Design And Optimization Of Nanostructured Optical Filters

Transcription

1 Univeity of Cental Floida lectonic Thee and ietation octoal ietation (Open Acce) eign And Optimiation Of Nanotuctued Optical Filte 8 Jeemiah Bown Univeity of Cental Floida Find imila wok at: Univeity of Cental Floida Libaie Pat of the lectomagnetic and Photonic Common and the Optic Common STARS Citation Bown Jeemiah "eign And Optimiation Of Nanotuctued Optical Filte" (8). lectonic Thee and ietation Thi octoal ietation (Open Acce) i bought to you fo fee and open acce by STARS. It ha been accepted fo incluion in lectonic Thee and ietation by an authoied adminitato of STARS. Fo moe infomation pleae contact lee.doton@ucf.edu.

2 SIGN AN OPTIMIZATION OF NANOSTRUCTUR OPTICAL FILTRS by JRMIA ANIL BROWN B.S. Phyic and Mathematic Univeity of Alabama in untville M.S. Optic Univeity of Cental Floida 6 A dietation ubmitted in patial fulfillment of the equiement fo the degee of octo of Philoophy in the epatment of Optic in the College of Optic and Photonic at the Univeity of Cental Floida Olando Floida Sping Tem 9 Majo Pofeo: ic G. Johnon

3 9 Jeemiah aniel Bown ii

4 ABSTRACT Optical filte encompa a vat aay of device and tuctue fo a wide vaiety of application. Geneally peaking an optical filte i ome tuctue that applie a deigned amplitude and phae tanfom to an incident ignal. iffeent clae of filte have vatly divegent chaacteitic and one of the challenge in the optical deign poce i identifying the ideal filte fo a given application and optimiing it to obtain a pecific epone. In paticula it i highly advantageou to obtain a filte that can be eamlely integated into an oveall device package without equiing exotic fabication tep extemely enitive alignment o complicated conveion between optical and electical ignal. Thi dietation exploe thee clae of nano-cale optical filte in an effot to obtain diffeent type of dipeive epone function. Fit dipeive waveguide ae deigned uing a ub-wavelength peiodic tuctue to tanmit a ingle T popagating mode with vey high econd ode dipeion. Next an innovative appoach fo decoupling waveguide tajectoie fom Bagg gating i outlined and ued to obtain a unifom econd-ode dipeion epone while minimiing fabication limitation. Finally high Q-facto micocavitie ae coupled into axiymmetic pilla tuctue that offe extemely high goup delay ove vey naow tanmiion bandwidth. While thee thee novel filte ae quite divee in thei opeation and taget application they offe extemely compact tuctue given the magnitude of the dipeion o goup delay they intoduce to an incident ignal. They ae alo deigned and tuctued a to be fomed on an optical wafe cale uing tandad integated cicuit fabication technique. iii

5 A numbe of fequency-domain numeical imulation method ae developed to fully chaacteie and model each of the diffeent filte. The complete filte epone which include the dipeion and delay chaacteitic and optical coupling i ued to evaluate each filte deign concept. oweve due to the complex natue of the tuctue geometie and electomagnetic inteaction an iteative optimiation appoach i equied to impove the tuctue deign and obtain a uitable epone. To thi end a Paticle Swam Optimiation algoithm i developed and applied to the imulated filte epone to geneate optimal filte deign. iv

6 To my beloved wife Megan fo he patience and encouagement though thi long endeavo. v

7 ACKNOWLGMNTS Fit and foemot I would like to thank my wondeful wife Megan whoe patience devotion and encouagement ha aided me moe than wod can expe. e love and attentivene have helped me focu on the final goal in thi wok and have povided me with the additional incentive to complete thi degee. I would alo like to expe my heatfelt appeciation to. ic Johnon whoe guidance inight and diection have taught me a geat deal. I am extemely gateful fo the chance to tudy unde hi tutelage and complete my eeach with hi aid. I am alo paticulaly thankful to my committee membe. Jim Mohaam. Geoge Stegeman. on Malocha and. Winton Schoenfeld fo thei time and aitance. Futhemoe the National Science Foundation waant paticula mention fo it financial aid in the completion of thi wok. Alok Mehta and Tippe Rumpf povided invaluable aitance and inight at vaiou tage of my eeach and I am exceedingly gateful. I would alo like to thank the othe membe of the Mico-Photonic Lab goup and othe aociate at CROL fo thei fiendhip and contibution thoughout thi poce. Finally I want to thank my fiend and family fo all that they have done along the way and paticulaly my fathe fo intilling in me a love fo phyic a deep appeciation fo the beauty of mathematic and a wonde at the myteie of electo-magnetic field. vi

8 TABL OF CONTNTS LIST OF FIGURS...xiii LIST OF TABLS... xvii LIST OF ABBRVIATIONS... xviii CAPTR INTROUCTION.... Baic of Optical Filteing.... Fequency-ependent Phae and ipeion..... Goup elay..... Quadatic ipeion ighe Ode ipeion Summay and eciption of ipeion Clae of Optical Filte ipeive Guiding Stuctue Type of ipeion in Waveguide ighly ipeive Guiding Stuctue eign of ipeive Waveguide....5 Bagg Gating Filte Bagg Gating Theoy Pinciple of Chiped Bagg Gating Goup Velocity in Bagg Gating eign of Bagg Stuctue... 8 vii

9 .6 Nano-Scale Reonant Stuctue Reonant Optical Filte Mode of Optical Reonato Lo Mechanim and Quality Facto Mode Volume Reonant nhancement eign of Reonant Cavitie Sign Convention Optical Filte eign Summay CAPTR RSARC OVRVIW Optical Filte eign eign and Optimiation Tool Numeical Analyi of Optical Filte Optimiation of Optical Filte ipeive Waveguide Filte eign Bagg Gating Filte eign Micocavity Filte eign Reeach Summay... 4 CAPTR 3 FRQUNCY OMAIN MOLING OF OPTICAL FILTRS igentate of Optical Filte Modeling Field on a Gid Finite iffeence Appoximation viii

10 3.. Field Repeentation Maxwell quation Fomulation Pefectly Matched Laye Bounday Condition Mateial Teno xpeion Cylindical Coodinate Modification of Maxwell quation igenfequency Calculation Poblem Symmety eivative Opeato and Bounday Condition Tanvee Mode ighe Ode Mode Quality Facto of igenfequencie Model Benchmak igenmode Calculation Mode xpeion ipeion Calculation Bent Waveguide Mode Method of Line Calculation Method of Line Fomulation eivation of Tanfe Matix valuation of Matix Function Laye Tanition ix

11 3.7.5 oubling Algoithm Fequency omain Model Summay CAPTR 4 OPTIMIZATION OF LCTRO MAGNTIC STRUCTURS Pinciple of Optimiation Method of Pobabilitic Seache Paticle Swam Mechanic Initialiation Solution valuation Paticle Motion Bounday Condition iect Compaion of Paticle Swam and Genetic Algoithm Appoache Paticle Swam Optimiation Summay... 9 CAPTR 5 ISPRSIV WAVGUI FILTRS ipeion in Standad Waveguide Nanotuctued Waveguide Analyi of ipeion Amplified Waveguide Theoetical Analyi Altenative Configuation Coupling Conideation Nano-AWGS a elay Line ipeive Waveguide Summay... CAPTR 6 SPIRAL BRAGG GRATING FILTRS... 3 x

12 6. Bagg Filte Chaacteitic Spial Bagg Stuctue eciption Linealy Chiped Bagg Stuctue Coupling Stength Conideation Additional eign Containt Quantification of Fabication Iue Altenative Goup elay Function Contant Peiod elay Line Long Peiod Gating Tailoed Phae elay fo ipeion Compenation Spial Bagg Stuctue a Amplitude Repone Filte Spial Bagg Filte Summay CAPTR 7 AXISYMMTRIC RSONANT CAVITY FILTRS Single Cavity Optimiation Paticle Swam Optimiation of Cavity eign Mode Choice etemination of Cavity Fitne igh Q-facto Cavity eign Coupling to Axiymmetic Mode Coupled Reonato Filte Coupled Reonato Optical Waveguide Coupling of Low Q-facto Reonato xi

13 7.3.3 GaA/AlA Cavity Filte Coupled igh-q Cavitie Filte Thee-imenional Filte Micocavity Filte Summay CAPTR 8 CONCLUSION Backgound Optical Filte eign and Modeling Numeical Modeling Tool Numeical Optimiation Appoach Summay of Reult Nano-AWG Spial Bagg Stuctue Coupled Cavity Filte Filte Compaion Recommendation fo Futue Wok RFRNCS xii

14 LIST OF FIGURS Figue -: Relative Phae elay of Fequency Component Caue Boadening of Signal Figue -: Pule Boadened fom p to n with iffeent Magnitude of Goup elay Ripple... 7 Figue -3: Reflective Bagg Gating Figue -4: Goup Velocity in a Fibe Bagg Gating Figue -5: ielectic Cylindical Cavity Figue -: Nano ipeion Amplified Waveguide Stuctue Figue -: Spial Waveguide Radial Gating and the Reulting Combination Figue -3: Multiplexing of GaA/AlA Cavity Filte Figue 3-: Field Poitioning fo (ab) -; (c) - T; (d) - TM; (e) 3-; (f) Axiymmetic Simulation Figue 3-: Field Component Location fo Axiymmetic Simulation Figue 3-3: Q-facto ependence on Cavity Length (a) Publihed and (b) Simulated Figue 3-4: Layeing of Pemittivity Region... 7 Figue 4-: Fitne Map of Ratigin Function... 8 Figue 4-: Initial Paticle itibution in Paamete Space Figue 4-3: Paticle Movement Figue 4-4: Standad Paticle Swam Bounday Condition Figue 4-5: Paticle Attempting to Leave Bounded Paamete Space Figue 4-6: Paticle Swam Bounday Tanfom... 9 Figue 4-7: (a) Genetic Algoithm Veu (b) Paticle Swam Convegence Rate Figue 5-: T Mode ipeion in Ridge Waveguide xiii

15 Figue 5-: Nano ipeion Amplified Waveguide Stuctue Figue 5-3: T Mode Pofile in Nano-AWG Stuctue Figue 5-4: negy enity fo (a) T Mode and (b) TM Mode in Nano-AWG Stuctue Figue 5-5: ipeion Cuve fo Nano-AWG Stuctue.... Figue 5-6: Second Nano-AWG Stuctue. (a) T Mode negy enity; (b) TM Mode negy enity... Figue 5-7: ipeion Cuve fo Second Nano-AWG Stuctue.... Figue 5-8: ffective Index and ipeion fo Tue T Mode in an Infinite Gating Figue 5-9: ipeion fo T Mode in a Similaly imenioned Ridge Waveguide Opeating Nea Cutoff Figue 5-: negy enity and lectic Field fo Altenative Fin Waveguide Layout... 6 Figue 5-: negy enity and lectic Field in (a) ual-laye and (b) Multi-Layeed Tench Bulge Waveguide Figue 5-3: (a) Symmetic and (b) Antiymmetic Mode fo Nano-AWG Coupling Appoach Figue 5-4: Goup Index fo (a) Fit and (b) Second Nano-AWG deign.... Figue 6-: Spial Waveguide Radial Gating and the Reulting Combination Figue 6-: Nonideal Bagg Gating Chip... 5 Figue 6-3: Pixilation of a Single Gating Peiod Figue 6-4: xpected Reflectance and Goup elay fo m Linealy Chiped Bagg Stuctue.7 Figue 6-5: xpected Goup elay Ripple fo Linealy Chiped Stuctue Witten with.5nm and 5nm Pixel... 7 Figue 6-6: Goup elay fom Contant Radiu Waveguide with (a). and (b).5 Index iffeential... 9 Figue 6-7: Goup elay Ripple fo (a) Fit-Ode Bagg Gating and (b) Thid-Ode Bagg Gating with nm Nomally itibuted Random Petubation to Peiod Sie xiv

16 Figue 6-8: epatue fom Linea Goup elay Cuve fo Waveguide with Linea Angula ependence... 3 Figue 6-9: Sinuoidal Waveguide Tajectoy (xaggeated fo ffect) Figue 6-: Sinuoidal Gating Peiod Figue 6-: Amplitude Repone fo Sinuoidal Waveguide Tajectoy Figue 7-: epiction of Single-Cavity Geometie Figue 7-: igh Q-facto Micocavity Figue 7-3: Cylindical-Bagg Micocavity and Amplitude Repone Figue 7-4: (a) Reonant Wavelength and (b) Q-facto fo Combination of Cavity Geometie Figue 7-5: Cavity with Longitudinal Confinement Povided by (a) BR Laye and (b) Optimied Reflecting Laye Figue 7-6: Coupling Methodology fo Micocavity Reonato Figue 7-7: SPAC Stuctue and Output Aimuthally Polaied Beam Figue 7-8: igh NA Lene (NA =.45) Figue 7-9: Aimuthally Polaied Field at igh NA Len Focu Figue 7-: (a) and (b) Component of Radially Polaied Beam at igh NA Len Focu Figue 7-: lectic Field Intenity of Radially Polaied Beam at igh NA Len Focu Figue 7-: Amplitude Repone of 6 Coupled Cylindical Cavitie Sepaated by (a) nm (b) nm and (c) 3 nm Figue 7-3: Co-Section and 3- Pofile of GaA/AlA Filte Unit Cell Figue 7-4: (a) Reonant igenmode and (b) Amplitude Repone of GaA/AlA Filte Figue 7-5: Multiplexing of GaA/AlA Cavity Filte Figue 7-6: Amplitude Repone of a evice with (a) 4 and (b) 64 Cavitie Figue 7-7: ffective Index of Popagating Mode in (a) Cavity and (b) Pot Region xv

17 Figue 7-8: Amplitude Repone with Cavity Spacing of (a) nm (b) 375nm (c) 45nm and (d) 475nm... 6 Figue 7-9: Amplitude Repone fo Cavity Radiu of (a) 357nm and (b) 49nm Figue 7-: Amplitude Repone fo 6-Cavity Filte Calculated via (a) Method of Line and (b) Finite iffeence-time omain... 6 Figue 7-: Radial Co-Section of Reonant Fequency Field Intenity in Filte... 6 Figue 7-: (a) Tanmitted and (b) Reflected Signal Figue 7-3: Amplitude Repone and Goup elay fo (a) Tanmitted and (b) Reflected Signal Figue 7-4: Coupled Cavity ielectic Pilla Filte Figue 7-5: Single Cavity Amplitude Repone with (a) 4 BR Pai; (b) 6 BR Pai; and (c) 8 BR Pai Figue 7-6: Coupled Cavity Pai Amplitude Repone with (a) 4 BR Pai; (b) 6 BR Pai; and (c) 8 BR Pai Figue 7-7: (a) 4 Coupled Cavitie and (b) 8 Coupled Cavitie Amplitude Repone with 6 BR Laye Pai Figue 7-8: (a) (b) (c) 4 and (d) 8 Coupled Cavitie with Inceaed Coupling Contant. 7 Figue 7-9: Goup elay fo Reduced Coupling Contant Stuctue... 7 Figue 7-3: igh Confinement Couple Cavitie Figue 7-3: Single Cavity Amplitude Repone with (a) 6 and (b) 8 BR Laye Pai... 7 Figue 7-3: Amplitude Repone a a Function of Coupling Contant Figue 7-33: Phae Repone of Coupled Cavitie Spaced by (a) (b) (c) 4 and (d) 6 BR Laye Pai Figue 7-34: Amplitude Repone Change ue to Sloped Sidewall fo (a) Cavity and (b) 4 Cavity Stuctue Figue 7-35: Amplitude Repone a a Function of Coupling Contant xvi

18 LIST OF TABLS Table 3-: Compaion of Tanvee Mode Reonance Table 3-: Compaion of Vaiou Mode Reonance Table 7-: Summay of PC-ncaed Coupled Cavitie Phae Repone Table 8-: Compaion of Optical Filte eign xvii

19 LIST OF ABBRVIATIONS CROW CW BR M FF FT FIR FSR FWM GA GR IIR MOL MPF NA Nano-AWG PC PC PCV PMF PML PSO QPM RCWA S SG SPAC T TM WM Coupled-Reonato Optical Waveguide Continuou Wave itibuted Bagg Reflecto lectomagnetic Finite iffeence-fequency omain Finite iffeence-time omain Finite Impule Repone Fee Spectal Range Full-Width alf Max Genetic Algoithm Goup elay Ripple Infinite Impule Repone Method of Line Minimum Phae Filte Numeical Apetue Nano ipeion-amplified Waveguide Photonic Cytal Pefect lectic Conducto Plama nhanced Chemical Vapo epoition Polaiation Maintaining Fibe Pefectly Matched Laye Paticle Swam Optimiation Quai-Phae Matching Rigoou Coupled Wave Analyi Spontaneou miion Second amonic Geneation Spatially Polaiing Autocloned Stuctue Tanvee lectic Tanvee Magnetic Wavelength iviion Multiplexing xviii

20 CAPTR INTROUCTION. Baic of Optical Filteing Optical tuctue of all hape and ie ely on thei ability to bend and contol fequencie of light. The tem optical filte i an exceedingly boad deciption that eentially include any tuctue that pupoely ditinguihe between diffeent fequency component of an incident ignal and teat them in diffeent way. The tanfomation applied to an input ignal can be in tem of amplitude phae o both. The fome i accomplihed pincipally though ome fom of eonance o intefeence while the latte involve dipeive and phae delay effect. While a numbe of tuctue uch a all-pa filte offe minimal amplitude ditotion ome degee of fequency-dependent phae i alway applied by the filte to the incident ignal. Ignoing nonlineaitie optical filte conit of linea time-invaiant ytem which may be chaacteied in the time domain in tem of an impule epone function h(t) []. Given an input ignal x(t) the output y(t) i defined a: t xt ht x ht In the fequency domain thi elationhip become whee y d (.) X Y (.) i e (.3)

21 The complex-valued tanfe function () i the pimay focu of filte deign poblem. It magnitude povide the amplitude ditotion of the incident ignal while the fequency-dependent phae () decibe the phae accumulated fo a given fequency component upon tanmiion though the filte. Some of the moe inteeting effect eulting fom filte ely upon the pectal dependence of thi phae function.. Fequency-ependent Phae and ipeion.. Goup elay The pinciple govening the induced pectally-dependent phae o dipeion ae well known though it i illutative to highlight a few baic detail. The phae may be witten in tem of a popagation contant of an optical mode inide the filte (.4) c neff whee n eff i the effective index of the guiding tuctue and decibe the popagation ditance. Auming that the amplitude ditotion may be ignoed and we ae able to teat the filte a a guiding tuctue along which a ignal tavel popagation of a pecific fequency hamonic a ditance though a filte take the fom a i a e (.5) Since we geneally conide optical pule with a eaonably mall fequency bandwidth popagating though media whoe optical epone ha finite deivative with epect to fequency it i convenient to expand qn..4 in a Taylo Seie about a cente fequency ω : While thi teatment doe not pecifically apply to filte baed on mechanim othe than guiding tuctue (uch a cavitie o gating) the mathematical teatment may till be ued to a cetain degee though the identified vaiable may take on a diffeent phyical meaning.

22 O 3 (.6) v g v g i known a the goup velocity and it invee i the goup delay pe unit of popagation ditance which decibe the phae accumulated by a popagating optical ignal. It i given by v g d d n eff c dneff c d (.7) Thi epeent a delay expeienced by optical ignal popagating though the tuctue. The actual goup delay fo a ignal popagating a ditance i given by g d (.8) v d g It i alo eaonable to define a goup index n g which povide a deciptive figue of meit fo evaluation of degee of delay a given tuctue povide. The goup index i defined accoding to n g c g dneff dneff c neff neff (.9) v d d g and offe a value in tem of delay pe unit length. Fom thi we ee that goup delay only become highly ignificant nea hap eonant peak in the effective index cuve. Thu obtaining a lage goup delay equie deigning a device eithe to opeate nea mateial eonance peak o to ely on a geomety that ceate a eonance condition o opeate nea a modal cutoff condition. 3

23 .. Quadatic ipeion ω in qn..6 decibe the peading o dipeion of the optical ignal a a function of fequency. It i often convenient to wok in tem of wavelength intead of fequency o we ue an equivalent definition fo dipeion : c c n eff (.) c Retuning to qn..5 we know that an optical ignal decibed in the fequency domain i elated to the time domain deciption via a Fouie Tanfom pai: A i a A te t d (.) it i t g t a e d e a e d 4 i t v (.) qn.. i analytically olvable fo a few diffeent type of input ignal. The mot common one i a Gauian pule of wait. Afte popagating a finite ditance the new beam wait ha a dependence on the dipeion of the medium and i given by whee (.3) c (.4) The dipeive tem in the exponent of qn.. induce fequency chip wheeby the pectal component of an optical ignal ae no longe centeed on top of each othe (Figue -). Thi can wok to one advantage a a pule incident on the dipeive tuctue aleady chiped

24 with a ign oppoite of the fequency deivative of the popagation contant fo the tuctue will expeience compeion intead of boadening ince the fequency component ae puhed back on top of each othe by the tuctue dipeion. Figue -: Relative Phae elay of Fequency Component Caue Boadening of Signal...3 ighe Ode ipeion An additional conideation when dealing with pule dipeion i the cae whee highe ode dipeion i peent. In ome cae it i quite ignificant though in mot cae involving imple pule compeion and expanion it how up a a light ipple in an othewie linea goup delay cuve. Thi can be epeented a an additional fequency-dependent phae tem incopoated into qn... The phae can geneally take the fom p a p GR co (.5) 5

25 whee ω p give the fequency of the ipple and τ a give it amplitude. define the phae between a ipple peak and the pectal cente of the ignal. Following the deivation of [] we note that thi phae eult in a tem of the fom e i co n i n J n e in (.6) accoding to the Jacobi-Ange expanion. J n () i a Beel function of the fit kind. We can then evaluate the new pule accoding to n n A t i J p p a e in A' t n n (.7) Thu goup delay ipple (GR) eult in a continuum of ovelapping ignal with diffeent amplitude and tempoal cente. The phae diffeence aco the ovelapping pule eult in a noticeable ditotion of the final ignal. Figue - demontate the effect of GR on a pule boadened fom p to n. The fequency of the ipple i taken to be T and the eult ae plotted fo diffeent value of the amplitude. A can be een GR caue ocillation in the output ignal though the magnitude ha to be a noticeable pecentage of the oveall goup delay befoe it ignificantly affect the output quality. 6

26 Figue -: Pule Boadened fom p to n with iffeent Magnitude of Goup elay Ripple...4 Summay and eciption of ipeion The fequency-dependent epone of optical tuctue eult in thee diffeent ode of dipeion that apply a phae tanfom to an incident ignal in dipeive waveguide and delay line filte. The fit i imply temed delay o goup delay and decibe a time delay applied to an optical ignal. Thi i the pincipal conideation fo nealy monochomatic ignal o tuctue that ae not dominated by quadatic dipeion. Thi i paticulaly applicable in a vaiety of ituation uch a time ynchoniation in communication ytem o optical buffeing of data until it may be poceed [3]. The econd type of dipeion known a quadatic dipeion o imply dipeion decibe a diffeence in goup delay fo diffeent fequency component of an optical ignal. A dicued above a chomatic pule impinging upon a dipeive guiding tuctue eceive a diffeent delay fo each component fequency and i thu boadened a it popagate. 7

27 Telecommunication imaging ytem militay ue fo high-powe lae and numeou othe optical application elying on ulta-hot pule equie mean fo compenating o contolling dipeive effect on uch optical ignal. In paticula thee method make ue of dipeive guiding tuctue to boaden pule to bette contol individual fequency component. Additionally in high-peed communication ytem dipeion-induced pule boadening i aleady peent and dipeive tuctue ae needed to compenate fo thi and to etoe the oiginal ignal. Othe application [4] make ue of dipeive delay fo time ynchoniation o data buffeing [3] o to enhance nonlinea inteaction by lowing the goup velocity [5]. To inceae the ate at which a pule i pead o compeed the dipeion mut be inceaed. Fo example conide a p Gauian pule at.55 micon. xpanding the pule to p within a mm length of a dipeive waveguide equie a tuctue with dipeion on the ode of 68 p nm - km -. A typical optical fibe ha a dipeion of aound - p nm - km - nea thi wavelength. To expand the ame pule up to n within the ame length of waveguide equie a dipeion of nealy 4 p nm - km -. The thid categoy of dipeion involve the highe ode tem in qn..6. Wheea quadatic dipeion eult fom a convolution of the fequency component of a pule with a linealy-vaying fequency-dependent goup velocity the highe ode tem eult in a futhe convolution of the fequency-domain phae-hifted pule with a et of Beel function. In geneal the highe ode dipeion eult in noie in the output ignal and i uually undeiable. Fo a elatively naow ignal bandwidth thee tem may have minimal effect though fo mot optical filte the highe ode tem ae minimied a much a poible [6]. 8

