interfere

1. G. Giuliani - INTERFEROMETRY_02 1
ELECTRO-OPTICAL
INSTRUMENTATION
Guido Giuliani
Università di Pavia
e-mail: guido.giuliani@unipv.it
Recommended textbook:
S. Donati, “'Electro-Optical
Instrumentation”, Prentice Hall, 2004

2. G. Giuliani - INTERFEROMETRY_02 2
INTERFEROMETRY - Summary
1. Principle
2. Applications
3. Basic configurations
 Michelson – dual beam
 Michelson – dual frequency
4. Extension to
mechanical/geometrical
measurements
5. Limit performance
6. Speckle Pattern & operation on
diffusive surfaces
7. Vibrometers: LDV
(Laser Doppler
Vibrometer)
8. Self-Mixing Interferometry
9. ESPI (Electronic Speckle
Pattern Interferometry)
10. White-light interferometry
(profilometry)
11. Interferometers for gravitational
waves

3. G. Giuliani - INTERFEROMETRY_02 3
5. Performance Limits
 The interferometer performance can be limited by several factors and
causes:
 Limits in the plane Displacement (Velocity) vs. Frequency
 Cosine error
 Limited coherence length of the laser source (temporal coherence limit)
 Quantum noise
 Spatial coherence and polarization effects
 Dispersion of the propagation medium
 Thermodinamic phase noise
 Speckle-Pattern errors

4. G. Giuliani - INTERFEROMETRY_02 4
Performance Limits
Limits in the plane Displacement (Velocity) vs. Frequency
 “Type 1” limits define the minimum
measureble displacement, which is
often coincident with the resolution.
Cause:
 Quantization (e.g.: /8 counting)
 Photodetection noise (typ.: quantum
noise)
 “Type 2” limits define the maximum
measurable velocity
Cause:
 Limited bandwidth (B), or limited real-
rime processing capabilities of the
electronics
  it limits the detection of the Doppler
frequency shift: fdoppler = 2(v/c) < B
 “Type 3” limits (low-frequency) is more
severe for the dual-beam scheme
 baseband signal processing  prone to
EMI disturbance
1 Hz 1 kHz 1 MHz
1 nm
1 m
1 mm
DISPLACEMENT
(peak
amplitude)
FREQUENCY
SPEED
(m
/s)
1 m
0.006
0.6
60
6000
NOISE or Quantization
HF
cutoff
Doppler lim
it
low-frequency
displacement interferometers
vibrometers
2
1
1
3

5. G. Giuliani - INTERFEROMETRY_02 5
Performance Limits
Consine error
 Output photocurrent: Iph = I0 {1 + Vcos[2k(sm-sr)]}
  The effectively measured pathlength is: ksmcos()
Where:  = angle due t residual alignment errors
systematic error on the scale factor (responsivity) of the interferometer
 In addition: variations of the amplitude of the fringes
and of the fringe visibility (0 < V < 1) can cause measurement errors
 Missed fringe countings
 Larger influence of EMI disturbance
He-Ne freq
stab laser
I
I
1
pol,0°
2
PD
PD
pol,90°

/8 plate
reference
corner cube
measurement
corner cube
sm
r
s
BS
x
y
linear pol
at 45°
k


6. G. Giuliani - INTERFEROMETRY_02 6
Performance Limits
Coherence length of the laser source - 1
 The e.m. field emitted by a real laser shows
phase jumps at random time instants
 The average duration of the interval between two subsequent phase jumps is
called coherence time (c)
 Definitions:
 Coherent length: Lc = cc [m]
 Linewidth:  = 1/c [Hz]
 is the FWHM of the power spectrum of the e.m. field
 Typical values
 He-Ne laser: Lc = 300 m  = 300 kHz (c = 1 s)
 Semiconductor laser (good) Lc = 30 m  = 3 MHz (c = 100 ns)
 Semiconductor laser (poor) Lc = 1.5 m  = 60 MHz (c = 5 ns)
E
t

