Ring Resonators for Integrated Optics Applications

Transcription

1 Ring Rsnatrs fr Intgratd Optics Applicatins by Michal Gad A thsis prsntd t th Univrsity f Watrl in fulfillmnt f th thsis rquirmnt fr th dgr f Dctr f Philsphy in Physics Watrl, Ontari, Canada, 0 Michal Gad 0

2 AUTHOR'S DECLARATION I hrby dclar that I am th sl authr f this thsis. This is a tru cpy f th thsis, including any rquird final rvisins, as accptd by my xaminrs. I undrstand that my thsis may b mad lctrnically availabl t th public. ii

3 Abstract Intgratd ring rsnatrs hav attractd a cnsidrabl intrst in ptical cmmunicatins bcaus f thir small siz and wid rang f applicability. Hr w cnsidr svral aspcts f ths dvics, bginning with a tunabl hybrid ring rsnatrs cnsisting f a silicn vr insulatr (SOI) ring cvrd with a plymr layr in a variabl lctric fild. Varying th fild changs th plymr rfractiv indx and cnsquntly th rsnanc cnditin f th cavity. This dvic ffrs a larg dgr f ptical cnfinmnt tgthr with a high mdulatin spd. Subsquntly, w dsign and prsnt fabricatin rsults fr a Wavlngth Divisin Multiplxing (WDM) multiplxr/dmultiplxr frmd frm a sris f ring rsnatrs with tw channls sparatd by50ghz ach that is prdictd t xhibit a fr spctral rang (FSR) f 00 GHz, signal disprsin lss than 30 ps / nm and a signal crss-talk lss than 3dB. Finally, w analyz th applicatin f th cupld ring wavguid circuit t rtatin snsrs basd n th Sagnac phas shift. Hr, hwvr ur analysis indicats that a singl ring, f th sam ara xhibits a highr dgr f snsitivity t rtatinal mtin than a multipl ring circuit. iii

4 Acknwldgmnts It was a grat pprtunity fr m t wrk undr th suprvisin f Prf. David Yvick, wh is always availabl, knwldgabl and supprtiv. H is always pn t any qustins and is abl f giving gd guidanc. With an xprt y, Prf. Yvick culd s what was usually hiddn t m, which was ssntial t gt t crativ idas. I wuld lik als t acknwldg th imprtant rl f Prf. Paul Jssp frm McMastr Univrsity spcially in th dvic masurmnt part that tk plac at his lab facilitis. Th fabricatin part was dn in cllabratin with CMC, IMEC and UBC. This wrk was supprtd by th Canadian Institut fr Phtnics Innvatins (CIPI). iv

5 Ddicatin I wuld lik t ddicat this ffrt t th sul f my fathr, th mmbrs f my supprtiv family, my mthr, Mrit, Jsph, Danil, Mina and Mna and als my frinds in Canada wh hlpd m a lt thrugh this trip, Albrt Wasif, Ayad Fkry, Grg Bassm, Grg Bskals, Grg Shakr, Grg Sliman, Michal Naim, Michl El-Naggar, Hany Lwis, Hany Samul, Jhn Saad, Mina Farid, Mina Salb and Salam Gabran. v

6 Tabl f Cntnts AUTHOR'S DECLARATION... ii Abstract... iii Acknwldgmnts... iv Ddicatin... v Tabl f Cntnts... vi List f Figurs... viii List f Tabls... xiii Chaptr Intrductin.... Elctrmagntic Backgrund..... Light prpagatin in a tw dimnsinal straight wavguid..... Effctiv indx mthd (EIM) Light prpagatin in ring rsnatrs Bam prpagatin mthd (BPM) Fild cupling and bnding lss calculatins Pwr cupling Prpagatin lss factr Th basic ring rsnatr circuit....4 Cmplx RR circuit analysis Transfr matrix mthd Cupling f mds in tim (CMT) Cnclusin... 9 Chaptr Tunabl hybrid ring rsnatrs Hybrid structur dsign Opratin and dimnsins Plymr thicknss Hybrid structur applicatin Pwr transmissin tuning Ring-bus cupling variatin vi

7 .3 Cnclusin Chaptr 3 Cmpund ring rsnatrs Transfr matrix apprach WDM cmpund ring rsnatr structur intrlavr circuit Cmparisn f intrlavrs CMT analysis CMS circuit paramtrs CMT circuit paramtrs Numrical rsults Dsign, fabricatin and charactrizatin Pst fabricatin study Cnclusin... 8 Chaptr 4 High snsitivity ring rsnatr Gyrscps Ovrviw Circuit analysis Sagnac ffct Crw gyrscp Lp f ring gyrscp Summary f prvius CROW and FOG rsults Numrical rsults: Cnclusin Chaptr 5 Cnclusin and futur wrk... 0 Rfrncs vii

8 List f Figurs FIGURE.: A D WAVEGUIDE WITH CORE WIDTH w, CORE RI n, CLADDING RI n AND SUBSTRATE RI n 3... FIGURE.: THREE DIMENSIONAL WAVEGUIDE... 6 FIGURE.3: (A) THE D WAVEGUIDES REPRESENTING THE (A) FIRST AND (B) SECOND STEP OF THE EIM... 7 FIGURE.4: WG STRUCTURE. DIFFERENT MATERIAL REGIONS ARE COLORED DIFFERENTLY... 8 FIGURE.5: WG STRUCTURE FIGURE.6: B-V DIAGRAM FOR THE REDUCED WG AS IN THE FIRST STEP OF THE EIM. SOLID (DOTTED) LINE CURVES REPRESENT TE (TM) MODES FIGURE.7: B-V DIAGRAM FOR THE REDUCED WG AS IN THE SECOND STEP OF THE EIM. SOLID (DOTTED) LINE CURVES REPRESENT TE (TM) MODES FIGURE.8: A RING RESONATOR WAVEGUIDE STRUCTURE... FIGURE.9: THE TE-LIKE MODE FOR WG WITH THE CHARACTERISTICS IN TABLE FIGURE.0: THE TE-LIKE MODE FOR WG WITH THE CHARACTERISTICS IN TABLE FIGURE.: TWO WAVEGUIDES WGI AND WGII, IN CLOSE PROXIMITY FIGURE.: COUPLING BETWEEN TWO STRAIGHT WG WAVEGUIDES. THE POWER PROFILE IS GIVEN BY THE LEFT PLOT WHILE THE POWER LEVEL VERSUS PROPAGATION DIRECTION IS SHOWN AS THE BLUE AND GREEN LINES FIGURE.3: POWER COUPLING BETWEEN A STRAIGHT AND A CURVED WAVEGUIDE FOR (A) TE- LIKE AND (B) TM-LIKE MODES FIGURE.4: NEGLIGIBLE BENDING LOSSES FOR BENDS WITH viii R 5 m FOR (A) TE-LIKE FIELDS IN WG, (B) TM-LIKE FIELDS IN WG, (C) TE-LIKE FIELDS IN WG AND (D) TM- LIKE FIELDS IN WG... FIGURE.5: SINGLE RR CIRCUIT.... FIGURE.6: A RING RESONATOR COUPLED TO TWO BUSES FIGURE.7: SERIES COUPLED RRS... 6 FIGURE.8: PARALLEL COUPLED RRS... 6 FIGURE.9: A LOOP OF RINGS COUPLED TO TWO BUSES.... 7

9 FIGURE.0: A SINGLE RR COUPLED TO TWO BUSES FIGURE.: A SIMPLE SINGLE RING RESONATOR CIRCUIT FIGURE.: A VERTICAL CROSS SECTION OF THE WAVEGUIDE STRUCTURE. THE COLOR MARK TO THE RIGHT DEFINES THE LAYER MATERIAL FIGURE.3: THE ABSORPTION LOSS INDUCED BY THE GOLD LAYER AS A FUNCTION OF POLYMER THICKNESS. THE INSET SHOWS THE LOSSES FOR t 0.8 m FIGURE.4: MODE REFRACTIVE INDEX ( n ) AS A FUNCTION OF THE FREE SPACE WAVELENGTH ( ) FOR DIFFERENT POLYMER RI ( n ) VALUES. (A) TE MODE WITH MODE WITH ix t 0.8 m (B) TM t m FIGURE.5: (A) VARIATION OF THE R n AND n WITH V (B) THE DEPENDENCE OF R REQUIRED ON V FOR FULL ON-OFF SWITCHING FIGURE.6. THE VARIATION OF THE FIELD TRANSFER COEFFICIENT r AND THE FIELD LOSS COEFFICIENT WITH MEAN RING RADIUS R R AND THE RING-BUS GAP g FIGURE.7: THE DEVICE NORMALIZED POWER TRANSMISSION (A) FIRST DESIGN POINT, (B) SECOND DESIGN POINT FIGURE.8: RACETRACK-BUS CONFIGURATION. THE SILICA LAYER SURROUNDING THE SILICON CORE IS NOT SHOWN FIGURE.9: VARIATION OF THE POWER TRANSMISSION FACTOR (T ) ON RESONANCE WITH THE FIELD COUPLING RATIO ( k ) FOR DIFFERENT ROUND-TRIP LOSSES... 4 FIGURE.0. W n ) (SOLID LINE) AND Z (m) (DASHED LINE) FOR L n /, (A) n, (B) ( n, (C) n 3, (D) n 4, (E) n 5, (F) n 6, (G) n 7, (H) n FIGURE.: W n ) (SOLID LINE) AND Z (m) (DASHED LINE) FOR (A) L 40.05m, (B) ( L 80m FIGURE 3.: THE COMPOUND RING RESONATOR CIRCUIT WITH (A) N 4, (B) 6 N RING RESONATORS FIGURE 3.: THE CIRCUIT RESPONSE WITH k k AND k INCREASING FROM. 5 0 TO 0.6 IN STEPS OF THE ARROWS INDICATE INCREASING PARAMETER VALUES. THE

10 ROUND TRIP POWER LOSS IS 0 %. (A) THE POWER SPECTRA. (B) THE PHASE VARIATION. (C) THE NORMALIZED GROUP DELAY. (D) THE THROUGH PORT DISPERSION. (E) THE DROP PORT DISPERSION. (F) THE THROUGH PORT POLE-ZERO DIAGRAM. (G) THE DROP PORT POLE- ZERO DIAGRAM FIGURE 3.3: THE CIRCUIT RESPONSE WITH k AND k k INCREASING FROM TO IN STEPS OF THE ARROWS INDICATE INCREASING PARAMETER VALUES. THE ROUND TRIP POWER LOSS IS 0 %. (A) THE POWER SPECTRA. (B) THE PHASE VARIATION. (C) THE NORMALIZED GROUP DELAY. (D) THE THROUGH PORT DISPERSION. (E) THE DROP PORT DISPERSION. (F) THE THROUGH PORT POLE-ZERO DIAGRAM. (G) THE DROP PORT POLE- ZERO DIAGRAM FIGURE 3.4: THE CIRCUIT RESPONSE WITH k AND k k FOR A ROUND TRIP POWER LOSS OF 0 %. (A) THE POWER SPECTRA. (B) THE PHASE VARIATION. (C) THE NORMALIZED GROUP DELAY. (D) THE THROUGH PORT DISPERSION. (E) THE DROP PORT DISPERSION. (F) THE THROUGH PORT POLE-ZERO DIAGRAM. (G) THE DROP PORT POLE-ZERO DIAGRAM FIGURE 3.5: A COMPOUND FOUR RING CIRCUIT ATTACHED TO A SINGLE RING STAGE FIGURE 3.6: THE SINGLE RING STAGE RESPONSE WITH k AND 0 k FOR A ROUND TRIP POWER LOSS OF 0 %. (A) THE POWER SPECTRA. (B) THE PHASE VARIATION. (C) THE NORMALIZED GROUP DELAY. (D) THE THROUGH PORT DISPERSION. (E) THE THROUGH PORT POLE-ZERO DIAGRAM FIGURE 3.7: THE DROP PORT RESPONSE WITH AN ADDITIONAL SINGLE RING STAGE FOR A ROUND TRIP POWER LOSS EQUAL TO 0 %. (A) THE POWER SPECTRA. (B) THE PHASE VARIATION. (C) THE NORMALIZED GROUP DELAY. (D) THE DROP PORT DISPERSION. (E) THE DROP PORT POLE-ZERO DIAGRAM FIGURE 3.8: THE CMT MODEL OF THE COMPOUND RING RESONATOR CIRCUIT FIGURE 3.9: (A) THE THROUGH PORT AND (B) THE DROP PORT TRANSMISSION CHARACTERISTICS FOR A LOSSLESS CIRCUIT, - - BY THE CMT MODEL (RED LINE), - BY THE CMS MODEL AND O BY THE FDTD MODEL. THE SMALL SHIFT OF RESULTS BY THE CMS x

11 AND THE CMT MODELS IS SHOWN IN THE INSET. THE RESONANCE WAVELENGTH CORRESPONDS TO m FIGURE 3.0: AS IN FIGURE (3.9) BUT FOR (A) THE THROUGH PORT AND (B) THE DROP PORT TRANSMISSION CHARACTERISTICS FOR A CIRCUIT WITH 5 % POWER LOSS PER ROUND TRIP FIGURE 3.: AS IN FIGURE 3.9 BUT FOR (A) THE THROUGH PORT AND (B) THE DROP PORT TRANSMISSION CHARACTERISTICS FOR A CIRCUIT WITH 0 % POWER LOSS PER ROUND TRIP FIGURE 3.: THE SINGLE-MODE SOI WAVEGUIDE CROSS-SECTION FIGURE 3.3: A SCHEMATIC OF THE PROPOSED CIRCUIT WITH RINGS REPLACED WITH RACETRACKS FIGURE 3.4: THE CHARACTERIZATION SETUP WITH DIFFERENT PARTS LABELED. POSITIONER () HOLDS THE INPUT FIBER; WHILE POSITIONER () HOLDS THE OUTPUT FIBER AND POSITIONER (3) HOLDS THE CHIP UNDER TEST FIGURE 3.5: (A) THE LAYOUT OF THE SIX COPIES OF THE PROPOSED CIRCUIT IN SECTION 3.. THE TWO DEVICES ON THE RIGHT MOST (, ), TWO IN THE MIDDLE (3, 4) AND TWO ON THE LEFT MOST (5, 6) OF THE CHIP CORRESPOND TO THE DIMENSIONS ON THE ST, ND AND 3 RD ENTRIES IN TABLE 3.3 RESPECTIVELY, (B) THE LAYOUT OF DEVICE () SHOWING THE INPUT, THROUGH AND DROP PORTS. ON THE RIGHT IS THE TAPERED WAVEGUIDES FOLLOWED BY THE GRATING COUPLERS FIGURE 3.6: SEM PICTURES FOR THE FABRICATED (A) DEVICE () WITH g 0.m, (B) DEVICE (3) WITH g 0.3m AND (C) DEVICE (5) WITH g 0.4m FIGURE 3.7: AN OPTICAL PHOTO FOR SOME THE FABRICATED CIRCUITS FIGURE 3.8: MEASURED AND THEORETICAL TRANSMISSION CHARACTERISTICS OF (A) THE THOUGH PORT OF DEVICE (A), (B) THE DROP PORT OF DEVICE (A), (C) THE THROUGH PORT OF DEVICE (B) AND (D) THE DROP PORT OF DEVICE (B) FIGURE 3.9: MEASURED AND THEORETICAL TRANSMISSION CHARACTERISTICS AS IN FIGURE 3.8 BUT WITH MODIFIED FIELD COUPLING COEFFICIENTS xi

12 FIGURE 4.: (A) A RING ROTATING ABOUT A CENTER OF ROTATION AT A DISTANCE R FROM ITS CENTER, (B) A FIBER OPTIC GYROSCOPE (FOG) AND (C) A SINGLE RING GYROSCOPE FIGURE 4.: CROW GYROSCOPE WITH N FIGURE 4.3: CLR GYROSCOPE WITH N FIGURE 4.4: A CROW CIRCUIT PERFORMANCE WITH.55 m (A) THE NORMALIZED OUTPUT POWER AT PORT B AS A FUNCTION OF ROTATIONAL SPEED ( ) FOR A CROW WITH R 5 m, 0 AND k k k 0., (B) THE RELATIVE SENSITIVITY AS A FUNCTION OF THE ROTATIONAL SPEED FOR THE SAME CROW WITH R 5 m, 0, AND k 0. k k, (C) THE CROW SENSITIVITY AS A FUNCTION OF THE POWER COUPLING COEFFICIENT ( ) WITH N 9, 000Hz 000 r / s, R 5m AND 0, (D) THE CROW SENSITIVITY AS A FUNCTION OF THE RING RADIUS ( R ) WITH N 9, 0, 000Hz 000 r / s AND k k k FIGURE 4.5: (A) THE NORMALIZED POWER TRANSMISSION OF A CROW GYROSCOPE WITH N 8, R 5cm, AND 0. db/ km AT. 55m. (B) THE SENSITIVITY OF THE CROW GYROSCOPE AND ITS EQUIVALENT FOG OF R f 45 cm, N 393 turns, f AND L f 093 m FIGURE 4.6: THE SENSITIVITY AS A FUNCTION OF N AND R WITH. 55m AND 0, (A) A CROW GYROSCOPE, (B) A CLR GYROSCOPE FIGURE 4.7: THE SENSITIVITY AS A FUNCTION OF N AND R WITH.55 m AND 0. db / km, (A) A CROW GYROSCOPE, (B) A CLR GYROSCOPE FIGURE 4.8: THE SENSITIVITY AS A FUNCTION OF N AND R WITH m AND 7 db / km, (A) A CROW GYROSCOPE, (B) A CLR GYROSCOPE FIGURE 4.9: THE SENSITIVITY OF THE EQUIVALENT FOG FOR (A) CROW GYROSCOPES AND (B) FOR CLR GYROSCOPES, WITH.55 m AND 0. db/ km FIGURE 4.0: THE SENSITIVITY OF THE EQUIVALENT SINGLE RING GYROSCOPE FOR (A) CROW GYROSCOPES AND (B) FOR CLR GYROSCOPES, WITH.55 m AND 0. db / km. 99 xii

