Диссертация (1102364), страница 27
Текст из файла (страница 27)
ðèñ 20 è ïðèëîæåíèå Ï.5.4.). Òàêèì îáðàçîì ïðè ÷¼òíîì m, p - ÷¼òíîå.ϕ2j = ϕ0j+1 − ϕ0j − ϕ1j − 2 sin(ϕ1j )gj2 + (πp)ϕ1 + ϕ1j2= − j+1− gj+1sin(ϕ1j+1 ) − gj2 sin(ϕ1j ) + (πp)2(0.5.22)ϕ2 = −ϕ1 − 2g 2 sin(ϕ1 ) + (πp)(0.5.24)(0.5.23)Íî òàê êàê g ìàëî, à ϕ1 ïîðÿäêà π , òî n ≥ 1 â (0.5.24) äëÿ îáåñïå÷åíèÿ íåîòðèöàòåëüíîéòîëùèíû ñëîÿ. Òàê êàê äëÿ ðàáîòû çåðêàëà íåîáõîäèìî ϕ0 ∈ [− π2 ; π2 ] (ò.å. ÷¼òíîå m), òîϕ2 = 2π − ϕ1 − 2 sin(ϕ1 )g 2Ï.5.3.Ê ðàçëîæåíèþ êîýôôèöèåíòà ïðîïóñêàíèÿÏóñòü çåðêàëî ñîäåðæèò 2E + 2N ñëî¼â (2E ñëî¼â ïîäãîíÿþò ïîä óñëîâèå óïðîùåíèÿ).Ïðåäñòàâèì ê.î.
â âèäå |Γ2N +2E | = 1 − αN . ge + Γ2N +2E e−iϕc 2|Γ2N +2E+e | = 1 + ge Γ2N +2E e−iϕc 2ge2 + 2ge |Γ2N +2E | cos(ϕ) + |Γ2N +2E |2=1 + 2ge |Γ2N +2E | cos(ϕ) + ge2 |Γ2N +2E |21 − ge2=1−2αN ,21 + 2ge cos(ϕ) + geãäå ge =ne −n2Nne +n2N- ïåðåõîä âî âíåøíþþ ñðåäó, ϕ = ϕc +π+ϕ12(0.5.25)+ g 2 sin(ϕ1 ). Òîãäà êîýôôèöèåíòïðîïóñêàíèÿ ïî ìîùíîñòèT = βαN β=21 − ge21 + 2ge cos(ϕ) + ge2(0.5.26)(0.5.27)Îáîçíà÷èì îòäåëüíî ÷èñëèòåëü è çíàìåíàòåëü äðîáè (0.5.17) êàê p2p24u(Γ) = g 2(1 − cos(ϕ1 )) ± Γ0 1 − 2g cos(ϕ1 ) + g2pp24v(Γ) = gΓ0 2(1 − cos(ϕ1 )) ± 1 − 2g cos(ϕ1 ) + g(0.5.28)(0.5.29)151Òîãäàu|1 = v|1 u 0u0 v − v 0 uu0 − v 0= 2α|1 =|1 =vv2v u 00u0 v 0 uv 00u00uv 02− 2 2 − 2 + 2 3 |1 =|1 =vvvvv00000 00u −vv u −vv0=−2= 2α 1 − 2vv vv(0.5.30)(0.5.31)(0.5.32)è(1 − g 2 )2α= p2p24g 2(1 − cos(ϕ1 )) ± 1 − 2g cos(ϕ1 ) + g(0.5.33)ÈòîãîΓ=Ï.5.4.u= 1 + 2α − α(1 − 2α)2v(0.5.34)Ê îïðåäåëåíèþ ôàçû ïîñëå ñëîÿÏóñòüZeiϕ =a + ibx + iy(0.5.35)ÒîãäàsZ=a2 + b 2x2 + y 2eiϕ = ei(ϕ↑ −ϕ↓ )bx − ayax + byax + bycos(ϕ) = p(x2 + y 2 )(a2 + b2 )tg(ϕ) =(0.5.36)(0.5.37)(0.5.38)(0.5.39)Åñëè ax + by > 0 òî ϕ ∈ [− π2 ; π2 ] è ò.