Gravitational wave interferometer OPTICS

Gravitational wave interferometer OPTICS

François BONDU

CNRS UMR 6162 ARTEMIS,

Observatoire de la Côte d’Azur, Nice, France

EGO, Cascina, Italy

Fabry-Perot cavity in practice Rules for optical design

Optical performances

Contents

I. Fabry-Perot cavity in practice

Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront

Length / Frequency measurement: cavity transfer function

II. Rules for gravitational wave interferometer optical design

Optimum values for mirror transmissions

“dark fringe”: contrast defect

“Mode Cleaner”

III. Optical performances

Actual performances:

Mirror metrology

Optical simulation

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

VIRGO optical design

Slave laser

Master laser

Hz /

10 . 3

~

2

 L

h  L Fabry-Perot cavity to detect gravitational wave

Suspended mirrors to cancel seismic noise

L=3 km

Long arms to divide mirror and suspension thermal noise Recycling mirror to reduce shot noise

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

L=144m

Output Mode Cleaner to filter output mode

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

1. Fabry-Perot cavity: A. parameters

REFLECTION TRANSMISSION

Can we understand these shapes?

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

Mirror 1 Mirror 2

E_in E_sto E_trans

E_ref

1. Fabry-Perot cavity: A. parameters

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

E_rt = r₁ P^-1 r₂ P E_sto

with 2

) 1

(

1 2 1

2 1

2 1

1 1 1

1 1

L L T

L T

L P

P

L L T

T r

RT

RT sto

rt





 





 









Round trip “losses”

1. Fabry-Perot cavity: A. parameters

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity

1. Fabry-Perot cavity: A. parameters

c E L

e r r E t

E E

t E

c L i

P

i in sto

RT in

sto

 





with 4 1

: solution state

steady

) / 2

exp(

delay n

propagatio

2 1

1

1

 



















 2



 L L

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

E_rt = r₁ P^-1 r₂ P E_sto

Period

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity

1. Fabry-Perot cavity: A. parameters

2

2 1 1 2

1 1

] 1 2 [ 0

1

 

 





 





 

r r P t

P e E

r r E t

in sto

i in sto







SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

Recycling gain

2 1

 

 

 t

G

RESONANCE CONDITION

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity

 



 4 2

that so

) ,

(

k

c

L  L 

1

) (

4

0





 











 



 c

L f

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

RESONANCE CONDITION

Suppose now

/ 1 if f E G

E

P in

sto

 

1. Fabry-Perot cavity: A. parameters

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

f

Finesse FSR

 2 _Finesse

1. Fabry-Perot cavity: A. parameters

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

on resonance reflectivity

) ( 2

1

) ( 1

2 1

1

) 1

) (

(

f i in

ref

e r r

e L r

r E

f E

r











 





 



 

2 1

1 2

1

1

) 1

) ( 0

( r r

L r

r r

)

0

(   R

1. Fabry-Perot cavity: A. parameters

Round Trip Losses Free Spectral Range Recycling gain

Cavity Pole Finesse

Cavity reflectivity

2nd order In T+P

1st order in T+P Finesse

On resonance reflection

transmission

1. Fabry-Perot cavity: A. parameters

p T 2  

) 1

2 (

1

 T  p  

 T



 



   



1 2

1

1 ( 1 ) r r p r

 r

2 2 1

1 



 



  r t r t

2 1

2 1

1 r r r

 r



 ^{T 4 T }

^{T p}



SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

1. Fabry-Perot cavity: A. parameters

T1 = 12%

T2 = 5%

L = 0

(finesse = 35)

REFLECTION TRANSMISSION

SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

1 0

) 1

) ( 0 (

2 1

1 2

1

 





  

r r

L r

r r

 0



Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer

 0



1. Fabry-Perot cavity: B. Matching

Optimal coupling

Over-coupling

Under-coupling

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer SCALAR MODEL:

“plane waves”

scalar transmissions, scalar losses of mirrors

Frequency/Length tuning

c L

0 0

4 

 

1. Fabry-Perot cavity: B. Matching

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer NON-SCALAR MODEL:

1. Fabry-Perot cavity: B. Matching

Mirror 1 Mirror 2

E_in E_sto E_trans

E_ref z axis

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer NON-SCALAR MODEL:

1. Fabry-Perot cavity: B. Matching

E_sto(x,y) = k E_in(x,y)

(k complex number)

E_sto E_in

Wavefront matching:

Superpose angles and lateral drifts

of incoming and resonating beam

NON-SCALAR MODEL:

1. Fabry-Perot cavity: B. Matching

E_sto(x,y) = k E_in(x,y)

(k complex number)

E_sto E_in

Wavefront matching:

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability

The Fabry-Perot interferometer

NON-SCALAR MODEL:

2 2

1 1

2 2 1 2

1

, ,

) , ,

(    



 

 

C

Definition of beam coupling:

Round trip coupling losses:

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer

) ,

(

1

rt

C E E

L  

1. Fabry-Perot cavity: B. Matching

NON-SCALAR MODEL:

2 1

1

1

T L L

T

L

   

Definition of stability:

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer

Definition of stability in case of perfect surface figures:

1 ) 1

)(

1 ( 0

2 1







 R

L R

L

1. Fabry-Perot cavity: B. Matching

1. Fabry-Perot cavity: B. Matching

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer Charles 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

Impedance matching

Frequency/length tuning (“lock”) Wavefront matching

alignment

beam size / position

surface defects - stability The Fabry-Perot interferometer

1. Fabry-Perot cavity: B. Matching

Phase modulated laser:

m phase modulation index fm modulation frequency

SB- C SB+

 

 



  



in

m e

m e

e

0 ^

1 2

2





 

 



    



ref

m e

R m e

R R

e

0

( f f ) 2

) 2 f f

( )

f

(





1. Fabry-Perot cavity: C. measurement

 

p p 0

0

f f 1

f f ) 1 ( )

f ( Im



 











c c c

P Pm R Pm

s 

f

f 

 ^{( f ) ( f f ) ( f ) ( f f )} 

2 Im

*

*

0 c c m c c m

P

m R R R R

P

s    

error signal:

Does not provide information about frequency behavior once locked

1. Fabry-Perot cavity: C. measurement

Modulated laser + measurement line:

n phase modulation index fn modulation frequency

p 0

) noise frequency

(

f f 1

) 1

( i m TF P



  

SB- C SB+

f << FSR, f ≠ fm

This pole

1. Fabry-Perot cavity: C. measurement

Contents

I. Fabry-Perot cavity in practice

Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront

Length / Frequency measurement: cavity transfer function

II. Rules for gravitational wave interferometer optical design

Optimum values for mirror transmissions

“dark fringe”: contrast defect

“Mode Cleaner”

III. Optical performances

Actual performances:

Mirror metrology

Optical simulation

2. Optical design: A. mirror transmissions

cavity RT

FP L G

R  1  

Fabry-Perot cavity with Rmax transmissions as end mirrors

Virgo mirrors: L

~500 ppm, G

~ 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

• Michelson simple :

BS

laser Pin

Pout

min max

min

max

P

P P C  P  

P

, P

= P

On black and white fringes

2. Optical design: B. dark fringe

Slave laser

Master laser

L=3 km

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

L=144m

2. Optical design: C. Mode Cleaners

Detection

Output Mode-Cleaner

Output Mode Cleaner on Suspended Bench

Photodiodes on Detection Bench

Beam

Contents

I. Fabry-Perot cavity in practice

Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront

Length / Frequency measurement: cavity transfer function

II. Rules for gravitational wave interferometer optical design

Optimum values for mirror transmissions

“dark fringe”: contrast defect

“Mode Cleaner”

III. Optical performances

Actual performances:

Mirror metrology

Measured optical parameters

16.7 W 7.1 W

F = 49±0.5 F = 51 ±1

G^carrier = 30-35 (exp. 50) G^SB ~ 20 (exp. 36)

1 – C < 10^-4

1 – C = 3.10^-3 (mean) Slave laser

Master laser

1 W T=10%

Losses in input Mode Cleaner?

Recycling gain?

Arm finesses?

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

Scatterometer CASI 400x400mm

Micromap 400x400 mm (local defects)

Absorption

Photothermal Deflection System

Phase shift interferometer

Diam Diam 35 cm 35 cm Rms Rms 2.3 nm 2.3 nm

Surface maps

Ex: a large flat mirror

-Good quality silica - Good polishing

- Control of coating deposition (DIBS) with no pollutants

- Surface correction

TWO optical programs:

- One that propagates wavefront with 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

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

Optical program: typical results (Modal simulation)

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

Details at F

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)

Input Mode Cleaner Losses

T=2427 ppm T=2457 ppm

T = 5.7 ppm

Roud-trip losses:

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

Mirror transmission measurements

+ transfer function details measurements

=> Mode mismatching 17%

=> Cavity transmissitivity for TEM00 83%

(september 2005)

Contents

I. Fabry-Perot cavity in practice

Scalar parameters – cavity reflectivity, mirror transmissions, losses Matching: impedance, frequency/length tuning, wavefront

Length / Frequency measurement: cavity transfer function

II. Rules for gravitational wave interferometer optical design

Optimum values for mirror transmissions

“dark fringe”: contrast defect

“Mode Cleaner”

III. Optical performances

Actual performances:

Mirror metrology

Optical simulation