0
Optical
power
spectrum


7. G. Giuliani - INTERFEROMETRY_02 7
Performance Limits
Coherence length of the laser source - 2
 The coherence length of the laser must be compared with the difference (unbalance)
between the reference path and measurement path of the interferometer L = |sm-sr|
 Interferometric signal:
Iph = I0 {1 + Vcos[2k(sm-sr)]}
 fringe visibility V = exp[-(L/Lc)]
 |sm-sr| >> Lc  no interferometric signal is available (only noise)
it is not possible to perform interferometric measurements
 |sm-sr| < Lc  interferometric measurements are OK
BUT: phase noise can be generated
 Non-zero linewidth (0)  the instantaneous laser frequency shows
temporal fluctuations  (t) = 0 + (t)
  also the interferometric phase shows temporal fluctuations:
(t) = 2k(sm-sr) = (4/)(sm-sr) = (4(t)/c)(sm-sr) = (4/c)(sm-sr)[0+(t)] = 0+(t)
 We have then a phase noise, whose variance is:
 = (4/c)(sm-sr) = 2  2(sm-sr)/0  /0
  we have an uncertainty s of the measurand (sm-sr):
s = /2k = (sm-sr)  (/0) = (0/)  (sm-sr)/Lc
V
L =|sm-sr|
Lc
1

8. G. Giuliani - INTERFEROMETRY_02 8
Performance Limits
Quantum noise - 1
 An interferometer can be used a Vibrometer: an instrument capable of measuring
small displacements (<< ), having zero-mean
 Technique:
 The “slow” (DC) phase difference between the
reference and measurement pathlengths
is kept constant (by some specific method)
 The interferometer operates
around the point of maximum sensitivity
(half-fringe, linear part of the Iph vs.  characteristic)
  the system acts a linear transducer
of small displacements (<< )
 The minimum measurable displacement is limited by the photocurrent noise
(NED = Noise Equivalent Displacement)
 Photodiode + load resistance R  the noise can be expressed as:
i2
n = 2q(Iph+Id)B + 4kTB/R  2qIphB = 2qI0B [A2]
(the term I0 = P = (q/h)P can be made arbitrarily large, by increasing P)
 The term 2qI0B is called quantum noise , or shot noise
 NOTE: the phase noise limit could be worse than the quantum noise limit !!!!
 = 2ks
t
t
Iph
I0

9. G. Giuliani - INTERFEROMETRY_02 9
Performance Limits
Quantum noise - 2
 Iph = I0 {1 + Vcos[2k(sm-sr)]}
 Signal term: Is = (I0V)2ksm
 Signal-to-noise ratio:
(S/N) = I2
s/i2
n = (I0V2ksm)2
/ 2qI0B
 The NED can be found by letting
(S/N) = 1 and solving for sm
 NED = (/2V)(qB/2I0)1/2 =
= (/2V)(hB/2P)1/2
 It is possible to calculate the
equivalent quantum phase noise:
n = 2kNED = V(2hB/P)1/2
 Letting:
P = 1 mW ; B = 1 Hz
 the performance limits are:
10-8 rad and 1 fm
10
10
10
10
-5
-3
-7
-9
1 100 10k 1M 100M
Measurement Bandwith B (Hz)
Phase
noise

(rad)
or
1/(S/N)
ratio
n
633 nm =0.9,
V=1
Equivalent
Power
P (W)
0.1 W
10 W
1 mW
100 mW
10 W
1 nm
1 pm
1 fm
NED
-
noise-eqiovalent-displacement
(m)

10. G. Giuliani - INTERFEROMETRY_02 10
Performance Limits
Comparison: phase noise / quantum noise
 Example 1
He-Ne Laser: P=1mW; =300kHz (Lc=300m);
 Phase noise: s = (0/)  (sm-sr)/Lc = 0.066 nm (for |sm-sr|= 0.1 m )
s = 6.6 nm (for |sm-sr|= 10 m )
 Quantum noise: NED = 1 fm (for B = 1 Hz)
NED = 1 pm (for B = 1 MHz)
  whenever the pathlength difference is NOT kept  0, then the phase noise is
the main limit to the interferometer performance
 Example 2
Semiconductor laser: P=10mW; =3MHz (Lc=30m);
 Phase noise: s = (0/2)  (sm-sr)/Lc = 0.33 nm (for |sm-sr|= 0.1 m )
s = 3.3 nm (for |sm-sr|= 1 m )
 Quantum noise: NED = 0.31 fm (for B = 1 Hz)
NED = 0.31 pm (for B = 1 MHz)

