JQE_ek_saby Mekg a ue saby ase aavy euey ua Y. P. R. Hee ew IEEE a. M. Nau ew OSA a APS Absa Usg he Maxwe-Bh euas we have sue yas assae wh a ee-ay uabe ase uae by a susa sga wh a euey ea he avy ee sea age. We ha FM ekg ay u whe a hghe e FM e aues he wes hesh. F sh-avy ases he use wave a ekg s ae a uke he se Gaussa shae suggese by evus sues. I e exa saby bseve ase FM ea we eve a ea saby aayss he asg FM e. Ou aayss shws ha hs ye saby s abue e e a ase eaxa yas. We he hese aay esus by uea suas. Iex Tes Maxwe Bh ua FM ekg FM sa saby h ase T I. INTRODUCTION he FM eke h ase [] has bee suggese as a vee eve geeag w se wave subae sgas a eag hgh see k sgas a aag--ga vees. Beause suh aas eue a sabe a a uses wh a hgh ee ae a a aw use wh s esabe have a heea e aabe eeg he haaess suh as he e-a wave a saby. Whe he ase s uae by susa sgas wh a euey se he avy ee sea age FSR [3-5] se FM sas saby he a sw ae wh he FM ua euey usa a as FM ekg ay u. The ase aavy FM yas was exesvey sue bh he e [3-5] a euey-as [6-8]. The s eys sesse euas aaye he seay sae wave he eke sae. Hweve aks he aby aaye he ase esse. O he he ha he exsg euey-a eh uses ue-e euas ha ae u sve a seuey es a ess eeve aaya as he ua euey aahes he Maus eeve O 3 5. Ths wk was sue a by he U.S. Deae Deese ue ga N4---78. Y. a P. Hee ae wh he Cee Mwave / ghwave Egeeg Dexe Uvesy Phaeha PA 94. Phe: 5-895- 94; ax: 5-895-4968; e-a: ye@.ee.exe.eu.. M. Nau s wh he Deae Physs Dexe Uvesy Phaeha PA 94. avy FSR. Nehe eh es a g es he as FM sa FM eke ea. Reey gh e a exae hs as usg Fue aayss he -asg sae [9]. Hweve s a sasay es he saby FM ea sees be avaabe ye. Pevusy we have ue a ew heea e [6] base Maxwe-Bh euas a ee- uabe h ase [7] wh he e aaye a euey swhg [8] a FM bawh. Cae wh he FM ase sues [3-5] hs e es a auae eae he aavy ee- ua se he FM ase. I hs ae we eve a e shsae ua suabe aayg he seay sae a FM ase uae by susa sgas. Usg hs ua we w suy he as FM ekg. Me a we w exa he uee saby wh he he a saby aayss. The vay u aah s sae a se avy egh s ha u es s aabe aaye h ases as we as he yes ases wh ge aves. Ths ae s gae as wg. I se II we ue he heea e. I se III we eve a se se-sse euas esbe he seay sae behav a sugges a uea ehue he su whh he ah FM sa FM ekg s ae. I se IV we aess he ssues saby a suss se V he uea suas agas whh we a es he heea esus bae he evus ses. Fay se VI we suae he a esus. II. THEORETICA MODE As se he heeay e ae Faby- Pe F-P avy u e s base a uea g avy as shw sheaay Fg.. The use a g avy ses he heea aayss by awg us ege he saa-he-bug ee whh yay eeges F-P sag wave esas. Ths sa s auay va a F-P ase whe he u s absbe he eg se e ase whee he ee he saa-he bug s e. T ve he suas whee avy es ay have ee eave es he saa ae hs suss easues he eeve a ah. The g he saa ae
JQE_ek_saby aks he begg he ga eu. The ga a ee- ug egs ae seaae by w eay seges. The exy hs avy e es he ee- ua. Pevus euey-a wks [6-8] a he Fue hey suy FM ases [9] ee he ee- ua ehe ug a aae e. We ee av hs beause he ee- ua a he ga eu ae hysay seaae. I e esbe hs be a e eas ash we se e he ee- ua as a e-vayg bu he buay he ave ga eu [6]. Pu Deay: τ 4 Ga Se E/O Mua 3 V s Paa ee: Deay: τ Ouu Fg.. A uea g avy wh a a-avy ee- ua We ege he ak asvese uy he avy e e a esbe he a e es a ae wave ae a a swy vayg e evee k E [ ξ e..] whee ξ s he evee he e π / Λ s he avy esae euey ha s ses he a esae s a ege Λ s he u-uae avy egh s he see gh vauu a k s he wave ube k /. Ou aayss he ga eu s base he Maxwe- Bh