方向性結合器のGreen-関数を用いた電力移行特性の解析とパラメータの最適設計

ํ޲ੑ݁߹ثͷ

Green-ؔ਺Λ༻͍ͨిྗҠߦಛੑͷղੳͱ

ύϥϝʔλͷ࠷దઃܭ

্ϲࢢɹ࣮ԝ

†

؛Ԭɹਗ਼

‡

Analysis and parameter optimum design on the power transfer

characteristics in the optical directional couplers

using the Green-function

Mio KAMIGAICHI

†

and Kiyoshi KISHIOKA

‡

Abstract: This paper describes an analyzing method in the optical directional couplers

with perturbed structure parameters, in which the Green-function is utilized in solving the coupled mode equations for the purpose of easily understanding physical meanings of the power transfer characteristics in the perturbed structures. Parameter designs in the couplers composed of the nonlinear waveguides are effectively performed using the advantages of the Green-function, and designed characteristics of an optical switch, power filter and power limiter are demonstrated.

1

·͕͖͑

ޫ௨৴γεςϜٕज़ͷൃలͱͦͷ༻్ͷ֦େʹ൐͍ɺγεςϜʹ༻͍ΒΕΔߴੑೳͳޫ σόΠεͷ։ൃͱɺ৴པੑͷ֬อ͕ӹʑॏཁͱͳ͍ͬͯΔɻ͜ͷΑ͏ͳٕज़తͳχʔζʹ ରͯ͠ɺಋ೾࿏Λ఻ൖ͢Δޫ೾ͷੑ࣭Λ޼Έʹར༻ͨ͠ߴػೳͳฏ໘ಋ೾࿏ܗޫσόΠε ͸ɺ௿ίετͰɺूੵ౓͕ߴ͘ɺ৴པੑͷߴ͔͞Β࠷΋༗ޮͳޫσόΠεͰ͋Δͱ͞Εͯ ͍Δɻ ํ޲ੑ݁߹ث (Directional Coupler) ͸ɺޫಋ೾࿏Ͱߏ੒͞ΕΔޫूੵճ࿏ૉࢠͷجຊߏ ੒ཁૉͷҰͭͱͯ͠ɺछʑͷߏ଄͕ఏҊ͞Εɺͦͷ੡࡞ํ๏ΛؚΊͯ੝Μʹݚڀ͞Ε͍ͯ Δ [1]ɻಋ೾࿏ؒͰى͜ΔύϫʔҠߦ͸ɺ۶ં཰ɺಋ೾࿏෯౳ͷߏ଄ύϥϝʔλʹ༩͑Δ ઁಈྔΛ੍ޚ͢Δ͜ͱʹΑͬͯɺॴ๬ͷಛੑ͕࣮ݱ͞Ε͍ͯΔɻ

݁߹ثͷಛੑͷղੳํ๏ͱͯ͠ɺBPM(Beam Propagation Method) ͷΑ͏ͳܭࢉػΛ ༻͍Δ਺஋ܭࢉ๏ [2] ͱɺ݁߹ཧ࿦Λ༻͍Δղੳతͳख๏ [3],[4] ͕޿͘༻͍ΒΕ͍ͯΔɻ BPM ͸ɺ݁߹ث಺Λ఻ൖ͢Δޫͷք෼෍ͷ༷ࢠΛՄࢹԽ͢Δͷʹ༗ޮͰ͋Δͱಉ࣌ʹɺ ߏ଄ύϥϝʔλʹର͢Δద༻ൣғ΋޿͍ɻܭࢉਫ਼౓΋ߴ͘ɺύϥϝʔλͷઁಈΛؚΉ൥ࡶ ͳߏ଄΁ͷద༻ՄೳͰ͋Δ͕ɺಛੑͷ֤ύϥϝʔλґଘੑͷ೺Ѳ͕೉͘͠ɺଟେͳܭࢉ࣌ ؒΛཁ͢Δ೉఺͕͋ΔɻҰํɺ݁߹ཧ࿦ʹΑΔղੳ͸ɺઁಈ͕ಛੑʹ༩͑Δ෺ཧతͳҙຯ ͷ೺Ѳ͕༰қͰ͋Δ͜ͱʹՃ͑ɺܭࢉίετ͸ BPM ΑΓང͔ʹ༏ҐͰ͋Δɻ †େࡕిؾ௨৴େֶɹֶज़ݚڀһ ‡େࡕిؾ௨৴େֶɹ޻ֶ෦ɾిࢠػց޻ֶՊ 大阪電気通信大学　研究論集 （自然科学編） 第 52 号 大阪電気通信大学　研究論集 （自然科学編） 第 52 号

ຊ࿦จͰ͸ɺߏ଄ύϥϝʔλʹઁಈΛؚΜͩνϟωϧಋ೾࿏Ͱߏ੒͞ΕΔ 2 ಋ೾࿏ํ޲ ੑ݁߹ث͕ཧ࿦తʹऔΓѻΘΕ͍ͯΔɻύϥϝʔλͷઁಈྔͱಛੑͱͷؔ܎ͷ೺ѲΛ༰қ ʹ͢Δ໨తͰɺಋ೾࿏ͷϞʔυৼ෯ͷؒʹ੒ཱ͢Δ݁߹ํఔࣜͷղͷ Green-ؔ਺ [5] Λ༻ ͍ͨఆࣜԽ͕ߦΘΕ͍ͯΔɻ͞ΒʹɺGreen-ؔ਺ʹΑΔදݱΛ׆͔ͯ͠ɺඇઢܗ݁߹ثͷ ύϥϝʔλͷઃܭྫ΋ࣔ͞Ε͍ͯΔɻ ݁߹ํఔࣜΛ Green-ؔ਺Λ༻͍ͯղ๏͢Δ͜ͱʹΑͬͯɺઁಈΛ࣋ͬͨ݁߹ܥͷղ͕ ղੳతʹ؆ܿʹදݱ͞Εɺಋ೾࿏ؒͰੜ͡ΔޫͷύϫʔҠߦಛੑͷύϥϝʔλґଘੑͷ೺ Ѳ͕༰қͱͳ͍ͬͯΔɻՃ͑ͯɺඇઢܗޫֶޮՌ (Kerr-ޮՌ) ʹΑΔ۶ં཰ͷઁಈͱɺಋ ೾࿏෯౳ͷزԿֶతͳઁಈΛ۠ผ͢Δ͜ͱͳ͘ɺ౷ҰతʹऔΓѻ͏͜ͱ͕ՄೳͰ͋Δ఺΋ େ͖ͳಛ௕ͱͳ͍ͬͯΔɻ

2

ํ޲ੑ݁߹ثͷಈ࡞

:DYHJXLGH 5*ODVV :DYHJXLGH P lc in Pout Pout 1 z=0

f

1( )

f

2( )

WG

1

WG

2 l z y

a

2

a

y y ਤ 1ɿνϟωϧಋ೾࿏ ਤ 2ɿ݁߹ث಺ͷύϫʔҠߦ ຊ࿦จͰऔΓѻΘΕΔํ޲ੑ݁߹ث͕ਤ 1 ʹࣔ͞Ε͍ͯΔɻΨϥεج൘தʹΠΦϯ֦ࢄ ʹΑͬͯ࡞ΒΕͨνϟωϧಋ೾࿏Ͱߏ੒͞ΕΔ 2 ಋ೾࿏ํ޲ੑ݁߹ثͷύϫʔҠߦಛੑ͕ औΓѻΘΕΔ [6]ɻ ਤ 2 ʹ͸ɺ2 ಋ೾࿏ؒͰى͖ΔύϫʔҠߦͷ༷ࢠ͕໛ࣜతʹඳ͔Ε͍ͯΔɻยํͷಋ೾ ࿏͔Βೖࣹͨ͠ޫ͕఻ൖ͢ΔʹͭΕͯ΋͏ยํͷಋ೾࿏΁ͱҠߦ͢Δɻ݁߹ثͷஅ໘ʢy ํ޲ʣͷք෼෍ F (y, z) Λɺಋ೾࿏͕୯ಠͰଘࡏ͍ͯͨ͠৔߹ͷͦΕͧΕͷಋ೾࿏ͷϞʔ υͷք෼෍ f1(y)ɺf2(y) ͷ࿨ͱͯ͠ɺ

