1. 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

2. 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
Accurate in-situ metrology

3. VIRGO optical design Fabry-Perot cavity to detect gravitational wave
Input <<Mode Cleaner>> to filter out input beam jitter and select mode
L=144m
Output Mode Cleaner to filter output mode
Recycling mirror to reduce shot noise
Suspended mirrors to cancel seismic noise
L=3 km
Long arms to divide mirror and suspension thermal noise
Michelson configuration at dark fringe + servo loop to cancel laser frequency noise
Slave laser
Master
laser

4. 1. Fabry-Perot cavity: A. parameters
SCALAR MODEL:
“plane waves”
scalar transmissions, scalar losses of mirrors
REFLECTION
TRANSMISSION
Can we understand these shapes?

5. 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
Ein
Esto
Etrans
Eref
Mirror 1
Mirror 2
Ert = r1 P-1 r2 P Esto

6. 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
Ert = r1 P-1 r2 P Esto
Round trip “losses”

7. 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
Ert = r1 P-1 r2 P Esto
Period:
Free spectral range

8. 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
RESONANCE CONDITION
Recycling gain

9. 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
RESONANCE CONDITION
Suppose now
Cavity pole

10. 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
Finesse

11. 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

12. 1. Fabry-Perot cavity: A. parameters
2nd order
In T+P
1st order
in T+P
Finesse
On resonance reflection
transmission

13. 1. Fabry-Perot cavity: A. parameters
SCALAR MODEL:
“plane waves”
scalar transmissions, scalar losses of mirrors
T1 = 12%
T2 = 5%
L = 0
(finesse = 35)
REFLECTION
TRANSMISSION

14. 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
SCALAR MODEL:
“plane waves”
scalar transmissions, scalar losses of mirrors
Optimal coupling
Over-coupling
Under-coupling

15. 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
SCALAR MODEL:
“plane waves”
scalar transmissions, scalar losses of mirrors
Frequency/Length tuning

16. 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:
Ein
Esto
Etrans
Eref
z axis
Mirror 1
Mirror 2
Ert = r1 P-1 r2 P Esto
Ein(x,y) ; Esto(x,y) ;
r1, P, r2 are operators

17. 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:
Wavefront matching:
Esto(x,y) = k Ein(x,y)
(k complex number)
Esto
Ein
Superpose angles and lateral drifts
of incoming and resonating beam
<<ALIGNMENT ACTIVITY>>

18. 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:
Wavefront matching:
Esto(x,y) = k Ein(x,y)
(k complex number)
Ein
Esto
Superpose beam positions and beam widths <<MATCHING ACTIVITY>>

19. 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:
Definition of beam coupling:
Round trip coupling losses:
Too small mirror diameters “clipping”
imperfect surface: local defects, random figures

20. 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:
Definition of stability:
Definition of stability in case of perfect surface figures:

21. 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 ( )
Alfred Perot ( )
Amédée Jobin (mirror manufacturer) ( )
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

22. 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

23. 1. Fabry-Perot cavity: C. measurement
Phase modulated laser:
SB C SB+
m phase modulation index
fm modulation frequency

24. 1. Fabry-Perot cavity: C. measurement
error signal:
Does not provide information about frequency behavior once locked

25. 1. Fabry-Perot cavity: C. measurement
Modulated laser + measurement line:
SB C SB+
n phase modulation index
fn modulation frequency
This pole
f << FSR, f ≠ fm

26. 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
Accurate in-situ metrology

27. 2. Optical design: A. mirror transmissions
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)

28. 2. Optical design: B. dark fringe
Michelson simple :
laser
Pin BS
Pmax, Pmin = Pout
On black and white fringes
Pout

29. 2. Optical design: C. Mode Cleaners
Input <<Mode Cleaner>> to filter out input beam jitter and select mode
L=3 km
L=144m
Slave laser
Master
laser
Output Mode Cleaner to filter output mode

30. Detection Beam Photodiodes on Detection Bench Output Mode Cleaner
on Suspended Bench
Output Mode-Cleaner
Beam

31. 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
Accurate in-situ metrology

32. Measured optical parameters
Losses in input Mode Cleaner?
Arm finesses?
Slave laser
Master
laser
1 W
F = 49±0.5
F = 51 ±1
Recycling gain?
16.7 W
7.1 W
Gcarrier = (exp. 50)
GSB ~ 20 (exp. 36)
T=10%
1 – C < 10-4
1 – C = (mean)
III. Optical performances

33. Absorption
Photothermal Deflection System
Scatterometer CASI 400x400mm
Mirror metrology
Micromap 400x400 mm
(local defects)
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
reproducibility 0.4 nm; step 0.35 mm
resolution 30 ppb/cm // 20 ppb
resolution of a few ppm
transmission map
Phase shift interferometer
Instruments: ESPCI, Paris
Coating, 140 m2 room class 1: LMA, Lyon
The VIRGO large mirrors: a challenge for low loss coatings, CQG 2004, 21

34. Surface maps Good quality silica Good polishing
Ex: a large flat mirror
Good quality silica
Good polishing
Control of coating deposition
(DIBS) with no pollutants
- Surface correction
Diam 35 cm
Rms 2.3 nm
p-p nm
III. Optical performances

35. Optical simulation 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…
TWO optical programs:
One that propagates wavefront
with FFT
One that decomposes beams
on TEM HG(m,n) base
III. Optical performances

36. Optical program: typical results (Modal simulation)
Scalar defects
Maps
Maps+thermal
Opt mod index
0.068
0.172±0.001
0.215 ±0.001
Opt demod phase
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

37. Virgo simulation with surface maps and with an incoming field of 20W
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
III. Optical performances

38. Fabry-Perot cavity transfer function measurements
Details at FFSR
Fit values with 95% confidence interval:
fp = 479 +/- 3.3 Hz
fz = /- 2.2 Hz
FSR = /- 2.2 Hz
L = /- 30 mm
Error bars: from measurement errors,
Not for constant biases.
(fit both real and imaginary parts simultaneously)
III. Optical performances

39. Input Mode Cleaner Losses
Roud-trip losses:
Computed from mirror maps: 115 ppm
From measurements: 846 +/- 5 ppm
T = 5.7 ppm
Mirror transmission measurements
+ transfer function details measurements
=> Mode mismatching 17%
=> Cavity transmissitivity for TEM00 83%
(september 2005)
T=2457 ppm
T=2427 ppm
III. Optical performances

40. 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
Accurate in-situ metrology