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1INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Laser Laser InterferometersInterferometers: : SummarySummary
SUMMARY
• Overview of Applications• The basic Laser Interferometer ◊ two-beams ◊ two-frequency• Performance parameters• Ultimate limits ◊ quantum-noise ◊ temporal coherence
◊ spatial coherence ◊ dispersion of the medium ◊ thermodynamical phase noise◊ Brownian motion ◊ speckle-related errors
• Read-out configurations of Interferometry◊ EXTERNAL ◊ INTERNAL ◊ INJECTION (or SELF-MIXING)
• Applications Laser Vibrometry ◊ short- ◊ medium- ◊ large-distance ◊ angle ◊ remote echoes ◊ absolute distance• White-light Interferometry ◊ profilometry
2INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
INTERFEROMETRY
a well-known technique (used in optics since the 1800s), now exploiting the long coherence length of lasers
The first ‘laser interferometer’ appears in 1965 (HP5526)since then, 2K .. 5K units sold per year (a big success . !)
The first scientific experiments (1968): earth crust tides
Doppler velocimeters follow in 1970s (another success. !)
ESPI vibration/strain analyzers in 1975
Interferometric Profilometry Microscopes (1990)
Interferometric antennas for GW in construction (2005)
widespread applications, a big tree growing up . . .
3INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
VIBRATION Analyzers
LASER INTERFEROMETRY
scient if ic usest echnical
uses
s≈1km
s≈1OOm
s≈1m
s≈<1cm
INTERFEROMETERS for MECHANICAL METROLOGY
DOPPLER VELOCIMETERS for ANEMOMETRY
ESPI (SPECKLE PATTERN)
RLG and FOG GYROSCOPES
avionics
indust rial uses
SPACE TELEMETRY of GEODETIC SATELLITES
EARTH TIDES SENSING
LARGE STRUCT VIBROMETRY
METROLOGY of LENGTH (and derived quantities)
BIOLOGICAL MOTILITY SENSING ACOUSTIC EMISSION and SAW SENSING
GEODETIC GRAVIMETRY
GRAVITATIONAL ANTENNAS
OPTICAL FIBER SENSORS
OPTICS ELECTRONICSWAVE and FIELDS
TECHNOLOGY
MEASUR.SCIENCE
Laser interferometryis a big tree withroots in optics, electronics, e.m.
fields, measurementscience, and technology.
Since the advent of the laser it has
spread out in manynew branches of engineering and
physics, for sensingand measurement
applications
4INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
(top): the conceptualscheme of aninterferometer iscreating two pathswith beam splittingand recombination; (bottom): dependenceon optical pathlengthdifferenceof the signal out fromthe photodetector
Interferometry: introduction
LASER
experiment
PHOTO- DETECTOR
REFERENCE ARM
MEASUREMENT ARM
frequency shifter (optional)
E0
E m
Er
I ph
φ − φm rπ/2 3π/2π 2π
I ph
I r
Iph = σ |Em+Er |2 = σ | Em exp iφm +Er exp iφr |2
Iph = σ [ Em2+Er
2+2EmEr Re{exp i(φm-φr)}] = Im +Ιr+2(√ImIr) cos(φm-φr)
5INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Common optical interferometers and their bulk optics configuration. M and R are the measurand and reference arms, respectively, when propagation pathsare in optical fiber
Back to basics: interferometers
R
Msource
detect
MICHELSON
source
M
partially reflecting mirror
beamsplitter
FABRY-PEROT
or
total reflecting mirrorCaption:
MACH-ZEHNDER SAGNAC
source
M
R
source
M
M2M1
PD
PD
PD1
PD2
PD
PD photodetector
detect
PD2
PD1
6INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Comparison of basic optical interferometers
Michelson Fabry-Perot Mach-Zehnder Sagnac___________________________________________________________________Responsivity R 2 F 1 2Balanced/Unbalancedconfiguration B or U U B or U B No. of channelsof basic setup 1 1 2 1Reference available yes no yes no Minimum number of beamsplitters (or partially 1 1 2 1 reflecting mirrors)Source and detectorfrom the same side yes yes/no no yesRetro-reflection to the source yes yes no yesMetallization required(for fiber versions) 2 2 0 0Cascadeability no yes yes no _________________________________ ____________________________________Notes: Michelson and Mach-Zehnder may be operated as either unbalanced or balanced (the latter being of course the preferred choice). Michelson and Fabry-Perot can have source and detector on the same side if the output is taken by a beamsplitter added on the input path .
7INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
the Laser Interferometer (LI)Laser Interferometer is used todesignate an instrument capableof measuring displacement of a target, with fraction-of-λresolution and 10-6 accuracy on a scale of meters, using acorner-cube as a target. Developed soon after the He-Ne(helium neon) laser in 1961, the LI has become well-establishedfor measurements and calibration.Precision is exceptionally goodbecause connected to the laser λ, easily stabilized to better thanΔλ/λ≈10-8 in commercial, cheap He-Ne units
best approaches to develop the LI are: dual-beam and two-frequency.
8INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Back to basics: prisms and cubes
α
β= 90°
β= 45°
prismpentaprism
corner cube
Retro-reflection in a 90° prism
and in a pentaprism with
a 45° vertex angle (top), and in a corner cube retro-reflector
(bottom)
9INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Evolution from the classicalMichelson interferometer (a), tothe Twyman-Green (b) usingangular-alignment tolerantcorner-cubes, and to (c), the modern two-beam laser interferometer measuring the displacement with its sign.After recombination we get twoequal beams, one of which iscollected by the photodiode PD. Signal out from the PD is:
from the Michelson to modern LI
He-Ne freq stab laser
I
I
1pol,0°
2
PD
PD
pol,90°
λ/8 plate slow axis oriented at 90°
reference corner cube
measurement corner cube
sm
rs
He-Ne freq stab laser
Iph
PD
reference corner cube
measurement corner cube
sm
rs
He-Ne laser
I ph
PD
reference mirror
measurement mirror
sm
rs
(a)
(b)
(c)
pellicle beamsplitter
cube splitter
BS
xy
linear pol at 45°Iph = 1/2 Im+1/2 Ιr +√ImIr cos 2k(sm-sr)
= Im [1+ cos 2k(sm-sr)]i.e., it is periodic of λ/2 in sm , butsign is missing. To recover sign, wego to the dual-beam scheme
10INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
By polarization diversity, we make available a pair of signals as:
Iph1 = 1/4Im+1/4Ιr+1/2√ImIr cos 2k(sm -sr) = 1/2Im [1+cos 2k(sm-sr)]
Iph2 = 1/4Im+1/4Ιr+1/2√ImIr cos 2k(sm-sr+λ/8) = 1/2 Im [1-sin 2k(sm-sr)]
and now sign of sm increment can be recovered
Dual-beam LI
He-Ne freq stab laser
I
I
1pol,0°
2
PD
PD
pol,90°
λ/8 plate slow axis oriented at 90°
reference corner cube
measurement corner cube
sm
rs
BS
xy
linear pol at 45°
11INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
cosine and sine signals are passed through discriminators, logic signals of amplitude(A’s) and slope (S’s) are obtained and an appropriate logic combination of them actsas the up/down command at the counter inputThe logic for up/down countsis found by inspection asU =SC*AS*+AC*SS +
+SCAS+ACSS*== SC*⊕ AS+AC⊕ SS
Signal handling in a dual-beam LI
s (t)m
cos 2ks sin 2ksIm
threshold of discriminators
time
Ac As
threshold crossing pulses (CNT)
U = Ac SsAs Sc +up/down control
Ss Sc
time
-
12INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
The up/down 7-decade decimal counter stores the displacementunder measurement in λ/8 units. A 6-digit multiplier brings the reading to metric units for the buffer register and the display
Output conditioning
U 7-digit up/down decade counter
6-digit multiplier by 0.791...
6-decade display
up/dn
clock
ex-or
or
As
Sc
Ac
Ss
CNT
6-decade buffer register
13INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Pulses Sc’ and Ss’ from the switching edges of the discriminator outputs are processed directly by the AND/OR gates, and the results are counted in a 7-digit up/down counter.
A more robust strategy for counting in a dual-beam LI
Sc
differentiated A and Ac s
timeSs'
'
discriminator thresh at Im
discriminator thresh at Im
I
I
c
s
time differentiator
time differentiator
A
A
c
s
pulse rectifier
pos
neg
Sc
Ss pulse rectifier
pos
neg
up counts
down counts
7-digit up/down decade counter
6-digit multiplier by 0.791...
6-decade display
up
dn
Ac As
s (t)m
'
'
buffer register
14INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Operation in the field of the dual-beam LI is satisfactory. But, it reveals drawbacks to be eventually corrected:- optical beam interruption loses counts and measurement shall be repeated- high-frequency (EMI) disturbances may leads to count errors; - ambient-induced vibrations can occasionally lead to incorrect counts.Strategies for improvement: - monitor the amplitudes of signal Iph1 and Iph2, and give a warning when
both fall below, say 5% of the time-averaged threshold. - good shielding from electromagnetic as well as mechanical disturbancesReason for criticality is that the dual-beam LI works on threshold crossingwith baseband signals: all the disturbances falling in a spectral range 0-B are indistinguishable from signals, and, may lead to incorrect switching of the discriminators
Operation of the dual-beam LI
15INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
The dual-frequency LI uses the two modes of a Zeeman laser, with a f1-f2≈ 5MHz offset and orthogonal polarizations. The Glan beamsplitter sends one mode to the reference path and the other mode to the measure path. Upon recombination, the phase difference 2k(sm-sr) is now found on the carrier frequency f1-f2.
