• RFQʢRadio-Frequency Quadrupoleʣ*2ཅࢠՃ଎‫ث‬ͷՃ଎؅• DTLʢDrift Tube Linacʣ*3• ACSʢAnnular-ring Coupled Structureʣ*4ɹͱෳ਺ͷछྨͷՃ଎؅ʹΑΓɺΠΦϯ‫Ͱݯ‬ੜ੒ͨ͠େ‫౓ڧ‬ͷཅࢠϏʔϜ*5 Λ 400MeV ·ͰՃ଎͢͸͡Ίʹిࢠ΍ཅࢠ౳ͷՙిཻࢠͷՃ଎ʹ༻͍ΒΕΔՃ
଎؅͸ɺਐߦ೾‫ܕ‬ʢtraveling-wave typeʣՃ଎؅ΔɻຊߨٛͰ͸ɺి࣓‫ֶؾ‬Λ‫ͯ͠ͱૅج‬ɺ͜ΕΒఆࡏ೾‫ܕ‬Ճ଎؅ͷ‫ૅج‬తࣄ߲Λղઆ͢Δɻ

ͱఆࡏ೾‫ܕ‬ʢstanding-wave typeʣՃ଎؅ʹ෼͚ɹΒΕΔɻਐߦ೾‫ܕ‬Ճ଎؅͸ओʹిࢠͷՃ଎ʹ༻͍ΒΕɺߴप೾ͷి࣓৔͕ిࢠͷ଎౓ͱಉ‫ͯ͠ظ‬Ճ଎؅ͷதΛਐߦ͢Δɻ*1 Ұํఆࡏ೾‫ܕ‬Ճ଎؅Ͱ͸ɺۭಎ‫ʹ಺ثৼڞ‬ੜ͡Δఆࡏ೾ͷి৔ΛՙిཻࢠͷՃ଎ʹ༻͍͍ͯΔɻిࢠͱཅࢠʹ͍ͭͯͦΕΒͷΤωϧΪʔͱ଎౓ β = v/c ͷؔ܎Λࣔͨ͠ਤ 1 ͔ΒΘ͔ΔΑ͏ʹɺΤωϧΪʔΛ༩͑ͯ΋ཅࢠͷ଎౓͸ిࢠ΄Ͳ‫͍ͳ্͔͍͕ͯͬʹܹٸ‬ɻ଎౓͕ޫ
଎ʹൺ΂ͯे෼খ͍͞ཅࢠͷՃ଎ʹ͓͍ͯ͸ɺ͜ͷఆࡏ೾‫ܕ‬Ճ଎؅͕༻͍ΒΕΔɻ

ਤ1ΤωϧΪʔͱ଎౓ β ͷؔ܎௨ৗɺཅࢠͷઢ‫ܗ‬Ճ଎‫ث‬ʢຊςΩετͰ͸ϦχΞοΫͱ‫Ϳݺ‬ʣͰ͸ɺෳ਺ͷՃ଎؅ߏ଄͕࠾༻͞
ΕΔɻ͜Ε͸ɺͦͷΤωϧΪʔྖҬʹΑͬͯޮ཰ͷྑ͍Ճ଎؅ߏ଄͕ҟͳΔ͔ΒͰ͋Δɻ೔ຊͷେ‫౓ڧ‬ཅࢠՃ଎‫ࢪث‬ઃ J-PARCʢJapan Proton

Accelerator Research ComplexʣͷϦχΞοΫʹ
͓͍ͯ͸ɺ
*1

*2

ಋ೾؅Λ఻ൖ͢Δి࣓೾ͷҐ૬଎౓ vp ͸ޫ଎ c ΑΓେ͖͍ʢ‫ޙ‬ड़ʣɻͦ͜Ͱɺಋ೾؅ʹ݀ͷۭ͍ͨԁ൫Λप‫ظ‬తʹઃஔ͢Δ͜ͱʹΑΓɺҐ૬଎౓Λిࢠͷ଎౓ ve ≈ cʹ߹ΘͤΔɻ

６－ 1ߴप೾࢛ॏ‫ܕۃ‬Ճ଎؅ɻຊςΩετͰ͸ղઆ͍ͯ͠ͳ
͍͕ɺେ‫౓ڧ‬ཅࢠϦχΞοΫʹ͓͍ͯඇৗʹॏཁͳՃ଎؅ɻ*3 ‫ޙ‬ड़͢ΔΞϧόϨ‫ܕ‬Ճ଎؅ɻ*4 ‫ޙ‬ड़͢Δ π/2 Ϟʔυ݁߹ۭಎ‫ܕ‬Ճ଎؅ͷҰͭɻ
*5 ਖ਼֬ʹ͸ɺෛਫૉΠΦϯ H− ϏʔϜɻΛ༻͍ͯɺ

1 ి࣓೾ͷ‫ج‬ຊੑ࣭c= √1.1 ϚΫε΢Σϧͷํఔࣜ࣍ͷ 4 ͭͷࣜ1= 2.99792458 × 108 [m/s]ε 0 µ0ͱද͢͜ͱ͕Ͱ͖Δɻ

⃗⃗ = − ∂B∇×E∂t⃗
⃗ = ⃗i + ∂ D∇×H∂t⃗∇·D =ρ
⃗ =0
∇·B

(1.1)
(1.2)

ਅۭதʢిྲྀີ౓ ⃗i = 0ɺిՙີ౓ ρ = 0ʣͷి࣓೾Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.1) ͓Αͼࣜ(1.2) ͸⃗⃗ = −µ0 ∂ H∇×E∂t⃗⃗ = ε0 ∂ E
∇×H
∂t

(1.3)
(1.4)

Λ·ͱΊͯɺϚΫε΢ΣϧͷํఔࣜʢMaxwell’s

(1.8)
(1.9)

ͱͳΔɻ͜ΕΒͷࣜΛ t Ͱඍ෼ͯ͠ɺϕΫτϧղequationsʣͱ͍͏ɻ*6 ֤‫߸ه‬͸ɺ⃗ = ∇(∇ · A)
⃗ − ∇2 A⃗ Λ༻͍Δੳͷެࣜ ∇ × ∇ × A⃗ ɿి৔ʢ‫౓ڧ‬ʣʢElectric field intensityʣ
Eͱɺ࣍ͷ೾ಈํఔࣜ

⃗ ɿ࣓৔ʢ‫౓ڧ‬ʣʢMagnetic field intensityʣ
H

( )
( )
⃗
⃗
∂2 E
E
−
ε
∇
µ
0
0
⃗ =0
⃗
∂t2 H
H
2

⃗ ɿిଋີ౓ʢElectric flux densityʣ
D⃗ ɿ࣓ଋີ౓ʢMagnetic flux densityʣ
B(1.10)⃗ H⃗ ∝ ejωt͕ಘΒΕΔɻ֯प೾਺ ω Λ༻͍ͯɺE,⃗i ɿిྲྀີ౓ʢElectric current densityʣͱ͢Δͱɺࣜ (1.10) ͸ɺϔϧϜϗϧπํఔࣜρ ɿిՙີ౓ʢElectric charge densityʣ

( )
( )
⃗
⃗
E
E
2
+ ω ε 0 µ0 ⃗ = 0
∇
⃗
H
H
2

Λද͢ɻ·ͨ͜ΕΒ͸ɺε ɿ༠ి཰ʢPermittivity of the dielectricʣͱͳΔɻµ ɿಁ࣓཰ʢPermeability of the materialʣɹ(1.11)σ ɿి‫఻ؾ‬ಋ཰ʢElectrical conductivityʣ
1.2 ਅۭதͷฏ໘ి࣓೾

ͱͯ͠௚ߦ࠲ඪ‫ܥ‬ͷதͰɺਅۭதΛ +z ํ޲ʹਐΉฏ⃗ = εE⃗D(1.5)⃗ = µH
⃗
B

(1.6)

⃗
⃗i = σ E

(1.7)

Λຬͨ͢ɻ

໘ి࣓೾Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.11) ͸

d2
dz 2

(1.12)

ͱͳΔɻΑͬͯɺ+z ํ޲ʹਐΉฏ໘ి࣓೾͸ɺ೾√ਅۭதͷޫͷ଎͞ c ͸ɺਅۭͷ༠ి཰ ε0

਺ k0 = ω ε0 µ0 Λ༻͍ͯɺ( ) ( )⃗
⃗ 0 j(ωt−k z)
E
E
0
=
⃗0 e
⃗
H
H

ε0 = 8.854 × 10−12 [F/m]͓Αͼɺਅۭͷಁ࣓཰ µ0µ0 = 4π × 10

[H/m]

͖෼͚Δͱɺ

⃗ =0
ࣜ (1.2), (1.3) ͱɺϕΫτϧղੳͷެࣜ ∇·(∇× A)Λ༻͍Δ͜ͱʹΑΓɺిՙͷอଘଇ∇ · ⃗i +(1.13)

ͱॻ͚Δɻ࣍ʹɺ͜ΕΛ௚ަ࠲ඪ‫ܥ‬ͷ֤੒෼ʹॻ
−7

*6

( )
( )
⃗
⃗
E
E
2
+
ω
ε
µ
0
0
⃗
⃗ =0
H
H

∂ρ
=0
∂t

Ex = E0x ej(ωt−k0 z) , Hx = H0x ej(ωt−k0 z)
Ey = E0y ej(ωt−k0 z) , Hy = H0y ej(ωt−k0 z)
Ez = E0z ej(ωt−k0 z) , Hz = H0z ej(ωt−k0 z)(1.14)͕ಘΒΕΔɻ

６－ 2

ͱͳΔɻ͜ΕΛࣜ (1.8), (1.9) ʹ୅ೖ͢Δ͜ͱʹΑ͋Δۭؒʢମੵ V ʣ಺ͷ࣓‫ؾ‬ΤωϧΪʔ Um ͸Γɺ࣍ͷؔ܎ΛಘΔɻ

Um =

√

√
ε0 E0x = µ0 H0y√√− ε0 E0y = µ0 H0x(1.15)∫∫∫1 ⃗ ⃗µH · H dV
V 2(1.22)

ͱ‫ه‬ड़͞ΕΔɻ

E0z = 0, H0z = 0ࣜ (1.17) Ͱද͞ΕΔϙΠϯςΟϯάϕΫτϧͷ

Αͬͯɺਅۭதͷฏ໘ి࣓೾͕ɺ࣍ͷ‫ج‬ຊੑ࣭Λൃࢄ΋ͭ͜ͱ͕Θ͔Δɻ

1. ి࣓৔͸ਐߦํ޲ʹରͯ͠ਨ௚Ͱ͋Γɺਐߦํ޲ʹͦͷ੒෼Λ΋ͨͳ͍ɻ͜ͷΑ͏ͳ೾Λɺ

TEM ೾ʢTransverse electromagnetic wavesʣͱ‫Ϳݺ‬ɻ

⃗ = ∇ · (E⃗ × H)⃗∇·S(1.23)

ʹ͍ͭͯߟ͑Α͏ɻ͜Ε͸ɺϕΫτϧղੳͷެࣜ⃗ × B)⃗ =B⃗ · (∇ × A)
⃗ −A
⃗ · (∇ × B)⃗ ɺ͓Α∇ · (Aͼࣜ (1.1), (1.2) Λ༻͍ͯɺ⃗ ·H⃗ = 0ʣɺ
2. ి৔ͱ࣓৔͸‫ʹ͍ޓ‬ਨ௚Ͱ͋ΓʢE

͜͜ͰɺZ0 ͸ಛੑΠϯϐʔμϯεʢCharac-⃗⃗⃗ · ⃗i − εE
⃗ · ∂E⃗ = −µH⃗ · ∂H − E
∇·S
∂t
( ∂t
)
∂ 1 ⃗ ⃗ 1 ⃗ ⃗⃗ · ⃗iεE · E + µH · H − E
=−
∂t 2
2
∂
⃗ · ⃗i= − (ue + um ) − E(1.24)∂tteristic impedanceʣͱ‫ݺ‬͹ΕɺਅۭதͰ͸

ͱॻ͚Δɻ͜ͷ྆ลΛดۭؒʢମੵ V ɺද໘ੵ S ′ ʣ

Z0 = 376.7 [Ohm] ≈ 120π [Ohm] Ͱ͋ΔɻͰੵ෼͢Δͱɺࠨล͸Ψ΢εͷੵ෼ఆཧΛ༻͍ͯ͜ΕΒ͸࣍ͷؔ܎Λ΋ͭɻ

Ex
Ey
=−
=
Hy
Hx

√

µ0
= Z0
ε0

(1.16)

·ͨɺి৔ͱ࣓৔͸ಉҐ૬Ͱ͋Δɻ∫∫∫

3. ి࣓೾ͷਐΉํ޲͸ɺϙΠϯςΟϯάϕΫτ
ϧʢPoynting vectorʣ

⃗=E
⃗ ×H
⃗
S

V

ω
1
=√
=c
k0
ε0 µ0

(1.17)(1.18)∫∫∫

V(ue + um ) dV −∂= − (Ue + Um ) −∂t−
1.3 ి࣓৔ͷΤωϧΪʔͱϙΠϯςΟϯάϕΫ

∂(Ue + Um )∂t ∫∫∫=

τϧV(1.19)

͋Δۭؒʢମੵ V ʣ಺ͷి‫ؾ‬ΤωϧΪʔ Ue ͸

Ue =

1 ⃗ ⃗
εE · E dV2V1 ⃗ ⃗µH · H2∫∫∫V⃗ · ⃗i dVEV⃗ · ⃗i dV (1.26)
E∫∫∫⃗ · ⃗i dV +
E

∫∫

S′

⃗ · ⃗n dS ′S

(1.27)

࣓৔ΤωϧΪʔΛɺӈลୈҰ߲͸୯Ґ࣌ؒʹดۭؒ಺Ͱδϡʔϧ೤ʢిѹ × ిྲྀʣͱࣦͯ͠ΘΕΔ
ΤωϧΪʔΛද͍ͯ͠ΔɻΤωϧΪʔͷอଘଇʹ(1.20)

ΑΓɺӈลୈೋ߲͸ɺ୯Ґ࣌ؒʹด‫ۂ‬໘͔Β֎ʹग़Δి࣓৔ͷΤωϧΪʔΛද͢ͱղऍ͢Δ͜ͱ͕
Ͱ͖Δɻͭ·ΓɺϙΠϯςΟϯάϕΫτϧ͸ɺ୯ͱ‫ه‬ड़͞ΕΔɻ·ͨɺ࣓‫ؾ‬ΤωϧΪʔີ౓ um ͸

um =(1.25)

ͱͳΓɺࠨล͸୯Ґ࣌ؒʹ‫ݮ‬গ͢Δดۭؒ಺ͷిి‫ؾ‬ΤωϧΪʔີ౓ ue ͸∫∫∫

⃗ · ⃗n dS ′Sͱද͞ΕΔɻΑͬͯɺ

ɹ

1 ⃗ ⃗
εE · E
2

S′τϧʣɺӈล͸∂−∂tͱͳΓɺਅۭதͷޫͷ଎͞ c ͱҰக͢Δɻ

ue =

∫∫

ͱද͞Εʢ⃗n ͸ด‫ۂ‬໘্Ͱ֎޲͖ͷ୯Ґ๏ઢϕΫͷ޲͖ͱҰக͢Δɻ·ͨͦͷ଎͞ vp ͸ɺ

vp =⃗ dV =∇·S(1.21)

Ґ࣌ؒʹ୯Ґ໘ੵΛ௨ա͢Δి࣓৔ͷΤωϧΪʔΛҙຯ͢Δ͜ͱ͕Θ͔Δɻ

６－ 3

ి࣓৔ͷΤωϧΪʔ͸࣌ؒͱͱ΋ʹมԽ͢Δͷ

ͱॻ͚Δɻ*8 ·ͨҰൠʹɺͰɺ͜ΕΛ࣌ؒฏ‫Ͱۉ‬දͦ͏ɻ֯प೾਺ ω Λ༻͍

⃗ ×H⃗ ∗} =
Re{Eͯɺి৔͓Αͼ࣓৔Λ( ) ( )⃗
⃗ 0 jωt
E
E
=
⃗0 e
⃗
H
H

(1.28)

ͱද͢ͱɺి‫ؾ‬ΤωϧΪʔີ౓ ue ͷ࣌ؒฏ‫ ۉ‬we
͸ɺ

we =

1 ⃗ ⃗∗
εE · E
4͕੒ΓཱͭͷͰɺϙΠϯςΟϯάϕΫτϧͷ࣌ؒ⟨⃗ฏ‫ ۉ‬S1 ⃗ ⃗∗µH · H
4

⟩

ͷൃࢄ͸ɺ

⟨ ⟩ 1
⃗ ×H
⃗ ∗ ) + ∇ · (E
⃗ = {∇ · (E⃗ ∗ × H)}
⃗
∇· S
4
(
)
⃗∗
⃗
∂
E
1
∂
E
∗
⃗ ·
⃗·
+E
=− ε E
4
∂t
∂t
(
)
⃗∗
⃗
∂
H
1
∂
H
⃗∗·
⃗ ·
+H
− µ H
4
∂t
∂t

(1.29)

࣓‫ؾ‬ΤωϧΪʔີ౓ um ͷ࣌ؒฏ‫ ۉ‬wm ͸ɺ

wm =(1.30)

