
1
2
E 0  E1  E 2
Field continuity:
E0
2
E0  E1  E2
Power conservation: P0  P1  P2 or,
E0 )
(2)
2
2
 E*0  E 0  ( E1*  E*2 )  ( E1  E 2 )  E1* E1  E*2 E 2  E1* E 2  E*2 E1  E1  E 2  E1* E 2  E*2 E1
Comparison with (1):

E1*E 2  E*2 E1  0

E1 e  j1 E 2 e j2  E 2 e  j21 E1 e j1  E1 E 2  e j( 1 2 )  e  j( 1 2 )  2 E1 E 2  cos(1  2 )  0 .
I.e., must be:
1  2  
Interferometry ITSS 2007


or 1  2  
2
2
Both output waves are in quadrature!
Friedemann Mohr  Pforzheim UnivApplSci
7
Components IV:
Lossless beamsplitters and
their technical
realisation
E0
E1
bulk
E2
E0
E1 
E2 
E0
e
2
E0
2
Interferometry ITSS 2007
j
E2

4

j
e 4
E1
E1
E0
E2
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fiber
optic
integrated
optic
8
Components V:
Lossy
beamsplitters
Results achieved
above are valid for
purely dielectric layer.
However: Metal has
complex refractive
index
n̂  n  ik .
Substrate refractive index assumed was nsub=1.5
(Raine, Downs, 1978)
Interferometry ITSS 2007
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9
y
Polarisation I:
Polarisation ellipse
and Jones vector

a
b
E-field of a wave can, at a fixed position

x
= arctan b/a


in space, be described by the Jones vector
  E x   Ê x  e j( t  x ) 
E 
j( t  y ) 

 E y   Ê y  e
a) Linear polarisation in x direction (elevation angle 0):
Ê y  0 ,
b) Linear polarisation in y direction (elevation angle 90°): Ê x  0 ,
c) Linear polarisation with 45° (135°) elevation angle:
d) Circular polarisation:
Interferometry ITSS 2007
, y  x (  y  x   )
Ê x  Ê y ,  y   x   2
Ê x  Ê y
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10
Propagation of a wave can, generally, be
described by the Jones matrix
Polarisation II:
Propagation and
Jones matrix
Polariser

 A B 
E 2  J  E1  
  E1
 C D
1 0
0 0
,
J (0)  
J
(
90

)


0 1 
 0 0


 cos2 
sin  cos 
J ( )  

sin 2  
sin  cos 
Retarder,
here with specific
retardation of
D
(quarterwave
plate)
Interferometry ITSS 2007
0  1 0 
1

J (0)  
j  
,
2
0 e  0  j 
0  1 0
1

J (90)  
j  

0 e 2  0 j 

1  1
J (45) 

2 e j 2



e 2   1 1 j 



2  j 1
1 
j
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11
Polarising beam splitters (PBS)
s
s
p
left: crystal type
(Wollaston prism)
Interferometry ITSS 2007
p
right: thin film type
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12
Some characteristic polarisation states
y
y
y
x
x
y
y
x
x
x
x
y
top:
linear || x,
linear || y
bottom: linear, +45°, linear -45°
Interferometry ITSS 2007
y
y
x
top:
circular, RH,
bottom: elliptical, RH
Friedemann Mohr  Pforzheim UnivApplSci
x
circular, LH
elliptical, LH
13
lost
radiation
rear
mirror
used radiation
glass tube
front
mirror
a
LR
laser
threshold
b
He-Ne
laser for
interferometry I
1,5 GHz
0
a tube design
c
600
MHz
b gain curve
M+2
M+1
M
M-1
M-2
c mode scheme
d
d real modes
due to b and c
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
14
He-Ne
laser for
interferometry II
0
Laser
heating
foil
NBS
f0+300MHz
f0-300MHz
QWP
PBS
D1
C
(POL)
+
-
Interferometry ITSS 2007
D2
NBS
POL
PBS
D1, D2
C
neutral beam splitter
polariser
polarising beam splitter
detectors
control circuit
Stability improvement:
~ 10-6  ~10-9
Coherence length:
km‘s
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15
1 Physical Basics and components
2 Interferometry for path measurement
3 Laser vibrometry for vibration
measurement
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
16
The historical Michelson-Morley experiment I
Aim: Proving the existence of the ether
Interferometry ITSS 2007
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17
The historical Michelson-Morley experiment II
N
N
z
Source
Source
x
Det
a
y
Det
S
b
S
Approach: Verification of Doppler
effect on speed of light using highresolution phase measurement
Interferometry ITSS 2007
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18
1
Laser
I0
2

