Thermal noise
- Mirror thermal noise
- Mirror coating structure damping (4枚鏡のtotal変位.[m/rtHz])
mirrorcoa[f_, tm_, dcoa_, phicoa_, w1_, w2_, E0_] := 5.5*10-19*(dcoa/15*10-6)1/2*(phicoa/1.2*10-2)1/2*Sqrt[(10-2/w1)2*2+ (10-2/w2)2*2]*(tm/300)1/2*(7.24*1010/E0)1/2*(100/f)1/2;
dcoa:thickness of coating[m], 7.5*10-6.
phicoa:loss angle of coating, 4*10-4.
w1:beam radius of front mirror[m], 4.89*10-3.
w2:beam radius of end mirror[m], 8.48*10-3.
- tm:temperature[K], 300K or 20K.
E0:Young's modulus of substrate[N/m2], 4*1011.
- Mirror substrate structure damping (4枚鏡のtotal変位.[m/rtHZ])
mirrorhomo[f_, tm_, sigma_, phimir_, w1_, w2_, E0_] := Sqrt[4*kb*tm*(1 - sigma2)*phimir/(Sqrt[Pi]*E0*2*Pi*f)]*Sqrt[1/w1*2 + 1/w2*2];
kb = 1.39 10-23; (*Boltzmann's constant [J/K]*)
- sigma:poisson ratio, 0.27.
phimir:loss angle of mirror material, 10-7 at 300K, 10-8 at 20K.
w1:beam radius of front mirror[m], 4.89*10-3.
w2:beam radius of end mirror[m], 8.48*10-3.
- tm:temperature[K], 300K or 20K.
E0:Young's modulus[N/m2], 4*1011^.
- Mirror substrate thermoelastic damping
- thermoelastic dampingの周波数依存性は、ビーム径、基材の熱拡散係数で決まるcur-off frequencyの前後で変化する。
f >> cut-off frequency -> thermoelastic dampingの周波数依存性 1/f. 室温では「観測帯域」>> cut-off frequencyが一般に成り立つ.
f << cut-off frequency -> thermoelastic dampingの周波数依存性 f. 低温では「観測帯域」<< cut-off frequencyが一般に成り立つ.
thelasticFreq[w_, kappa_, spheat_] := (kappa/(spheat w2/2)) (1/(2 Pi)) [Hz]
室温(f >> cut-off frequency)での熱雑音振幅 (鏡1枚分、変位。[m/rtHz])
thelasticRoom[f_, t_, alpha_, sigma_, kappa_, spheat_, w_] := (8/Sqrt[2 Pi]) alpha2 (1 + sigma)2 (kb t2 kappa)/((2 Pi f)2 spheat2) (Sqrt[2]/w)3
低温(f << cut-off frequency)での熱雑音振幅 (鏡1枚分、変位。[m/rtHz])
thelasticCryo[f_, t_, alpha_, sigma_, kappa_, spheat_] := (8/Sqrt[2 Pi]) alpha2 (1 + sigma)2 (kb t2 )/kappa (kappa/(2 Pi f spheat ))0.5
kb = 1.39 10-23 J/second; (*Boltzmann's constant [J/K]*)
- t : temperature [K], 300K or 20K.
alpha : linerar expansion ratio, 5.4*10-6 at 300K, 5.6*10-9 at 20K.
- sigma : poisson ratio, 0.27.
- kappa : thermal conductivity [W/K/m],46 at 300K, 15700 at 20K.
spheat : specific heat [J/K/m3], 3.09*106 at 300K, 2760 at 20K.
w : beam radius of front mirror [m], front: 4.89*10-3, end: 8.48*10-3.
- thermoelastic dampingの周波数依存性は、ビーム径、基材の熱拡散係数で決まるcur-off frequencyの前後で変化する。
- Mathematicaにコピペで利用可能...なはず。
mirrorcoa[f_, tm_, dcoa_, phicoa_, w1_, w2_, E0_] := 5.5*10(-19)*(dcoa/(15*10(-6)))(1/2)*(phicoa/(1.2*10(-2)))(1/2)*Sqrt[(10(-2)/w1)2*2 + (10(-2)/w2)2*2]*(tm/300)(1/2)*(7.24*1010/E0)(1/2)*(100/f)^(1/2)
- mirrorhomo[f_, tm_, sigma_, phimir_, w1_, w2_, E0_] := Sqrt[4*kb*tm*(1 - sigma^2)*phimir/(Sqrt[Pi]*E0*2*Pi*f)]*Sqrt[1/w1*2 + 1/w2*2]
- thelasticFreq[w_, kappa_, spheat_] := kappa/(spheat w^2/2) 1/(2 Pi)
thelasticRoom[f_, t_, alpha_, sigma_, kappa_, spheat_, w_] := (8/Sqrt[2 Pi]) alpha2 (1 + sigma)2 (kb t2 kappa)/((2 Pi f)2 spheat2) (Sqrt[2]/w)3
thelasticCryo[f_, t_, alpha_, sigma_, kappa_, spheat_] := (8/Sqrt[2 Pi]) alpha2 (1 + sigma)2 (kb t2 )/kappa (kappa/(2 Pi f spheat ))0.5
thelastic[f_, t_, alpha_, sigma_, kappa_, spheat_, w_] := If[f > thelasticFreq[w, kappa, spheat], thelasticRoom[f, t, alpha, sigma, kappa, spheat, w], thelasticCryo[f, t, alpha, sigma, kappa, spheat]] (*Cut-off frequencyを計算し、室温と低温のthermoelastic dampingの熱雑音計算式を選択し、変位を計算する関数。鏡一枚分の変位。[m/rtHz]*)
- Mirror coating structure damping (4枚鏡のtotal変位.[m/rtHz])