2018 TIGP: Advanced Nanotechnology (A) Part 1: Photonic Crystals and Devices

Transcription

1 08 TIGP: Advaced Naotecholog A Part : Photoc Crstals ad Devces Lecture # M-Hsug Shh Research Ceter for Appled Sceces RCAS Academa Sca Tawa Mar. 4 08

2 Comparso of Quatum Mechacs ad lectrdamcs Table teded comparso betwee quatum mechacs a perodc potetal ad electromagetsm a perodc delectrc. What s the ma fucto that cotas all of the formato? How do we separate out the tme depedece of the fucto to ormal modes? What s the master equato that determes the ormal modes of the sstem? Are there a other codtos o the ma fucto? Where does the perodct QUANTUM MCHANICS IN A PRIODIC POTNTIAL CRYSTAL The scalar wave fucto rt. r t C r e t / h pad a set of eerg egestates r p V r r r m The Schrödger equato. Yes t must be ormalable. LCTROMAGNTISM IN A PRIODIC DILCTRIC PHOTONIC CRYSTAL The magetc vector feld H r t. t H r t C H r e pad a set of harmoc modes H r. H r H r r c The Mawell equatos. Yes the feld must be both ormalable ad trasverse: H 0 The potetal: The delectrc: r r R

3 of the sstem eter? Is there a teracto betwee ormal modes? What mportat propertes do the ormal modes have commo? O what fact about the master equato do the mportat propertes rel? What s the varatoal theorem we use to determe the ormal modes ad frequeces? What s the heurstc that goes alog wth the QUANTUM MCHANICS IN A PRIODIC POTNTIAL CRYSTAL V r V r R for all lattce vectors R. Yes there s a electroelectro repulsve teracto that* maes large-scale computato dffcult. gestates wth dfferet eerges are orthogoal the have real egevalues ad ca be foud wth a varatoal prcple. The Hamltoa H s a lear Hermta operator. var var H s mmed whe s a egestate of HHamltoa. The wave fucto cocetrates regos of low potetal whle remag orthogoal to 3 LCTROMAGNTISM IN A PRIODIC DILCTRIC PHOTONIC CRYSTAL for all lattce vectors R. I the lear regme lght modes ca pass rght through oe aother udsturbed ad ca be calculated depedetl. Modes wth dfferet frequeces are orthogoal the have real postve egevalues ad ca be foud wth a varatoal prcple. The Mawell operator s a lear Hermta operator. var H H H H var s mmed whe H Feld s a ormal mode of. The felds cocetrate ther electrcal eerg hgh- regos whle remag orthogoal to lower modes.

4 varatoal theorem? What s the phscal eerg of the sstem? Is there a atural legth scale to the sstem? What s the mathematcal statemet that sas: A s a smmetr of the sstem? How do we classf the ormal modes usg a sstem s smmetres? What does the dscrete traslatoal smmetr of a crstal allow us to do? What are the allowable values for the wave v vector? What do we mea b the term bad QUANTUM MCHANICS IN A PRIODIC POTNTIAL CRYSTAL lower states. The egevalue of the Hamltoa. Usuall because costats such as the Bohr radus set the legth scale. A commutes wth the Hamltoa: A H 0. Dstgush them b how the trasform uder a smmetr operato A. r u r e r Wrte the wave fucto Bloch form. The le the Brllou oe recprocal space. The fuctos whch tell us the eerges 4 LCTROMAGNTISM IN A PRIODIC DILCTRIC PHOTONIC CRYSTAL 8 dr D H The electromagetc eerg. No solutos are scalable to a legth scale. A commutes wth the Mawell operator: A 0. Dstgush them b how the trasform uder a smmetr operato A. H r u r e r Wrte the harmoc modes Bloch form. The le the Brllou oe recprocal space. The fuctos whch tell us the frequeces of the