13 List f Tabls TABLE. : WG STRUCTURE TABLE. : WG STRUCTURE TABLE.; WAVEGUIDE REFRACTIVE INDICES TABLE.: WAVEGUIDE CROSS SECTION DIMENSIONS TABLE.3: COMPARISON OF FOUR TUNABLE RING RESONATOR CIRCUITS WHERE OUR SECOND DESIGN' PARAMETERS ARE SHOWN BETWEEN BRACKETS TABLE.4: CIRCUIT PARAMETERS FOR L m TABLE.5: CIRCUIT PARAMETERS FOR L 80 m TABLE 3.: THE PERFORMANCE OF THE OPTIMAL DESIGN IN [ 43] COMPARED TO THE COMPOUND RR CIRCUIT PERFORMANCE TABLE 3.: SINGLE MODE SOI WAVEGUIDE PARAMETERS TABLE 3.3: DIMENSIONS OF THE RACETRACKS IN m WITH L xiii

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15 Chaptr Intrductin Intgratd ptics is incrasingly mplyd in miniaturizing cmpnnts that prfrm fundamntal prcssing functins such as gnrating, dtcting, filtring, amplifying, ruting and multiplxing f signals in tlcmmunicatins and data prcssing systms. Whil standard intgratd ptics cmpnnts such as arrayd wavguids and multi-md intrfrnc (MMI) dvics ar wll stablishd in dvic applicatins, th ring rsnatr (RR) is bcming incrasingly cmptitiv in altrnativ dsigns bcaus f its small siz and functinality. Th bjctiv f this thsis is t prps nw RR basd circuits that ar ptimizd fr high spd applicatins. Hr w build n prvius wrk that has includd th applicatin f ring rsnatrs t intgratd RR cavitis, [ 0] hav bn mplyd in numrus cntxts such as plarizatin cnvrtrs [ ], filtrs [ 3] ptical dlay lins [ 4], dmultiplxrs [ 5], rflctrs [ 6], rtatinal mtin dtctrs [ 7-8] and lgic circuits [ 9]. Fabricatin platfrms fr RR dvics includ silicn vr insulatr (SOI) [ 0], plymrs [ - 3] and grup III-V smicnductrs [ 4-5]. Accrdingly, in th first chaptr f this thsis w vrviw th wavguid thry and thn dmnstrat th mannr in which RR dvic paramtrs ar calculatd in rdr t stablish th basic principls f th RR circuit pratin. In th nxt chaptr w prps a nvl hybrid ring rsnatr structur that mplys th bst faturs f tw wll stablishd platfrms, namly, plymr and SOI t btain a high tuning spd whil prsrving cmpatibility with CMOS tchnlgy. In Chaptr 3, w study a ring rsnatr (RR) structur with intrnal fdback that labl a cmpund ring rsnatr structur and dmnstrat th distinguishd faturs f this structur by incrprating it int a standard wavlngth divisin multiplxing (WDM) intrlavr/dintrlavr circuit. W thn analyz this circuit with thr numrical apprachs, th cupling f mds in spac (CMS), th cupling f mds in tim (CMT) and th finit diffrnc tim dmain (FDTD) simulatins. W furthr dsign th circuit layut and fabricat cpis f th dvic fr tsting. Th masurmnts agr wll with thrtical calculatins xcpt fr minr dviatins that prsumably culd b liminatd with a mr ptimizd dsign. In Chaptr 4 w apply ur RR dsign t gyrscps fr rtatinal mtin dtctin and cmpar th prfrmanc t prviusly prpsd RR basd gyrscps. W cnfirmd as thr authrs hav ntd prviusly that a singl rsnant ring prvids a highr snsitivity than any mr cmplx dsign prpsd t dat. W finally cnclud ur wrk and discuss pssibl futur avnus fr xplratin.. Elctrmagntic Backgrund T bgin, w discuss lctrmagntic fild prpagatin in a D wavguid [ 6] fllwd by th ffctiv indx mthd (EIM) [ 7-0], which is usful t rduc a 3D wavguid int D. Nxt, w discuss lctrmagntic fild prpagatin in a D ring wavguid [ ] and finally w vrviw th bam prpagatin mthd (BPM) that frms th basis fr th wavguid simulatr mplyd in much f th rmaindr f this thsis [ - 8].

16 .. Light prpagatin in a tw dimnsinal straight wavguid W nw valuat th cmplx prpagatin cnstant f th lctrmagntic mds in a wavguid, i /, whr is th pwr lss cfficint, n / is th md prpagatin cnstant, n is th md ffctiv rfractiv indx and is th fr spac wavlngth which is st t. 55m in this wrk unlss thrwis spcifid. In this sctin w furthr spcializ t lsslss wavguids with 0 and w fllw th tratmnt in [ 6]. Cnsidr th D wavguid shwn in Figur., whr th wavguid cr cnsists f a dilctric layr lying in th y z plan within w/ x w/. Th layr prmittivity is n with bing th fr spac prmittivity and n th cr rfractiv indx (RI). Th suprstrat layr cvrs th tp f th cr in w/ x with n whil th substrat layr xtnds in th rgin x w/ with 3 n3. Typically, th RI valus f th thr layrs hav th rlatin n n3 and n n. Th wavguid is cnsidrd a D wavguid sinc all layrs ar assumd infinit in th y dirctin and cnsquntly fr a wav prpagating in th z dirctin th filds d nt vary in this y dirctin. Sinc mtallic bundaris ar nt mplyd, th wavguid is trmd an pn wavguid fr which th filds vanish at x. Th lctric fild intnsity, E, and th magntic fild intnsity, H f mdal filds with a i t mnchrmatic frquncy dpndnc givn by, whr c is th fild angular 0 / frquncy, ar givn by: it iz it iz it iz E( x, y, z) E ( x) x E ( x) y E ( x) z (..a) x y it iz it iz it iz H( x, y, z) H ( x) x H ( x) y H ( x) z (..b) x y z z Figur.: A D wavguid with cr width w, cr RI n, cladding RI n and substrat RI n 3

17 Maxwll s quatins in a nn-magntic dilctric matrial ar: H E i H t (..a) E H ie t (..b) whr is th fr spac prmability. Fr a mdal fild, 0 and y z i s that: H x E y (.3.a) H z i E y x (.3.b) ih x H x z i j E y (.3.c) and E x H y (.4.a) j E z i j H y x (.4.b) ie x E x z i H y (.4.c) j whr j n rprsnts th layr j. Th first st rprsnts th transvrs lctric (TE) fild md whr th lctric fild lis in th y -dirctin and th tw magntic fild cmpnnts can b drivd frm E. Similarly, th scnd st f quatins rprsnts th transvrs magntic (TM) y md fr which th magntic fild is instad in th y-dirctin and th tw lctric fild cmpnnts can b cmputd frm H. In ithr cas th scalar wav quatin hlds s that: y 3

18 E y x ( k j ) E 0 (.5.a) y and H x y ( k j ) H 0 (.5.b) y rspctivly, whr k n k, j, and 3, k j j is th fr spac prpagatin cnstant, c and c is th vacuum spd f light. Fr a guidd md, k k3 k r quivalntly n n n and cnsquntly w xpct th slutin t th TE and TM quatins t tak th 3 n frm: E y kx ( x A A cs( k kx3( x A3 w/ ) x x) B w/ ) : : sin( k x x) : w/ x w/ x w/ x w/ (.6.a) and H y kx ( x A5 A4cs( k kx3 ( x A6 w/ ) x x) B w/ ) : : sin( k x x) : w/ x w/ x w/ x w/ (.6.b) rspctivly, whr k k x, k x k and k x 3 k3 ar th transvrs prpagatin cnstants in th rgins j, and 3 rspctivly. Als, A 6, B and B ar cnstant valus t b dtrmind thrugh th bundary cnditin and pwr nrmalizatin. Th bundary cnditins applid at x w/ rsult frm th cntinuity f th tangntial lctric and magntic fild cmpnnts, namly E y and H z fr th TE md and H y and Ez fr th TM md. This lads t th disprsin rlatin: v b b b m tan ( q3 ) tan ( q ) (.7) b b whr 4

19 q : fr thtemd j n / n j : fr thtmmd (.8.a) 3 n n (.8.b) n max n, n 3 n, n3 n, n n max b (.8.c) n max 3 n n w v k n max, 3 (.8.d) n n In th prsnt cas max n, 3 3 sinc n3 n. Th paramtrs b and v ar calld th nrmalizd prpagatin cnstant and th nrmalizd frquncy rspctivly whil m is an intgr trmd th md rdr such that m 0 dfins th fundamntal md with th largst, whil masurs th asymmtry f th cladding rfractiv indics with 0 fr a symmtric wavguid, i.. n3 n. Th disprsin rlatin can thn b slvd graphically r numrically. Whil in th D prblm th mds ar TE and/r TM, in 3D rctangular wavguids th mds ar hybrid. Whil apprximat analytic mthds can b applid, th simplifid ffctiv indx mthd (EIM) dtaild in th subsqunt sctin [ 7-0] yilds gd numrical stimats f th prpagatin cnstants fr many 3D wavguid structurs... Effctiv indx mthd (EIM) W cnsidr first th 3D rctangular wavguid f Figur. with cr dimnsins labld w t and RI n which divids th surrunding spac int 8 rgins with rfractiv indics n whr j,3,.., 9. Th substrat nrmally cmpriss th thr rgins givn by j 3,4, 9. j Whil pur TE r TM mds d nt xist, th mds ar ithr TE-lik mds with a dminant lctric fild cmpnnt E paralll t th substrat r TM-lik mds with E nrmal t th substrat [ 6]. x y 5

20 Figur.: Thr dimnsinal wavguid Th EIM calculatin f n f a TE-lik md [ 7-0] first xtnds th j,, 3 rgins t infinity in th x -dirctin as shwn in Figur.3.a and dtrmins th ffctiv RI f this structur n frm th disprsin rlatin f quatin (.7). Rcall that th TE-lik md f th 3D wavguid in Figur. has its dminant lctric fild paralll t th substrat, i.. in th x -dirctin mtivating th chic f th TE md fr th D wavguid calculatin. Nxt, th rgins givn by j,5, 8, ar xtndd t infinity in th y -dirctin whil nis rplacd with n s that th D wavguid f Figur.3.b is btaind. Th ffctiv rfractiv indx f th TE-lik md f th thr-dimnsinal structur is thn apprximatd as n f this D wavguid TM md, sinc th TM md in this stp crrspnds t th plarizatin f th TE-lik 3D md. 6

21 (a) (b) Figur.3: (a) Th D wavguids rprsnting th (a) first and (b) scnd stp f th EIM Th D wavguid shwn in Figur.3.b. can als b mplyd t calculat prprtis such as th cupling rati r th bnding lss. Th stps f th EIM ar slightly diffrnt fr thr 3D wavguid structurs such as th rib wavguid, c.f. [ 9-0]. Th tw 3D singl-md rctangular wavguids mplyd in this thsis ar labld WG and WG and ar dfind in Tabl. and Tabl. and Figur.4 and Figur.5 rspctivly. Pwr lss cfficints appar in ths tabls crrspnd t scattring lss, as will b discussd latr, sinc this is th main pwr lss mchanism in ths wavguids. Prprty Cr width ( w ) Cr hight ( t ) Valu 0.3 m [ 9] 0.3 m [ 9] Silicn cr RI ( n ) [ 9] Burid Silica RI ( n3 n4 n5 n8 n9 ). 444 [ 9] Plymr (CLD/APC) layr RI ( n ). 6 [ ] Air RI ( n6 n7 ) Scattring pwr lss cfficint ( ) 6 db/ cm[ 30] Tabl. : WG structur. 7

22 Prprty Cr width ( w ) Cr hight ( t ) Valu 0.5 m 0. m Silicn cr RI ( n ) [ 9] Burid Silica RI ( n3 n4 n9 ). 444 [ 9] Air RI ( n n5 n6 n7 n8 ) Scattring pwr lss cfficint ( ) Tabl. : WG structur..4 db/ cm [ 3-3] Figur.4: WG structur. Diffrnt matrial rgins ar clrd diffrntly 8

23 Figur.5: WG structur. As a numrical xampl f th EIM, w calculat n fr th TE-lik md f WG. In th first stp, th 3D structur is rducd t D as in Figur.3.a with th wavguid asymmtry n n3 cfficint 0.054, thicknss t 0.3m and nrmalizd frquncy n max n, n 3 n, n.87 w v k n max 3. Th crrspnding b-v diagram is shwn in Figur.6 whr th vrtical lin marks th valu f v. Th crrspnding valu f th nrmalizd prpagatin cnstant is givn by th intrsctin f th vrtical lin and th TE curv f th fundamntal md n maxn, n3 with m 0, yilding b and n.97. Nxt w cmput th TM n max n, n 3 md f th D wavguid shwn in Figur.3.b, with a cr RI n.965 and th tw cladding layrs with n n Th cr thicknss is w 0.3m. Th crrspnding b-v diagram is 5 8 shwn in Figur.7. Fr this symmtric D wavguid, 0 and v Similarly, b is givn by th intrsctin f th vrtical lin v.5747 and th TM md curv with m 0 yilding finally n.8. 9

24 Figur.6: b-v diagram fr th rducd WG as in th first stp f th EIM. Slid (dttd) lin curvs rprsnt TE (TM) mds. Figur.7: b-v diagram fr th rducd WG as in th scnd stp f th EIM. Slid (dttd) lin curvs rprsnt TE (TM) mds. 0

25 Fr WG, th n f th TE-lik md, is btaind first frm n f th TE md fr th fild in th cnfiguratin f Figur.3.a with RI valus frm Tabl. and t 0. m. This yilds 0.087, v.4089, b 0.556and hnc n.76. In th scnd stp w slv fr th TM md f th cnfiguratin in Figur.3.b mplying th valu f n frm th first stp, n n and w 0.5 m t btain 0, v.606, b and n Light prpagatin in ring rsnatrs W nxt slv Maxwll s quatins fr th ring rsnatr f Figur.8 with man radius R, cr width w and rfractiv indx (RI) n, surrundd by cladding layrs f RI valus n in 0 r R w/ and n 3 in r R w/ with r bing th radial distanc frm th ring cntr. Th structur is symmtric abut th y axis which is nrmal t th pag utward. W cnsidr hr th D prblm s that th rfractiv indx is invariant in bth y and. Figur.8: A ring rsnatr wavguid structur Mdal lctric and magntic fild intnsitis can b rspctivly xprssd as [ ]: E it ir ( r,, y) ( Er, E, Ey) (.9.a) H it ir ( r,, y) ( Hr, H, H y) (.9.b) whr i / is th cmplx prpagatin cnstant f th md with th ral part and th pwr lss cfficint. Substituting (.9) int (..a) yilds tw sparat sts f quatins:

26 r E y H r R (.0.a) i H r E y (.0. b) y j r E in H r R i r rh r (.0. c) and r j y E n H r R (..a) E in r H j y (..b) y r H i E r R i r re r (..c) whr has bn rplacd by j n with is th fr spac prmittivity and j n is th RI f layr j. Again fr a TE md th tw magntic fild cmpnnts can b xprssd in trms f th singl lctric fild cmpnnt y E whil fr a TM md th tw lctric fild cmpnnts ar givn in trms f y H. Fr a TE md, in ach f th thr rgins with cnstant RI, th lctric fild bys th Hlmhltz wav quatin: 0 ) ) ( ( y j E r R k n r r r (.) Apprpriat slutins t this quatin in th thr rgins ar givn by: / ) : ( / / ) : ( ) ( / 0 ): ( 3 () 3 r w R r n k H C w R r w R r n k D Y r n k C J w R r r n k J C E y (.3)

