Advanced Virgo Sensitivity Curve Study VIR-0073D-12 Authors

Advanced Virgo -

Advanced Virgo sensitivity curve study

VIR-0073D-12

Authors: M Punturo1

Issue: D Date: November 19, 2012

1 INFN - Sezione di Perugia, Italy,

AdV – Advanced Virgo – Sensitivity Curve Study Web: https://wwwcascina.virgo.infn.it/ Email: [email protected] VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 2 of 78

Abstract

This document describes the sensitivity curve modeling of the Advanced Virgo detector.

Table 1: Document history Issue Date Description A 09-03-2012 First description of the fundamental noises, as currently implemented in gwinc. Just started the computation of the AdV noise and missing any evaluation of technical noises (to be inserted in a future release). B 29-03-2012 Residual seismic noise introduced in gwinc and in the note. Losses budget introduced in the quantum noise chapter

C 24-07-2012 Detection distance corrected by Patrick Sutton, because of an incorrect integration constant, resulting in to an increment of about 6% in the horizon (function int73.m). Reference configuration file modified reducing the clear aperture for the clipping losses to 33 cm dominated by the baffles and not by the coating. Suspension thermal noise still to be modified because computed taking in account a reference mass anymore present in the AdV design. D 19-11-2012 Thermal noise model updated removing the reference mass. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 3 of 78

Contents

1 Seismic Noise 5 1.1 The Virgo seismic ﬁltering system ...... 5 1.2 Gravity Gradient Noise ...... 6

2 Test mass Thermal Noises 11 2.1 Mirror Brownian Noise ...... 11 2.1.1 The coating contribution ...... 12 2.1.2 Substrate loss angle ...... 12 2.1.3 Surface losses contribution ...... 13 2.2 Substrate thermo-elastic noise ...... 13 2.3 Thermo-optical noises ...... 14

3 Suspension Thermal Noise 17 3.1 Including the violin modes of the mirror suspension wires ...... 18 3.1.1 Optimized (Dumbbell–shaped) ﬁbres ...... 19 3.2 Vertical bouncing mode thermal noise ...... 19

4 Residual Gas 20

5 Quantum Noise 20 5.1 Initial interferometers ...... 20 5.2 Read–out schemes ...... 21 5.3 Signal Recycled interferometer ...... 21 5.4 Optical losses budget ...... 24

6 AdV Sensitivity 26

Bibliography 27

Nomenclature 30

Appendices 32

A IFO structure 32

B Brownian Substrate Code 41

C Seismic noise generation Code 42

D Brownian Substrate Finite Size Correction Code 43

E Brownian Coating parameters feeding Code 45

F Brownian Coating Code 47

G Thermo-elastic Substrate Code 50

H Thermo-elastic Substrate Finite Correction Code 51

I Thermo-optical coating core computation Code 52

J Thermo-optical coating core computation Code 55

K Thermo-elastic Coating Finite Correction Code 57 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 4 of 78

L Suspension Thermal Noise Code 60

M Optimised ﬁbre Code 67

N Residual Gas Code 71

O Quantum (optical read–out) noise 73

P Miscellanea of computations 76 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 5 of 78

1 Seismic Noise

The seismic noise, consisting in to the natural and anthropogenic shaking of soil, enters in the noise budget of AdV through both the residual vibration transmitted through the SA to the mirrors and the direct coupling due to the Newtonian attraction force of the suspended test masses to the soil (the so–called Newtonian or Gravity Gradient Noise [1–3]). Seismic activity at the Virgo site is obviously affected by the weather conditions (wind, sea excitation) and by the human activity (for few examples see [4] and [5]). In particular, at the Virgo site, the wind effect is visible under 0.1 Hz, the ocean between 0.1 and 0.2 Hz, the Tirreno sea between 0.2 and 1 Hz, the traffic between 1 and 5 Hz [6]; this generate a structured noise, mainly below the detection cut–in frequency. This noise (so called µ–seism) obviously fluctuates according to the environmental condition, as shown in Fig.1 and in Fig.2.

Figure 1: Spectral amplitude of the (µ–) seismic displacement noise measured in the Virgo Central Building (CB). The color level indicates the persistence of the noise amplitude in the correspondent range [7].

An accurate simulation of the seismic noise should take in account all these structures; this has been attempted trying to reproduce the main bumps at low frequency and the ≈ 10−7/f 2 behavior at high frequency

pj Y 2 2 fjf X xseism = a1 · f − f + i + a2 · bumps (1) j Q j=1,3 j Bump functions are coded inside groundSeism.m (see Appendix C). In gwinc these parameters are contained in ifo.Seismic. In Fig.3 the seismic noise measured in the Virgo CB is compared with the model of Eq.1 in case of quiet µ–seism. In Fig.4 the same comparison is made in case of noisy environmental conditions. 1.1 The Virgo seismic filtering system The most recent description of the filtering performance of the Virgo Super–Attenuator (SA) is in [8]. Essentially some reduction of the filtering capability has been seen, probably due to a magnetic short–cut in the vertical VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 6 of 78

Figure 2: Statistical property of the spectral amplitude of the (µ–) seismic displacement noise measured in the Virgo Central Building (CB). For each frequency bin, it is reconstructed the probability to find a seismic amplitude below the threshold indicated by each curve [7]. chain of filters, but it is still valid to approximate the transfer function (TF) of the whole chain, by the product of the TFs of each stage. A detailed description of the SA TF computation is in the Paolo Ruggi Thesis [9] and here only the results are reported. In Fig.5 the Horizontal and Vertical TFs are shown [6]. The Blue curve represents the TF from the filter 0 to the mirror for the horizontal displacement; the determine the effective filtering the inverted pendulum filtering should be considered. For technical reason, it is preferred to measure the seismic noise on the top of the IP and to consider the filtering properties of the chain F0–mirror. The Red curve represents the Vertical TF ground–to–mirror. It is clear that in the in–band frequency range, the role of the vertical–to–horizontal coupling is crucial to determine the effective filtering of the SA. A mechanical −3 coupling angle of about θc ' 10 is expected. Adopting such as coupling angle it is possible to compute the residual seismic noise at the level of the mirror, as shown in Fig.6.

1.2 Gravity Gradient Noise The Gravity gradient noise or Newtonian Noise [1–3] is due to the direct coupling, through the Newtonian attraction force, of the suspended test masses to the soil. The latest and more general evaluation of this noise source has been done by G. Cella in [10] (see Eq.50 in [10]); for low frequencies and surface√ detector that evaluation coincides with the previous estimation [2] of the spectral amplitude (in terms of h 1/ Hz): √ 2 π 4π ω 2 h (ω) = GG · x (ω); ω = pGρ (2) GGN 3 L ω seism GG soil where G is the (universal) gravity constant and ρsoil is the averaged density of the soil in the AdV site (ρ ' 2300 − 2700 [kg/m3] [11, 12]). VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 7 of 78

Virgo central building, quiet condition (1am and calm sea) −5 10 VERTICAL Virgo EW Virgo NS −6 10 model 2e−7/f2 AdV model

−7 10

−8 10

−9

m / sqrt(Hz) 10

−10 10

−11 10

−12 10 −1 0 1 2 10 10 10 10 Hz

Figure 3: Seismic noise in the Virgo CB [7], in a condition of quiet µ–seism. The green curve shows the spectral amplitude of the vertical seismic red curve the East–West horizontal vibration and the blue curve the North–South horizontal seismic vibration. The dashed black line is the 10−7/f 2 simple model. The magenta markers shows the prediction of Eq.1, computed with the low noise parameters. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 8 of 78

Virgo central building, noisy condition (8am, high microseism) −5 10 VERTICAL Virgo EW

−6 Virgo NS 10 model 2e−7/f2 AdV model

−7 10

−8 10

−9

m / sqrt(Hz) 10

−10 10

−11 10

−12 10 −1 0 1 2 10 10 10 10 Hz

Figure 4: Seismic noise in the Virgo CB [7], in a condition of noisy µ–seism. The green curve shows the spectral amplitude of the vertical seismic red curve the East–West horizontal vibration and the blue curve the North–South horizontal seismic vibration. The dashed black line is the 10−7/f 2 simple model. The magenta markers shows the prediction of Eq.1, computed with the high noise parameters. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 9 of 78

5 10 Horizontal TF Vertical TF

0 10

−5 10

−10 Transfer Function 10

−15 10

−20 10 −2 −1 0 1 2 10 10 10 10 10 Hz

Figure 5: Transfer Functions (TF) of the Virgo SA [6]: Blue curve, Horizontal TF from the filter 0 to the mirror. To determine the effective filtering the role of the inverted pendulum (IP) must be inserted. Red curve, Vertical transfer function from ground to mirror (in this plot two curves, relative to a different frequency sampling, have been overlapped). VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 10 of 78

AdV residual seimic noise (noisy condition) −5 10

−10 10

−15 10

−20 10 1 / sqrt(Hz)

−25 10

−30 10

−35 10 −1 0 1 2 10 10 10 10 Hz

Figure 6: Residual seismic noise [6] at the level of the mirror (dimensionless strain unit), computed using a −3 TFs similar to the one in Fig.5, a vertical–to–horizontal coupling angle coupling angle θc ' 10 and the input seismic noises of Eq.1. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 11 of 78

2 Test mass Thermal Noises

Four thermal noise contribution can be identified in the test masses of the interferometer: • Classical Brownian motion of the mirror • Thermo–Dynamical fluctuation of the mirror bulk • Thermo–Dynamical fluctuation of the mirror coating

• Thermo–refractive ﬂuctuation of the mirror coating

2.1 Classical Brownian motion of the mirror The evaluation of the Brownian fluctuation of the AdV mirror can be evaluated through the direct application of the fluctuation–dissipation theorem [13], as shown by Levin [14] and Bondu et. al. [15]. The displacement power spectrum at low frequency (well below the first resonant mode) of a single mirror is:

8k T hx (ω)i2 = B φ (f) · U (3) Brown ω mirr where φmirr is the eﬀective loss angle of the mirror and U the the elastic energy stored in the mirror under a normalized gaussian pressure having the shape of the laser beam. The energy U has been evaluated by Levin:

1 − σ2 U = √ FS (4) 2 π · EFS · wbeam where wbeam is the size of the beam illuminating the mirror. The formula 4 is valid for an inﬁnite size mirror and it has been evaluated in the case of a ﬁnite size mirror (thickness hm, radius Rm) by Vinet [16]:

U = U0 + ∆U (5) where 2 wbeam ∞ exp − ξk 2 2Rm 1 − σFS X U0 = Wk 2 (6) πRm · EFS ξkJ0(ξk) k=0 hm hm hm 1 − exp −4ξk R + 4 ξk R · exp −2ξk R W = m m m k h i2 2 hm hm hm 1 − exp −2ξk − 4 ξk exp −2ξk Rm Rm Rm where ξk are the zeros of the Bessel function J1. The ﬁnite size mirror correction ∆U is

2 " 4 2 # Rm hm hm ∆U = 3 + 12σFS · Σ · + 72 (1 − σFS)Σ (7) 6πhm · EFS Rm Rm 2 wbeam ∞ exp −0.5 ξk X 2Rm Σ = 2 ξ J0(ξk) k=0 k These formulas are implemented in the code shown in the Appendices B and D. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 12 of 78

2.1.1 The coating contribution

The eﬀective loss angle φmirr should take in account the substrate dissipation φbulk and all the contributions given by parassitic losses (surface, coatings,...); in particular the coating contribution can be modeled [17] as δU φ = φ (f) + d φ (8) mirr bulk coat U coat where δU is the energy density at the surface integrated over the surface and dcoat is the thickness of the coating. The factor δU/Uφcoat can be divided in the terms parallel (k) and perpendicular (⊥) to the optic face: δU δU δU δU φ = ⊥ · φ + k−⊥ · φ − φ + k · φ = (9) U coat U ⊥ U k ⊥ U k

EFS 1 = 2 √ · 1 − σFS πwbeam ( " 2 # 2 2 ) 1 2σ⊥Ek Ek σ⊥ (1 + σFS) (1 − 2σFS) Ek (1 + σFS) (1 − 2σFS) 1 − φ⊥ + φk − φ⊥ + 2 2 φk E⊥ E⊥ 1 − σk E⊥EFS 1 − σk EFS 1 + σk where the loss angles and Young’s modulus perpendicular and parallel to the coating plane are deﬁned by

En1φn1dn1 + En2φn2dn2 φk = (10) Ekdcoat E⊥ φn1dn1 φn2dn2 φ⊥ = + dcoat En1 En2 En1dn1 + En2dn2 Ek = dcoat dcoat E⊥ = dn1 + dn2 En1 En2

The two Poisson’s ratios σ1 and σ2 are deﬁned by σ + σ σ ' n1 n2 (11) 1 2 σn1En1dn1 + σn2En2dn2 σ2 = En1dn1 + En2dn2

Here dcoat = dn1 +dn2 is the coating thickness. The index n1 indicates the high refraction index coating material (Ta2O5) and n2 the low refraction index coating material (SiO2). The coating Brownian noise is computed in the Appendices E and F.

2.1.2 Substrate loss angle Penn and coll. [18] shown that the loss angle of fused silica samples (of volume V and surface S) measured in diﬀerent experiments by the GW community follows a model that takes in account three dissipation mechanisms: the surface (φsurf ), the bulk (φbulk(f)) and the thermo-elastic dissipation (φth). Suprasil loss angle can be modeled as: V φ f, = φ + φ (f) + φ = (12) sub S surf bulk th V −1 f C3 = C + C + C φ 1 S 2 1Hz 4 th

The coeﬃcients C2 and C3 are in gwinc respectively given by the variables ifo.Materials.Substrate.c2 and ifo.Materials.Substrate.MechanicalLossExponent (see A) and the computation is performed in Ap- pendix B. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 13 of 78

2.1.3 Surface losses contribution In gwinc (see subbrownian.m in Appendix B) is computed an additional dissipation on the Brownian noise given by surface losses and computed as: csurfETM = alphas ∗(1−2∗ sigma )/((1 − sigma )∗Y∗ pi ∗wETMˆ 2 ) ; csurfITM = alphas ∗(1−2∗ sigma )/((1 − sigma )∗Y∗ pi ∗wITMˆ 2 ) ; c s u r f = 8 ∗ kBT ∗ (csurfITM + csurfETM) ./ (2 ∗ pi ∗ f ) ; % account for 2 ITM and 2 ETM, and convert to strain whith 1/Lˆ2 n = 2 ∗ (csurf + cbulk) / Lˆ2; Currently is not clear to me the correctness of this evluation, because the mirror, essentially, has not free surface. Hence, either only the coating contribution must be accounted (considering also the AR coating dissipation) or I would use the modeling in formula (13) of [17].

2.2 Substrate thermo-elastic noise The modeling of the substrate thermo-elastic noise is fully described in [19]. The power spectrum of the single mirror displacement can be modeled as:

FTM 2 ITM S (ω) = CFTM S (ω) (13) where the superscripts stand for ITM - Inﬁnite Test Mass and FTM - Finite Test Mass (hence CFTM is a correction factor quite close to the unity). The thermo-elastic noise for an inﬁnite test mass is

2 2 2 ITM 8 (1 + σFS) κFSαFSkBT S (ω) = √ √ 3 (14) 2 2 2 2πCV ρFS wbeam/ 2 ω where κFS is the thermal conductivity of the fused silica, αFS is the linear thermal expansion coefficient, ρFS the density and CV is the specific heat per unit mass at constant volume. The infinite test mass thermo-elastic contribution is computed in Appendix G. The correction factor is (formula (46) of [19])

√ 3 2 ( πwbeam) CFTM = 3 × Rm    5 2 ∞ 5 2 2  R hmc X R knp (1 − Qn) J (ξn) h 2 i m e1 + m n 0 × (1 − Q ) (1 + Q ) + 8h k Q (1 − Q ) + 4h2 k2 (1 + Q ) 2 h i2 n n m n n n m n n  (1 + σ) 2 2   n=1 (1 − Qn) − 4hmknQn  (15) where kn = ξn/Rm (16a)

Qn = exp (−2knhm) (16b)

Z Rm 2 2 2 2 2 exp −r /r0 exp −knr0/4 pn = 2 2 2 J0 (knr) rdr ' 2 2 (16c) RmJ0 (ξm) 0 πr0 πRmJ0 (ξn) √ r0 = wbeam/ 2 (16d) where the approximation in Eq.16c stands for small values of n (see caveat in page 5 of [19]). The ﬁnite test mass correction to the thermo-elastic noise is computed in Appendix H. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 14 of 78

2.3 Thermo-optical noises Thanks to the Fluctuation-Dissipation theorem [13], the thermal dissipation in a material can be linked to the temperature fluctuation. In an “optical material” this can be related to an optical phase noise through two different couplings: the linear thermal expansion coefficient α (defined as thermo-elastic noise) and the refraction index themperature gradiend β = dn/dT (defined as thermo-refractive noise). Several authors have treated the two components separately. For example the thermo-elastic noise in the coating has been described in [20– 22] (as guided reading I suggest the Einstein Telescope internal note [23]); the corresponding power spectrum of the displacement noise induced by thermo-elastic fluctuation in the coating is expressed, for example, by formula (7) in [22]. Thermo-refractive noise is instead separately described in [24] (see, in that paper, for example formula (9) ). An complete view of the thermo-dynamical and thermo-refractive noises, unified under the name thermo-optical, has been performed by M. Evans and collaborators in [25] and hereafter I will use their results. Thermo-optical noise is driven by the thermal fluctuation in the coating; the power spectrum of the coating thermal fluctuation,investigated by a Gaussian beam, is √ 2 2 k T 2 ∆T √B STO = 2 (17) π wbeam κFSCFSω This thermal fluctuation determines a displacement noise through the thermal gradient: ∂∆z ∂∆z ∂∆z ∂∆z ∂φ ∂φ = TE + TR = = ∂z (18a) ∂T ∂T ∂T ∂φ ∂T φ ∂T ∂∆z λ = ∂z = − (18b) ∂φ φ 4π where the suffices “TE” and “TR” stand for “Thermo-Elastic” and “Thermo-Refractive” (in this subsection we will use also the suffices“s”and“c”to indicates“substrate”and“coating”related properties. Furthermore, we are here adopting the same formalism described in [25]: a bar used above a symbol indicates an“effective”coefficient, having the same unit and meaning of its barless counterpart, but averaged in a context o a semi–infinite medium). The full evaluation is performed in [25] and the final formula is

