繞軸旋轉電子束產生兆赫波電子迴旋脈射機制研究

Graduate Institute of Physics College of Science

National Taiwan University Doctoral dissertation

Terahertz Electron Cyclotron Maser Interactions with an Axis-encircling Electron Beam

Guan-De Li

Advisor: Kwo-Ray Chu, Ph.D.

104 11

November 2015

i

Acknowledgement

THEORETICAL NANOPHYSICS Group

Schrodinger's equation

R616

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2015.11.11 ii

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dt = Ωc = Ωe/γ URXRV

mec²dγ

dt = − eEx(t)vx(t) = eEx0v_⊥+Qb(ωt) bBM φ = −1

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γ = γ₀ URXkV

φ = Ωc0t + φ0 URXk#V

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dγ

dt = eEx0v_⊥0 2mec²

#− bBM!

(ω− Ωc0)t

bHQr − φ0

"

+bBM!

(ω + Ωc)t

7bi

+ φ0

"$

≃ eEx0v_⊥0 2mec² bBM φ0

URXjV

γ ≃ γ⁰+ eEx0v_⊥0t

2m_ec² bBM φ0, URXj#V

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dφ

dt = Ωc = Ωe

γ ≃ Ωe

γ0

%

1 + eEx0v_⊥0t 2γ0mec² bBM φ0

&₋₁

≃ Ωc0

!1− eEx0v_⊥0t 2γ0mec² bBM φ0

" URX9V

φ≃ Ωc0t− eEx0v_⊥0Ωc0t²

4γ0mec² bBM φ0+ φ0 = Ωc0

!1− eEx0v_⊥0t 4γ0mec² bBM φ0

"

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mec²dγ

dt = eEx0v_⊥0+Qb(ωt) bBM φ ≃ eE^x0v_⊥0+Qb(ωt) bBM!

Ωc0t + φ0− eEx0v_⊥0Ωc0t² 4γ₀m_ec² bBM φ0

"

= eEx0v_⊥0+Qb(ωt) bBM!

(Ωc0t + φ0)− ϵ"

= eEx0v_⊥0+Qb(ωt)%

bBM(Ωc0t + φ0)+Qb ϵ − +Qb(Ω^c0t + φ0)bBM ϵ&

≃ eEx0v_⊥0+Qb(ωt)%

bBM(Ωc0t + φ0)− +Qb(Ωc0t + φ0)ϵ&

≃ eEx0v_⊥0+Qb(ωt) bBM(Ωc0t + φ0)− e²E_x0² v_⊥0² Ωc0t²

4γ0mec² +Qb(ωt) +Qb(Ωc0t + φ0)bBM φ0

1 2

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< mec²dγ

dt >φ0= 1 2π

' 2π 0

dφ0

(mec²dγ dt

)= e²E_x0² v_⊥0² Ωc0t²

8γ0mec² +Qb(ωt) bBM(Ωc0t)

1 2

!bBM(ωt+Ωc0t)−bBM(ωt−Ωc0t)", URXeV r?2`2 i?2 `2HiBQM Q7 < bBM(nθ − θ0) >_θ 4 y Bb mb2/ UM Bb M BMi2;2`VX A;MQ`2 i?2 7bi p`BiBQM i2`K bBM(ωt + Ωct)-

< mec²dγ

dt >φ0=−e²E_x0² v_⊥0² Ωc0t² 16γ0mec² bBM!

(ω− Ω^c0)t"

URXdV

h?2 /2imMBM; #2ir22M T?b2b Q7 i?2 rp2 M/ i?2 2H2+i`QM /2+B/2b i?2 /B`2+iBQM Q7 2M2`;v ~QrX 1H2+i`QMb rBHH HQb2 2M2`;v B7 7`2[m2M+v Q7 `7 }2H/b Bb 7bi2` i?M 2H2+i`QM +v+HQi`QM 7`2[m2M+v- M/ pB+2 p2`b. ++Q`/BM; iQ +QMb2`piBQM Q7 2M2`;v- B7 2H2+i`QMb HQb2f;BM 2M2`;v BM p2`;2- `7 }2H/b rBHH ;`Qrf/KT i i?2 bK2 iBK2X

RXj xBKmi?H M/ tBH "mM+?BM;

Gbi b2+iBQM- r2 B;MQ`2 i?2 `7 "@}2H/ M/ ;2i  bB;MB}+Mi +QM+2Ti Q7 i?2 #bB+

K2+?MBbK rBi?Qmi /B{+mHi /2`BpiBQMbX LQr r2 iF2 i?2 `7 "@}2H/ #+F M/ MHvx2 8

Bib 2z2+iX(R) qBi? i?2 2tBbi2M+2 Q7 `7 "@}2H/- i?2 z /BbTH+2K2Mi Q7 2H2+i`QM Bb MQi x2`QX >2M+2- z /2T2M/2M+2 Q7 `7 }2H/b b?QmH/ #2 BM+Hm/2/ b

1_⊥ = E0! +Qb(ωt − k^zz)ˆ2x+bBM(ωt − kzz)ˆ2y

" URX3V

"_⊥ = k_zc

ω ˆ2z× 1⊥, URX3#V

M/  +B`+mH` THM2 rp2 BM 7`22 bT+2 Bb bbmK2/X "2bB/2b- i?2 i`Mbp2`b2 +QK@

TQM2Mi Q7 i?2 2H2+i`QM p2HQ+Biv Bb p_⊥ = v_⊥(

− bBM φˆ2^x++Qb φˆ2y) M/ z +QKTQM2Mi Bb x2`Q BMBiBHHvX *QMbB/2` i?2 ;v`iBQMH KQiBQM M/ i?2 tBH i`MbBi Q7 2H2+i`QMb- r2 /2}M2 M 2z2+iBp2 +v+HQi`QM T?b2 Ui.e. i?2 2H2+i`QM 2z2+iBp2 +v+HQi`QM T?b2V b φ₂₇₇ = kzz + φX >2M+2 Bib /2`BpiBp2

Ω₂₇₇ = dφ₂₇₇

dt ≃ kzv_z+ Ω_c URXNV

L2ti- #2+mb2 i?2 KQK2MimK M/ 2M2`;v +?M;2 rBi? iBK2 /m`BM; i?2 BMi2`+iBQM-

∆Ω₂₇₇ = kz∆vz+ Ωe∆(1

γ) = kz∆vz− Ωc

γ ∆γ. URXRyV

M/ i?2M- r2 i`v iQ Q#iBM ∆γ M/ ∆vz 7`QK i?2 2[miBQM Q7 KQiBQMbX d

dt(γmec²) = − e1⊥· p⊥ URXRRV

∆γ = − e

mec²(p_⊥· 1⊥

)∆t; URXRR#V

d

dt(γmevz) = γme(d

dtvz) + (d

dtγ)mevz

≃0

= − e!1_⊥+p

c × (B⁰ˆ2z+"_⊥)"

· ˆ2^z

= − e

c!p_⊥× (kzc

ω ˆ2z× 1⊥)"

· ˆ2^z

= − ekz

ω

!ˆ2z(p_⊥· 1⊥)− (pz· ˆ2z)"

· ˆ2z

= − ekz

ω (p_⊥· 1⊥

) URXRkV

∆vz = − ek_z

γmeω(p⊥· 1⊥)

