ୈI෦(a)ϚΠΫϩ೾ཧ࿦
1 ͸͡Ίʹ
ϚΠΫϩ೾͸ɺిࢠϨϯδ΍‫ܞ‬ଳి࿩౳Ͱ࢖ΘΕ͓ͯΓɺ‫ݱ‬୅ਓͷੜ‫ͯͬͱʹ׆‬ඞཁෆՄܽͳଘࡏͰ͋
Δɻ෺ཧతʹ͸ɺϚΠΫϩ೾͸ి࣓೾ͷҰछͰɺप೾(b)

਺ʢ·ͨ͸ɺ೾௕ʣ͕ɺ͓Αͦ 300 MHzʢ೾௕ 1 mʣ100 mm͘Β͍͔Β 300 GHzʢ೾௕ 1 mmʣ͘Β͍ͷؒͷి࣓೾ͷ͜ͱΛ‫͏ݴ‬ʢ‫ͳີݫ‬ఆٛ͸ͳ͍Α͏Ͱ͋Δʣɻ
ϚΠΫϩ೾Λ࢖ͬͨՙిཻࢠͷՃ଎͸ɺՃ଎‫͠ͱث‬ͯ࠷΋Ұൠతͳํ๏Ͱ͋ΔɻͦͷՃ଎૷ஔʢߴप೾
Ճ଎ۭಎʣͷେ͖͞͸ɺେࡶ೺ʹ‫ͦͯͬݴ‬ͷ೾௕͘Β͍ͷαΠζͱͳΔɻͭ·Γɺ300 MHz Λ࢖͑͹ 1 m

ਤ 1 ࣮ࡍͷՃ଎ߏ଄ͷྫɻ(a) SuperKEKB ཅిࢠμϯϐϯάϦϯάՃ଎‫༻ث‬ͷ 509 MHz ৗ఻ಋߴप೾Ճ଎ۭಎɻ(b) SLAC ͰϏʔϜࢼ‫ݧ‬Λߦͬͨ
200 GHz ৗ఻ಋՃ଎ߏ଄ʢࢀߟจ‫[ ݙ‬1]ʣɻ

ఔ౓ɺ300 GHz Λ࢖͑͹ 1 mm ఔ౓ͷαΠζͰ͋Δɻ͜Ε͘Β͍ͷαΠζͰ͋Ε͹ɺ੡଄্ɺେ͖ͳ
ࠔ೉͸ͳ͍ɻ‫͑ݴʹٯ‬͹ɺՃ଎૷ஔ 1 ୆Ͱ਺े m ΋ͷେ͖͕͋͞Ε͹‫ڊ‬େա͗ͯ੡࡞ࠔ೉Ͱ͋Δɻ·ͨɺ

0.1 mm ΑΓখ͍͞Ճ଎ۭಎͷ੡࡞͸໺৺తա͗Δɻ
ਤ 1 ʹɺ࣮ࡍʹ੡଄͞Εͨ 509 MHz ͱ 200 GHz ͷ

2 ৼಈ໰୊ͱෳૉ਺දࣔ

Ճ଎ߏ଄ͷྫΛࣔ͢ɻՃ଎ߏ଄ͷप೾਺͸ɺՃ଎‫ث‬ͷ໨తɺͦͷ࣌୅ͷٕज़ɺ‫ط‬ଘͷՃ଎‫࢖Ͱث‬ΘΕ͍ͯϚΠΫϩ೾͸ʢͦͷ໊ͷ௨Γʣ೾ʢৼಈʣͰ͋Δɻͦ

Δप೾਺౳ʹΑΓܾ·ΔɻϚΠΫϩ೾͸·ͨɺՃ଎‫ث‬͜Ͱɺ؆୯ͷͨΊɺ·ͣ 1 ࣍‫ݩ‬ͷৼಈ໰୊Λߟ͑ΔɿͷϏʔϜ਍அ΍ϏʔϜҐஔ‫ܭ‬ଌʹ΋࢖ΘΕΔɻ͍ͣΕ΋ɺ‫ͳͱૅج‬ΔϚΠΫϩ೾ཧ࿦ʹ͸‫ڞ‬௨఺͕ଟ͍ɻ

mຊॻͰ͸ɺԼ‫ه‬Λ࢖༻ɾԾఆ͢Δɿ

d2 x
= −kx
dt2

(1)

͜Ε͸ɺ࣭ྔ m ͷ࣭఺͕ɺ1 ࣍‫࠲ݩ‬ඪ x ্Ͱόω• ࠃࡍ୯Ґ‫ܥ‬ఆ਺ k ʹΑΔ୯ৼಈΛ͢ΔӡಈํఔࣜͰ͋Δɻ͜Ε• ‫਺ڏ‬୯ҐΛ i Ͱදࣔʹɺ଎౓ dx/dt ʹൺྫ͢Δ‫ݮ‬ਰྗʢൺྫఆ਺Λ α ͱ• ෳૉྔ͸ଠࣈͰදࣔ

͢Δʣɺ‫ͼٴ‬ɺ֯ৼಈ਺ ω Ͱৼಈ͢Δ֎ྗ͕͋Δͱ͢• ϕΫτϧྔ͸্෦ʹ໼ҹΛ෇͚ͯදࣔ⃗ ͰදΘ͢• ೚ҙͷෳૉϕΫτϧΛ F

Δͱɺӡಈํఔࣜ͸ɺ• ϑʔϦΤม‫ʹ׵‬͸ԼઢΛ෇͚ͯදࣔm• ࣌ؒΛ t Ͱදࣔ• ഔ࣭͸౳ํతͱ͠ɺ༠ి཰΍ಁ࣓཰͸ςϯιϧ

d2 x
dx+ F0 cos ωt= −kx − α
2
dt
dt

(2)

ͱͳΔɻ͜ͷํఔࣜ͸ɺ2 ֊ͷఆ਺܎਺ඇಉ࣍ઢ‫ܗ‬ৗʹ͸ͳΒͣɺεΧϥʔྔͱͯ͠ѻ͏ඍ෼ํఔࣜͰ͋Γɺ͜͜Ͱ͸ɺҎԼͷΑ͏ʹͯ͠ղ͍ຊॻͷ಺༰͸ɺϚΠΫϩ೾ͷ‫ૅج‬ɺ‫ͼٴ‬ɺՃ଎‫Ͱث‬ͯΈΔɻ·ͣɺ্‫ྗ֎ه‬ͷҐ૬Λ ωt ˠ ωt + π/2 ͱΑ͘࢖͏஌ࣝʹ‫͍ͯͬݶ‬Δɻ·ͨɺஶऀͷઐ໳΍‫ڵ‬

ͯ͠ 90◦ ͣΒͨ͠ −F0 sin(ωt) ʹର͢Δ 1 ࣍‫࠲ݩ‬ඪy ্ͷӡಈํఔࣜɿ

ຯͷ౎߹Ͱภͬͨ಺༰ʹͳ͍ͬͯΔͱࢥΘΕΔɻϚΠΫϩ೾ཧ࿦Λ໢ཏ͍ͯ͠Δ΋ͷͰ͸ͳ͍ͷͰɺͦͷ఺Λྃ͝ঝ͍͖͍ͨͩͨɻ

m
1

１－ 1

dy
d2 y
− F0 sin ωt= −ky − α
2
dt
dt

(3)

Λผ్ߟ͑ɺෳૉ਺ z = x + iy ʹର͢Δӡಈํఔࣜɿ

ঘɺࣜ (8) ͷ‫਺ڏ‬ղ y(t) ͸ɺ

dz
d2 z+ F0 e−iωt= −kz − α2dtdty(t) = ℑ{z(t)}m(4)Λղ͘͜ͱΛߟ͑Δɻ͜͜Ͱɺz ͱ i Ҏ֎͸࣮਺Ͱ͋ΓɺΦΠϥʔͷެࣜɿeiθ = cos θ + i sin θ− (k − mω 2 ) sin ωt

(5)

(12)

͍Δ͚ͩͰ͋Δɻैͬͯɺ‫਺ڏ‬ղΛ෺ཧղͱͯ͠΋໰୊ͳ͍ɻఆٛͷ໰୊Ͱ͋ΔɻຊॻͰ͸ɺಛ‫͖ͳه‬(6)৔߹͸ɺ࣮਺෦Λ෺ཧղͱ͢ΔɻΛߟ͑ΔʢA ͸ෳૉఆ਺ʣ
ɻ͜Ε͸ɺࣜ (4) ΛϑʔϦ3 ฏ໘೾ͱ‫܈‬଎౓Τม‫͠׵‬ɺ͋Δಛఆͷप೾਺੒෼ͷΈΛߟ͑Δ͜ͱ

ฏ໘೾ͱ͸ɺۭؒ࠲ඪ ⃗x ʹ͓͚Δ೾ͷৼ෯Λ ϕ ͱʹ૬౰͢Δɻࣜ (6) Λࣜ (4) ʹ୅ೖ͢Δͱɺ(k − mω 2 ) + iωαF0A=(k − mω 2 )2 + ω 2 α2}]Ͱ͋Γɺࣜ (10) ͷ࣮਺ղͱൺ΂ͯҐ૬͕ 90◦ ͣΕͯΛ࢖ͬͨɻࣜ (4) ͷղͱͯ͠ɺ

z(t) = Ae−iωt

(11)

F0
=
(k − mω 2 )2 + ω 2 α2[{ωα cos ωt

ͯ͠ɺ

(7)

⃗

ϕ(⃗x) = Aei(k·⃗x−ωt)(13)ΛಘΔɻͦͯ͠ɺࣜ (7) Λࣜ (6) ʹ୅ೖ͢Δͱɺෳૉͷ‫Ͱܗ‬දΘ͞ΕΔ೾ͷ͜ͱͰ͋Δɻ͜͜Ͱɺω ͸೾ղ z(t) ͸ɺͷ֯प೾਺Ͱ͋Δɻࣜ (13) ͷҐ૬ʢࢦ਺෦ʣ͕ҰఆF0z(t) =(k − mω 2 )2 + ω 2 α2[(k − mω 2 ) cos ωt + ωα sin ωt{+ i ωα cos ωt}]− (k − mω 2 ) sin ωtͷ࣌ɿ⃗k · ⃗x − ωt = const.

͜Ε͸ 3 ࣍‫಺ۭؒݩ‬ͷฏ໘ΛදΘ͢ɻಉҐ૬໘͕ฏ໘ʹͳΔͷͰɺࣜ (13) ͰදΘ͞ΕΔ೾Λฏ໘೾ͱ‫ݴ‬(8)

͏*2 ɻ͜ͷಉҐ૬໘͸ɺ଎͞ ω/|⃗k| ͰϕΫτϧ ⃗k ͷํ޲ʹਐΉɻ·ͨɺ|⃗k| ͸ɺ‫ ཭ڑ‬2π ͷؒʹ͋Δ೾ͷͱͳΔɻ෺ཧղ͸ɺࣜ (8) ͷ࣮਺෦Λͱͬͯɺx(t)= ℜ{z(t)}F0=(k − mω 2 )2 + ω 2 α2][(k − mω 2 ) cos ωt + ωα sin ωt(14)‫͓ͯͬͳͱ਺ݸ‬Γɺ⃗k Λ೾਺ϕΫτϧͱ‫͏ݴ‬ɻ࣮ۭؒͱ೾਺ۭؒͷؒͷؔ܎Ͱ͋Δ 3 ࣍‫ݩ‬ϑʔϦΤม‫׵‬

(9)

ͷࣜɿ

F (⃗x) =

(10)

∫ ∞

⃗
d3⃗k F (⃗k) eik·⃗x−∞(

F (⃗x) : ೚ҙͷʢੑ࣭ͷΑ͍ʣෳૉؔ਺

Ͱ͋Δɻ͜ͷΑ͏ʹɺৼಈ໰୊͸ෳૉ਺දࣔʹ͢Δ(15))ͱɺൺֱత؆୯ͳ୅਺‫Ͱࢉܭ‬ղ͚Δ͜ͱ͕ଟ͍*1 ɻ·͔Β΋Θ͔ΔΑ͏ʹɺ೚ҙͷ೾͸ฏ໘೾ͷॏͶ߹Θͨɺෳૉ਺ͷ଍͠ࢉ͸ෳૉϕΫτϧͷ࿨ɺ͔͚ࢉ͸ৼͤͰද‫ݱ‬ग़དྷΔɻ

෯ͷੵͱภ֯ͷ࿨ͰදͤΔͷͰɺ෺ཧతඳ૾΋ඳ͖࣍ʹɺࣜ (15) ʹ͓͍ͯɺੵ෼͢Δ೾਺ͷྖҬ͕ ⃗k΍͍͢ɻ͜ͷΑ͏ͳཧ༝͔Βɺి৔΍࣓৔΋ෳૉ਺= ⃗k0 ͷपΓͷ͘͝খ͍͞ྖҬ ∆k ͷΈͰ͋Δ৔߹ΛͰද͢͜ͱ͕͠͹͠͹͋ΔɻຊॻͰ΋ɺి৔ͱ࣓৔

ߟ͑Δɻฏ໘೾Ͱ͸ɺ֯प೾਺ ω ͸೾਺ʹൺྫʢω͸ෳૉ਺Ͱද͢ɻ

∝ |⃗k|ʣ͢Δ͕ɺ͜͜Ͱ͸ɺω(⃗k) ͱͯ͠೚ҙͷґଘੑ

*1 ୠ͠ɺ௨ৗͷి࣓‫ؾ‬ཧ࿦ͷΑ͏ͳઢ‫ܗ‬ཧ࿦ʹ͔͠ద༻ग़དྷ

*2 ฏ໘೾ͷଞɺಉҐ૬໘͕‫ٿ‬໘ʹͳΔ೾͕͋ΓɺͦΕΛ‫ٿ‬໘

ͳ͍ɻ

೾ͱ͍͏ɻ

2

１－ 2

1/∆k

ͱ͢Δɻ͜Ε͸ɺ෼ࢄੑഔ࣭΍ಋ೾؅಺Λ఻ΘΔϚΠΫϩ೾Λ૝ఆ͓ͯ͠Γɺͦͷ৔߹ɺω ͱ |⃗k| ͸ઢ‫ܗ‬ؔ܎ʹ͸ͳΒͳ͍ɻω = ω(⃗k) ͰදΘ͞ΕΔؔ܎Λ෼

0.75ࢄؔ܎ͱ‫͏ݴ‬ʢઢ‫ܗ‬ͷ৔߹͸෼ࢄແ͠ʣɻ⃗k Λ ⃗k0 ͷ

0.5पΓͰςʔϥʔల։ͯ͠ 1 ࣍ͷ߲·ͰऔΔͱɺ

0.25

振幅

�
⃗k) ��∂ω(· ∆⃗kω(⃗k) ≈ ω(⃗k0 ) +�∂⃗k �⃗k=⃗k

1

0

(16)

-0.25

(17)

-0.75

Ͱ͋Δɻ͜ΕΛɺ࣌ؒґଘ߲ e−iω(k)t ΋‫ؚ‬Ίͯࣜ (15)ʹ୅ೖ͢Δͱɺ

-1

-0.5

0

⃗k = ⃗k0 + ∆⃗k

⃗

⃗

0

0.5

1

1.5

2

2.5

ਤ2

೾ଋͷྫɻ

位置

3

3.5

⃗

F (⃗x) = ei(k0 ·⃗x−ω(k0 )t)
(
)
�
∫
∂ω(⃗
k) ��
i∆⃗
k· ⃗
x−
t
�
∂⃗
k
⃗
k=⃗
k0
× d3⃗k F (⃗k) e

(18)

∆k

4.1 ਅۭதͷϚΠΫϩ೾ΛಘΔɻ|∆⃗k| ≪ |⃗k| ͱԾఆͨ͠ͷͰɺࣜ (18) ͷੵਅۭதͷϚΫε΢Σϧํఔࣜ͸ɺ෼ͷதͷ೾͸ɺੵ෼ͷ֎ͷ೾ΑΓʢۭؒతʹʣͣͬͱΏͬ͘ΓมԽ͢Δɻ࣮ࡍɺࣜ (18) ͸ɺਤ 2 ʹ͋ΔΑ⃗⃗ x, t) + ∂ B(⃗x, t) = 0 ɹ⃗ × E(⃗∇∂t⃗ x, t)∂D(⃗⃗ x, t) −⃗ × H(⃗=0∇∂t⃗ x, t) = 0⃗ · D(⃗∇⃗ x, t) = 0⃗ · B(⃗∇

͏ͳ೾ଋΛදΘ͓ͯ͠Γɺੵ෼෦෼͕೾ଋͷแབྷઢʹͳΔɻैͬͯɺ೾ଋͷਐΉ଎͞ vg ͸ͦͷแབྷઢͷਐΉ଎͞Ͱ͋Γɺࣜ (18) ΑΓɺ⃗vg =∂ω(⃗k)∂⃗k(19)

Ͱ͋Δɻ͜ΕΛ‫܈‬଎౓ͱ‫͏ݴ‬ɻ·ͨɺࣜ (18) ͷੵ෼

(20)
(21)
(22)
(23)

ͷ֎ʹ͋Δ೾ͷਐΉ଎͞ ω(⃗k0 )/|⃗k0 | ΛҐ૬଎౓ vp⃗ ͱH⃗ ͸ͦΕͧΕి৔ͱ࣓৔ɺD⃗
Ͱ͋Δɻ͜͜ͰɺE೾ଋࣗମ͸‫܈‬଎౓ vg ͰਐΉɻ෼ࢄੑഔ࣭΍ಋ೾؅಺ϒϥ‫Ͱ߸ه‬ɺ۩ମతʹ͸ɺͱ‫͏ݴ‬ɻ೾ଋΛߏ੒͢Δ೾͸Ґ૬଎౓ vp ͰਐΉ͕ɺ⃗ ͸ͦΕͧΕిଋີ౓ͱ࣓ଋີ౓Ͱ͋Δɻ∇⃗ ͸φͱBΛ఻ΘΔϚΠΫϩ೾ͷ೾ଋͰ͸ɺҐ૬଎౓͸ޫ଎Λ௒͑Δ͕ɺ‫܈‬଎౓͸ޫ଎ҎԼͰ͋Δɻͭ·ΓɺҐ૬଎⃗ =∇౓͸‫͚͔ݟ‬ͷ଎͞Ͱ͋Δɻ࣮ࡍʹΤωϧΪʔ͕఻ΘΔ଎͞͸‫܈‬଎౓ͰɺҼՌ཯͸ഁΕͳ͍ɻ͜Ε͸ɺ‫࣮ݱ‬

(

∂ ∂ ∂
,
,
∂x ∂y ∂z)(24)ͷ೾͕‫׬‬શͳ୯৭ͷฏ໘೾Ͱ͸ͳ͘ɺඞͣप೾਺ʹͰ͋Γɺ͜Ε΋਺ֶతͳϕΫτϧͰ͋Δɻ͜͜Ͱɺਅ

෯͕͋Γɺ೾ଋͱͳ͍ͬͯΔࣄ࣮ͱໃ६͠ͳ͍ɻঘɺ

ۭͷ༠ి཰ͱಁ࣓཰ΛͦΕͧΕɺ∆⃗k ͕େ͖͍৔߹͸ɺࣜ (16) ͷۙࣅ͸ѱ͘ͳΓɺߴ࣍ͷ߲·Ͱߟྀ͠ͳ͚Ε͹ͳΒͳ͍ɻ͜ͷ৔߹ɺ‫܈‬

ϵ0 = 8.85418782 × 10−12 F/m଎౓ͷ֓೦͸ෆਖ਼֬ʹͳΔɻ

µ0 = 1.256637 × 10

4 ϚΫε΢Σϧํఔࣜ

−6

H/m

(25)
(26)

ͱදΘ͢ͱɺ

ϚΠΫϩ೾͸ి࣓೾Ͱ͋Γɺి࣓೾͸ϚΫε΢Σ
ϧํఔࣜͰ‫هʹີݫ‬ड़ग़དྷΔɻຊઅͰ͸ɺϚΫε΢Σ

⃗ x, t) = ϵ0 E(⃗⃗ x, t)D(⃗⃗ x, t)⃗ x, t) = µ0 H(⃗
B(⃗ϧํఔࣜΛ‫ʹج‬ɺϚΠΫϩ೾ཧ࿦ͷ‫ج‬ຊࣄ߲Λղઆ
͢Δɻ

3

１－ 3

(27)
(28)

ͱॻ͚Δ͜ͱ͔Βɺ্‫ه‬ϚΫε΢Σϧํఔࣜ (20)ʙ


k

⃗ ͱH
⃗ ͷ࿈ཱඍ෼ํఔࣜɿ
(23) ͸ɺE⃗⃗ x, t) + µ0 ∂ H(⃗x, t) = 0 ɹ⃗ × E(⃗∇∂t⃗ x, t)∂E(⃗⃗ x, t) − ϵ0
⃗ × H(⃗=0∇∂t⃗ x, t) = 0⃗ · E(⃗∇⃗ x, t) = 0⃗ · H(⃗
∇

(29)
(30)


H

(31)
(32)


E

⃗ ʣΛऔΓɺͦΕʹࣜ (30)ͱͳΔɻࣜ (29) ͷճసʢ∇×Λ୅ೖͯ͠ɺϕΫτϧղੳͷެࣜɿ

(
)
⃗
⃗ × ∇
⃗ ×F
∇
(
)
⃗ −∇
⃗
⃗ ∇
⃗ ·F
⃗ 2F
=∇

(33)

ਤ3

(34)
ঘɺແଛࣦͰඇ෼ࢄੑͷഔ࣭Ͱຬͨ͞ΕͨྖҬͷͱࣜ (31)ɺ(32) Λద༻͢Δͱɺ

(2⃗ 2 − ϵ 0 µ0 ∂
∇
∂t2

)

⃗ x, t) = 0
E(⃗৔߹͸ɺ୯ʹ ϵ0 → ϵɺµ0 → µ ͱஔ͖‫͑׵‬Δ͚ͩͰ

(35)

͋Δɻ͜͜Ͱɺϵ ͱ µ ͸ɺͦΕͧΕɺͦͷഔ࣭ͷ༠ి཰ͱಁ࣓཰Ͱ͋ΔɻΛಘΔɻಉ༷ͷํ๏Ͱɺ(2⃗ 2 − ϵ 0 µ0 ∂
∇
∂t2

)

⃗ ɺ࣓৔ H⃗ ɺฏ໘೾ͷਐߦํ޲ ⃗k ͷؔ܎ɻి৔ E

4.2 ಋମதͷϚΠΫϩ೾⃗ x, t) = 0
H(⃗ϚΠΫϩ೾͕ਅۭྖҬ͔ΒಋମʹೖΔͱͲ͏ͳΔ(36)

͔Λௐ΂Δɻಋମͱ͸ɺ۩ମతʹ͸ಔ΍εςϯϨεͷ

√΋ಘΔɻ͜ΕΒ͸ɺ଎͞ c0 = 1/ ϵ0 µ0 ͰਐΉ೾Α͏ͳۚଐͰ͋Δɻಋମ಺Ͱ͸ΦʔϜͷ๏ଇʹैͬΛද͢೾ಈํఔࣜͰ͋Γɺc0 ͸ਅۭதͷޫ଎ʢ≈ͯ఻ಋిྲྀ͕ྲྀΕΔɻ఻ಋిྲྀͷେ͖͞͸ి৔ʹൺྫ͢Δɻͦͷൺྫఆ਺͸ి‫఻ؾ‬ಋ཰ σ Ͱ͋ΓɺΦʔ