I phR
Dual-frequency LI
He-Ne Zeeman laser
IphM
polarizer at 45°
λ/4
ref.
meas.
sm
rs
Glan cube
PDm
PDr
1f
2f
1f 2
f
5%BS
16INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
With the dual frequency setup we get signals:
IphR = η Iav {1+cos 2π(f1-f2)t }
IphM = Iav {1+ cos[2π(f1-f2)t +ϕconst+2ksm]}
Now, phase 2ksm is superposed on a frequency carrier f1-f2 and to recover it, we have several possible choices. The best is to get a pulse per period fromsignals IphR , IphM through a comparator and differentiator, getting:(2π)-1d/dt [2π(f1-f2)t + const] = f1-f2 rate of pulses, REF
(2π)-1d/dt [2π(f1-f2)t +ϕt+2ksm] = f1-f2 +2kvm/(2π) rate of pulses, MEASPulses are counted on a time period T and the contents subtracted, yielding∫0-T f1-f2 dt = (f1-f2)T ∫0-T [f1-f2 +2kvm]dt = (f1-f2)T+[sm(Τ)− sm(0)] /(λ/2)
difference = [sm(Τ)− sm(0)] /(λ/2)
Reference and measurement signals in dual-f LI
17INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Signal processing for the two-frequency LI: measurement and reference signalsare counted on a period T, then transferred to buffers. Results are subtracted toyield displacement. A main adder gives Δs in λ/2 units, and a multiplierconverts the counts in decimal units for the 6-digit display
I phR
I phM
Signal processing in dual-frequency LI
comparator
comparator6-digit decadecounter
subtractor (6 digit and sign)
+
-
6-digit decadecounter
G1
6-digit decadebuffer
6-digit decadebuffer
G2
reset
main adder register
G1 T
G26-decade display
multiplier (by λ/2)
18INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
A typical LI instrument (HP 5526). Right: the He-Ne frequencystabilized laser, left: the 7-digit display unit, center and bottom: corner cubes and beamsplitters todefine the opticalpath
Dual-frequency LI commercial products
19INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
The concept of interpolation can be used to extend resolution to λ/200 in a two-frequency interferometer (they sell it, but user seldom do really need it!)
Interpolation in a dual-frequency LI
time
I =
cos 2π(f - f ) t phR
1 2
phM I =
cos 2π(f - f ) t + 2ks1 2
phMI
IphR
local oscillator cos 2π (f -f +f ) tIM21
cos(2π f t+2ks)
cos(2π f t)IM
IM
START STOP
ANALOG OUT
DIGITAL COUNTS
CLOCK at 100 fIM
downcorverted at fIM
f -f1 2
:100 DIVIDER
100 f IM
clock at
IM f
MIXERS
UNIPV:UNIPV:
20INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
To extend resolution of the measurement down to nm, channelsare triplicated, thanks to the 3-sector waveplate. Signals areIph = I0i[1+cos2k(sm-sr+nλ/3)] (n=1,3) and allow solving for 2ks
Extension to nm by analogue handling
He-Ne freq stab laser
Iph1,2,3
PD's
reference corner cube
measurement corner cube
sm
cube splitter retardance plate
0°120°
240°
ph1ph2
ph3
21INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Using 3 signals, S1 = a [1+cos2ks]S2 = a [1+cos(2ks+120°)] S3 = a [1+cos(2ks+240°)],
2ks can be solved for irrespective of a
Ambiguity of cos/sin pair
C
S
ks
the pair C= a [1+cos2ks ]S= a [1+sin2ks]
has ambiguity in the argument 2ks
22INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Measurement is readilyextended to three-axis with33% and 50% beamsplitters,while it needs only one laser
I phR
Extension to 3-axes
He-Ne Zeeman laser
IphMx
X-axismeas.
sm
PD r
I phMy
Y-axismeas.
sm
IphMz
Z-axismeas.
33% trans. Beamsplitter
50% BS
mirror
sm
23INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
A universal tool machineis equipped with a three-axis laser interferometerto control the x,y,z movement of the carriage(by courtesy of Hewlett-Packard)
Equipping a tool-machine with the LI
24INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Planarity measurement scheme: a bracket carrying a pair of corner cubes isused in coinnection to a beam splitter bracket realizing a Mach-Zehnderinterfero-meter. An angle error α gives a variation 2αL→ λ/4 of optical path. Angular resolution for a single count, with α1c = λ/8L, and for L=100 mm and λ=0.63μm, gives a typical α1c =0.8 10-6 rad or 0.16 arc-sec.