5 structure? What s the phscal org of the bad structure? What happes sde a gap the bad structure? What do we call the bads mmedatel above ad below the gap? How do we troduce defects to the sstem? What s the result of troducg a defect? How do we classf dfferet tpes of defects? QUANTUM MCHANICS IN A PRIODIC POTNTIAL CRYSTAL of the allowed egestates. The electro wave scatters coheretl from the dfferet potetal regos. No propagatg electros that eerg rage are allowed to est regardless of wave vector. The bad above the gap s the coducto bad; the bad below the gap s the valece bad. B addg foreg atoms to the crstal whch breas the traslatoal smmetr of the atomc potetal. It mght create a allowed state a bad gap thereb permttg a localed electro state aroud the defect. Door atoms pull states from the coducto bad to the gap; acceptor atoms pull states from the valece 5 LCTROMAGNTISM IN A PRIODIC DILCTRIC PHOTONIC CRYSTAL allowed harmoc modes. The electromagetc felds scatter coheretl at the terfaces betwee dfferet delectrc regos. No eteded modes that frequec rage are allowed to est regardless of wave vector. The bad above the gap s the ar bad; the bad below the gap s the delectrc bad. B chagg the delectrc costat of certa regos whch breas the traslatoal smmetr of r. It mght create a allowed state a bad gap thereb permttg a localed mode aroud the defect. Delectrc defects pull states from the ar bad to the gap; ar defects pull states from the delectrc bad to the gap.

6 I short wh s the stud of the phscs of the sstem mportat? QUANTUM MCHANICS IN A PRIODIC POTNTIAL CRYSTAL bad to the gap. We ca talor the electroc propertes of materals to our eeds. LCTROMAGNTISM IN A PRIODIC DILCTRIC PHOTONIC CRYSTAL We ca talor the optcal propertes of materals to our eeds. 6

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9 We had dscussed the bad structure for -D photoc crstals. Two popular tpes of lattces rectagular & tragular lattces had bee show the prevous otes. For most of applcato wth photoc crstals the bad gap the bad structure s oe of the most mportat parameters. A lot of applcatos use the optcal propertes sde or aroud the photoc bad gap so photoc crstal s oe tpe of photoc bad gap materals PBG wth the cotrol of the bad gap characterstcs. We ca egeer the optcal propertes of the optcal/photoc devces. It s qute smlar to the bad gap of semcoductor electrc materal. People ca egeer electroc propertes b cotrollg the bad gap of S GaAs or others. -D photoc crstals bad gap ca be decded b the lattce geometr ad the parameters to dscuss how those parameters tae some effect for the bad gap. r/a value of photoc crstal holes or rods Wthout chagg the materal or the lattce tpe we ca 9 a r. Now we are gog

10 ust var the hole or rod radus r to modf the bad gap frequec or wave legth ad wdth. Let s cosder a -D tragular lattce wth ar holes embedded the delectrc materal aga..0 a r Now we use =3.4 ad var r/a value of ar holes. The bad dagram of photoc lattces wth the dfferet r/a values are show below. We have the bad structure of T ad TM modes wth the r/a values. Apparetl the bads shft to hgher frequec as the r/a value creases. However the shfts mght be dfferet because of the dfferet polaratos..e. T TM. The bad gap whch s shaded the bad dagrams s ot alwas ested for all r/a values. I ths eample the T bad gap start to show up whe the r/a value s large tha 0.8. The bad gap s aroud 0

11 ormaled frequec 0.. We ca plot the photoc bad gap posto versus the r/a value whch s show the page 86. The hgher curve s the frequec posto of the top boudar of the bad gap whle the lower curve s frequec of bottom boudar of the bad gap. The rego surrouded b the two curves s the bad gap rego wth vared r/a values. Ths bad gap rego goes to hgher frequec area whe r/a value crease. The wdth of the photoc bad gap whch s the dfferece betwee two curves top-left fgure s plotted the bottom fgure. The bad gap wdth creases as r/a value crease ad t reaches the mamum 0.8 ormaled frequec at r/a value 0.4. After the mamum pot the gap wdth start to decrease as r/a value creases. For ths case we had realed The photoc bad gap ca be shfted b varg r/a value. It meas that the devce operated wave legth ca be cotrolled b the r/a value of photoc lattces.