2 ∆z ∆T z ∂φ Cc STO = STO × ∂φ − αsd (19) ∂T Cs Let ﬁrst focus on the terms within the parenthesis in Eq.19; the second term subtracts the thermo-elastic contribution of the substrate whereas the ﬁrst term contains both the thermo-elastic and thermo-refractive contributions of the coating. If we indicate with the index “0” the input medium (the “vacuum”) and “N” is the number of layers of the dielectric coating) we can write

N ∂φc X ∂φc ∂φk = (20) ∂T ∂φk ∂T k=0 where the first factor describes the overall reflection phase variation due to the round–trip phase change in each layer and the second factor describes the phase gradiend of each layer with the temperature. The computation of these terms is quite tricky and all the explanations can be found the in Appendix B in [25]; here we make an oversimplified summary. The reflectivity between the layer 1 and 2 of a coating is

n1 − n2 r1,2 = (21) n1 + n2 If we pile–up two transitions −iφ2 −r2,1 + r2,3e r1,2,3 = −iφ (22) 1 − r2,1r2,3e 2 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 15 of 78

where φ2 is the round-trip phase in layer 2. Noting that rk,k+1 = −rk+1,k and defining rk ≡ rk,k+1 it is possible to write the recursive formula for the effective reflectivity of a coating layer,including the round-trip in that layer: r + r −iφk −iφk k k+1 rk ≡ e rk,k+1,...,N = e (23) 1 + rkrk+1 z The overall reflection of the coating is then rc = r0 and its phase φ0 = ∆c/∂φ takes in account the total change of thickness ∆c of the coating. Now it is possible to compute the first factor of the Eq.20: ∂φ 1 ∂r c = = 0 (24a) ∂φk r0 ∂φk

2  −iφk 1−rk ∂rk+1 e 2 if k < j  (1+r r ) ∂φj ∂rk  k k+1 = −irk if k > j (24b) ∂φj  0 if k = j Finally, it is possible to compute the second factor of Eq.20 ∂φ 4π 4π k | = (β + α n ) d = B d (25a) ∂T k>0 λ k k k k λ k k

N ∂φ0 4π X d = − α d = α (25b) ∂T λ k k c ∂z k=1 φ Eq.25b represents the geometric variation of the whole coating, hence the Thermo-elastic contribution. Eq.25a is related to the optical path variation, hence it represents the thermo-refractive contribution. In the Eq.25a and 25b have been used the following deﬁnitions N X d = dk (26a) k=1 1 + σs 1 + σk Ek αk = αk + (1 − 2σs) (26b) 1 − σk 1 + σs Es N X dk α = α (26c) c k d k=1

Bk ≡ βk + αknk (26d) where d is the total thickness of the coating, αk is the thermal expansion coefficient for a given layer k and αc is the effective linear expansion coefficient of the whole coating. For sake of completness hereafter are shown the latest definition needed to perform the above computations:

N X dk C = C (27a) c k d k=1

N !−1 X 1 dk κc = (27b) κk d k=1

αs ' 2αs (1 + σs) (27c) where αs is the effective expansion coefficient of the substrate in a semi-infinite mirror. In gwinc this tricky computation is performed in getCoatTOPhase.m (see Appendix I). The previous computations have been performed under the hypothesis that the coating thickness is quite smaller than the thermal diffusion lenght rT r κ r = (28) T Cω VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 16 of 78

Removing this hypothesis and considering also a corrective factor due to the finite size of the mirror, Eq.19 becomes ∆z ∆T 2 STO ' STO × Γtc × ∆αfsmd − βλ (29) where Γtc is a correction factor related to the coating thickness, the first term within the parentesis describes the thermo-elastic contribution and the second term the thermo-refractive one, with the definition ∂z TR ≡ −βλ (30) ∂T The coating thickness correction factor is

2 2 2 pEΓ0 + pEpRςΓ1 + pRς Γ2 Γtc = 2 (31) Rς ΓD where Γ0 = 2 (sinh(ς) − sin(ς)) + 2R (cosh(ς) − cos(ς)) (32a)

Γ1 = 8 sin(ς/2) (R cosh(ς/2) + sinh(ς/2)) (32b) 2 2 Γ2 = 1 + R sinh(ς) + 1 − R sin(ς) + 2R cosh(ς) (32c) 2 2 ΓD = 1 + R cosh(ς) + 1 − R cos(ς) + 2R sinh(ς) (32d) r κ C R = c c (32e) κsCs −βλ ∆αd pR = ,PE = (32f) ∆αd − βλ ∆αd − βλ The parameter ς is a dimensionless, frequency dependent, scale factor deﬁned by √ 2d r2ωC ς = = c d (33) rT c κc that compares the overall thickness d of the coating to its thermal diﬀusion lenght rT . The parameter ς describes the novelty of this new description of the thermo-optic noise; in fact, the correction factor Γtc can be approximated as:  p2 +3 p −R2 p −3 1−R2  E ( R ) E ( ) 2 1 + 3R ς − 6 ς if ς 1 Γtc ' 2 2 2pE pR  R(1+R)ς2 + R if ς 1

Hence the thermo-refractive and thermo-elastic components have a relative negative sign if d rT c (thin coating), are partially coherent and partially canceling if d ' rT c and sum in quadrature if d rT c (thick coating). In gwinc the thickness correction is evaluated in getCoatThickCorr.m (see Appendix J) Let jump back to the understanding of the main thermo-optical noise equation 29. The ﬁrst term within the parenthesis, representing the thermo-elastic contribution, can be decomposed as:

∆αfsm = Cfsm∆α (34a)

Cc ∆α = αc − αs (34b) Cs where Cfsm is a correction factor, related to the ﬁnite size of the mirror. Its evaluation is done in [21]: v 2 u 12R2 √ u m −1 + 2 S1 r0 2u hm Cfsm = tS2 + 2 (35a) Rm 4 (1 − 2σs) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 17 of 78

−k2 r2 ∞ exp n 0 X 4 S = (35b) 1 ξ2 J (ξ ) n=1 n 0 n

−k2 r2 ∞ exp n 0 " 2 #2 X 2 (1 − Qn) S2 = (35c) J 2 (ξ ) 2 2 2 n=1 0 n (1 − Qn) − 4kmhmQn

(see Eq.16a and successive for the definition of kn, Qn and r0). In gwing the finite mirror size coating correction is computed in getCoatFiniteCorr.m (see Appendix K). It is useful to note that, in high reflectors, composed by quarter–wave doublets of high reflectivity material “H” and low reflectivity material “L” β is given by

2 2 nLBH + nH BL βQW ' 2 2 (36) 4 (nH − nL) and, if the last layer is a low reﬂectivity protection layer the previous equation is modiﬁed in

βQW BL β ' 2 + 2 (37) nL 4nL 3 Suspension Thermal Noise

The full description of the thermal noise due to the AdV suspension is performed in [26]; here we summarise the results. The novelty of the PPP model in [26] is the fact that it takes in account the contributions to the thermal noise due to the dissipations both in the Marionette and in the Reference Mass. In the early versions of the AdV suspension thermal noise modeling a payload design similar to the Virgo+ conﬁguration were adopted. For this reason the reference mass were taken in account and the following matrices were 3 × 3, as shown in [26] and in some residual code line in gwinc. Now, the reference mass has been removed in the payload in the current design of AdV and hence the model is now described by 2 × 2 matrices. Let suppose to indicate with the (1) the Mirror and (2) the marionette (in gwinc, instead, the notation is reversed); these are “coupled harmonic oscillators” and the equations of motion assume a matrix aspect (see Eq.32 in [26]):

˙ 1 2 Fe ≡ Z · Xe → Z = M · Ω + iωΓ − ω I (38) iω where Z is the mechanical impedance 2×2 matrix, M is the diagonal matrix of the masses, I the identity matrix, Ω the matrix of the pendulum (complex)frequencies and Γ is the matrix of the possible viscous dissipations:

m 0 M = 1 (39a) 0 m2

2 2 ! ω1 −ω1 Ω = µ ω2+ω2 (39b) − µ1 ω2 1 1 2 µ2 1 µ2 γ1 −γ1 Γ = µ1 µ1γ1+µ2γ2 (39c) − γ1 µ2 µ2

2 2 2 2 2 g ωj = ωp j + ωw j = ωp j [(1 + Dj) + iDjφj] , ωp j = (39d) Lj

The term ωw j is related to the elasticity of the suspension wires; neglecting for simplicity the index j: s b nEI ω2 = ω2 ≡ ω2D (40) w p 2L mg p VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 18 of 78

P where nj is the number of suspension wires (n1 = 4, n2 = 1), µj = mj/ j mj, bj is the number of ﬂexural 4 points (bj = 2), Eis the Young’s modulus of the material composing the wires, I ≡ (π/4)r is the cross–section momentum (r is the radius of the suspension wire, supposed cylindrical) and D is the so–called Dilution factor. In the present formulation of the mechanical impedance (Eq.38)it has been (a bit artiﬁcially) separated the viscous damping processes (γj) by the structural dissipation processes, represented by the term φj in Eq.39d. In the loss angle φj must be included the bulk structural dissipation of the suspension wire material φmat, the thermo-elastic contribution φth(ω) and the eventual surface losses φsurf :

φj(ω) = φmat j + φsurf j + φth j(ω) (41a) d 4 φ = φ η s = φ d (41b) surf mat V/S mat r s Λ 2 ET ωτ C (2r)2 φth(ω) = α − βE , τ = (41c) E · πr2 C 1 + (ωτ)2 2.16 · 2π · κ where (in Eq. 41b) η is a numerical factor depending on the mode shape (η = 2 for cylindrical fibres and pendulum mode) ds is the thickness of the lossy surface, V and S are respectively the volume (V = 2πrjLj) and 2 the lateral surface (S = πrj Lj) of the suspension wire. In Eq. 41c Λ is the suspension wire tension (Λ = mg/n −1 −3 in case of a mass m suspended by n wires), C = ρCv is the heat capacity per volume unit JK m , κ is the thermal conductivity and βE ≡ ∂ ln E/∂T (see [27]). The viscous dissipation coefficients γj are related to the mechanical quality factors Qj of these modes (if the dissipation mechanism is viscous only) by <(ωj) γj ≡ (42) Qj Now we have all the ingredients to compute the pendulum thermal noise: (" # ) 4k T 4k T 1 −1 S (ω) = B <Z−1 = B < M · Ω + iωΓ − ω2I (43) p,1 ω2 11 ω2 iω 11 3.1 Including the violin modes of the mirror suspension wires In the suspR AdV.m code of gwinc (see Appendix L), this computation is performed taking in account the results of the section 2 of [26]; the stiffness of the mirror suspension fibres is take in account to evaluate also the violin modes. Eq.38 can be modified in the following way: 1 Z = K − ω2M (44) iω including in the K matrix the role of the Ω and Γ matrices. Obviously K is the matrix of the elastic constants of the suspension wires: k (ω) −k (ω) K = pend pend (45) −kpend (ω) kpend (ω) + k2 2 where k2 = m2 ω2 + iωγ2 . Instead, kpend (ω) is the full elastic constant of the mirror suspension wires; in case of a simple cylindrical fibre [26, 28]:

3 2 2 3 p+ cos (p−L) + p+p− sin (p−L) + p+p− cos (p−L) + p− sin (p−L) kpend (ω) = n1EIp+p− 2 2 (46a) p+ − p− sin (p−L) − 2p+p− cos (p−L) s ±Λ + pΛ2 + 4EIρ (πr2) ω2 p = (46b) ± 2EI

Obviously, the violin modes are given by the poles of kpend (ω) and the pendulum angular frequency can 2 be computed as ωp ' limω→0 <(kpend)/m1. The “structural” dissipation is hidden in the Young’s modulus E = E0(1+iφ), whereas the eventual viscous damping can be added ad hoc from the measured Qp as kpend (ω) ← <(kpend (ω)) + i [=(kpend (ω)) + ω/ (ωpQ1) <(kpend (ω))] VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 19 of 78

3.1.1 Optimized (Dumbbell–shaped) fibres In [29] and in the Sec. 2.1 of [26] it is shown that to minimise simultaneously the vertical bouncing frequency (thinner fibres) and the thermo-elastic dissipation (larger fibre diameter, as dictated by Eq.41c) it is possible to adopt the so–called optimised fibres, having larger terminal and a thinner central body (3–segments fibre, j = 1 top, j = 2 body, j = 3 attached to the mirror). It is possible to solve the system of elastic equations that describes the transverse displacement X along the fibre axis y (Eq.49 of [26]) obtaining the three displacement solutions one for each segment of the fibre):

−p+, j −y −p+, j (Lj −y) Xj = Aje + Bje + Cj cos (p−, jy) + Dj sin (p−, jy) , j = 1, 2, 3. (47)

The twelve integration constants are determined by the 4 boundary conditions

X(0) = 0; X(i) (0) = 0; X(L) = δ; X(i) (L) = 0; (48) where δ is the displacement of the end terminal of the fibre and the suffix (i) indicates the derivative with respect the longitudinal coordinate y, and by the 8 joining conditions (2 interfaces times (X, X(i), force and moment)). It is the possible to write an effective loss angle weighting the diameter dependent loss angle φ (surface and thermo–elastic) with the elastic energy distribution:

R L φ (y) uelastic (y) dy φ = 0 (49a) R L 0 uelastic (y) dy

2 Z L 1 h (ii)i uelastic (y) = EI X ,Uelastic ≡ uelastic (y) dy (49b) 2 0 Hence, it is possible to compute an eﬀective elastic constant k by

(iii) Force EIj=3X (L) k ≡ = − j=3 (50) Dumbbell δ δ

Replacing kpend with kDumbbell in the matrix K of Eq.45 and then computing the impedance matrix Z as in Eq.44, it is possible to evaluate the thermal noise in Eq.43. The code that implements the optimised ﬁbre model is in Appendix M and the structure that contains the geometrical description of the dumbbell shaped ﬁbre is ifo.Suspension.Stage(1).

3.2 Vertical bouncing mode thermal noise The computation of the vertical thermal noise of the coupled oscillators system (Marionette–Reference Mass– Mirror) is performed in Sec.(1.5) of [26]. Essentially is applied again Eq.43 where the dissipation matrix Γv and the frequencies matrix Ωv are γv 1 −γv 1 Γv = µ1 µ2γv 2−µ1γv 1 (51a) − γv 1 µ2 µ2 2 2 ! ωv 1 −ωv 1 Ωv = µ ω2 +ω2 (51b) − µ1 ω2 1 v 1 v 2 µ2 v 1 µ2 where 2 ( Ej (πrj ) 2 if j = 1 ω = Lj (52) v j 2 (2πfv SA) if j = 2

Here fv SA is the vertical mode of the whole Super–Attenuator (SA) (in Virgo fv SA ' 0.4 Hz) dominated by the softness of the anti–spring system. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 20 of 78

4 Residual Gas phase noise computation

The evaluation of the residual gas phase noise is based on the computation made by Whitcomb and Weiss [30, 31], verified by Zucker [32] and finally adapted for a multi–gas residual contamination by Cella [33]. In the hypothesis of absence of collision between the gas particles and the boundaries, absence of delays and interaction with the beam, the power spectrum of the equivalent length fluctuation is (considering 2 arms):

2 Z L X 8ρA(2παA) 1 w(z) SL(f) = exp [−2πf ]dz (53) Vp,A w(z) Vp,A A 0 where Vp,A is the most probable velocity of the gas molecula of mass MA: r 2kBT Vp,A = (54) MA

ρA is the number density of the gas, related to the partial pressure PA of the gas by:

PA ρA = (55) kBT The summ of formula 53 is extended to all the gasses playing a role in determining the residual pressure. In AdV are taken in account H2O, H2,N2,O2 and the properties of thes gasses are included in the Matlab data structure named ifo.Infrastructure.ResidualGas and listed in Appendix A. The code implementing the Residual Gas computation is shown in the Appendix N.

5 Quantum Noise

In the initial interferometers, like Virgo, the description of the noise related to the discrete photon composition of the light has been treated in a classical way, by distinguishing the shot noise, due to the photon counting, from the radiation pressure noise, due the photon impulse transfer to the mirror. This naive description of the quantum noise fails in the case of the advanced detectors, because of the presence of the signal recycling mirror (SRM) at the output port. The eﬀect of the SRM is to correlate the shot noise to the radiation pressure through the mixing of the quadratures at the output of the interferometer; in this case it is better to use the appropriate name of quantum noise. The description of the quantum noise in an advanced interferometer is performed in two fundamental papers: H. J. Kimble and collaborators [34] (usually indicated as KLMTV), where the right formalism is introduced, and A. Buonanno and Y.Chen [35], where the full computation is performed. Here we will report only the results, using the same formalism and, as much as possible, symbols adopted in these articles.

