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/michele.punturo@pg.infn.it

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.