ͱॻ͚Δɻ*7 ͜͜Ͱɺࣜதͷ ∗ ͸ɺෳૉ‫ڞ‬໾Λද

1 ⃗∗ ⃗
⃗ ·E
⃗ ∗)
− σ(E
·E+E
4
(
)
∂ 1 ⃗ ⃗∗ 1 ⃗ ⃗ ∗εE · E + µH · H
=−
∂t 4
4
1 ⃗ ⃗∗− σE · E
2⟨⟩∂⃗ · ⃗i= − (we + wm ) − E(1.34)∂t͢ɻͭ·Γɺ೚ҙͷෳૉ਺ z ʹ͍ͭͯɺͦͷ࣮෦Λ Re{z} = xɺ‫ڏ‬෦Λ Im{z} = y ͱද͢ͱɺ

z = x + jyz ∗ = x − jy

(1.31)⃗ ͷ࣌ؒͰ͋Δɻ·ͨɺϙΠϯςΟϯάϕΫτϧ S⟨⃗ฏ‫ ۉ‬S

*7

⟩

͸ɺ

⟨

⟩
⃗ ×H
⃗ ∗}
⃗ = 1 Re{ES
21 ⃗⃗ ∗ ) + (E⃗⃗ ∗ × H)}{(E × H2
(1.33)(1.32)

ͱͳΔɻ͜ͷ྆ลΛดۭؒʢମੵ V ɺද໘ੵ S ′ ʣ*8⃗ H⃗ ͷ֤੒෼Λ
E,
  
 

|E0x |ejθex ejωt
E0x ejωtExEy  = E0y ejωt  = |E0y |ejθey ejωt Ez
E0z ejωt|E0z |ejθez ejωt
  



|H0x |ejθmx ejωt
H0x ejωt
Hx
Hy  = H0y ejωt  = |H0y |ejθmy ejωt HzH0z ejωt|H0z |ejθmz ejωt

⃗ ͷ x ੒෼ Sx ͸ɺྫ͑͹ɺS

Sx = Re{Ey }Re{Hz } − Re{Ez }Re{Hy }= |E0y ||H0z | cos(ωt + θey ) cos(ωt + θmz )− |E0z ||H0y | cos(ωt + θez ) cos(ωt + θmy )1= |E0y ||H0z |{cos(2ωt + θey + θmz )2+ cos(θey − θmz )}ͱॻ͘ͱɺue ͸)
1 (ε (Re{Ex })2 + (Re{Ey })2 + (Re{Ez })22
1 (= ε |E0x |2 cos2 (ωt + θex )2+ |E0y |2 cos2 (ωt + θey ))+ |E0z |2 cos2 (ωt + θez )

−

ue =

1
|E0z ||H0y |{cos(2ωt + θez + θmy )2+ cos(θez − θmy )}ͱͳΔɻΑͬͯɺSx ͷ࣌ؒฏ‫ۉ‬⟨

ͱͳΔɻΑͬͯɺue ͷ࣌ؒฏ‫ ۉ‬we ͸ɺ

)
(
1
1
1
1
ε
|E0x |2 + |E0y |2 + |E0z |22
222
1 ⃗ ⃗∗= εE · E
4
ͱදͤΔɻಉ༷ʹɺum ͷ࣌ؒฏ‫ ۉ‬wm ͸ɺ
)
(
1
1
1
1
wm = µ|H0x |2 + |H0y |2 + |H0z |22
222
1 ⃗ ⃗∗= µH · H
4
we =

Sx

⟩

=

⟨

Sx

⟩

͸ɺ

1
{|E0y ||H0z | cos(θey − θmz )2− |E0z ||H0y | cos(θez − θmy )}1(Re{Ey Hz∗ } − Re{Ez Hy∗ })2
1⃗ ×H⃗ ∗ )x }= Re{(E2=⃗ ͷ࣌ؒฏ‫ۉ‬
ͱॻ͚Δɻͭ·ΓɺS

ͱදͤΔɻ

ͱදͤΔɻ

６－ 4

⟨

⃗
S

⟩

=

⟨

⃗
S

1
⃗ ×H
⃗ ∗}
Re{E
2

⟩

͸ɺ

Ͱੵ෼͠ɺΨ΢εͷੵ෼ఆཧΛ༻͍Δͱɺ

−ͱ͢Δͱɺ্ͷඍ෼ํఔࣜͷղ͸

(∂(We + Wm )∂t ∫∫∫∫∫⟨⟩⟨ ⟩⃗⃗ · ⃗n dS ′⃗=E · i dV +
S
S′

V

Ex
Hy

)

(1.35)()
E0x j(ωt−nz)
=
e
H0y
)
(
E0x −αz j(ωt−βz)
e
=
·e
H0y

(1.41)͕ಘΒΕΔɻ͜Ε͸ɺ࣌ؒฏ‫͍ͯͭʹۉ‬΋ɺࣜͱॻ͚ɺి࣓৔͕ e−αz Ͱ‫ݮ‬ਰ͢Δ͜ͱ͕Θ͔Δɻ

(1.27) ͷΤωϧΪʔอଘଇ͕੒Γཱͭ͜ͱΛද͠·ͨɺࣜ (1.40) ͷ࣮਺෦ͱ‫਺ڏ‬෦͔Βɺα ͱ β ͕

͍ͯΔɻ

ͦΕͧΕ

ྫ͑͹ɺਅۭதΛ +z ํ޲ʹਐΉฏ໘ి࣓೾ʹͭ1α2 = ω 2 εµ2͍ͯɺͦͷి࣓৔ͷΤωϧΪʔͱϙΠϯςΟϯά
ϕΫτϧͷؔ܎Λௐ΂Α͏ɻి࣓৔ͷΤωϧΪʔ1β 2 = ω 2 εµ2ີ౓Λ u = ue + um ͱͯ͠ɺϙΠϯςΟϯάϕΫ
τϧ Sz ͕Sz = cu(1.36)

ి࣓৔ͷΤωϧΪʔີ౓ͷ࣌ؒฏ‫ۉ‬Λ w = we +wm ͱͯ͠ɺϙΠϯςΟϯάϕΫτϧͷ࣌ؒฏ‫ۉ‬
⟨ ⟩
Sz ͕
⟨ ⟩
Sz = cw(1.37)ͱද͞ΕΔ͜ͱ͕ɺ؆୯ͳ‫ʹࢉܭ‬ΑΓ͔֬ΊΒΕΔɻͭ·Γɺి࣓৔ͷΤωϧΪʔ͕ฏ໘೾ͷਐߦํ޲ʹޫ଎ c Ͱ఻ൖ͍ͯ͠Δ͜ͱΛද͍ͯ͠Δɻ
ɹ

(√
(√

)

σ2
+1−1
ω 2 ε2)σ2+1+1
ω 2 ε2(1.42)(1.43)

ͱॻ͚Δ͜ͱ͕Θ͔Δɻ
ಛੑΠϯϐʔμϯε Z ͸ɺࣜ (1.41) Λࣜ (1.1)ʹ୅ೖ͢Δ͜ͱʹΑΓɺ

Z=

Ex
ωµ
= 2
(β + jα)Hy
α + β2(1.44)ͱಘΒΕΔɻͭ·Γɺి৔ͷҐ૬͕࣓৔ͷҐ૬ʹൺ΂ͯ θ = tan−1 (α/β) ͚ͩਐΜͰ͍Δ͜ͱΛ͍ࣔͯ͠Δɻి‫఻ؾ‬ಋ཰ σ ͱҐ૬ࠩ θ ͷؔ܎Λɺਤ

2 ʹࣔ͢ɻਅۭதͰ͸ σ = 0 Ͱ͋Δ͔Βɺα = 0,√β = ω ε0 µ0 = k0 ͱͳΓɺి৔ͱ࣓৔͕ಉҐ૬Ͱɺ‫ݮ‬ਰ͸ੜ͡ͳ͍ɻҰํɺσ ͕े෼େ͖͘ͳΔ

1.4 දൽޮՌ+z ํ޲ʹਐΉฏ໘ి࣓೾ʹ͍ͭͯɺి‫఻ؾ‬ಋ཰σ ̸= 0 ͷ৔߹Λߟ͑Α͏ɻࣜ (1.1) ͓Αͼࣜ (1.2)⃗=Λ t Ͱඍ෼ͯ͠ɺϕΫτϧղੳͷެࣜ ∇×∇× Aͱɺి৔ͱ࣓৔ͷҐ૬͕ࠩ 45˃ͱͳΓɺి࣓೾͕
ਐΉʹͭΕͯ‫ݮ‬ਰ͢Δɻ⃗ Λ༻͍Δͱɺ⃗ − ∇2 A∇(∇ · A)⃗ − ∇2 E⃗⃗ = (ω 2 εµ − jωσµ)E∇(∇ · E)⃗
⃗ = (ω 2 εµ − jωσµ)H⃗ − ∇2 H∇(∇ · H)

(1.38)⃗ H⃗ ʹ͍ͭͯɺฏ໘೾Ͱ͋Δ͜ΛಘΔɻ͜͜ͰɺE,⃗ = ∇·H⃗ = 0 ͱͳΔɻͦͯ͠ɺx ࣠ͱ͔Βɺ∇ · E
ͱ y ࣠ΛͦΕͧΕి৔ͷ޲͖ͱ࣓৔ͷ޲͖ʹҰக͢ΔΑ͏ʹબͿͱɺ্ࣜ͸

d2
dz 2

(

Ex
Hy

)

(

Ex
+ (ω εµ − jωσµ)
Hy
2

)

ਤ2

= 0 (1.39)
௨ৗͷۚଐಋମͰ͸ɺσ ≃ 107 [S/m]ɺε ≃ ε0 ͰͱͳΔɻ࣍ʹɺෳૉ਺ n = β − jα Λ༻͍ͯ222

ω εµ − jωσµ = n = (β − jα)ి‫఻ؾ‬ಋ཰ͱҐ૬ࠩ

(1.40)

͋ΔͷͰɺσ/ωε ≃ 1018 /ω ͱͳΓɺω ͕ඇৗʹେ͖͘ͳΒͳ͍‫ݶ‬Γɺσ/ωε ≫ 1 ͷؔ܎͕੒Γཱͭɻ６－ 5Ώ͑ʹಋମதͰ͸ɺࣜ (1.42), (1.43) ͔Β

α=β=

√

1
ωµσ
=
2
δ

Λදൽ఍߅ʢskin resistanceʣͱ‫Ϳݺ‬ɻΑͬͯɺ͜ͷͱ͖ͷϙΠϯςΟϯάϕΫτϧͷ࣌ؒฏ‫ۉ‬(1.45)

͸ɺ⟨

ͱͳΔɻΑͬͯɺಋମதͷి࣓೾ͷి৔ͱ࣓৔͸ɺ

(

Ex
Hy

)

=

(

)
E0x −z/δ j(ωt−z/δ)
e
·e
H0y

(1.46)

ͱͳΔɻ͜͜Ͱɺ

δ=

√

2
ωµσ

(1.47)⟩ 1
Sz = Re{Ex Hy∗ }2
1= Re{(1 + j)Rs Hy Hy∗ }2
1= Rs Hy Hy∗
2

⟨

Sz

⟩

(1.50)

ͱͳΔɻɹΛදൽͷް͞ʢSkin depthʣͱ‫Ϳݺ‬ɻۚଐಋମͷ৔߹ɺి‫఻ؾ‬ಋ཰ σ ͕େ͖͍ͨΊɺදൽͷް͞ δ

1.5 ‫ڥ‬ք৚݅͸ඇৗʹখ͍͞ɻಋମ಺෦ʹೖࣹͨ͠ి࣓೾͸ɺి৔ͷ઀ઢ੒෼

e−z/δ ͷ߲ʹΑͬͯ‫ٸ‬଎ʹ‫ݮ‬ਰ͢Δɻ͜ͷΑ͏ͳ‫ݱ‬৅ΛදൽޮՌʢSkin eﬀectʣͱ‫Ϳݺ‬ɻදൽޮՌʹΑͬͯɺి࣓৔͸ z = δ ͷҐஔͰɺද໘ʢz = 0ʣʹൺ΂ͯ e−1 ‫ݮ‬গ͢Δɻಔʢσ = 5.8 × 107 ʣͱΞϧϛχ΢Ϝʢσ = 4.0 × 107 ʣʹ͍ͭͯɺप೾਺ fਤ4ి࣓৔ͷ઀ઢ੒෼

ͱදൽް͞ δ ͷؔ܎Λਤ 3 ʹࣔ͢ɻਤ 4 ͷΑ͏ʹɺ‫ڥ‬ք໘ʹฏߦͳ 2 ຊͷઢ෼ AB,

CD ΛͱΓɺ໘ʹަࠩ͢Δด‫ۂ‬ઢ ABCD Λ࡞Δɻͦͯ͠ɺ͜ͷઢ෼ AB, CD ͷ௕͞Λ ∆lɺઢ෼ BC,

DA ͷ௕͞Λ ∆d ͱ͢Δɻ·ͨɺ‫ڥ‬քʹ͓͚Δ୯
Ґ઀ઢϕΫτϧΛ ⃗tɺ୯Ґ๏ઢϕΫτϧΛ ⃗n ͱͯ͠ɺ୯ҐϕΫτϧ ⃗u Λɺ⃗u = ⃗t × ⃗n ͱ͢Δɻ͜ͷด⃗ ͷઢੵ෼Λߟ͑Α͏ɻ͜‫ۂ‬ઢ ABCD ্Ͱి৔ EΕ͸ɺ͜ͷด‫ۂ‬ઢ ABCD ্ͷ୯Ґ઀ઢϕΫτϧΛ
⃗t′ ͱͯ͠ετʔΫεͷఆཧͱࣜ (1.1) Λ༻͍Δ͜
ਤ3

ͱʹΑΓɺ

प೾਺ͱදൽͷް͞�ಋମதΛ +z ํ޲ʹਐΉฏ໘ి࣓೾ʢి৔ͷ޲͖ʹ x ࣠ɺ࣓৔ͷ޲͖ʹ y ࣠ΛબͿʣʹ͍ͭͯɺಛ
ੑΠϯϐʔμϯε Z ͸ɺࣜ (1.45) Λࣜ (1.44) ʹ୅ೖ͢Δ͜ͱʹΑΓɺExZ== (1 + j)Hy√ωµ= (1 + j)Rs
2σ

Rs =

S

ͱදͤΔɻΦ ͸‫ۂ‬໘ ABCD ΛͭΒ͵࣓͘ଋΛද

(1.48)

͢ɻ͜͜ͰɺEt1 , Et2 ΛͦΕͧΕഔ࣭ 1, 2 ʹ͓͚
Δి৔ͷ઀ઢ੒෼ͱͯ͠ɺ∆dʢઢ෼ BC, DAʣΛͱಘΒΕΔɻ͜͜Ͱɺ

√

ABCD

∫∫

⃗ · ⃗u dS(∇ × E)∫∫∂⃗ · ⃗u dS
B
=−
∂t S
∂Φ
(1.51)=−∂t⃗ · ⃗t′ ds =Eे෼খͯ͘͞͠΍Ε͹ɺࣜ (1.51) ͸

ωµ
1
=
2σ
δσ

(1.49)

６－ 6−Et1 ∆l + Et2 ∆l = 0(1.52)

ͱͳΔɻ͜ΕʹΑΓɺి৔ͷ઀ઢ੒෼ʹ͍ͭͯ‫ڥ‬͸ɺ∆d → 0 Ͱ 0 ͱ͸ͳΒͳ͍ɻ͜͜Ͱɺද໘ిྲྀີ౓ ⃗is = ⃗i∆d Λ༻͍Δͱɺࣜʢ1.55ʣ͸ք৚݅

Et1 = Et2

−Ht1 ∆l + Ht2 ∆l = ⃗is · ⃗u ∆l(1.53)(1.59)͕ಘΒΕΔɻ͢ͳΘͪɺి৔ͷ઀ઢ੒෼͸྆ഔ࣭ͱͳΔɻ·ͨɺ‫׬‬શಋମͷதʹ͸ి࣓೾͸৵ೖͤͷ‫ڥ‬ք໘Ͱ࿈ଓͰ͋Δɻ

ͣɺ࣓৔͸ 0 Ͱ͋Δɻͭ·ΓɺHt2 = 0 Ͱ͋Δ࣍ʹɺഔ࣭ 1 ͕ۭ‫ؾ‬ʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕͔Βɺ

σ = ∞ ͷ‫׬‬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ‫׬‬શಋମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓৔͸ 0 Ͱ͋Δɻͭ·ΓɺEt2 = 0 Ͱ͋Δ͔Βɺ

Ht1 = ⃗is · (−⃗u)(1.60)⃗ 1 ͷํ޲ʹେ͖͞ |Ht1 |ͱͳΓɺ࣓৔ʹΑͬͯ ⃗n×Hͷද໘ిྲྀ͕ྲྀΕΔ͜ͱ͕Θ͔Δɻ

Et1 = 0(1.54)

ɹͱͳΓɺి࣓೾ͷి৔͸ಋମද໘ʹฏߦͳ੒෼Λిଋີ౓ͷ๏ઢ੒෼

΋ͨͳ͍͜ͱ͕Θ͔Δɻɹ࣓৔ͷ઀ઢ੒෼⃗ ͷઢಉ༷ʹɺਤ 4 ͷด‫ۂ‬ઢ ABCD ্Ͱ࣓৔ Hੵ෼Λߟ͑Α͏ɻ͜Ε΋ɺετʔΫεͷఆཧͱࣜ

ਤ5ి࣓৔ͷ๏ઢ੒෼

(1.2) Λ༻͍Δ͜ͱʹΑΓɺ�
∫∫⃗ · ⃗u dS⃗ · ⃗t′ ds =(∇ × H)
H
ABCD
S
∫∫
∫∫
∂
⃗ · ⃗u dS⃗D=i · ⃗u dS +∂t S
S
(1.55)

