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The Virgo design Thermal noise curve is evaluated with a Q~10 6, and the maximum expected Q, due to coating losses, is Q~ Standard Quantum Limit is about a factor 100 below Virgo TN i.e. equivalent to a Q=10 10 ; consequently, for reaching SQL sensitivity Q should improve by / ~ 400- A bit difficult !! For this reason it is interesting to explore Coating-less Mirrors This argument will become even more important if we want to go below SQL.
Citation preview
Some Ideas on Coatingless all-reflective ITF
Adalberto Giazotto & Giancarlo Cella
INFN- Pisa
Why Total Reflecting Mirrors?
Experimental data from Yamamoto and Numata show a very high coating loss angle even at low temperature. Subsequent experiments
for reducing coating losses gave a 50% improvement but it is not obvious if the optical specifications are still good.
7
8
510.2~1
410~~
Total
CoatingB
CCoating
B
CBulkTotal
Q
EE
EE
EC/EB=d/w0~10-4
2/10
2/12/1
)(16)(~
wEEE
Tkx
CoatingB
CBulkB
Total
d w0MIRROR
1 10 100 1000 1000010-2710-2610-2510-2410-2310-2210-2110-2010-1910-1810-1710-1610-1510-1410-1310-12
Virgo 28-3-2001http://www.virgo.infn.it/[email protected]
Radiation Pressure Quantum Limit Wire Creep Absorption Asymmetry Acoustic Noise Magnetic Noise Distorsion by laser heating Coating phase reflectivity
h(f)
[1/s
qrt(H
z)]
Frequency [Hz]
Total Seismic Noise Newtonian (Cella-Cuoco) Thermal Noise (total) Thermal Noise (Pendulum) Thermal Noise (Mirror) Mirror thermoelastic noise Shot Noise
MLΩ1h~SQL
The Virgo design Thermal noise curve is evaluated with a Q~106, and the maximum expected Q, due to coating losses, is Q~2.5 107. Standard Quantum Limit is about a factor 100 below Virgo TN i.e. equivalent to a Q=1010 ; consequently, for reaching SQL sensitivity Q should improve by 1010/ 2.5 107~ 400- A bit difficult !! For this reason it is interesting to explore Coating-less Mirrors
This argument will become even more important if we want to go below SQL.
Some History
In 1970 Robert Forward (Hughes Lab.) build the first Interferometer for GW detection. It was equipped with Roof Prisms Mirrors. Arm length ~2m
Toraldo di Francia in 1965 proposes flat Roof Prism for creating a stable Radio Frequency cavity. Stability was succesfully experimentally tested.
Braginsky et al. Recently Published a paper on the use of Roof Prism and Corner Cube mirrors in Fabry Perot cavities using Antireflective Coatings .
A further point: Gratings Technology (see recent presentations at Einstein’s Week in Jena) it is improving
a-lot on losses (10-3) but does not work without Coatings
1) Capability of constructing a completely Coatingless FP Cavity
2) Capability of constructing a completely Coatingless Beam Injection system.
What are then the Missing things for creating a Very Low Thermal Noise Fabry Perot Cavity?
Rotation Parabolas as exact reflectors for closed geometrical optical trajectories
22
1 22 LzxL
y
L/2
L
y
x
22
1 22 LzxL
y
y
x
z
No need of AR Coatings No Beam Losses The spherical mirror surface is matched to the constant phase beam surface curvature
Equivalence to Spherical mirrors of Rotation Parabolas as reflectors for closed geometrical optics trajectories
Lbaxa
LbLy 2
xLzx
y
2
22
xLzx
y
2
22
LL/2
x
y
axaLbLy
b
A Rotation-Parabolic Reflector self-conjugating at distance L is equivalent to a spherical mirror with curvature radius L/2
Reflective cavity with Asymmetric Arms
L2
y
xL1
Beam Splitter for Power Injection in the Parabolic 4 elements all-Reflective Cavity
LASER
Output Beam
Vacuum in
Beam Splitter
Total Reflection
1) If α+β=π the trajectory inside the prism is closed and d=r. i.e. the trajectory is replicating itself.
2) If α+β=π and the trajectory A is parallel displaced, the inner trajectory is length invariant.
3) The only way for changing inner trajectory length is by rotating the prism. Then we still have d=r but the trajectory does not close and consequently does not replicate itself. This property can be used for making inner trajectory resonating or antiresonating
Prism B-S Properties
Total Reflection
α β
dr
A
Non Total Reflection
δL
Optical Diagram of a 4 Elements Coatingless all- reflective cavity
Lu
Ld
M1M2
M3
M4
outA1
αin
L1
R1,T1
B1
U
D1
(τ1,ρ1)
(1, 2)
(1, 2)
L0
inA1
11X
12X
1J
R2,T2
B2
B3
D2
L2
B4
δW
(λ1, λ2)
(1, 2) outA2inA2
21X
22X
2J
(1, 2)
δW Vacuum due to losses
1 ie2ie(τ2,ρ2)
22 RT , 11 RT ,
L3
ΤΓ, RΓ
Vin
The Classical Amplitudes U and B3
B3 =DδW+Cαin+EB4
U=AδW+Bαin+CB4
1222212
2221222
211
1012012
12
211
121
cdLLL
RcdLLL
ieTT
c
LdLLRc
LdLLi
ecdLuLRc
dLuLieRTRTT
A
1222212
2221222
211
1021202122
12
211
cdLLL
RcdLLL
ieTT
c
LLRc
LLi
ecdLuLRc
dLuLieRTR
B
1222212
2221222
211
2122212
2221212
121
1222212
2221222
211
12212
00
112
cdLLL
RcdLLL
ieTT
cdLLL
RcdLLL
ie
cdLuLRc
dLuLieTRR
cdLLL
RcdLLL
ieTT
cuLdLRc
uLdLieTRR
cdLuLRc
dLuLie
cdLL
RcdLL
ieTTC
1222212
2221222
211
2
122122
2222
cdLLL
RcdLLL
ieTT
cdLuLRc
dLuLieTRR
c
LRc
Li
e
E
1222212
2221222
211
12212
cdLLL
RcdLLL
ieTT
cuLdLRc
uLdLieTRR
D
αin
U
δW
B3
B4
Resonance conditions and its consequences
B4 is connected to U by means of C coefficient i.e. Since the vacuum Vn is also entering from the port B4 it is relevant to understand the coupling between Vn and U i.e. the behavior of C coefficient as a function of Ω.