299,792,458 m/sʣͰ͋ΔɻϜͷ๏ଇ͸ɺ

͜͜Ͱɺి৔Λฏ໘೾ʢࣜ (13)ʣͰදΘ͢ͱɺ⃗ x, t) = A⃗ E (⃗k, ω)ei(⃗k·⃗x−ωt)E(⃗⃗ = σE⃗J(37)

Ͱ͋Δɻࣜ (37) Λࣜ (31) ʹ୅ೖ͢Δͱɺ(41)⃗ ͸ిྲྀີ౓ϕΫτϧͰ͋Δ*3 ɻಋ
ͱͳΔɻ͜͜ͰɺJ⃗ E · ⃗k = 0AମதͷϚΫε΢Σϧํఔࣜ͸ɺ͜ͷ఻ಋిྲྀ͕มҐ(38)⃗ʹՃΘΓɺҎԼͷΑ͏ʹͳΔɿిྲྀ ∂ D/∂t⃗ ͷ޲͖ A⃗ E ͸ɺ೾ಈͷਐߦΛಘΔɻैͬͯɺి৔ E⃗ ʹ͍ͭํ޲ ⃗k ʹରͯ͠௚֯Ͱ͋Δɻ͜Ε͸ɺ࣓৔ H⃗⃗ x, t) + µ ∂ H(⃗x, t) = 0 ɹ(42)⃗ × E(⃗∇∂t⃗⃗ x, t) = 0 (43)⃗ x, t) − ϵ ∂ E(⃗x, t) − σ E(⃗⃗ × H(⃗∇∂t⃗ x, t) = 0 (44)
⃗ · E(⃗∇⃗ x, t) = 0 (45)⃗ · H(⃗∇

ͯ΋ಉ༷Ͱ͋Δɻ࣍ʹɺࣜ (37) Λࣜ (29) ʹ୅ೖ͠ɺ࣓৔͕ɺ⃗ x, t) = A⃗ H (⃗k, ω)ei(⃗k·⃗x−ωt)
H(⃗

(39)

ͱॻ͚Δ͜ͱʹ஫ҙ͢Δͱɺ⃗⃗ H ∥ ⃗k × E
H(40)

͜͜Ͱɺϵ ͱ µ ͸ɺͦΕͧΕɺಋମதͷ༠ి཰ͱಁ࣓

Ͱ͋Δ͜ͱ͕Θ͔Δɻͭ·Γɺਅۭதͷ೚ҙͷి࣓৔⃗ ʣΛ཰Ͱ͋ΔɻҎલͱಉ༷ʹɺࣜ (42) ͷճసʢ∇×√͸ɺਤ 3 ʹ͋Δؔ܎Ͱ ⃗k ͷํ޲ʹ଎͞ c0 = 1/ ϵ0 µ0*3 ిྲྀີ౓ϕΫτϧΛ໘ੵ෼͢Δͱɺͦͷੵ෼ྖҬΛ௨Δి

ͰਐΉԣ೾ͷॏͶ߹ΘͤͱͳΔɻ

ྲྀ஋ʹͳΔɻ

4

１－ 4

औΓɺͦΕʹࣜ (43) Λ୅ೖͯ͠ɺϕΫτϧղੳͷެࣜͱࣜ (44) Λద༻͢Δͱɺ

(2⃗ 2 − µσ ∂ − ϵµ ∂
∇
∂t
∂t2

)

⃗ x, t) = 0
E(⃗

(46)
(47)

ΛಘΔɻ͜Ε͸ి৴ํఔࣜͰ͋Γɺ࣌ؒͷ 1 ֊ඍ෼ͷ߲ʢࠨลୈೋ߲ʣ͸೾ʹ‫ݮ‬ਰΛ༩͑Δɻ؆୯ͷͨΊɺฏ໘೾͕ z ࣠ͷਖ਼ͷํ޲΁ਐΈɺz = 0 Ͱಋମʹɻి৔͸ x ํ޲੒෼ͷΈͱ
ೖΔͱ͠Α͏ʢਤ 4 ࢀরʣͯ͠ɺ

E x (t) = E 0 ei(kz−ωt)

(48)
ਤ4

ͱ͓͘ɻ͜͜Ͱɺk ͸ෳૉ਺Ͱ͋Δɻ͜ΕΛࣜ (46)22k = ϵµω + iµσω

Skin Depth [µm]ʹ୅ೖ͢Δͱɺ

(49)

ΛಘΔɻ௨ৗͷಋମͱϚΠΫϩ೾Ͱ͸ σ/(ϵω) ≫ 1͕੒ΓཱͭͷͰɺk ≈ ±(1 + i)

√

µσω
2

(50)

4
3.5
3
2.5
2
1.5

Ͱ͋Δɻ͜ΕΛɺࣜ (48) ʹ୅ೖͯ͠ɺ

|E x (t)| ≈ |E 0 | e−

√ µσω
2

൒ແ‫ݶ‬ಋମɻ

z

1
0.5

(51)

0

ͱͳΔɻΑͬͯɺϚΠΫϩ೾͕ಋମʹೖΔͱɺͦͷৼ

2

1

10

Frequency [GHz]෯͸ࢦ਺ؔ਺తʹ‫ݮ‬ਰ͠ɺਂ͕͞

δskin =

√

2
µσω

10

ਤ5

(52)

ಔͷදൽޮՌͷਂ͞ɻ

ͷͱ͜ΖͰৼ෯͕ 1/e ≈ 0.37 ഒʹ‫ݮ‬ਰ͢Δɻ͜ͷΑ

4.3 ଛࣦͷ͋ΔઈԑମதͷϚΠΫϩ೾͏ͳ‫ݱ‬৅ΛදൽޮՌͱ‫͍ݴ‬ɺ͜ͷ δskin ΛදൽޮՌͷ࣍ʹɺଛࣦͷ͋Δ౳ํత͔ͭҰ༷ͳઈԑମͰۭؒ

ਂ͞ʢSkin depthʣͱ‫͏ݴ‬ɻਤ 5 ʹɺϚΠΫϩ೾ྖ͕ຬͨ͞Ε͍ͯΔ৔߹Λߟ͑Δɻ͜Ε͸ɺՃ଎‫Ͱث‬Ҭʹ͓͚ΔಔͷදൽޮՌͷਂ͞Λࣔ͢ɻैͬͯɺແ

͸ɺϚΠΫϩ೾‫ٵ‬ऩମͷϞσϧͱͯ͠ॏཁͰ͋Δɻ

ࢎૉಔͰ࡞ΒΕΔৗ఻ಋߴप೾Ճ଎ۭಎͷి‫ؾ‬తͳઈԑମʹ͓͚Δଛࣦ͸ɺ༠ి཰΍ಁ࣓཰ʹෳૉ੒෼ੑೳ͸ɺۭಎ಺ද໘ͷബൽҰຕʢ਺ϛΫϩϯҎԼʣͰΛಋೖ͢Δ͜ͱͰද‫ݱ‬ग़དྷΔɻෳૉൺ༠ి཰ɺ‫ͼٴ‬ɺܾ·͍ͬͯΔͷͰ͋Δɻ

ෳૉൺಁ࣓཰ΛɺͦΕͧΕɺҎԼͰ͸ɺಋମ͸‫׬‬શಋମʢσ = ∞ʣͱͯ͠ి࣓

ϵr (ω) = ϵ′r (ω) + iϵ′′r (ω)µr (ω) = µ′r (ω) + iµ′′r (ω)৔Λ‫ٻ‬ΊΔɻຆͲͷ৔߹ɺಋମʹ༗‫ݶ‬ͷి‫఻ؾ‬ಋ཰

͕͋Δͱͯ͠‫ٻ‬Ίͨి࣓৔ͱɺ‫׬‬શಋମΛԾఆͯ͠

(53)
(54)

‫ٻ‬Ίͨి࣓৔ͱͰ༗ҙͳࠩ͸ແ͍ɻಋମද໘ʹ͓͚ͱఆٛ͢Δɻෳૉ༠ి཰ɺ‫ͼٴ‬ɺෳૉಁ࣓཰͸ɺͦΕΔϚΠΫϩ೾ͷน໘ଛࣦిྗΛ‫ٻ‬ΊΔ৔߹͸ɺ‫׬‬શͧΕɺಋମΛԾఆͯ͠‫ٻ‬Ίͨి࣓৔ͷಋମද໘ʹ͓͚Δ࣓

ϵ(ω) = ϵ0 ϵr (ω)µ(ω) = µ0 µr (ω)৔͔Β‫ٻ‬ΊΔઁಈ࿦తํ๏Λ࢖͏ͷ͕ҰൠతͰ͋Δɻ

5

１－ 5

(55)
(56)

Ͱ͋Δɻϵ′r ͱ µ′r ͸ 1 Ҏ্ʢਅۭͰ͸ 1ʣɺϵ′′r ͱ µ′′rͱදΘ͢ɻ͜ͷ͔ࣜΒ΋Θ͔ΔΑ͏ʹɺෳૉ༠ి཰ɾ

͸θϩʢແଛࣦʣ·ͨ͸ਖ਼஋ʢଛࣦ͋ΓʣͰ͋ΔɻҰෳૉಁ࣓཰͸ɺҹՃ৔ʢE ɺH ʣͱ༠ಋ৔ʢD ɺB ʣൠతʹ͸ɺ༠ి཰ͱಁ࣓཰͸प೾਺ͷؔ਺Ͱ͋ΔͷͷൺͰ͋Γɺి࣓৔ʢి৔ɺ࣓৔ʣͱഔ࣭ͷ૬‫࡞ޓ‬Ͱʢ෼ࢄੑഔ࣭ʣɺࣜ (27) ΍ (28) ͷΑ͏ͳؔ܎͸ਅ༻ΛදΘ͢ʢഔ࣭ͷి࣓৔ʹର͢ΔԠ౴Λ‫ؚ‬Ήʣɻࣜ

ۭதɺ·ͨ͸ɺඇ෼ࢄੑഔ࣭Ͱ͔͠੒Γཱͨͳ͍ɻͳ

(70) ͱ (71) ΛϚΫε΢Σϧํఔࣜ (65)ɺ(66) ʹ୅ͥͳΒɺࣜ (27) ͱ (28) ͸࣌ؒྖҬʹ͓͚Δ͕ࣜͩɺೖ͢Δͱɺप೾਺ʹґଘ͢Δ༠ి཰΍ಁ࣓཰͸प೾਺ྖҬʹ͓⃗ x, ω) = 0 ɹ⃗ x, ω) − iωµ(ω)H(⃗⃗ × E(⃗∇⃗ x, ω) = 0⃗ x, ω) + iωϵ(ω)E(⃗⃗ × H(⃗∇

͚Δ֓೦͔ͩΒͰ͋Δɻͦ͜Ͱɺਅిՙ΋ਅిྲྀ΋ແ͍৔߹ͷϚΫε΢Σϧํఔࣜɿ⃗⃗ x, t) + ∂ B(⃗x, t) = 0 ɹ⃗ × E(⃗∇∂t⃗ x, t)∂D(⃗⃗ x, t) −⃗ × H(⃗=0∇∂t⃗ x, t) = 0⃗ · D(⃗∇⃗ x, t) = 0⃗ · B(⃗∇⃗ x, t) =E(⃗⃗ x, t) =D(⃗⃗ x, t) =H(⃗⃗ x, t) =
B(⃗

−∞
∫ ∞
−∞
∫ ∞

−∞
∫ ∞

⃗ x, ω) edω E(⃗(73)⃗ ʣΛऔΔ
ͱͳΔɻ࠶ͼɺࣜ (72)ɺ(73) ͷճసʢ∇×(57)ͱɺి৔ͱ࣓৔ͦΕͧΕ͕ຬͨ͢΂͖ҎԼͷΑ͏ͳ
ํఔࣜɿ

(58)

(

)
⃗ x, ω) = 0⃗ 2 + k(ω)2 E(⃗∇()⃗ x, ω) = 0⃗ 2 + k(ω)2 H(⃗
∇

(59)
(60)

(74)
(75)

͕ಘΒΕΔɻ͜͜Ͱɺk ͸ɺʹ͓͚Δి࣓৔ʹϑʔϦΤ‫ٯ‬ม‫׵‬ɿ

∫ ∞

(72)

−iωt

(61)

⃗ x, ω) e−iωtdω D(⃗(62)⃗ x, ω) e−iωtdω H(⃗(63)⃗ x, ω) e−iωtdω B(⃗(64)√k(ω) = ±ω ϵ(ω)µ(ω)
(76)
ω√
=±
ϵr (ω)µr (ω)
(77)
c0
√
ω
ϵ′r µ′r − ϵ′′r µ′′r + i(ϵ′r µ′′r + ϵ′′r µ′r ) (78)=±c0Ͱ͋Γɺഔ࣭ͷ༠ి཰ͱಁ࣓཰͔Βܾ·ΔෳૉྔͰ͋
Δɻࣜ (74)ɺ(75) ͸ɺບ΍Ի೾౳ͷৼಈ໰୊Ͱ΋Α
͘ग़ͯ͘ΔϔϧϜϗϧπํఔࣜͰ͋Δɻ݁‫ہ‬ɺप೾−∞਺ྖҬͰϚΫε΢ΣϧํఔࣜΛղ͘͜ͱ͸ɺࣜ (74)ɺ

Λ୅ೖ͢Δͱɺ

(75) ͷ‫ݻ‬༗ํఔࣜΛ༩͑ΒΕͨ‫ڥ‬ք৚݅Ͱղ͘͜ͱ

⃗ x, ω) = 0 ɹ⃗ x, ω) − iω B(⃗⃗ × E(⃗∇⃗ x, ω) = 0⃗ x, ω) + iω D(⃗⃗ × H(⃗∇⃗ x, ω) = 0⃗ · D(⃗∇⃗ x, ω) = 0⃗ · B(⃗∇(65)ʹ‫ؼ‬ண͢Δɻ͜͜Ͱɺ‫ݻ‬༗஋͸‫ݻ‬༗Ϟʔυप೾਺ʹ

(66)

ରԠ͢Δɻ

(67)

༠ి཰΍ಁ࣓཰ͷ‫ڏ‬෦ͷޮՌΛ‫ͯݟ‬ΈΑ͏ɻ(68)⃗ x, ω) =؆୯ͷͨΊɺి৔͸ x ੒෼ͷΈ࣋ͪʢE(⃗(E x (⃗x, ω), 0, 0)ʣɺz ࣠ͷਖ਼ͷํ޲ʹਐΜͰ͍Δฏ໘ΛಘΔɻ࣌ؒґଘੑ͸ϑʔϦΤม‫׵‬ͷ e−iωt ͕୲͍ɺ೾Λߟ͑Δͱɺࣜ (74) ͸ɺϚΫε΢Σϧํఔ͔ࣜΒ࣌ؒඍ෼͕ফ͑ͯগ͠؆୯

(ʹͳͬͨɻ͜͜ͰɺϕΫτϧղੳͷެࣜɿ

(

⃗
⃗ · ∇
⃗ ×F
∇

)

=0

(69)

(79)

ͱͳΔɻ֯प೾਺ ω ͷ࣌ؒґଘੑ΋‫ؚ‬Ίͯɺ͜ͷղ͸ɺΑΓɺ্‫ه‬ͷࣜ (67)ɺ(68) ͸ɺࣜ (65)ɺ(66) ʹ‫·ؚ‬E x (⃗x, ω)e−iωt ∝ ei(k(ω)z−ωt)ΕΔ͜ͱ͕Θ͔ΔͷͰɺࣜ (67)ɺ(68) ͸ෆཁͱͳΔɻ࣍ʹɺࣜ (55) ͱ (56) Λ࢖͍ɺ⃗ x, ω)⃗ x, ω) = ϵ(ω)E(⃗D(⃗⃗ x, ω)⃗ x, ω) = µ(ω)H(⃗
B(⃗

)
∂2
2
+ k(ω) E x (⃗x, ω) = 0∂z 2i(ℜ{k(ω)}z−ωt)−ℑ{k(ω)}z

=e

(80)
(81)

(70)

Ͱ͋Δɻෳૉ༠ి཰ɾಁ࣓཰ͷ࣮෦ͱ‫ڏ‬෦͸θϩ·(71)ͨ͸ਖ਼஋Ͱఆٛͨ͠ͷͰɺ͜ͷฏ໘೾͕ z ࣠ͷਖ਼ͷ

6

１－ 6

ℑ

ํ޲ʹਐΉ͜ͱΑΓɺࣜ (76) ӈลͰਖ਼ͷූ߸͕બ͹
ΕΔɻ͜ͷ࣌ɺℑ{k} > 0 Ͱ͋Γɺe−ℑ{k(ω)}z ͸ഔ࣭ͱͷ૬‫ʹ༻࡞ޓ‬ΑΔϚΠΫϩ೾ͷ‫ݮ‬ਰΛදΘ͢ɻͭ

εr

ε r’’

·Γɺ1/ℑ{k} ͸‫ݮ‬ਰ௕Ͱ͋ΔʢͦΕ͚ͩਐΉͱৼ෯͕ 1/e ≈ 0.37 ഒʹ‫ݮ‬গʣɻ·ͨɺ͜ͷฏ໘೾ͷਐΉδε଎͞ c ͸ɺωc=ℜ{k(ω)}c0)= (√ℜϵr (ω)µr (ω)

ε r’

(82)
(83)

ਤ6

ℜ

ෳૉൺ༠ి཰ ϵr ͱ༠ిਖ਼઀ δϵ ɻͰ͋Δɻ͜Ε͸ɺෳૉ༠ి཰ɾಁ࣓཰ͷ‫ڏ‬෦΋ɺઈԑମ಺ʹ͓͚Δి࣓೾ͷ఻ൖ଎౓ʹؔ܎͢Δ͜ͱΛࣔ

5 ఻ૹઢ࿏

͍ͯ͠Δɻ΋ͬͱ؆୯ͳ৔߹ͱͯ͠ɺඇ࣓ੑʢµ′r =1ɺµ′′r = 0ʣͷ༠ిମʢϵ′r > 1ɺϵ′′r > 0ʣΛߟ͑ΔɻՃ଎‫ݶʹث‬ΒͣɺϚΠΫϩ೾Λ࢖ͬͨͲͷΑ͏ͳ͜Ε͸ɺϚΠΫϩ೾‫ٵ‬ऩମͱͯ͠Α͘࢖ΘΕΔਫ΍γεςϜͰ΋ɺ͋Δ৔ॴͰൃੜͤͨ͞ϚΠΫϩ೾Λผ୸ԽέΠૉηϥϛοΫε౳ͷϞσϧͰ͋Δɻ͜ͷ࣌ɺω√ ′k(ω) =ϵr (ω) + iϵ′′r (ω)
c0√ω
ϵ′r (ω) i δϵ (ω)
e 2=c0 cos δϵ (ω)ͷ৔ॴʹૹΔඞཁ͕͋ΔɻϚΠΫϩ೾ͷ৔߹͸ɺۚଐͰ‫ڥ‬ք৚݅Λ༩͑ͨઢ࿏Ͱ఻ૹ͢Δ͜ͱ͕ՄೳͰ

(84)

͋ΔɻຊઅͰ͸ɺ಺෦͕ਅۭͷ৔߹ͷ఻ૹઢ࿏Λ఻ΘΔϚΠΫϩ೾ʹ͍ͭͯߟ͑Δɻ

(85)

5.1 ಉ࣠ઢ

ͱͳΔɻ͜͜Ͱɺδϵʢ∈ [0, π/2)ʣ͸ɺਤ 6ɺ·ͨ͸ɺϵ′′tan δϵ = r′ϵrൺֱతখ͍͞ిྗͷϚΠΫϩ೾఻ૹʹΑ͘༻͍Δ
ͷ͕ɺಉ࣠ઢͰ͋Δɻಉ࣠ઢ͸֎ಋମͱ಺ಋମ͔Β(86)੒ΓɺͦͷؒΛਤ 7 ʹࣔ͢ TEM Ϟʔυ (TransverseͰఆٛ͞ΕΔྔͰɺ༠ిਖ਼઀ͱ‫͏ݴ‬ʢ࣮ࡍ͸ɺʮϩεɾElectroMagnetic mode) ͷ‫Ͱܗ‬ϚΠΫϩ೾Λ఻ૹλϯδΣϯτʯ·ͨ͸ʮλϯσϧλʯͱ‫͕ͱ͜͏ݴ‬ଟ͢Δɻ͍ʣɻδϵ = 0 ͸ແଛࣦͰɺ༠ిଛࣦ͕େ͖͍ఔ δϵ ͸֎ಋମͱ಺ಋମ͕‫׬‬શಋମʢ఍߅θϩʣͷ৔߹ɺಉ

େ͖͘ͳΔɻδϵ Λ࢖ͬͯ‫ݮ‬ਰ௕ΛදΘ͢ͱɺ

1
λ
√0
=
ℑ{k(ω)}
2π ϵ′r (ω)√࣠ઢͷ TEM Ϟʔυ͸‫׬‬શʹԣ೾ͱͳΔɻ͜ͷ৔߹ɺ⃗ԣํ޲ి࣓৔͸อଘྗͱͳΔ͜ͱΛࣔͦ͏ɻ·ͣɺ∇ͱి࣓৔Λਐߦํ޲ʢz ࣠ʣͱԣํ޲ʹ෼ղ͠ɺ֯

2 cos δϵ (ω)(87)
1 − cos δϵ (ω)प೾਺ ω ͷ୯৭೾ʢ·ͨ͸ɺϑʔϦΤม‫׵‬ʣΛߟ͑
Δͱɺ

Ͱ͋Δɻ͜͜Ͱɺλ0 = 2πc0 /ω ͸͜ͷप೾਺ʹ͓͚
Δਅۭதͷ೾௕Ͱ͋Δɻ্͔ࣜΒɺ༠ిਖ਼઀͕େ͖
͍ఔɺ‫ݮ‬ਰ௕͸୹͘ͳΓɺ୯Ґ௕͋ͨ͞Γͷ༠ిଛࣦ͕େ͖͘ͳΔ͜ͱ͕Θ͔Δɻ·ͨɺप೾਺͕ߴ͘
ͳΔʢ೾௕͕୹͘ͳΔʣ৔߹΋ɺ୯Ґ௕͋ͨ͞Γͷ༠ిଛࣦ͸େ͖͘ͳΔɻ༠ిମதͷޫ଎͸ɺࣜ (82)ɺ

c= √ϵ′r (ω)√2 cos δϵ (ω)
1 + cos δϵ (ω)(89)⃗ = (0, 0, E )
E
z
z
⃗
⃗
⃗
H =H +H

(92)

z

(93)

⃗ = (0, 0, H )
H
z
z

(94)

t

(85) ΑΓɺ
c0

⃗z
⃗ =∇
⃗t+∇
∇
)
(
∂
⃗
∇z = 0, 0,∂z⃗⃗⃗
E =E +E

t

(88)

z

(90)
(91)

ͱͳΓɺෳૉൺ༠ి཰ͷ࣮෦ʢϵ′r ʣ͕ಉ͡Ͱ΋ɺ༠ి
ͱͳΔɻ͜ΕΒΛɺࣜ (72)ɺ(73) ʹ୅ೖͯ͠·ͱΊਖ਼઀͕େ͖͍ఔɺ༠ిମதͷޫ଎͸஗͘ͳΔɻ

7

１－ 7

電場

磁場
ਤ7

𝑥𝑥𝑥𝑥

WX77D ಉ࣠ઢ಺Λ఻ൖ͢Δ 500 MHz ͷ TEM ϞʔυɻΛಘΔɻ‫׬‬શʹԣ೾ͷ࣌ɺ⃗ (⃗x, ω) = 0
E⃗z (⃗x, ω) = H
z

𝑟𝑟𝑟𝑟

θ

ਤ8

ͳͷͰɺ্ࣜ (95)ʙ(98) ͸ɺ⃗ (⃗x, ω) = 0 ɹ (100)
⃗t×E
∇
t
⃗
⃗
⃗
∇z × E (⃗x, ω) − iωµ0 H (⃗x, ω) = 0 ɹ (101)