F (y, z) = a1(z)f1(y) + a2(z)f2(y) (1)

ͷΑ͏ʹۙࣅతʹද͢ɻͦΕͧΕͷಋ೾࿏ͷք͸ɺ͓ޓ͍ͷӨڹΛड͚ͯ఻ൖํ޲ (z ํ޲) ʹԊͬͯมԽ͢Δ͕ɺͦͷมԽ͸ a1(z)ɺa2(z) ʹΑͬͯද͞Ε͍ͯΔɻ֤ಋ೾࿏ͷϞʔυ ͷք෼෍͕఻ૹిྗͰن֨͞Ε͍ͯΕ͹ɺن֨Խ͞Εͨք෼෍͸ɺ _|fi|2_{dy = 1, (i = 1, 2)} Λຬͨ͢ͷͰɺ֤ಋ೾࿏ͷ఻ૹిྗ Pi, (i = 1, 2) ͸ɺ Pi =|ai|2  ֤ಋ೾࿏அ໘|f| 2_{dy = |a} i|2, (i = 1, 2) (2)

ͷΑ͏ʹϞʔυৼ෯ aiͷઈର஋ͷ 2 ৐Ͱද͞ΕΔɻ Ϟʔυৼ෯ ai͸ɺF (y) Λ Maxwell ͷํఔࣜʹ୅ೖͯ͠ಘΒΕΔ z ʹؔ͢Δ࿈ཱඍ෼ํ ఔࣜΛຬͨ͠ɺͦΕΛղ͘͜ͱʹΑͬͯɺ֤ಋ೾࿏ͷޫύϫʔͷ z ґଘੑΛ஌Δ͜ͱ͕Ͱ ͖Δɻ

2.1

ઁಈ͕͋Δ݁߹ث

δ

1

+Δ

0

-Δ

0

e

−jβz

δ

2 1

a

2

a

ਤ 3ɿઁಈΛؚΉ݁߹ث ਤ 3 ʹࣔ͞ΕΔΑ͏ͳ 2 ಋ೾࿏ؒʹҐ૬ෆ੔߹ (఻ൖఆ਺ͷࠩ) Λ༩͑ͨ݁߹ثΛߟ͑ Δɻ఻ൖఆ਺͕ࠩੜ͡ΔཁҼͱͯ͠͸ಋ೾࿏෯ͷࠩҟɺ۶ં཰ͷมԽ౳͕૝ఆ͞ΕΔɻਤ 3 ʹ͸ɺҰ༷ͳ఻ൖఆ਺ࠩ Δ_{0ͱ఻ൖ࣠ʹԊͬͯมԽ͢Δ఻ൖఆ਺ࠩ δ1(z)ɺδ2(z) ͷ྆ํ} ͕ଘࡏ͢Δ݁߹ث͕ඳ͔Ε͍ͯΔɻ ਤ 3 ͷΑ͏ʹɺಋ೾࿏ؒʹҐ૬ෆ੔߹͕͋Δͱɺಋ೾࿏ؒͷύϫʔҠߦ͕๦͛ΒΕɺ׬ શͳύϫʔҠߦ͕ى͜Βͳ͘ͳΔɻҐ૬ෆ੔߹ྔͷ఻ൖ࣠ʹԊͬͨ෼෍Λௐ੔͢Δ͜ͱʹ ΑΓɺ೾௕ϑΟϧλɺύϫʔσόΠμɺ෼೾ث౳ͷػೳૉࢠʹɺॴ๬ͷಛੑΛ࣋ͨͤΔ͜ ͱ͕Ͱ͖Δɻ ୈ 4 અͰࣔ͞ΕΔܭࢉྫͰ͸ɺඇઢܗޫֶޮՌʢKerr-ޮՌʣʹΑͬͯޫڧ౓ʹൺྫ͠ ͯಋ೾࿏ͷ۶ં཰͕มԽ͠ɺͦΕʹΑͬͯϞʔυͷ఻ൖఆ਺͕มԽ͠ɺ఻ൖఆ਺ࠩΛੜ͡ Δ৔߹ͷύϫʔҠߦಛੑ΋ࣔ͞Ε͍ͯΔɻ 2.1.1 ఻ૹύϫʔʹର͢Δ࣮ޮ۶ં཰ͷมԽ ͜͜Ͱ͸ɺඇઢܗ݁߹ثͷಛੑͷղੳɾઃܭʹඞཁͳಋ೾࿏ͷ఻ૹύϫʔʹର͢ΔϞʔ υͷ࣮ޮ۶ં཰ͷมԽ͕ܭࢉ͞ΕΔ [7]ɻޫֶඇઢܗʢKerr-ޮՌʣࡐྉͱͯ͠ɺ൒ಋମυʔ ϓΨϥεʢHOYA ੡/R-66 Sharp-cutoff Filter Glassʣ͕૝ఆ͞Εɺͦͷதʹ K-ΠΦϯͷ ೤֦ࢄʹΑͬͯ࡞ΒΕΔඇઢܗνϟωϧಋ೾࿏ͷ఻ૹిྗ P ʹର͢ΔϞʔυͷ࣮ޮ۶ં

ද 1ɿܭࢉʹ༻͍ͨύϥϝʔλ ද໘۶ં཰มԽ ɹ _{Δn 1.868×10}−3 ج൘ͷ۶ં཰ _ns 1.5457 ಋ೾࿏෯ɹɹɹɹɹ _W 9 _{[μ m]} K+ͷ֦ࢄ࣌ؒ _{t 30} [min] ֦ࢄ܎਺ɹ _Dk 0.28 _{[μ m}2/min] ֦ࢄਂ͞ɹ _{Depth 5.797 [μ m]} ඇઢܗ۶ં཰ɹ _{n2 8.6 × 10}−15 _[m2_/W] ޫͷ೾௕ɹɹɹɹ _{λ 0.66 [μ m]} ಋ೾࿏ִؒ (݁߹ث) _{ɹ d 3.5 [μ m]} ڥք-ڥքִؒ 0 0.4 0.8 1.2 1.6 1.5458 1.5459 P [W] Nef f ns=1.5457 nc=1.0 N =A0+A1P A0=1.546 A1 n2=8.6E-15 [m /W]2 eff =9.347E-5 ਤ 4ɿ఻ૹిྗʹର͢ΔϞʔυͷ࣮ޮ۶ં཰ͷมԽ ઢܗྖҬͷ࣮ޮ۶ં཰ Neff = 1.54579 ਤ 4 ʹࠩ෼๏ʹΑͬͯٻΊΒΕͨඇઢܗνϟωϧಋ೾࿏Λ఻ൖ͢ΔجຊϞʔυͷ࣮ޮ۶ ં཰ͷ఻ૹిྗ P ʹର͢ΔมԽ͕ϓϩοτ͞Ε͍ͯΔɻ࣮ޮ۶ં཰ͷมԽྔ ΔNeff ͸఻ ૹిྗ P ʹରͯ͠ɺ΄΅ൺྫؔ܎ʹ͋ΓɺΔNeff = N2P ͕੒ཱ͍ͯ͠Δɻൺྫఆ਺ N2ͷ ஋͸ɺਤ 4 ʹϓϩοτ͞Εͨ௚ઢͷ܏͖ A1ͱͯ͠ಘΔ͜ͱ͕Ͱ͖ɺ9.347 × 10−5[1/W] ͱ ͳΔɻಘΒΕͨ N2ͷ஋ΑΓɺ఻ૹύϫʔ P ͕༩͑ΒΕͨ࣌ͷ఻ൖఆ਺ͷઁಈྔ δ ͸Ϟʔ υৼ෯Λ༻͍ͯɺ δ = ΔNeff · k0 = k0N2 · P = k0N2· |a|2 (3) ͷ༷ʹಘΒΕΔɻ͜͜Ͱɺk0͸ޫͷਅۭதͰͷ೾਺Ͱ͋Δɻ ܭࢉʹ࢖ΘΕͨ൒ಋମυʔϓΨϥεʢR-66ʣͷόϧΫͰͷඇઢܗ۶ં཰ n2ͷ஋͸ɺ8.6× 10−15[m2/W] Ͱ͋Δ [8]ɻͦͷଞͷ֦ࢄಋ೾࿏ͷߏ଄ύϥϝʔλ͸ද 1 ʹࣔ͞Ε͍ͯΔɻ