euas [7] a ae-wave e a a hgeeusy baee w-eve ave eu. I he swy vayg evee axa SVEA he euas he e evee a he a vaabes he ga eu have he wg [7]: F F P a P [ F. P ] b [ F P C. C. ] whee F s he esess Rab euey / F μξ / h h s Pak s sa a ae esevey he eay aes he aa a ua ves he ave eu P s he esess evee he a aa s he ua eee e a bewee we a ue asg eves s he usauae ga e u egh a / s he sae eug bewee he a A a he seee avy esaes. Nex we eve he buay s he ga eu wh he he a se- su he e ha agaes hugh he ee- se. Usuay he ee- eu s ea wh as a ue-eee hase ua [3-5]. Bu se we ae eese suas whee he hage he uag wave a be egee ug he as e wh he ee- se we ey a wave aga e whh he evee he a e se he ua s esbe by he wg aga eua [9]: F F e Vug F 3 h e whee s he eave ex he ee- aea s he ee- ee h s he hkess he ee- eu a s he ug vage. Ths V ug s a s-e aa eea eua whh a se su a be u yeg a ea bewee he e evees a he w buaes he ee- se: F Φ 3 F τ 3 e 4 I E. 4 τ 3 s he as e hugh he ee- eu Φ s he hase eay ve a sge Φ β V u u τ 3 ug whee β e / h. F a susa ua Φ akes he : Φ s φ whee β V s τ 3 / / s he ua ex e u V s he aue he susa vage sga a φ s eae he a hase he ua sga. Beag ha he w ue eay ahs y ue a e ag he e evee a ae usg E. 4 we he buay he e evee he ave ga eu be: s τ 4 φ F F τ e 5 whee τ τ 3 τ τ s he a aga e use 4 he ga eu a s he a e aue
JQE_ek_saby 3 eevy he uu. Ne ha φ s a abay ube; we se be eua τ τ a heeby 4 we exess he buay he e vee : F s τ F τ e. 6 The euas Es. a he buay E. 6 uuey eee he ase syse. We w suss he uea sus se V. Hee e aae he heea aayss we eve a aeave. Beause he ase syse ueges a e euba a a ae se he avy u ae he sus he g-e ae exee be e eay e. Thus we exess he e evee a he a vaabes as suess Fue es wh swy vayg ex aues: F ex 7a P ex 7b ex 7 whee a ae swy-vayg he sae he ua euey. U subsug Es. 7 he Maxwe-Bh Euas Es. we ba a ew axae se euas : 8a [. ] 8b [ ]. 8 The esg ew buay s τ e ae: τ Γ 9 whee Γ a s he J J h e Besse u. I he ex w ses we aaye he ase seay sae a s saby. I us be ehase ha E. 8 we ege he hghe e Fue ees he ves esy > beause he ves eay e s uh ge ha he avy u e he ases ue sea. III. ONG-TIME SOUTIONS AND TRANSITION TO FM MODEOCKING Ae he ases have e u he Fue ees a a sabe ase ae exee aue he ex a ex b whee s he euey se he g-e su. U subsug Es. Es. 8 we ha a us sasy he euas / a b / /[ / ] whee /. F sy we useves he sea ases wh hgh-q aves whh he saa vaa s sa. Cseuey ag E. a be axae as beg eee sae. I hs sua E. a yes e sa bewee he es a he u a uu buaes he ave eu: / e. Seaaey he buay E.9 ves ahe sa: e τ Γ. 3 U subsug E. 3 E. we ba a se se-sse euas he Fue aues : g e τ e / Γ 4 whee τ a g s he hesh ga he su. Pease / τ s he avy u e gh e ha s egave. The sa sga avy ga us be age ha g e he g-e su exs. Eua 4 esebes a egevaue be. Hweve u ase sea a we-ee egevaue we us
JQE_ek_saby 4 auae he ga g a he euey se. P sussg he su ehue s useu u ha whe he ase avy s esae wh he a ga eak E. 4 has egeeae sus esg a sae ga. I s easy vey ha s a seay sae su ha sases E. 4 he ' ' ' s as a egae su wh ' ' '. 