1 Gravitational wave interferometer OPTICS François BONDU CNRS UMR 6162 ARTEMIS, Observatoire de la...

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Gravitational wave interferometer OPTICS

François BONDU

CNRS UMR 6162 ARTEMIS,Observatoire de la Côte d’Azur, Nice, France

EGO, Cascina, Italy

May 2006

Fabry-Perot cavity in practice

Rules for optical design

Optical performances

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Contents

I. Fabry-Perot cavity in practiceScalar parameters – cavity reflectivity, mirror transmissions, lossesMatching: impedance, frequency/length tuning, wavefrontLength / Frequency measurement: cavity transfer function

II. Rules for gravitational wave interferometer optical designOptimum values for mirror transmissions“dark fringe”: contrast defect“Mode Cleaner”

III. Optical performancesActual performances:Mirror metrologyOptical simulationAccurate in-situ metrology

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Michelson configuration at dark fringe + servo loop to cancel laser frequency noise

VIRGO optical design

Slave laser

Masterlaser

Hz/10.3~2 23L

Lh

Fabry-Perot cavity to detect gravitational wave

Suspended mirrors to cancel seismic noise

L=3 km

Long arms to divide mirror and suspension thermal noiseRecycling mirror to reduce shot noiseInput <<Mode Cleaner>> to filter out input beam jitter and select mode

L=144m

Output Mode Cleaner to filter output mode

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SCALAR MODEL:“plane waves”scalar transmissions, scalar losses of mirrors

1. Fabry-Perot cavity: A. parameters

REFLECTION TRANSMISSION

Can we understand these shapes?

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Round Trip LossesFree Spectral RangeRecycling gainCavity PoleFinesseCavity reflectivity


Mirror 1 Mirror 2

Ein EstoEtrans

Eref

Ert = r1 P-1 r2 P Esto


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2with

)1(

211

2121

11111

LLTTL

LPP

LTLTr

RT

RTstort

Round trip “losses”



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c

LE

err

tE

EEtE

cLiP

inisto

RTinsto

4with

1

:solution statesteady

)/2exp(delaynpropagatio

21

1

1

2LL



Free spectral rangePeriod:

L

cFSRFSR

2,


8


2

21

1

21

1

1]2[0

1

rr

tPP

Eerr

tE

insto

inisto


Recycling gain

2

11

1

1

rr

tG

RESONANCE CONDITION


9

24

that so),( 00000 k

c

LL

1

)(4 000

c

Lf


RESONANCE CONDITION

Suppose now

21

21

2

1Maximum HalfLinewidth Half

/1

rr

rrFSRfwith

fif

GEE

P

Pinsto

Cavity pole



10


Pf

FSRFinesse

2 Finesse



11


on resonance reflectivity

)(21

)(121

1

)1()(

fi

fi

in

ref

err

eLrr

E

Efr

21

121

1

)1()0(

rr

Lrrr

2)0( R



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2nd order

In T+P

1st order

in T+P

Finesse

On resonance reflection

transmission


pT2

)1(2

1 pTT

21

121

1)1(

rrprr

2

21

21

1

rrtt

21

21

1 rrrr

2214pTTT

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T1 = 12%T2 = 5%L = 0(finesse = 35)

REFLECTION TRANSMISSION

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01

)1()0(

21

121

rr

Lrrr

0

Impedance matchingFrequency/length tuning (“lock”)Wavefront matching

alignmentbeam size / positionsurface defects - stability

The Fabry-Perot interferometer

0

1. Fabry-Perot cavity: B. Matching

Optimal coupling

Over-coupling

Under-coupling

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Frequency/Length tuning

c

L000

4


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NON-SCALAR MODEL:


Mirror 1 Mirror 2

Ein EstoEtrans

Eref

Ein(x,y) ; Esto(x,y) ;

r1, P, r2 are operators

z axis


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NON-SCALAR MODEL:


Esto(x,y) = k Ein(x,y) (k complex number)

EstoEin

Wavefront matching:

Superpose angles and lateral driftsof incoming and resonating beam

<<ALIGNMENT ACTIVITY>>

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NON-SCALAR MODEL:


Esto(x,y) = k Ein(x,y) (k complex number)

Esto

Ein

Wavefront matching:

Superpose beam positions and beam widths <<MATCHING ACTIVITY>>




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NON-SCALAR MODEL:

2211

2

2121

,,

,),(

C

Definition of beam coupling:

Round trip coupling losses:




Too small mirror diameters “clipping” imperfect surface: local defects, random figures

),(1 stortrt EECL


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NON-SCALAR MODEL:

2111 LLTTLrt

Definition of stability:




Definition of stability in case of perfect surface figures:

1)1)(1(021

R

L

R

L


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The Fabry-Perot interferometerCharles Fabry (1867-1945)Alfred Perot (1863-1925)Amédée Jobin (mirror manufacturer) (1861-1945)Gustave Yvon (>1911)Marseille – beginning of 20th century

“Les franges des lames minces argentées”,Annales de Chimie et de Physique, 7e série, t12, 12 décembre 1897

“A taste of Fabry and Perot’s Discoveries, Physica Scripta, T86,76-82, 2000

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Phase modulated laser:

m phase modulation indexfm modulation frequency

SB- C SB+

mmC iii

in em

em

e f2f2f20 22

1

ti

mCti

mCCti

ref em

Rem

RRe mmC f2f2f20 2

)ff(2

)ff()f(

1. Fabry-Perot cavity: C. measurement

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2

p

p00

ff1

ff

)1()f(Im

c

c

cP mPRmPs

mff

)ff()f()ff()f(Im2

**0 mccmccP RRRRm

Ps

error signal:

Does not provide information about frequency behavior once locked


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Modulated laser + measurement line:

n phase modulation indexfn modulation frequency

p

0)noisefrequency(

ff1

)1(

i

mPTF

SB- C SB+

f << FSR, f ≠ fm

This pole


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Contents




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2. Optical design: A. mirror transmissions

cavityRTFP GLR 1

Fabry-Perot cavity with Rmax transmissions as end mirrors

Virgo mirrors: LRT ~500 ppm, Gcavity ~ 32 reflectivity defect 1.5%

Was estimated 1-5 % at design

Have as much as possible power on beamsplitter

Match “losses” of cavities with recycling mirror

Was estimated 8 % at design (5.5 % recent refit)

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• Michelson simple :

BS

laser

Pin

Pout

minmax

minmax

PPPPC

Pmax, Pmin = Pout On black and white fringes

2. Optical design: B. dark fringe

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Slave laser

Masterlaser

L=3 km

Input <<Mode Cleaner>> to filter out input beam jitter and select mode

L=144m

Output Mode Cleaner to filter output mode

2. Optical design: C. Mode Cleaners

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Detection

Output Mode-CleanerOutput Mode Cleaneron Suspended Bench

Photodiodeson Detection Bench

Beam

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Contents




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Measured optical parameters

16.7 W 7.1 W

F = 49±0.5

F = 51 ±1

Gcarrier = 30-35 (exp. 50)GSB ~ 20 (exp. 36)

1 – C < 10-4

1 – C = 3.10-3 (mean)

Slave laser

Masterlaser

1 W T=10%

III. Optical performances

Losses in input Mode Cleaner?

Recycling gain?

Arm finesses?

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Mirror metrology

reproducibility 0.4 nm; step 0.35 mm

resolution 30 ppb/cm // 20 ppb

resolution of a few ppm

transmission map

Before and/or after the coating process, maps are measured:

- Mirror surface map (modified profilometer)

- bulk and coating absorption map (“mirage” bench)

- scatter map (commercial instrument)

- transmission map (commercial instrument)

- local defects measurements

- birefringency The VIRGO large mirrors: a challenge for low loss coatings, CQG 2004, 21

Instruments: ESPCI, ParisCoating, 140 m2 room class 1: LMA, Lyon

Scatterometer CASI 400x400mm

Micromap 400x400 mm(local defects)

Absorption Photothermal Deflection System

Phase shift interferometer

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DiamDiam 35 cm35 cmRms Rms 2.3 nm2.3 nmp-p p-p 11.5 nm11.5 nm

Surface maps

Ex: a large flat mirror

-Good quality silica

- Good polishing

- Control of coating deposition (DIBS) with no pollutants

- Surface correction


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TWO optical programs:

- One that propagates wavefrontwith FFT

- One that decomposes beams on TEM HG(m,n) base

- Check out cavity visibility (total losses)

- Check out expected recycling gain,for varying radii of curvature

- Check out expected contrast defect

- Check out modulation frequency

- Improve interferometer parameters…

Optical simulation


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Scalar defects Maps Maps+thermal

Opt mod index 0.068 0.172±0.001 0.215 ±0.001

Opt demod phase 0 2 ±0 17 ±1

Finesse N 49.26 49.1 ±0.2 49.3 ±0.2

Finesse W 49.79 49.6 ±0.2 49.7 ±0.2

dF/F [%] 0.27 0.23 ±0.12 0.24 ±0.12

Asymmetry [%] 1.05 1.00 ±0.3 2.78 ±0.5

Stored power N [kW] 15.38 10.82 ±0 11.15 ±0

Lost power N [W] 0.23 4.11 ±1 3.70 ±1

Surtension N 31.37 31.18 ±0.02 31.15 ±0.02

Stored power W [kW] 15.55 10.91 ±0 11.27 ±0.3

Lost power W [W] 0.19 6.05 ±0.02 5.85 ±0.04

Surtension W 31.70 31.42 ±0.01 31.48 ±0.1

Carrier power on BS [W] 978.5 684.5 ±0.5 725.1 ±2

Sideband power on BS [W] 1.70 8.56 ±0.1 10.9 ±0.2

Reflected carrier [W] 17.84 8.42 ±0.01 9.82 ±0.08

Reflected sb [W] 0.027 0.24 ±0 0.26 ±0.01

CITF surtension Carrier 49.04 34.74 ±0.03 37.10 ±0.08

CITF surtension SB 36.49 29.01 ±0.02 24.0 ±0.1

Transmitted (detected) carrier [mW] 0.064 (0.064) 359 ±6 (1.6 ±0) 324 ±40 (3.5 ±0.1)

Transmitted (detected) sb [mW] 18.7 (17.9) 93.0 ±0.8 (70.0 ±1) 125 ±2 (100 ±2)

Sensitivity [*1E-23] 2.48 2.87 ±0.01 2.96 ±0.02

Optical program: typical results (Modal simulation)

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Example:

Virgo simulation with surface maps and with an incoming field of 20W

Contrast defect= 0.94%

North arm amplification = 31.65

West arm amplification = 32.06

Recycling gain = 34.56


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Details at FFSR

Fabry-Perot cavity transfer function measurements

Fit values with 95% confidence interval:

fp = 479 +/- 3.3 Hz

fz = -177 +/- 2.2 Hz

FSR = 1044039 +/- 2.2 Hz

L = 143.573326 +/- 30 m

Error bars: from measurement errors,Not for constant biases.

(fit both real and imaginary parts simultaneously)


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Input Mode Cleaner Losses

T=2427 ppmT=2457 ppm

T = 5.7 ppm

Roud-trip losses:

Computed from mirror maps: 115 ppmFrom measurements: 846 +/- 5 ppm

Mirror transmission measurements+ transfer function details measurements

=> Mode mismatching 17%=> Cavity transmissitivity for TEM00 83%


(september 2005)

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