ï.a =Γ0 cos(ϕ0 + ϕ1 ) + g cos(ϕ1 ) − g − g 2 Γ0 cos(ϕ0 )(0.5.40)b = − Γ0 sin(ϕ0 + ϕ1 ) − g sin(ϕ1 ) + g 2 Γ0 sin(ϕ0 )(0.5.41)x = − gΓ0 cos(ϕ0 + ϕ1 ) − g 2 cos(ϕ1 ) + 1 + gΓ0 cos(ϕ0 )(0.5.42)y =gΓ0 sin(ϕ0 + ϕ1 ) + g 2 sin(ϕ1 ) − gΓ0 sin(ϕ0 )(0.5.43)152−Γ0 sin(ϕ0 + ϕ1 ) − g sin(ϕ1 ) + g 2 Γ0 sin(ϕ0 )−arg Γ2 = arctgΓ0 cos(ϕ0 + ϕ1 ) + g cos(ϕ1 ) − g − g 2 Γ0 cos(ϕ0 )gΓ0 sin(ϕ0 + ϕ1 ) + g 2 sin(ϕ1 ) − gΓ0 sin(ϕ0 )− arctg−gΓ0 cos(ϕ0 + ϕ1 ) − g 2 cos(ϕ1 ) + 1 + gΓ0 cos(ϕ0 )= arctg(0.5.44)(1 − g 2 )(−(Γ20 + 1)g sin(ϕ1 ) + g 2 Γ0 sin(ϕ0 − ϕ1 ) − Γ0 sin(ϕ0 + ϕ1 ))(1 + g 2 )((Γ20 + 1)g(cos(ϕ1 ) − 1) + Γ0 cos(ϕ1 + ϕ0 ) + g 2 Γ0 cos(ϕ1 − ϕ0 )) − 4g 2 Γ0 cos(ϕ0 )0,102KKKKKKKK4681012p0,10,20,30,40,50,60,70,8G0= 0.4G0= 0.34G0= 0.2G0= 0.06Ðèñ.
20: Çíàê Re[Γ2 ].  îáëàñòÿõ< 0 â ôîðìóëå (0.5.20) arg Γ2 ∈ [−3π/2; −π/2]. Òîëùèíû â λ/4.  ñëó÷àå m íå÷¼òíîå, ñäâèíóòü íà 2π .−Γ0 sin(ϕ1 /2 + (πm)) + (Γ20 + 1)(cos(ϕ1 ) − 1)(g + g 3 ) − 6Γ0 sin(ϕ1 /2 + (πm))3 g 2 > 0 (0.5.45)arctg(ctg(x/2)) = arctg(tg(π/2 − x/2)) =ϕ0g=0 =Ï.6.π−x+ (πn)2π − ϕ1+ (πn)2(0.5.46)(0.5.47)Ê îöåíêå èíòåãðàëîâ øóìà ðàññòåêëîâàíèÿÄëÿ ðàñ÷¼òà óêîðà÷èâàíèÿ ñòåðæíÿ è øóìà íåîáõîäèìî ïðîèçâåñòè óñðåäíåíèÿ ïî îáú¼ìó îòêëèêà íà ñõëîïûâàíèå ïóçûðüêà. Ïðè ýòîì îòêëèê â ñëó÷àå äëèííîãî ñòåðæíÿ ïîñòî-153ÿíåí è ðàâåí îòêëèêó ïîëóïðîñòðàíñòâà íà ïóçûð¼ê íà ãëóáèíå R, ðàâíîé ðàäèóñó ñòåðæíÿ. Òàêèì îáðàçîì, â îáîèõ ñëó÷àÿõ âû÷èñëåíèÿ ïðîèçâîäÿòñÿ íàä ôîðìóëîé äëÿ îòêëèêà(2.1.18), òîëüêî â ðàçíûõ ïðåäåëàõ.