5 I he e-a hs es ha F s a su F T / s as a su. Thus uess F a F ' ae ea w egeeae sus exs he sae ga. As we sha see se IV he e bewee he w sus F a F ' ay eas sabes. The egeeae sus se he egeeay whe he avy s eue away he a esae.e. %. I he evus sues FM ekg hs egeeay was eee as he π hase ueay [3-8]. F a bee uesag he as he FM ekg ege we see a ehue ha a ey a he sus whu he vegee bes e euee evus ea shees [6-8]. Beause E. 4 s sa a egevaue be we seek a su wh he he saa egevaue ehues. We ea g as a kw xe uay a exess E. 4 he a egevaue be. g e e τ τ e Γ 6 τ whee e ays he e he egevaue a s eessay a ue agay uay. I evg E. 6 we ege he eeee se he exea e E. 4. Ths s juse beause eeses he gwh ae whh us be uh sae ha a as evusy ague. Ou aah svg E. 4 a be suae as ws. Fs we sve E. 6 a gve g usg he saa egevaue ehue a eae a ua a bewee a g. A hs svg E. 4 s euvae g he : Re[ g ]. Ths aah s aabe g a he ssbe sus a has e vegee bes beause he ea s y use a se -seah. I svg E. 4 we have awe ee vaues he avy egh a he ua aaee a gae ew sghs uesag he se FM sa a he as FM ekg. Cssey wh evus sues he sus E. 4 ae euey-uae a ae sgas whe he ua euey s a way he gua e sag. Cseuey e sy he egy we ee hese sus as he FM es w. We ha he FM ekg ea evves a hghe e FM e whe he ua euey s ue wa he gua e sag. The as he se FM sa FM ekg a be eee as a sh he h e a hghe e FM e. T usae he ess Fg. shws he hesh ga vaus FM es whe he ua eug he gua e sag s vae -..5. We shu e ha FM eke ea us y sa vaues beause s a awba ess. Ne ha [ FSR / k] / s sae he eay ae he aa a FSR / k s he gua e sag gve by he we kw e ug ua [7]. I hs aua we assue ha he ase avy FSR s / 8 he uu has a eevy 99 % a he sge ass ua ex s eua.. I a we se he avy a esae wh he a ga ee % hus w egeeae sus ay u he sae hesh ga. Fay ave a a uea su we uae E. 6 a. The hesh ga uves shw Fg. ae hghy asye au. I he ua euey s suey eve he gua e sag he eee e ea he e wh he wes hesh s he h e FM e he seu whh s eee a he ga ee whe %. Ths e ea was eee as he FM sa [3-8]. Whe we eue he euey ua eses exg age FM sebas. F he h e FM e he FM sebas ae ushe away he ga ee. Thus hs su w see ess ga a exeee a ease he asg hesh. Hweve he hghe e FM es whh ay ae eve he ga ee se he FM sebas ae ushe se he a ga ee whee he FM e a see e ga. As a esu he heshs ae wee. Ths e he asg hesh s shw eay Fg.. Whe. 78 he s FM sa e aues he wes hesh a bees he eee e ea. Whe. 43 he FM sa e has he wes hesh a s wh he 3 FM sa e. A a eug -.95 he 4 h FM sa eeges as he wes hesh e. F eug vaues agg -..5 we bseve a e ahe a a aae hesh uves. I hs eg he e-a wave shws FM eke ea. The wes hesh uve he ekg saes hs exae s assae wh he 4 h FM sa e. I he eug eg.. a ae eeavg hesh uves us.
JQE_ek_saby 5 Bey hs eg he ase gauay swhes bak a FM sa. Fg. : Thesh he FM es as a u he ae ua euey eug / FSR / k. The ase aaees ae:. FSR / 8 a.99. T uhe usae he e ha he FM ekg ea gaes he hghe e FM es he sea a waves he 4h FM e ae e a e eug vaues Fg. 3 whh we bseve a sh as ess. Whe he ua euey s ue se he gua e sag he euey ua s ese ue he esae ees. The exe FM sebas ea he ga ee ae geay ehae whe hse a away he ga ee ae suesse eag ss he FM seu. The esu s ha he seu s ushe wa he ga ee as he ua eug s eue. I hs ess he e-a wave evves beg a sghy AM-uae ae a a sgy AMuae a sha uses. F sae vaues he sge ass ua ex we a ey he eaues assae wh he FM ekg whe he eug aahes. We esae hs eaue Fg. 4 a Fg. 5 whee s eue.5. The s e se FM e aues he wes ga he vy. Ahugh he wave ee Fg. 5 shws sg AM ua s eay a a sha uses as assue he ea sues ekg [3-5]. I a hs su we ey e aj esy eak a a sae use ha s axaey a ha-e aa eah e. I we uhe eue s key ha he h e e esg he a FM sa ea he wes hesh ve he ee age he ua eug. Theee he ase aas he FM sa ea. Ths s ue ee evus eseve [3-5]. Ths s ee Fg. 6 whee s se.. Fg. 3: Seay sae seu a wave ee euey eug : a. b. 3.. The es he aaees ae ea hse Fg.. Fg. 4: Thesh as a u euey eug aaee vaues:.5. The es aaees ae ea hse Fg..