Planarity measurement
I phR
He-Ne Zeeman laser
PD
mirror
50% BS
BS
IphMPD
mirror
50% BS
α
L
extra path: 2αL
25INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Rectangularity: resolution is the same as for planarity, α1c =λ/8L, whereas precision is affected by the pentaprism dihedral angle error ε=γ-45° (typ <1 arcsec in best units).
I phR
Rectangularity measurement
He-Ne Zeeman laser
PD
mirror
50% BS
BS
IphMPD
26INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Modification of the optical setup for operation on a diffusingtarget: a lens focuses the laser beam on the target and collectsbackscattered light for the detector
I phR
Scheme for operation on diffusing target
He-Ne Zeeman laser
IphM
rs
PD m
PDr
Fwt
wl
27INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
What is actually measured with the interferometer:Iph = I0 [1+V cos Rk.(sm-sr)]
where R =responsivity and V= fringe visibility depend on the optical interferometer used.□ Term k. sm reveals that the path actually measured is k smcos αks (cosine-error)□ Another error is the cyclic error, when V<1, coming fromminute spurious reflections on parasitic optical path. If ε isthe fraction of power leak to the path, an error of amplitudescyc= (λ/16)√ε is generated, which has a periodicity λ/2 versus displacement
Errors affecting measurements of the LI
28INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Wegel’s diagramfor interferometerperformances: operation is inside the thick-lineperimeterTypicalperformances of a displacement-measuringinterferometer and of a vibrometer are indicated.
Interferometer performances
1 Hz 1 kHz 1 MHz
1 nm
1 μm
1 mm
DIS
PLA
CE
ME
NT
(pea
k am
plitu
de)
FREQUENCY
SPEED (m/s)
1 m
0.006
0.6
60
6000
NOISE or Quantization
HF cutoff
Doppler limit
low
-freq
uenc
y
displacement interferometers
vibrometers
29INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Fundamental factors and mechanisms affecting the limits of performance of the interferometer:
• quantum noise
• temporal coherence
• spatial coherence and polarization effects
• dispersion of the medium
• thermodynamic phase noise
• brownian motion
• speckle-regime
Limits of Performance
30INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
◊ Photodiode output signal is Iph/I0 = 1+Vsin φ ≈ 1+Vφ,where φ=RkΔs is the small displacement to be measured.
◊ For a small deviation around Iph it is: ΔIph/I0 = V Δφ
◊ Added to Iph, we find a shot (or quantum) noise of current I0, with rms value in = (2eI0B)1/2
◊ The associated signal-to-noise ratio of the current amplitudemeasurement at I0 is (S/N)-1= in/I0 =(2eB/I0)1/2. Letting Δφ=φn
and ΔIph= in, we get: φn = in/VI0 = (2eB/I0)1/2/V = 1/(S/N)V
◊Noise-Equivalent-Displacement (NED) is the displacement Δsgiving the same effect as noise φn. By definition, RkΔs = φn or:
NED = φn/kR = (λ/2π)/(S/N) RV = (λ/2π) (2eB/I0)1/2/RV
Limits of Performance: quantum limit
31INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
phase noise (left) and NED (right) vsmeasurementbandwidth B, withthe detected power P as a parameter, at the quantum noiselimit of operation in an interferometer
Limits of Performance: quantum limit
10
10
10
10
-5
-3
-7
-9
1 100 10k 1M 100M
Measurement Bandwith B (Hz)
Pha
se n
oise
φ
(rad
) o
r 1/
(S/N
) rat
ion
λ = 633 nm, η=0.9, R=V=1
Equivalent Power P (W)
0.1 μW 10 μW
1 mW
100 mW
10 W
1 nm
1 pm
1 fm
NE
D -
nois
e-eq
iova
lent
-dis
plac
emen
t (m
)
φn = 1/(S/N)NED = (λ/2π)/(S/N)
= λ/2π√Nph
32INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Beams propagated on measurement sm and reference sr paths are superposedwith a finite time difference, T=(sm–sr)/c.