𝑧𝑧𝑧𝑧

𝑦𝑦𝑦𝑦

t

t

⃗ (⃗x, ω) = 0 ɹ (102)
⃗t×H
∇
t
⃗
⃗
⃗
(103)
∇z × H (⃗x, ω) + iωϵ0 E (⃗x, ω) = 0
t

t

⃗ ʣ
ͱͳΔɻΑͬͯɺࣜ (100)ɺ(102) ΑΓɺి৔ʢEt⃗ ʣ͕ʢԣํ޲ʹ‫ͯͬݶ‬ʣอଘྗͰ͋Δ͜ͱ࣓৔ʢHtԁ౵࠲ඪ‫( ܥ‬r, θ, z)ɻ

ͱ͕Θ͔ͬͨɻ͜͜Ͱɺಉ࣠ઢͷରশੑΑΓɺਤ 8 ʹ
͋Δԁ౵࠲ඪ‫( ܥ‬r, θ, z) Λ༻͍ɺΔͱɺ⃗ = E⃗ + E⃗
E
t
r
θ
⃗ +H
⃗
⃗ =H
H
t

⃗ (⃗x, ω) − iωµ0 H⃗ (⃗x, ω) = 0 ɹ (95)
⃗t×E
∇
t
z
⃗
⃗
⃗
⃗
∇t × E (⃗x, ω) + ∇z × E (⃗x, ω)
z

(99)

r

(104)
θ

(105)

⃗ ͸อଘྗͳͷͰɺ֎ಋମͱ಺ಋମͷ
ͱ෼ղ͢ΔɻEtt⃗ (⃗x, ω) = 0 ɹ (96)−iωµ0 Ht⃗⃗⃗∇t × H (⃗x, ω) + iωϵ0 E (⃗x, ω) = 0 ɹ (97)ؒʹిҐ͕ࠩ͋Δ৔߹ͷ੩ి৔ͱಉ͡Ͱ͋Δͱͯ͠ɺ

(98)

(107)

t

z

⃗ (⃗x, ω) + ∇⃗ (⃗x, ω)
⃗z ×H
⃗t×H
∇
z
t
⃗ (⃗x, ω) = 0+iωϵ0 EtE⃗ θ = 0
1
E⃗ r ∝
r
8

１－ 8

(106)

⃗ ͸ɺ಺ಋମʹఆৗిྲྀ͕
ͱ͓͚Δɻ·ͨɺ࣓৔ Ht

ΔҰपͷઢੵ෼͕ɺͦͷԁ಺෦ͷྖҬΛ௨ΔʹిྲྀྲྀΕ͍ͯΔ৔߹ͱಉ͡Ͱ͋Δͱͯ͠ɺʹ౳͍͜͠ͱΑΓɺ

⃗ ∝1
H
θ
r
⃗ =0
H
r

(108)

I = 2πr0 H θ
√
z
ϵ0
= 2πa
E 0 eiω( c0 −t)
µ0

(109)

ͱ͓͚ΔɻΑͬͯɺࣜ (101) ͱ (103) Λ͓‫ʹ͍ޓ‬୅

(118)ͱిྲྀ͕‫·ٻ‬ΔɻΑͬͯɺࣜ (116) ͱ (118) ΑΓɺಉ

ೖ͢Δͱɺࣜ (104)ʙ(109) ΑΓɺ()2ω
Er = 0
c0
( )2
∂2H θ
ω
+
Hθ = 0
∂z 2
c0
∂2Er
+
∂z 2

(117)

iωz

ΛಘΔɻࣜ (110)ɺ(111) ͷҰൠղ͸ e c0 ͱ e࣠ઢͷಛੑΠϯϐʔμϯε͸ɺ

(110)
1
Z0 =
2π

(111)

µ0
ln
ϵ0

( )
b
a

(119)

ͱͳΔɻଛࣦͷͳ͍৔߹ɺಛੑΠϯϐʔμϯε͸࣮਺−i cω z0Ͱ͋Δɻ·ͨɺಉ࣠ઢͷಛੑΠϯϐʔμϯε͸ɺେ͖ͷॏͶ߹ΘͤͰ͋Δ͕ɺ࣌ؒґଘΛ e−iωt ͱఆٛ͠ɺ

͞ʢεέʔϧʣʹ͸ґଘͤͣɺ‫ܘ‬ͷൺʢb/aʣͰܾ·Δɻ

TEM ೾͕ z ࣠ͷਖ਼ͷํ޲ʹਐΜͰ͍Δͱ͢Δͱྫ͑͹ɺWX77D ಉ࣠ઢͷ৔߹͸ɺa = 16.7 mmɺb

ωe−i c0 z ͸ফ͑Δɻ݁‫ہ‬ɺTEM ೾ͷղ͸ɺz
1
E r (r, z, t) = aE 0 eiω( c0 −t)
√ r
z
ϵ0
1
H θ (r, z, t) = a
E 0 eiω( c0 −t)µ0r√= 38.5 mm ͳͷͰɺࣜ (119)ɺ(25)ɺ(26) ΑΓɺ
(112)

Z 0 ≈ 50 Ω

(113)

(120)

Ͱ͋Δɻಉ࣠ઢͷ৔߹ɺྺ࢙తͳࣄ৘͔Βɺ50 Ω ͱͱͳΔɻ͜͜ͰɺE 0 ͸ɺz = t = 0ɺr = a ʹ͓͚

75 Ω ͷೋ௨ΓͷಛੑΠϯϐʔμϯε͕ੈͷதͰ࢖Θ

Δి৔Ͱ͋Γɺ࣓৔ͷ܎਺͸ɺࣜ (101)ɺ·ͨ͸ɺࣜΕ͍ͯΔɻ50 Ω ͷγεςϜʹ 75 Ω ͷಉ࣠ઢΛ‫͙ܨ‬(103) ͔Βܾ·Δɻి৔ͱ࣓৔ͷҐ૬͸ಉ͡Ͱ͋Γɺͱ൓ࣹ͕‫͖ى‬ɺ಺෦Ͱఆࡏ೾͕ൃੜͯ͠τϥϒϧͷ

ͱ΋ʹޫ଎ͰਐΉ͜ͱ͕Θ͔Δɻ

‫ͳͱݩ‬Δ͜ͱ͕͋ΔͷͰɺ஫ҙͯ͠΋Β͍͍ͨɻ఻ૹ࿏ઢͷಛੑΠϯϐʔμϯε Z 0 ͸ɺ

Z0 =

V
I

ঘɺTEM ೾͸‫ݪ‬ཧతʹ͍͔ͳΔप೾਺Ͱ΋ޫ଎Ͱ
఻ൖ͢Δɻ

(114)

5.2 ಋ೾؅

Ͱఆٛ͞ΕΔɻ͜͜ͰɺV ͱ I ͸ɺͦΕͧΕɺిѹۚଐͷμΫτͷதʹϚΠΫϩ೾Λ௨͢఻ૹํ͕ࣜ

ͱిྲྀͰ͋Δɻ఻ૹ࿏ઢͷҰൠతੑ࣭ͱͯ͠ɺಛੑ

͋ΔɻͦͷμΫτΛಋ೾؅ͱ͍͏ɻಉ࣠ઢʹൺ΂ͯɺΠϯϐʔμϯεͷ౳͍͠ҟͳΔ఻ૹ࿏ઢͰ‫͍ͩܨ‬Γɺಋ೾؅Ͱ͸୯Ґ௕͋ͨ͞Γͷిྗଛࣦ͕খ͞ͷͰɺಛੑΠϯϐʔμϯεʹ౳͍͠ΠϯϐʔμϯεͰ఻ૹେిྗϚΠΫϩ೾ͷ఻ૹʹΑ͘༻͍ΒΕΔɻ

ઢ࿏Λऴ୺͢Ε͹ɺ৴߸ͷ൓ࣹ͸‫͜ى‬Βͳ͍ɻͦΕϚΠΫϩ೾ͷ఻ൖํ޲ʹ z ࣠ͷਖ਼ͷํ޲ΛͱΓɺzͰ͸ɺಉ࣠ઢͷಛੑΠϯϐʔμϯεΛ‫ٻ‬ΊͯΈΑ͏ɻ࣠ํ޲ͱ z ࣠ʹ௚֯ͳํ޲Λ෼͚ͯߟ͑ΔͱศརͰ಺ಋମͷ֎൒‫ܘ‬Λ aɺ֎ಋମͷ಺൒‫ܘ‬Λ b ͱ͢Δͱɺ

͋Δɻ͜͜Ͱɺి࣓৔ͷ z ґଘੑΛ eγz ͱԾఆͯ͠ɺ

V ͸ి৔ E r Λੵ෼ͯ͠ɺ
V =

∫ b

drE r

⃗ (t) (x, y, ω)eγz⃗ x, ω) = E
E(⃗
(115)

⃗
⃗ x, ω) = H
H(⃗

a

z
b
= aE 0 eiω( c0 −t) ln( )
a

(116)

(t)

(x, y, ω)eγz

(121)
(122)

ͱ͓͘ɻγ ͸ෳૉ਺ͱ͠ɺͦΕ͕७‫਺ڏ‬ͷ৔߹ɺz

ͱͳΔɻ͜͜Ͱɺࣜ (112) Λ࢖ͬͨɻ࣍ʹɺετʔΫํ޲ʹ఻ൖ͢Δ೾ಈͱͳΔɻ্ࣜ (121)ɺ(122) ΛࣜεͷఆཧΑΓɺr = r0 ʹ͓͚Δԁप্ͷ H θ ʹର͢(74)ɺ(75) ʹ୅ೖ͢Δͱɺి࣓৔ͷ z ํ޲੒෼ͷํ
9

１－ 9

𝑦𝑦𝑦𝑦

ఔࣜɿ

[

(

)2 ]

ω
E (t)
z =0
c0
[
( )2 ]
22⃗ +γ + ω
H (t)
∇
t
z =0
c0
⃗ 2 + γ2 +
∇
t

(123)

𝑥𝑥𝑥𝑥

(124)

͕ಘΒΕΔɻ·ͨɺࣜ (121)ɺ(122) Λࣜ (72)ɺ(73)ʹ୅ೖ͢Δͱɺ

−iωϵ0 E (t)x =∂H (t)z− γH (t)y∂y∂H (t)(t)z=
γH−−iωϵ0 E (t)yx∂x∂E (t)ziωµ0 H (t)− γE (t)
=
x
y
∂y
(t)
iωµ0 H (t)y = γE x −∂E (t)
z
∂x

𝑏𝑏𝑏𝑏

𝑎𝑎𝑎𝑎

𝑧𝑧𝑧𝑧

(125)

ਤ 9 σΧϧτ࠲ඪ‫( ܥ‬x, y, z) ͱۣ‫ܗ‬ಋ೾؅ͷஅ໘ੇ๏ (a, b)ɻ

(126)
(127)

TE(Transverse Electric) ϞʔυશͯͷҐஔͰ E z = 0ɺ‫ͭ׌‬ɺશͯͷಋ೾؅಺ද໘

(128)

্Ͱ ∂H z /∂n = 0ɻ͜͜Ͱɺn ʹΑΔภඍ෼͸ɺಋ͕ಘΒΕɺ্ͷ̐ࣜΛ·ͱΊΔͱɺ௚֯ํ޲੒෼Λ‫ٻ‬೾؅಺ද໘ʹਨ௚ͳํ޲΁ͷඍ෼Ͱ͋ΔɻΊΔࣜɿ

5.2.1 ۣ‫ܗ‬ಋ೾؅ۣ‫ܗ‬ಋ೾؅͸ɺஅ໘͕௕ํ‫Ͱܗ‬ɺେిྗϚΠΫϩ೾

[ 2
] (t)
∂H (t)z2
ω ϵ0 µ0 + γ E x = iωµ0∂y

∂E (t)
z
+γ
∂x
[ 2
]
∂H (t)zω ϵ0 µ0 + γ 2 E (t)
=
−iωµ
0
y
∂x
∂E (t)
z
+γ
∂y
]
[ 2
∂E (t)z=−iωϵ
ω ϵ0 µ0 + γ 2 H (t)
0
x
∂y
+γ

∂H (t)z∂x[ 2]∂E (t)zω ϵ0 µ0 + γ 2 H (t)y = iωϵ0∂x∂H (t)z+γ∂yͷ఻ૹʹ࠷΋ଟ͘༻͍ΒΕ͍ͯΔɻۣ‫ܗ‬ಋ೾؅಺Λ఻ൖ͢ΔϞʔυΛ‫ٻ‬ΊΔͨΊɺਤ 9 ʹ͋ΔσΧϧτ
࠲ඪ‫ܥ‬Λߟ͑ɺಋ೾؅அ໘ͷྖҬΛ 0 < x < a ‫ͭ׌‬(129)

0 < y < b ͱ͢Δɻ·ͨɺϚΠΫϩ೾͕఻ൖ͢Δํ޲
ʹ z ࣠ΛͱΔɻ
௨ৗɺۣ‫ܗ‬ಋ೾؅ͰϚΠΫϩ೾Λ఻ૹ͢Δ৔߹͸

(130)

TE ϞʔυʢE z = 0ʣΛ࢖͏ɻͦ͜ͰɺTE ϞʔυΛ‫ٻ‬ΊͯΈΑ͏ɻࣜ (124) ΑΓɺ[⃗ 2 + γ2 +
∇
t

(131)

(

ω
c0

)2 ]

H (t)
z =0

(133)

Ͱ͋ΔͷͰɺ‫ڥ‬ք৚݅ɿ��∂H (t)
z �
=0
�
∂x �
x=0,a
�
(t) �
∂H z �
=0
�
∂y �

(132)

ΛಘΔɻैͬͯɺಋ೾؅ϞʔυΛ‫ٻ‬ΊΔʹ͸ɺ༩͑ΒΕͨ‫ڥ‬ք৚݅ͷ΋ͱͰࣜ (123)ɺ(124) Λղ͍ͯ z ํ

(134)

(135)

y=0,b

޲੒෼Λ‫͠ࢉܭ‬ɺͦͷ݁ՌΛࣜ (129)ʙ(132) ʹ୅ೖͯ͠ɺ௚֯ํ޲੒෼Λ‫͢ࢉܭ‬Ε͹Α͍ɻΛ՝͢ͱɺH (t)z ͷղ͸ɺ

ศ্ٓɺಋ೾؅Ϟʔυ͸̎छྨʹ෼͚ͯߟ͑Δɿ

(t)

H (t)
z (x, y) = H 0 cos

TM(Transverse Magnetic) Ϟʔυ(t)શͯͷҐஔͰ H z = 0ɺ‫ͭ׌‬ɺશͯͷಋ೾؅಺ද໘

( mπx )
acos

( nπy )b(136)

ͱͳΔɻ͜͜ͰɺH 0 ͸ɺx = y = 0 ʹ͓͚Δ஋Ͱ

্Ͱ E z = 0ɻ͋Δɻ·ͨɺm ͱ n ͸θϩҎ্ͷ੔਺͕ͩɺm ͱ n

10

１－ 10

ͷ྆ํ͕ಉ࣌ʹθϩͱͳΔ৔߹͸আ͘ɻࣜ (136) ΛͱͳΓɺk = γ/i ͕೾਺ʹͳ͍ͬͯΔ͜ͱ͕Θ͔Δɻࣜ (133) ʹ୅ೖ͢Δͱɺ͜ͷ೾͕ z ํ޲ʹਐΉ଎͞ vp ͸ɺγ=√( mπ )2a+( nπ )2

−

b

(

ω
c0

)2

ω
k
c0
=√
ω2
1 − ωc2(137)ΛಘΔɻγ ͕७‫਺ڏ‬ͷ৔߹ʹ೾͕఻ൖ͢ΔͷͰɺ఻√(m )2 ( n )2+ab

2π/k ͱͳΔɻ͙͢ʹΘ͔ΔΑ͏ʹɺvp ͸ޫ଎Λ௒͑
Δɻͦͯ͠ɺλwg ͸ࣗ༝ۭؒͷ೾௕ΑΓ΋௕͘ɺ೾ͷप೾਺͕ःஅप೾਺ʹۙͮ͘ʹͭΕൃࢄ͢Δɻ࣮͸ɺvp ͸ʮҐ૬଎౓ʯͱ‫ݺ‬͹Ε͍ͯΔ଎͞ͰɺΤωϧΪʔΛ఻͑Δ଎͞Ͱ͸ͳ͘ɺ‫্͚͔ݟ‬ɺి࣓৔ͷύλʔϯ͕ͦͷ଎͞ͰҠಈ͍ͯ͠ΔΑ͏ʹ‫͑ݟ‬Δ͚ͩ
Ͱ͋Δɻࣄ࣮ɺΤωϧΪʔͷ఻ൖ଎౓ vem ͸ɺి࣓(t) iω nπE (t)x = −H 0ωc2 ϵ0 b( mπx)( nπy )
cos
sin
(139)
a
b
(t) iω mπE (t)y = H0ωc2 ϵ0 a( mπx)( nπy )
sin
cos
(140)
a
b
(141)
E (t)
z = 0
−ik
mπ
(t)
H (t)
x = H0ωc2 ϵ0 µ0 a
( mπx )
( nπy )
sin
cos
(142)
a
b
(t) −ik nπH (t)y = H0ωc2 ϵ0 µ0 b
( mπx )
( nπy )
cos
sin
(143)
a(
b (
)
)
mπx
nπy
(t)
cos
(144)
H (t)
z (x, y) = H 0 cosab৔ͷΤωϧΪʔີ౓ Uem ɺ‫ͼٴ‬ɺΤωϧΪʔີ౓ͷ⃗ Λ࢖ͬͯɺ
ྲྀΕΛදΘ͢ϙΠϯςΟϯάϕΫτϧ P{ }⃗ = (0, 0, Uem vem )ℜ Pͱʢ్த‫ࢉܭ‬͸লུ͢Δ͕ʣɺ

vem = c0
< c0

=

=

ω
c0

)2

1−

−

ωc2
ω2

( mπ )2a−( nπ )2b( ω )2
c

ω

(153)
(154)

ࣜ (147) ͱࣜ (19) ΑΓɺ

1
dω
= dk
dk
√ dω( )
ωc 2
= c0 1 −
ω
< c0

vg =

(146)

(155)
(156)
(157)

ͱͳΓɺޫ଎Ҏ্ʹ͸ͳΒͳ͍ɻ͍ͣΕʹͤΑɺҼՌ཯͸ഁΕͳ͍ɻ
ۣ‫ܗ‬ಋ೾؅಺Λ఻ൖ͢ΔϚΠΫϩ೾ͷҐ૬଎౓ͱ‫܈‬଎౓ɺ‫ͼٴ‬ɺःஅप೾਺ʹ͍ͭͯɺϚΠΫϩ೾ͷ఻ૹʹ࠷΋Α͘࢖͏ TE10 Ϟʔυʢਤ 10ʣΛྫʹৄ͠ͱ͓͍ͨɻ͙͢‫Ͱޙ‬ड़΂Δ͕ɺ͜ͷࣜ (147) ͸ۣ‫ܗ‬ಋ೾؅಺Ϟʔυͷ෼ࢄؔ܎ΛදΘ͍ͯ͠Δɻz ґଘੑͱ࣌ؒґଘੑ͸ɺω1−Εͨ࣌ͷ‫܈‬଎౓ vg ͱಉ͡ʹͳΔɻ࣮ࡍɺ෼ࢄؔ܎ͷ(147)

eγz−iωt = eik(z− k t)√

ͱͳΔ͜ͱ͕Θ͔Δɻ͜Ε͸ɺಋ೾؅಺ʹ೾ଋΛೖ

(145)
ω
c0
√

(152)

⃗ Λ‫͢ࢉܭ‬Δ
ͱͳΔͷͰ*4 ɺి࣓৔ͷղ͔Β Uem ɺPͱͳΔɻ͜͜Ͱɺ

((151)

Ͱ͋Δɻ͜͜Ͱɺಋ೾؅಺೾௕ʢλwg ʣ͸ɺλwg =(138)

͕ଘࡏ͢Δɻ͜ΕΛःஅप೾਺ͱ‫͏ݴ‬ɻःஅप೾਺
ΑΓ௿͍೾ΛೖΕͨͱͯ͠΋ɺࢦ਺ؔ਺తʹ‫ݮ‬ਰ͢Δɻࣜ (136) Λࣜ (129)ʙ(132) ʹ୅ೖͯ͠௚֯ํ޲
੒෼͕ಘΒΕΔɻΑͬͯɺۣ‫ܗ‬ಋ೾؅ͷ TE Ϟʔυ͸ɺϞʔυͷΠϯσοΫε (m, n) ʹରͯ͠ɺ

γ
k=
i
√

(150)

> co

ൖϞʔυͷप೾਺ʹ͸࠷௿஋ɿπωc = √
ϵ 0 µ0

(149)

vp =

(148)

*4 ిՙີ౓ͱిྲྀີ౓ϕΫτϧͷؔ܎ͱಉ༷Ͱ͋Δɻ

11

１－ 11

(a)
電場
𝑥𝑥𝑥𝑥

(b)
磁場

𝑜𝑜𝑜𝑜

𝑧𝑧𝑧𝑧

(c)
表面
電流

ਤ 10 WR-1500 ۣ‫ܗ‬ಋ೾؅಺Λ఻ൖ͢Δ 500 MHz ͷ TE10 Ϟʔυɻ(a)ɺ(b)ɺ(c) ͸ಉ͡Ґ૬Ͱදࣔ͠
͍ͯΔɻ(c) ͷిྲྀ͸ɺಋ೾؅಺ද໘্ΛྲྀΕΔɻ

͘ߟ͑ͯΈΔɻࣜ (139)ʙ(148) ʹ m = 1ɺn = 0 Λ୅ೖ͢ΔͱɺTE10 Ϟʔυͷి࣓৔ͱͯ͠ɺ
Ex = 0(t) ωaµ0
sin
Ey = H 0 iπEz = 0(t)

H x = −H 0 i
Hy = 0( πx )ae

ikz−iωt( πx )kasineikz−iωtπa(t)
H z = H 0 cos( πx )ae

ikz−iωt޲ʹਐ·ͳ͍ɻz ํ޲ʹਐΉʹ͸ɺਤ 11 ʹ͋ΔΑ͏ʹɺx ࣠ʹରͯ͠ θ ͷ֯౓Λ΋ͬͯ൓ࣹ͠ͳ͕Βਐ

(158)

Ή͜ͱʹͳΔɻͱ͜Ζ͕ɺͲͷΑ͏ͳ֯౓Ͱ΋Α͍

(159)

Θ͚Ͱ͸ͳ͘ɺʢॏͶ߹Θͤͷ݁Ռʣࣜ (159) ͰදΘ

(160)

͞ΕΔ೾ʹͳΔ͜ͱ͕ඞཁͰ͋ΔɻͦͷͨΊͷ৚݅͸ɺਤ 11 ΑΓɺ

(161)

λ0
= a sin θ
2

(162)
(163)

(164)

Ͱ͋Δ͜ͱ͕Θ͔Δɻ͜ͷ࣌ɺΛಘΔɻ࣓৔͸ x-z ฏ໘಺ʹͷΈ੒෼Λ࣋ͪɺH x

x-y ฏ໘಺͚ͩΛ‫ݟ‬Ε͹ɺE y ͸ɺx = 0 ͱ x = a ͷ

vp = c0 / cos θ > c0vg = c0 cos θ < c0√
= c0 1 − sin2 θ
√
)2
(
c0
= c0 1 −2afؒΛߦ͖དྷ͢Δ࠷௿࣍ͷ೾Ͱ͋Δ͕ɺͦΕͰ͸ z ํͱͳΔ͜ͱ΋ɺਤ 11 ΑΓΘ͔Δɻःஅप೾਺ fc ɺ‫ٴ‬