5.1 Initial interferometers Let start from a non–signal recycled interferometer (like Virgo). The spectral amplitude of optical read-out noise, using the KLMTV formalism, can be written as: s hSQL (ω) 1 hor(ω) = √ + K (ω) (56) 2 K (ω) where r 8 h (ω) = ~ (57) SQL mω2L2 is the Standard Quantum Limit (SQL) (m here is the mirror mass),

4 IBS 2γFP K (ω) = 2 2 2 (58) ISQL ω (γFP + ω ) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 21 of 78

is the effective coupling constant between the motion of the test mass and the output signal, having defined IBS the input power impinging on the beam splitter (BS) mirror, c T · c γ = 2πf = 2π = (59) FP FP 4L ·F 4L is the decay factor of the Fabry–Perot (FP) cavity, where F is the finesse of the main cavities and T is the power transmittivity of the input mirrors. ISQL is the light power needed to reach the SQL at a (GW) frequency ω = γFP : 2 4 mL γFP 2πc ISQL = ; ωlaser = (60) 4ωlaser λ −1 By noting that K ∝ m and hSQL ∝ m it is easy to see that in Eq.56 the first term within the root gives the shot noise contribution (independent from m) and the second term gives the radiation pressure (∝ 1/mω2).

5.2 Read–out schemes In effect the read–out scheme of initial GW detectors was based on a Heterodyne configuration (see left panel of Fig.7) and the shot noise part should be increased by the non-stationary contribution given by the radio– frequency (RF) modulation (computed to be p3/2 in Virgo [36, 37]). In theory, the SQL is an effective limit for an interferometer with free floating mass, but thanks to the ponderomotive squeezing effect due to the fluctuating radiation pressure the SQL can be overcome, because of the correlation introduced between the radiation pressure and the phase (shot noise) components. This needs a different read-out scheme, the homodyne detection, shown in the central panel of Fig.7, being it one of the quantum nondemolition devices [38]. Using an homodyne detection scheme the optical read–out noise can be written [35] as

hSQL (ω)q 2 hor(ω) = √ 1 + (tan η − K) (61) 2K where η is the homodyne phase; it is evident that for η = arctan (K) the SQL is overcome. Because of the technical difficulties [39] to implement a complete homodyne read–out scheme in GW detectors, a special case of homodyne is implemented: the DC detection (see right panel of Fig.7). In a DC detection scheme the operating point of the interferometer is slightly shifted off the resonance by acting either on the MICH or on the DARM control loops [40]. The small amount of carrier light exiting at the output port because of this offset acts as local oscillator beating with GW-produced field from differential motion of the arms at GW (audio) frequencies. The dark–fringe offset acts as homodyne phase (disturbed by all the effects of unbalancing between the cavities).

5.3 Signal Recycled interferometer Let add a mirror at the exit port of the interferometer, at a distance l from the Beam Splitter mirror, as shown in Fig.8. This mirror reinject the electric fields di (the index i indicates the two quadratures) back in to the interferometer. Since one of these two quadrature contains the gravitational signal, this configuration is generically named “Signal Recycled” (SR). The full description of the of a SR interferometer is performed in two twin articles [35, 41] and here we report the results and few simplified considerations. Let consider i = 1 the amplitude quadrature and i = 2 the phase quadrature. In a conventional interferometer the antisimmetric gravitational signal is embedded in d2 (see Fig.8). The effect of the SR mirror is to reflect back the fields d1 and d2 mixing up their content [42]: ! ! d SR cavity cos 2φ − sin 2φ d vacuum fields c b1 −−−−−−→ ρ b1 + τ ⇔ b1 (62) db2 sin 2φ cos 2φ db2 from outside bc2 where ρ and τ here are the amplitude reflectivity and transmissivity of the SR mirror and ω l l φ ≡ laser = 2π (63) c mod2π λ mod2π VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 22 of 78

Figure 7: Figure extracted from [39]. Read–Out schemes for GW interferometers; from left to right: Heterodyne, Homodyne and DC detection.

is the phase gained by the carrier frequency ωlaser while travelling one way in the SR cavity. The corresponding phase gained by the GW sidebands is Φ ≡ [ωl/c]mod2π. Eq.62 suggests that, with in the SR interferometer, both the output quadratures bi contain the antisimmetric h signal and it is impossible to put the signal info in only one by a transformation. The role of the homodyne detection, then, is to extract a linear combination of the two quadratures: bη = b1 sin η + b2 cos η (64)

Figure 8: Figure extracted from [42]. In the left panel is schematised a signal recycles (SR) interferometer with all the electric ﬁelds (ai, bi, . . . , i = 1, 2). In the right panel the variables needed to deﬁne the antisymmetric mode (containing the gravitational signal) are shown: xbantisym = (xbn1 − xbn2) − (xbe1 − xbe2).

Now we are almost ready to jump to the conclusions of the Buonanno and Chen paper [35], but ﬁrst we need to introduce the losses in the interferometer (as done in KLMTV [34] and extended in [35]). The losses in the SR cavities are introduced as an extra ﬁeld pi entering from the SR mirror and the losses in the output VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 23 of 78

photo–detection process are introduced as an addition output field qi (see Fig.7 of [35]). As done in KLMTV, at the losses in arm cavities correspond a vacuum field (having quadratures ni) entering in the interferometer; these losses are quantified by the loss coefficient per round trip (RTL) L. The parameter

2L L ≡ = (65) T 2γFP L/c represents the fraction of carrier photons, impinging in each cavity, that gets lost. It is useful also to deﬁne the frequency dependent parameter 2 E = 2 (66) 1 + (ωlaser/γFP )

The photodetector loss is defined by λPD and the losses inside the SR cavity by the fraction of photons lost at L each bounce of the interior field off the SR mirror λSR. The input–output relations for the lossy field bi at the SRM are given by Eq. 5.6 in [35], hereafter copied:

L L L √ L b1 1 2iβ C11 C12 a1 iβ D1 h L = L e L L + 2Kτe L + b2 M C21 C22 a2 D2 hSQL P P p Q Q q N N n e2iβ 11 12 1 + e2iβ 11 12 1 + e2iβ 11 12 1 (67) P21 P22 p2 Q21 Q22 q2 N21 N22 n2

The quantum noise power spectrum is then given by the Eq.5.13 of [35], hereafter copied:

h2 Sη = SQL |CL sin η + CL cos η|2 + |CL sin η + CL cos η|2+ h 2 L L 2 11 21 12 22 2Kτ |D1 sin η + D2 cos η| 2 2 |P11 sin η + P21 cos η| + |P12 sin η + P22 cos η| + 2 2 |Q11 sin η + Q21 cos η| + |Q12 sin η + Q22 cos η| + 2 2 |N11 sin η + N21 cos η| + |N12 sin η + N22 cos η| (68)

The matrix elements used in Eq.67 and Eq.68 are deﬁned in the Eq.5.7–5.12 of [35], hereafter copied:

K K M L = 1 + ρ2e4iβ − 2ρ cos 2φ + sin 2φ e2iβ + λ ρ −ρe2iβ + cos 2φ + sin 2φ e2iβ 2 SR 2 K + ρ 2 (cos β)2 −ρe2iβ + cos 2φ 3 + e2iβ sin 2φ e2iβ (69a) 2

p K CL = CL = 1 − λ 1 + ρ2 cos 2φ + sin 2φ − 2ρ cos 2β 11 22 PD 2 1 h 2 i − −2 1 + e2iβ ρ + 4 1 + ρ2 cos2 β cos 2φ + 3 + e2iβ K 1 + ρ2 sin 2φ 4 1 K +λ e2iβρ − 1 + ρ2 cos 2φ + sin 2φ (69b) SR 2 2

L p 2 2 C12 = 1 − λPD τ − sin 2φ + K sin φ 1 1 + sin φ 3 + e2iβ K sin φ + 4 cos2 β cos φ + λ sin 2φ + K sin2 φ (69c) 2 2 SR VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 24 of 78

L p 2 2 C21 = 1 − λPD τ sin 2φ − K cos φ 1 1 + cos φ 3 + e2iβ K cos φ − 4 cos2 β sin φ + λ − sin 2φ + K cos2 φ (69d) 2 2 SR

p 1 1 DL = 1 − λ − 1 + ρe2iβ sin φ + 3 + ρ + 2ρe4iβ + e2iβ (1 + 5ρ) sin φ + λ e2iβρ sin φ (69e) 1 PD 4 2 SR

p 1 1 DL = 1 − λ − −1 + ρe2iβ cos φ + −3 + ρ + 2ρe4iβ + e2iβ (−1 + 5ρ) cos φ + λ e2iβρ cos φ 2 PD 4 2 SR (69f)

1p p P = P = 1 − λ λ τ −2ρe2iβ + 2 cos 2φ + K sin 2φ (69g) 11 22 2 PD SR p p P12 = − 1 − λPD λSR τ sin φ (2 cos φ + K sin φ) (69h) p p P21 = 1 − λPD λSR τ cos φ (2 sin φ − K cos φ) (69i)

p −2iβ 2 2iβ Q11 = Q22 = λPD e + ρ e − ρ (2 cos 2φ + K sin 2φ) 1 + ρ e−2iβ cos 2φ + e2iβ (−2ρ − 2ρ cos 2β + cos 2φ + K sin 2φ) + 2 cos 2φ + 3K sin 2φ 2 1 − λ ρ 2ρe2iβ − 2 cos 2φ − K sin 2φ (69j) 2 SR

Q12 = 0 = Q21 (69k)

r p N = 1 − λ τ K 1 + ρe2iβ sin φ + 2 cos β e−iβ cos φ − ρeiβ (cos φ + K sin φ)]} (69l) 11 PD 2 √ p −iβ iβ N22 = − 1 − λPD 2 τ −e + ρe cos β cos φ (69m) √ p −iβ iβ N12 = − 1 − λPD 2 τ e + ρe cos β sin φ (69n) r p N = 1 − λ τ {−K (1 + ρ) cos φ + 2 cos β ( e−iβ + ρeiβ )cos β sin φ} (69o) 21 PD 2 In the previous equations β indicates the phase shift of the GW sideband in the arm: ω β = arctan (70) γFP The computation of the quantum noise in the gwinc code is shown in Appendix O.

5.4 Optical losses budget To compute the quantum noise it is crucial to evaluate the RTL losses L, the photo–detection process losses λPD and the SRC losses λSR. In AdV the RTL losses, requirement is L ≤ 75 ppm (71)

(50 ppm linked to ﬂatness defects, 25 ppm due to transmission, absorption, scattering); this is coded in the variable ifo.Optics.Loss in the Appendix A that contains the single mirror losses (= L/2).In eﬀect in gwinc, to these losses is added contribution of the clipping losses (see Appendix P):

" ITM 2# " ETM 2# Rcoat Rcoat Lclip = exp −2 ITM + exp −2 ETM (72) Rbeam Rbeam VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 25 of 78

Table 2: Computation of the Photo-detection process loss budget (λPD) Process Comment losses Telescope aberrations ≤ 0.5% OMC losses mode mismatch, misalignment, surface losses ≤ 5.0% Faraday Isolator = 2.0% Sub–total optical losses ≤ 7.5% B1p photodiodes 2 × 2.5 mW (only 0.24 mW on TEM00) = 0.3% B1p quadrants 2 × 2.5 mW (only TEM00 + SB) ⇒ 1.6 mW = 2.0% on TEM00 B1 photodiodes with AC signals 0.8 mW ? ' 1.0% ? B1p phase camera 15 mW (only 0.8 mW on B1?) ' 1.0% ? Sub–total sensing losses ≤ 4.5% B1 DC Photodiodes QE ≥ 0.98 in GEO ≤ 2.0% Total PD losses λPD ≤ 14%

The computation of the λPD losses are based on [43], reproduced in Table 5.4 and coded in the variable ifo.Optics.PhotoDetectorEﬃciency in Appendix A. The SRC losses λSR are computed in Appendix O as the pile–up of 3 three contributions: the mismatch between the modes in the interferometer arms and in the SRC ifo.Optics.coupling , set to be 1−0.997 = 0.3 %; the losses around the BS mirror ifo.Optics.BSLoss , set to be 1510ppm (see Table 5.4); the losses due to the residual thermal lensing due to the limits of the thermal compensation system (TCS) ifo.TCS.SRCloss, currently set to 300ppm (see TCS chapter of [44]).

Table 3: Computation of the PRC loss budget (table extracted from the Optical Simulation and Design chapter of [44]) Loss origin Loss per round trip [ppm] Pick–Oﬀ Plate HR 2 × 300 Pick–Oﬀ Plate AR 2 × 300 2 Compensation Plates 2 × 200 Beam Splitter AR 100 Surface scattering 17 × 10 Absorption 40 Losses in PRC 1510 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 26 of 78

6 AdV Sensitivity

The ﬁnal AdV sensitivity, computed through gwinc, is shown in Fig.9. The expected performances, in this provisional conﬁguration, are:

• NS–NS detection distance: DNS ' 146 Mpc

• BH–BH detection distance: DBH ' 1161 Mpc

AdV Noise Curve: P = 125.0 W in −18 10 Quantum noise Gravity Gradients −19 10 Suspension thermal noise Coating Brownian noise

−20 Coating Thermo−optic noise 10 Substrate Brownian noise

Hz] Seismic noise √ −21 Excess Gas 10 Total noise

−22 10 Strain [1/

−23 10

−24 10

0 1 2 3 4 10 10 10 10 10 Frequency [Hz]

Figure 9: AdV sensitivity computed by gwinc with the parameters in Appendix A. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 27 of 78

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[40] B. Abbott et al., “Proposal for a Homodyne (DC) Detection Experiment at the LIGO Caltech 40-Meter Laboratory,” Tech. rep., LIGO internal note LIGO-T050086-00-R, 2005. 21 [41] A. Buonanno and Y. Chen, “Signal recycled laser-interferometer gravitational-wave detectors as optical springs,” Phys. Rev. D, vol. 65, p. 042001, 2002. 21 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 29 of 78

[42] A. Buonanno and Y. Chen, “Signal recycled laser-interferometer gravitational-wave detectors as optical springs,” arXiv:gr-qc/0107021v1, 2001. 21, 22 [43] R. Gouaty, “private communication,” 2012. 25 [44] The Virgo Collaboration, “Advanced Virgo Technical Design Report,” Tech. rep., Virgo internal note VIR- xxxA-12, 2012. 25, 39

[45] D. R. M. Crooks et al., “Experimental measurements of mechanical dissipation associated with dielectric coatings formed using SiO 2 , Ta 2 O 5 and Al 2 O 3,” Classical and Quantum Gravity, vol. 23 (15), p. 4953, 2006. 38 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 30 of 78

Nomenclature

Abbreviations AdV Advanced Virgo aLIGO Advanced LIGO BS Beam Splitter

CB Virgo Central Building FP Fabry–Perot IP Inverted Pendulum RTL Round Trip Losses

SA Super–Attenuator, the passive ﬁltering system of Virgo SQL Standard Quantum Limit SRM Signal Recycling Mirror

TF Transfer Function V+ Virgo+

Glossary Compact binary A compact binary is an astronomical binary consisting of a pair of compact stars

Compact star Throughout this document a compact star stands for a neutron star or a black hole Dual Recycling Using Power- and Signal-Recycling at the same time G1 The GEO600 detector near Hannover in Germany GEO600 The GEO600 detector located near Hannover in Germany

H1 The LIGO 4-kilometre interferometer at Hanford, USA H2 The LIGO 2-kilometre interferometer at Hanford, USA Horizon Distance The distance at which a gravitational wave detector would measure a matched-filter signal- to-noise ratio of 8 for an optimally oriented (i.e. face-on) compact binary source located in a direction perpendicular to the plane of the detector L1 The LIGO 4-kilometre interferometer at Livingston Parish, Louisiana, USA Power Recycling Re-using the light reflected back to the interferometer input by placing a mirror there and resonantly enhancing the circulating light power. Has the same effect as using a more powerful laser.

PPP model Thermal noise model describing the contribution of the whole last stage of the suspension (including the Marionette, the Reference Mass and the Mirror), developed by F. Piergiovanni, M. Punturo and P. Puppo VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 31 of 78

QND Typically the term quantum nondemolition (QND) measurement is used to describe a measurement of a quantum system which preserves the integrity of the system and the value of the measured observable. In the literature on gravitational wave detectors this term is often used to describe a variety of interferometer schemes in which shot noise and radiation pressure noise can be simultaneously suppressed. Such systems are typically not performing a strict QND measurement, thus they may more appropriately be referred to as Quantum Noise Reduction (QNR) systems.