∆t. URXRk#V

e

hF2 URXRR#V M/ URXRk#V BMiQ URXRyV- M/ ;2i

∆Ω₂₇₇ = − ek_z²

γmeω(p_⊥· 1⊥

)∆t + eΩc

γmec²(p_⊥· 1⊥

)∆t

≃ e

γmeω (ω²

c² − k²z)(p_⊥· 1⊥

)∆t,

URXRjV

r?2`2 bvM+?`QMBbK +QM/BiBQM ω ≃ Ω^c Bb mb2/X _2bmHib BM/B+i2 i?i i?2`2 `2 irQ KQ/mHi2/ [mMiBi2b- γ M/ vz- iQ +QMi`QH i?2 /2imMBM;X xBKmxH #mM+?BM; Bb 7Q`K2/ #v `2/Bbi`B#miBM; γ /m2 iQ i?2 `7 1@}2H/- r?BH2 tBH #mM+?BM; Bb 7Q`K2/

#v `2/Bbi`B#miBM; vz /m2 iQ i?2 `7 "@}2H/X AM URXRjV- ω<sup>2</sup>/c<sup>2</sup> M/ k<sub>z</sub><sup>2</sup> i2`Kb `2T`2@

b2Mi xBKmi?H M/ tBH #mM+?BM; `2bT2+iBp2HvX h?2 +QMi`B#miBQMb Q7 irQ #mM+?BM;

K2+?MBbKb `2 QTTQbBi2 /B`2+iBQMb- r?B+? `2bmHi BM +M+2HHiBQM Q7 xBKmi?H M/

tBH #mM+?BM;X

RX9 Pp2`pB2r

6Q` i?2 ;v`Qi`QM BM h>x `2;BK2- i?2 ?`KQMB+ BMi2`+iBQMb Bb  F2v Bbbm2 7Q`

+m``2Mi bim/vX q2 BK iQ mM/2`biM/ i?2 Q#bi+H2 Q7 ?B;? 7`2[m2M+v- /2;`/iBQM Q7 TQr2`- KQ/2 +QKT2iBiBQMb M/ +QM+2`M2/ iBK2@/2T2M/2Mi T?2MQK2MX AM *?Ti2`

k- bQK2 Bbbm2b bbQ+Bi2/ rBi? h>x ;v`Qi`QMb BM+Hm/BM; i?2 P?KB+ HQbb 2z2+i M/

i?2 +?QB+2 Q7 T`K2i2` bT+2 `2 BMp2biB;i2/X *?Ti2` j /2b+`B#2b 7Q`KmH2 Q7 MmK2`B+H K2i?Q/b b r2HH b bQK2 i`B+Fb Q7 ++2H2`iBM; +H+mHiBQMX *?Ti2` 9 2tKBM2b #Qi? HBM2` M/ MQMHBM2` ?`KQMB+ BMi2`+iBQMb- 7QHHQr2/ #v  +QM+HmbBQM BM *?Ti2` 8X

d

*?Ti2` k

Abbm2b Q7 i?2 :v`Qi`QM BM i?2 h>x _2;BK2

kXR liBHBiv Q7 Pp2`KQ/2/ ai`m+im`2b

PM2 Q7 i?2 F2v 7+iQ`b Q7 TQr2` /2;`/2/ BM h>x `2;BK2 Bb P?KB+ rHH HQbbX hQ

MHvx2 TQr2` /BbbBTiBQM- }`bi r2 BMi`Q/m+2 i?2 /2}MBiBQM Q7 ii2MmiBQM +QMbiMi 7Q` TQr2`

β =−1 P

dP

dz, UkXRV

r?2`2 P Bb 7Q`r`/ rp2 TQr2`- M/ dP /dz-  M2;iBp2 pHm2- Bb i?2 P?KB+ TQr2`

/BbbBTi2/ QM i?2 bB/2 rHH T2` mMBi H2M;i?X h?2 bm`7+2 +m``2Mi BM/m+2/ #v `7 }2H/

i 7`2[m2M+v f ~Qrb HQM; i?2 bB/2 rHH rBi?BM  72r bFBM /2Ti?b Uδ ∼ 1/√

fV `2bmHib BM P?KB+ TQr2` T2` mMBi `2 T`QTQ`iBQMH iQ √fX "2+mb2 +B`+mK72`2M+2 Q7  rp2;mB/2 Bb 2πrw- P?KB+ TQr2` T2` mMBi H2M;i? Bb T`QTQ`iBQMH iQ rw√

fX PM i?2 Qi?2` ?M/- +`Qbb@b2+iBQM Q7  rp2;mB/2 Bb πr_w²- r?B+? BKTHB2b i?i 7Q`r`/ rp2

3

0 5 10 15 xmn

0.5 1.0 1.5 2.0 2.5 3.0 3.5

(dB/cm)

0,1

0,2 1,1

1,2 2,1

2,2 3,1

3,2 4,1

4,2 5,1

5,2 6,1

6,2 7,1

7,2 8,1

8,2 9,1

9,2 10,1

10,2 11,1

11,2 12,1

12,2 13,1

14,1 15,1

16,1 17,1

n = 1 n = 2 n > 2

⇠

6B;m`2 kXR, ii2MmiBQM +QMbiMi 7Q` TQr2` pb xmn 7Q` 2+? h1mn KQ/2 i QT2`iBM;

7`2[m2M+v Ryyy :>xX bbmK2  +QTT2` rHH Q7 `2bBbiBpBiv ρ<sub>*m</sub> = 1.72× 10<sup>−8</sup> Ω@K 7Q`

+H+mHiBQMX PMHv KQ/2b rBi? n 4 R M/ k `2 /2MQi2/ #v irQ BM/B+2b Um, nVX PT2`iBM;

7`2[m2M+v Bb bHB;?iHv #Qp2 +miQz 7`2[m2M+v Uf = 1.06fcVX >2M+2- +miQz 7`2[m2M+v Bb  +QMbiMi- r?B+? Bb KBMiBM2/ 7Q` ?B;? Q`/2` KQ/2 #v  H`;2 rHH `/BmbX 6`QK U2.2V- r2 FMQr i?i β /2+`2b2b rBi? BM+`2bBM; rw UQ` xmn).