MachZehnder
interferometer I
1
1
I1
D1
2
1  k  z1 , 2  k  z 2
E11  E 0 e
jt
E12  E 0 e
jt
D2
 1 j 
 1  j  E
 jkz 1
4


e e
e 4   0 e jt e  jkz 1
 2

 2
 2
 1  j 
 1 j  E
 jkz 2
4



e
e
e 4   0 e jt e  jkz 2
 2

 2
 2
E1  E11  E12 
I1  E1
2

E 0 jt  jkz 1
e e
 e  jkz 2
2
 E1*

E 02
 E1 
 1  cos k ( z 1  z 2 )
2
Interferometry ITSS 2007
I2
E 21  E 0 e
jt
E 22  E 0 e
jt


 1 j 

j  E
j
E
 jkz1 1
0 jt 2  jkz1
4
4




e e
e 
e e e
 j 0 e jt e  jkz1
2
 2

 2
 2

 1  j 
 1  j  E
j
E
 jkz 2
jt
0
4
4


e e
e 
e e 2 e  jkz 2   j 0 e jt e  jkz 2
2
 2

 2
 2
E 2  E 21  E 22  j

E 0 jt  jkz 1
e e
 e  jkz 2
2
2
I 2  E 2  E*2  E 2 

E02
 1  cos k(z1  z 2 )
2
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19
1
Laser
I0
MachZehnder
Interferometer
1
2

1
I1
D1
2
D2
I2
I1
I2


Interferometry ITSS 2007

D
I1 
I0
 1  cos( D)
2
I2 
I0
 1  cos( D)
2

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20
MZI arrangement for path measurement
stabilized
Laser
D2
NBS
D1
x
Problem:
No directional
information !!
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
21
Mach-Zehnder interferometer with directional sensitivity I
PBS
Laser
phi 1
Basic Arrangement of
polarisation interferometer
PBS
phi 2
Superposition of
2 orthogonally
polarised waves
yields not only
output intensity
but polarisation
ellipse.
Polarisation ellipse carries two
informations:
Examples of polarisation states at output port
Shape and
elevation angle.
D
D
Interferometry ITSS 2007
D
D1
D1
Friedemann Mohr  Pforzheim UnivApplSci
Or:
Phase difference
and direction.
22
Mach-Zehnder interferometer with directional sensitivity II
PBS
PBS = polarising
Laser
beam splitter
NBS = neutral
beam splitter
QWP = quarter wave plate
phi 1
I2
I2'

-

D
PBS
phi 2
D2
NBS
D2'

PBS
QWP
D1'
D1
I1'
D1's detect circularity

-
Interferometry ITSS 2007
I1

D

of polarisation ellipse
D2's detect orientation
of polarisation ellipse
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23
Scalar case, from above:
Calculating
interference
taking into
account
polarisation
E11  E 0 e
jt
E12  E 0 e
jt
 1 j 
 1  j  E
 jkz 1
4


e e
e 4   0 e jt e  jkz 1
 2

 2
 2
 1  j 
 1 j  E
 jkz 2
4

e

e
e 4   0 e jt e  jkz 2
2
2



 2
E1  E11  E12 
2

E 0 jt  jkz 1
e e
 e  jkz 2
2
I1  E1  E1*  E1 

E 02
 1  cos k ( z 1  z 2 )
2
Vectorial case, requires Jones calculus:

E11 

E11 
  

J c  J b  J a  ...E0  e jt
  

J c  J b  J a  ...E0  e jt
 

E1  E11  E12
 2  
   
   
I1  E1  E1  E1  E0  J a  J b  J a  ...  J c  J b  J a  E0
Calculation of I2 in an analogous way….
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
24
Processing of directional signals I
Interferometry ITSS 2007
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25
Processing of directional signals II
Stabilized
Laser
D1
D1'
PBS
QWP
D2'
PBS
x
D2
1+cos
1-cos
+
-
2cos
+
-
2sin
1+sin
1-sin
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
26
Environmental factors
Path is measured in multiples (or fractions) of wavelength
Problem: Wavelength is dependent
2
on environmental factors:
D 
 Dz

- temperature, J
0
- atmospheric pressure, p
where


and n  n(J , p, F , G )
- humidity factor of air, F
n
- gas content of air, G
Solutions:
- measure all parameters (J, ...), calculate n, compensate arithmetically
- measure n directly
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
27
MZI for tooling machine calibration
inkrementales
Meßsystem
LaserInterferometer
X
Zähler
Zähler
N IS
N LI
B IS
B LI
X ist
Interferometry ITSS 2007
Istposition
Sollposition
X soll
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28
MZI based mask positioning
in semiconductor industry
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
29
1 Physical Basics and components
2 Interferometry for path measurement
3 Laser vibrometry for vibration
measurement
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
30
Heterodyne interferometer for laser vibrometry
PBS
QWP
target
Laser
fB
AOM
camera lens
D1
D2
NBS
PBS=polarising beam splitter
NBS=neutral beam splitter
QWP=quarter wave plate
AOM=acoustooptic modulator
Interferometry ITSS 2007
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31
Acoustooptic modulator (Bragg cell)
Interferometry ITSS 2007
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32
Doppler shift
f1 = f0 (1 + v/csound)
f1 = f0 (1 - v/csound)
f0
Laser
f1
f1 = f0 (1 + 2 v/clight) = f0 + fD
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
33
Heterodyne Interferometer: Calculation of Interference
PBS
QWP
target
Laser
fB
AOM
camera lens
D1
D2
NBS
PBS=polarising beam splitter
NBS=neutral beam splitter
j (  0  B ) t
0
QWP=quarter wave plate
E
E
1
1
; E12  0 e j (0 D )t 
e

