ITM noise only; Silica; 290 K; 30kg

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30kg Silica mirror, Silica suspension (d=550um)

ϕw and ϕm are derived later.

a=14.75cm, h=20cm, w=3.53153cm

14.75^2 π 20 2200/10^6.

1.Parameters (for LCGT)

1.1 Physical constants

hb=1.05457266912510183 (10^(-34));(* reduced Planck constant (J*s) *)
ω0=1.8 (10^15);
c=2.99792458000000117`*^8;(* speed of light (m/s) *)
Ω=2 π f;
kB=1.38*10^(-23);(* Boltzmann's constant (J/K) *)
g=9.8;(* acceleration of the gravity (m/s^2) *)
L=3000;

1.2 Laser

lambda=1064*10^(-9);(* wavelength (m) *)

1.3 Main cavity

l=3*10^3;(* length of cavity (m) *)
w0=3.53153*10^(-2); (* beam radius at mirrors (m); new *)

Mirror

rho=2200; (* density (kg/m^3) for silica *)
radius=14.75*10^(-2);(* radius (m) *)
height=20*10^(-2);(* height (m) *)
mass= rho*Pi*radius^2*height;(* mass (kg) *)
Tm=290;(* temperature of mirror (K) *)
alpha300=5.1*10^(-7);(* thermal expansion coefficient of silica at 300 K(/K) *)
Cth300=1.64 10^6/rho;(* specific heat of silica at 300 K(J/(m^3)*K) *)
kappa300=1.38;(* thermal conductivity of silica at 300 K (W/m*K) *)

T300=290;

1.6 Suspension

Parameters (m2 is the test mass)

kB=1.38*10^(-23);(* Boltzmann's constant (J/K) *)
ω=2π f;

m1=60;
m2=mass; 
m3=30;

mjoint=0.1; (* mini GAS joint *)

μ21=m2/m1;
μ31=m3/m1;

I1=(π/4)(r1^4);
I2=(π/4)(r2^4);
I3=(π/4)(r3^4);

r1=0.0006/2; (* This doesn't matter much *)
r2=0.00055/2; (* simply from strength *)
r3=0.0004/2;

Y1=161*(10^9); (* Tungsten *)
Y2=400*(10^9); (* Sapphire *)
Y3=130*(10^9); (* BeCu *)

N1=4;
N2=4;
N3=4;

g=9.8;

lsus1=0.5; 
lsus2=0.3; 
lsus3=0.3;

b1=2;
b2=2;
b3=2;

D1=((b1*N1(Sqrt[(m1 g/N1) Y1 I1]))/(2m1 g lsus1)); (* Saulson(1990); b is added. *)
D2=((b2*N2(Sqrt[(m2 g/N2) Y2 I2]))/(2m2 g lsus2));
D3=((b3*N3(Sqrt[(m3 g/N3) Y3 I3]))/(2m3 g lsus3));

ω1=(Sqrt[(g/lsus1)])(1+D1);
ω2=(Sqrt[(g/lsus2)])(1+D2);
ω3=(Sqrt[(g/lsus3)])(1+D3);

T1=290;
T2=290; (* temp *)
T3=290; (* temp *)

ϕp2=(10^(-4)); (* Tungsten *)
ϕp3=5*(10^(-6)); (* BeCu *)
ϕcl=(10^(-3));
ϕcl2=(10^(-3));

Q1=(((((ω1)/ω)ϕp2*D1+ϕcl))^(-1)); (* 1st term=structure damping, 2nd term=viscous damping *)
Q2=(((((ω2)/ω)ϕp*D2+0))^(-1));
Q3=(((((ω3)/ω)ϕp3*D3+0*(10^(-7))))^(-1)); (* Is it 1e-2 for magnetic damping? *)

Ksusp

lsus=lsus2; 
dwire=2*r2;
Ewire=Y2;
rho=4000;
mass=m2;

n=N2;

Icwire=I2;(* moment of cross section (m^4) *)

ksus=Sqrt[(-mass*g/n+Sqrt[(mass*g/n)^2+4*Ewire*(1+I ϕp)*Icwire*rho*Pi*(dwire/2)^2*(2*Pi*f)^2])/(2*Ewire*(1+I ϕp)*Icwire)];

kele=Sqrt[(mass*g/n+Sqrt[(mass*g/n)^2+4*Ewire*(1+I ϕp)*Icwire*rho*Pi*(dwire/2)^2*(2*Pi*f)^2])/(2*Ewire*(1+I ϕp)*Icwire)];