ඇઢܗνϟωϧಋ೾࿏ͷ఻ൖϞʔυͷ࣮ޮ۶ં཰͸ҎԼʹࣔ͢ܭࢉखॱʹΑͬͯಘΔ͜ ͱ͕Ͱ͖ΔɿKerr-ޮՌΛߟྀʹೖΕͨಋ೾࿏அ໘಺ͷ۶ં཰෼෍Λ n(x, y) ͱද͠ɺϞʔ υͷ࣮ޮ۶ં཰Λ Neff ͱද͢ͱɺϞʔυͷిք E ͸ɺ  ∂2 ∂x2 + ∂2 ∂y2  E + k02[n2(x, y) − Neff2]E = 0 (4) ͷ Helmholtz ํఔࣜΛຬͨ͢ɻKerr-ޮՌΛߟྀʹೖΕͨ۶ં཰෼෍ n(x, y) ͸ɺ఻ૹޫύ ϫʔ͕খ͘͞ Kerr-ޮՌʹΑΔ۶ં཰มԽ͕ແࢹͰ͖ΔઢܗྖҬͷ۶ં཰෼෍ ¯_{n(x, y)[9],[10]} ͱ Kerr-ޮՌʹΑΔมಈ෼ Δn(x, y) ͷ࿨ͱͯ͠ɺ n(x, y) = ¯n(x, y) + Δn(x, y), Δn(x, y) = 1 2n2 | E(x, y) | 2 n(x, y) Z0 (5) Ͱ༩͑ΒΕΔɻ͜͜ͰɺZ0͸ਅۭதͷ೾ಈΠϯϐʔμϯεΛද͍ͯ͠ΔɻE Λ x ٴͼ y ʹ ͍ͭͯࠩ෼Խ͠ɺࣜ (4) Λࠩ෼๏Ͱ਺஋తʹղ๏͢Δ͜ͱʹΑͬͯɺϞʔυͷք෼෍ʢE ͷ෼෍ʣͱ۶ં཰෼෍ n(x, y) ΛಘΔ͜ͱ͕Ͱ͖Δɻࠩ෼๏ʹΑΔղ๏ͷ E ͷ൓෮աఔʹ ԙ͍ͯɺ࣮ޮ۶ં཰๏ [7] ʹΑͬͯɺ൓෮ͷ֤εςοϓຖʹνϟωϧಋ೾࿏ͷ࣮ޮ۶ં཰ Neff Λܭࢉ͠ɺͦͷ஋ͷऩଋΛҎͯ൓෮Λऴྃͤͨ͞ɻE(x, y) ͸఻ૹύϫʔʹൺྫͨ͠ ೚ҙੑ͕͋ΔͷͰɺ൓෮ͷ౓ʹͦͷ௚લͰಘΒΕͨ E(x, y) ͱ n(x, y) ʹΑͬͯಋ೾࿏அ໘ ಺ͷ఻ૹύϫʔ P Λܭࢉ͠ɺͦͷ஋͕ಋ೾࿏΁ͷೖྗύϫʔ Pinʹ౳͘͠ͳΔΑ͏ʹɺE ʹ√_Pin/√_{P Λ৐͡ͳ͕Β൓෮͕܁Γฦ͞Εͨɻ} 2.1.2 ඇઢܗ݁߹ثͷύϥϝʔλͷ఻ૹύϫʔʹର͢ΔมԽ 0 0.4 0.8 1.2 1.6 0.00028 0.0003 0.00032 0.00034 0 0.001 P P [W] 1.0[W] 0.5[W] C

←

→

Δβ C [1/ μ m] Δβ [1/ μ m] 入射側導波路) [W] d=3.5[μm] 導波路間隔 -0.001 P ( #1+ #2＝1.5 #1 ਤ 5ɿPin(= P#1+ P#2) ʹର͢Δ݁߹܎਺ c ͱҐ૬ෆ੔߹ྔ Δβ ͷมԽ 0 0.4 0.8 1.2 1.6 1.5458 1.5459 P#1+P#2 =1.5[W] 1.0[W] 0.5[W] P (入射側導波路)[W] 0.5[W] 1.0[W] Neff P#1+P#2=1.5[W ] d=3.5[μm] 導波路間隔 Even-mode Odd-mode #1 ਤ 6ɿPin(= P#1+ P#2) ʹର͢Δۮରশͱ حରশϞʔυͷ࣮ޮ۶ં཰ͷมԽ

ਤ 5 ʹ͸ɺඇઢܗ݁߹ث΁ͷೖྗύϫʔ PinΛύϥϝʔλʹͯ͠ɺೖྗଆͷಋ೾࿏ͷύ ϫʔ P#1(΋͏Ұํͷಋ೾࿏ͷύϫʔɿP#2) ʹର͢Δ݁߹܎਺ c ͱҐ૬ෆ੔߹ྔ Δβ ͷม Խ͕ࣔ͞Ε͍ͯΔɻඇઢܗಋ೾࿏Ͱ͸ɺಋ೾࿏ͷ఻ૹύϫʔʹԠͯ͡Ϟʔυͷ࣮ޮ۶ં཰ ͕มԽ͢ΔͨΊɺಋ೾࿏ؒͷిྗҠߦྔʹԠͨ͡Ґ૬ෆ੔߹ྔ͕ൃੜ͢Δɻಉ࣌ʹɺಋ೾ ࿏ͷ۶ં཰ͷมԽʹ൐͍ಋ೾࿏΁ͷքͷूதͷ౓߹͍΋มԽ͠ɺͦΕʹ൐ͬͯɺ݁߹܎਺ ΋มԽ͢Δɻ Δβ ͸ɺ֤ಋ೾࿏ʹ఻ૹిྗ P#1ɺP#2Λ༩͑ͯɺࠩ෼๏ʹΑ֤ͬͯಋ೾࿏ͷϞʔυͷ ࣮ޮ۶ં཰Λܭࢉͯ͠ɺͦΕΑΓɺ֤ಋ೾࿏ͷϞʔυͷ఻ൖఆ਺ β1ͱ β2ΛٻΊɺͦͷࠩ ΑΓܭࢉ͞Ε͍ͯΔɻ Ұํɺc ͸ɺ֤ಋ೾࿏ͷ఻ൖఆ਺ β1ɺβ2ʹՃ͑ͯɺ2 ಋ೾࿏͔ΒͳΔ݁߹ثΛ఻ൖ͢Δ ۮରশϞʔυͱحରশϞʔυͷ఻ൖఆ਺ βeͱ βoΛࠩ෼๏ͰٻΊɺͦΕΒΑΓɺ c = 1 2  (βe− βo)2− (β1 − β2)2 (6) ʹΑͬͯܭࢉ͞Ε͍ͯΔɻܭࢉʹ͸ද 1 ʹࣔ͞Ε͍ͯΔߏ଄ύϥϝʔλͷ஋͕࢖༻͞Εͨɻ ਤ 6 ʹ͸ɺ݁߹ͨ͠ಋ೾࿏ͷجຊϞʔυͰ͋ΔۮରশϞʔυͱɺୈҰ࣍ϞʔυͰ͋Δ حରশϞʔυͷ࣮ޮ۶ં཰ Neffeͱ Neffoͷܭࢉ݁Ռ͕ɺਤ 5 ͱಉ༷ɺ݁߹ث΁ͷೖࣹύ ϫʔ PinΛύϥϝʔλʹͯ͠ɺҰํͷಋ೾࿏ͷ఻ૹύϫʔ P ʹର͢ΔมԽ͕ࣔ͞Ε͍ͯ ΔɻۮରশϞʔυͱحରশϞʔυͷ఻ൖఆ਺͸ɺβe = k0Neffeɺβo = k0NeffoʹΑͬͯܭ ࢉ͞Ε͍ͯΔɻ ਤ 5 ʹࣔͨ͠ܭࢉ݁Ռ͔ΒɺΔβ ͷมԽ (ӈॎ࣠) ʹൺ΂ͯ c ͷมԽ (ࠨॎ࣠) ͸े෼ʹখ ͍͞ࣄ͕൑Δɻैͬͯɺ6 અͰͷγϛϡϨʔγϣϯ΍ઃܭͰ͸ɺೖࣹύϫʔʹର͢Δ c ͷ มԽΛແࢹ͠ɺઢܗྖҬͷ஋ͱͯ͠औΓѻΘΕΔɻ