TQr2` Bb T`QTQ`iBQMH iQ r²_wX >2M+2- Bi Bb 2bv iQ b?Qr i?i(Ry)

β ∼

√f rw

. UkXkV

:`Qri? Q7 β rBi? BM+`2bBM; 7`2[m2M+v Bb /2i`BK2MiHX hQ `2K2/v T`Q#H2K Q7 2t@

+22/2/ P?KB+ HQbb BM h>x `2;BK2- M Qp2`KQ/2/ bi`m+im`2 Bb /QTi2/X β Bb `2/m+2/

iQ  bKHH pHm2 rBi? H`;2 bBx2 Q7 rwX hQ KBMiBM +miQz 7`2[m2M+v M2` h>x `2;BK2-

?B;? Q`/2` KQ/2 Bb 2bb2MiBHX *QMbB/2` M Qp2`KQ/2/ bi`m+im`2 QT2`i2/ 7Q` /Bp2`b2 h1mn KQ/2b rBi? +QMbiMi 7`2[m2M+v M/ f = 1.06fc- r?B+? Bb  ivTB+H QT2`iBM;

7`2[m2M+v 7Q`  ;v`Qi`QMX 6B;X kXRX /BbTHvb i?i ii2MmiBQM +QMbiMi β pb xmn 7Q`

T`QT;iBM;  rp2 i +QMbiMi 7`2[m2M+v Ryyy :>xX Ai Bb b?QrM i?i- BM ;2M2`H- N

i?2 ?B;?2` Q`/2` KQ/2 i?2 H`;2` rHH `/Bmb Bb- M/ ?2M+2 i?2 HQr2` P?KB+ HQbb BbX β Bb ∼j /" U∼ 50WV T2` +K 7Q` 7mM/K2MiH KQ/2 h111 KQ/2- r?2`2b β Bb #2HQr yX8 /" T2` +K 7Q` ?B;? Q`/2` KQ/2b UrBi? xmn >Ry M/ M > RVX AM //BiBQM- T`Q#H2Kb Q7 7#`B+iBQM M/ iQH2`M+2 +QmH/ #2 HH2pBi2/ rBi? H`;2` bBx2 Q7 rwX PM i?2 Qi?2`

?M/- HBM2` i?2Q`v T`2/B+ib i?i  ?B;? Q`/2` KQ/2 QT2`i2/ i  +2`iBM "@}2H/

Bb Q7i2M 2t+Bi2/ rBi? KMv +QKT2iBM; KQ/2b- r?B+? T`2p2Mi ?B;? +m``2Mi QT2`iBQM 7Q`  bi2/v bBM;H2 KQ/2 2t+BiiBQMX

Ai Bb rQ`i? MQiBM; i?i β Q7 bQK2 KQ/2b UrBi? m > 0 M/ n = 1V 7Q`Kb MQi?2`

i`2M/ rBi? H`;2 pHm2bX AM `2HBiv- bbmKBM; i?i i?2 7Q`r`/ rp2 T`QT;i2b BM

 mMB7Q`K rp2;mB/2 rBi?Qmi `2~2+iBQM- i?2 P?KB+ TQr2` T2` mMBi H2M;i? M/ rp2 TQr2` `2(e)

−dP dz = 1

8x²_mnKmnωk_mn² δ rw

(1 + m² x²_mn− m²

ω² ω_c²)*

*f**² UkXjV

Pnet = 1

8x²_mnKmnωkz

**f**², UkXj#V

r?2`2 Kmn = J_m²(xmn)(

1− _x^m2_mn² )- Jm(x) Bb mi? Q`/2` "2bb2H 7mM+iBQM- xmn Bb ni?

bQHmiBQM Q7 J_m^′ (x) = 0X h?2`27Q`2- i?2 ii2MmiBQM +QMbiMi +M #2 2tT`2bb2/ b

β = k_mn² kz

δ rw

(1 + m² x²_mn− m²

ω²

ω_c²), UkX9V

r?B+? Bb ;`22/ rBi? UkXkVX Ux<sup>2</sup><sub>mn</sub>− m<sup>2</sup>V i2`K BM UkX9V- r?B+? Bb bKHH r?2M KQ/2 BM/B+2b biBb7v m > 0 M/ n = 1- 2tTHBMb r?v P?KB+ HQbb Bb 2M?M+2/X h?2 `2bmHi bm;;2bib i?i QT2`iBQM 7Q` h>x `2;BK2 rBi?  KQ/2 biBb7vBM; n = 1 Q` iQQ bKHH xmn Bb BMTT`QT`Bi2X

Ry

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

?

10 ¹⁰ 10 ⁸ 10 ⁶ 10 ⁴ 10 ² 10⁰

J02 s(s?)

s = 1

s = 2

s = 3

s = 4

s = 5 s = 6

?0

exact

leading-order

6B;m`2 kXk, .2T2M/2M+2 Q7 #2K@rp2 +QmTHBM; bi`2M;i? Hsm QM MQ`KHBx2/ 2H2+i`QM i`Mbp2`b2 p2HQ+Biv β_⊥ rBi? /Bz2`2Mi sX "2K T`K2i2`b, Vb 4 jy Fo M/ α 4 RXRX aQHB/

M/ /b? +m`p2 `2 2t+i pHm2 M/ H2/BM;@Q`/2` i2`K pHm2 Q7 J<sup>′2</sup><sub>s</sub>(sβ<sub>⊥</sub>)`2bT2+iBp2HvX h?2 p2`iB+H /Qi HBM2 Bb H#2H Q7 BMBiBH β_⊥0X(RR)

kXk >`KQMB+ AMi2`+iBQM

hQ F22T bmbiBM2/ i?2 BMi2`+iBQM #2ir22M `7 }2H/ M/ 2H2+i`QMb BM ;v`iBQMH KQiBQM- i`MbBi M;H2

Θ = (ω− kzv_z− sΩc)τ, UkX8V

i?2 iQiH T?b2 Q7 rp2 b 2tT2`B2M+2/ #v i?2 2H2+i`QM- b?QmH/ #2  bKHH pHm2- r?2`2 kz Bb +QH/ +B`+mBi T`QT;iBQM +QMbiMi i 7`2[m2M+v ω M/ τ Bb 2H2+i`QM i`MbBi iBK2

i 2H2+i`QM bT22/ vzX 6Q` bBKTHB+Biv- Bi +M #2 2tT`2bb2/ b bvM+?`QMBbK +QM/BiBQM ω≃ sΩcX hQ T`Q/m+2 1*J `/BiBQM i R h>x #v 7mM/K2MiH +v+HQi`QM ?`KQMB+

BMi2`+iBQM s = 1- i?2 2H2+i`QM ;v`i2b i #Qmi R h>x b r2HH- r?B+? `2[mB`2b "@}2H/

∼ 40 hX >Qr2p2`- i?2 QMHv rv iQ bmTTHv  "@}2H/ ∼ 40 h MQr/vb Bb miBHBxBM;  RR

TmHb2/ K;M2i- r?B+? +M MQi #2 +QMiBMmQmb rp2 U+rV QT2`iBQMX 6Q` i?2 Tm`TQb2 Q7 h>x `/BiBQM- ?B;?@?`KQMB+ BMi2`+iBQM Us > 1V Bb 7pQ`#H2 iQ `2/m+2 `2[mB`2K2Mi Q7 "@}2H/ #v  7+iQ` sX Ai Bb 72bB#H2 iQ +?B2p2 h>x `2;BK2 7Q` +r QT2`iBQM Q7 ?B;?@

?`KQMB+ BMi2`+iBQM rBi?  "@}2H/ ∼ 20 h #v  +`vQ;2MB+ K;M2iX h?2 }`bi i?`22 2tT2`BK2Mib Q7 ;v`Q@KQMQi`QMb iQ #`2F i?2 h>x ;T `2 7mM/K2MiH- b2+QM/- M/

i?B`/ +v+HQi`QM ?`KQMB+ BMi2`+iBQMb- `2bT2+iBp2Hv(d- 3- N)X

hQ `2HBx2 ?Qr ?`KQMB+ BMi2`+iBQM Bb BM~m2M+2/ #v /Bz2`2Mi s- #2K@rp2 +Qm@

THBM; bi`2M;i? Bb

Hsm = J_s−m² (kmnrC)J^′2_s(kmnrL). UkXeV lbBM; i?2 QT2`iBM; M/ bvM+?`QMBbK +QM/BiBQMb ωc ≃ ω ≃ sΩ^c- i?2 kmnrLi2`K +M

#2 TT`QtBKi2/ b

kmnrL =(ωc

c )(v_⊥

Ωc

) ≃(sΩc

c )(v_⊥

Ωc

)= sβ_⊥. UkXdV

6B;X kXk /BbTHvb i?i /2T2M/2M+2 Q7 #2K@rp2 +QmTHBM; bi`2M;i? Hsm QM i?2 FBM2bii2 Q7 M 2H2+i`QM- r?B+? Bb MQ`KHBx2/ 2H2+i`QM i`Mbp2`b2 p2HQ+Biv β_⊥- rBi?

