ిࢠՃ଎‫ث‬ͷՃ଎؅
͸͡Ίʹ

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ຊߨٛͰ͸ओʹిࢠΛՃ଎͢Δ໨తͰ༻͍ΒΕΔ
ߴप೾ (radio freqency; RF) Ճ଎؅ΛऔΓ্͛ղઆ͢
Δɻ͜͜Ͱ͸ిࢠͷਐߦํ޲΁ͷ଎౓͸ޫ଎ͱ΄΅౳͍͠ (ve ∼= c) ͱԾఆ͢Δɻैͬͯɺ௚ྲྀ‫( ܕ‬direct

DC 電場

current; DC) ͷՃ଎؅΍ɺ಺෦Ͱ଎౓͕มԽ͢ΔΑ
͏ͳ࢖͍ํΛ͢ΔՃ଎؅͸ର৅֎ͱ͢Δɻ

RF 場
電場

磁場

RF Ճ଎؅͸‫ݱ‬୅ͷՃ଎‫ث‬Λࢧ͑ΔඞਢཁૉͰ͋
ΔͨΊɺ1984 ೥ͷॳճߴΤωϧΪʔՃ଎‫ث‬ηϛφʔਤ 1: DC ి৔ͱ RF ి৔ͷ໛ࣜਤɻDC ి৔ΛಘΔ(OHO) Ҏ߱ɺ౓ʑOHO ͷςʔϚͱͯ͠औΓ্͛ΒΕ͍ͯΔɻࠓճࢲʹ༩͑ΒΕͨςʔϚ͸ʮిࢠՃ଎
‫ث‬ͷՃ଎؅ʯͰ͋Γɺਐߦ೾ɾఆࡏ೾‫ܕ‬Ճ଎؅ͷ૒ํʹ͸੩తͳిҐ͕ࠩඞཁͱͳΔɻಈతͳి৔͕͋Ε͹Λղઆ͢ΔΑ͏ʹͱ͍͏͜ͱͰ͋ͬͨɻࢲࣗ਎͸‫ݱ‬τͳిࢠॐ૷ஔͷΈͰిࢠΛ૬ର࿦త଎౓·ͰՃ଎Ͱࡏ KEK ϑΥτϯϑΝΫτϦʔ (PF) Ͱ PF ‫ۭܕ‬ಎΛɺ͖Δɻ͜ͷΑ͏ʹ୹͍‫Ͱ཭ڑ‬ൺֱత؆୯ʹ૬ର࿦త଎

PF-AR(advanced ring) Ͱ APS ۭಎΛ༻͍ͨՃ଎‫ث‬ӡసʹ‫ܞ‬Θ͍ͬͯΔɻʮՃ଎‫ػ‬ӡసʹ‫ܞ‬Θ͍ͬͯΔʯͱ
ॻ͍ͨͷ͸͜ΕΒۭಎͷ։ൃʹࢲࣗ਎͸ؔ༩͍ͯ͠ͳ

౓Λୡ੒Ͱ͖Δ͜ͱ͕ిࢠՃ଎ͷಛ௃ͱ͍͑Δɻ͍͔ΒͰ͋Δɻ࣮ࡍɺࢲ͸ཻࢠՃ଎ʹඞཁͱ͞ΕΔେΓେ‫ن‬໛ͳՃ଎૷ஔ͕ඞཁʹͳΔɻྫ͑͹ɺཅࢠΛՃग़ྗͷ RF Ճ଎؅Λ։ൃͨ͠‫ݧܦ‬͸ແ͍ɻ଎͠ޫ଎ͷ 0.990 ഒͷӡಈ଎౓ΛಘΔʹ͸ɺ5.71 GeVͦ͜ͰຊςΩετͰ͸ɺ࣮ࡍͷՃ଎‫͍༻Ͱث‬ΒΕͯඞ࣓ͣ৔΋ଘࡏ͢Δ (RF ৔)ɻ

͜Εʹର͠ɺཅࢠ͸ిࢠͷ໿ 1836 ഒͷ੩ࢭ࣭ྔΤωϧΪʔΛ࣋ͭͨΊɺ૬ର࿦త଎౓ΛಘΔͨΊʹ͸ΑͷӡಈΤωϧΪʔΛ༩͑Δ͜ͱ͕ඞཁͱͳΔɻ

͍ΔՃ଎؅ͷઌ୺తɾٕज़తͳ಺༰ʹ͸‫ͣͤٴݴ‬ɺՃҎ্ͷߟ࡯͔ΒɺՃ଎؅Λ࢖ཻͬͯࢠΛՃ଎͢Δ৔଎؅ʹର͢Δ‫ૅج‬తɾ‫ڭ‬Պॻతͳ಺༰Λ঺հ͢Δ͜ͱ

߹ɺిࢠͱཅࢠ·ͨ͸ͦͷଞͷॏཻ͍ࢠͳͲର৅ͱͳ

Λ໨ඪͱ͢Δɻࢲʹͱͬͯ OHO ͱ‫͑ݴ‬͹Ճ଎‫ث‬ΛษΔཻࢠʹΑͬͯߟ͑ํɺઃ‫ࢥܭ‬૝ͳͲ͕େ͖͘ҟͳͬ‫͢ڧ‬ΔͨΊͷྑ࣭ͳ‫ڭ‬Պॻ‫͋Ͱ܈‬ΓɺࢲͷՃ଎‫ʹث‬ରͯ͘Δ͜ͱ͕༧૝Ͱ͖Δɻిࢠͷ৔߹͸ೖࣹ‫ث‬ͷॳஈ͢Δ஌ࣝͷଟ͘͸ OHO ςΩετ͔ΒಘΒΕͨ΋ͷͰ෦Λআ͍ͯͦͷਐߦํ޲΁ͷ଎౓Λޫ଎ͱͯ͠ઃ‫͠ܭ‬

͋ΔɻࠓճɺվΊͯՃ଎‫ث‬ͷ‫ૅج‬தͷ‫͋Ͱૅج‬Δʮి

ͯྑ͍͕ɺཅࢠͷ৔߹͸Ճ଎‫ث‬ͷେ෦෼ʢ਺ GeV ΫࢠՃ଎‫ث‬ͷՃ଎؅ʯΛࣥච͢Δʹ͋ͨΓɺࠓ‫ޙ‬Ճ଎‫ث‬

ϥε·ͰͷՃ଎‫ث‬ͷ৔߹ʣʹ͓͍ͯ଎౓͕ঃʑʹมԽ෼໺Ͱͷ‫׆‬༂Λૂ͏ਓͨͪͷཧղΛ΄Μͷগ͠Ͱ΋ॿ

͢ΔͨΊ͜ΕΛߟྀ͢Δඞཁ͕ग़ͯ͘ΔɻຊςΩετ

͚Δ͜ͱ͕Ͱ͖Ε͹ͱࢥ͏ɻͰ͸૬ର࿦త଎౓·ͰՃ଎͞Εɺ΄΅ޫ଎Ͱӡಈ͢Δ
ిࢠΛ೦಄ʹٞ࿦ΛਐΊΔɻ

1.1

ిࢠͷՃ଎

ిࢠͷՃ଎Λߟ͑Δࡍɺ΄ͱΜͲͷ৔߹ʹ͓͍ͯՃ

1.2

DC ͱ RF Ճ଎଎؅಺Ͱͷ଎౓มԽΛߟ͑ͳͯ͘΋ྑ͍ͱ͍͏఺͕

ՙిཻࢠΛՃ଎͢Δʹ͸ి৔͕༻͍ΒΕΔɻDC(੩)ଞͷཅࢠͳͲॏཻ͍ࢠΛՃ଎͢Δ৔߹ͱҟͳΔɻిࢠ

ి৔ͱ RF(ಈ) ి৔ͷ෺ཧతͳҧ͍͸ి৔͕࣌ؒత

͸੩ࢭ࣭ྔΤωϧΪʔ 511 keVɺిՙ e = −1.602 ×

ʹৼಈ͢Δ͔൱͔Ͱ͋Γɺి৔͕࣌ؒతʹৼಈ͢Δ৔

10−19 C Λ࣋ͭૉཻࢠͰ͋Γɺྫ͑͹ 3 MeV ·ͰՃ଎͢Δͱ଎౓͸ޫ଎ͷ 0.990 ഒʹୡ͢Δɻैͬͯɺి
ࢠͷՃ଎ʹ͓͍ͯ͸ɺҰ࿈ͷՃ଎‫ث‬ͷిࢠੜ੒௚‫͝ޙ‬߹ʹ͸ Maxwell ͷํఔࣜʹै͍࣓৔΋ଘࡏ͢Δ͜ͱ͘Ұ෦ͷΈ଎౓มԽΛߟྀ͢Ε͹ྑ͍͜ͱʹͳΔɻ࣮

΁ͷ RF ৔ͷಋೖ΍Ճ଎؅಺෦Ͱͷߴप೾ͷి‫ؾ‬తͳࡍͷՃ଎‫Ͱث‬͸ిࢠ‫ ͯ͠ͱݯ‬RF ిࢠॐΛ࠾༻͠ɺి

݁߹ͳͲʹ΋༻͍ΒΕΔɻ·ͨɺՃ଎؅಺ʹྭৼ͞Εࢠॐ෦ͷΈͰ਺ MeV ͷӡಈΤωϧΪʔΛ༩͑ΔΑ͏Δ࣓৔΋ి৔ͱಉ༷ʹిࢠϏʔϜͱ૬‫͢༻࡞ޓ‬Δɻ͜

ͳઃ‫͍ͯͬͳʹܭ‬Δ΋ͷ΋͋Δɻ͜ͷ৔߹ɺίϯύΫΕ͸ϏʔϜΛΩοΫ͢Δ (ਐߦํ޲Λ‫͛ۂ‬Δ) ໨తͰʹͳΔ (ਤ 1 ࢀর)ɻRF ࣓৔͸࣌ʹ͸Ճ଎؅ͷൃ೤ʹ‫د‬༩͠Ճ଎؅ͷੑೳΛ੍‫͢ݶ‬Δ͕ɺಋ೾؅͔ΒՃ଎؅࢖ΘΕΔ͜ͱ΋͋ΔɻΫϩ೾ʹ͍ͭͯ෺ཧతͳಛ௃Λ਺ࣜͱͱ΋ʹ঺հͯ͠RF ి৔ΛՃ଎ʹ༻͍Δ৔߹ɺՙిཻࢠΛՃ଎؅ʹෳ਺ճ௨աͤ͞Δ͜ͱͰ‫܁‬Γฦ͠Ճ଎͕ՄೳͱͳΔɻ

͍͘ɻྫ͑͹ɺRF Ճ଎؅Λ௚ྻʹෳ਺ฒ΂Δ͜ͱͰՙిཻ

Ճ଎؅಺ʹྭৼ͞ΕΔి࣓৔ͷ 3 ࣍‫ۭؒݩ‬෼෍ͱ࣌ࢠΛଟஈՃ଎͢Δ͜ͱ͕ՄೳͱͳΔɺ·ͨՙిཻࢠΛؒߏ଄Λཧղ͠ɺ໨తʹԠͨ͡࠷దͷՃ଎؅Λ੡࡞ɾԁ‫ي‬ಓΛඳ͘Α͏ʹपճͤ͞೚ҙҐஔͷ RF Ճ଎؅ʹ

ӡ༻͢Δඞཁ͕͋ΔɻిࢠՃ଎؅ͱҰ‫ͯͬݴͰݴ‬΋໨ෳ਺ճ௨͢͜ͱͰ΋ଟஈՃ଎͕ՄೳͱͳΔɻ͜ͷΑ͏తʹԠͯ͡ߏ଄΍ߟྀ͢΂͖෺ཧ͸༷ʑͰ͋Δɻྫ͑

ʹɺRF Ճ଎؅Λ༻͍Δ͜ͱͰ DC Ճ଎؅Ͱ͸ಘΒΕ͹ɺϦχΞοΫ౳ͰՃ଎ޯ഑Λॏࢹ͢Δ৔߹͸ਐߦ೾ͳ͍ඇৗʹେ͖ͳӡಈΤωϧΪʔΛޮ཰తʹཻࢠʹ༩‫ܕ‬Ճ଎؅Λ༻͍ɺϦϯά‫ܕ‬Ճ଎‫Ͳͳث‬Ճ଎ޯ഑ΑΓ΋

͑Δ͜ͱ͕ՄೳͱͳΔ 1 ɻཻࢠӡಈͷ҆ఆੑΛ௥‫͢ٻ‬Δ৔߹͸୯ηϧͷఆࡏ೾‫ܕ‬RF ʹΑΔཻࢠՃ଎Λ҆ఆɾޮ཰తʹߦ͏ͨΊʹ͸Ճ଎؅Λ༻͍Δ͜ͱ͕ଟ͍ɻ·ͨɺి࣓೾ͷप೾਺ɺ

1.3ৗ఻ಋͱ௒఻ಋ

Ճ଎‫܁‬Γฦ͠ (਺ Hz ͔Β CW ӡస)ɺඃՃ଎ిՙྔ
΋Ճ଎؅ઃ‫ܭ‬ͷॏཁͳύϥϝʔλͰ͋Δɻ͔͠͠ɺ͜

ৗ఻ಋͱ௒఻ಋͷҧ͍ʹ͍ͭͯ΋؆୯ʹड़΂͓ͯ͘ͷΑ͏ͳ࣮ࡍతͳՃ଎؅ͷઃ‫ܭ‬ɾ੡࡞ɾӡ༻ʹؔ͢ΔͱɺՃ଎؅ύϥϝʔλͱͯ͠ͷେ͖ͳҧ͍͸ Quality͜ͱʹ͸ຊςΩετͰ͸౿Έࠐ·ͳ͍ɻ͜ͷΑ͏ͳ಺factor (Q ஋) ʹ͋Δɻྫ͑͹ɺ500 MHz ଳͰ Q ஋ʹ͍࣮ͭͯࡍͷൺֱ͢ΔͱɺKEK-PF ‫ܕ‬ৗ఻ಋۭಎ

༰ʹ‫ڵ‬ຯͷ͋Δํ͸ա‫ڈ‬ͷςʔϚʹԊͬͨ OHO ςΩ͸ Q0 = 40000 [1], KEK-B ‫ܕ‬ͷ௒఻ಋۭಎ͸ Qext =ຊςΩετͰ͸ɺՃ଎؅ʹด͡ࠐΊΒΕͨϚΠΫϩ2 × 108 [2] ͱͦͷࠩ͸ 4 ܻʹ΋‫Ϳٴ‬ɻ·ͨɺγεςϜͱͯ͠ΈΔͳΒ͹ɺ‫ۃ‬௿Թ·ͰۭಎԹ౓ΛԼ͛ΔͨΊ೾ͷ‫ج‬ຊతͳ෺ཧΛཧղ͢΂͘ɺ·ͣಋ೾؅ʹ͍ͭͯͷྫྷౚ‫ػ‬γεςϜ͕ඞཁ͔Ͳ͏͔ɺࡐྉ͸௒఻ಋମ͔

଎؅ΛॱʹऔΓ্͛Δ͜ͱͰՃ଎؅શൠʹ͍ͭͯ၆ᛌ

Ͳ͏͔ͳͲͱ͍ͬͨ͜ͱ΋૬ҧ఺ͱͳΔɻతɾ‫ૅج‬తͳ஌ࣝΛಘΔ͜ͱΛ໨తͱ͢Δɻ

ຊςΩετͰ͸Ճ଎؅ͷ‫ૅج‬෺ཧΛղઆ͢Δͭ΋ΓͳͷͰɺ‫ج‬ຊతʹ͸ৗ఻ಋͱ௒఻ಋΛ۠ผ͠ͳ͍ɻղ
આ͸ৗ఻ಋՃ଎؅Λ૝ఆͯ͠ਐΊΔ͕ɺ͜͜Ͱ৮ΕΔ
಺༰͸௒఻ಋՃ଎؅Λษ‫͢ڧ‬Δʹ͓͍ͯ΋‫ͳͱૅج‬Δ
͸ͣͰ͋Δɻ

1.4

ຊςΩετͷྲྀΕຊςΩετͰ͸ࠓ೥౓ͷςʔϚʮϚΠΫϩ೾ͷ‫ج‬ૅʯͷଞͷςΩετ౳Ͱઆ໌͞ΕΔͰ͋Ζ͏ɺ‫ج‬ຊతͳϚΠΫϩ೾ͷཧ࿦ɺRF ۭಎͷ‫ج‬ຊతͳύϥϝʔλʹ͍ͭͯ͸‫ط‬஌Ͱ͋Δͱߟ͑ɺ৽ͨʹఆٛΛࣔͨ͠ΓৄࡉΛड़΂ͨΓ͸͠ͳ͍ɻ͜͜Ͱ͍͏ύϥϝʔλͱ͸Q ஋, γϟϯτΠϯϐʔμϯε౳Ͱ͋Γɺ΋͜͠ΕΒͷྔʹ͍ͭͯॳࣖͰ͋Ε͹ຊςΩετΛಡΈਐΊΔલʹগͳ͘ͱ΋ఆ͙ٛΒ͍͸཈͓͍͑ͯͯ௖͚Ε͹ͱ
ࢥ͏ɻ·ͨɺϚΠΫϩ೾ΛऔΓѻ͏ͷʹඞཁͳి࣓‫ؾ‬ֶͷ‫ج‬ຊɺMaxwell ํఔࣜʹ͍ͭͯ΋ֶ෦ଔ‫ۀ‬ੜఔ౓ͷ஌ࣝ͸͋Δ͜ͱ͕๬·͍͠ɻຊςΩετͰ͸͜ΕΒ஌ࣝΛલఏʹՃ଎؅ (ಋମ಺) ʹด͡ࠐΊΒΕͨϚΠ
1 DC ి৔Ͱ΋ଟஈԽ͢Δ͜ͱ͸ՄೳͰ͋Δ͕ɺάϥϯυϨϕϧ
ʹରͯ͠ߴ͍ిѹΛอͭඞཁ͕͋ΔͨΊ਺े MV ఔ౓͕‫ݶ‬քͱͳΔɻ͞Βʹ DC ͷՃ଎؅ͰߴిѹΛ࣮‫͢ݱ‬ΔʹͨΊʹ͸૷ஔ͕‫ۃ‬୺ʹେ‫ܕ‬Խ͢Δͱ͍͏ܽ఺΋͋ΔɻετΛࢀߟʹͯ͠௖͚Ε͹ྑ͍͔ͱࢥ͏ɻ

ղઆ͢Δɻ͜ΕΛ΋ͱʹਐߦ೾‫ܕ‬Ճ଎؅ɺఆࡏ೾‫ܕ‬Ճ

2

ಋ೾؅

2.1

ࣗ༝ۭؒͱಋ೾؅

ࣗ༝ۭؒΛ఻೻ (propagation)2 ͢Δి࣓೾͸ฏ໘೾y(plane wave) Ͱ͋Γ (ਤ 2 ࢀর)ɺΤωϧΪʔʹԠͨ͡प೾਺Ͱৼಈ͠ͳ͕ΒۭؒΛޫ଎ͰਐΉ (ਤ 2 ࢀর)ɻ͜ͷࡍɺి࣓৔͸ਐߦํ޲ʹਨ௚ͳฏ໘಺Ͱৼಈ͠ɺ

b

x

z

ಉํ޲ʹਐߦ͢ΔՙిཻࢠΛՃ଎͢Δ͜ͱ͸Ͱ͖ͳ͍ɻฏ໘೾͸ి࣓೾ͷਐߦํ޲ʹి࣓৔੒෼Λ࣋ͨͳa͍ͨΊɺTEM ೾ʢtransverse electric and magnetic

waveʣʹ෼ྨ͞ΕΔɻ·ͨɺి࣓೾͸ΤωϧΪʔΛ൐ͬͯਐߦ͢Δɻ୯Ґ໘ੵΛ୯Ґ࣌ؒ͋ͨΓʹ௨ա͢
Δి࣓৔ͷΤωϧΪʔ͸ɺి৔ϕΫτϧ Ẽ ͱ࣓৔ϕ

ਤ 3: ۣ‫ܗ‬ಋ೾؅ͷ໛ࣜਤɻి࣓೾͸ z ࣠ʹԊΓෛ͔Ϋτϧ H̃ ͷ֎ੵͰ༩͑ΒΕΔ (Poynting’s vector)ɻ

޲͸ແ‫͑ߟͱݶ‬Δɻ

Βਖ਼ํ޲ʹ޲͚఻೻͢Δ΋ͷͱ͢Δɻ͜͜Ͱ͸ z ࣠ํRF Ճ଎؅ͱ͸தۭͷಋମ 3 ʹి࣓৔ (ߴप೾) Λ఻೻ͤ͞ɺཻࢠͷӡಈΤωϧΪʔΛ૿Ճͤ͞Δ (Ճ଎)4ग़‫ޱ‬ଆʹ఻೻͢Δߴप೾Λೖࣹ೾΋͘͠͸ਐߦ೾ 5 ɺͨΊʹ༻͍ΒΕΔՃ଎‫૷ث‬ஔͰ͋ΔɻपғΛಋମͰऔಋମͰ൓ࣹ͞Εͨ೾Λ൓ࣹ೾ͱ‫Ϳݺ‬ɻߴप೾͸ಋମʹΓғΈɺ಺෦ʹߴप೾Λด͡ࠐΊͨ΋ͷΛۭಎ΋͘͠৵ೖɾಁաͰ͖ͳ͍ͨΊ 6 ɺҰൠతͳ೾ͷٞ࿦Ͱͯ͘

͸ RF ۭಎͱ‫Ϳݺ‬ɻRF Ճ଎؅, RF ۭಎΛཧղ͢Δ

Δ۶ંɾಁա೾͸ಋ೾؅Ͱ͸ߟ͑ͳ͍ɻ

ʹ͸ɺಋମ಺Ͱͷߴप೾ͷৼΔ෣͍ɾ༷ࢠΛ஌Δ͜ͱ

ಋ೾؅಺ʹ‫ܗ‬੒͞ΕΔ೾໘͸ಋ೾؅ͷ‫ܗ‬ঢ়ʹΑΓ͕ۙಓͩͱࢥΘΕΔɻ·ͣ͸͜ΕΛ໨తͱ͠ಋ೾؅ͱ

ܾఆ͞Εɺͦͷ༷ࢠʹΑΓ఻೻Ϟʔυ͕ఆٛ͞ΕΔɻͦͷதΛ఻೻͢Δి࣓೾ʹ͍ͭͯઆ໌͢Δɻ

ී௨ɺ఻೻Ϟʔυ͸ෳ਺ଘࡏ͠ɺ֤఻೻Ϟʔυʹ͓͚ಋ೾؅ (wave guide) ͱ͸ۭಎͷର޲͢ΔҰ࣠ํ๏Δ೾໘͸ಋ೾؅಺ͷ‫ڥ‬ք໘ (ۣ‫ܗ‬ͷ৔߹͸ 6 ໘த 2 ໘ͷಋମΛऔΓ෷͍ (ి࣓೾ͷೖ‫ͱޱ‬ग़‫ޱ‬Λ࡞Δ)ɺߴ

͕औΓআ͔Ε͍ͯΔͨΊɺ࢒Γͷ 4 ໘ʹ‫ڥ‬ք͕ଘࡏप೾Λ௨աͰ͖ΔΑ͏ʹͨ͠΋ͷͰ͋ΓɺՃ଎‫Ͱث‬͸͢Δ) ͔Βͷ൓ࣹ೾ΒͷॏͶ߹Θͤʢ‫ׯ‬বʣͷ݁Ռʹߴप೾‫ൃͰݯ‬ੜͨ͠ಛʹେ‫౓ڧ‬ͷߴप೾Λۭಎ౳ʹΑΓ‫ܗ‬੒͞ΕΔɻ͜Ε͸ಋ೾؅಺ߴप೾ͷ఻೻ํ޲ಋ͘ࡍʹ༻͍ΒΕΔɻಋ೾؅಺ͷߴप೾͸ಋମͷ‫ڥ‬քʹ௚ަ͢Δ໘Ͱͷ‫ڥ‬ք৚݅ʮಋମ໘Ͱͷి৔ͷ઀ઢ໘ͰҰఆͷ৚݅ (‫ڥ‬ք৚݅) Λຬͨ͠ͳ͕Β൓ࣹɾճ੒෼͕θϩʯΛߟྀͯ͠ Maxwell ํఔࣜ (͜ͷ৔߹ɺં͞Εɺ͜ΕΒ͕ॏͶ߹Θ͞Γʢ‫ׯ‬ব͠ʣͳ͕Β೾໘Helmholtz ํఔࣜʹͳΔ) Λղ͘͜ͱͰ‫ٻ‬ΊΒΕΔɻ(wave front) Λ‫ܗ‬੒͢Δɻ͜͜Ͱಋ೾؅಺Λೖ‫͔ޱ‬Βಋ೾؅Λ఻೻͢Δߴप೾͸ฏ໘೾ʢTEM ೾ʣͰ͸ͳ
͘ 7 ɺTEʢtransverse electric wave; ి৔͸ԣํ޲ͩ