࡞ΒΕΔԁப‫͑ߟ͍ͯͭʹܗ‬Α͏ɻ͜ΕΒͷ໘͸

ͱදͤΔɻ͜͜ͰɺHt1 , Ht2 ΛͦΕͧΕഔ࣭ 1, 2ཧΛ༻͍Δͱ

ʹ͓͚Δ࣓৔ͷ઀ઢ੒෼ͱͯ͠ɺ∆dʢઢ෼ BC,∫∫∫ਤ 5 ͷΑ͏ʹɺ‫ڥ‬ք໘ʹ·͕ͨͬͨ̎ͭͷ໘Ͱ‫ڥ‬ք໘ʹฏߦͰɺ໘ੵΛ ∆S ͱ͠ɺ͜ͷԁபͷߴ͞Λ ∆d ͱ͢Δɻ͜ͷԁப‫ܗ‬ด‫ۂ‬໘ʢମੵ V ɺද⃗ ʹΨ΢εͷൃࢄఆ໘ੵ S ′ ʣʹ͍ͭͯɺిଋີ౓ DDAʣΛे෼খͯ͘͞͠΍Ε͹ɺࣜ (1.55) ͸−Ht1 ∆l + Ht2 ∆l = 0

V(1.56)⃗ dV =∇·D∫∫∫V(1.57)͕ಘΒΕΔɻ͢ͳΘͪɺ࣓৔ͷ઀ઢ੒෼΋‫ڥ‬ք໘࣍ʹɺഔ࣭ 1 ͕ۭ‫ؾ‬ʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕

σ = ∞ ͷ‫׬‬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ͜ͷͱ͖ɺࣜ (1.55) ͷӈลୈҰ߲

S(1.61)(1.58)

ρ dV =∫∫

S′⃗ · ⃗n′ dS ′D

(1.62)

ͱͳΔɻԁப͕े෼ബ͍ʢ∆dʣͱͯͦ͠ͷଆ໘ΑΓग़ΔిଋΛແࢹ͢Δͱɺ্ࣜ͸ρ∆S∆d = Dn1 ∆S − Dn2 ∆SͰ࿈ଓͰ͋Δɻ⃗ · ⃗u ∆l∆d
⃗i · ⃗u dS = σ E⃗ · ⃗n′ dS ′Dϧʣɻ͜͜Ͱɺࣜ (1.3) Λ༻͍Δͱɺ͜Ε͸

ք৚݅

∫∫

S′

ͱͳΔʢ⃗
n′ ͸ด‫ۂ‬໘্Ͱ֎޲͖ͷ୯Ґ๏ઢϕΫτ

ͱͳΔɻ͜ΕʹΑΓɺ࣓৔ͷ઀ઢ੒෼ʹ͍ͭͯ‫ڥ‬

Ht1 = Ht2∫∫

(1.63)

ͱͳΔɻ͜͜Ͱɺද໘ిՙີ౓ ρs = ρ∆d Λ༻͍Δͱɺిଋີ౓ͷ๏ઢ੒෼ʹ͍ͭͯɺ‫ڥ‬ք৚݅Dn1 − Dn2 = ρs͕ಘΒΕΔɻ

６－ 7(1.64)

࣍ʹɺഔ࣭ 1 ͕ۭ‫ؾ‬ʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕

σ = ∞ ͷ‫׬‬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ‫׬‬શಋମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓ք͸ 0 Ͱ͋Δɻ2 ಋ೾؅
2.1 TE ೾ͱ TM ೾

ͭ·ΓɺDn2 = 0 Ͱ͋Δ͔Βɺ

Dn1 = ρsಋ೾؅ͷ؅࣠ํ޲ʢ+z ํ޲ʣʹejωt−γz = e−αz ej(ωt−βz)(1.65)(2.1)

Ͱ఻ൖ͢Δి࣓೾Λߟ͑Α͏ɻ͜͜Ͱɺ఻ൖఆ਺

ͱͳΔɻ

ʢpropagation constantʣγ ͸ɺ‫ݮ‬ਰఆ਺ʢattenu-

ɹation constantʣα ͱҐ૬ఆ਺ʢphase constantʣβ Λ༻͍ͯ

࣓ଋີ౓ͷ๏ઢ੒෼
ਤ 5 ͷԁப‫ܗ‬ด‫ۂ‬໘ʢମੵ V ɺද໘ੵ S ′ ʣʹͭ⃗ ʹΨ΢εͷൃࢄఆཧΛ༻͍Δͱ
͍ͯɺ࣓ଋີ౓ B∫∫∫

V⃗ dV =∇·B∫∫S′⃗ · ⃗n′ dS ′B

(1.66)

0=S′⃗ · ⃗n′ dS ′Bͱද͞ΕΔɻ
ಋ೾؅ʹ͓͚Δ‫ݮ‬ਰ͕ͳ͍৔߹ʢα = 0 ͷ৔⃗⃗߹ʣɺి৔ E(x,y, z, t) ͓Αͼ࣓৔ H(x,y, z, t) ͸(

(1.67)⃗
E(x,y, z, t)⃗H(x, y, z, t)ք৚݅(1.69)∂H0z∂H0y−= jωε0 E0x
∂y
∂z
∂H0z
∂H0x
−
= jωε0 E0y
∂z
∂x
∂H0x
∂H0y
−
= jωε0 E0z∂x∂y࣍ʹɺഔ࣭ 1 ͕ۭ‫ؾ‬ʢ·ͨ͸ਅۭʣɺഔ࣭ 2 ͕

σ = ∞ ͷ‫׬‬શಋମͰ͋Δ৔߹Λߟ͑Α͏ɻ‫׬‬શಋମͷதʹ͸ి࣓೾͸৵ೖͤͣɺి࣓ք͸ 0 Ͱ͋Δɻͭ·ΓɺBn2 = 0 Ͱ͋Δ͔Βɺ

(1.70)

(2.4)
(2.5)
(2.6)

͓Αͼɺ

͕ಘΒΕΔɻͱͳΔɻ

)⃗ 0 (x, y) j(ωt−βz)
E(2.3)⃗ 0 (x, y) e
H

∂E0y
∂E0z
−
= −jωµ0 H0x
∂y
∂z
∂E0z
∂E0x
−
= −jωµ0 H0y
∂z
∂x
∂E0x
∂E0y
−
= −jωµ0 H0z∂x∂y(1.68)

ͱͳΔɻΑͬͯɺ࣓ଋີ౓ͷ๏ઢ੒෼ʹ͍ͭͯɺ‫ڥ‬

Bn1 = 0=(Λ֤੒෼ʹ͍ͭͯॻ͖෼͚ΔͱɺΓग़Δ࣓ଋΛແࢹ͢Δͱɺ্ࣜ͸

Bn1 = Bn2)·ͣɺਅۭதͷϚΫε΢Σϧํఔࣜ (1.8), (1.9)

ͱͳΔɻԁப͕े෼ബ͍ʢ∆dʣͱͯͦ͠ͷଆ໘Α

0 = Bn1 ∆S − Bn2 ∆S(2.2)࣍ͷΑ͏ʹ‫ه‬ड़Ͱ͖Δɻ

ͱͳΔɻ͜͜Ͱɺࣜ (1.4) Λ༻͍Δͱɺ͜Ε͸∫∫γ = α + jβ

(2.7)
(2.8)
(2.9)

ͱͳΔɻ͜͜Ͱɺ∂/∂z = −jβ Ͱ͋Δ͔Βɺࣜ(2.4), (2.5), (2.7), (2.8) ΑΓɺਐߦํ޲ʹਨ௚ͳ
੒෼ E0x , E0y , H0x , H0y ͸ɺ

)
(
∂H0z
∂E0z
−j
+β
E0x = 2 ωµ0
kc
∂y
∂x
)
(
∂H0z
∂E0z
+j
E0y = 2 ωµ0
−β
kc
∂x
∂y
)
(
∂H0z
∂E0z
+j
H0x = 2 ωε0
−β
kc
∂y
∂x
)
(
∂E0z
∂H0z
−j
H0y = 2 ωε0
+β
kc
∂x
∂y

６－ 8

(2.10)(2.11)(2.12)(2.13)ͱॻ͚Δɻͨͩ͠ɺ

ͱॻ͘͜ͱ͕Ͱ͖Δɻ͜ͷղ͸ɺ

kc2 = ω 2 ε0 µ0 − β 2(2.14)

kc2 = kx2 + ky2(2.18)

Hz = 0ʣͱ͢Ε͹ɺEx , Ey , Hx , Hy ͸ͱ΋ʹҰX(x) = cos(kx x + θx )(2.19)

ఆͷղΛ΋ͨͳ͍ɻ্͕ࣜҙຯΛ΋ͭͨΊʹ͸ɺ

Y (y) = cos(ky y + θy )(2.20)

Ͱ͋Δɻࣜ (2.10)ʙ(2.13) ͔ΒΘ͔ΔΑ͏ʹɺి৔ ͱ ࣓ ৔ ͕ ؅ ࣠ ํ ޲ ʹ ੒ ෼ Λ ΋ ͨ ͳ ͍ʢEz =Ez , Hz ͷ͍ͣΕ͔Ұํ͕ 0 Ͱͳ͍৔߹ͷΈͰ͋
ΔɻEz = 0, Hz ̸= 0 ͷ৔߹Λ TE ೾ʢTransverseͱͯ͠ɺͱද͞ΕΔɻΑͬͯɺH0z (x, y) ͸

H0z = cos(kx x + θx ) cos(ky y + θy )electric wavesʣɺEz ̸= 0, Hz = 0 ͷ৔߹Λ TM ೾

ʢTransverse magnetic wavesʣͱ‫Ϳݺ‬ɻ

(2.21)

ͱॻ͚ΔͷͰɺ͜ΕΛࣜ (2.10)ʙ(2.13) ʹ୅ೖ͢

ɹΔ͜ͱʹΑΓɺE0x , E0y , H0x , H0y ͕‫·ٻ‬Δɻํ‫ܗ‬ಋ೾؅ͷ Ex , Ey ʹ͍ͭͯͷ‫ڥ‬ք৚݅

2.2 ํ‫ܗ‬ಋ೾؅
E0y (x = 0, y) = E0y (x = a, y) = 0(2.22)

E0x (x, y = 0) = E0x (x, y = b) = 0(2.23)

͔Βɺθx = θy = 0mπ(m = 0, 1, 2, ...)
kx =
a
nπ
ky =
(n = 0, 1, 2, ...)
b
ਤ6

)(

⃗0
E
⃗
H0

)

+ kc2

(

⃗0
E
⃗
H0

(2.26)͢Δ TE ೾͸ɺjωµ0 kycos(kx x) sin(ky y)
kc2−jωµ0 kx
E0y =sin(kx x) cos(ky y)
kc2
E0z = 0E0x =Δͱɺࣜ (2.14) ͷ kc Λ༻͍ͯ

∂2
∂2
+
∂x2
∂y 2

(2.25)

Ͱͳ͚Ε͹ͳΒͳ͍ɻΑͬͯɺํ‫ܗ‬ಋ೾؅Λ఻ൖํ‫ܗ‬ಋ೾؅

ࣜ (2.3) ΛϔϧϜϗϧπํఔࣜ (1.11) ʹ୅ೖ͢(

(2.24))=0(2.15)

ͱͳΔɻ͜ΕΛ‫׬‬શಋମͷ‫ڥ‬ք৚݅Λ༻͍ͯղ

jβkx
sin(kx x) cos(ky y)
kc2
jβky
H0y = 2 cos(kx x) sin(ky y)kc
H0z = cos(kx x) cos(ky y)

H0x =͘͜ͱʹΑΓɺTE ೾ʢEz = 0ʣ͓Αͼ TM ೾
ʢHz = 0ʣͷ৔߹ʹ͍ͭͯɺํ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ
ి࣓৔Λ‫ٻ‬ΊΑ͏ɻɹ

(2.27)(2.28)(2.29)(2.30)(2.31)(2.32)ͱද͞ΕΔɻ࠷௿࣍ͷϞʔυ͸ɺm = 1, n = 0
ͷɺTE10 ϞʔυͰ͋Δɻํ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ TE ೾ɹTE ೾ʢEz = 0ʣͷ৔߹ɺH0z (x, y) Λ
H0z (x, y) = X(x)Y (y)(2.16)

ͷΑ͏ʹม਺෼཭͢Δͱɺࣜ (2.15) ͸࣓৔ʹ͍ͭͯ

1 d2 X
1 d2 Y++ kc2 = 0
X dx2
Y dy 2ํ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ TM ೾

TM ೾ʢHz = 0ʣͷ৔߹ͷ E0z (x, y) ʹ͍ͭͯ΋ɺTE ೾ͷͱ͖ͷ H0z (x, y) ͱಉ༷ʹ

(2.17)

６－ 9

E0z (x, y) = X(x)Y (y)(2.33)

TM ೾͸ɺ−jβkx
cos(kx x) sin(ky y)
kc2
−jβky
E0y =
sin(kx x) cos(ky y)
kc2
E0z = sin(kx x) sin(ky y)E0x =jωε0 kysin(kx x) cos(ky y)
kc2−jωε0 kx
H0y =cos(kx x) sin(ky y)
kc2
H0z = 0H0x =(2.39)(2.40)(2.41)(2.42)(2.43)(2.44)ͱද͞ΕΔɻ࠷௿࣍ͷϞʔυ͸ɺm = 1, n = 1
ͷɺTM11 ϞʔυͰ͋Δɻਤ7ํ‫ܗ‬ಋ೾؅ͷ TE Ϟʔυి࣓৔ʢࢀߟจ‫ݙ‬[11] ΑΓʣͱม਺෼཭͢Δ͜ͱʹΑΓɺ

E0z = sin(kx x + θx ) sin(ky y + θy )(2.34)

ͱॻ͚Δɻͨͩ͠ɺ

kc2 = kx2 + ky2(2.35)

Ͱ͋ΔɻΑͬͯɺࣜ (2.34) Λࣜ (2.10)ʙ(2.13) ʹ୅ೖ͢Δ͜ͱʹΑΓɺE0x , E0y , H0x , H0y ͕‫·ٻ‬Δɻ‫ڥ‬ք৚݅ʢࣜ (2.22), (2.23)ʣΑΓθx = θy = 0mπ(m = 0, 1, 2, ...)
kx =
a
nπ
ky =
(n = 0, 1, 2, ...)
b(2.36)(2.37)

ਤ8(2.38)
ɹͰͳ͚Ε͹ͳΒͳ͍ͷͰɺํ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ

６－ 10ํ‫ܗ‬ಋ೾؅ͷ TM Ϟʔυి࣓৔ʢࢀߟจ‫[ ݙ‬11] ΑΓʣͱॻ͘͜ͱ͕Ͱ͖Δɻࣜ (2.51) ͷղ͸ɺ

Φ(ϕ) = cos(mϕ)(2.52)

ͱॻ͚Δɻͨͩ͠ɺप‫ظ‬৚݅ Φ(ϕ) = Φ(ϕ + 2π)
ΑΓɺm ͸੔਺Ͱͳ͚Ε͹ͳΒͳ͍ɻ·ͨɺࣜ(2.50) ͸

ਤ9

(
)
m2
d2 R
1 dR
+ 1−
R=0
+
d(kc r)2
kc r d(kc r)(kc r)2(2.53)

waveguide 2ͱॻ͖௚͢͜ͱʹΑΓɺϕοηϧͷඍ෼ํఔࣜͰ

2.3 ԁ‫ܗ‬ಋ೾؅

͋Δ͜ͱ͕Θ͔ΔɻΑͬͯɺ͜ͷղ͸ m ࣍ͷϕοࣜ (2.15) Λɺԁப࠲ඪηϧؔ਺ Jm (x) Λ༻͍ͯɺx = r cos ϕ(2.45)

y = r sin ϕ(2.46)

z=z(2.47)

1 ∂r ∂r(2.54)*10 ਤ 10 ʹɺϕοηϧؔ਺ y = J (x)ͱද͞ΕΔɻ
m*11 ͜ΕΒʹΑΓɺE
Λࣔ͢ɻ
0z ͸

Ͱද͢ͱɺ

(

R(r) = Jm (kc r)

(

∂
r
∂r

)

1 ∂2
+ 2 2
r ∂ϕ

)(

⃗0
E
⃗
H0

)

+ kc2

(

)
⃗0
E
⃗0 = 0H(2.48)*9 ํ‫ܗ‬ಋ೾؅ͷͱ͖ͱಉ༷ʹɺ‫׬‬શಋମͱॻ͚Δɻ

E0z (r, ϕ) = Jm (kc r) cos(mϕ)(2.55)

ͱॻ͘͜ͱ͕Ͱ͖ΔͷͰɺ͜ΕΛࣜ (2.10)ʙ(2.13)
ʹ୅ೖ͢Δ͜ͱʹΑΓɺE0r , E0ϕ , H0r , H0ϕ ͕‫ٻ‬ͷ‫ڥ‬ք৚݅Λ༻͍ͯ͜ΕΛղ͖ɺԁ‫ܗ‬ಋ೾؅Λ఻
ൖ͢Δి࣓৔Λ‫ٻ‬ΊΑ͏ɻɹԁ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ TM ೾

TM ೾ʢHz = 0ʣͷ৔߹ɺE0z (r, ϕ) Λ
E0z (r, ϕ) = R(r)Φ(ϕ)(2.49)