27 sinc th fild is vrywhr finit and dcays t zr as r. Th cnstantsc, C, D andc 3 ar dtrmind frm th fild nrmalizatin and th bundary cnditins at th intrfacs, r R w/ and r R w/, btaind frm th cntinuity f E y and H. Ths yild th disprsin rlatin [ ]: J ( kn ( R w/ )) q J ( kn ( R w/ )) Y ( kn ( R w/ )) q J ( k n ( R w/ )) J '( kn ( R w/ )) J '( kn ( R w/ )) Y '( kn ( R w/ )) J '( k n ( R w/ )) J ( kn ( R w/ )) J '( kn ( R w/ )) q () 3 () H ( k n ( R w/ )) H '( k n ( R w/ )) Y ( kn ( R w/ )) q () H ( k n ( R w/ )) Y '( k n ( R w/ )) H () '( k n ( R w/ )) 3 3 (.4) with with q n / n. Fr th TM md, H y and j j q j rplacd by q n j / n E ar cntinuus rsulting in th disprsin rlatin j. In bth cass / R and thrfr th mdal prpagatin cnstant is btaind numrically. Altrnativly, cnfrmal mapping [ 33] rplacs th bnt wavguid with an quivalnt gradd indx straight wavguid t which mthds such as th Wntzl-Kramrs- Brilluin (WKB) tchniqu [ 33-34] can b applid. Ths ar hwvr nt trivial fr bnt wavguids and spcially fr 3D wavguids which gnrally rquir th EIM fr a simplifid analysis. Hwvr, ring rsnatrs with R w, can b gnrally rplacd by a straight wavguid with th sam width and RI valus fr cr and cladding layrs [ 33-34], whil bnding cntributs ngligibly t cmpard t surfac scattring fr mst silicn vr insulatr (SOI) wavguids...4 Bam prpagatin mthd (BPM) Th bam prpagatin mthd (BPM) [ - 8] is a n-way marching algrithm fr th Hlmhltz scalar wav quatin fr small RI cntrast wavguids, which in 3D Cartsian crdinats taks th frm: ( x n If z is th prpagatin dirctin, fr paraxial filds, w can writ y z 3 k ) E 0 (.5) E E ~ ( x, y, z) itikz k k n whr n is nrmally chsn btwn th cr and th cladding RI valus [ 8]. Applying th slwly varying fild nvlp apprximatin, fr which ~ E ( z ik x y k E z ( n E yilds: z n ~ )) E with (.6) An implicit finit-diffrnc (FD) mthd basd n th wll-knwn Crank-Nichlsn schm [ 4], [ 6-7] can nw b applid as in th cmmrcial BamPROP cd mplyd latr in this thsis. Hr

28 th fild is discrtizd n a st f grid pints alng th transvrs plan. Frm th discrtizd fild at th wavguid ntranc, th fild prpagatin is numrically valuatd at z n z. Whil th abv quatin prtains t th prpagatin f scalar filds with small bam divrgncs, mr cmplicatd frmulatin applis t plarizd fild [ 35-36] and nn-cllimatd prpagatin [ 5], [ 37]. Sinc hwvr th mthd cannt b applid t tw-way prpagatin as in a ring rsnatr, ur calculatins ar applid t straight wavguids r ffctiv straight wavguid prfils fr which w calculat th md prfil, th prpagatin cnstant and fild cupling btwn nighbring wavguids and prpagatin lsss. An arc f cntral angl 0 is mplyd t stimat th bnding lsss r xtrnal cupling strngths f an RR as discussd latr. Th BamPrp rsults fr th n f th TE-lik md f WG and WG yild th md prfils and nrmalizd ffctiv indics f Figur.9 and Figur.0 rspctivly. Th nrmalizd RI indics diffr by apprximatly 0. 05fr WG and 0. fr WG btwn th EIM and th BPM. Sinc this yilds cnsidrabl rrrs fr quantitis such as th cupling lsss, 3D simulatins ar gnrally mplyd blw. Figur.9: Th TE-lik md fr WG with th charactristics in Tabl. 4.

29 Figur.0: Th TE-lik md fr WG with th charactristics in Tabl... Fild cupling and bnding lss calculatins W nw apply th mthds f th prvius sctin t valuat fild cupling and th prpagatin lsss... Pwr cupling Apprciabl pwr can b xchangd btwn tw wavguids that supprt mds with th sam r narly th sam prpagatin cnstants whn th mdal filds apprciably vrlap. Th fild cupling rati, k rprsnts th rati f th cupld filds btwn th tw wavguids and can b valuatd by th cupld md thry [ 3], [ 6], [ 38-40]. Hr w cnsidr tw wavguids, (WGI) and (WGII) in cls prximity as shwn in Figur.. 5

30 Figur.: Tw wavguids WGI and WGII, in cls prximity. Hr th cr RI, width and man radius f curvatur f WG j ar givn by R j rspctivly with n j, W j and j I, II whil th surrunding mdia hav an RI f niii and Th lctric fild cmpnnts arund th intractin rgin ar givn by a, b, c and d. If th wavguid cntr t cntr distanc is s ( z), th rati f th fild cupld frm WGII int WGI is thn [ 3], [ 38]: k sin( intractin rgin jz (s( z)) dz) (.7) whr k n n ) is th diffrnc f prpagatin cnstants f th wavguids I II ( I II and is th cupling cfficint givn by th vrlap intgral [ 3], [ 38]: whr E I (x) and (x) E II w / I ( s( z)) ( ni niii ) EI ( x) EII *( x) dx (.8) 4 wi / ar th lctric filds f th mds f prpagatin f WGI and WGII rspctivly. Fr th TE md dscribd by quatin (.6.a) and th crrspnding bundary cnditins w arriv at: 6

31 ( ni niii s( z)) ( k k ( xi xiv ( k xiv cs( k xi I w / )sinh( k xiv )cs( k I )( w xii w / ) k I w / ) / / k xi II sin( k k xiv( wii / s) xiii xi ) w / ) csh( k I xiv w I / )) (.9) whr j k j xi k I I, k k n k, j I, II and III. xii k II II, k xiii I III k, k xiv II III k, and Sinc substituting quatin (.9) int (.7) yilds a cmplicatd intgral [ 38], w fcus hr n tw simplr cass. Fr tw straight wavguids intracting vr a lngth L, th intgratin dmain in quatin (.7) is L/ z L/ whil s( z) is rplacd by th cnstant s in quatin (.9). Accrdingly, (s ) Lsin( X ) k sin( ) (.0) X with X L / (.) If th tw wavguids ar symmtric, 0 and quatin (.0) rducs t: k sin( L) (.) Thn w dfin th cupling distanc /( ) crrspnds t th intractin lngth fr cmplt pwr transfr frm n straight wavguid t th thr, i.. k 00%. Fr tw wavguids with idntical cr RI, i.. ni nii, th cntr t cntr sparatin can b apprximatd by s( z) s z /(R m ) with / R m / RI / RII [ 3], [ 38]. Th fild cupling rati is thn apprximatly [ 3]: k sin( (s ) l ) (.3) m whr l m R m nii niii (.4) Th EIM mthd can ftn b mplyd t rduc th prblm t D bfr applying th abv cupling frmula. Th ffctiv RI thn rplacs th cr RIs, ni r n II. 7

32 An altrnativ is t mply th BamPrp as shwn in Figur.. t calculat light cupling btwn tw paralll wavguids sparatd by s 0.7 m and xcitd by a TE plarizd Gaussian bam in th lft wavguid. Cmplt pwr transfr is bsrvd at th cupling distanc f 47.5 m, yilding /( ) 0.033/ m. Th maximum pwr n th mnitr plt is lss than 00% sinc th initial fild ds nt prfctly match th lft wavguid md. Figur.: Cupling btwn tw straight WG wavguids. Th pwr prfil is givn by th lft plt whil th pwr lvl vrsus prpagatin dirctin is shwn as th blu and grn lins. A similar prcdur can b mplyd t calculat th cupling btwn a bus and a ring. Whil th bam prpagatin mthd is difficult t apply t prpagatin alng a larg angular rgin, prpagating th fild vr a 0 accuratly rprsnts th intractin rgin in ur simulatin [ 9]. Launching light int th straight wavguid and mnitring th rmaining pwr in th straight wavguid aftr th intractin yilds th rsults f Figur.3 fr which th straight and curvd wavguids xchang pwr acrss a gap f g 0.3 m s that s w g 0.6m. Th man radius f th bnt wavguid is R 50 m. Th blu and grn lins n th mnitr sid display th pwr variatin with z in th straight and bnt wavguids, rspctivly. Bcaus f th mismatch lss btwn th Gaussian xcitatin and th md at th ntranc fac f th straight wavguid, th nrmalizd pwr lvl is again lss than 00 %. Th input pwr is cupld frm th straight wavguid t th curvd wavguid s that th pwr lvl dcrass frm apprximatly 88% t 8 % fr a TE-lik md and t 38% with a TM-lik md crrspnding t k 6/8. 8 fr th TE-lik and k 5/8. 8 fr th TM-lik md. 8

33 Figur.3: Pwr cupling btwn a straight and a curvd wavguid fr (a) TE-lik and (b) TM-lik mds... Prpagatin lss factr In straight wavguid sgmnts, lsss ccur thrugh scattring and intrinsic absrptin lsss. Bnding lsss ar als prsnt in curvd wavguids whil md mismatch lsss ar prsnt at intrfacs btwn straight and bnt sgmnts. Sinc SOI wavguids shuld pssss small crss sctins in rdr t xhibit small fabricatin ftprints, singl md pratin and small bnding lsss [ 0], sid-wall rughnss frm fabricatin imprfctins can b f significant ffct yilding scattring lsss typically btwn 0. db/ cm t 5 db/ cm [ 0] that ar xamind bth thrtically as wll as xprimntally in [ 0], [ 3], [ 9] and [ 4-4]. Additinally, pwr lakag int a lwr rfractiv indx substrat layr can ccur if th burid xid layr is nt thick nugh t supprss any vrlap btwn th prpagating md and th silicn substrat. Typically this rquirs a m thick silica layr [ 9]. Absrptin lsss ar ngligibl fr SOI wavguids prating at th tlcmmunicatins wavlngth.55 m sinc th nrgy gap f silicn is.v [ 4]. Als, sinc th mdal wav-frnt prpagats with fastr phas vlcitis far frm th cntr f rtatin f a bnt wavguid, at a crtain distanc frm th cntr, th wav-frnt wuld hav t prpagat with a phas vlcity gratr than th spd f light in th mdium. Sinc this is nt pssibl, bnding lsss ar instad inducd. T dcras bnding lsss, th lctric fild must b tightly cnfind insid th wavguid cr, which rquirs a larg rfractiv indx (RI) cntrast btwn th wavguid cr and cladding. Indd, SOI prvids largr cntrast than mst thr platfrms such as plymrs, sinc th RI cntrast is apprximatly fr SOI wavguids. This nabls small bnding radius wavguids ( R 5m ) with ngligibl bnding lsss [ 3] that can b mplyd in.g. larg fr spctral rang ( FSR) WDM multiplxrs as will b discussd in Chaptr 3. An illustratin using BamPROP is prsntd in Figur.4 9

34 (a) (b) 0

35 (c) (d) Figur.4: Ngligibl bnding lsss fr bnds with R 5 m fr (a) TE-lik filds in WG, (b) TM-lik filds in WG, (c) TE-lik filds in WG and (d) TM-lik filds in WG

36 Additinally, sinc th pak f th md prpagating thrugh a bnt wavguid sgmnt shifts tward th utr rim, md mismatch lsss tak plac at transitin btwn straight and curvd wavguids as vidncd in ractrack rsnatrs whr th fild is cnstantly travrsing btwn straight and curvd wavguids..3 Th basic ring rsnatr circuit In its simplst frm, a RR circuit cnsists f a straight wavguid, usually rfrrd t as th bus wavguid, adjacnt t a circular ring rsnatr wavguid as shwn in Figur.5.Th sparatin gap btwn th tw lmnts, th man radius f th ring and th width f bth wavguids ar dntd by g, R and w rspctivly. Th ffctiv lctric fild cmpnnts ntring and laving th intractin rgin ar th bus input (prt I) and th utput (prt II) filds givn by a and b rspctivly and th crrspnding filds in th ring, c and d. Th light cupld vanscntly frm th bus t th ring cycls nc arund th ring bfr subsquntly intrfring at th intractin rgin with th nwly cupld fild. Th intrfrnc cnditin dpnds n th gmtry, mainly R, th matrial f th wavguids and th fr spac wavlngth, and lads t a Lrnzian variatin f th rati f th utput fild t th input fild with wavlngth as discussd latr. Figur.5: Singl RR circuit. In th cupling f mds in spac (CMS) tchniqu [ 43], th fild cmpnnts arund th intractin rgin ar rlatd by: b r a ik d (.5) c r d ik a (.6)

37 in which k is th rati f th fild cupld btwn th ring and th bus, and k r. Th trm ik accunts fr th phas shift xprincd by th fild at cupling transitin. T avid multimd disprsin singl md wavguids ar rquird s that th ring fild cmpnnts satisfy: Hr d i c (.7) i l, is th rund trip cmplx phas factr, l is th rund trip phas shift, l R is th man circumfrnc f th ring, factr, n is th md lngitudinal prpagatin n is th md ffctiv rfractiv indx and is th pwr lss cfficint. Frm th abv thr quatins, th nrmalizd fild transmissin f th dvic is: b a r r (.8) i l / with yilding fr a nrmalizd pwr transmissin, T, b r r cs( ) T (.9) a r cs( ) r 3 r Accrdingly, T Tmax ( ) whn ( m ) and th ring is ff-rsnanc, whil r r T Tmin ( ) whn m and th ring is n-rsnanc, whr m 0,,.. is th md r lngitudinal rsnanc rdr. Physically, aftr n trip arund th ring, th lctric fild partially cupls back int th bus, with a ttal phas shift frm th tw cupling transitins. Thus if th ring is ff-rsnanc, th ttal phas shift is an vn multipl f and th intrfrnc btwn th bus and th ring fild is cnstructiv. Unlik travling wav rsnatrs such as Fabry-Prt rsnatrs, th RR can xtract a particular wavlngth cmpltly frm th input pwr spctrum, i.. T min 0 if th pwr cupld t th ring quals th rund trip pwr lss, i.. r, trmd critical cupling [ 3]. A mr cmplx circuit is rprsntd by Figur.6, whr a ring is cupld t tw buss. Such a cnfiguratin has fur prts: th input prt I (III), th crrspnding thrugh prt II (IV), and th drp prts IV (II). Th fild cmpnnts arund th bttm intractin rgin ar similarly givn by a, b, c and d. Th mathmatical analysis is similar t th abv and yilds th fllwing cupling: b r a ik d (.30)

38 c r d ik a (.3) b c r a ik d (.3) r d ik a (.33) and prpagatin: ( ) i d c (.34) d i c (.35) quatins, whr k is th rati f th fild cupld btwn th ring and th bttm bus, and k r Figur.6: A ring rsnatr cupld t tw buss. W calculat th utput fild cmpnnts b and b fr a singl input fild and thn rcalculat ths cmpnnts fr th scnd input. Th ttal utput fild frm any prt is thn th suprpsitin f th tw sts f utput filds. Fr a 0, w find th rsulting nrmalizd fild transmissin f th thrugh and drp prts rspctivly givn as: b a r r r i i r (.36) 4

39 b Th ratis a and b a with a 0 b a k k i / i r r can thn b fund by rplacing ( k, r ) by ( k (.37), r ) in quatins (.36) and (.37). Hwvr, cmplx structurs with multipl rings cannt b asily btaind in this fashin and th transfr matrix mthd intrducd blw is instad typically mplyd..4 Cmplx RR circuit analysis.4. Transfr matrix mthd Th st f quatins rprsnting th CMS tchniqu abv can b r-arrangd in th frm f tw grups f matrics, ths dscribing th fild cupling vr th intractin rgins, and phas matrics dscribing fild prpagatin btwn th cupling rgins. Multiplying ths matrics rsults in a transfr matrix that rlats th filds at th input and th utput f th circuit. Thus fr th circuit shwn in Figur.6 th cupling quatins ar: d Q c d c Q a b a b (.38) (.39) with cupling matrics Q ik r r and Q ik r r. As wll, d c d P c (.40) with a phas matrix, P i / 0 i / 0, Hnc a b a T b (.4) 5

40 whr th transfr matrix T is thnq PQ. Th bundary cnditin a 0, thn lads t quatins (.36) and (.37). This mthd is spcially usful fr cmplx circuits as in Figur.7 and Figur.8 that can b mplyd t mdify th RR transfr charactristics fr.g. WDM applicatins. Hwvr, whil th transfr matrix prcdur can b applid t any structur, fr a circuit with intrnal fdback btwn th rings as in Figur.9, furthr analysis is rquird, c.f. Chaptr 3. Figur.7: Sris cupld RRs Figur.8: Paralll cupld RRs 6