磁場

電場

向
進行方

͚ʣ·ͨ͸ TMʢtransverse magnetic wave; ࣓৔͸ԣํ޲͚ͩʣ೾ʹ෼ྨ͞ΕΔɻ

2.2

ಋ೾؅಺ͷి࣓৔

ྫͱۣͯ͠‫ܗ‬ಋ೾؅Λ఻೻͢Δ TE Ϟʔυ (Ez = 0)ͷి࣓৔ͷ࣌ؒ (t) ൃలͷ༷ࢠΛ਺ࣜΛ༻͍ͯॻ͖Լ
͢ɻͨͩ͠ɺۣ‫ܗ‬ಋ೾؅͸ xyz ࠲ඪʹ͓͍ͯ x ࣠ํ޲ʹ a, y ࣠ํ޲ʹ b ͷ௕ํ‫ܗ‬அ໘Λ࣋ͭ΋ͷͱ͠ɺిਤ 2: ࣗ༝ۭؒΛ఻೻͢Δి࣓೾ͷ໛ࣜਤɻ
2 ೾ಈ͕ഔ࣭ͷதΛ֦͕͍ͬͯ͘͜ͱΛʮ఻೻ʯ΋͘͠͸ʮ఻ൖʯͱ͍͍ɺ‫ࡏݱ‬͸ͲͪΒ΋࢖༻͞Ε͍ͯΔ͕ɺຊςΩετͰ͸ʮ఻೻ʯʹ౷Ұ͢Δɻ
3 Ұൠతʹ͸ۚଐ͕༻͍ΒΕΔɻ
4 ৔߹ʹΑͬͯ͸‫ݮ‬଎΋ߦ͏ɻ࣓೾͸ z ࣠ʹԊͬͯෛ͔Βਖ਼ํ޲ʹ఻೻͢Δͱߟ͑Δ
5 ‫ޙ‬ड़͢Δ෺ཧྔͰ͋ΔҐ૬଎౓ v ͱ‫܈‬଎౓ v Λ༻͍ͯɺpg
vp · vg > 0 ͷ৚݅Λຬͨ͢΋ͷ͕ʮਐߦ೾ʯɺvp · vg < 0 ͱͳΔ΋ͷ͕ʮ‫ޙ‬ਐ೾ʯͱఆٛ͞ΕΔɻ
6 ۚଐ͸‫׬‬શಋମͰ͸ͳ͍͕ɺ͜͜Ͱ͸ද໘఍߅͕े෼ʹখ͍͞ͱ͠ɺ‫׬‬શಋମͱಉ༷ʹߟ͑Δɻ
7 TEM ೾͸ಋ೾؅಺Λ఻೻Ͱ͖ͳ͍ [3]ɻ(ਤ 3 ࢀর)ɻTE-modeHz = Hmn cos(kx x) cos(ky y)eiωt−γziωµky
Hmn cos(kx x) sin(ky y)eiωt−γz
kc 2
iωµkx
Ey = −
Hmn sin(kx x) cos(ky y)eiωt−γz
kc 2γkx
Hx = 2 Hmn sin(kx x) cos(ky y)eiωt−γzkc
γky
Hy = 2 Hmn cos(kx x) sin(ky y)eiωt−γz
kc
Ex =

γ=

!
kc 2 − ω 2 /c2

kc 2 = kx 2 + ky 2
mπ
nπ
kx =
, ky =
a
b

(2.1)
(2.2)
(2.3)
(2.4)
(2.5)

(2.6)

y

(2.7)
(2.8)

x

͜͜Ͱɺµ ͸ಁ࣓཰ɺω ͸ి࣓೾ͷ֯प೾਺ɺi ͸‫਺ڏ‬୯ҐͰ͋Δɻm, n ͸ͦΕͧΕɺm(n) = 0, 1, 2, · · · ͷ஋Λ΋ͭਖ਼ͷ੔਺Ͱ͋Γɺ͜ΕΒͷ૊Έ߹Θͤ (m, n)ਤ 4: TEmn Ϟʔυͷ xy ฏ໘ʹ͓͚Δి৔ͷઈର஋ͷ༷ࢠɻʹΑΓ TEmn ͱͯ͠఻೻ϞʔυΛ۠ผ͢Δɻͨͩ͠ɺ

TE00 Ϟʔυ͸ଘࡏͤͣ 8 ɺm, n ͷ͏ͪগͳ͘ͱ΋ͲͪΒ͔Ұํ͸ 1 Ҏ্ͷ੔਺ͱ͢Δඞཁ͕͋ΔɻHmn
͸Ϟʔυຖʹ༩͑ΒΕΔఆ਺Ͱ͋Γɺ࣓৔ Hz ͷৼΔɻͦΕ΄ͲϐʔΫిྗ͕େ͖͘ͳ͍৔߹͸ۭ‫ؾ‬ͷ·෯ʹରԠ͍ͯ͠Δɻ·ͨɺkc ͕ಋ೾؅ͷΧοτΦϑ

ΒΕΔɻ͜Ε͸ಋ೾؅಺Ͱͷ์ిΛආ͚Δͷ͕໨తͰΛܾΊΔ೾਺Ͱ͋ΓɺΧοτΦϑʢःஅʣप೾਺͸͋Δɻ

fc = ckc /(2π) ͱͳΔɻ্Ͱۣࣔͨ͠‫ܗ‬ಋ೾؅಺ͷి࣓৔ͷࣜΛோΊΔͱɺEx , Ey ͸ͦΕͧΕ sin(ky y), sin(kx x) ͱ͍͏ؔ਺Λ

2.3‫ؚ‬Ή͜ͱ͕Θ͔Δɻ͜ΕΒࡾ֯ؔ਺͸ɺEx ͍ͭͯ͸y = 0, bɺEy ͍ͭͯ͸ x = 0, a ʹ͓͍ͯ੒෼͕θϩʹͳΔ͜ͱΛอূ͓ͯ͠Γɺઌʹड़΂ͨʮಋମ໘Ͱͷి৔ͷ઀ઢ੒෼͕θϩʯͱ͍͏‫ڥ‬ք৚݅Λຬͨ͢ɻ·ͨɺࢦ਺ m, n ͕େ͖͘ͳΔͱɺి࣓৔ͷ֤੒෼͕ͦ
ΕͧΕͷ࣠ํ޲ʹ࣋ͭઅͷ਺͕େ͖͘ͳΔ͜ͱΘ͔Δɻਤ 4 ʹ TEmn Ϟʔυͷ xy ฏ໘ʹ͓͚Δి৔ͷ༷ࢠΛࣔ͢ɻ
ࣜ (2.1 ͔Β 2.5) ʹ͸ಁ࣓཰ µ ͕໌ࣔతʹ‫·ؚ‬Ε͓ͯΓɺ༠ి཰ ε Λ༻͍Δͱഔ࣭தͰͷޫ଎ c ͸ɺ

1
c= √
εµ

(2.9)

ͱॻ͚Δɻಋ೾؅Ͱ͸఻೻ͤ͞Δి࣓೾ͷిྗʢಛʹϐʔΫిྗʣʹԠͯ͡಺෦ʹॆర͢ΔΨεछ͕બ͹Ε
8 (0, 0) ΛબͿ৔߹ɺk

x = ky = 0 ͱͳΓɺHz Λআ͘શͯͷ੒෼͸߃౳తʹθϩͱͳΔɻ͔͠͠ɺHz ͕࣌ؒతʹมಈ͢Δͱ
Maxwell ͷࣜʹै͍ɺx, y ํ޲ʹి৔͕ੜ͡ͳ͚Ε͹ͳΒͣໃ६͕ੜ͡Δɻ

·Ͱྑ͍͕ɺϐʔΫిྗ͕େ͖͍৔߹͸ SF6 Ψε౳ΛՃѹॆర͢Δ͔ɺ΋͘͠͸಺෦Λਅۭഉ‫͍༻ͯ͠ؾ‬఻೻܎਺ࣜ (2.1 ͔Β 2.5) Ͱࣔ͢ి࣓৔ͷ֤੒෼ʹ͓͍ͯม਺ z ʹର͢Δґଘੑ͸ɺ౳͘͠ exp (−γz) ͱ͍͏ؔ܎
Ͱ༩͑ΒΕΔɻγ ͸఻೻܎਺ͱ‫ݺ‬͹ΕΔྔͰ͋Γɺkc2
ͱ ω 2 /c2 ͷେখؔ܎ʹΑΓɺਖ਼ͷ࣮਺͔ਖ਼ූ߸ͷ७‫ڏ‬਺ͱͳΔɻલऀ͸ೖࣹ೾͕ +z ํ޲ʹਐΉʹͭΕͯ‫ݮ‬
ਰ 9 ͢Δ͜ͱΛɺ‫ऀޙ‬͸‫ݮ‬ਰ͢Δ͜ͱͳ͘ +z ํ޲ʹ఻೻͢Δ͜ͱΛҙຯ͢Δɻγ ͕७‫਺ڏ‬ͷ৔߹ɺγ = iβͱ͓͘ͱɺࣜ (2.6, 2.7) ΑΓɺ

ω2= kc 2 + β 2 = kx 2 + ky 2 + β 2c2(2.10)

ͱ͍͏ؔ܎͕ಋ͔Εɺ֤੒෼ͷ೾਺ͱ֯प೾਺ͷؔ܎ੑʹ͍ͭͯ‫ݟ‬௨͕͠Α͘ͳΔʢͱචऀ͸ࢥ͏ʣɻ·ͨɺ͜ͷ৔߹ʹࣜ (2.1 ͔Β 2.5) ΛோΊΔͱɺHz Ҏ֎ͷ੒෼ʹ͸‫਺ڏ‬୯Ґ i ͕͔͔͍ͬͯΔɻ͜Ε͸ɺHz ͱ͜
9 ͜͜Ͱͷ‫ݮ‬ਰͱ͸ɺδϡʔϧ೤ʹΑΔ΋ͷͰ͸ͳ͘ɺಋ೾؅ͷಋମ໘ʹ͓͍ͯ൓ࣹ͕‫͜ى‬Γి࣓೾͕఻ΘΒͳ͍͜ͱΛҙຯ͢ΔɻΕҎ֎ͷ੒෼Ͱ‫౓ڧ‬ͷϐʔΫΛ༩͑ΔҐ૬͕ π/2 ͣφθΕΔ͜ͱΛҙຯ͍ͯ͠Δɻ

λ

ω/c
θ
λg

ΧοτΦϑप೾਺

2.4γ ͕ਖ਼ූ߸ͷ७‫਺ڏ‬ͷͱ͖ʹɺి࣓೾͕఻೻ՄೳͰ

φω > ckc ≡ ωcͱ͍͏ؔ܎͕ಋ͔ΕΔ

10(2.11)

ɻ͢ͳΘͪɺಋ೾؅಺Λ఻aβθ
平面波の軌跡͋Δͱ͍͏৚͔݅Βɺkc

波面

ਤ 5: ฏ໘೾ͷ߹੒ͱͯ͠Έͨಋ೾؅಺Ͱͷి࣓೾ɻxz ฏ໘

೻͢Δి࣓೾͸ಋ೾؅ͷ‫ز‬Կֶతͳ‫ܗ‬ঢ়ʹΑΓܾ·ΔΧοτΦϑप೾਺ΑΓେ͖͘ͳ͚Ε͹ͳΒͳ͍ɻ͜Εͱ͍͏ؔ܎͕੒Γཱͭɻ͜Ε͸ z ࣠Λରশʹ θ ͚ͩ܏͕ΧοτΦϑप೾਺ͱ͍͏໊ͷ༝དྷͰ͋Δɻ

͍ͨํ޲ʹਐΉ 2 ͭͷฏ໘೾ͷॏͶ߹Θͤͱߟ͑Δ͜

ࣜʢ2.7, 2.8ʣ͔ΒΘ͔ΔΑ͏ʹɺಋ೾؅Λ఻೻͢ͱ͕Ͱ͖Δɻͭ·Γɺಋ೾؅಺Ͱ͸ฏ໘೾͸ಋମ໘ͰΔϞʔυʹΑΓΧοτΦϑप೾਺͸ҟͳΓɺࢦ਺ʢm,൓ࣹ͞Εͳ͕ΒδάβάʹਐΜͰ͍ΔͱཧղͰ͖Δɻnʣ͕େ͖͘ͳΔͱʢߴ࣍ͷϞʔυʹͳΔͱʣͦͷ஋͸େ͖͘ͳΔɻ͜͜Ͱɺਤ 3 ʹ͓͍ͯ a = 2b ΛԾ͜͜Ͱ͸ۣ‫ܗ‬ಋ೾؅ͷ TE10 Ϟʔυʹ͍ͭͯઆ໌ͨ͠

ఆ͢ΔͱɺΧοτΦϑप೾਺ͷҰ൪௿͍Ϟʔυ (‫ج‬ຊͯ΋ಉ༷Ͱ͋ΔɻҎ্Λਤ 5 ʹࣔ͢ɻϞʔυͱ‫ )Ϳݺ‬͸ TE10 Ͱ͋Γɺͦͷ࣍ʹେ͖͍ͷ͕
(10)

(01)

(20)

TE01 ͱ TE20 ϞʔυͱͳΔ (ωc < ωc = ωc )ɻ
͜͜ͰɺTE01 ͱ TE20 Ϟʔυ͸֤ʑ͕શ͘ҟͳΔి
࣓৔੒෼ʢxy ฏ໘಺ͷి࣓৔෼෍ʣΛ΋͕ͭɺಉ͡

͕ɺଞͷ‫ܗ‬ঢ়ͷಋ೾؅ɺଞͷ TE, TM Ϟʔυʹ͍ͭ͜͜Ͱɺθ ͷ‫ݶۃ‬Λߟ͑Δɻ·ͣɺθ ∼= 0 ͷ৔߹ɺ
∼∼kx = 0, ky = 0 ͱͳΓɺి࣓೾͸ TE Ϟʔυͱͯ͠͸఻೻Ͱ͖ͳ͘ͳΔɻ͜Ε͸ઌʹٞ࿦ͨ͠ TE00 Ϟʔυ͕ଘࡏ͠ͳ͍ͷͱಉ༷ͳঢ়‫͋Ͱگ‬ΓɺTE ϞʔυΛԾΧοτΦϑप೾਺Λ࣋ͭɻ͜ͷΑ͏ͳঢ়ଶΛϞʔυॖఆͯ͠ಘΒΕͨࣜ (2.1 ͔Β 2.5ʣΛ༻͍ͯ͸ٞ࿦Ͱ͖

ୀ͍ͯ͠Δͱ‫ͼݺ‬ɺ࣮ࡍͷಋ೾؅΍Ճ଎؅Λ࢖༻͢Δ

ͳ͍ɻ͜ͷ৔߹ɺಋ೾؅ʹೖࣹ͞Εͨి࣓೾͸ಋମ໘ͱ͖ʹߴप೾‫ݯ‬ͷप೾਺Λ੍‫ͯ͠ݶ‬΋ෳ਺ͷϞʔυ͕Λҙࣝ͢Δ͜ͱͳ͘ 11 ʢಋ೾؅Λྭৼ͢Δ͜ͱͳ͠ྭৼ͞ΕΔՄೳੑ͕͋ΔͨΊ஫ҙ͕ඞཁͱͳΔɻैͬ

ʹʣTEM ೾ͱͯ͠௚ਐΛଓ͚Δɻ͍Θ͹ɺ‫ڥ‬ք͕ᴴ

(10)
(01)
ͯɺωc
< ω < ωc ͱͳΔి࣓೾Λಋ೾؅ʹೖࣹ͔൴ํʹ͋ΔۭؒΛి࣓೾͕ਐΉΑ͏ͳঢ়ଶͱߟ͑Β͢Ε͹ɺಋ೾؅಺ʹ͸‫ج‬ຊϞʔυ͚͕ͩ఻೻͢Δ͜ͱΕΔɻ
࣍ʹಋ೾؅ΛਐΉ೾ͷप೾਺͕ΧοτΦϑप೾਺ͱʹͳΔɻ

Ұக͢Δ৔߹ɺω/c = kc Ͱ͋Δ͔Βɺθ = π/2 ͱͳΔɻ͜ͷ৔߹ɺਤ 5 ΑΓɺి࣓೾͸ x ࣠ํ޲Λԟ෮͢ฏ໘೾ͷ߹੒

2.5

ΔͷΈͰ +z ࣠ํ޲ʹ͸ਐ·ͳ͍ɻಋ೾؅Λ఻೻͢Δి࣓೾ͷ೾໘͸ಋମ໘Ͱͷ൓ࣹ೾ͷॏͶ߹ΘͤͰ‫ܗ‬੒͞ΕΔͱઌʹॻ͍ͨɻ͜Εʹ͍ͭͯ TE10 ϞʔυΛྫͱͯ͠આ໌͢Δɻͨͩ͠ɺଛࣦͷͳ͍൓ࣹ໘ΛԾఆ͢ΔɻTE10 Ϟʔυͷ Hz ͸ɺ࣌ؒʹґଘ͢Δ߲Λআ͍ͯҎԼͷΑ͏ʹද͞ΕΔɻ

2.6෼ࢄؔ܎ͱҐ૬଎౓ΧοτΦϑप೾਺ΑΓେ͖͍प೾਺Λ࣋ͭి࣓೾͸ಋ೾؅Λ఻೻Ͱ͖Δ͜ͱΛઌʹड़΂ͨɻಋ೾؅Λ఻೻
͢Δࡍɺप೾਺ͷҧ͍͸఻೻ʹͲͷΑ͏ͳӨ‫ڹ‬Λ༩͑

Hz = H10 cos(kx x)e−iβz#
H10 " i(kx x−βz)e=+ ei(−kx x−βz)2(2.12)

ΔͷͩΖ͏͔ʁਤ 6 ʹࣜ (2.10) Λ ω/c ʹ͍ͭͯղ͖ɺ

(2.13)β ͷؔ਺ͱͯ͠ϓϩοτͨ͠΋ͷʢ੺ͷ‫ۂ‬ઢʣΛࣔ͢ɻ੺ͷ࣮‫ۂ‬ઢ͕‫ج‬ຊϞʔυɺҰ఺࠯ઢ͕࣍ͷϞʔυͷ෼͜͜Ͱɺky = 0 ͱɺࣜ (2.10) ΑΓɺ

10 kc ͸ਖ਼ͱ͢Δ

ࢄؔ܎Λҙຯ͢Δɻͨͩ͠ɺॎ࣠͸ΧοτΦϑ֯प೾kx = (ω/c) sin θ(2.14)

β = (ω/c) cos θ(2.15)਺ ωc Ͱ‫֨ن‬Խ͠ɺβ < 0 ͷྖҬ΋‫ؚ‬Ίਤࣔͨ͠ɻ੨
৭ͷ࣮ઢ͸ z ࣠ํ޲ʹޫ଎ͰਐΉ೾Λද͢ɻ
11 Ұํͷಋମ໘Ͱ൓ࣹ͞Εͯ΋ɺଞͷಋମ໘͸೾௕ͱൺ΂े෼ԕ͘ʹ͋ΔͨΊɺଞͷಋମ໘Ͱ൓ࣹ͞Εͨ೾ͱ͸‫ׯ‬বͰ͖ͳ͍ɻ

Hz-field, f/fc=1.27

ω/ωc
4
3

ω/ωc
= 2.54

2

1.91
1.27

Ey-field, f/fc=1.91

Hz-field, f/fc=1.91

Ey-field, f/fc=2.54

Hz-field, f/fc=2.54

群速度

1

位相速度
0

-4

-3

-2

-1

0

1

2

3

4
β

x

ਤ 6: ಋ೾؅Λ఻೻͢Δి࣓೾ͷ෼ࢄؔ܎‫͔఺ݪ‬Β෼ࢄؔ܎Λࣔ͢‫ۂ‬ઢ (෼ࢄ‫ۂ‬ઢ) ͷҰ఺Λtͭͳ͍ͩͱ͖ɺͦͷ܏͖͕Ґ૬଎౓Λɺަ఺ʹ͓͚Δ
઀ઢͷޯ഑͕‫܈‬଎౓Λද͢ɻҐ૬଎౓͸೾໘ͷਐΉ଎

-1.0౓Ͱ͋ΓɺҰൠతͳಋ೾؅ʹ͓͍ͯి࣓೾ͷҐ૬଎౓͸͍ͣΕͷप೾਺ʹ͓͍ͯ΋ޫ଎ΑΓେ͖͍ɻ·ͨɺΧοτΦϑप೾਺ʹۙͮ͘ʹͭΕҐ૬଎౓͸͞Βʹେ͖͘ͳΓɺΧοτΦϑप೾਺ͱҰக͢Δ఺Ͱ͸ແ‫ݶ‬େ

0.0

1.0

ਤ 7: ͋Δ z Ґஔʹ͓͚Δ TE10 ϞʔυͷҟͳΔप೾
਺ʹ͓͚Δ tx ฏ໘Ͱͷ Ey , Hz ͷ఻೻ঢ়ଶͷมԽɻՃ଎؅‫ܗ‬ঢ়ɺਤࣔൣғ͸શϓϩοτͰ‫ڞ‬௨ɻʹͳΔɻ͜Εͱ൓ରʹɺप೾਺͕େ͖͘ͳΔͱɺҐ૬଎౓͸઴ۙతʹޫ଎ʹ͍͍ۙͮͯ͘ɻ‫܈‬଎౓͸ਐߦ೾ΛோΊͨͱ͖ͷҐ૬ͷ୯Ґ‫܁‬Γฦ͠ͷ‫཭ڑ‬Λ؅಺೾௕ͷΤωϧΪʔ͕఻ΘΔ଎౓Ͱ͋ΓɺΧοτΦϑप೾਺ͱΛҙຯ͠ɺઅ 2.4 Ͱ࿦ͨ͡Α͏ʹɺΧοτΦϑप೾

(guide wavelength) ͱ‫Ϳݺ‬ɻ͜͜Ͱɺ͜Ε·Ͱग़͖֤ͯͨ෺ཧྔʹ͍ͭͯ਺ࣜΛ͓ࣔͯ͘͠ɻ

਺ͱಉ͡प೾਺Λ࣋ͭ೾͸ z ࣠ํ޲ʹ͸ਐ·ͳ͍ͱ͍ɾҐ૬଎౓Ͱ͸θϩͱͳΔɻ͜Ε͸ΤωϧΪʔ͕఻ૹ͞Εͳ͍͜͏ߟ࡯ͱҰக͢Δɻ
ਤ 6 ʹ͓͍ͯɺβ > 0 ͷྖҬͰ͸‫܈‬଎౓͸ਖ਼ɺβ < 0Ͱ͸ෛͱͳΔ͜ͱ͕Θ͔Δɻ͜Ε͸ΤωϧΪʔͷ఻Θࢠ͕ҟͳΔ͜ͱ͕Θ͔Δɻ·ͨɺEy , Hz Ͱ࠷େ‫౓ڧ‬ΛऔΔ͕࣌ؒ π/2 Ґ૬ͣΕ͍ͯΔ͜ͱ΋֬ೝͰ͖Δɻ
ਤ 7 ͸ tx ฏ໘Λ zx ฏ໘ͱͯ͠΋ (z ࣠ͷਤࣔൣғ

(2.17)

dω
β
λ
=
c=
c
dβ
ω/c
λg

(2.18)

ɾ‫܈‬଎౓

vg =͠ 1.27, 1.91, 2.54 ഒͷप೾਺Λ࣋ͭߴप೾ͷ੒෼ Ey ,ฏ໘Ͱͷి࣓৔ͷ༷ࢠ͸ಉ͡Ͱ͋Δ͕ɺtx ฏ໘Ͱͷ༷

2π
vp
=
β
f

λg =

͍ͯΔɻ

Hz ʹ͍ͭͯɺtx ฏ໘Ͱͷ఻೻ͷ༷ࢠΛࣔ͢ɻ͍ͣΕͷप೾਺ʹ͓͍ͯ΋఻೻Ϟʔυ͕ಉҰͰ͋ΔͨΊ xy(2.16)

ɾ؅಺೾௕Δํ޲͕લऀ͸ +z, ‫ऀޙ‬͸ −z ํ޲Ͱ͋Δ͜ͱΛࣔ͠
ਤ 7 ʹ‫ج‬ຊ (TE10 ) ϞʔυͷΧοτΦϑप೾਺ʹର

ω
β

vp =

ͨͩ͠ɺω, λ ͳͲఴࣈͷແ͍ྔ͸ɺશͯಋ೾؅Λ఻
೻͢Δి࣓೾ʹؔ͢ΔྔͰ͋Δɻࣜʢ2.18ʣͷ 2 ͭΊҎ߱ͷ౳߸͸ࣜ (2.10) ‫( ࣜͼٴ‬2.16, 2.17) Λ༻͍ͯಋ͘͜ͱ͕Ͱ͖Δɻ͞Βʹɺࣜ (2.16, 2.17) ΑΓҐ૬଎౓ͱ‫܈‬଎౓͸ҎԼͷؔ܎Λຬͨ͢͜ͱ͕Θ͔Δɻ

v p v g = c2

Λ͏·͘બͿ͜ͱͰ) શ͘ಉ༷ͷඳ૾͕ಘΒΕΔɻzx(2.19)

ฏ໘ͱͯ͠ਤ 7 ΛோΊΔͱɺप೾਺͕ߴ͘ͳΔʹͭΕҐ૬ਐΈྔʢҐ૬଎౓ʣ͕খ͘͞ͳ͍ͬͯΔ͜ͱ͕Θ͔Δɻ͜ͷ݁Ռ͸ɺਤ 6 Ͱ֤఺ͷҐ૬଎౓Λൺֱͯ͠Έͯ΋੒Γཱͭ͜ͱ͕Θ͔Δɻzx ฏ໘Ͱి࣓৔෼෍

2.7

KEK-PF ͷۣ‫ܗ‬ಋ೾؅ۣ‫ܗ‬ಋ೾؅͸ओʹେిྗߴप೾ΛՃ଎؅ʹಋͨ͘Ίʹ༻͍ΒΕΔɻ௨ৗ༻͍ΒΕΔͷ͸‫ج‬ຊϞʔυͱͳΔyද 1: ۣ‫ܗ‬ಋ೾؅ WR-1500 ͷΧοτΦϑप೾਺m n ΧοτΦϑप೾਺

1
2

0
0

393.7 MHz
787.4 MHz

0
1

1
1

787.4 MHz
880.3 MHz

r
z
φ
a

x

ද 2: ۣ‫ܗ‬ಋ೾؅ WR-1500 ͷ TE10 ϞʔυΛҟͳΔप೾਺Ͱྭৼͨ͠৔߹ͷ఻೻ʹؔΘΔύϥϝʔλɻप೾਺Ґ૬଎౓ ‫܈‬଎౓ ؅಺೾௕ɹ

MHz
(f )

(f /fc ) ɹ(vp /c)(vg /c)

m
(λg )

500
750
1000

1.27
1.91
2.54

1.62
1.17
1.09

0.62
0.85
0.92

0.97
0.47
0.33

ਤ 8: ԁ౵‫ܕ‬ಋ೾؅ͷ໛ࣜਤɻి࣓೾͸ z ࣠ʹԊͬͯෛ͔Βਖ਼ํ޲ʹ޲͚఻೻͢Δ΋ͷͱ͢Δɻԁ౵෦ͷ಺൒‫ܘ‬Λ a ͱ͢Δɻ·ͨɺද 1 ͱද 2 Λ‫ݟ‬ൺ΂ΔͱΘ͔ΔΑ͏ʹද 2