JQE_ek_saby 6 eessay a Gaussa use a. Ths s aua ue sh avy ases. I he ex se we w e a ea saby aayss wh he bjeve exag he sabes ha ay u ue aavy euey ua. Ths s he a ehass hs ae. s Fg. 5: Wave he FM e whe.5 a.. The es aaees ae ea hse Fg.. Fg. 6: Thesh as a u euey eug whe.. The es aaees ae ea hse Fg.. We have as sue he ee vayg he avy egh he eke use wave. I Fg.7 we he uu waves a he esg sea ke he eke ea a vaey avy eghs whe keeg he es he aaees he sae as hse Fg.. The esus esg he ge aves Fg. 7a agee wh he evus ekg sues whee he uu wave s a use a wh a Gaussa shae a he seu has a Gaussa evee. Hweve as he avy egh eeases eve he use a ues be bseve bh s seu a use shae evae sgay a Gaussa. The use bees we a geeay w eaks wh ueua aues ae bseve eah e. Fuhe suy eveas ha sh avy ases a age sge ass ua ex es esy he use hg a ease he asg hesh ahe ha aw he use wh. Usg ehues bwe vea egevaue aayss we have sue he hesh seu a wave a ase ue aavy ee- ua. We ha FM ekg gaes hghe e FM es whh aue we heshs as he ua euey aahes he gua e sag. Ths s sa he us aw he Fue hey FM ekg [9] whh says ha FM ekg us whe ahe Fue e bas he wes hesh. I a we ha ekg es aways u whe aahes e a ha he uu wave whe ekg s ae a Fg. 7: Seu a wave as a u avy eghs. a FSR / 3 b FSR / 4 FSR / FSR. Ohe aaees ae ea Fg.. IV. INEAR STABIITY ANAYSIS As aeay bseve ay exees a FM ase ay exeee sabes whee e he FM es eve se III s sabe a he ase uu akes he sw e uas-e esy usas whee sw s eeee he FM ua euey. I hs se we ay u a ea saby aayss uesa he g hs saby. The sag hs aayss s ee by Es. 8. Ue he assu ha he ase ueges a sa euba he a FM e wh he saes hesh we e:
JQE_ek_saby 7 ex ] [ ex ] [ 7 whee { } s he seee a FM e s he euey se he FM e ha has he wes hesh see E. a { } ae he eubas he h Fue ees. I geea bh a have -eggbe saa eeee see Es.. As ease e ha { } s he FM e ha has he saes hesh a eessay he h e FM e. U subsug Es. 7 Es. 8 a ae egeg hghe e es we ba he wg se ea eea euas he eubas. 8a 8b [ ] / 8 [ ] / 8. 8e Fwg he saa eue we assue he eubas have he wg :. ex ex ex ex ex 9 The saby he a FM e eues ha a vaues have egave ea as. U subsug Es. 9 Es. 8 we ba: a b e a ae eag Es. a e E. a we have: ] [ whee / /. I evg E. we have assue ha <<. Ths s juse beause bh a ae aes hage he swy vayg ees. As we have ague he evus se he ase syse has a hgh-q avy a a sw ves eay ae he saa eeee he ua ves a be egee. Thus we e a ba he wg he ea bewee a by svg E. : e e /. 3 Seaaey u subsug Es. 7 a E. 9 he buay E. 5 we ba ahe se sas a he u a he uu buaes he ga eu: e Γ τ. 4 Fay ae eag E. 4 E. 3 we ba / / e e Γ τ. 5a Say he euba we have / / e e Γ τ. 5b We a eee Es. 5 a seuey vesgae he saby he ase. I he wg suss we use he a { } ae he
JQE_ek_saby 8 su he euba e E. 5 whee { } a { }. Bee sussg u su ehues we ehase he wg w eaues hs syse euas: a. I a { } s a su Es. 5 he a { } s as a su. Thus he vaue es ex jugae as whh s exee se he ua ves s a ea uay. b. Eua 5 aways aws he va su: a. The euba eeses he a hase ueay whh eas ha he ase a have abay a a hase. Ths su a be ge. F he s sae ases ue sea s sevea es ague sae ha a he eay aes. We a easy vey ha was eua e Es. 5 wu bee ea E. 6 whh s he egevaue euas use ey he FM es. Thus we se a ehue whee he sus E. 6 ae egae as he h e sus E. 5. Seay he h e axa he sus Es.5 ae: whee a s he h sae E. 6. Ne ha hs we use he e sae ee he sus E. 6. We wsh ehase ha hese saes ae eae he FM es a ha hey ae exe aseg e ag he ae evu Re. We a easy ve ha Re a ha wh he h e axa saby shu eve u whh s use ay he exeea bsevas. Thus we us se he ee a e vaues. F hs use we exess he euba e as a sues saes.e. he h e su ε 6a ε 6b wheeε a ε ae he ex ees esbg he h wegh he sae wh he euba e. U subsug E. 6 E. 5 a ag he ees he sae saes bh ses we ba a euvae E. 5 exesse es ε a ε τ ε / / τ ε / / ε ε ε ε 7a 7b whee a ae eee by he wg ea ageba eua T...... a a ae gve by: 8. 