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2008. Vol. 20, No. 19. P. 16001612.110. Bondu F., Hello P., Vinet J.-Y. Thermal noise in mirrors of interferometric gravitationalwave antennas // Physics Letters A. 1998. Vol. 246, No. 34. P. 227 236.129ÏðèëîæåíèåÏ.1.Ê ìåòîäó èìïåäàíñîâÇäåñü áóäóò ïîëó÷åíû íåâîçìóù¼ííûå êîýôôèöèåíòû îòðàæåíèÿ ÷åòâåðòüâîëíîâîãî ïîêðûòèÿ. Ðàññìîòðèì óðàâíåíèÿ (1.2.1)-(1.2.3) íà ãðàíèöå ðàçäåëà äâóõ äèýëåêòðèêîâ.
Î÷åâèäíî, ÷òî èìïåäàíñ íåïðåðûâåí âñþäó. ÒîãäàΓ(+0) = 0(0.1.1)Z(+0) = ηs(0.1.2)Γ(−0) =Z(−0) − η1Z(−0) + η1(0.1.3)(0.1.4)Z(−0) = Z(+0)Äàëåå âíóòðè ñëîÿ êîýôôèöèåíò îòðàæåíèÿ ìåíÿåòñÿ íåïðåðûâíî:E− eiωt+ik(z+dz)= Γ(z)ei2kdzΓ(z + dz) =iωt−ik(z+dz)E+ e(0.1.5)Òîãäàω(0.1.6)Γ(−d1 + 0) = Γ(−0)e−i2n1 c d1Z(−d1 + 0) = η1Γ(−d1 − 0) =−i2n1 ωc d11 + Γ(−0)e1 + Γ(−d1 + 0)= η1ω1 − Γ(−d1 + 0)1 − Γ(−0)e−i2n1 c d1(0.1.7)Z(−d1 − 0) − η2Z(−d1 + 0) − η2=Z(−d1 − 0) + η2Z(−d1 + 0) + η2(0.1.8)(0.1.9)PPPÈòîãî, îáîçíà÷àÿ Γi = Γ( i−1−di − 0) è Zi = Z( i1 −di − 0) = Z( i1 −di + 0), ïîëó÷èì1Zi−1 − ηiZi−1 + ηiω1 + Γi e−i2ni c diZi = ηiω1 − Γi e−i2ni c diΓi =(0.1.10)(0.1.11)Äëÿ ìíîãîñëîéíîãî ÷åòâåðòüâîëíîâîãî çåðêàëà φi = −π ,2ηi+1η2= i+1Zi−1Ziηi2 2 Nη0Z2N =ηsη12 2 N 2η1η1Z2N +1 =2η0ηsZi+1 =(0.1.12)(0.1.13)(0.1.14)130Γ2N +eΓ2N +1+eη 2N ηs − η12N ηe== 02Nη0 ηs + η12N ηeηsηeηsηe1−η12N +2 − η02N ηs ηe= 2N +2=η1+ η02N ηs ηe1+ 2Nη0η1−1(0.1.15) 2Nη0η1ηs ηeη12ηs ηeη12+1 2Nη0η1(0.1.16) 2Nη0η12Ñòîèò îòìåòèòü, ÷òî âñå èìïåäàíñû è êîýôôèöèåíòû îòðàæåíèÿ äåéñòâèòåëüíû.Ï.2.Ê ðàñ÷¼òó øóìîâàîé äîáàâêèÇäåñü áóäóò ïîëó÷åíû ôîðìóëû äëÿ âîçìóù¼ííûõ êîýôôèöèåíòîâ îòðàæåíèÿ â îòñóòñòâèå ôîòîóïðóãîñòè.