TE10 ϞʔυͰ͋Γɺஅ໘ͷ௕ล a ͱ୹ล b ͕ a = 2bͱͳΔ΋ͷ͕༻͍ΒΕΔɻઌʹਤࣔͨͦ͠ͷి࣓೾ͷͰࣔͨ͠࠷େप೾਺ 1000 MHz ͸ WR-1500 ͷ TE20 ,༷ࢠʹ͍ͭͯ͸શ্ͯ‫ه‬৚݅ʢa = 2bʣΛຬͨͯ͠TE01 , TE11 ϞʔυͷΧοτΦϑप೾਺ΑΓେ͖͍ɺ͜ͷ৔߹ɺຊདྷར༻͍ͨ͠Ϟʔυ TE10 ϞʔυҎ֎͍Δɻͷ఻೻Ϟʔυ΋ྭৼ͞ΕΔՄೳੑ͕͋Δɻ͜ͷͨΊɺ

KEK-PF Ͱ͸ EIA ‫ Ͱ֨ن‬WR-1500 ͱ‫ݺ‬͹ΕΔಋ೾؅Λ࢖༻͓ͯ͠Γɺa = 381 mm, b = 190.5 mm ͷ಺෦அ໘‫ܗ‬ঢ়Λ΋ͭɻ͜ͷಋ೾؅ͷ‫ج‬ຊϞʔυͱ‫ͭز‬

WR-1500 ಋ೾؅͸प೾਺ 787 MHz ʹ‫ز‬Β͔ͷ༨༟ΛΈͨ஋ҎԼͷप೾਺Λ΋ͭి࣓೾Λ఻ૹ͢Δͷʹ༻͔ͷߴ࣍Ϟʔυʹ͍ͭͯΧοτΦϑप೾਺Λද 1 ʹࣔ1500 ಋ೾؅ͷप೾਺ଳҬ͸ 0.49 ͔Β 0.75 GHz ͱͳͬ͢ɻ͜ͷಋ೾؅ͷ TE10 ϞʔυͷΧοτΦϑप೾਺͸

͍ͯΔɻ௿͍ํͷप೾਺ଳҬ͕ 0.49 GHz ͱΧοτΦ

393.7 MHz Ͱ͋ΓɺҰ্ͭͷϞʔυͰ͸ 787.4 MHz
ͱͳΔɻKEK-PF Ͱ༻͍ΒΕ͍ͯΔ RF ۭಎͷ‫ৼڞ‬प೾਺͸ 500.1 MHz Ͱ͋ΔͨΊɺߴ࣍ͷϞʔυΛྭϑप೾਺ 0.39 GHz ΑΓ 100MHz ఔ౓େ͖͘ͳͬͯ

ৼ͢Δ͜ͱແ͘‫ج‬ຊͷ TE10 ϞʔυͰߴप೾Λ఻೻Ͱω/ωc =1.27 ͷ఺ʣҎԼͷྖҬͰ͸‫܈‬଎౓͸‫ʹ୺ۃ‬௿Լ͢Δɻ

͖Δ͜ͱʹͳΔɻͱ͜ΖͰɺද 1 Ͱ͸ TE20 , TE01 ͷΒΕΔɻ࣮ࡍʹಋ೾؅ͷΧλϩάΛΈͯΈΔͱɺWR-͍Δͷ͸ɺ‫܈‬଎౓͕஗͘ͳΓ͗͢Δ͔ΒͩͱࢥΘΕΔɻਤ 6 ͔ΒΘ͔ΔΑ͏ʹɺप೾਺ 500 MHzʢਤͷΧοτΦϑप೾਺͸શ͘ಉ͡Ͱ͋Γ (ॖୀ͍ͯ͠Δ)ɺ
͞Βʹ TM Ϟʔυͷ‫ج‬ຊϞʔυͰ͋Δ TM11 Ϟʔυ΋ TE11 ΋ॖୀ͍ͯ͠Δɻ

2.8

ԁ౵‫ܕ‬ಋ೾؅

ۣ‫ܗ‬ಋ೾؅ʹؔ͢Δ‫ه‬ड़ͷ࠷‫ʹޙ‬ɺࢀߟͱͯ͠ WR-Ҏ্ɺిࢠՃ଎؅ʹ͍ͭͯͷઆ໌ͰલఏͱͳΔ֓೦

1500 ͷ TE10 ϞʔυΛҟͳΔप೾਺Ͱྭৼͨ͠৔߹

‫֤ͼٴ‬छ෺ཧྔʹ͍ͭͯઆ໌ͨͭ͠΋ΓͰ͋ΔɻՃ଎ͷ఻೻ʹؔΘΔύϥϝʔλʹ͍ͭͯද 2 ʹ·ͱΊͨɻ؅ͱͯ͠͸ۣ‫Ͱܗ‬͸ແ͘ɺߴप೾ಛੑ΍Ճ޻ੑͷ‫఺؍‬

͜͜·Ͱઆ໌͖ͯͨ͠Α͏ʹɺҎԼͷ߲໨͕֬ೝͰ

͔Βରশੑͷྑ͍ԁ౵‫ܗ‬ͷ΋ͷ͕Ұൠతʹ༻͍ΒΕ

͖Δɻ

Δɻ͜͜Ͱ͸ɺ࣍અҎ߱ͷͨΊʹԁ౵ಋ೾؅ʹ͍ͭͯ• ಋ೾؅಺ͰͷҐ૬଎౓͸ޫ଎ΑΓେ͖͍͜ͱɺి
࣓೾ͷप೾਺͕ߴ͘ͳΔʹ͍ͭͯޫ଎ʹۙͮ͘͜ͱ• ‫܈‬଎౓͸प೾਺͕ߴ͘ͳΔʹͭΕͯେ͖͘ͳΓɺޫ଎ʹۙͮ͘͜ͱ• Ґ૬଎౓ͱ‫܈‬଎౓ͷੵ͸ޫ଎ͷೋ৐ͱͳΔ͜ͱ

۩ମతͳి࣓৔੒෼ͷ‫ܗ‬Λ঺հ͓ͯ͘͠ɻԁ౵ಋ೾؅ʹ͓͍ͯ΋ɺΧοτΦϑप೾਺Λ kc ͱࣔ͢ɻۣ‫ܗ‬ಋ೾؅ͷ kc Λԁ౵‫ܕ‬ಋ೾؅ͱ kc ͱͯ͠ஔ͖‫͑׵‬Δ͜ͱ
Ͱɺ͜͜·Ͱʹۣ‫ܗ‬ಋ೾؅Λ೦಄ʹஔ͍ͯ঺հ֤ͨ͠
෺ཧྔ΍ͦΕΒͷؔ܎͸ͦͷ··దԠͰ͖Δɻ
ਤ 8 Ͱࣔͨ͠಺൒‫ ܘ‬a ͷԁ౵ಋ೾؅ͷ TM, TE Ϟʔυͷి࣓৔ͷ༷ࢠΛ਺ࣜΛ༻͍ͯҎԼʹॻ͖Լ͢ 12 ɻԁ౵ಋ೾؅ͷ৔߹ɺి࣓৔ͷ֤੒෼͸ϕοηϧؔ਺ͱࡾ֯ؔ਺Λ༻͍ͯද‫͖Ͱݱ‬ɺr ํ޲ͷґଘੑ͸ϕοηϧؔ਺ʹ φ ํ޲͸ࡾ֯ؔ਺ʹै͏ɻ

1
0.8

J0(x)
ρ11´=1.841

0.6
0.4

TM-modeiωt−γzEz = Emn Jm (kc r) cos(mφ)e−γ
Er =Emn Jm ′ (kc r) cos(mφ)eiωt−γzkc
γmJm (kc r)
Eφ = 2 Emnsin(mφ)eiωt−γz
r
kc
−iωεm
Jm (kc r)
Hr =Emnsin(mφ)eiωt−γz
r
kc 2
−iωε
Hφ =
Emn Jm ′ (kc r) cos(mφ)eiωt−γz
kc
ρmn
kc =
a

(2.20)

0.2(2.21)

0-0.2(2.22)(2.23)iωµm
Jm (kc r)
Hmnsin(mφ)eiωt−γz
r
kc 2
iωµ
Eφ =
Hmn Jm ′ (kc r) cos(mφ)eiωt−γz
kc
−γ
Hr =
Hmn Jm ′ (kc r) cos(mφ)eiωt−γzkc
γmJm (kc r)
Hφ = 2 Hmnsin(mφ)eiωt−γz
r
kc
Er =

ρ01=
2.405

ρ11=
3.832

2

4

-0.4
-0.6
0

x

6

8

10

(2.24)
ਤ 9: ϕοηϧؔ਺ J0 , J1 ͷ༷ࢠ(2.25)

TE-modeHz = Hmn Jm (kc r) cos(mφ)eiωt−γzJ1(x)(2.26)(2.27)(2.28)(2.29)(2.30)

ρmn ′(2.31)a͜͜ͰɺTM, TE Ϟʔυʹ͓͍ͯ γ ͱ kc ͷؔ܎ (෼kc =ද 3: ϕοηϧؔ਺ Jm ͷࠜ
❍❍
❍❍ m
0
1
2
❍❍
n
1
2.405 3.832
5.136

2
3

5.520
8.654

7.016
10.1738.417
11.620ද 4: ϕοηϧؔ਺ Jm ′ ͷࠜ
❍❍
m
❍
0
1
2
❍❍
n
❍

1
2
3

3.832
7.016
10.173

1.841
5.331
8.536

3.054
6.706
9.969

ࢄؔ܎) ͸ಉҰͷࣜ (2.6) Ͱ༩͑ΒΕΔɻ·ͨɺJm ͸
ୈ 1 छϕοηϧؔ਺Ͱ͋ΓɺϓϥΠϜ͸ม਺Ͱͷඍ෼Λҙຯ͢Δɻm = 0 ʹ͍ͭͯ͸ J0 ′ = −J1 ͷؔ܎͕͋Δɻρmn , ρmn ′ ͸ͦΕͧΕ Jm , Jm ′ ͷ n ൪໨ͷࠜͰ
͋Δɻ͜ΕΒͷ௿࣍ʹ͓͚Δ۩ମతͳ஋Λද 3, 4 ʹ
ͯɺJ0 (x), J1 (x) ͷ༷ࢠΛਤ 9 ʹࣔ͢ɻϕοηϧؔ਺ͷม਺͸શͯ kc r Ͱ༩͑ΒΕɺkc ͸ϕοηϧؔ਺ͷࠜʢେ͖͕͞θϩͱͳΔ஋ʣͱԁ౵൒೾਺͕௿͍ͷ͸ TE11 ϞʔυͰ͋Δ͜ͱ͕Θ͔Δɻ·ͨɺ࣍ʹΧοτΦϑप೾਺͕௿͍Ϟʔυ͸ TM01 ɺͦͷ࣍ʹ͸ TM11 ͱ TE01 ͕ ρ11 = ρ01 ′ ͱͳΓಉ͡஋Ͱଓ͘ɻ͜Ε͸ઌʹड़΂ͨ J0 ′ = −J1 ͱ͍͏ϕοηϧؔ਺ͷੑ࣭ʹΑΔ΋ͷͰ͋ΓɺҰൠతʹ TM1n ͱ TE0n͸‫ʹ͍ޓ‬ॖୀ͢Δؔ܎ʹ͋Δɻ͜ΕΒͷ఻೻Ϟʔυͷԁ౵அ໘Ͱͷి࣓৔ͷ༷ࢠΛਤ 10 ʹࣔ͢ɻ‫ ܘ‬a ͷൺͱͳΔɻ͜ͷͨΊɺTM, TE ͷ͍ͣΕͷϞʔ

ਤ 10 ʹࣔͨ͠఻೻Ϟʔυͷ͏ͪҰൠʹՃ଎؅Ͱి

υʹ͓͍ͯ΋ Ez ‫ ͼٴ‬Eφ ͸ r = a ͭ·Γԁ౵ͷ಺൒ࢠϏʔϜΛՃ଎͢Δͷʹ༻͍ΒΕΔͷ͕ TM01 Ϟʔ‫( ܘ‬ಋମ໘) ͰඞͣθϩͱͳΔɻ͜Ε͸ʮಋମ໘ͰͷυͰ͋ΓɺEz ͷ࠷େ஋͸ԁ౵෦த৺ʹଘࡏ͢Δɻ΋ి৔ͷ઀ઢ੒෼͕θϩʯͱ͍͏‫ڥ‬ք৚݅Λৗʹຬͨ͢͠ɺిࢠ଎౓ͱ TM01 ϞʔυͷҐ૬଎౓͕౳͚͠Ε͜ͱΛҙຯ͢Δɻ͹ɺిࢠϏʔϜΛՃ଎؅ʹ௨աͤ͞Δ͜ͱͰྦྷੵతͳ

ԁ౵ಋ೾؅ͷ఻೻Ϟʔυͷ͏ͪ࠷΋ΧοτΦϑप೾Ճ଎͕ՄೳͱͳΔɻ͔͠͠ɺਤ 6 Ͱࣔͨ͠Α͏ʹ௨ৗ਺͕௿͍ͷ͸ɺϕοηϧؔ਺ͷ͕ࠜ࠷΋খ͘͞ͳΔ৔ͷಋ೾؅͸શͯͷप೾਺ʹ͓͍ͯޫ଎Λ্ճΔҐ૬଎߹Ͱ͋Δɻैͬͯɺද 3, 4 ͔Βɺ࠷΋ΧοτΦϑप౓Λ͓࣋ͬͯΓɺిࢠ଎౓Λ͜Εͱ౳͘͢͠Δ͜ͱ͸

12 ຊࣜΛಋग़͢Δࡍɺੵ෼ఆ਺ͱͯ͠ φ

0 ͕ࡾ֯ؔ਺಺ʹಘΒΕΔ͕ɺຊ࣭తʹҙຯΛ੒͞ͳ͍ͨΊ͜͜Ͱ͸ φ0 = 0 ͱ͢Δɻ‫ݪ‬ཧతʹෆՄೳͰ͋Δɻͦ͜Ͱɺ࣮ࡍʹ͸Ճ଎؅಺Ͱ

ՌΛಋ͘͜ͱ͕Ͱ͖Δ [4]ɻ࣮ࡍɺTE01 Ϟʔυ͸प೾਺ʹର͠‫ݮ‬ਰ͕୯ௐ‫ݮ‬গ͢Δͱ͍͏ಛੑΛ࣋ͪɺϛϦ೾ͳͲߴ͍प೾਺Λ࣋ͭి࣓೾ͷ௕‫޲ʹૹ఻཭ڑ‬͍͍ͯΔɻ͜Ε͸ΑΓߴ࣍ͷ TE0n ϞʔυͰ΋ಉ༷Ͱ
͋Δɻ

y

x
-1.0

0.0

1.0

ਤ 10: ԁ౵ಋ೾؅ͷ୅දతͳ఻೻Ϟʔυͷి࣓৔ͷ༷ࢠɻΧοτΦϑप೾਺ͷ௿͍΋ͷΛॱʹ 4 ͭࣔ͢ɻͷి࣓೾ͷҐ૬଎౓Λ஗͘͢ΔΑ͏ͳ޻෉Λߨ͡Δɻ͜Εʹ͍ͭͯ͸࣍અҎ߱Ͱઆ໌͢Δɻ࠷‫ ʹޙ‬TE01 Ϟʔυʹ͍ͭͯগ͠ิ଍͓ͯ͘͠ɻਤ

10 ʹࣔͨ͠Α͏ʹɺTE01 ϞʔυͰ͸ి࣓৔͸‫ʹڞ‬ԁ౵؅಺෦ʹूத͓ͯ͠Γɺಋମͷ͋Δ֎प෦Ͱ͸ऑ͘ͳ͍ͬͯΔɻ͜Ε͸ଞͷ఻೻Ϟʔυͱൺ΂ಋମ໘Ͱͷଛࣦ͕গͳ͍͜ͱΛ͓ࣔࠦͯ͠Γɺղੳతʹ΋͜ͷ݁ਐߦ೾‫ܕ‬Ճ଎؅3lલઅͷಋ೾؅ʹ͍ͭͯͷٞ࿦Λ΋ͱʹɺిࢠՃ଎
؅ɺຊઅͰ͸·ͣਐߦ೾‫ܕ‬Ճ଎؅ʹ͍ͭͯઆ໌͢Δɻਐߦ೾‫ܕ‬Ճ଎؅ͷ໛ࣜਤΛਤ 11 ʹࣔ͢ɻਐߦ೾‫ܕ‬Ճ଎؅Ͱ͸ߴप೾‫͔ݯ‬Βಋ͔Εͨߴप೾͸‫ݮ‬ਰ͠ͳ͕ΒՃ଎؅ΛਐΈɺ୺෦ʹୡͨ͠‫ޙ‬͸Ճ଎؅͔ΒऔΓग़͞zΕμϛʔϩʔυʹࣺͯΒΕΔɻਤ 12: प‫ظ‬తʹ݀ͷ͋Δԁ൫ (σΟεΫ) Λ͍ΕͨՃ

3.1σΟεΫϩʔυՃ଎؅ͷి࣓೾ԁ౵ಋ೾؅Λ఻೻͢Δి࣓೾ͷ TM Ϟʔυ͸؅࣠ํ޲ʹి৔੒෼͕ଘࡏ͢ΔͨΊɺ͜ΕΛిࢠϏʔϜͱ࣌ؒతʹಉ‫ͤ͞ظ‬Δ͜ͱͰՃ଎͕ՄೳͱͳΔɻ͔͠଎؅ͷ໛ࣜਤɻz ࣠ํ޲ʹप‫ ظ‬l Ͱແ‫ʹݶ‬σΟεΫ͕ฒͿͱԾఆ͢Δɻ

3.2‫ج‬ຊ೾ͱۭؒߴௐ೾

͠ɺલઅͰઆ໌ͨ͠Α͏ʹಋ೾؅Λ఻೻͢Δి࣓৔ͷແ‫ʹݶ‬σΟεΫߏ଄͕࿈ͳΔՃ଎؅ʹ֯प೾਺ ω ͷҐ૬଎౓͸ޫ଎ΑΓେ͖͍ͨΊɺ͍͔ͳΔՙిཻࢠͱ

ి࣓೾Λ఻೻ͤ͞Δͱɺप‫ظ‬త‫ڥ‬ք৚݅Λຬͨ͠ಉ͡

΋ಉ‫͍ͳ͖Ͱظ‬ɻ͜ͷ໰୊Λղܾ͢ΔͨΊʹ࠷΋Α͘प೾਺ͱͳΔ͕Ґ૬଎౓ͷҟͳΔి࣓೾͕ແ਺ʹ‫ݱ‬༻͍ΒΕ͍ͯΔखஈ͕ɺத৺ʹ݀ͷۭ͍ͨۚଐͷ࢓੾Εɺి࣓৔ͷ֤੒෼͸͜ΕΒͷ࿨ͱͯ͠ද‫͞ݱ‬ΕΔɻΓ (σΟεΫ) Λ؅࣠ํ޲ʹप‫ظ‬తʹૠೖ͢Δํ๏Ͱ

ྫͱͯ͠ TMmn ϞʔυΛߟ͑Δͱɺͦͷ࣠ํ޲ి৔

͋Δɻਤ 12 ʹͦͷ໛ࣜਤΛࣔ͢ɻ͜ͷ̎ຕͷσΟεEz ͸ࣜ (2.20) ΑΓ࣍ͷΑ͏ʹॻ͚Δ 13 ɻΫʹ࢓੾ΒΕͨԁ౵‫ܗ‬ঢ়ͷ۠ؒΛηϧߏ଄ (ηϧ) ͱ‫ͼݺ‬ɺಛʹਤ 12 ͷΑ͏ͳՃ଎؅͸σΟεΫϩʔυ‫ܕ‬ߏ଄ମ (disk-loaded structure) ͱ‫ݺ‬͹ΕΔɻ؅࣠ํ޲ʹσΟεΫ͕ଘࡏ͢Δ৔߹ɺ఻೻͢Δి࣓೾͕σΟεΫΛ௨ա͢Δ౓ʹೖࣹ೾ͱ൓ରํ޲ʹਐΉ൓ࣹ೾͕ੜ͡Δ͜ͱʹͳΔɻͭ·Γɺಋ೾؅Ͱ͸અ

2.5 Ͱઆ໌ͨ͠Α͏ʹฏ໘೾͕δάβΫʹંΕ‫͕ۂ‬Γͳ͕Β +z ํ޲ʹਐΉͱߟ͑Δ͜ͱ͕Ͱ͖͕ͨɺσΟεΫϩʔυ‫ܕ‬Ճ଎؅Ͱ͸ −z ํ޲ʹਐΉ੒෼΋ߟྀ͢

Ez =

∞
$

as Jm (kc,s r) cos(mφ)ei(ωt−βs z)

(3.1)

s=−∞

2πs
, (s = 0, ±1, ±2, · · · )l2kc,s= (ω/c)2 − βs 2β ≡ β s = β0 +

(3.2)
(3.3)

͜Ε͸ϑʔϦΤ‫਺ڃ‬ͷ࿨ͱͳ͓ͬͯΓɺas ͸ϑʔϦΤ‫਺ڃ‬ͷ֤܎਺ɺβ0 ͸ୈ 1 ϒϦϧΞϯκʔϯ (brillouinzone) Ͱߟ͑ͨͱ͖ͷ఻೻܎਺ͱͳΔɻࣜ (3.1) ͸ۭΔඞཁ͕Ͱͯ͘ΔɻಛʹσΟεΫ͕ෳ਺ଘࡏͦ͠ͷؒؒతʹ l/s ͷप‫ظ‬Λ࣋ͬͨ೾ͷ߹੒ͱΈͳͤɺs = 0

ִ l ͕प‫ظ‬తͳ৔߹ɺ఻೻͞ΕΔి࣓೾ͷ೾໘͸ಋମͷ೾͸‫ج‬ຊ೾ɺͦͷଞͷ s = ±1, ±2, · · · Ͱ༩͑ΒΕଆ໘ʹՃ֤͑σΟεΫ͔Βͷ൓ࣹ೾ͱͷप‫ظ‬తͳॏͶΔ೾͸ۭؒߴௐ೾ (spatial harmonicsʣͱ‫ݺ‬͹ΕΔɻۭؒߴௐ೾͸શͯ‫ج‬ຊ೾ͱಉ͡प೾਺ɾ‫܈‬଎౓Λ࣋ͭ߹ΘͤͱͳΔɻ͕Ґ૬଎౓͸ҟͳΓɺҎԼͷࣜͰ༩͑ΒΕΔɻ

RF 源

ダミーロードvp,s =

ωω=βsβ0 + 2πs/l(3.4)·ͨɺࣜ (3.1, 3.2) Ͱ༩͑ΒΕΔؔ܎͸ฒਐରশੑΛRF 波࣋ͪɺFloquet ͷఆཧ·ͨ͸‫ݻ‬ମ෺ཧֶͰ͍͏ BlochͷఆཧΛຬͨ͢ 14 ɻҰൠʹۭؒߴௐ೾ͷৼ෯ as ͸‫ج‬ຊ೾ (s = 0) ͕࠷΋େ͖͘ɺ࣍ʹ (s = −1), (s = 1) ͱ࣍਺͕େ͖͘ͳ

電子ビームΔʹ࿈Εͯখ͘͞ͳΔɻैͬͯɺप‫ظ‬తͳՃ଎ߏ଄Λ加速管
ਤ 11: ਐߦ೾‫ܕ‬Ճ଎؅ͷ໛ࣜਤɻ΋ͬͨ΄ͱΜͲͷՃ଎؅Ͱ͸‫ج‬ຊ೾Λిࢠͱಉ‫ͤ͞ظ‬ΔΑ͏ઃ‫͞ܭ‬ΕΔɻ
13 ଞͷ੒෼΋ಉ༷ʹॻ͖Լ͢͜ͱ͕Ͱ͖Δɻ
14 Floquet ͷఆཧ,Bloch ͷఆཧ͸࢖༻͞ΕΔ෼໺͕ҟͳΔ͚ͩͰҙຯ͸ಉ͡Ͱ͋Δɻ΋ͷ͕ɺηϧ͕࣓ؒ৔Ͱ݁߹͢Δ৔߹ͷ෼ࢄ‫ۂ‬ઢͰω
周期的構造なし(導波管)

͋Δɻԣ࣠͸ɺηϧؒͷి࣓৔ͷҐ૬ਐΈʹରԠͯ͠

ディスクあり(高次モード)͓ΓɺҐ૬ਐΈྔʹԠͨ͡ి࣓৔ͷ఻೻ϞʔυΛҐ૬Ϟʔυͱͯ۠͠ผ͢Δɻࣜ (3.5) ΍ਤ 13 ͔ΒΘ͔ΔΑ͏ʹɺి৔Ͱ݁߹͢Δ৔߹͸θϩϞʔυ (β = 0 ͱ

ωπ

s=-1

ͷަ఺) ͕࠷΋प೾਺͕௿͘ɺπ Ϟʔυ (β = π/l) ͕

s=0

大ディスクあり(基本モード)ω0 = ωc
ω=-cβ

ω=cβ

࠷΋प೾਺͕ߴ͍ɻ࣓৔Ͱ݁߹͢Δ৔߹͸͜ͷ܏޲͸‫ٯ‬స͢Δɻ͜ΕΒͷ܏޲͸ి࣓৔෼෍ͷ༷ࢠʹΑΓɺσΟεΫΛִͯͨ྆ଆͷి࣓৔ͷΤωϧΪʔ͕૿Ճ͢
Δ͔‫ݮ‬গ͢Δ͔Ͱܾ·ΔɻՃ଎؅಺ͷి࣓೾͕ి৔ɾ

0
0
β

-π/l

π/l

࣓৔ͷͲͪΒͰ݁߹͢Δ͔͸֤఻೻Ϟʔυͷి࣓৔෼෍ʹΑͬͯҟͳΔɻਤ 13: σΟεΫϩʔυՃ଎؅ͷ෼ࢄؔ܎ɻ੺৭ͷ࣮

ਤ 13 ʹ͓͍ͯɺ૒‫ۂ‬ઢͰࣔ͞ΕΔ఺ઢ (੺৭) ͕ૉઢ͕σΟεΫϩʔυՃ଎؅ͷ‫ج‬ຊϞʔυͷ෼ࢄؔ܎Ͱ

ͷԁ౵ಋ೾؅ͷ෼ࢄؔ܎Ͱ͋Γɺ࣮ઢ (੺৭) ͕σΟ͋Γɺ੺৭ͷ఺ઢ͕ૉͷಋ೾؅ͷ΋ͷͰ͋Δɻ·ͨɺεΫϩʔυՃ଎؅ͷ‫ج‬ຊϞʔυͷ෼ࢄؔ܎Ͱ͋ΔɻୈҰ఺࠯ઢͷ‫ۂ‬ઢ͸ߴ࣍Ϟʔυͷ෼ࢄؔ܎ɺ੨৭ͷ࣮ઢ