41 Figur.9: A lp f rings cupld t tw buss..4. Cupling f mds in tim (CMT) In th CMS tchniqu, tim dpndnc is liminatd as all filds ar assumd t hav harmnic i t dpndnc n tim (i.. ~ ). Hwvr, th sam assumptin can b utilizd t study th circuit using a tim dmain apprach that dals with ach ring in th circuit as a lumpd (i.. dimnsinlss) scillatr, s that th phas variatin du t fild prpagatin arund th ring is ngligibl w.r.t. that f th cupling transitin. This tchniqu is knwn as th cupling f mds in tim (CMT) and was suggstd fr studying travlling wav rsnatrs in [ 3]. Th CMT tchniqu psssss th advantag f simpl frmulatin f th quatins dscribing th prblm and cnsquntly a quickr calculatin f th transfr charactristics [ 3]. On th thr hand, and unlik th CMS, th CMT is limitd t small cupling and small lsss. An xampl xpliting th CMT is givn by Figur.0, whr a singl RR is cupld t tw buss. Th fild amplituds f th first input, scnd input, thrugh and drp filds ar givn by S, S, S and S rspctivly and ar nrmalizd s that th crrspnding fild pwrs ar givn by i f t S i (t), S f (t), d S t (t) and S d (t) rspctivly [ 3]. Th rsnant md f th RR thugh is dscribd by th nrgy amplitud f and ttal nrgy f (t). 7

42 Th CMT cupling cfficints Figur.0: A singl RR cupld t tw buss. and rplac th CMS cupling cfficints k and, d and rspctivly. Thr dcay rats rprsnt th pwr scap frm th ring, namly l whr th first tw crrspnd t th dcay int th thrugh and th drp prts whil th third is th dcay du t wavguid lsss. Th rlatins btwn th cupling cfficints in bth mdls ar givn by [ 3] k v /( R) and k v /( R) whr v g c / ng is th grup vlcity and n g th grup rfractiv indx f th RR, Th quatins fr f ar thn: g g / and / d [ 3]. k df dt ( i ) f isi (.4) S t S i f (.43) i S d S i f (.44) f Hr, whil is th angular rsnanc frquncy f th RR md. Fr a d l singl input signal a 0 in th CMS mdl crrspnding t s 0 in CMT md. Fr a d harmnic tim dpndnc is rplacd by and hnc th thrugh and drp nrmalizd fild dt transmissins ar rspctivly: f 8

43 whr S S t i i i S S d i i is th angular frquncy shift frm rsnanc yilding th pwr transmissin (.45) (.46) S S t i ( ) (.47) S S d i 4 (.48) And hnc w s that th drp transmissin dscribs a Lrntzian functin f frquncy..5 Cnclusin In this intrductin w prsntd th basic principls f pratin f th RR circuit. In th rmaindr f this thsis w analyz th tuning and tailring f th RR Lrntzian transmissin charactristics with a viw t WDM applicatins. W als dtrmin th dvic paramtrs,.g. th lsss and th cupling r th mdal analysis using BamPROP simulatins. In Chaptr, w cnsidr a hybrid structur cmprising f tunabl plymr vr SOI and dmnstrat th applicatin f this structur t high spd switchs. In Chaptr 3, w dsign a clsd sris f ring rsnatrs and discuss its pssibl applicatin as a WDM multiplxr. W apply bth th CMS and CMT analysis tchniqus and shw hw th CMT givs quick and accurat rsults, undr th limitatin f small cupling and lsss. W als xplit th finit diffrnc tim dmain (FDTD) simulatins t analyz th sam circuit and cntrast th rsults t ths f th CMS and CMT. As wll, w prsnt fabricatin and tst rsults fr this circuit. Finally, w discuss th applicatin f ur structur t rtatinal mtin dtctrs and cmpar th structur t fibr ptic gyrscps in Chaptr 4. W finally cnclud and discuss th futur wrk in Chaptr 5. 9

44 Chaptr Tunabl hybrid ring rsnatrs Tunabl ring rsnatrs basd n diffrnt tuning tchniqus such as thrmal tuning [ 44], plasma injctin tuning [ 45-47] and lctr-ptic plymr tuning [ - ] has prviusly bn intrducd. Whil th spd f thrmal tuning is limitd t milliscnds [ 44], plasma injctin xhibits a tuning spd up t5 GHz, and can b asily implmntd in SOI by surrunding th ptical wavguid cr by a p-dpd and n-dpd rgins [ 45]. Elctrns and hls flw thrugh th rsulting p-i-n junctin, undr an xtrnally applid frward tuning vltag, t altr th ffctiv RI f th fild ncuntrs. Hwvr, th injctd carrirs ccupy th sam rgin as th prpagating light, rsulting in undsirabl absrptin lsss. Cnsquntly, th pwr transmissin quality factr at th rsnanc frquncis bcms dpndnt n th tuning vltag. In cntrast, lctr-ptic plymrs nabl tunabl ring rsnatrs with vry high tuning spds f up t00 GHz [ ]. Hwvr, th small rfractiv indx (RI) cntrast btwn th wavguid cr and cladding rquirs that th ring radius b larg t dcras th bnding lsss at th xpns f th ftprint and th fr spctral rang f transmissin. Hwvr, xtnding dsign cncpts that wr rcntly applid t a hybrid plymr-soi Bragg rflctr [ 48] and a plymr-cmpund smicnductr dirctinal cuplr [ 49], w hr prps a nvl tunabl RR dvic that lvrags th advantags f bth plymrs and smicnductrs.. Hybrid structur dsign.. Opratin and dimnsins A simpl RR circuit cnsists f n ring cupld t a bus wavguid is shwn in Figur., whr th fild cmpnnts arund th cupling rgin ar dntd by a, b, c and d whil th ring man radius, th bus width and th ring-bus cupling gap ar dntd by R, w and g rspctivly. Th cntral fr spac wavlngth in this wrk is takn t b.55 m. Th ring and th bus ar takn t pssss th sam wavguid structur fr which a plymr layr cvrs th tp sid f a cnvntinal silicn cr silica clad wavguid as shwn in Figur.. Whn an xtrnal tuning vltag (V ), is applid, th lctr-ptically inducd RI variatin in th plymr layr altrs th ffctiv RI, n, f th wavguid md and hnc th transmissin charactristics f th RR. Th tuning vltag is applid acrss tw lctrds. Th first lctrd is a gld layr with thicknss t4 0. m [ ] that rsids n tp f th plymr layr, which has a cmplx rfractiv indx n i [ 50], whr i is th imaginary unit rprsnting th absrptin lss in that 4 layr. Th scnd lctrd is burid undr th burid xid layr and is cmpsd f a dpd silicn layr, with a RI n5 n and thicknsst5 0. m. Hr, 3 t is chsn small nugh t supprss th pwr vrlap with th dpd silicn layr, dcrasing th pwr lakag, and als t ffctivly liminat th ptntial drp vr th lctrd. 30

45 Figur.: A simpl singl ring rsnatr circuit. 3 Gld Plymr Si Silica Dpd Si Figur.: A vrtical crss sctin f th wavguid structur. Th clr mark t th right dfins th layr matrial. Th wavguid cr shuld satisfy thr cnditins. First, it shuld prat in a singl md t avid multimd disprsin ffcts. Scnd, th prpagatin lss shuld b minimizd t nsur a high quality factr, Q. Finally, th dvic dimnsins shuld b small in rdr t minimiz th tuning vltagv and th ftprint, as discussd blw. Assuming a cnstant tmpratur and nglcting strsss as in [ 48-49], it is fund that all ths cnditins ar satisfid fr a Si-cr wavguid surrundd by silica with w t 0.3 m and a silica layr thicknss t3 m as in [ 9]. Th pwr prpagatin lss cfficint du t sidwall scattring and substrat lakag is 4 6 db/ cm.40 / m as fund xprimntally [ 30] and thrtically [ 9]. Th diffrnc btwn th prsnt structur and that f [ 9] is th rplacmnt f silica with plymr n n sid f th cr. Th plymr cating has a thicknss t and is frmd frm CLD/APC (crss linkd dgr /amrphus plycarbnat) as in [ ]. Th plymr rfractiv indx n, is a functin f V whr n n p.6 whn V 0, and n n p n whnv 0. T satisfy ths cnditins, w can adjust fur paramtrs. Th first tw ar, t and th fild plarizatin, which dtrmin th valus f

46 bth th absrptin lss du t th gld layr and V. Th third is R which dtrmins th bnding lsss and finally th ring-bus gap, g, that cntrls th ring-bus cupling alng with R. Th last tw paramtrs ar applicatin dpndnt as will b shwn blw... Plymr thicknss W nw dtrmin t accrding t a tradff btwn V and th absrptin lss in th gld layr. Sinc th lctric fild is cntinuus: E E 3E3 (.) whr E is th xtrnally applid lctric fild cmpnnt thrugh th prmittivity ( ) layr. j Nglcting fringing ffcts and th ptntial drp n th lctrds w hav j V t [ E (.) E te t3e3 t t t3 ] 3 Th variatin in th plymr RI, n, du t E is givn by [ 3]: 3 n n pr33e (.3) in which r33 is th plymr EO-cfficint. Assuming that n n p, w apprximat n n p n n p (.4.a) r quivalntly This yilds p (.4.b) V n [ ][ t t t 3 ] (.5) 3 r 3 33 n Th cutff frquncy f such circuits can rach 0 00GHz [ ], [ 5] fr which and [ 5] whr is th air prmittivity. Plymrs xhibit a small variatin in th RI with frquncy [ 53-54] which implis that th valu f n. 6 in th ptical rang can b usd as is in th micrwav frquncy rang ( 0 00GHz ), and hnc w can us.6. Thus, frm quatin (.5) t minimiz V, t must b rducd. On th thr hand, fr smallr t valus, th fild vrlaps mr with th gld lctrd augmnting th absrptin lss. W thrfr mply th smi-vctrial bam prpagatin mthd (BPM) [ - 8], [ 55] t cmput 3

47 th variatin f th lsss in th gld layr with t fr bth th TE and TM mds whr w mply th cmplx rfractiv indx valu f th gld layr, i [ 50]. Th absrptin lss in db/ cmdu t th gld layr is thn plttd vrsus t in Figur.3, frm which it is bvius that th absrptin lss f th TM md is gratr than that f th TE md fr th sam plymr thicknsst. This is xpctd sinc th TM md is mr wakly cnfind in th dirctin nrmal t th substrat and thrfr vrlaps mr with th plymr and gld layrs. Thrfr, in rdr t liminat th gld absrptin, t shuld b gratr than m fr th TM-lik and gratr than 0.8 m fr th TE-lik md. Figur.3: Th absrptin lss inducd by th gld layr as a functin f plymr thicknss. Th inst shws th lsss fr t 0.8 m Finally, t dtrmin which plarizatin and t valu t slct, w xamin th snsitivity f variatins in n. Hr w tak t 0.8m fr th TE-lik md and t m n t fr th TM-lik n fr diffrnt md as just discussd. Again, w mply th smi-vctrial BPM t calculat ( ) n yilding th rsults f Figur.4. W first bsrv that bth plarizatins ar narly valus f 33

48 n qually affctd by th wavlngth variatin as.6 / m fr th TE-lik md and n.3 / m fr th TM-lik md at cnstantv. Scndly, th rspns f th TM-lik md n t th plymr RI variatin ( 0. 8 ) is narly n rdr f magnitud largr than that f th n n TE-lik md ( ) at cnstant wavlngth. This again can b xplaind by th incrasd n vrlap f th TM md with th plymr. Thrfr, w d nt cnsidr th TE md in this chaptr. Th crrspnding layr rfractiv indics and thicknsss ar summarizd in Tabl. and Tabl. rspctivly whr w mply WG charactristics in Tabl.. a b Figur.4: Md rfractiv indx ( n ) as a functin f th fr spac wavlngth ( ) fr diffrnt plymr RI ( n ) valus. (a) TE md with t 0.8 m (b) TM md with t m n n n 3 n 4 n 5 n [ 9] n p. 6 [ 9]. 444 [ 9] i [ 50] n = n 3 Tabl.: Wavguid rfractiv indics. 34

49 t t t 3 t 4 t 5 0.3m [ 9] m m [ ] 0.m [ ] 0. m 0.3 m [ 9] Tabl.: Wavguid crss sctin dimnsins. At this pint w hav dducd th paramtrs f th structur xcpt th man radius f th ring ( R ) and th ring-bus gap ( g ), which ar dtrmind accrding t th dsird dvic pratin. In th fllwing sctins w tun th transmissin charactristics f th hybrid structur by altring th ring rsnanc r th ring-bus cupling.. Hybrid structur applicatin Th dvic transmissin charactristics f an lctr-ptic dvic ar tund thrugh th dpndnc f n n V. Hr w assum that R w and that th mds in th bus and th ring hav idntical valus f n as discussd in Chaptr. If bth th ring and th bus xprinc th sam V, th inducd rfractiv indx chang n is nrmally idntical in bth wavguids prvnting phasmismatch. This way, th rsnanc cnditin f th ring can b altrd. Anthr way is t apply V t th bus nly s that th phas mismatch altrs th cupling with th ring whil th ring rsnanc is maintaind. Eithr way, th transmissin charactristics f th dvic bcm tunabl... Pwr transmissin tuning... Dsign Th pwr transmissin cfficint fr th circuit in Figur. is dfind as th rati f th utput pwr t th input pwr, namly [ 56]: w T r r cs( ) (.6) r cs( ) ( r) whil th quality factr is [ 57 56]: Q Rn r (.7) ( r) Hr, r k, k is th fild cupling rati dfind as th rati f th lctric fild cupld l btwn th ring and th bus, xp( ) is th fild lss cfficint, l R is th ring man circumfrnc, is th rund trip phas shift that can b xprssd as l and is th 3 db-bandwidth (B.W.). Th transmissin cfficint is a maximum fr th ff-rsnanc stat with n 35

50 T r r max ( ) rati,, can b dfind as and a minimum at th n-rsnanc stat with T T m ax m in T r r min ( ). Th xtinctin [ 57] and is maximizd whn th critical cupling cnditin, r, is fulfilld [ 57]. Th wavlngth diffrnc,, btwn tw succssiv transmissin maxima is calld th fr spctral rang, FSR and is givn by [ ], [ 59]: whr n dn FSR (.8) Rn g n is th grup rfractiv indx [ 59]. Th shift in th transmissin d charactristics du t V, is calld th tuning rang, TR. Fr maximum tuning f th ring transmissin charactristics, R, is chsn t prvid full tuning btwn n-rsnanc and ff rsnanc stats, s that TR FSR. Cnsquntly, R shuld satisfy th cnditins g fr n-rsnanc pratin at a spcifid wavlngth and n R m (.9.a) n n R ( m 0.5) (.9.b) fr ff-rsnanc pratin at th sam wavlngth, whr m is intgr. W can subtract quatin (.9.a) frm (.9.b) t find that th ring radius shuld satisfy th fllwing cnditin: R R 4 n (.0) fr cmplt switching frm n-rsnanc t ff-rsnanc at th sam wavlngth. Equatins and (.0) rprsnt th primary tradff btwn V and R (.5) sinc incrasing th valu f V incrass bth n and n, whil rducing R. Fr a plymr with r pm / V which is xpctd t b shrtly availabl [ 60], w find th rsults fr n ( V ) and n (V ) displayd in Figur.5.a whil th crrspnding R(V ) is shwn in Figur.5.b. Th rang f V fr high spd switching applicatins is assumd t li in th rang f 0Vlt and th crrspnding radii ar cmputd t b m. Th prduct VR, whr R R, rmains narly qual t 496 Vlt. m vr 36

51 this rang f V, as culd b prdictd frm th narly linar dpndnc f and Figur.5.a. n n n in Figur.4.a Th BPM has als bn mplyd t calculat th bnding lsss fr this rang f R fllwing th tchniqu in [ 9] whr th calculatd md is launchd int an arc f th ring with a 0 cntral angl and th pwr is mnitrd. Thn th bnding lss is scald t 90 t find th lss aftr a quartr trip. Simulatins shw that th bnding lss is vry small cmpard t th scattring lss s that rmains at th scattring lss valu f 6 db / cm. a b Figur.5: (a) Variatin f n andn with V (b) th dpndnc f th V fr full ON-OFF switching. R R rquird n Th last paramtr t study is th ring-bus sparatin g which is mplyd t cntrl th fild cupling rati k. Again w apply ur BPM prgram t calculat k R, g ). W fllw th mthd f ( [ 9] in which th calculatd md is launchd int th bus which is nxt t an arc f th ring with 0 cntral angl. Th pwr in th bus is thn mnitrd and th utput pwr f th bus is idntifid with r. Th rsults fr R R valus fr which g varis frm 0. m t 0.6 m in stps f 0.m ar shwn in Figur.6 whr is als plttd t dmnstrat whn th critical cupling cnditin can b fulfilld. Fr instanc, tw pints ar markd n Figur.6 that yild a maximum transmissin xtinctin rati sinc th critical cupling cnditin is fulfilld. At th First dsign pint, w find, R 49.6 m, V 0Vlt, g 0.4 m, n , n 0. 0, n 0.005, and r implying that R is minimizd whil V is larg. Fr th scnd pint, R 3.4 m, V.6Vlt, 0. m, n. 6774, n 0.009, g n , and r Th dvic rspns in bth cass is studid in th nxt sctin. Th nrmalizd transmissin fr th dvic is shwn in Figur.7.a and Figur.7.b fr th first and scnd dsign pints rspctivly. Th first dsign has th fllwing transmissin 37