28 .3 Clae of Optical Filte Optical filte can lagely be divided into two baic clae baed on thei filteing mechanim [7]. Finite impule epone (FIR) filte ae eentially ingle-pa device. They do not ely on feedback mechanim o optical eflection. Thu ignal delay i limited and tictly detemined by the length of the tuctue. ipeive waveguide and Mach-Zende baed filte fit into thi categoy [4 8]. The altenative cla temed infinite impule epone (IIR) filte elie on multiple eflection and feedback mechanim. Filte in thi cla include Bagg gating eonato and many all-pa filte. Thei eonant natue allow them to poduce extemely lage goup delay fo fequencie nea eonance. The cla of IIR filte may futhe be divided into two categoie baed on the elation (if any) between the amplitude epone and phae epone of the filte. Specifically if ln and () fom a ilbet Tanfom pai eithe the phae o amplitude epone i ufficient to fully chaacteie the filte. Given one the othe i completely detemined [7]. Thee type of tuctue temed minimum phae filte (MPF) include Faby-Peot ytem and ome Bagg gating tuctue [4]. Non-MPF filte do not have the elationhip between the phae and amplitude epone and thu offe an additional degee of feedom in tem of the filte deign poce. The non-mpf categoy of IIR filte include chiped Bagg gating and all-pa filte. To impove the epone of the filte it i common pactice to couple multiple filte to each othe [9]. Thi can povide a naowe line width fo eonant tanmiion line fom cetain filte though coupling and fabication eo can diminih the pectal epone athe than impove it. 9

29 .4 ipeive Guiding Stuctue.4. Type of ipeion in Waveguide In bulk media dipeion i caued by the vaiation of mateial efactive index with epect to wavelength. In waveguide tuctue dipeion i caued by two additional effect: the diffeence in popagation contant fo diffeent mode and the dependence of a ingle mode popagation contant on fequency. So long a the tuctue i ingle-moded the fome play no pat and may be ignoed. Mateial dipeion i geneally quite mall in compaion with the dipeive value and i only dependent on the mateial ued o thi type of dipeion cannot be adjuted though modification to the waveguide hape. ence thi dicuion will be confined to conideation of waveguide dipeion and an exploation of the way diffeent fequencie inteact with the waveguide hape to allow the tuning of it phae epone into a deied optical filte..4. ighly ipeive Guiding Stuctue A waveguide with no longitudinal tuctual vaiation hould intoduce minimal amplitude ditotion of the incident ignal and mut ely olely on a phae epone a it filteing mechanim. Additionally ince the tuctue doe not ely on eonance effect the magnitude of the intoduced dipeion mut be a lage a poible to povide a eaonable filte epone within a elatively hot waveguide length. Thu we pecifically look fo tuctue that contain a vey high goup delay o conideable quadatic dipeion depending on the application. We ae pimaily concened with guiding tuctue of a wafe-ie cale that may be fomed uing tandad lithogaphic and nanofabication pocee and that poduce dipeive delay on the ode of p/nm. While bulk mateial may yield ubtantial dipeion nea

30 eonance they alo poduce a athe ignificant degee of aboptive lo. The focu of thi eeach will be on lage dipeion magnitude obtained though the actual tuctue of the waveguide. We alo etict ou conideation to low intenity incident light and neglect nonlinea effect in thee type of guiding tuctue..4.3 eign of ipeive Waveguide A numbe of diffeent appoache have been taken to vaying the co-ectional geomety of guiding tuctue to poduce ubtantial goup delay and dipeion. Although pai o combination of waveguide ae not geneally well known fo thei dipeion pimaily becaue mot baic deciption aume identical waveguide they can pove quite effective unde cetain cicumtance. When the tuctue diffe ubtantially diimila mode may actually fom coupled mode with dipeive magnitude dependent on the diffeence in goup delay fo the two individual guide. Thei opeation ha been exploed and dicued elewhee [ ]. Plamonic [ 3] and photonic cytal waveguide [4-6] make ue of combination of mateial and geometical popetie to achieve eonance effect uitable fo a tong fequency dependence. An additional appoach peented hee adjut the geometical co ection of waveguide in the manne of a ubwavelength gating o an effective index mateial puhing the mode cloe to cut-off to obtain highly dipeive behavio [7]. Neglecting catteing loe the eulting filte function fo a length L of the eulting waveguide will have a filte epone of the fom i L e (.8)

31 whee () i the popagation contant decibed ealie. The deign poblem cente on optimiing fo the deied epone. To adequately obtain a pecific magnitude of dipeion and goup delay one mut caefully tune the geometic hape of the tuctue. Thi i paticulaly tue when it opeate vey nea mode cut-off condition a the tuctue bandwidth become quite limited and a deign wavelength can be hifted into a catteing mode if the geomety change too much. The deign poce equie a mean to analye the co-ectional geomety detemine the effective efactive indice fo a ange of fequencie and fom thi infomation deive an etimate of goup delay and dipeion..5 Bagg Gating Filte Bagg tuctue and chiped Bagg gating epecially a ued in optical fibe ae ome of the moe ecogniable optical filte [8-] paticulaly in the ealm of pule compeion and tetching. They involve the coupling of light fom one popagating mode into anothe mode (eithe fowad o backwad popagating) [ ] in a manne highly dependent on fequency o moe pecifically on the atio of the gating fequency to the popagating wave fequency. One of the bigget dawback with thee tuctue i the inheent fabication limitation which inevitably eult in the intoduction of non-ideal dipeion in the fom of GR to the ytem. Vaiou tudie have conideed the magnitude of the impact of the GR on the quality of the ytem a a whole [ 3 4] while othe have conideed a vaiety of mean to compenate fo thi poblem [5 6]. Additional difficultie with Bagg tuctue include the ignificant length of the tuctue needed fo any ubtantial degee of dipeion and the coupling lo fo the typical Bagg eflecto aangement.

32 .5. Bagg Gating Theoy Thee ae a eveal way to appoach the opeational theoy of Bagg eflecto. We geneally conide a hallow gating fabicated on top of a waveguide (often an optical fibe) a demontated in Figue -3 in a fom known a a ditibuted Bagg eflecto (BR). Fo diffaction gating the well known Bagg condition i given by k m k mk (.9) inc whee k m and k inc give the popagation vecto fo the m th diffacted ode and the incident field epectively and K give the gating vecto. Fo ufficiently mall gating peiod the only popagating ode ae the tanmitted and eflected mode. The Bagg condition implifie to n B (.) which decibe the gating peiod L needed to diffact the maximum poible enegy into the eflected mode given an aveage waveguide index n and fee pace wavelength λ B. Figue -3: Reflective Bagg Gating. We geneally utilie a gating that povide a vey mall index contat which put it in the weak coupling egime. We can tun to coupled mode theoy to futhe decibe the behavio of the tuctue. If A give the field tength of the fowad popagating mode and B give the tength fo the eflected mode the coupled-mode equation can be witten a da ibe d db iae d i i (.) 3

33 4 whee κ i the coupling contant and i given by B n n (.) which obviouly goe to eo when the Bagg condition i met. The coupling contant fo the Bagg gating tuctue i given by [4 7] B n n n (.3) whee Δn i the index modulation depth and i the confinement facto of the mode inide the waveguide. By taightfowad olution of qn.. it may be hown [7] that the olution take the fom i i e L in i L co L in i A B e L in i L co L in i L co A A (.4) whee the bounday condition A() = A and B(L) = have been aumed (field incident fom the left only). L i taken to be the length of the gating and. Thu the complex filte epone fo the tanmitted and eflected beam ae given epectively:

34 L A A B A co L i inl i inl co L i inl e il (.5) Notice that if the incident light fequency fall within a top band defined by become imaginay and the field tength of the fowad popagating wave decay exponentially with ditance along the waveguide. xactly at the Bagg wavelength the eflectance become R tanh L. If the magnitude of the coupling contant i mall (uually due to the intoduction of a vey mall efactive index modulation) and the gating length i ufficiently long vey pecific band of fequencie may be electively eflected. Thu Bagg tuctue intoduce a definite amplitude tanfomation to the optical ignal in contat with the dipeive waveguide tuctue whee the filte epone conit pimaily of an induced phae vaiation..5. Pinciple of Chiped Bagg Gating We now note that if the gating peiod i gadually changed o chiped along it length diffeent fequency component may be given a elative phae delay popotional to the ditance between coeponding Bagg gating peiod. Thi ituation mean that both κ and ae function of. The exact olution of qn.. i nontivial and i not uually olved diectly. A bette olution i obtained by conideing each ection of unifom-peiod gating individually and identifying a x tanmiion-line matix fo it. The full filte epone i given by the poduct of each of thee matice [8]. Since the matice ae eldom tivial the poduct i uually calculated numeically and may be ued to evaluate the filte epone of abitay non- 5

35 unifom gating [8]. oweve an appoximate epone may be dicued fom a puely analytical pepective. To obtain unifom econd-ode dipeion fom a chiped Bagg tuctue we would apply a unifom linea chip to the gating peiod. The time delay between two pectal component eflected fom diffeent poition along the gating epaated by a ditance L i given by L t n (.6) c whee c i the peed of light and n i the effective index of the guiding tuctue. The pectal width between thee component can be imilaly detemined by making ue of qn..: (.7) n Thi give a value fo the dipeive delay of the tuctue a L t nl c (.8).5.3 Goup Velocity in Bagg Gating Anothe effect common to eonant tuctue uch a Bagg gating involve the change of the goup velocity of a mode. It wa mentioned ealie that the phae epone of a unifom Bagg gating i given by the ilbet tanfom of it amplitude epone. Thu lage change in the filte amplitude epone will eult in a coepondingly lage change in it phae epone. Such a vaiation in the phae coepond diectly to a lage goup delay and a eonant tuctue opeating nea the band edge of a filte ee goup velocity dop quickly to eo (ee Figue -4). Thi effect can alo be decibed in tem of the multiple eflection the 6

36 mode expeience befoe it i able to leak out of the tuctue. Fom eithe pepective the inteaction between the mode and the device geomety i ignificantly inceaed. Thi effect ha been pecifically employed to enhance the gain available to a given eonant cavity in the fomation of one-dimenional band-edge lae [9]. Simila effect have been obtained though caeful tuning of two- and thee-dimenional geometie [3 3]. Figue -4: Goup Velocity in a Fibe Bagg Gating. The lage goup delay intoduced nea the edge of the band can alo be ued to geat advantage in cetain delay line application. Although thee i only a vey mall bandwidth to which the lage delay value apply thi appoach offe a mean by which the Bagg tuctue may be ued in a tanmiion ytem obviating the need fo a potentially loy coupling cheme to epaate out the incident and eflected ignal [9]. The goup delay in the pa band i given by [3] nl g (.9) c which apidly divege a the wavelength appoache the top band. 7

37 .5.4 eign of Bagg Stuctue Analyi of qn..8 eveal a numbe of advantage and diadvantage of thee tuctue. Fit thee tuctue can have an extemely lage bandwidth given an appopiately lage chip. The dipeive delay can be inceaed by imply inceaing the length of the tuctue while actual dipeion i dependent tictly on the effective index of the tuctue and the total chip. A device in gla may have a bandwidth of 5 nm and dipeion on the ode of 6 p nm - km - quite eaily by uing a total gating chip of 5 nm. Unfotunately to make effective ue of uch magnitude of dipeion that chip mut be tetched out ove ten of centimete. At thi point fabication conideation come into play and one mut define ome unique manne in which to unifomly inceae the gating peiod by faction of a nanomete without intoducing ignificant GR. Vaiou appoache have been attempted to mitigate thee difficultie [6]. A vaiety of othe filteing application can alo be obtained though a nonlinea functional vaiation of the gating peiod along the gating length. Thi can be utilied to tune the filteing chaacteitic of the tuctue by intoducing pa- and top-band to vaiou fequency component in addition to the induced goup delay. While the epone may be evaluated fo an abitay gating peiod function detemining the neceay function to obtain a deied filte epone become a much moe challenging poblem..6 Nano-Scale Reonant Stuctue.6. Reonant Optical Filte Optical cavity eonato act much like thei analog in the acoutic wold whee tuning fok eonate at pecific fequencie baed on thei ie. Much of the teminology may alo be 8

38 boowed fom the aea of electonic whee eonato ae compoed of inducto-capacito cicuit. The popetie of micocavity eonato have been exploed in geat detail and ae exploited fo myiad application. Wheea waveguide ae baed pincipally on the concept of tanmitting ignal eonato ae device deigned to toe optical enegy and build up high field intenitie [33]. In ecent yea they have been mot ueful in quantum electodynamic expeiment and have povided excellent ouce and filte in optical communication [34]. While many type of cavitie have been fabicated they commonly fit into one of thee baic geometie. Faby-Peot cavitie ae baed on the concept of highly eflective coating (often in the fom of BR laye) at eithe end of a guiding tuctue typically foming a ot of pilla tuctue [35]. In contat whipeing galley cavitie make ue of a cicula o elliptical path fo the popagating component of the eonant mode. Thee often come in the fom of micophee o dik o ing eonato [36]. The final tandad eonato geomety conit of a defect inide a photonic cytal tuctue [37 38] whee a tanding wave take on a moe two- o thee-dimenional natue in compaion to the Faby Peot (whee a tanding wave i in the axial diection) and the whipeing galley (whee a tanding wave i oiented aimuthally) tuctue..6. Mode of Optical Reonato.6.. Wave quation fo Cavity Reonato The deivation of the eonant mode inide cavity eonato begin with Maxwell equation: i i (.3) 9

39 The tandad deivation of the wave equation involve taking the cul of Faaday Law and ubtituting Ampee Law into it aiving at (.3) k If the pemittivity in the egion can be teated a piecewie contant we may obtain a imila expeion fo the magnetic field in each of the epaate egion: k (.3) In ectangula coodinate the left-hand ide of both equation will imply futhe uing the vecto identity: (.33) Thi can be futhe implified by incopoating Gau Law: (.34) Since we ae again auming piecewie contant pemittivity the divegence of the electic and magnetic field ae both identically eo inide each egion of unifom pemittivity. Unfotunately in non-cateian coodinate ytem the Laplacian of the vecto field ha no independent definition and qn..3 and.3 cannot be implified in thi manne which would nomally poe a poblem fo cylindical geomety. oweve thi may be ovecome by expeing the wave equation in tem of the longitudinal component of the field [33]. Thu combining qn..33 with qn..3 and.3 we obtain k (.35)

40 whee the Laplacian opeato i expeed a (.36) The tanvee component ae obtained by diect application of Maxwell quation (qn..3) to thi eult and olving in tem of the longitudinal component. If the field vecto ae witten a expeed a [39] T ẑ and T ẑ the tanvee component may be k i T T T ẑ (.37) T k T i T ẑ (.38) whee T i the tanvee gadient opeato and we have aumed. Thi elie on ou ability to epeent an abitay field ditibution a a upepoition of plane wave though a Fouie Tanfom..6.. Cylindical Symmety The cavity geometie conideed in thi eeach ae pincipally otationally ymmetic. In uch cae the aimuthal dependence of all olution implifie geatly. The bounday condition equie that a given olution and all it deivative mut be continuou at =. Thu all eonant olution fo a otationally ymmetic cavity mut have the fom in e (.39) Theefoe the aimuthal deivative ued to detemine the vaiou field component may be expeed a

41 in (.4) Note that the ign of the aimuthal mode numbe may be witched without affecting the olution of the wave equation. Although the oppoite ign would eult in an independent olution et the adial dependence i identical and it may afely be ignoed without lo of geneality Solution fo ielectic Cylinde A a pecific example conide the geomety depicted in Figue -5. A dielectic cylinde of pemittivity with adiu a and length L i embedded in a emi-infinite egion of pemittivity c. The cylinde may be poitioned o that the bottom end it on a ubtate egion with pemittivity. The implicity of the geomety ugget that a epaable olution of the fom Z R hould be appopiate. Thu qn..35 become k n Z R (.4) and imilaly fo the field. To obtain ealitic behavio of the field at vey mall and vey lage adial value the appopiate olution to the adial equation take the fom a k K a k J c n n R (.4) The longitudinal olution ae given imply by i Z e (.43)

42 An equivalent olution et may be expeed fo the magnetic field [33]. Figue -5: ielectic Cylindical Cavity Radial Solution The eonant fequencie and hence wavenumbe k of the cavity ae one of the key qualitie we look fo when witing paticula olution to the equation above. To obtain the exact expeion fo the fequencie equie implementing bounday condition and matching tanvee field component at the edge of the cavity. Gau Law equie the electic and magnetic field component tangential to the adial uface of the dielectic cylinde to be continuou. ence the olution fo and fo adii le than a mut be continuou with the olution fo adii lage than a. To implify the expeion omewhat we make the following change of vaiable: h q k k c (.44) Thu fom qn..4 we have fo ah CK aq AJ n n (.45) and fo 3

43 ah K aq BJ n n (.46) whee A and B ae the amplitude of the longitudinal electic and magnetic field inide the cylinde and C and ae the amplitude outide the cylinde. To obtain the aimuthal field component the olution of qn..4 and.43 ae ubtituted into qn..37 and.38. The field-matching expeion fo the aimuthal electic field component become n AJ h n ah i hbj ' ah CK aq i qk ' aq q n n n n (.47) Similaly continuity of the aimuthal magnetic field component equie n BJ h n n ah i haj ' ah K aq i qck ' aq n q n c n (.48) The geneal olution to all fou continuity expeion eult in the following [33]: J n' ahj n ah ah Kn' aqk n aq J n' ah aq ahj n ah ckn' aqk n aq n aq ah aq k (.49) While the geneal cae i a athe convoluted expeion the imple tanvee cae baed on n= implify thing conideably. It can eaily be hown that the eonant fequencie fo T mp mode which aume eo longitudinal electic field come fom etting the leftmot tem in quae backet to eo while the fequencie fo the TM mp mode deived fom the aumption that = come fom etting the ightmot tem fom the left hand ide of qn..49 to eo. By ubtituting in qn..44 and olving numeically one obtain value fo k in tem of. Fo a complete numeical value one mut tun to the longitudinal olution to 4

44 detemine. In geneal thee will be multiple numeical olution to qn..49. Thee value coepond to the diffeent adial mode numbe m Longitudinal Solution The dicuion in the peviou ection povide the tandad olution fo popagating field in cylindical waveguide. oweve the eonant popety of a cylindical cavity involve tanding wave athe than taveling wave. Thi may be epeented by taking a fowad and backwad popagating mode of equal amplitude and umming the two to obtain a inuoidal field dependence [39]: Z in (.5) The tandad appoach at thi point i to aume pefect electical conducto (PC) at the top and bottom uface of the cylinde in which cae = and L = p whee p take on intege value and efe to the longitudinal mode numbe of the given eonance. A puely dielectic eonato doe not include the equiement that the electic field go to eo at the end of the cylinde and the mode will extend outwad into the uounding egion eulting in a lage value fo..6.3 Lo Mechanim and Quality Facto The eult of the peceding dicuion point to the peence of a et of dicete eonant mode in a given cavity. Futhe thoe mode take the fom of tanding wave. Fo uch olution the total enegy contained in the cavity will be a time-independent contant value. Specifically the enegy will tanition fom electical to magnetic and back again in an optical analog to the ideal cicuit eonato coniting of an inducto and a capacito. 5

45 Thi theoetical model lead to two conideation. Fit it i eaonable to peume that any non-ideal eonato peent in any ealitic envionment mut be able to contain optical ignal at fequencie othe than the pecified eonance. Thi i paticulaly tue in the cae of loaded eonato which ae coupled to othe tuctue by ome mean. Light of vaiou fequencie may be diectly injected into the eonant cavity. In loaded eonato the eonant mode may exit at ome fequency othe than the pecified eonance. The amplitude of the mode will deceae with the diffeence between the injected fequency and the ideal eonant fequency [4]. The econd iue of note i that tuctue that pemanently contain contant amount of optical enegy ae of no pactical value. A moe ueful tuctue i one into which we can couple a packet of enegy toe it fo ome length of time and then extact it again at ome late time peiod. Thi equie that ome mechanim exit by which the contained enegy in the cavity may deceae ove time. The pimay eaon fo thi enegy lo involve ome combination of intenal diipative aboptive loe into the cavity mateial and coupling between the intenal eonant cavity mode and extenal catteing and taveling wave. The eonant chaacteitic of optical cavitie ae temed in like manne to thei cicuit analog. In addition to the eonant fequencie the othe popety of note i the cavity quality facto (Q-facto) of the coeponding eonance. Thi unitle numbe i given a Q U du (.5) dt whee i the fequency of a given eonance and U i the toed enegy. Note that thi i a diffeential equation fo toed enegy whoe olution take the fom 6

46 U t t Q U e (.5) Thi ugget a uitable mean to incopoate the Q-facto diectly into the eonant fequency. If the fequency i aumed to be complex with the fom i the expeion fo the contained enegy in the cavity eult in the following [4]: U i t t t t e V V V t e V V V e i t (.53) Thu we may expe the Q-facto in tem of the atio between the eal and imaginay pat of the eonant fequency: Re Q (.54) Im Fom cicuit theoy the tanfe function of a eonant cicuit element i given by a Loentian function in tem of the input fequency: i iq (.55) Fom thi elation one can define a fequency bandwidth Δ coeponding to the full-width half-max (FWM) of the tanfe function. Thi i elated to the Q-facto in the following way: Q (.56) One additional apect that i not immediately evident i that the Q-facto often inceae fo highe-ode mode. Thi make ene if one viualie the highe-ode eonato a a 7

47 collection of malle eonato containing lowe-ode mode. Thi could be accomplihed by poitioning PC at each of the node in the oiginal eonato. Thu only the outemot eonato ha impefect loy idewall and the total powe diipation fom the tuctue i ignificantly le than the um of the enegy loe fom a collection of individual low-ode eonato with impefect idewall. On the othe hand the total enegy contained inide the tuctue i exactly equal to the um of the enegy in each of the malle eonato. Theefoe the atio of the contained enegy to powe diipation and hence the quality facto i ignificantly inceaed fo highe-ode eonance [4]..6.4 Mode Volume An additional chaacteitic of inteet fo eonant cavitie i the mode volume. While the contained cavity volume i immediately intuitive the mode volume i an integal of the volume of pace weighted by the field intenity. Thi value povide a bette indication of the oveall ie of the contained mode and can indicate it degee of confinement. Lage mode volume ae ueful in cavity amplifie a the field i pead ove a lage gain egion. On the othe hand cavity quantum electodynamic expeiment ely on vey mall mode volume to obtain an enhancement to the pontaneou emiion ate [ ]. The effective mode volume i given by the volume integal of the field intenity divided by the peak intenity [4]: V m max dd (.57) 8

48 .6.5 Reonant nhancement.6.5. Spontaneou miion and Pucell Facto Some of the majo application fo eonant micocavitie involve quantum electodynamic expeiment and luminecent ouce. Both cae typically make ue of a quantum dot o dipole located inide a cavity and coupled to it eonant mode [34]. When the coupling i vey tong the atomic ouce inteact coheently with the eonant mode eulting in an entangled tate and a plitting of the tanmiion peak [44]. The weak coupling ytem wa oiginally decibed by Pucell [45] who pedicted that if the diipation time fo a photon emitted by an atom o quantum dot inide a cavity wee hote than it adiative lifetime then eaboption would be minimal and the pontaneou emiion (S) ate would be enhanced [4]. In a eonant cavity with a lage Q-facto the mode denity i damatically inceaed which lead to the enhanced S ate [46]. The enhancement i quantified by the Pucell facto f n Q V m (.58) whee /n give the eonant wavelength inide the cavity Goup Velocity and Slow Light While the low goup velocity i typically obeved in eonant device incopoating a highly eflective tuctue at eithe end which i uually effected by mean of BR laye a imila eult may be obtained in cavitie with unifom geomety in the axial diection [47]. If a cylindical cavity i coated in the adial diection with highly eflective laye (typically eithe BR laye metallic film o a photonic cytal tuctue) a pai of eonant mode (uch a the 9

