Диссертация (1149766), страница 4
Текст из файла (страница 4)
Êðîìå ýëåêòðè÷åñêîãî ïîëÿ èìååòñÿ òàêæå ïîñòîÿííîå~:ìàãíèòíîå ïîëå HH1 =eax2ea(|x3 | + l)eax1,H=,H=,23(a2 + 1)ρ3(a2 + 1)ρ3(a2 + 1)ρ31ãäå ρ = (x21 +x22 +(|x3 |+l)2 ) 2 . Åñëè ìàòåðèàëüíàÿ ïîâåðõíîñòü ïðåäñòàâëÿåòñîáîé ïëîñêîñòü x3 = 0 è ïàðàëëåëüíî åé òå÷åò òîê j ïî ëèíèè x3 = l, x2 =0, òî â ýòîé ñèñòåìå ìàãíèòíîå ïîëå òàêîå æå, êàê ïîëå ñîçäàâàåìîå òåìæå òîêîì è ïîëóïðîñòðàíñòâîì x3 ≤ 0 ñ ìàãíèòíîé ïðîíèöàåìîñòüþ µef =1/(1 + 2a).
Òîê è ïëîñêîñòü ïîðîæäàþò òàêæå àíîìàëüíîå ýëåêòðè÷åñêîå~:ïîëå EE1 = 0, E2 =2ja x22ja |x3 | + l,E=,3a2 + 1 τ 2a2 + 1 τ 21ãäå τ = (x22 + (|x3 | + l)2 ) 2 .1.5.Ýôåêò Êàçèìèðà-ÏîëäåðàÏîòåíöèàë Êàçèìèðà-Ïîëäåðà îïèñûâàåò âçàèìîäåéñâèå íåéòðàëüíî-ãî àòîìà ñ ïîâåðõíîñòüþ ìàòåðèàëüíîãî òåëà, êîòîðîå ïðåäñòàâëÿåò ñîáîé24âçàèìîäåéñòâèå ôëóêòóèðóþùåãî ýëåêòðè÷åñêîãî äèïîëüíîãî ìîìåíòà àòîìà ñî ñâîèì èçîáðàæåíèåì. Âïåðâûå ðàñ÷åòû ýòîãî ïîòåíöèàëà ïðîâåäåíûÊàçèìèðîì è Ïîëäåðîì â 1948 ãîäó [44]. Äëÿ ìîäåëè ñ ïîòåíöèàëîì ×åðíàÑàéìîíñà ïîòåíöèàë Êàçèìèðà-Ïîëäåðà áûë âû÷èñëåí äëÿ ïëîñêîñòè. Äëÿýíåðãèè îñíîâíîãî ñîñòîÿíèÿ àòîìà, íàõîäÿùåãîñÿ íà ðàññòîÿíèè l îò ïëîñêîñòè, áûë ïîëó÷åí ñëåäóþùèé ðåçóëüòàò [40]:Z +∞1a2E=−dωe−2ωl 2(1 + 2ωl)α33 (iω)23264π l 1 + a0!Z +∞+dωe−2ωl (1 + 2ωl + 4ω 2 l2 ) α11 (iω) + α22 (iω) .01a+64π 2 l2 1 + a2Z+∞dωe−2ωl 2ω 1 + 2ωl α12 (iω) − α21 (iω)0 ïðåäåëå a → +∞ ýòî âûðàæåíèå ñîâïàäàåò ñ ðåçóëüòàòîì Êàçàìèðà èÏîëäåðà äëÿ èäåàëüíî ïðîâîäÿùåé ïëîñêîñòè [44].