Äîïóñòèì, â ðåçóëüòàòå øóìîâ, òîëùèíû ñëî¼â ïîêðûòèÿ èçìåíèëèñü.Òîãäà ýòî ìîæíî ó÷åñòü â èçìåíåíèè íàáåãà ôàç â (0.1.6), ââåäÿ φi → φi + ∆i . Áóäåì èñêàòüíîâûå èìïåäàíñû è êîýôôèöèåíòû îòðàæåíèÿ êàê âîçìóùåíèÿ ñòàðûõ:(0.2.1)Γ‘(−d1 + 0) = Γ(−0)eiφ1 +i∆1 = Γ(−0)eiφ1 (1 + i∆1 )1 + Γ(−d1 + 0)1 + Γ1 eiφ1 (1 + i∆1 )= η1=1 − Γ(−d1 + 0)1 − Γ1 eiφ1 (1 + i∆1 )2Γ1 eiφ1= Z1 1 +i∆ = Z1 (1 + z1 i∆1 )1 − Γ21 ei2φ1Z‘1 = η1Z‘1 − η2Z1 (1 + z1 i∆1 ) − η2==Z‘1 + η2Z1 (1 + z1 i∆1 ) + η22η2 Z1= Γ2 1 + 2z1 i∆1 = Γ2 (1 + g2 i∆1 )Z1 − η22(0.2.2)Γ‘2 =Çäåñü ìû èñïîëüçîâàëè ïðåäïîëîæåíèå, ÷òî |z1 i∆1 | = | −2Γ1 eiφ1i∆|1−Γ21 ei2φ1(0.2.3)<< 11 + Γ‘2 eiφ2 (1 + i∆2 )=1 − Γ‘2 eiφ2 (1 + i∆2 )1 + Γ2 eiφ2 (1 + g2 i∆1 )(1 + i∆2 )= η2=1 − Γ2 eiφ2 (1 + g2 i∆1 )(1 + i∆2 )2Γ2 eiφ2(i∆2 + g2 i∆1 ) == Z2 1 +1 − Γ22 ei2φ2Z‘2 = η2= Z2 (1 + z2 (i∆2 + g2 i∆1 ))(0.2.4)Çäåñü ìû èñïîëüçîâàëè, ÷òî g2 i∆1 + i∆2 << 1, g2 ∆1 ∆2 ≈ 0Z‘2 − η3Z2 (1 + z2 (i∆2 + g2 i∆1 )) − η3==Z‘2 + η3Z2 (1 + z2 (i∆2 + g2 i∆1 )) + η32η3 Z2= Γ3 1 + 2z2 (i∆2 + g2 i∆1 ) = Γ3 (1 + g3 i∆2 + g3 g2 i∆1 )Z2 − η32Γ‘3 =(0.2.5)131Èòîãî ïîëó÷èì22(Zk−1− ηk2 )eiφk2Γk eiφk=1 − Γ2k ei2φk(Zk−1 + ηk )2 − (Zk−1 − ηk )2 ei2φk2ηk Zk−1 2Γk−1 eiφk−12ηk Zk−1;g=fz=fk = 2kk k−12− ηk2 1 − Γ2k−1 ei2φk−1Zk−1 − ηk2Zk−1!!i−1 Yii−1XXΓ‘i = Γi 1 + igk ∆j = Γi 1 + iαij ∆jzk =j=1 k=j+1Z‘i = Zi1 + izi∆i +(0.2.6)(0.2.7)(0.2.8)j=1i−1 YiX!!gk ∆j(0.2.9)j=1 k=j+1èëè ðåêóðåíòíîΓ‘i+1 = Γi+1 (1 + gi+1 (Γ‘i /Γi − 1 + i∆i+1 )) = Γi+1 (1 + fi+1 (Z‘i /Zi − 1))(0.2.10)Z‘i+1 = Zi+1 (1 + zi+1 (Γ‘i+1 /Γi+1 − 1 + i∆i+1 )) = Zi+1 (1 + zi+1 (fi+1 (Z‘i /Zi − 1) + i∆i+1 ))(0.2.11)Îäíàêî â ñëó÷àå λ/4 - îòðàæàòåëÿ ïðè ïðèáëèæåíèè ê ïîâåðõíîñòè zi → ∞ è ðàçëîæåíèå(0.2.3) íå ìîæåò èìåòü ìåñòî ïðè ïðîèçâîëüíîì øóìå.