49 T and TM mode) will epel each othe and can eult in anomalou dipeion and even eo goup velocity unde the ight condition. Unde uch condition the Q-facto of the cavity i damatically enhanced if the cavity dimenion ae caefully balanced with the eflection at the end cap of the cavity. Without condition at the end eticting the field at the boundaie a pai of mode exit fo each eonance ode (tanding ine wave veu coine wave). By expeing the tanfe function fo the pai of mode in tem of cavity length and eflection at both end one may obtain it eigenvalue. The Q-facto fo the pai of eonance i given in tem of the cavity length and goup velocity a follow [48]: L Q (.59) v g At point whee the tanfe function eigenvalue ae cloet to (thei imaginay pat dop to eo) the Q-facto peak tongly. The ditance between the peak i cloely elated to the end cap eflection and cavity geomety [49] Nonlinea ffect In eonant optical filte electomagnetic enegy can become highly concentated in vey mall aea. Thi lead diectly to nonlinea effect which can eithe poe ignificant deign challenge o povide an inteeting and exploitable filte chaacteitic. Thid-ode nonlineaitie pincipally appea in the fom of the Ke effect in which the efactive index of a mateial vaie a a function of the incident field intenity: n I (.6) n 3

50 n i the mateial-dependent Ke contant and I i the incident field intenity. In tanmiion line thi effect eult in elf-phae modulation which can take the appeaance of tandad dipeion [5 5]. Fo optical cavitie expected to opeate unde pecified field tength thi effect may be calculated and applied a a petubation to the cavity pemittivity to fully analye a given deign. oweve the actual hape of a eonant mode tongly influence the tength of the Ke effect o it i difficult to include in the fit-ode deign phae. Anothe nonlinea effect that may be explicitly incopoated into the filte deign i econd hamonic geneation (SG). Thi effect eult in a polaiation popotional to the quae of the incident field: P (.6) () : Thi mean that an incident pump beam can poduce an output ignal at twice it initial fequency. Thi ha been expeimentally demontated in a vaiety of tuctue including photonic cytal [5] which may be deigned to incopoate a ign eveal of the nonlinea tem to povide quai-phae matching (QPM) [53 54]. In the cae of optical micocavitie we have the poibility of dual-eonance cavitie. By deigning the tuctue to be eonant at both fundamental and hamonic mode we obtain ignificantly enhanced SG eulting fom a diect coollay of the Pucell effect outlined above [55 56]. Seveal iue ae caue fo conideation when attempting to deign uch a cavity. Fit thee ae iue with phae and mode matching. SG can eult in a econd-ode pump polaiation eithe exactly in phae o out of phae with the eonant hamonic ignal. The fome i pefeed while the latte hould be uppeed by appopiate cavity deign [57 58]. 3

51 Futhe the mode mut ovelap ufficiently to ceate a phae-matched inteaction. The elevant ovelap integal may be expeed a [59] S (.6) A an additional conideation the polaiation of the elected eonant mode mut match the coeponding value of the nonlineaity teno. The teno i commonly witten in the fom d d d 3 d d d 3 d3 d 3 d 33 d eff (.63) d4 d 4 d 34 d 5 d 5 d 35 d6 d 6 d 36 whee the d ij tem of the teno ae given uch that i give the eulting econd-ode polaiation (x y and epectively) and J indicate the combination of the two fundamental field component to be multiplied (xx yy y x and xy epectively). Mot highly nonlinea mateial only have a few noneo teno tem thu the polaiation of the fundamental and hamonic mode fo cylindical cavity eonato mut be choen caefully. Fo example conide a eonato uing a T pump mode. If T mp hamonic mode ae expected the nonlinea mateial mut have lage d and d teno value (d 33 may alo be ued if the cytal i oiented popely). Unfotunately mot mateial do not exhibit uch popetie (though SiO could be ued if one i content with vey low conveion). On the othe hand GaA ha a vey lage value fo the d 4 teno element. Thi indicate an x-polaied 3

52 hamonic given a poduct of an y and fundamental. A T pump mode ha a polaiation given by xˆ in yˆ co (.64) Thi allow u to obtain a econd-ode polaiation of the fom P d ˆ in co d ˆ in 4 4 (.65) Notice that thi coepond to the -component of a econd-aimuthal-ode mode and thi then hould be the choice of the eonant hamonic mode to be optimied..6.6 eign of Reonant Cavitie Reonant optical cavitie offe a unique and fitting appoach to a wide vaiety of optical application. The ability to toe optical enegy fo ome length of time caue it to inteact with ome mateial geomety and then eleae it fom the bai fo vaiou optical filteing methodologie. Taking a boad pectum of input fequencie and tipping out a naow band centeed on a pecific eonance i eential to photoluminecent tuctue of vaiou type. oweve a thi ection ha highlighted the opeation and pectal chaacteitic of eonant cavitie vay damatically with hift in the geometical configuation of the tuctue. To make effective ue of uch cavitie fo the many poible application mentioned peviouly two deign tool ae cucial. Fit a method to quickly obtain the optical chaacteitic of the cavity i vital. While a baic analytic appoach ha been outlined in thi chapte a igoou model applicable to abitay cavity geometie i neceay. The econd tool of impot to the cavity deign poce i a mean to optimie the geometical configuation and tweak the tuctue in vaiou way to obtain pecific eonance 33

53 condition. Simple cylindical cavitie can lagely be deigned by an analytical method and even in moe complicated tuctue a ough idea of the neceay cavity ie can be etimated. oweve the toleance fo ome of thee paamete ae extemely limited if one i to obtain a high Q-facto eonance at a pecific fequency and they may be too difficult fo back-of-theenvelope method. Alo when ealitic geometie uch a tilted cavity idewall and vaiation in laye thicknee ae taken into account the equied paamete combination hift beyond the ange of pediction available to implitic deign method. Finally even when cavitie ae deigned fo pecific eonance condition a cavity-baed optical filte equie a complete epone function which involve coupling chaacteitic. In one-dimenional eonant filte o in thoe baed on a waveguide coupling light into and out of eonant cavitie the epone tanfe function ha hap peak aound the eonant fequencie and elatively flat epone away fom them [6 6]. In thee-dimenional cavitie light that doe not couple into the cavity i not neceaily e-coupled into a eflected mode in the manne of waveguide-baed eonato. A lage faction of the enegy away fom eonance can couple into cavity catteing mode and leak out of the device. Thu a complete analyi of cavity eonant filte equie a complete model fo light coupled into and out of the tuctue a a function of fequency..7 Sign Convention Thi eeach make epeated ue of Maxwell quation and vaiou deign and analyi method deived fom them. Solution to Maxwell quation ae typically expeed a taveling wave of the fom (.66) i t k e 34

54 Unfotunately the ign in the exponent i not conitent aco diffeent field. The - ign i moe common among phyicit while enginee migate towad the + ign (in addition to the ue of j in place of i ). While the diffeent convention have equivalent meaning one mut apply them conitently in any deivation paticulaly in the fomulation of numeical modeling appoache. Thi eeach make ue of the +i ign convention thoughout..8 Optical Filte eign Summay Optical filte and eonato include a vat pectum of device and application. To fully dicu optical filte in detail i well beyond the cope of thi eeach. oweve thi chapte ha povided a theoetical analyi of thee diffeent type of filte. ipeive waveguide tuctue fall within the categoy of finite impule epone tuctue and ely on length caling to intoduce a time delay to the optical ignal. Thei filte epone pimaily intoduce a fequency-dependent phae vaiation with little in the way of amplitude modulation (at leat fo fequencie above cut-off). Infinite impule epone filte include the two peented example of a Bagg gating tuctue and a eonant micocavity. Thee type of tuctue ely on eonance and multiple eflection and can poduce magnitude of delay that ae independent of the device ie. In addition thee two filte intoduce an amplitude vaiation to the output ignal and epaate pecific fequency component fo eithe tanmiion o eflection. Lage delay magnitude uually occu eithe at o vey cloe to the eonance of thee tuctue (a would be pedicted fom the ilbet tanfom of the amplitude epone). oweve in ome way thee two type of tuctue ae oppoite of each othe. Bagg tuctue ely on eonant popetie to eflect naow band of fequencie and lagely tanmit 35

55 the et. Micocavitie ae ued pimaily to elect and tanmit naow eonant fequencie while eflecting (o imply ejecting) off-eonant component. A baic theoetical famewok fo the deciption of each of thee filte ha been peented in thi chapte. Succeeding chapte will deal with the tool needed to adequately model the ealitic epone of each type of filte. Additionally we exploe a popoed method fo taking the actual filte epone and feeding it back into it geometical deciption. Thi allow u to optimie the filte deign fo a deied pectal epone. 36

56 CAPTR RSARC OVRVIW. Optical Filte eign Optical filte find ue in a vat aay of application fom telecommunication to ignal poceing and detecting. One of the mot ignificant challenge in obtaining a filte fo a given application i the ability to quickly deciphe the neceay tuctue geomety that will povide the given filte epone. In the optical filte deign poce one mut fit detemine the type of filteing capabilitie equied. A filte that imply eek to intoduce a phae delay ove a given fequency band will ue a vey diffeent device than one that need to electively emove vey naow fequency component fom a boadband ignal. Beyond that an appopiate method mut be devied both to model the epone of a deigned optical filte and to adjut it geomety to obtain a epone function that moe cloely matche that equied by the given application. Thee clae of optical filte have been identified and dicued in detail. Innovative deign appoache will be peented fo filte in each categoy and the neceay numeical modeling and optimiation tool equied to develop each filte will be dicued.. eign and Optimiation Tool In ode to adequately analye and deign any uch tuctue a numbe of numeical technique i equied. The fit categoy of deign tool conit of method that evaluate Maxwell quation fo a given geomety in an attempt to detemine the inteaction of electic and magnetic field with the tuctue. Thee tool allow one to analye a given tuctue and 37

57 detemine it oveall electomagnetic popetie and it epone to given bounday condition (expeed a optical ouce). The othe type of deign tool ued in thi eeach i a numeical optimiation method. Optical filte typically have a wide vaiety of geometical vaiable each affecting the filte epone in a diffeent manne. The goal of numeical optimiation i to find a combination of geometical paamete that allow the filte to pefom to ome pedefined citeia... Numeical Analyi of Optical Filte CAPTR 3 peent a numbe of fequency-domain numeical technique. Thee method aume monochomatic condition and evaluate Maxwell quation aco a given filte geomety fo a ingle optical fequency. To find the oveall pectal epone one mut ecalculate the epone at a lage numbe of cloely paced fequencie aco the band of inteet. The fequency-domain technique peented heein ae theefold. Fit an eigenfequency olve involve expeing Maxwell quation a an eigenvalue poblem with the optical fequency given by the eigenvalue of the ytem. Thi allow one to obtain the eonant fequencie and coeponding Q-facto of a given filte geomety. Since eonance condition ae inheently aumed thi method i pimaily ued to analye the pectal chaacteitic of eonant cavitie. The econd numeical method imilaly expee Maxwell quation a an eigenytem but in contat with the fit appoach aume that the field may be expeed a wave popagating along the -axi. In thi manne the eigenvalue of the ytem ae the wave 38

58 popagation contant fo a given optical fequency. Thi method lend itelf well to the analyi of dipeive waveguide. The final technique known a the method of line (MOL) i lagely an extenion of the econd method to geometie that vay in the popagation diection. Specifically popagation contant can be obtained in each dicete laye in the axial diection and then the et ae connected uing electomagnetic bounday condition to obtain a ingle tanfe matix decibing the epone of the entie tuctue. Thi appoach allow one to obtain both eflectance and tanmittance fo an abitay incident electic field and can povide both phae and amplitude epone. It can be ued fo mot type of optical filte and will be pecifically applied to the analyi of Bagg gating tuctue and optical cavity filte in thi eeach... Optimiation of Optical Filte valuation of the phae and amplitude epone fo a given geomety i eential to the oveall analyi of optical filte but the deign phae typically equie incemental impovement to deign to poduce a pecific filte epone. Fo ome imple filte and epone function equiite geometie may often be deived fom fit pinciple. oweve complex epone function and elaboate filte geometie may equie an iteative numeical optimiation method. CAPTR 4 outline the paticle wam optimiation (PSO) tool one of the moe ecently developed pobabilitic each algoithm. Thi method povide a eaonably fat and efficient way to adjut filte geometie to obtain deied filte epone. 39

59 .3 ipeive Waveguide Filte eign The fit majo categoy of optical filte dicued above involve waveguide with lage phae epone. In thee tuctue the geomety i deigned to inceae goup delay and dipeion. CAPTR 5 exploe a pecific example of a dipeive waveguide filteing mechanim (illutated in Figue -). Thi geomety opeate vey cloe to cutoff fo T-type popagating mode and thi combined with the peiodic natue of the tuctue eult in vey lage dipeive magnitude. The actual capabilitie and epone of the tuctue (temed a Nano ipeion Amplified Waveguide o Nano-AWG ) i dicued in detail. Coupling to the tuctue and altenative appoache ae alo exploed. Figue -: Nano ipeion Amplified Waveguide Stuctue..4 Bagg Gating Filte eign The econd majo aea of optical filte exploed in thi eeach make ue of Bagg gating to fom the filte epone. CAPTR 6 exploe an innovative appoach to the deign and fabication of Bagg gating waveguide tuctue. The gating vecto i decoupled fom the waveguide tajectoy intoducing an additional degee of flexibility in the deign of filte epone without uccumbing to fabication containt. Figue - illutate the baic appoach 4

60 wheeby a adial gating i fabicated on an optical ubtate and a waveguide with an abitay tajectoy i placed diectly on top. The combination of the two poduce ome filte epone. The decoupling appoach in addition to poviding a ignificant degee of flexibility in tem of poible epone allow one to e-ue the ame gating fo a vaiety of diffeent waveguide and coeponding filte epone. A pecial cae deign baed on a linea goup delay cuve i dicued in detail and the paamete and appoach neceay fo a moe geneal cae i outlined. + = Figue -: Spial Waveguide Radial Gating and the Reulting Combination..5 Micocavity Filte eign The final cla of optical filte of inteet to thi eeach depat fom the waveguide natue of the fit two filte clae and move into the ealm of thee-dimenional filte deign uing eonant micocavitie a the foundational element. CAPTR 7 decibe the deign and optimiation of axially ymmetic eonant micocavitie and a numbe of appoache to optical filteing uing uch tuctue. One appoach ue exotic cavity geometie to confine the eonant mode and obtain lage Q-facto to allow a ingle cavity to filte out naow 4

61 tanmiion band aound deign wavelength. A moe inteeting cae make ue of multiplexed cavitie with omewhat lowe Q-facto (a illutated in Figue -3). By coupling a lage numbe of thee tuctue togethe the Q-facto i inceaed although additional pectal featue begin to appea due to eonance and ocillation between cavitie. Additionally by making ue of highe Q cavitie uing BR laye fo confinement ignificantly highe goup delay magnitude ae obtained albeit ove a much naowe bandwidth. Thee featue and poible ue fo thee tuctue ae dicued in detail and ome pecific deign example ae demontated and analyed. Figue -3: Multiplexing of GaA/AlA Cavity Filte..6 Reeach Summay The fundamental goal of thi eeach i to povide the deign tool and appoache neceay fo the fomulation of vaiou type of nanotuctued optical filte. The applicable numeical tool fo deign analyi and optimiation ae deived and dicued in detail. In addition thee clae of optical filte ae exploed and innovative deign in each categoy ae peented and chaacteied. Coupling and fabication challenge ae noted and accounted fo in 4

62 the analyi and calculation and pediction fo the oveall filte epone fo the diffeent deign ae peented. 43

63 CAPTR 3 FRQUNCY OMAIN MOLING OF OPTICAL FILTRS 3. igentate of Optical Filte The peceding chapte outlined a few vaietie of optical filte each of which ha a unique epone function in the fequency domain. elay line ae highly dipeive device whoe geometical configuation ae ued to intoduce a lage hift in the popagation contant a a function of wavelength. The vaiation of the popagation contant of the fundamental guided mode povide a mean to calculate the tuctue dipeion. Optical eonato involve a cavity aound which electomagnetic field ciculate. The eonant mode of the cavity ae caued by tanding wave of the field aound the cicumfeence of the cavity. The geomety of the cavitie and the equiement that the accumulated phae aound the peimete be an intege multiple of tend to eult in a vey naow pectal width of the eonant mode fo micocavity eonato. Thi give them a vey ignificant quality facto which make them advantageou choice fo a numbe of optical filteing application. The difficulty i in deigning and pedicting the behavio of both of thee type of optical filte. Fequency domain imulation method ae baed fundamentally on olving Maxwell quation eithe at o fo a ingle fequency value. Thi i in diect contat to the time domain appoach peented in the next chapte. Thee model ae baed on teady-tate behavio of ytem. In ode to obtain boadband eult fequency domain imulation mut be un multiple time to acetain the epone at each individual pectal component. Thu thee ae cetain limitation to thee model. ipeion nonlinea effect and othe apect of ome 44

64 optical filte ae not diectly attainable fom thee model though they may be etimated baed on filte behavio calculated fo vaiou elevant wavelength. That aid fequency domain model ae typically quite fat. Futhe they ae abolutely eential if one wihe to obeve vey naow pectal linewidth in the optical epone. Time domain model mut be un fo exobitant amount of time to be able to even appoximate the hap fequency-dependent change in the epone function. While thee ae myiad fequency domain model thoe elevant to thi eeach ae all baed in pat on fomulating Maxwell quation into an eigenvalue poblem whee the eigenvalue will epeent eithe a popagation contant o eonant fequency depending on the fomulation. 3. Modeling Field on a Gid 3.. Finite iffeence Appoximation Citical to fomulating the equation fo any model and to epeenting the field and mateial paamete i the detemination of the bet numeical epeentation. The pinciple model ued in thi eeach ae baed on the finite diffeence epeentation fo deivative and a ectangula gid. In thi way continuou function ae fit to a gid whee the value ae pecified at dicete point. I make ue of a ectangula gid fo eae of implementation though othe have demontated impoved accuacy and efficiency with moe elaboate gidding cheme uch a vaiable gid pacing [6] and confomal uface gidding [63]. The tandad appoach fo the finite diffeence expeion involve a imple Taylo eie expanion allowing fo econd ode accuate deivative [64]. The fit deivative ae expeed accoding to 45

65 f ' x x f x f x x (3.) x x O Thi povide the fit deivative midway between two gid point epaated by a ditance Δx given the field value at both point. A econd deivative may be obtained by applying the fit deivative opeato twice. Thi equation i uitable fo mot ituation though additional tem may be included to povide deivative of highe-ode accuacy. Additionally othe [65 66] have demontated altenative fomulation of the deivative opeato that povide a ignificant impovement in the ode of accuacy unde cetain condition. 3.. Field Repeentation The numeical method ued thoughout thi eeach ae explicitly baed only on the cul equation athe than the full et of Maxwell quation. In geneal the divegence equation ae a diect conequence of a chage-fee medium. oweve when the field component ae napped to a finite gid exta cae mut be taken to enue the divegence condition ae atified paticulaly fo two- and thee-dimenional poblem. If the field component ae oiented incoectly o if they ae all co-located on the gid the divegence condition will not neceaily be atified. The mot common finite diffeence gidding algoithm wa intoduced by Yee [67] and i illutated in Figue 3-. The -field ae intelaced aound the -field o that they otate aound each othe. Thi method offe a numbe of convenient attibute [68]: ) The location of the field component lend itelf immediately to a taightfowad implementation of the cul expeion with econd-ode accuate deivative. 46

66 ) The tangential component ae natually maintained aco the inteface of diimila mateial without the need to apply additional equiement to tictly enfoce the bounday condition. 3) The divegence condition ae automatically atified. 4) The nonphyical numeical dipeion eulting fom the finite gid i ignificantly le with the Yee gid than with co-located gid [69]. (a) (b) (c) (d) 47

67 (e) (f) Figue 3-: Field Poitioning fo (ab) -; (c) - T; (d) - TM; (e) 3-; (f) Axiymmetic Simulation. To povide ufficiently accuate eult the gid pacing mut be ignificantly malle than the wavelength of inteet. On the othe hand if the gid pacing i too mall the computation time and compute memoy equied to complete the imulation become uneaonable. A eaonable compomie between peed and accuacy occu when the gid pacing i oughly between min n max and min 4n max whee min i the mallet wavelength of inteet and n max i the laget efactive index peent in the imulation egion. 3.3 Maxwell quation Fomulation A with mot numeical method we begin the deivation with Maxwell quation: i i The tandad deivation of the wave equation involve taking the cul of Faaday Law and ubtituting Ampee Law into it aiving at 48 (3.)

68 49 k (3.3) In ectangula coodinate the left hand ide will implify futhe uing the vecto identity (3.4) Thi can be futhe implified by incopoating Gau Law: (3.5) qn. 3.3 then implifie to k (3.6) In non-cateian coodinate ytem the vecto Laplacian i not defined apat fom the cul opeato (qn. 3.4) and qn. 3.3 cannot be implified in thi manne. Note that the cul opeato may be witten in matix fom accoding to y x A A A x y x y A (3.7) o long a we ae opeating in ectangula coodinate. The cae i imila with cylindical coodinate though the matix i lightly moe complicated:

69 5 A A A A (3.8) It will alo be convenient to epeent the divegence opeato in cylindical coodinate in matix fom: A A A A (3.9) 3.4 Pefectly Matched Laye Bounday Condition The bounday condition equie exta cae to eliminate non-phyical eflection fom the edge of the imulated aea. The mot common fomulation fo abobing boundaie i known a the pefectly matched laye (PML) and incopoate what i known a the tetched coodinate method [7 7]. In thi method each ucceive laye at the bounday ha a omewhat highe electical conductivity than the one befoe it while at the ame time a imulated magnetic conductivity i applied to the laye to enue impedance matching and to minimie eflection. In ome cae iichlet boundaie may be ued without adveely affecting the imulation eult. Thi i only the cae if the field i ufficiently mall at all boundaie o that eflection ae negligible. Thi doe allow ome flexibility in the fomulation of the expeion but alo equie exta caution a the had boundaie can caue the appeaance of non-phyical guided mode that would not exit apat fom the bounday condition.

70 5 Additionally the etimation of lo tem of both guided mode and eonant fequencie equie a mechanim fo aid lo to occu. PML boundaie allow enegy fom the imulated mode to leak out of the egion eulting in complex eigenvalue whoe imaginay pat povide a mechanim to etimate the lo Mateial Teno xpeion Pemeability and pemittivity both become teno epeenting uniaxial mateial inide PML egion [7]. Fo minimal eflection they can be hown [68] to take the fom of an invetible diagonal matix i u u u i u u u i u u u (3.) whee u u and u 3 ae the coodinate fo the thee dimenional egion (xy fo Cateian coodinate o fo cylindical coodinate). The and σ tem cale the eal and imaginay pat of the pemittivity (and pemeability) a a function of depth inide the PML. Thee ae diffeent method fo expeing the and σ tem [68] though I make ue of a polynomial gading of the fom m max m max d u u d u u (3.)