This causes a sub-unitary coherence factor μtc = ⟨cosφn ⟩ in detected signal:
Iph/I0 = 1+ cos[k(sm–sr) +ϕ(t)-ϕ(t+T)] = 1+ cos[k(sm–sr)+φn] =
= 1+ cos k(sm–sr) cosφn+ sin k(sm–sr) sinφn so that, by averaging
⟨Iph⟩/I0 = 1+ μtc cos k(sm–sr)
and ⟨in2⟩/I0
2 = sin2k(sm–sr) ⟨sin2φn⟩≈ sin2k(sm–sr) ⟨φn2⟩
Phase fluctuation φn2 is traced to the Allan’s variance, σφ
2 = ⟨φn2⟩, being
σφ =(2πτ) σν(2,T,τ), and also σν(2,τ,τ)= (τ/T) σν(2,T,τ) for T<< τ. Then
NEDtc = (λ/2π)φn = (λ/2π) (2πT) σν(2,τ,τ) = cT σν(2,τ,τ)/ ν
or, in conclusion, NEDtc= (sm–sr) (σν/ν)
Limits of performance: temporal coherence
33INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Superposing the spatial distributions Em(x,y) and Er(x,y) on the photodetector, we get a coherence factor
μsp =∫AEm(x,y) Er*(x,y)dxdy /[∫A⏐Em(x,y)⏐2dxdy ∫A⏐Er (x,y)⏐2dxdy]1/2
It is μsp ≈1 only for single-mode beam spatial distributions. If the beam is a mixture of N modes, only homologous modes willcontribute to μsp, and result is that μsp cannot be larger than 1/N. Similarly, we shall use the same State of Polarization (SOP) for bothmodes, either linear or circular, or generically elliptic.If not, the signal will be reduced by a factor:
μpol = Em.Er/⏐Em⏐⏐Er⏐
Limits of performance: spatial coherence and polarization state
34INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
As beams propagate in air, the unit of measure of a LI is λ/nair. In standard condition (T=15°C, p=760 mbar) air has an index of refraction
(nair –1)|st = 272.6 + 4.608/λ(μm) + 0.061/λ(μm)2 (ppm).
Correction is 280 ppm (λ=633nm), also weakly dependent on λ. The excess to 1 of the index of refraction is proportional to the mole numberper unit volume n/V, and thus equal to P/RT for the law of perfect gases:
nair –1 = (nair –1)|st (P/760) (288/T)
Temperature and pressure coefficients of the correction follow as
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
With ΔT=10°C and Δp=10 mbar, T and p affect the 5th and 6th decimal digitsof the displacement measurement. Correction is performed by two sensorswith outputs that change the multiplication factor used to convert the fraction-of-λ counts to a metric scale.
Limits of performance: dispersion of the medium
35INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
The thermodynamic fluctuation of index of refraction of the medium introduces a phase noise φth that may become comparable or even largerthan quantum noise. By an analysis of the phenomenon, the NEDth is found as:
NEDth = φth/k ≈ [ζ kBT2LB/κ]1/2
where T = absolute temperature, kB = Boltzmann constant, L =path length, κ =thermal conductivity, ζ =thermooptical coefficient (8.10-6 in silica)
This NED is important in OFSs
Because of the interference of counter-rotating waves sharing the samemedium, the Sagnac configuration provides a partial cancellation of φth and
the best immunity to thermodynamic phase noise
Limits of performance: Thermodynamic Phase Noise
36INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Brownian motion add fluctuation in interferometric measurements of vibrations or displacements on small masses. On the line of sight of the measurement, we find a thermodynamic degree of freedom and hence an energy (1/2)kT. This energy shall be equated to kinematic energy (1/2)m⟨v2⟩, thus obtaining:
⟨v2⟩=kT / m
In the same way, for a rotating target for which we want to measure the angular speed of rotation Ω, we have a kinematic energy (1/2)I⟨Ω2⟩, where I isthe inertia momentum of the target. Equating to (1/2)kT gives a variance of the angular speed:
⟨Ω2⟩= kT/I
Letting numbers in these equations reveals that even for not so small masses(e.g., 1 to 10 g), the Brownian-induced speed or displacement may becomecomparable to the quantum noise NED.
Limits of performance: Brownian motion
37INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
When the interferometer works on a diffusing surface, a random phaseerror originates form the speckle-pattern statistics, either due to a displacement on-axis (Δsl), transversal (Δst), or projection (Δw) .In all cases the error is
NEDtp = C λ Δx /x,
where Δx is the displacement relative to speckle size (Δsl/Sl, Δst/St) or spot size (Δw/w), and C is a parameter for the speckle statistics
Speckle-related errors
detect
source
Δsl
tΔs
Δw
38INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Read-Out Configurations ofInterferometry
• in external configuration, the laser feeds an opticalinterferometer external to the source, and provides anintensity signal I0 cos Φcarrying the phase information• the internal configurationuses the laser mirrors as the optical interferometer, and gives an output I0 cos Ωt, a frequency signal Ω proportionalto the optical phase shift 2ks• is the injection-modulation, or self-mixing interferometer, weallow the field returned fromtarget into the laser cavity fieldto produce AM and FM modulation carry phase 2ks.
LASER
LASER
LASER
s
s
s
PD
PD
PD
EXTERNAL INTERFEROMETRY
SELF-MIXING INTERFEROMETRY
I = I 0 cos 2ks
I = I 0 cos Ωt,Ω = (c/2L) 2ks
I = I 0 (1+m ) A F(1+m ) cos ωt,
FmAm = r(cT/2Lg) cos 2ks
= r(cT/2L) sin 2ks
INTERNAL INTERFEROMETRY
laser interferometers, Doppler velocimeters, etc.