9 I evg Es. 7 we use he a ha a ae uh sae ha / τ. Euas 7 have he a e ea egevaue be whh saa su ehues exs []. Thus by e ua we a veey auae ε a. ε We wsh e ha as a ee e su he ea egevaues we have as vewe he saby assae wh ase FM ea he eseve a syse see Aex. F hs suy we ha he saby s he esu he e eeee bewee he FM e ha ay saes a ahe sae ha s exe by euba. Whe hs eeee s ue he ase eaxa yas he saby ay hay. Oe eessay hs ye saby s ha he euey se bewee he w es us be se he ase eaxa sa euey. Whe saby us he eegy he a FM e s asee. Eveuay a esse eegy exhage eves bewee he w saes. Nw we ae eay sve he ea egevaues. I e esue auay Es. 7 ae uae a. I Fg. 8 we Re as a u he ua eug assug he ase s ay eag he FM e wh he wes hesh. As hs suy we se / a he ae sa sga ga C / be e-5 a esevey. The es he aaees ae he sae as hse Fg. 4 exe whh s se be e. as we ae usae he ea bewee he hewse egeeae saes as susse he evus se. I Fg. 8 he Re uve eves a suy we % -.6e-4 a.9e-4. Ths s beause a ee FM e aues he wes hesh a Es. 7 us hage seuey. Isaby
JQE_ek_saby 9 haes he wg hee egs: -e-4< % <-e-5 e- 5< % <.9e-4 a 3e-4< % <7.6e-4. Wh he he he syse es see Aex we ha he s eg he saby s ause by he e bewee he h a s e FM es. I he se eg he saby s eae by he e bewee he FM es e a - whh ae egeeae whe. I he h eg he saby s he esu he e bewee he h s a - s e FM es. we u eve see Fg. 9 ay saea a a hghe u eve whee he eaxa sa euey s sgay hghe ha. Fg. : Re as a u he CW sa sga ga C a he 4 ae ves esy eay ae % whee 3.3. A he aaees ae he sae as hse Fg. 8. Fg. 8 Re as a u ua euey eug % whee.5. a 5. The eag aaee vaues ae he sae as hse Fg. 4. Nex we suy he ee he sge-ass ua ex he ase saby. The FM ase es behave e saby age vaues. Ths s beause a age ushes a age vaue. Whe es sgay he eaxa sa euey saby s ukey u. Fgue 9 shws Re as a u he aaees use g. 8 exe whh s ease.. As ae wh Fg. 8 hs ase we u ha he sabes u sevea sa egs wh he age 8e 4< % <e4. Fg. 9 Re as a u ua euey eug % whee.. a 5. The eag aaee vaues ae he sae as hse Fg. Fay we aaye Fg. he a he u we a he ua ves eay ae he saby. We se a % eua.5 a 3.3e-4 esevey. The eaxa sa euey eases whe he u eve s ase. Thus he saby whh s ese a a I suay we have eve a aheaa ua as we as a syse e see Aex eee he saby a FM ase. Wh ehass syses havg a sw eay he ves esy we have exae he saby e see s sae ases ue FM ea. Nex we w e uea suas es he heea esus bae ses 3 a 4. V. NUMERICA SIMUATIONS AND DISCUSSION Wh he he saa egevaue ehues we have vesgae he as FM sa FM ekg. FM ekg us whe a hghe e FM e aues he wes hesh. We have as ee a ea saby aayss he FM ase whh ay bee usabe beause he e bewee saes wh w heshs whse euey eee s se he ase eaxa sa euees. Ths saby s aae by sw uua he ase uu we. I hs se we hese esus by uea sus he Maxwe-Bh euas E.. Fs we se he ase a sabe FM ase suy he bu-u a eke ea. As a as we usae hese esses a g a a sh avy ase Fgs. a a b esevey. The aaees use he aua Fg. a ae ea hse Fg. exe ha % 3e-4 e-5 a C he CW ae / u ga s se be eua 5 e esue sabe behav. As exee sag a vey sa a e he ase uu evves a sabe use-a whh agees we wh he heea es shw Fg. 3. A sa ue s e Fg. b a ase wh she avy egh. I hs ase we see he aaees he ase be he sae as hse Fg. 7 exe ha s eua / e-4. Ceay he seay sae uu wave as agees we wh he esus Fg. 7.