71 whee m i the ode of the gading and d i the width of the PML egion. A ignificant amount of expeimentation [7] ha been pefomed on vaiou combination of paamete though m º 3 d º gid point and 3. 5n max (whee n i efactive index and Δ i the gid pacing) appea to povide eaonably good pefomance. Optimal value of max vay accoding to the poblem geomety though it i uually between and Cylindical Coodinate Thee ae additional conideation in cylindical coodinate. The component might not appea to need any PML conideation due to a lack of any bounday in that diection. oweve the adial and aimuthal component ae not independent o a adially-dependent aimuthal PML tem i till equied [73 74]. qn. 3. i till uitable fo the and component ( and 3 ) but the aimthal tem i given by whee i the adiu at the edge of the PML egion. ' d' (3.) Modification of Maxwell quation Since the PML teno cale both the pemittivity and pemeability both of Mawell cul quation ae affected: i i (3.3) 5

72 The eulting expeion ae no longe a eay to implify into a uccinct fom a they wee peviouly. oweve the PML teno i an invetible diagonal matix and the matix fom fo the cul expeion allow u to olve the pai with a limited degee of manipulation. 3.5 igenfequency Calculation When attempting to model and deign mico-cavitie one need a method fo obtaining the eonant mode of the tuctue. The tuctue will typically eonate at a vaiety of diffeent fequencie each having it own field ditibution and Q-facto. Thee ae a numbe of diffeent method fo obtaining thee eonant mode [75-77] each method offeing a diffeent et of advantage and diadvantage. Fo ufficiently mall poblem o geometie whee ymmety may educe the neceay imulation point method baed on linea algeba [78-8] povide a fat and efficient algoithm fo obtaining the mode. Maxwell quation expe the field component in the cavity. The eonance condition and cavity geomety ae ued to combine and implify the equation into a pai of diffeential equation in tem of the electic field. Thee ae expeed in tem of an eigenvalue poblem whee the calculated eigenvalue ae intepeted a the eonant fequencie of the cavity. When we calculate eigenfequencie we ae looking fo two piece of infomation: the eonant fequencie of the cavity and the Q-facto of each eonance. PML boundaie ae eential to the etimation of Q-facto a mentioned above o qn. 3.3 ae cental to the fomulation of the model. Futhe the pola coodinate ytem add a degee of complexity to the opeato and the fomulation of the cul expeion in qn. 3.8 i ued. 53

73 3.5. Poblem Symmety We aume a otationally ymmetic cavity geomety which implie an angula ymmety fo the field a well. Fo a eonant mode inide uch a tuctue the bounday condition equie in n e in e n (3.4) whee n i an intege decibing the aimuthal vaiation of the mode. The mode equation ae epaately deived fo each n value and ae olved independently. The excited field in the egion hould be the upepoition of all the eonant mode fom each n value. Thi indicate that the angula deivative can be expeed analytically in (3.5) and imilaly fo the field. With n given the poblem can be educed to two dimenion by incopoating the divegence condition (auming a chage-fee egion). Since Gau Law i defined in tem of electic diplacement athe than electic field we make a change of vaiable at thi point: The divegence condition then give (3.6) in (3.7) The aimuthal component may now be eliminated by a convenient tanfom matix: 54

74 i i i (3.8) n n n Thi allow u to wite an eigenvalue poblem in tem of and. oweve the tem involving n in the denominato ugget that an altenative appoach mut be applied when woking with eo-ode mode. qn. 3.7 indicate that the aimuthal tem i independent of the adial and -diected field component in thi cae. The ame i tue with the magnetic field which ugget that the eo-ode mode may be epeented a a function of eithe o excluively. The electic diplacement vecto component ae co-located with the coeponding electic field component eivative Opeato and Bounday Condition Since eveal tem incopoate an invee of the adial coodinate (which become ingula on the -axi) pope poitioning of field component i citical. The gid i tuctued o that only field component that go to eo on the -axi ae poitioned on intege adial gid location [68]. Component that ae noneo on-axi ae located on the half-gid in the adial diection a illutated in Figue

75 56 Figue 3-: Field Component Location fo Axiymmetic Simulation qn. 3. povide the fomulation fo the deivative opeato but the poitioning of the field component equie light diffeence in the exact contuction of the coeponding deivative. Fo example: m m m m m m m m m m (3.9) Thu we define fou diffeent deivative opeato: A A A (3.) A A A (3.)

76 A A A (3.) A A A (3.3) The opeato ae identified o that the -labeled deivative opeato ae ued in the cul of the electic field in qn. 3.3 while the -labeled opeato ae incopoated in the cul of the magnetic field. qn. 3. and 3.3 opeate on intege gid and eult in value fo field component on the half-intege gid. qn. 3. and 3. opeate on the half intege gid and eult in value located on the intege gid. Thi allow each of the deivative opeato to be implemented a matice with noneo element tictly along the main diagonal and a ingle additional diagonal. The field component can then be epeented a a vecto allowing the ytem of equation to be olved via linea algebaic method Tanvee Mode A dicued ealie the eo-ode mode ae handled in a lightly diffeent manne than the highe-ode one. The divegence condition (qn. 3.7) indicate that the aimuthal component of the electic field i linealy independent fom the adial and axial component. Thi indicate that the eo-ode mode can be divided into two categoie: tanvee electic ( ) and tanvee magnetic ( ). Thee will be deignated a T mp and TM mp mode epectively T mp Mode Since the aimuthal deivative ae all eo the co-poduct opeato fom qn. 3.8 implifie to the following: 57

77 58 A A A A (3.4) Since we ae only concened with the and field component Maxwell quation in opeato fom become i i (3.5) Since i invetible the fit equation may be olved fo the magnetic field i (3.6) which i then ubtituted into the econd equation: (3.7) Afte ome implification and ubtitution of the appopiate PML matix element the opeato matix equation become k 33 (3.8)

78 We may implify thi expeion a bit futhe if we expand the PML tem. The tem ae expeed accoding to 33 (3.9) Some of the PML tem may be pulled out of the patial deivative eulting in k (3.3) The opeato on the left hand ide may all be combined into a ingle matix eulting in a tandad eigenvalue poblem. The eigenvalue of the ytem give the complex wavenumbe of each of the eonant T mp mode while the eigenvecto give the aimuthal electic field fo the mode TM mp Mode Fo the TM mode the fit concen i the bounday condition. A with the T cae the aimuthal field component i eo on-axi which ugget that the coodinate ytem indicated in Figue 3- would be bette eved if the and field component wee all wapped. Following the ame method a the deivation of the T mode we aume that the only field component peent ae and which intead eult in 59

79 6 i i (3.3) Solving fo the electic field and ubtituting give k (3.3) xpanding and implifying futhe poduce the following: k 33 (3.33) Once again canceling appopiate PML tem eult in the final eigenvalue expeion: k (3.34) Thi equation i olved in the ame manne a the T equation to obtain the TM mp eonant mode ighe Ode Mode The deivation of the eigenvalue expeion fo the highe ode mode cloely follow that of the T mode though with a geate degee of complexity. One majo diffeence i that the equation ae deived in tem of electic diplacement defined by qn. 3.6 a oppoed to electic field. A befoe Maxwell quation fom qn. 3.3 ae expeed in matix fom:

80 6 i i (3.35) Afte eplacing the deivative with the appopiate opeato and implifying the equation become i in in in in i (3.36) The two equation may now be combined and implified futhe eulting in

81 6 k in in in in (3.37) Combining matice once again and canceling PML tem yield the following: k in in n in in n (3.38) Thi matix i then combined with qn. 3.8 to eliminate the aimuthal tem:

82 63 k n n (3.39) qn i the full eigenvalue expeion fo the complete et of highe ode mode. Once the electic diplacement vecto fo each eonant mode ae calculated qn. 3.6 and 3.8 may be ued to obtain the full 3- vecto fom of the electic field fo each of thee mode Quality Facto of igenfequencie Popely implemented PML bounday condition will incopoate a lo mechanim into the eigenvalue computed fom the ytem of equation which povide an etimate of the enegy leakage out of the geomety. Thu the eonant fequencie calculated in the olve will in geneal be complex-valued. The imaginay pat give the lo and i diectly elated to the Q- facto of the cavity [8] accoding to Im Re Q (3.4) Model Benchmak To veify the pedicted eonant fequencie and Q-facto fom the model a vaiety of tet cae wee ued to benchmak the model againt publihed value. [73] give value fo a cylindical dielectic eonato uounded by ai. The cylinde adiu i 5.5mm it length i

83 4.6mm and it pemittivity i 38. The imulated eonance compaed to thoe epoted i demontated in Table 3-. [75] epot value fo a few highe ode mode fo a imila eonato except with a length of 4.6mm. Thee compaion ae peented in Table 3-. Table 3-: Compaion of Tanvee Mode Reonance. Mode Simulated feq. Repoted feq. Simulated Q Repoted Q T 4.846G 4.873G T 9.8G 9.99G TM 7.59G 7.583G The model wa alo benchmaked againt moe complicated geometie [8] with imila degee of accuacy. To enue the imulation accuacy i good ove a ange of geometie the peiodic enhancement of Q-facto a a function of cavity length decibed and imulated though a time domain calculation in [49] wa alo confimed by the eigenmode imulation a demontated in Figue 3-3. The eonant fequencie matched vey cloely with the publihed value and indicate excellent ageement paticulaly conideing the vat diffeence in imulation methodology and data gidding. The tong ageement between publihed data and imulation eult fo a ange of geometie and eonance povide geate confidence in the model eult. Table 3-: Compaion of Vaiou Mode Reonance. Mode Simulated feq. Repoted feq. Simulated Q Repoted Q T 4.855G 4.89G TM 7.577G 7.54G M 6.9G 6.333G M 7.73G 7.75G

84 (a) (b) Figue 3-3: Q-facto ependence on Cavity Length (a) Publihed and (b) Simulated. 3.6 igenmode Calculation A dicued peviouly delay line offe anothe type of optical filte in which the total delay cale linealy with length. In thi cae it i deiable to obtain a guiding co-ection that offe an extemely high dipeion aound the deied opeating fequency. To deign uch a tuctue one need a eliable method fo calculating the fequency epone of an abitailyhaped waveguide co-ection. While mateial and modal dipeion will have ome oveall effect we ae pimaily concened with deigning tuctue containing extemely lage waveguide dipeion which i elated to the fequency dependence of the effective index of the waveguide. A uch we imply need to calculate the effective index ove a ange of fequencie to obtain the dipeion of the delay line. While the peviou ection outlined a method fo obtaining the tanding wave eonant fequencie of a tuctue a light vaiation of that fomulation allow u intead to obtain the 65

85 popagation contant coeponding to taveling wave a a given fequency. In thi fomulation we make the aumption that light of a pecific fequency i being guided though the delay line pependicula to it co-ection with ome popagation contant. By a taightfowad deivation fom Maxwell quation we aive at a linea algebaic fomulation wheein the popagation contant ae given by the eigenvalue. We may olve thi expeion ove a band of cloely paced fequencie to obtain the pectal epone of the delay line. Thi appoach i known a an eigenmode fomulation wheea the appoach fo the eonato i temed the eigenfequency fomulation Mode xpeion Fo thee waveguide we aume that popagation i along the -axi o the electic field take the fom (3.4) i xy xye Thi allow u to eliminate the -deivative. qn. 3.6 then implifie to T k (3.4) Thi equation i alo an eigenmode expeion whee the eigenvalue ae the popagation contant of the coeponding eigenmode. A T mode may be obtained by auming thee i no electic field component in the diection of popagation ( =). Maxwell quation may then be olved in tem of : k (3.43) Similaly a TM mode may be obtained by auming = and olving in tem of : 66

86 k (3.44) Thee equation ae all available in Comol Femlab oftwae which due to it efficiency and eae of ue povided mot of the popagation contant fo the dipeive waveguide tuctue dicued in thi eeach. The value wee benchmaked againt a cutom olve cipt implemented accoding to the finite diffeence method decibed in [79] ipeion Calculation The eult of eigenmode imulation ae often given in tem of an effective efactive index fo the tuctue expeed accoding to n eff (3.45) whee i the calculated popagation contant and i the imulated wavelength. Once the popagation contant ae detemined at fine incement ove a band of wavelength the dipeion may alo be obtained. ipeion i expeed accoding to the following equation: c n eff (3.46) c We can then apply the finite diffeence expeion to the deivative tem to obtain the magnitude of dipeion fo the guiding tuctue ove a given fequency band Bent Waveguide Mode Anothe mode-olve poblem aie when we allow the waveguide tajectoy to bend along a cuved path. An angula popagation diection will ely on a athe diffeent mode hape than will popagation down a taight-line waveguide. Fo gentle cuvatue the diffeence in the popagation contant i not ubtantial but adial lo tem can become ignificant fo long 67

87 68 bent waveguide. It i wothwhile to implement a complex-valued mode olve to acetain the lo tem a a function of waveguide cuvatue in ode to detemine the limitation fo a cuved waveguide filte Model Fomulation We again tat with Maxwell quation (qn. 3.3) and aume that the popagating eigenmode take the fom R i e (3.47) whee R i the adiu of cuvatue of the waveguide and i the complex-valued popagation contant a dicued peviouly. Thu baed on the deivation in ection Maxwell quation in matix fom may be witten a follow: i R i R i R i R i i (3.48) Note that thee equation ae identical to thoe in ection with the light adjutment to the aimuthal deivative: R n. Thu we may wite the geneal expeion fo the mode in bent waveguide in the following manne:

88 69 R k k (3.49) Thi can be implified a bit futhe to obtain a convenient eigenvalue expeion: R R R k R R R R R R R k (3.5) While thi i accuate fo abitay hybid mode we typically aume that the idge waveguide contain eithe hoiontally o vetically polaied mode. We can conide each of thee cae individually Bent Waveguide T Mode In thi fomulation we define the T mode a thoe fo which the electic field i pependicula to the waveguide cuvatue. Thu = by aumption and the eigenvalue poblem implifie to the following:

89 7 k R (3.5) In imila manne to the eigenfequency calculation outlined above thi equation may be expeed in matix fom and olved numeically uing an eigenvalue olve Bent Waveguide TM Mode We define the TM mode a thoe with the electic field in the ame plane a the waveguide cuvatue. Thu = and we wite the eigenvalue poblem in tem of. The equation implifie to the following: k R (3.5) Lo Tem The eigenvalue olve etun complex eigenvalue that we intepet a the popagation contant: i n k i n k eff eff (3.53) The amplitude of the mode then decay accoding to R R n ik e e eff (3.54) The electic field intenity lo (in db) may then be witten a e R log log (3.55) The popagation ditance along the waveguide i given by R L o lo i expeed a

90 Lo (db pe unit length) = log Im e (3.56) 3.7 Method of Line Calculation While the above method allow one to pedict cetain chaacteitic of an optical filte epone it i often neceay to imulate the entie complex epone function fo a given incident ouce condition. Thee ae a vaiety of fequency domain popagation method available though thi wok will focu on the MOL appoach [83-86]. Thi method opeate unde imila aumption a the eigenmode imulation. Specifically popagation i aumed to be along the -axi. oweve intead of imply evaluating a cala popagation contant fo a ingle egion the MOL appoach involve calculating a tanmiion matix (baed on the eigenvalue fomulation expeed above) fo each individual egion and then building up matice that decibe the tanmiion and eflection fo an entie tuctue Method of Line Fomulation Thi method i baed on the aumption that pemittivity and pemeability ae piecewiecontant in the popagation diection. Thu the tuctue may be divided into individual laye each with it own pemittivity ditibution in the plane (ee Figue 3-4). In each laye the electic field take the fom (3.57) e i G ig A e B Note that the A tem efe to fowad-popagating wave (in the +i ign convention ued hee) and the B tem deal with backwad-popagating wave. 7

91 3 Figue 3-4: Layeing of Pemittivity Region eivation of Tanfe Matix Fom Maxwell quation (pe qn. 3.3) we can wite the following: i in in (3.58) and i in in (3.59) If we conide only eo-ode aimuthal mode the equation implify to 7

92 73 i (3.6) and i (3.6) The T mode can be witten in tem of the aimuthal electic field component. In thi cae Maxwell quation combine to poduce the following: k 33 (3.6) The longitudinal deivative tem may be eliminated uing qn. 3.57: k G 33 (3.63) Similaly the TM mode ae expeed in tem of the aimuthal magnetic field component. In uch a cae the following expeion may be obtained: k 33 (3.64) liminating the longitudinal deivative and implifying poduce the following: k G 33 (3.65)

93 In tem of the deivative opeato defined above thee equation may be witten in the following manne: T cae: G k 33 (3.66) TM cae: G k 33 (3.67) valuation of Matix Function A an aide one may note that we actually have the fomulation fo the quae of the popagation matix. Additionally the fomulation of the popoed olution fo the electic field i baed on an exponential function of the matix. Powe and exponential function of uch matice may be obtained in a taightfowad manne by method of diagonaliation. Specifically the matix may be expeed a G T T (3.68) whee and T ae the eigenvalue and eigenvecto of G. Then we may wite the following: and G T T (3.69) e ig Te i T (3.7) 74

94 Laye Tanition At an inteface between laye we know that the tangential field component ae continuou. Specifically fo and (o and ) at the bounday between the m th and (m+) th laye we obtain m m m m m m m m m m m m B G A G B G A G B A B A (3.7) If we aume that thee i no field incident fom the ight (B m+ = ) we can obtain eflection and tanmiion coefficient: m m m m m m m m m m m m m m m m m m m A R I A G G I A T A A G G I G G I A R B (3.7) Similaly by etting A m = we can deive tanmiion and eflection coefficient fo light incident on the oppoite ide of the inteface. The full et of coefficient ae given a m m m m m m m m m m m m m m m m m m R I T R R R I T G G I G G I R (3.73) Of geate inteet ae the eflection and tanmiion coefficient fo multiple laye. Thee i a vaiety of method to accuately account fo the tanition. The implet appoach known a the tanmiion matix fomulation [87] imply involve matching the fowad- and backwad-popagating amplitude at each bounday to fomulate a tanfe matix. oweve the exponential tem fo the backwad-popagating field can become numeically untable leading to eoneou imulation. A bette appoach i to diectly um the eflected and tanmitted

95 component in an infinite eie to account fo multiple eflection and popagation aco a given laye. Thi ultimately lead to the following expeion [88 89]: R T R T m AB m AB m BA m BA R T R T m m m AB e m BA m m T igmm T e m AB igm igmm m igmm igm me RAB e I Rm me igmm m igmm I Rm me RAB e Tm m igmm igmm m igmm e Rm me I RAB e m m igmm igmm m I RAB e Rm me TBA m m R R m AB m m e e igmm igmm T T m m m BA (3.74) ee m R AB and m T AB epeent the eflection and tanmiion coefficient fo a wave incident though the m th laye fom the left and m R BA and m T BA ae the coefficient fo a wave incident fom the ight; R mm+ T mm+ R m+m and T m+m ae the eflection and tanmiion coefficient fo the ingle inteface between the m and (m+) laye expeed in qn. 3.73; and m give the thickne of the m+ laye. Thu the coefficient matice may be built up baed on the matice defined fo ucceeding laye. The algoithm iteate backwad though the entie tuctue coniting of M laye to obtain a ingle oveall et of filte epone matice fo a given fequency. To begin one intoduce a ghot M+ laye with eo eflection and unity tanmiion coefficient. Additionally a imila m= laye may be intoduced with like coefficient to povide the appopiate tanition fom a ouce plane to the fit laye inteface oubling Algoithm In ome cae we wih to model the tanmiion and eflection though tuctue of a peiodic natue. In uch cae the eflection and tanmiion coefficient outlined above may be calculated fo a ingle peiod and then appopiately tanfomed though a doubling 76

96 algoithm to decibe the oveall tuctue. The moe geneal cae of thi algoithm allow u to titch togethe potentially dipaate adjacent egion whoe coefficient have been calculated peviouly. If C denote the leftmot egion and denote the ightmot egion the eflection and tanmiion coefficient fo the eulting combination ae given by the following: R T R T C AB C AB C BA C BA R T R T C AB AB BA C BA C C TBARAB I RBA C C I RBARAB TAB C TABRBAI RAB C I R R T AB BA BA R R AB C BA T T C BA BA (3.75) Obviouly fo peiodic tuctue egion C and ae identical. Thu once the coefficient fo the pai ae calculated the doubling algoithm may be applied a econd time to epeent 4 peiod a thid time to povide the coefficient fo 8 peiod and o on. Thi allow u to quickly and efficiently build up the tanition coefficient fo a lage peiodic tuctue. 3.8 Fequency omain Model Summay Thi chapte ha outlined thee baic fequency domain modeling tool. The eigenfequency calculation povide a fat and efficient way to evaluate eonant cavity filte. While they do not offe a mean to pedict the epone function fo an abitay incident ouce condition they do poduce the vaiou eonant chaacteitic of inteet. Specifically the pedicted complex eigenfequencie give the eential infomation to calculate the pectal poitioning and the bandwidth of the elevant eonance. The eigenmode imulation ae ued to detemine the dipeive chaacteitic of delay line filte. The eigenvalue epeent the popagation contant of the mode of inteet and the vaiation of the popagation contant with fequency allow u to pedict the filte dipeion. Finally the MOL imulation allow u to 77

97 imulate the eult of an abitay ouce condition incident on a given optical filte. It yield both eflectance and tanmittance and give u a mean to viualie how thee change with incident fequency and ouce condition. 78

98 CAPTR 4 OPTIMIZATION OF LCTRO MAGNTIC STRUCTURS 4. Pinciple of Optimiation Fo vaiou electomagnetic deign poblem it i uually neceay to make ue of an optimiation of the geomety of the phyical element of the optical ytem o device. epending on the complexity of the deign thi may be tivial o exceedingly difficult. Any optimiation poblem elie on two component. The fit i a collection of numeical value o vaiable offeing a complete deciption of the component to be optimied. The numbe of independent vaiable in thi collection define the dimenionality of the poible et of olution. Thi et temed paamete pace contain the entie et of poible deign o olution to the optimiation poblem expeed in vecto fom. The econd apect of the optimiation poblem i a function that map a vecto in paamete pace decibing a poible olution to a value expeing how cloely the behavio of the olution come to the abolute optimum. Thi mapping function i typically temed the fitne function o cot function. (Often the fome tem lend itelf moe natually to maximiation poblem and the latte to minimiation though they ae uually ued intechangeably.) A imple optimiation example could conit of an N-dimenional paamete pace with a e Jong cot function: f N x i i X (4.) One of the difficultie in optimiation i that the cot function aely ha a imple o obviou a olution a the example above. e Jong function ha a ingle minimum at x i =. oweve mot fitne function have a lage numbe of minima thoughout paamete pace. 79

99 Only one value povide the global minimum though imple optimiation algoithm elying on gadient and teepet decent will often get caught in one of the myiad local minima. A good example of thi i Ratigin function f N i x i X N x co (4.) i which i illutated in two dimenion in Figue Figue 4-: Fitne Map of Ratigin Function. A poblem with a lage numbe of minima (o maxima) equie a moe bute-foce each algoithm that can exhautively go though all of paamete pace and evaluate evey local minimum befoe ettling in the global bet. oweve a the dimenionality of paamete pace gow exhautive o deteminitic each algoithm become highly inefficient and we mut ely on pobabilitic method that combine apect of a global exhautive each with fat localied optimiation method. 8

100 4. Method of Pobabilitic Seache One of the pimay method of pobabilitic optimiation wa the Genetic Algoithm (GA) [9 9]. It developed into a vey obut each algoithm applicable to a wide vaiety of poblem and attacted ignificant attention and inteet becoming quite matue ove the yea. ventually a numbe of vaiation of GA wee applied to optical and electo magnetic deign poblem [9-96]. The cental concept of evolutionay optimiation algoithm i a goup o population of potential olution in paamete pace. The quality o fitne of the olution i evaluated uing whateve citeia i appopiate fo the given optimiation poblem. The olution ae then hifted in a quai-andom manne with a bia towad point in paamete pace peviouly identified to be moe optimal. In GA the bette olution ae elected to pa on thei infomation to futue geneation and diectly exchange infomation though a poce known a coove. The paamete of two olution fom the cuent geneation ae combined togethe to fom two childen olution fo the next geneation allowing the olution to quickly move aound paamete pace while exchanging infomation about optimal point. An additional opeation known a mutation take a ingle olution and hift it in paamete pace by ome elatively mall petubation allowing the moe optimal individual of the population to pefom a localied each and eo in on a globally optimal olution. The ucce of the GA appoach pued additional eeach into uing behavio fom natue to olve mathematical and phyical poblem. In 995 the waming o flocking natue of inect and bid wa fit applied to mathematical each poblem in what wa temed Paticle Swam Optimiation [97 98]. Intead of uing a genetic appoach whee gene compete 8

101 to pa on thei infomation to futue geneation the olution behave a ocial inect moving andomly though pace and adjuting thei velocitie towad othe olution that identify moe optimal poition. Thee i no pecific election opeato in PSO wheeby lee olution ae eliminated fom the gene pool. oweve ince the olution (temed paticle) ae continuouly moving they have a low pobability of etuning to non-optimal poition in pace. Intead each paticle i acceleated lightly towad the bet point in paamete pace it had peviouly located poviding a mean of localied each akin to the mutation opeato of the GA. Additionally the paticle communicate infomation to each othe about the cuent global optimum and each expeience an acceleation towad thi point [99]. PSO ha been applied to a wide vaiety of poblem and ha been found to pefom extaodinaily well in many cae locating an optimal olution fate and moe efficiently than GA and othe imila each method [99 ]. With the inteet uounding PSO it wa quickly applied to electical and electo magnetic deign poblem [ ] and vey ecently to poblem in the optical egion of the pectum [3 4]. 4.3 Paticle Swam Mechanic 4.3. Initialiation The PSO algoithm conit of a population of M individual paticle moving aound in N-dimenional paamete pace. In each iteation o geneation of the population all paticle ae updated accoding to thei cuent velocity vecto. The m th paticle of the n th geneation i defined accoding to n X m x x x3 8 N... x (4.3)

