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/m^{2], 4*10}11^.

  • 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.

  • 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*10^10/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]) alpha^{2 (1 + sigma)}2 (kb t^{2 kappa)/((2 Pi f)}2 spheat^{2) (Sqrt[2]/w)}3

    • thelasticCryo[f_, t_, alpha_, sigma_, kappa_, spheat_] := (8/Sqrt[2 Pi]) alpha^{2 (1 + sigma)}2 (kb t^{2 )/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]*)