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15 The wdth of photoc bad gap ca be modfed wth the dfferet r/a value. Larger bad gap gves us more degree of freedom to desg the photoc crstal devces ad more defect modes or bads from the photoc crstal defect cavtes or wavegudes. 5

16 Refractve de of photoc crstal lattces I et eample we are gog to eame the effect due to the varato of the refractve de of the photoc crstal lattces. We stll tae the tragular lattces wth ar holes as a eample. Now we f the r/a=0.3 ad de of ar holes.0. The bacgroud de s vared from.5 to a r r 0.3 a hole.0 6

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18 3 Complete bad gap for both polaratos I the prevous eamples some have the photoc bad gap for T modes some have bad gap ol for TM mode ad some eve have o bad gap for the both polaratos. Is t possble to have the photoc bad gap for both T ad TM polaratos smultaeousl? The aswer s Yes! Now let s tae a loo to a eample frst the we ll aale the reaso for T ad TM bad a r.0 gaps. Cosder a -D tragular photoc crstal lattces wth the ar holes aga. Ths tme we ll let =3.4 8

19 ad r/a value vared from 0.35 to The bad structures of T ad TM modes wth these r/a values are show below. 9

20 Accordg to the bad dagrams the prevous pages we ca plot the bad gap posto ormaled frequec versus r/a value le we dd the page 9. I the top fgure of page 9 0

21 the square-curves are the top & bottom boudares for T bad gap whle the crcle-curves are the bad gap boudares versus r/a value for TM mode. We also plot the wdth of bad gaps for T & TM modes versus r/a value of lattces the bottom fgure.

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23 What s mportat for the photoc crstal bad gap? We have doe several eamples for the dfferet photoc crstals ad plotted ther bad dagrams ad the photoc bad gaps. To appl photoc crstals the real devces ad egeer ther optcal propertes aroud the bad gap rego we should dscuss several e ssues for the photoc bad gap. Scalg propertes of photoc crstals I M there s o fudametal legth scale other tha the assumpto that the sstem s macroscopc. Therefore photoc crstals there s o fudametal costat wth the dmesos of legth. The relatoshp betwee M problems s the dfferece ol b a cotracto or epaso of all dstaces. Cosder the egevalue equato for Hr [ H r] H r r Now we mae a compresso or epaso for the structure ' r the r ' r s 3

24 where s s a scale factor. We ca also mae chages of the varables r sr ' s the the above equato becomes r' r' s ' [ s ' H ] H r' s s s r' r' ' [ ' H ] H ' r' s s s Ths s ust the same egevalue equato wth the mode r H r= H ad frequec =. Ths meas that we ca s s obta the ew mode ad ew frequec ust scale a factor s from the orgal solutos of the Mawell s equatos. To uderstad the scalg propertes of photoc crstals let s cosder the followed eample. 4

25 a.0 r r Now we fed & ad 0.3 a but we var the lattce costat a to stud the chages the bad structure & bad gaps. I page 97 we plot the bad dagrams for a =000 m ad a =000 m. We realed that bad structures loos same the ormale frequec ut a. It meas the frequec var learl wth the scale of the lattce cost..e. a The rato of dfferet delectrc costats. I the prevous dscusso for -D photoc crstal bad structures we had eamed the bads ad bad gaps var wth the dfferet delectrc costats. However s the absolutel value of a delectrc costat the e parameter for the bad gap? 5

26 Actuall what s reall mportat s the rato of the dfferet costats ot the actual values. Let s cosder ths eample. a r Now we fed a & r the lattces. We also fed the rato of two refractve dces 3 = to 6 =. but vared the actual values from The bad dagrams are plotted the page 98. 6

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28 So as log as the rato fed we ca ust scale the delectrc costats to whatever we wat ad the optcal propertes wll be same. 8

29 Cocept of the Lght Coe/Le I -D photoc crstal we assumed the perodc structure - plae ad fte thcess -as. However for most of the cases we have the fte thcess the photoc crstal slab structure. It s mportat to cosder whether lght comg from a outsde homogeeous medum ca couple to the sde modes through the etrace boudar plae. Ths d of couplg mode called the lea mode because t s ot a egemode t s ot a egemode of the structure ad t s eerg lea to the outsde space through the couplg. No-lea mode lea mode 9