102 and it velocity vecto i defined a n m N V v v... v (4.4) v 3 The initial paticle poition and velocitie ae geneated baed on a unifom andom ditibution aco paamete pace a indicated in Figue 4-. Figue 4-: Initial Paticle itibution in Paamete Space Solution valuation Pio to updating the paticle futhe one mut evaluate each popoed olution baed on the given deign citeia. A cot o fitne function accept the paticle poition vecto a input and etun a cala quantity decibing the pefomance of the given olution. The cot function i the only point of the PSO algoithm whee the paticle eceive any meaning and ae teated in anything othe than an abtact manne. ach deign poblem will equie a diffeent cot function incopoating vaiou evaluation citeia. Alo it i convenient to incopoate penaltie 83

103 of vaiou ot into the cot function to puh paticle away fom olution that might mathematically appea optimal but offe unealitic o undeiable behavio. In deigning the PSO algoithm one mut chooe whethe to maximie o minimie the cot function. The PSO ued hee minimie the fitne function of the population. If cetain apect of the deign ae expeed in tem of maximiing chaacteitic of the optical pefomance (Q-facto of eonato fo example) we will imply take the invee to obtain a fitne value that may be minimied. Pio to updating the velocitie and poition the PSO algoithm evaluate the fitne of each paticle: n m n m f F X (4.5) At thi point one mut detemine how the cuent geneation matche up againt peviou one. PSO keep tack of both the bet point located globally and the bet point located by the individual paticle: pbet n pbet X m X m f m f m pbet n m m n (4.6) f f The cuent global optimum X gbet i imply the bet of the X m pbet olution Paticle Motion Pio to allowing the paticle to move the velocity vecto ae petubed with mall acceleation towad optimal point in paamete pace. One component acceleate the paticle towad the cuent global optimum X gbet offeing the communication between paticle needed fo good global eaching of paamete pace. An additional acceleation component puhe 84

104 each paticle towad the bet poition it ha located on it own X m pbet. Thu the velocity update equation take the fom V n m wv n m c pbet n n X X c X X m m gbet m (4.7) whee w c and c ae contant decibing the elative contibution of each velocity component and and ae andom numbe between and which povide a degee of andomne to the each algoithm. The exact value of the paamete w c and c fo efficient optimiation vay omewhat fom poblem to poblem. oweve while vaying the paamete value may inceae the peed of the optimiation it aely effect the ability of the PSO to ultimately convege [99]. The fit of the coefficient w decibe the paticle inetial weight and define what faction of the initial velocity i maintained fom one time tep to the next. It effectively povide a balance between localied and global exploation. Lage value eult in fat moving paticle that exploe moe globally while malle value caue the paticle to focu moe on localied eache and infomation about local minima povided by othe paticle. Bette convegence may be obtained by initially etting thi paamete to a lage value (nea unity) and deceaing it ove time to allow the wam to move globally ealy on while gadually hifting towad local efinement late in the imulation [5]. The othe two coefficient c and c ae the cognitive and ocial coefficient epectively. They povide the acceleation towad the global and neaet local minima. xact value ae not uually citical though fine tuning can inceae the peed of convegence [6]. In the abence of fine tuning a tandad choice of value i c = c = [7]. 85

105 An additional containt found to aid convegence i a vecto that define maximum velocity. Since thee i no immediate containt on velocity in qn. 4.7 lage acceleation can induce the paticle to jump pat minima and mi them altogethe. Angeline [] found that impoing a maximum velocity thehold V max in each dimenion ignificantly aided convegence. The magnitude of each component of the velocity vecto i compaed to the coeponding component of V max and i et to that value if that component exceed the maximum. Lage value of V max eult in paticle moving aco paamete pace too quickly and miing olution while mall value impede the global each mechanim. Suitable value fo bounded poblem ae aound one quate to one half of the allowed ange in each dimenion. Once the velocity vecto i appopiately updated it i added to the cuent paticle poition: X n n n m X m Vm t (4.8) In geneal we may imply define the time tep Δt to be unity and incopoate any deied vaiation to thi into the velocity update paamete in qn. 4.7 []. Othe have hown that the time tep may be eplaced by a velocity contiction facto that cale the velocity to pevent too lage o too mall of a tep though paamete pace. oweve imila behavio may be obtained though the inetial weight and the maximum velocity thehold outlined above [99 ]. The poce of the velocity and poition update i illutated in Figue 4-3. The dak blue aow indicate the initial velocity vecto while the dahed aow denote acceleation component towad the paticle peviou bet location (light blue) and the cuent globally optimal olution (oange). The black aow indicate the eulting velocity vecto that i ued to 86

106 update the paticle poition. Thi alo become the initial velocity fo the next geneation (econd dak blue aow). Figue 4-3: Paticle Movement. 4.4 Bounday Condition An additional conideation of impotance in developing the PSO algoithm involve the ange of phyically ealiable geometie available fo the cavity o othe deign poblem. Standad PSO implementation offe no containt to bound paamete pace yet thi can lead to completely unealitic olution uch a negative laye thicknee o cavity adii too mall to be fabicated. Such behavio may lead the each algoithm towad undeiable olution and can poduce unexpected and meaningle eult fom the fitne function. Fo the type of poblem conideed in thi eeach and in othe elated electomagnetic deign poblem it i appopiate to contain the paticle to pecified limit []. The bound may typically be 87

107 expeed a pioi baed on known fabication capabilitie and fom ough etimation of cavity hape uitable fo the deign taget. The concept of bounding each pace fo electomagnetic poblem i not unique to thi eeach. Robinon [] expeed thee conceptual appoache to the poceing of paticle that co ove the bounday. Abobing boundaie topped inconguou paticle at the bounday and eoed out the nomal component of aid paticle velocity. Reflecting boundaie eulted in the paticle bouncing completely off the edge of paamete pace and eveed the ign of the nomal component of the paticle velocity. Inviible boundaie allowed the paticle to wande at will but pevented eoneou fitne calculation fom occuing on paticle that tayed outide the bound. Intead the fitne wa et to an extemely poo value to pevent it fom enticing additional paticle to co the edge of defined paamete pace. Figue 4-4 illutate the mechanic of the (a) abobing (b) eflecting and (c) inviible bounday condition. Robinon found modet diffeence in the ate of convegence between thee type of boundaie depending on the poblem conideed. (a) (b) (c) Figue 4-4: Standad Paticle Swam Bounday Condition. In thi eeach I developed a moe genealied bounday ytem that allow one to pecify how epulive the paamete pace wall ae to the paticle. In keeping with the concept of paticle moving though paamete pace they ae foced to bounce off the paamete bound 88

108 in the manne of Robinon eflecting boundaie. oweve I genealied the expeion to define an elaticity of the bounday. In like manne to the mechanical analog a fully elatic wall (elaticity paamete of ) wa pefectly eflecting while an inelatic wall (elaticity paamete of ) foced the paticle to top at the bounday and loe all momentum nomal to the bounday. laticity value between eo and one indicated eflection off the boundaie with lo of momentum. Thee ae alo occaion whee optimal olution ae expected to occu nea o outide of the pecified extent of paamete pace. In uch cae negative value of the elaticity paamete can be ued to allow the paticle to pa though the boundaie to vaying degee. When it i et to a value of - the effective eult i to allow the paticle to completely ignoe the boundaie altogethe. Smalle negative value eult in a deceleation of the paticle a it coe the bounday. Thi allow limited eaching outide the edge of paamete pace with acceleation component tending to pull the paticle back inide. Figue 4-5 illutate a paticle that i violating the defined boundaie a a eult of the poition update defined in qn The updated paticle ditance outide the boundaie may be expeed in vectoal fom by X n B (4.9) whee B give the vecto fom of the boundaie. The ign of the component of the ditance vecto ae given by d k x x l n k n k k u l x k k n k u k (4.) 89

109 whee u k ae the component of the uppe bound and l k ae the component of the lowe bound. Figue 4-5: Paticle Attempting to Leave Bounded Paamete Space. Afte defining the ditance vecto fo a taying paticle it poition and velocity vecto component coeponding with noneo value of the ditance vecto ae updated accoding to x n k n k d k x (4.) v n k n k v (4.) whee i the elaticity paamete fo the boundaie. Phyically peaking the poition i eet to the bounday and then puhed back inide (o outide if the ign of i negative) by a ditance popotional to the elaticity. The final poition of the paticle fo both the (a) poitive and the (b) negative cae of the elaticity coefficient i illutated in Figue V n d d X n+ (a) X n (b) Figue 4-6: Paticle Swam Bounday Tanfom. 9

110 4.5 iect Compaion of Paticle Swam and Genetic Algoithm Appoache A a tet of the optimiation peed of PSO in compaion to that of the GA appoach fo the optical filte unde conideation in thi eeach I fomulated a tet cae fo both algoithm. A mico-cavity filte geomety wa optimied to have a high Q-facto eonance fo the T mode at a pecified wavelength (a will be dicued in detail in late chapte). The two algoithm wee implemented and upplied with the ame fitne function. The eulting cuve fo optimal fitne a a function of iteation numbe ae illuminating. It wa immediately evident that the ate of convegence can vay geatly depending on how cloe the andom initial population of olution ae to optimal value. Additionally eticting the paamete pace a much a poible ignificantly peed the ate of convegence. With thee facto in mind both algoithm wee upplied with identical boundaie and initial population. The few paamete pecific to PSO wee et to default value a outlined ealie in thi chapte. The GA ued in thi compaion wa upplied by MathWok MATLAB oftwae. efault etting fo mot paamete wee alo ued fo thi cae with one exception. The intemediate value appoach wa ued fo the coove option which poduce childen fom a weighted aveage of two paent olution. Thi offe a bette appoach fo poblem compoed of a mall numbe of continuou (eal-valued) paamete a oppoed to long ting of binay o intege-valued data (in which cae a diect exchange of value between paent i the pefeed appoach [9]). Figue 4-7 how the eult of the compaion optimiation with (a) the optimal and aveage fitne a a function of geneation fo the GA and (b) the fitne of the bet paticle in the PSO. It i clea that the GA i able to quickly move the population into the egion of an 9

111 optimal olution much fate than the PSO. oweve it get caught in a local minimum (in thi cae a highe ode mode) and i neve able to fully jump out. In fact afte about geneation the entie population appea to conit of olution in the cloe neighbohood of thi non-optimal olution. In contat the PSO tep downwad towad an optimal olution at a moe gadual ate. oweve ince the paticle ae continuouly moving thoughout paamete pace they ae fa le likely to get pemanently tuck in a non-optimal local minimum. (a) (b) Figue 4-7: (a) Genetic Algoithm Veu (b) Paticle Swam Convegence Rate. 4.6 Paticle Swam Optimiation Summay PSO povide a unique and efficient appoach to a vaiety of pobabilitic optimiation poblem. It povide an extemely geneal famewok and a mean to efficiently obtain a global minimum making it paticulaly uitable fo a vaiety of electomagnetic deign poblem. It i quite efficient and demontatedly moe obut than the moe conventional GA appoach fo the type of poblem of inteet in thi eeach. Vaiou autho [ ] have thooughly exploed choice fo algoithm paamete of a numbe of diffeent optimiation poblem and have povided vey eaonable 9

112 value fo efficient optimiation. Thu all that emain i to detemine a well-defined fitne function that can take a vecto ting of numbe intepet it a a geometical deciption of an optical device and evaluate it optical pefomance etuning a cala numeical value expeing that pefomance. Thi i typically taightfowad a we might expe the geometical abeation of an optical ytem defined by the vecto [94] o the diffeence fom a deied diffaction patten eulting fom a given gating filte [4] hould that be the deign poblem in quetion. Moe petinent to thi eeach the vecto may be eaily ued to expe the geomety of a cavity eonato dipeive waveguide o Bagg tuctue while the cot function evaluate thee in tem of eonant fequency lo and pectal output. 93

113 CAPTR 5 ISPRSIV WAVGUI FILTRS ipeive waveguide and delay line have filte epone that ae pimaily phaebaed ove the ange of opeation of the device. In the abence of catteing and leakage loe the epone function take the fom i L e whee the popagation contant may be expeed a (5.) O 3 (5.) v g Thee expeion wee dicued in moe detail in Chapte.. The deign of thee tuctue involve the tak of tuning the goup velocity v g and the econd ode dipeion to obtain a pecific filte epone. The pecific focu of thi dicuion i to deign a waveguide with a vey lage dipeive magnitude which allow thee type of tuctue to povide potential olution to challenge in the aea of pule boadening and compeion. 5. ipeion in Standad Waveguide Ridge waveguide opeating nea cutoff tend to have a athe ignificant amount of dipeion. Figue 5- how the dipeion fo a 4nm x nm ilicon idge waveguide on a ilicon dioxide ubtate a detemined by the effective index method. The tuctue mut be thi mall in ode to maintain a ingle-mode pofile. The dipeive magnitude i elatively high Scatteing and leakage loe ae cetainly citical component of the filte epone function but ae difficult to contol and aely povide meaningful fequency-dependent effect in thee type of tuctue. We uually attempt to minimie thee effect athe than ty to tune them fo pecific filteing application. 94

114 and it can be inceaed by a facto of 5 o moe by hinking the idge down to 3 nm x 6 nm ince it puhe the opeation vey cloe to modal cutoff. Simila tuctue (temed nano-wie) have been tuned fo a dipeion peak a high a 8 p nm - km - albeit with a vey difficult and enitive fabication poce [8 9]. oweve uch tuctue have ome intinic dawback. With uch a minute waveguide ie coupling light into the tuctue become highly poblematic though not impoible. Additionally the high index contat eult in a highly confined mode in a vey mall aea. Confining a pule of any ignificant intenity to uch a tiny aea vitually guaantee that nonlinea effect will play a ignificant ole. epending on the application thi may be an advantage but fo pule compeion and tetching application nonlineaitie tend to hut much moe than they help. Figue 5-: T Mode ipeion in Ridge Waveguide. 5. Nanotuctued Waveguide While coupling and nonlinea iue can be accounted fo it would be advantageou to have a much moe eaonably ied waveguide while imultaneouly maintaining (o inceaing) 95

115 the oveall device dipeion. SiON ha a ignificantly lowe index than Si and could povide one altenative in the aea of nano-wie. oweve thi eeach peent anothe option. Intead of eticting ouelve to imple idge tuctue we make ue of a gating-baed tuctue in the hape of a idge [7]. The high dipeion of a idge-like waveguide can be taken advantage of but the gating natue lowe the effective index of the idge and foce the mode to pead ove a much lage aea. If the peiod i too lage the tuctue will act a a diffaction gating and pead the light in multiple ode. oweve fo an appopiate choice of tuctue ie and peiod only the fundamental diffaction ode will popagate and the tuctue fom a ingle-mode waveguide. Figue 5- illutate the geomety of uch a tuctue with a 75 nm peiod 5 nm fin and.5 micon depth. 6 peiod ae ued in thi tuctue. Figue 5-: Nano ipeion Amplified Waveguide Stuctue. 5.3 Analyi of ipeion Amplified Waveguide Comol FMLab oftwae implement a finite element olve which i ued to evaluate the waveguide popagation contant a a function of wavelength fo the two diffeent type of 96

116 mode T and TM. The T-like mode ae an expeion of the wave equation in tem of and ae baed on the aumption that thee i no component of the -field along the -axi. The TM-like mode ae olution of the fomulation of the wave equation in tem of with the aumption that the -field i entiely tanvee to the -axi. The fomulation fo the T-like cae i k (5.3) n n and fo the TM-like cae i n k (5.4) Note that thi definition fo the T and TM mode i not pecifically identical to the moe taditional definition in the waveguide community whee the T mode i defined in tem of - field paallel to the gating vecto fo thi tuctue (in the hoiontal diection in all the waveguide pofile diagam) and the TM mode i baed on -field paallel to the gating vecto fo thi tuctue (with the -field pimaily lying in the vetical diection in the waveguide pofile diagam). In contat thi definition i moe eminicent of the adial type of T and TM mode defined fo fibe. Fo the emainde of thi chapte T TM T-like and TM-like will efe to the FMLab definition of the mode while tue T and tue TM will efe to the taditional definition ued in the waveguide community a defined above. The mode equation wee expeed a eigenytem and wee olved fo the eigenmode popagation contant. The eult wee additionally veified uing an independent olve baed on tandad finite diffeence method [79]. The finite diffeence olve wa baed on a 97

117 fixed ectangula gid and thu had a ignificantly lowe accuacy than the FMLab eult but the eult ageed to within a few pecent and added confidence in the FMLab imulation. Additionally the imulation wee teted fo convegence baed on gid ampling to futhe enue the accuacy of the imulation. ipeion i calculated uing the finite diffeence appoximation to the econd deivative of the popagation contant. Although thi aume negligible mateial dipeion we know that mateial dipeion povide nowhee nea the dipeive magnitude we equie and hould be negligible in compaion to the deied lage waveguide dipeion. The mode pofile fo the popagating T mode in the deigned nano dipeion amplified waveguide (Nano-AWG) opeating at.55 micon i hown in Figue 5-4. The plot coloation indicate the field component while the aow illutate the -field. Figue 5-4 povide a gaph of the enegy denity of the popagating T and TM mode. It i inteeting to note that the majoity of the enegy fo the T cae i actually in the ubtate intead of the fin while the convee i tue of the TM mode. Figue 5-3: T Mode Pofile in Nano-AWG Stuctue. 98

118 In ome way thi eem counteintuitive but in eality it i conitent with what appea to be occuing. In the T cae the hot peiod of the fin eult in an effective medium above the ubtate laye that ha an effective efactive index too low to uppot a mode by itelf but it i ufficient to allow a mode to popagate jut beneath it in a imila manne to mode fomed in a ilicon laye jut beneath a mall ilicon idge. In the T cae the enegy i obviouly much moe tightly confined but it i till pead ove a ignificantly lage aea than the ingle moded idge waveguide mentioned ealie. Additionally one hould note that hote wavelength have popagation contant futhe fom cutoff and thu the enegy hift out of the ubtate and back into the fin eulting in a ituation moe eminicent of the enegy ditibution fo the peented TM cae. Inteetingly thi offe additional poibilitie becaue the tuctue could potentially be ued in an amplifie configuation whee the pump beam i ued to excite a ignal in a neighboing medium. (a) (b) Figue 5-4: negy enity fo (a) T Mode and (b) TM Mode in Nano-AWG Stuctue. Afte olving fo the popagation contant (and hence the effective efactive indice) at a ange of incident wavelength the oftwae evaluate dipeion though a diffeence appoximation to the econd deivative. The T mode ha emakably high calculated dipeion 99

119 while the TM cae although having eaonably high dipeion i not ingle moded. The dipeion fo the pinciple mode in both cae i hown in Figue 5-5. Figue 5-5: ipeion Cuve fo Nano-AWG Stuctue. Of additional inteet i the exteme biefingence thi tuctue exhibit. The efactive index fo the pinciple T mode at a wavelength of.55 micon i.4663 while the index fo the TM mode at that wavelength i.778. The equation fo the biefingent beat length L B B n TM n T (5.5) give a beat length magnitude of.5 micon. Typical polaiation-maintaining fibe (PMF) have beat length on the ode of -3 mm. The ilicon-ai inteface intoduce an index contat that i quite lage and hence the mode i highly confined. Uing SiON (index of.) fo the fin intead of Si (index of 3.47) allow the enegy to pead out aco the inteface to a much geate degee and futhe inceae the dipeion. Additionally the lowe guiding index mean that the fin mut be ignificantly lage to maintain a ingle-moded guiding tuctue which make fabication omewhat eaie. The T cae i ingle-moded fo the SiON Nano-AWG with a peiod of 85 nm. The duty cycle i maintained between the two cae (which give 45 nm width fin) a i

120 the gating height (.5 micon). The enegy denity fo the T and TM cae ae peented in Figue 5-6. (a) (b) Figue 5-6: Second Nano-AWG Stuctue. (a) T Mode negy enity; (b) TM Mode negy enity. Once again dipeion fo the T and TM mode i calculated fom the popagation contant. Figue 5-7 how a plot of dipeion fo each cae. Inteetingly the dipeion i double that of the fit Nano-AWG deign. Alo the calculated beat length fo thi deign at a wavelength of.55 micon i micon. A can be een the Nano-AWG dipeion i nealy a full ode of magnitude geate than that of the idge waveguide tuctue albeit about an ode of magnitude le than that available to chiped Bagg tuctue. Alo thee i a noticeable highe-ode dipeion component peent which limit the bandwidth a geat deal moe than fo the gating tuctue. oweve the bandwidth tuctue length and dipeive magnitude ae not o cloely intelinked fo thee tuctue. Inceaing the length of thee device to obtain a geate dipeive delay i imply a matte of fabicating a longe device wheea in the gating ytem

121 uch an inceae equie a coeponding deceae in gating chip pe unit length to maintain the magnitude of econd-ode dipeion. Figue 5-7: ipeion Cuve fo Second Nano-AWG Stuctue. 5.4 Theoetical Analyi Thee ae eveal inteeting chaacteitic in thee tuctue. Fit the gating povide an additional dipeion effect. The effective index fo the tue T mode in a gating tuctue include a highe ode dependency on the atio of wavelength to gating peiod [] 4 3 O c g P c T c g P eff n n n n n f f n (5.6) whee c g P c g T n f n f n f fn (5.7) f i the gating fill facto i the gating peiod n g i the efactive index of the gating and n c i the efactive index of the uounding egion. While thi expeion indicate the peence of a highe-ode gating dipeion it i not entiely accuate in thi cae due to the high contat

122 mateial and the extemely mall ie of the gating peiod. A uch to fully analye thee tuctue many moe tem fom the pemittivity expanion need to be incopoated. We tuned to a igoou coupled wave analyi (RCWA) [] to calculate an effective index fo the Nano- AWG tuctue. The effective index and coeponding dipeion fo an infinite gating uing the deign paamete fo the fit tuctue ae hown in Figue 5-8. Figue 5-8: ffective Index and ipeion fo Tue T Mode in an Infinite Gating. Although the dipeion i of a imila magnitude to the Nano-AWG dipeion it lope i in the oppoite diection. oweve a noted ealie and illutated in Figue 5-4(a) and Figue 5-6(a) much of the mode enegy i not actually contained in the idge. In addition the lope of the dipeion in Figue 5- i of the ame ign a what we ee fom the model of the Nano-AWG. Thi ugget that the tuctue ae actually acting much moe like idge waveguide vey cloe to cutoff which we would expect to have high dipeion. Fo compaion a model wa made of a idge waveguide of the ame dimenion a the fit Nano- AWG. The cladding and ubtate egion wee kept the ame a in the fin tuctue but the gating egion wa eplaced by a olid ectangula guiding egion with an index of.567 which i the effective index of the gating at a wavelength of.55 accoding to the RCWA. The 3

123 dipeion of thi tuctue give the limit fo the Nano-AWG tuctue whee the gating effect goe to eo. The dipeion i the dotted line in Figue 5-9. The hape of the cuve wa vey imila to the cuve fo the Nano-AWG though the magnitude wa noticeably malle. To analye thi futhe the contant efactive index of the idge tuctue wa eplaced by the index value pedicted fom the RCWA at each wavelength analyed. The popagation contant wee then found and analyed at each wavelength to detemine dipeion. The modal analyi pedicted that thee wee no popagating mode fo wavelength above.55 m. Below that cutoff point thi epeentation povided an uppe limit fo the Nano-AWG dipeion whee the gating effect eache the highet magnitude. ipeion fo the popagating mode i plotted a the dahed line in Figue 5-9. The dipeion fo the infinite gating limit i much highe than what wa pedicted fo the Nano-AWG. oweve the infinite gating tuctue i non-guiding at fequencie whee the Nano-AWG actually uppot a mode which ugget that the RCWA i not an entiely accuate indicato of the effective index in the guiding tuctue. Thi i a enible concluion ince the RCWA i pedicated on an infinite gating tuctue wheea the actual Nano-AWG contain only ix gating peiod. On the othe hand the infinite gating limit and contant index limit do povide an uppe and lowe bound to the expected dipeion fo the actual tuctue. A the figue demontate the modeled Nano-AWG dipeion fall comfotably between the two limit a we would expect. 4