Ksusp=n*Ewire*(1+I ϕp)Icwire*((ke ks ((ke^2)+(ks^2))(Sinh[keL]ke Cos[ksL]+Cosh[keL] ks Sin[ksL]))/(2 ke ks(1-Cos[ksL] Cosh[keL])+((ke^2)-(ks^2)) Sin[ksL] Sinh[keL]))/.{ke->kele,ks->ksus,keL->kele lsus,ksL->ksus lsus};

Impedance matrix

D11=-(ω^2)+((ω1)^2)+μ21 Ksusp/m2+μ31 ((ω2)^2)+I ω(((ω1)/Q1)+μ31((ω3)/Q3)); (* changed *)
D12=-μ21*Ksusp/m2; (* changed *)
D13=-μ31 ((ω3)^2)-I ω(μ31((ω3)/Q3));
D21=-Ksusp/m2;  (* changed *)
D22=-(ω^2)+Ksusp/m2; (* changed *)
D23=0;
D31=-((ω3)^2)-I ω(((ω3)/Q3));
D32=0;
D33=-(ω^2)+((ω3)^2)+I ω(((ω3)/Q3));

Z=(1/(I ω))(({{m1*D11, m1*D12, m1*D13}, {m2*D21, m2*D22, m2*D23}, {m3*D31, m3*D32, m3*D33}}));

Thermal noise spectrum (PPP Eq.(69))

Tmat=(({{T1, 0, 0}, {0, T2, 0}, {0, 0, T3}}));

Fsq=4kB Re[Tmat.Z];

invD=Inverse[({{D11, D12, D13}, {D21, D22, D23}, {D31, D32, D33}})];

m[1]=m1;
m[2]=m2;
m[3]=m3;

Sth=(∑_(j=1)^3)((Fsq[[j]][[j]])/((m[j])^2))Abs[invD[[2]][[j]]]^2+((Fsq[[1]][[2]])/(m[1]m[2]))2Re[invD[[2]][[1]]*Conjugate[invD[[2]][[2]]]]+((Fsq[[2]][[3]])/(m[2]m[3]))2Re[invD[[2]][[2]]*Conjugate[invD[[2]][[3]]]]+((Fsq[[1]][[3]])/(m[1]m[3]))2Re[invD[[2]][[1]]*Conjugate[invD[[2]][[3]]]];

no violin (for reference)

D11noV=-(ω^2)+((ω1)^2)+μ21 ((ω2)^2)+μ31 ((ω2)^2)+I ω(((ω1)/Q1)+μ21((ω2)/Q2)+μ31((ω3)/Q3));
D12noV=-μ21 ((ω2)^2)-I ω(μ21((ω2)/Q2));
D21noV=-((ω2)^2)-I ω(((ω2)/Q2));
D22noV=-(ω^2)+((ω2)^2)+I ω(((ω2)/Q2));

ZnoV=(1/(I ω))(({{m1*D11noV, m1*D12noV, m1*D13}, {m2*D21noV, m2*D22noV, m2*D23}, {m3*D31, m3*D32, m3*D33}}));

FsqnoV=4kB Re[Tm.ZnoV];

invDnoV=Inverse[({{D11noV, D12noV, D13}, {D21noV, D22noV, D23}, {D31, D32, D33}})];

SthnoV=(∑_(j=1)^3)((FsqnoV[[j]][[j]])/((m[j])^2))Abs[invDnoV[[2]][[j]]]^2+((FsqnoV[[1]][[2]])/(m[1]m[2]))2Re[invDnoV[[2]][[1]]*Conjugate[invDnoV[[2]][[2]]]]+((FsqnoV[[2]][[3]])/(m[2]m[3]))2Re[invDnoV[[2]][[2]]*Conjugate[invDnoV[[2]][[3]]]]+((FsqnoV[[1]][[3]])/(m[1]m[3]))2Re[invDnoV[[2]][[1]]*Conjugate[invDnoV[[2]][[3]]]];

Vertical thermal noise

μt=m1/(m1+m2+m3);

ωv1=(Sqrt[(((2π 0.4))^2)/μt]);
ωv2=(Sqrt[((π (r2^2)Y2/lsus2)/(m2/4))]);
ωv3=(Sqrt[((π (r3^2)Y3/lsus3)/(m3/4))]);

Qv1=(((((Sqrt[m1/mjoint])/(2ωv2*((ωv1)^2) ω))*(((ωv1)/ω)*ϕp2+ϕcl2)+((ϕcl)/(ω*Sqrt[m1/mjoint]))))^(-1)); (* see minigas2.nb *)

Qv2=(((((ωv2)/ω)*ϕp))^(-1)); (* no visvous damping *)
Qv3=(((((ωv3)/ω)*ϕp3+0*(10^(-7))))^(-1)); 
VHC=1/200;