Resonant Sideband Extraction The same technique as SR but operated anti-resonant, i.e. widening and/or detuning the bandwidth by reducing the reflectivity of the compound mirror formed by the inboard cavity mirror and the SR mirror. Signal Recycling Resonantly enhancing the GW signals exiting the interferometer through the output port by placing a mirror there. This increases the sensitivity at the cost of the bandwidth. With a different (anti-resonant) tuning the same technique can be used for widening the bandwidth at the expense of the sensitivity (RSE). SR can optimize the sensitivity for an arbitrary frequency. V1 The Virgo gravitational wave detector in Italy Vacuum fluctuations Fluctuations that result from the quantum nature of an electromagnetic field even at the lowest possible energy level (zero mean energy = vacuum). Virgo Virgo is a 3-kilometre gravitational wave detector located near Pisa, Italy

Symbols Please note that some symbols might stand for more than one quantity depending on the context

αA Polarizability of the residual gas molecula β dn/dT refraction index thermal gradiend C Heat capacity per volume unit JK−1m−3

DL Luminosity distance to the source

EFS Young modulus of the Fused Silica substrate F Finesse of a Fabry Perot cavity h Gravitational wave amplitude, usually denoting the detector response κ Thermal conductivity WK−1m−1

K Eﬀective coupling constant between the motion of the test mass and the output signal