30 terface // Now cosder medums wth de &. I medum the sgal has wave vector ad frequec. I medum the sgal has. I couplg process the eerg coservato ad mometum coservato eerg to terface must be vald so = ad / = / / / I a homogeeous medum wth de the lght coe ca be descrbed b the relato c // c Whe =.e. ar the the lght le c ŷ ˆ 30

31 Now let s use the laer structure to descrbe the cocept of lght le coe. ŷ ˆ Cosder the above structure medum s embedded b medum ad the refractve de of medum s larger tha the refractve de of medum. Assume a mode medum wth the wave vector ad a propagato agle. After pass through the terface to the medum the mode has the wave vector ad the propagato agle. Note we have the Sell s law or s s // = // or s s 3

32 For a mode cofed the medum t meas that we eed have a mode total reflectve from the terface..e. c where s c s s or c so oce there s o wave propagate to the c medum from the terface. However ths does t dcate the lght feld vashes completel the medum. See the lea ad o-lea modes the fgure of page 99. 3

33 Photoc Crstal Slab Structure So far we had dscussed the photoc crstals whch the lattce s perodc -D ad ftel eteded the ẑ as. Now we are gog to cosder the photoc crstals wth a fte thcess. a b Addg the fte thcess for the -D photoc crstal lattces meas we have to deal wth the propagato vector the ẑ -as. ve for propagato -plae - plae the delectrc slab wll have guded modes whch there s ot ero. There are some effects from ths fte thcess of the slab 33

34 structure. There s o loger a complete photoc bad gap. It s stll possble to have the bad gaps for the guded modes of the slab. Modes caot be classfed as pure T or TM modes. So let s dscuss the two ssues oe b oe. Frst thg s the dsappearace of a complete bad gap. Ths s due to the o-ero. ve two-dmesoal photoc crstals that are fte the thrd dmesoal Here s the ẑ -as the photoc bad gap dsappears whe we allow out-of-plae 0 propagato. Here we show the bad dagrams from -D photoc crstals. The are perodc - plae ad ftel the ẑ -as. 34

35 Fg. Bad dagrams for the photoc crstals from a Fg. a ad b Fg. b. The shaded rego dcates the frequeces of states troduced whe vertcal propagato.e. perpedcular to the plae of perodct s permtted. If we restrct the photoc lattce to a fte thcess the we epect that there wll be modes guded b the delectrc slab ad radato modes or lea modes. 35

36 The dsperso relato for a slab wavegude s followed radato modes c guded modes Lea modes c The bad structure for a slab photoc crstal lattce has the smlar stuato. Bads below the lght le are guded b the delectrc slab. Bads above the lght le are the lea waves. Ther eerg s 36

37 ot cofed sde the delectrc slab. There s also a cotuum of radato modes above the lght le. The lght le s gve as the case of the slab wavegude b the free propagato plae wave dsperso relato of the claddg. The pressure of the radato modes elmates the complete bad gap. It s stll possble however to obta gaps the guded spectrum. Lght the frequec rage correspodg to the gap the guded modes ca t propagate the plae of the slab. Accordg to the above argumets oce we have a fte thcess for the -D photoc crstals the bad dagrams of -D lattces the page 0 wll become the bad dagrams show the et page. 37

38 Fg.4. Proected bad dagrams correspodg to the two slabs Fg.3. Whether states are eve or odd wth respect to the horotal mrror plae of the slab s dcated b ope or flled crcles respectvel. The shaded regos above the lght coe are radato modes or lea modes for the -D photoc crstal slabs. Ol the bads below the lght coes are cosdered as the guded or 38

39 cofed modes the slab. We are most lel to use those modes for the real devces. Now let s tal about the polarato of slab photoc crstals. Sce the smmetr uder reflecto through - plae s o loger vald for all the slab modes ca ot be classfed as eve or odd modes for all. However t s possble to separate the slab modes to eve to odd at some pots. Cosder the two eamples I II.0.0 ˆ ẑ ample I s a suspeded membrae photoc crstal structure whch has ar claddg above ad bottom of the slab. I ths case we ll have reflecto smmetr at - plae of = 0.e. at ceter of the slab. We therefore have separated eve ad odd modes at = 0. I the - plae of the ceter of the slab the eve modes are T ad the odd modes are TM. However at all other the eve modes are ol T-le ad the odd modes are 39 ŷ.0