Ï.6.1.Óêîðà÷èâàíèå çåðêàëàÄëÿ èíòåãðàëà (2.1.20), ñ÷èòàÿ ýôôåêòèâíûé ðàäèóñ îáëàñòè êðèñòàëëèçàöèè b ìàëûìZ Zd3 k 32 22 2~ k⊥hδz(t)i =λξVa te−k⊥ w /4 e−ik~rj 2 e−k b /4d rjk(2π)3VZ ZZ332 w 2 /4 −i~2 w 2 /4 −i~2−k⊥k~rj k⊥ d k−k⊥k~rj k⊥ d kd3 rj(0.6.1)=λξVa t−beeee233k (2π)4 (2π)VÄëÿ ïåðâîãî ïîðÿäêà ïîëó÷èìZ Z21 −k⊥2 w2 /4 −ikz zj k⊥dk⊥ dkz 3eeIl (R/w, L/w) =J0 (k⊥ ρ)d rj2wk(2π)2ZV Zdk⊥ 3π −k⊥2 w2 /4 −k⊥ zjeek⊥ J0 (k⊥ ρ)d rj=w(2π)2V(0.6.2)Òîãäà â ñëó÷àå øèðîêîãî çåðêàëà (L = wY ≤ R), èìåþùåìñÿ â çåðêàëå LIGO, è óñðåäíåíèÿïî ãàóññó ïîëó÷èìIlY <X (X, YZ∞)=e−k2 /40XJ0 (kX)(1 − e−kY )dk.2k(0.6.3)Ïîäñòàâëÿÿ â (0.6.2) ρ = 0 ïîä èíòåãðàë ïî îáú¼ìó (ïîëó÷èì ïîä èíòåãðàëîì ðåçóëüòàò(2.1.22)) è z = R, èíòåãðèðóÿ ïî ÷àñòè öèëèíäðà çà ãëóáèíîé z = R ïîëó÷èì äîáàâêó êóêîðà÷èâàíèþ â îáëàñòè íåïðèìåíèìîñòè ïðèáëèæåíèÿ ïîëóïðîñòðàíñòâàIlconst (X, Y2) = X (Y − X) 1 −√X2πX (1 − erf(X)) e/2(0.6.4)Èòîãî äëÿ îöåíêè ñìåùåíèÿ äëèííîãî çåðêàëà ìîæíî èñïîëüçîâàòü Il (X, Y ) ≈ IlY <X (X, X) +Ilconst (X, Y ) < Ilconst (X, Y + X).Äëÿ îöåíêè óêîðà÷èâàíèÿ ñòåðæíÿ (òðóáêè) òàê æå ïðîèçâîäèòñÿ ïîäñòàíîâêà ρ = 0è z = R â (0.6.2).
Ïðè ýòîì ïîëàãàåòñÿ R w è ïðîèçâîäèòñÿ ðàçëîæåíèå â ðÿä äî ïåðâîãî ÷ëåíà, èñêëþ÷àÿ òàêèì îáðàçîì w. Äàëüíåéøåå èíòåãðèðîâàíèå ïî îáú¼ìó ñâåä¼òñÿ êäîìíîæåíèþ íà îáú¼ì ñòåðæíÿ (òðóáêè):hδz(t)i =λξVa tV.4πR2(0.6.5)Çäåñü V = πR2 L äëÿ ñòåðæíÿ è V = π(R2 − we2 )L äëÿ òðóáêè. Îòñþäà ïîëó÷àåòñÿ âûðàæåíèå (2.1.23) äëÿ íàõîæäåíèÿ λ. Î÷åâèäíî òàê æå, ÷òî äëÿ îáùíîñòè ìîæíî òàê æå ââåñòèIle (X, Y ) = Y /4 äëÿ ñòåðæíÿ è Ile (X, Y ) = (1 − 1/X 2 )Y /4 äëÿ ñòåðæíÿ.154Ï.6.2.