ͱ H z ͷҐ૬͸ 90 ౓ͣΕ͍ͯΔͷͰɺਤ 10(b) ʹ͋ΔΑ͏ʹɺӔΛ‫͘ר‬ɻి৔͸ y ํ޲ʹͷΈ੒෼Λ࣋ɻͪɺx = a/2ʢதؒ఺ʣͰ࠷େͱͳΔʢਤ 10(a) ࢀরʣ͜͜ͰɺE y ͷ෼෍Λฏ໘೾ͷ‫͔఺؍‬Βߟ͑ͯΈΔɻ

12

１－ 12

(165)
(166)
(167)
(168)

ͼɺःஅ೾௕ λc ͸ɺvg ͕࣮਺ʹͳΔ৚݅ΑΓɺc02a
λc = 2aͱͳΔɻ‫ڥ‬ք৚݅͸ɺE (t)z (r = R) = 0

(169)

fc =

(174)

Ͱ͋Δɻ͜͜Ͱɺม਺෼཭‫ܗ‬ΛԾఆͯ͠ɺ

(170)

Ͱ͋ΔɻϚΠΫϩ೾ͷप೾਺͕ःஅप೾਺ͷ࣌ɺฏE (t)z = R(r)Θ(θ)໘೾ͷ൒೾௕͕ಋ೾؅ͷ෯ a ͱ౳͘͠ͳΓɺθ = 0 ͱͱ͓͖ɺ͜ΕΛࣜ (173) ʹ୅ೖͯ͠ɺ྆ลΛ R(r)Θ(θ)ͳͬͯɺz ํ޲ʹ͸ਐ·ͳ͘ͳΔɻ

ͰׂΓɺ

[
1
∂2
∂
r2 2 + r
R(r)
∂r
∂r
{
(

ਤ 10(c) ʹ͸ɺಋ೾؅಺ද໘্ΛྲྀΕΔද໘ిྲྀΛࣔͨ͠ɻද໘ిྲྀ͸ɺද໘࣓৔ʹΑͬͯ‫͜ى‬Γɺ޲͖͸࣓৔ϕΫτϧʹਨ௚Ͱ͋Δɻͭ·Γɺද໘ిྲྀ

+r2

͸ɺ̚໘Ͱ͸ਐߦํ޲ɺ̝໘Ͱ͸௚֯ํ޲ʹྲྀΕΔɻ
ैͬͯɺಋ೾؅ͷ઀ଓ෦ʹ͓͍ͯɺ̚໘Ͱ͸ද໘ిྲྀγ2 ++͕ʢಋ೾؅ͷʣ઀ଓ໘Λ౉ΔͷͰɺ̚໘ʢͷಛʹதԝʣ

ω
c0

)2 }]

(175)

R(r)

1 ∂2
Θ(θ) = 0
Θ(θ) ∂θ2(176)

ͷ࿈݁Ϙϧτʹ؇Έͷແ͍Α͏஫ҙ͕ඞཁͰ͋ΔɻΛಘΔɻ্ࣜͷࠨลͰ r ͱ θ ͕෼཭ͨ͠ͷͰɺ೚ҙߴ࣍Ϟʔυʹ͍ͭͯ৮Ε͓ͯ͜͏ɻ࠷௿࣍ͷ TE10ͷ (r, θ) ʹର্͕ͯࣜ͠੒ΓཱͭͨΊʹ͸ɺC r ͱϞʔυͷͻͱ্ͭʹɺਤ 12 ʹࣔ͢ TE20 Ϟʔυ͕C θ Λఆ਺ͱͯ͠ɺ
[1∂2∂
r2 2 + r
R(r)
∂r
∂r
{
( )2 }]
ω+r2 γ 2 +R(r) = C rc0

͋ΔɻWR-1500 ಋ೾؅ͷ৔߹ɺa = 381.0 mm ͳͷͰɺ ࣜ (138) ΑΓɺःஅप೾਺͸ɺc0≈ 393.4 MHz
2ac0
fc (TE20 ) =≈ 786.9 MHzafc (TE10 ) =

(171)
(172)

(177)

1 ∂2
Θ(θ) = C θΘ(θ) ∂θ2
Cr + Cθ = 0

Ͱ͋Δɻैͬͯɺ786.9 MHz Ҏ্ͷप೾਺ͷϚΠΫϩ೾Λ WR-1500 ಋ೾؅ʹ௨ͨ͠৔߹ɺTE10 ͱ

(178)
(179)
√

TE20 ͷ̎ͭͷϞʔυ͕ࠞࡏͯ͠͠·͏Մೳੑ͕͋

Ͱͳ͚Ε͹ͳΒͳ͍ɻࣜ (178) ͷҰൠղ͸ ei√−C θ θ

ΔɻҟͳΔಋ೾؅಺೾௕ͷϞʔυ͕ࠞࡏ͢Δͱෆ౎ͱ e−i߹ͳͷͰɺ௨ৗɺΑΓߴ͍प೾਺ͷϚΠΫϩ೾Λ఻2π) = Θ(θ) ΑΓɺC θ = −m2 ʢm ͸θϩҎ্ͷ੔ૹ͢Δ৔߹͸ɺΑΓখ͍͞ಋ೾؅Λ࢖͍ɺa Λখ͘͞Α͏ʹղͷҐ૬Λఆٛ͢Δͱɺఆ਺܎਺͸লུͯ͠ɺ

ʹ͢Δɻಉ࣠ઢʹ͍ͭͯ΋ɺಉ༷Ͱ͋Δɻਤ 13 ʹɺΘ(θ) = cos mθ

509 MHz ͱ 11.4 GHz ͷ RF γεςϜͰ࢖༻͍ͯ͠

(180)

ͱఆ·Δɻैͬͯɺࣜ (177) ͸ɺΔۣ‫ܗ‬ಋ೾؅ͷྫΛࣔ͢ɻ

[

5.2.2 ԁ‫ܗ‬ಋ೾؅

∂2
∂
+r
2
∂r
∂r
{
]
( )2 }
ω22
2+r γ +− m R(r) = 0
c0͜͜Ͱ͸ɺՙిཻࢠͷՃ଎ʹؔ༩͠͏Δ TM ϞʔυʢશͯͷҐஔͰ H z = 0ɺ‫ͭ׌‬ɺશͯͷಋ೾؅಺ද
໘্Ͱ E z = 0ʣΛ‫ٻ‬ΊͯΈΔɻԁ‫ܗ‬ಋ೾؅ͷ಺൒‫ܘ‬Λ R ͱ͢Δɻಉ࣠ઢͱಉ༷ʹɺਤ 14 ʹ͋Δԁ౵࠲

r2

(181)

ͱͳΔɻ͜Ε͸ϕοηϧͷඍ෼ํఔࣜͳͷͰɺR(r)ඪ‫ܥ‬Λ༻͍Δͱɺࣜ (123) ͸ɺ2ͷઢ‫͕ͩ߹݁ܗ‬ɺप‫ڥظ‬ք৚݅ Θ(θ +਺ʣͰͳ͚Ε͹ͳΒͣɺ·ͨɺΘ(π/2) ͕࣮਺ͱͳΔ

ͯ͠ःஅप೾਺Λ্͛ɺγϯάϧϞʔυʹͳΔΑ͏

[−C θ θ

͸ m ࣍ͷϕοηϧؔ਺ Jm Λ࢖͍ɺఆ਺܎਺͸লུ
ͯ͠ɺ

2

1 ∂
∂
1 ∂
+ 2 2
+
2
∂r
r ∂r r ∂θ( )2 ]
ω
2
+γ +
E (t)
z =0
c0

R(r) = Jm (αr)
( )2
ω
2
2
α =γ +
c0

(173)
13

１－ 13

(182)
(183)

𝑥𝑥𝑥𝑥

𝑎𝑎𝑎𝑎
y

電場
（TE10モード）

θ
𝑎𝑎𝑎𝑎

𝑎𝑎𝑎𝑎

𝑥𝑥𝑥𝑥

𝑜𝑜𝑜𝑜

𝑧𝑧𝑧𝑧

θ
θ

λwg = λ0 / cosθ
ਤ 11ۣ‫ܗ‬ಋ೾؅಺ TE10 Ϟʔυʹ͓͚ΔҐ૬଎౓ vp ͱ‫܈‬଎౓ vg ɻ

電場

磁場
ਤ 12

WR-1500 ۣ‫ܗ‬ಋ೾؅಺Λ఻ൖ͢Δ 500 MHz ͷ TE20 Ϟʔυɻ

14

１－ 14

𝑣𝑣𝑣𝑣𝑔𝑔𝑔𝑔

𝑣𝑣𝑣𝑣𝑝𝑝𝑝𝑝

(a)ͱͳΔɻ͜ΕҎ্ͷप೾਺ͷϚΠΫϩ೾ͷΈ͕఻ൖ

(b)

同じＰＨＳ

͢Δɻ͜͜Ͱ΋ɺۣ‫ܗ‬ಋ೾؅ͱಉ༷ʹɺःஅप೾਺͸ಋ೾؅ͷαΠζʹ൓ൺྫ͢ΔɻҎ্ΑΓɺz ํ޲੒෼
ͷղ͸ɺ

(t)
E (t)
z = E 0 Jm

ਤ 13(a) UHF ଳʢ509 MHz ౳ʣͰ༻͍Δ WR1500 ۣ‫ܗ‬ಋ೾؅ʢஅ໘಺ੇ๏ɿ381.0×190.5 mmʣɻ(b) X όϯυʢ11.4 GHz ౳ʣͰ༻͍Δ WR-90 ۣ‫ܗ‬ಋ೾؅ʢஅ໘಺ੇ๏ɿ22.86×10.1 mmʣɻ(a) ͱ(b) Ͱɺಉ͡ PHS Λεέʔϧͱͯ͠ҰॹʹࡱӨɻ

(188)

ͱͳΔɻ͜ΕΛࣜ (129)ʙ(132) ʹ୅ೖ͢Ε͹ɺ௚֯ํ޲੒෼͕ಘΒΕΔͷ͕ͩɺH (t)z = 0 ͷ࣌ʹࣜ (129)γ ∂E (t)zα2 ∂r
γ 1 ∂E (t)(t)
zEθ = 2α r ∂θ
iωϵ0 1 ∂E (t)
z
H (t)
=
−
r
α2 r ∂θiωϵ0 ∂E (t)(t)
zHθ = 2
α
∂r
E (t)
r =

θ 𝑟𝑟𝑟𝑟𝑦𝑦𝑦𝑦𝑧𝑧𝑧𝑧

jmn ͱॻ͘ͱʢn ͸ 1 Ҏ্ͷ੔਺ʣɺ‫ڥ‬ք৚݅ͷࣜ(174) Λຬͨͨ͢Ίʹɺα ͷͱΓ͏Δ஋͕
jmn
R

(184)

ͱ੍‫͞ݶ‬ΕΔɻΑͬͯɺ্ࣜͱࣜ (183) ΑΓɺ

(

jmn
R

)2

−

(

ω
c0

)2

(185)

(192)

R

)

′
kJm

(

(198)

Ͱɺeikz−iωt Ͱ͋Γɺ

γ
k=ɹ
i
√(

(186)

=

Ͱ͋Δɻͭ·Γɺԁ‫ܗ‬ಋ೾؅ʹ͓͚Δ TM Ϟʔυͷःஅप೾਺͸ɺc0 jmn2π R(191)

ͱ‫·ٻ‬Δɻz ͱ࣌ؒͷґଘੑ͸ɺ͢΂ͯͷ੒෼ʹ‫ڞ‬௨

ͷͰɺ

fc =

(

H (t)
z = 0

ͱͳΔɻγ ͕७‫਺ڏ‬ͷ࣌ʹ೾͕ z ํ޲ʹ఻ൖ͢Δ

ω
jmn
>
c0
R

(190)

)
jmn
r cos mθ
(193)
jmn
R
(
)2
(
)
R
m
jmn
(t)
(t)
k Jm
r sin mθ (194)E θ = −E 0 i
jmn
r
R
(
)
j
mn
(t)
r cos mθ
(195)
E (t)
z = E 0 Jm
R
(
)2
(
)
m
R
jmn
(t)
H (t)
iωϵ
=
E
J
r
sin mθ (196)
0
m
0
r
jmn
r
R
)
(
(
)
jmn
R
(t)
(t)
′
Jm
r cos mθ(197)
H θ = E 0 iωϵ0
jmn
R
(t)
E (t)
r = E0 iԁ౵࠲ඪ‫( ܥ‬r, θ, z)ɻ

α=

(189)

Λ࢖͏ͱɺTM Ϟʔυͷి࣓৔͸ɺ

ͱॻ͚ΔɻJm ͷʢখ͍͞ํ͔Βʣn ൪໨ͷྵ఺Λ

γ2 =

)
jmn
r cos mθRʙ(132) Λԁ౵࠲ඪ‫ʹܥ‬ม‫ࣜͨ͠׵‬ʢ్த‫ࢉܭ‬লུʣ
ɿ

𝑥𝑥𝑥𝑥
𝑅𝑅𝑅𝑅

ਤ 14

(

(199)
ω
c0

)2

−

(

jmn
R

)2

(200)

Ͱ͋Δɻ
۩ମྫͱͯ͠ɺTM01 ϞʔυΛߟ͑Δɻࣜ (193)ʙ

(187)
15

１－ 15

(a)
電場

(b)
磁場
ਤ 15

಺‫ ܘ‬150 mm ͷԁ‫ܗ‬ಋ೾؅಺Λ఻ൖ͢Δ 500 MHz ͷ TM01 Ϟʔυɻ(a) ͱ (b) ͸ಉ͡Ґ૬Ͱද͍ࣔͯ͠Δɻ

(198) ʹ m = 0ɺn = 1 Λ୅ೖ͢Δͱɺ(t)

Er = E0 i
Eθ = 0(t)(

E z = E 0 J0
Hr = 0

R
j01

(

(t)
H θ = E 0 iωϵ0

Hz = 0

)

kJ0′

(

)

j01
r eikz−iωt
R

)

j01
r eikz−iωt
R
(

R
j01

)

J0′

(

ϩʔυ‫͢ʹܕ‬Δ͔ɺ·ͨ͸ɺ಺෦ʹ༠ిମͷ౵Λ͸ΊࠐΜͰɺҐ૬଎౓Λޫ଎ʢϏʔϜͷ଎͞ʣʹ౳͘͢͠(201)

Ε͹ɺԁ‫ܗ‬ಋ೾؅Ͱ΋ߴΤωϧΪʔిࢠͷ࿈ଓՃ଎(202)͕ՄೳͱͳΔ*5 ɻ(203)

6 ۭಎ‫ثৼڞ‬(204)ۭಎ‫ͱثৼڞ‬͸ɺۚଐ੡ͷ༰‫Ͱث‬ɺ಺෦͸ۭಎͰ͋)j01
r eikz−iωt (205)R(206)

Δɻͦͷۭಎ಺ʹɺͦͷ‫ৼڞ‬प೾਺ʹ౳͍͠ϚΠΫϩ೾Λ౤ೖ͢Δͱɺۭಎ಺ʹଟ͘ͷϚΠΫϩ೾ΤωϧΪʔΛஷΊΔ͜ͱ͕ग़དྷΔɻՃ଎‫͚͓ʹث‬ΔՃ଎

ͱͳΔɻ0 ࣍ͷϕοηϧؔ਺ͷ 1 ൪໨ͷྵ఺͸‫ʹط‬ۭಎͰ͸ɺ୔ࢁͨΊࠐΜͩϚΠΫϩ೾ʹΑΔߴ͍ి஌ΒΕ͓ͯΓɺj01 ≈ 2.4048 Ͱ͋Δɻશͯͷ੒෼ͰքͰՙిཻࢠΛՃ଎͢ΔɻຊઅͰ͸ɺଟ͘ͷՃ଎ۭθ ґଘੑ͕ফ͑ɺ࣠ରশͱͳͬͨɻE z ͱ E r ͰҐ૬ಎͷ‫ج‬ຊ‫͍ͯͬͳͱܗ‬ΔϐϧϘοΫε‫ۭܕ‬ಎΛྫʹ͕ 90˃ͣΕ͍ͯΔͷͰɺಉ͡ҐஔͰ͕࣌ؒ‫ͭܦ‬ɺ·ͱΓɺղઆ͢Δɻ

ͨ͸ɺ͋Δ࣌ࠁͰ z ࠲ඪʹԊͬͯҠಈ͢Δͱɺి৔

6.1 ‫ݻ‬༗ϞʔυϕΫτϧ͸ r − z ฏ໘಺Ͱճస͢Δɻ࣓৔ϕΫτϧϐϧϘοΫε‫ۭܕ‬ಎͱ͸ɺਤ 16 ʹ͋ΔΑ͏ͳɺԁ͸ θ ੒෼ͷΈΛ࣋ͭɻਤ 15 ʹ TM01 Ϟʔυͷి࣓‫ܗ‬ಋ೾؅ͷҰ෦Λ੾Γऔͬͨ‫ܗ‬ঢ়Ͱ͋Δɻۭಎ಺෦

৔ͷྫΛࣔ͢ɻߴΤωϧΪʔͷిࢠʢ଎͞͸΄΅ޫ͸ਅۭͱ͠ɺͦͷ಺൒‫ܘ‬Λ Rɺ௕͞Λ d ͱ͢Δɻԁ଎ʣΛ͜ͷి৔ͷ೾ʹ৐ͤɺৗʹిࢠͷपΓʹՃ଎ి‫ܗ‬ಋ೾؅Ϟʔυͱͷҧ͍͸ɺz = 0ɺ‫ͼٴ‬ɺz = d ʹ৔͕͋ΔΑ͏ʹͯ͠࿈ଓՃ଎͕ग़དྷͦ͏͕ͩɺి৔
ͷ೾ͷਐΉ଎͞͸Ґ૬଎౓Ͱޫ଎ΑΓ଎͍ͷͰɺ୯*5 ୯ ७ ͳ ߏ ଄ Ͱ ߴ ޯ ഑ Ճ ଎ Λ ࣮ ‫ ݱ‬ग़ དྷ Δ Ճ ଎ ߏ ଄ ͱ ͠ ͯ७ͳԁ‫ܗ‬ಋ೾؅Ͱ͸ߴΤωϧΪʔిࢠͷ࿈ଓՃ଎͸

ʮDielectric-loaded accelerating strcutreʯ͕‫͞ڀݚ‬Εͯ
͍Δ [2]ɻ

ग़དྷͳ͍ɻઢ‫ܗ‬Ճ଎‫Ͱث‬Α͘࢖ΘΕ͍ͯΔσΟεΫ

16

１－ 16

𝑥𝑥𝑥𝑥
𝑅𝑅𝑅𝑅

ͱͳΔɻ৽ͨͳ‫ڥ‬ք৚݅͸ɺ(−) (−)
A(+) E (+)
E r (z = 0) = 0
r (z = 0) + A
(+)

(220)

(+)
(−)
A E θ (z = 0) + A(−) E θ (z = 0) = 0(+)(−)
A(+) E θ (z = d) + A(−) E q (z = d) = 0

(221)

A

(+)

θ

𝑟𝑟𝑟𝑟

ਤ 16

𝑑𝑑𝑑𝑑

𝑦𝑦𝑦𝑦

f

𝑧𝑧𝑧𝑧

(219)

(−) (−)E (+)
E r (z = d) = 0
r (z = d) + A(222)

Ͱ͋Δɻࣜ (219) ͱ (221) ΑΓɺA(+) − A(−) = 0(223)

Ͱ͋Δɻͭ·Γɺ+z ํ޲ʹਐΉԁ‫ܗ‬ಋ೾؅Ϟʔυͱ−z ํ޲ʹਐΉϞʔυͷҐ૬ࠩ͸ 180˃Ͱ͋Δɻ͜Ε͸ɺz = 0 ʹ͓͚Δ‫ݻ‬ఆ୺൓ࣹʹ૬౰͢Δɻ͜ͷ݁ՌϐϧϘοΫεͱԁ౵࠲ඪ‫ܥ‬ɻͱɺࣜ (220)ɺ(222) ΑΓɺ͓͍ͯۚଐද໘͕͋ΔͨΊɺE r = E θ = 0 ͱ͍͏‫ڥ‬

eikd − e−ikd = 2i sin kd = 0ք৚͕݅՝͞ΕΔ͜ͱ͚ͩͰ͋Δɻ࣮ࡍɺ+z ํ޲ʹ͕՝͞ΕΔɻ͜ΕΛຬͨͨ͢Ίʹɺp ΛθϩҎ্ͷ੔ਐΉԁ‫ܗ‬ಋ೾؅Ϟʔυͱ −z ํ޲ʹਐΉԁ‫ܗ‬ಋ೾؅਺ͱͯ͠ɺϞʔυͷॏͶ߹ΘͤͰϐϧϘοΫε‫ۭܕ‬ಎ಺ͷ‫ݻ‬༗k=ϞʔυΛ‫ٻ‬ΊΔ͜ͱ͕ग़དྷΔɻ
·ͣ͸ɺTM ϞʔυΛ‫ٻ‬ΊΔɻ+z ํ޲ʹਐΉԁ‫ܗ‬ಋ೾؅Ϟʔυ͸ɺࣜ (193)ʙ(198) ΑΓɺ

pπ
d

(224)

(225)

Ͱͳ͚Ε͹ͳΒͳ͍ɻ͜ͷࣜͱࣜ (200) ΑΓɺ͕࣍ࣜಘΒΕɺϞʔυप೾਺͸੔਺ͷ૊ (m, n, p) ʹΑͬ

(

)

(

)

jmn
ikz−iωt(207)
r cos mθe
jmn
R
)2
(
(
)
jmn
R
m
(+)
(t)
ikz−iωtk Jm(208)
r sin mθe
E θ = −E 0 i
jmn
r
R
(
)
jmn
(t)
(+)
ikz−iωt
E z = E 0 Jm
(209)
r cos mθe
R
(
)2
(
)
R
jmn
m
(t)
(+)
ikz−iωt
H r = E 0 iωϵ0(210)
Jmr sin mθe
jmn
r
R
(
)
(
)
R
jmn
(+)
(t)
′
ikz−iωtJm
H θ = E 0 iωϵ0(211)
r cos mθe
jmn
R
(+)

Er

(+)

Hz

(t)

= E0 i

R

′

kJm

=0

(212)

= −E 0 i

(

(−)

=0

Hz

R

)

′

kJm

= ωmnp

jmn
R

)2

+

( pπ )2
d

(226)
(227)

͜Ε͸ɺz = d ʹ͓͍ͯ΋‫ݻ‬ఆ୺൓ࣹ͢ΔͨΊʹप೾਺΍ಋ೾؅಺೾௕͕௕͞ d ʹԠܾͯ͡·Δͱ͍͏
͜ͱͰ͋Δɻ࠷‫ʹޙ‬ɺ

1
2

(228)

ͱશମͷ܎਺ΛબͿͱɺϐϧϘοΫε‫ۭܕ‬ಎ಺ͷ TM
Ϟʔυ͸ɺ

(

)
jmn
−ikz−iωt(213)
r cos mθe
jmn
R
)2
(
(
)
jmn
R
m
(−)
(t)
−ikz−iωtk Jm(214)r sin mθeEθ = E0 i
jmn
r
R
(
)
jmn
(t)
(−)
−ikz−iωt
E z = E 0 Jm
(215)
r cos mθe
R
(
)2
(
)
R
jmn
m
(t)
−ikz−iωt
(−)
(216)
Jm
r sin mθe
H r = E 0 iωϵ0
jmn
r
R
(
)
(
)
R
jmn
(−)
(t)
−ikz−iωt′Jm
H θ = E 0 iωϵ0(217)
r cos mθe
jmn
R
(t)