RLG gyroscope
a new approach
optical pathlength is read on INTENSITY
optical pathlength is read on FREQUENCY
optical pathlength is on AMPLITUDE and FREQUENCY modulation
MM
RM
39INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Top: a scheme with polarization-split modes uses a Glan cube. Bottom: the theoretical response of frequency difference f1-f2 versusdisplacement Δs is linear, with a range of c/2L=750 MHz for a Δs=λ/2=316 nm displacement (valuesfor a 30-cm He-Ne). A dead band around f1-f2≈0 is found because of locking.With low-scatter mirrors and a high-quality Glan, we go down to a few MHz of locking range as comparedto several hundreds MHz of the basic scheme. Biasing operation off f2-f1=0, e.g., fbias=100 MHz, we mayalso detect the sign of Δs
Internal-interferometry configuration
LASER
PD
M
M
R
M
Glan BS 45° polarizer
f 1
f 2
f 1-f 2
(MHz)
250
500
sΔ
sΔ(nm)150 300-150-300
-250
-500
dead band (locking range)
linear range
injection modulation region
out
Δf = (c/2L) Δs/(λ/2)Rf = Δf/Δs = (c/λL)
40INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Rotating vector description of injection modulation (top), and the circuit analogy of the injectionphenomenon (bottom)• the cavity field E rotates in the phase plane at optical frequency ω. •a fraction αE is allowed out and propagates to a remote target, so the phase shift of the go-and-returnpath is Φ=2ks• returned field adds tracking the unperturbed field, and the result isa phasor addition, ie AM and FM modulation is generated• AM index is cos 2ks, FM deviation is proportional to sin2ks
Injection (or self-mixing) Interferometry
Φ=2ksE
aE expiΦ
AM, a cos Φ
FM, a sinΦ
attenuation a
delay T Φ= ωT
oscillator, ω
41INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
To describe the injection phenomena, we haveto distinguish three levels of injectionaccording to the fraction of power returned tothe cavity, and to the ratio of cavity length L totarget distance s. As it will be shown later, the dependence issummarized by a feedback parameter C definedas [12]: C = a nl L√(1+αen
2)/s
• 3 levels of injection are found, depending on power attenuation a and ratio L/s of cavity length to distance. • the relevant feedback parameter is: C = a s √(1+αen
2) / nl L
•weak feedback, (C<<1): most the HeNe case, the simple picture of rotatingvectors is applicable, the laser has nearly the same properties as in the unperturbed state, and interferometric waveforms are sinusoidal.
• moderate feedback, (C≈1): easily found in LDs (R≈ 0.05…0.3). The interferometric signal becomes a distorted-sinusoid waveform, and switchingbetween levels in each 2ks=2π period may occur. Both spectral line and coherence significantly deviate from the unperturbed state.
• strong feedback, (C>>1): interferometric waveform exhibits multiple switching in each 2ks=2π period, coherence length and line width are strongly affected, and the laser oscillates on the external cavity
Self-mixing Interferometry
42INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Waveforms of optical power in an injection interferometer. At C<<1, the normal dependencecos2ks from external phaseshift is found. Increasing the injection, the waveformbecomes progressivelydistorted and asymmetric. As C reaches unity, a switchingoccurs with period 2π. If the target reverses its motion, the phase shift decreases, and the waveform becomes time-inverted, except at the switching point where now itfollows the up arrow. BeyondC≥4.6, more than one switching per period occurs, and interferometer operationmay be affected by errors.
external-path phaseshift 2ks0 π 2π 3π 4π
C=0.05
0.1
0.5
0.8am
plitu
de m
odul
atio
n of
opt
ical
pow
er
1.5
weak-level injection regime
1.01
medium-level
C�4.6 high-level
5π
43INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
writing the laser cavity field E and returned signal field Es as rotatingvectors:
E = E exp iϕ and Es = a E exp i(2ks+ϕ),
with ϕ = Ωt+ψ=2πνt+ψLamb’s equations are written as:dE/dt = [(α−βE2)c -Γ] E + (c/2L) aE cos (ϕ+2ks)dϕ/dt = Ωc + ζ(α−βE2)c + (c/2L) a sin (ϕ+2ks)
where:a = At1
2η is the total field loss; t1=√(1-r12) is the field transmission of the input
mirror; α=λ2(n2-n1)/8πτ21Δνat is the active medium gain rate per unit length; n2-n1 is population inversion (cm-3); Δνat is the (atomic) gain line width and τ21 is the active level lifetime; β is the gain saturation coefficient; Γ= Ωc/2Q is the cavity field loss-rate; Ωc is the cavity resonant frequency; L is the cavity length, and ζ = (ν0-νat)/Δνat is the frequency detuning
Analysis of injection at weak-feedback level
44INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
• Letting (d/dt)E=0, we get the steady-state solution. • At a=0, quiescent values E0, Ω0 are readily found as E=E0 =√[(α−Γ/c)/β] and Ω=Ω0=Ωc+ζΓ/c. • For a≠0, we try a small-perturbation solution E=E0+ΔE and ΔE<<E0retaining only first-order terms in ΔE. • Neglecting pulling term ζ<<1 and dropping unessential phase ϕ0,, we get:
E = E0 [1+ (a/2Lγ0) cos 2ks] = E0 [1+mA]dϕ/dt = Ω0 + a (c/2L) sin 2ks = Ω0+Δ
where γ0=α-Γ/c is the net gain per unit length (equals βE2 as in the permanentregime of oscillation, α+βE2-Γ/c=0).• At first order of perturbation, t cavity field has an AM with modulationindex mA=(a/2Lγ0)cos2ks proportional to the cosine of the externalpathlength, and an FM with frequency deviation Δ/2π=a(c/4πL) sin2ks proportional to the sine of the external pathlength.