JQE_ek_saby Fg. : The buu he ase e a g e sabe su. a FSR / 8..3 a 5 ; b 4 FSR /.4 a. 4. The he aaees ase ae he sae as Fg.. saby s shw Fg. 3 whe s hse eua -5.e-4. As he ase he sa ua ex we bseve w euey usas bu wh eah use we e ay es waves ese ekg uses. Ths saby s ause by e bewee he w egeeae FM ekg es as suggese by he heea aayss se IV. We he vesgae he se saby wh he sae uea e. We usae sevea exaes usabe ea Fg. wh he sae aaees use Fg. 8. The saby bseve by uea sua s g ageee wh he heea aayss ee se IV. I Fg. a we se be eua -7e-5 a bseve saby he a sw usa he ase uu we. As we hage %.7e-4 Fg. b he usa hugh s exss hage s. A se exaa shws ha e use as ay es AM uae wave. I Fg. whe s eua 5.4e-4 we see a e ae usg behav wh eay ee usg e. Ths us beause hs ase hee FM es he h he s a he - s bue saby see Aex. Fg. 3: The buu sabe usa whe s ease. a -5.e-4. A he eag aaees ae sa ha Fg. 9. Fay we ay u uea suas exe he e he u eve he ase saby. Jus as e u se 4 he u eve aes he ase saby by hagg s eaxa sa euey. Ths s esae Fg. 4 whee saby us whe he ae CW sa sga ga C s hse eua 4 a vashes whe C bees 4. The aaees ae he sae as hse use Fg. 9 wh eua 5e-4. Fg. 4: Ia u eve ase saby whe -3.3e-4 a 5e 5. The eag aaees ae sa ha gue a C 4 b C4. Fg. : The buu he ase e he sabe ea whe a he aaees ae sa ha Fg. 8. a -7e-5 b.7e-4 5.4e-4. Nex we suy he ase saby whe he sge-ass ua ex s ease.. Jus as ee Fg. 9 he ase bees e sabe hese aaees ahugh we see evee se sa egs wh he age % -8e-4 < % < -e-4. A exae a ah VI. CONCUSION I hs ae we esee a heea e base Maxwe-Bh euas a ase uegg aavy FM ua wh ua euey ea he avy FSR. Wh he he hs e we aaye he seay sae waves FM ases a sue he as FM sa FM ekg. We u ha FM ekg us whe a hghe e FM e bees he eee su by aug he wes asg hesh. I a ekg sae he e-a wave s eessay a a aw Gaussa uses. Isea
JQE_ek_saby eeg he avy egh a he ua ex he aua use a ay be aw a ay have w asye eaks eah ua e. Fuhee we ee a ea saby aayss exa he uee saby ug FM ea. F a FM ase wh a hgh-q avy a a sw ves eay ae we abue he saby he e bewee he FM es. Ths hee eves whe w FM es ee a he eeee bewee he eaes a euba he ves esy ea he ase eaxa sa euees. The saby FM ea s geeay assae wh a sw e uas-e esy usa. Fay we bae uea sus Maxwe-Bh euas a e he heea esus ea saby aayss. APPENDIX A. INTERPRETATION OF INSTABIITY I hs aex we eve aa seas e ee he saby assae wh he ase FM ea he eseve a syse. We ve Es. 7a a 7b by τ / a esevey. The u sug he τ / esug euas we ha s he he wg ageba eua R s G s A whee s R s s s Aa Re s Re[ ] G s s s τ / Ab a whee a ae esevey he aua euey a he ag ae he ase eaxa sa gve by / % a g Re / τ /% ; s he eay ae he sae Re a s he euey se bewee he saes a I. The u Rs eaes he ase eaxa yas a G s