(−)

ω = c0√(

A(+) = A(−) =Ͱ͋Δɻ−z ํ޲ʹਐΉԁ‫ܗ‬ಋ೾؅Ϟʔυ͸ɺ+z ํ޲ʹਐΉ্‫ه‬Ϟʔυʹ͓͍ͯ k → −k ͱஔ͖‫͑׵‬Δ
͚ͩͰɺ

Er

ܾͯ·Δ஋ʹ཭ࢄԽ͞ΕΔɿ

(218)

(

)
jmn
nπz −iωt(229)
r cos mθ sin
e
jmn
R
d
(
)2
(
)
R
jmn
m
nπz −iωt
(t)
(230)
k Jm
r sin mθ sineEθ = E0
jmn
r
R
d
(
)
jmn
nπz −iωt(t)
E z = E 0 Jm
(231)
r cos mθ cos
e
R
d
(
)2
(
)
R
jmn
m
nπz −iωt
(t)
(232)
Jm
r sin mθ cose
H r = E 0 iωϵ0
jmn
r
R
d
(
)
jmn
R ′
nπz −iωt(t)
H θ = E 0 iωϵ0
J(233)
r cos mθ cos
e
jmn m
R
d
(t)

E r = −E 0

Hz = 0

17

１－ 17

R

′

kJm

(234)

ͱͳΓɺ+z ํ޲ʹਐΉԁ‫ܗ‬ಋ೾؅Ϟʔυͱ −z ํ޲

18 ʹࣔ͢ɻి৔͸த৺࣠ʢz ࣠ʣ্Ͱ࠷΋‫͘ڧ‬ɺԁʹਐΉϞʔυͷॏͶ߹ΘͤͰϐϧϘοΫε‫ۭܕ‬ಎ಺౵น໘ (r = R) ͰθϩͱͳΔɻҰํɺ࣓৔͸த৺࣠ͷ‫ݻ‬༗ϞʔυΛ‫ٻ‬ΊΔ͜ͱ͕ग़དྷͨɻ
TM Ϟʔυͱಉ༷ͷํ๏ͰɺTE ϞʔυʢશͯͷҐஔ
Ͱ E z = 0ɺ‫ͭ׌‬ɺશͯͷಋ೾؅಺ද໘্Ͱ ∂H z /∂n= 0ʣʹ͍ͭͯղ͘ͱʢ్த‫ࢉܭ‬লུʣɺ্ͰθϩͰ͋Δɻి৔ͱ࣓৔ͰҐ૬͕ 90˃ͣΕ͍ͯΔͷͰɺి৔͕࠷ߴͷॠؒɺ࣓৔͸θϩͰ͋Δɻ·ͨɺ࣓৔͕࠷ߴͷॠؒɺి৔͸θϩͱͳΓɺి৔ͱ࣓
৔͕͓‫͍ͯͬ͋͠ৼྭʹ͍ޓ‬Δɻ‫ݻ‬༗Ϟʔυप೾਺

(

)
′
jmn
nπz −iωt(235)
r sin mθ sin
e
R
d
( ′
)
′
jmn
nπz −iωt(t) iωµ0 jmn ′(236)
Jr cos mθ sineE θ = −E 0
k2
R m
R
d
(t) iωµ0 m
Jm
k2 r

E r = −E 0

Ez = 0͸ɺۭಎ൒‫ ܘ‬R ʹ൓ൺྫ͠ɺ௕͞ʢdʣʹ͸ґଘ͠ͳ
͍*7 ɻঘɺ(Super)KEKB ओϦϯάͷ RF प೾਺Ͱ͋
Δ 509MHz ʹରԠ͢ΔϐϧϘοΫε‫ܕ‬Ճ଎ۭಎͷ൒

(237)
)
′
jmn
pπz −iωt(238)
r cos mθ cos
e
R
d
( ′
)
jmn
pπz −iωt(t) 1 pπ m(239)Jmr sin mθ cose
H θ = −E 0
2
k d r
R
d
( ′
)
jmn
pπz −iωt(t)(240)
r cos mθ sine
H z = E 0 JmRd(t)
H r = E0′1 pπ jmn′
J2
k d R m(‫ܘ‬͸ R = 22.56 cm Ͱ͋Δɻ͜͜ͰɺQ ஋ʢQuality factorʣͷͻͱͭͰ͋Δ಺෦ Q ஋ʢແෛՙ Q ஋ͱ΋‫͏ݴ‬ʣΛఆٛ͢Δɻ௨ৗɺ

Q0 ͱ‫ه‬ड़͢Δɻ͜Ε͸ɺՃ଎ۭಎ౳ͷੑೳΛදΘ͢ॏཁͳແ࣍‫͋Ͱྔݩ‬ΔɻQ0 ஋͸ɺۭಎ಺ͷి࣓৔ͷ

ͱͳΔɻΤωϧΪʔʢU ʣͱน໘ଛࣦిྗʢPwall ʣͷൺʹ‫ৼڞ‬TEmnp ɺTMmnp ϞʔυͷΠϯσοΫε (m, n, p)

֯प೾਺*8 Λֻ͚ͨྔͰ͋Δɿͱ‫ݻ‬༗Ϟʔυి࣓৔ͷ‫ͱܗ‬ͷؔ܎Λ‫ͯݟ‬ΈΑ͏ɻ2m
͸ి࣓৔ͷ z ࣠·ΘΓํ޲ͷઅͷ‫͋Ͱ਺ݸ‬Γɺm = 0

Q0 = ω

Λ୯‫( ۃ‬monopole)ɺm = 1 Λ૒‫( ۃ‬dipole) ͱ‫͏ݴ‬ɻn ͸ಈ‫ܘ‬ʢrʣํ޲ͷઅͷ‫͍ͯͬͳʹ਺ݸ‬ΔʢTE Ϟʔຯ͢ΔɻQ0 ஋͸प೾਺ʹґଘ͠ɺ‫ݻ‬༗Ϟʔυ͝ͱʹྫΛਤ 17 ʹࣔ͢ɻϐϧϘοΫε‫ۭܕ‬ಎ಺ͷ೚ҙͷిҟͳΔ஋Λ࣋ͭɻಔ੡ͷৗ఻ಋՃ଎ۭಎͰ͸ɺ௨ৗɺ࣓৔͸ɺ͜͜Ͱ‫ٻ‬Ίͨ TMɺTE Ϟʔυղʢࣜ (229)Q0 ஋͸਺ສʙ਺ेສఔ౓Ͱ͋Δɻਤ 19 ʹɺ൒‫ ܘ‬Rʙ(234)ɺ‫ͼٴ‬ɺ(235)ʙ(240)ʣͷॏͶ߹ΘͤͰද‫ݱ‬
ग़དྷΔ*6 ɻ௨ৗཻࢠՃ଎ʹ࢖ΘΕΔͷ͸ʢTM ϞʔυͷதͰʣ࠷΋௿͍‫ݻ‬༗प೾਺Λ࣋ͭ TM010 ϞʔυͰ͋Δɻ۩
ମతͳղ͸ɺࣜ (229)ʙ(234) ΑΓɺ(t)

E z = E 0 J0
Hr = 0j01
r e−iωt
R

R ′
(t)
J0
H θ = E 0 ωϵ0j01
Hz = 0c0 j01
ω=
R

(

)
π
j01
r e−i(ωt− 2 )R= 22.56 cmɺ௕͞ d = 26.00 cm ͷ৔߹ͷಔ੡ϐϧϘοΫε‫ۭܕ‬ಎͷ Q0 ཧ࿦஋Λࣔ͢ɻҰൠతͳ܏޲
ͱͯ͠ɺཻࢠՃ଎ʹΑ͘༻͍Δ TM010 Ϟʔυ͸‫ཱݽ‬͓ͯ͠ΓɺͦΕΑΓߴ͍प೾਺ͷϞʔυʢߴ࣍Ϟʔυʣ͸ɺप೾਺ͷߴ͍ํʹߦ͚͹ߦ͘ఔɺີू͍ͯ͠

(241)
(242)

)

(248)

ΤωϧΪʔ͕น໘ଛࣦిྗʹൺ΂ͯେ͖͍͜ͱΛҙΈʣɻp ͸ z ࣠ํ޲ͷෲͷ‫͋Ͱ਺ݸ‬Δɻ‫ݻ‬༗Ϟʔυͷ

(

Pwall

Q0 ஋͕େ͖͍ͱ͍͏͜ͱ͸ɺ஝͑ΒΕΔి࣓৔ͷυͷ E θ ,H r ɺTM Ϟʔυͷ E θ ,E z ,H r ʹؔͯ͠ͷEr = 0Eθ = 0UΔɻ͜ͷ͜ͱ͸ɺTM010 ϞʔυΛՃ଎Ϟʔυͱͯ͠
࠾༻͢Δ͜ͱͷཧ༝ͷͻͱͭͰ͋Δɻ

(243)

6.2 ఻ૹઢ࿏ͱͷ݁߹(244)

લઅ·Ͱ͸୯ಠͷۭಎΛߟ͕͑ͨɺۭಎ಺ʹ‫ݻ‬༗(245)

ϞʔυΛྭৼ͢ΔͨΊʹ͸ɺ఻ૹઢ࿏Λۭಎʹ‫͗ܨ‬ɺ֎෦͔Β RF ύϫʔΛ౤ೖ͠ͳ͚Ε͹ͳΒͳ͍ɻͦ(246)

ͷͨΊͷํ๏ͱͯ͠ɺਤ 20 ʹ͋ΔΑ͏ʹۭಎʹ௚઀

(247)

ಋ೾؅Λ෇͚ͨΓɺਤ 21 ͋ΔΑ͏ͳΧοϓϥʔΛۭͰ͋ΓɺE z ͱ H θ ͷ 2 ੒෼ͷΈΛ࣋ͭɻి࣓৔͸ɺಎʹऔΓ෇͚ɺRF ύϫʔΛ౤ೖ͢Δʢ·ͨ͸ɺRF࣠ରশʢθ ґଘੑͳ͠ʣͰɺz ํ޲ʹҰఆʢz ґଘੑύϫʔΛऔΓग़͢ʣɻ௨ৗɺಋ೾؅΍ΧοϓϥʔΛ෇ͳ͠ʣͰ͋Δɻ|E z | ͱ |H θ | ͷಈ‫޲ํܘ‬ͷ෼෍Λਤ

*7 z ্࣠ʹϏʔϜύΠϓΛ෇͚ͯ݀Λ͚͋ͨ৔߹ɺTM010*6 ෳࡶͳ‫ܗ‬ঢ়ͷۭಎͰ͸ɺTMɺTE ϞʔυҎ֎ͷϋΠϒϦο

Ϟʔυͷ‫ݻ‬༗प೾਺͸௕͞ d ʹʢ‫ʹ͔ۇ‬ʣґଘ͢Δɻ
*8 ਖ਼֬ʹ͸ɺน໘ଛࣦ͕ແ͍ͱԾఆͨ࣌͠ͷ‫֯ৼڞ‬प೾਺υϞʔυ͕ଘࡏ͢Δ৔߹΋͋Δɻ

18

１－ 18

電場：

ਤ 17

磁場：

ϐϧϘοΫε‫ۭܕ‬ಎ಺ͷ‫ݻ‬༗Ϟʔυͷྫɻ

19

１－ 19

Arbitrary Scaleͱఆٛ͠ɺ಺෦ Q ஋ͱ֎෦ Q ஋ͷؔ܎͸ɺ

Q0 = QL (1 + β)

1
0.8

|Ez |

ͱͳΔɻβ = Pext /Pwall = 1 ͷ৔߹͸ɺPwall = Pext

|Hθ|

0.6

ͳͷͰɺ͜ͷ৚݅ͰۭಎʹϚΠΫϩ೾Λ౤ೖͨ͠৔
߹ʹ͸ɺೖྗిྗͱۭಎน໘ଛࣦిྗ͕௼Γ߹͍ɺ൓ࣹ͕ແ͘ͳΔɻ͜ΕΛ੔߹ͷͱΕͨঢ়ଶͱ‫͏ݴ‬ɻβ0.4

͕ 1 Ͱͳ͍৔߹͸ɺͦͷ௼Γ߹่͍͕Εɺ൓ࣹిྗ

0.2
0
0

(255)

͕ੜ͡Δɻβ < 1 ͷ৔߹Λૄ݁߹ɺβ > 1 ͷ৔߹Λ
0.5

1

1.5

2

‫͏ݴͱ߹݁ڧ‬ɻ݁߹౓ β ͷௐ੔͸ɺϧʔϓͷ֯౓΍

2.5

j01r/RΞϯςφͷಥ͖ग़͠ྔΛม͑Δ͜ͱʹΑΓߦ͏ɻ֤छ Q ஋΍݁߹౓͸ɺ‫ৼڞ‬प೾਺ͱͱ΋ʹɺۭಎͷ࠷

ਤ 18 ϐϧϘοΫε‫ۭܕ‬ಎ಺ͷ‫ݻ‬༗Ϟʔυ TM010
ͷి৔ E z ɺ‫ͼٴ‬ɺ࣓৔ H θ ͷ‫͞ڧ‬ͷಈ‫޲ํܘ‬෼෍ɻ͜ΕΒͷ৔͸ z ΍ θ ʹ͸ґଘ͠ͳ͍ɻj01 ≈ 2.4048
͸ 0 ࣍ͷϕοηϧؔ਺ͷ 1 ൪໨ͷθϩ఺ɻॎ࣠ͷεέʔϧ͸೚ҙͰ͋Δɻ΋‫ج‬ຊతͳύϥϝʔλͰ͋Δɻ࣮ࡍͷۭಎͰͦΕΒͷྔΛଌఆ͢Δํ๏͸༷ʑ͋Γɺࢀߟจ‫[ ݙ‬3] ౳Ͱֶ΂Δɻࣜ (252) ΑΓɺೖྗϚΠΫϩ೾ΛࢭΊͨॠؒҎ߱ͷి࣓৔ΤωϧΪʔ U ͷ࣌ؒมԽ͸ɺ͚ͨ͜ͱʹΑΔۭಎ಺ి࣓৔ʢಛʹՃ଎Ϟʔυʣ΁ͷ

U = U0 eӨ‫ڹ‬͸খ͘͞ͳΔΑ͏ʹ͢Δɻ
͜͜Ͱɺ৽ͨͳ Q ஋Λఆٛ͢Δɻ͋Δ‫ݻ‬༗Ϟʔυ− Qω t(256)

L

ͱͳΔ͜ͱ͕Θ͔Δɻి࣓৔ͷৼ෯͸ɺΤωϧΪʔΛ͋Δఔ౓ྭৼͨ͠ঢ়ଶͰɺೖྗ೾ΛࢭΊͨͱ͢Δɻͷฏํࠜʹൺྫ͢ΔͷͰɺͦͷॠؒɺۭಎน໘ଛࣦిྗ Pwall ͱۭಎ͔Βಋ೾؅

ω
− 2Q
t

(257)

− Tt
f

(258)

ʹग़ͯߦ͘ϚΠΫϩ೾ͷిྗ Pextʢਤ 20 ࢀরʣͷ࿨|E|, |H| ∝ e͸ɺۭಎ಺ి࣓৔ΤωϧΪʔ U ͷ࣌ؒมԽʹ౳͍͠

= e

ͷͰɺ

L

ʹैͬͯࢦ਺ؔ਺తʹ‫ݮ‬ਰ͢Δɻ͜͜ͰɺdU= − (Pwall + Pext )
dt

(249)

Tf =

ͱ͍͏ؔ܎͕੒Γཱͭɻ͜͜Ͱɺ֎෦ Q ஋ʢQext ʣΛQext = ω

U
Pext

2QL
ω

(259)

Λ Filling Time ͱ‫͍ݴ‬ɺೖྗϚΠΫϩ೾ΛࢭΊͨॠ͔ؒΒ Tf ͚͕ͩ࣌ؒ‫ۭͱͭܦ‬ಎ಺ి࣓৔ͷৼ෯͸

(250)

1/e ≈ 0.37 ഒʹ·Ͱ‫ݮ‬ਰ͢Δɻ͜Ε͸ɺϚΠΫϩ೾

ͱఆٛ͢Δɻࣜ (248)ɺ(250) Λࣜ (249) ʹ୅ೖͯ͠ɺ

Λ౤ೖۭͯ͠ಎ಺‫ݻ‬༗ϞʔυΛ࠷ߴৼ෯·Ͱྭৼ͢

(
)
1
dU
1
= −ωU
+
dt
Q0
Qext
ω
U
=−
QL

Δͷʹཁ͢Δ࣌ؒͷ໨҆Ͱ΋͋Δʢೖࣹ೾ͷిྗ͕(251)Ұఆͷ৔߹ʣɻਤ 22 ͸ɺ509 MHz ͷৗ఻ಋՃ଎ۭ
ಎʹେిྗϚΠΫϩ೾Λ౤ೖͯ͠ӡస͍ͯͨ࣌͠ʹ(252)

RF εΠονΛखಈͰ੾ͬͨॠؒͷ৴߸Ͱ͋ΔɻͦΛಘΔɻ͜͜ͰɺQL Λ֎෦ Q ஋ɺ·ͨ͸ɺෛՙ Q͜Ͱ͸ɺ509 MHz ͷ৴߸ΛΦγϩείʔϓͰ௚઀‫؍‬஋ͱ‫͍ݴ‬ɺଌͨ͠ɻ͜ͷۭಎͷՃ଎ϞʔυʢTM010 ʣͷ Q0 ஋͸

1
1
1
=
+
QL
Q0
Qext

໿̏ສͰɺ݁߹౓͸ β ≈ 1.3 Ͱ͋ͬͨͷͰɺFilling(253)

time ͸ɺࣜ (259) ΑΓɺͰఆٛ͢Δɻ͞Βʹɺ݁߹౓ β Λ

Q0
β=
Qext

Tf ≈

30000
2 × 1+1.3

2π × 509000000≈ 8.1 s

(254)
20

１－ 20

(260)
(261)

(1,4,0)(2,3,3)(3,3,2) (1,1,5)(0,3,4)(2,2,4) (0,1,5)(1,4,1)(4,2,3)(3,3,1)(4,1,4)(1,3,3)(3,3,0) (1,2,4)(0,4,2)(2,3,2)(3,2,3)(3,1,4)(4,2,2)(1,2,3)(0,1,4)(0,2,4)(2,1,4) (0,4,1)(2,3,1) (0,3,3)(2,2,3) (0,4,0)(2,3,0)(1,1,4)(1,3,2)(4,2,1)(3,2,2)(4,1,3)(4,2,0)(1,3,1)(3,1,3)(0,3,2)(3,2,1) (1,3,0)(2,2,2)(0,2,3)(3,2,0)(2,1,3)(4,1,2)(0,3,1)(1,1,3)(1,2,2)(2,2,1)(0,3,0)(0,1,3)(0,1,5)(3,3,3)(2,1,5)(1,3,4)(4,3,2)(1,1,5)(0,4,1)(3,2,4)(2,4,1)(0,1,4)(0,3,2)(1,2,4)(1,4,1)(1,3,3)(0,3,3)(4,3,1)(0,2,4)(1,4,2)(2,3,3)(2,2,4)(3,3,2)(4,2,3)(4,1,4)(3,1,4)(3,3,1)(0,2,3)(1,1,4)(3,2,3)(2,3,2)(2,1,4)(4,2,2)(1,2,3)(2,2,3) (0,3,1)(2,3,1)(1,3,2)(4,1,3)(3,2,2)(4,2,1)(3,1,3)(0,1,3)(1,3,1)(0,2,2)(2,1,3)(2,2,2)(3,2,1)(1,1,3)(1,2,2)(0,2,1)(2,2,1)

１－ 21(2,2,0)(3,1,2)(4,1,2)(0,1,2)(0,1,1)(1,1,2)(1,2,1)(2,1,2)(3,1,2)(4,1,1)(3,1,1)(2,1,1)(1,1,1)

21

3

Resonance Frequency [GHz]2.5

3Resonance Frequency [GHz]

2.5

2

1.5

(0,2,2)(1,2,1)(2,1,2)(1,2,0)(1,1,2)(0,2,1)(0,1,2)(0,2,0)TE(4,1,1)(4,1,0)(3,1,1)(2,1,1)(1,1,1)

4(2,1,0)(1,1,0)(0,1,1)

2

1

0.5

0

(p = 1 ~ 6)

14

(3,1,0)

4

2
1.5

1
0.5
0

(0,1,0)2(p = 0 ~ 6)

20 Copper Pillbox with= 22.56 cm
18 R
d = 26.00 cm(m = 0 ~ 4)16(n = 1 ~ 5)

Q0 /10000

14

Modes
20 Copper Pillbox with mnp
22.56 cm
18 R=
d = 26.00 cm(m = 0 ~ 4)16(n = 1 ~ 5)

Q0 /10000

TMmnp Modes

12

10

8

6

12

10

8

6

ਤ 19 ಔ੡ϐϧϘοΫε‫ۭܕ‬ಎʢ൒‫ ܘ‬22.56 cmɺ௕͞ 26.00 cmʣͷ Q0 ཧ࿦஋ʢॎ࣠ʣͱ‫ݻ‬༗Ϟʔυप೾਺ʢԣ࣠ʣɻಔͷి‫ؾ‬఍߅཰ͱͯ͠͸ IACS100%ʢ1.72 × 10−8 Ω·mʣΛ༻͍ͨɻ·ͨɺͦͷಁ࣓཰͸ਅۭதͱಉ͡ͱͨ͠ʢµ = µ0 = 1.25664 × 10−6 H/mʣɻ

Ch.1: Reflected Wave from the Cavity Ch.2: Input Wave to the Cavity
Ch.4: Pickup Wave from the Cavity
Ch.3: Control Voltage (Modulator)

空洞

2 µs

導波管

𝑈𝑈𝑈𝑈

𝑃𝑃𝑃𝑃ext

𝑃𝑃𝑃𝑃wallਤ 20 ۭಎ಺ి࣓৔Λྭৼ͢ΔͨΊʹۭಎଆ໘ʹಋ೾؅Λ෇͚ͨঢ়ଶͷ໛ࣜਤɻਤ 22 509 MHz ͷৗ఻ಋՃ଎ۭಎͷେిྗӡస࣌ʹ RF εΠονΛखಈͰ੾ͬͨॠؒͷ֤छ৴߸ɻ
੺৭ഁઢ͸ɺՃ଎Ϟʔυి࣓৔ͷϐοϓΞοϓ‫ݕ‬೾৴߸ʢCh. 4ʣͷแབྷઢɻE (ω )
ਤ 21 ϧʔϓʢࠨʣͱΞϯςφʢӈʣͷࣸਅɻϧʔϓɺ‫ͼٴ‬ɺΞϯςφ͸ͦΕͧΕۭಎ಺ి࣓৔ͷ࣓৔ɺ‫ͼٴ‬ɺి৔ʹ݁߹͢Δɻ͜ΕΒ͸௿ύϫʔͰͷQ ஋ଌఆʹ࢖༻ͨ͠ηϛϦδοτಉ࣠έʔϒϧͷઌ୺෦Ͱ͋Δɻ𝜞𝜞𝜞𝜞 =