1 ϒϦϧΞϯκʔϯ (−π/l ≤ β ≤ π/l ) ʹ͓͍ͯ྆ऀΛൺ΂ΔͱθϩϞʔυ෇ۙ (β ∼= 0) Ͱ͸෼ࢄؔ܎͸͋·ΓมԽ͠ͳ͍ɻ͔͠͠ɺπ Ϟʔυʹۙͮ͘ʹ͕ͭޫ଎ͷిࢠͷ‫ͳͱ੻ي‬Δɻ

3.3

ΕͯɺͦͷมԽ͸େ͖͘ͳΔɻ͜Ε͸؅಺೾௕ͷ൒෼σΟεΫϩʔυՃ଎؅ͷ෼ࢄؔ܎ࣜ (3.2, 3.3) ΑΓσΟεΫϩʔυՃ଎؅ͷ෼ࢄؔ܎͕
प‫ظ‬ੑΛ࣋ͭ͜ͱ͕Θ͔ΔɻՃ଎؅ͷ෼ࢄؔ܎͸౳Ձλg /2 ͕प‫ ִؒظ‬l ʹۙ͘ͳΓɺҰͭલͷσΟεΫͱ࣍ͷσΟεΫͰͷ൓ࣹ೾͕ಉҐ૬ʹ͍ۙঢ়ଶͰॏͳΓ߹͏Α͏ʹͳΔͨΊͰ͋Δɻ͜ͷ৔߹ɺଟ͘ͷσΟεճ࿏Λ༻͍ͨղੳʹΑΓൺֱత؆୯ʹ‫ٻ‬ΊΒΕΔ [5]ɻΫ͔Βͷ൓ࣹ೾͕‫ڧʹ͍ޓ‬Ί͍͋ɺ఻೻͢Δ೾ͷ‫܈‬଎ηϧؒͷి࣓೾͕ి৔ʹΑΓ݁߹͢Δ৔߹ [6–8] ͷ෼౓͕஗͘ͳΔɻࢄؔ܎͸ɺω = ω0%1 + κC (1 − cos βl)(3.5)࣓৔ʹΑΓ݁߹͢Δ৔߹͸

ω0ω=%1 + κL (1 − cos βl)3.4ύεόϯυͱετοϓόϯυ

ਤ 13 ʹ͓͍ͯి࣓೾͕‫ج‬ຊϞʔυͰ఻೻Ͱ͖Δप

(3.6)

ͱͳΔɻͦΕͧΕ κ ͸ηϧؒͷ݁߹ͷ‫͞ڧ‬Λද͢ఆ਺(݁߹ఆ਺) Ͱ͋ΓɺԼ෇จࣈͷ C,L ͸ͦΕͧΕ౳Ձճ࿏ͷ੩ి༰ྔ C ͱΠϯμΫλϯε L ʹ༝དྷ͍ͯ͠
Δɻω0 ͸θϩϞʔυͰͷ֯प೾਺Ͱ͋Γɺ͜ͷҙຯ͸‫ʹޙ‬આ໌͢Δɻ·ͨɺಉ༷ʹ π Ϟʔυͷ֯प೾਺Λωπ ͱද͢ɻ
݁߹ఆ਺ κ ͸ి৔ɾ࣓৔݁߹ͷ͍ͣΕͷ৔߹ʹ͓͍೾਺͸ ω0 ͔Β ωπ ·Ͱͷ۠ؒͰ͋Γɺ͜ΕΛύεόϯυ (pass band) ͱ͍͏ɻ͞Βʹप೾਺Λ্͍͛ͯ͘
ͱɺҰ఺࠯ઢͰࣔͨ࣍͠ͷ఻೻Ϟʔυͷ ωπ Ҏ্ʹͳΔ·Ͱి࣓೾͸఻೻Ͱ͖ͳ͍ɺ͜ͷ۠ؒΛετοϓό
ϯυ (stop band) ͱ͍͏ɻ͜ͷΑ͏ͳ෼ࢄؔ܎͕ੜ͡
Δͷ͸؅࣠ํ޲ (z ࣠) ʹઃ͚ΒΕͨप‫ظ‬ੑʹΑΔ΋ͷͰɺ‫ݻ‬ମ෺ཧֶʹ͓͍ͯ݁থ͕௒֨ࢠߏ଄Λ࣋ͭͱΤωϧΪʔ෼ࢄؔ܎ʹෆ࿈ଓੑ͕ੜ͡Δ‫ݱ‬৅ͱྨࣅ͍ͯ͠Δɻͯ΋ɺσΟεΫ؅ͷ։‫͕ޱ‬খ͍͞‫Ͱݶۃ‬͸ κ = 0 ͱͳΓɺω = ω0 ͱͳΔɻ͜ͷঢ়ଶͰ͸ηϧؒͷ݁߹͕ͳ͘ͳΓɺ֤ηϧ͸ಠཱͱͳΔɻ
σΟεΫͷ։‫͕ޱ‬ൺֱతେ͖͍ (͢ͳΘͪηϧؒͷ݁߹͕େ͖͍) ৔߹ʹ͍ͭͯɺσΟεΫϩʔυՃ଎؅ͷ෼ࢄؔ܎Λਤ 13 ʹࣔ͢ɻ‫ج‬ຊϞʔυͱ࣮ͯ͠ઢ (੺৭) Ͱࣔͨ͠΋ͷ͕ɺηϧ͕ؒి৔Ͱ݁߹͢Δ৔߹ͷ෼ࢄ‫ۂ‬ઢɺߴ࣍Ϟʔυ (੺ͷҰ఺࠯ઢ) ͱͯࣔͨ͠͠

3.5

π ϞʔυͱͦͷଞͷϞʔυ؅಺೾௕ͷ൒෼ λg /2 ͕ηϧִؒ l ͱҰக (β = π/l)ͨ͠ͱ͖ 15 ɺྡΓ߹͏ηϧؒͷి࣓৔ͷҐ૬ࠩ͸ π
ͱͳΔ (ਤ 14 ࢀর) ͨΊɺ൓ࣹ೾Ͳ͏͠͸ 2π ͷҐ૬ࠩΛ࣋ͭ͜ͱʹͳΔɻͭ·Γɺશͯͷ൓ࣹ೾͸ಉ͡Ґ
15 1 ೾௕ʹର͠ 2 ຕͷσΟεΫ͕ͪΐ͏Ͳ؅಺೾௕ͷ൒෼ͷҐஔؔ܎ͱͳΔͱ͖ɻ

Ez

3.6

λg

1

ిࢠͱͷಉ‫ظ‬ਤ 13 Ͱ੨ઢͰࣔͨ͠ͷ͕ ω = ±cβ ͱͳΔϒϦϧπモード

ΞϯκʔϯதͰͷޫ଎ͷిࢠϏʔϜͷ‫͋Ͱ੻ي‬Γɺ͜

0ͷ௚ઢͱ෼ࢄ‫ۂ‬ઢͱ͕ަΘΔ఺ʢਤதͷ੺‫ؙ‬ʣͷҐ૬Ϟʔυ͕ిࢠͱಉ‫͖Ͱظ‬Δɻͭ·Γɺ্‫ه‬Ґ૬Ϟʔυ(Ճ଎Ϟʔυͱ΋‫ݺ‬͹ΕΔ) ͕ిࢠͷྦྷੵతͳՃ଎ʹ‫د‬༩͢Δɻ͞Βʹɺిࢠͷ‫੻ي‬Λ୧͍ͬͯ͘ͱߴ࣍఻

-1
λg
1

೻Ϟʔυͷ෼ࢄ‫ۂ‬ઢͱ΋ަΘΔ͜ͱ͕Θ͔Δɻ͜Ε͸ߴ࣍Ϟʔυ͕ଘࡏ͢Δ৔߹ɺަ఺Ͱ༩͑ΒΕΔߴ࣍఻

π/2モード0೻ɾҐ૬ϞʔυͷӨ‫ڹ‬΋ిࢠ͸ྦྷੵతʹड͚Δ͜ͱΛҙຯ͠ɺՃ଎؅ʹߴ࣍Ϟʔυ͕ଘࡏ͢Δ৔߹͸஫ҙ͕ඞཁͱͳΔɻ-1λg࣍ʹɺ໨తͱ͢Δ఻೻ɾҐ૬ϞʔυΛ࣮‫͢ݱ‬Δप೾

1਺ͷి࣓೾ͷΈΛՃ଎؅ʹ఻೻ͤ͞Δ͜ͱΛߟ͑Δɻ
͜ͷ৔߹ɺલड़ۭͨؒ͠ߴௐ೾΋ಉ࣌ʹ఻೻͞ΕΔɻ

2π/3モード0ਤ 13 ʹ͓͍ͯ੺ന‫Ͱؙ‬දͨ͠఺͕ s = −1 ͷۭؒߴௐ೾Λࣔ͠ɺs ͷ஋ͷҟͳΔಉ༷ͳ఺͸ਤࣔͨ͠ൣғ֎ʹ΋ແ਺ʹଘࡏ͢Δɻલड़ͨ͠Α͏ʹɺۭؒߴௐ೾

-1
0

l

2l
z

3l

4l

͸Ճ଎Ϟʔυͱಉ͡‫܈‬଎౓ (‫ۂ‬ઢͷޯ഑) Λ΋͕ͭҐ૬଎౓͕ҟͳΔͨΊɺిࢠͱྦྷੵతͳ૬‫༻࡞ޓ‬Λ͢Δ

ਤ 14: ֤Ґ૬Ϟʔυʹ͓͚Δ Ez ͷ༷ࢠɻ͜ͱ͸Ͱ͖ͣɺे෼ͳՃ଎؅‫཭ڑ‬Λ૸Δ͏ͪʹۭؒߴ૬Ͱ‫׬‬શʹॏͳΓ͋͏ 16 ɻैͬͯɺՃ଎؅͕ແ‫ʹݶ‬௕

ਤ 15 ʹσΟεΫϩʔυ‫ܕ‬Ճ଎؅Λਐߦ͢Δి࣓೾

͘ి࣓೾ͷนଛࣦ͕ͳ͚Ε͹ɺ൓ࣹ೾͸ఆৗঢ়ଶͰ͸ͱ͜Εʹಉ‫ͯ͠ظ‬ਐΉిࢠϏʔϜͷ໛ࣜਤΛҰఆ࣌ؒೖࣹ೾ͱಉ͡ৼ෯ʹͳΔɻ͜Ε͸ɺೖࣹ೾ͱ൓ࣹ೾Ͱ

ຖͷεφοϓγϣοτͱͯࣔ͢͠ɻ࣌ؒൃలʹ൐͍ɺఆࡏ೾͕ੜ͡Δ͜ͱΛҙຯ͠ɺఆࡏ೾Ͱ͸೾͸ਐߦ͠ి࣓೾ͱిࢠϏʔϜ͕Ճ଎؅ΛਐΉ͕ɺిࢠϏʔϜͱͳ͍ͨΊɺ‫܈‬଎౓͸θϩͱͳΔɻ͜ͷҐ૬Ϟʔυ͸ πి࣓೾ͷ૬ରҐ૬͸ৗʹอͨΕΔɻ͜ͷͨΊిࢠϏʔϞʔυͱ‫ݺ‬͹Εɺఆࡏ೾‫ܕ‬Ճ଎؅ͰΑ͘༻͍ΒΕΔɻ

Ϝ͸Ճ଎؅Λ௨ա͍ͯ͠ΔؒɺྦྷੵతʹՃ଎͞ΕΔ͜

·ͨ π/2 Ϟʔυʹ͍ͭͯ΋ಉ༷ʹɺ؅಺೾௕ͷ 4 ෼ͱʹͳΔɻ·ͨɺਤ 15 Ͱ͸ۭؒߴௐ೾Λߟྀ͍ͯ͠

ͷ 1 (λg /4) ͕ηϧִؒ l ͱҰக͢Δͱ͖

17(ਤ 14 ࢀর)ɺηϧؒͷి࣓৔ͷҐ૬ࠩ͸ π/2 ͱͳΔɻ͜ͷ৔߹ɺྡΓ߹͏σΟεΫ͔Βͷ൓ࣹ೾ͷҐ૬ࠩ͸ π ͱͳௐ೾͔Βిࢠ͕ಘΔՃ଎ήΠϯ͸θϩͱͳΔɻ

ͳ͍ɻ
ͱ͜ΖͰɺࣜ (3.3) ʹ͓͍ͯɺ‫ج‬ຊ೾ (s = 0) ͷҐ૬଎౓͕ޫ଎‫ࢠిͼٴ‬଎౓ͱҰக͢Δ৔߹ɺΓɺ‫ʹ͍ޓ‬ଧͪফ͠߹͏͜ͱʹͳΔɻͭ·Γɺ൓ࣹ೾͸ଘࡏ͠ͳ͍΋ͷͱΈͳͤΔɻ͜Ε͕Ճ଎؅͕ਐߦ೾

2kc,0= (ω/c)2 − β0 2 = 0‫ܕ‬Ճ଎؅ͱͯ͠੒ཱ͢Δঢ়ଶͰ͋Δɻθϩͱ π ϞʔυͱͳΔɻैͬͯɺࣜ (3.1) ʹ͓͍ͯ m = 0 ͷ৔߹ɺҎ֎ͷશͯͷҐ૬Ϟʔυʹ͓͍ͯ΋ಉ༷ͷٞ࿦͸੒ཱ͠ɺ൓ࣹ೾͸ଧͪফ͞Εɺਐߦ೾‫ܕ‬Ճ଎؅ͱͯ͠੒ཱ

J0 (kc,0 r) = J0 (0) = 1 ͱ͍͏ؔ܎͕੒ΓཱͭͨΊɺEz
͸ r ʹґଘ͠ͳ͍͜ͱʹͳΔɻ͜ͷ৔߹ɺిࢠϏʔϜ͢Δɻͨͩ͠ɺҐ૬ϞʔυʹԠͯ͡൓ࣹ೾͕‫׬‬શʹଧ͕‫͋ʹ޲ํܘ‬Δେ͖͞Λ͍࣋ͬͯͯ΋ɺड͚ΔՃ଎ి

ͪফ͠߹͏ͨΊʹඞཁͳσΟεΫ਺͕ΑΓଟ͘ඞཁͱ

৔͸ಉ͡ͱͳΔɻͳΔɻ·ͨɺ‫܈‬଎౓͸ π/2 Ϟʔυ͕࠷΋େ͖͘ɺθϩ΍ π Ϟʔυʹۙͮ͘ʹͭΕখ͘͞ͳΔ (ਤ 13 ࢀর)ɻ

3.7

‫܈‬଎౓

‫܈‬଎౓͸Ճ଎؅ͷઃ‫͍͓ͯʹܭ‬प೾਺ͷ࣍ʹॏཁͳ
16 ‫ݻ‬ମ෺ཧֶʹग़ͯ͘Δ Bragg ৚݅Λຬͨ͢ɻ
17 1 ೾௕ʹର͠ 4 ຕͷσΟεΫΛ౳ִؒʹઃ͚Δͱ͖ύϥϝʔλͩͱ͍͑Δɻ؅಺ʹి࣓೾͕ॆర͞ΕΔͷEz
1

͘ͳΔɻ‫܈‬଎౓͸͜ͷΑ͏ͳࣄ৘Λ౿·͑όϥϯεྑ

t=0ܾ͘ఆ͞ΕΔ΂͖Ͱ͋Δɻ

0

3.8

γϟϯτΠϯϐʔμϯεՃ଎؅ͷ΋͏ҰͭͷύϥϝʔλɺσΟεΫִؒʢҐ-1
t=1/3T1૬Ϟʔυʣ͸୯Ґ௕͋ͨ͞ΓͷγϟϯτΠϯϐʔμϯε (shunt impedance) rsh ͕࠷େͱͳΔΑ͏બ͹ΕΔͷ͕๬·͍͠ɻ͜ͷ৚݅Λ‫ٻ‬ΊΔͨΊʹɺ࢝ΊʹՃ଎

0

؅͕ 1 ηϧͰߏ੒͞ΕΔ৔߹ɺͭ·ΓϐϧϘοΫε

(pillbox) ‫ܕ‬ͷ RF ۭಎ 18 ʹ͍ͭͯωοτͷγϟϯτ
Πϯϐʔμϯε Rsh Λߟ͑Δɻͦͯ࣍͠ʹଟηϧͷՃ-1
t=2/3T1଎؅ʹ͍ͭͯ࿩Λൃలͤ͞Δɻrsh ͱ Rsh ͷҧ͍͸ɺલऀ͕୯Ґ௕͋ͨ͞Γͷྔɺ‫ۭ͕ऀޙ‬ಎ (Ճ଎؅) શ0ମͱͯ͠ͷྔͱͯ͠ఆٛ͞ΕΔ఺Ͱ͋Δɻ

-1
1

3.8.1

t=T

ϐϧϘοΫε‫ۭܕ‬ಎຊςΩετͰ͸γϟϯτΠϯϐʔμϯε Rsh ͸ɺ

0P.B. Wilson [9] ͷఆٛΛ༻͍Δ 19 ɻ

-1

Rsh =

0

l

2l
z

3l

4l

ਤ 15: Ճ଎؅ΛਐΉి࣓೾ (੺ઢ) ͱిࢠϏʔϜ (੨‫ )ؙ‬ͷεφοϓγϣοτɻ఻೻͢Δి࣓೾͸Ґ૬Ϟʔυ 2π/3 ͷ‫ج‬ຊ೾ͷΈΛਤࣔͨ͠ɻ

Vc2(Ez T l)2
=
Pc
Pc

(3.9)

Pc ͱ l ͸ͦΕͧΕนফඅిྗͱηϧͷ௕͞Ͱ͋ΓɺVc
͸ਖ਼‫ݭ‬೾Ͱ࣌ؒมԽ͢Δి৔಺ΛిࢠϏʔϜ͕ηϧ 1ͭΛ௨Γൈ͚Δؒʹड͚ΔՃ଎ిѹͰ͋ΔɻT ͸ Vc
ͱ Ez Λ݁ͼ͚ͭΔ܎਺Ͱ͋ΓɺτϥϯδοτλΠϜ
ϑΝΫλʔ (transit time factor) ͱ‫ݺ‬͹Εɺৗʹ 1 ҎԼͱͳΔɻʹඞཁͳ࣌ؒɺϑΟϦϯάλΠϜ͸Ճ଎؅ͷ௕͞Λ L
ͱͯ͠τf = L/vg(3.7)

ͱ༩͑ΒΕΔɻϑΟϦϯάλΠϜ͕௕͍ͱి࣓೾͕Ճ
଎؅Λຬͨ͢ͷʹ͕͔͔࣌ؒΔͨΊɺೖࣹ೾ʹ͸௕͍ύϧε෯͕ඞཁͱ͞ΕΔɻ͞ΒʹɺՃ଎؅಺ͷߴप೾ిྗ P ͸୯Ґ௕͋ͨ͞Γͷ஝ੵΤωϧΪʔ w ͱҎԼͷؔ܎ʹ͋Δɻ

P = w · vg

(3.8)ैͬͯɺ͋Δߴप೾ిྗΛԾఆͨ͠ͱ͖ɺ‫܈‬଎౓͕খ͍͞ͱ୯Ґ௕͋ͨ͞Γͷ஝ੵΤωϧΪʔ͸େ͖͘ͳΓɺۭಎʹྭৼ͞ΕΔՃ଎ి৔΋େ͖͘ͳΔɻ͜Ε͸

Pc ͸ηϧ಺ͷ஝ੵΤωϧΪʔ U ͱෛՙͷແ͍৔߹ͷ Q ஋ Qo ʹର͠ɺҎԼͷؔ܎Λ࣋ͭɻ
Q0 =

ωU
Pc

(3.10)

ैͬͯɺࣜ (3.9) ΑΓɺT 2 ͱ Q0 ͕‫ʹڞ‬େ͖͚Ε͹ Rsh΋େ͖͘ͳΔͱ༧૝Ͱ͖Δɻ͜͜Ͱ TM0n0 ϞʔυΛԾఆ͠ɺT ʹ͍ͭͯม਺Λ໌ࣔͯ͠ఆٛ͢Δͱ࣍ͷΑ͏ʹͳΔɻ
&'
&
& l/2
&
& −l/2 Ez (r, z)eiωz/ve dz &
T ≡
' l/2
E (0, z)dz
−l/2 z(3.11)

ॳ‫ظ‬ͷՃ଎ޮ཰͸େ͖͍͕ɺߴిքΛྭৼ͢ΔͨΊͷนফඅిྗʹΑΓɺߴप೾ిྗ͕‫ٸ‬଎ʹࣦΘΕΔ͜ͱ
Λҙຯ͢ΔɻҰํɺ‫܈‬଎౓͕େ͖͍৔߹ɺ஝ੵΤωϧΪʔͱՃ଎ి৔͸খ͘͞ͳΓɺՃ଎ޮ཰͕૬ରతʹ௿18 ͨͩ͠ɺՃ଎؅ͱͷൺֱ͕໨తͰ͋ΔͨΊ z ࣠ํ޲ͷࢦ਺(TMmnp ͱͨ͠ͱ͖ͷ p) Λ 0 ͱ͢Δɻ19 R = V 2 /P ͱఆٛ͢Δ͜ͱ΋͋ΔͨΊ஫ҙ͕ඞཁɻ
c
c

1

1

Rsh /Rsh,max

T2

rsh /rsh,max
0.8

0.8

2π/3
0.6

Q0 /Q0,max

0.6

0.4

0.4
0.2

0

0
0

0.2

l=
0.29λ

0.2

l=
0.44λ

0.4

0.6

0.8

1

0

π/4

l/λ
ਤ 16: ϐϧϘοΫε‫ ܕ‬RF ۭಎͷγϟϯτΠϯϐʔμϯεͷηϧ௕ґଘੑɻ

π/2
3π/4
Phase shift per cell [rad]πਤ 17: ਐߦ೾‫ܕ‬Ճ଎؅ͷ୯Ґ‫ͨ͋཭ڑ‬ΓͷγϟϯτΠ
ϯϐʔμϯεɻԣ࣠͸ηϧ͋ͨΓͷҐ૬ਐΈͰ͋ΓɺҐ૬ϞʔυʹରԠ͢Δɻ

·ͨɺిࢠϏʔϜ͕؅্࣠ (r = 0) Λ௨ա͢Δ৔߹ɺT ͸ҎԼͷΑ͏ʹॻ͖Լ͢͜ͱ͕Ͱ͖Δɻ
sin{ωl/(2ve )}T =ωl/(2ve )sin{πl/(βe λ)}=πl/(βe λ)Ͱ͸ɺRsh ͷ୅ΘΓʹ rsh ͕༻͍ΒΕɺͦͷؔ܎͸

TM01 Ϟʔυͷ৔߹ɺҎԼͰ༩͑ΒΕΔɻ
rsh =(3.12)

ͨͩ͠ɺβe = ve /c Ͱ͋Δɻిࢠ଎౓Λޫ଎ͱΈͳ͠ɺม਺Λ x = πl/λ ͱͱΔͱ࠷దͳ Rsh Λ༩͑Δηϧ௕͸ి࣓೾ͷप೾਺ͱಠ

Rsh (x) = AR

4x2
T (x)2
ρ01 (ρ01 + 2x)

rsh (x) = Ar

4x
T (x)2
ρ01 (ρ01 + 2x)(3.15)

·ͨɺࣜ (3.10) Λਐߦ೾‫ܕ‬Ճ଎؅ͷ৔߹ʹॻ͖௚͢ͱ

Q0 = −(3.13)(3.14)

ैͬͯɺTM01 Ϟʔυͷ rsh (x) ͸࣍ͷΑ͏ʹͳΔɻཱʹܾΊΔ͜ͱ͕Ͱ͖Δ [8, 10, 11]ɻྫͱͯ͠ TM010
Ϟʔυͷ Rsh (x) ΛҎԼʹࣔ͢ɻ

Rsh
l

ωw
dP/dz

(3.16)

ͱॻ͚Δɻෛ߸͸ dP/dz < 0 ͱͳΔͨΊͰ͋Δɻࣜ (3.15) ʹ͍ͭͯηϧ͋ͨΓͷҐ૬ਐΈΛԣ࣠ΛAR ͸ Ez ΍ Q0 ͷಋग़࣌ʹදΕΔఆ਺Λ·ͱΊͨ΋ͷͰ͋Γɺx ʹ͸ґଘ͠ͳ͍ɻ

ͱͯ͠ɺؔ܎Λϓϩοτͨ͠ͷ͕ਤ 17 Ͱ͋Δɻਐߦ೾‫ܕ‬Ճ଎؅ͷ৔߹ɺrsh ͸ηϧ௕Λ l = 0.29λ ͱ͢Δࣜ (3.13) Λ༻͍ͯ Rsh /Rsh,max Λϓϩοτ͢Δͱ

͜ͱͰɺ࠷େͱͳΔ͜ͱ͕Θ͔Δ [10, 11]ɻਐߦ೾‫ܕ‬

ਤ 16 ͷΑ͏ʹͳΔɻͨͩ͠ɺηϧͷ௕͞ͱ೾௕ͷؔՃ଎؅Ͱ͸ਤதʹ੺‫ͨࣔ͠Ͱؙ‬೾௕ͷ໿ 3 ෼ͷ 1 Λη܎ΛΘ͔Γ΍͘͢͢ΔͨΊɺԣ࣠͸ π ʹ஋Λ୅ೖ͠ϧ௕ͱ͢Δ 2π/3 Ϟʔυ͕Α͘༻͍ΒΕΔɻ͜Ε͸্l/λ ͱͨ͠ɻRsh ͕࠷େΛ༩͑Δͷ͸ l = 0.44λ ͷͱ͖Ͱ͋ΓɺϐϧϘοΫε‫ ܕ‬RF ۭಎͷ TM010 Ϟʔυड़ͨ͜͠ͱ͕ཧ༝Ͱ͋Δɻ·ͨɺπ Ϟʔυͱ l = 0.29λͰ͸ߴप೾ͷ൒೾௕ΑΓΘ͔ͣʹ୹͍ηϧ௕ΛબͿ͜Ͱ͸ 1.40 ഒɺ2π/3 ϞʔυͰ͸ 1.38 ഒͷ rsh ͕ಘΒΕ