−23 kB Boltzman Constant kB = 1.380658 × 10 J/K L Geometric length

λ Laser wavelenght λ = 1064nm

σFS Poisson ration of the Fused Silica substrate w(z) beam radius proﬁle VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 32 of 78 H i l l : 1993) − 00 − T000012 − grounded figures of merit for − v00r01 2012/11/05 Optic Handbook, Waynant & Ediger (McGraw − 12 − ifo =IFOModel(varargin) (IFI/Plenum,1970) Vol 4 , Pt 2 1991 numbers for suprasil 0073C − d i e l e c t r i c s in a Vacuum , page 29 1. Electro 4. Quartz Glass5. Y.S. for Touloukian Optics6. Marvin Data (ed), J. and7. Thermophysical Weber Properties, R.S.8. (ed) P. Krishnan CRC Properties Handbook Heraeus Klocek, et9. of data al.,Thermal Handbook Rai of laser sheet, Weiss, Matter of science Expansion infrared electronic of and and technology, Crystals log optical from , Pergamon 5/10/2006 materials, Press Marcel Decker, 2. LIGO/GEO data/experience3. Suspension reference design , LIGO 10. Wikipedia11. D.K. online Davies,12. encyclopedia The Gretarsson13. Generation Fejer ,14. & 2006 Harry, Braginsky and Dissipation Gretarsson thesis of Static Charge on References: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ function Appendices A IFO structure % % % % % % % % % % % % % % % % % % % introduced% VIR actual AdV payload % IFOMODEL returns% benchmark a% structure package, programs and which describing noise provides simulator. an science IFO for Part use of in the bench % % % comparing% interferometric% Id: gravitational IFOModel.m,D wave detector designs. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 33 of 78 30]; − 26 ] ; − 10ˆ 8 ] ; ∗ 10ˆ − ∗ 30 1.49 26 5.3 − − 8 10ˆ − 10ˆ ∗ 10ˆ ∗ 10ˆ ∗ ~ 27 4.7 31 1.63 pi ) − − ∗ 10ˆ 10ˆ ∗ ∗ −−−−−−−−−−−−−−−−−−−−−−−−−−−− 7 4 − % m; % m % m; 26 3.34765 30 7.81917 − − 10ˆ 10ˆ ; ∗ ∗ λ −−−−−−−−−−−−−−−−−− mol); Gas Constant 7 10ˆ ∗ − 3.26 e6 ; ∗ s; (Plancks constant)/(2 s; Planks constant % m − − 6; Giancarlo Cella , 12/05/2009 %J %m/s;% Speed mˆ3/Kg/sˆ2;% of J/K; Grav.%J light Boltzman Constant%J/(K Constant in Vacuum % F/m; Permittivity of Free Space % K;% Temperature sec; of Seconds the in Vacuum a year % m/sˆ2;% grav. Sampling acceleration frequency (Hz) − ; −−−−−−−− pi ∗ ifo .Constants.c 2 ∗ ∗ 12; − 34; 23; 11; ifo .Constants.G/ifo .Constants.cˆ2; − − − ∗ 86400; ∗ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ifo.Infrastructure.Length =2999.8; ifo.Infrastructure.ResidualGas.pressures = [10ˆ ifo.Infrastructure.ResidualGas.masses = [3.01289 ifo.Constants.cifo.Constants.Gifo.Constants.kBifo .Constants.h =ifo 2.99792458e8; .Constants.R = 6.67259e = = 1.380658e ifo = .Constants.hbar 8.31447215; ifo .Constants.hbar = 1.054572e ifo .Constants.Tempifo.Constants.yrifo = .Constants.Mpc 290; = 365.2422 = ifo .Constants.yr ifo .Infrastructure.ResidualGas. polarizabilities = [1.43319 ifo .Constants.E0 = 8.8541878176e ifo .Constants.MSolifo.Constants.gifo.Constants.fs = 1.989e30 = 9.81; = 20000; ifo.Laser.Wavelength =1.064e %% Infrastructure %% Detailed model for residual gas noise %% Each%% quantity molecules,%% is%% in a the% vector Pa; following partial which contain order: pressures the GOAL: parameter for one kind of H2O H2 N2 O2 % kg; Mass of the molecule: %mˆ3; Gas polarizability:%% Physical and other constantMaterialss; All in SI units %% Laser VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 34 of 78 srm 5.3 i s 5.4 used L l a r g e r 5.4 5.4 and of Eq.63 × φ detection efficiency: PR d i s t a n c e − trip loss S in the SRC as Eq.71 − in Sec.5.3 2 Eq.65 is 2 − − PD is defined as the λ l %W (exiting from IMC); % see Section % photo % it% contains losses%PD, all after% ...).% the the computed Defined% SR average in (OMC, as % Table the per% RTL in mirror loss% Its power% the value loss.% power is% loss see stated% Section mismatch near in% beamsplitter btwn calculate arms & SRC an modes; effective used to r % m; ITM% to (in% SRM distance effect% BS in% Section% but it% the doesn’t phase introduced play directly any effective in the role code because 6/2; − 0.14; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− − s u b s p h i ∗ ˆ2 , where | 6; subs > − Psi | subs the specific lensing profiles phi ) ∗ c o a t + P subst the power absorbed in coating and substrate p h i exp ( i ∗ | c o a t & p h i coat Psi −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− < coat & P is the beam hitting the ITM and | > S = Psi − | ifo.Laser.Power =125; ifo.Optics.SRM.CavityLength = 24; ifo.Optics.PhotoDetectorEfficiency = 1 ifo.Optics.Lossifo.Optics.BSLossifo.Optics.coupling = 1510.0e = 0.997; =75e % and P %% Parameter% The describing% presumably mode% mismatch thermal This dominant% into 1 increase lensing effect% the is SRC,% of phi and estimated a = thus thermal P an by lens increased calculating in effective the the ITMs round is loss an increased of the SRC. % % This expression can be expanded to 2nd order and is given by %ifo.Laser.Power%% Optics =25; %W(exiting fromIMC); % with p h i %TBC VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 35 of 78 TCSeff )ˆ2 % Noisy% seism Noisy seism − (1 ∗ 3 44 2]; % Quiet seism 3 44 2 ] ; − − subst ˆ2 subst ˆ2 P P VIRGO ∗ ∗ s s s s % kg/mˆ3; density of the ground near the mirror −−−−−−−− subst + s subst + s P P ∗ ∗ coat coat 1= pole; 1 = zero − P P 2 2 2 % ∗ ∗ − − − % V.Fafone March 2012 8 0.1 3.4 0.05 0.04 8.0 0.1 5e c s c s 8 12.0 3.5 0.5 0.04 8.0 0.1 5e − 6; s s − 1 1 ] ; − ∗ ∗ − 8 4e % Wattˆ % Wattˆ % Wattˆ 8 4e 1 ss where calculated analytically by Phil Wilems (4/2007) − − − coat ˆ2 + 2 coat ˆ2 + 2 s s =7.631; c s =7.321; c s and s c c =7.024; c c P c c P cc , s ifo.Seismic.fi=[0.4 0.6 1.2]; ifo.Seismic.Rho = 2.3e3; ifo.Seismic.Qi=[3ifo . Seismic .PoWi=[ 3 3]; ifo .TCS.SRCloss=300.0e ifo .Seismic.sp=[4e i f o .TCS. s i f o .TCS. s i f o .TCS. s % Gravity Gradient section % % common part % % To avoid% and% iterative calculate TCS.SRCloss is TCSeff calculation incorporated as a we bench as define%% an output. Seismic additional TCS.SCRloss% and =% Gravity S reviewed loss% as Gradient an in (22/03/2012)% input Seismic the Parameters%ifo SRC by .Seismic.sp=[4e noise M.Punturo,%ifo.Seismic.fi=[0.40.60.6]; section I.Fiori and P.Ruggi %Quietseism %TBC % S= s % s % The% hardest compensating% TCS part efficiency% this a to% phase model loss. of TCSeff% distortion. is 1 The Thus Watt as how% above as S= absorbed the efficient s a formula E.g. reduction simple is thus TCSeff=0.99 the Ansatz eqivalent in becomes: TCS means we effective system that define to is the the the power in compensated uncompensated that distortion produces distortion of 10mWatt. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 36 of 78 1.4 − c o u p l i n g − 0.60, noisy times beta = 0.15 p o l e s ) − − mail to NAR 27April06mail dimensionless to NAR 27April06 units AdV .m. − − ’ , 0 ) ; horizontal x − % from G Harry e % from G Harry e % v e r t i c a l % (1/K), dlnE/dT % Kg/mˆ3; % J/Kg/K; % kg W/m/ ; % 1/K; % Pa; Youngs Modulus v00r01.txt’, ’ % quiet times beta = 0.35 TF % TF amplitude at fnorm 2; − 3; 4; 7; %% USED IN BENCH4 10; 7; − − − − − 017; % normalization frequency − ) / 3 ; pi ( −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sqrt ∗ . fnorm = 1 0 ; pz = importdata(’fullSA Anorm=6.1106e [26] 09 − 015A − ifo.Suspension.Silica.Y = 7.2e10; ifo.Suspension.Silica.Dissdepth = 1.5e ifo.Suspension.Silica.Phi = 4.1e ifo .Suspension.SAifo .Suspension.SA ifo.Seismic.Beta = 2 ifo .Suspension.Tempifo .Suspension.VHCoupling.theta = ifoifo.Suspension.Silica.Rho .Constants.Temp;ifo.Suspension.Silica.C =ifo.Suspension.Silica.K 1e ifo.Suspension.Silica.Alphaifo = .Suspension. 2.2e3; = Silica = 772; 3.9e = .dlnEdT 1.38; = 1.52e ifo .Suspension.SA %ifo.Suspension.Silica.Y = 7.27e10; % Young’s modulus IN %%USED BENCH 4 % Proper% documentation in VIR of this suspension% loaded thermal% by noise an external model can text be found file generated through Octopus by P.Ruggi % % last% stage section % % to% update% when The SA the modelling payload is design introduced will through be available its Transfer Function (zero %% Suspension:% SI This Units% LIGO. part This replaces set the of suspension orginially parameters used only suspension works part with from suspR Advanced %ifo.Suspension.Silica.Alpha = 5.1e VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 37 of 78 %”” %”” %”” % value found for UNS G10860, similar composition %”” % from producer communication spa) (WAGNER %diameter%diameter of%diameter the of%length first the of%length middle the of head %length second part the of head first the of%fibre middle the head ’s%fibre second part diameter ’s head length %M.Punturo, entry 22436 (20/03/2009) Virgo logbook %Mirror mass%unused in in%Mass kg; of the% marionette Viscous new% payload Q still% of design affects the% unmeasured frequency Marionette;% the and thermal behavior it very this noise low of parameter the is 6; 6; 6; 6; 4; − 6; − − − − % Marging steel data from P.Puppo 4; %WRONGNUMBER, true for maraging!!! − − 2.98e 4; 4 6; − − − 4; − 2.782e − − 1.3e − −−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−− %Switch 0=cylindrical fibres; 1=optimized fibres Cylindrical fibres Optimized fibres −−−−−−−−−− −−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ifo.Suspension.C85Steel.Rhoifo.Suspension.C85Steel.Cifo.Suspension.C85Steel.Kifo.Suspension.C85Steel.Alphaifo.Suspension.C85Steel.Y = 7900; = = = 502; 14e 50; = 200e9; ifo .Suspension.C85Steel.dlnEdT = ifo .Suspension.Stage(2).Massifo .Suspension.Stage(3).Massifo .Suspension.Mario.QV = 42; = = 1000; 140; ifo .Suspension.Optcyl=1; ifo .Suspension.Stage(1).WireDiameter2ifo .Suspension.Stage(1).WireDiameter3ifo .Suspension.Stage(1).Length1ifo .Suspension.Stage(1).Length2ifo = .Suspension.Stage(1).Length3 400e =ifo 800e = .Suspension.Stage(1).WireDiameter 0.05;ifo = .Suspension.Stage(1).Length 0.6; = 0.05; = 400e = 0.7; ifo .Suspension.MaragingSteel.Yifo .Suspension.MaragingSteel.Dissdepthifo .Suspension.Stage(1).Mass = 189e9; = 0; = 42; ifo .Suspension.Stage(1).WireDiameter1 = 800e ifo.Suspension.C85Steel.Phiifo .Suspension.MaragingSteel.Rhoifo.Suspension.MaragingSteel.Cifo.Suspension.MaragingSteel.Kifo = .Suspension.MaragingSteel.Alpha =ifo 1e 7425; .Suspension.MaragingSteel.dlnEdTifo.Suspension.MaragingSteel.Phi = 460; = 24.6; = = 17.925e = 1e % ifo.Suspension.C85Steel.dlnEdT = % % % VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 38 of 78 6 from Braginsky − [45] 27a 6 Fejer et al, 5e − in Eq.11 in Eq.11 1 1 n n E σ %Viscous Q of marionette for%Viscous horizontal Q of marionette motion for verical motion % Fejer e t a l % Crooks% et Fejer al, e t Fejer a l et al % % % Specific% used% heat in see Eq per . Crooks volume et unit al % 3.6 e % dn/dT, value Gretarrson (G070161) %Viscous Q of reference mass for horizontal motion %RM wire ’s%Marionette diameter wire’s%RM wire’s diameter%Marionette length wire’s%Numbers of length%Numbers wires of%Numbers wires in of mirror wires in%Viscous RM suspension suspension in Q marionette of mirror suspension for%Viscous horizontal%Viscous Q of Q motion mirror of reference for vertical mass for motion vertical motion 3; % Measured at LMA, Email from Laurent (5/3/2009) 6; − − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 6; 5; 7; − % dn/dT, (ref. 14) 4; − − − 6; − ifo .Materials.Coating.Sigmahighnifo .Materials.Coating.CVhighnifo = .Materials.Coating.Alphahighn = 0.23; 2.1e6; = 3.6e ifo .Materials.Coating.Yhighn = 140e9; ifo .Suspension.Stage(2).WireDiameterifo .Suspension.Stage(3).WireDiameterifo .Suspension.Stage(2).Lengthifo = .Suspension.Stage(3).Length 400e =ifo 1.85e .Suspension.Stage(1).NWiresifo = .Suspension.Stage(2).NWires 0.7;ifo = .Suspension.Stage(3).NWires 1.125; =ifo 4; .Suspension.Stage(1).Qvh=1e20; =ifo 4; .Suspension.Stage(2).Qvh=1e20; =ifo 1; .Suspension.Stage(3).Qvh=ifoifo .Suspension.Stage(1).Qvv=1e20;ifo .Suspension.Stage(2).Qvv=1e5; .Suspension.Mario.QV;ifo .Suspension.Stage(3).Qvv=ifo .Suspension.Mario.QV; ifo .Materials.Coating.ThermalDiffusivityhighnifo .Materials.Coating.Indexhighn = = 33; 2.035; ifo .Materials.Coating.Betahighnifo .Materials.Coating.Phihighn = 1.4e = 2.3e ifo .Materials.Coating.Alphalownifo .Materials.Coating.Betalown = 5.1e = 8e ifo .Materials.Coating.Ylownifo .Materials.Coating.Sigmalownifo .Materials.Coating.CVlown = 72e9; = = 0.17; 1.6412e6; %% Dielectric%% coating high index material material: parameters tantala % %% low index material: silica VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 39 of 78 12 f o r Sup2 − [44] [44] % Coeff% term of% for freq Exponent% bulk 0.822 depend. for mechanical for freq Sup2 dependence loss, of 7.15e silica loss, % Value form Heraeus, Email from Laurent (5/3/2009) % m; % m; % Surface% N/mˆ2; loss% Youngs Kg/mˆ3; modulus% limit Poisson Kg/mˆ3; (ref.% (ref. (ref. 1/K; ratio% thermal 4) J/Kg/K; 4) 12) (ref.% specific J/m/s/K; expansion 4) thermal coeff. heat conductivity (ref. (ref. 4) (ref. 4) 4) % ROC of% ROC ITM of ETM % baffle% Coating aperture diameter) (1 Sec.2.3.3 cm less of than the % 1/m;%W; bulk tolerable absorption heating coef power (factor 1 ATC) % Fejer e t a l % Measured at LMA, Email from Laurent (5/3/2009) 7; − 12; 4; 5; − − − 12; − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ifo .Optics.Curvature.ETM = 1683; ifo .Optics.Diameter.ETM = .33; ifo .Optics.Curvature.ITM = 1420; ifo .Optics.Diameter.ITM = .33; ifo .Materials.MassThickness = 0.200; ifo.Optics.pcrit = 10; ifo.Materials.Substrate.c2 = 7.6e ifo .Materials.Substrate.RefractiveIndexifo = .Materials.MassRadius 1.452; = 0.175; ifo .Materials.Coating.ThermalDiffusivitylownifo .Materials.Coating.Philownifo .Materials.Coating.Indexlown = 4.0e = = 1.38; 1.458; ifo . Materials .Substrate.MechanicalLossExponent=0.77;ifo .Materials.Substrate.Alphasifo.Materials.Substrate.MirrorYifo .Materials.Substrate.MirrorSigmaifo .Materials.Substrate.MassDensityifo = .Materials.Substrate.MassAlpha 5.2eifo .Materials.Substrate.MassCMifo .Materials.Substrate.MassKappa = = 0.167; 7.27e10; = 2.2e3; = = 3.9e 739; = 1.38; ifo .Optics.SubstrateAbsorption = 0.3e BA GEOMETRY %BEAM %ifo.Optics.Curvature.ITM=1530;%ifo.Optics.Curvature.ETM=1530;%ifo.Optics.Diameter.ITM=.34;%ifo.Optics.Diameter.ETM=.34; %ROCofITM %ROCofETM %Coating %Coating diameter diameter %%Substrate Material parameters %ifo.Materials.Substrate.RefractiveIndex% % The% following GEOMETRY %MIRROR = data 1.45; are extracted from the % MIR mevans chapter 25 Apr of 2008 the AdV TDR VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 40 of 78 64 in Eq.63 2 φ / ) in Eq . η T unephase − π = ( φ % absorption of ITM % Transmittance% PRM trasmittance% of SRM % SRM tuning% , used NSNS to optimized compute % Transmittance% Transmittance of ITM of ETM % % % demod/detection/homodyne% d e f i n% e d phase as 6; − mevans June 2008 − 6; − / 2 ; pi PR PARAMETERS − ifo .Optics.PRM.Transmittanceifo = .Optics.SRM.Tunephase 0.05; = 0.35; ifo .Optics.SRM.Transmittance = 0.20; ifo .Optics.ETM.Transmittance = 1e ifo .Optics.Quadrature.dc = ifo .Optics.ITM.CoatingThicknessLownifo .Optics.ITM.CoatingThicknessCapifo .Optics.ETM.CoatingThicknessLown =ifo 0.308; .Optics.ETM.CoatingThicknessCap = 0.5; = 0.27; = 0.5; ifo .Optics.ITM.Transmittance = 0.014; ifo .Optics.ITM.CoatingAbsorption = 1.0e %ifo.Optics.SRM.Transmittance = 1;%ifo.Optics.SRM.Tunephase= 0.0; % Transmittance of SRM %SRM tuning, wideband % These%ifo.Optics.ITM.BeamRadius= are%ifo.Optics.ETM.BeamRadius= now computed in 0.0554; precompIFO 0.065; ,%FINESSE SR %m; %m; 1/eˆ2 1/eˆ2 power%radius power radius % coating layer optical thicknesses VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 41 of 78 f ) ; ∗ f ) ; pi ∗ ∗ pi ∗ wITMˆ 2 ) ; EM 2 )wETMˆ ; ∗ ∗ pi pi ∗ ∗ Y Y ∗ ∗ phibulk ./ (2 sigma ) sigma ) ∗ − − wITMˆ 2 ) ; EM 2 )wETMˆ ; ∗ ∗ ifo .Constants.Temp; pi pi sigma )/((1 sigma )/((1 ∗ ∗ ∗ ∗ ∗ 2 2 − − (aITM + aETM) . (csurfITM + csurfETM) ./ (2 (1 (1 ∗ ∗ ∗ ∗ strain noise psd arising from the Brownian thermal noise f . ˆ n ; ∗ kBT kBT − ∗ ∗ (csurf + cbulk) / Lˆ2; n = subbrownian(f, ifo) ∗ [cITM, aITM][cETM, = aETM] = subbrownianFiniteCorr(ifo subbrownianFiniteCorr(ifo , , ’ITM’); ’ETM’); c s u r f = 8 alphas = ifo .Materials.Substrate.Alphas; phibulk = c2. cbulk = 8 c2 = ifo.Materials.Substrate.c2; sigma = ifo .Materials.Substrate.MirrorSigma; csurfITM = alphas csurfETM = alphas n = 2 n = ifo .Materials.Substrate.MechanicalLossExponent; L = ifo.Infrastructure.Length; kBT = ifo .Constants.kB wITM = ifo .Optics .ITM.BeamRadius; Y = ifo .Materials.Substrate.MirrorY; % csurfITM = alphas/(Y % Surface% csurfETM loss = alphas/(Y contribution % account for 2 ITM and 2 ETM, and convert to strain whith 1/Lˆ2 % Bulk substrate contribution EM=wETM ifo . Optics .ETM.BeamRadius; function B Brownian Substrate Code % due to mechanical loss in the substrate material % SUBBROWNIAN VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 42 of 78 % t h i r d bump % f i r s t bump % second bump sp(10))./sp(11)).ˆ2)); − f . / sp ( 5 ) ) ; ∗ ( ( ( f ∗ 0 . 5 . − 1= pole; 1 = zero ( bumps ; − ∗ % exp sp(7))./sp(8))); ∗ − sp(4)).ˆ2+sp(4). ( ( f − exp (TF) ; hfslope+sp(2). n = groundSeism(f, ifo) abs ∗ fi=ifo.Seismic. fi ; h f s l o p e= sp=ifo .Seismic.sp; function fnorm = 0 ; Qi=ifo .Seismic.Qi; b2=sp(6)./(1+ b3 = (sp(9)/sp(11)). b1=sp(3)./(( f.ˆ2 n=sp ( 1 ) . bumps=b1+b2+b3 ; C Seismic noise generation Code Anorm = 0 ; PoWi=i f o . Seismic . PoWi ; TF = TransFunct(f ,fi ,Qi,PoWi, fnorm, Anorm); % % % % M. Punturo% % (14/03/2012) % read% the seism parameters VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 43 of 78 aftm folding of power ∗ − qc/0002055 (hereafter LT) size test masses. − − Phi ( f ) ∗ % LT eq . 