3

Ұ༷ͳύϥϝʔλΛ࣋ͭ݁߹ํఔࣜͷ݁߹ํఔࣜͷղ

͜͜Ͱ͸ɺҰ༷ͳߏ଄ύϥϝʔλͱҰ༷ͳҐ૬ෆ੔߹ྔ Δ₀Λ࣋ͭ݁߹ث ʡඇઁಈܥ ʡ ͷ݁߹ํఔࣜͷղ͕ٻΊΒΕΔɻ

a

1

a

2

z

z=0

c

β+Δ

₀

β-Δ

₀ ਤ 7ɿҰ༷ͳύϥϝʔλΛ࣋ͭ݁߹ث

݁߹ํఔࣜͷղ

ਤ 7 ʹࣔ͢Α͏ʹɺҰ༷ͳҐ૬ෆ੔߹ྔ_±Δ0Λ࣋ͭ݁߹ثʹԙ͍ͯɺ֤ಋ೾࿏ͷքͷ z ํ޲ͷมԽΛ༩͑ΔϞʔυৼ෯Λ଎͍มԽͱ஗͍มԽͷੵͰද͠ɺai(z) · e−jβz ͱ͢Δ ͱɺ஗͍มԽΛද͢ ai(z), (i = 1, 2) ʹ͍ͭͯͷ݁߹ํఔࣜ͸ɺ ∂a ∂z =−j ⎛ ⎝ +Δ0 c c −Δ0 ⎞ ⎠_{a ≡ −j ˆCa (7)} Ͱ༩͑ΒΕΔɻ͜͜Ͱɺβ ͸ಋ೾࿏͕ಠཱͯ͋͠Δ࣌ͷϞʔυͷ఻ൖఆ਺Ͱ͋Δɻ ඇઁಈܥͷ݁߹ํఔࣜͷղ͸ɺ a(z) = ˆF a(0) (8) Ͱ༩͑ΒΕΔɻ͜͜Ͱɺ ˆ_{F ͸Ϟʔυৼ෯ͷ఻ૹߦྻͱݺ͹ΕΔϞʔυৼ෯ͷ఻ൖΛද͢ߦ} ྻͰɺ ˆ F = ˆT ⎛ ⎜ ⎝ e −jqz ₀ 0 _ejqz ⎞ ⎟ ⎠Tˆ−1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ cos qz − j  Δ₀ q  sin qz −j  c q  sin qz −j  c q  sin qz cos qz + j  Δ₀ q  sin qz ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (9) ˆ T = ⎛ ⎜ ⎝ c c q − Δ0 − (q + Δ0) ⎞ ⎟ ⎠, q =  c2+ Δ₀2 (10) ͱද͞ΕΔɻ

4

ઁಈܥͷ݁߹ํఔࣜͷ

Green-ؔ਺ʹΑΔղ

4.1

݁߹ํఔࣜͷ

Green-

ؔ਺ʹΑΔදݱ

͜͜Ͱ͸ɺύϥϝʔλʹઁಈΛؚΉ݁߹ํఔࣜͷ Green-ؔ਺ʹΑΔղ๏͕ࣔ͞ΕΔɻਤ 3 ʹࣔ͢Α͏ͳɺҰ༷ͳҐ૬ෆ੔߹ྔ±Δ_{0ʹՃ͑ͯɺz ͷؔ਺ͱͯ͠༩͑ΒΕΔҐ૬ෆ੔} ߹ྔ δi(z), (i = 1, 2) Λ࣋ͬͨ݁߹ثΛߟ͑Δɻ͜ͷ݁߹ثʹର͢Δ݁߹ํఔࣜ͸ɺ ∂a(z) ∂z + j ⎛ ⎝ δ1(z) + Δ0 c c δ2(z) − Δ0 ⎞ ⎠_{a(z) = 0} (11) Ͱ༩͑ΒΕΔɻӈล͕ઁಈ߲ʹͳΔΑ͏ʹมܗΛ͢Δͱɺ ∂a(z) ∂z + j ⎛ ⎝ Δ0 c c −Δ0 ⎞ ⎠_{a(z) = −j} ⎛ ⎝ δ1(z) 0 0 _δ2(z) ⎞ ⎠_a(z) (12)

ͱͳΔɻ͜ͷํఔࣜͷӈลΛ೾ݯ (ڧ੍߲) ͱݟͳ͠ɺͦΕΛ δ-ؔ਺ʹஔ͖׵͑ͨํఔࣜ ∂ ˆG0 ∂z + j ⎛ ⎝ Δ0 c c −Δ0 ⎞ ⎠_Gˆ 0 =−δ (z − ξ)  1 0 0 1  (13) Λߟ͑Δɻ͜ͷํఔࣜΛຬͨ͢ Green-ؔ਺ʢDiadic Green-ؔ਺ʣͱݺ͹ΕΔղΛ༻͍Δ ͱɺઁಈ߲ΛؚΉ݁߹ํఔࣜͷղΛٻΊΔ͜ͱ͕Ͱ͖Δɻ ͦͷղ͸ Green-ؔ਺ ˆ_G0Λ༻͍Δͱɺ a(z) = ¯a(z) + j z 0 ˆ G0(z; ξ) ⎛ ⎝ δ1(ξ) 0 0 _δ2(ξ) ⎞ ⎠_a(ξ)dξ (14) ͱද͢͜ͱ͕Ͱ͖Δɻ͜͜Ͱɺ¯_{a(z) ͸ઁಈ͕ແ͍݁߹ثͷϞʔυৼ෯Λද͠ɺࣜ (9) ͷ఻} ૹߦྻ ˆ_{F ʹΑͬͯ} ¯ a(z) = ˆF a(0) (15) ͱ༩͑ΒΕΔɻࣜ (14) ͕ઁಈ߲ΛؚΉ݁߹ํఔࣜͷղͰ͋Δ͜ͱ͸ɺࣜ (11) ຢ͸ (12) ʹ ௚઀୅ೖͯ͠ɺ༰қʹ͔֬ΊΔ͜ͱ͕Ͱ͖Δɻ