Ñïåöèôèêà ïîòåíöèàëà ×åðíà-Ñàéìîíñà ïðîÿâëÿåòñÿ â ïðèñóòñòâèè âêëàäîâ íåäèàãîíàëüíûõýëåìåíòîâ ìàòðèöû αjk (iω) ïîëÿðèçóåìîñòè àòîìà. Ýòè âêëàäû ïðè îïðåäåëåííûõ óñëîâèÿõ ìîãóò áûòü äîìèíèðóþùèìè, ÷òî äàåò âîçìîæíîñòüèõ ýêñïåðèìåíòàëüíîãî îáíàðóæåíèÿ è òåì ñàìûì ïðîâåðêè ïðèìåíèìîcòèìîäåëè âçàèìîäåéñòâèÿ ×åðíà-Ñàéìîíñà [40].252. Ðàññåÿíèå ýëåêòðîìàãíèòíûõ âîëí íà ïëîñêîéïîâåðõíîñòè â ìîäåëè ñ ïîòåíöèàëîì ×åðíà-Ñàéìîíñà2.1.Ïîñòàíîâêà çàäà÷è ìîäåëè ñ ïîòåíöèàëîì ×åðíà-Ñàéìîíñà óðàâíåíèÿ Ìàêñâåëëà ìîäè-ôèöèðóþòñÿ. Ýòî ïðîÿâëÿåòñÿ â èçìåíåíèè çàêîíîâ ðàñïðîñòðàíåíèÿ ýëåêòðîìàãíèòíûõ âîëí.  ðàìêàõ ìîäåëè ñ ïîòåíöèàëîì ×åðíà-Ñàéìîíñà [35]â äàííîé ãëàâå áóäåò èçó÷åíî ðàñïðîñòðàíåíèå âîëí â ðàçäåëåííîì ìàòåðèàëüíîé ïëîñêîñòüþ ïðîñòðàíñòâå.
Áóäåò ïîêàçàíî, ÷òî îäíîé èç îñîáåííîñòåé ýòèõ ïðîöåññîâ ÿâëÿåòñÿ íåçàâèñèìîñòü êîýôôèöèåíòîâ ïðîõîæäåíèÿè îòðàæåíèÿ îò óãëà ïàäåíèÿ âîëíû è åå ÷àñòîòû: åñëè a - êîíñòàíòà âçàèìîäåéñòâèÿ ïîâåðõíîñòè ñ ýëåêòðîìàãíèòíûì ïîëåì, òî êîýôèöèåíò îòðàæåíèÿ âîëíû a2 /(1 + a2 ), à êîýôôèöèåíò ïðîõîæäåíèÿ - 1/(1 + a2 ). Íàïðàâëåíèÿ ïðîøåäøåãî è ïàäàþùåãî ëó÷åé ñîâïàäàþò.
Ïðè ïðîõîæäåíèèè îòðàæåíèè ëó÷à ìåíÿåòñÿ åãî ïîëÿðèçàöèÿ. Ïðè ýòîì ÷åì ìåíüøå îòíîñèòåëüíàÿ èíòåíñèâíîñòü îòðàæåííîé èëè ïðîøåäøåé âîëí, òåì ñèëüíååìåíÿåòñÿ èõ ïîëÿðèçàöèÿ ïî ñðàâíåíèþ ñ ïîëÿðèçàöèåé ïàäàþùåé âîëíû.Ïðè ìàëûõ a, âåêòîðà ýëåêòðè÷åñêîãî ïîëÿ îòðàæåííîé è ïàäàþùåé âîëíïî÷òè îðòîãîíàëüíû, à ïàäþùåé è ïðîøåäøåé - ïî÷òè ïàðàëëåëüíû. Åñëèa 1 òî ïî÷òè îðòîãîíàëüíû äðóã-äðóãó âåêòîðà ïàäàþùåé è ïðîøåäøåéâîëí,à ó îòðàæåííîé è ïàäàþùåé îíè ïî÷òè ïàðàëëåëüíû.26Äëÿ ïëîñêîãî äåôåêòà Φ(x) = x3 ôóíêöèîíàë äåéñòâèÿ èìååò âèä1S(A) = − Fµν F µν + Sφ (A),4ãäåaSφ (A) =2Zε3µνρ Aµ (x)Fνρ (x)δ(Φ(x))dx, Fµν = ∂µ Aν − ∂µ Aν .Äëÿ ïëîñêîñòè x3 = 0 óðàâíåíèÿ Ýéëåðà-Ëàãðàíæà ðàñìàòðèâàåìîé ìîäåëè ïðåäñòàâëÿþò ñîáîé ìîäèôèöèðîâàííûå óðàâíåíèÿ Ìàêñâåëà:δS(A)= ∂µ F µν + aε3νσρ Fσρ δ(x3 ) = 0.δAν(2.1)Ìû ðåøèì èõ âîñïîëüçîâàâøèñü ïðåîáðàçîâàíèåì Ôóðüå ïî êîîðäèíàòàìx0 , x1 , x2 äëÿ âåêòîð-ïîòåíöèàëà Aµ :Z1Aµ (x) =eipx Aµ (p, x3 )dp.3(2π) 2Z1Aµ (p, x3 ) =eipx Aµ (p, x3 )dp.3(2π) 2(2.2)Çäåñü è â äàëüíåéøåì â ýòîé ãëàâå ìû èñïîëüçóåì îáîçíà÷åíèå p äëÿâåêòîðà p = (p0 , p1 , p2 ), p2 = p20 − p21 − p22 , px = p0 x0 − p1 x1 − p2 x2 .