Analysis of injection at weak-feedback level 2
45INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Polarizer P1 in front of PH is oriented at 45° to allow beating of propagatedmode with the mode kept in cavity. After FM and AM demodulation, signalssin 2ks and cos 2ks are obtained. The frequency signal is also used forfrequency stabilization through the thermal actuator. Through differentiationand cross-multiplying of sin 2ks and cos 2ks, the v and s are recovered.
Injection interferometrywith He-Ne. Two modes at f1-f2≈100kHz are generatedby Zeeman-splitting due toa transversal magnetic fieldapplied to the capillarytube. The modes are linearorthogonal polarized. One is allowed to propagate through polarizer P, the other is kept in cavity.
The He-Ne injection interferometer
46INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Waveforms of the electrical signal out from photodiode PH when the target is a diffuser (white paper on a loudspeaker driven at audio frequency). To ease comparison, bottom traces with injection are shown along with top traces with the beam blocked off. On a 20 μs/divscale, the waveform reveals the FM signal in form of a time jitter; on a 1ms/div scale, the AM signal showsup in form of a ripple. Using a corner cube target, the signals would bemuch larger and require an attenuationinserted on the beam
The He-Ne injection interferometer 2
47INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Top: waveforms of target driving signaland of the interferometric signalsS=sin 2ks and C=cos 2ks demodulatedfrom the photodiode output; bottom: the reconstructed velocity signalv=ds/dt and its integral, the displacements. Time scale is 1ms/div, displacementcan be read as 4.5 λ/2 ≈ 1.4 μm peak topeak.Note that ringing in the reconstructedwaveform is due to the loudspeakerresponse.
The He-Ne injection interferometer 3
48INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Pickup of biological motility with the He-Ne laser injection interferometer. The blood pulsation waveform is measured on a finger and reveals the leftventricular ejection (LVE) as well as the dicrotic incisure (DI). The verticalscale is 0.1 μm/div, and the horizontal scale is 0.2 s/div.
The He-Ne injection interferometer 4
49INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Considering the model, we get the balance equation:E r1 r2 exp 2α∗L exp i2kL + E a exp i2ks → Eloop gain is: Gloop= r1 r2 exp 2α∗L exp i2kL +a exp i2ks, whencea phase condition: r1 r2 exp 2α∗L sin 4πLnl(ν-ν0)/c+a sin 2ks =0Using r1r2 exp 2α∗L=1 (1stBC), and assuming that ν-ν0 is smallwe obtain: ν= ν0 + (c/4πLn1) a sin 4πs/λIf distance s changes by λ/2, the argument of the sine changes by2π, and ν covers a full swing of the sine, with peak-to-peak (c/2L) a/2πn1 ≈ C
Injection interferometer: three mirror model
L s
Er1r2
a
50INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
injection-perturbed frequency ν vs unperturbed frequencyν0: at increasing feedback level (C values), the curve hasthree intersections revealing bistability and hysteresis
Injection interferometer : three mirror model 2
actu
al fr
eque
ncy
ν
unperturbed frequency ν0
C-0
C=1C>1
increasing s
51INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
• about the AM signal, it is |Gloop| =1 from the 1st Barkhausencriterion, or Re2(Gloop)+Im2(Gloop)=1. Letting E=E0+ΔE, developing the modulus |Gloop| and considering ΔE<<E0 , wecan find:
ΔE = E0 (a cos 4πsν/c)/[ 2Lγ0(1+a cos4πsν/c)]• this coincides with the weak-feedback result for a<<1. • when factor a increases, waveform is distorted because of the
cosine dependence at the denominator. • at C=1, when frequency switches, ΔE and the AM signal will
switch too, because of the ν-dependence• by solving numerically the equation (or using the diagram), we
find a function ν=F(4πsν0/c) to replace ν in sin4πsν/c at moderate feedback level.