4.2

Diadic-Green

ؔ਺ͷಋग़

ࣜ (13) ͷ྆ลΛϑʔϦΤม׵ͯ͠੔ཧ͢Δͱɺ ˆ ˜ G0 = j ⎡ ⎣ Δ0+ α c c α − Δ0 ⎤ ⎦ −1 e−jαξ = j α2− q2 ⎡ ⎣ α − Δ0 −c −c Δ_{0+ α} ⎤ ⎦_e−jαξ (16) ͱͳΔɻ͜͜Ͱɺ ˆ_G˜ 0ͷΑ͏ʹ G ͷϑʔϦΤม׵Λ ˜G Ͱදͨ͠ɻ × ×

+q

-q

+∞

-∞

α- 面

ε

ਤ 8ɿෳૉੵ෼࿏

ˆ G0͸ɺࣜ (16) Λ֤੒෼͝ͱʹٯϑʔϦΤม׵͢Δ͜ͱʹΑͬͯٻΊΒΕΔɻٯϑʔϦ Τม׵ʹඞཁͳෳૉੵ෼ͷੵ෼࿏Λਤ 8 ʹࣔ͢ɻ্࣮࣠ʹ_{±q ͷۃ͕ଘࡏ͍ͯ͠ΔͨΊɺ} ੵ෼ྖҬ಺ʹۃؚ͕·ΕΔΑ͏ʹੵ෼࿏͕ઃఆ͞Ε͍ͯΔɻ·ͨɺੵ෼ͷऩଋΛߟྀ͠ ͯɺIm(α) > 0 ͷ৚͕݅ຬͨ͞ΕΔΑ͏ੵ෼࿏͕બ͹Εɺෳૉੵ෼͸ 2 ͭͷۃͷཹ਺ͷ࿨ ͱͯ͠ಘΔ͜ͱ͕Ͱ͖Δɻ Ҏ্ͷܭࢉ݁ՌΛవΊΔͱ ˆ_G0͸ɺ ˆ G0(z; ξ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ − cos q(z − ξ) + j  Δ₀ q  sin q(z − ξ) j  c q  sin q(z − ξ) j  c q  sin q(z − ξ) − cos q(z − ξ) − j  Δ₀ q  sin q(z − ξ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (17) ͱٻΊΒΕΔɻ

5

ಛੑͷղੳྫ

͜͜Ͱ͸ɺલઅ·ͰʹಘΒΕͨઁಈ߲ΛؚΉ݁߹ํఔࣜͷղΛ༻͍ͨ਺஋ܭࢉ݁ՌΛࣔ ͠ɺGreen-ؔ਺Λ༻͍ͨղͷදݱࣜͷ༗༻ੑ͕ࣔ͞ΕΔɻ ࣜ (14) ΛݟΔͱɺઁಈ߲Λ༩͑Δ Green-ؔ਺Λੵ෼߲ʹٻΊΔ΂͖Ϟʔυৼ෯_{a ؚ͕} ·Ε͍ͯΔɻ͢ͳΘͪɺGreen-ؔ਺Λ༻͍ͨղͷදݱࣜ͸ɺੵ෼ํఔࣜʹͳ͍ͬͯΔɻੵ ෼ํఔࣜͷղ๏ͷͨΊʹඞཁͳੵ෼߲ͷۙࣅܭࢉͱͯ͠ʀ (i) ੵ෼ʹؚ·ΕΔa Λඇઁಈղ ¯a Ͱۙࣅ͢Δํ๏ (ii) ੵ෼ʹؚ·Ε͍ͯΔa ʹ (i) ʹΑͬͯಘΒΕͨۙࣅղΛ࢖͍ɺੵ෼Λ܁Γฦ͢൓෮๏ (iii) ੵ෼ൣғΛখ۠ؒʹ෼͚ɺٻੵ๏Λ༻͍Δஞ࣍ۙࣅ๏ ౳͕ߟ͑ΒΕΔɻҎԼͷܭࢉͰ͸ɺઢܗ݁߹ثʹԙ͍ͯ͸ (ii) ͕ɺΑΓਫ਼౓͕ٻΊΒΕΔ ඇઢܗ݁߹ثʹ͸ (iii) ͕༻͍ΒΕΔɻ

5.1

ઢܗ݁߹ث

ਤ 9 ʹ͸ɺઢܗࡐྉͰߏ੒͞Εͨ݁߹ث (ઢܗ݁߹ث) ʹ൓෮๏Λద༻ͯ͠ಘΒΕͨύ ϫʔҠߦಛੑͷܭࢉྫ͕ࣔ͞ΕΔɻਤ (a) ʹ͸ɺෆ੔߹ྔͷઁಈ߲ δ1ͱ δ2ͷ z ʹԊͬͨ Ґஔ͕ඳ͔Ε͍ͯΔɻ·ͨɺਤ (b) ʹ͸ɺδ1ͱ δ2ͷ z ʹԊͬͨ step-ܗঢ়ͷ෼෍͕ࣔ͞Ε

͍ͯΔɻstep ͷ௕͞ ls ͸ɺlsi = l/(2i) ʹैܾͬͯΊΒΕ͍ͯΔɻ͜͜Ͱɺi ͸ z ʹԊͬͨ

۠ؒ൪߸Λද͍ͯ͠Δɻ ਤ (c) ͱ (d) ʹ͸ɺδ ͷৼ෯͕ҟͳΔܭࢉྫ͕ࣔ͞Ε͍ͯΔɻ(c) ʹ͸ৼ෯͕ 0.125 × c ͷ ݁Ռ͕ɺ·ͨɺ(d) ʹ͸ 0.5 × c ͷ৔߹͕ඳ͔Ε͍ͯΔɻԿΕͷ৔߹΋ɺ൓෮ճ਺Λม͑ͯ ܭࢉ͞Εͨ݁Ռ͕ಉ࣌ʹඳ͔Ε͓ͯΓɺࠇઢ͕൓෮ແ͠ͷ݁Ռɺ੺ઢ͕ 1 ճ͚ͩͷ൓෮Λ ߦͬͨ݁ՌͰ͋Δɻ൓෮Λ܁Γฦͯ͠΋ɺ݁Ռʹେ͖ͳࠩҟ͸ݱΕͳ͔ͬͨɻ൓෮ʹΑΔ ܭࢉਫ਼౓ͷվળΛࣔͨ͢Ίʹɺ྆ಋ೾࿏ͷ఻ૹిྗͷ࿨ Ptotal =|a1|2+|a2|2ͷ஋΋ϓϩο τ͞Ε͍ͯΔɻܭࢉ݁ՌΛݟΔͱɺઁಈྔͷৼ෯͕খ͍͞ߏ଄ (ਤ (c)) ʹԙ͍ͯ͸ɺ൓෮