Ñïåêòðàëüíàÿ ïëîòíîñòü øóìàÄëÿ ïîëó÷åíèÿ êâàäðàòóð ôîðìóëû (2.1.21) ïîëîæèì ýôôåêòèâíûé ðàäèóñ îáëàñòèêðèñòàëëèçàöèè b ìàëûì è ðàçëîæèì âûðàæåíèå â ðÿä, êàê â ïðåäûäóùåì ñëó÷àå.2Z Z3λ 2 22 w 2 /4 −i~−k⊥k~rj k⊥ −k2 b2 /4 d kSδz = 2 ξ Vaed3 rj(0.6.6)ee23ωk(2π)V2Z ZZ33λ 2 22 w 2 /4 −i~2 w 2 /4 −i~−k⊥k~rj k⊥ d k2−k⊥k~rj k⊥ d kd3 rj= 2 ξ Vaee−bee2 (2π)33ωk4(2π)ZV Z Z3λ 2 2d3 k222~~ k⊥ k⊥ d k= 2 ξ Vae−(k⊥ +k⊥ )w /4 e−ik~rj −ik~rj 2 2ωk k (2π)3 (2π)3VZZ33dkb2kdk2 +k 2 )w 2 /4 −i~~k~⊥ k⊥−(k⊥k~r−ir22jj⊥d3 rj(0.6.7)−ee(k + k )4k2 k2(2π)3 (2π)3Äëÿ ïåðâîãî èíòåãðàëà ïîëó÷èì2Z Z2dk⊥ dkz2 w 2 /4 −ik z k⊥−k⊥Sz jeeI0 =d3 rjJ (k ρ)2 0 ⊥2k(2π)V2Z Zdk⊥2 w 2 /4 −k z−k⊥⊥ j=πeek⊥ J0 (k⊥ ρ)d3 rj2(2π)ZV Z0dk2 +k 02 )w 2 /4 −(k +k 0 )z⊥ dk⊥2 −(k⊥00j⊥⊥⊥=π eek⊥ k⊥ J0 (k⊥ ρ)J0 (k⊥ ρ)d3 rj2 (2π)2(2π)V(0.6.8)~ = {k⊥ , k 0 }.
ÒîãäàÄëÿ äàëüíåéøåãî óïðîùåíèÿ ïîëîæèì K⊥Z Z ∞ Z π/22 200Sπ 2 e−K w /4 e−K(cos φ +sin φ )zj K 2 sin φ0 cos φ0 J0 (Kρj cos φ0 )J0 (Kρj sin φ0 )I0 =V00KdK dφ0 3d rj(2π)2 (2π)2(0.6.9)Òåïåðü ïðîèíòåãðèðóåì ïî îáú¼ìó è ïîëó÷èìZ π/2 Z ∞2 2RK sin(2φ0 )S− w 4K−KL(sin(φ0 )+cos(φ0 ))I0 =e(1−e)8 cos(2φ0 )(sin(φ0 ) + cos(φ0 ))00(cos(φ0 )J1 (KR cos(φ0 ))J0 (KR sin(φ0 )) − sin(φ0 )J1 (KR sin(φ0 ))J0 (KR cos(φ0 )))dKdφ02π(0.6.10)Âçÿòü äàííûé èíòåãðàë àíàëèòè÷åñêè íå ïðåäñòàâëÿåòñÿ âîçìîæíûì.Ï.7.Ê îöåíêå àìïëèòóä ìîäóëÿöèèÏðåïèøåì åù¼ ðàç óðàâíåíèå (3.2.6)Ȧk =Xj(−i∆k δkj + iδω∓kj µ± e±iωRF t − κkj )Aj + iF c2Xk2ω(0.7.1)155Çàìåòèì, ÷òî δω+kj âñåãäà èä¼ò ñ µ− è íàîáîðîò.