2ͷϞʔυ΋͘͠͸ 2π/3 ϞʔυΛൺ΂Δͱɺl = 0.29λͱͰ࠷େͷՃ଎ޮ཰ΛಘΒΕΔ͜ͱ͕Θ͔ΔɻT ͱ

Δɻઅ 3.5 Ͱ࿦ͨ͡Α͏ʹɺπ ϞʔυͰ͸Ճ଎؅Λ఻

Q0 ͷ l/λ ґଘੑ΋‫ڵ‬ຯਂ͍ͨΊɺࢀߟͱͯ͠ਤ 16 ʹࣔͨ͠ɻ೻͢Δి࣓೾͸ఆࡏ೾ͱͳΓɺ‫ޙ‬ड़͢ΔΑ͏ʹఆࡏ೾‫ܕ‬Ճ଎؅Ͱ͸ π Ϟʔυʹ͓͍ͯ࠷΋ߴ͍ rsh ͕ಘΒΕΔɻैͬͯɺߴ͍Ճ଎ޯ഑ΛಘΔʹ͸ਐߦ೾‫ܕ‬Ճ଎؅

3.8.2

ਐߦ೾‫ܕ‬Ճ଎؅ਐߦ೾‫ܕ‬Ճ଎؅ʹ͍ͭͯͷٞ࿦ʹ໭Δɻਐߦ೾‫ܕ‬Ճ଎؅ͷΑ͏ʹඇৗʹଟ͘ͷηϧ͕ଘࡏ͢ΔՃ଎؅

ͷํ͕ఆࡏ೾‫ܕ‬Ճ଎؅ΑΓ༗རͱͳΔɻߴ͍Ճ଎ޯ഑
ΛಘΑ͏ͱ͢ΔϦχΞοΫ (ઢ‫ܕ‬Ճ଎‫͚͓ʹ )ث‬Δओ
Ճ଎؅ͱͯ͠ఆࡏ೾‫Ͱܕ‬͸ͳ͘ਐߦ೾‫͞༻࠾͕ܕ‬ΕΔ͜ͱ͕ଟ͍ͷ͸͜ͷͨΊͰ͋Δɻ

3.9

Ճ଎‫ࣜܕ‬

લઅ·ͰͰਐߦ೾‫ܕ‬Ճ଎؅ʹ͍ͭͯҰൠతͳੑ࣭
Λઆ໌͕ͨ͠ɺՃ଎؅఻೻தͷߴप೾ిྗͷ‫ݮ‬ਰʹ͍ͭͯ͸ແࢹ͖ͯͨ͠ɻ͔͠͠ɺ࣮ࡍʹ͸Ճ଎؅ʹೖࣹ͞Εͨి࣓೾͸‫཭ڑ‬Λ͢͢ΉʹͭΕͯిྗΛࣦ͏ɻ‫ݮ‬ਰఆ਺ (total attenuation parameter) τ0 ͸‫ݮ‬ਰ܎਺

α(z) Λ༻͍ͯ࣍ͷΑ͏ʹॻ͚Δɻ
τ0 =

( L

α(z)dzैͬͯɺՃ଎؅શମΛ௨ͯ͠ͷՃ଎ిѹ͸ɺ
( L
VCI =
E0 e−τ0 z/L dz
0
)
%
+
2 *
= rsh P0 L
1 − e−τ0(3.23)
τ0ͱ‫·ٻ‬ΔɻCI ‫ܕ‬ͷՃ଎ిѹ͸ τ0 ʹ͍ͭͯϐʔΫ஋Λ࣋ͬͨؔ਺‫ͯ͠ͱܥ‬ද͞ΕΔɻ࠷େ஋͸ τ0 = (eτ0 − 1)/2 Λຬ(3.17)

ͨ͢ͱ͖Ͱ͋Γɺτ0 ≃ 1.26 ͷ৔߹Ͱ͋Δɻ0

L ͸࣮ޮతͳՃ଎؅ͷ௕͞Ͱ͋Δɻ·ͨɺ͜͜Ͱ΋؅࣠ํ޲Λ z ͱ͢Δɻ

3.9.2

ఆՃ଎ޯ഑‫ܕ‬CI ‫ʹܕ‬ର͠ɺՃ଎؅಺Ͱͷߴप೾ిྗ‫ݮ‬ਰʹ߹Θͤͯηϧ‫ܗ‬ঢ়Λมߋ͠ɺՃ଎؅શ௕ʹ౉ͬͯՃ଎ిѹ

3.9.1

ఆΠϯϐʔμϯε‫ܕ‬͕౳͘͠ͳΔΑ͏ʹઃ‫͞ܭ‬ΕΔՃ଎؅ΛఆՃ଎ޯ഑

Ճ଎؅ͷηϧ‫ܗ‬ঢ়͕શ௕ʹΘͨΓ౳͍͠৔߹ɺ୯Ґ௕͋ͨ͞ΓͷγϟϯτΠϯϐʔμϯε͸Ճ଎؅શ௕ʹΘͨΓҰఆͱͳΔɻ͜ͷ৔߹ɺߴप೾ిྗ͸Ճ଎؅ΛਐΉʹͭΕҰఆͷൺ཰Ͱ‫ݮ‬ਰ͢ΔͨΊɺ֤ηϧͰͷߴप೾ిྗ͸ࢦ਺ؔ਺తͳ‫ݮ‬ਰ܏޲Λ࣋ͭɻ͜ͷΑ͏ͳՃ଎‫ࣜܕ‬͸ఆΠϯϐʔμϯε (constant impedance :CI) ‫ݺͱܕ‬͹ΕΔɻ·ͣɺCI ‫ܕ‬ͷՃ଎؅ʹ͍ͭͯಛ௃Λઆ໌͢Δɻ
CI ‫Ͱܕ‬͸ηϧ‫ܗ‬ঢ়͕Ճ଎؅શ௕ʹ͓͍ͯҰఆͰ͋ΔͨΊɺα(z) ΋ z ʹରͯ͠ҰఆͱͳΓɺ‫ݮ‬ਰఆ਺͸
τ0 = αL(3.18)

ͱͳΔɻ·ͨɺ‫ݮ‬ਰ܎਺͸

α=

ω
2vg Q

(3.19)

Ͱ༩͑ΒΕɺvg ‫ ͼٴ‬Q ͕େ͖͍ఔখ͘͞ͳΔɻCI ‫ܕ‬Ͱͷߴप೾ిྗ PCI (z) ‫ిͼٴ‬৔ͷେ͖͞ ECI (z) ͸ࣜ%(3.8, 3.16, 3.19) ͱؔ܎ࣜ E(z) ∝ P (z) ͔Β‫ٻ‬ΊΒΕΔɻ

PCI (z) = P0 e−2τ0 z/L(3.20)ECI (z) = E0 e−τ0 z/L(3.21)

͜͜Ͱɺ୯Ґ௕͋ͨ͞ΓͷγϟϯτΠϯϐʔμϯεrsh ͱ z = 0 Ͱͷߴप೾ిྗ P0 ʹରͯ͠ɺE0 ͸ҎԼͷΑ͏ͳؔ܎͕͋Δɻ

E0

2

&
dP &&
2rsh P0 τ0= −rsh=&dz z=0L(3.22)(constant gradient : CG) ‫Ϳݺͱܕ‬ɻCG ‫ܕ‬ͷ৔߹ɺՃ଎؅௕͞ʹԠͯ͡σΟεΫ։‫ޱ‬ΛࡡΓిྗͷ‫ݮ‬ਰ܏޲ͱ‫܈‬଎౓ͷ‫ݮ‬গ܏޲ΛҰகͤ͞ΔΑ͏ઃ‫͢ܭ‬Δɻ͜ͷՃ଎‫͍ͯͭʹࣜܕ‬΋ಉ༷ʹՃ଎ిѹΛ‫ٻ‬ΊΔɻͨͩ͠ɺ‫܈‬଎౓ vg ͱ‫ݮ‬ਰ܎਺ α ͸σΟεΫ։‫ޱ‬ͷมԽʹහ‫͋Ͱײ‬Δ͕ɺQ ஋ͱ rsh ͸͋·Γහ‫Ͱײ‬͸ͳ͍ɻͦ
͜ͰɺQ ஋ ‫ ͼٴ‬rsh ͸ CG ‫ܕ‬ͷՃ଎؅ʹ͓͍ͯ΋Ұ
ఆͰ͋ΔͱԾఆ͢ΔɻCG ‫Ͱܕ‬͸Ճ଎؅શ௕ʹ౉ͬͯՃ଎ిѹ͕౳͍ͨ͠
Ίɺ୯Ґ௕͋ͨ͞Γͷߴप೾ిྗ‫ݮ‬ਰྔ͸Ұఆͱͳ
Δ 20 ɻैͬͯɺz = L Ͱͷߴप೾ిྗΛ PL ͱ͢ΔͱҎԼͷؔ܎͕੒ཱ͢ΔɻdP
PL − P0= P0 +z(3.24)dz
L·ͨɺPL ͸ࣜ (3.17) Ͱ༩͑ΒΕΔ‫ݮ‬ਰఆ਺Λ༻͍ͯɺPL = P0 e−2τ0 ͱ‫ٻ‬ΊΒΕΔͨΊɺ؅಺ͷߴप೾ిྗ
ͷ PCG (z) ͸ɺ

"
+#
z *
PCG (z) = P0 1 −1 − e−2τ0(3.25)
L
ͱॻ͚Δɻͭ·ΓɺCG ‫Ͱܕ‬͸ߴप೾ిྗ͸ z ʹର͠ࢦ਺ؔ਺Ͱ͸ͳ͘ઢ‫ܕ‬ͷؔ܎Ͱ‫ݮ‬ਰ͢ΔɻCG ‫ܕ‬ͷ৔߹ɺ୯Ґ௕͋ͨ͞ΓͷγϟϯτΠϯϐʔμϯε rsh ͱߴप೾ిྗ P0 ʹର͢Δ E0 ͷؔ܎͸ҎԼͷΑ͏ʹͳΔɻ

dP
rsh P0=(1 − e−2τ0 )(3.26)dz
LैͬͯɺՃ଎؅શମΛ௨ͯ͠ͷՃ଎ిѹ͸ɺ
( L
VCG =
ECG dz0%= rsh P0 L(1 − e−2τ0 )(3.27)ECG 2 = −rsh

20 ҰఆͱͳΔΑ͏ʹ‫ݮ‬ਰ܎਺ α Λௐ੔͢Δ

Relative voltage, V/V0

1.0଎؅ʹͱͬͯେ͖ͳσϝϦοτʹͳΔɻ·ͨɺCI ‫ܕ‬

CG-typeͰ͸Ճ଎؅಺Ͱͷߴप೾ଛࣦ͕Ұ༷Ͱͳ͍ͨΊɺՃ଎CI-type0.8؅ࣗମͷൃ೤΋Ұ༷Ͱ͸ͳ͘ͳΔɻ͜ͷͨΊྫྷ٫ʹ͍ͭͯ΋஫ҙ͕ඞཁͱͳΔɻ

0.6

࣮ࡍͷՃ଎؅Λௐ΂ͯΈΔͱɺถࠃ SLAC ͷ 3 m

0.4

Ճ଎؅ (प೾਺ 2856 MHz) [12] Ͱ͸ τ0 = 0.57ɺཧ‫ݚ‬

0.2

SACLA ͷ 1.8 m Ճ଎؅ (प೾਺ 5712 MHz) [13] Ͱ͸τ0 = 0.53 ͕બ͹Ε͍ͯΔɻ͜ΕΒ͸͢΂ͯ (४ 22 )CG0‫ܕ‬ͷՃ଎‫͋Ͱࣜܕ‬Δɻ

0

0.5
1.0
Total attenuation parameter, τ0

1.5

ਤ 18: ఆΠϯϐʔμϯε (CI) ‫ͱܕ‬ఆՃ଎ޯ഑ (CG)‫ܕ‬ਐߦ೾‫ܕ‬Ճ଎؅ͷՃ଎ిѹɻͱ‫·ٻ‬ΔɻCG ‫ܕ‬ͷ৔߹ɺՃ଎ిѹ͸ τ0 ʹ͍ͭͯ୯ௐ૿Ճͷ܏޲ΛͱΔɻ

3.9.3

Ճ଎‫ࣜܕ‬ͷൺֱCI ‫ ͼٴܕ‬CG ‫͍ͯͭʹܕ‬ಘΒΕͨՃ଎ిѹͷࣜ
(3.23, 3.27) ʹ͍ͭͯ τ0 ґଘੑΛਤ 18 ʹࣔ͢ɻͨͩ
͠ɺ૬ରతͳ܏޲Λ஌Δͷ͕໨తͰ͋ΔͨΊɺ‫ڞ‬௨ͷ√ఆ਺ rsh P0 L ≡ V0 Λ༻͍ͯ‫֨ن‬Խͯ͋͠Δɻ྆ऀͷՃ଎ిѹ͸ τ0 < 1 Ͱ͸΄ͱΜͲ͕ࠩͳ͍͕ɺτ0 > 1ΑΓେ͖͘ͳΔʹͭΕͯ CG ‫ܕ‬ͷํ͕Ճ଎ޮ཰͕Α͘
ͳΔ͜ͱ͕Θ͔ΔɻCG ‫ܕ‬ͷ৔߹ɺՃ଎ిѹ͸ τ0 ͷ૿Ճʹରͯ͠୯ௐ
૿Ճ͢ΔͨΊɺτ0 Λେ͖͘͢Δఔޮ཰͕ྑ͍ɻ͜ͷແ‫ݶ‬େͷ‫ݶۃ‬͸Ճ଎؅ʹશͯͷΤωϧΪʔΛҠಈͤ͞Δ͜ͱΛҙຯ͢Δɻ·ͨɺτ0 Λେ͖͘͢Δ͜ͱ͸ vg
Λখ͘͢͞Δ͜ͱΛҙຯ͠ɺ͜Ε͸ࣜ (3.7) ΑΓϑΟϦϯάλΠϜ͕େ͖͘ͳΔ͜ͱʹͭͳ͕Δɻैͬͯɺ
࣮ࡍͷՃ଎؅Ͱ τ0 Λେ͖͘͢Δ͜ͱ͸‫࣮ݱ‬తͰ͸ͳ͍ɻ͜ͷଞɺຊςΩετ͸ৄ͘͠ड़΂ͳ͍͕ɺϏʔϜෛՙɺ֤छ‫ʹࠩޡ‬ର͢Δ଱ੑΛߟྀ͢Δͱ τ0 ͕খ͞
͍ (vg ͕େ͖͍) ํ͕๬·͍͠ɻCG ‫ܕ‬ͷՃ଎؅͸ߏ଄͕ෳࡶʹͳΔͨΊ੡࡞্ͷίετ͕େ͖͘ͳΔ໰୊͸͋Δ͕ɺՃ଎؅ͱͯ͠͸ CI‫ܕ‬ΑΓ΋༏Ε͍ͯΔͱߟ͑ΒΕΔɻCI ‫Ͱܕ‬͸Ճ଎؅ೖ‫࠷Ͱޱ‬΋Ճ଎ిѹ͕େ͖͘ग़‫ޱ‬ଆͰ͸௿͍ɻ͜ͷͨ
ΊɺCI ‫ Ͱܕ‬CG ‫ͱܕ‬ಉఔ౓ͷՃ଎ޮ཰ΛಘΑ͏ͱ͢
Δͱɺೖ‫ޱ‬ଆͷిѹ͸ඞͣ CI ‫ܕ‬ͷํ͕େ͖͘ͳͬͯ
͠·͏ 21 ɻՃ଎؅಺ͷՃ଎ిѹ͸ߴిք͕Ҿ͖‫͢͜ى‬์ిͰ੍‫͞ݶ‬ΕΔ͜ͱ͕ଟ͍ͨΊɺ͜Ε͸ CI ‫ܕ‬ͷՃ
21 CG ‫ܕ‬ͷిѹ͕ CI ‫ܕ‬ͷฏ‫ۉ‬ͷిѹͱ౳͘͠ͳΔঢ়‫گ‬Λߟ͑ͯ

͍Δɻ

22ʮ४ʯͱ͍͏ͷ͸ɺίετ࡟‫Ͳͳݮ‬੡࡞্ͷ౎߹Ͱશͯͷηϧ‫ܗ‬ঢ়ΛมԽͤ͞ΔͷͰ͸ແ͘ɺηϧ‫ܗ‬ঢ়ͷมԽΛҰఆ਺ͷηϧຖʹ
͢ΔͳͲߏ଄Λ؆ུԽ͍ͯ͠Δ͜ͱΛҙຯ͢Δɻ

4ఆࡏ೾‫ܕ‬Ճ଎؅yఆࡏ೾؅ͷ໛ࣜਤΛਤ 19 ʹࣔ͢ɻఆࡏ೾؅Ͱ͸ߴrप೾‫͔ݯ‬Βಋ͔Εͨೖࣹ೾͸Ճ଎؅಺ΛԿ౓΋ԟ෮͠ɺཧ૝తͳ৔߹͸શͯͷߴप೾ిྗ͕Ճ଎؅಺Ͱফ
අ͞ΕΔɻͨͩ͠ɺ࣮ࡍͷӡసͰ͸ӡస৚݅ʹԠͯ͡zφগͳ͔Βۭͣಎ͔Βͷ൓ࣹ೾͕ଘࡏ͢ΔͨΊɺߴप೾a‫ثػ‬ͷอ‫ޢ‬Λ໨తͱͨ͠૷ஔ (μϛʔϩʔυ౳) ͸ඞ

x

ཁͱͳΔɻ
Ճ଎؅ͷதͰ͸ి࣓೾͸྆୺ͷนͰ൓ࣹ͞Εਐߦํ޲ͷҟͳΔ೾Ͳ͏͕͠ॏͶ߹Θͬͨ͞ʢ‫ׯ‬বͨ͠ʣঢ়
ଶʹͳΓɺఆࡏ೾Λ͍ͭͬͯ͘Δɻఆࡏ೾ͱ͸೾͕ॏͳΓ߹ͬͯɺ೾‫͕ܗ‬ਐߦͤͣͦͷ৔ʹཹ·ͬͯৼಈ͠
˙ Δ̇೾ͷ͜ͱͰ͋Δɻਐߦ೾‫ܕ‬Ճ଎؅
͍ͯΔΑ͏ʹΈ̇͑Ͱ͸‫ج‬ຊతʹҰํ޲ʹਐΉ೾ͷΈΛߟ͑Δ఺͕ɺఆࡏ
೾‫ܕ‬Ճ଎؅ͱҟͳ͍ͬͯΔɻຊઅͰ͸·ͣఆࡏ೾‫ܕ‬Ճ଎؅ͷ࠷΋୯७ͳྫͱ͠
ͯɺϐϧϘοΫε‫ܗ‬ঢ়Λ΋ͭ୯ηϧͷఆࡏ೾‫ܕ‬Ճ଎؅(ϐϧϘοΫε‫ ܕ‬RF ۭಎ) ʹ͍ͭͯड़΂ΔɻҰൠతͳ୯ηϧͷՃ଎؅Ͱ͸ηϧͷ௕͞ (؅࣠ํ޲‫͕ )཭ڑ‬༗‫͋Ͱݶ‬Γηϧ಺ʹߏ଄తͳप‫ظ‬ੑ͸ແ͍ͨΊɺલઅ·Ͱͷਐߦ೾‫ܕ‬Ճ଎؅ͷٞ࿦Ͱద༻ͨ͠ xy ࣠ʹର͢Δ‫ڥ‬ք৚݅ʹՃ͑ z ࣠ํ޲΋ಉ༷ʹߟྀ͢Δඞཁ͕lਤ 20: ϐϧϘοΫε‫ ܕ‬RF ۭಎͷ໛ࣜਤɻి࣓೾͸ z࣠ʹ͓͍ͯ΋ด͡ࠐΊΒΕΔɻԁ‫ܗ‬෦ͷ಺൒‫ܘ‬Λ aɺηϧ௕Λ l ͱ͢Δɻ૿΍ͨ͠σΟεΫϩʔυ‫ܕ‬ͷఆࡏ೾‫ܕ‬Ճ଎؅Λྫʹଟηϧͷ৔߹ʹ͍ͭͯऔΓѻ͏ɻ͜ͷ৔߹ɺ؅࣠ํ޲ʹ͸ηϧ਺ʹԠͨࣗ͡༝౓͕‫ڐ‬༰͞ΕΔ͜ͱʹͳΓɺ࠶౓ɺਐߦ೾‫ܕ‬Ճ଎؅ͷ৔߹ͱ΄΅ಉ༷ͷҐ૬Ϟʔυ(෼ࢄ‫ۂ‬ઢ) Λߟྀ͢Δඞཁ͕ੜ͡Δɻ

͋Δ 23 ɻ͜ͷ৔߹ɺ3 ࣍‫ཱݩ‬ମߏ଄ʹର͠ 3 ૊ͷଋറ৚͕݅ద༻͞ΕΔͨΊ෼ࢄ‫ۂ‬ઢ͸‫ۂ‬ઢͰ͸ͳ͘఺ͱͳ

4.1

୯ηϧఆࡏ೾‫ۭܕ‬ಎ (ϐϧϘοΫε)4.1.1‫ৼڞ‬Ϟʔυͷి࣓೾ΓɺϞʔυॖୀΛߟ͑ͳ͚Ε͹ 24 ి࣓৔ͷ֤‫ݻ‬༗Ϟʔ
υ 25 ʹର͠ 1 ͭͷ‫ৼڞ‬प೾਺ׂ͕Γ౰ͯΒΕΔ 26 ɻຊઅͰ͸ϐϧϘοΫε‫ ܕ‬RF ۭಎʹଓ͖ɺηϧ਺ΛϐϧϘοΫε‫ ܕ‬RF ۭಎ (ਤ 20 ࢀর; ಺൒‫ܘ‬Λ aɺηϧ௕Λ l) ʹ͍ͭͯ TM, TE Ϟʔυͷి࣓৔ͷ֤੒෼ͷ༷ࢠΛ਺ࣜͰࣔ͢ 27 ɻRF 源

TM-modeダミーロードRF 波

Ez = Emnp Jm (kc r) cos(mφ) cos(kz z)

(4.1)

kz
Emnp Jm ′ (kc r) cos(mφ) sin(kz z) (4.2)
kcmkz
Jm (kc r)
Eφ =Emnpsin(mφ) sin(kz z) (4.3)
r
kc 2
iωεm
Jm (kc r)
Hr = −Emnp
sin(mφ) cos(kz z)
r
kc 2
(4.4)
Er = −

電子ビーム
加速管
ਤ 19: ఆࡏ೾‫ܕ‬Ճ଎؅ͷ໛ࣜਤɻ23 ‫ڥ‬ք৚݅͸ 3 ࣠ʹରͯ͠ి‫ؾ‬త୹བྷ (ಋମ໘Ͱͷి৔ͷ઀ઢ੒෼͕θϩ) Ͱ͋Δɻ
24 Ϟʔυॖୀ͕͋Δ৔߹͸ɺෳ਺ͷ‫ݻ‬༗Ϟʔυ͕ಉ͡‫ৼڞ‬प೾਺Λ΋ͭ͜ͱʹͳΔɻ
25 ఆࡏ೾‫ܕ‬Ճ଎؅ͷ৔߹ɺ఻೻Ϟʔυͷ୅ΘΓʹ‫ݻ‬༗Ϟʔυͱ͍͏୯‫ޠ‬Λ༻͍Δɻ
26 ֤‫ݻ‬༗Ϟʔυʹର͢Δ‫ৼڞ‬प೾਺͸֤ Q ஋ʹԠͯ͡༗‫ݶ‬ͷप೾਺֦͕ΓΛ࣋ͭɻHφ = −iωεEmnp Jm ′ (kc r) cos(mφ) cos(kz z) (4.5)
kc

27 TE-mode ͷࢦ਺ p ͕ 1 ͔Β࢝·Δͷ͸ɺp = 0 ͷ৔߹ɺHzΛআ͘શͯͷ੒෼͕θϩͱͳΓɺಈి࣓৔ͱ͸Έͳͤͳ͘ͳΔ͔Β
Ͱ͋Δɻ

TE-mode

ρmn
pπ
kc =
, kz =al⎧⎪⎨ m = 0, 1, 2, · · ·n = 1, 2, 3, · · ·⎪⎩p = 0, 1, 2, · · ·Hz = Hmnp Jm (kc r) cos(mφ) sin(kz z)

(4.6)

4
TM01p
3
2
(4.7)

iωµm
Jm (kc r)
Hmnpsin(mφ) sin(kz z) (4.8)
r
kc 2
iωµ
Eφ =
Hmnp Jm ′ (kc r) cos(mφ) sin(kz z)
(4.9)
kc
kz
Hr = Hmnp Jm ′ (kc r) cos(mφ) cos(kz z)(4.10)kc
mkz
Jm (kc r)
Hφ = − 2 Hmnp
sin(mφ) cos(kz z)rkc(4.11)
Er =

ρmn ′
pπ
kc =
, kz =al⎧⎪⎨ m = 0, 1, 2, · · ·ω/ω0(4.12)

10-4-3ͨͩ͠ɺશͯͷ੒෼ʹ‫ৼڞ‬प೾਺ʹै͍࣌ؒతʹৼEz
12.24), ࣜ (2.26 ͔Β 2.30) ‫ͼٴ‬ɺਐߦ೾‫ܕ‬Ճ଎؅ (σΟ͑ΒΕΔ͜ͱͰ͋Γɺి࣓೾͸ z ํ޲ʹ఻೻ͤͣҰఆͷۭؒ಺Ͱৼಈ͢Δɻ

1

2

3

4

p=1
p=2

p=3

-0.5
p=4
-1
0

εΫϩʔυ‫ܕ‬Ճ଎؅) ͷࣜ (3.1) ͱൺֱͯ͠ɺ࠷΋ಛ௃తͳͷ͸؅࣠ (z ࣠) ํ޲ͷґଘੑ͕ࡾ֯ؔ਺Ͱ༩

0
p

p=0

0.5

ಈ͢Δ߲ eiωres t ͕͔͔Δɻ্ࣜͰ༩͑ͨϐϧϘοΫε‫ ܕ‬RF ۭಎͱԁ‫ܗ‬ಋ೾؅Ͱͷి࣓৔ͷࣜ (2.20 ͔Β

-1

ਤ 21: ϐϧϘοΫε‫ ܕ‬RF ۭಎͷ෼ࢄؔ܎ɻ

0n = 1, 2, 3, · · ·⎪⎩p = 1, 2, 3, · · ·

-2

l/4

l/2
Cavity Length

3l/4

l

ਤ 22: ϐϧϘοΫε‫ ܕ‬RF ۭಎ TM01p ϞʔυͷEz (p = 0, 1, 2, 3, 4) ͷ༷ࢠɻΫε‫ ܕ‬RF ۭಎͰ͸ kz ͸ࢦ਺ p ʹΑܾͬͯ·Γɺ෼ࢄ