/Bz2`2Mi sX Ai TT2`b i?i Hsm /2+`2b2b `TB/Hv rBi? BM+`2bBM; sX 6Q` 7m`i?2`

mM/2`biM/BM;- H2/BM;@Q`/2` i2`K Bb 2tKBM2/ b 7QHHQrbX

J_n^′(x) = d

dxJn(x) = d dx

% 1 n!

(x 2

)n

+?B;? Q`/2` i2`Kb&

≃ 1

2(n− 1)!

(x 2

)n−1 UkX3V

n! = n× (n − 1) × (n − 2) × ... × 3 × 2 × 1 Bb 7+iQ`BH 7mM+iBQMX >2M+2-

J^′2_s(sβ_⊥)≃! 1 2(s− 1)!

(sβ_⊥ 2

)s−1"2

= 1

2^2s!

(s− 1)!"2(sβ_⊥)^2s−2 = (s/2)^2s−2 4!

(s− 1)!"2β_⊥^2s−2 UkXNV

*QKT`BM; 2t+i pHm2 M/ H2/BM;@Q`/2` i2`K pHm2 BM 6B;X kXk- r2 Q#b2`p2 i?i irQ +m`p2b `2 +HQb2 7Q` 2+? sX b  `2bmHi- MHvbBb Q7 H2/BM;@Q`/2` #2?pBQ` Bb

?2HT7mH iQ BMp2biB;i2 ?`KQMB+ BMi2`+iBQM /2;`/iBQMX h#H2 kXR Bb H2/BM;@Q`/2`

Rk

h#H2 kXR, G2/BM;@Q`/2` i2`K Q7 /2T2M/2M+2 Q7 #2K@rp2 +QmTHBM; bi`2M;i? Hsm QM MQ`KHBx2/ 2H2+i`QM i`Mbp2`b2 p2HQ+Biv β_⊥ 7Q` i?2 }`bi bBt +v+HQi`QM ?`KQMB+ MmK#2` sX UkXNV `2 mb2/ 7Q` +H+mHiBM; +Q2{+B2MibX

+v+HQi`QM ?`KQMB+ MmK#2` s H2/BM;@Q`/2` i2`K Q7 J^′2_s(x) H2/BM;@Q`/2` i2`K Q7 J^′2_s(sβ_⊥)

R ¹₄ 4 2.50 × 10^{−1 1}₄ 4 2.50 × 10⁻¹

k ₁₆¹x² 4 6.25 × 10⁻² x^{2 1}₄β_⊥² 4 2.50 × 10⁻¹ β_⊥² j ₂₅₆¹ x⁴ 4 3.91 × 10⁻³ x⁴ ₂₅₆⁸¹β_⊥⁴ 4 3.16 × 10⁻¹ β_⊥⁴ 9 ₉₂₁₆¹ x⁶ 4 1.09 × 10⁻⁴ x^{6 4}₉β_⊥⁶ 4 4.44 × 10⁻¹ β_⊥⁶ 8 ₅₈₉₈₂₄¹ x⁸ 4 1.70 × 10⁻⁶ x^{8 390626}₅₈₉₈₂₄β_⊥⁸ 4 6.62 × 10⁻¹ β_⊥⁸ e _58982400¹ x¹⁰ 4 1.70 × 10⁻⁸ x^{10 6561}₆₄₀₀β_⊥¹⁰ 4 1.03 × 10⁰ β_⊥¹⁰

i2`K 7Q` }`bi bBt +v+HQi`QM ?`KQMB+ MmK#2` sX AM i?2 7mM+iBQMb Q7 β_⊥- i?2 K;MB@

im/2b Q7 H2/BM;@Q`/2` +Q2{+B2Mib `2 `Qm;?Hv i?2 bK2X h?2`27Q`2- /2T2M/2M+2 Q7 Hsm QM β_⊥ Bb #Qmi β_⊥^2s−2 7Q` +v+HQi`QM ?`KQMB+ MmK#2` sX 6Q` ?`KQMB+ BMi2`+@

iBQM s > 1- H2/BM;@Q`/2` i2`K bHB;?iHv Qp2`2biBKi2b i?2 2t+i pHm2 rBi? H`;2 β<sub>⊥</sub>X 6Q` 7mM/K2MiH@?`KQMB+ BMi2`+iBQM s = 1- H2/BM;@Q`/2` i2`K Bb yXk8- r?B+? Bb  +QMbiMic ?Qr2p2`- 2t+i pHm2 /2+`2b2b HBiiH2 rBi? BM+`2bBM; β_⊥X

G2/BM;@Q`/2` #2?pBQ` `2p2Hb i?i i?2 ?B;?2` i?2 +v+HQi`QM ?`KQMB+ s- i?2

?B;?2` i?2 b2MbBiBpBiv Q7 Hsm BbX .m`BM; BMi2`+iBQM- 2H2+i`QMb +M #2 bTHBi BMiQ irQ +i2;Q`B2b, 7pQ`#H2 2H2+i`QMb- r?B+? i`Mb72` i?2B` FBM2iB+ 2M2`;v BMiQ rp2-

M/ mM7pQ`#H2 2H2+i`QMb- r?B+? ;BM i?2B` FBM2iB+ 2M2`;v 7`QK rp2X AM i?2 MQM@

HBM2` bi;2- ?`KQMB+ BMi2`+iBQM #2K@rp2 +QmTHBM; bi`2M;i? #2+QK2b r2F2` M/

r2F2` 7Q` 7pQ`#H2 2H2+i`QMb- M/ bi`QM;2` M/ bi`QM;2` 7Q` mM7pQ`#H2 2H2+i`QMbX(RR) JQ`2Qp2`- ?`KQMB+ BMi2`+iBQM /2;`/iBQM Bb KQ`2 BMi2Mb2 7Q`  ?B;?2` s KQ/2- r?B+? ;Bp2b  BM?2`2Mi /pMi;2 7Q`  HQr2`@?`KQMB+ KQ/2 iQ Qp2`+QK2  ?B;?2`@

Rj

0 5 10 15 20 B (T)

10 ⁵ 10 ⁴ 10 ³ 10 ² 10 ¹ 10⁰

Hsm

s = 1 s = 2 s = 3 (a)

0 5 10 15 20

B (T)

s = 1 s = 2 s = 3 (b)