D11v=-(ω^2)+((ωv1)^2)+μ21 ((ωv2)^2)+μ31 ((ωv3)^2)+I ω(((ωv1)/Qv1)+μ21((ωv2)/Qv2)+μ31((ωv3)/Qv3));
D12v=-μ21 ((ωv2)^2)-I ω(μ21((ωv2)/Qv2));
D13v=-μ31 ((ωv3)^2)-I ω(μ31((ωv3)/Qv3));
D21v=-((ωv2)^2)-I ω(((ωv2)/Qv2));
D22v=-(ω^2)+((ωv2)^2)+I ω(((ωv2)/Qv2));
D23v=0;
D31v=-((ωv3)^2)-I ω(((ωv3)/Qv3));
D32v=0;
D33v=-(ω^2)+((ωv3)^2)+I ω(((ωv3)/Qv3));

Zv=(1/(I ω))(({{m1*D11v, m1*D12v, m1*D13v}, {m2*D21v, m2*D22v, m2*D23v}, {m3*D31v, m3*D32v, m3*D33v}}));

Thermal noise spectrum

Fsqv=4kB Re[Tmat.Zv];

invDv=Inverse[({{D11v, D12v, D13v}, {D21v, D22v, D23v}, {D31v, D32v, D33v}})];

Sthv=(∑_(j=1)^3)((Fsqv[[j]][[j]])/((m[j])^2))Abs[invDv[[2]][[j]]]^2+((Fsqv[[1]][[2]])/(m[1]m[2]))2Re[invDv[[2]][[1]]*Conjugate[invDv[[2]][[2]]]]+((Fsqv[[2]][[3]])/(m[2]m[3]))2Re[invDv[[2]][[2]]*Conjugate[invDv[[2]][[3]]]]+((Fsqv[[1]][[3]])/(m[1]m[3]))2Re[invDv[[2]][[1]]*Conjugate[invDv[[2]][[3]]]];

phi-p

ϕtherm=((Ewire (alpha300^2)T2)/(rho Cth300))((τd Ω)/(1+((τd)^2)(Ω^2)));
τd=(((dwire^2) rho Cth300)/(13.55*kappa300));
ϕbulk=4*(10^(-10));

ϕp=ϕbulk(1+8(ds/dwire))+ϕtherm/.ds->200*(10^(-6)); (* this ds is actually for silica *)

Sqrt[mass g/2/π/Ty]*2/.Ty->750.*10^6

LogLogPlot[{ϕbulk,ϕtherm},{f,1,100000}]

Coating

Ncoa1=9;

Ys=7.2*10^10;(* Young's modulus for silica (Pa=N/m^2) *)
Yt=1.4*10^11;(* Young's modulus for Ta2O5 (Pa=N/m^2) *)
σs = 0.17;(* Poisson ratio for silica *)
σt = 0.23;(* Poisson ratio for Ta2O5 or sapphire *)
n1=1.45; (* Refraction index of silica *)
n2=2.07; (* Refraction index of tantala *)
dcoa1s=(Ncoa1+1)/4*10^(-6)/n1;(* thickness of ITM silica coating (m) *)
dcoa1t=Ncoa1/4*10^(-6)/n2;(* thickness of ITM tantala coating (m) *)
ϕc1=4.0*10^(-5);(* loss angle of silica coating; new *)
ϕc2=2.0*10^(-4);(* loss angle of tantala coating; new *)

Cs=1.64 10^6;
κs=1.38/Cs;
αexps=5.1 10^-7;
Ct=2.1 10^6;
κt=33/Ct;
αexpt=3.6 10^-6;

Csap300=Cth300*rho;

lightwave=1064*(10^(-9));
β1=8*(10^(-6)); (* silica; taken from Evans 2008 *)
β2=14*(10^(-6)); (* tantala; taken from Evans 2008 *)

βref=(((n1^2)β2+(n2^2)β1)/(4((n2^2)-(n1^2))));

Averaging (R and γc are averaged without the notation "Av.")

CcAv:=Ct(coat/(coa+coat))+Cs(coa/(coa+coat));

κcAv:=(1/CcAv)((((1/(κt Ct))(coat/(coa+coat))+(1/(κs Cs))(coa/(coa+coat))))^(-1));

ξAv := Sqrt[2 Ω/κcAv]*(coa+coat); (* Averaged *)

R:=(((Sqrt[κcAv]) CcAv)/((Sqrt[κs]) Cs)); (* Averaged *)

3.Thermal noise

3.1 Suspension by PPP code

hsusp[f_]=(Sqrt[Abs[Sth]+Abs[Sthv](VHC^2)])*(Sqrt[2])/L;

3.2 Mirror

3.2.1 Substrate thermal noise

Loss angle

Φ1=(10^(-9)); (* frequency-independent *)
Φ2=6.5*(10^(-12))/height+7.6*(10^(-12))*f^0.77;  (* frequency-dependent *)