4.1.2

ؔ܎͸཭ࢄతͳ఺ͱͯ͠ද‫͞ݱ‬ΕΔɻ·ͨɺͦΕΒͭϐϧϘοΫε‫ ܕ‬RF ۭಎͷ෼ࢄؔ܎TM, TE Ϟʔυʹ͓͚Δ‫֯ৼڞ‬प೾਺ ωres ͸લઅͰࣔͨ͠ kc , kz Λ༻͍Δͱ‫ڞ‬௨ʹҎԼͷΑ͏ʹ༩͑ΒΕΔɻ!
ωres = c kc 2 + kz 2ͳ͙ۙࣅ‫ۂ‬ઢ͸ಋ೾؅ͷ৔߹ͱಉ༷ͳ૒‫ۂ‬ઢΛඳ͘ɻࢀߟͱͯ͠ɺਤ 21 ʹࣔͨ͠ɺp = 0, 1, 2, 3, 4 ʹ͍ͭͯ TM01p Ϟʔυͷ Ez ͷ༷ࢠΛਤ 22 ʹࣔ͢ɻp = 0͕Ճ଎؅ͷθϩϞʔυɺp = 1, 2, 3, 4 ͕Ճ଎؅ͷ pπ(4.13)

ϞʔυʹରԠ͍ͯ͠Δ͜ͱ͕Θ͔Δɻ͜ͷΑ͏ʹ‫֯ৼڞ‬प೾਺͸ɺ֤ mnp ͷ૊Έ߹Θͤʹ͖ͭ 1 ͭͷ஋ΛͱΔΑ͏ʹͳΔɻ·ͨɺηϧ਺͕ 1 Ͱ
͋ΔͨΊɺਐߦ೾‫ܕ‬Ճ଎؅ͷ৔߹ͷΑ͏ͳҐ૬Ϟʔυ

4.1.3

Ճ଎Ϟʔυͱߴ࣍Ϟʔυ΍ۭؒߴௐ೾ͱ͍͏֓೦͸ͳ͘ͳΓɺ୅ΘΓʹ kz ͕୯ηϧ RF ۭಎ͸਺ GeV ఔ౓ͷൺֱతΤωϧΪʔ؅࣠ํ޲ʹి࣓৔ͷઅ͕‫͋ͭز‬Δ͔ʹରԠ͢Δྔͱͷ௿͍Ϧϯά‫ࢠిܕ‬Ճ଎‫Ͱث‬Α͘༻͍ΒΕΔɻ͜Ε͸ͳΔɻඞཁͱ͞ΕΔՃ଎ిѹ͕ͦΕ΄Ͳߴ͘ͳ͍͜ͱ΍Ϗʔ

͜͜ͰɺTM01p Ϟʔυʹ͍ͭͯ p Λม਺ͱͯ͠ϓϜෆ҆ఆੑͷ໰୊Λߟྀ͢Δͱɺ୯ηϧ RF ۭಎͷํϩοτͨ͠෼ࢄؔ܎Λਤ 21 ʹ੺‫ࣔ͢Ͱؙ‬ɻϐϧϘο͕ରࡦΛߨ͡Δͷ͕༗རͰ͋Δ͔ΒͰ͋Δɻଟ͘ͷϦද 5: ϐϧϘοΫε‫ ܕ‬RF ۭಎͷ࠷దηϧ௕࣌ͷ TM010
ͱͦͷߴ࣍Ϟʔυͷछྨͱप೾਺ൺ
Ϟʔυ໊শ f /f(T M 010)

TM010

1.00

TE111
TM011

1.37
1.51

TM110
TE211
TM111

1.59
1.70
1.96

TE011
TM210

1.96
2.14

TM020
TE112
TM012

2.30
2.40
2.48

TE121

2.49

ϯά‫ࢠిܕ‬Ճ଎‫Ͱث‬͸ಉ͡ RF ۭಎΛେిྲྀͷిࢠϏʔϜ͕࿈ଓ (CW) తʹ௨ա͢ΔͨΊɺిࢠϏʔϜͷ΢ΣΠΫ৔ (wake field) ʹΑͬͯՃ଎ϞʔυҎ֎ͷ‫د‬ੜϞʔυ͕ۭಎ಺ʹྭৼ͞Ε 28 ϏʔϜࣗ਎ʹѱӨ‫ڹ‬Λ༩͑Δ͜ͱ͕໰୊ͱͳΔɻిࢠϏʔϜͷՃ଎؅಺ͷి࣓೾ͱͷ૬‫༻࡞ޓ‬΍ϏʔϜෆ҆ఆੑʹ͍ͭͯ͸จ‫[ ݙ‬14–16] ͳͲͰৄ͘͠આ໌͞Ε͍ͯΔɻैͬͯɺϦϯά‫ࢠిܕ‬Ճ଎‫ث‬ͷՃ଎؅Ͱ͸‫د‬ੜϞʔυΛޮՌతʹ‫ݮ‬ਰͤ͞Δ޻෉͕ॏཁͱͳΔɻ͜ͷΑ͏ͳ‫͔఺؍‬Β͸ߏ଄తʹγϯϓϧͳ୯ηϧ RF ۭಎΛ༻͍ͨํ͕༗
རͱͳΔɻ࣮ࡍʹ޻෉͕ࢪ͞Εͨ RF ۭಎ͸‫ݮ‬ਰۭಎ(damped-cavity) ͱ‫ݺ‬͹Εɺ਺ଟ͘ͷํ͕ࣜใࠂ͞Ε͍ͯΔ [1, 17–20]ɻ
ిࢠϏʔϜͷՃ଎ʹ͸ిࢠϏʔϜͷਐߦํ޲ͱ౳͠
͍ి৔੒෼Λ΋ͭి࣓৔Ϟʔυ͕ඞཁͰ͋ΓɺͰ͖Δ
͚ͩ௿࣍ͷϞʔυΛՃ଎ϞʔυʹબͿํ͕‫د‬ੜϞʔυΛආ͚Δ‫͔఺؍‬Β༗རͰ͋ΔɻͦͷͨΊɺՃ଎Ϟʔυʹ͸࠷΋௿࣍ͷ TM010 Ϟʔυ͕༻͍ΒΕΔ͜ͱ͕ଟ͍ɻRF Ճ଎ۭಎͷઃ‫Ͱܭ‬͸ϏʔϜෛՙΛߟ͑ͳ͚Ε͹ 29 ɺҰൠʹγϟϯτΠϯϐʔμϯεΛߴ͘͢ΔΑ͏࠷దԽ͢ΔɻϐϧϘοΫε‫ ܕ‬RF ۭಎͷ TM010 ͷ৔߹ɺઅ 3.8.1 Ͱઆ໌ͨ͠Α͏ʹηϧ௕Λ‫ৼڞ‬प೾਺ͷ೾௕ͷ 0.44 ഒʹબͿͱγϟϯτΠϯϐʔμϯε͸࠷େͱͳΔɻ
28 ‫ج‬ຊతʹՃ଎؅͸໨తͱ͢Δి࣓৔ϞʔυͷΈΛྭ‫͢ى‬Δͨ
Ίɺ໨తͷप೾਺ͷߴप೾ͷΈ͕‫͞څڙ‬ΕΔɻ͔͍͠ɺ΢ΣΠΫ৔͸ϏʔϜ‫ܗ‬ঢ়ʹԠͨ͡प೾਺੒෼Λ‫ؚ‬Έɺ͜Ε͕ۭಎͷߴ࣍ϞʔυΛྭৼ͢Δɻ
29 ϏʔϜෛՙʹΑΔۭಎಉௐ֯ʢσνϡʔχϯά֯ʣ௿‫ͭʹݮ‬͍ͯ͸ R/Q ͕খ͍͞ํ͕༗རͱͳΔɻप௕ͷ௕͍ԁ‫ܗ‬Ճ଎‫Ͱث‬͸ϏʔϜෆ҆ఆੑΛ཈੍͢ΔͨΊಉௐ֯Λ཈͑Δඞཁ͕͋Δɻ

ද 5 ʹϐϧϘοΫε‫ ܕ‬RF ۭಎʹ͓͍ͯηϧ௕Λ‫ڞ‬ৼ೾௕ͷ 0.44 ഒʹબΜͩ৔߹ʹ͓͚ΔɺՃ଎Ϟʔυ

(TM010 ) ͱଞͷ‫ݻ‬༗Ϟʔυͱͷप೾਺ൺΛ‫͔ࣔͭز‬͢ɻ࣮ࡍʹ RF ۭಎΛઃ‫͢ܭ‬Δͱ͖͸ϏʔϜύΠϓΛ
͚ͭͨΓɺ͞Βʹ Q ஋΍γϟϯτΠϯϐʔμϯεΛ
վળ͢ΔΑ͏ͳ޻෉Λ͢ΔͨΊɺϐϧϘοΫε‫ ܕ‬RFۭಎΛͦͷ··Ճ଎ۭಎͷ‫ج‬ຊߏ଄ͱ͢Δ͜ͱ͸গͳ͍͕ɺϐϧϘοΫε‫ ܕ‬RF ۭಎͷߴ࣍Ϟʔυʹ͍ͭͯ஌͓ͬͯ͘͜ͱ͸༗ӹͩͱࢥΘΕΔɻද 5 ʹࣔͨ͠‫ݻ‬༗Ϟʔυͷ͏ͪɺϏʔϜෆ҆ఆੑͷ‫͔఺؍‬Βಛʹ஫ҙ͕ඞཁͳ΋ͷ͸ࢦ਺ m ͕ 0(monopole mode) ·ͨ͸1(dipole mode) ΛͱΔ΋ͷͰ͋Δɻ͜ΕΒ஫ҙ͕ඞཁͳϞʔυʹ͍ͭͯ͸ɺਤ 23 ʹి࣓৔ͷ༷ࢠΛࣔ͢ɻۭಎΛઃ‫͢ܭ‬Δࡍ͸ɺ֤‫ݻ‬༗Ϟʔυͷి࣓৔ͷ༷ࢠΛ೺Ѳ͠ɺՃ଎ϞʔυҎ֎ͷ‫د‬ੜϞʔυΛे෼ʹ‫ݮ‬ਰͤ͞ΔΑ͏ͳߏ଄ (޻෉) Λ࣮૷͢Δ͜ͱ͕ॏཁʹͳΔɻ

TE,TMmnp-mode

y

y

2a

a

0

x

l

z

ਤ 23: ϐϧϘοΫε‫ۭܕ‬ಎͷ୅දతͳ‫ৼڞ‬Ϟʔυͷి࣓೾ͷ༷ࢠɻ4.2(N+1) セルଟηϧۭಎΑΓߴ͍Ճ଎ిѹΛ୹͍۠ؒͰಘΔʹ͸ɺۭಎͷη(N-1) フルセルϧ਺Λ૿΍͢͜ͱΛ‫ݕ‬౼͢ΔՁ஋͕͋Δɻ‫࣮ݱ‬తͳ͜ͱΛ๨Εͯ‫ͳ୺ۃ‬࿩Λ͢Δͱɺ1 ୆ͷۭಎͰ 2 ഒͷ

半セル

半セル

Ճ଎ిѹΛಘΔͨΊʹ͸ɺPc = Vc2 /Rsh ͷؔ܎͔Β

4 ഒͷ RF ిྗ͕ඞཁͱ͞ΕΔ͕ɺՃ଎‫ʹث‬ಋೖ͢Δ
RF ۭಎΛ 1 ୆૿΍ͤ͹ߴप೾ిྗ͸ 2 ഒͰ͢Ήɻ͞Βʹɺ2 ୆ͷۭಎΛ࿈݁͢Ε͹ɺRF ۭಎͷؒͷϏʔϜύΠϓͳͲۭؒతͳεϖʔεΛઅ໿Ͱ͖ΔͨΊɺ࣮࣭తͳՃ଎ޮ཰͸޲্͢Δɻ͜ͷΑ͏ʹ୯ηϧۭಎෳ਺୆ʹΑͬͯߏ੒͞ΕΔ RF ૷ஔΛଟηϧۭಎͷγε

n=0

n=1

n=N-1

n=N

l

(N-1)lNl0ςϜʹมߋ͢Δ͜ͱ͸ɺ֤छϦιʔεͷ‫ݶ‬ΒΕͨதͰ

空洞番号：nՄೳͳ‫ݶ‬Γߴ͍Ճ଎ੑೳΛಘΔ্Ͱ༗ޮͰ͋Δɻ

セル間の位相すすみ：πq /N ,q = 0, 1, 2 …, Nଟηϧͷఆࡏ೾‫ܕ‬Ճ଎؅͸୯ηϧͷ RF ۭಎ͕ߴप೾ (ి‫ )ؾ‬తʹͭͳ͕ͬͨߏ଄ͩͱߟ͑ΒΕΔɻఆࡏਤ 24: (N +1) ηϧఆࡏ೾‫ܕ‬Ճ଎؅ߏ଄ͷఆٛɻ(N −1)೾‫ܕ‬Ճ଎؅ͷਐߦ೾‫ܕ‬Ճ଎؅ͱͷҧ͍͸ɺ؅࣠ํ޲ͷ‫ݸ‬ͷϑϧηϧ (௕͞ l) ͱ 2 ͭͷ൒ηϧ ((௕͞ l/2) ͰલํͷΈͰ͸ͳ͘‫ʹํޙ‬ਐΉ೾͕ಉ࣌ʹଘࡏ͠ɺͦΕߏ੒͞ΕΔҰࣜͷՃ଎؅Λ (N + 1) ηϧͱ͢ΔɻΒͷ‫ׯ‬বʹΑΓఆࡏ೾ͱͯ͠೾໘͕‫ܗ‬੒͞ΕΔͱ͍͏ωq఺Ͱ͋Γɺ୯ηϧ RF ۭಎͱଟηϧ RF ۭಎͱͷҧ͍͸ηϧؒͷҐ૬ਐΈΛߟྀͨ͠؅࣠ํ޲ͷి࣓৔෼෍ωπΛߟ͑Δඞཁ͕͋Δͱ͍͏఺Ͱ͋Δɻ

4.2.1

Ճ଎؅ߏ଄ͷఆٛω0 = ωcఆࡏ೾؅Λٞ࿦͢Δ্ͰՃ଎؅ߏ଄ͷఆٛɺηϧ਺ͷ਺͑ํ౳ʹ஫ҙ͢Δඞཁ͕͋Δɻ͜ΕΒ͸ςΩετ

大΍࿦จʹΑͬͯ౷Ұ͞Ε͓ͯΒͣ࣌ʹ‫ޡ‬ղΛট͘ɻຊςΩετͰ͸ͦͷΑ͏ͳ‫ޡ‬ղΛ‫ྗۃ‬ආ͚ΔͨΊʹɺ·ͣՃ଎؅ߏ଄ʹ͍ͭͯఆ͓ٛͯ͘͠ɻ

00

π/(4l )

ਤ 24 ͰຊςΩετͰલఏͱ͢ΔՃ଎؅ߏ଄Λఆٛ

π/(2l )
β3π/(4l )π/l͢Δɻ(N +1) ηϧ͸௕͞ l/2 ͷ൒ηϧΛ྆୺ʹ࣋ͭ௕ਤ 25: ఆࡏ೾‫ܕ‬Ճ଎؅ͷ෼ࢄؔ܎ɻ੺‫ؙ‬͸ (4 + 1) η͞ l ͷϑϧηϧ (௨ৗηϧ)(N − 1) ‫ߏͰݸ‬੒͞ΕΔͱ

ϧͷۭಎΛԾఆͨ͠৔߹ͷҐ૬ϞʔυɻͷϑϧηϧΛ n = 1 ͱ͍͏Α͏ʹɺn = 0, 1, 2 · · · , N{0, l, · · · , (N − 1)l, N l} ʹ૬౰͢ΔҐஔʹ͋ͨΔɻηΕɺq ͸ q = 0, 1, 2 · · · N ͷൣғͱͳΔɻҐ૬ਐΈྔϞʔυʹ͓͚Δ‫֯ৼڞ‬प೾਺Ͱ͋Δɻ·ͨɺࣜ (4.14)

͢Δɻ͜ͷ৔߹ɺۭಎ൪߸͸୺ͷ൒ηϧΛ n = 0ɺྡͱ͢Δɻ·ͨɺηϧؒͷҐ૬ਐΈ͸ πq/N ͰදΘ͞͸Ґ૬Ϟʔυͷ໊শʹ΋༻͍ΒΕɺq = 0 ͸θϩϞʔυɺq = N ͸ π ϞʔυʹରԠ͢Δɻैͬͯɺ(N + 1)ηϧͰྭ‫͞ى‬ΕΔҐ૬Ϟʔυ਺͸ (N + 1) ‫͋Ͱݸ‬Δɻ

(N + 1) ηϧͰͷ n ൪໨ͷۭಎͰͷఆࡏ೾ి৔͸ɺ
nπq iωq t
E(n) = E0 coseNϧؒͷҐ૬ਐΈྔͷ਺͑ํ΋ಉ༷Ͱ͋Δɻωq ͸֤Ґ૬͸؅࣠ํ޲ͷલํ‫ʹํޙͼٴ‬ਐΉి࣓೾ͷ࿨ͱͯ͠ɺ
#E0 " i(ωq t−πqn/N )E(n) =e+ ei(ωq t+πqn/N ) (4.15)2ͱ΋͔͚Δɻ

(4.14)

ͱͳΓɺ͜Ε͸୺෦Λআۭ͘ಎ (ϑϧηϧ) Ͱ͸த৺

4.2.2

σΟεΫϩʔυ‫ܕ‬Ճ଎؅ͷ෼ࢄؔ܎Ґஔɺ୺෦ηϧ (൒ηϧ) Ͱ͸‫ڥ‬ք໘Ͱͷి৔ৼ෯ΛσΟεΫϩʔυ‫ܕ‬ఆࡏ೾‫ܕ‬Ճ଎؅͸ਐߦ೾‫ܕ‬Ճ଎؅ද͢ɻ͜ΕΒ͸ਤ 24 ʹ͓͚Δۭಎ໛ࣜਤԼ෦ͷ࠲ඪͱಉ༷ʹۭಎ಺෦Λ఻೻Ͱ͖Δి࣓೾ͷप೾਺͸͋Δ‫ݶ‬ΒΕͨൣғͱͳΓɺͦͷ෼ࢄ‫ۂ‬ઢ͸ਤ 13 ʹྨࣅ͠
͍ͯΔɻͨͩ͠ɺՃ଎؅ͷηϧ਺ʹԠͯ͡෼ࢄ‫ۂ‬ઢ͸࿈ଓతͰ͸ແ͘཭ࢄԽ͞ΕΔɻ࣍ࣜ͸ (N + 1) ηϧՃ଎؅ͷ෼ࢄؔ܎Ͱ͋Γɺωπ/2 ͸ π/2 ϞʔυͰͷप೾
਺Ͱ͋Δɻ

ωπ/2

ωq = %
1 + κ cos(πq/N )4.2.3

Ґ૬ϞʔυͱγϟϯτΠϯϐʔμϯεఆࡏ೾‫ܕ‬Ճ଎؅ͷ࠷దͳʢ୯Ґ௕͋ͨ͞Γͷγϟϯ
τΠϯϐʔμϯε͕࠷େͱͳΔʣҐ૬Ϟʔυʹ͍ͭ
ͯɺR.H. Miller ʹ฿ͬͯߟ͑ͯΈΔ [21]ɻ

(N + 1) ‫ݸ‬ͷηϧ͕࿈݁ͨ͠ఆࡏ೾‫ܕ‬Ճ଎؅Λߟ͑
(4.16)

Δɻ͜ͷͱ͖ɺn ‫ݸ‬Ίͷηϧͷిѹ͸ࣜ (4.15) Λ༻͍ͯɺκ ͸ྡ઀ͨ͠ηϧؒͷ݁߹ఆ਺Ͱ͋Δɻ
ྫͱͯ͠ɺηϧ਺͕ (4 + 1) ͷఆࡏ೾‫ܕ‬Ճ଎؅ʹ͍ͭͯɺ෼ࢄ‫ۂ‬ઢͷதͰऔΓಘΔҐ૬ϞʔυΛਤ 25 ʹnπqcos ωq t(4.20)
N
ͱͳΔɻηϧ௕Λ l ͱ͢ΔͱɺిࢠϏʔϜ͕ n ‫ݸ‬Ίͷۭಎʹ౸ண͢Δ࣌ؒ͸ɺ࣍ͷΑ͏ʹॻ͚Δɻ
V (n) = V0 cos੺‫ࣔ͢Ͱؙ‬ɻͨͩ͠ఆࡏ೾‫ܕ‬Ճ଎؅ͷ৔߹ɺ೾ͷਐߦํ޲͸ߟ͑Δඞཁ͕ແ͍ͨΊɺਤ͸ୈҰ৅‫ݶ‬ͷΈΛࣔt=ͨ͠ɻ͜͜Ͱɺκ ≪ 1 ͷ৔߹ɺύεόϯυͷ෯ δω ͸ɺδωq = ωπ/2 κ

z
l
=n
ve
ve

(4.21)

ैͬͯɺV (n) ͸࣍ͷΑ͏ʹॻ͖‫͑׵‬ΒΕΔɻ

V (n) = V0 cos(4.17)nπqnl
cos ωqNve(4.22)

ͱ‫ٻ‬ΊΒΕΔ 30 ɻ͜ͷ෯͸ηϧ਺ N ʹґଘ͠ͳ͍࠷େͷՃ଎ޮ཰͕ಘΒΕΔͷ͸ిࢠϏʔϜͷηϧؒͨΊɺηϧ਺͕૿͑Δͱ෼ࢄ‫ۂ‬ઢΛຒΊΔҐ૬ϞʔͷҠಈ࣌ؒͱి࣓৔ͷҐ૬ਐΈྔ͕ಉ‫ͳ͖ͱͨ͠ظ‬υ͕ࠞΈ߹͍ɺྡΓ߹͏Ϟʔυͷִ͕ؒ‫ͳ͘ڱ‬Δ͜

ͷͰɺ

0q
1
ωq l
=π
+ 2p
ve
N

ͱ͕༧૝Ͱ͖Δɻैͬͯɺ֤Ϟʔυ͕࣋ͭόϯυ෯δωq = ωq /Qq (Qq ͸֤Ґ૬Ϟʔυʹ͓͚Δ Q ஋) ͕ྡΓ߹͏ϞʔυִؒΑΓେ͖͘ͳΔͱɺຊདྷ࠾༻ͨ͠
͍ϞʔυʹՃ͑ྡ઀ͨ͠Ϟʔυ΋ྭৼ͞ΕΔ͜ͱʹͳͱͳΔɻ͜͜Ͱɺp ͸੔਺Ͱ͋ΔɻҎ্ΑΓɺՃ଎؅શମͰͷ߹‫ిܭ‬ѹ͸Δɻ·ͨɺՃ଎؅ͷ੡࡞‫ࠩޡ‬౳Ͱηϧຖͷप೾਺ʹΤV =ϥʔ͕ੜͨ͡৔߹΋ಉ༷Ͱ͋Γɺ͜ͷΑ͏ͳ৔߹ɺి

N$

V (n) = V0n=1৔ৼ෯΍ηϧؒͷҐ૬͕ࠩཚΕΔɻ࣍અͰৄ͘͠ड़΂Δ͕ π ϞʔυͰηϧؒͷҐ૬͕ࠩཚΕΔͱɺՃ଎ޮ

V =཰ͷେ෯ͳ‫ݮ‬গ (࠷େ൒෼) ͕‫͜ى‬ΔͨΊக໋తͳ໰
୊ͱͳΔɻ

(4.23)

2

N
$

cos2

n=1

nπq
N

N V0
2 , (q ̸= 0, N )N V0 , (q = 0, N )(4.24)(4.25)

ͱͳΔɻ͜ͷӨ‫ڹ‬͸෼ࢄ‫ۂ‬ઢͷ‫܈( ཰ۂ‬଎౓) ͕খ͘͞ͳΔπ Ϟʔυ෇ۙͰ‫ݦ‬ஶͱͳΓɺ࠷΋‫܈( ཰ۂ‬଎౓) ͷେ͖͍ π/2 Ϟʔυ෇ۙͰখ͘͞ͳΔɻκ ≪ 1 ͷ৔߹ɺπϞʔυ෇ۙͰྡ઀͢ΔϞʔυؒͷप೾਺ࠩ͸ɺδω
κ 0 π 12=(4.18)
ωq
2 N

π/2 Ϟʔυ෇ۙͰ͸
δω
κ0π1
=
ωq
2 N

Ճ଎؅શମͰͷนফඅిྗ Pt ͸ɺۭಎ 1 ηϧ෼ͷγϟϯτΠϯϐʔμϯε Rc ༻͍ͯ

Pt =

N
N
1 $ 2
1 $0
nπq 12V (n) =
V0 cos(4.26)
Rc n=1
Rc n=1
N
2
N V0 2
2Rc , (q ̸= 0, N )Pt =(4.27)N V0 2
Rc , (q = 0, N )ͱͳΔɻैͬͯɺՃ଎؅શମͰͷγϟϯτΠϯϐʔμ(4.19)

ϯε Rt ͸࣍ͷΑ͏ʹ‫ٻ‬ΊΒΕΔɻͱ‫ٻ‬ΊΒΕΔ 31 ɻ͜ΕΑΓɺπ ϞʔυͰ͸ηϧ਺ Nͷೋ৐Ͱ҆ఆੑ͕ࣦΘΕΔͷʹର͠ɺπ/2 ϞʔυͰ͸ηϧ਺ N ʹൺྫ͢Δ͜ͱ͕Θ͔Δɻ
!