ͷΑ͏ʹม਺෼཭͢Δͱɺࣜ (2.48) ͸ɺm2 Λఆ
਺ͱͯ͠

ਤ 10

(

)
2

d2 R 1 dRm+ kc2 − 2
+
2
dr
r dr
r
2
d Φ
+ m2 Φ = 0
dϕ2
*9

r=

√

R=0

(2.50)(2.51)*10ϕοηϧؔ਺ʢJ0 (x), J1 (x), J2 (x)ʣϕοηϧͷඍ෼ํఔࣜ

y ′′ +

1 ′
y +
x

(

1−

ν2
x2

)

y=0

ͷҰൠղ͸ɺϕοηϧؔ਺ Jν (x) ͱϊΠϚϯؔ਺ Yν (x)Λ༻͍ͯ

x2 + y 2 , ϕ = tan−1 (y/x) Ͱ͋Δ͔Βɺ

y = C1 Jν (x) + C2 Yν (x)∂∂sin ϕ ∂= cos ϕ
−
∂x
∂r
r ∂ϕ
∂
∂
cos ϕ ∂= sin ϕ
+
∂y
∂r
r ∂ϕ
(
)
2
2
∂
∂
1 ∂
∂
1 ∂2
+
=
r
+ 2
2
2
∂x
∂y
r ∂r
∂r
r ∂ϕ2

ͱද͞ΕΔ͕ɺx → 0 ͷͱ͖ Yν (x) → −∞ Ͱ͋ΔͨΊɺC2 = 0 ͱ͍ͯ͠Δɻ
*11 ϕοηϧؔ਺ Jm (x) ͸࣍਺ m ͕େ͖͘ͳΔ΄Ͳ্ཱΓͷ஗͍ؔ਺ͰɺҰ୴্ཱͬͯ͠·͏ͱɺࡾ֯ؔ਺ͷΑ͏ͳৼಈΛଓ͚Δɻͦͷप‫ظ‬͸΄΅ 2π Ͱ͋Γɺৼ෯͸
√
1/ x ʹൺྫͯ͠খ͘͞ͳ͍ͬͯ͘ɻ

ͱͳΔɻ

６－ 11

·Δɻ*12͜͜Ͱɺԁ‫ܗ‬ಋ೾؅ͷ Eϕ ʹ͍ͭͯͷ‫ڥ‬ք৚݅

E0ϕ (r = a, ϕ) = 0(2.56)Jm (kc a) = 0(2.57)

͔ΒɺͰͳ͚Ε͹ͳΒͳ͍ɻͭ·Γɺࣜ (2.57) ͷղΛ

kc a = ρmn ͱ͢Δͱɺ
kc =

ρmn
a

(2.58)

Ͱͳ͚Ε͹ͳΒͳ͍ɻද 1 ʹɺρmn ͷൺֱత௿࣍
ͷ͍͔ͭ͘ͷ஋Λࣔ͢ɻ
ද1

Jm (x) = 0 ͷࠜɿx = ρmn

n=1

n=2

n=3

m=0

2.405

5.520

8.654

m=1

3.832

7.016

10.173

m=2

5.136

8.417

11.620ਤ 11ԁ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ TM ೾͸ɺ

−jβ ′
E0r =
J (kc r) cos(mϕ)
kc mjβm Jm (kc r)sin(mϕ)E0ϕ = 2kcr
E0z = Jm (kc r) cos(mϕ)−jωε0 m Jm (kc r)sin(mϕ)kc2
r−jωε0 ′H0ϕ =Jm (kc r) cos(mϕ)
kc
H0z = 0H0r =ԁ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ TE ೾

TE ೾ʢEz = 0ʣͷ৔߹ͷ H0z (r, ϕ) ʹ͍ͭͯ
(2.59)

΋ɺTM ೾ͷͱ͖ͷ E0z (r, ϕ) ͱಉ༷ʹH0z (r, ϕ) = R(r)Φ(ϕ)(2.60)(2.61)(2.62)(2.63)(2.64)

H0z (r, ϕ) = Jm (kc r) cos(mϕ)(2.66)

ͱॻ͘͜ͱ͕Ͱ͖ΔɻΑͬͯɺ͜ΕΛࣜ (2.10)ʙ

(2.13) ʹ୅ೖ͢Δ͜ͱʹΑΓɺE0r , E0ϕ , H0r , H0ϕ͕‫·ٻ‬Δɻ͜͜ͰɺE0ϕ (r, ϕ) ʹ͍ͭͯॻ͘ͱɺ

E0ϕ (r, ϕ) =ͷɺTM01 ϞʔυͰ͋Δɻ

jωµ0 ′J (kc r) cos(mϕ)
kc m(2.67)

Ͱ͋Δ͕ɺ‫ڥ‬ք৚݅ʢࣜ (2.56)ʣʹΑΓ′Jm(kc a) = 0*12(2.65)

ͱม਺෼཭͢Δ͜ͱʹΑΓɺͱද͞ΕΔɻ࠷௿࣍ͷϞʔυ͸ɺm = 0, n = 1ɹԁ‫ܗ‬ಋ೾؅ͷ TM Ϟʔυి࣓৔ʢࢀߟจ‫[ ݙ‬11] ΑΓʣ⃗ ʹ͍ͭͯ੒Γཱͭ೚ҙͷϕΫτϧ V
Vr = Vx cos ϕ + Vy sin ϕ(2.68)

Ͱͳ͚Ε͹ͳΒͳ͍ɻͭ·Γɺࣜ (2.68) ͷղΛ

kc a = ρ′mn ͱ͢Δͱɺ

Vϕ = −Vx sin ϕ + Vy cos ϕkc =Λ༻͍Ε͹Α͍ɻ

６－ 12

ρ′mn
a

Ͱͳ͚Ε͹ͳΒͳ͍ɻද 2 ʹɺρ′mn ͷൺֱత௿࣍
ͷ͍͔ͭ͘ͷ஋Λࣔ͢ɻද2′Jm(x) = 0 ͷࠜɿx = ρ′mn

n=1

n=2

n=3

m=0

3.832

7.016

10.173

m=1

1.841

5.331

8.536

m=2

3.054

6.706

9.969

ԁ‫ܗ‬ಋ೾؅Λ఻ൖ͢Δ TE ೾͸ɺjωµ0 m Jm (kc r)sin(mϕ)kc2
r
jωµ0 ′E0ϕ =J (kc r) cos(mϕ)
kc m
E0z = 0
E0r =

−jβ ′
J (kc r) cos(mϕ)
kc mjβm Jm (kc r)sin(mϕ)H0ϕ = 2kcr
H0z = Jm (kc r) cos(mϕ)
H0r =(2.69)(2.70)(2.71)(2.72)(2.73)
ਤ 12(2.74)

ԁ‫ܗ‬ಋ೾؅ͷ TE Ϟʔυి࣓৔ʢࢀߟจ‫[ ݙ‬11] ΑΓʣͱද͞ΕΔɻ࠷௿࣍ͷϞʔυ͸ɺm = 0, n = 1β Λ༻͍ͯ

ͷɺTE01 ϞʔυͰ͋Δɻ
ɹ

λg =

(

ःஅप೾਺ͱ؅಺೾௕
Ґ૬ఆ਺ β = 0 ͷͱ͖ͷप೾਺Λ ω = ωc ͱ͢Δͱɺࣜ (2.14) ΑΓ

2π
λc

)2

=

(

2π
λ

)2

−

(

2π
λg

)2

(2.78)

Ͱ͋ΔͷͰɺ

kc
= ckc
ε0 µ0

1
1
1
= 2+ 22λλcλg(2.75)

ͱͳΓɺ͜ΕΛःஅ֯प೾਺ʢcutoﬀ angular fre਺ΑΓେ͖͍ͱ͖ɺి࣓೾͸ಋ೾؅Λ఻ൖ͢Δ͜
ͱ͕Ͱ͖Δɻःஅ೾௕ʢcutoﬀ wavelengthʣλc ͸ɺःஅप೾਺ʢcutoﬀ frequencyʣfc = ωc /2π Λ༻
͍ͯ

c
2π
=
fc
kc

(2.79)

ͱͳΓɺ؅಺೾௕ λg ͸

quencyʣͱ‫Ϳݺ‬ɻͭ·Γɺ֯प೾਺͕ःஅ֯प೾

λc =(2.77)ͱද͞ΕΔɻ͜͜Ͱɺࣜ (2.14) ΑΓ

2.4 ಋ೾؅Λ఻ൖ͢Δి࣓೾ͷੑ࣭ωc = √

2π
β

ͱॻ͚Δɻ

λg = √λ1 − (λ/λc )2(2.80)

Ґ૬଎౓ʢphase velocityʣvp ͸ɺҐ૬ఆ਺ β Λ
༻͍ͯ

vp =
(2.76)

ωβ(2.81)ͱද͞ΕΔɻͭ·ΓɺͱදͤΔɻ؅಺೾௕ʢguide wavelengthʣλg ͸ɺҐ૬ఆ਺６－ 13

v p = f λg = √c
1 − (λ/λc )2(2.82)ͱॻ͚Δɻ

্ͷి৔ͱ C’D ໘্ͷి৔͸ɺେ͖͕͞ಉ͡Ͱ޲

۩ମతʹɺํ‫ܗ‬ಋ೾؅Λ TEm0 ϞʔυͰ఻ൖ͖͕͢‫ͳʹٯ‬Δඞཁ͕͋ΔɻͦͷͨΊʹ͸ɺB ఺ͷ

Δి࣓೾ʹ͍ͭͯߟ͑Α͏ɻn = 0 Ͱ͋ΔͨΊɺి৔͕ C ఺Ͱ൓ࣹ͞ΕɺD ఺ʹ౸ୡ͢Δ·Ͱʹɺ

kc =

√(

mπ )2 ( nπ )2
mπ
+
=
a
b
a

Ґ૬͕ͪΐ͏ͲҰप‫ظ‬ɺ·ͨ͸ͦͷ੔਺ഒ͚ͩ‫ܦ‬(2.83)

ͱͳΓɺಋ೾؅Λ ej(ωt−βz) Ͱ఻ൖ͢Δి࣓৔͸( mπ )jωµ0 asinxE0y = −mπ
a
( mπ )
jβa
H0x =
sin
x
mπ(
a
)
mπ
H0z = cosxaE0x = Ez = Hy = 0ա͠ͳ͚Ε͹ͳΒͳ͍ɻͭ·ΓɺBC ؒɺCD ؒͷ‫཭ڑ‬Λ s ͱͯ͠ɺB ఺͔Β D ఺ʹ౸ୡ͢Δ·ͰͷҐ૬ͷਐΈ͕

(2.84)(2.85)(2.86)

2πss+ π + 2π = π + 2πmλλͰͳ͚Ε͹ͳΒͳ͍ɻ͜ΕʹΑΓɺC ఺ʹೖࣹ͢Δ֯౓ θ ͸ɺsin θ =

(2.87)ͱද͞ΕΔɻ͜ͷి৔͸ y ੒෼ͷΈͰɺి৔΋࣓৔

Δɻःஅ೾௕ λc ͸ɺ

λc =

2π
2a
=
kc
m

λ
λg = √( )2
1 − mλ2amλmλg
cos θ = s =22(2.91)(2.92)

ΑΓλg =(2.88)

Ͱ͋Δ͔Βɺ؅಺೾௕ λg ͸ɺ

mλ
s
=
′
CC
2a

ͱͳΔɻΑͬͯ؅಺೾௕ λg ͸ɺ΋ x ͷؔ਺Ͱ͋Δɻ·ͨ͜ͷͱ͖ɺm′ = 1, 2, ..., mͱͯ͠ x = a(2m′ − 1)/(2m) Ͱɺి৔͕࠷େͱͳ

(2.90)

λ
=√
cos θ

λ
( )2
1 − mλ2a(2.93)

ͱͳΓɺࣜ (2.80) ͱҰக͢Δ͜ͱ͕Θ͔Δɻ
ਤ 13 ͔Β͸·ͨɺҎԼͷ͜ͱ͕Θ͔Δɻฏ໘೾

(2.89)

͕؅࣠ํ޲ʢ+z ํ޲ʣʹਐΉ଎౓Λ v ′ ͱ͢Δͱɺ͜Ε͸ͱॻ͚Δɻ͜͜Ͱɺํ‫ܗ‬ಋ೾؅಺ΛਐΉฏ໘೾͸ɺv ′ = c cos θਤ 13 ͷΑ͏ʹඳ͘͜ͱ͕Ͱ͖Δɻਤ 13 Ͱɺి࣓(2.94)

೾ͷਐߦํ޲ͱਨ௚ͳ໘Ͱ͸ɺి৔ͷେ͖͞ɺ޲

Ͱ͋Δɻ͜Ε͸ి࣓৔ͷΤωϧΪʔͷ఻ൖ଎౓Ͱ

͖ͱ΋ʹಉ͡Ͱ͋Δɻ͜ͷ໘͕ AC ʹԊͬͯਐΈɺ͋Γʢ‫ޙ‬ड़ʣɺޫ଎ΑΓখ͍͞ɻ·ͨҐ૬଎౓ vp

C ఺Ͱ൓ࣹ͞Εͯ CF ʹԊͬͯਐΉɻC ఺Ͱ൓ࣹʹ͍ͭͯ͸͞ΕΔࡍɺి৔ͷ޲͖͸‫ͳͱٯ‬Δɻ͜ΕʹΑΓ C఺ͷి৔͕ 0 ͱͳΓɺ‫ڥ‬ք৚݅Λຬͨ͢͜ͱʹͳΔɻ·ͨɺC ఺ͷ޲͔͍ଆʹ͋Δ C’ ఺ʹ͓͍ͯ΋‫ڥ‬ք৚݅Λຬͨ͞ͳ͚Ε͹ͳΒͳ͍ͷͰɺC’B ໘

vp cos θ = c(2.95)

͕੒Γཱͪɺޫ଎ΑΓେ͖͍ɻ͔͜͠͠Ε͸ɺి࣓৔ͷϞʔυͷ‫఻͕ܗ‬ൖ͢Δ଎౓Ͱ͋Γɺ࣮ࡍͷి࣓৔ͷΤωϧΪʔ͕ͦͷ଎౓Ͱ఻ൖ͍ͯ͠
ΔͷͰ͸ͳ͍ɻप೾਺͕ःஅप೾਺ʹ͚ۙͮ͹ʢω → ωc ʣɺҐ૬଎౓ vp ͸ແ‫ݶ‬େͱͳΔɻ·ͨ͜ͷͱ͖ɺθ → π/2 ͱͳΓɺి࣓೾͸ಋ೾؅Λ఻ൖ͠ͳ͍ɻࣜ (2.94), (2.95) ΑΓɺv ′ ͱ vp ʹ͸v ′ vp = c 2ͷؔ܎͕͋Δ͜ͱ͕Θ͔Δɻਤ 13ํ‫ܗ‬ಋ೾؅Λ TEm0 ϞʔυͰ఻ൖ͢Δฏ໘೾

６－ 14

ɹ

(2.96)

ಋ೾؅ʹ͓͚Δి࣓৔ͷΤωϧΪʔ

ͱॻ͚Δɻ͜͜Ͱɺి࣓৔ͷΤωϧΪʔͷ఻ൖ଎ಋ೾؅Ͱ఻ൖ͞ΕΔిྗͷ࣌ؒฏ‫ۉ‬஋ P ͸ɺࣜ

(1.32) Ͱ༩͑ΒΕΔϙΠϯςΟϯάϕΫτϧͷ࣌⟨ ⟩⃗ Λɺಋ೾؅அ໘ੵ S ′ Ͱੵ෼͢Δؒ͜ฏ‫ۉ‬஋ SͱʹΑͬͯ࣍ͷΑ͏ʹ༩͑ΒΕΔɻ

P =

∫∫

⟨

S′

1
=
2
=

1
2

⟩

0 0

ͷؔ܎͕͋ΔͷͰɺ(2.102)ɹ

Re{−Ey Hx∗ } dxdy

0 0
2 ∫ b∫ a‫܈‬଎౓ͱ෼ࢄ‫ۂ‬ઢ֯प೾਺ ω ͱҐ૬଎౓ β ͷؔ܎ʢ෼ࢄؔ܎ʣΛදࣜ͢ (2.14) Λ༻͍ͯɺ‫܈‬଎౓ʢgroup velocityʣ

vg Λ‫ٻ‬ΊΑ͏ɻ͜Ε͸ɺ୯Ґ௕͋ͨ͞Γͷి‫ؾ‬ΤωϧΪʔͷ࣌ؒฏ‫ۉ‬஋vg =We ͸ɺࣜ (1.29) Λ୯Ґ௕͋ͨ͞Γͷମੵʹ͍ͭ
ͯੵ෼ͯ͠

βc2
c2
dω
=
=
dβ
ω
vp

(2.103)ͱͳΓɺࣜ (2.102) Ͱද͞ΕΔి࣓৔ͷΤωϧΪʔ

ε
We =
4

∫ b∫ aͷ఻ൖ଎౓ ven ͱҰக͍ͯ͠Δ͜ͱ͕Θ͔Δɻͭ

⃗ ·E⃗ ∗ dxdyE·Γɺి࣓৔ͷΤωϧΪʔ͸‫܈‬଎౓ vg Ͱ఻ൖ͠ɺ

0 0
2
2 2 ∫ b∫ aω εµ a4m2 π 2ω 2 εµ2 a3 b=
8m2 π 2
=

β
c2
P
=
=
W
ωεµ
vp

ೝͰ͖Δɻ

)
(
βωµa
2 mπ
x
dxdy
sin
2m2 π 2 0 0aβωµa3 b=(2.97)
4m2 π 2=(2.101)ͱͳΓɺven ͕ࣜ (2.96) ͷ v ′ ͱҰக͢Δ͜ͱ͕֬⃗ ×H⃗ ∗ )z } dxdy
Re{(E∫ b∫ aP = ven W

ven =⃗ · ⃗n dS ′S∫ b∫ a౓Λ ven ͱ͢Δͱɺ( mπ )
x dxdysin2
a0 0͜Ε͸ޫ଎ΑΓখ͍͞ɻࣜ (2.14) ͷ෼ࢄؔ܎Λਤࣔ͢Δͱɺਤ 14 ͷΑ(2.98)