Injection inteferometer : three mirror model 3
52INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
L-K equations rephrase Lamb’s equations for the LD, relatingamplitude E and phase ϕ, and add a third equation for the active-level population N:
dE(t)/dt = [gN(N-Ntr)-1/τp]E(t) + (ac/2nlL)E(t-τ) cos[ω0τ+ϕ(t)- ϕ(t-τ)]
dϕ/dt = αengN(N-NT) - (ac/2nlL) E(t-τ)/E(t) sin [ω0τ+ϕ(t)- ϕ(t-τ)]
(d/dt)N = J/ed - N/τs - gN (N-Ntr) E2(t)
where: gN =αc/(n2-n1)= λ2c/8πτ21 ; Δνat =gain, N=N(t)=carrier density; Ntr and NT are at transparency and at threshold, so gN(NT-Ntr) =1/τp; τp =photon lifetime; a = At1
2η= field loss ; τ =2s/c is external roundtrip time; αen= linewidth enhancement ; J=pump current density; τs = charge-carrier lifetime.
Analysis by the Lang and Kobayashi equations
53INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
• letting dE(t)/dt=0 and (d/dt)N =0 in LLK eqs., and solving for stationary valuesE0f , N0f in presence of feedback, we get: N0f =NT-(ac/2nlLgN) cosω0fτ• this shows that feedback induces a modulation of the carrier density N - a voltage is developed across the laser-diode junction• the perturbed oscillation frequency ω0f is found as:
ω0f = ω0 - (ac/2nlL) [αen cos ω0fτ + sin ω0fτ] • this equation can be brought to the Barkhausen phase-condition. • using the feedback parameter C, and with τ=2s/c, it becomes:
ω0fτ =ω0τ -C/√(1+αen2)[αencosω0f τ +sinω0f τ ] ,
or also: ω0fτ = ω0τ - C sin (ω0f τ + atan αen)
• in a 2π-period of ω0τ, this equation has just one solution for C<1 (weak-levelinjection), and three solutions for 1<C<4.6 (moderate-level injection). • one solution of the 3 is unstable, impling a switching on the stable solutions• for C>4.6, multiple switching is in a 2π-period of ω0fτ (strong-injection). Letting (d/dt)N=0, we get E0f
2 = [J/ed - N/τs]/gN(N0f -Ntr)
Analysis by the Lang and Kobayashi equations 2
54INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
• using this result and N0f, we obtainthe optical power variation ΔP as:
ΔP = P0 (acτp/nlL) cosω0fτ /[1+(acτp/nlL) cosω0fτ]
which becomes the familiar ΔP0mAcos2ks expression for a<<1.
• reconstruction of the displacement waveform s(t) from the AM signal: bycombining eqs of previous slide, we obtain the following expressionrelating ω0fτ=2ks to the instantaneous ΔP/ΔP0=F(t) modulation:
ω0fτ = ± arccos F(t) - C{-αenF(t) ±√[1-F(t)2]}/√(1+αen2) +2mπ
where the sign shall be taken ‘+/-’ for (dF(t)/dt)(ds/dt) </>0, respectively,and m shall be increased by 1 every two zero-crossings of F(t).• reconstruction algorithm gives the correct signal, even when it has anarbitrary waveform, with a typical 5 nm accuracy (at λ=633 nm)
Analysis by the Lang and Kobayashi equations 3
55INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Experimental signals, in responseto sinusoidal drive s(t) signal (top trace in both pictures). Top: at C=0.6, the cos 2ks waveform isslightly distorted, with fallingsemiperiods somewhat faster thanthe rising ones; bottom: at C largerthan 1, a switching occurs in the fall semi-period (bottom picture isfor C=2.2). wavelength: λ=850 nm, amplitude: ≈3 μm peak-to-peak, time scale: 2 ms/div.
Selfmix (or injection) waveforms: experiment
56INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Drive s(t) (uppermost) and theoretical ΔP/P waveformscorresponding to the experimental data of previousslide, for a few values of the feedback parameter C, rangingfrom weak to moderate levelof injection (C=0.01, 0.7, and 3.3 from top to bottom). The relative vertical scale of ΔP/ P signal is 0.5, 3, and 10 (top tobottom). Horizontal scale: 1ms/div.
Selfmix (or injection) waveforms from theory
57INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
Block scheme of the laser-diode self-mix interferometer working on the waveform switching to count displacement in steps of λ/2 withsign. We the SMI, only one channel is needed.!
LD self-mixing interferometer: block-scheme
laserPD
target (retroreflector, superdiffuser or diffuser)
variable attenuator
telescope
RC differentiator
preamp
polarity sorter
up/down counterup
dn
reset
multiplier by λ/2
6-digit display
58INTERFEROMETERSfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)
A simple experiment with the injection interferometer
see film of the output waveforms