1222212
2221222
211
2122212
2221212
121
1222212
2221222
211
12212
00
112
cdLLL
RcdLLL
ieTT
cdLLL
RcdLLL
ie
cdLuLRc
dLuLieTRR
cdLLL
RcdLLL
ieTT
cuLdLRc
uLdLieTRR
cdLuLRc
dLuLie
cdLL
RcdLL
ieTTC
αin
U
δW
B3
B4
Vn
The resonance conditions are:
C becomes :
+1 =Prism Resonance -1 =Prism
Antiresonance
Arm Resonance
1
11222212
cuLdLR
cdLLL
R
12d21
12d21
12d21
12d21
12d21
12d21
ΨΨc
2L2L2LΩi2
21
ΨΨc
2L2L2LΩi2
12
00ΨΨ
c2L2L2L
Ωi312
22
21
ΨΨc
2L2L2LΩi2
21
ΨΨc
2L2L2LΩi2
12
00ΨΨ
c2L2L2L
Ωi312
22
21
ee11
e1111
ee11
e1111
Re
RRRR
RRRR
Antires
RRRR
RRRR
s
cdLL
RcdLL
ie
cdLL
RcdLL
ie
C
C
If R1=R2 and Ω=0, CRes ≠0. While CA.Res =0 for any R1, R2 and Ω=0
If the prism is antiresonant and the arms are in resonance, it follows that at Ω=0 there is no coupling Vn-U i.e. C=0. Then only Vn sidebands may couple to U.
12d21
12d21
12d21
ΨΨc
2L2L2LΩi2
21
ΨΨc
2L2L2LΩi2
12
00ΨΨ
c2L2L2L
Ωi312
22
21
ee11
e1111
RRRR
RRRR
Antires
cdLL
RcdLL
ie
C
The Cavity Finesse
U=AδW+Bαin+CB4
Cavity Finesse can be evaluated by differentiating B with respect L1 and L2 and then making .For the sake of simplicity we set the prism both in resonance (-) and in antiresonance (+);
212
2122212
2121
212
2121
2102120212
221
2122212
2121
212
0
0
0 ,
i
cdLLL
ieRRRRRR
Rc
LLRc
LLi
eRRc
dLLLi
eRRRR
ciLB
At Ω=0 we obtain the finesse
Ponderomotive actions and thermal noise couple to U trough the term where δLi are both the mirror and prism displacements produced by these two phenomena.
212
102
221212
101
01212
212
RRcR
cLLRRRRR
RRciLB
iLiLB
212
1
RRF
11222212
cdLLL
R
Toward an interferometric application
If we close the port B3 with a mirror, then the vacuum B4 will be stopped and U will depend only by δW and αin
αin
U
δW
Vn B3
B4
ina
EcL
RcL
ieR
CcL
RcL
ieRE
cL
RcL
ieRB
EcL
RcL
ieR
inVTWDcL
RcL
ieR
CWAU
3232
1
23232
3232
1
3232
1
132
32
U=A’δW+C’αin
B3 Mirror Thermal Noise
We have seen that C=0 for Ω=0; this means that under the choosen resonance conditions ther is no carrier in B3 and if there is no carrier, sidebands, to first order, can not be produced. This is easily seen by differentiating U with respect to L3:
0
0
21
2
30
3232
1
23232
33
1
32
32
23
3
EcLReR
cLReR
LC
EcL
RcL
ieR
CcL
RcL
ieR
BLL
U
ina cL
i
cL
i
This means that we may put a normal mirror on B3 without affecting total reflecting cavity thermal noise.
The Interferometer
U
LASER
If we inject in U a Squeezed Vacuum, we can beat SQL and obtain, in priciple a very performant ITF.
Beam Splitter for Power Injection in the 3 elements Parabolic all-Reflective Cavity
Perhaps a more performant configuration
(1, 2)
Ldαin
UB1
B2 M1 Lu M2
M3 M4
M5
M6
R2
R1
12X
11X
outA1 inA1
1J
L1
δW
L2R3
1 ie
Conclusions
Some Difficulties1) The Parabolic and Beam Splitter prisms needs to have a
remarkable Refraction Index omogeneity for avoiding higher mode production. At the moment we may obtain λ/1000 over 10 cm thicknes of silica, enough for starting tests.
2) The Parabolic Prism edge should be very thin, some µm, otherwise both higher modes and beam losses will be produced. The design of a beam with zero intensity on the prism edge could be the solution.
A big Advantage Without coatings the Prisms mechanical Q can be enormous even at room temperature; this solution could be instrumental also for future SQL beating ITF’s.