AOM=acoustooptic
2 modulator 2
2
2
E11 
E1  E11  E12 

E0 j (  D   B ) t E 0 j (    D ) t
e

e
2
2


E 0 j D t
e
 e j B t e j 0 t
2
E02
I1  E1  E  E1 
 1  cos(B  D )t 
2
2
I 2  E2
Interferometry ITSS 2007
2
*
1
E02
 E  E2 
 1  cos(B  D )t 
2
*
2
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34
Operating range of heterodyne vibrometer
fB
f
fB+fD
f
fB+fD
f
•
Operating range depends on
Bragg frequency
•
typically, fB = 40 MHz
•
|v|=10 m/s corr. to |fD| = 32MHz
operating range
40MHz
40MHz +/- 32MHz
Interferometry ITSS 2007
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36
Vibrometer block diagram / velocity decoder block
Laser-interferometric
measurement head
fB
Master
oscillator
fB
Vibrating
target
(f B+fD)
Local
oscillator
fLO
i.f. = (fB + f D) - f LO
FM demodulator
Velocity
output
Mixer
Range
setting
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
38
Vibrometer block diagram / fringe counter block
Laser-interferometric
measurement head
fB
(f B+fD)
cos
Master
oscillator
fB
Local
oscillator
fLO
sin
Vibrating
target
i.f. = (fB + f D) - f LO
cos(i.f.)
up/down
counter
digital/
analog
converter
Displacement
output
sin(i.f.)
Range
setting
Interferometry ITSS 2007
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39
Vibrometer head Polytec design II
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
40
3
10
Operating
range diagram
of a laser
vibrometer
1
-3
10
10
3
displacement (m)
-6
10
10-9
1
velocity (m/s)
-12
10
9
10
-3
10
10
6
3
10
-6
10
1
acceleration (m/s²)
-3
10
-9
10
-3
10
1
10
3
10
6
vibration frequency (Hz)
grey area with red bounds: : operating range of velocity decoder (FM decoder)
white area with dotted bounds:: operating range of displacement decoder (fringe counter)
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
41
Measurement example #1 for single point mode
Laser
Vibrometer
Vibr.Generator
Spectrum Analyser
top: loudspeaker drive signal
center: velocity decoder output
bottom: displacement decoder output
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
42
Measurement example #2
HD drive dynamic measurements:
With stationary disk the R/W head
touches the disk. With rotating disk
the head is flying over the disk
(hydrodynamic lubrication)
Lowest possible flight height gives
best storage density. Optimum is h=0.
Vibrometer serves for measuring /
optimizing resonance characteristics
of flight control system
by courtesy of Polytec GmbH
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
43
Application example #3
Valve position
measurement in
automotive
industry using
fiberoptic vibrometers
Using two fiber heads,
differential velocity
measurement between
two points is possible.
Interferometry ITSS 2007
by courtesy of Polytec GmbH
Friedemann Mohr  Pforzheim UnivApplSci
44
More vibrometer application aspects








Measure body sound contactlessly and with high precision
Avoid mass loading of DUT
Acquire many data points in short time
Measure from points otherwise difficult accessible
Be widely independent from material properties
Measure from smooth, hot, minute, intricate structures
Measure high-frequency vibrations
Measure from large distances…..
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
45
Laser vibrometer in scanning mode
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
46
Fiber sensor coil deformation under forced vibration
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
47
Scanning vibrometer measurement example
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
48
Summary
1 Physical basics and components
2 Interferometry for path measurement:
Operating concepts and applications
3 Laser vibrometry for vibration measurement:
Operating concepts and applications
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
49
Vibrometers - More Applications
• Medicine – Ear drum, hearing functions, heart
• Zoologie – Elephants, insekts, spider webs
• Household – Washing machines, vacuum
cleaners, shavers
• Entertainment – Loudspeakers
• Military – Guns, mines
• Civil engineering – Buildings, bridges
• ..........and much more
by courtesy of Polytec GmbH
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
50
Processing of directional signals III
Forward
F  (C  S )  ( C  S)  ( C  S )  (C  S )
C
-2pi
pi
-pi

F / RCounter
2pi

S
R  ( C  S )  (C  S)  (C  S )  ( C  S)
-2pi
-pi
pi
2pi
Reverse
Interferometry ITSS 2007
Friedemann Mohr  Pforzheim UnivApplSci
51