𝝎𝝎𝝎𝝎𝟎𝟎𝟎𝟎𝑸𝑸𝑸𝑸𝐋𝐋𝐋𝐋

Ͱ͋Δɻਤ 22 Λ‫ͯݟ‬ΈΔͱɺՃ଎Ϟʔυͷϐοϓ
Ξοϓ‫ݕ‬೾৴߸ʢCh. 4ʣͷ‫ݮ‬ਰ࣌ؒ͸໿ 8

ω0

s Ͱ͋

Γɺࣜ (261) ͷ Filling time ʹ౳͍͠ɻਤ 23

͜Ε·Ͱ͸ۭಎͷ‫ৼڞ‬प೾਺্Ͱߟ͖͕͑ͯͨɺ͜

‫ܕৼڞ‬εϖΫτϧɻ͜Ͱ͸ɺप೾਺͕‫ৼڞ‬प೾਺͔Βগͣ͠Εͨ৔߹ʹిʹͳΔɻ͜ͷ࣌ɺਤ 23 ͷ͋ΔΑ͏ʹɺ࣓৔ͷৼ෯͕ͲͷΑ͏ʹͳΔ͔Λ‫ͯݟ‬ΈΔɻ‫ৼڞ‬प೾਺͕ ω0 ͷ‫ݻ‬༗ϞʔυΛߟ͑Δͱɺࣜ (257) ΑΓɺE(t) = E 0 eω− 2Q0 tLe−iω0 tΓ=(262)εϖΫτϧ͸ΑΓγϟʔϓʹͳΔɻ∫ ∞6.3 ۭಎిѹ1
dt E(t) eiωt
(263)
2π −∞
∫
)
(
ω
E0 ∞
− 2Q0 +i(ω0 −ω) t
L
dt e
(264)
=
2π 0
ω0
E 0 2QL − i(ω0 − ω)
(265)
=
(
)
2π ω0 22
+ (ω0 − ω)2QLਤ 24 ʹ͋ΔΑ͏ʹɺz ্࣠ʹࡉ͍ϏʔϜύΠϓΛ
෇͚ͨͱ͠Α͏ɻͦͷύΠϓͷଠ͞͸ۭಎͷ‫ͱܘ‬ൺ΂ͯඇৗʹখ͍͞ͱ͠ɺۭಎ಺ి࣓৔ͱͯ͠͸લʹ‫ٻ‬ΊͨϐϧϘοΫε‫ۭܕ‬ಎͱಉ͡ղ͕࢖͑Δͱ͢Δɻۭಎ͕ z = 0 ∼ d ͷؒʹ͋ΔΑ͏ʹϏʔϜύΠϓͷத৺্࣠ʹ z ࣠Λஔ͘ɻͦ͜ʹɺz < 0 ͔Β +zͱͳΓɺి৔‫౓ڧ‬͸‫ܕৼڞ‬εϖΫτϧɿ|E(ω)| ∝ (

ω0
2QL

)2

1
+ (ω0

(267)

͕൒஋෯ͱͳΓɺQL ͕େ͖͚Ε͹େ͖͍ఔɺ‫ܕৼڞ‬ͱͳΓɺ͜ͷϑʔϦΤม‫׵‬͸ɺE(ω) =ω0QLͷํ޲ʹ޲͔ͬͯ z ্࣠ΛߴΤωϧΪʔͷిࢠʢిՙɿe(> 0)ʣ͕ޫ଎ c0 Ͱ૸Γൈ͚Δ࣌ɺͦͷిࢠ͕

(266)
− ω)2

֯प೾਺ ω ͷ‫ݻ‬༗Ϟʔυͷి৔͔Βड͚ΔऔΔΤω

22

１－ 22

𝑧𝑧𝑧𝑧 = 0𝑧𝑧𝑧𝑧 = 𝑑𝑑𝑑𝑑

ℑ

Vc

f

φ

ℜ

𝑑𝑑𝑑𝑑

ℜ{Vc }

ਤ 24 ϐϧϘοΫε‫ۭܕ‬ಎʹࡉ͍ϏʔϜύΠϓΛ
औΓ෇͚ͨঢ়ଶɻ

= Vc cos φਤ 25ϧΪʔ W ͸ɺ

W =

∫ d
0

= eℜ

dz e ℜ {E z (0, 0, z, t0 + z/c0 )}
{∫
d

(268)
}

ͳ͍͜ͱΛূ໌͠Α͏ɻԁ౵࠲ඪ‫ܥ‬Λ࢖͍ɺẼ z (r, z)͸ z ʹؔ͢ΔϑʔϦΤม‫Ͱ׵‬දΘ͢ɿdz Ẽ z (0, 0, z) e−iω(t0 +z/c0 ) (269)0= e ℜ {V c }= e |V c | cos ϕẼ z (r, z) =

(270)
(271)

∫ ∞

(273)

͜ͷࣜΛࣜ (272) ʹ୅ೖ͢Δͱɺۭಎిѹ͸ɺ
V c (r) = e

−iωt0

∫ ∞

dz

−∞

−iω cz0

dk Ẽ z (r, k)eikz−∞

ͱॻ͚Δɻ͜͜Ͱɺ

∫ d

ۭಎిѹɻ

∫ ∞

−∞

dk Ẽ z (r, k) e()
i k− ω zc0(274)ԽΛදΘ͢ɻ·ͨɺẼ z ͸ E z ͔Β࣌ؒґଘੑΛআ͍

ͱͳΔɻ͜͜Ͱɺz ʹؔ͢Δੵ෼ൣғ͕ແ‫ͬͳʹݶ‬ͨ͜ͱʹ஫ҙ͢Δɻz ʹؔ͢Δੵ෼Λઌʹߦ͏ͱɺͦ
ͷੵ෼͸σΟϥοΫͷσϧλؔ਺ δ ͱͳΓɺ
(
)
∫ ∞
ω
−iωt0
V c (r) = e
2π
dk Ẽ z (r, k) δ k −
(275)
c0
−∞

ͨ΋ͷͰɺ࣌ࠁ t = t0 ʹిࢠ͕ z = 0 Λ௨աͨ͠ͱ

Ͱ͋ΔɻҰํɺి৔ɿ

V c = e−iωt0dz Ẽ z (0, 0, z) e

(272)

0

͸ۭಎిѹͱ‫͏ݴ‬ෳૉྔͰ͋Δʢਤ 25 ࢀরʣɻҼࢠze−iω c0 ͸ɺిࢠ͕ۭಎ಺Λ௨ա͢Δ్தͷి৔ͷมͨ͠ɻϕ = 0 ͱͳΔλΠϛϯάͰిࢠ͕ۭಎʹೖΕ

E z (r, z, t) = Ẽ z (r, z) e−iωt
(276)
∫ ∞
=
dk Ẽ z (r, k)ei(kz−ωt) (277)͹ɺW ͸࠷େͱͳΔɻۭಎిѹͷઈର஋ |V c | ͸ ϕ= 0 ʹରԠ͠ɺʢλΠϛϯάతʹʣ࠷େՃ଎ͷిѹͰ−∞͋Δɻ·ͨɺՃ଎ۭಎ͕ෳ਺୆͋Δ৔߹ʹ͸ɺۭಎి͕ਅۭதͷϚΫε΢Σϧํఔࣜ (35) Λຬͨ͢͜ͱ

ѹͷϕΫτϧ࿨ΛͱΔɻঘɺՃ଎΍‫ݮ‬଎ʹؔ܎͢ΔΑΓɺ‫ݻ‬༗Ϟʔυ͕ෳ਺͋Δ৔߹͸ɺͦΕͧΕͷϞʔυ͔

(2⃗ 2 − k2 + ω∇tc20

Βͷ‫د‬༩ͷ࿨ͱͳΔɻ࣮ࡍͷϏʔϜύΠϓ‫ܘ‬͸Ճ଎ۭಎͷ‫ʹܘ‬ൺ΂ͯඇ

)Ẽ z (r, k) = 0(278)

͕ಘΒΕΔɻࣜ (275)ɺ(278) ΑΓɺৗʹখ͍͞ͱ͍͏͜ͱ͸ɺ·ͣͳ͍ɻϏʔϜύΠϓ‫͕ܘ‬େ͖͍ͱɺۭಎ಺ి࣓৔͕ϏʔϜύΠϓʹછΈ

⃗ 2 V c (r) = e−iωt0 2π∇tग़͠ɺిࢠ͕ϏʔϜύΠϓͷதͷͲ͜Λ௨͔ͬͨͰۭ

−iωt0ಎిѹ͕มΘΓͦ͏Ͱ͋Δɻ͔͠͠ɺϏʔϓύΠϓ=eͱۭಎ͕࣠ରশͳΒ͹ɺۭಎిѹ͸௨ΔҐஔʹґଘ͠

=0

2π

∫ ∞

−∞

23

１－ 23

dk

∫ ∞

−∞

(

2

2

⃗ Ẽ (r, k) δ
dk ∇
t
z

k −

ω2
c20

)

Ẽ z (r, k) δ

(

k−

(

k−

ω
c0
ω
c0

)

)

(279)
(280)
(281)

ͭ·Γɺۭಎిѹ͸ௐ࿨ؔ਺Ͱ͋Δɻௐ࿨ؔ਺ͷੑΑΓ‫·ٻ‬Δ͜ͱΛ࢖͏ͱɺ�∫�
−iω cz �2
2� d
√
0 �
(0)
dze
J
�
0
0
2σ
Rsh = 2(287){
( 01 )}2∫ωµdS c0 ϵ0 J0′ jR
r
ۭಎ಺໘
√
c2
4 ω02 sin2 2cω0 d
2σ
)
(
(288)
=2
ωµ ∫ + ∫ dS ��c0 ϵ0 J1 ( j01 r)��2
R
୺൘
ଆ໘
√
2σ
1
8 ωµ
sin2 2cω0 d
ω 2 ϵ20=(289)( 01 )2∫R
4π 0 dr rJ1 jR
r + 2πRdJ1 (j01 )2࣭ΑΓɺ࣠ରশੑ͕͋Ε͹ɺ೚ҙͷҐஔ r ʹରͯ͠ɺV c (r) = V c (0)(282)͕੒Γཱͪɺۭಎిѹ͕௨աҐஔʹґଘ͠ͳ͍͜ͱ
͕Θ͔Δɻ͜Ε͸਺ֶతͳ‫͋Ͱ݁ؼ‬Γɺ࣠ରশͰ͋
Ε͹ɺͲͷΑ͏ͳ‫ܗ‬ঢ়Ͱ΋੒ΓཱͭʢϏʔϜύΠϓͷଠ͞΍ۭಎͷ௕͞ d ͕ͲͷΑ͏ͳ஋Ͱ΋ʣɻ௚‫؍‬తͱͳΔɻ͜͜ͰɺJ0 (0) = 1ɺJ0′ (x) = −J1 (x) Λ࢖ͬʹड͚ೖΕ೉͍͜ͱ͕ͩɺୈ II ෦Ͱ۩ମతͳγϛϡͨɻ্ͷࣜ (289) ͷ෼ࢠ͕࠷େͱͳΔͷ͸ɺ 2cω0 d =Ϩʔγϣϯͷ݁ՌΛ‫ͯݟ‬ɺཧղͯ͠ΈΑ͏ɻ

π/2 ͷ࣌ɺͭ·Γɺۭಎ௕ d ͕ TM010 Ϟʔυͷࣗ༝

6.4 γϟϯτΠϯϐʔμϯε0πʣͰۭ͋ؒʹ͓͚Δ೾௕ͷ൒෼ʹ౳͍࣌͠ʢd = cωۭಎిѹͷ 2 ৐Λน໘ଛࣦిྗͰׂͬͨൺΛγϟΔɻ͜Ε͸ɺ൒प‫࠷Ͱظ‬େͷՃ଎Λड͚Δ͜ͱʹ૬ϯτΠϯϐʔμϯεʢRsh ʣͱ͍͏ɿ

౰͢Δ͕ɺγϟϯτΠϯϐʔμϯε͕͜ͷ࣌ʹ࠷େͱͳΔ༁Ͱ͸ͳ͍ɻͳͥͳΒɺࣜ (289) ͷ෼฼ୈ 2

Rsh =|V c |2
Pwall߲ʢۭಎଆ໘ʹ͓͚Δน໘ଛࣦిྗʣ͕ۭಎ௕ d ʹൺ(283)

ྫ͢Δ͔ΒͰ͋Δɻ࣮ࡍ͸ɺۭಎ௕ d ͕ࣗ༝ۭؒʹ
͓͚Δ೾௕ͷ໿ 4 ׂ‫ڧ‬ͷͱ͜ΖͰγϟϯτΠϯϐʔγϟϯτΠϯϐʔμϯε΋ɺ֤छ Q ஋ͱಉ͘͡ɺۭ

μϯε͕࠷େͱͳΔɻϐϧϘοΫε‫ۭܕ‬ಎͷۭಎ௕ಎͷ֤ TM Ϟʔυ͝ͱʹܾΊΒΕΔɻ৔߹ʹΑͬͯ͸͜ͷΑ͏ʹܾͯ͠·͍ͬͯΔ*9 ɻ

͸ɺRsh = |V c |2 /2/Pwall ͱఆٛ͢Δ͜ͱ΋͋Δ͕ɺ

࣍ʹɺʢͦΕͧΕͷ TM Ϟʔυʹؔͯ͠ʣγϟϯՃ଎‫Ͱث‬͸೾ߴ஋ʢϐʔΫ஋ʣʹॏ఺Λ͓͘͜ͱ͕ଟ

ʢʮΞʔϧɾΦʔτΠϯϐʔμϯεͱ Q0 ஋ͷൺ RQsh0͍ͷͰɺຊॻͰ΋ࣜ (283) ͷఆٛΛ࠾༻͢Δɻ੔߹όʔɾΩϡʔʯͱ‫͏ݴ‬ʣΛߟ͑ΔɻQ0 ͷఆٛࣜ (248)͕ͱΕ͍ͯΔ৔߹ʢβ ≈ 1ʣɺۭಎ΁౤ೖ͢ΔϚΠΫͱۭಎిѹͷఆٛࣜ (272) ΑΓɺϩ೾ͷೖྗిྗΛ Pinp ͱ͢Δͱɺ্ࣜΑΓɺ√
Rsh Pwall√≈ Rsh Pinp

|V c | =

(284)
(285)

Ͱ͋Δɻͭ·ΓɺγϟϯτΠϯϐʔμϯε͕ߴ͍ఔɺ
খ͍͞ೖྗిྗͰߴ͍ۭಎిѹ͕ಘΒΕɺ͜ͷిѹ͕͸ۭηϧͯ͠ফ͑ͨ͜ͱ͔Β΋Θ͔ΔΑ͏ʹɺ RQsh
0∫∫·ͨɺω ∝ L−1 ɺ dV ∝ L3 ɺ dz ∝ L ͳͷͰɺࣜμϯε͸ɺࣜ (243)ɺ(272) Λ্ࣜʹ୅ೖ͠ɺ·ͨɺน(291) ͸ L ʹґଘ͠ͳ͍ɻͭ·Γɺۭಎͷେ͖͕͞໘ଛࣦిྗ͕ۭಎ಺໘ͷۚଐͷి‫఻ؾ‬ಋ཰ σ ͱಁ࣓L ഒʹͳͬͯ΋ɺۭಎͷ‫͕ܗ‬૬ࣅ‫ͳܗ‬Β͹ɺରԠ͢Δ཰ µ Λ༻͍ͯҎԼͷۭಎ಺໘্ͷ໘ੵ෼ɿ∫

ͱॻ͚Δɻน໘ଛࣦిྗ Pwall ͕෼฼෼ࢠͰΩϟϯωϧΪʔີ౓ʹൺྫ͢ΔͷͰɺ|Ẽ| ∝ L−3/2 Ͱ͋Δɻ

ͷ TM010 Ϟʔυͷ৔߹ɺz ্࣠ͷγϟϯτΠϯϐʔ

ωµ
2σ

(291)

ͷ୅දతͳαΠζΛ L ͱ͢Δͱɺ|Ẽ|2 ͸ి࣓৔ͷΤ଎ޮ཰ʯΛදΘ͢͜ͱʹͳΔɻϐϧϘοΫε‫ۭܕ‬ಎ

√

(290)

ಎ಺ද໘ͷ఍߅஋΍෺Թʹґଘ͠ͳ͍ɻ·ͨɺۭಎՃ଎ిѹͰ͋Ε͋͹ɺγϟϯτΠϯϐʔμϯε͸ʮՃ

1
Pwall =
2Rsh
|V c |2
=
Q0
ωU
�∫
�2
�
−iω cz �0 �
� dz Ẽ z (0, 0, z) e
=
�
�2
∫
�
�
ωϵ0
dV
Ẽ
(x,
y,
z)
�
�
z
2
ۭಎ಺

͸ಉ͡Ͱ͋Δɻैͬͯɺ RQsh͸ۭ‫ݻ‬༗Ϟʔυͷ RQsh00ಎͷ‫ز‬Կֶత‫ܗ‬ঢ়ͷΈͰܾ·Δۭಎ‫ݻ‬༗ͷྔͰ͋Γɺ

ۭಎ಺໘

⃗ 2
dS |H|(286)*9 Ұํɺۭಎͷ಺‫ܘ‬͸ɺप೾਺ʹΑܾͬͯ·Δɻ

24

１－ 24

ۭಎͷ‫ܗ‬ঢ়ʹؔ܎͢Δి‫ؾ‬తੑ࣭Λಛ௃͚ͮΔࢦඪͷ‫ܗ‬ঢ়Λࣗ༝ʹબ΂Δ͜ͱ͕ɺ͜ͷํ๏ͷେ͖ͳಛͱͯ͠͠͹͠͹࢖ΘΕΔɻͪͳΈʹɺϐϧϘοΫε

௃Ͱ͋Δɻ͜ΕʹΑΓɺෳࡶͳ‫ڥ‬ք৚݅Λਖ਼֬ʹදͷ஋͸ɺ100ʙ200 Ω ఔ‫ܕ‬୯ηϧۭಎͷయ‫ܕ‬తͳ RQsh0‫ݱ‬ग़དྷΔɻୠ͠ɺཁૉͷ࡞੒ʹ͕͔͔࣌ؒΔͷ͕໰

౓Ͱ͋Δɻ

୊Ͱ͋Δɻঘɺ‫ڥ‬ք৚݅ʹద߹ͤͯ͞ཁૉΛ࡞੒͢

Rsh
Q0ʹؔͯ͠΋͏ͻͱͭॏཁͳ͜ͱ͕͋Δɻࣜ

ΔͨΊ͔ɺʢՃ଎‫Ͱث‬Α͋͘Δʣՙిཻࢠͱͷ૬‫࡞ޓ‬(291) ΛΑ͘‫ͯݟ‬ΈΔͱɺʮத৺࣠ʢϏʔϜ࣠ʣ্ͷ༻Λ༗‫ݶ‬ཁૉ๏Ͱ‫͢ࢉܭ‬Δ͜ͱ͸؆୯Ͱ͸ͳ͍Α͏ి৔ͷશి৔ʹର͢Δׂ߹ʯͱ͍͏‫͍ͯͬͳʹܗ‬Δɻ

Ͱ͋ΔɻΑͬͯɺ͜ͷղ๏Λ࢖ͬͨγϛϡϨʔγϣ

͸ɺͦ
͜Ε͸ɺ
ʢۭಎΛՃ଎ۭಎͱͯ͠࢖͏৔߹ʣRQsh
0ϯιϑτͰ͸ɺϏʔϜྭৼΛѻ͑ͳ͍΋ͷ͕ଟ͍ʢશ

ͷ‫ݻ‬༗ϞʔυͱϏʔϜͱͷؒͰ΍ΓऔΓ͢ΔΤωϧ͘ແ͍͜ͱ͸ͳ͍ʣɻΪʔྔΛදΘ͍ͯ͠Δ͜ͱΛҙຯ͢Δɻྫ͑͹ɺՃ

8.2 ‫ڥ‬քཁૉ๏Λߴ͘͢ΔͱɺՃ଎ࣗମʹ͸༗རͩ଎Ϟʔυͷ RQsh0‫ڥ‬քཁૉ๏ʢBoundary Element Methodʣ͸ɺϚ͕ɺϏʔϜ͔Βͷ൓࡞༻͕େ͖͘ͳΔɻ

Ϋε΢Σϧํఔࣜͷੵ෼‫ܗ‬ʢ‫ڥ‬քੵ෼ํఔࣜʣʹ‫ͮج‬͖ɺͦͷ‫ڥ‬ք಺෦ͷి࣓৔Λ‫ٻ‬ΊΔํ๏Ͱ͋Δɻ‫ܭ‬ࢉ֨ࢠ͸‫ڥ‬քද໘ʹͷΈઃఆ͢Ε͹Α͍͜ͱ͕ɺ͜

ୈ II ෦

ͷํ๏ͷಛ௃Ͱ͋Δɻ·ͨɺ͜ͷํ๏Λ࢖͏ͱɺແ‫ݶ‬ϚΠΫϩ೾γϛϡϨʔγϣϯྖҬΛ‫ʹີݫ‬ѻ͏͜ͱ͕ՄೳͰ͋Δɻ͜ͷΑ͏ʹཧ࿦తʹ͸༏Ε͍ͯΔํ๏͕ͩɺ࣮ࡍͷ਺஋‫ʹࢉܭ‬͸͕࣌ؒʢଞͷํ๏ͱൺ΂ͯʣ͔͔Δ͜ͱ͕ଟ͘ɺՃ଎
‫ۀث‬քͰ࢖ΘΕ͍ͯΔγϛϡϨʔγϣϯιϑτͰ͜

7 ͸͡Ίʹ

ͷํ๏Λ࣮૷ͨ͠΋ͷ͸গͳ͍ɻ

8.3 ༗‫ࠩݶ‬෼๏ۙ೥ͷίϯϐϡʔλؔ࿈ٕज़ͷਐาʹΑΓɺϚΠΫϩ೾Λ‫ؚ‬Ή༷ʑͳి࣓৔ؔ࿈‫ݱ‬৅Λਫ਼ີʹ਺஋‫ܭ‬༗‫ࠩݶ‬෼๏ʢFinite Diﬀerence Methodʣ͸ɺϚΫࢉʢγϛϡϨʔγϣϯʣग़དྷΔΑ͏ʹͳ͖ͬͯͨɻ͜

ε΢Σϧํఔࣜͷඍ෼‫ܗ‬Λࠩ෼Ͱද‫͢ݱ‬Δํ๏Ͱ͋Ε͸ɺϚΫε΢Σϧํఔࣜʹैͬͨ‫͋Ͱࢉܭ‬Γɺ͍Θ

Δɻίϯϐϡʔλ͸ࠩ෼ʢҾ͖ࢉʣ͕ෆಘҙͳͷͰɺΏΔୈ̍‫ݪ‬ཧ‫͋Ͱࢉܭ‬Δɻ࣮ࡍɺୈ I ෦ʹ͋Δ಺༰͸ࠩ෼Λଟ਺ճߦ͏ͱ‫͕ࠩޡ‬஝ੵ͞ΕΔ͜ͱ͕͋Δɻ

શͯγϛϡϨʔγϣϯͰ਺஋‫ࢉܭ‬ՄೳͰ͋Δɻैͬ

͔͠͠ɺ‫Ͱࡏݱ‬͸ɺͦͷΑ͏ͳ‫ࠩޡ‬Λ཈͑Δํ๏͕

ͯɺਖ਼͍͠γϛϡϨʔγϣϯΛߦ͑͹ɺੲͷਓ͕Ճ଎͋Γɺ·ͨɺίϯϐϡʔλ͕୯ਫ਼౓ʢ໿ 7 ܻʣ͔Βഒۭಎͷ‫ڀݚ‬։ൃ౳Ͱߦ͍ͬͯͨ௿ిྗϞσϧʢίʔਫ਼౓ʢ໿ 16 ܻʣ΁Ҡߦͨ͜͠ͱ΋͋Γɺ༗‫ࠩݶ‬෼๏ϧυϞσϧͱ΋‫͏ݴ‬ʣͷ࡞੒΍ଌఆ͸ɺ‫ج‬ຊతʹෆཁ͸޿͘࢖ΘΕ͍ͯΔɻಛʹɺϏʔϜྭৼͷి࣓৔Λ