Ïîýòîìó íà âðåìÿ óáåð¼ì µ± â δω∓kj äëÿ+ +−óïðîùåíèÿ çàïèñè. Ïîäñòàâèì ðåøåíèå bk + a−k e + ak e ãäå ωRF t â ïîêàçàòåëÿõ ýêñïîíåíòûîïóùåíû äëÿ ñîêðàùåíèÿ. Òîãäà ïîëó÷èì+ +− −+ +−−iωRF a−k e + iωRF ak e = − i∆k (ak e + ak e + bk ) + iX+ +−−++−(a−j e + aj e + bj )(δωkj e + δωkj e )jX−+ +−κkj (a−j e + aj e + bj ) + iF Xkp =+ +−= − i∆k (a−k e + ak e + bk )+X+ 2+−2−−++−+ δω+kj a+i(δω−kj a−+ δω+kj a−j + δωkj aj ej ej + bj (δωkj e + δωkj e ))j−X(0.7.2)−+ +κkj (a−j e + a e + bj ) + iF XkpÒàê êàê èç îäíîé ôóíêöèè Aj ìû ñäåëàëè íåñêîëüêî, òî ìû äîëæíû áóäåì íàëîæèòü íà±2ωRFa±â âèäåj è bj óñëîâèÿ. Ïðèìåíèì ìåòîä ãàðìîíè÷åñêîãî áàëàíñà, ñîõðàíÿÿ ÷ëåíû ñ ee(±ωRF ).
Ýòîò ÷ëåí ïîìîæåò ó÷åñòü êâàçèñòàöèîíàðíûé ñëó÷àé, åñëè ââåä¼ííàÿ ôóíêöèÿìàëà íà âûñîêèõ ÷àñòîòàõ ìîäóëÿöèè, è ñòðåìèòñÿ ê åäèíèöå íà ìàëûõ.− i∆k bk + iX++(δω−kj a−j + δωkj aj ) −−−iωRF a−k = −i∆k ak + iµ−X+iωRF a+k = −i∆k ak + iµ+XXκkj bj + iF c2Xkp = 02ω(0.7.3)X(0.7.4)+(a−j e(−ωRF ) + bj )δωkj −κkj a−jj−(a+j e(ωRF ) + bj )δωkj −Xκkj a+j(0.7.5)jÈç âòîðîãî è òðåòüåãî óðàâíåíèÿ ïîëó÷èì, âîññòàíàâëèâàÿ µ± ïåðåä δω∓kj−1 ∓a±k = µ± M± δωkj bj(0.7.6)ãäå M± = ∆k ± ωRF − e(±ωRF )µ± δω∓kj − iκkj . Ïîäñòàâëÿÿ â ïåðâîå, ïîëó÷èìbk =F c2 −1B Xk2ω(0.7.7)ãäå B = ∆k − µ+ µ− (δω−km M−−1 δω+nj + δω+km M+−1 δω−nj ) − iκkj . Ñëàãàåìûå ñ M ìàëû ïðè ìàëîéìîùíîñòè ÑÂ×, îäíàêî ïðè å¼ ïîâûøåíèè ïðèâîäèò ê óìåíüøåíèþ (èëè íàñûùåíèþ [89])ìîäóëèðîâàííûõ êîìïîíåíò.Ñðàâíèì ïîëó÷åííûå ôîðìóëû ñ êâàçèñòàöèîíàðûì ñëó÷àåì, êîãäà ωRF → 0 (ïî÷òèïîñòîÿííîå ïîëå).
Òîãäà 2µ+ = 2µ− = ωM+ = M− = M = ∆k − ωδωkj − iκkjB = ∆k − ω 2 δωkj M −1 δωkj − iκkj(0.7.8)(0.7.9)156Òîãäà, ó÷èòûâàÿ ìàëîñòü íåäèàãîíàëüíûõ ÷ëåíîâ Xij è êîåôôèöèåíòîâ δωkj êàê ïðè âûâîäå(3.2.12)(3.2.13) ïîëó÷èìF c2−1 + ωM −1 δωkj B −1 Xl ≈A k = a++a+b=kkk2ω2ωδωkjFcXkF c2Xk≈1+=2ω∆k − ωδωkj − iκkj ∆k − iκkk2ω ∆k − ωδωkj − iκkj(0.7.10)Ñ äðóãîé ñòîðîíû â ñòàöèîíàðíîì ñëó÷àå óðàâíåíèå (3.2.6) ÿâëÿåòñÿ àëãåáðàè÷åñêèì è íàïðÿìóþ äà¼ò−Aj = a+j + aj + b j =F c2 −1D Xk ,2ωD−1 = (∆k − ωδωkj − iκkj )−1 ,(0.7.11)(0.7.12)÷òî ñîâïàäàåò ñ (0.7.10).