!
(1 − κ) − 1/ (1 + κ)} Λ k ≪ 1 Ͱ

30 ύεόϯυ෯ ω {1/0ల։͢Δ͜ͱͰ‫·ٻ‬Δɻ31 ࣜ (4.16) Λඍ෼͠ɺͦΕͧΕ q = N, N/2 ෇ۙͰల։͢Δ͜ͱͰ‫ٻ‬ΊΒΕΔ

Rt =

2

Rt ≡

V2
Pt

N Rc
2 , (q ̸= 0, N )N Rc , (q = 0, N )(4.28)(4.29)

͜ΕΒͷ݁ՌΑΓɺҐ૬Ϟʔυθϩ, π ϞʔυҎ֎ʹ͓͍ͯ͸Ճ଎؅શମͷγϟϯτΠϯϐʔμϯε Rt͸ɺۭಎ 1 ηϧ෼ͷγϟϯτΠϯϐʔμϯε Rc ͱηϧ਺ N ͱͷੵͷ൒෼ʹͳΔ͜ͱ͕Θ͔Δɻ͜ͷ݁Ռ͸θϩ, π ϞʔυҎ֎Ͱ͸Ճ଎؅಺Λ఻೻͍ͯ͠Δ೾

(؅࣠ํ޲Λ +z, −z ʹਐΉ੒෼) ͷ͏ͪɺిࢠϏʔϜͱಉ͡ํ޲ʹ͢͢Ή੒෼ͷΈ͕ిࢠϏʔϜͱಉ‫͠ظ‬Ճ଎ʹ‫د‬༩͍ͯ͠Δͱߟ͑ΔͱཧղͰ͖Δ 32 ɻҰํɺθϩ, π ϞʔυͰ͸‫܈‬଎౓͕θϩͱͳΔͨΊɺ෼ࢄ‫ۂ‬ઢ্Ͱ͸؅࣠ํ޲Λ +z, −z ʹਐΉ੒෼͸۠ผ͕͔ͭͳ͘ͳΔɻྫ͑͹ɺπ Ϟʔυͷ৔߹ʹࣜ (4.15)
ʹ͓͚Δ e ͷ܎਺͕ −πn, πn ͱͳΔ͕ɺ͜ΕΒ͸౳ՁͰ͋Δ 33 ɻैͬͯɺՃ଎؅શମͷγϟϯτΠϯϐʔμϯε Rt ͱͯ͠͸ఆࡏ೾Λ‫ܗ‬੒͢Δ྆ํͷ೾ͷޮՌ

0

l

2l

3l

4l

Λड͚ΔͨΊɺͦͷଞͷҐ૬ϞʔυͷΑ͏ʹ൒෼ͱ͸ͳΒͳ͍ɻ
ͱ͜ΖͰɺਐߦ೾‫ܕ‬Ճ଎؅ͷγϟϯτΠϯϐʔμϯεʹ͍ͭͯड़΂ͨઅ 3.8.2 ʹ͓͍ͯɺπ ϞʔυͰ͸Ճ଎؅಺ͷి࣓೾͸ఆࡏ೾ͱΈͳͤΔͱઆ໌͕ͨ͠ɺ͜Ε͸ π ϞʔυͰͷՃ଎؅શମ΋͘͠͸୯Ґ௕͋ͨ͞ΓͷγϟϯτΠϯϐʔμϯε͸Ճ଎؅಺ͷߴप೾ిྗ
͕ಉ͡৔߹ɺఆࡏ೾‫ܕ‬Ճ଎؅ͱਐߦ೾‫ܕ‬Ճ଎؅Ͱ౳͠
͘ͳΔ͜ͱΛҙຯ͢Δɻ
ϐϧϘοΫε‫ܕ‬ηϧΛ‫ج‬ຊप‫ߏظ‬଄ͱ͢Δਐߦ೾‫ܕ‬Ճ଎؅Ͱ͸ɺ0.58π(l = 0.29λ) ͷҐ૬ϞʔυΛ࠾༻͢
Δ͜ͱͰ࠷େՃ଎ޮ཰ (π Ϟʔυͷ໿ 1.4 ഒ) ͕ಘΒΕ͕ͨɺఆࡏ೾‫ܕ‬Ճ଎؅Ͱ࠷େՃ଎ޮ཰͕ಘΒΕΔͷ͸π ϞʔυͰ͋ΓɺͦͷγϟϯτΠϯϐʔμϯε͸ద౰ͳҐ૬ϞʔυΛબΜͩਐߦ೾‫ܕ‬Ճ଎؅ΑΓ΋খ͘͞ͳΔɻͭ·ΓɺϏʔϜͷ‫܁‬Γฦ͠౳ӡస৚݅ʹΑΔޮՌ
Λߟ͑ͳ͚Ε͹ਐߦ೾‫ܕ‬Ճ଎؅ͷํ͕ߴ͍Ճ଎ޮ཰͕
ಘΒΕΔ͜ͱΛҙຯ͢Δɻਤ 26: ఆࡏ೾‫ܕ‬Ճ଎؅ͷ‫ࢉܭ‬Ϟσϧɻ(4+1) ηϧߏ଄ɻͷϩε͸ͳ͍ͱԾఆ͠ɺి࣓৔͸‫ݻ‬༗஋ղੳ (Eigenmode) ιϧόʹΑΓ‫ٻ‬Ίͨɻ‫Ͱࢉܭ‬ಘΒΕͨఆࡏ೾‫ܕ‬Ճ଎؅ͷ π/4 ͔Β π ·Ͱͷ֤Ґ૬Ϟʔυʹ͍ͭͯ؅࣠ํ޲ͷి৔ Ez Λਤ 27
ʹࣔ͢ 34 ɻͨͩ͠ɺ‫ࢉܭ‬͸ର৅ͱ͢ΔҐ૬Ϟʔυͷ਺͚ͩߦ͍ɺͦͷ౓ຖʹηϧ௕ l ͱԁ౵‫ܘ‬Λௐ੔ͨ͠ɻ
͜Ε͸‫ࢉܭ‬ର৅ͱͳΔҐ૬ϞʔυͷҐ૬଎౓ͱ‫ݻ‬༗प೾਺Λશͯͷ৔߹Ͱ౳͘͢͠ΔͨΊͰ͋ΓɺҐ૬଎౓͸ޫ଎ͷిࢠϏʔϜͱಉ‫͢ظ‬ΔΑ͏બΜͩɻ
ਤ 27 ͸άϥϑ࿮ͱ಺෦ͷ࣮ઢͰσΟεΫΛ‫ؚ‬ΉՃ଎؅‫ڥ‬քΛ͍ࣔͯ͠Δɻ·ͨɺॎ࣠͸࠷େ஋Ͱ‫֨ن‬Խͨ͠ి৔ͷେ͖͞Λ͍ࣔͯ͠Δɻԣ࣠͸ηϧ௕ͷ੔਺ഒʹ໨੝Λࣔͯ͋͠Γɺ1 ηϧ෼ʹ઎ΊΔి৔ͷҐ૬ਐΈྔ͕Ґ૬ϞʔυʹରԠ͍ͯ͠Δ͜ͱ͕Θ͔Δɻ
ఆࡏ೾‫ܕ‬Ճ଎؅Ͱ͸੺࣮ઢͰࣔͨ͠ి৔‫ ͕౓ڧ‬eiωt

4.2.4ి࣓৔γϛϡϨʔγϣϯ݁Ռͱͷൺֱఆࡏ೾‫ܕ‬Ճ଎؅ʹ͍ͭͯ΋͏গ͠ཧղΛਂΊΔͨΊʹి࣓৔γϛϡϨʔγϣϯίʔυ CST MWstudio [22]Λ࢖ͬͯ‫۩ͨ͠ࢉܭ‬ମతͳྫΛ঺հ͢Δɻ
‫Ͱࢉܭ‬͸ਤ 26 ʹࣔ͢ϞσϧΛԾఆ͠ TM01 ϞʔυΛԾఆͨ͠‫ࢉܭ‬ΛߦͬͨɻϞσϧ͸ϑϧηϧͷ௕͞Λl ͱ͢Δ (4+1) ηϧͷՃ଎؅ߏ଄Λ༻͍ͨɻϞσϧͷ‫ڥ‬ք৚݅͸ xyz ͷ 3 ࣠ʹର͠ Et = 0(ಋମ໘Ͱͷి৔ͷ઀ઢ੒෼͕ଘࡏ͠ͳ͍) ͱͨ͠ɻ‫Ͱࢉܭ‬͸ಋମ໘Ͱ32 ‫ʹີݫ‬͸ɺ͜Ε͸ N Λແ‫ݶ‬େͱͨ͠৔߹Ͱ͋ΓɺN ͕༗‫ͳݶ‬৔߹͸ిࢠϏʔϜͱ൓ରํ޲ʹਐΉ੒෼΋Θ͔ͣͰ͋Δ͕Ճ଎ʹ‫د‬༩͢Δɻ
33 ਤ 25 ͷ෼ࢄ‫ۂ‬ઢͰ͸ɺπ Ϟʔυʹۙͮ͘ͱ 2 ͭͷ೾͕࠲ඪ(π/l) ʹ β < π/l ͱ β > π/l ͷ྆ํ͔Β͍͖ۙͮͯͯɺπ ϞʔυͰ͸ 2 ͭͷ೾͕࠲ඪ (π/l) Ͱ‫׬‬શʹॏͳΔͱઆ໌Ͱ͖Δɻ

ʹै͍࣌ؒతʹৼಈ͢Δɻਐߦ೾‫ܕ‬Ճ଎؅Ͱ͸ిࢠϏʔϜͱి࣓೾ͷҐ૬͕ಉ͡଎౓ͰਐΉͨΊɺҰ୴ɺ೾ͷ௖఺ (ΫϨετ) ʹిࢠΛ৐ͤΕ͹ྡͷηϧʹ͓͍ͯ΋ͦͷ··࠷େͷి৔Λड͚ͳ͕Βిࢠ͸Ճ଎͞
ΕΔɻҰํɺఆࡏ೾‫ܕ‬Ճ଎؅Ͱ͸֤ηϧͷి࣓৔ৼ෯
͸൓ࣹ೾ͱͷ߹੒Ͱ༩͑ΒΕ೾͸ਐߦ͠ͳ͍ͨΊɺਤ

27 ͷΑ͏ʹిࢠϏʔϜ͕ਐΉηϧຖʹి৔ͷৼ෯͕ҟͳΓՃ଎؅શମͰͷՃ଎ΤωϧΪʔ͸ਐߦ೾‫ܕ‬Ճ଎؅ΑΓ΋௿͘ͳΔɻͨͩ͠ɺπ Ϟʔυ͚ͩ͸ྫ֎ͰηϧؒͰͷి৔ৼ෯͸ҰఆͱͳΓɺ૝ఆ͞ΕΔ࠷େͷՃ଎ΤωϧΪʔ͕ಘΒΕΔɻ͜Ε͕લઅͰ਺ࣜΛ༻͍ͯߦͬͨߟ࡯ͷఆੑతͳઆ໌ͱͳΔɻ
34 ಉ༷ͳ݁Ռ͸਺஋‫ʹࢉܭ‬པΒͣͱ΋౳Ձճ࿏ϞσϧΛ༻͍ͨ‫Ͱࢉܭ‬΋ಘΒΕΔ [8, 23]ɻ+

π/4 モードrsh /rsh,max
1.0
Traveling wave

0.8

0

Ez

0.6

π/2 モード

0.4+Standing wave

0.2
0
0
0

π/4

+

3π/4 モード

3π/4π/2
Phase shift per cellπ

ਤ 28: ఆࡏ೾‫ܕ‬Ճ଎؅ͷ૬ରγϟϯτΠϯϐʔμϯεɻ੺‫ؙ‬ͷϓϩοτ͸൒ηϧ 2 ͭ (୯Ґप‫ظ‬, ௕͞ l)0

Λ 8 प‫ظ‬෼࿈݁ͨ͠ϞσϧͰͷ‫݁ࢉܭ‬Ռɻ࣮ઢ͸અ

3.8.2 Ͱ‫ٻ‬ΊͨγϟϯτΠϯϐʔμϯεͷղੳ‫݁ࢉܭ‬ՌͰి࣓৔γϛϡϨʔγϣϯ݁ՌΛۙࣅͨ͠‫ۂ‬ઢͰ͋-Γɺ఺ઢ͸͜ΕΛ 2 ഒͨ͠΋ͷɻ

πモード+

ͳ‫Ͱࢉܭ‬͸ηϧ਺Λແ‫͍ͯ͠ͱݶ‬Δ) ͜ͱ΍ి࣓৔γ

0ϛϡϨʔγϣϯͷϝογϡαΠζ͕ෆे෼Ͱ͋Δ͜ͱ͕Ө‫͍ͯ͠ڹ‬ΔͱࢥΘΕΔɻ

0

l

2l

3l

4l
4.2.5

ജप‫ߏظ‬଄Ճ଎؅ਤ 27: (4+1) ηϧఆࡏ೾‫ܕ‬Ճ଎؅ͷ֤Ґ૬Ϟʔυͷి

અ 4.2.3 ͱ 4.2.4 ʹ͓͍ͯɺఆࡏ೾‫ܕ‬Ճ଎؅Λ༻͍৔ (Ez )ɻਤ 26 Λ༻͍ͯి࣓৔ղੳʹΑΓಘΒΕͨɻ

Δ৔߹ʹ࠷΋Ճ଎ޮ཰͕ྑ͍ͷ͸ π ϞʔυΛ࠾༻ͨ͠ͱ͖Ͱ͋Δ͜ͱΛઆ໌ͨ͠ɻ͔͠͠ɺπ Ϟʔυʹ͸ਤ 28 ͸ಉ༷ͷ‫Ͱࢉܭ‬ಘΒΕͨి৔෼෍͔Βޫ଎Ͱ

અ 4.2.2 Ͱࢦఠͨ͠Α͏ʹ‫܈‬଎౓͕θϩʹͳΔ͜ͱʹ૸ΔిࢠͷγϟϯτΠϯϐʔμϯεΛ‫ٻ‬Ίϓϩοτ

‫ى‬Ҽͨ͠໰୊͕͋Γɺ࣮ࡍʹ͜ͷϞʔυΛͦͷ··Ճ(੺‫ͨ͠ )ؙ‬΋ͷͰ͋Δɻͨͩ͠ɺ4 ηϧͰ͸ϓϩοτ
఺͕গͳ͔ͬͨͨΊ 8 ηϧͷϞσϧͰ‫ࢉܭ‬Λߦ͍ɺπ଎؅ʹ࠾༻͢Δͷ͸ಘࡦͰ͸ͳ͍ɻಛʹηϧ਺͕େϞʔυͷ݁Ռʹ͍ͭͯ͸ɺಘΒΕͨγϟϯτΠϯϐʔࣜͰ͸ 5 ηϧఔ౓͕࣮༻্ͷ‫ݶ‬քͩͱߟ͑ΒΕ͍ͯμϯεͷ൒෼ͷ஋Λࢀߟͱͯ͠੺ࡾ֯Ͱࣔͨ͠ɻਤ 28

Δ [8, 24]35 ɻ͖͍৔߹ɺՃ଎ޮ཰͕‫ྼʹܹٸ‬Խ͢ΔͨΊৗ఻ಋํͰ͸ࣜ (3.15) ͷ܎਺ Ar Λ೚ҙʹௐ੔͠ɺ‫ࢉܭ‬஋ (πҰํɺπ/2 Ϟʔυ͸‫܈‬଎౓͕࠷େͰ͋ΔͨΊɺઅϞʔυΛআ͘఺ͰͷγϟϯτΠϯϐʔμϯε) ͷۙࣅ

4.2.2 Ͱࢦఠͨ͠ π Ϟʔυͷ໰୊Λආ͚ɺηϧ਺ΛΑ‫ۂ‬ઢ (࣮ઢ) ͱͯࣔͨ͠͠ɻࣜ (3.15) ͕ి࣓৔γϛϡΓ૿΍ͨ͠ఆࡏ೾‫ܕ‬Ճ଎؅͕࣮‫͖Ͱݱ‬Δɻ͔͠͠ɺఆ

ϨʔγϣϯͰಘΒΕͨγϟϯτΠϯϐʔμϯεͷมԽࡏ೾‫ܕ‬Ճ଎؅ͷ π/2 Ϟʔυ͸ɺਤ 27 Ͱ΋ࣔͨ͠Α͏܏޲Λྑ͘࠶‫͍ͯ͠ݱ‬Δ͜ͱ͕Θ͔Δɻ఺ઢ͸࣮ઢΛ

ʹ 1 ηϧ͓͖ʹి৔ৼ෯͕θϩͱͳΔͨΊɺγϟϯτ

2 ഒͨ͠‫ۂ‬ઢͰ͋Γɺਐߦ೾‫ܕ‬Ճ଎؅ͷγϟϯτΠϯ
ϐʔμϯεΛҙࣝͨ͠΋ͷͰ͋ΔɻਤΑΓɺి࣓৔γ
ϛϡϨʔγϣϯͰಘΒΕͨ݁Ռ͸ɺ͓͓Αͦղੳ‫ࢉܭ‬Πϯϐʔμϯε͸൒‫͢ݮ‬Δɻπ Ϟʔυͷߴ͍γϟϯτΠϯϐʔμϯεͱ π/2 Ϟʔυͷ҆ఆੑΛཱ྆͢΂͘ఏҊ͞Εͨͷ͕ജप‫ߏظ‬଄ (biperiodic structure) Ͱͱಉ͡܏޲Λ࠶‫͍ͯ͠ݱ‬Δ͜ͱ͕Θ͔Δɻͨͩ͠ɺҐ૬Ϟʔυ͕ π ʹۙͮ͘ʹͭΕ܏޲͕ҟͳ͖͍ͬͯͯ
Δɻ͜Ε͸‫ࢉܭ‬Ϟσϧͷۭಎ਺͕༗‫͋Ͱݶ‬Δ (ղੳత35 ௒఻ಋํࣜͷۭಎͰ͸ Q ஋͕ߴ͍ͨΊ͜ͷޮՌ͸؇࿨͞Εɺ‫ ʹ࣮ݱ‬9 ηϧͷ π Ϟʔυఆࡏ೾‫ܕ‬Ճ଎؅͕ଟ͘ͷՃ଎‫ث‬γεςϜͰ࠾༻͞Ε͍ͯΔɻ

加速セル : 2N+1 個

Disk loaded structure

Alternating periodic structure (APS)
結合セル : 2N 個
セル数合計：4N+1 個
Side coupled structure (SCS)

セル間の位相すすみ：πq /2N ,q = 0, 1, 2, …, 2N

ਤ 30: ্࣠݁߹ߏ଄Ճ଎؅ͷ‫ࢉܭ‬Ϟσϧɻ

w

結合セル

Annular coupled structure (ACS)
wc
wa
加速セル
Disk and washer structure (DWS)
0

π/4

π/2
πq/2N

3π/4

π

ਤ 29: ജप‫ߏظ‬଄Ճ଎؅ͷྫɺσΟεΫϩʔυ‫ܕ‬, APS

ਤ 31: ্࣠݁߹ߏ଄Ճ଎؅ͷ෼ࢄؔ܎ (Ճ଎Ϟʔυͱ‫ܕ‬, SCS ‫ܕ‬,ACS ‫ܕ‬, DW ‫ܕ‬ͷ໛ࣜਤɻਤதͷ໼ҹ͸݁߹Ϟʔυͷप೾਺͕ҟͳΔ৔߹)ɻ

π/2 Ϟʔυͷి৔ͷ༷ࢠΛ͍ࣔͯ͠Δɻి‫ؾ‬తʹ͸֎ଆͷηϧΛհͯ͠ π/2 ϞʔυͰ݁߹͋͠Γɺͦͷ୅දతͳ΋ͷ͕ APS(alternating periodic

structure) ߏ଄ [25, 26] Ͱ͋ΔɻAPS ߏ଄Ͱ͸ π/2 ϞʔυͰి৔ৼ෯͕θϩͱͳΔ

͍ͯΔͱߟ͑Δ͜ͱ͕Ͱ͖Δɻ

4.2.6

ജप‫ߏظ‬଄Ճ଎؅ͷ෼ࢄؔ܎ηϧͷ௕͞Λ΋͏Ұํͱൺֱͯ͠୹͘͢Δ͜ͱͰɺՃ
଎؅શମͱͯ͠ͷՃ଎ޮ཰Λ޲্͍ͤͯ͞Δɻ͜ͷജप‫ߏظ‬଄Ճ଎؅ʹ͍ͭͯཧղ͢ΔͨΊɺAPS ߏଞɺജप‫ߏظ‬଄ͷՃ଎؅ͱͯ͠͸αΠυΧοϓϧ‫ܕ‬଄ʹ୅ද͞ΕΔ্࣠݁߹ߏ଄ (on-axis coupled struc-ߏ଄ (SCS: side coupled structure) [27]ɺ‫؀‬ঢ়݁߹ΫɾΞϯυɾϫογϟʔߏ଄ (DAWS: disk-and-washerture) Λྫͱͯ͠෼ࢄؔ܎Λߟ͑Δɻ͜͜Ͱɺജप‫ظ‬ߏ଄Ճ଎؅ͷηϧͷ͏ͪɺి৔͕ଘࡏ͢ΔํͷηϧΛՃ଎ηϧ (accelerating cell)ɺଘࡏ͠ͳ͍ํΛ݁߹η

structure) [29] ͳͲ͕͋Δ (ਤ 29 ࢀর)ɻ͜ΕΒͷۭϧ (coupling cell) ͱ‫͢ʹͱ͜Ϳݺ‬Δɻ

ߏ଄ (ACS: annular coupled structure) [28]ɺσΟεಎͰ͸ి৔ৼ෯͕θϩͱͳΔηϧΛ APS ۭಎͷΑ͏্࣠݁߹ߏ଄Ճ଎؅ͷ‫ࢉܭ‬ϞσϧΛਤ 30 ʹࣔ͢ɻʹ୹͘͢ΔͷͰ͸ແ͘ϏʔϜ͔࣠Β֎ଆʹͣΒ͓ͯ͠Ճ଎؅͸ 2N ‫ݸ‬ͷ݁߹ηϧͱ 2N + 1 ‫ݸ‬ͷՃ଎ηϧͰΓɺిࢠϏʔϜ͕௨ա͢Δηϧ͸࣮࣭తʹ͸ π Ϟʔߏ੒͞ΕΔͱ͠ɺ߹‫ ܭ‬4N + 1 ‫ݸ‬ͷηϧΛߟ͑Δɻ͜υͱͳΔΑ͏ʹͳ͍ͬͯΔɻͭ·Γɺ͜ΕΒͷۭಎͰͷ৔߹ɺηϧؒͷҐ૬ਐΈ͸ πq/(2N ) Ͱද͞ΕΔɻͨ͸ిࢠϏʔϜ͔ΒΈΔͱ π ϞʔυͰྭৼ͍ͯ͠Δ͕ɺ

ͩ͠ɺq = 0, 1, 2, · · · , 2N Ͱ͋ΔɻΕΔɻҎ্ʹΑΓɺπ Ϟʔυͱಉ౳ͷՃ଎ޮ཰ͱ π/2Ϟʔυͷ҆ఆੑΛ࣋ͬͨఆࡏ೾Ճ଎؅͕࣮‫͢ݱ‬Δɻ͜
ͷΑ͏ͳ߹ྲྀ৚݅Λ࣮‫͢ݱ‬Δജप‫ߏظ‬଄Λ߹ྲྀ‫ܕ‬ജप‫ߏظ‬଄ͱ‫ͼݺ‬ɺҰൠʹ݁߹ۭಎ‫ܕ‬Ճ଎؅ͱ͍͏ͱ߹ྲྀ
‫ܕ‬ജप‫ߏظ‬଄Λඋ͑ͨՃ଎؅Λࢦ͢͜ͱ΋ଟ͍ɻ߹ྲྀ‫ܕ‬ജप‫ߏظ‬଄͸ෳࡶͰ͋Γ੡࡞΋ख͕͔͔ؒΔ
͕ɺπ/2 Ϟʔυͷ߹ྲྀ‫ܕ‬Ճ଎؅͸ߴ͍ੑೳ͕ಘΒΕΔ
ͨΊΑ͘༻͍ΒΕΔɻKEK ͷτϦελϯՃ଎‫Ͱث‬͸

9 ηϧͷ APS ۭಎ͕ 104 ୆༻͍ΒΕɺ‫ࡏݱ‬͸ KEKAR(6.5GeV) Ͱಉ༷ͷ΋ͷ͕ 6 ୆ӡ༻͞Ε͍ͯΔɻ·ͨɺཧ‫ ݚ‬SACLA ͷ௿ΤωϧΪʔ෦Ͱ͸ɺਐߦ೾‫ܕ‬Ճ଎؅ͷ৔߹ʹೖྗΧϓϥͷͭ͘Δి࣓৔ͷඇରশਤ 32: ্࣠݁߹ߏ଄Ճ଎؅ͷ෼ࢄؔ܎ (ωa = ωc ‫ͼٴ‬ωa ∼= ωc ) ɻੑΛආ͚ΔͨΊ L όϯυͷ APS ۭಎ͕༻͍ΒΕ͍ͯ
Δ [13]ɻ͜ͷଞɺߴग़ྗͱখ‫ܕ‬Խ͕ඞཁͳҩྍ༻Ճ଎‫ث‬౳Ͱ΋ଟ͘༻͍ΒΕ͍ͯΔɻ

෼ࢄؔ܎͸Ճ଎ηϧͷप೾਺Λ ωa ɺ݁߹ηϧͷप೾਺Λ ωc ͱ͢Δͱɺ͜ΕΒ྆ηϧؒͷ݁߹܎਺Λ κͱͯ͠ɺҎԼͷΑ͏ʹ༩͑ΒΕΔ 36 ɻ
3
43
4
πq
ωa 2
ωc 2
κ2 cos2= 1− 2
1− 2
2N
ωq
ωq

5

͓ΘΓʹ
Ҏ্ɺిࢠՃ଎؅ͱ͍͏୊໨ͰɺલఏͱͳΔಋ೾؅

(4.30)

ʹؔ͢Δ෺ཧΛ΋ͱʹɺਐߦ೾‫ܕ‬Ճ଎؅ɾఆࡏ೾‫ܕ‬Ճ଎؅ʹ͍ͭͯ‫ૅج‬తͳ஌ࣝΛ঺հͨ͠ɻຊςΩετͰωq ͸೚ҙͷҐ૬Ϟʔυʹ͓͚Δ‫ݻ‬༗प೾਺Ͱ͋Δɻ͜
͜Ͱ q = N ɺͭ·Γ π/2 ϞʔυͰ͸ɺࣜ (4.30) ͷࠨล͸͜ΕΒిࢠՃ଎؅ʹ͍ͭͯ၆ᛌతͳཧղΛಘΔ͜ͱ

