Photon-phonon interaction in photonic crystals

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1 IOP Cofr Srs: Mtrls S d Egrg Photo-phoo trto photo rystls To t ths rtl: T Ut IOP Cof. Sr.: Mtr. S. Eg. 3 Vw th rtl ol for updts d hmts. Rltd ott - study o optl proprts of photo rystls osstg of hollow rods G Fuj T Mtsumoto T Tkhsh t l. - Frto of Trhrtz Wv Rsotors wth lum Dmod Photo Crystls for Frquy mplfto Wtr Solvts N Oht T Nk d S Krhr - Squzg of Phoos Photo-Phoo Itrto v t-stoks Lght * Ch Jhu d Guo Gug Ths ott ws dowlodd from IP ddrss o 7 t 3:

2 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 Photo-phoo trto photo rystls T Ut Isttut of Md d Iformto Thology Ch Uvrsty -33 Yyo-ho Ig-ku Ch Jp E-ml: ut@fulty.h-u.jp strt. Photo-phoo trto o th logy of ltro-phoo trto s osdrd o-dmsol photo rystl. Wh ltt vrto s rtflly trodud to th photo rystl govrg quto of ltromgt fld s drvd. smpl modl s umrlly lysd d th followg ovl phom r foud out. Th ltt vrto grts th lght of frquy whh ddd th tgrl multpl of th vrto frquy to tht of th dt wv d lso mplfs th dt wv rsotly. O rso th mplfto ftor rss vry rpdly wth th umr of lyrs rss. Rso frqus hg wth th phss of ltt vrto. Th mplfto phomo s lytlly dsussd for low frquy of th ltt vrto.. Itroduto It s wdly kow tht ltromgt flds prodlly modultd dltr md s ltros rystls form rgy d struturs. Suh phom hv vstgtd rgorously d tsvly for mor th t yrs[-]. O th othr hd th ltro-phoo trto s othr rmrkl phomo of ltro mtl or smodutor. I som ss ltros odutors form polros y trtg wth phoos. Furthrmor ths uss v suprodutvty phomo. Now lt us osdr th logy of th ltro-phoo trto photo rystls. Nw physl phom ptd photo rystls wth ltt vrtos[3-4]. Nmly f th ltt of photo rystl s rtflly oslltd wht would hpp? I th prst study drt trto tw ltromgt flds d ltt vrtos s osdrd o-dmsol photo rystl ostrutd y prodlly rrgd dltr plts.. Formulto W osdr o-dmsol photo rystl mly mult-lyr of dltr plts. Th sptl vrto of th dltr ostt s ssumd s U d whr d s th ltt ostt d w do ot spfy th futo U. Th -th dltrs s ssumd to oslltd hrmolly wth mpltud ξ t. Th th sptl d tm vrto of th dltr ostt s prssd s follows: Pulshd udr l y IOP Pulshg Ltd

3 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 t U d ξ t && ξ t ξ. t I ordr to dsr hvour of th ltromgt fld E t ths systm w g wth th ufd Mwll quto. Th ltromgt fld s ssumd to propgt log th -s d to prlll to th lyrs. Th w hv E t E t whr th rltv prmlty s st to. I ths prolm th tm drvtv of t s sstl d s th org of th photo-phoo trto. ssumg th mpltud ξ t to sufftly smllr th th ltt ostt d tht s ξ t d << t pdd s U η t ξ t. η η d Th frst d sod drvtvs r otd strghtforwrd s t U η & ξ t t η η d η t U. ξ t t η η d y susttutg ths qutos th ufd Mwll quto oms U η t ξ t t η t η d η U η & ξ t. t η d Th vrl t s slr fld s rprsttv of th vtor fld. Suppos w wrt th -th mpltud ξ t s ξ t ξ os t th Fourr trsform of th govrg quto s otd s follows [3]: U η. ξ η η d Hr ξ d r th mpltud frquy d wvumr of th ltt vrto rsptvly. Th vrl s th frquy of th ltromgt fld. Th trms o th rght hd sd prss th fft of th photo-phoo trto. It s proportol to ξ. Ths ms tht ξ dos ot d to lrg f dor r lrg ordr to s th flu of th trto. Th fld s oupld wth ±. Ths quto s dffr quto frquy sp. Thrfor wh ltromgt wv wth frquy omg s dt towrd th photo rystl ltromgt flds wth frqus l wll td whr l s tgr. 3. Numrl Implmtto Hrftr w osdr th s of rtgulr vrto of dltr ostt s show Fg..