Âèä ýòèõ ôîðìóë îäíîçíà÷íî ïîêàçûâàåò, ÷òî δωkj , à òî÷íåå å¼ äåéñòâèòåëüíàÿ ÷àñòü, ÿâëÿåòñÿ îòíîñèòåëüíûì ñäâèãîì ÷àñòîòû ïðè ïðèëîæåíèè ïîñòîÿííîãîïîëÿ.Ï.7.1.Âûñøèå ãàðìîíèêèÏîëó÷åííûå ôîðìóëû ëåãêî îáîáùèòü íà áîëüøåå ÷èñëî ãàðìîíèê. Äëÿ ýòîãî ïðîñòî+ −2− 2−−2+äîáàâèì èõ â àíçàö: bk + a−+ a2++ .... Òîãäà ïîëó÷èìk e + ak e + ak ek e+ +2−2+−...
− i2ωRF a2−+ i2ωRF a2+− iωRF a−k ek ek e + iωRF ak e =+ +2−2+−= − i∆k (... + a2−+ a2++ a−k ek ek e + ak e + bk )+X+ +2−2+−−++−+i(... + a2−+ a2++ a−j ej ej e + aj e + bj )(δωkj e + δωkj e )j−X+ +2−2+−κkj (a2−+ a2++ a−j ej ej e + aj e + bj ) + iF Xkp =+ +2−2+−+ a2++ a−= − i∆k (a2−k ek ek e + ak e + bk )+X+ibj (δω−kj e+ + δω+kj e− )jX+ 2+−2−+i(δω−kj a−+ δω+kj a−+ δω+kj a+j + δωkj aj ej ej )jX2+ 3+−−3−++i(δω−kj a2−+ δω+kj a2−+ δω+kj a2+j e + δωkj aj ej ej e )j+ ...X+ +2−2+−−κkj (...
+ a2−+ a2++ a−j e + aj e + bj ) + iF Xkpk ek e(0.7.13)157Äàëåå, ïðèìåíÿÿ ìåòîä ãàðìîíè÷åñêîãî áàëàíñà ïîìíèì, ÷òî èìååì íåîãðàíè÷åííîå ÷èñëîãàðìîíèê:− i∆k bk + iXXF c2++aXkp = 0(δω−kj a−+δ)−κb+ikj jωkj jj2ω−−iωRF a−k = −i∆k ak + iX+iωRF a+k = −i∆k ak + iX(0.7.14)−+(a2−j δωkj + bj δωkj ) −Xκkj a−j(0.7.15)+−(a2+j δωkj + bj δωkj ) −Xκkj a+j(0.7.16)jj−i2ωRF a2−kXX3− −− += −i∆k a2−+i(aδ+aδ)−κkj a−ωkjjj ωkjjki2ωRF a2+kXX+ −3+ += −i∆k a2++i+a)−κkj a+(aδδωkjjj ωkjjk(0.7.17)j(0.7.18)j...Èç âòîðîãî è òðåòüåãî óðàâíåíèÿ ïîëó÷èì, âîññòàíàâëèâàÿ µ± ïåðåä δω∓kjank=Mn−1µ+ δω−kj ajn−1+µ− δω+kj an+1j(0.7.19)(0.7.20)Mn = ∆k + nωRF − iκkjÄëÿ ðåøåíèÿ ñèñòåìû ñëåäóåò ïîëîæèòü, ÷òî àìïëèòóäà íåêîåé ãàðìîíèêè ïîä íîìåðîì(±N )±N ìàëà è íå òðåáóåò ó÷¼òà. òîãäà ïîëó÷èì ÷òî ak(N ∓1)= MN−1∓1 µ± δω∓kj aj.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