͸θϩͱͳΓɺωq ͸ ωa ͱ ωc ͷ 2 ͭͷղΛऔΓಘΔɻ

Ұൠతͳ಺༰ʹཹΊ͖ͯͨͭ΋ΓͰ͋ΔɻిࢠՃ଎؅

ਤ 31 ʹ ωa < ωc ͷ৔߹ͷ෼ࢄؔ܎Λࣔ͢ɻ݁߹܎਺

ʹ͍ͭͯ͜Ε͔Βษ‫ڧ‬Λߦ͏ํʹ͸ຊςΩετͷ಺༰͸ k = 0.05 ͱ͠ɺηϧ਺ N ͸ແ‫ͨ͠ͱݶ‬ɻωa ̸= ωcΛ΋ͱʹͯ͠ɺΑΓਂ͘ઐ໳తͳ஌ࣝΛ਎ʹ͚ͭͯ௖ͭʹ෼཭͠ɺಛʹ ωa < ωc ͷ৔߹͸ɺՃ଎ηϧ಺ͰຊςΩετʹ͸ࢲͷཧղෆ଍ʹΑΔਖ਼֬Ͱ͸ແ͍‫ه‬0 ͔Β π/2 Ϟʔυ͕ɺ݁߹ηϧͰ π/2 ͔Β π Ϟʔυ͕ྭৼ͞ΕΔΑ͏ʹͳΔɻ͔͠͠ɺՃ଎Ϟʔυͱͯ͠ड़΍ؒҧͬͨղऍ΋ଘࡏ͢ΔͱࢥΘΕΔɻ͜Ε͔Β࣮ߟ͍͑ͯΔ π/2 ϞʔυͰͷ‫܈‬଎౓͸௨ৗͷՃ଎؅ͷʹͤͣੋඇࣗ෼ͷखͱ಄Λಈ͔ͯ͠‫͖ͨ׆‬஌ࣝΛ਎ʹπ Ϟʔυͱಉ༷ʹθϩͱͳͬͯ͠·͏ɻജप‫ߏظ‬଄Ճ଎؅Ͱ͸͜ͷঢ়‫͔گ‬Β͞Βʹηϧ‫ܗ‬ঢ়͚ͭͯ΋Β͍͍ͨɻ

ͷ৔߹ɺ(4N + 1) ηϧՃ଎؅ͱͯ͠͸෼ࢄ‫ۂ‬ઢ͕ 2Λɺωa = ωc ͱͳΔΑ͏ௐ੔͢Δɻωa = ωc Λ࣮‫͠ݱ‬ͨ৔߹ͷ෼ࢄ‫ۂ‬ઢΛਤ 32 ʹ੺৭ͷ࣮ઢͰࣔ͢ɻࢀߟͱͯ͠ ωa ∼= ωc ͷ৔߹ͷ‫ۂ‬ઢ΋߹Θͤͯࣔ͢ɻਤதɺ
੨৭ͷ࣮ઢ͕ਤ 31 Ͱࣔͨ͠෼ࢄ‫ۂ‬ઢͱಉ͡ύϥϝʔλͰͷղੳ‫݁ࢉܭ‬Ռ (εέʔϧ͸มΘ͍ͬͯΔ) Ͱ͋ΔɻਤΑΓɺπ/2 Ϟʔυʹ͓͚ΔՃ଎ηϧͱ݁߹ηϧͷ‫ݻ‬༗प೾਺Λ͚͍ۙͮͯ͘͜ͱͰɺঃʑʹ π/2 ෇ۙͷ෼ࢄ‫ۂ‬ઢ͕มԽ͠਱ʹ͸‫׈‬Β͔ʹͭͳ͕Δ͜ͱ͕Θ͔Δɻ·ͨɺ͜ͷ৚݅ʹ͓͚Δ π/2 Ϟʔυͷ‫܈‬଎౓͸ଞͷ఻೻ϞʔυதͰ࠷େͱͳ͍ͬͯΔ͜ͱ͕Θ͔Δɻ͜ͷ ωa = ωc ͷؔ܎͸߹ྲྀ (confluence) ৚݅ͱ‫ݺ‬͹
36 ࣜ (4.30) ΋౳Ձճ࿏Λ༻͍ͯ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻৄࡉ͸ [8]΍ [30] ͳͲΛࢀরͷ͜ͱɻΛ໨తʹɺ͋·Γ‫࣮ݱ‬ɾ࣮຿తͳ಺༰ʹ͸౿Έࠐ·ͣɺ͚Ε͹ͱࢥ͏ɻ

ࡍʹ RF ؔ࿈ͷษ‫ڧ‬Λ͢Δਓ͸ɺຊςΩετΛӏವΈ·ͨɺຊςΩετͷߏ૝ஈ֊Ͱ͸ɺ• ࣮ࡍͷՃ଎‫Ͱث‬ͷ RF Ճ଎؅ͷӡ༻ํ๏ʢిࢠ
ϏʔϜͱͷ૬‫༻࡞ޓ‬Λ‫ؚ‬Ίͯʣ• ࣮ࡍʹ։ൃɾӡ༻͞Εͨ RF Ճ଎؅ͷಠ૑తͳ޻෉΍ಛ௃ͳͲΛ‫ه‬ड़͢Δͭ΋ΓͰ͕͋ͬͨɺςΩετશମͷόϥϯε΍࣌ؒͷ౎߹ͰׂѪͨ͠ɻ͔͠͠ɺͦͷ݁Ռɺ50 ෼ 2 ίϚͷߨٛͱͯ͠͸ద੾ͳ಺༰ྔʹͳͬͨͷͰ͸ͳ͍͔ͱߟ͍͑ͯΔɻ‫ڵ‬ຯͷ͋Δಡऀ͸ຊςΩετͷ‫ૅج‬తͳ஌ࣝΛ΋ͱʹ࣮ࡍͷ RF Ճ଎؅Λௐ΂ͯΈΔͱΑΓਂ͘ษ‫͖Ͱڧ‬ΔͱࢥΘΕΔɻ

ँࣙྺ࢙͋Δ OHO Ճ଎‫ث‬ηϛφʔͰςΩετΛ࡞੒ͤͯ͞௖͘‫ػ‬ձΛԼͬͨ͞ߴΤωϧΪʔՃ଎‫ػڀݚث‬ߏͷখྛ޾ଇࢯɺಓԂਅҰ࿠ࢯʹ‫͢·͠ँײ‬ɻ·ͨɺ[9] P. B. Wilson, J. E. Griﬃn, RA. Carrigan,
FR. Huson, and M. Month. High energy electron linacs; application to storage ring rf systems and linear colliders. In SLAC-PUB-2884,
Vol. 87, pp. 450–555. AIP, 1982.

ߴΤωϧΪʔՃ଎‫ ߏػڀݚث‬Ճ଎‫ࢪڀݚث‬ઃͰಉάϧʔϓʹॴଐ͢Δࡔதষ‫ࢯޛ‬ɺߴ‫ʹࢯؽڮ‬͸ςΩετ
ࣥචʹଟେͳ‫ޚ‬ཧղΛ௖͖ɺ௨ৗ‫ۀ‬຿ͷܰ‫ݮ‬ɾࢿྉͷఏ‫ͯ͠ྗڠͲͳڙ‬௖͖·ͨ͜͠ͱ৺͔Β‫·͍ͯ͠ँײ‬͢ɻ͞Βʹɺࡔதষ‫ʹࢯޛ‬͸্‫ʹه‬Ճ͑ςΩετ಺༰ʹؔ͢Δ૬ஊɺ‫ߘݪ‬ͷνΣοΫͳͲଟେͳ‫ྗڠ‬Λ௖͖·ͨ͠ɻview of Scientific Instruments, Vol. 19, No. 2,
pp. 89–108, 1948.
[11] M. Chodorow, E. L. Ginzton, W. W. Hansen,
R. L. Kyhl, R. B. Neal, and W. K. H. Panofsky.
Stanford highʖenergy linear electron accelerator (mark iii). Review of Scientific Instruments,
Vol. 26, No. 2, pp. 134–204, 1955.

ࢀߟจ‫ݙ‬[1] T. Koseki, M. Izawa, and Y. Kamiya. Dampedstructure cavity for a high-brilliant vacuum ultraviolet and soft xray storage ring. Review of
Scientific Instruments, Vol. 66, No. 2, pp. 1926–
1929, 1995.
[2] T. Furuya, K. Akai, K. Hara, K. Hosoyama,
A. Kabe, Y. Kojima, S. Mitsunobu, Y. Morita,
H. Nakai, and T. Tajima. Recent status of thesuperconducting cavities for KEKB. In Proc. of
SRF9, pp. 31–36, 1999. paper MOP001.
[3] தౡকޫ. ϚΠΫϩ೾޻ֶ, ୈ 2 ষ, p. 44. ৿๺ग़൛, 1975.
[4] ٢ాޫ޺. ిࢠઢ‫ܗ‬Ճ଎‫͚͓ʹث‬Δߴप೾σόΠεͷ‫ૅج‬ʙγϛϡϨʔγϣϯʙ੡࡞ɾࢼ‫ݧ‬. ϏʔϜΤωϧΪʔճऩ‫ ݯޫޫࣹ์౓ًߴܕ‬- ERL -.

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[5] Thomas P. Wangler.

[10] E. L. Ginzton, W. W. Hansen, and W. R.
Kennedy. A linear electron accelerator. Re-

Principles of RF Lin-ear Accelerators, chapter 3, p. 66. A WileyInterscience, 1998.
[6] ࢁ‫ޱ‬੣࠸. Ճ଎؅ɾཱମճ࿏. OHO ςΩετ ిࢠઢ‫ܗ‬Ճ଎‫ث‬ͷ‫ૅج‬, ୈ 3.1.1 ষ, p. 5.4. KEK,2002.
[7] ‫ڷܙ‬തจ. ߴप೾Ճ଎؅. OHO ςΩετ ̭ઢࣗ༝ిࢠϨʔβʔ ʙ̨̡̖̘̖, ୈ 3.2 ষ. KEK,2013.
[8] ߴా ߞ࣏. ߴप೾Ճ଎ͷ‫ૅج‬ɹ-վగ൛-. In KEK
Report 2003-11. KEK, 2003.

[12] Gregory A. Loew and Richard Talman. Elementary principles of linear accelerators. AIP
Conference Proceedings, Vol. 105, No. 1, pp. 1–
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SCSS X-FEL

Conceptual Design Report. Technical report,
RIKEN Harima Institute/SPring-8, 2005.
[14] ‫ٱ‬อড়. ϏʔϜෆ҆ఆੑ. OHO ςΩετ ௿ΤϛολϯεϦϯάಛू, ୈ 3.1.1 ষ, p. 5.4. KEK,1991.
[15] ੺Ҫ࿨‫ݑ‬. ̧̛γεςϜ. OHO ςΩετ ̗ϑΝΫτϦʔͷՃ଎‫ث‬, ୈ 3.1.1 ষ, p. 5.4. KEK, 1994.
[16] Ѩ෦఩࿠. ߴप೾Ճ଎ೖ໳. OHO ςΩετ ̗ϑΝΫτϦʔͱՃ଎‫ث‬ͷ‫ૅج‬, ୈ 3.1.1 ষ, p. 5.4.
KEK, 2004.
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PEP-II b factory. In LBL-33360. 1992.
[18] S. Sakanaka, F. Hinode, K. Kubo, and
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[19] F. Marhauser, E. Weihreter, D. M. Dykes, and
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Vol. 2, pp. 846–848 vol.2, 2001.

[20] H. Ego, J. Watanabe, S. Kimura, and K. Sato.
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SPring-8-II storage ring. In Proc. of 11th PASJ
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MOOL14 in Japanese.

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[24] ੨‫޾׮‬. ߴप೾Ճ଎ͷ‫ૅج‬. OHO ςΩετ Ճ଎‫ث‬ͷ‫ͱૅج‬ҩྍԠ༻ʙҩྍͷͨΊͷՃ଎‫ث‬. KEK,2012.
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෇࿥ Aߴप೾ۭಎͷ౳Ձճ࿏

ͱͳΔɻ͜͜Ͱ V0 ͸ۭಎిѹͷৼ෯ʢϐʔΫ஋ʣͰ͋Γɺۭಎిѹ͸υϥΠϒిྲྀʹର͠ φ ͚ͩҐ૬͕ͣRF Ճ଎؅Λղੳ͢Δࡍɺ౳Ձճ࿏͸ඇৗʹศརͳπʔϧͱͳΔɻຊςΩετͰ͸‫͔ͭز‬ͷॏཁͳؔ܎ࣜ

Ε͍ͯΔͱͨ͠ɻ·ͨɺ‫ৼڞ‬प೾਺ ωr ͸ɺճ࿏ఆ਺
Λ༻͍ͯҎԼͷΑ͏ʹॻ͚Δɻ

Λূ໌ແ͠ʹ༻͍͕ͨɺ͜ΕΒͷ΄ͱΜͲ͸౳Ձճ࿏ωr = √

Λ༻͍Δ͜ͱͰൺֱతʹ؆୯ʹಋ͘͜ͱ͕Ͱ͖Δɻ·ͨɺ౳Ձճ࿏Λ༻͍Δ͜ͱͰɺՃ଎؅Λి‫ؾ‬తʹද‫ݱ‬͢Δ͜ͱ͕ՄೳͱͳΓɺՃ଎؅ʹର͢ΔΑΓਂ͍ཧղU=

͸ɺ࠷΋؆୯ͳ୯ηϧ RF ۭಎʹ͍ͭͯ౳Ձճ࿏Λ঺

1
CV0 2
2

P =

୯ηϧ RF ۭಎͷ౳Ձճ࿏ਤ 33 ʹ୯ηϧ RF ۭಎͷ౳Ձճ࿏Λࣔ͢ɻI(t) ͸֎෦͔Β‫͞څڙ‬ΕΔճ࿏ͷυϥΠϒిྲྀͰ͋Γɺ࣮ࡍʹ͸େిྗߴप೾‫͔ݯ‬Βͷ RF ిྗ‫څڙ‬Λҙຯ͢Δɻ

(෇࿥ A.5)఍߅Ͱͷ࣮ޮతͳফඅిྗ P ͸ɺ͓͘ɻ෇࿥ A.1(෇࿥ A.4)ίϯσϯαʹ஝͑ΒΕΔΤωϧΪʔ U ͸ɺͱൃలతͳΞΠσΞΛಘΔ͜ͱ΋‫ظ‬଴Ͱ͖Δɻ͜͜Ͱհ͠ɺRF ۭಎʹ܎ΘΔύϥϝʔλͱͷؔ܎Λࣔͯ͠

1
LC

1 V02
2 R

(෇࿥ A.6)

ͱͳΔɻ
ͱ͜ΖͰɺΑ͘༻͍ΒΕΔۭಎύϥϝʔλΛճ࿏ఆ਺Ͱॻ͖ද͢ͱҎԼͷΑ͏ʹͳΔɻ
ɾγϟϯτΠϯϐʔμϯεͦΕͧΕɺΠϯμΫλϯε L, ੩ి༰ྔ C, ఍߅ R ͱ
͍ͯ͠ΔɻRF ۭಎͰ͸֎෦ RF ‫ʹݯ‬ΑΓ‫ۦ‬ಈ͞Εۭಎిѹ Vc (t) ͕ྭৼ͞ΕΔɻ
'Vc (t)dt Vc
dVc
I(t) = C
+
+
dt
L
R

Rsh = 2R

(෇࿥ A.7)

ωr U= ωr RCP(෇࿥ A.8)

ɾQ ϑΝΫλʔ(෇࿥ A.1)
Q=ຊςΩετͰ͸͜ΕҎ্ཱͪೖΒͳ͍͕ɺ্‫ه‬ඍ෼ํఔࣜΛ༻͍ۭͯಎͷৼΔ෣͍Λ࣌ؒతʹղੳ͢Δ͜ͱ

ɾϑΟϦϯάλΠϜ

Ͱɺೖྗిྗ͕มಈͨ͠৔߹ͳͲͷա౉తͳۭಎԠ౴τf =ΛఆྔԽͯٞ͠࿦͢Δ͜ͱ͕ՄೳͱͳΔɻ͞Βʹɺਤ

33 ͷ౳Ձճ࿏Λ֦ு͢Δ͜ͱͰϏʔϜෛՙ΍ͦͷมಈΛ‫ؚ‬Ίͨղੳ΋ՄೳͱͳΔɻࣜ (෇࿥ A.1) ͷఆৗঢ়ଶʹ͓͍ͯ I(t), Vc (t) ͸I(t) = I0 eiωr t(෇࿥ A.2)

Vc (t) = V0 ei(ωr t+φ)(෇࿥ A.3)

2Q
= 2RC
ωr

(෇࿥ A.9)ͨͩ͠ɺγϟϯτΠϯϐʔμϯεʹ͍ͭͯ͸ిѹͷϐʔΫ஋Λ༻͍Δ͔࣮ޮ஋Λ༻͍Δ͔Ͱද‫͕ݱ‬มΘΓɺ࣮ࡍʹՃ଎‫ۀث‬քͰ͸྆ํͷఆ͕ٛࠞࡏͯ͠༻͍
ΒΕ͍ͯΔɻ্ࣜ (෇࿥ A.7) ͸ϐʔΫ஋Λ༻͍ͨ৔߹ͷ΋ͷ (P.B. Wilson ྲྀ) Ͱɺ࣮ޮ஋Λ༻͍ͨ৔߹
͸ɺRsh = R ͱͳΔɻ͜ͷͨΊɺγϟϯτΠϯϐʔVc(t)μϯε΍ۭಎ R/Q ʹ͍ͭͯ࿦͡Δࡍ͸஫ҙ͕ඞཁͰ͋Δɻ

෇࿥ A.2

R

L

C

I(t)

ճ࿏ఆ਺ͷҙຯ౳Ձճ࿏ͷཁૉ C, L, R ʹ͍࣮ͭͯࡍͷۭಎͱରൺ͓ͯ͘͠ͱཧղ͕ਂ·Δɻͦ͜Ͱɺ௨ৗՃ଎Ϟʔυͯ͠༻͍ΒΕΔ͜ͱͷଟ͍ TM010 Ϟʔυͷి࣓৔ΛྫʹͱΓઆ໌͓ͯ͘͠ɻਤ 34 ͱਤ 35 ʹ TM010 Ϟʔυͷ୯ηϧ RF ۭಎͰͷి࣓৔ (Ez , Hφ ) ෼෍Λࣔ͢ɻ͜ͷ 2 ͭͷి࣓৔੒෼ͷΈ͕ TM010 Ϟʔυʹଘࡏ͠ɺਤ 33: ୯ηϧ RF ۭಎͷ౳Ձճ࿏ɻ

ͦΕΒͷ࣌ؒతͳҐ૬͸ π/2 ͣΕ͍ͯΔɻ

131.7mm (0.44λ/2)

229.1mm

130mm

232mm

r
y

y

z

a

ਤ 36: PF ۭಎͷ‫ࢉܭ‬Ϟσϧ (1/4 அ໘)ɻ࣮ࡍͷۭಎ0l͸ਤԼଆͱࠨଆΛରশ໘ͱͨ࣠͠ରশߏ଄ͱͳΔɻz෦Λ‫͘͢ڱ‬Δ͜ͱͰ C Λେ͖͘Ͱ͖Δɻਤ 34: TM010 Ϟʔυͷ୯ηϧ RF ۭಎͰͷి࣓৔࣍ʹηϧͷ྆୺൘ (ίϯσϯα) ʹ‫͞څڙ‬ΕΔిՙ

(Ez ,Hφ ) ෼෍ɻ͸ۭಎଆ໘Λԟ෮ͯ͠ྲྀΕΔɻిྲྀ͕ྲྀΕΔͱΞϯϖʔϧ (Ampere) ͷ๏ଇʹै࣓ͬͯ৔͕ൃੜ͢Δɻ͜

Pillbox_TM010 mode, (z=0, y=0)Ε͕ɺ౳Ձճ࿏ͷΠϯμΫλϯε L Ͱ͋Γɺਤ 34 ͷ

1.0
0.8

Hφ ΛΈͯ΋ɺ֎प෦ͷ࣓৔‫͕౓ڧ‬େ͖͘ͳ͍ͬͯΔ
͜ͱ͕Θ͔ΔɻैͬͯɺΠϯμΫλϯε L ͸ۭಎଆ໘ͷ௕͞Λ୹͘͢Δ͜ͱͰখ͘͢͞Δ͜ͱ͕Ͱ͖Δɻ

0.6

͜Ε͸ϐϧϘοΫεͷ֯ͷ෦෼Λ‫ۂ‬ઢʹ͢Δ (R Λͭ

Ez/Emax

Hφ/Hmax͚Δ) ͜ͱΛҙຯ͢Δɻ

0.4

࠷‫ʹޙ‬఍߅ R Ͱ͋Δ͕ɺۚଐ͸༗‫ݶ‬ͷ఍߅Λ࣋ͭͨΊɺۭಎද໘Λߴप೾ిྲྀ͕ྲྀΕΔͱදൽ఍߅ʹ

0.2
0.0

ΑΓ೤͕ൃੜ͠ిྗଛࣦ͕‫͜ى‬Δɻ͜ͷిྗଛࣦ͕

0

a/2
Radial position

ਤ 35: TM010 Ϟʔυͷ୯ηϧ RF ۭಎͰͷి࣓৔(Ez /Emax ,Hφ /Hmax ) ෼෍ (z = 0, y = 0)ɻ
ి‫ؾ‬ճ࿏ʹ͓͍ͯίϯσϯα͸ 2 ຕͷి‫ిͰۃ‬ՙΛ஝ੵ͢Δɻίϯσϯαʹ஝͑ΒΕΔΤωϧΪʔ͸‫ۃ‬൘ؒͷిҐࠩͷೋ৐ʹൺྫ͢Δɻ͜Ε͸ TM010 ϞʔυΛྭৼ͍ͯ͠Δ RF ۭಎͰ͍͏ͱɺηϧͷ྆୺൘ͱಉ

a

Pc = V02 /(2R) = V02 /Rsh Ͱ͋Δɻͨͩ͠ɺ౳Ձճ࿏
ͷ఍߅ R ͸ۭಎͷ୯ͳΔද໘఍߅ͱ͸ҟͳΔɻ·ͨɺ
Vc (t) ͸ిࢠϏʔϜͷ‫ي‬ಓʹԊۭͬͯಎͷೖ‫͔ޱ‬Βग़‫ࢠిʹͰ·ޱ‬ϏʔϜ͕ड͚Δి৔Λੵ෼ͨ͠΋ͷͰ͋Δ (અ 3.8.1 ࢀর)ɻ෇࿥ A.3࣮ࡍͷۭಎ࣮ࡍͷۭಎΛྫʹճ࿏ఆ਺ʹ͍ͭͯ΋͏গ͠ิ଍
͢Δɻ

౳ͩͱߟ͑Δ͜ͱ͕Ͱ͖ΔɻTM010 ϞʔυͰ͸ి৔

ਤ 36 ʹ‫ ࡏݱ‬KEK-PF Ͱ༻͍ΒΕ͍ͯΔ RF ۭಎ [1]͸ηϧͷ྆୺൘ؒʹଘࡏ͠ɺԁஅ໘ͷத৺෦΁ߦ͘΄ͷ໛ࣜਤΛ੨ઢͰࣔ͢ɻPF Ͱ͸ຊۭಎͷ TM010 ϞʔͲి৔͸େ͖͍ɻैͬͯɺRF ۭಎͷ୺൘ͷಛʹத৺υΛՃ଎Ϟʔυͱͯ͠༻͍͓ͯΓɺ‫ܗ‬ঢ়͸ϐϧϘοΫ

ද 6: PF ‫ۭܕ‬ಎͷύϥϝʔλ
TM010 Ϟʔυͷ‫ৼڞ‬प೾਺
500 MHzQ0 ஋
40000γϟϯτΠϯϐʔμϯε RshՃ଎ిѹ V0 @ 36 kW

7.0 MΩ
500 kVճ࿏ఆ਺੩ి༰ྔ C

3.6 pFΠϯμΫλϯε L

27.9 nHε‫( ܕ‬ਤதͷ੺ઢ) Λϕʔεʹߴ࣍ϞʔυΛߟྀͭͭ͠γϟϯτΠϯϐʔμϯεͱ Q ஋Λ࠷దԽ͢ΔΑ͏मਖ਼͞Ε͍ͯΔɻ‫ৼڞ‬प೾਺͸໿ 500 MHz Ͱ͋Δɻ؆୯ʹઆ໌͢Δͱ؅࣠ํ޲ʹ͍ͭͯ͸ϊʔζίʔϯͱ‫ݺ‬͹ΕΔۭಎ಺෦ʹͤΓग़ͨ͠‫ܗ‬ঢ়Λ΋ͪɺి৔ΛΑΓूத͍ͤͯ͞Δɻ֎प෦͸ԁ‫ؙ͍ۙʹܗ‬ΈΛଳͼ͓ͯΓɺϩεΛ཈͑ Q ஋Λ޲্͍ͤͯ͞Δɻ·ͨɺηϧ௕͸ϏʔϜύΠϓ΁ͷి৔ͷછΈग़͠Λߟྀ͠ɺϐϧϘοΫεͷ࠷ద௕͞ 0.44λ ΑΓ΍΍୹͘ͳ͍ͬͯΔɻ
͜͜ͰɺPF ‫ۭܕ‬ಎͷύϥϝʔλΛճ࿏ఆ਺Ͱද͠
ͯΈΔɻද 6 ʹ PF ‫ۭܕ‬ಎͷେମͷύϥϝʔλΛࣔ͢ɻQ0 ஋͸ 40, 000, γϟϯτΠϯϐʔμϯε͸ 7.0 M Ω,ͦͯ͠‫ৼڞ‬प೾਺͸ 500 MHz Ͱ͋ΔͷͰɺ੩ి༰ྔ C
͸ 3.6 pFɺΠϯμΫλϯε͸ 27.9 nH ͱ‫͞ࢉܭ‬ΕΔɻ
ͱ͜ΖͰɺPF ‫ۭܕ‬ಎͰ͸࠷େͰ 500 kV(໿ 36 kW ͷফඅిྗΛԾఆͨ͠৔߹) ͷՃ଎ిѹΛ࣮‫͖Ͱݱ‬Δɻ্‫ه‬ͷΑ͏ͳ௿͍ΠϯμΫλϯεΛ࣋ͪɺ500 kV ΋ͷߴిѹΛ 500 MHz ͰҹՃͰ͖Δͱ͍͏ͷ͸ RF ۭಎͷେ͖ͳಛ௃ͱ͍͑Δɻ