6B;m`2 kXj, "2K@rp2 +QmTHBM; bi`2M;i? Hsm pb "@}2H/ 7Q` 2+? h1^(s)mn KQ/2 rBi? UV

tBb@2M+B`+HBM; 2H2+i`QM #2Kb UrC 4 y +KV M/ U#V Qz@tBb 2H2+i`QM #2Kb UrC 4 yXyj8 +KVX "2K M/ bi`m+im`2 T`K2i2`b, Vb 4 jy Fo- α 4 RXR M/ rw 4 yXkyk +KX

?`KQMB+ KQ/2X h?2 7mM/K2MiH@?`KQMB+ KQ/2 s = 1 TQbb2bb2b  i`2K2M/Qmb

/pMi;2 #2+mb2 Q7 i?2 ?B;?2bi Hsm M/ BMb2MbBiBp2M2bb Q7 Hsm iQ β_⊥X

kXj tBb@2M+B`+HBM; 1H2+i`QM "2Kb 7Q` >`KQMB+

AMi2`+iBQMb

AM //BiBQM iQ G`KQ` `/Bmb rL UQ` β_⊥V- Hsm HbQ /2T2M/b QM ;mB/BM; +2Mi2`

`/Bmb rC b J<sub>s</sub><sup>2</sup><sub>−m</sub>(kmnrC)X JtBKmK Q7 "2bb2H 7mM+iBQM Js−m(kmnrC)Q++m`b i s = mM/ rC = 0- r?B+? bm;;2bib mb2b Q7 M tBb@2M+B`+HBM; 2H2+i`QM #2KX 6m`i?2`KQ`2- i?2`2 Bb MQ 2t+BiiBQM rBi? s ̸= m KQ/2b- r?Qb2 H^sm Bb x2`QX 6Q`  K2MBM;7mH +T`BbQM- r2 `2Hi2 Hsm rBi? "@}2H/ #v mb2b Q7 i?2 QT2`iBM; M/ bvM+?`QMBbK +QM/BiBQMb ωc ≃ ω ≃ sΩc ;BM-

ωc = cxmn

rw ≃ sΩc = s eB

γmc, UkXRyV

R9

0 5 10 15 20 B (T)

10 ⁵ 10 ⁴ 10 ³ 10 ² 10 ¹ 10⁰

Hsm

s = 1 s = 2 s = 3 (a)

0 5 10 15 20

B (T)

s = 1 s = 2 s = 3 (b)

6B;m`2 kX9, "2K@rp2 +QmTHBM; bi`2M;i? Hsm pb "@}2H/ 7Q` 2+? h1^(s)mn KQ/2 rBi?

BKT2`72+i tBb@2M+B`+HBM; 2H2+i`QM #2Kb UV rC 4 yXyyR +K M/ U#V rC 4 yXyy8 +KX JQ/2b biBb7vBM; s ̸= m `2 /2MQi2/ #v bi`bX "2K M/ bi`m+im`2 T`K2i2`b, Vb 4 jy Fo- α 4 RXR M/ rw 4 yXkyk +KX

B ≃ γmc² erw

xmn

s = B`27xmn

s , UkXRRV

r?2`2 B`27 4 yX3Nj h Bb `272`2M+2 "@}2H/ T2` mMBi pHm2 Q7 xmn/sX 6B;X kXj /BbTHvb Hsm pb "@}2H/ 7Q` 2+? h1^(s)mnKQ/2 rBi? tBb@2M+B`+HBM; 2H2+i`QM #2Kb UrC 4 y +KV

M/ Qz@tBb 2H2+i`QM #2Kb UrC 4 yXyj8 +KVX b K2MiBQM2/ T`2pBQmbHv- Hsm7Q` #Qi?

2H2+i`QM #2Kb /2+`2b2b rBi? BM+`2bBM; sX *QKT`BM; irQ };m`2b BM 6B;X kXj- r2 }M/ irQ /pMi;2b Q7 M tBb@2M+B`+HBM; 2H2+i`QM #2K- r?B+? `2 bi`QM; Hsm M/

 bT`b2 KQ/2 TQTmHiBQMX ai`QM; Hsm MQi QMHv 2M?M+2b #BHBiv iQ bmTT`2bb Qi?2`

KQ/2b- #mi HbQ BM+`2b2b QmiTmi TQr2`X KQM;  bT`b2 KQ/2 TQTmHiBQM- i?2`2

`2 2tT2+i2/ KQ`2 rBM/Qrb 7Q` ?`KQMB+ QT2`iBQM- M/ i?2`2#v KQ/2 +QKT2iBiBQMb 7`QK Qp2`KQ/2/ QT2`iBQM +M #2 T`iBHHv pQB/2/X

6Q` M BKT2`72+i tBb@2M+B`+HBM; #2K- Hsm Q7 s = m KQ/2 Bb /2;`/2/- r?2`2b Hsm Q7 s ̸= m KQ/2 Bb BMi2MbB}2/X 6B;X kX9 b?Qrb i?i /2;`/iBQM Q7 /2bB`2/

R8

Hsm rBi? xmn Bb KQ`2 b2`BQmb r?2M s BM+`2b2bX h?2 KQ/2 TQTmHiBQM #2+QK2b /2Mb2- r?B+? Bb /2i`BK2MiH 7Q` ?B;? ?`KQMB+ QT2`iBQMX hQ pQB/ #Qp2@K2MiBQM2/

T`Q#H2Kb- iQH2`M+2 Q7 rC 7Q` M tBb@2M+B`+HBM; 2H2+i`QM #2K Bb bi`B+i2` rBi?  ?B;?2`

Q`/2` KQ/2X

Re

*?Ti2` j

LmK2`B+H bbmKTiBQM- JQ/2H

M/ H;Q`Bi?K

jXR 1H2+i`QK;M2iB+ qp2 1[miBQM rBi? aQm`+2b

q2 rBHH BMi`Q/m+2 ?Qr iQ KQ/2H M QT2`iBM; ;v`Qi`QM BM iBK2 /QKBM BM 7QHHQr@

BM; b2+iBQMbX(Rk)(e)(Rj) Jtr2HHǶb 2[miBQMb BM :mbbBM mMBi `2

∇ · 1 = 4πρ UjXRV

∇ · " = 0 UjXR#V

∇ × 1 = − 1 c

∂"

∂t UjXR+V

∇ × " = 4π c C +1

c

∂1

∂t. UjXR/V

PT2`i2 ∇× QM UjXR/V- mb2 UjXR#V- UjXR+V M/ p2+iQ` i`BTH2 T`Q/m+i B/2MiBiv

∇ × (∇ × ") = ∇(∇ · ") − ∇²", UjXkV

M/ i?2M r2 Q#iBM

∇(∇ · ")

=y − ∇²" = 4π

c ∇ × C + 1 c

∂

∂t(∇ × 1)