124 Figue 5-9: ipeion fo T Mode in a Similaly imenioned Ridge Waveguide Opeating Nea Cutoff. entially then thee ae two effect peent in the Nano-AWG tuctue. The dominant one i the waveguide effect fo a idge-like waveguiding tuctue with an extemely low index contat. The gating natue allow u to fabicate a idge with a ubtantially lowe index contat than could be obtained with tandad mateial. The econd effect i the gating dipeion. Thi induce a deceae in the effective index a a function of wavelength which amplifie the tendency of the waveguide to appoach cutoff at highe wavelength and caue the lope and magnitude of the dipeion to be inceaed even futhe. Thee combined effect offe boade inight into the natue of the dipeion plot in Figue 5-5(a) and Figue 5-7(a). The fit Nano-AWG deign actually ha an extema in the dipeion cuve. It eem that below.54 m the tuctue i fa enough fom cutoff that the waveguide effect i no longe dominant and the gating effect play a lage ole giving the cuve a negative lope. Alo the econd deign actually ha a ignificantly lage dipeion depite lowe-index mateial and a ignificantly lage tuctue. oweve thi make ene 5

125 when we note that the configuation appoximate a idge waveguide nea cutoff ince the enegy i pead ove a much lage aea than in the fit deign. ence in light of a low index contat idge-like waveguide appoximation the dipeion cuve do make a geat deal of ene. 5.5 Altenative Configuation One ignificant concen egading thee tuctue i the ability to accuately fabicate uch mall gating line with a lage apect atio. Thee ae a few altenative configuation that could alleviate thee fabication iue. If we otate the fin ection by 9 degee elative to the ubtate we can fabicate them by mean of depoition of laye of altenating mateial with diffeent efactive indice. epending on the natue of the mateial it would be poible to utilie diffeent etch epone and induce undecutting in altenating laye to inceae the index contat between them (Figue 5-). Figue 5-: negy enity and lectic Field fo Altenative Fin Waveguide Layout. The illutated hoiontally-layeed example Nano-AWG conit of a 4-peiod tuctue. The longe laye have a width of micon height of.um and efactive index of 6

126 .5. The malle laye have a width of.8um height of.um and efactive index of.6. Thi tuctue alo uppot only a ingle popagating T mode. oweve the peviou tuctue attained high dipeive magnitude when the mode wa puhed downwad into the ubtate egion a chaacteitic unavailable to the hoiontally-layeed tuctue. Indeed the calculated dipeion fo the hoiontal tuctue i on the ode of 8 p nm - km - nealy an ode of magnitude malle than the oiginal deign. It i poible that futhe optimiation of the geomety could yield a highe dipeive magnitude paticulaly if it opeate cloe to cut-off condition. oweve ince the mode cannot be puhed out into an adjacent ubtate egion o eaily (without loing the gating effect altogethe) the tuctue i unlikely to attain dipeive magnitude a lage a the peviou one. Thee i anothe fabication appoach that offe ignificant meit. When plamaenhanced chemical vapo depoition (PCV) i ued to gow an oxide laye in a naow tench the phyical dynamic of the poce eult in a laye that bulge lightly towad the cente of the tench and tape off towad the edge. Thi effect ha been exploited to poduce tuctue temed tench-bulge waveguide [ 3]. Becaue of the vey wide hallow natue of the waveguide they contain only a ingle popagating T mode. All othe ae cut off. They alo exhibit faily ignificant leakage into the eulting lab waveguide on top of the ubtate though fabication pocee can alleviate that omewhat. The deign of thee tuctue can become much moe involved becaue the available paamete conit olely of the efactive indice of the laye the depoition time and the dimenion of the tench. The exact laye thicknee have to be calculated fom thee paamete baed on PCV depoition model [4 5]. While thee tuctue wee not 7

127 actively exploed in the coue of thi eeach it i poible to etimate a poible geomety and imulate the popagating mode. The enegy denity and electic field fo an appoximation to the guide illutated in o! Refeence ouce not found. and a veion with a lage numbe of laye ae both peented in Figue 5-. When the laye thicknee ae deigned fo opeation cloe to cutoff the calculated dipeion i appoximately 6 p nm - km -. Thi imulation doe not account fo leakage loe which would cetainly poe a ignificant iue fo any application fo thee tuctue but the magnitude of the dipeion ugget that thi may be a valuable avenue to puue to obtain viable Nano-AWG tuctue. (a) (b) Figue 5-: negy enity and lectic Field in (a) ual-laye and (b) Multi-Layeed Tench Bulge Waveguide. 5.6 Coupling Conideation Pehap the othe mot ignificant concen fo thee tuctue i the ability to couple light into and out of them. With the tange hape fo the mode pofile it i expected that ignificant loe will be peent at the inteface o diect end-fie coupling i unlikely to povide a eaonable appoach. The mot likely candidate i an evanecent coupling cheme uing a popagation contant-matched idge waveguide tuctue. In thi cae a um quae idge waveguide compoed of a SiON laye with efactive index.75 i ufficiently matched to allow a pai of coupled mode to exit between it and the oiginal Nano-AWG tuctue. 8

128 The ymmetic and antiymmetic mode ae hown in Figue 5-. With a 3um cente-to-cente pacing the effective efactive indice of the two mode ae and.4664 epectively. The coupling length i given by L n n a (5.8) Fo thi configuation the coupling length i expected to be appoximately.mm. (a) (b) Figue 5-: (a) Symmetic and (b) Antiymmetic Mode fo Nano-AWG Coupling Appoach. Inteetingly thi configuation may povide an additional appoach of meit. Specifically coupling between highly diimila waveguide of othewie modet dipeion can eult in extemely high dipeion [ ]. The ymmetic an antiymmetic each exhibit ignificant dipeion but with oppoite ign. Thi effect occu when two waveguide with 9

129 diffeing goup velocitie ae coupled. At the wavelength whee both waveguide have identical popagation contant the dipeion fo the two coupled mode peak. The value at a given fequency can be etimated fom (5.9) whee i the aveage dipeion of the individual waveguide and ω i the fequency at which the individual popagation contant ae equal. The chaacteitic bandwidth i given by 3 (5.) v v whee i the coupling contant and and ae the goup velocity of the individual guide. By deceaing the coupling contant and inceaing the diffeence of the goup delay between the two waveguide one may obtain extemely lage dipeive magnitude. 5.7 Nano-AWGS a elay Line The Nano-AWG wee deigned and optimied fo lage econd-ode dipeion athe than a ubtantial goup delay. oweve it i woth exploing the delay-line application of thee tuctue a well. A dicued peviouly FIR filte including thee device lack a tuctual enhancement to the delay and pimaily achieve lage phae delay imply by inceaing the length of the tuctue. elay peak geneally occu nea band edge [4] which ae not peent in the Nano-AWG. Additionally lage goup delay i often aociated with mall bandwidth a wa illutated in ection.5.3. The ubtantial bandwidth (nm) of the Nano-AWG futhe ugget that the oveall goup delay will be limited.

130 Since delay cale with tuctue length it i advantageou to expe a goup delay figue of meit independent of length. The atio of the delay time to the tuctue length i (when multiplied by c) the tuctue goup index (qn..9). The goup index fo the two Nano- AWG deign i hown in Figue 5-3. A expected thee tuctue povide a elatively mino inceae of about 3% ove a imple optical fibe [7]. Thu while the Nano-AWG offe emakable dipeive popetie they ae only maginally bette the othe much imple waveguide geometie fo delay line pupoe. (a) (b) Figue 5-3: Goup Index fo (a) Fit and (b) Second Nano-AWG deign. 5.8 ipeive Waveguide Summay Thi chapte ha peented the deign and analyi of tanmiion-baed dipeive optical filte. Thee filte have a unifom co-ection in the longitudinal diection and wee deigned to exhibit a geat deal of econd-ode dipeion. The Nano-AWG tuctue opeate cloe to cutoff condition and thei filteing capability i enhanced by the gating natue of the tuctue.

131 Thee deign offe a numbe of challenge paticulaly in tem of fabication and coupling. In addition ince they opeate nea cutoff and fabication i likely to induce oughne and dicontinuitie in the tuctue one could eaonably expect a elatively ignificant degee of catteing lo. oweve Nano-AWGS ae not without thei benefit a well. While they have ignificantly lowe dipeion than imilaly ied Bagg gating tuctue they involve a unifom tuctue that may be wapped in a pialing manne [6] to fom a tuctue with ignificant goup delay. Additionally the tuctue dipeion and tuctue length ae independent of each othe unlike the gating device. Thi mean that total dipeive delay i fa le limited fo thee tuctue though loe cetainly povide an uppe bound on it. A fa a the coupling and fabication iue go a numbe of appoache have been peented. vanecent coupling may be obtained though a popely deigned idge waveguide though the coupling contant i faily mall which mean that a faily ignificant coupling length i equied. oweve the evanecent coupling may alo povide a mean to futhe inceae the oveall tuctue dipeion athe ubtantially. Altenative configuation fo Nano-AWGS have been peented including a hoiontally-oiented tuctue and a tench-bulge waveguide fomulation. Fabication i much moe taightfowad fo both altenative ince dielectic laye can be depoited with a geat deal of unifomity and alignment of epaate patten i not equied fo thi cae. ence the only fabication tep equiing caeful conideation and optimiation i the etching and poible undecutting of altenating laye though the method fo thee opeation ae faily well documented.

132 CAPTR 6 SPIRAL BRAGG GRATING FILTRS Bagg tuctue and chiped Bagg gating ae the eult of a peiodic (o lightly apeiodic) petubation of the efactive index along the length of a given optical guiding tuctue [7]. The fabication of pemanent Bagg gating in fibe wa demontated by ill in 978 [8] and the concept of linealy adjuting the gating peiod to yield a contant dipeion tuctue wa late noted by Ouellette []. Today thee tuctue find myiad application in telecommunication aea uch a dipeion compenation [9] wavelength election and multiplexing [9] and fibe lae [] a well a optical toage [] ening [] and delay line [4] ue. The peiodic cougation of the efactive index pofile couple an incident ignal fom one popagating mode into anothe mode (eithe fowad o backwad popagating) [ ] in a manne highly dependent on fequency o moe pecifically on the atio of the gating fequency to the popagating wave fequency. One of the bigget dawback with thee tuctue i the inheent fabication limitation which inevitably eult in the intoduction of non-ideal dipeion in the fom of goup delay ipple (GR) to the ytem. Vaiou tudie have conideed the magnitude of the impact of the GR on the quality of the ytem a a whole [ 3 4] while othe have conideed a vaiety of mean to compenate fo thi poblem [5 6]. Additional difficultie with chiped Bagg tuctue include the ignificant length of the tuctue needed to obtain a ufficient degee of dipeion and the coupling lo fo the typical Bagg eflecto aangement (whee eflected ignal have to be epaated fom the incident ignal). 3

133 Much of the attention given to Bagg tuctue focue on fibe-baed filte due to thei low loe and eady incopoation into othe telecommunication device. oweve thee i alo a geat deal of inteet in wafe-baed Bagg waveguide tuctue [3 4]. Such tuctue allow one to utilie a wide ange of fabication technique to moe diectly tailo the waveguide and gating hape to a pecific application and impove the filte epone [5] and to acce moe exotic guiding and egeneation mateial. Unfotunately waveguide typically have omewhat highe loe than optical fibe and the high peciion equied fom the gating fabication can intoduce ignificant challenge but the ability to diectly incopoate the Bagg tuctue into othe wafe-baed device without eoting to fibe-wafe coupling can lagely mitigate thee iue fo a vaiety of application. 6. Bagg Filte Chaacteitic Bagg gating tuctue povide one of the moe common appoache to optical filteing. Unlike dipeive waveguide Bagg tuctue intoduce both amplitude and phae change to the incident ignal. qn..4 give the complete complex filte epone fo a fowad and backwad popagating beam auming a unifom contant-peiod gating. In uch tuctue the amplitude epone i completely detemined by the phae epone by a ilbet Tanfom and vice vea. In the moe geneal cae whee the gating peiod can vay the epone can no longe be expeed analytically and the phae and amplitude epone become independent of each othe. When the eulting gating can be appoximated by a equence of contant-peiod gating (whee L fo each ection) the oveall epone can be fomulated in tem of the poduct of x tanfe matice whoe component ae given by a fom of the olution expeed in 4

134 qn..4 (with the caveat that the bounday condition that the amplitude of backwad popagating wave goe to eo at = L mut be emoved) [8]. oweve fo apid change in the gating peiod o index modulation a moe accuate epone equie a diffeent et of tanfe matice that conide a ingle gating peiod at a time [6]. In eithe cae the filte epone may be calculated numeically. The goal at thi point i to deign and optimie the gating peiod to tune the filte to a deied epone. 6. Spial Bagg Stuctue eciption In a wafe-baed Bagg tuctue attempting to accuately vay a gating peiod fom point to point along the waveguide in a quai-abitay manne can be vey difficult. oweve it i poible to completely decouple the waveguide fom the gating. In thi way the gating can be fabicated to an extemely high peciion and epoduced uing uch technique a nanoimpint. Aftewad a waveguide can be aligned to the gating and fabicated in uch a way that it tajectoy poduce the deied gating chip. Thi offe a numbe of advantage the bigget of which i the wide ange of dipeion cuve acceible fom a ingle gating tuctue. Additionally meauable eo in the gating tuctue can be compenated omewhat by applying an appopiate petubation to the waveguide tajectoy. The advantage become paticulaly evident when we conide the poibility of a adial gating combined with a pial waveguide tajectoy which allow u to obtain abitaily long waveguide (and coepondingly lage goup delay) on a ingle ubtate. The gating function P(q) and the waveguide tajectoy f(q) (both expeed in pola coodinate) can be defined completely independently of each othe. The two ae then combined to obtain an extemely flexible functional dependence of gating peiod on ditance 5

135 taveled in a waveguide in the manne illutated in Figue 6-. Although a tanmiive filte equie the waveguide to exit the cente of the pial by coing ove itelf the nealy pependicula natue of the inteection and mall waveguide width help minimie catteing loe at thoe point. + = Figue 6-: Spial Waveguide Radial Gating and the Reulting Combination. Thi decoupling alo allow one to deign a pecific gating tuctue and ue it epeatedly fo a numbe of completely diffeent filte epone function. In mot cae we will make ue of the implet cae fo the gating function and aume that it ha no aimuthal dependence P (6.) whee i ome initial adiu at which the actual peiod take on the initial value. If the deied filte epone can be tanlated to a imple analytic functional fom fo the gating peiod the exact waveguide tajectoy can often be obtained analytically. oweve the moe geneal cae involve an abitay numeically-expeed filte epone. The waveguide 6

136 tajectoy i then bet deigned though an iteative optimiation appoach (uch a PSO). The tajectoy i expeed in tem of the adiu and ha a functional dependence on the angula coodinate. Fo a ealitic waveguide the function mut be continuou and mooth (continuou in the fit deivative). The implet expeion would be a polynomial dependence expeed a a a f... (6.) whee and a i ae optimied paamete. i not an independent paamete but may be included to decibe a final angle. Thi i pecifically ueful if the deied filte epone would be bette decibed in piecewie fahion with diffeent tajectoie connected togethe at pecific adii and angle. In uch cae the epaate waveguide ection mut mege moothly to enue minimal catteing loe. Thi i ufficiently accomplihed by continuity of the function and thei fit deivative. If we aume the ection f n and f n+ ae to meet at angle n+ the appopiate containt on the econd waveguide ae given by n a n n n an a n n n an n n... a 3a n n n n 3 n n... (6.3) 6.3 Linealy Chiped Bagg Stuctue The tandad and mot obviou cae of a Bagg filte i a imple linealy-chiped Bagg gating. The gating peiod i given by (6.4) 7

137 whee i the initial gating peiod i the ditance along the waveguide and i a contant decibing the gating chip pe unit length. Fo convenience let u define anothe contant which we will label the chip paamete by L (6.5) whee decibe the total chip ove the length of the gating and L i the length of the gating. With thi definition we ewite qn. 6.4 in the following manne: (6.6) We can now equate qn. 6.6 and 6. to olve fo the waveguide tajectoy (6.7) whee i the total ac length of the waveguide which i given in calculu text in tem of the adial function a f f d (6.8) If we aume the total deied chip to be vey mall in compaion to the length of the gating the oveall change in adiu mut be quite mall in compaion to the adiu of cuvatue of the waveguide and we can expect f f (6.9) Theefoe 8

138 f d df (6.) d d by equating the ac length with qn. 6.7 and letting the adiu take the functional dependence f(q). The olution of thi equation i a imple exponential dependence of the waveguide adiu on angle f e (6.) whee q i given in adian and i allowed to take on poitive value. We can ubtitute qn. 6. back into qn. 6. to olve fo the angle at which a given final peiod - i eached: f ln (6.) By numeical compaion to the exact eult detemined fom qn. 6.8 the eo baed on the aumption expeed by qn. 6.9 i le than uad ove a total angula extent of ten of adian fo eaonable et of deign paamete. Thu the aumption may be conideed appopiate. A a final point it i woth noting the phyical ignificance of the chip paamete which i diectly elated to the elative time delay expeienced by two diffeent incident wavelength: nl c n L c c c (6.3) 9

139 6.4 Coupling Stength Conideation Section.5. howed that the complex filte epone fo the tanmitted and eflected beam ae given by L A A B A co L i inl i inl co L i inl e il (6.4) whee κ i the coupling contant expeed a n (6.5) B and the detuning paamete i given by n B (6.6) Again at the Bagg wavelength the eflectance become R tanh L but towad the edge of the top band defined by the eflectance tat to dop off. The hapne of the band edge and the oveall tuctue eflectance ae diectly elated to the poduct L though the tuctue bandwidth i detemined olely by the coupling contant and ultimately by the gating index contat. While lage bandwidth and eflectance ae geneally deiable thee ae a couple dawback to a highe index contat. Fit the vey natue of cougation on top of a

140 waveguide eult in ome degee of catteing paticulaly at fequencie that ae not Baggmatched. Fo a typical waveguide index contat on the ode of 5% eult in lo of about 3 db/mm while contat of % and le poduce loe moe than an ode of magnitude malle [7]. Secondly highe contat eult in a lage oveall impact on the optical ignal fom each gating peiod and any fabication eo due in uface oughne o peiod ie will have a moe ponounced influence on the optical ignal. Since the gating eflectance depend tongly on a combination of the gating length and the coupling tength the two facto can be ued to compenate fo each othe. Fo good band extinction the poduct hould be eaonably lage. oweve eflectance tat to dop off towad the edge of the band. At L = the eflectance at the cente of the band i about 93% but i only 9% of that at. 9. When L > 4 eflectance i ove 98% fo 9% of the ejection band. Moe of the band become ueable a the poduct inceae but L = 4 eve a a uitable woking minimum fo coupling tength and gating length. Thi lead to a limitation on the maximum poible chip (and bandwidth) of the pial gating tuctue. The difficulty come fom the length neceay to povide good eflectance ove the entie band. A the waveguide pial inwad the gating peiod and hence the effective Bagg wavelength deceae. We equie that a given incident wavelength emain in the eflection band fo a minimum ditance L. Baed on the analyi above it i eaonable to equie the wavelength in quetion to emain within the 9% of full bandwidth fo the tuctue meaning that the coupling contant (which detemine the bandwidth) play a ignificant ole.

141 Thu ove a waveguide length of L the Bagg wavelength i allowed to deceae fom to the point whee. 9. Fom qn. 6.6 and the definition of the coupling contant we find.9n n B (6.7) Fo mall index contat we may appoximate the maximum chip by.9n n (6.8) If we make ue of the aumption noted above that the minimum length i given by L > 4 we may expe the maximum chip pe unit length: L.45 n n. 77 nl n (6.9) which clealy inceae with the quae of the gating index contat. Baed on the analyi in qn. 6.3 we can obtain a minimum elative delay: 5.65 n min c n (6.) Fo eaonable index contat and confinement facto thi wok out to a minimum elative delay of appoximately. p/nm. 6.5 Additional eign Containt It i impotant at thi point to note that the pial cannot imply be made abitaily mall. The minimum adiu of cuvatue i tongly limited by bend leakage. A method fo etimating the leakage lo of thee guide a a function of cuvatue i outlined in ection Fo

142 dielectic waveguide in the pectal egion of inteet the lo gow exponentially with cuvatue and i on the ode of a few db pe mm fo adii of cuvatue aound -3mm [8]. Baed on thi we take 5mm a a eaonable minimum fo the adiu of cuvatue to have minimal loe. Thi give the minimum poible adiu fo the pial tuctue. Futhe if the device i to be ued in tanmiion the innemot pial loop will have to cuve 8 degee and exit though the cente of the pial. Thi final bend ha double the cuvatue of the highet cuvatue of the pial and thu if the adiu of cuvatue of any pat of the tuctue i to be no le than 5 mm the innemot adiu fo the pial can be no le than mm. Anothe impotant quantity to conide i the ditance between ucceive loop of the pial. To have minimal coupling between waveguide loop a minimum eaonable epaation ditance between them i about 5-um. Fom analyi of qn. 6. we find the adial ditance between two ing to be f f e f (6.) Thi ditance take on a minimum value when f(q) i at a minimum (i.e. the adiu fom qn. 6. at which the final gating peiod i obtained). Fo a linealy chiped gating given a deied gating peiod total chip initial waveguide adiu and minimum allowed adial ditance between ing qn. 6. in combination with qn. 6.5 can yield a maximum gating length. In mot cae thi i not a evee limitation. If the initial adiu i 3mm initial peiod i 5nm and total chip i nm tetched out ove a length of 5m the mallet adial epaation between ucceive loop i till um while the tuctue delay i.67n/nm. 3

143 In moe geneal tem we can ewite thi a a maximum gating chip paamete in tem of the initial adiu and gating peiod the minimum adial epaation between pial loop and the oveall tuctue chip: max (6.) min ln Since the minimum adial pacing i much malle than the initial pial adiu a Taylo eie expanion implifie thi to max (6.3) min Uing qn. 6.3 we can alo deive a maximum elative delay fo a tuctue (auming multiple pial loop ae equied): max c min (6.4) While a delay of.67n/nm wa noted fo a lage tuctue above if we intead aume a malle pial tuctue with an oute adiu of mm and a minimum loop pacing of 5um a peiod of 5nm with a chip of nm yield a maximum elative delay of 64p/nm. 6.6 Quantification of Fabication Iue An additional potential advantage of the decoupling of the gating function fom the waveguide tajectoy involve fabication accuacy. It can be quite difficult to poition gating line with an accuacy ignificantly bette than nm. Yet gating line napped to the neaet nanomete will not poduce a deiable unifom phae epone and in the exteme cae can eult in a dicete eflection pectum imila to that illutated in Figue 6-. The adial natue 4

144 of the gating and the cuved waveguide tajectoy allow one to ovecome any ot of finite gid iue intoduced by fabication limitation [6] Peiod Sie (nm) Reflectance (db) Peiod Numbe wavelength (micon) Figue 6-: Nonideal Bagg Gating Chip. In a taight-line gating the pixel napping would be unifom aco the width of the gating but becaue of the angula natue of the gating line in the pial method the gating peiod vaie ignificantly point by point aco the width of a ingle egment (ee Figue 6-3). oweve while the edge will theoetically be witten in a vey jagged manne the pacing between the point i on the ode of.5nm (depending on the eolution ued duing e-beam witing). Futhe the natue of the gating fabication poce will uually caue pixilation to wah out eulting in a mooth line. Thi eult in an aveaging of the peiod length and pead out any emaining eo ove the length of the tuctue. 5

145 Figue 6-3: Pixilation of a Single Gating Peiod. To model the gating tuctue we calculated the exact napped ditance between gating line fo pai of coeponding point in the waveguide. The calculation wee epeated and aveaged fo eveal hunded point panning the width of each ection of waveguide. Thi povided a numeical appoximation to the aveaging the light mode actually undegoe. Ideally the ucceive peiod would demontate a mooth linea cuve accoding to qn Simulation indicate that thee i till a degee of eo in the peiod due to e-beam pixel napping but the eo ae educed by eveal ode of magnitude fom the taight-line cae. In the cae of a gating witten with a.5nm eolution (the bet eolution achievable on the e- beam ytem) thee tandad deviation of the eo in the peiod fall within.35nm which i ubtantially bette than one might expect. Alo thi calculation wa baed on a vey udimentay etimation that oveetate the eo. A moe accuate eo detemination hould yield ignificantly bette eult. Sumetky [4] povide an analytic and quantitative analyi of 6