52 charactristics. scnd dsign w hav, FSR TR.85nm, 0 db, 4 Q 3.4 0, 45.9 pm whil fr th 4 FSR TR 0.9 nm, db, Q , 46. pm. Figur.6. Th variatin f th fild transfr cfficint r and th fild lss cfficint with man ring radius R R and th ring-bus gap g (a) (b) Figur.7: Th dvic nrmalizd pwr transmissin (a) first dsign pint, (b) scnd dsign pint 38

53 Whil th quality factr is high fr bth dsigns, th larg rati f th FSR in th tw dsigns is th rsult f th larg rati f th tw dsign radii.... Cmparisn with similar circuits Tabl.3 cmpars th charactristics f th tw dsigns fr th prpsd hybrid dvic t tw thr tunabl RR circuits that wr tstd xprimntally. Th first f ths mplys Si/SiO tchnlgy with plasma injctin tuning [ 45] and th scnd is a pur plymr structur [ ] that is tund thrugh th plymr EO cfficint (nt that th circuit in [ ] mplys diffrnt wavguids fr input and utput). Th faturs that ar th sam in ur dsign and th tw thr dsigns ar shadd. As w s, ur dsign is cmpatibl with silicn tchnlgy, xhibits a small wavguid crss sctin dimnsins cmpard t plymr wavguids [ ], and displays a high silicn/silica RI cntrast and hnc largr FSR and TR. Th tuning in [ 45] dpnds n carrir injctin int th wavguid cr which shifts th transmissin charactristics thrugh a chang in n. Thn, th tuning spd f this dvic is limitd by th carrir liftim and furthr rquirs a biplar V t fficintly injct and xtract carrirs frm th wavguid. As wll, th transmissin quality thn dpnds n th tuning. T achiv largr shifts mr lctrns must b injctd, dcrasing th quality factr. On th thr hand, fr th hybrid structur, th tuning dpnds n th lctr-ptic ffct in plymrs which is far fastr, can b achivd with a singl plarityv, and displays a quality factr that is narly indpndnt f th tuning vltag. Th main fabricatin difficulty is th dpd silicn layr bnath th silica layr. On f th availabl altrnativs is thrugh fabricating th dvic in hydrgnatd amrphus silicn with lw absrptin lss [ 6-6] sinc amrphus silicn can b grwn vr silica. A scnd pssibility is using implantatin f xygn ins and subsquntly annaling as in [ 63]. Finally, th substrat can b dpd bfr th layr transfr stp in th Smart-cut prcss t fabricat silicn vr xid SOI [ 64]. 39

54 Si/SiO [ 45] First dsign (scnd dsign) Full plymr [ ] rsnanc( m) Wavguid crss sctin dimnsins m 0.5 m 0.3 m 0.3 m 5 m m R( m) ( 3. 4) 750 V (Vlt) V pp = 3 {Diffrnt plaritis fr carrir injctin and xtractin} 0(. 6) {Singl plarity} 4.85 {Singl plarity} TR (nm) ( 0. 5) Q ( 3. 35) 6. (db) 5 0 ( ) Nt givn Mdulatin frquncy. Transmissin Charactristics Difficult t fabricat. Cmpatibility with Silicn Tchnlgy. 5 GHz Up t 0 00 GHz Mr carrir injctin incrass th lsss and altrs th transmissin charactristics. Standard Silicn Tchnlgy. Rquirs many fabricatin stps fr p-i-n junctin and lctrd frmatin. Cmpatibl. Ds nt dpnd n tuning. Silicn Tchnlgy with tw xtra stps fr th plymr layr and burid dpd silicn layr. Cmpatibl. Up t 0 00GHz [ ], [ 48], [ 65] Ds nt dpnd n tuning. Many stps fr diffrnt plymr layr tratmnt. Rquirs spcial tratmnt [ 65-66]. Tabl.3: Cmparisn f fur tunabl ring rsnatr circuits whr ur scnd dsign' paramtrs ar shwn btwn brackts... Ring-bus cupling variatin In th nxt prpsd circuit dsign, th tuning vltag is applid nly t th bus lctrds inducing a phas mismatch btwn th bus and ring mds that altrs th cupling and cnsquntly th transmissin charactristics. T simplify th cupling calculatins, w invstigat th ractrack-bus cnfiguratin in Figur.8 in plac f a RR systm. Th ractrack-bus cupling is thn apprximatd by th cupling btwn tw paralll bus wavguids, nglcting th cupling in th bnt rgins. 40

55 ... Principl f pratin W first slightly mdify svral circuit paramtrs. All th paramtrs dfind prviusly in this chaptr ar still prsnt but ring paramtrs ar rplacd by ractrack paramtrs. Thrfr, l is givn by, l R L (.) with L is th lngth f th ractrack sid rprsnting th intractin lngth btwn th tw straight wavguids (SWGs). Furthr, dnts th ractrack rund trip phas shift, and fr simplicity w liminat th dpndnc n by dsigning th dvic such that th rsnanc cnditin: m (.) Figur.8: Ractrack-bus cnfiguratin. Th silica layr surrunding th silicn cr is nt shwn. Is fulfilld, whr m is intgr. Th maximum T is T, which ccurs fr zr pwr cupling, max r, whil th minimum T min 0 is btaind at r, as prviusly mntind. Frm th variatin f T with k prsntd in Figur.9, w cnclud that small variatin in k can shift T frm a maximum t minimum, spcially fr lw lss circuits with. Als, fr V 0 w find n.605 at.55 m. 4

56 Figur.9: Variatin f th pwr transmissin factr (T ) n rsnanc with th fild cupling rati ( k ) fr diffrnt rund-trip lsss As discussd in Chaptr, fr tw paralll SWGs, k is givn by [ 38]: k sin( L/ L / jz dz) (.3) in which rprsnts th cupling cfficint btwn tw paralll SWGs, d n d is th phas mismatch and nd n n is th diffrnc in th ffctiv RI btwn th cupld wavguids whn pwr is cupld frm wavguid (), which is th bus, int wavguid (), which is th ractrack sid and z is th dirctin f prpagatin. Frm quatin (.3) w find: with L sin( X ) k sin( ) X (.4) X nd L d L / (.5) W labl X th 'tuning trm', sinc it quantifis th dgr t which k r quivalntly th nrmalizd pwr transmissin ( T ) has bn adjustd. Sinc sin( X ) / X is an vn functin f X, ths quantitis ar dpndnt n th abslut valu f. Fr 0, quatin (.0) rducs t th wll knwn rlatin: 4 nd d k sin( L) (.6)

57 Accrdingly, fr T T max, L shuld insur that with n phas mismatch, i.. d 0, th cupling vanishs crrspnding t r and k 0. Frm (.6), this yilds L n / (.7) fr intgr n,,... (Nt that n 0is unphysical sinc L 0). Hwvr, T T max as w will dmnstrat blw, is difficult t achiv and hnc this cnditin is nly applid t btain an initial stimat f L which is thn ptimizd. Fr an applid tuning vltag, V, th phas mismatch,, shuld satisfy T T min 0. This yilds th cnditin: d sin( X ) sin( X L ) (.8) which is ur cntral quatin. In th nxt sctin w dsign a dvic that ralizs this cnditin.... Dvic dsign T slv quatin ((.8), w first writ th lft and right hand sids f th quatin as: whr nd sin( X ) sin( nd L / ) L. H. S W (, L, n ) sin( L ) sin( L ) (.9) X n L / m R. H. S Z( m) xp( ) (.0) n and m ar indpndnt variabls and Z(m) is a discrt functin f m. Th bjctiv nw is t dtrmin th dimnsins L and R. W can thn mply a md slvr t find th valu f crrspnding t nd, alng with quatin (.5) in rdr t btain V, whil R can b calculatd n frm m by mplying quatins (.) and (.). W slct insuring a high switching spd, and m 000 t limit th valu f R. d 3 n d 0 t giv V 8.3Vlt, Nxt, mplying quatin (.7), w maximiz th cupling cfficint, in rdr t minimiz th ractrack dimnsins, by stting g 0. m which yilds a cupling lngth 9.75 m and n 0.6/ m. Figur.0 displays W and Z fr L and n t n 8. 43

58 (a) (b) (c) (d) () (f) 44

59 (g) Figur.0. W ( n ) (slid lin) and Z (m) (dashd lin) fr L n /, (a) n, (b) n, (c) n 3, (d) n 4, () n 5, (f) n 6, (g) n 7, (h) n 8 sin( X ) Fr typical valus r cs( n ) is ngativ with dd n. Whil many slutins f X quatin (.8) xist, fr ths slutins quatin (.6) implis that T T s that th xtinctin Tmax rati 0 lg( ) 0. Fr vn n, hwvr, W and cnsquntly k ar qual r slightly Tmin smallr than 0 whil r, s that n slutins t quatin (.8) xist. Thrfr, w rlax th cnditin f quatin (.7) by prmitting T max, but w still mply this quatin t stimat th valu fr L at which w bgin a sarch fr th xact slutin. W als st a 6 cnditin, L. H. S R. H. S 0, that dfins th accptabl valus fr L. Fr th first slutin w thrfr start with an L valu cls t / 39 m, and subsquntly btain L m. Figur..a thn shws thr xists pssibl slutins t quatin (.8), in th rang 0 m 90, whil Tabl.4 idntifis th crrspnding circuit. Hr w hav mplyd r pm / Vlt which shuld b achivabl in th nar futur [ 60]. W find th scnd rlvant valu fr L in th vicinity f 4 / 78 m, which yilds L 80 m. Again, Figur..b and Tabl.5 shw th crrspnding slutins and circuit paramtrs in th rang 97 m 000. Bth tabls indicat a trad-ff amng V and ( R andt m ax ) sinc ractracks with smallr radii xhibit a smallr ptical lss and cnsquntly a largr T, givn that th bnding lsss ar ngligibl. Th ptimum dsign is thrfr givn by th ninth ntry f Tabl.5, shwn in bld, sinc th last tw ntris hav R 0 and ar thrfr unphysical. As wll, fr typical SOI wavguids, th bnding lsss can b nglctd fr R 5 m [ 3], and w thrfr nglct slutins that vilat this assumptin. m ax min (h) max 45

60 a b Figur.: W n ) (slid lin) and Z (m) (dashd lin) fr (a) L 40.05m, (b) L 80m ( Slutin N. m R( m) n d 0 3 n 0 3 V (Vlt) T max( db) T min ( db) (db ) Tabl.4: Circuit paramtrs fr L m 46

61 Slutin N. m R( m) n d 0 3 n 0 3 V (Vlt) max( db) T T min ( db) (db) Tabl.5: Circuit paramtrs fr L 80 m.3 Cnclusin In this chaptr w prpsd a nvl tunabl ring rsnatr circuit and dmnstratd tw tchniqus fr tuning th transmissin charactristics, th first f which mdifis th rsnanc stats f th ring whil th scnd altrs th phas f th fild at th ring-bus cupling rgin. Our circuit is cmpatibl with SOI tchnlgy, is xpctd t xhibit a high switching spd f 0 00GHz [ ], [ 65] vn whn drivn with a singl vltag, whil th circuit in [ 45] rquirs diffrnt plaritis t nhanc th carrir-liftim limitd circuit spd that is in any cas far blw that f ur circuit. Furthr, th ptical lsss and thrfr th transmissin charactristics f ur circuit ar indpndnt f V, in cntrast t [ 45]. Our circuits incrprat th as f manufactur and th small wavguid dimnsins f silicn tchnlgy with th high switching spd f plymr tchnlgy. W thrfr bliv that ur dsign culd find applicatin in practical intgratd ptic structurs. Th circuit is als cmpatibl with silicn dvics and culd als b mplyd in WDM applicatins. Additinal wavlngth slctivity shuld furthr b achivabl by incrasing th vrlap btwn th prpagating pwr and th plymr layr prhaps as in [ 67] whr th mdal fild is shiftd twards th plymr layr by an intrmdiat thin high RI layr btwn th silicn and th plymr. Altrnativly, push-pull driving lctrds culd dcras th tuning vltag by half [ 65]. Multipl ring circuits with mr cmplx transmissin charactristics can als b dsignd if rquird. Finally, whil ring rsnatr circuits snsitiv t fabricatin tlranc thrugh th cupling and rsnanc cnditins, a tunabl structur can cmpnsat fr such a tlranc thrugh th adjustabl xtrnal vltag. 47

62 Chaptr 3 Cmpund ring rsnatrs Th singl ring circuit f th prvius chaptr xhibits a nar-lrntzian pwr transmissin which is inadquat fr sm applicatins such as cmplx filtrs. Ring rsnatr circuits with additinal rings cupld in sris r paralll hav prviusly bn prpsd t adapt th pwr transmissin t diffrnt applicatin rquirmnts [ 43], [ 68]. Hr w xamin a diffrnt structur cnsisting f a clsd lp f cupld rings which w trm a "cmpund ring rsnatr circuit". Th intrnal fdback btwn th rings facilitats th shaping f th transmissin charactristics as w dmnstrat thrugh th dsign f a signal intrlavr. W will analyz this circuit with th cupling f mds in spac (CMS), and tim (CMT) mthds as wll as thrugh finit diffrnc tim dmain (FDTD) simulatins and cntrast th accuracy f th thr prcdurs. W thn dsign, fabricat and charactriz a WDM intrlavr/ dintrlavr basd n this circuit. 3. Transfr matrix apprach In this sctin w intrduc th cmpund ring structur and calculat th lctric fild f th thrugh and drp prts. Th cmpund ring rsnatr (RR) circuit structur is shwn in Figur 3. whr idntical rings with man radius R ar, fr simplicity, vnly distributd within tw utr bus wavguids such that thir cntrs ar lcatd n th vrtics f a unifrm plygn. Th numbr f rings, N, is chsn t b vn t avid lctric fild rflctin at th input prts. Th ring fild cmpnnts ar dntd a, b, c and d with j N. Th prts f th uppr bus ar labld I and II fr th input, j a and utput j j j b filds whil th lwr bus prts ar labld III and IV with a and b rspctivly. Th gaps btwn tw nighbring crrspnding input and utput filds rings, btwn ring j and th first bus, and btwn ring j N / and th scnd bus ar dntd g, g and g rspctivly. Th crrspnding fild cupling ratis ar k, k and dfind as th rati f th fild cupld btwn tw nighbring cmpnnts. Th width f all wavguids is dntd by w. k 48

63 (a) (b) Figur 3.: Th cmpund ring rsnatr circuit with (a) N 4, (b) N 6 ring rsnatrs. T cmput th circuit transmissin charactristics w mply th transfr matrix mthd [ 43], [ 68] in which th lctric fild cmpnnts ar rlatd thrugh tw typs f matrics. Ths ar th d a d a d j a j cupling matrics: Q, c b Q, Q, j N, with c b c j b j r r r Q, Q ik r, Q ik r, k r, k r, and ik r a j d j a j d j k r and th phas matrics: P, j N / and P, b j c j b j c j i 0 i 0 j N / with P i, P. Hr i l, is th rund trip i 0 0 phas shift, l R, is th pwr lss cfficint,, ( ) and N fr a unifrm plygn. By symmtry w hav P P s that whn all cupling N cfficints ar qual th matrix P QP Q is unimdular. Nt that whil cupling matrics hav U bn intrducd fr all th rings in th circuit, th phas matrics quatins abv ar nly mplyd fr th rings that ar nt cupld t buss. Spcial matrics, V and V, ar ndd alng with th phas matrics fr th rings cupld t th buss as shwn blw. W cnsidr th cas that a 0 s that th input t th circuit is a 0, and th circuit functins as a dintrlavr with thrugh ( b ) and drp ( b ) prts. Again bcaus f th symmtry f th circuit, similar rsults 49

64 apply if a 0 and a 0 and k and k ar intrchangd. Oprating as an intrlavr bth input filds ar nnzr ( a 0 and a 0 ), which is a suprpsitin f th tw abv cass. Accrdingly, w tak b a d N ( 3..a) c N (3..b) d N a in which, and, ar dtrmind blw. Stting and w hav fr dd N / x x X ( 3..a) x3 x4 y y Y (3..b) y3 y4 d c a b N N N / N / a X b Y a b ( 3.3) ( 3.4.a) ( N )/ 4 Y Y V ( PQ ) U (3.4.b) 0 V (3.4.c) 0 / r If instad N / is vn, ( N )/ 4 X PU P Q) Y (3.4.d) d c N / N / Y ( a b ( 3.5.a) Y N / 4 Y PU (3.5.b) 50

65 5 0 0 r V (3.5.c) PY V PU X X N 4 / (3.5.d) Applying th bundary cnditin i N b d ( 3.6) tgthr with b x a x d N, 4 3 b x a x c N, w find fr 0 N d x x a b ( 3.7.a) 4 3 x x x x a c N (3.7.b) whil fr 0 a 4 x x d c N N ( 3.8.a) x d b N (3.8.b) Thrfr th intrnal dvic fild rflctin and transmissin cfficints ar i i a b ( 3.9.a) and ) ( i i N i a c (3.9.b)