11. G. Giuliani - INTERFEROMETRY_02 11
Performance Limits
Spatial coherence and polarization effects
 Need for spatial coherence: it is necessary that the transverse spatial distribution of the
fields Em(x,y) e Er(x,y) onto the photodetector be the same
 Spatial coherence factor:
sp = ∫AEm(x,y) Er*(x,y)dxdy /[∫A|Em(x,y)|2dxdy  ∫A|Er (x,y)|2dxdy]1/2
 sp  1 only for the case of single-mode beams with the same diameter (and radius of
curvature, in the gaussian beam approximation)
 With multi-mode beams (thet contain N modes each) only modes with the same spatial
distribution contribute to sp  sp  1/N
 The two beams must have the same polarization state (linear, circular, or elliptical)
 Polarization factor:
 pol = EmEr/(|Em||Er|)
 All the above effects, combined, define the final effective visibility:
V = (sp pol)exp[-(L/Lc)]

12. G. Giuliani - INTERFEROMETRY_02 12
Performance Limits
Dispersion of the propagation medium
 A laser interferometer with beam(s) propagation in air performs measurements with a
scale factor related to: /nair
 For a He-Ne laser it is possible to determine  with a precision of 8 digits
 Variations of nair ?
 In standard conditions (T = 15°C, P = 760 mbar):
(nair –1)st = (272.6 + 4.608/(m) + 0.061/(m)
2)10-6 = 0.000280 (@  = 632.8 nm)
 Effects of pressure: the quantity (nair –1) is proportional to the number of moles per unit
volume  n/V = P/RT:
nair –1 = (nair –1)st (P/760)(288/T)
 The coefficients that account for variations of nair upon temperature and pressure changes
are:
d(nair –1)/dT =- (nair –1)st (288/T2) ≈ -1 ppm/°C
d(nair –1)/dP =- (nair –1)st (1/760) ≈ + 0.36 ppm/mbar
 Variation of T=10°C and P=10mbar influence the 5th and the 6th digit of the
displacement measurement
 to achieve an acuravy better than 10-6
, temperature and pressure sensors must be used
to achieve the correct value of the scale factor of the interferometer

13. G. Giuliani - INTERFEROMETRY_02 13
6. Speckle-Pattern
Operation on diffusive surfaces
 In many practical cases it is impossible to use the interferometer onto a
cooperative target
 The operation of laser interferometers (and of most laser instrumentation)
on targets with rough, diffusive surfaces involve the phenomen of
speckle-pattern
 When light with high temporal and
spatial coherence is projected onto a
diffusive surface, the back-diffused light
has a granular structure, similar to a bi-
dimensional white noise
 This phenomenon is called
speckle-pattern
 Speckle = small point, or colored stain
He-Ne laser on paper

14. G. Giuliani - INTERFEROMETRY_02 14
Speckle-Pattern
Origin of the Speckles
 The speckle-pattern is the field emitted in a semi-space by a diffuser illuminated by
coherent light
 Diffusing surface: it has random height variations, with amplitude z >> 
laser beam
D
z
diffuser
z
s t
l
s
P
P+P'
P+P''
E(P)
E(P+P')
E(P+P'')

x
y
z
_
_
_
 The resulting field in point P results from
the sum of many vectors, each being
originated by a different point of the
diffuser  the phase relation between
the different contributions is
 A displacement from P towareds P+P’
or P+P’’, implies that the field in these
points gradually loses coherence with
respect to the field in P
 The spatial contour of a single speckle
(grain) is defined as the volume where
the fields correlation with point P is >
0.5
 Individual speckle grains take the
shape of an ellipsoid, with the major
axis aligned towards the center of the
diffuser area illuminated by the laser