Äëÿ ïîëÿA(x3 , p) óðàâíåíèÿ (2.1) çàïèñûâàþòñÿ â âèäå(2.3)ν=3:p2 A3 + ∂ 3 ipA = 0,ν 6= 3 :(−p2 − ∂32 )Aν − ipν (ipA − ∂3 A3 ) + 2aε3νηρ ipη Aρ δ(x3 ) = 0 (2.4) ñèëó âòîðîãî ðàâåíñòâà â (2.2), óñëîâèå âåùåñòâåííîñòè Aµ (x) = A∗µ (x)âåêòîð-ïîòåíöèàëà Aµ (x) äëÿ A(x3 , p) èìååò âèä A∗ (x3 , p) = A(x3 , −p). Âîñïîëüçîâàâøèñü ýòèì ñîîòíîøåíèåì ìû ïîëó÷àåì èíòåãðàëüíîå ïðåäñòàâëåíèåAµ (x) =Z13(2π) 2θ(p0 ) eipx Aµ (p, x3 ) + e−ipx A∗µ (p, x3 ) dp =27=Z2<(2π)32θ(p0 ) eipx Aµ (p, x3 ) dp(2.5)â êîòîðîì < îáîçíà÷àåò âåùåñòâåííóþ ÷àñòü è âåùåñòâåííîñòü Aµ (x) î÷åâèäíà.2.2.Âûáîð êàëèáðîâêèÓðàâíåíèå Ýéëåðà-Ëàãðàíæà (2.1) äëÿ ðàññìàòðèâàåìîé çàäà÷è èí-âàðèàíòíî îòíîñèòåëüíî êàëèáðîâî÷íîãî ïðåîáðàçîâàíèÿ Aµ (x) → Aµ (x) +∂µ ϕ(x), ïîýòîìó ðåøåíèå (2.1) íàõîäèòñÿ ñ òî÷íîñòüþ äî êàëèáðîâî÷íîãîïðåîáðàçîâàíèÿ, è ìû ìîæåì ïðè åãî ïîñòðîåíèè çàôèêñèâàòü êàëèáðîâêó.Ìû áóäåì ïðîâîäèòü ðàñ÷åòû â òåìïîðàëüíîé êàëèáðîâêåA0 = 0(2.6)Âîñïîëüçîâàâøèñü êàëèáðîâî÷íûì óñëîâèåì (2.6), ïåðåïèøåì óðàâíåíèÿ(2.3), (2.4) â âèäåp2 A3 − ∂ 3 (ip1 A1 + ip2 A2 ) = 0(2.7)p0 (ip1 A1 + ip2 A2 + ∂3 A3 ) − 2a(p1 A2 − p2 A1 )δ(x3 ) = 0(2.8)(−p2 − ∂32 )A1 + ip1 (ip1 A1 + ip2 A2 + ∂3 A3 ) + 2aip0 A2 δ(x3 ) = 0(2.9)(−p2 − ∂32 )A2 + ip2 (ip1 A1 + ip2 A2 + ∂3 A3 ) − 2aip0 A1 δ(x3 ) = 0(2.10)Ïîëîæèìip1 A1 + ip2 A2 + ∂3 A3 = α, A1 |x3 =0 = a1 , A2 |x3 =0 = a2 .