=−¹_c^∂"_∂t

= 4π

c ∇ × C − 1 c²

∂²

∂t²", UjXjV Rd

Q`- i?2 BM?QKQ;2M2Qmb 2H2+i`QK;M2iB+ rp2 2[miBQM

%∇²− 1 c²

∂²

∂t²

&

" = −4π

c ∇ × C. UjX9V

q2 MHvx2 i?Bb 2[miBQM BM +vHBM/`B+H +QQ`/BMi2 M/ 7Q+mb QM i?2 z +QKTQM2Mi,

%1 r

∂

∂r(r ∂

∂r) + 1 r²

∂²

∂θ² + ∂²

∂z² − 1 c²

∂²

∂t²

&

Bz =−4π c

1 r

% ∂

∂r(rJθ)−∂Jr

∂θ

&

. UjX8V

h?Bb Bb MQi  2bv T`Q#H2K r?2M +m``2Mi i2`K C Bb T`2b2MiX q2 bbmK2 i?i i?2`2 Bb MQ bQm`+2 iQ ;2i i?2 T`2HBKBM`v FMQrH2/;2 Q7 +QH/ KQ/2 }2H/,

%1 r

∂

∂r(r ∂

∂r) + 1 r²

∂²

∂θ² + ∂²

∂z² − 1 c²

∂²

∂t²

&

Bz(r, θ, z, t) = 0. UjXeV

qBi? ?`KQMB+ iBK2 /2T2M/2M+2 UBz ∼ e<sup>−iωt</sup>- ω Bb `2HV M/ `/BH #QmM/`v +QM/B@

iBQM Q7  +B`+mH` rp2;mB/2 rBi? mMB7Q`K rHH `/Bmb rw M/ rBi?Qmi rHH HQbb Ui.e.

ρ4 yV- Bz Bb 2bv iQ #2 bQHp2/ #v b2T`iBQM Q7 p`B#H2b K2i?Q/X 6Q` h1mn KQ/2-

Bz(r, θ, z, t) = Jm(kmnr)e^imθ(C+e^+ikzz+ C₋e^−ikzz)e^−iωt, UjXdV

r?2`2 C+ M/ C<sub>−</sub> `2 KTHBim/2b Q7 7Q`r`/ M/ #+Fr`/ rp2 `2bT2+iBp2Hv- m Bb BMi2;2`- Jm(x)Bb m@i? Q`/2` "2bb2H 7mM+iBQM M/ kmn= xmn/rw- xmnBb n@i? MQMx2`Q

`QQi Q7 J<sub>m</sub><sup>′</sup> (x) = 0X hQ bBKTHB7v 2tT`2bbBQM Q7 2H2+i`QK;M2iB+ }2H/b- T?vbB+H [mMiBiv UH27i@?M/ bB/2 Q7 UjXdVV Bb Q#iBM2/ 7i2` iFBM; `2H T`i Q7 T?bQ` U`B;?i@?M/ bB/2 Q7 UjXdVVX am#biBimi2 UjXdV BMiQ UjXeV M/ ;2i

% ∂²

∂r² +1 r

∂

∂r − m²

r² − kz²+ω² c²

&

Bz = 0. UjX3V

h?2 /Bz2`2MiBH 2[miBQM Q7 "2bb2H 7mM+iBQM Bb U`2TH+2 x rBi? kmnrV

x² d²

dx²J_m(x)+x d

dxJ_m(x)+(x²−m²)J_m(x) = r²% d² dr²+1

r d

dr+(k_mn² −m² r² )&

J_m(k_mnr) = 0.

UjXNV

R3

>2M+2 UjX3V #2+QK2b

%ω²

c² − kmn² − k²z

&

B_z = 0. UjXRyV

_2KQp2 i`BpBH bQHmiBQM Bz = 0 M/ i?2M r2 ;2i i?2 /BbT2`bBQM `2HiBQM

ω² = ω_cmn² + c²k_z², UjXRRV

r?2`2 ωcmn = ckmn Bb +miQz 7`2[m2M+vX "2+mb2 Q7 Jtr2HHǶb 2[miBQM- i`Mbp2`b2 +QKTQM2Mib Q7 7Q`r`/ M/ #+Fr`/ h1mn KQ/2 2H2+i`QK;M2iB+ }2H/b b?QmH/ #2

"_⊥ =± ik_z

k_mn² ∇tBz UjXRkV

1_⊥ =− iω

ck²_mnˆ2z × ∇tBz, UjXRk#V Q`- BM KQ`2 /2iBH-

Br =ikz

k_mnJ_m^′ (kmnr)e^imθ(C+e^+ikzz− C−e^−ikzz)e^−iωt UjXRjV B_θ =k_zm

k²_mnrJ_m(k_mnr)e^imθ(C₊e^+ikzz− C−e^−ikzz)e^−iωt UjXRj#V Er =− ωm

ck_mn² rJm(kmnr)e^imθ(C+e^+ikzz+ C₋e^−ikzz)e^−iωt UjXRj+V Eθ =− iω

ckmn

J_m^′ (kmnr)e^imθ(C+e^+ikzz + C₋e^−ikzz)e^−iωt. UjXRj/V

A7 rHH HQbb Bb BM+Hm/2/- Dmbi KQ/B7v k²_mn i2`K BM rp2 2[miBQM

k_mn² → kcmn²

%1− (1 + i) δ

r_w(1 + m² x²_mn− m²

ω² ω²_cmn)&

, UjXR9V

r?2`2 δ Bb bFBM /2Ti?X *QMbB/2`  bi`m+im`2 U+pBiv Q` iT2`2/ rp2;mB/2V rBi? MQM@

mMB7Q`K rHH `/Bmb rw(z)M/ rHH `2bBbiBpBiv ρ(z) HQM; x@tBb- `2H2b2 i?2 +QMbi`BMi QM iBK2 /2T2M/2M+2 e<sup>−iωt</sup>- M/ HHQr KmHiB@KQ/2 T`QT;iBQMbX A7 MQM@mMB7Q`KBiv Bb r2F 2MQm;? bm+? i?i KQ/2 +QMp2`bBQMb `2 M2;HB;B#H2- Dmbi KQ/B7v i?2 z M/ t

RN

/2T2M/2M+2b Q7 2H2+i`QK;M2iB+ }2H/b

(C+e^+ikzz−iωt+ C₋e^{−ikzz−iωt})→kmn² f (z, t) UjXR8V

ikzC+e^+ikzz−iωt− ik^zC₋e^{−ikzz−iωt} = ∂

∂z(C+e^+ikzz−iωt+ C₋e^{−ikzz−iωt})→kmn²

∂

∂zf (z, t) UjXR8#V

−iωC+e^+ikzz−iωt− iωC−e^{−ikzz−iωt} = ∂

∂t(C+e^+ikzz−iωt+ C₋e^{−ikzz−iωt})→kmn²

∂

∂tf (z, t).