ಉ͘͡ɺ୯Ґ௕͋ͨ͞Γͷ࣓‫ؾ‬ΤωϧΪʔͷ࣌ؒ͏ͳ෼ࢄ‫ۂ‬ઢʢdispersion curveʣ͕ඳ͚Δɻ෼ࢄ‫ۂ‬ઢ͸ɺω = ±cβ ͳΔ઴ۙઢʢਤதͷഁઢʣΛ΋ͪɺω ࣠ͱःஅ֯प೾਺ ωc = ckc ͰަΘΔ૒‫ۂ‬ઢฏ‫ۉ‬஋ Wm ͸ɺࣜ (1.30) Λ୯Ґ௕͋ͨ͞Γͷମੵ

ͱͳΔɻҐ૬଎౓ vp ͸ɺ‫͔఺ݪ‬Β෼ࢄ‫ۂ‬ઢ্ͷ఺ʹ͍ͭͯੵ෼ͯ͠

(β, ω) ʹҾ͍ͨ௚ઢͷ܏͖ ω/β ʹΑͬͯ༩͑ΒΕ

µ
Wm =
4

∫ b∫ aΔɻ·ͨ‫܈‬଎౓ vg ͸ɺ఺ (β, ω) ʹ͓͚Δ෼ࢄ‫ۂ‬ઢ

⃗∗⃗ · H dxdyHͷ઀ઢͷ܏͖ dω/dβ ʹ౳͍͠ɻͦͯ͠ɺ0 < β ͷ

0 0∫ b∫ a ()(β 2 a22 mπ
xsinm2 π 2a0 0( mπ ))+ cos2x dxdya()µab β 2 a2
+1=8m2 π 2ω 2 εµ2 a3 b
= We
(2.99)=
8m2 π 2=µ4ྖҬͰ͸ +z ํ޲΁ͷલਐ೾Λද͠ɺβ < 0 ͷྖҬͰ͸ −z ํ޲΁ͷ‫ޙ‬ਐ೾Λද͍ͯ͠Δɻ

Αͬͯɺ୯Ґ௕͋ͨ͞Γͷి࣓৔ͷΤωϧΪʔ W
͸ɺW = W e + Wm =

ω 2 εµ2 a3 b
4m2 π 2(2.100)

６－ 15

ਤ 14

෼ࢄ‫ۂ‬ઢʢkc2 = ω 2 εµ − β 2 ʣͭ·Γɺ

3 ۭಎ‫ثৼڞ‬3.1 ۭಎ‫ثৼڞ‬ͷ‫ৼڞ‬प೾਺1
λ0 = √(kc /2π)2 + (p/2d)2z ํ޲Λ؅࣠ํ޲ͱͯ͠ɺz = 0 ͓Αͼ z = d Ͱಋ೾؅ʹಋମͰϑλΛͯ͠ശΛ࡞Ζ͏ɻ͜ͷͱ͖ɺप೾਺ͱ؅࣠ํ޲௕͞ d ͕͋Δ৚݅Λຬͨ͢ͱ͖ɺ

(3.5)

͜ͷശͷதͰి࣓৔͕‫͢ৼڞ‬Δɻͭ·Γɺ+z ํ

ͱͳΔ͜ͱ͕Θ͔Δɻ͜ͷͱ͖ͷ‫ৼڞ‬प೾਺ f0޲ʹਐΉి࣓৔ͱ −z ํ޲ʹਐΉి࣓৔͕ॏͳΓ͸ɺ߹ͬͯɺఆࡏ೾Λੜͤ͡͞Δɻ͜ͷശΛۭಎ‫ৼڞ‬‫ث‬ʢcavity resonatorʣͱ‫ͼݺ‬ɺ͜ͷͱ͖ͷप೾਺

c
c
f0 =
=
λ0
2π

Λ‫ৼڞ‬प೾਺ʢresonant frequencyʣͱ‫Ϳݺ‬ɻۭಎ‫಺ثৼڞ‬ͷి࣓৔͸ɺ‫֯ৼڞ‬प೾਺ ω0 Λ༻

√

kc2 +

⃗ H
⃗ Λ
͍ͯ E,
(

⃗
E(x,
y, z, t)⃗H(x,y, z, t)

)

=

(

)
⃗ 0 (x, y, z) jω t
E
0
⃗ 0 (x, y, z) e
H

(3.1)

( pπ )2
d

(3.6)

*15
Ͱ͋Δɻ

ɹ

ͱද͠ɺ͜ΕΛ೾ಈํఔࣜ (1.10) ʹ୅ೖͯ͠ಘΒ

3.2 ௚ํମۭಎ‫ثৼڞ‬

ΕΔ࣍ͷϔϧϜϗϧπํఔࣜ

∇2

(

⃗0
E
⃗0
H

)

+ ω02 εµ

(

⃗0
E
⃗0
H

)

=0

(3.2)

Λղ͘͜ͱʹΑΓಘΒΕΔɻ*13 *14
ϑλ z = 0, d ʹ͓͚Δ‫ڥ‬ք৚͔݅Βɺ؅಺೾௕

λg ͸ɺp Λ੔਺ͱͯ͠
λg =

2d
p

(3.3)

Ͱͳ͚Ε͹ͳΒͳ͍ɻΑͬͯɺࣜʢ2.79ʣΑΓ‫ৼڞ‬೾௕ λ0 ͸

ਤ 15

1
1
1
= 2+ 22
λ0
λc
λg( k )2 ( p )2
c
=
+
2π
2d
*13

(3.4)

ɹ

Ͱ͋Δɻ
ಋ೾؅ͷͱ͖ͱಉ༷ʹɺ௚ํମۭಎ‫͍ͯͭʹثৼڞ‬͸ɺ

*15

͜͜Ͱɺ

∇2 =
*14

௚ํମۭಎ‫ثৼڞ‬

∂2
∂2
∂2
+
+
∂x2
∂y 2
∂z 2

H0z = X(x)Y (y)Z(z)TE Ϟʔυ

E0z = X(x)Y (y)Z(z)TM Ϟʔυ

ۭಎ‫಺ثৼڞ‬ͷి࣓৔ʢఆࡏ೾ʣ͸ɺಋ೾؅಺ͷਐߦ೾ʢβ → +β ʣͷి࣓৔ͱ‫ޙ‬ਐ೾ʢβ → −β ʣͷి࣓৔ΛॏͶ߹ΘͤΔ͜ͱʹΑͬͯ΋ಘΒΕΔɻ͜ͷͱ͖ͷҐ૬ఆ਺ β ͸ɺԁபۭಎ‫͍ͯͭʹثৼڞ‬͸ɺ

H0z = R(r)Φ(ϕ)Z(z)TE Ϟʔυ

E0z = R(r)Φ(ϕ)Z(z)TM Ϟʔυ

β=ͱม਺෼཭ͯ͠ɺ‫ڥ‬ք৚݅ʢಋ೾؅ͷͱ͖ͷ‫ڥ‬ք৚݅ͱϑλ z = 0, d ʹ͓͚Δ‫ڥ‬ք৚݅ʣΛ༻͍Ε͹Α͍ɻ

６－ 16

Ͱ͋Δɻ

pπ
2π
=
λg
d

ɹ
௚ํମۭಎ‫͚͓ʹثৼڞ‬Δ TM Ϟʔυ

E0x =

E0y =

−kz kxcos(kx x) sin(ky y) sin(kz z)
kc2(3.18)−kz kysin(kx x) cos(ky y) sin(kz z)
kc2(3.19)

E0z = sin(kx x) sin(ky y) cos(kz z)
H0x =
ਤ 16

௚ํମۭಎ‫ثৼڞ‬ͷి࣓৔ʢࢀߟจ‫[ ݙ‬11]

ΑΓʣ

H0y =
௚ํମۭಎ‫͚͓ʹثৼڞ‬Δ TE Ϟʔυ

E0x =

E0y =

jω0 µ0 kycos(kx x) sin(ky y) sin(kz z)
kc2
(3.7)

−jω0 ε0 kx
cos(kx x) sin(ky y) cos(kz z)
kc2(3.22)(3.9)

ω02 ε0 µ0 = kx2 + ky2 + kz2mπnπpπ, ky =, kz =kx =abd
kc2 = ω02 ε0 µ0 − kz2 = kx2 + ky2

c
f0 =
2π

√(
mπ )2
a

+

( nπ )2b+( pπ )2Ͱ͋Δɻɹ

3.3 ԁபۭಎ‫ثৼڞ‬ω02 ε0 µ0 = kx2 + ky2 + kz2mπnπpπ, ky =, kz =kx =abd
kc2 = ω02 ε0 µ0 − kz2 = kx2 + ky2(3.13)(3.14)(3.15)

Ͱ͋Δɻ‫ৼڞ‬प೾਺ f0 ͓Αͼ‫ৼڞ‬೾௕ λ0 ͸√(mπ )2 ( nπ )2 ( pπ )2++(3.16)abd1
λ0 = √(3.17)
2(m/2a) + (n/2b)2 + (p/2d)2

c
f0 =
2π

d

1
λ0 = √(m/2a)2 + (n/2b)2 + (p/2d)2(3.12)

͜͜Ͱɺ(3.24)(3.25)(3.26)

Ͱ͋Δɻ‫ৼڞ‬प೾਺ f0 ͓Αͼ‫ৼڞ‬೾௕ λ0 ͸

−kz kxsin(kx x) cos(ky y) cos(kz z)
kc2(3.10)−kz kycos(kx x) sin(ky y) cos(kz z)
H0y =
kc2
(3.11)(3.23)

͜͜Ͱɺ−jω0 µ0 kx
sin(kx x) cos(ky y) sin(kz z)
kc2
(3.8)

H0z = cos(kx x) cos(ky y) sin(ky y)jω0 ε0 ky
sin(kx x) cos(ky y) cos(kz z)
kc2(3.21)

H0z = 0E0z = 0H0x =(3.20)

Ͱ͋Δɻ

６－ 17

ਤ 17

ԁபۭಎ‫ثৼڞ‬(3.27)(3.28)

ɹԁபۭಎ‫͚͓ʹثৼڞ‬Δ TE Ϟʔυjωµm Jm (kc r)sin(mϕ) sin(kz z) (3.39)
kc2
r
jωµ ′
J (kc r) cos(mϕ) sin(kz z)(3.40)
E0ϕ =kc m
E0z = 0(3.41)E0r =kz ′J (kc r) cos(mϕ) cos(kz z)(3.42)
kc m−kz m Jm (kc r)sin(mϕ) cos(kz z) (3.43)
H0ϕ =
kc2
r
(3.44)
H0z = Jm (kc r) cos(mϕ) sin(kz z)
H0r =

͜͜Ͱɺ
ਤ 18

ρ′mn
pπ
, kz =adkc2 = ω02 ε0 µ0 − kz2ԁபۭಎ‫ثৼڞ‬ͷి࣓৔ʢࢀߟจ‫[ ݙ‬11] ΑΓʣ

ɹ

kc =

(3.45)(3.46)

Ͱ͋Δɻ‫ৼڞ‬प೾਺ f0 ͓Αͼ‫ৼڞ‬೾௕ λ0 ͸ԁபۭಎ‫͚͓ʹثৼڞ‬Δ TM Ϟʔυ

−kz ′
E0r =
J (kc r) cos(mϕ) sin(kz z)
kc m
mkz Jm (kc r)sin(mϕ) sin(kz z)
E0ϕ = 2kcr
E0z = Jm (kc r) cos(mϕ) cos(kz z)

c
f0 =
2π

−jωεm Jm (kc r)sin(mϕ) cos(kz z)
kc2r(3.32)−jωε ′J (kc r) cos(mϕ) cos(kz z) (3.33)
H0ϕ =
kc m
H0z = 0(3.34)c2πρmn )2 ( pπ )2+ad1

λ0 = √(ρmn /2πa)2 + (p/2d)2( pπ )2d(3.47)(3.48)

ɹ

3.4 ۭಎ‫ثৼڞ‬ͷ Q ஋ۭಎ‫ثৼڞ‬ͷ Q ஋͸ɺ

QL = ω0‫ثৼڞ‬தͷ஝ੵΤωϧΪʔ
1 ඵؒʹࣦΘΕΔΤωϧΪʔ(3.35)(3.36)(3.49)

Ͱఆٛ͞ΕΔɻͭ·Γɺ

QL = ω 0

Ͱ͋Δɻ‫ৼڞ‬प೾਺ f0 ͓Αͼ‫ৼڞ‬೾௕ λ0 ͸

f0 =

+

Ͱ͋Δɻ

͜͜Ͱɺ

√(

a

)2

(3.31)

H0r =

ρmn
pπ
, kz =
kc =
a
d
2
2
2
kc = ω0 ε0 µ0 − kz

mn1
λ0 = √′(ρmn /2πa)2 + (p/2d)2(3.29)(3.30)√(
ρ′W
W= ω0 dWP− dt(3.50)

Ͱ͋Δɻ
ۭಎͷนଛࣦʹΑΔফඅిྗΛ P0 ɺ݁߹޸ʢಋ
೾؅౳ʣ͔Βग़ΔిྗΛ Pext ͱ͢Δͱɺ

(3.37)(3.38)

P = P0 + Pext

(3.51)

1
1
1
=
+
QL
Q0
Qext

(3.52)

Ͱ͋Δ͔Βɺ

Ͱ͋Δɻ

６－ 18

ͱॻ͚Δɻ͜ͷͱ͖ɺ

W
Q0 = ω0
P0
W
Qext = ω0
Pext

4 ఆࡏ೾‫ܕ‬Ճ଎؅(3.53)

ɹ(3.54)
4.1 ϐϧϘοΫεۭಎͰ͋ΓɺQ0 ͸಺෦ Qʢinternal Qʣ΋͘͠͸ແ

·ͣ࠷ॳʹɺఆࡏ೾‫ܕ‬Ճ଎؅Ͱ࠷΋‫ج‬ຊͱͳΔɺෛՙ Qʢunloaded Qʣͱ‫ݺ‬͹ΕɺQext ͸֎෦ Q

ϐϧϘοΫεۭಎͷ TM010 Ϟʔυʹ͍ͭͯߟ͑

ʢexternal Qʣͱ‫ݺ‬͹ΕΔɻ·ͨɺQL ͸ෛՙ Q

Α͏ɻ͜͜ͰɺϐϧϘοΫεۭಎʢpillbox cavityʣʢloaded Qʣͱ‫ݺ‬͹ΕΔɻͱ͸લઅͷԁபۭಎ‫ثৼڞ‬ʢcylindrical cavityʣͷ‫Ͱ಺ثৼڞ‬ଛࣦ͕͋Δͱɺ౰વ஝ੵΤωϧΪʔW ͸‫ݮ‬ਰ͢Δɻࣜ (3.50) ΑΓɺ
dW
ω0
=−
W
dt
QL

͜ͱͰ͋ΓɺTM Ϟʔυʹ͓͚Δి৔ͷ؅࣠ํ޲ʢz ํ޲ʣ੒෼ Ez ΛՙిཻࢠͷՃ଎ʹ༻͍Δɻ
ϐϧϘοΫεۭಎʹ͓͍ͯ‫ج‬ຊϞʔυʢ࠷௿࣍(3.55)

Ͱ͋ΔͷͰɺ஝ੵΤωϧΪʔ W ͸ɺe−(ω0 /QL )t Ͱ‫ݮ‬ਰ͢Δ͜ͱ͕Θ͔ΔɻΑͬͯɺ‫಺ثৼڞ‬ͷి࣓ͷϞʔυʣͰ͋Δ TM010 Ϟʔυͷి࣓৔͸ɺࣜ(3.29)ʙ(3.34) Ͱ m = 0, n = 1, p = 1 ͱ͢Δ͜ͱʹΑΓ৔͸ɺe−(ω0 /2QL )t Ͱ‫ݮ‬ਰ͢ΔɻQ ͕ߴ͍ఔ‫͕ৼڞ‬E0z = J0 (kc r)√ε ′J (kc r)H0ϕ = −jµ 0
E0r = E0ϕ = H0z = H0r = 0

࣋ଓ͢Δ͜ͱ͕Θ͔Δɻ

(4.1)
(4.2)
(4.3)

ͱಘΒΕΔɻ͜͜Ͱɺ

kc =

ρ01
a

(4.4)

Ͱ͋Δɻి৔ E0z ͓Αͼ࣓৔ H0ϕ ͸ͱ΋ʹɺr ʹͷΈґଘ͠ɺz ʹΑΒͳ͍ɻE0z ͓Αͼ H0ϕ ͷ‫ܘ‬ํ޲ґଘੑΛਤ 19 ʹࣔ͢ɻE0z ͸؅্࣠ʢr = 0ʣͰ࠷େͱͳΓɺH0ϕ ͸ɺρ01 r/a = 1.841 Ͱ࠷େͱ
*16
ͳΔɻ

ਤ 19
*16

ϐϧϘοΫεۭಎ TM010 Ϟʔυͷి࣓৔J0′ (x) ʹ͍ͭͯɺ࠷େ஋͸J0′ (1.841) = 0.5819
·ͨɺr = aɺͭ·Γɺx = kc r = kc a = ρ01 = 2.405

６－ 19‫ৼڞ‬प೾਺ f0 ͸

f0 =ͱ‫ٻ‬ΊΒΕΔɻ

c ρ01
2π a

࣍ʹɺz = 0 ΋͘͠͸ z = d Ͱࣦ͏ిྗ P1 ͸

(4.5)

∫ a
1
Rs
Hϕ∗ Hϕ · 2πrdr
2
∫0 a
ε ′
1
J0 (kc r)2 · 2πrdr
= Rs
2
0 µ
1
ε 1 ρ2= Rs · 2π · 2 · 01 J1 (ρ01 )2
2
µ kc
2
ε
1
(4.9)
= Rs πa2 J1 (ρ01 )2
2
µ