15. G. Giuliani - INTERFEROMETRY_02 15
Speckle-Pattern
Speckle size - 1
 Transverse and longitidinal size of the speckle
grains are statistical variables
 we are interested in knowing the average
values of the longitudinal dimension sl (along
z) and of the transverse dimension st (in the xy
plane)
 For a diffuser with circular laser illuminating
spot with diameter D, it is:
st = z/D; sl = (2z/D)2
with z = distance from the center fo the diffuser
 The longitudinal dimension is much larger than
the transverse one
 The projection along the normal axis of
speckle grains that lie outside the normal is
identical to that of the in-axis speckle grains
laser beam
D
diffuser
z st
sl

x
y
z

16. G. Giuliani - INTERFEROMETRY_02 16
Speckle-Pattern
Speckle size - 2
 Each speckle can be considered as a volume corresponding to a single spatial
mode, with acceptance a=A = 2
Demonstration:
 Acceptance: a = Area  Solid Angle = A
 Solid angle under which the source (diffuser) is seen from point P: = (D/2z)2
 Area: A = (st/2)2
 Letting A = 2 (single mode condition)  2 = (st/2)2
(D/2z)2
 st = (4/)(z/D)
 The set of rays that define  is (trasversally) smaller than s for longitudinal extent
equal to: st/, con  = D/2z  si = (2/)(2z/D)2
 Formulae shown in previous slides are obtained (apart from multiplicative factors 1)
D
z
source

sl
s
t

17. G. Giuliani - INTERFEROMETRY_02 17
Speckle-Pattern
Speckle size - 3
 Example
Plaser = 1 mW
D = 2.5 mm
z = 0.5 m
 = 632.8 nm
 st = z/D = 126 m
 sl = (2z/D)2 = 25 mm
 A photodetector with diameter Dfot = 10mm receives a total power given by:
Pr = BA = (Plaser/A)A (Dfot/2z)2
= Plaser(Dfot/2z)2
= 0.1 W
 The photodetector receives N speckles:
N =Adet/Aspeckle = (Dfot/2)2/ (st/2)2 = (Dfot/st)2 = 6300,
 The useful power to generate the interferometric signal is the one that belongs to a
single spatial mode (that is, a single speckle grain)
 Puseful = 0.1 W / 6300 = 15 pW
laser beam
D
diffuser
z st
sl

x
y
z
Dfot

18. G. Giuliani - INTERFEROMETRY_02 18
Speckle-Pattern
Speckle size - 3
 Example
Plaser = 1 mW
D = 0.25 mm
z = 0.5 m
 = 632.8 nm
 st = z/D = 1.26 mm
 sl = (2z/D)2 = 25 mm
 A photodetector with diameter Dfot = 10mm receives a total power given by:
Pr = BA = (Plaser/A)A (Dfot/2z)2
= Plaser(Dfot/2z)2
= 0.1 W
 The photodetector receives N speckles:
N =Adet/Aspeckle = (Dfot/2)2/ (st/2)2 = (Dfot/st)2 = 63,
 The useful power to generate the interferometric signal is the one that belongs to a
single spatial mode (that is, a single speckle grain)
 Puseful = 0.1 W / 63 = 1.5 nW
laser beam
D
diffuser
z st
sl

x
y
z
Dfot

19. G. Giuliani - INTERFEROMETRY_02 19
Performance Limits
Speckle-Pattern
 When performing interferometric measurements on diffusive surfaces there are additional
error sources
 Intensity effect
 The field Em could represent a “dark” speckle  fading of the interferometric signal (“signal drop-
out”)
 Possible solutions:
 Improve the focusing on the target surface (make D as small as possible)
 speckle size increases (st  1/D)
 speckle number N decreases  the back-diffused power is distributed over a smaller
numberof speckles  larger signal-to-noise ratio
 Use of a second sensor in parallel (sensor diversity)  probability of signal drop-out decreases
 Move the laser spot onto the target surface in the transverse direction  a different area of the
diffuser is illuminated  the speckle distribution changes  a “bright speckle” may hit the
photodetector (“bright speckle-tracking”)
 Phase effect
 Within each speckle, a phase error ( 2) can occur
 General consequence:
 It is not possible to measure accurately large target displacements
 only vibration measuments are possible (laser vibrometry)
IphR
He-Ne
Zeeman
laser
IphM
r
s
PDm
PD r
F
D
wl