UjXR8+V h?mb- 2H2+i`QK;M2iB+ }2H/b `2

Bz = +

(m,n)

k_mn² fmn(z, t)Jm(kmnr)e^imθ UjXReV

Br = +

(m,n)

kmn

∂

∂zfmn(z, t)J_m^′ (kmnr)e^imθ UjXRe#V Bθ = +

(m,n)

im r

∂

∂zfmn(z, t)Jm(kmnr)e^imθ UjXRe+V Er =− +

(m,n)

im cr

∂

∂tfmn(z, t)Jm(kmnr)e^imθ UjXRe/V Eθ = +

(m,n)

kmn

c

∂

∂tfmn(z, t)J_m^′ (kmnr)e^imθ. UjXRe2V q2 i`2i +m``2Mi bQm`+2 i2`K b bbQ+Bi2/ #2K T2`im`#iBQM Ui.e. +QH/ KQ/2 2H2+@

i`QK;M2iB+ }2H/b TT`QtBKiBQMV- M/ bm#biBimi2 UjXReV BMiQ UjX8V M/ ;2i

_2# +

(m,n)

% ∂²

∂z² − k²mn− 1 c²

∂²

∂t²

&

k_mn² f (z, t)Jm(kmnr)e^imθ$

=−4π c

1 r

% ∂

∂r(rJθ)− ∂

∂θJr

&

. UjXRdV q2 7Q+mb QM h1m^′n^′ KQ/2 M/ QT2`i2

' rw

0

rJm^′(km^′n^′r)dr 1 2π

' 2π 0

e^−im′θdθ

ky

QM #Qi? bB/2b Q7 UjXRdV M/ BMi2;`i2 #v T`ib 1

2k²_mnr²_w

2 J_m²(x_mn)(1− m² x²_mn)% ∂²

∂z² − k²mn− 1 c²

∂²

∂t²

&

f_mn(z, t)

= 4π c

1 2π

' 2π 0

dθe^−imθ ' rw

0

rdr%

kmnJθJ_m^′ (kmnr) +im

r J_rJm(kmnr)&

= 2 ' 2π

0

dθ ' rw

0

rdr%

(C · 1^∗mn) ,∂

∂tf_mn^∗ (z, t)&

≡ 2 ' 2π

0

dθ ' rw

0

rdrC · 2^∗mn,

UjXR3V

r?2`2 r2 mb2/ UjXRe/V- UjXRe2V M/

' rw

0

rdr% m²

k_mn² r²J_m²(kmnr) + J^′2_m(kmnr)&

= r²_w

2 J_m²(xmn)-

1− m² x²_mn

.≡ r²_w

2 Kmn, UjXRNV

M/ 1mn Bb 2H2+i`B+ }2H/ Q7 h1mn KQ/2X "2+mb2 Q7 Q`i?Q;QMHBiB2b- QMHv (m^′, n^′) = (m, n)bm`pBp2bX 6Q` bBKTHB+Biv- (m^′, n^′)BM UjXR3V Bb `2TH+2/ rBi? (m, n)X 6BMHHv- r2 Q#iBM

% ∂²

∂z² − kmn² − 1 c²

∂²

∂t²

&

f_mn(z, t) = 8 x²_mnKmn

' 2π 0

dθ ' rw

0

rdrC · 2^∗mn, UjXkyV

Ai Bb rQ`i? MQiBM; i?i 1 Q7 h1 KQ/2 Bb p2`iB+H iQ x@tBb- bQ QMHv i`Mbp2`b2 +QK@

TQM2Mi Q7 +m``2Mi /2MbBiv C_⊥ +QMi`B#mi2 iQ ;2M2`i2 Q` #bQ`# 2H2+i`QK;M2iB+ }2H/b Q7 h1 KQ/2bX

*m``2Mi /2MbBiv C(`, t) Bb Q#iBM2/ 7`QK KQiBQMb Q7 2H2+i`QMbX 1H2+i`QM #2K Bb BM@

D2+i2/ BMiQ BMi2`+iBQM `2;BQM 7`QK z = 0X M 2H2+i`QM Bb +?`+i2`Bx2/ #v Bib BMBiBH TQbBiBQM M/ KQK2MimK X0j^′ ≡ (ψ0, Pz0, φ0)j^′(6B;X jXR)X *m``2Mi /2MbBiv +QmH/ #2 2tT`2bb2/ b

C(`, t) = ρ(`, t)p(`, t) → −1 r

Ne

+

j^′=1

eδ!

r−rj^′(t, X_0j′)"

δ!

θ−θj^′(t, X_0j′)"

δ!

z−zj^′(t, X_0j′)"pj^′(t, X_0j′), UjXkRV

r?2`2 Ne M/ −e `2 MmK#2` M/ mMBi +?`;2 Q7 2H2+i`QMbX *QMi`B#miBQM Q7 +m``2Mi /2MbBiv 7`QK  #mM+? Q7 2H2+i`QMb rBi? bBKBH` i`D2+iQ`B2b BM T?b2 bT+2 +M #2

kR

6B;m`2 jXR, h?2 2H2+i`QM Q`#Bi rBi? "@}2H/ BMbB/2  rHH `/Bmb rw +B`+mH` rp2;mB/2X SQbBiBQM M/ KQK2MimK Q7 i?2 2H2+i`QM BM +vHBM/`B+H +QQ`/BMi2 `2 U`-θ-xV M/ UP<sub>⊥</sub>, φ, P<sub>z</sub>VX h?2 2H2+i`QM T2`7Q`Kb +v+HQi`QM KQiBQM `QmM/ ;mB/BM; +2Mi2` HQ+i2/ i UrC, ψ- xV rBi?

G`KQ` `/Bmb rL = v<sub>⊥</sub>/Ωc = P<sub>⊥</sub>/meΩeX Ωc = Ωe/γ M/ Ωe = eBz0/mec `2 `2HiBpBbiB+

M/ `2bi 2H2+i`QM +v+HQi`QM 7`2[m2M+B2b `2bT2+iBp2HvX(e)

/2b+`B#2/ 2z2+iBp2Hv #v  K+`Q T`iB+H2-

Cj =− 1 r

+

j^′

eδ!

r− r^j′(t, X0j^′)"

δ!

θ− θ^j′(t, X0j^′)"

δ!

z− z^j′(t, X0j^′)"pj^′(t, X0j^′)

≃ − 1

reNbjδ!

r− rj(t, X0j)"

δ!

θ− θj(t, X0j)"

δ!

z− zj(t, X0j)"pj(t, X0j),

UjXkkV r?2`2 Nbj Bb MmK#2` Q7 2H2+i`QMb Q7 j@i? K+`Q T`iB+H2X h?2M

' 2π 0

dθ ' rw

0

rdrC(`, t) · 2<sup>∗</sup>mn(`, t)

= +N

j=1

' 2π 0

dθ ' rw

0

rdrCj· 2^∗mn(`, t)

≃ − +N

j=1

eN_bjδ%

z− zj(t, X_0j)&

pj(t)· 2^∗mnj,

UjXkjV

r?2`2 N Bb MmK#2` Q7 K`+Q T`iB+H2b M/ 2mnj =2mn(`j, t)X LQr r2 +QMbB/2` ?Qr iQ KQ/2H 2H2+i`QM #2K rBi? }MBi2 K+`Q T`iB+H2bX AM }MBi2 /Bz2`2M+2 iBK2 /QKBM

kk

U6.h.V K2i?Q/- iBK2 Tbb2b rBi? bi2T ∆t /Bb+`2i2HvX h?2`27Q`2- Ni`Mbp2`b2 K+`Q T`iB+H2b `2 BMD2+i2/ BMiQ BMi2`+iBQM `2;BQM i ∆t BMi2`pHbX *?`;2 BM+`2K2Mi 7i2`