P1 =

ͱͳΓɺԁபͷ‫ ܘ‬a ʹͷΈґଘ͠ɺ࣠ํ޲௕͞ dʹΑΒͳ͍ɻ
࣍ʹɺϐϧϘοΫεۭಎʹ͓͚Δ TM010 Ϟʔυ*17 ·ͣɺి‫ؾ‬ΤωϧΪʔͷ࣌ͷ Q ஋Λ‫ٻ‬ΊΑ͏ɻؒฏ‫ ۉ‬We ͸

∫∫∫
1
⃗ ·E
⃗ ∗ dVE
We = ε4∫ a
1
J0 (kc r)2 · 2πrdr
= ε·d
4
0
∫
2π ρ01
1= ε·d 2xJ0 (x)2 dx
4
kc 0
2π ρ2
1
= ε · d 2 · 01 J1 (ρ01 )2
4kc
2
1= επa2 dJ1 (ρ01 )2
4

r = a Ͱࣦ͏ిྗ P2 ͸
1
Rs · Hϕ∗ Hϕ · 2πad
2
1ε
= Rs · J0′ (kc a)2 · 2πad2µε
= Rs πad J1 (ρ01 )2
µ

P2 =

(4.6)

Αͬͯɺน͔ΒࣦΘΕΔిྗ P0 ͸࣓‫ؾ‬ΤωϧΪʔͷ࣌ؒฏ‫ ۉ‬Wm ͸

P0 = 2 × P1 + P2

∫∫∫

1
⃗ ·H
⃗ ∗ dV
µ
H
4
∫ a
ε ′
1
J0 (kc r)2 · 2πrdr
= µ·d
4
0 µ
∫
1
ε a
= µ · d · 2π
J1 (kc r)2 rdr
4
µ 0
∫ ρ01
1
ε 1
= µ · d · 2π · 2
xJ1 (x)2 dx
4µ kc 01
ε 1 ρ2= µ · d · 2π · 2 · 01 J1 (ρ01 )2
4
µ kc
2
1
= επa2 dJ1 (ρ01 )2 = We
(4.7)
4

Wm =

(4.10)

ε= Rs πa(a + d) J1 (ρ01 )2µ(4.11)

ͱ‫ٻ‬ΊΒΕΔɻΑͬͯɺࣜ (3.53) ΑΓɺQ0 ஋͸

ρ01
Q0 =
2Rs
=

√

µ
1
ε 1 + a/daδ(1 + a/d)

(4.12)

ͱ‫ٻ‬ΊΒΕΔɻ͜͜Ͱɺδ ͸ࣜ (1.47) Ͱ༩͑ΒΕΔදൽͷް͞Ͱ͋ΔɻɹͱͳΓɺWe ͱ౳͍͜͠ͱ͕Θ͔ΔɻΑͬͯɺ஝ੵ

4.2 ΞϧόϨ‫ܕ‬Ճ଎؅ΤωϧΪʔͷ࣌ؒฏ‫ ۉ‬W ͸

W = W e + Wm =1
επa2 dJ1 (ρ01 )2
2

ਤ 20 ͷΑ͏ʹɺϐϧϘοΫεۭಎͷ྆୺ͷนΛ

(4.8)

ತ‫͠ʹܕ‬Α͏ɻ͜ͷͱ͖ͷి৔෼෍Λਤ 20(a) ʹɺ࣓৔෼෍Λਤ 20(b) ʹࣔ͢ɻ͜͜Ͱɺࠨӈͷತ‫ܕ‬ͷͱ͖ɺJ0′ (x) ஋͸

J0′ (2.405) = 0.5191
Ͱ͋Δɻ
*17 ҎԼͷ‫Ͱࢉܭ‬͸ɺϕοηϧؔ਺ʹ͍ͭͯ੒ΓཱͭJ−n (x) = (−1)n Jn (x)d(xν Jν (x)) = xν Jν−1 (x)
dx
∫
)
x2 (
xJν (x)2 dx =Jν (x)2 − Jν+1 (x)Jν−1 (x)2Λ༻͍Δɻి‫ۃ‬ͷؒܺ g ΛΪϟοϓͱ‫Ϳݺ‬ɻ࣍ʹɺ͜ͷۭಎͷ z ্࣠ʹϏʔϜ͕௨ա͢ΔͨΊͷ݀ʢϏʔϜϗʔϧʣΛ։͚ɺ2 ‫ͯ͠઀ྡݸ‬ฒ΂ͯΈΑ͏ɻਤ 20 ͷۭಎ 1 ‫ݸ‬Ληϧͱ‫Ϳݺ‬ɻηϧʹྭৼ͞ΕΔ‫ج‬ຊϞʔυి࣓৔ͷ૊Έ߹Θͤ͸ɺਤ

21 ͷΑ͏ʹ 2 ௨Γ͋Γɺ্ਤΛ 2π Ϟʔυʢ0 ϞʔυʣɺԼਤΛ π Ϟʔυͱ‫Ϳݺ‬ɻ2π ϞʔυͰ͸ྡΓ߹͏ηϧͷి࣓৔ͷҐ૬͕ಉҐ૬Ͱ͋Γɺπ Ϟʔ

６－ 20

ਤ 20

ΞϧόϨ‫ܕ‬Ճ଎؅ͷϢχοτηϧͱͳΔ

ۭಎͷஅ໘ਤɻ(a) ి৔෼෍ɻ(b) (a) ͔Β 1/4प‫ޙظ‬ͷ࣓৔෼෍ɻਤ 22ΞϧόϨ‫ܕ‬Ճ଎؅ʹΑΔՙిཻࢠՃ଎ͷ‫ݪ‬ཧ͜ͷ౵ΛυϦϑτνϡʔϓʢdrift tubeʣͱ‫Ϳݺ‬ɻ͜ͷΑ͏ͳՃ଎؅ΛΞϧόϨ‫ܕ‬υϦϑτνϡʔϒϦχΞοΫʢAlvarez drift-tube linacʣͱ‫Ϳݺ‬ʢຊςΩετͰ͸ɺΞϧόϨ‫ܕ‬Ճ଎؅ͱ‫Ϳݺ‬ʣɻΞϧόϨ‫ܕ‬Ճ଎؅ʹΑΔՙిཻࢠՃ଎ͷ‫ݪ‬ཧΛਤ 22 ʹ
ࣔ͢ɻ
ਤ 21

্ਤɿ0 Ϟʔυʢ2π ϞʔυʣɻԼਤɿπ

ɹ

Ϟʔυɻ

4.3 γϟϯτΠϯϐʔμϯευͰ͸͜Ε͕‫ٯ‬Ґ૬ʢҐ૬͕ࠩ 180˃ʣͱͳͬͯ
͍Δɻ2π ϞʔυͰ͸ɺ‫ڞ‬௨ͷนͰిྲྀ͕θϩͱͳΓɺඞཁͰ͸ͳ͘ͳΔɻ͜ΕΛཅࢠͷՃ଎؅ͱ͠

ۭಎͷ্࣠Ճ଎ిѹ V0 Λɺۭಎ௕͞ Lɺ্࣠ి৔ Ez (z) Λ༻͍ͯɺ

ͯԠ༻͢ΔͨΊɺ1 ηϧͷ࣠ํ޲ʢz ํ޲ʣ௕͞ L

V0 =

∫ L

Ez (z)dz(4.14)

0

Λ

v
v c
L=
=
= βλ
f0
c f0

ͱఆٛ͢Δɻ͜ͷͱ͖ɺ্࣠ฏ‫ిۉ‬৔ E0 ͸ɺ(4.13)

1
E0 =Lͱ͠Α͏ɻ͜ͷͱ͖ͷ β ͸ɺޫͷ଎౓ʹର͢Δཻ
ࢠͷ଎౓ v/c Ͱ͋Γɺλ ͸ۭಎ಺Ͱ‫͢ৼڞ‬Δߴप೾

∫ L

Ez (z)dz =

0

V0
L

(4.15)

ͱॻ͚Δɻ

ͷ೾௕Ͱ͋Δɻి৔͕Ճ଎ํ޲Λ޲͍͍ͯΔͱ͖

γϟϯτΠϯϐʔμϯεʢshunt impedanceʣʹཅࢠ͕౵ͱ౵ͷؒΛ௨ա͠ɺి৔͕‫޲ํٯ‬Λ޲Rsh [MΩ] ͸ɺ͖͞΄Ͳͷ V0 ͱนଛࣦ P0 Λ༻

͍͍ͯΔͱ͖ʹཅࢠ͕౵ͷதΛ௨ա͢Ε͹ɺཅࢠ͍ͯɺ͸ߴप೾ʹಉ‫ͯ͠ظ‬Ճ଎ͷྗΛड͚Δ͜ͱʹͳΔɻ

６－ 21

Rsh =

V02
P0

(4.16)

ͱఆٛ͞ΕΔɻ͜Ε͸ɺ্࣠ฏ‫ిۉ‬৔ E0 ͱۭಎ
௕͞ L Λ༻͍ͯɺt = 0 ͷͱ͖ʹՃ଎Ϊϟοϓͷத৺ʢz = 0ʣΛ௨ա͢ΔՙిཻࢠΛߟ͑Δͱɺ͜ͷՙిཻࢠ͕Ճ

Rsh =

E02
P0 /L2଎Ϊϟοϓͷத৺ z = 0 Ͱड͚Δి৔͸ɺ

(4.17)
Ez (0, 0) = Ez (0) cos ϕͱॻ͘͜ͱ͕Ͱ͖Δɻۭಎ಺Ͱ࣓৔͕ྭৼ͍ͯΔͱ͖ʹ͸ɺۭಎ಺นΛిྲྀ͕ྲྀΕΔͨΊɺଛࣦ P0
͕ੜ͡ΔɻՃ଎ۭಎͱͯ͠͸ɺۭಎʹిྗΛ౤ೖͨ͠ࡍɺϏʔϜ্࣠ʹͲΕ͚ͩՃ଎ిѹΛੜͤ͡͞ΒΕΔ͔͕ॏཁͰ͋ΔɻγϟϯτΠϯϐʔμϯε Rsh ͸ɺ౤ೖిྗ P (= P0 ) ʹରͯ͠ɺͲΕ͚ͩ

Ͱ͋Δɻͭ·Γ ϕ ͸ɺ͜ͷՙిཻࢠ͕͜ͷՃ଎ΪϟοϓΛ௨ա͢Δͱ͖ͷి৔ͷՃ଎Ґ૬Λද͍ͯ͠Δɻ͜ͷՃ଎Ґ૬ ϕ Λ༻͍ͯɺՃ଎ΪϟοϓΛ௨ա͢Δిՙ q ͷՙిཻࢠ͕ಘΔΤωϧΪʔ∆W Λɺ
∆W = qV0 T cos ϕͷՃ଎ిѹΛੜͤ͡͞Δ͔Λද͍ͯ͠Δɻͭ·ΓɺՃ଎ۭಎͷ౤ೖిྗʹର͢ΔՃ଎ޮ཰ͷྑ͞Λද
͍ͯ͠Δɻ

∫ L/2

Ez (z)dz(4.25)∫ L/2

Ez (z)dz(4.26)

1E0 =Lε Zsh [MΩ/m] ͸ɺ

E02
Rsh
=
L
P0 /L

(4.24)

ͱදͦ͏ɻ͜͜Ͱ V0 ͸ɺ্࣠ฏ‫ిۉ‬৔

·ͨɺ୯Ґ௕͋ͨ͞ΓͷγϟϯτΠϯϐʔμϯZsh =(4.23)

−L/2

Λ༻͍ͯɺ

(4.18)
V0 = E0 L =

Ͱ͋Δɻ

−L/2

Rsh /Q0 [MΩ] ͸ͱఆٛ͞ΕΔɻՃ଎ΪϟοϓΛ௨ա͢ΔՙిཻࢠRsh
V2
V 2 /P0= 0= 0
Q0
ω0 W/P0ω0 W

͕ಘΔΤωϧΪʔ ∆W Λ͜ͷΑ͏ʹද͢ͱɺࣜ

(4.19)(4.22) ΑΓͰ͋ΓɺZsh /Q0 [MΩ/m] ͸ɺ

E02

Zsh
=
Q0
ω0 W/LqV0 T cos ϕ = q

T =ޮ཰Λද͍ͯ͠Δɻ
ɹ

=

∫ L/2

−L/2

Ez (z) cos(ωt(z) + ϕ)dz

∫ L/2

V0 cos ϕ

Ez (z) cos ωt(z)dz∫ L/2
E (z)dz−L/2 z∫ L/2
E (z) sin ωt(z)dz−L/2 z(4.28)− tan ϕ∫ L/2
E (z)dz−L/2 z−L/2

4.4 τϥϯδοτλΠϜϑΝΫλʔՙిཻࢠ͕ z ্࣠Ͱड͚Δి৔ Ez (z, t) ͸ɺ(4.21)(4.27)

Ͱ͋Δ͔ΒɺT ͸‫ܗ‬ͷΈͰܾ·Δɻͭ·Γɺۭಎͷ‫·ܾͰܗ‬ΔՃ଎

Ez (z, t) = Ez (z) cos(ωt(z) + ϕ)

Ez (z) cos(ωt(z) + ϕ)dz

−L/2

(4.20)

Ͱ͋Δɻ͜ΕΒ͸ۭಎͷࡐ࣭ʹґଘͤͣɺۭಎͷ∫ L/2

ͱͳΔɻ͜ͷ T ΛτϥϯδοτλΠϜϑΝΫλʔ

ͱॻ͚Δɻz = 0 ΛՃ଎Ϊϟοϓͷத৺ͱ͠ɺ

ʢtransit-time factorʣͱ‫Ϳݺ‬ɻ͜͜ͰɺՃ଎ΪϟοL = βλ ͱ͢ΔͱɺՃ଎ΪϟοϓΛ௨ա͢Δిՙ qϓͷத৺ʢz = 0ʣʹରͯ͠ɺී௨ Ez (z) ͸‫ؔۮ‬ͷՙిཻࢠ͕ಘΔΤωϧΪʔ ∆W ͸ɺ਺Ͱ͋ΔͷͰɺEz (z) sin ωt(z) ͸‫ͳͱ਺ؔح‬Δɻ

∆W = q

∫ L/2

Αͬͯɺ

Ez (z) cos(ωt(z) + ϕ)dz (4.22)

−L/2

ͱॻ͚Δɻ

∫ L/2

−L/2

６－ 22

Ez (z) sin ωt(z)dz = 0(4.29)

Ͱ͋Δ͔ΒɺτϥϯδοτλΠϜϑΝΫλʔ͸

T =

∫ L/2

−L/2

Ez (z) cos ωt(z)dz∫ L/2

E (z)dz−L/2 zΓɺΪϟοϓ௕ g ͕খ͍͞΄ͲɺτϥϯδοτλΠ
ϜϑΝΫλʔ͕େ͖͘ͳΓɺ࠷େ஋Ͱ͋Δ 1 ʹۙ

(4.30)

ͮ͘͜ͱ͕Θ͔ΔɻՃ଎Ϊϟοϓ g ͷܾΊํʹ͸̎ͭͷ੍໿͕͋Δɻୈ 1 ͸ɺ์ి‫ݶ‬քͰ͋Γɺୈ

ͱॻ͚Δɻ͞ΒʹɺՃ଎Ϊϟοϓʹ͓͚Δՙిཻ

2 ͸Ϣχοτηϧͷ‫ৼڞ‬प೾਺ f0 ΛҰఆʹอͭඞࢠͷ଎౓มԽ͸ඇৗʹখ͍ͨ͞Ίɺωt(z) ΛཁੑͰ͋Δɻ

ωt(z) ≈ ω

2πz
z
=
v
βλ

ɹ

(4.31)
ΩϧύτϦοΫ‫ݶ‬քͱ͢ΔͱɺτϥϯδοτλΠϜϑΝΫλʔ͸ɺ

T =

∫ L/2

−L/2

Ez (z) cos(2πz/βλ)dz∫ L/2
E (z)dz−L/2 z์ి‫ݶ‬քͷ̍ͭͷࢦඪͱͯ͠࢖ΘΕΔͷ͕ɺ࣍(4.32)

ͷΩϧύτϦοΫͷࣜʢKilpatrick formulaʣͰ͋Δɻ

f = 1.64Ek2 exp

ͱॻ͚Δɻͭ·Γɺి৔ Ez ͕ cos(ωt) ͰมԽ͢ΔͨΊʹɺՙిཻࢠ͕ಘΔΤωϧΪʔ ∆W ͕ɺ

qV0 cos ϕ ʹରͯ͠ɺͦͷ T ഒʹখ͘͞ͳΔ͜ͱΛ

(

8.5
−
Ek

)

(4.34)

͜͜Ͱɺҙຯ͍ͯ͠Δɻf ɿप೾਺ [MHz]ɹ

Ek ɿ์ి‫ݶ‬քΛ༩͑Δి৔ʢKilpatrick limitʣ[MV/m]
Ͱ͋Δɻ͜ΕΛਤࣔ͢Δͱɺਤ 24 ͷΑ͏ʹͳΔɻ

Ϊϟοϓ௕ͱτϥϯδοτλΠϜϑΝΫλʔΪϟοϓ௕ g ͷ಺ଆͰి৔ Ez (z) = Eg ɺ֎ଆ
Ͱి৔ Ez (z) = 0 Ͱ͋Δۣ‫ిܗ‬৔Λߟ͑Δɻ͜ͷͱ͖ͷτϥϯδοτλΠϜϑΝΫλʔ͸ɺ