Ïðîâåðèì óñëîâèå, èñïîëüçîâàâøååñÿïðè ðàçëîæåíèè∆i = 2k0 ni δdi <<Ïîëó÷èì, ïîëîæèâ µ = 1, η =1nq1zi(0.2.12)0µ0δdj <<nj1 ns nj−1102(j−1) 2k0 nj n2jns1 − n0(0.2.13)Äëÿ çåðêàëà, èñïîëüçóþùåãîñÿ â LIGO ïîëó÷èì δdj << 0.05. Òàêèì îáðàçîì ôîðìóëû(0.2.6)-(0.2.8) îêàçûâàþòñÿ ñïðàâåäëèâûìè.Ï.2.1.Óïðîùåíèÿ äëÿ ÷åòâåðòüâîëíîâîãî îòðàæàòåëÿÄëÿ λ/4-îòðàæàòåëÿ ìîæíî çàïèñàòüzk = −222(Zk−1− ηk2 )2(Zk−1− ηk2 )1=−=−22(Zk−1 + ηk ) − (Zk−1 − ηk )4ηk Zk−1fk(0.2.14)ÑîîòâåòñòâåííîZjηj−ηjZjη 4N − η 2 η04N −2= − 1 2N s 2N2η1 ηs η0 −1η 4N η 2 − η 4N +2= − 0 2Ns +1 1 2N2η1ηs η01zj =2z2Nz2N +1(0.2.15)(0.2.16)(0.2.17)132Ìîæíî çàìåòèòü, ÷òîzj =2(j−1) 2ηs − η12jj η0(−1)2η1j ηs η0j−1(−1)j=2ηsη0η0η1jη0−ηsη1η0j !(0.2.18)Òîãäà èñïîëüçóÿ (0.2.14) ïîëó÷èìiYαij =gk =k=j+1Z‘i = Zi1 + iziiY∆i +zk−1 fk =k=j+1i−1 YiXiYzk fkk=j+1!!zk−1 fk ∆j= Zizj= fi (−1)i−j−1 zjzii−1X1 + izi ∆i + i(−1)i−j zj ∆jj=1 k=j+1= Zi1+i(0.2.19)!j=1!iX(0.2.20)(−1)i−j zj ∆jj=1È òîãäà1 + i(−1)NZ‘N = ZNN2(j−1) 2Xηη − η 2j0j=1Γ‘N +e = ΓN +e1 + ifN +eNX1sjj−12η1 ηs η0!∆j(0.2.21)!(0.2.22)(−1)N −j zj ∆jj=1fN +e =Ï.2.2.2ηe ZN, αj = αN +ejZN2 − ηe2(0.2.23)Ê ðàñ÷¼òó íåîäíîðîäíîãî øóìà â ïîêðûòèèÏîëó÷åííûå âûøå ôîðìóëû ëåãêî ìîäèôèöèðîâàòü äëÿ ñëó÷àÿ íåîäíîðîäíîãî øóìàâ òîëñòîì ñëîå, íàïðèìåð ñâåòîäåëèòåëå èëè âõîäíîì (÷àñòè÷íî ïðîïóñêàþùåì) çåðêàëå,ãäå ñâåòîâàÿ ìîùíîñòü íàõîäèòñÿ â ïîäëîæêå.