esbes he eeee bewee he saes a. Thus he saby he ase s euvae he saby a syse shw Fg. A. F he eseve hs syse he se saby s he esu he e eeee bewee he a FM e a he saes exe by euba a he ase eaxa yas. Whe hs eeee s ue he ase eaxa yas saby ay hae. Wh he he a Nyus he eebak ga R s G s we a easy eee he saby: he ase s sabe a y hs Nyus es eass - he ex ae. As e ha he eebak Fg. A s ae u va a e aae ahs. Thus by suyg Nyus he ase u a sge ah R s G s we a juge a ea sae auses saby. G j R j G 3 j G j R j R j Fg. A. The euvae eebak syse. F he vew he Nyus aga we a as esae he s saby u. e s se a abay sae. Ne ha bh Rs a G s eese ae sas. I bh ae ghy ae a have a sa aua euey he syse s e saby. Ths s exae as ws. F a ghy ae sa esse s ague eses a s hase ueges as 8 egee hage ea he aua euey. Theee whe w suh esses wh sa aua euees uy ea he aua euey he hase he u hages eay by a u e 8 whe he ague eas age. Gahay hs eas ha he Nyus he eebak ga ay eass - he ex ae ag saby. F he ases ees ha have sa he eaxa sa Rs s aways ghy ae. Thus e a saby u hee us be a sae ha sasy a Re << suaeusy. The s eues he euey se bewee he w saes us be se ha ase eaxa sa euey a he se eues he sae have a w hesh. T ue hs aex we use Fg. A suy a ase usabe ase FM ea. The aaee use hee ae ea hse Fg. 8. Whe s -7e-5 he Nyus agas shw ha saby s ause by he e bewee w saes ha eaes he s a he h e FM es esevey. A.7e 4 he saby as exss ue he e bewee he s a he - s FM es whh ae egeeae whe. Fay we se be 5.4e-4 he sua bees e ae a agehe hee saes ae vve saby. Ag he Nyus agas a 3 bh he e bewee he s a he h e FM es a he e bewee - s a he h e FM es ause saby. Whe hs haes we exe see a uu saby he esy usa wh ee usg e.
JQE_ek_saby Iagay Axs Iagay Axs Iagay Axs Nyus Daga Nyus Daga Nyus Daga a.4 a.5 a3.3..5. -. -.5 - -. - -.3 -.5 - -.4 - -.5 - -.5 - -.5.5-3 - - - -.8 -.6 -.4 -.. Rea Axs Rea Axs Rea Axs Nyus Daga Nyus Daga Nyus Daga.5.5 b.5 b b3..5.5.5 -.5 -.5 -.5 - - -. -.5 -.5 -.5 - - - - - -.8 -.6 -.4 -.. Rea Axs Rea Axs Rea Axs Nyus Daga Nyus Daga Nyus Daga.8.5.6 3.4.5.5. -.5 -. -.5 - -.4 -.5 -.6 - - -.8 -.5 - -.5 - -.5 -. - -.8 -.6 -.4 -. - -.5 Rea Axs Rea Axs Rea Axs g y [3] P.F. Cuey A.I. Fegus Feuey uae ea a T:sahe ase Os Cuas v. 83. -. 85-9 May 99 [4] D.M. Kae S.R. Bawe A.I. Fegus FM ye ases Ae Physs B-Phhyss & ase Chesy v. B39. 3 7-78 Mah 986. [5] R. Tag J.M Feh J.M. Yabugh Feuey Saba a Nse Suess he Ag FM ase IEEE J. Quau Ee v QE-4. 644-648 O 968. [6] Y. S. M. Gwasse P. Hee. Nau Dyas a ee-ay uabe h ase sube IEEE J. Quau Ee. [7] N. B. Abaha P. Mae.M. Nau Dya saby a usas ases Pgess Os v. XXV.-89 988 [8] A. Z. Geak R.G. Bewe Oa hee ases by ase euey swhg Physa Revew A v. 7. 