ʹΑͬͯܭࢉਫ਼౓͕֬อ͞ΕΔ͜ͱ͕൑Δɻ͜Εʹରͯ͠ɺઁಈྔͷৼ෯͕େ͖͘ͳΔͱ (ਤ (d))ɺ൓෮ճ਺Λ૿΍ͯ͠΋ɺे෼ͳਫ਼౓͕ಘΒΕͳ͍͜ͱ͕൑Δɻ ͳ͓ɺ͜ͷܭࢉྫͰ͸ɺz ํ޲ʹରͯ͠Ұ༷ͳҐ૬ෆ੔߹ Δ0ͷ஋͸ɺͦΕͧΕɺਤ (c) Ͱ͸ Δ_{0 = 0.125 × cɺਤ (d) Ͱ͸ Δ0 = 0.5 × c ʹઃఆ͞Ε͍ͯΔɻ}

z

a

₁

a

₂ ・・・ ・・・ 2 4 8 ・・・ lc δ₁ δ₂ lc lc (a) δ ͷ z ʹԊͬͨҐஔ δ１ δ2 z z ・・・ ・・・ lsi (b) δ ͷ෼෍ Total Power 0 10 20

z

[mm] 0 1

|a

1

|

2

|a

2

|

2

|a

i

|

2

/

|a

1

|

2 (0) Δ0=0.125×c

_|a

1

|

2

|a

2

|

2 Δ0=0.5×c Total Power 0 1 0 10 20

z

[mm]

|a

i

|

2

/

|a

1

|

2 (0) (̲) δ ͕খ͍͞৔߹ (d) δ ͕େ͖͍৔߹ ਤ 9ɿઢܗ݁߹ثͷύϫʔҠߦಛੑͷܭࢉྫ

5.2

ඇઢܗ݁߹ث

ਤ 10 ʹ͸ɺඇઢܗ݁߹ثͷղੳͷࠔ೉͕֓͞೦తʹඳ͔Ε͍ͯΔɻඇઢܗ݁߹ث͸۶ ં཰͕ޫڧ౓ʹґଘͯ͠มԽ͢ΔͨΊɺಋ೾࿏ͷஅ໘಺ͷޫڧ౓͕൑Βͳ͚Ε͹ɺ఻ൖఆ ਺ β ΛܾΊΔ͜ͱ͕Ͱ͖ͳ͍ɻ൓ରʹಋ೾࿏அ໘ͷޫڧ౓͸ɺҐ૬ෆ੔߹ྔʹࢧ഑͞ΕΔ ύϫʔҠߦྔʹґଘ͢Δɻ͜ͷΑ͏ʹɺ۶ં཰ͷมԽΛ௨ͯ͠ɺҐ૬ෆ੔߹ྔͱύϫʔҠ ߦྔ͸͓ޓ͍ʹؔ܎͠߹͍ͬͯΔͷͰɺઢܗ݁߹ͷΑ͏ʹɺ֤ύϥϝʔλΛಠཱʹܾΊΔ ͜ͱ͕Ͱ͖ͳ͍ɻ

ͭ·ΓɺಛੑΛܾΊΔύϥϝʔλ͕ޓ͍ʹབྷΈ߹͏࠶ؼతͳؔ܎ʹ͋Δ͜ͱ͕ɺඇઢܗ ݁߹ثͷղੳΛࠔ೉ͳ΋ͷʹ͍ͯ͠Δ͕ɺGreen-ؔ਺Λ༻͍Δͱൺֱత؆୯ʹಛੑΛಘΔ ͜ͱ͕Ͱ͖Δɻ 結合係数 c 屈折率 n 伝搬定数ᶾ 3RZHU ਤ 10ɿඇઢܗ݁߹ثͷύϥϝʔλͷ࠶ؼతͳؔ܎ ਤ 11 ʹɺ྆ಋ೾࿏ͱ΋ඇઢܗੑΛ࣋ͭ݁߹ثͷύϫʔҠߦಛੑͷܭࢉྫ͕ࣔ͞Ε͍ͯ Δɻೖࣹύϫʔ PinΛ 0.1ʙ0.5[W] ͷൣғͰม͑ͯܭࢉ͞Ε͍ͯΔɻ݁߹ثͷ௕͞ lc ͸ɺ 5[mm] ʹઃఆ͞Ε͍ͯΔɻ͜ͷ௕͞͸ɺ݁߹ث͕ઢܗಈ࡞͢Δ৔߹ɺଈͪɺPin͕े෼খ ͘͞ɺಋ೾࿏ͷඇઢܗੑΛແࢹͰ͖Δঢ়ଶͰͷ݁߹௕ʹ౳͍͠ɻԿΕͷ৔߹΋ɺޫ͸ಋ೾ ࿏ 1 ͔Βೖྗ͞Ε͍ͯΔͱ૝ఆ͞ΕɺΔ₀ = 0 ʹઃఆ͞Ε͍ͯΔɻ ·ͨɺGreen-ؔ਺ʹΑΔղͷදࣔʹݱΕΔੵ෼ํఔࣜͷղ๏ʹ͸ɺஞ࣍ۙࣅ๏͕༻͍Β Ε͍ͯΔɻ

z

[mm] 0 2 4 0 1

|a

1

|

2

/

Pin ύ

|a

2

|

2

/

Pin

|a

1

|

2

|a

2

|

2 =0.1 [W] 0.3[W] 0.5[W] Total Power Pin 規格化 ਤ 11ɿஞ࣍ۙࣅ๏Λ༻͍ͨղੳ݁Ռ ೖࣹύϫʔ͕খ͍͞ྖҬͰ͸ɺ྆ಋ೾࿏ͷ PinͰن֨Խ͞Εͨύϫʔͷ߹ܭ஋ (Total-Power) ͷ஋͕ɺz ͷશܭࢉྖҬͰ 1 ʹอͨΕ͓ͯΓɺਫ਼౓ͷྑ͍݁Ռ͕ಘΒΕΔ͜ͱ͕ ൑Δɻ

Pinͷ૿Ճʹ൐ͬͯҐ૬ෆ੔߹ྔ δ(z) ͕૿Ճ͢ΔͨΊɺग़ྗ୺Ͱ͸׬શͳύϫʔҠߦ͕ ى͖ͳ͍͜ͱ͕൑Δɻ͞Βʹɺੵ෼ํఔࣜͷղ๏ͷۙࣅʹ൐͏ޡࠩͷͨΊʹɺن֨Խ͞Ε ͨύϫʔͷ߹ܭ஋͕ 1 ʹอͨΕͣɺਫ਼౓͕ѱԽ͍ͯ͠Δ͜ͱ΋൑Δɻ