Äëÿ ýòîãî èñïîëüçóåì ïðåäåëüíûé ïåðåõîänj = n(z) ≈ n íåâîçìóù¼ííûé ïîêàçàòåëü ïðåëîìëåíèÿ ñëîÿ, δnj = δn(z) è dj → dz → 0(ϕj → dϕ). ÒîãäàΓj+1 =ejPgj+1,j + Γej = Γj e−iϕj = Γ1 e−i j ϕk ,=Γej1 + gj+1,j ΓÄëÿ ñëîÿ òîëùèíîé L ΓN +1 = Γ1 e−iPNϕj(0.2.24)= Γ1 e−2ik0 nL . Äàëåå äëÿ èìïåäàíñíûõ êîåôôèöèåí-òîâPk−1Pk−1Pk−1ϕliϕlϕl1 − Γ21 e−2iΓ−1− Γ1 e−i1 e,zk ==Pk−1ϕl22Γ1 e−iPPk−1Pkϕl −i2ϕki k ϕl1 − Γ21 e−i2Γ−1− Γ1 e−i ϕle1 e=zek =.Pk−1ϕl e−iϕk22Γ1 e−i(0.2.25)(0.2.26)133Çàìåòèì, ÷òî zk = zek−1 . Óïðîñòèì äàëåå âûðàæåíèÿ èñïîëüçóÿ Γ1 = |Γ1 |e−i2ϕ1 è âñïîìèíàÿ,PP÷òî â ñëåäñòâèå ïðåäåëüíîãî ïåðåõîäà k−1 ϕl = k ϕl − ϕk = 2k0 nz − 2k0 ndz :zk = i|Γ1 | sin (2k0 n(z − dz) + 2ϕ1 ) ,(0.2.27)zek = i|Γ1 | sin (2k0 nz + 2ϕ1 ) .(0.2.28)Ïåðåõîäèì ê ñëåäóþùåìó êîåôôèöèåíòóζk = zek − zk = i|Γ1 |(sin (2k0 nz + 2ϕ1 ) − sin (2k0 n(z − dz) + 2ϕ1 )) == 2ik0 n|Γ1 | cos (2k0 nz + 2ϕ1 ) dz.(0.2.29)(0.2.30)Íàêîíåö, ÷òîáû ïåðåïèñàòü îñíîâíûå ôîðìóëû, íåîáõîäèìî çàìåòèòü, ÷òî ∆j = −2k0 δn(z)dzè δnN +1 = 0 òàê êàê i âíå L îáëàñòè.Γ0N +1 = ΓN +1 (1 + ε),N N +1δnjδnN +1 X Y zk+i∆j − ζj=ε = zN +1nN +1zenk−1jj=1 k=j+1Z Lδn(z)2ik0 n cos (2k0 nz + 2ϕ)e dz ==−i2k0 δn(z)dz −n0Z L= −2ik0δn(z) (1 + cos (2k0 nz + 2ϕ))e dz(0.2.31)(0.2.32)(0.2.33)0Òàêèì îáðàçîì, äëÿ ìàëîé íåîäíîðîäíîñòè â ñëîå òîëùèíîé L íà íà÷àëüíîì êîåôôèöèåíòå îòðàæåíèÿ Γ1 , îòðàæåíèå íà ïîâåðõíîñòèZ Lp 0−2ik0 nLΓN +1 =Γ1 e(1 − 4ik0δn(z) cos2 k0 nz + i Ln Γ1 dz) =0Z Lp=Γ1 e−2ik0 nL (1 − 4ik0δn(z) cos2 k0 nz + ϕ1 /2 + i ln |Γ1 | dz)0Z L=Γ1 e−2ik0 nL (1 − 2ik0δn(z)dz+0Z L1 + Γ211 − Γ21− ik0δn(z) cos(2k0 nz + ϕ1 )+ i sin(2k0 nz + ϕ1 )dz),Γ1Γ10(0.2.34)(0.2.35)ãäå Ln îáîçíà÷àåò êîìïëåêñíûé ëîãàðèôì.