Òîãäà èç óðàâíåíèÿ (2.8) ñëåäóåò, ÷òîα = 2ap 1 a2 − p 2 a1δ(x3 ),p0(2.11)28è, â ñèëó óðàâíåíèé (2.9), (2.10), ïîëÿ A1 , A2 , óäîâëåòâîðÿþò óðàâíåíèÿì(p2 + ∂32 )Ai + ci δ(x3 ) = 0, i = 1, 2,(2.12)â êîòîðûõc1 ≡ −2ia 22ia 2(p1 − p20 )a2 − p1 p2 a1 , c2 ≡(p2 − p20 )a1 − p1 p2 a2 .p0p0Òàêèì îáðàçîì â ðàññìàòðèâàåìîì íàìè ñëó÷àå òåìïîðàëüíîé êàëèáðîâêè íàì íóæíî íàéòè ðåøåíèÿ äâóõ îäíîòèïíûõ äèôôåðåíöèàëüíûõóðàâíåíèé (2.12).2.3.Ðåøåíèå óðàâíåíèÿ ∂t2 ψ + p2 ψ + cδ(t) = 0×òîáû íàéòè A(x3 , p) ðåøèì âíà÷àëå âñïîìîãàòåëüíóþ çàäà÷ó Íàé-äåì îáùåå ðåøåíèå óðàâíåíèÿ∂t2 ψ + p2 ψ + cδ(t) = 0(2.13)Ïóñòücce−ip|t|−iptiptf (t) ≡[θ(t)e+ θ(−t)e ] =2pi2piÏðîäèôôåðåíöèðîâàâ ýòó ôóíêöèþ ïî t, ïîëó÷èì :cf 0 (t) = − [θ(t)e−ipt − θ(−t)eipt ]2cpi[θ(t)e−ipt + θ(−t)eipt ] − cδ(t)f 00 (t) =2Òàêèì îáðàçîì, ôóíêöèÿ f (t) ÿâëÿåòñÿ îäíèì èç ðåøåíèé óðàâíåíèÿ (2.13).Îäíîðîäíîå óðàâíåíèå ∂t2 ϕ + p2 ϕ = 0 èìååò ðåøåíèå ϕ(t) = d1 eipt + d2 e−ipt ,ãäå d1 , d2 - ïðîèçâîëüíûå ïîñòîÿííûå.
Ñëåäîâàòåëüíî, îáùèì ðåøåíèåìóðàâíåíèÿ (2.13) ÿâëÿåòñÿiptψ(t) = d1 e−ipt+ d2 ece−ip|t|+,2pi(2.14)29ãäå d1 , d2 - ïðîèçâîëüíûå ïîñòîÿííûå. Ïîëó÷åííûé ðåçóëüòàò, êàê ìû ïîêàæåì äàëåå, ïîçâîëÿåò íàéòè ðåøåíèå ñèñòåìû óðàâíåíèé (2.7-2.10) â ÿâíîìâèäå.2.4.Ðåøåíèå óðàâíåíèé Ýéëåðà-ËàãðàæàÒàê êàê ôóíêöèÿ (2.14) ÿâëÿåòñÿ îáùèì ðåøåíèåì óðàâíåíèÿ (2.13),ìû ìîæåì çàïèñàòü îáùåå ðåøåíèå óðàâíåíèé äëÿ ïîëåé A1 , A2 â âèäåc1 e−iρ|x3 |A1 (x3 , p) =++,2iρc2 e−iρ|x3 |(2) iρx3(2) −iρx3A2 (x3 , p) = d1 e+ d2 e+.2iρ(1)d1 eiρx3ãäå ρ ≡p(1)d2 e−iρx3(2.15)(2.16)p2 .