Φ2/.f->100

The following equation is for the infinite-size mirror.

hmirrorstr300[f_]=((Sqrt[2])/L)(Sqrt[((Φ2)/(Φ1))])×(Sqrt[((4kB Tm(1-((σs)^2))Φ1)/((Sqrt[π])Ys w0 Ω))]);

3.2.2 Mirror (thermoelastic damping)

hmirrorthermo300simple[f_]=((Sqrt[2])/L)*Sqrt[8*2/Sqrt[Pi]*alpha300^2*(1+σs)^2*kB*T300^2*kappa300/(rho*Cth300)^2*(1/w0^3)/(2*Pi*f)^2];

3.2.2-b Mirror (thermoelastic damping); exact

Jhyper=(1/( ((Ωc)^2)))(1-Re[(E^((Ωc I)/2)) ((1-Ωc I)(1- Erf[(((Sqrt[Ωc]) (1+I))/2)]))])-(Sqrt[(1/(π ((Ωc)^3)))]);
Ωc=π Cs2 (w0^2)f/kappa300/.Cs2->Cth300*rho;

hmirrorthermo300[f_]=((Sqrt[2])/L)(Sqrt[((4 kB (Tm^2) (alpha300^2) (((1+σs))^2)w0)/((Sqrt[π]) kappa300))*Jhyper]);

3.2.3 Mirror (coating: structural damping)

hmirrorcoa300[f_]=((Sqrt[2])/L)√(((4 kB T300 dcoa1s ϕc1)/(π (w0^2)Ω))(((Ys^2)(((1+σs))^2)(((1-2σs))^2)+(Ys^2)(((1+σs))^2)(1-2σs))/((Ys^2)Ys(1-((σs)^2))))+((4 kB T300 dcoa1t ϕc2)/(π (w0^2)Ω))(((Yt^2)(((1+σs))^2)(((1-2σs))^2)+(Ys^2)(((1+σt))^2)(1-2σt))/((Ys^2)Yt(1-((σt)^2))))); (* infinite-size ITM *)

3.2.4 Thermo-optic noise

ΔαAv:=(αexpt((1+σs)/(1-σt))(((1+σt)/(1+σs))+(1-2σs)(Yt/Ys)))(coat/(coa+coat))+αexps((1+σs)/(1-σs))(((1+σs)/(1+σs))+(1-2σs)(Ys/Ys))(coa/(coa+coat))-2alpha300(1+σs)(CcAv/Csap300);
StoAv:=((2 kB (T300^2))/Ω)(1/(π (w0^2)))(1/CcAv)(1/(ξAv (coa+coat)))(1/(ΓD))(Γ0 (((ΔαAv (coa+coat)))^2)-Γ1 ΔαAv (coa+coat) βref lightwave ξAv+Γ2(((βref lightwave))^2)((ξAv)^2));
Γ0:=(2(Sinh[ξAv]-Sin[ξAv])+2R(Cosh[ξAv]-Cos[ξAv]));
Γ1:=8Sin[ξAv/2](R Cosh[ξAv/2]+Sinh[ξAv/2]);
Γ2:=((1+(R^2))Sinh[ξAv]+(1-(R^2))Sin[ξAv]+2R Cosh[ξAv]);
ΓD:=(1+(R^2))Cosh[ξAv]+(1-(R^2))Cos[ξAv]+2R Sinh[ξAv];

hTO1[f_]=((Sqrt[2])/L)(Sqrt[StoAv])/.{coa->dcoa1s,coat->dcoa1t}; (* ITM; infinite-size mirror *)

3.2.6 Total (room temperature)

hmirror300[f_]:=Sqrt[hmirrorstr300[f]^2+hmirrorthermo300[f]^2+hmirrorcoa300[f]^2+hTO1[f]^2];

Export

SetDirectory[NotebookDirectory[]];

Export["hmirrorcoaITMsilica30kg.m",hmirrorcoa300[f]];

Export["hmirrorstrITMsilica30kg.m",hmirrorstr300[f]];

Export["hmirrorthermoITMsilica30kg.m",hmirrorthermo300[f]];

Export["hmirrorTOITMsilica30kg.m",hTO1[f]];

Export["hsuspITMsilica30kg.m",hsusp300[f]];

LogLogPlot[{hmirrorcoa300[f],hmirrorthermo300simple[f],hmirrorthermo300[f],hmirrorstr300[f],hsusp300[f],hTO1[f]},{f,1,(10^4)},PlotRange->{(1/(10^25)),(1/(10^20))}]

LogLogPlot[{hmirrorthermo300simple[f],hmirrorthermo300[f]},{f,(10^(-4)),(10^3)},PlotRange->All]