Ýòîò ðåçóëüòàò ñîâïàäàåò ñ ðåçóëüòàòàìè èç [36]ñ óñðåäíåíèåì ïî sin2 , òàê êàê îíè ïî-ñóòè ïîëîæèëè Γ1 = −1.Çàìå÷ó, ÷òî Γ1 ïîëó÷åíî èç Γ0 áåç øóìà, ò.å ôîðìóëà îïèñûâàåò ïåðåõîä îò áåñøóìíîãîê áåñøóìíîìó ñëîþ ïðè ïîñòîÿííîì íåâîçìóù¼ííîì êîýôôèöèåíòå ïðåëîìëåíèÿ.Åñëè íàì íåîáõîäèìî îñòàòüñÿ âíóòðè øóìÿùåé îáëàñòè ÷òîáû îïèñàòü ïåðåõîä âíóòðèñëîÿ ñ äàííûì ïîêàçàòåëåì ïðåëîìëåíèÿ, íåîáõîäèìî îáðàòèòü ñêà÷îê ïîêàçàòåëÿ ïðåëîì-134ëåíèÿ íà ëåâîé (ïåðåäíåé)δn(dj )1 − Γ2j+1 δn(dj )Γ+Γ−gj+1j+1j+1,j2nej === Γj+1 1 +Γδn(d )1 − gj+1,j Γj+12Γj+1n1 + 2nj Γj+1δn(dj )δn(dj )= Γj+1 1 + i sin(i Ln Γj+1 )= Γj+1 1 + i sin(i Ln Γj + 2k0 nj dj )=nnδn(dj )δn(dj )= Γj+1 1 + zj+1= Γj+1 1 + zej(0.2.36)nnè ïðàâîé (çàäíåé) ãðàíèöå ñëîÿ1 − Γ2j δn(0)δn(0)ej = Γej 1 −ej 1 − i sin(i Ln Γj )Γ=Γ2Γjnnδn(0).= Γj+1 1 − zjn(0.2.37)ãäå δn(dj ) îçíà÷àåò ëåâóþ (ïåðåäíþþ) ñòîðîíó.
Çàìåòèì, ÷òî â ñëó÷àå ïîñòîÿííîãî δnZp δn(0)δn(dj )2− izj+ iezj− 4ik0 δn(z) cos k0 nz + i Ln Γj dz =nndj= −i sin(i Ln Γj )δn(dj )δn(0)+ i sin(i Ln Γj + 2k0 nj dj )nnZ− 2ik0 dj δnj − 2ik0δn(z) cos (2k0 nz + i Ln Γj ) dz = −2ik0 dj δnj(0.2.38)djÍàêîíåö ïîëó÷èìΓ0N +1=Γ1 e=Γ1 e=Γ1 e−2ik0 nL−2ik0 nL−2ik0 nLZ− ik00Ï.2.3.L(1 +(1 +(1 +δnj (0)iµ0jnjδnj (0)iµ0jnjδnj (0)iµ0jnjZL− 4ik0p δn(z) cos2 k0 nz + i Ln Γ1 dz) =0ZL2− 4ik0δn(z) cospk0 nz + ϕ1 /2 + i ln |Γ1 | dz)0Z− 2ik0Lδn(z)dz+01 − Γ211 + Γ21δn(z) cos(2k0 nz + ϕ1 )+ i sin(2k0 nz + ϕ1 )dz),Γ1Γ1(0.2.39)Îöåíêà ñïåêòðàëüîé ïëîòíîñòè óïðóãèõ øóìîâÎöåíèì äèñïåðñèþ øóìà, ñ÷èòàÿ øóì êàæäîãî ñëîÿ δd íåçàâèñèìûì, íî ðàçíûì äëÿäëÿ ÷¼òíûõ è íå÷¼òíûõ ñëî¼â (îïðåäåëÿþùèìñÿ ìàòåðèàëîì) , ò.å.