Íåòðóäíî óáåäèòüñÿ, ÷òî èç ïðåäïîëîæåíèÿ îá îãðàíè÷åí-íîñòè Aj (x3 , p) ïðè ëþáûõ çíà÷åíèÿõ x3 ñëåäóåò, ÷òî p2 > 0, ïîýòîìó ìûðàññìîòðèì òîëüêî ñëó÷àé ρ > 0. Ïîëå A3 íàõîäèòñÿ íåïîñðåäñòâåííî èçóðàâíåíèÿ (2.7):A3 (x3 , p) =(3)d1 eiρx3+(3)d2 e−iρx3c3 e−iρ|x3 |+ (x3 ),2iρ(2.17)ãäå111(1)(2)(3)(1)(2)(3)d1 = − (p1 d1 + p2 d1 ), d2 = (p1 d2 + p2 d2 ), c3 = (p1 c1 + p2 c2 ),ρρρè (x3 ) ≡ x3 /|x3 |.Ïîëîæèâ x3 = 0 â (2.15), (2.16) è âîñïîëüçîâàâøèñü îáîçíà÷åíèÿìè(2.11), ìû ïîëó÷èì ñîîòíîøåíèÿ(j)(j)aj = d1 + d2 +(j)(j)cj, j = 1, 2.2iρÏóñòü Dj = (d1 + d2 )ρ, j = 1, 2. Òîãäà ñèñòåìà ëèíåéíûõ óðàâíåíèé äëÿ30a1 , a2 çàïèøåòñÿ â âèäåa1 (p0 ρ − ap1 p2 ) + aa2 (p21 − p20 ) = D1 p0 ,aa1 (p22 − p20 ) − a2 (p0 ρ + ap1 p2 ) = −D2 p0 .Ðåøèâ åe, ïîëó÷èìaD2 (p20 − p21 ) + D1 (ap1 p2 + p0 ρ),p0 ρ2 (a2 + 1)aD1 (p20 − p22 ) + D2 (ap1 p2 − p0 ρ)a2 = −,p0 ρ2 (a2 + 1)a1 =(2.18)(2.19)è2ai[D1 (ap0 ρ − p1 p2 ) − D2 (p20 − p21 )],p0 ρ(a2 + 1)2ai[D2 (ap0 ρ + p1 p2 )] + D1 (p20 − p22 )].c2 = −p0 ρ(a2 + 1)c1 = −Ïîëîæèâ~ = (A1 , A2 , A3 ), ~a = (a1 , a2 , a3 ), f~ = (c1 , c2 , c3 )/(2iρ),A(1)(2)(2)d~1 = (dj j, dj , dj ) j = 1, 2ïðåäñòàâèì ôîðìóëû (2.15 - 2.17) â êîìïàêòíîé ôîðìå:~ 3 , p) = d~1 (p)eiρx3 + d~2 (p)e−iρx3 + R(x3 )f~(p)e−iρ|x3 | .A(x(2.20)ãäå R(x3 ) äèàãîíàëüíàÿ ìàòðèöà ñ ýëåìåíòàìè R11 (x3 ) = R22 (x3 ) =1, R33 (x3 ) = (x3 ).
Òàêèì îáðàçîì, âîñïîëüçîâàâøèñü ñîîòíîøåíèÿìè (2.5),(2.20), ìû ïîëó÷àåì ñëåäóþùåå ïðåäñòàâëåíèå ðåøåíèÿ óðàâíåíèé ÝéëåðàËàãðàíæà äëÿ ðàññìàòðèâàåìîé ìîäåëè~A(x)=+θ(x3 )2<3(2π) 2Zθ(−x3 )2<3(2π) 2Znoi(px+ρx3 )i(px−ρx3 )~~~θ(p0 ) d1 (p) e+ [d2 (p) + f (p)] edp +noi(px+ρx3 )i(px−ρx3 )~~~θ(p0 ) [d1 (p) + T f (p)] e+ d2 (p) edp.
(2.21)31Çäåñü T = R(−1) äèàãîíàëüíàÿ ìàòðèöà ñ ýëåìåíòàìè T11 (x3 ) = T22 (x3 ) =1, T33 (x3 ) = −1. Ïåðâûå ñëàãàåìûå â ïîäûíòåãðàëüíûõ âûðàæåíèÿõ â(2.21) îïèñûâàþò âîëíû, äâèæóùèåñÿ â îòðèöàòåëüíîì íàïðàâëåíèè òðåòüåé îñè, à âòîðûå - â ïîëîæèòåëüíîì.2.5.Paccåÿíèå âîëí íà ïëîñêîñòèÄëÿ çàäà÷è ðàññåÿíèÿ âîëíû ñ âîëíîâûì âåêòîðîì ~k = (k1 , k2 , k3 ),ïàäàþùåé íà ïëîñêîñòü èç ïîëóïðîñòðàíñòâà ñ îòðèöàòåëüíîé êîîðäèíàòîéx3 ìû äîëæíû èìåòü â ïîëóïðîñòðàíñòâå x3 > 0 òîëüêî ïðîõîäÿùóþ âîëíó,äâèæóùóþñÿ îò ïëîñêîñòè x3 = 0 â ïîëîæèòåëüíîì íàïðàâëåíèè òðåòüåéîñè.