Ͱ͋Δɻୠ͠ɺͦΕ͸γϛϡϨʔγϣϯ͕ਖ਼͚͠ΕγϛϡϨʔγϣϯ͢Δ৔߹͸ɺ࣌ؒྖҬ༗‫ࠩݶ‬෼๏͹ͷ͜ͱͰ͋ΔɻຊઅͰ͸ɺϚΠΫϩ೾γϛϡϨʔ

ʢFinite-Diﬀerence Time-Domain method: FDTDʣγϣϯͷ‫͍ͯͭʹૅج‬ղઆ͢Δɻ

͕༗༻Ͱ͋Δɻ

8.4 ༗‫ੵݶ‬෼๏

8 ‫ํࢉܭ‬๏ͷछྨ༗‫ੵݶ‬෼๏ʢFinite Integration Technique: FITʣϚΫε΢ΣϧํఔࣜΛ਺஋తʹղ͘ํ๏͸ෳ਺͋͸ɺϚΫε΢Σϧํఔࣜͷੵ෼‫ܗ‬Λ཭ࢄԽ͠ɺάϦοΔɻ͍ͣΕͷํ๏Ͱ΋ɺۭؒ΍࣌ؒΛ཭ࢄԽ͢Δɻ࣮υԽ͞Ε্ۭͨؒͷࢧ഑ํఔࣜΛߏ੒ɺͦΕΛ਺஋ࡍʹγϛϡϨʔγϣϯΛߦ͏্Ͱղ๏ͷৄࡉ͸ͦΕతʹղ͘ํ๏Ͱ͋Δɻ௚ަάϦου্Ͱ͸ɺFIT ͱఔॏཁͰ͸ͳ͍ͷͰɺ͜͜Ͱ͸ɺ୅දతͳղ๏ͷ֓ཁ

FDTD ͸౳ՁͰ͋ΔɻFIT ͸ɺ੩తͳ৔͔Βߴप೾ͷΈղઆ͢Δɻ·Ͱɺͦͯ͠ɺप೾਺ྖҬͱ࣌ؒྖҬͷ྆ํʹద༻ग़

8.1 ༗‫ݶ‬ཁૉ๏

དྷΔԠ༻ੑͷ޿͍ํ๏Ͱ͋Δɻ༗‫ݶ‬ཁૉ๏ʢFinite Element MethodʣͰ͸ɺ‫ࢉܭ‬ྖҬΛখ͞ͳྖҬʢཁૉʣʹ෼ׂ͢Δɻͦͷࡍɺཁૉ

25

１－ 25
２－ 1

ϦΤม‫ͨ͠׵‬΋ͷͳͷͰɺप೾਺ͱ࣌ؒͷؒʹෆ֬ఆ

9 ‫ྖࢉܭ‬Ҭͷछྨੑ͕͋Δɻྫ͑͹ɺ࣌ؒྖҬͰ t = 0 ͔Β t = tmax·Ͱ‫ࢉܭ‬Λͨ͠ͱ͢Δͱɺͦͷ݁ՌΛϑʔϦΤม‫׵‬௨ৗΑ͘‫͚͔ݟ‬ΔϚΫε΢Σϧํఔࣜɿ⃗⃗ x, t) + ∂ B(⃗x, t) = 0 ɹ⃗ × E(⃗∇∂t⃗ x, t)∂D(⃗⃗ x, t) −⃗ × H(⃗=0∇∂t⃗ x, t) = 0⃗ · D(⃗∇⃗ x, t) = 0⃗ · B(⃗∇

ͯ͠ಘΒΕΔεϖΫτϧͷղ૾౓ ∆f ͸ɺ

(292)
∆f ≈

(293)

Q ஋͸ɺQL = Q0 /2 = 15000 Ͱ͋Δɻࣜ (267) ΑΓɺ‫ܕৼڞ‬εϖΫτϧͷ൒஋෯͸ɺΓ ≈ 200 kHz ͱͳΔɻͭ·Γɺtmax ≈ 1/(200 kHz) ≈ 5 ϛϦඵΑΓ΢Σϧํఔ͔ࣜΒಋ͍ͨϔϧϜϗϧπํఔࣜɿ⃗ x, ω) = 0⃗ 2 + k(ω)2 E(⃗∇()⃗ x, ω) = 0⃗ 2 + k(ω)2 H(⃗
∇

(298)

͠Α͏ɻ੔߹͕ͱΕ͍ͯΔʢβ = 1ʣͱ͢Δͱɺෛՙ

(295)

ྖҬʯʹ͓͚Δද‫͋Ͱݱ‬ΔɻҰํɺୈ I ෦ͰϚΫε

)

tmax

ͱͳΔɻྫ͑͹ɺQ0 = 30000 ͷՃ଎ۭಎ͕͋ͬͨͱ

(294)

͸ɺి࣓৔ͷ࣌ؒൃలΛ‫ه‬ड़͢Δ΋ͷͰ͋Γɺʮ࣌ؒ(1΋௕͘‫͚ͳ͠ࢉܭ‬Ε͹ͳΒͳ͍ɻप೾਺ͷղ૾౓ΛN ഒྑ͘͢ΔͨΊʹ͸ɺ࣌ؒྖҬͷ‫ࢉܭ‬Λ N ഒ௕͘(296)

͢Δඞཁ͕͋Δɻͭ·Γɺ࣌ؒྖҬͷ‫ࢉܭ‬͸ɺ‫ڱ‬ଳҬ໰୊ʹ͸޲͍͓ͯΒͣɺ޿ଳҬ໰୊ʹద͍ͯ͠Δɻ

(297)

9.3 ‫ݻ‬༗Ϟʔυղੳ͸ɺϑʔϦΤม‫ͯ͠׵‬ɺ͋Δಛఆͷप೾਺ʹؔͯ͠‫ه‬ϚΫε΢Σϧํఔ͔ࣜΒಋ͍ͨϔϧϜϗϧπํఔड़ͨ͠ʮप೾਺ྖҬʯʹ͓͚Δද‫͋Ͱݱ‬ΔɻͲͪΒ΋਺ֶతʹ͸౳ՁͰ͋Δ͕ɺͦͷඳ૾͸ਖ਼൓ରͰ͋Δɻ͜͜Ͱ͸ɺ‫ྖࢉܭ‬Ҭʹ͍ͭͯ؆୯ʹղઆ͢Δɻʢେ‫ن‬໛ʣ‫ݻ‬༗஋ࣜ (74)ɺ(75) Λ‫ݻ‬༗ํఔࣜͱͯ͠ɺ
໰୊Λ਺஋తʹղ͘ํ๏Ͱ͋Δɻཧ࿦తʹ͸प೾਺ྖҬͰ͋Δ͕ɺྭৼ‫ݯ‬͸ແ͘ɺଘࡏՄೳͳి࣓৔Λ‫ٻ‬

9.1 प೾਺ྖҬΊΔͱ͍͏ʢ͋Δҙຯʣಛघͳ໰୊ͳͷͰɺຊॻͰ͸Ճ଎‫ث‬͸प೾਺ྖҬͷ૷ஔͱ΋‫͑ݴ‬ΔͷͰɺप೾प೾਺ྖҬͱ۠ผ͢Δɻ‫ݻ‬༗ϞʔυղੳͰඞཁͳͷ਺ྖҬͷ‫ࢉܭ‬͸ඞਢͰ͋Δɻಛʹɺߴ͍ Q ஋Λ࣋ͬ͸ɺϔϧϜϗϧπํఔࣜͱ‫׬‬શͳ‫ڥ‬ք৚݅ʢྫ͑͹ɺۭͨಎͰ͸ɺୈ I ෦ͷࣜ (266) Ͱࣔͨ͠Α͏ʹɺۭۚଐͰғ·ΕۭͨಎʣͷΈͰ͋Δɻ֎քͱϚΠΫϩಎ಺ి࣓৔ͷ‫౓ڧ‬͸Ӷ͍‫ܕৼڞ‬εϖΫτϧͱͳΔͷ೾ͷग़ೖΓ͕͋Δϙʔτ͕͋ͬͯ͸ͳΒͳ͍ɻैͬ

ͰɺҰ఺ͷप೾਺ʹ͓͚Δ‫͕ࢉܭ‬ॏཁͱͳΔɻͭ·ͯɺ͋Δϙʔτͷ֎෦ Q ஋Λ‫ݻ‬༗ϞʔυղੳͰ‫ٻ‬ΊΓɺप೾਺ྖҬͷ‫ࢉܭ‬͸ɺ‫ڱ‬ଳҬ໰୊ʹద͍ͯ͠Δɻ

Δ৔߹ʹ͸஫ҙ͕ඞཁͰ͋Δ*10 ɻप೾਺ྖҬͰ͸ɺϑʔϦΤม‫ʹ׵‬ΑΓ t = −∞ ͔

10 ιϑτ΢ΤΞͷྫʮ࣌ؒʯ͸ແ͍ɻैͬ
Β t = +∞ ·Ͱੵ෼ͨͨ͠Ίɺͯɺॳ‫ظ‬৚݅ͱ͍͏΋ͷ΋ͳ͘ɺి࣓৔ΛܾΊΔͨՃ଎‫ۀث‬քͰ࢖ΘΕ͍ͯΔి࣓৔γϛϡϨʔγϣΊʹɺྭৼ‫͕ݯ‬ඞཁͰ͋Δɻྭৼ‫࠷ͯ͠ͱݯ‬΋Α͋͘Δͷ͸ɺϙʔτϞʔυʢಉ࣠ઢ΍ಋ೾؅ͷϞʔυʣͰ͋Δɻϯ༻ιϑτ͸ଟ਺͋ΓɺશͯΛྻ‫͢ڍ‬Δͷ͸೉͍͠ͷͰɺ͜͜Ͱ͸ɺ୅දతͳ̏ͭʹ͍ͭͯ؆୯ʹղઆ͢Δɻ

9.2 ࣌ؒྖҬ10.1

࣌ؒྖҬͰ͸ɺి࣓৔ͷ࣌ؒൃలΛ௥͏ͷͰɺॳHFSS

HFSS [5] ͸ɺ༗‫ݶ‬ཁૉ๏ʢFEMʣʹ‫ࢉܭ͍ͨͮج‬‫ظ‬৚͕݅ඞཁͰ͋Δʢ͞Βʹɺྭৼ‫͕ݯ‬ՃΘΔ৔߹

Λߦ͏γϛϡϨʔγϣϯιϑτͰ͋Δɻߴ͍ਫ਼౓ͷ΋͋Δʣɻผ్‫ٻ‬Ίͨి࣓৔Λ t = 0 ʹ͓͚Δॳ‫ిظ‬࣓৔ͱ͠ɺͦͷ‫ޙ‬ͷ࣌ؒൃలΛ‫͢ࢉܭ‬Δ͜ͱ΋͋Δɻ
ྭৼ‫ͯ͠ͱݯ‬Α͋͘Δ͸ɺϙʔτϞʔυͱϏʔϜͰ‫͓͍ͯ޲ʹࢉܭ‬ΓɺՃ଎‫ۀث‬քͰ͸ੲ͔Βࠓ·Ͱɺଟ͘ͷϢʔβʔ͕͍Δɻୠ͠ɺϏʔϜྭৼ໰୊͸ѻ͑ͳ͍ɻ͋ΔɻϑʔϦΤม‫׵‬ͷؔ܎ʹ͋Δ̎ͭͷྔͷؒʹ͸ɺඞͣ*10 ਖ਼͘͠͸ Slater ͷಉௐ‫ۂ‬ઢ๏Λ࢖͏ɻৄ͘͠͸ɺࢀߟจ‫ݙ‬ෆ֬ఆੑ͕ଘࡏ͢Δɻप೾਺ྖҬ͸࣌ؒྖҬΛϑʔ[4] ͷୈ V ষࢀরɻ

26

２－ 2

10.2

CST STUDIO SUITECST STUDIO SUITE [6] ͸ɺ༗‫ੵݶ‬෼๏ʢFITʣʹ‫ࢉܭ͍ͨͮج‬Λߦ͏γϛϡϨʔγϣϯιϑτͰ͋Δɻ
໊લʹʮSuiteʯͱ͋Δ͜ͱ͔Β΋Θ͔ΔΑ͏ʹɺϏʔϜྭৼ໰୊΋‫ؚ‬Ίͯɺి࣓৔ؔ܎ͷ‫ࢉܭ‬ͷຆͲΛѻ͍ͬͯΔɻ࠷ۙͰ͸ɺϚΠΫϩ೾ʹΑΔൃ೤΍ɺͦͷൃ೤͔Βੜ͡ΔԠྗ౳΋࿈‫ࢉܭͯ͠ܞ‬ग़དྷΔΑ͏ʹͳͬͨɻஶऀͷ‫ݸ‬ਓతͳ‫ײ‬૝Ͱ͸ɺ࢖͍΍͢͞Λ༏ઌ͍ͯ͠ΔΑ͏Ͱ͋Γɺਫ਼౓ͷߴ͍‫ࢉܭ‬Λߦ͏৔߹ਤ 26͸ɺ஫ҙ͕ඞཁͱͳΔ͜ͱ͕͋Δɻ໪࿦ɺߴਫ਼౓ͷ‫ܭ‬CST STUDIO SUITE ͷ Modeling ‫ػ‬ೳɻࢉ͸ग़དྷΔ͕ɺଟগɺγϛϡϨʔγϣϯɾύϥϝʔλͷௐ੔͕ඞཁͳ͜ͱ΋͋Δɻ‫ۂ‬໘࢛໘ମϝογϡΛ

Λྫʹղઆ͢Δɻ࢖͑͹ɺෳࡶͳ‫ܗ‬ঢ়΋Αۙ͘ࣅग़དྷΔɻ

11.1 ιϦουͷ࡞੒

10.3

GdfidLҰൠతɺ‫ͭ׌‬ɺ؆қͳํ๏͸ɺਤ 26 ͷ੺৭ഁઢ࿮

ʮGitter
GdfidLʢδʔɾσΟʔɾϑΟυΡϧͱಡΉʀʹ͋Δ‫ػ‬ೳΛ࢖͏ํ๏Ͱ͋Δɻ௚ํମ΍౵ͳͲɺ‫ج‬drüber, fertig ist die Laubeʯͷུ‫ޠ‬ʣ[7] ͸ɺ࣌ؒྖຊతͳ‫ܗ‬ঢ়ͷιϦουͱɺͦͷՃ޻౳͕ߦ͑Δɻ2 ࣍Ҭ༗‫ࠩݶ‬෼๏ʢFDTDʣʹ‫ࢉܭ͍ͨͮج‬Λߦ͏γϛϡ‫ݩ‬ਤ‫͔ܗ‬Β 3 ࣍‫ݩ‬ιϦουΛ࡞Δ͜ͱ΋ՄೳͰ͋ΔɻϨʔγϣϯιϑτͰ͋Δɻϝογϡ͸̒໘ମͷΈͰߦͬͨશͯͷૢ࡞͸ɺਤ 26 ͷ྘৭࿮ʹ͋ΔʮHistory͋Γɺෳࡶͳ‫ܗ‬ঢ়Λѻ͏৔߹͸ϝογϡαΠζΛখListʯʹ࢒Δɻ͜ͷํ๏͸؆୯ɾखܰͰ͸͋Δ͕ɺෳ

͘͢͞Δඞཁ͕͋ΔɻGdfidL ͸ PC ΫϥελΛ࢖ͬࡶ‫ͭ׌‬େ‫ن‬໛ͳιϦουΛ࡞Δ࣌ʹ͸໰୊͕͋Δɻͨฒྻ‫ࢉܭ‬Λ૝ఆͯ͠։ൃ͞Ε͓ͯΓɺେ‫ن‬໛ͳ PC৚݅ʹΑͬͯ৔߹෼͚ͨ͠Γɺෳࡶͳ‫݁ࢉܭ‬ՌΛ൓ΫϥελΛ͍࣋ͬͯΕ͹ɺେ‫ن‬໛໰୊ΛγϛϡϨʔөͤͨ͞Γ͍ͨ͠৔߹͸ɺͦͷ౎౓࡞Γ௚͠Ͱ͋Δɻ

γϣϯ͢Δ͜ͱ͕ՄೳͰ͋ΔɻGdfidL ͕࠷΋ҖྗΛͦ͜ͰɺϚΫϩ‫ޠݴ‬Λ࢖͏ํ๏Λ঺հ͠Α͏ɻCSTൃ‫͢ش‬Δͷ͸ɺ΢ΣʔΫ৔‫ࢉܭ‬౳ͷϏʔϜྭৼ໰୊STUDIO SUITE Ͱ͸ VBA (Visual Basic for Appli-

Ͱ͋ΔɻʮϦελʔτ‫ػ‬ೳʯ͕͋Γɺ͋Δஈ֊ͰҰ୴

cations) ͕αϙʔτ͞Ε͍ͯΔɻਤ 27 ͷ੺৭໼ҹʹ‫ࢉܭ‬Λதஅ͠ɺ‫ޙ‬೔ɺதஅͨ࣌͠఺͔Β‫ࢉܭ‬Λ࠶։͋ΔϚΫϩΤσΟλʔΛ։͖ɺϏδϡΞϧϕʔγοग़དྷΔɻ͜ͷ‫ػ‬ೳ͸ɺඇৗʹ௕͍‫཭ڑ‬ʢྫ͑͹ɺԁ‫ܗ‬Ϋͷจ๏ͰιϦου౳Λ‫ه‬ड़͢ΔͷͰ͋Δɻྫ͑͹ɺՃ଎‫̍ث‬प෼Ҏ্ʣ·Ͱ΢ΣʔΫɾϙςϯγϟϧΛ

௚ํମιϦουΛ࡞Δͷ͸ɺ‫͍ͨ͠ࢉܭ‬৔߹ʹ༗ޮͰ͋ΔʢPC ΫϥελΛ࢖ͬͯ΋਺ि͔͔ؒΔ‫ࢉܭ‬ʣɻ·ͨɺGdfidL Ͱ͸ɺMoving

With BrickMesh ๏ [8] ʹΑΔ΢ΣʔΫɾϙςϯγϟϧ‫͕ࢉܭ‬Մ

.ResetೳͰ͋Δɻ͜Ε͸ɺ‫؍‬ଌ఺ۙ๣ͷΈϝογϡੜ੒͠ɺ.Name "solid1"

ͦͷϝογϡྖҬΛϏʔϜͷਐߦʹ߹ΘͤͯਐΊͯ.Component "component1"

͍͘ํ๏ͰɺϏʔϜɾόϯν಺ͷۙ‫཭ڑ‬΢ΣʔΫɾϙ.Material "Vacuum"ςϯγϟϧʢྫ͑͹ɺόϯν௕ͷ 100 ෼ͷ 1ʣͷ‫ࢉܭ‬.Xrange "0", "100"͕গͳ͍ϝϞϦͰߦ͑Δɻ

.Yrange "0", "100".Zrange "0", "100"11 ϫʔΫϑϩʔ.CreateEnd WithࡢࠓͷγϛϡϨʔγϣϯιϑτʢಛʹ঎༻ʣ͸ඇৗʹΑ͘Ͱ͖͓ͯΓɺ‫͢ࢉܭ‬Δର৅ʢ3 ࣍‫ݩ‬ιϦουϞ

౳ͱೖྗ͢Ε͹Α͍ɻHistory List ʹ͸ VBA ͷ‫ݴ‬σϧʣΛ࡞੒ͯ͠͠·͑͹ɺ̓ʙׂ̔ํग़དྷͨͱ‫ͬݴ‬‫Ͱޠ‬ཤྺ͕࢒͍ͬͯΔͷͰɺͦͷ಺༰͕ࢀߟʹͳΔɻ

ͯΑ͍Ͱ͋Ζ͏ɻ͜ͷઅͰ͸ɺCST STUDIO SUITE͜ͷ VBA Ͱ͸ɺϏδϡΞϧϕʔγοΫͰ࢖͑Δຆ

27

２－ 3

ਤ 27

CST STUDIO SUITE ͷϚΫϩ‫ػ‬ೳɻ

②出力したいステップを選択

ਤ 29

①

CST STUDIO SUITE ͷιϧόʔબ୒ɻ③

ਤ 28 CST STUDIO SUITE ͷ History List ͔ΒɺͦΕ·Ͱʹߦͬͨखଓ͖ΛҰ‫ׅ‬ग़ྗ͢Δํ๏ɻͲͷ਺ֶؔ਺͕࢖͑Δɻ

CST STUDIO SUITE ͷ VBA Ͱ͸ΦϒδΣΫτࢦ޲ϓϩάϥϛϯά͕αϙʔτ͞Ε͍ͯΔɻྫͱͯ͠ɺCST STUDIO SUITE ͷιϦουΛ GdfidL ͕ΠϯϙʔτՄೳͳ‫ʹࣜܗ‬ม‫͢׵‬ΔͨΊͷϚΫϩ [9] ͕ࢀ

ਤ 30

CST STUDIO SUITE ͷϙετϓϩηοαʵɻߟʹͳΔɻͦ͜Ͱ࢖͍ͬͯΔϚΫϩͰ͸ɺΫϥεΛ࢖༻͍ͯ͠Δɻஶऀͷ͓નΊ͸ɺ·ͣɺҰൠతͳํ๏ͰιϦουΛ࡞Γग़͠ɺ͋Δஈ֊ͰϚΫϩ‫ʹ༻࢖ޠݴ‬Ҡߦ͢Δ

͍͠ɻํ๏Ͱ͋ΔɻHistory List ʹͨΊࠐ·Εͨखଓ͖͸ɺ

11.3 ϙετϓϩηε
͋·Γ࣌ؒͷ͔͔Βͳ͍‫ࢉܭ‬͸ɺʢιϧόʔ‫ޙࢉܭ‬ਤ 28 ʹ͋Δํ๏Λ࢖͑͹ɺҰ‫ͭͱͻׅͯ͠‬ͷϚΫϩͱͯ͠ग़ྗग़དྷΔɻͷʣϙετϓϩηεͱͯ͠ѻΘΕΔɻ‫ݻ‬༗Ϟʔυ‫ܭ‬11.2 ιϧόʔͷબ୒ɾ࣮ߦ

ࢉ‫ޙ‬ͷ Q0 ஋‫ࢉܭ‬౳Ͱ͋ΔɻCST STUDIO SUITE Ͱ

CST STUDIO SUITE Ͱ͸ɺਤ 29 ʹ͋Δը໘Ͱιϧ͸ඇৗʹଟ͘ͷϙετϓϩηε༻ϚΫϩ͕༻ҙ͞ΕόʔΛબ୒͢ΔɻιϦουͷ࡞੒͸ɺιϧόʔબ୒ɻΦϯϥΠϯϚχϡΞϧ΍࣮ࡍʹ࢖ͬ
͍ͯΔʢਤ 30ʣͷલͰ΋‫Ͱޙ‬΋ߏΘͳ͍ɻҰ୴ιϧόʔΛબ୒͢ΕͯΈ͔ͯ֬Ί͍͖͍ͯͨͩͨɻ