QM2 BMD2+iBQM Bb

Nb = Ib(t)∆t

e = Ib(t)∆z

ev_z0 , UjXk9V

r?2`2 ∆z = vz0∆tX M/ i?2M r2 /2}M2 p2HQ+Biv r2B;?i 7mM+iBQM b

Wjt ≡ Nbjt/Nb ≡ Cjt2tT%−(Pz0jt − ¯Pz0)² 2∆P_z²

&

, UjXk8V

r?2`2 Pz0jt M/ ¯Pz0 `2 BMBiBH KQK2MimKb Q7 jt@i? K+`Q T`iB+H2 M/ p2`;2 Q7

HH T`iB+H2bX ∆Pz0/ ¯Pz0 = ∆vz0/¯vz0 Bb `272``2/ iQ b i?2 #2K p2HQ+Biv bT`2/X Cjt

BM UjXk8V biBb}2b MQ`KHBxiBQM +QM/BiBQM

Ni`Mbp2`b2+

jt=1

W_jt = 1. UjXkeV

aQ 7`- r2 /Bb+mbb i?Bb KQ/2H BM Q`/BM`v +vHBM/`B+H +QQ`/BMi2 (r, θ, z, P<sub>⊥</sub>, φ, Pz, γ, t) r?B+? Bb bmBi#H2 iQ bQHp2 }2H/b QM +B`+mH` #QmM/`v +QM/BiBQMX >Qr2p2`-  KQ`2 mb27mH MHvbBb Bb #mBHi QM bHQr@iBK2 b+H2 +QQ`/BMi2 (rC, ψ, z, r_L, φ, P_z, γ, t)(6B;X jXR)X

TTHvBM; i?2 `2+m`bBQM `2HiBQMb M/ bmKKiBQM i?2Q`2K(6B;X jXk) Q7 "2bb2H 7mM+iBQM

J_m^′ (x) + m

xJm(x) = Jm−1(x) UjXkdV Jm−1(x) + Jm+1(x) = 2J_m^′ (x) UjXkd#V e^±imθ1Jm(x1) =

+∞ q=−∞

Jm+q(x2)Jq(x3)e^±iqθ2, UjXkd+V

i?2 bQm`+2 i2`K p_⊥j· 2^∗mnj BM rp2 2[miBQM UjXkyV +M #2 2tT`2bb2/ b

p_⊥j · 2^∗mnj = kmnv_⊥j

c (`, t) +^∞

s=−∞

Js−m(kmnrCj)J_s^′(kmnrLj)e^iAsj, UjXk3V

r?2`2

1 = E_⊥ˆ2_⊥+Eφˆ2φ=%

Er+Qb(φ−θ)+EθbBM(φ−θ)&

ˆ2_⊥+%

−ErbBM(φ−θ)+Eθ+Qb(φ−θ)&

ˆ2φ, kj

6B;m`2 jXk, .B;`K Q7 i`BM;H2 7Q` bmKKiBQM i?2Q`2K Q7 "2bb2H 7mM+iBQMX

ˆ2_⊥ = p_⊥/v_⊥- ˆ2φ = ˆ2z × ˆ2⊥ M/ Asj = −sφ + (s − m)ψ + (m − ^s₂)π `2 mb2/X Ai Bb ?2HT7mH i?i i?2 T?b2 i2`K Q7 UjXk3V Bb 6Qm`B2` 2tTM/2/ BMiQ BM}MBi2 bBMmbQB/H

?`KQMB+b iQ ?M/H2 i?2 `2bQMMi ?`KQMB+X h?2 MQM@`2bQMMi ?`KQMB+b +QmH/ #2 i`2i2/ b MQBb2 i2`Kb Q` ;Bp2M mT BM +H+mHiBQMX .Bb+`2i2 7Q`K Q7 rp2 2[miBQM UjXkyV 7i2` mbBM; UjXk3V Bb

% ∂²

∂z² − k²mn(Zi)− 1 c²

∂²

∂t²

&

fmn(Zi, tp)

=−8kmneN_b x²_mnKmnc

+N j

+∞ s=−∞

Wjδ(Zi− zj)v_⊥jJ_s−m(kmnrCj)J_s^′(kmnrLj)e^iAsj ≡ source(Zi, tp), UjXkNV r?2`2 i M/ p /2MQi2 i?2 H#2Hb Q7 mMB7Q`K bTiBH M/ i2KTQ`H ;`B/b `2bT2+iBp2HvX AM }MBi2 /Bz2`2M+2 K2i?Q/- r2 +QKTmi2 i?2 }`bi M/ b2+QM/ /2`BpiBp2 Q7 }2H/ #v BM7Q`KiBQM QM i?2 ;`B/bX

d

dxf (x)→ f (x + ∆x)− f(x − ∆x)

2∆x UjXjyV

d²

dx²f (x)→ 1

∆x

% d

dxf (x + ∆x/2)− d

dxf (x− ∆x/2)&

→ 1

∆x

%f(x + ∆x) − f(x)

∆x − f (x)− f(x − ∆x)

∆x

&

= f (x + ∆x)− 2f(x) + f(x − ∆x)

∆x² UjXjy#V

k9

>2M+2

∂²

∂z²f_i^p → f_i+1^p − 2fi^p + f_i−1^p

∆z² UjXjRV

∂²

∂t²f_i^p → f_i^p+1− 2fi^p + f_i^p−1

∆t² , UjXjR#V

r?2`2 f<sub>i</sub><sup>p</sup> Bb h1mn KQ/2 }2H/ fmn(Zi, tp)- 7Q` +QMp2MB2M+2X h?2`27Q`2

f_i^p+1→%

2− c²∆t²k²_mn&

f_i^p− fi^p−1+c²∆t²

∆z²

%f_i+1^p − 2fi^p+ f_i^p₋₁&

− c²∆t²source^p_i. UjXjkV

hQ bBKmHi2  Qb+BHHiQ`- i?2 `7 }2H/ ;`Qrb 7`QK  MQBb2 H2p2H iQ  bB;MB}+Mi

KTHBim/2X "2HQr  }2H/ i?`2b?QH/- }2H/ 7mM+iBQM f(Zi, tp)Bb /Bbim`#2/ #v  `M/QK

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7i2` /Bbim`#BM; M/ /2HBp2`B2b 2M2`;v iQ Bib }2H/bX

LQi2 i?i M2Bi?2` K+`Q T`iB+H2 MQ` rp2 +M KQp2 KQ`2 i?M QM2 bTiBH bi2T

∆z 7i2` QM2 iBK2 bi2T ∆tX hQ pQB/ MmK2`B+H 2``Q`- r2 b?QmH/ Q#2v *Qm`Mi bi#BHBiv +QM/BiBQM

c∆t

∆z < 1, UjXjjV

r?B+? HBKBib i?2 mTT2` #QmM/ Q7 iBK2 bi2T M/ BM~m2M+2b bBKmHiBQM T`Q+2bb iBK2X

jXk S`iB+H2@i`+BM; Q7 1H2+i`QM JQiBQMb

6`QK L2riQMǶb Hr Q7 KQiBQM d

dtS = −e(1 + p

c × "), UjXj9V

r?2`2 /2`BpiBp2 Q7 KQK2MimK M/ 2H2+i`QK;M2iB+ }2H/b +M #2 2tT`2bb2/ b d

dtS = d dt

%P_⊥2ˆ_⊥+ Pzˆ2z

&

= dP_⊥

dt 2ˆ_⊥+ P_⊥dφ

dtˆ2φ+ dPz

dt ˆ2z UjXj8V k8