Ñëåäîâàòåëüíî, â (2.21) ìû äîëæíû ïîëîæèòü d~1 = 0.  ðåçóëüòàòåïîëó÷èì~A(x)=+θ(−x3 )2<3(2π) 2θ(x3 )2<(2π)Z32Z~ tr (p) ei(px−ρx3 ) dp+θ(p0 )Anoi(px+ρx3 )i(px−ρx3 )~~θ(p0 ) Ar e+ Ain edp.~ in (p), A~ r (p), A~ tr (p) ïàäàþùåé, îòðàæåííîé èãäå âåêòîðíûå àìïëèòóäû Aïðîõîäÿùåé âîëí çàïèñûâàþòñÿ â âèäå~ in (p) = d~2 (p), A~ r (p) = T f~ (p), A~ tr (p) = d~2 (p) + f~ (p).Aè òåì ñàìûì óäîâëåòâîðÿþò ñîîòíîøåíèþ~ r = T (A~ tr − A~ in ).A(2.22)Òàêèì îáðàçîì, âåêòîðíàÿ àìïëèòóäà îòðàæåííîé âîëíû ïîëó÷àåòñÿ èçðàçíîñòè àìïëèòóä ïàäàþùåé è ïðîõîäÿùåé âîëí èçìåíåíèåì â íåé çíàêàó òðåòüåé êîìïîíåíòû.322.6.Ñîáñòâåííûå ìîäûÑîáñòâåííûìè ìîäàìè ìû íàçîâåì âîëíû äëÿ êîòîðûõ àìïëèòóäûïàäàþùåé è ïðîõîäÿùåé âîëí ïðîïîðöèîíàëüíû äðóã äðóãó:~ tr = λA~ in .A(2.23)Äëÿ íèõ èç (2.22),(2.23) ñëåäóåò, ÷òî~ r = (λ − 1)T A~ in , a1 = λd(1) , a2 = λd(2) .A22(2.24)Âîñïîëüçîâàâøèñü (2.18),(2.19), íàéäåì ñîîòíîøåíèÿ, êîòîðûì â ñèëó äâóõ(1)(2)ïîñëåäíèõ ðàâåíñòâ â (2.24), äîëæíû óäîâëåòâîðÿòü d2 , d2 :(1)(2)(1)λd2ad (p2 − p21 ) + d2 (ap1 p2 + p0 ρ)= 2 0,p0 ρ(a2 + 1)(2)(1)(2)λd2ad (p2 − p22 ) + d2 (ap1 p2 − p0 ρ)=− 2 0.p0 ρ(a2 + 1)(1)(2)Äàííàÿ ñèñòåìà óðàâíåíèé èìååò íåíóëåâûå ðåøåíèÿ d2 , d2 , åñëè(a2 + 1)λ2 − 2λ + 1 = 0.Ýòî óðàâíåíèå èìååò äâà ðåøåíèÿ:λ=ii≡ λ1 , λ =≡ λ2 .i−ai+aÒàêèì îáðàçîì, èìåþòñÿ äâå ñîáñòâåííûå ìîäû, äëÿ êîòîðûõλ = λ1 ,λ = λ2 ,~ (1) = g1 V~1 ,Ain~ (2) = g2 V~2 ,AinãäåV~1 ≡ (p20 − p21 , −ip0 ρ − p1 p2 , −ip0 p2 + p1 ρ),V~2 ≡ (p20 − p21 , ip0 ρ − p1 p2 , ip0 p2 + p1 ρ),33è g1 , g2 - ïðîèçâîëüíûå ôóíêöèè îò p0 , p1 , p2 .