66 Stting, a a i / b ikd r a, c ika r d, i / d c N and c a ik b d d d c N a. With i / i / r ik and ir a a a c N a a i / k r i / i / i ir b which lads t th thrugh-prt transmissin, a w btain w hav b a r i r i i i i ( 3.0) This is idntical t th xprssin fr rflctin frm a singl ring if th rund trip cmplx phas i shift trm, i b, rplacs th trm i b d bn / a. Fr N / dd s a d b a a that thrfr, in which N /, b i / a ik ( y3 y4 i ) a a and N /. Th drp-prt transmissin fr th circuit is b kk( y3 y4i ) i i ( 3.) a r i rprsnt th phas shift f th filds rlativ t th input fild phas. Fr vn b d c a b N / again yilding quatin ( 3.) with th matrics f a d c a a N / quatins ( 3.5.a) (3.5.d). T cnclud this sctin, th cmpund ring transmissin charactristics ar givn by quatins ( 3.0) and ( 3.) fr th thrugh and drp prts rspctivly. In th fllwing sctin, ths charactristics ar ptimizd t match th standard WDM intrlavr / dintrlavr circuit spcificatins. 3. WDM cmpund ring rsnatr structur intrlavr circuit In this sctin w mply th cmpund RR structur prsntd in th prvius sctin t build a standard WDM dintrlavr circuit. An intrlavr circuit cmbins signals frm tw diffrnt ptical channls carrying dd and vn signals int n stram with half th channl spacing, whil a dintrlavr splits n stram int tw [ 69]. Th rquirmnts n this circuit ar as fllws [ 43], [ 69]: Th channl spacing was takn t b 50 GHz and th fr spctral rang, FSR, f a channl was st t 00 GHz. Th crss-talk, dfind as th maximum transmissin f a channl within 0GHz f th maximum transmissin frquncy f th nighbring channl shuld b 3dB. Th abslut valu f th signal disprsin shuld likwis b limitd t 30ps / nm within th channl bandwidth ( 0GHz ). Finally, th shap factr f th pass-band, arbitrarily dfind as th rati f th db bandwidth t th 0dB bandwidth, is prfrably gratr than

67 W study tw cass t illustrat th dynamics f th circuit rspns bfr w discuss th ptimum dsign. W first transfrm th fild transmissin f th tw utput prts int th Z-dmain [ 70-7]. Th Z-transfrm analysis nabls an imprvd undrstanding f th circuit pl-zr dynamics thrfr facilitats dsign ptimizatin as illustratd by th thr xampls blw. Hr w i first substitut z in quatins ( 3.0) and ( 3.) such that l rprsnts a nrmalizd frquncy. W fllw th sam assumptins as in [ 43] and [ 69], s that th pwr lss aftr a rund trip arund a ring is 0 % [ 43] whil th straight wavguids ar cnsidrd lsslss. Th fr spac wavlngth is takn as. 55m. This lavs thr adjustabl paramtrs, k, k and k f which w tak k k fr simplicity and, mr imprtantly, fr symmtry. Fr ach chic f th tw rmaining cupling cfficints, w can thn valuat and numrically in trms f Th zrs ar th slutins f 0 and 0, whil th pls ar cmputd frm th slutins t / 0 and / 0. Sinc n rvlutin arund th unit circl in Z-dmain crrspnds t a FSR 00GHz, whil th passband f ach channl is 0GHz abut th channl maximum transmissin, th passband f a channl is / 5 36 arund th angl crrspnding t th channl maximum transmissin. In th fllwing rsults, th slid (dashd) lin n th graphs rprsnts th rsults fr th thrugh (drp) channl. W hav studid th filtr rspns fr numrus valus f th cupling cfficints. Tw illustrativ xampls f th filtr dynamics ar prsntd in cass A and B blw whil th ptimal dsign is givn in cas C Cas A: W st k k , whil incrasing k frm 0. 5t 0. 6 in stps f This lads t th circuit rspns f Figur 3..a thrugh 3..g. Fr ur cmpund ring rsnatr circuit, as all RR circuits, th thrugh (drp) prt has th spctrum maximum (minimum) cntrd at /, 3,.. and th minimum (maximum) cntrd at / 0,,... sinc th lctric fild intrfrs dstructivly within an ff-rsnanc ring i.. /, 3,.. and cnstructivly within an n-rsnanc ring, / 0,,... Th transmissin spctra f th tw channls ar cmplmntary fr lsslss rings. Als, whn k incrass, mr pwr is xchangd btwn th tw channls and cnsquntly th crss-talk is highr as in Figur 3..a. Bth channls rprsnt autrgrssiv mving avrag (ARMA) filtrs sinc bth hav pls and zrs, as shwn in Figur 3..f and 3..g. Th numbrs in th diagrams indicat th multiplicity f th pls and zrs. Th thrugh prt has th pls and zrs utsid th passband, / 0., whil th drp prt has its pls and zrs insid r cls t th passband / Th pl-zr dynamics clarly xplain th disprsin curv sinc incrasing k displacs th pls and zrs twards th passband f th thrugh prt and away frm th passband f th drp prt. Cnsquntly, th abslut valu f th disprsin incrass fr th thrugh prt fild and dcrass fr th drp prt fild within th crrspnding passbands. z. 53

68 (a) (b) (c) (d) () (f) 54

69 (g) Figur 3.: Th circuit rspns with k k and k incrasing frm 0. 5 t 0. 6 in stps f Th arrws indicat incrasing paramtr valus. Th rund trip pwr lss is 0 %. (a) Th pwr spctra. (b) Th phas variatin. (c) Th nrmalizd grup dlay. (d) Th thrugh prt disprsin. () Th drp prt disprsin. (f) Th thrugh prt pl-zr diagram. (g) Th drp prt pl-zr diagram. Cas B: Hr w st k 0. 55, whil k k is incrasd frm t in stps f Th circuit rspns is nw that f Figur 3.3.a thrugh Figur 3.3.g. Again, incrasing th cupling cfficints, k and k incrass th crss-talk fr th channls. Fr th thrugh prt, w bsrv frm Figur 3.3.a and 3.3.f that th channl psssss thr zrs, a ral n lying at angl 0 that gnrats th minimum at / 0 n Figur 3.3.a and a cmplx cnjugat pair that is assciatd with th lcal minima and th sid lbs n th sids f th passband. Th cmplx zrs ar displacd twards th ral axis, as k incrass, and thn divid s that th zrs psss ral and rciprcal valus. In this cas, nly n minimum appars in th pwr spctrum which thn lacks f sid-lbs. Th mtin f th zrs and pls in Figur 3.3.f agrs with th dcras in th abslut valu f th disprsin f th thrugh channl as shwn in Figur 3.3.d sinc thy mv away frm th passband as k and k incras. Additinally, Figur 3.3.f and 3.3.g indicat that th distanc f th pl psitins frm th rigin dcrass with incrasd k s that th influnc f th pls n th pwr spctrum is diminishd. This als xplains th dcrasd disprsin magnitud fr th drp prt vidnt in Figur

70 (a) (b) (c) (d) () (f) 56

71 Figur 3.3: Th circuit rspns with k 0. 55and k k incrasing frm t in stps f Th arrws indicat incrasing paramtr valus. Th rund trip pwr lss is 0 %. (a) Th pwr spctra. (b) Th phas variatin. (c) Th nrmalizd grup dlay. (d) Th thrugh prt disprsin. () Th drp prt disprsin. (f) Th thrugh prt pl-zr diagram. (g) Th drp prt pl-zr diagram. (g) Cas C: Frm th discussin f cass A and B, th ptimum valus f th cupling cfficints in th circuit ar k and k 0 k with th circuit rspns prsntd in Figur 3.4.a thrugh 3.4.g. Th crss-talk fr bth channls is fund t b 4dB whil th maximum disprsin f th thrugh channl is ps/ nm which satisfis ur statd dsign rquirmnts. Hwvr th drp prt xhibits a maximum disprsin ( 93 ps / nm) that xcds th maximum allwd disprsin ( 30 ps/ nm). Thrfr, a scnd stag must b mplyd at th drp prt frmd frm a singl ring rsnatr circuit, as shwn in Figur 3.5. Th ring is cupld t th drp prt and is dsignd t act cls t a unity filtr, whr th transmissin amplitud is narly qual t0 db, as shwn in Figur 3.6.a. This ds nt add significant rippls t th drp channl spctrum. Th pl and zr f th singl ring pssss ral and rciprcal valus as displayd in Figur Nt that k 0 fr this stag sinc nly n bus is cupld t th fifth ring. Th cmbind circuit rspns at th drp prt is shwn in Figur 3.7.a thrugh Sinc th disprsin f th tw stags is qual in magnitud but ppsit in sign, th abslut valu f th ttal disprsin is dcrasd. W hav als studid th pssibility f using a Mach Zndr Intrfrmtr (MZI) t rduc th disprsin at th drp prt. Frm Figur 3.4.g w bsrv that by placing tw MZIs in sris with th drp prt f ur circuit w can crat tw zrs in th Z-transfrm dmain that ar lcatd in cls prximity t th tw cmplx pls thus dcrasing th disprsin. Unfrtunatly, w hav fund that such a prcdur gnrats rippls (transmissin amplitud variatins) in th passband and f curs als incrass th circuit ara. 57

72 (a) (b) (c) (d) () (f) 58

73 (g) Figur 3.4: Th circuit rspns with k and k k fr a rund trip pwr lss f 0 %. (a) Th pwr spctra. (b) Th phas variatin. (c) Th nrmalizd grup dlay. (d) Th thrugh prt disprsin. () Th drp prt disprsin. (f) Th thrugh prt pl-zr diagram. (g) Th drp prt pl-zr diagram. Figur 3.5: A cmpund fur ring circuit attachd t a singl ring stag. 59

74 (a) (b) (c) (d) () Figur 3.6: Th singl ring stag rspns with k and k 0 fr a rund trip pwr lss f 0 %. (a) Th pwr spctra. (b) Th phas variatin. (c) Th nrmalizd grup dlay. (d) Th thrugh prt disprsin. () Th thrugh prt pl-zr diagram. 60

75 (a) (b) (c) (d) () Figur 3.7: Th drp prt rspns with an additinal singl ring stag fr a rund trip pwr lss qual t 0 %. (a) Th pwr spctra. (b) Th phas variatin. (c) Th nrmalizd grup dlay. (d) Th drp prt disprsin. () Th drp prt pl-zr diagram. 6

76 3.3 Cmparisn f intrlavrs W fund that th bst circuit f th sam natur, i.. a RR basd intrlavr circuit; t cmpar with is fund in [ 43] whr many RR basd intrlavr circuits wr studid bfr th authrs gt t thir ptimal dsign. Hnc w prsnt in Tabl 3. a cmparisn f ur prpsd WDM intrlavr circuit, basd n th cmpund RR structur, t th prviusly publishd [ 43] tw stag intrlavr circuit frmd frm fur rings in paralll fllwd by thr rings in sris which matchs th sam WDM rquirmnts. Th circuit ara in [ 43] and in this wrk ar btaind by nglcting th gaps btwn th rings and th bus-lins with rspct t th ring radius. W als assum that fr th circuit in [ 43], th distanc btwn th tw stags is half th ring circumfrnc in th sam mannr as th distanc btwn tw cnscutiv ring cntrs in th first stag. Finally, th circuit ara crrspnds t th ara f a rctangl f tw sids rprsnting th largst tw prpndicular dimnsins f th circuit. Our nw circuit clarly satisfis th intrlavr/dintrlavr circuit rquirmnts but ffrs additinal advantags cmpard t th thr circuit [ 43]. Namly, ur RR circuit ccupis 38% f th ara f [ 43] and nly rquirs 5 rings instad f 7, simplifying th dsign and fabricatin. Whil th valus f k frm n nd f th ring array t th thr wr varid symmtrically (apdizatin) in [ 43], th fabricatin must b xtrmly accurat fr th gap distanc btwn cupld rings t b sufficintly prcis nugh that th dsird valu f k is btaind. Mrvr, sinc th diffrnc btwn th cupling valus is abut [ 43], fabricatin tlrancs in th rang f nanmtrs wuld affct th circuit prfrmanc. Mrvr, apdizatin incrass th dsign paramtrs and cnsquntly th dsign cmplxity. On th thr hand, in ur cmpund ring circuit, apdizatin is unncssary and in fact w fund in additinal calculatins that it did nt significantly affct ur rsults. Thrfr, ur cmpund RR basd intrlavr has a gratr fabricatin tlranc cmpard with th ptimal dsign in [ 43]. Finally, th cmpund circuit xhibits cmptitiv valus f th disprsin, th insrtin lss and th passband shap factr as prsntd in Tabl 3.. 6

77 Crsstalk db (thrugh/drp) Disprsin ps / nm (thrugh/drp) Ara Shap factr (thrugh/drp) Insrtin lss db (thrugh/drp) Rquirmnts [ 43], [ 60] 3/ 3 30/ 30 min 0. 6 Circuit in [ 43] 37/ 35 5/ R Minimum 0.55/ / 0.5 Cmpund RR circuit 4/ 4 / 33R.66/ /. 7 Tabl 3.: Th prfrmanc f th ptimal dsign in [ 43] cmpard t th cmpund RR circuit prfrmanc. As a cnclusin f this sctin, th nw WDM intrlavr xcds th prfrmanc and simplicity f prviusly rprtd ring rsnatr intrlavr circuits. Th layut f th rings is als apprximatly circular, which rducs fabricatin ara and thus incrass th packag dnsity with a gratr fabricatin tlranc. Additinally, such a dsign might als functin as a building blck in thr applicatins such as ptical dlay lins. 3.4 CMT analysis In this sctin, w analyz th cmpund ring rsnatr structur fr small cupling, with bth th CMT and th FDTD mthds. With th CMT, th RR is mdld as a lumpd scillatr [ 3], [ 7] such that, fr sufficintly small cupling, i.. th nrgy cupling cfficint dfind blw is much smallr than th rsnanc frquncy f th ring, th lctric fild amplitud changs ngligibly acrss th rgin vr which tw lmnts in th circuit ar cupld. This lads t simplr quatins than in th CMS. Hwvr, this prcdur rquirs that th pwr cupling rati btwn tw adjacnt lmnts is small, th pwr lss is small, and furthr nly valuats th circuit transmissin charactristics vr narrw frquncy bands arund th rsnanc frquncis. W als shw that, whil th FDTD is a rbust tchniqu [ 3], it dmands substantial cmputatinal rsurcs spcially fr thr dimnsinal (3D) cmplicatd structurs with larg RR radii. Althugh th ffctiv indx tchniqu [ 7] can in crtain cass b applid t gnrat an quivalnt D wavguid prfil, fr dsign purpss, th dvic transmissin spctra must b calculatd fr numrus valus f th lss and cupling cfficints which is vry tim cnsuming with th FDTD simulatins. Th shap factr and th insrtin lss f th prsntd dsign in [ 43] ar nt spcifid. Thrfr, w calculatd ths valus basd n th quatins prsntd in [ 43]. Th passband shap factr fr th drp prt is dfind hr as th rati f th db bandwidth t th db bandwidth bcaus f th additinal stag lss. 63

78 W als shw that fr small cupling and lsss, th rsults f ths tchniqus agr with ach thr and with th CMS prcdur mplyd prviusly. W als find th rlatinship btwn th lss factrs in th CMS and th CMT mdls and apply th CMT t a lssy circuit. W find that th CMT yilds rapid and accurat rsults fr th transfr charactristics abut th rsnanc frquncis dspit th cmplxity f ur xampl, which is cnsidrably mr dmanding than that analyzd in [ 3] CMS circuit paramtrs T cmpar th CMS and CMT mdls, w nw xamin th cmpund RR structur in Figur 3.8 but with th cupling and lss cfficints givn in [ 3] in D, which satisfy th limits f small cupling and lsss. Th bus and ring structurs pssss a cr and cladding rfractiv indx (RI) f n 3, and n, and a width f w 0. m. Fr ths valus, a md slvr yilds n fr th transvrs lctric (TE) fild md in th vicinity f.334 m, frm which n. 37 at.334 m. Th grup RI is givn by dn ng n, which yilds n g at.334 m. d W nxt dtrmin th cupling cfficints fr R.7m and g g g 0. m. First w apply th analytic rsult, basd n cupld md thry, in th appndix f [ 3], fr th cupling cfficint btwn tw straight paralll wavguids (buss),, which yilds, k sin( l ) fr th fractin f th cupld fild whr l b is th intractin lngth. Fr th cupling btwn a bus and a ring r btwn tw rings, w mply l m R m n n, with R m RR /( R R ) whr R and R ar th radii f th tw cupld lmnts, instad f l b. Additinally, fr small cupling w can apprximat km sin( l m ) l m. In this mannr, w find 5.3 / m, k k 8% and k 3%. W hav als rpatd this calculatin with th BPM, by launching pwr int n f th tw straight wavguids sparatd by a distanc f 0. m. Mnitring th pwr in th tw wavguids yilds a cupling cfficint 6.8 / m. Similarly, w invstigatd th cupling btwn a straight wavguid and 0 dgrs f a nighbring ring [ 9]. Equating th pwr cupld int th ring t th pwr lst frm th straight wavguid, yilds k k 4.4%. Hwvr, w adpt th rsult f th cupld md thry tchniqu, just as in [3], in rdr t cmpar mr dirctly th CMT mdl f [ 3] with th FDTD and CMS prcdurs. Finally, w valuat th frmulas f th CMS mdl studid prviusly fr th thrugh-prt transmissin b i i, and th drp prt transmissin a a rprsnt th phas shift f th filds rlativ t th input fild phas. b, in which and b b 64