35a % LT 57 % LT 56 (BHV eq. 3) % LT uses e %LT 53 (BHV eq. btwn 29 & 30) f )) ∗ pi Y); ∗ ∗ Qm) ; a ∗ ∗ pi Qm; ∗ T / (2 h ) . ˆ 2 . ∗ ∗ Estimate amplitude coefficient of j0m . ˆ 2 ) ) ; km. ∗ ∗ (km h kB − ∗ ∗ j0m ) ) ; 4 ∗ ∗ − (1+sigma)/( ∗ r0/a).ˆ2/4); h ) ; ∗ ∗ ( 2 ) ; x2 . / ( zeta . Qm).ˆ2 (1+Qm)+4 ∗ km ∗ − sigma ) ∗ 2 x ( f ) = (8 − ( zeta sqrt − (Um. ( x ; (1 − Qm). [cftm, aftm] = subbrownianFiniteCorr(ifo , opticName) (x./(zeta.ˆ2. ( ∗ ∗ − exp sum 236 (hereafter BHV) and Liu and Thorne gr exp sum − zeta = ifo.Constants.BesselZeros; s = r0 = w / sigma = ifo .Materials.Substrate.MirrorSigma; x2 = x . a = ifo .Materials.MassRadius; j0m = besselj(0,zeta); x = h = ifo .Materials.MassThickness; U0 = U0 U0 = km = zeta /a ; w = ifo .Optics.(opticName).BeamRadius; Y = ifo .Materials.Substrate.MirrorY; % get some numbers % do the work Um = (1 Um = Um./((1 Qm = function D Brownian Substrate Finite Size Correction Code % % [cftm,% cftm aftm]% = aftm finite =% = subbrownianFiniteCorr(ifo amplitude%S mirror coefficient correction , opticName) thermal factor for noise thermal contribution noise: to displacement noise is % subbrownianFiniteCorr% mirror thermal noise contribution for finite % % Equation% 227 references to Bondu, et al. Physics Letters A 246 (1998) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 44 of 78 % LT 59 f ) r0 ) ; ∗ ∗ pi Y ∗ ∗ ) % LT 28 % LT 54 % LT 58 (eq. following BHV 31) s ; pi ∗ ∗ (2 Phi ) / (2 s ˆ 2 ; sigma ∗ ∗ ∗ Y); ∗ sqrt p0 T ∗ ∗ ∗ hˆ3 ∗ sigma ) hˆ2 pi ∗ kB − ∗ pi ∗ (1 ∗ ∗ p0 ) ˆ 2 ; a ˆ2/(6 ∗ ∗ hˆ2 sigmaˆ2) / (2 ∗ a ˆ 2 ) ; pi ∗ − pi cftm = aftm / aitm; aitm = (1 aftm = DeltaU + U0; DeltaU = ( DeltaU = DeltaU + 12 DeltaU = DeltaU + 72 DeltaU = DeltaU p0 = 1/( % finite mirror correction % amplitude% coef factored for infinite out: (8 TM VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 45 of 78 6); 6); 6); 6); 6); 6); − − − − − − 6, 1.156, e 2.5 e 6, 1.186, e 2.61 e 6, 1.05 e 6, 2.61 e − − − − − − wave layers , mevans 25 Apr 2008 − quater − ADV(f,ADV ( f ifo, , i f o ’ITM’, , ’ETM’ 1.6e , 3.5 e ADV(f,ADV(f, ifo, ifo ’ITM’, , 1.82e ’ETM’, 3.83e ADV(f, ifoADV(f, , ’ITM’, ifo 1.64e , ’ETM’, 3.83e ADV ( f , i f o ) effect of optical coating on Brownian thermal noise n = coatbrownian − SbrITM = getCoatBrownian L = ifo.Infrastructure.Length; SbrETM = getCoatBrownian % Constants % compute% noise [email protected],% Overall power from thickness% one The ITM% 30/08/2008 values and SbrITM of one% SbrETM given coating = ETM getCoatBrownian = here getCoatBrownian layers% are Values% is derived Finesse calculated%SbrITM given from 888 = as getCoatBrownian with Raffaeles an method input. Email from (25/08/2008) Laurent (Email 06/03/2009) %SbrETM = getCoatBrownian % Values% Finesse calculated 444 with method from% Laurent Uncomment%SbrITM (Email for =%SbrETM 06/03/2009) getCoatBrownian(f, ’optimized = getCoatBrownian(f ’ coatings ifo , , ifo , ’ITM’); ’ETM’); function E Brownian Coating parameters feeding Code % COAT %% returns% Added strain% by Expanded G% Harry noise Modified to% 8/3/02 reduce power Modified to from spectrum approximation return work to in by accept , strain 1 GMH Nakagawa, 8/03 / coating noise, Hz Gretarsson, with PF 4/07 non et al. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 46 of 78 (SbrITM + SbrETM) / Lˆ2; ∗ n = 2 %% f i g u% r e loglog(f,% legend(’single f sqrt(SbrITM)/L, i g u r e ITM’, f, ’single sqrt(SbrETM)/L, ETM’, f, ’total sqrt(n)) coating noise’) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 47 of 78 n − 415 − ADV(f, ifo , wBeam, dlown, dhighn) ifo .Constants.Temp; n f i r s t . − ∗ 4 < SbrZ = getCoatBrownian n layers, with low − [wBeam, dOpt] = getCoatParFromName(ifo , wBeam); % wBeam has been used as the name of an optic sigmasub = ifo .Materials.Substrate.MirrorSigma; Yhighn = ifo .Materials.Coating.Yhighn; lambda = ifo .Laser.Wavelength; Ysub = ifo .Materials.Substrate.MirrorY; kBT = ifo .Constants.kB % end % Constants % check% arguments i f nargin function F Brownian Coating Code % SbrZ% SbrZ =% getCoatBrownian(f, =% returns getCoatBrownian(f, coating the ifo , coating layers. ifo opticName) , brownian wBeam, dOpt) The noise layers for are a assumed given to collection be alernating of low % high% % f% = ifo frequency% = parameter% vector opticName% struct in = name% Hz wBeam of from =% ifoArg.Optics.(opticName).BeamRadius the IFOmodel.m dOpt% = Optic wBeam ifoArg.Optics.(opticName).CoatLayerOpticalThickness =% beam struct dOpt radius% = coating to% (at use% layer 1 for SbrZ / =% = wBeam eˆ2 thickness the Brownian and power)% dOpt optical adapted noise vector% based from spectra thickness, (Nlayer bench62 on Harry for x normalized et one 1) mirror al., by in lambda, Class mˆ2 / Quant of Hz Grav each 24 coating (2007) layer. 405 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 48 of 78 dlown ) ; ∗ dhighn+Ylown dhighn ) ; ∗ c o a t i n g ∗ − phihighn ∗ phihighn/Yhighn); dlown)/(Yhighn ∗ ∗ Ylown ∗ dlown + Yhighn phipara+phiperp/Yperp); ∗ dlown ) ; ∗ ∗ coating layer and the highn − philown lambda / nH; ∗ lambda / nL ; ∗ philown/Ylown + dhighn ∗ dhighn+sigmalown ∗ ∗ (Ypara/Ysubˆ2 ( Ylown ∗ ∗ dhighn + Ylown ∗ ( dlown ∗ Yhighn phihighn ; ∗ Ypara ) philown ; ∗ ∗ wITMˆ2) ∗ ( Yhighn ∗ ∗ (sigmahighn+sigmalown ); ∗ phiperp = Yperp/dCoat phipara = 1/(dCoat phihighn = ifo .Materials.Coating.Phihighn; sigmahighn = ifo .Materials.Coating.Sigmahighn; philown = ifo .Materials.Coating.Philown; sigma2 = (sigmahighn sigma1 = 1/2 sigmalown = ifo .Materials.Coating.Sigmalown; dCoat = dlown + dhighn; Yperp = dCoat/(dhighn/Yhighn+dlown/Ylown); Ypara = 1/dCoat Ylown = ifo .Materials.Coating.Ylown; nL = ifo .Materials.Coating.Indexlown; nH = ifo .Materials.Coating.Indexhighn; % Brownian% cITM contribution = dITM/( pi to coating thermal noise, low Poisson ratio limit % This is exact % dhighn = sum(dOpt(2:2:end)) % This% is average a works kludge, really the well real formula is very complicted but this % The% following the overall lines thickness ar obsulete of the for lown Advanced Virgo as we give directly %%%%%%%% t h i s part i s%%%%%%%%%%%%%%%%% d i r e c t l y from bench62 %%%%%%%%%%%%%%%%% % l a y e% r . [email protected],% compute% dlown thickness = 30/08/2008 sum(dOpt(1:2:end)) of each material in the coating % for% debugging,% Llown = this Lhighn dlown = is dhighn a rough but direct estimate %% Lsum = Llown Lall + = Lhighn; [Lsum, Llown, Lhighn] VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 49 of 78 phiperp)+... − phipara ) ; ∗ ( phipara ∗ ... − sigmasub ) ) − (1 ∗ phiperp +... sigmasub ) ) ∗ − (1 sigmasubˆ2)) ∗ − sigma1 ˆ2) (1 ∗ − sigma1 ) ) ) (1 sigma1 ) ∗ − − (1 (1 ∗ phipara+phiperp/Yperp); ∗ ∗ ( ( 1 / ( Yperp ∗ Ysub ∗ f ) ; sigmasubˆ2) ∗ − wBeamˆ 2 ) ∗ (1 (Ypara/Ysubˆ2 pi ∗ sigmasub)ˆ2/(Ysubˆ2 pi ∗ ∗ ∗ 2 − (1 sigmasub)/(Yperp ∗ ∗ wETMˆ2) 2 ∗ c . / (2 − ∗ (1 ∗ sigmasubˆ2)/( Ypara/(Yperpˆ2 ∗ − kBT (1 ∗ ∗ sigma2 (1+sigmasub) ∗ ∗ sigma2 ˆ2 ∗ 2 Ypara Ypara c = dCoat SbrZ = 4 % cETM = dCoat/(pi % Brownian contribution to coating thermal% noise noise, power full formula VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 50 of 78 % % thermal% thermal conductivity% heat expansion capacity @% constant note kBT mass has factor Temp kBT; % temperature % LT 18% LT less 18 less factor factor 1/omegaˆ2 1/omegaˆ2 ∗ Temp ∗ L ) . ˆ 2 ; ∗ f ∗ alpha ˆ2 rho ) ˆ 2 ) ; pi ∗ ifo .Constants.Temp; ∗ ∗ ∗ ( 2 ) ) ˆ 3 ; ( 2 ) ) ˆ 3 ; (CM kappa ∗ ) ∗ sqrt sqrt pi ∗ subthermFiniteCorr(ifo , ’ITM’); subthermFiniteCorr(ifo , ’ETM’); (2 ∗ ∗ noise psd arising from thermoelastic fluctuations in mirror sqrt − n = subtherm(f, ifo) (1+sigma)ˆ2 (SITM + . SETM) / ( 2 ∗ ∗ alpha = ifo .Materials.Substrate.MassAlpha; rho = ifo .Materials.Substrate.MassDensity; sigma = ifo .Materials.Substrate.MirrorSigma; kappa = ifo .Materials.Substrate.MassKappa; S0 = S0 /( S0 = 8 n = 2 L = ifo.Infrastructure.Length; Temp = ifo .Constants.Temp; SITM = S0 /(wITM/ SITM = SITM kBT = ifo .Constants.kB wITM = ifo .Optics .ITM.BeamRadius; SETM = S0 /(wETM/ SETM = SETM % Corrections for finite test masses: % 2 mirrors each type, factor omegaˆ2, dimensionless strain EM=wETM ifo . Optics .ETM.BeamRadius; CM = ifo .Materials.Substrate.MassCM; function G Thermo-elastic Substrate Code % SUBTHERM VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 51 of 78 f o l d i n g − % LT 37 qc/0002055 (hereafter LT) − Qm) ; ∗ Qm) . ˆ 2 ; ∗ % LT 40 % LT eq . 35a % LT 32 % LT 32 % LT 28 % LT 46 % LT uses power e km. Qm) ; ∗ − h ∗ (1+Qm) ; (1 a ˆ 2 ; j0m ) . ˆ 2 ) ; ∗ ∗ ∗ ∗ km) . ˆ 2 . ∗ ( a Qm. (h ∗ ∗ ∗ 4 r0 )ˆ3 pi ∗ − (1+Qm)+8 ) j0m ).ˆ2. pm./zeta.ˆ2); ∗ c1ˆ2/(1+sigma)ˆ2; ∗ ∗ pi ∗ ∗ km) . ˆ 2 . ∗ Qm). (2 (pm. Qm).ˆ2 − (h ( j0m . ∗ − ∗ qc/0002055 equation 46) h ) ; − ((1 sqrt h ) ; ∗ km. sum ∗ ( r0).ˆ2/4)./( ∗ ∗ ∗ ∗ ∗ km ( 2 ) ; ( c o e f f ) + h ∗ a ˆ 2 ) ; Qm). (km 2 ∗ − finite size test mass correction to noise amplitude coefficient − − c0 /h ; pi ( ( sum coeff = subthermFiniteCorr(ifo , opticName) ( a/h)ˆ2 ∗ sqrt − 2 ∗ − exp exp 236 (hereafter BHV) or Liu and Thorne gr − c o e f f = (1 coeff = coeff. coeff = coeffcoeff + = 4 coeff./((1 coeff = coeff c o e f f = zeta = ifo.Constants.BesselZeros; r0 = w/ c1 = c1 = c1 + p0/(2 c0 = 6 sigma = ifo .Materials.Substrate.MirrorSigma; a = ifo .Materials.MassRadius; p0 = 1/( j0m = besselj(0,zeta); h = ifo .Materials.MassThickness; km = zeta /a ; w = ifo .Optics.(opticName).BeamRadius; pm = % extract some numbers % do the work Qm = function H Thermo-elastic Substrate Finite Correction Code % % Equation% 227 references to Bondu, et al. Physics Letters A 246 (1998) % CFTM20 % (Liu & Thorne gr VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 52 of 78 )); end 1) + nAll ( 2 : − end refractive index / lambda ∗ TR, rCoat] = ... )) ./ (nAll(1: end TR TR, rCoat] = ... optic phase derivative with respect to temperature − TE , dphi Ta2O5 a SiO2 nAll ( 2 : ∗ elastic phase derivative (dphi / dT) refractive phase derivative (dphi / dT) n − alpha − − alpha − TE + dphi 1) TE , dphi ∗ ∗ − dT , dphi end [ dphi dT , dphi dT = total thermo TE = thermo TR = thermo getCoatTOPhase(nIn, nOut, nLayer, dOpt, aLayer, bLayer) SiO2 ˜ 2.3 Ta2O5 ˜ 3.5 nAll = [nIn; nLayer(:); nOut]; r = ( nAll ( 1 : function I Thermo-optical coating core computation Code % % dphi % aLayer% = change% bLayer in% = geometrical change% = in the thickness refractive effective = with dn/dT index = temperature thermal dd/dT with expansion temperature coeffient of the coating layer % combine reflectivities %% getCoatTOPhase(nIn,% returns% nOut, nIn coating% = nLayer, nOut refractive% = reflection nLayer dOpt, refractive% = dOpt aLayer, index refractive% phase index of bLayer) = derivatives input optical of index output medium of thickness (e.g., wrt medium each = (e.g., temperature geometrical layer, vacuum / lambda = SiO2 1) ordered thickness of = 1.45231 each input @ layer to 1064nm) output (N x 1) % vector of all% reflectivity refractive indexs of each interface % = dphi % a % dphi % [ dphi % rCoat% = amplitude% Note% about reflectivity a aLayer: on of a SiO2 coating substrate (complex) , % dphi % % see% getCoatTOPos (see also for more T080101) information VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 53 of 78 rbar(n + 1)); dGeo ) ; ∗ ∗ nLayer ) . ∗ ... 1 ) ) ; ∗ − )); rbar(m + 1)).ˆ2; dGeo ) ; ∗ end ∗ ephi (m) ); dOpt (n ∗ ∗ dOpt ( dphi / rbar(1)); end (bLayer + aLayer . ∗ r ( pi ( dr ∗ ( aLayer . (r(n) + rbar(n + 1)) / (1 + r(n) pi ∗ ∗ 1:1 1:1 rbar (n + 1 ) ; ( dOpt ) ) ; ) ∗ ∗ dphi (n) − − 4 i imag sum dd . ∗ − ( r ) ) ; ( r ) ) ; 4 i ∗ ∗ ( end i size − ( ( − pi pi exp 1:1 size size ( dphi ( ( ∗ ∗ exp − r(m).ˆ2) / (1 + r(m) zeros sum e l a s t i c refractive coupling 1 − ) = ) = ephi ( way phase in this layer − − zeros zeros dphi (n) = dr − > (1 m = n: n dphi (n) = end end dd = 4 TE = 4 TR = dr ephi (n) = 1 ; ephi (n) = n = numel(dOpt): n = numel(dOpt): dphi = i f else dr rbar(n) = ephi(n) for end end % one % accumulate reflecitivity dr ephi = ephi ( rbar = rbar ( for for dphi dphi dphi dGeo = dOpt ./ nLayer; end end % geometrical distances% phase derivatives % thermo % thermo % t o t a l % reflectivity derivatives VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 54 of 78 TE; TR + dphi dT = dphi rCoat = rbar(1); dphi % coating reflectivity VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 55 of 78 d ) } alpha {\ bar \ c )) ./ . . . − Delta \ ( ch ∗ c ) ) ; − s + R . 0 > − ( ch lambda, dTE = − } ∗ R)); ( sh ∗ 1 becomes ∗ beta Ks ) ) ; << {\ 1 as x i ∗ Cc / Kc ) ; > bar sh + Rˆ2 ∗ 1 / (3 − ∗ −\ w − x i . ˆ 2 ) ) . ∗ pS.MassDensity; R (R ∗ ∗ ∗ (2 ∗ Kc / ( Cs f ; ∗ x i ∗ sqrt ( x i ) ; ( x i ) ; − (Cc pi ∗ ( x i ) ; ( x i ) ; gTC = getCoatThickCorr(f , ifo , dOpt, dTE, dTR) ∗ sinh cosh sin sqrt cos [dc, Cc, Kc] = getCoatAvg(ifo , dOpt); x i = dc s = sh = c = ch = pS = ifo.Materials.Substrate; Cs = pS .MassCM Ks = pS.MassKappa; w = 2 R = % parameter extraction % compute coating average% R and parameters xi (from T080101, Thick Coating Correction) % trig functions of xi % pR and pE (dTR = function J Thermo-optical coating core computation Code %% which (% ch in + gTC c the = + 1 2 limit of xi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % gTC =% getCoatThickCorr(f,% finite% ifo% coating Uses (see , dOpt, correction thickness getCoatThermoOptic%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dTE, dTR) % factor correction For for% comparison example from equation% T080101, in usage) modified from the ”Thick Fejer so bTR Coating = that (PRD70, 0 2004) limit, Correction” gFC the (Evans) % gTC= (2 ./ (R VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 56 of 78 gD ) ; ∗ x i . ˆ 2 . ∗ g2 ) . / (R ∗ ( x i / 2 ) ) ; x i . ˆ 2 . sh ; ch ; ∗ sinh ∗ ∗ R R ∗ ∗ g1 + pRˆ2 s + 2 c ) ; c + 2 ( x i / 2) + ∗ ∗ ∗ − x i . cosh ( ch ∗ ∗ Rˆ2) Rˆ2) ∗ − − (R pR R ∗ ∗ ∗ sh + (1 ch + (1 s ) + 2 ∗ ∗ g0 + pE − ( x i / 2) . ∗ ( sh sin ∗ ∗ g0 =g1 2 = 8 g2 = (1 + Rˆ2) pE = dTE / (dTR + dTE ) ; gD = (1 + Rˆ2) pR = dTR / (dTR + dTE ) ; gTC = (pEˆ2 % various parts of gTC % and finally , the correction factor end VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 57 of 78 255 − %substrate%substrate radius thickness% zeros of 1st order bessel function (J1) refractive index / lambda ∗ mat/0302617 − 3 < Cfsm = getCoatFiniteCorr(ifo , wBeam, dOpt) [wBeam, dOpt] = getCoatParFromName(ifo , wBeam); % wBeam has been used as the name of an optic i f nargin zeta = ifo.Constants.BesselZeros; lambda = ifo .Laser.Wavelength; pS = ifo.Materials.Substrate; end pC = ifo.Materials.Coating; R = ifo .Materials.MassRadius; H = ifo .Materials.MassThickness; % check arguments % parameter extraction function K Thermo-elastic Coating Finite Correction Code % Cfsm =% Cfsm getCoatFiniteCorr(ifo =% getCoatFiniteCorr(ifo% finite , opticName) , mirror Uses wBeam, size correction dOpt) correction factor from PLA 2003 vol 312 pg 244 %%% ”Thermodynamical by V. fluctuations B. http://arxiv.org/abs/cond Braginsky in and optical S. P. mirror Vyatchanin coatings” % % (see% getCoatTOPos% version for example 1 by usage) Sam Wald, 2008 % % ifo% = opticName parameter% = name% struct wBeam of =% ifoArg.Optics.(opticName).BeamRadius the dOpt from% = Optic IFOmodel.m wBeam ifoArg.Optics.(opticName).CoatLayerOpticalThickness =% beam struct dOpt radius% to = (at use optical 1 for / wBeam eˆ2 thickness and = power) dOpt geometrical / lambda thickness of each layer VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 58 of 78 POISSONf) avg − POISSONF) avg − YOUNGSF/(1 (1+POISSONf)/(1 ∗ ∗ sigL ) ; sigH ) ; POISSONF) avg − − − ) ) /) ) nL ; / nH; end end sigL ) ; sigH ) ; − − dH) / dc ; dH) / dc ; dH) / dc ; S); ∗ ∗ ∗ Y H ∗ pS.MassDensity; ( dOpt ( 1 : 2 : ( dOpt ( 2 : 2 : L / (1 H / (1 ∗ sigL ) ; (1 + sigL) / (1 (1 + sigH) / (1 Y Y sum sum ∗ ∗ ∗ ∗ − ∗ ∗ S; dL + yyH dL + xxH dL + C ∗ ∗ ∗ L S = pS.MirrorY; S = pS .MassCM L = pC.Ylown; L = pC.CVlown; H = pC.Yhighn; H = pC.CVhighn; sigS = pS.MirrorSigma; sigL = pC.Sigmalown; sigH = pC.Sigmahighn; alphaS = pS.MassAlpha; alphaL = pC.Alphalown; zzL = 1 / (1 dc = dH + dL ; alphaH = pC.Alphahighn; Cf = (C Yf = ( yyL Xf = ( xxL C xxL = alphaL yyL = alphaL Cr = Cf / C Y Yr = Yf / (alphaS Xr = Xf / alphaS; C nL = pC.Indexlown; dL = lambda Y xxH = alphaH yyH = alphaH C Y nH = pC.Indexhighn; dH = lambda % VALUE Z AVERAGE COATING = 1/(1 % VALUE X AVERAGE COATING = ALPHAF % VALUE Y AVERAGE COATING = ALPHAF % coating sums % SPECIFIC AVERAGE HEAT (simple volume average for coating) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 59 of 78 ... ∗ Qm) ; Rˆ2 in denominator ∗ ∗ Cr ) ∗ Hˆ2 . 2 ∗ − sigS)))ˆ2 + S2; ABA 2 ) )LAMBDAˆ ; − ∗ km. ˆ 2 s i g S ) ) / 2 ; (1 ∗ ∗ 4 ∗ 2 s i g S ) + Zf − Yr − ∗ − (1 2 ∗ % left out factor of pi − ( Cr (1 + s i g S )ˆ2 Qm) . ˆ 2 (1 ∗ ∗ − ∗ Rˆ2 ∗ (pm ./ zeta.ˆ2); Cr + S1 sum − dH) / dc ; ∗ ∗ Yr P) / (2 Qm).ˆ2 ./ ((1 ∗ H); r0ˆ2 / 4) ./ j0m; ∗ Lm. ˆ 2 ) ; − ( 2 ) ; ∗ ∗ ∗ (1 + sigS) + (Yr (1 sigH ) ; s i g S km sqrt ∗ ∗ − ∗ ∗ dL + zzH ( ( r0 ˆ2 Rˆ2 / Hˆ2) Zf 2 ∗ 2 Cr + (Xr / (1 + sigS) +Yr km. ˆ 2 ∗ − − − (pm. ˆ 2 . ( ( − − sqrt exp exp sum (1 + s i g S ) r0 = wBeam / Zf = ( zzL S1 = (12 S2 = j0m = besselj(0, zeta); zzH = 1 / (1 Cfsm = P = (Xr km = zeta / R; pm = Lm = Xr % values of J0% at between zeros eq of 77 J1 and 78 %%%%%%%%%%%%%%%%%%FINITE SIZE CALCULATION CORRECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % beam size parameter used by Braginsky % eq 88 % eq 90 and 91 % eq 92 % eq 60 and 70 ABA= LAMBDA Qm = end VIR-0073D-12 issue :D AdV sensitivity 09). date : November 19, 2012