T =

∫ g/2

−g/2

Eg cos(2πz/βλ)dz
∫ g/2
E dz
−g/2 gsin(πg/βλ)=
πg/βλf ͕େ͖͘ͳΔ΄ͲɺEk ͕େ͖͘ͳΔ͜ͱ͕Θ͔Δɻ͜ͷΩϧύτϦοΫ‫ݶ‬ք͸ɺࠓ೔΄Ͳͷߴਅۭ͕࣮‫͞ݱ‬ΕΔΑΓલͷ࣌୅ʹɺ࣮‫ʹݧ‬ΑΓಘΒΕͨ΋ͷͰ͋Δɻ‫Ͱࡏݱ‬͸ɺ͜Ε͸ઈରతͳ‫ڐ‬༰‫ݶ‬ք஋Ͱ͸ͳ͘ɺ”ΩϧύτϦοΫͷԿഒͷిք‫ʹ౓ڧ‬૬౰”ͳͲͱɺ̍ͭͷ໨҆ͱͯ͠࢖ΘΕ͍ͯ

(4.33)Δɻ*18

ͱͳΔɻ͜ͷͱ͖ɺԣ࣠Ληϧ௕ʹର͢ΔΪϟοϓ௕ g/βλɺॎ࣠ΛτϥϯδοτλΠϜϑΝΫλʔT ͱͯ͠ਤࣔ͢Δͱɺਤ 23 ͷΑ͏ʹͳΔɻͭ·

ਤ 24

*18

ਤ 23

ΩϧύτϦοΫͷ์ి‫ݶ‬ք࠷େද໘ి৔ Es Λ Ek Λ༻͍ͯ

Es = bEk

ۣ‫ిܗ‬৔ͷͱ͖ͷτϥϯδοτλΠϜ

ͱද͠ɺb ͸”bravery factor”ͱ‫ݺ‬͹ΕΔʢࢀߟจ‫ݙ‬

ϑΝΫλʔ

[1]ʣɻ

６－ 23

4.5 ༗ޮγϟϯτΠϯϐʔμϯεΤωϧΪʔ͕ߴ͘ͳΔʹ͕ͨͬͯ͠ β ͕େ͖ۭ͘ಎͷ্࣠Ճ଎ిѹ V0 ͓Αͼ্࣠Ճ଎ి৔ E0

ͳΔͨΊɺηϧ௕͸௕͘ͳΔɻͦ͜Ͱɺβ ʹ͋Θ͸ɺcos(ωt) Ͱมಈ͢ΔɻΑ࣮ͬͯࡍʹϏʔϜʹ‫د‬ͤͯ g Λ௕͘͢Δ͜ͱʹΑͬͯɺg/βλ ΛҰఆʹ༩͢Δͷ͸ɺτϥϯδοτλΠϜϑΝΫλʔ T Λ

อͭɻ͢Δͱ্ͷΑ͏ͳ୯७ͳۙࣅͰ͸ɺd ΍ D༻͍ͯɺͦΕͧΕ V0 T ͓Αͼ E0 T Ͱ͋Δɻͦ͜

Λม͑ͳͯ͘΋ۭಎͷ‫ৼڞ‬प೾਺ΛҰఆʹอͭ͜ͰɺγϟϯτΠϯϐʔμϯε Rsh Λͱ͕Ͱ͖Δɻ͔࣮͠͠ࡍʹ͸ɺd ʹൺ΂ͯ g ͕େ

(V0 T )2
(E0 T )2=
= Rsh T 2
P0
P0 /L2

͖͘ͳΔʹ͕ͨͬͯ͠ɺՃ଎Ϊϟοϓͷి৔͕͠(4.35)

ͱॻ͖௚ͦ͏ɻ͜ͷ Rsh T 2 [MΩ] Λɺ༗ޮγϟϯτΠϯϐʔμϯεʢeﬀective shunt impedanceʣͱ‫Ϳݺ‬ɻಉ༷ʹɺ୯Ґ௕͋ͨ͞Γͷ༗ޮγϟϯτΠ
ϯϐʔμϯε ZT 2 [MΩ/m] ͸ɺ2͕ͨͬͯ͠ɺd ΍ D ΛҰఆʹอͬͨ··‫ৼڞ‬प೾਺ΛҰఆʹอͭͨΊʹ͸ɺg/βλ Λେ͖͘͢Δඞཁ
͕ੜ͡ɺ͜Ε͸τϥϯδοτλΠϜϑΝΫλʔͷ௿ԼΛট͘ɻͭ·ΓɺՃ଎ޮ཰ʢ༗ޮγϟϯτΠ
ϯϐʔμϯεʣ͕௿Լ͢Δ͜ͱΛҙຯ͢Δɻਤ 25

2(E0 T )
Rsh T
=
L
P0 /L

Zsh T 2 =Έͩ͠ɺࣜ (4.39) ͷ‫ࢉܭ‬ΑΓ΋ C ͕େ͖͘ͳΔɻ

(4.36)

ʹɺཅࢠΤωϧΪʔͱ୯Ґ௕͋ͨ͞Γͷ༗ޮγϟϯτΠϯϐʔμϯεͷؔ܎Λࣔ͢ɻΞϧόϨ‫ܕ‬ՃͱͳΔɻ଎؅ͷ༗ޮγϟϯτΠϯϐʔμϯε͕খ͘͞ͳΔ

ΞϧόϨ‫ܕ‬Ճ଎؅ͷ༗ޮγϟϯτΠϯϐʔμϯεʹ͍ͭͯߟ͑Α͏ɻ·ͣɺϢχοτηϧͷ‫ৼڞ‬पߴΤωϧΪʔଆͰ͸ɺ࣍ʹड़΂Δ݁߹ۭಎ‫ܕ‬ͷՃ଎؅͕༻͍ΒΕΔɻ

೾਺ f0 ΛҰఆʹอͭඞཁੑ͔Βɺηϧ௕ L = βλͱΪϟοϓ௕ g ͷؔ܎Λௐ΂Α͏ɻਤ 20 ΑΓɺϢχοτηϧ͸ LC ‫ৼڞ‬ճ࿏ͱΈͳ͢͜ͱ͕Ͱ͖Δ
ͷͰɺͦͷ‫ৼڞ‬प೾਺͸ ω02 = 1/LC ͱද͢͜ͱ
͕Ͱ͖Δɻg ͕ βλ ʹൺ΂ͯे෼খ͍͞ͱͯ͠ɺϢχοτηϧΛಉ࣠؅ͱΈͳ͢͜ͱʹΑΓɺ࣓ଋ Φ
͸

Φ=

∫∫

S

⃗ · ⃗n dS
B∼ µ0 βλ

∫ D/2
d/2

I
µ0 Iβλ D
dr =
ln
2πr
2π
d

(4.37)
ਤ 25

ͱද͞ΕΔɻΑͬͯɺΦ = LI Ͱ͋Δ͔ΒɺL∼µ0 βλ Dln
2πdཅࢠΤωϧΪʔͱ༗ޮγϟϯτΠϯϐʔμϯεʢࢀߟจ‫[ ݙ‬2] ΑΓʣ

(4.38)
ɹͱ‫·ٻ‬Δɻ·ͨ੩ి༰ྔ C ʹ͍ͭͯ͸ɺΪϟοϓΛฏߦฏ൘ͱΈͳ͢͜ͱʹΑΓ

4.6 ݁߹ۭಎ‫ܕ‬Ճ଎؅πd2
C ∼ ε04g(4.39)

࿈ۭಎͱ΋‫ݺ‬͹Εɺෳ਺ͷՃ଎ۭಎΛϏʔϜ࣠ํͱಘΒΕΔɻΑͬͯɺ‫ৼڞ‬प೾਺͸

ω02 =
ͱॻ͚Δɻ

޲ʹ࿈݁ͨ͠΋ͷͰ͋Δɻ͜ͷ࿈݁͞Εۭͨಎͷ

2

g
8c
1
∼
· 2 D
LC
βλ d ln d݁߹ۭಎ‫ܕ‬Ճ଎؅ʢcoupled-cavity linacʣ͸ɺଟ(4.40)

ͻͱͭͻͱͭΛηϧͱ‫Ϳݺ‬ɻ֤ηϧʹྭৼ͞ΕΔ
ి࣓৔͸ɺϏʔϜϗʔϧͳͲΛ௨ͯ͠ి࣓‫ؾ‬తʹ݁߹͍ͯ͠Δɻ݁߹ۭಎ‫ܕ‬Ճ଎؅ʹྭৼ͞ΕΔి

６－ 24

͜͜Ͱɺk = M/L ͱͨ͠ɻ࣍ʹɺߦྻ M Λ



ਤ 26

1
k/2

M =
 0
 0
0

૬‫ޓ‬ΠϯμΫλϯε M ʹΑͬͯ݁߹͞

Εͨ 5 ݁߹‫ثৼڞ‬

k
0
0
1
k/2
0
k/2
1
k/2
0
k/2
1
0
0
k

‫ݻ‬༗ϕΫτϧ X Λ࣓৔ͷϞʔυ͸ɺਤ 26 ͷΑ͏ʹɺෳ਺ͷ‫ৼڞ‬ճ࿏͕૬‫ޓ‬ΠϯμΫλϯε M ʹΑͬͯ݁߹ͨ݁͠
߹‫ثৼڞ‬Λߟ͑Δͱཧղ͠΍͍͢ɻ͜͜Ͱ͸̑ͭͷ LC ‫ৼڞ‬ճ࿏Λ૝ఆ͠ɺͦΕͧΕ͕ಉ͡‫ৼڞ‬प೾਺ ω0 Λ΋ۭͭಎ‫ثৼڞ‬Λද͢ͱߟ͑Α͏ɻ྆୺ͷηϧʹ͍ͭͯ΋ಉ͡‫ৼڞ‬प೾਺ ω02 = 1/LC


0
0 

0 

k/2
1

(4.52)

 
x0
x1 
 

X=
x2 
x3 
x4

(4.53)

ω02XΩ2(4.54)

ͱ͢Δͱɺࣜ (4.47)ʙ(4.51) ͸Λ΋ͨͤΔͨΊɺ྆୺ͷ‫ৼڞ‬ճ࿏ͷ੩ి༰ྔΛMX =2C ʹ͍ͯ͠Δɻ‫ৼڞ‬ճ࿏ʹྲྀΕΔిྲྀΛɺࠨ͔Β
I0 , I1 , I2 , I3 , I4 ͱ͢Δͱɺ͜ΕΒʹؔͯ࣍͠ͷ̑ͱॻ͘͜ͱ͕Ͱ͖Δɻ͜͜Ͱɺߦྻ M ͷ‫ݻ‬༗஋ͭͷํఔ͕ࣜ੒ΓཱͭɻΩq (q = 0, 1, 2, 3, 4) ͸ɺE Λ 5 × 5 ୯Ґߦྻͱ1
d2 I0
d2 I 1I0 = L 2 + M 2
2C
dt
dt
d2 I1
d2 I 0d 2 I2
1− I1 = 2L 2 + M 2 + M 2
C
dt
dt
dt
d2 I2
d2 I 1d 2 I3
1− I2 = 2L 2 + M 2 + M 2
C
dt
dt
dt
d2 I3
1
d2 I 2d 2 I4− I3 = 2L 2 + M 2 + M 2
C
dt
dt
dt
d2 I4
d2 I 31
I4 = L 2 + M 2
−
2C
dt
dt

−

ͯ͠

(4.41)()
ω02
det M − 2 E = 0Ω(4.42)(4.43)

Λ‫͢ࢉܭ‬Δ͜ͱʹΑΓɺ࣍ͷΑ͏ʹಘΒΕΔɻ

(4.44)• 0 Ϟʔυ‫ݻ‬༗ϕΫτϧ X0 ͸

 
1
1
 

X0 = 
1
1
1(4.45)

͜ΕΒ̑ͭͷ‫ ͨ͠߹͕݁ثৼڞ‬5 ݁߹‫ثৼڞ‬ͷ‫ৼڞ‬प೾਺Λ Ω ͱ͠ɺn = 0, 1, 2, 3, 4 ͱͯࣜ͠(4.41)ʙ(4.45) ͷղΛ
In = xn ejΩt

(4.46)

ͱॻ͘ͱɺ͜ΕΒͷํఔࣜ͸࣍ͷΑ͏ʹॻ͖௚͢Ͱ͋Δɻ·ͨ͜ͷͱ͖ͷ‫ৼڞ‬प೾਺ Ω0 ͸ɺ

Ω0 = √(4.47)

ͱͳΔɻ

(4.48)• π/4 Ϟʔυ

(4.49)(4.50)(4.51)

６－ 25(4.56)

Ͱ͋ΓɺྡΓ߹͏ηϧͷి࣓৔ͷҐ૬ࠩ͸ɺ0͜ͱ͕Ͱ͖Δɻω02x0 = x0 + kx1
Ω2
ω02
k
k
x1 = x 1 + x 0 + x 2
2
Ω
2
2
ω02
k
k
x2 = x 2 + x 1 + x 3
Ω2
2
2
ω02
k
k
x3 = x 3 + x 2 + x 4
Ω2
2
2
ω02
x4 = x4 + kx3Ω2(4.55)

ω0
1+k(4.57)‫ݻ‬༗ϕΫτϧ X1 ͸1√
 1/ 2 
X1 = 
 0√ 
−1/ 2
−1
(4.58)

Ͱ͋ΓɺྡΓ߹͏ηϧͷి࣓৔ͷҐ૬ࠩ͸ɺ

π/4 Ͱ͋Δɻ·ͨ͜ͷͱ͖ͷ‫ৼڞ‬प೾਺ Ω1
͸ɺ

ͱͳΔɻ

ω0
Ω1 = √√
1 + k/ 2(4.59)• π/2 Ϟʔυ‫ݻ‬༗ϕΫτϧ X2 ͸


1
0
 

X2 = 
−1
0
1


ਤ 27

5 ݁߹ۭಎͷϞʔυɻ˙ɿX0ʢ0 Ϟʔυʣɻ
˔ɿX1 ʢπ/4 Ϟʔυʣɻ˛ɿX2 ʢπ/2 Ϟʔυʣɻ
ɻ˗ɿX4 ʢπ Ϟʔυʣɻ
˝ɿX3 ʢ3π/4 Ϟʔυʣ(4.60)

Ͱ͋ΓɺྡΓ߹͏ηϧి࣓৔ͷҐ૬ࠩ͸ɺπ/2
Ͱ͋Δɻ·ͨ͜ͷͱ͖ͷ‫ৼڞ‬प೾਺ Ω2 ͸ɺ

Ω2 = ω0(4.61)

͜ΕΒ̑ͭͷ‫ݻ‬༗ϕΫτϧ Xq (q = 0, 1, 2, 3, 4)ͷ੒෼Λਤࣔ͢Δͱɺਤ 27 ͷΑ͏ʹͳΔɻ͜͜·Ͱ͸ 5 ݁߹‫ثৼڞ‬Λߟ͕͑ͨɺ࣍ʹ N + 1‫ݸ‬ͷۭಎͷ݁߹Λߟ͑Α͏ɻq ΛϞʔυ൪߸ʢq =ͱͳΔɻ

0, 1, 2, ..., N ʣͱͯ͠ɺ‫ݻ‬༗஋ Ωq ͸

• 3π/4 Ϟʔυ‫ݻ‬༗ϕΫτϧ X3 ͸





1√
−1/ 20
X3 =  √ 
 1/ 2 −1(4.62)Ωq = √

ω0
1 + k cos(πq/N )ͱͳΔɻ·ͨͦͷ‫ݻ‬༗ϕΫτϧ Xq ʹ͍ͭͯɺͦ
ͷ n ൪໨ʢn = 0, 1, 2, ..., N ʣͷηϧͷ੒෼ Xq,n͸ɺͰ͋ΓɺྡΓ߹͏ηϧͷి࣓৔ͷҐ૬ࠩ͸ɺ

3π/4 Ͱ͋Δɻ·ͨ͜ͷͱ͖ͷ‫ৼڞ‬प೾਺ Ω3

Xq,n = cos

͸ɺ

ͱͳΔɻ

ω0
Ω3 = √√
1 − k/ 2( πq )
nN(4.67)

ͱͳΔɻ͜͜ͰɺྡΓ߹͏ηϧͷҐ૬ࠩ ϕ ͸ɺ

(4.63)
ϕ=πqN(4.68)

Ͱ͋Δɻྫ͑͹ɺq = 0 ͷͱ͖ 0 Ϟʔυɺq = N/2• π Ϟʔυͷͱ͖ π/2 Ϟʔυɺq = N ͷͱ͖ π Ϟʔυͱͳ‫ݻ‬༗ϕΫτϧ X4 ͸


1
−1
 

X4 = 
1
−1
1


Δɻ0 Ϟʔυɺπ/2 Ϟʔυɺπ Ϟʔυʹ͓͚Δ֤ηϧͷి৔෼෍Λਤࣔ͢Δͱɺਤ 28 ͷΑ͏ʹͳΔɻ

(4.64)

Ͱ͋Δɻ·ͨ͜ͷͱ͖ͷ‫ৼڞ‬प೾਺ Ω4 ͸ɺω0Ω4 = √
1−k

π/2 ϞʔυͰ͸ɺి৔ͷൃੜ͢Δηϧͱ͠ͳ͍ηϧ͕ަ‫ݱʹޓ‬Εɺ͜ΕΒ͸ͦΕͧΕՃ଎ηϧɺ݁Ͱ͋ΓɺྡΓ߹͏ηϧͷి࣓৔ͷҐ૬ࠩ͸ɺπͱͳΔɻ

(4.66)