6

ඇઢܗ݁߹ثͷύϥϝʔλઃܭ

͜͜Ͱ͸ɺ݁߹ثΛ఻ൖ࣠ʹԊͬͯ 10 ݸͷখ۠ؒʹ෼͚ɺॴ๬ͷύϫʔ෼ذൺΛಘΔ ͨΊʹඞཁͳ֤খ۠ؒͷҐ૬ෆ੔߹ྔ δi1ɺδi2, (i = 1, 10) ͱɺҰ༷ͳෆ੔߹ྔ Δ0ͷ߹ܭ 21 ݸͷ஋ͷઃܭΛߦ͍ͬͯΔɻ ඇઢܗ݁߹ثΛߏ੒͢Δඇઢܗಋ೾࿏Ͱ͸ɺ֤఺Ͱͷ఻ૹύϫʔ_|ai(z)|2_{ʹൺྫ࣮ͯ͠} ޮ۶ં཰͕มԽ͠ɺͦΕʹΑͬͯɺҐ૬ෆ੔߹ྔ΋มԽ͢Δ͜ͱʹͳΔɻैͬͯɺ֤ z ͷ ҐஔͰͷϞʔυৼ෯Λ༩͑Δࣜ (14) ʹؚ·ΕΔ δ1(z) ͱ δ2(z) ͸ɺ఻ૹύϫʔ͕े෼ʹখ ͍͞ઢܗྖҬͰͷ஋ δiν, (i = 1, 10; ν = 1, 2) ͱɺඇઢܗੑʹىҼ͢Δ஋ N2|a(z)|2ͷ࿨ͱ ͯ͠, ಋ೾࿏ 1 ٴͼ 2 ʹԙ͍ͯɺ δ1(z) = δi1+ N2|a1(z)|2, δ2(z) = δi2+ N2|a2(z)|2 (i = 1, 10) ͱද͞ΕΔɻ͜͜Ͱɺઃܭ͞ΕΔͷ͸ɺઢܗྖҬͷҐ૬ෆ੔߹ྔͰ͋Δ δi1ͱ δi2ͱ Δ0ͷ ஋Ͱ͋Δɻ ࠷దͳҐ૬ෆ੔߹ྔͷ஋͸ɺNewton ๏ [11] Λ༻͍ͯɺॴ๬ͷ෼ذൺͱܭࢉͰಘΒΕΔ ஋ͱͷࠩͷ 2 ৐͕࠷খ (ۃখ) ͱͳΔΑ͏ʹܾఆ͞ΕΔɻ ҎԼʹࣔ͞ΕΔઃܭͰ͸ɺઢܗྖҬͰͷ݁߹௕ l Λ༩͑ɺͦΕΑΓ݁߹܎਺Λ c = π/2l ʹΑͬͯ༩͍͑ͯΔɻ·ͨɺ݁߹ثͷ௕͞ lc΋֤ྫͰͦΕͧΕ༧Ί༩͍͑ͯΔɻ ઃܭྫ͸ʀ(1)3dB-ύϫʔσόΠμɺ(2) ޫεΠονɺ(3) ύϫʔϑΟϧλɺٴͼ (4) ύϫʔ Ϧϛολͷ 4 ͭͰ͋Δɻશͯͷઃܭྫʹԙ͍ͯɺޫ͸ಋ೾࿏ 1 ʹͷΈʹೖࣹ͞ΕΔͱ૝ఆ ͞Ε͍ͯΔɻ·ͨɺಋ೾࿏ͷඇઢܗੑʹ͍ͭͯ͸ɺઃܭྫ (1) Ͱ͸ɺยํͷΈ͕ඇઢܗͷ ৔߹ͱɺ྆ํ͕ඇઢܗͷ৔߹͕ࣔ͞ΕΔɻҰํɺઃܭྫ (2)ʙ(4) Ͱ͸ɺยํͷΈ͕ඇઢܗ ͱ૝ఆ͞Ε͍ͯΔɻ

6.1

3dB-

ύϫʔσόΠμ

ਤ 12ɺ13 ʹ 3dB-ύϫʔσόΠμͷઃܭྫ͕ࣔ͞Ε͍ͯΔɻਤ 12 ͸ɺಋ೾࿏ 1(ೖྗଆ) ͷΈ͕ඇઢܗಋ೾࿏ͷ৔߹Ͱ͋Δɻਤ 13 ͸ɺ྆ํ͕ඇઢܗͷ৔߹Ͱ͋ΔɻԿΕͷྫʹԙ ͍ͯ΋ɺ(a) ʹ͸ɺઃܭ͞ΕͨҐ૬ෆ੔߹ྔ δi1,i2ͷ఻ൖ࣠ʹԊͬͨ෼෍͕ࣔ͞Εɺ(b) ʹ ͸෼ذൺͷೖࣹޫύϫʔґଘੑ͕ࣔ͞Ε͍ͯΔɻਤ 12ɺ13 ͷ݁Ռͱ΋ɺೖࣹޫύϫʔ Pin ͷ޿ൣғͳมԽʹରͯ͠ɺग़ྗύϫʔͷ౳෼഑͕࣮ݱ͞Ε͍ͯΔɻ͜ΕΒͷઃܭྫͰ͸ɺ l = 4.0[mm]ɺlc = 5.0[mm] ʹઃఆ͞Ε͍ͯΔɻҐ૬ෆ੔߹ྔ δi1ɺδi2ɺٴͼ Δ0ͷ஋͸ɺ݁ ߹܎਺ͷ஋Ͱن֨Խ͞Εͯࣔ͞Ε͍ͯΔɻ

0 1 -1

ț

i1

ț

i2

,

/

c

/

c 4 2 0

z

[mm] Δ0

/

c= 0.238

ț

i1

ț

i2 0.5 1 Pout /P in 0 0.2 0.4 l=4.0[mm] lc=5.0[mm]

|

_{a |}

2 WG1: 非線形 1 WG2: 線形

|a

2

|

2

P

in[W] (a)δ ͷઃܭ஋ (b) ෼ذൺͷೖࣹޫڧ౓ґଘੑ ਤ 12ɿ3dB-ύϫʔσόΠμͷઃܭྫʢಋ೾࿏ 1 ͕ඇઢܗͷ৔߹ʣ 0 0.5

ț

i1

ț

i2

,

/

c

/

c 4 2 0

z

[mm] Δ0

/

c= - 0.925

ț

i1

ț

i2 -0.5 0 0.5 1 Pout /P in 0 0.2 0.4 l=4.0[mm] lc=5.0[mm]

|

_{a |}

2 WG1: 非線形 1 WG2: 非線形

|a

2

|

2

P

in[W] (a)δ ͷઃܭ஋ (b) ෼ذൺͷೖࣹޫڧ౓ґଘੑ ਤ 13ɿ3dB-ύϫʔσόΠμͷઃܭྫʢ྆ಋ೾࿏͕ඇઢܗͷ৔߹ʣ

6.2

ޫεΠον

0 0.2 20 10 0

z

[mm]

ț

i1

ț

i2

,

/

c

/

c Δ0

/

c =0.442 -0.2

ț

i1

ț

i2

|a

2

|

2 WG2: 線形 Δ0

/

c= 0.44209 WG1: 非線形 =0.31416[1/mm]

|a

1

|

2 0 0.2 0.4 0 1

P

out

/

P

in Cross-state Bar-state switching

c

P

in[W] (a)δ ͷઃܭ஋ (b) εΠονϯάಛੑ ਤ 14ɿޫεΠονͷઃܭྫ ݁߹ث௕ lc = 18.1[mm]ɺ݁߹௕ l = 5.0[mm] ਤ 14 ʹޫεΠονͷઃܭྫ͕ࣔ͞Ε͍ͯΔɻ෼ذൺͷઃఆ஋͸ʀCross-state @Pin=0.2[W]ɺ Bar-state @Pin=0.5[W] Ͱ͋Δɻl = 5.0[mm]ɺlc = 18.1[mm] ʹઃఆ͞Ε͍ͯΔɻઃܭ͞ Εͨ݁߹ثͷಛੑ͔Β൑ΔΑ͏ʹɺύϥϝʔλΛద੾ʹઃఆ͢Δ͜ͱͰɺ”ޫʹΑΔޫͷ εΠον”͕ՄೳͰ͋Δ͜ͱ͕ظ଴͞ΕΔɻ

6.3

ύϫʔϑΟϧλ

͜͜Ͱݴ͏ύϫʔϑΟϧλͱ͸ɺೖࣹύϫʔͷ஋ʹΑͬͯग़ྗޫڧ౓ʹॴ๬ͷมԽΛ༩ ͑Δಈ࡞Λݴ͏ɻ͜͜Ͱࣔ͢ྫͰ͸ɺCross-port ͷن֨Խग़ྗ Pout/Pin͕ೖྗύϫʔ Pin ʹରͯ͠ɺ༩͑ΒΕͨ Gauss-෼෍ʹैͬͯมԽ͢Δ༷ʹઃܭ͞ΕΔɻ ྫͰ͸ɺCross-port ͷن֨Խग़ྗ Pout/Pin͕