79 3.4. CMT circuit paramtrs W nw mply th RR circuit mdl f Figur 3.8, with amplitud in ring j with N 4. Th quantity Furthr s i and s f rprsnt th input filds whil f, j N, rprsnting th nrgy j f j (t) is nrmalizd t th nrgy strd in ring j. s t and s d rprsnt th thrugh and drp prt transmittd filds rspctivly and ar nrmalizd such that th crrspnding fild pwrs ar givn by S i (t), S f (t), S t (t) and dscribd by th nrgy amplitud f and ttal nrgy S d (t) rspctivly [ 3]. Th rsnant md f th RR thugh is f (t). Th CMT cupling cfficints ar, and j in plac f th CMS cupling cfficints k, k and k j rspctivly. As abv, w st and j. Th amunt f pwr cupld ut f a ring is paramtrizd by j thr dcay rats, d and l whr th first tw rprsnt th dcay int th thrugh and th drp prts rspctivly, whil th third is th dcay du t wavguid lsss. Th rlatins btwn th cupling cfficints in bth mdls ar givn by [ 3], whr k v /( R) and j k jvg /( R) with v g c / ng th grup vlcity, / and / d [ 3], and in th prsnt cas. As in th CMS mdl, w xamin th dvic transmissin fr a singl input d signal, s that a 0 and s 0 in th CMS and CMT mdls rspctivly. f g Figur 3.8: Th CMT mdl f th cmpund ring rsnatr circuit. Whil n rlatinship btwn and l is spcifid in [ 3], w can btain this fllwing th sam prcdur mplyd in [ 3] t rlat, k and. Hnc, fr an islatd ring, th pwr dcay rsults purly frm wavguid lsss. Cnsquntly, in th CMT mdl, th pwr flwing in a lssy ring t / l dcays as, with t l / vg fr n turn whil in th CMS mdl, th pwr dcays as l fr 65

80 66 n turn. Accrdingly, g l v. Fllwing th tchniqu in [ 3], w hav fr th cnfiguratin f Figur 3.8 s A A f ( 3..a) 4 3 f f f f f T (3..b) s i s (3..c) l l d l l i i i i i i i i i i i i A (3..d) 000 T i A (3..) in which j j with th input signal angular frquncy, j is th th j ring rsnanc angular frquncy givn by l n c m j, with,..., m is th azimuthal rsnanc rdr, and c is vlcity f light in fr spac. Th thrugh and drp prt transmissin is furthr L i L i t L s s ( L i L i d L s s ) with: f i s s i t ( 3.3.a) f 3 i s s f d (3.3.b) and L and L rprsnt th phas shifts f th filds rlativ t th input.

81 3.4.3 Numrical rsults W nw calculat th transmissin accrding t th CMT mdl which is thn cmpard t th rsults frm th CMS and FDTD tchniqus fr thr rprsntativ ring wavguids. In ur first calculatin, w st th lss cfficint t zr, 0 in th CMS mdl s that accrdingly / l 0 in th CMT mdl. Th transmissin charactristics f th tw prts in bth mdls is thn valuatd and plttd against th rund trip phas shift, l with l R in Figur 3.9. Nxt w cnsidr th lss cfficints crrspnding t 5 % and 0 % pwr lss pr turn, rprsnting diffrnt pssibl lsss du t fabricatin prcss tlranc in sidwall tching, xid layr thicknss r matrial intrinsic absrptin, which yild th graphs f Figur 3.0 and Figur 3. rspctivly. Th FDTD rsults ar hr fittd, fr simplicity, t ratinal functins thrugh quatins (9) and () f [ 3] fr th thrugh and drp prt transmissin rspctivly. (a) 67

82 Figur 3.9: (a) Th thrugh prt and (b) th drp prt transmissin charactristics fr a lsslss circuit, - - by th CMT mdl (rd lin), - by th CMS mdl and by th FDTD mdl. Th small shift f rsults by th CMS and th CMT mdls is shwn in th inst. Th rsnanc wavlngth crrspnds t m 9 (b) (a) 68

83 (b) Figur 3.0: As in Figur (3.9) but fr (a) Th thrugh prt and (b) th drp prt transmissin charactristics fr a circuit with 5 % pwr lss pr rund trip (a) 69

84 Figur 3.: As in Figur 3.9 but fr (a) Th thrugh prt and (b) th drp prt transmissin charactristics fr a circuit with 0 % pwr lss pr rund trip. 70 (b) In ur FDTD simulatins, th nrmalizd md f th slab wavguid is launchd int th input prt and th vrlap btwn th utput prt pwr and th launch pwr is valuatd at th utput. Th tim stp is st t m / c and th prfctly matchd layr (PML) bundary cnditins ar mplyd. Th grid siz is varid in bth th latral dirctin (x) and th lngitudinal dirctin (z) starting with 0.08 m dwn t 0.0m whn th valus f pwr transmissin vr th rang f wavlngth f intrst, i.. arund rsnanc, start t strngly saturat. Th FDTD rsnanc wavlngths shwn in Figur 3.9, Figur 3.0 and Figur 3. ar shiftd frm ths gnratd by th thr tchniqus by lss than nm. This is primarily th rsult f th finit grid pint spacing as w hav fund that th FDTD transmissin curvs apprachs ths f th thr tchniqus as th intrval btwn succssiv grid-pints dcrass. This tndncy is vidnt fr th thr valus f lsss that w hav xamind. Hwvr, sinc th tim rquird fr a calculatin fr ur twdimnsinal squar msh implmntatin rapidly incrass with th numbr f grid pints, th CPU tim vn fr ur small ring radius f R.7 m rachs svral days fr th smallst grid pint spacing f 0.0m. This ffctivly prcluds th applicatin f th FTDT mthd t structurs which rquirs a ring rsnatr circumfrnc f abut m t achiv th fr spctral rang f FSR 00GHz typically assciatd with WDM applicatins. Othrwis, th bandwidth and th transmissin paks ar in xcllnt agrmnt amng th thr tchniqus which thrfr als vrifis ur rlatinships btwn th lss cfficints in th CMS and th CMT mdls.

85 Accrdingly, w hav analyzd a cmpund RR circuit basd n th CMT tchniqu and hav cmpard ur rsults t ths f th CMS and FDTD mthds fr bth lsslss and lssy circuits. Hr w bsrvd that whil th CMS yilds a cmplicatd analytic dscriptin f th prblm, th accuracy f th FDTD, can hwvr b insufficint fr larg RR circuits unlss xtnsiv cmputatinal rsurcs ar availabl. Additinally, whil th CMT can b wll-suitd t ring rsnatr basd circuit analysis as suggstd in [ 3], th xampls chsn in this rfrnc f rings cupld in sris r in paralll d nt cnclusivly stablish th rlativ advantags f th CMT sinc ths structurs can b mdld simply by multiplying th paramtric matrics apparing in th CMS mthd. In cntrast, ur circuit includs an intrnal fdback path fr pwr prpagatin that cnsidrably cmplicats th frmalism as dtaild prviusly in this chaptr. In ur mr invlvd xampl, th rlativ simplicity f th CMT analysis bcms far mr vidnt sinc tw sts f CMT quatins rplac sts f CMS quatins whn th cupling and lsss ar small. W als xamind th rlatinship btwn th lss cfficint f th CMT and CMS mdls. Finally, w btaind FDTD rsults that agr wll with th CMS and CMT mthds, and additinally dmnstratd that th ffcts f radiatin mds ar ngligibl fr ur structur. Nt that w hav mplyd th circuit paramtrs in [ 3] in rdr t facilitat ur cmparisn btwn th CMS, CMT and FDTD tchniqus fr th cmpund RR circuit cnfiguratin sinc fr ths paramtrs th cupling and lsss ar sufficintly small that th CMT is applicabl. Hwvr, th CMT mthd cannt analyz th cmpund RR basd intrlavr circuit that w invstigatd abv with th CMS mthd, sinc th cupling cfficints ar t larg. In th fllwing sctin w discuss th dsign, fabricatin and charactrizatin f a intrlavr circuit that implmnts ur CMS mthd rsults. 3.5 Dsign, fabricatin and charactrizatin T satisfy th WDM intrlavr/dintrlavr spcificatins f sctin 3. w apply th fllwing dsign stps. First, th fr spctral rang FSR 00GHz and hnc th ring circumfrnc,l, is fund frm: FSR c ( Hz) ( m) n l n l 8 whr c 30 m/ s is th spd f light in spac, n g th rquird FSR. Th dvic is thn n-rsnanc at nm, with n.4309 and rsnanc rdr m 84 whil th wavguid pwr lss is.4 db/ cm [ 3-3], with ngligibl 7 g ( 3.4) g n dn, is th md grup RI and d n is th md ffctiv RI. T avid multimd bradning, singl-md silicn vr insulatr (SOI) wavguid, with th crss sctin shwn in Figur 3. and paramtrs f Tabl 3. is mplyd. Thrugh a bam prpagatin mthd (BPM) simulatin [ 73] w find, nglcting matrial disprsin, that th TE-lik md, with th dminant lctric fild cmpnnt paralll t th substrat, dn 6 has n.43056, n g 3.98, 0 / m at 550 nm and thus l 755 m fr d

86 bnding lsss as R 5 m [ 3]. Th m thick silica layr supprsss pwr lakag t th substrat lading t an vrall pwr lss pr cycl f dB r 4 % pr cycl. Additinal lsss in ral dvics as a rsult f imprfctins can thn furthr incras lsss up t th prmittd 0% pr cycl [ 43]. T simplify th cupling rati calculatins w rplacd th circular ring with a straight-sidd ractrack and mplyd ur BPM simulatr t dtrmin th cupling cfficint,, btwn tw straight wavguids sparatd by a gap, g, by first dtrmining th rquird cupling lngth as dtaild bfr in Chaptr. w h Si Air h SiO Si substrat Figur 3.: Th singl-md SOI wavguid crss-sctin. Paramtr Silicn RI ( Silica RI ( Si Valu n ). 474 Silica 3 [ 9] n ). 444 [ 9] Air RI ( n a ) w h h 0.5m 0.m m Tabl 3.: Singl md SOI wavguid paramtrs. Th ractrack dimnsins ar dtrmind by first cnsidring th mutually cupld ractracks f Figur 3.3. If L, L rprsnt th lngth f th straight sids and R is th crnr radius, sctin 3. sts th shrtst path n th ractrack btwn th intractin rgins with nighbring ractracks qual t 5 % f th ractrack circumfrnc yilding th dsign ruls: L R L L 755 m ( 3.5.a) L R s l (3.5.b) L L l R (3.5.c) 4 7

87 k k sin( l) (3.5.d) k sin( l) (3.5.) whr l, l and s ar th ractrack-bus and ractrack-ractrack intractin lngths and th sparatin btwn th tw sid ractracks, c.f. Figur 3.3. Fr th additinal ractrack at prt IV, th circumfrnc shuld satisfy L 755 m yilding an intractin lngth with th bus l a givn by : k a sin( l ) (3.5.f) a Th shap f th additinal stag is thrfr dpndnt n th availabl ara n th chip. Tabl 3.3 displays th ractrack dimnsins that satisfy th cnditins g g g with g 0. m, 0.3 m and 0.4 m, whr w hav st L 0 t dcras th md mismatch lss in th transitin btwn th bnt and straight wavguids and als t dcras th circuit ara. Prt I Prt II g L w s L l R Prt IV l Prt III Figur 3.3: A schmatic f th prpsd circuit with rings rplacd with ractracks. g l l 73 L s R la Tabl 3.3: Dimnsins f th ractracks in m with L 0 Th layut was dsignd using Dsign Wrkshp 000 [ 74]. Th dvics wr fabricatd with93 nm phtlithgraphy by PIXfab at IMEC. Intgratd grating cuplrs wr mplyd t cupl light int and ut f th TE wavguid md.

88 In th stup shwn in Figur 3.4, light frm a tunabl lasr pwr is launchd int a plarizatin maintaining fibr tiltd by 0 with rspct t th vrtical. Th grating cuplr is cnnctd t th input prt f th dintrlavr undr tst thrugh a taprd wavguid. A similar arrangmnt, with a rgular fibr thugh, cupls pwr ut f th thrugh and th drp prts f th dvic. Micrscp Whit lamp Th utput fibr Th input fibr Psitinr Chip undr tst Psitinr Psitinr 3 Figur 3.4: Th charactrizatin stup with diffrnt parts labld. Psitinr () hlds th input fibr; whil psitinr () hlds th utput fibr and psitinr (3) hlds th chip undr tst. Aftr aligning th input and utput fibrs, th lasr wavlngth is swpt vr a rang f a fw nanmtrs arund 550 nm with a 0.0nm stp, and th transmissin pwr is rcrdd using a dtctr cnnctd t a PC. Th pwr transmissin is nrmalizd with rspct t its pak valu and plttd as a functin f wavlngth fr ach dvic. Tw cpis f ach circuit as in th layut shwn in Figur 3.5, all with thrtically quivalnt prfrmanc, wr cpid nt 6 diffrnt psitins n th wafr ach with a diffrnt xpsur. Th psitins ar indicatd by a rw numbr: -4,-,-,0,+,+4, with +4, 0, -4 bing th psitins with th maximum, idal (crrspnding t w 500nm) and minimum xpsur lvl rspctivly. Thrfr, thr ar 36 dvics t tst. SEM picturs f thr f th fabricatd dvics ar prsntd in Figur 3.6. A phtgraph f sm f th fabricatd dvics is shwn in Figur 3.7. Masurmnt rsults fr th tw dvics with bst prfrmanc, labld as dvic (A) and dvic (B), ar rprtd hr and displayd in (d) Figur 3.8. Dvic (A) is dvic numbr (4) with xpsur dgr (-) whil Dvic (B) is dvic numbr (6) with xpsur dgr (0). 74

89 Th utput prts 6 Th input prts (a) (b) Figur 3.5: (a) th layut f th six cpis f th prpsd circuit in sctin 3.. Th tw dvics n th right mst (, ), tw in th middl (3, 4) and tw n th lft mst (5, 6) f th chip crrspnd t th dimnsins n th st, nd and 3 rd ntris in Tabl 3.3 rspctivly, (b) th layut f dvic () shwing th input, thrugh and drp prts. On th right is th taprd wavguids fllwd by th grating cuplrs. 75

90 Figur 3.6: SEM picturs fr th fabricatd (a) dvic () with g 0.3m and (c) dvic (5) with g 0.4m g 0.m, (b) dvic (3) with Figur 3.7: An ptical pht fr sm th fabricatd circuits. 76

91 (a) (b) 77

92 (c) (d) Figur 3.8: Masurd and thrtical transmissin charactristics f (a) th thugh prt f dvic (A), (b) th drp prt f dvic (A), (c) th thrugh prt f dvic (B) and (d) th drp prt f dvic (B). 78

93 3.6 Pst fabricatin study Th masurd fr spctral rang fr ths dvics is FSR 0.7 nm, as ppsd t th thrtically calculatd FSR 0. 8nm, indicating that th actual grup rfractiv indx is n 4. 5 nt n 4. Cnsquntly fr a FSR 0.8 nm th ptimum ractrack circumfrnc shuld hav g bn L 670 m instad f L 755m. Th rsnanc wavlngth was additinally shiftd frm th thrtically calculatd nm FSR. Hnc w hav shiftd th thrtical transmissin curvs t bst match th xprimntal transmissin curvs as indicatd in Figur 3.9. Th dviatins frm th thrtical xpctatins als includ rippls in th transmissin band that prbably rsult frm slight diffrncs in th dimnsins f th fiv ractracks in ach dvic which lad t small rsnanc shifts. Th stp rll ff f th curvs nar th transmissin minima rquirs a mr accurat scanning stp than 0.0nm t display th thrtically prdictd minimum valus. Hwvr th gnral faturs f th transmissin bandwidth and rll ff agr with th thrtical curvs. Th apparnt dviatins frm th thrtical xpctatins may cm as a rsult f th fabricatin tlranc f th wavguid and th gap dimnsins as wll as th thrtical assumptins that w fllwd t simplify th calculatins [ 75], [ 76-79]. In a prvius study [ 75], a ring rsnatr circuit basd n idntical wavguid dimnsins, i.. w h 500 nm 0 nm, was fabricatd using th sam tchnlgy but th actual wavguid width was fund t b w 40nm instad f th dsird w 500nm. Thrugh BPM simulatins, w find that such a fabricatin dimnsinal rrr f 80 nm rsults in a chang f n g by 0. and f th pwr cupling cfficint by Similarly, changing w by nm shifts th transmissin spctrum by a wavlngth f 0.96 nm, r quivalntly 76 GHz. Mrvr, as in [ 75] th wavguid walls might b slantd with an angl up t 9 and finally, w hav nglctd th cupling t th rund prtins f th ractracks. A dtaild study n th ffct f fabricatin tlranc n th phas disrdr and th cupling disrdr in ring rsnatr circuits is givn in [ 79]. Accrdingly, w intrducd fabricatin tlranc int ur simulatins by first adding a randm, unifrmly distributd rrr within [ 0.,0. ] t th fild cupling cfficints, whil rstricting th cupling t valus, thn mplying th xprimntal valu, 0.7 nm f th FSRin plac f0.8 nm, r quivalntly n 4. 5 instad f n 4, and finally shifting th transmissin spctra g g fr bst matching btwn thrtical and xprimntal curvs. Th crrspnding simulatin rsults fr ths dvics ar givn in Figur 3.9, dmnstrating a far bttr agrmnt btwn th numrical and th xprimntal rsults. Th cupling ratis ar hr: k k 0. 97, k 0. 95, fr bth dvics, whil k 0. 43, k 0. 57, k and k fr Dvic (A) and k 0.45, k 0. 6, k and k fr Dvic (B), whr k, k, k 3 and k 4 ar rspctivly th fild cupling cfficints acrss th tp-right, bttm-right, bttm-lft and tp-lft gaps in Figur 3.3. a g 79