146 how uch eo in the gating peiod will affect the goup delay cuve of the ytem. Figue 6-4 illutate the calculated eflectance and goup delay fo the deigned tuctue. Figue 6-5 how the goup delay ipple expected fo tuctue witten with.5nm and 5nm pixel eflectance (db) goup delay (n) wavelength (micon) wavelength (micon) Figue 6-4: xpected Reflectance and Goup elay fo m Linealy Chiped Bagg Stuctue. 5 goup delay ipple (p) 5-5 goup delay ipple (p) wavelength (micon) wavelength (micon) Figue 6-5: xpected Goup elay Ripple fo Linealy Chiped Stuctue Witten with.5nm and 5nm Pixel. 7

147 6.7 Altenative Goup elay Function 6.7. Contant Peiod elay Line Anothe application of Bagg gating tuctue i in the aea of delay line. Although the elative delay aco the eflection band i not paticulaly ubtantial tanmitted fequencie at the edge of the band expeience damatic goup delay a pedicted by [3]: nl g (5) c Thee tuctue have a vey mall bandwidth and the gating hould maintain a fixed peiod to keep the pecified fequency at the band edge. Thi would be accomplihed via a waveguide with contant adiu. Note howeve that the geometical concept i amenable eithe to multiple waveguide acting in paallel o to a ingle waveguide that cuve inwad a hot ditance to yield a diffeent effective gating peiod fo ubequent waveguide ection. Fo tuctue of thi ot it i advantageou to poduce a much delay a poible ove a hot ditance. Thu we employ a eaonably lage index contat without going o lage a to incu ubtantial catteing penaltie. The cental Bagg wavelength wa taken to be.55um and the gating laye had indice of.57 and.56. Thi give an expected coupling contant of.3um - and a eflection bandwidth of 6.3nm. A mm length tuctue i theefoe ufficient to yield a vey hap bandedge at.5468um. If we take the expected waveguide lo to be.db/mm and the catteing lo due to the gating to be anothe.3db/mm and the bend lo to be db/mm the oveall lo i le than 3dB. The tuctue offe a delay of.7n ove the mm gating length which give a goup index of 5. On the othe hand if the index contat i educed by 5% the coupling i cut in half and the tuctue mut be doubled in length to 8

148 obtain the ame hap bandedge albeit at a lightly longe wavelength. In thi cae the delay i twice a much but the oveall goup index emain the ame. In addition ince adiation loe due to the cuvatue of the waveguide ae expected to dominate the hote device expeience ignificantly lowe lo. Goup delay fo both tuctue i illutated in Figue 6-6. (a) (b) Figue 6-6: Goup elay fom Contant Radiu Waveguide with (a). and (b).5 Index iffeential Long Peiod Gating An additional deign conideation involve longe peiod gating. At cetain peiod longe peiod gating have been ued to couple light into fowad-popagating cladding mode fo band ejection and dipeion pupoe [9] but they may alo be deigned to couple to evee-popagating mode at a highe diffaction ode. In paticula when the gating peiod i given by the Bagg condition i till met fo the backwad popagating mode. B 3 (6) n 9

149 The lage peiod eae fabication toleance omewhat but doe intoduce the poibility of catteed diffaction ode (though thee don t meet the Bagg condition fo maximum diffaction efficiency). Thi alo mean that the coupling tength i educed and the minimum waveguide length i inceaed but fo a dipeion-type application we ae aely woking nea the minimum gating length fo eaonable eflectance. Thee i alo ignificantly highe catteing lo a the popagating mode can couple to adiating diffaction ode. On the othe hand thee ae ignificant advantage to thi appoach paticulaly when fabication iue ae conideed. Small eo in the peiod ie poe a much moe ignificant poblem when the gating peiod themelve ae mall. Figue 6-7 illutate thi ituation. A nm amplitude nomally ditibuted andom petubation wa applied to nomal fit-ode gating and longpeiod thid-ode gating (both baed on the ame deign a the dipeive Bagg tuctue outlined in the peviou ection). Clealy the GR i ubtantially woe fo the hot-peiod tuctue and ufficiently o a to have a lage impact on device pefomance than the additional lo tem intoduced by the long-peiod tuctue. (a) (b) Figue 6-7: Goup elay Ripple fo (a) Fit-Ode Bagg Gating and (b) Thid-Ode Bagg Gating with nm Nomally itibuted Random Petubation to Peiod Sie. 3

150 6.7.3 Tailoed Phae elay fo ipeion Compenation alie thi chapte indicated that a unifom linea delay lope equie an exponential dependence in the waveguide tajectoy. oweve it i illutative to conide the filte epone fo a waveguide with a linea dependence on angle: f (6.7) Baed on the ac length fomula above the path length a a function of angle become (6.8) Thu intead of obtaining a linea elationhip between the gating peiod and gating length we intead have (6.9) which ugget that the eulting goup delay cuve will be of a quadatic natue. Indeed Figue 6-8 demontate the diffeence between the eulting goup delay and the linea cuve poduced by the exponential pial. It i woth noting that thi deviation i only about.5% and would likely be wallowed up in the GR noie fom a ealitic gating tuctue. oweve tuctue with lage oveall goup delay and geate bandwidth could expeience a much lage impact fom thi effect. On the othe hand with appopiate optimiation the dipeion intoduced by a tuctue of thi fom could eaily be ued to compenate fo othe device that intoduce unwanted dipeion and phae ditotion. 3

151 Figue 6-8: epatue fom Linea Goup elay Cuve fo Waveguide with Linea Angula ependence Spial Bagg Stuctue a Amplitude Repone Filte Anothe potential application fo thee tuctue involve amplitude filteing of cloely paced pectal band fo wavelength diviion multiplexing (WM) elated device. In thi cae we ae not looking fo a continuou goup delay but athe a band of cloely paced tanmiion (o eflection) band. A i dicued in CAPTR 7 coupled eonant cavitie yield hap cloely paced eonant tanmiion line. While it i difficult to obtain the ame behavio with Bagg waveguide tuctue without intoducing extemely lage waveguide cuvatue the concept i uggetive of a potential appoach utiliing the decoupling method. Intead of pialing the waveguide inwad in a continuou manne to obtain a mooth goup delay cuve we deigned the waveguide to have a contant adiu with the exception of a peiodic petubation of the fom 8in e f (6.3) 3

152 Thi eult in a tajectoy of the fom hown in Figue 6-9. Note that the peiodic ipple ha been exaggeated ignificantly to demontate the oveall hape. With an initial adiu of 3.9mm the actual total adial hift wa appoximately 3um fo thi tuctue. Figue 6-9: Sinuoidal Waveguide Tajectoy (xaggeated fo ffect). A may be expected the tajectoy poduce a peiodic vaiation in the gating peiod a hown in Figue 6- and the amplitude epone of the tuctue i hown in Figue 6-. The eflection band ae paced by appoximately.nm though they ae not paticulaly unifom in thi cae. oweve a moe complete optimiation of the tuctue hould be uitable to obtain a bette oveall epone. Figue 6-: Sinuoidal Gating Peiod. 33

153 Figue 6-: Amplitude Repone fo Sinuoidal Waveguide Tajectoy. 6.8 Spial Bagg Filte Summay Thi chapte peented a unique appoach to Bagg gating filte deign in which the functional fom of both the waveguide tajectoie and gating tuctue ae fully independent and quite abitay. The method povide fo eue of gating and eaed fabication toleance while offeing a geat deal of flexibility in tem of application and optical delay cuve. eign containt included a minimum adiu of cuvatue to pevent ignificant adial leakage and a minimum gating chip to pevent coupling between ucceive pial loop. Coupling tength a a eult of cougation depth play a ignificant ole in the oveall tuctue epone by intoducing catteing lo and detemining the width of the ejection band. Additionally by incopoating highe-ode gating peiod into the device deign fabication eo can be ubtantially mitigated. Baed on the deivation peented heein thi appoach i capable of poviding elative delay between pectal component of anywhee between p/nm and n/nm depending on the choen deign containt. While dipeion compenation i one 34

154 diect application of thi appoach thi ange of elative delay value offe a unique appoach fo WM-elated application and tunable ouce. The adial natue of the gating line and the cuved waveguide tajectoy combine though vey taightfowad fabication pocee. The angula natue of the tuctue component aveage out fabication eo and offe a ubtantially educed GR than can be obtained with linea gating fabicated with the ame pocee. Thi chapte demontated the deign poce and the epone model fo a tuctue incopoated a m long waveguide onto a tandad wafe. The tuctue had a flat eflectance band ove a bandwidth of about 4nm and a dipeive delay of 5p/nm. In addition we povided a delay line appoach utiliing high index contat gating to obtain an extemely lage goup delay with an effective goup index of 5 with expected leakage well below 3 db. Altenative tajectoy deign wee alo peented a example demontating the wide ange of filteing application attainable with thi appoach. Long peiod gating offe educed fabication complexity. Othe imple functional fom of the waveguide tajectoy may be ued to obtain a vaiety of nonlinea delay cuve which lend themelve well to coecting and offetting fequency delay aleady peent in an incident ignal. On the othe hand an appopiately deigned peiodic petubation to the waveguide tajectoy can be ued to obtain unique tanmiion band fo electing pecific fequency component. PSO o othe optimiation appoache ae eaily applied to the functional fom to obtain pecific filte epone tailoed to a wide ange of application. 35

155 CAPTR 7 AXISYMMTRIC RSONANT CAVITY FILTRS Optical cavitie offe anothe type of eonant tuctue that find ue in many diffeent application. While moe commonly ued fo optical ouce and quantum electodynamic expeiment they may alo be ued in filteing application [6]. The tandad appoach fo eonant filte involve a ing tuctue coupled to a waveguide. In uch condition the epone of the ignal coupled into and tanmitted though the eonato ha a naow bandwidth centeed on the tuctue eonance. The non-eonant band do not couple into the tuctue and emain in the waveguide. The filte phae epone alo involve lage goup delay nea the eonant fequencie. In thi chapte we exploe the application of thee-dimenional axiymmetic cavity tuctue to optical filteing and diffeent way to appoach thee type of tuctue. In geneal a good eonance-baed filte mut have a lage Q-facto. Cetain geometie may be exploited to obtain thi with a ingle cavity though uch tuctue ae often bette uited fo optical ouce and vaiou othe electodynamic application than to diect filteing of tanmitted and eflected ignal. On the othe hand chain of weakly coupled low-q cavitie can be built up into a guiding tuctue with ignificant filteing popetie. 7. Single Cavity Optimiation alie in thi wok method fo analying and modeling micocavity eonato wee outlined and dicued in detail. Of inteet in thi ection i to apply thoe tool and the optimiation appoach dicued peviouly to deign and optimie eonato fo pecific 36

156 pectal output and eonance chaacteitic. In paticula we would like to povide an optimiation pogam with a et of deign citeia that pecify that a deigned cavity hould eonate with the highet Q-facto poible at a given fequency and with a pecified mode numbe (e.g. T ). The pogam then etun the cavity geomety neceay to obtain thoe chaacteitic. In geneal the deign and optimiation of eonant cavitie i nontivial. Although cetain apect can be detemined analytically uch a the dimenion equied fo a baic cylindical tuctue to eonate at a pecified fequency an optimal eonato deign may eaonably be expected to come only though a numeical optimiation appoach. Futhe the poce can be athe time conuming. ach newly popoed deign uggeted by the optimiation algoithm mut be modeled and teted fo the deied eonant chaacteitic. In a PSO appoach an aveage wam ie of 3 paticle mut be e-evaluated at evey iteation. A typical optimiation that may take iteation to convege then equie 6 epaate cavity model. Time domain imulation which can take anywhee fom a few minute to eveal hou to povide a full diagnotic of a given cavity become completely unuable in thi cenaio. On the othe hand the eigenmode analyi dicued ealie in thi wok ha been hown to wok well with PSO fo the cavity deign poce [3]. 7.. Paticle Swam Optimiation of Cavity eign One of the mot ignificant difficultie in the deign poblem come when we conide the ange of geometic paamete available to the optimiation algoithm. Fo example conide the cavity geometie illutated in Figue 7-. The cavity adiu and length would nealy alway be optimied to obtain the deied eonant fequency o wavelength. But a 37

157 dielectic cavity by itelf ha a athe low Q-facto and one mut additionally optimie the uounding egion to obtain naow eonance. In the pilla tuctue one might optimie the individual laye to adjut fo a tapeing of the tuctue width intoduced duing fabication [35 4 3]. In pilla adial confinement i povided imply by the index contat in a longitudinal guiding ituation. One altenative ue elie on efactive index contat in the vetical diection while intoducing a highly eflective confinement medium in the adial diection. Cavitie encaed by metallic film ae common in micowave application but the loy natue of metal at optical fequencie tend to be fa too geat to be ueful in thee type of configuation. oweve highly eflective adial confinement tuctue with no intinic aboptive lo ae capable of uppoting extemely high Q-facto unde cetain condition [47 49]. It i ueful to optimie the geomety to obtain thi effect which may be futhe accentuated by optimiing laye epaating the cavity fom a bulk ubtate egion. Figue 7-: epiction of Single-Cavity Geometie. 38

158 With uch a lage paamete pace to conide paticulaly when we have noted peviouly that the eonance chaacteitic can have vey naow peak in paamete pace (ee Figue 3-3) an exhautive o deteminitic each algoithm would be hopelely inefficient. Futhe it i vey eaonable to aume that paamete pace will contain a numbe of locallyoptimal olution that pefom fa woe than ome globally-optimal cavity deign which mean that calculu-baed each algoithm will inevitably fail. Thi make the cavity eonato deign an ideal poblem fo the PSO algoithm dicued ealie. 7.. Mode Choice It ha been hown [3] that cylindical waveguide uounded by metallic cladding laye offe ubtantially educed aboptive lo to the T popagating mode. Thi eult fom the electic field being polaied tangential to the dielectic-metallic inteface. In like manne a T p mode maintain a fixed elationhip between electic polaiation and the inteface between the cavity and the confinement egion. The difficulty of coupling to and fom the cavity equie that the light be focued to a mall a pot a poible. Radially ymmetic mode may be focued to a ignificantly malle pot than can linealy polaied o othe hybid mode [33 34]. Additionally mode of thi type ae ignificantly malle than the higheode hybid mode which allow fo malle cavitie and educed mode volume. One of the pinciple eaon thi mode i lagely ignoed i the difficulty in coupling to it. The mode involve linealy polaied light which i eay to obtain. Thu mode matching to cavitie baed on that mode i faily taightfowad allowing one to couple light in and out of uch tuctue without temendou difficulty. On the othe hand obtaining aimuthally (T ) and adially (TM ) polaied popagating mode i not nealy o tivial. An appoach fo 39

159 coupling to aimuthally and adially polaied eonant mode will be exploed in ubequent ection etemination of Cavity Fitne To povide a obut optimiation appoach fo the contuction of eonant filte all impotant deign citeia have to be appopiately epeented and given uitable weight. An impopely defined fitne function may eult in undeiable olution that may behave well in ome apect and vey pooly in othe. In the cae of cavity eonato the mot cucial equiement i that the cavity eonate at a pecified wavelength with a high a Q-facto a poible. It i alo wothwhile to include a component to foce thi wavelength to coepond to a pecific mode numbe. Thu the eigenmode olve outlined in an ealie chapte offe a uitable tool to obtain the metic needed fo the fitne function. The eigenmode olve will take the cavity geomety a an input and etun the eonant wavelength Q-facto and electic field ditibution. Since the poblem i aumed to be axiymmetic and the aimuthal mode numbe i pecified a an input to the eigenmode olve it need not facto into the fitne function. On the othe hand the field ditibution can be ued to detemine the adial and axial mode numbe. Thee may all be combined into a ingle fitne mapping function of the following fom: F a a a A 3 Q M M M M (7.) In thi expeion give the deign wavelength and M and M pecify the deied adial and axial mode numbe epectively (typically both unity). M and M ae the coeponding chaacteitic of the neaet eonant mode of given cavity geomety. Q i the calculated quality 4

160 facto of the cavity. The thee paamete in the exponent a a and a 3 (all aumed to be poitive-valued) povide weight fo each of the cavity chaacteitic. The emaining coefficient A i imply a cale facto to emove unit fom the expeion and detemine what magnitude of diffeence between the deied and actual wavelength i ignificant. Fo example a value of um - eentially mean that deviation of the eonant wavelength fom deign ae not ignificant until they ae of the magnitude of nm-nm. Thi fomulation of the cavity fitne function povide eveal featue. The function i deigned to be minimied o value malle than neceitate high Q-facto. The diffeent cavity chaacteitic ae combined geometically athe than algebaically which pevent the function fom elying olely on a ingle apect while neglecting the othe. Since the mode numbe i intege-valued it make ene that it hould have no impact on the oveall cavity fitne o long a it matche the deied mode. A uch in thi fomulation the potion dealing with mode numbe become unity when they ae coect. Likewie the eonant wavelength contibution alo appoache unity a it nea the deign wavelength. Thi leave the Q-facto alone to poduce the equied mall fitne value. A vey epectable eonant cavity deign would have a Q-facto about. The deign citeia ued fo optimiation in thi eeach aumed that a ufficiently optimied cavity would have a T eonant mode at a wavelength nea.5um with a Q-facto ove. Baed on tial and eo the exponent weight in qn. 7. a and a fo the Q-facto and eonant wavelength wee et to. Thi allowed the fitne function to be ufficiently naowed in the egion of an optimal olution o a to foce the paticle down into it quickly while till being ufficiently wide that the paticle did not peed by without locating it. The wavelength 4

161 caling coefficient A wa et to um - which foced the eonance to be within about 5nm of the deign wavelength. The exponent weight on the mode numbe a 3 wa et to 8 which inceaed the fitne value by ove two ode of magnitude fo cavity eonant mode othe than the pecified deign mode numbe. With thee fitne function paamete the deign citeia wee atified with a fitne value malle than -. Thi wa choen a the convegence thehold fo the PSO appoach igh Q-facto Cavity eign Unde vey pecific combination of cavity dimenion a vey tong enhancement in the Q-facto i obeved if a lole highly eflective egion uound the cavity in the adial diection [47 49]. Thi effect wa dicued by Ibenecu and demontated numeically uing PC a a uounding medium. The baic concept i demontated in Figue 7- and the eonance wee calculated and plotted in Figue 3-3 a a benchmak fo the eigenmode olve. entially fo thi type of ituation the geomety caue a epulion between polaiation tate eulting in a egion with vey mall goup velocity at ome point in the dipeion elationhip [49]. Thi occu at vey pecific value of cavity adiu and geomety depending on the eflectivitie at eithe end of the cavity. PC ielectic Cavity PC Figue 7-: igh Q-facto Micocavity. 4

162 While PC ued fo adial confinement allow the Q-facto to each 4-5 eplacing them with eal metal film eult in a damatic eduction of the Q-facto. It hould be noted that metal film wok well in the micowave egion of the pectum (and have been ued to fom micocavitie fo decade) whee the fequency-dependent pemittivity appoache that of a PC. Futhe when media with ufficiently high gain ae peent uch tuctue have been hown to function a low-thehold lae [35]. Thee i an altenative appoach to the geomety that offe a imila effect. Yaiv oiginally popoed the idea of a adial Bagg gating encaing optical fibe [36] and the concept ha alo been ued to aid confinement in micocavitie [37 38]. Thi configuation ha imila peak of the Q-facto fo cetain combination of geometical paamete a in the PC-encaed cavity cenaio [47]. Becaue of the naow ange of paamete offeing enhanced Q-facto PSO offe a convenient appoach a a micocavity deign tool [3 39]. An example of an optimied BRencaed cavity and it coeponding amplitude epone i hown in Figue 7-3 and the eonant wavelength and Q-facto fo vaiou combination of cavity length and adii ae hown in Figue 7-4. Notice that the eonant wavelength depend pimaily on the adiu while the Q-facto i affected moe tongly by the cavity length. Figue 7-3: Cylindical-Bagg Micocavity and Amplitude Repone. 43

163 (a) (b) Figue 7-4: (a) Reonant Wavelength and (b) Q-facto fo Combination of Cavity Geometie. To impove confinement futhe it i woth exploing a mean of educing leakage into the ubtate. A conventional appoach i to place a et of BR laye between the ubtate and the cavity (a hown in Figue 7-5(a)) to educe downwad popagation. Indeed in ou imulation thi impoved the peak Q-facto fom aound 3 to about 5. oweve it ha been noted peviouly [39 4] that the mall ie of the cavitie eult in a geat deal of diffaction at the end of the cavity. Thu the longitudinal leakage conit of a lage ange of wave vecto which educe the effectivene of the BR. The tuctue may be optimied omewhat to povide the highet eflection fo the peak wave vecto but thi offe only a mall impovement. An altenative i to optimie each eflecting laye independently. Thi wa een to inceae the Q-facto by nealy a facto of to appoximately 9. 44

164 (a) (b) Figue 7-5: Cavity with Longitudinal Confinement Povided by (a) BR Laye and (b) Optimied Reflecting Laye. 7. Coupling to Axiymmetic Mode While we may optimie ingle cavitie fo high Q-facto at pecific eonance fequencie we would like to apply thee cavitie to the field of optical filteing. A uch it i abolutely eential to be able to couple an incident ignal into the tuctue and obtain a welldefined epone fo the ignal tanmitted though and eflected fom the device. In the cae of axiymmetic cavity mode thi involve conveting a ignal to eithe a adial o aimuthal polaiation tate and focuing it down to a ufficient ie fo mode matching with the cavity in the manne illutated in Figue

165 Micocavity LP Beam T Beam Polaiation Conveting lement Focuing Len Figue 7-6: Coupling Methodology fo Micocavity Reonato. A uitable polaiation-conveting element wa oiginally fabicated by Mohammed [3] and wa developed into an appopiate tuctue fo oxide wafe by Rumpf and Mehta [4]. The latte cae a patially polaiing autocloned element (SPAC) baed on a patiallyvaied effective-index gating tuctue ha been ued expeimentally to convet a linealy polaied incident beam into both aimuthal and adial polaiation. The tuctue and it output beam ae illutated in Figue 7-7. Figue 7-7: SPAC Stuctue and Output Aimuthally Polaied Beam. Once the incident beam i in the coect polaiation tate it mut alo be focued down to the appopiate ie to couple into the cavity filte. Notice that fo the appopiate cavity adii 46

166 fo the eonance we have decibed the pot ie i below the geometical diffaction limit of mot lene. Thu we need a high-na len and we hould numeically calculate the eulting pot ie diectly uing the full diffaction integal ince polaiation tate will tongly influence the eulting pot ie and hape fo uch lene [4 43]. With moden fabication method it i poible to fomulate an ulta-high NA len on a mall cale. An example of uch a len (coutey of P. Sinivaan) i hown in Figue 7-8. Figue 7-8: igh NA Lene (NA =.45). To obtain the eulting field ditibution fo the pot at the focu of uch a len we tun to vecto diffaction theoy [33]. In the cae of lage Fenel numbe we can ignoe diffaction effect fom the edge of the len and decompoe the incident ignal into a eie of plane wave. At the focu of the len the eulting field ditibution i calculated accoding to the upepoition of each plane wave A k x k y ik dk xdk y e k (7.) whee A i the complex vecto amplitude of the plane wave on tanmiion though the len apetue. If we aume an aimuthally polaied beam (T ) incident on the len given by 47

167 inc ˆ e w (7.3) the eulting field at the pot may be expeed accoding to ˆ f w in ik co kf in co e e J k in d (7.4) whee i the axial ditance between the obevation plane and the len focu f i the focal length of the len and i the maximum angula extent of the len a een fom the focal point. Thi ditibution i illutated in Figue 7-9 fo eveal value of the focuing len numeical apetue. Similaly if the polaiation of the incident field i witched to the adial diection the eulting field ditibution become J k f w ik co in in co kf in coe e in J k in ˆ d ˆ (7.5) Clealy the TM cae ha both longitudinal and adial polaiation component. Thee ae hown in Figue 7- while the eulting electic field intenity i demontated in Figue 7-. Figue 7-9: Aimuthally Polaied Field at igh NA Len Focu. 48