Pout/Pin= Pmax exp  −Pin− P0 w 2 Ͱ༩͑ΒΕΔ Pinʹର͢ΔґଘੑΛ࣋ͭΑ͏ʹҐ૬ෆ੔߹ྔ͕ઃܭ͞ΕΔɻ͜͜ͰɺP0͸ Gauss-෼෍͕ۃେͱͳΔ Pinͷ஋Ͱ͋Δɻ2w ͸෼෍ͷ൒஋෯Λ༩͑ΔύϥϝʔλͰ͋Δɻ ਤ 15 ʹࣔ͞ΕͨྫͰ͸ɺl = 4.5[mm]ɺlc = 15[mm]ɺP0 = 0.25[W]ɺw = 0.12[W] ʹ ͦΕͧΕઃఆ͞Ε͍ͯΔɻઃఆ஋ͱͯ͠ɺGauss-෼෍্ʹ͋Δ̓ͭͷ఺ (ؙҹ) Λ༩͑ɺ Newton ๏ʹΑͬͯ࠷దύϥϝʔλͷ஋͕୳ࡧ͞Εͨɻઃܭ͞Εͨ δi1ͱ δi2ͷ෼෍͕ਤ (a) ʹɺͦΕΛ༻͍ͯܭࢉ͞ΕͨϑΟϧλಛੑ͕ਤ (b) ʹͦΕͧΕඳ͔Ε͍ͯΔɻ

0 1 0 10

z

[mm]

ț

i1

ț

i2

,

/

c

/

c Δ0

/

c= 0.487

ț

i1

ț

i2 0 1 Pout

/

Pin 0.5 ●：設定値 WG1: 非線形 WG2: 線形

|a

1

|

2

|a

2

|

2 ① ② ③ ④ ⑤ ⑥ _⑦ ● ● ● ● ● ● ● 0 0.1 0.2 0.3 0.4 0.5

P

in[W] (a)δ ͷઃܭ஋ (b) ϑΟϧλϦϯάಛੑ ਤ 15ɿύϫʔϑΟϧλͷઃܭྫ (ಋ೾࿏ 1 ͕ඇઢܗͷ৔߹)

6.4

ύϫʔϦϛολ

͜͜Ͱ͸ɺύϫʔϦϛολͷઃܭྫ͕ࣔ͞ΕΔɻCross-port ͷن֨Խग़ྗ Pout/Pin͕

Pout/Pin = Pmax[1− exp(−αPin)] , α = −ln 0.1

P0

ʹैͬͯมԽ͢Δ݁߹ث͕ઃܭ͞ΕΔɻ͜͜ͰɺPmax͸ Pout/Pinͷ্ݶ஋

(Limitted-Value)ɺ_P0͸ P_{out/Pin = 0.9 × Pmax}ͱͳΔ P_inͷ஋Ͱ͋Δɻ

ਤ 16 ʹ͸ɺPmax = 0.25ɺP0 = 0.2[W] ͱ༩͑ͯɺ໨ඪؔ਺ʹΑͬͯ 5 ͭͷ໨ඪ஋Λ༩

͑ͯઃܭ͞Εͨಛੑ͕ࣔ͞Ε͍ͯΔɻਤ (a) ʹ͸ δi1ͱ δi2ͷ෼෍͕ɺਤ (b) ʹ͸෼ذൺͷ

Pinґଘੑ͕ࣔ͞Ε͍ͯΔɻਤதͷؙҹ͸໨ඪ஋Ͱ͋Δɻઃܭ͞ΕͨύϥϝʔλʹΑͬͯɺ

݁߹ث͸Ϧϛολͱͯ͠ͷಈ࡞Λ͍ͯ͠Δ͜ͱ͕൑Δɻl = 4.5[mm]ɺlc = 5.0[mm] ʹઃ

0 1 Δ0

/

c= 1.135

ț

i1

ț

i2

ț

i1

ț

i2

,

/

c

/

c 4 2 0

z

[mm] -1 0 0.5 1 Pout /P in 0 0.2 0.4 Pin[W] lc=5.0[mm] l=4.5[mm]

|a

1

|

2 WG1: 非線形 WG2: 非線形

|a

2

|

2 ● ● ● ● ● ●_{： 設定値} 0.25 (a)δ ͷઃܭ஋ (b) ύϫʔϦϛολͷಛੑ ਤ 16ɿύϫʔϦϛολͷઃܭྫ (྆ಋ೾࿏͕ඇઢܗͷ৔߹)

7

·ͱΊ

ຊใࠂͰ͸ɺ෺ཧతҙຯͷ೺ѲΛ໨తʹɺํ޲ੑ݁߹ثͷ݁߹ํఔࣜͷղͷ Green-ؔ ਺ʹΑΔఆࣜԽΛߦͬͨɻಋग़͞Εͨղͷදࣔࣜʹ͸ɺߏ଄ύϥϝʔλ͕෺ཧతͳҙຯ͕ ཧղ͠қ͍ܗͰؚ·Ε͓ͯΓɺಛੑͷཧղʹ༗༻Ͱ͋Δͱಉ࣌ʹɺύϥϝʔλͷ࠷దઃܭ ʹ΋༗ޮͰ͋Δ͜ͱΛࣔͨ͠ɻ

ࢀߟจݙ

[1] ࢁޱɺ؛Ԭʀʠ Ϧοξಋ೾࿏ඇઢܗํ޲ੑ݁߹ث ʡ, ฏ੒ 25 ೥౓ిؾؔ܎ֶձؔ੢࿈ ߹େձɺG8-2(2013). [2] େ໺ɺࢁޱɺ؛Ԭʀʠ ඇઢܗϦοξಋ೾࿏ํ޲ੑ݁߹ثͷύϫʔҠߦಛੑ ʡɺฏ੒ 25 ೥౓ిؾؔ܎ֶձؔ੢࿈߹େձɺG8-11(2013). [3] ৿ࢁɺ؛Ԭʀʡඇઢܗํ޲ੑ݁߹ثͷ൓෮๏ʹΑΔղੳͱͦͷಛੑ ʡɺ େࡕిؾ௨৴ େֶلཁ (ࣗવՊֶฤ)ɺୈ 43 ߸ɺpp.13-26 (2008).

[4] K.Yasumoto, H.Maeda and N.Maekawa; ”Couple-Mode analysis of an Asymmet-ric Nonlinear Coupling Medium”, IEEE,J.Quantum Electron.,Vol.27,No.3,pp.708-795 (1991).

[5] ࣰ࡚ɺएྛɺ໦ଜʀʡݱ୅޻ֶͷͨΊͷภඍ෼ํఔࣜͱάϦʔϯؔ਺ ʡɺݱ୅޻ֶࣾ (1987).

[6] ୔ɺ؛ԬɺཬଜɺԼ୅ʀʡޫ޻ֶ֓࿦ ʡɺே૔ॻళɺpp.142-144 (1995).

[7] G.B.Hocker and W.K.Burns; ”Mode dispersion in diffused channel waveguides by the effective index method”, Appl. Opt., Vol.16, pp.113-118 (1977).

[8] ੢࿬ɺ؛Ԭʀʡ൒ಋମυʔϓΨϥεඇઢܗޫಋ೾࿏ʹؔ͢Δݚڀ ʡɺେࡕిؾ௨৴େ ֶमֶ࢜Ґ࿦จ (1998).

[9] K.Kishioka; ”Determination of the diffusion-parameter values in K-ion exchange waveguides made by diluted KNO₃ in soda-lime glass”, IEICE Japan, Trans. Elec-tron. Vol.E78-C, No.10, pp.1409-1418 (1995).

[10] ཱ઒ɺ؛Ԭʀʠ رऍ AgNO3 Ͱ࡞ΒΕΔ֦ࢄಋ೾࿏ͷύϥϝʔλਪఆͱ۶ં཰෼෍ͷ

ܭࢉ ʡɺେࡕిؾ௨৴େֶݚڀ࿦ूʢࣗવՊֶฤʣɺୈ 37 ߸ɺpp.31-44ʢ2002,Mayʣ. [11] ෋࢜௨αϒϧʔνϯύοΫ SSLII(Scientific Subroutine Library II) ϚχϡΞϧɺχϡʔ