− page : 60 of 78 015A − gravitational energy) is considered for marionette and RF stages. << gw . i t ) − −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− AdV(f ,ifopar) mechanical properties − Mirror pendulum noise=suspR2 =ifo .Suspension. Silica .dlnEdT; −−−−−−−−−− ifo=ifopar ; phi0=ifo .Suspension. Silica .Phi ; alpha=ifo .Suspension. Silica .Alpha; rho=ifo .Suspension. Silica .Rho; ds=ifo .Suspension. Silica .Dissdepth; function thetaHV=ifo . Suspension .VHCoupling. theta ; g=ifo .Constants.g; beta kb=ifo .Constants.kB; E0=ifo .Suspension. Silica .Y; L Suspension Thermal Noise Code Temp=ifo . Suspension .Temp; C=ifo .Suspension. Silica .C; K=ifo .Suspension. Silica .K; % Francesco% Paola Piergiovanni% Puppo Michele (paola.puppo@ego ([email protected]) Punturo% ([email protected]. September% Suspension 2012. Modified thermal it) by noise P. is Puppo. obtained by the fluctuation dissipation theorem approach(see Virgo note VIR % A% complete In% the elastic as% new a A% design cascade low computation INPUT:% frequency of frequency of OUTPUT: for suspension the pendula. approximation the% (f), Last In thermal mirror Stage this list (elastic stage noise suspension routine, of PSD is energy (noise). parameters(ifopar). the performed% indexes General reaction (violin 1 parameter refers mass modes is to are suppressed marionette. taken% and Horizontal into Mirror’s only account). the to stage% Vertical marionette parameter%Fused and coupling Silica has the no mirror thermo index. are suspended to the filter 7 VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 61 of 78 %Switch cylindrical/optimized fibres ( f ) ) ; ( f ) ) ; ( f ) ) ; ( f ) , 3 0 0 0 ) ; length (1 , length ( f ) , 3 0( 0 0 f ) ; ) , 3 0 0 0 ) ; length length (1 , ( r ˆ 4 ; (1 , ∗ zeros 3; /4 %Tension per fibre length length − f ; ( ( pi zeros zeros zeros ∗ I I= r=d / 2 ; r1=d1 / 2 ; r2=d2 / 2 ; r3=d3 / 2 ; pi g/N; d1=ifo .Suspension.Stage(1).WireDiameter1;d2=ifo .Suspension.Stage(1).WireDiameter2;d3=ifo .Suspension.Stage(1).WireDiameter3; L1=ifo .Suspension.Stage(1).Length1;L2=ifo .Suspension.Stage(1).Length2;L3=ifo .Suspension.Stage(1).Length3; d=ifo .Suspension.Stage(1).WireDiameter; L=ifo .Suspension.Stage(1).Length; =1L+3; L=L1+L2+L3 ∗ Optcyl==0, ∗ zeros zeros i f d e l t a=1e else Optcyl=ifo .Suspension.Optcyl; phithT= z= Energy= TotEnergy= phiST= x= end w=2 T=m N=ifo .Suspension.Stage(1).NWires; m=ifo .Suspension.Stage(1).Mass; VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 62 of 78 Energy(i ,j); ∗ tau ˆ2) ∗ C); C); ∗ ∗ 3(f(i),0,ifo); suspn tau ˆ 2 ) ; ∗ Temp/( rho Temp/( rho ∗ ∗ tau/(1+w( i )ˆ2 ∗ E0 ) ) . ˆ 2 E0 ) ) . ˆ 2 w( i ) ∗ ∗ ∗ Energy(i ,j); ∗ tau/(1+w( i )ˆ2 ds ) ; T/(S T/(S r )ˆ2/K; r )ˆ2/K; ∗ ∗ ∗ ∗ ∗ ∗ (2 (2 ds ) ; (Energy(i ,:)); ∗ ∗ beta beta ∗ w( i ) ∗ L1+L2 , C C − − ( z ( i , : ) ) , ∗ ∗ (1+mu. < ∗ sum rho rho L1 , (w) , ∗ ∗ (1+mu. % Cylindrical fibres < ( alpha ( alpha ∗ ∗ ∗ z ( i , j ) length r ˆ 2 ; r ˆ 2 ; j =1: z ( i , j ) length ∗ ∗ r=r2 ; r=r3 ; r=r1 ; pi pi % Optimized fibres i f e l s e i f else end [z(i ,:) x(i ,:) dummy Energy(i ,:)]=An i =1: Delta=E0 Delta=E0 for phiS=phi0 tau =0.0737 tau =0.0737 phithT( i)=Delta phithT( i)=phithT( i)+Delta phithT(i)=0; TotEnergy( i)= phiST( i)=phiST( i)+phiS phiST( i)=0; phiST( i)=phi0 S0=S ; S= S= mu=4/r ; mu=4./ r ; % Optimized fibres routine is called % Surface% the contribution bending energy and thermoelastic contribution are weighted by %Surface%motion contribution ) for circular cross section fibres (horizontal %Thermoelastic contribution Optcyl==0; i f else for VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 63 of 78 C1 ) ; ∗ tau1 ˆ 2 ) ; ∗ Temp/( rho1 ∗ −−−−−−−−−−−−−−−−−−−−− L3)/L ; ∗ E1 ) ) . ˆ 2 ∗ tau1/(1+w( i )ˆ2 ∗ L2+r3 ˆ2 T1/( S1 Marionette ∗ ∗ w( i ) (dd1)ˆ2/K1; ∗ ∗ beta1 C1 − ∗ L1+r2 ˆ2 ∗ (w) , rho1 ∗ ( alpha1 ( r1 ˆ2 g/N1 ; ∗ ∗ ∗ length pi ( dd1 / 2 ) ˆ 2 ; ∗ i =1: phith1(i)=Delta1 phithT(i)=phithT(i)/TotEnergy(i ); phiST(i)=phiST(i)/TotEnergy(i ); S0= end pi −−−−−−−−−−−−−−−−−−−−− Delta1=E1 phi1=ifo .Suspension.MaragingSteel.Phi; beta1=ifo .Suspension.MaragingSteel.dlnEdT; alpha1=ifo .Suspension.MaragingSteel.Alpha; for tau1=0.0737 rho1=ifo .Suspension.MaragingSteel.Rho; phiT=phiST+phithT ; S1= dd1=ifo .Suspension.Stage(3).WireDiameter; LL1=ifo .Suspension.Stage(3).Length; E1=ifo .Suspension.MaragingSteel.Y; end end C1=ifo .Suspension.MaragingSteel.C; T1=(m1+m) N1=ifo .Suspension.Stage(3).NWires; K1=ifo .Suspension.MaragingSteel.K; m1=ifo .Suspension.Stage(3).Mass; % Marionette and RM stages parameter are loaded. % Thermoelastic contribution % VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 64 of 78 09 p . . . − ∗ L) ∗ 0.15A (p − sin S/Tˆ 2 ) ) ) ; ∗ ∗ rho S/Tˆ 2 ) ) ) ; ∗ ∗ rho ∗ w( i )ˆ2 ∗ L)+lambdaˆ2 ∗ II (p ∗ w( i )ˆ2 ∗ E ∗ II cos ∗ ∗ E (1+4 ∗ sqrt (1+4 ∗ ( lambdaˆ3 ∗ sqrt (T+T ∗ p ∗ ∗ w01 ) ; T+T II) ∗ − ∗ ( E ∗ lambda ∗ ∗ II) II ∗ %Cylindrical fibres ∗ E (1/(2 phiT ( i ) ) ; N1/M1/g/LL1ˆ2); ∗ E phi1 to t ) ; ∗ ∗ ∗ (w) , ∗ (1+ d i l 1 ) ; I1 phiT1+w/(Q1 ∗ sqrt ∗ ∗ (1/(2 (1+1 i (E1 ∗ (1+1 i (dd1/2).ˆ4; ( k0p1/M1) ; length ∗ g/LL1 ∗ Optcyl==0, sqrt ∗ /4 sqrt lambda= p= Kpend( i )=4 E=E0 i f i =1: sqrt pi d i l 1= phi1tot=dil1 I1= for phiT1=phith1+phi1 ; k0p1=M1 kp1=k0p1 w01= end Q1=ifo .Suspension.Stage(3).Qvh; M1=m+m1; Q=ifo .Suspension.Stage(1).Qvh; % Dilution factors for marionette and RM stages % Gravitational reaction constants % sec . 2 % Horizontal viscous Q’s % Total loss angle internal+viscous % Reaction constants for mirror stage fused silica fibres VIR VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 65 of 78 spring system − 09 − 0.15A − %viscous dissipation is added p ˆ 2 ) ; ∗ (Kpend ) ) ) ; L) ∗ (p real %horizontal suspension thermal noise PSD ( sin − abs L) ∗ ∗ (p Q). ∗ 4/3000ˆ2; sin ∗ ∗ p ˆ 2 ) . . . 3(f(i),phiT(i),ifo); ∗ L) ∗ suspn (p p+lambdaˆ2 Xh=B. The admittance is calculated. VIR (Kpend)+w./(w0 cos ∗ ∗ ∗ (admitt)./w.ˆ2 L) ∗ w. ˆ 2 ; ( f ) ] ) ; ∗ imag ( (p real ∗ ds ) ; m1 ∗ ∗ − cos v w. ˆ 2 ; ∗ ∗ length Fend/delta ; L)+lambda m 1.3 ∗ ∗ Temp − (w) , ∗ − % 0.4Hz is the resonance frequency of the marionette stage due to the anti (p Xh ( 2 , : ) ; (Kpend(1))/m); kb ∗ 1.2 lambda ∗ Kpend ; Kpend ; ( [ 2 , 2 , (Kpend)+1 i (1 + mu sin ∗ w. − − − 0 . 4 ; 2 ∗ ∗ ∗ % Optimized fibres ∗ real length − ( real pi /( h = 4 [z(i ,:) x(i ,:) Fend]=An ∗ zeros Kpend( i )=4 +pˆ3 i =1: else end Xh(:,i)=linsolve(Ah(: ,: , i),B’); sqrt v=2/r ; n o i s e for phiv=phi0 admitt=1 i Kpend= mu wv1=2 end end w0= Ah = Ah(1 ,1 ,:)=kp1+KpendAh(1 ,2 ,:)= Ah(2 ,1 ,:)= Ah(2 ,2 ,:)=Kpend B=[0 1 ] ; % sec . 1.1 % Thermal% noise in a is martix obtained formalism from Ah FDT. The equations of motion are written % VERTICAL MOTION %Surface contribution for circular cross section fibres (vertical motion) %Vertical resonance frequency for mirror wires (from Hook’s law) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 66 of 78 09 − 0.15A − %vertical suspension thermal noise PSD 4/3000ˆ2; ∗ v ) . /w.ˆ2 %vertical coupled to horizontal noise Xv=B. The admittance is calculated. VIR v ; ( admitt ∗ % Total th noise PSD ( f ) ] ) ; real n o i s e ∗ ∗ c v ; w. ˆ 2 ; phiv1 ) ; ∗ ∗ wv1 ) ; ∗ Temp m1 phiv ) ; length ∗ wv ) ; − ∗ ∗ Xv ( 2 , : ) ; w. ˆ 2 ; (w) , ∗ kb ∗ S0/L/m) ; ∗ (1+1 i m ∗ ∗ w. − h+n o i s e ∗ kv ; kv ; ( [ 2 , 2 , (1+1 i E0 M1 − − ∗ ∗ ∗ length m (4 ∗ v=1 i cv = thetaHVˆ2 v = 4 zeros i =1: Xv(:,i)=linsolve(Av(: ,: ,i),B’); sqrt n o i s e noise=noise n o i s e admitt for phiv1=phi1+w/(Qv1 phiv=phiv+w/(Qv kv1=wv1ˆ2 kv=wvˆ2 end end Qv1=ifo .Suspension.Stage(3).Qvv; wv= Av(1 ,2 ,:)= Av = Av(1 ,1 ,:)=kv1+kv Av(2 ,1 ,:)= Av(2,2,:)=kv Qv=ifo .Suspension.Stage(1).Qvv; B=[0 1 ] ; % sec . 1.5 % Thermal% noise in a is martix obtained formalism from Av FDT. The equations of motion are written %Vertical viscous Q’s % Total loss angle internal+viscous % Elastic constants for vertical motion VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 67 of 78 09 s e c t i o n 2 . 1 . − 015A − Pa ] { 3(f,phi,ifopar) % Number of suspension’s fibres AdV.m calls this function two times: % FS density [kg/mˆ3]% Mirror mass [kg] suspn % Gravitational field [N/kg] % Fused Silica Young Modulus phi ) ; ∗ [z x Fend Energy]=An r1 ˆ 4 ; ∗ r1 ˆ 2 ; /4 ∗ (1+1 i ∗ pi pi ifo=ifopar ; I1= r1=d1 / 2 ; r2=d2 / 2 ; r3=d3 / 2 ; rho=ifo .Suspension. Silica .Rho; function g=ifo .Constants.g; d1=ifo .Suspension.Stage(1).WireDiameter1;d2=ifo .Suspension.Stage(1).WireDiameter2;d3=ifo .Suspension.Stage(1).WireDiameter3; S1= L1=ifo .Suspension.Stage(1).Length1;L2=ifo .Suspension.Stage(1).Length2;L3=ifo .Suspension.Stage(1).Length3; E0=ifo .Suspension. Silica .Y; M Optimised ﬁbre Code E=E0 N=ifo .Suspension.Stage(1).NWires; % first% and , then, to find to the find energy the imaginary distribution reaction that costant allow of to the compute fibre. the energy weighted loss angle % INPUT:%OUTPUT: frequency, z% coordinate In loss% this fibres) along angle, routine the is list the fibre analytically elastic of axis, parameter equation solved fibre according for bending, an with optimized force VIR at fibre the (3 end segments of the fibre , bending energy distribution % For any frequency value, suspR m=ifo .Suspension.Stage(1).Mass; VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 68 of 78 09 − 015A − I1 ) ) ; I2 ) ) ; I3 ) ) ; ∗ ∗ ∗ E E E ∗ ∗ ∗ I1 ) ) ; I2 ) ) ; I3 ) ) ; ∗ ∗ ∗ E E E ∗ ∗ ∗ S1 ) ) / ( 2 S2 ) ) / ( 2 S3 ) ) / ( 2 ∗ ∗ ∗ rho rho rho S1 ) ) / ( 2 S2 ) ) / ( 2 S3 ) ) / ( 2 ∗ ∗ ∗ ∗ ∗ ∗ rho rho rho wˆ2 wˆ2 wˆ2 ∗ ∗ ∗ ∗ ∗ ∗ I1 I2 I3 ∗ ∗ ∗ wˆ2 wˆ2 wˆ2 ∗ ∗ ∗ E E E ex1 0p10000000 0]; ∗ ∗ ∗ I1 I2 I3 ∗ ∗ ∗ ∗ E E E ∗ ∗ ∗ (Tˆ2+4 (Tˆ2+4 (Tˆ2+4 L1 ) ; L3 ) ; (Tˆ2+4 (Tˆ2+4 (Tˆ2+4 sqrt sqrt sqrt ∗ ∗ ( (T+ ( (T+ ( (T+ sqrt sqrt sqrt L1 ) ; L1 ) ; L2 ) ; L2 ) ; L3 ) ; L3 ) ; % shift at the end of the fibre lambda1 lambda1 ∗ ∗ ∗ ∗ ∗ ∗ T+ T+ T+ 3; lambda1 lambda3 % Fibre tension − − − − ( 1 2 , 1 2 ) ; r2 ˆ 4 ; r3 ˆ 4 ; sqrt sqrt sqrt − − − f ; ( p1 ( p2 ( p3 ( p2 ( p3 ( p1 (( (( (( ∗ ∗ ( ( r2 ˆ 2 ; r3 ˆ 2 ; ∗ /4 /4 ∗ ∗ sin sin sin pi cos cos cos g/N; exp exp ∗ sqrt sqrt sqrt pi pi pi pi zeros ∗ d e l t a=1e I2= I3= sn2= sn3= sn1= ex3= ex1= cn1= cn2= cn3= lambda2= lambda3= lambda1= S2= p1= p2= p3= S3= w=2 T=m AA(1,:)=[1 ex1 1 0AA(2 ,:)=[ 0 0 0 0 0 0 0 0]; AA= % eq1: x1(0)=0 % eq2: x1’(0)=0 %the solutions%eq(50)).%Twelve coeff for%in equation the the is are 3 diameter a segments needed. vector discontinuity contain They that are 4 contains the integration points. 4 the boundary twelve constants conditions integration each and (see the constants. VIR 8 joining conditions (x, x’,force, moment) VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 69 of 78 p3 ˆ 3 ] ; ∗ p3ˆ2 0 ] ; ∗ p2ˆ3 0 0 0 0 ] ; ∗ p2ˆ2 0 0 0 0 0]; ∗ ex3 I3 ex3 0 I3 ∗ ∗ lambda3 ˆ2 lambda3 ˆ3 ∗ ∗ I3 I3 lambda2ˆ2 0 I2 lambda2ˆ3 0 0 I2 ∗ − − ∗ I2 − p3 ] ; − sn1 ) cn1 ) I2 ∗ ∗ lambda3ˆ2 lambda3ˆ3 ∗ ∗ I3 ex3 0 p1ˆ2 p1ˆ3 cn3 ] ; ∗ p2 0 0 0 0 ] ; − − − ∗ ( ( − ∗ ∗ z1 ) ) . . . sn2 cn2 ) I3 − ∗ ∗ (L1 sn3 p3 lambda3 ∗ sn1 I1 ∗ − cn1 ) I1 p2ˆ2 p2ˆ3 ∗ ∗ p3 ∗ − ( − I2 ∗ p1ˆ3 − ∗ p1ˆ2 lambda1 − ( − cn2 cn1 lambda2 0 0 ∗ ( sn2 I2 ∗ ∗ 1 0 ] ; ∗ cn2 lambda3 exp − ∗ ∗ p2ˆ2 ex3 lambda3 ∗ p2ˆ3 sn1 p1 ex3 ∗ ∗ I2 ∗ − lambda1ˆ3 I1 − 1 sn2 p2 ∗ p1 ∗ − − 1 0 0 0 0 0 ] ; lambda1ˆ2 I1 ∗ p2 − lambda3 − z1)+coef(2) − ∗ 1 0 ex1 ) I1 ∗ − ex1 I1 lambda2ˆ3 I2 lambda2ˆ2 ∗ ∗ ∗ ex1 lambda1 lambda1 ∗ − ( lambda1 ˆ3 exp − lambda1 ˆ2 ∗ ( (0,L1,1000); ∗ ∗ lambda1 − linspace coef=linsolve (AA,BB’); z1= x1 = c o e f (1) BB=[0 0 0 0 0 0 0 0 0 0 0 delta]; AA(9,:)=[0 0 0 0 0 lambda2 AA(8,:)=[0 0 0 0 0 I2 AA(3,:)=[ex1 1 cn1AA(4 sn1 ,:)=[ AA(6 ,:)=[I1 AA(7,:)=[0 0 0 0 0 I2 AA(10,:)=[0 0 0 0 0 1 cn2 sn2 AA(11,:)=[0 0 0 0AA(12,:)=[0 0 0 0 0 0 0 0 0 0 0 0 ex3 1 cn3 sn3]; AA(5 ,:)=[I1 % eq8: F2(L2)=F3(0) % eq3: x1(L1)=x2(0) % eq4: x1’(L1)=x2’(0) % eq6: M1(L1)=M2(0) % eq7: M2(L2)=M3(0) % eq5: F1(L1)=F2(0) % eq9: x2’(L2)=x3’(0) % eq10: x2(L2)=x3(0) % first fibre ’s segment % eq11: x3’(L3)=0 % eq12: x3(L3)’=delta %Force VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 70 of 78 cn3 ; ∗ p3 ˆ3) − z3 ) ) . . . ( z1 ) ) . . . z2 ) ) . . . − ∗ − − (L3 (L1 (L2 ∗ ∗ ∗ lambda3 sn3+coef(12) lambda1 lambda2 − ∗ ( − − ( ( exp p3ˆ3 ∗ exp exp ∗ ∗ ∗ z3 ) ; z3 ) ) . . . ∗ z1 ) ; z2 ) ) . . . z2 ) ; − ∗ ∗ − ( p3 (L3 (L2 ( p1 ( p2 ∗ ∗ sin lambda3 ˆ2 lambda1 ˆ2 lambda2 ˆ2 ∗ ∗ sin sin ∗ ∗ ∗ ∗ lambda3 p3 ˆ2) lambda2 − − p1 ˆ2) p2 ˆ2) ( − ( z3 ) ; ( ∗ − − lambda3ˆ3+coef (11) ∗ ( ( z1 ) ; z2 ) ; exp ∗ ∗ ∗ ∗ ∗ exp ∗ ( p3 z3)+coef(10) z1)+coef(2) z2)+coef(6) ∗ ∗ ∗ ∗ ( p1 ( p2 )]; sin %Energy distribution along the fibre ∗ sin sin end ∗ ∗ z3)+coef(12) lambda3 lambda1 lambda2 z1)+coef(4) z2)+coef(8) ∗ − − − ex3+coef(10) ∗ ∗ ( ( ( z2)+coef(6) z3)+coef(10) )+z1 ( ∗ ( p3 ∗ ∗ %bending energy E=1/2 E I x’’ˆ2 %bending energy E=1/2 E I x’’ˆ2 ( p1 ( p2 exp exp exp %bending