168 (a) (b) Figue 7-: (a) and (b) Component of Radially Polaied Beam at igh NA Len Focu. Figue 7-: lectic Field Intenity of Radially Polaied Beam at igh NA Len Focu. 7.3 Coupled Reonato Filte So fa we have demontated a eaonable appoach to deigning and optimiing ingle cavitie fo naow eonance and we have outlined a feaible method to couple an incident ignal into uch tuctue. The final tep needed to contuct a ueful optical filte i to expe the effect intoduced by coupling multiple cavitie to each othe in a chain. In uch a ituation it i eadily appaent that the exotic geometie dicued ealie in thi chapte do not lend themelve well to a multiplexed device. oweve the chaining of a lage numbe of cavitie togethe allow u to make ue of much weake eonato without intoducing lage lo tem to the epone function. 49

169 7.3. Coupled Reonato Optical Waveguide Thee ae a numbe of way to appoach a tuctue compoed of coupled eonato. A vey implitic fit-ode appoximation may be obtained by noting that each ucceive eonato eceive a input the filteed ignal output fom the peviou eonato in the line. In othe wod if we ignoe backwad popagation and ocillation between neighboing cavitie the oveall amplitude tanmiion function i appoximated by N TN T (7.6) If the impule epone of a ingle filte i given by a Loentian function centeed at the cavity eonant fequency the output of the econd filte will be the Loent filte function applied to the boadband ignal twice and o on. Thu it i eaonable to aume that the Q-facto fo an N-cavity filte i appoximately equal to NQ whee Q i the Q-facto of a ingle eonant cavity. Now thi appoach obviouly ignoe coupling and ocillation which ae extemely ignificant facto and have paticula implication fo the eonant fequency. The imple cae of a pai of eonato i compaable to a coupled waveguide tuctue in which the nominally identical popagating mode of two waveguide combine and plit into a ymmetic mode and an antiymmetic mode accoding to coupled-mode theoy. The diffeence between the popagation contant of the two mode i diectly popotional to the coupling contant of the waveguide pai [7]. In like manne a pai of identical coupled eonato will eonate at two diffeent mode centeed on the oiginal eonant fequency. The fequency epaation between the two mode i diectly popotional to the coupling contant between the two cavitie [44]. 5

170 Yaiv oiginally popoed that a line of coupled eonato could act a a waveguide with minimal lo ove a tanmiion band even if hap bend occu in the popagation diection [45]. Auming a quai-infinite linea combination (paallel to the -axi) of high-q eonato one may aume that the olution fo the eigenmode atify the Bloch theoem. Thu K ˆ K i t inkl t e e nl n whee i the eigenmode of the individual eonato i the ingle-cavity eonant fequency and L i the pacing between eonato. Unde the aumption that the individual eigenmode i nomalied to and that the full filte mode Yaiv how that the following dipeion elation hold: e K (7.7) atifie Maxwell quation inkl n n K inkl ne (7.8) n whee n and n ae coupling contant given by (n ) n n ' nlˆ 3 d nlˆ nlˆ 3 ' d 3 d (7.9) and whee i the pemittivity ditibution fo a ingle eonato and i the pemittivity due to the additional eonato. In the weak coupling limit (and auming identical cavitie o that = ) the dipeion elation educe to co KL (7.) K 5

171 whee = -. It hould be noted fo completene that the + in qn. 7. hould in eality be eplaced by to account fo poitive and negative goup velocitie though the latte i unlikely to occu phyically [46]. Fom thi elationhip we can obtain a goup index: c (7.) L n G Note that thi indicate an inceae in the initial goup delay by a facto of appoximately. Fom qn. 7. we can eaily deive the full fequency bandwidth of the tuctue: ue (7.) The eigenmode of the coupled eonato tuctue ae paced evenly in K-pace with value anging fom to [47]. Thu if the wave vecto ae paced out by L K N L (7.3) the fequency pacing of the mode i appoximated by inkl (7.4) N Thi mean that the eigenfequencie hould be cloely paced towad the edge of the band and pead out moe towad the middle. oweve when a elatively mall numbe of cavitie i peent the epaation i le eaily expeed and the aveage pacing can offe geate meaning. Baed on the deived goup index we can calculate the time equied to tavee a ingle eonato: 5

172 Ln G (7.5) c Conequently the leakage out of one eonato into the next which coepond to it effective Q-facto can be expeed in the following manne [48]: Q eff (7.6) The oveall Q-facto fo a eonance of the entie device i alo eaily obtainable. The tanit time though the device i imply the ingle-cavity lifetime caled by the numbe of cavitie N. Thu the Q-facto fo a device eonance line may be expeed a N Q dev (7.7) Thi deivation doe not diectly account fo adial loe out of the oveall tuctue o potential intinic aboptive loe which may be expeed in tem of an intinic Q-facto. A moe detailed accounting fo thee loe may be found elewhee [48] though it i woth noting that o long a the intinic cavity lifetime i ubtantially lage than the lifetime due to coupling ( fom qn. 7.5 above) which occu when Q Q the effective Q-facto will int eff dominate in the expeion of oveall epone function. In othe wod a long a ignal enegy leak out of the device ignificantly lowe than it i allowed to move between ucceive eonant cavitie the cavity coupling dominate the oveall filte epone Coupling of Low Q-facto Reonato Thi analyi ha aumed elatively lage Q-facto fo the coupled eonato. The tandad coupled-eonato optical waveguide (CROW) appoache make ue of eithe ing 53

173 eonato [46 49] o photonic cytal (PC) [48 5] tuctue. One limitation of both appoache i thei confinement to a two-dimenional plane: the filte may be tiled in only a ingle diection. A moe flexible appoach would enable fee-pace optic to couple incident light into a two-dimenional aay of thee-dimenional optical filte. The implet appoach to thi poblem i to fabicate an aay of cylindical coupledeonato filte. Unfotunately imple cylindical dielectic eonant cavitie tend to have elatively low Q-facto. Thi intoduce a numbe of vaiation to the peviou analyi. Fit and mot obviouly the low Q-facto mean that enegy i not toed in individual eonato fo a long meaning that the oveall tuctue delay i expected to be ignificantly educed. Additionally the lage fequency bandwidth of the eonato mean that the filte tanmiion band will be ignificantly lage than if high-q eonato ae ued. oweve a noted above the Q-facto fo device eonance can be expected to cale with the numbe of eonato ued. Anothe iue eult fom the low modal confinement. Thi lead diectly to much highe coupling than one would expect fom high-q filte. Fom qn. 7.8 and 7.9 we expect to find multiple pectal ode in the tanmiion and eflection band. On the othe hand a the coupling tength i educed one may eaonably expect fewe and naowe eonance line. A an example conide an iolated (in ai) cylinde of SiON (efactive index.936) with adiu 4nm and height 3nm. The eonant wavelength i appoximately.4um and the Q-facto i appoximately. The amplitude epone fo a tack of 6 unifomly paced identical cylinde a the cente-to-cente pacing vaie fom 4nm to 6nm i demontated in Figue 7-. Notice the widely epaated band that cloe in and naow a the epaation inceae and the coupling contant deceae. 54

174 (a) (b) (c) Figue 7-: Amplitude Repone of 6 Coupled Cylindical Cavitie Sepaated by (a) nm (b) nm and (c) 3 nm. A final conideation fo thee 3- tuctue involve the egion epaating the eonant cavitie. In the cae of ing eonato and PC cavitie thee i no guiding povided in the epaating egion. oweve if we ae making ue of pilla compoed of laye of diffeent dielectic mateial it i likely that cetain ange of tuctue geometie will povide ubtantial confinement and guiding in the egion between cavitie. Such tuctue will till have CROW apect but will additionally act in pat a Bagg gating tuctue. Thi mean that in addition to the eonance peak we can expect lage band ejection egion coeponding to the vaiou Bagg gating ode in the tanmiion pectum. While thi complicate a full analyi of the tuctue it alo ugget a mean fo eliminating ome of the tanmiion line intoduced by tong coupling between eonato GaA/AlA Cavity Filte A the bai fo a coupled eonato optical filte we begin with a cylindical GaA cavity (efactive index 3.5) uppoted by an AlA pot (efactive index 3.) with a lightly malle adiu (a hown in Figue 7-3). The cavity wa optimied fo a eonance nea.5um and the eulting adiu wa 383nm and the cavity length wa 53nm. The AlA pot adiu wa 55

175 nm which ha a cutoff fo popagating mode at appoximately.5um. The eulting tuctue had a eonant wavelength of.48um with a Q-facto of 6. A adial co ection of the eonant aimuthal (T mn ) eigenmode at thi wavelength i illutated in Figue 7-4(a) while Figue 7-4(b) demontate the filte amplitude epone given an incident ouce baed on the eult of the high-na coupling decibed above. Figue 7-3: Co-Section and 3- Pofile of GaA/AlA Filte Unit Cell. (a) (b) Figue 7-4: (a) Reonant igenmode and (b) Amplitude Repone of GaA/AlA Filte. 56

176 Multiplexed Filte evice It i eadily appaent that a ignificant potion of the incident enegy i catteed adially out of the filte and only light within a pecific band pae though the tuctue to any ignificant degee. The moe inteeting cae occu when we begin to build a coupled netwok of thee tuctue a i illutated in Figue 7-5. In thi cae the cente-to-cente pacing between cavitie wa 93nm (pilla length wa 4nm). Figue 7-5: Multiplexing of GaA/AlA Cavity Filte. A wa outlined above eveal diffeent effect begin to occu. Fit the ubequent tanmiion of a filteed ignal fom one cavity to the next eult in a teady naowing of the tanmiion peak. Additionally the altenating egion of high and low effective index poduce a Bagg gating effect which mean we hould expect to ee a tong eflection band in the amplitude epone a the numbe of cavitie gow. Finally tong coupling between the cavitie eult in a plitting of the eonant mode. Thi poduce multiple cloely-paced peak in the tuctue epone function. The amplitude epone of a tuctue with (a) 4 and (b) 64 cavitie i hown in Figue 7-6. All thee of thee featue ae clealy evident in the epone cuve. 57

177 (a) (b) Figue 7-6: Amplitude Repone of a evice with (a) 4 and (b) 64 Cavitie Bagg Reflection and Cavity Spacing ependence The Bagg eflection may be explained quantitatively if we look at the effective index of the popagating mode in the two diffeent filte egion. A i hown in Figue 7-7 the uppot pot opeate faily cloe to cutoff in the egion of inteet. Thi ugget a ubtle connection to the Nano-AWG filte exploed peviouly and might imply that dipeive effect could play an impotant ole in thee tuctue. Additionally note that in the wavelength band nea the Bagg eflection peak (centeed at appoximately.37um) the effective index of the pot i about. and that of the cavity egion i oughly 3.. Thi eult in a Bagg wavelength of 4.4um which ha a thid diffaction-ode eflection peak at.38um. 58

178 (a) (b) Figue 7-7: ffective Index of Popagating Mode in (a) Cavity and (b) Pot Region. Thu the effective index appoach to the calculation of the eflectance pectum i in ageement with the obeved value. It i alo uggetive of a mean to tailo the epone function; if the pacing between cavitie i educed two effect will occu. Fit the cavitie will couple moe tongly eulting in a lage epaation between eigenfequencie which hould lead to a wide tanmiion bandwidth. Additionally the Bagg eflection will hift to hote wavelength. Fo a light inceae in cavity pacing the oppoite will occu fo both phenomena. Baed on the effective index calculation it i eaonable to expect that a 5nm hift in the pacing will yield a hift in the thid diffaction-ode Bagg eflection peak of oughly 4nm. Amplitude epone fo cavity pacing of (a) nm (b) 375nm (c) 45nm and (d) 475nm ae hown in Figue

179 (a) (b) (c) (d) Figue 7-8: Amplitude Repone with Cavity Spacing of (a) nm (b) 375nm (c) 45nm and (d) 475nm Reonance Shift and Cavity Sie ependence A final inteeting petubation that may be applied to the cavity geomety to tune the oveall epone involve a vaiation in the cavity adiu. Thi i a bit moe complex than the cavity pacing ituation a it diectly change the eonant fequencie a well a the cavity effective index (which will hift the Bagg eflection peak). Figue 7-9 how the change in the filte epone when the adiu i deceaed o inceaed by 6nm. It i vey clea that the eonant wavelength hift by about.7um in each cae while the Bagg eflection band hift 6

180 much le. What i paticulaly inteeting i that in the cae of a 357nm adiu the tanmiion peak move inide the eflection band. The damatic change in the amplitude epone fo a elatively mall change in the cavity geomety indicate that thee tuctue equie vey caeful tuning to obtain a deied epone function though it i alo highly uggetive of a poible application. Specifically an optically active mateial in the cavity egion could eceive a mall electically- (o tempeatue-) induced vaiation to it efactive index (which offe a imila effect a a change in cavity adiu) and could eaily witch the eonant tanmiion line on and off by puhing it into o out of the eflection band. (a) (b) Figue 7-9: Amplitude Repone fo Cavity Radiu of (a) 357nm and (b) 49nm Feaible Stuctue Repone While the tuctue epone ha been demontated and evaluated with 64 cavitie peent uch a device i extemely difficult to fabicate. Since the peiod i nealy um the tuctue would conit of an 8nm diamete fee-tanding pilla nealy 6um tall. Thi would eveely tetch the bound of eaonable fabication pocee and i theefoe unealitic. A moe appopiate deign would conit of 6 peiod. Such a tuctue would be lightly le 6

181 than 5um tall. While thi i till difficult to ceate it i not impoible (and a lightly hote tuctue coniting of -4 peiod would not have a ignificantly diffeent epone function). The amplitude epone of thi tuctue calculated via (a) MOL and (b) FT i hown in Figue 7-. Note that the time domain equie a much lowe patial eolution to pefom the calculation in a eaonable time fame and i thu not quite a accuate. oweve the ageement between the two imulation offe ome eauance that the pectal eult ae coect. The time domain imulation i alo capable of plotting the ingle-fequency CW field inide the tuctue at peak tanmiion a hown in Figue 7-. (a) (b) Figue 7-: Amplitude Repone fo 6-Cavity Filte Calculated via (a) Method of Line and (b) Finite iffeence-time omain. Figue 7-: Radial Co-Section of Reonant Fequency Field Intenity in Filte. 6

182 The FWM bandwidth of the tanmiion line i appoximately 3nm which coepond to a device Q-facto of 5. While thi i not paticulaly high in tem of pectal election the natue of the tuctue i uggetive of eaonable delay line functionality. The magnitude and phae of the tanmitted and eflected mode ae plotted in Figue 7-. The tanmitted ignal ha a eaonably flat phae aco it which could be uitable fo calculating phae delay a a function of fequency. (a) (b) Figue 7-: (a) Tanmitted and (b) Reflected Signal. A bette appoach i to apply a ankel Tanfom to both the incident and tanmitted ignal. The tanfom pai take the fom F f k f J k F k J k d kdk (7.8) whee F i the tanfomed ignal and J i a Beel function of the fit kind with ode. Thi appoach i the cylindical analog to uing the Fouie Tanfom to expe a ignal in tem of plane wave. ee we expe the ignal in tem of an infinite um of Beel function which i appopiate fo cylindical coodinate ince the olution to the wave equation natually take thi fom. 63

183 Fo aimuthal polaiation it i appopiate to make ue of fit-ode Beel function and efe to the popagating Beel-wave with peak amplitude. The fequency-dependent vaiation of the phae on thi wave povide an adequate mean of etimating the tuctue dipeion. Figue 7-3 how the amplitude epone and goup delay fo (a) the tanmitted and (b) eflected ignal. Note that the delay in the eflected ignal gow quite lage at the band edge while the tanmitted ignal delay i oughly flat aco the entie bandwidth a wa peviouly pedicted and obeved in the cae of ing-eonato and photonic cytal CROW [49 5]. Fom qn..9 we may etimate the goup index of the tuctue to be 8.. By educing the inte-cavity coupling and inceaing the Q-facto of the individual eonato it i eaonable to expect that the delay could be inceaed ignificantly. (a) (b) Figue 7-3: Amplitude Repone and Goup elay fo (a) Tanmitted and (b) Reflected Signal Coupled igh-q Cavitie Filte Fo a vaiety of othe application it i deiable to deign tuctue with ignificantly inceaed goup delay ove a faily mall tanmiion bandwidth. To obtain uch tuctue without acificing the eae of fabication and the ability to multiplex them in a two-dimenional 64

184 aay it i ueful to etun to micopot cavitie compoed of monolithic pilla of altenating dielectic laye (in the fom of BR fo confinement pupoe) uounding a mall cavity laye [35 3]. Thee tuctue do not equie any exotic fabication method beyond tandad depoition lithogaphy and etching. Futhe coupled cavity filte may be fomed by adding thicke cavity laye afte a pecified numbe of BR laye pai a illutated in Figue 7-4. The pinciple fabication concen involve the apect atio of the eulting pilla. A additional cavitie ae tacked the pilla height gow which can poe a ubtantial challenge fo accuate fabication pocee. Figue 7-4: Coupled Cavity ielectic Pilla Filte Single Cavity Repone Reonant cavity deign lagely follow the method outlined ealie in thi chapte. Baed on chaacteied fabication pocee we choe to ue dielectic film with efactive indice of.936 (SiON) and.4496 (SiO ). Additionally the lage index contat offeed highe eflectance and confinement which povided highe Q-facto without equiing exta BR laye. The pilla adiu wa et to um and the optimal cavity laye thickne wa found to be 65

185 .53um. Thi eult in a T eonant mode at.55um. With thi opeating wavelength one would nomally expect the quate-wave thicknee fo the two Bagg laye to be nm and 67nm. oweve the naow tuctue diamete deceae the effective index of the vaiou laye ubtantially. To compenate and obtain the deied eflectance the laye thicknee wee inceaed to 45nm and 38nm epectively. Figue 7-5 demontate the amplitude epone of thi cavity with (a) 4 (b) 6 and (c) 8 BR laye pai on eithe ide of it. Clealy a BR laye ae added the bandwidth of the eonance line deceae eulting in an inceaed cavity Q-facto. The calculated value of the Q-facto fo each cae i (a) 65 (b) and (c) 53. Note that the epone i baed on coupling a ealitic ouce condition into the eonato and back out again. Thu the educed eflectance ceiling i due to catteing loe a the incident ignal couple to a popagating mode. (a) (b) (c) Figue 7-5: Single Cavity Amplitude Repone with (a) 4 BR Pai; (b) 6 BR Pai; and (c) 8 BR Pai Coupled Reonato Stuctue To fom a coupled cavity ytem the tuctue decibed in the peviou ection eceive an additional cavity and et of BR laye diectly on top. Thu the 6 BR coupled tuctue ha two cavitie epaated by 6 BR laye pai with additional et of 6 pai on eithe end fo a total of laye (the middle BR ection equie an exta low-index laye fo ymmety). 66

186 Note that the coupling contant in the cae of the ingle cavity (which decibed coupling the incident ignal to the tuctue) i now identical to that decibing coupling between the cavity pai. Thu baed on the analyi of ection 7.3. we hould expect the two cavity tuctue to have Q-facto appoximately double that of the ingle cavity cae and the pacing between eonance to deceae a additional BR laye ae ued ince the eulting coupling contant i educed. Figue 7-6 illutate the amplitude epone of the two cavity tuctue. (a) how the 4 BR laye pai cae and ha a eonance pacing of 46.3nm and an aveage Q-facto of 3; (b) ha 6 BR pai a pectal epaation of 4.9nm and an aveage Q-facto of 45; in (c) the 8 BR pai cae ha a wavelength epaation of 4.5nm and a Q-facto of. (a) (b) (c) Figue 7-6: Coupled Cavity Pai Amplitude Repone with (a) 4 BR Pai; (b) 6 BR Pai; and (c) 8 BR Pai. Although thee eult have meit a moe applicable tuctue would have ignificantly moe cavitie peent. Thu we extend the tuctue patten and demontate the epone fo 4- and 8-cavity device in Figue 7-7. Notice that the tuctue bandwidth matche cloely with that of Figue 7-6(b). In thi cae the aveage Q-facto wee 8 and epectively and the aveage eonance pacing wee 3.8nm and 7.nm. Thi matche well with the theoetical 67

187 analyi outlined above. The Q-facto cale linealy with the numbe of cavitie peent while the line pacing i inveely popotional to the ame. (a) (b) Figue 7-7: (a) 4 Coupled Cavitie and (b) 8 Coupled Cavitie Amplitude Repone with 6 BR Laye Pai Stonge Coupling fo Flat Tanmiion Band While the eult thu fa have matched well with theoetical pediction and offe combfunction tanmiion filte uitable fo a vaiety of application we ae pecifically tageting a device with a elatively boad tanmiion band and extemely high goup delay. Unfotunately by making the inte-cavity coupling contant the ame a the end-coupling contant we have pevented the tanmiion peak fom ovelapping. In othe wod a additional BR laye wee added to epaate the cavitie theeby educing the inte-cavity coupling and naowing the fee pectal ange (FSR) the ingle-cavity Q-facto coepondingly inceaed and the tanmiion line naowed. A bette appoach would allow one to tune the coupling contant independent of the intinic cavity chaacteitic. 68

188 To thi end we began by ineting a low-index (.4496) laye midway between each of the eonance cavitie and attempted to optimie thi laye thickne to obtain a elatively flat tanmiion band. Unfotunately thi had the undeied effect of ceating an additional et of eonance cavitie. The ovelapping et of coupled eonance eulted in a athe convoluted filte epone. A bette appoach would involve a epaation laye that only uppoted leaky mode thu educing the chance fo a econday Faby-Peot effect to occu. The new deign had a pilla adiu of.65um and a cavity laye thickne of.55um. The low-index BR laye had a efactive index of.65 and a thickne of 88nm while the high-index (.936) laye wa 45nm thick. The epaation laye had a efactive index of.4496 and a thickne of.3um. With BR laye pai peent the ingle-cavity tuctue had a eonance at.5um with a Q-facto of 47. The amplitude epone of the tuctue with (a) (b) (c) 4 and (d) 8 coupled cavitie i hown in Figue 7-8. Note that the tanmiion band i elatively flat without ignificant ipple due to the additional cavitie peent. A cavitie ae added the band edge become much hape in like manne to Bagg eflection gating. (a) (b) 69

189 (c) (d) Figue 7-8: (a) (b) (c) 4 and (d) 8 Coupled Cavitie with Inceaed Coupling Contant. Since the bandwidth fo thee tuctue i contant obtaining a Q-facto fo the tuctue i not immediately intuitive and ha limited meaning beide. oweve the chaacteitic of inteet i the tuctue goup delay which i hown in Figue 7-9. Notice that the delay peak tongly at the band edge and i eaonably flat though omewhat educed aco the middle. Baed on the laye thicknee and the calculated delay qn..9 give a goup index of 6 fo the tuctue nea the middle of the tanmiion band and 4 fo fequencie at the edge of the band. oweve it i eaonable to expect that a mino inceae in the pace laye thickne hould ubtantially inceae the delay and the tuctue goup index. Note though that thi degee of delay compae quite favoably to that obtained by othe CROW tuctue. Fo example a CROW coniting of ing eonato offeed a total delay of p fo a goup index of 3 but with a bandwidth of le than.nm [49]. 7

190 Inceaed Mode Confinement fo ighe Goup elay The delay fo the demontated pilla cavitie i ubtantial but one would eaonably expect that it could be inceaed futhe if the Q-facto of the individual cavitie could be ubtantially inceaed. Section 7..4 outlined a mean fo inceaing modal confinement and demontated a maked impovement in the Q-facto fo uch tuctue. Specifically encaing the pilla tuctue adially in a highly-eflective laye damatically inceae it Q-facto. Figue 7-9: Goup elay fo Reduced Coupling Contant Stuctue. To exploit thi effect I ued the ame baic pilla tuctue uounded adially by a PC laye. Simila eult hould be attainable fom ealitic eflecting tuctue uch a highly eflective metallic film (epecially in the micowave egion of the pectum) and the adial Bagg gating dicued ealie. I etuned to the.936/.4496 index laye deign and eoptimied the cavity adiu and thickne and the oveall thickne of the Bagg laye to obtain a high-q tuctue eonant nea.5um. The eulting tuctue (illutated in Figue 7-3) had a diamete of.6um a cavity thickne of 45nm and BR laye thicknee of 7nm and 36nm. With 8 BR pai on eithe ide of the cavity the ingle-cavity pilla eonated at 7

191 .469um with a Q-facto of. ven with only 6 BR pai peent the tuctue till had a Q-facto of 5. The ingle cavity amplitude epone fo both cae i hown in Figue 7-3. Figue 7-3: igh Confinement Couple Cavitie. (a) (b) Figue 7-3: Single Cavity Amplitude Repone with (a) 6 and (b) 8 BR Laye Pai. 7