Èñïîëüçóÿ ýòè îáîçíà÷åíèÿ,~ in (p), A~ r (p), A~ tr (p) cîáñòâåíìû ìîæåì çàïèñàòü âåêòîðûå àìïëèòóäû Aíûõ ìîä ïàäàþùåé, îòðàæåííîé è ïðîõîäÿøåé âîëí â ñëåäóþùåì âèäå:(j) ~~ (j) = gj (p)V~j (p), A~ (j)~ (j)~Ar = gj (p)Kr T Vj (p), Atr = gj (p)Ktr Vj (p), j = 1, 2.inÇäåñü ìû èñïîëüçîâàëè îáîçíà÷åíèÿKr(1)−ia + a21 − ia1 + iaia + a2(1)(2)(2),K=,K=,K=.=trtrr1 + a21 + a21 + a21 + a2Ïîëó÷åííûå íàìè õàðàêòåðèñòèêè ñîáñòâåííûõ ìîä óäîâëåòâîðÿþò ñîîò-~2 = V~1∗ , Ktr(2) = Ktr(1)∗ , Kr(2) = Kr(1)∗ .íîøåíèÿì V2.7.Ðàññåÿíèå ïëîñêèõ âîëíÂûáåðåì â êà÷åñòâå ñîáñòâåííûõ ìîä ïëîñêèå âîëíû ñ âåêòîðíûìè(j)~ (p) = V~j (p), j = 1, 2. Ìû ðàññìîòðèì òàêæå èõàìïëèòóäàìè âèäà Ain~ (2) ñ êîýôôèöèåíòàìè C1 , C2 .~ (1) + C2 A~ in = C1 Aëèíåéíóþ êîìáèíàöèþ Ainin~ r (p), A~ tr (p) îòðàæåííîé è ïðîõîäÿùåé âîëí ìûÄëÿ âåêòîðíûõ àìïëèòóä Aèìååì:~ r = C1 Kr(1) T V~1 + C2 T Kr(1)∗ V~1 ∗ , A~ tr = C1 Ktr(1) V~1 + C2 Ktr(1)∗ V~1∗ .A(2.25)Âåêòîð V~1 ìîæíî ïðåäñòàâèòü â âèäå~ 1 + iU~ 2, U~ 1 ≡ (p20 − p21 , −p1 p2 , p1 ρ), U~ 2 ≡ (0, −p0 ρ, −p0 p2 ),V~1 = Uïîýòîìóa~1 − U~ 2 , Y~2 ≡ aU~2 + U~ 1,(Y~1 + iY~2 ), Y~1 ≡ aU1 + a2~ 1 + iZ~ 2 ), Z~1 ≡ U~ 1 + aU~ 2 = Y~2 , Z~2 ≡ U~ 2 − aU~ 1 = −Y~1 .(ZKr(1) V~1 =(1)Ktr V~1 =11 + a234Ñëåäîâàòåëüíî,~ in = (C1 + C2 )U~ 1 + i(C1 − C2 )U~ 2,Aa((C1 + C2 )T Y~1 + i(C1 − C2 )T Y~2 ),1 + a2~ tr = 1 ((C1 + C2 )Z~ 1 + i(C1 − C2 )Z~ 2 ).A1 + a2~r =A(2.26)Åñëè îáîçíà÷èòüκ1 = |C1 + C2 |, κ2 = |C1 − C2 |, φ1 = −i lnC1 + C2C2 − C1, φ2 = −i ln,|C1 + C2 ||C2 − C1 |òî âåêòîð-ïîòåíöèàëû ðàññìàòðèâàåìûõ íàìè ïëîñêèõ âîëí çàïèøóòñÿ ââèäå~ 2,~ in (p, x) = σ<A~ in (p)ei(px−ρx3 ) = κ1 αin U~ 1 + κ2 βin UA~ r (p, x) = σ<A~ r (p)ei(px+ρx3 ) = κ1 αr T Y~1 + κ2 βr T Y~2 ,A~ 2,~ 1 − κ2 βtr Z~ tr (p, x) = σ<A~ tr (p)ei(px−ρx3 ) = κ1 αtr ZAãäå σ = 2(2π)−3/2 ,αin = σ cos(px − ρx3 + φ1 ), βin = σ sin(px − ρx3 + φ2 ),αr =aσaσcos(px+ρx+φ),β=sin(px + ρx3 + φ2 ),31r1 + a21 + a2aaαtr =α,β=βin .intr1 + a21 + a2 ðàññìàòðèâàåìîé íàìè êàëèáðîâêå íàïðÿæåííîñòü ýëåêòðè÷åñêîãî ïîëÿ~ ñîâïàäàåò ñ ïðîèçâîäíîé ïî x0 îò âåêòîð ïîòåíöèàëà A~E~ in = σ< ip0 A~ in (p)e~ 1 + κ2 αin U~ 2 ),E= −p0 (κ1 βin Ui(px+ρx)3~ r = σ< ip0 A~ r (p)eE= −p0 (κ1 βr T Y~1 + κ2 αr T Y~2 ),i(px−ρx3 )~~~ 1 + κ2 αtr Z~ 2 ).Etr = σ< ip0 Atr (p)e= −p0 (κ1 βtr Zi(px−ρx3 )(2.27)35~ = ∂~ × A,~ èç êîòîðîé íåïîñðåäÌàãíèòíîå ïîëå âû÷èñëÿåòñÿ ïî ôîðìóëå Hñòâåííî ïîëó÷àåòñÿ ñëåäóþùèé ðåçóëüòàò~~~~~~~ in = − Pin × Ein , H~ r = − Pr × Er , H~ tr = − Ptr × Etr ,Hp0p0p0(2.28)~ in , H~ r, H~ tr - âåêòîðà íàïðÿæåííîñòåé ìàãíèòíîãî ïîëÿ ïàäàþùåé,ãäå H~in = P~tr = (p1 , p2 , ρ), P~r = (p1 , p2 , −ρ).îòðàæåííîé, ïðîõîäÿùåé âîëí è PÏîäñòàâèâ â (2.28) âåêòîðû íàïðÿæåííîñòåé ýëåêòðè÷åñêèõ ïîëåé (2.27),ïîëó÷èì~ in = p0 (κ2 αin U~ 1 − κ1 βin U~ 2 ),H~ r = −p0 (κ2 αr T Y~1 − κ1 βr T Y~2 ),H~ tr = p0 (κ2 αtr Z~ 1 − κ1 βtr Z~ 2 ).HÄëÿ èíòåíñèâíîñòåé Iin , Ir , Itr ïàäàþùåé, îòðàæåííîé è ïðîõîäÿùåéâîëí ìû èìååì ñëåäóþùèå âûðàæåíèÿ~ r (p))|2 , I~tr = |σ A~ tr (p))|2~ in (p)|2 , I~r = |σ A~ in (p)ei(px−ρx3 ) |2 = |σ AIin ≡ |σ A~ j , Y~j , Z~ j (2.25) è ìàòðèöû TÍåïîñðåäñòâåííî èç îïðåäåëåíèé âåêòîðîâ Uïîëó÷àåì:~ 1U~ 2 = Y~1 Y~2 = Z~ 1Z~ 2 = 0, U~ 1U~1 = U~ 2U~ 2 = p20 (p20 − p21 ).U(2.29)~ 1Z~1 = Z~ 2Z~ 2 = p20 (p20 − p21 )(1 + a2 ), T T T = T 2 = 1Y~1 Y~1 = Y~2 Y~2 = Z(2.30) ñèëó (2.26), (2.29), (2.30)a21Ir =I,I=Iin .intr1 + a21 + a2Ñäåäîâàòåëüíî, êîýôôèöèåíòû îòðàæåíèÿ Kr ≡ Ir /Iin è ïðîõîæäåíèÿKtr ≡ Itr /Iin ïëîñêîé âîëíû ïðè åå ðàññåÿíèè íà ïëîñêîñòè íå çàâèñÿò îò36÷àñòîòû, óãëà ïàäåíèÿ è âûðàæàþòñÿ ÷åðåç êîíñòàíòó a âçàèìîäåéñòâèÿýëåêòðîìàãíèòíîãî ïîëÿ ñ ïîâåðõíîñòüþ, õàðàêòåðèçóþùóþ ñâîéñòâà ååìàòåðèàëà:Kr =a21,K=.tr1 + a21 + a2Ðàññìîòðèì äâèæåíèÿ âîëí âäîëü îñè z3 .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