energy E=1/2 E I x’’ˆ2 ∗ ∗ ∗ end cos z3)+coef(12) cos cos ∗ z1)+coef(4) z2)+coef(8) ∗ ∗ ∗ ∗ ∗ lambda2 lambda3 ( p3 x32 . ˆ 2 ; x12 . ˆ 2 ; x22 . ˆ 2 ; − − ( p1 ( p2 lambda3 ˆ3) ∗ ∗ ∗ p3 ˆ2) ( ( ) z3+z2 ( p1 ˆ2) p2 ˆ2) %Force at the end of the fibre − − I3 I1 I2 cos lambda1 ˆ2 lambda2 ˆ2 lambda3 ˆ2 ( ( − − cos cos ∗ ∗ ∗ exp exp ∗ ∗ ∗ ∗ ∗ ∗ ( ( end ∗ ∗ ∗ ∗ E E E ∗ ∗ (0,L2,1000); (0,L3,1000); x33 ; ∗ ∗ ∗ ∗ I3 ∗ E − +c o e f (3) +c o e f (7) +c o e f (7) +c o e f (11) +c o e f (11) +c o e f (3) linspace linspace z2= z3= z=[z1 z2+z1( Energy3=0.5 Energy1=0.5 Energy2=0.5 Energy=[Energy1 Energy2 Energy3 ]; x12 = coef(1) x22 = coef(5) x32 = coef(9) x33 = coef(9) x2 = c o e f (5) x3 = c o e f (9) Fend= x=[x1 x2 x3]; end % second fibre ’s segment % third fibre ’s segment VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 71 of 78 parameter of the ETM parameter of the ITM − − % Pressure inside the vacuum % Mean speed of Gas % Radius% Radius of curvature of curvature of ITM of% number ETM density of Gas g2 ) ˆ 2 ) ) ; ∗ g1 ∗ 2 % first% second resonator resonator g % g gaussian% Rayleigh beam waist range size % location% location of ITM of relative ETM relative to to the the waist waist − L; L; ∗ g2))/((g1+g2 ∗ ∗ g1 g2 ) ) g2 ) ) ∗ − ∗ g1 (1 ∗ g1 ∗ 2 ∗ 2 − g2 ) − ∗ ; pi ( ( ( g1 adv(f, ifo, mol) T) / (M) ) ; sqrt g1))/(g1+g2 ∗ g2))/(g1+g2 ∗ k T); − ( waist ) ; ∗ ∗ − Lambda)/ (1 ∗ ∗ (1 ((2 ∗ waistˆ2/Lambda; sqrt n = gas (L/R1 ) ; (L/R2 ) ; ∗ ((g2 − − − sqrt pi zr = waist = waist waist = (L waist = alpha = ifo.Infrastructure.ResidualGas. polarizabilities(mol); rho = (P) / ( k z1 = z2 = ( ( g1 g1 = 1 g2 = 1 v0 = k = ifo.Constants.kB; Lambda = ifo .Laser.Wavelength; R1 =ifo .Optics.Curvature.ITM; R2 =ifo .Optics.Curvature.ETM; L = ifo.Infrastructure.Length; P = ifo .Infrastructure.ResidualGas.pressures(mol); T = ifo .Constants.Temp; % The% exponential can be integrated of Eq. 1 analytically of P940008 is expanded to first order; this M = ifo .Infrastructure.ResidualGas.masses(mol) ; function N Residual Gas Code % The% following residual% interferometer. function% gas Micheal% molecules models Pathlength E.% Zucker, Added The the through% method Noise to noise Cleaned and the Bench used% in Stanley spectrum substituted up by laser here Sensitive by Zhigang E. caused PF, is beams% Whitcomb This Apr Pan, Interferometers first presented by in% in 07; Summer function The the the 2006 their eliminated modifications order by arms passage is Due Rainer of expansion paper to of numerical based Weiss, the have Optical Residual of on been integration exp, the Gas. performed standard to and by speed ’gas.m’ Giancarlo it function up Cella. form 12/05/2009. gwinc v1. VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 72 of 78 (z1ˆ2 + zrˆ2)); sqrt t z2: %8.3f’,waist,z1,z2)); \ ( z1 + log − t z1 : %8.3 f \ alpha)ˆ2)/v0); ∗ pi ∗ f /v0 ; (z2ˆ2 + zrˆ2)) (2 ∗ ∗ L ∗ rho sqrt pi ∗ ∗ 2 ((4 zr / waist ; 0) = 0 ; − ∗ ∗ ( z2 + < log z i n t /Lˆ 2 ; ∗ z i n t = z i n t z i n t = z i n t =z i n t z i n t = z i n t z i n t ( z i n t n = 2 % optical path length for one arm % eliminate any negative values due to first order approx. % account for both arms & turn into strain noise power % disp(sprintf(’waist: %8.5f, VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 73 of 78 % Loss in PD Process [Power] % Mirror mass [kg] % Mismatch % Mismatch % BS Loss [Power] % [BnC, table 1]% Laser Length% SRC angular of Length% arm ITM frequency Transmittance [m] cavities [rads/s] [Power] [m] % SRM Reflectivity% [amplitude] SRC Detunning% Homodyne Readout% phase [BnC,% between [BnC, between 2.14 & 2.14 2.15] & 2.15] SR Detuning SRC one pass phase shift % Laser% Wavelength Plancks% SOL Constant [m] [m/ s% ] [BnC, [Js] table 1] Signal angular frequency [rads/s] % SR cavity loss [Power] % SRM Transmittance [amplitude] SR); ); lambda pi ∗ − tau ˆ2 − ifo .Optics.PhotoDetectorEfficiency; Omega/c , 2 ifo .Optics.coupling; f ; c/lambda ; ∗ ( ifo .Optics.SRM.Transmittance); (1 ds ) / 2 ; ∗ ∗ − − − pi pi pi ∗ ∗ sqrt sqrt n = shotrad(f, ifo) SR = mismatch + bsloss; PD = 1 0 = 2 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− l = ifo.Optics.SRM.CavityLength; bsloss = ifo.Optics.BSLoss; eta = ifo.Optics.Quadrature.dc; phi = ( tau = rho = ds = ifo.Optics.SRM.Tunephase; hbar = ifo.Constants.hbar; c = ifo.Constants.c; function Phi = mod( l omega mismatch = mismatch + ifo .TCS.SRCloss; mismatch = 1 lambda = ifo .Laser.Wavelength; lambda lambda O Quantum (optical read–out) noise Omega = 2 L = ifo.Infrastructure.Length; T = ifo.Optics.ITM.Transmittance; % BSloss + mismatch has been incorporated into a SRC Loss % % % % % Quantum% noise All% model Updated references to%f to include Buonanno & losses Chen PRD 64 DEC 2006 042006 Kirk (2001) McKenzie (hereafter using BnC notation BnC) %SignalFreq.[Hz] m = ifo.Materials.MirrorMass; VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 74 of 78 ... ∗ ( phi)+ . . . ( phi ) ) + . . . phi ) ) ) ) ; phi ) ) + . . . ∗ ( phi ) ) + . . . ∗ (2 cos cos (1+rho ˆ2) ... ( phi)+ . . . (2 − prfactor) [W] sin ∗ ∗ ∗ sin ∗ ∗ sin sin ∗ ∗ ∗ ) . ˆ 2 ) . ˆ 2 phi ) ) rho ) ) rho + 4 ∗ ∗ beta ( (2 ∗ beta ( rho ) ) 1+5 ∗ − sin cos ( cos ∗ 4 ∗ ∗ (1+rho . ˆ 2 ) ∗ ) ) . ˆ 2 (phi).ˆ2 )+... phi)+Kappa/2 (1+5 ) ∗ ∗ ∗ − ) ( phi ) + . . . (2 sin ( phi ) + . . . ∗ beta (phi).ˆ2 ) + ... ∗ beta i cos cos ∗ beta i ( phi ) sin ∗ Kappa ∗ i ( ( phi ) + 4 cos ∗ ∗ (2 ∗ ∗ ∗ ∗ (2 sin ∗ (2 sin ∗ PRD2001] Arm cavity half bandwidth [1/s] )) exp − )) ∗ exp )). phi) + Kappa/2 Kappa exp (1+ ∗ − phi ) + Kappa beta Kappa ) + ( phi ) ) ; ( phi ) ) ; (2 ∗ ∗ beta ∗ ∗ i ) + ∗ beta i 2 phi) (2 ∗ Kappa ∗ (phi).ˆ2) ); sin cos cos 2 (1+rho ˆ2) ∗ ∗ − % [BnC, Table 1] Power at% [BnC, BS% (Power after [BnC, 2.12] 2.11] SQL Phase Strain shift of GWSB in arm % [BnC,% after [KLMTV % [BnC, 5.2]% after [BnC, Round 5.3] Trip 5.2] loss Loss Loss% Parameter in [BnC, coefficent arm 2.14] [Power]% Power [BnC for 2.13] to arm reach cavity Effective free Radiation mass SQL Pressure Coupling )). (2 ∗ beta ∗ ( ( (2 (2 sin ∗ ∗ ∗ ∗ beta i ( i cos ∗ i ( ∗ ∗ ∗ exp ∗ )). sin ∗ beta exp ∗ 1/2 (4 rho rho ∗ ∗ exp − i (phi).ˆ2) ); (4 − ( (( ∗ ∗ 0 ) ; ∗ ac ) . ˆ 2 ) ; beta exp ) ) (2 sin rho ∗ ∗ ∗ ∗ exp 1+rho i ∗ ∗ ∗ − e p s i l o n ) (1+rho ∗ ( rho ∗ exp L ) . ˆ 2 ) ) ; L/c ) ; (2 ∗ rho omega ∗ ∗ beta beta − − ∗ ∗ ∗ ∗ ac ac ˆ 4 ) . / . . . i i ( (1+rho ˆ2) ( ( phi ) + Kappa tau ˆ2 tau ˆ2 1/4 beta 2 ; phi ) + ( 3+ ∗ exp ∗ ∗ ∗ ac ) ; i ∗ ( (3+ ∗ ∗ ∗ ∗ ∗ ∗ − acˆ2+Omega . ˆ 2 ) ) ; (2 (2 (2 ∗ (Omega ) (2 ∗ ∗ phi)+Kappa (2 3+rho+2 ( (3+ sin gamma gamma exp exp ∗ − PD) PD) PD) PD) PD) ac ˆ4)/(4 cos ∗ ∗ − ( ( 3+rho+2 (2 ∗ ∗ ∗ beta exp ∗ ( ∗ ( ( phi ) ∗ ∗ ( gamma (2 sin ∗ ∗ SR SR SQL) ) . ˆ 2 ∗ L); ( hbar . / (m cos lambda lambda lambda lambda lambda gamma ∗ arm /(2 ( phi ) cos ∗ ∗ ∗ SR − − − − − ∗ ∗ 0 /I SR (Omega . / gamma (8 beta (1 (1 (1 (1 (1 L; ( sin c /(4 Lˆ2 e p s i l o n epsilon ./(1+(Omega/gamma e p s i l o n rho ∗ ∗ ∗ lambda lambda SR ((I ∗ ∗ ∗ ∗ ∗ ∗ ∗ sqrt sqrt sqrt sqrt sqrt sqrt 2 cos atan e p s i l o n (Omega . ˆ 2 . lambda 1/2 1/4 1/4 1/2 lambda ∗ ∗ = arm = ifo.Optics.Loss ac = T e p s i l o n 1/2 1/2 lambda ∗ ∗ L = L = C11 L = L = L = L = 0 = ifo.gwinc.pbs; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− SQL = SQL = (m 1/2 1/2 epsilon = lambda I Epsilon = 2 lambda I beta C21 C12 C11 C22 gamma D1 D2 Kappa = 2 h % % Coefficients% [BnC, Equations 5.8 to 5.12] % % VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 75 of 78 ( phi ) + ...... ( phi ) ) ) ) ; ... ∗ ( phi ) ) ; ( phi ) ) ; sin ∗ ∗ sin )). )) phi ) ) + . . . sin cos ∗ )). phi ) ) + . . . ∗ ∗ ∗ ∗ (2 (2 beta beta ∗ ∗ beta i sin ∗ Kappa ( i sin ∗ ∗ − ... ( i ∗ ∗ (2 (... ( phi ) ) ; exp ). ∗ ( phi)+Kappa ( phi ) + . . . ∗ exp ( phi)+Kappa ( phi) (eta)).ˆ2+... ∗ exp sin ∗ cos cos ∗ <<<<<<<<<<<<<<<<< ) sin cos cos beta ( ∗ (eta)).ˆ2+... (eta)).ˆ2+... ∗ ∗ )+rho ∗ ∗ ∗ i phi)+Kappa )+rho ... ). phi)+Kappa (2 (2 ∗ ∗ ∗ phi ) ) ) ; cos cos beta ∗ ∗ ∗ (1+ rho ( phi ) ) ; (2 (2 ∗ ∗ beta ∗ ∗ ... (2 ∗ beta ∗ ∗ i (eta)).ˆ2). (2 ∗ beta (1+ rho ) i (2 rho cos cos ∗ ∗ exp − ( phi ) ( phi ) cos ∗ ( ∗ − ( i ( eta)+P21 sin cos )+ ( ∗ sin tau ∗ ∗ SR )). (Kappa. (2 sin cos ∗ exp L ∗ phi)+ ∗ sin ∗ ∗ ∗ Kappa exp exp − ( ( eta)+Q21 ( eta)+N21 ∗ ∗ beta ∗ ( − ∗ ∗ ∗ ( (2 tau tau rho beta ∗ Kappa (2 sin sin tau rho ∗ ∗ ∗ − − SR) ( P11 tau tau ∗ ∗ lambda ) ∗ − ( i tau cos ∗ ∗ ∗ cos ∗ ∗ SR) SR) ( eta)+D2 ) phi) ∗ abs 1/2 (Q11 (N11 phi)+Kappa beta exp ∗ ( phi) ∗ ∗ ∗ − i (2 sin rho ( lambda (2 ∗ ∗ beta abs abs ∗ 2 cos ∗ L i e p s i l o n ) e p s i l o n ) (2 cos − ∗ e p s i l o n ) ( lambda ( lambda ) phi )) cos ∗ ∗ )+rho ∗ ∗ 2 2 (epsilon/2) sqrt ∗ ∗ ∗ (D1 (2 (2 rho exp − − (eta)).ˆ2+...( eta )).ˆ2+ (2 (2 ∗ ∗ ) ( ∗ 2 sqrt sqrt )+2 beta beta sqrt ∗ ∗ abs − i i sin sqrt sqrt cos cos ∗ ∗ exp ∗ ( eta )).ˆ2+ ( ( ( eta )).ˆ2+ (eta)).ˆ2); sqrt ∗ ∗ ∗ ∗ beta ∗ ∗ − − ... PD) ∗ beta ( ( ∗ L L ∗ i ∗ ∗ i cos cos cos )+rho ˆ2 ∗ PD) PD) PD) PD) ( phi ) ; ( phi ) ; ∗ ∗ ∗ exp exp ∗ PD) PD) (2 tau ˆ2. ( ( Kappa (2 rho ∗ ∗ ∗ ∗ PD) sin cos ∗ beta lambda ). ). ∗ exp ∗ ∗ − i ) ) ∗ exp ∗ ∗ lambda lambda lambda lambda (1 2 ( eta)+C21 ( eta)+C22 lambda lambda Kappa − − − − phi )+3 rho beta beta ∗ − − − ( ( eta)+P22 ( ( eta)+Q22 ( eta)+N22 rho beta beta ( ∗ (1 (1 (1 (1 ∗ e p s i l o n ( ( ( lambda sin sin ∗ sqrt (1 (1 2 (2 ∗ ∗ ∗ ∗ UNU NOISE DENSITY SPECTRAL POWER QUANTUM [BnC, 5 . 1 3 ] ( 2 sin sin sin cos cos − exp L L sqrt sqrt sqrt sqrt cos cos ∗ ∗ ∗ ∗ ∗ ( ( sqrt sqrt sqrt 1/2 2 2 cos − − − ∗ 2 ( C11 ( P12 ( C12 (Q12 (N12 SQL . ˆ 2 . / ( 2 abs abs abs abs abs −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− >>>>>>>> P21 = P11 = 1/2 P22P12 = P11 ; = n = h N11 = N12 = N22 = N21 = Q22Q12Q21 = Q11 ; = 0 ; = 0 ; Q11 = % % % VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 76 of 78 asym ( . . . ROC % Kg ... ∗ ifo .Materials.MassThickness; ∗ (ifo .Optics.Diameter.ITM/2/ifo .Optics.ITM.BeamRadius).ˆ2); ∗ 2 − ( exp ifo = precompIFO(ifo , PRfixed) ifo . Materials .MassRadiusˆ2 ifo . Infrastructure .Length, ifo .Optics.Curvature.ITM, ifo .Optics.Curvature.ETM, ifo .Laser.Wavelength); ∗ isfield(ifo , ’gwinc’) && isfield(ifo.gwinc, ’PRfixed’) && ... ifo .gwinc.PRfixed == PRfixed ifo .Materials.Substrate.MassDensity; pi return ifo .gwinc.PRfixed = PRfixed; ifo.Materials.MirrorMass = ...ifo .Optics.ITM.Thickness = ifo .Materials.MassThickness; ifo .Materials.MirrorMass = ifo .Materials.MirrorMass i f clippingloss .ITM = [waist , ifo .Optics.ITM.BeamRadius, ifo .Optics.ETM.BeamRadius,waistpos] = OSD end % check DERIVED OPTICS VALES PRfixed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculate optics ’ parameters % Use a function to% compute Uncomment beam% if disp(sprintf([’The diameter you want (this% Calculate to% is see 2010) arm cavity a clipping the good calculated waist for losses ”near is beam from %8.5f concentric” parameters. finite and is cavities) mirror at coating %8.2f’],waist,waistpos)); diameter (RLW May function P Miscellanea of computations % ifo% = precompIFO(ifo% add% , precomputed To PRfixed) prevent% ifo data% recomputation the argument to% the argument contains% of IFO PRfixed, (mevans model these ifo.gwinc.PRfixed, June no precomputed 2008) changes data, are and made. this if the matches VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 77 of 78 x ) asym(L,Rc1,Rc2,lambda) mevans 2 May 2008 − f ; ROC x ; ( darmseis x ; (ifo . Optics .Diameter.ETM/2/ifo . Optics .ETM.BeamRadius).ˆ2); isempty ∗ 2 | | − ( f = d a r m s e i s x = darmseis f ) exp f darmseis besselzeros [w0, w1, w2, z0] = OSD (besselzeros) ( d a r m s e i s s e i s m i c besselzeros = besselzero(1, 300, 1); load besselzeros; d a r m s e i s try catch load end % load saved values, or just compute them function ifo.Constants.BesselZeros = besselzeros; ifo .Optics.ITM.CoatLayerOpticalThicknessifo .Optics.ETM.CoatLayerOpticalThickness =ifo.gwinc.pbs getCoatDopt(ifo = getCoatDopt(ifoifo.gwinc.finesse ,ifo.gwinc.prfactor = , ’ITM’); pbs; ifo ’ETM’); .Optics.PRM.Transmittance = finesse; = prfactor; = Tpr; ifo.Optics.Loss = ifo.Optics.Loss + (clippingloss.ITM + clippingloss.ETM)/2; ifo .Seismic.darmseis ifo .Seismic.darmseis i f isempty i f isempty clippingloss .ETM = [pbs, finesse , prfactor , Tpr] = BSPower(ifo , PRfixed); global global end end % load seismic info % compute LOAD SAVED DATA power on BS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % precompute bessels zeros for finite mirror corrections % coating layer optical thicknesses VIR-0073D-12 issue :D AdV sensitivity date : November 19, 2012 page : 78 of 78 % giving the input mirror a negative curvature L); ∗ Rc2+2 − ); pi z1 ) ; ∗ z1 ) (1 +(z1/zr)ˆ2);(1 +(z2/zr)ˆ2); Lˆ2)/( Rc1 − lambda/ ∗ − sqrt sqrt L ( ( Rc1 ( zr ∗ ∗ ∗ z1 ; Rc1 ; − sqrt sqrt − zr = z2 = L + z1 ; z1 = ( Rc2 z0 = Rc1 = w0 = w1 = w0 w2 = w0