͹ɺ‫ޙ‬͸ɺ‫͔ͭز‬ͷΦϓγϣϯΛબΜͰɺιϧόʔ

12 ࣮ࡍͷ‫ྫࢉܭ‬Λ࣮ߦ͢Δ͚ͩͰ͋ΔɻͲͷιϧόʔΛ࢖͑͹Α͍͜ͷઅͰ͸ɺCST STUDIO SUITE Λ࢖ͬͨγϛϡ͔ʹ͍ͭͯ͸ɺ෺ཧతཧղ͕ඞཁͳ৔߹͕গͳ͔ΒϨʔγϣϯྫΛ‫঺͔ͭز‬հ͠Α͏ɻ

ͣ͋Δɻ۩ମྫΛ࣍અʹࡌͤͨͷͰɺࢀߟʹͯ͠ཉ

28

２－ 4

WR-90矩形導波管内TE10モード

銅の電気伝導率を持ったソリッド

電場の強さ

WC-90円形導波管内
TM01モード∅12.7

電場ベクトル∅22.86ਤ 31

ਤ 32

γϛϡϨʔγϣϯ༻ιϦουϞσϧͷྫɻप೾਺ྖҬ‫ࢉܭ‬ͷ݁Ռྫɻࣜ (267) ΑΓɺω0QL
2π × 11.424 GHz≈4500≈ 1.6 MHzΓ=12.1 प೾਺ྖҬ‫ࢉܭ‬ͷྫ
ਤ 31 ʹ͋ΔϞσϧʹରͯ͠ Frequency Domain

(299)
(300)
(301)

Ͱ͋Δɻैͬͯɺ൒஋෯ͷे෼ͷҰͷ෼ղೳ͕ཉ͠

Solver Λ૸Βͤɺۣ‫ܗ‬ಋ೾؅্୺ͷ੺৭ྖҬͰ TE10

͍ͱ͢Δͱɺࣜ (298) ΑΓɺϞʔυʢୈ I ෦ͷਤ 10ʣΛྭৼ͠ɺϞʔυม‫ث׵‬Λ

10.16 MHz≈ 6.3 s࢖ͬͯɺԁ‫ܗ‬ಋ೾؅಺ TM01 Ϟʔυʢୈ I ෦ͷਤ 15ʣ

tmax ≈

ʹม‫ͯ͠׵‬ɺ3 ‫ݸ‬ͷηϧ͔ΒͳΔࢼ‫ۭݧ‬ಎʹ RF ύϫʔΛ౤ೖ͢Δɻೖࣹ೾ͷप೾਺͸ 11.4 GHz ͘Β

(302)
(303)

͍ͳͷͰɺ಺‫ ܘ‬22.86 mm ͷԁ‫ܗ‬ಋ೾؅಺͸఻ൖग़͘Β͍·ͰͷʢγϛϡϨʔγϣϯͷੈքͷதʹ͓͚དྷΔ͕ɺࢼ‫ۭݧ‬ಎͷऴ୺ʹ͋Δ಺‫ ܘ‬12.7 mm ͷԁ‫ܗ‬Δʣ‫͕ؒ࣌ࢉܭ‬ඞཁͰ͋Δɻ΋͠ɺҰ൩‫ࢉܭ‬δϣϒΛಋ೾؅಺͸ʢःஅप೾਺ΑΓ௿͍ͷͰʣ఻ൖ͠ͳ͍ɻ

૸Βͤͯɺ਺ ns ͘Β͍͔͠‫͕ࢉܭ‬ਐ·ͳ͔ͬͨͱ͠

ೖࣹ೾ͷप೾਺͕ࢼ‫ۭݧ‬ಎͷ‫ৼڞ‬प೾਺ͱ߹͍ͬͯ

ͯΒɺ࣌ؒྖҬͰͷ‫ࢉܭ‬͸೉͍͠ͱΘ͔Δɻ

Δ৔߹ɺࢼ‫ۭݧ‬ಎ಺ʹϚΠΫϩ೾ͷΤωϧΪʔ͕ஷ12.2 ࣌ؒྖҬ‫ࢉܭ‬ͷྫΊΒΕΔɻ͜ͷιϦουͰ͸ɺ11.424 GHz ʹ π Ϟʔ࣌ؒྖҬͰ͸ɺి࣓৔ͷ࣌ؒൃలΛѻ͏ͷͰɺա౉υʢྡ߹͏ۭಎͰϞʔυͷҐ૬͕ π ͣΕ͍ͯΔʣ͕‫ݱ‬৅Λ‫ݟ‬Δ͜ͱ͕ग़དྷΔɻ͜͜Ͱ͸ɺ௚ઢՃ଎‫Ͱث‬

͋Δɻਤ 32 ʹɺ11.424 GHz ͷϚΠΫϩ೾ΛϙʔτΑ͋͘ΔύϧεӡసΛྫʹͱΔɻ௨ৗͷύϧε͸ۣͰྭৼͨ͠৔߹ͷ‫݁ࢉܭ‬ՌΛࣔ͢ɻͲ͜ʹ‫ৼڞ‬प೾‫͋Ͱܗ‬Δ͕ɺਤ 33(a) ͷ੺৭೾‫͋ʹܗ‬ΔΑ͏ͳೋஈ

਺͕͋Δ͔Θ͔Βͳ͍͜ͱ͕ଟ͍ͷͰɺ௨ৗ͸ɺ͋Δ

֊εςοϓύϧεʢ11.424 GHz ͷ୯৭೾ʣΛਤ 31प೾਺ྖҬ಺ΛεΩϟϯ͢Δɻͭ·Γɺ͋Δप೾਺

ͷϞσϧʹ౤ೖ͢ΔɻCST STUDIO SUITE Ͱ͜ΕΛεςοϓͰԿճ΋‫͢ࢉܭ‬ΔͷͰ͋Δɻ

ྭৼ‫͢ͱݯ‬Δʹ͸ɺTime Domain Solver ‫ى‬ಈ‫ޙ‬ɺφ্ͷ‫ࢉܭ‬Λ࣌ؒྖҬͰߦ͏͜ͱ΋ՄೳͰ͋Δɻ୯ϏήʔγϣϯπϦʔͷʮExcitation SignalsʯΛӈΫҰͷप೾਺Ͱ͸ͳ͘ɺ͋Δप೾਺෼෍Λ΋ͬͨ೾ଋϦοΫ͠ɺʮNew Excitation SignalʯΛબ୒ɻSignal

Λೖࣹͯ͠ɺ݁ՌΛϑʔϦΤม‫͢׵‬Δɻਤ 31 ʹ͋Δ

Type ͱͯ͠ʮUser definedʯΛબΜͰ͔ΒɺEdit Ϙࢼ‫ۭݧ‬ಎͷ π Ϟʔυͷ Q0 ஋͸ 9000 Ͱ͋Δɻ੔߹͸λϯΛԡ͢ͱɺϢʔβʔ͕ఆٛग़དྷΔ΢Πϯυ΢͕‫ݱ‬ͱΕ͍ͯΔͷͰɺෛՙ Q ஋͸ͦͷ൒෼Ͱ͋ΔɻैͬΕɺͦͷதͰ VBA ‫ޠݴ‬Λ࢖ͬͯྭৼ೾Λ‫ه‬ड़͢Δɻͯɺπ ϞʔυͷεϖΫτϧ‫౓ڧ‬ͷ൒஋෯͸ɺୈ I ෦ͷ

ਤ 31 ʹ͋ΔϞσϧͷϙʔτʹεςοϓύϧεΛೖྗ

29

２－ 5

Normalized Amplitude

反射波（緑色）1000 mm1000 mm

r=70mmMaintaining
Step

(a)

r=0

z=-1000

Charging
Step

Amplitude [a.u]255 mm∅150 mm∅444 mm

入力波（赤色）z=0zz=1255

ਤ 34 ϐϧϘοΫε‫ܕ‬Ճ଎ۭಎͷྫɻ੨৭ྖҬ͸ਅۭɻͦΕҎ֎͸ಋମɻ

試験空洞内に励振されるモードの振幅（紫色）ͷͰɺQL = 9000/2 = 4500ɻΑͬͯɺୈ I ෦ͷࣜ(259) ΑΓɺ2 × 4500
2π × 11.424 GHz≈ 125 ns

Tf =

(b)

(304)
(305)

Ͱ͋Δɻैͬͯɺۣ‫ܗ‬೾Ͱύϧε෯͕਺ඦ ns ҎԼTime [ns]ͷ৔߹͸ɺࢼ‫ۭݧ‬ಎ಺ʹྭৼ͞ΕΔϞʔυͷৼ෯͸ҰఆʹͳΒͳ͍ʢ্ཱ͕ͪΓ్தʣɻ͜ͷγϛϡϨʔਤ 33 ࣌ؒྖҬ‫݁ࢉܭ‬Ռͷྫɻ11.424 GHz ͷ୯৭೾εςοϓύϧεʢ(a) ͷ੺৭ೖྗ೾ʣΛਤ 31 ͷϞσϧʹ౤ೖͨ͠ɻ

γϣϯ͸ɺεςοϓύϧεӡస͕࣮ࡍʹ໰୊ͳ͘ߦ
͑Δ͔ΛγϛϡϨʔγϣϯ্Ͱ֬ೝ͢ΔͨΊʹஶऀ͕࡞ͬͨ΋ͷͰ͋Δɻ͜ΕʹΑΓɺ࣮ࡍͷߴిքࢼ‫࣌ݧ‬͸εϜʔζʹεςοϓύϧεӡసΛߦ͑ͨɻ‫ݱ‬

͢Δ࣌ؒྖҬ‫ࢉܭ‬Λߦ͏ͱɺਤ 33 ʹ͋Δ݁Ռ͕ಘΒ୅ͷి࣓৔γϛϡϨʔγϣϯͷҙٛͷͻͱͭʹɺՃ

ΕΔɻਤ 33(a) ͷ྘৭೾‫ܗ‬͸ࢼ‫ۭݧ‬ಎ͔Βͷ൓ࣹ೾଎‫ث‬౳ͷ࣮ࡍͷӡసΛϑϧγϛϡϨʔγϣϯͯ͠ɺ

Ͱ͋Δɻύϧεͷ্ཱ͕ͪΓͱཱͪԼ͕ΓͰେ͖ͳ໰୊ɾෆ۩߹ͷࣄલൃ‫ݟ‬΍࠷దͳӡసύϥϝʔλͷ൓ࣹ͕ग़͍ͯΔɻ͜Ε͸ɺ੔߹ͷऔΕͨঢ়ଶͰ΋ɺύൃ‫ݟ‬౳Λߦ͑Δ͜ͱʹ͋Δɻ

ϧεͷ্ཱ͕ͪΓͱཱͪԼ͕ΓͰ͸‫ৼڞ‬प೾਺Ҏ֎

12.3 ‫ݻ‬༗Ϟʔυ‫ࢉܭ‬ͷྫͷ੒෼͕ൃੜ͢ΔͨΊͰ͋Δɻਤ 33(b) ͷࢵ৭೾‫ܗ‬

ਤ 34 ʹ͋ΔϐϧϘοΫε‫ܕ‬Ճ଎ۭಎʹରͯ͠͸ɺࢼ‫ۭݧ‬ಎͷதʹྭৼ͞ΕΔ‫ݻ‬༗Ϟʔυʢ͜ͷ৔߹

Eigenmode Solver Λ૸ΒͤͯΈΔɻ͜ͷ࣌ɺ‫ٻ‬Ίͨ͸ π Ϟʔυʣͷৼ෯Ͱ͋Δɻ͜ͷγϛϡϨʔγϣϯ͍‫ݻ‬༗Ϟʔυͷ࠷௿प೾਺ͱɺ‫ݻ‬༗Ϟʔυͷ‫਺ݸ‬Λઃͷೖྗ೾ͷεςοϓൺʢεςοϓલ൒ͱεςοϓ‫ޙ‬

ఆ͢Δɻͦ͏͢Δͱɺઃఆͨ͠࠷௿प೾਺Ҏ্ͷ‫ݻ‬༗൒ͷৼ෯ൺʣ͸༧Ίௐ੔ͯ͋͠Γɺεςοϓ‫ޙ‬൒ͰϞʔυΛઃఆͨ͠‫ͯ͘͠ࢉܭͰ·਺ݸ‬ΕΔɻ͜ͷྫࢼ‫ۭݧ‬ಎͷதʹྭৼ͞ΕΔϞʔυͷৼ෯͕ҰఆʹͳͰ͸ɺ࠷௿प೾਺ͱͯ͠ 500 MHzɺ‫ݻ‬༗Ϟʔυͷ‫ݸ‬ΔΑ͏ʹͯ͋͠Δɻ͜Ε͸ɺఆࡏ೾‫ܕ‬ͷύϧεӡస਺͸̐‫ͨ͠ͱݸ‬ɻ‫݁ࢉܭ‬ՌΛਤ 35 ʹࣔ͢ɻ‫ݻ‬༗ϞʔͰɺύϧε෯͕ Filling time ఔ౓Ͱ͋Δ৔߹ʹ࢖Θυप೾਺ͷ௿͍ॱʹղ͕‫·ٻ‬Δɻ͜͜Ͱɺ‫ݻ‬༗ϞʔΕΔςΫχοΫͰ͋Δɻ࣮ࡍɺFiling time Λ‫ٻ‬Ίͯυ̍͸ɺ௨ৗΑཻ͘ࢠՃ଎ʹ࢖͏ TM010 ϞʔυͰ͋ΈΑ͏ɻ͜ͷࢼ‫ۭݧ‬ಎͷ Q0 ஋͸ 9000 Ͱɺ‫ݻ‬༗ϞʔΔɻͦͷి৔‫౓ڧ‬෼෍Λਤ 36 ʹࣔ͢ɻ‫ݻ‬༗Ϟʔυ̐υप೾਺͸ 11.424 GHz Ͱ͋Δɻ੔߹͕ͱΕ͍ͯΔ͸ɺΑ͘ϏʔϜෆ҆ఆੑͷ‫ݪ‬ҼͱͳΔ TM110 Ϟʔυ

30

２－ 6

Ͱ͋Δɻ

ࢥΘΕΔɿ

‫ٻ‬Ίͨ‫ݻ‬༗Ϟʔυʹରͯ͠ɺϙετϓϩηεͰ Q0

1. γϛϡϨʔγϣϯ͸࣮‫Ͱݧ‬͸ͳ͍ͷͰɺߟྀ͞

஋౳Λ‫ࢉܭ‬ग़དྷΔɻ͜͜Ͱ͸ɺϞʔυ̍ʹରۭͯ͠Ε͍ͯΔ෺ཧ͸‫ݶ‬ఆ͞Ε͓ͯΓɺͦΕΛे෼ཧಎిѹͷҐஔඇґଘੑΛ֬ೝͯ͠ΈΔɻୈ I ෦ͷࣜղ͓ͯ͘͠ɻ

(282) ʹΑΕ͹ɺ࣠ରশੑ͕͋Δ৔߹ɺͲͷΑ͏ͳ‫ܗ‬√ঢ়Ͱ΋ɺۭಎిѹ͸ੵ෼͢ΔҐஔʢr = x2 + y 2 ʣ

2. όάͷͳ͍γϛϡϨʔγϣϯιϑτ͸ଘࡏ͠ͳ͍͜ͱΛ಄ͷย۱ʹৗʹஔ͍͓ͯ͘ɻ

ʹґΒͳ͍͸ͣͰ͋Δ͕ɺਤ 36 ʹ͋Δి৔ͷ‫͞ڧ‬෼

3. ‫݁ࢉܭ‬ՌͷϝογϡαΠζґଘੑ͸ඞͣ֬ೝ෍Λ‫ݟ‬Δ‫ݶ‬ΓͰ͸ɺ|V c (r = 0)| > |V c (r = 70mm)|

͢ΔɻͱͳΓͦ͏Ͱ͋Δɻ͜͜Ͱɺz = −1000 mm ͔Β z

4. प೾਺ྖҬͱ‫ݻ‬༗Ϟʔυ‫ࢉܭ‬͸ஞ࣍తํ๏Ͱ෺

= ζ ·ͰͷΤωϧΪʔརಘ W̃ Λɺ
�
�∫
�
� ζ
�
−iω cz �
0 � (306)
dz Ẽ z (r, z) eW̃ (ζ, r) = ��� −1000 mmཧղΛ‫ٻ‬ΊΔͷͰɺʢಛʹෳࡶͳ‫ܗ‬ঢ়Ͱ͸ʣඇ෺ཧղʹؕΔ͜ͱ΋͋Δɻ‫݁ࢉܭ‬Ռ͕෺ཧతʹଥ౰͔Ͳ͏͔Λ֬ೝ͢Δɻ

5. ՄೳͳΒ͹ɺෳ਺ͷιϧόʔ΍ɺෳ਺ͷγϛϡ

ͱఆٛ͢ΔɻW̃ (1000 mm, r) = |V c (r)| Ͱ͋Δɻ‫ݻ‬ϨʔγϣϯιϑτΛ࢖ͬͯΫϩενΣοΫ͢Δɻ༗ϞʔυιϧόʔͷϙετϓϩηοαʔΛ࢖ͬͯ W̃Λ਺஋‫݁ͨ͠ࢉܭ‬ՌΛਤ 37 ʹࣔ͢ɻ͔֬ʹɺζ <14 ͓‫͍ئ‬255 mm Ͱ͸ɺ r = 70 mm ΑΓ r = 0 Λ௨Δํ͕ΑΓଟ͘ͷΤωϧΪʔΛಘ͍ͯΔ͕ɺr = 0 Ͱ͸ۭಎΛຊॻ͸ߴΤωϧΪʔՃ଎‫ث‬ηϛφʔ OHO’17 ༻ʹա͗ͨลΓ͔Β‫ݮ‬଎Ґ૬ͷޮՌ͕͋Γɺ݁‫ہ‬ɺۭಎిࣥචͨ͠΋ͷͰ͕͢ɺࠓ‫ޙ‬΋ਵ࣌ߋ৽ͯ͠ɺ‫׬‬੒౓ѹ͸Ґஔ r ʹґଘ͠ͳ͍͜ͱ͕Θ͔ΔɻͰ͸ɺ‫ݮ‬଎ҐΛߴΊ͍ͨͱࢥ͍·͢ɻຊॻΛಡΜͰɺෆ۩߹΍ཁ૬ͷޮՌ͕ͳ͍͘Β͍ۭಎΪϟοϓ௕͕୹͍৔߹͸๬౳͋Γ·ͨ͠Βɺஶऀʢtetsuo.abe@kek.jpʣ·ͲͷΑ͏ʹͳΔͷͰ͋Ζ͏͔ɻਤ 38 ʹɺۭಎΪϟοͰ͝࿈བྷ͍͚ͨͩΔͱ޾͍Ͱ͢ɻߋ৽൛͸Πϯλʔϓ௕͕ 50 mm ͷ৔߹ͷ‫ݻ‬༗Ϟʔυ̍ͷి৔ͷ‫͞ڧ‬Λωοτ্Ͱެ։༧ఆͰ͢ɻ

ࣔ͢ɻͲ͏‫ͯݟ‬΋ |V c (r = 0)| < |V c (r = 70mm)|ࢀߟจ‫ݙ‬

͔ΒΘ͔ΔΑ͏ʹɺۭಎ಺ྖҬͰ͸ r = 0 ͷํ͕ r[1] M. Dal Forno, et al.: “rf breakdown measure-= 70 mm ΑΓΤωϧΪʔརಘ͕গͳ͍͕ɺϏʔϜύ

ments in electron beam driven 200 GHz cop-Πϓ΁ͷ࿙Εి৔͸ r = 70 mm ΑΓ r = 0 ͷํ͕ଟper and copper-silver accelerating structures”,͍ͷͰɺे෼௕͍ྖҬͰੵ෼͢Ε͹ɺ݁‫ہ‬ɺ͜ͷ৔߹

Phys. Rev. Accel. Beams, 19, 11, p. 111301

΋ۭಎిѹ͸Ґஔʹґଘ͠ͳ͍͜ͱ͕͔ΘΔɻҎ্

(2016).

ͱͳΓͦ͏Ͱ͋Δɻ͔͠͠ɺਤ 39 ʹࣔͨ͠‫݁ࢉܭ‬Ռ

[2] C. Jing, et al.: “Experiment on Multipactorͷ͜ͱ͔ΒɺϚΫε΢Σϧํఔ͔ࣜΒಋ͍ͨࣜ (282)Suppression in Dielectric-loaded AcceleratingΛʮ࣮‫ײ‬ʯग़དྷͨͱࢥ͏ɻ

Structures with a Solenoid Field”, Proceedings,

13 ࠷‫ʹޙ‬

4th International Particle Accelerator Conference (IPAC 2013): Shanghai, China, May 12-‫ݱ‬୅ͷʢಛʹ঎༻ͷʣϚΠΫϩ೾γϛϡϨʔγϣ

17, 2013, p. TUPEA087 (2013).

ϯιϑτ͸‫׬‬੒౓͕ߴ͘ɺ૷ஔ։ൃͷͨΊͷσβΠ

[3] E. L. Ginzton: “Microwave Measurements”,ϯͷΈͳΒͣɺՃ଎‫ث‬ͷӡస΍૷ஔͷࢼ‫ݧ‬ͷঢ়ଶΛ

Ann Arbor, Michigan (1957).

ਖ਼֬ʹ໛ٖͨ͠ϑϧγϛϡϨʔγϣϯΛߦ͏ͱ͜ʹ[4] J. C. Slater: “Microwave Electronics”, D. VanΑΓɺ໰୊ɺෆ۩߹ɺ࠷దͳӡసύϥϝʔλ౳ͷࣄNostrand (1950).

લൃ‫ݟ‬΋ՄೳͰ͋Δɻ·ͨɺϚΠΫϩ೾ཧ࿦ͷ෺ཧతཧղͷิॿʹ΋ͳΔͱ͍͏ଘࡏҙٛ΋͋Δɻୠ͠ɺ

[5] http://www.cybernet.co.jp/ansys/product/lineup/hfss/.

ஶऀͷ‫͔ݧܦ‬ΒɺԼ‫ʹ߲ࣄه‬͸஫ҙ͕ඞཁͰ͋Δͱ

[6] http://www.aetjapan.com/software/CST Overview.php.
31

２－ 7

モード１
（528.0 MHz）

モード２
（680.8 MHz）

モード３
（796.3 MHz）

モード４
（802.0 MHz）

ਤ 35

‫ݻ‬༗Ϟʔυ‫ࢉܭ‬ͷ݁Ռྫɻి৔ϕΫτϧͷΈදࣔɻ

32

２－ 8

空洞領域

r=70mmr=0

ζ [mm]
ਤ 36 ‫ݻ‬༗Ϟʔυ̍ͷి৔ͷ‫͞ڧ‬ΛΧʔϖοτͷߴ͞ͰදࣔʢۭಎΪϟοϓ௕͕ 255 mm ͷ৔߹ʣɻਤ 39 ΤωϧΪʔརಘ‫ࢉܭ‬ͷ݁ՌʢۭಎΪϟοϓ௕͕ 50 mm ͷ৔߹ʣ
ɻ[7] http://www.gdfidl.de/.
[8] K. L. Bane and T. Weiland: “Wake Force Com-

空洞領域

putation in the Time Domain for Long Structures”, Conf. Proc., C830811, pp. 314–316(1983).
[9] http://research.kek.jp/people/tabe/SS2GD Manual.pdf.

ζ [mm]
ਤ 37 ΤωϧΪʔརಘ‫ࢉܭ‬ͷ݁ՌʢۭಎΪϟοϓ௕͕ 255 mm ͷ৔߹ʣɻ

r=0 r=70mm

r=70mmr=0

ਤ 38 Ϟʔυ̍ͷి৔ͷ‫͞ڧ‬ΛΧʔϖοτͷߴ͞ͰදࣔʢۭಎΪϟοϓ௕͕ 50 mm ͷ৔߹ʣɻ33

２－ 9