94 Nting that th fild cupling ratis in th riginal dsign wr: k 0 k , k a and k k k k k 0. 55, w cnclud that small cupling cfficint changs can 3 4 strngly affct th dvic prfrmanc, which mr gnrally prsnts a srius challng in fabricating cmplx dvics such as th intrlavr. Hwvr, mr carful subsqunt fabricatin runs culd prsumably yild imprvd dvics. (a) (b) 80

95 (c) (d) Figur 3.9: Masurd and thrtical transmissin charactristics as in Figur 3.8 but with mdifid fild cupling cfficints. 8

96 3.7 Cnclusin In this chaptr w studid in dtail ur "cmpund ring rsnatr circuit". W custmizd th dsign fr WDM intrlavr/dintrlavr applicatins, gnrating a layut with simplr dsign ruls, smallr ara and cmptitiv prfrmanc cmpard t thr circuits f th sam natur. This circuit was thn mplyd t bnchmark th CMS, CMT and FDTD mdling tchniqus. Dspit th gnrality f th CMS prcdur and th accuracy f th FDTD, th CMT mdl yilds rapid and accurat rsults fr small cupling and small lsss. W thn dsignd, fabricatd and charactrizd an intrlavr/dintrlavr circuit fr WDM pratin. Many cpis f th circuit wr fabricatd with diffrnt wavguid sparatin gaps and bnding radii t stablish th fabricatin tlranc. Our xprimntal masurmnts ar in qualitativ agrmnt with thrtical prdictins fr th circuit prfrmanc. Dviatins btwn th tw sts f rsults rsulting frm fabricatin rrrs culd prsumably b largly liminatd thrugh multipl dsign-tst cycls that wuld clarly stablish ptical prprtis such as th ffctiv indics and lsss f th wavguids and cuplrs and idntify th ptimal dsign paramtrs that wuld cmpnsat, fr xampl, lithgraphy prximity ffcts. 8

97 Chaptr 4 High snsitivity ring rsnatr Gyrscps In th prvius chaptr w analyzd th cmpund ring rsnatr circuit which is simply a clsd lp f rings (CLR), and intrducd a dsign that matchs th rquirmnts f a standard WDM intrlavr/ dintrlavr circuit. Nxt w custmiz th sam structur fr a wavguid gyrscp that dtcts rtatinal mtin thrugh th Sagnac Effct [ 80]. Hr w first utlin rtatinal mtin dtctin with ring wavguids and thn vrviw prvius mathmatical tchniqus fr invstigating ring gyrscps including th fibr ptic gyrscp (FOG), th rsnant FOG (RFOG), and th cupld rsnant ptical wavguid (CROW) gyrscp. Finally, w analyz ur CLR gyr and cmpar ur rsults t prviusly publishd CROW and FOG rsults. 4. Ovrviw Rtatinal mtin can b dtctd by launching tw cuntr prpagating wavs int a rtating lp wavguid thrugh th Sagnac ffct as th rtatinal cntributin t th phas accumulatd by th tw wavs is qual and ppsit [ 80]. Cnsquntly, th intrfrnc signal gnratd by mixing th tw wavs at th utput is a functin f th rtatinal mtin. Whil ring rsnatr circuits hav bn prpsd fr rtatinal mtin dtctin [ 8], th authrs f [ 7] dmnstrat that a cnvntinal FOG with th sam ftprint and transmissin lsss is still mr snsitiv t rtatinal mtin than th crrspnding CROW structur, whr th snsitivity is dfind as th rat f variatin f th circuit utput pwr with rtatinal spd. Hwvr, a standard FOG rquirs lng fibr lngths. In this wrk, w prvid a cmparisn f CROW and FOG gyrscps t th CLR structur discussd blw by varying th wavlngth, cupling cfficints, wavguid lsss, numbr f rings and ring radius. W find that th prfrmanc f th CLR dvic xcds that f th thr structurs; hwvr, th ptimal structur crrspnds t th n in which th fild circls arund th rings f th dvic with maximum cupling btwn rings. This crrspnds ffctivly t a singl ring, and indd, w subsquntly dmnstrat that a simpl ring structur yilds imprvd prfrmanc. Thus, th apprach in this wrk prvids an altrnat mthd t stablish that a singl lp rsnant gyr displays gratr snsitivity t rtatin than thr prpsd structurs f th sam ara, as alrady ntd in [ 8]. 4. Circuit analysis 4.. Sagnac ffct Cnsidr a ring rsnatr with a man radius R rtating at an angular vlcity with a cntr lcatd at a distanc R frm th cntr f rtatin as shwn in Figur 4..a. If an lctric fild prpagats thrugh th ring frm a psitin at angl with th hrizntal t a psitin at angl, th rati f th lctric fild at th lattr and frmr lcatins is i is whr i l, 83

98 l is th rund trip phas shift, l R, n is th fild prpagatin cnstant, n is th fild ffctiv indx, is th fr spac wavlngth, is th pwr lss cfficint in m and th Sagnac phas shift inducd by th rtatinal mtin [ 80-8] is s d s, with d s V. dr c whr V ( R R ) is th linar vlcity f th sgmnt dr. Hr c is th vacuum spd f c light and is th angular frquncy f th fild. Frm Figur 4..a, R s ( ) v (sin( ) sin( )), with R V. Th first trm c ( ) is assciatd with th phas shift du t th rtatinal mtin if th ring is cntrd at th cntr f rtatin, i.. R 0, whil v (sin( ) sin( )) is th additinal phas shift du t th shift btwn th ring cntr and th cntr f rtatin. In th fllwing w assum a clckwis rtatin dirctin and dnt th input fild and pwr by a and p whil th filds and pwrs at th utput prts A and B ar dntd by a, p and a, p B rspctivly. W assum furthr that th rings f all gyrscps ar n-rsnanc as in [ 7] t maximiz th snsitivity t rtatinal mtin, which implis that l m, with m an intgr, whr w mply th cnvntin that fr a frward travlling wav bth phas trms ar ngativ fr th cas f Figur 4..a, whn th fild travls in th clckwis dirctin f rtatin, s that dr. V 0 fr. If th wav travls ppsit th dirctin f rtatin, and th trm ( ) is psitiv, whil if th fild prpagats such that dr. V 0, th trm v(sin( ) sin( )) is psitiv. f Fr an FOG, as shwn in Figur 4..b, with numbr f turns f f R N f, lp radius A A R f and ttal lngth l R N th nrmalizd fild transmissin f th tw prts can b asily prvn t b fa fii a A i ab ( fi fii ) and fb ( fi fii ), whr a a l / il in f f f. fi l / il in Fr a singl ring with radius R s btwn th tw arms f a Mach-Zndr cntrd at rtatin axis, ab c.f. Figur 4..c, th nrmalizd transmittd fild thrugh is thn s ( si sii ), with a r r l s / ils i si ls / ils i r r k r whr k, r r ls / ils i sii ls / ils i r r, l s R s f f f B and, k r, k ( ) is th fild rati cupld btwn th ring and th uppr (lwr) bus. 84

99 85 Th nrmalizd fild drp transmissin is similarly ) ( sii si A s a a with / / i l i l i l i l si s s s s r r k k and / / i l i l i l i l sii s s s s r r k k. (a) (b) (c) Figur 4.: (a) A ring rtating abut a cntr f rtatin at a distanc R frm its cntr, (b) a fibr ptic gyrscp (FOG) and (c) a singl ring gyrscp.

100 4.. Crw gyrscp Th cupld rsnatr ptical wavguid (CROW) gyrscp prpsd in [ 8] and [ 7] cnsists f a clsd lp with an dd numbr, N [ 7], f cascadd rings cupld in sris and fd thrugh 3dB cuplrs as in Figur 4.. Th rati f th fild cupld btwn th lftmst (rightmst) rings and th nighbring buss is dntd by k ( k ), whil that btwn tw nighbring rings is dntd by k. Nt that in [ 8] th dfinitin f th fild cupling rati is th squar rt f th pwr cupling cfficint. W thn dfin th cupling matrics, r r r Q, Q ik r, Q ik r and th phas matrics: ik r 0 p P 0, p 0 p4 P 0, p 0 p3 P 3 3 0, p 0 p P 4 4 p 0. whr k r, i i ( ) iv sin( / ) ii iv sin( / ) i i ( ) iv sin( / ) p, p, p, i i iv sin( / ) N p4,, ( ) and fr a unifrm N plygn th vrtics f which ar th cntrs f th rings as shwn in Figur 4.. Thn ai ai aii aii T and bi b T, with a I and a II as dfind in Figur 4. prducing tw I bii bii utput fild cmpnnts, th thrugh b ( b II ) and th drp b I ( b II ), whil a I a II 0. N QPQ ( P QPQ ) I Thn, w hav T Q Q and T QPQ 3 ( P4 QP3 Q) Q Q. Hnc it is bi T bii T asy t prv that ci and cii ar th fild drp-prt ai T(,) aii T (,) bi T (,) bii T (,) transmissin cfficints, whil ci and cii ar th fild a T (,) a T (,) I thrugh-prt transmissin cfficints. Finally, th nrmalizd fild transmissin at th tw utput a A i ab prts A and B ar givn by ca ( ci cii ) and cb ( ci cii ) rspctivly. a a 3 II N 86

101 Figur 4.: CROW gyrscp with N Lp f ring gyrscp A CROW gyrscp cntains a cmplt circl f wavguid rings trminatd by a 3dB cuplr. Thrfr, th fild can prpagat multipl tims thrugh ach ring accumulating additinal Sagnac phas shift in th cas f wak cupling btwn rings. If th 3 db cuplr is rplacd by an additinal ring w thrfr arriv at th CLR gyrscp in which th fild can prpagat multipl tims arund th ntir structur, accumulating an additinal Sagnac shift. W accrdingly cnsidr an vn numbr f rings; N cupld arund a circl, as shwn in Figur 4.3. Th cupling btwn th tp (bttm) ring and th nighbring bus is rprsntd by Q ( Q ) and th cupling btwn tw nighbring rings is Q. Th phas matrics ar p P 6 0, with p7 i i iv sin( / ) p and p 8 p i i iv sin( / ) 5, i i ( ) iv sin( / ) p 0 5 p P 5 0 and p6 i i ( ) iv sin( / ) 6,. W thrfr mdify th frmalism f Chaptr 3 by substituting P 5 and P 6 in plac f P and P. This yilds a fild thrugh prt bi r i p8 transmissin cfficint I and drp-prt transmissin cfficint a r p I b a I I k k 4 p Chaptr 3 xcpt that ( y 3 y4i ) r i p I i and 7 i 8. Th transmissin cfficint xprssins ar thn as givn in ( ) p 7 i in plac f p7 i i 87

102 i ( ) and i rspctivly. Th transmissin cfficints fr cuntr clckwis i bii bii lctric fild flw, i.. II and II, ar givn by th sam xprssins but with a a II II a A ab rplacd by. Finally, ( I II ) and ( I II ). Th dsign fr a a dd valus f N / yilds mr cmplx circuit layuts sinc th utput signal prts must b cupld thrugh crss-vr wavguids, hnc w rstrict ur attntin t N 4,8,... Figur 4.3: CLR gyrscp with N Summary f prvius CROW and FOG rsults Bfr prsnting rsults n ur nw structurs, w first lucidat th faturs f th CROW and FOG dsigns f [ 8] and [ 7] that ar rquird in th cmparisn with th additinal wavguid structurs xamind in th nxt sctin. First, w nt that if th CROW gyrscp calculatins lading t Figur (3) in [ 8], which w bliv wr carrid ut fr.55 m, 0. 0 and 0 ar graphd in S.I. units, w arriv at th rsults f Figur 4.4 whr is givn in rad / s such that Hz rad / s. As xpctd, th magnitud f th CROW gyrscp snsitivity S incrass as ( N ) whil frm Figur 4.4.a, th nrmalizd pwr lvl at th utput prt (B) is fr a singl ring CROW and 0 fr a -ring structur. Nt that vn with th mr physical input valus f [ 7], S varis as ( N ), but whil quatin 3 in [ 7] indicats that th utput pwr at prt B varis as sin ( s ) s ( N ) fr small and mdrat, Figur 5 f this rfrnc dmnstrats immdiatly that this ds nt apply t lssy wavguids. 88

103 Thus dtcting th utput signal is challnging fr a lsslss structur with many rings. This prblm is nt vidnt in [ 8] sinc mplying arbitrary units fr physical quantitis such as th utput fild intnsity at prt B masks S valus in th rdr f 0 8 [/ Hz ]. T prv that ur units crrspnd t ths mplyd in th calculatins f [ 8], in Figur 4.4.b, w plt instad th rlativ snsitivity S S against which cincids with Figur 3.b in [ 8]. Finally, w nt that N N whil S is a functin f, Figurs 3.c and 3.d f [ 8] graph S against and R rspctivly withut spcifying. Hwvr, if w idntify th crrspnding with th magnitud f th maximum rtatinal angular vlcity in Figur 3.a f [ 8], namly 000Hz 000 rad / s, and plt S against and R, w arriv at Figur 4.4.c and Figur 4.4.d rspctivly fr unit input pwr which indd again agrs with [ 8]. Hwvr, an input pwr f mw yilds S 0 4 Ws / rad 0 4 W / Hz, which implis that th dvic is clarly impractical spcially if w cnsidr ralistic valus fr 0 and << 000 Hz. W bliv this is th ffct f mplying miniatur radius ( R 5m ) in this calculatin as wll f th pwr lss thrugh th unusd prts in th circuit as ntd in [ 7]. p (a) 89

104 (b) (c) 90

105 (d) Figur 4.4: A CROW circuit prfrmanc with.55 m (a) Th nrmalizd utput pwr at prt B as a functin f rtatinal spd ( ) fr a CROW with R 5 m, 0 and k k k 0., (b) Th rlativ snsitivity as a functin f th rtatinal spd fr th sam CROW with R 5 m, 0, and k k k 0., (c) Th CROW snsitivity as a functin f th pwr cupling cfficint ( ) with N 9, 000Hz 000 r / s, R 5m and 0, (d) Th CROW snsitivity as a functin f th ring radius ( R ) with N 9, 0, 000Hz 000 r / s and k k k 0. In [ 7] a prcdur is givn fr calculating th CROW gyrscp dimnsins that yild th sam L f ftprint and lss as a givn FOG; namly, R f R N, and pa / p, rspctivly whr p A / p is th nrmalizd dtctd pwr at prt A f a CROW. Sinc, fr th cas f Figur (5.c) f [ 7] N 8, and R 5cm, th quivalnt FOG has 45 cm and L f ln( pa / p) ln(0.6) 0. /000/ Lf 093m, yilding N f 393 fibr turns. R Th CROW pwr transmissin and snsitivity ar shwn in Figur 4.5.a and Figur 4.5.b, whr th lattr is sn t cincid with th Figur 5.c f [ 7]. Evidntly thn, th CROW gyrscp, whil nt as snsitiv as a FOG, th far smallr dimnsins favr this structur in intgratd gyrscp applicatins. f R f 9

106 (a) (b) Figur 4.5: (a) Th nrmalizd pwr transmissin f a CROW gyrscp with N 8, R 5cm, and 0. db/ km at. 55m. (b) Th snsitivity f th CROW gyrscp and its quivalnt FOG f R f 45 cm, N 393 turns, and L f 093 m. f 4.4 Numrical rsults: In this sctin w cmpar th CLR gyrscp t quivalnt CROW and FOG structurs and thn dmnstrat that a gyrscp cnsisting f a singl ring in fact gnrally dmnstrats suprir prfrmanc. T d this, in cntrast t [ 7],[ 8] whr th ring-ring and th ring-bus cupling cfficints ar idntical w nly assum qual cupling btwn th buss and thir nighbring rings,.g. k k but rgard th ring-ring fild cupling rati k as an indpndnt valu that rangs hr frm t In ur calculatins, w dfin th snsitivity at prts A and B fr 9