߹ηϧͱ‫ݺ‬͹ΕΔɻ֤ηϧͷ࣠ํ޲௕͞Λ dɺྭৼ͞ΕΔϞʔυͷ೾௕Λ λg ͱ͢ΔͱɺྡΓ߹͏ηϧͷి࣓৔ͷҐ૬ࠩ ϕ ͸ɺ

(4.65)
ϕ=

６－ 26

2π
d = βdλg(4.69)ਤ 29݁߹ۭಎ‫ܕ‬Ճ଎؅ʹ͓͚Δ෼ࢄ‫ۂ‬ઢʢ˙ɿ

5 ηϧͷ৔߹ʣɻ

‫܈‬଎౓͕͋Δͱ͍͏͜ͱ͸ɺి࣓৔ͷΤωϧΪʔʹྲྀΕ͕͋Δͱ͍͏͜ͱͰ͋Δɻ‫܈‬଎౓͕େ͖͚
Ε͹ɺͳΜΒ͔ͷ‫ݪ‬ҼͰͲ͔͜ͷηϧͷ஝ੵΤωϧΪʔ͕খ͘͞ͳͬͨͱ͖ɺ଎΍͔ʹ͜ͷηϧʹਤ 28ΤωϧΪʔΛ‫͢څڙ‬Δ͜ͱ͕Ͱ͖Δɻͭ·ΓɺՃ݁߹ۭಎ‫ܕ‬Ճ଎؅ͷి৔෼෍଎؅಺ʹ҆ఆͳి࣓৔Λྭৼͤ͞Δ͜ͱ͕Ͱ͖Δɻ
Ͱ͋Δ͜ͱ͕Θ͔Δɻ͜͜Ͱɺβ = 2π/λg ͸Ґ૬ఆ਺Ͱ͋Δɻ*19Ґ૬ࠩ ϕ ͸Ґ૬ఆ਺ β ʹൺྫ͍ͯ͠ΔͨΊɺҐ૬ࠩ ϕ ͱ‫ৼڞ‬प೾਺ Ω ͷؔ܎Λදࣜ͢ (4.66) ΑΓɺਤ 29 ͷΑ͏ʹ෼ࢄ‫ۂ‬ઢΛඳ͘͜ͱ͕Ͱ͖Δɻਤதͷ̑ͭͷ఺͸ 5 ݁߹‫ثৼڞ‬ͷͱ͖ͷϞʔυΛද͍ͯ͠Δɻਤ 29 ΑΓɺ෼ࢄ‫ۂ‬ઢͷ܏͖ɺͭ·Γ‫܈‬଎౓͸ɺ0 Ϟʔυ͓Αͼ π ϞʔυͰ 0 Ͱ͋Γɺ

π/2 ϞʔυͰ࠷େͱͳΔ͜ͱ͕Θ͔Δɻ·ͨɺόϯυ෯ʢbandwidthʣδω ͸ɺk ≪ 1 ͷͱ͖ɺδω = ΩN − Ω0
ω0
ω0
−√
=√
1−k
1+k
∼
= ω0 kΑͬͯɺπ/2 ϞʔυͰ‫܈‬଎౓͕࠷େʹͳΔͱ͍͏͜ͱ͸ɺπ/2 ϞʔυͰ࠷΋҆ఆʹి࣓৔ΛྭৼͰ
͖Δ͜ͱΛҙຯ͍ͯ͠Δɻ͜ͷ΄͔ʹ΋ π/2 Ϟʔυʹ͸ɺྡ઀͢ΔϞʔυͷप೾਺͕ࠩ࠷େɺ‫ݸ‬ʑͷηϧͷ‫ৼڞ‬प೾਺ͷͣΕ͕༩͑Δి৔෼෍ͷ࿪ΈʢςΟϧτʣ͕খ͍͞ɺ౳ͷ௕ॴ͕͋ΔɻҰํ π/2ϞʔυͰ͸ɺి৔͕ൃੜͤͣϏʔϜՃ଎ʹ‫د‬༩͠ͳ͍݁߹ηϧ͕͓͖̍ͭʹଘࡏ͢ΔͨΊɺ0 Ϟʔυ΍ π Ϟʔυʹൺ΂ͯ୯Ґ௕͋ͨ͞ΓͷՃ଎ޮ཰͕
ѱ͍ɻͦ͜Ͱɺ҆ఆͳి࣓৔Λྭৼͤ͞Δ͜ͱͷͰ͖Δ π/2 ϞʔυՃ଎؅ʹ͍ͭͯɺͦͷՃ଎ޮ཰ΛߴΊΔͨΊͷվྑ͕ਐΊΒΕ͖ͯͨɻ͜ΕΒͷ

π/2 Ϟʔυ݁߹ۭಎ‫ܕ‬Ճ଎؅ͷྫΛਤ 30 ʹࣔ͢ɻ
(4.70)APSʢAlternating Periodic Structureʣ‫ܕ‬Ճ଎؅Ͱ͸ɺ݁߹ηϧͷ࣠ํ޲௕͞Λ୹͘͢Δ͜ͱʹΑͱͳΔɻͭ·Γɺόϯυ෯͸ N ʹΑΒͣɺ‫܈‬଎౓Γɺ୯Ґ௕͋ͨ͞ΓͷՃ଎ి৔ͷׂ߹Λେ͖͘͠͸݁߹܎਺ k ͕େ͖͍΄Ͳେ͖͘ͳΔɻ

͍ͯΔɻSCSʢSide Coupled Structureʣ‫ܕ‬Ճ଎؅

*19

ࣜ (4.67) ͸ఆࡏ೾Λද͔͢Βɺ࣌ؒมԽΛߟྀͯ͠

Xq,n ejΩq t =πqn )1 ( j(Ωq t+ πqn )N+ ej(Ωq t− N )e2ͱॻ͖௚͢͜ͱʹΑΓɺ

πqn
πq
ϕ
β=
=
=
Nz
Nd
d
Ͱ͋Δ͜ͱ͕Θ͔Δɻ

Ͱ͸ɺ݁߹ηϧΛϏʔϜ͔࣠Βԣʹ֎͠ɺ্࣠ʹ͸Ճ଎ηϧ͚͕ͩฒΜͰ͍ΔɻACSʢAnnular-ring

Coupled Structureʣ‫ܕ‬Ճ଎؅Ͱ͸ɺϏʔϜ࣠Λத৺ͱͯ݁͠߹ηϧΛ‫ܗ؀‬ঢ়ͱ͢Δ͜ͱʹΑΓϏʔϜ͔࣠Β֎͠ɺ্࣠ʹ͸Ճ଎ηϧ͚͕ͩฒΜͰ͍ΔɻAPS Ͱ͸ɺՃ଎ηϧͱ݁߹ηϧ͕ϏʔϜϗʔ

６－ 27

Disk loaded structure

Alternating periodic structure (APS)ਤ 31

Side coupled structure (SCS)

wg-cavity

ͱॻ͚Δɻਤ 31 ΑΓɺ࣍ࣜͷΤωϧΪʔอଘଇdW= Pin − Pref − P0
dt

(4.73)

͕੒Γཱͭ͜ͱ͕Θ͔Δɻ͜͜Ͱɺಋ೾؅ʹ͓͚ΔΞυϛολϯε Gwg ɺ͓Αͼۭಎʹ͓͚ΔΞυ
ϛολϯε G0 Λ༻͍Δͱɺࠨล͸Annular-ring coupled structure (ACS)

ਤ 30

(
)
dW
d Q0 P0=dtdt
ω0(1 + β)QL d=· (G0 V02 )ω0dt(1 + β)QL
dV0· 2G0 V0=ω0dt

(4.74)

ͱॻ͚Δɻ·ͨӈล͸ɺ

π/2 Ϟʔυ݁߹ۭಎ‫ܕ‬Ճ଎؅Pin −Pref − P0ϧΛ௨ͯ͠ి࣓‫ؾ‬తʹ݁߹͞ΕΔɻҰํɺSCS ͓

2= Gwg Vin2 − Gwg Vref
− G0 V02

Αͼ ACS Ͱ͸ɺՃ଎ηϧͱ݁߹ηϧͷؒʹઃ͚Β

= Gwg Vin2 − Gwg (V0 − Vin )2 − G0 V02

Εͨ݁߹εϩοτΛ௨ͯ݁͠߹͞ΕΔɻ

= −Gwg V02 + 2Gwg Vin V0 − G0 V02ɹ

4.7 ۭಎ΁ͷిྗ౤ೖۭಎ΁ͷిྗ౤ೖΛߟ͑Α͏ɻߴप೾‫ൃͰݯ‬ੜ= −βG0 V02 + 2βG0 Vin V0 − G0 V02{}(4.75)= G0 V0 −(1 + β)V0 + 2βVin

ͱॻ͚Δɻ͜͜Ͱɺ

ͤͨ͞ߴप೾ిྗ͸ɺಋ೾؅౳ͷཱମճ࿏Λ௨ͯ͡

Vref = V0 − Vin
Gwg
β=
G0

ۭಎ·Ͱ‫͞څڙ‬ΕΔɻಋ೾؅ͱۭಎ͸ɺΧϓϥʔͱ‫ݺ‬͹ΕΔ݁߹‫ʹث‬ΑΓ݁߹͞ΕΔɻ͜ͷͱ͖ɺ

(4.76)(4.77)

Λ༻͍ͨɻࣜ (4.74), (4.75) ΑΓɺࣜ (4.73) ͷΤ݁߹܎਺ β Λ

β=

Pext
P0

ωϧΪʔอଘଇ͸(4.71)

2QL dV0
2β+ V0 =
Vin
ω0 dt
1+β

ʹΑΓఆٛ͢Δɻ͜Εͱɺࣜ (3.52)ʙ(3.54) ʹΑ(4.78)

ͱॻ͖௚ͤΔɻV0 ʹ͍ͭͯɺࣜ (4.78) Λղ͘ͱɺ

Γɺ

QL =

1
Q0
1+β

(4.72)

６－ 28

V0 (t) =

)
ω
2β (
− 0 t
1 − e 2QL Vin
1+β(4.79)

*20
ͱͳΔɻ

0 < t < t1 ͷͱ͖ɺVref ͸)(ω02β − 2Q
1−β
t
L
+
e
Vin
Vref = −1+β1+β(4.80)

ͱͳΔɻ͜ͷͱ͖ͷ൓ࣹ܎਺ Γ(t) = Vref /Vin ͸ɺΓ(t) =

ω02β − 2Qβ−1tL−e
β+1 1+β(4.81)

Ͱ͋Δɻt1 < t Ͱ͸ɺ−ω0(t−t )1
V0 (t) = V0 (t1 )e 2QL)ω0ω
2β ( 2Qt− 0 t
e L 1 − 1 e 2QL Vin (4.82)=
1+βͱͳΔɻ·ͨɺVin = 0 Ͱ͋Δ͔ΒɺVref ͸

Vref (t) = V0 (t)(4.83)
(
)
ω
ω
0 t
2β
− 0 t
e 2QL 1 − 1 e 2QL Vin (4.84)
=
1+β

ਤ 32

ిྗ Pref ʢࠨਤʣͱ஝ੵΤωϧΪʔ W ∝ V02ʢӈਤʣɻ

ͱͳΔɻਤ 32 ʹɺβ = 0.2, 1.0, 2.0 ͱͨ͠ͱ͖ͷ൓ࣹిྗ Pref ͓Αͼ஝ੵΤωϧΪʔ W∝ V02 Λࣔ͢ɻt = 0 ͰɺPin = Pref ɺͭ·Γશ൓ࣹ͕‫͜ى‬

Ͱ͋Δ͔Βɺ

ΔɻPin = 1MWʙ1.5MW ͱେ͖͍ͷͰɺ࢖༻͠

1 + |Γ|1 − |Γ|1 − |Γ|β=
1 + |Γ|β=͍ͯΔߴप೾‫ثػ‬ͷ଱ѹ౳ʹ஫ҙ͕ඞཁͰ͋Δɻ

0 < t < t1 Ͱɺt ͕े෼େ͖͍ͱɺ
Γ=

β−1
β+1

݁߹܎਺ β = 0.2, 1.0, 2.0 ͷͱ͖ͷ൓ࣹ

(4.85)(1 < β ɿover coupling)(4.86)(β < 1 ɿunder coupling) (4.87)*21 β = 1ɿcritical couplingʣͱͳΔɻͷͱ͖ɺΓ = 0ͱͳΓɺશಁաΛද͍ͯ͠Δɻ

*20
ω

V0 = Ṽ e− 2Q0 tLͱͯ͠ɺࣜ (4.78) ʹ୅ೖ͢Δͱ
ω0
dṼ
2β
ω0
t
·
=
Vin e 2QLdt
2QL 1 + βͱͳΔɻ͜ΕΛ t Ͱੵ෼͢Δ͜ͱʹΑΓɺC Λੵ෼ఆ਺
ͱͯ͠

Ṽ =

ω0
2β
t
Vin e 2QL + C
1+β͕ಘΒΕΔɻΑͬͯɺ

V0 =

ω
2β
− 0 t
Vin + Ce 2QL
1+β

*21

ͱͳΔɻ͜͜Ͱɺॳ‫ظ‬৚݅ V0 (t = 0) = 0 ʹΑΓɺC ͸C=−ͭ·ΓɺVSWRʢVoltage Standing Wave RatioʣΛ
༻͍ͯ

2β
Vin
1+β

ͱͳΔɻΑͬͯɺࣜ (4.78) ͷղͱͯ͠ɺࣜ (4.79) ͕ಘ
ΒΕΔɻ

６－ 29

ͱॻ͚Δɻ

VSWR = β(1 < β)1
VSWR =
β(β < 1)

[16] ٢ాޫ޺, ”ిࢠઢ‫ܗ‬Ճ଎‫͚͓ʹث‬Δߴप೾σ͓ΘΓʹόΠεͷ‫”ૅج‬, OHO’08.

Ҏ্ɺཅࢠϦχΞοΫͷॳֶऀΛର৅ʹɺఆࡏ[17] ੨‫޾׮‬, ”ߴप೾Ճ଎ͷ‫”ૅج‬, OHO’12.

೾‫ܕ‬Ճ଎؅ͷ‫ج‬ຊతͳ಺༰ʹ͍ͭͯղઆ͠·ͨ͠ɻຊςΩετΛ·ͱΊΔʹ͋ͨͬͯɺओʹԼ‫ه‬ͷ [1]ʙ[17] Λࢀߟʹ͍͖ͤͯͨͩ͞·ͨ͠ɻ͜ΕΒΛॻ͞Ε·ͨ͠ॾઌੜํʹ৺ΑΓ‫ँײ‬Λਃ্͛͠·͢ɻཅࢠ༻Ճ଎ۭಎͷΘ͔Γ΍͍͢ೖ໳ॻͱͳΔΑ͏ҙࣝͯ͠ॻ͖࢝Ί·͕ͨ͠ɺචऀͷྗෆ଍ʹΑΓɺ͜ΕΒࢀߟจ‫ݙ‬ͷग़དྷͷѱ͍ൈਮʹͳͬͯ
͠·ͬͨ‫ײ‬΋͋Γ·͢ɻ෺ཧతɾ޻ֶతҙຯͷઆ໌͕ෆे෼ͳՕॴ΍ɺΑΓৄࡉͳ಺༰౳ʹ͍ͭͯ͸ɺա‫ڈ‬ͷ OHO ςΩετ౳ͷྑॻΛࢀߟʹ֤ࣗิ଍͍ͯͩ͘͠͞ɻ
࠷‫ʹޙ‬ɺྺ࢙͋ΔຊηϛφʔͷߨࢣΛͤͯ͞௖͘‫ػ‬ձΛ༩͑ͯԼ͍͞·ͨ͠খྛ޾ଇࢯɺಓԂਅҰ࿠ࢯɺ಺౻෋࢜༤ࢯʹ৺͔Β͓ྱΛਃ্͛͠·͢ɻɹࢀߟจ‫ݙ‬[1] Thomas P. Wangler, ”RF Linear Accelerators”, WILEY-VCH.
[2] ُҪ‫ږ‬, ໦‫ݩݪ‬ԝ, ”Ճ଎‫ث‬Պֶ”, ‫ؙ‬ળ.
[3] ‫ޙ‬౻‫ݑ‬Ұ, ࢁ࡚मҰ࿠, ”ৄղ ి࣓‫ֶؾ‬ԋश”,‫ཱڞ‬ग़൛.

[4] খ໺ࣉՅ޹ɺ”෺ཧͷͨΊͷԠ༻਺ֶ”, ী՚๪.
[5] అҪ৴ྗ, ”ి࣓೾ͷ‫”ૅج‬, ಺ా࿝௽ะ.
[6] தౡকޫ, ”ϚΠΫϩ೾޻ֶ”, ৿๺ग़൛.
[7] Robert

E.

Collin,”Foundations

Microwave Engineering,for2nd Edition”,

McGraw-Hill, Inc.
[8] Ճ౻ོ෉, ”ϥΠφοΫͱͦͷपล”, OHO’84.
[9] ࢁ࡚ྑ੒, ”ߴप೾Ճ଎૷ஔ”, OHO’84.
[10] ං‫ޙ‬णହ, ”ߴप೾ۭಎͷઃ‫”ܭ‬, OHO’85.
[11] ߴ࡚ӫҰ, ”ཅࢠϦχΞοΫ”, OHO’89.
[12] ෱ຊఃٛ, ”ϦχΞʔΞΫηϥϨʔλೖ໳”,OHO’97.
[13] ߴాߞ࣏, ”ߴप೾Ճ଎”, OHO’97.
[14] ஑্խ‫ل‬, ”ઢ‫ܗ‬Ճ଎‫( ث‬II)”, OHO’01.
[15] ࢁ‫ޱ‬੣࠸, ”Ճ଎؅ɾཱମճ࿏”, OHO’05.

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