4 463-473 A. 978. [9] A. Yav Iu Oa Ees New Yk 976 [] W. Pess S. Teuksky W. Veeg a B. Faey Nue ees C Cabge Uvesy Pess E 99 46-463. Fg. A. Nyus agas a eebak ga a e bewee he - s a he s e FM es a a e bewee he h a he s e FM es a3 whe s -7e-5; b eebak ga b e bewee he - s a he s e FM es b a e bewee he h a he s e FM es b3 whe s.7e-4; eebak ga e bewee he s a he h e FM es a e bewee he - s a he h e FM es 3 whe s 5.4e-4. The es aaees ae he sae as hse Fg. 8. REFERENCES [] J. C. Vea P.R. Hee Me-ke Mh ase Oa Tase Fbe Ra IEEE Tas. Mwave They a Teh. v. 49..88-887 O.. [] A. Maja D. Y P. R. Hee W. D. Jes S. M. Gwasse Y.. Mwave Geea by Regeeave Me-kg a N:YVO4/MgO:NbO3 Mh ase IEEE Ieaa Mwave Sysu Dges Phaeha Pesyvaa 95-98 Jue 3 [3] S. gh P. aa Te-a aayss euey uae ase sa Ae hyss ee v. 73. 6.7-75 Aug. 998. [4] D.J. Kuega A.E. Sega FM-ase ea he N:YAG ase IEEE J. Quau Ee v. QE-6.. 673-677 Nv. 97 [5] A.E. Sega D. J. Kuega Mua euey eug ees he FM e-ke ase IEEE J. Quau Ee v. QE-6.. 83-88 De. 97 [6] S.E. Has O.P. MDu FM ase sa-hey Ae Physs ee v. 5.5-6 N. 964 [7] S.E. Has O.P. MDu FM ase sa-exee Ae Physs ee v. 5.-4 964 [8] S.E. Has O.P. MDu They FM ase Osa IEEE J. Quau Ee v. QE-.6. 45-6 Se. 965 [9] S. gh P. aa Fue hey aavy ase euey ua Physa Revew A v. 6 5. 46-47 Nv. 999 [] H.A. Haus A They Fe Me kg IEEE J. Quau Ee. v. QE-. 7.33-39 Juy 975. [] A. M T. Ok A. Sga T. Kbayash FM ase ea a T:sahe ase IEICE Tas. O Ees v. E78-C.. 88-9 Ja. 995. [] A.P. Ws D.M. Kae. Mua ue heee ase: he ase se e suess exea avy e ases Os Cuas v.. 5-6. 577-588 O. 994. Y. eeve hs B.Eg. Oees Huahg Uvesy See a Tehgy Cha 996. He eeve hs Ph.D. Eea Egeeg Dexe Uvesy Phaeha PA 3. He s uey a Reseah Asssa Pess a he Cee Mwave ghwave Egeeg Eea a Cue Egeeg Deae Dexe Uvesy. Se 998 he has bee eg eseah he aea wave hs a he Cee. Hs eseah eess ue: uabe h ases hyb a/aa be a syse hee a uas hgh ya age wave be k a ase ea yas. P. R. Hee S 66-M 67-SM 89-F 9 was b Buaes Hugay 936. He eeve he M.S. egee hyss a Ph.D. egee eea egeeg he Uvesy Mesa a Meas 963 a 967 esevey. Se 967 he has bee he auy Dexe Uvesy Phaeha PA whee he s uey he ese Kaus Pess Eea a Cue Egeeg. He s he De he Cee Mwave-ghwave Egeeg Dexe Uvesy. He has auhe -auhe e ha 4 aes s-sae ees waves hs sa eegy a bea egeeg. He has as seve as a sua ueus age a sa as He s a ew he IEEE. He s a ee he IEEE Meu Mea a seve as he Dsgushe eue he IEEE Mwave They a Tehues Sey IEEE MTT-S.. M. Nau was b T Iay 94. He eeve he D s egee hyss he Uvesy Ma Iay 964. He s uey he Fas K. Davs ess hyss Dexe Uvesy. He has auhe -auhe e ha eha aes a sx bks he aeas Quau s ase hyss a ea yaa syses. He s a Few he Aea Physa Sey a he Oa Sey Aea. He s he ee he 99 Ese Mea a he 999 ab Mea hs svey he subaua e awg saeus ess.