Laser Interferometers: Summary - unipv 4-a.pdf · the Laser Interferometer (LI) Laser...

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


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 . . .


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


(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)


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


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 .


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.


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)


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


Iph

PD



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


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


I

I

1pol,0°

2

PD

PD

pol,90°

λ/8 plate slow axis oriented at 90°



sm

rs

BS

xy

linear pol at 45°


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

-


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


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


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


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


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


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)


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


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:


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


Iph1,2,3

PD's



sm

cube splitter retardance plate

0°120°

240°

ph1ph2

ph3


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


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


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


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


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


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


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


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


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


◊ 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


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


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


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


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


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


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


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


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


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)


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


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


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


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


• 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


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


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


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.



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.



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


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


• 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


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


• 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


• 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


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


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


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


A simple experiment with the injection interferometer

see film of the output waveforms

Laser Interferometers: Summary - unipv 4-a.pdf · the Laser Interferometer (LI) Laser...

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Transcript of Laser Interferometers: Summary - unipv 4-a.pdf · the Laser Interferometer (LI) Laser...