hδd2j i → S(Ω)j = ξj (ω)δd(1.2.48), hδdj δdk i = 0. Èñïîëüçóÿ òàáë. 1.2 ïîëó÷èì äëÿ êâàðöåâîé ïîäëîæêèr2π ì−20√S(Ω)si = 0.60 ∗ 10ωÃör2π ì√ .S(Ω)ta = 1.26 ∗ 10−20ωÃö(0.2.40)(0.2.41)135Äàëåå ìíîæèòåëè 4k02 n2e èp2π/ω â ÷èñëåííûõ ðàñ÷åòàõ äëÿ ïðîñòîòû ñðàâíåíèÿ áóäåì îïóñ√êàòü, è ïîëó÷àòü äèñïåðñèè“ øóìà â ì/ Ãö íà ÷àñòîòå 1 Ãö. Ïðè æåëàíèè èñòèííûå ñïåê”òðàëüíûå ïëîòíîñòè ôëóêòóàöèé ôàçû ìîæíî ïîëó÷èòü, äîìíîæèâ ðåçóëüòàòû íà ýòè ìíîæèòåëè.Sφ2 = 4k02NX2 2fN +e (−1)N −j zj nj − ne Sδd=jj=1#2 j j !N(−1)ηηηηs0012= 4k02fN +e−nj − ne Sδdj2ηηηη01s0j=1 NXfN2 +e η02 η1 j 2 2fN2 +e 2 2fN2 +e ηs2 η02 j 2 22 2nS+nS−nS= 4k0 Sd[j δdjj δdj4 η02 η124 ηs2 η02 j δdjj=1 j jη0η1NN2 η022 ηs− fN +e ne (−1) nj Sδdj+ fN +e ne (−1) nj Sδdj+ Sδdn2 ]j eη0 η1ηs η0NX"(0.2.42)Ðàññìîòðèì ïðåäåëû Z2N → 0 è Z2N → ∞. Îòìåòèì, ÷òî ïðè ýòîì f2N +e → 0• Z2N → 0 (n1 < n0 )n20n40−2+ N ]+n40 − n41n20 − n21n4n22+ 4k02 n2e Sd1[ 4 1 4 − 2 2 1 2 + N]n0 − n1n0 − n12Sφ2 =4k02 n2e Sd0[(0.2.43)• Z2N → ∞ (n1 > n0 )22Sφ2 =4k02 (Sd0+ Sd1)n2e [−n41 n402n21 n20++ N]n4e n40 − n41n2e n20 − n21(0.2.44)Òåïåðü äëÿ 2N + 1 ñëîÿS12φ=Sφ2+f224k02 Sd1[n21 2N +1+e+ ne n1 f2N +1+e4Z2Nη12Z2Nη12+ 2η12Z 2Nf2η1−+ n2e − 2N +1+e n21 ] =Z2N2(0.2.45)ãäå â Sφ âìåñòî f2N +e èñïîëüçóåòñÿ −f2N +1+e =2ne n21 Zv Z2N2 −n2 Z 2 .n41 Z2Ne vÒîãäà• Z2N → 0 (n1 < n0 ) (òîåñòü Z2N +1 → ∞)22S12φ =4k02 n2e (Sd0+ Sd1)[n40 n41n20 n212−2+ N ] + 4k02 n2e Sd144 422 2n0 − n1 nen0 − n1 ne(0.2.46)136• Z2N → ∞ (n1 > n0 ) (òîåñòü Z2N +1 → 0)n20n4022222(S+S)+2(S 2 + Sd1) + (Sd0+ Sd1)N ]+d1n40 − n41 d0n20 − n21 d022Z2N2 22 f2N +1+e Z2N4k0 Sd1 [n1+nnf+ n2e ] =e12N+1+e24η1η142nn22)[− 4 0 4 + 2 2 0 2 + N ]+ Sd1=4k02 n2e (Sd0(0.2.47)n0 − n1n0 − n1S12φ =4k02 n2e [−√Èòîãî äëÿ êâàðö-òàíòàëàòíîãî çåðêàëà èç 42 ñëî¼â ïîëó÷èì Snorm = 6.04 × 10−20 ì/ Ãö, ñ√43 ñëîÿìè Snorm = 6.18 × 10−20 ì/ Ãö.
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