40 ol TM-le. The are ot rgorousl T or TM. I eample II we caot have a whch has the reflecto smmetr for the photoc crstal sstem. For ths case we ol have T-le or eve-le modes ad TM-le or odd-le modes. 40

41 Normaled frequec a/ Defects ad defect modes from the photoc crstals After the dscusso of o-defect photoc crstal lattces we are gog to stud the photoc crstals wth the defects or the defect les. For a tragular photoc crstal wth the defect we have the bad dagram for T mode. Defect modes Defect bads Complete Bad Gap 0. M M K I-plae Wavevector Now f we create a defect pot b removg a ar hole from the lattces to form a defect cavt. * Cofe photo test sde the defect 4

42 Removed hole The we ll smultaeousl have some defect modes the photoc bad gap regos. The modes descrbe the photo / wave behavor assocated wth the defect we created. Removed le If we remove a le from the lattce to form a wavegude we ll have the defect bads the gap rego to descrbe the character of the wave propagato sde or aroud the defect le area. There are several otes we should meto for the defect modes / bads of the photoc crstals The defect modes whch are formed b removg some 4

43 lattce pots are ol the solated modes the bad gaps. It meas that there s o allowed modes / states betwee those defect modes for the photoc crstal cavt. The defect bads whch are created b removg a le of lattces are composed b ma defect modes the gaps. Sce the defects the lattces ope oe-dmesoal degree of freedom a pot or state o the defect bads s the allowed state for the photoc crstal wavegude. 3 The defect modes ad bads are ot ol located sde the completel bad gap regos but also the gap rego outsde the complete bad gap. 43

44 Cavt The D 4 photoc crstal defect cavt o a SOI substrate. Wavegude The wth photoc crstal wavegude a GaAs suspeded membrae. 44

45 Qualt factor Q ad photo lfetme Qualt factor Q s a mportat parameter for a cavt. It mal descrbes the photo lfe tme sde the cavt or the eerg stored the cavt. There are several deftos or descrptos for the qualt factor of a cavt. eerg stored the sstem at resoace Q eerg lost a ccle of oscllato or W T -dw dt W -dw dt 0 0 Wt W0 e 0t - Q W : Stored eerg : Resoace freq. We ca also have Q related to the photo lfetme τ p or p Q 0 Q 0 P 45

46 3 I epereces we ca obta the Q value from the spectrum Q 0 0 I photoc crstals we use the photoc crstal defect structure serve as the resoat cavt for the desged defect modes. The qualt factor Q wll represet a measure of how ma oscllatos tae place sde the cavt before the ected photo eerg dsspate out of the cavt. It meas that the photo eerg for arrow frequec bads be trapped the cavt for loger perod of tme f we have a hgh-q cavt. Now the mportat questo becomes how the defect cavt should be desged for troducg the hgh-q modes to the structure. There are several eamples of the hgh-q photoc crstal cavtes the followg pages. 46

47 Photoc Crstal D4 Lasers IGaAsP Membrae 47

48 Fte-dfferece tme-doma FDTD smulato method The fte-dfferece tme-doma FDTD method s wdel used to drectl solve tme-depedet Mawells equatos ow. FDTD was orgall proposed for M waves wth log wavelegths such as mcrowaves because the spatal dscrmato requremet s small ~ 0 0. I 966 Yee descrbed the bass of FDTD umercal techque for M wave the tme doma o a space grd. K.S. Yee I Trasacto of Ateas ad Propagato Recetl FDTD method s popular smulatg photoc bad gap structure ad other photoc devces. For 3-D smulato of the Mawell s equatos t become more complcated. The tme-depedet Mawell s equatos are H 0 t H t Ther { } compoet represetatos are 48

49 49 t H 3 t H 4 t H 5 H H t 6 H H t 7 H H t We assume that are spatal dscretato ad that t s a tme step the fucto F t s dscreted as 8 F t F t F