4 - Fgur. Illustrto of th umrl modl of o-dmsol photo rystl. Th lyr of th dltr ostt hs thkss of d th ltt ostt s d. Th th futo η U s wrtt y th ut stp futo s { } U η η η so th drvtv ossts of two dlt futos. Howvr th otuty of th fld holds. Th oudry odtos t ± r otd s follows rsptvly; { } { }. ξ ξ Th fld r otuous t y oudrs of lyrs ut th frst drvtv of th fld hs jumps. Th flds wth th -th lyr d th lyr r prssd y lr omto of two dpdt fudmtl solutos wth h lyr for h frquy s p p p p D C whr s th vloty of lght vuum. Th offts dpd ot oly o ut lso o. y ms of th oudry odtos w ot st of quto of th offts s { } { } [ ] { } D C D C ξ WCCMPCOM IOP Pulshg IOP Cof. Srs: Mtrls S d Egrg 3 do: x3 3

5 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 C D { C D } ξ [ { } { }]. Now w osdr ft systm of N lyrs d dt wv of pl wv of frquy p[ ]. If w ssum th frquy of th fld tw l m d l m th trsmsso sptr r otd y solvg st of 4N lm qutos. Th th trsmsso prolty T l of frquy l s dfd y T l l N l for th tgr l stsfyg l >. 4. Numrl Rsults t frst th trsmsso sptr for th lght of frquy of ± d so o for 3 lyrs N 3 r show Fg.. Th prmtrs r st s 8 π ξ. d l m 4. I ths s th whol photo rystl s just oly shk. W s th fld of frquy dffrt from th dt o r grtd. Th tsts rflt th photo d strutur of th systm wthout ltt vrto d grow proporto to th squr of th frquy. Hr w must py ttto splly to th trsmsso sptrum of th dt frquy. W s th dt wv s rsotly mplfd [5]. Th trsmsso sptrum T th s of lyrs N s plottd Fg. 3 whr th vlus of prmtrs rm. Th rsot mplfto s rpdly hd s th umr of lyrs s rsd. Th trvl tw rsot pks s out π. It sms tht th rsot pks r lotd t photo d dgs. Th trsmsso sptrum T for π s plottd Fg.4. Othr prmtrs r ot hgd. Th trvl oms out π 4 d th pks ppr wth th d. Th trsmsso sptrum T for π 3 d s plottd Fg.5. Othr prmtrs r ot hgd. I ths s w s modfto of d strutur.. th ppr of w d gps. Th mplfto ours t th dgs of th w ds. Wh s st to π w srly fd th mplfto for. For π π 3 d π 4 howvr th mplfto s rtly osrvd th hgh frquy rgo. 4

6 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 T ê T μ μ μ -6. μ ê T3.6.4 T-3.5. T T ê ê ê T- T ê ê ê.5 T ê Fgur. Th trsmsso sptr of 3 lyrs. Th prmtrs r st s 8 π d ξ.. 5

7 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 4 3 T ê Fgur 3. Th trsmsso sptrum T of lyrs for. 4 3 T ê Fgur 4. Th trsmsso sptrum T of lyrs for π. 4 3 T ê Fgur 5. Th trsmsso sptrum d. T of lyrs for π 3 6

8 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 5. lytl Trtmt of th mplfto W osdr th s tht th vlu of th rto s smll ough ordr to dv lytl trtmt of th mplfto. Th w pd th flds ± srs so tht w ot U η ξ { } η ξ ξ η U η η U η η d η d η d Ths quto rwrtt s ~ os s. η l ~ U ξ s η η d U d ξ os. Now th dltr ostt hs th mgry prt. Th mgry prt dpds o th proprts of th fld th photo rystl. Wh th mgry prt s gtv th dt fld s mplfd. For th mgry prt vshs so tht w s tht ltt vrto ot shk of whol systm mks photo rystl tv mdum [6]. Th rl prt of th dltr ostt U d ξ os shows tht ths quto s fftvly for dul-prod multlyr strutur [7 8] whh th ltt ostt s modultd lk d ξ s s. Th trsmsso prolty T l ths s rflts th d strutur of dul-prod o-dmsol photo rystl. 6. Summry d Colusos I th prst ppr w hv osdrd photo-phoo trto o th logy of ltro-phoo trto d drvd th govrg quto. Wh ltt vrto s trodud th ltt vrto dos ot oly grt th wvs wth vrous frqus ut lso mplfs th dt wv rsotly. O rso th mplfto ftor rss vry rpdly s th umr of th plts rss. Rso frqus hg wth th phss of vrto of h lyr. Suh mplfto ours pt for th s tht ll lyrs osllt wth th sm phs mly tht of shkg th systm v f th frquy of th ltt vrto s vry smll. Ltt vrto mks photo rystl tv md. Th mplfto s osrvl. 7

9 WCCMPCOM IOP Cof. Srs: Mtrls S d Egrg 3 IOP Pulshg do: x3 Rfrs [] K. Ohtk: Phys. Rv. Vol [] Joh D. Joopoulos Rort D. Md Joshu N. W : Photo Crystls: Moldg th Flow of Lght Prto Uv. Prss 995 [3] T. Ut: Mtg strts of th Physl Soty of Jp Vol. 54 Issu Prt p Jps. [4] Y. Tk t l. Ntur Mtrls Vol [5] T. Ut d K. Ohtk: Mtg strts of th Physl Soty of Jp Vol. 54 Issu Prt p Jps. [6] T. Ut d K. Ohtk: Mtg strts of th Physl Soty of Jp Vol. 55 Issu Prt p.86 Jps. [7] R. Shmd T. Kod T. Ut d K. Ohtk : J. Phys. So. Jp Vol. 67 o. pp [8] R. Shmd T. Kod T. Ut d K. Ohtk : J. ppl. Phys. Vol. 9 o. 8 pp

Section 5.1/5.2: Areas and Distances the Definite Integral

Section 5.1/5.2: Areas and Distances the Definite Integral Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto: Th Dt Itgrl Suppos w