50 I the paper of 966 Yee orgated a set of fte-dfferece equatos for the tme-depedet Mawell s curl equatos for the lossless materals case 0 0. The above fgure shows Yee space lattce whch llustrates the Yee algorthm. Ths algorthm ceters ts ad H compoets 3-D space so that ever compoet s surrouded b four crculatg H compoets ad each H compoet s surrouded four crculatg compoet. 50

51 5 The Yee algorthm also ceters ts ad H compoets tme. All of the computato the modeled space are completed ad stored memor for partcular tme pot usg prevousl stored H data. The all of the H computatos the space are completed ad stored memor usg the data ust computed. The ccle begs aga wth the recomputato of the compoets based o the ew obtaed H. Ths process cotues utl tme-step fshed. Yee used cetered fte-dfferece or cetral-dfferece epressos for the space ad tme dervatves. Ths wa ca obta smpler programg ad secod-order accurac. For eample the frst spatal dervatve of the fucto F t F t F wll be ] [ O F F F

52 F F Ad the frst tme dervatve of s F F O[ t ] t t There are two otes we should uderstad that Yee chose the cetral-dfferece space ad tme.. Wth the cetral-dfferece space tme Yee s goal s the secod-order accurate cetral dfferecg O[ ] { O[ t ]} but the dfferece s ol stead of.. Yee chose he cetral-dfferece spacetme because we ca terleave ad H compoets the space lattce at tervals of t. Wth Yee algorthm we ca re-wrte 3-D Mawell s equatos. 5

53 53 quato become t H H equato 3 become t H H equato 4 become t H H

54 54 quato 5 become H H H H t equato 6 become H H H H t quato 7 become

55 55 H H H H t

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58 The followg llustrato shows how we appl FDTD to real structure.. Frst we costruct the structure we wat to smulate. The materal propertes are assged to correspoded locatos. For eample refractve de r [.e. delectrc fucto [ r ]. After settg-up the r detectors the desged structure to collect the sgal we ca lauch the tal sources ad let them 58

59 propagatg followg the FDTD algorthm.. The detectors wll collect the sgal vared wth tme durg the FDTD calculato process. Fgure shows a eample of the collected test versus tme from a detector. The mpulse tal ectato s usuall appled the smulato. Ad the collected sgal goes stable or sa coverge after amout of smulato tmeor smulato steps. 3. After the smulato fshg the sgal s collected sa It b the detectors. The fourer trasform wll be appled to trasfer tme-doma spectrum to frequec-doma spectrum. Sce tme t ad frequec are the cougates the Fourer trasformato. Fgure 3 s a frequec-doma spectrum whch shows the domat frequeces durg the FDTD propagato the smulated structure. 4. Not ol frequec-doma formato wll be obtaed from the FDTD smulato we also ca have more detals about ths structure le mode profles 59

60 see attachmets spectrum wavelegth ad etc. for eample fgure 4 s a bad dagram or bad structure of the smulated structure from the frequec-doma spectrum ad the tal propagato of the sources. 60

61 FDTD for D4 cavt Slab de =3.67 Hole de=.0 *From Y.C. Yag. 6

62 Qualt factor Q ad photo lfetme Qualt factor Q s a mportat parameter for a cavt. It mal descrbes the photo lfe tme sde the cavt or the eerg stored the cavt. There are several deftos or descrptos for the qualt factor of a cavt. eerg stored the sstem at resoace Q eerg lost a ccle of oscllato or W T -dw dt W -dw dt 0 0 Wt W0 e 0t - Q W : Stored eerg : Resoace freq. We ca also have Q related to the photo lfetme τ p or p Q 0 Q 0 P 6

63 spectrum 3 I epereces we ca obta the Q value from the Q 0 0 I photoc crstals we use the photoc crstal defect structure serve as the resoat cavt for the desged defect modes. The qualt factor Q wll represet a measure of how ma oscllatos tae place sde the cavt before the ected photo eerg dsspate out of the cavt. It meas that the photo eerg for arrow frequec bads be trapped the cavt for loger perod of tme f we have a hgh-q cavt. Now the mportat questo becomes how the defect cavt should be desged for troducg the hgh-q modes to the structure. There are several eamples of the hgh-q photoc crstal cavtes the followg pages. 63